LMFDB - Embedded newform 2025.4.a.bd.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2025,4,Mod(1,2025)] 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("2025.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2025, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 2025 = 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2025.a (trivial)

Newform invariants

comment:select newform

sage:traces = [8,0,0,24,0,0,0,0,0,0,0,0,0,42] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

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

Self dual:	yes
Analytic conductor:	$119.478867762$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 44x^{6} + 567x^{4} - 2024x^{2} + 1900 $ x^8 - 44x^6 + 567x^4 - 2024*x^2 + 1900
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.22227$ of defining polynomial
Character	$\chi$	$=$	2025.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.22227 q^{2} -6.50606 q^{4} -33.1668 q^{7} -17.7303 q^{8} +23.6760 q^{11} +30.9094 q^{13} -40.5387 q^{14} +30.3772 q^{16} -48.7539 q^{17} +34.3436 q^{19} +28.9385 q^{22} +181.561 q^{23} +37.7796 q^{26} +215.785 q^{28} -4.29174 q^{29} -270.521 q^{31} +178.972 q^{32} -59.5905 q^{34} +163.355 q^{37} +41.9772 q^{38} +7.40836 q^{41} +378.631 q^{43} -154.038 q^{44} +221.917 q^{46} +177.089 q^{47} +757.035 q^{49} -201.098 q^{52} -587.283 q^{53} +588.057 q^{56} -5.24566 q^{58} -844.470 q^{59} +590.383 q^{61} -330.649 q^{62} -24.2662 q^{64} +494.174 q^{67} +317.196 q^{68} +53.0514 q^{71} +427.365 q^{73} +199.664 q^{74} -223.441 q^{76} -785.258 q^{77} -708.463 q^{79} +9.05501 q^{82} +243.242 q^{83} +462.790 q^{86} -419.784 q^{88} -558.336 q^{89} -1025.17 q^{91} -1181.25 q^{92} +216.450 q^{94} +471.121 q^{97} +925.301 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64}+ \cdots - 1238 q^{94}+O(q^{100}) $ 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * q^94

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 $

$2$	1.22227	0.432138	0.216069	−	0.976378i	$-0.430676\pi$
$2$	1.22227	0.432138	0.216069	+	0.976378i	$0.430676\pi$
$3$	0	0
$3$	0	0
$4$	−6.50606	−0.813257
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−33.1668	−1.79084	−0.895419	−	0.445225i	$-0.853124\pi$
$7$	−33.1668	−1.79084	−0.895419	+	0.445225i	$0.853124\pi$
$8$	−17.7303	−0.783577
$9$	0	0
$10$	0	0
$11$	23.6760	0.648963	0.324482	−	0.945892i	$-0.394810\pi$
$11$	23.6760	0.648963	0.324482	+	0.945892i	$0.394810\pi$
$12$	0	0
$13$	30.9094	0.659440	0.329720	−	0.944079i	$-0.393046\pi$
$13$	30.9094	0.659440	0.329720	+	0.944079i	$0.393046\pi$
$14$	−40.5387	−0.773888
$15$	0	0
$16$	30.3772	0.474644
$17$	−48.7539	−0.695563	−0.347781	−	0.937576i	$-0.613065\pi$
$17$	−48.7539	−0.695563	−0.347781	+	0.937576i	$0.613065\pi$
$18$	0	0
$19$	34.3436	0.414682	0.207341	−	0.978269i	$-0.433519\pi$
$19$	34.3436	0.414682	0.207341	+	0.978269i	$0.433519\pi$
$20$	0	0
$21$	0	0
$22$	28.9385	0.280441
$23$	181.561	1.64601	0.823004	−	0.568036i	$-0.192297\pi$
$23$	181.561	1.64601	0.823004	+	0.568036i	$0.192297\pi$
$24$	0	0
$25$	0	0
$26$	37.7796	0.284969
$27$	0	0
$28$	215.785	1.45641
$29$	−4.29174	−0.0274812	−0.0137406	−	0.999906i	$-0.504374\pi$
$29$	−4.29174	−0.0274812	−0.0137406	+	0.999906i	$0.504374\pi$
$30$	0	0
$31$	−270.521	−1.56732	−0.783660	−	0.621190i	$-0.786650\pi$
$31$	−270.521	−1.56732	−0.783660	+	0.621190i	$0.786650\pi$
$32$	178.972	0.988688
$33$	0	0
$34$	−59.5905	−0.300579
$35$	0	0
$36$	0	0
$37$	163.355	0.725821	0.362910	−	0.931824i	$-0.381783\pi$
$37$	163.355	0.725821	0.362910	+	0.931824i	$0.381783\pi$
$38$	41.9772	0.179200
$39$	0	0
$40$	0	0
$41$	7.40836	0.0282193	0.0141097	−	0.999900i	$-0.495509\pi$
$41$	7.40836	0.0282193	0.0141097	+	0.999900i	$0.495509\pi$
$42$	0	0
$43$	378.631	1.34281	0.671404	−	0.741091i	$-0.265691\pi$
$43$	378.631	1.34281	0.671404	+	0.741091i	$0.265691\pi$
$44$	−154.038	−0.527774
$45$	0	0
$46$	221.917	0.711302
$47$	177.089	0.549596	0.274798	−	0.961502i	$-0.411389\pi$
$47$	177.089	0.549596	0.274798	+	0.961502i	$0.411389\pi$
$48$	0	0
$49$	757.035	2.20710
$50$	0	0
$51$	0	0
$52$	−201.098	−0.536294
$53$	−587.283	−1.52207	−0.761033	−	0.648713i	$-0.775308\pi$
$53$	−587.283	−1.52207	−0.761033	+	0.648713i	$0.775308\pi$
$54$	0	0
$55$	0	0
$56$	588.057	1.40326
$57$	0	0
$58$	−5.24566	−0.0118757
$59$	−844.470	−1.86340	−0.931700	−	0.363228i	$-0.881675\pi$
$59$	−844.470	−1.86340	−0.931700	+	0.363228i	$0.881675\pi$
$60$	0	0
$61$	590.383	1.23919	0.619596	−	0.784921i	$-0.287296\pi$
$61$	590.383	1.23919	0.619596	+	0.784921i	$0.287296\pi$
$62$	−330.649	−0.677298
$63$	0	0
$64$	−24.2662	−0.0473949
$65$	0	0
$66$	0	0
$67$	494.174	0.901090	0.450545	−	0.892754i	$-0.351230\pi$
$67$	494.174	0.901090	0.450545	+	0.892754i	$0.351230\pi$
$68$	317.196	0.565671
$69$	0	0
$70$	0	0
$71$	53.0514	0.0886767	0.0443383	−	0.999017i	$-0.485882\pi$
$71$	53.0514	0.0886767	0.0443383	+	0.999017i	$0.485882\pi$
$72$	0	0
$73$	427.365	0.685196	0.342598	−	0.939482i	$-0.388693\pi$
$73$	427.365	0.685196	0.342598	+	0.939482i	$0.388693\pi$
$74$	199.664	0.313654
$75$	0	0
$76$	−223.441	−0.337243
$77$	−785.258	−1.16219
$78$	0	0
$79$	−708.463	−1.00897	−0.504483	−	0.863422i	$-0.668317\pi$
$79$	−708.463	−1.00897	−0.504483	+	0.863422i	$0.668317\pi$
$80$	0	0
$81$	0	0
$82$	9.05501	0.0121946
$83$	243.242	0.321678	0.160839	−	0.986981i	$-0.448580\pi$
$83$	243.242	0.321678	0.160839	+	0.986981i	$0.448580\pi$
$84$	0	0
$85$	0	0
$86$	462.790	0.580278
$87$	0	0
$88$	−419.784	−0.508512
$89$	−558.336	−0.664983	−0.332491	−	0.943106i	$-0.607889\pi$
$89$	−558.336	−0.664983	−0.332491	+	0.943106i	$0.607889\pi$
$90$	0	0
$91$	−1025.17	−1.18095
$92$	−1181.25	−1.33863
$93$	0	0
$94$	216.450	0.237501
$95$	0	0
$96$	0	0
$97$	471.121	0.493145	0.246572	−	0.969124i	$-0.420696\pi$
$97$	471.121	0.493145	0.246572	+	0.969124i	$0.420696\pi$
$98$	925.301	0.953771
$99$	0	0
$100$	0	0
$101$	1173.91	1.15652	0.578258	−	0.815854i	$-0.303733\pi$
$101$	1173.91	1.15652	0.578258	+	0.815854i	$0.303733\pi$
$102$	0	0
$103$	−1495.18	−1.43034	−0.715169	−	0.698952i	$-0.753650\pi$
$103$	−1495.18	−1.43034	−0.715169	+	0.698952i	$0.753650\pi$
$104$	−548.033	−0.516722
$105$	0	0
$106$	−717.818	−0.657742
$107$	717.918	0.648633	0.324317	−	0.945949i	$-0.394866\pi$
$107$	717.918	0.648633	0.324317	+	0.945949i	$0.394866\pi$
$108$	0	0
$109$	282.308	0.248075	0.124038	−	0.992278i	$-0.460416\pi$
$109$	282.308	0.248075	0.124038	+	0.992278i	$0.460416\pi$
$110$	0	0
$111$	0	0
$112$	−1007.51	−0.850011
$113$	−2328.58	−1.93854	−0.969269	−	0.246002i	$-0.920883\pi$
$113$	−2328.58	−1.93854	−0.969269	+	0.246002i	$0.920883\pi$
$114$	0	0
$115$	0	0
$116$	27.9223	0.0223493
$117$	0	0
$118$	−1032.17	−0.805246
$119$	1617.01	1.24564
$120$	0	0
$121$	−770.445	−0.578847
$122$	721.607	0.535502
$123$	0	0
$124$	1760.02	1.27463
$125$	0	0
$126$	0	0
$127$	−162.215	−0.113341	−0.0566704	−	0.998393i	$-0.518048\pi$
$127$	−162.215	−0.113341	−0.0566704	+	0.998393i	$0.518048\pi$
$128$	−1461.43	−1.00917
$129$	0	0
$130$	0	0
$131$	−2239.21	−1.49344	−0.746720	−	0.665138i	$-0.768373\pi$
$131$	−2239.21	−1.49344	−0.746720	+	0.665138i	$0.768373\pi$
$132$	0	0
$133$	−1139.07	−0.742629
$134$	604.014	0.389395
$135$	0	0
$136$	864.423	0.545027
$137$	1001.81	0.624748	0.312374	−	0.949959i	$-0.398876\pi$
$137$	1001.81	0.624748	0.312374	+	0.949959i	$0.398876\pi$
$138$	0	0
$139$	−1793.56	−1.09445	−0.547224	−	0.836986i	$-0.684315\pi$
$139$	−1793.56	−1.09445	−0.547224	+	0.836986i	$0.684315\pi$
$140$	0	0
$141$	0	0
$142$	64.8431	0.0383205
$143$	731.812	0.427952
$144$	0	0
$145$	0	0
$146$	522.355	0.296099
$147$	0	0
$148$	−1062.80	−0.590279
$149$	1986.20	1.09205	0.546027	−	0.837768i	$-0.316140\pi$
$149$	1986.20	1.09205	0.546027	+	0.837768i	$0.316140\pi$
$150$	0	0
$151$	−1868.36	−1.00692	−0.503460	−	0.864018i	$-0.667940\pi$
$151$	−1868.36	−1.00692	−0.503460	+	0.864018i	$0.667940\pi$
$152$	−608.923	−0.324935
$153$	0	0
$154$	−959.797	−0.502225
$155$	0	0
$156$	0	0
$157$	−1961.80	−0.997255	−0.498627	−	0.866816i	$-0.666162\pi$
$157$	−1961.80	−0.997255	−0.498627	+	0.866816i	$0.666162\pi$
$158$	−865.933	−0.436012
$159$	0	0
$160$	0	0
$161$	−6021.81	−2.94773
$162$	0	0
$163$	1586.28	0.762254	0.381127	−	0.924523i	$-0.375536\pi$
$163$	1586.28	0.762254	0.381127	+	0.924523i	$0.375536\pi$
$164$	−48.1992	−0.0229495
$165$	0	0
$166$	297.307	0.139009
$167$	−2442.69	−1.13186	−0.565932	−	0.824452i	$-0.691484\pi$
$167$	−2442.69	−1.13186	−0.565932	+	0.824452i	$0.691484\pi$
$168$	0	0
$169$	−1241.61	−0.565139
$170$	0	0
$171$	0	0
$172$	−2463.40	−1.09205
$173$	−1520.90	−0.668391	−0.334196	−	0.942504i	$-0.608465\pi$
$173$	−1520.90	−0.668391	−0.334196	+	0.942504i	$0.608465\pi$
$174$	0	0
$175$	0	0
$176$	719.212	0.308027
$177$	0	0
$178$	−682.437	−0.287364
$179$	902.128	0.376694	0.188347	−	0.982103i	$-0.439687\pi$
$179$	902.128	0.376694	0.188347	+	0.982103i	$0.439687\pi$
$180$	0	0
$181$	−3472.27	−1.42592	−0.712962	−	0.701203i	$-0.752647\pi$
$181$	−3472.27	−1.42592	−0.712962	+	0.701203i	$0.752647\pi$
$182$	−1253.03	−0.510333
$183$	0	0
$184$	−3219.14	−1.28977
$185$	0	0
$186$	0	0
$187$	−1154.30	−0.451394
$188$	−1152.15	−0.446963
$189$	0	0
$190$	0	0
$191$	1160.63	0.439688	0.219844	−	0.975535i	$-0.429445\pi$
$191$	1160.63	0.439688	0.219844	+	0.975535i	$0.429445\pi$
$192$	0	0
$193$	2061.38	0.768815	0.384408	−	0.923164i	$-0.374406\pi$
$193$	2061.38	0.768815	0.384408	+	0.923164i	$0.374406\pi$
$194$	575.837	0.213106
$195$	0	0
$196$	−4925.31	−1.79494
$197$	743.457	0.268879	0.134439	−	0.990922i	$-0.457077\pi$
$197$	743.457	0.268879	0.134439	+	0.990922i	$0.457077\pi$
$198$	0	0
$199$	350.592	0.124888	0.0624442	−	0.998048i	$-0.480110\pi$
$199$	350.592	0.124888	0.0624442	+	0.998048i	$0.480110\pi$
$200$	0	0
$201$	0	0
$202$	1434.83	0.499774
$203$	142.343	0.0492144
$204$	0	0
$205$	0	0
$206$	−1827.52	−0.618103
$207$	0	0
$208$	938.942	0.313000
$209$	813.121	0.269114
$210$	0	0
$211$	151.733	0.0495059	0.0247529	−	0.999694i	$-0.492120\pi$
$211$	151.733	0.0495059	0.0247529	+	0.999694i	$0.492120\pi$
$212$	3820.90	1.23783
$213$	0	0
$214$	877.490	0.280299
$215$	0	0
$216$	0	0
$217$	8972.30	2.80682
$218$	345.057	0.107203
$219$	0	0
$220$	0	0
$221$	−1506.95	−0.458682
$222$	0	0
$223$	2528.95	0.759423	0.379711	−	0.925105i	$-0.376023\pi$
$223$	2528.95	0.759423	0.379711	+	0.925105i	$0.376023\pi$
$224$	−5935.91	−1.77058
$225$	0	0
$226$	−2846.16	−0.837715
$227$	4898.39	1.43224	0.716118	−	0.697979i	$-0.245917\pi$
$227$	4898.39	1.43224	0.716118	+	0.697979i	$0.245917\pi$
$228$	0	0
$229$	−2865.32	−0.826838	−0.413419	−	0.910541i	$-0.635666\pi$
$229$	−2865.32	−0.826838	−0.413419	+	0.910541i	$0.635666\pi$
$230$	0	0
$231$	0	0
$232$	76.0939	0.0215337
$233$	−4330.31	−1.21754	−0.608772	−	0.793345i	$-0.708338\pi$
$233$	−4330.31	−1.21754	−0.608772	+	0.793345i	$0.708338\pi$
$234$	0	0
$235$	0	0
$236$	5494.17	1.51542
$237$	0	0
$238$	1976.42	0.538288
$239$	3760.32	1.01772	0.508860	−	0.860850i	$-0.330067\pi$
$239$	3760.32	1.01772	0.508860	+	0.860850i	$0.330067\pi$
$240$	0	0
$241$	−3105.62	−0.830087	−0.415043	−	0.909802i	$-0.636234\pi$
$241$	−3105.62	−0.830087	−0.415043	+	0.909802i	$0.636234\pi$
$242$	−941.692	−0.250142
$243$	0	0
$244$	−3841.06	−1.00778
$245$	0	0
$246$	0	0
$247$	1061.54	0.273458
$248$	4796.41	1.22812
$249$	0	0
$250$	0	0
$251$	−2618.73	−0.658536	−0.329268	−	0.944236i	$-0.606802\pi$
$251$	−2618.73	−0.658536	−0.329268	+	0.944236i	$0.606802\pi$
$252$	0	0
$253$	4298.65	1.06820
$254$	−198.271	−0.0489788
$255$	0	0
$256$	−1592.14	−0.388705
$257$	1144.76	0.277853	0.138926	−	0.990303i	$-0.455635\pi$
$257$	1144.76	0.277853	0.138926	+	0.990303i	$0.455635\pi$
$258$	0	0
$259$	−5417.95	−1.29983
$260$	0	0
$261$	0	0
$262$	−2736.92	−0.645372
$263$	2950.25	0.691713	0.345856	−	0.938287i	$-0.387588\pi$
$263$	2950.25	0.691713	0.345856	+	0.938287i	$0.387588\pi$
$264$	0	0
$265$	0	0
$266$	−1392.25	−0.320918
$267$	0	0
$268$	−3215.13	−0.732818
$269$	3069.15	0.695647	0.347824	−	0.937560i	$-0.386921\pi$
$269$	3069.15	0.695647	0.347824	+	0.937560i	$0.386921\pi$
$270$	0	0
$271$	−6181.73	−1.38566	−0.692829	−	0.721102i	$-0.743636\pi$
$271$	−6181.73	−1.38566	−0.692829	+	0.721102i	$0.743636\pi$
$272$	−1481.01	−0.330145
$273$	0	0
$274$	1224.48	0.269977
$275$	0	0
$276$	0	0
$277$	5044.68	1.09424	0.547122	−	0.837053i	$-0.315724\pi$
$277$	5044.68	1.09424	0.547122	+	0.837053i	$0.315724\pi$
$278$	−2192.22	−0.472952
$279$	0	0
$280$	0	0
$281$	6352.35	1.34857	0.674287	−	0.738469i	$-0.264451\pi$
$281$	6352.35	1.34857	0.674287	+	0.738469i	$0.264451\pi$
$282$	0	0
$283$	−1303.15	−0.273726	−0.136863	−	0.990590i	$-0.543702\pi$
$283$	−1303.15	−0.273726	−0.136863	+	0.990590i	$0.543702\pi$
$284$	−345.156	−0.0721169
$285$	0	0
$286$	894.472	0.184934
$287$	−245.711	−0.0505362
$288$	0	0
$289$	−2536.05	−0.516193
$290$	0	0
$291$	0	0
$292$	−2780.46	−0.557240
$293$	2433.97	0.485303	0.242652	−	0.970113i	$-0.421983\pi$
$293$	2433.97	0.485303	0.242652	+	0.970113i	$0.421983\pi$
$294$	0	0
$295$	0	0
$296$	−2896.33	−0.568736
$297$	0	0
$298$	2427.67	0.471917
$299$	5611.95	1.08544
$300$	0	0
$301$	−12558.0	−2.40475
$302$	−2283.64	−0.435128
$303$	0	0
$304$	1043.26	0.196827
$305$	0	0
$306$	0	0
$307$	−5629.59	−1.04657	−0.523286	−	0.852157i	$-0.675294\pi$
$307$	−5629.59	−1.04657	−0.523286	+	0.852157i	$0.675294\pi$
$308$	5108.93	0.945157
$309$	0	0
$310$	0	0
$311$	4607.52	0.840092	0.420046	−	0.907503i	$-0.362014\pi$
$311$	4607.52	0.840092	0.420046	+	0.907503i	$0.362014\pi$
$312$	0	0
$313$	7214.85	1.30290	0.651450	−	0.758692i	$-0.274161\pi$
$313$	7214.85	1.30290	0.651450	+	0.758692i	$0.274161\pi$
$314$	−2397.85	−0.430951
$315$	0	0
$316$	4609.30	0.820549
$317$	−6183.53	−1.09559	−0.547794	−	0.836613i	$-0.684532\pi$
$317$	−6183.53	−1.09559	−0.547794	+	0.836613i	$0.684532\pi$
$318$	0	0
$319$	−101.611	−0.0178343
$320$	0	0
$321$	0	0
$322$	−7360.27	−1.27383
$323$	−1674.39	−0.288438
$324$	0	0
$325$	0	0
$326$	1938.87	0.329398
$327$	0	0
$328$	−131.353	−0.0221120
$329$	−5873.46	−0.984238
$330$	0	0
$331$	1657.31	0.275208	0.137604	−	0.990487i	$-0.456060\pi$
$331$	1657.31	0.275208	0.137604	+	0.990487i	$0.456060\pi$
$332$	−1582.54	−0.261607
$333$	0	0
$334$	−2985.63	−0.489121
$335$	0	0
$336$	0	0
$337$	8802.58	1.42287	0.711435	−	0.702752i	$-0.248045\pi$
$337$	8802.58	1.42287	0.711435	+	0.702752i	$0.248045\pi$
$338$	−1517.58	−0.244218
$339$	0	0
$340$	0	0
$341$	−6404.86	−1.01713
$342$	0	0
$343$	−13732.2	−2.16172
$344$	−6713.26	−1.05219
$345$	0	0
$346$	−1858.95	−0.288837
$347$	3024.89	0.467967	0.233983	−	0.972241i	$-0.424824\pi$
$347$	3024.89	0.467967	0.233983	+	0.972241i	$0.424824\pi$
$348$	0	0
$349$	−8247.55	−1.26499	−0.632494	−	0.774565i	$-0.717969\pi$
$349$	−8247.55	−1.26499	−0.632494	+	0.774565i	$0.717969\pi$
$350$	0	0
$351$	0	0
$352$	4237.34	0.641622
$353$	−10224.9	−1.54169	−0.770847	−	0.637021i	$-0.780167\pi$
$353$	−10224.9	−1.54169	−0.770847	+	0.637021i	$0.780167\pi$
$354$	0	0
$355$	0	0
$356$	3632.56	0.540802
$357$	0	0
$358$	1102.64	0.162784
$359$	−9178.35	−1.34934	−0.674672	−	0.738118i	$-0.735715\pi$
$359$	−9178.35	−1.34934	−0.674672	+	0.738118i	$0.735715\pi$
$360$	0	0
$361$	−5679.52	−0.828039
$362$	−4244.05	−0.616195
$363$	0	0
$364$	6669.78	0.960416
$365$	0	0
$366$	0	0
$367$	640.156	0.0910514	0.0455257	−	0.998963i	$-0.485504\pi$
$367$	640.156	0.0910514	0.0455257	+	0.998963i	$0.485504\pi$
$368$	5515.33	0.781268
$369$	0	0
$370$	0	0
$371$	19478.3	2.72577
$372$	0	0
$373$	−4807.89	−0.667408	−0.333704	−	0.942678i	$-0.608299\pi$
$373$	−4807.89	−0.667408	−0.333704	+	0.942678i	$0.608299\pi$
$374$	−1410.87	−0.195065
$375$	0	0
$376$	−3139.84	−0.430651
$377$	−132.655	−0.0181222
$378$	0	0
$379$	4449.52	0.603051	0.301526	−	0.953458i	$-0.402504\pi$
$379$	4449.52	0.603051	0.301526	+	0.953458i	$0.402504\pi$
$380$	0	0
$381$	0	0
$382$	1418.61	0.190006
$383$	−6289.84	−0.839153	−0.419577	−	0.907720i	$-0.637821\pi$
$383$	−6289.84	−0.839153	−0.419577	+	0.907720i	$0.637821\pi$
$384$	0	0
$385$	0	0
$386$	2519.56	0.332234
$387$	0	0
$388$	−3065.14	−0.401054
$389$	5650.19	0.736442	0.368221	−	0.929738i	$-0.379967\pi$
$389$	5650.19	0.736442	0.368221	+	0.929738i	$0.379967\pi$
$390$	0	0
$391$	−8851.83	−1.14490
$392$	−13422.5	−1.72943
$393$	0	0
$394$	908.705	0.116193
$395$	0	0
$396$	0	0
$397$	−11922.6	−1.50725	−0.753623	−	0.657307i	$-0.771696\pi$
$397$	−11922.6	−1.50725	−0.753623	+	0.657307i	$0.771696\pi$
$398$	428.517	0.0539689
$399$	0	0
$400$	0	0
$401$	−14042.9	−1.74881	−0.874403	−	0.485201i	$-0.838746\pi$
$401$	−14042.9	−1.74881	−0.874403	+	0.485201i	$0.838746\pi$
$402$	0	0
$403$	−8361.63	−1.03355
$404$	−7637.50	−0.940544
$405$	0	0
$406$	173.982	0.0212674
$407$	3867.59	0.471031
$408$	0	0
$409$	4015.80	0.485498	0.242749	−	0.970089i	$-0.421951\pi$
$409$	4015.80	0.485498	0.242749	+	0.970089i	$0.421951\pi$
$410$	0	0
$411$	0	0
$412$	9727.75	1.16323
$413$	28008.4	3.33705
$414$	0	0
$415$	0	0
$416$	5531.91	0.651981
$417$	0	0
$418$	993.853	0.116294
$419$	−15881.9	−1.85175	−0.925873	−	0.377835i	$-0.876669\pi$
$419$	−15881.9	−1.85175	−0.925873	+	0.377835i	$0.876669\pi$
$420$	0	0
$421$	3401.87	0.393817	0.196908	−	0.980422i	$-0.436910\pi$
$421$	3401.87	0.393817	0.196908	+	0.980422i	$0.436910\pi$
$422$	185.459	0.0213934
$423$	0	0
$424$	10412.7	1.19266
$425$	0	0
$426$	0	0
$427$	−19581.1	−2.21919
$428$	−4670.82	−0.527506
$429$	0	0
$430$	0	0
$431$	−6067.90	−0.678145	−0.339072	−	0.940760i	$-0.610113\pi$
$431$	−6067.90	−0.678145	−0.339072	+	0.940760i	$0.610113\pi$
$432$	0	0
$433$	−2368.69	−0.262891	−0.131446	−	0.991323i	$-0.541962\pi$
$433$	−2368.69	−0.262891	−0.131446	+	0.991323i	$0.541962\pi$
$434$	10966.6	1.21293
$435$	0	0
$436$	−1836.71	−0.201749
$437$	6235.47	0.682570
$438$	0	0
$439$	−3731.54	−0.405687	−0.202844	−	0.979211i	$-0.565018\pi$
$439$	−3731.54	−0.405687	−0.202844	+	0.979211i	$0.565018\pi$
$440$	0	0
$441$	0	0
$442$	−1841.90	−0.198214
$443$	15917.5	1.70714	0.853572	−	0.520975i	$-0.174431\pi$
$443$	15917.5	1.70714	0.853572	+	0.520975i	$0.174431\pi$
$444$	0	0
$445$	0	0
$446$	3091.06	0.328175
$447$	0	0
$448$	804.831	0.0848766
$449$	9088.41	0.955252	0.477626	−	0.878563i	$-0.341497\pi$
$449$	9088.41	0.955252	0.477626	+	0.878563i	$0.341497\pi$
$450$	0	0
$451$	175.401	0.0183133
$452$	15149.9	1.57653
$453$	0	0
$454$	5987.16	0.618923
$455$	0	0
$456$	0	0
$457$	−11877.8	−1.21580	−0.607901	−	0.794013i	$-0.707988\pi$
$457$	−11877.8	−1.21580	−0.607901	+	0.794013i	$0.707988\pi$
$458$	−3502.20	−0.357308
$459$	0	0
$460$	0	0
$461$	−7251.15	−0.732581	−0.366291	−	0.930501i	$-0.619372\pi$
$461$	−7251.15	−0.732581	−0.366291	+	0.930501i	$0.619372\pi$
$462$	0	0
$463$	1778.78	0.178547	0.0892733	−	0.996007i	$-0.471546\pi$
$463$	1778.78	0.178547	0.0892733	+	0.996007i	$0.471546\pi$
$464$	−130.371	−0.0130438
$465$	0	0
$466$	−5292.80	−0.526147
$467$	5153.79	0.510683	0.255342	−	0.966851i	$-0.417812\pi$
$467$	5153.79	0.510683	0.255342	+	0.966851i	$0.417812\pi$
$468$	0	0
$469$	−16390.2	−1.61371
$470$	0	0
$471$	0	0
$472$	14972.7	1.46012
$473$	8964.49	0.871433
$474$	0	0
$475$	0	0
$476$	−10520.4	−1.01303
$477$	0	0
$478$	4596.13	0.439795
$479$	−7850.07	−0.748808	−0.374404	−	0.927266i	$-0.622153\pi$
$479$	−7850.07	−0.748808	−0.374404	+	0.927266i	$0.622153\pi$
$480$	0	0
$481$	5049.20	0.478635
$482$	−3795.91	−0.358712
$483$	0	0
$484$	5012.56	0.470751
$485$	0	0
$486$	0	0
$487$	8659.46	0.805744	0.402872	−	0.915256i	$-0.368012\pi$
$487$	8659.46	0.805744	0.402872	+	0.915256i	$0.368012\pi$
$488$	−10467.7	−0.971002
$489$	0	0
$490$	0	0
$491$	1990.75	0.182977	0.0914883	−	0.995806i	$-0.470838\pi$
$491$	1990.75	0.182977	0.0914883	+	0.995806i	$0.470838\pi$
$492$	0	0
$493$	209.239	0.0191149
$494$	1297.49	0.118172
$495$	0	0
$496$	−8217.67	−0.743920
$497$	−1759.54	−0.158806
$498$	0	0
$499$	−11262.0	−1.01033	−0.505166	−	0.863022i	$-0.668569\pi$
$499$	−11262.0	−1.01033	−0.505166	+	0.863022i	$0.668569\pi$
$500$	0	0
$501$	0	0
$502$	−3200.79	−0.284578
$503$	−9837.79	−0.872058	−0.436029	−	0.899933i	$-0.643616\pi$
$503$	−9837.79	−0.872058	−0.436029	+	0.899933i	$0.643616\pi$
$504$	0	0
$505$	0	0
$506$	5254.11	0.461608
$507$	0	0
$508$	1055.38	0.0921753
$509$	9403.17	0.818837	0.409419	−	0.912347i	$-0.365732\pi$
$509$	9403.17	0.818837	0.409419	+	0.912347i	$0.365732\pi$
$510$	0	0
$511$	−14174.3	−1.22707
$512$	9745.45	0.841195
$513$	0	0
$514$	1399.21	0.120071
$515$	0	0
$516$	0	0
$517$	4192.76	0.356668
$518$	−6622.20	−0.561704
$519$	0	0
$520$	0	0
$521$	6915.28	0.581505	0.290752	−	0.956798i	$-0.406094\pi$
$521$	6915.28	0.581505	0.290752	+	0.956798i	$0.406094\pi$
$522$	0	0
$523$	−9137.40	−0.763959	−0.381980	−	0.924171i	$-0.624758\pi$
$523$	−9137.40	−0.763959	−0.381980	+	0.924171i	$0.624758\pi$
$524$	14568.4	1.21455
$525$	0	0
$526$	3606.01	0.298915
$527$	13188.9	1.09017
$528$	0	0
$529$	20797.5	1.70934
$530$	0	0
$531$	0	0
$532$	7410.83	0.603948
$533$	228.988	0.0186089
$534$	0	0
$535$	0	0
$536$	−8761.87	−0.706073
$537$	0	0
$538$	3751.32	0.300615
$539$	17923.6	1.43233
$540$	0	0
$541$	4862.56	0.386429	0.193214	−	0.981157i	$-0.438109\pi$
$541$	4862.56	0.386429	0.193214	+	0.981157i	$0.438109\pi$
$542$	−7555.74	−0.598795
$543$	0	0
$544$	−8725.57	−0.687695
$545$	0	0
$546$	0	0
$547$	1037.26	0.0810791	0.0405395	−	0.999178i	$-0.487092\pi$
$547$	1037.26	0.0810791	0.0405395	+	0.999178i	$0.487092\pi$
$548$	−6517.84	−0.508081
$549$	0	0
$550$	0	0
$551$	−147.394	−0.0113960
$552$	0	0
$553$	23497.4	1.80689
$554$	6165.96	0.472864
$555$	0	0
$556$	11669.0	0.890067
$557$	−86.7829	−0.00660164	−0.00330082	−	0.999995i	$-0.501051\pi$
$557$	−86.7829	−0.00660164	−0.00330082	+	0.999995i	$0.501051\pi$
$558$	0	0
$559$	11703.3	0.885502
$560$	0	0
$561$	0	0
$562$	7764.28	0.582770
$563$	15914.5	1.19133	0.595664	−	0.803234i	$-0.296889\pi$
$563$	15914.5	1.19133	0.595664	+	0.803234i	$0.296889\pi$
$564$	0	0
$565$	0	0
$566$	−1592.81	−0.118287
$567$	0	0
$568$	−940.618	−0.0694850
$569$	14177.4	1.04454	0.522272	−	0.852779i	$-0.325084\pi$
$569$	14177.4	1.04454	0.522272	+	0.852779i	$0.325084\pi$
$570$	0	0
$571$	21676.4	1.58867	0.794333	−	0.607483i	$-0.207821\pi$
$571$	21676.4	1.58867	0.794333	+	0.607483i	$0.207821\pi$
$572$	−4761.21	−0.348035
$573$	0	0
$574$	−300.326	−0.0218386
$575$	0	0
$576$	0	0
$577$	−6083.49	−0.438924	−0.219462	−	0.975621i	$-0.570430\pi$
$577$	−6083.49	−0.438924	−0.219462	+	0.975621i	$0.570430\pi$
$578$	−3099.74	−0.223066
$579$	0	0
$580$	0	0
$581$	−8067.54	−0.576072
$582$	0	0
$583$	−13904.5	−0.987765
$584$	−7577.32	−0.536903
$585$	0	0
$586$	2974.96	0.209718
$587$	−8266.51	−0.581253	−0.290626	−	0.956837i	$-0.593864\pi$
$587$	−8266.51	−0.581253	−0.290626	+	0.956837i	$0.593864\pi$
$588$	0	0
$589$	−9290.65	−0.649940
$590$	0	0
$591$	0	0
$592$	4962.27	0.344507
$593$	−18155.4	−1.25725	−0.628627	−	0.777707i	$-0.716383\pi$
$593$	−18155.4	−1.25725	−0.628627	+	0.777707i	$0.716383\pi$
$594$	0	0
$595$	0	0
$596$	−12922.3	−0.888120
$597$	0	0
$598$	6859.32	0.469061
$599$	−6899.65	−0.470637	−0.235319	−	0.971918i	$-0.575613\pi$
$599$	−6899.65	−0.470637	−0.235319	+	0.971918i	$0.575613\pi$
$600$	0	0
$601$	−8402.89	−0.570318	−0.285159	−	0.958480i	$-0.592046\pi$
$601$	−8402.89	−0.570318	−0.285159	+	0.958480i	$0.592046\pi$
$602$	−15349.2	−1.03918
$603$	0	0
$604$	12155.7	0.818886
$605$	0	0
$606$	0	0
$607$	−16342.9	−1.09281	−0.546407	−	0.837520i	$-0.684005\pi$
$607$	−16342.9	−1.09281	−0.546407	+	0.837520i	$0.684005\pi$
$608$	6146.53	0.409992
$609$	0	0
$610$	0	0
$611$	5473.70	0.362426
$612$	0	0
$613$	15671.0	1.03254	0.516268	−	0.856427i	$-0.327321\pi$
$613$	15671.0	1.03254	0.516268	+	0.856427i	$0.327321\pi$
$614$	−6880.87	−0.452263
$615$	0	0
$616$	13922.9	0.910663
$617$	30226.8	1.97226	0.986132	−	0.165962i	$-0.0530728\pi$
$617$	30226.8	1.97226	0.986132	+	0.165962i	$0.0530728\pi$
$618$	0	0
$619$	4890.59	0.317560	0.158780	−	0.987314i	$-0.449244\pi$
$619$	4890.59	0.317560	0.158780	+	0.987314i	$0.449244\pi$
$620$	0	0
$621$	0	0
$622$	5631.64	0.363035
$623$	18518.2	1.19088
$624$	0	0
$625$	0	0
$626$	8818.49	0.563032
$627$	0	0
$628$	12763.6	0.811025
$629$	−7964.19	−0.504854
$630$	0	0
$631$	−18164.8	−1.14600	−0.573001	−	0.819554i	$-0.694221\pi$
$631$	−18164.8	−1.14600	−0.573001	+	0.819554i	$0.694221\pi$
$632$	12561.3	0.790602
$633$	0	0
$634$	−7557.94	−0.473445
$635$	0	0
$636$	0	0
$637$	23399.5	1.45545
$638$	−124.197	−0.00770688
$639$	0	0
$640$	0	0
$641$	−17122.9	−1.05509	−0.527546	−	0.849526i	$-0.676888\pi$
$641$	−17122.9	−1.05509	−0.527546	+	0.849526i	$0.676888\pi$
$642$	0	0
$643$	−17652.2	−1.08263	−0.541317	−	0.840819i	$-0.682074\pi$
$643$	−17652.2	−1.08263	−0.541317	+	0.840819i	$0.682074\pi$
$644$	39178.2	2.39726
$645$	0	0
$646$	−2046.55	−0.124645
$647$	−6134.20	−0.372736	−0.186368	−	0.982480i	$-0.559672\pi$
$647$	−6134.20	−0.372736	−0.186368	+	0.982480i	$0.559672\pi$
$648$	0	0
$649$	−19993.7	−1.20928
$650$	0	0
$651$	0	0
$652$	−10320.5	−0.619908
$653$	12879.6	0.771850	0.385925	−	0.922530i	$-0.373882\pi$
$653$	12879.6	0.771850	0.385925	+	0.922530i	$0.373882\pi$
$654$	0	0
$655$	0	0
$656$	225.045	0.0133941
$657$	0	0
$658$	−7178.95	−0.425326
$659$	−28522.0	−1.68598	−0.842990	−	0.537929i	$-0.819207\pi$
$659$	−28522.0	−1.68598	−0.842990	+	0.537929i	$0.819207\pi$
$660$	0	0
$661$	−3738.93	−0.220011	−0.110006	−	0.993931i	$-0.535087\pi$
$661$	−3738.93	−0.220011	−0.110006	+	0.993931i	$0.535087\pi$
$662$	2025.67	0.118928
$663$	0	0
$664$	−4312.75	−0.252059
$665$	0	0
$666$	0	0
$667$	−779.214	−0.0452343
$668$	15892.3	0.920497
$669$	0	0
$670$	0	0
$671$	13977.9	0.804190
$672$	0	0
$673$	22142.8	1.26827	0.634133	−	0.773224i	$-0.281357\pi$
$673$	22142.8	1.26827	0.634133	+	0.773224i	$0.281357\pi$
$674$	10759.1	0.614876
$675$	0	0
$676$	8077.98	0.459603
$677$	11631.7	0.660329	0.330164	−	0.943923i	$-0.392896\pi$
$677$	11631.7	0.660329	0.330164	+	0.943923i	$0.392896\pi$
$678$	0	0
$679$	−15625.6	−0.883143
$680$	0	0
$681$	0	0
$682$	−7828.46	−0.439541
$683$	−28361.6	−1.58891	−0.794456	−	0.607321i	$-0.792244\pi$
$683$	−28361.6	−1.58891	−0.794456	+	0.607321i	$0.792244\pi$
$684$	0	0
$685$	0	0
$686$	−16784.5	−0.934160
$687$	0	0
$688$	11501.8	0.637356
$689$	−18152.6	−1.00371
$690$	0	0
$691$	−8292.49	−0.456529	−0.228264	−	0.973599i	$-0.573305\pi$
$691$	−8292.49	−0.456529	−0.228264	+	0.973599i	$0.573305\pi$
$692$	9895.05	0.543574
$693$	0	0
$694$	3697.23	0.202226
$695$	0	0
$696$	0	0
$697$	−361.187	−0.0196283
$698$	−10080.7	−0.546649
$699$	0	0
$700$	0	0
$701$	−586.036	−0.0315753	−0.0157876	−	0.999875i	$-0.505026\pi$
$701$	−586.036	−0.0315753	−0.0157876	+	0.999875i	$0.505026\pi$
$702$	0	0
$703$	5610.19	0.300985
$704$	−574.527	−0.0307575
$705$	0	0
$706$	−12497.6	−0.666224
$707$	−38934.7	−2.07113
$708$	0	0
$709$	−17673.2	−0.936149	−0.468075	−	0.883689i	$-0.655052\pi$
$709$	−17673.2	−0.936149	−0.468075	+	0.883689i	$0.655052\pi$
$710$	0	0
$711$	0	0
$712$	9899.46	0.521065
$713$	−49116.1	−2.57982
$714$	0	0
$715$	0	0
$716$	−5869.30	−0.306349
$717$	0	0
$718$	−11218.4	−0.583102
$719$	2714.68	0.140807	0.0704037	−	0.997519i	$-0.477571\pi$
$719$	2714.68	0.140807	0.0704037	+	0.997519i	$0.477571\pi$
$720$	0	0
$721$	49590.4	2.56150
$722$	−6941.90	−0.357827
$723$	0	0
$724$	22590.8	1.15964
$725$	0	0
$726$	0	0
$727$	−2595.19	−0.132394	−0.0661969	−	0.997807i	$-0.521087\pi$
$727$	−2595.19	−0.132394	−0.0661969	+	0.997807i	$0.521087\pi$
$728$	18176.5	0.925365
$729$	0	0
$730$	0	0
$731$	−18459.8	−0.934007
$732$	0	0
$733$	−8334.71	−0.419986	−0.209993	−	0.977703i	$-0.567344\pi$
$733$	−8334.71	−0.419986	−0.209993	+	0.977703i	$0.567344\pi$
$734$	782.443	0.0393467
$735$	0	0
$736$	32494.3	1.62739
$737$	11700.1	0.584774
$738$	0	0
$739$	−18970.8	−0.944317	−0.472159	−	0.881514i	$-0.656525\pi$
$739$	−18970.8	−0.944317	−0.472159	+	0.881514i	$0.656525\pi$
$740$	0	0
$741$	0	0
$742$	23807.7	1.17791
$743$	−15043.1	−0.742770	−0.371385	−	0.928479i	$-0.621117\pi$
$743$	−15043.1	−0.742770	−0.371385	+	0.928479i	$0.621117\pi$
$744$	0	0
$745$	0	0
$746$	−5876.54	−0.288412
$747$	0	0
$748$	7509.94	0.367100
$749$	−23811.0	−1.16160
$750$	0	0
$751$	7098.11	0.344892	0.172446	−	0.985019i	$-0.444833\pi$
$751$	7098.11	0.344892	0.172446	+	0.985019i	$0.444833\pi$
$752$	5379.46	0.260863
$753$	0	0
$754$	−162.140	−0.00783130
$755$	0	0
$756$	0	0
$757$	−25376.1	−1.21837	−0.609187	−	0.793027i	$-0.708504\pi$
$757$	−25376.1	−1.21837	−0.609187	+	0.793027i	$0.708504\pi$
$758$	5438.51	0.260601
$759$	0	0
$760$	0	0
$761$	7489.36	0.356753	0.178377	−	0.983962i	$-0.442915\pi$
$761$	7489.36	0.356753	0.178377	+	0.983962i	$0.442915\pi$
$762$	0	0
$763$	−9363.25	−0.444263
$764$	−7551.14	−0.357580
$765$	0	0
$766$	−7687.88	−0.362630
$767$	−26102.1	−1.22880
$768$	0	0
$769$	35506.8	1.66503	0.832515	−	0.554002i	$-0.186900\pi$
$769$	35506.8	1.66503	0.832515	+	0.554002i	$0.186900\pi$
$770$	0	0
$771$	0	0
$772$	−13411.4	−0.625244
$773$	37535.4	1.74651	0.873257	−	0.487259i	$-0.162003\pi$
$773$	37535.4	1.74651	0.873257	+	0.487259i	$0.162003\pi$
$774$	0	0
$775$	0	0
$776$	−8353.12	−0.386417
$777$	0	0
$778$	6906.05	0.318244
$779$	254.430	0.0117020
$780$	0	0
$781$	1256.05	0.0575479
$782$	−10819.3	−0.494755
$783$	0	0
$784$	22996.6	1.04759
$785$	0	0
$786$	0	0
$787$	2496.63	0.113082	0.0565408	−	0.998400i	$-0.481993\pi$
$787$	2496.63	0.113082	0.0565408	+	0.998400i	$0.481993\pi$
$788$	−4836.97	−0.218668
$789$	0	0
$790$	0	0
$791$	77231.7	3.47161
$792$	0	0
$793$	18248.4	0.817174
$794$	−14572.6	−0.651338
$795$	0	0
$796$	−2280.97	−0.101566
$797$	−31119.0	−1.38305	−0.691526	−	0.722352i	$-0.743061\pi$
$797$	−31119.0	−1.38305	−0.691526	+	0.722352i	$0.743061\pi$
$798$	0	0
$799$	−8633.77	−0.382279
$800$	0	0
$801$	0	0
$802$	−17164.3	−0.755724
$803$	10118.3	0.444667
$804$	0	0
$805$	0	0
$806$	−10220.2	−0.446638
$807$	0	0
$808$	−20813.7	−0.906218
$809$	26410.8	1.14778	0.573890	−	0.818932i	$-0.305434\pi$
$809$	26410.8	1.14778	0.573890	+	0.818932i	$0.305434\pi$
$810$	0	0
$811$	2617.02	0.113312	0.0566559	−	0.998394i	$-0.481956\pi$
$811$	2617.02	0.113312	0.0566559	+	0.998394i	$0.481956\pi$
$812$	−926.093	−0.0400240
$813$	0	0
$814$	4727.24	0.203550
$815$	0	0
$816$	0	0
$817$	13003.6	0.556839
$818$	4908.40	0.209802
$819$	0	0
$820$	0	0
$821$	−11728.9	−0.498589	−0.249295	−	0.968428i	$-0.580199\pi$
$821$	−11728.9	−0.498589	−0.249295	+	0.968428i	$0.580199\pi$
$822$	0	0
$823$	−33171.7	−1.40497	−0.702486	−	0.711698i	$-0.747926\pi$
$823$	−33171.7	−1.40497	−0.702486	+	0.711698i	$0.747926\pi$
$824$	26510.1	1.12078
$825$	0	0
$826$	34233.8	1.44206
$827$	−37989.2	−1.59736	−0.798679	−	0.601757i	$-0.794468\pi$
$827$	−37989.2	−1.59736	−0.798679	+	0.601757i	$0.794468\pi$
$828$	0	0
$829$	−9051.33	−0.379211	−0.189605	−	0.981860i	$-0.560721\pi$
$829$	−9051.33	−0.379211	−0.189605	+	0.981860i	$0.560721\pi$
$830$	0	0
$831$	0	0
$832$	−750.053	−0.0312541
$833$	−36908.4	−1.53518
$834$	0	0
$835$	0	0
$836$	−5290.21	−0.218858
$837$	0	0
$838$	−19412.0	−0.800209
$839$	−29458.7	−1.21219	−0.606094	−	0.795393i	$-0.707265\pi$
$839$	−29458.7	−1.21219	−0.606094	+	0.795393i	$0.707265\pi$
$840$	0	0
$841$	−24370.6	−0.999245
$842$	4158.00	0.170183
$843$	0	0
$844$	−987.185	−0.0402610
$845$	0	0
$846$	0	0
$847$	25553.2	1.03662
$848$	−17840.0	−0.722440
$849$	0	0
$850$	0	0
$851$	29658.9	1.19471
$852$	0	0
$853$	10163.5	0.407960	0.203980	−	0.978975i	$-0.434612\pi$
$853$	10163.5	0.407960	0.203980	+	0.978975i	$0.434612\pi$
$854$	−23933.4	−0.958997
$855$	0	0
$856$	−12728.9	−0.508254
$857$	19520.2	0.778059	0.389030	−	0.921225i	$-0.372810\pi$
$857$	19520.2	0.778059	0.389030	+	0.921225i	$0.372810\pi$
$858$	0	0
$859$	−23043.9	−0.915308	−0.457654	−	0.889130i	$-0.651310\pi$
$859$	−23043.9	−0.915308	−0.457654	+	0.889130i	$0.651310\pi$
$860$	0	0
$861$	0	0
$862$	−7416.61	−0.293052
$863$	32489.6	1.28153	0.640764	−	0.767738i	$-0.278618\pi$
$863$	32489.6	1.28153	0.640764	+	0.767738i	$0.278618\pi$
$864$	0	0
$865$	0	0
$866$	−2895.18	−0.113605
$867$	0	0
$868$	−58374.3	−2.28266
$869$	−16773.6	−0.654782
$870$	0	0
$871$	15274.6	0.594215
$872$	−5005.41	−0.194386
$873$	0	0
$874$	7621.43	0.294964
$875$	0	0
$876$	0	0
$877$	−40252.9	−1.54988	−0.774939	−	0.632036i	$-0.782220\pi$
$877$	−40252.9	−1.54988	−0.774939	+	0.632036i	$0.782220\pi$
$878$	−4560.95	−0.175313
$879$	0	0
$880$	0	0
$881$	49327.8	1.88637	0.943187	−	0.332263i	$-0.107812\pi$
$881$	49327.8	1.88637	0.943187	+	0.332263i	$0.107812\pi$
$882$	0	0
$883$	−31936.4	−1.21715	−0.608577	−	0.793495i	$-0.708259\pi$
$883$	−31936.4	−1.21715	−0.608577	+	0.793495i	$0.708259\pi$
$884$	9804.33	0.373026
$885$	0	0
$886$	19455.5	0.737721
$887$	−3174.52	−0.120169	−0.0600845	−	0.998193i	$-0.519137\pi$
$887$	−3174.52	−0.120169	−0.0600845	+	0.998193i	$0.519137\pi$
$888$	0	0
$889$	5380.16	0.202975
$890$	0	0
$891$	0	0
$892$	−16453.5	−0.617606
$893$	6081.86	0.227908
$894$	0	0
$895$	0	0
$896$	48471.0	1.80726
$897$	0	0
$898$	11108.5	0.412801
$899$	1161.00	0.0430719
$900$	0	0
$901$	28632.4	1.05869
$902$	214.387	0.00791386
$903$	0	0
$904$	41286.5	1.51899
$905$	0	0
$906$	0	0
$907$	−27550.1	−1.00858	−0.504292	−	0.863533i	$-0.668246\pi$
$907$	−27550.1	−1.00858	−0.504292	+	0.863533i	$0.668246\pi$
$908$	−31869.2	−1.16478
$909$	0	0
$910$	0	0
$911$	−5634.64	−0.204922	−0.102461	−	0.994737i	$-0.532672\pi$
$911$	−5634.64	−0.204922	−0.102461	+	0.994737i	$0.532672\pi$
$912$	0	0
$913$	5759.00	0.208757
$914$	−14517.9	−0.525394
$915$	0	0
$916$	18642.0	0.672432
$917$	74267.4	2.67451
$918$	0	0
$919$	8518.91	0.305781	0.152891	−	0.988243i	$-0.451142\pi$
$919$	8518.91	0.305781	0.152891	+	0.988243i	$0.451142\pi$
$920$	0	0
$921$	0	0
$922$	−8862.87	−0.316576
$923$	1639.79	0.0584770
$924$	0	0
$925$	0	0
$926$	2174.15	0.0771567
$927$	0	0
$928$	−768.100	−0.0271704
$929$	−6640.06	−0.234503	−0.117252	−	0.993102i	$-0.537408\pi$
$929$	−6640.06	−0.234503	−0.117252	+	0.993102i	$0.537408\pi$
$930$	0	0
$931$	25999.3	0.915245
$932$	28173.2	0.990176
$933$	0	0
$934$	6299.32	0.220685
$935$	0	0
$936$	0	0
$937$	44972.1	1.56795	0.783977	−	0.620789i	$-0.213188\pi$
$937$	44972.1	1.56795	0.783977	+	0.620789i	$0.213188\pi$
$938$	−20033.2	−0.697343
$939$	0	0
$940$	0	0
$941$	−6264.93	−0.217036	−0.108518	−	0.994094i	$-0.534611\pi$
$941$	−6264.93	−0.217036	−0.108518	+	0.994094i	$0.534611\pi$
$942$	0	0
$943$	1345.07	0.0464492
$944$	−25652.7	−0.884452
$945$	0	0
$946$	10957.0	0.376579
$947$	9672.34	0.331899	0.165950	−	0.986134i	$-0.446931\pi$
$947$	9672.34	0.331899	0.165950	+	0.986134i	$0.446931\pi$
$948$	0	0
$949$	13209.6	0.451846
$950$	0	0
$951$	0	0
$952$	−28670.1	−0.976054
$953$	14173.3	0.481761	0.240880	−	0.970555i	$-0.422564\pi$
$953$	14173.3	0.481761	0.240880	+	0.970555i	$0.422564\pi$
$954$	0	0
$955$	0	0
$956$	−24464.9	−0.827667
$957$	0	0
$958$	−9594.91	−0.323588
$959$	−33226.8	−1.11882
$960$	0	0
$961$	43390.4	1.45649
$962$	6171.48	0.206836
$963$	0	0
$964$	20205.4	0.675074
$965$	0	0
$966$	0	0
$967$	43588.7	1.44955	0.724776	−	0.688984i	$-0.241943\pi$
$967$	43588.7	1.44955	0.724776	+	0.688984i	$0.241943\pi$
$968$	13660.2	0.453571
$969$	0	0
$970$	0	0
$971$	20974.8	0.693216	0.346608	−	0.938010i	$-0.387333\pi$
$971$	20974.8	0.693216	0.346608	+	0.938010i	$0.387333\pi$
$972$	0	0
$973$	59486.8	1.95998
$974$	10584.2	0.348192
$975$	0	0
$976$	17934.2	0.588176
$977$	8765.10	0.287022	0.143511	−	0.989649i	$-0.454161\pi$
$977$	8765.10	0.287022	0.143511	+	0.989649i	$0.454161\pi$
$978$	0	0
$979$	−13219.2	−0.431549
$980$	0	0
$981$	0	0
$982$	2433.24	0.0790710
$983$	39341.3	1.27649	0.638247	−	0.769832i	$-0.279660\pi$
$983$	39341.3	1.27649	0.638247	+	0.769832i	$0.279660\pi$
$984$	0	0
$985$	0	0
$986$	255.747	0.00826028
$987$	0	0
$988$	−6906.44	−0.222392
$989$	68744.9	2.21027
$990$	0	0
$991$	5063.64	0.162313	0.0811563	−	0.996701i	$-0.474139\pi$
$991$	5063.64	0.162313	0.0811563	+	0.996701i	$0.474139\pi$
$992$	−48415.5	−1.54959
$993$	0	0
$994$	−2150.64	−0.0686258
$995$	0	0
$996$	0	0
$997$	29030.8	0.922181	0.461090	−	0.887353i	$-0.347458\pi$
$997$	29030.8	0.922181	0.461090	+	0.887353i	$0.347458\pi$
$998$	−13765.2	−0.436603
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2025.4.a.bd.1.5		8
3.2	odd	2		2025.4.a.bc.1.4		8
5.2	odd	4		405.4.b.c.244.5	yes	8
5.3	odd	4		405.4.b.c.244.4	yes	8
5.4	even	2	inner	2025.4.a.bd.1.4		8
15.2	even	4		405.4.b.b.244.4	✓	8
15.8	even	4		405.4.b.b.244.5	yes	8
15.14	odd	2		2025.4.a.bc.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.4.b.b.244.4	✓	8	15.2	even	4
405.4.b.b.244.5	yes	8	15.8	even	4
405.4.b.c.244.4	yes	8	5.3	odd	4
405.4.b.c.244.5	yes	8	5.2	odd	4
2025.4.a.bc.1.4		8	3.2	odd	2
2025.4.a.bc.1.5		8	15.14	odd	2
2025.4.a.bd.1.4		8	5.4	even	2	inner
2025.4.a.bd.1.5		8	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:	\( N \)	\(=\)	\( 2025 = 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.22227 q^{2} -6.50606 q^{4} -33.1668 q^{7} -17.7303 q^{8} +23.6760 q^{11} +30.9094 q^{13} -40.5387 q^{14} +30.3772 q^{16} -48.7539 q^{17} +34.3436 q^{19} +28.9385 q^{22} +181.561 q^{23} +37.7796 q^{26} +215.785 q^{28} -4.29174 q^{29} -270.521 q^{31} +178.972 q^{32} -59.5905 q^{34} +163.355 q^{37} +41.9772 q^{38} +7.40836 q^{41} +378.631 q^{43} -154.038 q^{44} +221.917 q^{46} +177.089 q^{47} +757.035 q^{49} -201.098 q^{52} -587.283 q^{53} +588.057 q^{56} -5.24566 q^{58} -844.470 q^{59} +590.383 q^{61} -330.649 q^{62} -24.2662 q^{64} +494.174 q^{67} +317.196 q^{68} +53.0514 q^{71} +427.365 q^{73} +199.664 q^{74} -223.441 q^{76} -785.258 q^{77} -708.463 q^{79} +9.05501 q^{82} +243.242 q^{83} +462.790 q^{86} -419.784 q^{88} -558.336 q^{89} -1025.17 q^{91} -1181.25 q^{92} +216.450 q^{94} +471.121 q^{97} +925.301 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 24 q^{4} + 42 q^{14} + 4 q^{16} - 118 q^{19} - 36 q^{26} - 318 q^{29} - 416 q^{31} - 638 q^{34} + 486 q^{41} - 852 q^{44} - 598 q^{46} - 350 q^{49} + 1530 q^{56} + 1146 q^{59} - 398 q^{61} - 1640 q^{64}+ \cdots - 1238 q^{94}+O(q^{100}) \) 8 * q + 24 * q^4 + 42 * q^14 + 4 * q^16 - 118 * q^19 - 36 * q^26 - 318 * q^29 - 416 * q^31 - 638 * q^34 + 486 * q^41 - 852 * q^44 - 598 * q^46 - 350 * q^49 + 1530 * q^56 + 1146 * q^59 - 398 * q^61 - 1640 * q^64 - 1728 * q^71 - 1218 * q^74 - 3498 * q^76 - 2596 * q^79 + 480 * q^86 - 1086 * q^89 - 2574 * q^91 - 1238 * q^94

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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