LMFDB - Embedded newform 5025.2.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5025,2,Mod(1,5025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("5025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5025 = 3 \cdot 5^{2} \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5025.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:	$40.1248270157$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1005)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	5025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} -1.00000 q^{4} -1.00000 q^{6} +3.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{16} +6.00000 q^{17} -1.00000 q^{18} +8.00000 q^{19} +4.00000 q^{22} +2.00000 q^{23} +3.00000 q^{24} -6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{31} -5.00000 q^{32} -4.00000 q^{33} -6.00000 q^{34} -1.00000 q^{36} -8.00000 q^{38} +6.00000 q^{39} +12.0000 q^{41} +12.0000 q^{43} +4.00000 q^{44} -2.00000 q^{46} -2.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} +6.00000 q^{51} -6.00000 q^{52} -10.0000 q^{53} -1.00000 q^{54} +8.00000 q^{57} +6.00000 q^{58} -12.0000 q^{59} -2.00000 q^{61} +2.00000 q^{62} +7.00000 q^{64} +4.00000 q^{66} +1.00000 q^{67} -6.00000 q^{68} +2.00000 q^{69} +8.00000 q^{71} +3.00000 q^{72} -8.00000 q^{73} -8.00000 q^{76} -6.00000 q^{78} +2.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} -14.0000 q^{83} -12.0000 q^{86} -6.00000 q^{87} -12.0000 q^{88} +14.0000 q^{89} -2.00000 q^{92} -2.00000 q^{93} +2.00000 q^{94} -5.00000 q^{96} -2.00000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})\)

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 $

$2$	−1.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−1.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.00000	−0.500000
$5$	0	0
$5$	0	0
$6$	−1.00000	−0.408248
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	3.00000	1.06066
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	−1.00000	−0.288675
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	0	0
$15$	0	0
$16$	−1.00000	−0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	−1.00000	−0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	0	0
$21$	0	0
$22$	4.00000	0.852803
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	3.00000	0.612372
$25$	0	0
$26$	−6.00000	−1.17670
$27$	1.00000	0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	−5.00000	−0.883883
$33$	−4.00000	−0.696311
$34$	−6.00000	−1.02899
$35$	0	0
$36$	−1.00000	−0.166667
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	−8.00000	−1.29777
$39$	6.00000	0.960769
$40$	0	0
$41$	12.0000	1.87409	0.937043	−	0.349215i	$-0.113552\pi$
$41$	12.0000	1.87409	0.937043	+	0.349215i	$0.113552\pi$
$42$	0	0
$43$	12.0000	1.82998	0.914991	−	0.403473i	$-0.132197\pi$
$43$	12.0000	1.82998	0.914991	+	0.403473i	$0.132197\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	−2.00000	−0.294884
$47$	−2.00000	−0.291730	−0.145865	−	0.989305i	$-0.546597\pi$
$47$	−2.00000	−0.291730	−0.145865	+	0.989305i	$0.546597\pi$
$48$	−1.00000	−0.144338
$49$	−7.00000	−1.00000
$50$	0	0
$51$	6.00000	0.840168
$52$	−6.00000	−0.832050
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	8.00000	1.05963
$58$	6.00000	0.787839
$59$	−12.0000	−1.56227	−0.781133	−	0.624364i	$-0.785358\pi$
$59$	−12.0000	−1.56227	−0.781133	+	0.624364i	$0.785358\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	7.00000	0.875000
$65$	0	0
$66$	4.00000	0.492366
$67$	1.00000	0.122169
$67$	1.00000	0.122169
$68$	−6.00000	−0.727607
$69$	2.00000	0.240772
$70$	0	0
$71$	8.00000	0.949425	0.474713	−	0.880141i	$-0.342552\pi$
$71$	8.00000	0.949425	0.474713	+	0.880141i	$0.342552\pi$
$72$	3.00000	0.353553
$73$	−8.00000	−0.936329	−0.468165	−	0.883641i	$-0.655085\pi$
$73$	−8.00000	−0.936329	−0.468165	+	0.883641i	$0.655085\pi$
$74$	0	0
$75$	0	0
$76$	−8.00000	−0.917663
$77$	0	0
$78$	−6.00000	−0.679366
$79$	2.00000	0.225018	0.112509	−	0.993651i	$-0.464111\pi$
$79$	2.00000	0.225018	0.112509	+	0.993651i	$0.464111\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−12.0000	−1.32518
$83$	−14.0000	−1.53670	−0.768350	−	0.640030i	$-0.778922\pi$
$83$	−14.0000	−1.53670	−0.768350	+	0.640030i	$0.778922\pi$
$84$	0	0
$85$	0	0
$86$	−12.0000	−1.29399
$87$	−6.00000	−0.643268
$88$	−12.0000	−1.27920
$89$	14.0000	1.48400	0.741999	−	0.670402i	$-0.233878\pi$
$89$	14.0000	1.48400	0.741999	+	0.670402i	$0.233878\pi$
$90$	0	0
$91$	0	0
$92$	−2.00000	−0.208514
$93$	−2.00000	−0.207390
$94$	2.00000	0.206284
$95$	0	0
$96$	−5.00000	−0.510310
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	7.00000	0.707107
$99$	−4.00000	−0.402015
$100$	0	0
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	−6.00000	−0.594089
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	18.0000	1.76505
$105$	0	0
$106$	10.0000	0.971286
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	−8.00000	−0.749269
$115$	0	0
$116$	6.00000	0.557086
$117$	6.00000	0.554700
$118$	12.0000	1.10469
$119$	0	0
$120$	0	0
$121$	5.00000	0.454545
$122$	2.00000	0.181071
$123$	12.0000	1.08200
$124$	2.00000	0.179605
$125$	0	0
$126$	0	0
$127$	12.0000	1.06483	0.532414	−	0.846484i	$-0.321285\pi$
$127$	12.0000	1.06483	0.532414	+	0.846484i	$0.321285\pi$
$128$	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	$0.444091\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−1.00000	−0.0863868
$135$	0	0
$136$	18.0000	1.54349
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	−2.00000	−0.170251
$139$	−2.00000	−0.169638	−0.0848189	−	0.996396i	$-0.527031\pi$
$139$	−2.00000	−0.169638	−0.0848189	+	0.996396i	$0.527031\pi$
$140$	0	0
$141$	−2.00000	−0.168430
$142$	−8.00000	−0.671345
$143$	−24.0000	−2.00698
$144$	−1.00000	−0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	−7.00000	−0.577350
$148$	0	0
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	0	0
$151$	12.0000	0.976546	0.488273	−	0.872691i	$-0.337627\pi$
$151$	12.0000	0.976546	0.488273	+	0.872691i	$0.337627\pi$
$152$	24.0000	1.94666
$153$	6.00000	0.485071
$154$	0	0
$155$	0	0
$156$	−6.00000	−0.480384
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	−2.00000	−0.159111
$159$	−10.0000	−0.793052
$160$	0	0
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−12.0000	−0.937043
$165$	0	0
$166$	14.0000	1.08661
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	0	0
$171$	8.00000	0.611775
$172$	−12.0000	−0.914991
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	−12.0000	−0.901975
$178$	−14.0000	−1.04934
$179$	−16.0000	−1.19590	−0.597948	−	0.801535i	$-0.704017\pi$
$179$	−16.0000	−1.19590	−0.597948	+	0.801535i	$0.704017\pi$
$180$	0	0
$181$	22.0000	1.63525	0.817624	−	0.575753i	$-0.195291\pi$
$181$	22.0000	1.63525	0.817624	+	0.575753i	$0.195291\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	6.00000	0.442326
$185$	0	0
$186$	2.00000	0.146647
$187$	−24.0000	−1.75505
$188$	2.00000	0.145865
$189$	0	0
$190$	0	0
$191$	4.00000	0.289430	0.144715	−	0.989473i	$-0.453773\pi$
$191$	4.00000	0.289430	0.144715	+	0.989473i	$0.453773\pi$
$192$	7.00000	0.505181
$193$	−12.0000	−0.863779	−0.431889	−	0.901927i	$-0.642153\pi$
$193$	−12.0000	−0.863779	−0.431889	+	0.901927i	$0.642153\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	7.00000	0.500000
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	4.00000	0.284268
$199$	20.0000	1.41776	0.708881	−	0.705328i	$-0.249200\pi$
$199$	20.0000	1.41776	0.708881	+	0.705328i	$0.249200\pi$
$200$	0	0
$201$	1.00000	0.0705346
$202$	0	0
$203$	0	0
$204$	−6.00000	−0.420084
$205$	0	0
$206$	−8.00000	−0.557386
$207$	2.00000	0.139010
$208$	−6.00000	−0.416025
$209$	−32.0000	−2.21349
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	10.0000	0.686803
$213$	8.00000	0.548151
$214$	6.00000	0.410152
$215$	0	0
$216$	3.00000	0.204124
$217$	0	0
$218$	10.0000	0.677285
$219$	−8.00000	−0.540590
$220$	0	0
$221$	36.0000	2.42162
$222$	0	0
$223$	12.0000	0.803579	0.401790	−	0.915732i	$-0.368388\pi$
$223$	12.0000	0.803579	0.401790	+	0.915732i	$0.368388\pi$
$224$	0	0
$225$	0	0
$226$	−6.00000	−0.399114
$227$	−14.0000	−0.929213	−0.464606	−	0.885517i	$-0.653804\pi$
$227$	−14.0000	−0.929213	−0.464606	+	0.885517i	$0.653804\pi$
$228$	−8.00000	−0.529813
$229$	−30.0000	−1.98246	−0.991228	−	0.132164i	$-0.957808\pi$
$229$	−30.0000	−1.98246	−0.991228	+	0.132164i	$0.957808\pi$
$230$	0	0
$231$	0	0
$232$	−18.0000	−1.18176
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	−6.00000	−0.392232
$235$	0	0
$236$	12.0000	0.781133
$237$	2.00000	0.129914
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	0	0
$246$	−12.0000	−0.765092
$247$	48.0000	3.05417
$248$	−6.00000	−0.381000
$249$	−14.0000	−0.887214
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	−8.00000	−0.502956
$254$	−12.0000	−0.752947
$255$	0	0
$256$	−17.0000	−1.06250
$257$	−22.0000	−1.37232	−0.686161	−	0.727450i	$-0.740706\pi$
$257$	−22.0000	−1.37232	−0.686161	+	0.727450i	$0.740706\pi$
$258$	−12.0000	−0.747087
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	−4.00000	−0.247121
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	−12.0000	−0.738549
$265$	0	0
$266$	0	0
$267$	14.0000	0.856786
$268$	−1.00000	−0.0610847
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	14.0000	0.850439	0.425220	−	0.905090i	$-0.360197\pi$
$271$	14.0000	0.850439	0.425220	+	0.905090i	$0.360197\pi$
$272$	−6.00000	−0.363803
$273$	0	0
$274$	−6.00000	−0.362473
$275$	0	0
$276$	−2.00000	−0.120386
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	2.00000	0.119952
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	2.00000	0.119098
$283$	16.0000	0.951101	0.475551	−	0.879688i	$-0.342249\pi$
$283$	16.0000	0.951101	0.475551	+	0.879688i	$0.342249\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	24.0000	1.41915
$287$	0	0
$288$	−5.00000	−0.294628
$289$	19.0000	1.11765
$290$	0	0
$291$	−2.00000	−0.117242
$292$	8.00000	0.468165
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	0	0
$297$	−4.00000	−0.232104
$298$	−18.0000	−1.04271
$299$	12.0000	0.693978
$300$	0	0
$301$	0	0
$302$	−12.0000	−0.690522
$303$	0	0
$304$	−8.00000	−0.458831
$305$	0	0
$306$	−6.00000	−0.342997
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	0	0
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	18.0000	1.01905
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	4.00000	0.225733
$315$	0	0
$316$	−2.00000	−0.112509
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	10.0000	0.560772
$319$	24.0000	1.34374
$320$	0	0
$321$	−6.00000	−0.334887
$322$	0	0
$323$	48.0000	2.67079
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	−4.00000	−0.221540
$327$	−10.0000	−0.553001
$328$	36.0000	1.98777
$329$	0	0
$330$	0	0
$331$	−2.00000	−0.109930	−0.0549650	−	0.998488i	$-0.517505\pi$
$331$	−2.00000	−0.109930	−0.0549650	+	0.998488i	$0.517505\pi$
$332$	14.0000	0.768350
$333$	0	0
$334$	−18.0000	−0.984916
$335$	0	0
$336$	0	0
$337$	−14.0000	−0.762629	−0.381314	−	0.924445i	$-0.624528\pi$
$337$	−14.0000	−0.762629	−0.381314	+	0.924445i	$0.624528\pi$
$338$	−23.0000	−1.25104
$339$	6.00000	0.325875
$340$	0	0
$341$	8.00000	0.433224
$342$	−8.00000	−0.432590
$343$	0	0
$344$	36.0000	1.94099
$345$	0	0
$346$	6.00000	0.322562
$347$	−16.0000	−0.858925	−0.429463	−	0.903085i	$-0.641297\pi$
$347$	−16.0000	−0.858925	−0.429463	+	0.903085i	$0.641297\pi$
$348$	6.00000	0.321634
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	6.00000	0.320256
$352$	20.0000	1.06600
$353$	14.0000	0.745145	0.372572	−	0.928003i	$-0.378476\pi$
$353$	14.0000	0.745145	0.372572	+	0.928003i	$0.378476\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−14.0000	−0.741999
$357$	0	0
$358$	16.0000	0.845626
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	45.0000	2.36842
$362$	−22.0000	−1.15629
$363$	5.00000	0.262432
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	−2.00000	−0.104257
$369$	12.0000	0.624695
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	24.0000	1.24101
$375$	0	0
$376$	−6.00000	−0.309426
$377$	−36.0000	−1.85409
$378$	0	0
$379$	34.0000	1.74646	0.873231	−	0.487306i	$-0.162020\pi$
$379$	34.0000	1.74646	0.873231	+	0.487306i	$0.162020\pi$
$380$	0	0
$381$	12.0000	0.614779
$382$	−4.00000	−0.204658
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	3.00000	0.153093
$385$	0	0
$386$	12.0000	0.610784
$387$	12.0000	0.609994
$388$	2.00000	0.101535
$389$	−6.00000	−0.304212	−0.152106	−	0.988364i	$-0.548606\pi$
$389$	−6.00000	−0.304212	−0.152106	+	0.988364i	$0.548606\pi$
$390$	0	0
$391$	12.0000	0.606866
$392$	−21.0000	−1.06066
$393$	4.00000	0.201773
$394$	−10.0000	−0.503793
$395$	0	0
$396$	4.00000	0.201008
$397$	36.0000	1.80679	0.903394	−	0.428811i	$-0.141067\pi$
$397$	36.0000	1.80679	0.903394	+	0.428811i	$0.141067\pi$
$398$	−20.0000	−1.00251
$399$	0	0
$400$	0	0
$401$	8.00000	0.399501	0.199750	−	0.979847i	$-0.435987\pi$
$401$	8.00000	0.399501	0.199750	+	0.979847i	$0.435987\pi$
$402$	−1.00000	−0.0498755
$403$	−12.0000	−0.597763
$404$	0	0
$405$	0	0
$406$	0	0
$407$	0	0
$408$	18.0000	0.891133
$409$	30.0000	1.48340	0.741702	−	0.670729i	$-0.234019\pi$
$409$	30.0000	1.48340	0.741702	+	0.670729i	$0.234019\pi$
$410$	0	0
$411$	6.00000	0.295958
$412$	−8.00000	−0.394132
$413$	0	0
$414$	−2.00000	−0.0982946
$415$	0	0
$416$	−30.0000	−1.47087
$417$	−2.00000	−0.0979404
$418$	32.0000	1.56517
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	4.00000	0.194717
$423$	−2.00000	−0.0972433
$424$	−30.0000	−1.45693
$425$	0	0
$426$	−8.00000	−0.387601
$427$	0	0
$428$	6.00000	0.290021
$429$	−24.0000	−1.15873
$430$	0	0
$431$	8.00000	0.385346	0.192673	−	0.981263i	$-0.438284\pi$
$431$	8.00000	0.385346	0.192673	+	0.981263i	$0.438284\pi$
$432$	−1.00000	−0.0481125
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	10.0000	0.478913
$437$	16.0000	0.765384
$438$	8.00000	0.382255
$439$	4.00000	0.190910	0.0954548	−	0.995434i	$-0.469569\pi$
$439$	4.00000	0.190910	0.0954548	+	0.995434i	$0.469569\pi$
$440$	0	0
$441$	−7.00000	−0.333333
$442$	−36.0000	−1.71235
$443$	−32.0000	−1.52037	−0.760183	−	0.649709i	$-0.774891\pi$
$443$	−32.0000	−1.52037	−0.760183	+	0.649709i	$0.774891\pi$
$444$	0	0
$445$	0	0
$446$	−12.0000	−0.568216
$447$	18.0000	0.851371
$448$	0	0
$449$	−22.0000	−1.03824	−0.519122	−	0.854700i	$-0.673741\pi$
$449$	−22.0000	−1.03824	−0.519122	+	0.854700i	$0.673741\pi$
$450$	0	0
$451$	−48.0000	−2.26023
$452$	−6.00000	−0.282216
$453$	12.0000	0.563809
$454$	14.0000	0.657053
$455$	0	0
$456$	24.0000	1.12390
$457$	16.0000	0.748448	0.374224	−	0.927338i	$-0.377909\pi$
$457$	16.0000	0.748448	0.374224	+	0.927338i	$0.377909\pi$
$458$	30.0000	1.40181
$459$	6.00000	0.280056
$460$	0	0
$461$	−30.0000	−1.39724	−0.698620	−	0.715493i	$-0.746202\pi$
$461$	−30.0000	−1.39724	−0.698620	+	0.715493i	$0.746202\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−6.00000	−0.277945
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	−6.00000	−0.277350
$469$	0	0
$470$	0	0
$471$	−4.00000	−0.184310
$472$	−36.0000	−1.65703
$473$	−48.0000	−2.20704
$474$	−2.00000	−0.0918630
$475$	0	0
$476$	0	0
$477$	−10.0000	−0.457869
$478$	−16.0000	−0.731823
$479$	8.00000	0.365529	0.182765	−	0.983157i	$-0.441495\pi$
$479$	8.00000	0.365529	0.182765	+	0.983157i	$0.441495\pi$
$480$	0	0
$481$	0	0
$482$	10.0000	0.455488
$483$	0	0
$484$	−5.00000	−0.227273
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−6.00000	−0.271607
$489$	4.00000	0.180886
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	−12.0000	−0.541002
$493$	−36.0000	−1.62136
$494$	−48.0000	−2.15962
$495$	0	0
$496$	2.00000	0.0898027
$497$	0	0
$498$	14.0000	0.627355
$499$	−30.0000	−1.34298	−0.671492	−	0.741012i	$-0.734346\pi$
$499$	−30.0000	−1.34298	−0.671492	+	0.741012i	$0.734346\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	−12.0000	−0.535586
$503$	8.00000	0.356702	0.178351	−	0.983967i	$-0.442924\pi$
$503$	8.00000	0.356702	0.178351	+	0.983967i	$0.442924\pi$
$504$	0	0
$505$	0	0
$506$	8.00000	0.355643
$507$	23.0000	1.02147
$508$	−12.0000	−0.532414
$509$	6.00000	0.265945	0.132973	−	0.991120i	$-0.457548\pi$
$509$	6.00000	0.265945	0.132973	+	0.991120i	$0.457548\pi$
$510$	0	0
$511$	0	0
$512$	11.0000	0.486136
$513$	8.00000	0.353209
$514$	22.0000	0.970378
$515$	0	0
$516$	−12.0000	−0.528271
$517$	8.00000	0.351840
$518$	0	0
$519$	−6.00000	−0.263371
$520$	0	0
$521$	12.0000	0.525730	0.262865	−	0.964833i	$-0.415333\pi$
$521$	12.0000	0.525730	0.262865	+	0.964833i	$0.415333\pi$
$522$	6.00000	0.262613
$523$	16.0000	0.699631	0.349816	−	0.936819i	$-0.386244\pi$
$523$	16.0000	0.699631	0.349816	+	0.936819i	$0.386244\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	18.0000	0.784837
$527$	−12.0000	−0.522728
$528$	4.00000	0.174078
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−12.0000	−0.520756
$532$	0	0
$533$	72.0000	3.11867
$534$	−14.0000	−0.605839
$535$	0	0
$536$	3.00000	0.129580
$537$	−16.0000	−0.690451
$538$	6.00000	0.258678
$539$	28.0000	1.20605
$540$	0	0
$541$	−26.0000	−1.11783	−0.558914	−	0.829226i	$-0.688782\pi$
$541$	−26.0000	−1.11783	−0.558914	+	0.829226i	$0.688782\pi$
$542$	−14.0000	−0.601351
$543$	22.0000	0.944110
$544$	−30.0000	−1.28624
$545$	0	0
$546$	0	0
$547$	28.0000	1.19719	0.598597	−	0.801050i	$-0.295725\pi$
$547$	28.0000	1.19719	0.598597	+	0.801050i	$0.295725\pi$
$548$	−6.00000	−0.256307
$549$	−2.00000	−0.0853579
$550$	0	0
$551$	−48.0000	−2.04487
$552$	6.00000	0.255377
$553$	0	0
$554$	8.00000	0.339887
$555$	0	0
$556$	2.00000	0.0848189
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	2.00000	0.0846668
$559$	72.0000	3.04528
$560$	0	0
$561$	−24.0000	−1.01328
$562$	24.0000	1.01238
$563$	32.0000	1.34864	0.674320	−	0.738440i	$-0.264437\pi$
$563$	32.0000	1.34864	0.674320	+	0.738440i	$0.264437\pi$
$564$	2.00000	0.0842152
$565$	0	0
$566$	−16.0000	−0.672530
$567$	0	0
$568$	24.0000	1.00702
$569$	−18.0000	−0.754599	−0.377300	−	0.926091i	$-0.623147\pi$
$569$	−18.0000	−0.754599	−0.377300	+	0.926091i	$0.623147\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	24.0000	1.00349
$573$	4.00000	0.167102
$574$	0	0
$575$	0	0
$576$	7.00000	0.291667
$577$	18.0000	0.749350	0.374675	−	0.927156i	$-0.377754\pi$
$577$	18.0000	0.749350	0.374675	+	0.927156i	$0.377754\pi$
$578$	−19.0000	−0.790296
$579$	−12.0000	−0.498703
$580$	0	0
$581$	0	0
$582$	2.00000	0.0829027
$583$	40.0000	1.65663
$584$	−24.0000	−0.993127
$585$	0	0
$586$	30.0000	1.23929
$587$	12.0000	0.495293	0.247647	−	0.968850i	$-0.420343\pi$
$587$	12.0000	0.495293	0.247647	+	0.968850i	$0.420343\pi$
$588$	7.00000	0.288675
$589$	−16.0000	−0.659269
$590$	0	0
$591$	10.0000	0.411345
$592$	0	0
$593$	−10.0000	−0.410651	−0.205325	−	0.978694i	$-0.565825\pi$
$593$	−10.0000	−0.410651	−0.205325	+	0.978694i	$0.565825\pi$
$594$	4.00000	0.164122
$595$	0	0
$596$	−18.0000	−0.737309
$597$	20.0000	0.818546
$598$	−12.0000	−0.490716
$599$	20.0000	0.817178	0.408589	−	0.912719i	$-0.366021\pi$
$599$	20.0000	0.817178	0.408589	+	0.912719i	$0.366021\pi$
$600$	0	0
$601$	−42.0000	−1.71322	−0.856608	−	0.515968i	$-0.827432\pi$
$601$	−42.0000	−1.71322	−0.856608	+	0.515968i	$0.827432\pi$
$602$	0	0
$603$	1.00000	0.0407231
$604$	−12.0000	−0.488273
$605$	0	0
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	−40.0000	−1.62221
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	−6.00000	−0.242536
$613$	0	0		−	1.00000i	$-0.5\pi$
$613$	0	0			1.00000i	$0.5\pi$
$614$	−8.00000	−0.322854
$615$	0	0
$616$	0	0
$617$	30.0000	1.20775	0.603877	−	0.797077i	$-0.293622\pi$
$617$	30.0000	1.20775	0.603877	+	0.797077i	$0.293622\pi$
$618$	−8.00000	−0.321807
$619$	40.0000	1.60774	0.803868	−	0.594808i	$-0.202772\pi$
$619$	40.0000	1.60774	0.803868	+	0.594808i	$0.202772\pi$
$620$	0	0
$621$	2.00000	0.0802572
$622$	−24.0000	−0.962312
$623$	0	0
$624$	−6.00000	−0.240192
$625$	0	0
$626$	10.0000	0.399680
$627$	−32.0000	−1.27796
$628$	4.00000	0.159617
$629$	0	0
$630$	0	0
$631$	−30.0000	−1.19428	−0.597141	−	0.802137i	$-0.703697\pi$
$631$	−30.0000	−1.19428	−0.597141	+	0.802137i	$0.703697\pi$
$632$	6.00000	0.238667
$633$	−4.00000	−0.158986
$634$	−6.00000	−0.238290
$635$	0	0
$636$	10.0000	0.396526
$637$	−42.0000	−1.66410
$638$	−24.0000	−0.950169
$639$	8.00000	0.316475
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	6.00000	0.236801
$643$	12.0000	0.473234	0.236617	−	0.971603i	$-0.423961\pi$
$643$	12.0000	0.473234	0.236617	+	0.971603i	$0.423961\pi$
$644$	0	0
$645$	0	0
$646$	−48.0000	−1.88853
$647$	−12.0000	−0.471769	−0.235884	−	0.971781i	$-0.575799\pi$
$647$	−12.0000	−0.471769	−0.235884	+	0.971781i	$0.575799\pi$
$648$	3.00000	0.117851
$649$	48.0000	1.88416
$650$	0	0
$651$	0	0
$652$	−4.00000	−0.156652
$653$	46.0000	1.80012	0.900060	−	0.435767i	$-0.143523\pi$
$653$	46.0000	1.80012	0.900060	+	0.435767i	$0.143523\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	−12.0000	−0.468521
$657$	−8.00000	−0.312110
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	0	0
$661$	22.0000	0.855701	0.427850	−	0.903850i	$-0.359271\pi$
$661$	22.0000	0.855701	0.427850	+	0.903850i	$0.359271\pi$
$662$	2.00000	0.0777322
$663$	36.0000	1.39812
$664$	−42.0000	−1.62992
$665$	0	0
$666$	0	0
$667$	−12.0000	−0.464642
$668$	−18.0000	−0.696441
$669$	12.0000	0.463947
$670$	0	0
$671$	8.00000	0.308837
$672$	0	0
$673$	6.00000	0.231283	0.115642	−	0.993291i	$-0.463108\pi$
$673$	6.00000	0.231283	0.115642	+	0.993291i	$0.463108\pi$
$674$	14.0000	0.539260
$675$	0	0
$676$	−23.0000	−0.884615
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	−6.00000	−0.230429
$679$	0	0
$680$	0	0
$681$	−14.0000	−0.536481
$682$	−8.00000	−0.306336
$683$	0	0		−	1.00000i	$-0.5\pi$
$683$	0	0			1.00000i	$0.5\pi$
$684$	−8.00000	−0.305888
$685$	0	0
$686$	0	0
$687$	−30.0000	−1.14457
$688$	−12.0000	−0.457496
$689$	−60.0000	−2.28582
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	16.0000	0.607352
$695$	0	0
$696$	−18.0000	−0.682288
$697$	72.0000	2.72719
$698$	−10.0000	−0.378506
$699$	6.00000	0.226941
$700$	0	0
$701$	0	0		−	1.00000i	$-0.5\pi$
$701$	0	0			1.00000i	$0.5\pi$
$702$	−6.00000	−0.226455
$703$	0	0
$704$	−28.0000	−1.05529
$705$	0	0
$706$	−14.0000	−0.526897
$707$	0	0
$708$	12.0000	0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	42.0000	1.57402
$713$	−4.00000	−0.149801
$714$	0	0
$715$	0	0
$716$	16.0000	0.597948
$717$	16.0000	0.597531
$718$	−24.0000	−0.895672
$719$	−32.0000	−1.19340	−0.596699	−	0.802465i	$-0.703521\pi$
$719$	−32.0000	−1.19340	−0.596699	+	0.802465i	$0.703521\pi$
$720$	0	0
$721$	0	0
$722$	−45.0000	−1.67473
$723$	−10.0000	−0.371904
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−5.00000	−0.185567
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	72.0000	2.66302
$732$	2.00000	0.0739221
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−16.0000	−0.590571
$735$	0	0
$736$	−10.0000	−0.368605
$737$	−4.00000	−0.147342
$738$	−12.0000	−0.441726
$739$	2.00000	0.0735712	0.0367856	−	0.999323i	$-0.488288\pi$
$739$	2.00000	0.0735712	0.0367856	+	0.999323i	$0.488288\pi$
$740$	0	0
$741$	48.0000	1.76332
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	−6.00000	−0.219971
$745$	0	0
$746$	−22.0000	−0.805477
$747$	−14.0000	−0.512233
$748$	24.0000	0.877527
$749$	0	0
$750$	0	0
$751$	−24.0000	−0.875772	−0.437886	−	0.899030i	$-0.644273\pi$
$751$	−24.0000	−0.875772	−0.437886	+	0.899030i	$0.644273\pi$
$752$	2.00000	0.0729325
$753$	12.0000	0.437304
$754$	36.0000	1.31104
$755$	0	0
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−34.0000	−1.23494
$759$	−8.00000	−0.290382
$760$	0	0
$761$	−22.0000	−0.797499	−0.398750	−	0.917060i	$-0.630556\pi$
$761$	−22.0000	−0.797499	−0.398750	+	0.917060i	$0.630556\pi$
$762$	−12.0000	−0.434714
$763$	0	0
$764$	−4.00000	−0.144715
$765$	0	0
$766$	12.0000	0.433578
$767$	−72.0000	−2.59977
$768$	−17.0000	−0.613435
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−22.0000	−0.792311
$772$	12.0000	0.431889
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−12.0000	−0.431331
$775$	0	0
$776$	−6.00000	−0.215387
$777$	0	0
$778$	6.00000	0.215110
$779$	96.0000	3.43956
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−12.0000	−0.429119
$783$	−6.00000	−0.214423
$784$	7.00000	0.250000
$785$	0	0
$786$	−4.00000	−0.142675
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−10.0000	−0.356235
$789$	−18.0000	−0.640817
$790$	0	0
$791$	0	0
$792$	−12.0000	−0.426401
$793$	−12.0000	−0.426132
$794$	−36.0000	−1.27759
$795$	0	0
$796$	−20.0000	−0.708881
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	−12.0000	−0.424529
$800$	0	0
$801$	14.0000	0.494666
$802$	−8.00000	−0.282490
$803$	32.0000	1.12926
$804$	−1.00000	−0.0352673
$805$	0	0
$806$	12.0000	0.422682
$807$	−6.00000	−0.211210
$808$	0	0
$809$	−16.0000	−0.562530	−0.281265	−	0.959630i	$-0.590754\pi$
$809$	−16.0000	−0.562530	−0.281265	+	0.959630i	$0.590754\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	0	0
$813$	14.0000	0.491001
$814$	0	0
$815$	0	0
$816$	−6.00000	−0.210042
$817$	96.0000	3.35861
$818$	−30.0000	−1.04893
$819$	0	0
$820$	0	0
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	−6.00000	−0.209274
$823$	8.00000	0.278862	0.139431	−	0.990232i	$-0.455473\pi$
$823$	8.00000	0.278862	0.139431	+	0.990232i	$0.455473\pi$
$824$	24.0000	0.836080
$825$	0	0
$826$	0	0
$827$	−6.00000	−0.208640	−0.104320	−	0.994544i	$-0.533267\pi$
$827$	−6.00000	−0.208640	−0.104320	+	0.994544i	$0.533267\pi$
$828$	−2.00000	−0.0695048
$829$	18.0000	0.625166	0.312583	−	0.949890i	$-0.398806\pi$
$829$	18.0000	0.625166	0.312583	+	0.949890i	$0.398806\pi$
$830$	0	0
$831$	−8.00000	−0.277517
$832$	42.0000	1.45609
$833$	−42.0000	−1.45521
$834$	2.00000	0.0692543
$835$	0	0
$836$	32.0000	1.10674
$837$	−2.00000	−0.0691301
$838$	12.0000	0.414533
$839$	8.00000	0.276191	0.138095	−	0.990419i	$-0.455902\pi$
$839$	8.00000	0.276191	0.138095	+	0.990419i	$0.455902\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	2.00000	0.0689246
$843$	−24.0000	−0.826604
$844$	4.00000	0.137686
$845$	0	0
$846$	2.00000	0.0687614
$847$	0	0
$848$	10.0000	0.343401
$849$	16.0000	0.549119
$850$	0	0
$851$	0	0
$852$	−8.00000	−0.274075
$853$	44.0000	1.50653	0.753266	−	0.657716i	$-0.228477\pi$
$853$	44.0000	1.50653	0.753266	+	0.657716i	$0.228477\pi$
$854$	0	0
$855$	0	0
$856$	−18.0000	−0.615227
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	24.0000	0.819346
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	0	0
$861$	0	0
$862$	−8.00000	−0.272481
$863$	−6.00000	−0.204242	−0.102121	−	0.994772i	$-0.532563\pi$
$863$	−6.00000	−0.204242	−0.102121	+	0.994772i	$0.532563\pi$
$864$	−5.00000	−0.170103
$865$	0	0
$866$	−2.00000	−0.0679628
$867$	19.0000	0.645274
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	6.00000	0.203302
$872$	−30.0000	−1.01593
$873$	−2.00000	−0.0676897
$874$	−16.0000	−0.541208
$875$	0	0
$876$	8.00000	0.270295
$877$	4.00000	0.135070	0.0675352	−	0.997717i	$-0.478487\pi$
$877$	4.00000	0.135070	0.0675352	+	0.997717i	$0.478487\pi$
$878$	−4.00000	−0.134993
$879$	−30.0000	−1.01187
$880$	0	0
$881$	46.0000	1.54978	0.774890	−	0.632096i	$-0.217805\pi$
$881$	46.0000	1.54978	0.774890	+	0.632096i	$0.217805\pi$
$882$	7.00000	0.235702
$883$	−20.0000	−0.673054	−0.336527	−	0.941674i	$-0.609252\pi$
$883$	−20.0000	−0.673054	−0.336527	+	0.941674i	$0.609252\pi$
$884$	−36.0000	−1.21081
$885$	0	0
$886$	32.0000	1.07506
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−4.00000	−0.134005
$892$	−12.0000	−0.401790
$893$	−16.0000	−0.535420
$894$	−18.0000	−0.602010
$895$	0	0
$896$	0	0
$897$	12.0000	0.400668
$898$	22.0000	0.734150
$899$	12.0000	0.400222
$900$	0	0
$901$	−60.0000	−1.99889
$902$	48.0000	1.59823
$903$	0	0
$904$	18.0000	0.598671
$905$	0	0
$906$	−12.0000	−0.398673
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	−8.00000	−0.264906
$913$	56.0000	1.85333
$914$	−16.0000	−0.529233
$915$	0	0
$916$	30.0000	0.991228
$917$	0	0
$918$	−6.00000	−0.198030
$919$	30.0000	0.989609	0.494804	−	0.869004i	$-0.335240\pi$
$919$	30.0000	0.989609	0.494804	+	0.869004i	$0.335240\pi$
$920$	0	0
$921$	8.00000	0.263609
$922$	30.0000	0.987997
$923$	48.0000	1.57994
$924$	0	0
$925$	0	0
$926$	32.0000	1.05159
$927$	8.00000	0.262754
$928$	30.0000	0.984798
$929$	−28.0000	−0.918650	−0.459325	−	0.888268i	$-0.651909\pi$
$929$	−28.0000	−0.918650	−0.459325	+	0.888268i	$0.651909\pi$
$930$	0	0
$931$	−56.0000	−1.83533
$932$	−6.00000	−0.196537
$933$	24.0000	0.785725
$934$	2.00000	0.0654420
$935$	0	0
$936$	18.0000	0.588348
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−16.0000	−0.521585	−0.260793	−	0.965395i	$-0.583984\pi$
$941$	−16.0000	−0.521585	−0.260793	+	0.965395i	$0.583984\pi$
$942$	4.00000	0.130327
$943$	24.0000	0.781548
$944$	12.0000	0.390567
$945$	0	0
$946$	48.0000	1.56061
$947$	−18.0000	−0.584921	−0.292461	−	0.956278i	$-0.594474\pi$
$947$	−18.0000	−0.584921	−0.292461	+	0.956278i	$0.594474\pi$
$948$	−2.00000	−0.0649570
$949$	−48.0000	−1.55815
$950$	0	0
$951$	6.00000	0.194563
$952$	0	0
$953$	−30.0000	−0.971795	−0.485898	−	0.874016i	$-0.661507\pi$
$953$	−30.0000	−0.971795	−0.485898	+	0.874016i	$0.661507\pi$
$954$	10.0000	0.323762
$955$	0	0
$956$	−16.0000	−0.517477
$957$	24.0000	0.775810
$958$	−8.00000	−0.258468
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	0	0
$963$	−6.00000	−0.193347
$964$	10.0000	0.322078
$965$	0	0
$966$	0	0
$967$	−24.0000	−0.771788	−0.385894	−	0.922543i	$-0.626107\pi$
$967$	−24.0000	−0.771788	−0.385894	+	0.922543i	$0.626107\pi$
$968$	15.0000	0.482118
$969$	48.0000	1.54198
$970$	0	0
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−8.00000	−0.256337
$975$	0	0
$976$	2.00000	0.0640184
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−4.00000	−0.127906
$979$	−56.0000	−1.78977
$980$	0	0
$981$	−10.0000	−0.319275
$982$	20.0000	0.638226
$983$	−60.0000	−1.91370	−0.956851	−	0.290578i	$-0.906153\pi$
$983$	−60.0000	−1.91370	−0.956851	+	0.290578i	$0.906153\pi$
$984$	36.0000	1.14764
$985$	0	0
$986$	36.0000	1.14647
$987$	0	0
$988$	−48.0000	−1.52708
$989$	24.0000	0.763156
$990$	0	0
$991$	−10.0000	−0.317660	−0.158830	−	0.987306i	$-0.550772\pi$
$991$	−10.0000	−0.317660	−0.158830	+	0.987306i	$0.550772\pi$
$992$	10.0000	0.317500
$993$	−2.00000	−0.0634681
$994$	0	0
$995$	0	0
$996$	14.0000	0.443607
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	30.0000	0.949633
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5025.2.a.a.1.1		1
5.2	odd	4		1005.2.c.a.604.1	✓	2
5.3	odd	4		1005.2.c.a.604.2	yes	2
5.4	even	2		5025.2.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1005.2.c.a.604.1	✓	2	5.2	odd	4
1005.2.c.a.604.2	yes	2	5.3	odd	4
5025.2.a.a.1.1		1	1.1	even	1	trivial
5025.2.a.g.1.1		1	5.4	even	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:	\( N \)	\(=\)	\( 5025 = 3 \cdot 5^{2} \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5025.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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