LMFDB - Embedded newform 3330.2.a.v.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3330,2,Mod(1,3330)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3330, 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("3330.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3330.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:	$26.5901838731$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 370)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3330.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

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

\(q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} +2.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -6.00000 q^{23} +1.00000 q^{25} -4.00000 q^{26} -1.00000 q^{28} -3.00000 q^{29} +5.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} -1.00000 q^{35} +1.00000 q^{37} +2.00000 q^{38} +1.00000 q^{40} -3.00000 q^{41} -1.00000 q^{43} -3.00000 q^{44} -6.00000 q^{46} -12.0000 q^{47} -6.00000 q^{49} +1.00000 q^{50} -4.00000 q^{52} -3.00000 q^{53} -3.00000 q^{55} -1.00000 q^{56} -3.00000 q^{58} -1.00000 q^{61} +5.00000 q^{62} +1.00000 q^{64} -4.00000 q^{65} -4.00000 q^{67} -3.00000 q^{68} -1.00000 q^{70} -6.00000 q^{71} -16.0000 q^{73} +1.00000 q^{74} +2.00000 q^{76} +3.00000 q^{77} +8.00000 q^{79} +1.00000 q^{80} -3.00000 q^{82} +12.0000 q^{83} -3.00000 q^{85} -1.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} +4.00000 q^{91} -6.00000 q^{92} -12.0000 q^{94} +2.00000 q^{95} +17.0000 q^{97} -6.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	−1.00000	−0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.00000	−0.727607	−0.363803	−	0.931476i	$-0.618522\pi$
$17$	−3.00000	−0.727607	−0.363803	+	0.931476i	$0.618522\pi$
$18$	0	0
$19$	2.00000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	2.00000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	1.00000	0.223607
$21$	0	0
$22$	−3.00000	−0.639602
$23$	−6.00000	−1.25109	−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000	−1.25109	−0.625543	+	0.780189i	$0.715123\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	0	0
$28$	−1.00000	−0.188982
$29$	−3.00000	−0.557086	−0.278543	−	0.960424i	$-0.589851\pi$
$29$	−3.00000	−0.557086	−0.278543	+	0.960424i	$0.589851\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−1.00000	−0.169031
$36$	0	0
$37$	1.00000	0.164399
$37$	1.00000	0.164399
$38$	2.00000	0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	−1.00000	−0.152499	−0.0762493	−	0.997089i	$-0.524294\pi$
$43$	−1.00000	−0.152499	−0.0762493	+	0.997089i	$0.524294\pi$
$44$	−3.00000	−0.452267
$45$	0	0
$46$	−6.00000	−0.884652
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	1.00000	0.141421
$51$	0	0
$52$	−4.00000	−0.554700
$53$	−3.00000	−0.412082	−0.206041	−	0.978543i	$-0.566058\pi$
$53$	−3.00000	−0.412082	−0.206041	+	0.978543i	$0.566058\pi$
$54$	0	0
$55$	−3.00000	−0.404520
$56$	−1.00000	−0.133631
$57$	0	0
$58$	−3.00000	−0.393919
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	−1.00000	−0.128037	−0.0640184	−	0.997949i	$-0.520392\pi$
$61$	−1.00000	−0.128037	−0.0640184	+	0.997949i	$0.520392\pi$
$62$	5.00000	0.635001
$63$	0	0
$64$	1.00000	0.125000
$65$	−4.00000	−0.496139
$66$	0	0
$67$	−4.00000	−0.488678	−0.244339	−	0.969690i	$-0.578571\pi$
$67$	−4.00000	−0.488678	−0.244339	+	0.969690i	$0.578571\pi$
$68$	−3.00000	−0.363803
$69$	0	0
$70$	−1.00000	−0.119523
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	0	0
$73$	−16.0000	−1.87266	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	−16.0000	−1.87266	−0.936329	+	0.351123i	$0.885800\pi$
$74$	1.00000	0.116248
$75$	0	0
$76$	2.00000	0.229416
$77$	3.00000	0.341882
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	1.00000	0.111803
$81$	0	0
$82$	−3.00000	−0.331295
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−3.00000	−0.325396
$86$	−1.00000	−0.107833
$87$	0	0
$88$	−3.00000	−0.319801
$89$	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
$91$	4.00000	0.419314
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−12.0000	−1.23771
$95$	2.00000	0.205196
$96$	0	0
$97$	17.0000	1.72609	0.863044	−	0.505128i	$-0.168555\pi$
$97$	17.0000	1.72609	0.863044	+	0.505128i	$0.168555\pi$
$98$	−6.00000	−0.606092
$99$	0	0
$100$	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	$0.403510\pi$
$102$	0	0
$103$	−4.00000	−0.394132	−0.197066	−	0.980390i	$-0.563141\pi$
$103$	−4.00000	−0.394132	−0.197066	+	0.980390i	$0.563141\pi$
$104$	−4.00000	−0.392232
$105$	0	0
$106$	−3.00000	−0.291386
$107$	−6.00000	−0.580042	−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000	−0.580042	−0.290021	+	0.957020i	$0.593662\pi$
$108$	0	0
$109$	11.0000	1.05361	0.526804	−	0.849987i	$-0.323390\pi$
$109$	11.0000	1.05361	0.526804	+	0.849987i	$0.323390\pi$
$110$	−3.00000	−0.286039
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	−9.00000	−0.846649	−0.423324	−	0.905978i	$-0.639137\pi$
$113$	−9.00000	−0.846649	−0.423324	+	0.905978i	$0.639137\pi$
$114$	0	0
$115$	−6.00000	−0.559503
$116$	−3.00000	−0.278543
$117$	0	0
$118$	0	0
$119$	3.00000	0.275010
$120$	0	0
$121$	−2.00000	−0.181818
$122$	−1.00000	−0.0905357
$123$	0	0
$124$	5.00000	0.449013
$125$	1.00000	0.0894427
$126$	0	0
$127$	20.0000	1.77471	0.887357	−	0.461084i	$-0.152539\pi$
$127$	20.0000	1.77471	0.887357	+	0.461084i	$0.152539\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−4.00000	−0.350823
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	−2.00000	−0.173422
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−12.0000	−1.02523	−0.512615	−	0.858619i	$-0.671323\pi$
$137$	−12.0000	−1.02523	−0.512615	+	0.858619i	$0.671323\pi$
$138$	0	0
$139$	−13.0000	−1.10265	−0.551323	−	0.834292i	$-0.685877\pi$
$139$	−13.0000	−1.10265	−0.551323	+	0.834292i	$0.685877\pi$
$140$	−1.00000	−0.0845154
$141$	0	0
$142$	−6.00000	−0.503509
$143$	12.0000	1.00349
$144$	0	0
$145$	−3.00000	−0.249136
$146$	−16.0000	−1.32417
$147$	0	0
$148$	1.00000	0.0821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	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
$154$	3.00000	0.241747
$155$	5.00000	0.401610
$156$	0	0
$157$	−13.0000	−1.03751	−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000	−1.03751	−0.518756	+	0.854922i	$0.673605\pi$
$158$	8.00000	0.636446
$159$	0	0
$160$	1.00000	0.0790569
$161$	6.00000	0.472866
$162$	0	0
$163$	11.0000	0.861586	0.430793	−	0.902451i	$-0.358234\pi$
$163$	11.0000	0.861586	0.430793	+	0.902451i	$0.358234\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	12.0000	0.931381
$167$	−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
$169$	3.00000	0.230769
$170$	−3.00000	−0.230089
$171$	0	0
$172$	−1.00000	−0.0762493
$173$	−15.0000	−1.14043	−0.570214	−	0.821496i	$-0.693140\pi$
$173$	−15.0000	−1.14043	−0.570214	+	0.821496i	$0.693140\pi$
$174$	0	0
$175$	−1.00000	−0.0755929
$176$	−3.00000	−0.226134
$177$	0	0
$178$	6.00000	0.449719
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	4.00000	0.296500
$183$	0	0
$184$	−6.00000	−0.442326
$185$	1.00000	0.0735215
$186$	0	0
$187$	9.00000	0.658145
$188$	−12.0000	−0.875190
$189$	0	0
$190$	2.00000	0.145095
$191$	−3.00000	−0.217072	−0.108536	−	0.994092i	$-0.534616\pi$
$191$	−3.00000	−0.217072	−0.108536	+	0.994092i	$0.534616\pi$
$192$	0	0
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	17.0000	1.22053
$195$	0	0
$196$	−6.00000	−0.428571
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	−16.0000	−1.13421	−0.567105	−	0.823646i	$-0.691937\pi$
$199$	−16.0000	−1.13421	−0.567105	+	0.823646i	$0.691937\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	6.00000	0.422159
$203$	3.00000	0.210559
$204$	0	0
$205$	−3.00000	−0.209529
$206$	−4.00000	−0.278693
$207$	0	0
$208$	−4.00000	−0.277350
$209$	−6.00000	−0.415029
$210$	0	0
$211$	5.00000	0.344214	0.172107	−	0.985078i	$-0.444942\pi$
$211$	5.00000	0.344214	0.172107	+	0.985078i	$0.444942\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−1.00000	−0.0681994
$216$	0	0
$217$	−5.00000	−0.339422
$218$	11.0000	0.745014
$219$	0	0
$220$	−3.00000	−0.202260
$221$	12.0000	0.807207
$222$	0	0
$223$	17.0000	1.13840	0.569202	−	0.822198i	$-0.307252\pi$
$223$	17.0000	1.13840	0.569202	+	0.822198i	$0.307252\pi$
$224$	−1.00000	−0.0668153
$225$	0	0
$226$	−9.00000	−0.598671
$227$	27.0000	1.79205	0.896026	−	0.444001i	$-0.146441\pi$
$227$	27.0000	1.79205	0.896026	+	0.444001i	$0.146441\pi$
$228$	0	0
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−3.00000	−0.196960
$233$	0	0		−	1.00000i	$-0.5\pi$
$233$	0	0			1.00000i	$0.5\pi$
$234$	0	0
$235$	−12.0000	−0.782794
$236$	0	0
$237$	0	0
$238$	3.00000	0.194461
$239$	9.00000	0.582162	0.291081	−	0.956698i	$-0.405985\pi$
$239$	9.00000	0.582162	0.291081	+	0.956698i	$0.405985\pi$
$240$	0	0
$241$	−28.0000	−1.80364	−0.901819	−	0.432113i	$-0.857768\pi$
$241$	−28.0000	−1.80364	−0.901819	+	0.432113i	$0.857768\pi$
$242$	−2.00000	−0.128565
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	−6.00000	−0.383326
$246$	0	0
$247$	−8.00000	−0.509028
$248$	5.00000	0.317500
$249$	0	0
$250$	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
$253$	18.0000	1.13165
$254$	20.0000	1.25491
$255$	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	$0.689739\pi$
$258$	0	0
$259$	−1.00000	−0.0621370
$260$	−4.00000	−0.248069
$261$	0	0
$262$	0	0
$263$	−9.00000	−0.554964	−0.277482	−	0.960731i	$-0.589500\pi$
$263$	−9.00000	−0.554964	−0.277482	+	0.960731i	$0.589500\pi$
$264$	0	0
$265$	−3.00000	−0.184289
$266$	−2.00000	−0.122628
$267$	0	0
$268$	−4.00000	−0.244339
$269$	18.0000	1.09748	0.548740	−	0.835993i	$-0.315108\pi$
$269$	18.0000	1.09748	0.548740	+	0.835993i	$0.315108\pi$
$270$	0	0
$271$	2.00000	0.121491	0.0607457	−	0.998153i	$-0.480652\pi$
$271$	2.00000	0.121491	0.0607457	+	0.998153i	$0.480652\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−12.0000	−0.724947
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−28.0000	−1.68236	−0.841178	−	0.540758i	$-0.818138\pi$
$277$	−28.0000	−1.68236	−0.841178	+	0.540758i	$0.818138\pi$
$278$	−13.0000	−0.779688
$279$	0	0
$280$	−1.00000	−0.0597614
$281$	−24.0000	−1.43172	−0.715860	−	0.698244i	$-0.753965\pi$
$281$	−24.0000	−1.43172	−0.715860	+	0.698244i	$0.753965\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−6.00000	−0.356034
$285$	0	0
$286$	12.0000	0.709575
$287$	3.00000	0.177084
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−3.00000	−0.176166
$291$	0	0
$292$	−16.0000	−0.936329
$293$	−21.0000	−1.22683	−0.613417	−	0.789760i	$-0.710205\pi$
$293$	−21.0000	−1.22683	−0.613417	+	0.789760i	$0.710205\pi$
$294$	0	0
$295$	0	0
$296$	1.00000	0.0581238
$297$	0	0
$298$	−6.00000	−0.347571
$299$	24.0000	1.38796
$300$	0	0
$301$	1.00000	0.0576390
$302$	8.00000	0.460348
$303$	0	0
$304$	2.00000	0.114708
$305$	−1.00000	−0.0572598
$306$	0	0
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	3.00000	0.170941
$309$	0	0
$310$	5.00000	0.283981
$311$	9.00000	0.510343	0.255172	−	0.966896i	$-0.417868\pi$
$311$	9.00000	0.510343	0.255172	+	0.966896i	$0.417868\pi$
$312$	0	0
$313$	26.0000	1.46961	0.734803	−	0.678280i	$-0.237274\pi$
$313$	26.0000	1.46961	0.734803	+	0.678280i	$0.237274\pi$
$314$	−13.0000	−0.733632
$315$	0	0
$316$	8.00000	0.450035
$317$	21.0000	1.17948	0.589739	−	0.807594i	$-0.299231\pi$
$317$	21.0000	1.17948	0.589739	+	0.807594i	$0.299231\pi$
$318$	0	0
$319$	9.00000	0.503903
$320$	1.00000	0.0559017
$321$	0	0
$322$	6.00000	0.334367
$323$	−6.00000	−0.333849
$324$	0	0
$325$	−4.00000	−0.221880
$326$	11.0000	0.609234
$327$	0	0
$328$	−3.00000	−0.165647
$329$	12.0000	0.661581
$330$	0	0
$331$	26.0000	1.42909	0.714545	−	0.699590i	$-0.246634\pi$
$331$	26.0000	1.42909	0.714545	+	0.699590i	$0.246634\pi$
$332$	12.0000	0.658586
$333$	0	0
$334$	−12.0000	−0.656611
$335$	−4.00000	−0.218543
$336$	0	0
$337$	−4.00000	−0.217894	−0.108947	−	0.994048i	$-0.534748\pi$
$337$	−4.00000	−0.217894	−0.108947	+	0.994048i	$0.534748\pi$
$338$	3.00000	0.163178
$339$	0	0
$340$	−3.00000	−0.162698
$341$	−15.0000	−0.812296
$342$	0	0
$343$	13.0000	0.701934
$344$	−1.00000	−0.0539164
$345$	0	0
$346$	−15.0000	−0.806405
$347$	12.0000	0.644194	0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000	0.644194	0.322097	+	0.946707i	$0.395612\pi$
$348$	0	0
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−3.00000	−0.159901
$353$	21.0000	1.11772	0.558859	−	0.829263i	$-0.311239\pi$
$353$	21.0000	1.11772	0.558859	+	0.829263i	$0.311239\pi$
$354$	0	0
$355$	−6.00000	−0.318447
$356$	6.00000	0.317999
$357$	0	0
$358$	0	0
$359$	36.0000	1.90001	0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000	1.90001	0.950004	+	0.312239i	$0.101079\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	2.00000	0.105118
$363$	0	0
$364$	4.00000	0.209657
$365$	−16.0000	−0.837478
$366$	0	0
$367$	35.0000	1.82699	0.913493	−	0.406855i	$-0.133375\pi$
$367$	35.0000	1.82699	0.913493	+	0.406855i	$0.133375\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	1.00000	0.0519875
$371$	3.00000	0.155752
$372$	0	0
$373$	−22.0000	−1.13912	−0.569558	−	0.821951i	$-0.692886\pi$
$373$	−22.0000	−1.13912	−0.569558	+	0.821951i	$0.692886\pi$
$374$	9.00000	0.465379
$375$	0	0
$376$	−12.0000	−0.618853
$377$	12.0000	0.618031
$378$	0	0
$379$	−16.0000	−0.821865	−0.410932	−	0.911666i	$-0.634797\pi$
$379$	−16.0000	−0.821865	−0.410932	+	0.911666i	$0.634797\pi$
$380$	2.00000	0.102598
$381$	0	0
$382$	−3.00000	−0.153493
$383$	24.0000	1.22634	0.613171	−	0.789950i	$-0.289894\pi$
$383$	24.0000	1.22634	0.613171	+	0.789950i	$0.289894\pi$
$384$	0	0
$385$	3.00000	0.152894
$386$	14.0000	0.712581
$387$	0	0
$388$	17.0000	0.863044
$389$	−9.00000	−0.456318	−0.228159	−	0.973624i	$-0.573271\pi$
$389$	−9.00000	−0.456318	−0.228159	+	0.973624i	$0.573271\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	−6.00000	−0.303046
$393$	0	0
$394$	6.00000	0.302276
$395$	8.00000	0.402524
$396$	0	0
$397$	−34.0000	−1.70641	−0.853206	−	0.521575i	$-0.825345\pi$
$397$	−34.0000	−1.70641	−0.853206	+	0.521575i	$0.825345\pi$
$398$	−16.0000	−0.802008
$399$	0	0
$400$	1.00000	0.0500000
$401$	−6.00000	−0.299626	−0.149813	−	0.988714i	$-0.547867\pi$
$401$	−6.00000	−0.299626	−0.149813	+	0.988714i	$0.547867\pi$
$402$	0	0
$403$	−20.0000	−0.996271
$404$	6.00000	0.298511
$405$	0	0
$406$	3.00000	0.148888
$407$	−3.00000	−0.148704
$408$	0	0
$409$	32.0000	1.58230	0.791149	−	0.611623i	$-0.209483\pi$
$409$	32.0000	1.58230	0.791149	+	0.611623i	$0.209483\pi$
$410$	−3.00000	−0.148159
$411$	0	0
$412$	−4.00000	−0.197066
$413$	0	0
$414$	0	0
$415$	12.0000	0.589057
$416$	−4.00000	−0.196116
$417$	0	0
$418$	−6.00000	−0.293470
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	−10.0000	−0.487370	−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000	−0.487370	−0.243685	+	0.969854i	$0.578356\pi$
$422$	5.00000	0.243396
$423$	0	0
$424$	−3.00000	−0.145693
$425$	−3.00000	−0.145521
$426$	0	0
$427$	1.00000	0.0483934
$428$	−6.00000	−0.290021
$429$	0	0
$430$	−1.00000	−0.0482243
$431$	−15.0000	−0.722525	−0.361262	−	0.932464i	$-0.617654\pi$
$431$	−15.0000	−0.722525	−0.361262	+	0.932464i	$0.617654\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	−5.00000	−0.240008
$435$	0	0
$436$	11.0000	0.526804
$437$	−12.0000	−0.574038
$438$	0	0
$439$	35.0000	1.67046	0.835229	−	0.549902i	$-0.185335\pi$
$439$	35.0000	1.67046	0.835229	+	0.549902i	$0.185335\pi$
$440$	−3.00000	−0.143019
$441$	0	0
$442$	12.0000	0.570782
$443$	−30.0000	−1.42534	−0.712672	−	0.701498i	$-0.752515\pi$
$443$	−30.0000	−1.42534	−0.712672	+	0.701498i	$0.752515\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	17.0000	0.804973
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−6.00000	−0.283158	−0.141579	−	0.989927i	$-0.545218\pi$
$449$	−6.00000	−0.283158	−0.141579	+	0.989927i	$0.545218\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	−9.00000	−0.423324
$453$	0	0
$454$	27.0000	1.26717
$455$	4.00000	0.187523
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	14.0000	0.654177
$459$	0	0
$460$	−6.00000	−0.279751
$461$	−33.0000	−1.53696	−0.768482	−	0.639872i	$-0.778987\pi$
$461$	−33.0000	−1.53696	−0.768482	+	0.639872i	$0.778987\pi$
$462$	0	0
$463$	−4.00000	−0.185896	−0.0929479	−	0.995671i	$-0.529629\pi$
$463$	−4.00000	−0.185896	−0.0929479	+	0.995671i	$0.529629\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	0	0
$467$	−27.0000	−1.24941	−0.624705	−	0.780860i	$-0.714781\pi$
$467$	−27.0000	−1.24941	−0.624705	+	0.780860i	$0.714781\pi$
$468$	0	0
$469$	4.00000	0.184703
$470$	−12.0000	−0.553519
$471$	0	0
$472$	0	0
$473$	3.00000	0.137940
$474$	0	0
$475$	2.00000	0.0917663
$476$	3.00000	0.137505
$477$	0	0
$478$	9.00000	0.411650
$479$	−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
$484$	−2.00000	−0.0909091
$485$	17.0000	0.771930
$486$	0	0
$487$	2.00000	0.0906287	0.0453143	−	0.998973i	$-0.485571\pi$
$487$	2.00000	0.0906287	0.0453143	+	0.998973i	$0.485571\pi$
$488$	−1.00000	−0.0452679
$489$	0	0
$490$	−6.00000	−0.271052
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	0	0
$493$	9.00000	0.405340
$494$	−8.00000	−0.359937
$495$	0	0
$496$	5.00000	0.224507
$497$	6.00000	0.269137
$498$	0	0
$499$	14.0000	0.626726	0.313363	−	0.949633i	$-0.398544\pi$
$499$	14.0000	0.626726	0.313363	+	0.949633i	$0.398544\pi$
$500$	1.00000	0.0447214
$501$	0	0
$502$	−18.0000	−0.803379
$503$	−12.0000	−0.535054	−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000	−0.535054	−0.267527	+	0.963550i	$0.586206\pi$
$504$	0	0
$505$	6.00000	0.266996
$506$	18.0000	0.800198
$507$	0	0
$508$	20.0000	0.887357
$509$	0	0		−	1.00000i	$-0.5\pi$
$509$	0	0			1.00000i	$0.5\pi$
$510$	0	0
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	0	0
$514$	−18.0000	−0.793946
$515$	−4.00000	−0.176261
$516$	0	0
$517$	36.0000	1.58328
$518$	−1.00000	−0.0439375
$519$	0	0
$520$	−4.00000	−0.175412
$521$	39.0000	1.70862	0.854311	−	0.519763i	$-0.173980\pi$
$521$	39.0000	1.70862	0.854311	+	0.519763i	$0.173980\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	0	0
$525$	0	0
$526$	−9.00000	−0.392419
$527$	−15.0000	−0.653410
$528$	0	0
$529$	13.0000	0.565217
$530$	−3.00000	−0.130312
$531$	0	0
$532$	−2.00000	−0.0867110
$533$	12.0000	0.519778
$534$	0	0
$535$	−6.00000	−0.259403
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	18.0000	0.775315
$540$	0	0
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	2.00000	0.0859074
$543$	0	0
$544$	−3.00000	−0.128624
$545$	11.0000	0.471188
$546$	0	0
$547$	−1.00000	−0.0427569	−0.0213785	−	0.999771i	$-0.506805\pi$
$547$	−1.00000	−0.0427569	−0.0213785	+	0.999771i	$0.506805\pi$
$548$	−12.0000	−0.512615
$549$	0	0
$550$	−3.00000	−0.127920
$551$	−6.00000	−0.255609
$552$	0	0
$553$	−8.00000	−0.340195
$554$	−28.0000	−1.18961
$555$	0	0
$556$	−13.0000	−0.551323
$557$	−12.0000	−0.508456	−0.254228	−	0.967144i	$-0.581821\pi$
$557$	−12.0000	−0.508456	−0.254228	+	0.967144i	$0.581821\pi$
$558$	0	0
$559$	4.00000	0.169182
$560$	−1.00000	−0.0422577
$561$	0	0
$562$	−24.0000	−1.01238
$563$	3.00000	0.126435	0.0632175	−	0.998000i	$-0.479864\pi$
$563$	3.00000	0.126435	0.0632175	+	0.998000i	$0.479864\pi$
$564$	0	0
$565$	−9.00000	−0.378633
$566$	−4.00000	−0.168133
$567$	0	0
$568$	−6.00000	−0.251754
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	12.0000	0.501745
$573$	0	0
$574$	3.00000	0.125218
$575$	−6.00000	−0.250217
$576$	0	0
$577$	−34.0000	−1.41544	−0.707719	−	0.706494i	$-0.750276\pi$
$577$	−34.0000	−1.41544	−0.707719	+	0.706494i	$0.750276\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−3.00000	−0.124568
$581$	−12.0000	−0.497844
$582$	0	0
$583$	9.00000	0.372742
$584$	−16.0000	−0.662085
$585$	0	0
$586$	−21.0000	−0.867502
$587$	−27.0000	−1.11441	−0.557205	−	0.830375i	$-0.688126\pi$
$587$	−27.0000	−1.11441	−0.557205	+	0.830375i	$0.688126\pi$
$588$	0	0
$589$	10.0000	0.412043
$590$	0	0
$591$	0	0
$592$	1.00000	0.0410997
$593$	36.0000	1.47834	0.739171	−	0.673517i	$-0.235217\pi$
$593$	36.0000	1.47834	0.739171	+	0.673517i	$0.235217\pi$
$594$	0	0
$595$	3.00000	0.122988
$596$	−6.00000	−0.245770
$597$	0	0
$598$	24.0000	0.981433
$599$	−12.0000	−0.490307	−0.245153	−	0.969484i	$-0.578838\pi$
$599$	−12.0000	−0.490307	−0.245153	+	0.969484i	$0.578838\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	1.00000	0.0407570
$603$	0	0
$604$	8.00000	0.325515
$605$	−2.00000	−0.0813116
$606$	0	0
$607$	−22.0000	−0.892952	−0.446476	−	0.894795i	$-0.647321\pi$
$607$	−22.0000	−0.892952	−0.446476	+	0.894795i	$0.647321\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	−1.00000	−0.0404888
$611$	48.0000	1.94187
$612$	0	0
$613$	29.0000	1.17130	0.585649	−	0.810564i	$-0.300840\pi$
$613$	29.0000	1.17130	0.585649	+	0.810564i	$0.300840\pi$
$614$	−34.0000	−1.37213
$615$	0	0
$616$	3.00000	0.120873
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	35.0000	1.40677	0.703384	−	0.710810i	$-0.251671\pi$
$619$	35.0000	1.40677	0.703384	+	0.710810i	$0.251671\pi$
$620$	5.00000	0.200805
$621$	0	0
$622$	9.00000	0.360867
$623$	−6.00000	−0.240385
$624$	0	0
$625$	1.00000	0.0400000
$626$	26.0000	1.03917
$627$	0	0
$628$	−13.0000	−0.518756
$629$	−3.00000	−0.119618
$630$	0	0
$631$	−7.00000	−0.278666	−0.139333	−	0.990246i	$-0.544496\pi$
$631$	−7.00000	−0.278666	−0.139333	+	0.990246i	$0.544496\pi$
$632$	8.00000	0.318223
$633$	0	0
$634$	21.0000	0.834017
$635$	20.0000	0.793676
$636$	0	0
$637$	24.0000	0.950915
$638$	9.00000	0.356313
$639$	0	0
$640$	1.00000	0.0395285
$641$	−15.0000	−0.592464	−0.296232	−	0.955116i	$-0.595730\pi$
$641$	−15.0000	−0.592464	−0.296232	+	0.955116i	$0.595730\pi$
$642$	0	0
$643$	−13.0000	−0.512670	−0.256335	−	0.966588i	$-0.582515\pi$
$643$	−13.0000	−0.512670	−0.256335	+	0.966588i	$0.582515\pi$
$644$	6.00000	0.236433
$645$	0	0
$646$	−6.00000	−0.236067
$647$	48.0000	1.88707	0.943537	−	0.331266i	$-0.107476\pi$
$647$	48.0000	1.88707	0.943537	+	0.331266i	$0.107476\pi$
$648$	0	0
$649$	0	0
$650$	−4.00000	−0.156893
$651$	0	0
$652$	11.0000	0.430793
$653$	−30.0000	−1.17399	−0.586995	−	0.809590i	$-0.699689\pi$
$653$	−30.0000	−1.17399	−0.586995	+	0.809590i	$0.699689\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	12.0000	0.467809
$659$	36.0000	1.40236	0.701180	−	0.712984i	$-0.252657\pi$
$659$	36.0000	1.40236	0.701180	+	0.712984i	$0.252657\pi$
$660$	0	0
$661$	5.00000	0.194477	0.0972387	−	0.995261i	$-0.468999\pi$
$661$	5.00000	0.194477	0.0972387	+	0.995261i	$0.468999\pi$
$662$	26.0000	1.01052
$663$	0	0
$664$	12.0000	0.465690
$665$	−2.00000	−0.0775567
$666$	0	0
$667$	18.0000	0.696963
$668$	−12.0000	−0.464294
$669$	0	0
$670$	−4.00000	−0.154533
$671$	3.00000	0.115814
$672$	0	0
$673$	−10.0000	−0.385472	−0.192736	−	0.981251i	$-0.561736\pi$
$673$	−10.0000	−0.385472	−0.192736	+	0.981251i	$0.561736\pi$
$674$	−4.00000	−0.154074
$675$	0	0
$676$	3.00000	0.115385
$677$	42.0000	1.61419	0.807096	−	0.590421i	$-0.201038\pi$
$677$	42.0000	1.61419	0.807096	+	0.590421i	$0.201038\pi$
$678$	0	0
$679$	−17.0000	−0.652400
$680$	−3.00000	−0.115045
$681$	0	0
$682$	−15.0000	−0.574380
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	0	0
$685$	−12.0000	−0.458496
$686$	13.0000	0.496342
$687$	0	0
$688$	−1.00000	−0.0381246
$689$	12.0000	0.457164
$690$	0	0
$691$	−19.0000	−0.722794	−0.361397	−	0.932412i	$-0.617700\pi$
$691$	−19.0000	−0.722794	−0.361397	+	0.932412i	$0.617700\pi$
$692$	−15.0000	−0.570214
$693$	0	0
$694$	12.0000	0.455514
$695$	−13.0000	−0.493118
$696$	0	0
$697$	9.00000	0.340899
$698$	26.0000	0.984115
$699$	0	0
$700$	−1.00000	−0.0377964
$701$	−30.0000	−1.13308	−0.566542	−	0.824033i	$-0.691719\pi$
$701$	−30.0000	−1.13308	−0.566542	+	0.824033i	$0.691719\pi$
$702$	0	0
$703$	2.00000	0.0754314
$704$	−3.00000	−0.113067
$705$	0	0
$706$	21.0000	0.790345
$707$	−6.00000	−0.225653
$708$	0	0
$709$	35.0000	1.31445	0.657226	−	0.753693i	$-0.271730\pi$
$709$	35.0000	1.31445	0.657226	+	0.753693i	$0.271730\pi$
$710$	−6.00000	−0.225176
$711$	0	0
$712$	6.00000	0.224860
$713$	−30.0000	−1.12351
$714$	0	0
$715$	12.0000	0.448775
$716$	0	0
$717$	0	0
$718$	36.0000	1.34351
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	4.00000	0.148968
$722$	−15.0000	−0.558242
$723$	0	0
$724$	2.00000	0.0743294
$725$	−3.00000	−0.111417
$726$	0	0
$727$	−28.0000	−1.03846	−0.519231	−	0.854634i	$-0.673782\pi$
$727$	−28.0000	−1.03846	−0.519231	+	0.854634i	$0.673782\pi$
$728$	4.00000	0.148250
$729$	0	0
$730$	−16.0000	−0.592187
$731$	3.00000	0.110959
$732$	0	0
$733$	−31.0000	−1.14501	−0.572506	−	0.819901i	$-0.694029\pi$
$733$	−31.0000	−1.14501	−0.572506	+	0.819901i	$0.694029\pi$
$734$	35.0000	1.29187
$735$	0	0
$736$	−6.00000	−0.221163
$737$	12.0000	0.442026
$738$	0	0
$739$	11.0000	0.404642	0.202321	−	0.979319i	$-0.435152\pi$
$739$	11.0000	0.404642	0.202321	+	0.979319i	$0.435152\pi$
$740$	1.00000	0.0367607
$741$	0	0
$742$	3.00000	0.110133
$743$	51.0000	1.87101	0.935504	−	0.353315i	$-0.114946\pi$
$743$	51.0000	1.87101	0.935504	+	0.353315i	$0.114946\pi$
$744$	0	0
$745$	−6.00000	−0.219823
$746$	−22.0000	−0.805477
$747$	0	0
$748$	9.00000	0.329073
$749$	6.00000	0.219235
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	12.0000	0.437014
$755$	8.00000	0.291150
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−16.0000	−0.581146
$759$	0	0
$760$	2.00000	0.0725476
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	−11.0000	−0.398227
$764$	−3.00000	−0.108536
$765$	0	0
$766$	24.0000	0.867155
$767$	0	0
$768$	0	0
$769$	−40.0000	−1.44244	−0.721218	−	0.692708i	$-0.756418\pi$
$769$	−40.0000	−1.44244	−0.721218	+	0.692708i	$0.756418\pi$
$770$	3.00000	0.108112
$771$	0	0
$772$	14.0000	0.503871
$773$	39.0000	1.40273	0.701366	−	0.712801i	$-0.252574\pi$
$773$	39.0000	1.40273	0.701366	+	0.712801i	$0.252574\pi$
$774$	0	0
$775$	5.00000	0.179605
$776$	17.0000	0.610264
$777$	0	0
$778$	−9.00000	−0.322666
$779$	−6.00000	−0.214972
$780$	0	0
$781$	18.0000	0.644091
$782$	18.0000	0.643679
$783$	0	0
$784$	−6.00000	−0.214286
$785$	−13.0000	−0.463990
$786$	0	0
$787$	−22.0000	−0.784215	−0.392108	−	0.919919i	$-0.628254\pi$
$787$	−22.0000	−0.784215	−0.392108	+	0.919919i	$0.628254\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	8.00000	0.284627
$791$	9.00000	0.320003
$792$	0	0
$793$	4.00000	0.142044
$794$	−34.0000	−1.20661
$795$	0	0
$796$	−16.0000	−0.567105
$797$	18.0000	0.637593	0.318796	−	0.947823i	$-0.396721\pi$
$797$	18.0000	0.637593	0.318796	+	0.947823i	$0.396721\pi$
$798$	0	0
$799$	36.0000	1.27359
$800$	1.00000	0.0353553
$801$	0	0
$802$	−6.00000	−0.211867
$803$	48.0000	1.69388
$804$	0	0
$805$	6.00000	0.211472
$806$	−20.0000	−0.704470
$807$	0	0
$808$	6.00000	0.211079
$809$	30.0000	1.05474	0.527372	−	0.849635i	$-0.323177\pi$
$809$	30.0000	1.05474	0.527372	+	0.849635i	$0.323177\pi$
$810$	0	0
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	3.00000	0.105279
$813$	0	0
$814$	−3.00000	−0.105150
$815$	11.0000	0.385313
$816$	0	0
$817$	−2.00000	−0.0699711
$818$	32.0000	1.11885
$819$	0	0
$820$	−3.00000	−0.104765
$821$	−36.0000	−1.25641	−0.628204	−	0.778048i	$-0.716210\pi$
$821$	−36.0000	−1.25641	−0.628204	+	0.778048i	$0.716210\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−4.00000	−0.139347
$825$	0	0
$826$	0	0
$827$	3.00000	0.104320	0.0521601	−	0.998639i	$-0.483389\pi$
$827$	3.00000	0.104320	0.0521601	+	0.998639i	$0.483389\pi$
$828$	0	0
$829$	47.0000	1.63238	0.816189	−	0.577785i	$-0.196083\pi$
$829$	47.0000	1.63238	0.816189	+	0.577785i	$0.196083\pi$
$830$	12.0000	0.416526
$831$	0	0
$832$	−4.00000	−0.138675
$833$	18.0000	0.623663
$834$	0	0
$835$	−12.0000	−0.415277
$836$	−6.00000	−0.207514
$837$	0	0
$838$	36.0000	1.24360
$839$	48.0000	1.65714	0.828572	−	0.559883i	$-0.189154\pi$
$839$	48.0000	1.65714	0.828572	+	0.559883i	$0.189154\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	−10.0000	−0.344623
$843$	0	0
$844$	5.00000	0.172107
$845$	3.00000	0.103203
$846$	0	0
$847$	2.00000	0.0687208
$848$	−3.00000	−0.103020
$849$	0	0
$850$	−3.00000	−0.102899
$851$	−6.00000	−0.205677
$852$	0	0
$853$	−10.0000	−0.342393	−0.171197	−	0.985237i	$-0.554763\pi$
$853$	−10.0000	−0.342393	−0.171197	+	0.985237i	$0.554763\pi$
$854$	1.00000	0.0342193
$855$	0	0
$856$	−6.00000	−0.205076
$857$	15.0000	0.512390	0.256195	−	0.966625i	$-0.417531\pi$
$857$	15.0000	0.512390	0.256195	+	0.966625i	$0.417531\pi$
$858$	0	0
$859$	−40.0000	−1.36478	−0.682391	−	0.730987i	$-0.739060\pi$
$859$	−40.0000	−1.36478	−0.682391	+	0.730987i	$0.739060\pi$
$860$	−1.00000	−0.0340997
$861$	0	0
$862$	−15.0000	−0.510902
$863$	−39.0000	−1.32758	−0.663788	−	0.747921i	$-0.731052\pi$
$863$	−39.0000	−1.32758	−0.663788	+	0.747921i	$0.731052\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	2.00000	0.0679628
$867$	0	0
$868$	−5.00000	−0.169711
$869$	−24.0000	−0.814144
$870$	0	0
$871$	16.0000	0.542139
$872$	11.0000	0.372507
$873$	0	0
$874$	−12.0000	−0.405906
$875$	−1.00000	−0.0338062
$876$	0	0
$877$	−13.0000	−0.438979	−0.219489	−	0.975615i	$-0.570439\pi$
$877$	−13.0000	−0.438979	−0.219489	+	0.975615i	$0.570439\pi$
$878$	35.0000	1.18119
$879$	0	0
$880$	−3.00000	−0.101130
$881$	−15.0000	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$881$	−15.0000	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$882$	0	0
$883$	29.0000	0.975928	0.487964	−	0.872864i	$-0.337740\pi$
$883$	29.0000	0.975928	0.487964	+	0.872864i	$0.337740\pi$
$884$	12.0000	0.403604
$885$	0	0
$886$	−30.0000	−1.00787
$887$	−33.0000	−1.10803	−0.554016	−	0.832506i	$-0.686905\pi$
$887$	−33.0000	−1.10803	−0.554016	+	0.832506i	$0.686905\pi$
$888$	0	0
$889$	−20.0000	−0.670778
$890$	6.00000	0.201120
$891$	0	0
$892$	17.0000	0.569202
$893$	−24.0000	−0.803129
$894$	0	0
$895$	0	0
$896$	−1.00000	−0.0334077
$897$	0	0
$898$	−6.00000	−0.200223
$899$	−15.0000	−0.500278
$900$	0	0
$901$	9.00000	0.299833
$902$	9.00000	0.299667
$903$	0	0
$904$	−9.00000	−0.299336
$905$	2.00000	0.0664822
$906$	0	0
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	27.0000	0.896026
$909$	0	0
$910$	4.00000	0.132599
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	17.0000	0.562310
$915$	0	0
$916$	14.0000	0.462573
$917$	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	$0.585007\pi$
$920$	−6.00000	−0.197814
$921$	0	0
$922$	−33.0000	−1.08680
$923$	24.0000	0.789970
$924$	0	0
$925$	1.00000	0.0328798
$926$	−4.00000	−0.131448
$927$	0	0
$928$	−3.00000	−0.0984798
$929$	−45.0000	−1.47640	−0.738201	−	0.674581i	$-0.764324\pi$
$929$	−45.0000	−1.47640	−0.738201	+	0.674581i	$0.764324\pi$
$930$	0	0
$931$	−12.0000	−0.393284
$932$	0	0
$933$	0	0
$934$	−27.0000	−0.883467
$935$	9.00000	0.294331
$936$	0	0
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−12.0000	−0.391397
$941$	18.0000	0.586783	0.293392	−	0.955992i	$-0.405216\pi$
$941$	18.0000	0.586783	0.293392	+	0.955992i	$0.405216\pi$
$942$	0	0
$943$	18.0000	0.586161
$944$	0	0
$945$	0	0
$946$	3.00000	0.0975384
$947$	−33.0000	−1.07236	−0.536178	−	0.844105i	$-0.680132\pi$
$947$	−33.0000	−1.07236	−0.536178	+	0.844105i	$0.680132\pi$
$948$	0	0
$949$	64.0000	2.07753
$950$	2.00000	0.0648886
$951$	0	0
$952$	3.00000	0.0972306
$953$	60.0000	1.94359	0.971795	−	0.235826i	$-0.0757795\pi$
$953$	60.0000	1.94359	0.971795	+	0.235826i	$0.0757795\pi$
$954$	0	0
$955$	−3.00000	−0.0970777
$956$	9.00000	0.291081
$957$	0	0
$958$	−24.0000	−0.775405
$959$	12.0000	0.387500
$960$	0	0
$961$	−6.00000	−0.193548
$962$	−4.00000	−0.128965
$963$	0	0
$964$	−28.0000	−0.901819
$965$	14.0000	0.450676
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	−2.00000	−0.0642824
$969$	0	0
$970$	17.0000	0.545837
$971$	−57.0000	−1.82922	−0.914609	−	0.404341i	$-0.867501\pi$
$971$	−57.0000	−1.82922	−0.914609	+	0.404341i	$0.867501\pi$
$972$	0	0
$973$	13.0000	0.416761
$974$	2.00000	0.0640841
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	45.0000	1.43968	0.719839	−	0.694141i	$-0.244216\pi$
$977$	45.0000	1.43968	0.719839	+	0.694141i	$0.244216\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	−6.00000	−0.191663
$981$	0	0
$982$	12.0000	0.382935
$983$	−51.0000	−1.62665	−0.813324	−	0.581811i	$-0.802344\pi$
$983$	−51.0000	−1.62665	−0.813324	+	0.581811i	$0.802344\pi$
$984$	0	0
$985$	6.00000	0.191176
$986$	9.00000	0.286618
$987$	0	0
$988$	−8.00000	−0.254514
$989$	6.00000	0.190789
$990$	0	0
$991$	47.0000	1.49300	0.746502	−	0.665383i	$-0.231732\pi$
$991$	47.0000	1.49300	0.746502	+	0.665383i	$0.231732\pi$
$992$	5.00000	0.158750
$993$	0	0
$994$	6.00000	0.190308
$995$	−16.0000	−0.507234
$996$	0	0
$997$	−28.0000	−0.886769	−0.443384	−	0.896332i	$-0.646222\pi$
$997$	−28.0000	−0.886769	−0.443384	+	0.896332i	$0.646222\pi$
$998$	14.0000	0.443162
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3330.2.a.v.1.1		1
3.2	odd	2		370.2.a.a.1.1	✓	1
12.11	even	2		2960.2.a.j.1.1		1
15.2	even	4		1850.2.b.g.149.1		2
15.8	even	4		1850.2.b.g.149.2		2
15.14	odd	2		1850.2.a.o.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
370.2.a.a.1.1	✓	1	3.2	odd	2
1850.2.a.o.1.1		1	15.14	odd	2
1850.2.b.g.149.1		2	15.2	even	4
1850.2.b.g.149.2		2	15.8	even	4
2960.2.a.j.1.1		1	12.11	even	2
3330.2.a.v.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3330.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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