LMFDB - Embedded newform 5625.2.a.j.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5625,2,Mod(1,5625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5625, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5625 = 3^{2} \cdot 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5625.a (trivial)

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:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{15})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 4x + 1 $ x^4 - x^3 - 4x^2 + 4x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1875)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.82709$ of defining polynomial
Character	$\chi$	$=$	5625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.95630 q^{2} +1.82709 q^{4} +4.57433 q^{7} -0.338261 q^{8} +O(q^{10})$	$q+1.95630 q^{2} +1.82709 q^{4} +4.57433 q^{7} -0.338261 q^{8} +4.16535 q^{11} +3.75638 q^{13} +8.94874 q^{14} -4.31592 q^{16} +3.26755 q^{17} -0.243625 q^{19} +8.14866 q^{22} +0.654182 q^{23} +7.34858 q^{26} +8.35772 q^{28} +8.54732 q^{29} -3.90694 q^{31} -7.76669 q^{32} +6.39228 q^{34} -10.6151 q^{37} -0.476602 q^{38} -0.769579 q^{41} -3.90345 q^{43} +7.61048 q^{44} +1.27977 q^{46} +7.67555 q^{47} +13.9245 q^{49} +6.86324 q^{52} -12.1197 q^{53} -1.54732 q^{56} +16.7211 q^{58} -4.90248 q^{59} -14.7035 q^{61} -7.64313 q^{62} -6.56210 q^{64} -0.316897 q^{67} +5.97010 q^{68} -0.446705 q^{71} +11.1172 q^{73} -20.7664 q^{74} -0.445125 q^{76} +19.0537 q^{77} +9.80731 q^{79} -1.50552 q^{82} -8.86482 q^{83} -7.63631 q^{86} -1.40898 q^{88} +7.81040 q^{89} +17.1829 q^{91} +1.19525 q^{92} +15.0156 q^{94} -8.15938 q^{97} +27.2404 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8	$ 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8} + 6 q^{11} + 7 q^{13} + 10 q^{14} - 9 q^{16} + 7 q^{17} - 9 q^{19} + 6 q^{22} - 10 q^{23} + 2 q^{26} + 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} - 10 q^{37} + 6 q^{38} + q^{43} + 9 q^{44} + 5 q^{46} + 23 q^{47} - 3 q^{49} + 13 q^{52} - 2 q^{58} - 4 q^{59} - 43 q^{61} + 10 q^{62} - 7 q^{64} + 8 q^{67} + 3 q^{68} + 27 q^{71} + 15 q^{73} - 5 q^{74} + 9 q^{76} + 15 q^{77} + 10 q^{79} - 20 q^{82} - 3 q^{83} - 24 q^{86} - 3 q^{88} + 9 q^{89} + 5 q^{91} + 15 q^{92} - 22 q^{94} + 13 q^{97} + 42 q^{98}+O(q^{100}) $ 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8 + 6 * q^11 + 7 * q^13 + 10 * q^14 - 9 * q^16 + 7 * q^17 - 9 * q^19 + 6 * q^22 - 10 * q^23 + 2 * q^26 + 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 - 10 * q^37 + 6 * q^38 + q^43 + 9 * q^44 + 5 * q^46 + 23 * q^47 - 3 * q^49 + 13 * q^52 - 2 * q^58 - 4 * q^59 - 43 * q^61 + 10 * q^62 - 7 * q^64 + 8 * q^67 + 3 * q^68 + 27 * q^71 + 15 * q^73 - 5 * q^74 + 9 * q^76 + 15 * q^77 + 10 * q^79 - 20 * q^82 - 3 * q^83 - 24 * q^86 - 3 * q^88 + 9 * q^89 + 5 * q^91 + 15 * q^92 - 22 * q^94 + 13 * q^97 + 42 * q^98

Display 100 coefficients 10 coefficients

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 $

$2$	1.95630	1.38331	0.691655	−	0.722228i	$-0.256882\pi$
$2$	1.95630	1.38331	0.691655	+	0.722228i	$0.256882\pi$
$3$	0	0
$3$	0	0
$4$	1.82709	0.913545
$5$	0	0
$5$	0	0
$6$	0	0
$7$	4.57433	1.72893	0.864467	−	0.502690i	$-0.167656\pi$
$7$	4.57433	1.72893	0.864467	+	0.502690i	$0.167656\pi$
$8$	−0.338261	−0.119593
$9$	0	0
$10$	0	0
$11$	4.16535	1.25590	0.627950	−	0.778253i	$-0.283894\pi$
$11$	4.16535	1.25590	0.627950	+	0.778253i	$0.283894\pi$
$12$	0	0
$13$	3.75638	1.04183	0.520915	−	0.853608i	$-0.325591\pi$
$13$	3.75638	1.04183	0.520915	+	0.853608i	$0.325591\pi$
$14$	8.94874	2.39165
$15$	0	0
$16$	−4.31592	−1.07898
$17$	3.26755	0.792496	0.396248	−	0.918143i	$-0.370312\pi$
$17$	3.26755	0.792496	0.396248	+	0.918143i	$0.370312\pi$
$18$	0	0
$19$	−0.243625	−0.0558914	−0.0279457	−	0.999609i	$-0.508897\pi$
$19$	−0.243625	−0.0558914	−0.0279457	+	0.999609i	$0.508897\pi$
$20$	0	0
$21$	0	0
$22$	8.14866	1.73730
$23$	0.654182	0.136406	0.0682032	−	0.997671i	$-0.478273\pi$
$23$	0.654182	0.136406	0.0682032	+	0.997671i	$0.478273\pi$
$24$	0	0
$25$	0	0
$26$	7.34858	1.44117
$27$	0	0
$28$	8.35772	1.57946
$29$	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	$0.614109\pi$
$32$	−7.76669	−1.37297
$33$	0	0
$34$	6.39228	1.09627
$35$	0	0
$36$	0	0
$37$	−10.6151	−1.74512	−0.872560	−	0.488507i	$-0.837542\pi$
$37$	−10.6151	−1.74512	−0.872560	+	0.488507i	$0.837542\pi$
$38$	−0.476602	−0.0773151
$39$	0	0
$40$	0	0
$41$	−0.769579	−0.120188	−0.0600940	−	0.998193i	$-0.519140\pi$
$41$	−0.769579	−0.120188	−0.0600940	+	0.998193i	$0.519140\pi$
$42$	0	0
$43$	−3.90345	−0.595271	−0.297636	−	0.954680i	$-0.596198\pi$
$43$	−3.90345	−0.595271	−0.297636	+	0.954680i	$0.596198\pi$
$44$	7.61048	1.14732
$45$	0	0
$46$	1.27977	0.188692
$47$	7.67555	1.11959	0.559797	−	0.828630i	$-0.310879\pi$
$47$	7.67555	1.11959	0.559797	+	0.828630i	$0.310879\pi$
$48$	0	0
$49$	13.9245	1.98921
$50$	0	0
$51$	0	0
$52$	6.86324	0.951760
$53$	−12.1197	−1.66477	−0.832387	−	0.554195i	$-0.813026\pi$
$53$	−12.1197	−1.66477	−0.832387	+	0.554195i	$0.813026\pi$
$54$	0	0
$55$	0	0
$56$	−1.54732	−0.206769
$57$	0	0
$58$	16.7211	2.19559
$59$	−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	$0.890392\pi$
$62$	−7.64313	−0.970679
$63$	0	0
$64$	−6.56210	−0.820263
$65$	0	0
$66$	0	0
$67$	−0.316897	−0.0387151	−0.0193576	−	0.999813i	$-0.506162\pi$
$67$	−0.316897	−0.0387151	−0.0193576	+	0.999813i	$0.506162\pi$
$68$	5.97010	0.723981
$69$	0	0
$70$	0	0
$71$	−0.446705	−0.0530141	−0.0265070	−	0.999649i	$-0.508438\pi$
$71$	−0.446705	−0.0530141	−0.0265070	+	0.999649i	$0.508438\pi$
$72$	0	0
$73$	11.1172	1.30117	0.650584	−	0.759434i	$-0.274524\pi$
$73$	11.1172	1.30117	0.650584	+	0.759434i	$0.274524\pi$
$74$	−20.7664	−2.41404
$75$	0	0
$76$	−0.445125	−0.0510593
$77$	19.0537	2.17137
$78$	0	0
$79$	9.80731	1.10341	0.551704	−	0.834040i	$-0.313978\pi$
$79$	9.80731	1.10341	0.551704	+	0.834040i	$0.313978\pi$
$80$	0	0
$81$	0	0
$82$	−1.50552	−0.166257
$83$	−8.86482	−0.973040	−0.486520	−	0.873669i	$-0.661734\pi$
$83$	−8.86482	−0.973040	−0.486520	+	0.873669i	$0.661734\pi$
$84$	0	0
$85$	0	0
$86$	−7.63631	−0.823444
$87$	0	0
$88$	−1.40898	−0.150197
$89$	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
$92$	1.19525	0.124613
$93$	0	0
$94$	15.0156	1.54874
$95$	0	0
$96$	0	0
$97$	−8.15938	−0.828459	−0.414230	−	0.910172i	$-0.635949\pi$
$97$	−8.15938	−0.828459	−0.414230	+	0.910172i	$0.635949\pi$
$98$	27.2404	2.75170
$99$	0	0
$100$	0	0
$101$	9.54383	0.949646	0.474823	−	0.880081i	$-0.342512\pi$
$101$	9.54383	0.949646	0.474823	+	0.880081i	$0.342512\pi$
$102$	0	0
$103$	−9.86698	−0.972222	−0.486111	−	0.873897i	$-0.661585\pi$
$103$	−9.86698	−0.972222	−0.486111	+	0.873897i	$0.661585\pi$
$104$	−1.27064	−0.124596
$105$	0	0
$106$	−23.7098	−2.30290
$107$	16.1273	1.55908	0.779542	−	0.626350i	$-0.215452\pi$
$107$	16.1273	1.55908	0.779542	+	0.626350i	$0.215452\pi$
$108$	0	0
$109$	−2.44236	−0.233936	−0.116968	−	0.993136i	$-0.537318\pi$
$109$	−2.44236	−0.233936	−0.116968	+	0.993136i	$0.537318\pi$
$110$	0	0
$111$	0	0
$112$	−19.7424	−1.86549
$113$	10.9293	1.02814	0.514070	−	0.857748i	$-0.328137\pi$
$113$	10.9293	1.02814	0.514070	+	0.857748i	$0.328137\pi$
$114$	0	0
$115$	0	0
$116$	15.6167	1.44998
$117$	0	0
$118$	−9.59069	−0.882895
$119$	14.9468	1.37017
$120$	0	0
$121$	6.35016	0.577287
$122$	−28.7645	−2.60421
$123$	0	0
$124$	−7.13834	−0.641042
$125$	0	0
$126$	0	0
$127$	12.8613	1.14126	0.570629	−	0.821208i	$-0.306699\pi$
$127$	12.8613	1.14126	0.570629	+	0.821208i	$0.306699\pi$
$128$	2.69598	0.238293
$129$	0	0
$130$	0	0
$131$	−2.37441	−0.207453	−0.103727	−	0.994606i	$-0.533077\pi$
$131$	−2.37441	−0.207453	−0.103727	+	0.994606i	$0.533077\pi$
$132$	0	0
$133$	−1.11442	−0.0966325
$134$	−0.619944	−0.0535550
$135$	0	0
$136$	−1.10528	−0.0947773
$137$	−12.5026	−1.06817	−0.534086	−	0.845430i	$-0.679344\pi$
$137$	−12.5026	−1.06817	−0.534086	+	0.845430i	$0.679344\pi$
$138$	0	0
$139$	−0.356135	−0.0302070	−0.0151035	−	0.999886i	$-0.504808\pi$
$139$	−0.356135	−0.0302070	−0.0151035	+	0.999886i	$0.504808\pi$
$140$	0	0
$141$	0	0
$142$	−0.873886	−0.0733349
$143$	15.6466	1.30844
$144$	0	0
$145$	0	0
$146$	21.7485	1.79992
$147$	0	0
$148$	−19.3948	−1.59425
$149$	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	$0.573915\pi$
$152$	0.0824089	0.00668424
$153$	0	0
$154$	37.2746	3.00368
$155$	0	0
$156$	0	0
$157$	6.69480	0.534303	0.267151	−	0.963655i	$-0.413918\pi$
$157$	6.69480	0.534303	0.267151	+	0.963655i	$0.413918\pi$
$158$	19.1860	1.52636
$159$	0	0
$160$	0	0
$161$	2.99244	0.235838
$162$	0	0
$163$	−13.8314	−1.08336	−0.541681	−	0.840584i	$-0.682212\pi$
$163$	−13.8314	−1.08336	−0.541681	+	0.840584i	$0.682212\pi$
$164$	−1.40609	−0.109797
$165$	0	0
$166$	−17.3422	−1.34602
$167$	13.5017	1.04479	0.522397	−	0.852703i	$-0.325038\pi$
$167$	13.5017	1.04479	0.522397	+	0.852703i	$0.325038\pi$
$168$	0	0
$169$	1.11035	0.0854118
$170$	0	0
$171$	0	0
$172$	−7.13196	−0.543807
$173$	−13.7259	−1.04356	−0.521779	−	0.853080i	$-0.674732\pi$
$173$	−13.7259	−1.04356	−0.521779	+	0.853080i	$0.674732\pi$
$174$	0	0
$175$	0	0
$176$	−17.9773	−1.35509
$177$	0	0
$178$	15.2794	1.14524
$179$	−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	$0.674226\pi$
$182$	33.6148	2.49170
$183$	0	0
$184$	−0.221284	−0.0163133
$185$	0	0
$186$	0	0
$187$	13.6105	0.995297
$188$	14.0239	1.02280
$189$	0	0
$190$	0	0
$191$	4.29614	0.310858	0.155429	−	0.987847i	$-0.450324\pi$
$191$	4.29614	0.310858	0.155429	+	0.987847i	$0.450324\pi$
$192$	0	0
$193$	13.0436	0.938897	0.469449	−	0.882960i	$-0.344453\pi$
$193$	13.0436	0.938897	0.469449	+	0.882960i	$0.344453\pi$
$194$	−15.9621	−1.14602
$195$	0	0
$196$	25.4413	1.81724
$197$	9.17856	0.653945	0.326973	−	0.945034i	$-0.393972\pi$
$197$	9.17856	0.653945	0.326973	+	0.945034i	$0.393972\pi$
$198$	0	0
$199$	0.677499	0.0480266	0.0240133	−	0.999712i	$-0.492356\pi$
$199$	0.677499	0.0480266	0.0240133	+	0.999712i	$0.492356\pi$
$200$	0	0
$201$	0	0
$202$	18.6705	1.31365
$203$	39.0982	2.74416
$204$	0	0
$205$	0	0
$206$	−19.3027	−1.34488
$207$	0	0
$208$	−16.2122	−1.12411
$209$	−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	$0.687842\pi$
$212$	−22.1439	−1.52085
$213$	0	0
$214$	31.5497	2.15670
$215$	0	0
$216$	0	0
$217$	−17.8716	−1.21321
$218$	−4.77799	−0.323606
$219$	0	0
$220$	0	0
$221$	12.2741	0.825647
$222$	0	0
$223$	−9.96064	−0.667013	−0.333507	−	0.942748i	$-0.608232\pi$
$223$	−9.96064	−0.667013	−0.333507	+	0.942748i	$0.608232\pi$
$224$	−35.5274	−2.37377
$225$	0	0
$226$	21.3809	1.42224
$227$	−24.0646	−1.59722	−0.798612	−	0.601846i	$-0.794432\pi$
$227$	−24.0646	−1.59722	−0.798612	+	0.601846i	$0.794432\pi$
$228$	0	0
$229$	−15.3847	−1.01665	−0.508326	−	0.861165i	$-0.669735\pi$
$229$	−15.3847	−1.01665	−0.508326	+	0.861165i	$0.669735\pi$
$230$	0	0
$231$	0	0
$232$	−2.89123	−0.189818
$233$	−6.58907	−0.431664	−0.215832	−	0.976430i	$-0.569246\pi$
$233$	−6.58907	−0.431664	−0.215832	+	0.976430i	$0.569246\pi$
$234$	0	0
$235$	0	0
$236$	−8.95727	−0.583069
$237$	0	0
$238$	29.2404	1.89537
$239$	−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	$0.435755\pi$
$242$	12.4228	0.798567
$243$	0	0
$244$	−26.8647	−1.71984
$245$	0	0
$246$	0	0
$247$	−0.915147	−0.0582294
$248$	1.32157	0.0839196
$249$	0	0
$250$	0	0
$251$	7.11580	0.449145	0.224573	−	0.974457i	$-0.427901\pi$
$251$	7.11580	0.449145	0.224573	+	0.974457i	$0.427901\pi$
$252$	0	0
$253$	2.72490	0.171313
$254$	25.1606	1.57871
$255$	0	0
$256$	18.3983	1.14990
$257$	−7.15589	−0.446372	−0.223186	−	0.974776i	$-0.571646\pi$
$257$	−7.15589	−0.446372	−0.223186	+	0.974776i	$0.571646\pi$
$258$	0	0
$259$	−48.5572	−3.01720
$260$	0	0
$261$	0	0
$262$	−4.64505	−0.286972
$263$	1.30967	0.0807577	0.0403789	−	0.999184i	$-0.487144\pi$
$263$	1.30967	0.0807577	0.0403789	+	0.999184i	$0.487144\pi$
$264$	0	0
$265$	0	0
$266$	−2.18014	−0.133673
$267$	0	0
$268$	−0.579000	−0.0353680
$269$	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	$0.405288\pi$
$272$	−14.1025	−0.855088
$273$	0	0
$274$	−24.4588	−1.47761
$275$	0	0
$276$	0	0
$277$	20.0455	1.20442	0.602210	−	0.798338i	$-0.294287\pi$
$277$	20.0455	1.20442	0.602210	+	0.798338i	$0.294287\pi$
$278$	−0.696706	−0.0417857
$279$	0	0
$280$	0	0
$281$	4.42685	0.264084	0.132042	−	0.991244i	$-0.457847\pi$
$281$	4.42685	0.264084	0.132042	+	0.991244i	$0.457847\pi$
$282$	0	0
$283$	11.2028	0.665938	0.332969	−	0.942938i	$-0.391950\pi$
$283$	11.2028	0.665938	0.332969	+	0.942938i	$0.391950\pi$
$284$	−0.816170	−0.0484308
$285$	0	0
$286$	30.6094	1.80997
$287$	−3.52031	−0.207797
$288$	0	0
$289$	−6.32315	−0.371950
$290$	0	0
$291$	0	0
$292$	20.3121	1.18868
$293$	26.6504	1.55693	0.778465	−	0.627688i	$-0.215998\pi$
$293$	26.6504	1.55693	0.778465	+	0.627688i	$0.215998\pi$
$294$	0	0
$295$	0	0
$296$	3.59069	0.208705
$297$	0	0
$298$	29.4064	1.70347
$299$	2.45735	0.142112
$300$	0	0
$301$	−17.8557	−1.02918
$302$	−11.0643	−0.636681
$303$	0	0
$304$	1.05147	0.0603057
$305$	0	0
$306$	0	0
$307$	−9.48870	−0.541549	−0.270774	−	0.962643i	$-0.587280\pi$
$307$	−9.48870	−0.541549	−0.270774	+	0.962643i	$0.587280\pi$
$308$	34.8128	1.98364
$309$	0	0
$310$	0	0
$311$	−11.5152	−0.652969	−0.326485	−	0.945203i	$-0.605864\pi$
$311$	−11.5152	−0.652969	−0.326485	+	0.945203i	$0.605864\pi$
$312$	0	0
$313$	−2.14459	−0.121219	−0.0606097	−	0.998162i	$-0.519305\pi$
$313$	−2.14459	−0.121219	−0.0606097	+	0.998162i	$0.519305\pi$
$314$	13.0970	0.739106
$315$	0	0
$316$	17.9188	1.00801
$317$	0.735614	0.0413162	0.0206581	−	0.999787i	$-0.493424\pi$
$317$	0.735614	0.0413162	0.0206581	+	0.999787i	$0.493424\pi$
$318$	0	0
$319$	35.6026	1.99336
$320$	0	0
$321$	0	0
$322$	5.85410	0.326236
$323$	−0.796056	−0.0442937
$324$	0	0
$325$	0	0
$326$	−27.0584	−1.49862
$327$	0	0
$328$	0.260319	0.0143737
$329$	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	$0.401162\pi$
$332$	−16.1968	−0.888917
$333$	0	0
$334$	26.4133	1.44527
$335$	0	0
$336$	0	0
$337$	−14.2506	−0.776282	−0.388141	−	0.921600i	$-0.626883\pi$
$337$	−14.2506	−0.776282	−0.388141	+	0.921600i	$0.626883\pi$
$338$	2.17218	0.118151
$339$	0	0
$340$	0	0
$341$	−16.2738	−0.881275
$342$	0	0
$343$	31.6749	1.71028
$344$	1.32039	0.0711905
$345$	0	0
$346$	−26.8519	−1.44357
$347$	0.174615	0.00937383	0.00468691	−	0.999989i	$-0.498508\pi$
$347$	0.174615	0.00937383	0.00468691	+	0.999989i	$0.498508\pi$
$348$	0	0
$349$	−18.6640	−0.999059	−0.499530	−	0.866297i	$-0.666494\pi$
$349$	−18.6640	−0.999059	−0.499530	+	0.866297i	$0.666494\pi$
$350$	0	0
$351$	0	0
$352$	−32.3510	−1.72431
$353$	8.67239	0.461585	0.230792	−	0.973003i	$-0.425868\pi$
$353$	8.67239	0.461585	0.230792	+	0.973003i	$0.425868\pi$
$354$	0	0
$355$	0	0
$356$	14.2703	0.756325
$357$	0	0
$358$	−8.73468	−0.461642
$359$	−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
$362$	−27.3943	−1.43981
$363$	0	0
$364$	31.3947	1.64553
$365$	0	0
$366$	0	0
$367$	0.530351	0.0276841	0.0138421	−	0.999904i	$-0.495594\pi$
$367$	0.530351	0.0276841	0.0138421	+	0.999904i	$0.495594\pi$
$368$	−2.82340	−0.147180
$369$	0	0
$370$	0	0
$371$	−55.4397	−2.87828
$372$	0	0
$373$	29.3489	1.51963	0.759813	−	0.650142i	$-0.225290\pi$
$373$	29.3489	1.51963	0.759813	+	0.650142i	$0.225290\pi$
$374$	26.6261	1.37680
$375$	0	0
$376$	−2.59634	−0.133896
$377$	32.1069	1.65359
$378$	0	0
$379$	4.84634	0.248940	0.124470	−	0.992223i	$-0.460277\pi$
$379$	4.84634	0.248940	0.124470	+	0.992223i	$0.460277\pi$
$380$	0	0
$381$	0	0
$382$	8.40451	0.430012
$383$	21.3587	1.09138	0.545689	−	0.837988i	$-0.316268\pi$
$383$	21.3587	1.09138	0.545689	+	0.837988i	$0.316268\pi$
$384$	0	0
$385$	0	0
$386$	25.5171	1.29879
$387$	0	0
$388$	−14.9079	−0.756835
$389$	−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
$392$	−4.71011	−0.237897
$393$	0	0
$394$	17.9560	0.904608
$395$	0	0
$396$	0	0
$397$	−16.3146	−0.818806	−0.409403	−	0.912354i	$-0.634263\pi$
$397$	−16.3146	−0.818806	−0.409403	+	0.912354i	$0.634263\pi$
$398$	1.32539	0.0664357
$399$	0	0
$400$	0	0
$401$	−10.3272	−0.515718	−0.257859	−	0.966183i	$-0.583017\pi$
$401$	−10.3272	−0.515718	−0.257859	+	0.966183i	$0.583017\pi$
$402$	0	0
$403$	−14.6759	−0.731061
$404$	17.4374	0.867545
$405$	0	0
$406$	76.4877	3.79602
$407$	−44.2158	−2.19170
$408$	0	0
$409$	26.8421	1.32726	0.663629	−	0.748062i	$-0.269015\pi$
$409$	26.8421	1.32726	0.663629	+	0.748062i	$0.269015\pi$
$410$	0	0
$411$	0	0
$412$	−18.0279	−0.888169
$413$	−22.4255	−1.10349
$414$	0	0
$415$	0	0
$416$	−29.1746	−1.43040
$417$	0	0
$418$	−1.98522	−0.0971001
$419$	−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	$0.691283\pi$
$422$	−31.6259	−1.53952
$423$	0	0
$424$	4.09964	0.199096
$425$	0	0
$426$	0	0
$427$	−67.2588	−3.25488
$428$	29.4660	1.42429
$429$	0	0
$430$	0	0
$431$	39.7692	1.91561	0.957807	−	0.287413i	$-0.0927953\pi$
$431$	39.7692	1.91561	0.957807	+	0.287413i	$0.0927953\pi$
$432$	0	0
$433$	−18.3272	−0.880750	−0.440375	−	0.897814i	$-0.645154\pi$
$433$	−18.3272	−0.880750	−0.440375	+	0.897814i	$0.645154\pi$
$434$	−34.9622	−1.67824
$435$	0	0
$436$	−4.46242	−0.213711
$437$	−0.159375	−0.00762394
$438$	0	0
$439$	−37.2170	−1.77627	−0.888135	−	0.459582i	$-0.847999\pi$
$439$	−37.2170	−1.77627	−0.888135	+	0.459582i	$0.847999\pi$
$440$	0	0
$441$	0	0
$442$	24.0118	1.14213
$443$	−8.41969	−0.400032	−0.200016	−	0.979793i	$-0.564099\pi$
$443$	−8.41969	−0.400032	−0.200016	+	0.979793i	$0.564099\pi$
$444$	0	0
$445$	0	0
$446$	−19.4859	−0.922686
$447$	0	0
$448$	−30.0172	−1.41818
$449$	−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
$452$	19.9688	0.939253
$453$	0	0
$454$	−47.0775	−2.20946
$455$	0	0
$456$	0	0
$457$	−2.57124	−0.120277	−0.0601387	−	0.998190i	$-0.519154\pi$
$457$	−2.57124	−0.120277	−0.0601387	+	0.998190i	$0.519154\pi$
$458$	−30.0971	−1.40634
$459$	0	0
$460$	0	0
$461$	−34.8861	−1.62481	−0.812404	−	0.583095i	$-0.801841\pi$
$461$	−34.8861	−1.62481	−0.812404	+	0.583095i	$0.801841\pi$
$462$	0	0
$463$	8.24992	0.383406	0.191703	−	0.981453i	$-0.438599\pi$
$463$	8.24992	0.383406	0.191703	+	0.981453i	$0.438599\pi$
$464$	−36.8895	−1.71255
$465$	0	0
$466$	−12.8902	−0.597125
$467$	24.4499	1.13140	0.565702	−	0.824609i	$-0.308605\pi$
$467$	24.4499	1.13140	0.565702	+	0.824609i	$0.308605\pi$
$468$	0	0
$469$	−1.44959	−0.0669359
$470$	0	0
$471$	0	0
$472$	1.65832	0.0763303
$473$	−16.2593	−0.747602
$474$	0	0
$475$	0	0
$476$	27.3092	1.25172
$477$	0	0
$478$	−34.9930	−1.60054
$479$	−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
$482$	12.1761	0.554605
$483$	0	0
$484$	11.6023	0.527378
$485$	0	0
$486$	0	0
$487$	20.2715	0.918589	0.459294	−	0.888284i	$-0.348102\pi$
$487$	20.2715	0.918589	0.459294	+	0.888284i	$0.348102\pi$
$488$	4.97364	0.225146
$489$	0	0
$490$	0	0
$491$	−36.2638	−1.63656	−0.818281	−	0.574818i	$-0.805073\pi$
$491$	−36.2638	−1.63656	−0.818281	+	0.574818i	$0.805073\pi$
$492$	0	0
$493$	27.9287	1.25785
$494$	−1.79030	−0.0805493
$495$	0	0
$496$	16.8621	0.757129
$497$	−2.04337	−0.0916579
$498$	0	0
$499$	2.39550	0.107237	0.0536187	−	0.998561i	$-0.482924\pi$
$499$	2.39550	0.107237	0.0536187	+	0.998561i	$0.482924\pi$
$500$	0	0
$501$	0	0
$502$	13.9206	0.621307
$503$	−22.7079	−1.01249	−0.506247	−	0.862389i	$-0.668968\pi$
$503$	−22.7079	−1.01249	−0.506247	+	0.862389i	$0.668968\pi$
$504$	0	0
$505$	0	0
$506$	5.33070	0.236979
$507$	0	0
$508$	23.4988	1.04259
$509$	−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
$512$	30.6006	1.35237
$513$	0	0
$514$	−13.9990	−0.617470
$515$	0	0
$516$	0	0
$517$	31.9714	1.40610
$518$	−94.9922	−4.17372
$519$	0	0
$520$	0	0
$521$	5.56462	0.243790	0.121895	−	0.992543i	$-0.461103\pi$
$521$	5.56462	0.243790	0.121895	+	0.992543i	$0.461103\pi$
$522$	0	0
$523$	22.6133	0.988811	0.494405	−	0.869231i	$-0.335386\pi$
$523$	22.6133	0.988811	0.494405	+	0.869231i	$0.335386\pi$
$524$	−4.33826	−0.189518
$525$	0	0
$526$	2.56210	0.111713
$527$	−12.7661	−0.556101
$528$	0	0
$529$	−22.5720	−0.981393
$530$	0	0
$531$	0	0
$532$	−2.03615	−0.0882782
$533$	−2.89083	−0.125216
$534$	0	0
$535$	0	0
$536$	0.107194	0.00463007
$537$	0	0
$538$	41.0323	1.76903
$539$	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	$0.586868\pi$
$542$	18.8833	0.811109
$543$	0	0
$544$	−25.3780	−1.08807
$545$	0	0
$546$	0	0
$547$	−5.45889	−0.233405	−0.116703	−	0.993167i	$-0.537232\pi$
$547$	−5.45889	−0.233405	−0.116703	+	0.993167i	$0.537232\pi$
$548$	−22.8435	−0.975824
$549$	0	0
$550$	0	0
$551$	−2.08234	−0.0887107
$552$	0	0
$553$	44.8618	1.90772
$554$	39.2150	1.66608
$555$	0	0
$556$	−0.650692	−0.0275955
$557$	4.84648	0.205352	0.102676	−	0.994715i	$-0.467260\pi$
$557$	4.84648	0.205352	0.102676	+	0.994715i	$0.467260\pi$
$558$	0	0
$559$	−14.6628	−0.620172
$560$	0	0
$561$	0	0
$562$	8.66023	0.365310
$563$	−37.5755	−1.58362	−0.791809	−	0.610769i	$-0.790861\pi$
$563$	−37.5755	−1.58362	−0.791809	+	0.610769i	$0.790861\pi$
$564$	0	0
$565$	0	0
$566$	21.9160	0.921198
$567$	0	0
$568$	0.151103	0.00634014
$569$	−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	$0.704405\pi$
$572$	28.5878	1.19532
$573$	0	0
$574$	−6.88676	−0.287448
$575$	0	0
$576$	0	0
$577$	2.22388	0.0925815	0.0462907	−	0.998928i	$-0.485260\pi$
$577$	2.22388	0.0925815	0.0462907	+	0.998928i	$0.485260\pi$
$578$	−12.3699	−0.514522
$579$	0	0
$580$	0	0
$581$	−40.5506	−1.68232
$582$	0	0
$583$	−50.4830	−2.09079
$584$	−3.76051	−0.155611
$585$	0	0
$586$	52.1360	2.15372
$587$	29.8085	1.23033	0.615164	−	0.788399i	$-0.289090\pi$
$587$	29.8085	1.23033	0.615164	+	0.788399i	$0.289090\pi$
$588$	0	0
$589$	0.951829	0.0392194
$590$	0	0
$591$	0	0
$592$	45.8141	1.88295
$593$	43.9251	1.80379	0.901895	−	0.431956i	$-0.142177\pi$
$593$	43.9251	1.80379	0.901895	+	0.431956i	$0.142177\pi$
$594$	0	0
$595$	0	0
$596$	27.4642	1.12498
$597$	0	0
$598$	4.80731	0.196585
$599$	−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	$0.220429\pi$
$602$	−34.9310	−1.42368
$603$	0	0
$604$	−10.3336	−0.420468
$605$	0	0
$606$	0	0
$607$	45.6549	1.85308	0.926538	−	0.376202i	$-0.122770\pi$
$607$	45.6549	1.85308	0.926538	+	0.376202i	$0.122770\pi$
$608$	1.89216	0.0767372
$609$	0	0
$610$	0	0
$611$	28.8322	1.16643
$612$	0	0
$613$	−39.0685	−1.57796	−0.788980	−	0.614418i	$-0.789391\pi$
$613$	−39.0685	−1.57796	−0.788980	+	0.614418i	$0.789391\pi$
$614$	−18.5627	−0.749130
$615$	0	0
$616$	−6.44512	−0.259681
$617$	27.4125	1.10359	0.551794	−	0.833981i	$-0.313944\pi$
$617$	27.4125	1.10359	0.551794	+	0.833981i	$0.313944\pi$
$618$	0	0
$619$	−22.8242	−0.917384	−0.458692	−	0.888595i	$-0.651682\pi$
$619$	−22.8242	−0.917384	−0.458692	+	0.888595i	$0.651682\pi$
$620$	0	0
$621$	0	0
$622$	−22.5272	−0.903259
$623$	35.7273	1.43139
$624$	0	0
$625$	0	0
$626$	−4.19545	−0.167684
$627$	0	0
$628$	12.2320	0.488110
$629$	−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	$0.435803\pi$
$632$	−3.31743	−0.131960
$633$	0	0
$634$	1.43908	0.0571531
$635$	0	0
$636$	0	0
$637$	52.3056	2.07242
$638$	69.6492	2.75744
$639$	0	0
$640$	0	0
$641$	42.7831	1.68983	0.844916	−	0.534899i	$-0.179650\pi$
$641$	42.7831	1.68983	0.844916	+	0.534899i	$0.179650\pi$
$642$	0	0
$643$	10.1067	0.398571	0.199285	−	0.979941i	$-0.436138\pi$
$643$	10.1067	0.398571	0.199285	+	0.979941i	$0.436138\pi$
$644$	5.46747	0.215448
$645$	0	0
$646$	−1.55732	−0.0612719
$647$	16.7580	0.658826	0.329413	−	0.944186i	$-0.393149\pi$
$647$	16.7580	0.658826	0.329413	+	0.944186i	$0.393149\pi$
$648$	0	0
$649$	−20.4205	−0.801576
$650$	0	0
$651$	0	0
$652$	−25.2713	−0.989700
$653$	−20.7798	−0.813175	−0.406588	−	0.913612i	$-0.633281\pi$
$653$	−20.7798	−0.813175	−0.406588	+	0.913612i	$0.633281\pi$
$654$	0	0
$655$	0	0
$656$	3.32144	0.129680
$657$	0	0
$658$	68.6865	2.67768
$659$	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	$0.479103\pi$
$662$	21.7497	0.845325
$663$	0	0
$664$	2.99862	0.116369
$665$	0	0
$666$	0	0
$667$	5.59150	0.216504
$668$	24.6688	0.954466
$669$	0	0
$670$	0	0
$671$	−61.2454	−2.36435
$672$	0	0
$673$	−35.2852	−1.36015	−0.680073	−	0.733144i	$-0.738052\pi$
$673$	−35.2852	−1.36015	−0.680073	+	0.733144i	$0.738052\pi$
$674$	−27.8785	−1.07384
$675$	0	0
$676$	2.02872	0.0780276
$677$	31.2943	1.20274	0.601369	−	0.798971i	$-0.294622\pi$
$677$	31.2943	1.20274	0.601369	+	0.798971i	$0.294622\pi$
$678$	0	0
$679$	−37.3237	−1.43235
$680$	0	0
$681$	0	0
$682$	−31.8363	−1.21908
$683$	−10.8131	−0.413751	−0.206876	−	0.978367i	$-0.566330\pi$
$683$	−10.8131	−0.413751	−0.206876	+	0.978367i	$0.566330\pi$
$684$	0	0
$685$	0	0
$686$	61.9654	2.36585
$687$	0	0
$688$	16.8470	0.642286
$689$	−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	$0.921471\pi$
$692$	−25.0784	−0.953338
$693$	0	0
$694$	0.341599	0.0129669
$695$	0	0
$696$	0	0
$697$	−2.51463	−0.0952485
$698$	−36.5122	−1.38201
$699$	0	0
$700$	0	0
$701$	−5.20728	−0.196676	−0.0983382	−	0.995153i	$-0.531353\pi$
$701$	−5.20728	−0.196676	−0.0983382	+	0.995153i	$0.531353\pi$
$702$	0	0
$703$	2.58611	0.0975372
$704$	−27.3335	−1.03017
$705$	0	0
$706$	16.9657	0.638514
$707$	43.6566	1.64188
$708$	0	0
$709$	−37.4108	−1.40499	−0.702497	−	0.711687i	$-0.747931\pi$
$709$	−37.4108	−1.40499	−0.702497	+	0.711687i	$0.747931\pi$
$710$	0	0
$711$	0	0
$712$	−2.64195	−0.0990114
$713$	−2.55585	−0.0957174
$714$	0	0
$715$	0	0
$716$	−8.15780	−0.304871
$717$	0	0
$718$	−26.8364	−1.00153
$719$	−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
$722$	−37.0535	−1.37899
$723$	0	0
$724$	−25.5850	−0.950861
$725$	0	0
$726$	0	0
$727$	1.21933	0.0452225	0.0226113	−	0.999744i	$-0.492802\pi$
$727$	1.21933	0.0452225	0.0226113	+	0.999744i	$0.492802\pi$
$728$	−5.81231	−0.215418
$729$	0	0
$730$	0	0
$731$	−12.7547	−0.471750
$732$	0	0
$733$	−13.3535	−0.493221	−0.246611	−	0.969115i	$-0.579317\pi$
$733$	−13.3535	−0.493221	−0.246611	+	0.969115i	$0.579317\pi$
$734$	1.03752	0.0382957
$735$	0	0
$736$	−5.08083	−0.187282
$737$	−1.31999	−0.0486224
$738$	0	0
$739$	−5.24691	−0.193011	−0.0965054	−	0.995332i	$-0.530766\pi$
$739$	−5.24691	−0.193011	−0.0965054	+	0.995332i	$0.530766\pi$
$740$	0	0
$741$	0	0
$742$	−108.456	−3.98156
$743$	−42.9203	−1.57459	−0.787296	−	0.616575i	$-0.788520\pi$
$743$	−42.9203	−1.57459	−0.787296	+	0.616575i	$0.788520\pi$
$744$	0	0
$745$	0	0
$746$	57.4150	2.10211
$747$	0	0
$748$	24.8676	0.909249
$749$	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	$0.620435\pi$
$752$	−33.1270	−1.20802
$753$	0	0
$754$	62.8106	2.28743
$755$	0	0
$756$	0	0
$757$	10.3260	0.375306	0.187653	−	0.982235i	$-0.439912\pi$
$757$	10.3260	0.375306	0.187653	+	0.982235i	$0.439912\pi$
$758$	9.48087	0.344361
$759$	0	0
$760$	0	0
$761$	−29.2923	−1.06185	−0.530923	−	0.847420i	$-0.678155\pi$
$761$	−29.2923	−1.06185	−0.530923	+	0.847420i	$0.678155\pi$
$762$	0	0
$763$	−11.1722	−0.404460
$764$	7.84943	0.283982
$765$	0	0
$766$	41.7839	1.50971
$767$	−18.4155	−0.664947
$768$	0	0
$769$	−22.9834	−0.828803	−0.414402	−	0.910094i	$-0.636009\pi$
$769$	−22.9834	−0.828803	−0.414402	+	0.910094i	$0.636009\pi$
$770$	0	0
$771$	0	0
$772$	23.8318	0.857725
$773$	54.2216	1.95022	0.975108	−	0.221732i	$-0.0711709\pi$
$773$	54.2216	1.95022	0.975108	+	0.221732i	$0.0711709\pi$
$774$	0	0
$775$	0	0
$776$	2.76000	0.0990782
$777$	0	0
$778$	−20.4099	−0.731731
$779$	0.187489	0.00671748
$780$	0	0
$781$	−1.86068	−0.0665805
$782$	4.18172	0.149538
$783$	0	0
$784$	−60.0970	−2.14632
$785$	0	0
$786$	0	0
$787$	6.60711	0.235518	0.117759	−	0.993042i	$-0.462429\pi$
$787$	6.60711	0.235518	0.117759	+	0.993042i	$0.462429\pi$
$788$	16.7701	0.597409
$789$	0	0
$790$	0	0
$791$	49.9941	1.77759
$792$	0	0
$793$	−55.2320	−1.96135
$794$	−31.9161	−1.13266
$795$	0	0
$796$	1.23785	0.0438745
$797$	2.34163	0.0829446	0.0414723	−	0.999140i	$-0.486795\pi$
$797$	2.34163	0.0829446	0.0414723	+	0.999140i	$0.486795\pi$
$798$	0	0
$799$	25.0802	0.887274
$800$	0	0
$801$	0	0
$802$	−20.2031	−0.713397
$803$	46.3070	1.63414
$804$	0	0
$805$	0	0
$806$	−28.7105	−1.01128
$807$	0	0
$808$	−3.22831	−0.113571
$809$	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	$0.673875\pi$
$812$	71.4361	2.50691
$813$	0	0
$814$	−86.4992	−3.03180
$815$	0	0
$816$	0	0
$817$	0.950979	0.0332705
$818$	52.5112	1.83601
$819$	0	0
$820$	0	0
$821$	−33.3855	−1.16516	−0.582580	−	0.812773i	$-0.697957\pi$
$821$	−33.3855	−1.16516	−0.582580	+	0.812773i	$0.697957\pi$
$822$	0	0
$823$	−17.5383	−0.611345	−0.305672	−	0.952137i	$-0.598881\pi$
$823$	−17.5383	−0.611345	−0.305672	+	0.952137i	$0.598881\pi$
$824$	3.33762	0.116271
$825$	0	0
$826$	−43.8710	−1.52647
$827$	−26.6114	−0.925369	−0.462684	−	0.886523i	$-0.653114\pi$
$827$	−26.6114	−0.925369	−0.462684	+	0.886523i	$0.653114\pi$
$828$	0	0
$829$	20.3212	0.705784	0.352892	−	0.935664i	$-0.385198\pi$
$829$	20.3212	0.705784	0.352892	+	0.935664i	$0.385198\pi$
$830$	0	0
$831$	0	0
$832$	−24.6497	−0.854575
$833$	45.4989	1.57644
$834$	0	0
$835$	0	0
$836$	−1.85410	−0.0641255
$837$	0	0
$838$	−9.87698	−0.341195
$839$	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
$842$	−45.3910	−1.56428
$843$	0	0
$844$	−29.5371	−1.01671
$845$	0	0
$846$	0	0
$847$	29.0477	0.998091
$848$	52.3078	1.79626
$849$	0	0
$850$	0	0
$851$	−6.94424	−0.238045
$852$	0	0
$853$	−9.81934	−0.336208	−0.168104	−	0.985769i	$-0.553764\pi$
$853$	−9.81934	−0.336208	−0.168104	+	0.985769i	$0.553764\pi$
$854$	−131.578	−4.50251
$855$	0	0
$856$	−5.45524	−0.186456
$857$	15.9866	0.546093	0.273047	−	0.962001i	$-0.411969\pi$
$857$	15.9866	0.546093	0.273047	+	0.962001i	$0.411969\pi$
$858$	0	0
$859$	−20.5913	−0.702567	−0.351284	−	0.936269i	$-0.614255\pi$
$859$	−20.5913	−0.702567	−0.351284	+	0.936269i	$0.614255\pi$
$860$	0	0
$861$	0	0
$862$	77.8002	2.64989
$863$	−51.5169	−1.75366	−0.876828	−	0.480803i	$-0.840345\pi$
$863$	−51.5169	−1.75366	−0.876828	+	0.480803i	$0.840345\pi$
$864$	0	0
$865$	0	0
$866$	−35.8534	−1.21835
$867$	0	0
$868$	−32.6531	−1.10832
$869$	40.8509	1.38577
$870$	0	0
$871$	−1.19038	−0.0403346
$872$	0.826157	0.0279772
$873$	0	0
$874$	−0.311785	−0.0105463
$875$	0	0
$876$	0	0
$877$	17.6050	0.594480	0.297240	−	0.954803i	$-0.403934\pi$
$877$	17.6050	0.594480	0.297240	+	0.954803i	$0.403934\pi$
$878$	−72.8075	−2.45713
$879$	0	0
$880$	0	0
$881$	47.1453	1.58837	0.794183	−	0.607679i	$-0.207899\pi$
$881$	47.1453	1.58837	0.794183	+	0.607679i	$0.207899\pi$
$882$	0	0
$883$	23.2300	0.781751	0.390875	−	0.920444i	$-0.372172\pi$
$883$	23.2300	0.781751	0.390875	+	0.920444i	$0.372172\pi$
$884$	22.4259	0.754266
$885$	0	0
$886$	−16.4714	−0.553368
$887$	20.5226	0.689082	0.344541	−	0.938771i	$-0.388035\pi$
$887$	20.5226	0.689082	0.344541	+	0.938771i	$0.388035\pi$
$888$	0	0
$889$	58.8320	1.97316
$890$	0	0
$891$	0	0
$892$	−18.1990	−0.609347
$893$	−1.86995	−0.0625756
$894$	0	0
$895$	0	0
$896$	12.3323	0.411993
$897$	0	0
$898$	−6.48566	−0.216429
$899$	−33.3939	−1.11375
$900$	0	0
$901$	−39.6018	−1.31933
$902$	−6.27104	−0.208803
$903$	0	0
$904$	−3.69695	−0.122959
$905$	0	0
$906$	0	0
$907$	13.1950	0.438133	0.219066	−	0.975710i	$-0.429699\pi$
$907$	13.1950	0.438133	0.219066	+	0.975710i	$0.429699\pi$
$908$	−43.9682	−1.45914
$909$	0	0
$910$	0	0
$911$	−27.1021	−0.897932	−0.448966	−	0.893549i	$-0.648208\pi$
$911$	−27.1021	−0.897932	−0.448966	+	0.893549i	$0.648208\pi$
$912$	0	0
$913$	−36.9251	−1.22204
$914$	−5.03010	−0.166381
$915$	0	0
$916$	−28.1093	−0.928757
$917$	−10.8613	−0.358673
$918$	0	0
$919$	−3.80568	−0.125538	−0.0627690	−	0.998028i	$-0.519993\pi$
$919$	−3.80568	−0.125538	−0.0627690	+	0.998028i	$0.519993\pi$
$920$	0	0
$921$	0	0
$922$	−68.2475	−2.24761
$923$	−1.67799	−0.0552317
$924$	0	0
$925$	0	0
$926$	16.1393	0.530369
$927$	0	0
$928$	−66.3844	−2.17917
$929$	−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
$932$	−12.0388	−0.394345
$933$	0	0
$934$	47.8312	1.56508
$935$	0	0
$936$	0	0
$937$	8.55321	0.279421	0.139711	−	0.990192i	$-0.455383\pi$
$937$	8.55321	0.279421	0.139711	+	0.990192i	$0.455383\pi$
$938$	−2.83583	−0.0925931
$939$	0	0
$940$	0	0
$941$	48.3669	1.57672	0.788358	−	0.615216i	$-0.210931\pi$
$941$	48.3669	1.57672	0.788358	+	0.615216i	$0.210931\pi$
$942$	0	0
$943$	−0.503445	−0.0163944
$944$	21.1587	0.688657
$945$	0	0
$946$	−31.8079	−1.03416
$947$	8.71018	0.283043	0.141521	−	0.989935i	$-0.454801\pi$
$947$	8.71018	0.283043	0.141521	+	0.989935i	$0.454801\pi$
$948$	0	0
$949$	41.7603	1.35560
$950$	0	0
$951$	0	0
$952$	−5.05593	−0.163864
$953$	−43.4534	−1.40760	−0.703798	−	0.710401i	$-0.748514\pi$
$953$	−43.4534	−1.40760	−0.703798	+	0.710401i	$0.748514\pi$
$954$	0	0
$955$	0	0
$956$	−32.6819	−1.05701
$957$	0	0
$958$	−9.38497	−0.303215
$959$	−57.1912	−1.84680
$960$	0	0
$961$	−15.7358	−0.507606
$962$	−78.0062	−2.51502
$963$	0	0
$964$	11.3719	0.366264
$965$	0	0
$966$	0	0
$967$	60.2789	1.93844	0.969220	−	0.246197i	$-0.0791810\pi$
$967$	60.2789	1.93844	0.969220	+	0.246197i	$0.0791810\pi$
$968$	−2.14801	−0.0690397
$969$	0	0
$970$	0	0
$971$	−5.42583	−0.174123	−0.0870616	−	0.996203i	$-0.527748\pi$
$971$	−5.42583	−0.174123	−0.0870616	+	0.996203i	$0.527748\pi$
$972$	0	0
$973$	−1.62908	−0.0522259
$974$	39.6570	1.27069
$975$	0	0
$976$	63.4593	2.03128
$977$	−36.5741	−1.17011	−0.585054	−	0.810994i	$-0.698927\pi$
$977$	−36.5741	−1.17011	−0.585054	+	0.810994i	$0.698927\pi$
$978$	0	0
$979$	32.5331	1.03976
$980$	0	0
$981$	0	0
$982$	−70.9427	−2.26387
$983$	−41.0896	−1.31056	−0.655278	−	0.755388i	$-0.727449\pi$
$983$	−41.0896	−1.31056	−0.655278	+	0.755388i	$0.727449\pi$
$984$	0	0
$985$	0	0
$986$	54.6369	1.73999
$987$	0	0
$988$	−1.67206	−0.0531952
$989$	−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	$0.521263\pi$
$992$	30.3440	0.963424
$993$	0	0
$994$	−3.99744	−0.126791
$995$	0	0
$996$	0	0
$997$	10.1577	0.321698	0.160849	−	0.986979i	$-0.448577\pi$
$997$	10.1577	0.321698	0.160849	+	0.986979i	$0.448577\pi$
$998$	4.68631	0.148342
$999$	0	0

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.f.1.4	✓	4	15.14	odd	2
1875.2.a.g.1.1	yes	4	3.2	odd	2
1875.2.b.d.1249.1		8	15.2	even	4
1875.2.b.d.1249.8		8	15.8	even	4
5625.2.a.j.1.4		4	1.1	even	1	trivial
5625.2.a.m.1.1		4	5.4	even	2

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:	\( N \)	\(=\)	\( 5625 = 3^{2} \cdot 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5625.a (trivial)

\(f(q)\)	\(=\)	\(q+1.95630 q^{2} +1.82709 q^{4} +4.57433 q^{7} -0.338261 q^{8} +O(q^{10})\)	\(q+1.95630 q^{2} +1.82709 q^{4} +4.57433 q^{7} -0.338261 q^{8} +4.16535 q^{11} +3.75638 q^{13} +8.94874 q^{14} -4.31592 q^{16} +3.26755 q^{17} -0.243625 q^{19} +8.14866 q^{22} +0.654182 q^{23} +7.34858 q^{26} +8.35772 q^{28} +8.54732 q^{29} -3.90694 q^{31} -7.76669 q^{32} +6.39228 q^{34} -10.6151 q^{37} -0.476602 q^{38} -0.769579 q^{41} -3.90345 q^{43} +7.61048 q^{44} +1.27977 q^{46} +7.67555 q^{47} +13.9245 q^{49} +6.86324 q^{52} -12.1197 q^{53} -1.54732 q^{56} +16.7211 q^{58} -4.90248 q^{59} -14.7035 q^{61} -7.64313 q^{62} -6.56210 q^{64} -0.316897 q^{67} +5.97010 q^{68} -0.446705 q^{71} +11.1172 q^{73} -20.7664 q^{74} -0.445125 q^{76} +19.0537 q^{77} +9.80731 q^{79} -1.50552 q^{82} -8.86482 q^{83} -7.63631 q^{86} -1.40898 q^{88} +7.81040 q^{89} +17.1829 q^{91} +1.19525 q^{92} +15.0156 q^{94} -8.15938 q^{97} +27.2404 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8	\( 4 q - q^{2} + q^{4} + 5 q^{7} + 3 q^{8} + 6 q^{11} + 7 q^{13} + 10 q^{14} - 9 q^{16} + 7 q^{17} - 9 q^{19} + 6 q^{22} - 10 q^{23} + 2 q^{26} + 5 q^{28} + 28 q^{29} - 10 q^{31} + 7 q^{34} - 10 q^{37} + 6 q^{38} + q^{43} + 9 q^{44} + 5 q^{46} + 23 q^{47} - 3 q^{49} + 13 q^{52} - 2 q^{58} - 4 q^{59} - 43 q^{61} + 10 q^{62} - 7 q^{64} + 8 q^{67} + 3 q^{68} + 27 q^{71} + 15 q^{73} - 5 q^{74} + 9 q^{76} + 15 q^{77} + 10 q^{79} - 20 q^{82} - 3 q^{83} - 24 q^{86} - 3 q^{88} + 9 q^{89} + 5 q^{91} + 15 q^{92} - 22 q^{94} + 13 q^{97} + 42 q^{98}+O(q^{100}) \) 4 * q - q^2 + q^4 + 5 * q^7 + 3 * q^8 + 6 * q^11 + 7 * q^13 + 10 * q^14 - 9 * q^16 + 7 * q^17 - 9 * q^19 + 6 * q^22 - 10 * q^23 + 2 * q^26 + 5 * q^28 + 28 * q^29 - 10 * q^31 + 7 * q^34 - 10 * q^37 + 6 * q^38 + q^43 + 9 * q^44 + 5 * q^46 + 23 * q^47 - 3 * q^49 + 13 * q^52 - 2 * q^58 - 4 * q^59 - 43 * q^61 + 10 * q^62 - 7 * q^64 + 8 * q^67 + 3 * q^68 + 27 * q^71 + 15 * q^73 - 5 * q^74 + 9 * q^76 + 15 * q^77 + 10 * q^79 - 20 * q^82 - 3 * q^83 - 24 * q^86 - 3 * q^88 + 9 * q^89 + 5 * q^91 + 15 * q^92 - 22 * q^94 + 13 * q^97 + 42 * q^98

Introduction

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