LMFDB - Embedded newform 336.6.a.g.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(1,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 6, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("336.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.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:	$53.8889634572$
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	$\chi$	$=$	336.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-9.00000 q^{3} +24.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} +O(q^{10})$

$q-9.00000 q^{3} +24.0000 q^{5} -49.0000 q^{7} +81.0000 q^{9} -66.0000 q^{11} +98.0000 q^{13} -216.000 q^{15} -216.000 q^{17} +340.000 q^{19} +441.000 q^{21} +1038.00 q^{23} -2549.00 q^{25} -729.000 q^{27} -2490.00 q^{29} +7048.00 q^{31} +594.000 q^{33} -1176.00 q^{35} -12238.0 q^{37} -882.000 q^{39} +6468.00 q^{41} +15412.0 q^{43} +1944.00 q^{45} -20604.0 q^{47} +2401.00 q^{49} +1944.00 q^{51} +32490.0 q^{53} -1584.00 q^{55} -3060.00 q^{57} -34224.0 q^{59} +35654.0 q^{61} -3969.00 q^{63} +2352.00 q^{65} -12680.0 q^{67} -9342.00 q^{69} +42642.0 q^{71} +33734.0 q^{73} +22941.0 q^{75} +3234.00 q^{77} +85108.0 q^{79} +6561.00 q^{81} +106764. q^{83} -5184.00 q^{85} +22410.0 q^{87} +34884.0 q^{89} -4802.00 q^{91} -63432.0 q^{93} +8160.00 q^{95} +18662.0 q^{97} -5346.00 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−9.00000	−0.577350
$3$	−9.00000	−0.577350
$4$	0	0
$5$	24.0000	0.429325	0.214663	−	0.976688i	$-0.431135\pi$
$5$	24.0000	0.429325	0.214663	+	0.976688i	$0.431135\pi$
$6$	0	0
$7$	−49.0000	−0.377964
$7$	−49.0000	−0.377964
$8$	0	0
$9$	81.0000	0.333333
$10$	0	0
$11$	−66.0000	−0.164461	−0.0822304	−	0.996613i	$-0.526204\pi$
$11$	−66.0000	−0.164461	−0.0822304	+	0.996613i	$0.526204\pi$
$12$	0	0
$13$	98.0000	0.160830	0.0804151	−	0.996761i	$-0.474375\pi$
$13$	98.0000	0.160830	0.0804151	+	0.996761i	$0.474375\pi$
$14$	0	0
$15$	−216.000	−0.247871
$16$	0	0
$17$	−216.000	−0.181272	−0.0906362	−	0.995884i	$-0.528890\pi$
$17$	−216.000	−0.181272	−0.0906362	+	0.995884i	$0.528890\pi$
$18$	0	0
$19$	340.000	0.216070	0.108035	−	0.994147i	$-0.465544\pi$
$19$	340.000	0.216070	0.108035	+	0.994147i	$0.465544\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	1038.00	0.409145	0.204573	−	0.978851i	$-0.434420\pi$
$23$	1038.00	0.409145	0.204573	+	0.978851i	$0.434420\pi$
$24$	0	0
$25$	−2549.00	−0.815680
$26$	0	0
$27$	−729.000	−0.192450
$28$	0	0
$29$	−2490.00	−0.549800	−0.274900	−	0.961473i	$-0.588645\pi$
$29$	−2490.00	−0.549800	−0.274900	+	0.961473i	$0.588645\pi$
$30$	0	0
$31$	7048.00	1.31723	0.658615	−	0.752480i	$-0.271143\pi$
$31$	7048.00	1.31723	0.658615	+	0.752480i	$0.271143\pi$
$32$	0	0
$33$	594.000	0.0949514
$34$	0	0
$35$	−1176.00	−0.162270
$36$	0	0
$37$	−12238.0	−1.46962	−0.734812	−	0.678271i	$-0.762730\pi$
$37$	−12238.0	−1.46962	−0.734812	+	0.678271i	$0.762730\pi$
$38$	0	0
$39$	−882.000	−0.0928554
$40$	0	0
$41$	6468.00	0.600911	0.300456	−	0.953796i	$-0.402861\pi$
$41$	6468.00	0.600911	0.300456	+	0.953796i	$0.402861\pi$
$42$	0	0
$43$	15412.0	1.27112	0.635562	−	0.772050i	$-0.280768\pi$
$43$	15412.0	1.27112	0.635562	+	0.772050i	$0.280768\pi$
$44$	0	0
$45$	1944.00	0.143108
$46$	0	0
$47$	−20604.0	−1.36053	−0.680263	−	0.732968i	$-0.738134\pi$
$47$	−20604.0	−1.36053	−0.680263	+	0.732968i	$0.738134\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	1944.00	0.104658
$52$	0	0
$53$	32490.0	1.58877	0.794383	−	0.607417i	$-0.207794\pi$
$53$	32490.0	1.58877	0.794383	+	0.607417i	$0.207794\pi$
$54$	0	0
$55$	−1584.00	−0.0706071
$56$	0	0
$57$	−3060.00	−0.124748
$58$	0	0
$59$	−34224.0	−1.27997	−0.639986	−	0.768386i	$-0.721060\pi$
$59$	−34224.0	−1.27997	−0.639986	+	0.768386i	$0.721060\pi$
$60$	0	0
$61$	35654.0	1.22683	0.613414	−	0.789762i	$-0.289796\pi$
$61$	35654.0	1.22683	0.613414	+	0.789762i	$0.289796\pi$
$62$	0	0
$63$	−3969.00	−0.125988
$64$	0	0
$65$	2352.00	0.0690484
$66$	0	0
$67$	−12680.0	−0.345090	−0.172545	−	0.985002i	$-0.555199\pi$
$67$	−12680.0	−0.345090	−0.172545	+	0.985002i	$0.555199\pi$
$68$	0	0
$69$	−9342.00	−0.236220
$70$	0	0
$71$	42642.0	1.00390	0.501951	−	0.864896i	$-0.332616\pi$
$71$	42642.0	1.00390	0.501951	+	0.864896i	$0.332616\pi$
$72$	0	0
$73$	33734.0	0.740902	0.370451	−	0.928852i	$-0.379203\pi$
$73$	33734.0	0.740902	0.370451	+	0.928852i	$0.379203\pi$
$74$	0	0
$75$	22941.0	0.470933
$76$	0	0
$77$	3234.00	0.0621603
$78$	0	0
$79$	85108.0	1.53427	0.767137	−	0.641484i	$-0.221681\pi$
$79$	85108.0	1.53427	0.767137	+	0.641484i	$0.221681\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	106764.	1.70110	0.850550	−	0.525895i	$-0.176270\pi$
$83$	106764.	1.70110	0.850550	+	0.525895i	$0.176270\pi$
$84$	0	0
$85$	−5184.00	−0.0778247
$86$	0	0
$87$	22410.0	0.317427
$88$	0	0
$89$	34884.0	0.466822	0.233411	−	0.972378i	$-0.425011\pi$
$89$	34884.0	0.466822	0.233411	+	0.972378i	$0.425011\pi$
$90$	0	0
$91$	−4802.00	−0.0607881
$92$	0	0
$93$	−63432.0	−0.760503
$94$	0	0
$95$	8160.00	0.0927644
$96$	0	0
$97$	18662.0	0.201386	0.100693	−	0.994918i	$-0.467894\pi$
$97$	18662.0	0.201386	0.100693	+	0.994918i	$0.467894\pi$
$98$	0	0
$99$	−5346.00	−0.0548202
$100$	0	0
$101$	153084.	1.49323	0.746614	−	0.665257i	$-0.231678\pi$
$101$	153084.	1.49323	0.746614	+	0.665257i	$0.231678\pi$
$102$	0	0
$103$	−35864.0	−0.333093	−0.166547	−	0.986034i	$-0.553262\pi$
$103$	−35864.0	−0.333093	−0.166547	+	0.986034i	$0.553262\pi$
$104$	0	0
$105$	10584.0	0.0936864
$106$	0	0
$107$	95454.0	0.805999	0.403000	−	0.915200i	$-0.367968\pi$
$107$	95454.0	0.805999	0.403000	+	0.915200i	$0.367968\pi$
$108$	0	0
$109$	212222.	1.71090	0.855449	−	0.517887i	$-0.173281\pi$
$109$	212222.	1.71090	0.855449	+	0.517887i	$0.173281\pi$
$110$	0	0
$111$	110142.	0.848488
$112$	0	0
$113$	62106.0	0.457549	0.228774	−	0.973479i	$-0.426528\pi$
$113$	62106.0	0.457549	0.228774	+	0.973479i	$0.426528\pi$
$114$	0	0
$115$	24912.0	0.175656
$116$	0	0
$117$	7938.00	0.0536101
$118$	0	0
$119$	10584.0	0.0685145
$120$	0	0
$121$	−156695.	−0.972953
$122$	0	0
$123$	−58212.0	−0.346936
$124$	0	0
$125$	−136176.	−0.779517
$126$	0	0
$127$	53044.0	0.291828	0.145914	−	0.989297i	$-0.453388\pi$
$127$	53044.0	0.291828	0.145914	+	0.989297i	$0.453388\pi$
$128$	0	0
$129$	−138708.	−0.733884
$130$	0	0
$131$	−69324.0	−0.352944	−0.176472	−	0.984306i	$-0.556468\pi$
$131$	−69324.0	−0.352944	−0.176472	+	0.984306i	$0.556468\pi$
$132$	0	0
$133$	−16660.0	−0.0816669
$134$	0	0
$135$	−17496.0	−0.0826236
$136$	0	0
$137$	129846.	0.591054	0.295527	−	0.955334i	$-0.404505\pi$
$137$	129846.	0.591054	0.295527	+	0.955334i	$0.404505\pi$
$138$	0	0
$139$	104356.	0.458121	0.229061	−	0.973412i	$-0.426435\pi$
$139$	104356.	0.458121	0.229061	+	0.973412i	$0.426435\pi$
$140$	0	0
$141$	185436.	0.785500
$142$	0	0
$143$	−6468.00	−0.0264503
$144$	0	0
$145$	−59760.0	−0.236043
$146$	0	0
$147$	−21609.0	−0.0824786
$148$	0	0
$149$	217194.	0.801461	0.400730	−	0.916196i	$-0.368756\pi$
$149$	217194.	0.801461	0.400730	+	0.916196i	$0.368756\pi$
$150$	0	0
$151$	−221000.	−0.788769	−0.394385	−	0.918945i	$-0.629042\pi$
$151$	−221000.	−0.788769	−0.394385	+	0.918945i	$0.629042\pi$
$152$	0	0
$153$	−17496.0	−0.0604241
$154$	0	0
$155$	169152.	0.565520
$156$	0	0
$157$	−378370.	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−378370.	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	0	0
$159$	−292410.	−0.917275
$160$	0	0
$161$	−50862.0	−0.154642
$162$	0	0
$163$	−104816.	−0.309000	−0.154500	−	0.987993i	$-0.549377\pi$
$163$	−104816.	−0.309000	−0.154500	+	0.987993i	$0.549377\pi$
$164$	0	0
$165$	14256.0	0.0407650
$166$	0	0
$167$	426972.	1.18470	0.592350	−	0.805681i	$-0.298200\pi$
$167$	426972.	1.18470	0.592350	+	0.805681i	$0.298200\pi$
$168$	0	0
$169$	−361689.	−0.974134
$170$	0	0
$171$	27540.0	0.0720234
$172$	0	0
$173$	331068.	0.841012	0.420506	−	0.907290i	$-0.361853\pi$
$173$	331068.	0.841012	0.420506	+	0.907290i	$0.361853\pi$
$174$	0	0
$175$	124901.	0.308298
$176$	0	0
$177$	308016.	0.738993
$178$	0	0
$179$	400194.	0.933551	0.466775	−	0.884376i	$-0.345416\pi$
$179$	400194.	0.933551	0.466775	+	0.884376i	$0.345416\pi$
$180$	0	0
$181$	588098.	1.33430	0.667150	−	0.744924i	$-0.267514\pi$
$181$	588098.	1.33430	0.667150	+	0.744924i	$0.267514\pi$
$182$	0	0
$183$	−320886.	−0.708309
$184$	0	0
$185$	−293712.	−0.630946
$186$	0	0
$187$	14256.0	0.0298122
$188$	0	0
$189$	35721.0	0.0727393
$190$	0	0
$191$	−939342.	−1.86312	−0.931559	−	0.363590i	$-0.881551\pi$
$191$	−939342.	−1.86312	−0.931559	+	0.363590i	$0.881551\pi$
$192$	0	0
$193$	338390.	0.653919	0.326960	−	0.945038i	$-0.393976\pi$
$193$	338390.	0.653919	0.326960	+	0.945038i	$0.393976\pi$
$194$	0	0
$195$	−21168.0	−0.0398651
$196$	0	0
$197$	−237942.	−0.436823	−0.218412	−	0.975857i	$-0.570088\pi$
$197$	−237942.	−0.436823	−0.218412	+	0.975857i	$0.570088\pi$
$198$	0	0
$199$	−204464.	−0.366003	−0.183001	−	0.983113i	$-0.558581\pi$
$199$	−204464.	−0.366003	−0.183001	+	0.983113i	$0.558581\pi$
$200$	0	0
$201$	114120.	0.199238
$202$	0	0
$203$	122010.	0.207805
$204$	0	0
$205$	155232.	0.257986
$206$	0	0
$207$	84078.0	0.136382
$208$	0	0
$209$	−22440.0	−0.0355351
$210$	0	0
$211$	348724.	0.539232	0.269616	−	0.962968i	$-0.413103\pi$
$211$	348724.	0.539232	0.269616	+	0.962968i	$0.413103\pi$
$212$	0	0
$213$	−383778.	−0.579604
$214$	0	0
$215$	369888.	0.545725
$216$	0	0
$217$	−345352.	−0.497866
$218$	0	0
$219$	−303606.	−0.427760
$220$	0	0
$221$	−21168.0	−0.0291541
$222$	0	0
$223$	−1.47006e6	−1.97957	−0.989787	−	0.142554i	$-0.954468\pi$
$223$	−1.47006e6	−1.97957	−0.989787	+	0.142554i	$0.954468\pi$
$224$	0	0
$225$	−206469.	−0.271893
$226$	0	0
$227$	589560.	0.759387	0.379694	−	0.925112i	$-0.376029\pi$
$227$	589560.	0.759387	0.379694	+	0.925112i	$0.376029\pi$
$228$	0	0
$229$	−1.04534e6	−1.31725	−0.658627	−	0.752469i	$-0.728863\pi$
$229$	−1.04534e6	−1.31725	−0.658627	+	0.752469i	$0.728863\pi$
$230$	0	0
$231$	−29106.0	−0.0358883
$232$	0	0
$233$	651222.	0.785849	0.392925	−	0.919571i	$-0.371463\pi$
$233$	651222.	0.785849	0.392925	+	0.919571i	$0.371463\pi$
$234$	0	0
$235$	−494496.	−0.584108
$236$	0	0
$237$	−765972.	−0.885813
$238$	0	0
$239$	513462.	0.581452	0.290726	−	0.956806i	$-0.406103\pi$
$239$	513462.	0.581452	0.290726	+	0.956806i	$0.406103\pi$
$240$	0	0
$241$	−694714.	−0.770484	−0.385242	−	0.922816i	$-0.625882\pi$
$241$	−694714.	−0.770484	−0.385242	+	0.922816i	$0.625882\pi$
$242$	0	0
$243$	−59049.0	−0.0641500
$244$	0	0
$245$	57624.0	0.0613322
$246$	0	0
$247$	33320.0	0.0347506
$248$	0	0
$249$	−960876.	−0.982130
$250$	0	0
$251$	1.39608e6	1.39870	0.699352	−	0.714777i	$-0.253472\pi$
$251$	1.39608e6	1.39870	0.699352	+	0.714777i	$0.253472\pi$
$252$	0	0
$253$	−68508.0	−0.0672884
$254$	0	0
$255$	46656.0	0.0449321
$256$	0	0
$257$	−1.00520e6	−0.949339	−0.474670	−	0.880164i	$-0.657432\pi$
$257$	−1.00520e6	−0.949339	−0.474670	+	0.880164i	$0.657432\pi$
$258$	0	0
$259$	599662.	0.555466
$260$	0	0
$261$	−201690.	−0.183267
$262$	0	0
$263$	−1.25301e6	−1.11703	−0.558515	−	0.829494i	$-0.688629\pi$
$263$	−1.25301e6	−1.11703	−0.558515	+	0.829494i	$0.688629\pi$
$264$	0	0
$265$	779760.	0.682097
$266$	0	0
$267$	−313956.	−0.269520
$268$	0	0
$269$	−1.76069e6	−1.48355	−0.741774	−	0.670650i	$-0.766015\pi$
$269$	−1.76069e6	−1.48355	−0.741774	+	0.670650i	$0.766015\pi$
$270$	0	0
$271$	−770528.	−0.637331	−0.318666	−	0.947867i	$-0.603235\pi$
$271$	−770528.	−0.637331	−0.318666	+	0.947867i	$0.603235\pi$
$272$	0	0
$273$	43218.0	0.0350960
$274$	0	0
$275$	168234.	0.134147
$276$	0	0
$277$	707738.	0.554208	0.277104	−	0.960840i	$-0.410625\pi$
$277$	707738.	0.554208	0.277104	+	0.960840i	$0.410625\pi$
$278$	0	0
$279$	570888.	0.439077
$280$	0	0
$281$	2.30432e6	1.74091	0.870456	−	0.492247i	$-0.163824\pi$
$281$	2.30432e6	1.74091	0.870456	+	0.492247i	$0.163824\pi$
$282$	0	0
$283$	−1.60903e6	−1.19426	−0.597128	−	0.802146i	$-0.703692\pi$
$283$	−1.60903e6	−1.19426	−0.597128	+	0.802146i	$0.703692\pi$
$284$	0	0
$285$	−73440.0	−0.0535575
$286$	0	0
$287$	−316932.	−0.227123
$288$	0	0
$289$	−1.37320e6	−0.967140
$290$	0	0
$291$	−167958.	−0.116270
$292$	0	0
$293$	517020.	0.351834	0.175917	−	0.984405i	$-0.443711\pi$
$293$	517020.	0.351834	0.175917	+	0.984405i	$0.443711\pi$
$294$	0	0
$295$	−821376.	−0.549524
$296$	0	0
$297$	48114.0	0.0316505
$298$	0	0
$299$	101724.	0.0658030
$300$	0	0
$301$	−755188.	−0.480440
$302$	0	0
$303$	−1.37776e6	−0.862116
$304$	0	0
$305$	855696.	0.526708
$306$	0	0
$307$	−1.35002e6	−0.817512	−0.408756	−	0.912644i	$-0.634037\pi$
$307$	−1.35002e6	−0.817512	−0.408756	+	0.912644i	$0.634037\pi$
$308$	0	0
$309$	322776.	0.192311
$310$	0	0
$311$	−1.34538e6	−0.788758	−0.394379	−	0.918948i	$-0.629040\pi$
$311$	−1.34538e6	−0.788758	−0.394379	+	0.918948i	$0.629040\pi$
$312$	0	0
$313$	256154.	0.147788	0.0738942	−	0.997266i	$-0.476457\pi$
$313$	256154.	0.147788	0.0738942	+	0.997266i	$0.476457\pi$
$314$	0	0
$315$	−95256.0	−0.0540899
$316$	0	0
$317$	1.84629e6	1.03193	0.515967	−	0.856609i	$-0.327433\pi$
$317$	1.84629e6	1.03193	0.515967	+	0.856609i	$0.327433\pi$
$318$	0	0
$319$	164340.	0.0904204
$320$	0	0
$321$	−859086.	−0.465344
$322$	0	0
$323$	−73440.0	−0.0391675
$324$	0	0
$325$	−249802.	−0.131186
$326$	0	0
$327$	−1.91000e6	−0.987788
$328$	0	0
$329$	1.00960e6	0.514231
$330$	0	0
$331$	3.33238e6	1.67180	0.835900	−	0.548881i	$-0.184946\pi$
$331$	3.33238e6	1.67180	0.835900	+	0.548881i	$0.184946\pi$
$332$	0	0
$333$	−991278.	−0.489875
$334$	0	0
$335$	−304320.	−0.148156
$336$	0	0
$337$	−1.63481e6	−0.784136	−0.392068	−	0.919936i	$-0.628240\pi$
$337$	−1.63481e6	−0.784136	−0.392068	+	0.919936i	$0.628240\pi$
$338$	0	0
$339$	−558954.	−0.264166
$340$	0	0
$341$	−465168.	−0.216633
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	−224208.	−0.101415
$346$	0	0
$347$	841530.	0.375185	0.187593	−	0.982247i	$-0.439932\pi$
$347$	841530.	0.375185	0.187593	+	0.982247i	$0.439932\pi$
$348$	0	0
$349$	−977242.	−0.429476	−0.214738	−	0.976672i	$-0.568890\pi$
$349$	−977242.	−0.429476	−0.214738	+	0.976672i	$0.568890\pi$
$350$	0	0
$351$	−71442.0	−0.0309518
$352$	0	0
$353$	3.45857e6	1.47727	0.738634	−	0.674106i	$-0.235471\pi$
$353$	3.45857e6	1.47727	0.738634	+	0.674106i	$0.235471\pi$
$354$	0	0
$355$	1.02341e6	0.431001
$356$	0	0
$357$	−95256.0	−0.0395569
$358$	0	0
$359$	3.47301e6	1.42223	0.711115	−	0.703076i	$-0.248190\pi$
$359$	3.47301e6	1.42223	0.711115	+	0.703076i	$0.248190\pi$
$360$	0	0
$361$	−2.36050e6	−0.953314
$362$	0	0
$363$	1.41026e6	0.561734
$364$	0	0
$365$	809616.	0.318088
$366$	0	0
$367$	−3.11994e6	−1.20915	−0.604575	−	0.796548i	$-0.706657\pi$
$367$	−3.11994e6	−1.20915	−0.604575	+	0.796548i	$0.706657\pi$
$368$	0	0
$369$	523908.	0.200304
$370$	0	0
$371$	−1.59201e6	−0.600497
$372$	0	0
$373$	−2.01673e6	−0.750543	−0.375272	−	0.926915i	$-0.622451\pi$
$373$	−2.01673e6	−0.750543	−0.375272	+	0.926915i	$0.622451\pi$
$374$	0	0
$375$	1.22558e6	0.450054
$376$	0	0
$377$	−244020.	−0.0884244
$378$	0	0
$379$	5.38083e6	1.92420	0.962102	−	0.272690i	$-0.0879134\pi$
$379$	5.38083e6	1.92420	0.962102	+	0.272690i	$0.0879134\pi$
$380$	0	0
$381$	−477396.	−0.168487
$382$	0	0
$383$	−807432.	−0.281261	−0.140630	−	0.990062i	$-0.544913\pi$
$383$	−807432.	−0.281261	−0.140630	+	0.990062i	$0.544913\pi$
$384$	0	0
$385$	77616.0	0.0266870
$386$	0	0
$387$	1.24837e6	0.423708
$388$	0	0
$389$	891390.	0.298671	0.149336	−	0.988787i	$-0.452286\pi$
$389$	891390.	0.298671	0.149336	+	0.988787i	$0.452286\pi$
$390$	0	0
$391$	−224208.	−0.0741667
$392$	0	0
$393$	623916.	0.203772
$394$	0	0
$395$	2.04259e6	0.658702
$396$	0	0
$397$	1.12345e6	0.357749	0.178875	−	0.983872i	$-0.442754\pi$
$397$	1.12345e6	0.357749	0.178875	+	0.983872i	$0.442754\pi$
$398$	0	0
$399$	149940.	0.0471504
$400$	0	0
$401$	1.72037e6	0.534271	0.267136	−	0.963659i	$-0.413923\pi$
$401$	1.72037e6	0.534271	0.267136	+	0.963659i	$0.413923\pi$
$402$	0	0
$403$	690704.	0.211850
$404$	0	0
$405$	157464.	0.0477028
$406$	0	0
$407$	807708.	0.241695
$408$	0	0
$409$	77246.0	0.0228332	0.0114166	−	0.999935i	$-0.496366\pi$
$409$	77246.0	0.0228332	0.0114166	+	0.999935i	$0.496366\pi$
$410$	0	0
$411$	−1.16861e6	−0.341245
$412$	0	0
$413$	1.67698e6	0.483784
$414$	0	0
$415$	2.56234e6	0.730324
$416$	0	0
$417$	−939204.	−0.264496
$418$	0	0
$419$	5.20615e6	1.44871	0.724356	−	0.689427i	$-0.242137\pi$
$419$	5.20615e6	1.44871	0.724356	+	0.689427i	$0.242137\pi$
$420$	0	0
$421$	1.71847e6	0.472539	0.236270	−	0.971688i	$-0.424075\pi$
$421$	1.71847e6	0.472539	0.236270	+	0.971688i	$0.424075\pi$
$422$	0	0
$423$	−1.66892e6	−0.453509
$424$	0	0
$425$	550584.	0.147860
$426$	0	0
$427$	−1.74705e6	−0.463697
$428$	0	0
$429$	58212.0	0.0152711
$430$	0	0
$431$	580626.	0.150558	0.0752789	−	0.997163i	$-0.476015\pi$
$431$	580626.	0.150558	0.0752789	+	0.997163i	$0.476015\pi$
$432$	0	0
$433$	4.15087e6	1.06395	0.531973	−	0.846761i	$-0.321451\pi$
$433$	4.15087e6	1.06395	0.531973	+	0.846761i	$0.321451\pi$
$434$	0	0
$435$	537840.	0.136279
$436$	0	0
$437$	352920.	0.0884042
$438$	0	0
$439$	−3.88407e6	−0.961891	−0.480946	−	0.876750i	$-0.659707\pi$
$439$	−3.88407e6	−0.961891	−0.480946	+	0.876750i	$0.659707\pi$
$440$	0	0
$441$	194481.	0.0476190
$442$	0	0
$443$	2.31499e6	0.560453	0.280226	−	0.959934i	$-0.409590\pi$
$443$	2.31499e6	0.560453	0.280226	+	0.959934i	$0.409590\pi$
$444$	0	0
$445$	837216.	0.200418
$446$	0	0
$447$	−1.95475e6	−0.462723
$448$	0	0
$449$	−1.92281e6	−0.450113	−0.225056	−	0.974346i	$-0.572257\pi$
$449$	−1.92281e6	−0.450113	−0.225056	+	0.974346i	$0.572257\pi$
$450$	0	0
$451$	−426888.	−0.0988263
$452$	0	0
$453$	1.98900e6	0.455396
$454$	0	0
$455$	−115248.	−0.0260979
$456$	0	0
$457$	6.86215e6	1.53699	0.768493	−	0.639858i	$-0.221007\pi$
$457$	6.86215e6	1.53699	0.768493	+	0.639858i	$0.221007\pi$
$458$	0	0
$459$	157464.	0.0348859
$460$	0	0
$461$	2.97167e6	0.651250	0.325625	−	0.945499i	$-0.394425\pi$
$461$	2.97167e6	0.651250	0.325625	+	0.945499i	$0.394425\pi$
$462$	0	0
$463$	−4.87423e6	−1.05670	−0.528352	−	0.849025i	$-0.677190\pi$
$463$	−4.87423e6	−1.05670	−0.528352	+	0.849025i	$0.677190\pi$
$464$	0	0
$465$	−1.52237e6	−0.326503
$466$	0	0
$467$	8.17301e6	1.73416	0.867081	−	0.498167i	$-0.165993\pi$
$467$	8.17301e6	1.73416	0.867081	+	0.498167i	$0.165993\pi$
$468$	0	0
$469$	621320.	0.130432
$470$	0	0
$471$	3.40533e6	0.707305
$472$	0	0
$473$	−1.01719e6	−0.209050
$474$	0	0
$475$	−866660.	−0.176244
$476$	0	0
$477$	2.63169e6	0.529589
$478$	0	0
$479$	−2.34397e6	−0.466782	−0.233391	−	0.972383i	$-0.574982\pi$
$479$	−2.34397e6	−0.466782	−0.233391	+	0.972383i	$0.574982\pi$
$480$	0	0
$481$	−1.19932e6	−0.236360
$482$	0	0
$483$	457758.	0.0892829
$484$	0	0
$485$	447888.	0.0864600
$486$	0	0
$487$	−316928.	−0.0605534	−0.0302767	−	0.999542i	$-0.509639\pi$
$487$	−316928.	−0.0605534	−0.0302767	+	0.999542i	$0.509639\pi$
$488$	0	0
$489$	943344.	0.178401
$490$	0	0
$491$	5.20041e6	0.973495	0.486748	−	0.873543i	$-0.338183\pi$
$491$	5.20041e6	0.973495	0.486748	+	0.873543i	$0.338183\pi$
$492$	0	0
$493$	537840.	0.0996634
$494$	0	0
$495$	−128304.	−0.0235357
$496$	0	0
$497$	−2.08946e6	−0.379440
$498$	0	0
$499$	4.86773e6	0.875135	0.437568	−	0.899185i	$-0.355840\pi$
$499$	4.86773e6	0.875135	0.437568	+	0.899185i	$0.355840\pi$
$500$	0	0
$501$	−3.84275e6	−0.683987
$502$	0	0
$503$	−426888.	−0.0752305	−0.0376153	−	0.999292i	$-0.511976\pi$
$503$	−426888.	−0.0752305	−0.0376153	+	0.999292i	$0.511976\pi$
$504$	0	0
$505$	3.67402e6	0.641081
$506$	0	0
$507$	3.25520e6	0.562416
$508$	0	0
$509$	−9.41621e6	−1.61095	−0.805474	−	0.592631i	$-0.798089\pi$
$509$	−9.41621e6	−1.61095	−0.805474	+	0.592631i	$0.798089\pi$
$510$	0	0
$511$	−1.65297e6	−0.280035
$512$	0	0
$513$	−247860.	−0.0415827
$514$	0	0
$515$	−860736.	−0.143005
$516$	0	0
$517$	1.35986e6	0.223753
$518$	0	0
$519$	−2.97961e6	−0.485558
$520$	0	0
$521$	1.84039e6	0.297041	0.148520	−	0.988909i	$-0.452549\pi$
$521$	1.84039e6	0.297041	0.148520	+	0.988909i	$0.452549\pi$
$522$	0	0
$523$	979108.	0.156522	0.0782612	−	0.996933i	$-0.475063\pi$
$523$	979108.	0.156522	0.0782612	+	0.996933i	$0.475063\pi$
$524$	0	0
$525$	−1.12411e6	−0.177996
$526$	0	0
$527$	−1.52237e6	−0.238777
$528$	0	0
$529$	−5.35890e6	−0.832600
$530$	0	0
$531$	−2.77214e6	−0.426658
$532$	0	0
$533$	633864.	0.0966447
$534$	0	0
$535$	2.29090e6	0.346036
$536$	0	0
$537$	−3.60175e6	−0.538986
$538$	0	0
$539$	−158466.	−0.0234944
$540$	0	0
$541$	5.96117e6	0.875666	0.437833	−	0.899056i	$-0.355746\pi$
$541$	5.96117e6	0.875666	0.437833	+	0.899056i	$0.355746\pi$
$542$	0	0
$543$	−5.29288e6	−0.770358
$544$	0	0
$545$	5.09333e6	0.734531
$546$	0	0
$547$	−8.73025e6	−1.24755	−0.623775	−	0.781604i	$-0.714402\pi$
$547$	−8.73025e6	−1.24755	−0.623775	+	0.781604i	$0.714402\pi$
$548$	0	0
$549$	2.88797e6	0.408943
$550$	0	0
$551$	−846600.	−0.118795
$552$	0	0
$553$	−4.17029e6	−0.579901
$554$	0	0
$555$	2.64341e6	0.364277
$556$	0	0
$557$	−3.01066e6	−0.411172	−0.205586	−	0.978639i	$-0.565910\pi$
$557$	−3.01066e6	−0.411172	−0.205586	+	0.978639i	$0.565910\pi$
$558$	0	0
$559$	1.51038e6	0.204435
$560$	0	0
$561$	−128304.	−0.0172121
$562$	0	0
$563$	−1.17573e7	−1.56327	−0.781637	−	0.623733i	$-0.785615\pi$
$563$	−1.17573e7	−1.56327	−0.781637	+	0.623733i	$0.785615\pi$
$564$	0	0
$565$	1.49054e6	0.196437
$566$	0	0
$567$	−321489.	−0.0419961
$568$	0	0
$569$	1.31578e7	1.70374	0.851870	−	0.523754i	$-0.175469\pi$
$569$	1.31578e7	1.70374	0.851870	+	0.523754i	$0.175469\pi$
$570$	0	0
$571$	1.03344e7	1.32647	0.663234	−	0.748412i	$-0.269183\pi$
$571$	1.03344e7	1.32647	0.663234	+	0.748412i	$0.269183\pi$
$572$	0	0
$573$	8.45408e6	1.07567
$574$	0	0
$575$	−2.64586e6	−0.333732
$576$	0	0
$577$	−7.88133e6	−0.985508	−0.492754	−	0.870169i	$-0.664010\pi$
$577$	−7.88133e6	−0.985508	−0.492754	+	0.870169i	$0.664010\pi$
$578$	0	0
$579$	−3.04551e6	−0.377541
$580$	0	0
$581$	−5.23144e6	−0.642955
$582$	0	0
$583$	−2.14434e6	−0.261290
$584$	0	0
$585$	190512.	0.0230161
$586$	0	0
$587$	554568.	0.0664293	0.0332146	−	0.999448i	$-0.489426\pi$
$587$	554568.	0.0664293	0.0332146	+	0.999448i	$0.489426\pi$
$588$	0	0
$589$	2.39632e6	0.284614
$590$	0	0
$591$	2.14148e6	0.252200
$592$	0	0
$593$	−9.20369e6	−1.07479	−0.537397	−	0.843329i	$-0.680592\pi$
$593$	−9.20369e6	−1.07479	−0.537397	+	0.843329i	$0.680592\pi$
$594$	0	0
$595$	254016.	0.0294150
$596$	0	0
$597$	1.84018e6	0.211312
$598$	0	0
$599$	−8.54295e6	−0.972839	−0.486419	−	0.873725i	$-0.661697\pi$
$599$	−8.54295e6	−0.972839	−0.486419	+	0.873725i	$0.661697\pi$
$600$	0	0
$601$	−9.61555e6	−1.08590	−0.542948	−	0.839767i	$-0.682692\pi$
$601$	−9.61555e6	−1.08590	−0.542948	+	0.839767i	$0.682692\pi$
$602$	0	0
$603$	−1.02708e6	−0.115030
$604$	0	0
$605$	−3.76068e6	−0.417713
$606$	0	0
$607$	−2.21264e6	−0.243747	−0.121873	−	0.992546i	$-0.538890\pi$
$607$	−2.21264e6	−0.243747	−0.121873	+	0.992546i	$0.538890\pi$
$608$	0	0
$609$	−1.09809e6	−0.119976
$610$	0	0
$611$	−2.01919e6	−0.218814
$612$	0	0
$613$	−7.96215e6	−0.855814	−0.427907	−	0.903823i	$-0.640749\pi$
$613$	−7.96215e6	−0.855814	−0.427907	+	0.903823i	$0.640749\pi$
$614$	0	0
$615$	−1.39709e6	−0.148948
$616$	0	0
$617$	−1.37397e7	−1.45299	−0.726497	−	0.687170i	$-0.758853\pi$
$617$	−1.37397e7	−1.45299	−0.726497	+	0.687170i	$0.758853\pi$
$618$	0	0
$619$	8.70113e6	0.912744	0.456372	−	0.889789i	$-0.349149\pi$
$619$	8.70113e6	0.912744	0.456372	+	0.889789i	$0.349149\pi$
$620$	0	0
$621$	−756702.	−0.0787401
$622$	0	0
$623$	−1.70932e6	−0.176442
$624$	0	0
$625$	4.69740e6	0.481014
$626$	0	0
$627$	201960.	0.0205162
$628$	0	0
$629$	2.64341e6	0.266402
$630$	0	0
$631$	−445412.	−0.0445337	−0.0222668	−	0.999752i	$-0.507088\pi$
$631$	−445412.	−0.0445337	−0.0222668	+	0.999752i	$0.507088\pi$
$632$	0	0
$633$	−3.13852e6	−0.311326
$634$	0	0
$635$	1.27306e6	0.125289
$636$	0	0
$637$	235298.	0.0229757
$638$	0	0
$639$	3.45400e6	0.334634
$640$	0	0
$641$	−8.00119e6	−0.769147	−0.384573	−	0.923094i	$-0.625651\pi$
$641$	−8.00119e6	−0.769147	−0.384573	+	0.923094i	$0.625651\pi$
$642$	0	0
$643$	1.58402e7	1.51090	0.755448	−	0.655209i	$-0.227419\pi$
$643$	1.58402e7	1.51090	0.755448	+	0.655209i	$0.227419\pi$
$644$	0	0
$645$	−3.32899e6	−0.315075
$646$	0	0
$647$	−1.30187e6	−0.122266	−0.0611331	−	0.998130i	$-0.519471\pi$
$647$	−1.30187e6	−0.122266	−0.0611331	+	0.998130i	$0.519471\pi$
$648$	0	0
$649$	2.25878e6	0.210505
$650$	0	0
$651$	3.10817e6	0.287443
$652$	0	0
$653$	7.34149e6	0.673753	0.336877	−	0.941549i	$-0.390629\pi$
$653$	7.34149e6	0.673753	0.336877	+	0.941549i	$0.390629\pi$
$654$	0	0
$655$	−1.66378e6	−0.151528
$656$	0	0
$657$	2.73245e6	0.246967
$658$	0	0
$659$	6.18934e6	0.555176	0.277588	−	0.960700i	$-0.410465\pi$
$659$	6.18934e6	0.555176	0.277588	+	0.960700i	$0.410465\pi$
$660$	0	0
$661$	−1.96690e7	−1.75097	−0.875484	−	0.483248i	$-0.839457\pi$
$661$	−1.96690e7	−1.75097	−0.875484	+	0.483248i	$0.839457\pi$
$662$	0	0
$663$	190512.	0.0168321
$664$	0	0
$665$	−399840.	−0.0350616
$666$	0	0
$667$	−2.58462e6	−0.224948
$668$	0	0
$669$	1.32305e7	1.14291
$670$	0	0
$671$	−2.35316e6	−0.201765
$672$	0	0
$673$	7.18259e6	0.611285	0.305642	−	0.952146i	$-0.401129\pi$
$673$	7.18259e6	0.611285	0.305642	+	0.952146i	$0.401129\pi$
$674$	0	0
$675$	1.85822e6	0.156978
$676$	0	0
$677$	−1.89192e7	−1.58647	−0.793234	−	0.608917i	$-0.791604\pi$
$677$	−1.89192e7	−1.58647	−0.793234	+	0.608917i	$0.791604\pi$
$678$	0	0
$679$	−914438.	−0.0761167
$680$	0	0
$681$	−5.30604e6	−0.438432
$682$	0	0
$683$	−2.12204e7	−1.74061	−0.870306	−	0.492512i	$-0.836079\pi$
$683$	−2.12204e7	−1.74061	−0.870306	+	0.492512i	$0.836079\pi$
$684$	0	0
$685$	3.11630e6	0.253754
$686$	0	0
$687$	9.40808e6	0.760517
$688$	0	0
$689$	3.18402e6	0.255522
$690$	0	0
$691$	−1.63276e7	−1.30085	−0.650424	−	0.759571i	$-0.725409\pi$
$691$	−1.63276e7	−1.30085	−0.650424	+	0.759571i	$0.725409\pi$
$692$	0	0
$693$	261954.	0.0207201
$694$	0	0
$695$	2.50454e6	0.196683
$696$	0	0
$697$	−1.39709e6	−0.108929
$698$	0	0
$699$	−5.86100e6	−0.453710
$700$	0	0
$701$	−5.40470e6	−0.415409	−0.207705	−	0.978192i	$-0.566599\pi$
$701$	−5.40470e6	−0.415409	−0.207705	+	0.978192i	$0.566599\pi$
$702$	0	0
$703$	−4.16092e6	−0.317542
$704$	0	0
$705$	4.45046e6	0.337235
$706$	0	0
$707$	−7.50112e6	−0.564387
$708$	0	0
$709$	2.21195e7	1.65257	0.826284	−	0.563253i	$-0.190450\pi$
$709$	2.21195e7	1.65257	0.826284	+	0.563253i	$0.190450\pi$
$710$	0	0
$711$	6.89375e6	0.511424
$712$	0	0
$713$	7.31582e6	0.538939
$714$	0	0
$715$	−155232.	−0.0113558
$716$	0	0
$717$	−4.62116e6	−0.335701
$718$	0	0
$719$	−2.55819e7	−1.84548	−0.922742	−	0.385418i	$-0.874057\pi$
$719$	−2.55819e7	−1.84548	−0.922742	+	0.385418i	$0.874057\pi$
$720$	0	0
$721$	1.75734e6	0.125897
$722$	0	0
$723$	6.25243e6	0.444839
$724$	0	0
$725$	6.34701e6	0.448460
$726$	0	0
$727$	9.29438e6	0.652205	0.326103	−	0.945334i	$-0.394265\pi$
$727$	9.29438e6	0.652205	0.326103	+	0.945334i	$0.394265\pi$
$728$	0	0
$729$	531441.	0.0370370
$730$	0	0
$731$	−3.32899e6	−0.230420
$732$	0	0
$733$	3.40699e6	0.234213	0.117107	−	0.993119i	$-0.462638\pi$
$733$	3.40699e6	0.234213	0.117107	+	0.993119i	$0.462638\pi$
$734$	0	0
$735$	−518616.	−0.0354101
$736$	0	0
$737$	836880.	0.0567537
$738$	0	0
$739$	−2.18135e7	−1.46932	−0.734658	−	0.678438i	$-0.762657\pi$
$739$	−2.18135e7	−1.46932	−0.734658	+	0.678438i	$0.762657\pi$
$740$	0	0
$741$	−299880.	−0.0200633
$742$	0	0
$743$	−3.79246e6	−0.252028	−0.126014	−	0.992028i	$-0.540218\pi$
$743$	−3.79246e6	−0.252028	−0.126014	+	0.992028i	$0.540218\pi$
$744$	0	0
$745$	5.21266e6	0.344087
$746$	0	0
$747$	8.64788e6	0.567033
$748$	0	0
$749$	−4.67725e6	−0.304639
$750$	0	0
$751$	2.01483e7	1.30358	0.651790	−	0.758400i	$-0.274018\pi$
$751$	2.01483e7	1.30358	0.651790	+	0.758400i	$0.274018\pi$
$752$	0	0
$753$	−1.25647e7	−0.807542
$754$	0	0
$755$	−5.30400e6	−0.338638
$756$	0	0
$757$	1.18427e7	0.751126	0.375563	−	0.926797i	$-0.377449\pi$
$757$	1.18427e7	0.751126	0.375563	+	0.926797i	$0.377449\pi$
$758$	0	0
$759$	616572.	0.0388490
$760$	0	0
$761$	2.97791e6	0.186402	0.0932008	−	0.995647i	$-0.470290\pi$
$761$	2.97791e6	0.186402	0.0932008	+	0.995647i	$0.470290\pi$
$762$	0	0
$763$	−1.03989e7	−0.646659
$764$	0	0
$765$	−419904.	−0.0259416
$766$	0	0
$767$	−3.35395e6	−0.205858
$768$	0	0
$769$	−2.02441e7	−1.23447	−0.617237	−	0.786777i	$-0.711748\pi$
$769$	−2.02441e7	−1.23447	−0.617237	+	0.786777i	$0.711748\pi$
$770$	0	0
$771$	9.04684e6	0.548101
$772$	0	0
$773$	−7.37953e6	−0.444202	−0.222101	−	0.975024i	$-0.571291\pi$
$773$	−7.37953e6	−0.444202	−0.222101	+	0.975024i	$0.571291\pi$
$774$	0	0
$775$	−1.79654e7	−1.07444
$776$	0	0
$777$	−5.39696e6	−0.320698
$778$	0	0
$779$	2.19912e6	0.129839
$780$	0	0
$781$	−2.81437e6	−0.165103
$782$	0	0
$783$	1.81521e6	0.105809
$784$	0	0
$785$	−9.08088e6	−0.525961
$786$	0	0
$787$	−1.36289e7	−0.784377	−0.392188	−	0.919885i	$-0.628282\pi$
$787$	−1.36289e7	−0.784377	−0.392188	+	0.919885i	$0.628282\pi$
$788$	0	0
$789$	1.12771e7	0.644918
$790$	0	0
$791$	−3.04319e6	−0.172937
$792$	0	0
$793$	3.49409e6	0.197311
$794$	0	0
$795$	−7.01784e6	−0.393809
$796$	0	0
$797$	−1.49548e7	−0.833938	−0.416969	−	0.908921i	$-0.636908\pi$
$797$	−1.49548e7	−0.833938	−0.416969	+	0.908921i	$0.636908\pi$
$798$	0	0
$799$	4.45046e6	0.246626
$800$	0	0
$801$	2.82560e6	0.155607
$802$	0	0
$803$	−2.22644e6	−0.121849
$804$	0	0
$805$	−1.22069e6	−0.0663919
$806$	0	0
$807$	1.58462e7	0.856527
$808$	0	0
$809$	2.87242e7	1.54304	0.771519	−	0.636206i	$-0.219497\pi$
$809$	2.87242e7	1.54304	0.771519	+	0.636206i	$0.219497\pi$
$810$	0	0
$811$	1.52265e7	0.812922	0.406461	−	0.913668i	$-0.366763\pi$
$811$	1.52265e7	0.812922	0.406461	+	0.913668i	$0.366763\pi$
$812$	0	0
$813$	6.93475e6	0.367963
$814$	0	0
$815$	−2.51558e6	−0.132661
$816$	0	0
$817$	5.24008e6	0.274652
$818$	0	0
$819$	−388962.	−0.0202627
$820$	0	0
$821$	−3.31001e7	−1.71384	−0.856921	−	0.515447i	$-0.827626\pi$
$821$	−3.31001e7	−1.71384	−0.856921	+	0.515447i	$0.827626\pi$
$822$	0	0
$823$	1.35915e7	0.699470	0.349735	−	0.936849i	$-0.386272\pi$
$823$	1.35915e7	0.699470	0.349735	+	0.936849i	$0.386272\pi$
$824$	0	0
$825$	−1.51411e6	−0.0774500
$826$	0	0
$827$	−3.13936e6	−0.159616	−0.0798082	−	0.996810i	$-0.525431\pi$
$827$	−3.13936e6	−0.159616	−0.0798082	+	0.996810i	$0.525431\pi$
$828$	0	0
$829$	1.27081e7	0.642234	0.321117	−	0.947040i	$-0.395942\pi$
$829$	1.27081e7	0.642234	0.321117	+	0.947040i	$0.395942\pi$
$830$	0	0
$831$	−6.36964e6	−0.319972
$832$	0	0
$833$	−518616.	−0.0258960
$834$	0	0
$835$	1.02473e7	0.508621
$836$	0	0
$837$	−5.13799e6	−0.253501
$838$	0	0
$839$	2.98312e7	1.46307	0.731536	−	0.681803i	$-0.238804\pi$
$839$	2.98312e7	1.46307	0.731536	+	0.681803i	$0.238804\pi$
$840$	0	0
$841$	−1.43110e7	−0.697720
$842$	0	0
$843$	−2.07389e7	−1.00512
$844$	0	0
$845$	−8.68054e6	−0.418220
$846$	0	0
$847$	7.67806e6	0.367742
$848$	0	0
$849$	1.44813e7	0.689504
$850$	0	0
$851$	−1.27030e7	−0.601290
$852$	0	0
$853$	−1.92215e7	−0.904515	−0.452257	−	0.891888i	$-0.649381\pi$
$853$	−1.92215e7	−0.904515	−0.452257	+	0.891888i	$0.649381\pi$
$854$	0	0
$855$	660960.	0.0309215
$856$	0	0
$857$	−2.65655e7	−1.23556	−0.617782	−	0.786349i	$-0.711969\pi$
$857$	−2.65655e7	−1.23556	−0.617782	+	0.786349i	$0.711969\pi$
$858$	0	0
$859$	9.16844e6	0.423948	0.211974	−	0.977275i	$-0.432011\pi$
$859$	9.16844e6	0.423948	0.211974	+	0.977275i	$0.432011\pi$
$860$	0	0
$861$	2.85239e6	0.131130
$862$	0	0
$863$	2.92196e7	1.33551	0.667755	−	0.744381i	$-0.267255\pi$
$863$	2.92196e7	1.33551	0.667755	+	0.744381i	$0.267255\pi$
$864$	0	0
$865$	7.94563e6	0.361067
$866$	0	0
$867$	1.23588e7	0.558379
$868$	0	0
$869$	−5.61713e6	−0.252328
$870$	0	0
$871$	−1.24264e6	−0.0555009
$872$	0	0
$873$	1.51162e6	0.0671286
$874$	0	0
$875$	6.67262e6	0.294630
$876$	0	0
$877$	9.71286e6	0.426430	0.213215	−	0.977005i	$-0.431606\pi$
$877$	9.71286e6	0.426430	0.213215	+	0.977005i	$0.431606\pi$
$878$	0	0
$879$	−4.65318e6	−0.203132
$880$	0	0
$881$	1.65372e7	0.717833	0.358917	−	0.933370i	$-0.383146\pi$
$881$	1.65372e7	0.717833	0.358917	+	0.933370i	$0.383146\pi$
$882$	0	0
$883$	2.39487e7	1.03367	0.516833	−	0.856086i	$-0.327111\pi$
$883$	2.39487e7	1.03367	0.516833	+	0.856086i	$0.327111\pi$
$884$	0	0
$885$	7.39238e6	0.317268
$886$	0	0
$887$	4.62846e6	0.197527	0.0987637	−	0.995111i	$-0.468511\pi$
$887$	4.62846e6	0.197527	0.0987637	+	0.995111i	$0.468511\pi$
$888$	0	0
$889$	−2.59916e6	−0.110301
$890$	0	0
$891$	−433026.	−0.0182734
$892$	0	0
$893$	−7.00536e6	−0.293969
$894$	0	0
$895$	9.60466e6	0.400797
$896$	0	0
$897$	−915516.	−0.0379914
$898$	0	0
$899$	−1.75495e7	−0.724212
$900$	0	0
$901$	−7.01784e6	−0.287999
$902$	0	0
$903$	6.79669e6	0.277382
$904$	0	0
$905$	1.41144e7	0.572848
$906$	0	0
$907$	−2.06126e7	−0.831983	−0.415991	−	0.909369i	$-0.636565\pi$
$907$	−2.06126e7	−0.831983	−0.415991	+	0.909369i	$0.636565\pi$
$908$	0	0
$909$	1.23998e7	0.497743
$910$	0	0
$911$	3.46749e6	0.138427	0.0692133	−	0.997602i	$-0.477951\pi$
$911$	3.46749e6	0.138427	0.0692133	+	0.997602i	$0.477951\pi$
$912$	0	0
$913$	−7.04642e6	−0.279764
$914$	0	0
$915$	−7.70126e6	−0.304095
$916$	0	0
$917$	3.39688e6	0.133400
$918$	0	0
$919$	3.61227e7	1.41088	0.705442	−	0.708767i	$-0.250748\pi$
$919$	3.61227e7	1.41088	0.705442	+	0.708767i	$0.250748\pi$
$920$	0	0
$921$	1.21502e7	0.471991
$922$	0	0
$923$	4.17892e6	0.161458
$924$	0	0
$925$	3.11947e7	1.19874
$926$	0	0
$927$	−2.90498e6	−0.111031
$928$	0	0
$929$	1.29366e7	0.491792	0.245896	−	0.969296i	$-0.420918\pi$
$929$	1.29366e7	0.491792	0.245896	+	0.969296i	$0.420918\pi$
$930$	0	0
$931$	816340.	0.0308672
$932$	0	0
$933$	1.21084e7	0.455390
$934$	0	0
$935$	342144.	0.0127991
$936$	0	0
$937$	5.01394e7	1.86565	0.932824	−	0.360332i	$-0.117336\pi$
$937$	5.01394e7	1.86565	0.932824	+	0.360332i	$0.117336\pi$
$938$	0	0
$939$	−2.30539e6	−0.0853257
$940$	0	0
$941$	−1.05568e7	−0.388651	−0.194325	−	0.980937i	$-0.562252\pi$
$941$	−1.05568e7	−0.388651	−0.194325	+	0.980937i	$0.562252\pi$
$942$	0	0
$943$	6.71378e6	0.245860
$944$	0	0
$945$	857304.	0.0312288
$946$	0	0
$947$	3.14684e6	0.114025	0.0570124	−	0.998373i	$-0.481843\pi$
$947$	3.14684e6	0.114025	0.0570124	+	0.998373i	$0.481843\pi$
$948$	0	0
$949$	3.30593e6	0.119159
$950$	0	0
$951$	−1.66166e7	−0.595787
$952$	0	0
$953$	5.22829e7	1.86478	0.932389	−	0.361455i	$-0.117720\pi$
$953$	5.22829e7	1.86478	0.932389	+	0.361455i	$0.117720\pi$
$954$	0	0
$955$	−2.25442e7	−0.799883
$956$	0	0
$957$	−1.47906e6	−0.0522043
$958$	0	0
$959$	−6.36245e6	−0.223397
$960$	0	0
$961$	2.10452e7	0.735095
$962$	0	0
$963$	7.73177e6	0.268666
$964$	0	0
$965$	8.12136e6	0.280744
$966$	0	0
$967$	2.48235e7	0.853682	0.426841	−	0.904327i	$-0.359626\pi$
$967$	2.48235e7	0.853682	0.426841	+	0.904327i	$0.359626\pi$
$968$	0	0
$969$	660960.	0.0226134
$970$	0	0
$971$	−1.33077e7	−0.452956	−0.226478	−	0.974016i	$-0.572721\pi$
$971$	−1.33077e7	−0.452956	−0.226478	+	0.974016i	$0.572721\pi$
$972$	0	0
$973$	−5.11344e6	−0.173154
$974$	0	0
$975$	2.24822e6	0.0757403
$976$	0	0
$977$	8.17705e6	0.274069	0.137035	−	0.990566i	$-0.456243\pi$
$977$	8.17705e6	0.274069	0.137035	+	0.990566i	$0.456243\pi$
$978$	0	0
$979$	−2.30234e6	−0.0767739
$980$	0	0
$981$	1.71900e7	0.570299
$982$	0	0
$983$	1.32465e7	0.437238	0.218619	−	0.975810i	$-0.429845\pi$
$983$	1.32465e7	0.437238	0.218619	+	0.975810i	$0.429845\pi$
$984$	0	0
$985$	−5.71061e6	−0.187539
$986$	0	0
$987$	−9.08636e6	−0.296891
$988$	0	0
$989$	1.59977e7	0.520075
$990$	0	0
$991$	1.48550e7	0.480494	0.240247	−	0.970712i	$-0.422772\pi$
$991$	1.48550e7	0.480494	0.240247	+	0.970712i	$0.422772\pi$
$992$	0	0
$993$	−2.99914e7	−0.965215
$994$	0	0
$995$	−4.90714e6	−0.157134
$996$	0	0
$997$	−3.33769e6	−0.106343	−0.0531714	−	0.998585i	$-0.516933\pi$
$997$	−3.33769e6	−0.106343	−0.0531714	+	0.998585i	$0.516933\pi$
$998$	0	0
$999$	8.92150e6	0.282829

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.a.g.1.1		1
3.2	odd	2		1008.6.a.k.1.1		1
4.3	odd	2		42.6.a.f.1.1	✓	1
12.11	even	2		126.6.a.b.1.1		1
20.3	even	4		1050.6.g.m.799.1		2
20.7	even	4		1050.6.g.m.799.2		2
20.19	odd	2		1050.6.a.a.1.1		1
28.3	even	6		294.6.e.f.79.1		2
28.11	odd	6		294.6.e.b.79.1		2
28.19	even	6		294.6.e.f.67.1		2
28.23	odd	6		294.6.e.b.67.1		2
28.27	even	2		294.6.a.i.1.1		1
84.83	odd	2		882.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.f.1.1	✓	1	4.3	odd	2
126.6.a.b.1.1		1	12.11	even	2
294.6.a.i.1.1		1	28.27	even	2
294.6.e.b.67.1		2	28.23	odd	6
294.6.e.b.79.1		2	28.11	odd	6
294.6.e.f.67.1		2	28.19	even	6
294.6.e.f.79.1		2	28.3	even	6
336.6.a.g.1.1		1	1.1	even	1	trivial
882.6.a.i.1.1		1	84.83	odd	2
1008.6.a.k.1.1		1	3.2	odd	2
1050.6.a.a.1.1		1	20.19	odd	2
1050.6.g.m.799.1		2	20.3	even	4
1050.6.g.m.799.2		2	20.7	even	4

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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