LMFDB - Embedded newform 9025.2.a.ct.1.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 95)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.8
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.22159 q^{2} -0.804421 q^{3} -0.507728 q^{4} +0.982669 q^{6} -3.79180 q^{7} +3.06341 q^{8} -2.35291 q^{9} +O(q^{10})$	\(q-1.22159 q^{2} -0.804421 q^{3} -0.507728 q^{4} +0.982669 q^{6} -3.79180 q^{7} +3.06341 q^{8} -2.35291 q^{9} -1.23733 q^{11} +0.408427 q^{12} +3.45598 q^{13} +4.63201 q^{14} -2.72676 q^{16} -3.00695 q^{17} +2.87428 q^{18} +3.05020 q^{21} +1.51150 q^{22} -6.14780 q^{23} -2.46427 q^{24} -4.22178 q^{26} +4.30599 q^{27} +1.92520 q^{28} +4.28727 q^{29} -5.10134 q^{31} -2.79584 q^{32} +0.995330 q^{33} +3.67325 q^{34} +1.19464 q^{36} +9.13084 q^{37} -2.78006 q^{39} -5.33021 q^{41} -3.72608 q^{42} +9.12495 q^{43} +0.628225 q^{44} +7.51007 q^{46} -7.30333 q^{47} +2.19346 q^{48} +7.37774 q^{49} +2.41886 q^{51} -1.75470 q^{52} -3.33235 q^{53} -5.26014 q^{54} -11.6158 q^{56} -5.23727 q^{58} +0.817318 q^{59} +6.40201 q^{61} +6.23172 q^{62} +8.92175 q^{63} +8.86888 q^{64} -1.21588 q^{66} -1.01092 q^{67} +1.52671 q^{68} +4.94542 q^{69} -13.6795 q^{71} -7.20791 q^{72} +11.0627 q^{73} -11.1541 q^{74} +4.69169 q^{77} +3.39609 q^{78} +1.38420 q^{79} +3.59490 q^{81} +6.51132 q^{82} +2.52052 q^{83} -1.54867 q^{84} -11.1469 q^{86} -3.44877 q^{87} -3.79043 q^{88} -2.69842 q^{89} -13.1044 q^{91} +3.12141 q^{92} +4.10362 q^{93} +8.92164 q^{94} +2.24903 q^{96} +3.06253 q^{97} -9.01254 q^{98} +2.91131 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	$ 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) $ 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.22159	−0.863792	−0.431896	−	0.901923i	$-0.642155\pi$
$2$	−1.22159	−0.863792	−0.431896	+	0.901923i	$0.642155\pi$
$3$	−0.804421	−0.464432	−0.232216	−	0.972664i	$-0.574598\pi$
$3$	−0.804421	−0.464432	−0.232216	+	0.972664i	$0.574598\pi$
$4$	−0.507728	−0.253864
$5$	0	0
$5$	0	0
$6$	0.982669	0.401173
$7$	−3.79180	−1.43317	−0.716583	−	0.697502i	$-0.754295\pi$
$7$	−3.79180	−1.43317	−0.716583	+	0.697502i	$0.754295\pi$
$8$	3.06341	1.08308
$9$	−2.35291	−0.784303
$10$	0	0
$11$	−1.23733	−0.373068	−0.186534	−	0.982449i	$-0.559725\pi$
$11$	−1.23733	−0.373068	−0.186534	+	0.982449i	$0.559725\pi$
$12$	0.408427	0.117903
$13$	3.45598	0.958517	0.479259	−	0.877674i	$-0.340906\pi$
$13$	3.45598	0.958517	0.479259	+	0.877674i	$0.340906\pi$
$14$	4.63201	1.23796
$15$	0	0
$16$	−2.72676	−0.681689
$17$	−3.00695	−0.729293	−0.364647	−	0.931146i	$-0.618810\pi$
$17$	−3.00695	−0.729293	−0.364647	+	0.931146i	$0.618810\pi$
$18$	2.87428	0.677474
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.05020	0.665608
$22$	1.51150	0.322253
$23$	−6.14780	−1.28191	−0.640953	−	0.767580i	$-0.721461\pi$
$23$	−6.14780	−1.28191	−0.640953	+	0.767580i	$0.721461\pi$
$24$	−2.46427	−0.503016
$25$	0	0
$26$	−4.22178	−0.827959
$27$	4.30599	0.828688
$28$	1.92520	0.363829
$29$	4.28727	0.796126	0.398063	−	0.917358i	$-0.369683\pi$
$29$	4.28727	0.796126	0.398063	+	0.917358i	$0.369683\pi$
$30$	0	0
$31$	−5.10134	−0.916227	−0.458114	−	0.888894i	$-0.651475\pi$
$31$	−5.10134	−0.916227	−0.458114	+	0.888894i	$0.651475\pi$
$32$	−2.79584	−0.494240
$33$	0.995330	0.173265
$34$	3.67325	0.629958
$35$	0	0
$36$	1.19464	0.199106
$37$	9.13084	1.50110	0.750550	−	0.660813i	$-0.229789\pi$
$37$	9.13084	1.50110	0.750550	+	0.660813i	$0.229789\pi$
$38$	0	0
$39$	−2.78006	−0.445167
$40$	0	0
$41$	−5.33021	−0.832440	−0.416220	−	0.909264i	$-0.636645\pi$
$41$	−5.33021	−0.832440	−0.416220	+	0.909264i	$0.636645\pi$
$42$	−3.72608	−0.574947
$43$	9.12495	1.39154	0.695771	−	0.718264i	$-0.255063\pi$
$43$	9.12495	1.39154	0.695771	+	0.718264i	$0.255063\pi$
$44$	0.628225	0.0947084
$45$	0	0
$46$	7.51007	1.10730
$47$	−7.30333	−1.06530	−0.532650	−	0.846336i	$-0.678804\pi$
$47$	−7.30333	−1.06530	−0.532650	+	0.846336i	$0.678804\pi$
$48$	2.19346	0.316599
$49$	7.37774	1.05396
$50$	0	0
$51$	2.41886	0.338707
$52$	−1.75470	−0.243333
$53$	−3.33235	−0.457734	−0.228867	−	0.973458i	$-0.573502\pi$
$53$	−3.33235	−0.457734	−0.228867	+	0.973458i	$0.573502\pi$
$54$	−5.26014	−0.715814
$55$	0	0
$56$	−11.6158	−1.55223
$57$	0	0
$58$	−5.23727	−0.687687
$59$	0.817318	0.106406	0.0532029	−	0.998584i	$-0.483057\pi$
$59$	0.817318	0.106406	0.0532029	+	0.998584i	$0.483057\pi$
$60$	0	0
$61$	6.40201	0.819693	0.409847	−	0.912154i	$-0.365582\pi$
$61$	6.40201	0.819693	0.409847	+	0.912154i	$0.365582\pi$
$62$	6.23172	0.791430
$63$	8.92175	1.12404
$64$	8.86888	1.10861
$65$	0	0
$66$	−1.21588	−0.149665
$67$	−1.01092	−0.123504	−0.0617520	−	0.998092i	$-0.519669\pi$
$67$	−1.01092	−0.123504	−0.0617520	+	0.998092i	$0.519669\pi$
$68$	1.52671	0.185141
$69$	4.94542	0.595359
$70$	0	0
$71$	−13.6795	−1.62346	−0.811731	−	0.584032i	$-0.801474\pi$
$71$	−13.6795	−1.62346	−0.811731	+	0.584032i	$0.801474\pi$
$72$	−7.20791	−0.849460
$73$	11.0627	1.29479	0.647395	−	0.762155i	$-0.275858\pi$
$73$	11.0627	1.29479	0.647395	+	0.762155i	$0.275858\pi$
$74$	−11.1541	−1.29664
$75$	0	0
$76$	0	0
$77$	4.69169	0.534668
$78$	3.39609	0.384531
$79$	1.38420	0.155735	0.0778673	−	0.996964i	$-0.475189\pi$
$79$	1.38420	0.155735	0.0778673	+	0.996964i	$0.475189\pi$
$80$	0	0
$81$	3.59490	0.399433
$82$	6.51132	0.719054
$83$	2.52052	0.276663	0.138332	−	0.990386i	$-0.455826\pi$
$83$	2.52052	0.276663	0.138332	+	0.990386i	$0.455826\pi$
$84$	−1.54867	−0.168974
$85$	0	0
$86$	−11.1469	−1.20200
$87$	−3.44877	−0.369747
$88$	−3.79043	−0.404061
$89$	−2.69842	−0.286032	−0.143016	−	0.989720i	$-0.545680\pi$
$89$	−2.69842	−0.286032	−0.143016	+	0.989720i	$0.545680\pi$
$90$	0	0
$91$	−13.1044	−1.37371
$92$	3.12141	0.325430
$93$	4.10362	0.425526
$94$	8.92164	0.920197
$95$	0	0
$96$	2.24903	0.229541
$97$	3.06253	0.310953	0.155476	−	0.987840i	$-0.450309\pi$
$97$	3.06253	0.310953	0.155476	+	0.987840i	$0.450309\pi$
$98$	−9.01254	−0.910404
$99$	2.91131	0.292598
$100$	0	0
$101$	3.72705	0.370856	0.185428	−	0.982658i	$-0.440633\pi$
$101$	3.72705	0.370856	0.185428	+	0.982658i	$0.440633\pi$
$102$	−2.95484	−0.292573
$103$	13.2086	1.30149	0.650743	−	0.759298i	$-0.274457\pi$
$103$	13.2086	1.30149	0.650743	+	0.759298i	$0.274457\pi$
$104$	10.5871	1.03815
$105$	0	0
$106$	4.07076	0.395387
$107$	5.13305	0.496230	0.248115	−	0.968731i	$-0.420189\pi$
$107$	5.13305	0.496230	0.248115	+	0.968731i	$0.420189\pi$
$108$	−2.18627	−0.210374
$109$	1.58777	0.152081	0.0760404	−	0.997105i	$-0.475772\pi$
$109$	1.58777	0.152081	0.0760404	+	0.997105i	$0.475772\pi$
$110$	0	0
$111$	−7.34503	−0.697160
$112$	10.3393	0.976973
$113$	20.0832	1.88927	0.944634	−	0.328126i	$-0.106417\pi$
$113$	20.0832	1.88927	0.944634	+	0.328126i	$0.106417\pi$
$114$	0	0
$115$	0	0
$116$	−2.17676	−0.202108
$117$	−8.13161	−0.751768
$118$	−0.998424	−0.0919124
$119$	11.4018	1.04520
$120$	0	0
$121$	−9.46903	−0.860821
$122$	−7.82061	−0.708044
$123$	4.28773	0.386612
$124$	2.59009	0.232597
$125$	0	0
$126$	−10.8987	−0.970932
$127$	3.45304	0.306408	0.153204	−	0.988195i	$-0.451041\pi$
$127$	3.45304	0.306408	0.153204	+	0.988195i	$0.451041\pi$
$128$	−5.24241	−0.463368
$129$	−7.34030	−0.646277
$130$	0	0
$131$	−7.58746	−0.662920	−0.331460	−	0.943469i	$-0.607541\pi$
$131$	−7.58746	−0.662920	−0.331460	+	0.943469i	$0.607541\pi$
$132$	−0.505357	−0.0439857
$133$	0	0
$134$	1.23493	0.106682
$135$	0	0
$136$	−9.21152	−0.789881
$137$	3.68633	0.314944	0.157472	−	0.987523i	$-0.449666\pi$
$137$	3.68633	0.314944	0.157472	+	0.987523i	$0.449666\pi$
$138$	−6.04126	−0.514266
$139$	−18.6273	−1.57994	−0.789972	−	0.613143i	$-0.789905\pi$
$139$	−18.6273	−1.57994	−0.789972	+	0.613143i	$0.789905\pi$
$140$	0	0
$141$	5.87494	0.494760
$142$	16.7107	1.40233
$143$	−4.27618	−0.357592
$144$	6.41581	0.534651
$145$	0	0
$146$	−13.5140	−1.11843
$147$	−5.93480	−0.489494
$148$	−4.63598	−0.381075
$149$	1.68785	0.138274	0.0691369	−	0.997607i	$-0.477975\pi$
$149$	1.68785	0.138274	0.0691369	+	0.997607i	$0.477975\pi$
$150$	0	0
$151$	−1.33890	−0.108958	−0.0544791	−	0.998515i	$-0.517350\pi$
$151$	−1.33890	−0.108958	−0.0544791	+	0.998515i	$0.517350\pi$
$152$	0	0
$153$	7.07508	0.571987
$154$	−5.73130	−0.461841
$155$	0	0
$156$	1.41152	0.113012
$157$	−6.04267	−0.482258	−0.241129	−	0.970493i	$-0.577518\pi$
$157$	−6.04267	−0.482258	−0.241129	+	0.970493i	$0.577518\pi$
$158$	−1.69092	−0.134522
$159$	2.68061	0.212587
$160$	0	0
$161$	23.3112	1.83718
$162$	−4.39147	−0.345027
$163$	9.79728	0.767382	0.383691	−	0.923462i	$-0.374653\pi$
$163$	9.79728	0.767382	0.383691	+	0.923462i	$0.374653\pi$
$164$	2.70630	0.211326
$165$	0	0
$166$	−3.07903	−0.238979
$167$	3.87632	0.299959	0.149979	−	0.988689i	$-0.452079\pi$
$167$	3.87632	0.299959	0.149979	+	0.988689i	$0.452079\pi$
$168$	9.34400	0.720905
$169$	−1.05618	−0.0812444
$170$	0	0
$171$	0	0
$172$	−4.63299	−0.353262
$173$	−4.73220	−0.359782	−0.179891	−	0.983687i	$-0.557575\pi$
$173$	−4.73220	−0.359782	−0.179891	+	0.983687i	$0.557575\pi$
$174$	4.21296	0.319384
$175$	0	0
$176$	3.37389	0.254316
$177$	−0.657468	−0.0494183
$178$	3.29635	0.247072
$179$	9.37815	0.700956	0.350478	−	0.936571i	$-0.386019\pi$
$179$	9.37815	0.700956	0.350478	+	0.936571i	$0.386019\pi$
$180$	0	0
$181$	19.5008	1.44948	0.724741	−	0.689022i	$-0.241960\pi$
$181$	19.5008	1.44948	0.724741	+	0.689022i	$0.241960\pi$
$182$	16.0081	1.18660
$183$	−5.14991	−0.380692
$184$	−18.8332	−1.38840
$185$	0	0
$186$	−5.01293	−0.367566
$187$	3.72058	0.272076
$188$	3.70810	0.270441
$189$	−16.3274	−1.18765
$190$	0	0
$191$	15.5024	1.12171	0.560857	−	0.827913i	$-0.310472\pi$
$191$	15.5024	1.12171	0.560857	+	0.827913i	$0.310472\pi$
$192$	−7.13431	−0.514874
$193$	19.8432	1.42834	0.714172	−	0.699970i	$-0.246804\pi$
$193$	19.8432	1.42834	0.714172	+	0.699970i	$0.246804\pi$
$194$	−3.74114	−0.268598
$195$	0	0
$196$	−3.74588	−0.267563
$197$	25.9708	1.85034	0.925170	−	0.379553i	$-0.123922\pi$
$197$	25.9708	1.85034	0.925170	+	0.379553i	$0.123922\pi$
$198$	−3.55642	−0.252744
$199$	−5.41593	−0.383925	−0.191963	−	0.981402i	$-0.561485\pi$
$199$	−5.41593	−0.383925	−0.191963	+	0.981402i	$0.561485\pi$
$200$	0	0
$201$	0.813208	0.0573593
$202$	−4.55291	−0.320342
$203$	−16.2565	−1.14098
$204$	−1.22812	−0.0859856
$205$	0	0
$206$	−16.1355	−1.12421
$207$	14.4652	1.00540
$208$	−9.42363	−0.653411
$209$	0	0
$210$	0	0
$211$	20.1197	1.38510	0.692550	−	0.721370i	$-0.256487\pi$
$211$	20.1197	1.38510	0.692550	+	0.721370i	$0.256487\pi$
$212$	1.69193	0.116202
$213$	11.0041	0.753988
$214$	−6.27046	−0.428640
$215$	0	0
$216$	13.1910	0.897533
$217$	19.3432	1.31311
$218$	−1.93960	−0.131366
$219$	−8.89905	−0.601342
$220$	0	0
$221$	−10.3920	−0.699040
$222$	8.97259	0.602201
$223$	−10.8496	−0.726546	−0.363273	−	0.931683i	$-0.618341\pi$
$223$	−10.8496	−0.726546	−0.363273	+	0.931683i	$0.618341\pi$
$224$	10.6013	0.708327
$225$	0	0
$226$	−24.5333	−1.63193
$227$	8.78226	0.582899	0.291449	−	0.956586i	$-0.405862\pi$
$227$	8.78226	0.582899	0.291449	+	0.956586i	$0.405862\pi$
$228$	0	0
$229$	18.0824	1.19492	0.597459	−	0.801899i	$-0.296177\pi$
$229$	18.0824	1.19492	0.597459	+	0.801899i	$0.296177\pi$
$230$	0	0
$231$	−3.77409	−0.248317
$232$	13.1336	0.862265
$233$	−26.3658	−1.72728	−0.863640	−	0.504109i	$-0.831821\pi$
$233$	−26.3658	−1.72728	−0.863640	+	0.504109i	$0.831821\pi$
$234$	9.93346	0.649371
$235$	0	0
$236$	−0.414975	−0.0270126
$237$	−1.11348	−0.0723282
$238$	−13.9282	−0.902833
$239$	−9.16897	−0.593091	−0.296546	−	0.955019i	$-0.595835\pi$
$239$	−9.16897	−0.593091	−0.296546	+	0.955019i	$0.595835\pi$
$240$	0	0
$241$	−26.5167	−1.70809	−0.854045	−	0.520199i	$-0.825858\pi$
$241$	−26.5167	−1.70809	−0.854045	+	0.520199i	$0.825858\pi$
$242$	11.5672	0.743570
$243$	−15.8098	−1.01420
$244$	−3.25048	−0.208091
$245$	0	0
$246$	−5.23784	−0.333952
$247$	0	0
$248$	−15.6275	−0.992345
$249$	−2.02756	−0.128491
$250$	0	0
$251$	−14.9471	−0.943451	−0.471726	−	0.881745i	$-0.656369\pi$
$251$	−14.9471	−0.943451	−0.471726	+	0.881745i	$0.656369\pi$
$252$	−4.52982	−0.285352
$253$	7.60684	0.478238
$254$	−4.21819	−0.264673
$255$	0	0
$256$	−11.3337	−0.708356
$257$	−10.7988	−0.673610	−0.336805	−	0.941574i	$-0.609346\pi$
$257$	−10.7988	−0.673610	−0.336805	+	0.941574i	$0.609346\pi$
$258$	8.96680	0.558249
$259$	−34.6223	−2.15132
$260$	0	0
$261$	−10.0875	−0.624403
$262$	9.26874	0.572624
$263$	0.549168	0.0338632	0.0169316	−	0.999857i	$-0.494610\pi$
$263$	0.549168	0.0338632	0.0169316	+	0.999857i	$0.494610\pi$
$264$	3.04910	0.187659
$265$	0	0
$266$	0	0
$267$	2.17066	0.132842
$268$	0.513275	0.0313532
$269$	−21.9270	−1.33691	−0.668455	−	0.743752i	$-0.733044\pi$
$269$	−21.9270	−1.33691	−0.668455	+	0.743752i	$0.733044\pi$
$270$	0	0
$271$	−5.28538	−0.321064	−0.160532	−	0.987031i	$-0.551321\pi$
$271$	−5.28538	−0.321064	−0.160532	+	0.987031i	$0.551321\pi$
$272$	8.19923	0.497151
$273$	10.5414	0.637997
$274$	−4.50316	−0.272046
$275$	0	0
$276$	−2.51093	−0.151140
$277$	17.4391	1.04781	0.523906	−	0.851776i	$-0.324474\pi$
$277$	17.4391	1.04781	0.523906	+	0.851776i	$0.324474\pi$
$278$	22.7548	1.36474
$279$	12.0030	0.718599
$280$	0	0
$281$	23.1841	1.38305	0.691524	−	0.722354i	$-0.256940\pi$
$281$	23.1841	1.38305	0.691524	+	0.722354i	$0.256940\pi$
$282$	−7.17675	−0.427369
$283$	4.59388	0.273078	0.136539	−	0.990635i	$-0.456402\pi$
$283$	4.59388	0.273078	0.136539	+	0.990635i	$0.456402\pi$
$284$	6.94548	0.412138
$285$	0	0
$286$	5.22372	0.308885
$287$	20.2111	1.19302
$288$	6.57836	0.387633
$289$	−7.95823	−0.468131
$290$	0	0
$291$	−2.46356	−0.144416
$292$	−5.61684	−0.328700
$293$	18.1326	1.05932	0.529659	−	0.848211i	$-0.322320\pi$
$293$	18.1326	1.05932	0.529659	+	0.848211i	$0.322320\pi$
$294$	7.24987	0.422821
$295$	0	0
$296$	27.9715	1.62581
$297$	−5.32791	−0.309157
$298$	−2.06185	−0.119440
$299$	−21.2467	−1.22873
$300$	0	0
$301$	−34.6000	−1.99431
$302$	1.63558	0.0941171
$303$	−2.99812	−0.172237
$304$	0	0
$305$	0	0
$306$	−8.64282	−0.494077
$307$	−4.41407	−0.251924	−0.125962	−	0.992035i	$-0.540202\pi$
$307$	−4.41407	−0.251924	−0.125962	+	0.992035i	$0.540202\pi$
$308$	−2.38210	−0.135733
$309$	−10.6253	−0.604452
$310$	0	0
$311$	15.0338	0.852487	0.426243	−	0.904609i	$-0.359837\pi$
$311$	15.0338	0.852487	0.426243	+	0.904609i	$0.359837\pi$
$312$	−8.51646	−0.482150
$313$	7.80549	0.441193	0.220596	−	0.975365i	$-0.429200\pi$
$313$	7.80549	0.441193	0.220596	+	0.975365i	$0.429200\pi$
$314$	7.38164	0.416570
$315$	0	0
$316$	−0.702796	−0.0395354
$317$	−27.2310	−1.52944	−0.764722	−	0.644361i	$-0.777123\pi$
$317$	−27.2310	−1.52944	−0.764722	+	0.644361i	$0.777123\pi$
$318$	−3.27460	−0.183631
$319$	−5.30474	−0.297009
$320$	0	0
$321$	−4.12913	−0.230466
$322$	−28.4767	−1.58694
$323$	0	0
$324$	−1.82523	−0.101402
$325$	0	0
$326$	−11.9682	−0.662858
$327$	−1.27724	−0.0706313
$328$	−16.3286	−0.901596
$329$	27.6927	1.52675
$330$	0	0
$331$	11.9800	0.658480	0.329240	−	0.944246i	$-0.393208\pi$
$331$	11.9800	0.658480	0.329240	+	0.944246i	$0.393208\pi$
$332$	−1.27974	−0.0702348
$333$	−21.4840	−1.17732
$334$	−4.73526	−0.259102
$335$	0	0
$336$	−8.31716	−0.453738
$337$	−4.14343	−0.225707	−0.112853	−	0.993612i	$-0.535999\pi$
$337$	−4.14343	−0.225707	−0.112853	+	0.993612i	$0.535999\pi$
$338$	1.29021	0.0701782
$339$	−16.1553	−0.877437
$340$	0	0
$341$	6.31201	0.341815
$342$	0	0
$343$	−1.43230	−0.0773370
$344$	27.9534	1.50715
$345$	0	0
$346$	5.78078	0.310777
$347$	−9.16935	−0.492237	−0.246118	−	0.969240i	$-0.579155\pi$
$347$	−9.16935	−0.492237	−0.246118	+	0.969240i	$0.579155\pi$
$348$	1.75103	0.0938653
$349$	−32.6883	−1.74976	−0.874881	−	0.484337i	$-0.839061\pi$
$349$	−32.6883	−1.74976	−0.874881	+	0.484337i	$0.839061\pi$
$350$	0	0
$351$	14.8814	0.794312
$352$	3.45937	0.184385
$353$	26.4701	1.40886	0.704431	−	0.709773i	$-0.251202\pi$
$353$	26.4701	1.40886	0.704431	+	0.709773i	$0.251202\pi$
$354$	0.803153	0.0426871
$355$	0	0
$356$	1.37006	0.0726131
$357$	−9.17181	−0.485424
$358$	−11.4562	−0.605480
$359$	12.4121	0.655083	0.327542	−	0.944837i	$-0.393780\pi$
$359$	12.4121	0.655083	0.327542	+	0.944837i	$0.393780\pi$
$360$	0	0
$361$	0	0
$362$	−23.8219	−1.25205
$363$	7.61708	0.399793
$364$	6.65347	0.348736
$365$	0	0
$366$	6.29106	0.328839
$367$	−5.32121	−0.277765	−0.138882	−	0.990309i	$-0.544351\pi$
$367$	−5.32121	−0.277765	−0.138882	+	0.990309i	$0.544351\pi$
$368$	16.7636	0.873861
$369$	12.5415	0.652885
$370$	0	0
$371$	12.6356	0.656009
$372$	−2.08352	−0.108026
$373$	−19.3050	−0.999575	−0.499788	−	0.866148i	$-0.666589\pi$
$373$	−19.3050	−0.999575	−0.499788	+	0.866148i	$0.666589\pi$
$374$	−4.54501	−0.235017
$375$	0	0
$376$	−22.3730	−1.15380
$377$	14.8167	0.763100
$378$	19.9454	1.02588
$379$	−20.5109	−1.05357	−0.526787	−	0.849997i	$-0.676603\pi$
$379$	−20.5109	−1.05357	−0.526787	+	0.849997i	$0.676603\pi$
$380$	0	0
$381$	−2.77770	−0.142306
$382$	−18.9375	−0.968927
$383$	−22.7253	−1.16121	−0.580605	−	0.814186i	$-0.697184\pi$
$383$	−22.7253	−1.16121	−0.580605	+	0.814186i	$0.697184\pi$
$384$	4.21710	0.215203
$385$	0	0
$386$	−24.2402	−1.23379
$387$	−21.4702	−1.09139
$388$	−1.55493	−0.0789396
$389$	5.04493	0.255788	0.127894	−	0.991788i	$-0.459178\pi$
$389$	5.04493	0.255788	0.127894	+	0.991788i	$0.459178\pi$
$390$	0	0
$391$	18.4862	0.934885
$392$	22.6010	1.14152
$393$	6.10351	0.307881
$394$	−31.7255	−1.59831
$395$	0	0
$396$	−1.47815	−0.0742800
$397$	−22.9429	−1.15147	−0.575735	−	0.817636i	$-0.695284\pi$
$397$	−22.9429	−1.15147	−0.575735	+	0.817636i	$0.695284\pi$
$398$	6.61603	0.331632
$399$	0	0
$400$	0	0
$401$	22.7649	1.13683	0.568413	−	0.822744i	$-0.307558\pi$
$401$	22.7649	1.13683	0.568413	+	0.822744i	$0.307558\pi$
$402$	−0.993404	−0.0495465
$403$	−17.6301	−0.878220
$404$	−1.89233	−0.0941468
$405$	0	0
$406$	19.8587	0.985568
$407$	−11.2978	−0.560012
$408$	7.40993	0.366846
$409$	−23.8028	−1.17697	−0.588486	−	0.808507i	$-0.700276\pi$
$409$	−23.8028	−1.17697	−0.588486	+	0.808507i	$0.700276\pi$
$410$	0	0
$411$	−2.96536	−0.146270
$412$	−6.70640	−0.330400
$413$	−3.09911	−0.152497
$414$	−17.6705	−0.868458
$415$	0	0
$416$	−9.66239	−0.473737
$417$	14.9842	0.733777
$418$	0	0
$419$	−15.9374	−0.778593	−0.389296	−	0.921113i	$-0.627282\pi$
$419$	−15.9374	−0.778593	−0.389296	+	0.921113i	$0.627282\pi$
$420$	0	0
$421$	2.65354	0.129326	0.0646629	−	0.997907i	$-0.479403\pi$
$421$	2.65354	0.129326	0.0646629	+	0.997907i	$0.479403\pi$
$422$	−24.5780	−1.19644
$423$	17.1840	0.835517
$424$	−10.2084	−0.495761
$425$	0	0
$426$	−13.4424	−0.651289
$427$	−24.2751	−1.17476
$428$	−2.60619	−0.125975
$429$	3.43984	0.166077
$430$	0	0
$431$	−33.3222	−1.60507	−0.802536	−	0.596604i	$-0.796516\pi$
$431$	−33.3222	−1.60507	−0.802536	+	0.596604i	$0.796516\pi$
$432$	−11.7414	−0.564908
$433$	22.3800	1.07552	0.537758	−	0.843099i	$-0.319272\pi$
$433$	22.3800	1.07552	0.537758	+	0.843099i	$0.319272\pi$
$434$	−23.6294	−1.13425
$435$	0	0
$436$	−0.806155	−0.0386078
$437$	0	0
$438$	10.8710	0.519435
$439$	17.4561	0.833132	0.416566	−	0.909105i	$-0.363233\pi$
$439$	17.4561	0.833132	0.416566	+	0.909105i	$0.363233\pi$
$440$	0	0
$441$	−17.3591	−0.826625
$442$	12.6947	0.603825
$443$	−1.49859	−0.0712001	−0.0356001	−	0.999366i	$-0.511334\pi$
$443$	−1.49859	−0.0712001	−0.0356001	+	0.999366i	$0.511334\pi$
$444$	3.72928	0.176984
$445$	0	0
$446$	13.2538	0.627584
$447$	−1.35774	−0.0642189
$448$	−33.6290	−1.58882
$449$	−11.9911	−0.565896	−0.282948	−	0.959135i	$-0.591312\pi$
$449$	−11.9911	−0.565896	−0.282948	+	0.959135i	$0.591312\pi$
$450$	0	0
$451$	6.59521	0.310556
$452$	−10.1968	−0.479617
$453$	1.07704	0.0506037
$454$	−10.7283	−0.503503
$455$	0	0
$456$	0	0
$457$	12.9472	0.605646	0.302823	−	0.953047i	$-0.402071\pi$
$457$	12.9472	0.605646	0.302823	+	0.953047i	$0.402071\pi$
$458$	−22.0892	−1.03216
$459$	−12.9479	−0.604357
$460$	0	0
$461$	13.8680	0.645899	0.322949	−	0.946416i	$-0.395326\pi$
$461$	13.8680	0.645899	0.322949	+	0.946416i	$0.395326\pi$
$462$	4.61038	0.214494
$463$	4.18473	0.194481	0.0972405	−	0.995261i	$-0.468998\pi$
$463$	4.18473	0.194481	0.0972405	+	0.995261i	$0.468998\pi$
$464$	−11.6903	−0.542710
$465$	0	0
$466$	32.2081	1.49201
$467$	−23.3019	−1.07828	−0.539141	−	0.842216i	$-0.681251\pi$
$467$	−23.3019	−1.07828	−0.539141	+	0.842216i	$0.681251\pi$
$468$	4.12864	0.190847
$469$	3.83322	0.177002
$470$	0	0
$471$	4.86085	0.223976
$472$	2.50378	0.115246
$473$	−11.2905	−0.519139
$474$	1.36021	0.0624765
$475$	0	0
$476$	−5.78899	−0.265338
$477$	7.84072	0.359002
$478$	11.2007	0.512307
$479$	35.1170	1.60454	0.802268	−	0.596963i	$-0.203626\pi$
$479$	35.1170	1.60454	0.802268	+	0.596963i	$0.203626\pi$
$480$	0	0
$481$	31.5560	1.43883
$482$	32.3924	1.47543
$483$	−18.7520	−0.853247
$484$	4.80769	0.218531
$485$	0	0
$486$	19.3130	0.876055
$487$	−33.4209	−1.51445	−0.757223	−	0.653156i	$-0.773444\pi$
$487$	−33.4209	−1.51445	−0.757223	+	0.653156i	$0.773444\pi$
$488$	19.6120	0.887791
$489$	−7.88113	−0.356397
$490$	0	0
$491$	−17.5387	−0.791511	−0.395755	−	0.918356i	$-0.629517\pi$
$491$	−17.5387	−0.791511	−0.395755	+	0.918356i	$0.629517\pi$
$492$	−2.17700	−0.0981468
$493$	−12.8916	−0.580609
$494$	0	0
$495$	0	0
$496$	13.9101	0.624582
$497$	51.8700	2.32669
$498$	2.47684	0.110990
$499$	32.6830	1.46309	0.731545	−	0.681793i	$-0.238799\pi$
$499$	32.6830	1.46309	0.731545	+	0.681793i	$0.238799\pi$
$500$	0	0
$501$	−3.11819	−0.139310
$502$	18.2591	0.814946
$503$	−17.5243	−0.781370	−0.390685	−	0.920524i	$-0.627762\pi$
$503$	−17.5243	−0.781370	−0.390685	+	0.920524i	$0.627762\pi$
$504$	27.3309	1.21742
$505$	0	0
$506$	−9.29240	−0.413098
$507$	0.849611	0.0377325
$508$	−1.75321	−0.0777860
$509$	−29.9995	−1.32970	−0.664852	−	0.746975i	$-0.731505\pi$
$509$	−29.9995	−1.32970	−0.664852	+	0.746975i	$0.731505\pi$
$510$	0	0
$511$	−41.9475	−1.85565
$512$	24.3299	1.07524
$513$	0	0
$514$	13.1916	0.581858
$515$	0	0
$516$	3.72687	0.164066
$517$	9.03659	0.397429
$518$	42.2941	1.85830
$519$	3.80668	0.167094
$520$	0	0
$521$	10.4875	0.459464	0.229732	−	0.973254i	$-0.426215\pi$
$521$	10.4875	0.459464	0.229732	+	0.973254i	$0.426215\pi$
$522$	12.3228	0.539354
$523$	32.7586	1.43244	0.716218	−	0.697877i	$-0.245872\pi$
$523$	32.7586	1.43244	0.716218	+	0.697877i	$0.245872\pi$
$524$	3.85237	0.168291
$525$	0	0
$526$	−0.670856	−0.0292507
$527$	15.3395	0.668198
$528$	−2.71402	−0.118113
$529$	14.7955	0.643283
$530$	0	0
$531$	−1.92307	−0.0834543
$532$	0	0
$533$	−18.4211	−0.797908
$534$	−2.65165	−0.114748
$535$	0	0
$536$	−3.09687	−0.133764
$537$	−7.54398	−0.325547
$538$	26.7857	1.15481
$539$	−9.12866	−0.393199
$540$	0	0
$541$	14.7958	0.636122	0.318061	−	0.948070i	$-0.396968\pi$
$541$	14.7958	0.636122	0.318061	+	0.948070i	$0.396968\pi$
$542$	6.45655	0.277333
$543$	−15.6868	−0.673186
$544$	8.40697	0.360446
$545$	0	0
$546$	−12.8773	−0.551097
$547$	−1.61001	−0.0688390	−0.0344195	−	0.999407i	$-0.510958\pi$
$547$	−1.61001	−0.0688390	−0.0344195	+	0.999407i	$0.510958\pi$
$548$	−1.87165	−0.0799529
$549$	−15.0633	−0.642888
$550$	0	0
$551$	0	0
$552$	15.1498	0.644819
$553$	−5.24860	−0.223193
$554$	−21.3033	−0.905092
$555$	0	0
$556$	9.45758	0.401091
$557$	−22.3165	−0.945581	−0.472790	−	0.881175i	$-0.656753\pi$
$557$	−22.3165	−0.945581	−0.472790	+	0.881175i	$0.656753\pi$
$558$	−14.6627	−0.620720
$559$	31.5357	1.33382
$560$	0	0
$561$	−2.99291	−0.126361
$562$	−28.3214	−1.19467
$563$	2.42120	0.102042	0.0510208	−	0.998698i	$-0.483753\pi$
$563$	2.42120	0.102042	0.0510208	+	0.998698i	$0.483753\pi$
$564$	−2.98287	−0.125602
$565$	0	0
$566$	−5.61182	−0.235882
$567$	−13.6311	−0.572453
$568$	−41.9059	−1.75833
$569$	−25.3556	−1.06296	−0.531481	−	0.847070i	$-0.678364\pi$
$569$	−25.3556	−1.06296	−0.531481	+	0.847070i	$0.678364\pi$
$570$	0	0
$571$	3.79252	0.158712	0.0793561	−	0.996846i	$-0.474714\pi$
$571$	3.79252	0.158712	0.0793561	+	0.996846i	$0.474714\pi$
$572$	2.17113	0.0907797
$573$	−12.4704	−0.520960
$574$	−24.6896	−1.03052
$575$	0	0
$576$	−20.8676	−0.869485
$577$	−36.8138	−1.53258	−0.766289	−	0.642496i	$-0.777899\pi$
$577$	−36.8138	−1.53258	−0.766289	+	0.642496i	$0.777899\pi$
$578$	9.72166	0.404368
$579$	−15.9623	−0.663369
$580$	0	0
$581$	−9.55731	−0.396504
$582$	3.00945	0.124746
$583$	4.12321	0.170766
$584$	33.8895	1.40236
$585$	0	0
$586$	−22.1505	−0.915030
$587$	−27.5764	−1.13820	−0.569100	−	0.822268i	$-0.692708\pi$
$587$	−27.5764	−1.13820	−0.569100	+	0.822268i	$0.692708\pi$
$588$	3.01326	0.124265
$589$	0	0
$590$	0	0
$591$	−20.8914	−0.859358
$592$	−24.8976	−1.02328
$593$	7.43391	0.305274	0.152637	−	0.988282i	$-0.451223\pi$
$593$	7.43391	0.305274	0.152637	+	0.988282i	$0.451223\pi$
$594$	6.50850	0.267047
$595$	0	0
$596$	−0.856967	−0.0351027
$597$	4.35669	0.178307
$598$	25.9547	1.06137
$599$	−7.48833	−0.305965	−0.152982	−	0.988229i	$-0.548888\pi$
$599$	−7.48833	−0.305965	−0.152982	+	0.988229i	$0.548888\pi$
$600$	0	0
$601$	24.9173	1.01640	0.508199	−	0.861240i	$-0.330312\pi$
$601$	24.9173	1.01640	0.508199	+	0.861240i	$0.330312\pi$
$602$	42.2668	1.72267
$603$	2.37861	0.0968646
$604$	0.679797	0.0276605
$605$	0	0
$606$	3.66246	0.148777
$607$	−13.5201	−0.548764	−0.274382	−	0.961621i	$-0.588473\pi$
$607$	−13.5201	−0.548764	−0.274382	+	0.961621i	$0.588473\pi$
$608$	0	0
$609$	13.0770	0.529908
$610$	0	0
$611$	−25.2402	−1.02111
$612$	−3.59222	−0.145207
$613$	5.65406	0.228365	0.114183	−	0.993460i	$-0.463575\pi$
$613$	5.65406	0.228365	0.114183	+	0.993460i	$0.463575\pi$
$614$	5.39217	0.217610
$615$	0	0
$616$	14.3725	0.579086
$617$	18.4368	0.742239	0.371120	−	0.928585i	$-0.378974\pi$
$617$	18.4368	0.742239	0.371120	+	0.928585i	$0.378974\pi$
$618$	12.9797	0.522121
$619$	−6.99902	−0.281314	−0.140657	−	0.990058i	$-0.544922\pi$
$619$	−6.99902	−0.281314	−0.140657	+	0.990058i	$0.544922\pi$
$620$	0	0
$621$	−26.4724	−1.06230
$622$	−18.3650	−0.736371
$623$	10.2319	0.409931
$624$	7.58056	0.303465
$625$	0	0
$626$	−9.53508	−0.381099
$627$	0	0
$628$	3.06803	0.122428
$629$	−27.4560	−1.09474
$630$	0	0
$631$	28.0613	1.11710	0.558551	−	0.829470i	$-0.311358\pi$
$631$	28.0613	1.11710	0.558551	+	0.829470i	$0.311358\pi$
$632$	4.24036	0.168672
$633$	−16.1847	−0.643285
$634$	33.2650	1.32112
$635$	0	0
$636$	−1.36102	−0.0539681
$637$	25.4973	1.01024
$638$	6.48020	0.256554
$639$	32.1867	1.27328
$640$	0	0
$641$	−26.8330	−1.05984	−0.529921	−	0.848047i	$-0.677778\pi$
$641$	−26.8330	−1.05984	−0.529921	+	0.848047i	$0.677778\pi$
$642$	5.04409	0.199074
$643$	26.8783	1.05998	0.529988	−	0.848005i	$-0.322196\pi$
$643$	26.8783	1.05998	0.529988	+	0.848005i	$0.322196\pi$
$644$	−11.8358	−0.466394
$645$	0	0
$646$	0	0
$647$	8.88424	0.349275	0.174638	−	0.984633i	$-0.444125\pi$
$647$	8.88424	0.349275	0.174638	+	0.984633i	$0.444125\pi$
$648$	11.0126	0.432617
$649$	−1.01129	−0.0396965
$650$	0	0
$651$	−15.5601	−0.609849
$652$	−4.97435	−0.194811
$653$	−39.4102	−1.54224	−0.771121	−	0.636689i	$-0.780303\pi$
$653$	−39.4102	−1.54224	−0.771121	+	0.636689i	$0.780303\pi$
$654$	1.56025	0.0610107
$655$	0	0
$656$	14.5342	0.567465
$657$	−26.0295	−1.01551
$658$	−33.8291	−1.31879
$659$	−17.3119	−0.674377	−0.337189	−	0.941437i	$-0.609476\pi$
$659$	−17.3119	−0.674377	−0.337189	+	0.941437i	$0.609476\pi$
$660$	0	0
$661$	−14.2421	−0.553952	−0.276976	−	0.960877i	$-0.589332\pi$
$661$	−14.2421	−0.553952	−0.276976	+	0.960877i	$0.589332\pi$
$662$	−14.6346	−0.568789
$663$	8.35952	0.324657
$664$	7.72138	0.299647
$665$	0	0
$666$	26.2446	1.01696
$667$	−26.3573	−1.02056
$668$	−1.96811	−0.0761486
$669$	8.72768	0.337431
$670$	0	0
$671$	−7.92137	−0.305801
$672$	−8.52788	−0.328970
$673$	3.86700	0.149062	0.0745310	−	0.997219i	$-0.476254\pi$
$673$	3.86700	0.149062	0.0745310	+	0.997219i	$0.476254\pi$
$674$	5.06155	0.194964
$675$	0	0
$676$	0.536250	0.0206250
$677$	15.7102	0.603790	0.301895	−	0.953341i	$-0.402381\pi$
$677$	15.7102	0.603790	0.301895	+	0.953341i	$0.402381\pi$
$678$	19.7351	0.757923
$679$	−11.6125	−0.445646
$680$	0	0
$681$	−7.06463	−0.270717
$682$	−7.71067	−0.295257
$683$	11.3613	0.434727	0.217364	−	0.976091i	$-0.430254\pi$
$683$	11.3613	0.434727	0.217364	+	0.976091i	$0.430254\pi$
$684$	0	0
$685$	0	0
$686$	1.74968	0.0668031
$687$	−14.5458	−0.554959
$688$	−24.8815	−0.948599
$689$	−11.5166	−0.438746
$690$	0	0
$691$	−2.01088	−0.0764974	−0.0382487	−	0.999268i	$-0.512178\pi$
$691$	−2.01088	−0.0764974	−0.0382487	+	0.999268i	$0.512178\pi$
$692$	2.40267	0.0913357
$693$	−11.0391	−0.419341
$694$	11.2012	0.425190
$695$	0	0
$696$	−10.5650	−0.400464
$697$	16.0277	0.607093
$698$	39.9315	1.51143
$699$	21.2092	0.802205
$700$	0	0
$701$	12.4628	0.470713	0.235357	−	0.971909i	$-0.424374\pi$
$701$	12.4628	0.470713	0.235357	+	0.971909i	$0.424374\pi$
$702$	−18.1789	−0.686120
$703$	0	0
$704$	−10.9737	−0.413586
$705$	0	0
$706$	−32.3355	−1.21696
$707$	−14.1322	−0.531497
$708$	0.333815	0.0125455
$709$	−3.52250	−0.132290	−0.0661452	−	0.997810i	$-0.521070\pi$
$709$	−3.52250	−0.132290	−0.0661452	+	0.997810i	$0.521070\pi$
$710$	0	0
$711$	−3.25689	−0.122143
$712$	−8.26634	−0.309794
$713$	31.3620	1.17452
$714$	11.2042	0.419305
$715$	0	0
$716$	−4.76155	−0.177947
$717$	7.37571	0.275451
$718$	−15.1624	−0.565855
$719$	−23.9065	−0.891563	−0.445781	−	0.895142i	$-0.647074\pi$
$719$	−23.9065	−0.891563	−0.445781	+	0.895142i	$0.647074\pi$
$720$	0	0
$721$	−50.0845	−1.86524
$722$	0	0
$723$	21.3306	0.793293
$724$	−9.90109	−0.367971
$725$	0	0
$726$	−9.30492	−0.345338
$727$	5.47926	0.203214	0.101607	−	0.994825i	$-0.467602\pi$
$727$	5.47926	0.203214	0.101607	+	0.994825i	$0.467602\pi$
$728$	−40.1441	−1.48784
$729$	1.93302	0.0715933
$730$	0	0
$731$	−27.4383	−1.01484
$732$	2.61475	0.0966440
$733$	−18.8893	−0.697691	−0.348846	−	0.937180i	$-0.613426\pi$
$733$	−18.8893	−0.697691	−0.348846	+	0.937180i	$0.613426\pi$
$734$	6.50031	0.239931
$735$	0	0
$736$	17.1883	0.633569
$737$	1.25084	0.0460754
$738$	−15.3205	−0.563956
$739$	34.9585	1.28597	0.642985	−	0.765879i	$-0.277696\pi$
$739$	34.9585	1.28597	0.642985	+	0.765879i	$0.277696\pi$
$740$	0	0
$741$	0	0
$742$	−15.4355	−0.566655
$743$	27.6384	1.01395	0.506976	−	0.861960i	$-0.330763\pi$
$743$	27.6384	1.01395	0.506976	+	0.861960i	$0.330763\pi$
$744$	12.5711	0.460877
$745$	0	0
$746$	23.5827	0.863425
$747$	−5.93055	−0.216988
$748$	−1.88904	−0.0690702
$749$	−19.4635	−0.711180
$750$	0	0
$751$	−53.7087	−1.95986	−0.979929	−	0.199348i	$-0.936118\pi$
$751$	−53.7087	−1.95986	−0.979929	+	0.199348i	$0.936118\pi$
$752$	19.9144	0.726203
$753$	12.0237	0.438169
$754$	−18.0999	−0.659160
$755$	0	0
$756$	8.28990	0.301501
$757$	24.5798	0.893368	0.446684	−	0.894692i	$-0.352605\pi$
$757$	24.5798	0.893368	0.446684	+	0.894692i	$0.352605\pi$
$758$	25.0558	0.910068
$759$	−6.11909	−0.222109
$760$	0	0
$761$	−0.906887	−0.0328746	−0.0164373	−	0.999865i	$-0.505232\pi$
$761$	−0.906887	−0.0328746	−0.0164373	+	0.999865i	$0.505232\pi$
$762$	3.39320	0.122923
$763$	−6.02051	−0.217957
$764$	−7.87099	−0.284763
$765$	0	0
$766$	27.7609	1.00304
$767$	2.82464	0.101992
$768$	9.11706	0.328984
$769$	21.3193	0.768792	0.384396	−	0.923168i	$-0.374410\pi$
$769$	21.3193	0.768792	0.384396	+	0.923168i	$0.374410\pi$
$770$	0	0
$771$	8.68676	0.312846
$772$	−10.0749	−0.362605
$773$	−16.6037	−0.597192	−0.298596	−	0.954380i	$-0.596518\pi$
$773$	−16.6037	−0.597192	−0.298596	+	0.954380i	$0.596518\pi$
$774$	26.2277	0.942733
$775$	0	0
$776$	9.38176	0.336786
$777$	27.8509	0.999145
$778$	−6.16282	−0.220948
$779$	0	0
$780$	0	0
$781$	16.9260	0.605661
$782$	−22.5824	−0.807546
$783$	18.4609	0.659740
$784$	−20.1173	−0.718475
$785$	0	0
$786$	−7.45596	−0.265945
$787$	−41.3474	−1.47388	−0.736938	−	0.675960i	$-0.763729\pi$
$787$	−41.3474	−1.47388	−0.736938	+	0.675960i	$0.763729\pi$
$788$	−13.1861	−0.469735
$789$	−0.441762	−0.0157271
$790$	0	0
$791$	−76.1514	−2.70763
$792$	8.91853	0.316906
$793$	22.1252	0.785690
$794$	28.0267	0.994631
$795$	0	0
$796$	2.74982	0.0974648
$797$	−22.7002	−0.804083	−0.402042	−	0.915621i	$-0.631699\pi$
$797$	−22.7002	−0.804083	−0.402042	+	0.915621i	$0.631699\pi$
$798$	0	0
$799$	21.9608	0.776916
$800$	0	0
$801$	6.34913	0.224335
$802$	−27.8093	−0.981980
$803$	−13.6881	−0.483044
$804$	−0.412889	−0.0145615
$805$	0	0
$806$	21.5367	0.758599
$807$	17.6385	0.620905
$808$	11.4175	0.401665
$809$	−27.1508	−0.954571	−0.477285	−	0.878748i	$-0.658379\pi$
$809$	−27.1508	−0.954571	−0.477285	+	0.878748i	$0.658379\pi$
$810$	0	0
$811$	45.2598	1.58929	0.794644	−	0.607076i	$-0.207658\pi$
$811$	45.2598	1.58929	0.794644	+	0.607076i	$0.207658\pi$
$812$	8.25385	0.289653
$813$	4.25167	0.149113
$814$	13.8013	0.483734
$815$	0	0
$816$	−6.59563	−0.230893
$817$	0	0
$818$	29.0771	1.01666
$819$	30.8334	1.07741
$820$	0	0
$821$	14.7796	0.515813	0.257907	−	0.966170i	$-0.416967\pi$
$821$	14.7796	0.515813	0.257907	+	0.966170i	$0.416967\pi$
$822$	3.62244	0.126347
$823$	−38.1769	−1.33076	−0.665381	−	0.746504i	$-0.731731\pi$
$823$	−38.1769	−1.33076	−0.665381	+	0.746504i	$0.731731\pi$
$824$	40.4634	1.40961
$825$	0	0
$826$	3.78582	0.131726
$827$	45.3277	1.57620	0.788099	−	0.615548i	$-0.211065\pi$
$827$	45.3277	1.57620	0.788099	+	0.615548i	$0.211065\pi$
$828$	−7.34439	−0.255235
$829$	−37.9276	−1.31728	−0.658640	−	0.752458i	$-0.728868\pi$
$829$	−37.9276	−1.31728	−0.658640	+	0.752458i	$0.728868\pi$
$830$	0	0
$831$	−14.0284	−0.486638
$832$	30.6507	1.06262
$833$	−22.1845	−0.768648
$834$	−18.3044	−0.633831
$835$	0	0
$836$	0	0
$837$	−21.9663	−0.759267
$838$	19.4689	0.672542
$839$	7.14266	0.246592	0.123296	−	0.992370i	$-0.460654\pi$
$839$	7.14266	0.246592	0.123296	+	0.992370i	$0.460654\pi$
$840$	0	0
$841$	−10.6193	−0.366184
$842$	−3.24153	−0.111710
$843$	−18.6498	−0.642332
$844$	−10.2153	−0.351627
$845$	0	0
$846$	−20.9918	−0.721713
$847$	35.9046	1.23370
$848$	9.08652	0.312032
$849$	−3.69541	−0.126826
$850$	0	0
$851$	−56.1346	−1.92427
$852$	−5.58709	−0.191410
$853$	−51.0391	−1.74755	−0.873773	−	0.486335i	$-0.838334\pi$
$853$	−51.0391	−1.74755	−0.873773	+	0.486335i	$0.838334\pi$
$854$	29.6542	1.01474
$855$	0	0
$856$	15.7246	0.537456
$857$	−13.6042	−0.464710	−0.232355	−	0.972631i	$-0.574643\pi$
$857$	−13.6042	−0.464710	−0.232355	+	0.972631i	$0.574643\pi$
$858$	−4.20207	−0.143456
$859$	−38.4375	−1.31147	−0.655735	−	0.754991i	$-0.727641\pi$
$859$	−38.4375	−1.31147	−0.655735	+	0.754991i	$0.727641\pi$
$860$	0	0
$861$	−16.2582	−0.554079
$862$	40.7059	1.38645
$863$	−54.3455	−1.84994	−0.924970	−	0.380039i	$-0.875910\pi$
$863$	−54.3455	−1.84994	−0.924970	+	0.380039i	$0.875910\pi$
$864$	−12.0389	−0.409571
$865$	0	0
$866$	−27.3391	−0.929021
$867$	6.40176	0.217415
$868$	−9.82110	−0.333350
$869$	−1.71270	−0.0580995
$870$	0	0
$871$	−3.49374	−0.118381
$872$	4.86398	0.164715
$873$	−7.20584	−0.243881
$874$	0	0
$875$	0	0
$876$	4.51830	0.152659
$877$	13.5325	0.456961	0.228481	−	0.973548i	$-0.426624\pi$
$877$	13.5325	0.456961	0.228481	+	0.973548i	$0.426624\pi$
$878$	−21.3241	−0.719653
$879$	−14.5862	−0.491982
$880$	0	0
$881$	−27.9577	−0.941920	−0.470960	−	0.882155i	$-0.656092\pi$
$881$	−27.9577	−0.941920	−0.470960	+	0.882155i	$0.656092\pi$
$882$	21.2057	0.714032
$883$	−54.5487	−1.83571	−0.917855	−	0.396915i	$-0.870081\pi$
$883$	−54.5487	−1.83571	−0.917855	+	0.396915i	$0.870081\pi$
$884$	5.27630	0.177461
$885$	0	0
$886$	1.83066	0.0615021
$887$	−35.9778	−1.20802	−0.604008	−	0.796979i	$-0.706430\pi$
$887$	−35.9778	−1.20802	−0.604008	+	0.796979i	$0.706430\pi$
$888$	−22.5008	−0.755078
$889$	−13.0932	−0.439133
$890$	0	0
$891$	−4.44806	−0.149016
$892$	5.50867	0.184444
$893$	0	0
$894$	1.65860	0.0554717
$895$	0	0
$896$	19.8782	0.664083
$897$	17.0913	0.570662
$898$	14.6482	0.488817
$899$	−21.8708	−0.729432
$900$	0	0
$901$	10.0202	0.333822
$902$	−8.05662	−0.268256
$903$	27.8329	0.926222
$904$	61.5230	2.04622
$905$	0	0
$906$	−1.31570	−0.0437110
$907$	−4.40371	−0.146223	−0.0731114	−	0.997324i	$-0.523293\pi$
$907$	−4.40371	−0.146223	−0.0731114	+	0.997324i	$0.523293\pi$
$908$	−4.45900	−0.147977
$909$	−8.76941	−0.290863
$910$	0	0
$911$	14.2563	0.472333	0.236166	−	0.971713i	$-0.424109\pi$
$911$	14.2563	0.472333	0.236166	+	0.971713i	$0.424109\pi$
$912$	0	0
$913$	−3.11870	−0.103214
$914$	−15.8162	−0.523152
$915$	0	0
$916$	−9.18093	−0.303347
$917$	28.7701	0.950073
$918$	15.8170	0.522038
$919$	−28.7276	−0.947637	−0.473819	−	0.880622i	$-0.657125\pi$
$919$	−28.7276	−0.947637	−0.473819	+	0.880622i	$0.657125\pi$
$920$	0	0
$921$	3.55077	0.117002
$922$	−16.9410	−0.557922
$923$	−47.2762	−1.55612
$924$	1.91621	0.0630387
$925$	0	0
$926$	−5.11201	−0.167991
$927$	−31.0787	−1.02076
$928$	−11.9865	−0.393477
$929$	−30.6928	−1.00700	−0.503499	−	0.863996i	$-0.667954\pi$
$929$	−30.6928	−1.00700	−0.503499	+	0.863996i	$0.667954\pi$
$930$	0	0
$931$	0	0
$932$	13.3866	0.438494
$933$	−12.0935	−0.395923
$934$	28.4652	0.931411
$935$	0	0
$936$	−24.9104	−0.814222
$937$	50.4315	1.64752	0.823762	−	0.566936i	$-0.191871\pi$
$937$	50.4315	1.64752	0.823762	+	0.566936i	$0.191871\pi$
$938$	−4.68261	−0.152893
$939$	−6.27890	−0.204904
$940$	0	0
$941$	−8.54594	−0.278590	−0.139295	−	0.990251i	$-0.544484\pi$
$941$	−8.54594	−0.278590	−0.139295	+	0.990251i	$0.544484\pi$
$942$	−5.93795	−0.193469
$943$	32.7691	1.06711
$944$	−2.22863	−0.0725357
$945$	0	0
$946$	13.7924	0.448428
$947$	−20.5825	−0.668841	−0.334421	−	0.942424i	$-0.608541\pi$
$947$	−20.5825	−0.668841	−0.334421	+	0.942424i	$0.608541\pi$
$948$	0.565344	0.0183615
$949$	38.2325	1.24108
$950$	0	0
$951$	21.9051	0.710323
$952$	34.9282	1.13203
$953$	59.1921	1.91742	0.958711	−	0.284384i	$-0.0917889\pi$
$953$	59.1921	1.91742	0.958711	+	0.284384i	$0.0917889\pi$
$954$	−9.57812	−0.310103
$955$	0	0
$956$	4.65534	0.150564
$957$	4.26725	0.137940
$958$	−42.8984	−1.38599
$959$	−13.9778	−0.451367
$960$	0	0
$961$	−4.97635	−0.160528
$962$	−38.5484	−1.24285
$963$	−12.0776	−0.389195
$964$	13.4633	0.433623
$965$	0	0
$966$	22.9072	0.737028
$967$	−10.9970	−0.353639	−0.176819	−	0.984243i	$-0.556581\pi$
$967$	−10.9970	−0.353639	−0.176819	+	0.984243i	$0.556581\pi$
$968$	−29.0075	−0.932335
$969$	0	0
$970$	0	0
$971$	−32.3692	−1.03878	−0.519389	−	0.854538i	$-0.673840\pi$
$971$	−32.3692	−1.03878	−0.519389	+	0.854538i	$0.673840\pi$
$972$	8.02706	0.257468
$973$	70.6309	2.26432
$974$	40.8265	1.30817
$975$	0	0
$976$	−17.4567	−0.558776
$977$	−11.1038	−0.355241	−0.177621	−	0.984099i	$-0.556840\pi$
$977$	−11.1038	−0.355241	−0.177621	+	0.984099i	$0.556840\pi$
$978$	9.62748	0.307853
$979$	3.33882	0.106709
$980$	0	0
$981$	−3.73588	−0.119277
$982$	21.4250	0.683701
$983$	12.5322	0.399716	0.199858	−	0.979825i	$-0.435952\pi$
$983$	12.5322	0.399716	0.199858	+	0.979825i	$0.435952\pi$
$984$	13.1351	0.418731
$985$	0	0
$986$	15.7482	0.501525
$987$	−22.2766	−0.709072
$988$	0	0
$989$	−56.0984	−1.78383
$990$	0	0
$991$	−36.7597	−1.16771	−0.583856	−	0.811857i	$-0.698457\pi$
$991$	−36.7597	−1.16771	−0.583856	+	0.811857i	$0.698457\pi$
$992$	14.2625	0.452836
$993$	−9.63695	−0.305819
$994$	−63.3637	−2.00977
$995$	0	0
$996$	1.02945	0.0326193
$997$	−24.4801	−0.775291	−0.387646	−	0.921808i	$-0.626712\pi$
$997$	−24.4801	−0.775291	−0.387646	+	0.921808i	$0.626712\pi$
$998$	−39.9251	−1.26381
$999$	39.3173	1.24394

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.ct.1.8		24
5.2	odd	4		1805.2.b.l.1084.8		24
5.3	odd	4		1805.2.b.l.1084.17		24
5.4	even	2	inner	9025.2.a.ct.1.17		24
19.3	odd	18		475.2.l.f.351.6		48
19.13	odd	18		475.2.l.f.226.6		48
19.18	odd	2		9025.2.a.cu.1.17		24
95.3	even	36		95.2.p.a.9.6	yes	48
95.13	even	36		95.2.p.a.74.3	yes	48
95.18	even	4		1805.2.b.k.1084.8		24
95.22	even	36		95.2.p.a.9.3	✓	48
95.32	even	36		95.2.p.a.74.6	yes	48
95.37	even	4		1805.2.b.k.1084.17		24
95.79	odd	18		475.2.l.f.351.3		48
95.89	odd	18		475.2.l.f.226.3		48
95.94	odd	2		9025.2.a.cu.1.8		24
285.32	odd	36		855.2.da.b.739.3		48
285.98	odd	36		855.2.da.b.199.3		48
285.203	odd	36		855.2.da.b.739.6		48
285.212	odd	36		855.2.da.b.199.6		48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
95.2.p.a.9.3	✓	48	95.22	even	36
95.2.p.a.9.6	yes	48	95.3	even	36
95.2.p.a.74.3	yes	48	95.13	even	36
95.2.p.a.74.6	yes	48	95.32	even	36
475.2.l.f.226.3		48	95.89	odd	18
475.2.l.f.226.6		48	19.13	odd	18
475.2.l.f.351.3		48	95.79	odd	18
475.2.l.f.351.6		48	19.3	odd	18
855.2.da.b.199.3		48	285.98	odd	36
855.2.da.b.199.6		48	285.212	odd	36
855.2.da.b.739.3		48	285.32	odd	36
855.2.da.b.739.6		48	285.203	odd	36
1805.2.b.k.1084.8		24	95.18	even	4
1805.2.b.k.1084.17		24	95.37	even	4
1805.2.b.l.1084.8		24	5.2	odd	4
1805.2.b.l.1084.17		24	5.3	odd	4
9025.2.a.ct.1.8		24	1.1	even	1	trivial
9025.2.a.ct.1.17		24	5.4	even	2	inner
9025.2.a.cu.1.8		24	95.94	odd	2
9025.2.a.cu.1.17		24	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22159 q^{2} -0.804421 q^{3} -0.507728 q^{4} +0.982669 q^{6} -3.79180 q^{7} +3.06341 q^{8} -2.35291 q^{9} +O(q^{10})\)	\(q-1.22159 q^{2} -0.804421 q^{3} -0.507728 q^{4} +0.982669 q^{6} -3.79180 q^{7} +3.06341 q^{8} -2.35291 q^{9} -1.23733 q^{11} +0.408427 q^{12} +3.45598 q^{13} +4.63201 q^{14} -2.72676 q^{16} -3.00695 q^{17} +2.87428 q^{18} +3.05020 q^{21} +1.51150 q^{22} -6.14780 q^{23} -2.46427 q^{24} -4.22178 q^{26} +4.30599 q^{27} +1.92520 q^{28} +4.28727 q^{29} -5.10134 q^{31} -2.79584 q^{32} +0.995330 q^{33} +3.67325 q^{34} +1.19464 q^{36} +9.13084 q^{37} -2.78006 q^{39} -5.33021 q^{41} -3.72608 q^{42} +9.12495 q^{43} +0.628225 q^{44} +7.51007 q^{46} -7.30333 q^{47} +2.19346 q^{48} +7.37774 q^{49} +2.41886 q^{51} -1.75470 q^{52} -3.33235 q^{53} -5.26014 q^{54} -11.6158 q^{56} -5.23727 q^{58} +0.817318 q^{59} +6.40201 q^{61} +6.23172 q^{62} +8.92175 q^{63} +8.86888 q^{64} -1.21588 q^{66} -1.01092 q^{67} +1.52671 q^{68} +4.94542 q^{69} -13.6795 q^{71} -7.20791 q^{72} +11.0627 q^{73} -11.1541 q^{74} +4.69169 q^{77} +3.39609 q^{78} +1.38420 q^{79} +3.59490 q^{81} +6.51132 q^{82} +2.52052 q^{83} -1.54867 q^{84} -11.1469 q^{86} -3.44877 q^{87} -3.79043 q^{88} -2.69842 q^{89} -13.1044 q^{91} +3.12141 q^{92} +4.10362 q^{93} +8.92164 q^{94} +2.24903 q^{96} +3.06253 q^{97} -9.01254 q^{98} +2.91131 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9}+O(q^{10}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9	\( 24 q + 18 q^{4} - 12 q^{6} + 12 q^{9} + 12 q^{11} - 24 q^{14} + 6 q^{16} - 6 q^{21} - 42 q^{24} - 12 q^{26} - 36 q^{29} - 42 q^{31} - 6 q^{34} - 6 q^{36} + 24 q^{39} - 60 q^{41} - 30 q^{44} - 6 q^{46} + 12 q^{49} - 30 q^{51} - 24 q^{54} - 18 q^{56} - 60 q^{59} + 30 q^{61} + 36 q^{66} - 66 q^{69} - 96 q^{71} + 24 q^{74} - 72 q^{79} - 96 q^{81} + 54 q^{84} - 108 q^{86} - 84 q^{89} - 96 q^{91} - 36 q^{94} - 120 q^{96}+O(q^{100}) \) 24 * q + 18 * q^4 - 12 * q^6 + 12 * q^9 + 12 * q^11 - 24 * q^14 + 6 * q^16 - 6 * q^21 - 42 * q^24 - 12 * q^26 - 36 * q^29 - 42 * q^31 - 6 * q^34 - 6 * q^36 + 24 * q^39 - 60 * q^41 - 30 * q^44 - 6 * q^46 + 12 * q^49 - 30 * q^51 - 24 * q^54 - 18 * q^56 - 60 * q^59 + 30 * q^61 + 36 * q^66 - 66 * q^69 - 96 * q^71 + 24 * q^74 - 72 * q^79 - 96 * q^81 + 54 * q^84 - 108 * q^86 - 84 * q^89 - 96 * q^91 - 36 * q^94 - 120 * q^96

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more