LMFDB - Embedded newform 690.2.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [690,2,Mod(1,690)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(690, 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("690.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 690 = 2 \cdot 3 \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	690.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:	$5.50967773947$
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	$\chi$	$=$	690.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$

\(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +4.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -2.00000 q^{11} -1.00000 q^{12} -4.00000 q^{14} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} -1.00000 q^{20} -4.00000 q^{21} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.00000 q^{28} -4.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +2.00000 q^{33} -2.00000 q^{34} -4.00000 q^{35} +1.00000 q^{36} +10.0000 q^{37} +1.00000 q^{40} +6.00000 q^{41} +4.00000 q^{42} +2.00000 q^{43} -2.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} +12.0000 q^{47} -1.00000 q^{48} +9.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{55} -4.00000 q^{56} +4.00000 q^{58} +12.0000 q^{59} +1.00000 q^{60} -14.0000 q^{61} +4.00000 q^{63} +1.00000 q^{64} -2.00000 q^{66} +2.00000 q^{67} +2.00000 q^{68} -1.00000 q^{69} +4.00000 q^{70} -2.00000 q^{71} -1.00000 q^{72} +6.00000 q^{73} -10.0000 q^{74} -1.00000 q^{75} -8.00000 q^{77} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +8.00000 q^{83} -4.00000 q^{84} -2.00000 q^{85} -2.00000 q^{86} +4.00000 q^{87} +2.00000 q^{88} -8.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -12.0000 q^{94} +1.00000 q^{96} -9.00000 q^{98} -2.00000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−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
$7$	4.00000	1.51186	0.755929	−	0.654654i	$-0.227186\pi$
$7$	4.00000	1.51186	0.755929	+	0.654654i	$0.227186\pi$
$8$	−1.00000	−0.353553
$9$	1.00000	0.333333
$10$	1.00000	0.316228
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	−1.00000	−0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	−4.00000	−1.06904
$15$	1.00000	0.258199
$16$	1.00000	0.250000
$17$	2.00000	0.485071	0.242536	−	0.970143i	$-0.422021\pi$
$17$	2.00000	0.485071	0.242536	+	0.970143i	$0.422021\pi$
$18$	−1.00000	−0.235702
$19$	0	0		−	1.00000i	$-0.5\pi$
$19$	0	0			1.00000i	$0.5\pi$
$20$	−1.00000	−0.223607
$21$	−4.00000	−0.872872
$22$	2.00000	0.426401
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.00000	0.755929
$29$	−4.00000	−0.742781	−0.371391	−	0.928477i	$-0.621119\pi$
$29$	−4.00000	−0.742781	−0.371391	+	0.928477i	$0.621119\pi$
$30$	−1.00000	−0.182574
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	2.00000	0.348155
$34$	−2.00000	−0.342997
$35$	−4.00000	−0.676123
$36$	1.00000	0.166667
$37$	10.0000	1.64399	0.821995	−	0.569495i	$-0.192861\pi$
$37$	10.0000	1.64399	0.821995	+	0.569495i	$0.192861\pi$
$38$	0	0
$39$	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	$0.344787\pi$
$42$	4.00000	0.617213
$43$	2.00000	0.304997	0.152499	−	0.988304i	$-0.451268\pi$
$43$	2.00000	0.304997	0.152499	+	0.988304i	$0.451268\pi$
$44$	−2.00000	−0.301511
$45$	−1.00000	−0.149071
$46$	−1.00000	−0.147442
$47$	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
$49$	9.00000	1.28571
$50$	−1.00000	−0.141421
$51$	−2.00000	−0.280056
$52$	0	0
$53$	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
$55$	2.00000	0.269680
$56$	−4.00000	−0.534522
$57$	0	0
$58$	4.00000	0.525226
$59$	12.0000	1.56227	0.781133	−	0.624364i	$-0.214642\pi$
$59$	12.0000	1.56227	0.781133	+	0.624364i	$0.214642\pi$
$60$	1.00000	0.129099
$61$	−14.0000	−1.79252	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−14.0000	−1.79252	−0.896258	+	0.443533i	$0.853725\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	0	0
$66$	−2.00000	−0.246183
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	2.00000	0.242536
$69$	−1.00000	−0.120386
$70$	4.00000	0.478091
$71$	−2.00000	−0.237356	−0.118678	−	0.992933i	$-0.537866\pi$
$71$	−2.00000	−0.237356	−0.118678	+	0.992933i	$0.537866\pi$
$72$	−1.00000	−0.117851
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	−10.0000	−1.16248
$75$	−1.00000	−0.115470
$76$	0	0
$77$	−8.00000	−0.911685
$78$	0	0
$79$	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
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−4.00000	−0.436436
$85$	−2.00000	−0.216930
$86$	−2.00000	−0.215666
$87$	4.00000	0.428845
$88$	2.00000	0.213201
$89$	−8.00000	−0.847998	−0.423999	−	0.905663i	$-0.639374\pi$
$89$	−8.00000	−0.847998	−0.423999	+	0.905663i	$0.639374\pi$
$90$	1.00000	0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	−12.0000	−1.23771
$95$	0	0
$96$	1.00000	0.102062
$97$	0	0		−	1.00000i	$-0.5\pi$
$97$	0	0			1.00000i	$0.5\pi$
$98$	−9.00000	−0.909137
$99$	−2.00000	−0.201008
$100$	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	$0.703650\pi$
$102$	2.00000	0.198030
$103$	4.00000	0.394132	0.197066	−	0.980390i	$-0.436859\pi$
$103$	4.00000	0.394132	0.197066	+	0.980390i	$0.436859\pi$
$104$	0	0
$105$	4.00000	0.390360
$106$	−6.00000	−0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	−1.00000	−0.0962250
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−2.00000	−0.190693
$111$	−10.0000	−0.949158
$112$	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
$115$	−1.00000	−0.0932505
$116$	−4.00000	−0.371391
$117$	0	0
$118$	−12.0000	−1.10469
$119$	8.00000	0.733359
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	14.0000	1.26750
$123$	−6.00000	−0.541002
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	−4.00000	−0.356348
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−1.00000	−0.0883883
$129$	−2.00000	−0.176090
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−2.00000	−0.172774
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	−2.00000	−0.170872	−0.0854358	−	0.996344i	$-0.527228\pi$
$137$	−2.00000	−0.170872	−0.0854358	+	0.996344i	$0.527228\pi$
$138$	1.00000	0.0851257
$139$	20.0000	1.69638	0.848189	−	0.529694i	$-0.177693\pi$
$139$	20.0000	1.69638	0.848189	+	0.529694i	$0.177693\pi$
$140$	−4.00000	−0.338062
$141$	−12.0000	−1.01058
$142$	2.00000	0.167836
$143$	0	0
$144$	1.00000	0.0833333
$145$	4.00000	0.332182
$146$	−6.00000	−0.496564
$147$	−9.00000	−0.742307
$148$	10.0000	0.821995
$149$	−14.0000	−1.14692	−0.573462	−	0.819232i	$-0.694400\pi$
$149$	−14.0000	−1.14692	−0.573462	+	0.819232i	$0.694400\pi$
$150$	1.00000	0.0816497
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	8.00000	0.644658
$155$	0	0
$156$	0	0
$157$	−22.0000	−1.75579	−0.877896	−	0.478852i	$-0.841053\pi$
$157$	−22.0000	−1.75579	−0.877896	+	0.478852i	$0.841053\pi$
$158$	−8.00000	−0.636446
$159$	−6.00000	−0.475831
$160$	1.00000	0.0790569
$161$	4.00000	0.315244
$162$	−1.00000	−0.0785674
$163$	−16.0000	−1.25322	−0.626608	−	0.779334i	$-0.715557\pi$
$163$	−16.0000	−1.25322	−0.626608	+	0.779334i	$0.715557\pi$
$164$	6.00000	0.468521
$165$	−2.00000	−0.155700
$166$	−8.00000	−0.620920
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	4.00000	0.308607
$169$	−13.0000	−1.00000
$170$	2.00000	0.153393
$171$	0	0
$172$	2.00000	0.152499
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	−4.00000	−0.303239
$175$	4.00000	0.302372
$176$	−2.00000	−0.150756
$177$	−12.0000	−0.901975
$178$	8.00000	0.599625
$179$	4.00000	0.298974	0.149487	−	0.988764i	$-0.452238\pi$
$179$	4.00000	0.298974	0.149487	+	0.988764i	$0.452238\pi$
$180$	−1.00000	−0.0745356
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	0	0
$183$	14.0000	1.03491
$184$	−1.00000	−0.0737210
$185$	−10.0000	−0.735215
$186$	0	0
$187$	−4.00000	−0.292509
$188$	12.0000	0.875190
$189$	−4.00000	−0.290957
$190$	0	0
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	−1.00000	−0.0721688
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	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	$0.568565\pi$
$198$	2.00000	0.142134
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	−1.00000	−0.0707107
$201$	−2.00000	−0.141069
$202$	12.0000	0.844317
$203$	−16.0000	−1.12298
$204$	−2.00000	−0.140028
$205$	−6.00000	−0.419058
$206$	−4.00000	−0.278693
$207$	1.00000	0.0695048
$208$	0	0
$209$	0	0
$210$	−4.00000	−0.276026
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	6.00000	0.412082
$213$	2.00000	0.137038
$214$	12.0000	0.820303
$215$	−2.00000	−0.136399
$216$	1.00000	0.0680414
$217$	0	0
$218$	−14.0000	−0.948200
$219$	−6.00000	−0.405442
$220$	2.00000	0.134840
$221$	0	0
$222$	10.0000	0.671156
$223$	2.00000	0.133930	0.0669650	−	0.997755i	$-0.478668\pi$
$223$	2.00000	0.133930	0.0669650	+	0.997755i	$0.478668\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	−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
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	1.00000	0.0659380
$231$	8.00000	0.526361
$232$	4.00000	0.262613
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	12.0000	0.781133
$237$	−8.00000	−0.519656
$238$	−8.00000	−0.518563
$239$	22.0000	1.42306	0.711531	−	0.702655i	$-0.248002\pi$
$239$	22.0000	1.42306	0.711531	+	0.702655i	$0.248002\pi$
$240$	1.00000	0.0645497
$241$	−14.0000	−0.901819	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	−14.0000	−0.901819	−0.450910	+	0.892570i	$0.648900\pi$
$242$	7.00000	0.449977
$243$	−1.00000	−0.0641500
$244$	−14.0000	−0.896258
$245$	−9.00000	−0.574989
$246$	6.00000	0.382546
$247$	0	0
$248$	0	0
$249$	−8.00000	−0.506979
$250$	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	$0.307688\pi$
$252$	4.00000	0.251976
$253$	−2.00000	−0.125739
$254$	−6.00000	−0.376473
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	2.00000	0.124757	0.0623783	−	0.998053i	$-0.480131\pi$
$257$	2.00000	0.124757	0.0623783	+	0.998053i	$0.480131\pi$
$258$	2.00000	0.124515
$259$	40.0000	2.48548
$260$	0	0
$261$	−4.00000	−0.247594
$262$	0	0
$263$	−32.0000	−1.97320	−0.986602	−	0.163144i	$-0.947836\pi$
$263$	−32.0000	−1.97320	−0.986602	+	0.163144i	$0.947836\pi$
$264$	−2.00000	−0.123091
$265$	−6.00000	−0.368577
$266$	0	0
$267$	8.00000	0.489592
$268$	2.00000	0.122169
$269$	24.0000	1.46331	0.731653	−	0.681677i	$-0.238749\pi$
$269$	24.0000	1.46331	0.731653	+	0.681677i	$0.238749\pi$
$270$	−1.00000	−0.0608581
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	2.00000	0.120824
$275$	−2.00000	−0.120605
$276$	−1.00000	−0.0601929
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−20.0000	−1.19952
$279$	0	0
$280$	4.00000	0.239046
$281$	32.0000	1.90896	0.954480	−	0.298275i	$-0.0964112\pi$
$281$	32.0000	1.90896	0.954480	+	0.298275i	$0.0964112\pi$
$282$	12.0000	0.714590
$283$	−10.0000	−0.594438	−0.297219	−	0.954809i	$-0.596059\pi$
$283$	−10.0000	−0.594438	−0.297219	+	0.954809i	$0.596059\pi$
$284$	−2.00000	−0.118678
$285$	0	0
$286$	0	0
$287$	24.0000	1.41668
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	0	0
$292$	6.00000	0.351123
$293$	14.0000	0.817889	0.408944	−	0.912559i	$-0.365897\pi$
$293$	14.0000	0.817889	0.408944	+	0.912559i	$0.365897\pi$
$294$	9.00000	0.524891
$295$	−12.0000	−0.698667
$296$	−10.0000	−0.581238
$297$	2.00000	0.116052
$298$	14.0000	0.810998
$299$	0	0
$300$	−1.00000	−0.0577350
$301$	8.00000	0.461112
$302$	12.0000	0.690522
$303$	12.0000	0.689382
$304$	0	0
$305$	14.0000	0.801638
$306$	−2.00000	−0.114332
$307$	4.00000	0.228292	0.114146	−	0.993464i	$-0.463587\pi$
$307$	4.00000	0.228292	0.114146	+	0.993464i	$0.463587\pi$
$308$	−8.00000	−0.455842
$309$	−4.00000	−0.227552
$310$	0	0
$311$	−2.00000	−0.113410	−0.0567048	−	0.998391i	$-0.518059\pi$
$311$	−2.00000	−0.113410	−0.0567048	+	0.998391i	$0.518059\pi$
$312$	0	0
$313$	20.0000	1.13047	0.565233	−	0.824931i	$-0.308786\pi$
$313$	20.0000	1.13047	0.565233	+	0.824931i	$0.308786\pi$
$314$	22.0000	1.24153
$315$	−4.00000	−0.225374
$316$	8.00000	0.450035
$317$	−10.0000	−0.561656	−0.280828	−	0.959758i	$-0.590609\pi$
$317$	−10.0000	−0.561656	−0.280828	+	0.959758i	$0.590609\pi$
$318$	6.00000	0.336463
$319$	8.00000	0.447914
$320$	−1.00000	−0.0559017
$321$	12.0000	0.669775
$322$	−4.00000	−0.222911
$323$	0	0
$324$	1.00000	0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	−14.0000	−0.774202
$328$	−6.00000	−0.331295
$329$	48.0000	2.64633
$330$	2.00000	0.110096
$331$	−28.0000	−1.53902	−0.769510	−	0.638635i	$-0.779499\pi$
$331$	−28.0000	−1.53902	−0.769510	+	0.638635i	$0.779499\pi$
$332$	8.00000	0.439057
$333$	10.0000	0.547997
$334$	0	0
$335$	−2.00000	−0.109272
$336$	−4.00000	−0.218218
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	13.0000	0.707107
$339$	−6.00000	−0.325875
$340$	−2.00000	−0.108465
$341$	0	0
$342$	0	0
$343$	8.00000	0.431959
$344$	−2.00000	−0.107833
$345$	1.00000	0.0538382
$346$	6.00000	0.322562
$347$	4.00000	0.214731	0.107366	−	0.994220i	$-0.465758\pi$
$347$	4.00000	0.214731	0.107366	+	0.994220i	$0.465758\pi$
$348$	4.00000	0.214423
$349$	−30.0000	−1.60586	−0.802932	−	0.596071i	$-0.796728\pi$
$349$	−30.0000	−1.60586	−0.802932	+	0.596071i	$0.796728\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	2.00000	0.106600
$353$	−34.0000	−1.80964	−0.904819	−	0.425797i	$-0.859994\pi$
$353$	−34.0000	−1.80964	−0.904819	+	0.425797i	$0.859994\pi$
$354$	12.0000	0.637793
$355$	2.00000	0.106149
$356$	−8.00000	−0.423999
$357$	−8.00000	−0.423405
$358$	−4.00000	−0.211407
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	1.00000	0.0527046
$361$	−19.0000	−1.00000
$362$	6.00000	0.315353
$363$	7.00000	0.367405
$364$	0	0
$365$	−6.00000	−0.314054
$366$	−14.0000	−0.731792
$367$	4.00000	0.208798	0.104399	−	0.994535i	$-0.466708\pi$
$367$	4.00000	0.208798	0.104399	+	0.994535i	$0.466708\pi$
$368$	1.00000	0.0521286
$369$	6.00000	0.312348
$370$	10.0000	0.519875
$371$	24.0000	1.24602
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	4.00000	0.206835
$375$	1.00000	0.0516398
$376$	−12.0000	−0.618853
$377$	0	0
$378$	4.00000	0.205738
$379$	4.00000	0.205466	0.102733	−	0.994709i	$-0.467241\pi$
$379$	4.00000	0.205466	0.102733	+	0.994709i	$0.467241\pi$
$380$	0	0
$381$	−6.00000	−0.307389
$382$	0	0
$383$	−24.0000	−1.22634	−0.613171	−	0.789950i	$-0.710106\pi$
$383$	−24.0000	−1.22634	−0.613171	+	0.789950i	$0.710106\pi$
$384$	1.00000	0.0510310
$385$	8.00000	0.407718
$386$	14.0000	0.712581
$387$	2.00000	0.101666
$388$	0	0
$389$	22.0000	1.11544	0.557722	−	0.830028i	$-0.311675\pi$
$389$	22.0000	1.11544	0.557722	+	0.830028i	$0.311675\pi$
$390$	0	0
$391$	2.00000	0.101144
$392$	−9.00000	−0.454569
$393$	0	0
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	−2.00000	−0.100504
$397$	32.0000	1.60603	0.803017	−	0.595956i	$-0.203227\pi$
$397$	32.0000	1.60603	0.803017	+	0.595956i	$0.203227\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−12.0000	−0.599251	−0.299626	−	0.954057i	$-0.596862\pi$
$401$	−12.0000	−0.599251	−0.299626	+	0.954057i	$0.596862\pi$
$402$	2.00000	0.0997509
$403$	0	0
$404$	−12.0000	−0.597022
$405$	−1.00000	−0.0496904
$406$	16.0000	0.794067
$407$	−20.0000	−0.991363
$408$	2.00000	0.0990148
$409$	−10.0000	−0.494468	−0.247234	−	0.968956i	$-0.579522\pi$
$409$	−10.0000	−0.494468	−0.247234	+	0.968956i	$0.579522\pi$
$410$	6.00000	0.296319
$411$	2.00000	0.0986527
$412$	4.00000	0.197066
$413$	48.0000	2.36193
$414$	−1.00000	−0.0491473
$415$	−8.00000	−0.392705
$416$	0	0
$417$	−20.0000	−0.979404
$418$	0	0
$419$	−26.0000	−1.27018	−0.635092	−	0.772437i	$-0.719038\pi$
$419$	−26.0000	−1.27018	−0.635092	+	0.772437i	$0.719038\pi$
$420$	4.00000	0.195180
$421$	−6.00000	−0.292422	−0.146211	−	0.989253i	$-0.546708\pi$
$421$	−6.00000	−0.292422	−0.146211	+	0.989253i	$0.546708\pi$
$422$	−4.00000	−0.194717
$423$	12.0000	0.583460
$424$	−6.00000	−0.291386
$425$	2.00000	0.0970143
$426$	−2.00000	−0.0969003
$427$	−56.0000	−2.71003
$428$	−12.0000	−0.580042
$429$	0	0
$430$	2.00000	0.0964486
$431$	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
$433$	24.0000	1.15337	0.576683	−	0.816968i	$-0.304347\pi$
$433$	24.0000	1.15337	0.576683	+	0.816968i	$0.304347\pi$
$434$	0	0
$435$	−4.00000	−0.191785
$436$	14.0000	0.670478
$437$	0	0
$438$	6.00000	0.286691
$439$	20.0000	0.954548	0.477274	−	0.878755i	$-0.341625\pi$
$439$	20.0000	0.954548	0.477274	+	0.878755i	$0.341625\pi$
$440$	−2.00000	−0.0953463
$441$	9.00000	0.428571
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	−10.0000	−0.474579
$445$	8.00000	0.379236
$446$	−2.00000	−0.0947027
$447$	14.0000	0.662177
$448$	4.00000	0.188982
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	−1.00000	−0.0471405
$451$	−12.0000	−0.565058
$452$	6.00000	0.282216
$453$	12.0000	0.563809
$454$	−12.0000	−0.563188
$455$	0	0
$456$	0	0
$457$	−24.0000	−1.12267	−0.561336	−	0.827588i	$-0.689713\pi$
$457$	−24.0000	−1.12267	−0.561336	+	0.827588i	$0.689713\pi$
$458$	26.0000	1.21490
$459$	−2.00000	−0.0933520
$460$	−1.00000	−0.0466252
$461$	20.0000	0.931493	0.465746	−	0.884918i	$-0.345786\pi$
$461$	20.0000	0.931493	0.465746	+	0.884918i	$0.345786\pi$
$462$	−8.00000	−0.372194
$463$	18.0000	0.836531	0.418265	−	0.908325i	$-0.362638\pi$
$463$	18.0000	0.836531	0.418265	+	0.908325i	$0.362638\pi$
$464$	−4.00000	−0.185695
$465$	0	0
$466$	−10.0000	−0.463241
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	0	0
$469$	8.00000	0.369406
$470$	12.0000	0.553519
$471$	22.0000	1.01371
$472$	−12.0000	−0.552345
$473$	−4.00000	−0.183920
$474$	8.00000	0.367452
$475$	0	0
$476$	8.00000	0.366679
$477$	6.00000	0.274721
$478$	−22.0000	−1.00626
$479$	−40.0000	−1.82765	−0.913823	−	0.406112i	$-0.866884\pi$
$479$	−40.0000	−1.82765	−0.913823	+	0.406112i	$0.866884\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	14.0000	0.637683
$483$	−4.00000	−0.182006
$484$	−7.00000	−0.318182
$485$	0	0
$486$	1.00000	0.0453609
$487$	18.0000	0.815658	0.407829	−	0.913058i	$-0.366286\pi$
$487$	18.0000	0.815658	0.407829	+	0.913058i	$0.366286\pi$
$488$	14.0000	0.633750
$489$	16.0000	0.723545
$490$	9.00000	0.406579
$491$	−24.0000	−1.08310	−0.541552	−	0.840667i	$-0.682163\pi$
$491$	−24.0000	−1.08310	−0.541552	+	0.840667i	$0.682163\pi$
$492$	−6.00000	−0.270501
$493$	−8.00000	−0.360302
$494$	0	0
$495$	2.00000	0.0898933
$496$	0	0
$497$	−8.00000	−0.358849
$498$	8.00000	0.358489
$499$	−28.0000	−1.25345	−0.626726	−	0.779240i	$-0.715605\pi$
$499$	−28.0000	−1.25345	−0.626726	+	0.779240i	$0.715605\pi$
$500$	−1.00000	−0.0447214
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−8.00000	−0.356702	−0.178351	−	0.983967i	$-0.557076\pi$
$503$	−8.00000	−0.356702	−0.178351	+	0.983967i	$0.557076\pi$
$504$	−4.00000	−0.178174
$505$	12.0000	0.533993
$506$	2.00000	0.0889108
$507$	13.0000	0.577350
$508$	6.00000	0.266207
$509$	28.0000	1.24108	0.620539	−	0.784176i	$-0.286914\pi$
$509$	28.0000	1.24108	0.620539	+	0.784176i	$0.286914\pi$
$510$	−2.00000	−0.0885615
$511$	24.0000	1.06170
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−2.00000	−0.0882162
$515$	−4.00000	−0.176261
$516$	−2.00000	−0.0880451
$517$	−24.0000	−1.05552
$518$	−40.0000	−1.75750
$519$	6.00000	0.263371
$520$	0	0
$521$	32.0000	1.40195	0.700973	−	0.713188i	$-0.252749\pi$
$521$	32.0000	1.40195	0.700973	+	0.713188i	$0.252749\pi$
$522$	4.00000	0.175075
$523$	22.0000	0.961993	0.480996	−	0.876723i	$-0.340275\pi$
$523$	22.0000	0.961993	0.480996	+	0.876723i	$0.340275\pi$
$524$	0	0
$525$	−4.00000	−0.174574
$526$	32.0000	1.39527
$527$	0	0
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	6.00000	0.260623
$531$	12.0000	0.520756
$532$	0	0
$533$	0	0
$534$	−8.00000	−0.346194
$535$	12.0000	0.518805
$536$	−2.00000	−0.0863868
$537$	−4.00000	−0.172613
$538$	−24.0000	−1.03471
$539$	−18.0000	−0.775315
$540$	1.00000	0.0430331
$541$	−22.0000	−0.945854	−0.472927	−	0.881102i	$-0.656803\pi$
$541$	−22.0000	−0.945854	−0.472927	+	0.881102i	$0.656803\pi$
$542$	−8.00000	−0.343629
$543$	6.00000	0.257485
$544$	−2.00000	−0.0857493
$545$	−14.0000	−0.599694
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	−2.00000	−0.0854358
$549$	−14.0000	−0.597505
$550$	2.00000	0.0852803
$551$	0	0
$552$	1.00000	0.0425628
$553$	32.0000	1.36078
$554$	−8.00000	−0.339887
$555$	10.0000	0.424476
$556$	20.0000	0.848189
$557$	26.0000	1.10166	0.550828	−	0.834619i	$-0.314312\pi$
$557$	26.0000	1.10166	0.550828	+	0.834619i	$0.314312\pi$
$558$	0	0
$559$	0	0
$560$	−4.00000	−0.169031
$561$	4.00000	0.168880
$562$	−32.0000	−1.34984
$563$	8.00000	0.337160	0.168580	−	0.985688i	$-0.446082\pi$
$563$	8.00000	0.337160	0.168580	+	0.985688i	$0.446082\pi$
$564$	−12.0000	−0.505291
$565$	−6.00000	−0.252422
$566$	10.0000	0.420331
$567$	4.00000	0.167984
$568$	2.00000	0.0839181
$569$	−12.0000	−0.503066	−0.251533	−	0.967849i	$-0.580935\pi$
$569$	−12.0000	−0.503066	−0.251533	+	0.967849i	$0.580935\pi$
$570$	0	0
$571$	−36.0000	−1.50655	−0.753277	−	0.657704i	$-0.771528\pi$
$571$	−36.0000	−1.50655	−0.753277	+	0.657704i	$0.771528\pi$
$572$	0	0
$573$	0	0
$574$	−24.0000	−1.00174
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	13.0000	0.540729
$579$	14.0000	0.581820
$580$	4.00000	0.166091
$581$	32.0000	1.32758
$582$	0	0
$583$	−12.0000	−0.496989
$584$	−6.00000	−0.248282
$585$	0	0
$586$	−14.0000	−0.578335
$587$	−4.00000	−0.165098	−0.0825488	−	0.996587i	$-0.526306\pi$
$587$	−4.00000	−0.165098	−0.0825488	+	0.996587i	$0.526306\pi$
$588$	−9.00000	−0.371154
$589$	0	0
$590$	12.0000	0.494032
$591$	6.00000	0.246807
$592$	10.0000	0.410997
$593$	−30.0000	−1.23195	−0.615976	−	0.787765i	$-0.711238\pi$
$593$	−30.0000	−1.23195	−0.615976	+	0.787765i	$0.711238\pi$
$594$	−2.00000	−0.0820610
$595$	−8.00000	−0.327968
$596$	−14.0000	−0.573462
$597$	−16.0000	−0.654836
$598$	0	0
$599$	−30.0000	−1.22577	−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000	−1.22577	−0.612883	+	0.790173i	$0.709990\pi$
$600$	1.00000	0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	−8.00000	−0.326056
$603$	2.00000	0.0814463
$604$	−12.0000	−0.488273
$605$	7.00000	0.284590
$606$	−12.0000	−0.487467
$607$	−14.0000	−0.568242	−0.284121	−	0.958788i	$-0.591702\pi$
$607$	−14.0000	−0.568242	−0.284121	+	0.958788i	$0.591702\pi$
$608$	0	0
$609$	16.0000	0.648353
$610$	−14.0000	−0.566843
$611$	0	0
$612$	2.00000	0.0808452
$613$	14.0000	0.565455	0.282727	−	0.959200i	$-0.408761\pi$
$613$	14.0000	0.565455	0.282727	+	0.959200i	$0.408761\pi$
$614$	−4.00000	−0.161427
$615$	6.00000	0.241943
$616$	8.00000	0.322329
$617$	−14.0000	−0.563619	−0.281809	−	0.959470i	$-0.590935\pi$
$617$	−14.0000	−0.563619	−0.281809	+	0.959470i	$0.590935\pi$
$618$	4.00000	0.160904
$619$	−24.0000	−0.964641	−0.482321	−	0.875995i	$-0.660206\pi$
$619$	−24.0000	−0.964641	−0.482321	+	0.875995i	$0.660206\pi$
$620$	0	0
$621$	−1.00000	−0.0401286
$622$	2.00000	0.0801927
$623$	−32.0000	−1.28205
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.0000	−0.799361
$627$	0	0
$628$	−22.0000	−0.877896
$629$	20.0000	0.797452
$630$	4.00000	0.159364
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	−8.00000	−0.318223
$633$	−4.00000	−0.158986
$634$	10.0000	0.397151
$635$	−6.00000	−0.238103
$636$	−6.00000	−0.237915
$637$	0	0
$638$	−8.00000	−0.316723
$639$	−2.00000	−0.0791188
$640$	1.00000	0.0395285
$641$	−8.00000	−0.315981	−0.157991	−	0.987441i	$-0.550502\pi$
$641$	−8.00000	−0.315981	−0.157991	+	0.987441i	$0.550502\pi$
$642$	−12.0000	−0.473602
$643$	14.0000	0.552106	0.276053	−	0.961142i	$-0.410973\pi$
$643$	14.0000	0.552106	0.276053	+	0.961142i	$0.410973\pi$
$644$	4.00000	0.157622
$645$	2.00000	0.0787499
$646$	0	0
$647$	12.0000	0.471769	0.235884	−	0.971781i	$-0.424201\pi$
$647$	12.0000	0.471769	0.235884	+	0.971781i	$0.424201\pi$
$648$	−1.00000	−0.0392837
$649$	−24.0000	−0.942082
$650$	0	0
$651$	0	0
$652$	−16.0000	−0.626608
$653$	−26.0000	−1.01746	−0.508729	−	0.860927i	$-0.669885\pi$
$653$	−26.0000	−1.01746	−0.508729	+	0.860927i	$0.669885\pi$
$654$	14.0000	0.547443
$655$	0	0
$656$	6.00000	0.234261
$657$	6.00000	0.234082
$658$	−48.0000	−1.87123
$659$	−18.0000	−0.701180	−0.350590	−	0.936529i	$-0.614019\pi$
$659$	−18.0000	−0.701180	−0.350590	+	0.936529i	$0.614019\pi$
$660$	−2.00000	−0.0778499
$661$	6.00000	0.233373	0.116686	−	0.993169i	$-0.462773\pi$
$661$	6.00000	0.233373	0.116686	+	0.993169i	$0.462773\pi$
$662$	28.0000	1.08825
$663$	0	0
$664$	−8.00000	−0.310460
$665$	0	0
$666$	−10.0000	−0.387492
$667$	−4.00000	−0.154881
$668$	0	0
$669$	−2.00000	−0.0773245
$670$	2.00000	0.0772667
$671$	28.0000	1.08093
$672$	4.00000	0.154303
$673$	−18.0000	−0.693849	−0.346925	−	0.937893i	$-0.612774\pi$
$673$	−18.0000	−0.693849	−0.346925	+	0.937893i	$0.612774\pi$
$674$	4.00000	0.154074
$675$	−1.00000	−0.0384900
$676$	−13.0000	−0.500000
$677$	−22.0000	−0.845529	−0.422764	−	0.906240i	$-0.638940\pi$
$677$	−22.0000	−0.845529	−0.422764	+	0.906240i	$0.638940\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	2.00000	0.0766965
$681$	−12.0000	−0.459841
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	0	0
$685$	2.00000	0.0764161
$686$	−8.00000	−0.305441
$687$	26.0000	0.991962
$688$	2.00000	0.0762493
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	−20.0000	−0.760836	−0.380418	−	0.924815i	$-0.624220\pi$
$691$	−20.0000	−0.760836	−0.380418	+	0.924815i	$0.624220\pi$
$692$	−6.00000	−0.228086
$693$	−8.00000	−0.303895
$694$	−4.00000	−0.151838
$695$	−20.0000	−0.758643
$696$	−4.00000	−0.151620
$697$	12.0000	0.454532
$698$	30.0000	1.13552
$699$	−10.0000	−0.378235
$700$	4.00000	0.151186
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	0	0
$704$	−2.00000	−0.0753778
$705$	12.0000	0.451946
$706$	34.0000	1.27961
$707$	−48.0000	−1.80523
$708$	−12.0000	−0.450988
$709$	22.0000	0.826227	0.413114	−	0.910679i	$-0.364441\pi$
$709$	22.0000	0.826227	0.413114	+	0.910679i	$0.364441\pi$
$710$	−2.00000	−0.0750587
$711$	8.00000	0.300023
$712$	8.00000	0.299813
$713$	0	0
$714$	8.00000	0.299392
$715$	0	0
$716$	4.00000	0.149487
$717$	−22.0000	−0.821605
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	−1.00000	−0.0372678
$721$	16.0000	0.595871
$722$	19.0000	0.707107
$723$	14.0000	0.520666
$724$	−6.00000	−0.222988
$725$	−4.00000	−0.148556
$726$	−7.00000	−0.259794
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	4.00000	0.147945
$732$	14.0000	0.517455
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	−4.00000	−0.147643
$735$	9.00000	0.331970
$736$	−1.00000	−0.0368605
$737$	−4.00000	−0.147342
$738$	−6.00000	−0.220863
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	−10.0000	−0.367607
$741$	0	0
$742$	−24.0000	−0.881068
$743$	16.0000	0.586983	0.293492	−	0.955962i	$-0.405183\pi$
$743$	16.0000	0.586983	0.293492	+	0.955962i	$0.405183\pi$
$744$	0	0
$745$	14.0000	0.512920
$746$	22.0000	0.805477
$747$	8.00000	0.292705
$748$	−4.00000	−0.146254
$749$	−48.0000	−1.75388
$750$	−1.00000	−0.0365148
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	12.0000	0.437595
$753$	−18.0000	−0.655956
$754$	0	0
$755$	12.0000	0.436725
$756$	−4.00000	−0.145479
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	−4.00000	−0.145287
$759$	2.00000	0.0725954
$760$	0	0
$761$	2.00000	0.0724999	0.0362500	−	0.999343i	$-0.488459\pi$
$761$	2.00000	0.0724999	0.0362500	+	0.999343i	$0.488459\pi$
$762$	6.00000	0.217357
$763$	56.0000	2.02734
$764$	0	0
$765$	−2.00000	−0.0723102
$766$	24.0000	0.867155
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	6.00000	0.216366	0.108183	−	0.994131i	$-0.465497\pi$
$769$	6.00000	0.216366	0.108183	+	0.994131i	$0.465497\pi$
$770$	−8.00000	−0.288300
$771$	−2.00000	−0.0720282
$772$	−14.0000	−0.503871
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	−2.00000	−0.0718885
$775$	0	0
$776$	0	0
$777$	−40.0000	−1.43499
$778$	−22.0000	−0.788738
$779$	0	0
$780$	0	0
$781$	4.00000	0.143131
$782$	−2.00000	−0.0715199
$783$	4.00000	0.142948
$784$	9.00000	0.321429
$785$	22.0000	0.785214
$786$	0	0
$787$	−34.0000	−1.21197	−0.605985	−	0.795476i	$-0.707221\pi$
$787$	−34.0000	−1.21197	−0.605985	+	0.795476i	$0.707221\pi$
$788$	−6.00000	−0.213741
$789$	32.0000	1.13923
$790$	8.00000	0.284627
$791$	24.0000	0.853342
$792$	2.00000	0.0710669
$793$	0	0
$794$	−32.0000	−1.13564
$795$	6.00000	0.212798
$796$	16.0000	0.567105
$797$	50.0000	1.77109	0.885545	−	0.464553i	$-0.153785\pi$
$797$	50.0000	1.77109	0.885545	+	0.464553i	$0.153785\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−1.00000	−0.0353553
$801$	−8.00000	−0.282666
$802$	12.0000	0.423735
$803$	−12.0000	−0.423471
$804$	−2.00000	−0.0705346
$805$	−4.00000	−0.140981
$806$	0	0
$807$	−24.0000	−0.844840
$808$	12.0000	0.422159
$809$	−30.0000	−1.05474	−0.527372	−	0.849635i	$-0.676823\pi$
$809$	−30.0000	−1.05474	−0.527372	+	0.849635i	$0.676823\pi$
$810$	1.00000	0.0351364
$811$	52.0000	1.82597	0.912983	−	0.407997i	$-0.133772\pi$
$811$	52.0000	1.82597	0.912983	+	0.407997i	$0.133772\pi$
$812$	−16.0000	−0.561490
$813$	−8.00000	−0.280572
$814$	20.0000	0.701000
$815$	16.0000	0.560456
$816$	−2.00000	−0.0700140
$817$	0	0
$818$	10.0000	0.349642
$819$	0	0
$820$	−6.00000	−0.209529
$821$	−16.0000	−0.558404	−0.279202	−	0.960232i	$-0.590070\pi$
$821$	−16.0000	−0.558404	−0.279202	+	0.960232i	$0.590070\pi$
$822$	−2.00000	−0.0697580
$823$	−10.0000	−0.348578	−0.174289	−	0.984695i	$-0.555763\pi$
$823$	−10.0000	−0.348578	−0.174289	+	0.984695i	$0.555763\pi$
$824$	−4.00000	−0.139347
$825$	2.00000	0.0696311
$826$	−48.0000	−1.67013
$827$	44.0000	1.53003	0.765015	−	0.644013i	$-0.222732\pi$
$827$	44.0000	1.53003	0.765015	+	0.644013i	$0.222732\pi$
$828$	1.00000	0.0347524
$829$	10.0000	0.347314	0.173657	−	0.984806i	$-0.444442\pi$
$829$	10.0000	0.347314	0.173657	+	0.984806i	$0.444442\pi$
$830$	8.00000	0.277684
$831$	−8.00000	−0.277517
$832$	0	0
$833$	18.0000	0.623663
$834$	20.0000	0.692543
$835$	0	0
$836$	0	0
$837$	0	0
$838$	26.0000	0.898155
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	−4.00000	−0.138013
$841$	−13.0000	−0.448276
$842$	6.00000	0.206774
$843$	−32.0000	−1.10214
$844$	4.00000	0.137686
$845$	13.0000	0.447214
$846$	−12.0000	−0.412568
$847$	−28.0000	−0.962091
$848$	6.00000	0.206041
$849$	10.0000	0.343199
$850$	−2.00000	−0.0685994
$851$	10.0000	0.342796
$852$	2.00000	0.0685189
$853$	56.0000	1.91740	0.958702	−	0.284413i	$-0.0917988\pi$
$853$	56.0000	1.91740	0.958702	+	0.284413i	$0.0917988\pi$
$854$	56.0000	1.91628
$855$	0	0
$856$	12.0000	0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	0	0
$859$	36.0000	1.22830	0.614152	−	0.789188i	$-0.289498\pi$
$859$	36.0000	1.22830	0.614152	+	0.789188i	$0.289498\pi$
$860$	−2.00000	−0.0681994
$861$	−24.0000	−0.817918
$862$	−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
$866$	−24.0000	−0.815553
$867$	13.0000	0.441503
$868$	0	0
$869$	−16.0000	−0.542763
$870$	4.00000	0.135613
$871$	0	0
$872$	−14.0000	−0.474100
$873$	0	0
$874$	0	0
$875$	−4.00000	−0.135225
$876$	−6.00000	−0.202721
$877$	−56.0000	−1.89099	−0.945493	−	0.325643i	$-0.894419\pi$
$877$	−56.0000	−1.89099	−0.945493	+	0.325643i	$0.894419\pi$
$878$	−20.0000	−0.674967
$879$	−14.0000	−0.472208
$880$	2.00000	0.0674200
$881$	40.0000	1.34763	0.673817	−	0.738898i	$-0.264654\pi$
$881$	40.0000	1.34763	0.673817	+	0.738898i	$0.264654\pi$
$882$	−9.00000	−0.303046
$883$	−48.0000	−1.61533	−0.807664	−	0.589643i	$-0.799269\pi$
$883$	−48.0000	−1.61533	−0.807664	+	0.589643i	$0.799269\pi$
$884$	0	0
$885$	12.0000	0.403376
$886$	20.0000	0.671913
$887$	−36.0000	−1.20876	−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000	−1.20876	−0.604381	+	0.796696i	$0.706579\pi$
$888$	10.0000	0.335578
$889$	24.0000	0.804934
$890$	−8.00000	−0.268161
$891$	−2.00000	−0.0670025
$892$	2.00000	0.0669650
$893$	0	0
$894$	−14.0000	−0.468230
$895$	−4.00000	−0.133705
$896$	−4.00000	−0.133631
$897$	0	0
$898$	34.0000	1.13459
$899$	0	0
$900$	1.00000	0.0333333
$901$	12.0000	0.399778
$902$	12.0000	0.399556
$903$	−8.00000	−0.266223
$904$	−6.00000	−0.199557
$905$	6.00000	0.199447
$906$	−12.0000	−0.398673
$907$	42.0000	1.39459	0.697294	−	0.716786i	$-0.254387\pi$
$907$	42.0000	1.39459	0.697294	+	0.716786i	$0.254387\pi$
$908$	12.0000	0.398234
$909$	−12.0000	−0.398015
$910$	0	0
$911$	−12.0000	−0.397578	−0.198789	−	0.980042i	$-0.563701\pi$
$911$	−12.0000	−0.397578	−0.198789	+	0.980042i	$0.563701\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	24.0000	0.793849
$915$	−14.0000	−0.462826
$916$	−26.0000	−0.859064
$917$	0	0
$918$	2.00000	0.0660098
$919$	−32.0000	−1.05558	−0.527791	−	0.849374i	$-0.676980\pi$
$919$	−32.0000	−1.05558	−0.527791	+	0.849374i	$0.676980\pi$
$920$	1.00000	0.0329690
$921$	−4.00000	−0.131804
$922$	−20.0000	−0.658665
$923$	0	0
$924$	8.00000	0.263181
$925$	10.0000	0.328798
$926$	−18.0000	−0.591517
$927$	4.00000	0.131377
$928$	4.00000	0.131306
$929$	30.0000	0.984268	0.492134	−	0.870519i	$-0.336217\pi$
$929$	30.0000	0.984268	0.492134	+	0.870519i	$0.336217\pi$
$930$	0	0
$931$	0	0
$932$	10.0000	0.327561
$933$	2.00000	0.0654771
$934$	−24.0000	−0.785304
$935$	4.00000	0.130814
$936$	0	0
$937$	48.0000	1.56809	0.784046	−	0.620703i	$-0.213153\pi$
$937$	48.0000	1.56809	0.784046	+	0.620703i	$0.213153\pi$
$938$	−8.00000	−0.261209
$939$	−20.0000	−0.652675
$940$	−12.0000	−0.391397
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	−22.0000	−0.716799
$943$	6.00000	0.195387
$944$	12.0000	0.390567
$945$	4.00000	0.130120
$946$	4.00000	0.130051
$947$	−52.0000	−1.68977	−0.844886	−	0.534946i	$-0.820332\pi$
$947$	−52.0000	−1.68977	−0.844886	+	0.534946i	$0.820332\pi$
$948$	−8.00000	−0.259828
$949$	0	0
$950$	0	0
$951$	10.0000	0.324272
$952$	−8.00000	−0.259281
$953$	14.0000	0.453504	0.226752	−	0.973952i	$-0.427189\pi$
$953$	14.0000	0.453504	0.226752	+	0.973952i	$0.427189\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	22.0000	0.711531
$957$	−8.00000	−0.258603
$958$	40.0000	1.29234
$959$	−8.00000	−0.258333
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	0	0
$963$	−12.0000	−0.386695
$964$	−14.0000	−0.450910
$965$	14.0000	0.450676
$966$	4.00000	0.128698
$967$	58.0000	1.86515	0.932577	−	0.360971i	$-0.117555\pi$
$967$	58.0000	1.86515	0.932577	+	0.360971i	$0.117555\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	0	0
$971$	50.0000	1.60458	0.802288	−	0.596937i	$-0.203616\pi$
$971$	50.0000	1.60458	0.802288	+	0.596937i	$0.203616\pi$
$972$	−1.00000	−0.0320750
$973$	80.0000	2.56468
$974$	−18.0000	−0.576757
$975$	0	0
$976$	−14.0000	−0.448129
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−16.0000	−0.511624
$979$	16.0000	0.511362
$980$	−9.00000	−0.287494
$981$	14.0000	0.446986
$982$	24.0000	0.765871
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	6.00000	0.191273
$985$	6.00000	0.191176
$986$	8.00000	0.254772
$987$	−48.0000	−1.52786
$988$	0	0
$989$	2.00000	0.0635963
$990$	−2.00000	−0.0635642
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	0	0
$993$	28.0000	0.888553
$994$	8.00000	0.253745
$995$	−16.0000	−0.507234
$996$	−8.00000	−0.253490
$997$	40.0000	1.26681	0.633406	−	0.773819i	$-0.281656\pi$
$997$	40.0000	1.26681	0.633406	+	0.773819i	$0.281656\pi$
$998$	28.0000	0.886325
$999$	−10.0000	−0.316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	690.2.a.b.1.1	✓	1
3.2	odd	2		2070.2.a.s.1.1		1
4.3	odd	2		5520.2.a.r.1.1		1
5.2	odd	4		3450.2.d.n.2899.1		2
5.3	odd	4		3450.2.d.n.2899.2		2
5.4	even	2		3450.2.a.t.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.b.1.1	✓	1	1.1	even	1	trivial
2070.2.a.s.1.1		1	3.2	odd	2
3450.2.a.t.1.1		1	5.4	even	2
3450.2.d.n.2899.1		2	5.2	odd	4
3450.2.d.n.2899.2		2	5.3	odd	4
5520.2.a.r.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:	\( N \)	\(=\)	\( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	690.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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