LMFDB - Embedded newform 864.4.f.b.431.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,4,Mod(431,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("864.431");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	864.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:	$50.9776502450$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.5
Character	$\chi$	$=$	864.431
Dual form			864.4.f.b.431.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-12.8173 q^{5} -12.3152i q^{7} +O(q^{10})$	$q-12.8173 q^{5} -12.3152i q^{7} +24.6924i q^{11} -2.63962i q^{13} -93.1513i q^{17} +133.638 q^{19} +17.4864 q^{23} +39.2827 q^{25} -181.008 q^{29} -304.497i q^{31} +157.848i q^{35} +81.2372i q^{37} +314.098i q^{41} -333.068 q^{43} +52.6362 q^{47} +191.335 q^{49} -227.189 q^{53} -316.489i q^{55} +256.400i q^{59} +808.924i q^{61} +33.8328i q^{65} +110.747 q^{67} -582.985 q^{71} -290.848 q^{73} +304.093 q^{77} +485.514i q^{79} +1135.21i q^{83} +1193.95i q^{85} +1017.71i q^{89} -32.5076 q^{91} -1712.87 q^{95} -1265.30 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) $ 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$-1$	$-1$

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	0
$3$	0	0
$4$	0	0
$5$	−12.8173	−1.14641	−0.573206	−	0.819411i	$-0.694301\pi$
$5$	−12.8173	−1.14641	−0.573206	+	0.819411i	$0.694301\pi$
$6$	0	0
$7$	− 12.3152i	− 0.664961i	−0.943110	−	0.332480i	$-0.892115\pi$
$7$	− 12.3152i	− 0.664961i	0.943110	−	0.332480i	$-0.107885\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	24.6924i	0.676821i	0.940999	+	0.338411i	$0.109889\pi$
$11$	24.6924i	0.676821i	−0.940999	+	0.338411i	$0.890111\pi$
$12$	0	0
$13$	− 2.63962i	− 0.0563154i	−0.999603	−	0.0281577i	$-0.991036\pi$
$13$	− 2.63962i	− 0.0563154i	0.999603	−	0.0281577i	$-0.00896405\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 93.1513i	− 1.32897i	−0.747301	−	0.664485i	$-0.768651\pi$
$17$	− 93.1513i	− 1.32897i	0.747301	−	0.664485i	$-0.231349\pi$
$18$	0	0
$19$	133.638	1.61361	0.806805	−	0.590817i	$-0.201195\pi$
$19$	133.638	1.61361	0.806805	+	0.590817i	$0.201195\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	17.4864	0.158529	0.0792647	−	0.996854i	$-0.474743\pi$
$23$	17.4864	0.158529	0.0792647	+	0.996854i	$0.474743\pi$
$24$	0	0
$25$	39.2827	0.314262
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−181.008	−1.15905	−0.579523	−	0.814956i	$-0.696761\pi$
$29$	−181.008	−1.15905	−0.579523	+	0.814956i	$0.696761\pi$
$30$	0	0
$31$	− 304.497i	− 1.76417i	−0.471092	−	0.882084i	$-0.656140\pi$
$31$	− 304.497i	− 1.76417i	0.471092	−	0.882084i	$-0.343860\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	157.848i	0.762319i
$36$	0	0
$37$	81.2372i	0.360954i	0.983579	+	0.180477i	$0.0577642\pi$
$37$	81.2372i	0.360954i	−0.983579	+	0.180477i	$0.942236\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	314.098i	1.19644i	0.801333	+	0.598218i	$0.204124\pi$
$41$	314.098i	1.19644i	−0.801333	+	0.598218i	$0.795876\pi$
$42$	0	0
$43$	−333.068	−1.18122	−0.590610	−	0.806957i	$-0.701113\pi$
$43$	−333.068	−1.18122	−0.590610	+	0.806957i	$0.701113\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	52.6362	0.163357	0.0816785	−	0.996659i	$-0.473972\pi$
$47$	52.6362	0.163357	0.0816785	+	0.996659i	$0.473972\pi$
$48$	0	0
$49$	191.335	0.557828
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−227.189	−0.588807	−0.294403	−	0.955681i	$-0.595121\pi$
$53$	−227.189	−0.588807	−0.294403	+	0.955681i	$0.595121\pi$
$54$	0	0
$55$	− 316.489i	− 0.775916i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	256.400i	0.565771i	0.959154	+	0.282885i	$0.0912916\pi$
$59$	256.400i	0.565771i	−0.959154	+	0.282885i	$0.908708\pi$
$60$	0	0
$61$	808.924i	1.69790i	0.528470	+	0.848952i	$0.322766\pi$
$61$	808.924i	1.69790i	−0.528470	+	0.848952i	$0.677234\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	33.8328i	0.0645606i
$66$	0	0
$67$	110.747	0.201940	0.100970	−	0.994889i	$-0.467805\pi$
$67$	110.747	0.201940	0.100970	+	0.994889i	$0.467805\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−582.985	−0.974474	−0.487237	−	0.873270i	$-0.661995\pi$
$71$	−582.985	−0.974474	−0.487237	+	0.873270i	$0.661995\pi$
$72$	0	0
$73$	−290.848	−0.466318	−0.233159	−	0.972439i	$-0.574906\pi$
$73$	−290.848	−0.466318	−0.233159	+	0.972439i	$0.574906\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	304.093	0.450059
$78$	0	0
$79$	485.514i	0.691451i	0.938336	+	0.345726i	$0.112367\pi$
$79$	485.514i	0.691451i	−0.938336	+	0.345726i	$0.887633\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1135.21i	1.50127i	0.660715	+	0.750637i	$0.270253\pi$
$83$	1135.21i	1.50127i	−0.660715	+	0.750637i	$0.729747\pi$
$84$	0	0
$85$	1193.95i	1.52355i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1017.71i	1.21210i	0.795426	+	0.606051i	$0.207247\pi$
$89$	1017.71i	1.21210i	−0.795426	+	0.606051i	$0.792753\pi$
$90$	0	0
$91$	−32.5076	−0.0374475
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1712.87	−1.84986
$96$	0	0
$97$	−1265.30	−1.32445	−0.662224	−	0.749306i	$-0.730387\pi$
$97$	−1265.30	−1.32445	−0.662224	+	0.749306i	$0.730387\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1974.84	−1.94558	−0.972790	−	0.231689i	$-0.925575\pi$
$101$	−1974.84	−1.94558	−0.972790	+	0.231689i	$0.925575\pi$
$102$	0	0
$103$	− 1832.79i	− 1.75331i	−0.481122	−	0.876654i	$-0.659771\pi$
$103$	− 1832.79i	− 1.75331i	0.481122	−	0.876654i	$-0.340229\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1238.47i	− 1.11894i	−0.828849	−	0.559472i	$-0.811004\pi$
$107$	− 1238.47i	− 1.11894i	0.828849	−	0.559472i	$-0.188996\pi$
$108$	0	0
$109$	− 921.156i	− 0.809456i	−0.914437	−	0.404728i	$-0.867366\pi$
$109$	− 921.156i	− 0.809456i	0.914437	−	0.404728i	$-0.132634\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1871.75i	1.55822i	0.626886	+	0.779111i	$0.284329\pi$
$113$	1871.75i	1.55822i	−0.626886	+	0.779111i	$0.715671\pi$
$114$	0	0
$115$	−224.129	−0.181740
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1147.18	−0.883713
$120$	0	0
$121$	721.287	0.541913
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1098.66	0.786139
$126$	0	0
$127$	1474.51i	1.03025i	0.857115	+	0.515125i	$0.172255\pi$
$127$	1474.51i	1.03025i	−0.857115	+	0.515125i	$0.827745\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	388.379i	0.259029i	0.991578	+	0.129514i	$0.0413419\pi$
$131$	388.379i	0.259029i	−0.991578	+	0.129514i	$0.958658\pi$
$132$	0	0
$133$	− 1645.78i	− 1.07299i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	850.882i	0.530626i	0.964162	+	0.265313i	$0.0854753\pi$
$137$	850.882i	0.530626i	−0.964162	+	0.265313i	$0.914525\pi$
$138$	0	0
$139$	910.252	0.555443	0.277721	−	0.960662i	$-0.410421\pi$
$139$	910.252	0.555443	0.277721	+	0.960662i	$0.410421\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	65.1786	0.0381154
$144$	0	0
$145$	2320.03	1.32874
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2997.09	−1.64786	−0.823930	−	0.566692i	$-0.808223\pi$
$149$	−2997.09	−1.64786	−0.823930	+	0.566692i	$0.808223\pi$
$150$	0	0
$151$	574.759i	0.309756i	0.987934	+	0.154878i	$0.0494985\pi$
$151$	574.759i	0.309756i	−0.987934	+	0.154878i	$0.950502\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3902.82i	2.02247i
$156$	0	0
$157$	479.930i	0.243966i	0.992532	+	0.121983i	$0.0389253\pi$
$157$	479.930i	0.243966i	−0.992532	+	0.121983i	$0.961075\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 215.350i	− 0.105416i
$162$	0	0
$163$	−693.359	−0.333178	−0.166589	−	0.986026i	$-0.553275\pi$
$163$	−693.359	−0.333178	−0.166589	+	0.986026i	$0.553275\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−863.270	−0.400011	−0.200006	−	0.979795i	$-0.564096\pi$
$167$	−863.270	−0.400011	−0.200006	+	0.979795i	$0.564096\pi$
$168$	0	0
$169$	2190.03	0.996829
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−808.671	−0.355388	−0.177694	−	0.984086i	$-0.556864\pi$
$173$	−808.671	−0.355388	−0.177694	+	0.984086i	$0.556864\pi$
$174$	0	0
$175$	− 483.776i	− 0.208972i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	2790.92i	1.16538i	0.812694	+	0.582691i	$0.198000\pi$
$179$	2790.92i	1.16538i	−0.812694	+	0.582691i	$0.802000\pi$
$180$	0	0
$181$	− 3693.88i	− 1.51693i	−0.651715	−	0.758464i	$-0.725950\pi$
$181$	− 3693.88i	− 1.51693i	0.651715	−	0.758464i	$-0.274050\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 1041.24i	− 0.413802i
$186$	0	0
$187$	2300.13	0.899476
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−74.7511	−0.0283183	−0.0141592	−	0.999900i	$-0.504507\pi$
$191$	−74.7511	−0.0283183	−0.0141592	+	0.999900i	$0.504507\pi$
$192$	0	0
$193$	−3585.24	−1.33716	−0.668578	−	0.743642i	$-0.733097\pi$
$193$	−3585.24	−1.33716	−0.668578	+	0.743642i	$0.733097\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−1294.54	−0.468185	−0.234092	−	0.972214i	$-0.575212\pi$
$197$	−1294.54	−0.468185	−0.234092	+	0.972214i	$0.575212\pi$
$198$	0	0
$199$	4107.09i	1.46303i	0.681823	+	0.731517i	$0.261187\pi$
$199$	4107.09i	1.46303i	−0.681823	+	0.731517i	$0.738813\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2229.16i	0.770720i
$204$	0	0
$205$	− 4025.88i	− 1.37161i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3299.83i	1.09213i
$210$	0	0
$211$	−2186.50	−0.713389	−0.356695	−	0.934221i	$-0.616096\pi$
$211$	−2186.50	−0.713389	−0.356695	+	0.934221i	$0.616096\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4269.03	1.35416
$216$	0	0
$217$	−3749.95	−1.17310
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−245.884	−0.0748415
$222$	0	0
$223$	2662.76i	0.799604i	0.916602	+	0.399802i	$0.130921\pi$
$223$	2662.76i	0.799604i	−0.916602	+	0.399802i	$0.869079\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3617.15i	− 1.05762i	−0.848741	−	0.528808i	$-0.822639\pi$
$227$	− 3617.15i	− 1.05762i	0.848741	−	0.528808i	$-0.177361\pi$
$228$	0	0
$229$	4612.45i	1.33100i	0.746397	+	0.665501i	$0.231782\pi$
$229$	4612.45i	1.33100i	−0.746397	+	0.665501i	$0.768218\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 5501.45i	− 1.54683i	−0.633899	−	0.773416i	$-0.718546\pi$
$233$	− 5501.45i	− 1.54683i	0.633899	−	0.773416i	$-0.281454\pi$
$234$	0	0
$235$	−674.653	−0.187274
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−1009.38	−0.273186	−0.136593	−	0.990627i	$-0.543615\pi$
$239$	−1009.38	−0.273186	−0.136593	+	0.990627i	$0.543615\pi$
$240$	0	0
$241$	109.326	0.0292211	0.0146106	−	0.999893i	$-0.495349\pi$
$241$	109.326	0.0292211	0.0146106	+	0.999893i	$0.495349\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2452.39	−0.639501
$246$	0	0
$247$	− 352.753i	− 0.0908711i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5215.38i	− 1.31152i	−0.754969	−	0.655760i	$-0.772348\pi$
$251$	− 5215.38i	− 1.31152i	0.754969	−	0.655760i	$-0.227652\pi$
$252$	0	0
$253$	431.782i	0.107296i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	1700.22i	0.412673i	0.978481	+	0.206336i	$0.0661541\pi$
$257$	1700.22i	0.412673i	−0.978481	+	0.206336i	$0.933846\pi$
$258$	0	0
$259$	1000.46	0.240020
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−7993.02	−1.87403	−0.937017	−	0.349284i	$-0.886425\pi$
$263$	−7993.02	−1.87403	−0.937017	+	0.349284i	$0.886425\pi$
$264$	0	0
$265$	2911.94	0.675015
$266$	0	0
$267$	0	0
$268$	0	0
$269$	7670.20	1.73851	0.869257	−	0.494360i	$-0.164597\pi$
$269$	7670.20	1.73851	0.869257	+	0.494360i	$0.164597\pi$
$270$	0	0
$271$	3749.62i	0.840492i	0.907410	+	0.420246i	$0.138056\pi$
$271$	3749.62i	0.840492i	−0.907410	+	0.420246i	$0.861944\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	969.984i	0.212699i
$276$	0	0
$277$	− 824.479i	− 0.178838i	−0.995994	−	0.0894190i	$-0.971499\pi$
$277$	− 824.479i	− 0.178838i	0.995994	−	0.0894190i	$-0.0285010\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6914.40i	1.46790i	0.679206	+	0.733948i	$0.262324\pi$
$281$	6914.40i	1.46790i	−0.679206	+	0.733948i	$0.737676\pi$
$282$	0	0
$283$	−4752.59	−0.998277	−0.499138	−	0.866522i	$-0.666350\pi$
$283$	−4752.59	−0.998277	−0.499138	+	0.866522i	$0.666350\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3868.19	0.795583
$288$	0	0
$289$	−3764.16	−0.766164
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1680.22	−0.335016	−0.167508	−	0.985871i	$-0.553572\pi$
$293$	−1680.22	−0.335016	−0.167508	+	0.985871i	$0.553572\pi$
$294$	0	0
$295$	− 3286.35i	− 0.648607i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 46.1576i	− 0.00892764i
$300$	0	0
$301$	4101.82i	0.785464i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 10368.2i	− 1.94650i
$306$	0	0
$307$	4357.18	0.810025	0.405012	−	0.914311i	$-0.367267\pi$
$307$	4357.18	0.810025	0.405012	+	0.914311i	$0.367267\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	4751.34	0.866315	0.433158	−	0.901318i	$-0.357399\pi$
$311$	4751.34	0.866315	0.433158	+	0.901318i	$0.357399\pi$
$312$	0	0
$313$	3567.09	0.644166	0.322083	−	0.946712i	$-0.395617\pi$
$313$	3567.09	0.644166	0.322083	+	0.946712i	$0.395617\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6856.87	1.21489	0.607445	−	0.794362i	$-0.292194\pi$
$317$	6856.87	1.21489	0.607445	+	0.794362i	$0.292194\pi$
$318$	0	0
$319$	− 4469.51i	− 0.784466i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12448.5i	− 2.14444i
$324$	0	0
$325$	− 103.692i	− 0.0176978i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 648.227i	− 0.108626i
$330$	0	0
$331$	−4586.60	−0.761639	−0.380819	−	0.924649i	$-0.624358\pi$
$331$	−4586.60	−0.761639	−0.380819	+	0.924649i	$0.624358\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−1419.48	−0.231506
$336$	0	0
$337$	920.913	0.148859	0.0744293	−	0.997226i	$-0.476286\pi$
$337$	920.913	0.148859	0.0744293	+	0.997226i	$0.476286\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	7518.75	1.19403
$342$	0	0
$343$	− 6580.46i	− 1.03589i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1702.71i	0.263419i	0.991288	+	0.131709i	$0.0420465\pi$
$347$	1702.71i	0.263419i	−0.991288	+	0.131709i	$0.957953\pi$
$348$	0	0
$349$	1423.17i	0.218283i	0.994026	+	0.109141i	$0.0348102\pi$
$349$	1423.17i	0.218283i	−0.994026	+	0.109141i	$0.965190\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 4478.03i	− 0.675188i	−0.941292	−	0.337594i	$-0.890387\pi$
$353$	− 4478.03i	− 0.675188i	0.941292	−	0.337594i	$-0.109613\pi$
$354$	0	0
$355$	7472.29	1.11715
$356$	0	0
$357$	0	0
$358$	0	0
$359$	10723.3	1.57647	0.788234	−	0.615376i	$-0.210996\pi$
$359$	10723.3	1.57647	0.788234	+	0.615376i	$0.210996\pi$
$360$	0	0
$361$	11000.1	1.60374
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3727.88	0.534592
$366$	0	0
$367$	− 7336.51i	− 1.04350i	−0.853100	−	0.521748i	$-0.825280\pi$
$367$	− 7336.51i	− 1.04350i	0.853100	−	0.521748i	$-0.174720\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	2797.88i	0.391533i
$372$	0	0
$373$	− 908.936i	− 0.126174i	−0.998008	−	0.0630870i	$-0.979905\pi$
$373$	− 908.936i	− 0.126174i	0.998008	−	0.0630870i	$-0.0200946\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	477.793i	0.0652721i
$378$	0	0
$379$	−409.994	−0.0555673	−0.0277836	−	0.999614i	$-0.508845\pi$
$379$	−409.994	−0.0555673	−0.0277836	+	0.999614i	$0.508845\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4186.92	0.558594	0.279297	−	0.960205i	$-0.409899\pi$
$383$	4186.92	0.558594	0.279297	+	0.960205i	$0.409899\pi$
$384$	0	0
$385$	−3897.64	−0.515954
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−114.977	−0.0149860	−0.00749299	−	0.999972i	$-0.502385\pi$
$389$	−114.977	−0.0149860	−0.00749299	+	0.999972i	$0.502385\pi$
$390$	0	0
$391$	− 1628.88i	− 0.210681i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 6222.98i	− 0.792688i
$396$	0	0
$397$	6669.68i	0.843178i	0.906787	+	0.421589i	$0.138528\pi$
$397$	6669.68i	0.843178i	−0.906787	+	0.421589i	$0.861472\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 2541.57i	− 0.316508i	−0.987398	−	0.158254i	$-0.949414\pi$
$401$	− 2541.57i	− 0.316508i	0.987398	−	0.158254i	$-0.0505865\pi$
$402$	0	0
$403$	−803.756	−0.0993498
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2005.94	−0.244301
$408$	0	0
$409$	−532.629	−0.0643932	−0.0321966	−	0.999482i	$-0.510250\pi$
$409$	−532.629	−0.0643932	−0.0321966	+	0.999482i	$0.510250\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3157.63	0.376215
$414$	0	0
$415$	− 14550.3i	− 1.72108i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1083.70i	− 0.126354i	−0.998002	−	0.0631770i	$-0.979877\pi$
$419$	− 1083.70i	− 0.126354i	0.998002	−	0.0631770i	$-0.0201233\pi$
$420$	0	0
$421$	12096.1i	1.40031i	0.713993	+	0.700153i	$0.246885\pi$
$421$	12096.1i	1.40031i	−0.713993	+	0.700153i	$0.753115\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 3659.24i	− 0.417645i
$426$	0	0
$427$	9962.10	1.12904
$428$	0	0
$429$	0	0
$430$	0	0
$431$	5175.97	0.578463	0.289232	−	0.957259i	$-0.406600\pi$
$431$	5175.97	0.578463	0.289232	+	0.957259i	$0.406600\pi$
$432$	0	0
$433$	−13147.5	−1.45919	−0.729595	−	0.683880i	$-0.760291\pi$
$433$	−13147.5	−1.45919	−0.729595	+	0.683880i	$0.760291\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2336.85	0.255805
$438$	0	0
$439$	10374.0i	1.12785i	0.825827	+	0.563924i	$0.190709\pi$
$439$	10374.0i	1.12785i	−0.825827	+	0.563924i	$0.809291\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3872.10i	− 0.415280i	−0.978205	−	0.207640i	$-0.933422\pi$
$443$	− 3872.10i	− 0.415280i	0.978205	−	0.207640i	$-0.0665782\pi$
$444$	0	0
$445$	− 13044.3i	− 1.38957i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 13015.7i	− 1.36803i	−0.729466	−	0.684017i	$-0.760231\pi$
$449$	− 13015.7i	− 1.36803i	0.729466	−	0.684017i	$-0.239769\pi$
$450$	0	0
$451$	−7755.83	−0.809773
$452$	0	0
$453$	0	0
$454$	0	0
$455$	416.659	0.0429303
$456$	0	0
$457$	−6889.55	−0.705206	−0.352603	−	0.935773i	$-0.614703\pi$
$457$	−6889.55	−0.705206	−0.352603	+	0.935773i	$0.614703\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−16451.8	−1.66212	−0.831059	−	0.556184i	$-0.812265\pi$
$461$	−16451.8	−1.66212	−0.831059	+	0.556184i	$0.812265\pi$
$462$	0	0
$463$	1164.65i	0.116903i	0.998290	+	0.0584515i	$0.0186163\pi$
$463$	1164.65i	0.116903i	−0.998290	+	0.0584515i	$0.981384\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12193.5i	− 1.20824i	−0.796893	−	0.604120i	$-0.793525\pi$
$467$	− 12193.5i	− 1.20824i	0.796893	−	0.604120i	$-0.206475\pi$
$468$	0	0
$469$	− 1363.88i	− 0.134282i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 8224.25i	− 0.799474i
$474$	0	0
$475$	5249.66	0.507096
$476$	0	0
$477$	0	0
$478$	0	0
$479$	19819.1	1.89051	0.945257	−	0.326328i	$-0.105811\pi$
$479$	19819.1	1.89051	0.945257	+	0.326328i	$0.105811\pi$
$480$	0	0
$481$	214.435	0.0203273
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16217.7	1.51836
$486$	0	0
$487$	8416.52i	0.783140i	0.920148	+	0.391570i	$0.128068\pi$
$487$	8416.52i	0.783140i	−0.920148	+	0.391570i	$0.871932\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	− 4141.68i	− 0.380675i	−0.981719	−	0.190338i	$-0.939042\pi$
$491$	− 4141.68i	− 0.380675i	0.981719	−	0.190338i	$-0.0609582\pi$
$492$	0	0
$493$	16861.1i	1.54034i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	7179.61i	0.647987i
$498$	0	0
$499$	−14802.8	−1.32799	−0.663993	−	0.747739i	$-0.731139\pi$
$499$	−14802.8	−1.32799	−0.663993	+	0.747739i	$0.731139\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	17652.9	1.56482	0.782409	−	0.622765i	$-0.213991\pi$
$503$	17652.9	1.56482	0.782409	+	0.622765i	$0.213991\pi$
$504$	0	0
$505$	25312.0	2.23044
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12050.2	−1.04934	−0.524672	−	0.851305i	$-0.675812\pi$
$509$	−12050.2	−1.04934	−0.524672	+	0.851305i	$0.675812\pi$
$510$	0	0
$511$	3581.86i	0.310083i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	23491.4i	2.01001i
$516$	0	0
$517$	1299.71i	0.110563i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 19423.6i	− 1.63333i	−0.577114	−	0.816663i	$-0.695821\pi$
$521$	− 19423.6i	− 1.63333i	0.577114	−	0.816663i	$-0.304179\pi$
$522$	0	0
$523$	−2056.97	−0.171979	−0.0859895	−	0.996296i	$-0.527405\pi$
$523$	−2056.97	−0.171979	−0.0859895	+	0.996296i	$0.527405\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28364.3	−2.34453
$528$	0	0
$529$	−11861.2	−0.974868
$530$	0	0
$531$	0	0
$532$	0	0
$533$	829.101	0.0673777
$534$	0	0
$535$	15873.8i	1.28277i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	4724.51i	0.377549i
$540$	0	0
$541$	8763.73i	0.696455i	0.937410	+	0.348227i	$0.113216\pi$
$541$	8763.73i	0.696455i	−0.937410	+	0.348227i	$0.886784\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	11806.7i	0.927971i
$546$	0	0
$547$	−14324.5	−1.11969	−0.559844	−	0.828598i	$-0.689139\pi$
$547$	−14324.5	−1.11969	−0.559844	+	0.828598i	$0.689139\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−24189.5	−1.87025
$552$	0	0
$553$	5979.23	0.459788
$554$	0	0
$555$	0	0
$556$	0	0
$557$	4559.27	0.346827	0.173413	−	0.984849i	$-0.444520\pi$
$557$	4559.27	0.346827	0.173413	+	0.984849i	$0.444520\pi$
$558$	0	0
$559$	879.174i	0.0665208i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	36.8282i	0.00275688i	0.999999	+	0.00137844i	$0.000438772\pi$
$563$	36.8282i	0.00275688i	−0.999999	+	0.00137844i	$0.999561\pi$
$564$	0	0
$565$	− 23990.7i	− 1.78636i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	541.911i	0.0399264i	0.999801	+	0.0199632i	$0.00635490\pi$
$569$	541.911i	0.0399264i	−0.999801	+	0.0199632i	$0.993645\pi$
$570$	0	0
$571$	−3687.02	−0.270223	−0.135111	−	0.990830i	$-0.543139\pi$
$571$	−3687.02	−0.270223	−0.135111	+	0.990830i	$0.543139\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	686.915	0.0498197
$576$	0	0
$577$	−11835.5	−0.853929	−0.426965	−	0.904268i	$-0.640417\pi$
$577$	−11835.5	−0.853929	−0.426965	+	0.904268i	$0.640417\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	13980.4	0.998288
$582$	0	0
$583$	− 5609.83i	− 0.398517i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 4372.06i	− 0.307418i	−0.988116	−	0.153709i	$-0.950878\pi$
$587$	− 4372.06i	− 0.307418i	0.988116	−	0.153709i	$-0.0491218\pi$
$588$	0	0
$589$	− 40692.3i	− 2.84668i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	15064.4i	1.04321i	0.853189	+	0.521603i	$0.174666\pi$
$593$	15064.4i	1.04321i	−0.853189	+	0.521603i	$0.825334\pi$
$594$	0	0
$595$	14703.7	1.01310
$596$	0	0
$597$	0	0
$598$	0	0
$599$	17841.5	1.21700	0.608500	−	0.793554i	$-0.291772\pi$
$599$	17841.5	1.21700	0.608500	+	0.793554i	$0.291772\pi$
$600$	0	0
$601$	9819.62	0.666474	0.333237	−	0.942843i	$-0.391859\pi$
$601$	9819.62	0.666474	0.333237	+	0.942843i	$0.391859\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9244.93	−0.621256
$606$	0	0
$607$	22375.3i	1.49618i	0.663595	+	0.748092i	$0.269030\pi$
$607$	22375.3i	1.49618i	−0.663595	+	0.748092i	$0.730970\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 138.940i	− 0.00919950i
$612$	0	0
$613$	5013.85i	0.330354i	0.986264	+	0.165177i	$0.0528196\pi$
$613$	5013.85i	0.330354i	−0.986264	+	0.165177i	$0.947180\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24967.3i	1.62908i	0.580105	+	0.814542i	$0.303011\pi$
$617$	24967.3i	1.62908i	−0.580105	+	0.814542i	$0.696989\pi$
$618$	0	0
$619$	9789.39	0.635652	0.317826	−	0.948149i	$-0.397047\pi$
$619$	9789.39	0.635652	0.317826	+	0.948149i	$0.397047\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	12533.4	0.806000
$624$	0	0
$625$	−18992.2	−1.21550
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7567.35	0.479698
$630$	0	0
$631$	− 15488.0i	− 0.977125i	−0.872529	−	0.488562i	$-0.837521\pi$
$631$	− 15488.0i	− 0.977125i	0.872529	−	0.488562i	$-0.162479\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 18899.2i	− 1.18109i
$636$	0	0
$637$	− 505.052i	− 0.0314143i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8605.51i	0.530261i	0.964213	+	0.265130i	$0.0854150\pi$
$641$	8605.51i	0.530261i	−0.964213	+	0.265130i	$0.914585\pi$
$642$	0	0
$643$	21084.5	1.29315	0.646573	−	0.762852i	$-0.276201\pi$
$643$	21084.5	1.29315	0.646573	+	0.762852i	$0.276201\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−28639.0	−1.74021	−0.870105	−	0.492867i	$-0.835949\pi$
$647$	−28639.0	−1.74021	−0.870105	+	0.492867i	$0.835949\pi$
$648$	0	0
$649$	−6331.13	−0.382925
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−8008.99	−0.479963	−0.239982	−	0.970777i	$-0.577141\pi$
$653$	−8008.99	−0.479963	−0.239982	+	0.970777i	$0.577141\pi$
$654$	0	0
$655$	− 4977.96i	− 0.296954i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 834.545i	− 0.0493312i	−0.999696	−	0.0246656i	$-0.992148\pi$
$659$	− 834.545i	− 0.0493312i	0.999696	−	0.0246656i	$-0.00785210\pi$
$660$	0	0
$661$	− 5999.98i	− 0.353059i	−0.984295	−	0.176530i	$-0.943513\pi$
$661$	− 5999.98i	− 0.353059i	0.984295	−	0.176530i	$-0.0564872\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	21094.4i	1.23009i
$666$	0	0
$667$	−3165.18	−0.183743
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−19974.3	−1.14918
$672$	0	0
$673$	16463.5	0.942976	0.471488	−	0.881873i	$-0.343717\pi$
$673$	16463.5	0.942976	0.471488	+	0.881873i	$0.343717\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−3479.36	−0.197522	−0.0987612	−	0.995111i	$-0.531488\pi$
$677$	−3479.36	−0.197522	−0.0987612	+	0.995111i	$0.531488\pi$
$678$	0	0
$679$	15582.4i	0.880705i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	8012.99i	0.448914i	0.974484	+	0.224457i	$0.0720609\pi$
$683$	8012.99i	0.448914i	−0.974484	+	0.224457i	$0.927939\pi$
$684$	0	0
$685$	− 10906.0i	− 0.608317i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	599.692i	0.0331589i
$690$	0	0
$691$	9886.24	0.544270	0.272135	−	0.962259i	$-0.412270\pi$
$691$	9886.24	0.544270	0.272135	+	0.962259i	$0.412270\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−11667.0	−0.636767
$696$	0	0
$697$	29258.7	1.59003
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−18757.0	−1.01062	−0.505309	−	0.862938i	$-0.668622\pi$
$701$	−18757.0	−1.01062	−0.505309	+	0.862938i	$0.668622\pi$
$702$	0	0
$703$	10856.4i	0.582440i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	24320.6i	1.29373i
$708$	0	0
$709$	− 6728.64i	− 0.356416i	−0.983993	−	0.178208i	$-0.942970\pi$
$709$	− 6728.64i	− 0.356416i	0.983993	−	0.178208i	$-0.0570300\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5324.56i	− 0.279672i
$714$	0	0
$715$	−835.412	−0.0436960
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−18415.0	−0.955164	−0.477582	−	0.878587i	$-0.658487\pi$
$719$	−18415.0	−0.955164	−0.477582	+	0.878587i	$0.658487\pi$
$720$	0	0
$721$	−22571.3	−1.16588
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7110.48	−0.364244
$726$	0	0
$727$	− 14150.0i	− 0.721861i	−0.932593	−	0.360931i	$-0.882459\pi$
$727$	− 14150.0i	− 0.721861i	0.932593	−	0.360931i	$-0.117541\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	31025.7i	1.56981i
$732$	0	0
$733$	24275.1i	1.22322i	0.791159	+	0.611610i	$0.209478\pi$
$733$	24275.1i	1.22322i	−0.791159	+	0.611610i	$0.790522\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2734.62i	0.136677i
$738$	0	0
$739$	11548.3	0.574848	0.287424	−	0.957803i	$-0.407201\pi$
$739$	11548.3	0.574848	0.287424	+	0.957803i	$0.407201\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−9610.61	−0.474534	−0.237267	−	0.971444i	$-0.576252\pi$
$743$	−9610.61	−0.474534	−0.237267	+	0.971444i	$0.576252\pi$
$744$	0	0
$745$	38414.5	1.88913
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−15252.0	−0.744054
$750$	0	0
$751$	11904.8i	0.578444i	0.957262	+	0.289222i	$0.0933965\pi$
$751$	11904.8i	0.578444i	−0.957262	+	0.289222i	$0.906604\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 7366.84i	− 0.355109i
$756$	0	0
$757$	899.610i	0.0431927i	0.999767	+	0.0215964i	$0.00687487\pi$
$757$	899.610i	0.0431927i	−0.999767	+	0.0215964i	$0.993125\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 9242.47i	− 0.440262i	−0.975470	−	0.220131i	$-0.929352\pi$
$761$	− 9242.47i	− 0.440262i	0.975470	−	0.220131i	$-0.0706485\pi$
$762$	0	0
$763$	−11344.3	−0.538256
$764$	0	0
$765$	0	0
$766$	0	0
$767$	676.800	0.0318616
$768$	0	0
$769$	−27775.4	−1.30248	−0.651239	−	0.758872i	$-0.725751\pi$
$769$	−27775.4	−1.30248	−0.651239	+	0.758872i	$0.725751\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−27780.2	−1.29260	−0.646302	−	0.763082i	$-0.723685\pi$
$773$	−27780.2	−1.29260	−0.646302	+	0.763082i	$0.723685\pi$
$774$	0	0
$775$	− 11961.5i	− 0.554411i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	41975.4i	1.93058i
$780$	0	0
$781$	− 14395.3i	− 0.659544i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 6151.40i	− 0.279685i
$786$	0	0
$787$	−41572.7	−1.88298	−0.941490	−	0.337040i	$-0.890574\pi$
$787$	−41572.7	−1.88298	−0.941490	+	0.337040i	$0.890574\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	23051.0	1.03616
$792$	0	0
$793$	2135.25	0.0956181
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−2407.07	−0.106980	−0.0534899	−	0.998568i	$-0.517035\pi$
$797$	−2407.07	−0.106980	−0.0534899	+	0.998568i	$0.517035\pi$
$798$	0	0
$799$	− 4903.13i	− 0.217097i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 7181.73i	− 0.315614i
$804$	0	0
$805$	2760.20i	0.120850i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2008.27i	0.0872769i	0.999047	+	0.0436385i	$0.0138950\pi$
$809$	2008.27i	0.0872769i	−0.999047	+	0.0436385i	$0.986105\pi$
$810$	0	0
$811$	−33619.7	−1.45567	−0.727833	−	0.685754i	$-0.759473\pi$
$811$	−33619.7	−1.45567	−0.727833	+	0.685754i	$0.759473\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	8886.97	0.381960
$816$	0	0
$817$	−44510.5	−1.90603
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3163.91	0.134496	0.0672480	−	0.997736i	$-0.478578\pi$
$821$	3163.91	0.134496	0.0672480	+	0.997736i	$0.478578\pi$
$822$	0	0
$823$	− 9816.28i	− 0.415764i	−0.978154	−	0.207882i	$-0.933343\pi$
$823$	− 9816.28i	− 0.415764i	0.978154	−	0.207882i	$-0.0666570\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1020.59i	− 0.0429135i	−0.999770	−	0.0214568i	$-0.993170\pi$
$827$	− 1020.59i	− 0.0429135i	0.999770	−	0.0214568i	$-0.00683043\pi$
$828$	0	0
$829$	24566.1i	1.02921i	0.857428	+	0.514605i	$0.172061\pi$
$829$	24566.1i	1.02921i	−0.857428	+	0.514605i	$0.827939\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 17823.1i	− 0.741337i
$834$	0	0
$835$	11064.8	0.458578
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−27158.7	−1.11755	−0.558773	−	0.829321i	$-0.688728\pi$
$839$	−27158.7	−1.11755	−0.558773	+	0.829321i	$0.688728\pi$
$840$	0	0
$841$	8374.86	0.343387
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−28070.3	−1.14278
$846$	0	0
$847$	− 8882.82i	− 0.360351i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	1420.55i	0.0572218i
$852$	0	0
$853$	34573.8i	1.38779i	0.720076	+	0.693895i	$0.244107\pi$
$853$	34573.8i	1.38779i	−0.720076	+	0.693895i	$0.755893\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	23596.3i	0.940531i	0.882525	+	0.470265i	$0.155842\pi$
$857$	23596.3i	0.940531i	−0.882525	+	0.470265i	$0.844158\pi$
$858$	0	0
$859$	−284.040	−0.0112821	−0.00564105	−	0.999984i	$-0.501796\pi$
$859$	−284.040	−0.0112821	−0.00564105	+	0.999984i	$0.501796\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−10351.3	−0.408299	−0.204149	−	0.978940i	$-0.565443\pi$
$863$	−10351.3	−0.408299	−0.204149	+	0.978940i	$0.565443\pi$
$864$	0	0
$865$	10365.0	0.407421
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−11988.5	−0.467989
$870$	0	0
$871$	− 292.331i	− 0.0113723i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 13530.3i	− 0.522751i
$876$	0	0
$877$	29383.3i	1.13136i	0.824625	+	0.565680i	$0.191386\pi$
$877$	29383.3i	1.13136i	−0.824625	+	0.565680i	$0.808614\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	29740.6i	1.13733i	0.822570	+	0.568663i	$0.192539\pi$
$881$	29740.6i	1.13733i	−0.822570	+	0.568663i	$0.807461\pi$
$882$	0	0
$883$	7250.82	0.276341	0.138171	−	0.990408i	$-0.455878\pi$
$883$	7250.82	0.276341	0.138171	+	0.990408i	$0.455878\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35040.1	1.32641	0.663207	−	0.748436i	$-0.269195\pi$
$887$	35040.1	1.32641	0.663207	+	0.748436i	$0.269195\pi$
$888$	0	0
$889$	18159.0	0.685076
$890$	0	0
$891$	0	0
$892$	0	0
$893$	7034.18	0.263594
$894$	0	0
$895$	− 35772.1i	− 1.33601i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	55116.3i	2.04475i
$900$	0	0
$901$	21162.9i	0.782507i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	47345.5i	1.73903i
$906$	0	0
$907$	23140.4	0.847149	0.423575	−	0.905861i	$-0.360775\pi$
$907$	23140.4	0.847149	0.423575	+	0.905861i	$0.360775\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−6.16618	−0.000224253 0	−0.000112127	−	1.00000i	$-0.500036\pi$
$911$	−6.16618	−0.000224253 0	−0.000112127		1.00000i	$0.500036\pi$
$912$	0	0
$913$	−28031.1	−1.01609
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4782.98	0.172244
$918$	0	0
$919$	39266.8i	1.40946i	0.709477	+	0.704729i	$0.248931\pi$
$919$	39266.8i	1.40946i	−0.709477	+	0.704729i	$0.751069\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1538.86i	0.0548778i
$924$	0	0
$925$	3191.22i	0.113434i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4172.41i	0.147355i	0.997282	+	0.0736773i	$0.0234735\pi$
$929$	4172.41i	0.147355i	−0.997282	+	0.0736773i	$0.976527\pi$
$930$	0	0
$931$	25569.6	0.900117
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−29481.4	−1.03117
$936$	0	0
$937$	33899.4	1.18190	0.590952	−	0.806707i	$-0.298752\pi$
$937$	33899.4	1.18190	0.590952	+	0.806707i	$0.298752\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−10061.5	−0.348562	−0.174281	−	0.984696i	$-0.555760\pi$
$941$	−10061.5	−0.348562	−0.174281	+	0.984696i	$0.555760\pi$
$942$	0	0
$943$	5492.46i	0.189670i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 40149.5i	− 1.37770i	−0.724903	−	0.688851i	$-0.758115\pi$
$947$	− 40149.5i	− 1.37770i	0.724903	−	0.688851i	$-0.241885\pi$
$948$	0	0
$949$	767.729i	0.0262608i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 20507.0i	− 0.697047i	−0.937300	−	0.348524i	$-0.886683\pi$
$953$	− 20507.0i	− 0.697047i	0.937300	−	0.348524i	$-0.113317\pi$
$954$	0	0
$955$	958.105	0.0324645
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10478.8	0.352845
$960$	0	0
$961$	−62927.3	−2.11229
$962$	0	0
$963$	0	0
$964$	0	0
$965$	45953.0	1.53293
$966$	0	0
$967$	− 50805.8i	− 1.68956i	−0.535113	−	0.844781i	$-0.679731\pi$
$967$	− 50805.8i	− 1.68956i	0.535113	−	0.844781i	$-0.320269\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	40091.8i	1.32503i	0.749047	+	0.662517i	$0.230512\pi$
$971$	40091.8i	1.32503i	−0.749047	+	0.662517i	$0.769488\pi$
$972$	0	0
$973$	− 11210.0i	− 0.369348i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 50207.4i	− 1.64409i	−0.569421	−	0.822046i	$-0.692833\pi$
$977$	− 50207.4i	− 1.64409i	0.569421	−	0.822046i	$-0.307167\pi$
$978$	0	0
$979$	−25129.7	−0.820377
$980$	0	0
$981$	0	0
$982$	0	0
$983$	20394.2	0.661724	0.330862	−	0.943679i	$-0.392660\pi$
$983$	20394.2	0.661724	0.330862	+	0.943679i	$0.392660\pi$
$984$	0	0
$985$	16592.5	0.536733
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−5824.18	−0.187258
$990$	0	0
$991$	− 6537.44i	− 0.209554i	−0.994496	−	0.104777i	$-0.966587\pi$
$991$	− 6537.44i	− 0.209554i	0.994496	−	0.104777i	$-0.0334130\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 52641.7i	− 1.67724i
$996$	0	0
$997$	8180.90i	0.259871i	0.991522	+	0.129936i	$0.0414771\pi$
$997$	8180.90i	0.259871i	−0.991522	+	0.129936i	$0.958523\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.4.f.b.431.5		24
3.2	odd	2	inner	864.4.f.b.431.19		24
4.3	odd	2		216.4.f.b.107.18	yes	24
8.3	odd	2	inner	864.4.f.b.431.20		24
8.5	even	2		216.4.f.b.107.8	yes	24
12.11	even	2		216.4.f.b.107.7	✓	24
24.5	odd	2		216.4.f.b.107.17	yes	24
24.11	even	2	inner	864.4.f.b.431.6		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.4.f.b.107.7	✓	24	12.11	even	2
216.4.f.b.107.8	yes	24	8.5	even	2
216.4.f.b.107.17	yes	24	24.5	odd	2
216.4.f.b.107.18	yes	24	4.3	odd	2
864.4.f.b.431.5		24	1.1	even	1	trivial
864.4.f.b.431.6		24	24.11	even	2	inner
864.4.f.b.431.19		24	3.2	odd	2	inner
864.4.f.b.431.20		24	8.3	odd	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	864.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-12.8173 q^{5} -12.3152i q^{7} +O(q^{10})\)	\(q-12.8173 q^{5} -12.3152i q^{7} +24.6924i q^{11} -2.63962i q^{13} -93.1513i q^{17} +133.638 q^{19} +17.4864 q^{23} +39.2827 q^{25} -181.008 q^{29} -304.497i q^{31} +157.848i q^{35} +81.2372i q^{37} +314.098i q^{41} -333.068 q^{43} +52.6362 q^{47} +191.335 q^{49} -227.189 q^{53} -316.489i q^{55} +256.400i q^{59} +808.924i q^{61} +33.8328i q^{65} +110.747 q^{67} -582.985 q^{71} -290.848 q^{73} +304.093 q^{77} +485.514i q^{79} +1135.21i q^{83} +1193.95i q^{85} +1017.71i q^{89} -32.5076 q^{91} -1712.87 q^{95} -1265.30 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 48 q^{19} + 600 q^{25} + 432 q^{43} - 816 q^{49} - 1632 q^{67} - 216 q^{73} - 3600 q^{91} + 2280 q^{97}+O(q^{100}) \) 24 * q + 48 * q^19 + 600 * q^25 + 432 * q^43 - 816 * q^49 - 1632 * q^67 - 216 * q^73 - 3600 * q^91 + 2280 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more