LMFDB - Embedded newform 390.2.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [390,2,Mod(1,390)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(390, 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("390.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 390 = 2 \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	390.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:	$3.11416567883$
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	$\chi$	$=$	390.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} +1.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +1.00000 q^{18} +2.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} -6.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} -4.00000 q^{31} +1.00000 q^{32} -2.00000 q^{35} +1.00000 q^{36} +2.00000 q^{37} +2.00000 q^{38} +1.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} -4.00000 q^{43} -1.00000 q^{45} -6.00000 q^{46} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +1.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +2.00000 q^{57} -1.00000 q^{60} -10.0000 q^{61} -4.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} -1.00000 q^{65} +8.00000 q^{67} -6.00000 q^{69} -2.00000 q^{70} +1.00000 q^{72} +8.00000 q^{73} +2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{76} +1.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -12.0000 q^{83} +2.00000 q^{84} -4.00000 q^{86} +6.00000 q^{89} -1.00000 q^{90} +2.00000 q^{91} -6.00000 q^{92} -4.00000 q^{93} -2.00000 q^{95} +1.00000 q^{96} +8.00000 q^{97} -3.00000 q^{98} +O(q^{100})\)

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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	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
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	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
$21$	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
$25$	1.00000	0.200000
$26$	1.00000	0.196116
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	0	0
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	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
$39$	1.00000	0.160128
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	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
$45$	−1.00000	−0.149071
$46$	−6.00000	−0.884652
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	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	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	2.00000	0.264906
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	−10.0000	−1.28037	−0.640184	−	0.768221i	$-0.721142\pi$
$61$	−10.0000	−1.28037	−0.640184	+	0.768221i	$0.721142\pi$
$62$	−4.00000	−0.508001
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	−1.00000	−0.124035
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	−6.00000	−0.722315
$70$	−2.00000	−0.239046
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	2.00000	0.232495
$75$	1.00000	0.115470
$76$	2.00000	0.229416
$77$	0	0
$78$	1.00000	0.113228
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	2.00000	0.218218
$85$	0	0
$86$	−4.00000	−0.431331
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	−1.00000	−0.105409
$91$	2.00000	0.209657
$92$	−6.00000	−0.625543
$93$	−4.00000	−0.414781
$94$	0	0
$95$	−2.00000	−0.205196
$96$	1.00000	0.102062
$97$	8.00000	0.812277	0.406138	−	0.913812i	$-0.366875\pi$
$97$	8.00000	0.812277	0.406138	+	0.913812i	$0.366875\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−12.0000	−1.19404	−0.597022	−	0.802225i	$-0.703650\pi$
$101$	−12.0000	−1.19404	−0.597022	+	0.802225i	$0.703650\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	1.00000	0.0980581
$105$	−2.00000	−0.195180
$106$	−6.00000	−0.582772
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	−16.0000	−1.53252	−0.766261	−	0.642529i	$-0.777885\pi$
$109$	−16.0000	−1.53252	−0.766261	+	0.642529i	$0.777885\pi$
$110$	0	0
$111$	2.00000	0.189832
$112$	2.00000	0.188982
$113$	−12.0000	−1.12887	−0.564433	−	0.825479i	$-0.690905\pi$
$113$	−12.0000	−1.12887	−0.564433	+	0.825479i	$0.690905\pi$
$114$	2.00000	0.187317
$115$	6.00000	0.559503
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	−10.0000	−0.905357
$123$	−6.00000	−0.541002
$124$	−4.00000	−0.359211
$125$	−1.00000	−0.0894427
$126$	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
$130$	−1.00000	−0.0877058
$131$	−18.0000	−1.57267	−0.786334	−	0.617802i	$-0.788023\pi$
$131$	−18.0000	−1.57267	−0.786334	+	0.617802i	$0.788023\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	8.00000	0.691095
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	18.0000	1.53784	0.768922	−	0.639343i	$-0.220793\pi$
$137$	18.0000	1.53784	0.768922	+	0.639343i	$0.220793\pi$
$138$	−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	$0.389818\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	0	0
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	−3.00000	−0.247436
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	1.00000	0.0816497
$151$	20.0000	1.62758	0.813788	−	0.581161i	$-0.197401\pi$
$151$	20.0000	1.62758	0.813788	+	0.581161i	$0.197401\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	0	0
$155$	4.00000	0.321288
$156$	1.00000	0.0800641
$157$	2.00000	0.159617	0.0798087	−	0.996810i	$-0.474569\pi$
$157$	2.00000	0.159617	0.0798087	+	0.996810i	$0.474569\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	−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
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	2.00000	0.154303
$169$	1.00000	0.0769231
$170$	0	0
$171$	2.00000	0.152944
$172$	−4.00000	−0.304997
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	0	0
$178$	6.00000	0.449719
$179$	18.0000	1.34538	0.672692	−	0.739923i	$-0.265138\pi$
$179$	18.0000	1.34538	0.672692	+	0.739923i	$0.265138\pi$
$180$	−1.00000	−0.0745356
$181$	−22.0000	−1.63525	−0.817624	−	0.575753i	$-0.804709\pi$
$181$	−22.0000	−1.63525	−0.817624	+	0.575753i	$0.804709\pi$
$182$	2.00000	0.148250
$183$	−10.0000	−0.739221
$184$	−6.00000	−0.442326
$185$	−2.00000	−0.147043
$186$	−4.00000	−0.293294
$187$	0	0
$188$	0	0
$189$	2.00000	0.145479
$190$	−2.00000	−0.145095
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	1.00000	0.0721688
$193$	20.0000	1.43963	0.719816	−	0.694165i	$-0.244226\pi$
$193$	20.0000	1.43963	0.719816	+	0.694165i	$0.244226\pi$
$194$	8.00000	0.574367
$195$	−1.00000	−0.0716115
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	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
$201$	8.00000	0.564276
$202$	−12.0000	−0.844317
$203$	0	0
$204$	0	0
$205$	6.00000	0.419058
$206$	8.00000	0.557386
$207$	−6.00000	−0.417029
$208$	1.00000	0.0693375
$209$	0	0
$210$	−2.00000	−0.138013
$211$	−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
$214$	12.0000	0.820303
$215$	4.00000	0.272798
$216$	1.00000	0.0680414
$217$	−8.00000	−0.543075
$218$	−16.0000	−1.08366
$219$	8.00000	0.540590
$220$	0	0
$221$	0	0
$222$	2.00000	0.134231
$223$	14.0000	0.937509	0.468755	−	0.883328i	$-0.344703\pi$
$223$	14.0000	0.937509	0.468755	+	0.883328i	$0.344703\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−12.0000	−0.798228
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	2.00000	0.132453
$229$	−16.0000	−1.05731	−0.528655	−	0.848837i	$-0.677303\pi$
$229$	−16.0000	−1.05731	−0.528655	+	0.848837i	$0.677303\pi$
$230$	6.00000	0.395628
$231$	0	0
$232$	0	0
$233$	24.0000	1.57229	0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000	1.57229	0.786146	+	0.618041i	$0.212073\pi$
$234$	1.00000	0.0653720
$235$	0	0
$236$	0	0
$237$	8.00000	0.519656
$238$	0	0
$239$	−24.0000	−1.55243	−0.776215	−	0.630468i	$-0.782863\pi$
$239$	−24.0000	−1.55243	−0.776215	+	0.630468i	$0.782863\pi$
$240$	−1.00000	−0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	−10.0000	−0.640184
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	2.00000	0.127257
$248$	−4.00000	−0.254000
$249$	−12.0000	−0.760469
$250$	−1.00000	−0.0632456
$251$	18.0000	1.13615	0.568075	−	0.822977i	$-0.307688\pi$
$251$	18.0000	1.13615	0.568075	+	0.822977i	$0.307688\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	12.0000	0.748539	0.374270	−	0.927320i	$-0.377893\pi$
$257$	12.0000	0.748539	0.374270	+	0.927320i	$0.377893\pi$
$258$	−4.00000	−0.249029
$259$	4.00000	0.248548
$260$	−1.00000	−0.0620174
$261$	0	0
$262$	−18.0000	−1.11204
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	4.00000	0.245256
$267$	6.00000	0.367194
$268$	8.00000	0.488678
$269$	12.0000	0.731653	0.365826	−	0.930683i	$-0.380786\pi$
$269$	12.0000	0.731653	0.365826	+	0.930683i	$0.380786\pi$
$270$	−1.00000	−0.0608581
$271$	−4.00000	−0.242983	−0.121491	−	0.992592i	$-0.538768\pi$
$271$	−4.00000	−0.242983	−0.121491	+	0.992592i	$0.538768\pi$
$272$	0	0
$273$	2.00000	0.121046
$274$	18.0000	1.08742
$275$	0	0
$276$	−6.00000	−0.361158
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	8.00000	0.479808
$279$	−4.00000	−0.239474
$280$	−2.00000	−0.119523
$281$	6.00000	0.357930	0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000	0.357930	0.178965	+	0.983855i	$0.442725\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−2.00000	−0.118470
$286$	0	0
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	8.00000	0.468968
$292$	8.00000	0.468165
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	0	0
$296$	2.00000	0.116248
$297$	0	0
$298$	6.00000	0.347571
$299$	−6.00000	−0.346989
$300$	1.00000	0.0577350
$301$	−8.00000	−0.461112
$302$	20.0000	1.15087
$303$	−12.0000	−0.689382
$304$	2.00000	0.114708
$305$	10.0000	0.572598
$306$	0	0
$307$	−16.0000	−0.913168	−0.456584	−	0.889680i	$-0.650927\pi$
$307$	−16.0000	−0.913168	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	8.00000	0.455104
$310$	4.00000	0.227185
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	1.00000	0.0566139
$313$	14.0000	0.791327	0.395663	−	0.918396i	$-0.370515\pi$
$313$	14.0000	0.791327	0.395663	+	0.918396i	$0.370515\pi$
$314$	2.00000	0.112867
$315$	−2.00000	−0.112687
$316$	8.00000	0.450035
$317$	−30.0000	−1.68497	−0.842484	−	0.538721i	$-0.818908\pi$
$317$	−30.0000	−1.68497	−0.842484	+	0.538721i	$0.818908\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	−12.0000	−0.668734
$323$	0	0
$324$	1.00000	0.0555556
$325$	1.00000	0.0554700
$326$	20.0000	1.10770
$327$	−16.0000	−0.884802
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	−12.0000	−0.658586
$333$	2.00000	0.109599
$334$	12.0000	0.656611
$335$	−8.00000	−0.437087
$336$	2.00000	0.109109
$337$	−34.0000	−1.85210	−0.926049	−	0.377403i	$-0.876817\pi$
$337$	−34.0000	−1.85210	−0.926049	+	0.377403i	$0.876817\pi$
$338$	1.00000	0.0543928
$339$	−12.0000	−0.651751
$340$	0	0
$341$	0	0
$342$	2.00000	0.108148
$343$	−20.0000	−1.07990
$344$	−4.00000	−0.215666
$345$	6.00000	0.323029
$346$	−6.00000	−0.322562
$347$	24.0000	1.28839	0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000	1.28839	0.644194	+	0.764862i	$0.277193\pi$
$348$	0	0
$349$	−16.0000	−0.856460	−0.428230	−	0.903670i	$-0.640863\pi$
$349$	−16.0000	−0.856460	−0.428230	+	0.903670i	$0.640863\pi$
$350$	2.00000	0.106904
$351$	1.00000	0.0533761
$352$	0	0
$353$	18.0000	0.958043	0.479022	−	0.877803i	$-0.340992\pi$
$353$	18.0000	0.958043	0.479022	+	0.877803i	$0.340992\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	18.0000	0.951330
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	−1.00000	−0.0527046
$361$	−15.0000	−0.789474
$362$	−22.0000	−1.15629
$363$	−11.0000	−0.577350
$364$	2.00000	0.104828
$365$	−8.00000	−0.418739
$366$	−10.0000	−0.522708
$367$	−16.0000	−0.835193	−0.417597	−	0.908633i	$-0.637127\pi$
$367$	−16.0000	−0.835193	−0.417597	+	0.908633i	$0.637127\pi$
$368$	−6.00000	−0.312772
$369$	−6.00000	−0.312348
$370$	−2.00000	−0.103975
$371$	−12.0000	−0.623009
$372$	−4.00000	−0.207390
$373$	−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
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	0	0
$378$	2.00000	0.102869
$379$	−10.0000	−0.513665	−0.256833	−	0.966456i	$-0.582679\pi$
$379$	−10.0000	−0.513665	−0.256833	+	0.966456i	$0.582679\pi$
$380$	−2.00000	−0.102598
$381$	8.00000	0.409852
$382$	24.0000	1.22795
$383$	−36.0000	−1.83951	−0.919757	−	0.392488i	$-0.871614\pi$
$383$	−36.0000	−1.83951	−0.919757	+	0.392488i	$0.871614\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	20.0000	1.01797
$387$	−4.00000	−0.203331
$388$	8.00000	0.406138
$389$	12.0000	0.608424	0.304212	−	0.952604i	$-0.401607\pi$
$389$	12.0000	0.608424	0.304212	+	0.952604i	$0.401607\pi$
$390$	−1.00000	−0.0506370
$391$	0	0
$392$	−3.00000	−0.151523
$393$	−18.0000	−0.907980
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	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
$399$	4.00000	0.200250
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	8.00000	0.399004
$403$	−4.00000	−0.199254
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	0	0
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	6.00000	0.296319
$411$	18.0000	0.887875
$412$	8.00000	0.394132
$413$	0	0
$414$	−6.00000	−0.294884
$415$	12.0000	0.589057
$416$	1.00000	0.0490290
$417$	8.00000	0.391762
$418$	0	0
$419$	−18.0000	−0.879358	−0.439679	−	0.898155i	$-0.644908\pi$
$419$	−18.0000	−0.879358	−0.439679	+	0.898155i	$0.644908\pi$
$420$	−2.00000	−0.0975900
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−4.00000	−0.194717
$423$	0	0
$424$	−6.00000	−0.291386
$425$	0	0
$426$	0	0
$427$	−20.0000	−0.967868
$428$	12.0000	0.580042
$429$	0	0
$430$	4.00000	0.192897
$431$	24.0000	1.15604	0.578020	−	0.816023i	$-0.303826\pi$
$431$	24.0000	1.15604	0.578020	+	0.816023i	$0.303826\pi$
$432$	1.00000	0.0481125
$433$	−34.0000	−1.63394	−0.816968	−	0.576683i	$-0.804347\pi$
$433$	−34.0000	−1.63394	−0.816968	+	0.576683i	$0.804347\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	−16.0000	−0.766261
$437$	−12.0000	−0.574038
$438$	8.00000	0.382255
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	2.00000	0.0949158
$445$	−6.00000	−0.284427
$446$	14.0000	0.662919
$447$	6.00000	0.283790
$448$	2.00000	0.0944911
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−12.0000	−0.564433
$453$	20.0000	0.939682
$454$	12.0000	0.563188
$455$	−2.00000	−0.0937614
$456$	2.00000	0.0936586
$457$	32.0000	1.49690	0.748448	−	0.663193i	$-0.230799\pi$
$457$	32.0000	1.49690	0.748448	+	0.663193i	$0.230799\pi$
$458$	−16.0000	−0.747631
$459$	0	0
$460$	6.00000	0.279751
$461$	18.0000	0.838344	0.419172	−	0.907907i	$-0.362320\pi$
$461$	18.0000	0.838344	0.419172	+	0.907907i	$0.362320\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	0	0
$465$	4.00000	0.185496
$466$	24.0000	1.11178
$467$	−12.0000	−0.555294	−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000	−0.555294	−0.277647	+	0.960683i	$0.589555\pi$
$468$	1.00000	0.0462250
$469$	16.0000	0.738811
$470$	0	0
$471$	2.00000	0.0921551
$472$	0	0
$473$	0	0
$474$	8.00000	0.367452
$475$	2.00000	0.0917663
$476$	0	0
$477$	−6.00000	−0.274721
$478$	−24.0000	−1.09773
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	−1.00000	−0.0456435
$481$	2.00000	0.0911922
$482$	2.00000	0.0910975
$483$	−12.0000	−0.546019
$484$	−11.0000	−0.500000
$485$	−8.00000	−0.363261
$486$	1.00000	0.0453609
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−10.0000	−0.452679
$489$	20.0000	0.904431
$490$	3.00000	0.135526
$491$	6.00000	0.270776	0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000	0.270776	0.135388	+	0.990793i	$0.456772\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	2.00000	0.0899843
$495$	0	0
$496$	−4.00000	−0.179605
$497$	0	0
$498$	−12.0000	−0.537733
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	−1.00000	−0.0447214
$501$	12.0000	0.536120
$502$	18.0000	0.803379
$503$	6.00000	0.267527	0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000	0.267527	0.133763	+	0.991013i	$0.457294\pi$
$504$	2.00000	0.0890871
$505$	12.0000	0.533993
$506$	0	0
$507$	1.00000	0.0444116
$508$	8.00000	0.354943
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	12.0000	0.529297
$515$	−8.00000	−0.352522
$516$	−4.00000	−0.176090
$517$	0	0
$518$	4.00000	0.175750
$519$	−6.00000	−0.263371
$520$	−1.00000	−0.0438529
$521$	−18.0000	−0.788594	−0.394297	−	0.918983i	$-0.629012\pi$
$521$	−18.0000	−0.788594	−0.394297	+	0.918983i	$0.629012\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	−18.0000	−0.786334
$525$	2.00000	0.0872872
$526$	6.00000	0.261612
$527$	0	0
$528$	0	0
$529$	13.0000	0.565217
$530$	6.00000	0.260623
$531$	0	0
$532$	4.00000	0.173422
$533$	−6.00000	−0.259889
$534$	6.00000	0.259645
$535$	−12.0000	−0.518805
$536$	8.00000	0.345547
$537$	18.0000	0.776757
$538$	12.0000	0.517357
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	20.0000	0.859867	0.429934	−	0.902861i	$-0.358537\pi$
$541$	20.0000	0.859867	0.429934	+	0.902861i	$0.358537\pi$
$542$	−4.00000	−0.171815
$543$	−22.0000	−0.944110
$544$	0	0
$545$	16.0000	0.685365
$546$	2.00000	0.0855921
$547$	−4.00000	−0.171028	−0.0855138	−	0.996337i	$-0.527253\pi$
$547$	−4.00000	−0.171028	−0.0855138	+	0.996337i	$0.527253\pi$
$548$	18.0000	0.768922
$549$	−10.0000	−0.426790
$550$	0	0
$551$	0	0
$552$	−6.00000	−0.255377
$553$	16.0000	0.680389
$554$	−22.0000	−0.934690
$555$	−2.00000	−0.0848953
$556$	8.00000	0.339276
$557$	−6.00000	−0.254228	−0.127114	−	0.991888i	$-0.540571\pi$
$557$	−6.00000	−0.254228	−0.127114	+	0.991888i	$0.540571\pi$
$558$	−4.00000	−0.169334
$559$	−4.00000	−0.169182
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	6.00000	0.253095
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	12.0000	0.504844
$566$	−4.00000	−0.168133
$567$	2.00000	0.0839921
$568$	0	0
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	−2.00000	−0.0837708
$571$	−16.0000	−0.669579	−0.334790	−	0.942293i	$-0.608665\pi$
$571$	−16.0000	−0.669579	−0.334790	+	0.942293i	$0.608665\pi$
$572$	0	0
$573$	24.0000	1.00261
$574$	−12.0000	−0.500870
$575$	−6.00000	−0.250217
$576$	1.00000	0.0416667
$577$	−4.00000	−0.166522	−0.0832611	−	0.996528i	$-0.526534\pi$
$577$	−4.00000	−0.166522	−0.0832611	+	0.996528i	$0.526534\pi$
$578$	−17.0000	−0.707107
$579$	20.0000	0.831172
$580$	0	0
$581$	−24.0000	−0.995688
$582$	8.00000	0.331611
$583$	0	0
$584$	8.00000	0.331042
$585$	−1.00000	−0.0413449
$586$	6.00000	0.247858
$587$	36.0000	1.48588	0.742940	−	0.669359i	$-0.233431\pi$
$587$	36.0000	1.48588	0.742940	+	0.669359i	$0.233431\pi$
$588$	−3.00000	−0.123718
$589$	−8.00000	−0.329634
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	0	0
$596$	6.00000	0.245770
$597$	−16.0000	−0.654836
$598$	−6.00000	−0.245358
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	1.00000	0.0408248
$601$	38.0000	1.55005	0.775026	−	0.631929i	$-0.217737\pi$
$601$	38.0000	1.55005	0.775026	+	0.631929i	$0.217737\pi$
$602$	−8.00000	−0.326056
$603$	8.00000	0.325785
$604$	20.0000	0.813788
$605$	11.0000	0.447214
$606$	−12.0000	−0.487467
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	10.0000	0.404888
$611$	0	0
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	−16.0000	−0.645707
$615$	6.00000	0.241943
$616$	0	0
$617$	18.0000	0.724653	0.362326	−	0.932051i	$-0.381983\pi$
$617$	18.0000	0.724653	0.362326	+	0.932051i	$0.381983\pi$
$618$	8.00000	0.321807
$619$	14.0000	0.562708	0.281354	−	0.959604i	$-0.409217\pi$
$619$	14.0000	0.562708	0.281354	+	0.959604i	$0.409217\pi$
$620$	4.00000	0.160644
$621$	−6.00000	−0.240772
$622$	0	0
$623$	12.0000	0.480770
$624$	1.00000	0.0400320
$625$	1.00000	0.0400000
$626$	14.0000	0.559553
$627$	0	0
$628$	2.00000	0.0798087
$629$	0	0
$630$	−2.00000	−0.0796819
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	−4.00000	−0.158986
$634$	−30.0000	−1.19145
$635$	−8.00000	−0.317470
$636$	−6.00000	−0.237915
$637$	−3.00000	−0.118864
$638$	0	0
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	42.0000	1.65890	0.829450	−	0.558581i	$-0.188654\pi$
$641$	42.0000	1.65890	0.829450	+	0.558581i	$0.188654\pi$
$642$	12.0000	0.473602
$643$	−40.0000	−1.57745	−0.788723	−	0.614749i	$-0.789257\pi$
$643$	−40.0000	−1.57745	−0.788723	+	0.614749i	$0.789257\pi$
$644$	−12.0000	−0.472866
$645$	4.00000	0.157500
$646$	0	0
$647$	−18.0000	−0.707653	−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000	−0.707653	−0.353827	+	0.935311i	$0.615120\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	1.00000	0.0392232
$651$	−8.00000	−0.313545
$652$	20.0000	0.783260
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	−16.0000	−0.625650
$655$	18.0000	0.703318
$656$	−6.00000	−0.234261
$657$	8.00000	0.312110
$658$	0	0
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	0	0
$661$	−28.0000	−1.08907	−0.544537	−	0.838737i	$-0.683295\pi$
$661$	−28.0000	−1.08907	−0.544537	+	0.838737i	$0.683295\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	−12.0000	−0.465690
$665$	−4.00000	−0.155113
$666$	2.00000	0.0774984
$667$	0	0
$668$	12.0000	0.464294
$669$	14.0000	0.541271
$670$	−8.00000	−0.309067
$671$	0	0
$672$	2.00000	0.0771517
$673$	26.0000	1.00223	0.501113	−	0.865382i	$-0.332924\pi$
$673$	26.0000	1.00223	0.501113	+	0.865382i	$0.332924\pi$
$674$	−34.0000	−1.30963
$675$	1.00000	0.0384900
$676$	1.00000	0.0384615
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	−12.0000	−0.460857
$679$	16.0000	0.614024
$680$	0	0
$681$	12.0000	0.459841
$682$	0	0
$683$	36.0000	1.37750	0.688751	−	0.724998i	$-0.258159\pi$
$683$	36.0000	1.37750	0.688751	+	0.724998i	$0.258159\pi$
$684$	2.00000	0.0764719
$685$	−18.0000	−0.687745
$686$	−20.0000	−0.763604
$687$	−16.0000	−0.610438
$688$	−4.00000	−0.152499
$689$	−6.00000	−0.228582
$690$	6.00000	0.228416
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	24.0000	0.911028
$695$	−8.00000	−0.303457
$696$	0	0
$697$	0	0
$698$	−16.0000	−0.605609
$699$	24.0000	0.907763
$700$	2.00000	0.0755929
$701$	−36.0000	−1.35970	−0.679851	−	0.733351i	$-0.737955\pi$
$701$	−36.0000	−1.35970	−0.679851	+	0.733351i	$0.737955\pi$
$702$	1.00000	0.0377426
$703$	4.00000	0.150863
$704$	0	0
$705$	0	0
$706$	18.0000	0.677439
$707$	−24.0000	−0.902613
$708$	0	0
$709$	8.00000	0.300446	0.150223	−	0.988652i	$-0.452001\pi$
$709$	8.00000	0.300446	0.150223	+	0.988652i	$0.452001\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	18.0000	0.672692
$717$	−24.0000	−0.896296
$718$	0	0
$719$	36.0000	1.34257	0.671287	−	0.741198i	$-0.265742\pi$
$719$	36.0000	1.34257	0.671287	+	0.741198i	$0.265742\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	−15.0000	−0.558242
$723$	2.00000	0.0743808
$724$	−22.0000	−0.817624
$725$	0	0
$726$	−11.0000	−0.408248
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	2.00000	0.0741249
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	0	0
$732$	−10.0000	−0.369611
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	−16.0000	−0.590571
$735$	3.00000	0.110657
$736$	−6.00000	−0.221163
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−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
$741$	2.00000	0.0734718
$742$	−12.0000	−0.440534
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	−4.00000	−0.146647
$745$	−6.00000	−0.219823
$746$	−22.0000	−0.805477
$747$	−12.0000	−0.439057
$748$	0	0
$749$	24.0000	0.876941
$750$	−1.00000	−0.0365148
$751$	32.0000	1.16770	0.583848	−	0.811863i	$-0.301546\pi$
$751$	32.0000	1.16770	0.583848	+	0.811863i	$0.301546\pi$
$752$	0	0
$753$	18.0000	0.655956
$754$	0	0
$755$	−20.0000	−0.727875
$756$	2.00000	0.0727393
$757$	14.0000	0.508839	0.254419	−	0.967094i	$-0.418116\pi$
$757$	14.0000	0.508839	0.254419	+	0.967094i	$0.418116\pi$
$758$	−10.0000	−0.363216
$759$	0	0
$760$	−2.00000	−0.0725476
$761$	−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
$763$	−32.0000	−1.15848
$764$	24.0000	0.868290
$765$	0	0
$766$	−36.0000	−1.30073
$767$	0	0
$768$	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
$771$	12.0000	0.432169
$772$	20.0000	0.719816
$773$	−42.0000	−1.51064	−0.755318	−	0.655359i	$-0.772517\pi$
$773$	−42.0000	−1.51064	−0.755318	+	0.655359i	$0.772517\pi$
$774$	−4.00000	−0.143777
$775$	−4.00000	−0.143684
$776$	8.00000	0.287183
$777$	4.00000	0.143499
$778$	12.0000	0.430221
$779$	−12.0000	−0.429945
$780$	−1.00000	−0.0358057
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−2.00000	−0.0713831
$786$	−18.0000	−0.642039
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	6.00000	0.213741
$789$	6.00000	0.213606
$790$	−8.00000	−0.284627
$791$	−24.0000	−0.853342
$792$	0	0
$793$	−10.0000	−0.355110
$794$	−22.0000	−0.780751
$795$	6.00000	0.212798
$796$	−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	$0.396721\pi$
$798$	4.00000	0.141598
$799$	0	0
$800$	1.00000	0.0353553
$801$	6.00000	0.212000
$802$	−6.00000	−0.211867
$803$	0	0
$804$	8.00000	0.282138
$805$	12.0000	0.422944
$806$	−4.00000	−0.140894
$807$	12.0000	0.422420
$808$	−12.0000	−0.422159
$809$	18.0000	0.632846	0.316423	−	0.948618i	$-0.397518\pi$
$809$	18.0000	0.632846	0.316423	+	0.948618i	$0.397518\pi$
$810$	−1.00000	−0.0351364
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	0	0
$813$	−4.00000	−0.140286
$814$	0	0
$815$	−20.0000	−0.700569
$816$	0	0
$817$	−8.00000	−0.279885
$818$	14.0000	0.489499
$819$	2.00000	0.0698857
$820$	6.00000	0.209529
$821$	30.0000	1.04701	0.523504	−	0.852023i	$-0.324625\pi$
$821$	30.0000	1.04701	0.523504	+	0.852023i	$0.324625\pi$
$822$	18.0000	0.627822
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	8.00000	0.278693
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	−6.00000	−0.208514
$829$	−34.0000	−1.18087	−0.590434	−	0.807086i	$-0.701044\pi$
$829$	−34.0000	−1.18087	−0.590434	+	0.807086i	$0.701044\pi$
$830$	12.0000	0.416526
$831$	−22.0000	−0.763172
$832$	1.00000	0.0346688
$833$	0	0
$834$	8.00000	0.277017
$835$	−12.0000	−0.415277
$836$	0	0
$837$	−4.00000	−0.138260
$838$	−18.0000	−0.621800
$839$	−24.0000	−0.828572	−0.414286	−	0.910147i	$-0.635969\pi$
$839$	−24.0000	−0.828572	−0.414286	+	0.910147i	$0.635969\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	−4.00000	−0.137849
$843$	6.00000	0.206651
$844$	−4.00000	−0.137686
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−22.0000	−0.755929
$848$	−6.00000	−0.206041
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−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	$0.353164\pi$
$854$	−20.0000	−0.684386
$855$	−2.00000	−0.0683986
$856$	12.0000	0.410152
$857$	12.0000	0.409912	0.204956	−	0.978771i	$-0.434295\pi$
$857$	12.0000	0.409912	0.204956	+	0.978771i	$0.434295\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	4.00000	0.136399
$861$	−12.0000	−0.408959
$862$	24.0000	0.817443
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	1.00000	0.0340207
$865$	6.00000	0.204006
$866$	−34.0000	−1.15537
$867$	−17.0000	−0.577350
$868$	−8.00000	−0.271538
$869$	0	0
$870$	0	0
$871$	8.00000	0.271070
$872$	−16.0000	−0.541828
$873$	8.00000	0.270759
$874$	−12.0000	−0.405906
$875$	−2.00000	−0.0676123
$876$	8.00000	0.270295
$877$	−34.0000	−1.14810	−0.574049	−	0.818821i	$-0.694628\pi$
$877$	−34.0000	−1.14810	−0.574049	+	0.818821i	$0.694628\pi$
$878$	8.00000	0.269987
$879$	6.00000	0.202375
$880$	0	0
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	−3.00000	−0.101015
$883$	−52.0000	−1.74994	−0.874970	−	0.484178i	$-0.839119\pi$
$883$	−52.0000	−1.74994	−0.874970	+	0.484178i	$0.839119\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	18.0000	0.604381	0.302190	−	0.953248i	$-0.402282\pi$
$887$	18.0000	0.604381	0.302190	+	0.953248i	$0.402282\pi$
$888$	2.00000	0.0671156
$889$	16.0000	0.536623
$890$	−6.00000	−0.201120
$891$	0	0
$892$	14.0000	0.468755
$893$	0	0
$894$	6.00000	0.200670
$895$	−18.0000	−0.601674
$896$	2.00000	0.0668153
$897$	−6.00000	−0.200334
$898$	−18.0000	−0.600668
$899$	0	0
$900$	1.00000	0.0333333
$901$	0	0
$902$	0	0
$903$	−8.00000	−0.266223
$904$	−12.0000	−0.399114
$905$	22.0000	0.731305
$906$	20.0000	0.664455
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	12.0000	0.398234
$909$	−12.0000	−0.398015
$910$	−2.00000	−0.0662994
$911$	−36.0000	−1.19273	−0.596367	−	0.802712i	$-0.703390\pi$
$911$	−36.0000	−1.19273	−0.596367	+	0.802712i	$0.703390\pi$
$912$	2.00000	0.0662266
$913$	0	0
$914$	32.0000	1.05847
$915$	10.0000	0.330590
$916$	−16.0000	−0.528655
$917$	−36.0000	−1.18882
$918$	0	0
$919$	56.0000	1.84727	0.923635	−	0.383274i	$-0.125203\pi$
$919$	56.0000	1.84727	0.923635	+	0.383274i	$0.125203\pi$
$920$	6.00000	0.197814
$921$	−16.0000	−0.527218
$922$	18.0000	0.592798
$923$	0	0
$924$	0	0
$925$	2.00000	0.0657596
$926$	14.0000	0.460069
$927$	8.00000	0.262754
$928$	0	0
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	4.00000	0.131165
$931$	−6.00000	−0.196642
$932$	24.0000	0.786146
$933$	0	0
$934$	−12.0000	−0.392652
$935$	0	0
$936$	1.00000	0.0326860
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	16.0000	0.522419
$939$	14.0000	0.456873
$940$	0	0
$941$	6.00000	0.195594	0.0977972	−	0.995206i	$-0.468820\pi$
$941$	6.00000	0.195594	0.0977972	+	0.995206i	$0.468820\pi$
$942$	2.00000	0.0651635
$943$	36.0000	1.17232
$944$	0	0
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	36.0000	1.16984	0.584921	−	0.811090i	$-0.301125\pi$
$947$	36.0000	1.16984	0.584921	+	0.811090i	$0.301125\pi$
$948$	8.00000	0.259828
$949$	8.00000	0.259691
$950$	2.00000	0.0648886
$951$	−30.0000	−0.972817
$952$	0	0
$953$	48.0000	1.55487	0.777436	−	0.628962i	$-0.216520\pi$
$953$	48.0000	1.55487	0.777436	+	0.628962i	$0.216520\pi$
$954$	−6.00000	−0.194257
$955$	−24.0000	−0.776622
$956$	−24.0000	−0.776215
$957$	0	0
$958$	−24.0000	−0.775405
$959$	36.0000	1.16250
$960$	−1.00000	−0.0322749
$961$	−15.0000	−0.483871
$962$	2.00000	0.0644826
$963$	12.0000	0.386695
$964$	2.00000	0.0644157
$965$	−20.0000	−0.643823
$966$	−12.0000	−0.386094
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	−11.0000	−0.353553
$969$	0	0
$970$	−8.00000	−0.256865
$971$	18.0000	0.577647	0.288824	−	0.957382i	$-0.406736\pi$
$971$	18.0000	0.577647	0.288824	+	0.957382i	$0.406736\pi$
$972$	1.00000	0.0320750
$973$	16.0000	0.512936
$974$	2.00000	0.0640841
$975$	1.00000	0.0320256
$976$	−10.0000	−0.320092
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	20.0000	0.639529
$979$	0	0
$980$	3.00000	0.0958315
$981$	−16.0000	−0.510841
$982$	6.00000	0.191468
$983$	−12.0000	−0.382741	−0.191370	−	0.981518i	$-0.561293\pi$
$983$	−12.0000	−0.382741	−0.191370	+	0.981518i	$0.561293\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	0	0
$987$	0	0
$988$	2.00000	0.0636285
$989$	24.0000	0.763156
$990$	0	0
$991$	32.0000	1.01651	0.508257	−	0.861206i	$-0.330290\pi$
$991$	32.0000	1.01651	0.508257	+	0.861206i	$0.330290\pi$
$992$	−4.00000	−0.127000
$993$	26.0000	0.825085
$994$	0	0
$995$	16.0000	0.507234
$996$	−12.0000	−0.380235
$997$	−46.0000	−1.45683	−0.728417	−	0.685134i	$-0.759744\pi$
$997$	−46.0000	−1.45683	−0.728417	+	0.685134i	$0.759744\pi$
$998$	14.0000	0.443162
$999$	2.00000	0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	390.2.a.g.1.1	✓	1
3.2	odd	2		1170.2.a.g.1.1		1
4.3	odd	2		3120.2.a.b.1.1		1
5.2	odd	4		1950.2.e.k.1249.2		2
5.3	odd	4		1950.2.e.k.1249.1		2
5.4	even	2		1950.2.a.b.1.1		1
12.11	even	2		9360.2.a.bg.1.1		1
13.5	odd	4		5070.2.b.n.1351.1		2
13.8	odd	4		5070.2.b.n.1351.2		2
13.12	even	2		5070.2.a.k.1.1		1
15.2	even	4		5850.2.e.r.5149.1		2
15.8	even	4		5850.2.e.r.5149.2		2
15.14	odd	2		5850.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
390.2.a.g.1.1	✓	1	1.1	even	1	trivial
1170.2.a.g.1.1		1	3.2	odd	2
1950.2.a.b.1.1		1	5.4	even	2
1950.2.e.k.1249.1		2	5.3	odd	4
1950.2.e.k.1249.2		2	5.2	odd	4
3120.2.a.b.1.1		1	4.3	odd	2
5070.2.a.k.1.1		1	13.12	even	2
5070.2.b.n.1351.1		2	13.5	odd	4
5070.2.b.n.1351.2		2	13.8	odd	4
5850.2.a.bk.1.1		1	15.14	odd	2
5850.2.e.r.5149.1		2	15.2	even	4
5850.2.e.r.5149.2		2	15.8	even	4
9360.2.a.bg.1.1		1	12.11	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:	\( N \)	\(=\)	\( 390 = 2 \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	390.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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