LMFDB - Embedded newform 3234.2.a.q.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3234,2,Mod(1,3234)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3234, 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("3234.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3234.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:	$25.8236200137$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3234.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{12} -2.00000 q^{13} +1.00000 q^{16} +1.00000 q^{18} -2.00000 q^{19} -1.00000 q^{22} -1.00000 q^{24} -5.00000 q^{25} -2.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -2.00000 q^{31} +1.00000 q^{32} +1.00000 q^{33} +1.00000 q^{36} +2.00000 q^{37} -2.00000 q^{38} +2.00000 q^{39} -4.00000 q^{43} -1.00000 q^{44} +6.00000 q^{47} -1.00000 q^{48} -5.00000 q^{50} -2.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} +2.00000 q^{57} -6.00000 q^{58} -2.00000 q^{61} -2.00000 q^{62} +1.00000 q^{64} +1.00000 q^{66} -4.00000 q^{67} -12.0000 q^{71} +1.00000 q^{72} +4.00000 q^{73} +2.00000 q^{74} +5.00000 q^{75} -2.00000 q^{76} +2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} -6.00000 q^{83} -4.00000 q^{86} +6.00000 q^{87} -1.00000 q^{88} +6.00000 q^{89} +2.00000 q^{93} +6.00000 q^{94} -1.00000 q^{96} -2.00000 q^{97} -1.00000 q^{99} +O(q^{100})$

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
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	−1.00000	−0.408248
$7$	0	0
$7$	0	0
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	−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
$15$	0	0
$16$	1.00000	0.250000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	1.00000	0.235702
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	0	0
$21$	0	0
$22$	−1.00000	−0.213201
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	−1.00000	−0.204124
$25$	−5.00000	−1.00000
$26$	−2.00000	−0.392232
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.00000	−1.11417	−0.557086	−	0.830455i	$-0.688081\pi$
$29$	−6.00000	−1.11417	−0.557086	+	0.830455i	$0.688081\pi$
$30$	0	0
$31$	−2.00000	−0.359211	−0.179605	−	0.983739i	$-0.557482\pi$
$31$	−2.00000	−0.359211	−0.179605	+	0.983739i	$0.557482\pi$
$32$	1.00000	0.176777
$33$	1.00000	0.174078
$34$	0	0
$35$	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	$0.447432\pi$
$38$	−2.00000	−0.324443
$39$	2.00000	0.320256
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	−1.00000	−0.150756
$45$	0	0
$46$	0	0
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	−1.00000	−0.144338
$49$	0	0
$50$	−5.00000	−0.707107
$51$	0	0
$52$	−2.00000	−0.277350
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	−1.00000	−0.136083
$55$	0	0
$56$	0	0
$57$	2.00000	0.264906
$58$	−6.00000	−0.787839
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−2.00000	−0.254000
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	1.00000	0.123091
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−12.0000	−1.42414	−0.712069	−	0.702109i	$-0.752242\pi$
$71$	−12.0000	−1.42414	−0.712069	+	0.702109i	$0.752242\pi$
$72$	1.00000	0.117851
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	2.00000	0.232495
$75$	5.00000	0.577350
$76$	−2.00000	−0.229416
$77$	0	0
$78$	2.00000	0.226455
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−6.00000	−0.658586	−0.329293	−	0.944228i	$-0.606810\pi$
$83$	−6.00000	−0.658586	−0.329293	+	0.944228i	$0.606810\pi$
$84$	0	0
$85$	0	0
$86$	−4.00000	−0.431331
$87$	6.00000	0.643268
$88$	−1.00000	−0.106600
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	2.00000	0.207390
$94$	6.00000	0.618853
$95$	0	0
$96$	−1.00000	−0.102062
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	−5.00000	−0.500000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	0	0
$103$	−14.0000	−1.37946	−0.689730	−	0.724066i	$-0.742271\pi$
$103$	−14.0000	−1.37946	−0.689730	+	0.724066i	$0.742271\pi$
$104$	−2.00000	−0.196116
$105$	0	0
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	−1.00000	−0.0962250
$109$	−10.0000	−0.957826	−0.478913	−	0.877862i	$-0.658969\pi$
$109$	−10.0000	−0.957826	−0.478913	+	0.877862i	$0.658969\pi$
$110$	0	0
$111$	−2.00000	−0.189832
$112$	0	0
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	2.00000	0.187317
$115$	0	0
$116$	−6.00000	−0.557086
$117$	−2.00000	−0.184900
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	−2.00000	−0.181071
$123$	0	0
$124$	−2.00000	−0.179605
$125$	0	0
$126$	0	0
$127$	8.00000	0.709885	0.354943	−	0.934888i	$-0.384500\pi$
$127$	8.00000	0.709885	0.354943	+	0.934888i	$0.384500\pi$
$128$	1.00000	0.0883883
$129$	4.00000	0.352180
$130$	0	0
$131$	6.00000	0.524222	0.262111	−	0.965038i	$-0.415581\pi$
$131$	6.00000	0.524222	0.262111	+	0.965038i	$0.415581\pi$
$132$	1.00000	0.0870388
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	0	0
$137$	−18.0000	−1.53784	−0.768922	−	0.639343i	$-0.779207\pi$
$137$	−18.0000	−1.53784	−0.768922	+	0.639343i	$0.779207\pi$
$138$	0	0
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	−6.00000	−0.505291
$142$	−12.0000	−1.00702
$143$	2.00000	0.167248
$144$	1.00000	0.0833333
$145$	0	0
$146$	4.00000	0.331042
$147$	0	0
$148$	2.00000	0.164399
$149$	6.00000	0.491539	0.245770	−	0.969328i	$-0.420959\pi$
$149$	6.00000	0.491539	0.245770	+	0.969328i	$0.420959\pi$
$150$	5.00000	0.408248
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	0	0
$155$	0	0
$156$	2.00000	0.160128
$157$	16.0000	1.27694	0.638470	−	0.769647i	$-0.279568\pi$
$157$	16.0000	1.27694	0.638470	+	0.769647i	$0.279568\pi$
$158$	8.00000	0.636446
$159$	6.00000	0.475831
$160$	0	0
$161$	0	0
$162$	1.00000	0.0785674
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	0	0
$165$	0	0
$166$	−6.00000	−0.465690
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	−2.00000	−0.152944
$172$	−4.00000	−0.304997
$173$	−18.0000	−1.36851	−0.684257	−	0.729241i	$-0.739873\pi$
$173$	−18.0000	−1.36851	−0.684257	+	0.729241i	$0.739873\pi$
$174$	6.00000	0.454859
$175$	0	0
$176$	−1.00000	−0.0753778
$177$	0	0
$178$	6.00000	0.449719
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	−20.0000	−1.48659	−0.743294	−	0.668965i	$-0.766738\pi$
$181$	−20.0000	−1.48659	−0.743294	+	0.668965i	$0.766738\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	0	0
$186$	2.00000	0.146647
$187$	0	0
$188$	6.00000	0.437595
$189$	0	0
$190$	0	0
$191$	24.0000	1.73658	0.868290	−	0.496058i	$-0.165220\pi$
$191$	24.0000	1.73658	0.868290	+	0.496058i	$0.165220\pi$
$192$	−1.00000	−0.0721688
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	0	0
$197$	−6.00000	−0.427482	−0.213741	−	0.976890i	$-0.568565\pi$
$197$	−6.00000	−0.427482	−0.213741	+	0.976890i	$0.568565\pi$
$198$	−1.00000	−0.0710669
$199$	10.0000	0.708881	0.354441	−	0.935079i	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	−5.00000	−0.353553
$201$	4.00000	0.282138
$202$	−6.00000	−0.422159
$203$	0	0
$204$	0	0
$205$	0	0
$206$	−14.0000	−0.975426
$207$	0	0
$208$	−2.00000	−0.138675
$209$	2.00000	0.138343
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	12.0000	0.822226
$214$	0	0
$215$	0	0
$216$	−1.00000	−0.0680414
$217$	0	0
$218$	−10.0000	−0.677285
$219$	−4.00000	−0.270295
$220$	0	0
$221$	0	0
$222$	−2.00000	−0.134231
$223$	−2.00000	−0.133930	−0.0669650	−	0.997755i	$-0.521332\pi$
$223$	−2.00000	−0.133930	−0.0669650	+	0.997755i	$0.521332\pi$
$224$	0	0
$225$	−5.00000	−0.333333
$226$	−18.0000	−1.19734
$227$	−18.0000	−1.19470	−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000	−1.19470	−0.597351	+	0.801980i	$0.703780\pi$
$228$	2.00000	0.132453
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	0	0
$231$	0	0
$232$	−6.00000	−0.393919
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−2.00000	−0.130744
$235$	0	0
$236$	0	0
$237$	−8.00000	−0.519656
$238$	0	0
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	28.0000	1.80364	0.901819	−	0.432113i	$-0.142232\pi$
$241$	28.0000	1.80364	0.901819	+	0.432113i	$0.142232\pi$
$242$	1.00000	0.0642824
$243$	−1.00000	−0.0641500
$244$	−2.00000	−0.128037
$245$	0	0
$246$	0	0
$247$	4.00000	0.254514
$248$	−2.00000	−0.127000
$249$	6.00000	0.380235
$250$	0	0
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	0	0
$253$	0	0
$254$	8.00000	0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−18.0000	−1.12281	−0.561405	−	0.827541i	$-0.689739\pi$
$257$	−18.0000	−1.12281	−0.561405	+	0.827541i	$0.689739\pi$
$258$	4.00000	0.249029
$259$	0	0
$260$	0	0
$261$	−6.00000	−0.371391
$262$	6.00000	0.370681
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	1.00000	0.0615457
$265$	0	0
$266$	0	0
$267$	−6.00000	−0.367194
$268$	−4.00000	−0.244339
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	0	0
$271$	−20.0000	−1.21491	−0.607457	−	0.794353i	$-0.707810\pi$
$271$	−20.0000	−1.21491	−0.607457	+	0.794353i	$0.707810\pi$
$272$	0	0
$273$	0	0
$274$	−18.0000	−1.08742
$275$	5.00000	0.301511
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−14.0000	−0.839664
$279$	−2.00000	−0.119737
$280$	0	0
$281$	−6.00000	−0.357930	−0.178965	−	0.983855i	$-0.557275\pi$
$281$	−6.00000	−0.357930	−0.178965	+	0.983855i	$0.557275\pi$
$282$	−6.00000	−0.357295
$283$	−14.0000	−0.832214	−0.416107	−	0.909316i	$-0.636606\pi$
$283$	−14.0000	−0.832214	−0.416107	+	0.909316i	$0.636606\pi$
$284$	−12.0000	−0.712069
$285$	0	0
$286$	2.00000	0.118262
$287$	0	0
$288$	1.00000	0.0589256
$289$	−17.0000	−1.00000
$290$	0	0
$291$	2.00000	0.117242
$292$	4.00000	0.234082
$293$	−18.0000	−1.05157	−0.525786	−	0.850617i	$-0.676229\pi$
$293$	−18.0000	−1.05157	−0.525786	+	0.850617i	$0.676229\pi$
$294$	0	0
$295$	0	0
$296$	2.00000	0.116248
$297$	1.00000	0.0580259
$298$	6.00000	0.347571
$299$	0	0
$300$	5.00000	0.288675
$301$	0	0
$302$	8.00000	0.460348
$303$	6.00000	0.344691
$304$	−2.00000	−0.114708
$305$	0	0
$306$	0	0
$307$	34.0000	1.94048	0.970241	−	0.242140i	$-0.0778494\pi$
$307$	34.0000	1.94048	0.970241	+	0.242140i	$0.0778494\pi$
$308$	0	0
$309$	14.0000	0.796432
$310$	0	0
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	2.00000	0.113228
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	16.0000	0.902932
$315$	0	0
$316$	8.00000	0.450035
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	6.00000	0.336463
$319$	6.00000	0.335936
$320$	0	0
$321$	0	0
$322$	0	0
$323$	0	0
$324$	1.00000	0.0555556
$325$	10.0000	0.554700
$326$	20.0000	1.10770
$327$	10.0000	0.553001
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	−6.00000	−0.329293
$333$	2.00000	0.109599
$334$	−12.0000	−0.656611
$335$	0	0
$336$	0	0
$337$	14.0000	0.762629	0.381314	−	0.924445i	$-0.375472\pi$
$337$	14.0000	0.762629	0.381314	+	0.924445i	$0.375472\pi$
$338$	−9.00000	−0.489535
$339$	18.0000	0.977626
$340$	0	0
$341$	2.00000	0.108306
$342$	−2.00000	−0.108148
$343$	0	0
$344$	−4.00000	−0.215666
$345$	0	0
$346$	−18.0000	−0.967686
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	6.00000	0.321634
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	0	0
$351$	2.00000	0.106752
$352$	−1.00000	−0.0533002
$353$	−18.0000	−0.958043	−0.479022	−	0.877803i	$-0.659008\pi$
$353$	−18.0000	−0.958043	−0.479022	+	0.877803i	$0.659008\pi$
$354$	0	0
$355$	0	0
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−20.0000	−1.05118
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	0	0
$366$	2.00000	0.104542
$367$	22.0000	1.14839	0.574195	−	0.818718i	$-0.305315\pi$
$367$	22.0000	1.14839	0.574195	+	0.818718i	$0.305315\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	2.00000	0.103695
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	6.00000	0.309426
$377$	12.0000	0.618031
$378$	0	0
$379$	20.0000	1.02733	0.513665	−	0.857991i	$-0.328287\pi$
$379$	20.0000	1.02733	0.513665	+	0.857991i	$0.328287\pi$
$380$	0	0
$381$	−8.00000	−0.409852
$382$	24.0000	1.22795
$383$	−6.00000	−0.306586	−0.153293	−	0.988181i	$-0.548988\pi$
$383$	−6.00000	−0.306586	−0.153293	+	0.988181i	$0.548988\pi$
$384$	−1.00000	−0.0510310
$385$	0	0
$386$	2.00000	0.101797
$387$	−4.00000	−0.203331
$388$	−2.00000	−0.101535
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	−6.00000	−0.302660
$394$	−6.00000	−0.302276
$395$	0	0
$396$	−1.00000	−0.0502519
$397$	16.0000	0.803017	0.401508	−	0.915855i	$-0.368486\pi$
$397$	16.0000	0.803017	0.401508	+	0.915855i	$0.368486\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	−5.00000	−0.250000
$401$	6.00000	0.299626	0.149813	−	0.988714i	$-0.452133\pi$
$401$	6.00000	0.299626	0.149813	+	0.988714i	$0.452133\pi$
$402$	4.00000	0.199502
$403$	4.00000	0.199254
$404$	−6.00000	−0.298511
$405$	0	0
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	40.0000	1.97787	0.988936	−	0.148340i	$-0.0473931\pi$
$409$	40.0000	1.97787	0.988936	+	0.148340i	$0.0473931\pi$
$410$	0	0
$411$	18.0000	0.887875
$412$	−14.0000	−0.689730
$413$	0	0
$414$	0	0
$415$	0	0
$416$	−2.00000	−0.0980581
$417$	14.0000	0.685583
$418$	2.00000	0.0978232
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	20.0000	0.973585
$423$	6.00000	0.291730
$424$	−6.00000	−0.291386
$425$	0	0
$426$	12.0000	0.581402
$427$	0	0
$428$	0	0
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	−1.00000	−0.0481125
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	0	0
$436$	−10.0000	−0.478913
$437$	0	0
$438$	−4.00000	−0.191127
$439$	28.0000	1.33637	0.668184	−	0.743996i	$-0.267072\pi$
$439$	28.0000	1.33637	0.668184	+	0.743996i	$0.267072\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−12.0000	−0.570137	−0.285069	−	0.958507i	$-0.592016\pi$
$443$	−12.0000	−0.570137	−0.285069	+	0.958507i	$0.592016\pi$
$444$	−2.00000	−0.0949158
$445$	0	0
$446$	−2.00000	−0.0947027
$447$	−6.00000	−0.283790
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	−5.00000	−0.235702
$451$	0	0
$452$	−18.0000	−0.846649
$453$	−8.00000	−0.375873
$454$	−18.0000	−0.844782
$455$	0	0
$456$	2.00000	0.0936586
$457$	−22.0000	−1.02912	−0.514558	−	0.857455i	$-0.672044\pi$
$457$	−22.0000	−1.02912	−0.514558	+	0.857455i	$0.672044\pi$
$458$	−20.0000	−0.934539
$459$	0	0
$460$	0	0
$461$	−6.00000	−0.279448	−0.139724	−	0.990190i	$-0.544622\pi$
$461$	−6.00000	−0.279448	−0.139724	+	0.990190i	$0.544622\pi$
$462$	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
$466$	18.0000	0.833834
$467$	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
$470$	0	0
$471$	−16.0000	−0.737241
$472$	0	0
$473$	4.00000	0.183920
$474$	−8.00000	−0.367452
$475$	10.0000	0.458831
$476$	0	0
$477$	−6.00000	−0.274721
$478$	0	0
$479$	−24.0000	−1.09659	−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000	−1.09659	−0.548294	+	0.836286i	$0.684723\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	28.0000	1.27537
$483$	0	0
$484$	1.00000	0.0454545
$485$	0	0
$486$	−1.00000	−0.0453609
$487$	−4.00000	−0.181257	−0.0906287	−	0.995885i	$-0.528888\pi$
$487$	−4.00000	−0.181257	−0.0906287	+	0.995885i	$0.528888\pi$
$488$	−2.00000	−0.0905357
$489$	−20.0000	−0.904431
$490$	0	0
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	0	0
$494$	4.00000	0.179969
$495$	0	0
$496$	−2.00000	−0.0898027
$497$	0	0
$498$	6.00000	0.268866
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	0	0
$501$	12.0000	0.536120
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	9.00000	0.399704
$508$	8.00000	0.354943
$509$	24.0000	1.06378	0.531891	−	0.846813i	$-0.321482\pi$
$509$	24.0000	1.06378	0.531891	+	0.846813i	$0.321482\pi$
$510$	0	0
$511$	0	0
$512$	1.00000	0.0441942
$513$	2.00000	0.0883022
$514$	−18.0000	−0.793946
$515$	0	0
$516$	4.00000	0.176090
$517$	−6.00000	−0.263880
$518$	0	0
$519$	18.0000	0.790112
$520$	0	0
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	−6.00000	−0.262613
$523$	10.0000	0.437269	0.218635	−	0.975807i	$-0.429840\pi$
$523$	10.0000	0.437269	0.218635	+	0.975807i	$0.429840\pi$
$524$	6.00000	0.262111
$525$	0	0
$526$	24.0000	1.04645
$527$	0	0
$528$	1.00000	0.0435194
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	−6.00000	−0.259645
$535$	0	0
$536$	−4.00000	−0.172774
$537$	12.0000	0.517838
$538$	0	0
$539$	0	0
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−20.0000	−0.859074
$543$	20.0000	0.858282
$544$	0	0
$545$	0	0
$546$	0	0
$547$	8.00000	0.342055	0.171028	−	0.985266i	$-0.445291\pi$
$547$	8.00000	0.342055	0.171028	+	0.985266i	$0.445291\pi$
$548$	−18.0000	−0.768922
$549$	−2.00000	−0.0853579
$550$	5.00000	0.213201
$551$	12.0000	0.511217
$552$	0	0
$553$	0	0
$554$	−10.0000	−0.424859
$555$	0	0
$556$	−14.0000	−0.593732
$557$	18.0000	0.762684	0.381342	−	0.924434i	$-0.375462\pi$
$557$	18.0000	0.762684	0.381342	+	0.924434i	$0.375462\pi$
$558$	−2.00000	−0.0846668
$559$	8.00000	0.338364
$560$	0	0
$561$	0	0
$562$	−6.00000	−0.253095
$563$	−18.0000	−0.758610	−0.379305	−	0.925272i	$-0.623837\pi$
$563$	−18.0000	−0.758610	−0.379305	+	0.925272i	$0.623837\pi$
$564$	−6.00000	−0.252646
$565$	0	0
$566$	−14.0000	−0.588464
$567$	0	0
$568$	−12.0000	−0.503509
$569$	−6.00000	−0.251533	−0.125767	−	0.992060i	$-0.540139\pi$
$569$	−6.00000	−0.251533	−0.125767	+	0.992060i	$0.540139\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	2.00000	0.0836242
$573$	−24.0000	−1.00261
$574$	0	0
$575$	0	0
$576$	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	$0.433255\pi$
$578$	−17.0000	−0.707107
$579$	−2.00000	−0.0831172
$580$	0	0
$581$	0	0
$582$	2.00000	0.0829027
$583$	6.00000	0.248495
$584$	4.00000	0.165521
$585$	0	0
$586$	−18.0000	−0.743573
$587$	24.0000	0.990586	0.495293	−	0.868726i	$-0.335061\pi$
$587$	24.0000	0.990586	0.495293	+	0.868726i	$0.335061\pi$
$588$	0	0
$589$	4.00000	0.164817
$590$	0	0
$591$	6.00000	0.246807
$592$	2.00000	0.0821995
$593$	−24.0000	−0.985562	−0.492781	−	0.870153i	$-0.664020\pi$
$593$	−24.0000	−0.985562	−0.492781	+	0.870153i	$0.664020\pi$
$594$	1.00000	0.0410305
$595$	0	0
$596$	6.00000	0.245770
$597$	−10.0000	−0.409273
$598$	0	0
$599$	−36.0000	−1.47092	−0.735460	−	0.677568i	$-0.763034\pi$
$599$	−36.0000	−1.47092	−0.735460	+	0.677568i	$0.763034\pi$
$600$	5.00000	0.204124
$601$	−8.00000	−0.326327	−0.163163	−	0.986599i	$-0.552170\pi$
$601$	−8.00000	−0.326327	−0.163163	+	0.986599i	$0.552170\pi$
$602$	0	0
$603$	−4.00000	−0.162893
$604$	8.00000	0.325515
$605$	0	0
$606$	6.00000	0.243733
$607$	40.0000	1.62355	0.811775	−	0.583970i	$-0.198502\pi$
$607$	40.0000	1.62355	0.811775	+	0.583970i	$0.198502\pi$
$608$	−2.00000	−0.0811107
$609$	0	0
$610$	0	0
$611$	−12.0000	−0.485468
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	34.0000	1.37213
$615$	0	0
$616$	0	0
$617$	42.0000	1.69086	0.845428	−	0.534089i	$-0.179345\pi$
$617$	42.0000	1.69086	0.845428	+	0.534089i	$0.179345\pi$
$618$	14.0000	0.563163
$619$	−20.0000	−0.803868	−0.401934	−	0.915669i	$-0.631662\pi$
$619$	−20.0000	−0.803868	−0.401934	+	0.915669i	$0.631662\pi$
$620$	0	0
$621$	0	0
$622$	−6.00000	−0.240578
$623$	0	0
$624$	2.00000	0.0800641
$625$	25.0000	1.00000
$626$	10.0000	0.399680
$627$	−2.00000	−0.0798723
$628$	16.0000	0.638470
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	8.00000	0.318223
$633$	−20.0000	−0.794929
$634$	18.0000	0.714871
$635$	0	0
$636$	6.00000	0.237915
$637$	0	0
$638$	6.00000	0.237542
$639$	−12.0000	−0.474713
$640$	0	0
$641$	−30.0000	−1.18493	−0.592464	−	0.805597i	$-0.701845\pi$
$641$	−30.0000	−1.18493	−0.592464	+	0.805597i	$0.701845\pi$
$642$	0	0
$643$	−32.0000	−1.26196	−0.630978	−	0.775800i	$-0.717346\pi$
$643$	−32.0000	−1.26196	−0.630978	+	0.775800i	$0.717346\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−6.00000	−0.235884	−0.117942	−	0.993020i	$-0.537630\pi$
$647$	−6.00000	−0.235884	−0.117942	+	0.993020i	$0.537630\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	10.0000	0.392232
$651$	0	0
$652$	20.0000	0.783260
$653$	30.0000	1.17399	0.586995	−	0.809590i	$-0.300311\pi$
$653$	30.0000	1.17399	0.586995	+	0.809590i	$0.300311\pi$
$654$	10.0000	0.391031
$655$	0	0
$656$	0	0
$657$	4.00000	0.156055
$658$	0	0
$659$	0	0		−	1.00000i	$-0.5\pi$
$659$	0	0			1.00000i	$0.5\pi$
$660$	0	0
$661$	−44.0000	−1.71140	−0.855701	−	0.517471i	$-0.826874\pi$
$661$	−44.0000	−1.71140	−0.855701	+	0.517471i	$0.826874\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	−6.00000	−0.232845
$665$	0	0
$666$	2.00000	0.0774984
$667$	0	0
$668$	−12.0000	−0.464294
$669$	2.00000	0.0773245
$670$	0	0
$671$	2.00000	0.0772091
$672$	0	0
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	14.0000	0.539260
$675$	5.00000	0.192450
$676$	−9.00000	−0.346154
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	18.0000	0.691286
$679$	0	0
$680$	0	0
$681$	18.0000	0.689761
$682$	2.00000	0.0765840
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	−2.00000	−0.0764719
$685$	0	0
$686$	0	0
$687$	20.0000	0.763048
$688$	−4.00000	−0.152499
$689$	12.0000	0.457164
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−18.0000	−0.684257
$693$	0	0
$694$	0	0
$695$	0	0
$696$	6.00000	0.227429
$697$	0	0
$698$	10.0000	0.378506
$699$	−18.0000	−0.680823
$700$	0	0
$701$	−18.0000	−0.679851	−0.339925	−	0.940452i	$-0.610402\pi$
$701$	−18.0000	−0.679851	−0.339925	+	0.940452i	$0.610402\pi$
$702$	2.00000	0.0754851
$703$	−4.00000	−0.150863
$704$	−1.00000	−0.0376889
$705$	0	0
$706$	−18.0000	−0.677439
$707$	0	0
$708$	0	0
$709$	26.0000	0.976450	0.488225	−	0.872718i	$-0.337644\pi$
$709$	26.0000	0.976450	0.488225	+	0.872718i	$0.337644\pi$
$710$	0	0
$711$	8.00000	0.300023
$712$	6.00000	0.224860
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	18.0000	0.671287	0.335643	−	0.941989i	$-0.391046\pi$
$719$	18.0000	0.671287	0.335643	+	0.941989i	$0.391046\pi$
$720$	0	0
$721$	0	0
$722$	−15.0000	−0.558242
$723$	−28.0000	−1.04133
$724$	−20.0000	−0.743294
$725$	30.0000	1.11417
$726$	−1.00000	−0.0371135
$727$	46.0000	1.70605	0.853023	−	0.521874i	$-0.174767\pi$
$727$	46.0000	1.70605	0.853023	+	0.521874i	$0.174767\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	0	0
$732$	2.00000	0.0739221
$733$	−26.0000	−0.960332	−0.480166	−	0.877178i	$-0.659424\pi$
$733$	−26.0000	−0.960332	−0.480166	+	0.877178i	$0.659424\pi$
$734$	22.0000	0.812035
$735$	0	0
$736$	0	0
$737$	4.00000	0.147342
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	0	0
$741$	−4.00000	−0.146944
$742$	0	0
$743$	48.0000	1.76095	0.880475	−	0.474093i	$-0.157224\pi$
$743$	48.0000	1.76095	0.880475	+	0.474093i	$0.157224\pi$
$744$	2.00000	0.0733236
$745$	0	0
$746$	2.00000	0.0732252
$747$	−6.00000	−0.219529
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−4.00000	−0.145962	−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000	−0.145962	−0.0729810	+	0.997333i	$0.523251\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	12.0000	0.437014
$755$	0	0
$756$	0	0
$757$	26.0000	0.944986	0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000	0.944986	0.472493	+	0.881334i	$0.343354\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	0	0
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−8.00000	−0.289809
$763$	0	0
$764$	24.0000	0.868290
$765$	0	0
$766$	−6.00000	−0.216789
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	16.0000	0.576975	0.288487	−	0.957484i	$-0.406848\pi$
$769$	16.0000	0.576975	0.288487	+	0.957484i	$0.406848\pi$
$770$	0	0
$771$	18.0000	0.648254
$772$	2.00000	0.0719816
$773$	−36.0000	−1.29483	−0.647415	−	0.762138i	$-0.724150\pi$
$773$	−36.0000	−1.29483	−0.647415	+	0.762138i	$0.724150\pi$
$774$	−4.00000	−0.143777
$775$	10.0000	0.359211
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	6.00000	0.215110
$779$	0	0
$780$	0	0
$781$	12.0000	0.429394
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	0	0
$786$	−6.00000	−0.214013
$787$	−50.0000	−1.78231	−0.891154	−	0.453701i	$-0.850103\pi$
$787$	−50.0000	−1.78231	−0.891154	+	0.453701i	$0.850103\pi$
$788$	−6.00000	−0.213741
$789$	−24.0000	−0.854423
$790$	0	0
$791$	0	0
$792$	−1.00000	−0.0355335
$793$	4.00000	0.142044
$794$	16.0000	0.567819
$795$	0	0
$796$	10.0000	0.354441
$797$	−36.0000	−1.27519	−0.637593	−	0.770374i	$-0.720070\pi$
$797$	−36.0000	−1.27519	−0.637593	+	0.770374i	$0.720070\pi$
$798$	0	0
$799$	0	0
$800$	−5.00000	−0.176777
$801$	6.00000	0.212000
$802$	6.00000	0.211867
$803$	−4.00000	−0.141157
$804$	4.00000	0.141069
$805$	0	0
$806$	4.00000	0.140894
$807$	0	0
$808$	−6.00000	−0.211079
$809$	−42.0000	−1.47664	−0.738321	−	0.674450i	$-0.764381\pi$
$809$	−42.0000	−1.47664	−0.738321	+	0.674450i	$0.764381\pi$
$810$	0	0
$811$	10.0000	0.351147	0.175574	−	0.984466i	$-0.443822\pi$
$811$	10.0000	0.351147	0.175574	+	0.984466i	$0.443822\pi$
$812$	0	0
$813$	20.0000	0.701431
$814$	−2.00000	−0.0701000
$815$	0	0
$816$	0	0
$817$	8.00000	0.279885
$818$	40.0000	1.39857
$819$	0	0
$820$	0	0
$821$	−42.0000	−1.46581	−0.732905	−	0.680331i	$-0.761836\pi$
$821$	−42.0000	−1.46581	−0.732905	+	0.680331i	$0.761836\pi$
$822$	18.0000	0.627822
$823$	20.0000	0.697156	0.348578	−	0.937280i	$-0.386665\pi$
$823$	20.0000	0.697156	0.348578	+	0.937280i	$0.386665\pi$
$824$	−14.0000	−0.487713
$825$	−5.00000	−0.174078
$826$	0	0
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	52.0000	1.80603	0.903017	−	0.429604i	$-0.141347\pi$
$829$	52.0000	1.80603	0.903017	+	0.429604i	$0.141347\pi$
$830$	0	0
$831$	10.0000	0.346896
$832$	−2.00000	−0.0693375
$833$	0	0
$834$	14.0000	0.484780
$835$	0	0
$836$	2.00000	0.0691714
$837$	2.00000	0.0691301
$838$	12.0000	0.414533
$839$	−6.00000	−0.207143	−0.103572	−	0.994622i	$-0.533027\pi$
$839$	−6.00000	−0.207143	−0.103572	+	0.994622i	$0.533027\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	−10.0000	−0.344623
$843$	6.00000	0.206651
$844$	20.0000	0.688428
$845$	0	0
$846$	6.00000	0.206284
$847$	0	0
$848$	−6.00000	−0.206041
$849$	14.0000	0.480479
$850$	0	0
$851$	0	0
$852$	12.0000	0.411113
$853$	−2.00000	−0.0684787	−0.0342393	−	0.999414i	$-0.510901\pi$
$853$	−2.00000	−0.0684787	−0.0342393	+	0.999414i	$0.510901\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−36.0000	−1.22974	−0.614868	−	0.788630i	$-0.710791\pi$
$857$	−36.0000	−1.22974	−0.614868	+	0.788630i	$0.710791\pi$
$858$	−2.00000	−0.0682789
$859$	40.0000	1.36478	0.682391	−	0.730987i	$-0.260940\pi$
$859$	40.0000	1.36478	0.682391	+	0.730987i	$0.260940\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−24.0000	−0.816970	−0.408485	−	0.912765i	$-0.633943\pi$
$863$	−24.0000	−0.816970	−0.408485	+	0.912765i	$0.633943\pi$
$864$	−1.00000	−0.0340207
$865$	0	0
$866$	22.0000	0.747590
$867$	17.0000	0.577350
$868$	0	0
$869$	−8.00000	−0.271381
$870$	0	0
$871$	8.00000	0.271070
$872$	−10.0000	−0.338643
$873$	−2.00000	−0.0676897
$874$	0	0
$875$	0	0
$876$	−4.00000	−0.135147
$877$	14.0000	0.472746	0.236373	−	0.971662i	$-0.424041\pi$
$877$	14.0000	0.472746	0.236373	+	0.971662i	$0.424041\pi$
$878$	28.0000	0.944954
$879$	18.0000	0.607125
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	−12.0000	−0.403148
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−2.00000	−0.0671156
$889$	0	0
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	−2.00000	−0.0669650
$893$	−12.0000	−0.401565
$894$	−6.00000	−0.200670
$895$	0	0
$896$	0	0
$897$	0	0
$898$	−18.0000	−0.600668
$899$	12.0000	0.400222
$900$	−5.00000	−0.166667
$901$	0	0
$902$	0	0
$903$	0	0
$904$	−18.0000	−0.598671
$905$	0	0
$906$	−8.00000	−0.265782
$907$	−52.0000	−1.72663	−0.863316	−	0.504664i	$-0.831616\pi$
$907$	−52.0000	−1.72663	−0.863316	+	0.504664i	$0.831616\pi$
$908$	−18.0000	−0.597351
$909$	−6.00000	−0.199007
$910$	0	0
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	2.00000	0.0662266
$913$	6.00000	0.198571
$914$	−22.0000	−0.727695
$915$	0	0
$916$	−20.0000	−0.660819
$917$	0	0
$918$	0	0
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	−34.0000	−1.12034
$922$	−6.00000	−0.197599
$923$	24.0000	0.789970
$924$	0	0
$925$	−10.0000	−0.328798
$926$	8.00000	0.262896
$927$	−14.0000	−0.459820
$928$	−6.00000	−0.196960
$929$	6.00000	0.196854	0.0984268	−	0.995144i	$-0.468619\pi$
$929$	6.00000	0.196854	0.0984268	+	0.995144i	$0.468619\pi$
$930$	0	0
$931$	0	0
$932$	18.0000	0.589610
$933$	6.00000	0.196431
$934$	36.0000	1.17796
$935$	0	0
$936$	−2.00000	−0.0653720
$937$	52.0000	1.69877	0.849383	−	0.527777i	$-0.176974\pi$
$937$	52.0000	1.69877	0.849383	+	0.527777i	$0.176974\pi$
$938$	0	0
$939$	−10.0000	−0.326338
$940$	0	0
$941$	−18.0000	−0.586783	−0.293392	−	0.955992i	$-0.594784\pi$
$941$	−18.0000	−0.586783	−0.293392	+	0.955992i	$0.594784\pi$
$942$	−16.0000	−0.521308
$943$	0	0
$944$	0	0
$945$	0	0
$946$	4.00000	0.130051
$947$	−12.0000	−0.389948	−0.194974	−	0.980808i	$-0.562462\pi$
$947$	−12.0000	−0.389948	−0.194974	+	0.980808i	$0.562462\pi$
$948$	−8.00000	−0.259828
$949$	−8.00000	−0.259691
$950$	10.0000	0.324443
$951$	−18.0000	−0.583690
$952$	0	0
$953$	−42.0000	−1.36051	−0.680257	−	0.732974i	$-0.738132\pi$
$953$	−42.0000	−1.36051	−0.680257	+	0.732974i	$0.738132\pi$
$954$	−6.00000	−0.194257
$955$	0	0
$956$	0	0
$957$	−6.00000	−0.193952
$958$	−24.0000	−0.775405
$959$	0	0
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−4.00000	−0.128965
$963$	0	0
$964$	28.0000	0.901819
$965$	0	0
$966$	0	0
$967$	−16.0000	−0.514525	−0.257263	−	0.966342i	$-0.582821\pi$
$967$	−16.0000	−0.514525	−0.257263	+	0.966342i	$0.582821\pi$
$968$	1.00000	0.0321412
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	−1.00000	−0.0320750
$973$	0	0
$974$	−4.00000	−0.128168
$975$	−10.0000	−0.320256
$976$	−2.00000	−0.0640184
$977$	18.0000	0.575871	0.287936	−	0.957650i	$-0.407031\pi$
$977$	18.0000	0.575871	0.287936	+	0.957650i	$0.407031\pi$
$978$	−20.0000	−0.639529
$979$	−6.00000	−0.191761
$980$	0	0
$981$	−10.0000	−0.319275
$982$	−12.0000	−0.382935
$983$	6.00000	0.191370	0.0956851	−	0.995412i	$-0.469496\pi$
$983$	6.00000	0.191370	0.0956851	+	0.995412i	$0.469496\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	4.00000	0.127257
$989$	0	0
$990$	0	0
$991$	−40.0000	−1.27064	−0.635321	−	0.772248i	$-0.719132\pi$
$991$	−40.0000	−1.27064	−0.635321	+	0.772248i	$0.719132\pi$
$992$	−2.00000	−0.0635001
$993$	4.00000	0.126936
$994$	0	0
$995$	0	0
$996$	6.00000	0.190117
$997$	10.0000	0.316703	0.158352	−	0.987383i	$-0.449382\pi$
$997$	10.0000	0.316703	0.158352	+	0.987383i	$0.449382\pi$
$998$	−4.00000	−0.126618
$999$	−2.00000	−0.0632772

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3234.2.a.q.1.1		1
3.2	odd	2		9702.2.a.n.1.1		1
7.6	odd	2		462.2.a.f.1.1	✓	1
21.20	even	2		1386.2.a.c.1.1		1
28.27	even	2		3696.2.a.i.1.1		1
77.76	even	2		5082.2.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.2.a.f.1.1	✓	1	7.6	odd	2
1386.2.a.c.1.1		1	21.20	even	2
3234.2.a.q.1.1		1	1.1	even	1	trivial
3696.2.a.i.1.1		1	28.27	even	2
5082.2.a.l.1.1		1	77.76	even	2
9702.2.a.n.1.1		1	3.2	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:	\( N \)	\(=\)	\( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3234.a (trivial)

Introduction

L-functions

Modular forms

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Representations

Groups

Database

Properties

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