LMFDB - Embedded newform 800.6.f.d.49.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,6,Mod(49,800)]

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([0, 1, 1]))

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

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

chi := DirichletCharacter("800.49");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	800.f (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:	$128.307055850$
Analytic rank:	$0$
Dimension:	$40$
Twist minimal:	no (minimal twist has level 200)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			49.20
Character	$\chi$	$=$	800.49
Dual form			800.6.f.d.49.19

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.457817 q^{3} -195.837i q^{7} -242.790 q^{9} +358.513i q^{11} +951.826 q^{13} -2101.46i q^{17} +2882.77i q^{19} +89.6574i q^{21} +1537.16i q^{23} +222.403 q^{27} +3875.11i q^{29} -5182.92 q^{31} -164.133i q^{33} -9494.97 q^{37} -435.762 q^{39} +3802.93 q^{41} +4330.63 q^{43} +12506.7i q^{47} -21545.1 q^{49} +962.084i q^{51} +25075.5 q^{53} -1319.78i q^{57} -18453.8i q^{59} +17488.7i q^{61} +47547.3i q^{63} +7490.73 q^{67} -703.739i q^{69} -1056.69 q^{71} -55286.5i q^{73} +70210.1 q^{77} +89387.0 q^{79} +58896.2 q^{81} -103847. q^{83} -1774.09i q^{87} +38137.1 q^{89} -186403. i q^{91} +2372.83 q^{93} +22739.9i q^{97} -87043.5i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) $ 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

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
$3$	−0.457817	−0.0293689	−0.0146845	−	0.999892i	$-0.504674\pi$
$3$	−0.457817	−0.0293689	−0.0146845	+	0.999892i	$0.504674\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 195.837i	− 1.51060i	−0.655379	−	0.755300i	$-0.727491\pi$
$7$	− 195.837i	− 1.51060i	0.655379	−	0.755300i	$-0.272509\pi$
$8$	0	0
$9$	−242.790	−0.999137
$10$	0	0
$11$	358.513i	0.893353i	0.894696	+	0.446676i	$0.147392\pi$
$11$	358.513i	0.893353i	−0.894696	+	0.446676i	$0.852608\pi$
$12$	0	0
$13$	951.826	1.56207	0.781033	−	0.624490i	$-0.214693\pi$
$13$	951.826	1.56207	0.781033	+	0.624490i	$0.214693\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2101.46i	− 1.76360i	−0.471628	−	0.881798i	$-0.656333\pi$
$17$	− 2101.46i	− 1.76360i	0.471628	−	0.881798i	$-0.343667\pi$
$18$	0	0
$19$	2882.77i	1.83200i	0.401178	+	0.916000i	$0.368601\pi$
$19$	2882.77i	1.83200i	−0.401178	+	0.916000i	$0.631399\pi$
$20$	0	0
$21$	89.6574i	0.0443648i
$22$	0	0
$23$	1537.16i	0.605899i	0.953007	+	0.302950i	$0.0979714\pi$
$23$	1537.16i	0.605899i	−0.953007	+	0.302950i	$0.902029\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	222.403	0.0587126
$28$	0	0
$29$	3875.11i	0.855636i	0.903865	+	0.427818i	$0.140718\pi$
$29$	3875.11i	0.855636i	−0.903865	+	0.427818i	$0.859282\pi$
$30$	0	0
$31$	−5182.92	−0.968658	−0.484329	−	0.874886i	$-0.660936\pi$
$31$	−5182.92	−0.968658	−0.484329	+	0.874886i	$0.660936\pi$
$32$	0	0
$33$	− 164.133i	− 0.0262368i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−9494.97	−1.14022	−0.570111	−	0.821568i	$-0.693100\pi$
$37$	−9494.97	−1.14022	−0.570111	+	0.821568i	$0.693100\pi$
$38$	0	0
$39$	−435.762	−0.0458762
$40$	0	0
$41$	3802.93	0.353312	0.176656	−	0.984273i	$-0.443472\pi$
$41$	3802.93	0.353312	0.176656	+	0.984273i	$0.443472\pi$
$42$	0	0
$43$	4330.63	0.357174	0.178587	−	0.983924i	$-0.442847\pi$
$43$	4330.63	0.357174	0.178587	+	0.983924i	$0.442847\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	12506.7i	0.825842i	0.910767	+	0.412921i	$0.135492\pi$
$47$	12506.7i	0.825842i	−0.910767	+	0.412921i	$0.864508\pi$
$48$	0	0
$49$	−21545.1	−1.28191
$50$	0	0
$51$	962.084i	0.0517949i
$52$	0	0
$53$	25075.5	1.22620	0.613099	−	0.790006i	$-0.289923\pi$
$53$	25075.5	1.22620	0.613099	+	0.790006i	$0.289923\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 1319.78i	− 0.0538039i
$58$	0	0
$59$	− 18453.8i	− 0.690171i	−0.938571	−	0.345085i	$-0.887850\pi$
$59$	− 18453.8i	− 0.690171i	0.938571	−	0.345085i	$-0.112150\pi$
$60$	0	0
$61$	17488.7i	0.601774i	0.953660	+	0.300887i	$0.0972827\pi$
$61$	17488.7i	0.601774i	−0.953660	+	0.300887i	$0.902717\pi$
$62$	0	0
$63$	47547.3i	1.50930i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	7490.73	0.203862	0.101931	−	0.994791i	$-0.467498\pi$
$67$	7490.73	0.203862	0.101931	+	0.994791i	$0.467498\pi$
$68$	0	0
$69$	− 703.739i	− 0.0177946i
$70$	0	0
$71$	−1056.69	−0.0248773	−0.0124387	−	0.999923i	$-0.503959\pi$
$71$	−1056.69	−0.0248773	−0.0124387	+	0.999923i	$0.503959\pi$
$72$	0	0
$73$	− 55286.5i	− 1.21426i	−0.794602	−	0.607130i	$-0.792321\pi$
$73$	− 55286.5i	− 1.21426i	0.794602	−	0.607130i	$-0.207679\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	70210.1	1.34950
$78$	0	0
$79$	89387.0	1.61141	0.805706	−	0.592316i	$-0.201786\pi$
$79$	89387.0	1.61141	0.805706	+	0.592316i	$0.201786\pi$
$80$	0	0
$81$	58896.2	0.997413
$82$	0	0
$83$	−103847.	−1.65462	−0.827310	−	0.561746i	$-0.810130\pi$
$83$	−103847.	−1.65462	−0.827310	+	0.561746i	$0.810130\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 1774.09i	− 0.0251291i
$88$	0	0
$89$	38137.1	0.510355	0.255178	−	0.966894i	$-0.417866\pi$
$89$	38137.1	0.510355	0.255178	+	0.966894i	$0.417866\pi$
$90$	0	0
$91$	− 186403.i	− 2.35966i
$92$	0	0
$93$	2372.83	0.0284485
$94$	0	0
$95$	0	0
$96$	0	0
$97$	22739.9i	0.245391i	0.992444	+	0.122696i	$0.0391539\pi$
$97$	22739.9i	0.245391i	−0.992444	+	0.122696i	$0.960846\pi$
$98$	0	0
$99$	− 87043.5i	− 0.892582i
$100$	0	0
$101$	− 52711.9i	− 0.514168i	−0.966389	−	0.257084i	$-0.917238\pi$
$101$	− 52711.9i	− 0.514168i	0.966389	−	0.257084i	$-0.0827618\pi$
$102$	0	0
$103$	− 38301.0i	− 0.355727i	−0.984055	−	0.177863i	$-0.943081\pi$
$103$	− 38301.0i	− 0.355727i	0.984055	−	0.177863i	$-0.0569185\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	151431.	1.27866	0.639329	−	0.768933i	$-0.279212\pi$
$107$	151431.	1.27866	0.639329	+	0.768933i	$0.279212\pi$
$108$	0	0
$109$	146420.i	1.18041i	0.807253	+	0.590206i	$0.200953\pi$
$109$	146420.i	1.18041i	−0.807253	+	0.590206i	$0.799047\pi$
$110$	0	0
$111$	4346.96	0.0334871
$112$	0	0
$113$	− 59368.9i	− 0.437384i	−0.975794	−	0.218692i	$-0.929821\pi$
$113$	− 59368.9i	− 0.437384i	0.975794	−	0.218692i	$-0.0701789\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−231094.	−1.56072
$118$	0	0
$119$	−411544.	−2.66409
$120$	0	0
$121$	32519.6	0.201921
$122$	0	0
$123$	−1741.05	−0.0103764
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 13242.9i	− 0.0728577i	−0.999336	−	0.0364288i	$-0.988402\pi$
$127$	− 13242.9i	− 0.0728577i	0.999336	−	0.0364288i	$-0.0115982\pi$
$128$	0	0
$129$	−1982.63	−0.0104898
$130$	0	0
$131$	29434.9i	0.149860i	0.997189	+	0.0749298i	$0.0238733\pi$
$131$	29434.9i	0.149860i	−0.997189	+	0.0749298i	$0.976127\pi$
$132$	0	0
$133$	564553.	2.76742
$134$	0	0
$135$	0	0
$136$	0	0
$137$	226821.i	1.03248i	0.856443	+	0.516241i	$0.172669\pi$
$137$	226821.i	1.03248i	−0.856443	+	0.516241i	$0.827331\pi$
$138$	0	0
$139$	206068.i	0.904637i	0.891856	+	0.452319i	$0.149403\pi$
$139$	206068.i	0.904637i	−0.891856	+	0.452319i	$0.850597\pi$
$140$	0	0
$141$	− 5725.76i	− 0.0242541i
$142$	0	0
$143$	341242.i	1.39548i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	9863.72	0.0376485
$148$	0	0
$149$	276062.i	1.01869i	0.860563	+	0.509344i	$0.170112\pi$
$149$	276062.i	1.01869i	−0.860563	+	0.509344i	$0.829888\pi$
$150$	0	0
$151$	88898.4	0.317287	0.158643	−	0.987336i	$-0.449288\pi$
$151$	88898.4	0.317287	0.158643	+	0.987336i	$0.449288\pi$
$152$	0	0
$153$	510215.i	1.76207i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	273937.	0.886954	0.443477	−	0.896286i	$-0.353745\pi$
$157$	273937.	0.886954	0.443477	+	0.896286i	$0.353745\pi$
$158$	0	0
$159$	−11480.0	−0.0360122
$160$	0	0
$161$	301034.	0.915272
$162$	0	0
$163$	22520.3	0.0663902	0.0331951	−	0.999449i	$-0.489432\pi$
$163$	22520.3	0.0663902	0.0331951	+	0.999449i	$0.489432\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4216.06i	0.0116981i	0.999983	+	0.00584906i	$0.00186182\pi$
$167$	4216.06i	0.0116981i	−0.999983	+	0.00584906i	$0.998138\pi$
$168$	0	0
$169$	534680.	1.44005
$170$	0	0
$171$	− 699908.i	− 1.83042i
$172$	0	0
$173$	240365.	0.610599	0.305299	−	0.952256i	$-0.401243\pi$
$173$	240365.	0.610599	0.305299	+	0.952256i	$0.401243\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	8448.47i	0.0202696i
$178$	0	0
$179$	− 245501.i	− 0.572691i	−0.958126	−	0.286346i	$-0.907559\pi$
$179$	− 245501.i	− 0.572691i	0.958126	−	0.286346i	$-0.0924406\pi$
$180$	0	0
$181$	16588.7i	0.0376370i	0.999823	+	0.0188185i	$0.00599048\pi$
$181$	16588.7i	0.0376370i	−0.999823	+	0.0188185i	$0.994010\pi$
$182$	0	0
$183$	− 8006.62i	− 0.0176735i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	753401.	1.57551
$188$	0	0
$189$	− 43554.7i	− 0.0886912i
$190$	0	0
$191$	555884.	1.10256	0.551278	−	0.834322i	$-0.314140\pi$
$191$	555884.	1.10256	0.551278	+	0.834322i	$0.314140\pi$
$192$	0	0
$193$	− 397154.i	− 0.767477i	−0.923442	−	0.383738i	$-0.874636\pi$
$193$	− 397154.i	− 0.767477i	0.923442	−	0.383738i	$-0.125364\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	722243.	1.32592	0.662961	−	0.748654i	$-0.269300\pi$
$197$	722243.	1.32592	0.662961	+	0.748654i	$0.269300\pi$
$198$	0	0
$199$	−129890.	−0.232511	−0.116256	−	0.993219i	$-0.537089\pi$
$199$	−129890.	−0.232511	−0.116256	+	0.993219i	$0.537089\pi$
$200$	0	0
$201$	−3429.38	−0.00598722
$202$	0	0
$203$	758890.	1.29252
$204$	0	0
$205$	0	0
$206$	0	0
$207$	− 373209.i	− 0.605377i
$208$	0	0
$209$	−1.03351e6	−1.63662
$210$	0	0
$211$	− 608872.i	− 0.941499i	−0.882267	−	0.470749i	$-0.843984\pi$
$211$	− 608872.i	− 0.941499i	0.882267	−	0.470749i	$-0.156016\pi$
$212$	0	0
$213$	483.772	0.000730620 0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	1.01501e6i	1.46325i
$218$	0	0
$219$	25311.1i	0.0356615i
$220$	0	0
$221$	− 2.00023e6i	− 2.75485i
$222$	0	0
$223$	− 22540.8i	− 0.0303534i	−0.999885	−	0.0151767i	$-0.995169\pi$
$223$	− 22540.8i	− 0.0303534i	0.999885	−	0.0151767i	$-0.00483108\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−150889.	−0.194354	−0.0971770	−	0.995267i	$-0.530981\pi$
$227$	−150889.	−0.194354	−0.0971770	+	0.995267i	$0.530981\pi$
$228$	0	0
$229$	724516.i	0.912976i	0.889730	+	0.456488i	$0.150893\pi$
$229$	724516.i	0.912976i	−0.889730	+	0.456488i	$0.849107\pi$
$230$	0	0
$231$	−32143.3	−0.0396334
$232$	0	0
$233$	336561.i	0.406138i	0.979164	+	0.203069i	$0.0650916\pi$
$233$	336561.i	0.406138i	−0.979164	+	0.203069i	$0.934908\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−40922.8	−0.0473255
$238$	0	0
$239$	330500.	0.374262	0.187131	−	0.982335i	$-0.440081\pi$
$239$	330500.	0.374262	0.187131	+	0.982335i	$0.440081\pi$
$240$	0	0
$241$	900033.	0.998196	0.499098	−	0.866546i	$-0.333665\pi$
$241$	900033.	0.998196	0.499098	+	0.866546i	$0.333665\pi$
$242$	0	0
$243$	−81007.6	−0.0880055
$244$	0	0
$245$	0	0
$246$	0	0
$247$	2.74389e6i	2.86171i
$248$	0	0
$249$	47542.8	0.0485944
$250$	0	0
$251$	− 963725.i	− 0.965537i	−0.875748	−	0.482768i	$-0.839631\pi$
$251$	− 963725.i	− 0.965537i	0.875748	−	0.482768i	$-0.160369\pi$
$252$	0	0
$253$	−551093.	−0.541282
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 1.50847e6i	− 1.42463i	−0.701857	−	0.712317i	$-0.747646\pi$
$257$	− 1.50847e6i	− 1.42463i	0.701857	−	0.712317i	$-0.252354\pi$
$258$	0	0
$259$	1.85947e6i	1.72242i
$260$	0	0
$261$	− 940839.i	− 0.854898i
$262$	0	0
$263$	721202.i	0.642936i	0.946920	+	0.321468i	$0.104176\pi$
$263$	721202.i	0.642936i	−0.946920	+	0.321468i	$0.895824\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−17459.8	−0.0149886
$268$	0	0
$269$	− 1.43076e6i	− 1.20556i	−0.797909	−	0.602778i	$-0.794061\pi$
$269$	− 1.43076e6i	− 1.20556i	0.797909	−	0.602778i	$-0.205939\pi$
$270$	0	0
$271$	1.94059e6	1.60514	0.802568	−	0.596561i	$-0.203467\pi$
$271$	1.94059e6	1.60514	0.802568	+	0.596561i	$0.203467\pi$
$272$	0	0
$273$	85338.3i	0.0693007i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	1.71081e6	1.33968	0.669842	−	0.742504i	$-0.266362\pi$
$277$	1.71081e6	1.33968	0.669842	+	0.742504i	$0.266362\pi$
$278$	0	0
$279$	1.25836e6	0.967822
$280$	0	0
$281$	1.44573e6	1.09225	0.546126	−	0.837703i	$-0.316102\pi$
$281$	1.44573e6	1.09225	0.546126	+	0.837703i	$0.316102\pi$
$282$	0	0
$283$	1.07186e6	0.795555	0.397777	−	0.917482i	$-0.369782\pi$
$283$	1.07186e6	0.795555	0.397777	+	0.917482i	$0.369782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 744755.i	− 0.533714i
$288$	0	0
$289$	−2.99628e6	−2.11027
$290$	0	0
$291$	− 10410.7i	− 0.00720688i
$292$	0	0
$293$	176850.	0.120347	0.0601735	−	0.998188i	$-0.480835\pi$
$293$	176850.	0.120347	0.0601735	+	0.998188i	$0.480835\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	79734.3i	0.0524510i
$298$	0	0
$299$	1.46311e6i	0.946455i
$300$	0	0
$301$	− 848097.i	− 0.539547i
$302$	0	0
$303$	24132.4i	0.0151006i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2.76503e6	1.67438	0.837189	−	0.546914i	$-0.184198\pi$
$307$	2.76503e6	1.67438	0.837189	+	0.546914i	$0.184198\pi$
$308$	0	0
$309$	17534.8i	0.0104473i
$310$	0	0
$311$	−748055.	−0.438564	−0.219282	−	0.975662i	$-0.570371\pi$
$311$	−748055.	−0.438564	−0.219282	+	0.975662i	$0.570371\pi$
$312$	0	0
$313$	2.84259e6i	1.64003i	0.572340	+	0.820017i	$0.306036\pi$
$313$	2.84259e6i	1.64003i	−0.572340	+	0.820017i	$0.693964\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	97492.6	0.0544908	0.0272454	−	0.999629i	$-0.491326\pi$
$317$	97492.6	0.0544908	0.0272454	+	0.999629i	$0.491326\pi$
$318$	0	0
$319$	−1.38928e6	−0.764384
$320$	0	0
$321$	−69327.5	−0.0375528
$322$	0	0
$323$	6.05802e6	3.23091
$324$	0	0
$325$	0	0
$326$	0	0
$327$	− 67033.4i	− 0.0346674i
$328$	0	0
$329$	2.44927e6	1.24752
$330$	0	0
$331$	2.71471e6i	1.36193i	0.732317	+	0.680964i	$0.238439\pi$
$331$	2.71471e6i	1.36193i	−0.732317	+	0.680964i	$0.761561\pi$
$332$	0	0
$333$	2.30529e6	1.13924
$334$	0	0
$335$	0	0
$336$	0	0
$337$	495886.i	0.237852i	0.992903	+	0.118926i	$0.0379452\pi$
$337$	495886.i	0.237852i	−0.992903	+	0.118926i	$0.962055\pi$
$338$	0	0
$339$	27180.0i	0.0128455i
$340$	0	0
$341$	− 1.85814e6i	− 0.865353i
$342$	0	0
$343$	927903.i	0.425860i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−3.14504e6	−1.40217	−0.701087	−	0.713076i	$-0.747302\pi$
$347$	−3.14504e6	−1.40217	−0.701087	+	0.713076i	$0.747302\pi$
$348$	0	0
$349$	2.62573e6i	1.15395i	0.816762	+	0.576975i	$0.195767\pi$
$349$	2.62573e6i	1.15395i	−0.816762	+	0.576975i	$0.804233\pi$
$350$	0	0
$351$	211689.	0.0917129
$352$	0	0
$353$	536996.i	0.229369i	0.993402	+	0.114684i	$0.0365857\pi$
$353$	536996.i	0.229369i	−0.993402	+	0.114684i	$0.963414\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	188412.	0.0782415
$358$	0	0
$359$	−369981.	−0.151511	−0.0757553	−	0.997126i	$-0.524137\pi$
$359$	−369981.	−0.151511	−0.0757553	+	0.997126i	$0.524137\pi$
$360$	0	0
$361$	−5.83425e6	−2.35623
$362$	0	0
$363$	−14888.0	−0.00593021
$364$	0	0
$365$	0	0
$366$	0	0
$367$	1.41712e6i	0.549214i	0.961557	+	0.274607i	$0.0885478\pi$
$367$	1.41712e6i	0.549214i	−0.961557	+	0.274607i	$0.911452\pi$
$368$	0	0
$369$	−923315.	−0.353008
$370$	0	0
$371$	− 4.91072e6i	− 1.85230i
$372$	0	0
$373$	−2.16053e6	−0.804060	−0.402030	−	0.915626i	$-0.631695\pi$
$373$	−2.16053e6	−0.804060	−0.402030	+	0.915626i	$0.631695\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.68843e6i	1.33656i
$378$	0	0
$379$	− 1.60069e6i	− 0.572412i	−0.958168	−	0.286206i	$-0.907606\pi$
$379$	− 1.60069e6i	− 0.572412i	0.958168	−	0.286206i	$-0.0923942\pi$
$380$	0	0
$381$	6062.84i	0.00213975i
$382$	0	0
$383$	− 3.07943e6i	− 1.07269i	−0.844000	−	0.536344i	$-0.819805\pi$
$383$	− 3.07943e6i	− 1.07269i	0.844000	−	0.536344i	$-0.180195\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−1.05143e6	−0.356866
$388$	0	0
$389$	− 2.71294e6i	− 0.909006i	−0.890745	−	0.454503i	$-0.849817\pi$
$389$	− 2.71294e6i	− 0.909006i	0.890745	−	0.454503i	$-0.150183\pi$
$390$	0	0
$391$	3.23029e6	1.06856
$392$	0	0
$393$	− 13475.8i	− 0.00440122i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	4.05795e6	1.29220	0.646101	−	0.763252i	$-0.276399\pi$
$397$	4.05795e6	1.29220	0.646101	+	0.763252i	$0.276399\pi$
$398$	0	0
$399$	−258462.	−0.0812762
$400$	0	0
$401$	5.22811e6	1.62362	0.811809	−	0.583923i	$-0.198483\pi$
$401$	5.22811e6	1.62362	0.811809	+	0.583923i	$0.198483\pi$
$402$	0	0
$403$	−4.93324e6	−1.51311
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 3.40407e6i	− 1.01862i
$408$	0	0
$409$	1.89554e6	0.560306	0.280153	−	0.959955i	$-0.409615\pi$
$409$	1.89554e6	0.560306	0.280153	+	0.959955i	$0.409615\pi$
$410$	0	0
$411$	− 103843.i	− 0.0303229i
$412$	0	0
$413$	−3.61394e6	−1.04257
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 94341.5i	− 0.0265682i
$418$	0	0
$419$	663430.i	0.184612i	0.995731	+	0.0923060i	$0.0294238\pi$
$419$	663430.i	0.184612i	−0.995731	+	0.0923060i	$0.970576\pi$
$420$	0	0
$421$	− 1.14330e6i	− 0.314381i	−0.987568	−	0.157190i	$-0.949756\pi$
$421$	− 1.14330e6i	− 0.314381i	0.987568	−	0.157190i	$-0.0502436\pi$
$422$	0	0
$423$	− 3.03650e6i	− 0.825130i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3.42494e6	0.909040
$428$	0	0
$429$	− 156226.i	− 0.0409837i
$430$	0	0
$431$	−559165.	−0.144993	−0.0724964	−	0.997369i	$-0.523097\pi$
$431$	−559165.	−0.144993	−0.0724964	+	0.997369i	$0.523097\pi$
$432$	0	0
$433$	− 4.04780e6i	− 1.03753i	−0.854918	−	0.518764i	$-0.826392\pi$
$433$	− 4.04780e6i	− 1.03753i	0.854918	−	0.518764i	$-0.173608\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.43129e6	−1.11001
$438$	0	0
$439$	−4.17879e6	−1.03488	−0.517439	−	0.855720i	$-0.673114\pi$
$439$	−4.17879e6	−1.03488	−0.517439	+	0.855720i	$0.673114\pi$
$440$	0	0
$441$	5.23095e6	1.28081
$442$	0	0
$443$	−6.19142e6	−1.49893	−0.749464	−	0.662045i	$-0.769689\pi$
$443$	−6.19142e6	−1.49893	−0.749464	+	0.662045i	$0.769689\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 126386.i	− 0.0299178i
$448$	0	0
$449$	3.11618e6	0.729469	0.364735	−	0.931111i	$-0.381160\pi$
$449$	3.11618e6	0.729469	0.364735	+	0.931111i	$0.381160\pi$
$450$	0	0
$451$	1.36340e6i	0.315633i
$452$	0	0
$453$	−40699.2	−0.00931837
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 4.44694e6i	− 0.996026i	−0.867170	−	0.498013i	$-0.834063\pi$
$457$	− 4.44694e6i	− 0.996026i	0.867170	−	0.498013i	$-0.165937\pi$
$458$	0	0
$459$	− 467371.i	− 0.103545i
$460$	0	0
$461$	7.51985e6i	1.64800i	0.566590	+	0.824000i	$0.308262\pi$
$461$	7.51985e6i	1.64800i	−0.566590	+	0.824000i	$0.691738\pi$
$462$	0	0
$463$	4.34253e6i	0.941434i	0.882284	+	0.470717i	$0.156005\pi$
$463$	4.34253e6i	0.941434i	−0.882284	+	0.470717i	$0.843995\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	890290.	0.188903	0.0944515	−	0.995529i	$-0.469890\pi$
$467$	890290.	0.188903	0.0944515	+	0.995529i	$0.469890\pi$
$468$	0	0
$469$	− 1.46696e6i	− 0.307955i
$470$	0	0
$471$	−125413.	−0.0260489
$472$	0	0
$473$	1.55258e6i	0.319082i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−6.08810e6	−1.22514
$478$	0	0
$479$	3.93305e6	0.783233	0.391617	−	0.920128i	$-0.371916\pi$
$479$	3.93305e6	0.783233	0.391617	+	0.920128i	$0.371916\pi$
$480$	0	0
$481$	−9.03757e6	−1.78110
$482$	0	0
$483$	−137818.	−0.0268806
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 3.91119e6i	− 0.747286i	−0.927573	−	0.373643i	$-0.878109\pi$
$487$	− 3.91119e6i	− 0.747286i	0.927573	−	0.373643i	$-0.121891\pi$
$488$	0	0
$489$	−10310.1	−0.00194981
$490$	0	0
$491$	4.43336e6i	0.829907i	0.909843	+	0.414954i	$0.136202\pi$
$491$	4.43336e6i	0.829907i	−0.909843	+	0.414954i	$0.863798\pi$
$492$	0	0
$493$	8.14339e6	1.50900
$494$	0	0
$495$	0	0
$496$	0	0
$497$	206940.i	0.0375797i
$498$	0	0
$499$	− 4.84134e6i	− 0.870391i	−0.900336	−	0.435196i	$-0.856679\pi$
$499$	− 4.84134e6i	− 0.870391i	0.900336	−	0.435196i	$-0.143321\pi$
$500$	0	0
$501$	− 1930.18i	0 0.000343561i
$502$	0	0
$503$	1.68379e6i	0.296735i	0.988932	+	0.148367i	$0.0474018\pi$
$503$	1.68379e6i	0.296735i	−0.988932	+	0.148367i	$0.952598\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	−244786.	−0.0422927
$508$	0	0
$509$	9.81220e6i	1.67869i	0.543595	+	0.839347i	$0.317063\pi$
$509$	9.81220e6i	1.67869i	−0.543595	+	0.839347i	$0.682937\pi$
$510$	0	0
$511$	−1.08271e7	−1.83426
$512$	0	0
$513$	641136.i	0.107561i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−4.48380e6	−0.737768
$518$	0	0
$519$	−110043.	−0.0179326
$520$	0	0
$521$	−2.02601e6	−0.327000	−0.163500	−	0.986543i	$-0.552278\pi$
$521$	−2.02601e6	−0.327000	−0.163500	+	0.986543i	$0.552278\pi$
$522$	0	0
$523$	402702.	0.0643768	0.0321884	−	0.999482i	$-0.489752\pi$
$523$	402702.	0.0643768	0.0321884	+	0.999482i	$0.489752\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.08917e7i	1.70832i
$528$	0	0
$529$	4.07347e6	0.632886
$530$	0	0
$531$	4.48041e6i	0.689575i
$532$	0	0
$533$	3.61973e6	0.551897
$534$	0	0
$535$	0	0
$536$	0	0
$537$	112394.i	0.0168193i
$538$	0	0
$539$	− 7.72421e6i	− 1.14520i
$540$	0	0
$541$	8.47881e6i	1.24549i	0.782423	+	0.622747i	$0.213984\pi$
$541$	8.47881e6i	1.24549i	−0.782423	+	0.622747i	$0.786016\pi$
$542$	0	0
$543$	− 7594.57i	− 0.00110536i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−7.10206e6	−1.01488	−0.507441	−	0.861686i	$-0.669409\pi$
$547$	−7.10206e6	−1.01488	−0.507441	+	0.861686i	$0.669409\pi$
$548$	0	0
$549$	− 4.24609e6i	− 0.601255i
$550$	0	0
$551$	−1.11710e7	−1.56752
$552$	0	0
$553$	− 1.75053e7i	− 2.43420i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	2.42337e6	0.330965	0.165482	−	0.986213i	$-0.447082\pi$
$557$	2.42337e6	0.330965	0.165482	+	0.986213i	$0.447082\pi$
$558$	0	0
$559$	4.12200e6	0.557929
$560$	0	0
$561$	−344919.	−0.0462712
$562$	0	0
$563$	−1.03666e7	−1.37837	−0.689184	−	0.724586i	$-0.742031\pi$
$563$	−1.03666e7	−1.37837	−0.689184	+	0.724586i	$0.742031\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 1.15341e7i	− 1.50669i
$568$	0	0
$569$	−1.33323e7	−1.72633	−0.863165	−	0.504921i	$-0.831521\pi$
$569$	−1.33323e7	−1.72633	−0.863165	+	0.504921i	$0.831521\pi$
$570$	0	0
$571$	− 4.77970e6i	− 0.613494i	−0.951791	−	0.306747i	$-0.900760\pi$
$571$	− 4.77970e6i	− 0.613494i	0.951791	−	0.306747i	$-0.0992405\pi$
$572$	0	0
$573$	−254493.	−0.0323809
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 5.73641e6i	− 0.717300i	−0.933472	−	0.358650i	$-0.883237\pi$
$577$	− 5.73641e6i	− 0.717300i	0.933472	−	0.358650i	$-0.116763\pi$
$578$	0	0
$579$	181823.i	0.0225400i
$580$	0	0
$581$	2.03371e7i	2.49947i
$582$	0	0
$583$	8.98990e6i	1.09543i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.32016e6	−0.637279	−0.318639	−	0.947876i	$-0.603226\pi$
$587$	−5.32016e6	−0.637279	−0.318639	+	0.947876i	$0.603226\pi$
$588$	0	0
$589$	− 1.49412e7i	− 1.77458i
$590$	0	0
$591$	−330655.	−0.0389409
$592$	0	0
$593$	828623.i	0.0967654i	0.998829	+	0.0483827i	$0.0154067\pi$
$593$	828623.i	0.0967654i	−0.998829	+	0.0483827i	$0.984593\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	59465.9	0.00682861
$598$	0	0
$599$	5.87196e6	0.668676	0.334338	−	0.942453i	$-0.391487\pi$
$599$	5.87196e6	0.668676	0.334338	+	0.942453i	$0.391487\pi$
$600$	0	0
$601$	3.38777e6	0.382584	0.191292	−	0.981533i	$-0.438732\pi$
$601$	3.38777e6	0.382584	0.191292	+	0.981533i	$0.438732\pi$
$602$	0	0
$603$	−1.81868e6	−0.203686
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 4.02635e6i	− 0.443547i	−0.975098	−	0.221774i	$-0.928815\pi$
$607$	− 4.02635e6i	− 0.443547i	0.975098	−	0.221774i	$-0.0711846\pi$
$608$	0	0
$609$	−347432.	−0.0379601
$610$	0	0
$611$	1.19042e7i	1.29002i
$612$	0	0
$613$	−5.30633e6	−0.570352	−0.285176	−	0.958475i	$-0.592052\pi$
$613$	−5.30633e6	−0.570352	−0.285176	+	0.958475i	$0.592052\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.56854e6i	− 1.01189i	−0.862566	−	0.505944i	$-0.831144\pi$
$617$	− 9.56854e6i	− 1.01189i	0.862566	−	0.505944i	$-0.168856\pi$
$618$	0	0
$619$	6.58401e6i	0.690659i	0.938481	+	0.345330i	$0.112233\pi$
$619$	6.58401e6i	0.690659i	−0.938481	+	0.345330i	$0.887767\pi$
$620$	0	0
$621$	341870.i	0.0355739i
$622$	0	0
$623$	− 7.46866e6i	− 0.770943i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	473157.	0.0480659
$628$	0	0
$629$	1.99533e7i	2.01089i
$630$	0	0
$631$	1.34839e7	1.34816	0.674079	−	0.738659i	$-0.264541\pi$
$631$	1.34839e7	1.34816	0.674079	+	0.738659i	$0.264541\pi$
$632$	0	0
$633$	278752.i	0.0276508i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−2.05072e7	−2.00244
$638$	0	0
$639$	256555.	0.0248559
$640$	0	0
$641$	−7.72929e6	−0.743010	−0.371505	−	0.928431i	$-0.621158\pi$
$641$	−7.72929e6	−0.743010	−0.371505	+	0.928431i	$0.621158\pi$
$642$	0	0
$643$	−9.41828e6	−0.898347	−0.449174	−	0.893444i	$-0.648282\pi$
$643$	−9.41828e6	−0.898347	−0.449174	+	0.893444i	$0.648282\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.44122e6i	− 0.323185i	−0.986858	−	0.161593i	$-0.948337\pi$
$647$	− 3.44122e6i	− 0.323185i	0.986858	−	0.161593i	$-0.0516631\pi$
$648$	0	0
$649$	6.61593e6	0.616566
$650$	0	0
$651$	− 464687.i	− 0.0429743i
$652$	0	0
$653$	−1.39531e7	−1.28053	−0.640263	−	0.768155i	$-0.721175\pi$
$653$	−1.39531e7	−1.28053	−0.640263	+	0.768155i	$0.721175\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	1.34230e7i	1.21321i
$658$	0	0
$659$	− 1.16197e7i	− 1.04227i	−0.853474	−	0.521135i	$-0.825509\pi$
$659$	− 1.16197e7i	− 1.04227i	0.853474	−	0.521135i	$-0.174491\pi$
$660$	0	0
$661$	2.02640e6i	0.180394i	0.995924	+	0.0901970i	$0.0287497\pi$
$661$	2.02640e6i	0.180394i	−0.995924	+	0.0901970i	$0.971250\pi$
$662$	0	0
$663$	915737.i	0.0809071i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−5.95668e6	−0.518429
$668$	0	0
$669$	10319.6i	0 0.000891447i
$670$	0	0
$671$	−6.26993e6	−0.537596
$672$	0	0
$673$	5.84126e6i	0.497129i	0.968615	+	0.248564i	$0.0799587\pi$
$673$	5.84126e6i	0.497129i	−0.968615	+	0.248564i	$0.920041\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.28265e7	1.07557	0.537784	−	0.843083i	$-0.319262\pi$
$677$	1.28265e7	1.07557	0.537784	+	0.843083i	$0.319262\pi$
$678$	0	0
$679$	4.45331e6	0.370688
$680$	0	0
$681$	69079.6	0.00570797
$682$	0	0
$683$	−7.24686e6	−0.594426	−0.297213	−	0.954811i	$-0.596057\pi$
$683$	−7.24686e6	−0.594426	−0.297213	+	0.954811i	$0.596057\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 331695.i	− 0.0268131i
$688$	0	0
$689$	2.38676e7	1.91540
$690$	0	0
$691$	2.08754e7i	1.66318i	0.555388	+	0.831592i	$0.312570\pi$
$691$	2.08754e7i	1.66318i	−0.555388	+	0.831592i	$0.687430\pi$
$692$	0	0
$693$	−1.70463e7	−1.34834
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 7.99171e6i	− 0.623100i
$698$	0	0
$699$	− 154083.i	− 0.0119279i
$700$	0	0
$701$	− 1.73225e7i	− 1.33142i	−0.746209	−	0.665712i	$-0.768128\pi$
$701$	− 1.73225e7i	− 1.33142i	0.746209	−	0.665712i	$-0.231872\pi$
$702$	0	0
$703$	− 2.73718e7i	− 2.08889i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−1.03229e7	−0.776703
$708$	0	0
$709$	− 1.99424e7i	− 1.48991i	−0.667113	−	0.744956i	$-0.732470\pi$
$709$	− 1.99424e7i	− 1.48991i	0.667113	−	0.744956i	$-0.267530\pi$
$710$	0	0
$711$	−2.17023e7	−1.61002
$712$	0	0
$713$	− 7.96700e6i	− 0.586909i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−151308.	−0.0109917
$718$	0	0
$719$	−2.44269e7	−1.76217	−0.881083	−	0.472962i	$-0.843185\pi$
$719$	−2.44269e7	−1.76217	−0.881083	+	0.472962i	$0.843185\pi$
$720$	0	0
$721$	−7.50075e6	−0.537361
$722$	0	0
$723$	−412050.	−0.0293160
$724$	0	0
$725$	0	0
$726$	0	0
$727$	1.52812e7i	1.07231i	0.844119	+	0.536155i	$0.180124\pi$
$727$	1.52812e7i	1.07231i	−0.844119	+	0.536155i	$0.819876\pi$
$728$	0	0
$729$	−1.42747e7	−0.994829
$730$	0	0
$731$	− 9.10064e6i	− 0.629910i
$732$	0	0
$733$	−7.69123e6	−0.528732	−0.264366	−	0.964422i	$-0.585163\pi$
$733$	−7.69123e6	−0.528732	−0.264366	+	0.964422i	$0.585163\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.68552e6i	0.182121i
$738$	0	0
$739$	− 7.32363e6i	− 0.493305i	−0.969104	−	0.246652i	$-0.920669\pi$
$739$	− 7.32363e6i	− 0.493305i	0.969104	−	0.246652i	$-0.0793306\pi$
$740$	0	0
$741$	− 1.25620e6i	− 0.0840453i
$742$	0	0
$743$	− 1.61613e7i	− 1.07400i	−0.843582	−	0.537001i	$-0.819557\pi$
$743$	− 1.61613e7i	− 1.07400i	0.843582	−	0.537001i	$-0.180443\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	2.52130e7	1.65319
$748$	0	0
$749$	− 2.96557e7i	− 1.93154i
$750$	0	0
$751$	3.01764e7	1.95240	0.976199	−	0.216879i	$-0.0695876\pi$
$751$	3.01764e7	1.95240	0.976199	+	0.216879i	$0.0695876\pi$
$752$	0	0
$753$	441209.i	0.0283568i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−1.40193e7	−0.889171	−0.444586	−	0.895736i	$-0.646649\pi$
$757$	−1.40193e7	−0.889171	−0.444586	+	0.895736i	$0.646649\pi$
$758$	0	0
$759$	252299.	0.0158969
$760$	0	0
$761$	−1.39222e7	−0.871456	−0.435728	−	0.900078i	$-0.643509\pi$
$761$	−1.39222e7	−0.871456	−0.435728	+	0.900078i	$0.643509\pi$
$762$	0	0
$763$	2.86744e7	1.78313
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 1.75648e7i	− 1.07809i
$768$	0	0
$769$	1.41369e7	0.862063	0.431031	−	0.902337i	$-0.358150\pi$
$769$	1.41369e7	0.862063	0.431031	+	0.902337i	$0.358150\pi$
$770$	0	0
$771$	690602.i	0.0418400i
$772$	0	0
$773$	1.39310e7	0.838558	0.419279	−	0.907858i	$-0.362283\pi$
$773$	1.39310e7	0.838558	0.419279	+	0.907858i	$0.362283\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	− 851295.i	− 0.0505857i
$778$	0	0
$779$	1.09630e7i	0.647268i
$780$	0	0
$781$	− 378838.i	− 0.0222242i
$782$	0	0
$783$	861835.i	0.0502366i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−9.60345e6	−0.552701	−0.276351	−	0.961057i	$-0.589125\pi$
$787$	−9.60345e6	−0.552701	−0.276351	+	0.961057i	$0.589125\pi$
$788$	0	0
$789$	− 330178.i	− 0.0188823i
$790$	0	0
$791$	−1.16266e7	−0.660712
$792$	0	0
$793$	1.66462e7i	0.940010i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	1.97451e7	1.10107	0.550534	−	0.834813i	$-0.314424\pi$
$797$	1.97451e7	1.10107	0.550534	+	0.834813i	$0.314424\pi$
$798$	0	0
$799$	2.62823e7	1.45645
$800$	0	0
$801$	−9.25932e6	−0.509915
$802$	0	0
$803$	1.98209e7	1.08476
$804$	0	0
$805$	0	0
$806$	0	0
$807$	655027.i	0.0354059i
$808$	0	0
$809$	−1.73178e7	−0.930296	−0.465148	−	0.885233i	$-0.653999\pi$
$809$	−1.73178e7	−0.930296	−0.465148	+	0.885233i	$0.653999\pi$
$810$	0	0
$811$	3.16102e7i	1.68762i	0.536642	+	0.843810i	$0.319693\pi$
$811$	3.16102e7i	1.68762i	−0.536642	+	0.843810i	$0.680307\pi$
$812$	0	0
$813$	−888436.	−0.0471411
$814$	0	0
$815$	0	0
$816$	0	0
$817$	1.24842e7i	0.654342i
$818$	0	0
$819$	4.52568e7i	2.35762i
$820$	0	0
$821$	1.86667e7i	0.966519i	0.875477	+	0.483260i	$0.160547\pi$
$821$	1.86667e7i	0.966519i	−0.875477	+	0.483260i	$0.839453\pi$
$822$	0	0
$823$	3.88081e7i	1.99720i	0.0528555	+	0.998602i	$0.483168\pi$
$823$	3.88081e7i	1.99720i	−0.0528555	+	0.998602i	$0.516832\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.98829e7	−1.51935	−0.759677	−	0.650301i	$-0.774643\pi$
$827$	−2.98829e7	−1.51935	−0.759677	+	0.650301i	$0.774643\pi$
$828$	0	0
$829$	2.10001e7i	1.06129i	0.847594	+	0.530645i	$0.178050\pi$
$829$	2.10001e7i	1.06129i	−0.847594	+	0.530645i	$0.821950\pi$
$830$	0	0
$831$	−783237.	−0.0393451
$832$	0	0
$833$	4.52763e7i	2.26078i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.15270e6	−0.0568724
$838$	0	0
$839$	−7.61546e6	−0.373501	−0.186750	−	0.982407i	$-0.559796\pi$
$839$	−7.61546e6	−0.373501	−0.186750	+	0.982407i	$0.559796\pi$
$840$	0	0
$841$	5.49468e6	0.267888
$842$	0	0
$843$	−661881.	−0.0320783
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 6.36854e6i	− 0.305022i
$848$	0	0
$849$	−490713.	−0.0233646
$850$	0	0
$851$	− 1.45953e7i	− 0.690860i
$852$	0	0
$853$	9.56553e6	0.450128	0.225064	−	0.974344i	$-0.427741\pi$
$853$	9.56553e6	0.450128	0.225064	+	0.974344i	$0.427741\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 2.34556e7i	− 1.09092i	−0.838136	−	0.545462i	$-0.816354\pi$
$857$	− 2.34556e7i	− 1.09092i	0.838136	−	0.545462i	$-0.183646\pi$
$858$	0	0
$859$	− 9.13964e6i	− 0.422616i	−0.977420	−	0.211308i	$-0.932228\pi$
$859$	− 9.13964e6i	− 0.422616i	0.977420	−	0.211308i	$-0.0677723\pi$
$860$	0	0
$861$	340961.i	0.0156746i
$862$	0	0
$863$	− 1.04151e7i	− 0.476034i	−0.971261	−	0.238017i	$-0.923503\pi$
$863$	− 1.04151e7i	− 0.476034i	0.971261	−	0.238017i	$-0.0764974\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	1.37175e6	0.0619764
$868$	0	0
$869$	3.20464e7i	1.43956i
$870$	0	0
$871$	7.12987e6	0.318446
$872$	0	0
$873$	− 5.52102e6i	− 0.245179i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	2.02537e7	0.889210	0.444605	−	0.895727i	$-0.353344\pi$
$877$	2.02537e7	0.889210	0.444605	+	0.895727i	$0.353344\pi$
$878$	0	0
$879$	−80964.7	−0.00353447
$880$	0	0
$881$	2.74936e7	1.19341	0.596707	−	0.802459i	$-0.296475\pi$
$881$	2.74936e7	1.19341	0.596707	+	0.802459i	$0.296475\pi$
$882$	0	0
$883$	3.50971e7	1.51485	0.757425	−	0.652922i	$-0.226457\pi$
$883$	3.50971e7	1.51485	0.757425	+	0.652922i	$0.226457\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.77301e7i	0.756660i	0.925671	+	0.378330i	$0.123502\pi$
$887$	1.77301e7i	0.756660i	−0.925671	+	0.378330i	$0.876498\pi$
$888$	0	0
$889$	−2.59346e6	−0.110059
$890$	0	0
$891$	2.11151e7i	0.891042i
$892$	0	0
$893$	−3.60538e7	−1.51294
$894$	0	0
$895$	0	0
$896$	0	0
$897$	− 669837.i	− 0.0277964i
$898$	0	0
$899$	− 2.00844e7i	− 0.828818i
$900$	0	0
$901$	− 5.26953e7i	− 2.16252i
$902$	0	0
$903$	388273.i	0.0158459i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1.04686e7	0.422542	0.211271	−	0.977428i	$-0.432240\pi$
$907$	1.04686e7	0.422542	0.211271	+	0.977428i	$0.432240\pi$
$908$	0	0
$909$	1.27979e7i	0.513725i
$910$	0	0
$911$	4.87147e7	1.94475	0.972376	−	0.233420i	$-0.0749918\pi$
$911$	4.87147e7	1.94475	0.972376	+	0.233420i	$0.0749918\pi$
$912$	0	0
$913$	− 3.72304e7i	− 1.47816i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	5.76445e6	0.226378
$918$	0	0
$919$	−2.63015e7	−1.02729	−0.513643	−	0.858004i	$-0.671704\pi$
$919$	−2.63015e7	−1.02729	−0.513643	+	0.858004i	$0.671704\pi$
$920$	0	0
$921$	−1.26587e6	−0.0491747
$922$	0	0
$923$	−1.00579e6	−0.0388600
$924$	0	0
$925$	0	0
$926$	0	0
$927$	9.29911e6i	0.355420i
$928$	0	0
$929$	−4.91045e7	−1.86673	−0.933366	−	0.358925i	$-0.883143\pi$
$929$	−4.91045e7	−1.86673	−0.933366	+	0.358925i	$0.883143\pi$
$930$	0	0
$931$	− 6.21096e7i	− 2.34847i
$932$	0	0
$933$	342472.	0.0128802
$934$	0	0
$935$	0	0
$936$	0	0
$937$	3.24439e7i	1.20721i	0.797282	+	0.603607i	$0.206271\pi$
$937$	3.24439e7i	1.20721i	−0.797282	+	0.603607i	$0.793729\pi$
$938$	0	0
$939$	− 1.30138e6i	− 0.0481660i
$940$	0	0
$941$	− 5.58017e6i	− 0.205435i	−0.994711	−	0.102717i	$-0.967246\pi$
$941$	− 5.58017e6i	− 0.205435i	0.994711	−	0.102717i	$-0.0327537\pi$
$942$	0	0
$943$	5.84573e6i	0.214072i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	2.29821e7	0.832748	0.416374	−	0.909193i	$-0.363301\pi$
$947$	2.29821e7	0.832748	0.416374	+	0.909193i	$0.363301\pi$
$948$	0	0
$949$	− 5.26231e7i	− 1.89675i
$950$	0	0
$951$	−44633.7	−0.00160034
$952$	0	0
$953$	− 2.98158e7i	− 1.06344i	−0.846920	−	0.531721i	$-0.821545\pi$
$953$	− 2.98158e7i	− 1.06344i	0.846920	−	0.531721i	$-0.178455\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	636034.	0.0224492
$958$	0	0
$959$	4.44200e7	1.55967
$960$	0	0
$961$	−1.76649e6	−0.0617026
$962$	0	0
$963$	−3.67659e7	−1.27756
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 4.13822e6i	− 0.142314i	−0.997465	−	0.0711570i	$-0.977331\pi$
$967$	− 4.13822e6i	− 0.142314i	0.997465	−	0.0711570i	$-0.0226691\pi$
$968$	0	0
$969$	−2.77346e6	−0.0948884
$970$	0	0
$971$	− 5.43256e7i	− 1.84908i	−0.381081	−	0.924542i	$-0.624448\pi$
$971$	− 5.43256e7i	− 1.84908i	0.381081	−	0.924542i	$-0.375552\pi$
$972$	0	0
$973$	4.03558e7	1.36655
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 2.00334e7i	− 0.671456i	−0.941959	−	0.335728i	$-0.891018\pi$
$977$	− 2.00334e7i	− 0.671456i	0.941959	−	0.335728i	$-0.108982\pi$
$978$	0	0
$979$	1.36726e7i	0.455927i
$980$	0	0
$981$	− 3.55493e7i	− 1.17939i
$982$	0	0
$983$	− 2.92947e6i	− 0.0966953i	−0.998831	−	0.0483476i	$-0.984604\pi$
$983$	− 2.92947e6i	− 0.0966953i	0.998831	−	0.0483476i	$-0.0153955\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1.12132e6	−0.0366383
$988$	0	0
$989$	6.65688e6i	0.216411i
$990$	0	0
$991$	−5.27467e7	−1.70613	−0.853064	−	0.521807i	$-0.825258\pi$
$991$	−5.27467e7	−1.70613	−0.853064	+	0.521807i	$0.825258\pi$
$992$	0	0
$993$	− 1.24284e6i	− 0.0399984i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−1.04084e7	−0.331625	−0.165812	−	0.986157i	$-0.553025\pi$
$997$	−1.04084e7	−0.331625	−0.165812	+	0.986157i	$0.553025\pi$
$998$	0	0
$999$	−2.11171e6	−0.0669454

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.6.f.d.49.20		40
4.3	odd	2		200.6.f.d.149.31		40
5.2	odd	4		800.6.d.d.401.11		20
5.3	odd	4		800.6.d.b.401.10		20
5.4	even	2	inner	800.6.f.d.49.21		40
8.3	odd	2		200.6.f.d.149.9		40
8.5	even	2	inner	800.6.f.d.49.22		40
20.3	even	4		200.6.d.d.101.5	yes	20
20.7	even	4		200.6.d.c.101.16	yes	20
20.19	odd	2		200.6.f.d.149.10		40
40.3	even	4		200.6.d.d.101.6	yes	20
40.13	odd	4		800.6.d.b.401.11		20
40.19	odd	2		200.6.f.d.149.32		40
40.27	even	4		200.6.d.c.101.15	✓	20
40.29	even	2	inner	800.6.f.d.49.19		40
40.37	odd	4		800.6.d.d.401.10		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
200.6.d.c.101.15	✓	20	40.27	even	4
200.6.d.c.101.16	yes	20	20.7	even	4
200.6.d.d.101.5	yes	20	20.3	even	4
200.6.d.d.101.6	yes	20	40.3	even	4
200.6.f.d.149.9		40	8.3	odd	2
200.6.f.d.149.10		40	20.19	odd	2
200.6.f.d.149.31		40	4.3	odd	2
200.6.f.d.149.32		40	40.19	odd	2
800.6.d.b.401.10		20	5.3	odd	4
800.6.d.b.401.11		20	40.13	odd	4
800.6.d.d.401.10		20	40.37	odd	4
800.6.d.d.401.11		20	5.2	odd	4
800.6.f.d.49.19		40	40.29	even	2	inner
800.6.f.d.49.20		40	1.1	even	1	trivial
800.6.f.d.49.21		40	5.4	even	2	inner
800.6.f.d.49.22		40	8.5	even	2	inner

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:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	800.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-0.457817 q^{3} -195.837i q^{7} -242.790 q^{9} +358.513i q^{11} +951.826 q^{13} -2101.46i q^{17} +2882.77i q^{19} +89.6574i q^{21} +1537.16i q^{23} +222.403 q^{27} +3875.11i q^{29} -5182.92 q^{31} -164.133i q^{33} -9494.97 q^{37} -435.762 q^{39} +3802.93 q^{41} +4330.63 q^{43} +12506.7i q^{47} -21545.1 q^{49} +962.084i q^{51} +25075.5 q^{53} -1319.78i q^{57} -18453.8i q^{59} +17488.7i q^{61} +47547.3i q^{63} +7490.73 q^{67} -703.739i q^{69} -1056.69 q^{71} -55286.5i q^{73} +70210.1 q^{77} +89387.0 q^{79} +58896.2 q^{81} -103847. q^{83} -1774.09i q^{87} +38137.1 q^{89} -186403. i q^{91} +2372.83 q^{93} +22739.9i q^{97} -87043.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 40 q + 3240 q^{9} - 14320 q^{31} + 44904 q^{39} - 11608 q^{41} - 125304 q^{49} + 15448 q^{71} - 15560 q^{79} + 193968 q^{81} + 6320 q^{89}+O(q^{100}) \) 40 * q + 3240 * q^9 - 14320 * q^31 + 44904 * q^39 - 11608 * q^41 - 125304 * q^49 + 15448 * q^71 - 15560 * q^79 + 193968 * q^81 + 6320 * q^89

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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