LMFDB - Embedded newform 126.2.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,2,Mod(1,126)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("126.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$1.00611506547$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	126.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-1.00000 q^{2} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{7} -1.00000 q^{8} -2.00000 q^{10} +4.00000 q^{11} +6.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -4.00000 q^{19} +2.00000 q^{20} -4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{25} -6.00000 q^{26} -1.00000 q^{28} +2.00000 q^{29} -1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} -10.0000 q^{37} +4.00000 q^{38} -2.00000 q^{40} +6.00000 q^{41} -4.00000 q^{43} +4.00000 q^{44} +8.00000 q^{46} +1.00000 q^{49} +1.00000 q^{50} +6.00000 q^{52} -6.00000 q^{53} +8.00000 q^{55} +1.00000 q^{56} -2.00000 q^{58} -4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{64} +12.0000 q^{65} +4.00000 q^{67} -2.00000 q^{68} +2.00000 q^{70} -8.00000 q^{71} +10.0000 q^{73} +10.0000 q^{74} -4.00000 q^{76} -4.00000 q^{77} +2.00000 q^{80} -6.00000 q^{82} +4.00000 q^{83} -4.00000 q^{85} +4.00000 q^{86} -4.00000 q^{88} +6.00000 q^{89} -6.00000 q^{91} -8.00000 q^{92} -8.00000 q^{95} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$

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
$5$	2.00000	0.894427	0.447214	−	0.894427i	$-0.352416\pi$
$5$	2.00000	0.894427	0.447214	+	0.894427i	$0.352416\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−2.00000	−0.632456
$11$	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
$13$	6.00000	1.66410	0.832050	−	0.554700i	$-0.187167\pi$
$13$	6.00000	1.66410	0.832050	+	0.554700i	$0.187167\pi$
$14$	1.00000	0.267261
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	0	0
$19$	−4.00000	−0.917663	−0.458831	−	0.888523i	$-0.651732\pi$
$19$	−4.00000	−0.917663	−0.458831	+	0.888523i	$0.651732\pi$
$20$	2.00000	0.447214
$21$	0	0
$22$	−4.00000	−0.852803
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	−6.00000	−1.17670
$27$	0	0
$28$	−1.00000	−0.188982
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.00000	0.342997
$35$	−2.00000	−0.338062
$36$	0	0
$37$	−10.0000	−1.64399	−0.821995	−	0.569495i	$-0.807139\pi$
$37$	−10.0000	−1.64399	−0.821995	+	0.569495i	$0.807139\pi$
$38$	4.00000	0.648886
$39$	0	0
$40$	−2.00000	−0.316228
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	4.00000	0.603023
$45$	0	0
$46$	8.00000	1.17954
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.00000	0.141421
$51$	0	0
$52$	6.00000	0.832050
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	0	0
$55$	8.00000	1.07872
$56$	1.00000	0.133631
$57$	0	0
$58$	−2.00000	−0.262613
$59$	−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
$61$	6.00000	0.768221	0.384111	−	0.923287i	$-0.374508\pi$
$61$	6.00000	0.768221	0.384111	+	0.923287i	$0.374508\pi$
$62$	0	0
$63$	0	0
$64$	1.00000	0.125000
$65$	12.0000	1.48842
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	−2.00000	−0.242536
$69$	0	0
$70$	2.00000	0.239046
$71$	−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
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	10.0000	1.16248
$75$	0	0
$76$	−4.00000	−0.458831
$77$	−4.00000	−0.455842
$78$	0	0
$79$	0	0		−	1.00000i	$-0.5\pi$
$79$	0	0			1.00000i	$0.5\pi$
$80$	2.00000	0.223607
$81$	0	0
$82$	−6.00000	−0.662589
$83$	4.00000	0.439057	0.219529	−	0.975606i	$-0.429548\pi$
$83$	4.00000	0.439057	0.219529	+	0.975606i	$0.429548\pi$
$84$	0	0
$85$	−4.00000	−0.433861
$86$	4.00000	0.431331
$87$	0	0
$88$	−4.00000	−0.426401
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	−6.00000	−0.628971
$92$	−8.00000	−0.834058
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	−1.00000	−0.101015
$99$	0	0
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	0	0
$103$	8.00000	0.788263	0.394132	−	0.919054i	$-0.371045\pi$
$103$	8.00000	0.788263	0.394132	+	0.919054i	$0.371045\pi$
$104$	−6.00000	−0.588348
$105$	0	0
$106$	6.00000	0.582772
$107$	−12.0000	−1.16008	−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000	−1.16008	−0.580042	+	0.814587i	$0.696964\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	−8.00000	−0.762770
$111$	0	0
$112$	−1.00000	−0.0944911
$113$	14.0000	1.31701	0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000	1.31701	0.658505	+	0.752577i	$0.271189\pi$
$114$	0	0
$115$	−16.0000	−1.49201
$116$	2.00000	0.185695
$117$	0	0
$118$	4.00000	0.368230
$119$	2.00000	0.183340
$120$	0	0
$121$	5.00000	0.454545
$122$	−6.00000	−0.543214
$123$	0	0
$124$	0	0
$125$	−12.0000	−1.07331
$126$	0	0
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−12.0000	−1.05247
$131$	20.0000	1.74741	0.873704	−	0.486458i	$-0.161711\pi$
$131$	20.0000	1.74741	0.873704	+	0.486458i	$0.161711\pi$
$132$	0	0
$133$	4.00000	0.346844
$134$	−4.00000	−0.345547
$135$	0	0
$136$	2.00000	0.171499
$137$	−10.0000	−0.854358	−0.427179	−	0.904167i	$-0.640493\pi$
$137$	−10.0000	−0.854358	−0.427179	+	0.904167i	$0.640493\pi$
$138$	0	0
$139$	4.00000	0.339276	0.169638	−	0.985506i	$-0.445740\pi$
$139$	4.00000	0.339276	0.169638	+	0.985506i	$0.445740\pi$
$140$	−2.00000	−0.169031
$141$	0	0
$142$	8.00000	0.671345
$143$	24.0000	2.00698
$144$	0	0
$145$	4.00000	0.332182
$146$	−10.0000	−0.827606
$147$	0	0
$148$	−10.0000	−0.821995
$149$	−6.00000	−0.491539	−0.245770	−	0.969328i	$-0.579041\pi$
$149$	−6.00000	−0.491539	−0.245770	+	0.969328i	$0.579041\pi$
$150$	0	0
$151$	−8.00000	−0.651031	−0.325515	−	0.945537i	$-0.605538\pi$
$151$	−8.00000	−0.651031	−0.325515	+	0.945537i	$0.605538\pi$
$152$	4.00000	0.324443
$153$	0	0
$154$	4.00000	0.322329
$155$	0	0
$156$	0	0
$157$	−10.0000	−0.798087	−0.399043	−	0.916932i	$-0.630658\pi$
$157$	−10.0000	−0.798087	−0.399043	+	0.916932i	$0.630658\pi$
$158$	0	0
$159$	0	0
$160$	−2.00000	−0.158114
$161$	8.00000	0.630488
$162$	0	0
$163$	20.0000	1.56652	0.783260	−	0.621694i	$-0.213555\pi$
$163$	20.0000	1.56652	0.783260	+	0.621694i	$0.213555\pi$
$164$	6.00000	0.468521
$165$	0	0
$166$	−4.00000	−0.310460
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	23.0000	1.76923
$170$	4.00000	0.306786
$171$	0	0
$172$	−4.00000	−0.304997
$173$	−22.0000	−1.67263	−0.836315	−	0.548250i	$-0.815294\pi$
$173$	−22.0000	−1.67263	−0.836315	+	0.548250i	$0.815294\pi$
$174$	0	0
$175$	1.00000	0.0755929
$176$	4.00000	0.301511
$177$	0	0
$178$	−6.00000	−0.449719
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	6.00000	0.444750
$183$	0	0
$184$	8.00000	0.589768
$185$	−20.0000	−1.47043
$186$	0	0
$187$	−8.00000	−0.585018
$188$	0	0
$189$	0	0
$190$	8.00000	0.580381
$191$	0	0		−	1.00000i	$-0.5\pi$
$191$	0	0			1.00000i	$0.5\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	14.0000	1.00514
$195$	0	0
$196$	1.00000	0.0714286
$197$	10.0000	0.712470	0.356235	−	0.934396i	$-0.384060\pi$
$197$	10.0000	0.712470	0.356235	+	0.934396i	$0.384060\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	0	0
$202$	−2.00000	−0.140720
$203$	−2.00000	−0.140372
$204$	0	0
$205$	12.0000	0.838116
$206$	−8.00000	−0.557386
$207$	0	0
$208$	6.00000	0.416025
$209$	−16.0000	−1.10674
$210$	0	0
$211$	20.0000	1.37686	0.688428	−	0.725304i	$-0.258301\pi$
$211$	20.0000	1.37686	0.688428	+	0.725304i	$0.258301\pi$
$212$	−6.00000	−0.412082
$213$	0	0
$214$	12.0000	0.820303
$215$	−8.00000	−0.545595
$216$	0	0
$217$	0	0
$218$	2.00000	0.135457
$219$	0	0
$220$	8.00000	0.539360
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−16.0000	−1.07144	−0.535720	−	0.844396i	$-0.679960\pi$
$223$	−16.0000	−1.07144	−0.535720	+	0.844396i	$0.679960\pi$
$224$	1.00000	0.0668153
$225$	0	0
$226$	−14.0000	−0.931266
$227$	−12.0000	−0.796468	−0.398234	−	0.917284i	$-0.630377\pi$
$227$	−12.0000	−0.796468	−0.398234	+	0.917284i	$0.630377\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	16.0000	1.05501
$231$	0	0
$232$	−2.00000	−0.131306
$233$	22.0000	1.44127	0.720634	−	0.693316i	$-0.243851\pi$
$233$	22.0000	1.44127	0.720634	+	0.693316i	$0.243851\pi$
$234$	0	0
$235$	0	0
$236$	−4.00000	−0.260378
$237$	0	0
$238$	−2.00000	−0.129641
$239$	0	0		−	1.00000i	$-0.5\pi$
$239$	0	0			1.00000i	$0.5\pi$
$240$	0	0
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−5.00000	−0.321412
$243$	0	0
$244$	6.00000	0.384111
$245$	2.00000	0.127775
$246$	0	0
$247$	−24.0000	−1.52708
$248$	0	0
$249$	0	0
$250$	12.0000	0.758947
$251$	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
$253$	−32.0000	−2.01182
$254$	0	0
$255$	0	0
$256$	1.00000	0.0625000
$257$	30.0000	1.87135	0.935674	−	0.352865i	$-0.114792\pi$
$257$	30.0000	1.87135	0.935674	+	0.352865i	$0.114792\pi$
$258$	0	0
$259$	10.0000	0.621370
$260$	12.0000	0.744208
$261$	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
$265$	−12.0000	−0.737154
$266$	−4.00000	−0.245256
$267$	0	0
$268$	4.00000	0.244339
$269$	−22.0000	−1.34136	−0.670682	−	0.741745i	$-0.733998\pi$
$269$	−22.0000	−1.34136	−0.670682	+	0.741745i	$0.733998\pi$
$270$	0	0
$271$	0	0		−	1.00000i	$-0.5\pi$
$271$	0	0			1.00000i	$0.5\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	10.0000	0.604122
$275$	−4.00000	−0.241209
$276$	0	0
$277$	−10.0000	−0.600842	−0.300421	−	0.953807i	$-0.597127\pi$
$277$	−10.0000	−0.600842	−0.300421	+	0.953807i	$0.597127\pi$
$278$	−4.00000	−0.239904
$279$	0	0
$280$	2.00000	0.119523
$281$	−26.0000	−1.55103	−0.775515	−	0.631329i	$-0.782510\pi$
$281$	−26.0000	−1.55103	−0.775515	+	0.631329i	$0.782510\pi$
$282$	0	0
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	−8.00000	−0.474713
$285$	0	0
$286$	−24.0000	−1.41915
$287$	−6.00000	−0.354169
$288$	0	0
$289$	−13.0000	−0.764706
$290$	−4.00000	−0.234888
$291$	0	0
$292$	10.0000	0.585206
$293$	−30.0000	−1.75262	−0.876309	−	0.481749i	$-0.840002\pi$
$293$	−30.0000	−1.75262	−0.876309	+	0.481749i	$0.840002\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	10.0000	0.581238
$297$	0	0
$298$	6.00000	0.347571
$299$	−48.0000	−2.77591
$300$	0	0
$301$	4.00000	0.230556
$302$	8.00000	0.460348
$303$	0	0
$304$	−4.00000	−0.229416
$305$	12.0000	0.687118
$306$	0	0
$307$	28.0000	1.59804	0.799022	−	0.601302i	$-0.205351\pi$
$307$	28.0000	1.59804	0.799022	+	0.601302i	$0.205351\pi$
$308$	−4.00000	−0.227921
$309$	0	0
$310$	0	0
$311$	8.00000	0.453638	0.226819	−	0.973937i	$-0.427167\pi$
$311$	8.00000	0.453638	0.226819	+	0.973937i	$0.427167\pi$
$312$	0	0
$313$	10.0000	0.565233	0.282617	−	0.959233i	$-0.408798\pi$
$313$	10.0000	0.565233	0.282617	+	0.959233i	$0.408798\pi$
$314$	10.0000	0.564333
$315$	0	0
$316$	0	0
$317$	18.0000	1.01098	0.505490	−	0.862832i	$-0.331312\pi$
$317$	18.0000	1.01098	0.505490	+	0.862832i	$0.331312\pi$
$318$	0	0
$319$	8.00000	0.447914
$320$	2.00000	0.111803
$321$	0	0
$322$	−8.00000	−0.445823
$323$	8.00000	0.445132
$324$	0	0
$325$	−6.00000	−0.332820
$326$	−20.0000	−1.10770
$327$	0	0
$328$	−6.00000	−0.331295
$329$	0	0
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	4.00000	0.219529
$333$	0	0
$334$	−8.00000	−0.437741
$335$	8.00000	0.437087
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	−23.0000	−1.25104
$339$	0	0
$340$	−4.00000	−0.216930
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	4.00000	0.215666
$345$	0	0
$346$	22.0000	1.18273
$347$	−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
$349$	22.0000	1.17763	0.588817	−	0.808267i	$-0.299594\pi$
$349$	22.0000	1.17763	0.588817	+	0.808267i	$0.299594\pi$
$350$	−1.00000	−0.0534522
$351$	0	0
$352$	−4.00000	−0.213201
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	0	0
$355$	−16.0000	−0.849192
$356$	6.00000	0.317999
$357$	0	0
$358$	−12.0000	−0.634220
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	18.0000	0.946059
$363$	0	0
$364$	−6.00000	−0.314485
$365$	20.0000	1.04685
$366$	0	0
$367$	32.0000	1.67039	0.835193	−	0.549957i	$-0.185356\pi$
$367$	32.0000	1.67039	0.835193	+	0.549957i	$0.185356\pi$
$368$	−8.00000	−0.417029
$369$	0	0
$370$	20.0000	1.03975
$371$	6.00000	0.311504
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	8.00000	0.413670
$375$	0	0
$376$	0	0
$377$	12.0000	0.618031
$378$	0	0
$379$	−20.0000	−1.02733	−0.513665	−	0.857991i	$-0.671713\pi$
$379$	−20.0000	−1.02733	−0.513665	+	0.857991i	$0.671713\pi$
$380$	−8.00000	−0.410391
$381$	0	0
$382$	0	0
$383$	16.0000	0.817562	0.408781	−	0.912633i	$-0.365954\pi$
$383$	16.0000	0.817562	0.408781	+	0.912633i	$0.365954\pi$
$384$	0	0
$385$	−8.00000	−0.407718
$386$	−2.00000	−0.101797
$387$	0	0
$388$	−14.0000	−0.710742
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	16.0000	0.809155
$392$	−1.00000	−0.0505076
$393$	0	0
$394$	−10.0000	−0.503793
$395$	0	0
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−8.00000	−0.401004
$399$	0	0
$400$	−1.00000	−0.0500000
$401$	−18.0000	−0.898877	−0.449439	−	0.893311i	$-0.648376\pi$
$401$	−18.0000	−0.898877	−0.449439	+	0.893311i	$0.648376\pi$
$402$	0	0
$403$	0	0
$404$	2.00000	0.0995037
$405$	0	0
$406$	2.00000	0.0992583
$407$	−40.0000	−1.98273
$408$	0	0
$409$	−22.0000	−1.08783	−0.543915	−	0.839140i	$-0.683059\pi$
$409$	−22.0000	−1.08783	−0.543915	+	0.839140i	$0.683059\pi$
$410$	−12.0000	−0.592638
$411$	0	0
$412$	8.00000	0.394132
$413$	4.00000	0.196827
$414$	0	0
$415$	8.00000	0.392705
$416$	−6.00000	−0.294174
$417$	0	0
$418$	16.0000	0.782586
$419$	36.0000	1.75872	0.879358	−	0.476162i	$-0.157972\pi$
$419$	36.0000	1.75872	0.879358	+	0.476162i	$0.157972\pi$
$420$	0	0
$421$	6.00000	0.292422	0.146211	−	0.989253i	$-0.453292\pi$
$421$	6.00000	0.292422	0.146211	+	0.989253i	$0.453292\pi$
$422$	−20.0000	−0.973585
$423$	0	0
$424$	6.00000	0.291386
$425$	2.00000	0.0970143
$426$	0	0
$427$	−6.00000	−0.290360
$428$	−12.0000	−0.580042
$429$	0	0
$430$	8.00000	0.385794
$431$	0	0		−	1.00000i	$-0.5\pi$
$431$	0	0			1.00000i	$0.5\pi$
$432$	0	0
$433$	2.00000	0.0961139	0.0480569	−	0.998845i	$-0.484697\pi$
$433$	2.00000	0.0961139	0.0480569	+	0.998845i	$0.484697\pi$
$434$	0	0
$435$	0	0
$436$	−2.00000	−0.0957826
$437$	32.0000	1.53077
$438$	0	0
$439$	−24.0000	−1.14546	−0.572729	−	0.819745i	$-0.694115\pi$
$439$	−24.0000	−1.14546	−0.572729	+	0.819745i	$0.694115\pi$
$440$	−8.00000	−0.381385
$441$	0	0
$442$	12.0000	0.570782
$443$	4.00000	0.190046	0.0950229	−	0.995475i	$-0.469708\pi$
$443$	4.00000	0.190046	0.0950229	+	0.995475i	$0.469708\pi$
$444$	0	0
$445$	12.0000	0.568855
$446$	16.0000	0.757622
$447$	0	0
$448$	−1.00000	−0.0472456
$449$	−34.0000	−1.60456	−0.802280	−	0.596948i	$-0.796380\pi$
$449$	−34.0000	−1.60456	−0.802280	+	0.596948i	$0.796380\pi$
$450$	0	0
$451$	24.0000	1.13012
$452$	14.0000	0.658505
$453$	0	0
$454$	12.0000	0.563188
$455$	−12.0000	−0.562569
$456$	0	0
$457$	10.0000	0.467780	0.233890	−	0.972263i	$-0.424854\pi$
$457$	10.0000	0.467780	0.233890	+	0.972263i	$0.424854\pi$
$458$	2.00000	0.0934539
$459$	0	0
$460$	−16.0000	−0.746004
$461$	−22.0000	−1.02464	−0.512321	−	0.858794i	$-0.671214\pi$
$461$	−22.0000	−1.02464	−0.512321	+	0.858794i	$0.671214\pi$
$462$	0	0
$463$	−32.0000	−1.48717	−0.743583	−	0.668644i	$-0.766875\pi$
$463$	−32.0000	−1.48717	−0.743583	+	0.668644i	$0.766875\pi$
$464$	2.00000	0.0928477
$465$	0	0
$466$	−22.0000	−1.01913
$467$	−28.0000	−1.29569	−0.647843	−	0.761774i	$-0.724329\pi$
$467$	−28.0000	−1.29569	−0.647843	+	0.761774i	$0.724329\pi$
$468$	0	0
$469$	−4.00000	−0.184703
$470$	0	0
$471$	0	0
$472$	4.00000	0.184115
$473$	−16.0000	−0.735681
$474$	0	0
$475$	4.00000	0.183533
$476$	2.00000	0.0916698
$477$	0	0
$478$	0	0
$479$	16.0000	0.731059	0.365529	−	0.930800i	$-0.380888\pi$
$479$	16.0000	0.731059	0.365529	+	0.930800i	$0.380888\pi$
$480$	0	0
$481$	−60.0000	−2.73576
$482$	−2.00000	−0.0910975
$483$	0	0
$484$	5.00000	0.227273
$485$	−28.0000	−1.27141
$486$	0	0
$487$	8.00000	0.362515	0.181257	−	0.983436i	$-0.441983\pi$
$487$	8.00000	0.362515	0.181257	+	0.983436i	$0.441983\pi$
$488$	−6.00000	−0.271607
$489$	0	0
$490$	−2.00000	−0.0903508
$491$	−12.0000	−0.541552	−0.270776	−	0.962642i	$-0.587280\pi$
$491$	−12.0000	−0.541552	−0.270776	+	0.962642i	$0.587280\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	24.0000	1.07981
$495$	0	0
$496$	0	0
$497$	8.00000	0.358849
$498$	0	0
$499$	−44.0000	−1.96971	−0.984855	−	0.173379i	$-0.944532\pi$
$499$	−44.0000	−1.96971	−0.984855	+	0.173379i	$0.944532\pi$
$500$	−12.0000	−0.536656
$501$	0	0
$502$	−12.0000	−0.535586
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	32.0000	1.42257
$507$	0	0
$508$	0	0
$509$	−6.00000	−0.265945	−0.132973	−	0.991120i	$-0.542452\pi$
$509$	−6.00000	−0.265945	−0.132973	+	0.991120i	$0.542452\pi$
$510$	0	0
$511$	−10.0000	−0.442374
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−30.0000	−1.32324
$515$	16.0000	0.705044
$516$	0	0
$517$	0	0
$518$	−10.0000	−0.439375
$519$	0	0
$520$	−12.0000	−0.526235
$521$	6.00000	0.262865	0.131432	−	0.991325i	$-0.458042\pi$
$521$	6.00000	0.262865	0.131432	+	0.991325i	$0.458042\pi$
$522$	0	0
$523$	20.0000	0.874539	0.437269	−	0.899331i	$-0.355946\pi$
$523$	20.0000	0.874539	0.437269	+	0.899331i	$0.355946\pi$
$524$	20.0000	0.873704
$525$	0	0
$526$	−24.0000	−1.04645
$527$	0	0
$528$	0	0
$529$	41.0000	1.78261
$530$	12.0000	0.521247
$531$	0	0
$532$	4.00000	0.173422
$533$	36.0000	1.55933
$534$	0	0
$535$	−24.0000	−1.03761
$536$	−4.00000	−0.172774
$537$	0	0
$538$	22.0000	0.948487
$539$	4.00000	0.172292
$540$	0	0
$541$	30.0000	1.28980	0.644900	−	0.764267i	$-0.276899\pi$
$541$	30.0000	1.28980	0.644900	+	0.764267i	$0.276899\pi$
$542$	0	0
$543$	0	0
$544$	2.00000	0.0857493
$545$	−4.00000	−0.171341
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	−10.0000	−0.427179
$549$	0	0
$550$	4.00000	0.170561
$551$	−8.00000	−0.340811
$552$	0	0
$553$	0	0
$554$	10.0000	0.424859
$555$	0	0
$556$	4.00000	0.169638
$557$	2.00000	0.0847427	0.0423714	−	0.999102i	$-0.486509\pi$
$557$	2.00000	0.0847427	0.0423714	+	0.999102i	$0.486509\pi$
$558$	0	0
$559$	−24.0000	−1.01509
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	26.0000	1.09674
$563$	−44.0000	−1.85438	−0.927189	−	0.374593i	$-0.877783\pi$
$563$	−44.0000	−1.85438	−0.927189	+	0.374593i	$0.877783\pi$
$564$	0	0
$565$	28.0000	1.17797
$566$	−4.00000	−0.168133
$567$	0	0
$568$	8.00000	0.335673
$569$	6.00000	0.251533	0.125767	−	0.992060i	$-0.459861\pi$
$569$	6.00000	0.251533	0.125767	+	0.992060i	$0.459861\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	24.0000	1.00349
$573$	0	0
$574$	6.00000	0.250435
$575$	8.00000	0.333623
$576$	0	0
$577$	34.0000	1.41544	0.707719	−	0.706494i	$-0.249724\pi$
$577$	34.0000	1.41544	0.707719	+	0.706494i	$0.249724\pi$
$578$	13.0000	0.540729
$579$	0	0
$580$	4.00000	0.166091
$581$	−4.00000	−0.165948
$582$	0	0
$583$	−24.0000	−0.993978
$584$	−10.0000	−0.413803
$585$	0	0
$586$	30.0000	1.23929
$587$	28.0000	1.15568	0.577842	−	0.816149i	$-0.303895\pi$
$587$	28.0000	1.15568	0.577842	+	0.816149i	$0.303895\pi$
$588$	0	0
$589$	0	0
$590$	8.00000	0.329355
$591$	0	0
$592$	−10.0000	−0.410997
$593$	−18.0000	−0.739171	−0.369586	−	0.929197i	$-0.620500\pi$
$593$	−18.0000	−0.739171	−0.369586	+	0.929197i	$0.620500\pi$
$594$	0	0
$595$	4.00000	0.163984
$596$	−6.00000	−0.245770
$597$	0	0
$598$	48.0000	1.96287
$599$	−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
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	−4.00000	−0.163028
$603$	0	0
$604$	−8.00000	−0.325515
$605$	10.0000	0.406558
$606$	0	0
$607$	48.0000	1.94826	0.974130	−	0.225989i	$-0.0725612\pi$
$607$	48.0000	1.94826	0.974130	+	0.225989i	$0.0725612\pi$
$608$	4.00000	0.162221
$609$	0	0
$610$	−12.0000	−0.485866
$611$	0	0
$612$	0	0
$613$	−42.0000	−1.69636	−0.848182	−	0.529705i	$-0.822303\pi$
$613$	−42.0000	−1.69636	−0.848182	+	0.529705i	$0.822303\pi$
$614$	−28.0000	−1.12999
$615$	0	0
$616$	4.00000	0.161165
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	−44.0000	−1.76851	−0.884255	−	0.467005i	$-0.845333\pi$
$619$	−44.0000	−1.76851	−0.884255	+	0.467005i	$0.845333\pi$
$620$	0	0
$621$	0	0
$622$	−8.00000	−0.320771
$623$	−6.00000	−0.240385
$624$	0	0
$625$	−19.0000	−0.760000
$626$	−10.0000	−0.399680
$627$	0	0
$628$	−10.0000	−0.399043
$629$	20.0000	0.797452
$630$	0	0
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	0	0
$633$	0	0
$634$	−18.0000	−0.714871
$635$	0	0
$636$	0	0
$637$	6.00000	0.237729
$638$	−8.00000	−0.316723
$639$	0	0
$640$	−2.00000	−0.0790569
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	0	0
$643$	−4.00000	−0.157745	−0.0788723	−	0.996885i	$-0.525132\pi$
$643$	−4.00000	−0.157745	−0.0788723	+	0.996885i	$0.525132\pi$
$644$	8.00000	0.315244
$645$	0	0
$646$	−8.00000	−0.314756
$647$	−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
$649$	−16.0000	−0.628055
$650$	6.00000	0.235339
$651$	0	0
$652$	20.0000	0.783260
$653$	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
$655$	40.0000	1.56293
$656$	6.00000	0.234261
$657$	0	0
$658$	0	0
$659$	28.0000	1.09073	0.545363	−	0.838200i	$-0.316392\pi$
$659$	28.0000	1.09073	0.545363	+	0.838200i	$0.316392\pi$
$660$	0	0
$661$	−2.00000	−0.0777910	−0.0388955	−	0.999243i	$-0.512384\pi$
$661$	−2.00000	−0.0777910	−0.0388955	+	0.999243i	$0.512384\pi$
$662$	4.00000	0.155464
$663$	0	0
$664$	−4.00000	−0.155230
$665$	8.00000	0.310227
$666$	0	0
$667$	−16.0000	−0.619522
$668$	8.00000	0.309529
$669$	0	0
$670$	−8.00000	−0.309067
$671$	24.0000	0.926510
$672$	0	0
$673$	2.00000	0.0770943	0.0385472	−	0.999257i	$-0.487727\pi$
$673$	2.00000	0.0770943	0.0385472	+	0.999257i	$0.487727\pi$
$674$	−18.0000	−0.693334
$675$	0	0
$676$	23.0000	0.884615
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	0	0
$679$	14.0000	0.537271
$680$	4.00000	0.153393
$681$	0	0
$682$	0	0
$683$	−12.0000	−0.459167	−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000	−0.459167	−0.229584	+	0.973289i	$0.573736\pi$
$684$	0	0
$685$	−20.0000	−0.764161
$686$	1.00000	0.0381802
$687$	0	0
$688$	−4.00000	−0.152499
$689$	−36.0000	−1.37149
$690$	0	0
$691$	−4.00000	−0.152167	−0.0760836	−	0.997101i	$-0.524242\pi$
$691$	−4.00000	−0.152167	−0.0760836	+	0.997101i	$0.524242\pi$
$692$	−22.0000	−0.836315
$693$	0	0
$694$	12.0000	0.455514
$695$	8.00000	0.303457
$696$	0	0
$697$	−12.0000	−0.454532
$698$	−22.0000	−0.832712
$699$	0	0
$700$	1.00000	0.0377964
$701$	2.00000	0.0755390	0.0377695	−	0.999286i	$-0.487975\pi$
$701$	2.00000	0.0755390	0.0377695	+	0.999286i	$0.487975\pi$
$702$	0	0
$703$	40.0000	1.50863
$704$	4.00000	0.150756
$705$	0	0
$706$	−30.0000	−1.12906
$707$	−2.00000	−0.0752177
$708$	0	0
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	16.0000	0.600469
$711$	0	0
$712$	−6.00000	−0.224860
$713$	0	0
$714$	0	0
$715$	48.0000	1.79510
$716$	12.0000	0.448461
$717$	0	0
$718$	−8.00000	−0.298557
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	−8.00000	−0.297936
$722$	3.00000	0.111648
$723$	0	0
$724$	−18.0000	−0.668965
$725$	−2.00000	−0.0742781
$726$	0	0
$727$	−8.00000	−0.296704	−0.148352	−	0.988935i	$-0.547397\pi$
$727$	−8.00000	−0.296704	−0.148352	+	0.988935i	$0.547397\pi$
$728$	6.00000	0.222375
$729$	0	0
$730$	−20.0000	−0.740233
$731$	8.00000	0.295891
$732$	0	0
$733$	6.00000	0.221615	0.110808	−	0.993842i	$-0.464656\pi$
$733$	6.00000	0.221615	0.110808	+	0.993842i	$0.464656\pi$
$734$	−32.0000	−1.18114
$735$	0	0
$736$	8.00000	0.294884
$737$	16.0000	0.589368
$738$	0	0
$739$	−12.0000	−0.441427	−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000	−0.441427	−0.220714	+	0.975339i	$0.570839\pi$
$740$	−20.0000	−0.735215
$741$	0	0
$742$	−6.00000	−0.220267
$743$	−24.0000	−0.880475	−0.440237	−	0.897881i	$-0.645106\pi$
$743$	−24.0000	−0.880475	−0.440237	+	0.897881i	$0.645106\pi$
$744$	0	0
$745$	−12.0000	−0.439646
$746$	−22.0000	−0.805477
$747$	0	0
$748$	−8.00000	−0.292509
$749$	12.0000	0.438470
$750$	0	0
$751$	48.0000	1.75154	0.875772	−	0.482724i	$-0.160353\pi$
$751$	48.0000	1.75154	0.875772	+	0.482724i	$0.160353\pi$
$752$	0	0
$753$	0	0
$754$	−12.0000	−0.437014
$755$	−16.0000	−0.582300
$756$	0	0
$757$	6.00000	0.218074	0.109037	−	0.994038i	$-0.465223\pi$
$757$	6.00000	0.218074	0.109037	+	0.994038i	$0.465223\pi$
$758$	20.0000	0.726433
$759$	0	0
$760$	8.00000	0.290191
$761$	22.0000	0.797499	0.398750	−	0.917060i	$-0.369444\pi$
$761$	22.0000	0.797499	0.398750	+	0.917060i	$0.369444\pi$
$762$	0	0
$763$	2.00000	0.0724049
$764$	0	0
$765$	0	0
$766$	−16.0000	−0.578103
$767$	−24.0000	−0.866590
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	8.00000	0.288300
$771$	0	0
$772$	2.00000	0.0719816
$773$	2.00000	0.0719350	0.0359675	−	0.999353i	$-0.488549\pi$
$773$	2.00000	0.0719350	0.0359675	+	0.999353i	$0.488549\pi$
$774$	0	0
$775$	0	0
$776$	14.0000	0.502571
$777$	0	0
$778$	−26.0000	−0.932145
$779$	−24.0000	−0.859889
$780$	0	0
$781$	−32.0000	−1.14505
$782$	−16.0000	−0.572159
$783$	0	0
$784$	1.00000	0.0357143
$785$	−20.0000	−0.713831
$786$	0	0
$787$	−36.0000	−1.28326	−0.641631	−	0.767014i	$-0.721742\pi$
$787$	−36.0000	−1.28326	−0.641631	+	0.767014i	$0.721742\pi$
$788$	10.0000	0.356235
$789$	0	0
$790$	0	0
$791$	−14.0000	−0.497783
$792$	0	0
$793$	36.0000	1.27840
$794$	−6.00000	−0.212932
$795$	0	0
$796$	8.00000	0.283552
$797$	−6.00000	−0.212531	−0.106265	−	0.994338i	$-0.533889\pi$
$797$	−6.00000	−0.212531	−0.106265	+	0.994338i	$0.533889\pi$
$798$	0	0
$799$	0	0
$800$	1.00000	0.0353553
$801$	0	0
$802$	18.0000	0.635602
$803$	40.0000	1.41157
$804$	0	0
$805$	16.0000	0.563926
$806$	0	0
$807$	0	0
$808$	−2.00000	−0.0703598
$809$	−10.0000	−0.351581	−0.175791	−	0.984428i	$-0.556248\pi$
$809$	−10.0000	−0.351581	−0.175791	+	0.984428i	$0.556248\pi$
$810$	0	0
$811$	−44.0000	−1.54505	−0.772524	−	0.634985i	$-0.781006\pi$
$811$	−44.0000	−1.54505	−0.772524	+	0.634985i	$0.781006\pi$
$812$	−2.00000	−0.0701862
$813$	0	0
$814$	40.0000	1.40200
$815$	40.0000	1.40114
$816$	0	0
$817$	16.0000	0.559769
$818$	22.0000	0.769212
$819$	0	0
$820$	12.0000	0.419058
$821$	−38.0000	−1.32621	−0.663105	−	0.748527i	$-0.730762\pi$
$821$	−38.0000	−1.32621	−0.663105	+	0.748527i	$0.730762\pi$
$822$	0	0
$823$	−56.0000	−1.95204	−0.976019	−	0.217687i	$-0.930149\pi$
$823$	−56.0000	−1.95204	−0.976019	+	0.217687i	$0.930149\pi$
$824$	−8.00000	−0.278693
$825$	0	0
$826$	−4.00000	−0.139178
$827$	36.0000	1.25184	0.625921	−	0.779886i	$-0.284723\pi$
$827$	36.0000	1.25184	0.625921	+	0.779886i	$0.284723\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	−8.00000	−0.277684
$831$	0	0
$832$	6.00000	0.208013
$833$	−2.00000	−0.0692959
$834$	0	0
$835$	16.0000	0.553703
$836$	−16.0000	−0.553372
$837$	0	0
$838$	−36.0000	−1.24360
$839$	−56.0000	−1.93333	−0.966667	−	0.256036i	$-0.917584\pi$
$839$	−56.0000	−1.93333	−0.966667	+	0.256036i	$0.917584\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	−6.00000	−0.206774
$843$	0	0
$844$	20.0000	0.688428
$845$	46.0000	1.58245
$846$	0	0
$847$	−5.00000	−0.171802
$848$	−6.00000	−0.206041
$849$	0	0
$850$	−2.00000	−0.0685994
$851$	80.0000	2.74236
$852$	0	0
$853$	14.0000	0.479351	0.239675	−	0.970853i	$-0.422959\pi$
$853$	14.0000	0.479351	0.239675	+	0.970853i	$0.422959\pi$
$854$	6.00000	0.205316
$855$	0	0
$856$	12.0000	0.410152
$857$	−42.0000	−1.43469	−0.717346	−	0.696717i	$-0.754643\pi$
$857$	−42.0000	−1.43469	−0.717346	+	0.696717i	$0.754643\pi$
$858$	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
$862$	0	0
$863$	32.0000	1.08929	0.544646	−	0.838666i	$-0.316664\pi$
$863$	32.0000	1.08929	0.544646	+	0.838666i	$0.316664\pi$
$864$	0	0
$865$	−44.0000	−1.49604
$866$	−2.00000	−0.0679628
$867$	0	0
$868$	0	0
$869$	0	0
$870$	0	0
$871$	24.0000	0.813209
$872$	2.00000	0.0677285
$873$	0	0
$874$	−32.0000	−1.08242
$875$	12.0000	0.405674
$876$	0	0
$877$	−2.00000	−0.0675352	−0.0337676	−	0.999430i	$-0.510751\pi$
$877$	−2.00000	−0.0675352	−0.0337676	+	0.999430i	$0.510751\pi$
$878$	24.0000	0.809961
$879$	0	0
$880$	8.00000	0.269680
$881$	−18.0000	−0.606435	−0.303218	−	0.952921i	$-0.598061\pi$
$881$	−18.0000	−0.606435	−0.303218	+	0.952921i	$0.598061\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	−12.0000	−0.403604
$885$	0	0
$886$	−4.00000	−0.134383
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	0	0
$889$	0	0
$890$	−12.0000	−0.402241
$891$	0	0
$892$	−16.0000	−0.535720
$893$	0	0
$894$	0	0
$895$	24.0000	0.802232
$896$	1.00000	0.0334077
$897$	0	0
$898$	34.0000	1.13459
$899$	0	0
$900$	0	0
$901$	12.0000	0.399778
$902$	−24.0000	−0.799113
$903$	0	0
$904$	−14.0000	−0.465633
$905$	−36.0000	−1.19668
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	−12.0000	−0.398234
$909$	0	0
$910$	12.0000	0.397796
$911$	0	0		−	1.00000i	$-0.5\pi$
$911$	0	0			1.00000i	$0.5\pi$
$912$	0	0
$913$	16.0000	0.529523
$914$	−10.0000	−0.330771
$915$	0	0
$916$	−2.00000	−0.0660819
$917$	−20.0000	−0.660458
$918$	0	0
$919$	40.0000	1.31948	0.659739	−	0.751495i	$-0.270667\pi$
$919$	40.0000	1.31948	0.659739	+	0.751495i	$0.270667\pi$
$920$	16.0000	0.527504
$921$	0	0
$922$	22.0000	0.724531
$923$	−48.0000	−1.57994
$924$	0	0
$925$	10.0000	0.328798
$926$	32.0000	1.05159
$927$	0	0
$928$	−2.00000	−0.0656532
$929$	−18.0000	−0.590561	−0.295280	−	0.955411i	$-0.595413\pi$
$929$	−18.0000	−0.590561	−0.295280	+	0.955411i	$0.595413\pi$
$930$	0	0
$931$	−4.00000	−0.131095
$932$	22.0000	0.720634
$933$	0	0
$934$	28.0000	0.916188
$935$	−16.0000	−0.523256
$936$	0	0
$937$	−22.0000	−0.718709	−0.359354	−	0.933201i	$-0.617003\pi$
$937$	−22.0000	−0.718709	−0.359354	+	0.933201i	$0.617003\pi$
$938$	4.00000	0.130605
$939$	0	0
$940$	0	0
$941$	26.0000	0.847576	0.423788	−	0.905761i	$-0.360700\pi$
$941$	26.0000	0.847576	0.423788	+	0.905761i	$0.360700\pi$
$942$	0	0
$943$	−48.0000	−1.56310
$944$	−4.00000	−0.130189
$945$	0	0
$946$	16.0000	0.520205
$947$	−4.00000	−0.129983	−0.0649913	−	0.997886i	$-0.520702\pi$
$947$	−4.00000	−0.129983	−0.0649913	+	0.997886i	$0.520702\pi$
$948$	0	0
$949$	60.0000	1.94768
$950$	−4.00000	−0.129777
$951$	0	0
$952$	−2.00000	−0.0648204
$953$	−26.0000	−0.842223	−0.421111	−	0.907009i	$-0.638360\pi$
$953$	−26.0000	−0.842223	−0.421111	+	0.907009i	$0.638360\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	−16.0000	−0.516937
$959$	10.0000	0.322917
$960$	0	0
$961$	−31.0000	−1.00000
$962$	60.0000	1.93448
$963$	0	0
$964$	2.00000	0.0644157
$965$	4.00000	0.128765
$966$	0	0
$967$	8.00000	0.257263	0.128631	−	0.991692i	$-0.458942\pi$
$967$	8.00000	0.257263	0.128631	+	0.991692i	$0.458942\pi$
$968$	−5.00000	−0.160706
$969$	0	0
$970$	28.0000	0.899026
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−4.00000	−0.128234
$974$	−8.00000	−0.256337
$975$	0	0
$976$	6.00000	0.192055
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	0	0
$979$	24.0000	0.767043
$980$	2.00000	0.0638877
$981$	0	0
$982$	12.0000	0.382935
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	0	0
$985$	20.0000	0.637253
$986$	4.00000	0.127386
$987$	0	0
$988$	−24.0000	−0.763542
$989$	32.0000	1.01754
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	−8.00000	−0.253745
$995$	16.0000	0.507234
$996$	0	0
$997$	14.0000	0.443384	0.221692	−	0.975117i	$-0.428842\pi$
$997$	14.0000	0.443384	0.221692	+	0.975117i	$0.428842\pi$
$998$	44.0000	1.39280
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.2.a.a.1.1		1
3.2	odd	2		42.2.a.a.1.1	✓	1
4.3	odd	2		1008.2.a.j.1.1		1
5.2	odd	4		3150.2.g.r.2899.1		2
5.3	odd	4		3150.2.g.r.2899.2		2
5.4	even	2		3150.2.a.bo.1.1		1
7.2	even	3		882.2.g.h.361.1		2
7.3	odd	6		882.2.g.j.667.1		2
7.4	even	3		882.2.g.h.667.1		2
7.5	odd	6		882.2.g.j.361.1		2
7.6	odd	2		882.2.a.b.1.1		1
8.3	odd	2		4032.2.a.m.1.1		1
8.5	even	2		4032.2.a.e.1.1		1
9.2	odd	6		1134.2.f.g.757.1		2
9.4	even	3		1134.2.f.j.379.1		2
9.5	odd	6		1134.2.f.g.379.1		2
9.7	even	3		1134.2.f.j.757.1		2
12.11	even	2		336.2.a.d.1.1		1
15.2	even	4		1050.2.g.a.799.2		2
15.8	even	4		1050.2.g.a.799.1		2
15.14	odd	2		1050.2.a.i.1.1		1
21.2	odd	6		294.2.e.c.67.1		2
21.5	even	6		294.2.e.a.67.1		2
21.11	odd	6		294.2.e.c.79.1		2
21.17	even	6		294.2.e.a.79.1		2
21.20	even	2		294.2.a.g.1.1		1
24.5	odd	2		1344.2.a.q.1.1		1
24.11	even	2		1344.2.a.i.1.1		1
28.27	even	2		7056.2.a.k.1.1		1
33.32	even	2		5082.2.a.d.1.1		1
39.38	odd	2		7098.2.a.f.1.1		1
48.5	odd	4		5376.2.c.bc.2689.2		2
48.11	even	4		5376.2.c.e.2689.1		2
48.29	odd	4		5376.2.c.bc.2689.1		2
48.35	even	4		5376.2.c.e.2689.2		2
60.59	even	2		8400.2.a.k.1.1		1
84.11	even	6		2352.2.q.i.961.1		2
84.23	even	6		2352.2.q.i.1537.1		2
84.47	odd	6		2352.2.q.n.1537.1		2
84.59	odd	6		2352.2.q.n.961.1		2
84.83	odd	2		2352.2.a.l.1.1		1
105.104	even	2		7350.2.a.f.1.1		1
168.83	odd	2		9408.2.a.bw.1.1		1
168.125	even	2		9408.2.a.n.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.2.a.a.1.1	✓	1	3.2	odd	2
126.2.a.a.1.1		1	1.1	even	1	trivial
294.2.a.g.1.1		1	21.20	even	2
294.2.e.a.67.1		2	21.5	even	6
294.2.e.a.79.1		2	21.17	even	6
294.2.e.c.67.1		2	21.2	odd	6
294.2.e.c.79.1		2	21.11	odd	6
336.2.a.d.1.1		1	12.11	even	2
882.2.a.b.1.1		1	7.6	odd	2
882.2.g.h.361.1		2	7.2	even	3
882.2.g.h.667.1		2	7.4	even	3
882.2.g.j.361.1		2	7.5	odd	6
882.2.g.j.667.1		2	7.3	odd	6
1008.2.a.j.1.1		1	4.3	odd	2
1050.2.a.i.1.1		1	15.14	odd	2
1050.2.g.a.799.1		2	15.8	even	4
1050.2.g.a.799.2		2	15.2	even	4
1134.2.f.g.379.1		2	9.5	odd	6
1134.2.f.g.757.1		2	9.2	odd	6
1134.2.f.j.379.1		2	9.4	even	3
1134.2.f.j.757.1		2	9.7	even	3
1344.2.a.i.1.1		1	24.11	even	2
1344.2.a.q.1.1		1	24.5	odd	2
2352.2.a.l.1.1		1	84.83	odd	2
2352.2.q.i.961.1		2	84.11	even	6
2352.2.q.i.1537.1		2	84.23	even	6
2352.2.q.n.961.1		2	84.59	odd	6
2352.2.q.n.1537.1		2	84.47	odd	6
3150.2.a.bo.1.1		1	5.4	even	2
3150.2.g.r.2899.1		2	5.2	odd	4
3150.2.g.r.2899.2		2	5.3	odd	4
4032.2.a.e.1.1		1	8.5	even	2
4032.2.a.m.1.1		1	8.3	odd	2
5082.2.a.d.1.1		1	33.32	even	2
5376.2.c.e.2689.1		2	48.11	even	4
5376.2.c.e.2689.2		2	48.35	even	4
5376.2.c.bc.2689.1		2	48.29	odd	4
5376.2.c.bc.2689.2		2	48.5	odd	4
7056.2.a.k.1.1		1	28.27	even	2
7098.2.a.f.1.1		1	39.38	odd	2
7350.2.a.f.1.1		1	105.104	even	2
8400.2.a.k.1.1		1	60.59	even	2
9408.2.a.n.1.1		1	168.125	even	2
9408.2.a.bw.1.1		1	168.83	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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