LMFDB - Embedded newform 3042.2.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3042 = 2 \cdot 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 26)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	3042.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{10} +4.00000 q^{11} -4.00000 q^{14} +1.00000 q^{16} -3.00000 q^{17} -1.00000 q^{20} +4.00000 q^{22} +4.00000 q^{23} -4.00000 q^{25} -4.00000 q^{28} +1.00000 q^{29} -4.00000 q^{31} +1.00000 q^{32} -3.00000 q^{34} +4.00000 q^{35} -3.00000 q^{37} -1.00000 q^{40} -9.00000 q^{41} -8.00000 q^{43} +4.00000 q^{44} +4.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} -4.00000 q^{50} +9.00000 q^{53} -4.00000 q^{55} -4.00000 q^{56} +1.00000 q^{58} -4.00000 q^{59} +7.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -4.00000 q^{67} -3.00000 q^{68} +4.00000 q^{70} -8.00000 q^{71} -11.0000 q^{73} -3.00000 q^{74} -16.0000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -9.00000 q^{82} +3.00000 q^{85} -8.00000 q^{86} +4.00000 q^{88} -6.00000 q^{89} +4.00000 q^{92} -8.00000 q^{94} -2.00000 q^{97} +9.00000 q^{98} +O(q^{100})$

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	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−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	$0.293963\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−4.00000	−1.06904
$15$	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
$19$	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
$23$	4.00000	0.834058	0.417029	−	0.908893i	$-0.363071\pi$
$23$	4.00000	0.834058	0.417029	+	0.908893i	$0.363071\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	−4.00000	−0.755929
$29$	1.00000	0.185695	0.0928477	−	0.995680i	$-0.470403\pi$
$29$	1.00000	0.185695	0.0928477	+	0.995680i	$0.470403\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	4.00000	0.676123
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	0	0
$39$	0	0
$40$	−1.00000	−0.158114
$41$	−9.00000	−1.40556	−0.702782	−	0.711405i	$-0.748059\pi$
$41$	−9.00000	−1.40556	−0.702782	+	0.711405i	$0.748059\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	4.00000	0.589768
$47$	−8.00000	−1.16692	−0.583460	−	0.812142i	$-0.698301\pi$
$47$	−8.00000	−1.16692	−0.583460	+	0.812142i	$0.698301\pi$
$48$	0	0
$49$	9.00000	1.28571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	0	0
$53$	9.00000	1.23625	0.618123	−	0.786082i	$-0.287894\pi$
$53$	9.00000	1.23625	0.618123	+	0.786082i	$0.287894\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	−4.00000	−0.534522
$57$	0	0
$58$	1.00000	0.131306
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	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
$70$	4.00000	0.478091
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	−3.00000	−0.348743
$75$	0	0
$76$	0	0
$77$	−16.0000	−1.82337
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−1.00000	−0.111803
$81$	0	0
$82$	−9.00000	−0.993884
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	3.00000	0.325396
$86$	−8.00000	−0.862662
$87$	0	0
$88$	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	$0.603011\pi$
$90$	0	0
$91$	0	0
$92$	4.00000	0.417029
$93$	0	0
$94$	−8.00000	−0.825137
$95$	0	0
$96$	0	0
$97$	−2.00000	−0.203069	−0.101535	−	0.994832i	$-0.532375\pi$
$97$	−2.00000	−0.203069	−0.101535	+	0.994832i	$0.532375\pi$
$98$	9.00000	0.909137
$99$	0	0
$100$	−4.00000	−0.400000
$101$	−7.00000	−0.696526	−0.348263	−	0.937397i	$-0.613228\pi$
$101$	−7.00000	−0.696526	−0.348263	+	0.937397i	$0.613228\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	9.00000	0.874157
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−4.00000	−0.381385
$111$	0	0
$112$	−4.00000	−0.377964
$113$	1.00000	0.0940721	0.0470360	−	0.998893i	$-0.485022\pi$
$113$	1.00000	0.0940721	0.0470360	+	0.998893i	$0.485022\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	1.00000	0.0928477
$117$	0	0
$118$	−4.00000	−0.368230
$119$	12.0000	1.10004
$120$	0	0
$121$	5.00000	0.454545
$122$	7.00000	0.633750
$123$	0	0
$124$	−4.00000	−0.359211
$125$	9.00000	0.804984
$126$	0	0
$127$	−8.00000	−0.709885	−0.354943	−	0.934888i	$-0.615500\pi$
$127$	−8.00000	−0.709885	−0.354943	+	0.934888i	$0.615500\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−20.0000	−1.74741	−0.873704	−	0.486458i	$-0.838289\pi$
$131$	−20.0000	−1.74741	−0.873704	+	0.486458i	$0.838289\pi$
$132$	0	0
$133$	0	0
$134$	−4.00000	−0.345547
$135$	0	0
$136$	−3.00000	−0.257248
$137$	−9.00000	−0.768922	−0.384461	−	0.923141i	$-0.625613\pi$
$137$	−9.00000	−0.768922	−0.384461	+	0.923141i	$0.625613\pi$
$138$	0	0
$139$	16.0000	1.35710	0.678551	−	0.734553i	$-0.262608\pi$
$139$	16.0000	1.35710	0.678551	+	0.734553i	$0.262608\pi$
$140$	4.00000	0.338062
$141$	0	0
$142$	−8.00000	−0.671345
$143$	0	0
$144$	0	0
$145$	−1.00000	−0.0830455
$146$	−11.0000	−0.910366
$147$	0	0
$148$	−3.00000	−0.246598
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	0	0
$153$	0	0
$154$	−16.0000	−1.28932
$155$	4.00000	0.321288
$156$	0	0
$157$	11.0000	0.877896	0.438948	−	0.898513i	$-0.355351\pi$
$157$	11.0000	0.877896	0.438948	+	0.898513i	$0.355351\pi$
$158$	−4.00000	−0.318223
$159$	0	0
$160$	−1.00000	−0.0790569
$161$	−16.0000	−1.26098
$162$	0	0
$163$	−8.00000	−0.626608	−0.313304	−	0.949653i	$-0.601436\pi$
$163$	−8.00000	−0.626608	−0.313304	+	0.949653i	$0.601436\pi$
$164$	−9.00000	−0.702782
$165$	0	0
$166$	0	0
$167$	12.0000	0.928588	0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000	0.928588	0.464294	+	0.885681i	$0.346308\pi$
$168$	0	0
$169$	0	0
$170$	3.00000	0.230089
$171$	0	0
$172$	−8.00000	−0.609994
$173$	−14.0000	−1.06440	−0.532200	−	0.846619i	$-0.678635\pi$
$173$	−14.0000	−1.06440	−0.532200	+	0.846619i	$0.678635\pi$
$174$	0	0
$175$	16.0000	1.20949
$176$	4.00000	0.301511
$177$	0	0
$178$	−6.00000	−0.449719
$179$	24.0000	1.79384	0.896922	−	0.442189i	$-0.145798\pi$
$179$	24.0000	1.79384	0.896922	+	0.442189i	$0.145798\pi$
$180$	0	0
$181$	−21.0000	−1.56092	−0.780459	−	0.625207i	$-0.785014\pi$
$181$	−21.0000	−1.56092	−0.780459	+	0.625207i	$0.785014\pi$
$182$	0	0
$183$	0	0
$184$	4.00000	0.294884
$185$	3.00000	0.220564
$186$	0	0
$187$	−12.0000	−0.877527
$188$	−8.00000	−0.583460
$189$	0	0
$190$	0	0
$191$	20.0000	1.44715	0.723575	−	0.690246i	$-0.242498\pi$
$191$	20.0000	1.44715	0.723575	+	0.690246i	$0.242498\pi$
$192$	0	0
$193$	−11.0000	−0.791797	−0.395899	−	0.918294i	$-0.629567\pi$
$193$	−11.0000	−0.791797	−0.395899	+	0.918294i	$0.629567\pi$
$194$	−2.00000	−0.143592
$195$	0	0
$196$	9.00000	0.642857
$197$	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
$199$	−4.00000	−0.283552	−0.141776	−	0.989899i	$-0.545281\pi$
$199$	−4.00000	−0.283552	−0.141776	+	0.989899i	$0.545281\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	−7.00000	−0.492518
$203$	−4.00000	−0.280745
$204$	0	0
$205$	9.00000	0.628587
$206$	−8.00000	−0.557386
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	9.00000	0.618123
$213$	0	0
$214$	4.00000	0.273434
$215$	8.00000	0.545595
$216$	0	0
$217$	16.0000	1.08615
$218$	2.00000	0.135457
$219$	0	0
$220$	−4.00000	−0.269680
$221$	0	0
$222$	0	0
$223$	−12.0000	−0.803579	−0.401790	−	0.915732i	$-0.631612\pi$
$223$	−12.0000	−0.803579	−0.401790	+	0.915732i	$0.631612\pi$
$224$	−4.00000	−0.267261
$225$	0	0
$226$	1.00000	0.0665190
$227$	24.0000	1.59294	0.796468	−	0.604681i	$-0.206699\pi$
$227$	24.0000	1.59294	0.796468	+	0.604681i	$0.206699\pi$
$228$	0	0
$229$	−6.00000	−0.396491	−0.198246	−	0.980152i	$-0.563524\pi$
$229$	−6.00000	−0.396491	−0.198246	+	0.980152i	$0.563524\pi$
$230$	−4.00000	−0.263752
$231$	0	0
$232$	1.00000	0.0656532
$233$	−10.0000	−0.655122	−0.327561	−	0.944830i	$-0.606227\pi$
$233$	−10.0000	−0.655122	−0.327561	+	0.944830i	$0.606227\pi$
$234$	0	0
$235$	8.00000	0.521862
$236$	−4.00000	−0.260378
$237$	0	0
$238$	12.0000	0.777844
$239$	−12.0000	−0.776215	−0.388108	−	0.921614i	$-0.626871\pi$
$239$	−12.0000	−0.776215	−0.388108	+	0.921614i	$0.626871\pi$
$240$	0	0
$241$	−3.00000	−0.193247	−0.0966235	−	0.995321i	$-0.530804\pi$
$241$	−3.00000	−0.193247	−0.0966235	+	0.995321i	$0.530804\pi$
$242$	5.00000	0.321412
$243$	0	0
$244$	7.00000	0.448129
$245$	−9.00000	−0.574989
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	9.00000	0.569210
$251$	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
$253$	16.0000	1.00591
$254$	−8.00000	−0.501965
$255$	0	0
$256$	1.00000	0.0625000
$257$	−15.0000	−0.935674	−0.467837	−	0.883815i	$-0.654967\pi$
$257$	−15.0000	−0.935674	−0.467837	+	0.883815i	$0.654967\pi$
$258$	0	0
$259$	12.0000	0.745644
$260$	0	0
$261$	0	0
$262$	−20.0000	−1.23560
$263$	24.0000	1.47990	0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000	1.47990	0.739952	+	0.672660i	$0.234848\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	0	0
$267$	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
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	−3.00000	−0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	−16.0000	−0.964836
$276$	0	0
$277$	−9.00000	−0.540758	−0.270379	−	0.962754i	$-0.587149\pi$
$277$	−9.00000	−0.540758	−0.270379	+	0.962754i	$0.587149\pi$
$278$	16.0000	0.959616
$279$	0	0
$280$	4.00000	0.239046
$281$	−5.00000	−0.298275	−0.149137	−	0.988816i	$-0.547650\pi$
$281$	−5.00000	−0.298275	−0.149137	+	0.988816i	$0.547650\pi$
$282$	0	0
$283$	28.0000	1.66443	0.832214	−	0.554455i	$-0.187073\pi$
$283$	28.0000	1.66443	0.832214	+	0.554455i	$0.187073\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	0	0
$287$	36.0000	2.12501
$288$	0	0
$289$	−8.00000	−0.470588
$290$	−1.00000	−0.0587220
$291$	0	0
$292$	−11.0000	−0.643726
$293$	−5.00000	−0.292103	−0.146052	−	0.989277i	$-0.546657\pi$
$293$	−5.00000	−0.292103	−0.146052	+	0.989277i	$0.546657\pi$
$294$	0	0
$295$	4.00000	0.232889
$296$	−3.00000	−0.174371
$297$	0	0
$298$	15.0000	0.868927
$299$	0	0
$300$	0	0
$301$	32.0000	1.84445
$302$	−12.0000	−0.690522
$303$	0	0
$304$	0	0
$305$	−7.00000	−0.400819
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−16.0000	−0.911685
$309$	0	0
$310$	4.00000	0.227185
$311$	24.0000	1.36092	0.680458	−	0.732787i	$-0.261781\pi$
$311$	24.0000	1.36092	0.680458	+	0.732787i	$0.261781\pi$
$312$	0	0
$313$	−22.0000	−1.24351	−0.621757	−	0.783210i	$-0.713581\pi$
$313$	−22.0000	−1.24351	−0.621757	+	0.783210i	$0.713581\pi$
$314$	11.0000	0.620766
$315$	0	0
$316$	−4.00000	−0.225018
$317$	3.00000	0.168497	0.0842484	−	0.996445i	$-0.473151\pi$
$317$	3.00000	0.168497	0.0842484	+	0.996445i	$0.473151\pi$
$318$	0	0
$319$	4.00000	0.223957
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	−16.0000	−0.891645
$323$	0	0
$324$	0	0
$325$	0	0
$326$	−8.00000	−0.443079
$327$	0	0
$328$	−9.00000	−0.496942
$329$	32.0000	1.76422
$330$	0	0
$331$	−8.00000	−0.439720	−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000	−0.439720	−0.219860	+	0.975531i	$0.570560\pi$
$332$	0	0
$333$	0	0
$334$	12.0000	0.656611
$335$	4.00000	0.218543
$336$	0	0
$337$	23.0000	1.25289	0.626445	−	0.779466i	$-0.284509\pi$
$337$	23.0000	1.25289	0.626445	+	0.779466i	$0.284509\pi$
$338$	0	0
$339$	0	0
$340$	3.00000	0.162698
$341$	−16.0000	−0.866449
$342$	0	0
$343$	−8.00000	−0.431959
$344$	−8.00000	−0.431331
$345$	0	0
$346$	−14.0000	−0.752645
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	16.0000	0.855236
$351$	0	0
$352$	4.00000	0.213201
$353$	7.00000	0.372572	0.186286	−	0.982496i	$-0.440355\pi$
$353$	7.00000	0.372572	0.186286	+	0.982496i	$0.440355\pi$
$354$	0	0
$355$	8.00000	0.424596
$356$	−6.00000	−0.317999
$357$	0	0
$358$	24.0000	1.26844
$359$	24.0000	1.26667	0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000	1.26667	0.633336	+	0.773877i	$0.281685\pi$
$360$	0	0
$361$	−19.0000	−1.00000
$362$	−21.0000	−1.10374
$363$	0	0
$364$	0	0
$365$	11.0000	0.575766
$366$	0	0
$367$	−28.0000	−1.46159	−0.730794	−	0.682598i	$-0.760850\pi$
$367$	−28.0000	−1.46159	−0.730794	+	0.682598i	$0.760850\pi$
$368$	4.00000	0.208514
$369$	0	0
$370$	3.00000	0.155963
$371$	−36.0000	−1.86903
$372$	0	0
$373$	−13.0000	−0.673114	−0.336557	−	0.941663i	$-0.609263\pi$
$373$	−13.0000	−0.673114	−0.336557	+	0.941663i	$0.609263\pi$
$374$	−12.0000	−0.620505
$375$	0	0
$376$	−8.00000	−0.412568
$377$	0	0
$378$	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
$381$	0	0
$382$	20.0000	1.02329
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	0	0
$385$	16.0000	0.815436
$386$	−11.0000	−0.559885
$387$	0	0
$388$	−2.00000	−0.101535
$389$	9.00000	0.456318	0.228159	−	0.973624i	$-0.426729\pi$
$389$	9.00000	0.456318	0.228159	+	0.973624i	$0.426729\pi$
$390$	0	0
$391$	−12.0000	−0.606866
$392$	9.00000	0.454569
$393$	0	0
$394$	6.00000	0.302276
$395$	4.00000	0.201262
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	−4.00000	−0.200502
$399$	0	0
$400$	−4.00000	−0.200000
$401$	3.00000	0.149813	0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000	0.149813	0.0749064	+	0.997191i	$0.476134\pi$
$402$	0	0
$403$	0	0
$404$	−7.00000	−0.348263
$405$	0	0
$406$	−4.00000	−0.198517
$407$	−12.0000	−0.594818
$408$	0	0
$409$	−31.0000	−1.53285	−0.766426	−	0.642333i	$-0.777967\pi$
$409$	−31.0000	−1.53285	−0.766426	+	0.642333i	$0.777967\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	−8.00000	−0.394132
$413$	16.0000	0.787309
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	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
$421$	5.00000	0.243685	0.121843	−	0.992549i	$-0.461120\pi$
$421$	5.00000	0.243685	0.121843	+	0.992549i	$0.461120\pi$
$422$	20.0000	0.973585
$423$	0	0
$424$	9.00000	0.437079
$425$	12.0000	0.582086
$426$	0	0
$427$	−28.0000	−1.35501
$428$	4.00000	0.193347
$429$	0	0
$430$	8.00000	0.385794
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	−5.00000	−0.240285	−0.120142	−	0.992757i	$-0.538335\pi$
$433$	−5.00000	−0.240285	−0.120142	+	0.992757i	$0.538335\pi$
$434$	16.0000	0.768025
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	0	0
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	−4.00000	−0.190693
$441$	0	0
$442$	0	0
$443$	−24.0000	−1.14027	−0.570137	−	0.821549i	$-0.693110\pi$
$443$	−24.0000	−1.14027	−0.570137	+	0.821549i	$0.693110\pi$
$444$	0	0
$445$	6.00000	0.284427
$446$	−12.0000	−0.568216
$447$	0	0
$448$	−4.00000	−0.188982
$449$	34.0000	1.60456	0.802280	−	0.596948i	$-0.203620\pi$
$449$	34.0000	1.60456	0.802280	+	0.596948i	$0.203620\pi$
$450$	0	0
$451$	−36.0000	−1.69517
$452$	1.00000	0.0470360
$453$	0	0
$454$	24.0000	1.12638
$455$	0	0
$456$	0	0
$457$	−31.0000	−1.45012	−0.725059	−	0.688686i	$-0.758188\pi$
$457$	−31.0000	−1.45012	−0.725059	+	0.688686i	$0.758188\pi$
$458$	−6.00000	−0.280362
$459$	0	0
$460$	−4.00000	−0.186501
$461$	−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
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	1.00000	0.0464238
$465$	0	0
$466$	−10.0000	−0.463241
$467$	−20.0000	−0.925490	−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000	−0.925490	−0.462745	+	0.886492i	$0.653135\pi$
$468$	0	0
$469$	16.0000	0.738811
$470$	8.00000	0.369012
$471$	0	0
$472$	−4.00000	−0.184115
$473$	−32.0000	−1.47136
$474$	0	0
$475$	0	0
$476$	12.0000	0.550019
$477$	0	0
$478$	−12.0000	−0.548867
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	−3.00000	−0.136646
$483$	0	0
$484$	5.00000	0.227273
$485$	2.00000	0.0908153
$486$	0	0
$487$	16.0000	0.725029	0.362515	−	0.931978i	$-0.381918\pi$
$487$	16.0000	0.725029	0.362515	+	0.931978i	$0.381918\pi$
$488$	7.00000	0.316875
$489$	0	0
$490$	−9.00000	−0.406579
$491$	8.00000	0.361035	0.180517	−	0.983572i	$-0.442223\pi$
$491$	8.00000	0.361035	0.180517	+	0.983572i	$0.442223\pi$
$492$	0	0
$493$	−3.00000	−0.135113
$494$	0	0
$495$	0	0
$496$	−4.00000	−0.179605
$497$	32.0000	1.43540
$498$	0	0
$499$	32.0000	1.43252	0.716258	−	0.697835i	$-0.245853\pi$
$499$	32.0000	1.43252	0.716258	+	0.697835i	$0.245853\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	12.0000	0.535586
$503$	−16.0000	−0.713405	−0.356702	−	0.934218i	$-0.616099\pi$
$503$	−16.0000	−0.713405	−0.356702	+	0.934218i	$0.616099\pi$
$504$	0	0
$505$	7.00000	0.311496
$506$	16.0000	0.711287
$507$	0	0
$508$	−8.00000	−0.354943
$509$	43.0000	1.90594	0.952971	−	0.303062i	$-0.0980090\pi$
$509$	43.0000	1.90594	0.952971	+	0.303062i	$0.0980090\pi$
$510$	0	0
$511$	44.0000	1.94645
$512$	1.00000	0.0441942
$513$	0	0
$514$	−15.0000	−0.661622
$515$	8.00000	0.352522
$516$	0	0
$517$	−32.0000	−1.40736
$518$	12.0000	0.527250
$519$	0	0
$520$	0	0
$521$	−39.0000	−1.70862	−0.854311	−	0.519763i	$-0.826020\pi$
$521$	−39.0000	−1.70862	−0.854311	+	0.519763i	$0.826020\pi$
$522$	0	0
$523$	36.0000	1.57417	0.787085	−	0.616844i	$-0.211589\pi$
$523$	36.0000	1.57417	0.787085	+	0.616844i	$0.211589\pi$
$524$	−20.0000	−0.873704
$525$	0	0
$526$	24.0000	1.04645
$527$	12.0000	0.522728
$528$	0	0
$529$	−7.00000	−0.304348
$530$	−9.00000	−0.390935
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−4.00000	−0.172935
$536$	−4.00000	−0.172774
$537$	0	0
$538$	18.0000	0.776035
$539$	36.0000	1.55063
$540$	0	0
$541$	25.0000	1.07483	0.537417	−	0.843317i	$-0.319400\pi$
$541$	25.0000	1.07483	0.537417	+	0.843317i	$0.319400\pi$
$542$	20.0000	0.859074
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	16.0000	0.684111	0.342055	−	0.939680i	$-0.388877\pi$
$547$	16.0000	0.684111	0.342055	+	0.939680i	$0.388877\pi$
$548$	−9.00000	−0.384461
$549$	0	0
$550$	−16.0000	−0.682242
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	−9.00000	−0.382373
$555$	0	0
$556$	16.0000	0.678551
$557$	39.0000	1.65248	0.826242	−	0.563316i	$-0.190475\pi$
$557$	39.0000	1.65248	0.826242	+	0.563316i	$0.190475\pi$
$558$	0	0
$559$	0	0
$560$	4.00000	0.169031
$561$	0	0
$562$	−5.00000	−0.210912
$563$	4.00000	0.168580	0.0842900	−	0.996441i	$-0.473138\pi$
$563$	4.00000	0.168580	0.0842900	+	0.996441i	$0.473138\pi$
$564$	0	0
$565$	−1.00000	−0.0420703
$566$	28.0000	1.17693
$567$	0	0
$568$	−8.00000	−0.335673
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	36.0000	1.50261
$575$	−16.0000	−0.667246
$576$	0	0
$577$	−39.0000	−1.62359	−0.811796	−	0.583942i	$-0.801510\pi$
$577$	−39.0000	−1.62359	−0.811796	+	0.583942i	$0.801510\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−1.00000	−0.0415227
$581$	0	0
$582$	0	0
$583$	36.0000	1.49097
$584$	−11.0000	−0.455183
$585$	0	0
$586$	−5.00000	−0.206548
$587$	−16.0000	−0.660391	−0.330195	−	0.943913i	$-0.607115\pi$
$587$	−16.0000	−0.660391	−0.330195	+	0.943913i	$0.607115\pi$
$588$	0	0
$589$	0	0
$590$	4.00000	0.164677
$591$	0	0
$592$	−3.00000	−0.123299
$593$	−1.00000	−0.0410651	−0.0205325	−	0.999789i	$-0.506536\pi$
$593$	−1.00000	−0.0410651	−0.0205325	+	0.999789i	$0.506536\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	15.0000	0.614424
$597$	0	0
$598$	0	0
$599$	44.0000	1.79779	0.898896	−	0.438163i	$-0.144371\pi$
$599$	44.0000	1.79779	0.898896	+	0.438163i	$0.144371\pi$
$600$	0	0
$601$	19.0000	0.775026	0.387513	−	0.921864i	$-0.373334\pi$
$601$	19.0000	0.775026	0.387513	+	0.921864i	$0.373334\pi$
$602$	32.0000	1.30422
$603$	0	0
$604$	−12.0000	−0.488273
$605$	−5.00000	−0.203279
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	−7.00000	−0.283422
$611$	0	0
$612$	0	0
$613$	−11.0000	−0.444286	−0.222143	−	0.975014i	$-0.571305\pi$
$613$	−11.0000	−0.444286	−0.222143	+	0.975014i	$0.571305\pi$
$614$	28.0000	1.12999
$615$	0	0
$616$	−16.0000	−0.644658
$617$	−29.0000	−1.16750	−0.583748	−	0.811935i	$-0.698414\pi$
$617$	−29.0000	−1.16750	−0.583748	+	0.811935i	$0.698414\pi$
$618$	0	0
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	4.00000	0.160644
$621$	0	0
$622$	24.0000	0.962312
$623$	24.0000	0.961540
$624$	0	0
$625$	11.0000	0.440000
$626$	−22.0000	−0.879297
$627$	0	0
$628$	11.0000	0.438948
$629$	9.00000	0.358854
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	−4.00000	−0.159111
$633$	0	0
$634$	3.00000	0.119145
$635$	8.00000	0.317470
$636$	0	0
$637$	0	0
$638$	4.00000	0.158362
$639$	0	0
$640$	−1.00000	−0.0395285
$641$	−19.0000	−0.750455	−0.375227	−	0.926933i	$-0.622435\pi$
$641$	−19.0000	−0.750455	−0.375227	+	0.926933i	$0.622435\pi$
$642$	0	0
$643$	−44.0000	−1.73519	−0.867595	−	0.497271i	$-0.834335\pi$
$643$	−44.0000	−1.73519	−0.867595	+	0.497271i	$0.834335\pi$
$644$	−16.0000	−0.630488
$645$	0	0
$646$	0	0
$647$	−28.0000	−1.10079	−0.550397	−	0.834903i	$-0.685524\pi$
$647$	−28.0000	−1.10079	−0.550397	+	0.834903i	$0.685524\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	−8.00000	−0.313304
$653$	18.0000	0.704394	0.352197	−	0.935926i	$-0.385435\pi$
$653$	18.0000	0.704394	0.352197	+	0.935926i	$0.385435\pi$
$654$	0	0
$655$	20.0000	0.781465
$656$	−9.00000	−0.351391
$657$	0	0
$658$	32.0000	1.24749
$659$	−48.0000	−1.86981	−0.934907	−	0.354892i	$-0.884518\pi$
$659$	−48.0000	−1.86981	−0.934907	+	0.354892i	$0.884518\pi$
$660$	0	0
$661$	49.0000	1.90588	0.952940	−	0.303160i	$-0.0980418\pi$
$661$	49.0000	1.90588	0.952940	+	0.303160i	$0.0980418\pi$
$662$	−8.00000	−0.310929
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	4.00000	0.154881
$668$	12.0000	0.464294
$669$	0	0
$670$	4.00000	0.154533
$671$	28.0000	1.08093
$672$	0	0
$673$	−29.0000	−1.11787	−0.558934	−	0.829212i	$-0.688789\pi$
$673$	−29.0000	−1.11787	−0.558934	+	0.829212i	$0.688789\pi$
$674$	23.0000	0.885927
$675$	0	0
$676$	0	0
$677$	−6.00000	−0.230599	−0.115299	−	0.993331i	$-0.536783\pi$
$677$	−6.00000	−0.230599	−0.115299	+	0.993331i	$0.536783\pi$
$678$	0	0
$679$	8.00000	0.307012
$680$	3.00000	0.115045
$681$	0	0
$682$	−16.0000	−0.612672
$683$	−44.0000	−1.68361	−0.841807	−	0.539779i	$-0.818508\pi$
$683$	−44.0000	−1.68361	−0.841807	+	0.539779i	$0.818508\pi$
$684$	0	0
$685$	9.00000	0.343872
$686$	−8.00000	−0.305441
$687$	0	0
$688$	−8.00000	−0.304997
$689$	0	0
$690$	0	0
$691$	20.0000	0.760836	0.380418	−	0.924815i	$-0.375780\pi$
$691$	20.0000	0.760836	0.380418	+	0.924815i	$0.375780\pi$
$692$	−14.0000	−0.532200
$693$	0	0
$694$	−12.0000	−0.455514
$695$	−16.0000	−0.606915
$696$	0	0
$697$	27.0000	1.02270
$698$	2.00000	0.0757011
$699$	0	0
$700$	16.0000	0.604743
$701$	−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
$703$	0	0
$704$	4.00000	0.150756
$705$	0	0
$706$	7.00000	0.263448
$707$	28.0000	1.05305
$708$	0	0
$709$	13.0000	0.488225	0.244113	−	0.969747i	$-0.421503\pi$
$709$	13.0000	0.488225	0.244113	+	0.969747i	$0.421503\pi$
$710$	8.00000	0.300235
$711$	0	0
$712$	−6.00000	−0.224860
$713$	−16.0000	−0.599205
$714$	0	0
$715$	0	0
$716$	24.0000	0.896922
$717$	0	0
$718$	24.0000	0.895672
$719$	28.0000	1.04422	0.522112	−	0.852877i	$-0.325144\pi$
$719$	28.0000	1.04422	0.522112	+	0.852877i	$0.325144\pi$
$720$	0	0
$721$	32.0000	1.19174
$722$	−19.0000	−0.707107
$723$	0	0
$724$	−21.0000	−0.780459
$725$	−4.00000	−0.148556
$726$	0	0
$727$	28.0000	1.03846	0.519231	−	0.854634i	$-0.326218\pi$
$727$	28.0000	1.03846	0.519231	+	0.854634i	$0.326218\pi$
$728$	0	0
$729$	0	0
$730$	11.0000	0.407128
$731$	24.0000	0.887672
$732$	0	0
$733$	1.00000	0.0369358	0.0184679	−	0.999829i	$-0.494121\pi$
$733$	1.00000	0.0369358	0.0184679	+	0.999829i	$0.494121\pi$
$734$	−28.0000	−1.03350
$735$	0	0
$736$	4.00000	0.147442
$737$	−16.0000	−0.589368
$738$	0	0
$739$	24.0000	0.882854	0.441427	−	0.897297i	$-0.354472\pi$
$739$	24.0000	0.882854	0.441427	+	0.897297i	$0.354472\pi$
$740$	3.00000	0.110282
$741$	0	0
$742$	−36.0000	−1.32160
$743$	−8.00000	−0.293492	−0.146746	−	0.989174i	$-0.546880\pi$
$743$	−8.00000	−0.293492	−0.146746	+	0.989174i	$0.546880\pi$
$744$	0	0
$745$	−15.0000	−0.549557
$746$	−13.0000	−0.475964
$747$	0	0
$748$	−12.0000	−0.438763
$749$	−16.0000	−0.584627
$750$	0	0
$751$	24.0000	0.875772	0.437886	−	0.899030i	$-0.355727\pi$
$751$	24.0000	0.875772	0.437886	+	0.899030i	$0.355727\pi$
$752$	−8.00000	−0.291730
$753$	0	0
$754$	0	0
$755$	12.0000	0.436725
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	8.00000	0.290573
$759$	0	0
$760$	0	0
$761$	42.0000	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$761$	42.0000	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	20.0000	0.723575
$765$	0	0
$766$	0	0
$767$	0	0
$768$	0	0
$769$	30.0000	1.08183	0.540914	−	0.841078i	$-0.318079\pi$
$769$	30.0000	1.08183	0.540914	+	0.841078i	$0.318079\pi$
$770$	16.0000	0.576600
$771$	0	0
$772$	−11.0000	−0.395899
$773$	38.0000	1.36677	0.683383	−	0.730061i	$-0.260508\pi$
$773$	38.0000	1.36677	0.683383	+	0.730061i	$0.260508\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	9.00000	0.322666
$779$	0	0
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−12.0000	−0.429119
$783$	0	0
$784$	9.00000	0.321429
$785$	−11.0000	−0.392607
$786$	0	0
$787$	4.00000	0.142585	0.0712923	−	0.997455i	$-0.477288\pi$
$787$	4.00000	0.142585	0.0712923	+	0.997455i	$0.477288\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	4.00000	0.142314
$791$	−4.00000	−0.142224
$792$	0	0
$793$	0	0
$794$	−14.0000	−0.496841
$795$	0	0
$796$	−4.00000	−0.141776
$797$	−30.0000	−1.06265	−0.531327	−	0.847167i	$-0.678307\pi$
$797$	−30.0000	−1.06265	−0.531327	+	0.847167i	$0.678307\pi$
$798$	0	0
$799$	24.0000	0.849059
$800$	−4.00000	−0.141421
$801$	0	0
$802$	3.00000	0.105934
$803$	−44.0000	−1.55273
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	−7.00000	−0.246259
$809$	−39.0000	−1.37117	−0.685583	−	0.727994i	$-0.740453\pi$
$809$	−39.0000	−1.37117	−0.685583	+	0.727994i	$0.740453\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−4.00000	−0.140372
$813$	0	0
$814$	−12.0000	−0.420600
$815$	8.00000	0.280228
$816$	0	0
$817$	0	0
$818$	−31.0000	−1.08389
$819$	0	0
$820$	9.00000	0.314294
$821$	38.0000	1.32621	0.663105	−	0.748527i	$-0.269238\pi$
$821$	38.0000	1.32621	0.663105	+	0.748527i	$0.269238\pi$
$822$	0	0
$823$	12.0000	0.418294	0.209147	−	0.977884i	$-0.432931\pi$
$823$	12.0000	0.418294	0.209147	+	0.977884i	$0.432931\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	16.0000	0.556711
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	7.00000	0.243120	0.121560	−	0.992584i	$-0.461210\pi$
$829$	7.00000	0.243120	0.121560	+	0.992584i	$0.461210\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−27.0000	−0.935495
$834$	0	0
$835$	−12.0000	−0.415277
$836$	0	0
$837$	0	0
$838$	−12.0000	−0.414533
$839$	−40.0000	−1.38095	−0.690477	−	0.723355i	$-0.742599\pi$
$839$	−40.0000	−1.38095	−0.690477	+	0.723355i	$0.742599\pi$
$840$	0	0
$841$	−28.0000	−0.965517
$842$	5.00000	0.172311
$843$	0	0
$844$	20.0000	0.688428
$845$	0	0
$846$	0	0
$847$	−20.0000	−0.687208
$848$	9.00000	0.309061
$849$	0	0
$850$	12.0000	0.411597
$851$	−12.0000	−0.411355
$852$	0	0
$853$	−7.00000	−0.239675	−0.119838	−	0.992793i	$-0.538237\pi$
$853$	−7.00000	−0.239675	−0.119838	+	0.992793i	$0.538237\pi$
$854$	−28.0000	−0.958140
$855$	0	0
$856$	4.00000	0.136717
$857$	17.0000	0.580709	0.290354	−	0.956919i	$-0.406227\pi$
$857$	17.0000	0.580709	0.290354	+	0.956919i	$0.406227\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	8.00000	0.272798
$861$	0	0
$862$	−24.0000	−0.817443
$863$	52.0000	1.77010	0.885050	−	0.465495i	$-0.154124\pi$
$863$	52.0000	1.77010	0.885050	+	0.465495i	$0.154124\pi$
$864$	0	0
$865$	14.0000	0.476014
$866$	−5.00000	−0.169907
$867$	0	0
$868$	16.0000	0.543075
$869$	−16.0000	−0.542763
$870$	0	0
$871$	0	0
$872$	2.00000	0.0677285
$873$	0	0
$874$	0	0
$875$	−36.0000	−1.21702
$876$	0	0
$877$	−15.0000	−0.506514	−0.253257	−	0.967399i	$-0.581502\pi$
$877$	−15.0000	−0.506514	−0.253257	+	0.967399i	$0.581502\pi$
$878$	8.00000	0.269987
$879$	0	0
$880$	−4.00000	−0.134840
$881$	9.00000	0.303218	0.151609	−	0.988441i	$-0.451555\pi$
$881$	9.00000	0.303218	0.151609	+	0.988441i	$0.451555\pi$
$882$	0	0
$883$	16.0000	0.538443	0.269221	−	0.963078i	$-0.413234\pi$
$883$	16.0000	0.538443	0.269221	+	0.963078i	$0.413234\pi$
$884$	0	0
$885$	0	0
$886$	−24.0000	−0.806296
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	0	0
$889$	32.0000	1.07325
$890$	6.00000	0.201120
$891$	0	0
$892$	−12.0000	−0.401790
$893$	0	0
$894$	0	0
$895$	−24.0000	−0.802232
$896$	−4.00000	−0.133631
$897$	0	0
$898$	34.0000	1.13459
$899$	−4.00000	−0.133407
$900$	0	0
$901$	−27.0000	−0.899500
$902$	−36.0000	−1.19867
$903$	0	0
$904$	1.00000	0.0332595
$905$	21.0000	0.698064
$906$	0	0
$907$	−24.0000	−0.796907	−0.398453	−	0.917189i	$-0.630453\pi$
$907$	−24.0000	−0.796907	−0.398453	+	0.917189i	$0.630453\pi$
$908$	24.0000	0.796468
$909$	0	0
$910$	0	0
$911$	12.0000	0.397578	0.198789	−	0.980042i	$-0.436299\pi$
$911$	12.0000	0.397578	0.198789	+	0.980042i	$0.436299\pi$
$912$	0	0
$913$	0	0
$914$	−31.0000	−1.02539
$915$	0	0
$916$	−6.00000	−0.198246
$917$	80.0000	2.64183
$918$	0	0
$919$	−48.0000	−1.58337	−0.791687	−	0.610927i	$-0.790797\pi$
$919$	−48.0000	−1.58337	−0.791687	+	0.610927i	$0.790797\pi$
$920$	−4.00000	−0.131876
$921$	0	0
$922$	−33.0000	−1.08680
$923$	0	0
$924$	0	0
$925$	12.0000	0.394558
$926$	−16.0000	−0.525793
$927$	0	0
$928$	1.00000	0.0328266
$929$	−21.0000	−0.688988	−0.344494	−	0.938789i	$-0.611949\pi$
$929$	−21.0000	−0.688988	−0.344494	+	0.938789i	$0.611949\pi$
$930$	0	0
$931$	0	0
$932$	−10.0000	−0.327561
$933$	0	0
$934$	−20.0000	−0.654420
$935$	12.0000	0.392442
$936$	0	0
$937$	−21.0000	−0.686040	−0.343020	−	0.939328i	$-0.611450\pi$
$937$	−21.0000	−0.686040	−0.343020	+	0.939328i	$0.611450\pi$
$938$	16.0000	0.522419
$939$	0	0
$940$	8.00000	0.260931
$941$	46.0000	1.49956	0.749779	−	0.661689i	$-0.230160\pi$
$941$	46.0000	1.49956	0.749779	+	0.661689i	$0.230160\pi$
$942$	0	0
$943$	−36.0000	−1.17232
$944$	−4.00000	−0.130189
$945$	0	0
$946$	−32.0000	−1.04041
$947$	24.0000	0.779895	0.389948	−	0.920837i	$-0.372493\pi$
$947$	24.0000	0.779895	0.389948	+	0.920837i	$0.372493\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	12.0000	0.388922
$953$	22.0000	0.712650	0.356325	−	0.934362i	$-0.384030\pi$
$953$	22.0000	0.712650	0.356325	+	0.934362i	$0.384030\pi$
$954$	0	0
$955$	−20.0000	−0.647185
$956$	−12.0000	−0.388108
$957$	0	0
$958$	0	0
$959$	36.0000	1.16250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	−3.00000	−0.0966235
$965$	11.0000	0.354103
$966$	0	0
$967$	40.0000	1.28631	0.643157	−	0.765735i	$-0.277624\pi$
$967$	40.0000	1.28631	0.643157	+	0.765735i	$0.277624\pi$
$968$	5.00000	0.160706
$969$	0	0
$970$	2.00000	0.0642161
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	−64.0000	−2.05175
$974$	16.0000	0.512673
$975$	0	0
$976$	7.00000	0.224065
$977$	−45.0000	−1.43968	−0.719839	−	0.694141i	$-0.755784\pi$
$977$	−45.0000	−1.43968	−0.719839	+	0.694141i	$0.755784\pi$
$978$	0	0
$979$	−24.0000	−0.767043
$980$	−9.00000	−0.287494
$981$	0	0
$982$	8.00000	0.255290
$983$	−52.0000	−1.65854	−0.829271	−	0.558846i	$-0.811244\pi$
$983$	−52.0000	−1.65854	−0.829271	+	0.558846i	$0.811244\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	−3.00000	−0.0955395
$987$	0	0
$988$	0	0
$989$	−32.0000	−1.01754
$990$	0	0
$991$	−24.0000	−0.762385	−0.381193	−	0.924496i	$-0.624487\pi$
$991$	−24.0000	−0.762385	−0.381193	+	0.924496i	$0.624487\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	32.0000	1.01498
$995$	4.00000	0.126809
$996$	0	0
$997$	−25.0000	−0.791758	−0.395879	−	0.918303i	$-0.629560\pi$
$997$	−25.0000	−0.791758	−0.395879	+	0.918303i	$0.629560\pi$
$998$	32.0000	1.01294
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.k.1.1		1
3.2	odd	2		338.2.a.c.1.1		1
12.11	even	2		2704.2.a.i.1.1		1
13.4	even	6		234.2.h.c.55.1		2
13.5	odd	4		3042.2.b.e.1351.1		2
13.8	odd	4		3042.2.b.e.1351.2		2
13.10	even	6		234.2.h.c.217.1		2
13.12	even	2		3042.2.a.e.1.1		1
15.14	odd	2		8450.2.a.s.1.1		1
39.2	even	12		338.2.e.b.147.1		4
39.5	even	4		338.2.b.b.337.2		2
39.8	even	4		338.2.b.b.337.1		2
39.11	even	12		338.2.e.b.147.2		4
39.17	odd	6		26.2.c.a.3.1	✓	2
39.20	even	12		338.2.e.b.23.1		4
39.23	odd	6		26.2.c.a.9.1	yes	2
39.29	odd	6		338.2.c.e.191.1		2
39.32	even	12		338.2.e.b.23.2		4
39.35	odd	6		338.2.c.e.315.1		2
39.38	odd	2		338.2.a.e.1.1		1
52.23	odd	6		1872.2.t.k.1153.1		2
52.43	odd	6		1872.2.t.k.289.1		2
156.23	even	6		208.2.i.b.113.1		2
156.47	odd	4		2704.2.f.g.337.2		2
156.83	odd	4		2704.2.f.g.337.1		2
156.95	even	6		208.2.i.b.81.1		2
156.155	even	2		2704.2.a.h.1.1		1
195.17	even	12		650.2.o.c.549.2		4
195.23	even	12		650.2.o.c.399.2		4
195.62	even	12		650.2.o.c.399.1		4
195.134	odd	6		650.2.e.c.601.1		2
195.173	even	12		650.2.o.c.549.1		4
195.179	odd	6		650.2.e.c.451.1		2
195.194	odd	2		8450.2.a.f.1.1		1
273.17	even	6		1274.2.e.m.471.1		2
273.23	odd	6		1274.2.e.n.165.1		2
273.62	even	6		1274.2.g.a.295.1		2
273.95	odd	6		1274.2.e.n.471.1		2
273.101	even	6		1274.2.h.a.373.1		2
273.173	even	6		1274.2.h.a.263.1		2
273.179	odd	6		1274.2.h.b.373.1		2
273.212	odd	6		1274.2.h.b.263.1		2
273.251	even	6		1274.2.g.a.393.1		2
273.257	even	6		1274.2.e.m.165.1		2
312.101	odd	6		832.2.i.e.321.1		2
312.173	odd	6		832.2.i.e.705.1		2
312.179	even	6		832.2.i.f.321.1		2
312.251	even	6		832.2.i.f.705.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
26.2.c.a.3.1	✓	2	39.17	odd	6
26.2.c.a.9.1	yes	2	39.23	odd	6
208.2.i.b.81.1		2	156.95	even	6
208.2.i.b.113.1		2	156.23	even	6
234.2.h.c.55.1		2	13.4	even	6
234.2.h.c.217.1		2	13.10	even	6
338.2.a.c.1.1		1	3.2	odd	2
338.2.a.e.1.1		1	39.38	odd	2
338.2.b.b.337.1		2	39.8	even	4
338.2.b.b.337.2		2	39.5	even	4
338.2.c.e.191.1		2	39.29	odd	6
338.2.c.e.315.1		2	39.35	odd	6
338.2.e.b.23.1		4	39.20	even	12
338.2.e.b.23.2		4	39.32	even	12
338.2.e.b.147.1		4	39.2	even	12
338.2.e.b.147.2		4	39.11	even	12
650.2.e.c.451.1		2	195.179	odd	6
650.2.e.c.601.1		2	195.134	odd	6
650.2.o.c.399.1		4	195.62	even	12
650.2.o.c.399.2		4	195.23	even	12
650.2.o.c.549.1		4	195.173	even	12
650.2.o.c.549.2		4	195.17	even	12
832.2.i.e.321.1		2	312.101	odd	6
832.2.i.e.705.1		2	312.173	odd	6
832.2.i.f.321.1		2	312.179	even	6
832.2.i.f.705.1		2	312.251	even	6
1274.2.e.m.165.1		2	273.257	even	6
1274.2.e.m.471.1		2	273.17	even	6
1274.2.e.n.165.1		2	273.23	odd	6
1274.2.e.n.471.1		2	273.95	odd	6
1274.2.g.a.295.1		2	273.62	even	6
1274.2.g.a.393.1		2	273.251	even	6
1274.2.h.a.263.1		2	273.173	even	6
1274.2.h.a.373.1		2	273.101	even	6
1274.2.h.b.263.1		2	273.212	odd	6
1274.2.h.b.373.1		2	273.179	odd	6
1872.2.t.k.289.1		2	52.43	odd	6
1872.2.t.k.1153.1		2	52.23	odd	6
2704.2.a.h.1.1		1	156.155	even	2
2704.2.a.i.1.1		1	12.11	even	2
2704.2.f.g.337.1		2	156.83	odd	4
2704.2.f.g.337.2		2	156.47	odd	4
3042.2.a.e.1.1		1	13.12	even	2
3042.2.a.k.1.1		1	1.1	even	1	trivial
3042.2.b.e.1351.1		2	13.5	odd	4
3042.2.b.e.1351.2		2	13.8	odd	4
8450.2.a.f.1.1		1	195.194	odd	2
8450.2.a.s.1.1		1	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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