LMFDB - Embedded newform 800.5.e.e.399.8

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			399.8
Character	$\chi$	$=$	800.399
Dual form			800.5.e.e.399.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+15.2196i q^{3} +47.5956 q^{7} -150.635 q^{9} -84.0101 q^{11} -236.405 q^{13} -407.645i q^{17} +124.159 q^{19} +724.383i q^{21} -198.689 q^{23} -1059.81i q^{27} +1313.51i q^{29} -1243.21i q^{31} -1278.60i q^{33} +854.861 q^{37} -3597.98i q^{39} +3072.37 q^{41} -1108.03i q^{43} +284.880 q^{47} -135.663 q^{49} +6204.18 q^{51} -2964.13 q^{53} +1889.64i q^{57} +657.496 q^{59} -4209.41i q^{61} -7169.54 q^{63} -2607.14i q^{67} -3023.96i q^{69} -4545.99i q^{71} +822.870i q^{73} -3998.51 q^{77} +3047.75i q^{79} +3928.41 q^{81} -11276.1i q^{83} -19991.1 q^{87} +4387.83 q^{89} -11251.8 q^{91} +18921.0 q^{93} -4448.77i q^{97} +12654.8 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	15.2196i	1.69106i	0.533927	+	0.845531i	$0.320716\pi$
$3$	15.2196i	1.69106i	−0.533927	+	0.845531i	$0.679284\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	47.5956	0.971338	0.485669	−	0.874143i	$-0.338576\pi$
$7$	47.5956	0.971338	0.485669	+	0.874143i	$0.338576\pi$
$8$	0	0
$9$	−150.635	−1.85969
$10$	0	0
$11$	−84.0101	−0.694299	−0.347149	−	0.937810i	$-0.612850\pi$
$11$	−84.0101	−0.694299	−0.347149	+	0.937810i	$0.612850\pi$
$12$	0	0
$13$	−236.405	−1.39885	−0.699423	−	0.714708i	$-0.746560\pi$
$13$	−236.405	−1.39885	−0.699423	+	0.714708i	$0.746560\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 407.645i	− 1.41054i	−0.708940	−	0.705269i	$-0.750826\pi$
$17$	− 407.645i	− 1.41054i	0.708940	−	0.705269i	$-0.249174\pi$
$18$	0	0
$19$	124.159	0.343930	0.171965	−	0.985103i	$-0.444988\pi$
$19$	124.159	0.343930	0.171965	+	0.985103i	$0.444988\pi$
$20$	0	0
$21$	724.383i	1.64259i
$22$	0	0
$23$	−198.689	−0.375594	−0.187797	−	0.982208i	$-0.560135\pi$
$23$	−198.689	−0.375594	−0.187797	+	0.982208i	$0.560135\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1059.81i	− 1.45378i
$28$	0	0
$29$	1313.51i	1.56185i	0.624626	+	0.780924i	$0.285252\pi$
$29$	1313.51i	1.56185i	−0.624626	+	0.780924i	$0.714748\pi$
$30$	0	0
$31$	− 1243.21i	− 1.29366i	−0.762635	−	0.646829i	$-0.776095\pi$
$31$	− 1243.21i	− 1.29366i	0.762635	−	0.646829i	$-0.223905\pi$
$32$	0	0
$33$	− 1278.60i	− 1.17410i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	854.861	0.624442	0.312221	−	0.950009i	$-0.398927\pi$
$37$	854.861	0.624442	0.312221	+	0.950009i	$0.398927\pi$
$38$	0	0
$39$	− 3597.98i	− 2.36553i
$40$	0	0
$41$	3072.37	1.82771	0.913853	−	0.406045i	$-0.133092\pi$
$41$	3072.37	1.82771	0.913853	+	0.406045i	$0.133092\pi$
$42$	0	0
$43$	− 1108.03i	− 0.599259i	−0.954056	−	0.299629i	$-0.903137\pi$
$43$	− 1108.03i	− 0.599259i	0.954056	−	0.299629i	$-0.0968630\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	284.880	0.128963	0.0644816	−	0.997919i	$-0.479461\pi$
$47$	284.880	0.128963	0.0644816	+	0.997919i	$0.479461\pi$
$48$	0	0
$49$	−135.663	−0.0565029
$50$	0	0
$51$	6204.18	2.38530
$52$	0	0
$53$	−2964.13	−1.05523	−0.527613	−	0.849485i	$-0.676913\pi$
$53$	−2964.13	−1.05523	−0.527613	+	0.849485i	$0.676913\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	1889.64i	0.581606i
$58$	0	0
$59$	657.496	0.188881	0.0944406	−	0.995530i	$-0.469894\pi$
$59$	657.496	0.188881	0.0944406	+	0.995530i	$0.469894\pi$
$60$	0	0
$61$	− 4209.41i	− 1.13126i	−0.824660	−	0.565629i	$-0.808633\pi$
$61$	− 4209.41i	− 1.13126i	0.824660	−	0.565629i	$-0.191367\pi$
$62$	0	0
$63$	−7169.54	−1.80639
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 2607.14i	− 0.580785i	−0.956908	−	0.290393i	$-0.906214\pi$
$67$	− 2607.14i	− 0.580785i	0.956908	−	0.290393i	$-0.0937859\pi$
$68$	0	0
$69$	− 3023.96i	− 0.635152i
$70$	0	0
$71$	− 4545.99i	− 0.901804i	−0.892574	−	0.450902i	$-0.851102\pi$
$71$	− 4545.99i	− 0.901804i	0.892574	−	0.450902i	$-0.148898\pi$
$72$	0	0
$73$	822.870i	0.154414i	0.997015	+	0.0772068i	$0.0246002\pi$
$73$	822.870i	0.154414i	−0.997015	+	0.0772068i	$0.975400\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3998.51	−0.674398
$78$	0	0
$79$	3047.75i	0.488343i	0.969732	+	0.244172i	$0.0785160\pi$
$79$	3047.75i	0.488343i	−0.969732	+	0.244172i	$0.921484\pi$
$80$	0	0
$81$	3928.41	0.598751
$82$	0	0
$83$	− 11276.1i	− 1.63683i	−0.574627	−	0.818415i	$-0.694853\pi$
$83$	− 11276.1i	− 1.63683i	0.574627	−	0.818415i	$-0.305147\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−19991.1	−2.64118
$88$	0	0
$89$	4387.83	0.553949	0.276974	−	0.960877i	$-0.410668\pi$
$89$	4387.83	0.553949	0.276974	+	0.960877i	$0.410668\pi$
$90$	0	0
$91$	−11251.8	−1.35875
$92$	0	0
$93$	18921.0	2.18766
$94$	0	0
$95$	0	0
$96$	0	0
$97$	− 4448.77i	− 0.472821i	−0.971653	−	0.236410i	$-0.924029\pi$
$97$	− 4448.77i	− 0.472821i	0.971653	−	0.236410i	$-0.0759710\pi$
$98$	0	0
$99$	12654.8	1.29118
$100$	0	0
$101$	− 9105.46i	− 0.892605i	−0.894882	−	0.446302i	$-0.852741\pi$
$101$	− 9105.46i	− 0.892605i	0.894882	−	0.446302i	$-0.147259\pi$
$102$	0	0
$103$	11234.1	1.05892	0.529459	−	0.848336i	$-0.322395\pi$
$103$	11234.1	1.05892	0.529459	+	0.848336i	$0.322395\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1475.53i	− 0.128878i	−0.997922	−	0.0644391i	$-0.979474\pi$
$107$	− 1475.53i	− 0.128878i	0.997922	−	0.0644391i	$-0.0205258\pi$
$108$	0	0
$109$	11699.2i	0.984696i	0.870399	+	0.492348i	$0.163861\pi$
$109$	11699.2i	0.984696i	−0.870399	+	0.492348i	$0.836139\pi$
$110$	0	0
$111$	13010.6i	1.05597i
$112$	0	0
$113$	4107.55i	0.321681i	0.986980	+	0.160841i	$0.0514205\pi$
$113$	4107.55i	0.321681i	−0.986980	+	0.160841i	$0.948580\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	35610.8	2.60142
$118$	0	0
$119$	− 19402.1i	− 1.37011i
$120$	0	0
$121$	−7583.30	−0.517949
$122$	0	0
$123$	46760.2i	3.09076i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−1191.31	−0.0738613	−0.0369307	−	0.999318i	$-0.511758\pi$
$127$	−1191.31	−0.0738613	−0.0369307	+	0.999318i	$0.511758\pi$
$128$	0	0
$129$	16863.7	1.01338
$130$	0	0
$131$	23101.7	1.34617	0.673086	−	0.739564i	$-0.264968\pi$
$131$	23101.7	1.34617	0.673086	+	0.739564i	$0.264968\pi$
$132$	0	0
$133$	5909.40	0.334072
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12201.4i	0.650085i	0.945699	+	0.325042i	$0.105379\pi$
$137$	12201.4i	0.650085i	−0.945699	+	0.325042i	$0.894621\pi$
$138$	0	0
$139$	32137.7	1.66335	0.831677	−	0.555260i	$-0.187381\pi$
$139$	32137.7	1.66335	0.831677	+	0.555260i	$0.187381\pi$
$140$	0	0
$141$	4335.74i	0.218085i
$142$	0	0
$143$	19860.4	0.971217
$144$	0	0
$145$	0	0
$146$	0	0
$147$	− 2064.74i	− 0.0955498i
$148$	0	0
$149$	− 22165.6i	− 0.998407i	−0.866485	−	0.499203i	$-0.833626\pi$
$149$	− 22165.6i	− 0.998407i	0.866485	−	0.499203i	$-0.166374\pi$
$150$	0	0
$151$	− 11899.8i	− 0.521898i	−0.965353	−	0.260949i	$-0.915965\pi$
$151$	− 11899.8i	− 0.521898i	0.965353	−	0.260949i	$-0.0840354\pi$
$152$	0	0
$153$	61405.5i	2.62316i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−13908.6	−0.564266	−0.282133	−	0.959375i	$-0.591042\pi$
$157$	−13908.6	−0.564266	−0.282133	+	0.959375i	$0.591042\pi$
$158$	0	0
$159$	− 45112.7i	− 1.78445i
$160$	0	0
$161$	−9456.72	−0.364829
$162$	0	0
$163$	15868.2i	0.597247i	0.954371	+	0.298623i	$0.0965274\pi$
$163$	15868.2i	0.597247i	−0.954371	+	0.298623i	$0.903473\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	34210.3	1.22666	0.613329	−	0.789828i	$-0.289830\pi$
$167$	34210.3	1.22666	0.613329	+	0.789828i	$0.289830\pi$
$168$	0	0
$169$	27326.3	0.956770
$170$	0	0
$171$	−18702.6	−0.639602
$172$	0	0
$173$	−11608.3	−0.387861	−0.193930	−	0.981015i	$-0.562124\pi$
$173$	−11608.3	−0.387861	−0.193930	+	0.981015i	$0.562124\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10006.8i	0.319410i
$178$	0	0
$179$	6801.56	0.212277	0.106138	−	0.994351i	$-0.466151\pi$
$179$	6801.56	0.212277	0.106138	+	0.994351i	$0.466151\pi$
$180$	0	0
$181$	9082.98i	0.277250i	0.990345	+	0.138625i	$0.0442682\pi$
$181$	9082.98i	0.277250i	−0.990345	+	0.138625i	$0.955732\pi$
$182$	0	0
$183$	64065.4	1.91303
$184$	0	0
$185$	0	0
$186$	0	0
$187$	34246.3i	0.979334i
$188$	0	0
$189$	− 50442.2i	− 1.41212i
$190$	0	0
$191$	33432.5i	0.916435i	0.888840	+	0.458217i	$0.151512\pi$
$191$	33432.5i	0.916435i	−0.888840	+	0.458217i	$0.848488\pi$
$192$	0	0
$193$	29676.4i	0.796702i	0.917233	+	0.398351i	$0.130417\pi$
$193$	29676.4i	0.796702i	−0.917233	+	0.398351i	$0.869583\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−44590.9	−1.14898	−0.574492	−	0.818510i	$-0.694800\pi$
$197$	−44590.9	−1.14898	−0.574492	+	0.818510i	$0.694800\pi$
$198$	0	0
$199$	38999.4i	0.984808i	0.870367	+	0.492404i	$0.163882\pi$
$199$	38999.4i	0.984808i	−0.870367	+	0.492404i	$0.836118\pi$
$200$	0	0
$201$	39679.6	0.982143
$202$	0	0
$203$	62517.5i	1.51708i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	29929.5	0.698487
$208$	0	0
$209$	−10430.6	−0.238790
$210$	0	0
$211$	−12374.7	−0.277952	−0.138976	−	0.990296i	$-0.544381\pi$
$211$	−12374.7	−0.277952	−0.138976	+	0.990296i	$0.544381\pi$
$212$	0	0
$213$	69188.0	1.52501
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 59171.1i	− 1.25658i
$218$	0	0
$219$	−12523.7	−0.261123
$220$	0	0
$221$	96369.4i	1.97312i
$222$	0	0
$223$	−87223.5	−1.75398	−0.876989	−	0.480511i	$-0.840451\pi$
$223$	−87223.5	−1.75398	−0.876989	+	0.480511i	$0.840451\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 61793.9i	− 1.19921i	−0.800298	−	0.599603i	$-0.795325\pi$
$227$	− 61793.9i	− 1.19921i	0.800298	−	0.599603i	$-0.204675\pi$
$228$	0	0
$229$	− 65329.0i	− 1.24576i	−0.782316	−	0.622881i	$-0.785962\pi$
$229$	− 65329.0i	− 1.24576i	0.782316	−	0.622881i	$-0.214038\pi$
$230$	0	0
$231$	− 60855.5i	− 1.14045i
$232$	0	0
$233$	17377.5i	0.320093i	0.987110	+	0.160046i	$0.0511644\pi$
$233$	17377.5i	0.320093i	−0.987110	+	0.160046i	$0.948836\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−46385.4	−0.825819
$238$	0	0
$239$	− 30364.0i	− 0.531573i	−0.964032	−	0.265786i	$-0.914368\pi$
$239$	− 30364.0i	− 0.531573i	0.964032	−	0.265786i	$-0.0856316\pi$
$240$	0	0
$241$	−27940.4	−0.481059	−0.240530	−	0.970642i	$-0.577321\pi$
$241$	−27940.4	−0.481059	−0.240530	+	0.970642i	$0.577321\pi$
$242$	0	0
$243$	− 26056.0i	− 0.441260i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−29351.7	−0.481105
$248$	0	0
$249$	171618.	2.76798
$250$	0	0
$251$	45124.9	0.716257	0.358128	−	0.933672i	$-0.383415\pi$
$251$	45124.9	0.716257	0.358128	+	0.933672i	$0.383415\pi$
$252$	0	0
$253$	16691.9	0.260774
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 30570.0i	− 0.462838i	−0.972854	−	0.231419i	$-0.925663\pi$
$257$	− 30570.0i	− 0.462838i	0.972854	−	0.231419i	$-0.0743367\pi$
$258$	0	0
$259$	40687.6	0.606544
$260$	0	0
$261$	− 197861.i	− 2.90455i
$262$	0	0
$263$	20426.5	0.295313	0.147656	−	0.989039i	$-0.452827\pi$
$263$	20426.5	0.295313	0.147656	+	0.989039i	$0.452827\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	66780.8i	0.936761i
$268$	0	0
$269$	− 18131.8i	− 0.250574i	−0.992121	−	0.125287i	$-0.960015\pi$
$269$	− 18131.8i	− 0.250574i	0.992121	−	0.125287i	$-0.0399851\pi$
$270$	0	0
$271$	− 115432.i	− 1.57177i	−0.618375	−	0.785883i	$-0.712209\pi$
$271$	− 115432.i	− 1.57177i	0.618375	−	0.785883i	$-0.287791\pi$
$272$	0	0
$273$	− 171248.i	− 2.29773i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−109401.	−1.42581	−0.712904	−	0.701262i	$-0.752620\pi$
$277$	−109401.	−1.42581	−0.712904	+	0.701262i	$0.752620\pi$
$278$	0	0
$279$	187270.i	2.40580i
$280$	0	0
$281$	115428.	1.46184	0.730918	−	0.682465i	$-0.239092\pi$
$281$	115428.	1.46184	0.730918	+	0.682465i	$0.239092\pi$
$282$	0	0
$283$	− 92630.1i	− 1.15659i	−0.815828	−	0.578295i	$-0.803718\pi$
$283$	− 92630.1i	− 1.15659i	0.815828	−	0.578295i	$-0.196282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	146231.	1.77532
$288$	0	0
$289$	−82653.7	−0.989615
$290$	0	0
$291$	67708.3	0.799569
$292$	0	0
$293$	−135366.	−1.57679	−0.788396	−	0.615168i	$-0.789088\pi$
$293$	−135366.	−1.57679	−0.788396	+	0.615168i	$0.789088\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	89034.7i	1.00936i
$298$	0	0
$299$	46971.1	0.525398
$300$	0	0
$301$	− 52737.3i	− 0.582083i
$302$	0	0
$303$	138581.	1.50945
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 10701.4i	− 0.113544i	−0.998387	−	0.0567721i	$-0.981919\pi$
$307$	− 10701.4i	− 0.113544i	0.998387	−	0.0567721i	$-0.0180809\pi$
$308$	0	0
$309$	170977.i	1.79069i
$310$	0	0
$311$	− 22690.1i	− 0.234594i	−0.993097	−	0.117297i	$-0.962577\pi$
$311$	− 22690.1i	− 0.234594i	0.993097	−	0.117297i	$-0.0374229\pi$
$312$	0	0
$313$	− 12521.3i	− 0.127808i	−0.997956	−	0.0639042i	$-0.979645\pi$
$313$	− 12521.3i	− 0.127808i	0.997956	−	0.0639042i	$-0.0203552\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	74285.6	0.739241	0.369621	−	0.929183i	$-0.379488\pi$
$317$	74285.6	0.739241	0.369621	+	0.929183i	$0.379488\pi$
$318$	0	0
$319$	− 110349.i	− 1.08439i
$320$	0	0
$321$	22456.8	0.217941
$322$	0	0
$323$	− 50612.7i	− 0.485126i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−178056.	−1.66518
$328$	0	0
$329$	13559.0	0.125267
$330$	0	0
$331$	116833.	1.06637	0.533187	−	0.845998i	$-0.320994\pi$
$331$	116833.	1.06637	0.533187	+	0.845998i	$0.320994\pi$
$332$	0	0
$333$	−128772.	−1.16127
$334$	0	0
$335$	0	0
$336$	0	0
$337$	− 32007.6i	− 0.281834i	−0.990021	−	0.140917i	$-0.954995\pi$
$337$	− 32007.6i	− 0.281834i	0.990021	−	0.140917i	$-0.0450051\pi$
$338$	0	0
$339$	−62515.0	−0.543982
$340$	0	0
$341$	104442.i	0.898185i
$342$	0	0
$343$	−120734.	−1.02622
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 116131.i	− 0.964474i	−0.876041	−	0.482237i	$-0.839824\pi$
$347$	− 116131.i	− 0.964474i	0.876041	−	0.482237i	$-0.160176\pi$
$348$	0	0
$349$	− 45650.0i	− 0.374792i	−0.982284	−	0.187396i	$-0.939995\pi$
$349$	− 45650.0i	− 0.374792i	0.982284	−	0.187396i	$-0.0600047\pi$
$350$	0	0
$351$	250544.i	2.03362i
$352$	0	0
$353$	− 68096.3i	− 0.546479i	−0.961946	−	0.273240i	$-0.911905\pi$
$353$	− 68096.3i	− 0.546479i	0.961946	−	0.273240i	$-0.0880952\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	295291.	2.31694
$358$	0	0
$359$	− 216133.i	− 1.67700i	−0.544904	−	0.838498i	$-0.683434\pi$
$359$	− 216133.i	− 1.67700i	0.544904	−	0.838498i	$-0.316566\pi$
$360$	0	0
$361$	−114906.	−0.881712
$362$	0	0
$363$	− 115414.i	− 0.875884i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−167653.	−1.24474	−0.622369	−	0.782724i	$-0.713830\pi$
$367$	−167653.	−1.24474	−0.622369	+	0.782724i	$0.713830\pi$
$368$	0	0
$369$	−462806.	−3.39896
$370$	0	0
$371$	−141079.	−1.02498
$372$	0	0
$373$	67649.7	0.486237	0.243118	−	0.969997i	$-0.421830\pi$
$373$	67649.7	0.486237	0.243118	+	0.969997i	$0.421830\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 310521.i	− 2.18479i
$378$	0	0
$379$	−133585.	−0.929995	−0.464998	−	0.885312i	$-0.653945\pi$
$379$	−133585.	−0.929995	−0.464998	+	0.885312i	$0.653945\pi$
$380$	0	0
$381$	− 18131.2i	− 0.124904i
$382$	0	0
$383$	−155150.	−1.05768	−0.528839	−	0.848722i	$-0.677372\pi$
$383$	−155150.	−1.05768	−0.528839	+	0.848722i	$0.677372\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	166908.i	1.11443i
$388$	0	0
$389$	− 66829.6i	− 0.441641i	−0.975314	−	0.220820i	$-0.929127\pi$
$389$	− 66829.6i	− 0.441641i	0.975314	−	0.220820i	$-0.0708735\pi$
$390$	0	0
$391$	80994.7i	0.529789i
$392$	0	0
$393$	351597.i	2.27646i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−127027.	−0.805961	−0.402981	−	0.915209i	$-0.632026\pi$
$397$	−127027.	−0.805961	−0.402981	+	0.915209i	$0.632026\pi$
$398$	0	0
$399$	89938.4i	0.564936i
$400$	0	0
$401$	−138416.	−0.860791	−0.430396	−	0.902640i	$-0.641626\pi$
$401$	−138416.	−0.860791	−0.430396	+	0.902640i	$0.641626\pi$
$402$	0	0
$403$	293900.i	1.80963i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−71817.0	−0.433549
$408$	0	0
$409$	328530.	1.96394	0.981971	−	0.189030i	$-0.0605344\pi$
$409$	328530.	1.96394	0.981971	+	0.189030i	$0.0605344\pi$
$410$	0	0
$411$	−185700.	−1.09933
$412$	0	0
$413$	31293.9	0.183467
$414$	0	0
$415$	0	0
$416$	0	0
$417$	489121.i	2.81283i
$418$	0	0
$419$	2314.68	0.0131845	0.00659224	−	0.999978i	$-0.497902\pi$
$419$	2314.68	0.0131845	0.00659224	+	0.999978i	$0.497902\pi$
$420$	0	0
$421$	220238.i	1.24259i	0.783576	+	0.621297i	$0.213394\pi$
$421$	220238.i	1.24259i	−0.783576	+	0.621297i	$0.786606\pi$
$422$	0	0
$423$	−42912.8	−0.239831
$424$	0	0
$425$	0	0
$426$	0	0
$427$	− 200349.i	− 1.09883i
$428$	0	0
$429$	302267.i	1.64239i
$430$	0	0
$431$	105642.i	0.568696i	0.958721	+	0.284348i	$0.0917772\pi$
$431$	105642.i	0.568696i	−0.958721	+	0.284348i	$0.908223\pi$
$432$	0	0
$433$	− 300880.i	− 1.60479i	−0.596795	−	0.802394i	$-0.703559\pi$
$433$	− 300880.i	− 1.60479i	0.596795	−	0.802394i	$-0.296441\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−24669.0	−0.129178
$438$	0	0
$439$	322911.i	1.67554i	0.546027	+	0.837768i	$0.316140\pi$
$439$	322911.i	1.67554i	−0.546027	+	0.837768i	$0.683860\pi$
$440$	0	0
$441$	20435.6	0.105078
$442$	0	0
$443$	− 32565.1i	− 0.165938i	−0.996552	−	0.0829688i	$-0.973560\pi$
$443$	− 32565.1i	− 0.165938i	0.996552	−	0.0829688i	$-0.0264402\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	337351.	1.68837
$448$	0	0
$449$	−182782.	−0.906651	−0.453325	−	0.891345i	$-0.649762\pi$
$449$	−182782.	−0.906651	−0.453325	+	0.891345i	$0.649762\pi$
$450$	0	0
$451$	−258111.	−1.26897
$452$	0	0
$453$	181109.	0.882561
$454$	0	0
$455$	0	0
$456$	0	0
$457$	241527.i	1.15647i	0.815872	+	0.578233i	$0.196258\pi$
$457$	241527.i	1.15647i	−0.815872	+	0.578233i	$0.803742\pi$
$458$	0	0
$459$	−432026.	−2.05062
$460$	0	0
$461$	− 140223.i	− 0.659808i	−0.944015	−	0.329904i	$-0.892984\pi$
$461$	− 140223.i	− 0.659808i	0.944015	−	0.329904i	$-0.107016\pi$
$462$	0	0
$463$	−248909.	−1.16112	−0.580561	−	0.814217i	$-0.697167\pi$
$463$	−248909.	−1.16112	−0.580561	+	0.814217i	$0.697167\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	59048.6i	0.270755i	0.990794	+	0.135377i	$0.0432247\pi$
$467$	59048.6i	0.270755i	−0.990794	+	0.135377i	$0.956775\pi$
$468$	0	0
$469$	− 124088.i	− 0.564139i
$470$	0	0
$471$	− 211682.i	− 0.954208i
$472$	0	0
$473$	93085.7i	0.416064i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	446501.	1.96239
$478$	0	0
$479$	− 72996.3i	− 0.318148i	−0.987267	−	0.159074i	$-0.949149\pi$
$479$	− 72996.3i	− 0.318148i	0.987267	−	0.159074i	$-0.0508509\pi$
$480$	0	0
$481$	−202093.	−0.873498
$482$	0	0
$483$	− 143927.i	− 0.616947i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	192677.	0.812402	0.406201	−	0.913784i	$-0.366853\pi$
$487$	192677.	0.812402	0.406201	+	0.913784i	$0.366853\pi$
$488$	0	0
$489$	−241508.	−1.00998
$490$	0	0
$491$	108537.	0.450210	0.225105	−	0.974335i	$-0.427728\pi$
$491$	108537.	0.450210	0.225105	+	0.974335i	$0.427728\pi$
$492$	0	0
$493$	535448.	2.20305
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 216369.i	− 0.875956i
$498$	0	0
$499$	382138.	1.53469	0.767343	−	0.641237i	$-0.221578\pi$
$499$	382138.	1.53469	0.767343	+	0.641237i	$0.221578\pi$
$500$	0	0
$501$	520665.i	2.07435i
$502$	0	0
$503$	390847.	1.54480	0.772398	−	0.635139i	$-0.219057\pi$
$503$	390847.	1.54480	0.772398	+	0.635139i	$0.219057\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	415894.i	1.61796i
$508$	0	0
$509$	380507.i	1.46868i	0.678783	+	0.734339i	$0.262508\pi$
$509$	380507.i	1.46868i	−0.678783	+	0.734339i	$0.737492\pi$
$510$	0	0
$511$	39165.0i	0.149988i
$512$	0	0
$513$	− 131584.i	− 0.500000i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−23932.8	−0.0895390
$518$	0	0
$519$	− 176673.i	− 0.655896i
$520$	0	0
$521$	225469.	0.830636	0.415318	−	0.909676i	$-0.363670\pi$
$521$	225469.	0.830636	0.415318	+	0.909676i	$0.363670\pi$
$522$	0	0
$523$	− 445823.i	− 1.62989i	−0.579537	−	0.814946i	$-0.696767\pi$
$523$	− 445823.i	− 1.62989i	0.579537	−	0.814946i	$-0.303233\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−506787.	−1.82475
$528$	0	0
$529$	−240364.	−0.858929
$530$	0	0
$531$	−99041.7	−0.351260
$532$	0	0
$533$	−726325.	−2.55668
$534$	0	0
$535$	0	0
$536$	0	0
$537$	103517.i	0.358973i
$538$	0	0
$539$	11397.1	0.0392299
$540$	0	0
$541$	− 151646.i	− 0.518129i	−0.965860	−	0.259064i	$-0.916586\pi$
$541$	− 151646.i	− 0.518129i	0.965860	−	0.259064i	$-0.0834142\pi$
$542$	0	0
$543$	−138239.	−0.468846
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 522917.i	− 1.74766i	−0.486227	−	0.873832i	$-0.661627\pi$
$547$	− 522917.i	− 1.74766i	0.486227	−	0.873832i	$-0.338373\pi$
$548$	0	0
$549$	634084.i	2.10379i
$550$	0	0
$551$	163084.i	0.537166i
$552$	0	0
$553$	145059.i	0.474346i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−295824.	−0.953504	−0.476752	−	0.879038i	$-0.658186\pi$
$557$	−295824.	−0.953504	−0.476752	+	0.879038i	$0.658186\pi$
$558$	0	0
$559$	261944.i	0.838271i
$560$	0	0
$561$	−521214.	−1.65611
$562$	0	0
$563$	− 36177.5i	− 0.114136i	−0.998370	−	0.0570679i	$-0.981825\pi$
$563$	− 36177.5i	− 0.114136i	0.998370	−	0.0570679i	$-0.0181751\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	186975.	0.581590
$568$	0	0
$569$	631690.	1.95110	0.975550	−	0.219777i	$-0.0705331\pi$
$569$	631690.	1.95110	0.975550	+	0.219777i	$0.0705331\pi$
$570$	0	0
$571$	−145796.	−0.447170	−0.223585	−	0.974684i	$-0.571776\pi$
$571$	−145796.	−0.447170	−0.223585	+	0.974684i	$0.571776\pi$
$572$	0	0
$573$	−508827.	−1.54975
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 333473.i	− 1.00163i	−0.865553	−	0.500817i	$-0.833033\pi$
$577$	− 333473.i	− 1.00163i	0.865553	−	0.500817i	$-0.166967\pi$
$578$	0	0
$579$	−451661.	−1.34727
$580$	0	0
$581$	− 536693.i	− 1.58992i
$582$	0	0
$583$	249017.	0.732642
$584$	0	0
$585$	0	0
$586$	0	0
$587$	410391.i	1.19103i	0.803345	+	0.595514i	$0.203051\pi$
$587$	410391.i	1.19103i	−0.803345	+	0.595514i	$0.796949\pi$
$588$	0	0
$589$	− 154355.i	− 0.444928i
$590$	0	0
$591$	− 678654.i	− 1.94300i
$592$	0	0
$593$	− 649441.i	− 1.84684i	−0.383787	−	0.923422i	$-0.625380\pi$
$593$	− 649441.i	− 1.84684i	0.383787	−	0.923422i	$-0.374620\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−593553.	−1.66537
$598$	0	0
$599$	− 217117.i	− 0.605119i	−0.953131	−	0.302559i	$-0.902159\pi$
$599$	− 217117.i	− 0.605119i	0.953131	−	0.302559i	$-0.0978411\pi$
$600$	0	0
$601$	−39014.7	−0.108014	−0.0540069	−	0.998541i	$-0.517199\pi$
$601$	−39014.7	−0.108014	−0.0540069	+	0.998541i	$0.517199\pi$
$602$	0	0
$603$	392727.i	1.08008i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−405423.	−1.10035	−0.550175	−	0.835050i	$-0.685439\pi$
$607$	−405423.	−1.10035	−0.550175	+	0.835050i	$0.685439\pi$
$608$	0	0
$609$	−951488.	−2.56548
$610$	0	0
$611$	−67347.0	−0.180400
$612$	0	0
$613$	149169.	0.396970	0.198485	−	0.980104i	$-0.436398\pi$
$613$	149169.	0.396970	0.198485	+	0.980104i	$0.436398\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	334637.i	0.879030i	0.898235	+	0.439515i	$0.144850\pi$
$617$	334637.i	0.879030i	−0.898235	+	0.439515i	$0.855150\pi$
$618$	0	0
$619$	−839.713	−0.00219154	−0.00109577	−	0.999999i	$-0.500349\pi$
$619$	−839.713	−0.00219154	−0.00109577	+	0.999999i	$0.500349\pi$
$620$	0	0
$621$	210573.i	0.546033i
$622$	0	0
$623$	208841.	0.538071
$624$	0	0
$625$	0	0
$626$	0	0
$627$	− 158749.i	− 0.403809i
$628$	0	0
$629$	− 348480.i	− 0.880799i
$630$	0	0
$631$	502081.i	1.26100i	0.776189	+	0.630500i	$0.217150\pi$
$631$	502081.i	1.26100i	−0.776189	+	0.630500i	$0.782850\pi$
$632$	0	0
$633$	− 188337.i	− 0.470033i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	32071.5	0.0790388
$638$	0	0
$639$	684784.i	1.67707i
$640$	0	0
$641$	362864.	0.883137	0.441569	−	0.897227i	$-0.354422\pi$
$641$	362864.	0.883137	0.441569	+	0.897227i	$0.354422\pi$
$642$	0	0
$643$	− 648692.i	− 1.56898i	−0.620144	−	0.784488i	$-0.712926\pi$
$643$	− 648692.i	− 1.56898i	0.620144	−	0.784488i	$-0.287074\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	242110.	0.578369	0.289184	−	0.957273i	$-0.406616\pi$
$647$	242110.	0.578369	0.289184	+	0.957273i	$0.406616\pi$
$648$	0	0
$649$	−55236.3	−0.131140
$650$	0	0
$651$	900557.	2.12495
$652$	0	0
$653$	−542264.	−1.27170	−0.635850	−	0.771813i	$-0.719350\pi$
$653$	−542264.	−1.27170	−0.635850	+	0.771813i	$0.719350\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	− 123953.i	− 0.287161i
$658$	0	0
$659$	−145840.	−0.335820	−0.167910	−	0.985802i	$-0.553702\pi$
$659$	−145840.	−0.335820	−0.167910	+	0.985802i	$0.553702\pi$
$660$	0	0
$661$	278680.i	0.637828i	0.947784	+	0.318914i	$0.103318\pi$
$661$	278680.i	0.637828i	−0.947784	+	0.318914i	$0.896682\pi$
$662$	0	0
$663$	−1.46670e6	−3.33667
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 260981.i	− 0.586621i
$668$	0	0
$669$	− 1.32750e6i	− 2.96608i
$670$	0	0
$671$	353633.i	0.785431i
$672$	0	0
$673$	− 431869.i	− 0.953504i	−0.879038	−	0.476752i	$-0.841814\pi$
$673$	− 431869.i	− 0.953504i	0.879038	−	0.476752i	$-0.158186\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	147.083	0.000320911 0	0.000160456	−	1.00000i	$-0.499949\pi$
$677$	147.083	0.000320911 0	0.000160456		1.00000i	$0.499949\pi$
$678$	0	0
$679$	− 211742.i	− 0.459269i
$680$	0	0
$681$	940475.	2.02793
$682$	0	0
$683$	313526.i	0.672097i	0.941845	+	0.336049i	$0.109091\pi$
$683$	313526.i	0.672097i	−0.941845	+	0.336049i	$0.890909\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	994279.	2.10666
$688$	0	0
$689$	700735.	1.47610
$690$	0	0
$691$	669052.	1.40121	0.700606	−	0.713548i	$-0.252913\pi$
$691$	669052.	1.40121	0.700606	+	0.713548i	$0.252913\pi$
$692$	0	0
$693$	602314.	1.25417
$694$	0	0
$695$	0	0
$696$	0	0
$697$	− 1.25244e6i	− 2.57805i
$698$	0	0
$699$	−264478.	−0.541296
$700$	0	0
$701$	− 722704.i	− 1.47070i	−0.677687	−	0.735350i	$-0.737018\pi$
$701$	− 722704.i	− 1.47070i	0.677687	−	0.735350i	$-0.262982\pi$
$702$	0	0
$703$	106138.	0.214764
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 433379.i	− 0.867021i
$708$	0	0
$709$	− 206901.i	− 0.411596i	−0.978595	−	0.205798i	$-0.934021\pi$
$709$	− 206901.i	− 0.411596i	0.978595	−	0.205798i	$-0.0659790\pi$
$710$	0	0
$711$	− 459097.i	− 0.908166i
$712$	0	0
$713$	247012.i	0.485890i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	462126.	0.898922
$718$	0	0
$719$	226077.i	0.437319i	0.975801	+	0.218659i	$0.0701683\pi$
$719$	226077.i	0.437319i	−0.975801	+	0.218659i	$0.929832\pi$
$720$	0	0
$721$	534691.	1.02857
$722$	0	0
$723$	− 425240.i	− 0.813501i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−236940.	−0.448301	−0.224151	−	0.974555i	$-0.571961\pi$
$727$	−236940.	−0.448301	−0.224151	+	0.974555i	$0.571961\pi$
$728$	0	0
$729$	714761.	1.34495
$730$	0	0
$731$	−451683.	−0.845277
$732$	0	0
$733$	−722039.	−1.34386	−0.671928	−	0.740616i	$-0.734534\pi$
$733$	−722039.	−1.34386	−0.671928	+	0.740616i	$0.734534\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	219027.i	0.403238i
$738$	0	0
$739$	371832.	0.680860	0.340430	−	0.940270i	$-0.389427\pi$
$739$	371832.	0.680860	0.340430	+	0.940270i	$0.389427\pi$
$740$	0	0
$741$	− 446720.i	− 0.813578i
$742$	0	0
$743$	239765.	0.434318	0.217159	−	0.976136i	$-0.430321\pi$
$743$	239765.	0.434318	0.217159	+	0.976136i	$0.430321\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	1.69858e6i	3.04399i
$748$	0	0
$749$	− 70228.4i	− 0.125184i
$750$	0	0
$751$	229750.i	0.407357i	0.979038	+	0.203679i	$0.0652898\pi$
$751$	229750.i	0.407357i	−0.979038	+	0.203679i	$0.934710\pi$
$752$	0	0
$753$	686781.i	1.21123i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	328293.	0.572888	0.286444	−	0.958097i	$-0.407527\pi$
$757$	328293.	0.572888	0.286444	+	0.958097i	$0.407527\pi$
$758$	0	0
$759$	254043.i	0.440985i
$760$	0	0
$761$	−481340.	−0.831155	−0.415578	−	0.909558i	$-0.636421\pi$
$761$	−481340.	−0.831155	−0.415578	+	0.909558i	$0.636421\pi$
$762$	0	0
$763$	556828.i	0.956472i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−155435.	−0.264216
$768$	0	0
$769$	142240.	0.240530	0.120265	−	0.992742i	$-0.461625\pi$
$769$	142240.	0.240530	0.120265	+	0.992742i	$0.461625\pi$
$770$	0	0
$771$	465261.	0.782687
$772$	0	0
$773$	445222.	0.745105	0.372552	−	0.928011i	$-0.378483\pi$
$773$	445222.	0.745105	0.372552	+	0.928011i	$0.378483\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	619247.i	1.02570i
$778$	0	0
$779$	381462.	0.628603
$780$	0	0
$781$	381909.i	0.626121i
$782$	0	0
$783$	1.39207e6	2.27059
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 625717.i	− 1.01025i	−0.863047	−	0.505124i	$-0.831447\pi$
$787$	− 625717.i	− 1.01025i	0.863047	−	0.505124i	$-0.168553\pi$
$788$	0	0
$789$	310882.i	0.499392i
$790$	0	0
$791$	195501.i	0.312461i
$792$	0	0
$793$	995126.i	1.58246i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−563906.	−0.887749	−0.443875	−	0.896089i	$-0.646396\pi$
$797$	−563906.	−0.887749	−0.443875	+	0.896089i	$0.646396\pi$
$798$	0	0
$799$	− 116130.i	− 0.181907i
$800$	0	0
$801$	−660959.	−1.03017
$802$	0	0
$803$	− 69129.4i	− 0.107209i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	275957.	0.423736
$808$	0	0
$809$	−326208.	−0.498423	−0.249212	−	0.968449i	$-0.580171\pi$
$809$	−326208.	−0.498423	−0.249212	+	0.968449i	$0.580171\pi$
$810$	0	0
$811$	−1.29032e6	−1.96181	−0.980904	−	0.194492i	$-0.937694\pi$
$811$	−1.29032e6	−1.96181	−0.980904	+	0.194492i	$0.937694\pi$
$812$	0	0
$813$	1.75682e6	2.65795
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 137571.i	− 0.206103i
$818$	0	0
$819$	1.69492e6	2.52685
$820$	0	0
$821$	− 681025.i	− 1.01036i	−0.863014	−	0.505180i	$-0.831426\pi$
$821$	− 681025.i	− 1.01036i	0.863014	−	0.505180i	$-0.168574\pi$
$822$	0	0
$823$	−261315.	−0.385802	−0.192901	−	0.981218i	$-0.561790\pi$
$823$	−261315.	−0.385802	−0.192901	+	0.981218i	$0.561790\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 424512.i	− 0.620697i	−0.950623	−	0.310348i	$-0.899554\pi$
$827$	− 424512.i	− 0.620697i	0.950623	−	0.310348i	$-0.100446\pi$
$828$	0	0
$829$	53453.9i	0.0777804i	0.999243	+	0.0388902i	$0.0123823\pi$
$829$	53453.9i	0.0777804i	−0.999243	+	0.0388902i	$0.987618\pi$
$830$	0	0
$831$	− 1.66503e6i	− 2.41113i
$832$	0	0
$833$	55302.6i	0.0796994i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	−1.31756e6	−1.88070
$838$	0	0
$839$	− 277775.i	− 0.394611i	−0.980342	−	0.197305i	$-0.936781\pi$
$839$	− 277775.i	− 0.394611i	0.980342	−	0.197305i	$-0.0632190\pi$
$840$	0	0
$841$	−1.01804e6	−1.43937
$842$	0	0
$843$	1.75676e6i	2.47205i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−360931.	−0.503104
$848$	0	0
$849$	1.40979e6	1.95586
$850$	0	0
$851$	−169852.	−0.234537
$852$	0	0
$853$	−403889.	−0.555090	−0.277545	−	0.960713i	$-0.589521\pi$
$853$	−403889.	−0.555090	−0.277545	+	0.960713i	$0.589521\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	214464.i	0.292006i	0.989284	+	0.146003i	$0.0466409\pi$
$857$	214464.i	0.292006i	−0.989284	+	0.146003i	$0.953359\pi$
$858$	0	0
$859$	−1.21203e6	−1.64258	−0.821288	−	0.570513i	$-0.806744\pi$
$859$	−1.21203e6	−1.64258	−0.821288	+	0.570513i	$0.806744\pi$
$860$	0	0
$861$	2.22558e6i	3.00217i
$862$	0	0
$863$	1.37798e6	1.85021	0.925107	−	0.379707i	$-0.123975\pi$
$863$	1.37798e6	1.85021	0.925107	+	0.379707i	$0.123975\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 1.25795e6i	− 1.67350i
$868$	0	0
$869$	− 256042.i	− 0.339056i
$870$	0	0
$871$	616342.i	0.812429i
$872$	0	0
$873$	670139.i	0.879299i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−424259.	−0.551610	−0.275805	−	0.961214i	$-0.588944\pi$
$877$	−424259.	−0.551610	−0.275805	+	0.961214i	$0.588944\pi$
$878$	0	0
$879$	− 2.06021e6i	− 2.66645i
$880$	0	0
$881$	566909.	0.730401	0.365200	−	0.930929i	$-0.381000\pi$
$881$	566909.	0.730401	0.365200	+	0.930929i	$0.381000\pi$
$882$	0	0
$883$	323624.i	0.415068i	0.978228	+	0.207534i	$0.0665437\pi$
$883$	323624.i	0.415068i	−0.978228	+	0.207534i	$0.933456\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1.06589e6	1.35477	0.677385	−	0.735628i	$-0.263113\pi$
$887$	1.06589e6	1.35477	0.677385	+	0.735628i	$0.263113\pi$
$888$	0	0
$889$	−56701.0	−0.0717443
$890$	0	0
$891$	−330026.	−0.415712
$892$	0	0
$893$	35370.3	0.0443543
$894$	0	0
$895$	0	0
$896$	0	0
$897$	714879.i	0.888480i
$898$	0	0
$899$	1.63297e6	2.02050
$900$	0	0
$901$	1.20831e6i	1.48844i
$902$	0	0
$903$	802637.	0.984337
$904$	0	0
$905$	0	0
$906$	0	0
$907$	852471.i	1.03625i	0.855305	+	0.518125i	$0.173370\pi$
$907$	852471.i	1.03625i	−0.855305	+	0.518125i	$0.826630\pi$
$908$	0	0
$909$	1.37160e6i	1.65997i
$910$	0	0
$911$	251245.i	0.302734i	0.988478	+	0.151367i	$0.0483675\pi$
$911$	251245.i	0.302734i	−0.988478	+	0.151367i	$0.951632\pi$
$912$	0	0
$913$	947309.i	1.13645i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.09954e6	1.30759
$918$	0	0
$919$	− 1.19030e6i	− 1.40937i	−0.709520	−	0.704685i	$-0.751088\pi$
$919$	− 1.19030e6i	− 1.40937i	0.709520	−	0.704685i	$-0.248912\pi$
$920$	0	0
$921$	162871.	0.192010
$922$	0	0
$923$	1.07470e6i	1.26148i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−1.69224e6	−1.96926
$928$	0	0
$929$	−542484.	−0.628572	−0.314286	−	0.949328i	$-0.601765\pi$
$929$	−542484.	−0.628572	−0.314286	+	0.949328i	$0.601765\pi$
$930$	0	0
$931$	−16843.8	−0.0194330
$932$	0	0
$933$	345334.	0.396712
$934$	0	0
$935$	0	0
$936$	0	0
$937$	441434.i	0.502790i	0.967885	+	0.251395i	$0.0808893\pi$
$937$	441434.i	0.502790i	−0.967885	+	0.251395i	$0.919111\pi$
$938$	0	0
$939$	190568.	0.216132
$940$	0	0
$941$	− 1.51534e6i	− 1.71132i	−0.517539	−	0.855660i	$-0.673152\pi$
$941$	− 1.51534e6i	− 1.71132i	0.517539	−	0.855660i	$-0.326848\pi$
$942$	0	0
$943$	−610447.	−0.686475
$944$	0	0
$945$	0	0
$946$	0	0
$947$	1.20629e6i	1.34509i	0.740058	+	0.672543i	$0.234798\pi$
$947$	1.20629e6i	1.34509i	−0.740058	+	0.672543i	$0.765202\pi$
$948$	0	0
$949$	− 194531.i	− 0.216001i
$950$	0	0
$951$	1.13059e6i	1.25010i
$952$	0	0
$953$	− 1.70875e6i	− 1.88145i	−0.339166	−	0.940727i	$-0.610145\pi$
$953$	− 1.70875e6i	− 1.88145i	0.339166	−	0.940727i	$-0.389855\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	1.67946e6	1.83377
$958$	0	0
$959$	580734.i	0.631452i
$960$	0	0
$961$	−622040.	−0.673552
$962$	0	0
$963$	222265.i	0.239673i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−1.21487e6	−1.29921	−0.649604	−	0.760273i	$-0.725065\pi$
$967$	−1.21487e6	−1.29921	−0.649604	+	0.760273i	$0.725065\pi$
$968$	0	0
$969$	770302.	0.820377
$970$	0	0
$971$	1.18211e6	1.25378	0.626888	−	0.779110i	$-0.284329\pi$
$971$	1.18211e6	1.25378	0.626888	+	0.779110i	$0.284329\pi$
$972$	0	0
$973$	1.52961e6	1.61568
$974$	0	0
$975$	0	0
$976$	0	0
$977$	403569.i	0.422794i	0.977400	+	0.211397i	$0.0678012\pi$
$977$	403569.i	0.422794i	−0.977400	+	0.211397i	$0.932199\pi$
$978$	0	0
$979$	−368622.	−0.384606
$980$	0	0
$981$	− 1.76230e6i	− 1.83123i
$982$	0	0
$983$	109135.	0.112943	0.0564713	−	0.998404i	$-0.482015\pi$
$983$	109135.	0.112943	0.0564713	+	0.998404i	$0.482015\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	206362.i	0.211834i
$988$	0	0
$989$	220153.i	0.225078i
$990$	0	0
$991$	− 1.72512e6i	− 1.75660i	−0.478111	−	0.878299i	$-0.658678\pi$
$991$	− 1.72512e6i	− 1.75660i	0.478111	−	0.878299i	$-0.341322\pi$
$992$	0	0
$993$	1.77814e6i	1.80330i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	1.35890e6	1.36709	0.683543	−	0.729910i	$-0.260438\pi$
$997$	1.35890e6	1.36709	0.683543	+	0.729910i	$0.260438\pi$
$998$	0	0
$999$	− 905990.i	− 0.907805i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	800.5.e.e.399.8		32
4.3	odd	2		200.5.e.e.99.10		32
5.2	odd	4		160.5.g.a.111.16		16
5.3	odd	4		800.5.g.h.751.1		16
5.4	even	2	inner	800.5.e.e.399.25		32
8.3	odd	2	inner	800.5.e.e.399.26		32
8.5	even	2		200.5.e.e.99.24		32
15.2	even	4		1440.5.g.a.271.6		16
20.3	even	4		200.5.g.h.51.12		16
20.7	even	4		40.5.g.a.11.5	✓	16
20.19	odd	2		200.5.e.e.99.23		32
40.3	even	4		800.5.g.h.751.2		16
40.13	odd	4		200.5.g.h.51.11		16
40.19	odd	2	inner	800.5.e.e.399.7		32
40.27	even	4		160.5.g.a.111.15		16
40.29	even	2		200.5.e.e.99.9		32
40.37	odd	4		40.5.g.a.11.6	yes	16
60.47	odd	4		360.5.g.a.91.12		16
120.77	even	4		360.5.g.a.91.11		16
120.107	odd	4		1440.5.g.a.271.11		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.5	✓	16	20.7	even	4
40.5.g.a.11.6	yes	16	40.37	odd	4
160.5.g.a.111.15		16	40.27	even	4
160.5.g.a.111.16		16	5.2	odd	4
200.5.e.e.99.9		32	40.29	even	2
200.5.e.e.99.10		32	4.3	odd	2
200.5.e.e.99.23		32	20.19	odd	2
200.5.e.e.99.24		32	8.5	even	2
200.5.g.h.51.11		16	40.13	odd	4
200.5.g.h.51.12		16	20.3	even	4
360.5.g.a.91.11		16	120.77	even	4
360.5.g.a.91.12		16	60.47	odd	4
800.5.e.e.399.7		32	40.19	odd	2	inner
800.5.e.e.399.8		32	1.1	even	1	trivial
800.5.e.e.399.25		32	5.4	even	2	inner
800.5.e.e.399.26		32	8.3	odd	2	inner
800.5.g.h.751.1		16	5.3	odd	4
800.5.g.h.751.2		16	40.3	even	4
1440.5.g.a.271.6		16	15.2	even	4
1440.5.g.a.271.11		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)

\(f(q)\)	\(=\)	\(q+15.2196i q^{3} +47.5956 q^{7} -150.635 q^{9} -84.0101 q^{11} -236.405 q^{13} -407.645i q^{17} +124.159 q^{19} +724.383i q^{21} -198.689 q^{23} -1059.81i q^{27} +1313.51i q^{29} -1243.21i q^{31} -1278.60i q^{33} +854.861 q^{37} -3597.98i q^{39} +3072.37 q^{41} -1108.03i q^{43} +284.880 q^{47} -135.663 q^{49} +6204.18 q^{51} -2964.13 q^{53} +1889.64i q^{57} +657.496 q^{59} -4209.41i q^{61} -7169.54 q^{63} -2607.14i q^{67} -3023.96i q^{69} -4545.99i q^{71} +822.870i q^{73} -3998.51 q^{77} +3047.75i q^{79} +3928.41 q^{81} -11276.1i q^{83} -19991.1 q^{87} +4387.83 q^{89} -11251.8 q^{91} +18921.0 q^{93} -4448.77i q^{97} +12654.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 864 q^{9} - 384 q^{11} - 1408 q^{19} - 4416 q^{41} + 4960 q^{49} + 35584 q^{51} + 28032 q^{59} + 20768 q^{81} - 13632 q^{89} - 49152 q^{91} + 5248 q^{99}+O(q^{100}) \) 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