LMFDB - Embedded newform 936.2.g.b.469.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [936,2,Mod(469,936)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(936, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("936.469");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 936 = 2^{3} \cdot 3^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	936.g (of order $2$, degree $1$, 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:	$7.47399762919$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			469.3
Root			$-0.866025 - 0.500000i$ of defining polynomial
Character	$\chi$	$=$	936.469
Dual form			936.2.g.b.469.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.366025 - 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +3.46410i q^{5} -1.26795 q^{7} +(-2.00000 + 2.00000i) q^{8} +O(q^{10})$	\(q+(0.366025 - 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +3.46410i q^{5} -1.26795 q^{7} +(-2.00000 + 2.00000i) q^{8} +(4.73205 + 1.26795i) q^{10} -4.73205i q^{11} -1.00000i q^{13} +(-0.464102 + 1.73205i) q^{14} +(2.00000 + 3.46410i) q^{16} -5.46410 q^{17} -0.732051i q^{19} +(3.46410 - 6.00000i) q^{20} +(-6.46410 - 1.73205i) q^{22} -4.00000 q^{23} -7.00000 q^{25} +(-1.36603 - 0.366025i) q^{26} +(2.19615 + 1.26795i) q^{28} +2.00000i q^{29} -6.73205 q^{31} +(5.46410 - 1.46410i) q^{32} +(-2.00000 + 7.46410i) q^{34} -4.39230i q^{35} +8.92820i q^{37} +(-1.00000 - 0.267949i) q^{38} +(-6.92820 - 6.92820i) q^{40} -8.92820 q^{41} +0.535898i q^{43} +(-4.73205 + 8.19615i) q^{44} +(-1.46410 + 5.46410i) q^{46} -6.73205 q^{47} -5.39230 q^{49} +(-2.56218 + 9.56218i) q^{50} +(-1.00000 + 1.73205i) q^{52} -2.92820i q^{53} +16.3923 q^{55} +(2.53590 - 2.53590i) q^{56} +(2.73205 + 0.732051i) q^{58} -10.1962i q^{59} -2.92820i q^{61} +(-2.46410 + 9.19615i) q^{62} -8.00000i q^{64} +3.46410 q^{65} -0.732051i q^{67} +(9.46410 + 5.46410i) q^{68} +(-6.00000 - 1.60770i) q^{70} +8.19615 q^{71} -7.46410 q^{73} +(12.1962 + 3.26795i) q^{74} +(-0.732051 + 1.26795i) q^{76} +6.00000i q^{77} +5.46410 q^{79} +(-12.0000 + 6.92820i) q^{80} +(-3.26795 + 12.1962i) q^{82} +3.26795i q^{83} -18.9282i q^{85} +(0.732051 + 0.196152i) q^{86} +(9.46410 + 9.46410i) q^{88} +17.3205 q^{89} +1.26795i q^{91} +(6.92820 + 4.00000i) q^{92} +(-2.46410 + 9.19615i) q^{94} +2.53590 q^{95} +6.39230 q^{97} +(-1.97372 + 7.36603i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 - 12 * q^7 - 8 * q^8	$ 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8} + 12 q^{10} + 12 q^{14} + 8 q^{16} - 8 q^{17} - 12 q^{22} - 16 q^{23} - 28 q^{25} - 2 q^{26} - 12 q^{28} - 20 q^{31} + 8 q^{32} - 8 q^{34} - 4 q^{38} - 8 q^{41} - 12 q^{44} + 8 q^{46} - 20 q^{47} + 20 q^{49} + 14 q^{50} - 4 q^{52} + 24 q^{55} + 24 q^{56} + 4 q^{58} + 4 q^{62} + 24 q^{68} - 24 q^{70} + 12 q^{71} - 16 q^{73} + 28 q^{74} + 4 q^{76} + 8 q^{79} - 48 q^{80} - 20 q^{82} - 4 q^{86} + 24 q^{88} + 4 q^{94} + 24 q^{95} - 16 q^{97} - 46 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 - 12 * q^7 - 8 * q^8 + 12 * q^10 + 12 * q^14 + 8 * q^16 - 8 * q^17 - 12 * q^22 - 16 * q^23 - 28 * q^25 - 2 * q^26 - 12 * q^28 - 20 * q^31 + 8 * q^32 - 8 * q^34 - 4 * q^38 - 8 * q^41 - 12 * q^44 + 8 * q^46 - 20 * q^47 + 20 * q^49 + 14 * q^50 - 4 * q^52 + 24 * q^55 + 24 * q^56 + 4 * q^58 + 4 * q^62 + 24 * q^68 - 24 * q^70 + 12 * q^71 - 16 * q^73 + 28 * q^74 + 4 * q^76 + 8 * q^79 - 48 * q^80 - 20 * q^82 - 4 * q^86 + 24 * q^88 + 4 * q^94 + 24 * q^95 - 16 * q^97 - 46 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/936\mathbb{Z}\right)^\times$.

$n$	$145$	$209$	$469$	$703$
$\chi(n)$	$1$	$1$	$-1$	$1$

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 $

$2$	0.366025	−	1.36603i	0.258819	−	0.965926i
$2$	0.366025	−	1.36603i	0.258819	−	0.965926i
$3$		0			0
$3$		0			0
$4$	−1.73205	−	1.00000i	−0.866025	−	0.500000i
$5$			3.46410i			1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$			3.46410i			1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$		0			0
$7$	−1.26795			−0.479240			−0.239620	−	0.970867i	$-0.577023\pi$
$7$	−1.26795			−0.479240			−0.239620	+	0.970867i	$0.577023\pi$
$8$	−2.00000	+	2.00000i	−0.707107	+	0.707107i
$9$		0			0
$10$	4.73205	+	1.26795i	1.49641	+	0.400961i
$11$		−	4.73205i		−	1.42677i	−0.700774	−	0.713384i	$-0.747162\pi$
$11$		−	4.73205i		−	1.42677i	0.700774	−	0.713384i	$-0.252838\pi$
$12$		0			0
$13$		−	1.00000i		−	0.277350i
$13$		−	1.00000i		−	0.277350i
$14$	−0.464102	+	1.73205i	−0.124036	+	0.462910i
$15$		0			0
$16$	2.00000	+	3.46410i	0.500000	+	0.866025i
$17$	−5.46410			−1.32524			−0.662620	−	0.748956i	$-0.730555\pi$
$17$	−5.46410			−1.32524			−0.662620	+	0.748956i	$0.730555\pi$
$18$		0			0
$19$		−	0.732051i		−	0.167944i	−0.996468	−	0.0839720i	$-0.973239\pi$
$19$		−	0.732051i		−	0.167944i	0.996468	−	0.0839720i	$-0.0267606\pi$
$20$	3.46410	−	6.00000i	0.774597	−	1.34164i
$21$		0			0
$22$	−6.46410	−	1.73205i	−1.37815	−	0.369274i
$23$	−4.00000			−0.834058			−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000			−0.834058			−0.417029	+	0.908893i	$0.636929\pi$
$24$		0			0
$25$	−7.00000			−1.40000
$26$	−1.36603	−	0.366025i	−0.267900	−	0.0717835i
$27$		0			0
$28$	2.19615	+	1.26795i	0.415034	+	0.239620i
$29$			2.00000i			0.371391i	0.982607	+	0.185695i	$0.0594537\pi$
$29$			2.00000i			0.371391i	−0.982607	+	0.185695i	$0.940546\pi$
$30$		0			0
$31$	−6.73205			−1.20911			−0.604556	−	0.796563i	$-0.706649\pi$
$31$	−6.73205			−1.20911			−0.604556	+	0.796563i	$0.706649\pi$
$32$	5.46410	−	1.46410i	0.965926	−	0.258819i
$33$		0			0
$34$	−2.00000	+	7.46410i	−0.342997	+	1.28008i
$35$		−	4.39230i		−	0.742435i
$36$		0			0
$37$			8.92820i			1.46779i	0.679264	+	0.733894i	$0.262299\pi$
$37$			8.92820i			1.46779i	−0.679264	+	0.733894i	$0.737701\pi$
$38$	−1.00000	−	0.267949i	−0.162221	−	0.0434671i
$39$		0			0
$40$	−6.92820	−	6.92820i	−1.09545	−	1.09545i
$41$	−8.92820			−1.39435			−0.697176	−	0.716900i	$-0.745560\pi$
$41$	−8.92820			−1.39435			−0.697176	+	0.716900i	$0.745560\pi$
$42$		0			0
$43$			0.535898i			0.0817237i	0.999165	+	0.0408619i	$0.0130104\pi$
$43$			0.535898i			0.0817237i	−0.999165	+	0.0408619i	$0.986990\pi$
$44$	−4.73205	+	8.19615i	−0.713384	+	1.23562i
$45$		0			0
$46$	−1.46410	+	5.46410i	−0.215870	+	0.805638i
$47$	−6.73205			−0.981971			−0.490985	−	0.871168i	$-0.663363\pi$
$47$	−6.73205			−0.981971			−0.490985	+	0.871168i	$0.663363\pi$
$48$		0			0
$49$	−5.39230			−0.770329
$50$	−2.56218	+	9.56218i	−0.362347	+	1.35230i
$51$		0			0
$52$	−1.00000	+	1.73205i	−0.138675	+	0.240192i
$53$		−	2.92820i		−	0.402220i	−0.979569	−	0.201110i	$-0.935545\pi$
$53$		−	2.92820i		−	0.402220i	0.979569	−	0.201110i	$-0.0644548\pi$
$54$		0			0
$55$	16.3923			2.21034
$56$	2.53590	−	2.53590i	0.338874	−	0.338874i
$57$		0			0
$58$	2.73205	+	0.732051i	0.358736	+	0.0961230i
$59$		−	10.1962i		−	1.32743i	−0.747987	−	0.663713i	$-0.768980\pi$
$59$		−	10.1962i		−	1.32743i	0.747987	−	0.663713i	$-0.231020\pi$
$60$		0			0
$61$		−	2.92820i		−	0.374918i	−0.982272	−	0.187459i	$-0.939975\pi$
$61$		−	2.92820i		−	0.374918i	0.982272	−	0.187459i	$-0.0600252\pi$
$62$	−2.46410	+	9.19615i	−0.312941	+	1.16791i
$63$		0			0
$64$		−	8.00000i		−	1.00000i
$65$	3.46410			0.429669
$66$		0			0
$67$		−	0.732051i		−	0.0894342i	−0.999000	−	0.0447171i	$-0.985761\pi$
$67$		−	0.732051i		−	0.0894342i	0.999000	−	0.0447171i	$-0.0142386\pi$
$68$	9.46410	+	5.46410i	1.14769	+	0.662620i
$69$		0			0
$70$	−6.00000	−	1.60770i	−0.717137	−	0.192156i
$71$	8.19615			0.972704			0.486352	−	0.873763i	$-0.338327\pi$
$71$	8.19615			0.972704			0.486352	+	0.873763i	$0.338327\pi$
$72$		0			0
$73$	−7.46410			−0.873607			−0.436804	−	0.899557i	$-0.643889\pi$
$73$	−7.46410			−0.873607			−0.436804	+	0.899557i	$0.643889\pi$
$74$	12.1962	+	3.26795i	1.41777	+	0.379891i
$75$		0			0
$76$	−0.732051	+	1.26795i	−0.0839720	+	0.145444i
$77$			6.00000i			0.683763i
$78$		0			0
$79$	5.46410			0.614759			0.307380	−	0.951587i	$-0.400548\pi$
$79$	5.46410			0.614759			0.307380	+	0.951587i	$0.400548\pi$
$80$	−12.0000	+	6.92820i	−1.34164	+	0.774597i
$81$		0			0
$82$	−3.26795	+	12.1962i	−0.360885	+	1.34684i
$83$			3.26795i			0.358704i	0.983785	+	0.179352i	$0.0574001\pi$
$83$			3.26795i			0.358704i	−0.983785	+	0.179352i	$0.942600\pi$
$84$		0			0
$85$		−	18.9282i		−	2.05305i
$86$	0.732051	+	0.196152i	0.0789391	+	0.0211517i
$87$		0			0
$88$	9.46410	+	9.46410i	1.00888	+	1.00888i
$89$	17.3205			1.83597			0.917985	−	0.396615i	$-0.129815\pi$
$89$	17.3205			1.83597			0.917985	+	0.396615i	$0.129815\pi$
$90$		0			0
$91$			1.26795i			0.132917i
$92$	6.92820	+	4.00000i	0.722315	+	0.417029i
$93$		0			0
$94$	−2.46410	+	9.19615i	−0.254153	+	0.948511i
$95$	2.53590			0.260178
$96$		0			0
$97$	6.39230			0.649040			0.324520	−	0.945879i	$-0.394797\pi$
$97$	6.39230			0.649040			0.324520	+	0.945879i	$0.394797\pi$
$98$	−1.97372	+	7.36603i	−0.199376	+	0.744081i
$99$		0			0
$100$	12.1244	+	7.00000i	1.21244	+	0.700000i
$101$		−	12.0000i		−	1.19404i	−0.802225	−	0.597022i	$-0.796350\pi$
$101$		−	12.0000i		−	1.19404i	0.802225	−	0.597022i	$-0.203650\pi$
$102$		0			0
$103$	6.92820			0.682656			0.341328	−	0.939944i	$-0.389123\pi$
$103$	6.92820			0.682656			0.341328	+	0.939944i	$0.389123\pi$
$104$	2.00000	+	2.00000i	0.196116	+	0.196116i
$105$		0			0
$106$	−4.00000	−	1.07180i	−0.388514	−	0.104102i
$107$			4.92820i			0.476427i	0.971213	+	0.238214i	$0.0765619\pi$
$107$			4.92820i			0.476427i	−0.971213	+	0.238214i	$0.923438\pi$
$108$		0			0
$109$			2.00000i			0.191565i	0.995402	+	0.0957826i	$0.0305354\pi$
$109$			2.00000i			0.191565i	−0.995402	+	0.0957826i	$0.969465\pi$
$110$	6.00000	−	22.3923i	0.572078	−	2.13502i
$111$		0			0
$112$	−2.53590	−	4.39230i	−0.239620	−	0.415034i
$113$	−2.53590			−0.238557			−0.119279	−	0.992861i	$-0.538058\pi$
$113$	−2.53590			−0.238557			−0.119279	+	0.992861i	$0.538058\pi$
$114$		0			0
$115$		−	13.8564i		−	1.29212i
$116$	2.00000	−	3.46410i	0.185695	−	0.321634i
$117$		0			0
$118$	−13.9282	−	3.73205i	−1.28220	−	0.343563i
$119$	6.92820			0.635107
$120$		0			0
$121$	−11.3923			−1.03566
$122$	−4.00000	−	1.07180i	−0.362143	−	0.0970359i
$123$		0			0
$124$	11.6603	+	6.73205i	1.04712	+	0.604556i
$125$		−	6.92820i		−	0.619677i
$126$		0			0
$127$	−4.00000			−0.354943			−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000			−0.354943			−0.177471	+	0.984126i	$0.556792\pi$
$128$	−10.9282	−	2.92820i	−0.965926	−	0.258819i
$129$		0			0
$130$	1.26795	−	4.73205i	0.111207	−	0.415028i
$131$			19.8564i			1.73486i	0.497557	+	0.867431i	$0.334230\pi$
$131$			19.8564i			1.73486i	−0.497557	+	0.867431i	$0.665770\pi$
$132$		0			0
$133$			0.928203i			0.0804854i
$134$	−1.00000	−	0.267949i	−0.0863868	−	0.0231473i
$135$		0			0
$136$	10.9282	−	10.9282i	0.937086	−	0.937086i
$137$	−12.9282			−1.10453			−0.552265	−	0.833668i	$-0.686237\pi$
$137$	−12.9282			−1.10453			−0.552265	+	0.833668i	$0.686237\pi$
$138$		0			0
$139$			10.0000i			0.848189i	0.905618	+	0.424094i	$0.139408\pi$
$139$			10.0000i			0.848189i	−0.905618	+	0.424094i	$0.860592\pi$
$140$	−4.39230	+	7.60770i	−0.371218	+	0.642968i
$141$		0			0
$142$	3.00000	−	11.1962i	0.251754	−	0.939560i
$143$	−4.73205			−0.395714
$144$		0			0
$145$	−6.92820			−0.575356
$146$	−2.73205	+	10.1962i	−0.226106	+	0.843840i
$147$		0			0
$148$	8.92820	−	15.4641i	0.733894	−	1.27114i
$149$		−	12.9282i		−	1.05912i	−0.848273	−	0.529560i	$-0.822357\pi$
$149$		−	12.9282i		−	1.05912i	0.848273	−	0.529560i	$-0.177643\pi$
$150$		0			0
$151$	−7.12436			−0.579772			−0.289886	−	0.957061i	$-0.593617\pi$
$151$	−7.12436			−0.579772			−0.289886	+	0.957061i	$0.593617\pi$
$152$	1.46410	+	1.46410i	0.118754	+	0.118754i
$153$		0			0
$154$	8.19615	+	2.19615i	0.660465	+	0.176971i
$155$		−	23.3205i		−	1.87315i
$156$		0			0
$157$			16.9282i			1.35102i	0.737352	+	0.675509i	$0.236076\pi$
$157$			16.9282i			1.35102i	−0.737352	+	0.675509i	$0.763924\pi$
$158$	2.00000	−	7.46410i	0.159111	−	0.593812i
$159$		0			0
$160$	5.07180	+	18.9282i	0.400961	+	1.49641i
$161$	5.07180			0.399714
$162$		0			0
$163$		−	16.7321i		−	1.31056i	−0.755388	−	0.655278i	$-0.772552\pi$
$163$		−	16.7321i		−	1.31056i	0.755388	−	0.655278i	$-0.227448\pi$
$164$	15.4641	+	8.92820i	1.20754	+	0.697176i
$165$		0			0
$166$	4.46410	+	1.19615i	0.346481	+	0.0928394i
$167$	5.66025			0.438004			0.219002	−	0.975724i	$-0.429720\pi$
$167$	5.66025			0.438004			0.219002	+	0.975724i	$0.429720\pi$
$168$		0			0
$169$	−1.00000			−0.0769231
$170$	−25.8564	−	6.92820i	−1.98310	−	0.531369i
$171$		0			0
$172$	0.535898	−	0.928203i	0.0408619	−	0.0707748i
$173$			6.92820i			0.526742i	0.964695	+	0.263371i	$0.0848343\pi$
$173$			6.92820i			0.526742i	−0.964695	+	0.263371i	$0.915166\pi$
$174$		0			0
$175$	8.87564			0.670936
$176$	16.3923	−	9.46410i	1.23562	−	0.713384i
$177$		0			0
$178$	6.33975	−	23.6603i	0.475184	−	1.77341i
$179$		−	10.3923i		−	0.776757i	−0.921500	−	0.388379i	$-0.873035\pi$
$179$		−	10.3923i		−	0.776757i	0.921500	−	0.388379i	$-0.126965\pi$
$180$		0			0
$181$			8.92820i			0.663628i	0.943345	+	0.331814i	$0.107661\pi$
$181$			8.92820i			0.663628i	−0.943345	+	0.331814i	$0.892339\pi$
$182$	1.73205	+	0.464102i	0.128388	+	0.0344015i
$183$		0			0
$184$	8.00000	−	8.00000i	0.589768	−	0.589768i
$185$	−30.9282			−2.27389
$186$		0			0
$187$			25.8564i			1.89081i
$188$	11.6603	+	6.73205i	0.850411	+	0.490985i
$189$		0			0
$190$	0.928203	−	3.46410i	0.0673389	−	0.251312i
$191$	14.5359			1.05178			0.525890	−	0.850552i	$-0.323732\pi$
$191$	14.5359			1.05178			0.525890	+	0.850552i	$0.323732\pi$
$192$		0			0
$193$	−1.60770			−0.115724			−0.0578622	−	0.998325i	$-0.518428\pi$
$193$	−1.60770			−0.115724			−0.0578622	+	0.998325i	$0.518428\pi$
$194$	2.33975	−	8.73205i	0.167984	−	0.626925i
$195$		0			0
$196$	9.33975	+	5.39230i	0.667125	+	0.385165i
$197$		−	3.07180i		−	0.218856i	−0.993995	−	0.109428i	$-0.965098\pi$
$197$		−	3.07180i		−	0.218856i	0.993995	−	0.109428i	$-0.0349020\pi$
$198$		0			0
$199$	16.7846			1.18983			0.594915	−	0.803789i	$-0.297186\pi$
$199$	16.7846			1.18983			0.594915	+	0.803789i	$0.297186\pi$
$200$	14.0000	−	14.0000i	0.989949	−	0.989949i
$201$		0			0
$202$	−16.3923	−	4.39230i	−1.15336	−	0.309041i
$203$		−	2.53590i		−	0.177985i
$204$		0			0
$205$		−	30.9282i		−	2.16012i
$206$	2.53590	−	9.46410i	0.176684	−	0.659395i
$207$		0			0
$208$	3.46410	−	2.00000i	0.240192	−	0.138675i
$209$	−3.46410			−0.239617
$210$		0			0
$211$			7.85641i			0.540857i	0.962740	+	0.270429i	$0.0871654\pi$
$211$			7.85641i			0.540857i	−0.962740	+	0.270429i	$0.912835\pi$
$212$	−2.92820	+	5.07180i	−0.201110	+	0.348332i
$213$		0			0
$214$	6.73205	+	1.80385i	0.460194	+	0.123308i
$215$	−1.85641			−0.126606
$216$		0			0
$217$	8.53590			0.579455
$218$	2.73205	+	0.732051i	0.185038	+	0.0495807i
$219$		0			0
$220$	−28.3923	−	16.3923i	−1.91421	−	1.10517i
$221$			5.46410i			0.367555i
$222$		0			0
$223$	−0.196152			−0.0131353			−0.00656767	−	0.999978i	$-0.502091\pi$
$223$	−0.196152			−0.0131353			−0.00656767	+	0.999978i	$0.502091\pi$
$224$	−6.92820	+	1.85641i	−0.462910	+	0.124036i
$225$		0			0
$226$	−0.928203	+	3.46410i	−0.0617432	+	0.230429i
$227$			2.87564i			0.190863i	0.995436	+	0.0954316i	$0.0304231\pi$
$227$			2.87564i			0.190863i	−0.995436	+	0.0954316i	$0.969577\pi$
$228$		0			0
$229$			5.32051i			0.351589i	0.984427	+	0.175795i	$0.0562494\pi$
$229$			5.32051i			0.351589i	−0.984427	+	0.175795i	$0.943751\pi$
$230$	−18.9282	−	5.07180i	−1.24809	−	0.334424i
$231$		0			0
$232$	−4.00000	−	4.00000i	−0.262613	−	0.262613i
$233$	−16.9282			−1.10900			−0.554502	−	0.832183i	$-0.687091\pi$
$233$	−16.9282			−1.10900			−0.554502	+	0.832183i	$0.687091\pi$
$234$		0			0
$235$		−	23.3205i		−	1.52126i
$236$	−10.1962	+	17.6603i	−0.663713	+	1.14958i
$237$		0			0
$238$	2.53590	−	9.46410i	0.164378	−	0.613467i
$239$	−18.7321			−1.21168			−0.605838	−	0.795588i	$-0.707162\pi$
$239$	−18.7321			−1.21168			−0.605838	+	0.795588i	$0.707162\pi$
$240$		0			0
$241$	−30.3923			−1.95774			−0.978870	−	0.204482i	$-0.934449\pi$
$241$	−30.3923			−1.95774			−0.978870	+	0.204482i	$0.934449\pi$
$242$	−4.16987	+	15.5622i	−0.268050	+	1.00037i
$243$		0			0
$244$	−2.92820	+	5.07180i	−0.187459	+	0.324689i
$245$		−	18.6795i		−	1.19339i
$246$		0			0
$247$	−0.732051			−0.0465793
$248$	13.4641	−	13.4641i	0.854971	−	0.854971i
$249$		0			0
$250$	−9.46410	−	2.53590i	−0.598562	−	0.160384i
$251$			6.39230i			0.403479i	0.979439	+	0.201739i	$0.0646594\pi$
$251$			6.39230i			0.403479i	−0.979439	+	0.201739i	$0.935341\pi$
$252$		0			0
$253$			18.9282i			1.19001i
$254$	−1.46410	+	5.46410i	−0.0918659	+	0.342848i
$255$		0			0
$256$	−8.00000	+	13.8564i	−0.500000	+	0.866025i
$257$	23.8564			1.48812			0.744061	−	0.668112i	$-0.232897\pi$
$257$	23.8564			1.48812			0.744061	+	0.668112i	$0.232897\pi$
$258$		0			0
$259$		−	11.3205i		−	0.703422i
$260$	−6.00000	−	3.46410i	−0.372104	−	0.214834i
$261$		0			0
$262$	27.1244	+	7.26795i	1.67575	+	0.449015i
$263$	27.3205			1.68465			0.842327	−	0.538966i	$-0.181185\pi$
$263$	27.3205			1.68465			0.842327	+	0.538966i	$0.181185\pi$
$264$		0			0
$265$	10.1436			0.623116
$266$	1.26795	+	0.339746i	0.0777430	+	0.0208312i
$267$		0			0
$268$	−0.732051	+	1.26795i	−0.0447171	+	0.0774523i
$269$		−	7.85641i		−	0.479014i	−0.970895	−	0.239507i	$-0.923014\pi$
$269$		−	7.85641i		−	0.479014i	0.970895	−	0.239507i	$-0.0769857\pi$
$270$		0			0
$271$	−20.1962			−1.22683			−0.613414	−	0.789761i	$-0.710204\pi$
$271$	−20.1962			−1.22683			−0.613414	+	0.789761i	$0.710204\pi$
$272$	−10.9282	−	18.9282i	−0.662620	−	1.14769i
$273$		0			0
$274$	−4.73205	+	17.6603i	−0.285874	+	1.06689i
$275$			33.1244i			1.99747i
$276$		0			0
$277$		−	1.85641i		−	0.111541i	−0.998444	−	0.0557703i	$-0.982239\pi$
$277$		−	1.85641i		−	0.111541i	0.998444	−	0.0557703i	$-0.0177615\pi$
$278$	13.6603	+	3.66025i	0.819288	+	0.219527i
$279$		0			0
$280$	8.78461	+	8.78461i	0.524981	+	0.524981i
$281$	−9.32051			−0.556015			−0.278007	−	0.960579i	$-0.589674\pi$
$281$	−9.32051			−0.556015			−0.278007	+	0.960579i	$0.589674\pi$
$282$		0			0
$283$		−	19.4641i		−	1.15702i	−0.815675	−	0.578510i	$-0.803634\pi$
$283$		−	19.4641i		−	1.15702i	0.815675	−	0.578510i	$-0.196366\pi$
$284$	−14.1962	−	8.19615i	−0.842387	−	0.486352i
$285$		0			0
$286$	−1.73205	+	6.46410i	−0.102418	+	0.382230i
$287$	11.3205			0.668228
$288$		0			0
$289$	12.8564			0.756259
$290$	−2.53590	+	9.46410i	−0.148913	+	0.555751i
$291$		0			0
$292$	12.9282	+	7.46410i	0.756566	+	0.436804i
$293$			32.9282i			1.92369i	0.273602	+	0.961843i	$0.411785\pi$
$293$			32.9282i			1.92369i	−0.273602	+	0.961843i	$0.588215\pi$
$294$		0			0
$295$	35.3205			2.05644
$296$	−17.8564	−	17.8564i	−1.03788	−	1.03788i
$297$		0			0
$298$	−17.6603	−	4.73205i	−1.02303	−	0.274120i
$299$			4.00000i			0.231326i
$300$		0			0
$301$		−	0.679492i		−	0.0391653i
$302$	−2.60770	+	9.73205i	−0.150056	+	0.560017i
$303$		0			0
$304$	2.53590	−	1.46410i	0.145444	−	0.0839720i
$305$	10.1436			0.580820
$306$		0			0
$307$		−	0.732051i		−	0.0417803i	−0.999782	−	0.0208902i	$-0.993350\pi$
$307$		−	0.732051i		−	0.0417803i	0.999782	−	0.0208902i	$-0.00665003\pi$
$308$	6.00000	−	10.3923i	0.341882	−	0.592157i
$309$		0			0
$310$	−31.8564	−	8.53590i	−1.80932	−	0.484806i
$311$	−1.07180			−0.0607760			−0.0303880	−	0.999538i	$-0.509674\pi$
$311$	−1.07180			−0.0607760			−0.0303880	+	0.999538i	$0.509674\pi$
$312$		0			0
$313$	0.392305			0.0221744			0.0110872	−	0.999939i	$-0.496471\pi$
$313$	0.392305			0.0221744			0.0110872	+	0.999939i	$0.496471\pi$
$314$	23.1244	+	6.19615i	1.30498	+	0.349669i
$315$		0			0
$316$	−9.46410	−	5.46410i	−0.532397	−	0.307380i
$317$			3.46410i			0.194563i	0.995257	+	0.0972817i	$0.0310148\pi$
$317$			3.46410i			0.194563i	−0.995257	+	0.0972817i	$0.968985\pi$
$318$		0			0
$319$	9.46410			0.529888
$320$	27.7128			1.54919
$321$		0			0
$322$	1.85641	−	6.92820i	0.103453	−	0.386094i
$323$			4.00000i			0.222566i
$324$		0			0
$325$			7.00000i			0.388290i
$326$	−22.8564	−	6.12436i	−1.26590	−	0.339197i
$327$		0			0
$328$	17.8564	−	17.8564i	0.985955	−	0.985955i
$329$	8.53590			0.470599
$330$		0			0
$331$		−	17.5167i		−	0.962803i	−0.876500	−	0.481401i	$-0.840128\pi$
$331$		−	17.5167i		−	0.962803i	0.876500	−	0.481401i	$-0.159872\pi$
$332$	3.26795	−	5.66025i	0.179352	−	0.310647i
$333$		0			0
$334$	2.07180	−	7.73205i	0.113364	−	0.423079i
$335$	2.53590			0.138551
$336$		0			0
$337$	1.46410			0.0797547			0.0398773	−	0.999205i	$-0.487303\pi$
$337$	1.46410			0.0797547			0.0398773	+	0.999205i	$0.487303\pi$
$338$	−0.366025	+	1.36603i	−0.0199092	+	0.0743020i
$339$		0			0
$340$	−18.9282	+	32.7846i	−1.02653	+	1.77800i
$341$			31.8564i			1.72512i
$342$		0			0
$343$	15.7128			0.848412
$344$	−1.07180	−	1.07180i	−0.0577874	−	0.0577874i
$345$		0			0
$346$	9.46410	+	2.53590i	0.508793	+	0.136331i
$347$		−	7.07180i		−	0.379634i	−0.981819	−	0.189817i	$-0.939211\pi$
$347$		−	7.07180i		−	0.379634i	0.981819	−	0.189817i	$-0.0607895\pi$
$348$		0			0
$349$		−	9.60770i		−	0.514288i	−0.966373	−	0.257144i	$-0.917219\pi$
$349$		−	9.60770i		−	0.514288i	0.966373	−	0.257144i	$-0.0827815\pi$
$350$	3.24871	−	12.1244i	0.173651	−	0.648074i
$351$		0			0
$352$	−6.92820	−	25.8564i	−0.369274	−	1.37815i
$353$	11.0718			0.589292			0.294646	−	0.955606i	$-0.404798\pi$
$353$	11.0718			0.589292			0.294646	+	0.955606i	$0.404798\pi$
$354$		0			0
$355$			28.3923i			1.50691i
$356$	−30.0000	−	17.3205i	−1.59000	−	0.917985i
$357$		0			0
$358$	−14.1962	−	3.80385i	−0.750290	−	0.201040i
$359$	−31.5167			−1.66339			−0.831693	−	0.555236i	$-0.812628\pi$
$359$	−31.5167			−1.66339			−0.831693	+	0.555236i	$0.812628\pi$
$360$		0			0
$361$	18.4641			0.971795
$362$	12.1962	+	3.26795i	0.641016	+	0.171760i
$363$		0			0
$364$	1.26795	−	2.19615i	0.0664586	−	0.115110i
$365$		−	25.8564i		−	1.35339i
$366$		0			0
$367$	−22.2487			−1.16137			−0.580687	−	0.814127i	$-0.697216\pi$
$367$	−22.2487			−1.16137			−0.580687	+	0.814127i	$0.697216\pi$
$368$	−8.00000	−	13.8564i	−0.417029	−	0.722315i
$369$		0			0
$370$	−11.3205	+	42.2487i	−0.588525	+	2.19641i
$371$			3.71281i			0.192760i
$372$		0			0
$373$			14.7846i			0.765518i	0.923848	+	0.382759i	$0.125026\pi$
$373$			14.7846i			0.765518i	−0.923848	+	0.382759i	$0.874974\pi$
$374$	35.3205	+	9.46410i	1.82638	+	0.489377i
$375$		0			0
$376$	13.4641	−	13.4641i	0.694358	−	0.694358i
$377$	2.00000			0.103005
$378$		0			0
$379$		−	14.5885i		−	0.749359i	−0.927154	−	0.374679i	$-0.877753\pi$
$379$		−	14.5885i		−	0.749359i	0.927154	−	0.374679i	$-0.122247\pi$
$380$	−4.39230	−	2.53590i	−0.225320	−	0.130089i
$381$		0			0
$382$	5.32051	−	19.8564i	0.272221	−	1.01594i
$383$	10.3397			0.528336			0.264168	−	0.964477i	$-0.414903\pi$
$383$	10.3397			0.528336			0.264168	+	0.964477i	$0.414903\pi$
$384$		0			0
$385$	−20.7846			−1.05928
$386$	−0.588457	+	2.19615i	−0.0299517	+	0.111781i
$387$		0			0
$388$	−11.0718	−	6.39230i	−0.562085	−	0.324520i
$389$			2.00000i			0.101404i	0.998714	+	0.0507020i	$0.0161459\pi$
$389$			2.00000i			0.101404i	−0.998714	+	0.0507020i	$0.983854\pi$
$390$		0			0
$391$	21.8564			1.10533
$392$	10.7846	−	10.7846i	0.544705	−	0.544705i
$393$		0			0
$394$	−4.19615	−	1.12436i	−0.211399	−	0.0566442i
$395$			18.9282i			0.952381i
$396$		0			0
$397$			4.53590i			0.227650i	0.993501	+	0.113825i	$0.0363104\pi$
$397$			4.53590i			0.227650i	−0.993501	+	0.113825i	$0.963690\pi$
$398$	6.14359	−	22.9282i	0.307951	−	1.14929i
$399$		0			0
$400$	−14.0000	−	24.2487i	−0.700000	−	1.21244i
$401$	−4.53590			−0.226512			−0.113256	−	0.993566i	$-0.536128\pi$
$401$	−4.53590			−0.226512			−0.113256	+	0.993566i	$0.536128\pi$
$402$		0			0
$403$			6.73205i			0.335347i
$404$	−12.0000	+	20.7846i	−0.597022	+	1.03407i
$405$		0			0
$406$	−3.46410	−	0.928203i	−0.171920	−	0.0460660i
$407$	42.2487			2.09419
$408$		0			0
$409$	12.9282			0.639259			0.319629	−	0.947543i	$-0.396442\pi$
$409$	12.9282			0.639259			0.319629	+	0.947543i	$0.396442\pi$
$410$	−42.2487	−	11.3205i	−2.08652	−	0.559080i
$411$		0			0
$412$	−12.0000	−	6.92820i	−0.591198	−	0.341328i
$413$			12.9282i			0.636155i
$414$		0			0
$415$	−11.3205			−0.555702
$416$	−1.46410	−	5.46410i	−0.0717835	−	0.267900i
$417$		0			0
$418$	−1.26795	+	4.73205i	−0.0620174	+	0.231452i
$419$		−	30.7846i		−	1.50393i	−0.659205	−	0.751963i	$-0.729107\pi$
$419$		−	30.7846i		−	1.50393i	0.659205	−	0.751963i	$-0.270893\pi$
$420$		0			0
$421$		−	24.2487i		−	1.18181i	−0.806741	−	0.590905i	$-0.798771\pi$
$421$		−	24.2487i		−	1.18181i	0.806741	−	0.590905i	$-0.201229\pi$
$422$	10.7321	+	2.87564i	0.522428	+	0.139984i
$423$		0			0
$424$	5.85641	+	5.85641i	0.284412	+	0.284412i
$425$	38.2487			1.85534
$426$		0			0
$427$			3.71281i			0.179676i
$428$	4.92820	−	8.53590i	0.238214	−	0.412598i
$429$		0			0
$430$	−0.679492	+	2.53590i	−0.0327680	+	0.122292i
$431$	−39.1244			−1.88455			−0.942277	−	0.334835i	$-0.891320\pi$
$431$	−39.1244			−1.88455			−0.942277	+	0.334835i	$0.891320\pi$
$432$		0			0
$433$	−6.78461			−0.326048			−0.163024	−	0.986622i	$-0.552125\pi$
$433$	−6.78461			−0.326048			−0.163024	+	0.986622i	$0.552125\pi$
$434$	3.12436	−	11.6603i	0.149974	−	0.559710i
$435$		0			0
$436$	2.00000	−	3.46410i	0.0957826	−	0.165900i
$437$			2.92820i			0.140075i
$438$		0			0
$439$	2.92820			0.139756			0.0698778	−	0.997556i	$-0.477739\pi$
$439$	2.92820			0.139756			0.0698778	+	0.997556i	$0.477739\pi$
$440$	−32.7846	+	32.7846i	−1.56294	+	1.56294i
$441$		0			0
$442$	7.46410	+	2.00000i	0.355031	+	0.0951303i
$443$		−	30.0000i		−	1.42534i	−0.701498	−	0.712672i	$-0.747485\pi$
$443$		−	30.0000i		−	1.42534i	0.701498	−	0.712672i	$-0.252515\pi$
$444$		0			0
$445$			60.0000i			2.84427i
$446$	−0.0717968	+	0.267949i	−0.00339968	+	0.0126878i
$447$		0			0
$448$			10.1436i			0.479240i
$449$	−17.6077			−0.830959			−0.415479	−	0.909603i	$-0.636386\pi$
$449$	−17.6077			−0.830959			−0.415479	+	0.909603i	$0.636386\pi$
$450$		0			0
$451$			42.2487i			1.98941i
$452$	4.39230	+	2.53590i	0.206597	+	0.119279i
$453$		0			0
$454$	3.92820	+	1.05256i	0.184360	+	0.0493990i
$455$	−4.39230			−0.205914
$456$		0			0
$457$	−14.0000			−0.654892			−0.327446	−	0.944870i	$-0.606188\pi$
$457$	−14.0000			−0.654892			−0.327446	+	0.944870i	$0.606188\pi$
$458$	7.26795	+	1.94744i	0.339609	+	0.0909979i
$459$		0			0
$460$	−13.8564	+	24.0000i	−0.646058	+	1.11901i
$461$			20.9282i			0.974724i	0.873200	+	0.487362i	$0.162041\pi$
$461$			20.9282i			0.974724i	−0.873200	+	0.487362i	$0.837959\pi$
$462$		0			0
$463$	1.66025			0.0771585			0.0385793	−	0.999256i	$-0.487717\pi$
$463$	1.66025			0.0771585			0.0385793	+	0.999256i	$0.487717\pi$
$464$	−6.92820	+	4.00000i	−0.321634	+	0.185695i
$465$		0			0
$466$	−6.19615	+	23.1244i	−0.287031	+	1.07122i
$467$			36.2487i			1.67739i	0.544601	+	0.838695i	$0.316681\pi$
$467$			36.2487i			1.67739i	−0.544601	+	0.838695i	$0.683319\pi$
$468$		0			0
$469$			0.928203i			0.0428604i
$470$	−31.8564	−	8.53590i	−1.46943	−	0.393732i
$471$		0			0
$472$	20.3923	+	20.3923i	0.938632	+	0.938632i
$473$	2.53590			0.116601
$474$		0			0
$475$			5.12436i			0.235122i
$476$	−12.0000	−	6.92820i	−0.550019	−	0.317554i
$477$		0			0
$478$	−6.85641	+	25.5885i	−0.313605	+	1.17039i
$479$	−21.2679			−0.971757			−0.485879	−	0.874026i	$-0.661500\pi$
$479$	−21.2679			−0.971757			−0.485879	+	0.874026i	$0.661500\pi$
$480$		0			0
$481$	8.92820			0.407091
$482$	−11.1244	+	41.5167i	−0.506701	+	1.89103i
$483$		0			0
$484$	19.7321	+	11.3923i	0.896911	+	0.517832i
$485$			22.1436i			1.00549i
$486$		0			0
$487$	−42.4449			−1.92336			−0.961680	−	0.274174i	$-0.911596\pi$
$487$	−42.4449			−1.92336			−0.961680	+	0.274174i	$0.911596\pi$
$488$	5.85641	+	5.85641i	0.265107	+	0.265107i
$489$		0			0
$490$	−25.5167	−	6.83717i	−1.15273	−	0.308872i
$491$		−	24.2487i		−	1.09433i	−0.837025	−	0.547165i	$-0.815707\pi$
$491$		−	24.2487i		−	1.09433i	0.837025	−	0.547165i	$-0.184293\pi$
$492$		0			0
$493$		−	10.9282i		−	0.492182i
$494$	−0.267949	+	1.00000i	−0.0120556	+	0.0449921i
$495$		0			0
$496$	−13.4641	−	23.3205i	−0.604556	−	1.04712i
$497$	−10.3923			−0.466159
$498$		0			0
$499$			7.26795i			0.325358i	0.986679	+	0.162679i	$0.0520135\pi$
$499$			7.26795i			0.325358i	−0.986679	+	0.162679i	$0.947986\pi$
$500$	−6.92820	+	12.0000i	−0.309839	+	0.536656i
$501$		0			0
$502$	8.73205	+	2.33975i	0.389731	+	0.104428i
$503$	−30.5359			−1.36153			−0.680764	−	0.732503i	$-0.738352\pi$
$503$	−30.5359			−1.36153			−0.680764	+	0.732503i	$0.738352\pi$
$504$		0			0
$505$	41.5692			1.84981
$506$	25.8564	+	6.92820i	1.14946	+	0.307996i
$507$		0			0
$508$	6.92820	+	4.00000i	0.307389	+	0.177471i
$509$			14.3923i			0.637928i	0.947767	+	0.318964i	$0.103335\pi$
$509$			14.3923i			0.637928i	−0.947767	+	0.318964i	$0.896665\pi$
$510$		0			0
$511$	9.46410			0.418667
$512$	16.0000	+	16.0000i	0.707107	+	0.707107i
$513$		0			0
$514$	8.73205	−	32.5885i	0.385154	−	1.43742i
$515$			24.0000i			1.05757i
$516$		0			0
$517$			31.8564i			1.40104i
$518$	−15.4641	−	4.14359i	−0.679454	−	0.182059i
$519$		0			0
$520$	−6.92820	+	6.92820i	−0.303822	+	0.303822i
$521$	25.1769			1.10302			0.551510	−	0.834168i	$-0.314052\pi$
$521$	25.1769			1.10302			0.551510	+	0.834168i	$0.314052\pi$
$522$		0			0
$523$		−	14.0000i		−	0.612177i	−0.952003	−	0.306089i	$-0.900980\pi$
$523$		−	14.0000i		−	0.612177i	0.952003	−	0.306089i	$-0.0990204\pi$
$524$	19.8564	−	34.3923i	0.867431	−	1.50243i
$525$		0			0
$526$	10.0000	−	37.3205i	0.436021	−	1.62725i
$527$	36.7846			1.60236
$528$		0			0
$529$	−7.00000			−0.304348
$530$	3.71281	−	13.8564i	0.161274	−	0.601884i
$531$		0			0
$532$	0.928203	−	1.60770i	0.0402427	−	0.0697024i
$533$			8.92820i			0.386723i
$534$		0			0
$535$	−17.0718			−0.738078
$536$	1.46410	+	1.46410i	0.0632396	+	0.0632396i
$537$		0			0
$538$	−10.7321	−	2.87564i	−0.462692	−	0.123978i
$539$			25.5167i			1.09908i
$540$		0			0
$541$			3.07180i			0.132067i	0.997817	+	0.0660334i	$0.0210344\pi$
$541$			3.07180i			0.132067i	−0.997817	+	0.0660334i	$0.978966\pi$
$542$	−7.39230	+	27.5885i	−0.317527	+	1.18503i
$543$		0			0
$544$	−29.8564	+	8.00000i	−1.28008	+	0.342997i
$545$	−6.92820			−0.296772
$546$		0			0
$547$			11.8564i			0.506943i	0.967343	+	0.253472i	$0.0815725\pi$
$547$			11.8564i			0.506943i	−0.967343	+	0.253472i	$0.918428\pi$
$548$	22.3923	+	12.9282i	0.956552	+	0.552265i
$549$		0			0
$550$	45.2487	+	12.1244i	1.92941	+	0.516984i
$551$	1.46410			0.0623728
$552$		0			0
$553$	−6.92820			−0.294617
$554$	−2.53590	−	0.679492i	−0.107740	−	0.0288688i
$555$		0			0
$556$	10.0000	−	17.3205i	0.424094	−	0.734553i
$557$			34.7846i			1.47387i	0.675963	+	0.736936i	$0.263728\pi$
$557$			34.7846i			1.47387i	−0.675963	+	0.736936i	$0.736272\pi$
$558$		0			0
$559$	0.535898			0.0226661
$560$	15.2154	−	8.78461i	0.642968	−	0.371218i
$561$		0			0
$562$	−3.41154	+	12.7321i	−0.143907	+	0.537069i
$563$		−	7.46410i		−	0.314574i	−0.987553	−	0.157287i	$-0.949725\pi$
$563$		−	7.46410i		−	0.314574i	0.987553	−	0.157287i	$-0.0502748\pi$
$564$		0			0
$565$		−	8.78461i		−	0.369571i
$566$	−26.5885	−	7.12436i	−1.11760	−	0.299459i
$567$		0			0
$568$	−16.3923	+	16.3923i	−0.687806	+	0.687806i
$569$	14.0000			0.586911			0.293455	−	0.955973i	$-0.405195\pi$
$569$	14.0000			0.586911			0.293455	+	0.955973i	$0.405195\pi$
$570$		0			0
$571$		−	2.67949i		−	0.112133i	−0.998427	−	0.0560666i	$-0.982144\pi$
$571$		−	2.67949i		−	0.112133i	0.998427	−	0.0560666i	$-0.0178559\pi$
$572$	8.19615	+	4.73205i	0.342698	+	0.197857i
$573$		0			0
$574$	4.14359	−	15.4641i	0.172950	−	0.645459i
$575$	28.0000			1.16768
$576$		0			0
$577$	−7.07180			−0.294403			−0.147201	−	0.989107i	$-0.547027\pi$
$577$	−7.07180			−0.294403			−0.147201	+	0.989107i	$0.547027\pi$
$578$	4.70577	−	17.5622i	0.195734	−	0.730490i
$579$		0			0
$580$	12.0000	+	6.92820i	0.498273	+	0.287678i
$581$		−	4.14359i		−	0.171905i
$582$		0			0
$583$	−13.8564			−0.573874
$584$	14.9282	−	14.9282i	0.617733	−	0.617733i
$585$		0			0
$586$	44.9808	+	12.0526i	1.85814	+	0.497887i
$587$		−	4.33975i		−	0.179120i	−0.995981	−	0.0895602i	$-0.971454\pi$
$587$		−	4.33975i		−	0.179120i	0.995981	−	0.0895602i	$-0.0285462\pi$
$588$		0			0
$589$			4.92820i			0.203063i
$590$	12.9282	−	48.2487i	0.532246	−	1.98637i
$591$		0			0
$592$	−30.9282	+	17.8564i	−1.27114	+	0.733894i
$593$	7.85641			0.322624			0.161312	−	0.986903i	$-0.448427\pi$
$593$	7.85641			0.322624			0.161312	+	0.986903i	$0.448427\pi$
$594$		0			0
$595$			24.0000i			0.983904i
$596$	−12.9282	+	22.3923i	−0.529560	+	0.917225i
$597$		0			0
$598$	5.46410	+	1.46410i	0.223444	+	0.0598716i
$599$	45.1769			1.84588			0.922939	−	0.384945i	$-0.125780\pi$
$599$	45.1769			1.84588			0.922939	+	0.384945i	$0.125780\pi$
$600$		0			0
$601$	−4.39230			−0.179166			−0.0895829	−	0.995979i	$-0.528553\pi$
$601$	−4.39230			−0.179166			−0.0895829	+	0.995979i	$0.528553\pi$
$602$	−0.928203	−	0.248711i	−0.0378307	−	0.0101367i
$603$		0			0
$604$	12.3397	+	7.12436i	0.502097	+	0.289886i
$605$		−	39.4641i		−	1.60444i
$606$		0			0
$607$	7.32051			0.297130			0.148565	−	0.988903i	$-0.452535\pi$
$607$	7.32051			0.297130			0.148565	+	0.988903i	$0.452535\pi$
$608$	−1.07180	−	4.00000i	−0.0434671	−	0.162221i
$609$		0			0
$610$	3.71281	−	13.8564i	0.150327	−	0.561029i
$611$			6.73205i			0.272350i
$612$		0			0
$613$		−	24.6410i		−	0.995241i	−0.867395	−	0.497621i	$-0.834207\pi$
$613$		−	24.6410i		−	0.995241i	0.867395	−	0.497621i	$-0.165793\pi$
$614$	−1.00000	−	0.267949i	−0.0403567	−	0.0108135i
$615$		0			0
$616$	−12.0000	−	12.0000i	−0.483494	−	0.483494i
$617$	−41.3205			−1.66350			−0.831751	−	0.555150i	$-0.812661\pi$
$617$	−41.3205			−1.66350			−0.831751	+	0.555150i	$0.812661\pi$
$618$		0			0
$619$			18.5885i			0.747133i	0.927603	+	0.373567i	$0.121865\pi$
$619$			18.5885i			0.747133i	−0.927603	+	0.373567i	$0.878135\pi$
$620$	−23.3205	+	40.3923i	−0.936574	+	1.62219i
$621$		0			0
$622$	−0.392305	+	1.46410i	−0.0157300	+	0.0587051i
$623$	−21.9615			−0.879870
$624$		0			0
$625$	−11.0000			−0.440000
$626$	0.143594	−	0.535898i	0.00573915	−	0.0214188i
$627$		0			0
$628$	16.9282	−	29.3205i	0.675509	−	1.17002i
$629$		−	48.7846i		−	1.94517i
$630$		0			0
$631$	−42.0526			−1.67409			−0.837043	−	0.547137i	$-0.815718\pi$
$631$	−42.0526			−1.67409			−0.837043	+	0.547137i	$0.815718\pi$
$632$	−10.9282	+	10.9282i	−0.434701	+	0.434701i
$633$		0			0
$634$	4.73205	+	1.26795i	0.187934	+	0.0503567i
$635$		−	13.8564i		−	0.549875i
$636$		0			0
$637$			5.39230i			0.213651i
$638$	3.46410	−	12.9282i	0.137145	−	0.511832i
$639$		0			0
$640$	10.1436	−	37.8564i	0.400961	−	1.49641i
$641$	22.2487			0.878771			0.439386	−	0.898299i	$-0.355196\pi$
$641$	22.2487			0.878771			0.439386	+	0.898299i	$0.355196\pi$
$642$		0			0
$643$			12.7321i			0.502103i	0.967974	+	0.251052i	$0.0807764\pi$
$643$			12.7321i			0.502103i	−0.967974	+	0.251052i	$0.919224\pi$
$644$	−8.78461	−	5.07180i	−0.346162	−	0.199857i
$645$		0			0
$646$	5.46410	+	1.46410i	0.214982	+	0.0576043i
$647$	−37.8564			−1.48829			−0.744144	−	0.668019i	$-0.767143\pi$
$647$	−37.8564			−1.48829			−0.744144	+	0.668019i	$0.767143\pi$
$648$		0			0
$649$	−48.2487			−1.89393
$650$	9.56218	+	2.56218i	0.375059	+	0.100497i
$651$		0			0
$652$	−16.7321	+	28.9808i	−0.655278	+	1.13497i
$653$		−	3.07180i		−	0.120209i	−0.998192	−	0.0601043i	$-0.980857\pi$
$653$		−	3.07180i		−	0.120209i	0.998192	−	0.0601043i	$-0.0191433\pi$
$654$		0			0
$655$	−68.7846			−2.68764
$656$	−17.8564	−	30.9282i	−0.697176	−	1.20754i
$657$		0			0
$658$	3.12436	−	11.6603i	0.121800	−	0.454564i
$659$			14.0000i			0.545363i	0.962104	+	0.272681i	$0.0879105\pi$
$659$			14.0000i			0.545363i	−0.962104	+	0.272681i	$0.912090\pi$
$660$		0			0
$661$		−	17.3205i		−	0.673690i	−0.941560	−	0.336845i	$-0.890640\pi$
$661$		−	17.3205i		−	0.673690i	0.941560	−	0.336845i	$-0.109360\pi$
$662$	−23.9282	−	6.41154i	−0.929996	−	0.249192i
$663$		0			0
$664$	−6.53590	−	6.53590i	−0.253642	−	0.253642i
$665$	−3.21539			−0.124687
$666$		0			0
$667$		−	8.00000i		−	0.309761i
$668$	−9.80385	−	5.66025i	−0.379322	−	0.219002i
$669$		0			0
$670$	0.928203	−	3.46410i	0.0358596	−	0.133830i
$671$	−13.8564			−0.534921
$672$		0			0
$673$	33.1769			1.27888			0.639438	−	0.768843i	$-0.279167\pi$
$673$	33.1769			1.27888			0.639438	+	0.768843i	$0.279167\pi$
$674$	0.535898	−	2.00000i	0.0206420	−	0.0770371i
$675$		0			0
$676$	1.73205	+	1.00000i	0.0666173	+	0.0384615i
$677$			21.0718i			0.809855i	0.914349	+	0.404927i	$0.132703\pi$
$677$			21.0718i			0.809855i	−0.914349	+	0.404927i	$0.867297\pi$
$678$		0			0
$679$	−8.10512			−0.311046
$680$	37.8564	+	37.8564i	1.45173	+	1.45173i
$681$		0			0
$682$	43.5167	+	11.6603i	1.66634	+	0.446494i
$683$		−	17.8038i		−	0.681245i	−0.940200	−	0.340623i	$-0.889362\pi$
$683$		−	17.8038i		−	0.681245i	0.940200	−	0.340623i	$-0.110638\pi$
$684$		0			0
$685$		−	44.7846i		−	1.71113i
$686$	5.75129	−	21.4641i	0.219585	−	0.819503i
$687$		0			0
$688$	−1.85641	+	1.07180i	−0.0707748	+	0.0408619i
$689$	−2.92820			−0.111556
$690$		0			0
$691$			32.8372i			1.24918i	0.780951	+	0.624592i	$0.214735\pi$
$691$			32.8372i			1.24918i	−0.780951	+	0.624592i	$0.785265\pi$
$692$	6.92820	−	12.0000i	0.263371	−	0.456172i
$693$		0			0
$694$	−9.66025	−	2.58846i	−0.366698	−	0.0982565i
$695$	−34.6410			−1.31401
$696$		0			0
$697$	48.7846			1.84785
$698$	−13.1244	−	3.51666i	−0.496764	−	0.133108i
$699$		0			0
$700$	−15.3731	−	8.87564i	−0.581047	−	0.335468i
$701$		−	40.6410i		−	1.53499i	−0.641055	−	0.767495i	$-0.721503\pi$
$701$		−	40.6410i		−	1.53499i	0.641055	−	0.767495i	$-0.278497\pi$
$702$		0			0
$703$	6.53590			0.246506
$704$	−37.8564			−1.42677
$705$		0			0
$706$	4.05256	−	15.1244i	0.152520	−	0.569213i
$707$			15.2154i			0.572234i
$708$		0			0
$709$		−	29.3205i		−	1.10115i	−0.834784	−	0.550577i	$-0.814408\pi$
$709$		−	29.3205i		−	1.10115i	0.834784	−	0.550577i	$-0.185592\pi$
$710$	38.7846	+	10.3923i	1.45556	+	0.390016i
$711$		0			0
$712$	−34.6410	+	34.6410i	−1.29823	+	1.29823i
$713$	26.9282			1.00847
$714$		0			0
$715$		−	16.3923i		−	0.613037i
$716$	−10.3923	+	18.0000i	−0.388379	+	0.672692i
$717$		0			0
$718$	−11.5359	+	43.0526i	−0.430516	+	1.60671i
$719$	−30.9282			−1.15343			−0.576714	−	0.816946i	$-0.695665\pi$
$719$	−30.9282			−1.15343			−0.576714	+	0.816946i	$0.695665\pi$
$720$		0			0
$721$	−8.78461			−0.327156
$722$	6.75833	−	25.2224i	0.251519	−	0.938682i
$723$		0			0
$724$	8.92820	−	15.4641i	0.331814	−	0.574719i
$725$		−	14.0000i		−	0.519947i
$726$		0			0
$727$	−22.9282			−0.850360			−0.425180	−	0.905109i	$-0.639789\pi$
$727$	−22.9282			−0.850360			−0.425180	+	0.905109i	$0.639789\pi$
$728$	−2.53590	−	2.53590i	−0.0939866	−	0.0939866i
$729$		0			0
$730$	−35.3205	−	9.46410i	−1.30727	−	0.350282i
$731$		−	2.92820i		−	0.108304i
$732$		0			0
$733$		−	37.3205i		−	1.37846i	−0.724541	−	0.689232i	$-0.757948\pi$
$733$		−	37.3205i		−	1.37846i	0.724541	−	0.689232i	$-0.242052\pi$
$734$	−8.14359	+	30.3923i	−0.300586	+	1.12180i
$735$		0			0
$736$	−21.8564	+	5.85641i	−0.805638	+	0.215870i
$737$	−3.46410			−0.127602
$738$		0			0
$739$		−	39.7654i		−	1.46279i	−0.681953	−	0.731396i	$-0.738869\pi$
$739$		−	39.7654i		−	1.46279i	0.681953	−	0.731396i	$-0.261131\pi$
$740$	53.5692	+	30.9282i	1.96924	+	1.13694i
$741$		0			0
$742$	5.07180	+	1.35898i	0.186192	+	0.0498899i
$743$	32.1962			1.18116			0.590581	−	0.806978i	$-0.298899\pi$
$743$	32.1962			1.18116			0.590581	+	0.806978i	$0.298899\pi$
$744$		0			0
$745$	44.7846			1.64078
$746$	20.1962	+	5.41154i	0.739434	+	0.198131i
$747$		0			0
$748$	25.8564	−	44.7846i	0.945404	−	1.63749i
$749$		−	6.24871i		−	0.228323i
$750$		0			0
$751$	1.07180			0.0391104			0.0195552	−	0.999809i	$-0.493775\pi$
$751$	1.07180			0.0391104			0.0195552	+	0.999809i	$0.493775\pi$
$752$	−13.4641	−	23.3205i	−0.490985	−	0.850411i
$753$		0			0
$754$	0.732051	−	2.73205i	0.0266597	−	0.0994954i
$755$		−	24.6795i		−	0.898179i
$756$		0			0
$757$		−	40.7846i		−	1.48234i	−0.671316	−	0.741171i	$-0.734271\pi$
$757$		−	40.7846i		−	1.48234i	0.671316	−	0.741171i	$-0.265729\pi$
$758$	−19.9282	−	5.33975i	−0.723825	−	0.193948i
$759$		0			0
$760$	−5.07180	+	5.07180i	−0.183973	+	0.183973i
$761$	−18.7846			−0.680942			−0.340471	−	0.940255i	$-0.610586\pi$
$761$	−18.7846			−0.680942			−0.340471	+	0.940255i	$0.610586\pi$
$762$		0			0
$763$		−	2.53590i		−	0.0918057i
$764$	−25.1769	−	14.5359i	−0.910869	−	0.525890i
$765$		0			0
$766$	3.78461	−	14.1244i	0.136744	−	0.510334i
$767$	−10.1962			−0.368162
$768$		0			0
$769$	31.4641			1.13462			0.567312	−	0.823503i	$-0.307983\pi$
$769$	31.4641			1.13462			0.567312	+	0.823503i	$0.307983\pi$
$770$	−7.60770	+	28.3923i	−0.274162	+	1.02319i
$771$		0			0
$772$	2.78461	+	1.60770i	0.100220	+	0.0578622i
$773$			27.4641i			0.987815i	0.869514	+	0.493908i	$0.164432\pi$
$773$			27.4641i			0.987815i	−0.869514	+	0.493908i	$0.835568\pi$
$774$		0			0
$775$	47.1244			1.69276
$776$	−12.7846	+	12.7846i	−0.458941	+	0.458941i
$777$		0			0
$778$	2.73205	+	0.732051i	0.0979488	+	0.0262453i
$779$			6.53590i			0.234173i
$780$		0			0
$781$		−	38.7846i		−	1.38782i
$782$	8.00000	−	29.8564i	0.286079	−	1.06766i
$783$		0			0
$784$	−10.7846	−	18.6795i	−0.385165	−	0.667125i
$785$	−58.6410

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:	\( N \)	\(=\)	\( 936 = 2^{3} \cdot 3^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	936.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(0.366025 - 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +3.46410i q^{5} -1.26795 q^{7} +(-2.00000 + 2.00000i) q^{8} +O(q^{10})\)	\(q+(0.366025 - 1.36603i) q^{2} +(-1.73205 - 1.00000i) q^{4} +3.46410i q^{5} -1.26795 q^{7} +(-2.00000 + 2.00000i) q^{8} +(4.73205 + 1.26795i) q^{10} -4.73205i q^{11} -1.00000i q^{13} +(-0.464102 + 1.73205i) q^{14} +(2.00000 + 3.46410i) q^{16} -5.46410 q^{17} -0.732051i q^{19} +(3.46410 - 6.00000i) q^{20} +(-6.46410 - 1.73205i) q^{22} -4.00000 q^{23} -7.00000 q^{25} +(-1.36603 - 0.366025i) q^{26} +(2.19615 + 1.26795i) q^{28} +2.00000i q^{29} -6.73205 q^{31} +(5.46410 - 1.46410i) q^{32} +(-2.00000 + 7.46410i) q^{34} -4.39230i q^{35} +8.92820i q^{37} +(-1.00000 - 0.267949i) q^{38} +(-6.92820 - 6.92820i) q^{40} -8.92820 q^{41} +0.535898i q^{43} +(-4.73205 + 8.19615i) q^{44} +(-1.46410 + 5.46410i) q^{46} -6.73205 q^{47} -5.39230 q^{49} +(-2.56218 + 9.56218i) q^{50} +(-1.00000 + 1.73205i) q^{52} -2.92820i q^{53} +16.3923 q^{55} +(2.53590 - 2.53590i) q^{56} +(2.73205 + 0.732051i) q^{58} -10.1962i q^{59} -2.92820i q^{61} +(-2.46410 + 9.19615i) q^{62} -8.00000i q^{64} +3.46410 q^{65} -0.732051i q^{67} +(9.46410 + 5.46410i) q^{68} +(-6.00000 - 1.60770i) q^{70} +8.19615 q^{71} -7.46410 q^{73} +(12.1962 + 3.26795i) q^{74} +(-0.732051 + 1.26795i) q^{76} +6.00000i q^{77} +5.46410 q^{79} +(-12.0000 + 6.92820i) q^{80} +(-3.26795 + 12.1962i) q^{82} +3.26795i q^{83} -18.9282i q^{85} +(0.732051 + 0.196152i) q^{86} +(9.46410 + 9.46410i) q^{88} +17.3205 q^{89} +1.26795i q^{91} +(6.92820 + 4.00000i) q^{92} +(-2.46410 + 9.19615i) q^{94} +2.53590 q^{95} +6.39230 q^{97} +(-1.97372 + 7.36603i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 - 12 * q^7 - 8 * q^8	\( 4 q - 2 q^{2} - 12 q^{7} - 8 q^{8} + 12 q^{10} + 12 q^{14} + 8 q^{16} - 8 q^{17} - 12 q^{22} - 16 q^{23} - 28 q^{25} - 2 q^{26} - 12 q^{28} - 20 q^{31} + 8 q^{32} - 8 q^{34} - 4 q^{38} - 8 q^{41} - 12 q^{44} + 8 q^{46} - 20 q^{47} + 20 q^{49} + 14 q^{50} - 4 q^{52} + 24 q^{55} + 24 q^{56} + 4 q^{58} + 4 q^{62} + 24 q^{68} - 24 q^{70} + 12 q^{71} - 16 q^{73} + 28 q^{74} + 4 q^{76} + 8 q^{79} - 48 q^{80} - 20 q^{82} - 4 q^{86} + 24 q^{88} + 4 q^{94} + 24 q^{95} - 16 q^{97} - 46 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 - 12 * q^7 - 8 * q^8 + 12 * q^10 + 12 * q^14 + 8 * q^16 - 8 * q^17 - 12 * q^22 - 16 * q^23 - 28 * q^25 - 2 * q^26 - 12 * q^28 - 20 * q^31 + 8 * q^32 - 8 * q^34 - 4 * q^38 - 8 * q^41 - 12 * q^44 + 8 * q^46 - 20 * q^47 + 20 * q^49 + 14 * q^50 - 4 * q^52 + 24 * q^55 + 24 * q^56 + 4 * q^58 + 4 * q^62 + 24 * q^68 - 24 * q^70 + 12 * q^71 - 16 * q^73 + 28 * q^74 + 4 * q^76 + 8 * q^79 - 48 * q^80 - 20 * q^82 - 4 * q^86 + 24 * q^88 + 4 * q^94 + 24 * q^95 - 16 * q^97 - 46 * q^98

Introduction

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