LMFDB - Embedded newform 102.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [102,2,Mod(1,102)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(102, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("102.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 102 = 2 \cdot 3 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	102.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:	$0.814474100617$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	102.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{12} +2.00000 q^{13} -2.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +2.00000 q^{21} -6.00000 q^{23} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -10.0000 q^{31} -1.00000 q^{32} +1.00000 q^{34} +1.00000 q^{36} +8.00000 q^{37} +4.00000 q^{38} +2.00000 q^{39} +6.00000 q^{41} -2.00000 q^{42} -4.00000 q^{43} +6.00000 q^{46} +12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +5.00000 q^{50} -1.00000 q^{51} +2.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} -2.00000 q^{56} -4.00000 q^{57} -12.0000 q^{59} +8.00000 q^{61} +10.0000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -4.00000 q^{67} -1.00000 q^{68} -6.00000 q^{69} +6.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -8.00000 q^{74} -5.00000 q^{75} -4.00000 q^{76} -2.00000 q^{78} -10.0000 q^{79} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} +4.00000 q^{86} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -10.0000 q^{93} -12.0000 q^{94} -1.00000 q^{96} +14.0000 q^{97} +3.00000 q^{98} +O(q^{100})\)

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	−1.00000	−0.235702
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	0	0
$21$	2.00000	0.436436
$22$	0	0
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	−10.0000	−1.79605	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	−10.0000	−1.79605	−0.898027	+	0.439941i	$0.854999\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	1.00000	0.171499
$35$	0	0
$36$	1.00000	0.166667
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	4.00000	0.648886
$39$	2.00000	0.320256
$40$	0	0
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	−2.00000	−0.308607
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	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	$0.160736\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	5.00000	0.707107
$51$	−1.00000	−0.140028
$52$	2.00000	0.277350
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	−2.00000	−0.267261
$57$	−4.00000	−0.529813
$58$	0	0
$59$	−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
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	10.0000	1.27000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−1.00000	−0.121268
$69$	−6.00000	−0.722315
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	−1.00000	−0.117851
$73$	2.00000	0.234082	0.117041	−	0.993127i	$-0.462659\pi$
$73$	2.00000	0.234082	0.117041	+	0.993127i	$0.462659\pi$
$74$	−8.00000	−0.929981
$75$	−5.00000	−0.577350
$76$	−4.00000	−0.458831
$77$	0	0
$78$	−2.00000	−0.226455
$79$	−10.0000	−1.12509	−0.562544	−	0.826767i	$-0.690177\pi$
$79$	−10.0000	−1.12509	−0.562544	+	0.826767i	$0.690177\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	4.00000	0.431331
$87$	0	0
$88$	0	0
$89$	−18.0000	−1.90800	−0.953998	−	0.299813i	$-0.903076\pi$
$89$	−18.0000	−1.90800	−0.953998	+	0.299813i	$0.903076\pi$
$90$	0	0
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	−10.0000	−1.03695
$94$	−12.0000	−1.23771
$95$	0	0
$96$	−1.00000	−0.102062
$97$	14.0000	1.42148	0.710742	−	0.703452i	$-0.248359\pi$
$97$	14.0000	1.42148	0.710742	+	0.703452i	$0.248359\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−5.00000	−0.500000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	1.00000	0.0990148
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	20.0000	1.91565	0.957826	−	0.287348i	$-0.0927736\pi$
$109$	20.0000	1.91565	0.957826	+	0.287348i	$0.0927736\pi$
$110$	0	0
$111$	8.00000	0.759326
$112$	2.00000	0.188982
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	4.00000	0.374634
$115$	0	0
$116$	0	0
$117$	2.00000	0.184900
$118$	12.0000	1.10469
$119$	−2.00000	−0.183340
$120$	0	0
$121$	−11.0000	−1.00000
$122$	−8.00000	−0.724286
$123$	6.00000	0.541002
$124$	−10.0000	−0.898027
$125$	0	0
$126$	−2.00000	−0.178174
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	−1.00000	−0.0883883
$129$	−4.00000	−0.352180
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−8.00000	−0.693688
$134$	4.00000	0.345547
$135$	0	0
$136$	1.00000	0.0857493
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	6.00000	0.510754
$139$	8.00000	0.678551	0.339276	−	0.940687i	$-0.389818\pi$
$139$	8.00000	0.678551	0.339276	+	0.940687i	$0.389818\pi$
$140$	0	0
$141$	12.0000	1.01058
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−2.00000	−0.165521
$147$	−3.00000	−0.247436
$148$	8.00000	0.657596
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	5.00000	0.408248
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	4.00000	0.324443
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	10.0000	0.795557
$159$	6.00000	0.475831
$160$	0	0
$161$	−12.0000	−0.945732
$162$	−1.00000	−0.0785674
$163$	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
$166$	−12.0000	−0.931381
$167$	18.0000	1.39288	0.696441	−	0.717614i	$-0.254766\pi$
$167$	18.0000	1.39288	0.696441	+	0.717614i	$0.254766\pi$
$168$	−2.00000	−0.154303
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−4.00000	−0.305888
$172$	−4.00000	−0.304997
$173$	0	0		−	1.00000i	$-0.5\pi$
$173$	0	0			1.00000i	$0.5\pi$
$174$	0	0
$175$	−10.0000	−0.755929
$176$	0	0
$177$	−12.0000	−0.901975
$178$	18.0000	1.34916
$179$	−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
$181$	8.00000	0.594635	0.297318	−	0.954779i	$-0.403908\pi$
$181$	8.00000	0.594635	0.297318	+	0.954779i	$0.403908\pi$
$182$	−4.00000	−0.296500
$183$	8.00000	0.591377
$184$	6.00000	0.442326
$185$	0	0
$186$	10.0000	0.733236
$187$	0	0
$188$	12.0000	0.875190
$189$	2.00000	0.145479
$190$	0	0
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−14.0000	−1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	−12.0000	−0.854965	−0.427482	−	0.904024i	$-0.640599\pi$
$197$	−12.0000	−0.854965	−0.427482	+	0.904024i	$0.640599\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	5.00000	0.353553
$201$	−4.00000	−0.282138
$202$	−6.00000	−0.422159
$203$	0	0
$204$	−1.00000	−0.0700140
$205$	0	0
$206$	4.00000	0.278693
$207$	−6.00000	−0.417029
$208$	2.00000	0.138675
$209$	0	0
$210$	0	0
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	6.00000	0.412082
$213$	6.00000	0.411113
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	−20.0000	−1.35769
$218$	−20.0000	−1.35457
$219$	2.00000	0.135147
$220$	0	0
$221$	−2.00000	−0.134535
$222$	−8.00000	−0.536925
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	−2.00000	−0.133631
$225$	−5.00000	−0.333333
$226$	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	$0.369623\pi$
$228$	−4.00000	−0.264906
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−30.0000	−1.96537	−0.982683	−	0.185296i	$-0.940675\pi$
$233$	−30.0000	−1.96537	−0.982683	+	0.185296i	$0.940675\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	−12.0000	−0.781133
$237$	−10.0000	−0.649570
$238$	2.00000	0.129641
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	11.0000	0.707107
$243$	1.00000	0.0641500
$244$	8.00000	0.512148
$245$	0	0
$246$	−6.00000	−0.382546
$247$	−8.00000	−0.509028
$248$	10.0000	0.635001
$249$	12.0000	0.760469
$250$	0	0
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	18.0000	1.12281	0.561405	−	0.827541i	$-0.310261\pi$
$257$	18.0000	1.12281	0.561405	+	0.827541i	$0.310261\pi$
$258$	4.00000	0.249029
$259$	16.0000	0.994192
$260$	0	0
$261$	0	0
$262$	0	0
$263$	12.0000	0.739952	0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000	0.739952	0.369976	+	0.929041i	$0.379366\pi$
$264$	0	0
$265$	0	0
$266$	8.00000	0.490511
$267$	−18.0000	−1.10158
$268$	−4.00000	−0.244339
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	0	0
$271$	−16.0000	−0.971931	−0.485965	−	0.873978i	$-0.661532\pi$
$271$	−16.0000	−0.971931	−0.485965	+	0.873978i	$0.661532\pi$
$272$	−1.00000	−0.0606339
$273$	4.00000	0.242091
$274$	6.00000	0.362473
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−4.00000	−0.240337	−0.120168	−	0.992754i	$-0.538343\pi$
$277$	−4.00000	−0.240337	−0.120168	+	0.992754i	$0.538343\pi$
$278$	−8.00000	−0.479808
$279$	−10.0000	−0.598684
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	−12.0000	−0.714590
$283$	−16.0000	−0.951101	−0.475551	−	0.879688i	$-0.657751\pi$
$283$	−16.0000	−0.951101	−0.475551	+	0.879688i	$0.657751\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	12.0000	0.708338
$288$	−1.00000	−0.0589256
$289$	1.00000	0.0588235
$290$	0	0
$291$	14.0000	0.820695
$292$	2.00000	0.117041
$293$	18.0000	1.05157	0.525786	−	0.850617i	$-0.323771\pi$
$293$	18.0000	1.05157	0.525786	+	0.850617i	$0.323771\pi$
$294$	3.00000	0.174964
$295$	0	0
$296$	−8.00000	−0.464991
$297$	0	0
$298$	6.00000	0.347571
$299$	−12.0000	−0.693978
$300$	−5.00000	−0.288675
$301$	−8.00000	−0.461112
$302$	−8.00000	−0.460348
$303$	6.00000	0.344691
$304$	−4.00000	−0.229416
$305$	0	0
$306$	1.00000	0.0571662
$307$	20.0000	1.14146	0.570730	−	0.821138i	$-0.306660\pi$
$307$	20.0000	1.14146	0.570730	+	0.821138i	$0.306660\pi$
$308$	0	0
$309$	−4.00000	−0.227552
$310$	0	0
$311$	18.0000	1.02069	0.510343	−	0.859971i	$-0.329518\pi$
$311$	18.0000	1.02069	0.510343	+	0.859971i	$0.329518\pi$
$312$	−2.00000	−0.113228
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	−10.0000	−0.562544
$317$	12.0000	0.673987	0.336994	−	0.941507i	$-0.390590\pi$
$317$	12.0000	0.673987	0.336994	+	0.941507i	$0.390590\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	0	0
$321$	0	0
$322$	12.0000	0.668734
$323$	4.00000	0.222566
$324$	1.00000	0.0555556
$325$	−10.0000	−0.554700
$326$	−20.0000	−1.10770
$327$	20.0000	1.10600
$328$	−6.00000	−0.331295
$329$	24.0000	1.32316
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	12.0000	0.658586
$333$	8.00000	0.438397
$334$	−18.0000	−0.984916
$335$	0	0
$336$	2.00000	0.109109
$337$	−22.0000	−1.19842	−0.599208	−	0.800593i	$-0.704518\pi$
$337$	−22.0000	−1.19842	−0.599208	+	0.800593i	$0.704518\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	0	0
$341$	0	0
$342$	4.00000	0.216295
$343$	−20.0000	−1.07990
$344$	4.00000	0.215666
$345$	0	0
$346$	0	0
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	10.0000	0.534522
$351$	2.00000	0.106752
$352$	0	0
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	12.0000	0.637793
$355$	0	0
$356$	−18.0000	−0.953998
$357$	−2.00000	−0.105851
$358$	12.0000	0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	−8.00000	−0.420471
$363$	−11.0000	−0.577350
$364$	4.00000	0.209657
$365$	0	0
$366$	−8.00000	−0.418167
$367$	−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
$369$	6.00000	0.312348
$370$	0	0
$371$	12.0000	0.623009
$372$	−10.0000	−0.518476
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	0	0
$375$	0	0
$376$	−12.0000	−0.618853
$377$	0	0
$378$	−2.00000	−0.102869
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	8.00000	0.409852
$382$	12.0000	0.613973
$383$	−12.0000	−0.613171	−0.306586	−	0.951843i	$-0.599187\pi$
$383$	−12.0000	−0.613171	−0.306586	+	0.951843i	$0.599187\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−4.00000	−0.203331
$388$	14.0000	0.710742
$389$	−30.0000	−1.52106	−0.760530	−	0.649303i	$-0.775061\pi$
$389$	−30.0000	−1.52106	−0.760530	+	0.649303i	$0.775061\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	3.00000	0.151523
$393$	0	0
$394$	12.0000	0.604551
$395$	0	0
$396$	0	0
$397$	−16.0000	−0.803017	−0.401508	−	0.915855i	$-0.631514\pi$
$397$	−16.0000	−0.803017	−0.401508	+	0.915855i	$0.631514\pi$
$398$	−2.00000	−0.100251
$399$	−8.00000	−0.400501
$400$	−5.00000	−0.250000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	4.00000	0.199502
$403$	−20.0000	−0.996271
$404$	6.00000	0.298511
$405$	0	0
$406$	0	0
$407$	0	0
$408$	1.00000	0.0495074
$409$	38.0000	1.87898	0.939490	−	0.342578i	$-0.111300\pi$
$409$	38.0000	1.87898	0.939490	+	0.342578i	$0.111300\pi$
$410$	0	0
$411$	−6.00000	−0.295958
$412$	−4.00000	−0.197066
$413$	−24.0000	−1.18096
$414$	6.00000	0.294884
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	8.00000	0.391762
$418$	0	0
$419$	−36.0000	−1.75872	−0.879358	−	0.476162i	$-0.842028\pi$
$419$	−36.0000	−1.75872	−0.879358	+	0.476162i	$0.842028\pi$
$420$	0	0
$421$	−22.0000	−1.07221	−0.536107	−	0.844150i	$-0.680106\pi$
$421$	−22.0000	−1.07221	−0.536107	+	0.844150i	$0.680106\pi$
$422$	−8.00000	−0.389434
$423$	12.0000	0.583460
$424$	−6.00000	−0.291386
$425$	5.00000	0.242536
$426$	−6.00000	−0.290701
$427$	16.0000	0.774294
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−30.0000	−1.44505	−0.722525	−	0.691345i	$-0.757018\pi$
$431$	−30.0000	−1.44505	−0.722525	+	0.691345i	$0.757018\pi$
$432$	1.00000	0.0481125
$433$	14.0000	0.672797	0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000	0.672797	0.336399	+	0.941720i	$0.390791\pi$
$434$	20.0000	0.960031
$435$	0	0
$436$	20.0000	0.957826
$437$	24.0000	1.14808
$438$	−2.00000	−0.0955637
$439$	26.0000	1.24091	0.620456	−	0.784241i	$-0.286947\pi$
$439$	26.0000	1.24091	0.620456	+	0.784241i	$0.286947\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	2.00000	0.0951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	8.00000	0.379663
$445$	0	0
$446$	28.0000	1.32584
$447$	−6.00000	−0.283790
$448$	2.00000	0.0944911
$449$	6.00000	0.283158	0.141579	−	0.989927i	$-0.454782\pi$
$449$	6.00000	0.283158	0.141579	+	0.989927i	$0.454782\pi$
$450$	5.00000	0.235702
$451$	0	0
$452$	−6.00000	−0.282216
$453$	8.00000	0.375873
$454$	−12.0000	−0.563188
$455$	0	0
$456$	4.00000	0.187317
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	−14.0000	−0.654177
$459$	−1.00000	−0.0466760
$460$	0	0
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	20.0000	0.929479	0.464739	−	0.885448i	$-0.346148\pi$
$463$	20.0000	0.929479	0.464739	+	0.885448i	$0.346148\pi$
$464$	0	0
$465$	0	0
$466$	30.0000	1.38972
$467$	12.0000	0.555294	0.277647	−	0.960683i	$-0.410445\pi$
$467$	12.0000	0.555294	0.277647	+	0.960683i	$0.410445\pi$
$468$	2.00000	0.0924500
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−10.0000	−0.460776
$472$	12.0000	0.552345
$473$	0	0
$474$	10.0000	0.459315
$475$	20.0000	0.917663
$476$	−2.00000	−0.0916698
$477$	6.00000	0.274721
$478$	12.0000	0.548867
$479$	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
$481$	16.0000	0.729537
$482$	10.0000	0.455488
$483$	−12.0000	−0.546019
$484$	−11.0000	−0.500000
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−10.0000	−0.453143	−0.226572	−	0.973995i	$-0.572752\pi$
$487$	−10.0000	−0.453143	−0.226572	+	0.973995i	$0.572752\pi$
$488$	−8.00000	−0.362143
$489$	20.0000	0.904431
$490$	0	0
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	6.00000	0.270501
$493$	0	0
$494$	8.00000	0.359937
$495$	0	0
$496$	−10.0000	−0.449013
$497$	12.0000	0.538274
$498$	−12.0000	−0.537733
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	0	0
$501$	18.0000	0.804181
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	−2.00000	−0.0890871
$505$	0	0
$506$	0	0
$507$	−9.00000	−0.399704
$508$	8.00000	0.354943
$509$	−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
$511$	4.00000	0.176950
$512$	−1.00000	−0.0441942
$513$	−4.00000	−0.176604
$514$	−18.0000	−0.793946
$515$	0	0
$516$	−4.00000	−0.176090
$517$	0	0
$518$	−16.0000	−0.703000
$519$	0	0
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	−10.0000	−0.436436
$526$	−12.0000	−0.523225
$527$	10.0000	0.435607
$528$	0	0
$529$	13.0000	0.565217
$530$	0	0
$531$	−12.0000	−0.520756
$532$	−8.00000	−0.346844
$533$	12.0000	0.519778
$534$	18.0000	0.778936
$535$	0	0
$536$	4.00000	0.172774
$537$	−12.0000	−0.517838
$538$	−24.0000	−1.03471
$539$	0	0
$540$	0	0
$541$	−16.0000	−0.687894	−0.343947	−	0.938989i	$-0.611764\pi$
$541$	−16.0000	−0.687894	−0.343947	+	0.938989i	$0.611764\pi$
$542$	16.0000	0.687259
$543$	8.00000	0.343313
$544$	1.00000	0.0428746
$545$	0	0
$546$	−4.00000	−0.171184
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−6.00000	−0.256307
$549$	8.00000	0.341432
$550$	0	0
$551$	0	0
$552$	6.00000	0.255377
$553$	−20.0000	−0.850487
$554$	4.00000	0.169944
$555$	0	0
$556$	8.00000	0.339276
$557$	−42.0000	−1.77960	−0.889799	−	0.456354i	$-0.849155\pi$
$557$	−42.0000	−1.77960	−0.889799	+	0.456354i	$0.849155\pi$
$558$	10.0000	0.423334
$559$	−8.00000	−0.338364
$560$	0	0
$561$	0	0
$562$	−30.0000	−1.26547
$563$	−12.0000	−0.505740	−0.252870	−	0.967500i	$-0.581374\pi$
$563$	−12.0000	−0.505740	−0.252870	+	0.967500i	$0.581374\pi$
$564$	12.0000	0.505291
$565$	0	0
$566$	16.0000	0.672530
$567$	2.00000	0.0839921
$568$	−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
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−12.0000	−0.501307
$574$	−12.0000	−0.500870
$575$	30.0000	1.25109
$576$	1.00000	0.0416667
$577$	2.00000	0.0832611	0.0416305	−	0.999133i	$-0.486745\pi$
$577$	2.00000	0.0832611	0.0416305	+	0.999133i	$0.486745\pi$
$578$	−1.00000	−0.0415945
$579$	14.0000	0.581820
$580$	0	0
$581$	24.0000	0.995688
$582$	−14.0000	−0.580319
$583$	0	0
$584$	−2.00000	−0.0827606
$585$	0	0
$586$	−18.0000	−0.743573
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−3.00000	−0.123718
$589$	40.0000	1.64817
$590$	0	0
$591$	−12.0000	−0.493614
$592$	8.00000	0.328798
$593$	18.0000	0.739171	0.369586	−	0.929197i	$-0.379500\pi$
$593$	18.0000	0.739171	0.369586	+	0.929197i	$0.379500\pi$
$594$	0	0
$595$	0	0
$596$	−6.00000	−0.245770
$597$	2.00000	0.0818546
$598$	12.0000	0.490716
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	5.00000	0.204124
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	8.00000	0.326056
$603$	−4.00000	−0.162893
$604$	8.00000	0.325515
$605$	0	0
$606$	−6.00000	−0.243733
$607$	−10.0000	−0.405887	−0.202944	−	0.979190i	$-0.565051\pi$
$607$	−10.0000	−0.405887	−0.202944	+	0.979190i	$0.565051\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	0	0
$611$	24.0000	0.970936
$612$	−1.00000	−0.0404226
$613$	−34.0000	−1.37325	−0.686624	−	0.727013i	$-0.740908\pi$
$613$	−34.0000	−1.37325	−0.686624	+	0.727013i	$0.740908\pi$
$614$	−20.0000	−0.807134
$615$	0	0
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	4.00000	0.160904
$619$	−28.0000	−1.12542	−0.562708	−	0.826656i	$-0.690240\pi$
$619$	−28.0000	−1.12542	−0.562708	+	0.826656i	$0.690240\pi$
$620$	0	0
$621$	−6.00000	−0.240772
$622$	−18.0000	−0.721734
$623$	−36.0000	−1.44231
$624$	2.00000	0.0800641
$625$	25.0000	1.00000
$626$	−14.0000	−0.559553
$627$	0	0
$628$	−10.0000	−0.399043
$629$	−8.00000	−0.318981
$630$	0	0
$631$	−4.00000	−0.159237	−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000	−0.159237	−0.0796187	+	0.996825i	$0.525370\pi$
$632$	10.0000	0.397779
$633$	8.00000	0.317971
$634$	−12.0000	−0.476581
$635$	0	0
$636$	6.00000	0.237915
$637$	−6.00000	−0.237729
$638$	0	0
$639$	6.00000	0.237356
$640$	0	0
$641$	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
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	−4.00000	−0.157378
$647$	36.0000	1.41531	0.707653	−	0.706560i	$-0.249754\pi$
$647$	36.0000	1.41531	0.707653	+	0.706560i	$0.249754\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	10.0000	0.392232
$651$	−20.0000	−0.783862
$652$	20.0000	0.783260
$653$	−24.0000	−0.939193	−0.469596	−	0.882881i	$-0.655601\pi$
$653$	−24.0000	−0.939193	−0.469596	+	0.882881i	$0.655601\pi$
$654$	−20.0000	−0.782062
$655$	0	0
$656$	6.00000	0.234261
$657$	2.00000	0.0780274
$658$	−24.0000	−0.935617
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	−22.0000	−0.855701	−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000	−0.855701	−0.427850	+	0.903850i	$0.640729\pi$
$662$	−20.0000	−0.777322
$663$	−2.00000	−0.0776736
$664$	−12.0000	−0.465690
$665$	0	0
$666$	−8.00000	−0.309994
$667$	0	0
$668$	18.0000	0.696441
$669$	−28.0000	−1.08254
$670$	0	0
$671$	0	0
$672$	−2.00000	−0.0771517
$673$	−34.0000	−1.31060	−0.655302	−	0.755367i	$-0.727459\pi$
$673$	−34.0000	−1.31060	−0.655302	+	0.755367i	$0.727459\pi$
$674$	22.0000	0.847408
$675$	−5.00000	−0.192450
$676$	−9.00000	−0.346154
$677$	−12.0000	−0.461197	−0.230599	−	0.973049i	$-0.574068\pi$
$677$	−12.0000	−0.461197	−0.230599	+	0.973049i	$0.574068\pi$
$678$	6.00000	0.230429
$679$	28.0000	1.07454
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−4.00000	−0.152944
$685$	0	0
$686$	20.0000	0.763604
$687$	14.0000	0.534133
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	32.0000	1.21734	0.608669	−	0.793424i	$-0.291704\pi$
$691$	32.0000	1.21734	0.608669	+	0.793424i	$0.291704\pi$
$692$	0	0
$693$	0	0
$694$	12.0000	0.455514
$695$	0	0
$696$	0	0
$697$	−6.00000	−0.227266
$698$	−26.0000	−0.984115
$699$	−30.0000	−1.13470
$700$	−10.0000	−0.377964
$701$	−42.0000	−1.58632	−0.793159	−	0.609015i	$-0.791565\pi$
$701$	−42.0000	−1.58632	−0.793159	+	0.609015i	$0.791565\pi$
$702$	−2.00000	−0.0754851
$703$	−32.0000	−1.20690
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	12.0000	0.451306
$708$	−12.0000	−0.450988
$709$	20.0000	0.751116	0.375558	−	0.926799i	$-0.377451\pi$
$709$	20.0000	0.751116	0.375558	+	0.926799i	$0.377451\pi$
$710$	0	0
$711$	−10.0000	−0.375029
$712$	18.0000	0.674579
$713$	60.0000	2.24702
$714$	2.00000	0.0748481
$715$	0	0
$716$	−12.0000	−0.448461
$717$	−12.0000	−0.448148
$718$	0	0
$719$	6.00000	0.223762	0.111881	−	0.993722i	$-0.464312\pi$
$719$	6.00000	0.223762	0.111881	+	0.993722i	$0.464312\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	−10.0000	−0.371904
$724$	8.00000	0.297318
$725$	0	0
$726$	11.0000	0.408248
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−4.00000	−0.148250
$729$	1.00000	0.0370370
$730$	0	0
$731$	4.00000	0.147945
$732$	8.00000	0.295689
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	10.0000	0.369107
$735$	0	0
$736$	6.00000	0.221163
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−4.00000	−0.147142	−0.0735712	−	0.997290i	$-0.523440\pi$
$739$	−4.00000	−0.147142	−0.0735712	+	0.997290i	$0.523440\pi$
$740$	0	0
$741$	−8.00000	−0.293887
$742$	−12.0000	−0.440534
$743$	42.0000	1.54083	0.770415	−	0.637542i	$-0.220049\pi$
$743$	42.0000	1.54083	0.770415	+	0.637542i	$0.220049\pi$
$744$	10.0000	0.366618
$745$	0	0
$746$	22.0000	0.805477
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	0	0
$751$	2.00000	0.0729810	0.0364905	−	0.999334i	$-0.488382\pi$
$751$	2.00000	0.0729810	0.0364905	+	0.999334i	$0.488382\pi$
$752$	12.0000	0.437595
$753$	12.0000	0.437304
$754$	0	0
$755$	0	0
$756$	2.00000	0.0727393
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	28.0000	1.01701
$759$	0	0
$760$	0	0
$761$	−54.0000	−1.95750	−0.978749	−	0.205061i	$-0.934261\pi$
$761$	−54.0000	−1.95750	−0.978749	+	0.205061i	$0.934261\pi$
$762$	−8.00000	−0.289809
$763$	40.0000	1.44810
$764$	−12.0000	−0.434145
$765$	0	0
$766$	12.0000	0.433578
$767$	−24.0000	−0.866590
$768$	1.00000	0.0360844
$769$	−22.0000	−0.793340	−0.396670	−	0.917961i	$-0.629834\pi$
$769$	−22.0000	−0.793340	−0.396670	+	0.917961i	$0.629834\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	14.0000	0.503871
$773$	−6.00000	−0.215805	−0.107903	−	0.994161i	$-0.534413\pi$
$773$	−6.00000	−0.215805	−0.107903	+	0.994161i	$0.534413\pi$
$774$	4.00000	0.143777
$775$	50.0000	1.79605
$776$	−14.0000	−0.502571
$777$	16.0000	0.573997
$778$	30.0000	1.07555
$779$	−24.0000	−0.859889
$780$	0	0
$781$	0	0
$782$	−6.00000	−0.214560
$783$	0	0
$784$	−3.00000	−0.107143
$785$	0	0
$786$	0	0
$787$	32.0000	1.14068	0.570338	−	0.821410i	$-0.306812\pi$
$787$	32.0000	1.14068	0.570338	+	0.821410i	$0.306812\pi$
$788$	−12.0000	−0.427482
$789$	12.0000	0.427211
$790$	0	0
$791$	−12.0000	−0.426671
$792$	0	0
$793$	16.0000	0.568177
$794$	16.0000	0.567819
$795$	0	0
$796$	2.00000	0.0708881
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	8.00000	0.283197
$799$	−12.0000	−0.424529
$800$	5.00000	0.176777
$801$	−18.0000	−0.635999
$802$	18.0000	0.635602
$803$	0	0
$804$	−4.00000	−0.141069
$805$	0	0
$806$	20.0000	0.704470
$807$	24.0000	0.844840
$808$	−6.00000	−0.211079
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−4.00000	−0.140459	−0.0702295	−	0.997531i	$-0.522373\pi$
$811$	−4.00000	−0.140459	−0.0702295	+	0.997531i	$0.522373\pi$
$812$	0	0
$813$	−16.0000	−0.561144
$814$	0	0
$815$	0	0
$816$	−1.00000	−0.0350070
$817$	16.0000	0.559769
$818$	−38.0000	−1.32864
$819$	4.00000	0.139771
$820$	0	0
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	6.00000	0.209274
$823$	−34.0000	−1.18517	−0.592583	−	0.805510i	$-0.701892\pi$
$823$	−34.0000	−1.18517	−0.592583	+	0.805510i	$0.701892\pi$
$824$	4.00000	0.139347
$825$	0	0
$826$	24.0000	0.835067
$827$	48.0000	1.66912	0.834562	−	0.550914i	$-0.185721\pi$
$827$	48.0000	1.66912	0.834562	+	0.550914i	$0.185721\pi$
$828$	−6.00000	−0.208514
$829$	−10.0000	−0.347314	−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000	−0.347314	−0.173657	+	0.984806i	$0.555558\pi$
$830$	0	0
$831$	−4.00000	−0.138758
$832$	2.00000	0.0693375
$833$	3.00000	0.103944
$834$	−8.00000	−0.277017
$835$	0	0
$836$	0	0
$837$	−10.0000	−0.345651
$838$	36.0000	1.24360
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	22.0000	0.758170
$843$	30.0000	1.03325
$844$	8.00000	0.275371
$845$	0	0
$846$	−12.0000	−0.412568
$847$	−22.0000	−0.755929
$848$	6.00000	0.206041
$849$	−16.0000	−0.549119
$850$	−5.00000	−0.171499
$851$	−48.0000	−1.64542
$852$	6.00000	0.205557
$853$	−28.0000	−0.958702	−0.479351	−	0.877623i	$-0.659128\pi$
$853$	−28.0000	−0.958702	−0.479351	+	0.877623i	$0.659128\pi$
$854$	−16.0000	−0.547509
$855$	0	0
$856$	0	0
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	0	0
$859$	−28.0000	−0.955348	−0.477674	−	0.878537i	$-0.658520\pi$
$859$	−28.0000	−0.955348	−0.477674	+	0.878537i	$0.658520\pi$
$860$	0	0
$861$	12.0000	0.408959
$862$	30.0000	1.02180
$863$	48.0000	1.63394	0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000	1.63394	0.816970	+	0.576681i	$0.195652\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	−14.0000	−0.475739
$867$	1.00000	0.0339618
$868$	−20.0000	−0.678844
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	−20.0000	−0.677285
$873$	14.0000	0.473828
$874$	−24.0000	−0.811812
$875$	0	0
$876$	2.00000	0.0675737
$877$	−4.00000	−0.135070	−0.0675352	−	0.997717i	$-0.521513\pi$
$877$	−4.00000	−0.135070	−0.0675352	+	0.997717i	$0.521513\pi$
$878$	−26.0000	−0.877457
$879$	18.0000	0.607125
$880$	0	0
$881$	−54.0000	−1.81931	−0.909653	−	0.415369i	$-0.863653\pi$
$881$	−54.0000	−1.81931	−0.909653	+	0.415369i	$0.863653\pi$
$882$	3.00000	0.101015
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−2.00000	−0.0672673
$885$	0	0
$886$	36.0000	1.20944
$887$	−18.0000	−0.604381	−0.302190	−	0.953248i	$-0.597718\pi$
$887$	−18.0000	−0.604381	−0.302190	+	0.953248i	$0.597718\pi$
$888$	−8.00000	−0.268462
$889$	16.0000	0.536623
$890$	0	0
$891$	0	0
$892$	−28.0000	−0.937509
$893$	−48.0000	−1.60626
$894$	6.00000	0.200670
$895$	0	0
$896$	−2.00000	−0.0668153
$897$	−12.0000	−0.400668
$898$	−6.00000	−0.200223
$899$	0	0
$900$	−5.00000	−0.166667
$901$	−6.00000	−0.199889
$902$	0	0
$903$	−8.00000	−0.266223
$904$	6.00000	0.199557
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−40.0000	−1.32818	−0.664089	−	0.747653i	$-0.731180\pi$
$907$	−40.0000	−1.32818	−0.664089	+	0.747653i	$0.731180\pi$
$908$	12.0000	0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	6.00000	0.198789	0.0993944	−	0.995048i	$-0.468309\pi$
$911$	6.00000	0.198789	0.0993944	+	0.995048i	$0.468309\pi$
$912$	−4.00000	−0.132453
$913$	0	0
$914$	10.0000	0.330771
$915$	0	0
$916$	14.0000	0.462573
$917$	0	0
$918$	1.00000	0.0330049
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	20.0000	0.659022
$922$	−30.0000	−0.987997
$923$	12.0000	0.394985
$924$	0	0
$925$	−40.0000	−1.31519
$926$	−20.0000	−0.657241
$927$	−4.00000	−0.131377
$928$	0	0
$929$	−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
$931$	12.0000	0.393284
$932$	−30.0000	−0.982683
$933$	18.0000	0.589294
$934$	−12.0000	−0.392652
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	−10.0000	−0.326686	−0.163343	−	0.986569i	$-0.552228\pi$
$937$	−10.0000	−0.326686	−0.163343	+	0.986569i	$0.552228\pi$
$938$	8.00000	0.261209
$939$	14.0000	0.456873
$940$	0	0
$941$	12.0000	0.391189	0.195594	−	0.980685i	$-0.437336\pi$
$941$	12.0000	0.391189	0.195594	+	0.980685i	$0.437336\pi$
$942$	10.0000	0.325818
$943$	−36.0000	−1.17232
$944$	−12.0000	−0.390567
$945$	0	0
$946$	0	0
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	−10.0000	−0.324785
$949$	4.00000	0.129845
$950$	−20.0000	−0.648886
$951$	12.0000	0.389127
$952$	2.00000	0.0648204
$953$	54.0000	1.74923	0.874616	−	0.484817i	$-0.161114\pi$
$953$	54.0000	1.74923	0.874616	+	0.484817i	$0.161114\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	−12.0000	−0.388108
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−12.0000	−0.387500
$960$	0	0
$961$	69.0000	2.22581
$962$	−16.0000	−0.515861
$963$	0	0
$964$	−10.0000	−0.322078
$965$	0	0
$966$	12.0000	0.386094
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	11.0000	0.353553
$969$	4.00000	0.128499
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	10.0000	0.320421
$975$	−10.0000	−0.320256
$976$	8.00000	0.256074
$977$	−30.0000	−0.959785	−0.479893	−	0.877327i	$-0.659324\pi$
$977$	−30.0000	−0.959785	−0.479893	+	0.877327i	$0.659324\pi$
$978$	−20.0000	−0.639529
$979$	0	0
$980$	0	0
$981$	20.0000	0.638551
$982$	36.0000	1.14881
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	−6.00000	−0.191273
$985$	0	0
$986$	0	0
$987$	24.0000	0.763928
$988$	−8.00000	−0.254514
$989$	24.0000	0.763156
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	10.0000	0.317500
$993$	20.0000	0.634681
$994$	−12.0000	−0.380617
$995$	0	0
$996$	12.0000	0.380235
$997$	8.00000	0.253363	0.126681	−	0.991943i	$-0.459567\pi$
$997$	8.00000	0.253363	0.126681	+	0.991943i	$0.459567\pi$
$998$	−32.0000	−1.01294
$999$	8.00000	0.253109

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	102.2.a.b.1.1	✓	1
3.2	odd	2		306.2.a.c.1.1		1
4.3	odd	2		816.2.a.d.1.1		1
5.2	odd	4		2550.2.d.g.2449.1		2
5.3	odd	4		2550.2.d.g.2449.2		2
5.4	even	2		2550.2.a.u.1.1		1
7.6	odd	2		4998.2.a.d.1.1		1
8.3	odd	2		3264.2.a.w.1.1		1
8.5	even	2		3264.2.a.i.1.1		1
12.11	even	2		2448.2.a.i.1.1		1
15.14	odd	2		7650.2.a.j.1.1		1
17.2	even	8		1734.2.f.b.1483.2		4
17.4	even	4		1734.2.b.f.577.1		2
17.8	even	8		1734.2.f.b.829.1		4
17.9	even	8		1734.2.f.b.829.2		4
17.13	even	4		1734.2.b.f.577.2		2
17.15	even	8		1734.2.f.b.1483.1		4
17.16	even	2		1734.2.a.b.1.1		1
24.5	odd	2		9792.2.a.bg.1.1		1
24.11	even	2		9792.2.a.ba.1.1		1
51.50	odd	2		5202.2.a.j.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
102.2.a.b.1.1	✓	1	1.1	even	1	trivial
306.2.a.c.1.1		1	3.2	odd	2
816.2.a.d.1.1		1	4.3	odd	2
1734.2.a.b.1.1		1	17.16	even	2
1734.2.b.f.577.1		2	17.4	even	4
1734.2.b.f.577.2		2	17.13	even	4
1734.2.f.b.829.1		4	17.8	even	8
1734.2.f.b.829.2		4	17.9	even	8
1734.2.f.b.1483.1		4	17.15	even	8
1734.2.f.b.1483.2		4	17.2	even	8
2448.2.a.i.1.1		1	12.11	even	2
2550.2.a.u.1.1		1	5.4	even	2
2550.2.d.g.2449.1		2	5.2	odd	4
2550.2.d.g.2449.2		2	5.3	odd	4
3264.2.a.i.1.1		1	8.5	even	2
3264.2.a.w.1.1		1	8.3	odd	2
4998.2.a.d.1.1		1	7.6	odd	2
5202.2.a.j.1.1		1	51.50	odd	2
7650.2.a.j.1.1		1	15.14	odd	2
9792.2.a.ba.1.1		1	24.11	even	2
9792.2.a.bg.1.1		1	24.5	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:	\( N \)	\(=\)	\( 102 = 2 \cdot 3 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	102.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more