LMFDB - Embedded newform 325.2.b.f.274.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [325,2,Mod(274,325)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(325, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("325.274");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 325 = 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	325.b (of order $2$, degree $1$, not 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:	$2.59513806569$
Analytic rank:	$0$
Dimension:	$4$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 65)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			274.4
Root			$0.707107 + 0.707107i$ of defining polynomial
Character	$\chi$	$=$	325.274
Dual form			325.2.b.f.274.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.41421i q^{2} -1.41421i q^{3} -3.82843 q^{4} +3.41421 q^{6} -4.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.41421i q^{2} -1.41421i q^{3} -3.82843 q^{4} +3.41421 q^{6} -4.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +3.41421 q^{11} +5.41421i q^{12} -1.00000i q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427i q^{17} +2.41421i q^{18} -0.585786 q^{19} -6.82843 q^{21} +8.24264i q^{22} +1.41421i q^{23} -6.24264 q^{24} +2.41421 q^{26} -5.65685i q^{27} +18.4853i q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579i q^{32} -4.82843i q^{33} +2.00000 q^{34} -3.82843 q^{36} +8.48528i q^{37} -1.41421i q^{38} -1.41421 q^{39} -3.17157 q^{41} -16.4853i q^{42} -11.0711i q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843i q^{47} -4.24264i q^{48} -16.3137 q^{49} -1.17157 q^{51} +3.82843i q^{52} +2.48528i q^{53} +13.6569 q^{54} -21.3137 q^{56} +0.828427i q^{57} +13.6569i q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264i q^{62} -4.82843i q^{63} +9.82843 q^{64} +11.6569 q^{66} +2.00000i q^{67} +3.17157i q^{68} +2.00000 q^{69} +11.8995 q^{71} -4.41421i q^{72} +8.48528i q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853i q^{77} -3.41421i q^{78} +8.48528 q^{79} -5.00000 q^{81} -7.65685i q^{82} -3.17157i q^{83} +26.1421 q^{84} +26.7279 q^{86} -8.00000i q^{87} -15.0711i q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421i q^{92} -2.48528i q^{93} -11.6569 q^{94} -2.24264 q^{96} +7.65685i q^{97} -39.3848i q^{98} +3.41421 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9	$ 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{11} + 24 q^{14} + 12 q^{16} - 8 q^{19} - 16 q^{21} - 8 q^{24} + 4 q^{26} + 24 q^{31} + 8 q^{34} - 4 q^{36} - 24 q^{41} - 24 q^{44} - 8 q^{46} - 20 q^{49} - 16 q^{51} + 32 q^{54} - 40 q^{56} - 24 q^{59} - 32 q^{61} + 28 q^{64} + 24 q^{66} + 8 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 20 q^{81} + 48 q^{84} + 56 q^{86} - 24 q^{89} - 8 q^{91} - 24 q^{94} + 8 q^{96} + 8 q^{99}+O(q^{100}) $ 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9 + 8 * q^11 + 24 * q^14 + 12 * q^16 - 8 * q^19 - 16 * q^21 - 8 * q^24 + 4 * q^26 + 24 * q^31 + 8 * q^34 - 4 * q^36 - 24 * q^41 - 24 * q^44 - 8 * q^46 - 20 * q^49 - 16 * q^51 + 32 * q^54 - 40 * q^56 - 24 * q^59 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 8 * q^69 + 8 * q^71 - 48 * q^74 - 8 * q^76 - 20 * q^81 + 48 * q^84 + 56 * q^86 - 24 * q^89 - 8 * q^91 - 24 * q^94 + 8 * q^96 + 8 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/325\mathbb{Z}\right)^\times$.

$n$	$27$	$301$
$\chi(n)$	$-1$	$1$

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 $

$2$	2.41421i	1.70711i	0.521005	+	0.853553i	$0.325557\pi$
$2$	2.41421i	1.70711i	−0.521005	+	0.853553i	$0.674443\pi$
$3$	− 1.41421i	− 0.816497i	−0.912871	−	0.408248i	$-0.866140\pi$
$3$	− 1.41421i	− 0.816497i	0.912871	−	0.408248i	$-0.133860\pi$
$4$	−3.82843	−1.91421
$5$	0	0
$5$	0	0
$6$	3.41421	1.39385
$7$	− 4.82843i	− 1.82497i	−0.409106	−	0.912487i	$-0.634159\pi$
$7$	− 4.82843i	− 1.82497i	0.409106	−	0.912487i	$-0.365841\pi$
$8$	− 4.41421i	− 1.56066i
$9$	1.00000	0.333333
$10$	0	0
$11$	3.41421	1.02942	0.514712	−	0.857363i	$-0.327899\pi$
$11$	3.41421	1.02942	0.514712	+	0.857363i	$0.327899\pi$
$12$	5.41421i	1.56295i
$13$	− 1.00000i	− 0.277350i
$13$	− 1.00000i	− 0.277350i
$14$	11.6569	3.11543
$15$	0	0
$16$	3.00000	0.750000
$17$	− 0.828427i	− 0.200923i	−0.994941	−	0.100462i	$-0.967968\pi$
$17$	− 0.828427i	− 0.200923i	0.994941	−	0.100462i	$-0.0320319\pi$
$18$	2.41421i	0.569036i
$19$	−0.585786	−0.134389	−0.0671943	−	0.997740i	$-0.521405\pi$
$19$	−0.585786	−0.134389	−0.0671943	+	0.997740i	$0.521405\pi$
$20$	0	0
$21$	−6.82843	−1.49008
$22$	8.24264i	1.75734i
$23$	1.41421i	0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$	1.41421i	0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$	−6.24264	−1.27427
$25$	0	0
$26$	2.41421	0.473466
$27$	− 5.65685i	− 1.08866i
$28$	18.4853i	3.49339i
$29$	5.65685	1.05045	0.525226	−	0.850963i	$-0.323981\pi$
$29$	5.65685	1.05045	0.525226	+	0.850963i	$0.323981\pi$
$30$	0	0
$31$	1.75736	0.315631	0.157816	−	0.987469i	$-0.449555\pi$
$31$	1.75736	0.315631	0.157816	+	0.987469i	$0.449555\pi$
$32$	− 1.58579i	− 0.280330i
$33$	− 4.82843i	− 0.840521i
$34$	2.00000	0.342997
$35$	0	0
$36$	−3.82843	−0.638071
$37$	8.48528i	1.39497i	0.716599	+	0.697486i	$0.245698\pi$
$37$	8.48528i	1.39497i	−0.716599	+	0.697486i	$0.754302\pi$
$38$	− 1.41421i	− 0.229416i
$39$	−1.41421	−0.226455
$40$	0	0
$41$	−3.17157	−0.495316	−0.247658	−	0.968847i	$-0.579661\pi$
$41$	−3.17157	−0.495316	−0.247658	+	0.968847i	$0.579661\pi$
$42$	− 16.4853i	− 2.54373i
$43$	− 11.0711i	− 1.68832i	−0.536090	−	0.844161i	$-0.680099\pi$
$43$	− 11.0711i	− 1.68832i	0.536090	−	0.844161i	$-0.319901\pi$
$44$	−13.0711	−1.97054
$45$	0	0
$46$	−3.41421	−0.503398
$47$	4.82843i	0.704298i	0.935944	+	0.352149i	$0.114549\pi$
$47$	4.82843i	0.704298i	−0.935944	+	0.352149i	$0.885451\pi$
$48$	− 4.24264i	− 0.612372i
$49$	−16.3137	−2.33053
$50$	0	0
$51$	−1.17157	−0.164053
$52$	3.82843i	0.530907i
$53$	2.48528i	0.341380i	0.985325	+	0.170690i	$0.0545996\pi$
$53$	2.48528i	0.341380i	−0.985325	+	0.170690i	$0.945400\pi$
$54$	13.6569	1.85846
$55$	0	0
$56$	−21.3137	−2.84816
$57$	0.828427i	0.109728i
$58$	13.6569i	1.79323i
$59$	−1.75736	−0.228789	−0.114394	−	0.993435i	$-0.536493\pi$
$59$	−1.75736	−0.228789	−0.114394	+	0.993435i	$0.536493\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	4.24264i	0.538816i
$63$	− 4.82843i	− 0.608325i
$64$	9.82843	1.22855
$65$	0	0
$66$	11.6569	1.43486
$67$	2.00000i	0.244339i	0.992509	+	0.122169i	$0.0389851\pi$
$67$	2.00000i	0.244339i	−0.992509	+	0.122169i	$0.961015\pi$
$68$	3.17157i	0.384610i
$69$	2.00000	0.240772
$70$	0	0
$71$	11.8995	1.41221	0.706105	−	0.708107i	$-0.250451\pi$
$71$	11.8995	1.41221	0.706105	+	0.708107i	$0.250451\pi$
$72$	− 4.41421i	− 0.520220i
$73$	8.48528i	0.993127i	0.868000	+	0.496564i	$0.165405\pi$
$73$	8.48528i	0.993127i	−0.868000	+	0.496564i	$0.834595\pi$
$74$	−20.4853	−2.38137
$75$	0	0
$76$	2.24264	0.257249
$77$	− 16.4853i	− 1.87867i
$78$	− 3.41421i	− 0.386584i
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	0	0
$81$	−5.00000	−0.555556
$82$	− 7.65685i	− 0.845558i
$83$	− 3.17157i	− 0.348125i	−0.984735	−	0.174063i	$-0.944310\pi$
$83$	− 3.17157i	− 0.348125i	0.984735	−	0.174063i	$-0.0556895\pi$
$84$	26.1421	2.85234
$85$	0	0
$86$	26.7279	2.88215
$87$	− 8.00000i	− 0.857690i
$88$	− 15.0711i	− 1.60658i
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	0	0
$91$	−4.82843	−0.506157
$92$	− 5.41421i	− 0.564471i
$93$	− 2.48528i	− 0.257712i
$94$	−11.6569	−1.20231
$95$	0	0
$96$	−2.24264	−0.228889
$97$	7.65685i	0.777436i	0.921357	+	0.388718i	$0.127082\pi$
$97$	7.65685i	0.777436i	−0.921357	+	0.388718i	$0.872918\pi$
$98$	− 39.3848i	− 3.97846i
$99$	3.41421	0.343141
$100$	0	0
$101$	−3.65685	−0.363871	−0.181935	−	0.983311i	$-0.558236\pi$
$101$	−3.65685	−0.363871	−0.181935	+	0.983311i	$0.558236\pi$
$102$	− 2.82843i	− 0.280056i
$103$	14.5858i	1.43718i	0.695434	+	0.718590i	$0.255212\pi$
$103$	14.5858i	1.43718i	−0.695434	+	0.718590i	$0.744788\pi$
$104$	−4.41421	−0.432849
$105$	0	0
$106$	−6.00000	−0.582772
$107$	9.41421i	0.910106i	0.890464	+	0.455053i	$0.150380\pi$
$107$	9.41421i	0.910106i	−0.890464	+	0.455053i	$0.849620\pi$
$108$	21.6569i	2.08393i
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	12.0000	1.13899
$112$	− 14.4853i	− 1.36873i
$113$	− 8.82843i	− 0.830509i	−0.909705	−	0.415254i	$-0.863693\pi$
$113$	− 8.82843i	− 0.830509i	0.909705	−	0.415254i	$-0.136307\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−21.6569	−2.01079
$117$	− 1.00000i	− 0.0924500i
$118$	− 4.24264i	− 0.390567i
$119$	−4.00000	−0.366679
$120$	0	0
$121$	0.656854	0.0597140
$122$	− 19.3137i	− 1.74858i
$123$	4.48528i	0.404424i
$124$	−6.72792	−0.604185
$125$	0	0
$126$	11.6569	1.03848
$127$	6.58579i	0.584394i	0.956358	+	0.292197i	$0.0943863\pi$
$127$	6.58579i	0.584394i	−0.956358	+	0.292197i	$0.905614\pi$
$128$	20.5563i	1.81694i
$129$	−15.6569	−1.37851
$130$	0	0
$131$	−16.9706	−1.48272	−0.741362	−	0.671105i	$-0.765820\pi$
$131$	−16.9706	−1.48272	−0.741362	+	0.671105i	$0.765820\pi$
$132$	18.4853i	1.60894i
$133$	2.82843i	0.245256i
$134$	−4.82843	−0.417113
$135$	0	0
$136$	−3.65685	−0.313573
$137$	17.3137i	1.47921i	0.673041	+	0.739605i	$0.264988\pi$
$137$	17.3137i	1.47921i	−0.673041	+	0.739605i	$0.735012\pi$
$138$	4.82843i	0.411023i
$139$	−4.48528	−0.380437	−0.190218	−	0.981742i	$-0.560920\pi$
$139$	−4.48528	−0.380437	−0.190218	+	0.981742i	$0.560920\pi$
$140$	0	0
$141$	6.82843	0.575057
$142$	28.7279i	2.41079i
$143$	− 3.41421i	− 0.285511i
$144$	3.00000	0.250000
$145$	0	0
$146$	−20.4853	−1.69537
$147$	23.0711i	1.90287i
$148$	− 32.4853i	− 2.67027i
$149$	11.6569	0.954967	0.477483	−	0.878641i	$-0.341549\pi$
$149$	11.6569	0.954967	0.477483	+	0.878641i	$0.341549\pi$
$150$	0	0
$151$	9.75736	0.794043	0.397021	−	0.917809i	$-0.370044\pi$
$151$	9.75736	0.794043	0.397021	+	0.917809i	$0.370044\pi$
$152$	2.58579i	0.209735i
$153$	− 0.828427i	− 0.0669744i
$154$	39.7990	3.20709
$155$	0	0
$156$	5.41421	0.433484
$157$	− 18.0000i	− 1.43656i	−0.695756	−	0.718278i	$-0.744931\pi$
$157$	− 18.0000i	− 1.43656i	0.695756	−	0.718278i	$-0.255069\pi$
$158$	20.4853i	1.62972i
$159$	3.51472	0.278735
$160$	0	0
$161$	6.82843	0.538155
$162$	− 12.0711i	− 0.948393i
$163$	18.9706i	1.48589i	0.669353	+	0.742945i	$0.266571\pi$
$163$	18.9706i	1.48589i	−0.669353	+	0.742945i	$0.733429\pi$
$164$	12.1421	0.948141
$165$	0	0
$166$	7.65685	0.594287
$167$	3.17157i	0.245424i	0.992442	+	0.122712i	$0.0391591\pi$
$167$	3.17157i	0.245424i	−0.992442	+	0.122712i	$0.960841\pi$
$168$	30.1421i	2.32552i
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	−0.585786	−0.0447962
$172$	42.3848i	3.23181i
$173$	16.8284i	1.27944i	0.768607	+	0.639721i	$0.220950\pi$
$173$	16.8284i	1.27944i	−0.768607	+	0.639721i	$0.779050\pi$
$174$	19.3137	1.46417
$175$	0	0
$176$	10.2426	0.772068
$177$	2.48528i	0.186805i
$178$	− 14.4853i	− 1.08572i
$179$	−5.65685	−0.422813	−0.211407	−	0.977398i	$-0.567804\pi$
$179$	−5.65685	−0.422813	−0.211407	+	0.977398i	$0.567804\pi$
$180$	0	0
$181$	0	0		−	1.00000i	$-0.5\pi$
$181$	0	0			1.00000i	$0.5\pi$
$182$	− 11.6569i	− 0.864064i
$183$	11.3137i	0.836333i
$184$	6.24264	0.460214
$185$	0	0
$186$	6.00000	0.439941
$187$	− 2.82843i	− 0.206835i
$188$	− 18.4853i	− 1.34818i
$189$	−27.3137	−1.98678
$190$	0	0
$191$	2.34315	0.169544	0.0847720	−	0.996400i	$-0.472984\pi$
$191$	2.34315	0.169544	0.0847720	+	0.996400i	$0.472984\pi$
$192$	− 13.8995i	− 1.00311i
$193$	4.34315i	0.312626i	0.987708	+	0.156313i	$0.0499609\pi$
$193$	4.34315i	0.312626i	−0.987708	+	0.156313i	$0.950039\pi$
$194$	−18.4853	−1.32717
$195$	0	0
$196$	62.4558	4.46113
$197$	− 10.9706i	− 0.781620i	−0.920471	−	0.390810i	$-0.872195\pi$
$197$	− 10.9706i	− 0.781620i	0.920471	−	0.390810i	$-0.127805\pi$
$198$	8.24264i	0.585779i
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	0	0
$201$	2.82843	0.199502
$202$	− 8.82843i	− 0.621166i
$203$	− 27.3137i	− 1.91705i
$204$	4.48528	0.314033
$205$	0	0
$206$	−35.2132	−2.45342
$207$	1.41421i	0.0982946i
$208$	− 3.00000i	− 0.208013i
$209$	−2.00000	−0.138343
$210$	0	0
$211$	3.31371	0.228125	0.114063	−	0.993474i	$-0.463614\pi$
$211$	3.31371	0.228125	0.114063	+	0.993474i	$0.463614\pi$
$212$	− 9.51472i	− 0.653474i
$213$	− 16.8284i	− 1.15306i
$214$	−22.7279	−1.55365
$215$	0	0
$216$	−24.9706	−1.69903
$217$	− 8.48528i	− 0.576018i
$218$	4.82843i	0.327022i
$219$	12.0000	0.810885
$220$	0	0
$221$	−0.828427	−0.0557260
$222$	28.9706i	1.94438i
$223$	9.51472i	0.637153i	0.947897	+	0.318576i	$0.103205\pi$
$223$	9.51472i	0.637153i	−0.947897	+	0.318576i	$0.896795\pi$
$224$	−7.65685	−0.511595
$225$	0	0
$226$	21.3137	1.41777
$227$	16.3431i	1.08473i	0.840142	+	0.542366i	$0.182472\pi$
$227$	16.3431i	1.08473i	−0.840142	+	0.542366i	$0.817528\pi$
$228$	− 3.17157i	− 0.210043i
$229$	4.82843	0.319071	0.159536	−	0.987192i	$-0.449000\pi$
$229$	4.82843	0.319071	0.159536	+	0.987192i	$0.449000\pi$
$230$	0	0
$231$	−23.3137	−1.53393
$232$	− 24.9706i	− 1.63940i
$233$	− 20.6274i	− 1.35135i	−0.737201	−	0.675674i	$-0.763853\pi$
$233$	− 20.6274i	− 1.35135i	0.737201	−	0.675674i	$-0.236147\pi$
$234$	2.41421	0.157822
$235$	0	0
$236$	6.72792	0.437950
$237$	− 12.0000i	− 0.779484i
$238$	− 9.65685i	− 0.625961i
$239$	3.41421	0.220847	0.110424	−	0.993885i	$-0.464779\pi$
$239$	3.41421	0.220847	0.110424	+	0.993885i	$0.464779\pi$
$240$	0	0
$241$	−14.4853	−0.933079	−0.466539	−	0.884500i	$-0.654499\pi$
$241$	−14.4853	−0.933079	−0.466539	+	0.884500i	$0.654499\pi$
$242$	1.58579i	0.101938i
$243$	− 9.89949i	− 0.635053i
$244$	30.6274	1.96072
$245$	0	0
$246$	−10.8284	−0.690395
$247$	0.585786i	0.0372727i
$248$	− 7.75736i	− 0.492593i
$249$	−4.48528	−0.284243
$250$	0	0
$251$	19.7990	1.24970	0.624851	−	0.780744i	$-0.285160\pi$
$251$	19.7990	1.24970	0.624851	+	0.780744i	$0.285160\pi$
$252$	18.4853i	1.16446i
$253$	4.82843i	0.303561i
$254$	−15.8995	−0.997623
$255$	0	0
$256$	−29.9706	−1.87316
$257$	− 27.6569i	− 1.72519i	−0.505898	−	0.862594i	$-0.668839\pi$
$257$	− 27.6569i	− 1.72519i	0.505898	−	0.862594i	$-0.331161\pi$
$258$	− 37.7990i	− 2.35326i
$259$	40.9706	2.54579
$260$	0	0
$261$	5.65685	0.350150
$262$	− 40.9706i	− 2.53117i
$263$	− 10.5858i	− 0.652748i	−0.945241	−	0.326374i	$-0.894173\pi$
$263$	− 10.5858i	− 0.652748i	0.945241	−	0.326374i	$-0.105827\pi$
$264$	−21.3137	−1.31177
$265$	0	0
$266$	−6.82843	−0.418678
$267$	8.48528i	0.519291i
$268$	− 7.65685i	− 0.467717i
$269$	25.3137	1.54340	0.771702	−	0.635984i	$-0.219406\pi$
$269$	25.3137	1.54340	0.771702	+	0.635984i	$0.219406\pi$
$270$	0	0
$271$	26.7279	1.62361	0.811803	−	0.583932i	$-0.198486\pi$
$271$	26.7279	1.62361	0.811803	+	0.583932i	$0.198486\pi$
$272$	− 2.48528i	− 0.150692i
$273$	6.82843i	0.413275i
$274$	−41.7990	−2.52517
$275$	0	0
$276$	−7.65685	−0.460888
$277$	12.8284i	0.770785i	0.922753	+	0.385393i	$0.125934\pi$
$277$	12.8284i	0.770785i	−0.922753	+	0.385393i	$0.874066\pi$
$278$	− 10.8284i	− 0.649446i
$279$	1.75736	0.105210
$280$	0	0
$281$	21.7990	1.30042	0.650209	−	0.759755i	$-0.274681\pi$
$281$	21.7990	1.30042	0.650209	+	0.759755i	$0.274681\pi$
$282$	16.4853i	0.981684i
$283$	− 16.7279i	− 0.994372i	−0.867644	−	0.497186i	$-0.834367\pi$
$283$	− 16.7279i	− 0.994372i	0.867644	−	0.497186i	$-0.165633\pi$
$284$	−45.5563	−2.70327
$285$	0	0
$286$	8.24264	0.487398
$287$	15.3137i	0.903940i
$288$	− 1.58579i	− 0.0934434i
$289$	16.3137	0.959630
$290$	0	0
$291$	10.8284	0.634774
$292$	− 32.4853i	− 1.90106i
$293$	26.1421i	1.52724i	0.645666	+	0.763620i	$0.276580\pi$
$293$	26.1421i	1.52724i	−0.645666	+	0.763620i	$0.723420\pi$
$294$	−55.6985	−3.24840
$295$	0	0
$296$	37.4558	2.17708
$297$	− 19.3137i	− 1.12070i
$298$	28.1421i	1.63023i
$299$	1.41421	0.0817861
$300$	0	0
$301$	−53.4558	−3.08114
$302$	23.5563i	1.35552i
$303$	5.17157i	0.297099i
$304$	−1.75736	−0.100791
$305$	0	0
$306$	2.00000	0.114332
$307$	− 24.8284i	− 1.41703i	−0.705694	−	0.708517i	$-0.749365\pi$
$307$	− 24.8284i	− 1.41703i	0.705694	−	0.708517i	$-0.250635\pi$
$308$	63.1127i	3.59618i
$309$	20.6274	1.17345
$310$	0	0
$311$	8.48528	0.481156	0.240578	−	0.970630i	$-0.422663\pi$
$311$	8.48528	0.481156	0.240578	+	0.970630i	$0.422663\pi$
$312$	6.24264i	0.353420i
$313$	− 4.82843i	− 0.272919i	−0.990646	−	0.136459i	$-0.956428\pi$
$313$	− 4.82843i	− 0.272919i	0.990646	−	0.136459i	$-0.0435723\pi$
$314$	43.4558	2.45236
$315$	0	0
$316$	−32.4853	−1.82744
$317$	2.14214i	0.120314i	0.998189	+	0.0601572i	$0.0191602\pi$
$317$	2.14214i	0.120314i	−0.998189	+	0.0601572i	$0.980840\pi$
$318$	8.48528i	0.475831i
$319$	19.3137	1.08136
$320$	0	0
$321$	13.3137	0.743099
$322$	16.4853i	0.918689i
$323$	0.485281i	0.0270018i
$324$	19.1421	1.06345
$325$	0	0
$326$	−45.7990	−2.53657
$327$	− 2.82843i	− 0.156412i
$328$	14.0000i	0.773021i
$329$	23.3137	1.28533
$330$	0	0
$331$	−26.0416	−1.43138	−0.715689	−	0.698419i	$-0.753887\pi$
$331$	−26.0416	−1.43138	−0.715689	+	0.698419i	$0.753887\pi$
$332$	12.1421i	0.666386i
$333$	8.48528i	0.464991i
$334$	−7.65685	−0.418964
$335$	0	0
$336$	−20.4853	−1.11756
$337$	− 12.8284i	− 0.698809i	−0.936972	−	0.349404i	$-0.886384\pi$
$337$	− 12.8284i	− 0.698809i	0.936972	−	0.349404i	$-0.113616\pi$
$338$	− 2.41421i	− 0.131316i
$339$	−12.4853	−0.678107
$340$	0	0
$341$	6.00000	0.324918
$342$	− 1.41421i	− 0.0764719i
$343$	44.9706i	2.42818i
$344$	−48.8701	−2.63490
$345$	0	0
$346$	−40.6274	−2.18414
$347$	4.24264i	0.227757i	0.993495	+	0.113878i	$0.0363274\pi$
$347$	4.24264i	0.227757i	−0.993495	+	0.113878i	$0.963673\pi$
$348$	30.6274i	1.64180i
$349$	−18.4853	−0.989494	−0.494747	−	0.869037i	$-0.664739\pi$
$349$	−18.4853	−0.989494	−0.494747	+	0.869037i	$0.664739\pi$
$350$	0	0
$351$	−5.65685	−0.301941
$352$	− 5.41421i	− 0.288579i
$353$	14.8284i	0.789238i	0.918845	+	0.394619i	$0.129123\pi$
$353$	14.8284i	0.789238i	−0.918845	+	0.394619i	$0.870877\pi$
$354$	−6.00000	−0.318896
$355$	0	0
$356$	22.9706	1.21744
$357$	5.65685i	0.299392i
$358$	− 13.6569i	− 0.721787i
$359$	8.10051	0.427528	0.213764	−	0.976885i	$-0.431428\pi$
$359$	8.10051	0.427528	0.213764	+	0.976885i	$0.431428\pi$
$360$	0	0
$361$	−18.6569	−0.981940
$362$	0	0
$363$	− 0.928932i	− 0.0487563i
$364$	18.4853	0.968892
$365$	0	0
$366$	−27.3137	−1.42771
$367$	− 35.5563i	− 1.85603i	−0.372547	−	0.928013i	$-0.621516\pi$
$367$	− 35.5563i	− 1.85603i	0.372547	−	0.928013i	$-0.378484\pi$
$368$	4.24264i	0.221163i
$369$	−3.17157	−0.165105
$370$	0	0
$371$	12.0000	0.623009
$372$	9.51472i	0.493315i
$373$	− 2.68629i	− 0.139091i	−0.997579	−	0.0695455i	$-0.977845\pi$
$373$	− 2.68629i	− 0.139091i	0.997579	−	0.0695455i	$-0.0221549\pi$
$374$	6.82843	0.353090
$375$	0	0
$376$	21.3137	1.09917
$377$	− 5.65685i	− 0.291343i
$378$	− 65.9411i	− 3.39165i
$379$	−29.0711	−1.49328	−0.746640	−	0.665228i	$-0.768334\pi$
$379$	−29.0711	−1.49328	−0.746640	+	0.665228i	$0.768334\pi$
$380$	0	0
$381$	9.31371	0.477156
$382$	5.65685i	0.289430i
$383$	− 29.1127i	− 1.48759i	−0.668408	−	0.743795i	$-0.733024\pi$
$383$	− 29.1127i	− 1.48759i	0.668408	−	0.743795i	$-0.266976\pi$
$384$	29.0711	1.48353
$385$	0	0
$386$	−10.4853	−0.533687
$387$	− 11.0711i	− 0.562774i
$388$	− 29.3137i	− 1.48818i
$389$	−28.6274	−1.45147	−0.725734	−	0.687976i	$-0.758500\pi$
$389$	−28.6274	−1.45147	−0.725734	+	0.687976i	$0.758500\pi$
$390$	0	0
$391$	1.17157	0.0592490
$392$	72.0122i	3.63717i
$393$	24.0000i	1.21064i
$394$	26.4853	1.33431
$395$	0	0
$396$	−13.0711	−0.656846
$397$	− 11.7990i	− 0.592174i	−0.955161	−	0.296087i	$-0.904318\pi$
$397$	− 11.7990i	− 0.592174i	0.955161	−	0.296087i	$-0.0956819\pi$
$398$	− 9.65685i	− 0.484054i
$399$	4.00000	0.200250
$400$	0	0
$401$	−5.31371	−0.265354	−0.132677	−	0.991159i	$-0.542357\pi$
$401$	−5.31371	−0.265354	−0.132677	+	0.991159i	$0.542357\pi$
$402$	6.82843i	0.340571i
$403$	− 1.75736i	− 0.0875403i
$404$	14.0000	0.696526
$405$	0	0
$406$	65.9411	3.27260
$407$	28.9706i	1.43602i
$408$	5.17157i	0.256031i
$409$	−7.17157	−0.354611	−0.177306	−	0.984156i	$-0.556738\pi$
$409$	−7.17157	−0.354611	−0.177306	+	0.984156i	$0.556738\pi$
$410$	0	0
$411$	24.4853	1.20777
$412$	− 55.8406i	− 2.75107i
$413$	8.48528i	0.417533i
$414$	−3.41421	−0.167799
$415$	0	0
$416$	−1.58579	−0.0777496
$417$	6.34315i	0.310625i
$418$	− 4.82843i	− 0.236166i
$419$	−10.8284	−0.529003	−0.264502	−	0.964385i	$-0.585207\pi$
$419$	−10.8284	−0.529003	−0.264502	+	0.964385i	$0.585207\pi$
$420$	0	0
$421$	−34.9706	−1.70436	−0.852180	−	0.523248i	$-0.824720\pi$
$421$	−34.9706	−1.70436	−0.852180	+	0.523248i	$0.824720\pi$
$422$	8.00000i	0.389434i
$423$	4.82843i	0.234766i
$424$	10.9706	0.532778
$425$	0	0
$426$	40.6274	1.96840
$427$	38.6274i	1.86931i
$428$	− 36.0416i	− 1.74214i
$429$	−4.82843	−0.233119
$430$	0	0
$431$	40.3848	1.94527	0.972633	−	0.232346i	$-0.0746403\pi$
$431$	40.3848	1.94527	0.972633	+	0.232346i	$0.0746403\pi$
$432$	− 16.9706i	− 0.816497i
$433$	7.65685i	0.367965i	0.982930	+	0.183982i	$0.0588990\pi$
$433$	7.65685i	0.367965i	−0.982930	+	0.183982i	$0.941101\pi$
$434$	20.4853	0.983325
$435$	0	0
$436$	−7.65685	−0.366697
$437$	− 0.828427i	− 0.0396290i
$438$	28.9706i	1.38427i
$439$	−0.970563	−0.0463224	−0.0231612	−	0.999732i	$-0.507373\pi$
$439$	−0.970563	−0.0463224	−0.0231612	+	0.999732i	$0.507373\pi$
$440$	0	0
$441$	−16.3137	−0.776843
$442$	− 2.00000i	− 0.0951303i
$443$	− 9.41421i	− 0.447283i	−0.974671	−	0.223641i	$-0.928206\pi$
$443$	− 9.41421i	− 0.447283i	0.974671	−	0.223641i	$-0.0717944\pi$
$444$	−45.9411	−2.18027
$445$	0	0
$446$	−22.9706	−1.08769
$447$	− 16.4853i	− 0.779727i
$448$	− 47.4558i	− 2.24208i
$449$	33.1127	1.56268	0.781342	−	0.624103i	$-0.214535\pi$
$449$	33.1127	1.56268	0.781342	+	0.624103i	$0.214535\pi$
$450$	0	0
$451$	−10.8284	−0.509891
$452$	33.7990i	1.58977i
$453$	− 13.7990i	− 0.648333i
$454$	−39.4558	−1.85175
$455$	0	0
$456$	3.65685	0.171248
$457$	18.0000i	0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$	18.0000i	0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$	11.6569i	0.544689i
$459$	−4.68629	−0.218737
$460$	0	0
$461$	9.51472	0.443145	0.221572	−	0.975144i	$-0.428881\pi$
$461$	9.51472	0.443145	0.221572	+	0.975144i	$0.428881\pi$
$462$	− 56.2843i	− 2.61858i
$463$	− 4.34315i	− 0.201843i	−0.994894	−	0.100922i	$-0.967821\pi$
$463$	− 4.34315i	− 0.201843i	0.994894	−	0.100922i	$-0.0321791\pi$
$464$	16.9706	0.787839
$465$	0	0
$466$	49.7990	2.30689
$467$	13.4142i	0.620736i	0.950617	+	0.310368i	$0.100452\pi$
$467$	13.4142i	0.620736i	−0.950617	+	0.310368i	$0.899548\pi$
$468$	3.82843i	0.176969i
$469$	9.65685	0.445912
$470$	0	0
$471$	−25.4558	−1.17294
$472$	7.75736i	0.357061i
$473$	− 37.7990i	− 1.73800i
$474$	28.9706	1.33066
$475$	0	0
$476$	15.3137	0.701903
$477$	2.48528i	0.113793i
$478$	8.24264i	0.377010i
$479$	30.7279	1.40399	0.701997	−	0.712180i	$-0.252292\pi$
$479$	30.7279	1.40399	0.701997	+	0.712180i	$0.252292\pi$
$480$	0	0
$481$	8.48528	0.386896
$482$	− 34.9706i	− 1.59287i
$483$	− 9.65685i	− 0.439402i
$484$	−2.51472	−0.114305
$485$	0	0
$486$	23.8995	1.08410
$487$	10.9706i	0.497124i	0.968616	+	0.248562i	$0.0799579\pi$
$487$	10.9706i	0.497124i	−0.968616	+	0.248562i	$0.920042\pi$
$488$	35.3137i	1.59858i
$489$	26.8284	1.21322
$490$	0	0
$491$	5.17157	0.233390	0.116695	−	0.993168i	$-0.462770\pi$
$491$	5.17157	0.233390	0.116695	+	0.993168i	$0.462770\pi$
$492$	− 17.1716i	− 0.774154i
$493$	− 4.68629i	− 0.211060i
$494$	−1.41421	−0.0636285
$495$	0	0
$496$	5.27208	0.236723
$497$	− 57.4558i	− 2.57725i
$498$	− 10.8284i	− 0.485233i
$499$	−41.5563	−1.86032	−0.930159	−	0.367157i	$-0.880331\pi$
$499$	−41.5563	−1.86032	−0.930159	+	0.367157i	$0.880331\pi$
$500$	0	0
$501$	4.48528	0.200388
$502$	47.7990i	2.13337i
$503$	37.8995i	1.68985i	0.534881	+	0.844927i	$0.320356\pi$
$503$	37.8995i	1.68985i	−0.534881	+	0.844927i	$0.679644\pi$
$504$	−21.3137	−0.949388
$505$	0	0
$506$	−11.6569	−0.518210
$507$	1.41421i	0.0628074i
$508$	− 25.2132i	− 1.11866i
$509$	−41.1127	−1.82229	−0.911144	−	0.412088i	$-0.864800\pi$
$509$	−41.1127	−1.82229	−0.911144	+	0.412088i	$0.864800\pi$
$510$	0	0
$511$	40.9706	1.81243
$512$	− 31.2426i	− 1.38074i
$513$	3.31371i	0.146304i
$514$	66.7696	2.94508
$515$	0	0
$516$	59.9411	2.63876
$517$	16.4853i	0.725022i
$518$	98.9117i	4.34593i
$519$	23.7990	1.04466
$520$	0	0
$521$	−17.6569	−0.773561	−0.386780	−	0.922172i	$-0.626413\pi$
$521$	−17.6569	−0.773561	−0.386780	+	0.922172i	$0.626413\pi$
$522$	13.6569i	0.597744i
$523$	− 19.7574i	− 0.863929i	−0.901891	−	0.431965i	$-0.857821\pi$
$523$	− 19.7574i	− 0.863929i	0.901891	−	0.431965i	$-0.142179\pi$
$524$	64.9706	2.83825
$525$	0	0
$526$	25.5563	1.11431
$527$	− 1.45584i	− 0.0634176i
$528$	− 14.4853i	− 0.630391i
$529$	21.0000	0.913043
$530$	0	0
$531$	−1.75736	−0.0762629
$532$	− 10.8284i	− 0.469472i
$533$	3.17157i	0.137376i
$534$	−20.4853	−0.886485
$535$	0	0
$536$	8.82843	0.381330
$537$	8.00000i	0.345225i
$538$	61.1127i	2.63476i
$539$	−55.6985	−2.39910
$540$	0	0
$541$	−7.17157	−0.308330	−0.154165	−	0.988045i	$-0.549269\pi$
$541$	−7.17157	−0.308330	−0.154165	+	0.988045i	$0.549269\pi$
$542$	64.5269i	2.77167i
$543$	0	0
$544$	−1.31371	−0.0563248
$545$	0	0
$546$	−16.4853	−0.705505
$547$	− 13.2132i	− 0.564956i	−0.959274	−	0.282478i	$-0.908844\pi$
$547$	− 13.2132i	− 0.564956i	0.959274	−	0.282478i	$-0.0911564\pi$
$548$	− 66.2843i	− 2.83152i
$549$	−8.00000	−0.341432
$550$	0	0
$551$	−3.31371	−0.141169
$552$	− 8.82843i	− 0.375763i
$553$	− 40.9706i	− 1.74225i
$554$	−30.9706	−1.31581
$555$	0	0
$556$	17.1716	0.728237
$557$	35.7990i	1.51685i	0.651759	+	0.758426i	$0.274031\pi$
$557$	35.7990i	1.51685i	−0.651759	+	0.758426i	$0.725969\pi$
$558$	4.24264i	0.179605i
$559$	−11.0711	−0.468256
$560$	0	0
$561$	−4.00000	−0.168880
$562$	52.6274i	2.21995i
$563$	− 7.75736i	− 0.326934i	−0.986549	−	0.163467i	$-0.947732\pi$
$563$	− 7.75736i	− 0.326934i	0.986549	−	0.163467i	$-0.0522677\pi$
$564$	−26.1421	−1.10078
$565$	0	0
$566$	40.3848	1.69750
$567$	24.1421i	1.01387i
$568$	− 52.5269i	− 2.20398i
$569$	10.3431	0.433607	0.216804	−	0.976215i	$-0.430437\pi$
$569$	10.3431	0.433607	0.216804	+	0.976215i	$0.430437\pi$
$570$	0	0
$571$	−11.5147	−0.481876	−0.240938	−	0.970541i	$-0.577455\pi$
$571$	−11.5147	−0.481876	−0.240938	+	0.970541i	$0.577455\pi$
$572$	13.0711i	0.546529i
$573$	− 3.31371i	− 0.138432i
$574$	−36.9706	−1.54312
$575$	0	0
$576$	9.82843	0.409518
$577$	34.8284i	1.44993i	0.688788	+	0.724963i	$0.258143\pi$
$577$	34.8284i	1.44993i	−0.688788	+	0.724963i	$0.741857\pi$
$578$	39.3848i	1.63819i
$579$	6.14214	0.255258
$580$	0	0
$581$	−15.3137	−0.635320
$582$	26.1421i	1.08363i
$583$	8.48528i	0.351424i
$584$	37.4558	1.54993
$585$	0	0
$586$	−63.1127	−2.60716
$587$	− 20.3431i	− 0.839651i	−0.907605	−	0.419826i	$-0.862091\pi$
$587$	− 20.3431i	− 0.839651i	0.907605	−	0.419826i	$-0.137909\pi$
$588$	− 88.3259i	− 3.64250i
$589$	−1.02944	−0.0424172
$590$	0	0
$591$	−15.5147	−0.638190
$592$	25.4558i	1.04623i
$593$	− 24.6274i	− 1.01133i	−0.862731	−	0.505663i	$-0.831248\pi$
$593$	− 24.6274i	− 1.01133i	0.862731	−	0.505663i	$-0.168752\pi$
$594$	46.6274	1.91315
$595$	0	0
$596$	−44.6274	−1.82801
$597$	5.65685i	0.231520i
$598$	3.41421i	0.139618i
$599$	−25.4558	−1.04010	−0.520049	−	0.854137i	$-0.674086\pi$
$599$	−25.4558	−1.04010	−0.520049	+	0.854137i	$0.674086\pi$
$600$	0	0
$601$	−44.6274	−1.82039	−0.910195	−	0.414180i	$-0.864069\pi$
$601$	−44.6274	−1.82039	−0.910195	+	0.414180i	$0.864069\pi$
$602$	− 129.054i	− 5.25984i
$603$	2.00000i	0.0814463i
$604$	−37.3553	−1.51997
$605$	0	0
$606$	−12.4853	−0.507180
$607$	− 31.7574i	− 1.28899i	−0.764608	−	0.644496i	$-0.777067\pi$
$607$	− 31.7574i	− 1.28899i	0.764608	−	0.644496i	$-0.222933\pi$
$608$	0.928932i	0.0376732i
$609$	−38.6274	−1.56526
$610$	0	0
$611$	4.82843	0.195337
$612$	3.17157i	0.128203i
$613$	− 14.6863i	− 0.593174i	−0.955006	−	0.296587i	$-0.904152\pi$
$613$	− 14.6863i	− 0.593174i	0.955006	−	0.296587i	$-0.0958484\pi$
$614$	59.9411	2.41903
$615$	0	0
$616$	−72.7696	−2.93197
$617$	− 10.9706i	− 0.441658i	−0.975313	−	0.220829i	$-0.929124\pi$
$617$	− 10.9706i	− 0.441658i	0.975313	−	0.220829i	$-0.0708763\pi$
$618$	49.7990i	2.00321i
$619$	−1.75736	−0.0706342	−0.0353171	−	0.999376i	$-0.511244\pi$
$619$	−1.75736	−0.0706342	−0.0353171	+	0.999376i	$0.511244\pi$
$620$	0	0
$621$	8.00000	0.321029
$622$	20.4853i	0.821385i
$623$	28.9706i	1.16068i
$624$	−4.24264	−0.169842
$625$	0	0
$626$	11.6569	0.465902
$627$	2.82843i	0.112956i
$628$	68.9117i	2.74988i
$629$	7.02944	0.280282
$630$	0	0
$631$	−9.75736	−0.388434	−0.194217	−	0.980959i	$-0.562217\pi$
$631$	−9.75736	−0.388434	−0.194217	+	0.980959i	$0.562217\pi$
$632$	− 37.4558i	− 1.48991i
$633$	− 4.68629i	− 0.186263i
$634$	−5.17157	−0.205389
$635$	0	0
$636$	−13.4558	−0.533559
$637$	16.3137i	0.646373i
$638$	46.6274i	1.84600i
$639$	11.8995	0.470737
$640$	0	0
$641$	47.6569	1.88233	0.941166	−	0.337944i	$-0.109731\pi$
$641$	47.6569	1.88233	0.941166	+	0.337944i	$0.109731\pi$
$642$	32.1421i	1.26855i
$643$	9.51472i	0.375224i	0.982243	+	0.187612i	$0.0600747\pi$
$643$	9.51472i	0.375224i	−0.982243	+	0.187612i	$0.939925\pi$
$644$	−26.1421	−1.03014
$645$	0	0
$646$	−1.17157	−0.0460949
$647$	− 9.41421i	− 0.370111i	−0.982728	−	0.185055i	$-0.940754\pi$
$647$	− 9.41421i	− 0.370111i	0.982728	−	0.185055i	$-0.0592465\pi$
$648$	22.0711i	0.867033i
$649$	−6.00000	−0.235521
$650$	0	0
$651$	−12.0000	−0.470317
$652$	− 72.6274i	− 2.84431i
$653$	46.9706i	1.83810i	0.394141	+	0.919050i	$0.371042\pi$
$653$	46.9706i	1.83810i	−0.394141	+	0.919050i	$0.628958\pi$
$654$	6.82843	0.267013
$655$	0	0
$656$	−9.51472	−0.371487
$657$	8.48528i	0.331042i
$658$	56.2843i	2.19419i
$659$	−17.8579	−0.695644	−0.347822	−	0.937561i	$-0.613079\pi$
$659$	−17.8579	−0.695644	−0.347822	+	0.937561i	$0.613079\pi$
$660$	0	0
$661$	29.5980	1.15123	0.575614	−	0.817722i	$-0.304763\pi$
$661$	29.5980	1.15123	0.575614	+	0.817722i	$0.304763\pi$
$662$	− 62.8701i	− 2.44351i
$663$	1.17157i	0.0455001i
$664$	−14.0000	−0.543305
$665$	0	0
$666$	−20.4853	−0.793789
$667$	8.00000i	0.309761i
$668$	− 12.1421i	− 0.469793i
$669$	13.4558	0.520233
$670$	0	0
$671$	−27.3137	−1.05443
$672$	10.8284i	0.417716i
$673$	6.48528i	0.249989i	0.992157	+	0.124995i	$0.0398914\pi$
$673$	6.48528i	0.249989i	−0.992157	+	0.124995i	$0.960109\pi$
$674$	30.9706	1.19294
$675$	0	0
$676$	3.82843	0.147247
$677$	20.1421i	0.774125i	0.922053	+	0.387063i	$0.126510\pi$
$677$	20.1421i	0.774125i	−0.922053	+	0.387063i	$0.873490\pi$
$678$	− 30.1421i	− 1.15760i
$679$	36.9706	1.41880
$680$	0	0
$681$	23.1127	0.885681
$682$	14.4853i	0.554670i
$683$	10.6863i	0.408900i	0.978877	+	0.204450i	$0.0655405\pi$
$683$	10.6863i	0.408900i	−0.978877	+	0.204450i	$0.934459\pi$
$684$	2.24264	0.0857495
$685$	0	0
$686$	−108.569	−4.14517
$687$	− 6.82843i	− 0.260521i
$688$	− 33.2132i	− 1.26624i
$689$	2.48528	0.0946817
$690$	0	0
$691$	6.92893	0.263589	0.131795	−	0.991277i	$-0.457926\pi$
$691$	6.92893	0.263589	0.131795	+	0.991277i	$0.457926\pi$
$692$	− 64.4264i	− 2.44912i
$693$	− 16.4853i	− 0.626224i
$694$	−10.2426	−0.388805
$695$	0	0
$696$	−35.3137	−1.33856
$697$	2.62742i	0.0995205i
$698$	− 44.6274i	− 1.68917i
$699$	−29.1716	−1.10337
$700$	0	0
$701$	14.6863	0.554694	0.277347	−	0.960770i	$-0.410545\pi$
$701$	14.6863	0.554694	0.277347	+	0.960770i	$0.410545\pi$
$702$	− 13.6569i	− 0.515445i
$703$	− 4.97056i	− 0.187468i
$704$	33.5563	1.26470
$705$	0	0
$706$	−35.7990	−1.34731
$707$	17.6569i	0.664054i
$708$	− 9.51472i	− 0.357585i
$709$	−45.1127	−1.69424	−0.847121	−	0.531399i	$-0.821666\pi$
$709$	−45.1127	−1.69424	−0.847121	+	0.531399i	$0.821666\pi$
$710$	0	0
$711$	8.48528	0.318223
$712$	26.4853i	0.992578i
$713$	2.48528i	0.0930745i
$714$	−13.6569	−0.511095
$715$	0	0
$716$	21.6569	0.809355
$717$	− 4.82843i	− 0.180321i
$718$	19.5563i	0.729836i
$719$	−28.9706	−1.08042	−0.540210	−	0.841530i	$-0.681655\pi$
$719$	−28.9706	−1.08042	−0.540210	+	0.841530i	$0.681655\pi$
$720$	0	0
$721$	70.4264	2.62282
$722$	− 45.0416i	− 1.67628i
$723$	20.4853i	0.761856i
$724$	0	0
$725$	0	0
$726$	2.24264	0.0832322
$727$	51.3553i	1.90466i	0.305063	+	0.952332i	$0.401322\pi$
$727$	51.3553i	1.90466i	−0.305063	+	0.952332i	$0.598678\pi$
$728$	21.3137i	0.789939i
$729$	−29.0000	−1.07407
$730$	0	0
$731$	−9.17157	−0.339223
$732$	− 43.3137i	− 1.60092i
$733$	21.3137i	0.787240i	0.919273	+	0.393620i	$0.128777\pi$
$733$	21.3137i	0.787240i	−0.919273	+	0.393620i	$0.871223\pi$
$734$	85.8406	3.16844
$735$	0	0
$736$	2.24264	0.0826648
$737$	6.82843i	0.251528i
$738$	− 7.65685i	− 0.281853i
$739$	−5.27208	−0.193937	−0.0969683	−	0.995287i	$-0.530915\pi$
$739$	−5.27208	−0.193937	−0.0969683	+	0.995287i	$0.530915\pi$
$740$	0	0
$741$	0.828427	0.0304330
$742$	28.9706i	1.06354i
$743$	− 21.5147i	− 0.789298i	−0.918832	−	0.394649i	$-0.870866\pi$
$743$	− 21.5147i	− 0.789298i	0.918832	−	0.394649i	$-0.129134\pi$
$744$	−10.9706	−0.402200
$745$	0	0
$746$	6.48528	0.237443
$747$	− 3.17157i	− 0.116042i
$748$	10.8284i	0.395927i
$749$	45.4558	1.66092
$750$	0	0
$751$	−27.5147	−1.00403	−0.502013	−	0.864860i	$-0.667407\pi$
$751$	−27.5147	−1.00403	−0.502013	+	0.864860i	$0.667407\pi$
$752$	14.4853i	0.528224i
$753$	− 28.0000i	− 1.02038i
$754$	13.6569	0.497353
$755$	0	0
$756$	104.569	3.80312
$757$	24.1421i	0.877461i	0.898619	+	0.438730i	$0.144572\pi$
$757$	24.1421i	0.877461i	−0.898619	+	0.438730i	$0.855428\pi$
$758$	− 70.1838i	− 2.54919i
$759$	6.82843	0.247856
$760$	0	0
$761$	8.62742	0.312744	0.156372	−	0.987698i	$-0.450020\pi$
$761$	8.62742	0.312744	0.156372	+	0.987698i	$0.450020\pi$
$762$	22.4853i	0.814556i
$763$	− 9.65685i	− 0.349602i
$764$	−8.97056	−0.324544
$765$	0	0
$766$	70.2843	2.53947
$767$	1.75736i	0.0634546i
$768$	42.3848i	1.52943i
$769$	22.9706	0.828340	0.414170	−	0.910200i	$-0.364072\pi$
$769$	22.9706	0.828340	0.414170	+	0.910200i	$0.364072\pi$
$770$	0	0
$771$	−39.1127	−1.40861
$772$	− 16.6274i	− 0.598434i
$773$	− 22.1421i	− 0.796397i	−0.917299	−	0.398199i	$-0.869635\pi$
$773$	− 22.1421i	− 0.796397i	0.917299	−	0.398199i	$-0.130365\pi$
$774$	26.7279	0.960715
$775$	0	0
$776$	33.7990	1.21331
$777$	− 57.9411i	− 2.07863i
$778$	− 69.1127i	− 2.47781i
$779$	1.85786	0.0665649
$780$	0	0
$781$	40.6274	1.45376
$782$	2.82843i	0.101144i
$783$	− 32.0000i	− 1.14359i
$784$	−48.9411	−1.74790
$785$	0	0
$786$	−57.9411	−2.06669
$787$	− 22.4853i	− 0.801514i	−0.916184	−	0.400757i	$-0.868747\pi$
$787$	− 22.4853i	− 0.801514i	0.916184	−	0.400757i	$-0.131253\pi$
$788$	42.0000i	1.49619i
$789$	−14.9706	−0.532966
$790$	0	0
$791$	−42.6274	−1.51566
$792$	− 15.0711i	− 0.535527i
$793$	8.00000i	0.284088i
$794$	28.4853	1.01090
$795$	0	0
$796$	15.3137	0.542780
$797$	22.9706i	0.813659i	0.913504	+	0.406830i	$0.133366\pi$
$797$	22.9706i	0.813659i	−0.913504	+	0.406830i	$0.866634\pi$
$798$	9.65685i	0.341849i
$799$	4.00000	0.141510
$800$	0	0
$801$	−6.00000	−0.212000
$802$	− 12.8284i	− 0.452988i
$803$	28.9706i	1.02235i
$804$	−10.8284	−0.381889
$805$	0	0
$806$	4.24264	0.149441
$807$	− 35.7990i	− 1.26018i
$808$	16.1421i	0.567878i
$809$	−45.2548	−1.59108	−0.795538	−	0.605904i	$-0.792811\pi$
$809$	−45.2548	−1.59108	−0.795538	+	0.605904i	$0.792811\pi$
$810$	0	0
$811$	−28.3848	−0.996724	−0.498362	−	0.866969i	$-0.666065\pi$
$811$	−28.3848	−0.996724	−0.498362	+	0.866969i	$0.666065\pi$
$812$	104.569i	3.66964i
$813$	− 37.7990i	− 1.32567i
$814$	−69.9411	−2.45144
$815$	0	0
$816$	−3.51472	−0.123040
$817$	6.48528i	0.226891i
$818$	− 17.3137i	− 0.605360i
$819$	−4.82843	−0.168719
$820$	0	0
$821$	−51.2548	−1.78881	−0.894403	−	0.447262i	$-0.852399\pi$
$821$	−51.2548	−1.78881	−0.894403	+	0.447262i	$0.852399\pi$
$822$	59.1127i	2.06179i
$823$	− 2.38478i	− 0.0831281i	−0.999136	−	0.0415640i	$-0.986766\pi$
$823$	− 2.38478i	− 0.0831281i	0.999136	−	0.0415640i	$-0.0132341\pi$
$824$	64.3848	2.24295
$825$	0	0
$826$	−20.4853	−0.712774
$827$	− 56.1421i	− 1.95225i	−0.217202	−	0.976127i	$-0.569693\pi$
$827$	− 56.1421i	− 1.95225i	0.217202	−	0.976127i	$-0.430307\pi$
$828$	− 5.41421i	− 0.188157i
$829$	−40.9706	−1.42297	−0.711483	−	0.702703i	$-0.751976\pi$
$829$	−40.9706	−1.42297	−0.711483	+	0.702703i	$0.751976\pi$
$830$	0	0
$831$	18.1421	0.629344
$832$	− 9.82843i	− 0.340739i
$833$	13.5147i	0.468257i
$834$	−15.3137	−0.530270
$835$	0	0
$836$	7.65685	0.264818
$837$	− 9.94113i	− 0.343616i
$838$	− 26.1421i	− 0.903065i
$839$	−6.72792	−0.232274	−0.116137	−	0.993233i	$-0.537051\pi$
$839$	−6.72792	−0.232274	−0.116137	+	0.993233i	$0.537051\pi$
$840$	0	0
$841$	3.00000	0.103448
$842$	− 84.4264i	− 2.90953i
$843$	− 30.8284i	− 1.06179i
$844$	−12.6863	−0.436680
$845$	0	0
$846$	−11.6569	−0.400771
$847$	− 3.17157i	− 0.108977i
$848$	7.45584i	0.256035i
$849$	−23.6569	−0.811901
$850$	0	0
$851$	−12.0000	−0.411355
$852$	64.4264i	2.20721i
$853$	− 13.4558i	− 0.460719i	−0.973106	−	0.230360i	$-0.926010\pi$
$853$	− 13.4558i	− 0.460719i	0.973106	−	0.230360i	$-0.0739903\pi$
$854$	−93.2548	−3.19111
$855$	0	0
$856$	41.5563	1.42037
$857$	− 11.6569i	− 0.398191i	−0.979980	−	0.199095i	$-0.936200\pi$
$857$	− 11.6569i	− 0.398191i	0.979980	−	0.199095i	$-0.0638003\pi$
$858$	− 11.6569i	− 0.397958i
$859$	27.7990	0.948489	0.474245	−	0.880393i	$-0.342721\pi$
$859$	27.7990	0.948489	0.474245	+	0.880393i	$0.342721\pi$
$860$	0	0
$861$	21.6569	0.738064
$862$	97.4975i	3.32078i
$863$	− 31.4558i	− 1.07077i	−0.844608	−	0.535385i	$-0.820167\pi$
$863$	− 31.4558i	− 1.07077i	0.844608	−	0.535385i	$-0.179833\pi$
$864$	−8.97056	−0.305185
$865$	0	0
$866$	−18.4853	−0.628155
$867$	− 23.0711i	− 0.783535i
$868$	32.4853i	1.10262i
$869$	28.9706	0.982759
$870$	0	0
$871$	2.00000	0.0677674
$872$	− 8.82843i	− 0.298968i
$873$	7.65685i	0.259145i
$874$	2.00000	0.0676510
$875$	0	0
$876$	−45.9411	−1.55221
$877$	− 25.3137i	− 0.854783i	−0.904067	−	0.427392i	$-0.859433\pi$
$877$	− 25.3137i	− 0.854783i	0.904067	−	0.427392i	$-0.140567\pi$
$878$	− 2.34315i	− 0.0790773i
$879$	36.9706	1.24699
$880$	0	0
$881$	19.0294	0.641118	0.320559	−	0.947229i	$-0.396129\pi$
$881$	19.0294	0.641118	0.320559	+	0.947229i	$0.396129\pi$
$882$	− 39.3848i	− 1.32615i
$883$	− 23.7574i	− 0.799499i	−0.916624	−	0.399749i	$-0.869097\pi$
$883$	− 23.7574i	− 0.799499i	0.916624	−	0.399749i	$-0.130903\pi$
$884$	3.17157	0.106672
$885$	0	0
$886$	22.7279	0.763559
$887$	22.3848i	0.751607i	0.926699	+	0.375804i	$0.122633\pi$
$887$	22.3848i	0.751607i	−0.926699	+	0.375804i	$0.877367\pi$
$888$	− 52.9706i	− 1.77758i
$889$	31.7990	1.06650
$890$	0	0
$891$	−17.0711	−0.571902
$892$	− 36.4264i	− 1.21965i
$893$	− 2.82843i	− 0.0946497i
$894$	39.7990	1.33108
$895$	0	0
$896$	99.2548	3.31587
$897$	− 2.00000i	− 0.0667781i
$898$	79.9411i	2.66767i
$899$	9.94113	0.331555
$900$	0	0
$901$	2.05887	0.0685911
$902$	− 26.1421i	− 0.870438i
$903$	75.5980i	2.51574i
$904$	−38.9706	−1.29614
$905$	0	0
$906$	33.3137	1.10677
$907$	− 9.21320i	− 0.305919i	−0.988232	−	0.152960i	$-0.951120\pi$
$907$	− 9.21320i	− 0.305919i	0.988232	−	0.152960i	$-0.0488805\pi$
$908$	− 62.5685i	− 2.07641i
$909$	−3.65685	−0.121290
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	2.48528i	0.0822959i
$913$	− 10.8284i	− 0.358369i
$914$	−43.4558	−1.43739
$915$	0	0
$916$	−18.4853	−0.610771
$917$	81.9411i	2.70593i
$918$	− 11.3137i	− 0.373408i
$919$	−0.485281	−0.0160080	−0.00800398	−	0.999968i	$-0.502548\pi$
$919$	−0.485281	−0.0160080	−0.00800398	+	0.999968i	$0.502548\pi$
$920$	0	0
$921$	−35.1127	−1.15700
$922$	22.9706i	0.756495i
$923$	− 11.8995i	− 0.391677i
$924$	89.2548	2.93627
$925$	0	0
$926$	10.4853	0.344568
$927$	14.5858i	0.479060i
$928$	− 8.97056i	− 0.294473i
$929$	−16.8284	−0.552123	−0.276061	−	0.961140i	$-0.589029\pi$
$929$	−16.8284	−0.552123	−0.276061	+	0.961140i	$0.589029\pi$
$930$	0	0
$931$	9.55635	0.313197
$932$	78.9706i	2.58677i
$933$	− 12.0000i	− 0.392862i
$934$	−32.3848	−1.05966
$935$	0	0
$936$	−4.41421	−0.144283
$937$	22.9706i	0.750416i	0.926941	+	0.375208i	$0.122429\pi$
$937$	22.9706i	0.750416i	−0.926941	+	0.375208i	$0.877571\pi$
$938$	23.3137i	0.761220i
$939$	−6.82843	−0.222837
$940$	0	0
$941$	18.7696	0.611870	0.305935	−	0.952052i	$-0.401031\pi$
$941$	18.7696	0.611870	0.305935	+	0.952052i	$0.401031\pi$
$942$	− 61.4558i	− 2.00234i
$943$	− 4.48528i	− 0.146061i
$944$	−5.27208	−0.171592
$945$	0	0
$946$	91.2548	2.96695
$947$	− 17.1127i	− 0.556088i	−0.960568	−	0.278044i	$-0.910314\pi$
$947$	− 17.1127i	− 0.556088i	0.960568	−	0.278044i	$-0.0896861\pi$
$948$	45.9411i	1.49210i
$949$	8.48528	0.275444
$950$	0	0
$951$	3.02944	0.0982362
$952$	17.6569i	0.572262i
$953$	35.2548i	1.14202i	0.820944	+	0.571008i	$0.193447\pi$
$953$	35.2548i	1.14202i	−0.820944	+	0.571008i	$0.806553\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−13.0711	−0.422749
$957$	− 27.3137i	− 0.882927i
$958$	74.1838i	2.39677i
$959$	83.5980	2.69952
$960$	0	0
$961$	−27.9117	−0.900377
$962$	20.4853i	0.660472i
$963$	9.41421i	0.303369i
$964$	55.4558	1.78611
$965$	0	0
$966$	23.3137	0.750106
$967$	− 47.9411i	− 1.54168i	−0.637027	−	0.770841i	$-0.719836\pi$
$967$	− 47.9411i	− 1.54168i	0.637027	−	0.770841i	$-0.280164\pi$
$968$	− 2.89949i	− 0.0931933i
$969$	0.686292	0.0220469
$970$	0	0
$971$	44.2843	1.42115	0.710575	−	0.703622i	$-0.248435\pi$
$971$	44.2843	1.42115	0.710575	+	0.703622i	$0.248435\pi$
$972$	37.8995i	1.21563i
$973$	21.6569i	0.694287i
$974$	−26.4853	−0.848643
$975$	0	0
$976$	−24.0000	−0.768221
$977$	39.5147i	1.26419i	0.774892	+	0.632094i	$0.217804\pi$
$977$	39.5147i	1.26419i	−0.774892	+	0.632094i	$0.782196\pi$
$978$	64.7696i	2.07110i
$979$	−20.4853	−0.654712
$980$	0	0
$981$	2.00000	0.0638551
$982$	12.4853i	0.398421i
$983$	− 1.02944i	− 0.0328339i	−0.999865	−	0.0164170i	$-0.994774\pi$
$983$	− 1.02944i	− 0.0328339i	0.999865	−	0.0164170i	$-0.00522592\pi$
$984$	19.7990	0.631169
$985$	0	0
$986$	11.3137	0.360302
$987$	− 32.9706i	− 1.04946i
$988$	− 2.24264i	− 0.0713479i
$989$	15.6569	0.497859
$990$	0	0
$991$	48.9706	1.55560	0.777801	−	0.628511i	$-0.216335\pi$
$991$	48.9706	1.55560	0.777801	+	0.628511i	$0.216335\pi$
$992$	− 2.78680i	− 0.0884809i
$993$	36.8284i	1.16871i
$994$	138.711	4.39964
$995$	0	0
$996$	17.1716	0.544102
$997$	− 28.8284i	− 0.913005i	−0.889722	−	0.456503i	$-0.849102\pi$
$997$	− 28.8284i	− 0.913005i	0.889722	−	0.456503i	$-0.150898\pi$
$998$	− 100.326i	− 3.17576i
$999$	48.0000	1.51865

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	325.2.b.f.274.4		4
3.2	odd	2		2925.2.c.r.2224.1		4
5.2	odd	4		65.2.a.b.1.1	✓	2
5.3	odd	4		325.2.a.i.1.2		2
5.4	even	2	inner	325.2.b.f.274.1		4
15.2	even	4		585.2.a.m.1.2		2
15.8	even	4		2925.2.a.u.1.1		2
15.14	odd	2		2925.2.c.r.2224.4		4
20.3	even	4		5200.2.a.bu.1.1		2
20.7	even	4		1040.2.a.j.1.2		2
35.27	even	4		3185.2.a.j.1.1		2
40.27	even	4		4160.2.a.z.1.1		2
40.37	odd	4		4160.2.a.bf.1.2		2
55.32	even	4		7865.2.a.j.1.2		2
60.47	odd	4		9360.2.a.cd.1.1		2
65.2	even	12		845.2.m.f.316.1		8
65.7	even	12		845.2.m.f.361.1		8
65.12	odd	4		845.2.a.g.1.2		2
65.17	odd	12		845.2.e.c.146.1		4
65.22	odd	12		845.2.e.h.146.2		4
65.32	even	12		845.2.m.f.361.4		8
65.37	even	12		845.2.m.f.316.4		8
65.38	odd	4		4225.2.a.r.1.1		2
65.42	odd	12		845.2.e.h.191.2		4
65.47	even	4		845.2.c.b.506.1		4
65.57	even	4		845.2.c.b.506.4		4
65.62	odd	12		845.2.e.c.191.1		4
195.77	even	4		7605.2.a.x.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.2.a.b.1.1	✓	2	5.2	odd	4
325.2.a.i.1.2		2	5.3	odd	4
325.2.b.f.274.1		4	5.4	even	2	inner
325.2.b.f.274.4		4	1.1	even	1	trivial
585.2.a.m.1.2		2	15.2	even	4
845.2.a.g.1.2		2	65.12	odd	4
845.2.c.b.506.1		4	65.47	even	4
845.2.c.b.506.4		4	65.57	even	4
845.2.e.c.146.1		4	65.17	odd	12
845.2.e.c.191.1		4	65.62	odd	12
845.2.e.h.146.2		4	65.22	odd	12
845.2.e.h.191.2		4	65.42	odd	12
845.2.m.f.316.1		8	65.2	even	12
845.2.m.f.316.4		8	65.37	even	12
845.2.m.f.361.1		8	65.7	even	12
845.2.m.f.361.4		8	65.32	even	12
1040.2.a.j.1.2		2	20.7	even	4
2925.2.a.u.1.1		2	15.8	even	4
2925.2.c.r.2224.1		4	3.2	odd	2
2925.2.c.r.2224.4		4	15.14	odd	2
3185.2.a.j.1.1		2	35.27	even	4
4160.2.a.z.1.1		2	40.27	even	4
4160.2.a.bf.1.2		2	40.37	odd	4
4225.2.a.r.1.1		2	65.38	odd	4
5200.2.a.bu.1.1		2	20.3	even	4
7605.2.a.x.1.1		2	195.77	even	4
7865.2.a.j.1.2		2	55.32	even	4
9360.2.a.cd.1.1		2	60.47	odd	4

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:	\( N \)	\(=\)	\( 325 = 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	325.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.41421i q^{2} -1.41421i q^{3} -3.82843 q^{4} +3.41421 q^{6} -4.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.41421i q^{2} -1.41421i q^{3} -3.82843 q^{4} +3.41421 q^{6} -4.82843i q^{7} -4.41421i q^{8} +1.00000 q^{9} +3.41421 q^{11} +5.41421i q^{12} -1.00000i q^{13} +11.6569 q^{14} +3.00000 q^{16} -0.828427i q^{17} +2.41421i q^{18} -0.585786 q^{19} -6.82843 q^{21} +8.24264i q^{22} +1.41421i q^{23} -6.24264 q^{24} +2.41421 q^{26} -5.65685i q^{27} +18.4853i q^{28} +5.65685 q^{29} +1.75736 q^{31} -1.58579i q^{32} -4.82843i q^{33} +2.00000 q^{34} -3.82843 q^{36} +8.48528i q^{37} -1.41421i q^{38} -1.41421 q^{39} -3.17157 q^{41} -16.4853i q^{42} -11.0711i q^{43} -13.0711 q^{44} -3.41421 q^{46} +4.82843i q^{47} -4.24264i q^{48} -16.3137 q^{49} -1.17157 q^{51} +3.82843i q^{52} +2.48528i q^{53} +13.6569 q^{54} -21.3137 q^{56} +0.828427i q^{57} +13.6569i q^{58} -1.75736 q^{59} -8.00000 q^{61} +4.24264i q^{62} -4.82843i q^{63} +9.82843 q^{64} +11.6569 q^{66} +2.00000i q^{67} +3.17157i q^{68} +2.00000 q^{69} +11.8995 q^{71} -4.41421i q^{72} +8.48528i q^{73} -20.4853 q^{74} +2.24264 q^{76} -16.4853i q^{77} -3.41421i q^{78} +8.48528 q^{79} -5.00000 q^{81} -7.65685i q^{82} -3.17157i q^{83} +26.1421 q^{84} +26.7279 q^{86} -8.00000i q^{87} -15.0711i q^{88} -6.00000 q^{89} -4.82843 q^{91} -5.41421i q^{92} -2.48528i q^{93} -11.6569 q^{94} -2.24264 q^{96} +7.65685i q^{97} -39.3848i q^{98} +3.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9	\( 4 q - 4 q^{4} + 8 q^{6} + 4 q^{9} + 8 q^{11} + 24 q^{14} + 12 q^{16} - 8 q^{19} - 16 q^{21} - 8 q^{24} + 4 q^{26} + 24 q^{31} + 8 q^{34} - 4 q^{36} - 24 q^{41} - 24 q^{44} - 8 q^{46} - 20 q^{49} - 16 q^{51} + 32 q^{54} - 40 q^{56} - 24 q^{59} - 32 q^{61} + 28 q^{64} + 24 q^{66} + 8 q^{69} + 8 q^{71} - 48 q^{74} - 8 q^{76} - 20 q^{81} + 48 q^{84} + 56 q^{86} - 24 q^{89} - 8 q^{91} - 24 q^{94} + 8 q^{96} + 8 q^{99}+O(q^{100}) \) 4 * q - 4 * q^4 + 8 * q^6 + 4 * q^9 + 8 * q^11 + 24 * q^14 + 12 * q^16 - 8 * q^19 - 16 * q^21 - 8 * q^24 + 4 * q^26 + 24 * q^31 + 8 * q^34 - 4 * q^36 - 24 * q^41 - 24 * q^44 - 8 * q^46 - 20 * q^49 - 16 * q^51 + 32 * q^54 - 40 * q^56 - 24 * q^59 - 32 * q^61 + 28 * q^64 + 24 * q^66 + 8 * q^69 + 8 * q^71 - 48 * q^74 - 8 * q^76 - 20 * q^81 + 48 * q^84 + 56 * q^86 - 24 * q^89 - 8 * q^91 - 24 * q^94 + 8 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

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Database

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