LMFDB - Embedded newform 4225.2.a.cb.1.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4225,2,Mod(1,4225)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [18,0,0,16,0,16,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$33.7367948540$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 26x^{16} + 281x^{14} - 1632x^{12} + 5482x^{10} - 10620x^{8} + 11052x^{6} - 5165x^{4} + 760x^{2} - 29 $ x^18 - 26x^16 + 281x^14 - 1632x^12 + 5482x^10 - 10620x^8 + 11052x^6 - 5165x^4 + 760x^2 - 29
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 845)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.887876$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.887876 q^{2} +1.26907 q^{3} -1.21168 q^{4} -1.12678 q^{6} +2.28128 q^{7} +2.85157 q^{8} -1.38947 q^{9} -3.13829 q^{11} -1.53770 q^{12} -2.02549 q^{14} -0.108488 q^{16} +1.72423 q^{17} +1.23367 q^{18} -4.27007 q^{19} +2.89510 q^{21} +2.78641 q^{22} +8.63424 q^{23} +3.61884 q^{24} -5.57053 q^{27} -2.76417 q^{28} -7.94250 q^{29} +4.17039 q^{31} -5.60682 q^{32} -3.98270 q^{33} -1.53090 q^{34} +1.68358 q^{36} +8.80437 q^{37} +3.79129 q^{38} +2.94700 q^{41} -2.57049 q^{42} -0.153762 q^{43} +3.80259 q^{44} -7.66614 q^{46} +6.54786 q^{47} -0.137679 q^{48} -1.79577 q^{49} +2.18817 q^{51} -1.93126 q^{53} +4.94594 q^{54} +6.50522 q^{56} -5.41901 q^{57} +7.05195 q^{58} +5.51115 q^{59} -9.90052 q^{61} -3.70279 q^{62} -3.16976 q^{63} +5.19513 q^{64} +3.53615 q^{66} -6.45526 q^{67} -2.08921 q^{68} +10.9574 q^{69} -4.74432 q^{71} -3.96216 q^{72} +13.1891 q^{73} -7.81719 q^{74} +5.17394 q^{76} -7.15931 q^{77} +13.4510 q^{79} -2.90099 q^{81} -2.61657 q^{82} +6.32511 q^{83} -3.50792 q^{84} +0.136522 q^{86} -10.0796 q^{87} -8.94905 q^{88} +15.7373 q^{89} -10.4619 q^{92} +5.29250 q^{93} -5.81368 q^{94} -7.11543 q^{96} +12.0815 q^{97} +1.59442 q^{98} +4.36055 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) $ 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

Coefficient data

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

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

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

$2$	−0.887876	−0.627823	−0.313912	−	0.949452i	$-0.601640\pi$
$2$	−0.887876	−0.627823	−0.313912	+	0.949452i	$0.601640\pi$
$3$	1.26907	0.732697	0.366348	−	0.930478i	$-0.380608\pi$
$3$	1.26907	0.732697	0.366348	+	0.930478i	$0.380608\pi$
$4$	−1.21168	−0.605838
$5$	0	0
$5$	0	0
$6$	−1.12678	−0.460004
$7$	2.28128	0.862242	0.431121	−	0.902294i	$-0.358118\pi$
$7$	2.28128	0.862242	0.431121	+	0.902294i	$0.358118\pi$
$8$	2.85157	1.00818
$9$	−1.38947	−0.463155
$10$	0	0
$11$	−3.13829	−0.946230	−0.473115	−	0.881001i	$-0.656870\pi$
$11$	−3.13829	−0.946230	−0.473115	+	0.881001i	$0.656870\pi$
$12$	−1.53770	−0.443896
$13$	0	0
$13$	0	0
$14$	−2.02549	−0.541335
$15$	0	0
$16$	−0.108488	−0.0271220
$17$	1.72423	0.418187	0.209094	−	0.977896i	$-0.432949\pi$
$17$	1.72423	0.418187	0.209094	+	0.977896i	$0.432949\pi$
$18$	1.23367	0.290780
$19$	−4.27007	−0.979621	−0.489810	−	0.871829i	$-0.662934\pi$
$19$	−4.27007	−0.979621	−0.489810	+	0.871829i	$0.662934\pi$
$20$	0	0
$21$	2.89510	0.631762
$22$	2.78641	0.594065
$23$	8.63424	1.80036	0.900182	−	0.435514i	$-0.143433\pi$
$23$	8.63424	1.80036	0.900182	+	0.435514i	$0.143433\pi$
$24$	3.61884	0.738692
$25$	0	0
$26$	0	0
$27$	−5.57053	−1.07205
$28$	−2.76417	−0.522379
$29$	−7.94250	−1.47488	−0.737442	−	0.675410i	$-0.763967\pi$
$29$	−7.94250	−1.47488	−0.737442	+	0.675410i	$0.763967\pi$
$30$	0	0
$31$	4.17039	0.749024	0.374512	−	0.927222i	$-0.377810\pi$
$31$	4.17039	0.749024	0.374512	+	0.927222i	$0.377810\pi$
$32$	−5.60682	−0.991154
$33$	−3.98270	−0.693300
$34$	−1.53090	−0.262548
$35$	0	0
$36$	1.68358	0.280597
$37$	8.80437	1.44743	0.723715	−	0.690099i	$-0.242433\pi$
$37$	8.80437	1.44743	0.723715	+	0.690099i	$0.242433\pi$
$38$	3.79129	0.615028
$39$	0	0
$40$	0	0
$41$	2.94700	0.460244	0.230122	−	0.973162i	$-0.426088\pi$
$41$	2.94700	0.460244	0.230122	+	0.973162i	$0.426088\pi$
$42$	−2.57049	−0.396635
$43$	−0.153762	−0.0234486	−0.0117243	−	0.999931i	$-0.503732\pi$
$43$	−0.153762	−0.0234486	−0.0117243	+	0.999931i	$0.503732\pi$
$44$	3.80259	0.573262
$45$	0	0
$46$	−7.66614	−1.13031
$47$	6.54786	0.955103	0.477552	−	0.878604i	$-0.341524\pi$
$47$	6.54786	0.955103	0.477552	+	0.878604i	$0.341524\pi$
$48$	−0.137679	−0.0198722
$49$	−1.79577	−0.256539
$50$	0	0
$51$	2.18817	0.306404
$52$	0	0
$53$	−1.93126	−0.265279	−0.132640	−	0.991164i	$-0.542345\pi$
$53$	−1.93126	−0.265279	−0.132640	+	0.991164i	$0.542345\pi$
$54$	4.94594	0.673057
$55$	0	0
$56$	6.50522	0.869297
$57$	−5.41901	−0.717765
$58$	7.05195	0.925967
$59$	5.51115	0.717490	0.358745	−	0.933436i	$-0.383205\pi$
$59$	5.51115	0.717490	0.358745	+	0.933436i	$0.383205\pi$
$60$	0	0
$61$	−9.90052	−1.26763	−0.633816	−	0.773484i	$-0.718512\pi$
$61$	−9.90052	−1.26763	−0.633816	+	0.773484i	$0.718512\pi$
$62$	−3.70279	−0.470254
$63$	−3.16976	−0.399352
$64$	5.19513	0.649392
$65$	0	0
$66$	3.53615	0.435270
$67$	−6.45526	−0.788635	−0.394318	−	0.918974i	$-0.629019\pi$
$67$	−6.45526	−0.788635	−0.394318	+	0.918974i	$0.629019\pi$
$68$	−2.08921	−0.253354
$69$	10.9574	1.31912
$70$	0	0
$71$	−4.74432	−0.563047	−0.281524	−	0.959554i	$-0.590840\pi$
$71$	−4.74432	−0.563047	−0.281524	+	0.959554i	$0.590840\pi$
$72$	−3.96216	−0.466945
$73$	13.1891	1.54367	0.771834	−	0.635824i	$-0.219340\pi$
$73$	13.1891	1.54367	0.771834	+	0.635824i	$0.219340\pi$
$74$	−7.81719	−0.908730
$75$	0	0
$76$	5.17394	0.593492
$77$	−7.15931	−0.815879
$78$	0	0
$79$	13.4510	1.51335	0.756676	−	0.653791i	$-0.226822\pi$
$79$	13.4510	1.51335	0.756676	+	0.653791i	$0.226822\pi$
$80$	0	0
$81$	−2.90099	−0.322332
$82$	−2.61657	−0.288952
$83$	6.32511	0.694271	0.347136	−	0.937815i	$-0.387154\pi$
$83$	6.32511	0.694271	0.347136	+	0.937815i	$0.387154\pi$
$84$	−3.50792	−0.382745
$85$	0	0
$86$	0.136522	0.0147215
$87$	−10.0796	−1.08064
$88$	−8.94905	−0.953972
$89$	15.7373	1.66815	0.834077	−	0.551649i	$-0.186001\pi$
$89$	15.7373	1.66815	0.834077	+	0.551649i	$0.186001\pi$
$90$	0	0
$91$	0	0
$92$	−10.4619	−1.09073
$93$	5.29250	0.548807
$94$	−5.81368	−0.599636
$95$	0	0
$96$	−7.11543	−0.726216
$97$	12.0815	1.22669	0.613345	−	0.789815i	$-0.289823\pi$
$97$	12.0815	1.22669	0.613345	+	0.789815i	$0.289823\pi$
$98$	1.59442	0.161061
$99$	4.36055	0.438251
$100$	0	0
$101$	3.79100	0.377218	0.188609	−	0.982052i	$-0.439602\pi$
$101$	3.79100	0.377218	0.188609	+	0.982052i	$0.439602\pi$
$102$	−1.94282	−0.192368
$103$	−14.6259	−1.44113	−0.720564	−	0.693388i	$-0.756117\pi$
$103$	−14.6259	−1.44113	−0.720564	+	0.693388i	$0.756117\pi$
$104$	0	0
$105$	0	0
$106$	1.71472	0.166548
$107$	4.19100	0.405159	0.202580	−	0.979266i	$-0.435067\pi$
$107$	4.19100	0.405159	0.202580	+	0.979266i	$0.435067\pi$
$108$	6.74968	0.649488
$109$	16.1460	1.54650	0.773252	−	0.634099i	$-0.218629\pi$
$109$	16.1460	1.54650	0.773252	+	0.634099i	$0.218629\pi$
$110$	0	0
$111$	11.1733	1.06053
$112$	−0.247491	−0.0233857
$113$	3.51935	0.331072	0.165536	−	0.986204i	$-0.447065\pi$
$113$	3.51935	0.331072	0.165536	+	0.986204i	$0.447065\pi$
$114$	4.81141	0.450629
$115$	0	0
$116$	9.62373	0.893541
$117$	0	0
$118$	−4.89322	−0.450457
$119$	3.93345	0.360579
$120$	0	0
$121$	−1.15114	−0.104649
$122$	8.79043	0.795848
$123$	3.73994	0.337219
$124$	−5.05316	−0.453787
$125$	0	0
$126$	2.81435	0.250722
$127$	2.42242	0.214955	0.107478	−	0.994208i	$-0.465723\pi$
$127$	2.42242	0.214955	0.107478	+	0.994208i	$0.465723\pi$
$128$	6.60100	0.583451
$129$	−0.195135	−0.0171807
$130$	0	0
$131$	−5.00647	−0.437418	−0.218709	−	0.975790i	$-0.570184\pi$
$131$	−5.00647	−0.437418	−0.218709	+	0.975790i	$0.570184\pi$
$132$	4.82575	0.420027
$133$	−9.74121	−0.844670
$134$	5.73147	0.495124
$135$	0	0
$136$	4.91676	0.421609
$137$	−12.8154	−1.09489	−0.547447	−	0.836840i	$-0.684400\pi$
$137$	−12.8154	−1.09489	−0.547447	+	0.836840i	$0.684400\pi$
$138$	−9.72885	−0.828175
$139$	−15.5289	−1.31714	−0.658571	−	0.752519i	$-0.728839\pi$
$139$	−15.5289	−1.31714	−0.658571	+	0.752519i	$0.728839\pi$
$140$	0	0
$141$	8.30968	0.699801
$142$	4.21237	0.353494
$143$	0	0
$144$	0.150740	0.0125617
$145$	0	0
$146$	−11.7103	−0.969151
$147$	−2.27896	−0.187965
$148$	−10.6681	−0.876908
$149$	−0.0413812	−0.00339008	−0.00169504	−	0.999999i	$-0.500540\pi$
$149$	−0.0413812	−0.00339008	−0.00169504	+	0.999999i	$0.500540\pi$
$150$	0	0
$151$	−6.01321	−0.489348	−0.244674	−	0.969605i	$-0.578681\pi$
$151$	−6.01321	−0.489348	−0.244674	+	0.969605i	$0.578681\pi$
$152$	−12.1764	−0.987636
$153$	−2.39576	−0.193686
$154$	6.35658	0.512228
$155$	0	0
$156$	0	0
$157$	22.7846	1.81841	0.909205	−	0.416348i	$-0.136690\pi$
$157$	22.7846	1.81841	0.909205	+	0.416348i	$0.136690\pi$
$158$	−11.9428	−0.950117
$159$	−2.45090	−0.194369
$160$	0	0
$161$	19.6971	1.55235
$162$	2.57572	0.202367
$163$	−1.52126	−0.119154	−0.0595772	−	0.998224i	$-0.518975\pi$
$163$	−1.52126	−0.119154	−0.0595772	+	0.998224i	$0.518975\pi$
$164$	−3.57081	−0.278833
$165$	0	0
$166$	−5.61592	−0.435880
$167$	−0.0491982	−0.00380707	−0.00190353	−	0.999998i	$-0.500606\pi$
$167$	−0.0491982	−0.00380707	−0.00190353	+	0.999998i	$0.500606\pi$
$168$	8.25557	0.636931
$169$	0	0
$170$	0	0
$171$	5.93311	0.453717
$172$	0.186310	0.0142060
$173$	−9.00465	−0.684611	−0.342305	−	0.939589i	$-0.611208\pi$
$173$	−9.00465	−0.684611	−0.342305	+	0.939589i	$0.611208\pi$
$174$	8.94941	0.678453
$175$	0	0
$176$	0.340467	0.0256637
$177$	6.99402	0.525703
$178$	−13.9728	−1.04731
$179$	22.8604	1.70866	0.854332	−	0.519727i	$-0.173966\pi$
$179$	22.8604	1.70866	0.854332	+	0.519727i	$0.173966\pi$
$180$	0	0
$181$	−17.7138	−1.31666	−0.658328	−	0.752731i	$-0.728736\pi$
$181$	−17.7138	−1.31666	−0.658328	+	0.752731i	$0.728736\pi$
$182$	0	0
$183$	−12.5644	−0.928789
$184$	24.6211	1.81509
$185$	0	0
$186$	−4.69909	−0.344554
$187$	−5.41113	−0.395701
$188$	−7.93388	−0.578638
$189$	−12.7079	−0.924366
$190$	0	0
$191$	9.73516	0.704411	0.352206	−	0.935923i	$-0.385432\pi$
$191$	9.73516	0.704411	0.352206	+	0.935923i	$0.385432\pi$
$192$	6.59298	0.475807
$193$	10.0009	0.719883	0.359942	−	0.932975i	$-0.382797\pi$
$193$	10.0009	0.719883	0.359942	+	0.932975i	$0.382797\pi$
$194$	−10.7269	−0.770145
$195$	0	0
$196$	2.17590	0.155421
$197$	−5.99999	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−5.99999	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−3.87162	−0.275144
$199$	5.81284	0.412061	0.206031	−	0.978546i	$-0.433945\pi$
$199$	5.81284	0.412061	0.206031	+	0.978546i	$0.433945\pi$
$200$	0	0
$201$	−8.19216	−0.577831
$202$	−3.36593	−0.236826
$203$	−18.1190	−1.27171
$204$	−2.65135	−0.185632
$205$	0	0
$206$	12.9859	0.904774
$207$	−11.9970	−0.833848
$208$	0	0
$209$	13.4007	0.926946
$210$	0	0
$211$	8.71262	0.599801	0.299901	−	0.953970i	$-0.403046\pi$
$211$	8.71262	0.599801	0.299901	+	0.953970i	$0.403046\pi$
$212$	2.34006	0.160716
$213$	−6.02087	−0.412543
$214$	−3.72109	−0.254368
$215$	0	0
$216$	−15.8848	−1.08082
$217$	9.51381	0.645839
$218$	−14.3356	−0.970930
$219$	16.7379	1.13104
$220$	0	0
$221$	0	0
$222$	−9.92055	−0.665824
$223$	0.0454220	0.00304168	0.00152084	−	0.999999i	$-0.499516\pi$
$223$	0.0454220	0.00304168	0.00152084	+	0.999999i	$0.499516\pi$
$224$	−12.7907	−0.854615
$225$	0	0
$226$	−3.12474	−0.207855
$227$	−6.42258	−0.426281	−0.213141	−	0.977022i	$-0.568369\pi$
$227$	−6.42258	−0.426281	−0.213141	+	0.977022i	$0.568369\pi$
$228$	6.56608	0.434849
$229$	−7.27972	−0.481057	−0.240529	−	0.970642i	$-0.577321\pi$
$229$	−7.27972	−0.481057	−0.240529	+	0.970642i	$0.577321\pi$
$230$	0	0
$231$	−9.08565	−0.597792
$232$	−22.6486	−1.48695
$233$	4.56107	0.298806	0.149403	−	0.988776i	$-0.452265\pi$
$233$	4.56107	0.298806	0.149403	+	0.988776i	$0.452265\pi$
$234$	0	0
$235$	0	0
$236$	−6.67773	−0.434683
$237$	17.0702	1.10883
$238$	−3.49241	−0.226380
$239$	16.9338	1.09536	0.547680	−	0.836688i	$-0.315511\pi$
$239$	16.9338	1.09536	0.547680	+	0.836688i	$0.315511\pi$
$240$	0	0
$241$	6.13059	0.394906	0.197453	−	0.980312i	$-0.436733\pi$
$241$	6.13059	0.394906	0.197453	+	0.980312i	$0.436733\pi$
$242$	1.02207	0.0657010
$243$	13.0300	0.835878
$244$	11.9962	0.767979
$245$	0	0
$246$	−3.32060	−0.211714
$247$	0	0
$248$	11.8922	0.755152
$249$	8.02700	0.508690
$250$	0	0
$251$	11.5843	0.731192	0.365596	−	0.930774i	$-0.380865\pi$
$251$	11.5843	0.731192	0.365596	+	0.930774i	$0.380865\pi$
$252$	3.84072	0.241943
$253$	−27.0967	−1.70356
$254$	−2.15081	−0.134954
$255$	0	0
$256$	−16.2511	−1.01570
$257$	12.4669	0.777665	0.388832	−	0.921309i	$-0.372879\pi$
$257$	12.4669	0.777665	0.388832	+	0.921309i	$0.372879\pi$
$258$	0.173256	0.0107864
$259$	20.0852	1.24803
$260$	0	0
$261$	11.0358	0.683101
$262$	4.44513	0.274621
$263$	28.3288	1.74683	0.873416	−	0.486975i	$-0.161900\pi$
$263$	28.3288	1.74683	0.873416	+	0.486975i	$0.161900\pi$
$264$	−11.3570	−0.698972
$265$	0	0
$266$	8.64898	0.530303
$267$	19.9717	1.22225
$268$	7.82168	0.477785
$269$	−12.1886	−0.743154	−0.371577	−	0.928402i	$-0.621183\pi$
$269$	−12.1886	−0.743154	−0.371577	+	0.928402i	$0.621183\pi$
$270$	0	0
$271$	17.6547	1.07245	0.536224	−	0.844076i	$-0.319850\pi$
$271$	17.6547	1.07245	0.536224	+	0.844076i	$0.319850\pi$
$272$	−0.187058	−0.0113421
$273$	0	0
$274$	11.3785	0.687400
$275$	0	0
$276$	−13.2769	−0.799174
$277$	11.7250	0.704489	0.352245	−	0.935908i	$-0.385419\pi$
$277$	11.7250	0.704489	0.352245	+	0.935908i	$0.385419\pi$
$278$	13.7877	0.826932
$279$	−5.79461	−0.346914
$280$	0	0
$281$	20.2642	1.20886	0.604430	−	0.796659i	$-0.293401\pi$
$281$	20.2642	1.20886	0.604430	+	0.796659i	$0.293401\pi$
$282$	−7.37796	−0.439351
$283$	8.24330	0.490013	0.245007	−	0.969521i	$-0.421210\pi$
$283$	8.24330	0.490013	0.245007	+	0.969521i	$0.421210\pi$
$284$	5.74858	0.341116
$285$	0	0
$286$	0	0
$287$	6.72292	0.396841
$288$	7.79048	0.459059
$289$	−14.0270	−0.825119
$290$	0	0
$291$	15.3322	0.898792
$292$	−15.9809	−0.935213
$293$	21.3215	1.24562	0.622808	−	0.782375i	$-0.285992\pi$
$293$	21.3215	1.24562	0.622808	+	0.782375i	$0.285992\pi$
$294$	2.02343	0.118009
$295$	0	0
$296$	25.1063	1.45927
$297$	17.4819	1.01441
$298$	0.0367414	0.00212837
$299$	0	0
$300$	0	0
$301$	−0.350775	−0.0202183
$302$	5.33898	0.307224
$303$	4.81103	0.276387
$304$	0.463251	0.0265693
$305$	0	0
$306$	2.12714	0.121600
$307$	−8.38560	−0.478591	−0.239296	−	0.970947i	$-0.576917\pi$
$307$	−8.38560	−0.478591	−0.239296	+	0.970947i	$0.576917\pi$
$308$	8.67476	0.494291
$309$	−18.5612	−1.05591
$310$	0	0
$311$	−2.82308	−0.160082	−0.0800412	−	0.996792i	$-0.525505\pi$
$311$	−2.82308	−0.160082	−0.0800412	+	0.996792i	$0.525505\pi$
$312$	0	0
$313$	19.1965	1.08505	0.542525	−	0.840040i	$-0.317468\pi$
$313$	19.1965	1.08505	0.542525	+	0.840040i	$0.317468\pi$
$314$	−20.2299	−1.14164
$315$	0	0
$316$	−16.2982	−0.916846
$317$	−5.76453	−0.323768	−0.161884	−	0.986810i	$-0.551757\pi$
$317$	−5.76453	−0.323768	−0.161884	+	0.986810i	$0.551757\pi$
$318$	2.17610	0.122029
$319$	24.9259	1.39558
$320$	0	0
$321$	5.31867	0.296859
$322$	−17.4886	−0.974600
$323$	−7.36258	−0.409665
$324$	3.51506	0.195281
$325$	0	0
$326$	1.35069	0.0748079
$327$	20.4903	1.13312
$328$	8.40357	0.464010
$329$	14.9375	0.823530
$330$	0	0
$331$	−4.06009	−0.223163	−0.111581	−	0.993755i	$-0.535592\pi$
$331$	−4.06009	−0.223163	−0.111581	+	0.993755i	$0.535592\pi$
$332$	−7.66399	−0.420616
$333$	−12.2334	−0.670385
$334$	0.0436819	0.00239017
$335$	0	0
$336$	−0.314083	−0.0171346
$337$	24.8339	1.35279	0.676395	−	0.736539i	$-0.263541\pi$
$337$	24.8339	1.35279	0.676395	+	0.736539i	$0.263541\pi$
$338$	0	0
$339$	4.46629	0.242576
$340$	0	0
$341$	−13.0879	−0.708749
$342$	−5.26787	−0.284854
$343$	−20.0656	−1.08344
$344$	−0.438465	−0.0236404
$345$	0	0
$346$	7.99501	0.429815
$347$	29.1599	1.56538	0.782692	−	0.622409i	$-0.213846\pi$
$347$	29.1599	1.56538	0.782692	+	0.622409i	$0.213846\pi$
$348$	12.2132	0.654695
$349$	7.29040	0.390246	0.195123	−	0.980779i	$-0.437489\pi$
$349$	7.29040	0.390246	0.195123	+	0.980779i	$0.437489\pi$
$350$	0	0
$351$	0	0
$352$	17.5958	0.937860
$353$	13.9534	0.742665	0.371333	−	0.928500i	$-0.378901\pi$
$353$	13.9534	0.742665	0.371333	+	0.928500i	$0.378901\pi$
$354$	−6.20982	−0.330048
$355$	0	0
$356$	−19.0685	−1.01063
$357$	4.99181	0.264195
$358$	−20.2972	−1.07274
$359$	16.2637	0.858366	0.429183	−	0.903218i	$-0.358802\pi$
$359$	16.2637	0.858366	0.429183	+	0.903218i	$0.358802\pi$
$360$	0	0
$361$	−0.766527	−0.0403435
$362$	15.7276	0.826627
$363$	−1.46087	−0.0766759
$364$	0	0
$365$	0	0
$366$	11.1557	0.583115
$367$	−34.4480	−1.79817	−0.899085	−	0.437774i	$-0.855767\pi$
$367$	−34.4480	−1.79817	−0.899085	+	0.437774i	$0.855767\pi$
$368$	−0.936712	−0.0488295
$369$	−4.09475	−0.213164
$370$	0	0
$371$	−4.40574	−0.228735
$372$	−6.41280	−0.332488
$373$	−23.2179	−1.20218	−0.601088	−	0.799183i	$-0.705266\pi$
$373$	−23.2179	−1.20218	−0.601088	+	0.799183i	$0.705266\pi$
$374$	4.80442	0.248430
$375$	0	0
$376$	18.6717	0.962918
$377$	0	0
$378$	11.2831	0.580338
$379$	12.2666	0.630094	0.315047	−	0.949076i	$-0.397980\pi$
$379$	12.2666	0.630094	0.315047	+	0.949076i	$0.397980\pi$
$380$	0	0
$381$	3.07422	0.157497
$382$	−8.64361	−0.442246
$383$	1.88332	0.0962329	0.0481165	−	0.998842i	$-0.484678\pi$
$383$	1.88332	0.0962329	0.0481165	+	0.998842i	$0.484678\pi$
$384$	8.37712	0.427493
$385$	0	0
$386$	−8.87959	−0.451959
$387$	0.213648	0.0108603
$388$	−14.6389	−0.743176
$389$	−20.0438	−1.01626	−0.508131	−	0.861280i	$-0.669663\pi$
$389$	−20.0438	−1.01626	−0.508131	+	0.861280i	$0.669663\pi$
$390$	0	0
$391$	14.8874	0.752889
$392$	−5.12078	−0.258638
$393$	−6.35356	−0.320494
$394$	5.32725	0.268383
$395$	0	0
$396$	−5.28357	−0.265509
$397$	20.6134	1.03456	0.517279	−	0.855817i	$-0.326945\pi$
$397$	20.6134	1.03456	0.517279	+	0.855817i	$0.326945\pi$
$398$	−5.16108	−0.258702
$399$	−12.3623	−0.618887
$400$	0	0
$401$	−35.8148	−1.78851	−0.894254	−	0.447560i	$-0.852293\pi$
$401$	−35.8148	−1.78851	−0.894254	+	0.447560i	$0.852293\pi$
$402$	7.27362	0.362775
$403$	0	0
$404$	−4.59346	−0.228533
$405$	0	0
$406$	16.0875	0.798407
$407$	−27.6307	−1.36960
$408$	6.23971	0.308912
$409$	−0.221949	−0.0109747	−0.00548733	−	0.999985i	$-0.501747\pi$
$409$	−0.221949	−0.0109747	−0.00548733	+	0.999985i	$0.501747\pi$
$410$	0	0
$411$	−16.2636	−0.802226
$412$	17.7218	0.873091
$413$	12.5725	0.618650
$414$	10.6518	0.523509
$415$	0	0
$416$	0	0
$417$	−19.7072	−0.965065
$418$	−11.8982	−0.581958
$419$	0.680369	0.0332382	0.0166191	−	0.999862i	$-0.494710\pi$
$419$	0.680369	0.0332382	0.0166191	+	0.999862i	$0.494710\pi$
$420$	0	0
$421$	−2.01932	−0.0984155	−0.0492078	−	0.998789i	$-0.515670\pi$
$421$	−2.01932	−0.0984155	−0.0492078	+	0.998789i	$0.515670\pi$
$422$	−7.73572	−0.376569
$423$	−9.09803	−0.442361
$424$	−5.50712	−0.267450
$425$	0	0
$426$	5.34578	0.259004
$427$	−22.5858	−1.09300
$428$	−5.07814	−0.245461
$429$	0	0
$430$	0	0
$431$	6.85397	0.330144	0.165072	−	0.986281i	$-0.447214\pi$
$431$	6.85397	0.330144	0.165072	+	0.986281i	$0.447214\pi$
$432$	0.604336	0.0290761
$433$	0.520934	0.0250345	0.0125172	−	0.999922i	$-0.496016\pi$
$433$	0.520934	0.0250345	0.0125172	+	0.999922i	$0.496016\pi$
$434$	−8.44708	−0.405473
$435$	0	0
$436$	−19.5637	−0.936931
$437$	−36.8688	−1.76367
$438$	−14.8612	−0.710094
$439$	−27.5889	−1.31675	−0.658374	−	0.752691i	$-0.728756\pi$
$439$	−27.5889	−1.31675	−0.658374	+	0.752691i	$0.728756\pi$
$440$	0	0
$441$	2.49517	0.118818
$442$	0	0
$443$	7.20684	0.342408	0.171204	−	0.985236i	$-0.445234\pi$
$443$	7.20684	0.342408	0.171204	+	0.985236i	$0.445234\pi$
$444$	−13.5385	−0.642508
$445$	0	0
$446$	−0.0403291	−0.00190964
$447$	−0.0525156	−0.00248390
$448$	11.8515	0.559933
$449$	0.135634	0.00640098	0.00320049	−	0.999995i	$-0.498981\pi$
$449$	0.135634	0.00640098	0.00320049	+	0.999995i	$0.498981\pi$
$450$	0	0
$451$	−9.24853	−0.435496
$452$	−4.26431	−0.200576
$453$	−7.63117	−0.358544
$454$	5.70245	0.267629
$455$	0	0
$456$	−15.4527	−0.723638
$457$	−0.308011	−0.0144081	−0.00720406	−	0.999974i	$-0.502293\pi$
$457$	−0.308011	−0.0144081	−0.00720406	+	0.999974i	$0.502293\pi$
$458$	6.46349	0.302019
$459$	−9.60488	−0.448317
$460$	0	0
$461$	28.6721	1.33539	0.667696	−	0.744434i	$-0.267281\pi$
$461$	28.6721	1.33539	0.667696	+	0.744434i	$0.267281\pi$
$462$	8.06693	0.375308
$463$	16.1023	0.748339	0.374169	−	0.927360i	$-0.377928\pi$
$463$	16.1023	0.748339	0.374169	+	0.927360i	$0.377928\pi$
$464$	0.861666	0.0400018
$465$	0	0
$466$	−4.04967	−0.187597
$467$	−3.47589	−0.160845	−0.0804224	−	0.996761i	$-0.525627\pi$
$467$	−3.47589	−0.160845	−0.0804224	+	0.996761i	$0.525627\pi$
$468$	0	0
$469$	−14.7262	−0.679994
$470$	0	0
$471$	28.9152	1.33234
$472$	15.7154	0.723361
$473$	0.482551	0.0221877
$474$	−15.1562	−0.696148
$475$	0	0
$476$	−4.76606	−0.218452
$477$	2.68342	0.122865
$478$	−15.0352	−0.687692
$479$	−14.4576	−0.660587	−0.330293	−	0.943878i	$-0.607148\pi$
$479$	−14.4576	−0.660587	−0.330293	+	0.943878i	$0.607148\pi$
$480$	0	0
$481$	0	0
$482$	−5.44320	−0.247931
$483$	24.9970	1.13740
$484$	1.39481	0.0634003
$485$	0	0
$486$	−11.5691	−0.524783
$487$	−2.91278	−0.131991	−0.0659954	−	0.997820i	$-0.521022\pi$
$487$	−2.91278	−0.131991	−0.0659954	+	0.997820i	$0.521022\pi$
$488$	−28.2320	−1.27800
$489$	−1.93058	−0.0873040
$490$	0	0
$491$	−7.60764	−0.343328	−0.171664	−	0.985156i	$-0.554914\pi$
$491$	−7.60764	−0.343328	−0.171664	+	0.985156i	$0.554914\pi$
$492$	−4.53160	−0.204300
$493$	−13.6947	−0.616778
$494$	0	0
$495$	0	0
$496$	−0.452437	−0.0203150
$497$	−10.8231	−0.485483
$498$	−7.12698	−0.319368
$499$	−4.07246	−0.182309	−0.0911543	−	0.995837i	$-0.529056\pi$
$499$	−4.07246	−0.182309	−0.0911543	+	0.995837i	$0.529056\pi$
$500$	0	0
$501$	−0.0624358	−0.00278943
$502$	−10.2854	−0.459059
$503$	34.0821	1.51965	0.759823	−	0.650130i	$-0.225286\pi$
$503$	34.0821	1.51965	0.759823	+	0.650130i	$0.225286\pi$
$504$	−9.03879	−0.402620
$505$	0	0
$506$	24.0586	1.06953
$507$	0	0
$508$	−2.93519	−0.130228
$509$	1.30664	0.0579159	0.0289579	−	0.999581i	$-0.490781\pi$
$509$	1.30664	0.0579159	0.0289579	+	0.999581i	$0.490781\pi$
$510$	0	0
$511$	30.0880	1.33102
$512$	1.22700	0.0542260
$513$	23.7865	1.05020
$514$	−11.0691	−0.488236
$515$	0	0
$516$	0.236441	0.0104087
$517$	−20.5491	−0.903747
$518$	−17.8332	−0.783545
$519$	−11.4275	−0.501612
$520$	0	0
$521$	−0.550102	−0.0241004	−0.0120502	−	0.999927i	$-0.503836\pi$
$521$	−0.550102	−0.0241004	−0.0120502	+	0.999927i	$0.503836\pi$
$522$	−9.79845	−0.428866
$523$	−17.0531	−0.745679	−0.372840	−	0.927896i	$-0.621616\pi$
$523$	−17.0531	−0.745679	−0.372840	+	0.927896i	$0.621616\pi$
$524$	6.06622	0.265004
$525$	0	0
$526$	−25.1525	−1.09670
$527$	7.19071	0.313232
$528$	0.432076	0.0188037
$529$	51.5501	2.24131
$530$	0	0
$531$	−7.65755	−0.332309
$532$	11.8032	0.511733
$533$	0	0
$534$	−17.7324	−0.767357
$535$	0	0
$536$	−18.4076	−0.795088
$537$	29.0114	1.25193
$538$	10.8220	0.466569
$539$	5.63566	0.242745
$540$	0	0
$541$	30.1330	1.29552	0.647759	−	0.761845i	$-0.275706\pi$
$541$	30.1330	1.29552	0.647759	+	0.761845i	$0.275706\pi$
$542$	−15.6752	−0.673307
$543$	−22.4800	−0.964709
$544$	−9.66744	−0.414488
$545$	0	0
$546$	0	0
$547$	5.36791	0.229515	0.114758	−	0.993394i	$-0.463391\pi$
$547$	5.36791	0.229515	0.114758	+	0.993394i	$0.463391\pi$
$548$	15.5281	0.663329
$549$	13.7564	0.587110
$550$	0	0
$551$	33.9150	1.44483
$552$	31.2459	1.32991
$553$	30.6854	1.30487
$554$	−10.4104	−0.442294
$555$	0	0
$556$	18.8160	0.797974
$557$	−7.05530	−0.298943	−0.149471	−	0.988766i	$-0.547757\pi$
$557$	−7.05530	−0.298943	−0.149471	+	0.988766i	$0.547757\pi$
$558$	5.14490	0.217801
$559$	0	0
$560$	0	0
$561$	−6.86710	−0.289929
$562$	−17.9921	−0.758950
$563$	31.6656	1.33455	0.667273	−	0.744814i	$-0.267462\pi$
$563$	31.6656	1.33455	0.667273	+	0.744814i	$0.267462\pi$
$564$	−10.0686	−0.423966
$565$	0	0
$566$	−7.31903	−0.307642
$567$	−6.61795	−0.277928
$568$	−13.5288	−0.567655
$569$	13.5818	0.569380	0.284690	−	0.958620i	$-0.408109\pi$
$569$	13.5818	0.569380	0.284690	+	0.958620i	$0.408109\pi$
$570$	0	0
$571$	−34.7338	−1.45356	−0.726782	−	0.686868i	$-0.758985\pi$
$571$	−34.7338	−1.45356	−0.726782	+	0.686868i	$0.758985\pi$
$572$	0	0
$573$	12.3546	0.516120
$574$	−5.96912	−0.249146
$575$	0	0
$576$	−7.21846	−0.300769
$577$	43.8511	1.82555	0.912773	−	0.408467i	$-0.133937\pi$
$577$	43.8511	1.82555	0.912773	+	0.408467i	$0.133937\pi$
$578$	12.4543	0.518029
$579$	12.6919	0.527456
$580$	0	0
$581$	14.4293	0.598630
$582$	−13.6131	−0.564283
$583$	6.06085	0.251015
$584$	37.6097	1.55630
$585$	0	0
$586$	−18.9308	−0.782026
$587$	−4.11991	−0.170047	−0.0850235	−	0.996379i	$-0.527097\pi$
$587$	−4.11991	−0.170047	−0.0850235	+	0.996379i	$0.527097\pi$
$588$	2.76136	0.113877
$589$	−17.8078	−0.733759
$590$	0	0
$591$	−7.61440	−0.313215
$592$	−0.955169	−0.0392572
$593$	2.16800	0.0890292	0.0445146	−	0.999009i	$-0.485826\pi$
$593$	2.16800	0.0890292	0.0445146	+	0.999009i	$0.485826\pi$
$594$	−15.5218	−0.636867
$595$	0	0
$596$	0.0501407	0.00205384
$597$	7.37689	0.301916
$598$	0	0
$599$	−37.1445	−1.51768	−0.758841	−	0.651276i	$-0.774234\pi$
$599$	−37.1445	−1.51768	−0.758841	+	0.651276i	$0.774234\pi$
$600$	0	0
$601$	−1.11320	−0.0454085	−0.0227043	−	0.999742i	$-0.507228\pi$
$601$	−1.11320	−0.0454085	−0.0227043	+	0.999742i	$0.507228\pi$
$602$	0.311445	0.0126935
$603$	8.96936	0.365261
$604$	7.28606	0.296466
$605$	0	0
$606$	−4.27160	−0.173522
$607$	−8.21780	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$607$	−8.21780	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$608$	23.9415	0.970955
$609$	−22.9943	−0.931776
$610$	0	0
$611$	0	0
$612$	2.90289	0.117342
$613$	−28.6108	−1.15558	−0.577790	−	0.816185i	$-0.696085\pi$
$613$	−28.6108	−1.15558	−0.577790	+	0.816185i	$0.696085\pi$
$614$	7.44537	0.300471
$615$	0	0
$616$	−20.4153	−0.822555
$617$	0.844622	0.0340032	0.0170016	−	0.999855i	$-0.494588\pi$
$617$	0.844622	0.0340032	0.0170016	+	0.999855i	$0.494588\pi$
$618$	16.4801	0.662925
$619$	−9.06143	−0.364210	−0.182105	−	0.983279i	$-0.558291\pi$
$619$	−9.06143	−0.364210	−0.182105	+	0.983279i	$0.558291\pi$
$620$	0	0
$621$	−48.0973	−1.93008
$622$	2.50655	0.100503
$623$	35.9012	1.43835
$624$	0	0
$625$	0	0
$626$	−17.0441	−0.681219
$627$	17.0064	0.679171
$628$	−27.6076	−1.10166
$629$	15.1808	0.605297
$630$	0	0
$631$	14.1174	0.562003	0.281002	−	0.959707i	$-0.409333\pi$
$631$	14.1174	0.562003	0.281002	+	0.959707i	$0.409333\pi$
$632$	38.3564	1.52573
$633$	11.0569	0.439472
$634$	5.11819	0.203269
$635$	0	0
$636$	2.96970	0.117756
$637$	0	0
$638$	−22.1311	−0.876177
$639$	6.59208	0.260778
$640$	0	0
$641$	−9.19011	−0.362988	−0.181494	−	0.983392i	$-0.558093\pi$
$641$	−9.19011	−0.362988	−0.181494	+	0.983392i	$0.558093\pi$
$642$	−4.72232	−0.186375
$643$	−4.14150	−0.163325	−0.0816624	−	0.996660i	$-0.526023\pi$
$643$	−4.14150	−0.163325	−0.0816624	+	0.996660i	$0.526023\pi$
$644$	−23.8665	−0.940472
$645$	0	0
$646$	6.53706	0.257197
$647$	−4.86077	−0.191097	−0.0955483	−	0.995425i	$-0.530460\pi$
$647$	−4.86077	−0.191097	−0.0955483	+	0.995425i	$0.530460\pi$
$648$	−8.27236	−0.324969
$649$	−17.2956	−0.678911
$650$	0	0
$651$	12.0737	0.473204
$652$	1.84328	0.0721883
$653$	−42.8456	−1.67668	−0.838340	−	0.545148i	$-0.816473\pi$
$653$	−42.8456	−1.67668	−0.838340	+	0.545148i	$0.816473\pi$
$654$	−18.1929	−0.711398
$655$	0	0
$656$	−0.319714	−0.0124827
$657$	−18.3258	−0.714958
$658$	−13.2626	−0.517031
$659$	0.355635	0.0138536	0.00692678	−	0.999976i	$-0.497795\pi$
$659$	0.355635	0.0138536	0.00692678	+	0.999976i	$0.497795\pi$
$660$	0	0
$661$	−12.9399	−0.503303	−0.251651	−	0.967818i	$-0.580974\pi$
$661$	−12.9399	−0.503303	−0.251651	+	0.967818i	$0.580974\pi$
$662$	3.60485	0.140107
$663$	0	0
$664$	18.0365	0.699952
$665$	0	0
$666$	10.8617	0.420883
$667$	−68.5774	−2.65533
$668$	0.0596123	0.00230647
$669$	0.0576436	0.00222863
$670$	0	0
$671$	31.0707	1.19947
$672$	−16.2323	−0.626174
$673$	27.5011	1.06009	0.530045	−	0.847970i	$-0.322175\pi$
$673$	27.5011	1.06009	0.530045	+	0.847970i	$0.322175\pi$
$674$	−22.0494	−0.849312
$675$	0	0
$676$	0	0
$677$	12.6270	0.485297	0.242648	−	0.970114i	$-0.421984\pi$
$677$	12.6270	0.485297	0.242648	+	0.970114i	$0.421984\pi$
$678$	−3.96551	−0.152295
$679$	27.5613	1.05770
$680$	0	0
$681$	−8.15069	−0.312335
$682$	11.6204	0.444969
$683$	−29.0442	−1.11135	−0.555673	−	0.831401i	$-0.687539\pi$
$683$	−29.0442	−1.11135	−0.555673	+	0.831401i	$0.687539\pi$
$684$	−7.18901	−0.274879
$685$	0	0
$686$	17.8158	0.680209
$687$	−9.23846	−0.352469
$688$	0.0166814	0.000635972 0
$689$	0	0
$690$	0	0
$691$	5.54360	0.210889	0.105444	−	0.994425i	$-0.466374\pi$
$691$	5.54360	0.210889	0.105444	+	0.994425i	$0.466374\pi$
$692$	10.9107	0.414763
$693$	9.94762	0.377879
$694$	−25.8904	−0.982785
$695$	0	0
$696$	−28.7426	−1.08949
$697$	5.08130	0.192468
$698$	−6.47297	−0.245006
$699$	5.78831	0.218934
$700$	0	0
$701$	−7.14386	−0.269820	−0.134910	−	0.990858i	$-0.543075\pi$
$701$	−7.14386	−0.269820	−0.134910	+	0.990858i	$0.543075\pi$
$702$	0	0
$703$	−37.5953	−1.41793
$704$	−16.3038	−0.614474
$705$	0	0
$706$	−12.3889	−0.466262
$707$	8.64831	0.325253
$708$	−8.47449	−0.318491
$709$	−33.2696	−1.24947	−0.624733	−	0.780838i	$-0.714792\pi$
$709$	−33.2696	−1.24947	−0.624733	+	0.780838i	$0.714792\pi$
$710$	0	0
$711$	−18.6897	−0.700917
$712$	44.8761	1.68180
$713$	36.0081	1.34851
$714$	−4.43211	−0.165868
$715$	0	0
$716$	−27.6994	−1.03517
$717$	21.4902	0.802567
$718$	−14.4402	−0.538902
$719$	2.93123	0.109317	0.0546583	−	0.998505i	$-0.482593\pi$
$719$	2.93123	0.109317	0.0546583	+	0.998505i	$0.482593\pi$
$720$	0	0
$721$	−33.3656	−1.24260
$722$	0.680581	0.0253286
$723$	7.78013	0.289346
$724$	21.4634	0.797680
$725$	0	0
$726$	1.29707	0.0481389
$727$	−14.9385	−0.554039	−0.277019	−	0.960864i	$-0.589347\pi$
$727$	−14.9385	−0.554039	−0.277019	+	0.960864i	$0.589347\pi$
$728$	0	0
$729$	25.2390	0.934777
$730$	0	0
$731$	−0.265122	−0.00980589
$732$	15.2240	0.562696
$733$	−9.64485	−0.356241	−0.178120	−	0.984009i	$-0.557002\pi$
$733$	−9.64485	−0.356241	−0.178120	+	0.984009i	$0.557002\pi$
$734$	30.5855	1.12893
$735$	0	0
$736$	−48.4106	−1.78444
$737$	20.2585	0.746230
$738$	3.63563	0.133830
$739$	−38.6006	−1.41995	−0.709973	−	0.704229i	$-0.751293\pi$
$739$	−38.6006	−1.41995	−0.709973	+	0.704229i	$0.751293\pi$
$740$	0	0
$741$	0	0
$742$	3.91175	0.143605
$743$	−43.3909	−1.59186	−0.795928	−	0.605391i	$-0.793017\pi$
$743$	−43.3909	−1.59186	−0.795928	+	0.605391i	$0.793017\pi$
$744$	15.0919	0.553298
$745$	0	0
$746$	20.6146	0.754754
$747$	−8.78853	−0.321555
$748$	6.55654	0.239731
$749$	9.56084	0.349345
$750$	0	0
$751$	−34.8770	−1.27268	−0.636341	−	0.771408i	$-0.719553\pi$
$751$	−34.8770	−1.27268	−0.636341	+	0.771408i	$0.719553\pi$
$752$	−0.710364	−0.0259043
$753$	14.7012	0.535742
$754$	0	0
$755$	0	0
$756$	15.3979	0.560016
$757$	11.6518	0.423493	0.211746	−	0.977325i	$-0.432085\pi$
$757$	11.6518	0.423493	0.211746	+	0.977325i	$0.432085\pi$
$758$	−10.8912	−0.395588
$759$	−34.3876	−1.24819
$760$	0	0
$761$	49.0684	1.77873	0.889364	−	0.457200i	$-0.151148\pi$
$761$	49.0684	1.77873	0.889364	+	0.457200i	$0.151148\pi$
$762$	−2.72952	−0.0988803
$763$	36.8334	1.33346
$764$	−11.7959	−0.426759
$765$	0	0
$766$	−1.67215	−0.0604172
$767$	0	0
$768$	−20.6238	−0.744197
$769$	−40.1202	−1.44677	−0.723386	−	0.690444i	$-0.757415\pi$
$769$	−40.1202	−1.44677	−0.723386	+	0.690444i	$0.757415\pi$
$770$	0	0
$771$	15.8214	0.569792
$772$	−12.1179	−0.436133
$773$	−11.6461	−0.418883	−0.209441	−	0.977821i	$-0.567165\pi$
$773$	−11.6461	−0.418883	−0.209441	+	0.977821i	$0.567165\pi$
$774$	−0.189693	−0.00681836
$775$	0	0
$776$	34.4512	1.23673
$777$	25.4895	0.914431
$778$	17.7964	0.638032
$779$	−12.5839	−0.450864
$780$	0	0
$781$	14.8891	0.532772
$782$	−13.2182	−0.472681
$783$	44.2439	1.58115
$784$	0.194820	0.00695786
$785$	0	0
$786$	5.64117	0.201214
$787$	−25.5641	−0.911262	−0.455631	−	0.890169i	$-0.650586\pi$
$787$	−25.5641	−0.911262	−0.455631	+	0.890169i	$0.650586\pi$
$788$	7.27005	0.258985
$789$	35.9512	1.27990
$790$	0	0
$791$	8.02860	0.285464
$792$	12.4344	0.441837
$793$	0	0
$794$	−18.3022	−0.649519
$795$	0	0
$796$	−7.04328	−0.249642
$797$	45.7347	1.62001	0.810003	−	0.586425i	$-0.199465\pi$
$797$	45.7347	1.62001	0.810003	+	0.586425i	$0.199465\pi$
$798$	10.9761	0.388551
$799$	11.2900	0.399412
$800$	0	0
$801$	−21.8665	−0.772614
$802$	31.7991	1.12287
$803$	−41.3912	−1.46067
$804$	9.92625	0.350072
$805$	0	0
$806$	0	0
$807$	−15.4682	−0.544506
$808$	10.8103	0.380305
$809$	31.4562	1.10594	0.552970	−	0.833201i	$-0.313494\pi$
$809$	31.4562	1.10594	0.552970	+	0.833201i	$0.313494\pi$
$810$	0	0
$811$	−38.0128	−1.33481	−0.667406	−	0.744694i	$-0.732595\pi$
$811$	−38.0128	−1.33481	−0.667406	+	0.744694i	$0.732595\pi$
$812$	21.9544	0.770449
$813$	22.4050	0.785779
$814$	24.5326	0.859868
$815$	0	0
$816$	−0.237390	−0.00831031
$817$	0.656576	0.0229707
$818$	0.197063	0.00689015
$819$	0	0
$820$	0	0
$821$	−55.7593	−1.94601	−0.973006	−	0.230778i	$-0.925873\pi$
$821$	−55.7593	−1.94601	−0.973006	+	0.230778i	$0.925873\pi$
$822$	14.4401	0.503656
$823$	−30.7203	−1.07084	−0.535420	−	0.844586i	$-0.679847\pi$
$823$	−30.7203	−1.07084	−0.535420	+	0.844586i	$0.679847\pi$
$824$	−41.7067	−1.45292
$825$	0	0
$826$	−11.1628	−0.388403
$827$	37.7024	1.31104	0.655520	−	0.755177i	$-0.272449\pi$
$827$	37.7024	1.31104	0.655520	+	0.755177i	$0.272449\pi$
$828$	14.5365	0.505177
$829$	−13.8208	−0.480015	−0.240008	−	0.970771i	$-0.577150\pi$
$829$	−13.8208	−0.480015	−0.240008	+	0.970771i	$0.577150\pi$
$830$	0	0
$831$	14.8799	0.516177
$832$	0	0
$833$	−3.09633	−0.107281
$834$	17.4975	0.605890
$835$	0	0
$836$	−16.2373	−0.561579
$837$	−23.2313	−0.802990
$838$	−0.604083	−0.0208677
$839$	16.0902	0.555495	0.277747	−	0.960654i	$-0.410412\pi$
$839$	16.0902	0.555495	0.277747	+	0.960654i	$0.410412\pi$
$840$	0	0
$841$	34.0833	1.17528
$842$	1.79290	0.0617875
$843$	25.7166	0.885727
$844$	−10.5569	−0.363382
$845$	0	0
$846$	8.07792	0.277725
$847$	−2.62607	−0.0902327
$848$	0.209519	0.00719490
$849$	10.4613	0.359031
$850$	0	0
$851$	76.0191	2.60590
$852$	7.29534	0.249934
$853$	48.9861	1.67725	0.838625	−	0.544709i	$-0.183360\pi$
$853$	48.9861	1.67725	0.838625	+	0.544709i	$0.183360\pi$
$854$	20.0534	0.686213
$855$	0	0
$856$	11.9509	0.408475
$857$	−8.57696	−0.292983	−0.146492	−	0.989212i	$-0.546798\pi$
$857$	−8.57696	−0.292983	−0.146492	+	0.989212i	$0.546798\pi$
$858$	0	0
$859$	−17.9983	−0.614095	−0.307048	−	0.951694i	$-0.599341\pi$
$859$	−17.9983	−0.614095	−0.307048	+	0.951694i	$0.599341\pi$
$860$	0	0
$861$	8.53184	0.290764
$862$	−6.08548	−0.207272
$863$	2.18400	0.0743444	0.0371722	−	0.999309i	$-0.488165\pi$
$863$	2.18400	0.0743444	0.0371722	+	0.999309i	$0.488165\pi$
$864$	31.2329	1.06257
$865$	0	0
$866$	−0.462525	−0.0157172
$867$	−17.8013	−0.604562
$868$	−11.5277	−0.391274
$869$	−42.2130	−1.43198
$870$	0	0
$871$	0	0
$872$	46.0414	1.55916
$873$	−16.7868	−0.568148
$874$	32.7349	1.10727
$875$	0	0
$876$	−20.2809	−0.685228
$877$	−27.9605	−0.944158	−0.472079	−	0.881556i	$-0.656496\pi$
$877$	−27.9605	−0.944158	−0.472079	+	0.881556i	$0.656496\pi$
$878$	24.4956	0.826685
$879$	27.0584	0.912658
$880$	0	0
$881$	−3.38544	−0.114058	−0.0570292	−	0.998373i	$-0.518163\pi$
$881$	−3.38544	−0.114058	−0.0570292	+	0.998373i	$0.518163\pi$
$882$	−2.21540	−0.0745964
$883$	−9.67005	−0.325423	−0.162712	−	0.986674i	$-0.552024\pi$
$883$	−9.67005	−0.325423	−0.162712	+	0.986674i	$0.552024\pi$
$884$	0	0
$885$	0	0
$886$	−6.39878	−0.214971
$887$	−10.1393	−0.340445	−0.170223	−	0.985406i	$-0.554449\pi$
$887$	−10.1393	−0.340445	−0.170223	+	0.985406i	$0.554449\pi$
$888$	31.8616	1.06920
$889$	5.52622	0.185343
$890$	0	0
$891$	9.10413	0.305000
$892$	−0.0550368	−0.00184277
$893$	−27.9598	−0.935639
$894$	0.0466273	0.00155945
$895$	0	0
$896$	15.0587	0.503076
$897$	0	0
$898$	−0.120427	−0.00401869
$899$	−33.1233	−1.10472
$900$	0	0
$901$	−3.32994	−0.110936
$902$	8.21155	0.273415
$903$	−0.445157	−0.0148139
$904$	10.0357	0.333781
$905$	0	0
$906$	6.77553	0.225102
$907$	48.7631	1.61915	0.809577	−	0.587014i	$-0.199697\pi$
$907$	48.7631	1.61915	0.809577	+	0.587014i	$0.199697\pi$
$908$	7.78208	0.258258
$909$	−5.26746	−0.174711
$910$	0	0
$911$	−53.5943	−1.77566	−0.887829	−	0.460174i	$-0.847787\pi$
$911$	−53.5943	−1.77566	−0.887829	+	0.460174i	$0.847787\pi$
$912$	0.587897	0.0194672
$913$	−19.8500	−0.656940
$914$	0.273475	0.00904575
$915$	0	0
$916$	8.82066	0.291443
$917$	−11.4212	−0.377160
$918$	8.52794	0.281464
$919$	−28.2142	−0.930701	−0.465350	−	0.885127i	$-0.654072\pi$
$919$	−28.2142	−0.930701	−0.465350	+	0.885127i	$0.654072\pi$
$920$	0	0
$921$	−10.6419	−0.350662
$922$	−25.4572	−0.838390
$923$	0	0
$924$	11.0089	0.362165
$925$	0	0
$926$	−14.2969	−0.469824
$927$	20.3221	0.667466
$928$	44.5321	1.46184
$929$	−2.90575	−0.0953346	−0.0476673	−	0.998863i	$-0.515179\pi$
$929$	−2.90575	−0.0953346	−0.0476673	+	0.998863i	$0.515179\pi$
$930$	0	0
$931$	7.66808	0.251311
$932$	−5.52654	−0.181028
$933$	−3.58269	−0.117292
$934$	3.08616	0.100982
$935$	0	0
$936$	0	0
$937$	−44.6821	−1.45970	−0.729850	−	0.683607i	$-0.760410\pi$
$937$	−44.6821	−1.45970	−0.729850	+	0.683607i	$0.760410\pi$
$938$	13.0751	0.426916
$939$	24.3616	0.795012
$940$	0	0
$941$	5.09938	0.166235	0.0831175	−	0.996540i	$-0.473512\pi$
$941$	5.09938	0.166235	0.0831175	+	0.996540i	$0.473512\pi$
$942$	−25.6731	−0.836476
$943$	25.4451	0.828606
$944$	−0.597894	−0.0194598
$945$	0	0
$946$	−0.428446	−0.0139300
$947$	−6.85210	−0.222663	−0.111332	−	0.993783i	$-0.535512\pi$
$947$	−6.85210	−0.222663	−0.111332	+	0.993783i	$0.535512\pi$
$948$	−20.6835	−0.671770
$949$	0	0
$950$	0	0
$951$	−7.31558	−0.237224
$952$	11.2165	0.363529
$953$	31.9155	1.03385	0.516923	−	0.856032i	$-0.327077\pi$
$953$	31.9155	1.03385	0.516923	+	0.856032i	$0.327077\pi$
$954$	−2.38254	−0.0771377
$955$	0	0
$956$	−20.5183	−0.663611
$957$	31.6326	1.02254
$958$	12.8366	0.414732
$959$	−29.2355	−0.944064
$960$	0	0
$961$	−13.6079	−0.438964
$962$	0	0
$963$	−5.82325	−0.187652
$964$	−7.42829	−0.239249
$965$	0	0
$966$	−22.1942	−0.714087
$967$	−15.8568	−0.509920	−0.254960	−	0.966952i	$-0.582062\pi$
$967$	−15.8568	−0.509920	−0.254960	+	0.966952i	$0.582062\pi$
$968$	−3.28255	−0.105505
$969$	−9.34361	−0.300160
$970$	0	0
$971$	18.3168	0.587813	0.293907	−	0.955834i	$-0.405045\pi$
$971$	18.3168	0.587813	0.293907	+	0.955834i	$0.405045\pi$
$972$	−15.7882	−0.506407
$973$	−35.4256	−1.13569
$974$	2.58619	0.0828669
$975$	0	0
$976$	1.07409	0.0343807
$977$	13.5382	0.433127	0.216563	−	0.976269i	$-0.430515\pi$
$977$	13.5382	0.433127	0.216563	+	0.976269i	$0.430515\pi$
$978$	1.71412	0.0548115
$979$	−49.3883	−1.57846
$980$	0	0
$981$	−22.4343	−0.716271
$982$	6.75464	0.215549
$983$	−34.5313	−1.10138	−0.550688	−	0.834711i	$-0.685635\pi$
$983$	−34.5313	−1.10138	−0.550688	+	0.834711i	$0.685635\pi$
$984$	10.6647	0.339978
$985$	0	0
$986$	12.1592	0.387227
$987$	18.9567	0.603398
$988$	0	0
$989$	−1.32762	−0.0422159
$990$	0	0
$991$	43.5272	1.38269	0.691343	−	0.722526i	$-0.257019\pi$
$991$	43.5272	1.38269	0.691343	+	0.722526i	$0.257019\pi$
$992$	−23.3826	−0.742398
$993$	−5.15253	−0.163510
$994$	9.60958	0.304797
$995$	0	0
$996$	−9.72612	−0.308184
$997$	−45.3802	−1.43720	−0.718602	−	0.695422i	$-0.755218\pi$
$997$	−45.3802	−1.43720	−0.718602	+	0.695422i	$0.755218\pi$
$998$	3.61584	0.114457
$999$	−49.0450	−1.55172

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.cb.1.7		18
5.2	odd	4		845.2.b.h.339.7	yes	18
5.3	odd	4		845.2.b.h.339.12	yes	18
5.4	even	2	inner	4225.2.a.cb.1.12		18
13.12	even	2		4225.2.a.ca.1.12		18
65.2	even	12		845.2.l.g.654.24		72
65.3	odd	12		845.2.n.h.529.7		36
65.7	even	12		845.2.l.g.699.13		72
65.8	even	4		845.2.d.e.844.14		36
65.12	odd	4		845.2.b.g.339.12	yes	18
65.17	odd	12		845.2.n.i.484.12		36
65.18	even	4		845.2.d.e.844.24		36
65.22	odd	12		845.2.n.h.484.7		36
65.23	odd	12		845.2.n.i.529.12		36
65.28	even	12		845.2.l.g.654.13		72
65.32	even	12		845.2.l.g.699.23		72
65.33	even	12		845.2.l.g.699.24		72
65.37	even	12		845.2.l.g.654.14		72
65.38	odd	4		845.2.b.g.339.7	✓	18
65.42	odd	12		845.2.n.h.529.12		36
65.43	odd	12		845.2.n.i.484.7		36
65.47	even	4		845.2.d.e.844.23		36
65.48	odd	12		845.2.n.h.484.12		36
65.57	even	4		845.2.d.e.844.13		36
65.58	even	12		845.2.l.g.699.14		72
65.62	odd	12		845.2.n.i.529.7		36
65.63	even	12		845.2.l.g.654.23		72
65.64	even	2		4225.2.a.ca.1.7		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.b.g.339.7	✓	18	65.38	odd	4
845.2.b.g.339.12	yes	18	65.12	odd	4
845.2.b.h.339.7	yes	18	5.2	odd	4
845.2.b.h.339.12	yes	18	5.3	odd	4
845.2.d.e.844.13		36	65.57	even	4
845.2.d.e.844.14		36	65.8	even	4
845.2.d.e.844.23		36	65.47	even	4
845.2.d.e.844.24		36	65.18	even	4
845.2.l.g.654.13		72	65.28	even	12
845.2.l.g.654.14		72	65.37	even	12
845.2.l.g.654.23		72	65.63	even	12
845.2.l.g.654.24		72	65.2	even	12
845.2.l.g.699.13		72	65.7	even	12
845.2.l.g.699.14		72	65.58	even	12
845.2.l.g.699.23		72	65.32	even	12
845.2.l.g.699.24		72	65.33	even	12
845.2.n.h.484.7		36	65.22	odd	12
845.2.n.h.484.12		36	65.48	odd	12
845.2.n.h.529.7		36	65.3	odd	12
845.2.n.h.529.12		36	65.42	odd	12
845.2.n.i.484.7		36	65.43	odd	12
845.2.n.i.484.12		36	65.17	odd	12
845.2.n.i.529.7		36	65.62	odd	12
845.2.n.i.529.12		36	65.23	odd	12
4225.2.a.ca.1.7		18	65.64	even	2
4225.2.a.ca.1.12		18	13.12	even	2
4225.2.a.cb.1.7		18	1.1	even	1	trivial
4225.2.a.cb.1.12		18	5.4	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q-0.887876 q^{2} +1.26907 q^{3} -1.21168 q^{4} -1.12678 q^{6} +2.28128 q^{7} +2.85157 q^{8} -1.38947 q^{9} -3.13829 q^{11} -1.53770 q^{12} -2.02549 q^{14} -0.108488 q^{16} +1.72423 q^{17} +1.23367 q^{18} -4.27007 q^{19} +2.89510 q^{21} +2.78641 q^{22} +8.63424 q^{23} +3.61884 q^{24} -5.57053 q^{27} -2.76417 q^{28} -7.94250 q^{29} +4.17039 q^{31} -5.60682 q^{32} -3.98270 q^{33} -1.53090 q^{34} +1.68358 q^{36} +8.80437 q^{37} +3.79129 q^{38} +2.94700 q^{41} -2.57049 q^{42} -0.153762 q^{43} +3.80259 q^{44} -7.66614 q^{46} +6.54786 q^{47} -0.137679 q^{48} -1.79577 q^{49} +2.18817 q^{51} -1.93126 q^{53} +4.94594 q^{54} +6.50522 q^{56} -5.41901 q^{57} +7.05195 q^{58} +5.51115 q^{59} -9.90052 q^{61} -3.70279 q^{62} -3.16976 q^{63} +5.19513 q^{64} +3.53615 q^{66} -6.45526 q^{67} -2.08921 q^{68} +10.9574 q^{69} -4.74432 q^{71} -3.96216 q^{72} +13.1891 q^{73} -7.81719 q^{74} +5.17394 q^{76} -7.15931 q^{77} +13.4510 q^{79} -2.90099 q^{81} -2.61657 q^{82} +6.32511 q^{83} -3.50792 q^{84} +0.136522 q^{86} -10.0796 q^{87} -8.94905 q^{88} +15.7373 q^{89} -10.4619 q^{92} +5.29250 q^{93} -5.81368 q^{94} -7.11543 q^{96} +12.0815 q^{97} +1.59442 q^{98} +4.36055 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 16 q^{4} + 16 q^{6} + 18 q^{9} + 22 q^{11} + 4 q^{14} - 12 q^{16} + 28 q^{19} + 26 q^{21} + 34 q^{24} - 20 q^{29} + 32 q^{31} + 18 q^{34} + 32 q^{36} + 52 q^{41} + 50 q^{44} + 30 q^{46} + 44 q^{49}+ \cdots + 76 q^{99}+O(q^{100}) \) 18 * q + 16 * q^4 + 16 * q^6 + 18 * q^9 + 22 * q^11 + 4 * q^14 - 12 * q^16 + 28 * q^19 + 26 * q^21 + 34 * q^24 - 20 * q^29 + 32 * q^31 + 18 * q^34 + 32 * q^36 + 52 * q^41 + 50 * q^44 + 30 * q^46 + 44 * q^49 - 40 * q^51 + 90 * q^54 - 20 * q^56 + 76 * q^59 + 8 * q^61 - 68 * q^64 + 8 * q^66 + 30 * q^69 + 72 * q^71 + 30 * q^74 + 4 * q^76 + 16 * q^79 - 30 * q^81 + 78 * q^84 - 30 * q^86 + 94 * q^89 - 128 * q^94 - 18 * q^96 + 76 * q^99

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