LMFDB - Embedded newform 3042.2.a.h.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:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +3.00000 q^{5} -2.00000 q^{7} -1.00000 q^{8} -3.00000 q^{10} +6.00000 q^{11} +2.00000 q^{14} +1.00000 q^{16} +3.00000 q^{17} -2.00000 q^{19} +3.00000 q^{20} -6.00000 q^{22} +6.00000 q^{23} +4.00000 q^{25} -2.00000 q^{28} -3.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} -3.00000 q^{34} -6.00000 q^{35} +7.00000 q^{37} +2.00000 q^{38} -3.00000 q^{40} -3.00000 q^{41} -10.0000 q^{43} +6.00000 q^{44} -6.00000 q^{46} +6.00000 q^{47} -3.00000 q^{49} -4.00000 q^{50} -3.00000 q^{53} +18.0000 q^{55} +2.00000 q^{56} +3.00000 q^{58} -7.00000 q^{61} -4.00000 q^{62} +1.00000 q^{64} +10.0000 q^{67} +3.00000 q^{68} +6.00000 q^{70} +6.00000 q^{71} +13.0000 q^{73} -7.00000 q^{74} -2.00000 q^{76} -12.0000 q^{77} -4.00000 q^{79} +3.00000 q^{80} +3.00000 q^{82} -6.00000 q^{83} +9.00000 q^{85} +10.0000 q^{86} -6.00000 q^{88} +18.0000 q^{89} +6.00000 q^{92} -6.00000 q^{94} -6.00000 q^{95} -14.0000 q^{97} +3.00000 q^{98} +O(q^{100})$

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
$5$	3.00000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	3.00000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−2.00000	−0.755929	−0.377964	−	0.925820i	$-0.623376\pi$
$7$	−2.00000	−0.755929	−0.377964	+	0.925820i	$0.623376\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−3.00000	−0.948683
$11$	6.00000	1.80907	0.904534	−	0.426401i	$-0.140219\pi$
$11$	6.00000	1.80907	0.904534	+	0.426401i	$0.140219\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.00000	0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	3.00000	0.727607	0.363803	−	0.931476i	$-0.381478\pi$
$17$	3.00000	0.727607	0.363803	+	0.931476i	$0.381478\pi$
$18$	0	0
$19$	−2.00000	−0.458831	−0.229416	−	0.973329i	$-0.573682\pi$
$19$	−2.00000	−0.458831	−0.229416	+	0.973329i	$0.573682\pi$
$20$	3.00000	0.670820
$21$	0	0
$22$	−6.00000	−1.27920
$23$	6.00000	1.25109	0.625543	−	0.780189i	$-0.284877\pi$
$23$	6.00000	1.25109	0.625543	+	0.780189i	$0.284877\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	−2.00000	−0.377964
$29$	−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
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−3.00000	−0.514496
$35$	−6.00000	−1.01419
$36$	0	0
$37$	7.00000	1.15079	0.575396	−	0.817875i	$-0.304848\pi$
$37$	7.00000	1.15079	0.575396	+	0.817875i	$0.304848\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−3.00000	−0.474342
$41$	−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
$43$	−10.0000	−1.52499	−0.762493	−	0.646997i	$-0.776025\pi$
$43$	−10.0000	−1.52499	−0.762493	+	0.646997i	$0.776025\pi$
$44$	6.00000	0.904534
$45$	0	0
$46$	−6.00000	−0.884652
$47$	6.00000	0.875190	0.437595	−	0.899172i	$-0.355830\pi$
$47$	6.00000	0.875190	0.437595	+	0.899172i	$0.355830\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	−4.00000	−0.565685
$51$	0	0
$52$	0	0
$53$	−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
$55$	18.0000	2.42712
$56$	2.00000	0.267261
$57$	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
$61$	−7.00000	−0.896258	−0.448129	−	0.893969i	$-0.647910\pi$
$61$	−7.00000	−0.896258	−0.448129	+	0.893969i	$0.647910\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	10.0000	1.22169	0.610847	−	0.791748i	$-0.290829\pi$
$67$	10.0000	1.22169	0.610847	+	0.791748i	$0.290829\pi$
$68$	3.00000	0.363803
$69$	0	0
$70$	6.00000	0.717137
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	−7.00000	−0.813733
$75$	0	0
$76$	−2.00000	−0.229416
$77$	−12.0000	−1.36753
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	3.00000	0.335410
$81$	0	0
$82$	3.00000	0.331295
$83$	−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
$85$	9.00000	0.976187
$86$	10.0000	1.07833
$87$	0	0
$88$	−6.00000	−0.639602
$89$	18.0000	1.90800	0.953998	−	0.299813i	$-0.0969242\pi$
$89$	18.0000	1.90800	0.953998	+	0.299813i	$0.0969242\pi$
$90$	0	0
$91$	0	0
$92$	6.00000	0.625543
$93$	0	0
$94$	−6.00000	−0.618853
$95$	−6.00000	−0.615587
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	4.00000	0.400000
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	14.0000	1.37946	0.689730	−	0.724066i	$-0.257729\pi$
$103$	14.0000	1.37946	0.689730	+	0.724066i	$0.257729\pi$
$104$	0	0
$105$	0	0
$106$	3.00000	0.291386
$107$	6.00000	0.580042	0.290021	−	0.957020i	$-0.406338\pi$
$107$	6.00000	0.580042	0.290021	+	0.957020i	$0.406338\pi$
$108$	0	0
$109$	−14.0000	−1.34096	−0.670478	−	0.741929i	$-0.733911\pi$
$109$	−14.0000	−1.34096	−0.670478	+	0.741929i	$0.733911\pi$
$110$	−18.0000	−1.71623
$111$	0	0
$112$	−2.00000	−0.188982
$113$	3.00000	0.282216	0.141108	−	0.989994i	$-0.454933\pi$
$113$	3.00000	0.282216	0.141108	+	0.989994i	$0.454933\pi$
$114$	0	0
$115$	18.0000	1.67851
$116$	−3.00000	−0.278543
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	25.0000	2.27273
$122$	7.00000	0.633750
$123$	0	0
$124$	4.00000	0.359211
$125$	−3.00000	−0.268328
$126$	0	0
$127$	−4.00000	−0.354943	−0.177471	−	0.984126i	$-0.556792\pi$
$127$	−4.00000	−0.354943	−0.177471	+	0.984126i	$0.556792\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	0	0		−	1.00000i	$-0.5\pi$
$131$	0	0			1.00000i	$0.5\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−10.0000	−0.863868
$135$	0	0
$136$	−3.00000	−0.257248
$137$	9.00000	0.768922	0.384461	−	0.923141i	$-0.374387\pi$
$137$	9.00000	0.768922	0.384461	+	0.923141i	$0.374387\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−6.00000	−0.507093
$141$	0	0
$142$	−6.00000	−0.503509
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	−13.0000	−1.07589
$147$	0	0
$148$	7.00000	0.575396
$149$	−9.00000	−0.737309	−0.368654	−	0.929567i	$-0.620181\pi$
$149$	−9.00000	−0.737309	−0.368654	+	0.929567i	$0.620181\pi$
$150$	0	0
$151$	10.0000	0.813788	0.406894	−	0.913475i	$-0.366612\pi$
$151$	10.0000	0.813788	0.406894	+	0.913475i	$0.366612\pi$
$152$	2.00000	0.162221
$153$	0	0
$154$	12.0000	0.966988
$155$	12.0000	0.963863
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	−3.00000	−0.237171
$161$	−12.0000	−0.945732
$162$	0	0
$163$	4.00000	0.313304	0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000	0.313304	0.156652	+	0.987654i	$0.449930\pi$
$164$	−3.00000	−0.234261
$165$	0	0
$166$	6.00000	0.465690
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	0	0
$170$	−9.00000	−0.690268
$171$	0	0
$172$	−10.0000	−0.762493
$173$	6.00000	0.456172	0.228086	−	0.973641i	$-0.426753\pi$
$173$	6.00000	0.456172	0.228086	+	0.973641i	$0.426753\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	6.00000	0.452267
$177$	0	0
$178$	−18.0000	−1.34916
$179$	−6.00000	−0.448461	−0.224231	−	0.974536i	$-0.571987\pi$
$179$	−6.00000	−0.448461	−0.224231	+	0.974536i	$0.571987\pi$
$180$	0	0
$181$	−7.00000	−0.520306	−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000	−0.520306	−0.260153	+	0.965567i	$0.583773\pi$
$182$	0	0
$183$	0	0
$184$	−6.00000	−0.442326
$185$	21.0000	1.54395
$186$	0	0
$187$	18.0000	1.31629
$188$	6.00000	0.437595
$189$	0	0
$190$	6.00000	0.435286
$191$	12.0000	0.868290	0.434145	−	0.900843i	$-0.357051\pi$
$191$	12.0000	0.868290	0.434145	+	0.900843i	$0.357051\pi$
$192$	0	0
$193$	−23.0000	−1.65558	−0.827788	−	0.561041i	$-0.810401\pi$
$193$	−23.0000	−1.65558	−0.827788	+	0.561041i	$0.810401\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	−3.00000	−0.214286
$197$	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
$199$	−10.0000	−0.708881	−0.354441	−	0.935079i	$-0.615329\pi$
$199$	−10.0000	−0.708881	−0.354441	+	0.935079i	$0.615329\pi$
$200$	−4.00000	−0.282843
$201$	0	0
$202$	15.0000	1.05540
$203$	6.00000	0.421117
$204$	0	0
$205$	−9.00000	−0.628587
$206$	−14.0000	−0.975426
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	−16.0000	−1.10149	−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000	−1.10149	−0.550743	+	0.834675i	$0.685655\pi$
$212$	−3.00000	−0.206041
$213$	0	0
$214$	−6.00000	−0.410152
$215$	−30.0000	−2.04598
$216$	0	0
$217$	−8.00000	−0.543075
$218$	14.0000	0.948200
$219$	0	0
$220$	18.0000	1.21356
$221$	0	0
$222$	0	0
$223$	−8.00000	−0.535720	−0.267860	−	0.963458i	$-0.586316\pi$
$223$	−8.00000	−0.535720	−0.267860	+	0.963458i	$0.586316\pi$
$224$	2.00000	0.133631
$225$	0	0
$226$	−3.00000	−0.199557
$227$	18.0000	1.19470	0.597351	−	0.801980i	$-0.296220\pi$
$227$	18.0000	1.19470	0.597351	+	0.801980i	$0.296220\pi$
$228$	0	0
$229$	22.0000	1.45380	0.726900	−	0.686743i	$-0.240960\pi$
$229$	22.0000	1.45380	0.726900	+	0.686743i	$0.240960\pi$
$230$	−18.0000	−1.18688
$231$	0	0
$232$	3.00000	0.196960
$233$	6.00000	0.393073	0.196537	−	0.980497i	$-0.437031\pi$
$233$	6.00000	0.393073	0.196537	+	0.980497i	$0.437031\pi$
$234$	0	0
$235$	18.0000	1.17419
$236$	0	0
$237$	0	0
$238$	6.00000	0.388922
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	0	0
$241$	1.00000	0.0644157	0.0322078	−	0.999481i	$-0.489746\pi$
$241$	1.00000	0.0644157	0.0322078	+	0.999481i	$0.489746\pi$
$242$	−25.0000	−1.60706
$243$	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
$250$	3.00000	0.189737
$251$	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
$253$	36.0000	2.26330
$254$	4.00000	0.250982
$255$	0	0
$256$	1.00000	0.0625000
$257$	3.00000	0.187135	0.0935674	−	0.995613i	$-0.470173\pi$
$257$	3.00000	0.187135	0.0935674	+	0.995613i	$0.470173\pi$
$258$	0	0
$259$	−14.0000	−0.869918
$260$	0	0
$261$	0	0
$262$	0	0
$263$	6.00000	0.369976	0.184988	−	0.982741i	$-0.440775\pi$
$263$	6.00000	0.369976	0.184988	+	0.982741i	$0.440775\pi$
$264$	0	0
$265$	−9.00000	−0.552866
$266$	−4.00000	−0.245256
$267$	0	0
$268$	10.0000	0.610847
$269$	−18.0000	−1.09748	−0.548740	−	0.835993i	$-0.684892\pi$
$269$	−18.0000	−1.09748	−0.548740	+	0.835993i	$0.684892\pi$
$270$	0	0
$271$	16.0000	0.971931	0.485965	−	0.873978i	$-0.338468\pi$
$271$	16.0000	0.971931	0.485965	+	0.873978i	$0.338468\pi$
$272$	3.00000	0.181902
$273$	0	0
$274$	−9.00000	−0.543710
$275$	24.0000	1.44725
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	4.00000	0.239904
$279$	0	0
$280$	6.00000	0.358569
$281$	9.00000	0.536895	0.268447	−	0.963294i	$-0.413489\pi$
$281$	9.00000	0.536895	0.268447	+	0.963294i	$0.413489\pi$
$282$	0	0
$283$	14.0000	0.832214	0.416107	−	0.909316i	$-0.363394\pi$
$283$	14.0000	0.832214	0.416107	+	0.909316i	$0.363394\pi$
$284$	6.00000	0.356034
$285$	0	0
$286$	0	0
$287$	6.00000	0.354169
$288$	0	0
$289$	−8.00000	−0.470588
$290$	9.00000	0.528498
$291$	0	0
$292$	13.0000	0.760767
$293$	−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
$296$	−7.00000	−0.406867
$297$	0	0
$298$	9.00000	0.521356
$299$	0	0
$300$	0	0
$301$	20.0000	1.15278
$302$	−10.0000	−0.575435
$303$	0	0
$304$	−2.00000	−0.114708
$305$	−21.0000	−1.20246
$306$	0	0
$307$	10.0000	0.570730	0.285365	−	0.958419i	$-0.407885\pi$
$307$	10.0000	0.570730	0.285365	+	0.958419i	$0.407885\pi$
$308$	−12.0000	−0.683763
$309$	0	0
$310$	−12.0000	−0.681554
$311$	30.0000	1.70114	0.850572	−	0.525859i	$-0.176256\pi$
$311$	30.0000	1.70114	0.850572	+	0.525859i	$0.176256\pi$
$312$	0	0
$313$	−10.0000	−0.565233	−0.282617	−	0.959233i	$-0.591202\pi$
$313$	−10.0000	−0.565233	−0.282617	+	0.959233i	$0.591202\pi$
$314$	−5.00000	−0.282166
$315$	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
$319$	−18.0000	−1.00781
$320$	3.00000	0.167705
$321$	0	0
$322$	12.0000	0.668734
$323$	−6.00000	−0.333849
$324$	0	0
$325$	0	0
$326$	−4.00000	−0.221540
$327$	0	0
$328$	3.00000	0.165647
$329$	−12.0000	−0.661581
$330$	0	0
$331$	4.00000	0.219860	0.109930	−	0.993939i	$-0.464937\pi$
$331$	4.00000	0.219860	0.109930	+	0.993939i	$0.464937\pi$
$332$	−6.00000	−0.329293
$333$	0	0
$334$	0	0
$335$	30.0000	1.63908
$336$	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
$340$	9.00000	0.488094
$341$	24.0000	1.29967
$342$	0	0
$343$	20.0000	1.07990
$344$	10.0000	0.539164
$345$	0	0
$346$	−6.00000	−0.322562
$347$	30.0000	1.61048	0.805242	−	0.592946i	$-0.202035\pi$
$347$	30.0000	1.61048	0.805242	+	0.592946i	$0.202035\pi$
$348$	0	0
$349$	10.0000	0.535288	0.267644	−	0.963518i	$-0.413755\pi$
$349$	10.0000	0.535288	0.267644	+	0.963518i	$0.413755\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	−6.00000	−0.319801
$353$	−15.0000	−0.798369	−0.399185	−	0.916871i	$-0.630707\pi$
$353$	−15.0000	−0.798369	−0.399185	+	0.916871i	$0.630707\pi$
$354$	0	0
$355$	18.0000	0.955341
$356$	18.0000	0.953998
$357$	0	0
$358$	6.00000	0.317110
$359$	6.00000	0.316668	0.158334	−	0.987386i	$-0.449388\pi$
$359$	6.00000	0.316668	0.158334	+	0.987386i	$0.449388\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	7.00000	0.367912
$363$	0	0
$364$	0	0
$365$	39.0000	2.04135
$366$	0	0
$367$	2.00000	0.104399	0.0521996	−	0.998637i	$-0.483377\pi$
$367$	2.00000	0.104399	0.0521996	+	0.998637i	$0.483377\pi$
$368$	6.00000	0.312772
$369$	0	0
$370$	−21.0000	−1.09174
$371$	6.00000	0.311504
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	−18.0000	−0.930758
$375$	0	0
$376$	−6.00000	−0.309426
$377$	0	0
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−6.00000	−0.307794
$381$	0	0
$382$	−12.0000	−0.613973
$383$	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
$385$	−36.0000	−1.83473
$386$	23.0000	1.17067
$387$	0	0
$388$	−14.0000	−0.710742
$389$	−39.0000	−1.97738	−0.988689	−	0.149979i	$-0.952080\pi$
$389$	−39.0000	−1.97738	−0.988689	+	0.149979i	$0.952080\pi$
$390$	0	0
$391$	18.0000	0.910299
$392$	3.00000	0.151523
$393$	0	0
$394$	−6.00000	−0.302276
$395$	−12.0000	−0.603786
$396$	0	0
$397$	−14.0000	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−14.0000	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	10.0000	0.501255
$399$	0	0
$400$	4.00000	0.200000
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	0	0
$404$	−15.0000	−0.746278
$405$	0	0
$406$	−6.00000	−0.297775
$407$	42.0000	2.08186
$408$	0	0
$409$	1.00000	0.0494468	0.0247234	−	0.999694i	$-0.492129\pi$
$409$	1.00000	0.0494468	0.0247234	+	0.999694i	$0.492129\pi$
$410$	9.00000	0.444478
$411$	0	0
$412$	14.0000	0.689730
$413$	0	0
$414$	0	0
$415$	−18.0000	−0.883585
$416$	0	0
$417$	0	0
$418$	12.0000	0.586939
$419$	−24.0000	−1.17248	−0.586238	−	0.810139i	$-0.699392\pi$
$419$	−24.0000	−1.17248	−0.586238	+	0.810139i	$0.699392\pi$
$420$	0	0
$421$	−29.0000	−1.41337	−0.706687	−	0.707527i	$-0.749811\pi$
$421$	−29.0000	−1.41337	−0.706687	+	0.707527i	$0.749811\pi$
$422$	16.0000	0.778868
$423$	0	0
$424$	3.00000	0.145693
$425$	12.0000	0.582086
$426$	0	0
$427$	14.0000	0.677507
$428$	6.00000	0.290021
$429$	0	0
$430$	30.0000	1.44673
$431$	−6.00000	−0.289010	−0.144505	−	0.989504i	$-0.546159\pi$
$431$	−6.00000	−0.289010	−0.144505	+	0.989504i	$0.546159\pi$
$432$	0	0
$433$	−13.0000	−0.624740	−0.312370	−	0.949960i	$-0.601123\pi$
$433$	−13.0000	−0.624740	−0.312370	+	0.949960i	$0.601123\pi$
$434$	8.00000	0.384012
$435$	0	0
$436$	−14.0000	−0.670478
$437$	−12.0000	−0.574038
$438$	0	0
$439$	14.0000	0.668184	0.334092	−	0.942541i	$-0.391570\pi$
$439$	14.0000	0.668184	0.334092	+	0.942541i	$0.391570\pi$
$440$	−18.0000	−0.858116
$441$	0	0
$442$	0	0
$443$	36.0000	1.71041	0.855206	−	0.518289i	$-0.173431\pi$
$443$	36.0000	1.71041	0.855206	+	0.518289i	$0.173431\pi$
$444$	0	0
$445$	54.0000	2.55985
$446$	8.00000	0.378811
$447$	0	0
$448$	−2.00000	−0.0944911
$449$	18.0000	0.849473	0.424736	−	0.905317i	$-0.360367\pi$
$449$	18.0000	0.849473	0.424736	+	0.905317i	$0.360367\pi$
$450$	0	0
$451$	−18.0000	−0.847587
$452$	3.00000	0.141108
$453$	0	0
$454$	−18.0000	−0.844782
$455$	0	0
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	−22.0000	−1.02799
$459$	0	0
$460$	18.0000	0.839254
$461$	15.0000	0.698620	0.349310	−	0.937007i	$-0.386416\pi$
$461$	15.0000	0.698620	0.349310	+	0.937007i	$0.386416\pi$
$462$	0	0
$463$	−38.0000	−1.76601	−0.883005	−	0.469364i	$-0.844483\pi$
$463$	−38.0000	−1.76601	−0.883005	+	0.469364i	$0.844483\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−6.00000	−0.277945
$467$	18.0000	0.832941	0.416470	−	0.909149i	$-0.363267\pi$
$467$	18.0000	0.832941	0.416470	+	0.909149i	$0.363267\pi$
$468$	0	0
$469$	−20.0000	−0.923514
$470$	−18.0000	−0.830278
$471$	0	0
$472$	0	0
$473$	−60.0000	−2.75880
$474$	0	0
$475$	−8.00000	−0.367065
$476$	−6.00000	−0.275010
$477$	0	0
$478$	−6.00000	−0.274434
$479$	0	0		−	1.00000i	$-0.5\pi$
$479$	0	0			1.00000i	$0.5\pi$
$480$	0	0
$481$	0	0
$482$	−1.00000	−0.0455488
$483$	0	0
$484$	25.0000	1.13636
$485$	−42.0000	−1.90712
$486$	0	0
$487$	−2.00000	−0.0906287	−0.0453143	−	0.998973i	$-0.514429\pi$
$487$	−2.00000	−0.0906287	−0.0453143	+	0.998973i	$0.514429\pi$
$488$	7.00000	0.316875
$489$	0	0
$490$	9.00000	0.406579
$491$	18.0000	0.812329	0.406164	−	0.913800i	$-0.366866\pi$
$491$	18.0000	0.812329	0.406164	+	0.913800i	$0.366866\pi$
$492$	0	0
$493$	−9.00000	−0.405340
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	−12.0000	−0.538274
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	−3.00000	−0.134164
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−6.00000	−0.267527	−0.133763	−	0.991013i	$-0.542706\pi$
$503$	−6.00000	−0.267527	−0.133763	+	0.991013i	$0.542706\pi$
$504$	0	0
$505$	−45.0000	−2.00247
$506$	−36.0000	−1.60040
$507$	0	0
$508$	−4.00000	−0.177471
$509$	3.00000	0.132973	0.0664863	−	0.997787i	$-0.478821\pi$
$509$	3.00000	0.132973	0.0664863	+	0.997787i	$0.478821\pi$
$510$	0	0
$511$	−26.0000	−1.15017
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.00000	−0.132324
$515$	42.0000	1.85074
$516$	0	0
$517$	36.0000	1.58328
$518$	14.0000	0.615125
$519$	0	0
$520$	0	0
$521$	−33.0000	−1.44576	−0.722878	−	0.690976i	$-0.757181\pi$
$521$	−33.0000	−1.44576	−0.722878	+	0.690976i	$0.757181\pi$
$522$	0	0
$523$	−34.0000	−1.48672	−0.743358	−	0.668894i	$-0.766768\pi$
$523$	−34.0000	−1.48672	−0.743358	+	0.668894i	$0.766768\pi$
$524$	0	0
$525$	0	0
$526$	−6.00000	−0.261612
$527$	12.0000	0.522728
$528$	0	0
$529$	13.0000	0.565217
$530$	9.00000	0.390935
$531$	0	0
$532$	4.00000	0.173422
$533$	0	0
$534$	0	0
$535$	18.0000	0.778208
$536$	−10.0000	−0.431934
$537$	0	0
$538$	18.0000	0.776035
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−29.0000	−1.24681	−0.623404	−	0.781900i	$-0.714251\pi$
$541$	−29.0000	−1.24681	−0.623404	+	0.781900i	$0.714251\pi$
$542$	−16.0000	−0.687259
$543$	0	0
$544$	−3.00000	−0.128624
$545$	−42.0000	−1.79908
$546$	0	0
$547$	−34.0000	−1.45374	−0.726868	−	0.686778i	$-0.759025\pi$
$547$	−34.0000	−1.45374	−0.726868	+	0.686778i	$0.759025\pi$
$548$	9.00000	0.384461
$549$	0	0
$550$	−24.0000	−1.02336
$551$	6.00000	0.255609
$552$	0	0
$553$	8.00000	0.340195
$554$	−17.0000	−0.722261
$555$	0	0
$556$	−4.00000	−0.169638
$557$	3.00000	0.127114	0.0635570	−	0.997978i	$-0.479756\pi$
$557$	3.00000	0.127114	0.0635570	+	0.997978i	$0.479756\pi$
$558$	0	0
$559$	0	0
$560$	−6.00000	−0.253546
$561$	0	0
$562$	−9.00000	−0.379642
$563$	0	0		−	1.00000i	$-0.5\pi$
$563$	0	0			1.00000i	$0.5\pi$
$564$	0	0
$565$	9.00000	0.378633
$566$	−14.0000	−0.588464
$567$	0	0
$568$	−6.00000	−0.251754
$569$	−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
$571$	−22.0000	−0.920671	−0.460336	−	0.887745i	$-0.652271\pi$
$571$	−22.0000	−0.920671	−0.460336	+	0.887745i	$0.652271\pi$
$572$	0	0
$573$	0	0
$574$	−6.00000	−0.250435
$575$	24.0000	1.00087
$576$	0	0
$577$	−11.0000	−0.457936	−0.228968	−	0.973434i	$-0.573535\pi$
$577$	−11.0000	−0.457936	−0.228968	+	0.973434i	$0.573535\pi$
$578$	8.00000	0.332756
$579$	0	0
$580$	−9.00000	−0.373705
$581$	12.0000	0.497844
$582$	0	0
$583$	−18.0000	−0.745484
$584$	−13.0000	−0.537944
$585$	0	0
$586$	21.0000	0.867502
$587$	0	0		−	1.00000i	$-0.5\pi$
$587$	0	0			1.00000i	$0.5\pi$
$588$	0	0
$589$	−8.00000	−0.329634
$590$	0	0
$591$	0	0
$592$	7.00000	0.287698
$593$	9.00000	0.369586	0.184793	−	0.982777i	$-0.440839\pi$
$593$	9.00000	0.369586	0.184793	+	0.982777i	$0.440839\pi$
$594$	0	0
$595$	−18.0000	−0.737928
$596$	−9.00000	−0.368654
$597$	0	0
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	0	0
$601$	−37.0000	−1.50926	−0.754631	−	0.656150i	$-0.772184\pi$
$601$	−37.0000	−1.50926	−0.754631	+	0.656150i	$0.772184\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	10.0000	0.406894
$605$	75.0000	3.04918
$606$	0	0
$607$	32.0000	1.29884	0.649420	−	0.760430i	$-0.275012\pi$
$607$	32.0000	1.29884	0.649420	+	0.760430i	$0.275012\pi$
$608$	2.00000	0.0811107
$609$	0	0
$610$	21.0000	0.850265
$611$	0	0
$612$	0	0
$613$	31.0000	1.25208	0.626039	−	0.779792i	$-0.284675\pi$
$613$	31.0000	1.25208	0.626039	+	0.779792i	$0.284675\pi$
$614$	−10.0000	−0.403567
$615$	0	0
$616$	12.0000	0.483494
$617$	−15.0000	−0.603877	−0.301939	−	0.953327i	$-0.597634\pi$
$617$	−15.0000	−0.603877	−0.301939	+	0.953327i	$0.597634\pi$
$618$	0	0
$619$	−8.00000	−0.321547	−0.160774	−	0.986991i	$-0.551399\pi$
$619$	−8.00000	−0.321547	−0.160774	+	0.986991i	$0.551399\pi$
$620$	12.0000	0.481932
$621$	0	0
$622$	−30.0000	−1.20289
$623$	−36.0000	−1.44231
$624$	0	0
$625$	−29.0000	−1.16000
$626$	10.0000	0.399680
$627$	0	0
$628$	5.00000	0.199522
$629$	21.0000	0.837325
$630$	0	0
$631$	−20.0000	−0.796187	−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000	−0.796187	−0.398094	+	0.917345i	$0.630328\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	−3.00000	−0.119145
$635$	−12.0000	−0.476205
$636$	0	0
$637$	0	0
$638$	18.0000	0.712627
$639$	0	0
$640$	−3.00000	−0.118585
$641$	3.00000	0.118493	0.0592464	−	0.998243i	$-0.481130\pi$
$641$	3.00000	0.118493	0.0592464	+	0.998243i	$0.481130\pi$
$642$	0	0
$643$	16.0000	0.630978	0.315489	−	0.948929i	$-0.397831\pi$
$643$	16.0000	0.630978	0.315489	+	0.948929i	$0.397831\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	6.00000	0.236067
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0	0
$652$	4.00000	0.156652
$653$	−42.0000	−1.64359	−0.821794	−	0.569785i	$-0.807026\pi$
$653$	−42.0000	−1.64359	−0.821794	+	0.569785i	$0.807026\pi$
$654$	0	0
$655$	0	0
$656$	−3.00000	−0.117130
$657$	0	0
$658$	12.0000	0.467809
$659$	−24.0000	−0.934907	−0.467454	−	0.884018i	$-0.654829\pi$
$659$	−24.0000	−0.934907	−0.467454	+	0.884018i	$0.654829\pi$
$660$	0	0
$661$	−5.00000	−0.194477	−0.0972387	−	0.995261i	$-0.531001\pi$
$661$	−5.00000	−0.194477	−0.0972387	+	0.995261i	$0.531001\pi$
$662$	−4.00000	−0.155464
$663$	0	0
$664$	6.00000	0.232845
$665$	12.0000	0.465340
$666$	0	0
$667$	−18.0000	−0.696963
$668$	0	0
$669$	0	0
$670$	−30.0000	−1.15900
$671$	−42.0000	−1.62139
$672$	0	0
$673$	−13.0000	−0.501113	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	−13.0000	−0.501113	−0.250557	+	0.968102i	$0.580614\pi$
$674$	−23.0000	−0.885927
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	28.0000	1.07454
$680$	−9.00000	−0.345134
$681$	0	0
$682$	−24.0000	−0.919007
$683$	−48.0000	−1.83667	−0.918334	−	0.395805i	$-0.870466\pi$
$683$	−48.0000	−1.83667	−0.918334	+	0.395805i	$0.870466\pi$
$684$	0	0
$685$	27.0000	1.03162
$686$	−20.0000	−0.763604
$687$	0	0
$688$	−10.0000	−0.381246
$689$	0	0
$690$	0	0
$691$	−26.0000	−0.989087	−0.494543	−	0.869153i	$-0.664665\pi$
$691$	−26.0000	−0.989087	−0.494543	+	0.869153i	$0.664665\pi$
$692$	6.00000	0.228086
$693$	0	0
$694$	−30.0000	−1.13878
$695$	−12.0000	−0.455186
$696$	0	0
$697$	−9.00000	−0.340899
$698$	−10.0000	−0.378506
$699$	0	0
$700$	−8.00000	−0.302372
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	0	0
$703$	−14.0000	−0.528020
$704$	6.00000	0.226134
$705$	0	0
$706$	15.0000	0.564532
$707$	30.0000	1.12827
$708$	0	0
$709$	−5.00000	−0.187779	−0.0938895	−	0.995583i	$-0.529930\pi$
$709$	−5.00000	−0.187779	−0.0938895	+	0.995583i	$0.529930\pi$
$710$	−18.0000	−0.675528
$711$	0	0
$712$	−18.0000	−0.674579
$713$	24.0000	0.898807
$714$	0	0
$715$	0	0
$716$	−6.00000	−0.224231
$717$	0	0
$718$	−6.00000	−0.223918
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−28.0000	−1.04277
$722$	15.0000	0.558242
$723$	0	0
$724$	−7.00000	−0.260153
$725$	−12.0000	−0.445669
$726$	0	0
$727$	14.0000	0.519231	0.259616	−	0.965712i	$-0.416404\pi$
$727$	14.0000	0.519231	0.259616	+	0.965712i	$0.416404\pi$
$728$	0	0
$729$	0	0
$730$	−39.0000	−1.44345
$731$	−30.0000	−1.10959
$732$	0	0
$733$	31.0000	1.14501	0.572506	−	0.819901i	$-0.305971\pi$
$733$	31.0000	1.14501	0.572506	+	0.819901i	$0.305971\pi$
$734$	−2.00000	−0.0738213
$735$	0	0
$736$	−6.00000	−0.221163
$737$	60.0000	2.21013
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	21.0000	0.771975
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−36.0000	−1.32071	−0.660356	−	0.750953i	$-0.729595\pi$
$743$	−36.0000	−1.32071	−0.660356	+	0.750953i	$0.729595\pi$
$744$	0	0
$745$	−27.0000	−0.989203
$746$	−29.0000	−1.06177
$747$	0	0
$748$	18.0000	0.658145
$749$	−12.0000	−0.438470
$750$	0	0
$751$	14.0000	0.510867	0.255434	−	0.966827i	$-0.417782\pi$
$751$	14.0000	0.510867	0.255434	+	0.966827i	$0.417782\pi$
$752$	6.00000	0.218797
$753$	0	0
$754$	0	0
$755$	30.0000	1.09181
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	6.00000	0.217643
$761$	−30.0000	−1.08750	−0.543750	−	0.839248i	$-0.682996\pi$
$761$	−30.0000	−1.08750	−0.543750	+	0.839248i	$0.682996\pi$
$762$	0	0
$763$	28.0000	1.01367
$764$	12.0000	0.434145
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	36.0000	1.29735
$771$	0	0
$772$	−23.0000	−0.827788
$773$	−30.0000	−1.07903	−0.539513	−	0.841978i	$-0.681391\pi$
$773$	−30.0000	−1.07903	−0.539513	+	0.841978i	$0.681391\pi$
$774$	0	0
$775$	16.0000	0.574737
$776$	14.0000	0.502571
$777$	0	0
$778$	39.0000	1.39822
$779$	6.00000	0.214972
$780$	0	0
$781$	36.0000	1.28818
$782$	−18.0000	−0.643679
$783$	0	0
$784$	−3.00000	−0.107143
$785$	15.0000	0.535373
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	12.0000	0.426941
$791$	−6.00000	−0.213335
$792$	0	0
$793$	0	0
$794$	14.0000	0.496841
$795$	0	0
$796$	−10.0000	−0.354441
$797$	−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
$799$	18.0000	0.636794
$800$	−4.00000	−0.141421
$801$	0	0
$802$	3.00000	0.105934
$803$	78.0000	2.75256
$804$	0	0
$805$	−36.0000	−1.26883
$806$	0	0
$807$	0	0
$808$	15.0000	0.527698
$809$	51.0000	1.79306	0.896532	−	0.442978i	$-0.146078\pi$
$809$	51.0000	1.79306	0.896532	+	0.442978i	$0.146078\pi$
$810$	0	0
$811$	4.00000	0.140459	0.0702295	−	0.997531i	$-0.477627\pi$
$811$	4.00000	0.140459	0.0702295	+	0.997531i	$0.477627\pi$
$812$	6.00000	0.210559
$813$	0	0
$814$	−42.0000	−1.47210
$815$	12.0000	0.420342
$816$	0	0
$817$	20.0000	0.699711
$818$	−1.00000	−0.0349642
$819$	0	0
$820$	−9.00000	−0.314294
$821$	18.0000	0.628204	0.314102	−	0.949389i	$-0.398297\pi$
$821$	18.0000	0.628204	0.314102	+	0.949389i	$0.398297\pi$
$822$	0	0
$823$	−40.0000	−1.39431	−0.697156	−	0.716919i	$-0.745552\pi$
$823$	−40.0000	−1.39431	−0.697156	+	0.716919i	$0.745552\pi$
$824$	−14.0000	−0.487713
$825$	0	0
$826$	0	0
$827$	−48.0000	−1.66912	−0.834562	−	0.550914i	$-0.814279\pi$
$827$	−48.0000	−1.66912	−0.834562	+	0.550914i	$0.814279\pi$
$828$	0	0
$829$	17.0000	0.590434	0.295217	−	0.955430i	$-0.404608\pi$
$829$	17.0000	0.590434	0.295217	+	0.955430i	$0.404608\pi$
$830$	18.0000	0.624789
$831$	0	0
$832$	0	0
$833$	−9.00000	−0.311832
$834$	0	0
$835$	0	0
$836$	−12.0000	−0.415029
$837$	0	0
$838$	24.0000	0.829066
$839$	−12.0000	−0.414286	−0.207143	−	0.978311i	$-0.566417\pi$
$839$	−12.0000	−0.414286	−0.207143	+	0.978311i	$0.566417\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	29.0000	0.999406
$843$	0	0
$844$	−16.0000	−0.550743
$845$	0	0
$846$	0	0
$847$	−50.0000	−1.71802
$848$	−3.00000	−0.103020
$849$	0	0
$850$	−12.0000	−0.411597
$851$	42.0000	1.43974
$852$	0	0
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	−14.0000	−0.479070
$855$	0	0
$856$	−6.00000	−0.205076
$857$	−21.0000	−0.717346	−0.358673	−	0.933463i	$-0.616771\pi$
$857$	−21.0000	−0.717346	−0.358673	+	0.933463i	$0.616771\pi$
$858$	0	0
$859$	26.0000	0.887109	0.443554	−	0.896248i	$-0.353717\pi$
$859$	26.0000	0.887109	0.443554	+	0.896248i	$0.353717\pi$
$860$	−30.0000	−1.02299
$861$	0	0
$862$	6.00000	0.204361
$863$	18.0000	0.612727	0.306364	−	0.951915i	$-0.400888\pi$
$863$	18.0000	0.612727	0.306364	+	0.951915i	$0.400888\pi$
$864$	0	0
$865$	18.0000	0.612018
$866$	13.0000	0.441758
$867$	0	0
$868$	−8.00000	−0.271538
$869$	−24.0000	−0.814144
$870$	0	0
$871$	0	0
$872$	14.0000	0.474100
$873$	0	0
$874$	12.0000	0.405906
$875$	6.00000	0.202837
$876$	0	0
$877$	−41.0000	−1.38447	−0.692236	−	0.721671i	$-0.743374\pi$
$877$	−41.0000	−1.38447	−0.692236	+	0.721671i	$0.743374\pi$
$878$	−14.0000	−0.472477
$879$	0	0
$880$	18.0000	0.606780
$881$	−33.0000	−1.11180	−0.555899	−	0.831250i	$-0.687626\pi$
$881$	−33.0000	−1.11180	−0.555899	+	0.831250i	$0.687626\pi$
$882$	0	0
$883$	8.00000	0.269221	0.134611	−	0.990899i	$-0.457022\pi$
$883$	8.00000	0.269221	0.134611	+	0.990899i	$0.457022\pi$
$884$	0	0
$885$	0	0
$886$	−36.0000	−1.20944
$887$	48.0000	1.61168	0.805841	−	0.592132i	$-0.201714\pi$
$887$	48.0000	1.61168	0.805841	+	0.592132i	$0.201714\pi$
$888$	0	0
$889$	8.00000	0.268311
$890$	−54.0000	−1.81008
$891$	0	0
$892$	−8.00000	−0.267860
$893$	−12.0000	−0.401565
$894$	0	0
$895$	−18.0000	−0.601674
$896$	2.00000	0.0668153
$897$	0	0
$898$	−18.0000	−0.600668
$899$	−12.0000	−0.400222
$900$	0	0
$901$	−9.00000	−0.299833
$902$	18.0000	0.599334
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	−21.0000	−0.698064
$906$	0	0
$907$	44.0000	1.46100	0.730498	−	0.682915i	$-0.239288\pi$
$907$	44.0000	1.46100	0.730498	+	0.682915i	$0.239288\pi$
$908$	18.0000	0.597351
$909$	0	0
$910$	0	0
$911$	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
$914$	11.0000	0.363848
$915$	0	0
$916$	22.0000	0.726900
$917$	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	$0.585007\pi$
$920$	−18.0000	−0.593442
$921$	0	0
$922$	−15.0000	−0.493999
$923$	0	0
$924$	0	0
$925$	28.0000	0.920634
$926$	38.0000	1.24876
$927$	0	0
$928$	3.00000	0.0984798
$929$	33.0000	1.08269	0.541347	−	0.840799i	$-0.317914\pi$
$929$	33.0000	1.08269	0.541347	+	0.840799i	$0.317914\pi$
$930$	0	0
$931$	6.00000	0.196642
$932$	6.00000	0.196537
$933$	0	0
$934$	−18.0000	−0.588978
$935$	54.0000	1.76599
$936$	0	0
$937$	47.0000	1.53542	0.767712	−	0.640796i	$-0.221395\pi$
$937$	47.0000	1.53542	0.767712	+	0.640796i	$0.221395\pi$
$938$	20.0000	0.653023
$939$	0	0
$940$	18.0000	0.587095
$941$	42.0000	1.36916	0.684580	−	0.728937i	$-0.259985\pi$
$941$	42.0000	1.36916	0.684580	+	0.728937i	$0.259985\pi$
$942$	0	0
$943$	−18.0000	−0.586161
$944$	0	0
$945$	0	0
$946$	60.0000	1.95077
$947$	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
$950$	8.00000	0.259554
$951$	0	0
$952$	6.00000	0.194461
$953$	−54.0000	−1.74923	−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000	−1.74923	−0.874616	+	0.484817i	$0.838886\pi$
$954$	0	0
$955$	36.0000	1.16493
$956$	6.00000	0.194054
$957$	0	0
$958$	0	0
$959$	−18.0000	−0.581250
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	1.00000	0.0322078
$965$	−69.0000	−2.22119
$966$	0	0
$967$	22.0000	0.707472	0.353736	−	0.935345i	$-0.384911\pi$
$967$	22.0000	0.707472	0.353736	+	0.935345i	$0.384911\pi$
$968$	−25.0000	−0.803530
$969$	0	0
$970$	42.0000	1.34854
$971$	−60.0000	−1.92549	−0.962746	−	0.270408i	$-0.912841\pi$
$971$	−60.0000	−1.92549	−0.962746	+	0.270408i	$0.912841\pi$
$972$	0	0
$973$	8.00000	0.256468
$974$	2.00000	0.0640841
$975$	0	0
$976$	−7.00000	−0.224065
$977$	−3.00000	−0.0959785	−0.0479893	−	0.998848i	$-0.515281\pi$
$977$	−3.00000	−0.0959785	−0.0479893	+	0.998848i	$0.515281\pi$
$978$	0	0
$979$	108.000	3.45169
$980$	−9.00000	−0.287494
$981$	0	0
$982$	−18.0000	−0.574403
$983$	−36.0000	−1.14822	−0.574111	−	0.818778i	$-0.694652\pi$
$983$	−36.0000	−1.14822	−0.574111	+	0.818778i	$0.694652\pi$
$984$	0	0
$985$	18.0000	0.573528
$986$	9.00000	0.286618
$987$	0	0
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	38.0000	1.20711	0.603555	−	0.797321i	$-0.293750\pi$
$991$	38.0000	1.20711	0.603555	+	0.797321i	$0.293750\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	12.0000	0.380617
$995$	−30.0000	−0.951064
$996$	0	0
$997$	5.00000	0.158352	0.0791758	−	0.996861i	$-0.474771\pi$
$997$	5.00000	0.158352	0.0791758	+	0.996861i	$0.474771\pi$
$998$	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.h.1.1		1
3.2	odd	2		1014.2.a.f.1.1		1
12.11	even	2		8112.2.a.c.1.1		1
13.4	even	6		234.2.h.a.55.1		2
13.5	odd	4		3042.2.b.h.1351.2		2
13.8	odd	4		3042.2.b.h.1351.1		2
13.10	even	6		234.2.h.a.217.1		2
13.12	even	2		3042.2.a.i.1.1		1
39.2	even	12		1014.2.i.b.823.2		4
39.5	even	4		1014.2.b.c.337.1		2
39.8	even	4		1014.2.b.c.337.2		2
39.11	even	12		1014.2.i.b.823.1		4
39.17	odd	6		78.2.e.a.55.1	✓	2
39.20	even	12		1014.2.i.b.361.2		4
39.23	odd	6		78.2.e.a.61.1	yes	2
39.29	odd	6		1014.2.e.a.529.1		2
39.32	even	12		1014.2.i.b.361.1		4
39.35	odd	6		1014.2.e.a.991.1		2
39.38	odd	2		1014.2.a.c.1.1		1
52.23	odd	6		1872.2.t.c.1153.1		2
52.43	odd	6		1872.2.t.c.289.1		2
156.23	even	6		624.2.q.g.529.1		2
156.95	even	6		624.2.q.g.289.1		2
156.155	even	2		8112.2.a.m.1.1		1
195.17	even	12		1950.2.z.g.1849.1		4
195.23	even	12		1950.2.z.g.1699.1		4
195.62	even	12		1950.2.z.g.1699.2		4
195.134	odd	6		1950.2.i.m.601.1		2
195.173	even	12		1950.2.z.g.1849.2		4
195.179	odd	6		1950.2.i.m.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.a.55.1	✓	2	39.17	odd	6
78.2.e.a.61.1	yes	2	39.23	odd	6
234.2.h.a.55.1		2	13.4	even	6
234.2.h.a.217.1		2	13.10	even	6
624.2.q.g.289.1		2	156.95	even	6
624.2.q.g.529.1		2	156.23	even	6
1014.2.a.c.1.1		1	39.38	odd	2
1014.2.a.f.1.1		1	3.2	odd	2
1014.2.b.c.337.1		2	39.5	even	4
1014.2.b.c.337.2		2	39.8	even	4
1014.2.e.a.529.1		2	39.29	odd	6
1014.2.e.a.991.1		2	39.35	odd	6
1014.2.i.b.361.1		4	39.32	even	12
1014.2.i.b.361.2		4	39.20	even	12
1014.2.i.b.823.1		4	39.11	even	12
1014.2.i.b.823.2		4	39.2	even	12
1872.2.t.c.289.1		2	52.43	odd	6
1872.2.t.c.1153.1		2	52.23	odd	6
1950.2.i.m.451.1		2	195.179	odd	6
1950.2.i.m.601.1		2	195.134	odd	6
1950.2.z.g.1699.1		4	195.23	even	12
1950.2.z.g.1699.2		4	195.62	even	12
1950.2.z.g.1849.1		4	195.17	even	12
1950.2.z.g.1849.2		4	195.173	even	12
3042.2.a.h.1.1		1	1.1	even	1	trivial
3042.2.a.i.1.1		1	13.12	even	2
3042.2.b.h.1351.1		2	13.8	odd	4
3042.2.b.h.1351.2		2	13.5	odd	4
8112.2.a.c.1.1		1	12.11	even	2
8112.2.a.m.1.1		1	156.155	even	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:	\( N \)	\(=\)	\( 3042 = 2 \cdot 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3042.a (trivial)

Introduction

L-functions

Modular forms

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Representations

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