LMFDB - Embedded newform 5625.2.a.r.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5625,2,Mod(1,5625)] 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("5625.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [6,1,0,11,0,0,2,6,0,0,0,0,0,4,0,17] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	yes
Analytic conductor:	$44.9158511370$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.46840000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 11x^{4} + 8x^{3} + 31x^{2} - 15x - 9 $ x^6 - x^5 - 11x^4 + 8x^3 + 31x^2 - 15x - 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			1.3
Root			$-0.364088$ 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)

$f(q)$	$=$	$q-0.364088 q^{2} -1.86744 q^{4} -2.24941 q^{7} +1.40809 q^{8} -4.63962 q^{11} -3.75730 q^{13} +0.818981 q^{14} +3.22221 q^{16} -5.36779 q^{17} -5.66369 q^{19} +1.68923 q^{22} +1.32214 q^{23} +1.36799 q^{26} +4.20063 q^{28} -8.86744 q^{29} +1.21200 q^{31} -3.98934 q^{32} +1.95435 q^{34} +2.15975 q^{37} +2.06208 q^{38} -3.49965 q^{41} -1.16800 q^{43} +8.66420 q^{44} -0.481374 q^{46} -2.36614 q^{47} -1.94017 q^{49} +7.01653 q^{52} -14.1656 q^{53} -3.16736 q^{56} +3.22853 q^{58} -0.367089 q^{59} +5.29590 q^{61} -0.441273 q^{62} -4.99195 q^{64} -8.06723 q^{67} +10.0240 q^{68} -11.4185 q^{71} +13.7580 q^{73} -0.786340 q^{74} +10.5766 q^{76} +10.4364 q^{77} +11.6247 q^{79} +1.27418 q^{82} +11.1027 q^{83} +0.425253 q^{86} -6.53299 q^{88} -16.6526 q^{89} +8.45169 q^{91} -2.46901 q^{92} +0.861482 q^{94} -14.4790 q^{97} +0.706393 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38}+ \cdots + 15 q^{98}+O(q^{100}) $ 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

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 $

$2$	−0.364088	−0.257449	−0.128724	−	0.991680i	$-0.541088\pi$
$2$	−0.364088	−0.257449	−0.128724	+	0.991680i	$0.541088\pi$
$3$	0	0
$3$	0	0
$4$	−1.86744	−0.933720
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−2.24941	−0.850196	−0.425098	−	0.905147i	$-0.639760\pi$
$7$	−2.24941	−0.850196	−0.425098	+	0.905147i	$0.639760\pi$
$8$	1.40809	0.497834
$9$	0	0
$10$	0	0
$11$	−4.63962	−1.39890	−0.699448	−	0.714683i	$-0.746571\pi$
$11$	−4.63962	−1.39890	−0.699448	+	0.714683i	$0.746571\pi$
$12$	0	0
$13$	−3.75730	−1.04209	−0.521044	−	0.853530i	$-0.674457\pi$
$13$	−3.75730	−1.04209	−0.521044	+	0.853530i	$0.674457\pi$
$14$	0.818981	0.218882
$15$	0	0
$16$	3.22221	0.805553
$17$	−5.36779	−1.30188	−0.650940	−	0.759129i	$-0.725625\pi$
$17$	−5.36779	−1.30188	−0.650940	+	0.759129i	$0.725625\pi$
$18$	0	0
$19$	−5.66369	−1.29934	−0.649669	−	0.760217i	$-0.725093\pi$
$19$	−5.66369	−1.29934	−0.649669	+	0.760217i	$0.725093\pi$
$20$	0	0
$21$	0	0
$22$	1.68923	0.360144
$23$	1.32214	0.275685	0.137842	−	0.990454i	$-0.455983\pi$
$23$	1.32214	0.275685	0.137842	+	0.990454i	$0.455983\pi$
$24$	0	0
$25$	0	0
$26$	1.36799	0.268284
$27$	0	0
$28$	4.20063	0.793845
$29$	−8.86744	−1.64664	−0.823321	−	0.567576i	$-0.807881\pi$
$29$	−8.86744	−1.64664	−0.823321	+	0.567576i	$0.807881\pi$
$30$	0	0
$31$	1.21200	0.217681	0.108841	−	0.994059i	$-0.465286\pi$
$31$	1.21200	0.217681	0.108841	+	0.994059i	$0.465286\pi$
$32$	−3.98934	−0.705223
$33$	0	0
$34$	1.95435	0.335168
$35$	0	0
$36$	0	0
$37$	2.15975	0.355061	0.177531	−	0.984115i	$-0.443189\pi$
$37$	2.15975	0.355061	0.177531	+	0.984115i	$0.443189\pi$
$38$	2.06208	0.334513
$39$	0	0
$40$	0	0
$41$	−3.49965	−0.546553	−0.273277	−	0.961935i	$-0.588107\pi$
$41$	−3.49965	−0.546553	−0.273277	+	0.961935i	$0.588107\pi$
$42$	0	0
$43$	−1.16800	−0.178118	−0.0890589	−	0.996026i	$-0.528386\pi$
$43$	−1.16800	−0.178118	−0.0890589	+	0.996026i	$0.528386\pi$
$44$	8.66420	1.30618
$45$	0	0
$46$	−0.481374	−0.0709748
$47$	−2.36614	−0.345137	−0.172568	−	0.984998i	$-0.555207\pi$
$47$	−2.36614	−0.345137	−0.172568	+	0.984998i	$0.555207\pi$
$48$	0	0
$49$	−1.94017	−0.277167
$50$	0	0
$51$	0	0
$52$	7.01653	0.973018
$53$	−14.1656	−1.94580	−0.972901	−	0.231223i	$-0.925727\pi$
$53$	−14.1656	−1.94580	−0.972901	+	0.231223i	$0.925727\pi$
$54$	0	0
$55$	0	0
$56$	−3.16736	−0.423256
$57$	0	0
$58$	3.22853	0.423926
$59$	−0.367089	−0.0477909	−0.0238955	−	0.999714i	$-0.507607\pi$
$59$	−0.367089	−0.0477909	−0.0238955	+	0.999714i	$0.507607\pi$
$60$	0	0
$61$	5.29590	0.678070	0.339035	−	0.940774i	$-0.389899\pi$
$61$	5.29590	0.678070	0.339035	+	0.940774i	$0.389899\pi$
$62$	−0.441273	−0.0560417
$63$	0	0
$64$	−4.99195	−0.623994
$65$	0	0
$66$	0	0
$67$	−8.06723	−0.985570	−0.492785	−	0.870151i	$-0.664021\pi$
$67$	−8.06723	−0.985570	−0.492785	+	0.870151i	$0.664021\pi$
$68$	10.0240	1.21559
$69$	0	0
$70$	0	0
$71$	−11.4185	−1.35513	−0.677563	−	0.735465i	$-0.736964\pi$
$71$	−11.4185	−1.35513	−0.677563	+	0.735465i	$0.736964\pi$
$72$	0	0
$73$	13.7580	1.61025	0.805126	−	0.593104i	$-0.202098\pi$
$73$	13.7580	1.61025	0.805126	+	0.593104i	$0.202098\pi$
$74$	−0.786340	−0.0914101
$75$	0	0
$76$	10.5766	1.21322
$77$	10.4364	1.18934
$78$	0	0
$79$	11.6247	1.30789	0.653943	−	0.756544i	$-0.273114\pi$
$79$	11.6247	1.30789	0.653943	+	0.756544i	$0.273114\pi$
$80$	0	0
$81$	0	0
$82$	1.27418	0.140710
$83$	11.1027	1.21868	0.609338	−	0.792910i	$-0.291435\pi$
$83$	11.1027	1.21868	0.609338	+	0.792910i	$0.291435\pi$
$84$	0	0
$85$	0	0
$86$	0.425253	0.0458562
$87$	0	0
$88$	−6.53299	−0.696419
$89$	−16.6526	−1.76518	−0.882588	−	0.470148i	$-0.844201\pi$
$89$	−16.6526	−1.76518	−0.882588	+	0.470148i	$0.844201\pi$
$90$	0	0
$91$	8.45169	0.885978
$92$	−2.46901	−0.257412
$93$	0	0
$94$	0.861482	0.0888551
$95$	0	0
$96$	0	0
$97$	−14.4790	−1.47012	−0.735061	−	0.678001i	$-0.762847\pi$
$97$	−14.4790	−1.47012	−0.735061	+	0.678001i	$0.762847\pi$
$98$	0.706393	0.0713565
$99$	0	0
$100$	0	0
$101$	−2.16089	−0.215017	−0.107508	−	0.994204i	$-0.534287\pi$
$101$	−2.16089	−0.215017	−0.107508	+	0.994204i	$0.534287\pi$
$102$	0	0
$103$	7.13512	0.703044	0.351522	−	0.936180i	$-0.385664\pi$
$103$	7.13512	0.703044	0.351522	+	0.936180i	$0.385664\pi$
$104$	−5.29061	−0.518787
$105$	0	0
$106$	5.15754	0.500944
$107$	−14.3985	−1.39196	−0.695979	−	0.718062i	$-0.745029\pi$
$107$	−14.3985	−1.39196	−0.695979	+	0.718062i	$0.745029\pi$
$108$	0	0
$109$	16.4693	1.57748	0.788738	−	0.614730i	$-0.210735\pi$
$109$	16.4693	1.57748	0.788738	+	0.614730i	$0.210735\pi$
$110$	0	0
$111$	0	0
$112$	−7.24806	−0.684878
$113$	16.2639	1.52998	0.764990	−	0.644042i	$-0.222744\pi$
$113$	16.2639	1.52998	0.764990	+	0.644042i	$0.222744\pi$
$114$	0	0
$115$	0	0
$116$	16.5594	1.53750
$117$	0	0
$118$	0.133653	0.0123037
$119$	12.0743	1.10685
$120$	0	0
$121$	10.5260	0.956912
$122$	−1.92817	−0.174568
$123$	0	0
$124$	−2.26333	−0.203253
$125$	0	0
$126$	0	0
$127$	−5.14048	−0.456144	−0.228072	−	0.973644i	$-0.573242\pi$
$127$	−5.14048	−0.456144	−0.228072	+	0.973644i	$0.573242\pi$
$128$	9.79620	0.865870
$129$	0	0
$130$	0	0
$131$	−6.52438	−0.570038	−0.285019	−	0.958522i	$-0.592000\pi$
$131$	−6.52438	−0.570038	−0.285019	+	0.958522i	$0.592000\pi$
$132$	0	0
$133$	12.7399	1.10469
$134$	2.93718	0.253734
$135$	0	0
$136$	−7.55832	−0.648121
$137$	13.5071	1.15399	0.576993	−	0.816749i	$-0.304226\pi$
$137$	13.5071	1.15399	0.576993	+	0.816749i	$0.304226\pi$
$138$	0	0
$139$	−21.1376	−1.79287	−0.896435	−	0.443176i	$-0.853852\pi$
$139$	−21.1376	−1.79287	−0.896435	+	0.443176i	$0.853852\pi$
$140$	0	0
$141$	0	0
$142$	4.15733	0.348876
$143$	17.4324	1.45777
$144$	0	0
$145$	0	0
$146$	−5.00912	−0.414558
$147$	0	0
$148$	−4.03321	−0.331528
$149$	3.22853	0.264491	0.132246	−	0.991217i	$-0.457781\pi$
$149$	3.22853	0.264491	0.132246	+	0.991217i	$0.457781\pi$
$150$	0	0
$151$	−15.9240	−1.29588	−0.647939	−	0.761692i	$-0.724369\pi$
$151$	−15.9240	−1.29588	−0.647939	+	0.761692i	$0.724369\pi$
$152$	−7.97497	−0.646855
$153$	0	0
$154$	−3.79976	−0.306193
$155$	0	0
$156$	0	0
$157$	−3.21251	−0.256387	−0.128193	−	0.991749i	$-0.540918\pi$
$157$	−3.21251	−0.256387	−0.128193	+	0.991749i	$0.540918\pi$
$158$	−4.23243	−0.336714
$159$	0	0
$160$	0	0
$161$	−2.97403	−0.234386
$162$	0	0
$163$	13.7482	1.07685	0.538423	−	0.842675i	$-0.319020\pi$
$163$	13.7482	1.07685	0.538423	+	0.842675i	$0.319020\pi$
$164$	6.53538	0.510328
$165$	0	0
$166$	−4.04235	−0.313747
$167$	0.787486	0.0609375	0.0304688	−	0.999536i	$-0.490300\pi$
$167$	0.787486	0.0609375	0.0304688	+	0.999536i	$0.490300\pi$
$168$	0	0
$169$	1.11729	0.0859456
$170$	0	0
$171$	0	0
$172$	2.18116	0.166312
$173$	2.58986	0.196904	0.0984518	−	0.995142i	$-0.468611\pi$
$173$	2.58986	0.196904	0.0984518	+	0.995142i	$0.468611\pi$
$174$	0	0
$175$	0	0
$176$	−14.9498	−1.12689
$177$	0	0
$178$	6.06302	0.454443
$179$	−15.0630	−1.12586	−0.562930	−	0.826505i	$-0.690326\pi$
$179$	−15.0630	−1.12586	−0.562930	+	0.826505i	$0.690326\pi$
$180$	0	0
$181$	−12.5492	−0.932771	−0.466386	−	0.884582i	$-0.654444\pi$
$181$	−12.5492	−0.932771	−0.466386	+	0.884582i	$0.654444\pi$
$182$	−3.07716	−0.228094
$183$	0	0
$184$	1.86169	0.137245
$185$	0	0
$186$	0	0
$187$	24.9045	1.82120
$188$	4.41862	0.322261
$189$	0	0
$190$	0	0
$191$	−16.0684	−1.16267	−0.581334	−	0.813665i	$-0.697469\pi$
$191$	−16.0684	−1.16267	−0.581334	+	0.813665i	$0.697469\pi$
$192$	0	0
$193$	−8.48878	−0.611036	−0.305518	−	0.952186i	$-0.598830\pi$
$193$	−8.48878	−0.611036	−0.305518	+	0.952186i	$0.598830\pi$
$194$	5.27164	0.378481
$195$	0	0
$196$	3.62316	0.258797
$197$	−4.68061	−0.333479	−0.166740	−	0.986001i	$-0.553324\pi$
$197$	−4.68061	−0.333479	−0.166740	+	0.986001i	$0.553324\pi$
$198$	0	0
$199$	1.45880	0.103411	0.0517057	−	0.998662i	$-0.483534\pi$
$199$	1.45880	0.103411	0.0517057	+	0.998662i	$0.483534\pi$
$200$	0	0
$201$	0	0
$202$	0.786753	0.0553558
$203$	19.9465	1.39997
$204$	0	0
$205$	0	0
$206$	−2.59781	−0.180998
$207$	0	0
$208$	−12.1068	−0.839457
$209$	26.2773	1.81764
$210$	0	0
$211$	12.8107	0.881926	0.440963	−	0.897525i	$-0.354637\pi$
$211$	12.8107	0.881926	0.440963	+	0.897525i	$0.354637\pi$
$212$	26.4535	1.81683
$213$	0	0
$214$	5.24233	0.358358
$215$	0	0
$216$	0	0
$217$	−2.72627	−0.185071
$218$	−5.99628	−0.406119
$219$	0	0
$220$	0	0
$221$	20.1684	1.35667
$222$	0	0
$223$	0.223725	0.0149817	0.00749086	−	0.999972i	$-0.497616\pi$
$223$	0.223725	0.0149817	0.00749086	+	0.999972i	$0.497616\pi$
$224$	8.97365	0.599577
$225$	0	0
$226$	−5.92149	−0.393892
$227$	28.2672	1.87616	0.938081	−	0.346416i	$-0.112602\pi$
$227$	28.2672	1.87616	0.938081	+	0.346416i	$0.112602\pi$
$228$	0	0
$229$	−16.6618	−1.10104	−0.550521	−	0.834821i	$-0.685571\pi$
$229$	−16.6618	−1.10104	−0.550521	+	0.834821i	$0.685571\pi$
$230$	0	0
$231$	0	0
$232$	−12.4861	−0.819755
$233$	19.9807	1.30898	0.654491	−	0.756070i	$-0.272883\pi$
$233$	19.9807	1.30898	0.654491	+	0.756070i	$0.272883\pi$
$234$	0	0
$235$	0	0
$236$	0.685517	0.0446233
$237$	0	0
$238$	−4.39612	−0.284958
$239$	−18.6586	−1.20692	−0.603462	−	0.797392i	$-0.706212\pi$
$239$	−18.6586	−1.20692	−0.603462	+	0.797392i	$0.706212\pi$
$240$	0	0
$241$	7.10919	0.457943	0.228972	−	0.973433i	$-0.426464\pi$
$241$	7.10919	0.457943	0.228972	+	0.973433i	$0.426464\pi$
$242$	−3.83240	−0.246356
$243$	0	0
$244$	−9.88977	−0.633128
$245$	0	0
$246$	0	0
$247$	21.2802	1.35402
$248$	1.70660	0.108369
$249$	0	0
$250$	0	0
$251$	10.5636	0.666767	0.333384	−	0.942791i	$-0.391810\pi$
$251$	10.5636	0.666767	0.333384	+	0.942791i	$0.391810\pi$
$252$	0	0
$253$	−6.13421	−0.385655
$254$	1.87159	0.117434
$255$	0	0
$256$	6.41723	0.401077
$257$	−2.94764	−0.183869	−0.0919344	−	0.995765i	$-0.529305\pi$
$257$	−2.94764	−0.183869	−0.0919344	+	0.995765i	$0.529305\pi$
$258$	0	0
$259$	−4.85816	−0.301872
$260$	0	0
$261$	0	0
$262$	2.37545	0.146756
$263$	−0.220418	−0.0135916	−0.00679578	−	0.999977i	$-0.502163\pi$
$263$	−0.220418	−0.0135916	−0.00679578	+	0.999977i	$0.502163\pi$
$264$	0	0
$265$	0	0
$266$	−4.63845	−0.284402
$267$	0	0
$268$	15.0651	0.920246
$269$	−10.4148	−0.634999	−0.317499	−	0.948258i	$-0.602843\pi$
$269$	−10.4148	−0.634999	−0.317499	+	0.948258i	$0.602843\pi$
$270$	0	0
$271$	−23.4416	−1.42397	−0.711987	−	0.702193i	$-0.752205\pi$
$271$	−23.4416	−1.42397	−0.711987	+	0.702193i	$0.752205\pi$
$272$	−17.2962	−1.04873
$273$	0	0
$274$	−4.91775	−0.297092
$275$	0	0
$276$	0	0
$277$	18.4064	1.10593	0.552966	−	0.833204i	$-0.313496\pi$
$277$	18.4064	1.10593	0.552966	+	0.833204i	$0.313496\pi$
$278$	7.69595	0.461572
$279$	0	0
$280$	0	0
$281$	−0.120288	−0.00717577	−0.00358789	−	0.999994i	$-0.501142\pi$
$281$	−0.120288	−0.00717577	−0.00358789	+	0.999994i	$0.501142\pi$
$282$	0	0
$283$	−0.0913058	−0.00542756	−0.00271378	−	0.999996i	$-0.500864\pi$
$283$	−0.0913058	−0.00542756	−0.00271378	+	0.999996i	$0.500864\pi$
$284$	21.3234	1.26531
$285$	0	0
$286$	−6.34693	−0.375302
$287$	7.87213	0.464677
$288$	0	0
$289$	11.8132	0.694893
$290$	0	0
$291$	0	0
$292$	−25.6922	−1.50352
$293$	−19.6709	−1.14919	−0.574593	−	0.818439i	$-0.694840\pi$
$293$	−19.6709	−1.14919	−0.574593	+	0.818439i	$0.694840\pi$
$294$	0	0
$295$	0	0
$296$	3.04112	0.176762
$297$	0	0
$298$	−1.17547	−0.0680930
$299$	−4.96767	−0.287288
$300$	0	0
$301$	2.62730	0.151435
$302$	5.79774	0.333623
$303$	0	0
$304$	−18.2496	−1.04669
$305$	0	0
$306$	0	0
$307$	−9.01250	−0.514371	−0.257185	−	0.966362i	$-0.582795\pi$
$307$	−9.01250	−0.514371	−0.257185	+	0.966362i	$0.582795\pi$
$308$	−19.4893	−1.11051
$309$	0	0
$310$	0	0
$311$	23.0844	1.30899	0.654497	−	0.756065i	$-0.272881\pi$
$311$	23.0844	1.30899	0.654497	+	0.756065i	$0.272881\pi$
$312$	0	0
$313$	6.87528	0.388614	0.194307	−	0.980941i	$-0.437754\pi$
$313$	6.87528	0.388614	0.194307	+	0.980941i	$0.437754\pi$
$314$	1.16964	0.0660064
$315$	0	0
$316$	−21.7085	−1.22120
$317$	−12.9741	−0.728696	−0.364348	−	0.931263i	$-0.618708\pi$
$317$	−12.9741	−0.728696	−0.364348	+	0.931263i	$0.618708\pi$
$318$	0	0
$319$	41.1415	2.30348
$320$	0	0
$321$	0	0
$322$	1.08281	0.0603424
$323$	30.4015	1.69158
$324$	0	0
$325$	0	0
$326$	−5.00557	−0.277233
$327$	0	0
$328$	−4.92781	−0.272093
$329$	5.32241	0.293434
$330$	0	0
$331$	−5.77297	−0.317311	−0.158656	−	0.987334i	$-0.550716\pi$
$331$	−5.77297	−0.317311	−0.158656	+	0.987334i	$0.550716\pi$
$332$	−20.7336	−1.13790
$333$	0	0
$334$	−0.286714	−0.0156883
$335$	0	0
$336$	0	0
$337$	13.1047	0.713860	0.356930	−	0.934131i	$-0.383823\pi$
$337$	13.1047	0.713860	0.356930	+	0.934131i	$0.383823\pi$
$338$	−0.406792	−0.0221266
$339$	0	0
$340$	0	0
$341$	−5.62320	−0.304513
$342$	0	0
$343$	20.1101	1.08584
$344$	−1.64464	−0.0886731
$345$	0	0
$346$	−0.942937	−0.0506926
$347$	30.8690	1.65713	0.828567	−	0.559890i	$-0.189157\pi$
$347$	30.8690	1.65713	0.828567	+	0.559890i	$0.189157\pi$
$348$	0	0
$349$	28.0753	1.50284	0.751419	−	0.659825i	$-0.229370\pi$
$349$	28.0753	1.50284	0.751419	+	0.659825i	$0.229370\pi$
$350$	0	0
$351$	0	0
$352$	18.5090	0.986534
$353$	13.5595	0.721697	0.360849	−	0.932624i	$-0.382487\pi$
$353$	13.5595	0.721697	0.360849	+	0.932624i	$0.382487\pi$
$354$	0	0
$355$	0	0
$356$	31.0978	1.64818
$357$	0	0
$358$	5.48425	0.289851
$359$	1.81642	0.0958672	0.0479336	−	0.998851i	$-0.484736\pi$
$359$	1.81642	0.0958672	0.0479336	+	0.998851i	$0.484736\pi$
$360$	0	0
$361$	13.0773	0.688282
$362$	4.56899	0.240141
$363$	0	0
$364$	−15.7830	−0.827255
$365$	0	0
$366$	0	0
$367$	31.2892	1.63328	0.816640	−	0.577147i	$-0.195834\pi$
$367$	31.2892	1.63328	0.816640	+	0.577147i	$0.195834\pi$
$368$	4.26021	0.222079
$369$	0	0
$370$	0	0
$371$	31.8643	1.65431
$372$	0	0
$373$	−24.2034	−1.25321	−0.626603	−	0.779338i	$-0.715555\pi$
$373$	−24.2034	−1.25321	−0.626603	+	0.779338i	$0.715555\pi$
$374$	−9.06742	−0.468865
$375$	0	0
$376$	−3.33173	−0.171821
$377$	33.3176	1.71594
$378$	0	0
$379$	0.676661	0.0347577	0.0173789	−	0.999849i	$-0.494468\pi$
$379$	0.676661	0.0347577	0.0173789	+	0.999849i	$0.494468\pi$
$380$	0	0
$381$	0	0
$382$	5.85030	0.299327
$383$	−8.32588	−0.425433	−0.212716	−	0.977114i	$-0.568231\pi$
$383$	−8.32588	−0.425433	−0.212716	+	0.977114i	$0.568231\pi$
$384$	0	0
$385$	0	0
$386$	3.09066	0.157311
$387$	0	0
$388$	27.0387	1.37268
$389$	−34.8213	−1.76551	−0.882754	−	0.469835i	$-0.844313\pi$
$389$	−34.8213	−1.76551	−0.882754	+	0.469835i	$0.844313\pi$
$390$	0	0
$391$	−7.09696	−0.358909
$392$	−2.73193	−0.137983
$393$	0	0
$394$	1.70415	0.0858539
$395$	0	0
$396$	0	0
$397$	−24.3197	−1.22057	−0.610286	−	0.792182i	$-0.708945\pi$
$397$	−24.3197	−1.22057	−0.610286	+	0.792182i	$0.708945\pi$
$398$	−0.531130	−0.0266232
$399$	0	0
$400$	0	0
$401$	21.0996	1.05367	0.526833	−	0.849969i	$-0.323379\pi$
$401$	21.0996	1.05367	0.526833	+	0.849969i	$0.323379\pi$
$402$	0	0
$403$	−4.55383	−0.226843
$404$	4.03533	0.200765
$405$	0	0
$406$	−7.26227	−0.360420
$407$	−10.0204	−0.496694
$408$	0	0
$409$	1.03583	0.0512184	0.0256092	−	0.999672i	$-0.491847\pi$
$409$	1.03583	0.0512184	0.0256092	+	0.999672i	$0.491847\pi$
$410$	0	0
$411$	0	0
$412$	−13.3244	−0.656447
$413$	0.825732	0.0406316
$414$	0	0
$415$	0	0
$416$	14.9892	0.734904
$417$	0	0
$418$	−9.56725	−0.467950
$419$	−2.57098	−0.125601	−0.0628004	−	0.998026i	$-0.520003\pi$
$419$	−2.57098	−0.125601	−0.0628004	+	0.998026i	$0.520003\pi$
$420$	0	0
$421$	19.4929	0.950025	0.475012	−	0.879979i	$-0.342444\pi$
$421$	19.4929	0.950025	0.475012	+	0.879979i	$0.342444\pi$
$422$	−4.66423	−0.227051
$423$	0	0
$424$	−19.9465	−0.968686
$425$	0	0
$426$	0	0
$427$	−11.9126	−0.576492
$428$	26.8884	1.29970
$429$	0	0
$430$	0	0
$431$	10.8376	0.522026	0.261013	−	0.965335i	$-0.415943\pi$
$431$	10.8376	0.522026	0.261013	+	0.965335i	$0.415943\pi$
$432$	0	0
$433$	−29.8688	−1.43540	−0.717701	−	0.696351i	$-0.754806\pi$
$433$	−29.8688	−1.43540	−0.717701	+	0.696351i	$0.754806\pi$
$434$	0.992603	0.0476464
$435$	0	0
$436$	−30.7555	−1.47292
$437$	−7.48818	−0.358208
$438$	0	0
$439$	13.0354	0.622148	0.311074	−	0.950386i	$-0.399311\pi$
$439$	13.0354	0.622148	0.311074	+	0.950386i	$0.399311\pi$
$440$	0	0
$441$	0	0
$442$	−7.34307	−0.349274
$443$	18.1333	0.861539	0.430770	−	0.902462i	$-0.358242\pi$
$443$	18.1333	0.861539	0.430770	+	0.902462i	$0.358242\pi$
$444$	0	0
$445$	0	0
$446$	−0.0814555	−0.00385703
$447$	0	0
$448$	11.2289	0.530517
$449$	−13.7087	−0.646954	−0.323477	−	0.946236i	$-0.604852\pi$
$449$	−13.7087	−0.646954	−0.323477	+	0.946236i	$0.604852\pi$
$450$	0	0
$451$	16.2370	0.764572
$452$	−30.3719	−1.42857
$453$	0	0
$454$	−10.2918	−0.483016
$455$	0	0
$456$	0	0
$457$	17.8434	0.834680	0.417340	−	0.908750i	$-0.362962\pi$
$457$	17.8434	0.834680	0.417340	+	0.908750i	$0.362962\pi$
$458$	6.06635	0.283462
$459$	0	0
$460$	0	0
$461$	−1.25912	−0.0586430	−0.0293215	−	0.999570i	$-0.509335\pi$
$461$	−1.25912	−0.0586430	−0.0293215	+	0.999570i	$0.509335\pi$
$462$	0	0
$463$	22.6971	1.05482	0.527412	−	0.849610i	$-0.323162\pi$
$463$	22.6971	1.05482	0.527412	+	0.849610i	$0.323162\pi$
$464$	−28.5728	−1.32646
$465$	0	0
$466$	−7.27474	−0.336996
$467$	10.1460	0.469501	0.234751	−	0.972056i	$-0.424573\pi$
$467$	10.1460	0.469501	0.234751	+	0.972056i	$0.424573\pi$
$468$	0	0
$469$	18.1465	0.837927
$470$	0	0
$471$	0	0
$472$	−0.516894	−0.0237920
$473$	5.41906	0.249168
$474$	0	0
$475$	0	0
$476$	−22.5481	−1.03349
$477$	0	0
$478$	6.79337	0.310721
$479$	8.86088	0.404864	0.202432	−	0.979296i	$-0.435116\pi$
$479$	8.86088	0.404864	0.202432	+	0.979296i	$0.435116\pi$
$480$	0	0
$481$	−8.11484	−0.370005
$482$	−2.58837	−0.117897
$483$	0	0
$484$	−19.6567	−0.893488
$485$	0	0
$486$	0	0
$487$	−8.37280	−0.379408	−0.189704	−	0.981841i	$-0.560753\pi$
$487$	−8.37280	−0.379408	−0.189704	+	0.981841i	$0.560753\pi$
$488$	7.45709	0.337566
$489$	0	0
$490$	0	0
$491$	−3.44669	−0.155547	−0.0777734	−	0.996971i	$-0.524781\pi$
$491$	−3.44669	−0.155547	−0.0777734	+	0.996971i	$0.524781\pi$
$492$	0	0
$493$	47.5986	2.14373
$494$	−7.74785	−0.348592
$495$	0	0
$496$	3.90531	0.175354
$497$	25.6848	1.15212
$498$	0	0
$499$	−0.476153	−0.0213155	−0.0106578	−	0.999943i	$-0.503393\pi$
$499$	−0.476153	−0.0213155	−0.0106578	+	0.999943i	$0.503393\pi$
$500$	0	0
$501$	0	0
$502$	−3.84607	−0.171658
$503$	9.90779	0.441766	0.220883	−	0.975300i	$-0.429106\pi$
$503$	9.90779	0.441766	0.220883	+	0.975300i	$0.429106\pi$
$504$	0	0
$505$	0	0
$506$	2.23339	0.0992864
$507$	0	0
$508$	9.59955	0.425911
$509$	11.8211	0.523963	0.261981	−	0.965073i	$-0.415624\pi$
$509$	11.8211	0.523963	0.261981	+	0.965073i	$0.415624\pi$
$510$	0	0
$511$	−30.9473	−1.36903
$512$	−21.9288	−0.969126
$513$	0	0
$514$	1.07320	0.0473368
$515$	0	0
$516$	0	0
$517$	10.9780	0.482811
$518$	1.76880	0.0777165
$519$	0	0
$520$	0	0
$521$	−25.0561	−1.09773	−0.548863	−	0.835912i	$-0.684939\pi$
$521$	−25.0561	−1.09773	−0.548863	+	0.835912i	$0.684939\pi$
$522$	0	0
$523$	19.5306	0.854012	0.427006	−	0.904249i	$-0.359568\pi$
$523$	19.5306	0.854012	0.427006	+	0.904249i	$0.359568\pi$
$524$	12.1839	0.532256
$525$	0	0
$526$	0.0802515	0.00349913
$527$	−6.50574	−0.283395
$528$	0	0
$529$	−21.2520	−0.923998
$530$	0	0
$531$	0	0
$532$	−23.7911	−1.03147
$533$	13.1492	0.569556
$534$	0	0
$535$	0	0
$536$	−11.3594	−0.490650
$537$	0	0
$538$	3.79188	0.163480
$539$	9.00165	0.387729
$540$	0	0
$541$	−8.96076	−0.385253	−0.192627	−	0.981272i	$-0.561701\pi$
$541$	−8.96076	−0.385253	−0.192627	+	0.981272i	$0.561701\pi$
$542$	8.53479	0.366601
$543$	0	0
$544$	21.4140	0.918116
$545$	0	0
$546$	0	0
$547$	6.82992	0.292026	0.146013	−	0.989283i	$-0.453356\pi$
$547$	6.82992	0.292026	0.146013	+	0.989283i	$0.453356\pi$
$548$	−25.2236	−1.07750
$549$	0	0
$550$	0	0
$551$	50.2224	2.13955
$552$	0	0
$553$	−26.1488	−1.11196
$554$	−6.70153	−0.284721
$555$	0	0
$556$	39.4732	1.67404
$557$	5.53535	0.234540	0.117270	−	0.993100i	$-0.462586\pi$
$557$	5.53535	0.234540	0.117270	+	0.993100i	$0.462586\pi$
$558$	0	0
$559$	4.38851	0.185614
$560$	0	0
$561$	0	0
$562$	0.0437954	0.00184740
$563$	−39.1974	−1.65198	−0.825988	−	0.563688i	$-0.809382\pi$
$563$	−39.1974	−1.65198	−0.825988	+	0.563688i	$0.809382\pi$
$564$	0	0
$565$	0	0
$566$	0.0332433	0.00139732
$567$	0	0
$568$	−16.0782	−0.674628
$569$	−19.8092	−0.830444	−0.415222	−	0.909720i	$-0.636296\pi$
$569$	−19.8092	−0.830444	−0.415222	+	0.909720i	$0.636296\pi$
$570$	0	0
$571$	−22.9768	−0.961548	−0.480774	−	0.876845i	$-0.659644\pi$
$571$	−22.9768	−0.961548	−0.480774	+	0.876845i	$0.659644\pi$
$572$	−32.5540	−1.36115
$573$	0	0
$574$	−2.86615	−0.119631
$575$	0	0
$576$	0	0
$577$	−12.1833	−0.507198	−0.253599	−	0.967309i	$-0.581614\pi$
$577$	−12.1833	−0.507198	−0.253599	+	0.967309i	$0.581614\pi$
$578$	−4.30103	−0.178899
$579$	0	0
$580$	0	0
$581$	−24.9744	−1.03611
$582$	0	0
$583$	65.7232	2.72198
$584$	19.3725	0.801639
$585$	0	0
$586$	7.16194	0.295857
$587$	12.0991	0.499384	0.249692	−	0.968325i	$-0.419671\pi$
$587$	12.0991	0.499384	0.249692	+	0.968325i	$0.419671\pi$
$588$	0	0
$589$	−6.86437	−0.282841
$590$	0	0
$591$	0	0
$592$	6.95918	0.286021
$593$	−26.8465	−1.10245	−0.551227	−	0.834355i	$-0.685840\pi$
$593$	−26.8465	−1.10245	−0.551227	+	0.834355i	$0.685840\pi$
$594$	0	0
$595$	0	0
$596$	−6.02908	−0.246961
$597$	0	0
$598$	1.80867	0.0739619
$599$	−5.16011	−0.210836	−0.105418	−	0.994428i	$-0.533618\pi$
$599$	−5.16011	−0.210836	−0.105418	+	0.994428i	$0.533618\pi$
$600$	0	0
$601$	−20.5481	−0.838173	−0.419086	−	0.907946i	$-0.637650\pi$
$601$	−20.5481	−0.838173	−0.419086	+	0.907946i	$0.637650\pi$
$602$	−0.956567	−0.0389868
$603$	0	0
$604$	29.7372	1.20999
$605$	0	0
$606$	0	0
$607$	−7.83380	−0.317964	−0.158982	−	0.987281i	$-0.550821\pi$
$607$	−7.83380	−0.317964	−0.158982	+	0.987281i	$0.550821\pi$
$608$	22.5944	0.916324
$609$	0	0
$610$	0	0
$611$	8.89029	0.359663
$612$	0	0
$613$	−21.4304	−0.865564	−0.432782	−	0.901499i	$-0.642468\pi$
$613$	−21.4304	−0.865564	−0.432782	+	0.901499i	$0.642468\pi$
$614$	3.28134	0.132424
$615$	0	0
$616$	14.6953	0.592092
$617$	−7.08270	−0.285139	−0.142569	−	0.989785i	$-0.545536\pi$
$617$	−7.08270	−0.285139	−0.142569	+	0.989785i	$0.545536\pi$
$618$	0	0
$619$	−38.5317	−1.54872	−0.774360	−	0.632745i	$-0.781928\pi$
$619$	−38.5317	−1.54872	−0.774360	+	0.632745i	$0.781928\pi$
$620$	0	0
$621$	0	0
$622$	−8.40473	−0.336999
$623$	37.4585	1.50074
$624$	0	0
$625$	0	0
$626$	−2.50321	−0.100048
$627$	0	0
$628$	5.99918	0.239393
$629$	−11.5931	−0.462247
$630$	0	0
$631$	−35.1731	−1.40022	−0.700110	−	0.714035i	$-0.746866\pi$
$631$	−35.1731	−1.40022	−0.700110	+	0.714035i	$0.746866\pi$
$632$	16.3687	0.651110
$633$	0	0
$634$	4.72370	0.187602
$635$	0	0
$636$	0	0
$637$	7.28981	0.288833
$638$	−14.9791	−0.593029
$639$	0	0
$640$	0	0
$641$	12.9089	0.509872	0.254936	−	0.966958i	$-0.417946\pi$
$641$	12.9089	0.509872	0.254936	+	0.966958i	$0.417946\pi$
$642$	0	0
$643$	−42.0703	−1.65909	−0.829546	−	0.558438i	$-0.811401\pi$
$643$	−42.0703	−1.65909	−0.829546	+	0.558438i	$0.811401\pi$
$644$	5.55381	0.218851
$645$	0	0
$646$	−11.0688	−0.435497
$647$	2.45766	0.0966207	0.0483103	−	0.998832i	$-0.484616\pi$
$647$	2.45766	0.0966207	0.0483103	+	0.998832i	$0.484616\pi$
$648$	0	0
$649$	1.70315	0.0668546
$650$	0	0
$651$	0	0
$652$	−25.6740	−1.00547
$653$	−42.7727	−1.67382	−0.836912	−	0.547337i	$-0.815641\pi$
$653$	−42.7727	−1.67382	−0.836912	+	0.547337i	$0.815641\pi$
$654$	0	0
$655$	0	0
$656$	−11.2766	−0.440278
$657$	0	0
$658$	−1.93782	−0.0755442
$659$	−3.41659	−0.133092	−0.0665458	−	0.997783i	$-0.521198\pi$
$659$	−3.41659	−0.133092	−0.0665458	+	0.997783i	$0.521198\pi$
$660$	0	0
$661$	42.6380	1.65843	0.829213	−	0.558933i	$-0.188789\pi$
$661$	42.6380	1.65843	0.829213	+	0.558933i	$0.188789\pi$
$662$	2.10187	0.0816914
$663$	0	0
$664$	15.6335	0.606699
$665$	0	0
$666$	0	0
$667$	−11.7240	−0.453954
$668$	−1.47058	−0.0568986
$669$	0	0
$670$	0	0
$671$	−24.5709	−0.948550
$672$	0	0
$673$	7.02233	0.270691	0.135345	−	0.990798i	$-0.456786\pi$
$673$	7.02233	0.270691	0.135345	+	0.990798i	$0.456786\pi$
$674$	−4.77127	−0.183782
$675$	0	0
$676$	−2.08648	−0.0802491
$677$	4.96936	0.190988	0.0954941	−	0.995430i	$-0.469557\pi$
$677$	4.96936	0.190988	0.0954941	+	0.995430i	$0.469557\pi$
$678$	0	0
$679$	32.5692	1.24989
$680$	0	0
$681$	0	0
$682$	2.04734	0.0783966
$683$	−2.27348	−0.0869922	−0.0434961	−	0.999054i	$-0.513850\pi$
$683$	−2.27348	−0.0869922	−0.0434961	+	0.999054i	$0.513850\pi$
$684$	0	0
$685$	0	0
$686$	−7.32183	−0.279549
$687$	0	0
$688$	−3.76353	−0.143483
$689$	53.2246	2.02769
$690$	0	0
$691$	−16.7141	−0.635836	−0.317918	−	0.948118i	$-0.602984\pi$
$691$	−16.7141	−0.635836	−0.317918	+	0.948118i	$0.602984\pi$
$692$	−4.83641	−0.183853
$693$	0	0
$694$	−11.2390	−0.426627
$695$	0	0
$696$	0	0
$697$	18.7854	0.711547
$698$	−10.2219	−0.386904
$699$	0	0
$700$	0	0
$701$	−10.4650	−0.395256	−0.197628	−	0.980277i	$-0.563324\pi$
$701$	−10.4650	−0.395256	−0.197628	+	0.980277i	$0.563324\pi$
$702$	0	0
$703$	−12.2322	−0.461345
$704$	23.1607	0.872904
$705$	0	0
$706$	−4.93683	−0.185800
$707$	4.86072	0.182806
$708$	0	0
$709$	10.4906	0.393984	0.196992	−	0.980405i	$-0.436883\pi$
$709$	10.4906	0.393984	0.196992	+	0.980405i	$0.436883\pi$
$710$	0	0
$711$	0	0
$712$	−23.4484	−0.878765
$713$	1.60243	0.0600114
$714$	0	0
$715$	0	0
$716$	28.1292	1.05124
$717$	0	0
$718$	−0.661338	−0.0246809
$719$	51.8986	1.93549	0.967746	−	0.251929i	$-0.0810648\pi$
$719$	51.8986	1.93549	0.967746	+	0.251929i	$0.0810648\pi$
$720$	0	0
$721$	−16.0498	−0.597725
$722$	−4.76130	−0.177197
$723$	0	0
$724$	23.4348	0.870947
$725$	0	0
$726$	0	0
$727$	36.1382	1.34029	0.670145	−	0.742230i	$-0.266232\pi$
$727$	36.1382	1.34029	0.670145	+	0.742230i	$0.266232\pi$
$728$	11.9007	0.441070
$729$	0	0
$730$	0	0
$731$	6.26956	0.231888
$732$	0	0
$733$	−49.1978	−1.81716	−0.908581	−	0.417709i	$-0.862833\pi$
$733$	−49.1978	−1.81716	−0.908581	+	0.417709i	$0.862833\pi$
$734$	−11.3920	−0.420486
$735$	0	0
$736$	−5.27446	−0.194419
$737$	37.4289	1.37871
$738$	0	0
$739$	−0.931024	−0.0342483	−0.0171241	−	0.999853i	$-0.505451\pi$
$739$	−0.931024	−0.0342483	−0.0171241	+	0.999853i	$0.505451\pi$
$740$	0	0
$741$	0	0
$742$	−11.6014	−0.425901
$743$	7.46765	0.273962	0.136981	−	0.990574i	$-0.456260\pi$
$743$	7.46765	0.273962	0.136981	+	0.990574i	$0.456260\pi$
$744$	0	0
$745$	0	0
$746$	8.81218	0.322637
$747$	0	0
$748$	−46.5076	−1.70049
$749$	32.3881	1.18344
$750$	0	0
$751$	−11.8157	−0.431160	−0.215580	−	0.976486i	$-0.569164\pi$
$751$	−11.8157	−0.431160	−0.215580	+	0.976486i	$0.569164\pi$
$752$	−7.62420	−0.278026
$753$	0	0
$754$	−12.1305	−0.441768
$755$	0	0
$756$	0	0
$757$	−36.5733	−1.32928	−0.664640	−	0.747163i	$-0.731415\pi$
$757$	−36.5733	−1.32928	−0.664640	+	0.747163i	$0.731415\pi$
$758$	−0.246364	−0.00894834
$759$	0	0
$760$	0	0
$761$	−12.9848	−0.470700	−0.235350	−	0.971911i	$-0.575624\pi$
$761$	−12.9848	−0.470700	−0.235350	+	0.971911i	$0.575624\pi$
$762$	0	0
$763$	−37.0462	−1.34116
$764$	30.0067	1.08561
$765$	0	0
$766$	3.03135	0.109527
$767$	1.37926	0.0498023
$768$	0	0
$769$	0.337323	0.0121642	0.00608208	−	0.999982i	$-0.498064\pi$
$769$	0.337323	0.0121642	0.00608208	+	0.999982i	$0.498064\pi$
$770$	0	0
$771$	0	0
$772$	15.8523	0.570536
$773$	−32.7655	−1.17849	−0.589246	−	0.807953i	$-0.700575\pi$
$773$	−32.7655	−1.17849	−0.589246	+	0.807953i	$0.700575\pi$
$774$	0	0
$775$	0	0
$776$	−20.3877	−0.731877
$777$	0	0
$778$	12.6780	0.454528
$779$	19.8209	0.710158
$780$	0	0
$781$	52.9774	1.89568
$782$	2.58392	0.0924007
$783$	0	0
$784$	−6.25165	−0.223273
$785$	0	0
$786$	0	0
$787$	24.6230	0.877715	0.438857	−	0.898557i	$-0.355383\pi$
$787$	24.6230	0.877715	0.438857	+	0.898557i	$0.355383\pi$
$788$	8.74075	0.311376
$789$	0	0
$790$	0	0
$791$	−36.5842	−1.30078
$792$	0	0
$793$	−19.8983	−0.706608
$794$	8.85451	0.314235
$795$	0	0
$796$	−2.72422	−0.0965573
$797$	−28.1988	−0.998852	−0.499426	−	0.866357i	$-0.666456\pi$
$797$	−28.1988	−0.998852	−0.499426	+	0.866357i	$0.666456\pi$
$798$	0	0
$799$	12.7009	0.449327
$800$	0	0
$801$	0	0
$802$	−7.68212	−0.271265
$803$	−63.8318	−2.25258
$804$	0	0
$805$	0	0
$806$	1.65799	0.0584004
$807$	0	0
$808$	−3.04272	−0.107043
$809$	−46.0975	−1.62070	−0.810352	−	0.585944i	$-0.800724\pi$
$809$	−46.0975	−1.62070	−0.810352	+	0.585944i	$0.800724\pi$
$810$	0	0
$811$	19.6551	0.690183	0.345092	−	0.938569i	$-0.387848\pi$
$811$	19.6551	0.690183	0.345092	+	0.938569i	$0.387848\pi$
$812$	−37.2488	−1.30718
$813$	0	0
$814$	3.64831	0.127873
$815$	0	0
$816$	0	0
$817$	6.61517	0.231435
$818$	−0.377132	−0.0131861
$819$	0	0
$820$	0	0
$821$	−28.8469	−1.00676	−0.503382	−	0.864064i	$-0.667911\pi$
$821$	−28.8469	−1.00676	−0.503382	+	0.864064i	$0.667911\pi$
$822$	0	0
$823$	4.32008	0.150589	0.0752943	−	0.997161i	$-0.476010\pi$
$823$	4.32008	0.150589	0.0752943	+	0.997161i	$0.476010\pi$
$824$	10.0469	0.349999
$825$	0	0
$826$	−0.300639	−0.0104606
$827$	−54.9133	−1.90952	−0.954762	−	0.297371i	$-0.903890\pi$
$827$	−54.9133	−1.90952	−0.954762	+	0.297371i	$0.903890\pi$
$828$	0	0
$829$	−15.8046	−0.548916	−0.274458	−	0.961599i	$-0.588498\pi$
$829$	−15.8046	−0.548916	−0.274458	+	0.961599i	$0.588498\pi$
$830$	0	0
$831$	0	0
$832$	18.7563	0.650256
$833$	10.4144	0.360839
$834$	0	0
$835$	0	0
$836$	−49.0713	−1.69717
$837$	0	0
$838$	0.936064	0.0323358
$839$	20.7086	0.714939	0.357469	−	0.933925i	$-0.383640\pi$
$839$	20.7086	0.714939	0.357469	+	0.933925i	$0.383640\pi$
$840$	0	0
$841$	49.6315	1.71143
$842$	−7.09712	−0.244583
$843$	0	0
$844$	−23.9233	−0.823472
$845$	0	0
$846$	0	0
$847$	−23.6773	−0.813562
$848$	−45.6447	−1.56745
$849$	0	0
$850$	0	0
$851$	2.85549	0.0978850
$852$	0	0
$853$	−8.27523	−0.283338	−0.141669	−	0.989914i	$-0.545247\pi$
$853$	−8.27523	−0.283338	−0.141669	+	0.989914i	$0.545247\pi$
$854$	4.33724	0.148417
$855$	0	0
$856$	−20.2744	−0.692964
$857$	−6.29920	−0.215177	−0.107588	−	0.994196i	$-0.534313\pi$
$857$	−6.29920	−0.215177	−0.107588	+	0.994196i	$0.534313\pi$
$858$	0	0
$859$	26.4130	0.901200	0.450600	−	0.892726i	$-0.351210\pi$
$859$	26.4130	0.901200	0.450600	+	0.892726i	$0.351210\pi$
$860$	0	0
$861$	0	0
$862$	−3.94582	−0.134395
$863$	34.5355	1.17560	0.587802	−	0.809005i	$-0.299993\pi$
$863$	34.5355	1.17560	0.587802	+	0.809005i	$0.299993\pi$
$864$	0	0
$865$	0	0
$866$	10.8749	0.369543
$867$	0	0
$868$	5.09115	0.172805
$869$	−53.9343	−1.82960
$870$	0	0
$871$	30.3110	1.02705
$872$	23.1903	0.785321
$873$	0	0
$874$	2.72635	0.0922203
$875$	0	0
$876$	0	0
$877$	−44.1083	−1.48943	−0.744716	−	0.667382i	$-0.767415\pi$
$877$	−44.1083	−1.48943	−0.744716	+	0.667382i	$0.767415\pi$
$878$	−4.74605	−0.160171
$879$	0	0
$880$	0	0
$881$	−34.5724	−1.16477	−0.582386	−	0.812912i	$-0.697881\pi$
$881$	−34.5724	−1.16477	−0.582386	+	0.812912i	$0.697881\pi$
$882$	0	0
$883$	−3.56120	−0.119844	−0.0599220	−	0.998203i	$-0.519085\pi$
$883$	−3.56120	−0.119844	−0.0599220	+	0.998203i	$0.519085\pi$
$884$	−37.6633	−1.26675
$885$	0	0
$886$	−6.60212	−0.221802
$887$	−53.6149	−1.80021	−0.900106	−	0.435671i	$-0.856511\pi$
$887$	−53.6149	−1.80021	−0.900106	+	0.435671i	$0.856511\pi$
$888$	0	0
$889$	11.5630	0.387812
$890$	0	0
$891$	0	0
$892$	−0.417793	−0.0139887
$893$	13.4011	0.448450
$894$	0	0
$895$	0	0
$896$	−22.0356	−0.736159
$897$	0	0
$898$	4.99117	0.166558
$899$	−10.7473	−0.358443
$900$	0	0
$901$	76.0382	2.53320
$902$	−5.91170	−0.196838
$903$	0	0
$904$	22.9010	0.761677
$905$	0	0
$906$	0	0
$907$	41.7962	1.38782	0.693909	−	0.720062i	$-0.255887\pi$
$907$	41.7962	1.38782	0.693909	+	0.720062i	$0.255887\pi$
$908$	−52.7874	−1.75181
$909$	0	0
$910$	0	0
$911$	45.8484	1.51903	0.759513	−	0.650492i	$-0.225437\pi$
$911$	45.8484	1.51903	0.759513	+	0.650492i	$0.225437\pi$
$912$	0	0
$913$	−51.5121	−1.70480
$914$	−6.49657	−0.214887
$915$	0	0
$916$	31.1149	1.02806
$917$	14.6760	0.484644
$918$	0	0
$919$	−28.3017	−0.933588	−0.466794	−	0.884366i	$-0.654591\pi$
$919$	−28.3017	−0.933588	−0.466794	+	0.884366i	$0.654591\pi$
$920$	0	0
$921$	0	0
$922$	0.458429	0.0150976
$923$	42.9027	1.41216
$924$	0	0
$925$	0	0
$926$	−8.26374	−0.271563
$927$	0	0
$928$	35.3753	1.16125
$929$	−14.0835	−0.462064	−0.231032	−	0.972946i	$-0.574210\pi$
$929$	−14.0835	−0.462064	−0.231032	+	0.972946i	$0.574210\pi$
$930$	0	0
$931$	10.9885	0.360134
$932$	−37.3128	−1.22222
$933$	0	0
$934$	−3.69404	−0.120873
$935$	0	0
$936$	0	0
$937$	−17.7320	−0.579280	−0.289640	−	0.957136i	$-0.593536\pi$
$937$	−17.7320	−0.579280	−0.289640	+	0.957136i	$0.593536\pi$
$938$	−6.60691	−0.215723
$939$	0	0
$940$	0	0
$941$	13.8982	0.453068	0.226534	−	0.974003i	$-0.427261\pi$
$941$	13.8982	0.453068	0.226534	+	0.974003i	$0.427261\pi$
$942$	0	0
$943$	−4.62702	−0.150676
$944$	−1.18284	−0.0384981
$945$	0	0
$946$	−1.97301	−0.0641482
$947$	−28.7291	−0.933571	−0.466786	−	0.884371i	$-0.654588\pi$
$947$	−28.7291	−0.933571	−0.466786	+	0.884371i	$0.654588\pi$
$948$	0	0
$949$	−51.6929	−1.67802
$950$	0	0
$951$	0	0
$952$	17.0017	0.551029
$953$	−40.5303	−1.31291	−0.656453	−	0.754367i	$-0.727944\pi$
$953$	−40.5303	−1.31291	−0.656453	+	0.754367i	$0.727944\pi$
$954$	0	0
$955$	0	0
$956$	34.8438	1.12693
$957$	0	0
$958$	−3.22614	−0.104232
$959$	−30.3829	−0.981114
$960$	0	0
$961$	−29.5311	−0.952615
$962$	2.95451	0.0952573
$963$	0	0
$964$	−13.2760	−0.427591
$965$	0	0
$966$	0	0
$967$	21.1771	0.681010	0.340505	−	0.940243i	$-0.389402\pi$
$967$	21.1771	0.681010	0.340505	+	0.940243i	$0.389402\pi$
$968$	14.8216	0.476383
$969$	0	0
$970$	0	0
$971$	41.7880	1.34104	0.670520	−	0.741891i	$-0.266071\pi$
$971$	41.7880	1.34104	0.670520	+	0.741891i	$0.266071\pi$
$972$	0	0
$973$	47.5471	1.52429
$974$	3.04843	0.0976781
$975$	0	0
$976$	17.0645	0.546221
$977$	−20.1059	−0.643245	−0.321623	−	0.946868i	$-0.604228\pi$
$977$	−20.1059	−0.643245	−0.321623	+	0.946868i	$0.604228\pi$
$978$	0	0
$979$	77.2618	2.46930
$980$	0	0
$981$	0	0
$982$	1.25490	0.0400454
$983$	1.74028	0.0555063	0.0277531	−	0.999615i	$-0.491165\pi$
$983$	1.74028	0.0555063	0.0277531	+	0.999615i	$0.491165\pi$
$984$	0	0
$985$	0	0
$986$	−17.3301	−0.551901
$987$	0	0
$988$	−39.7394	−1.26428
$989$	−1.54425	−0.0491044
$990$	0	0
$991$	−13.8725	−0.440676	−0.220338	−	0.975424i	$-0.570716\pi$
$991$	−13.8725	−0.440676	−0.220338	+	0.975424i	$0.570716\pi$
$992$	−4.83507	−0.153514
$993$	0	0
$994$	−9.35153	−0.296613
$995$	0	0
$996$	0	0
$997$	23.7256	0.751399	0.375699	−	0.926742i	$-0.377403\pi$
$997$	23.7256	0.751399	0.375699	+	0.926742i	$0.377403\pi$
$998$	0.173362	0.00548766
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5625.2.a.r.1.3		6
3.2	odd	2		1875.2.a.i.1.4	✓	6
5.4	even	2		5625.2.a.o.1.4		6
15.2	even	4		1875.2.b.e.1249.7		12
15.8	even	4		1875.2.b.e.1249.6		12
15.14	odd	2		1875.2.a.l.1.3	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1875.2.a.i.1.4	✓	6	3.2	odd	2
1875.2.a.l.1.3	yes	6	15.14	odd	2
1875.2.b.e.1249.6		12	15.8	even	4
1875.2.b.e.1249.7		12	15.2	even	4
5625.2.a.o.1.4		6	5.4	even	2
5625.2.a.r.1.3		6	1.1	even	1	trivial

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

\(f(q)\)	\(=\)	\(q-0.364088 q^{2} -1.86744 q^{4} -2.24941 q^{7} +1.40809 q^{8} -4.63962 q^{11} -3.75730 q^{13} +0.818981 q^{14} +3.22221 q^{16} -5.36779 q^{17} -5.66369 q^{19} +1.68923 q^{22} +1.32214 q^{23} +1.36799 q^{26} +4.20063 q^{28} -8.86744 q^{29} +1.21200 q^{31} -3.98934 q^{32} +1.95435 q^{34} +2.15975 q^{37} +2.06208 q^{38} -3.49965 q^{41} -1.16800 q^{43} +8.66420 q^{44} -0.481374 q^{46} -2.36614 q^{47} -1.94017 q^{49} +7.01653 q^{52} -14.1656 q^{53} -3.16736 q^{56} +3.22853 q^{58} -0.367089 q^{59} +5.29590 q^{61} -0.441273 q^{62} -4.99195 q^{64} -8.06723 q^{67} +10.0240 q^{68} -11.4185 q^{71} +13.7580 q^{73} -0.786340 q^{74} +10.5766 q^{76} +10.4364 q^{77} +11.6247 q^{79} +1.27418 q^{82} +11.1027 q^{83} +0.425253 q^{86} -6.53299 q^{88} -16.6526 q^{89} +8.45169 q^{91} -2.46901 q^{92} +0.861482 q^{94} -14.4790 q^{97} +0.706393 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + q^{2} + 11 q^{4} + 2 q^{7} + 6 q^{8} + 4 q^{14} + 17 q^{16} + 2 q^{17} - 2 q^{19} - 9 q^{22} - q^{23} - 37 q^{26} + 44 q^{28} - 31 q^{29} - 2 q^{31} + 33 q^{32} + 37 q^{34} + 22 q^{37} + 27 q^{38}+ \cdots + 15 q^{98}+O(q^{100}) \) 6 * q + q^2 + 11 * q^4 + 2 * q^7 + 6 * q^8 + 4 * q^14 + 17 * q^16 + 2 * q^17 - 2 * q^19 - 9 * q^22 - q^23 - 37 * q^26 + 44 * q^28 - 31 * q^29 - 2 * q^31 + 33 * q^32 + 37 * q^34 + 22 * q^37 + 27 * q^38 - 33 * q^41 + 3 * q^43 + 11 * q^44 - 12 * q^46 - 6 * q^47 + 4 * q^49 + 33 * q^52 + 14 * q^53 + 30 * q^56 + q^58 + 8 * q^59 + 34 * q^61 + 31 * q^62 + 12 * q^64 - 2 * q^67 + 27 * q^68 + 3 * q^71 + 36 * q^73 - 36 * q^74 + 27 * q^76 + 16 * q^77 + 25 * q^79 - 36 * q^82 - 12 * q^83 + 30 * q^86 - 56 * q^88 - 18 * q^89 + 28 * q^91 + 3 * q^92 - 50 * q^94 - 7 * q^97 + 15 * q^98

Introduction

L-functions

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