LMFDB - Embedded newform 690.2.a.i.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)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +2.00000 q^{11} +1.00000 q^{12} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +2.00000 q^{22} +1.00000 q^{23} +1.00000 q^{24} +1.00000 q^{25} +1.00000 q^{27} -1.00000 q^{30} -8.00000 q^{31} +1.00000 q^{32} +2.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -1.00000 q^{40} -2.00000 q^{41} -2.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} +1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -2.00000 q^{53} +1.00000 q^{54} -2.00000 q^{55} +4.00000 q^{57} -1.00000 q^{60} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +2.00000 q^{66} -2.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} -10.0000 q^{71} +1.00000 q^{72} -10.0000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} -6.00000 q^{85} -2.00000 q^{86} +2.00000 q^{88} +4.00000 q^{89} -1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{93} +4.00000 q^{94} -4.00000 q^{95} +1.00000 q^{96} +16.0000 q^{97} -7.00000 q^{98} +2.00000 q^{99} +O(q^{100})\)

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
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	1.00000	0.288675
$13$	0	0		−	1.00000i	$-0.5\pi$
$13$	0	0			1.00000i	$0.5\pi$
$14$	0	0
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	−1.00000	−0.223607
$21$	0	0
$22$	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
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	−8.00000	−1.43684	−0.718421	−	0.695608i	$-0.755135\pi$
$31$	−8.00000	−1.43684	−0.718421	+	0.695608i	$0.755135\pi$
$32$	1.00000	0.176777
$33$	2.00000	0.348155
$34$	6.00000	1.02899
$35$	0	0
$36$	1.00000	0.166667
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−2.00000	−0.312348	−0.156174	−	0.987730i	$-0.549916\pi$
$41$	−2.00000	−0.312348	−0.156174	+	0.987730i	$0.549916\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	2.00000	0.301511
$45$	−1.00000	−0.149071
$46$	1.00000	0.147442
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	1.00000	0.136083
$55$	−2.00000	−0.269680
$56$	0	0
$57$	4.00000	0.529813
$58$	0	0
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−1.00000	−0.129099
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−8.00000	−1.01600
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	2.00000	0.246183
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	6.00000	0.727607
$69$	1.00000	0.120386
$70$	0	0
$71$	−10.0000	−1.18678	−0.593391	−	0.804914i	$-0.702211\pi$
$71$	−10.0000	−1.18678	−0.593391	+	0.804914i	$0.702211\pi$
$72$	1.00000	0.117851
$73$	−10.0000	−1.17041	−0.585206	−	0.810885i	$-0.698986\pi$
$73$	−10.0000	−1.17041	−0.585206	+	0.810885i	$0.698986\pi$
$74$	−6.00000	−0.697486
$75$	1.00000	0.115470
$76$	4.00000	0.458831
$77$	0	0
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−2.00000	−0.220863
$83$	−4.00000	−0.439057	−0.219529	−	0.975606i	$-0.570452\pi$
$83$	−4.00000	−0.439057	−0.219529	+	0.975606i	$0.570452\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	−2.00000	−0.215666
$87$	0	0
$88$	2.00000	0.213201
$89$	4.00000	0.423999	0.212000	−	0.977270i	$-0.432002\pi$
$89$	4.00000	0.423999	0.212000	+	0.977270i	$0.432002\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	−8.00000	−0.829561
$94$	4.00000	0.412568
$95$	−4.00000	−0.410391
$96$	1.00000	0.102062
$97$	16.0000	1.62455	0.812277	−	0.583272i	$-0.198228\pi$
$97$	16.0000	1.62455	0.812277	+	0.583272i	$0.198228\pi$
$98$	−7.00000	−0.707107
$99$	2.00000	0.201008
$100$	1.00000	0.100000
$101$	0	0		−	1.00000i	$-0.5\pi$
$101$	0	0			1.00000i	$0.5\pi$
$102$	6.00000	0.594089
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	0	0
$106$	−2.00000	−0.194257
$107$	8.00000	0.773389	0.386695	−	0.922208i	$-0.373617\pi$
$107$	8.00000	0.773389	0.386695	+	0.922208i	$0.373617\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−2.00000	−0.190693
$111$	−6.00000	−0.569495
$112$	0	0
$113$	10.0000	0.940721	0.470360	−	0.882474i	$-0.344124\pi$
$113$	10.0000	0.940721	0.470360	+	0.882474i	$0.344124\pi$
$114$	4.00000	0.374634
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	−7.00000	−0.636364
$122$	−2.00000	−0.181071
$123$	−2.00000	−0.180334
$124$	−8.00000	−0.718421
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	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
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	2.00000	0.174078
$133$	0	0
$134$	−2.00000	−0.172774
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	2.00000	0.170872	0.0854358	−	0.996344i	$-0.472772\pi$
$137$	2.00000	0.170872	0.0854358	+	0.996344i	$0.472772\pi$
$138$	1.00000	0.0851257
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	4.00000	0.336861
$142$	−10.0000	−0.839181
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	−10.0000	−0.827606
$147$	−7.00000	−0.577350
$148$	−6.00000	−0.493197
$149$	10.0000	0.819232	0.409616	−	0.912258i	$-0.365663\pi$
$149$	10.0000	0.819232	0.409616	+	0.912258i	$0.365663\pi$
$150$	1.00000	0.0816497
$151$	−20.0000	−1.62758	−0.813788	−	0.581161i	$-0.802599\pi$
$151$	−20.0000	−1.62758	−0.813788	+	0.581161i	$0.802599\pi$
$152$	4.00000	0.324443
$153$	6.00000	0.485071
$154$	0	0
$155$	8.00000	0.642575
$156$	0	0
$157$	−6.00000	−0.478852	−0.239426	−	0.970915i	$-0.576959\pi$
$157$	−6.00000	−0.478852	−0.239426	+	0.970915i	$0.576959\pi$
$158$	0	0
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−2.00000	−0.156174
$165$	−2.00000	−0.155700
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−13.0000	−1.00000
$170$	−6.00000	−0.460179
$171$	4.00000	0.305888
$172$	−2.00000	−0.152499
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	0	0
$176$	2.00000	0.150756
$177$	0	0
$178$	4.00000	0.299813
$179$	−24.0000	−1.79384	−0.896922	−	0.442189i	$-0.854202\pi$
$179$	−24.0000	−1.79384	−0.896922	+	0.442189i	$0.854202\pi$
$180$	−1.00000	−0.0745356
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−2.00000	−0.147844
$184$	1.00000	0.0737210
$185$	6.00000	0.441129
$186$	−8.00000	−0.586588
$187$	12.0000	0.877527
$188$	4.00000	0.291730
$189$	0	0
$190$	−4.00000	−0.290191
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	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	$0.668091\pi$
$194$	16.0000	1.14873
$195$	0	0
$196$	−7.00000	−0.500000
$197$	14.0000	0.997459	0.498729	−	0.866758i	$-0.333800\pi$
$197$	14.0000	0.997459	0.498729	+	0.866758i	$0.333800\pi$
$198$	2.00000	0.142134
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	1.00000	0.0707107
$201$	−2.00000	−0.141069
$202$	0	0
$203$	0	0
$204$	6.00000	0.420084
$205$	2.00000	0.139686
$206$	−16.0000	−1.11477
$207$	1.00000	0.0695048
$208$	0	0
$209$	8.00000	0.553372
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−2.00000	−0.137361
$213$	−10.0000	−0.685189
$214$	8.00000	0.546869
$215$	2.00000	0.136399
$216$	1.00000	0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	−10.0000	−0.675737
$220$	−2.00000	−0.134840
$221$	0	0
$222$	−6.00000	−0.402694
$223$	−14.0000	−0.937509	−0.468755	−	0.883328i	$-0.655297\pi$
$223$	−14.0000	−0.937509	−0.468755	+	0.883328i	$0.655297\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	10.0000	0.665190
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	4.00000	0.264906
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	0	0
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	−4.00000	−0.260931
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.0000	−1.68180	−0.840900	−	0.541190i	$-0.817974\pi$
$239$	−26.0000	−1.68180	−0.840900	+	0.541190i	$0.817974\pi$
$240$	−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	$0.479482\pi$
$242$	−7.00000	−0.449977
$243$	1.00000	0.0641500
$244$	−2.00000	−0.128037
$245$	7.00000	0.447214
$246$	−2.00000	−0.127515
$247$	0	0
$248$	−8.00000	−0.508001
$249$	−4.00000	−0.253490
$250$	−1.00000	−0.0632456
$251$	−18.0000	−1.13615	−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000	−1.13615	−0.568075	+	0.822977i	$0.692312\pi$
$252$	0	0
$253$	2.00000	0.125739
$254$	6.00000	0.376473
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	−2.00000	−0.124515
$259$	0	0
$260$	0	0
$261$	0	0
$262$	−4.00000	−0.247121
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	2.00000	0.123091
$265$	2.00000	0.122859
$266$	0	0
$267$	4.00000	0.244796
$268$	−2.00000	−0.122169
$269$	28.0000	1.70719	0.853595	−	0.520937i	$-0.174417\pi$
$269$	28.0000	1.70719	0.853595	+	0.520937i	$0.174417\pi$
$270$	−1.00000	−0.0608581
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	6.00000	0.363803
$273$	0	0
$274$	2.00000	0.120824
$275$	2.00000	0.120605
$276$	1.00000	0.0601929
$277$	24.0000	1.44202	0.721010	−	0.692925i	$-0.243678\pi$
$277$	24.0000	1.44202	0.721010	+	0.692925i	$0.243678\pi$
$278$	12.0000	0.719712
$279$	−8.00000	−0.478947
$280$	0	0
$281$	12.0000	0.715860	0.357930	−	0.933748i	$-0.383483\pi$
$281$	12.0000	0.715860	0.357930	+	0.933748i	$0.383483\pi$
$282$	4.00000	0.238197
$283$	10.0000	0.594438	0.297219	−	0.954809i	$-0.403941\pi$
$283$	10.0000	0.594438	0.297219	+	0.954809i	$0.403941\pi$
$284$	−10.0000	−0.593391
$285$	−4.00000	−0.236940
$286$	0	0
$287$	0	0
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	16.0000	0.937937
$292$	−10.0000	−0.585206
$293$	22.0000	1.28525	0.642627	−	0.766179i	$-0.277845\pi$
$293$	22.0000	1.28525	0.642627	+	0.766179i	$0.277845\pi$
$294$	−7.00000	−0.408248
$295$	0	0
$296$	−6.00000	−0.348743
$297$	2.00000	0.116052
$298$	10.0000	0.579284
$299$	0	0
$300$	1.00000	0.0577350
$301$	0	0
$302$	−20.0000	−1.15087
$303$	0	0
$304$	4.00000	0.229416
$305$	2.00000	0.114520
$306$	6.00000	0.342997
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	−16.0000	−0.910208
$310$	8.00000	0.454369
$311$	−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	$-0.691214\pi$
$313$	−20.0000	−1.13047	−0.565233	+	0.824931i	$0.691214\pi$
$314$	−6.00000	−0.338600
$315$	0	0
$316$	0	0
$317$	−6.00000	−0.336994	−0.168497	−	0.985702i	$-0.553891\pi$
$317$	−6.00000	−0.336994	−0.168497	+	0.985702i	$0.553891\pi$
$318$	−2.00000	−0.112154
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	8.00000	0.446516
$322$	0	0
$323$	24.0000	1.33540
$324$	1.00000	0.0555556
$325$	0	0
$326$	−8.00000	−0.443079
$327$	2.00000	0.110600
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−2.00000	−0.110096
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−4.00000	−0.219529
$333$	−6.00000	−0.328798
$334$	8.00000	0.437741
$335$	2.00000	0.109272
$336$	0	0
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	−13.0000	−0.707107
$339$	10.0000	0.543125
$340$	−6.00000	−0.325396
$341$	−16.0000	−0.866449
$342$	4.00000	0.216295
$343$	0	0
$344$	−2.00000	−0.107833
$345$	−1.00000	−0.0538382
$346$	6.00000	0.322562
$347$	−4.00000	−0.214731	−0.107366	−	0.994220i	$-0.534242\pi$
$347$	−4.00000	−0.214731	−0.107366	+	0.994220i	$0.534242\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	0	0
$351$	0	0
$352$	2.00000	0.106600
$353$	−6.00000	−0.319348	−0.159674	−	0.987170i	$-0.551044\pi$
$353$	−6.00000	−0.319348	−0.159674	+	0.987170i	$0.551044\pi$
$354$	0	0
$355$	10.0000	0.530745
$356$	4.00000	0.212000
$357$	0	0
$358$	−24.0000	−1.26844
$359$	−32.0000	−1.68890	−0.844448	−	0.535638i	$-0.820071\pi$
$359$	−32.0000	−1.68890	−0.844448	+	0.535638i	$0.820071\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	6.00000	0.315353
$363$	−7.00000	−0.367405
$364$	0	0
$365$	10.0000	0.523424
$366$	−2.00000	−0.104542
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	1.00000	0.0521286
$369$	−2.00000	−0.104116
$370$	6.00000	0.311925
$371$	0	0
$372$	−8.00000	−0.414781
$373$	−6.00000	−0.310668	−0.155334	−	0.987862i	$-0.549645\pi$
$373$	−6.00000	−0.310668	−0.155334	+	0.987862i	$0.549645\pi$
$374$	12.0000	0.620505
$375$	−1.00000	−0.0516398
$376$	4.00000	0.206284
$377$	0	0
$378$	0	0
$379$	−8.00000	−0.410932	−0.205466	−	0.978664i	$-0.565871\pi$
$379$	−8.00000	−0.410932	−0.205466	+	0.978664i	$0.565871\pi$
$380$	−4.00000	−0.205196
$381$	6.00000	0.307389
$382$	−8.00000	−0.409316
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	−14.0000	−0.712581
$387$	−2.00000	−0.101666
$388$	16.0000	0.812277
$389$	−18.0000	−0.912636	−0.456318	−	0.889817i	$-0.650832\pi$
$389$	−18.0000	−0.912636	−0.456318	+	0.889817i	$0.650832\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−7.00000	−0.353553
$393$	−4.00000	−0.201773
$394$	14.0000	0.705310
$395$	0	0
$396$	2.00000	0.100504
$397$	8.00000	0.401508	0.200754	−	0.979642i	$-0.435661\pi$
$397$	8.00000	0.401508	0.200754	+	0.979642i	$0.435661\pi$
$398$	24.0000	1.20301
$399$	0	0
$400$	1.00000	0.0500000
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	−2.00000	−0.0997509
$403$	0	0
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	−12.0000	−0.594818
$408$	6.00000	0.297044
$409$	22.0000	1.08783	0.543915	−	0.839140i	$-0.316941\pi$
$409$	22.0000	1.08783	0.543915	+	0.839140i	$0.316941\pi$
$410$	2.00000	0.0987730
$411$	2.00000	0.0986527
$412$	−16.0000	−0.788263
$413$	0	0
$414$	1.00000	0.0491473
$415$	4.00000	0.196352
$416$	0	0
$417$	12.0000	0.587643
$418$	8.00000	0.391293
$419$	−6.00000	−0.293119	−0.146560	−	0.989202i	$-0.546820\pi$
$419$	−6.00000	−0.293119	−0.146560	+	0.989202i	$0.546820\pi$
$420$	0	0
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−20.0000	−0.973585
$423$	4.00000	0.194487
$424$	−2.00000	−0.0971286
$425$	6.00000	0.291043
$426$	−10.0000	−0.484502
$427$	0	0
$428$	8.00000	0.386695
$429$	0	0
$430$	2.00000	0.0964486
$431$	−40.0000	−1.92673	−0.963366	−	0.268190i	$-0.913575\pi$
$431$	−40.0000	−1.92673	−0.963366	+	0.268190i	$0.913575\pi$
$432$	1.00000	0.0481125
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	0	0
$436$	2.00000	0.0957826
$437$	4.00000	0.191346
$438$	−10.0000	−0.477818
$439$	−12.0000	−0.572729	−0.286364	−	0.958121i	$-0.592447\pi$
$439$	−12.0000	−0.572729	−0.286364	+	0.958121i	$0.592447\pi$
$440$	−2.00000	−0.0953463
$441$	−7.00000	−0.333333
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	−6.00000	−0.284747
$445$	−4.00000	−0.189618
$446$	−14.0000	−0.662919
$447$	10.0000	0.472984
$448$	0	0
$449$	−2.00000	−0.0943858	−0.0471929	−	0.998886i	$-0.515028\pi$
$449$	−2.00000	−0.0943858	−0.0471929	+	0.998886i	$0.515028\pi$
$450$	1.00000	0.0471405
$451$	−4.00000	−0.188353
$452$	10.0000	0.470360
$453$	−20.0000	−0.939682
$454$	8.00000	0.375459
$455$	0	0
$456$	4.00000	0.187317
$457$	−8.00000	−0.374224	−0.187112	−	0.982339i	$-0.559913\pi$
$457$	−8.00000	−0.374224	−0.187112	+	0.982339i	$0.559913\pi$
$458$	−14.0000	−0.654177
$459$	6.00000	0.280056
$460$	−1.00000	−0.0466252
$461$	32.0000	1.49039	0.745194	−	0.666847i	$-0.232357\pi$
$461$	32.0000	1.49039	0.745194	+	0.666847i	$0.232357\pi$
$462$	0	0
$463$	34.0000	1.58011	0.790057	−	0.613033i	$-0.210051\pi$
$463$	34.0000	1.58011	0.790057	+	0.613033i	$0.210051\pi$
$464$	0	0
$465$	8.00000	0.370991
$466$	22.0000	1.01913
$467$	−4.00000	−0.185098	−0.0925490	−	0.995708i	$-0.529501\pi$
$467$	−4.00000	−0.185098	−0.0925490	+	0.995708i	$0.529501\pi$
$468$	0	0
$469$	0	0
$470$	−4.00000	−0.184506
$471$	−6.00000	−0.276465
$472$	0	0
$473$	−4.00000	−0.183920
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	−26.0000	−1.18921
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	−1.00000	−0.0456435
$481$	0	0
$482$	2.00000	0.0910975
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−16.0000	−0.726523
$486$	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	$0.485571\pi$
$488$	−2.00000	−0.0905357
$489$	−8.00000	−0.361773
$490$	7.00000	0.316228
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	−2.00000	−0.0901670
$493$	0	0
$494$	0	0
$495$	−2.00000	−0.0898933
$496$	−8.00000	−0.359211
$497$	0	0
$498$	−4.00000	−0.179244
$499$	−36.0000	−1.61158	−0.805791	−	0.592200i	$-0.798259\pi$
$499$	−36.0000	−1.61158	−0.805791	+	0.592200i	$0.798259\pi$
$500$	−1.00000	−0.0447214
$501$	8.00000	0.357414
$502$	−18.0000	−0.803379
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	0	0
$506$	2.00000	0.0889108
$507$	−13.0000	−0.577350
$508$	6.00000	0.266207
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	−6.00000	−0.265684
$511$	0	0
$512$	1.00000	0.0441942
$513$	4.00000	0.176604
$514$	6.00000	0.264649
$515$	16.0000	0.705044
$516$	−2.00000	−0.0880451
$517$	8.00000	0.351840
$518$	0	0
$519$	6.00000	0.263371
$520$	0	0
$521$	36.0000	1.57719	0.788594	−	0.614914i	$-0.210809\pi$
$521$	36.0000	1.57719	0.788594	+	0.614914i	$0.210809\pi$
$522$	0	0
$523$	34.0000	1.48672	0.743358	−	0.668894i	$-0.233232\pi$
$523$	34.0000	1.48672	0.743358	+	0.668894i	$0.233232\pi$
$524$	−4.00000	−0.174741
$525$	0	0
$526$	16.0000	0.697633
$527$	−48.0000	−2.09091
$528$	2.00000	0.0870388
$529$	1.00000	0.0434783
$530$	2.00000	0.0868744
$531$	0	0
$532$	0	0
$533$	0	0
$534$	4.00000	0.173097
$535$	−8.00000	−0.345870
$536$	−2.00000	−0.0863868
$537$	−24.0000	−1.03568
$538$	28.0000	1.20717
$539$	−14.0000	−0.603023
$540$	−1.00000	−0.0430331
$541$	26.0000	1.11783	0.558914	−	0.829226i	$-0.311218\pi$
$541$	26.0000	1.11783	0.558914	+	0.829226i	$0.311218\pi$
$542$	−8.00000	−0.343629
$543$	6.00000	0.257485
$544$	6.00000	0.257248
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	20.0000	0.855138	0.427569	−	0.903983i	$-0.359370\pi$
$547$	20.0000	0.855138	0.427569	+	0.903983i	$0.359370\pi$
$548$	2.00000	0.0854358
$549$	−2.00000	−0.0853579
$550$	2.00000	0.0852803
$551$	0	0
$552$	1.00000	0.0425628
$553$	0	0
$554$	24.0000	1.01966
$555$	6.00000	0.254686
$556$	12.0000	0.508913
$557$	−30.0000	−1.27114	−0.635570	−	0.772043i	$-0.719235\pi$
$557$	−30.0000	−1.27114	−0.635570	+	0.772043i	$0.719235\pi$
$558$	−8.00000	−0.338667
$559$	0	0
$560$	0	0
$561$	12.0000	0.506640
$562$	12.0000	0.506189
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	4.00000	0.168430
$565$	−10.0000	−0.420703
$566$	10.0000	0.420331
$567$	0	0
$568$	−10.0000	−0.419591
$569$	0	0		−	1.00000i	$-0.5\pi$
$569$	0	0			1.00000i	$0.5\pi$
$570$	−4.00000	−0.167542
$571$	−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
$573$	−8.00000	−0.334205
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	−2.00000	−0.0832611	−0.0416305	−	0.999133i	$-0.513255\pi$
$577$	−2.00000	−0.0832611	−0.0416305	+	0.999133i	$0.513255\pi$
$578$	19.0000	0.790296
$579$	−14.0000	−0.581820
$580$	0	0
$581$	0	0
$582$	16.0000	0.663221
$583$	−4.00000	−0.165663
$584$	−10.0000	−0.413803
$585$	0	0
$586$	22.0000	0.908812
$587$	−28.0000	−1.15568	−0.577842	−	0.816149i	$-0.696105\pi$
$587$	−28.0000	−1.15568	−0.577842	+	0.816149i	$0.696105\pi$
$588$	−7.00000	−0.288675
$589$	−32.0000	−1.31854
$590$	0	0
$591$	14.0000	0.575883
$592$	−6.00000	−0.246598
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	2.00000	0.0820610
$595$	0	0
$596$	10.0000	0.409616
$597$	24.0000	0.982255
$598$	0	0
$599$	26.0000	1.06233	0.531166	−	0.847268i	$-0.321754\pi$
$599$	26.0000	1.06233	0.531166	+	0.847268i	$0.321754\pi$
$600$	1.00000	0.0408248
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	0	0
$603$	−2.00000	−0.0814463
$604$	−20.0000	−0.813788
$605$	7.00000	0.284590
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	2.00000	0.0809776
$611$	0	0
$612$	6.00000	0.242536
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	12.0000	0.484281
$615$	2.00000	0.0806478
$616$	0	0
$617$	−42.0000	−1.69086	−0.845428	−	0.534089i	$-0.820655\pi$
$617$	−42.0000	−1.69086	−0.845428	+	0.534089i	$0.820655\pi$
$618$	−16.0000	−0.643614
$619$	20.0000	0.803868	0.401934	−	0.915669i	$-0.368338\pi$
$619$	20.0000	0.803868	0.401934	+	0.915669i	$0.368338\pi$
$620$	8.00000	0.321288
$621$	1.00000	0.0401286
$622$	−2.00000	−0.0801927
$623$	0	0
$624$	0	0
$625$	1.00000	0.0400000
$626$	−20.0000	−0.799361
$627$	8.00000	0.319489
$628$	−6.00000	−0.239426
$629$	−36.0000	−1.43541
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	−20.0000	−0.794929
$634$	−6.00000	−0.238290
$635$	−6.00000	−0.238103
$636$	−2.00000	−0.0793052
$637$	0	0
$638$	0	0
$639$	−10.0000	−0.395594
$640$	−1.00000	−0.0395285
$641$	4.00000	0.157991	0.0789953	−	0.996875i	$-0.474829\pi$
$641$	4.00000	0.157991	0.0789953	+	0.996875i	$0.474829\pi$
$642$	8.00000	0.315735
$643$	−14.0000	−0.552106	−0.276053	−	0.961142i	$-0.589027\pi$
$643$	−14.0000	−0.552106	−0.276053	+	0.961142i	$0.589027\pi$
$644$	0	0
$645$	2.00000	0.0787499
$646$	24.0000	0.944267
$647$	−36.0000	−1.41531	−0.707653	−	0.706560i	$-0.750246\pi$
$647$	−36.0000	−1.41531	−0.707653	+	0.706560i	$0.750246\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	−14.0000	−0.547862	−0.273931	−	0.961749i	$-0.588324\pi$
$653$	−14.0000	−0.547862	−0.273931	+	0.961749i	$0.588324\pi$
$654$	2.00000	0.0782062
$655$	4.00000	0.156293
$656$	−2.00000	−0.0780869
$657$	−10.0000	−0.390137
$658$	0	0
$659$	18.0000	0.701180	0.350590	−	0.936529i	$-0.385981\pi$
$659$	18.0000	0.701180	0.350590	+	0.936529i	$0.385981\pi$
$660$	−2.00000	−0.0778499
$661$	10.0000	0.388955	0.194477	−	0.980907i	$-0.437699\pi$
$661$	10.0000	0.388955	0.194477	+	0.980907i	$0.437699\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−4.00000	−0.155230
$665$	0	0
$666$	−6.00000	−0.232495
$667$	0	0
$668$	8.00000	0.309529
$669$	−14.0000	−0.541271
$670$	2.00000	0.0772667
$671$	−4.00000	−0.154418
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	28.0000	1.07852
$675$	1.00000	0.0384900
$676$	−13.0000	−0.500000
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	10.0000	0.384048
$679$	0	0
$680$	−6.00000	−0.230089
$681$	8.00000	0.306561
$682$	−16.0000	−0.612672
$683$	20.0000	0.765279	0.382639	−	0.923898i	$-0.375015\pi$
$683$	20.0000	0.765279	0.382639	+	0.923898i	$0.375015\pi$
$684$	4.00000	0.152944
$685$	−2.00000	−0.0764161
$686$	0	0
$687$	−14.0000	−0.534133
$688$	−2.00000	−0.0762493
$689$	0	0
$690$	−1.00000	−0.0380693
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−4.00000	−0.151838
$695$	−12.0000	−0.455186
$696$	0	0
$697$	−12.0000	−0.454532
$698$	26.0000	0.984115
$699$	22.0000	0.832116
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	0	0
$703$	−24.0000	−0.905177
$704$	2.00000	0.0753778
$705$	−4.00000	−0.150649
$706$	−6.00000	−0.225813
$707$	0	0
$708$	0	0
$709$	−38.0000	−1.42712	−0.713560	−	0.700594i	$-0.752918\pi$
$709$	−38.0000	−1.42712	−0.713560	+	0.700594i	$0.752918\pi$
$710$	10.0000	0.375293
$711$	0	0
$712$	4.00000	0.149906
$713$	−8.00000	−0.299602
$714$	0	0
$715$	0	0
$716$	−24.0000	−0.896922
$717$	−26.0000	−0.970988
$718$	−32.0000	−1.19423
$719$	−22.0000	−0.820462	−0.410231	−	0.911982i	$-0.634552\pi$
$719$	−22.0000	−0.820462	−0.410231	+	0.911982i	$0.634552\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	2.00000	0.0743808
$724$	6.00000	0.222988
$725$	0	0
$726$	−7.00000	−0.259794
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	10.0000	0.370117
$731$	−12.0000	−0.443836
$732$	−2.00000	−0.0739221
$733$	−30.0000	−1.10808	−0.554038	−	0.832492i	$-0.686914\pi$
$733$	−30.0000	−1.10808	−0.554038	+	0.832492i	$0.686914\pi$
$734$	16.0000	0.590571
$735$	7.00000	0.258199
$736$	1.00000	0.0368605
$737$	−4.00000	−0.147342
$738$	−2.00000	−0.0736210
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	6.00000	0.220564
$741$	0	0
$742$	0	0
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	−8.00000	−0.293294
$745$	−10.0000	−0.366372
$746$	−6.00000	−0.219676
$747$	−4.00000	−0.146352
$748$	12.0000	0.438763
$749$	0	0
$750$	−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	$0.301546\pi$
$752$	4.00000	0.145865
$753$	−18.0000	−0.655956
$754$	0	0
$755$	20.0000	0.727875
$756$	0	0
$757$	−42.0000	−1.52652	−0.763258	−	0.646094i	$-0.776401\pi$
$757$	−42.0000	−1.52652	−0.763258	+	0.646094i	$0.776401\pi$
$758$	−8.00000	−0.290573
$759$	2.00000	0.0725954
$760$	−4.00000	−0.145095
$761$	−14.0000	−0.507500	−0.253750	−	0.967270i	$-0.581664\pi$
$761$	−14.0000	−0.507500	−0.253750	+	0.967270i	$0.581664\pi$
$762$	6.00000	0.217357
$763$	0	0
$764$	−8.00000	−0.289430
$765$	−6.00000	−0.216930
$766$	16.0000	0.578103
$767$	0	0
$768$	1.00000	0.0360844
$769$	−42.0000	−1.51456	−0.757279	−	0.653091i	$-0.773472\pi$
$769$	−42.0000	−1.51456	−0.757279	+	0.653091i	$0.773472\pi$
$770$	0	0
$771$	6.00000	0.216085
$772$	−14.0000	−0.503871
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	−2.00000	−0.0718885
$775$	−8.00000	−0.287368
$776$	16.0000	0.574367
$777$	0	0
$778$	−18.0000	−0.645331
$779$	−8.00000	−0.286630
$780$	0	0
$781$	−20.0000	−0.715656
$782$	6.00000	0.214560
$783$	0	0
$784$	−7.00000	−0.250000
$785$	6.00000	0.214149
$786$	−4.00000	−0.142675
$787$	18.0000	0.641631	0.320815	−	0.947142i	$-0.396043\pi$
$787$	18.0000	0.641631	0.320815	+	0.947142i	$0.396043\pi$
$788$	14.0000	0.498729
$789$	16.0000	0.569615
$790$	0	0
$791$	0	0
$792$	2.00000	0.0710669
$793$	0	0
$794$	8.00000	0.283909
$795$	2.00000	0.0709327
$796$	24.0000	0.850657
$797$	42.0000	1.48772	0.743858	−	0.668338i	$-0.232994\pi$
$797$	42.0000	1.48772	0.743858	+	0.668338i	$0.232994\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	1.00000	0.0353553
$801$	4.00000	0.141333
$802$	0	0
$803$	−20.0000	−0.705785
$804$	−2.00000	−0.0705346
$805$	0	0
$806$	0	0
$807$	28.0000	0.985647
$808$	0	0
$809$	−6.00000	−0.210949	−0.105474	−	0.994422i	$-0.533636\pi$
$809$	−6.00000	−0.210949	−0.105474	+	0.994422i	$0.533636\pi$
$810$	−1.00000	−0.0351364
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−12.0000	−0.420600
$815$	8.00000	0.280228
$816$	6.00000	0.210042
$817$	−8.00000	−0.279885
$818$	22.0000	0.769212
$819$	0	0
$820$	2.00000	0.0698430
$821$	44.0000	1.53561	0.767805	−	0.640683i	$-0.221349\pi$
$821$	44.0000	1.53561	0.767805	+	0.640683i	$0.221349\pi$
$822$	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	$0.555763\pi$
$824$	−16.0000	−0.557386
$825$	2.00000	0.0696311
$826$	0	0
$827$	−32.0000	−1.11275	−0.556375	−	0.830932i	$-0.687808\pi$
$827$	−32.0000	−1.11275	−0.556375	+	0.830932i	$0.687808\pi$
$828$	1.00000	0.0347524
$829$	−6.00000	−0.208389	−0.104194	−	0.994557i	$-0.533226\pi$
$829$	−6.00000	−0.208389	−0.104194	+	0.994557i	$0.533226\pi$
$830$	4.00000	0.138842
$831$	24.0000	0.832551
$832$	0	0
$833$	−42.0000	−1.45521
$834$	12.0000	0.415526
$835$	−8.00000	−0.276851
$836$	8.00000	0.276686
$837$	−8.00000	−0.276520
$838$	−6.00000	−0.207267
$839$	12.0000	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$839$	12.0000	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	22.0000	0.758170
$843$	12.0000	0.413302
$844$	−20.0000	−0.688428
$845$	13.0000	0.447214
$846$	4.00000	0.137523
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	10.0000	0.343199
$850$	6.00000	0.205798
$851$	−6.00000	−0.205677
$852$	−10.0000	−0.342594
$853$	−16.0000	−0.547830	−0.273915	−	0.961754i	$-0.588319\pi$
$853$	−16.0000	−0.547830	−0.273915	+	0.961754i	$0.588319\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	8.00000	0.273434
$857$	18.0000	0.614868	0.307434	−	0.951569i	$-0.400530\pi$
$857$	18.0000	0.614868	0.307434	+	0.951569i	$0.400530\pi$
$858$	0	0
$859$	−20.0000	−0.682391	−0.341196	−	0.939992i	$-0.610832\pi$
$859$	−20.0000	−0.682391	−0.341196	+	0.939992i	$0.610832\pi$
$860$	2.00000	0.0681994
$861$	0	0
$862$	−40.0000	−1.36241
$863$	−12.0000	−0.408485	−0.204242	−	0.978920i	$-0.565473\pi$
$863$	−12.0000	−0.408485	−0.204242	+	0.978920i	$0.565473\pi$
$864$	1.00000	0.0340207
$865$	−6.00000	−0.204006
$866$	32.0000	1.08740
$867$	19.0000	0.645274
$868$	0	0
$869$	0	0
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	16.0000	0.541518
$874$	4.00000	0.135302
$875$	0	0
$876$	−10.0000	−0.337869
$877$	32.0000	1.08056	0.540282	−	0.841484i	$-0.318318\pi$
$877$	32.0000	1.08056	0.540282	+	0.841484i	$0.318318\pi$
$878$	−12.0000	−0.404980
$879$	22.0000	0.742042
$880$	−2.00000	−0.0674200
$881$	36.0000	1.21287	0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000	1.21287	0.606435	+	0.795133i	$0.292599\pi$
$882$	−7.00000	−0.235702
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	36.0000	1.20944
$887$	−28.0000	−0.940148	−0.470074	−	0.882627i	$-0.655773\pi$
$887$	−28.0000	−0.940148	−0.470074	+	0.882627i	$0.655773\pi$
$888$	−6.00000	−0.201347
$889$	0	0
$890$	−4.00000	−0.134080
$891$	2.00000	0.0670025
$892$	−14.0000	−0.468755
$893$	16.0000	0.535420
$894$	10.0000	0.334450
$895$	24.0000	0.802232
$896$	0	0
$897$	0	0
$898$	−2.00000	−0.0667409
$899$	0	0
$900$	1.00000	0.0333333
$901$	−12.0000	−0.399778
$902$	−4.00000	−0.133185
$903$	0	0
$904$	10.0000	0.332595
$905$	−6.00000	−0.199447
$906$	−20.0000	−0.664455
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	8.00000	0.265489
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	4.00000	0.132453
$913$	−8.00000	−0.264761
$914$	−8.00000	−0.264616
$915$	2.00000	0.0661180
$916$	−14.0000	−0.462573
$917$	0	0
$918$	6.00000	0.198030
$919$	48.0000	1.58337	0.791687	−	0.610927i	$-0.209203\pi$
$919$	48.0000	1.58337	0.791687	+	0.610927i	$0.209203\pi$
$920$	−1.00000	−0.0329690
$921$	12.0000	0.395413
$922$	32.0000	1.05386
$923$	0	0
$924$	0	0
$925$	−6.00000	−0.197279
$926$	34.0000	1.11731
$927$	−16.0000	−0.525509
$928$	0	0
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	8.00000	0.262330
$931$	−28.0000	−0.917663
$932$	22.0000	0.720634
$933$	−2.00000	−0.0654771
$934$	−4.00000	−0.130884
$935$	−12.0000	−0.392442
$936$	0	0
$937$	40.0000	1.30674	0.653372	−	0.757037i	$-0.273354\pi$
$937$	40.0000	1.30674	0.653372	+	0.757037i	$0.273354\pi$
$938$	0	0
$939$	−20.0000	−0.652675
$940$	−4.00000	−0.130466
$941$	−38.0000	−1.23876	−0.619382	−	0.785090i	$-0.712617\pi$
$941$	−38.0000	−1.23876	−0.619382	+	0.785090i	$0.712617\pi$
$942$	−6.00000	−0.195491
$943$	−2.00000	−0.0651290
$944$	0	0
$945$	0	0
$946$	−4.00000	−0.130051
$947$	52.0000	1.68977	0.844886	−	0.534946i	$-0.179668\pi$
$947$	52.0000	1.68977	0.844886	+	0.534946i	$0.179668\pi$
$948$	0	0
$949$	0	0
$950$	4.00000	0.129777
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−22.0000	−0.712650	−0.356325	−	0.934362i	$-0.615970\pi$
$953$	−22.0000	−0.712650	−0.356325	+	0.934362i	$0.615970\pi$
$954$	−2.00000	−0.0647524
$955$	8.00000	0.258874
$956$	−26.0000	−0.840900
$957$	0	0
$958$	0	0
$959$	0	0
$960$	−1.00000	−0.0322749
$961$	33.0000	1.06452
$962$	0	0
$963$	8.00000	0.257796
$964$	2.00000	0.0644157
$965$	14.0000	0.450676
$966$	0	0
$967$	−38.0000	−1.22200	−0.610999	−	0.791632i	$-0.709232\pi$
$967$	−38.0000	−1.22200	−0.610999	+	0.791632i	$0.709232\pi$
$968$	−7.00000	−0.224989
$969$	24.0000	0.770991
$970$	−16.0000	−0.513729
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	2.00000	0.0640841
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	38.0000	1.21573	0.607864	−	0.794041i	$-0.292027\pi$
$977$	38.0000	1.21573	0.607864	+	0.794041i	$0.292027\pi$
$978$	−8.00000	−0.255812
$979$	8.00000	0.255681
$980$	7.00000	0.223607
$981$	2.00000	0.0638551
$982$	−12.0000	−0.382935
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−2.00000	−0.0637577
$985$	−14.0000	−0.446077
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.00000	−0.0635963
$990$	−2.00000	−0.0635642
$991$	−36.0000	−1.14358	−0.571789	−	0.820401i	$-0.693750\pi$
$991$	−36.0000	−1.14358	−0.571789	+	0.820401i	$0.693750\pi$
$992$	−8.00000	−0.254000
$993$	4.00000	0.126936
$994$	0	0
$995$	−24.0000	−0.760851
$996$	−4.00000	−0.126745
$997$	−16.0000	−0.506725	−0.253363	−	0.967371i	$-0.581537\pi$
$997$	−16.0000	−0.506725	−0.253363	+	0.967371i	$0.581537\pi$
$998$	−36.0000	−1.13956
$999$	−6.00000	−0.189832

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
690.2.a.i.1.1	✓	1	1.1	even	1	trivial
2070.2.a.g.1.1		1	3.2	odd	2
3450.2.a.c.1.1		1	5.4	even	2
3450.2.d.r.2899.1		2	5.3	odd	4
3450.2.d.r.2899.2		2	5.2	odd	4
5520.2.a.d.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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