LMFDB - Embedded newform 1170.2.a.i.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1170,2,Mod(1,1170)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1170, 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("1170.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1170.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:	$9.34249703649$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 130)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1170.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +6.00000 q^{11} +1.00000 q^{13} -4.00000 q^{14} +1.00000 q^{16} +6.00000 q^{17} +2.00000 q^{19} -1.00000 q^{20} +6.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -4.00000 q^{28} +6.00000 q^{29} +2.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} +4.00000 q^{35} +2.00000 q^{37} +2.00000 q^{38} -1.00000 q^{40} +6.00000 q^{41} +2.00000 q^{43} +6.00000 q^{44} -6.00000 q^{46} +12.0000 q^{47} +9.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} -6.00000 q^{55} -4.00000 q^{56} +6.00000 q^{58} -6.00000 q^{59} +2.00000 q^{61} +2.00000 q^{62} +1.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} +6.00000 q^{68} +4.00000 q^{70} +6.00000 q^{71} -10.0000 q^{73} +2.00000 q^{74} +2.00000 q^{76} -24.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} +6.00000 q^{82} -6.00000 q^{85} +2.00000 q^{86} +6.00000 q^{88} +6.00000 q^{89} -4.00000 q^{91} -6.00000 q^{92} +12.0000 q^{94} -2.00000 q^{95} +2.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

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
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−1.00000	−0.316228
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.00000	−1.06904
$15$	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	$0.240633\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	6.00000	1.27920
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	0	0
$28$	−4.00000	−0.755929
$29$	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
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	4.00000	0.676123
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−1.00000	−0.158114
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−6.00000	−0.884652
$47$	12.0000	1.75038	0.875190	−	0.483779i	$-0.160736\pi$
$47$	12.0000	1.75038	0.875190	+	0.483779i	$0.160736\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	0	0
$52$	1.00000	0.138675
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	−6.00000	−0.809040
$56$	−4.00000	−0.534522
$57$	0	0
$58$	6.00000	0.787839
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	2.00000	0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	4.00000	0.478091
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	2.00000	0.232495
$75$	0	0
$76$	2.00000	0.229416
$77$	−24.0000	−2.73505
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	6.00000	0.662589
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	2.00000	0.215666
$87$	0	0
$88$	6.00000	0.639602
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−4.00000	−0.419314
$92$	−6.00000	−0.625543
$93$	0	0
$94$	12.0000	1.23771
$95$	−2.00000	−0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	1.00000	0.0980581
$105$	0	0
$106$	−6.00000	−0.582772
$107$	−18.0000	−1.74013	−0.870063	−	0.492941i	$-0.835922\pi$
$107$	−18.0000	−1.74013	−0.870063	+	0.492941i	$0.835922\pi$
$108$	0	0
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	−6.00000	−0.572078
$111$	0	0
$112$	−4.00000	−0.377964
$113$	6.00000	0.564433	0.282216	−	0.959351i	$-0.408930\pi$
$113$	6.00000	0.564433	0.282216	+	0.959351i	$0.408930\pi$
$114$	0	0
$115$	6.00000	0.559503
$116$	6.00000	0.557086
$117$	0	0
$118$	−6.00000	−0.552345
$119$	−24.0000	−2.20008
$120$	0	0
$121$	25.0000	2.27273
$122$	2.00000	0.181071
$123$	0	0
$124$	2.00000	0.179605
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.00000	0.177471	0.0887357	−	0.996055i	$-0.471717\pi$
$127$	2.00000	0.177471	0.0887357	+	0.996055i	$0.471717\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−1.00000	−0.0877058
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	−4.00000	−0.345547
$135$	0	0
$136$	6.00000	0.514496
$137$	6.00000	0.512615	0.256307	−	0.966595i	$-0.417494\pi$
$137$	6.00000	0.512615	0.256307	+	0.966595i	$0.417494\pi$
$138$	0	0
$139$	−16.0000	−1.35710	−0.678551	−	0.734553i	$-0.737392\pi$
$139$	−16.0000	−1.35710	−0.678551	+	0.734553i	$0.737392\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	6.00000	0.503509
$143$	6.00000	0.501745
$144$	0	0
$145$	−6.00000	−0.498273
$146$	−10.0000	−0.827606
$147$	0	0
$148$	2.00000	0.164399
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−10.0000	−0.813788	−0.406894	−	0.913475i	$-0.633388\pi$
$151$	−10.0000	−0.813788	−0.406894	+	0.913475i	$0.633388\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	−24.0000	−1.93398
$155$	−2.00000	−0.160644
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	24.0000	1.89146
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−6.00000	−0.460179
$171$	0	0
$172$	2.00000	0.152499
$173$	18.0000	1.36851	0.684257	−	0.729241i	$-0.260127\pi$
$173$	18.0000	1.36851	0.684257	+	0.729241i	$0.260127\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	6.00000	0.452267
$177$	0	0
$178$	6.00000	0.449719
$179$	−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
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	−6.00000	−0.442326
$185$	−2.00000	−0.147043
$186$	0	0
$187$	36.0000	2.63258
$188$	12.0000	0.875190
$189$	0	0
$190$	−2.00000	−0.145095
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	2.00000	0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−6.00000	−0.422159
$203$	−24.0000	−1.68447
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−10.0000	−0.696733
$207$	0	0
$208$	1.00000	0.0693375
$209$	12.0000	0.830057
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−2.00000	−0.136399
$216$	0	0
$217$	−8.00000	−0.543075
$218$	−10.0000	−0.677285
$219$	0	0
$220$	−6.00000	−0.404520
$221$	6.00000	0.403604
$222$	0	0
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	6.00000	0.399114
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	6.00000	0.393919
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	−6.00000	−0.390567
$237$	0	0
$238$	−24.0000	−1.55569
$239$	−30.0000	−1.94054	−0.970269	−	0.242028i	$-0.922188\pi$
$239$	−30.0000	−1.94054	−0.970269	+	0.242028i	$0.922188\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	25.0000	1.60706
$243$	0	0
$244$	2.00000	0.128037
$245$	−9.00000	−0.574989
$246$	0	0
$247$	2.00000	0.127257
$248$	2.00000	0.127000
$249$	0	0
$250$	−1.00000	−0.0632456
$251$	−24.0000	−1.51487	−0.757433	−	0.652913i	$-0.773547\pi$
$251$	−24.0000	−1.51487	−0.757433	+	0.652913i	$0.773547\pi$
$252$	0	0
$253$	−36.0000	−2.26330
$254$	2.00000	0.125491
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−8.00000	−0.497096
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	12.0000	0.741362
$263$	−18.0000	−1.10993	−0.554964	−	0.831875i	$-0.687268\pi$
$263$	−18.0000	−1.10993	−0.554964	+	0.831875i	$0.687268\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	−8.00000	−0.490511
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	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
$275$	6.00000	0.361814
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−16.0000	−0.959616
$279$	0	0
$280$	4.00000	0.239046
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	0	0
$283$	2.00000	0.118888	0.0594438	−	0.998232i	$-0.481067\pi$
$283$	2.00000	0.118888	0.0594438	+	0.998232i	$0.481067\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	6.00000	0.354787
$287$	−24.0000	−1.41668
$288$	0	0
$289$	19.0000	1.11765
$290$	−6.00000	−0.352332
$291$	0	0
$292$	−10.0000	−0.585206
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	6.00000	0.349334
$296$	2.00000	0.116248
$297$	0	0
$298$	−6.00000	−0.347571
$299$	−6.00000	−0.346989
$300$	0	0
$301$	−8.00000	−0.461112
$302$	−10.0000	−0.575435
$303$	0	0
$304$	2.00000	0.114708
$305$	−2.00000	−0.114520
$306$	0	0
$307$	8.00000	0.456584	0.228292	−	0.973593i	$-0.426686\pi$
$307$	8.00000	0.456584	0.228292	+	0.973593i	$0.426686\pi$
$308$	−24.0000	−1.36753
$309$	0	0
$310$	−2.00000	−0.113592
$311$	12.0000	0.680458	0.340229	−	0.940343i	$-0.389495\pi$
$311$	12.0000	0.680458	0.340229	+	0.940343i	$0.389495\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−10.0000	−0.564333
$315$	0	0
$316$	−4.00000	−0.225018
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	0	0
$319$	36.0000	2.01561
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	24.0000	1.33747
$323$	12.0000	0.667698
$324$	0	0
$325$	1.00000	0.0554700
$326$	20.0000	1.10770
$327$	0	0
$328$	6.00000	0.331295
$329$	−48.0000	−2.64633
$330$	0	0
$331$	14.0000	0.769510	0.384755	−	0.923019i	$-0.374286\pi$
$331$	14.0000	0.769510	0.384755	+	0.923019i	$0.374286\pi$
$332$	0	0
$333$	0	0
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	0	0
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	1.00000	0.0543928
$339$	0	0
$340$	−6.00000	−0.325396
$341$	12.0000	0.649836
$342$	0	0
$343$	−8.00000	−0.431959
$344$	2.00000	0.107833
$345$	0	0
$346$	18.0000	0.967686
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	14.0000	0.749403	0.374701	−	0.927146i	$-0.377745\pi$
$349$	14.0000	0.749403	0.374701	+	0.927146i	$0.377745\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	6.00000	0.319801
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	0	0
$364$	−4.00000	−0.209657
$365$	10.0000	0.523424
$366$	0	0
$367$	−10.0000	−0.521996	−0.260998	−	0.965339i	$-0.584052\pi$
$367$	−10.0000	−0.521996	−0.260998	+	0.965339i	$0.584052\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	−2.00000	−0.103975
$371$	24.0000	1.24602
$372$	0	0
$373$	−34.0000	−1.76045	−0.880227	−	0.474554i	$-0.842610\pi$
$373$	−34.0000	−1.76045	−0.880227	+	0.474554i	$0.842610\pi$
$374$	36.0000	1.86152
$375$	0	0
$376$	12.0000	0.618853
$377$	6.00000	0.309016
$378$	0	0
$379$	2.00000	0.102733	0.0513665	−	0.998680i	$-0.483642\pi$
$379$	2.00000	0.102733	0.0513665	+	0.998680i	$0.483642\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	0	0
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	0	0
$385$	24.0000	1.22315
$386$	2.00000	0.101797
$387$	0	0
$388$	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
$391$	−36.0000	−1.82060
$392$	9.00000	0.454569
$393$	0	0
$394$	−6.00000	−0.302276
$395$	4.00000	0.201262
$396$	0	0
$397$	−22.0000	−1.10415	−0.552074	−	0.833795i	$-0.686163\pi$
$397$	−22.0000	−1.10415	−0.552074	+	0.833795i	$0.686163\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	2.00000	0.0996271
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−24.0000	−1.19110
$407$	12.0000	0.594818
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−6.00000	−0.296319
$411$	0	0
$412$	−10.0000	−0.492665
$413$	24.0000	1.18096
$414$	0	0
$415$	0	0
$416$	1.00000	0.0490290
$417$	0	0
$418$	12.0000	0.586939
$419$	24.0000	1.17248	0.586238	−	0.810139i	$-0.300608\pi$
$419$	24.0000	1.17248	0.586238	+	0.810139i	$0.300608\pi$
$420$	0	0
$421$	38.0000	1.85201	0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000	1.85201	0.926003	+	0.377515i	$0.123221\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	6.00000	0.291043
$426$	0	0
$427$	−8.00000	−0.387147
$428$	−18.0000	−0.870063
$429$	0	0
$430$	−2.00000	−0.0964486
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−10.0000	−0.478913
$437$	−12.0000	−0.574038
$438$	0	0
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	−6.00000	−0.286039
$441$	0	0
$442$	6.00000	0.285391
$443$	−6.00000	−0.285069	−0.142534	−	0.989790i	$-0.545525\pi$
$443$	−6.00000	−0.285069	−0.142534	+	0.989790i	$0.545525\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	−28.0000	−1.32584
$447$	0	0
$448$	−4.00000	−0.188982
$449$	30.0000	1.41579	0.707894	−	0.706319i	$-0.249646\pi$
$449$	30.0000	1.41579	0.707894	+	0.706319i	$0.249646\pi$
$450$	0	0
$451$	36.0000	1.69517
$452$	6.00000	0.282216
$453$	0	0
$454$	12.0000	0.563188
$455$	4.00000	0.187523
$456$	0	0
$457$	26.0000	1.21623	0.608114	−	0.793849i	$-0.291926\pi$
$457$	26.0000	1.21623	0.608114	+	0.793849i	$0.291926\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	6.00000	0.279751
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	−18.0000	−0.833834
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	−12.0000	−0.553519
$471$	0	0
$472$	−6.00000	−0.276172
$473$	12.0000	0.551761
$474$	0	0
$475$	2.00000	0.0917663
$476$	−24.0000	−1.10004
$477$	0	0
$478$	−30.0000	−1.37217
$479$	6.00000	0.274147	0.137073	−	0.990561i	$-0.456230\pi$
$479$	6.00000	0.274147	0.137073	+	0.990561i	$0.456230\pi$
$480$	0	0
$481$	2.00000	0.0911922
$482$	26.0000	1.18427
$483$	0	0
$484$	25.0000	1.13636
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	−16.0000	−0.725029	−0.362515	−	0.931978i	$-0.618082\pi$
$487$	−16.0000	−0.725029	−0.362515	+	0.931978i	$0.618082\pi$
$488$	2.00000	0.0905357
$489$	0	0
$490$	−9.00000	−0.406579
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	36.0000	1.62136
$494$	2.00000	0.0899843
$495$	0	0
$496$	2.00000	0.0898027
$497$	−24.0000	−1.07655
$498$	0	0
$499$	−22.0000	−0.984855	−0.492428	−	0.870353i	$-0.663890\pi$
$499$	−22.0000	−0.984855	−0.492428	+	0.870353i	$0.663890\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−24.0000	−1.07117
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−36.0000	−1.60040
$507$	0	0
$508$	2.00000	0.0887357
$509$	−30.0000	−1.32973	−0.664863	−	0.746965i	$-0.731510\pi$
$509$	−30.0000	−1.32973	−0.664863	+	0.746965i	$0.731510\pi$
$510$	0	0
$511$	40.0000	1.76950
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.00000	0.264649
$515$	10.0000	0.440653
$516$	0	0
$517$	72.0000	3.16656
$518$	−8.00000	−0.351500
$519$	0	0
$520$	−1.00000	−0.0438529
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	12.0000	0.524222
$525$	0	0
$526$	−18.0000	−0.784837
$527$	12.0000	0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	6.00000	0.260623
$531$	0	0
$532$	−8.00000	−0.346844
$533$	6.00000	0.259889
$534$	0	0
$535$	18.0000	0.778208
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	54.0000	2.32594
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	14.0000	0.601351
$543$	0	0
$544$	6.00000	0.257248
$545$	10.0000	0.428353
$546$	0	0
$547$	−22.0000	−0.940652	−0.470326	−	0.882493i	$-0.655864\pi$
$547$	−22.0000	−0.940652	−0.470326	+	0.882493i	$0.655864\pi$
$548$	6.00000	0.256307
$549$	0	0
$550$	6.00000	0.255841
$551$	12.0000	0.511217
$552$	0	0
$553$	16.0000	0.680389
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−16.0000	−0.678551
$557$	6.00000	0.254228	0.127114	−	0.991888i	$-0.459429\pi$
$557$	6.00000	0.254228	0.127114	+	0.991888i	$0.459429\pi$
$558$	0	0
$559$	2.00000	0.0845910
$560$	4.00000	0.169031
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	0	0
$565$	−6.00000	−0.252422
$566$	2.00000	0.0840663
$567$	0	0
$568$	6.00000	0.251754
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	6.00000	0.250873
$573$	0	0
$574$	−24.0000	−1.00174
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	19.0000	0.790296
$579$	0	0
$580$	−6.00000	−0.249136
$581$	0	0
$582$	0	0
$583$	−36.0000	−1.49097
$584$	−10.0000	−0.413803
$585$	0	0
$586$	6.00000	0.247858
$587$	−36.0000	−1.48588	−0.742940	−	0.669359i	$-0.766569\pi$
$587$	−36.0000	−1.48588	−0.742940	+	0.669359i	$0.766569\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	6.00000	0.247016
$591$	0	0
$592$	2.00000	0.0821995
$593$	30.0000	1.23195	0.615976	−	0.787765i	$-0.288762\pi$
$593$	30.0000	1.23195	0.615976	+	0.787765i	$0.288762\pi$
$594$	0	0
$595$	24.0000	0.983904
$596$	−6.00000	−0.245770
$597$	0	0
$598$	−6.00000	−0.245358
$599$	12.0000	0.490307	0.245153	−	0.969484i	$-0.421162\pi$
$599$	12.0000	0.490307	0.245153	+	0.969484i	$0.421162\pi$
$600$	0	0
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−8.00000	−0.326056
$603$	0	0
$604$	−10.0000	−0.406894
$605$	−25.0000	−1.01639
$606$	0	0
$607$	14.0000	0.568242	0.284121	−	0.958788i	$-0.408298\pi$
$607$	14.0000	0.568242	0.284121	+	0.958788i	$0.408298\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	12.0000	0.485468
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	8.00000	0.322854
$615$	0	0
$616$	−24.0000	−0.966988
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	0	0
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	−2.00000	−0.0803219
$621$	0	0
$622$	12.0000	0.481156
$623$	−24.0000	−0.961540
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−10.0000	−0.399043
$629$	12.0000	0.478471
$630$	0	0
$631$	−34.0000	−1.35352	−0.676759	−	0.736204i	$-0.736616\pi$
$631$	−34.0000	−1.35352	−0.676759	+	0.736204i	$0.736616\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	30.0000	1.19145
$635$	−2.00000	−0.0793676
$636$	0	0
$637$	9.00000	0.356593
$638$	36.0000	1.42525
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	30.0000	1.18493	0.592464	−	0.805597i	$-0.298155\pi$
$641$	30.0000	1.18493	0.592464	+	0.805597i	$0.298155\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	24.0000	0.945732
$645$	0	0
$646$	12.0000	0.472134
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	0	0
$649$	−36.0000	−1.41312
$650$	1.00000	0.0392232
$651$	0	0
$652$	20.0000	0.783260
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	0	0
$655$	−12.0000	−0.468879
$656$	6.00000	0.234261
$657$	0	0
$658$	−48.0000	−1.87123
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	14.0000	0.544125
$663$	0	0
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−36.0000	−1.39393
$668$	12.0000	0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	12.0000	0.463255
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	−22.0000	−0.847408
$675$	0	0
$676$	1.00000	0.0384615
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−8.00000	−0.307012
$680$	−6.00000	−0.230089
$681$	0	0
$682$	12.0000	0.459504
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−6.00000	−0.229248
$686$	−8.00000	−0.305441
$687$	0	0
$688$	2.00000	0.0762493
$689$	−6.00000	−0.228582
$690$	0	0
$691$	−10.0000	−0.380418	−0.190209	−	0.981744i	$-0.560917\pi$
$691$	−10.0000	−0.380418	−0.190209	+	0.981744i	$0.560917\pi$
$692$	18.0000	0.684257
$693$	0	0
$694$	−18.0000	−0.683271
$695$	16.0000	0.606915
$696$	0	0
$697$	36.0000	1.36360
$698$	14.0000	0.529908
$699$	0	0
$700$	−4.00000	−0.151186
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	4.00000	0.150863
$704$	6.00000	0.226134
$705$	0	0
$706$	−6.00000	−0.225813
$707$	24.0000	0.902613
$708$	0	0
$709$	14.0000	0.525781	0.262891	−	0.964826i	$-0.415324\pi$
$709$	14.0000	0.525781	0.262891	+	0.964826i	$0.415324\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	6.00000	0.224860
$713$	−12.0000	−0.449404
$714$	0	0
$715$	−6.00000	−0.224387
$716$	−12.0000	−0.448461
$717$	0	0
$718$	30.0000	1.11959
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	40.0000	1.48968
$722$	−15.0000	−0.558242
$723$	0	0
$724$	2.00000	0.0743294
$725$	6.00000	0.222834
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	−4.00000	−0.148250
$729$	0	0
$730$	10.0000	0.370117
$731$	12.0000	0.443836
$732$	0	0
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−10.0000	−0.369107
$735$	0	0
$736$	−6.00000	−0.221163
$737$	−24.0000	−0.884051
$738$	0	0
$739$	−34.0000	−1.25071	−0.625355	−	0.780340i	$-0.715046\pi$
$739$	−34.0000	−1.25071	−0.625355	+	0.780340i	$0.715046\pi$
$740$	−2.00000	−0.0735215
$741$	0	0
$742$	24.0000	0.881068
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	6.00000	0.219823
$746$	−34.0000	−1.24483
$747$	0	0
$748$	36.0000	1.31629
$749$	72.0000	2.63082
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	12.0000	0.437595
$753$	0	0
$754$	6.00000	0.218507
$755$	10.0000	0.363937
$756$	0	0
$757$	38.0000	1.38113	0.690567	−	0.723269i	$-0.257361\pi$
$757$	38.0000	1.38113	0.690567	+	0.723269i	$0.257361\pi$
$758$	2.00000	0.0726433
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	54.0000	1.95750	0.978749	−	0.205061i	$-0.0657392\pi$
$761$	54.0000	1.95750	0.978749	+	0.205061i	$0.0657392\pi$
$762$	0	0
$763$	40.0000	1.44810
$764$	0	0
$765$	0	0
$766$	−12.0000	−0.433578
$767$	−6.00000	−0.216647
$768$	0	0
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	24.0000	0.864900
$771$	0	0
$772$	2.00000	0.0719816
$773$	−18.0000	−0.647415	−0.323708	−	0.946157i	$-0.604929\pi$
$773$	−18.0000	−0.647415	−0.323708	+	0.946157i	$0.604929\pi$
$774$	0	0
$775$	2.00000	0.0718421
$776$	2.00000	0.0717958
$777$	0	0
$778$	−6.00000	−0.215110
$779$	12.0000	0.429945
$780$	0	0
$781$	36.0000	1.28818
$782$	−36.0000	−1.28736
$783$	0	0
$784$	9.00000	0.321429
$785$	10.0000	0.356915
$786$	0	0
$787$	8.00000	0.285169	0.142585	−	0.989783i	$-0.454459\pi$
$787$	8.00000	0.285169	0.142585	+	0.989783i	$0.454459\pi$
$788$	−6.00000	−0.213741
$789$	0	0
$790$	4.00000	0.142314
$791$	−24.0000	−0.853342
$792$	0	0
$793$	2.00000	0.0710221
$794$	−22.0000	−0.780751
$795$	0	0
$796$	−16.0000	−0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	72.0000	2.54718
$800$	1.00000	0.0353553
$801$	0	0
$802$	−18.0000	−0.635602
$803$	−60.0000	−2.11735
$804$	0	0
$805$	−24.0000	−0.845889
$806$	2.00000	0.0704470
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	0	0
$811$	14.0000	0.491606	0.245803	−	0.969320i	$-0.420948\pi$
$811$	14.0000	0.491606	0.245803	+	0.969320i	$0.420948\pi$
$812$	−24.0000	−0.842235
$813$	0	0
$814$	12.0000	0.420600
$815$	−20.0000	−0.700569
$816$	0	0
$817$	4.00000	0.139942
$818$	−22.0000	−0.769212
$819$	0	0
$820$	−6.00000	−0.209529
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	14.0000	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$823$	14.0000	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	24.0000	0.835067
$827$	−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
$829$	−46.0000	−1.59765	−0.798823	−	0.601566i	$-0.794544\pi$
$829$	−46.0000	−1.59765	−0.798823	+	0.601566i	$0.794544\pi$
$830$	0	0
$831$	0	0
$832$	1.00000	0.0346688
$833$	54.0000	1.87099
$834$	0	0
$835$	−12.0000	−0.415277
$836$	12.0000	0.415029
$837$	0	0
$838$	24.0000	0.829066
$839$	6.00000	0.207143	0.103572	−	0.994622i	$-0.466973\pi$
$839$	6.00000	0.207143	0.103572	+	0.994622i	$0.466973\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	38.0000	1.30957
$843$	0	0
$844$	−4.00000	−0.137686
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−100.000	−3.43604
$848$	−6.00000	−0.206041
$849$	0	0
$850$	6.00000	0.205798
$851$	−12.0000	−0.411355
$852$	0	0
$853$	26.0000	0.890223	0.445112	−	0.895475i	$-0.353164\pi$
$853$	26.0000	0.890223	0.445112	+	0.895475i	$0.353164\pi$
$854$	−8.00000	−0.273754
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	8.00000	0.272956	0.136478	−	0.990643i	$-0.456422\pi$
$859$	8.00000	0.272956	0.136478	+	0.990643i	$0.456422\pi$
$860$	−2.00000	−0.0681994
$861$	0	0
$862$	−18.0000	−0.613082
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	0	0
$865$	−18.0000	−0.612018
$866$	26.0000	0.883516
$867$	0	0
$868$	−8.00000	−0.271538
$869$	−24.0000	−0.814144
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−10.0000	−0.338643
$873$	0	0
$874$	−12.0000	−0.405906
$875$	4.00000	0.135225
$876$	0	0
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	−16.0000	−0.539974
$879$	0	0
$880$	−6.00000	−0.202260
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	0	0
$883$	50.0000	1.68263	0.841317	−	0.540542i	$-0.181781\pi$
$883$	50.0000	1.68263	0.841317	+	0.540542i	$0.181781\pi$
$884$	6.00000	0.201802
$885$	0	0
$886$	−6.00000	−0.201574
$887$	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
$889$	−8.00000	−0.268311
$890$	−6.00000	−0.201120
$891$	0	0
$892$	−28.0000	−0.937509
$893$	24.0000	0.803129
$894$	0	0
$895$	12.0000	0.401116
$896$	−4.00000	−0.133631
$897$	0	0
$898$	30.0000	1.00111
$899$	12.0000	0.400222
$900$	0	0
$901$	−36.0000	−1.19933
$902$	36.0000	1.19867
$903$	0	0
$904$	6.00000	0.199557
$905$	−2.00000	−0.0664822
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	12.0000	0.398234
$909$	0	0
$910$	4.00000	0.132599
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	0	0
$914$	26.0000	0.860004
$915$	0	0
$916$	14.0000	0.462573
$917$	−48.0000	−1.58510
$918$	0	0
$919$	20.0000	0.659739	0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000	0.659739	0.329870	+	0.944027i	$0.392995\pi$
$920$	6.00000	0.197814
$921$	0	0
$922$	−6.00000	−0.197599
$923$	6.00000	0.197492
$924$	0	0
$925$	2.00000	0.0657596
$926$	−16.0000	−0.525793
$927$	0	0
$928$	6.00000	0.196960
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	18.0000	0.589926
$932$	−18.0000	−0.589610
$933$	0	0
$934$	18.0000	0.588978
$935$	−36.0000	−1.17733
$936$	0	0
$937$	2.00000	0.0653372	0.0326686	−	0.999466i	$-0.489599\pi$
$937$	2.00000	0.0653372	0.0326686	+	0.999466i	$0.489599\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	−12.0000	−0.391397
$941$	−6.00000	−0.195594	−0.0977972	−	0.995206i	$-0.531180\pi$
$941$	−6.00000	−0.195594	−0.0977972	+	0.995206i	$0.531180\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	−6.00000	−0.195283
$945$	0	0
$946$	12.0000	0.390154
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	2.00000	0.0648886
$951$	0	0
$952$	−24.0000	−0.777844
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	0	0
$955$	0	0
$956$	−30.0000	−0.970269
$957$	0	0
$958$	6.00000	0.193851
$959$	−24.0000	−0.775000
$960$	0	0
$961$	−27.0000	−0.870968
$962$	2.00000	0.0644826
$963$	0	0
$964$	26.0000	0.837404
$965$	−2.00000	−0.0643823
$966$	0	0
$967$	32.0000	1.02905	0.514525	−	0.857475i	$-0.327968\pi$
$967$	32.0000	1.02905	0.514525	+	0.857475i	$0.327968\pi$
$968$	25.0000	0.803530
$969$	0	0
$970$	−2.00000	−0.0642161
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	64.0000	2.05175
$974$	−16.0000	−0.512673
$975$	0	0
$976$	2.00000	0.0640184
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	0	0
$979$	36.0000	1.15056
$980$	−9.00000	−0.287494
$981$	0	0
$982$	24.0000	0.765871
$983$	48.0000	1.53096	0.765481	−	0.643458i	$-0.222501\pi$
$983$	48.0000	1.53096	0.765481	+	0.643458i	$0.222501\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	36.0000	1.14647
$987$	0	0
$988$	2.00000	0.0636285
$989$	−12.0000	−0.381578
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	2.00000	0.0635001
$993$	0	0
$994$	−24.0000	−0.761234
$995$	16.0000	0.507234
$996$	0	0
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−22.0000	−0.696398
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1170.2.a.i.1.1		1
3.2	odd	2		130.2.a.a.1.1	✓	1
4.3	odd	2		9360.2.a.z.1.1		1
5.2	odd	4		5850.2.e.bg.5149.2		2
5.3	odd	4		5850.2.e.bg.5149.1		2
5.4	even	2		5850.2.a.ba.1.1		1
12.11	even	2		1040.2.a.g.1.1		1
15.2	even	4		650.2.b.f.599.1		2
15.8	even	4		650.2.b.f.599.2		2
15.14	odd	2		650.2.a.l.1.1		1
21.20	even	2		6370.2.a.h.1.1		1
24.5	odd	2		4160.2.a.o.1.1		1
24.11	even	2		4160.2.a.b.1.1		1
39.2	even	12		1690.2.l.h.1161.1		4
39.5	even	4		1690.2.d.b.1351.2		2
39.8	even	4		1690.2.d.b.1351.1		2
39.11	even	12		1690.2.l.h.1161.2		4
39.17	odd	6		1690.2.e.d.991.1		2
39.20	even	12		1690.2.l.h.361.1		4
39.23	odd	6		1690.2.e.d.191.1		2
39.29	odd	6		1690.2.e.j.191.1		2
39.32	even	12		1690.2.l.h.361.2		4
39.35	odd	6		1690.2.e.j.991.1		2
39.38	odd	2		1690.2.a.f.1.1		1
60.59	even	2		5200.2.a.e.1.1		1
195.194	odd	2		8450.2.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
130.2.a.a.1.1	✓	1	3.2	odd	2
650.2.a.l.1.1		1	15.14	odd	2
650.2.b.f.599.1		2	15.2	even	4
650.2.b.f.599.2		2	15.8	even	4
1040.2.a.g.1.1		1	12.11	even	2
1170.2.a.i.1.1		1	1.1	even	1	trivial
1690.2.a.f.1.1		1	39.38	odd	2
1690.2.d.b.1351.1		2	39.8	even	4
1690.2.d.b.1351.2		2	39.5	even	4
1690.2.e.d.191.1		2	39.23	odd	6
1690.2.e.d.991.1		2	39.17	odd	6
1690.2.e.j.191.1		2	39.29	odd	6
1690.2.e.j.991.1		2	39.35	odd	6
1690.2.l.h.361.1		4	39.20	even	12
1690.2.l.h.361.2		4	39.32	even	12
1690.2.l.h.1161.1		4	39.2	even	12
1690.2.l.h.1161.2		4	39.11	even	12
4160.2.a.b.1.1		1	24.11	even	2
4160.2.a.o.1.1		1	24.5	odd	2
5200.2.a.e.1.1		1	60.59	even	2
5850.2.a.ba.1.1		1	5.4	even	2
5850.2.e.bg.5149.1		2	5.3	odd	4
5850.2.e.bg.5149.2		2	5.2	odd	4
6370.2.a.h.1.1		1	21.20	even	2
8450.2.a.i.1.1		1	195.194	odd	2
9360.2.a.z.1.1		1	4.3	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:	\( N \)	\(=\)	\( 1170 = 2 \cdot 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1170.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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