LMFDB - Embedded newform 800.5.g.h.751.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [800,5,Mod(751,800)]

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, 0]))

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

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

chi := DirichletCharacter("800.751");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 800 = 2^{5} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	800.g (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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 6 x^{15} + 14 x^{14} - 84 x^{13} + 628 x^{12} - 1392 x^{11} + 2016 x^{10} - 18048 x^{9} + \cdots + 4294967296 $ x^16 - 6x^15 + 14x^14 - 84x^13 + 628x^12 - 1392x^11 + 2016x^10 - 18048x^9 + 116992x^8 - 288768x^7 + 516096x^6 - 5701632x^5 + 41156608x^4 - 88080384x^3 + 234881024x^2 - 1610612736*x + 4294967296
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{58}\cdot 3^{2}\cdot 5^{5} $
Twist minimal:	no (minimal twist has level 40)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			751.3
Root			$-1.30255 + 3.78198i$ of defining polynomial
Character	$\chi$	$=$	800.751
Dual form			800.5.g.h.751.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-10.2034 q^{3} -43.3025i q^{7} +23.1094 q^{9} +165.050 q^{11} -201.647i q^{13} -172.778 q^{17} +640.745 q^{19} +441.833i q^{21} +497.626i q^{23} +590.681 q^{27} -509.953i q^{29} +492.136i q^{31} -1684.07 q^{33} -678.519i q^{37} +2057.49i q^{39} -613.848 q^{41} -1392.38 q^{43} -965.658i q^{47} +525.894 q^{49} +1762.92 q^{51} +4611.29i q^{53} -6537.78 q^{57} +822.630 q^{59} +6917.98i q^{61} -1000.70i q^{63} +4929.83 q^{67} -5077.48i q^{69} +4236.85i q^{71} +8507.29 q^{73} -7147.07i q^{77} -10205.9i q^{79} -7898.82 q^{81} -818.969 q^{83} +5203.25i q^{87} -10451.0 q^{89} -8731.82 q^{91} -5021.46i q^{93} +11691.7 q^{97} +3814.21 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 432 q^{9} - 192 q^{11} + 704 q^{19} + 3648 q^{27} + 992 q^{33} - 2208 q^{41} + 5568 q^{43} - 2480 q^{49} + 17792 q^{51} - 8608 q^{57} - 14016 q^{59} - 18880 q^{67} + 7360 q^{73} + 10384 q^{81} + 10560 q^{83}+ \cdots - 2624 q^{99}+O(q^{100}) $ 16 * q + 432 * q^9 - 192 * q^11 + 704 * q^19 + 3648 * q^27 + 992 * q^33 - 2208 * q^41 + 5568 * q^43 - 2480 * q^49 + 17792 * q^51 - 8608 * q^57 - 14016 * q^59 - 18880 * q^67 + 7360 * q^73 + 10384 * q^81 + 10560 * q^83 + 6816 * q^89 - 24576 * q^91 - 448 * q^97 - 2624 * q^99

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−10.2034	−1.13371	−0.566856	−	0.823817i	$-0.691840\pi$
$3$	−10.2034	−1.13371	−0.566856	+	0.823817i	$0.691840\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 43.3025i	− 0.883724i	−0.897083	−	0.441862i	$-0.854318\pi$
$7$	− 43.3025i	− 0.883724i	0.897083	−	0.441862i	$-0.145682\pi$
$8$	0	0
$9$	23.1094	0.285301
$10$	0	0
$11$	165.050	1.36405	0.682024	−	0.731329i	$-0.261100\pi$
$11$	165.050	1.36405	0.682024	+	0.731329i	$0.261100\pi$
$12$	0	0
$13$	− 201.647i	− 1.19318i	−0.802547	−	0.596589i	$-0.796522\pi$
$13$	− 201.647i	− 1.19318i	0.802547	−	0.596589i	$-0.203478\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−172.778	−0.597848	−0.298924	−	0.954277i	$-0.596628\pi$
$17$	−172.778	−0.597848	−0.298924	+	0.954277i	$0.596628\pi$
$18$	0	0
$19$	640.745	1.77492	0.887458	−	0.460888i	$-0.152469\pi$
$19$	640.745	1.77492	0.887458	+	0.460888i	$0.152469\pi$
$20$	0	0
$21$	441.833i	1.00189i
$22$	0	0
$23$	497.626i	0.940692i	0.882482	+	0.470346i	$0.155871\pi$
$23$	497.626i	0.940692i	−0.882482	+	0.470346i	$0.844129\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	590.681	0.810262
$28$	0	0
$29$	− 509.953i	− 0.606365i	−0.952933	−	0.303182i	$-0.901951\pi$
$29$	− 509.953i	− 0.606365i	0.952933	−	0.303182i	$-0.0980491\pi$
$30$	0	0
$31$	492.136i	0.512108i	0.966662	+	0.256054i	$0.0824225\pi$
$31$	492.136i	0.512108i	−0.966662	+	0.256054i	$0.917578\pi$
$32$	0	0
$33$	−1684.07	−1.54644
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 678.519i	− 0.495631i	−0.968807	−	0.247816i	$-0.920287\pi$
$37$	− 678.519i	− 0.495631i	0.968807	−	0.247816i	$-0.0797127\pi$
$38$	0	0
$39$	2057.49i	1.35272i
$40$	0	0
$41$	−613.848	−0.365168	−0.182584	−	0.983190i	$-0.558446\pi$
$41$	−613.848	−0.365168	−0.182584	+	0.983190i	$0.558446\pi$
$42$	0	0
$43$	−1392.38	−0.753043	−0.376522	−	0.926408i	$-0.622880\pi$
$43$	−1392.38	−0.753043	−0.376522	+	0.926408i	$0.622880\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 965.658i	− 0.437147i	−0.975820	−	0.218574i	$-0.929860\pi$
$47$	− 965.658i	− 0.437147i	0.975820	−	0.218574i	$-0.0701404\pi$
$48$	0	0
$49$	525.894	0.219031
$50$	0	0
$51$	1762.92	0.677787
$52$	0	0
$53$	4611.29i	1.64161i	0.571206	+	0.820807i	$0.306476\pi$
$53$	4611.29i	1.64161i	−0.571206	+	0.820807i	$0.693524\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−6537.78	−2.01224
$58$	0	0
$59$	822.630	0.236320	0.118160	−	0.992995i	$-0.462300\pi$
$59$	822.630	0.236320	0.118160	+	0.992995i	$0.462300\pi$
$60$	0	0
$61$	6917.98i	1.85917i	0.368606	+	0.929586i	$0.379835\pi$
$61$	6917.98i	1.85917i	−0.368606	+	0.929586i	$0.620165\pi$
$62$	0	0
$63$	− 1000.70i	− 0.252128i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	4929.83	1.09820	0.549101	−	0.835756i	$-0.314970\pi$
$67$	4929.83	1.09820	0.549101	+	0.835756i	$0.314970\pi$
$68$	0	0
$69$	− 5077.48i	− 1.06647i
$70$	0	0
$71$	4236.85i	0.840479i	0.907413	+	0.420239i	$0.138054\pi$
$71$	4236.85i	0.840479i	−0.907413	+	0.420239i	$0.861946\pi$
$72$	0	0
$73$	8507.29	1.59641	0.798207	−	0.602384i	$-0.205782\pi$
$73$	8507.29	1.59641	0.798207	+	0.602384i	$0.205782\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7147.07i	− 1.20544i
$78$	0	0
$79$	− 10205.9i	− 1.63530i	−0.575716	−	0.817650i	$-0.695276\pi$
$79$	− 10205.9i	− 1.63530i	0.575716	−	0.817650i	$-0.304724\pi$
$80$	0	0
$81$	−7898.82	−1.20390
$82$	0	0
$83$	−818.969	−0.118881	−0.0594403	−	0.998232i	$-0.518932\pi$
$83$	−818.969	−0.118881	−0.0594403	+	0.998232i	$0.518932\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5203.25i	0.687443i
$88$	0	0
$89$	−10451.0	−1.31940	−0.659700	−	0.751529i	$-0.729317\pi$
$89$	−10451.0	−1.31940	−0.659700	+	0.751529i	$0.729317\pi$
$90$	0	0
$91$	−8731.82	−1.05444
$92$	0	0
$93$	− 5021.46i	− 0.580583i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11691.7	1.24261	0.621306	−	0.783568i	$-0.286603\pi$
$97$	11691.7	1.24261	0.621306	+	0.783568i	$0.286603\pi$
$98$	0	0
$99$	3814.21	0.389165
$100$	0	0
$101$	− 13430.4i	− 1.31657i	−0.752767	−	0.658287i	$-0.771281\pi$
$101$	− 13430.4i	− 1.31657i	0.752767	−	0.658287i	$-0.228719\pi$
$102$	0	0
$103$	− 5546.66i	− 0.522826i	−0.965227	−	0.261413i	$-0.915812\pi$
$103$	− 5546.66i	− 0.522826i	0.965227	−	0.261413i	$-0.0841885\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	7485.33	0.653797	0.326899	−	0.945059i	$-0.393996\pi$
$107$	7485.33	0.653797	0.326899	+	0.945059i	$0.393996\pi$
$108$	0	0
$109$	− 13391.1i	− 1.12710i	−0.826082	−	0.563549i	$-0.809435\pi$
$109$	− 13391.1i	− 1.12710i	0.826082	−	0.563549i	$-0.190565\pi$
$110$	0	0
$111$	6923.20i	0.561903i
$112$	0	0
$113$	2822.55	0.221047	0.110523	−	0.993874i	$-0.464747\pi$
$113$	2822.55	0.221047	0.110523	+	0.993874i	$0.464747\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 4659.95i	− 0.340415i
$118$	0	0
$119$	7481.73i	0.528333i
$120$	0	0
$121$	12600.5	0.860629
$122$	0	0
$123$	6263.34	0.413995
$124$	0	0
$125$	0	0
$126$	0	0
$127$	15755.3i	0.976831i	0.872611	+	0.488416i	$0.162425\pi$
$127$	15755.3i	0.976831i	−0.872611	+	0.488416i	$0.837575\pi$
$128$	0	0
$129$	14207.0	0.853733
$130$	0	0
$131$	−4033.25	−0.235024	−0.117512	−	0.993071i	$-0.537492\pi$
$131$	−4033.25	−0.235024	−0.117512	+	0.993071i	$0.537492\pi$
$132$	0	0
$133$	− 27745.9i	− 1.56854i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−15035.7	−0.801090	−0.400545	−	0.916277i	$-0.631179\pi$
$137$	−15035.7	−0.801090	−0.400545	+	0.916277i	$0.631179\pi$
$138$	0	0
$139$	31639.4	1.63756	0.818782	−	0.574105i	$-0.194650\pi$
$139$	31639.4	1.63756	0.818782	+	0.574105i	$0.194650\pi$
$140$	0	0
$141$	9853.00i	0.495599i
$142$	0	0
$143$	− 33281.8i	− 1.62755i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−5365.90	−0.248318
$148$	0	0
$149$	− 34271.9i	− 1.54371i	−0.635798	−	0.771856i	$-0.719329\pi$
$149$	− 34271.9i	− 1.54371i	0.635798	−	0.771856i	$-0.280671\pi$
$150$	0	0
$151$	478.378i	0.0209806i	0.999945	+	0.0104903i	$0.00333922\pi$
$151$	478.378i	0.0209806i	−0.999945	+	0.0104903i	$0.996661\pi$
$152$	0	0
$153$	−3992.80	−0.170567
$154$	0	0
$155$	0	0
$156$	0	0
$157$	13731.7i	0.557088i	0.960423	+	0.278544i	$0.0898518\pi$
$157$	13731.7i	0.557088i	−0.960423	+	0.278544i	$0.910148\pi$
$158$	0	0
$159$	− 47050.9i	− 1.86112i
$160$	0	0
$161$	21548.4	0.831312
$162$	0	0
$163$	23891.6	0.899228	0.449614	−	0.893223i	$-0.351561\pi$
$163$	23891.6	0.899228	0.449614	+	0.893223i	$0.351561\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 27650.3i	− 0.991442i	−0.868482	−	0.495721i	$-0.834904\pi$
$167$	− 27650.3i	− 0.991442i	0.868482	−	0.495721i	$-0.165096\pi$
$168$	0	0
$169$	−12100.5	−0.423673
$170$	0	0
$171$	14807.2	0.506386
$172$	0	0
$173$	− 49203.7i	− 1.64401i	−0.569477	−	0.822007i	$-0.692854\pi$
$173$	− 49203.7i	− 1.64401i	0.569477	−	0.822007i	$-0.307146\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−8393.62	−0.267919
$178$	0	0
$179$	11247.1	0.351022	0.175511	−	0.984477i	$-0.443842\pi$
$179$	11247.1	0.351022	0.175511	+	0.984477i	$0.443842\pi$
$180$	0	0
$181$	9919.57i	0.302786i	0.988474	+	0.151393i	$0.0483759\pi$
$181$	9919.57i	0.302786i	−0.988474	+	0.151393i	$0.951624\pi$
$182$	0	0
$183$	− 70586.9i	− 2.10776i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−28517.0	−0.815494
$188$	0	0
$189$	− 25578.0i	− 0.716048i
$190$	0	0
$191$	− 36233.6i	− 0.993219i	−0.867974	−	0.496609i	$-0.834578\pi$
$191$	− 36233.6i	− 0.993219i	0.867974	−	0.496609i	$-0.165422\pi$
$192$	0	0
$193$	37612.9	1.00977	0.504885	−	0.863187i	$-0.331535\pi$
$193$	37612.9	1.00977	0.504885	+	0.863187i	$0.331535\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 27374.4i	− 0.705361i	−0.935744	−	0.352681i	$-0.885270\pi$
$197$	− 27374.4i	− 0.705361i	0.935744	−	0.352681i	$-0.114730\pi$
$198$	0	0
$199$	− 53766.9i	− 1.35772i	−0.734270	−	0.678858i	$-0.762475\pi$
$199$	− 53766.9i	− 1.35772i	0.734270	−	0.678858i	$-0.237525\pi$
$200$	0	0
$201$	−50301.0	−1.24504
$202$	0	0
$203$	−22082.2	−0.535859
$204$	0	0
$205$	0	0
$206$	0	0
$207$	11499.8i	0.268381i
$208$	0	0
$209$	105755.	2.42107
$210$	0	0
$211$	19829.6	0.445399	0.222700	−	0.974887i	$-0.428513\pi$
$211$	19829.6	0.445399	0.222700	+	0.974887i	$0.428513\pi$
$212$	0	0
$213$	− 43230.3i	− 0.952860i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	21310.7	0.452562
$218$	0	0
$219$	−86803.3	−1.80987
$220$	0	0
$221$	34840.2i	0.713339i
$222$	0	0
$223$	44470.0i	0.894247i	0.894472	+	0.447123i	$0.147552\pi$
$223$	44470.0i	0.894247i	−0.894472	+	0.447123i	$0.852448\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−41012.9	−0.795919	−0.397959	−	0.917403i	$-0.630281\pi$
$227$	−41012.9	−0.795919	−0.397959	+	0.917403i	$0.630281\pi$
$228$	0	0
$229$	39184.8i	0.747217i	0.927587	+	0.373608i	$0.121880\pi$
$229$	39184.8i	0.747217i	−0.927587	+	0.373608i	$0.878120\pi$
$230$	0	0
$231$	72924.5i	1.36662i
$232$	0	0
$233$	−64283.0	−1.18409	−0.592045	−	0.805905i	$-0.701679\pi$
$233$	−64283.0	−1.18409	−0.592045	+	0.805905i	$0.701679\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	104135.i	1.85396i
$238$	0	0
$239$	− 33481.2i	− 0.586145i	−0.956090	−	0.293073i	$-0.905322\pi$
$239$	− 33481.2i	− 0.586145i	0.956090	−	0.293073i	$-0.0946778\pi$
$240$	0	0
$241$	−28929.0	−0.498080	−0.249040	−	0.968493i	$-0.580115\pi$
$241$	−28929.0	−0.498080	−0.249040	+	0.968493i	$0.580115\pi$
$242$	0	0
$243$	32749.7	0.554618
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 129204.i	− 2.11779i
$248$	0	0
$249$	8356.27	0.134776
$250$	0	0
$251$	3632.83	0.0576630	0.0288315	−	0.999584i	$-0.490821\pi$
$251$	3632.83	0.0576630	0.0288315	+	0.999584i	$0.490821\pi$
$252$	0	0
$253$	82133.1i	1.28315i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	55530.5	0.840747	0.420373	−	0.907351i	$-0.361899\pi$
$257$	55530.5	0.840747	0.420373	+	0.907351i	$0.361899\pi$
$258$	0	0
$259$	−29381.6	−0.438001
$260$	0	0
$261$	− 11784.7i	− 0.172997i
$262$	0	0
$263$	4573.35i	0.0661185i	0.999453	+	0.0330593i	$0.0105250\pi$
$263$	4573.35i	0.0661185i	−0.999453	+	0.0330593i	$0.989475\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	106635.	1.49582
$268$	0	0
$269$	− 74452.8i	− 1.02891i	−0.857518	−	0.514454i	$-0.827995\pi$
$269$	− 74452.8i	− 1.02891i	0.857518	−	0.514454i	$-0.172005\pi$
$270$	0	0
$271$	− 13623.9i	− 0.185508i	−0.995689	−	0.0927541i	$-0.970433\pi$
$271$	− 13623.9i	− 0.185508i	0.995689	−	0.0927541i	$-0.0295670\pi$
$272$	0	0
$273$	89094.3	1.19543
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 18506.7i	− 0.241195i	−0.992701	−	0.120598i	$-0.961519\pi$
$277$	− 18506.7i	− 0.241195i	0.992701	−	0.120598i	$-0.0384811\pi$
$278$	0	0
$279$	11373.0i	0.146105i
$280$	0	0
$281$	26912.2	0.340830	0.170415	−	0.985372i	$-0.445489\pi$
$281$	26912.2	0.340830	0.170415	+	0.985372i	$0.445489\pi$
$282$	0	0
$283$	91580.2	1.14348	0.571740	−	0.820435i	$-0.306269\pi$
$283$	91580.2	1.14348	0.571740	+	0.820435i	$0.306269\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	26581.2i	0.322708i
$288$	0	0
$289$	−53668.7	−0.642577
$290$	0	0
$291$	−119295.	−1.40876
$292$	0	0
$293$	− 11242.9i	− 0.130961i	−0.997854	−	0.0654805i	$-0.979142\pi$
$293$	− 11242.9i	− 0.130961i	0.997854	−	0.0654805i	$-0.0208580\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	97491.8	1.10524
$298$	0	0
$299$	100345.	1.12241
$300$	0	0
$301$	60293.4i	0.665483i
$302$	0	0
$303$	137035.i	1.49261i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	−6583.47	−0.0698519	−0.0349260	−	0.999390i	$-0.511120\pi$
$307$	−6583.47	−0.0698519	−0.0349260	+	0.999390i	$0.511120\pi$
$308$	0	0
$309$	56594.8i	0.592734i
$310$	0	0
$311$	127524.i	1.31847i	0.751937	+	0.659235i	$0.229120\pi$
$311$	127524.i	1.31847i	−0.751937	+	0.659235i	$0.770880\pi$
$312$	0	0
$313$	−93160.2	−0.950915	−0.475457	−	0.879739i	$-0.657717\pi$
$313$	−93160.2	−0.950915	−0.475457	+	0.879739i	$0.657717\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 69292.6i	− 0.689554i	−0.938685	−	0.344777i	$-0.887954\pi$
$317$	− 69292.6i	− 0.689554i	0.938685	−	0.344777i	$-0.112046\pi$
$318$	0	0
$319$	− 84167.6i	− 0.827111i
$320$	0	0
$321$	−76375.8	−0.741217
$322$	0	0
$323$	−110707.	−1.06113
$324$	0	0
$325$	0	0
$326$	0	0
$327$	136634.i	1.27780i
$328$	0	0
$329$	−41815.4	−0.386318
$330$	0	0
$331$	−106686.	−0.973763	−0.486882	−	0.873468i	$-0.661866\pi$
$331$	−106686.	−0.973763	−0.486882	+	0.873468i	$0.661866\pi$
$332$	0	0
$333$	− 15680.2i	− 0.141404i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−34707.0	−0.305603	−0.152801	−	0.988257i	$-0.548829\pi$
$337$	−34707.0	−0.305603	−0.152801	+	0.988257i	$0.548829\pi$
$338$	0	0
$339$	−28799.6	−0.250603
$340$	0	0
$341$	81226.9i	0.698540i
$342$	0	0
$343$	− 126742.i	− 1.07729i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−191096.	−1.58706	−0.793529	−	0.608532i	$-0.791759\pi$
$347$	−191096.	−1.58706	−0.793529	+	0.608532i	$0.791759\pi$
$348$	0	0
$349$	19760.1i	0.162233i	0.996705	+	0.0811165i	$0.0258486\pi$
$349$	19760.1i	0.162233i	−0.996705	+	0.0811165i	$0.974151\pi$
$350$	0	0
$351$	− 119109.i	− 0.966787i
$352$	0	0
$353$	−65214.7	−0.523355	−0.261677	−	0.965155i	$-0.584276\pi$
$353$	−65214.7	−0.523355	−0.261677	+	0.965155i	$0.584276\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	− 76339.1i	− 0.598977i
$358$	0	0
$359$	183096.i	1.42066i	0.703871	+	0.710328i	$0.251453\pi$
$359$	183096.i	1.42066i	−0.703871	+	0.710328i	$0.748547\pi$
$360$	0	0
$361$	280233.	2.15033
$362$	0	0
$363$	−128568.	−0.975705
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 11615.6i	− 0.0862401i	−0.999070	−	0.0431200i	$-0.986270\pi$
$367$	− 11615.6i	− 0.0862401i	0.999070	−	0.0431200i	$-0.0137298\pi$
$368$	0	0
$369$	−14185.7	−0.104183
$370$	0	0
$371$	199680.	1.45073
$372$	0	0
$373$	192501.i	1.38361i	0.722083	+	0.691806i	$0.243185\pi$
$373$	192501.i	1.38361i	−0.722083	+	0.691806i	$0.756815\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−102830.	−0.723501
$378$	0	0
$379$	108067.	0.752342	0.376171	−	0.926550i	$-0.377240\pi$
$379$	108067.	0.752342	0.376171	+	0.926550i	$0.377240\pi$
$380$	0	0
$381$	− 160758.i	− 1.10744i
$382$	0	0
$383$	188520.i	1.28517i	0.766215	+	0.642585i	$0.222138\pi$
$383$	188520.i	1.28517i	−0.766215	+	0.642585i	$0.777862\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−32177.0	−0.214844
$388$	0	0
$389$	27057.1i	0.178806i	0.995996	+	0.0894030i	$0.0284959\pi$
$389$	27057.1i	0.178806i	−0.995996	+	0.0894030i	$0.971504\pi$
$390$	0	0
$391$	− 85978.9i	− 0.562391i
$392$	0	0
$393$	41152.8	0.266449
$394$	0	0
$395$	0	0
$396$	0	0
$397$	− 156094.i	− 0.990388i	−0.868782	−	0.495194i	$-0.835097\pi$
$397$	− 156094.i	− 0.990388i	0.868782	−	0.495194i	$-0.164903\pi$
$398$	0	0
$399$	283102.i	1.77827i
$400$	0	0
$401$	−77777.2	−0.483686	−0.241843	−	0.970315i	$-0.577752\pi$
$401$	−77777.2	−0.483686	−0.241843	+	0.970315i	$0.577752\pi$
$402$	0	0
$403$	99237.7	0.611036
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 111989.i	− 0.676065i
$408$	0	0
$409$	−247426.	−1.47911	−0.739553	−	0.673098i	$-0.764963\pi$
$409$	−247426.	−1.47911	−0.739553	+	0.673098i	$0.764963\pi$
$410$	0	0
$411$	153415.	0.908205
$412$	0	0
$413$	− 35621.9i	− 0.208842i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−322829.	−1.85652
$418$	0	0
$419$	−62888.2	−0.358213	−0.179107	−	0.983830i	$-0.557321\pi$
$419$	−62888.2	−0.358213	−0.179107	+	0.983830i	$0.557321\pi$
$420$	0	0
$421$	− 321321.i	− 1.81290i	−0.422311	−	0.906451i	$-0.638781\pi$
$421$	− 321321.i	− 1.81290i	0.422311	−	0.906451i	$-0.361219\pi$
$422$	0	0
$423$	− 22315.8i	− 0.124719i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	299566.	1.64300
$428$	0	0
$429$	339588.i	1.84518i
$430$	0	0
$431$	66812.9i	0.359672i	0.983697	+	0.179836i	$0.0575566\pi$
$431$	66812.9i	0.359672i	−0.983697	+	0.179836i	$0.942443\pi$
$432$	0	0
$433$	171149.	0.912846	0.456423	−	0.889763i	$-0.349130\pi$
$433$	171149.	0.912846	0.456423	+	0.889763i	$0.349130\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	318851.i	1.66965i
$438$	0	0
$439$	82871.1i	0.430005i	0.976613	+	0.215003i	$0.0689760\pi$
$439$	82871.1i	0.430005i	−0.976613	+	0.215003i	$0.931024\pi$
$440$	0	0
$441$	12153.1	0.0624899
$442$	0	0
$443$	−71237.1	−0.362994	−0.181497	−	0.983392i	$-0.558094\pi$
$443$	−71237.1	−0.362994	−0.181497	+	0.983392i	$0.558094\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	349690.i	1.75012i
$448$	0	0
$449$	306433.	1.52000	0.759999	−	0.649925i	$-0.225200\pi$
$449$	306433.	1.52000	0.759999	+	0.649925i	$0.225200\pi$
$450$	0	0
$451$	−101316.	−0.498107
$452$	0	0
$453$	− 4881.08i	− 0.0237859i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−31625.5	−0.151428	−0.0757138	−	0.997130i	$-0.524124\pi$
$457$	−31625.5	−0.151428	−0.0757138	+	0.997130i	$0.524124\pi$
$458$	0	0
$459$	−102057.	−0.484414
$460$	0	0
$461$	− 66080.1i	− 0.310934i	−0.987841	−	0.155467i	$-0.950312\pi$
$461$	− 66080.1i	− 0.310934i	0.987841	−	0.155467i	$-0.0496883\pi$
$462$	0	0
$463$	− 303583.i	− 1.41617i	−0.706126	−	0.708086i	$-0.749559\pi$
$463$	− 303583.i	− 1.41617i	0.706126	−	0.708086i	$-0.250441\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	121986.	0.559341	0.279670	−	0.960096i	$-0.409775\pi$
$467$	121986.	0.559341	0.279670	+	0.960096i	$0.409775\pi$
$468$	0	0
$469$	− 213474.i	− 0.970508i
$470$	0	0
$471$	− 140110.i	− 0.631577i
$472$	0	0
$473$	−229812.	−1.02719
$474$	0	0
$475$	0	0
$476$	0	0
$477$	106564.i	0.468355i
$478$	0	0
$479$	− 144501.i	− 0.629794i	−0.949126	−	0.314897i	$-0.898030\pi$
$479$	− 144501.i	− 0.629794i	0.949126	−	0.314897i	$-0.101970\pi$
$480$	0	0
$481$	−136821.	−0.591376
$482$	0	0
$483$	−219867.	−0.942468
$484$	0	0
$485$	0	0
$486$	0	0
$487$	7828.30i	0.0330073i	0.999864	+	0.0165036i	$0.00525351\pi$
$487$	7828.30i	0.0330073i	−0.999864	+	0.0165036i	$0.994746\pi$
$488$	0	0
$489$	−243776.	−1.01947
$490$	0	0
$491$	254423.	1.05534	0.527671	−	0.849449i	$-0.323066\pi$
$491$	254423.	1.05534	0.527671	+	0.849449i	$0.323066\pi$
$492$	0	0
$493$	88108.7i	0.362514i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	183466.	0.742752
$498$	0	0
$499$	19859.3	0.0797560	0.0398780	−	0.999205i	$-0.487303\pi$
$499$	19859.3	0.0797560	0.0398780	+	0.999205i	$0.487303\pi$
$500$	0	0
$501$	282127.i	1.12401i
$502$	0	0
$503$	289925.i	1.14591i	0.819587	+	0.572954i	$0.194203\pi$
$503$	289925.i	1.14591i	−0.819587	+	0.572954i	$0.805797\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	123467.	0.480323
$508$	0	0
$509$	101676.i	0.392447i	0.980559	+	0.196224i	$0.0628679\pi$
$509$	101676.i	0.392447i	−0.980559	+	0.196224i	$0.937132\pi$
$510$	0	0
$511$	− 368387.i	− 1.41079i
$512$	0	0
$513$	378476.	1.43815
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 159382.i	− 0.596290i
$518$	0	0
$519$	502045.i	1.86384i
$520$	0	0
$521$	−302599.	−1.11479	−0.557393	−	0.830248i	$-0.688198\pi$
$521$	−302599.	−1.11479	−0.557393	+	0.830248i	$0.688198\pi$
$522$	0	0
$523$	−61255.9	−0.223947	−0.111973	−	0.993711i	$-0.535717\pi$
$523$	−61255.9	−0.223947	−0.111973	+	0.993711i	$0.535717\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 85030.3i	− 0.306163i
$528$	0	0
$529$	32209.5	0.115099
$530$	0	0
$531$	19010.5	0.0674224
$532$	0	0
$533$	123781.i	0.435711i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	−114759.	−0.397958
$538$	0	0
$539$	86798.7	0.298769
$540$	0	0
$541$	− 223113.i	− 0.762307i	−0.924512	−	0.381154i	$-0.875527\pi$
$541$	− 223113.i	− 0.762307i	0.924512	−	0.381154i	$-0.124473\pi$
$542$	0	0
$543$	− 101213.i	− 0.343272i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	56408.6	0.188526	0.0942629	−	0.995547i	$-0.469951\pi$
$547$	56408.6	0.188526	0.0942629	+	0.995547i	$0.469951\pi$
$548$	0	0
$549$	159870.i	0.530424i
$550$	0	0
$551$	− 326750.i	− 1.07625i
$552$	0	0
$553$	−441941.	−1.44515
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 288826.i	− 0.930950i	−0.885061	−	0.465475i	$-0.845883\pi$
$557$	− 288826.i	− 0.930950i	0.885061	−	0.465475i	$-0.154117\pi$
$558$	0	0
$559$	280769.i	0.898514i
$560$	0	0
$561$	290971.	0.924535
$562$	0	0
$563$	611795.	1.93014	0.965071	−	0.261988i	$-0.0843782\pi$
$563$	611795.	1.93014	0.965071	+	0.261988i	$0.0843782\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	342039.i	1.06392i
$568$	0	0
$569$	−633137.	−1.95557	−0.977784	−	0.209613i	$-0.932780\pi$
$569$	−633137.	−1.95557	−0.977784	+	0.209613i	$0.932780\pi$
$570$	0	0
$571$	−279945.	−0.858618	−0.429309	−	0.903158i	$-0.641243\pi$
$571$	−279945.	−0.858618	−0.429309	+	0.903158i	$0.641243\pi$
$572$	0	0
$573$	369706.i	1.12602i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	371982.	1.11730	0.558651	−	0.829403i	$-0.311319\pi$
$577$	371982.	1.11730	0.558651	+	0.829403i	$0.311319\pi$
$578$	0	0
$579$	−383780.	−1.14479
$580$	0	0
$581$	35463.4i	0.105058i
$582$	0	0
$583$	761093.i	2.23924i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	44751.2	0.129876	0.0649380	−	0.997889i	$-0.479315\pi$
$587$	44751.2	0.129876	0.0649380	+	0.997889i	$0.479315\pi$
$588$	0	0
$589$	315333.i	0.908949i
$590$	0	0
$591$	279312.i	0.799676i
$592$	0	0
$593$	140260.	0.398863	0.199431	−	0.979912i	$-0.436090\pi$
$593$	140260.	0.398863	0.199431	+	0.979912i	$0.436090\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	548605.i	1.53926i
$598$	0	0
$599$	− 326102.i	− 0.908866i	−0.890781	−	0.454433i	$-0.849842\pi$
$599$	− 326102.i	− 0.908866i	0.890781	−	0.454433i	$-0.150158\pi$
$600$	0	0
$601$	111831.	0.309608	0.154804	−	0.987945i	$-0.450525\pi$
$601$	111831.	0.309608	0.154804	+	0.987945i	$0.450525\pi$
$602$	0	0
$603$	113925.	0.313319
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 442565.i	− 1.20116i	−0.799566	−	0.600579i	$-0.794937\pi$
$607$	− 442565.i	− 1.20116i	0.799566	−	0.600579i	$-0.205063\pi$
$608$	0	0
$609$	225314.	0.607510
$610$	0	0
$611$	−194722.	−0.521594
$612$	0	0
$613$	235020.i	0.625436i	0.949846	+	0.312718i	$0.101240\pi$
$613$	235020.i	0.625436i	−0.949846	+	0.312718i	$0.898760\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−212842.	−0.559098	−0.279549	−	0.960131i	$-0.590185\pi$
$617$	−212842.	−0.559098	−0.279549	+	0.960131i	$0.590185\pi$
$618$	0	0
$619$	30660.8	0.0800207	0.0400103	−	0.999199i	$-0.487261\pi$
$619$	30660.8	0.0800207	0.0400103	+	0.999199i	$0.487261\pi$
$620$	0	0
$621$	293938.i	0.762207i
$622$	0	0
$623$	452553.i	1.16599i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−1.07906e6	−2.74480
$628$	0	0
$629$	117233.i	0.296312i
$630$	0	0
$631$	398063.i	0.999755i	0.866096	+	0.499877i	$0.166622\pi$
$631$	398063.i	0.999755i	−0.866096	+	0.499877i	$0.833378\pi$
$632$	0	0
$633$	−202330.	−0.504954
$634$	0	0
$635$	0	0
$636$	0	0
$637$	− 106045.i	− 0.261343i
$638$	0	0
$639$	97911.2i	0.239790i
$640$	0	0
$641$	75692.2	0.184219	0.0921096	−	0.995749i	$-0.470639\pi$
$641$	75692.2	0.184219	0.0921096	+	0.995749i	$0.470639\pi$
$642$	0	0
$643$	−786008.	−1.90110	−0.950551	−	0.310570i	$-0.899480\pi$
$643$	−786008.	−1.90110	−0.950551	+	0.310570i	$0.899480\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 448043.i	− 1.07031i	−0.844752	−	0.535157i	$-0.820252\pi$
$647$	− 448043.i	− 1.07031i	0.844752	−	0.535157i	$-0.179748\pi$
$648$	0	0
$649$	135775.	0.322352
$650$	0	0
$651$	−217442.	−0.513075
$652$	0	0
$653$	− 396846.i	− 0.930671i	−0.885134	−	0.465335i	$-0.845934\pi$
$653$	− 396846.i	− 0.930671i	0.885134	−	0.465335i	$-0.154066\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	196598.	0.455459
$658$	0	0
$659$	−290177.	−0.668177	−0.334089	−	0.942542i	$-0.608428\pi$
$659$	−290177.	−0.668177	−0.334089	+	0.942542i	$0.608428\pi$
$660$	0	0
$661$	− 43962.0i	− 0.100618i	−0.998734	−	0.0503088i	$-0.983979\pi$
$661$	− 43962.0i	− 0.100618i	0.998734	−	0.0503088i	$-0.0160206\pi$
$662$	0	0
$663$	− 355489.i	− 0.808721i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	253766.	0.570402
$668$	0	0
$669$	− 453745.i	− 1.01382i
$670$	0	0
$671$	1.14181e6i	2.53600i
$672$	0	0
$673$	−314835.	−0.695110	−0.347555	−	0.937660i	$-0.612988\pi$
$673$	−314835.	−0.695110	−0.347555	+	0.937660i	$0.612988\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 317143.i	− 0.691954i	−0.938243	−	0.345977i	$-0.887548\pi$
$677$	− 317143.i	− 0.691954i	0.938243	−	0.345977i	$-0.112452\pi$
$678$	0	0
$679$	− 506281.i	− 1.09813i
$680$	0	0
$681$	418471.	0.902342
$682$	0	0
$683$	−64946.3	−0.139224	−0.0696118	−	0.997574i	$-0.522176\pi$
$683$	−64946.3	−0.139224	−0.0696118	+	0.997574i	$0.522176\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 399818.i	− 0.847128i
$688$	0	0
$689$	929854.	1.95874
$690$	0	0
$691$	225176.	0.471591	0.235796	−	0.971803i	$-0.424230\pi$
$691$	225176.	0.471591	0.235796	+	0.971803i	$0.424230\pi$
$692$	0	0
$693$	− 165165.i	− 0.343915i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	106060.	0.218315
$698$	0	0
$699$	655906.	1.34242
$700$	0	0
$701$	− 720970.i	− 1.46717i	−0.679597	−	0.733586i	$-0.737845\pi$
$701$	− 720970.i	− 1.46717i	0.679597	−	0.733586i	$-0.262155\pi$
$702$	0	0
$703$	− 434758.i	− 0.879704i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−581568.	−1.16349
$708$	0	0
$709$	616674.i	1.22677i	0.789784	+	0.613385i	$0.210192\pi$
$709$	616674.i	1.22677i	−0.789784	+	0.613385i	$0.789808\pi$
$710$	0	0
$711$	− 235853.i	− 0.466553i
$712$	0	0
$713$	−244899.	−0.481736
$714$	0	0
$715$	0	0
$716$	0	0
$717$	341622.i	0.664520i
$718$	0	0
$719$	− 369408.i	− 0.714577i	−0.933994	−	0.357288i	$-0.883701\pi$
$719$	− 369408.i	− 0.714577i	0.933994	−	0.357288i	$-0.116299\pi$
$720$	0	0
$721$	−240184.	−0.462034
$722$	0	0
$723$	295174.	0.564679
$724$	0	0
$725$	0	0
$726$	0	0
$727$	402715.i	0.761954i	0.924585	+	0.380977i	$0.124412\pi$
$727$	402715.i	0.761954i	−0.924585	+	0.380977i	$0.875588\pi$
$728$	0	0
$729$	305646.	0.575127
$730$	0	0
$731$	240572.	0.450206
$732$	0	0
$733$	− 22951.5i	− 0.0427172i	−0.999772	−	0.0213586i	$-0.993201\pi$
$733$	− 22951.5i	− 0.0427172i	0.999772	−	0.0213586i	$-0.00679918\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	813668.	1.49800
$738$	0	0
$739$	−161609.	−0.295922	−0.147961	−	0.988993i	$-0.547271\pi$
$739$	−161609.	−0.295922	−0.147961	+	0.988993i	$0.547271\pi$
$740$	0	0
$741$	1.31832e6i	2.40096i
$742$	0	0
$743$	194165.i	0.351718i	0.984415	+	0.175859i	$0.0562702\pi$
$743$	194165.i	0.351718i	−0.984415	+	0.175859i	$0.943730\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−18925.9	−0.0339168
$748$	0	0
$749$	− 324133.i	− 0.577777i
$750$	0	0
$751$	− 595741.i	− 1.05628i	−0.849158	−	0.528138i	$-0.822890\pi$
$751$	− 595741.i	− 1.05628i	0.849158	−	0.528138i	$-0.177110\pi$
$752$	0	0
$753$	−37067.2	−0.0653732
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 566219.i	− 0.988081i	−0.869439	−	0.494040i	$-0.835519\pi$
$757$	− 566219.i	− 0.988081i	0.869439	−	0.494040i	$-0.164481\pi$
$758$	0	0
$759$	− 838037.i	− 1.45472i
$760$	0	0
$761$	−268262.	−0.463223	−0.231612	−	0.972808i	$-0.574400\pi$
$761$	−268262.	−0.463223	−0.231612	+	0.972808i	$0.574400\pi$
$762$	0	0
$763$	−579866.	−0.996045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 165881.i	− 0.281972i
$768$	0	0
$769$	495755.	0.838329	0.419165	−	0.907910i	$-0.362323\pi$
$769$	495755.	0.838329	0.419165	+	0.907910i	$0.362323\pi$
$770$	0	0
$771$	−566600.	−0.953164
$772$	0	0
$773$	− 437740.i	− 0.732584i	−0.930500	−	0.366292i	$-0.880627\pi$
$773$	− 437740.i	− 0.732584i	0.930500	−	0.366292i	$-0.119373\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	299792.	0.496567
$778$	0	0
$779$	−393320.	−0.648143
$780$	0	0
$781$	699292.i	1.14645i
$782$	0	0
$783$	− 301219.i	− 0.491314i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−16629.0	−0.0268484	−0.0134242	−	0.999910i	$-0.504273\pi$
$787$	−16629.0	−0.0268484	−0.0134242	+	0.999910i	$0.504273\pi$
$788$	0	0
$789$	− 46663.8i	− 0.0749593i
$790$	0	0
$791$	− 122223.i	− 0.195345i
$792$	0	0
$793$	1.39499e6	2.21832
$794$	0	0
$795$	0	0
$796$	0	0
$797$	998541.i	1.57199i	0.618234	+	0.785994i	$0.287848\pi$
$797$	998541.i	1.57199i	−0.618234	+	0.785994i	$0.712152\pi$
$798$	0	0
$799$	166845.i	0.261348i
$800$	0	0
$801$	−241516.	−0.376427
$802$	0	0
$803$	1.40413e6	2.17759
$804$	0	0
$805$	0	0
$806$	0	0
$807$	759672.i	1.16648i
$808$	0	0
$809$	−523592.	−0.800011	−0.400006	−	0.916513i	$-0.630992\pi$
$809$	−523592.	−0.800011	−0.400006	+	0.916513i	$0.630992\pi$
$810$	0	0
$811$	741785.	1.12781	0.563905	−	0.825839i	$-0.309298\pi$
$811$	741785.	1.12781	0.563905	+	0.825839i	$0.309298\pi$
$812$	0	0
$813$	139010.i	0.210313i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−892158.	−1.33659
$818$	0	0
$819$	−201787.	−0.300833
$820$	0	0
$821$	− 971065.i	− 1.44066i	−0.693630	−	0.720331i	$-0.743990\pi$
$821$	− 971065.i	− 1.44066i	0.693630	−	0.720331i	$-0.256010\pi$
$822$	0	0
$823$	− 932619.i	− 1.37691i	−0.725280	−	0.688454i	$-0.758290\pi$
$823$	− 932619.i	− 1.37691i	0.725280	−	0.688454i	$-0.241710\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	1.11332e6	1.62782	0.813911	−	0.580989i	$-0.197334\pi$
$827$	1.11332e6	1.62782	0.813911	+	0.580989i	$0.197334\pi$
$828$	0	0
$829$	− 625811.i	− 0.910613i	−0.890335	−	0.455307i	$-0.849530\pi$
$829$	− 625811.i	− 0.910613i	0.890335	−	0.455307i	$-0.150470\pi$
$830$	0	0
$831$	188831.i	0.273446i
$832$	0	0
$833$	−90862.9	−0.130947
$834$	0	0
$835$	0	0
$836$	0	0
$837$	290695.i	0.414942i
$838$	0	0
$839$	− 3902.74i	− 0.00554428i	−0.999996	−	0.00277214i	$-0.999118\pi$
$839$	− 3902.74i	− 0.00554428i	0.999996	−	0.00277214i	$-0.000882401\pi$
$840$	0	0
$841$	447229.	0.632322
$842$	0	0
$843$	−274596.	−0.386402
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 545632.i	− 0.760559i
$848$	0	0
$849$	−934429.	−1.29638
$850$	0	0
$851$	337649.	0.466236
$852$	0	0
$853$	− 858678.i	− 1.18014i	−0.807353	−	0.590068i	$-0.799101\pi$
$853$	− 858678.i	− 1.18014i	0.807353	−	0.590068i	$-0.200899\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−143233.	−0.195021	−0.0975103	−	0.995235i	$-0.531088\pi$
$857$	−143233.	−0.195021	−0.0975103	+	0.995235i	$0.531088\pi$
$858$	0	0
$859$	892238.	1.20919	0.604595	−	0.796533i	$-0.293335\pi$
$859$	892238.	1.20919	0.604595	+	0.796533i	$0.293335\pi$
$860$	0	0
$861$	− 271218.i	− 0.365858i
$862$	0	0
$863$	− 838426.i	− 1.12575i	−0.826541	−	0.562877i	$-0.809695\pi$
$863$	− 838426.i	− 1.12575i	0.826541	−	0.562877i	$-0.190305\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	547603.	0.728497
$868$	0	0
$869$	− 1.68448e6i	− 2.23063i
$870$	0	0
$871$	− 994085.i	− 1.31035i
$872$	0	0
$873$	270189.	0.354519
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1.16386e6i	− 1.51322i	−0.653868	−	0.756609i	$-0.726855\pi$
$877$	− 1.16386e6i	− 1.51322i	0.653868	−	0.756609i	$-0.273145\pi$
$878$	0	0
$879$	114715.i	0.148472i
$880$	0	0
$881$	−1.22677e6	−1.58056	−0.790282	−	0.612744i	$-0.790066\pi$
$881$	−1.22677e6	−1.58056	−0.790282	+	0.612744i	$0.790066\pi$
$882$	0	0
$883$	443400.	0.568689	0.284344	−	0.958722i	$-0.408224\pi$
$883$	443400.	0.568689	0.284344	+	0.958722i	$0.408224\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 33686.5i	− 0.0428162i	−0.999771	−	0.0214081i	$-0.993185\pi$
$887$	− 33686.5i	− 0.0428162i	0.999771	−	0.0214081i	$-0.00681493\pi$
$888$	0	0
$889$	682244.	0.863249
$890$	0	0
$891$	−1.30370e6	−1.64218
$892$	0	0
$893$	− 618740.i	− 0.775900i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−1.02386e6	−1.27249
$898$	0	0
$899$	250966.	0.310524
$900$	0	0
$901$	− 796731.i	− 0.981436i
$902$	0	0
$903$	− 615198.i	− 0.754465i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	593935.	0.721979	0.360990	−	0.932570i	$-0.382439\pi$
$907$	593935.	0.721979	0.360990	+	0.932570i	$0.382439\pi$
$908$	0	0
$909$	− 310368.i	− 0.375620i
$910$	0	0
$911$	− 105485.i	− 0.127102i	−0.997979	−	0.0635510i	$-0.979757\pi$
$911$	− 105485.i	− 0.127102i	0.997979	−	0.0635510i	$-0.0202426\pi$
$912$	0	0
$913$	−135171.	−0.162159
$914$	0	0
$915$	0	0
$916$	0	0
$917$	174650.i	0.207696i
$918$	0	0
$919$	− 1.49149e6i	− 1.76600i	−0.469373	−	0.883000i	$-0.655520\pi$
$919$	− 1.49149e6i	− 1.76600i	0.469373	−	0.883000i	$-0.344480\pi$
$920$	0	0
$921$	67173.8	0.0791919
$922$	0	0
$923$	854349.	1.00284
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 128180.i	− 0.149163i
$928$	0	0
$929$	−1.27994e6	−1.48306	−0.741528	−	0.670922i	$-0.765898\pi$
$929$	−1.27994e6	−1.48306	−0.741528	+	0.670922i	$0.765898\pi$
$930$	0	0
$931$	336964.	0.388762
$932$	0	0
$933$	− 1.30118e6i	− 1.49476i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	−225182.	−0.256480	−0.128240	−	0.991743i	$-0.540933\pi$
$937$	−225182.	−0.256480	−0.128240	+	0.991743i	$0.540933\pi$
$938$	0	0
$939$	950551.	1.07806
$940$	0	0
$941$	1.51647e6i	1.71260i	0.516482	+	0.856298i	$0.327241\pi$
$941$	1.51647e6i	1.71260i	−0.516482	+	0.856298i	$0.672759\pi$
$942$	0	0
$943$	− 305467.i	− 0.343511i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−475706.	−0.530443	−0.265222	−	0.964187i	$-0.585445\pi$
$947$	−475706.	−0.530443	−0.265222	+	0.964187i	$0.585445\pi$
$948$	0	0
$949$	− 1.71547e6i	− 1.90481i
$950$	0	0
$951$	707021.i	0.781756i
$952$	0	0
$953$	−644868.	−0.710044	−0.355022	−	0.934858i	$-0.615527\pi$
$953$	−644868.	−0.710044	−0.355022	+	0.934858i	$0.615527\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	858796.i	0.937705i
$958$	0	0
$959$	651082.i	0.707943i
$960$	0	0
$961$	681323.	0.737745
$962$	0	0
$963$	172981.	0.186529
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 250168.i	− 0.267533i	−0.991013	−	0.133767i	$-0.957293\pi$
$967$	− 250168.i	− 0.267533i	0.991013	−	0.133767i	$-0.0427073\pi$
$968$	0	0
$969$	1.12959e6	1.20302
$970$	0	0
$971$	373022.	0.395636	0.197818	−	0.980239i	$-0.436614\pi$
$971$	373022.	0.395636	0.197818	+	0.980239i	$0.436614\pi$
$972$	0	0
$973$	− 1.37006e6i	− 1.44715i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−57277.7	−0.0600063	−0.0300031	−	0.999550i	$-0.509552\pi$
$977$	−57277.7	−0.0600063	−0.0300031	+	0.999550i	$0.509552\pi$
$978$	0	0
$979$	−1.72493e6	−1.79973
$980$	0	0
$981$	− 309460.i	− 0.321563i
$982$	0	0
$983$	351082.i	0.363330i	0.983360	+	0.181665i	$0.0581487\pi$
$983$	351082.i	0.363330i	−0.983360	+	0.181665i	$0.941851\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	426659.	0.437973
$988$	0	0
$989$	− 692883.i	− 0.708381i
$990$	0	0
$991$	196492.i	0.200077i	0.994984	+	0.100038i	$0.0318965\pi$
$991$	196492.i	0.200077i	−0.994984	+	0.100038i	$0.968103\pi$
$992$	0	0
$993$	1.08857e6	1.10397
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 736136.i	− 0.740572i	−0.928918	−	0.370286i	$-0.879260\pi$
$997$	− 736136.i	− 0.740572i	0.928918	−	0.370286i	$-0.120740\pi$
$998$	0	0
$999$	− 400788.i	− 0.401591i

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
40.5.g.a.11.9	✓	16	40.29	even	2
40.5.g.a.11.10	yes	16	20.19	odd	2
160.5.g.a.111.13		16	5.4	even	2
160.5.g.a.111.14		16	40.19	odd	2
200.5.e.e.99.1		32	40.37	odd	4
200.5.e.e.99.2		32	20.3	even	4
200.5.e.e.99.31		32	20.7	even	4
200.5.e.e.99.32		32	40.13	odd	4
200.5.g.h.51.7		16	4.3	odd	2
200.5.g.h.51.8		16	8.5	even	2
360.5.g.a.91.7		16	60.59	even	2
360.5.g.a.91.8		16	120.29	odd	2
800.5.e.e.399.9		32	40.3	even	4
800.5.e.e.399.10		32	5.2	odd	4
800.5.e.e.399.23		32	5.3	odd	4
800.5.e.e.399.24		32	40.27	even	4
800.5.g.h.751.3		16	1.1	even	1	trivial
800.5.g.h.751.4		16	8.3	odd	2	inner
1440.5.g.a.271.3		16	120.59	even	2
1440.5.g.a.271.14		16	15.14	odd	2

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.g (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-10.2034 q^{3} -43.3025i q^{7} +23.1094 q^{9} +165.050 q^{11} -201.647i q^{13} -172.778 q^{17} +640.745 q^{19} +441.833i q^{21} +497.626i q^{23} +590.681 q^{27} -509.953i q^{29} +492.136i q^{31} -1684.07 q^{33} -678.519i q^{37} +2057.49i q^{39} -613.848 q^{41} -1392.38 q^{43} -965.658i q^{47} +525.894 q^{49} +1762.92 q^{51} +4611.29i q^{53} -6537.78 q^{57} +822.630 q^{59} +6917.98i q^{61} -1000.70i q^{63} +4929.83 q^{67} -5077.48i q^{69} +4236.85i q^{71} +8507.29 q^{73} -7147.07i q^{77} -10205.9i q^{79} -7898.82 q^{81} -818.969 q^{83} +5203.25i q^{87} -10451.0 q^{89} -8731.82 q^{91} -5021.46i q^{93} +11691.7 q^{97} +3814.21 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 432 q^{9} - 192 q^{11} + 704 q^{19} + 3648 q^{27} + 992 q^{33} - 2208 q^{41} + 5568 q^{43} - 2480 q^{49} + 17792 q^{51} - 8608 q^{57} - 14016 q^{59} - 18880 q^{67} + 7360 q^{73} + 10384 q^{81} + 10560 q^{83}+ \cdots - 2624 q^{99}+O(q^{100}) \) 16 * q + 432 * q^9 - 192 * q^11 + 704 * q^19 + 3648 * q^27 + 992 * q^33 - 2208 * q^41 + 5568 * q^43 - 2480 * q^49 + 17792 * q^51 - 8608 * q^57 - 14016 * q^59 - 18880 * q^67 + 7360 * q^73 + 10384 * q^81 + 10560 * q^83 + 6816 * q^89 - 24576 * q^91 - 448 * q^97 - 2624 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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