LMFDB - Embedded newform 3042.2.a.d.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)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +O(q^{10})$

$q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +2.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +2.00000 q^{11} -2.00000 q^{14} +1.00000 q^{16} -5.00000 q^{17} +2.00000 q^{19} -1.00000 q^{20} -2.00000 q^{22} -6.00000 q^{23} -4.00000 q^{25} +2.00000 q^{28} +9.00000 q^{29} +4.00000 q^{31} -1.00000 q^{32} +5.00000 q^{34} -2.00000 q^{35} +11.0000 q^{37} -2.00000 q^{38} +1.00000 q^{40} +5.00000 q^{41} +10.0000 q^{43} +2.00000 q^{44} +6.00000 q^{46} +2.00000 q^{47} -3.00000 q^{49} +4.00000 q^{50} +1.00000 q^{53} -2.00000 q^{55} -2.00000 q^{56} -9.00000 q^{58} -8.00000 q^{59} -11.0000 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.00000 q^{67} -5.00000 q^{68} +2.00000 q^{70} -14.0000 q^{71} +13.0000 q^{73} -11.0000 q^{74} +2.00000 q^{76} +4.00000 q^{77} -4.00000 q^{79} -1.00000 q^{80} -5.00000 q^{82} +6.00000 q^{83} +5.00000 q^{85} -10.0000 q^{86} -2.00000 q^{88} +2.00000 q^{89} -6.00000 q^{92} -2.00000 q^{94} -2.00000 q^{95} +2.00000 q^{97} +3.00000 q^{98} +O(q^{100})$

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	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	1.00000	0.316228
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	−2.00000	−0.534522
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.00000	−1.21268	−0.606339	−	0.795206i	$-0.707363\pi$
$17$	−5.00000	−1.21268	−0.606339	+	0.795206i	$0.707363\pi$
$18$	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
$22$	−2.00000	−0.426401
$23$	−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
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	2.00000	0.377964
$29$	9.00000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	9.00000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	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
$34$	5.00000	0.857493
$35$	−2.00000	−0.338062
$36$	0	0
$37$	11.0000	1.80839	0.904194	−	0.427121i	$-0.140472\pi$
$37$	11.0000	1.80839	0.904194	+	0.427121i	$0.140472\pi$
$38$	−2.00000	−0.324443
$39$	0	0
$40$	1.00000	0.158114
$41$	5.00000	0.780869	0.390434	−	0.920631i	$-0.372325\pi$
$41$	5.00000	0.780869	0.390434	+	0.920631i	$0.372325\pi$
$42$	0	0
$43$	10.0000	1.52499	0.762493	−	0.646997i	$-0.223975\pi$
$43$	10.0000	1.52499	0.762493	+	0.646997i	$0.223975\pi$
$44$	2.00000	0.301511
$45$	0	0
$46$	6.00000	0.884652
$47$	2.00000	0.291730	0.145865	−	0.989305i	$-0.453403\pi$
$47$	2.00000	0.291730	0.145865	+	0.989305i	$0.453403\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	4.00000	0.565685
$51$	0	0
$52$	0	0
$53$	1.00000	0.137361	0.0686803	−	0.997639i	$-0.478121\pi$
$53$	1.00000	0.137361	0.0686803	+	0.997639i	$0.478121\pi$
$54$	0	0
$55$	−2.00000	−0.269680
$56$	−2.00000	−0.267261
$57$	0	0
$58$	−9.00000	−1.18176
$59$	−8.00000	−1.04151	−0.520756	−	0.853706i	$-0.674350\pi$
$59$	−8.00000	−1.04151	−0.520756	+	0.853706i	$0.674350\pi$
$60$	0	0
$61$	−11.0000	−1.40841	−0.704203	−	0.709999i	$-0.748695\pi$
$61$	−11.0000	−1.40841	−0.704203	+	0.709999i	$0.748695\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	−5.00000	−0.606339
$69$	0	0
$70$	2.00000	0.239046
$71$	−14.0000	−1.66149	−0.830747	−	0.556650i	$-0.812086\pi$
$71$	−14.0000	−1.66149	−0.830747	+	0.556650i	$0.812086\pi$
$72$	0	0
$73$	13.0000	1.52153	0.760767	−	0.649025i	$-0.224823\pi$
$73$	13.0000	1.52153	0.760767	+	0.649025i	$0.224823\pi$
$74$	−11.0000	−1.27872
$75$	0	0
$76$	2.00000	0.229416
$77$	4.00000	0.455842
$78$	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
$82$	−5.00000	−0.552158
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	5.00000	0.542326
$86$	−10.0000	−1.07833
$87$	0	0
$88$	−2.00000	−0.213201
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	0	0
$92$	−6.00000	−0.625543
$93$	0	0
$94$	−2.00000	−0.206284
$95$	−2.00000	−0.205196
$96$	0	0
$97$	2.00000	0.203069	0.101535	−	0.994832i	$-0.467625\pi$
$97$	2.00000	0.203069	0.101535	+	0.994832i	$0.467625\pi$
$98$	3.00000	0.303046
$99$	0	0
$100$	−4.00000	−0.400000
$101$	5.00000	0.497519	0.248759	−	0.968565i	$-0.419977\pi$
$101$	5.00000	0.497519	0.248759	+	0.968565i	$0.419977\pi$
$102$	0	0
$103$	10.0000	0.985329	0.492665	−	0.870219i	$-0.336023\pi$
$103$	10.0000	0.985329	0.492665	+	0.870219i	$0.336023\pi$
$104$	0	0
$105$	0	0
$106$	−1.00000	−0.0971286
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	2.00000	0.190693
$111$	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
$115$	6.00000	0.559503
$116$	9.00000	0.835629
$117$	0	0
$118$	8.00000	0.736460
$119$	−10.0000	−0.916698
$120$	0	0
$121$	−7.00000	−0.636364
$122$	11.0000	0.995893
$123$	0	0
$124$	4.00000	0.359211
$125$	9.00000	0.804984
$126$	0	0
$127$	−12.0000	−1.06483	−0.532414	−	0.846484i	$-0.678715\pi$
$127$	−12.0000	−1.06483	−0.532414	+	0.846484i	$0.678715\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	0	0
$131$	8.00000	0.698963	0.349482	−	0.936943i	$-0.386358\pi$
$131$	8.00000	0.698963	0.349482	+	0.936943i	$0.386358\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	2.00000	0.172774
$135$	0	0
$136$	5.00000	0.428746
$137$	17.0000	1.45241	0.726204	−	0.687479i	$-0.241283\pi$
$137$	17.0000	1.45241	0.726204	+	0.687479i	$0.241283\pi$
$138$	0	0
$139$	−12.0000	−1.01783	−0.508913	−	0.860818i	$-0.669953\pi$
$139$	−12.0000	−1.01783	−0.508913	+	0.860818i	$0.669953\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	14.0000	1.17485
$143$	0	0
$144$	0	0
$145$	−9.00000	−0.747409
$146$	−13.0000	−1.07589
$147$	0	0
$148$	11.0000	0.904194
$149$	3.00000	0.245770	0.122885	−	0.992421i	$-0.460785\pi$
$149$	3.00000	0.245770	0.122885	+	0.992421i	$0.460785\pi$
$150$	0	0
$151$	6.00000	0.488273	0.244137	−	0.969741i	$-0.421495\pi$
$151$	6.00000	0.488273	0.244137	+	0.969741i	$0.421495\pi$
$152$	−2.00000	−0.162221
$153$	0	0
$154$	−4.00000	−0.322329
$155$	−4.00000	−0.321288
$156$	0	0
$157$	−7.00000	−0.558661	−0.279330	−	0.960195i	$-0.590112\pi$
$157$	−7.00000	−0.558661	−0.279330	+	0.960195i	$0.590112\pi$
$158$	4.00000	0.318223
$159$	0	0
$160$	1.00000	0.0790569
$161$	−12.0000	−0.945732
$162$	0	0
$163$	−20.0000	−1.56652	−0.783260	−	0.621694i	$-0.786445\pi$
$163$	−20.0000	−1.56652	−0.783260	+	0.621694i	$0.786445\pi$
$164$	5.00000	0.390434
$165$	0	0
$166$	−6.00000	−0.465690
$167$	24.0000	1.85718	0.928588	−	0.371113i	$-0.121024\pi$
$167$	24.0000	1.85718	0.928588	+	0.371113i	$0.121024\pi$
$168$	0	0
$169$	0	0
$170$	−5.00000	−0.383482
$171$	0	0
$172$	10.0000	0.762493
$173$	22.0000	1.67263	0.836315	−	0.548250i	$-0.184706\pi$
$173$	22.0000	1.67263	0.836315	+	0.548250i	$0.184706\pi$
$174$	0	0
$175$	−8.00000	−0.604743
$176$	2.00000	0.150756
$177$	0	0
$178$	−2.00000	−0.149906
$179$	6.00000	0.448461	0.224231	−	0.974536i	$-0.428013\pi$
$179$	6.00000	0.448461	0.224231	+	0.974536i	$0.428013\pi$
$180$	0	0
$181$	5.00000	0.371647	0.185824	−	0.982583i	$-0.440505\pi$
$181$	5.00000	0.371647	0.185824	+	0.982583i	$0.440505\pi$
$182$	0	0
$183$	0	0
$184$	6.00000	0.442326
$185$	−11.0000	−0.808736
$186$	0	0
$187$	−10.0000	−0.731272
$188$	2.00000	0.145865
$189$	0	0
$190$	2.00000	0.145095
$191$	−4.00000	−0.289430	−0.144715	−	0.989473i	$-0.546227\pi$
$191$	−4.00000	−0.289430	−0.144715	+	0.989473i	$0.546227\pi$
$192$	0	0
$193$	17.0000	1.22369	0.611843	−	0.790979i	$-0.290428\pi$
$193$	17.0000	1.22369	0.611843	+	0.790979i	$0.290428\pi$
$194$	−2.00000	−0.143592
$195$	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	$-0.384671\pi$
$199$	10.0000	0.708881	0.354441	+	0.935079i	$0.384671\pi$
$200$	4.00000	0.282843
$201$	0	0
$202$	−5.00000	−0.351799
$203$	18.0000	1.26335
$204$	0	0
$205$	−5.00000	−0.349215
$206$	−10.0000	−0.696733
$207$	0	0
$208$	0	0
$209$	4.00000	0.276686
$210$	0	0
$211$	24.0000	1.65223	0.826114	−	0.563503i	$-0.190547\pi$
$211$	24.0000	1.65223	0.826114	+	0.563503i	$0.190547\pi$
$212$	1.00000	0.0686803
$213$	0	0
$214$	−18.0000	−1.23045
$215$	−10.0000	−0.681994
$216$	0	0
$217$	8.00000	0.543075
$218$	−2.00000	−0.135457
$219$	0	0
$220$	−2.00000	−0.134840
$221$	0	0
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	−2.00000	−0.133631
$225$	0	0
$226$	−3.00000	−0.199557
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	0	0
$229$	−10.0000	−0.660819	−0.330409	−	0.943838i	$-0.607187\pi$
$229$	−10.0000	−0.660819	−0.330409	+	0.943838i	$0.607187\pi$
$230$	−6.00000	−0.395628
$231$	0	0
$232$	−9.00000	−0.590879
$233$	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
$235$	−2.00000	−0.130466
$236$	−8.00000	−0.520756
$237$	0	0
$238$	10.0000	0.648204
$239$	−6.00000	−0.388108	−0.194054	−	0.980991i	$-0.562164\pi$
$239$	−6.00000	−0.388108	−0.194054	+	0.980991i	$0.562164\pi$
$240$	0	0
$241$	−7.00000	−0.450910	−0.225455	−	0.974254i	$-0.572387\pi$
$241$	−7.00000	−0.450910	−0.225455	+	0.974254i	$0.572387\pi$
$242$	7.00000	0.449977
$243$	0	0
$244$	−11.0000	−0.704203
$245$	3.00000	0.191663
$246$	0	0
$247$	0	0
$248$	−4.00000	−0.254000
$249$	0	0
$250$	−9.00000	−0.569210
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	0	0
$253$	−12.0000	−0.754434
$254$	12.0000	0.752947
$255$	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
$259$	22.0000	1.36701
$260$	0	0
$261$	0	0
$262$	−8.00000	−0.494242
$263$	−14.0000	−0.863277	−0.431638	−	0.902047i	$-0.642064\pi$
$263$	−14.0000	−0.863277	−0.431638	+	0.902047i	$0.642064\pi$
$264$	0	0
$265$	−1.00000	−0.0614295
$266$	−4.00000	−0.245256
$267$	0	0
$268$	−2.00000	−0.122169
$269$	14.0000	0.853595	0.426798	−	0.904347i	$-0.359642\pi$
$269$	14.0000	0.853595	0.426798	+	0.904347i	$0.359642\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−5.00000	−0.303170
$273$	0	0
$274$	−17.0000	−1.02701
$275$	−8.00000	−0.482418
$276$	0	0
$277$	−11.0000	−0.660926	−0.330463	−	0.943819i	$-0.607205\pi$
$277$	−11.0000	−0.660926	−0.330463	+	0.943819i	$0.607205\pi$
$278$	12.0000	0.719712
$279$	0	0
$280$	2.00000	0.119523
$281$	25.0000	1.49137	0.745687	−	0.666296i	$-0.232121\pi$
$281$	25.0000	1.49137	0.745687	+	0.666296i	$0.232121\pi$
$282$	0	0
$283$	26.0000	1.54554	0.772770	−	0.634686i	$-0.218871\pi$
$283$	26.0000	1.54554	0.772770	+	0.634686i	$0.218871\pi$
$284$	−14.0000	−0.830747
$285$	0	0
$286$	0	0
$287$	10.0000	0.590281
$288$	0	0
$289$	8.00000	0.470588
$290$	9.00000	0.528498
$291$	0	0
$292$	13.0000	0.760767
$293$	−1.00000	−0.0584206	−0.0292103	−	0.999573i	$-0.509299\pi$
$293$	−1.00000	−0.0584206	−0.0292103	+	0.999573i	$0.509299\pi$
$294$	0	0
$295$	8.00000	0.465778
$296$	−11.0000	−0.639362
$297$	0	0
$298$	−3.00000	−0.173785
$299$	0	0
$300$	0	0
$301$	20.0000	1.15278
$302$	−6.00000	−0.345261
$303$	0	0
$304$	2.00000	0.114708
$305$	11.0000	0.629858
$306$	0	0
$307$	14.0000	0.799022	0.399511	−	0.916728i	$-0.369180\pi$
$307$	14.0000	0.799022	0.399511	+	0.916728i	$0.369180\pi$
$308$	4.00000	0.227921
$309$	0	0
$310$	4.00000	0.227185
$311$	−6.00000	−0.340229	−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000	−0.340229	−0.170114	+	0.985424i	$0.554414\pi$
$312$	0	0
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	7.00000	0.395033
$315$	0	0
$316$	−4.00000	−0.225018
$317$	−33.0000	−1.85346	−0.926732	−	0.375722i	$-0.877395\pi$
$317$	−33.0000	−1.85346	−0.926732	+	0.375722i	$0.877395\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	12.0000	0.668734
$323$	−10.0000	−0.556415
$324$	0	0
$325$	0	0
$326$	20.0000	1.10770
$327$	0	0
$328$	−5.00000	−0.276079
$329$	4.00000	0.220527
$330$	0	0
$331$	28.0000	1.53902	0.769510	−	0.638635i	$-0.220501\pi$
$331$	28.0000	1.53902	0.769510	+	0.638635i	$0.220501\pi$
$332$	6.00000	0.329293
$333$	0	0
$334$	−24.0000	−1.31322
$335$	2.00000	0.109272
$336$	0	0
$337$	−9.00000	−0.490261	−0.245131	−	0.969490i	$-0.578831\pi$
$337$	−9.00000	−0.490261	−0.245131	+	0.969490i	$0.578831\pi$
$338$	0	0
$339$	0	0
$340$	5.00000	0.271163
$341$	8.00000	0.433224
$342$	0	0
$343$	−20.0000	−1.07990
$344$	−10.0000	−0.539164
$345$	0	0
$346$	−22.0000	−1.18273
$347$	−6.00000	−0.322097	−0.161048	−	0.986947i	$-0.551488\pi$
$347$	−6.00000	−0.322097	−0.161048	+	0.986947i	$0.551488\pi$
$348$	0	0
$349$	−6.00000	−0.321173	−0.160586	−	0.987022i	$-0.551338\pi$
$349$	−6.00000	−0.321173	−0.160586	+	0.987022i	$0.551338\pi$
$350$	8.00000	0.427618
$351$	0	0
$352$	−2.00000	−0.106600
$353$	17.0000	0.904819	0.452409	−	0.891810i	$-0.350565\pi$
$353$	17.0000	0.904819	0.452409	+	0.891810i	$0.350565\pi$
$354$	0	0
$355$	14.0000	0.743043
$356$	2.00000	0.106000
$357$	0	0
$358$	−6.00000	−0.317110
$359$	−30.0000	−1.58334	−0.791670	−	0.610949i	$-0.790788\pi$
$359$	−30.0000	−1.58334	−0.791670	+	0.610949i	$0.790788\pi$
$360$	0	0
$361$	−15.0000	−0.789474
$362$	−5.00000	−0.262794
$363$	0	0
$364$	0	0
$365$	−13.0000	−0.680451
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	−6.00000	−0.312772
$369$	0	0
$370$	11.0000	0.571863
$371$	2.00000	0.103835
$372$	0	0
$373$	9.00000	0.466002	0.233001	−	0.972476i	$-0.425145\pi$
$373$	9.00000	0.466002	0.233001	+	0.972476i	$0.425145\pi$
$374$	10.0000	0.517088
$375$	0	0
$376$	−2.00000	−0.103142
$377$	0	0
$378$	0	0
$379$	−12.0000	−0.616399	−0.308199	−	0.951322i	$-0.599726\pi$
$379$	−12.0000	−0.616399	−0.308199	+	0.951322i	$0.599726\pi$
$380$	−2.00000	−0.102598
$381$	0	0
$382$	4.00000	0.204658
$383$	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
$385$	−4.00000	−0.203859
$386$	−17.0000	−0.865277
$387$	0	0
$388$	2.00000	0.101535
$389$	−19.0000	−0.963338	−0.481669	−	0.876353i	$-0.659969\pi$
$389$	−19.0000	−0.963338	−0.481669	+	0.876353i	$0.659969\pi$
$390$	0	0
$391$	30.0000	1.51717
$392$	3.00000	0.151523
$393$	0	0
$394$	−6.00000	−0.302276
$395$	4.00000	0.201262
$396$	0	0
$397$	18.0000	0.903394	0.451697	−	0.892171i	$-0.350819\pi$
$397$	18.0000	0.903394	0.451697	+	0.892171i	$0.350819\pi$
$398$	−10.0000	−0.501255
$399$	0	0
$400$	−4.00000	−0.200000
$401$	−27.0000	−1.34832	−0.674158	−	0.738587i	$-0.735493\pi$
$401$	−27.0000	−1.34832	−0.674158	+	0.738587i	$0.735493\pi$
$402$	0	0
$403$	0	0
$404$	5.00000	0.248759
$405$	0	0
$406$	−18.0000	−0.893325
$407$	22.0000	1.09050
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	5.00000	0.246932
$411$	0	0
$412$	10.0000	0.492665
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−6.00000	−0.294528
$416$	0	0
$417$	0	0
$418$	−4.00000	−0.195646
$419$	32.0000	1.56330	0.781651	−	0.623716i	$-0.214378\pi$
$419$	32.0000	1.56330	0.781651	+	0.623716i	$0.214378\pi$
$420$	0	0
$421$	23.0000	1.12095	0.560476	−	0.828171i	$-0.310618\pi$
$421$	23.0000	1.12095	0.560476	+	0.828171i	$0.310618\pi$
$422$	−24.0000	−1.16830
$423$	0	0
$424$	−1.00000	−0.0485643
$425$	20.0000	0.970143
$426$	0	0
$427$	−22.0000	−1.06465
$428$	18.0000	0.870063
$429$	0	0
$430$	10.0000	0.482243
$431$	−2.00000	−0.0963366	−0.0481683	−	0.998839i	$-0.515338\pi$
$431$	−2.00000	−0.0963366	−0.0481683	+	0.998839i	$0.515338\pi$
$432$	0	0
$433$	−21.0000	−1.00920	−0.504598	−	0.863355i	$-0.668359\pi$
$433$	−21.0000	−1.00920	−0.504598	+	0.863355i	$0.668359\pi$
$434$	−8.00000	−0.384012
$435$	0	0
$436$	2.00000	0.0957826
$437$	−12.0000	−0.574038
$438$	0	0
$439$	10.0000	0.477274	0.238637	−	0.971109i	$-0.423299\pi$
$439$	10.0000	0.477274	0.238637	+	0.971109i	$0.423299\pi$
$440$	2.00000	0.0953463
$441$	0	0
$442$	0	0
$443$	−20.0000	−0.950229	−0.475114	−	0.879924i	$-0.657593\pi$
$443$	−20.0000	−0.950229	−0.475114	+	0.879924i	$0.657593\pi$
$444$	0	0
$445$	−2.00000	−0.0948091
$446$	−16.0000	−0.757622
$447$	0	0
$448$	2.00000	0.0944911
$449$	−30.0000	−1.41579	−0.707894	−	0.706319i	$-0.750354\pi$
$449$	−30.0000	−1.41579	−0.707894	+	0.706319i	$0.750354\pi$
$450$	0	0
$451$	10.0000	0.470882
$452$	3.00000	0.141108
$453$	0	0
$454$	−14.0000	−0.657053
$455$	0	0
$456$	0	0
$457$	−3.00000	−0.140334	−0.0701670	−	0.997535i	$-0.522353\pi$
$457$	−3.00000	−0.140334	−0.0701670	+	0.997535i	$0.522353\pi$
$458$	10.0000	0.467269
$459$	0	0
$460$	6.00000	0.279751
$461$	3.00000	0.139724	0.0698620	−	0.997557i	$-0.477744\pi$
$461$	3.00000	0.139724	0.0698620	+	0.997557i	$0.477744\pi$
$462$	0	0
$463$	14.0000	0.650635	0.325318	−	0.945605i	$-0.394529\pi$
$463$	14.0000	0.650635	0.325318	+	0.945605i	$0.394529\pi$
$464$	9.00000	0.417815
$465$	0	0
$466$	−6.00000	−0.277945
$467$	22.0000	1.01804	0.509019	−	0.860755i	$-0.330008\pi$
$467$	22.0000	1.01804	0.509019	+	0.860755i	$0.330008\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	2.00000	0.0922531
$471$	0	0
$472$	8.00000	0.368230
$473$	20.0000	0.919601
$474$	0	0
$475$	−8.00000	−0.367065
$476$	−10.0000	−0.458349
$477$	0	0
$478$	6.00000	0.274434
$479$	32.0000	1.46212	0.731059	−	0.682315i	$-0.239027\pi$
$479$	32.0000	1.46212	0.731059	+	0.682315i	$0.239027\pi$
$480$	0	0
$481$	0	0
$482$	7.00000	0.318841
$483$	0	0
$484$	−7.00000	−0.318182
$485$	−2.00000	−0.0908153
$486$	0	0
$487$	26.0000	1.17817	0.589086	−	0.808070i	$-0.299488\pi$
$487$	26.0000	1.17817	0.589086	+	0.808070i	$0.299488\pi$
$488$	11.0000	0.497947
$489$	0	0
$490$	−3.00000	−0.135526
$491$	30.0000	1.35388	0.676941	−	0.736038i	$-0.263305\pi$
$491$	30.0000	1.35388	0.676941	+	0.736038i	$0.263305\pi$
$492$	0	0
$493$	−45.0000	−2.02670
$494$	0	0
$495$	0	0
$496$	4.00000	0.179605
$497$	−28.0000	−1.25597
$498$	0	0
$499$	0	0		−	1.00000i	$-0.5\pi$
$499$	0	0			1.00000i	$0.5\pi$
$500$	9.00000	0.402492
$501$	0	0
$502$	4.00000	0.178529
$503$	14.0000	0.624229	0.312115	−	0.950044i	$-0.398963\pi$
$503$	14.0000	0.624229	0.312115	+	0.950044i	$0.398963\pi$
$504$	0	0
$505$	−5.00000	−0.222497
$506$	12.0000	0.533465
$507$	0	0
$508$	−12.0000	−0.532414
$509$	15.0000	0.664863	0.332432	−	0.943127i	$-0.392131\pi$
$509$	15.0000	0.664863	0.332432	+	0.943127i	$0.392131\pi$
$510$	0	0
$511$	26.0000	1.15017
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−3.00000	−0.132324
$515$	−10.0000	−0.440653
$516$	0	0
$517$	4.00000	0.175920
$518$	−22.0000	−0.966625
$519$	0	0
$520$	0	0
$521$	−25.0000	−1.09527	−0.547635	−	0.836717i	$-0.684472\pi$
$521$	−25.0000	−1.09527	−0.547635	+	0.836717i	$0.684472\pi$
$522$	0	0
$523$	−38.0000	−1.66162	−0.830812	−	0.556553i	$-0.812124\pi$
$523$	−38.0000	−1.66162	−0.830812	+	0.556553i	$0.812124\pi$
$524$	8.00000	0.349482
$525$	0	0
$526$	14.0000	0.610429
$527$	−20.0000	−0.871214
$528$	0	0
$529$	13.0000	0.565217
$530$	1.00000	0.0434372
$531$	0	0
$532$	4.00000	0.173422
$533$	0	0
$534$	0	0
$535$	−18.0000	−0.778208
$536$	2.00000	0.0863868
$537$	0	0
$538$	−14.0000	−0.603583
$539$	−6.00000	−0.258438
$540$	0	0
$541$	7.00000	0.300954	0.150477	−	0.988614i	$-0.451919\pi$
$541$	7.00000	0.300954	0.150477	+	0.988614i	$0.451919\pi$
$542$	8.00000	0.343629
$543$	0	0
$544$	5.00000	0.214373
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	17.0000	0.726204
$549$	0	0
$550$	8.00000	0.341121
$551$	18.0000	0.766826
$552$	0	0
$553$	−8.00000	−0.340195
$554$	11.0000	0.467345
$555$	0	0
$556$	−12.0000	−0.508913
$557$	−9.00000	−0.381342	−0.190671	−	0.981654i	$-0.561066\pi$
$557$	−9.00000	−0.381342	−0.190671	+	0.981654i	$0.561066\pi$
$558$	0	0
$559$	0	0
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−25.0000	−1.05456
$563$	−40.0000	−1.68580	−0.842900	−	0.538071i	$-0.819153\pi$
$563$	−40.0000	−1.68580	−0.842900	+	0.538071i	$0.819153\pi$
$564$	0	0
$565$	−3.00000	−0.126211
$566$	−26.0000	−1.09286
$567$	0	0
$568$	14.0000	0.587427
$569$	−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
$571$	−2.00000	−0.0836974	−0.0418487	−	0.999124i	$-0.513325\pi$
$571$	−2.00000	−0.0836974	−0.0418487	+	0.999124i	$0.513325\pi$
$572$	0	0
$573$	0	0
$574$	−10.0000	−0.417392
$575$	24.0000	1.00087
$576$	0	0
$577$	−27.0000	−1.12402	−0.562012	−	0.827129i	$-0.689973\pi$
$577$	−27.0000	−1.12402	−0.562012	+	0.827129i	$0.689973\pi$
$578$	−8.00000	−0.332756
$579$	0	0
$580$	−9.00000	−0.373705
$581$	12.0000	0.497844
$582$	0	0
$583$	2.00000	0.0828315
$584$	−13.0000	−0.537944
$585$	0	0
$586$	1.00000	0.0413096
$587$	−32.0000	−1.32078	−0.660391	−	0.750922i	$-0.729609\pi$
$587$	−32.0000	−1.32078	−0.660391	+	0.750922i	$0.729609\pi$
$588$	0	0
$589$	8.00000	0.329634
$590$	−8.00000	−0.329355
$591$	0	0
$592$	11.0000	0.452097
$593$	−39.0000	−1.60154	−0.800769	−	0.598973i	$-0.795576\pi$
$593$	−39.0000	−1.60154	−0.800769	+	0.598973i	$0.795576\pi$
$594$	0	0
$595$	10.0000	0.409960
$596$	3.00000	0.122885
$597$	0	0
$598$	0	0
$599$	0	0		−	1.00000i	$-0.5\pi$
$599$	0	0			1.00000i	$0.5\pi$
$600$	0	0
$601$	11.0000	0.448699	0.224350	−	0.974509i	$-0.427974\pi$
$601$	11.0000	0.448699	0.224350	+	0.974509i	$0.427974\pi$
$602$	−20.0000	−0.815139
$603$	0	0
$604$	6.00000	0.244137
$605$	7.00000	0.284590
$606$	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
$610$	−11.0000	−0.445377
$611$	0	0
$612$	0	0
$613$	−13.0000	−0.525065	−0.262533	−	0.964923i	$-0.584558\pi$
$613$	−13.0000	−0.525065	−0.262533	+	0.964923i	$0.584558\pi$
$614$	−14.0000	−0.564994
$615$	0	0
$616$	−4.00000	−0.161165
$617$	−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
$619$	−32.0000	−1.28619	−0.643094	−	0.765787i	$-0.722350\pi$
$619$	−32.0000	−1.28619	−0.643094	+	0.765787i	$0.722350\pi$
$620$	−4.00000	−0.160644
$621$	0	0
$622$	6.00000	0.240578
$623$	4.00000	0.160257
$624$	0	0
$625$	11.0000	0.440000
$626$	−6.00000	−0.239808
$627$	0	0
$628$	−7.00000	−0.279330
$629$	−55.0000	−2.19299
$630$	0	0
$631$	−12.0000	−0.477712	−0.238856	−	0.971055i	$-0.576772\pi$
$631$	−12.0000	−0.477712	−0.238856	+	0.971055i	$0.576772\pi$
$632$	4.00000	0.159111
$633$	0	0
$634$	33.0000	1.31060
$635$	12.0000	0.476205
$636$	0	0
$637$	0	0
$638$	−18.0000	−0.712627
$639$	0	0
$640$	1.00000	0.0395285
$641$	−5.00000	−0.197488	−0.0987441	−	0.995113i	$-0.531483\pi$
$641$	−5.00000	−0.197488	−0.0987441	+	0.995113i	$0.531483\pi$
$642$	0	0
$643$	−8.00000	−0.315489	−0.157745	−	0.987480i	$-0.550422\pi$
$643$	−8.00000	−0.315489	−0.157745	+	0.987480i	$0.550422\pi$
$644$	−12.0000	−0.472866
$645$	0	0
$646$	10.0000	0.393445
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	0	0
$649$	−16.0000	−0.628055
$650$	0	0
$651$	0	0
$652$	−20.0000	−0.783260
$653$	22.0000	0.860927	0.430463	−	0.902608i	$-0.358350\pi$
$653$	22.0000	0.860927	0.430463	+	0.902608i	$0.358350\pi$
$654$	0	0
$655$	−8.00000	−0.312586
$656$	5.00000	0.195217
$657$	0	0
$658$	−4.00000	−0.155936
$659$	−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
$661$	−25.0000	−0.972387	−0.486194	−	0.873851i	$-0.661615\pi$
$661$	−25.0000	−0.972387	−0.486194	+	0.873851i	$0.661615\pi$
$662$	−28.0000	−1.08825
$663$	0	0
$664$	−6.00000	−0.232845
$665$	−4.00000	−0.155113
$666$	0	0
$667$	−54.0000	−2.09089
$668$	24.0000	0.928588
$669$	0	0
$670$	−2.00000	−0.0772667
$671$	−22.0000	−0.849301
$672$	0	0
$673$	43.0000	1.65753	0.828764	−	0.559598i	$-0.189045\pi$
$673$	43.0000	1.65753	0.828764	+	0.559598i	$0.189045\pi$
$674$	9.00000	0.346667
$675$	0	0
$676$	0	0
$677$	46.0000	1.76792	0.883962	−	0.467559i	$-0.154866\pi$
$677$	46.0000	1.76792	0.883962	+	0.467559i	$0.154866\pi$
$678$	0	0
$679$	4.00000	0.153506
$680$	−5.00000	−0.191741
$681$	0	0
$682$	−8.00000	−0.306336
$683$	−40.0000	−1.53056	−0.765279	−	0.643699i	$-0.777399\pi$
$683$	−40.0000	−1.53056	−0.765279	+	0.643699i	$0.777399\pi$
$684$	0	0
$685$	−17.0000	−0.649537
$686$	20.0000	0.763604
$687$	0	0
$688$	10.0000	0.381246
$689$	0	0
$690$	0	0
$691$	2.00000	0.0760836	0.0380418	−	0.999276i	$-0.487888\pi$
$691$	2.00000	0.0760836	0.0380418	+	0.999276i	$0.487888\pi$
$692$	22.0000	0.836315
$693$	0	0
$694$	6.00000	0.227757
$695$	12.0000	0.455186
$696$	0	0
$697$	−25.0000	−0.946943
$698$	6.00000	0.227103
$699$	0	0
$700$	−8.00000	−0.302372
$701$	−34.0000	−1.28416	−0.642081	−	0.766637i	$-0.721929\pi$
$701$	−34.0000	−1.28416	−0.642081	+	0.766637i	$0.721929\pi$
$702$	0	0
$703$	22.0000	0.829746
$704$	2.00000	0.0753778
$705$	0	0
$706$	−17.0000	−0.639803
$707$	10.0000	0.376089
$708$	0	0
$709$	15.0000	0.563337	0.281668	−	0.959512i	$-0.409112\pi$
$709$	15.0000	0.563337	0.281668	+	0.959512i	$0.409112\pi$
$710$	−14.0000	−0.525411
$711$	0	0
$712$	−2.00000	−0.0749532
$713$	−24.0000	−0.898807
$714$	0	0
$715$	0	0
$716$	6.00000	0.224231
$717$	0	0
$718$	30.0000	1.11959
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	0	0
$721$	20.0000	0.744839
$722$	15.0000	0.558242
$723$	0	0
$724$	5.00000	0.185824
$725$	−36.0000	−1.33701
$726$	0	0
$727$	2.00000	0.0741759	0.0370879	−	0.999312i	$-0.488192\pi$
$727$	2.00000	0.0741759	0.0370879	+	0.999312i	$0.488192\pi$
$728$	0	0
$729$	0	0
$730$	13.0000	0.481152
$731$	−50.0000	−1.84932
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	2.00000	0.0738213
$735$	0	0
$736$	6.00000	0.221163
$737$	−4.00000	−0.147342
$738$	0	0
$739$	−16.0000	−0.588570	−0.294285	−	0.955718i	$-0.595081\pi$
$739$	−16.0000	−0.588570	−0.294285	+	0.955718i	$0.595081\pi$
$740$	−11.0000	−0.404368
$741$	0	0
$742$	−2.00000	−0.0734223
$743$	−12.0000	−0.440237	−0.220119	−	0.975473i	$-0.570644\pi$
$743$	−12.0000	−0.440237	−0.220119	+	0.975473i	$0.570644\pi$
$744$	0	0
$745$	−3.00000	−0.109911
$746$	−9.00000	−0.329513
$747$	0	0
$748$	−10.0000	−0.365636
$749$	36.0000	1.31541
$750$	0	0
$751$	26.0000	0.948753	0.474377	−	0.880322i	$-0.342673\pi$
$751$	26.0000	0.948753	0.474377	+	0.880322i	$0.342673\pi$
$752$	2.00000	0.0729325
$753$	0	0
$754$	0	0
$755$	−6.00000	−0.218362
$756$	0	0
$757$	−18.0000	−0.654221	−0.327111	−	0.944986i	$-0.606075\pi$
$757$	−18.0000	−0.654221	−0.327111	+	0.944986i	$0.606075\pi$
$758$	12.0000	0.435860
$759$	0	0
$760$	2.00000	0.0725476
$761$	34.0000	1.23250	0.616250	−	0.787551i	$-0.288651\pi$
$761$	34.0000	1.23250	0.616250	+	0.787551i	$0.288651\pi$
$762$	0	0
$763$	4.00000	0.144810
$764$	−4.00000	−0.144715
$765$	0	0
$766$	−24.0000	−0.867155
$767$	0	0
$768$	0	0
$769$	34.0000	1.22607	0.613036	−	0.790055i	$-0.289948\pi$
$769$	34.0000	1.22607	0.613036	+	0.790055i	$0.289948\pi$
$770$	4.00000	0.144150
$771$	0	0
$772$	17.0000	0.611843
$773$	18.0000	0.647415	0.323708	−	0.946157i	$-0.395071\pi$
$773$	18.0000	0.647415	0.323708	+	0.946157i	$0.395071\pi$
$774$	0	0
$775$	−16.0000	−0.574737
$776$	−2.00000	−0.0717958
$777$	0	0
$778$	19.0000	0.681183
$779$	10.0000	0.358287
$780$	0	0
$781$	−28.0000	−1.00192
$782$	−30.0000	−1.07280
$783$	0	0
$784$	−3.00000	−0.107143
$785$	7.00000	0.249841
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−4.00000	−0.142314
$791$	6.00000	0.213335
$792$	0	0
$793$	0	0
$794$	−18.0000	−0.638796
$795$	0	0
$796$	10.0000	0.354441
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	0	0
$799$	−10.0000	−0.353775
$800$	4.00000	0.141421
$801$	0	0
$802$	27.0000	0.953403
$803$	26.0000	0.917520
$804$	0	0
$805$	12.0000	0.422944
$806$	0	0
$807$	0	0
$808$	−5.00000	−0.175899
$809$	−5.00000	−0.175791	−0.0878953	−	0.996130i	$-0.528014\pi$
$809$	−5.00000	−0.175791	−0.0878953	+	0.996130i	$0.528014\pi$
$810$	0	0
$811$	36.0000	1.26413	0.632065	−	0.774915i	$-0.282207\pi$
$811$	36.0000	1.26413	0.632065	+	0.774915i	$0.282207\pi$
$812$	18.0000	0.631676
$813$	0	0
$814$	−22.0000	−0.771100
$815$	20.0000	0.700569
$816$	0	0
$817$	20.0000	0.699711
$818$	23.0000	0.804176
$819$	0	0
$820$	−5.00000	−0.174608
$821$	−30.0000	−1.04701	−0.523504	−	0.852023i	$-0.675375\pi$
$821$	−30.0000	−1.04701	−0.523504	+	0.852023i	$0.675375\pi$
$822$	0	0
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	16.0000	0.556711
$827$	−8.00000	−0.278187	−0.139094	−	0.990279i	$-0.544419\pi$
$827$	−8.00000	−0.278187	−0.139094	+	0.990279i	$0.544419\pi$
$828$	0	0
$829$	−35.0000	−1.21560	−0.607800	−	0.794090i	$-0.707948\pi$
$829$	−35.0000	−1.21560	−0.607800	+	0.794090i	$0.707948\pi$
$830$	6.00000	0.208263
$831$	0	0
$832$	0	0
$833$	15.0000	0.519719
$834$	0	0
$835$	−24.0000	−0.830554
$836$	4.00000	0.138343
$837$	0	0
$838$	−32.0000	−1.10542
$839$	44.0000	1.51905	0.759524	−	0.650479i	$-0.225432\pi$
$839$	44.0000	1.51905	0.759524	+	0.650479i	$0.225432\pi$
$840$	0	0
$841$	52.0000	1.79310
$842$	−23.0000	−0.792632
$843$	0	0
$844$	24.0000	0.826114
$845$	0	0
$846$	0	0
$847$	−14.0000	−0.481046
$848$	1.00000	0.0343401
$849$	0	0
$850$	−20.0000	−0.685994
$851$	−66.0000	−2.26245
$852$	0	0
$853$	−49.0000	−1.67773	−0.838864	−	0.544341i	$-0.816780\pi$
$853$	−49.0000	−1.67773	−0.838864	+	0.544341i	$0.816780\pi$
$854$	22.0000	0.752825
$855$	0	0
$856$	−18.0000	−0.615227
$857$	−45.0000	−1.53717	−0.768585	−	0.639747i	$-0.779039\pi$
$857$	−45.0000	−1.53717	−0.768585	+	0.639747i	$0.779039\pi$
$858$	0	0
$859$	−50.0000	−1.70598	−0.852989	−	0.521929i	$-0.825213\pi$
$859$	−50.0000	−1.70598	−0.852989	+	0.521929i	$0.825213\pi$
$860$	−10.0000	−0.340997
$861$	0	0
$862$	2.00000	0.0681203
$863$	46.0000	1.56586	0.782929	−	0.622111i	$-0.213725\pi$
$863$	46.0000	1.56586	0.782929	+	0.622111i	$0.213725\pi$
$864$	0	0
$865$	−22.0000	−0.748022
$866$	21.0000	0.713609
$867$	0	0
$868$	8.00000	0.271538
$869$	−8.00000	−0.271381
$870$	0	0
$871$	0	0
$872$	−2.00000	−0.0677285
$873$	0	0
$874$	12.0000	0.405906
$875$	18.0000	0.608511
$876$	0	0
$877$	−37.0000	−1.24940	−0.624701	−	0.780864i	$-0.714779\pi$
$877$	−37.0000	−1.24940	−0.624701	+	0.780864i	$0.714779\pi$
$878$	−10.0000	−0.337484
$879$	0	0
$880$	−2.00000	−0.0674200
$881$	−17.0000	−0.572745	−0.286372	−	0.958118i	$-0.592449\pi$
$881$	−17.0000	−0.572745	−0.286372	+	0.958118i	$0.592449\pi$
$882$	0	0
$883$	−8.00000	−0.269221	−0.134611	−	0.990899i	$-0.542978\pi$
$883$	−8.00000	−0.269221	−0.134611	+	0.990899i	$0.542978\pi$
$884$	0	0
$885$	0	0
$886$	20.0000	0.671913
$887$	−24.0000	−0.805841	−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000	−0.805841	−0.402921	+	0.915235i	$0.632005\pi$
$888$	0	0
$889$	−24.0000	−0.804934
$890$	2.00000	0.0670402
$891$	0	0
$892$	16.0000	0.535720
$893$	4.00000	0.133855
$894$	0	0
$895$	−6.00000	−0.200558
$896$	−2.00000	−0.0668153
$897$	0	0
$898$	30.0000	1.00111
$899$	36.0000	1.20067
$900$	0	0
$901$	−5.00000	−0.166574
$902$	−10.0000	−0.332964
$903$	0	0
$904$	−3.00000	−0.0997785
$905$	−5.00000	−0.166206
$906$	0	0
$907$	−44.0000	−1.46100	−0.730498	−	0.682915i	$-0.760712\pi$
$907$	−44.0000	−1.46100	−0.730498	+	0.682915i	$0.760712\pi$
$908$	14.0000	0.464606
$909$	0	0
$910$	0	0
$911$	32.0000	1.06021	0.530104	−	0.847933i	$-0.322153\pi$
$911$	32.0000	1.06021	0.530104	+	0.847933i	$0.322153\pi$
$912$	0	0
$913$	12.0000	0.397142
$914$	3.00000	0.0992312
$915$	0	0
$916$	−10.0000	−0.330409
$917$	16.0000	0.528367
$918$	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
$922$	−3.00000	−0.0987997
$923$	0	0
$924$	0	0
$925$	−44.0000	−1.44671
$926$	−14.0000	−0.460069
$927$	0	0
$928$	−9.00000	−0.295439
$929$	−23.0000	−0.754606	−0.377303	−	0.926090i	$-0.623148\pi$
$929$	−23.0000	−0.754606	−0.377303	+	0.926090i	$0.623148\pi$
$930$	0	0
$931$	−6.00000	−0.196642
$932$	6.00000	0.196537
$933$	0	0
$934$	−22.0000	−0.719862
$935$	10.0000	0.327035
$936$	0	0
$937$	−1.00000	−0.0326686	−0.0163343	−	0.999867i	$-0.505200\pi$
$937$	−1.00000	−0.0326686	−0.0163343	+	0.999867i	$0.505200\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	−2.00000	−0.0652328
$941$	−22.0000	−0.717180	−0.358590	−	0.933495i	$-0.616742\pi$
$941$	−22.0000	−0.717180	−0.358590	+	0.933495i	$0.616742\pi$
$942$	0	0
$943$	−30.0000	−0.976934
$944$	−8.00000	−0.260378
$945$	0	0
$946$	−20.0000	−0.650256
$947$	8.00000	0.259965	0.129983	−	0.991516i	$-0.458508\pi$
$947$	8.00000	0.259965	0.129983	+	0.991516i	$0.458508\pi$
$948$	0	0
$949$	0	0
$950$	8.00000	0.259554
$951$	0	0
$952$	10.0000	0.324102
$953$	−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
$955$	4.00000	0.129437
$956$	−6.00000	−0.194054
$957$	0	0
$958$	−32.0000	−1.03387
$959$	34.0000	1.09792
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	0	0
$964$	−7.00000	−0.225455
$965$	−17.0000	−0.547249
$966$	0	0
$967$	50.0000	1.60789	0.803946	−	0.594703i	$-0.202730\pi$
$967$	50.0000	1.60789	0.803946	+	0.594703i	$0.202730\pi$
$968$	7.00000	0.224989
$969$	0	0
$970$	2.00000	0.0642161
$971$	−20.0000	−0.641831	−0.320915	−	0.947108i	$-0.603990\pi$
$971$	−20.0000	−0.641831	−0.320915	+	0.947108i	$0.603990\pi$
$972$	0	0
$973$	−24.0000	−0.769405
$974$	−26.0000	−0.833094
$975$	0	0
$976$	−11.0000	−0.352101
$977$	21.0000	0.671850	0.335925	−	0.941889i	$-0.390951\pi$
$977$	21.0000	0.671850	0.335925	+	0.941889i	$0.390951\pi$
$978$	0	0
$979$	4.00000	0.127841
$980$	3.00000	0.0958315
$981$	0	0
$982$	−30.0000	−0.957338
$983$	60.0000	1.91370	0.956851	−	0.290578i	$-0.0938475\pi$
$983$	60.0000	1.91370	0.956851	+	0.290578i	$0.0938475\pi$
$984$	0	0
$985$	−6.00000	−0.191176
$986$	45.0000	1.43309
$987$	0	0
$988$	0	0
$989$	−60.0000	−1.90789
$990$	0	0
$991$	18.0000	0.571789	0.285894	−	0.958261i	$-0.407709\pi$
$991$	18.0000	0.571789	0.285894	+	0.958261i	$0.407709\pi$
$992$	−4.00000	−0.127000
$993$	0	0
$994$	28.0000	0.888106
$995$	−10.0000	−0.317021
$996$	0	0
$997$	−23.0000	−0.728417	−0.364209	−	0.931317i	$-0.618661\pi$
$997$	−23.0000	−0.728417	−0.364209	+	0.931317i	$0.618661\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3042.2.a.d.1.1		1
3.2	odd	2		1014.2.a.e.1.1		1
12.11	even	2		8112.2.a.bb.1.1		1
13.4	even	6		234.2.h.b.55.1		2
13.5	odd	4		3042.2.b.d.1351.2		2
13.8	odd	4		3042.2.b.d.1351.1		2
13.10	even	6		234.2.h.b.217.1		2
13.12	even	2		3042.2.a.m.1.1		1
39.2	even	12		1014.2.i.e.823.2		4
39.5	even	4		1014.2.b.a.337.1		2
39.8	even	4		1014.2.b.a.337.2		2
39.11	even	12		1014.2.i.e.823.1		4
39.17	odd	6		78.2.e.b.55.1	✓	2
39.20	even	12		1014.2.i.e.361.2		4
39.23	odd	6		78.2.e.b.61.1	yes	2
39.29	odd	6		1014.2.e.d.529.1		2
39.32	even	12		1014.2.i.e.361.1		4
39.35	odd	6		1014.2.e.d.991.1		2
39.38	odd	2		1014.2.a.a.1.1		1
52.23	odd	6		1872.2.t.i.1153.1		2
52.43	odd	6		1872.2.t.i.289.1		2
156.23	even	6		624.2.q.b.529.1		2
156.95	even	6		624.2.q.b.289.1		2
156.155	even	2		8112.2.a.x.1.1		1
195.17	even	12		1950.2.z.b.1849.1		4
195.23	even	12		1950.2.z.b.1699.1		4
195.62	even	12		1950.2.z.b.1699.2		4
195.134	odd	6		1950.2.i.b.601.1		2
195.173	even	12		1950.2.z.b.1849.2		4
195.179	odd	6		1950.2.i.b.451.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.e.b.55.1	✓	2	39.17	odd	6
78.2.e.b.61.1	yes	2	39.23	odd	6
234.2.h.b.55.1		2	13.4	even	6
234.2.h.b.217.1		2	13.10	even	6
624.2.q.b.289.1		2	156.95	even	6
624.2.q.b.529.1		2	156.23	even	6
1014.2.a.a.1.1		1	39.38	odd	2
1014.2.a.e.1.1		1	3.2	odd	2
1014.2.b.a.337.1		2	39.5	even	4
1014.2.b.a.337.2		2	39.8	even	4
1014.2.e.d.529.1		2	39.29	odd	6
1014.2.e.d.991.1		2	39.35	odd	6
1014.2.i.e.361.1		4	39.32	even	12
1014.2.i.e.361.2		4	39.20	even	12
1014.2.i.e.823.1		4	39.11	even	12
1014.2.i.e.823.2		4	39.2	even	12
1872.2.t.i.289.1		2	52.43	odd	6
1872.2.t.i.1153.1		2	52.23	odd	6
1950.2.i.b.451.1		2	195.179	odd	6
1950.2.i.b.601.1		2	195.134	odd	6
1950.2.z.b.1699.1		4	195.23	even	12
1950.2.z.b.1699.2		4	195.62	even	12
1950.2.z.b.1849.1		4	195.17	even	12
1950.2.z.b.1849.2		4	195.173	even	12
3042.2.a.d.1.1		1	1.1	even	1	trivial
3042.2.a.m.1.1		1	13.12	even	2
3042.2.b.d.1351.1		2	13.8	odd	4
3042.2.b.d.1351.2		2	13.5	odd	4
8112.2.a.x.1.1		1	156.155	even	2
8112.2.a.bb.1.1		1	12.11	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

Varieties

Fields

Representations

Groups

Database

Properties

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