LMFDB - Embedded newform 690.2.a.f.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)

$f(q)$

$=$

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

\(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +4.00000 q^{11} +1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} -1.00000 q^{18} +1.00000 q^{20} -4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +2.00000 q^{26} +1.00000 q^{27} +6.00000 q^{29} -1.00000 q^{30} -1.00000 q^{32} +4.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{39} -1.00000 q^{40} +2.00000 q^{41} -12.0000 q^{43} +4.00000 q^{44} +1.00000 q^{45} -1.00000 q^{46} +8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} +2.00000 q^{51} -2.00000 q^{52} -2.00000 q^{53} -1.00000 q^{54} +4.00000 q^{55} -6.00000 q^{58} +1.00000 q^{60} +6.00000 q^{61} +1.00000 q^{64} -2.00000 q^{65} -4.00000 q^{66} +12.0000 q^{67} +2.00000 q^{68} +1.00000 q^{69} +12.0000 q^{71} -1.00000 q^{72} -6.00000 q^{73} -2.00000 q^{74} +1.00000 q^{75} +2.00000 q^{78} +4.00000 q^{79} +1.00000 q^{80} +1.00000 q^{81} -2.00000 q^{82} -4.00000 q^{83} +2.00000 q^{85} +12.0000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{94} -1.00000 q^{96} -10.0000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)

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
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	1.00000	0.288675
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	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
$21$	0	0
$22$	−4.00000	−0.852803
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	−1.00000	−0.204124
$25$	1.00000	0.200000
$26$	2.00000	0.392232
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−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
$33$	4.00000	0.696311
$34$	−2.00000	−0.342997
$35$	0	0
$36$	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	$0.447432\pi$
$38$	0	0
$39$	−2.00000	−0.320256
$40$	−1.00000	−0.158114
$41$	2.00000	0.312348	0.156174	−	0.987730i	$-0.450084\pi$
$41$	2.00000	0.312348	0.156174	+	0.987730i	$0.450084\pi$
$42$	0	0
$43$	−12.0000	−1.82998	−0.914991	−	0.403473i	$-0.867803\pi$
$43$	−12.0000	−1.82998	−0.914991	+	0.403473i	$0.867803\pi$
$44$	4.00000	0.603023
$45$	1.00000	0.149071
$46$	−1.00000	−0.147442
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	1.00000	0.144338
$49$	−7.00000	−1.00000
$50$	−1.00000	−0.141421
$51$	2.00000	0.280056
$52$	−2.00000	−0.277350
$53$	−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
$55$	4.00000	0.539360
$56$	0	0
$57$	0	0
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	1.00000	0.129099
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.00000	−0.248069
$66$	−4.00000	−0.492366
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	2.00000	0.242536
$69$	1.00000	0.120386
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	−1.00000	−0.117851
$73$	−6.00000	−0.702247	−0.351123	−	0.936329i	$-0.614200\pi$
$73$	−6.00000	−0.702247	−0.351123	+	0.936329i	$0.614200\pi$
$74$	−2.00000	−0.232495
$75$	1.00000	0.115470
$76$	0	0
$77$	0	0
$78$	2.00000	0.226455
$79$	4.00000	0.450035	0.225018	−	0.974355i	$-0.427756\pi$
$79$	4.00000	0.450035	0.225018	+	0.974355i	$0.427756\pi$
$80$	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
$85$	2.00000	0.216930
$86$	12.0000	1.29399
$87$	6.00000	0.643268
$88$	−4.00000	−0.426401
$89$	−6.00000	−0.635999	−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000	−0.635999	−0.317999	+	0.948091i	$0.603011\pi$
$90$	−1.00000	−0.105409
$91$	0	0
$92$	1.00000	0.104257
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	7.00000	0.707107
$99$	4.00000	0.402015
$100$	1.00000	0.100000
$101$	6.00000	0.597022	0.298511	−	0.954406i	$-0.403510\pi$
$101$	6.00000	0.597022	0.298511	+	0.954406i	$0.403510\pi$
$102$	−2.00000	−0.198030
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	2.00000	0.196116
$105$	0	0
$106$	2.00000	0.194257
$107$	−4.00000	−0.386695	−0.193347	−	0.981130i	$-0.561934\pi$
$107$	−4.00000	−0.386695	−0.193347	+	0.981130i	$0.561934\pi$
$108$	1.00000	0.0962250
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−4.00000	−0.381385
$111$	2.00000	0.189832
$112$	0	0
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	6.00000	0.557086
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	−1.00000	−0.0912871
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	2.00000	0.180334
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	2.00000	0.175412
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	4.00000	0.348155
$133$	0	0
$134$	−12.0000	−1.03664
$135$	1.00000	0.0860663
$136$	−2.00000	−0.171499
$137$	−14.0000	−1.19610	−0.598050	−	0.801459i	$-0.704058\pi$
$137$	−14.0000	−1.19610	−0.598050	+	0.801459i	$0.704058\pi$
$138$	−1.00000	−0.0851257
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	−12.0000	−1.00702
$143$	−8.00000	−0.668994
$144$	1.00000	0.0833333
$145$	6.00000	0.498273
$146$	6.00000	0.496564
$147$	−7.00000	−0.577350
$148$	2.00000	0.164399
$149$	−10.0000	−0.819232	−0.409616	−	0.912258i	$-0.634337\pi$
$149$	−10.0000	−0.819232	−0.409616	+	0.912258i	$0.634337\pi$
$150$	−1.00000	−0.0816497
$151$	−16.0000	−1.30206	−0.651031	−	0.759051i	$-0.725663\pi$
$151$	−16.0000	−1.30206	−0.651031	+	0.759051i	$0.725663\pi$
$152$	0	0
$153$	2.00000	0.161690
$154$	0	0
$155$	0	0
$156$	−2.00000	−0.160128
$157$	−14.0000	−1.11732	−0.558661	−	0.829396i	$-0.688685\pi$
$157$	−14.0000	−1.11732	−0.558661	+	0.829396i	$0.688685\pi$
$158$	−4.00000	−0.318223
$159$	−2.00000	−0.158610
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	−1.00000	−0.0785674
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	2.00000	0.156174
$165$	4.00000	0.311400
$166$	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
$169$	−9.00000	−0.692308
$170$	−2.00000	−0.153393
$171$	0	0
$172$	−12.0000	−0.914991
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	−6.00000	−0.454859
$175$	0	0
$176$	4.00000	0.301511
$177$	0	0
$178$	6.00000	0.449719
$179$	−8.00000	−0.597948	−0.298974	−	0.954261i	$-0.596644\pi$
$179$	−8.00000	−0.597948	−0.298974	+	0.954261i	$0.596644\pi$
$180$	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
$183$	6.00000	0.443533
$184$	−1.00000	−0.0737210
$185$	2.00000	0.147043
$186$	0	0
$187$	8.00000	0.585018
$188$	8.00000	0.583460
$189$	0	0
$190$	0	0
$191$	16.0000	1.15772	0.578860	−	0.815427i	$-0.303498\pi$
$191$	16.0000	1.15772	0.578860	+	0.815427i	$0.303498\pi$
$192$	1.00000	0.0721688
$193$	−6.00000	−0.431889	−0.215945	−	0.976406i	$-0.569283\pi$
$193$	−6.00000	−0.431889	−0.215945	+	0.976406i	$0.569283\pi$
$194$	10.0000	0.717958
$195$	−2.00000	−0.143223
$196$	−7.00000	−0.500000
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	−4.00000	−0.284268
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−1.00000	−0.0707107
$201$	12.0000	0.846415
$202$	−6.00000	−0.422159
$203$	0	0
$204$	2.00000	0.140028
$205$	2.00000	0.139686
$206$	−8.00000	−0.557386
$207$	1.00000	0.0695048
$208$	−2.00000	−0.138675
$209$	0	0
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.00000	−0.137361
$213$	12.0000	0.822226
$214$	4.00000	0.273434
$215$	−12.0000	−0.818393
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	2.00000	0.135457
$219$	−6.00000	−0.405442
$220$	4.00000	0.269680
$221$	−4.00000	−0.269069
$222$	−2.00000	−0.134231
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	6.00000	0.399114
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−1.00000	−0.0659380
$231$	0	0
$232$	−6.00000	−0.393919
$233$	−26.0000	−1.70332	−0.851658	−	0.524097i	$-0.824403\pi$
$233$	−26.0000	−1.70332	−0.851658	+	0.524097i	$0.824403\pi$
$234$	2.00000	0.130744
$235$	8.00000	0.521862
$236$	0	0
$237$	4.00000	0.259828
$238$	0	0
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	1.00000	0.0645497
$241$	−22.0000	−1.41714	−0.708572	−	0.705638i	$-0.750660\pi$
$241$	−22.0000	−1.41714	−0.708572	+	0.705638i	$0.750660\pi$
$242$	−5.00000	−0.321412
$243$	1.00000	0.0641500
$244$	6.00000	0.384111
$245$	−7.00000	−0.447214
$246$	−2.00000	−0.127515
$247$	0	0
$248$	0	0
$249$	−4.00000	−0.253490
$250$	−1.00000	−0.0632456
$251$	12.0000	0.757433	0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000	0.757433	0.378717	+	0.925513i	$0.376365\pi$
$252$	0	0
$253$	4.00000	0.251478
$254$	16.0000	1.00393
$255$	2.00000	0.125245
$256$	1.00000	0.0625000
$257$	22.0000	1.37232	0.686161	−	0.727450i	$-0.259294\pi$
$257$	22.0000	1.37232	0.686161	+	0.727450i	$0.259294\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	−2.00000	−0.124035
$261$	6.00000	0.371391
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	−4.00000	−0.246183
$265$	−2.00000	−0.122859
$266$	0	0
$267$	−6.00000	−0.367194
$268$	12.0000	0.733017
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	−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	$0.578126\pi$
$272$	2.00000	0.121268
$273$	0	0
$274$	14.0000	0.845771
$275$	4.00000	0.241209
$276$	1.00000	0.0601929
$277$	14.0000	0.841178	0.420589	−	0.907251i	$-0.361823\pi$
$277$	14.0000	0.841178	0.420589	+	0.907251i	$0.361823\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	−8.00000	−0.476393
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	12.0000	0.712069
$285$	0	0
$286$	8.00000	0.473050
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	−13.0000	−0.764706
$290$	−6.00000	−0.352332
$291$	−10.0000	−0.586210
$292$	−6.00000	−0.351123
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	7.00000	0.408248
$295$	0	0
$296$	−2.00000	−0.116248
$297$	4.00000	0.232104
$298$	10.0000	0.579284
$299$	−2.00000	−0.115663
$300$	1.00000	0.0577350
$301$	0	0
$302$	16.0000	0.920697
$303$	6.00000	0.344691
$304$	0	0
$305$	6.00000	0.343559
$306$	−2.00000	−0.114332
$307$	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
$309$	8.00000	0.455104
$310$	0	0
$311$	−12.0000	−0.680458	−0.340229	−	0.940343i	$-0.610505\pi$
$311$	−12.0000	−0.680458	−0.340229	+	0.940343i	$0.610505\pi$
$312$	2.00000	0.113228
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	14.0000	0.790066
$315$	0	0
$316$	4.00000	0.225018
$317$	−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
$319$	24.0000	1.34374
$320$	1.00000	0.0559017
$321$	−4.00000	−0.223258
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	−2.00000	−0.110940
$326$	20.0000	1.10770
$327$	−2.00000	−0.110600
$328$	−2.00000	−0.110432
$329$	0	0
$330$	−4.00000	−0.220193
$331$	−12.0000	−0.659580	−0.329790	−	0.944054i	$-0.606978\pi$
$331$	−12.0000	−0.659580	−0.329790	+	0.944054i	$0.606978\pi$
$332$	−4.00000	−0.219529
$333$	2.00000	0.109599
$334$	−8.00000	−0.437741
$335$	12.0000	0.655630
$336$	0	0
$337$	22.0000	1.19842	0.599208	−	0.800593i	$-0.295482\pi$
$337$	22.0000	1.19842	0.599208	+	0.800593i	$0.295482\pi$
$338$	9.00000	0.489535
$339$	−6.00000	−0.325875
$340$	2.00000	0.108465
$341$	0	0
$342$	0	0
$343$	0	0
$344$	12.0000	0.646997
$345$	1.00000	0.0538382
$346$	14.0000	0.752645
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	6.00000	0.321634
$349$	−2.00000	−0.107058	−0.0535288	−	0.998566i	$-0.517047\pi$
$349$	−2.00000	−0.107058	−0.0535288	+	0.998566i	$0.517047\pi$
$350$	0	0
$351$	−2.00000	−0.106752
$352$	−4.00000	−0.213201
$353$	−10.0000	−0.532246	−0.266123	−	0.963939i	$-0.585743\pi$
$353$	−10.0000	−0.532246	−0.266123	+	0.963939i	$0.585743\pi$
$354$	0	0
$355$	12.0000	0.636894
$356$	−6.00000	−0.317999
$357$	0	0
$358$	8.00000	0.422813
$359$	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
$363$	5.00000	0.262432
$364$	0	0
$365$	−6.00000	−0.314054
$366$	−6.00000	−0.313625
$367$	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
$370$	−2.00000	−0.103975
$371$	0	0
$372$	0	0
$373$	10.0000	0.517780	0.258890	−	0.965907i	$-0.416643\pi$
$373$	10.0000	0.517780	0.258890	+	0.965907i	$0.416643\pi$
$374$	−8.00000	−0.413670
$375$	1.00000	0.0516398
$376$	−8.00000	−0.412568
$377$	−12.0000	−0.618031
$378$	0	0
$379$	8.00000	0.410932	0.205466	−	0.978664i	$-0.434129\pi$
$379$	8.00000	0.410932	0.205466	+	0.978664i	$0.434129\pi$
$380$	0	0
$381$	−16.0000	−0.819705
$382$	−16.0000	−0.818631
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	6.00000	0.305392
$387$	−12.0000	−0.609994
$388$	−10.0000	−0.507673
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	2.00000	0.101274
$391$	2.00000	0.101144
$392$	7.00000	0.353553
$393$	0	0
$394$	22.0000	1.10834
$395$	4.00000	0.201262
$396$	4.00000	0.201008
$397$	−10.0000	−0.501886	−0.250943	−	0.968002i	$-0.580741\pi$
$397$	−10.0000	−0.501886	−0.250943	+	0.968002i	$0.580741\pi$
$398$	4.00000	0.200502
$399$	0	0
$400$	1.00000	0.0500000
$401$	18.0000	0.898877	0.449439	−	0.893311i	$-0.351624\pi$
$401$	18.0000	0.898877	0.449439	+	0.893311i	$0.351624\pi$
$402$	−12.0000	−0.598506
$403$	0	0
$404$	6.00000	0.298511
$405$	1.00000	0.0496904
$406$	0	0
$407$	8.00000	0.396545
$408$	−2.00000	−0.0990148
$409$	10.0000	0.494468	0.247234	−	0.968956i	$-0.420478\pi$
$409$	10.0000	0.494468	0.247234	+	0.968956i	$0.420478\pi$
$410$	−2.00000	−0.0987730
$411$	−14.0000	−0.690569
$412$	8.00000	0.394132
$413$	0	0
$414$	−1.00000	−0.0491473
$415$	−4.00000	−0.196352
$416$	2.00000	0.0980581
$417$	4.00000	0.195881
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	−2.00000	−0.0974740	−0.0487370	−	0.998812i	$-0.515520\pi$
$421$	−2.00000	−0.0974740	−0.0487370	+	0.998812i	$0.515520\pi$
$422$	−4.00000	−0.194717
$423$	8.00000	0.388973
$424$	2.00000	0.0971286
$425$	2.00000	0.0970143
$426$	−12.0000	−0.581402
$427$	0	0
$428$	−4.00000	−0.193347
$429$	−8.00000	−0.386244
$430$	12.0000	0.578691
$431$	32.0000	1.54139	0.770693	−	0.637207i	$-0.219910\pi$
$431$	32.0000	1.54139	0.770693	+	0.637207i	$0.219910\pi$
$432$	1.00000	0.0481125
$433$	−26.0000	−1.24948	−0.624740	−	0.780833i	$-0.714795\pi$
$433$	−26.0000	−1.24948	−0.624740	+	0.780833i	$0.714795\pi$
$434$	0	0
$435$	6.00000	0.287678
$436$	−2.00000	−0.0957826
$437$	0	0
$438$	6.00000	0.286691
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	−4.00000	−0.190693
$441$	−7.00000	−0.333333
$442$	4.00000	0.190261
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	2.00000	0.0949158
$445$	−6.00000	−0.284427
$446$	−16.0000	−0.757622
$447$	−10.0000	−0.472984
$448$	0	0
$449$	−14.0000	−0.660701	−0.330350	−	0.943858i	$-0.607167\pi$
$449$	−14.0000	−0.660701	−0.330350	+	0.943858i	$0.607167\pi$
$450$	−1.00000	−0.0471405
$451$	8.00000	0.376705
$452$	−6.00000	−0.282216
$453$	−16.0000	−0.751746
$454$	12.0000	0.563188
$455$	0	0
$456$	0	0
$457$	−2.00000	−0.0935561	−0.0467780	−	0.998905i	$-0.514895\pi$
$457$	−2.00000	−0.0935561	−0.0467780	+	0.998905i	$0.514895\pi$
$458$	−14.0000	−0.654177
$459$	2.00000	0.0933520
$460$	1.00000	0.0466252
$461$	30.0000	1.39724	0.698620	−	0.715493i	$-0.253798\pi$
$461$	30.0000	1.39724	0.698620	+	0.715493i	$0.253798\pi$
$462$	0	0
$463$	8.00000	0.371792	0.185896	−	0.982569i	$-0.440481\pi$
$463$	8.00000	0.371792	0.185896	+	0.982569i	$0.440481\pi$
$464$	6.00000	0.278543
$465$	0	0
$466$	26.0000	1.20443
$467$	36.0000	1.66588	0.832941	−	0.553362i	$-0.186655\pi$
$467$	36.0000	1.66588	0.832941	+	0.553362i	$0.186655\pi$
$468$	−2.00000	−0.0924500
$469$	0	0
$470$	−8.00000	−0.369012
$471$	−14.0000	−0.645086
$472$	0	0
$473$	−48.0000	−2.20704
$474$	−4.00000	−0.183726
$475$	0	0
$476$	0	0
$477$	−2.00000	−0.0915737
$478$	12.0000	0.548867
$479$	−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
$481$	−4.00000	−0.182384
$482$	22.0000	1.00207
$483$	0	0
$484$	5.00000	0.227273
$485$	−10.0000	−0.454077
$486$	−1.00000	−0.0453609
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	−6.00000	−0.271607
$489$	−20.0000	−0.904431
$490$	7.00000	0.316228
$491$	−32.0000	−1.44414	−0.722070	−	0.691820i	$-0.756809\pi$
$491$	−32.0000	−1.44414	−0.722070	+	0.691820i	$0.756809\pi$
$492$	2.00000	0.0901670
$493$	12.0000	0.540453
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	0	0
$498$	4.00000	0.179244
$499$	12.0000	0.537194	0.268597	−	0.963253i	$-0.413440\pi$
$499$	12.0000	0.537194	0.268597	+	0.963253i	$0.413440\pi$
$500$	1.00000	0.0447214
$501$	8.00000	0.357414
$502$	−12.0000	−0.535586
$503$	0	0		−	1.00000i	$-0.5\pi$
$503$	0	0			1.00000i	$0.5\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	−4.00000	−0.177822
$507$	−9.00000	−0.399704
$508$	−16.0000	−0.709885
$509$	−26.0000	−1.15243	−0.576215	−	0.817298i	$-0.695471\pi$
$509$	−26.0000	−1.15243	−0.576215	+	0.817298i	$0.695471\pi$
$510$	−2.00000	−0.0885615
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.0000	−0.970378
$515$	8.00000	0.352522
$516$	−12.0000	−0.528271
$517$	32.0000	1.40736
$518$	0	0
$519$	−14.0000	−0.614532
$520$	2.00000	0.0877058
$521$	−22.0000	−0.963837	−0.481919	−	0.876216i	$-0.660060\pi$
$521$	−22.0000	−0.963837	−0.481919	+	0.876216i	$0.660060\pi$
$522$	−6.00000	−0.262613
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	0	0
$526$	−8.00000	−0.348817
$527$	0	0
$528$	4.00000	0.174078
$529$	1.00000	0.0434783
$530$	2.00000	0.0868744
$531$	0	0
$532$	0	0
$533$	−4.00000	−0.173259
$534$	6.00000	0.259645
$535$	−4.00000	−0.172935
$536$	−12.0000	−0.518321
$537$	−8.00000	−0.345225
$538$	18.0000	0.776035
$539$	−28.0000	−1.20605
$540$	1.00000	0.0430331
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	8.00000	0.343629
$543$	6.00000	0.257485
$544$	−2.00000	−0.0857493
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	−36.0000	−1.53925	−0.769624	−	0.638497i	$-0.779557\pi$
$547$	−36.0000	−1.53925	−0.769624	+	0.638497i	$0.779557\pi$
$548$	−14.0000	−0.598050
$549$	6.00000	0.256074
$550$	−4.00000	−0.170561
$551$	0	0
$552$	−1.00000	−0.0425628
$553$	0	0
$554$	−14.0000	−0.594803
$555$	2.00000	0.0848953
$556$	4.00000	0.169638
$557$	−26.0000	−1.10166	−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000	−1.10166	−0.550828	+	0.834619i	$0.685688\pi$
$558$	0	0
$559$	24.0000	1.01509
$560$	0	0
$561$	8.00000	0.337760
$562$	−10.0000	−0.421825
$563$	−20.0000	−0.842900	−0.421450	−	0.906852i	$-0.638479\pi$
$563$	−20.0000	−0.842900	−0.421450	+	0.906852i	$0.638479\pi$
$564$	8.00000	0.336861
$565$	−6.00000	−0.252422
$566$	−28.0000	−1.17693
$567$	0	0
$568$	−12.0000	−0.503509
$569$	42.0000	1.76073	0.880366	−	0.474295i	$-0.157297\pi$
$569$	42.0000	1.76073	0.880366	+	0.474295i	$0.157297\pi$
$570$	0	0
$571$	−40.0000	−1.67395	−0.836974	−	0.547243i	$-0.815677\pi$
$571$	−40.0000	−1.67395	−0.836974	+	0.547243i	$0.815677\pi$
$572$	−8.00000	−0.334497
$573$	16.0000	0.668410
$574$	0	0
$575$	1.00000	0.0417029
$576$	1.00000	0.0416667
$577$	10.0000	0.416305	0.208153	−	0.978096i	$-0.433255\pi$
$577$	10.0000	0.416305	0.208153	+	0.978096i	$0.433255\pi$
$578$	13.0000	0.540729
$579$	−6.00000	−0.249351
$580$	6.00000	0.249136
$581$	0	0
$582$	10.0000	0.414513
$583$	−8.00000	−0.331326
$584$	6.00000	0.248282
$585$	−2.00000	−0.0826898
$586$	18.0000	0.743573
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	−7.00000	−0.288675
$589$	0	0
$590$	0	0
$591$	−22.0000	−0.904959
$592$	2.00000	0.0821995
$593$	−42.0000	−1.72473	−0.862367	−	0.506284i	$-0.831019\pi$
$593$	−42.0000	−1.72473	−0.862367	+	0.506284i	$0.831019\pi$
$594$	−4.00000	−0.164122
$595$	0	0
$596$	−10.0000	−0.409616
$597$	−4.00000	−0.163709
$598$	2.00000	0.0817861
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	−1.00000	−0.0408248
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	0	0
$603$	12.0000	0.488678
$604$	−16.0000	−0.651031
$605$	5.00000	0.203279
$606$	−6.00000	−0.243733
$607$	−8.00000	−0.324710	−0.162355	−	0.986732i	$-0.551909\pi$
$607$	−8.00000	−0.324710	−0.162355	+	0.986732i	$0.551909\pi$
$608$	0	0
$609$	0	0
$610$	−6.00000	−0.242933
$611$	−16.0000	−0.647291
$612$	2.00000	0.0808452
$613$	10.0000	0.403896	0.201948	−	0.979396i	$-0.435273\pi$
$613$	10.0000	0.403896	0.201948	+	0.979396i	$0.435273\pi$
$614$	−12.0000	−0.484281
$615$	2.00000	0.0806478
$616$	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
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	0	0
$621$	1.00000	0.0401286
$622$	12.0000	0.481156
$623$	0	0
$624$	−2.00000	−0.0800641
$625$	1.00000	0.0400000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	−14.0000	−0.558661
$629$	4.00000	0.159490
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	−4.00000	−0.159111
$633$	4.00000	0.158986
$634$	6.00000	0.238290
$635$	−16.0000	−0.634941
$636$	−2.00000	−0.0793052
$637$	14.0000	0.554700
$638$	−24.0000	−0.950169
$639$	12.0000	0.474713
$640$	−1.00000	−0.0395285
$641$	18.0000	0.710957	0.355479	−	0.934684i	$-0.384318\pi$
$641$	18.0000	0.710957	0.355479	+	0.934684i	$0.384318\pi$
$642$	4.00000	0.157867
$643$	28.0000	1.10421	0.552106	−	0.833774i	$-0.313824\pi$
$643$	28.0000	1.10421	0.552106	+	0.833774i	$0.313824\pi$
$644$	0	0
$645$	−12.0000	−0.472500
$646$	0	0
$647$	16.0000	0.629025	0.314512	−	0.949253i	$-0.398159\pi$
$647$	16.0000	0.629025	0.314512	+	0.949253i	$0.398159\pi$
$648$	−1.00000	−0.0392837
$649$	0	0
$650$	2.00000	0.0784465
$651$	0	0
$652$	−20.0000	−0.783260
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	2.00000	0.0782062
$655$	0	0
$656$	2.00000	0.0780869
$657$	−6.00000	−0.234082
$658$	0	0
$659$	12.0000	0.467454	0.233727	−	0.972302i	$-0.424908\pi$
$659$	12.0000	0.467454	0.233727	+	0.972302i	$0.424908\pi$
$660$	4.00000	0.155700
$661$	30.0000	1.16686	0.583432	−	0.812162i	$-0.301709\pi$
$661$	30.0000	1.16686	0.583432	+	0.812162i	$0.301709\pi$
$662$	12.0000	0.466393
$663$	−4.00000	−0.155347
$664$	4.00000	0.155230
$665$	0	0
$666$	−2.00000	−0.0774984
$667$	6.00000	0.232321
$668$	8.00000	0.309529
$669$	16.0000	0.618596
$670$	−12.0000	−0.463600
$671$	24.0000	0.926510
$672$	0	0
$673$	34.0000	1.31060	0.655302	−	0.755367i	$-0.272541\pi$
$673$	34.0000	1.31060	0.655302	+	0.755367i	$0.272541\pi$
$674$	−22.0000	−0.847408
$675$	1.00000	0.0384900
$676$	−9.00000	−0.346154
$677$	6.00000	0.230599	0.115299	−	0.993331i	$-0.463217\pi$
$677$	6.00000	0.230599	0.115299	+	0.993331i	$0.463217\pi$
$678$	6.00000	0.230429
$679$	0	0
$680$	−2.00000	−0.0766965
$681$	−12.0000	−0.459841
$682$	0	0
$683$	44.0000	1.68361	0.841807	−	0.539779i	$-0.181492\pi$
$683$	44.0000	1.68361	0.841807	+	0.539779i	$0.181492\pi$
$684$	0	0
$685$	−14.0000	−0.534913
$686$	0	0
$687$	14.0000	0.534133
$688$	−12.0000	−0.457496
$689$	4.00000	0.152388
$690$	−1.00000	−0.0380693
$691$	−44.0000	−1.67384	−0.836919	−	0.547326i	$-0.815646\pi$
$691$	−44.0000	−1.67384	−0.836919	+	0.547326i	$0.815646\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	−12.0000	−0.455514
$695$	4.00000	0.151729
$696$	−6.00000	−0.227429
$697$	4.00000	0.151511
$698$	2.00000	0.0757011
$699$	−26.0000	−0.983410
$700$	0	0
$701$	−2.00000	−0.0755390	−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000	−0.0755390	−0.0377695	+	0.999286i	$0.512025\pi$
$702$	2.00000	0.0754851
$703$	0	0
$704$	4.00000	0.150756
$705$	8.00000	0.301297
$706$	10.0000	0.376355
$707$	0	0
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	−12.0000	−0.450352
$711$	4.00000	0.150012
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	−8.00000	−0.299183
$716$	−8.00000	−0.298974
$717$	−12.0000	−0.448148
$718$	0	0
$719$	−4.00000	−0.149175	−0.0745874	−	0.997214i	$-0.523764\pi$
$719$	−4.00000	−0.149175	−0.0745874	+	0.997214i	$0.523764\pi$
$720$	1.00000	0.0372678
$721$	0	0
$722$	19.0000	0.707107
$723$	−22.0000	−0.818189
$724$	6.00000	0.222988
$725$	6.00000	0.222834
$726$	−5.00000	−0.185567
$727$	32.0000	1.18681	0.593407	−	0.804902i	$-0.297782\pi$
$727$	32.0000	1.18681	0.593407	+	0.804902i	$0.297782\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	6.00000	0.222070
$731$	−24.0000	−0.887672
$732$	6.00000	0.221766
$733$	−14.0000	−0.517102	−0.258551	−	0.965998i	$-0.583245\pi$
$733$	−14.0000	−0.517102	−0.258551	+	0.965998i	$0.583245\pi$
$734$	−16.0000	−0.590571
$735$	−7.00000	−0.258199
$736$	−1.00000	−0.0368605
$737$	48.0000	1.76810
$738$	−2.00000	−0.0736210
$739$	−20.0000	−0.735712	−0.367856	−	0.929883i	$-0.619908\pi$
$739$	−20.0000	−0.735712	−0.367856	+	0.929883i	$0.619908\pi$
$740$	2.00000	0.0735215
$741$	0	0
$742$	0	0
$743$	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
$745$	−10.0000	−0.366372
$746$	−10.0000	−0.366126
$747$	−4.00000	−0.146352
$748$	8.00000	0.292509
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−44.0000	−1.60558	−0.802791	−	0.596260i	$-0.796653\pi$
$751$	−44.0000	−1.60558	−0.802791	+	0.596260i	$0.796653\pi$
$752$	8.00000	0.291730
$753$	12.0000	0.437304
$754$	12.0000	0.437014
$755$	−16.0000	−0.582300
$756$	0	0
$757$	50.0000	1.81728	0.908640	−	0.417579i	$-0.137121\pi$
$757$	50.0000	1.81728	0.908640	+	0.417579i	$0.137121\pi$
$758$	−8.00000	−0.290573
$759$	4.00000	0.145191
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	16.0000	0.579619
$763$	0	0
$764$	16.0000	0.578860
$765$	2.00000	0.0723102
$766$	−24.0000	−0.867155
$767$	0	0
$768$	1.00000	0.0360844
$769$	10.0000	0.360609	0.180305	−	0.983611i	$-0.442292\pi$
$769$	10.0000	0.360609	0.180305	+	0.983611i	$0.442292\pi$
$770$	0	0
$771$	22.0000	0.792311
$772$	−6.00000	−0.215945
$773$	6.00000	0.215805	0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000	0.215805	0.107903	+	0.994161i	$0.465587\pi$
$774$	12.0000	0.431331
$775$	0	0
$776$	10.0000	0.358979
$777$	0	0
$778$	−6.00000	−0.215110
$779$	0	0
$780$	−2.00000	−0.0716115
$781$	48.0000	1.71758
$782$	−2.00000	−0.0715199
$783$	6.00000	0.214423
$784$	−7.00000	−0.250000
$785$	−14.0000	−0.499681
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	−22.0000	−0.783718
$789$	8.00000	0.284808
$790$	−4.00000	−0.142314
$791$	0	0
$792$	−4.00000	−0.142134
$793$	−12.0000	−0.426132
$794$	10.0000	0.354887
$795$	−2.00000	−0.0709327
$796$	−4.00000	−0.141776
$797$	38.0000	1.34603	0.673015	−	0.739629i	$-0.264999\pi$
$797$	38.0000	1.34603	0.673015	+	0.739629i	$0.264999\pi$
$798$	0	0
$799$	16.0000	0.566039
$800$	−1.00000	−0.0353553
$801$	−6.00000	−0.212000
$802$	−18.0000	−0.635602
$803$	−24.0000	−0.846942
$804$	12.0000	0.423207
$805$	0	0
$806$	0	0
$807$	−18.0000	−0.633630
$808$	−6.00000	−0.211079
$809$	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
$811$	−28.0000	−0.983213	−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000	−0.983213	−0.491606	+	0.870817i	$0.663590\pi$
$812$	0	0
$813$	−8.00000	−0.280572
$814$	−8.00000	−0.280400
$815$	−20.0000	−0.700569
$816$	2.00000	0.0700140
$817$	0	0
$818$	−10.0000	−0.349642
$819$	0	0
$820$	2.00000	0.0698430
$821$	−2.00000	−0.0698005	−0.0349002	−	0.999391i	$-0.511111\pi$
$821$	−2.00000	−0.0698005	−0.0349002	+	0.999391i	$0.511111\pi$
$822$	14.0000	0.488306
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	−8.00000	−0.278693
$825$	4.00000	0.139262
$826$	0	0
$827$	−20.0000	−0.695468	−0.347734	−	0.937593i	$-0.613049\pi$
$827$	−20.0000	−0.695468	−0.347734	+	0.937593i	$0.613049\pi$
$828$	1.00000	0.0347524
$829$	−2.00000	−0.0694629	−0.0347314	−	0.999397i	$-0.511058\pi$
$829$	−2.00000	−0.0694629	−0.0347314	+	0.999397i	$0.511058\pi$
$830$	4.00000	0.138842
$831$	14.0000	0.485655
$832$	−2.00000	−0.0693375
$833$	−14.0000	−0.485071
$834$	−4.00000	−0.138509
$835$	8.00000	0.276851
$836$	0	0
$837$	0	0
$838$	12.0000	0.414533
$839$	56.0000	1.93333	0.966667	−	0.256036i	$-0.0824164\pi$
$839$	56.0000	1.93333	0.966667	+	0.256036i	$0.0824164\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	2.00000	0.0689246
$843$	10.0000	0.344418
$844$	4.00000	0.137686
$845$	−9.00000	−0.309609
$846$	−8.00000	−0.275046
$847$	0	0
$848$	−2.00000	−0.0686803
$849$	28.0000	0.960958
$850$	−2.00000	−0.0685994
$851$	2.00000	0.0685591
$852$	12.0000	0.411113
$853$	30.0000	1.02718	0.513590	−	0.858036i	$-0.328315\pi$
$853$	30.0000	1.02718	0.513590	+	0.858036i	$0.328315\pi$
$854$	0	0
$855$	0	0
$856$	4.00000	0.136717
$857$	6.00000	0.204956	0.102478	−	0.994735i	$-0.467323\pi$
$857$	6.00000	0.204956	0.102478	+	0.994735i	$0.467323\pi$
$858$	8.00000	0.273115
$859$	−52.0000	−1.77422	−0.887109	−	0.461561i	$-0.847290\pi$
$859$	−52.0000	−1.77422	−0.887109	+	0.461561i	$0.847290\pi$
$860$	−12.0000	−0.409197
$861$	0	0
$862$	−32.0000	−1.08992
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	−1.00000	−0.0340207
$865$	−14.0000	−0.476014
$866$	26.0000	0.883516
$867$	−13.0000	−0.441503
$868$	0	0
$869$	16.0000	0.542763
$870$	−6.00000	−0.203419
$871$	−24.0000	−0.813209
$872$	2.00000	0.0677285
$873$	−10.0000	−0.338449
$874$	0	0
$875$	0	0
$876$	−6.00000	−0.202721
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	0	0
$879$	−18.0000	−0.607125
$880$	4.00000	0.134840
$881$	2.00000	0.0673817	0.0336909	−	0.999432i	$-0.489274\pi$
$881$	2.00000	0.0673817	0.0336909	+	0.999432i	$0.489274\pi$
$882$	7.00000	0.235702
$883$	12.0000	0.403832	0.201916	−	0.979403i	$-0.435283\pi$
$883$	12.0000	0.403832	0.201916	+	0.979403i	$0.435283\pi$
$884$	−4.00000	−0.134535
$885$	0	0
$886$	−36.0000	−1.20944
$887$	8.00000	0.268614	0.134307	−	0.990940i	$-0.457119\pi$
$887$	8.00000	0.268614	0.134307	+	0.990940i	$0.457119\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	6.00000	0.201120
$891$	4.00000	0.134005
$892$	16.0000	0.535720
$893$	0	0
$894$	10.0000	0.334450
$895$	−8.00000	−0.267411
$896$	0	0
$897$	−2.00000	−0.0667781
$898$	14.0000	0.467186
$899$	0	0
$900$	1.00000	0.0333333
$901$	−4.00000	−0.133259
$902$	−8.00000	−0.266371
$903$	0	0
$904$	6.00000	0.199557
$905$	6.00000	0.199447
$906$	16.0000	0.531564
$907$	−20.0000	−0.664089	−0.332045	−	0.943264i	$-0.607738\pi$
$907$	−20.0000	−0.664089	−0.332045	+	0.943264i	$0.607738\pi$
$908$	−12.0000	−0.398234
$909$	6.00000	0.199007
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	0	0
$913$	−16.0000	−0.529523
$914$	2.00000	0.0661541
$915$	6.00000	0.198354
$916$	14.0000	0.462573
$917$	0	0
$918$	−2.00000	−0.0660098
$919$	36.0000	1.18753	0.593765	−	0.804638i	$-0.297641\pi$
$919$	36.0000	1.18753	0.593765	+	0.804638i	$0.297641\pi$
$920$	−1.00000	−0.0329690
$921$	12.0000	0.395413
$922$	−30.0000	−0.987997
$923$	−24.0000	−0.789970
$924$	0	0
$925$	2.00000	0.0657596
$926$	−8.00000	−0.262896
$927$	8.00000	0.262754
$928$	−6.00000	−0.196960
$929$	−30.0000	−0.984268	−0.492134	−	0.870519i	$-0.663783\pi$
$929$	−30.0000	−0.984268	−0.492134	+	0.870519i	$0.663783\pi$
$930$	0	0
$931$	0	0
$932$	−26.0000	−0.851658
$933$	−12.0000	−0.392862
$934$	−36.0000	−1.17796
$935$	8.00000	0.261628
$936$	2.00000	0.0653720
$937$	22.0000	0.718709	0.359354	−	0.933201i	$-0.382997\pi$
$937$	22.0000	0.718709	0.359354	+	0.933201i	$0.382997\pi$
$938$	0	0
$939$	22.0000	0.717943
$940$	8.00000	0.260931
$941$	−50.0000	−1.62995	−0.814977	−	0.579494i	$-0.803250\pi$
$941$	−50.0000	−1.62995	−0.814977	+	0.579494i	$0.803250\pi$
$942$	14.0000	0.456145
$943$	2.00000	0.0651290
$944$	0	0
$945$	0	0
$946$	48.0000	1.56061
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	4.00000	0.129914
$949$	12.0000	0.389536
$950$	0	0
$951$	−6.00000	−0.194563
$952$	0	0
$953$	−6.00000	−0.194359	−0.0971795	−	0.995267i	$-0.530982\pi$
$953$	−6.00000	−0.194359	−0.0971795	+	0.995267i	$0.530982\pi$
$954$	2.00000	0.0647524
$955$	16.0000	0.517748
$956$	−12.0000	−0.388108
$957$	24.0000	0.775810
$958$	24.0000	0.775405
$959$	0	0
$960$	1.00000	0.0322749
$961$	−31.0000	−1.00000
$962$	4.00000	0.128965
$963$	−4.00000	−0.128898
$964$	−22.0000	−0.708572
$965$	−6.00000	−0.193147
$966$	0	0
$967$	−32.0000	−1.02905	−0.514525	−	0.857475i	$-0.672032\pi$
$967$	−32.0000	−1.02905	−0.514525	+	0.857475i	$0.672032\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	10.0000	0.321081
$971$	−36.0000	−1.15529	−0.577647	−	0.816286i	$-0.696029\pi$
$971$	−36.0000	−1.15529	−0.577647	+	0.816286i	$0.696029\pi$
$972$	1.00000	0.0320750
$973$	0	0
$974$	−16.0000	−0.512673
$975$	−2.00000	−0.0640513
$976$	6.00000	0.192055
$977$	26.0000	0.831814	0.415907	−	0.909407i	$-0.363464\pi$
$977$	26.0000	0.831814	0.415907	+	0.909407i	$0.363464\pi$
$978$	20.0000	0.639529
$979$	−24.0000	−0.767043
$980$	−7.00000	−0.223607
$981$	−2.00000	−0.0638551
$982$	32.0000	1.02116
$983$	56.0000	1.78612	0.893061	−	0.449935i	$-0.148553\pi$
$983$	56.0000	1.78612	0.893061	+	0.449935i	$0.148553\pi$
$984$	−2.00000	−0.0637577
$985$	−22.0000	−0.700978
$986$	−12.0000	−0.382158
$987$	0	0
$988$	0	0
$989$	−12.0000	−0.381578
$990$	−4.00000	−0.127128
$991$	16.0000	0.508257	0.254128	−	0.967170i	$-0.418211\pi$
$991$	16.0000	0.508257	0.254128	+	0.967170i	$0.418211\pi$
$992$	0	0
$993$	−12.0000	−0.380808
$994$	0	0
$995$	−4.00000	−0.126809
$996$	−4.00000	−0.126745
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	−12.0000	−0.379853
$999$	2.00000	0.0632772

Twists

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

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