LMFDB - Embedded newform 825.2.k.h.518.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [825,2,Mod(518,825)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(825, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([2, 3, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("825.518");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 825 = 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	825.k (of order $4$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$6.58765816676$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{8})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 1 $ x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			518.1
Root			$-0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	825.518
Dual form			825.2.k.h.782.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.292893 + 0.292893i) q^{2} +(1.70711 - 0.292893i) q^{3} -1.82843i q^{4} +(0.585786 + 0.414214i) q^{6} +(0.585786 - 0.585786i) q^{7} +(1.12132 - 1.12132i) q^{8} +(2.82843 - 1.00000i) q^{9} +O(q^{10})$	\(q+(0.292893 + 0.292893i) q^{2} +(1.70711 - 0.292893i) q^{3} -1.82843i q^{4} +(0.585786 + 0.414214i) q^{6} +(0.585786 - 0.585786i) q^{7} +(1.12132 - 1.12132i) q^{8} +(2.82843 - 1.00000i) q^{9} -1.00000i q^{11} +(-0.535534 - 3.12132i) q^{12} +(3.41421 + 3.41421i) q^{13} +0.343146 q^{14} -3.00000 q^{16} +(-2.00000 - 2.00000i) q^{17} +(1.12132 + 0.535534i) q^{18} -0.828427i q^{19} +(0.828427 - 1.17157i) q^{21} +(0.292893 - 0.292893i) q^{22} +(-0.828427 + 0.828427i) q^{23} +(1.58579 - 2.24264i) q^{24} +2.00000i q^{26} +(4.53553 - 2.53553i) q^{27} +(-1.07107 - 1.07107i) q^{28} -8.82843 q^{29} +4.00000 q^{31} +(-3.12132 - 3.12132i) q^{32} +(-0.292893 - 1.70711i) q^{33} -1.17157i q^{34} +(-1.82843 - 5.17157i) q^{36} +(5.65685 - 5.65685i) q^{37} +(0.242641 - 0.242641i) q^{38} +(6.82843 + 4.82843i) q^{39} +4.82843i q^{41} +(0.585786 - 0.100505i) q^{42} +(4.58579 + 4.58579i) q^{43} -1.82843 q^{44} -0.485281 q^{46} +(-4.82843 - 4.82843i) q^{47} +(-5.12132 + 0.878680i) q^{48} +6.31371i q^{49} +(-4.00000 - 2.82843i) q^{51} +(6.24264 - 6.24264i) q^{52} +(8.48528 - 8.48528i) q^{53} +(2.07107 + 0.585786i) q^{54} -1.31371i q^{56} +(-0.242641 - 1.41421i) q^{57} +(-2.58579 - 2.58579i) q^{58} -2.34315 q^{59} -6.00000 q^{61} +(1.17157 + 1.17157i) q^{62} +(1.07107 - 2.24264i) q^{63} +4.17157i q^{64} +(0.414214 - 0.585786i) q^{66} +(-2.58579 + 2.58579i) q^{67} +(-3.65685 + 3.65685i) q^{68} +(-1.17157 + 1.65685i) q^{69} +9.65685i q^{71} +(2.05025 - 4.29289i) q^{72} +(4.58579 + 4.58579i) q^{73} +3.31371 q^{74} -1.51472 q^{76} +(-0.585786 - 0.585786i) q^{77} +(0.585786 + 3.41421i) q^{78} +4.82843i q^{79} +(7.00000 - 5.65685i) q^{81} +(-1.41421 + 1.41421i) q^{82} +(-4.24264 + 4.24264i) q^{83} +(-2.14214 - 1.51472i) q^{84} +2.68629i q^{86} +(-15.0711 + 2.58579i) q^{87} +(-1.12132 - 1.12132i) q^{88} -15.3137 q^{89} +4.00000 q^{91} +(1.51472 + 1.51472i) q^{92} +(6.82843 - 1.17157i) q^{93} -2.82843i q^{94} +(-6.24264 - 4.41421i) q^{96} +(-1.65685 + 1.65685i) q^{97} +(-1.84924 + 1.84924i) q^{98} +(-1.00000 - 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{7} - 4 q^{8}+O(q^{10}) $ 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^6 + 8 * q^7 - 4 * q^8	$ 4 q + 4 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{7} - 4 q^{8} + 12 q^{12} + 8 q^{13} + 24 q^{14} - 12 q^{16} - 8 q^{17} - 4 q^{18} - 8 q^{21} + 4 q^{22} + 8 q^{23} + 12 q^{24} + 4 q^{27} + 24 q^{28} - 24 q^{29} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} - 16 q^{38} + 16 q^{39} + 8 q^{42} + 24 q^{43} + 4 q^{44} + 32 q^{46} - 8 q^{47} - 12 q^{48} - 16 q^{51} + 8 q^{52} - 20 q^{54} + 16 q^{57} - 16 q^{58} - 32 q^{59} - 24 q^{61} + 16 q^{62} - 24 q^{63} - 4 q^{66} - 16 q^{67} + 8 q^{68} - 16 q^{69} + 28 q^{72} + 24 q^{73} - 32 q^{74} - 40 q^{76} - 8 q^{77} + 8 q^{78} + 28 q^{81} + 48 q^{84} - 32 q^{87} + 4 q^{88} - 16 q^{89} + 16 q^{91} + 40 q^{92} + 16 q^{93} - 8 q^{96} + 16 q^{97} + 52 q^{98} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^6 + 8 * q^7 - 4 * q^8 + 12 * q^12 + 8 * q^13 + 24 * q^14 - 12 * q^16 - 8 * q^17 - 4 * q^18 - 8 * q^21 + 4 * q^22 + 8 * q^23 + 12 * q^24 + 4 * q^27 + 24 * q^28 - 24 * q^29 + 16 * q^31 - 4 * q^32 - 4 * q^33 + 4 * q^36 - 16 * q^38 + 16 * q^39 + 8 * q^42 + 24 * q^43 + 4 * q^44 + 32 * q^46 - 8 * q^47 - 12 * q^48 - 16 * q^51 + 8 * q^52 - 20 * q^54 + 16 * q^57 - 16 * q^58 - 32 * q^59 - 24 * q^61 + 16 * q^62 - 24 * q^63 - 4 * q^66 - 16 * q^67 + 8 * q^68 - 16 * q^69 + 28 * q^72 + 24 * q^73 - 32 * q^74 - 40 * q^76 - 8 * q^77 + 8 * q^78 + 28 * q^81 + 48 * q^84 - 32 * q^87 + 4 * q^88 - 16 * q^89 + 16 * q^91 + 40 * q^92 + 16 * q^93 - 8 * q^96 + 16 * q^97 + 52 * q^98 - 4 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/825\mathbb{Z}\right)^\times$.

$n$	$376$	$551$	$727$
$\chi(n)$	$1$	$-1$	$e\left(\frac{3}{4}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	0.292893	+	0.292893i	0.207107	+	0.207107i	0.803037	−	0.595930i	$-0.203216\pi$
$2$	0.292893	+	0.292893i	0.207107	+	0.207107i	−0.595930	+	0.803037i	$0.703216\pi$
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$3$	1.70711	−	0.292893i	0.985599	−	0.169102i
$4$		−	1.82843i		−	0.914214i
$5$		0			0
$5$		0			0
$6$	0.585786	+	0.414214i	0.239146	+	0.169102i
$7$	0.585786	−	0.585786i	0.221406	−	0.221406i	−0.587684	−	0.809091i	$-0.699960\pi$
$7$	0.585786	−	0.585786i	0.221406	−	0.221406i	0.809091	+	0.587684i	$0.199960\pi$
$8$	1.12132	−	1.12132i	0.396447	−	0.396447i
$9$	2.82843	−	1.00000i	0.942809	−	0.333333i
$10$		0			0
$11$		−	1.00000i		−	0.301511i
$11$		−	1.00000i		−	0.301511i
$12$	−0.535534	−	3.12132i	−0.154595	−	0.901048i
$13$	3.41421	+	3.41421i	0.946932	+	0.946932i	0.998661	−	0.0517287i	$-0.0164731\pi$
$13$	3.41421	+	3.41421i	0.946932	+	0.946932i	−0.0517287	+	0.998661i	$0.516473\pi$
$14$	0.343146			0.0917096
$15$		0			0
$16$	−3.00000			−0.750000
$17$	−2.00000	−	2.00000i	−0.485071	−	0.485071i	0.421676	−	0.906747i	$-0.361442\pi$
$17$	−2.00000	−	2.00000i	−0.485071	−	0.485071i	−0.906747	+	0.421676i	$0.861442\pi$
$18$	1.12132	+	0.535534i	0.264298	+	0.126227i
$19$		−	0.828427i		−	0.190054i	−0.995475	−	0.0950271i	$-0.969706\pi$
$19$		−	0.828427i		−	0.190054i	0.995475	−	0.0950271i	$-0.0302938\pi$
$20$		0			0
$21$	0.828427	−	1.17157i	0.180778	−	0.255658i
$22$	0.292893	−	0.292893i	0.0624450	−	0.0624450i
$23$	−0.828427	+	0.828427i	−0.172739	+	0.172739i	−0.788182	−	0.615443i	$-0.788977\pi$
$23$	−0.828427	+	0.828427i	−0.172739	+	0.172739i	0.615443	+	0.788182i	$0.288977\pi$
$24$	1.58579	−	2.24264i	0.323697	−	0.457777i
$25$		0			0
$26$			2.00000i			0.392232i
$27$	4.53553	−	2.53553i	0.872864	−	0.487964i
$28$	−1.07107	−	1.07107i	−0.202413	−	0.202413i
$29$	−8.82843			−1.63940			−0.819699	−	0.572795i	$-0.805859\pi$
$29$	−8.82843			−1.63940			−0.819699	+	0.572795i	$0.805859\pi$
$30$		0			0
$31$	4.00000			0.718421			0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000			0.718421			0.359211	+	0.933257i	$0.383046\pi$
$32$	−3.12132	−	3.12132i	−0.551777	−	0.551777i
$33$	−0.292893	−	1.70711i	−0.0509862	−	0.297169i
$34$		−	1.17157i		−	0.200923i
$35$		0			0
$36$	−1.82843	−	5.17157i	−0.304738	−	0.861929i
$37$	5.65685	−	5.65685i	0.929981	−	0.929981i	−0.0677230	−	0.997704i	$-0.521573\pi$
$37$	5.65685	−	5.65685i	0.929981	−	0.929981i	0.997704	+	0.0677230i	$0.0215734\pi$
$38$	0.242641	−	0.242641i	0.0393615	−	0.0393615i
$39$	6.82843	+	4.82843i	1.09342	+	0.773167i
$40$		0			0
$41$			4.82843i			0.754074i	0.926198	+	0.377037i	$0.123057\pi$
$41$			4.82843i			0.754074i	−0.926198	+	0.377037i	$0.876943\pi$
$42$	0.585786	−	0.100505i	0.0903888	−	0.0155083i
$43$	4.58579	+	4.58579i	0.699326	+	0.699326i	0.964265	−	0.264939i	$-0.0853519\pi$
$43$	4.58579	+	4.58579i	0.699326	+	0.699326i	−0.264939	+	0.964265i	$0.585352\pi$
$44$	−1.82843			−0.275646
$45$		0			0
$46$	−0.485281			−0.0715508
$47$	−4.82843	−	4.82843i	−0.704298	−	0.704298i	0.261032	−	0.965330i	$-0.415937\pi$
$47$	−4.82843	−	4.82843i	−0.704298	−	0.704298i	−0.965330	+	0.261032i	$0.915937\pi$
$48$	−5.12132	+	0.878680i	−0.739199	+	0.126826i
$49$			6.31371i			0.901958i
$50$		0			0
$51$	−4.00000	−	2.82843i	−0.560112	−	0.396059i
$52$	6.24264	−	6.24264i	0.865699	−	0.865699i
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.182300	−	0.983243i	$-0.441646\pi$
$53$	8.48528	−	8.48528i	1.16554	−	1.16554i	0.983243	−	0.182300i	$-0.0583542\pi$
$54$	2.07107	+	0.585786i	0.281837	+	0.0797154i
$55$		0			0
$56$		−	1.31371i		−	0.175552i
$57$	−0.242641	−	1.41421i	−0.0321385	−	0.187317i
$58$	−2.58579	−	2.58579i	−0.339530	−	0.339530i
$59$	−2.34315			−0.305052			−0.152526	−	0.988299i	$-0.548741\pi$
$59$	−2.34315			−0.305052			−0.152526	+	0.988299i	$0.548741\pi$
$60$		0			0
$61$	−6.00000			−0.768221			−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000			−0.768221			−0.384111	+	0.923287i	$0.625492\pi$
$62$	1.17157	+	1.17157i	0.148790	+	0.148790i
$63$	1.07107	−	2.24264i	0.134942	−	0.282546i
$64$			4.17157i			0.521447i
$65$		0			0
$66$	0.414214	−	0.585786i	0.0509862	−	0.0721053i
$67$	−2.58579	+	2.58579i	−0.315904	+	0.315904i	−0.847192	−	0.531287i	$-0.821708\pi$
$67$	−2.58579	+	2.58579i	−0.315904	+	0.315904i	0.531287	+	0.847192i	$0.321708\pi$
$68$	−3.65685	+	3.65685i	−0.443459	+	0.443459i
$69$	−1.17157	+	1.65685i	−0.141041	+	0.199462i
$70$		0			0
$71$			9.65685i			1.14606i	0.819535	+	0.573029i	$0.194232\pi$
$71$			9.65685i			1.14606i	−0.819535	+	0.573029i	$0.805768\pi$
$72$	2.05025	−	4.29289i	0.241625	−	0.505922i
$73$	4.58579	+	4.58579i	0.536726	+	0.536726i	0.922566	−	0.385840i	$-0.126088\pi$
$73$	4.58579	+	4.58579i	0.536726	+	0.536726i	−0.385840	+	0.922566i	$0.626088\pi$
$74$	3.31371			0.385211
$75$		0			0
$76$	−1.51472			−0.173750
$77$	−0.585786	−	0.585786i	−0.0667566	−	0.0667566i
$78$	0.585786	+	3.41421i	0.0663273	+	0.386584i
$79$			4.82843i			0.543240i	0.962405	+	0.271620i	$0.0875595\pi$
$79$			4.82843i			0.543240i	−0.962405	+	0.271620i	$0.912441\pi$
$80$		0			0
$81$	7.00000	−	5.65685i	0.777778	−	0.628539i
$82$	−1.41421	+	1.41421i	−0.156174	+	0.156174i
$83$	−4.24264	+	4.24264i	−0.465690	+	0.465690i	−0.900515	−	0.434825i	$-0.856810\pi$
$83$	−4.24264	+	4.24264i	−0.465690	+	0.465690i	0.434825	+	0.900515i	$0.356810\pi$
$84$	−2.14214	−	1.51472i	−0.233726	−	0.165269i
$85$		0			0
$86$			2.68629i			0.289670i
$87$	−15.0711	+	2.58579i	−1.61579	+	0.277225i
$88$	−1.12132	−	1.12132i	−0.119533	−	0.119533i
$89$	−15.3137			−1.62325			−0.811625	−	0.584179i	$-0.801417\pi$
$89$	−15.3137			−1.62325			−0.811625	+	0.584179i	$0.801417\pi$
$90$		0			0
$91$	4.00000			0.419314
$92$	1.51472	+	1.51472i	0.157920	+	0.157920i
$93$	6.82843	−	1.17157i	0.708075	−	0.121486i
$94$		−	2.82843i		−	0.291730i
$95$		0			0
$96$	−6.24264	−	4.41421i	−0.637137	−	0.450524i
$97$	−1.65685	+	1.65685i	−0.168228	+	0.168228i	−0.786200	−	0.617972i	$-0.787954\pi$
$97$	−1.65685	+	1.65685i	−0.168228	+	0.168228i	0.617972	+	0.786200i	$0.287954\pi$
$98$	−1.84924	+	1.84924i	−0.186802	+	0.186802i
$99$	−1.00000	−	2.82843i	−0.100504	−	0.284268i
$100$		0			0
$101$			0.828427i			0.0824316i	0.999150	+	0.0412158i	$0.0131231\pi$
$101$			0.828427i			0.0824316i	−0.999150	+	0.0412158i	$0.986877\pi$
$102$	−0.343146	−	2.00000i	−0.0339765	−	0.198030i
$103$	4.24264	+	4.24264i	0.418040	+	0.418040i	0.884528	−	0.466488i	$-0.154481\pi$
$103$	4.24264	+	4.24264i	0.418040	+	0.418040i	−0.466488	+	0.884528i	$0.654481\pi$
$104$	7.65685			0.750816
$105$		0			0
$106$	4.97056			0.482784
$107$	13.8995	+	13.8995i	1.34371	+	1.34371i	0.892329	+	0.451386i	$0.149070\pi$
$107$	13.8995	+	13.8995i	1.34371	+	1.34371i	0.451386	+	0.892329i	$0.350930\pi$
$108$	−4.63604	−	8.29289i	−0.446103	−	0.797984i
$109$		−	2.00000i		−	0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$		−	2.00000i		−	0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$		0			0
$111$	8.00000	−	11.3137i	0.759326	−	1.07385i
$112$	−1.75736	+	1.75736i	−0.166055	+	0.166055i
$113$	8.48528	−	8.48528i	0.798228	−	0.798228i	−0.184588	−	0.982816i	$-0.559095\pi$
$113$	8.48528	−	8.48528i	0.798228	−	0.798228i	0.982816	+	0.184588i	$0.0590950\pi$
$114$	0.343146	−	0.485281i	0.0321385	−	0.0454508i
$115$		0			0
$116$			16.1421i			1.49876i
$117$	13.0711	+	6.24264i	1.20842	+	0.577132i
$118$	−0.686292	−	0.686292i	−0.0631783	−	0.0631783i
$119$	−2.34315			−0.214796
$120$		0			0
$121$	−1.00000			−0.0909091
$122$	−1.75736	−	1.75736i	−0.159104	−	0.159104i
$123$	1.41421	+	8.24264i	0.127515	+	0.743214i
$124$		−	7.31371i		−	0.656790i
$125$		0			0
$126$	0.970563	−	0.343146i	0.0864646	−	0.0305699i
$127$	−10.2426	+	10.2426i	−0.908887	+	0.908887i	−0.996182	−	0.0872951i	$-0.972178\pi$
$127$	−10.2426	+	10.2426i	−0.908887	+	0.908887i	0.0872951	+	0.996182i	$0.472178\pi$
$128$	−7.46447	+	7.46447i	−0.659772	+	0.659772i
$129$	9.17157	+	6.48528i	0.807512	+	0.570997i
$130$		0			0
$131$			17.6569i			1.54269i	0.636419	+	0.771343i	$0.280415\pi$
$131$			17.6569i			1.54269i	−0.636419	+	0.771343i	$0.719585\pi$
$132$	−3.12132	+	0.535534i	−0.271676	+	0.0466122i
$133$	−0.485281	−	0.485281i	−0.0420792	−	0.0420792i
$134$	−1.51472			−0.130852
$135$		0			0
$136$	−4.48528			−0.384610
$137$	1.65685	+	1.65685i	0.141555	+	0.141555i	0.774333	−	0.632778i	$-0.218086\pi$
$137$	1.65685	+	1.65685i	0.141555	+	0.141555i	−0.632778	+	0.774333i	$0.718086\pi$
$138$	−0.828427	+	0.142136i	−0.0705204	+	0.0120994i
$139$		−	16.8284i		−	1.42737i	−0.700468	−	0.713684i	$-0.747025\pi$
$139$		−	16.8284i		−	1.42737i	0.700468	−	0.713684i	$-0.252975\pi$
$140$		0			0
$141$	−9.65685	−	6.82843i	−0.813254	−	0.575057i
$142$	−2.82843	+	2.82843i	−0.237356	+	0.237356i
$143$	3.41421	−	3.41421i	0.285511	−	0.285511i
$144$	−8.48528	+	3.00000i	−0.707107	+	0.250000i
$145$		0			0
$146$			2.68629i			0.222319i
$147$	1.84924	+	10.7782i	0.152523	+	0.888969i
$148$	−10.3431	−	10.3431i	−0.850201	−	0.850201i
$149$	17.7990			1.45815			0.729075	−	0.684434i	$-0.239951\pi$
$149$	17.7990			1.45815			0.729075	+	0.684434i	$0.239951\pi$
$150$		0			0
$151$	14.4853			1.17880			0.589398	−	0.807843i	$-0.299365\pi$
$151$	14.4853			1.17880			0.589398	+	0.807843i	$0.299365\pi$
$152$	−0.928932	−	0.928932i	−0.0753463	−	0.0753463i
$153$	−7.65685	−	3.65685i	−0.619020	−	0.295639i
$154$		−	0.343146i		−	0.0276515i
$155$		0			0
$156$	8.82843	−	12.4853i	0.706840	−	0.999623i
$157$	−15.3137	+	15.3137i	−1.22217	+	1.22217i	−0.255307	+	0.966860i	$0.582177\pi$
$157$	−15.3137	+	15.3137i	−1.22217	+	1.22217i	−0.966860	+	0.255307i	$0.917823\pi$
$158$	−1.41421	+	1.41421i	−0.112509	+	0.112509i
$159$	12.0000	−	16.9706i	0.951662	−	1.34585i
$160$		0			0
$161$			0.970563i			0.0764911i
$162$	3.70711	+	0.393398i	0.291258	+	0.0309083i
$163$	−9.89949	−	9.89949i	−0.775388	−	0.775388i	0.203655	−	0.979043i	$-0.434718\pi$
$163$	−9.89949	−	9.89949i	−0.775388	−	0.775388i	−0.979043	+	0.203655i	$0.934718\pi$
$164$	8.82843			0.689384
$165$		0			0
$166$	−2.48528			−0.192895
$167$	8.24264	+	8.24264i	0.637835	+	0.637835i	0.950021	−	0.312186i	$-0.101061\pi$
$167$	8.24264	+	8.24264i	0.637835	+	0.637835i	−0.312186	+	0.950021i	$0.601061\pi$
$168$	−0.384776	−	2.24264i	−0.0296861	−	0.173023i
$169$			10.3137i			0.793362i
$170$		0			0
$171$	−0.828427	−	2.34315i	−0.0633514	−	0.179185i
$172$	8.38478	−	8.38478i	0.639333	−	0.639333i
$173$	−4.82843	+	4.82843i	−0.367099	+	0.367099i	−0.866418	−	0.499319i	$-0.833583\pi$
$173$	−4.82843	+	4.82843i	−0.367099	+	0.367099i	0.499319	+	0.866418i	$0.333583\pi$
$174$	−5.17157	−	3.65685i	−0.392056	−	0.277225i
$175$		0			0
$176$			3.00000i			0.226134i
$177$	−4.00000	+	0.686292i	−0.300658	+	0.0515848i
$178$	−4.48528	−	4.48528i	−0.336186	−	0.336186i
$179$	−19.3137			−1.44357			−0.721787	−	0.692115i	$-0.756679\pi$
$179$	−19.3137			−1.44357			−0.721787	+	0.692115i	$0.756679\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$	1.17157	+	1.17157i	0.0868428	+	0.0868428i
$183$	−10.2426	+	1.75736i	−0.757158	+	0.129908i
$184$			1.85786i			0.136964i
$185$		0			0
$186$	2.34315	+	1.65685i	0.171808	+	0.121486i
$187$	−2.00000	+	2.00000i	−0.146254	+	0.146254i
$188$	−8.82843	+	8.82843i	−0.643879	+	0.643879i
$189$	1.17157	−	4.14214i	0.0852194	−	0.301296i
$190$		0			0
$191$		−	11.3137i		−	0.818631i	−0.912393	−	0.409316i	$-0.865768\pi$
$191$		−	11.3137i		−	0.818631i	0.912393	−	0.409316i	$-0.134232\pi$
$192$	1.22183	+	7.12132i	0.0881777	+	0.513937i
$193$	9.07107	+	9.07107i	0.652950	+	0.652950i	0.953702	−	0.300752i	$-0.0972378\pi$
$193$	9.07107	+	9.07107i	0.652950	+	0.652950i	−0.300752	+	0.953702i	$0.597238\pi$
$194$	−0.970563			−0.0696823
$195$		0			0
$196$	11.5442			0.824583
$197$	−14.4853	−	14.4853i	−1.03203	−	1.03203i	−0.999470	−	0.0325639i	$-0.989633\pi$
$197$	−14.4853	−	14.4853i	−1.03203	−	1.03203i	−0.0325639	−	0.999470i	$-0.510367\pi$
$198$	0.535534	−	1.12132i	0.0380587	−	0.0796888i
$199$		−	12.9706i		−	0.919459i	−0.888059	−	0.459729i	$-0.847946\pi$
$199$		−	12.9706i		−	0.919459i	0.888059	−	0.459729i	$-0.152054\pi$
$200$		0			0
$201$	−3.65685	+	5.17157i	−0.257935	+	0.364775i
$202$	−0.242641	+	0.242641i	−0.0170721	+	0.0170721i
$203$	−5.17157	+	5.17157i	−0.362973	+	0.362973i
$204$	−5.17157	+	7.31371i	−0.362083	+	0.512062i
$205$		0			0
$206$			2.48528i			0.173158i
$207$	−1.51472	+	3.17157i	−0.105280	+	0.220440i
$208$	−10.2426	−	10.2426i	−0.710199	−	0.710199i
$209$	−0.828427			−0.0573035
$210$		0			0
$211$	−10.4853			−0.721837			−0.360918	−	0.932597i	$-0.617537\pi$
$211$	−10.4853			−0.721837			−0.360918	+	0.932597i	$0.617537\pi$
$212$	−15.5147	−	15.5147i	−1.06556	−	1.06556i
$213$	2.82843	+	16.4853i	0.193801	+	1.12955i
$214$			8.14214i			0.556585i
$215$		0			0
$216$	2.24264	−	7.92893i	0.152592	−	0.539496i
$217$	2.34315	−	2.34315i	0.159063	−	0.159063i
$218$	0.585786	−	0.585786i	0.0396745	−	0.0396745i
$219$	9.17157	+	6.48528i	0.619757	+	0.438235i
$220$		0			0
$221$		−	13.6569i		−	0.918659i
$222$	5.65685	−	0.970563i	0.379663	−	0.0651399i
$223$	−16.2426	−	16.2426i	−1.08769	−	1.08769i	−0.995766	−	0.0919214i	$-0.970699\pi$
$223$	−16.2426	−	16.2426i	−1.08769	−	1.08769i	−0.0919214	−	0.995766i	$-0.529301\pi$
$224$	−3.65685			−0.244334
$225$		0			0
$226$	4.97056			0.330637
$227$	20.2426	+	20.2426i	1.34355	+	1.34355i	0.892496	+	0.451055i	$0.148952\pi$
$227$	20.2426	+	20.2426i	1.34355	+	1.34355i	0.451055	+	0.892496i	$0.351048\pi$
$228$	−2.58579	+	0.443651i	−0.171248	+	0.0293815i
$229$			18.0000i			1.18947i	0.803921	+	0.594737i	$0.202744\pi$
$229$			18.0000i			1.18947i	−0.803921	+	0.594737i	$0.797256\pi$
$230$		0			0
$231$	−1.17157	−	0.828427i	−0.0770838	−	0.0545065i
$232$	−9.89949	+	9.89949i	−0.649934	+	0.649934i
$233$	−10.0000	+	10.0000i	−0.655122	+	0.655122i	−0.954222	−	0.299100i	$-0.903314\pi$
$233$	−10.0000	+	10.0000i	−0.655122	+	0.655122i	0.299100	+	0.954222i	$0.403314\pi$
$234$	2.00000	+	5.65685i	0.130744	+	0.369800i
$235$		0			0
$236$			4.28427i			0.278882i
$237$	1.41421	+	8.24264i	0.0918630	+	0.535417i
$238$	−0.686292	−	0.686292i	−0.0444857	−	0.0444857i
$239$	2.34315			0.151565			0.0757827	−	0.997124i	$-0.475854\pi$
$239$	2.34315			0.151565			0.0757827	+	0.997124i	$0.475854\pi$
$240$		0			0
$241$	−26.9706			−1.73733			−0.868663	−	0.495403i	$-0.835020\pi$
$241$	−26.9706			−1.73733			−0.868663	+	0.495403i	$0.835020\pi$
$242$	−0.292893	−	0.292893i	−0.0188279	−	0.0188279i
$243$	10.2929	−	11.7071i	0.660289	−	0.751011i
$244$			10.9706i			0.702318i
$245$		0			0
$246$	−2.00000	+	2.82843i	−0.127515	+	0.180334i
$247$	2.82843	−	2.82843i	0.179969	−	0.179969i
$248$	4.48528	−	4.48528i	0.284816	−	0.284816i
$249$	−6.00000	+	8.48528i	−0.380235	+	0.537733i
$250$		0			0
$251$			1.65685i			0.104580i	0.998632	+	0.0522899i	$0.0166520\pi$
$251$			1.65685i			0.104580i	−0.998632	+	0.0522899i	$0.983348\pi$
$252$	−4.10051	−	1.95837i	−0.258308	−	0.123366i
$253$	0.828427	+	0.828427i	0.0520828	+	0.0520828i
$254$	−6.00000			−0.376473
$255$		0			0
$256$	3.97056			0.248160
$257$	16.0000	+	16.0000i	0.998053	+	0.998053i	0.999998	−	0.00194553i	$-0.000619281\pi$
$257$	16.0000	+	16.0000i	0.998053	+	0.998053i	−0.00194553	+	0.999998i	$0.500619\pi$
$258$	0.786797	+	4.58579i	0.0489838	+	0.285499i
$259$		−	6.62742i		−	0.411808i
$260$		0			0
$261$	−24.9706	+	8.82843i	−1.54564	+	0.546466i
$262$	−5.17157	+	5.17157i	−0.319501	+	0.319501i
$263$	16.2426	−	16.2426i	1.00156	−	1.00156i	0.00156536	−	0.999999i	$-0.499502\pi$
$263$	16.2426	−	16.2426i	1.00156	−	1.00156i	0.999999	−	0.00156536i	$-0.000498269\pi$
$264$	−2.24264	−	1.58579i	−0.138025	−	0.0975984i
$265$		0			0
$266$		−	0.284271i		−	0.0174298i
$267$	−26.1421	+	4.48528i	−1.59987	+	0.274495i
$268$	4.72792	+	4.72792i	0.288804	+	0.288804i
$269$	−13.6569			−0.832673			−0.416337	−	0.909211i	$-0.636686\pi$
$269$	−13.6569			−0.832673			−0.416337	+	0.909211i	$0.636686\pi$
$270$		0			0
$271$	−9.51472			−0.577978			−0.288989	−	0.957332i	$-0.593319\pi$
$271$	−9.51472			−0.577978			−0.288989	+	0.957332i	$0.593319\pi$
$272$	6.00000	+	6.00000i	0.363803	+	0.363803i
$273$	6.82843	−	1.17157i	0.413275	−	0.0709068i
$274$			0.970563i			0.0586338i
$275$		0			0
$276$	3.02944	+	2.14214i	0.182351	+	0.128941i
$277$	17.0711	−	17.0711i	1.02570	−	1.02570i	0.0260402	−	0.999661i	$-0.491710\pi$
$277$	17.0711	−	17.0711i	1.02570	−	1.02570i	0.999661	−	0.0260402i	$-0.00828978\pi$
$278$	4.92893	−	4.92893i	0.295618	−	0.295618i
$279$	11.3137	−	4.00000i	0.677334	−	0.239474i
$280$		0			0
$281$		−	8.14214i		−	0.485719i	−0.970061	−	0.242860i	$-0.921915\pi$
$281$		−	8.14214i		−	0.485719i	0.970061	−	0.242860i	$-0.0780854\pi$
$282$	−0.828427	−	4.82843i	−0.0493321	−	0.287529i
$283$	4.58579	+	4.58579i	0.272597	+	0.272597i	0.830145	−	0.557548i	$-0.188258\pi$
$283$	4.58579	+	4.58579i	0.272597	+	0.272597i	−0.557548	+	0.830145i	$0.688258\pi$
$284$	17.6569			1.04774
$285$		0			0
$286$	2.00000			0.118262
$287$	2.82843	+	2.82843i	0.166957	+	0.166957i
$288$	−11.9497	−	5.70711i	−0.704146	−	0.336294i
$289$		−	9.00000i		−	0.529412i
$290$		0			0
$291$	−2.34315	+	3.31371i	−0.137358	+	0.194253i
$292$	8.38478	−	8.38478i	0.490682	−	0.490682i
$293$	−2.48528	+	2.48528i	−0.145192	+	0.145192i	−0.775966	−	0.630775i	$-0.782737\pi$
$293$	−2.48528	+	2.48528i	−0.145192	+	0.145192i	0.630775	+	0.775966i	$0.282737\pi$
$294$	−2.61522	+	3.69848i	−0.152523	+	0.215700i
$295$		0			0
$296$		−	12.6863i		−	0.737376i
$297$	−2.53553	−	4.53553i	−0.147127	−	0.263178i
$298$	5.21320	+	5.21320i	0.301993	+	0.301993i
$299$	−5.65685			−0.327144
$300$		0			0
$301$	5.37258			0.309671
$302$	4.24264	+	4.24264i	0.244137	+	0.244137i
$303$	0.242641	+	1.41421i	0.0139393	+	0.0812444i
$304$			2.48528i			0.142541i
$305$		0			0
$306$	−1.17157	−	3.31371i	−0.0669744	−	0.189432i
$307$	−18.7279	+	18.7279i	−1.06886	+	1.06886i	−0.0714121	+	0.997447i	$0.522751\pi$
$307$	−18.7279	+	18.7279i	−1.06886	+	1.06886i	−0.997447	+	0.0714121i	$0.977249\pi$
$308$	−1.07107	+	1.07107i	−0.0610298	+	0.0610298i
$309$	8.48528	+	6.00000i	0.482711	+	0.341328i
$310$		0			0
$311$		−	10.3431i		−	0.586506i	−0.956035	−	0.293253i	$-0.905262\pi$
$311$		−	10.3431i		−	0.586506i	0.956035	−	0.293253i	$-0.0947378\pi$
$312$	13.0711	−	2.24264i	0.740003	−	0.126965i
$313$	18.8284	+	18.8284i	1.06425	+	1.06425i	0.997789	+	0.0664563i	$0.0211693\pi$
$313$	18.8284	+	18.8284i	1.06425	+	1.06425i	0.0664563	+	0.997789i	$0.478831\pi$
$314$	−8.97056			−0.506238
$315$		0			0
$316$	8.82843			0.496638
$317$	−18.1421	−	18.1421i	−1.01896	−	1.01896i	−0.999817	−	0.0191472i	$-0.993905\pi$
$317$	−18.1421	−	18.1421i	−1.01896	−	1.01896i	−0.0191472	−	0.999817i	$-0.506095\pi$
$318$	8.48528	−	1.45584i	0.475831	−	0.0816397i
$319$			8.82843i			0.494297i
$320$		0			0
$321$	27.7990	+	19.6569i	1.55159	+	1.09714i
$322$	−0.284271	+	0.284271i	−0.0158418	+	0.0158418i
$323$	−1.65685	+	1.65685i	−0.0921898	+	0.0921898i
$324$	−10.3431	−	12.7990i	−0.574619	−	0.711055i
$325$		0			0
$326$		−	5.79899i		−	0.321176i
$327$	−0.585786	−	3.41421i	−0.0323941	−	0.188806i
$328$	5.41421	+	5.41421i	0.298950	+	0.298950i
$329$	−5.65685			−0.311872
$330$		0			0
$331$	1.65685			0.0910689			0.0455345	−	0.998963i	$-0.485501\pi$
$331$	1.65685			0.0910689			0.0455345	+	0.998963i	$0.485501\pi$
$332$	7.75736	+	7.75736i	0.425740	+	0.425740i
$333$	10.3431	−	21.6569i	0.566801	−	1.18679i
$334$			4.82843i			0.264200i
$335$		0			0
$336$	−2.48528	+	3.51472i	−0.135583	+	0.191744i
$337$	15.8995	−	15.8995i	0.866101	−	0.866101i	−0.125938	−	0.992038i	$-0.540194\pi$
$337$	15.8995	−	15.8995i	0.866101	−	0.866101i	0.992038	+	0.125938i	$0.0401939\pi$
$338$	−3.02082	+	3.02082i	−0.164311	+	0.164311i
$339$	12.0000	−	16.9706i	0.651751	−	0.921714i
$340$		0			0
$341$		−	4.00000i		−	0.216612i
$342$	0.443651	−	0.928932i	0.0239899	−	0.0502309i
$343$	7.79899	+	7.79899i	0.421106	+	0.421106i
$344$	10.2843			0.554491
$345$		0			0
$346$	−2.82843			−0.152057
$347$	−7.75736	−	7.75736i	−0.416437	−	0.416437i	0.467537	−	0.883974i	$-0.345142\pi$
$347$	−7.75736	−	7.75736i	−0.416437	−	0.416437i	−0.883974	+	0.467537i	$0.845142\pi$
$348$	4.72792	+	27.5563i	0.253443	+	1.47718i
$349$		−	10.9706i		−	0.587241i	−0.955922	−	0.293620i	$-0.905140\pi$
$349$		−	10.9706i		−	0.587241i	0.955922	−	0.293620i	$-0.0948602\pi$
$350$		0			0
$351$	24.1421	+	6.82843i	1.28861	+	0.364474i
$352$	−3.12132	+	3.12132i	−0.166367	+	0.166367i
$353$	12.4853	−	12.4853i	0.664524	−	0.664524i	−0.291919	−	0.956443i	$-0.594294\pi$
$353$	12.4853	−	12.4853i	0.664524	−	0.664524i	0.956443	+	0.291919i	$0.0942937\pi$
$354$	−1.37258	−	0.970563i	−0.0729520	−	0.0515848i
$355$		0			0
$356$			28.0000i			1.48400i
$357$	−4.00000	+	0.686292i	−0.211702	+	0.0363224i
$358$	−5.65685	−	5.65685i	−0.298974	−	0.298974i
$359$	−16.9706			−0.895672			−0.447836	−	0.894116i	$-0.647805\pi$
$359$	−16.9706			−0.895672			−0.447836	+	0.894116i	$0.647805\pi$
$360$		0			0
$361$	18.3137			0.963879
$362$	−1.75736	−	1.75736i	−0.0923648	−	0.0923648i
$363$	−1.70711	+	0.292893i	−0.0895999	+	0.0153729i
$364$		−	7.31371i		−	0.383342i
$365$		0			0
$366$	−3.51472	−	2.48528i	−0.183717	−	0.129908i
$367$	7.75736	−	7.75736i	0.404931	−	0.404931i	−0.475036	−	0.879967i	$-0.657565\pi$
$367$	7.75736	−	7.75736i	0.404931	−	0.404931i	0.879967	+	0.475036i	$0.157565\pi$
$368$	2.48528	−	2.48528i	0.129554	−	0.129554i
$369$	4.82843	+	13.6569i	0.251358	+	0.710947i
$370$		0			0
$371$		−	9.94113i		−	0.516118i
$372$	−2.14214	−	12.4853i	−0.111065	−	0.647332i
$373$	−15.8995	−	15.8995i	−0.823245	−	0.823245i	0.163327	−	0.986572i	$-0.447777\pi$
$373$	−15.8995	−	15.8995i	−0.823245	−	0.823245i	−0.986572	+	0.163327i	$0.947777\pi$
$374$	−1.17157			−0.0605806
$375$		0			0
$376$	−10.8284			−0.558433
$377$	−30.1421	−	30.1421i	−1.55240	−	1.55240i
$378$	1.55635	−	0.870058i	0.0800500	−	0.0447509i
$379$			18.6274i			0.956826i	0.878135	+	0.478413i	$0.158788\pi$
$379$			18.6274i			0.956826i	−0.878135	+	0.478413i	$0.841212\pi$
$380$		0			0
$381$	−14.4853	+	20.4853i	−0.742103	+	1.04949i
$382$	3.31371	−	3.31371i	0.169544	−	0.169544i
$383$	−9.31371	+	9.31371i	−0.475908	+	0.475908i	−0.903820	−	0.427912i	$-0.859249\pi$
$383$	−9.31371	+	9.31371i	−0.475908	+	0.475908i	0.427912	+	0.903820i	$0.359249\pi$
$384$	−10.5563	+	14.9289i	−0.538701	+	0.761839i
$385$		0			0
$386$			5.31371i			0.270461i
$387$	17.5563	+	8.38478i	0.892439	+	0.426222i
$388$	3.02944	+	3.02944i	0.153796	+	0.153796i
$389$	12.0000			0.608424			0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000			0.608424			0.304212	+	0.952604i	$0.401607\pi$
$390$		0			0
$391$	3.31371			0.167581
$392$	7.07969	+	7.07969i	0.357578	+	0.357578i
$393$	5.17157	+	30.1421i	0.260871	+	1.52047i
$394$		−	8.48528i		−	0.427482i
$395$		0			0
$396$	−5.17157	+	1.82843i	−0.259881	+	0.0918819i
$397$	22.8284	−	22.8284i	1.14573	−	1.14573i	0.158342	−	0.987384i	$-0.449385\pi$
$397$	22.8284	−	22.8284i	1.14573	−	1.14573i	0.987384	−	0.158342i	$-0.0506147\pi$
$398$	3.79899	−	3.79899i	0.190426	−	0.190426i
$399$	−0.970563	−	0.686292i	−0.0485889	−	0.0343575i
$400$		0			0
$401$		−	20.2843i		−	1.01295i	−0.862255	−	0.506474i	$-0.830949\pi$
$401$		−	20.2843i		−	1.01295i	0.862255	−	0.506474i	$-0.169051\pi$
$402$	−2.58579	+	0.443651i	−0.128967	+	0.0221273i
$403$	13.6569	+	13.6569i	0.680296	+	0.680296i
$404$	1.51472			0.0753601
$405$		0			0
$406$	−3.02944			−0.150348
$407$	−5.65685	−	5.65685i	−0.280400	−	0.280400i
$408$	−7.65685	+	1.31371i	−0.379071	+	0.0650383i
$409$		−	16.3431i		−	0.808117i	−0.914733	−	0.404058i	$-0.867599\pi$
$409$		−	16.3431i		−	0.808117i	0.914733	−	0.404058i	$-0.132401\pi$
$410$		0			0
$411$	3.31371	+	2.34315i	0.163453	+	0.115579i
$412$	7.75736	−	7.75736i	0.382178	−	0.382178i
$413$	−1.37258	+	1.37258i	−0.0675404	+	0.0675404i
$414$	−1.37258	+	0.485281i	−0.0674588	+	0.0238503i
$415$		0			0
$416$		−	21.3137i		−	1.04499i
$417$	−4.92893	−	28.7279i	−0.241371	−	1.40681i
$418$	−0.242641	−	0.242641i	−0.0118679	−	0.0118679i
$419$	35.3137			1.72519			0.862594	−	0.505897i	$-0.168838\pi$
$419$	35.3137			1.72519			0.862594	+	0.505897i	$0.168838\pi$
$420$		0			0
$421$	13.3137			0.648870			0.324435	−	0.945908i	$-0.394826\pi$
$421$	13.3137			0.648870			0.324435	+	0.945908i	$0.394826\pi$
$422$	−3.07107	−	3.07107i	−0.149497	−	0.149497i
$423$	−18.4853	−	8.82843i	−0.898785	−	0.429253i
$424$		−	19.0294i		−	0.924151i
$425$		0			0
$426$	−4.00000	+	5.65685i	−0.193801	+	0.274075i
$427$	−3.51472	+	3.51472i	−0.170089	+	0.170089i
$428$	25.4142	−	25.4142i	1.22844	−	1.22844i
$429$	4.82843	−	6.82843i	0.233119	−	0.329680i
$430$		0			0
$431$			5.65685i			0.272481i	0.990676	+	0.136241i	$0.0435020\pi$
$431$			5.65685i			0.272481i	−0.990676	+	0.136241i	$0.956498\pi$
$432$	−13.6066	+	7.60660i	−0.654648	+	0.365973i
$433$	−7.31371	−	7.31371i	−0.351474	−	0.351474i	0.509184	−	0.860658i	$-0.329947\pi$
$433$	−7.31371	−	7.31371i	−0.351474	−	0.351474i	−0.860658	+	0.509184i	$0.829947\pi$
$434$	1.37258			0.0658861
$435$		0			0
$436$	−3.65685			−0.175132
$437$	0.686292	+	0.686292i	0.0328298	+	0.0328298i
$438$	0.786797	+	4.58579i	0.0375946	+	0.219117i
$439$		−	24.1421i		−	1.15224i	−0.817365	−	0.576121i	$-0.804566\pi$
$439$		−	24.1421i		−	1.15224i	0.817365	−	0.576121i	$-0.195434\pi$
$440$		0			0
$441$	6.31371	+	17.8579i	0.300653	+	0.850374i
$442$	4.00000	−	4.00000i	0.190261	−	0.190261i
$443$	13.3137	−	13.3137i	0.632553	−	0.632553i	−0.316154	−	0.948708i	$-0.602392\pi$
$443$	13.3137	−	13.3137i	0.632553	−	0.632553i	0.948708	+	0.316154i	$0.102392\pi$
$444$	−20.6863	−	14.6274i	−0.981728	−	0.694186i
$445$		0			0
$446$		−	9.51472i		−	0.450535i
$447$	30.3848	−	5.21320i	1.43715	−	0.246576i
$448$	2.44365	+	2.44365i	0.115452	+	0.115452i
$449$	−14.3431			−0.676895			−0.338447	−	0.940985i	$-0.609902\pi$
$449$	−14.3431			−0.676895			−0.338447	+	0.940985i	$0.609902\pi$
$450$		0			0
$451$	4.82843			0.227362
$452$	−15.5147	−	15.5147i	−0.729751	−	0.729751i
$453$	24.7279	−	4.24264i	1.16182	−	0.199337i
$454$			11.8579i			0.556517i
$455$		0			0
$456$	−1.85786	−	1.31371i	−0.0870025	−	0.0615200i
$457$	19.4142	−	19.4142i	0.908159	−	0.908159i	−0.0879650	−	0.996124i	$-0.528036\pi$
$457$	19.4142	−	19.4142i	0.908159	−	0.908159i	0.996124	+	0.0879650i	$0.0280364\pi$
$458$	−5.27208	+	5.27208i	−0.246348	+	0.246348i
$459$	−14.1421	−	4.00000i	−0.660098	−	0.186704i
$460$		0			0
$461$		−	23.1716i		−	1.07921i	−0.841919	−	0.539604i	$-0.818574\pi$
$461$		−	23.1716i		−	1.07921i	0.841919	−	0.539604i	$-0.181426\pi$
$462$	−0.100505	−	0.585786i	−0.00467592	−	0.0272533i
$463$	−22.3848	−	22.3848i	−1.04031	−	1.04031i	−0.999153	−	0.0411560i	$-0.986896\pi$
$463$	−22.3848	−	22.3848i	−1.04031	−	1.04031i	−0.0411560	−	0.999153i	$-0.513104\pi$
$464$	26.4853			1.22955
$465$		0			0
$466$	−5.85786			−0.271360
$467$	−12.8284	−	12.8284i	−0.593629	−	0.593629i	0.344981	−	0.938610i	$-0.387885\pi$
$467$	−12.8284	−	12.8284i	−0.593629	−	0.593629i	−0.938610	+	0.344981i	$0.887885\pi$
$468$	11.4142	−	23.8995i	0.527622	−	1.10475i
$469$			3.02944i			0.139886i
$470$		0			0
$471$	−21.6569	+	30.6274i	−0.997895	+	1.41124i
$472$	−2.62742	+	2.62742i	−0.120937	+	0.120937i
$473$	4.58579	−	4.58579i	0.210855	−	0.210855i
$474$	−2.00000	+	2.82843i	−0.0918630	+	0.129914i
$475$		0			0
$476$			4.28427i			0.196369i
$477$	15.5147	−	32.4853i	0.710370	−	1.48740i
$478$	0.686292	+	0.686292i	0.0313902	+	0.0313902i
$479$	−26.3431			−1.20365			−0.601825	−	0.798628i	$-0.705559\pi$
$479$	−26.3431			−1.20365			−0.601825	+	0.798628i	$0.705559\pi$
$480$		0			0
$481$	38.6274			1.76126
$482$	−7.89949	−	7.89949i	−0.359812	−	0.359812i
$483$	0.284271	+	1.65685i	0.0129348	+	0.0753895i
$484$			1.82843i			0.0831103i
$485$		0			0
$486$	6.44365	−	0.414214i	0.292290	−	0.0187891i
$487$	−10.5858	+	10.5858i	−0.479688	+	0.479688i	−0.905032	−	0.425344i	$-0.860153\pi$
$487$	−10.5858	+	10.5858i	−0.479688	+	0.479688i	0.425344	+	0.905032i	$0.360153\pi$
$488$	−6.72792	+	6.72792i	−0.304559	+	0.304559i
$489$	−19.7990	−	14.0000i	−0.895341	−	0.633102i
$490$		0			0
$491$			18.6274i			0.840644i	0.907375	+	0.420322i	$0.138083\pi$
$491$			18.6274i			0.840644i	−0.907375	+	0.420322i	$0.861917\pi$
$492$	15.0711	−	2.58579i	0.679456	−	0.116576i
$493$	17.6569	+	17.6569i	0.795225	+	0.795225i
$494$	1.65685			0.0745454
$495$		0			0
$496$	−12.0000			−0.538816
$497$	5.65685	+	5.65685i	0.253745	+	0.253745i
$498$	−4.24264	+	0.727922i	−0.190117	+	0.0326190i
$499$		−	2.34315i		−	0.104894i	−0.998624	−	0.0524468i	$-0.983298\pi$
$499$		−	2.34315i		−	0.104894i	0.998624	−	0.0524468i	$-0.0167020\pi$
$500$		0			0
$501$	16.4853	+	11.6569i	0.736508	+	0.520790i
$502$	−0.485281	+	0.485281i	−0.0216592	+	0.0216592i
$503$	−26.3848	+	26.3848i	−1.17644	+	1.17644i	−0.195794	+	0.980645i	$0.562728\pi$
$503$	−26.3848	+	26.3848i	−1.17644	+	1.17644i	−0.980645	+	0.195794i	$0.937272\pi$
$504$	−1.31371	−	3.71573i	−0.0585172	−	0.165512i
$505$		0			0
$506$			0.485281i			0.0215734i
$507$	3.02082	+	17.6066i	0.134159	+	0.781937i
$508$	18.7279	+	18.7279i	0.830917	+	0.830917i
$509$	35.3137			1.56525			0.782626	−	0.622492i	$-0.213880\pi$
$509$	35.3137			1.56525			0.782626	+	0.622492i	$0.213880\pi$
$510$		0			0
$511$	5.37258			0.237669
$512$	16.0919	+	16.0919i	0.711167	+	0.711167i
$513$	−2.10051	−	3.75736i	−0.0927396	−	0.165891i
$514$			9.37258i			0.413407i
$515$		0			0
$516$	11.8579	−	16.7696i	0.522013	−	0.738238i
$517$	−4.82843	+	4.82843i	−0.212354	+	0.212354i
$518$	1.94113	−	1.94113i	0.0852882	−	0.0852882i
$519$	−6.82843	+	9.65685i	−0.299735	+	0.423889i
$520$		0			0
$521$		−	40.9706i		−	1.79495i	−0.441062	−	0.897476i	$-0.645398\pi$
$521$		−	40.9706i		−	1.79495i	0.441062	−	0.897476i	$-0.354602\pi$
$522$	−9.89949	−	4.72792i	−0.433289	−	0.206936i
$523$	−2.92893	−	2.92893i	−0.128073	−	0.128073i	0.640165	−	0.768238i	$-0.278866\pi$
$523$	−2.92893	−	2.92893i	−0.128073	−	0.128073i	−0.768238	+	0.640165i	$0.778866\pi$
$524$	32.2843			1.41034
$525$		0			0
$526$	9.51472			0.414861
$527$	−8.00000	−	8.00000i	−0.348485	−	0.348485i
$528$	0.878680	+	5.12132i	0.0382396	+	0.222877i
$529$			21.6274i			0.940322i
$530$		0			0
$531$	−6.62742	+	2.34315i	−0.287605	+	0.101684i
$532$	−0.887302	+	0.887302i	−0.0384694	+	0.0384694i
$533$	−16.4853	+	16.4853i	−0.714057	+	0.714057i
$534$	−8.97056	−	6.34315i	−0.388194	−	0.274495i
$535$		0			0
$536$			5.79899i			0.250478i
$537$	−32.9706	+	5.65685i	−1.42278	+	0.244111i
$538$	−4.00000	−	4.00000i	−0.172452	−	0.172452i
$539$	6.31371			0.271951
$540$		0			0
$541$	−34.9706			−1.50350			−0.751751	−	0.659447i	$-0.770790\pi$
$541$	−34.9706			−1.50350			−0.751751	+	0.659447i	$0.770790\pi$
$542$	−2.78680	−	2.78680i	−0.119703	−	0.119703i
$543$	−10.2426	+	1.75736i	−0.439554	+	0.0754155i
$544$			12.4853i			0.535302i
$545$		0			0
$546$	2.34315	+	1.65685i	0.100277	+	0.0709068i
$547$	6.72792	−	6.72792i	0.287665	−	0.287665i	−0.548491	−	0.836156i	$-0.684798\pi$
$547$	6.72792	−	6.72792i	0.287665	−	0.287665i	0.836156	+	0.548491i	$0.184798\pi$
$548$	3.02944	−	3.02944i	0.129411	−	0.129411i
$549$	−16.9706	+	6.00000i	−0.724286	+	0.256074i
$550$		0			0
$551$			7.31371i			0.311574i
$552$	0.544156	+	3.17157i	0.0231608	+	0.134991i
$553$	2.82843	+	2.82843i	0.120277	+	0.120277i
$554$	10.0000			0.424859
$555$		0			0
$556$	−30.7696			−1.30492
$557$	19.4558	+	19.4558i	0.824371	+	0.824371i	0.986731	−	0.162361i	$-0.0519109\pi$
$557$	19.4558	+	19.4558i	0.824371	+	0.824371i	−0.162361	+	0.986731i	$0.551911\pi$
$558$	4.48528	+	2.14214i	0.189877	+	0.0906838i
$559$			31.3137i			1.32443i
$560$		0			0
$561$	−2.82843	+	4.00000i	−0.119416	+	0.168880i
$562$	2.38478	−	2.38478i	0.100596	−	0.100596i
$563$	−0.242641	+	0.242641i	−0.0102261	+	0.0102261i	−0.712201	−	0.701975i	$-0.752302\pi$
$563$	−0.242641	+	0.242641i	−0.0102261	+	0.0102261i	0.701975	+	0.712201i	$0.252302\pi$
$564$	−12.4853	+	17.6569i	−0.525725	+	0.743488i
$565$		0			0
$566$			2.68629i			0.112913i
$567$	0.786797	−	7.41421i	0.0330423	−	0.311368i
$568$	10.8284	+	10.8284i	0.454351	+	0.454351i
$569$	−12.8284			−0.537796			−0.268898	−	0.963169i	$-0.586659\pi$
$569$	−12.8284			−0.537796			−0.268898	+	0.963169i	$0.586659\pi$
$570$		0			0
$571$	−21.5147			−0.900363			−0.450181	−	0.892937i	$-0.648641\pi$
$571$	−21.5147			−0.900363			−0.450181	+	0.892937i	$0.648641\pi$
$572$	−6.24264	−	6.24264i	−0.261018	−	0.261018i
$573$	−3.31371	−	19.3137i	−0.138432	−	0.806842i
$574$			1.65685i			0.0691558i
$575$		0			0
$576$	4.17157	+	11.7990i	0.173816	+	0.491625i
$577$	3.51472	−	3.51472i	0.146320	−	0.146320i	−0.630152	−	0.776472i	$-0.717007\pi$
$577$	3.51472	−	3.51472i	0.146320	−	0.146320i	0.776472	+	0.630152i	$0.217007\pi$
$578$	2.63604	−	2.63604i	0.109645	−	0.109645i
$579$	18.1421	+	12.8284i	0.753961	+	0.533131i
$580$		0			0
$581$			4.97056i			0.206214i
$582$	−1.65685	+	0.284271i	−0.0686788	+	0.0117834i
$583$	−8.48528	−	8.48528i	−0.351424	−	0.351424i
$584$	10.2843			0.425566
$585$		0			0
$586$	−1.45584			−0.0601404
$587$	22.4853	+	22.4853i	0.928067	+	0.928067i	0.997581	−	0.0695141i	$-0.0221449\pi$
$587$	22.4853	+	22.4853i	0.928067	+	0.928067i	−0.0695141	+	0.997581i	$0.522145\pi$
$588$	19.7071	−	3.38120i	0.812707	−	0.139439i
$589$		−	3.31371i		−	0.136539i
$590$		0			0
$591$	−28.9706	−	20.4853i	−1.19169	−	0.842652i
$592$	−16.9706	+	16.9706i	−0.697486	+	0.697486i
$593$	19.6569	−	19.6569i	0.807210	−	0.807210i	−0.177001	−	0.984211i	$-0.556639\pi$
$593$	19.6569	−	19.6569i	0.807210	−	0.807210i	0.984211	+	0.177001i	$0.0566394\pi$
$594$	0.585786	−	2.07107i	0.0240351	−	0.0849769i
$595$		0			0
$596$		−	32.5442i		−	1.33306i
$597$	−3.79899	−	22.1421i	−0.155482	−	0.906217i
$598$	−1.65685	−	1.65685i	−0.0677538	−	0.0677538i
$599$	23.3137			0.952572			0.476286	−	0.879290i	$-0.341983\pi$
$599$	23.3137			0.952572			0.476286	+	0.879290i	$0.341983\pi$
$600$		0			0
$601$	22.9706			0.936989			0.468494	−	0.883466i	$-0.344797\pi$
$601$	22.9706			0.936989			0.468494	+	0.883466i	$0.344797\pi$
$602$	1.57359	+	1.57359i	0.0641349	+	0.0641349i
$603$	−4.72792	+	9.89949i	−0.192536	+	0.403139i
$604$		−	26.4853i		−	1.07767i
$605$		0			0
$606$	−0.343146	+	0.485281i	−0.0139393	+	0.0197132i
$607$	−8.10051	+	8.10051i	−0.328789	+	0.328789i	−0.852126	−	0.523337i	$-0.824687\pi$
$607$	−8.10051	+	8.10051i	−0.328789	+	0.328789i	0.523337	+	0.852126i	$0.324687\pi$
$608$	−2.58579	+	2.58579i	−0.104867	+	0.104867i
$609$	−7.31371	+	10.3431i	−0.296366	+	0.419125i
$610$		0			0
$611$		−	32.9706i		−	1.33385i
$612$	−6.68629	+	14.0000i	−0.270277	+	0.565916i
$613$	18.0416	+	18.0416i	0.728695	+	0.728695i	0.970360	−	0.241665i	$-0.0776935\pi$
$613$	18.0416	+	18.0416i	0.728695	+	0.728695i	−0.241665	+	0.970360i	$0.577694\pi$
$614$	−10.9706			−0.442736
$615$		0			0
$616$	−1.31371			−0.0529308
$617$	7.51472	+	7.51472i	0.302531	+	0.302531i	0.842003	−	0.539472i	$-0.181376\pi$
$617$	7.51472	+	7.51472i	0.302531	+	0.302531i	−0.539472	+	0.842003i	$0.681376\pi$
$618$	0.727922	+	4.24264i	0.0292813	+	0.170664i
$619$			4.97056i			0.199784i	0.994998	+	0.0998919i	$0.0318497\pi$
$619$			4.97056i			0.199784i	−0.994998	+	0.0998919i	$0.968150\pi$
$620$		0			0
$621$	−1.65685	+	5.85786i	−0.0664873	+	0.235068i
$622$	3.02944	−	3.02944i	0.121469	−	0.121469i
$623$	−8.97056	+	8.97056i	−0.359398	+	0.359398i
$624$	−20.4853	−	14.4853i	−0.820068	−	0.579875i
$625$		0			0
$626$			11.0294i			0.440825i
$627$	−1.41421	+	0.242641i	−0.0564782	+	0.00969014i
$628$	28.0000	+	28.0000i	1.11732	+	1.11732i
$629$	−22.6274			−0.902214
$630$		0			0
$631$	1.65685			0.0659583			0.0329792	−	0.999456i	$-0.489501\pi$
$631$	1.65685			0.0659583			0.0329792	+	0.999456i	$0.489501\pi$
$632$	5.41421	+	5.41421i	0.215366	+	0.215366i
$633$	−17.8995	+	3.07107i	−0.711441	+	0.122064i
$634$		−	10.6274i		−	0.422069i
$635$		0			0
$636$	−31.0294	−	21.9411i	−1.23040	−	0.870022i
$637$	−21.5563	+	21.5563i	−0.854094	+	0.854094i
$638$	−2.58579	+	2.58579i	−0.102372	+	0.102372i
$639$	9.65685	+	27.3137i	0.382019	+	1.08051i
$640$		0			0
$641$			6.34315i			0.250539i	0.992123	+	0.125270i	$0.0399796\pi$
$641$			6.34315i			0.250539i	−0.992123	+	0.125270i	$0.960020\pi$
$642$	2.38478	+	13.8995i	0.0941196	+	0.548569i
$643$	1.41421	+	1.41421i	0.0557711	+	0.0557711i	0.734442	−	0.678671i	$-0.237444\pi$
$643$	1.41421	+	1.41421i	0.0557711	+	0.0557711i	−0.678671	+	0.734442i	$0.737444\pi$
$644$	1.77460			0.0699292
$645$		0			0
$646$	−0.970563			−0.0381863
$647$	−6.00000	−	6.00000i	−0.235884	−	0.235884i	0.579259	−	0.815143i	$-0.303342\pi$
$647$	−6.00000	−	6.00000i	−0.235884	−	0.235884i	−0.815143	+	0.579259i	$0.803342\pi$
$648$	1.50610	−	14.1924i	0.0591651	−	0.557530i
$649$			2.34315i			0.0919765i
$650$		0			0
$651$	3.31371	−	4.68629i	0.129874	−	0.183670i
$652$	−18.1005	+	18.1005i	−0.708870	+	0.708870i
$653$	−4.00000	+	4.00000i	−0.156532	+	0.156532i	−0.781028	−	0.624496i	$-0.785304\pi$
$653$	−4.00000	+	4.00000i	−0.156532	+	0.156532i	0.624496	+	0.781028i	$0.285304\pi$
$654$	0.828427	−	1.17157i	0.0323941	−	0.0458121i
$655$		0			0
$656$		−	14.4853i		−	0.565555i
$657$	17.5563	+	8.38478i	0.684938	+	0.327121i
$658$	−1.65685	−	1.65685i	−0.0645909	−	0.0645909i
$659$	−30.3431			−1.18200			−0.591001	−	0.806671i	$-0.701267\pi$
$659$	−30.3431			−1.18200			−0.591001	+	0.806671i	$0.701267\pi$
$660$		0			0
$661$	−44.6274			−1.73581			−0.867903	−	0.496734i	$-0.834532\pi$
$661$	−44.6274			−1.73581			−0.867903	+	0.496734i	$0.834532\pi$
$662$	0.485281	+	0.485281i	0.0188610	+	0.0188610i
$663$	−4.00000	−	23.3137i	−0.155347	−	0.905429i
$664$			9.51472i			0.369243i
$665$		0			0
$666$	9.37258	−	3.31371i	0.363180	−	0.128404i
$667$	7.31371	−	7.31371i	0.283188	−	0.283188i
$668$	15.0711	−	15.0711i	0.583117	−	0.583117i
$669$	−32.4853	−	22.9706i	−1.25595	−	0.888093i
$670$		0			0
$671$			6.00000i			0.231627i
$672$	−6.24264	+	1.07107i	−0.240815	+	0.0413173i
$673$	−30.7279	−	30.7279i	−1.18447	−	1.18447i	−0.978573	−	0.205902i	$-0.933987\pi$
$673$	−30.7279	−	30.7279i	−1.18447	−	1.18447i	−0.205902	−	0.978573i	$-0.566013\pi$
$674$	9.31371			0.358751
$675$		0			0
$676$	18.8579			0.725302
$677$	−14.4853	−	14.4853i	−0.556715	−	0.556715i	0.371656	−	0.928371i	$-0.378790\pi$
$677$	−14.4853	−	14.4853i	−0.556715	−	0.556715i	−0.928371	+	0.371656i	$0.878790\pi$
$678$	8.48528	−	1.45584i	0.325875	−	0.0559114i
$679$			1.94113i			0.0744936i
$680$		0			0
$681$	40.4853	+	28.6274i	1.55140	+	1.09701i
$682$	1.17157	−	1.17157i	0.0448618	−	0.0448618i
$683$	10.4853	−	10.4853i	0.401208	−	0.401208i	−0.477451	−	0.878659i	$-0.658439\pi$
$683$	10.4853	−	10.4853i	0.401208	−	0.401208i	0.878659	+	0.477451i	$0.158439\pi$
$684$	−4.28427	+	1.51472i	−0.163813	+	0.0579167i
$685$		0			0
$686$			4.56854i			0.174428i
$687$	5.27208	+	30.7279i	0.201142	+	1.17234i
$688$	−13.7574	−	13.7574i	−0.524494	−	0.524494i
$689$	57.9411			2.20738
$690$		0			0
$691$	29.9411			1.13901			0.569507	−	0.821986i	$-0.307134\pi$
$691$	29.9411			1.13901			0.569507	+	0.821986i	$0.307134\pi$
$692$	8.82843	+	8.82843i	0.335606	+	0.335606i
$693$	−2.24264	−	1.07107i	−0.0851909	−	0.0406865i
$694$		−	4.54416i		−	0.172494i
$695$		0			0
$696$	−14.0000	+	19.7990i	−0.530669	+	0.750479i
$697$	9.65685	−	9.65685i	0.365779	−	0.365779i
$698$	3.21320	−	3.21320i	0.121622	−	0.121622i
$699$	−14.1421	+	20.0000i	−0.534905	+	0.756469i
$700$		0			0
$701$		−	33.5147i		−	1.26583i	−0.774220	−	0.632917i	$-0.781858\pi$
$701$		−	33.5147i		−	1.26583i	0.774220	−	0.632917i	$-0.218142\pi$
$702$	5.07107	+	9.07107i	0.191395	+	0.342365i
$703$	−4.68629	−	4.68629i	−0.176747	−	0.176747i
$704$	4.17157			0.157222
$705$		0			0
$706$	7.31371			0.275255
$707$	0.485281	+	0.485281i	0.0182509	+	0.0182509i
$708$	1.25483	+	7.31371i	0.0471595	+	0.274866i
$709$		−	10.0000i		−	0.375558i	−0.982211	−	0.187779i	$-0.939871\pi$
$709$		−	10.0000i		−	0.375558i	0.982211	−	0.187779i	$-0.0601289\pi$
$710$		0			0
$711$	4.82843	+	13.6569i	0.181080	+	0.512172i
$712$	−17.1716	+	17.1716i	−0.643532	+	0.643532i
$713$	−3.31371	+	3.31371i	−0.124099	+	0.124099i
$714$	−1.37258	−	0.970563i	−0.0513676	−	0.0363224i
$715$		0			0
$716$			35.3137i			1.31974i
$717$	4.00000	−	0.686292i	0.149383	−	0.0256300i
$718$	−4.97056	−	4.97056i	−0.185500	−	0.185500i
$719$	−4.68629			−0.174769			−0.0873846	−	0.996175i	$-0.527851\pi$
$719$	−4.68629			−0.174769			−0.0873846	+	0.996175i	$0.527851\pi$
$720$		0			0
$721$	4.97056			0.185113
$722$	5.36396	+	5.36396i	0.199626	+	0.199626i
$723$	−46.0416	+	7.89949i	−1.71231	+	0.293785i
$724$			10.9706i			0.407718i
$725$		0			0
$726$	−0.585786	−	0.414214i	−0.0217406	−	0.0153729i
$727$	−9.89949	+	9.89949i	−0.367152	+	0.367152i	−0.866437	−	0.499286i	$-0.833596\pi$
$727$	−9.89949	+	9.89949i	−0.367152	+	0.367152i	0.499286	+	0.866437i	$0.333596\pi$
$728$	4.48528	−	4.48528i	0.166236	−	0.166236i
$729$	14.1421	−	23.0000i	0.523783	−	0.851852i
$730$		0			0
$731$		−	18.3431i		−	0.678446i
$732$	3.21320	+	18.7279i	0.118763	+	0.692204i
$733$	−7.89949	−	7.89949i	−0.291775	−	0.291775i	0.546006	−	0.837781i	$-0.316147\pi$
$733$	−7.89949	−	7.89949i	−0.291775	−	0.291775i	−0.837781	+	0.546006i	$0.816147\pi$
$734$	4.54416			0.167728
$735$		0			0
$736$	5.17157			0.190627
$737$	2.58579	+	2.58579i	0.0952487	+	0.0952487i
$738$	−2.58579	+	5.41421i	−0.0951841	+	0.199300i
$739$			5.51472i			0.202862i	0.994843	+	0.101431i	$0.0323421\pi$
$739$			5.51472i			0.202862i	−0.994843	+	0.101431i	$0.967658\pi$
$740$		0			0
$741$	4.00000	−	5.65685i	0.146944	−	0.207810i
$742$	2.91169	−	2.91169i	0.106891	−	0.106891i
$743$	−16.7279	+	16.7279i	−0.613688	+	0.613688i	−0.943905	−	0.330217i	$-0.892878\pi$
$743$	−16.7279	+	16.7279i	−0.613688	+	0.613688i	0.330217	+	0.943905i	$0.392878\pi$
$744$	6.34315	−	8.97056i	0.232551	−	0.328877i
$745$		0			0
$746$		−	9.31371i		−	0.340999i
$747$	−7.75736	+	16.2426i	−0.283827	+	0.594287i
$748$	3.65685	+	3.65685i	0.133708	+	0.133708i
$749$	16.2843			0.595014
$750$		0			0
$751$	−32.2843			−1.17807			−0.589035	−	0.808108i	$-0.700492\pi$
$751$	−32.2843			−1.17807			−0.589035	+	0.808108i	$0.700492\pi$
$752$	14.4853	+	14.4853i	0.528224	+	0.528224i
$753$	0.485281	+	2.82843i	0.0176846	+	0.103074i
$754$		−	17.6569i		−	0.643025i
$755$		0			0
$756$	−7.57359	−	2.14214i	−0.275449	−	0.0779087i
$757$	0.686292	−	0.686292i	0.0249437	−	0.0249437i	−0.694525	−	0.719469i	$-0.744385\pi$
$757$	0.686292	−	0.686292i	0.0249437	−	0.0249437i	0.719469	+	0.694525i	$0.244385\pi$
$758$	−5.45584	+	5.45584i	−0.198165	+	0.198165i
$759$	1.65685	+	1.17157i	0.0601400	+	0.0425254i
$760$		0			0
$761$			13.7990i			0.500213i	0.968218	+	0.250106i	$0.0804656\pi$
$761$			13.7990i			0.500213i	−0.968218	+	0.250106i	$0.919534\pi$
$762$	−10.2426	+	1.75736i	−0.371052	+	0.0636624i
$763$	−1.17157	−	1.17157i	−0.0424138	−	0.0424138i
$764$	−20.6863			−0.748404
$765$		0			0
$766$	−5.45584			−0.197128
$767$	−8.00000	−	8.00000i	−0.288863	−	0.288863i
$768$	6.77817	−	1.16295i	0.244586	−	0.0419644i
$769$		−	1.31371i		−	0.0473735i	−0.999719	−	0.0236868i	$-0.992460\pi$
$769$		−	1.31371i		−	0.0473735i	0.999719	−	0.0236868i	$-0.00754044\pi$
$770$		0			0
$771$	32.0000	+	22.6274i	1.15245	+	0.814907i
$772$	16.5858	−	16.5858i	0.596936	−	0.596936i
$773$	−4.97056	+	4.97056i	−0.178779	+	0.178779i	−0.790823	−	0.612045i	$-0.790347\pi$
$773$	−4.97056	+	4.97056i	−0.178779	+	0.178779i	0.612045	+	0.790823i	$0.290347\pi$
$774$	2.68629	+	7.59798i	0.0965568	+	0.273104i
$775$		0			0
$776$			3.71573i			0.133387i
$777$	−1.94113	−	11.3137i	−0.0696375	−	0.405877i
$778$	3.51472	+	3.51472i	0.126009	+	0.126009i
$779$	4.00000			0.143315
$780$		0			0
$781$	9.65685			0.345549
$782$	0.970563	+	0.970563i	0.0347073	+	0.0347073i
$783$	−40.0416	+	22.3848i	−1.43097	+	0.799967i
$784$		−	18.9411i		−	0.676469i
$785$		0			0
$786$	−7.31371	+	10.3431i	−0.260871	+	0.368928i
$787$	−39.2132	+	39.2132i	−1.39780	+	1.39780i	−0.591480	+	0.806319i	$0.701456\pi$
$787$	−39.2132	+	39.2132i	−1.39780	+	1.39780i	−0.806319	+	0.591480i	$0.798544\pi$
$788$	−26.4853	+	26.4853i	−0.943499	+	0.943499i
$789$	22.9706	−	32.4853i	0.817774	−	1.15651i
$790$		0			0
$791$		−	9.94113i		−	0.353466i
$792$	−4.29289	−	2.05025i	−0.152541	−	0.0728526i
$793$	−20.4853	−	20.4853i	−0.727454	−	0.727454i
$794$	13.3726			0.474575
$795$		0			0
$796$	−23.7157			−0.840582
$797$	−6.82843	−	6.82843i	−0.241875	−	0.241875i	0.575750	−	0.817626i	$-0.304710\pi$
$797$	−6.82843	−	6.82843i	−0.241875	−	0.241875i	−0.817626	+	0.575750i	$0.804710\pi$
$798$	−0.0832611	−	0.485281i	−0.00294741	−	0.0171788i
$799$			19.3137i			0.683270i
$800$		0			0
$801$	−43.3137	+	15.3137i	−1.53041	+	0.541083i
$802$	5.94113	−	5.94113i	0.209788	−	0.209788i
$803$	4.58579	−	4.58579i	0.161829	−	0.161829i
$804$	9.45584	+	6.68629i	0.333482	+	0.235807i
$805$		0			0
$806$			8.00000i			0.281788i
$807$	−23.3137	+	4.00000i	−0.820681	+	0.140807i
$808$	0.928932	+	0.928932i	0.0326797	+	0.0326797i
$809$	−2.20101			−0.0773834			−0.0386917	−	0.999251i	$-0.512319\pi$
$809$	−2.20101			−0.0773834			−0.0386917	+	0.999251i	$0.512319\pi$
$810$		0			0
$811$	31.1716			1.09458			0.547291	−	0.836942i	$-0.315659\pi$
$811$	31.1716			1.09458			0.547291	+	0.836942i	$0.315659\pi$
$812$	9.45584	+	9.45584i	0.331835	+	0.331835i
$813$	−16.2426	+	2.78680i	−0.569654	+	0.0977372i
$814$		−	3.31371i		−	0.116145i
$815$		0			0
$816$	12.0000	+	8.48528i	0.420084	+	0.297044i
$817$	3.79899	−	3.79899i	0.132910	−	0.132910i
$818$	4.78680	−	4.78680i	0.167366	−	0.167366i
$819$	11.3137	−	4.00000i	0.395333	−	0.139771i
$820$		0			0
$821$			3.17157i			0.110689i	0.998467	+	0.0553443i	$0.0176257\pi$
$821$			3.17157i			0.110689i	−0.998467	+	0.0553443i	$0.982374\pi$
$822$	0.284271	+	1.65685i	0.00991510	+	0.0577894i
$823$	−39.5563	−	39.5563i	−1.37885	−	1.37885i	−0.846570	−	0.532278i	$-0.821336\pi$
$823$	−39.5563	−	39.5563i	−1.37885	−	1.37885i	−0.532278	−	0.846570i	$-0.678664\pi$
$824$	9.51472			0.331461
$825$		0			0
$826$	−0.804041			−0.0279761
$827$	23.7574	+	23.7574i	0.826124	+	0.826124i	0.986978	−	0.160854i	$-0.0514248\pi$
$827$	23.7574	+	23.7574i	0.826124	+	0.826124i	−0.160854	+	0.986978i	$0.551425\pi$
$828$	5.79899	+	2.76955i	0.201529	+	0.0962486i
$829$		−	45.3137i		−	1.57381i	−0.617074	−	0.786905i	$-0.711682\pi$
$829$		−	45.3137i		−	1.57381i	0.617074	−	0.786905i	$-0.288318\pi$
$830$		0			0
$831$	24.1421	−	34.1421i	0.837481	−	1.18438i
$832$	−14.2426	+	14.2426i	−0.493775	+	0.493775i
$833$	12.6274	−	12.6274i	0.437514	−	0.437514i
$834$	6.97056	−	9.85786i	0.241371	−	0.341350i
$835$		0			0
$836$			1.51472i			0.0523876i
$837$	18.1421	−	10.1421i	0.627084	−	0.350563i
$838$	10.3431	+	10.3431i	0.357298	+	0.357298i
$839$	−39.5980			−1.36707			−0.683537	−	0.729916i	$-0.739559\pi$
$839$	−39.5980			−1.36707			−0.683537	+	0.729916i	$0.739559\pi$
$840$		0			0
$841$	48.9411			1.68763
$842$	3.89949	+	3.89949i	0.134385	+	0.134385i
$843$	−2.38478	−	13.8995i	−0.0821361	−	0.478724i
$844$			19.1716i			0.659913i
$845$		0			0
$846$	−2.82843	−	8.00000i	−0.0972433	−	0.275046i
$847$	−0.585786	+	0.585786i	−0.0201279	+	0.0201279i
$848$	−25.4558	+	25.4558i	−0.874157	+	0.874157i
$849$	9.17157	+	6.48528i	0.314768	+	0.222574i
$850$		0			0
$851$			9.37258i			0.321288i
$852$	30.1421	−	5.17157i	1.03265	−	0.177175i
$853$	4.58579	+	4.58579i	0.157014	+	0.157014i	0.781242	−	0.624228i	$-0.214586\pi$
$853$	4.58579	+	4.58579i	0.157014	+	0.157014i	−0.624228	+	0.781242i	$0.714586\pi$
$854$	−2.05887			−0.0704532
$855$		0			0
$856$	31.1716			1.06542
$857$	−19.6569	−	19.6569i	−0.671465	−	0.671465i	0.286588	−	0.958054i	$-0.407479\pi$
$857$	−19.6569	−	19.6569i	−0.671465	−	0.671465i	−0.958054	+	0.286588i	$0.907479\pi$
$858$	3.41421	−	0.585786i	0.116559	−	0.0199984i
$859$		−	28.9706i		−	0.988463i	−0.869330	−	0.494231i	$-0.835450\pi$
$859$		−	28.9706i		−	0.988463i	0.869330	−	0.494231i	$-0.164550\pi$
$860$		0			0
$861$	5.65685	+	4.00000i	0.192785	+	0.136320i
$862$	−1.65685	+	1.65685i	−0.0564327	+	0.0564327i
$863$	−30.9706	+	30.9706i	−1.05425	+	1.05425i	−0.0558087	+	0.998441i	$0.517774\pi$
$863$	−30.9706	+	30.9706i	−1.05425	+	1.05425i	−0.998441	+	0.0558087i	$0.982226\pi$
$864$	−22.0711	−	6.24264i	−0.750873	−	0.212379i
$865$		0			0
$866$		−	4.28427i		−	0.145585i
$867$	−2.63604	−	15.3640i	−0.0895246	−	0.521787i
$868$	−4.28427	−	4.28427i	−0.145418	−	0.145418i
$869$	4.82843			0.163793
$870$		0			0
$871$	−17.6569			−0.598280
$872$	−2.24264	−	2.24264i	−0.0759454	−	0.0759454i
$873$	−3.02944	+	6.34315i	−0.102531	+	0.214683i
$874$			0.402020i			0.0135985i
$875$		0			0
$876$	11.8579	−	16.7696i	0.400640	−	0.566591i
$877$	9.07107	−	9.07107i	0.306308	−	0.306308i	−0.537167	−	0.843476i	$-0.680506\pi$
$877$	9.07107	−	9.07107i	0.306308	−	0.306308i	0.843476	+	0.537167i	$0.180506\pi$
$878$	7.07107	−	7.07107i	0.238637	−	0.238637i
$879$	−3.51472	+	4.97056i	−0.118549	+	0.167653i
$880$		0			0
$881$		−	12.0000i		−	0.404290i	−0.979356	−	0.202145i	$-0.935209\pi$
$881$		−	12.0000i		−	0.404290i	0.979356	−	0.202145i	$-0.0647913\pi$
$882$	−3.38120	+	7.07969i	−0.113851	+	0.238386i
$883$	−6.58579	−	6.58579i	−0.221629	−	0.221629i	0.587555	−	0.809184i	$-0.300091\pi$
$883$	−6.58579	−	6.58579i	−0.221629	−	0.221629i	−0.809184	+	0.587555i	$0.800091\pi$
$884$	−24.9706			−0.839851
$885$		0			0
$886$	7.79899			0.262012
$887$	8.72792	+	8.72792i	0.293055	+	0.293055i	0.838286	−	0.545231i	$-0.183558\pi$
$887$	8.72792	+	8.72792i	0.293055	+	0.293055i	−0.545231	+	0.838286i	$0.683558\pi$
$888$	−3.71573	−	21.6569i	−0.124692	−	0.726756i
$889$			12.0000i			0.402467i
$890$		0			0
$891$	−5.65685	−	7.00000i	−0.189512	−	0.234509i
$892$	−29.6985	+	29.6985i	−0.994379	+	0.994379i
$893$	−4.00000	+	4.00000i	−0.133855	+	0.133855i
$894$	10.4264	+	7.37258i	0.348711	+	0.246576i
$895$		0			0
$896$			8.74517i			0.292155i
$897$	−9.65685	+	1.65685i	−0.322433	+	0.0553208i
$898$	−4.20101	−	4.20101i	−0.140190	−	0.140190i
$899$	−35.3137			−1.17778
$900$		0			0
$901$	−33.9411			−1.13074
$902$	1.41421	+	1.41421i	0.0470882	+	0.0470882i
$903$	9.17157	−	1.57359i	0.305211	−	0.0523659i
$904$		−	19.0294i		−	0.632910i
$905$		0			0
$906$	8.48528	+	6.00000i	0.281905	+	0.199337i
$907$	−24.7279	+	24.7279i	−0.821077	+	0.821077i	−0.986263	−	0.165185i	$-0.947178\pi$
$907$	−24.7279	+	24.7279i	−0.821077	+	0.821077i	0.165185	+	0.986263i	$0.447178\pi$
$908$	37.0122	−	37.0122i	1.22829	−	1.22829i
$909$	0.828427	+	2.34315i	0.0274772	+	0.0777172i
$910$		0			0
$911$			30.6274i			1.01473i	0.861731	+	0.507366i	$0.169381\pi$
$911$			30.6274i			1.01473i	−0.861731	+	0.507366i	$0.830619\pi$
$912$	0.727922	+	4.24264i	0.0241039	+	0.140488i
$913$	4.24264	+	4.24264i	0.140411	+	0.140411i
$914$	11.3726			0.376172
$915$		0			0
$916$	32.9117			1.08743
$917$	10.3431	+	10.3431i	0.341561	+	0.341561i
$918$	−2.97056	−	5.31371i	−0.0980432	−	0.175379i
$919$		−	14.4853i		−	0.477825i	−0.971041	−	0.238913i	$-0.923209\pi$
$919$		−	14.4853i		−	0.477825i	0.971041	−	0.238913i	$-0.0767910\pi$
$920$		0			0
$921$	−26.4853	+	37.4558i	−0.872720	+	1.23421i
$922$	6.78680	−	6.78680i	0.223511	−	0.223511i
$923$	−32.9706	+	32.9706i	−1.08524	+	1.08524i
$924$	−1.51472	+	2.14214i	−0.0498306	+	0.0704711i
$925$		0			0
$926$		−	13.1127i		−	0.430910i
$927$	16.2426	+	7.75736i	0.533478	+	0.254785i
$928$	27.5563	+	27.5563i	0.904581	+	0.904581i
$929$	−45.9411			−1.50728			−0.753640	−	0.657288i	$-0.771704\pi$
$929$	−45.9411			−1.50728			−0.753640	+	0.657288i	$0.771704\pi$
$930$		0			0
$931$	5.23045			0.171421
$932$	18.2843	+	18.2843i	0.598921	+	0.598921i
$933$	−3.02944	−	17.6569i	−0.0991793	−	0.578059i
$934$		−	7.51472i		−	0.245889i
$935$		0			0
$936$	21.6569	−	7.65685i	0.707876	−	0.250272i
$937$	−10.2426	+	10.2426i	−0.334612	+	0.334612i	−0.854335	−	0.519723i	$-0.826035\pi$
$937$	−10.2426	+	10.2426i	−0.334612	+	0.334612i	0.519723	+	0.854335i	$0.326035\pi$
$938$	−0.887302	+	0.887302i	−0.0289714	+	0.0289714i
$939$	37.6569	+	26.6274i	1.22888	+	0.868953i
$940$		0			0
$941$			13.7990i			0.449834i	0.974378	+	0.224917i	$0.0722111\pi$
$941$			13.7990i			0.449834i	−0.974378	+	0.224917i	$0.927789\pi$
$942$	−15.3137	+	2.62742i	−0.498948	+	0.0856059i
$943$	−4.00000	−	4.00000i	−0.130258	−	0.130258i
$944$	7.02944			0.228789
$945$		0			0
$946$	2.68629			0.0873389
$947$	−11.4558	−	11.4558i	−0.372265	−	0.372265i	0.496037	−	0.868302i	$-0.334788\pi$
$947$	−11.4558	−	11.4558i	−0.372265	−	0.372265i	−0.868302	+	0.496037i	$0.834788\pi$
$948$	15.0711	−	2.58579i	0.489486	−	0.0839824i
$949$			31.3137i			1.01649i
$950$		0			0
$951$	−36.2843	−	25.6569i	−1.17660	−	0.831981i
$952$	−2.62742	+	2.62742i	−0.0851551	+	0.0851551i
$953$	26.0000	−	26.0000i	0.842223	−	0.842223i	−0.146925	−	0.989148i	$-0.546938\pi$
$953$	26.0000	−	26.0000i	0.842223	−	0.842223i	0.989148	+	0.146925i	$0.0469376\pi$
$954$	14.0589	−	4.97056i	0.455173	−	0.160928i
$955$		0			0
$956$		−	4.28427i		−	0.138563i
$957$	2.58579	+	15.0711i	0.0835866	+	0.487178i
$958$	−7.71573	−	7.71573i	−0.249284	−	0.249284i
$959$	1.94113			0.0626822
$960$		0			0
$961$	−15.0000			−0.483871
$962$	11.3137	+	11.3137i	0.364769	+	0.364769i
$963$	53.2132	+	25.4142i	1.71477	+	0.818962i
$964$			49.3137i			1.58829i
$965$		0			0
$966$	−0.402020	+	0.568542i	−0.0129348	+	0.0182926i
$967$	25.0711	−	25.0711i	0.806231	−	0.806231i	−0.177830	−	0.984061i	$-0.556908\pi$
$967$	25.0711	−	25.0711i	0.806231	−	0.806231i	0.984061	+	0.177830i	$0.0569077\pi$
$968$	−1.12132	+	1.12132i	−0.0360406	+	0.0360406i
$969$	−2.34315	+	3.31371i	−0.0752727	+	0.106452i
$970$		0			0
$971$		−	10.6274i		−	0.341050i	−0.985353	−	0.170525i	$-0.945454\pi$
$971$		−	10.6274i		−	0.341050i	0.985353	−	0.170525i	$-0.0545464\pi$
$972$	−21.4056	−	18.8198i	−0.686585	−	0.603646i
$973$	−9.85786	−	9.85786i	−0.316029	−	0.316029i
$974$	−6.20101			−0.198693
$975$		0			0
$976$	18.0000			0.576166
$977$	20.0000	+	20.0000i	0.639857	+	0.639857i	0.950520	−	0.310663i	$-0.100551\pi$
$977$	20.0000	+	20.0000i	0.639857	+	0.639857i	−0.310663	+	0.950520i	$0.600551\pi$
$978$	−1.69848	−	9.89949i	−0.0543116	−	0.316551i
$979$			15.3137i			0.489428i
$980$		0			0
$981$	−2.00000	−	5.65685i	−0.0638551	−	0.180609i
$982$	−5.45584	+	5.45584i	−0.174103	+	0.174103i
$983$	8.34315	−	8.34315i	0.266105	−	0.266105i	−0.561424	−	0.827529i	$-0.689746\pi$
$983$	8.34315	−	8.34315i	0.266105	−	0.266105i	0.827529	+	0.561424i	$0.189746\pi$
$984$	10.8284	+	7.65685i	0.345198	+	0.244092i
$985$		0			0
$986$			10.3431i			0.329393i
$987$	−9.65685	+	1.65685i	−0.307381	+	0.0527383i
$988$	−5.17157	−	5.17157i	−0.164530	−	0.164530i
$989$	−7.59798			−0.241602
$990$		0			0
$991$	−37.9411			−1.20524			−0.602620	−	0.798028i	$-0.705876\pi$
$991$	−37.9411			−1.20524			−0.602620	+	0.798028i	$0.705876\pi$
$992$	−12.4853	−	12.4853i	−0.396408	−	0.396408i
$993$	2.82843	−	0.485281i	0.0897574	−	0.0153999i
$994$			3.31371i			0.105104i
$995$		0			0
$996$	15.5147	+	10.9706i	0.491603	+	0.347616i
$997$	21.5563	−	21.5563i	0.682696	−	0.682696i	−0.277911	−	0.960607i	$-0.589642\pi$
$997$	21.5563	−	21.5563i	0.682696	−	0.682696i	0.960607	+	0.277911i	$0.0896419\pi$
$998$	0.686292	−	0.686292i	0.0217242	−	0.0217242i
$999$	11.3137	−	40.0000i	0.357950	−	1.26554i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	825.2.k.h.518.1	yes	4
3.2	odd	2		825.2.k.b.518.2	yes	4
5.2	odd	4		825.2.k.b.782.2	yes	4
5.3	odd	4		825.2.k.g.782.1	yes	4
5.4	even	2		825.2.k.a.518.2	✓	4
15.2	even	4	inner	825.2.k.h.782.1	yes	4
15.8	even	4		825.2.k.a.782.2	yes	4
15.14	odd	2		825.2.k.g.518.1	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
825.2.k.a.518.2	✓	4	5.4	even	2
825.2.k.a.782.2	yes	4	15.8	even	4
825.2.k.b.518.2	yes	4	3.2	odd	2
825.2.k.b.782.2	yes	4	5.2	odd	4
825.2.k.g.518.1	yes	4	15.14	odd	2
825.2.k.g.782.1	yes	4	5.3	odd	4
825.2.k.h.518.1	yes	4	1.1	even	1	trivial
825.2.k.h.782.1	yes	4	15.2	even	4	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	825.k (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.292893 + 0.292893i) q^{2} +(1.70711 - 0.292893i) q^{3} -1.82843i q^{4} +(0.585786 + 0.414214i) q^{6} +(0.585786 - 0.585786i) q^{7} +(1.12132 - 1.12132i) q^{8} +(2.82843 - 1.00000i) q^{9} +O(q^{10})\)	\(q+(0.292893 + 0.292893i) q^{2} +(1.70711 - 0.292893i) q^{3} -1.82843i q^{4} +(0.585786 + 0.414214i) q^{6} +(0.585786 - 0.585786i) q^{7} +(1.12132 - 1.12132i) q^{8} +(2.82843 - 1.00000i) q^{9} -1.00000i q^{11} +(-0.535534 - 3.12132i) q^{12} +(3.41421 + 3.41421i) q^{13} +0.343146 q^{14} -3.00000 q^{16} +(-2.00000 - 2.00000i) q^{17} +(1.12132 + 0.535534i) q^{18} -0.828427i q^{19} +(0.828427 - 1.17157i) q^{21} +(0.292893 - 0.292893i) q^{22} +(-0.828427 + 0.828427i) q^{23} +(1.58579 - 2.24264i) q^{24} +2.00000i q^{26} +(4.53553 - 2.53553i) q^{27} +(-1.07107 - 1.07107i) q^{28} -8.82843 q^{29} +4.00000 q^{31} +(-3.12132 - 3.12132i) q^{32} +(-0.292893 - 1.70711i) q^{33} -1.17157i q^{34} +(-1.82843 - 5.17157i) q^{36} +(5.65685 - 5.65685i) q^{37} +(0.242641 - 0.242641i) q^{38} +(6.82843 + 4.82843i) q^{39} +4.82843i q^{41} +(0.585786 - 0.100505i) q^{42} +(4.58579 + 4.58579i) q^{43} -1.82843 q^{44} -0.485281 q^{46} +(-4.82843 - 4.82843i) q^{47} +(-5.12132 + 0.878680i) q^{48} +6.31371i q^{49} +(-4.00000 - 2.82843i) q^{51} +(6.24264 - 6.24264i) q^{52} +(8.48528 - 8.48528i) q^{53} +(2.07107 + 0.585786i) q^{54} -1.31371i q^{56} +(-0.242641 - 1.41421i) q^{57} +(-2.58579 - 2.58579i) q^{58} -2.34315 q^{59} -6.00000 q^{61} +(1.17157 + 1.17157i) q^{62} +(1.07107 - 2.24264i) q^{63} +4.17157i q^{64} +(0.414214 - 0.585786i) q^{66} +(-2.58579 + 2.58579i) q^{67} +(-3.65685 + 3.65685i) q^{68} +(-1.17157 + 1.65685i) q^{69} +9.65685i q^{71} +(2.05025 - 4.29289i) q^{72} +(4.58579 + 4.58579i) q^{73} +3.31371 q^{74} -1.51472 q^{76} +(-0.585786 - 0.585786i) q^{77} +(0.585786 + 3.41421i) q^{78} +4.82843i q^{79} +(7.00000 - 5.65685i) q^{81} +(-1.41421 + 1.41421i) q^{82} +(-4.24264 + 4.24264i) q^{83} +(-2.14214 - 1.51472i) q^{84} +2.68629i q^{86} +(-15.0711 + 2.58579i) q^{87} +(-1.12132 - 1.12132i) q^{88} -15.3137 q^{89} +4.00000 q^{91} +(1.51472 + 1.51472i) q^{92} +(6.82843 - 1.17157i) q^{93} -2.82843i q^{94} +(-6.24264 - 4.41421i) q^{96} +(-1.65685 + 1.65685i) q^{97} +(-1.84924 + 1.84924i) q^{98} +(-1.00000 - 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{7} - 4 q^{8}+O(q^{10}) \) 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^6 + 8 * q^7 - 4 * q^8	\( 4 q + 4 q^{2} + 4 q^{3} + 8 q^{6} + 8 q^{7} - 4 q^{8} + 12 q^{12} + 8 q^{13} + 24 q^{14} - 12 q^{16} - 8 q^{17} - 4 q^{18} - 8 q^{21} + 4 q^{22} + 8 q^{23} + 12 q^{24} + 4 q^{27} + 24 q^{28} - 24 q^{29} + 16 q^{31} - 4 q^{32} - 4 q^{33} + 4 q^{36} - 16 q^{38} + 16 q^{39} + 8 q^{42} + 24 q^{43} + 4 q^{44} + 32 q^{46} - 8 q^{47} - 12 q^{48} - 16 q^{51} + 8 q^{52} - 20 q^{54} + 16 q^{57} - 16 q^{58} - 32 q^{59} - 24 q^{61} + 16 q^{62} - 24 q^{63} - 4 q^{66} - 16 q^{67} + 8 q^{68} - 16 q^{69} + 28 q^{72} + 24 q^{73} - 32 q^{74} - 40 q^{76} - 8 q^{77} + 8 q^{78} + 28 q^{81} + 48 q^{84} - 32 q^{87} + 4 q^{88} - 16 q^{89} + 16 q^{91} + 40 q^{92} + 16 q^{93} - 8 q^{96} + 16 q^{97} + 52 q^{98} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^2 + 4 * q^3 + 8 * q^6 + 8 * q^7 - 4 * q^8 + 12 * q^12 + 8 * q^13 + 24 * q^14 - 12 * q^16 - 8 * q^17 - 4 * q^18 - 8 * q^21 + 4 * q^22 + 8 * q^23 + 12 * q^24 + 4 * q^27 + 24 * q^28 - 24 * q^29 + 16 * q^31 - 4 * q^32 - 4 * q^33 + 4 * q^36 - 16 * q^38 + 16 * q^39 + 8 * q^42 + 24 * q^43 + 4 * q^44 + 32 * q^46 - 8 * q^47 - 12 * q^48 - 16 * q^51 + 8 * q^52 - 20 * q^54 + 16 * q^57 - 16 * q^58 - 32 * q^59 - 24 * q^61 + 16 * q^62 - 24 * q^63 - 4 * q^66 - 16 * q^67 + 8 * q^68 - 16 * q^69 + 28 * q^72 + 24 * q^73 - 32 * q^74 - 40 * q^76 - 8 * q^77 + 8 * q^78 + 28 * q^81 + 48 * q^84 - 32 * q^87 + 4 * q^88 - 16 * q^89 + 16 * q^91 + 40 * q^92 + 16 * q^93 - 8 * q^96 + 16 * q^97 + 52 * q^98 - 4 * q^99

Introduction

L-functions

Modular forms

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