LMFDB - Embedded newform 800.5.e.e.399.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,5,Mod(399,800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(800, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("800.399");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	800.e (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$82.6959704671$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			399.10
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.9

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+10.2034i q^{3} +43.3025 q^{7} -23.1094 q^{9} +165.050 q^{11} -201.647 q^{13} -172.778i q^{17} -640.745 q^{19} +441.833i q^{21} +497.626 q^{23} +590.681i q^{27} +509.953i q^{29} +492.136i q^{31} +1684.07i q^{33} +678.519 q^{37} -2057.49i q^{39} -613.848 q^{41} +1392.38i q^{43} +965.658 q^{47} -525.894 q^{49} +1762.92 q^{51} +4611.29 q^{53} -6537.78i q^{57} -822.630 q^{59} +6917.98i q^{61} -1000.70 q^{63} +4929.83i q^{67} +5077.48i q^{69} +4236.85i q^{71} -8507.29i q^{73} +7147.07 q^{77} +10205.9i q^{79} -7898.82 q^{81} +818.969i q^{83} -5203.25 q^{87} +10451.0 q^{89} -8731.82 q^{91} -5021.46 q^{93} +11691.7i q^{97} -3814.21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) $ 32 * q - 864 * q^9 - 384 * q^11 - 1408 * q^19 - 4416 * q^41 + 4960 * q^49 + 35584 * q^51 + 28032 * q^59 + 20768 * q^81 - 13632 * q^89 - 49152 * q^91 + 5248 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/800\mathbb{Z}\right)^\times$.

$n$	$101$	$351$	$577$
$\chi(n)$	$-1$	$-1$	$-1$

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
$3$	10.2034i	1.13371i	0.823817	+	0.566856i	$0.191840\pi$
$3$	10.2034i	1.13371i	−0.823817	+	0.566856i	$0.808160\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	43.3025	0.883724	0.441862	−	0.897083i	$-0.354318\pi$
$7$	43.3025	0.883724	0.441862	+	0.897083i	$0.354318\pi$
$8$	0	0
$9$	−23.1094	−0.285301
$10$	0	0
$11$	165.050	1.36405	0.682024	−	0.731329i	$-0.261100\pi$
$11$	165.050	1.36405	0.682024	+	0.731329i	$0.261100\pi$
$12$	0	0
$13$	−201.647	−1.19318	−0.596589	−	0.802547i	$-0.703478\pi$
$13$	−201.647	−1.19318	−0.596589	+	0.802547i	$0.703478\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 172.778i	− 0.597848i	−0.954277	−	0.298924i	$-0.903372\pi$
$17$	− 172.778i	− 0.597848i	0.954277	−	0.298924i	$-0.0966278\pi$
$18$	0	0
$19$	−640.745	−1.77492	−0.887458	−	0.460888i	$-0.847531\pi$
$19$	−640.745	−1.77492	−0.887458	+	0.460888i	$0.847531\pi$
$20$	0	0
$21$	441.833i	1.00189i
$22$	0	0
$23$	497.626	0.940692	0.470346	−	0.882482i	$-0.344129\pi$
$23$	497.626	0.940692	0.470346	+	0.882482i	$0.344129\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	590.681i	0.810262i
$28$	0	0
$29$	509.953i	0.606365i	0.952933	+	0.303182i	$0.0980491\pi$
$29$	509.953i	0.606365i	−0.952933	+	0.303182i	$0.901951\pi$
$30$	0	0
$31$	492.136i	0.512108i	0.966662	+	0.256054i	$0.0824225\pi$
$31$	492.136i	0.512108i	−0.966662	+	0.256054i	$0.917578\pi$
$32$	0	0
$33$	1684.07i	1.54644i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	678.519	0.495631	0.247816	−	0.968807i	$-0.420287\pi$
$37$	678.519	0.495631	0.247816	+	0.968807i	$0.420287\pi$
$38$	0	0
$39$	− 2057.49i	− 1.35272i
$40$	0	0
$41$	−613.848	−0.365168	−0.182584	−	0.983190i	$-0.558446\pi$
$41$	−613.848	−0.365168	−0.182584	+	0.983190i	$0.558446\pi$
$42$	0	0
$43$	1392.38i	0.753043i	0.926408	+	0.376522i	$0.122880\pi$
$43$	1392.38i	0.753043i	−0.926408	+	0.376522i	$0.877120\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	965.658	0.437147	0.218574	−	0.975820i	$-0.429860\pi$
$47$	965.658	0.437147	0.218574	+	0.975820i	$0.429860\pi$
$48$	0	0
$49$	−525.894	−0.219031
$50$	0	0
$51$	1762.92	0.677787
$52$	0	0
$53$	4611.29	1.64161	0.820807	−	0.571206i	$-0.193524\pi$
$53$	4611.29	1.64161	0.820807	+	0.571206i	$0.193524\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 6537.78i	− 2.01224i
$58$	0	0
$59$	−822.630	−0.236320	−0.118160	−	0.992995i	$-0.537700\pi$
$59$	−822.630	−0.236320	−0.118160	+	0.992995i	$0.537700\pi$
$60$	0	0
$61$	6917.98i	1.85917i	0.368606	+	0.929586i	$0.379835\pi$
$61$	6917.98i	1.85917i	−0.368606	+	0.929586i	$0.620165\pi$
$62$	0	0
$63$	−1000.70	−0.252128
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4929.83i	1.09820i	0.835756	+	0.549101i	$0.185030\pi$
$67$	4929.83i	1.09820i	−0.835756	+	0.549101i	$0.814970\pi$
$68$	0	0
$69$	5077.48i	1.06647i
$70$	0	0
$71$	4236.85i	0.840479i	0.907413	+	0.420239i	$0.138054\pi$
$71$	4236.85i	0.840479i	−0.907413	+	0.420239i	$0.861946\pi$
$72$	0	0
$73$	− 8507.29i	− 1.59641i	−0.602384	−	0.798207i	$-0.705782\pi$
$73$	− 8507.29i	− 1.59641i	0.602384	−	0.798207i	$-0.294218\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	7147.07	1.20544
$78$	0	0
$79$	10205.9i	1.63530i	0.575716	+	0.817650i	$0.304724\pi$
$79$	10205.9i	1.63530i	−0.575716	+	0.817650i	$0.695276\pi$
$80$	0	0
$81$	−7898.82	−1.20390
$82$	0	0
$83$	818.969i	0.118881i	0.998232	+	0.0594403i	$0.0189316\pi$
$83$	818.969i	0.118881i	−0.998232	+	0.0594403i	$0.981068\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−5203.25	−0.687443
$88$	0	0
$89$	10451.0	1.31940	0.659700	−	0.751529i	$-0.270683\pi$
$89$	10451.0	1.31940	0.659700	+	0.751529i	$0.270683\pi$
$90$	0	0
$91$	−8731.82	−1.05444
$92$	0	0
$93$	−5021.46	−0.580583
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11691.7i	1.24261i	0.783568	+	0.621306i	$0.213397\pi$
$97$	11691.7i	1.24261i	−0.783568	+	0.621306i	$0.786603\pi$
$98$	0	0
$99$	−3814.21	−0.389165
$100$	0	0
$101$	− 13430.4i	− 1.31657i	−0.752767	−	0.658287i	$-0.771281\pi$
$101$	− 13430.4i	− 1.31657i	0.752767	−	0.658287i	$-0.228719\pi$
$102$	0	0
$103$	−5546.66	−0.522826	−0.261413	−	0.965227i	$-0.584188\pi$
$103$	−5546.66	−0.522826	−0.261413	+	0.965227i	$0.584188\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7485.33i	0.653797i	0.945059	+	0.326899i	$0.106004\pi$
$107$	7485.33i	0.653797i	−0.945059	+	0.326899i	$0.893996\pi$
$108$	0	0
$109$	13391.1i	1.12710i	0.826082	+	0.563549i	$0.190565\pi$
$109$	13391.1i	1.12710i	−0.826082	+	0.563549i	$0.809435\pi$
$110$	0	0
$111$	6923.20i	0.561903i
$112$	0	0
$113$	− 2822.55i	− 0.221047i	−0.993874	−	0.110523i	$-0.964747\pi$
$113$	− 2822.55i	− 0.221047i	0.993874	−	0.110523i	$-0.0352527\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	4659.95	0.340415
$118$	0	0
$119$	− 7481.73i	− 0.528333i
$120$	0	0
$121$	12600.5	0.860629
$122$	0	0
$123$	− 6263.34i	− 0.413995i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−15755.3	−0.976831	−0.488416	−	0.872611i	$-0.662425\pi$
$127$	−15755.3	−0.976831	−0.488416	+	0.872611i	$0.662425\pi$
$128$	0	0
$129$	−14207.0	−0.853733
$130$	0	0
$131$	−4033.25	−0.235024	−0.117512	−	0.993071i	$-0.537492\pi$
$131$	−4033.25	−0.235024	−0.117512	+	0.993071i	$0.537492\pi$
$132$	0	0
$133$	−27745.9	−1.56854
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 15035.7i	− 0.801090i	−0.916277	−	0.400545i	$-0.868821\pi$
$137$	− 15035.7i	− 0.801090i	0.916277	−	0.400545i	$-0.131179\pi$
$138$	0	0
$139$	−31639.4	−1.63756	−0.818782	−	0.574105i	$-0.805350\pi$
$139$	−31639.4	−1.63756	−0.818782	+	0.574105i	$0.805350\pi$
$140$	0	0
$141$	9853.00i	0.495599i
$142$	0	0
$143$	−33281.8	−1.62755
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 5365.90i	− 0.248318i
$148$	0	0
$149$	34271.9i	1.54371i	0.635798	+	0.771856i	$0.280671\pi$
$149$	34271.9i	1.54371i	−0.635798	+	0.771856i	$0.719329\pi$
$150$	0	0
$151$	478.378i	0.0209806i	0.999945	+	0.0104903i	$0.00333922\pi$
$151$	478.378i	0.0209806i	−0.999945	+	0.0104903i	$0.996661\pi$
$152$	0	0
$153$	3992.80i	0.170567i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13731.7	−0.557088	−0.278544	−	0.960423i	$-0.589852\pi$
$157$	−13731.7	−0.557088	−0.278544	+	0.960423i	$0.589852\pi$
$158$	0	0
$159$	47050.9i	1.86112i
$160$	0	0
$161$	21548.4	0.831312
$162$	0	0
$163$	− 23891.6i	− 0.899228i	−0.893223	−	0.449614i	$-0.851561\pi$
$163$	− 23891.6i	− 0.899228i	0.893223	−	0.449614i	$-0.148439\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	27650.3	0.991442	0.495721	−	0.868482i	$-0.334904\pi$
$167$	27650.3	0.991442	0.495721	+	0.868482i	$0.334904\pi$
$168$	0	0
$169$	12100.5	0.423673
$170$	0	0
$171$	14807.2	0.506386
$172$	0	0
$173$	−49203.7	−1.64401	−0.822007	−	0.569477i	$-0.807146\pi$
$173$	−49203.7	−1.64401	−0.822007	+	0.569477i	$0.807146\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 8393.62i	− 0.267919i
$178$	0	0
$179$	−11247.1	−0.351022	−0.175511	−	0.984477i	$-0.556158\pi$
$179$	−11247.1	−0.351022	−0.175511	+	0.984477i	$0.556158\pi$
$180$	0	0
$181$	9919.57i	0.302786i	0.988474	+	0.151393i	$0.0483759\pi$
$181$	9919.57i	0.302786i	−0.988474	+	0.151393i	$0.951624\pi$
$182$	0	0
$183$	−70586.9	−2.10776
$184$	0	0
$185$	0	0
$186$	0	0
$187$	− 28517.0i	− 0.815494i
$188$	0	0
$189$	25578.0i	0.716048i
$190$	0	0
$191$	− 36233.6i	− 0.993219i	−0.867974	−	0.496609i	$-0.834578\pi$
$191$	− 36233.6i	− 0.993219i	0.867974	−	0.496609i	$-0.165422\pi$
$192$	0	0
$193$	− 37612.9i	− 1.00977i	−0.863187	−	0.504885i	$-0.831535\pi$
$193$	− 37612.9i	− 1.00977i	0.863187	−	0.504885i	$-0.168465\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	27374.4	0.705361	0.352681	−	0.935744i	$-0.385270\pi$
$197$	27374.4	0.705361	0.352681	+	0.935744i	$0.385270\pi$
$198$	0	0
$199$	53766.9i	1.35772i	0.734270	+	0.678858i	$0.237525\pi$
$199$	53766.9i	1.35772i	−0.734270	+	0.678858i	$0.762475\pi$
$200$	0	0
$201$	−50301.0	−1.24504
$202$	0	0
$203$	22082.2i	0.535859i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−11499.8	−0.268381
$208$	0	0
$209$	−105755.	−2.42107
$210$	0	0
$211$	19829.6	0.445399	0.222700	−	0.974887i	$-0.428513\pi$
$211$	19829.6	0.445399	0.222700	+	0.974887i	$0.428513\pi$
$212$	0	0
$213$	−43230.3	−0.952860
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21310.7i	0.452562i
$218$	0	0
$219$	86803.3	1.80987
$220$	0	0
$221$	34840.2i	0.713339i
$222$	0	0
$223$	44470.0	0.894247	0.447123	−	0.894472i	$-0.352448\pi$
$223$	44470.0	0.894247	0.447123	+	0.894472i	$0.352448\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 41012.9i	− 0.795919i	−0.917403	−	0.397959i	$-0.869719\pi$
$227$	− 41012.9i	− 0.795919i	0.917403	−	0.397959i	$-0.130281\pi$
$228$	0	0
$229$	− 39184.8i	− 0.747217i	−0.927587	−	0.373608i	$-0.878120\pi$
$229$	− 39184.8i	− 0.747217i	0.927587	−	0.373608i	$-0.121880\pi$
$230$	0	0
$231$	72924.5i	1.36662i
$232$	0	0
$233$	64283.0i	1.18409i	0.805905	+	0.592045i	$0.201679\pi$
$233$	64283.0i	1.18409i	−0.805905	+	0.592045i	$0.798321\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−104135.	−1.85396
$238$	0	0
$239$	33481.2i	0.586145i	0.956090	+	0.293073i	$0.0946778\pi$
$239$	33481.2i	0.586145i	−0.956090	+	0.293073i	$0.905322\pi$
$240$	0	0
$241$	−28929.0	−0.498080	−0.249040	−	0.968493i	$-0.580115\pi$
$241$	−28929.0	−0.498080	−0.249040	+	0.968493i	$0.580115\pi$
$242$	0	0
$243$	− 32749.7i	− 0.554618i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	129204.	2.11779
$248$	0	0
$249$	−8356.27	−0.134776
$250$	0	0
$251$	3632.83	0.0576630	0.0288315	−	0.999584i	$-0.490821\pi$
$251$	3632.83	0.0576630	0.0288315	+	0.999584i	$0.490821\pi$
$252$	0	0
$253$	82133.1	1.28315
$254$	0	0
$255$	0	0
$256$	0	0
$257$	55530.5i	0.840747i	0.907351	+	0.420373i	$0.138101\pi$
$257$	55530.5i	0.840747i	−0.907351	+	0.420373i	$0.861899\pi$
$258$	0	0
$259$	29381.6	0.438001
$260$	0	0
$261$	− 11784.7i	− 0.172997i
$262$	0	0
$263$	4573.35	0.0661185	0.0330593	−	0.999453i	$-0.489475\pi$
$263$	4573.35	0.0661185	0.0330593	+	0.999453i	$0.489475\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	106635.i	1.49582i
$268$	0	0
$269$	74452.8i	1.02891i	0.857518	+	0.514454i	$0.172005\pi$
$269$	74452.8i	1.02891i	−0.857518	+	0.514454i	$0.827995\pi$
$270$	0	0
$271$	− 13623.9i	− 0.185508i	−0.995689	−	0.0927541i	$-0.970433\pi$
$271$	− 13623.9i	− 0.185508i	0.995689	−	0.0927541i	$-0.0295670\pi$
$272$	0	0
$273$	− 89094.3i	− 1.19543i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	18506.7	0.241195	0.120598	−	0.992701i	$-0.461519\pi$
$277$	18506.7	0.241195	0.120598	+	0.992701i	$0.461519\pi$
$278$	0	0
$279$	− 11373.0i	− 0.146105i
$280$	0	0
$281$	26912.2	0.340830	0.170415	−	0.985372i	$-0.445489\pi$
$281$	26912.2	0.340830	0.170415	+	0.985372i	$0.445489\pi$
$282$	0	0
$283$	− 91580.2i	− 1.14348i	−0.820435	−	0.571740i	$-0.806269\pi$
$283$	− 91580.2i	− 1.14348i	0.820435	−	0.571740i	$-0.193731\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26581.2	−0.322708
$288$	0	0
$289$	53668.7	0.642577
$290$	0	0
$291$	−119295.	−1.40876
$292$	0	0
$293$	−11242.9	−0.130961	−0.0654805	−	0.997854i	$-0.520858\pi$
$293$	−11242.9	−0.130961	−0.0654805	+	0.997854i	$0.520858\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	97491.8i	1.10524i
$298$	0	0
$299$	−100345.	−1.12241
$300$	0	0
$301$	60293.4i	0.665483i
$302$	0	0
$303$	137035.	1.49261
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 6583.47i	− 0.0698519i	−0.999390	−	0.0349260i	$-0.988880\pi$
$307$	− 6583.47i	− 0.0698519i	0.999390	−	0.0349260i	$-0.0111195\pi$
$308$	0	0
$309$	− 56594.8i	− 0.592734i
$310$	0	0
$311$	127524.i	1.31847i	0.751937	+	0.659235i	$0.229120\pi$
$311$	127524.i	1.31847i	−0.751937	+	0.659235i	$0.770880\pi$
$312$	0	0
$313$	93160.2i	0.950915i	0.879739	+	0.475457i	$0.157717\pi$
$313$	93160.2i	0.950915i	−0.879739	+	0.475457i	$0.842283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	69292.6	0.689554	0.344777	−	0.938685i	$-0.387954\pi$
$317$	69292.6	0.689554	0.344777	+	0.938685i	$0.387954\pi$
$318$	0	0
$319$	84167.6i	0.827111i
$320$	0	0
$321$	−76375.8	−0.741217
$322$	0	0
$323$	110707.i	1.06113i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−136634.	−1.27780
$328$	0	0
$329$	41815.4	0.386318
$330$	0	0
$331$	−106686.	−0.973763	−0.486882	−	0.873468i	$-0.661866\pi$
$331$	−106686.	−0.973763	−0.486882	+	0.873468i	$0.661866\pi$
$332$	0	0
$333$	−15680.2	−0.141404
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 34707.0i	− 0.305603i	−0.988257	−	0.152801i	$-0.951171\pi$
$337$	− 34707.0i	− 0.305603i	0.988257	−	0.152801i	$-0.0488294\pi$
$338$	0	0
$339$	28799.6	0.250603
$340$	0	0
$341$	81226.9i	0.698540i
$342$	0	0
$343$	−126742.	−1.07729
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 191096.i	− 1.58706i	−0.608532	−	0.793529i	$-0.708241\pi$
$347$	− 191096.i	− 1.58706i	0.608532	−	0.793529i	$-0.291759\pi$
$348$	0	0
$349$	− 19760.1i	− 0.162233i	−0.996705	−	0.0811165i	$-0.974151\pi$
$349$	− 19760.1i	− 0.162233i	0.996705	−	0.0811165i	$-0.0258486\pi$
$350$	0	0
$351$	− 119109.i	− 0.966787i
$352$	0	0
$353$	65214.7i	0.523355i	0.965155	+	0.261677i	$0.0842757\pi$
$353$	65214.7i	0.523355i	−0.965155	+	0.261677i	$0.915724\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	76339.1	0.598977
$358$	0	0
$359$	− 183096.i	− 1.42066i	−0.703871	−	0.710328i	$-0.748547\pi$
$359$	− 183096.i	− 1.42066i	0.703871	−	0.710328i	$-0.251453\pi$
$360$	0	0
$361$	280233.	2.15033
$362$	0	0
$363$	128568.i	0.975705i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	11615.6	0.0862401	0.0431200	−	0.999070i	$-0.486270\pi$
$367$	11615.6	0.0862401	0.0431200	+	0.999070i	$0.486270\pi$
$368$	0	0
$369$	14185.7	0.104183
$370$	0	0
$371$	199680.	1.45073
$372$	0	0
$373$	192501.	1.38361	0.691806	−	0.722083i	$-0.256815\pi$
$373$	192501.	1.38361	0.691806	+	0.722083i	$0.256815\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 102830.i	− 0.723501i
$378$	0	0
$379$	−108067.	−0.752342	−0.376171	−	0.926550i	$-0.622760\pi$
$379$	−108067.	−0.752342	−0.376171	+	0.926550i	$0.622760\pi$
$380$	0	0
$381$	− 160758.i	− 1.10744i
$382$	0	0
$383$	188520.	1.28517	0.642585	−	0.766215i	$-0.277862\pi$
$383$	188520.	1.28517	0.642585	+	0.766215i	$0.277862\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 32177.0i	− 0.214844i
$388$	0	0
$389$	− 27057.1i	− 0.178806i	−0.995996	−	0.0894030i	$-0.971504\pi$
$389$	− 27057.1i	− 0.178806i	0.995996	−	0.0894030i	$-0.0284959\pi$
$390$	0	0
$391$	− 85978.9i	− 0.562391i
$392$	0	0
$393$	− 41152.8i	− 0.266449i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	156094.	0.990388	0.495194	−	0.868782i	$-0.335097\pi$
$397$	156094.	0.990388	0.495194	+	0.868782i	$0.335097\pi$
$398$	0	0
$399$	− 283102.i	− 1.77827i
$400$	0	0
$401$	−77777.2	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$401$	−77777.2	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$402$	0	0
$403$	− 99237.7i	− 0.611036i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	111989.	0.676065
$408$	0	0
$409$	247426.	1.47911	0.739553	−	0.673098i	$-0.235037\pi$
$409$	247426.	1.47911	0.739553	+	0.673098i	$0.235037\pi$
$410$	0	0
$411$	153415.	0.908205
$412$	0	0
$413$	−35621.9	−0.208842
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 322829.i	− 1.85652i
$418$	0	0
$419$	62888.2	0.358213	0.179107	−	0.983830i	$-0.442679\pi$
$419$	62888.2	0.358213	0.179107	+	0.983830i	$0.442679\pi$
$420$	0	0
$421$	− 321321.i	− 1.81290i	−0.422311	−	0.906451i	$-0.638781\pi$
$421$	− 321321.i	− 1.81290i	0.422311	−	0.906451i	$-0.361219\pi$
$422$	0	0
$423$	−22315.8	−0.124719
$424$	0	0
$425$	0	0
$426$	0	0
$427$	299566.i	1.64300i
$428$	0	0
$429$	− 339588.i	− 1.84518i
$430$	0	0
$431$	66812.9i	0.359672i	0.983697	+	0.179836i	$0.0575566\pi$
$431$	66812.9i	0.359672i	−0.983697	+	0.179836i	$0.942443\pi$
$432$	0	0
$433$	− 171149.i	− 0.912846i	−0.889763	−	0.456423i	$-0.849130\pi$
$433$	− 171149.i	− 0.912846i	0.889763	−	0.456423i	$-0.150870\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−318851.	−1.66965
$438$	0	0
$439$	− 82871.1i	− 0.430005i	−0.976613	−	0.215003i	$-0.931024\pi$
$439$	− 82871.1i	− 0.430005i	0.976613	−	0.215003i	$-0.0689760\pi$
$440$	0	0
$441$	12153.1	0.0624899
$442$	0	0
$443$	71237.1i	0.362994i	0.983392	+	0.181497i	$0.0580942\pi$
$443$	71237.1i	0.362994i	−0.983392	+	0.181497i	$0.941906\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−349690.	−1.75012
$448$	0	0
$449$	−306433.	−1.52000	−0.759999	−	0.649925i	$-0.774800\pi$
$449$	−306433.	−1.52000	−0.759999	+	0.649925i	$0.774800\pi$
$450$	0	0
$451$	−101316.	−0.498107
$452$	0	0
$453$	−4881.08	−0.0237859
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 31625.5i	− 0.151428i	−0.997130	−	0.0757138i	$-0.975876\pi$
$457$	− 31625.5i	− 0.151428i	0.997130	−	0.0757138i	$-0.0241235\pi$
$458$	0	0
$459$	102057.	0.484414
$460$	0	0
$461$	− 66080.1i	− 0.310934i	−0.987841	−	0.155467i	$-0.950312\pi$
$461$	− 66080.1i	− 0.310934i	0.987841	−	0.155467i	$-0.0496883\pi$
$462$	0	0
$463$	−303583.	−1.41617	−0.708086	−	0.706126i	$-0.750441\pi$
$463$	−303583.	−1.41617	−0.708086	+	0.706126i	$0.750441\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	121986.i	0.559341i	0.960096	+	0.279670i	$0.0902252\pi$
$467$	121986.i	0.559341i	−0.960096	+	0.279670i	$0.909775\pi$
$468$	0	0
$469$	213474.i	0.970508i
$470$	0	0
$471$	− 140110.i	− 0.631577i
$472$	0	0
$473$	229812.i	1.02719i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−106564.	−0.468355
$478$	0	0
$479$	144501.i	0.629794i	0.949126	+	0.314897i	$0.101970\pi$
$479$	144501.i	0.629794i	−0.949126	+	0.314897i	$0.898030\pi$
$480$	0	0
$481$	−136821.	−0.591376
$482$	0	0
$483$	219867.i	0.942468i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−7828.30	−0.0330073	−0.0165036	−	0.999864i	$-0.505254\pi$
$487$	−7828.30	−0.0330073	−0.0165036	+	0.999864i	$0.505254\pi$
$488$	0	0
$489$	243776.	1.01947
$490$	0	0
$491$	254423.	1.05534	0.527671	−	0.849449i	$-0.323066\pi$
$491$	254423.	1.05534	0.527671	+	0.849449i	$0.323066\pi$
$492$	0	0
$493$	88108.7	0.362514
$494$	0	0
$495$	0	0
$496$	0	0
$497$	183466.i	0.742752i
$498$	0	0
$499$	−19859.3	−0.0797560	−0.0398780	−	0.999205i	$-0.512697\pi$
$499$	−19859.3	−0.0797560	−0.0398780	+	0.999205i	$0.512697\pi$
$500$	0	0
$501$	282127.i	1.12401i
$502$	0	0
$503$	289925.	1.14591	0.572954	−	0.819587i	$-0.305797\pi$
$503$	289925.	1.14591	0.572954	+	0.819587i	$0.305797\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	123467.i	0.480323i
$508$	0	0
$509$	− 101676.i	− 0.392447i	−0.980559	−	0.196224i	$-0.937132\pi$
$509$	− 101676.i	− 0.392447i	0.980559	−	0.196224i	$-0.0628679\pi$
$510$	0	0
$511$	− 368387.i	− 1.41079i
$512$	0	0
$513$	− 378476.i	− 1.43815i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	159382.	0.596290
$518$	0	0
$519$	− 502045.i	− 1.86384i
$520$	0	0
$521$	−302599.	−1.11479	−0.557393	−	0.830248i	$-0.688198\pi$
$521$	−302599.	−1.11479	−0.557393	+	0.830248i	$0.688198\pi$
$522$	0	0
$523$	61255.9i	0.223947i	0.993711	+	0.111973i	$0.0357171\pi$
$523$	61255.9i	0.223947i	−0.993711	+	0.111973i	$0.964283\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	85030.3	0.306163
$528$	0	0
$529$	−32209.5	−0.115099
$530$	0	0
$531$	19010.5	0.0674224
$532$	0	0
$533$	123781.	0.435711
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 114759.i	− 0.397958i
$538$	0	0
$539$	−86798.7	−0.298769
$540$	0	0
$541$	− 223113.i	− 0.762307i	−0.924512	−	0.381154i	$-0.875527\pi$
$541$	− 223113.i	− 0.762307i	0.924512	−	0.381154i	$-0.124473\pi$
$542$	0	0
$543$	−101213.	−0.343272
$544$	0	0
$545$	0	0
$546$	0	0
$547$	56408.6i	0.188526i	0.995547	+	0.0942629i	$0.0300494\pi$
$547$	56408.6i	0.188526i	−0.995547	+	0.0942629i	$0.969951\pi$
$548$	0	0
$549$	− 159870.i	− 0.530424i
$550$	0	0
$551$	− 326750.i	− 1.07625i
$552$	0	0
$553$	441941.i	1.44515i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	288826.	0.930950	0.465475	−	0.885061i	$-0.345883\pi$
$557$	288826.	0.930950	0.465475	+	0.885061i	$0.345883\pi$
$558$	0	0
$559$	− 280769.i	− 0.898514i
$560$	0	0
$561$	290971.	0.924535
$562$	0	0
$563$	− 611795.i	− 1.93014i	−0.261988	−	0.965071i	$-0.584378\pi$
$563$	− 611795.i	− 1.93014i	0.261988	−	0.965071i	$-0.415622\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−342039.	−1.06392
$568$	0	0
$569$	633137.	1.95557	0.977784	−	0.209613i	$-0.0672204\pi$
$569$	633137.	1.95557	0.977784	+	0.209613i	$0.0672204\pi$
$570$	0	0
$571$	−279945.	−0.858618	−0.429309	−	0.903158i	$-0.641243\pi$
$571$	−279945.	−0.858618	−0.429309	+	0.903158i	$0.641243\pi$
$572$	0	0
$573$	369706.	1.12602
$574$	0	0
$575$	0	0
$576$	0	0
$577$	371982.i	1.11730i	0.829403	+	0.558651i	$0.188681\pi$
$577$	371982.i	1.11730i	−0.829403	+	0.558651i	$0.811319\pi$
$578$	0	0
$579$	383780.	1.14479
$580$	0	0
$581$	35463.4i	0.105058i
$582$	0	0
$583$	761093.	2.23924
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44751.2i	0.129876i	0.997889	+	0.0649380i	$0.0206850\pi$
$587$	44751.2i	0.129876i	−0.997889	+	0.0649380i	$0.979315\pi$
$588$	0	0
$589$	− 315333.i	− 0.908949i
$590$	0	0
$591$	279312.i	0.799676i
$592$	0	0
$593$	− 140260.i	− 0.398863i	−0.979912	−	0.199431i	$-0.936090\pi$
$593$	− 140260.i	− 0.398863i	0.979912	−	0.199431i	$-0.0639095\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−548605.	−1.53926
$598$	0	0
$599$	326102.i	0.908866i	0.890781	+	0.454433i	$0.150158\pi$
$599$	326102.i	0.908866i	−0.890781	+	0.454433i	$0.849842\pi$
$600$	0	0
$601$	111831.	0.309608	0.154804	−	0.987945i	$-0.450525\pi$
$601$	111831.	0.309608	0.154804	+	0.987945i	$0.450525\pi$
$602$	0	0
$603$	− 113925.i	− 0.313319i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	442565.	1.20116	0.600579	−	0.799566i	$-0.294937\pi$
$607$	442565.	1.20116	0.600579	+	0.799566i	$0.294937\pi$
$608$	0	0
$609$	−225314.	−0.607510
$610$	0	0
$611$	−194722.	−0.521594
$612$	0	0
$613$	235020.	0.625436	0.312718	−	0.949846i	$-0.398760\pi$
$613$	235020.	0.625436	0.312718	+	0.949846i	$0.398760\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 212842.i	− 0.559098i	−0.960131	−	0.279549i	$-0.909815\pi$
$617$	− 212842.i	− 0.559098i	0.960131	−	0.279549i	$-0.0901850\pi$
$618$	0	0
$619$	−30660.8	−0.0800207	−0.0400103	−	0.999199i	$-0.512739\pi$
$619$	−30660.8	−0.0800207	−0.0400103	+	0.999199i	$0.512739\pi$
$620$	0	0
$621$	293938.i	0.762207i
$622$	0	0
$623$	452553.	1.16599
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 1.07906e6i	− 2.74480i
$628$	0	0
$629$	− 117233.i	− 0.296312i
$630$	0	0
$631$	398063.i	0.999755i	0.866096	+	0.499877i	$0.166622\pi$
$631$	398063.i	0.999755i	−0.866096	+	0.499877i	$0.833378\pi$
$632$	0	0
$633$	202330.i	0.504954i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	106045.	0.261343
$638$	0	0
$639$	− 97911.2i	− 0.239790i
$640$	0	0
$641$	75692.2	0.184219	0.0921096	−	0.995749i	$-0.470639\pi$
$641$	75692.2	0.184219	0.0921096	+	0.995749i	$0.470639\pi$
$642$	0	0
$643$	786008.i	1.90110i	0.310570	+	0.950551i	$0.399480\pi$
$643$	786008.i	1.90110i	−0.310570	+	0.950551i	$0.600520\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	448043.	1.07031	0.535157	−	0.844752i	$-0.320252\pi$
$647$	448043.	1.07031	0.535157	+	0.844752i	$0.320252\pi$
$648$	0	0
$649$	−135775.	−0.322352
$650$	0	0
$651$	−217442.	−0.513075
$652$	0	0
$653$	−396846.	−0.930671	−0.465335	−	0.885134i	$-0.654066\pi$
$653$	−396846.	−0.930671	−0.465335	+	0.885134i	$0.654066\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	196598.i	0.455459i
$658$	0	0
$659$	290177.	0.668177	0.334089	−	0.942542i	$-0.391572\pi$
$659$	290177.	0.668177	0.334089	+	0.942542i	$0.391572\pi$
$660$	0	0
$661$	− 43962.0i	− 0.100618i	−0.998734	−	0.0503088i	$-0.983979\pi$
$661$	− 43962.0i	− 0.100618i	0.998734	−	0.0503088i	$-0.0160206\pi$
$662$	0	0
$663$	−355489.	−0.808721
$664$	0	0
$665$	0	0
$666$	0	0
$667$	253766.i	0.570402i
$668$	0	0
$669$	453745.i	1.01382i
$670$	0	0
$671$	1.14181e6i	2.53600i
$672$	0	0
$673$	314835.i	0.695110i	0.937660	+	0.347555i	$0.112988\pi$
$673$	314835.i	0.695110i	−0.937660	+	0.347555i	$0.887012\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	317143.	0.691954	0.345977	−	0.938243i	$-0.387548\pi$
$677$	317143.	0.691954	0.345977	+	0.938243i	$0.387548\pi$
$678$	0	0
$679$	506281.i	1.09813i
$680$	0	0
$681$	418471.	0.902342
$682$	0	0
$683$	64946.3i	0.139224i	0.997574	+	0.0696118i	$0.0221761\pi$
$683$	64946.3i	0.139224i	−0.997574	+	0.0696118i	$0.977824\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	399818.	0.847128
$688$	0	0
$689$	−929854.	−1.95874
$690$	0	0
$691$	225176.	0.471591	0.235796	−	0.971803i	$-0.424230\pi$
$691$	225176.	0.471591	0.235796	+	0.971803i	$0.424230\pi$
$692$	0	0
$693$	−165165.	−0.343915
$694$	0	0
$695$	0	0
$696$	0	0
$697$	106060.i	0.218315i
$698$	0	0
$699$	−655906.	−1.34242
$700$	0	0
$701$	− 720970.i	− 1.46717i	−0.679597	−	0.733586i	$-0.737845\pi$
$701$	− 720970.i	− 1.46717i	0.679597	−	0.733586i	$-0.262155\pi$
$702$	0	0
$703$	−434758.	−0.879704
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 581568.i	− 1.16349i
$708$	0	0
$709$	− 616674.i	− 1.22677i	−0.789784	−	0.613385i	$-0.789808\pi$
$709$	− 616674.i	− 1.22677i	0.789784	−	0.613385i	$-0.210192\pi$
$710$	0	0
$711$	− 235853.i	− 0.466553i
$712$	0	0
$713$	244899.i	0.481736i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−341622.	−0.664520
$718$	0	0
$719$	369408.i	0.714577i	0.933994	+	0.357288i	$0.116299\pi$
$719$	369408.i	0.714577i	−0.933994	+	0.357288i	$0.883701\pi$
$720$	0	0
$721$	−240184.	−0.462034
$722$	0	0
$723$	− 295174.i	− 0.564679i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−402715.	−0.761954	−0.380977	−	0.924585i	$-0.624412\pi$
$727$	−402715.	−0.761954	−0.380977	+	0.924585i	$0.624412\pi$
$728$	0	0
$729$	−305646.	−0.575127
$730$	0	0
$731$	240572.	0.450206
$732$	0	0
$733$	−22951.5	−0.0427172	−0.0213586	−	0.999772i	$-0.506799\pi$
$733$	−22951.5	−0.0427172	−0.0213586	+	0.999772i	$0.506799\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	813668.i	1.49800i
$738$	0	0
$739$	161609.	0.295922	0.147961	−	0.988993i	$-0.452729\pi$
$739$	161609.	0.295922	0.147961	+	0.988993i	$0.452729\pi$
$740$	0	0
$741$	1.31832e6i	2.40096i
$742$	0	0
$743$	194165.	0.351718	0.175859	−	0.984415i	$-0.443730\pi$
$743$	194165.	0.351718	0.175859	+	0.984415i	$0.443730\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 18925.9i	− 0.0339168i
$748$	0	0
$749$	324133.i	0.577777i
$750$	0	0
$751$	− 595741.i	− 1.05628i	−0.849158	−	0.528138i	$-0.822890\pi$
$751$	− 595741.i	− 1.05628i	0.849158	−	0.528138i	$-0.177110\pi$
$752$	0	0
$753$	37067.2i	0.0653732i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	566219.	0.988081	0.494040	−	0.869439i	$-0.335519\pi$
$757$	566219.	0.988081	0.494040	+	0.869439i	$0.335519\pi$
$758$	0	0
$759$	838037.i	1.45472i
$760$	0	0
$761$	−268262.	−0.463223	−0.231612	−	0.972808i	$-0.574400\pi$
$761$	−268262.	−0.463223	−0.231612	+	0.972808i	$0.574400\pi$
$762$	0	0
$763$	579866.i	0.996045i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	165881.	0.281972
$768$	0	0
$769$	−495755.	−0.838329	−0.419165	−	0.907910i	$-0.637677\pi$
$769$	−495755.	−0.838329	−0.419165	+	0.907910i	$0.637677\pi$
$770$	0	0
$771$	−566600.	−0.953164
$772$	0	0
$773$	−437740.	−0.732584	−0.366292	−	0.930500i	$-0.619373\pi$
$773$	−437740.	−0.732584	−0.366292	+	0.930500i	$0.619373\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	299792.i	0.496567i
$778$	0	0
$779$	393320.	0.648143
$780$	0	0
$781$	699292.i	1.14645i
$782$	0	0
$783$	−301219.	−0.491314
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 16629.0i	− 0.0268484i	−0.999910	−	0.0134242i	$-0.995727\pi$
$787$	− 16629.0i	− 0.0268484i	0.999910	−	0.0134242i	$-0.00427318\pi$
$788$	0	0
$789$	46663.8i	0.0749593i
$790$	0	0
$791$	− 122223.i	− 0.195345i
$792$	0	0
$793$	− 1.39499e6i	− 2.21832i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−998541.	−1.57199	−0.785994	−	0.618234i	$-0.787848\pi$
$797$	−998541.	−1.57199	−0.785994	+	0.618234i	$0.787848\pi$
$798$	0	0
$799$	− 166845.i	− 0.261348i
$800$	0	0
$801$	−241516.	−0.376427
$802$	0	0
$803$	− 1.40413e6i	− 2.17759i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−759672.	−1.16648
$808$	0	0
$809$	523592.	0.800011	0.400006	−	0.916513i	$-0.369008\pi$
$809$	523592.	0.800011	0.400006	+	0.916513i	$0.369008\pi$
$810$	0	0
$811$	741785.	1.12781	0.563905	−	0.825839i	$-0.309298\pi$
$811$	741785.	1.12781	0.563905	+	0.825839i	$0.309298\pi$
$812$	0	0
$813$	139010.	0.210313
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 892158.i	− 1.33659i
$818$	0	0
$819$	201787.	0.300833
$820$	0	0
$821$	− 971065.i	− 1.44066i	−0.693630	−	0.720331i	$-0.743990\pi$
$821$	− 971065.i	− 1.44066i	0.693630	−	0.720331i	$-0.256010\pi$
$822$	0	0
$823$	−932619.	−1.37691	−0.688454	−	0.725280i	$-0.741710\pi$
$823$	−932619.	−1.37691	−0.688454	+	0.725280i	$0.741710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.11332e6i	1.62782i	0.580989	+	0.813911i	$0.302666\pi$
$827$	1.11332e6i	1.62782i	−0.580989	+	0.813911i	$0.697334\pi$
$828$	0	0
$829$	625811.i	0.910613i	0.890335	+	0.455307i	$0.150470\pi$
$829$	625811.i	0.910613i	−0.890335	+	0.455307i	$0.849530\pi$
$830$	0	0
$831$	188831.i	0.273446i
$832$	0	0
$833$	90862.9i	0.130947i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−290695.	−0.414942
$838$	0	0
$839$	3902.74i	0.00554428i	0.999996	+	0.00277214i	$0.000882401\pi$
$839$	3902.74i	0.00554428i	−0.999996	+	0.00277214i	$0.999118\pi$
$840$	0	0
$841$	447229.	0.632322
$842$	0	0
$843$	274596.i	0.386402i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	545632.	0.760559
$848$	0	0
$849$	934429.	1.29638
$850$	0	0
$851$	337649.	0.466236
$852$	0	0
$853$	−858678.	−1.18014	−0.590068	−	0.807353i	$-0.700899\pi$
$853$	−858678.	−1.18014	−0.590068	+	0.807353i	$0.700899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 143233.i	− 0.195021i	−0.995235	−	0.0975103i	$-0.968912\pi$
$857$	− 143233.i	− 0.195021i	0.995235	−	0.0975103i	$-0.0310879\pi$
$858$	0	0
$859$	−892238.	−1.20919	−0.604595	−	0.796533i	$-0.706665\pi$
$859$	−892238.	−1.20919	−0.604595	+	0.796533i	$0.706665\pi$
$860$	0	0
$861$	− 271218.i	− 0.365858i
$862$	0	0
$863$	−838426.	−1.12575	−0.562877	−	0.826541i	$-0.690305\pi$
$863$	−838426.	−1.12575	−0.562877	+	0.826541i	$0.690305\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	547603.i	0.728497i
$868$	0	0
$869$	1.68448e6i	2.23063i
$870$	0	0
$871$	− 994085.i	− 1.31035i
$872$	0	0
$873$	− 270189.i	− 0.354519i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	1.16386e6	1.51322	0.756609	−	0.653868i	$-0.226855\pi$
$877$	1.16386e6	1.51322	0.756609	+	0.653868i	$0.226855\pi$
$878$	0	0
$879$	− 114715.i	− 0.148472i
$880$	0	0
$881$	−1.22677e6	−1.58056	−0.790282	−	0.612744i	$-0.790066\pi$
$881$	−1.22677e6	−1.58056	−0.790282	+	0.612744i	$0.790066\pi$
$882$	0	0
$883$	− 443400.i	− 0.568689i	−0.958722	−	0.284344i	$-0.908224\pi$
$883$	− 443400.i	− 0.568689i	0.958722	−	0.284344i	$-0.0917759\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	33686.5	0.0428162	0.0214081	−	0.999771i	$-0.493185\pi$
$887$	33686.5	0.0428162	0.0214081	+	0.999771i	$0.493185\pi$
$888$	0	0
$889$	−682244.	−0.863249
$890$	0	0
$891$	−1.30370e6	−1.64218
$892$	0	0
$893$	−618740.	−0.775900
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 1.02386e6i	− 1.27249i
$898$	0	0
$899$	−250966.	−0.310524
$900$	0	0
$901$	− 796731.i	− 0.981436i
$902$	0	0
$903$	−615198.	−0.754465
$904$	0	0
$905$	0	0
$906$	0	0
$907$	593935.i	0.721979i	0.932570	+	0.360990i	$0.117561\pi$
$907$	593935.i	0.721979i	−0.932570	+	0.360990i	$0.882439\pi$
$908$	0	0
$909$	310368.i	0.375620i
$910$	0	0
$911$	− 105485.i	− 0.127102i	−0.997979	−	0.0635510i	$-0.979757\pi$
$911$	− 105485.i	− 0.127102i	0.997979	−	0.0635510i	$-0.0202426\pi$
$912$	0	0
$913$	135171.i	0.162159i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−174650.	−0.207696
$918$	0	0
$919$	1.49149e6i	1.76600i	0.469373	+	0.883000i	$0.344480\pi$
$919$	1.49149e6i	1.76600i	−0.469373	+	0.883000i	$0.655520\pi$
$920$	0	0
$921$	67173.8	0.0791919
$922$	0	0
$923$	− 854349.i	− 1.00284i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	128180.	0.149163
$928$	0	0
$929$	1.27994e6	1.48306	0.741528	−	0.670922i	$-0.234102\pi$
$929$	1.27994e6	1.48306	0.741528	+	0.670922i	$0.234102\pi$
$930$	0	0
$931$	336964.	0.388762
$932$	0	0
$933$	−1.30118e6	−1.49476
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 225182.i	− 0.256480i	−0.991743	−	0.128240i	$-0.959067\pi$
$937$	− 225182.i	− 0.256480i	0.991743	−	0.128240i	$-0.0409328\pi$
$938$	0	0
$939$	−950551.	−1.07806
$940$	0	0
$941$	1.51647e6i	1.71260i	0.516482	+	0.856298i	$0.327241\pi$
$941$	1.51647e6i	1.71260i	−0.516482	+	0.856298i	$0.672759\pi$
$942$	0	0
$943$	−305467.	−0.343511
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 475706.i	− 0.530443i	−0.964187	−	0.265222i	$-0.914555\pi$
$947$	− 475706.i	− 0.530443i	0.964187	−	0.265222i	$-0.0854451\pi$
$948$	0	0
$949$	1.71547e6i	1.90481i
$950$	0	0
$951$	707021.i	0.781756i
$952$	0	0
$953$	644868.i	0.710044i	0.934858	+	0.355022i	$0.115527\pi$
$953$	644868.i	0.710044i	−0.934858	+	0.355022i	$0.884473\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−858796.	−0.937705
$958$	0	0
$959$	− 651082.i	− 0.707943i
$960$	0	0
$961$	681323.	0.737745
$962$	0	0
$963$	− 172981.i	− 0.186529i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	250168.	0.267533	0.133767	−	0.991013i	$-0.457293\pi$
$967$	250168.	0.267533	0.133767	+	0.991013i	$0.457293\pi$
$968$	0	0
$969$	−1.12959e6	−1.20302
$970$	0	0
$971$	373022.	0.395636	0.197818	−	0.980239i	$-0.436614\pi$
$971$	373022.	0.395636	0.197818	+	0.980239i	$0.436614\pi$
$972$	0	0
$973$	−1.37006e6	−1.44715
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 57277.7i	− 0.0600063i	−0.999550	−	0.0300031i	$-0.990448\pi$
$977$	− 57277.7i	− 0.0600063i	0.999550	−	0.0300031i	$-0.00955173\pi$
$978$	0	0
$979$	1.72493e6	1.79973
$980$	0	0
$981$	− 309460.i	− 0.321563i
$982$	0	0
$983$	351082.	0.363330	0.181665	−	0.983360i	$-0.441851\pi$
$983$	351082.	0.363330	0.181665	+	0.983360i	$0.441851\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	426659.i	0.437973i
$988$	0	0
$989$	692883.i	0.708381i
$990$	0	0
$991$	196492.i	0.200077i	0.994984	+	0.100038i	$0.0318965\pi$
$991$	196492.i	0.200077i	−0.994984	+	0.100038i	$0.968103\pi$
$992$	0	0
$993$	− 1.08857e6i	− 1.10397i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	736136.	0.740572	0.370286	−	0.928918i	$-0.379260\pi$
$997$	736136.	0.740572	0.370286	+	0.928918i	$0.379260\pi$
$998$	0	0
$999$	400788.i	0.401591i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.10		32
4.3	odd	2		200.5.e.e.99.31		32
5.2	odd	4		160.5.g.a.111.13		16
5.3	odd	4		800.5.g.h.751.3		16
5.4	even	2	inner	800.5.e.e.399.23		32
8.3	odd	2	inner	800.5.e.e.399.24		32
8.5	even	2		200.5.e.e.99.1		32
15.2	even	4		1440.5.g.a.271.14		16
20.3	even	4		200.5.g.h.51.7		16
20.7	even	4		40.5.g.a.11.10	yes	16
20.19	odd	2		200.5.e.e.99.2		32
40.3	even	4		800.5.g.h.751.4		16
40.13	odd	4		200.5.g.h.51.8		16
40.19	odd	2	inner	800.5.e.e.399.9		32
40.27	even	4		160.5.g.a.111.14		16
40.29	even	2		200.5.e.e.99.32		32
40.37	odd	4		40.5.g.a.11.9	✓	16
60.47	odd	4		360.5.g.a.91.7		16
120.77	even	4		360.5.g.a.91.8		16
120.107	odd	4		1440.5.g.a.271.3		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.9	✓	16	40.37	odd	4
40.5.g.a.11.10	yes	16	20.7	even	4
160.5.g.a.111.13		16	5.2	odd	4
160.5.g.a.111.14		16	40.27	even	4
200.5.e.e.99.1		32	8.5	even	2
200.5.e.e.99.2		32	20.19	odd	2
200.5.e.e.99.31		32	4.3	odd	2
200.5.e.e.99.32		32	40.29	even	2
200.5.g.h.51.7		16	20.3	even	4
200.5.g.h.51.8		16	40.13	odd	4
360.5.g.a.91.7		16	60.47	odd	4
360.5.g.a.91.8		16	120.77	even	4
800.5.e.e.399.9		32	40.19	odd	2	inner
800.5.e.e.399.10		32	1.1	even	1	trivial
800.5.e.e.399.23		32	5.4	even	2	inner
800.5.e.e.399.24		32	8.3	odd	2	inner
800.5.g.h.751.3		16	5.3	odd	4
800.5.g.h.751.4		16	40.3	even	4
1440.5.g.a.271.3		16	120.107	odd	4
1440.5.g.a.271.14		16	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:	\( N \)	\(=\)	\( 800 = 2^{5} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	800.e (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+10.2034i q^{3} +43.3025 q^{7} -23.1094 q^{9} +165.050 q^{11} -201.647 q^{13} -172.778i q^{17} -640.745 q^{19} +441.833i q^{21} +497.626 q^{23} +590.681i q^{27} +509.953i q^{29} +492.136i q^{31} +1684.07i q^{33} +678.519 q^{37} -2057.49i q^{39} -613.848 q^{41} +1392.38i q^{43} +965.658 q^{47} -525.894 q^{49} +1762.92 q^{51} +4611.29 q^{53} -6537.78i q^{57} -822.630 q^{59} +6917.98i q^{61} -1000.70 q^{63} +4929.83i q^{67} +5077.48i q^{69} +4236.85i q^{71} -8507.29i q^{73} +7147.07 q^{77} +10205.9i q^{79} -7898.82 q^{81} +818.969i q^{83} -5203.25 q^{87} +10451.0 q^{89} -8731.82 q^{91} -5021.46 q^{93} +11691.7i q^{97} -3814.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \) 32 * q - 864 * q^9 - 384 * q^11 - 1408 * q^19 - 4416 * q^41 + 4960 * q^49 + 35584 * q^51 + 28032 * q^59 + 20768 * q^81 - 13632 * q^89 - 49152 * q^91 + 5248 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more