LMFDB - Embedded newform 1875.2.b.d.1249.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1875,2,Mod(1249,1875)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1875, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1875.1249");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1875 = 3 \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1875.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:	$14.9719503790$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.324000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 $ x^8 + 9x^6 + 26x^4 + 24*x^2 + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.1
Root			$0.209057i$ of defining polynomial
Character	$\chi$	$=$	1875.1249
Dual form			1875.2.b.d.1249.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.95630i q^{2} +1.00000i q^{3} -1.82709 q^{4} +1.95630 q^{6} +4.57433i q^{7} -0.338261i q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.95630i q^{2} +1.00000i q^{3} -1.82709 q^{4} +1.95630 q^{6} +4.57433i q^{7} -0.338261i q^{8} -1.00000 q^{9} -4.16535 q^{11} -1.82709i q^{12} -3.75638i q^{13} +8.94874 q^{14} -4.31592 q^{16} -3.26755i q^{17} +1.95630i q^{18} +0.243625 q^{19} -4.57433 q^{21} +8.14866i q^{22} +0.654182i q^{23} +0.338261 q^{24} -7.34858 q^{26} -1.00000i q^{27} -8.35772i q^{28} +8.54732 q^{29} -3.90694 q^{31} +7.76669i q^{32} -4.16535i q^{33} -6.39228 q^{34} +1.82709 q^{36} -10.6151i q^{37} -0.476602i q^{38} +3.75638 q^{39} +0.769579 q^{41} +8.94874i q^{42} +3.90345i q^{43} +7.61048 q^{44} +1.27977 q^{46} -7.67555i q^{47} -4.31592i q^{48} -13.9245 q^{49} +3.26755 q^{51} +6.86324i q^{52} -12.1197i q^{53} -1.95630 q^{54} +1.54732 q^{56} +0.243625i q^{57} -16.7211i q^{58} -4.90248 q^{59} -14.7035 q^{61} +7.64313i q^{62} -4.57433i q^{63} +6.56210 q^{64} -8.14866 q^{66} -0.316897i q^{67} +5.97010i q^{68} -0.654182 q^{69} +0.446705 q^{71} +0.338261i q^{72} -11.1172i q^{73} -20.7664 q^{74} -0.445125 q^{76} -19.0537i q^{77} -7.34858i q^{78} -9.80731 q^{79} +1.00000 q^{81} -1.50552i q^{82} -8.86482i q^{83} +8.35772 q^{84} +7.63631 q^{86} +8.54732i q^{87} +1.40898i q^{88} +7.81040 q^{89} +17.1829 q^{91} -1.19525i q^{92} -3.90694i q^{93} -15.0156 q^{94} -7.76669 q^{96} -8.15938i q^{97} +27.2404i q^{98} +4.16535 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	$ 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) $ 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times$.

$n$	$626$	$1252$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.95630i	− 1.38331i	−0.722228	−	0.691655i	$-0.756882\pi$
$2$	− 1.95630i	− 1.38331i	0.722228	−	0.691655i	$-0.243118\pi$
$3$	1.00000i	0.577350i
$3$	1.00000i	0.577350i
$4$	−1.82709	−0.913545
$5$	0	0
$5$	0	0
$6$	1.95630	0.798654
$7$	4.57433i	1.72893i	0.502690	+	0.864467i	$0.332344\pi$
$7$	4.57433i	1.72893i	−0.502690	+	0.864467i	$0.667656\pi$
$8$	− 0.338261i	− 0.119593i
$9$	−1.00000	−0.333333
$10$	0	0
$11$	−4.16535	−1.25590	−0.627950	−	0.778253i	$-0.716106\pi$
$11$	−4.16535	−1.25590	−0.627950	+	0.778253i	$0.716106\pi$
$12$	− 1.82709i	− 0.527436i
$13$	− 3.75638i	− 1.04183i	−0.853608	−	0.520915i	$-0.825591\pi$
$13$	− 3.75638i	− 1.04183i	0.853608	−	0.520915i	$-0.174409\pi$
$14$	8.94874	2.39165
$15$	0	0
$16$	−4.31592	−1.07898
$17$	− 3.26755i	− 0.792496i	−0.918143	−	0.396248i	$-0.870312\pi$
$17$	− 3.26755i	− 0.792496i	0.918143	−	0.396248i	$-0.129688\pi$
$18$	1.95630i	0.461103i
$19$	0.243625	0.0558914	0.0279457	−	0.999609i	$-0.491103\pi$
$19$	0.243625	0.0558914	0.0279457	+	0.999609i	$0.491103\pi$
$20$	0	0
$21$	−4.57433	−0.998200
$22$	8.14866i	1.73730i
$23$	0.654182i	0.136406i	0.997671	+	0.0682032i	$0.0217266\pi$
$23$	0.654182i	0.136406i	−0.997671	+	0.0682032i	$0.978273\pi$
$24$	0.338261	0.0690473
$25$	0	0
$26$	−7.34858	−1.44117
$27$	− 1.00000i	− 0.192450i
$28$	− 8.35772i	− 1.57946i
$29$	8.54732	1.58720	0.793599	−	0.608442i	$-0.208205\pi$
$29$	8.54732	1.58720	0.793599	+	0.608442i	$0.208205\pi$
$30$	0	0
$31$	−3.90694	−0.701708	−0.350854	−	0.936430i	$-0.614109\pi$
$31$	−3.90694	−0.701708	−0.350854	+	0.936430i	$0.614109\pi$
$32$	7.76669i	1.37297i
$33$	− 4.16535i	− 0.725095i
$34$	−6.39228	−1.09627
$35$	0	0
$36$	1.82709	0.304515
$37$	− 10.6151i	− 1.74512i	−0.488507	−	0.872560i	$-0.662458\pi$
$37$	− 10.6151i	− 1.74512i	0.488507	−	0.872560i	$-0.337542\pi$
$38$	− 0.476602i	− 0.0773151i
$39$	3.75638	0.601501
$40$	0	0
$41$	0.769579	0.120188	0.0600940	−	0.998193i	$-0.480860\pi$
$41$	0.769579	0.120188	0.0600940	+	0.998193i	$0.480860\pi$
$42$	8.94874i	1.38082i
$43$	3.90345i	0.595271i	0.954680	+	0.297636i	$0.0961980\pi$
$43$	3.90345i	0.595271i	−0.954680	+	0.297636i	$0.903802\pi$
$44$	7.61048	1.14732
$45$	0	0
$46$	1.27977	0.188692
$47$	− 7.67555i	− 1.11959i	−0.828630	−	0.559797i	$-0.810879\pi$
$47$	− 7.67555i	− 1.11959i	0.828630	−	0.559797i	$-0.189121\pi$
$48$	− 4.31592i	− 0.622949i
$49$	−13.9245	−1.98921
$50$	0	0
$51$	3.26755	0.457548
$52$	6.86324i	0.951760i
$53$	− 12.1197i	− 1.66477i	−0.554195	−	0.832387i	$-0.686974\pi$
$53$	− 12.1197i	− 1.66477i	0.554195	−	0.832387i	$-0.313026\pi$
$54$	−1.95630	−0.266218
$55$	0	0
$56$	1.54732	0.206769
$57$	0.243625i	0.0322689i
$58$	− 16.7211i	− 2.19559i
$59$	−4.90248	−0.638248	−0.319124	−	0.947713i	$-0.603389\pi$
$59$	−4.90248	−0.638248	−0.319124	+	0.947713i	$0.603389\pi$
$60$	0	0
$61$	−14.7035	−1.88259	−0.941297	−	0.337579i	$-0.890392\pi$
$61$	−14.7035	−1.88259	−0.941297	+	0.337579i	$0.890392\pi$
$62$	7.64313i	0.970679i
$63$	− 4.57433i	− 0.576311i
$64$	6.56210	0.820263
$65$	0	0
$66$	−8.14866	−1.00303
$67$	− 0.316897i	− 0.0387151i	−0.999813	−	0.0193576i	$-0.993838\pi$
$67$	− 0.316897i	− 0.0387151i	0.999813	−	0.0193576i	$-0.00616209\pi$
$68$	5.97010i	0.723981i
$69$	−0.654182	−0.0787542
$70$	0	0
$71$	0.446705	0.0530141	0.0265070	−	0.999649i	$-0.491562\pi$
$71$	0.446705	0.0530141	0.0265070	+	0.999649i	$0.491562\pi$
$72$	0.338261i	0.0398645i
$73$	− 11.1172i	− 1.30117i	−0.759434	−	0.650584i	$-0.774524\pi$
$73$	− 11.1172i	− 1.30117i	0.759434	−	0.650584i	$-0.225476\pi$
$74$	−20.7664	−2.41404
$75$	0	0
$76$	−0.445125	−0.0510593
$77$	− 19.0537i	− 2.17137i
$78$	− 7.34858i	− 0.832063i
$79$	−9.80731	−1.10341	−0.551704	−	0.834040i	$-0.686022\pi$
$79$	−9.80731	−1.10341	−0.551704	+	0.834040i	$0.686022\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	− 1.50552i	− 0.166257i
$83$	− 8.86482i	− 0.973040i	−0.873669	−	0.486520i	$-0.838266\pi$
$83$	− 8.86482i	− 0.973040i	0.873669	−	0.486520i	$-0.161734\pi$
$84$	8.35772	0.911902
$85$	0	0
$86$	7.63631	0.823444
$87$	8.54732i	0.916369i
$88$	1.40898i	0.150197i
$89$	7.81040	0.827900	0.413950	−	0.910300i	$-0.364149\pi$
$89$	7.81040	0.827900	0.413950	+	0.910300i	$0.364149\pi$
$90$	0	0
$91$	17.1829	1.80126
$92$	− 1.19525i	− 0.124613i
$93$	− 3.90694i	− 0.405131i
$94$	−15.0156	−1.54874
$95$	0	0
$96$	−7.76669	−0.792685
$97$	− 8.15938i	− 0.828459i	−0.910172	−	0.414230i	$-0.864051\pi$
$97$	− 8.15938i	− 0.828459i	0.910172	−	0.414230i	$-0.135949\pi$
$98$	27.2404i	2.75170i
$99$	4.16535	0.418634
$100$	0	0
$101$	−9.54383	−0.949646	−0.474823	−	0.880081i	$-0.657488\pi$
$101$	−9.54383	−0.949646	−0.474823	+	0.880081i	$0.657488\pi$
$102$	− 6.39228i	− 0.632930i
$103$	9.86698i	0.972222i	0.873897	+	0.486111i	$0.161585\pi$
$103$	9.86698i	0.972222i	−0.873897	+	0.486111i	$0.838415\pi$
$104$	−1.27064	−0.124596
$105$	0	0
$106$	−23.7098	−2.30290
$107$	− 16.1273i	− 1.55908i	−0.626350	−	0.779542i	$-0.715452\pi$
$107$	− 16.1273i	− 1.55908i	0.626350	−	0.779542i	$-0.284548\pi$
$108$	1.82709i	0.175812i
$109$	2.44236	0.233936	0.116968	−	0.993136i	$-0.462682\pi$
$109$	2.44236	0.233936	0.116968	+	0.993136i	$0.462682\pi$
$110$	0	0
$111$	10.6151	1.00755
$112$	− 19.7424i	− 1.86549i
$113$	10.9293i	1.02814i	0.857748	+	0.514070i	$0.171863\pi$
$113$	10.9293i	1.02814i	−0.857748	+	0.514070i	$0.828137\pi$
$114$	0.476602	0.0446379
$115$	0	0
$116$	−15.6167	−1.44998
$117$	3.75638i	0.347277i
$118$	9.59069i	0.882895i
$119$	14.9468	1.37017
$120$	0	0
$121$	6.35016	0.577287
$122$	28.7645i	2.60421i
$123$	0.769579i	0.0693906i
$124$	7.13834	0.641042
$125$	0	0
$126$	−8.94874	−0.797217
$127$	12.8613i	1.14126i	0.821208	+	0.570629i	$0.193301\pi$
$127$	12.8613i	1.14126i	−0.821208	+	0.570629i	$0.806699\pi$
$128$	2.69598i	0.238293i
$129$	−3.90345	−0.343680
$130$	0	0
$131$	2.37441	0.207453	0.103727	−	0.994606i	$-0.466923\pi$
$131$	2.37441	0.207453	0.103727	+	0.994606i	$0.466923\pi$
$132$	7.61048i	0.662407i
$133$	1.11442i	0.0966325i
$134$	−0.619944	−0.0535550
$135$	0	0
$136$	−1.10528	−0.0947773
$137$	12.5026i	1.06817i	0.845430	+	0.534086i	$0.179344\pi$
$137$	12.5026i	1.06817i	−0.845430	+	0.534086i	$0.820656\pi$
$138$	1.27977i	0.108941i
$139$	0.356135	0.0302070	0.0151035	−	0.999886i	$-0.495192\pi$
$139$	0.356135	0.0302070	0.0151035	+	0.999886i	$0.495192\pi$
$140$	0	0
$141$	7.67555	0.646398
$142$	− 0.873886i	− 0.0733349i
$143$	15.6466i	1.30844i
$144$	4.31592	0.359660
$145$	0	0
$146$	−21.7485	−1.79992
$147$	− 13.9245i	− 1.14847i
$148$	19.3948i	1.59425i
$149$	15.0317	1.23144	0.615722	−	0.787964i	$-0.288865\pi$
$149$	15.0317	1.23144	0.615722	+	0.787964i	$0.288865\pi$
$150$	0	0
$151$	−5.65576	−0.460259	−0.230130	−	0.973160i	$-0.573915\pi$
$151$	−5.65576	−0.460259	−0.230130	+	0.973160i	$0.573915\pi$
$152$	− 0.0824089i	− 0.00668424i
$153$	3.26755i	0.264165i
$154$	−37.2746	−3.00368
$155$	0	0
$156$	−6.86324	−0.549499
$157$	6.69480i	0.534303i	0.963655	+	0.267151i	$0.0860824\pi$
$157$	6.69480i	0.534303i	−0.963655	+	0.267151i	$0.913918\pi$
$158$	19.1860i	1.52636i
$159$	12.1197	0.961158
$160$	0	0
$161$	−2.99244	−0.235838
$162$	− 1.95630i	− 0.153701i
$163$	13.8314i	1.08336i	0.840584	+	0.541681i	$0.182212\pi$
$163$	13.8314i	1.08336i	−0.840584	+	0.541681i	$0.817788\pi$
$164$	−1.40609	−0.109797
$165$	0	0
$166$	−17.3422	−1.34602
$167$	− 13.5017i	− 1.04479i	−0.852703	−	0.522397i	$-0.825038\pi$
$167$	− 13.5017i	− 1.04479i	0.852703	−	0.522397i	$-0.174962\pi$
$168$	1.54732i	0.119378i
$169$	−1.11035	−0.0854118
$170$	0	0
$171$	−0.243625	−0.0186305
$172$	− 7.13196i	− 0.543807i
$173$	− 13.7259i	− 1.04356i	−0.853080	−	0.521779i	$-0.825268\pi$
$173$	− 13.7259i	− 1.04356i	0.853080	−	0.521779i	$-0.174732\pi$
$174$	16.7211	1.26762
$175$	0	0
$176$	17.9773	1.35509
$177$	− 4.90248i	− 0.368493i
$178$	− 15.2794i	− 1.14524i
$179$	−4.46491	−0.333723	−0.166861	−	0.985980i	$-0.553363\pi$
$179$	−4.46491	−0.333723	−0.166861	+	0.985980i	$0.553363\pi$
$180$	0	0
$181$	−14.0032	−1.04085	−0.520423	−	0.853908i	$-0.674226\pi$
$181$	−14.0032	−1.04085	−0.520423	+	0.853908i	$0.674226\pi$
$182$	− 33.6148i	− 2.49170i
$183$	− 14.7035i	− 1.08692i
$184$	0.221284	0.0163133
$185$	0	0
$186$	−7.64313	−0.560422
$187$	13.6105i	0.995297i
$188$	14.0239i	1.02280i
$189$	4.57433	0.332733
$190$	0	0
$191$	−4.29614	−0.310858	−0.155429	−	0.987847i	$-0.549676\pi$
$191$	−4.29614	−0.310858	−0.155429	+	0.987847i	$0.549676\pi$
$192$	6.56210i	0.473579i
$193$	− 13.0436i	− 0.938897i	−0.882960	−	0.469449i	$-0.844453\pi$
$193$	− 13.0436i	− 0.938897i	0.882960	−	0.469449i	$-0.155547\pi$
$194$	−15.9621	−1.14602
$195$	0	0
$196$	25.4413	1.81724
$197$	− 9.17856i	− 0.653945i	−0.945034	−	0.326973i	$-0.893972\pi$
$197$	− 9.17856i	− 0.653945i	0.945034	−	0.326973i	$-0.106028\pi$
$198$	− 8.14866i	− 0.579100i
$199$	−0.677499	−0.0480266	−0.0240133	−	0.999712i	$-0.507644\pi$
$199$	−0.677499	−0.0480266	−0.0240133	+	0.999712i	$0.507644\pi$
$200$	0	0
$201$	0.316897	0.0223522
$202$	18.6705i	1.31365i
$203$	39.0982i	2.74416i
$204$	−5.97010	−0.417991
$205$	0	0
$206$	19.3027	1.34488
$207$	− 0.654182i	− 0.0454688i
$208$	16.2122i	1.12411i
$209$	−1.01478	−0.0701941
$210$	0	0
$211$	−16.1662	−1.11293	−0.556464	−	0.830872i	$-0.687842\pi$
$211$	−16.1662	−1.11293	−0.556464	+	0.830872i	$0.687842\pi$
$212$	22.1439i	1.52085i
$213$	0.446705i	0.0306077i
$214$	−31.5497	−2.15670
$215$	0	0
$216$	−0.338261	−0.0230158
$217$	− 17.8716i	− 1.21321i
$218$	− 4.77799i	− 0.323606i
$219$	11.1172	0.751229
$220$	0	0
$221$	−12.2741	−0.825647
$222$	− 20.7664i	− 1.39375i
$223$	9.96064i	0.667013i	0.942748	+	0.333507i	$0.108232\pi$
$223$	9.96064i	0.667013i	−0.942748	+	0.333507i	$0.891768\pi$
$224$	−35.5274	−2.37377
$225$	0	0
$226$	21.3809	1.42224
$227$	24.0646i	1.59722i	0.601846	+	0.798612i	$0.294432\pi$
$227$	24.0646i	1.59722i	−0.601846	+	0.798612i	$0.705568\pi$
$228$	− 0.445125i	− 0.0294791i
$229$	15.3847	1.01665	0.508326	−	0.861165i	$-0.330265\pi$
$229$	15.3847	1.01665	0.508326	+	0.861165i	$0.330265\pi$
$230$	0	0
$231$	19.0537	1.25364
$232$	− 2.89123i	− 0.189818i
$233$	− 6.58907i	− 0.431664i	−0.976430	−	0.215832i	$-0.930754\pi$
$233$	− 6.58907i	− 0.431664i	0.976430	−	0.215832i	$-0.0692464\pi$
$234$	7.34858	0.480392
$235$	0	0
$236$	8.95727	0.583069
$237$	− 9.80731i	− 0.637053i
$238$	− 29.2404i	− 1.89537i
$239$	−17.8874	−1.15704	−0.578520	−	0.815668i	$-0.696370\pi$
$239$	−17.8874	−1.15704	−0.578520	+	0.815668i	$0.696370\pi$
$240$	0	0
$241$	6.22404	0.400926	0.200463	−	0.979701i	$-0.435755\pi$
$241$	6.22404	0.400926	0.200463	+	0.979701i	$0.435755\pi$
$242$	− 12.4228i	− 0.798567i
$243$	1.00000i	0.0641500i
$244$	26.8647	1.71984
$245$	0	0
$246$	1.50552	0.0959887
$247$	− 0.915147i	− 0.0582294i
$248$	1.32157i	0.0839196i
$249$	8.86482	0.561785
$250$	0	0
$251$	−7.11580	−0.449145	−0.224573	−	0.974457i	$-0.572099\pi$
$251$	−7.11580	−0.449145	−0.224573	+	0.974457i	$0.572099\pi$
$252$	8.35772i	0.526487i
$253$	− 2.72490i	− 0.171313i
$254$	25.1606	1.57871
$255$	0	0
$256$	18.3983	1.14990
$257$	7.15589i	0.446372i	0.974776	+	0.223186i	$0.0716457\pi$
$257$	7.15589i	0.446372i	−0.974776	+	0.223186i	$0.928354\pi$
$258$	7.63631i	0.475416i
$259$	48.5572	3.01720
$260$	0	0
$261$	−8.54732	−0.529066
$262$	− 4.64505i	− 0.286972i
$263$	1.30967i	0.0807577i	0.999184	+	0.0403789i	$0.0128565\pi$
$263$	1.30967i	0.0807577i	−0.999184	+	0.0403789i	$0.987144\pi$
$264$	−1.40898	−0.0867165
$265$	0	0
$266$	2.18014	0.133673
$267$	7.81040i	0.477989i
$268$	0.579000i	0.0353680i
$269$	20.9745	1.27884	0.639418	−	0.768859i	$-0.279175\pi$
$269$	20.9745	1.27884	0.639418	+	0.768859i	$0.279175\pi$
$270$	0	0
$271$	9.65260	0.586354	0.293177	−	0.956058i	$-0.405288\pi$
$271$	9.65260	0.586354	0.293177	+	0.956058i	$0.405288\pi$
$272$	14.1025i	0.855088i
$273$	17.1829i	1.03996i
$274$	24.4588	1.47761
$275$	0	0
$276$	1.19525	0.0719456
$277$	20.0455i	1.20442i	0.798338	+	0.602210i	$0.205713\pi$
$277$	20.0455i	1.20442i	−0.798338	+	0.602210i	$0.794287\pi$
$278$	− 0.696706i	− 0.0417857i
$279$	3.90694	0.233903
$280$	0	0
$281$	−4.42685	−0.264084	−0.132042	−	0.991244i	$-0.542153\pi$
$281$	−4.42685	−0.264084	−0.132042	+	0.991244i	$0.542153\pi$
$282$	− 15.0156i	− 0.894168i
$283$	− 11.2028i	− 0.665938i	−0.942938	−	0.332969i	$-0.891950\pi$
$283$	− 11.2028i	− 0.665938i	0.942938	−	0.332969i	$-0.108050\pi$
$284$	−0.816170	−0.0484308
$285$	0	0
$286$	30.6094	1.80997
$287$	3.52031i	0.207797i
$288$	− 7.76669i	− 0.457657i
$289$	6.32315	0.371950
$290$	0	0
$291$	8.15938	0.478311
$292$	20.3121i	1.18868i
$293$	26.6504i	1.55693i	0.627688	+	0.778465i	$0.284002\pi$
$293$	26.6504i	1.55693i	−0.627688	+	0.778465i	$0.715998\pi$
$294$	−27.2404	−1.58869
$295$	0	0
$296$	−3.59069	−0.208705
$297$	4.16535i	0.241698i
$298$	− 29.4064i	− 1.70347i
$299$	2.45735	0.142112
$300$	0	0
$301$	−17.8557	−1.02918
$302$	11.0643i	0.636681i
$303$	− 9.54383i	− 0.548279i
$304$	−1.05147	−0.0603057
$305$	0	0
$306$	6.39228	0.365423
$307$	− 9.48870i	− 0.541549i	−0.962643	−	0.270774i	$-0.912720\pi$
$307$	− 9.48870i	− 0.541549i	0.962643	−	0.270774i	$-0.0872798\pi$
$308$	34.8128i	1.98364i
$309$	−9.86698	−0.561313
$310$	0	0
$311$	11.5152	0.652969	0.326485	−	0.945203i	$-0.394136\pi$
$311$	11.5152	0.652969	0.326485	+	0.945203i	$0.394136\pi$
$312$	− 1.27064i	− 0.0719356i
$313$	2.14459i	0.121219i	0.998162	+	0.0606097i	$0.0193045\pi$
$313$	2.14459i	0.121219i	−0.998162	+	0.0606097i	$0.980695\pi$
$314$	13.0970	0.739106
$315$	0	0
$316$	17.9188	1.00801
$317$	− 0.735614i	− 0.0413162i	−0.999787	−	0.0206581i	$-0.993424\pi$
$317$	− 0.735614i	− 0.0413162i	0.999787	−	0.0206581i	$-0.00657615\pi$
$318$	− 23.7098i	− 1.32958i
$319$	−35.6026	−1.99336
$320$	0	0
$321$	16.1273	0.900138
$322$	5.85410i	0.326236i
$323$	− 0.796056i	− 0.0442937i
$324$	−1.82709	−0.101505
$325$	0	0
$326$	27.0584	1.49862
$327$	2.44236i	0.135063i
$328$	− 0.260319i	− 0.0143737i
$329$	35.1105	1.93570
$330$	0	0
$331$	11.1178	0.611089	0.305544	−	0.952178i	$-0.401162\pi$
$331$	11.1178	0.611089	0.305544	+	0.952178i	$0.401162\pi$
$332$	16.1968i	0.888917i
$333$	10.6151i	0.581706i
$334$	−26.4133	−1.44527
$335$	0	0
$336$	19.7424	1.07704
$337$	− 14.2506i	− 0.776282i	−0.921600	−	0.388141i	$-0.873117\pi$
$337$	− 14.2506i	− 0.776282i	0.921600	−	0.388141i	$-0.126883\pi$
$338$	2.17218i	0.118151i
$339$	−10.9293	−0.593597
$340$	0	0
$341$	16.2738	0.881275
$342$	0.476602i	0.0257717i
$343$	− 31.6749i	− 1.71028i
$344$	1.32039	0.0711905
$345$	0	0
$346$	−26.8519	−1.44357
$347$	− 0.174615i	− 0.00937383i	−0.999989	−	0.00468691i	$-0.998508\pi$
$347$	− 0.174615i	− 0.00937383i	0.999989	−	0.00468691i	$-0.00149190\pi$
$348$	− 15.6167i	− 0.837144i
$349$	18.6640	0.999059	0.499530	−	0.866297i	$-0.333506\pi$
$349$	18.6640	0.999059	0.499530	+	0.866297i	$0.333506\pi$
$350$	0	0
$351$	−3.75638	−0.200500
$352$	− 32.3510i	− 1.72431i
$353$	8.67239i	0.461585i	0.973003	+	0.230792i	$0.0741318\pi$
$353$	8.67239i	0.461585i	−0.973003	+	0.230792i	$0.925868\pi$
$354$	−9.59069	−0.509740
$355$	0	0
$356$	−14.2703	−0.756325
$357$	14.9468i	0.791070i
$358$	8.73468i	0.461642i
$359$	−13.7180	−0.724008	−0.362004	−	0.932177i	$-0.617907\pi$
$359$	−13.7180	−0.724008	−0.362004	+	0.932177i	$0.617907\pi$
$360$	0	0
$361$	−18.9406	−0.996876
$362$	27.3943i	1.43981i
$363$	6.35016i	0.333297i
$364$	−31.3947	−1.64553
$365$	0	0
$366$	−28.7645	−1.50354
$367$	0.530351i	0.0276841i	0.999904	+	0.0138421i	$0.00440620\pi$
$367$	0.530351i	0.0276841i	−0.999904	+	0.0138421i	$0.995594\pi$
$368$	− 2.82340i	− 0.147180i
$369$	−0.769579	−0.0400627
$370$	0	0
$371$	55.4397	2.87828
$372$	7.13834i	0.370106i
$373$	− 29.3489i	− 1.51963i	−0.650142	−	0.759813i	$-0.725290\pi$
$373$	− 29.3489i	− 1.51963i	0.650142	−	0.759813i	$-0.274710\pi$
$374$	26.6261	1.37680
$375$	0	0
$376$	−2.59634	−0.133896
$377$	− 32.1069i	− 1.65359i
$378$	− 8.94874i	− 0.460273i
$379$	−4.84634	−0.248940	−0.124470	−	0.992223i	$-0.539723\pi$
$379$	−4.84634	−0.248940	−0.124470	+	0.992223i	$0.539723\pi$
$380$	0	0
$381$	−12.8613	−0.658906
$382$	8.40451i	0.430012i
$383$	21.3587i	1.09138i	0.837988	+	0.545689i	$0.183732\pi$
$383$	21.3587i	1.09138i	−0.837988	+	0.545689i	$0.816268\pi$
$384$	−2.69598	−0.137578
$385$	0	0
$386$	−25.5171	−1.29879
$387$	− 3.90345i	− 0.198424i
$388$	14.9079i	0.756835i
$389$	−10.4329	−0.528971	−0.264486	−	0.964390i	$-0.585202\pi$
$389$	−10.4329	−0.528971	−0.264486	+	0.964390i	$0.585202\pi$
$390$	0	0
$391$	2.13757	0.108102
$392$	4.71011i	0.237897i
$393$	2.37441i	0.119773i
$394$	−17.9560	−0.904608
$395$	0	0
$396$	−7.61048	−0.382441
$397$	− 16.3146i	− 0.818806i	−0.912354	−	0.409403i	$-0.865737\pi$
$397$	− 16.3146i	− 0.818806i	0.912354	−	0.409403i	$-0.134263\pi$
$398$	1.32539i	0.0664357i
$399$	−1.11442	−0.0557908
$400$	0	0
$401$	10.3272	0.515718	0.257859	−	0.966183i	$-0.416983\pi$
$401$	10.3272	0.515718	0.257859	+	0.966183i	$0.416983\pi$
$402$	− 0.619944i	− 0.0309200i
$403$	14.6759i	0.731061i
$404$	17.4374	0.867545
$405$	0	0
$406$	76.4877	3.79602
$407$	44.2158i	2.19170i
$408$	− 1.10528i	− 0.0547197i
$409$	−26.8421	−1.32726	−0.663629	−	0.748062i	$-0.730985\pi$
$409$	−26.8421	−1.32726	−0.663629	+	0.748062i	$0.730985\pi$
$410$	0	0
$411$	−12.5026	−0.616710
$412$	− 18.0279i	− 0.888169i
$413$	− 22.4255i	− 1.10349i
$414$	−1.27977	−0.0628974
$415$	0	0
$416$	29.1746	1.43040
$417$	0.356135i	0.0174400i
$418$	1.98522i	0.0971001i
$419$	−5.04882	−0.246651	−0.123325	−	0.992366i	$-0.539356\pi$
$419$	−5.04882	−0.246651	−0.123325	+	0.992366i	$0.539356\pi$
$420$	0	0
$421$	−23.2026	−1.13082	−0.565412	−	0.824809i	$-0.691283\pi$
$421$	−23.2026	−1.13082	−0.565412	+	0.824809i	$0.691283\pi$
$422$	31.6259i	1.53952i
$423$	7.67555i	0.373198i
$424$	−4.09964	−0.199096
$425$	0	0
$426$	0.873886	0.0423399
$427$	− 67.2588i	− 3.25488i
$428$	29.4660i	1.42429i
$429$	−15.6466	−0.755426
$430$	0	0
$431$	−39.7692	−1.91561	−0.957807	−	0.287413i	$-0.907205\pi$
$431$	−39.7692	−1.91561	−0.957807	+	0.287413i	$0.907205\pi$
$432$	4.31592i	0.207650i
$433$	18.3272i	0.880750i	0.897814	+	0.440375i	$0.145154\pi$
$433$	18.3272i	0.880750i	−0.897814	+	0.440375i	$0.854846\pi$
$434$	−34.9622	−1.67824
$435$	0	0
$436$	−4.46242	−0.213711
$437$	0.159375i	0.00762394i
$438$	− 21.7485i	− 1.03918i
$439$	37.2170	1.77627	0.888135	−	0.459582i	$-0.152001\pi$
$439$	37.2170	1.77627	0.888135	+	0.459582i	$0.152001\pi$
$440$	0	0
$441$	13.9245	0.663071
$442$	24.0118i	1.14213i
$443$	− 8.41969i	− 0.400032i	−0.979793	−	0.200016i	$-0.935901\pi$
$443$	− 8.41969i	− 0.400032i	0.979793	−	0.200016i	$-0.0640994\pi$
$444$	−19.3948	−0.920438
$445$	0	0
$446$	19.4859	0.922686
$447$	15.0317i	0.710974i
$448$	30.0172i	1.41818i
$449$	−3.31527	−0.156457	−0.0782287	−	0.996935i	$-0.524926\pi$
$449$	−3.31527	−0.156457	−0.0782287	+	0.996935i	$0.524926\pi$
$450$	0	0
$451$	−3.20557	−0.150944
$452$	− 19.9688i	− 0.939253i
$453$	− 5.65576i	− 0.265731i
$454$	47.0775	2.20946
$455$	0	0
$456$	0.0824089	0.00385915
$457$	− 2.57124i	− 0.120277i	−0.998190	−	0.0601387i	$-0.980846\pi$
$457$	− 2.57124i	− 0.120277i	0.998190	−	0.0601387i	$-0.0191543\pi$
$458$	− 30.0971i	− 1.40634i
$459$	−3.26755	−0.152516
$460$	0	0
$461$	34.8861	1.62481	0.812404	−	0.583095i	$-0.198159\pi$
$461$	34.8861	1.62481	0.812404	+	0.583095i	$0.198159\pi$
$462$	− 37.2746i	− 1.73417i
$463$	− 8.24992i	− 0.383406i	−0.981453	−	0.191703i	$-0.938599\pi$
$463$	− 8.24992i	− 0.383406i	0.981453	−	0.191703i	$-0.0614011\pi$
$464$	−36.8895	−1.71255
$465$	0	0
$466$	−12.8902	−0.597125
$467$	− 24.4499i	− 1.13140i	−0.824609	−	0.565702i	$-0.808605\pi$
$467$	− 24.4499i	− 1.13140i	0.824609	−	0.565702i	$-0.191395\pi$
$468$	− 6.86324i	− 0.317253i
$469$	1.44959	0.0669359
$470$	0	0
$471$	−6.69480	−0.308480
$472$	1.65832i	0.0763303i
$473$	− 16.2593i	− 0.747602i
$474$	−19.1860	−0.881242
$475$	0	0
$476$	−27.3092	−1.25172
$477$	12.1197i	0.554925i
$478$	34.9930i	1.60054i
$479$	−4.79732	−0.219195	−0.109598	−	0.993976i	$-0.534956\pi$
$479$	−4.79732	−0.219195	−0.109598	+	0.993976i	$0.534956\pi$
$480$	0	0
$481$	−39.8745	−1.81812
$482$	− 12.1761i	− 0.554605i
$483$	− 2.99244i	− 0.136161i
$484$	−11.6023	−0.527378
$485$	0	0
$486$	1.95630	0.0887394
$487$	20.2715i	0.918589i	0.888284	+	0.459294i	$0.151898\pi$
$487$	20.2715i	0.918589i	−0.888284	+	0.459294i	$0.848102\pi$
$488$	4.97364i	0.225146i
$489$	−13.8314	−0.625479
$490$	0	0
$491$	36.2638	1.63656	0.818281	−	0.574818i	$-0.194927\pi$
$491$	36.2638	1.63656	0.818281	+	0.574818i	$0.194927\pi$
$492$	− 1.40609i	− 0.0633915i
$493$	− 27.9287i	− 1.25785i
$494$	−1.79030	−0.0805493
$495$	0	0
$496$	16.8621	0.757129
$497$	2.04337i	0.0916579i
$498$	− 17.3422i	− 0.777123i
$499$	−2.39550	−0.107237	−0.0536187	−	0.998561i	$-0.517076\pi$
$499$	−2.39550	−0.107237	−0.0536187	+	0.998561i	$0.517076\pi$
$500$	0	0
$501$	13.5017	0.603212
$502$	13.9206i	0.621307i
$503$	− 22.7079i	− 1.01249i	−0.862389	−	0.506247i	$-0.831032\pi$
$503$	− 22.7079i	− 1.01249i	0.862389	−	0.506247i	$-0.168968\pi$
$504$	−1.54732	−0.0689230
$505$	0	0
$506$	−5.33070	−0.236979
$507$	− 1.11035i	− 0.0493125i
$508$	− 23.4988i	− 1.04259i
$509$	−14.9257	−0.661569	−0.330784	−	0.943706i	$-0.607313\pi$
$509$	−14.9257	−0.661569	−0.330784	+	0.943706i	$0.607313\pi$
$510$	0	0
$511$	50.8536	2.24963
$512$	− 30.6006i	− 1.35237i
$513$	− 0.243625i	− 0.0107563i
$514$	13.9990	0.617470
$515$	0	0
$516$	7.13196	0.313967
$517$	31.9714i	1.40610i
$518$	− 94.9922i	− 4.17372i
$519$	13.7259	0.602499
$520$	0	0
$521$	−5.56462	−0.243790	−0.121895	−	0.992543i	$-0.538897\pi$
$521$	−5.56462	−0.243790	−0.121895	+	0.992543i	$0.538897\pi$
$522$	16.7211i	0.731862i
$523$	− 22.6133i	− 0.988811i	−0.869231	−	0.494405i	$-0.835386\pi$
$523$	− 22.6133i	− 0.988811i	0.869231	−	0.494405i	$-0.164614\pi$
$524$	−4.33826	−0.189518
$525$	0	0
$526$	2.56210	0.111713
$527$	12.7661i	0.556101i
$528$	17.9773i	0.782363i
$529$	22.5720	0.981393
$530$	0	0
$531$	4.90248	0.212749
$532$	− 2.03615i	− 0.0882782i
$533$	− 2.89083i	− 0.125216i
$534$	15.2794	0.661206
$535$	0	0
$536$	−0.107194	−0.00463007
$537$	− 4.46491i	− 0.192675i
$538$	− 41.0323i	− 1.76903i
$539$	58.0004	2.49825
$540$	0	0
$541$	−12.5382	−0.539059	−0.269529	−	0.962992i	$-0.586868\pi$
$541$	−12.5382	−0.539059	−0.269529	+	0.962992i	$0.586868\pi$
$542$	− 18.8833i	− 0.811109i
$543$	− 14.0032i	− 0.600933i
$544$	25.3780	1.08807
$545$	0	0
$546$	33.6148	1.43858
$547$	− 5.45889i	− 0.233405i	−0.993167	−	0.116703i	$-0.962768\pi$
$547$	− 5.45889i	− 0.233405i	0.993167	−	0.116703i	$-0.0372324\pi$
$548$	− 22.8435i	− 0.975824i
$549$	14.7035	0.627532
$550$	0	0
$551$	2.08234	0.0887107
$552$	0.221284i	0.00941849i
$553$	− 44.8618i	− 1.90772i
$554$	39.2150	1.66608
$555$	0	0
$556$	−0.650692	−0.0275955
$557$	− 4.84648i	− 0.205352i	−0.994715	−	0.102676i	$-0.967260\pi$
$557$	− 4.84648i	− 0.205352i	0.994715	−	0.102676i	$-0.0327404\pi$
$558$	− 7.64313i	− 0.323560i
$559$	14.6628	0.620172
$560$	0	0
$561$	−13.6105	−0.574635
$562$	8.66023i	0.365310i
$563$	− 37.5755i	− 1.58362i	−0.610769	−	0.791809i	$-0.709139\pi$
$563$	− 37.5755i	− 1.58362i	0.610769	−	0.791809i	$-0.290861\pi$
$564$	−14.0239	−0.590514
$565$	0	0
$566$	−21.9160	−0.921198
$567$	4.57433i	0.192104i
$568$	− 0.151103i	− 0.00634014i
$569$	−17.0619	−0.715272	−0.357636	−	0.933861i	$-0.616417\pi$
$569$	−17.0619	−0.715272	−0.357636	+	0.933861i	$0.616417\pi$
$570$	0	0
$571$	−28.6234	−1.19785	−0.598925	−	0.800805i	$-0.704405\pi$
$571$	−28.6234	−1.19785	−0.598925	+	0.800805i	$0.704405\pi$
$572$	− 28.5878i	− 1.19532i
$573$	− 4.29614i	− 0.179474i
$574$	6.88676	0.287448
$575$	0	0
$576$	−6.56210	−0.273421
$577$	2.22388i	0.0925815i	0.998928	+	0.0462907i	$0.0147401\pi$
$577$	2.22388i	0.0925815i	−0.998928	+	0.0462907i	$0.985260\pi$
$578$	− 12.3699i	− 0.514522i
$579$	13.0436	0.542073
$580$	0	0
$581$	40.5506	1.68232
$582$	− 15.9621i	− 0.661652i
$583$	50.4830i	2.09079i
$584$	−3.76051	−0.155611
$585$	0	0
$586$	52.1360	2.15372
$587$	− 29.8085i	− 1.23033i	−0.788399	−	0.615164i	$-0.789090\pi$
$587$	− 29.8085i	− 1.23033i	0.788399	−	0.615164i	$-0.210910\pi$
$588$	25.4413i	1.04918i
$589$	−0.951829	−0.0392194
$590$	0	0
$591$	9.17856	0.377555
$592$	45.8141i	1.88295i
$593$	43.9251i	1.80379i	0.431956	+	0.901895i	$0.357823\pi$
$593$	43.9251i	1.80379i	−0.431956	+	0.901895i	$0.642177\pi$
$594$	8.14866	0.334344
$595$	0	0
$596$	−27.4642	−1.12498
$597$	− 0.677499i	− 0.0277282i
$598$	− 4.80731i	− 0.196585i
$599$	−43.1539	−1.76322	−0.881610	−	0.471978i	$-0.843540\pi$
$599$	−43.1539	−1.76322	−0.881610	+	0.471978i	$0.843540\pi$
$600$	0	0
$601$	37.7365	1.53931	0.769653	−	0.638463i	$-0.220429\pi$
$601$	37.7365	1.53931	0.769653	+	0.638463i	$0.220429\pi$
$602$	34.9310i	1.42368i
$603$	0.316897i	0.0129050i
$604$	10.3336	0.420468
$605$	0	0
$606$	−18.6705	−0.758439
$607$	45.6549i	1.85308i	0.376202	+	0.926538i	$0.377230\pi$
$607$	45.6549i	1.85308i	−0.376202	+	0.926538i	$0.622770\pi$
$608$	1.89216i	0.0767372i
$609$	−39.0982	−1.58434
$610$	0	0
$611$	−28.8322	−1.16643
$612$	− 5.97010i	− 0.241327i
$613$	39.0685i	1.57796i	0.614418	+	0.788980i	$0.289391\pi$
$613$	39.0685i	1.57796i	−0.614418	+	0.788980i	$0.710609\pi$
$614$	−18.5627	−0.749130
$615$	0	0
$616$	−6.44512	−0.259681
$617$	− 27.4125i	− 1.10359i	−0.833981	−	0.551794i	$-0.813944\pi$
$617$	− 27.4125i	− 1.10359i	0.833981	−	0.551794i	$-0.186056\pi$
$618$	19.3027i	0.776469i
$619$	22.8242	0.917384	0.458692	−	0.888595i	$-0.348318\pi$
$619$	22.8242	0.917384	0.458692	+	0.888595i	$0.348318\pi$
$620$	0	0
$621$	0.654182	0.0262514
$622$	− 22.5272i	− 0.903259i
$623$	35.7273i	1.43139i
$624$	−16.2122	−0.649008
$625$	0	0
$626$	4.19545	0.167684
$627$	− 1.01478i	− 0.0405266i
$628$	− 12.2320i	− 0.488110i
$629$	−34.6855	−1.38300
$630$	0	0
$631$	10.0638	0.400632	0.200316	−	0.979731i	$-0.435803\pi$
$631$	10.0638	0.400632	0.200316	+	0.979731i	$0.435803\pi$
$632$	3.31743i	0.131960i
$633$	− 16.1662i	− 0.642549i
$634$	−1.43908	−0.0571531
$635$	0	0
$636$	−22.1439	−0.878061
$637$	52.3056i	2.07242i
$638$	69.6492i	2.75744i
$639$	−0.446705	−0.0176714
$640$	0	0
$641$	−42.7831	−1.68983	−0.844916	−	0.534899i	$-0.820350\pi$
$641$	−42.7831	−1.68983	−0.844916	+	0.534899i	$0.820350\pi$
$642$	− 31.5497i	− 1.24517i
$643$	− 10.1067i	− 0.398571i	−0.979941	−	0.199285i	$-0.936138\pi$
$643$	− 10.1067i	− 0.398571i	0.979941	−	0.199285i	$-0.0638621\pi$
$644$	5.46747	0.215448
$645$	0	0
$646$	−1.55732	−0.0612719
$647$	− 16.7580i	− 0.658826i	−0.944186	−	0.329413i	$-0.893149\pi$
$647$	− 16.7580i	− 0.658826i	0.944186	−	0.329413i	$-0.106851\pi$
$648$	− 0.338261i	− 0.0132882i
$649$	20.4205	0.801576
$650$	0	0
$651$	17.8716	0.700445
$652$	− 25.2713i	− 0.989700i
$653$	− 20.7798i	− 0.813175i	−0.913612	−	0.406588i	$-0.866719\pi$
$653$	− 20.7798i	− 0.813175i	0.913612	−	0.406588i	$-0.133281\pi$
$654$	4.77799	0.186834
$655$	0	0
$656$	−3.32144	−0.129680
$657$	11.1172i	0.433723i
$658$	− 68.6865i	− 2.67768i
$659$	15.9362	0.620785	0.310393	−	0.950608i	$-0.399539\pi$
$659$	15.9362	0.620785	0.310393	+	0.950608i	$0.399539\pi$
$660$	0	0
$661$	3.37323	0.131203	0.0656017	−	0.997846i	$-0.479103\pi$
$661$	3.37323	0.131203	0.0656017	+	0.997846i	$0.479103\pi$
$662$	− 21.7497i	− 0.845325i
$663$	− 12.2741i	− 0.476688i
$664$	−2.99862	−0.116369
$665$	0	0
$666$	20.7664	0.804680
$667$	5.59150i	0.216504i
$668$	24.6688i	0.954466i
$669$	−9.96064	−0.385100
$670$	0	0
$671$	61.2454	2.36435
$672$	− 35.5274i	− 1.37050i
$673$	35.2852i	1.36015i	0.733144	+	0.680073i	$0.238052\pi$
$673$	35.2852i	1.36015i	−0.733144	+	0.680073i	$0.761948\pi$
$674$	−27.8785	−1.07384
$675$	0	0
$676$	2.02872	0.0780276
$677$	− 31.2943i	− 1.20274i	−0.798971	−	0.601369i	$-0.794622\pi$
$677$	− 31.2943i	− 1.20274i	0.798971	−	0.601369i	$-0.205378\pi$
$678$	21.3809i	0.821129i
$679$	37.3237	1.43235
$680$	0	0
$681$	−24.0646	−0.922158
$682$	− 31.8363i	− 1.21908i
$683$	− 10.8131i	− 0.413751i	−0.978367	−	0.206876i	$-0.933670\pi$
$683$	− 10.8131i	− 0.413751i	0.978367	−	0.206876i	$-0.0663296\pi$
$684$	0.445125	0.0170198
$685$	0	0
$686$	−61.9654	−2.36585
$687$	15.3847i	0.586964i
$688$	− 16.8470i	− 0.642286i
$689$	−45.5263	−1.73441
$690$	0	0
$691$	−50.9819	−1.93944	−0.969722	−	0.244210i	$-0.921471\pi$
$691$	−50.9819	−1.93944	−0.969722	+	0.244210i	$0.921471\pi$
$692$	25.0784i	0.953338i
$693$	19.0537i	0.723790i
$694$	−0.341599	−0.0129669
$695$	0	0
$696$	2.89123	0.109592
$697$	− 2.51463i	− 0.0952485i
$698$	− 36.5122i	− 1.38201i
$699$	6.58907	0.249222
$700$	0	0
$701$	5.20728	0.196676	0.0983382	−	0.995153i	$-0.468647\pi$
$701$	5.20728	0.196676	0.0983382	+	0.995153i	$0.468647\pi$
$702$	7.34858i	0.277354i
$703$	− 2.58611i	− 0.0975372i
$704$	−27.3335	−1.03017
$705$	0	0
$706$	16.9657	0.638514
$707$	− 43.6566i	− 1.64188i
$708$	8.95727i	0.336635i
$709$	37.4108	1.40499	0.702497	−	0.711687i	$-0.252069\pi$
$709$	37.4108	1.40499	0.702497	+	0.711687i	$0.252069\pi$
$710$	0	0
$711$	9.80731	0.367803
$712$	− 2.64195i	− 0.0990114i
$713$	− 2.55585i	− 0.0957174i
$714$	29.2404	1.09429
$715$	0	0
$716$	8.15780	0.304871
$717$	− 17.8874i	− 0.668017i
$718$	26.8364i	1.00153i
$719$	−31.2527	−1.16553	−0.582764	−	0.812641i	$-0.698029\pi$
$719$	−31.2527	−1.16553	−0.582764	+	0.812641i	$0.698029\pi$
$720$	0	0
$721$	−45.1348	−1.68091
$722$	37.0535i	1.37899i
$723$	6.22404i	0.231475i
$724$	25.5850	0.950861
$725$	0	0
$726$	12.4228	0.461053
$727$	1.21933i	0.0452225i	0.999744	+	0.0226113i	$0.00719800\pi$
$727$	1.21933i	0.0452225i	−0.999744	+	0.0226113i	$0.992802\pi$
$728$	− 5.81231i	− 0.215418i
$729$	−1.00000	−0.0370370
$730$	0	0
$731$	12.7547	0.471750
$732$	26.8647i	0.992948i
$733$	13.3535i	0.493221i	0.969115	+	0.246611i	$0.0793168\pi$
$733$	13.3535i	0.493221i	−0.969115	+	0.246611i	$0.920683\pi$
$734$	1.03752	0.0382957
$735$	0	0
$736$	−5.08083	−0.187282
$737$	1.31999i	0.0486224i
$738$	1.50552i	0.0554191i
$739$	5.24691	0.193011	0.0965054	−	0.995332i	$-0.469234\pi$
$739$	5.24691	0.193011	0.0965054	+	0.995332i	$0.469234\pi$
$740$	0	0
$741$	0.915147	0.0336188
$742$	− 108.456i	− 3.98156i
$743$	− 42.9203i	− 1.57459i	−0.616575	−	0.787296i	$-0.711480\pi$
$743$	− 42.9203i	− 1.57459i	0.616575	−	0.787296i	$-0.288520\pi$
$744$	−1.32157	−0.0484510
$745$	0	0
$746$	−57.4150	−2.10211
$747$	8.86482i	0.324347i
$748$	− 24.8676i	− 0.909249i
$749$	73.7716	2.69555
$750$	0	0
$751$	−20.2461	−0.738792	−0.369396	−	0.929272i	$-0.620435\pi$
$751$	−20.2461	−0.738792	−0.369396	+	0.929272i	$0.620435\pi$
$752$	33.1270i	1.20802i
$753$	− 7.11580i	− 0.259314i
$754$	−62.8106	−2.28743
$755$	0	0
$756$	−8.35772	−0.303967
$757$	10.3260i	0.375306i	0.982235	+	0.187653i	$0.0600881\pi$
$757$	10.3260i	0.375306i	−0.982235	+	0.187653i	$0.939912\pi$
$758$	9.48087i	0.344361i
$759$	2.72490	0.0989075
$760$	0	0
$761$	29.2923	1.06185	0.530923	−	0.847420i	$-0.321845\pi$
$761$	29.2923	1.06185	0.530923	+	0.847420i	$0.321845\pi$
$762$	25.1606i	0.911471i
$763$	11.1722i	0.404460i
$764$	7.84943	0.283982
$765$	0	0
$766$	41.7839	1.50971
$767$	18.4155i	0.664947i
$768$	18.3983i	0.663893i
$769$	22.9834	0.828803	0.414402	−	0.910094i	$-0.363991\pi$
$769$	22.9834	0.828803	0.414402	+	0.910094i	$0.363991\pi$
$770$	0	0
$771$	−7.15589	−0.257713
$772$	23.8318i	0.857725i
$773$	54.2216i	1.95022i	0.221732	+	0.975108i	$0.428829\pi$
$773$	54.2216i	1.95022i	−0.221732	+	0.975108i	$0.571171\pi$
$774$	−7.63631	−0.274481
$775$	0	0
$776$	−2.76000	−0.0990782
$777$	48.5572i	1.74198i
$778$	20.4099i	0.731731i
$779$	0.187489	0.00671748
$780$	0	0
$781$	−1.86068	−0.0665805
$782$	− 4.18172i	− 0.149538i
$783$	− 8.54732i	− 0.305456i
$784$	60.0970	2.14632
$785$	0	0
$786$	4.64505	0.165683
$787$	6.60711i	0.235518i	0.993042	+	0.117759i	$0.0375711\pi$
$787$	6.60711i	0.235518i	−0.993042	+	0.117759i	$0.962429\pi$
$788$	16.7701i	0.597409i
$789$	−1.30967	−0.0466255
$790$	0	0
$791$	−49.9941	−1.77759
$792$	− 1.40898i	− 0.0500658i
$793$	55.2320i	1.96135i
$794$	−31.9161	−1.13266
$795$	0	0
$796$	1.23785	0.0438745
$797$	− 2.34163i	− 0.0829446i	−0.999140	−	0.0414723i	$-0.986795\pi$
$797$	− 2.34163i	− 0.0829446i	0.999140	−	0.0414723i	$-0.0132048\pi$
$798$	2.18014i	0.0771760i
$799$	−25.0802	−0.887274
$800$	0	0
$801$	−7.81040	−0.275967
$802$	− 20.2031i	− 0.713397i
$803$	46.3070i	1.63414i
$804$	−0.579000	−0.0204197
$805$	0	0
$806$	28.7105	1.01128
$807$	20.9745i	0.738337i
$808$	3.22831i	0.113571i
$809$	53.7248	1.88886	0.944432	−	0.328707i	$-0.106613\pi$
$809$	53.7248	1.88886	0.944432	+	0.328707i	$0.106613\pi$
$810$	0	0
$811$	−29.5877	−1.03897	−0.519483	−	0.854481i	$-0.673875\pi$
$811$	−29.5877	−1.03897	−0.519483	+	0.854481i	$0.673875\pi$
$812$	− 71.4361i	− 2.50691i
$813$	9.65260i	0.338532i
$814$	86.4992	3.03180
$815$	0	0
$816$	−14.1025	−0.493685
$817$	0.950979i	0.0332705i
$818$	52.5112i	1.83601i
$819$	−17.1829	−0.600419
$820$	0	0
$821$	33.3855	1.16516	0.582580	−	0.812773i	$-0.302043\pi$
$821$	33.3855	1.16516	0.582580	+	0.812773i	$0.302043\pi$
$822$	24.4588i	0.853100i
$823$	17.5383i	0.611345i	0.952137	+	0.305672i	$0.0988813\pi$
$823$	17.5383i	0.611345i	−0.952137	+	0.305672i	$0.901119\pi$
$824$	3.33762	0.116271
$825$	0	0
$826$	−43.8710	−1.52647
$827$	26.6114i	0.925369i	0.886523	+	0.462684i	$0.153114\pi$
$827$	26.6114i	0.925369i	−0.886523	+	0.462684i	$0.846886\pi$
$828$	1.19525i	0.0415378i
$829$	−20.3212	−0.705784	−0.352892	−	0.935664i	$-0.614802\pi$
$829$	−20.3212	−0.705784	−0.352892	+	0.935664i	$0.614802\pi$
$830$	0	0
$831$	−20.0455	−0.695372
$832$	− 24.6497i	− 0.854575i
$833$	45.4989i	1.57644i
$834$	0.696706	0.0241250
$835$	0	0
$836$	1.85410	0.0641255
$837$	3.90694i	0.135044i
$838$	9.87698i	0.341195i
$839$	23.3881	0.807448	0.403724	−	0.914881i	$-0.367716\pi$
$839$	23.3881	0.807448	0.403724	+	0.914881i	$0.367716\pi$
$840$	0	0
$841$	44.0566	1.51919
$842$	45.3910i	1.56428i
$843$	− 4.42685i	− 0.152469i
$844$	29.5371	1.01671
$845$	0	0
$846$	15.0156	0.516248
$847$	29.0477i	0.998091i
$848$	52.3078i	1.79626i
$849$	11.2028	0.384479
$850$	0	0
$851$	6.94424	0.238045
$852$	− 0.816170i	− 0.0279615i
$853$	9.81934i	0.336208i	0.985769	+	0.168104i	$0.0537644\pi$
$853$	9.81934i	0.336208i	−0.985769	+	0.168104i	$0.946236\pi$
$854$	−131.578	−4.50251
$855$	0	0
$856$	−5.45524	−0.186456
$857$	− 15.9866i	− 0.546093i	−0.962001	−	0.273047i	$-0.911969\pi$
$857$	− 15.9866i	− 0.546093i	0.962001	−	0.273047i	$-0.0880313\pi$
$858$	30.6094i	1.04499i
$859$	20.5913	0.702567	0.351284	−	0.936269i	$-0.385745\pi$
$859$	20.5913	0.702567	0.351284	+	0.936269i	$0.385745\pi$
$860$	0	0
$861$	−3.52031	−0.119972
$862$	77.8002i	2.64989i
$863$	− 51.5169i	− 1.75366i	−0.480803	−	0.876828i	$-0.659655\pi$
$863$	− 51.5169i	− 1.75366i	0.480803	−	0.876828i	$-0.340345\pi$
$864$	7.76669	0.264228
$865$	0	0
$866$	35.8534	1.21835
$867$	6.32315i	0.214745i
$868$	32.6531i	1.10832i
$869$	40.8509	1.38577
$870$	0	0
$871$	−1.19038	−0.0403346
$872$	− 0.826157i	− 0.0279772i
$873$	8.15938i	0.276153i
$874$	0.311785	0.0105463
$875$	0	0
$876$	−20.3121	−0.686282
$877$	17.6050i	0.594480i	0.954803	+	0.297240i	$0.0960661\pi$
$877$	17.6050i	0.594480i	−0.954803	+	0.297240i	$0.903934\pi$
$878$	− 72.8075i	− 2.45713i
$879$	−26.6504	−0.898894
$880$	0	0
$881$	−47.1453	−1.58837	−0.794183	−	0.607679i	$-0.792101\pi$
$881$	−47.1453	−1.58837	−0.794183	+	0.607679i	$0.792101\pi$
$882$	− 27.2404i	− 0.917232i
$883$	− 23.2300i	− 0.781751i	−0.920444	−	0.390875i	$-0.872172\pi$
$883$	− 23.2300i	− 0.781751i	0.920444	−	0.390875i	$-0.127828\pi$
$884$	22.4259	0.754266
$885$	0	0
$886$	−16.4714	−0.553368
$887$	− 20.5226i	− 0.689082i	−0.938771	−	0.344541i	$-0.888035\pi$
$887$	− 20.5226i	− 0.689082i	0.938771	−	0.344541i	$-0.111965\pi$
$888$	− 3.59069i	− 0.120496i
$889$	−58.8320	−1.97316
$890$	0	0
$891$	−4.16535	−0.139545
$892$	− 18.1990i	− 0.609347i
$893$	− 1.86995i	− 0.0625756i
$894$	29.4064	0.983497
$895$	0	0
$896$	−12.3323	−0.411993
$897$	2.45735i	0.0820486i
$898$	6.48566i	0.216429i
$899$	−33.3939	−1.11375
$900$	0	0
$901$	−39.6018	−1.31933
$902$	6.27104i	0.208803i
$903$	− 17.8557i	− 0.594200i
$904$	3.69695	0.122959
$905$	0	0
$906$	−11.0643	−0.367588
$907$	13.1950i	0.438133i	0.975710	+	0.219066i	$0.0703011\pi$
$907$	13.1950i	0.438133i	−0.975710	+	0.219066i	$0.929699\pi$
$908$	− 43.9682i	− 1.45914i
$909$	9.54383	0.316549
$910$	0	0
$911$	27.1021	0.897932	0.448966	−	0.893549i	$-0.351792\pi$
$911$	27.1021	0.897932	0.448966	+	0.893549i	$0.351792\pi$
$912$	− 1.05147i	− 0.0348175i
$913$	36.9251i	1.22204i
$914$	−5.03010	−0.166381
$915$	0	0
$916$	−28.1093	−0.928757
$917$	10.8613i	0.358673i
$918$	6.39228i	0.210977i
$919$	3.80568	0.125538	0.0627690	−	0.998028i	$-0.480007\pi$
$919$	3.80568	0.125538	0.0627690	+	0.998028i	$0.480007\pi$
$920$	0	0
$921$	9.48870	0.312663
$922$	− 68.2475i	− 2.24761i
$923$	− 1.67799i	− 0.0552317i
$924$	−34.8128	−1.14526
$925$	0	0
$926$	−16.1393	−0.530369
$927$	− 9.86698i	− 0.324074i
$928$	66.3844i	2.17917i
$929$	−41.8941	−1.37450	−0.687250	−	0.726421i	$-0.741182\pi$
$929$	−41.8941	−1.37450	−0.687250	+	0.726421i	$0.741182\pi$
$930$	0	0
$931$	−3.39235	−0.111180
$932$	12.0388i	0.394345i
$933$	11.5152i	0.376992i
$934$	−47.8312	−1.56508
$935$	0	0
$936$	1.27064	0.0415320
$937$	8.55321i	0.279421i	0.990192	+	0.139711i	$0.0446172\pi$
$937$	8.55321i	0.279421i	−0.990192	+	0.139711i	$0.955383\pi$
$938$	− 2.83583i	− 0.0925931i
$939$	−2.14459	−0.0699861
$940$	0	0
$941$	−48.3669	−1.57672	−0.788358	−	0.615216i	$-0.789069\pi$
$941$	−48.3669	−1.57672	−0.788358	+	0.615216i	$0.789069\pi$
$942$	13.0970i	0.426723i
$943$	0.503445i	0.0163944i
$944$	21.1587	0.688657
$945$	0	0
$946$	−31.8079	−1.03416
$947$	− 8.71018i	− 0.283043i	−0.989935	−	0.141521i	$-0.954801\pi$
$947$	− 8.71018i	− 0.283043i	0.989935	−	0.141521i	$-0.0451994\pi$
$948$	17.9188i	0.581977i
$949$	−41.7603	−1.35560
$950$	0	0
$951$	0.735614	0.0238539
$952$	− 5.05593i	− 0.163864i
$953$	− 43.4534i	− 1.40760i	−0.710401	−	0.703798i	$-0.751486\pi$
$953$	− 43.4534i	− 1.40760i	0.710401	−	0.703798i	$-0.248514\pi$
$954$	23.7098	0.767633
$955$	0	0
$956$	32.6819	1.05701
$957$	− 35.6026i	− 1.15087i
$958$	9.38497i	0.303215i
$959$	−57.1912	−1.84680
$960$	0	0
$961$	−15.7358	−0.507606
$962$	78.0062i	2.51502i
$963$	16.1273i	0.519695i
$964$	−11.3719	−0.366264
$965$	0	0
$966$	−5.85410	−0.188353
$967$	60.2789i	1.93844i	0.246197	+	0.969220i	$0.420819\pi$
$967$	60.2789i	1.93844i	−0.246197	+	0.969220i	$0.579181\pi$
$968$	− 2.14801i	− 0.0690397i
$969$	0.796056	0.0255730
$970$	0	0
$971$	5.42583	0.174123	0.0870616	−	0.996203i	$-0.472252\pi$
$971$	5.42583	0.174123	0.0870616	+	0.996203i	$0.472252\pi$
$972$	− 1.82709i	− 0.0586040i
$973$	1.62908i	0.0522259i
$974$	39.6570	1.27069
$975$	0	0
$976$	63.4593	2.03128
$977$	36.5741i	1.17011i	0.810994	+	0.585054i	$0.198927\pi$
$977$	36.5741i	1.17011i	−0.810994	+	0.585054i	$0.801073\pi$
$978$	27.0584i	0.865231i
$979$	−32.5331	−1.03976
$980$	0	0
$981$	−2.44236	−0.0779787
$982$	− 70.9427i	− 2.26387i
$983$	− 41.0896i	− 1.31056i	−0.755388	−	0.655278i	$-0.772551\pi$
$983$	− 41.0896i	− 1.31056i	0.755388	−	0.655278i	$-0.227449\pi$
$984$	0.260319	0.00829866
$985$	0	0
$986$	−54.6369	−1.73999
$987$	35.1105i	1.11758i
$988$	1.67206i	0.0531952i
$989$	−2.55357	−0.0811988
$990$	0	0
$991$	−4.20264	−0.133501	−0.0667506	−	0.997770i	$-0.521263\pi$
$991$	−4.20264	−0.133501	−0.0667506	+	0.997770i	$0.521263\pi$
$992$	− 30.3440i	− 0.963424i
$993$	11.1178i	0.352812i
$994$	3.99744	0.126791
$995$	0	0
$996$	−16.1968	−0.513216
$997$	10.1577i	0.321698i	0.986979	+	0.160849i	$0.0514233\pi$
$997$	10.1577i	0.321698i	−0.986979	+	0.160849i	$0.948577\pi$
$998$	4.68631i	0.148342i
$999$	−10.6151	−0.335848

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1875.2.b.d.1249.1		8
5.2	odd	4		1875.2.a.f.1.4	✓	4
5.3	odd	4		1875.2.a.g.1.1	yes	4
5.4	even	2	inner	1875.2.b.d.1249.8		8
15.2	even	4		5625.2.a.m.1.1		4
15.8	even	4		5625.2.a.j.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.4	✓	4	5.2	odd	4
1875.2.a.g.1.1	yes	4	5.3	odd	4
1875.2.b.d.1249.1		8	1.1	even	1	trivial
1875.2.b.d.1249.8		8	5.4	even	2	inner
5625.2.a.j.1.4		4	15.8	even	4
5625.2.a.m.1.1		4	15.2	even	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:	\( N \)	\(=\)	\( 1875 = 3 \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1875.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.95630i q^{2} +1.00000i q^{3} -1.82709 q^{4} +1.95630 q^{6} +4.57433i q^{7} -0.338261i q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.95630i q^{2} +1.00000i q^{3} -1.82709 q^{4} +1.95630 q^{6} +4.57433i q^{7} -0.338261i q^{8} -1.00000 q^{9} -4.16535 q^{11} -1.82709i q^{12} -3.75638i q^{13} +8.94874 q^{14} -4.31592 q^{16} -3.26755i q^{17} +1.95630i q^{18} +0.243625 q^{19} -4.57433 q^{21} +8.14866i q^{22} +0.654182i q^{23} +0.338261 q^{24} -7.34858 q^{26} -1.00000i q^{27} -8.35772i q^{28} +8.54732 q^{29} -3.90694 q^{31} +7.76669i q^{32} -4.16535i q^{33} -6.39228 q^{34} +1.82709 q^{36} -10.6151i q^{37} -0.476602i q^{38} +3.75638 q^{39} +0.769579 q^{41} +8.94874i q^{42} +3.90345i q^{43} +7.61048 q^{44} +1.27977 q^{46} -7.67555i q^{47} -4.31592i q^{48} -13.9245 q^{49} +3.26755 q^{51} +6.86324i q^{52} -12.1197i q^{53} -1.95630 q^{54} +1.54732 q^{56} +0.243625i q^{57} -16.7211i q^{58} -4.90248 q^{59} -14.7035 q^{61} +7.64313i q^{62} -4.57433i q^{63} +6.56210 q^{64} -8.14866 q^{66} -0.316897i q^{67} +5.97010i q^{68} -0.654182 q^{69} +0.446705 q^{71} +0.338261i q^{72} -11.1172i q^{73} -20.7664 q^{74} -0.445125 q^{76} -19.0537i q^{77} -7.34858i q^{78} -9.80731 q^{79} +1.00000 q^{81} -1.50552i q^{82} -8.86482i q^{83} +8.35772 q^{84} +7.63631 q^{86} +8.54732i q^{87} +1.40898i q^{88} +7.81040 q^{89} +17.1829 q^{91} -1.19525i q^{92} -3.90694i q^{93} -15.0156 q^{94} -7.76669 q^{96} -8.15938i q^{97} +27.2404i q^{98} +4.16535 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9}+O(q^{10}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9	\( 8 q - 2 q^{4} - 2 q^{6} - 8 q^{9} - 12 q^{11} + 20 q^{14} - 18 q^{16} + 18 q^{19} - 10 q^{21} - 6 q^{24} - 4 q^{26} + 56 q^{29} - 20 q^{31} - 14 q^{34} + 2 q^{36} + 14 q^{39} + 18 q^{44} + 10 q^{46} + 6 q^{49} + 14 q^{51} + 2 q^{54} - 8 q^{59} - 86 q^{61} + 14 q^{64} - 12 q^{66} + 20 q^{69} - 54 q^{71} - 10 q^{74} + 18 q^{76} - 20 q^{79} + 8 q^{81} + 10 q^{84} + 48 q^{86} + 18 q^{89} + 10 q^{91} + 44 q^{94} + 12 q^{99}+O(q^{100}) \) 8 * q - 2 * q^4 - 2 * q^6 - 8 * q^9 - 12 * q^11 + 20 * q^14 - 18 * q^16 + 18 * q^19 - 10 * q^21 - 6 * q^24 - 4 * q^26 + 56 * q^29 - 20 * q^31 - 14 * q^34 + 2 * q^36 + 14 * q^39 + 18 * q^44 + 10 * q^46 + 6 * q^49 + 14 * q^51 + 2 * q^54 - 8 * q^59 - 86 * q^61 + 14 * q^64 - 12 * q^66 + 20 * q^69 - 54 * q^71 - 10 * q^74 + 18 * q^76 - 20 * q^79 + 8 * q^81 + 10 * q^84 + 48 * q^86 + 18 * q^89 + 10 * q^91 + 44 * q^94 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more