LMFDB - Embedded newform 800.5.e.e.399.3

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			399.3
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-13.1346i q^{3} -61.6422 q^{7} -91.5179 q^{9} -182.209 q^{11} +141.781 q^{13} -40.3284i q^{17} -321.326 q^{19} +809.647i q^{21} -816.505 q^{23} +138.148i q^{27} -1199.96i q^{29} +361.487i q^{31} +2393.25i q^{33} -0.0209397 q^{37} -1862.23i q^{39} -872.793 q^{41} +3126.84i q^{43} +2659.89 q^{47} +1398.77 q^{49} -529.697 q^{51} +2951.01 q^{53} +4220.49i q^{57} +5410.19 q^{59} +2259.85i q^{61} +5641.37 q^{63} -2351.02i q^{67} +10724.5i q^{69} -660.927i q^{71} +5948.02i q^{73} +11231.8 q^{77} -7620.25i q^{79} -5598.43 q^{81} +3526.95i q^{83} -15760.9 q^{87} -5948.60 q^{89} -8739.68 q^{91} +4747.98 q^{93} -679.848i q^{97} +16675.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	− 13.1346i	− 1.45940i	−0.683767	−	0.729700i	$-0.739660\pi$
$3$	− 13.1346i	− 1.45940i	0.683767	−	0.729700i	$-0.260340\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−61.6422	−1.25800	−0.629002	−	0.777403i	$-0.716536\pi$
$7$	−61.6422	−1.25800	−0.629002	+	0.777403i	$0.716536\pi$
$8$	0	0
$9$	−91.5179	−1.12985
$10$	0	0
$11$	−182.209	−1.50586	−0.752932	−	0.658099i	$-0.771361\pi$
$11$	−182.209	−1.50586	−0.752932	+	0.658099i	$0.771361\pi$
$12$	0	0
$13$	141.781	0.838939	0.419470	−	0.907769i	$-0.362216\pi$
$13$	141.781	0.838939	0.419470	+	0.907769i	$0.362216\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 40.3284i	− 0.139545i	−0.997563	−	0.0697723i	$-0.977773\pi$
$17$	− 40.3284i	− 0.139545i	0.997563	−	0.0697723i	$-0.0222273\pi$
$18$	0	0
$19$	−321.326	−0.890100	−0.445050	−	0.895506i	$-0.646814\pi$
$19$	−321.326	−0.890100	−0.445050	+	0.895506i	$0.646814\pi$
$20$	0	0
$21$	809.647i	1.83593i
$22$	0	0
$23$	−816.505	−1.54349	−0.771744	−	0.635934i	$-0.780615\pi$
$23$	−816.505	−1.54349	−0.771744	+	0.635934i	$0.780615\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	138.148i	0.189503i
$28$	0	0
$29$	− 1199.96i	− 1.42682i	−0.700747	−	0.713410i	$-0.747150\pi$
$29$	− 1199.96i	− 1.42682i	0.700747	−	0.713410i	$-0.252850\pi$
$30$	0	0
$31$	361.487i	0.376157i	0.982154	+	0.188078i	$0.0602259\pi$
$31$	361.487i	0.376157i	−0.982154	+	0.188078i	$0.939774\pi$
$32$	0	0
$33$	2393.25i	2.19766i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−0.0209397	−1.52956e−5 0	−7.64781e−6	−	1.00000i	$-0.500002\pi$
$37$	−0.0209397	−1.52956e−5 0	−7.64781e−6		1.00000i	$0.500002\pi$
$38$	0	0
$39$	− 1862.23i	− 1.22435i
$40$	0	0
$41$	−872.793	−0.519211	−0.259605	−	0.965715i	$-0.583592\pi$
$41$	−872.793	−0.519211	−0.259605	+	0.965715i	$0.583592\pi$
$42$	0	0
$43$	3126.84i	1.69110i	0.533899	+	0.845548i	$0.320726\pi$
$43$	3126.84i	1.69110i	−0.533899	+	0.845548i	$0.679274\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2659.89	1.20411	0.602057	−	0.798453i	$-0.294348\pi$
$47$	2659.89	1.20411	0.602057	+	0.798453i	$0.294348\pi$
$48$	0	0
$49$	1398.77	0.582576
$50$	0	0
$51$	−529.697	−0.203651
$52$	0	0
$53$	2951.01	1.05056	0.525278	−	0.850931i	$-0.323961\pi$
$53$	2951.01	1.05056	0.525278	+	0.850931i	$0.323961\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	4220.49i	1.29901i
$58$	0	0
$59$	5410.19	1.55421	0.777103	−	0.629373i	$-0.216688\pi$
$59$	5410.19	1.55421	0.777103	+	0.629373i	$0.216688\pi$
$60$	0	0
$61$	2259.85i	0.607324i	0.952780	+	0.303662i	$0.0982094\pi$
$61$	2259.85i	0.607324i	−0.952780	+	0.303662i	$0.901791\pi$
$62$	0	0
$63$	5641.37	1.42136
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2351.02i	− 0.523730i	−0.965105	−	0.261865i	$-0.915662\pi$
$67$	− 2351.02i	− 0.523730i	0.965105	−	0.261865i	$-0.0843375\pi$
$68$	0	0
$69$	10724.5i	2.25257i
$70$	0	0
$71$	− 660.927i	− 0.131110i	−0.997849	−	0.0655551i	$-0.979118\pi$
$71$	− 660.927i	− 0.131110i	0.997849	−	0.0655551i	$-0.0208818\pi$
$72$	0	0
$73$	5948.02i	1.11616i	0.829787	+	0.558081i	$0.188462\pi$
$73$	5948.02i	1.11616i	−0.829787	+	0.558081i	$0.811538\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	11231.8	1.89438
$78$	0	0
$79$	− 7620.25i	− 1.22100i	−0.792017	−	0.610499i	$-0.790969\pi$
$79$	− 7620.25i	− 1.22100i	0.792017	−	0.610499i	$-0.209031\pi$
$80$	0	0
$81$	−5598.43	−0.853289
$82$	0	0
$83$	3526.95i	0.511969i	0.966681	+	0.255984i	$0.0823995\pi$
$83$	3526.95i	0.511969i	−0.966681	+	0.255984i	$0.917600\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−15760.9	−2.08230
$88$	0	0
$89$	−5948.60	−0.750991	−0.375495	−	0.926824i	$-0.622527\pi$
$89$	−5948.60	−0.750991	−0.375495	+	0.926824i	$0.622527\pi$
$90$	0	0
$91$	−8739.68	−1.05539
$92$	0	0
$93$	4747.98	0.548963
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 679.848i	− 0.0722551i	−0.999347	−	0.0361275i	$-0.988498\pi$
$97$	− 679.848i	− 0.0722551i	0.999347	−	0.0361275i	$-0.0115023\pi$
$98$	0	0
$99$	16675.4	1.70140
$100$	0	0
$101$	− 7870.10i	− 0.771503i	−0.922603	−	0.385751i	$-0.873942\pi$
$101$	− 7870.10i	− 0.771503i	0.922603	−	0.385751i	$-0.126058\pi$
$102$	0	0
$103$	3164.29	0.298265	0.149132	−	0.988817i	$-0.452352\pi$
$103$	3164.29	0.298265	0.149132	+	0.988817i	$0.452352\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 2977.72i	− 0.260086i	−0.991508	−	0.130043i	$-0.958489\pi$
$107$	− 2977.72i	− 0.260086i	0.991508	−	0.130043i	$-0.0415115\pi$
$108$	0	0
$109$	− 12111.3i	− 1.01938i	−0.860358	−	0.509690i	$-0.829760\pi$
$109$	− 12111.3i	− 1.01938i	0.860358	−	0.509690i	$-0.170240\pi$
$110$	0	0
$111$	0.275035i	0 2.23224e-5i
$112$	0	0
$113$	6297.71i	0.493203i	0.969117	+	0.246601i	$0.0793138\pi$
$113$	6297.71i	0.493203i	−0.969117	+	0.246601i	$0.920686\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−12975.5	−0.947876
$118$	0	0
$119$	2485.93i	0.175548i
$120$	0	0
$121$	18559.3	1.26763
$122$	0	0
$123$	11463.8i	0.757736i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−19296.8	−1.19640	−0.598201	−	0.801346i	$-0.704118\pi$
$127$	−19296.8	−1.19640	−0.598201	+	0.801346i	$0.704118\pi$
$128$	0	0
$129$	41069.8	2.46799
$130$	0	0
$131$	30503.1	1.77747	0.888734	−	0.458422i	$-0.151585\pi$
$131$	30503.1	1.77747	0.888734	+	0.458422i	$0.151585\pi$
$132$	0	0
$133$	19807.3	1.11975
$134$	0	0
$135$	0	0
$136$	0	0
$137$	19714.6i	1.05038i	0.850985	+	0.525190i	$0.176006\pi$
$137$	19714.6i	1.05038i	−0.850985	+	0.525190i	$0.823994\pi$
$138$	0	0
$139$	3780.14	0.195649	0.0978246	−	0.995204i	$-0.468812\pi$
$139$	3780.14	0.195649	0.0978246	+	0.995204i	$0.468812\pi$
$140$	0	0
$141$	− 34936.6i	− 1.75729i
$142$	0	0
$143$	−25833.8	−1.26333
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 18372.2i	− 0.850212i
$148$	0	0
$149$	− 26261.3i	− 1.18289i	−0.806347	−	0.591443i	$-0.798558\pi$
$149$	− 26261.3i	− 1.18289i	0.806347	−	0.591443i	$-0.201442\pi$
$150$	0	0
$151$	2021.58i	0.0886618i	0.999017	+	0.0443309i	$0.0141156\pi$
$151$	2021.58i	0.0886618i	−0.999017	+	0.0443309i	$0.985884\pi$
$152$	0	0
$153$	3690.77i	0.157664i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	30027.5	1.21820	0.609101	−	0.793093i	$-0.291530\pi$
$157$	30027.5	1.21820	0.609101	+	0.793093i	$0.291530\pi$
$158$	0	0
$159$	− 38760.4i	− 1.53318i
$160$	0	0
$161$	50331.2	1.94171
$162$	0	0
$163$	− 37292.9i	− 1.40362i	−0.712362	−	0.701812i	$-0.752374\pi$
$163$	− 37292.9i	− 1.40362i	0.712362	−	0.701812i	$-0.247626\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	28870.3	1.03519	0.517594	−	0.855627i	$-0.326828\pi$
$167$	28870.3	1.03519	0.517594	+	0.855627i	$0.326828\pi$
$168$	0	0
$169$	−8459.22	−0.296181
$170$	0	0
$171$	29407.1	1.00568
$172$	0	0
$173$	−44583.4	−1.48964	−0.744819	−	0.667267i	$-0.767464\pi$
$173$	−44583.4	−1.48964	−0.744819	+	0.667267i	$0.767464\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 71060.7i	− 2.26821i
$178$	0	0
$179$	8655.69	0.270144	0.135072	−	0.990836i	$-0.456873\pi$
$179$	8655.69	0.270144	0.135072	+	0.990836i	$0.456873\pi$
$180$	0	0
$181$	40289.9i	1.22981i	0.788600	+	0.614906i	$0.210806\pi$
$181$	40289.9i	1.22981i	−0.788600	+	0.614906i	$0.789194\pi$
$182$	0	0
$183$	29682.3	0.886330
$184$	0	0
$185$	0	0
$186$	0	0
$187$	7348.21i	0.210135i
$188$	0	0
$189$	− 8515.74i	− 0.238396i
$190$	0	0
$191$	3633.00i	0.0995862i	0.998760	+	0.0497931i	$0.0158562\pi$
$191$	3633.00i	0.0995862i	−0.998760	+	0.0497931i	$0.984144\pi$
$192$	0	0
$193$	15199.5i	0.408051i	0.978966	+	0.204025i	$0.0654025\pi$
$193$	15199.5i	0.408051i	−0.978966	+	0.204025i	$0.934598\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−38028.3	−0.979884	−0.489942	−	0.871755i	$-0.662982\pi$
$197$	−38028.3	−0.979884	−0.489942	+	0.871755i	$0.662982\pi$
$198$	0	0
$199$	49239.7i	1.24339i	0.783258	+	0.621697i	$0.213557\pi$
$199$	49239.7i	1.24339i	−0.783258	+	0.621697i	$0.786443\pi$
$200$	0	0
$201$	−30879.8	−0.764332
$202$	0	0
$203$	73967.9i	1.79495i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	74724.8	1.74391
$208$	0	0
$209$	58548.7	1.34037
$210$	0	0
$211$	8876.19	0.199371	0.0996855	−	0.995019i	$-0.468216\pi$
$211$	8876.19	0.199371	0.0996855	+	0.995019i	$0.468216\pi$
$212$	0	0
$213$	−8681.01	−0.191342
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 22282.8i	− 0.473207i
$218$	0	0
$219$	78124.9	1.62893
$220$	0	0
$221$	− 5717.78i	− 0.117069i
$222$	0	0
$223$	30807.0	0.619498	0.309749	−	0.950818i	$-0.399755\pi$
$223$	30807.0	0.619498	0.309749	+	0.950818i	$0.399755\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4525.34i	0.0878212i	0.999035	+	0.0439106i	$0.0139817\pi$
$227$	4525.34i	0.0878212i	−0.999035	+	0.0439106i	$0.986018\pi$
$228$	0	0
$229$	− 6977.92i	− 0.133062i	−0.997784	−	0.0665311i	$-0.978807\pi$
$229$	− 6977.92i	− 0.133062i	0.997784	−	0.0665311i	$-0.0211932\pi$
$230$	0	0
$231$	− 147525.i	− 2.76466i
$232$	0	0
$233$	− 6262.53i	− 0.115355i	−0.998335	−	0.0576777i	$-0.981630\pi$
$233$	− 6262.53i	− 0.115355i	0.998335	−	0.0576777i	$-0.0183696\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−100089.	−1.78193
$238$	0	0
$239$	− 60581.1i	− 1.06057i	−0.847818	−	0.530287i	$-0.822084\pi$
$239$	− 60581.1i	− 1.06057i	0.847818	−	0.530287i	$-0.177916\pi$
$240$	0	0
$241$	−6283.81	−0.108190	−0.0540952	−	0.998536i	$-0.517227\pi$
$241$	−6283.81	−0.108190	−0.0540952	+	0.998536i	$0.517227\pi$
$242$	0	0
$243$	84723.1i	1.43479i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−45557.9	−0.746740
$248$	0	0
$249$	46325.1	0.747167
$250$	0	0
$251$	−40906.3	−0.649296	−0.324648	−	0.945835i	$-0.605246\pi$
$251$	−40906.3	−0.649296	−0.324648	+	0.945835i	$0.605246\pi$
$252$	0	0
$253$	148775.	2.32428
$254$	0	0
$255$	0	0
$256$	0	0
$257$	71127.0i	1.07688i	0.842663	+	0.538441i	$0.180987\pi$
$257$	71127.0i	1.07688i	−0.842663	+	0.538441i	$0.819013\pi$
$258$	0	0
$259$	1.29077	1.92420e−5 0
$260$	0	0
$261$	109817.i	1.61209i
$262$	0	0
$263$	108240.	1.56486	0.782430	−	0.622738i	$-0.213980\pi$
$263$	108240.	1.56486	0.782430	+	0.622738i	$0.213980\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	78132.5i	1.09600i
$268$	0	0
$269$	− 31561.3i	− 0.436165i	−0.975930	−	0.218083i	$-0.930020\pi$
$269$	− 31561.3i	− 0.436165i	0.975930	−	0.218083i	$-0.0699802\pi$
$270$	0	0
$271$	106238.i	1.44657i	0.690549	+	0.723286i	$0.257369\pi$
$271$	106238.i	1.44657i	−0.690549	+	0.723286i	$0.742631\pi$
$272$	0	0
$273$	114792.i	1.54024i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−10816.7	−0.140973	−0.0704863	−	0.997513i	$-0.522455\pi$
$277$	−10816.7	−0.140973	−0.0704863	+	0.997513i	$0.522455\pi$
$278$	0	0
$279$	− 33082.5i	− 0.425001i
$280$	0	0
$281$	−45497.1	−0.576197	−0.288099	−	0.957601i	$-0.593023\pi$
$281$	−45497.1	−0.576197	−0.288099	+	0.957601i	$0.593023\pi$
$282$	0	0
$283$	29609.1i	0.369703i	0.982766	+	0.184851i	$0.0591804\pi$
$283$	29609.1i	0.369703i	−0.982766	+	0.184851i	$0.940820\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	53800.9	0.653169
$288$	0	0
$289$	81894.6	0.980527
$290$	0	0
$291$	−8929.54	−0.105449
$292$	0	0
$293$	−85118.8	−0.991495	−0.495747	−	0.868467i	$-0.665106\pi$
$293$	−85118.8	−0.991495	−0.495747	+	0.868467i	$0.665106\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	− 25171.9i	− 0.285366i
$298$	0	0
$299$	−115765.	−1.29489
$300$	0	0
$301$	− 192745.i	− 2.12741i
$302$	0	0
$303$	−103371.	−1.12593
$304$	0	0
$305$	0	0
$306$	0	0
$307$	76798.2i	0.814843i	0.913240	+	0.407422i	$0.133572\pi$
$307$	76798.2i	0.814843i	−0.913240	+	0.407422i	$0.866428\pi$
$308$	0	0
$309$	− 41561.7i	− 0.435288i
$310$	0	0
$311$	110813.i	1.14570i	0.819659	+	0.572851i	$0.194163\pi$
$311$	110813.i	1.14570i	−0.819659	+	0.572851i	$0.805837\pi$
$312$	0	0
$313$	− 59571.3i	− 0.608062i	−0.952662	−	0.304031i	$-0.901667\pi$
$313$	− 59571.3i	− 0.608062i	0.952662	−	0.304031i	$-0.0983327\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	101309.	1.00816	0.504081	−	0.863656i	$-0.331831\pi$
$317$	101309.	1.00816	0.504081	+	0.863656i	$0.331831\pi$
$318$	0	0
$319$	218643.i	2.14860i
$320$	0	0
$321$	−39111.2	−0.379569
$322$	0	0
$323$	12958.6i	0.124209i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−159077.	−1.48768
$328$	0	0
$329$	−163962.	−1.51478
$330$	0	0
$331$	−54230.8	−0.494982	−0.247491	−	0.968890i	$-0.579606\pi$
$331$	−54230.8	−0.494982	−0.247491	+	0.968890i	$0.579606\pi$
$332$	0	0
$333$	1.91636	1.72817e−5 0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	117299.i	1.03284i	0.856335	+	0.516421i	$0.172736\pi$
$337$	117299.i	1.03284i	−0.856335	+	0.516421i	$0.827264\pi$
$338$	0	0
$339$	82717.9	0.719780
$340$	0	0
$341$	− 65866.3i	− 0.566441i
$342$	0	0
$343$	61779.9	0.525121
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 29463.2i	− 0.244693i	−0.992487	−	0.122347i	$-0.960958\pi$
$347$	− 29463.2i	− 0.244693i	0.992487	−	0.122347i	$-0.0390419\pi$
$348$	0	0
$349$	− 79572.3i	− 0.653297i	−0.945146	−	0.326649i	$-0.894081\pi$
$349$	− 79572.3i	− 0.653297i	0.945146	−	0.326649i	$-0.105919\pi$
$350$	0	0
$351$	19586.7i	0.158982i
$352$	0	0
$353$	233242.i	1.87179i	0.352275	+	0.935896i	$0.385408\pi$
$353$	233242.i	1.87179i	−0.352275	+	0.935896i	$0.614592\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	32651.7	0.256194
$358$	0	0
$359$	52120.7i	0.404409i	0.979343	+	0.202205i	$0.0648106\pi$
$359$	52120.7i	0.404409i	−0.979343	+	0.202205i	$0.935189\pi$
$360$	0	0
$361$	−27070.5	−0.207722
$362$	0	0
$363$	− 243769.i	− 1.84997i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	154981.	1.15066	0.575330	−	0.817921i	$-0.304874\pi$
$367$	154981.	1.15066	0.575330	+	0.817921i	$0.304874\pi$
$368$	0	0
$369$	79876.1	0.586630
$370$	0	0
$371$	−181907.	−1.32160
$372$	0	0
$373$	105747.	0.760064	0.380032	−	0.924973i	$-0.375913\pi$
$373$	105747.	0.760064	0.380032	+	0.924973i	$0.375913\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 170131.i	− 1.19701i
$378$	0	0
$379$	−123203.	−0.857712	−0.428856	−	0.903373i	$-0.641083\pi$
$379$	−123203.	−0.857712	−0.428856	+	0.903373i	$0.641083\pi$
$380$	0	0
$381$	253455.i	1.74603i
$382$	0	0
$383$	−41797.4	−0.284939	−0.142469	−	0.989799i	$-0.545504\pi$
$383$	−41797.4	−0.284939	−0.142469	+	0.989799i	$0.545504\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 286161.i	− 1.91069i
$388$	0	0
$389$	175902.i	1.16244i	0.813746	+	0.581220i	$0.197425\pi$
$389$	175902.i	1.16244i	−0.813746	+	0.581220i	$0.802575\pi$
$390$	0	0
$391$	32928.3i	0.215385i
$392$	0	0
$393$	− 400647.i	− 2.59404i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	167811.	1.06473	0.532365	−	0.846515i	$-0.321303\pi$
$397$	167811.	1.06473	0.532365	+	0.846515i	$0.321303\pi$
$398$	0	0
$399$	− 260161.i	− 1.63416i
$400$	0	0
$401$	208614.	1.29734	0.648670	−	0.761070i	$-0.275326\pi$
$401$	208614.	1.29734	0.648670	+	0.761070i	$0.275326\pi$
$402$	0	0
$403$	51251.8i	0.315573i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.81541	2.30331e−5 0
$408$	0	0
$409$	−283543.	−1.69501	−0.847504	−	0.530789i	$-0.821896\pi$
$409$	−283543.	−1.69501	−0.847504	+	0.530789i	$0.821896\pi$
$410$	0	0
$411$	258944.	1.53293
$412$	0	0
$413$	−333496.	−1.95520
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 49650.6i	− 0.285531i
$418$	0	0
$419$	−54938.6	−0.312932	−0.156466	−	0.987683i	$-0.550010\pi$
$419$	−54938.6	−0.312932	−0.156466	+	0.987683i	$0.550010\pi$
$420$	0	0
$421$	− 95542.7i	− 0.539055i	−0.962993	−	0.269528i	$-0.913132\pi$
$421$	− 95542.7i	− 0.539055i	0.962993	−	0.269528i	$-0.0868675\pi$
$422$	0	0
$423$	−243427.	−1.36047
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 139302.i	− 0.764017i
$428$	0	0
$429$	339317.i	1.84370i
$430$	0	0
$431$	90739.7i	0.488475i	0.969715	+	0.244238i	$0.0785377\pi$
$431$	90739.7i	0.488475i	−0.969715	+	0.244238i	$0.921462\pi$
$432$	0	0
$433$	− 65987.8i	− 0.351956i	−0.984394	−	0.175978i	$-0.943691\pi$
$433$	− 65987.8i	− 0.351956i	0.984394	−	0.175978i	$-0.0563087\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	262364.	1.37386
$438$	0	0
$439$	− 75268.9i	− 0.390559i	−0.980748	−	0.195280i	$-0.937439\pi$
$439$	− 75268.9i	− 0.390559i	0.980748	−	0.195280i	$-0.0625614\pi$
$440$	0	0
$441$	−128012.	−0.658224
$442$	0	0
$443$	− 52370.2i	− 0.266856i	−0.991059	−	0.133428i	$-0.957402\pi$
$443$	− 52370.2i	− 0.266856i	0.991059	−	0.133428i	$-0.0425985\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−344931.	−1.72631
$448$	0	0
$449$	−282713.	−1.40234	−0.701169	−	0.712995i	$-0.747338\pi$
$449$	−282713.	−1.40234	−0.701169	+	0.712995i	$0.747338\pi$
$450$	0	0
$451$	159031.	0.781860
$452$	0	0
$453$	26552.6	0.129393
$454$	0	0
$455$	0	0
$456$	0	0
$457$	379431.i	1.81677i	0.418133	+	0.908386i	$0.362685\pi$
$457$	379431.i	1.81677i	−0.418133	+	0.908386i	$0.637315\pi$
$458$	0	0
$459$	5571.28	0.0264441
$460$	0	0
$461$	203541.i	0.957744i	0.877885	+	0.478872i	$0.158954\pi$
$461$	203541.i	0.957744i	−0.877885	+	0.478872i	$0.841046\pi$
$462$	0	0
$463$	−400318.	−1.86743	−0.933713	−	0.358023i	$-0.883451\pi$
$463$	−400318.	−1.86743	−0.933713	+	0.358023i	$0.883451\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	227026.i	1.04098i	0.853869	+	0.520488i	$0.174250\pi$
$467$	227026.i	1.04098i	−0.853869	+	0.520488i	$0.825750\pi$
$468$	0	0
$469$	144922.i	0.658855i
$470$	0	0
$471$	− 394399.i	− 1.77784i
$472$	0	0
$473$	− 569739.i	− 2.54656i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−270070.	−1.18697
$478$	0	0
$479$	− 258233.i	− 1.12549i	−0.826631	−	0.562744i	$-0.809746\pi$
$479$	− 258233.i	− 1.12549i	0.826631	−	0.562744i	$-0.190254\pi$
$480$	0	0
$481$	−2.96884	−1.28321e−5 0
$482$	0	0
$483$	− 661080.i	− 2.83374i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−30343.6	−0.127941	−0.0639705	−	0.997952i	$-0.520376\pi$
$487$	−30343.6	−0.127941	−0.0639705	+	0.997952i	$0.520376\pi$
$488$	0	0
$489$	−489828.	−2.04845
$490$	0	0
$491$	82979.2	0.344196	0.172098	−	0.985080i	$-0.444945\pi$
$491$	82979.2	0.344196	0.172098	+	0.985080i	$0.444945\pi$
$492$	0	0
$493$	−48392.2	−0.199105
$494$	0	0
$495$	0	0
$496$	0	0
$497$	40741.0i	0.164937i
$498$	0	0
$499$	−212545.	−0.853593	−0.426796	−	0.904348i	$-0.640358\pi$
$499$	−212545.	−0.853593	−0.426796	+	0.904348i	$0.640358\pi$
$500$	0	0
$501$	− 379201.i	− 1.51075i
$502$	0	0
$503$	143873.	0.568648	0.284324	−	0.958728i	$-0.408231\pi$
$503$	143873.	0.568648	0.284324	+	0.958728i	$0.408231\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	111109.i	0.432247i
$508$	0	0
$509$	137724.i	0.531588i	0.964030	+	0.265794i	$0.0856341\pi$
$509$	137724.i	0.531588i	−0.964030	+	0.265794i	$0.914366\pi$
$510$	0	0
$511$	− 366650.i	− 1.40414i
$512$	0	0
$513$	− 44390.5i	− 0.168677i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−484657.	−1.81323
$518$	0	0
$519$	585585.i	2.17398i
$520$	0	0
$521$	33788.5	0.124478	0.0622392	−	0.998061i	$-0.480176\pi$
$521$	33788.5	0.124478	0.0622392	+	0.998061i	$0.480176\pi$
$522$	0	0
$523$	− 314081.i	− 1.14826i	−0.818766	−	0.574128i	$-0.805341\pi$
$523$	− 314081.i	− 1.14826i	0.818766	−	0.574128i	$-0.194659\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	14578.2	0.0524906
$528$	0	0
$529$	386839.	1.38235
$530$	0	0
$531$	−495129.	−1.75602
$532$	0	0
$533$	−123745.	−0.435586
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 113689.i	− 0.394249i
$538$	0	0
$539$	−254868.	−0.877281
$540$	0	0
$541$	385898.i	1.31849i	0.751927	+	0.659246i	$0.229125\pi$
$541$	385898.i	1.31849i	−0.751927	+	0.659246i	$0.770875\pi$
$542$	0	0
$543$	529192.	1.79479
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 98962.2i	− 0.330746i	−0.986231	−	0.165373i	$-0.947117\pi$
$547$	− 98962.2i	− 0.330746i	0.986231	−	0.165373i	$-0.0528828\pi$
$548$	0	0
$549$	− 206817.i	− 0.686185i
$550$	0	0
$551$	385577.i	1.27001i
$552$	0	0
$553$	469729.i	1.53602i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−150468.	−0.484992	−0.242496	−	0.970152i	$-0.577966\pi$
$557$	−150468.	−0.484992	−0.242496	+	0.970152i	$0.577966\pi$
$558$	0	0
$559$	443325.i	1.41873i
$560$	0	0
$561$	96515.8	0.306671
$562$	0	0
$563$	351815.i	1.10993i	0.831872	+	0.554967i	$0.187269\pi$
$563$	351815.i	1.10993i	−0.831872	+	0.554967i	$0.812731\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	345100.	1.07344
$568$	0	0
$569$	−200171.	−0.618269	−0.309135	−	0.951018i	$-0.600039\pi$
$569$	−200171.	−0.618269	−0.309135	+	0.951018i	$0.600039\pi$
$570$	0	0
$571$	−310946.	−0.953701	−0.476851	−	0.878984i	$-0.658222\pi$
$571$	−310946.	−0.953701	−0.476851	+	0.878984i	$0.658222\pi$
$572$	0	0
$573$	47718.1	0.145336
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 309106.i	− 0.928443i	−0.885719	−	0.464222i	$-0.846334\pi$
$577$	− 309106.i	− 0.928443i	0.885719	−	0.464222i	$-0.153666\pi$
$578$	0	0
$579$	199639.	0.595509
$580$	0	0
$581$	− 217409.i	− 0.644059i
$582$	0	0
$583$	−537702.	−1.58199
$584$	0	0
$585$	0	0
$586$	0	0
$587$	328226.i	0.952569i	0.879291	+	0.476285i	$0.158017\pi$
$587$	328226.i	0.952569i	−0.879291	+	0.476285i	$0.841983\pi$
$588$	0	0
$589$	− 116155.i	− 0.334817i
$590$	0	0
$591$	499487.i	1.43004i
$592$	0	0
$593$	26286.3i	0.0747516i	0.999301	+	0.0373758i	$0.0118999\pi$
$593$	26286.3i	0.0747516i	−0.999301	+	0.0373758i	$0.988100\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	646744.	1.81461
$598$	0	0
$599$	− 195579.i	− 0.545090i	−0.962143	−	0.272545i	$-0.912135\pi$
$599$	− 195579.i	− 0.545090i	0.962143	−	0.272545i	$-0.0878654\pi$
$600$	0	0
$601$	−693475.	−1.91991	−0.959957	−	0.280147i	$-0.909617\pi$
$601$	−693475.	−1.91991	−0.959957	+	0.280147i	$0.909617\pi$
$602$	0	0
$603$	215161.i	0.591736i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−46263.1	−0.125562	−0.0627809	−	0.998027i	$-0.519997\pi$
$607$	−46263.1	−0.125562	−0.0627809	+	0.998027i	$0.519997\pi$
$608$	0	0
$609$	971539.	2.61955
$610$	0	0
$611$	377121.	1.01018
$612$	0	0
$613$	341759.	0.909491	0.454745	−	0.890621i	$-0.349730\pi$
$613$	341759.	0.909491	0.454745	+	0.890621i	$0.349730\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	422368.i	1.10948i	0.832023	+	0.554741i	$0.187183\pi$
$617$	422368.i	1.10948i	−0.832023	+	0.554741i	$0.812817\pi$
$618$	0	0
$619$	−546574.	−1.42649	−0.713244	−	0.700916i	$-0.752775\pi$
$619$	−546574.	−1.42649	−0.713244	+	0.700916i	$0.752775\pi$
$620$	0	0
$621$	− 112798.i	− 0.292496i
$622$	0	0
$623$	366685.	0.944750
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 769014.i	− 1.95614i
$628$	0	0
$629$	0.844463i	0 2.13442e-6i
$630$	0	0
$631$	267266.i	0.671251i	0.941995	+	0.335626i	$0.108948\pi$
$631$	267266.i	0.671251i	−0.941995	+	0.335626i	$0.891052\pi$
$632$	0	0
$633$	− 116585.i	− 0.290962i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	198318.	0.488746
$638$	0	0
$639$	60486.6i	0.148135i
$640$	0	0
$641$	−39388.3	−0.0958629	−0.0479315	−	0.998851i	$-0.515263\pi$
$641$	−39388.3	−0.0958629	−0.0479315	+	0.998851i	$0.515263\pi$
$642$	0	0
$643$	228811.i	0.553420i	0.960954	+	0.276710i	$0.0892441\pi$
$643$	228811.i	0.553420i	−0.960954	+	0.276710i	$0.910756\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	382095.	0.912772	0.456386	−	0.889782i	$-0.349144\pi$
$647$	382095.	0.912772	0.456386	+	0.889782i	$0.349144\pi$
$648$	0	0
$649$	−985788.	−2.34042
$650$	0	0
$651$	−292676.	−0.690599
$652$	0	0
$653$	752435.	1.76458	0.882292	−	0.470702i	$-0.155999\pi$
$653$	752435.	1.76458	0.882292	+	0.470702i	$0.155999\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 544350.i	− 1.26110i
$658$	0	0
$659$	317213.	0.730432	0.365216	−	0.930923i	$-0.380995\pi$
$659$	317213.	0.730432	0.365216	+	0.930923i	$0.380995\pi$
$660$	0	0
$661$	491003.i	1.12378i	0.827212	+	0.561889i	$0.189925\pi$
$661$	491003.i	1.12378i	−0.827212	+	0.561889i	$0.810075\pi$
$662$	0	0
$663$	−75100.8	−0.170851
$664$	0	0
$665$	0	0
$666$	0	0
$667$	979769.i	2.20228i
$668$	0	0
$669$	− 404638.i	− 0.904096i
$670$	0	0
$671$	− 411767.i	− 0.914548i
$672$	0	0
$673$	469121.i	1.03575i	0.855456	+	0.517875i	$0.173277\pi$
$673$	469121.i	1.03575i	−0.855456	+	0.517875i	$0.826723\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	233625.	0.509732	0.254866	−	0.966976i	$-0.417969\pi$
$677$	233625.	0.509732	0.254866	+	0.966976i	$0.417969\pi$
$678$	0	0
$679$	41907.4i	0.0908973i
$680$	0	0
$681$	59438.5	0.128166
$682$	0	0
$683$	780956.i	1.67411i	0.547116	+	0.837057i	$0.315726\pi$
$683$	780956.i	1.67411i	−0.547116	+	0.837057i	$0.684274\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−91652.2	−0.194191
$688$	0	0
$689$	418397.	0.881353
$690$	0	0
$691$	−283619.	−0.593990	−0.296995	−	0.954879i	$-0.595984\pi$
$691$	−283619.	−0.593990	−0.296995	+	0.954879i	$0.595984\pi$
$692$	0	0
$693$	−1.02791e6	−2.14037
$694$	0	0
$695$	0	0
$696$	0	0
$697$	35198.3i	0.0724530i
$698$	0	0
$699$	−82255.8	−0.168350
$700$	0	0
$701$	− 448693.i	− 0.913088i	−0.889701	−	0.456544i	$-0.849087\pi$
$701$	− 448693.i	− 0.913088i	0.889701	−	0.456544i	$-0.150913\pi$
$702$	0	0
$703$	6.72847	1.36146e−5 0
$704$	0	0
$705$	0	0
$706$	0	0
$707$	485130.i	0.970554i
$708$	0	0
$709$	− 191749.i	− 0.381453i	−0.981643	−	0.190727i	$-0.938916\pi$
$709$	− 191749.i	− 0.381453i	0.981643	−	0.190727i	$-0.0610844\pi$
$710$	0	0
$711$	697389.i	1.37955i
$712$	0	0
$713$	− 295156.i	− 0.580593i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−795708.	−1.54780
$718$	0	0
$719$	674986.i	1.30568i	0.757495	+	0.652840i	$0.226423\pi$
$719$	674986.i	1.30568i	−0.757495	+	0.652840i	$0.773577\pi$
$720$	0	0
$721$	−195054.	−0.375218
$722$	0	0
$723$	82535.4i	0.157893i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	32697.5	0.0618652	0.0309326	−	0.999521i	$-0.490152\pi$
$727$	32697.5	0.0618652	0.0309326	+	0.999521i	$0.490152\pi$
$728$	0	0
$729$	659332.	1.24065
$730$	0	0
$731$	126100.	0.235983
$732$	0	0
$733$	−836629.	−1.55713	−0.778565	−	0.627564i	$-0.784052\pi$
$733$	−836629.	−1.55713	−0.778565	+	0.627564i	$0.784052\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	428379.i	0.788666i
$738$	0	0
$739$	274618.	0.502852	0.251426	−	0.967877i	$-0.419100\pi$
$739$	274618.	0.502852	0.251426	+	0.967877i	$0.419100\pi$
$740$	0	0
$741$	598384.i	1.08979i
$742$	0	0
$743$	758534.	1.37403	0.687017	−	0.726641i	$-0.258920\pi$
$743$	758534.	1.37403	0.687017	+	0.726641i	$0.258920\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 322779.i	− 0.578448i
$748$	0	0
$749$	183553.i	0.327189i
$750$	0	0
$751$	− 692253.i	− 1.22740i	−0.789540	−	0.613699i	$-0.789681\pi$
$751$	− 692253.i	− 1.22740i	0.789540	−	0.613699i	$-0.210319\pi$
$752$	0	0
$753$	537288.i	0.947583i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−191429.	−0.334053	−0.167027	−	0.985952i	$-0.553417\pi$
$757$	−191429.	−0.334053	−0.167027	+	0.985952i	$0.553417\pi$
$758$	0	0
$759$	− 1.95410e6i	− 3.39206i
$760$	0	0
$761$	955895.	1.65060	0.825299	−	0.564697i	$-0.191007\pi$
$761$	955895.	1.65060	0.825299	+	0.564697i	$0.191007\pi$
$762$	0	0
$763$	746565.i	1.28239i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	767061.	1.30388
$768$	0	0
$769$	−512422.	−0.866513	−0.433256	−	0.901271i	$-0.642636\pi$
$769$	−512422.	−0.866513	−0.433256	+	0.901271i	$0.642636\pi$
$770$	0	0
$771$	934225.	1.57160
$772$	0	0
$773$	−198501.	−0.332203	−0.166101	−	0.986109i	$-0.553118\pi$
$773$	−198501.	−0.332203	−0.166101	+	0.986109i	$0.553118\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 16.9537i	0 2.80817e-5i
$778$	0	0
$779$	280451.	0.462149
$780$	0	0
$781$	120427.i	0.197434i
$782$	0	0
$783$	165771.	0.270387
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 730116.i	− 1.17881i	−0.807839	−	0.589403i	$-0.799363\pi$
$787$	− 730116.i	− 1.17881i	0.807839	−	0.589403i	$-0.200637\pi$
$788$	0	0
$789$	− 1.42169e6i	− 2.28376i
$790$	0	0
$791$	− 388205.i	− 0.620451i
$792$	0	0
$793$	320404.i	0.509508i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−403135.	−0.634649	−0.317325	−	0.948317i	$-0.602784\pi$
$797$	−403135.	−0.634649	−0.317325	+	0.948317i	$0.602784\pi$
$798$	0	0
$799$	− 107269.i	− 0.168028i
$800$	0	0
$801$	544403.	0.848507
$802$	0	0
$803$	− 1.08379e6i	− 1.68079i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−414546.	−0.636539
$808$	0	0
$809$	767840.	1.17320	0.586602	−	0.809875i	$-0.300465\pi$
$809$	767840.	1.17320	0.586602	+	0.809875i	$0.300465\pi$
$810$	0	0
$811$	98686.7	0.150043	0.0750217	−	0.997182i	$-0.476097\pi$
$811$	98686.7	0.150043	0.0750217	+	0.997182i	$0.476097\pi$
$812$	0	0
$813$	1.39539e6	2.11113
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 1.00473e6i	− 1.50525i
$818$	0	0
$819$	799837.	1.19243
$820$	0	0
$821$	− 578722.i	− 0.858586i	−0.903165	−	0.429293i	$-0.858763\pi$
$821$	− 578722.i	− 0.858586i	0.903165	−	0.429293i	$-0.141237\pi$
$822$	0	0
$823$	262419.	0.387432	0.193716	−	0.981058i	$-0.437946\pi$
$823$	262419.	0.387432	0.193716	+	0.981058i	$0.437946\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.18411e6i	1.73133i	0.500624	+	0.865665i	$0.333104\pi$
$827$	1.18411e6i	1.73133i	−0.500624	+	0.865665i	$0.666896\pi$
$828$	0	0
$829$	86230.4i	0.125473i	0.998030	+	0.0627367i	$0.0199828\pi$
$829$	86230.4i	0.125473i	−0.998030	+	0.0627367i	$0.980017\pi$
$830$	0	0
$831$	142073.i	0.205735i
$832$	0	0
$833$	− 56409.9i	− 0.0812953i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−49938.6	−0.0712829
$838$	0	0
$839$	468085.i	0.664967i	0.943109	+	0.332484i	$0.107887\pi$
$839$	468085.i	0.664967i	−0.943109	+	0.332484i	$0.892113\pi$
$840$	0	0
$841$	−732611.	−1.03581
$842$	0	0
$843$	597587.i	0.840903i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1.14404e6	−1.59468
$848$	0	0
$849$	388904.	0.539545
$850$	0	0
$851$	17.0974	2.36086e−5 0
$852$	0	0
$853$	991144.	1.36219	0.681096	−	0.732194i	$-0.261503\pi$
$853$	991144.	1.36219	0.681096	+	0.732194i	$0.261503\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 579579.i	− 0.789134i	−0.918867	−	0.394567i	$-0.870895\pi$
$857$	− 579579.i	− 0.789134i	0.918867	−	0.394567i	$-0.129105\pi$
$858$	0	0
$859$	477342.	0.646910	0.323455	−	0.946244i	$-0.395156\pi$
$859$	477342.	0.646910	0.323455	+	0.946244i	$0.395156\pi$
$860$	0	0
$861$	− 706654.i	− 0.953236i
$862$	0	0
$863$	87877.1	0.117992	0.0589962	−	0.998258i	$-0.481210\pi$
$863$	87877.1	0.117992	0.0589962	+	0.998258i	$0.481210\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.07565e6i	− 1.43098i
$868$	0	0
$869$	1.38848e6i	1.83866i
$870$	0	0
$871$	− 333330.i	− 0.439378i
$872$	0	0
$873$	62218.2i	0.0816374i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	315735.	0.410510	0.205255	−	0.978709i	$-0.434198\pi$
$877$	315735.	0.410510	0.205255	+	0.978709i	$0.434198\pi$
$878$	0	0
$879$	1.11800e6i	1.44699i
$880$	0	0
$881$	−233732.	−0.301138	−0.150569	−	0.988599i	$-0.548111\pi$
$881$	−233732.	−0.301138	−0.150569	+	0.988599i	$0.548111\pi$
$882$	0	0
$883$	− 1.12959e6i	− 1.44876i	−0.689399	−	0.724382i	$-0.742125\pi$
$883$	− 1.12959e6i	− 1.44876i	0.689399	−	0.724382i	$-0.257875\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	32714.8	0.0415812	0.0207906	−	0.999784i	$-0.493382\pi$
$887$	32714.8	0.0415812	0.0207906	+	0.999784i	$0.493382\pi$
$888$	0	0
$889$	1.18950e6	1.50508
$890$	0	0
$891$	1.02009e6	1.28494
$892$	0	0
$893$	−854692.	−1.07178
$894$	0	0
$895$	0	0
$896$	0	0
$897$	1.52052e6i	1.88977i
$898$	0	0
$899$	433768.	0.536708
$900$	0	0
$901$	− 119009.i	− 0.146599i
$902$	0	0
$903$	−2.53163e6	−3.10474
$904$	0	0
$905$	0	0
$906$	0	0
$907$	− 1.11093e6i	− 1.35043i	−0.737620	−	0.675216i	$-0.764050\pi$
$907$	− 1.11093e6i	− 1.35043i	0.737620	−	0.675216i	$-0.235950\pi$
$908$	0	0
$909$	720254.i	0.871682i
$910$	0	0
$911$	− 1.12408e6i	− 1.35445i	−0.735777	−	0.677223i	$-0.763183\pi$
$911$	− 1.12408e6i	− 1.35445i	0.735777	−	0.677223i	$-0.236817\pi$
$912$	0	0
$913$	− 642644.i	− 0.770955i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−1.88028e6	−2.23606
$918$	0	0
$919$	− 494243.i	− 0.585207i	−0.956234	−	0.292604i	$-0.905478\pi$
$919$	− 494243.i	− 0.585207i	0.956234	−	0.292604i	$-0.0945217\pi$
$920$	0	0
$921$	1.00871e6	1.18918
$922$	0	0
$923$	− 93706.7i	− 0.109994i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−289589.	−0.336994
$928$	0	0
$929$	19109.1	0.0221415	0.0110708	−	0.999939i	$-0.496476\pi$
$929$	19109.1	0.0221415	0.0110708	+	0.999939i	$0.496476\pi$
$930$	0	0
$931$	−449460.	−0.518551
$932$	0	0
$933$	1.45549e6	1.67204
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 595577.i	− 0.678357i	−0.940722	−	0.339179i	$-0.889851\pi$
$937$	− 595577.i	− 0.678357i	0.940722	−	0.339179i	$-0.110149\pi$
$938$	0	0
$939$	−782445.	−0.887407
$940$	0	0
$941$	− 901249.i	− 1.01781i	−0.860823	−	0.508904i	$-0.830051\pi$
$941$	− 901249.i	− 1.01781i	0.860823	−	0.508904i	$-0.169949\pi$
$942$	0	0
$943$	712640.	0.801395
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 1.00306e6i	− 1.11848i	−0.829006	−	0.559240i	$-0.811093\pi$
$947$	− 1.00306e6i	− 1.11848i	0.829006	−	0.559240i	$-0.188907\pi$
$948$	0	0
$949$	843315.i	0.936392i
$950$	0	0
$951$	− 1.33066e6i	− 1.47131i
$952$	0	0
$953$	− 352429.i	− 0.388048i	−0.980997	−	0.194024i	$-0.937846\pi$
$953$	− 352429.i	− 0.388048i	0.980997	−	0.194024i	$-0.0621540\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	2.87179e6	3.13566
$958$	0	0
$959$	− 1.21525e6i	− 1.32138i
$960$	0	0
$961$	792848.	0.858506
$962$	0	0
$963$	272515.i	0.293858i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	1.50153e6	1.60577	0.802883	−	0.596137i	$-0.203298\pi$
$967$	1.50153e6	1.60577	0.802883	+	0.596137i	$0.203298\pi$
$968$	0	0
$969$	170206.	0.181270
$970$	0	0
$971$	−909255.	−0.964378	−0.482189	−	0.876067i	$-0.660158\pi$
$971$	−909255.	−0.964378	−0.482189	+	0.876067i	$0.660158\pi$
$972$	0	0
$973$	−233016.	−0.246128
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.01442e6i	1.06274i	0.847139	+	0.531371i	$0.178323\pi$
$977$	1.01442e6i	1.06274i	−0.847139	+	0.531371i	$0.821677\pi$
$978$	0	0
$979$	1.08389e6	1.13089
$980$	0	0
$981$	1.10840e6i	1.15175i
$982$	0	0
$983$	1.44750e6	1.49800	0.748998	−	0.662573i	$-0.230535\pi$
$983$	1.44750e6	1.49800	0.748998	+	0.662573i	$0.230535\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	2.15357e6i	2.21067i
$988$	0	0
$989$	− 2.55308e6i	− 2.61019i
$990$	0	0
$991$	1.20661e6i	1.22862i	0.789064	+	0.614311i	$0.210566\pi$
$991$	1.20661e6i	1.22862i	−0.789064	+	0.614311i	$0.789434\pi$
$992$	0	0
$993$	712300.i	0.722378i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	774945.	0.779616	0.389808	−	0.920896i	$-0.372541\pi$
$997$	774945.	0.779616	0.389808	+	0.920896i	$0.372541\pi$
$998$	0	0
$999$	− 2.89277i	0 2.89857e-6i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.3		32
4.3	odd	2		200.5.e.e.99.12		32
5.2	odd	4		160.5.g.a.111.4		16
5.3	odd	4		800.5.g.h.751.14		16
5.4	even	2	inner	800.5.e.e.399.30		32
8.3	odd	2	inner	800.5.e.e.399.29		32
8.5	even	2		200.5.e.e.99.22		32
15.2	even	4		1440.5.g.a.271.1		16
20.3	even	4		200.5.g.h.51.14		16
20.7	even	4		40.5.g.a.11.3	✓	16
20.19	odd	2		200.5.e.e.99.21		32
40.3	even	4		800.5.g.h.751.13		16
40.13	odd	4		200.5.g.h.51.13		16
40.19	odd	2	inner	800.5.e.e.399.4		32
40.27	even	4		160.5.g.a.111.3		16
40.29	even	2		200.5.e.e.99.11		32
40.37	odd	4		40.5.g.a.11.4	yes	16
60.47	odd	4		360.5.g.a.91.14		16
120.77	even	4		360.5.g.a.91.13		16
120.107	odd	4		1440.5.g.a.271.16		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.3	✓	16	20.7	even	4
40.5.g.a.11.4	yes	16	40.37	odd	4
160.5.g.a.111.3		16	40.27	even	4
160.5.g.a.111.4		16	5.2	odd	4
200.5.e.e.99.11		32	40.29	even	2
200.5.e.e.99.12		32	4.3	odd	2
200.5.e.e.99.21		32	20.19	odd	2
200.5.e.e.99.22		32	8.5	even	2
200.5.g.h.51.13		16	40.13	odd	4
200.5.g.h.51.14		16	20.3	even	4
360.5.g.a.91.13		16	120.77	even	4
360.5.g.a.91.14		16	60.47	odd	4
800.5.e.e.399.3		32	1.1	even	1	trivial
800.5.e.e.399.4		32	40.19	odd	2	inner
800.5.e.e.399.29		32	8.3	odd	2	inner
800.5.e.e.399.30		32	5.4	even	2	inner
800.5.g.h.751.13		16	40.3	even	4
800.5.g.h.751.14		16	5.3	odd	4
1440.5.g.a.271.1		16	15.2	even	4
1440.5.g.a.271.16		16	120.107	odd	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)

\(f(q)\)	\(=\)	\(q-13.1346i q^{3} -61.6422 q^{7} -91.5179 q^{9} -182.209 q^{11} +141.781 q^{13} -40.3284i q^{17} -321.326 q^{19} +809.647i q^{21} -816.505 q^{23} +138.148i q^{27} -1199.96i q^{29} +361.487i q^{31} +2393.25i q^{33} -0.0209397 q^{37} -1862.23i q^{39} -872.793 q^{41} +3126.84i q^{43} +2659.89 q^{47} +1398.77 q^{49} -529.697 q^{51} +2951.01 q^{53} +4220.49i q^{57} +5410.19 q^{59} +2259.85i q^{61} +5641.37 q^{63} -2351.02i q^{67} +10724.5i q^{69} -660.927i q^{71} +5948.02i q^{73} +11231.8 q^{77} -7620.25i q^{79} -5598.43 q^{81} +3526.95i q^{83} -15760.9 q^{87} -5948.60 q^{89} -8739.68 q^{91} +4747.98 q^{93} -679.848i q^{97} +16675.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \) 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

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