LMFDB - Embedded newform 5225.2.a.c.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5225,2,Mod(1,5225)] 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("5225.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5225, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 5225 = 5^{2} \cdot 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5225.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,2,-1,0,2,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$41.7218350561$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1045)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5225.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +2.00000 q^{3} -1.00000 q^{4} +2.00000 q^{6} +2.00000 q^{7} -3.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -6.00000 q^{13} +2.00000 q^{14} -1.00000 q^{16} +1.00000 q^{18} -1.00000 q^{19} +4.00000 q^{21} -1.00000 q^{22} -6.00000 q^{24} -6.00000 q^{26} -4.00000 q^{27} -2.00000 q^{28} +6.00000 q^{29} +4.00000 q^{31} +5.00000 q^{32} -2.00000 q^{33} -1.00000 q^{36} -1.00000 q^{38} -12.0000 q^{39} -6.00000 q^{41} +4.00000 q^{42} +6.00000 q^{43} +1.00000 q^{44} +4.00000 q^{47} -2.00000 q^{48} -3.00000 q^{49} +6.00000 q^{52} -12.0000 q^{53} -4.00000 q^{54} -6.00000 q^{56} -2.00000 q^{57} +6.00000 q^{58} -8.00000 q^{59} -14.0000 q^{61} +4.00000 q^{62} +2.00000 q^{63} +7.00000 q^{64} -2.00000 q^{66} -6.00000 q^{67} -8.00000 q^{71} -3.00000 q^{72} -16.0000 q^{73} +1.00000 q^{76} -2.00000 q^{77} -12.0000 q^{78} -8.00000 q^{79} -11.0000 q^{81} -6.00000 q^{82} -6.00000 q^{83} -4.00000 q^{84} +6.00000 q^{86} +12.0000 q^{87} +3.00000 q^{88} -2.00000 q^{89} -12.0000 q^{91} +8.00000 q^{93} +4.00000 q^{94} +10.0000 q^{96} -4.00000 q^{97} -3.00000 q^{98} -1.00000 q^{99} +O(q^{100})\)

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 $

$2$	1.00000	0.707107	0.353553	−	0.935414i	$-0.384973\pi$
$2$	1.00000	0.707107	0.353553	+	0.935414i	$0.384973\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	2.00000	0.816497
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−3.00000	−1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−2.00000	−0.577350
$13$	−6.00000	−1.66410	−0.832050	−	0.554700i	$-0.812833\pi$
$13$	−6.00000	−1.66410	−0.832050	+	0.554700i	$0.812833\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	−1.00000	−0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	4.00000	0.872872
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−6.00000	−1.22474
$25$	0	0
$26$	−6.00000	−1.17670
$27$	−4.00000	−0.769800
$28$	−2.00000	−0.377964
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	5.00000	0.883883
$33$	−2.00000	−0.348155
$34$	0	0
$35$	0	0
$36$	−1.00000	−0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−1.00000	−0.162221
$39$	−12.0000	−1.92154
$40$	0	0
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	4.00000	0.617213
$43$	6.00000	0.914991	0.457496	−	0.889212i	$-0.348747\pi$
$43$	6.00000	0.914991	0.457496	+	0.889212i	$0.348747\pi$
$44$	1.00000	0.150756
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	−2.00000	−0.288675
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	6.00000	0.832050
$53$	−12.0000	−1.64833	−0.824163	−	0.566352i	$-0.808354\pi$
$53$	−12.0000	−1.64833	−0.824163	+	0.566352i	$0.808354\pi$
$54$	−4.00000	−0.544331
$55$	0	0
$56$	−6.00000	−0.801784
$57$	−2.00000	−0.264906
$58$	6.00000	0.787839
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	4.00000	0.508001
$63$	2.00000	0.251976
$64$	7.00000	0.875000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	−6.00000	−0.733017	−0.366508	−	0.930415i	$-0.619447\pi$
$67$	−6.00000	−0.733017	−0.366508	+	0.930415i	$0.619447\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	−3.00000	−0.353553
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	0	0
$75$	0	0
$76$	1.00000	0.114708
$77$	−2.00000	−0.227921
$78$	−12.0000	−1.35873
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	−6.00000	−0.662589
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	−4.00000	−0.436436
$85$	0	0
$86$	6.00000	0.646997
$87$	12.0000	1.28654
$88$	3.00000	0.319801
$89$	−2.00000	−0.212000	−0.106000	−	0.994366i	$-0.533804\pi$
$89$	−2.00000	−0.212000	−0.106000	+	0.994366i	$0.533804\pi$
$90$	0	0
$91$	−12.0000	−1.25794
$92$	0	0
$93$	8.00000	0.829561
$94$	4.00000	0.412568
$95$	0	0
$96$	10.0000	1.02062
$97$	−4.00000	−0.406138	−0.203069	−	0.979164i	$-0.565092\pi$
$97$	−4.00000	−0.406138	−0.203069	+	0.979164i	$0.565092\pi$
$98$	−3.00000	−0.303046
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−18.0000	−1.79107	−0.895533	−	0.444994i	$-0.853206\pi$
$101$	−18.0000	−1.79107	−0.895533	+	0.444994i	$0.853206\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	18.0000	1.76505
$105$	0	0
$106$	−12.0000	−1.16554
$107$	20.0000	1.93347	0.966736	−	0.255774i	$-0.0823304\pi$
$107$	20.0000	1.93347	0.966736	+	0.255774i	$0.0823304\pi$
$108$	4.00000	0.384900
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	−2.00000	−0.188982
$113$	−4.00000	−0.376288	−0.188144	−	0.982141i	$-0.560247\pi$
$113$	−4.00000	−0.376288	−0.188144	+	0.982141i	$0.560247\pi$
$114$	−2.00000	−0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−6.00000	−0.554700
$118$	−8.00000	−0.736460
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−14.0000	−1.26750
$123$	−12.0000	−1.08200
$124$	−4.00000	−0.359211
$125$	0	0
$126$	2.00000	0.178174
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−3.00000	−0.265165
$129$	12.0000	1.05654
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	2.00000	0.174078
$133$	−2.00000	−0.173422
$134$	−6.00000	−0.518321
$135$	0	0
$136$	0	0
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−8.00000	−0.671345
$143$	6.00000	0.501745
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	−16.0000	−1.32417
$147$	−6.00000	−0.494872
$148$	0	0
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	3.00000	0.243332
$153$	0	0
$154$	−2.00000	−0.161165
$155$	0	0
$156$	12.0000	0.960769
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	−8.00000	−0.636446
$159$	−24.0000	−1.90332
$160$	0	0
$161$	0	0
$162$	−11.0000	−0.864242
$163$	−4.00000	−0.313304	−0.156652	−	0.987654i	$-0.550070\pi$
$163$	−4.00000	−0.313304	−0.156652	+	0.987654i	$0.550070\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−6.00000	−0.465690
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	−12.0000	−0.925820
$169$	23.0000	1.76923
$170$	0	0
$171$	−1.00000	−0.0764719
$172$	−6.00000	−0.457496
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	12.0000	0.909718
$175$	0	0
$176$	1.00000	0.0753778
$177$	−16.0000	−1.20263
$178$	−2.00000	−0.149906
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	−12.0000	−0.889499
$183$	−28.0000	−2.06982
$184$	0	0
$185$	0	0
$186$	8.00000	0.586588
$187$	0	0
$188$	−4.00000	−0.291730
$189$	−8.00000	−0.581914
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	14.0000	1.01036
$193$	6.00000	0.431889	0.215945	−	0.976406i	$-0.430717\pi$
$193$	6.00000	0.431889	0.215945	+	0.976406i	$0.430717\pi$
$194$	−4.00000	−0.287183
$195$	0	0
$196$	3.00000	0.214286
$197$	24.0000	1.70993	0.854965	−	0.518686i	$-0.173579\pi$
$197$	24.0000	1.70993	0.854965	+	0.518686i	$0.173579\pi$
$198$	−1.00000	−0.0710669
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−12.0000	−0.846415
$202$	−18.0000	−1.26648
$203$	12.0000	0.842235
$204$	0	0
$205$	0	0
$206$	10.0000	0.696733
$207$	0	0
$208$	6.00000	0.416025
$209$	1.00000	0.0691714
$210$	0	0
$211$	28.0000	1.92760	0.963800	−	0.266627i	$-0.0859092\pi$
$211$	28.0000	1.92760	0.963800	+	0.266627i	$0.0859092\pi$
$212$	12.0000	0.824163
$213$	−16.0000	−1.09630
$214$	20.0000	1.36717
$215$	0	0
$216$	12.0000	0.816497
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	−32.0000	−2.16236
$220$	0	0
$221$	0	0
$222$	0	0
$223$	22.0000	1.47323	0.736614	−	0.676313i	$-0.236423\pi$
$223$	22.0000	1.47323	0.736614	+	0.676313i	$0.236423\pi$
$224$	10.0000	0.668153
$225$	0	0
$226$	−4.00000	−0.266076
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	2.00000	0.132453
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	0	0
$231$	−4.00000	−0.263181
$232$	−18.0000	−1.18176
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	8.00000	0.520756
$237$	−16.0000	−1.03931
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	0	0
$241$	14.0000	0.901819	0.450910	−	0.892570i	$-0.351100\pi$
$241$	14.0000	0.901819	0.450910	+	0.892570i	$0.351100\pi$
$242$	1.00000	0.0642824
$243$	−10.0000	−0.641500
$244$	14.0000	0.896258
$245$	0	0
$246$	−12.0000	−0.765092
$247$	6.00000	0.381771
$248$	−12.0000	−0.762001
$249$	−12.0000	−0.760469
$250$	0	0
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	−2.00000	−0.125988
$253$	0	0
$254$	−4.00000	−0.250982
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−20.0000	−1.24757	−0.623783	−	0.781598i	$-0.714405\pi$
$257$	−20.0000	−1.24757	−0.623783	+	0.781598i	$0.714405\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	0	0
$261$	6.00000	0.371391
$262$	4.00000	0.247121
$263$	−10.0000	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$263$	−10.0000	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$264$	6.00000	0.369274
$265$	0	0
$266$	−2.00000	−0.122628
$267$	−4.00000	−0.244796
$268$	6.00000	0.366508
$269$	−10.0000	−0.609711	−0.304855	−	0.952399i	$-0.598608\pi$
$269$	−10.0000	−0.609711	−0.304855	+	0.952399i	$0.598608\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	−24.0000	−1.45255
$274$	−2.00000	−0.120824
$275$	0	0
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	12.0000	0.719712
$279$	4.00000	0.239474
$280$	0	0
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	8.00000	0.476393
$283$	−6.00000	−0.356663	−0.178331	−	0.983970i	$-0.557070\pi$
$283$	−6.00000	−0.356663	−0.178331	+	0.983970i	$0.557070\pi$
$284$	8.00000	0.474713
$285$	0	0
$286$	6.00000	0.354787
$287$	−12.0000	−0.708338
$288$	5.00000	0.294628
$289$	−17.0000	−1.00000
$290$	0	0
$291$	−8.00000	−0.468968
$292$	16.0000	0.936329
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	−6.00000	−0.349927
$295$	0	0
$296$	0	0
$297$	4.00000	0.232104
$298$	6.00000	0.347571
$299$	0	0
$300$	0	0
$301$	12.0000	0.691669
$302$	−8.00000	−0.460348
$303$	−36.0000	−2.06815
$304$	1.00000	0.0573539
$305$	0	0
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	2.00000	0.113961
$309$	20.0000	1.13776
$310$	0	0
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	36.0000	2.03810
$313$	−26.0000	−1.46961	−0.734803	−	0.678280i	$-0.762726\pi$
$313$	−26.0000	−1.46961	−0.734803	+	0.678280i	$0.762726\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	8.00000	0.450035
$317$	20.0000	1.12331	0.561656	−	0.827371i	$-0.310164\pi$
$317$	20.0000	1.12331	0.561656	+	0.827371i	$0.310164\pi$
$318$	−24.0000	−1.34585
$319$	−6.00000	−0.335936
$320$	0	0
$321$	40.0000	2.23258
$322$	0	0
$323$	0	0
$324$	11.0000	0.611111
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−4.00000	−0.221201
$328$	18.0000	0.993884
$329$	8.00000	0.441054
$330$	0	0
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	8.00000	0.437741
$335$	0	0
$336$	−4.00000	−0.218218
$337$	−10.0000	−0.544735	−0.272367	−	0.962193i	$-0.587807\pi$
$337$	−10.0000	−0.544735	−0.272367	+	0.962193i	$0.587807\pi$
$338$	23.0000	1.25104
$339$	−8.00000	−0.434500
$340$	0	0
$341$	−4.00000	−0.216612
$342$	−1.00000	−0.0540738
$343$	−20.0000	−1.07990
$344$	−18.0000	−0.970495
$345$	0	0
$346$	−6.00000	−0.322562
$347$	26.0000	1.39575	0.697877	−	0.716218i	$-0.254128\pi$
$347$	26.0000	1.39575	0.697877	+	0.716218i	$0.254128\pi$
$348$	−12.0000	−0.643268
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	0	0
$351$	24.0000	1.28103
$352$	−5.00000	−0.266501
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	−16.0000	−0.850390
$355$	0	0
$356$	2.00000	0.106000
$357$	0	0
$358$	−12.0000	−0.634220
$359$	12.0000	0.633336	0.316668	−	0.948536i	$-0.397436\pi$
$359$	12.0000	0.633336	0.316668	+	0.948536i	$0.397436\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	14.0000	0.735824
$363$	2.00000	0.104973
$364$	12.0000	0.628971
$365$	0	0
$366$	−28.0000	−1.46358
$367$	28.0000	1.46159	0.730794	−	0.682598i	$-0.239150\pi$
$367$	28.0000	1.46159	0.730794	+	0.682598i	$0.239150\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	0	0
$371$	−24.0000	−1.24602
$372$	−8.00000	−0.414781
$373$	−14.0000	−0.724893	−0.362446	−	0.932005i	$-0.618058\pi$
$373$	−14.0000	−0.724893	−0.362446	+	0.932005i	$0.618058\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	−36.0000	−1.85409
$378$	−8.00000	−0.411476
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	−8.00000	−0.409316
$383$	30.0000	1.53293	0.766464	−	0.642287i	$-0.222014\pi$
$383$	30.0000	1.53293	0.766464	+	0.642287i	$0.222014\pi$
$384$	−6.00000	−0.306186
$385$	0	0
$386$	6.00000	0.305392
$387$	6.00000	0.304997
$388$	4.00000	0.203069
$389$	−26.0000	−1.31825	−0.659126	−	0.752032i	$-0.729074\pi$
$389$	−26.0000	−1.31825	−0.659126	+	0.752032i	$0.729074\pi$
$390$	0	0
$391$	0	0
$392$	9.00000	0.454569
$393$	8.00000	0.403547
$394$	24.0000	1.20910
$395$	0	0
$396$	1.00000	0.0502519
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	16.0000	0.802008
$399$	−4.00000	−0.200250
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	−12.0000	−0.598506
$403$	−24.0000	−1.19553
$404$	18.0000	0.895533
$405$	0	0
$406$	12.0000	0.595550
$407$	0	0
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	−10.0000	−0.492665
$413$	−16.0000	−0.787309
$414$	0	0
$415$	0	0
$416$	−30.0000	−1.47087
$417$	24.0000	1.17529
$418$	1.00000	0.0489116
$419$	−20.0000	−0.977064	−0.488532	−	0.872546i	$-0.662467\pi$
$419$	−20.0000	−0.977064	−0.488532	+	0.872546i	$0.662467\pi$
$420$	0	0
$421$	−14.0000	−0.682318	−0.341159	−	0.940006i	$-0.610819\pi$
$421$	−14.0000	−0.682318	−0.341159	+	0.940006i	$0.610819\pi$
$422$	28.0000	1.36302
$423$	4.00000	0.194487
$424$	36.0000	1.74831
$425$	0	0
$426$	−16.0000	−0.775203
$427$	−28.0000	−1.35501
$428$	−20.0000	−0.966736
$429$	12.0000	0.579365
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	4.00000	0.192450
$433$	4.00000	0.192228	0.0961139	−	0.995370i	$-0.469359\pi$
$433$	4.00000	0.192228	0.0961139	+	0.995370i	$0.469359\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	−32.0000	−1.52902
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−8.00000	−0.380091	−0.190046	−	0.981775i	$-0.560864\pi$
$443$	−8.00000	−0.380091	−0.190046	+	0.981775i	$0.560864\pi$
$444$	0	0
$445$	0	0
$446$	22.0000	1.04173
$447$	12.0000	0.567581
$448$	14.0000	0.661438
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	0	0
$451$	6.00000	0.282529
$452$	4.00000	0.188144
$453$	−16.0000	−0.751746
$454$	8.00000	0.375459
$455$	0	0
$456$	6.00000	0.280976
$457$	−32.0000	−1.49690	−0.748448	−	0.663193i	$-0.769201\pi$
$457$	−32.0000	−1.49690	−0.748448	+	0.663193i	$0.769201\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	−4.00000	−0.186097
$463$	−8.00000	−0.371792	−0.185896	−	0.982569i	$-0.559519\pi$
$463$	−8.00000	−0.371792	−0.185896	+	0.982569i	$0.559519\pi$
$464$	−6.00000	−0.278543
$465$	0	0
$466$	24.0000	1.11178
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	6.00000	0.277350
$469$	−12.0000	−0.554109
$470$	0	0
$471$	20.0000	0.921551
$472$	24.0000	1.10469
$473$	−6.00000	−0.275880
$474$	−16.0000	−0.734904
$475$	0	0
$476$	0	0
$477$	−12.0000	−0.549442
$478$	−24.0000	−1.09773
$479$	28.0000	1.27935	0.639676	−	0.768644i	$-0.279068\pi$
$479$	28.0000	1.27935	0.639676	+	0.768644i	$0.279068\pi$
$480$	0	0
$481$	0	0
$482$	14.0000	0.637683
$483$	0	0
$484$	−1.00000	−0.0454545
$485$	0	0
$486$	−10.0000	−0.453609
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	42.0000	1.90125
$489$	−8.00000	−0.361773
$490$	0	0
$491$	−16.0000	−0.722070	−0.361035	−	0.932552i	$-0.617576\pi$
$491$	−16.0000	−0.722070	−0.361035	+	0.932552i	$0.617576\pi$
$492$	12.0000	0.541002
$493$	0	0
$494$	6.00000	0.269953
$495$	0	0
$496$	−4.00000	−0.179605
$497$	−16.0000	−0.717698
$498$	−12.0000	−0.537733
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	0	0
$501$	16.0000	0.714827
$502$	−4.00000	−0.178529
$503$	10.0000	0.445878	0.222939	−	0.974832i	$-0.428435\pi$
$503$	10.0000	0.445878	0.222939	+	0.974832i	$0.428435\pi$
$504$	−6.00000	−0.267261
$505$	0	0
$506$	0	0
$507$	46.0000	2.04293
$508$	4.00000	0.177471
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−32.0000	−1.41560
$512$	−11.0000	−0.486136
$513$	4.00000	0.176604
$514$	−20.0000	−0.882162
$515$	0	0
$516$	−12.0000	−0.528271
$517$	−4.00000	−0.175920
$518$	0	0
$519$	−12.0000	−0.526742
$520$	0	0
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	6.00000	0.262613
$523$	28.0000	1.22435	0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000	1.22435	0.612177	+	0.790721i	$0.290294\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	−10.0000	−0.436021
$527$	0	0
$528$	2.00000	0.0870388
$529$	−23.0000	−1.00000
$530$	0	0
$531$	−8.00000	−0.347170
$532$	2.00000	0.0867110
$533$	36.0000	1.55933
$534$	−4.00000	−0.173097
$535$	0	0
$536$	18.0000	0.777482
$537$	−24.0000	−1.03568
$538$	−10.0000	−0.431131
$539$	3.00000	0.129219
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	20.0000	0.859074
$543$	28.0000	1.20160
$544$	0	0
$545$	0	0
$546$	−24.0000	−1.02711
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	2.00000	0.0854358
$549$	−14.0000	−0.597505
$550$	0	0
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−16.0000	−0.680389
$554$	−8.00000	−0.339887
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−28.0000	−1.18640	−0.593199	−	0.805056i	$-0.702135\pi$
$557$	−28.0000	−1.18640	−0.593199	+	0.805056i	$0.702135\pi$
$558$	4.00000	0.169334
$559$	−36.0000	−1.52264
$560$	0	0
$561$	0	0
$562$	6.00000	0.253095
$563$	36.0000	1.51722	0.758610	−	0.651546i	$-0.225879\pi$
$563$	36.0000	1.51722	0.758610	+	0.651546i	$0.225879\pi$
$564$	−8.00000	−0.336861
$565$	0	0
$566$	−6.00000	−0.252199
$567$	−22.0000	−0.923913
$568$	24.0000	1.00702
$569$	46.0000	1.92842	0.964210	−	0.265139i	$-0.0854179\pi$
$569$	46.0000	1.92842	0.964210	+	0.265139i	$0.0854179\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	−6.00000	−0.250873
$573$	−16.0000	−0.668410
$574$	−12.0000	−0.500870
$575$	0	0
$576$	7.00000	0.291667
$577$	−14.0000	−0.582828	−0.291414	−	0.956597i	$-0.594126\pi$
$577$	−14.0000	−0.582828	−0.291414	+	0.956597i	$0.594126\pi$
$578$	−17.0000	−0.707107
$579$	12.0000	0.498703
$580$	0	0
$581$	−12.0000	−0.497844
$582$	−8.00000	−0.331611
$583$	12.0000	0.496989
$584$	48.0000	1.98625
$585$	0	0
$586$	14.0000	0.578335
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	6.00000	0.247436
$589$	−4.00000	−0.164817
$590$	0	0
$591$	48.0000	1.97446
$592$	0	0
$593$	4.00000	0.164260	0.0821302	−	0.996622i	$-0.473828\pi$
$593$	4.00000	0.164260	0.0821302	+	0.996622i	$0.473828\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−6.00000	−0.245770
$597$	32.0000	1.30967
$598$	0	0
$599$	40.0000	1.63436	0.817178	−	0.576386i	$-0.195537\pi$
$599$	40.0000	1.63436	0.817178	+	0.576386i	$0.195537\pi$
$600$	0	0
$601$	−46.0000	−1.87638	−0.938190	−	0.346122i	$-0.887498\pi$
$601$	−46.0000	−1.87638	−0.938190	+	0.346122i	$0.887498\pi$
$602$	12.0000	0.489083
$603$	−6.00000	−0.244339
$604$	8.00000	0.325515
$605$	0	0
$606$	−36.0000	−1.46240
$607$	−40.0000	−1.62355	−0.811775	−	0.583970i	$-0.801498\pi$
$607$	−40.0000	−1.62355	−0.811775	+	0.583970i	$0.801498\pi$
$608$	−5.00000	−0.202777
$609$	24.0000	0.972529
$610$	0	0
$611$	−24.0000	−0.970936
$612$	0	0
$613$	36.0000	1.45403	0.727013	−	0.686624i	$-0.240908\pi$
$613$	36.0000	1.45403	0.727013	+	0.686624i	$0.240908\pi$
$614$	12.0000	0.484281
$615$	0	0
$616$	6.00000	0.241747
$617$	−26.0000	−1.04672	−0.523360	−	0.852111i	$-0.675322\pi$
$617$	−26.0000	−1.04672	−0.523360	+	0.852111i	$0.675322\pi$
$618$	20.0000	0.804518
$619$	4.00000	0.160774	0.0803868	−	0.996764i	$-0.474384\pi$
$619$	4.00000	0.160774	0.0803868	+	0.996764i	$0.474384\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−4.00000	−0.160257
$624$	12.0000	0.480384
$625$	0	0
$626$	−26.0000	−1.03917
$627$	2.00000	0.0798723
$628$	−10.0000	−0.399043
$629$	0	0
$630$	0	0
$631$	24.0000	0.955425	0.477712	−	0.878516i	$-0.341466\pi$
$631$	24.0000	0.955425	0.477712	+	0.878516i	$0.341466\pi$
$632$	24.0000	0.954669
$633$	56.0000	2.22580
$634$	20.0000	0.794301
$635$	0	0
$636$	24.0000	0.951662
$637$	18.0000	0.713186
$638$	−6.00000	−0.237542
$639$	−8.00000	−0.316475
$640$	0	0
$641$	−26.0000	−1.02694	−0.513469	−	0.858108i	$-0.671640\pi$
$641$	−26.0000	−1.02694	−0.513469	+	0.858108i	$0.671640\pi$
$642$	40.0000	1.57867
$643$	40.0000	1.57745	0.788723	−	0.614749i	$-0.210743\pi$
$643$	40.0000	1.57745	0.788723	+	0.614749i	$0.210743\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	33.0000	1.29636
$649$	8.00000	0.314027
$650$	0	0
$651$	16.0000	0.627089
$652$	4.00000	0.156652
$653$	−34.0000	−1.33052	−0.665261	−	0.746611i	$-0.731680\pi$
$653$	−34.0000	−1.33052	−0.665261	+	0.746611i	$0.731680\pi$
$654$	−4.00000	−0.156412
$655$	0	0
$656$	6.00000	0.234261
$657$	−16.0000	−0.624219
$658$	8.00000	0.311872
$659$	−44.0000	−1.71400	−0.856998	−	0.515319i	$-0.827673\pi$
$659$	−44.0000	−1.71400	−0.856998	+	0.515319i	$0.827673\pi$
$660$	0	0
$661$	18.0000	0.700119	0.350059	−	0.936727i	$-0.386161\pi$
$661$	18.0000	0.700119	0.350059	+	0.936727i	$0.386161\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	18.0000	0.698535
$665$	0	0
$666$	0	0
$667$	0	0
$668$	−8.00000	−0.309529
$669$	44.0000	1.70114
$670$	0	0
$671$	14.0000	0.540464
$672$	20.0000	0.771517
$673$	−14.0000	−0.539660	−0.269830	−	0.962908i	$-0.586968\pi$
$673$	−14.0000	−0.539660	−0.269830	+	0.962908i	$0.586968\pi$
$674$	−10.0000	−0.385186
$675$	0	0
$676$	−23.0000	−0.884615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−8.00000	−0.307238
$679$	−8.00000	−0.307012
$680$	0	0
$681$	16.0000	0.613121
$682$	−4.00000	−0.153168
$683$	18.0000	0.688751	0.344375	−	0.938832i	$-0.388091\pi$
$683$	18.0000	0.688751	0.344375	+	0.938832i	$0.388091\pi$
$684$	1.00000	0.0382360
$685$	0	0
$686$	−20.0000	−0.763604
$687$	−12.0000	−0.457829
$688$	−6.00000	−0.228748
$689$	72.0000	2.74298
$690$	0	0
$691$	28.0000	1.06517	0.532585	−	0.846376i	$-0.321221\pi$
$691$	28.0000	1.06517	0.532585	+	0.846376i	$0.321221\pi$
$692$	6.00000	0.228086
$693$	−2.00000	−0.0759737
$694$	26.0000	0.986947
$695$	0	0
$696$	−36.0000	−1.36458
$697$	0	0
$698$	−30.0000	−1.13552
$699$	48.0000	1.81553
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	24.0000	0.905822
$703$	0	0
$704$	−7.00000	−0.263822
$705$	0	0
$706$	18.0000	0.677439
$707$	−36.0000	−1.35392
$708$	16.0000	0.601317
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	0	0
$711$	−8.00000	−0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	12.0000	0.448461
$717$	−48.0000	−1.79259
$718$	12.0000	0.447836
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	1.00000	0.0372161
$723$	28.0000	1.04133
$724$	−14.0000	−0.520306
$725$	0	0
$726$	2.00000	0.0742270
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	36.0000	1.33425
$729$	13.0000	0.481481
$730$	0	0
$731$	0	0
$732$	28.0000	1.03491
$733$	−24.0000	−0.886460	−0.443230	−	0.896408i	$-0.646168\pi$
$733$	−24.0000	−0.886460	−0.443230	+	0.896408i	$0.646168\pi$
$734$	28.0000	1.03350
$735$	0	0
$736$	0	0
$737$	6.00000	0.221013
$738$	−6.00000	−0.220863
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	0	0
$741$	12.0000	0.440831
$742$	−24.0000	−0.881068
$743$	−40.0000	−1.46746	−0.733729	−	0.679442i	$-0.762222\pi$
$743$	−40.0000	−1.46746	−0.733729	+	0.679442i	$0.762222\pi$
$744$	−24.0000	−0.879883
$745$	0	0
$746$	−14.0000	−0.512576
$747$	−6.00000	−0.219529
$748$	0	0
$749$	40.0000	1.46157
$750$	0	0
$751$	4.00000	0.145962	0.0729810	−	0.997333i	$-0.476749\pi$
$751$	4.00000	0.145962	0.0729810	+	0.997333i	$0.476749\pi$
$752$	−4.00000	−0.145865
$753$	−8.00000	−0.291536
$754$	−36.0000	−1.31104
$755$	0	0
$756$	8.00000	0.290957
$757$	−2.00000	−0.0726912	−0.0363456	−	0.999339i	$-0.511572\pi$
$757$	−2.00000	−0.0726912	−0.0363456	+	0.999339i	$0.511572\pi$
$758$	4.00000	0.145287
$759$	0	0
$760$	0	0
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	−8.00000	−0.289809
$763$	−4.00000	−0.144810
$764$	8.00000	0.289430
$765$	0	0
$766$	30.0000	1.08394
$767$	48.0000	1.73318
$768$	−34.0000	−1.22687
$769$	−10.0000	−0.360609	−0.180305	−	0.983611i	$-0.557708\pi$
$769$	−10.0000	−0.360609	−0.180305	+	0.983611i	$0.557708\pi$
$770$	0	0
$771$	−40.0000	−1.44056
$772$	−6.00000	−0.215945
$773$	−24.0000	−0.863220	−0.431610	−	0.902060i	$-0.642054\pi$
$773$	−24.0000	−0.863220	−0.431610	+	0.902060i	$0.642054\pi$
$774$	6.00000	0.215666
$775$	0	0
$776$	12.0000	0.430775
$777$	0	0
$778$	−26.0000	−0.932145
$779$	6.00000	0.214972
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	−24.0000	−0.857690
$784$	3.00000	0.107143
$785$	0	0
$786$	8.00000	0.285351
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	−24.0000	−0.854965
$789$	−20.0000	−0.712019
$790$	0	0
$791$	−8.00000	−0.284447
$792$	3.00000	0.106600
$793$	84.0000	2.98293
$794$	18.0000	0.638796
$795$	0	0
$796$	−16.0000	−0.567105
$797$	−48.0000	−1.70025	−0.850124	−	0.526583i	$-0.823473\pi$
$797$	−48.0000	−1.70025	−0.850124	+	0.526583i	$0.823473\pi$
$798$	−4.00000	−0.141598
$799$	0	0
$800$	0	0
$801$	−2.00000	−0.0706665
$802$	−22.0000	−0.776847
$803$	16.0000	0.564628
$804$	12.0000	0.423207
$805$	0	0
$806$	−24.0000	−0.845364
$807$	−20.0000	−0.704033
$808$	54.0000	1.89971
$809$	46.0000	1.61727	0.808637	−	0.588308i	$-0.200206\pi$
$809$	46.0000	1.61727	0.808637	+	0.588308i	$0.200206\pi$
$810$	0	0
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	−12.0000	−0.421117
$813$	40.0000	1.40286
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−6.00000	−0.209913
$818$	−22.0000	−0.769212
$819$	−12.0000	−0.419314
$820$	0	0
$821$	2.00000	0.0698005	0.0349002	−	0.999391i	$-0.488889\pi$
$821$	2.00000	0.0698005	0.0349002	+	0.999391i	$0.488889\pi$
$822$	−4.00000	−0.139516
$823$	−24.0000	−0.836587	−0.418294	−	0.908312i	$-0.637372\pi$
$823$	−24.0000	−0.836587	−0.418294	+	0.908312i	$0.637372\pi$
$824$	−30.0000	−1.04510
$825$	0	0
$826$	−16.0000	−0.556711
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	−16.0000	−0.555034
$832$	−42.0000	−1.45609
$833$	0	0
$834$	24.0000	0.831052
$835$	0	0
$836$	−1.00000	−0.0345857
$837$	−16.0000	−0.553041
$838$	−20.0000	−0.690889
$839$	−36.0000	−1.24286	−0.621429	−	0.783470i	$-0.713448\pi$
$839$	−36.0000	−1.24286	−0.621429	+	0.783470i	$0.713448\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−14.0000	−0.482472
$843$	12.0000	0.413302
$844$	−28.0000	−0.963800
$845$	0	0
$846$	4.00000	0.137523
$847$	2.00000	0.0687208
$848$	12.0000	0.412082
$849$	−12.0000	−0.411839
$850$	0	0
$851$	0	0
$852$	16.0000	0.548151
$853$	24.0000	0.821744	0.410872	−	0.911693i	$-0.365224\pi$
$853$	24.0000	0.821744	0.410872	+	0.911693i	$0.365224\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	−60.0000	−2.05076
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	12.0000	0.409673
$859$	−4.00000	−0.136478	−0.0682391	−	0.997669i	$-0.521738\pi$
$859$	−4.00000	−0.136478	−0.0682391	+	0.997669i	$0.521738\pi$
$860$	0	0
$861$	−24.0000	−0.817918
$862$	−8.00000	−0.272481
$863$	−10.0000	−0.340404	−0.170202	−	0.985409i	$-0.554442\pi$
$863$	−10.0000	−0.340404	−0.170202	+	0.985409i	$0.554442\pi$
$864$	−20.0000	−0.680414
$865$	0	0
$866$	4.00000	0.135926
$867$	−34.0000	−1.15470
$868$	−8.00000	−0.271538
$869$	8.00000	0.271381
$870$	0	0
$871$	36.0000	1.21981
$872$	6.00000	0.203186
$873$	−4.00000	−0.135379
$874$	0	0
$875$	0	0
$876$	32.0000	1.08118
$877$	−38.0000	−1.28317	−0.641584	−	0.767052i	$-0.721723\pi$
$877$	−38.0000	−1.28317	−0.641584	+	0.767052i	$0.721723\pi$
$878$	−16.0000	−0.539974
$879$	28.0000	0.944417
$880$	0	0
$881$	14.0000	0.471672	0.235836	−	0.971793i	$-0.424217\pi$
$881$	14.0000	0.471672	0.235836	+	0.971793i	$0.424217\pi$
$882$	−3.00000	−0.101015
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	0	0
$885$	0	0
$886$	−8.00000	−0.268765
$887$	−12.0000	−0.402921	−0.201460	−	0.979497i	$-0.564569\pi$
$887$	−12.0000	−0.402921	−0.201460	+	0.979497i	$0.564569\pi$
$888$	0	0
$889$	−8.00000	−0.268311
$890$	0	0
$891$	11.0000	0.368514
$892$	−22.0000	−0.736614
$893$	−4.00000	−0.133855
$894$	12.0000	0.401340
$895$	0	0
$896$	−6.00000	−0.200446
$897$	0	0
$898$	−14.0000	−0.467186
$899$	24.0000	0.800445
$900$	0	0
$901$	0	0
$902$	6.00000	0.199778
$903$	24.0000	0.798670
$904$	12.0000	0.399114
$905$	0	0
$906$	−16.0000	−0.531564
$907$	−50.0000	−1.66022	−0.830111	−	0.557598i	$-0.811723\pi$
$907$	−50.0000	−1.66022	−0.830111	+	0.557598i	$0.811723\pi$
$908$	−8.00000	−0.265489
$909$	−18.0000	−0.597022
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	2.00000	0.0662266
$913$	6.00000	0.198571
$914$	−32.0000	−1.05847
$915$	0	0
$916$	6.00000	0.198246
$917$	8.00000	0.264183
$918$	0	0
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	0	0
$921$	24.0000	0.790827
$922$	30.0000	0.987997
$923$	48.0000	1.57994
$924$	4.00000	0.131590
$925$	0	0
$926$	−8.00000	−0.262896
$927$	10.0000	0.328443
$928$	30.0000	0.984798
$929$	18.0000	0.590561	0.295280	−	0.955411i	$-0.404587\pi$
$929$	18.0000	0.590561	0.295280	+	0.955411i	$0.404587\pi$
$930$	0	0
$931$	3.00000	0.0983210
$932$	−24.0000	−0.786146
$933$	0	0
$934$	36.0000	1.17796
$935$	0	0
$936$	18.0000	0.588348
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	−12.0000	−0.391814
$939$	−52.0000	−1.69696
$940$	0	0
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	20.0000	0.651635
$943$	0	0
$944$	8.00000	0.260378
$945$	0	0
$946$	−6.00000	−0.195077
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	16.0000	0.519656
$949$	96.0000	3.11629
$950$	0	0
$951$	40.0000	1.29709
$952$	0	0
$953$	−10.0000	−0.323932	−0.161966	−	0.986796i	$-0.551783\pi$
$953$	−10.0000	−0.323932	−0.161966	+	0.986796i	$0.551783\pi$
$954$	−12.0000	−0.388514
$955$	0	0
$956$	24.0000	0.776215
$957$	−12.0000	−0.387905
$958$	28.0000	0.904639
$959$	−4.00000	−0.129167
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	20.0000	0.644491
$964$	−14.0000	−0.450910
$965$	0	0
$966$	0	0
$967$	−14.0000	−0.450210	−0.225105	−	0.974335i	$-0.572272\pi$
$967$	−14.0000	−0.450210	−0.225105	+	0.974335i	$0.572272\pi$
$968$	−3.00000	−0.0964237
$969$	0	0
$970$	0	0
$971$	40.0000	1.28366	0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000	1.28366	0.641831	+	0.766846i	$0.278175\pi$
$972$	10.0000	0.320750
$973$	24.0000	0.769405
$974$	−2.00000	−0.0640841
$975$	0	0
$976$	14.0000	0.448129
$977$	−32.0000	−1.02377	−0.511885	−	0.859054i	$-0.671053\pi$
$977$	−32.0000	−1.02377	−0.511885	+	0.859054i	$0.671053\pi$
$978$	−8.00000	−0.255812
$979$	2.00000	0.0639203
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	−16.0000	−0.510581
$983$	38.0000	1.21201	0.606006	−	0.795460i	$-0.292771\pi$
$983$	38.0000	1.21201	0.606006	+	0.795460i	$0.292771\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	0	0
$987$	16.0000	0.509286
$988$	−6.00000	−0.190885
$989$	0	0
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	20.0000	0.635001
$993$	−56.0000	−1.77711
$994$	−16.0000	−0.507489
$995$	0	0
$996$	12.0000	0.380235
$997$	−12.0000	−0.380044	−0.190022	−	0.981780i	$-0.560856\pi$
$997$	−12.0000	−0.380044	−0.190022	+	0.981780i	$0.560856\pi$
$998$	4.00000	0.126618
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5225.2.a.c.1.1		1
5.4	even	2		1045.2.a.a.1.1	✓	1
15.14	odd	2		9405.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1045.2.a.a.1.1	✓	1	5.4	even	2
5225.2.a.c.1.1		1	1.1	even	1	trivial
9405.2.a.j.1.1		1	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5225 = 5^{2} \cdot 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5225.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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