LMFDB - Embedded newform 126.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [126,6,Mod(1,126)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(126, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("126.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	126.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:	$20.2083612964$
Analytic rank:	$1$
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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} -24.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +O(q^{10})$

\(q-4.00000 q^{2} +16.0000 q^{4} -24.0000 q^{5} +49.0000 q^{7} -64.0000 q^{8} +96.0000 q^{10} -66.0000 q^{11} +98.0000 q^{13} -196.000 q^{14} +256.000 q^{16} +216.000 q^{17} -340.000 q^{19} -384.000 q^{20} +264.000 q^{22} +1038.00 q^{23} -2549.00 q^{25} -392.000 q^{26} +784.000 q^{28} +2490.00 q^{29} -7048.00 q^{31} -1024.00 q^{32} -864.000 q^{34} -1176.00 q^{35} -12238.0 q^{37} +1360.00 q^{38} +1536.00 q^{40} -6468.00 q^{41} -15412.0 q^{43} -1056.00 q^{44} -4152.00 q^{46} -20604.0 q^{47} +2401.00 q^{49} +10196.0 q^{50} +1568.00 q^{52} -32490.0 q^{53} +1584.00 q^{55} -3136.00 q^{56} -9960.00 q^{58} -34224.0 q^{59} +35654.0 q^{61} +28192.0 q^{62} +4096.00 q^{64} -2352.00 q^{65} +12680.0 q^{67} +3456.00 q^{68} +4704.00 q^{70} +42642.0 q^{71} +33734.0 q^{73} +48952.0 q^{74} -5440.00 q^{76} -3234.00 q^{77} -85108.0 q^{79} -6144.00 q^{80} +25872.0 q^{82} +106764. q^{83} -5184.00 q^{85} +61648.0 q^{86} +4224.00 q^{88} -34884.0 q^{89} +4802.00 q^{91} +16608.0 q^{92} +82416.0 q^{94} +8160.00 q^{95} +18662.0 q^{97} -9604.00 q^{98} +O(q^{100})\)

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	−24.0000	−0.429325	−0.214663	−	0.976688i	$-0.568865\pi$
$5$	−24.0000	−0.429325	−0.214663	+	0.976688i	$0.568865\pi$
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	−64.0000	−0.353553
$9$	0	0
$10$	96.0000	0.303579
$11$	−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	$0.474375\pi$
$14$	−196.000	−0.267261
$15$	0	0
$16$	256.000	0.250000
$17$	216.000	0.181272	0.0906362	−	0.995884i	$-0.471110\pi$
$17$	216.000	0.181272	0.0906362	+	0.995884i	$0.471110\pi$
$18$	0	0
$19$	−340.000	−0.216070	−0.108035	−	0.994147i	$-0.534456\pi$
$19$	−340.000	−0.216070	−0.108035	+	0.994147i	$0.534456\pi$
$20$	−384.000	−0.214663
$21$	0	0
$22$	264.000	0.116291
$23$	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
$26$	−392.000	−0.113724
$27$	0	0
$28$	784.000	0.188982
$29$	2490.00	0.549800	0.274900	−	0.961473i	$-0.411355\pi$
$29$	2490.00	0.549800	0.274900	+	0.961473i	$0.411355\pi$
$30$	0	0
$31$	−7048.00	−1.31723	−0.658615	−	0.752480i	$-0.728857\pi$
$31$	−7048.00	−1.31723	−0.658615	+	0.752480i	$0.728857\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	−864.000	−0.128179
$35$	−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	$0.762730\pi$
$38$	1360.00	0.152785
$39$	0	0
$40$	1536.00	0.151789
$41$	−6468.00	−0.600911	−0.300456	−	0.953796i	$-0.597139\pi$
$41$	−6468.00	−0.600911	−0.300456	+	0.953796i	$0.597139\pi$
$42$	0	0
$43$	−15412.0	−1.27112	−0.635562	−	0.772050i	$-0.719232\pi$
$43$	−15412.0	−1.27112	−0.635562	+	0.772050i	$0.719232\pi$
$44$	−1056.00	−0.0822304
$45$	0	0
$46$	−4152.00	−0.289310
$47$	−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
$50$	10196.0	0.576773
$51$	0	0
$52$	1568.00	0.0804151
$53$	−32490.0	−1.58877	−0.794383	−	0.607417i	$-0.792206\pi$
$53$	−32490.0	−1.58877	−0.794383	+	0.607417i	$0.792206\pi$
$54$	0	0
$55$	1584.00	0.0706071
$56$	−3136.00	−0.133631
$57$	0	0
$58$	−9960.00	−0.388767
$59$	−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	$0.289796\pi$
$62$	28192.0	0.931422
$63$	0	0
$64$	4096.00	0.125000
$65$	−2352.00	−0.0690484
$66$	0	0
$67$	12680.0	0.345090	0.172545	−	0.985002i	$-0.444801\pi$
$67$	12680.0	0.345090	0.172545	+	0.985002i	$0.444801\pi$
$68$	3456.00	0.0906362
$69$	0	0
$70$	4704.00	0.114742
$71$	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	$0.379203\pi$
$74$	48952.0	1.03918
$75$	0	0
$76$	−5440.00	−0.108035
$77$	−3234.00	−0.0621603
$78$	0	0
$79$	−85108.0	−1.53427	−0.767137	−	0.641484i	$-0.778319\pi$
$79$	−85108.0	−1.53427	−0.767137	+	0.641484i	$0.778319\pi$
$80$	−6144.00	−0.107331
$81$	0	0
$82$	25872.0	0.424908
$83$	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
$86$	61648.0	0.898820
$87$	0	0
$88$	4224.00	0.0581456
$89$	−34884.0	−0.466822	−0.233411	−	0.972378i	$-0.574989\pi$
$89$	−34884.0	−0.466822	−0.233411	+	0.972378i	$0.574989\pi$
$90$	0	0
$91$	4802.00	0.0607881
$92$	16608.0	0.204573
$93$	0	0
$94$	82416.0	0.962037
$95$	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	$0.467894\pi$
$98$	−9604.00	−0.101015
$99$	0	0
$100$	−40784.0	−0.407840
$101$	−153084.	−1.49323	−0.746614	−	0.665257i	$-0.768322\pi$
$101$	−153084.	−1.49323	−0.746614	+	0.665257i	$0.768322\pi$
$102$	0	0
$103$	35864.0	0.333093	0.166547	−	0.986034i	$-0.446738\pi$
$103$	35864.0	0.333093	0.166547	+	0.986034i	$0.446738\pi$
$104$	−6272.00	−0.0568621
$105$	0	0
$106$	129960.	1.12343
$107$	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	$0.173281\pi$
$110$	−6336.00	−0.0499268
$111$	0	0
$112$	12544.0	0.0944911
$113$	−62106.0	−0.457549	−0.228774	−	0.973479i	$-0.573472\pi$
$113$	−62106.0	−0.457549	−0.228774	+	0.973479i	$0.573472\pi$
$114$	0	0
$115$	−24912.0	−0.175656
$116$	39840.0	0.274900
$117$	0	0
$118$	136896.	0.905077
$119$	10584.0	0.0685145
$120$	0	0
$121$	−156695.	−0.972953
$122$	−142616.	−0.867498
$123$	0	0
$124$	−112768.	−0.658615
$125$	136176.	0.779517
$126$	0	0
$127$	−53044.0	−0.291828	−0.145914	−	0.989297i	$-0.546612\pi$
$127$	−53044.0	−0.291828	−0.145914	+	0.989297i	$0.546612\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	9408.00	0.0488246
$131$	−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
$134$	−50720.0	−0.244015
$135$	0	0
$136$	−13824.0	−0.0640894
$137$	−129846.	−0.591054	−0.295527	−	0.955334i	$-0.595495\pi$
$137$	−129846.	−0.591054	−0.295527	+	0.955334i	$0.595495\pi$
$138$	0	0
$139$	−104356.	−0.458121	−0.229061	−	0.973412i	$-0.573565\pi$
$139$	−104356.	−0.458121	−0.229061	+	0.973412i	$0.573565\pi$
$140$	−18816.0	−0.0811348
$141$	0	0
$142$	−170568.	−0.709867
$143$	−6468.00	−0.0264503
$144$	0	0
$145$	−59760.0	−0.236043
$146$	−134936.	−0.523897
$147$	0	0
$148$	−195808.	−0.734812
$149$	−217194.	−0.801461	−0.400730	−	0.916196i	$-0.631244\pi$
$149$	−217194.	−0.801461	−0.400730	+	0.916196i	$0.631244\pi$
$150$	0	0
$151$	221000.	0.788769	0.394385	−	0.918945i	$-0.370958\pi$
$151$	221000.	0.788769	0.394385	+	0.918945i	$0.370958\pi$
$152$	21760.0	0.0763924
$153$	0	0
$154$	12936.0	0.0439540
$155$	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	$0.709854\pi$
$158$	340432.	1.08489
$159$	0	0
$160$	24576.0	0.0758947
$161$	50862.0	0.154642
$162$	0	0
$163$	104816.	0.309000	0.154500	−	0.987993i	$-0.450623\pi$
$163$	104816.	0.309000	0.154500	+	0.987993i	$0.450623\pi$
$164$	−103488.	−0.300456
$165$	0	0
$166$	−427056.	−1.20286
$167$	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
$170$	20736.0	0.0550304
$171$	0	0
$172$	−246592.	−0.635562
$173$	−331068.	−0.841012	−0.420506	−	0.907290i	$-0.638147\pi$
$173$	−331068.	−0.841012	−0.420506	+	0.907290i	$0.638147\pi$
$174$	0	0
$175$	−124901.	−0.308298
$176$	−16896.0	−0.0411152
$177$	0	0
$178$	139536.	0.330093
$179$	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	$0.267514\pi$
$182$	−19208.0	−0.0429837
$183$	0	0
$184$	−66432.0	−0.144655
$185$	293712.	0.630946
$186$	0	0
$187$	−14256.0	−0.0298122
$188$	−329664.	−0.680263
$189$	0	0
$190$	−32640.0	−0.0655943
$191$	−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	$0.393976\pi$
$194$	−74648.0	−0.142401
$195$	0	0
$196$	38416.0	0.0714286
$197$	237942.	0.436823	0.218412	−	0.975857i	$-0.429912\pi$
$197$	237942.	0.436823	0.218412	+	0.975857i	$0.429912\pi$
$198$	0	0
$199$	204464.	0.366003	0.183001	−	0.983113i	$-0.441419\pi$
$199$	204464.	0.366003	0.183001	+	0.983113i	$0.441419\pi$
$200$	163136.	0.288386
$201$	0	0
$202$	612336.	1.05587
$203$	122010.	0.207805
$204$	0	0
$205$	155232.	0.257986
$206$	−143456.	−0.235532
$207$	0	0
$208$	25088.0	0.0402076
$209$	22440.0	0.0355351
$210$	0	0
$211$	−348724.	−0.539232	−0.269616	−	0.962968i	$-0.586897\pi$
$211$	−348724.	−0.539232	−0.269616	+	0.962968i	$0.586897\pi$
$212$	−519840.	−0.794383
$213$	0	0
$214$	−381816.	−0.569928
$215$	369888.	0.545725
$216$	0	0
$217$	−345352.	−0.497866
$218$	−848888.	−1.20979
$219$	0	0
$220$	25344.0	0.0353036
$221$	21168.0	0.0291541
$222$	0	0
$223$	1.47006e6	1.97957	0.989787	−	0.142554i	$-0.0455316\pi$
$223$	1.47006e6	1.97957	0.989787	+	0.142554i	$0.0455316\pi$
$224$	−50176.0	−0.0668153
$225$	0	0
$226$	248424.	0.323536
$227$	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	$0.728863\pi$
$230$	99648.0	0.124208
$231$	0	0
$232$	−159360.	−0.194383
$233$	−651222.	−0.785849	−0.392925	−	0.919571i	$-0.628537\pi$
$233$	−651222.	−0.785849	−0.392925	+	0.919571i	$0.628537\pi$
$234$	0	0
$235$	494496.	0.584108
$236$	−547584.	−0.639986
$237$	0	0
$238$	−42336.0	−0.0484471
$239$	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	$0.625882\pi$
$242$	626780.	0.687981
$243$	0	0
$244$	570464.	0.613414
$245$	−57624.0	−0.0613322
$246$	0	0
$247$	−33320.0	−0.0347506
$248$	451072.	0.465711
$249$	0	0
$250$	−544704.	−0.551202
$251$	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
$254$	212176.	0.206354
$255$	0	0
$256$	65536.0	0.0625000
$257$	1.00520e6	0.949339	0.474670	−	0.880164i	$-0.342568\pi$
$257$	1.00520e6	0.949339	0.474670	+	0.880164i	$0.342568\pi$
$258$	0	0
$259$	−599662.	−0.555466
$260$	−37632.0	−0.0345242
$261$	0	0
$262$	277296.	0.249569
$263$	−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
$266$	66640.0	0.0577472
$267$	0	0
$268$	202880.	0.172545
$269$	1.76069e6	1.48355	0.741774	−	0.670650i	$-0.233985\pi$
$269$	1.76069e6	1.48355	0.741774	+	0.670650i	$0.233985\pi$
$270$	0	0
$271$	770528.	0.637331	0.318666	−	0.947867i	$-0.396765\pi$
$271$	770528.	0.637331	0.318666	+	0.947867i	$0.396765\pi$
$272$	55296.0	0.0453181
$273$	0	0
$274$	519384.	0.417938
$275$	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	$0.410625\pi$
$278$	417424.	0.323941
$279$	0	0
$280$	75264.0	0.0573710
$281$	−2.30432e6	−1.74091	−0.870456	−	0.492247i	$-0.836176\pi$
$281$	−2.30432e6	−1.74091	−0.870456	+	0.492247i	$0.836176\pi$
$282$	0	0
$283$	1.60903e6	1.19426	0.597128	−	0.802146i	$-0.296308\pi$
$283$	1.60903e6	1.19426	0.597128	+	0.802146i	$0.296308\pi$
$284$	682272.	0.501951
$285$	0	0
$286$	25872.0	0.0187032
$287$	−316932.	−0.227123
$288$	0	0
$289$	−1.37320e6	−0.967140
$290$	239040.	0.166907
$291$	0	0
$292$	539744.	0.370451
$293$	−517020.	−0.351834	−0.175917	−	0.984405i	$-0.556289\pi$
$293$	−517020.	−0.351834	−0.175917	+	0.984405i	$0.556289\pi$
$294$	0	0
$295$	821376.	0.549524
$296$	783232.	0.519590
$297$	0	0
$298$	868776.	0.566718
$299$	101724.	0.0658030
$300$	0	0
$301$	−755188.	−0.480440
$302$	−884000.	−0.557744
$303$	0	0
$304$	−87040.0	−0.0540176
$305$	−855696.	−0.526708
$306$	0	0
$307$	1.35002e6	0.817512	0.408756	−	0.912644i	$-0.365963\pi$
$307$	1.35002e6	0.817512	0.408756	+	0.912644i	$0.365963\pi$
$308$	−51744.0	−0.0310802
$309$	0	0
$310$	−676608.	−0.399883
$311$	−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	$0.476457\pi$
$314$	1.51348e6	0.866269
$315$	0	0
$316$	−1.36173e6	−0.767137
$317$	−1.84629e6	−1.03193	−0.515967	−	0.856609i	$-0.672567\pi$
$317$	−1.84629e6	−1.03193	−0.515967	+	0.856609i	$0.672567\pi$
$318$	0	0
$319$	−164340.	−0.0904204
$320$	−98304.0	−0.0536656
$321$	0	0
$322$	−203448.	−0.109349
$323$	−73440.0	−0.0391675
$324$	0	0
$325$	−249802.	−0.131186
$326$	−419264.	−0.218496
$327$	0	0
$328$	413952.	0.212454
$329$	−1.00960e6	−0.514231
$330$	0	0
$331$	−3.33238e6	−1.67180	−0.835900	−	0.548881i	$-0.815054\pi$
$331$	−3.33238e6	−1.67180	−0.835900	+	0.548881i	$0.815054\pi$
$332$	1.70822e6	0.850550
$333$	0	0
$334$	−1.70789e6	−0.837709
$335$	−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	$0.628240\pi$
$338$	1.44676e6	0.688816
$339$	0	0
$340$	−82944.0	−0.0389124
$341$	465168.	0.216633
$342$	0	0
$343$	117649.	0.0539949
$344$	986368.	0.449410
$345$	0	0
$346$	1.32427e6	0.594685
$347$	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	$0.568890\pi$
$350$	499604.	0.218000
$351$	0	0
$352$	67584.0	0.0290728
$353$	−3.45857e6	−1.47727	−0.738634	−	0.674106i	$-0.764529\pi$
$353$	−3.45857e6	−1.47727	−0.738634	+	0.674106i	$0.764529\pi$
$354$	0	0
$355$	−1.02341e6	−0.431001
$356$	−558144.	−0.233411
$357$	0	0
$358$	−1.60078e6	−0.660120
$359$	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
$362$	−2.35239e6	−0.943492
$363$	0	0
$364$	76832.0	0.0303941
$365$	−809616.	−0.318088
$366$	0	0
$367$	3.11994e6	1.20915	0.604575	−	0.796548i	$-0.293343\pi$
$367$	3.11994e6	1.20915	0.604575	+	0.796548i	$0.293343\pi$
$368$	265728.	0.102286
$369$	0	0
$370$	−1.17485e6	−0.446146
$371$	−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	$0.622451\pi$
$374$	57024.0	0.0210804
$375$	0	0
$376$	1.31866e6	0.481019
$377$	244020.	0.0884244
$378$	0	0
$379$	−5.38083e6	−1.92420	−0.962102	−	0.272690i	$-0.912087\pi$
$379$	−5.38083e6	−1.92420	−0.962102	+	0.272690i	$0.912087\pi$
$380$	130560.	0.0463822
$381$	0	0
$382$	3.75737e6	1.31742
$383$	−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
$386$	−1.35356e6	−0.462391
$387$	0	0
$388$	298592.	0.100693
$389$	−891390.	−0.298671	−0.149336	−	0.988787i	$-0.547714\pi$
$389$	−891390.	−0.298671	−0.149336	+	0.988787i	$0.547714\pi$
$390$	0	0
$391$	224208.	0.0741667
$392$	−153664.	−0.0505076
$393$	0	0
$394$	−951768.	−0.308881
$395$	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	$0.442754\pi$
$398$	−817856.	−0.258803
$399$	0	0
$400$	−652544.	−0.203920
$401$	−1.72037e6	−0.534271	−0.267136	−	0.963659i	$-0.586077\pi$
$401$	−1.72037e6	−0.534271	−0.267136	+	0.963659i	$0.586077\pi$
$402$	0	0
$403$	−690704.	−0.211850
$404$	−2.44934e6	−0.746614
$405$	0	0
$406$	−488040.	−0.146940
$407$	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	$0.496366\pi$
$410$	−620928.	−0.182424
$411$	0	0
$412$	573824.	0.166547
$413$	−1.67698e6	−0.483784
$414$	0	0
$415$	−2.56234e6	−0.730324
$416$	−100352.	−0.0284310
$417$	0	0
$418$	−89760.0	−0.0251271
$419$	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	$0.424075\pi$
$422$	1.39490e6	0.381295
$423$	0	0
$424$	2.07936e6	0.561714
$425$	−550584.	−0.147860
$426$	0	0
$427$	1.74705e6	0.463697
$428$	1.52726e6	0.403000
$429$	0	0
$430$	−1.47955e6	−0.385886
$431$	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	$0.321451\pi$
$434$	1.38141e6	0.352045
$435$	0	0
$436$	3.39555e6	0.855449
$437$	−352920.	−0.0884042
$438$	0	0
$439$	3.88407e6	0.961891	0.480946	−	0.876750i	$-0.340293\pi$
$439$	3.88407e6	0.961891	0.480946	+	0.876750i	$0.340293\pi$
$440$	−101376.	−0.0249634
$441$	0	0
$442$	−84672.0	−0.0206150
$443$	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
$446$	−5.88022e6	−1.39977
$447$	0	0
$448$	200704.	0.0472456
$449$	1.92281e6	0.450113	0.225056	−	0.974346i	$-0.427743\pi$
$449$	1.92281e6	0.450113	0.225056	+	0.974346i	$0.427743\pi$
$450$	0	0
$451$	426888.	0.0988263
$452$	−993696.	−0.228774
$453$	0	0
$454$	−2.35824e6	−0.536968
$455$	−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	$0.221007\pi$
$458$	4.18137e6	0.931440
$459$	0	0
$460$	−398592.	−0.0878282
$461$	−2.97167e6	−0.651250	−0.325625	−	0.945499i	$-0.605575\pi$
$461$	−2.97167e6	−0.651250	−0.325625	+	0.945499i	$0.605575\pi$
$462$	0	0
$463$	4.87423e6	1.05670	0.528352	−	0.849025i	$-0.322810\pi$
$463$	4.87423e6	1.05670	0.528352	+	0.849025i	$0.322810\pi$
$464$	637440.	0.137450
$465$	0	0
$466$	2.60489e6	0.555679
$467$	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
$470$	−1.97798e6	−0.413027
$471$	0	0
$472$	2.19034e6	0.452539
$473$	1.01719e6	0.209050
$474$	0	0
$475$	866660.	0.176244
$476$	169344.	0.0342572
$477$	0	0
$478$	−2.05385e6	−0.411148
$479$	−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
$482$	2.77886e6	0.544814
$483$	0	0
$484$	−2.50712e6	−0.486476
$485$	−447888.	−0.0864600
$486$	0	0
$487$	316928.	0.0605534	0.0302767	−	0.999542i	$-0.490361\pi$
$487$	316928.	0.0605534	0.0302767	+	0.999542i	$0.490361\pi$
$488$	−2.28186e6	−0.433749
$489$	0	0
$490$	230496.	0.0433684
$491$	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
$494$	133280.	0.0245724
$495$	0	0
$496$	−1.80429e6	−0.329308
$497$	2.08946e6	0.379440
$498$	0	0
$499$	−4.86773e6	−0.875135	−0.437568	−	0.899185i	$-0.644160\pi$
$499$	−4.86773e6	−0.875135	−0.437568	+	0.899185i	$0.644160\pi$
$500$	2.17882e6	0.389758
$501$	0	0
$502$	−5.58432e6	−0.989034
$503$	−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
$506$	274032.	0.0475801
$507$	0	0
$508$	−848704.	−0.145914
$509$	9.41621e6	1.61095	0.805474	−	0.592631i	$-0.201911\pi$
$509$	9.41621e6	1.61095	0.805474	+	0.592631i	$0.201911\pi$
$510$	0	0
$511$	1.65297e6	0.280035
$512$	−262144.	−0.0441942
$513$	0	0
$514$	−4.02082e6	−0.671284
$515$	−860736.	−0.143005
$516$	0	0
$517$	1.35986e6	0.223753
$518$	2.39865e6	0.392773
$519$	0	0
$520$	150528.	0.0244123
$521$	−1.84039e6	−0.297041	−0.148520	−	0.988909i	$-0.547451\pi$
$521$	−1.84039e6	−0.297041	−0.148520	+	0.988909i	$0.547451\pi$
$522$	0	0
$523$	−979108.	−0.156522	−0.0782612	−	0.996933i	$-0.524937\pi$
$523$	−979108.	−0.156522	−0.0782612	+	0.996933i	$0.524937\pi$
$524$	−1.10918e6	−0.176472
$525$	0	0
$526$	5.01204e6	0.789860
$527$	−1.52237e6	−0.238777
$528$	0	0
$529$	−5.35890e6	−0.832600
$530$	−3.11904e6	−0.482316
$531$	0	0
$532$	−266560.	−0.0408334
$533$	−633864.	−0.0966447
$534$	0	0
$535$	−2.29090e6	−0.346036
$536$	−811520.	−0.122008
$537$	0	0
$538$	−7.04275e6	−1.04903
$539$	−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	$0.355746\pi$
$542$	−3.08211e6	−0.450661
$543$	0	0
$544$	−221184.	−0.0320447
$545$	−5.09333e6	−0.734531
$546$	0	0
$547$	8.73025e6	1.24755	0.623775	−	0.781604i	$-0.285598\pi$
$547$	8.73025e6	1.24755	0.623775	+	0.781604i	$0.285598\pi$
$548$	−2.07754e6	−0.295527
$549$	0	0
$550$	−672936.	−0.0948565
$551$	−846600.	−0.118795
$552$	0	0
$553$	−4.17029e6	−0.579901
$554$	−2.83095e6	−0.391885
$555$	0	0
$556$	−1.66970e6	−0.229061
$557$	3.01066e6	0.411172	0.205586	−	0.978639i	$-0.434090\pi$
$557$	3.01066e6	0.411172	0.205586	+	0.978639i	$0.434090\pi$
$558$	0	0
$559$	−1.51038e6	−0.204435
$560$	−301056.	−0.0405674
$561$	0	0
$562$	9.21727e6	1.23101
$563$	−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
$566$	−6.43611e6	−0.844467
$567$	0	0
$568$	−2.72909e6	−0.354933
$569$	−1.31578e7	−1.70374	−0.851870	−	0.523754i	$-0.824531\pi$
$569$	−1.31578e7	−1.70374	−0.851870	+	0.523754i	$0.824531\pi$
$570$	0	0
$571$	−1.03344e7	−1.32647	−0.663234	−	0.748412i	$-0.730817\pi$
$571$	−1.03344e7	−1.32647	−0.663234	+	0.748412i	$0.730817\pi$
$572$	−103488.	−0.0132251
$573$	0	0
$574$	1.26773e6	0.160600
$575$	−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	$0.664010\pi$
$578$	5.49280e6	0.683872
$579$	0	0
$580$	−956160.	−0.118021
$581$	5.23144e6	0.642955
$582$	0	0
$583$	2.14434e6	0.261290
$584$	−2.15898e6	−0.261948
$585$	0	0
$586$	2.06808e6	0.248784
$587$	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
$590$	−3.28550e6	−0.388572
$591$	0	0
$592$	−3.13293e6	−0.367406
$593$	9.20369e6	1.07479	0.537397	−	0.843329i	$-0.319408\pi$
$593$	9.20369e6	1.07479	0.537397	+	0.843329i	$0.319408\pi$
$594$	0	0
$595$	−254016.	−0.0294150
$596$	−3.47510e6	−0.400730
$597$	0	0
$598$	−406896.	−0.0465297
$599$	−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	$0.682692\pi$
$602$	3.02075e6	0.339722
$603$	0	0
$604$	3.53600e6	0.394385
$605$	3.76068e6	0.417713
$606$	0	0
$607$	2.21264e6	0.243747	0.121873	−	0.992546i	$-0.461110\pi$
$607$	2.21264e6	0.243747	0.121873	+	0.992546i	$0.461110\pi$
$608$	348160.	0.0381962
$609$	0	0
$610$	3.42278e6	0.372439
$611$	−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	$0.640749\pi$
$614$	−5.40008e6	−0.578068
$615$	0	0
$616$	206976.	0.0219770
$617$	1.37397e7	1.45299	0.726497	−	0.687170i	$-0.241147\pi$
$617$	1.37397e7	1.45299	0.726497	+	0.687170i	$0.241147\pi$
$618$	0	0
$619$	−8.70113e6	−0.912744	−0.456372	−	0.889789i	$-0.650851\pi$
$619$	−8.70113e6	−0.912744	−0.456372	+	0.889789i	$0.650851\pi$
$620$	2.70643e6	0.282760
$621$	0	0
$622$	5.38152e6	0.557736
$623$	−1.70932e6	−0.176442
$624$	0	0
$625$	4.69740e6	0.481014
$626$	−1.02462e6	−0.104502
$627$	0	0
$628$	−6.05392e6	−0.612544
$629$	−2.64341e6	−0.266402
$630$	0	0
$631$	445412.	0.0445337	0.0222668	−	0.999752i	$-0.492912\pi$
$631$	445412.	0.0445337	0.0222668	+	0.999752i	$0.492912\pi$
$632$	5.44691e6	0.542447
$633$	0	0
$634$	7.38516e6	0.729687
$635$	1.27306e6	0.125289
$636$	0	0
$637$	235298.	0.0229757
$638$	657360.	0.0639369
$639$	0	0
$640$	393216.	0.0379473
$641$	8.00119e6	0.769147	0.384573	−	0.923094i	$-0.374349\pi$
$641$	8.00119e6	0.769147	0.384573	+	0.923094i	$0.374349\pi$
$642$	0	0
$643$	−1.58402e7	−1.51090	−0.755448	−	0.655209i	$-0.772581\pi$
$643$	−1.58402e7	−1.51090	−0.755448	+	0.655209i	$0.772581\pi$
$644$	813792.	0.0773212
$645$	0	0
$646$	293760.	0.0276956
$647$	−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
$650$	999208.	0.0927625
$651$	0	0
$652$	1.67706e6	0.154500
$653$	−7.34149e6	−0.673753	−0.336877	−	0.941549i	$-0.609371\pi$
$653$	−7.34149e6	−0.673753	−0.336877	+	0.941549i	$0.609371\pi$
$654$	0	0
$655$	1.66378e6	0.151528
$656$	−1.65581e6	−0.150228
$657$	0	0
$658$	4.03838e6	0.363616
$659$	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	$0.839457\pi$
$662$	1.33295e7	1.18214
$663$	0	0
$664$	−6.83290e6	−0.601429
$665$	399840.	0.0350616
$666$	0	0
$667$	2.58462e6	0.224948
$668$	6.83155e6	0.592350
$669$	0	0
$670$	1.21728e6	0.104762
$671$	−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	$0.401129\pi$
$674$	6.53922e6	0.554468
$675$	0	0
$676$	−5.78702e6	−0.487067
$677$	1.89192e7	1.58647	0.793234	−	0.608917i	$-0.208396\pi$
$677$	1.89192e7	1.58647	0.793234	+	0.608917i	$0.208396\pi$
$678$	0	0
$679$	914438.	0.0761167
$680$	331776.	0.0275152
$681$	0	0
$682$	−1.86067e6	−0.153182
$683$	−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
$686$	−470596.	−0.0381802
$687$	0	0
$688$	−3.94547e6	−0.317781
$689$	−3.18402e6	−0.255522
$690$	0	0
$691$	1.63276e7	1.30085	0.650424	−	0.759571i	$-0.274591\pi$
$691$	1.63276e7	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$692$	−5.29709e6	−0.420506
$693$	0	0
$694$	−3.36612e6	−0.265296
$695$	2.50454e6	0.196683
$696$	0	0
$697$	−1.39709e6	−0.108929
$698$	3.90897e6	0.303685
$699$	0	0
$700$	−1.99842e6	−0.154149
$701$	5.40470e6	0.415409	0.207705	−	0.978192i	$-0.433401\pi$
$701$	5.40470e6	0.415409	0.207705	+	0.978192i	$0.433401\pi$
$702$	0	0
$703$	4.16092e6	0.317542
$704$	−270336.	−0.0205576
$705$	0	0
$706$	1.38343e7	1.04459
$707$	−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	$0.190450\pi$
$710$	4.09363e6	0.304763
$711$	0	0
$712$	2.23258e6	0.165046
$713$	−7.31582e6	−0.538939
$714$	0	0
$715$	155232.	0.0113558
$716$	6.40310e6	0.466775
$717$	0	0
$718$	−1.38920e7	−1.00567
$719$	−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
$722$	9.44200e6	0.674095
$723$	0	0
$724$	9.40957e6	0.667150
$725$	−6.34701e6	−0.448460
$726$	0	0
$727$	−9.29438e6	−0.652205	−0.326103	−	0.945334i	$-0.605735\pi$
$727$	−9.29438e6	−0.652205	−0.326103	+	0.945334i	$0.605735\pi$
$728$	−307328.	−0.0214918
$729$	0	0
$730$	3.23846e6	0.224922
$731$	−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	$0.462638\pi$
$734$	−1.24797e7	−0.854999
$735$	0	0
$736$	−1.06291e6	−0.0723274
$737$	−836880.	−0.0567537
$738$	0	0
$739$	2.18135e7	1.46932	0.734658	−	0.678438i	$-0.237343\pi$
$739$	2.18135e7	1.46932	0.734658	+	0.678438i	$0.237343\pi$
$740$	4.69939e6	0.315473
$741$	0	0
$742$	6.36804e6	0.424616
$743$	−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
$746$	8.06692e6	0.530714
$747$	0	0
$748$	−228096.	−0.0149061
$749$	4.67725e6	0.304639
$750$	0	0
$751$	−2.01483e7	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$751$	−2.01483e7	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$752$	−5.27462e6	−0.340132
$753$	0	0
$754$	−976080.	−0.0625255
$755$	−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	$0.377449\pi$
$758$	2.15233e7	1.36062
$759$	0	0
$760$	−522240.	−0.0327972
$761$	−2.97791e6	−0.186402	−0.0932008	−	0.995647i	$-0.529710\pi$
$761$	−2.97791e6	−0.186402	−0.0932008	+	0.995647i	$0.529710\pi$
$762$	0	0
$763$	1.03989e7	0.646659
$764$	−1.50295e7	−0.931559
$765$	0	0
$766$	3.22973e6	0.198881
$767$	−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	$0.711748\pi$
$770$	−310464.	−0.0188705
$771$	0	0
$772$	5.41424e6	0.326960
$773$	7.37953e6	0.444202	0.222101	−	0.975024i	$-0.428709\pi$
$773$	7.37953e6	0.444202	0.222101	+	0.975024i	$0.428709\pi$
$774$	0	0
$775$	1.79654e7	1.07444
$776$	−1.19437e6	−0.0712006
$777$	0	0
$778$	3.56556e6	0.211193
$779$	2.19912e6	0.129839
$780$	0	0
$781$	−2.81437e6	−0.165103
$782$	−896832.	−0.0524438
$783$	0	0
$784$	614656.	0.0357143
$785$	9.08088e6	0.525961
$786$	0	0
$787$	1.36289e7	0.784377	0.392188	−	0.919885i	$-0.371718\pi$
$787$	1.36289e7	0.784377	0.392188	+	0.919885i	$0.371718\pi$
$788$	3.80707e6	0.218412
$789$	0	0
$790$	−8.17037e6	−0.465773
$791$	−3.04319e6	−0.172937
$792$	0	0
$793$	3.49409e6	0.197311
$794$	−4.49382e6	−0.252967
$795$	0	0
$796$	3.27142e6	0.183001
$797$	1.49548e7	0.833938	0.416969	−	0.908921i	$-0.363092\pi$
$797$	1.49548e7	0.833938	0.416969	+	0.908921i	$0.363092\pi$
$798$	0	0
$799$	−4.45046e6	−0.246626
$800$	2.61018e6	0.144193
$801$	0	0
$802$	6.88150e6	0.377787
$803$	−2.22644e6	−0.121849
$804$	0	0
$805$	−1.22069e6	−0.0663919
$806$	2.76282e6	0.149801
$807$	0	0
$808$	9.79738e6	0.527936
$809$	−2.87242e7	−1.54304	−0.771519	−	0.636206i	$-0.780503\pi$
$809$	−2.87242e7	−1.54304	−0.771519	+	0.636206i	$0.780503\pi$
$810$	0	0
$811$	−1.52265e7	−0.812922	−0.406461	−	0.913668i	$-0.633237\pi$
$811$	−1.52265e7	−0.812922	−0.406461	+	0.913668i	$0.633237\pi$
$812$	1.95216e6	0.103902
$813$	0	0
$814$	−3.23083e6	−0.170904
$815$	−2.51558e6	−0.132661
$816$	0	0
$817$	5.24008e6	0.274652
$818$	−308984.	−0.0161455
$819$	0	0
$820$	2.48371e6	0.128993
$821$	3.31001e7	1.71384	0.856921	−	0.515447i	$-0.172374\pi$
$821$	3.31001e7	1.71384	0.856921	+	0.515447i	$0.172374\pi$
$822$	0	0
$823$	−1.35915e7	−0.699470	−0.349735	−	0.936849i	$-0.613728\pi$
$823$	−1.35915e7	−0.699470	−0.349735	+	0.936849i	$0.613728\pi$
$824$	−2.29530e6	−0.117766
$825$	0	0
$826$	6.70790e6	0.342087
$827$	−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	$0.395942\pi$
$830$	1.02493e7	0.516417
$831$	0	0
$832$	401408.	0.0201038
$833$	518616.	0.0258960
$834$	0	0
$835$	−1.02473e7	−0.508621
$836$	359040.	0.0177675
$837$	0	0
$838$	−2.08246e7	−1.02439
$839$	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
$842$	−6.87390e6	−0.334136
$843$	0	0
$844$	−5.57958e6	−0.269616
$845$	8.68054e6	0.418220
$846$	0	0
$847$	−7.67806e6	−0.367742
$848$	−8.31744e6	−0.397192
$849$	0	0
$850$	2.20234e6	0.104553
$851$	−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	$0.649381\pi$
$854$	−6.98818e6	−0.327884
$855$	0	0
$856$	−6.10906e6	−0.284964
$857$	2.65655e7	1.23556	0.617782	−	0.786349i	$-0.288031\pi$
$857$	2.65655e7	1.23556	0.617782	+	0.786349i	$0.288031\pi$
$858$	0	0
$859$	−9.16844e6	−0.423948	−0.211974	−	0.977275i	$-0.567989\pi$
$859$	−9.16844e6	−0.423948	−0.211974	+	0.977275i	$0.567989\pi$
$860$	5.91821e6	0.272863
$861$	0	0
$862$	−2.32250e6	−0.106460
$863$	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
$866$	−1.66035e7	−0.752324
$867$	0	0
$868$	−5.52563e6	−0.248933
$869$	5.61713e6	0.252328
$870$	0	0
$871$	1.24264e6	0.0555009
$872$	−1.35822e7	−0.604894
$873$	0	0
$874$	1.41168e6	0.0625112
$875$	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	$0.431606\pi$
$878$	−1.55363e7	−0.680160
$879$	0	0
$880$	405504.	0.0176518
$881$	−1.65372e7	−0.717833	−0.358917	−	0.933370i	$-0.616854\pi$
$881$	−1.65372e7	−0.717833	−0.358917	+	0.933370i	$0.616854\pi$
$882$	0	0
$883$	−2.39487e7	−1.03367	−0.516833	−	0.856086i	$-0.672889\pi$
$883$	−2.39487e7	−1.03367	−0.516833	+	0.856086i	$0.672889\pi$
$884$	338688.	0.0145770
$885$	0	0
$886$	−9.25994e6	−0.396300
$887$	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
$890$	−3.34886e6	−0.141717
$891$	0	0
$892$	2.35209e7	0.989787
$893$	7.00536e6	0.293969
$894$	0	0
$895$	−9.60466e6	−0.400797
$896$	−802816.	−0.0334077
$897$	0	0
$898$	−7.69126e6	−0.318278
$899$	−1.75495e7	−0.724212
$900$	0	0
$901$	−7.01784e6	−0.287999
$902$	−1.70755e6	−0.0698808
$903$	0	0
$904$	3.97478e6	0.161768
$905$	−1.41144e7	−0.572848
$906$	0	0
$907$	2.06126e7	0.831983	0.415991	−	0.909369i	$-0.363435\pi$
$907$	2.06126e7	0.831983	0.415991	+	0.909369i	$0.363435\pi$
$908$	9.43296e6	0.379694
$909$	0	0
$910$	460992.	0.0184540
$911$	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
$914$	−2.74486e7	−1.08681
$915$	0	0
$916$	−1.67255e7	−0.658627
$917$	−3.39688e6	−0.133400
$918$	0	0
$919$	−3.61227e7	−1.41088	−0.705442	−	0.708767i	$-0.749252\pi$
$919$	−3.61227e7	−1.41088	−0.705442	+	0.708767i	$0.749252\pi$
$920$	1.59437e6	0.0621039
$921$	0	0
$922$	1.18867e7	0.460504
$923$	4.17892e6	0.161458
$924$	0	0
$925$	3.11947e7	1.19874
$926$	−1.94969e7	−0.747203
$927$	0	0
$928$	−2.54976e6	−0.0971917
$929$	−1.29366e7	−0.491792	−0.245896	−	0.969296i	$-0.579082\pi$
$929$	−1.29366e7	−0.491792	−0.245896	+	0.969296i	$0.579082\pi$
$930$	0	0
$931$	−816340.	−0.0308672
$932$	−1.04196e7	−0.392925
$933$	0	0
$934$	−3.26920e7	−1.22624
$935$	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	$0.117336\pi$
$938$	−2.48528e6	−0.0922292
$939$	0	0
$940$	7.91194e6	0.292054
$941$	1.05568e7	0.388651	0.194325	−	0.980937i	$-0.437748\pi$
$941$	1.05568e7	0.388651	0.194325	+	0.980937i	$0.437748\pi$
$942$	0	0
$943$	−6.71378e6	−0.245860
$944$	−8.76134e6	−0.319993
$945$	0	0
$946$	−4.06877e6	−0.147821
$947$	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
$950$	−3.46664e6	−0.124623
$951$	0	0
$952$	−677376.	−0.0242235
$953$	−5.22829e7	−1.86478	−0.932389	−	0.361455i	$-0.882280\pi$
$953$	−5.22829e7	−1.86478	−0.932389	+	0.361455i	$0.882280\pi$
$954$	0	0
$955$	2.25442e7	0.799883
$956$	8.21539e6	0.290726
$957$	0	0
$958$	9.37589e6	0.330064
$959$	−6.36245e6	−0.223397
$960$	0	0
$961$	2.10452e7	0.735095
$962$	4.79730e6	0.167132
$963$	0	0
$964$	−1.11154e7	−0.385242
$965$	−8.12136e6	−0.280744
$966$	0	0
$967$	−2.48235e7	−0.853682	−0.426841	−	0.904327i	$-0.640374\pi$
$967$	−2.48235e7	−0.853682	−0.426841	+	0.904327i	$0.640374\pi$
$968$	1.00285e7	0.343991
$969$	0	0
$970$	1.79155e6	0.0611364
$971$	−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
$974$	−1.26771e6	−0.0428177
$975$	0	0
$976$	9.12742e6	0.306707
$977$	−8.17705e6	−0.274069	−0.137035	−	0.990566i	$-0.543757\pi$
$977$	−8.17705e6	−0.274069	−0.137035	+	0.990566i	$0.543757\pi$
$978$	0	0
$979$	2.30234e6	0.0767739
$980$	−921984.	−0.0306661
$981$	0	0
$982$	−2.08016e7	−0.688365
$983$	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
$986$	−2.15136e6	−0.0704727
$987$	0	0
$988$	−533120.	−0.0173753
$989$	−1.59977e7	−0.520075
$990$	0	0
$991$	−1.48550e7	−0.480494	−0.240247	−	0.970712i	$-0.577228\pi$
$991$	−1.48550e7	−0.480494	−0.240247	+	0.970712i	$0.577228\pi$
$992$	7.21715e6	0.232856
$993$	0	0
$994$	−8.35783e6	−0.268304
$995$	−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	$0.516933\pi$
$998$	1.94709e7	0.618814
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	126.6.a.b.1.1		1
3.2	odd	2		42.6.a.f.1.1	✓	1
4.3	odd	2		1008.6.a.k.1.1		1
7.6	odd	2		882.6.a.i.1.1		1
12.11	even	2		336.6.a.g.1.1		1
15.2	even	4		1050.6.g.m.799.2		2
15.8	even	4		1050.6.g.m.799.1		2
15.14	odd	2		1050.6.a.a.1.1		1
21.2	odd	6		294.6.e.b.67.1		2
21.5	even	6		294.6.e.f.67.1		2
21.11	odd	6		294.6.e.b.79.1		2
21.17	even	6		294.6.e.f.79.1		2
21.20	even	2		294.6.a.i.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.6.a.f.1.1	✓	1	3.2	odd	2
126.6.a.b.1.1		1	1.1	even	1	trivial
294.6.a.i.1.1		1	21.20	even	2
294.6.e.b.67.1		2	21.2	odd	6
294.6.e.b.79.1		2	21.11	odd	6
294.6.e.f.67.1		2	21.5	even	6
294.6.e.f.79.1		2	21.17	even	6
336.6.a.g.1.1		1	12.11	even	2
882.6.a.i.1.1		1	7.6	odd	2
1008.6.a.k.1.1		1	4.3	odd	2
1050.6.a.a.1.1		1	15.14	odd	2
1050.6.g.m.799.1		2	15.8	even	4
1050.6.g.m.799.2		2	15.2	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:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more