LMFDB - Embedded newform 1040.6.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1040,6,Mod(1,1040)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1040, 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("1040.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 1040 = 2^{4} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	1040.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:	$166.799172605$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 65)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1040.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-6.00000 q^{3} -25.0000 q^{5} +244.000 q^{7} -207.000 q^{9} -794.000 q^{11} -169.000 q^{13} +150.000 q^{15} -1534.00 q^{17} -2706.00 q^{19} -1464.00 q^{21} +702.000 q^{23} +625.000 q^{25} +2700.00 q^{27} -5038.00 q^{29} +3634.00 q^{31} +4764.00 q^{33} -6100.00 q^{35} -7058.00 q^{37} +1014.00 q^{39} -294.000 q^{41} -7618.00 q^{43} +5175.00 q^{45} +3020.00 q^{47} +42729.0 q^{49} +9204.00 q^{51} +626.000 q^{53} +19850.0 q^{55} +16236.0 q^{57} +30066.0 q^{59} -5806.00 q^{61} -50508.0 q^{63} +4225.00 q^{65} +12436.0 q^{67} -4212.00 q^{69} -4734.00 q^{71} -14694.0 q^{73} -3750.00 q^{75} -193736. q^{77} +39804.0 q^{79} +34101.0 q^{81} +41776.0 q^{83} +38350.0 q^{85} +30228.0 q^{87} +7970.00 q^{89} -41236.0 q^{91} -21804.0 q^{93} +67650.0 q^{95} -78050.0 q^{97} +164358. q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	−6.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−6.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	−25.0000	−0.447214
$5$	−25.0000	−0.447214
$6$	0	0
$7$	244.000	1.88211	0.941054	−	0.338255i	$-0.109837\pi$
$7$	244.000	1.88211	0.941054	+	0.338255i	$0.109837\pi$
$8$	0	0
$9$	−207.000	−0.851852
$10$	0	0
$11$	−794.000	−1.97851	−0.989256	−	0.146192i	$-0.953298\pi$
$11$	−794.000	−1.97851	−0.989256	+	0.146192i	$0.953298\pi$
$12$	0	0
$13$	−169.000	−0.277350
$13$	−169.000	−0.277350
$14$	0	0
$15$	150.000	0.172133
$16$	0	0
$17$	−1534.00	−1.28737	−0.643685	−	0.765291i	$-0.722595\pi$
$17$	−1534.00	−1.28737	−0.643685	+	0.765291i	$0.722595\pi$
$18$	0	0
$19$	−2706.00	−1.71966	−0.859832	−	0.510576i	$-0.829432\pi$
$19$	−2706.00	−1.71966	−0.859832	+	0.510576i	$0.829432\pi$
$20$	0	0
$21$	−1464.00	−0.724424
$22$	0	0
$23$	702.000	0.276705	0.138353	−	0.990383i	$-0.455819\pi$
$23$	702.000	0.276705	0.138353	+	0.990383i	$0.455819\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	2700.00	0.712778
$28$	0	0
$29$	−5038.00	−1.11241	−0.556203	−	0.831047i	$-0.687742\pi$
$29$	−5038.00	−1.11241	−0.556203	+	0.831047i	$0.687742\pi$
$30$	0	0
$31$	3634.00	0.679173	0.339587	−	0.940575i	$-0.389713\pi$
$31$	3634.00	0.679173	0.339587	+	0.940575i	$0.389713\pi$
$32$	0	0
$33$	4764.00	0.761530
$34$	0	0
$35$	−6100.00	−0.841705
$36$	0	0
$37$	−7058.00	−0.847573	−0.423787	−	0.905762i	$-0.639299\pi$
$37$	−7058.00	−0.847573	−0.423787	+	0.905762i	$0.639299\pi$
$38$	0	0
$39$	1014.00	0.106752
$40$	0	0
$41$	−294.000	−0.0273141	−0.0136571	−	0.999907i	$-0.504347\pi$
$41$	−294.000	−0.0273141	−0.0136571	+	0.999907i	$0.504347\pi$
$42$	0	0
$43$	−7618.00	−0.628304	−0.314152	−	0.949373i	$-0.601720\pi$
$43$	−7618.00	−0.628304	−0.314152	+	0.949373i	$0.601720\pi$
$44$	0	0
$45$	5175.00	0.380960
$46$	0	0
$47$	3020.00	0.199417	0.0997085	−	0.995017i	$-0.468209\pi$
$47$	3020.00	0.199417	0.0997085	+	0.995017i	$0.468209\pi$
$48$	0	0
$49$	42729.0	2.54233
$50$	0	0
$51$	9204.00	0.495509
$52$	0	0
$53$	626.000	0.0306115	0.0153058	−	0.999883i	$-0.495128\pi$
$53$	626.000	0.0306115	0.0153058	+	0.999883i	$0.495128\pi$
$54$	0	0
$55$	19850.0	0.884818
$56$	0	0
$57$	16236.0	0.661899
$58$	0	0
$59$	30066.0	1.12446	0.562232	−	0.826979i	$-0.309943\pi$
$59$	30066.0	1.12446	0.562232	+	0.826979i	$0.309943\pi$
$60$	0	0
$61$	−5806.00	−0.199780	−0.0998901	−	0.994998i	$-0.531849\pi$
$61$	−5806.00	−0.199780	−0.0998901	+	0.994998i	$0.531849\pi$
$62$	0	0
$63$	−50508.0	−1.60328
$64$	0	0
$65$	4225.00	0.124035
$66$	0	0
$67$	12436.0	0.338449	0.169225	−	0.985577i	$-0.445874\pi$
$67$	12436.0	0.338449	0.169225	+	0.985577i	$0.445874\pi$
$68$	0	0
$69$	−4212.00	−0.106504
$70$	0	0
$71$	−4734.00	−0.111451	−0.0557253	−	0.998446i	$-0.517747\pi$
$71$	−4734.00	−0.111451	−0.0557253	+	0.998446i	$0.517747\pi$
$72$	0	0
$73$	−14694.0	−0.322725	−0.161363	−	0.986895i	$-0.551589\pi$
$73$	−14694.0	−0.322725	−0.161363	+	0.986895i	$0.551589\pi$
$74$	0	0
$75$	−3750.00	−0.0769800
$76$	0	0
$77$	−193736.	−3.72378
$78$	0	0
$79$	39804.0	0.717561	0.358781	−	0.933422i	$-0.383193\pi$
$79$	39804.0	0.717561	0.358781	+	0.933422i	$0.383193\pi$
$80$	0	0
$81$	34101.0	0.577503
$82$	0	0
$83$	41776.0	0.665628	0.332814	−	0.942992i	$-0.392002\pi$
$83$	41776.0	0.665628	0.332814	+	0.942992i	$0.392002\pi$
$84$	0	0
$85$	38350.0	0.575729
$86$	0	0
$87$	30228.0	0.428165
$88$	0	0
$89$	7970.00	0.106656	0.0533278	−	0.998577i	$-0.483017\pi$
$89$	7970.00	0.106656	0.0533278	+	0.998577i	$0.483017\pi$
$90$	0	0
$91$	−41236.0	−0.522003
$92$	0	0
$93$	−21804.0	−0.261414
$94$	0	0
$95$	67650.0	0.769057
$96$	0	0
$97$	−78050.0	−0.842255	−0.421127	−	0.907001i	$-0.638366\pi$
$97$	−78050.0	−0.842255	−0.421127	+	0.907001i	$0.638366\pi$
$98$	0	0
$99$	164358.	1.68540
$100$	0	0
$101$	−23010.0	−0.224447	−0.112223	−	0.993683i	$-0.535797\pi$
$101$	−23010.0	−0.224447	−0.112223	+	0.993683i	$0.535797\pi$
$102$	0	0
$103$	−121706.	−1.13037	−0.565183	−	0.824966i	$-0.691194\pi$
$103$	−121706.	−1.13037	−0.565183	+	0.824966i	$0.691194\pi$
$104$	0	0
$105$	36600.0	0.323972
$106$	0	0
$107$	70142.0	0.592269	0.296134	−	0.955146i	$-0.404302\pi$
$107$	70142.0	0.592269	0.296134	+	0.955146i	$0.404302\pi$
$108$	0	0
$109$	−195878.	−1.57914	−0.789568	−	0.613663i	$-0.789695\pi$
$109$	−195878.	−1.57914	−0.789568	+	0.613663i	$0.789695\pi$
$110$	0	0
$111$	42348.0	0.326231
$112$	0	0
$113$	−100238.	−0.738476	−0.369238	−	0.929335i	$-0.620381\pi$
$113$	−100238.	−0.738476	−0.369238	+	0.929335i	$0.620381\pi$
$114$	0	0
$115$	−17550.0	−0.123746
$116$	0	0
$117$	34983.0	0.236261
$118$	0	0
$119$	−374296.	−2.42297
$120$	0	0
$121$	469385.	2.91451
$122$	0	0
$123$	1764.00	0.0105132
$124$	0	0
$125$	−15625.0	−0.0894427
$126$	0	0
$127$	−39286.0	−0.216137	−0.108068	−	0.994143i	$-0.534467\pi$
$127$	−39286.0	−0.216137	−0.108068	+	0.994143i	$0.534467\pi$
$128$	0	0
$129$	45708.0	0.241834
$130$	0	0
$131$	−211460.	−1.07659	−0.538295	−	0.842757i	$-0.680931\pi$
$131$	−211460.	−1.07659	−0.538295	+	0.842757i	$0.680931\pi$
$132$	0	0
$133$	−660264.	−3.23660
$134$	0	0
$135$	−67500.0	−0.318764
$136$	0	0
$137$	26302.0	0.119726	0.0598628	−	0.998207i	$-0.480934\pi$
$137$	26302.0	0.119726	0.0598628	+	0.998207i	$0.480934\pi$
$138$	0	0
$139$	−1344.00	−0.00590014	−0.00295007	−	0.999996i	$-0.500939\pi$
$139$	−1344.00	−0.00590014	−0.00295007	+	0.999996i	$0.500939\pi$
$140$	0	0
$141$	−18120.0	−0.0767557
$142$	0	0
$143$	134186.	0.548741
$144$	0	0
$145$	125950.	0.497483
$146$	0	0
$147$	−256374.	−0.978545
$148$	0	0
$149$	−49086.0	−0.181131	−0.0905653	−	0.995891i	$-0.528867\pi$
$149$	−49086.0	−0.181131	−0.0905653	+	0.995891i	$0.528867\pi$
$150$	0	0
$151$	357998.	1.27773	0.638864	−	0.769320i	$-0.279405\pi$
$151$	357998.	1.27773	0.638864	+	0.769320i	$0.279405\pi$
$152$	0	0
$153$	317538.	1.09665
$154$	0	0
$155$	−90850.0	−0.303736
$156$	0	0
$157$	45450.0	0.147158	0.0735791	−	0.997289i	$-0.476558\pi$
$157$	45450.0	0.147158	0.0735791	+	0.997289i	$0.476558\pi$
$158$	0	0
$159$	−3756.00	−0.0117824
$160$	0	0
$161$	171288.	0.520790
$162$	0	0
$163$	−5892.00	−0.0173698	−0.00868488	−	0.999962i	$-0.502765\pi$
$163$	−5892.00	−0.0173698	−0.00868488	+	0.999962i	$0.502765\pi$
$164$	0	0
$165$	−119100.	−0.340566
$166$	0	0
$167$	−212772.	−0.590369	−0.295184	−	0.955440i	$-0.595381\pi$
$167$	−212772.	−0.590369	−0.295184	+	0.955440i	$0.595381\pi$
$168$	0	0
$169$	28561.0	0.0769231
$170$	0	0
$171$	560142.	1.46490
$172$	0	0
$173$	503178.	1.27822	0.639111	−	0.769114i	$-0.279302\pi$
$173$	503178.	1.27822	0.639111	+	0.769114i	$0.279302\pi$
$174$	0	0
$175$	152500.	0.376422
$176$	0	0
$177$	−180396.	−0.432806
$178$	0	0
$179$	−581724.	−1.35701	−0.678507	−	0.734594i	$-0.737373\pi$
$179$	−581724.	−1.35701	−0.678507	+	0.734594i	$0.737373\pi$
$180$	0	0
$181$	202202.	0.458764	0.229382	−	0.973337i	$-0.426330\pi$
$181$	202202.	0.458764	0.229382	+	0.973337i	$0.426330\pi$
$182$	0	0
$183$	34836.0	0.0768954
$184$	0	0
$185$	176450.	0.379046
$186$	0	0
$187$	1.21800e6	2.54708
$188$	0	0
$189$	658800.	1.34153
$190$	0	0
$191$	340608.	0.675572	0.337786	−	0.941223i	$-0.390322\pi$
$191$	340608.	0.675572	0.337786	+	0.941223i	$0.390322\pi$
$192$	0	0
$193$	275614.	0.532608	0.266304	−	0.963889i	$-0.414197\pi$
$193$	275614.	0.532608	0.266304	+	0.963889i	$0.414197\pi$
$194$	0	0
$195$	−25350.0	−0.0477410
$196$	0	0
$197$	538218.	0.988081	0.494041	−	0.869439i	$-0.335519\pi$
$197$	538218.	0.988081	0.494041	+	0.869439i	$0.335519\pi$
$198$	0	0
$199$	853840.	1.52842	0.764212	−	0.644965i	$-0.223128\pi$
$199$	853840.	1.52842	0.764212	+	0.644965i	$0.223128\pi$
$200$	0	0
$201$	−74616.0	−0.130269
$202$	0	0
$203$	−1.22927e6	−2.09367
$204$	0	0
$205$	7350.00	0.0122153
$206$	0	0
$207$	−145314.	−0.235712
$208$	0	0
$209$	2.14856e6	3.40238
$210$	0	0
$211$	1.00112e6	1.54804	0.774019	−	0.633162i	$-0.218243\pi$
$211$	1.00112e6	1.54804	0.774019	+	0.633162i	$0.218243\pi$
$212$	0	0
$213$	28404.0	0.0428974
$214$	0	0
$215$	190450.	0.280986
$216$	0	0
$217$	886696.	1.27828
$218$	0	0
$219$	88164.0	0.124217
$220$	0	0
$221$	259246.	0.357052
$222$	0	0
$223$	−21364.0	−0.0287687	−0.0143844	−	0.999897i	$-0.504579\pi$
$223$	−21364.0	−0.0287687	−0.0143844	+	0.999897i	$0.504579\pi$
$224$	0	0
$225$	−129375.	−0.170370
$226$	0	0
$227$	880748.	1.13445	0.567227	−	0.823561i	$-0.308016\pi$
$227$	880748.	1.13445	0.567227	+	0.823561i	$0.308016\pi$
$228$	0	0
$229$	−13030.0	−0.0164193	−0.00820967	−	0.999966i	$-0.502613\pi$
$229$	−13030.0	−0.0164193	−0.00820967	+	0.999966i	$0.502613\pi$
$230$	0	0
$231$	1.16242e6	1.43328
$232$	0	0
$233$	−1.20700e6	−1.45652	−0.728260	−	0.685300i	$-0.759671\pi$
$233$	−1.20700e6	−1.45652	−0.728260	+	0.685300i	$0.759671\pi$
$234$	0	0
$235$	−75500.0	−0.0891820
$236$	0	0
$237$	−238824.	−0.276189
$238$	0	0
$239$	187038.	0.211804	0.105902	−	0.994377i	$-0.466227\pi$
$239$	187038.	0.211804	0.105902	+	0.994377i	$0.466227\pi$
$240$	0	0
$241$	271690.	0.301322	0.150661	−	0.988585i	$-0.451860\pi$
$241$	271690.	0.301322	0.150661	+	0.988585i	$0.451860\pi$
$242$	0	0
$243$	−860706.	−0.935059
$244$	0	0
$245$	−1.06822e6	−1.13697
$246$	0	0
$247$	457314.	0.476949
$248$	0	0
$249$	−250656.	−0.256200
$250$	0	0
$251$	−102648.	−0.102841	−0.0514205	−	0.998677i	$-0.516375\pi$
$251$	−102648.	−0.102841	−0.0514205	+	0.998677i	$0.516375\pi$
$252$	0	0
$253$	−557388.	−0.547465
$254$	0	0
$255$	−230100.	−0.221598
$256$	0	0
$257$	−221182.	−0.208890	−0.104445	−	0.994531i	$-0.533307\pi$
$257$	−221182.	−0.208890	−0.104445	+	0.994531i	$0.533307\pi$
$258$	0	0
$259$	−1.72215e6	−1.59523
$260$	0	0
$261$	1.04287e6	0.947605
$262$	0	0
$263$	−1.40317e6	−1.25090	−0.625449	−	0.780265i	$-0.715084\pi$
$263$	−1.40317e6	−1.25090	−0.625449	+	0.780265i	$0.715084\pi$
$264$	0	0
$265$	−15650.0	−0.0136899
$266$	0	0
$267$	−47820.0	−0.0410517
$268$	0	0
$269$	−582954.	−0.491195	−0.245597	−	0.969372i	$-0.578984\pi$
$269$	−582954.	−0.491195	−0.245597	+	0.969372i	$0.578984\pi$
$270$	0	0
$271$	1.04690e6	0.865930	0.432965	−	0.901411i	$-0.357467\pi$
$271$	1.04690e6	0.865930	0.432965	+	0.901411i	$0.357467\pi$
$272$	0	0
$273$	247416.	0.200919
$274$	0	0
$275$	−496250.	−0.395702
$276$	0	0
$277$	1.10461e6	0.864987	0.432493	−	0.901637i	$-0.357634\pi$
$277$	1.10461e6	0.864987	0.432493	+	0.901637i	$0.357634\pi$
$278$	0	0
$279$	−752238.	−0.578555
$280$	0	0
$281$	908826.	0.686618	0.343309	−	0.939223i	$-0.388452\pi$
$281$	908826.	0.686618	0.343309	+	0.939223i	$0.388452\pi$
$282$	0	0
$283$	449254.	0.333446	0.166723	−	0.986004i	$-0.446681\pi$
$283$	449254.	0.333446	0.166723	+	0.986004i	$0.446681\pi$
$284$	0	0
$285$	−405900.	−0.296010
$286$	0	0
$287$	−71736.0	−0.0514082
$288$	0	0
$289$	933299.	0.657319
$290$	0	0
$291$	468300.	0.324184
$292$	0	0
$293$	−1.96083e6	−1.33435	−0.667175	−	0.744901i	$-0.732497\pi$
$293$	−1.96083e6	−1.33435	−0.667175	+	0.744901i	$0.732497\pi$
$294$	0	0
$295$	−751650.	−0.502876
$296$	0	0
$297$	−2.14380e6	−1.41024
$298$	0	0
$299$	−118638.	−0.0767442
$300$	0	0
$301$	−1.85879e6	−1.18254
$302$	0	0
$303$	138060.	0.0863896
$304$	0	0
$305$	145150.	0.0893444
$306$	0	0
$307$	1.79385e6	1.08627	0.543137	−	0.839644i	$-0.317236\pi$
$307$	1.79385e6	1.08627	0.543137	+	0.839644i	$0.317236\pi$
$308$	0	0
$309$	730236.	0.435078
$310$	0	0
$311$	−2.41233e6	−1.41428	−0.707141	−	0.707072i	$-0.750015\pi$
$311$	−2.41233e6	−1.41428	−0.707141	+	0.707072i	$0.750015\pi$
$312$	0	0
$313$	−2.15436e6	−1.24296	−0.621480	−	0.783430i	$-0.713468\pi$
$313$	−2.15436e6	−1.24296	−0.621480	+	0.783430i	$0.713468\pi$
$314$	0	0
$315$	1.26270e6	0.717008
$316$	0	0
$317$	2.59616e6	1.45105	0.725526	−	0.688195i	$-0.241597\pi$
$317$	2.59616e6	1.45105	0.725526	+	0.688195i	$0.241597\pi$
$318$	0	0
$319$	4.00017e6	2.20091
$320$	0	0
$321$	−420852.	−0.227964
$322$	0	0
$323$	4.15100e6	2.21384
$324$	0	0
$325$	−105625.	−0.0554700
$326$	0	0
$327$	1.17527e6	0.607810
$328$	0	0
$329$	736880.	0.375325
$330$	0	0
$331$	917226.	0.460157	0.230079	−	0.973172i	$-0.426102\pi$
$331$	917226.	0.460157	0.230079	+	0.973172i	$0.426102\pi$
$332$	0	0
$333$	1.46101e6	0.722007
$334$	0	0
$335$	−310900.	−0.151359
$336$	0	0
$337$	2.23894e6	1.07391	0.536954	−	0.843611i	$-0.319575\pi$
$337$	2.23894e6	1.07391	0.536954	+	0.843611i	$0.319575\pi$
$338$	0	0
$339$	601428.	0.284239
$340$	0	0
$341$	−2.88540e6	−1.34375
$342$	0	0
$343$	6.32497e6	2.90284
$344$	0	0
$345$	105300.	0.0476300
$346$	0	0
$347$	−3.41808e6	−1.52391	−0.761954	−	0.647631i	$-0.775760\pi$
$347$	−3.41808e6	−1.52391	−0.761954	+	0.647631i	$0.775760\pi$
$348$	0	0
$349$	2.35691e6	1.03581	0.517905	−	0.855438i	$-0.326712\pi$
$349$	2.35691e6	1.03581	0.517905	+	0.855438i	$0.326712\pi$
$350$	0	0
$351$	−456300.	−0.197689
$352$	0	0
$353$	−3.76395e6	−1.60771	−0.803854	−	0.594827i	$-0.797221\pi$
$353$	−3.76395e6	−1.60771	−0.803854	+	0.594827i	$0.797221\pi$
$354$	0	0
$355$	118350.	0.0498422
$356$	0	0
$357$	2.24578e6	0.932601
$358$	0	0
$359$	−3.28216e6	−1.34407	−0.672037	−	0.740517i	$-0.734581\pi$
$359$	−3.28216e6	−1.34407	−0.672037	+	0.740517i	$0.734581\pi$
$360$	0	0
$361$	4.84634e6	1.95725
$362$	0	0
$363$	−2.81631e6	−1.12180
$364$	0	0
$365$	367350.	0.144327
$366$	0	0
$367$	2.42605e6	0.940233	0.470116	−	0.882604i	$-0.344212\pi$
$367$	2.42605e6	0.940233	0.470116	+	0.882604i	$0.344212\pi$
$368$	0	0
$369$	60858.0	0.0232676
$370$	0	0
$371$	152744.	0.0576142
$372$	0	0
$373$	2.80635e6	1.04441	0.522204	−	0.852820i	$-0.325110\pi$
$373$	2.80635e6	1.04441	0.522204	+	0.852820i	$0.325110\pi$
$374$	0	0
$375$	93750.0	0.0344265
$376$	0	0
$377$	851422.	0.308526
$378$	0	0
$379$	−3.15392e6	−1.12785	−0.563927	−	0.825825i	$-0.690710\pi$
$379$	−3.15392e6	−1.12785	−0.563927	+	0.825825i	$0.690710\pi$
$380$	0	0
$381$	235716.	0.0831911
$382$	0	0
$383$	475044.	0.165477	0.0827384	−	0.996571i	$-0.473633\pi$
$383$	475044.	0.165477	0.0827384	+	0.996571i	$0.473633\pi$
$384$	0	0
$385$	4.84340e6	1.66532
$386$	0	0
$387$	1.57693e6	0.535222
$388$	0	0
$389$	150566.	0.0504490	0.0252245	−	0.999682i	$-0.491970\pi$
$389$	150566.	0.0504490	0.0252245	+	0.999682i	$0.491970\pi$
$390$	0	0
$391$	−1.07687e6	−0.356222
$392$	0	0
$393$	1.26876e6	0.414379
$394$	0	0
$395$	−995100.	−0.320903
$396$	0	0
$397$	241686.	0.0769618	0.0384809	−	0.999259i	$-0.487748\pi$
$397$	241686.	0.0769618	0.0384809	+	0.999259i	$0.487748\pi$
$398$	0	0
$399$	3.96158e6	1.24577
$400$	0	0
$401$	−3.19679e6	−0.992780	−0.496390	−	0.868100i	$-0.665341\pi$
$401$	−3.19679e6	−0.992780	−0.496390	+	0.868100i	$0.665341\pi$
$402$	0	0
$403$	−614146.	−0.188369
$404$	0	0
$405$	−852525.	−0.258267
$406$	0	0
$407$	5.60405e6	1.67693
$408$	0	0
$409$	423282.	0.125119	0.0625593	−	0.998041i	$-0.480074\pi$
$409$	423282.	0.125119	0.0625593	+	0.998041i	$0.480074\pi$
$410$	0	0
$411$	−157812.	−0.0460824
$412$	0	0
$413$	7.33610e6	2.11636
$414$	0	0
$415$	−1.04440e6	−0.297678
$416$	0	0
$417$	8064.00	0.00227096
$418$	0	0
$419$	1.13159e6	0.314887	0.157444	−	0.987528i	$-0.449675\pi$
$419$	1.13159e6	0.314887	0.157444	+	0.987528i	$0.449675\pi$
$420$	0	0
$421$	3.47699e6	0.956088	0.478044	−	0.878336i	$-0.341346\pi$
$421$	3.47699e6	0.956088	0.478044	+	0.878336i	$0.341346\pi$
$422$	0	0
$423$	−625140.	−0.169874
$424$	0	0
$425$	−958750.	−0.257474
$426$	0	0
$427$	−1.41666e6	−0.376008
$428$	0	0
$429$	−805116.	−0.211210
$430$	0	0
$431$	−3.41044e6	−0.884335	−0.442168	−	0.896932i	$-0.645790\pi$
$431$	−3.41044e6	−0.884335	−0.442168	+	0.896932i	$0.645790\pi$
$432$	0	0
$433$	−3.40722e6	−0.873335	−0.436667	−	0.899623i	$-0.643841\pi$
$433$	−3.40722e6	−0.873335	−0.436667	+	0.899623i	$0.643841\pi$
$434$	0	0
$435$	−755700.	−0.191481
$436$	0	0
$437$	−1.89961e6	−0.475840
$438$	0	0
$439$	7.09114e6	1.75612	0.878061	−	0.478549i	$-0.158837\pi$
$439$	7.09114e6	1.75612	0.878061	+	0.478549i	$0.158837\pi$
$440$	0	0
$441$	−8.84490e6	−2.16569
$442$	0	0
$443$	8.23508e6	1.99369	0.996847	−	0.0793445i	$-0.0252827\pi$
$443$	8.23508e6	1.99369	0.996847	+	0.0793445i	$0.0252827\pi$
$444$	0	0
$445$	−199250.	−0.0476978
$446$	0	0
$447$	294516.	0.0697172
$448$	0	0
$449$	−1.29601e6	−0.303383	−0.151691	−	0.988428i	$-0.548472\pi$
$449$	−1.29601e6	−0.303383	−0.151691	+	0.988428i	$0.548472\pi$
$450$	0	0
$451$	233436.	0.0540414
$452$	0	0
$453$	−2.14799e6	−0.491798
$454$	0	0
$455$	1.03090e6	0.233447
$456$	0	0
$457$	1.68196e6	0.376725	0.188363	−	0.982100i	$-0.439682\pi$
$457$	1.68196e6	0.376725	0.188363	+	0.982100i	$0.439682\pi$
$458$	0	0
$459$	−4.14180e6	−0.917608
$460$	0	0
$461$	−3.20663e6	−0.702743	−0.351372	−	0.936236i	$-0.614285\pi$
$461$	−3.20663e6	−0.702743	−0.351372	+	0.936236i	$0.614285\pi$
$462$	0	0
$463$	5.26370e6	1.14114	0.570570	−	0.821249i	$-0.306722\pi$
$463$	5.26370e6	1.14114	0.570570	+	0.821249i	$0.306722\pi$
$464$	0	0
$465$	545100.	0.116908
$466$	0	0
$467$	8.26813e6	1.75435	0.877173	−	0.480175i	$-0.159427\pi$
$467$	8.26813e6	1.75435	0.877173	+	0.480175i	$0.159427\pi$
$468$	0	0
$469$	3.03438e6	0.636999
$470$	0	0
$471$	−272700.	−0.0566413
$472$	0	0
$473$	6.04869e6	1.24311
$474$	0	0
$475$	−1.69125e6	−0.343933
$476$	0	0
$477$	−129582.	−0.0260765
$478$	0	0
$479$	−3.65468e6	−0.727797	−0.363899	−	0.931439i	$-0.618555\pi$
$479$	−3.65468e6	−0.727797	−0.363899	+	0.931439i	$0.618555\pi$
$480$	0	0
$481$	1.19280e6	0.235075
$482$	0	0
$483$	−1.02773e6	−0.200452
$484$	0	0
$485$	1.95125e6	0.376668
$486$	0	0
$487$	−7.13084e6	−1.36244	−0.681221	−	0.732077i	$-0.738551\pi$
$487$	−7.13084e6	−1.36244	−0.681221	+	0.732077i	$0.738551\pi$
$488$	0	0
$489$	35352.0	0.00668562
$490$	0	0
$491$	−5.72551e6	−1.07179	−0.535896	−	0.844284i	$-0.680026\pi$
$491$	−5.72551e6	−1.07179	−0.535896	+	0.844284i	$0.680026\pi$
$492$	0	0
$493$	7.72829e6	1.43208
$494$	0	0
$495$	−4.10895e6	−0.753734
$496$	0	0
$497$	−1.15510e6	−0.209762
$498$	0	0
$499$	7.17251e6	1.28950	0.644748	−	0.764395i	$-0.276962\pi$
$499$	7.17251e6	1.28950	0.644748	+	0.764395i	$0.276962\pi$
$500$	0	0
$501$	1.27663e6	0.227233
$502$	0	0
$503$	2.90611e6	0.512143	0.256072	−	0.966658i	$-0.417572\pi$
$503$	2.90611e6	0.512143	0.256072	+	0.966658i	$0.417572\pi$
$504$	0	0
$505$	575250.	0.100376
$506$	0	0
$507$	−171366.	−0.0296077
$508$	0	0
$509$	−8.37125e6	−1.43217	−0.716087	−	0.698011i	$-0.754069\pi$
$509$	−8.37125e6	−1.43217	−0.716087	+	0.698011i	$0.754069\pi$
$510$	0	0
$511$	−3.58534e6	−0.607404
$512$	0	0
$513$	−7.30620e6	−1.22574
$514$	0	0
$515$	3.04265e6	0.505515
$516$	0	0
$517$	−2.39788e6	−0.394549
$518$	0	0
$519$	−3.01907e6	−0.491988
$520$	0	0
$521$	5.37332e6	0.867258	0.433629	−	0.901092i	$-0.357233\pi$
$521$	5.37332e6	0.867258	0.433629	+	0.901092i	$0.357233\pi$
$522$	0	0
$523$	−5.26875e6	−0.842274	−0.421137	−	0.906997i	$-0.638369\pi$
$523$	−5.26875e6	−0.842274	−0.421137	+	0.906997i	$0.638369\pi$
$524$	0	0
$525$	−915000.	−0.144885
$526$	0	0
$527$	−5.57456e6	−0.874347
$528$	0	0
$529$	−5.94354e6	−0.923434
$530$	0	0
$531$	−6.22366e6	−0.957877
$532$	0	0
$533$	49686.0	0.00757558
$534$	0	0
$535$	−1.75355e6	−0.264871
$536$	0	0
$537$	3.49034e6	0.522315
$538$	0	0
$539$	−3.39268e7	−5.03004
$540$	0	0
$541$	−6.07956e6	−0.893056	−0.446528	−	0.894770i	$-0.647340\pi$
$541$	−6.07956e6	−0.893056	−0.446528	+	0.894770i	$0.647340\pi$
$542$	0	0
$543$	−1.21321e6	−0.176578
$544$	0	0
$545$	4.89695e6	0.706211
$546$	0	0
$547$	7.88715e6	1.12707	0.563536	−	0.826091i	$-0.309441\pi$
$547$	7.88715e6	1.12707	0.563536	+	0.826091i	$0.309441\pi$
$548$	0	0
$549$	1.20184e6	0.170183
$550$	0	0
$551$	1.36328e7	1.91296
$552$	0	0
$553$	9.71218e6	1.35053
$554$	0	0
$555$	−1.05870e6	−0.145895
$556$	0	0
$557$	−5.88545e6	−0.803788	−0.401894	−	0.915686i	$-0.631648\pi$
$557$	−5.88545e6	−0.803788	−0.401894	+	0.915686i	$0.631648\pi$
$558$	0	0
$559$	1.28744e6	0.174260
$560$	0	0
$561$	−7.30798e6	−0.980370
$562$	0	0
$563$	3.91526e6	0.520583	0.260291	−	0.965530i	$-0.416181\pi$
$563$	3.91526e6	0.520583	0.260291	+	0.965530i	$0.416181\pi$
$564$	0	0
$565$	2.50595e6	0.330256
$566$	0	0
$567$	8.32064e6	1.08692
$568$	0	0
$569$	9.78180e6	1.26660	0.633298	−	0.773908i	$-0.281701\pi$
$569$	9.78180e6	1.26660	0.633298	+	0.773908i	$0.281701\pi$
$570$	0	0
$571$	1.08198e7	1.38877	0.694386	−	0.719603i	$-0.255676\pi$
$571$	1.08198e7	1.38877	0.694386	+	0.719603i	$0.255676\pi$
$572$	0	0
$573$	−2.04365e6	−0.260028
$574$	0	0
$575$	438750.	0.0553411
$576$	0	0
$577$	1.48792e7	1.86055	0.930274	−	0.366865i	$-0.119569\pi$
$577$	1.48792e7	1.86055	0.930274	+	0.366865i	$0.119569\pi$
$578$	0	0
$579$	−1.65368e6	−0.205001
$580$	0	0
$581$	1.01933e7	1.25278
$582$	0	0
$583$	−497044.	−0.0605652
$584$	0	0
$585$	−874575.	−0.105659
$586$	0	0
$587$	−1.22649e7	−1.46916	−0.734578	−	0.678525i	$-0.762620\pi$
$587$	−1.22649e7	−1.46916	−0.734578	+	0.678525i	$0.762620\pi$
$588$	0	0
$589$	−9.83360e6	−1.16795
$590$	0	0
$591$	−3.22931e6	−0.380313
$592$	0	0
$593$	−1.54878e7	−1.80864	−0.904320	−	0.426856i	$-0.859621\pi$
$593$	−1.54878e7	−1.80864	−0.904320	+	0.426856i	$0.859621\pi$
$594$	0	0
$595$	9.35740e6	1.08358
$596$	0	0
$597$	−5.12304e6	−0.588291
$598$	0	0
$599$	−9.75710e6	−1.11110	−0.555551	−	0.831483i	$-0.687493\pi$
$599$	−9.75710e6	−1.11110	−0.555551	+	0.831483i	$0.687493\pi$
$600$	0	0
$601$	−7.57967e6	−0.855981	−0.427990	−	0.903783i	$-0.640778\pi$
$601$	−7.57967e6	−0.855981	−0.427990	+	0.903783i	$0.640778\pi$
$602$	0	0
$603$	−2.57425e6	−0.288309
$604$	0	0
$605$	−1.17346e7	−1.30341
$606$	0	0
$607$	−1.36231e7	−1.50073	−0.750367	−	0.661022i	$-0.770123\pi$
$607$	−1.36231e7	−1.50073	−0.750367	+	0.661022i	$0.770123\pi$
$608$	0	0
$609$	7.37563e6	0.805853
$610$	0	0
$611$	−510380.	−0.0553083
$612$	0	0
$613$	−1.20366e7	−1.29376	−0.646880	−	0.762592i	$-0.723927\pi$
$613$	−1.20366e7	−1.29376	−0.646880	+	0.762592i	$0.723927\pi$
$614$	0	0
$615$	−44100.0	−0.00470166
$616$	0	0
$617$	8.55509e6	0.904715	0.452358	−	0.891837i	$-0.350583\pi$
$617$	8.55509e6	0.904715	0.452358	+	0.891837i	$0.350583\pi$
$618$	0	0
$619$	1.33018e7	1.39535	0.697675	−	0.716414i	$-0.254218\pi$
$619$	1.33018e7	1.39535	0.697675	+	0.716414i	$0.254218\pi$
$620$	0	0
$621$	1.89540e6	0.197230
$622$	0	0
$623$	1.94468e6	0.200737
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−1.28914e7	−1.30958
$628$	0	0
$629$	1.08270e7	1.09114
$630$	0	0
$631$	−9.16681e6	−0.916526	−0.458263	−	0.888817i	$-0.651528\pi$
$631$	−9.16681e6	−0.916526	−0.458263	+	0.888817i	$0.651528\pi$
$632$	0	0
$633$	−6.00674e6	−0.595840
$634$	0	0
$635$	982150.	0.0966593
$636$	0	0
$637$	−7.22120e6	−0.705116
$638$	0	0
$639$	979938.	0.0949394
$640$	0	0
$641$	9.96437e6	0.957866	0.478933	−	0.877851i	$-0.341024\pi$
$641$	9.96437e6	0.957866	0.478933	+	0.877851i	$0.341024\pi$
$642$	0	0
$643$	−6.64194e6	−0.633530	−0.316765	−	0.948504i	$-0.602597\pi$
$643$	−6.64194e6	−0.633530	−0.316765	+	0.948504i	$0.602597\pi$
$644$	0	0
$645$	−1.14270e6	−0.108152
$646$	0	0
$647$	−844766.	−0.0793370	−0.0396685	−	0.999213i	$-0.512630\pi$
$647$	−844766.	−0.0793370	−0.0396685	+	0.999213i	$0.512630\pi$
$648$	0	0
$649$	−2.38724e7	−2.22477
$650$	0	0
$651$	−5.32018e6	−0.492010
$652$	0	0
$653$	5.79681e6	0.531993	0.265997	−	0.963974i	$-0.414299\pi$
$653$	5.79681e6	0.531993	0.265997	+	0.963974i	$0.414299\pi$
$654$	0	0
$655$	5.28650e6	0.481465
$656$	0	0
$657$	3.04166e6	0.274914
$658$	0	0
$659$	1.12406e7	1.00827	0.504136	−	0.863624i	$-0.331811\pi$
$659$	1.12406e7	1.00827	0.504136	+	0.863624i	$0.331811\pi$
$660$	0	0
$661$	−1.54928e7	−1.37920	−0.689599	−	0.724191i	$-0.742213\pi$
$661$	−1.54928e7	−1.37920	−0.689599	+	0.724191i	$0.742213\pi$
$662$	0	0
$663$	−1.55548e6	−0.137429
$664$	0	0
$665$	1.65066e7	1.44745
$666$	0	0
$667$	−3.53668e6	−0.307809
$668$	0	0
$669$	128184.	0.0110731
$670$	0	0
$671$	4.60996e6	0.395268
$672$	0	0
$673$	−723294.	−0.0615570	−0.0307785	−	0.999526i	$-0.509799\pi$
$673$	−723294.	−0.0615570	−0.0307785	+	0.999526i	$0.509799\pi$
$674$	0	0
$675$	1.68750e6	0.142556
$676$	0	0
$677$	−7.57359e6	−0.635082	−0.317541	−	0.948244i	$-0.602857\pi$
$677$	−7.57359e6	−0.635082	−0.317541	+	0.948244i	$0.602857\pi$
$678$	0	0
$679$	−1.90442e7	−1.58522
$680$	0	0
$681$	−5.28449e6	−0.436652
$682$	0	0
$683$	1.65552e7	1.35794	0.678972	−	0.734164i	$-0.262426\pi$
$683$	1.65552e7	1.35794	0.678972	+	0.734164i	$0.262426\pi$
$684$	0	0
$685$	−657550.	−0.0535430
$686$	0	0
$687$	78180.0	0.00631981
$688$	0	0
$689$	−105794.	−0.00849010
$690$	0	0
$691$	2.04593e7	1.63003	0.815016	−	0.579438i	$-0.196728\pi$
$691$	2.04593e7	1.63003	0.815016	+	0.579438i	$0.196728\pi$
$692$	0	0
$693$	4.01034e7	3.17211
$694$	0	0
$695$	33600.0	0.00263862
$696$	0	0
$697$	450996.	0.0351634
$698$	0	0
$699$	7.24199e6	0.560615
$700$	0	0
$701$	1.52050e7	1.16867	0.584334	−	0.811514i	$-0.301356\pi$
$701$	1.52050e7	1.16867	0.584334	+	0.811514i	$0.301356\pi$
$702$	0	0
$703$	1.90989e7	1.45754
$704$	0	0
$705$	453000.	0.0343262
$706$	0	0
$707$	−5.61444e6	−0.422433
$708$	0	0
$709$	−1.80833e7	−1.35102	−0.675509	−	0.737351i	$-0.736076\pi$
$709$	−1.80833e7	−1.35102	−0.675509	+	0.737351i	$0.736076\pi$
$710$	0	0
$711$	−8.23943e6	−0.611256
$712$	0	0
$713$	2.55107e6	0.187931
$714$	0	0
$715$	−3.35465e6	−0.245404
$716$	0	0
$717$	−1.12223e6	−0.0815236
$718$	0	0
$719$	2.08096e7	1.50121	0.750604	−	0.660752i	$-0.229763\pi$
$719$	2.08096e7	1.50121	0.750604	+	0.660752i	$0.229763\pi$
$720$	0	0
$721$	−2.96963e7	−2.12747
$722$	0	0
$723$	−1.63014e6	−0.115979
$724$	0	0
$725$	−3.14875e6	−0.222481
$726$	0	0
$727$	2.59006e7	1.81750	0.908749	−	0.417344i	$-0.137039\pi$
$727$	2.59006e7	1.81750	0.908749	+	0.417344i	$0.137039\pi$
$728$	0	0
$729$	−3.12231e6	−0.217599
$730$	0	0
$731$	1.16860e7	0.808859
$732$	0	0
$733$	−1.96307e7	−1.34951	−0.674754	−	0.738043i	$-0.735750\pi$
$733$	−1.96307e7	−1.34951	−0.674754	+	0.738043i	$0.735750\pi$
$734$	0	0
$735$	6.40935e6	0.437618
$736$	0	0
$737$	−9.87418e6	−0.669626
$738$	0	0
$739$	1.67436e7	1.12781	0.563906	−	0.825839i	$-0.309298\pi$
$739$	1.67436e7	1.12781	0.563906	+	0.825839i	$0.309298\pi$
$740$	0	0
$741$	−2.74388e6	−0.183578
$742$	0	0
$743$	−5.57725e6	−0.370637	−0.185318	−	0.982679i	$-0.559332\pi$
$743$	−5.57725e6	−0.370637	−0.185318	+	0.982679i	$0.559332\pi$
$744$	0	0
$745$	1.22715e6	0.0810041
$746$	0	0
$747$	−8.64763e6	−0.567016
$748$	0	0
$749$	1.71146e7	1.11471
$750$	0	0
$751$	−1.24035e7	−0.802499	−0.401250	−	0.915969i	$-0.631424\pi$
$751$	−1.24035e7	−0.802499	−0.401250	+	0.915969i	$0.631424\pi$
$752$	0	0
$753$	615888.	0.0395835
$754$	0	0
$755$	−8.94995e6	−0.571417
$756$	0	0
$757$	4.37170e6	0.277275	0.138637	−	0.990343i	$-0.455728\pi$
$757$	4.37170e6	0.277275	0.138637	+	0.990343i	$0.455728\pi$
$758$	0	0
$759$	3.34433e6	0.210719
$760$	0	0
$761$	−2.10490e7	−1.31756	−0.658780	−	0.752335i	$-0.728927\pi$
$761$	−2.10490e7	−1.31756	−0.658780	+	0.752335i	$0.728927\pi$
$762$	0	0
$763$	−4.77942e7	−2.97210
$764$	0	0
$765$	−7.93845e6	−0.490436
$766$	0	0
$767$	−5.08115e6	−0.311870
$768$	0	0
$769$	2.26551e7	1.38150	0.690748	−	0.723096i	$-0.257282\pi$
$769$	2.26551e7	1.38150	0.690748	+	0.723096i	$0.257282\pi$
$770$	0	0
$771$	1.32709e6	0.0804017
$772$	0	0
$773$	1.15053e7	0.692545	0.346272	−	0.938134i	$-0.387447\pi$
$773$	1.15053e7	0.692545	0.346272	+	0.938134i	$0.387447\pi$
$774$	0	0
$775$	2.27125e6	0.135835
$776$	0	0
$777$	1.03329e7	0.614003
$778$	0	0
$779$	795564.	0.0469712
$780$	0	0
$781$	3.75880e6	0.220506
$782$	0	0
$783$	−1.36026e7	−0.792898
$784$	0	0
$785$	−1.13625e6	−0.0658112
$786$	0	0
$787$	−967112.	−0.0556596	−0.0278298	−	0.999613i	$-0.508860\pi$
$787$	−967112.	−0.0556596	−0.0278298	+	0.999613i	$0.508860\pi$
$788$	0	0
$789$	8.41904e6	0.481471
$790$	0	0
$791$	−2.44581e7	−1.38989
$792$	0	0
$793$	981214.	0.0554091
$794$	0	0
$795$	93900.0	0.00526924
$796$	0	0
$797$	−2.85072e7	−1.58968	−0.794838	−	0.606821i	$-0.792444\pi$
$797$	−2.85072e7	−1.58968	−0.794838	+	0.606821i	$0.792444\pi$
$798$	0	0
$799$	−4.63268e6	−0.256723
$800$	0	0
$801$	−1.64979e6	−0.0908547
$802$	0	0
$803$	1.16670e7	0.638516
$804$	0	0
$805$	−4.28220e6	−0.232904
$806$	0	0
$807$	3.49772e6	0.189061
$808$	0	0
$809$	1.08912e7	0.585065	0.292533	−	0.956256i	$-0.405502\pi$
$809$	1.08912e7	0.585065	0.292533	+	0.956256i	$0.405502\pi$
$810$	0	0
$811$	−1.28535e7	−0.686228	−0.343114	−	0.939294i	$-0.611482\pi$
$811$	−1.28535e7	−0.686228	−0.343114	+	0.939294i	$0.611482\pi$
$812$	0	0
$813$	−6.28141e6	−0.333297
$814$	0	0
$815$	147300.	0.00776799
$816$	0	0
$817$	2.06143e7	1.08047
$818$	0	0
$819$	8.53585e6	0.444669
$820$	0	0
$821$	−9.60605e6	−0.497378	−0.248689	−	0.968583i	$-0.580000\pi$
$821$	−9.60605e6	−0.497378	−0.248689	+	0.968583i	$0.580000\pi$
$822$	0	0
$823$	−1.42909e7	−0.735460	−0.367730	−	0.929933i	$-0.619865\pi$
$823$	−1.42909e7	−0.735460	−0.367730	+	0.929933i	$0.619865\pi$
$824$	0	0
$825$	2.97750e6	0.152306
$826$	0	0
$827$	−2.40317e7	−1.22186	−0.610930	−	0.791685i	$-0.709204\pi$
$827$	−2.40317e7	−1.22186	−0.610930	+	0.791685i	$0.709204\pi$
$828$	0	0
$829$	1.10830e7	0.560107	0.280053	−	0.959984i	$-0.409648\pi$
$829$	1.10830e7	0.560107	0.280053	+	0.959984i	$0.409648\pi$
$830$	0	0
$831$	−6.62766e6	−0.332934
$832$	0	0
$833$	−6.55463e7	−3.27292
$834$	0	0
$835$	5.31930e6	0.264021
$836$	0	0
$837$	9.81180e6	0.484100
$838$	0	0
$839$	6.89303e6	0.338069	0.169034	−	0.985610i	$-0.445935\pi$
$839$	6.89303e6	0.338069	0.169034	+	0.985610i	$0.445935\pi$
$840$	0	0
$841$	4.87030e6	0.237446
$842$	0	0
$843$	−5.45296e6	−0.264279
$844$	0	0
$845$	−714025.	−0.0344010
$846$	0	0
$847$	1.14530e8	5.48543
$848$	0	0
$849$	−2.69552e6	−0.128344
$850$	0	0
$851$	−4.95472e6	−0.234528
$852$	0	0
$853$	−683466.	−0.0321621	−0.0160810	−	0.999871i	$-0.505119\pi$
$853$	−683466.	−0.0321621	−0.0160810	+	0.999871i	$0.505119\pi$
$854$	0	0
$855$	−1.40035e7	−0.655123
$856$	0	0
$857$	−7.89742e6	−0.367310	−0.183655	−	0.982991i	$-0.558793\pi$
$857$	−7.89742e6	−0.367310	−0.183655	+	0.982991i	$0.558793\pi$
$858$	0	0
$859$	−3.52556e7	−1.63021	−0.815107	−	0.579310i	$-0.803322\pi$
$859$	−3.52556e7	−1.63021	−0.815107	+	0.579310i	$0.803322\pi$
$860$	0	0
$861$	430416.	0.0197870
$862$	0	0
$863$	−1.76565e7	−0.807007	−0.403503	−	0.914978i	$-0.632208\pi$
$863$	−1.76565e7	−0.807007	−0.403503	+	0.914978i	$0.632208\pi$
$864$	0	0
$865$	−1.25794e7	−0.571638
$866$	0	0
$867$	−5.59979e6	−0.253002
$868$	0	0
$869$	−3.16044e7	−1.41970
$870$	0	0
$871$	−2.10168e6	−0.0938690
$872$	0	0
$873$	1.61564e7	0.717476
$874$	0	0
$875$	−3.81250e6	−0.168341
$876$	0	0
$877$	−6.40016e6	−0.280991	−0.140495	−	0.990081i	$-0.544869\pi$
$877$	−6.40016e6	−0.280991	−0.140495	+	0.990081i	$0.544869\pi$
$878$	0	0
$879$	1.17650e7	0.513592
$880$	0	0
$881$	−1.14571e7	−0.497318	−0.248659	−	0.968591i	$-0.579990\pi$
$881$	−1.14571e7	−0.497318	−0.248659	+	0.968591i	$0.579990\pi$
$882$	0	0
$883$	−2.42296e7	−1.04579	−0.522896	−	0.852397i	$-0.675148\pi$
$883$	−2.42296e7	−1.04579	−0.522896	+	0.852397i	$0.675148\pi$
$884$	0	0
$885$	4.50990e6	0.193557
$886$	0	0
$887$	−8.66087e6	−0.369617	−0.184809	−	0.982775i	$-0.559167\pi$
$887$	−8.66087e6	−0.369617	−0.184809	+	0.982775i	$0.559167\pi$
$888$	0	0
$889$	−9.58578e6	−0.406793
$890$	0	0
$891$	−2.70762e7	−1.14260
$892$	0	0
$893$	−8.17212e6	−0.342930
$894$	0	0
$895$	1.45431e7	0.606875
$896$	0	0
$897$	711828.	0.0295389
$898$	0	0
$899$	−1.83081e7	−0.755516
$900$	0	0
$901$	−960284.	−0.0394083
$902$	0	0
$903$	1.11528e7	0.455159
$904$	0	0
$905$	−5.05505e6	−0.205165
$906$	0	0
$907$	7.84287e6	0.316561	0.158280	−	0.987394i	$-0.449405\pi$
$907$	7.84287e6	0.316561	0.158280	+	0.987394i	$0.449405\pi$
$908$	0	0
$909$	4.76307e6	0.191195
$910$	0	0
$911$	942576.	0.0376288	0.0188144	−	0.999823i	$-0.494011\pi$
$911$	942576.	0.0376288	0.0188144	+	0.999823i	$0.494011\pi$
$912$	0	0
$913$	−3.31701e7	−1.31695
$914$	0	0
$915$	−870900.	−0.0343887
$916$	0	0
$917$	−5.15962e7	−2.02626
$918$	0	0
$919$	2.00734e7	0.784030	0.392015	−	0.919959i	$-0.371778\pi$
$919$	2.00734e7	0.784030	0.392015	+	0.919959i	$0.371778\pi$
$920$	0	0
$921$	−1.07631e7	−0.418107
$922$	0	0
$923$	800046.	0.0309108
$924$	0	0
$925$	−4.41125e6	−0.169515
$926$	0	0
$927$	2.51931e7	0.962904
$928$	0	0
$929$	1.10181e7	0.418858	0.209429	−	0.977824i	$-0.432840\pi$
$929$	1.10181e7	0.418858	0.209429	+	0.977824i	$0.432840\pi$
$930$	0	0
$931$	−1.15625e8	−4.37196
$932$	0	0
$933$	1.44740e7	0.544358
$934$	0	0
$935$	−3.04499e7	−1.13909
$936$	0	0
$937$	3.59532e7	1.33779	0.668896	−	0.743356i	$-0.266767\pi$
$937$	3.59532e7	1.33779	0.668896	+	0.743356i	$0.266767\pi$
$938$	0	0
$939$	1.29261e7	0.478415
$940$	0	0
$941$	1.28845e7	0.474345	0.237172	−	0.971468i	$-0.423779\pi$
$941$	1.28845e7	0.474345	0.237172	+	0.971468i	$0.423779\pi$
$942$	0	0
$943$	−206388.	−0.00755797
$944$	0	0
$945$	−1.64700e7	−0.599949
$946$	0	0
$947$	1.18911e7	0.430871	0.215436	−	0.976518i	$-0.430883\pi$
$947$	1.18911e7	0.430871	0.215436	+	0.976518i	$0.430883\pi$
$948$	0	0
$949$	2.48329e6	0.0895079
$950$	0	0
$951$	−1.55769e7	−0.558510
$952$	0	0
$953$	4.40094e7	1.56969	0.784844	−	0.619694i	$-0.212743\pi$
$953$	4.40094e7	1.56969	0.784844	+	0.619694i	$0.212743\pi$
$954$	0	0
$955$	−8.51520e6	−0.302125
$956$	0	0
$957$	−2.40010e7	−0.847130
$958$	0	0
$959$	6.41769e6	0.225337
$960$	0	0
$961$	−1.54232e7	−0.538723
$962$	0	0
$963$	−1.45194e7	−0.504525
$964$	0	0
$965$	−6.89035e6	−0.238190
$966$	0	0
$967$	−2.11144e7	−0.726128	−0.363064	−	0.931764i	$-0.618269\pi$
$967$	−2.11144e7	−0.726128	−0.363064	+	0.931764i	$0.618269\pi$
$968$	0	0
$969$	−2.49060e7	−0.852109
$970$	0	0
$971$	−2.44293e7	−0.831502	−0.415751	−	0.909478i	$-0.636481\pi$
$971$	−2.44293e7	−0.831502	−0.415751	+	0.909478i	$0.636481\pi$
$972$	0	0
$973$	−327936.	−0.0111047
$974$	0	0
$975$	633750.	0.0213504
$976$	0	0
$977$	5.15549e7	1.72796	0.863980	−	0.503527i	$-0.167964\pi$
$977$	5.15549e7	1.72796	0.863980	+	0.503527i	$0.167964\pi$
$978$	0	0
$979$	−6.32818e6	−0.211019
$980$	0	0
$981$	4.05467e7	1.34519
$982$	0	0
$983$	1.38938e7	0.458604	0.229302	−	0.973355i	$-0.426356\pi$
$983$	1.38938e7	0.458604	0.229302	+	0.973355i	$0.426356\pi$
$984$	0	0
$985$	−1.34554e7	−0.441883
$986$	0	0
$987$	−4.42128e6	−0.144463
$988$	0	0
$989$	−5.34784e6	−0.173855
$990$	0	0
$991$	−3.31496e7	−1.07225	−0.536123	−	0.844140i	$-0.680112\pi$
$991$	−3.31496e7	−1.07225	−0.536123	+	0.844140i	$0.680112\pi$
$992$	0	0
$993$	−5.50336e6	−0.177115
$994$	0	0
$995$	−2.13460e7	−0.683532
$996$	0	0
$997$	9.45871e6	0.301366	0.150683	−	0.988582i	$-0.451853\pi$
$997$	9.45871e6	0.301366	0.150683	+	0.988582i	$0.451853\pi$
$998$	0	0
$999$	−1.90566e7	−0.604132

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1040.6.a.a.1.1		1
4.3	odd	2		65.6.a.a.1.1	✓	1
12.11	even	2		585.6.a.a.1.1		1
20.3	even	4		325.6.b.a.274.1		2
20.7	even	4		325.6.b.a.274.2		2
20.19	odd	2		325.6.a.a.1.1		1
52.51	odd	2		845.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
65.6.a.a.1.1	✓	1	4.3	odd	2
325.6.a.a.1.1		1	20.19	odd	2
325.6.b.a.274.1		2	20.3	even	4
325.6.b.a.274.2		2	20.7	even	4
585.6.a.a.1.1		1	12.11	even	2
845.6.a.a.1.1		1	52.51	odd	2
1040.6.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 1040 = 2^{4} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	1040.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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