LMFDB - Embedded newform 864.3.e.a.161.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(161,864)]

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([0, 0, 1]))

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("864.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.e (of order $2$, degree $1$, 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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-5})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} + 9 $ x^4 - 4*x^2 + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.4
Root			$1.58114 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.3.e.a.161.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+4.47214i q^{5} +3.32456 q^{7} +O(q^{10})$	$q+4.47214i q^{5} +3.32456 q^{7} +7.75955i q^{11} +13.9737 q^{13} +3.55415i q^{17} -3.64911 q^{19} -13.4164i q^{23} +5.00000 q^{25} +24.9969i q^{29} -31.2982 q^{31} +14.8679i q^{35} -11.9737 q^{37} +78.6625i q^{41} +68.5964 q^{43} -51.5629i q^{47} -37.9473 q^{49} +76.8265i q^{53} -34.7018 q^{55} +11.9650i q^{59} -39.9737 q^{61} +62.4921i q^{65} +19.7018 q^{67} +110.234i q^{71} +4.94733 q^{73} +25.7971i q^{77} -45.3246 q^{79} -94.7151i q^{83} -15.8947 q^{85} +53.5479i q^{89} +46.4562 q^{91} -16.3193i q^{95} +80.9473 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{7}+O(q^{10}) $ 4 * q - 12 * q^7	$ 4 q - 12 q^{7} - 20 q^{13} + 36 q^{19} + 20 q^{25} - 24 q^{31} + 28 q^{37} + 72 q^{43} - 240 q^{55} - 84 q^{61} + 180 q^{67} - 132 q^{73} - 156 q^{79} + 240 q^{85} + 540 q^{91} + 172 q^{97}+O(q^{100}) $ 4 * q - 12 * q^7 - 20 * q^13 + 36 * q^19 + 20 * q^25 - 24 * q^31 + 28 * q^37 + 72 * q^43 - 240 * q^55 - 84 * q^61 + 180 * q^67 - 132 * q^73 - 156 * q^79 + 240 * q^85 + 540 * q^91 + 172 * 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
$5$	4.47214i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	4.47214i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	0	0
$7$	3.32456	0.474936	0.237468	−	0.971395i	$-0.423682\pi$
$7$	3.32456	0.474936	0.237468	+	0.971395i	$0.423682\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	7.75955i	0.705414i	0.935734	+	0.352707i	$0.114739\pi$
$11$	7.75955i	0.705414i	−0.935734	+	0.352707i	$0.885261\pi$
$12$	0	0
$13$	13.9737	1.07490	0.537449	−	0.843296i	$-0.319388\pi$
$13$	13.9737	1.07490	0.537449	+	0.843296i	$0.319388\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.55415i	0.209068i	0.994521	+	0.104534i	$0.0333351\pi$
$17$	3.55415i	0.209068i	−0.994521	+	0.104534i	$0.966665\pi$
$18$	0	0
$19$	−3.64911	−0.192058	−0.0960292	−	0.995379i	$-0.530614\pi$
$19$	−3.64911	−0.192058	−0.0960292	+	0.995379i	$0.530614\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 13.4164i	− 0.583322i	−0.956522	−	0.291661i	$-0.905792\pi$
$23$	− 13.4164i	− 0.583322i	0.956522	−	0.291661i	$-0.0942079\pi$
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	24.9969i	0.861960i	0.902361	+	0.430980i	$0.141832\pi$
$29$	24.9969i	0.861960i	−0.902361	+	0.430980i	$0.858168\pi$
$30$	0	0
$31$	−31.2982	−1.00962	−0.504810	−	0.863230i	$-0.668437\pi$
$31$	−31.2982	−1.00962	−0.504810	+	0.863230i	$0.668437\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	14.8679i	0.424796i
$36$	0	0
$37$	−11.9737	−0.323613	−0.161806	−	0.986823i	$-0.551732\pi$
$37$	−11.9737	−0.323613	−0.161806	+	0.986823i	$0.551732\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	78.6625i	1.91860i	0.282394	+	0.959299i	$0.408872\pi$
$41$	78.6625i	1.91860i	−0.282394	+	0.959299i	$0.591128\pi$
$42$	0	0
$43$	68.5964	1.59527	0.797633	−	0.603143i	$-0.206085\pi$
$43$	68.5964	1.59527	0.797633	+	0.603143i	$0.206085\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 51.5629i	− 1.09708i	−0.836123	−	0.548542i	$-0.815183\pi$
$47$	− 51.5629i	− 1.09708i	0.836123	−	0.548542i	$-0.184817\pi$
$48$	0	0
$49$	−37.9473	−0.774435
$50$	0	0
$51$	0	0
$52$	0	0
$53$	76.8265i	1.44956i	0.688982	+	0.724779i	$0.258058\pi$
$53$	76.8265i	1.44956i	−0.688982	+	0.724779i	$0.741942\pi$
$54$	0	0
$55$	−34.7018	−0.630941
$56$	0	0
$57$	0	0
$58$	0	0
$59$	11.9650i	0.202796i	0.994846	+	0.101398i	$0.0323315\pi$
$59$	11.9650i	0.202796i	−0.994846	+	0.101398i	$0.967668\pi$
$60$	0	0
$61$	−39.9737	−0.655306	−0.327653	−	0.944798i	$-0.606258\pi$
$61$	−39.9737	−0.655306	−0.327653	+	0.944798i	$0.606258\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	62.4921i	0.961417i
$66$	0	0
$67$	19.7018	0.294056	0.147028	−	0.989132i	$-0.453029\pi$
$67$	19.7018	0.294056	0.147028	+	0.989132i	$0.453029\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	110.234i	1.55259i	0.630367	+	0.776297i	$0.282904\pi$
$71$	110.234i	1.55259i	−0.630367	+	0.776297i	$0.717096\pi$
$72$	0	0
$73$	4.94733	0.0677717	0.0338858	−	0.999426i	$-0.489212\pi$
$73$	4.94733	0.0677717	0.0338858	+	0.999426i	$0.489212\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	25.7971i	0.335027i
$78$	0	0
$79$	−45.3246	−0.573729	−0.286864	−	0.957971i	$-0.592613\pi$
$79$	−45.3246	−0.573729	−0.286864	+	0.957971i	$0.592613\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 94.7151i	− 1.14115i	−0.821247	−	0.570573i	$-0.806721\pi$
$83$	− 94.7151i	− 1.14115i	0.821247	−	0.570573i	$-0.193279\pi$
$84$	0	0
$85$	−15.8947	−0.186996
$86$	0	0
$87$	0	0
$88$	0	0
$89$	53.5479i	0.601661i	0.953678	+	0.300831i	$0.0972639\pi$
$89$	53.5479i	0.601661i	−0.953678	+	0.300831i	$0.902736\pi$
$90$	0	0
$91$	46.4562	0.510508
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 16.3193i	− 0.171782i
$96$	0	0
$97$	80.9473	0.834509	0.417254	−	0.908790i	$-0.362992\pi$
$97$	80.9473	0.834509	0.417254	+	0.908790i	$0.362992\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	141.272i	1.39874i	0.714762	+	0.699368i	$0.246535\pi$
$101$	141.272i	1.39874i	−0.714762	+	0.699368i	$0.753465\pi$
$102$	0	0
$103$	−104.026	−1.00996	−0.504982	−	0.863130i	$-0.668501\pi$
$103$	−104.026	−1.00996	−0.504982	+	0.863130i	$0.668501\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	128.005i	1.19631i	0.801381	+	0.598154i	$0.204099\pi$
$107$	128.005i	1.19631i	−0.801381	+	0.598154i	$0.795901\pi$
$108$	0	0
$109$	89.8947	0.824722	0.412361	−	0.911021i	$-0.364704\pi$
$109$	89.8947	0.824722	0.412361	+	0.911021i	$0.364704\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	89.3249i	0.790486i	0.918577	+	0.395243i	$0.129340\pi$
$113$	89.3249i	0.790486i	−0.918577	+	0.395243i	$0.870660\pi$
$114$	0	0
$115$	60.0000	0.521739
$116$	0	0
$117$	0	0
$118$	0	0
$119$	11.8160i	0.0992940i
$120$	0	0
$121$	60.7893	0.502391
$122$	0	0
$123$	0	0
$124$	0	0
$125$	134.164i	1.07331i
$126$	0	0
$127$	−39.4036	−0.310264	−0.155132	−	0.987894i	$-0.549580\pi$
$127$	−39.4036	−0.310264	−0.155132	+	0.987894i	$0.549580\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 210.457i	− 1.60654i	−0.595613	−	0.803271i	$-0.703091\pi$
$131$	− 210.457i	− 1.60654i	0.595613	−	0.803271i	$-0.296909\pi$
$132$	0	0
$133$	−12.1317	−0.0912156
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 23.2787i	− 0.169917i	−0.996384	−	0.0849586i	$-0.972924\pi$
$137$	− 23.2787i	− 0.169917i	0.996384	−	0.0849586i	$-0.0270758\pi$
$138$	0	0
$139$	255.491	1.83807	0.919033	−	0.394181i	$-0.128972\pi$
$139$	255.491	1.83807	0.919033	+	0.394181i	$0.128972\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	108.429i	0.758248i
$144$	0	0
$145$	−111.789	−0.770961
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 127.056i	− 0.852723i	−0.904553	−	0.426362i	$-0.859795\pi$
$149$	− 127.056i	− 0.852723i	0.904553	−	0.426362i	$-0.140205\pi$
$150$	0	0
$151$	−291.816	−1.93255	−0.966277	−	0.257505i	$-0.917100\pi$
$151$	−291.816	−1.93255	−0.966277	+	0.257505i	$0.917100\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 139.970i	− 0.903032i
$156$	0	0
$157$	113.895	0.725444	0.362722	−	0.931897i	$-0.381848\pi$
$157$	113.895	0.725444	0.362722	+	0.931897i	$0.381848\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 44.6036i	− 0.277041i
$162$	0	0
$163$	200.035	1.22721	0.613604	−	0.789614i	$-0.289719\pi$
$163$	200.035	1.22721	0.613604	+	0.789614i	$0.289719\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 58.6712i	− 0.351325i	−0.984450	−	0.175662i	$-0.943793\pi$
$167$	− 58.6712i	− 0.351325i	0.984450	−	0.175662i	$-0.0562067\pi$
$168$	0	0
$169$	26.2633	0.155404
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 134.164i	− 0.775515i	−0.921761	−	0.387757i	$-0.873250\pi$
$173$	− 134.164i	− 0.775515i	0.921761	−	0.387757i	$-0.126750\pi$
$174$	0	0
$175$	16.6228	0.0949873
$176$	0	0
$177$	0	0
$178$	0	0
$179$	330.703i	1.84750i	0.382996	+	0.923750i	$0.374892\pi$
$179$	330.703i	1.84750i	−0.382996	+	0.923750i	$0.625108\pi$
$180$	0	0
$181$	−21.9210	−0.121110	−0.0605552	−	0.998165i	$-0.519287\pi$
$181$	−21.9210	−0.121110	−0.0605552	+	0.998165i	$0.519287\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 53.5479i	− 0.289448i
$186$	0	0
$187$	−27.5787	−0.147479
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 270.729i	− 1.41743i	−0.705496	−	0.708714i	$-0.749276\pi$
$191$	− 270.729i	− 1.41743i	0.705496	−	0.708714i	$-0.250724\pi$
$192$	0	0
$193$	−192.895	−0.999454	−0.499727	−	0.866183i	$-0.666566\pi$
$193$	−192.895	−0.999454	−0.499727	+	0.866183i	$0.666566\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 224.407i	− 1.13912i	−0.821949	−	0.569561i	$-0.807113\pi$
$197$	− 224.407i	− 1.13912i	0.821949	−	0.569561i	$-0.192887\pi$
$198$	0	0
$199$	−281.114	−1.41263	−0.706316	−	0.707896i	$-0.749644\pi$
$199$	−281.114	−1.41263	−0.706316	+	0.707896i	$0.749644\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	83.1034i	0.409376i
$204$	0	0
$205$	−351.789	−1.71605
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 28.3155i	− 0.135481i
$210$	0	0
$211$	−69.4911	−0.329342	−0.164671	−	0.986349i	$-0.552656\pi$
$211$	−69.4911	−0.329342	−0.164671	+	0.986349i	$0.552656\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	306.773i	1.42685i
$216$	0	0
$217$	−104.053	−0.479505
$218$	0	0
$219$	0	0
$220$	0	0
$221$	49.6646i	0.224727i
$222$	0	0
$223$	99.0875	0.444339	0.222169	−	0.975008i	$-0.428686\pi$
$223$	99.0875	0.444339	0.222169	+	0.975008i	$0.428686\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 169.706i	− 0.747602i	−0.927509	−	0.373801i	$-0.878054\pi$
$227$	− 169.706i	− 0.747602i	0.927509	−	0.373801i	$-0.121946\pi$
$228$	0	0
$229$	146.000	0.637555	0.318777	−	0.947830i	$-0.396728\pi$
$229$	146.000	0.637555	0.318777	+	0.947830i	$0.396728\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 155.489i	− 0.667335i	−0.942691	−	0.333667i	$-0.891714\pi$
$233$	− 155.489i	− 0.667335i	0.942691	−	0.333667i	$-0.108286\pi$
$234$	0	0
$235$	230.596	0.981261
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 149.981i	− 0.627536i	−0.949500	−	0.313768i	$-0.898409\pi$
$239$	− 149.981i	− 0.627536i	0.949500	−	0.313768i	$-0.101591\pi$
$240$	0	0
$241$	342.684	1.42193	0.710963	−	0.703230i	$-0.248259\pi$
$241$	342.684	1.42193	0.710963	+	0.703230i	$0.248259\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 169.706i	− 0.692676i
$246$	0	0
$247$	−50.9915	−0.206443
$248$	0	0
$249$	0	0
$250$	0	0
$251$	346.520i	1.38056i	0.723544	+	0.690278i	$0.242512\pi$
$251$	346.520i	1.38056i	−0.723544	+	0.690278i	$0.757488\pi$
$252$	0	0
$253$	104.105	0.411484
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 353.863i	− 1.37690i	−0.725284	−	0.688450i	$-0.758291\pi$
$257$	− 353.863i	− 1.37690i	0.725284	−	0.688450i	$-0.241709\pi$
$258$	0	0
$259$	−39.8071	−0.153695
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 336.210i	− 1.27837i	−0.769054	−	0.639183i	$-0.779273\pi$
$263$	− 336.210i	− 1.27837i	0.769054	−	0.639183i	$-0.220727\pi$
$264$	0	0
$265$	−343.579	−1.29652
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 290.453i	− 1.07975i	−0.841745	−	0.539876i	$-0.818471\pi$
$269$	− 290.453i	− 1.07975i	0.841745	−	0.539876i	$-0.181529\pi$
$270$	0	0
$271$	170.026	0.627403	0.313702	−	0.949522i	$-0.398431\pi$
$271$	170.026	0.627403	0.313702	+	0.949522i	$0.398431\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	38.7978i	0.141083i
$276$	0	0
$277$	−142.105	−0.513016	−0.256508	−	0.966542i	$-0.582572\pi$
$277$	−142.105	−0.513016	−0.256508	+	0.966542i	$0.582572\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 430.925i	− 1.53354i	−0.641920	−	0.766771i	$-0.721862\pi$
$281$	− 430.925i	− 1.53354i	0.641920	−	0.766771i	$-0.278138\pi$
$282$	0	0
$283$	188.596	0.666419	0.333209	−	0.942853i	$-0.391868\pi$
$283$	188.596	0.666419	0.333209	+	0.942853i	$0.391868\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	261.518i	0.911212i
$288$	0	0
$289$	276.368	0.956291
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 22.1251i	− 0.0755124i	−0.999287	−	0.0377562i	$-0.987979\pi$
$293$	− 22.1251i	− 0.0755124i	0.999287	−	0.0377562i	$-0.0120210\pi$
$294$	0	0
$295$	−53.5089	−0.181386
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 187.476i	− 0.627011i
$300$	0	0
$301$	228.053	0.757650
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 178.768i	− 0.586124i
$306$	0	0
$307$	−373.789	−1.21755	−0.608777	−	0.793341i	$-0.708340\pi$
$307$	−373.789	−1.21755	−0.608777	+	0.793341i	$0.708340\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 274.636i	− 0.883075i	−0.897243	−	0.441537i	$-0.854433\pi$
$311$	− 274.636i	− 0.883075i	0.897243	−	0.441537i	$-0.145567\pi$
$312$	0	0
$313$	582.684	1.86161	0.930805	−	0.365516i	$-0.119107\pi$
$313$	582.684	1.86161	0.930805	+	0.365516i	$0.119107\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 318.086i	− 1.00343i	−0.865034	−	0.501713i	$-0.832703\pi$
$317$	− 318.086i	− 1.00343i	0.865034	−	0.501713i	$-0.167297\pi$
$318$	0	0
$319$	−193.964	−0.608039
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 12.9695i	− 0.0401533i
$324$	0	0
$325$	69.8683	0.214979
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 171.424i	− 0.521045i
$330$	0	0
$331$	483.175	1.45974	0.729872	−	0.683584i	$-0.239580\pi$
$331$	483.175	1.45974	0.729872	+	0.683584i	$0.239580\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	88.1090i	0.263012i
$336$	0	0
$337$	−111.263	−0.330158	−0.165079	−	0.986280i	$-0.552788\pi$
$337$	−111.263	−0.330158	−0.165079	+	0.986280i	$0.552788\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 242.860i	− 0.712200i
$342$	0	0
$343$	−289.061	−0.842744
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 381.465i	− 1.09932i	−0.835387	−	0.549662i	$-0.814757\pi$
$347$	− 381.465i	− 1.09932i	0.835387	−	0.549662i	$-0.185243\pi$
$348$	0	0
$349$	−382.026	−1.09463	−0.547316	−	0.836926i	$-0.684350\pi$
$349$	−382.026	−1.09463	−0.547316	+	0.836926i	$0.684350\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	304.105i	0.861488i	0.902474	+	0.430744i	$0.141749\pi$
$353$	304.105i	0.861488i	−0.902474	+	0.430744i	$0.858251\pi$
$354$	0	0
$355$	−492.982	−1.38868
$356$	0	0
$357$	0	0
$358$	0	0
$359$	315.984i	0.880177i	0.897954	+	0.440089i	$0.145053\pi$
$359$	315.984i	0.880177i	−0.897954	+	0.440089i	$0.854947\pi$
$360$	0	0
$361$	−347.684	−0.963114
$362$	0	0
$363$	0	0
$364$	0	0
$365$	22.1251i	0.0606168i
$366$	0	0
$367$	490.132	1.33551	0.667754	−	0.744382i	$-0.267256\pi$
$367$	490.132	1.33551	0.667754	+	0.744382i	$0.267256\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	255.414i	0.688448i
$372$	0	0
$373$	451.605	1.21074	0.605369	−	0.795945i	$-0.293026\pi$
$373$	451.605	1.21074	0.605369	+	0.795945i	$0.293026\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	349.298i	0.926519i
$378$	0	0
$379$	184.140	0.485858	0.242929	−	0.970044i	$-0.421892\pi$
$379$	184.140	0.485858	0.242929	+	0.970044i	$0.421892\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	231.782i	0.605175i	0.953122	+	0.302588i	$0.0978505\pi$
$383$	231.782i	0.605175i	−0.953122	+	0.302588i	$0.902150\pi$
$384$	0	0
$385$	−115.368	−0.299657
$386$	0	0
$387$	0	0
$388$	0	0
$389$	140.472i	0.361111i	0.983565	+	0.180555i	$0.0577895\pi$
$389$	140.472i	0.361111i	−0.983565	+	0.180555i	$0.942210\pi$
$390$	0	0
$391$	47.6840	0.121954
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 202.698i	− 0.513158i
$396$	0	0
$397$	154.316	0.388705	0.194353	−	0.980932i	$-0.437739\pi$
$397$	154.316	0.388705	0.194353	+	0.980932i	$0.437739\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 500.879i	− 1.24908i	−0.780995	−	0.624538i	$-0.785287\pi$
$401$	− 500.879i	− 1.24908i	0.780995	−	0.624538i	$-0.214713\pi$
$402$	0	0
$403$	−437.351	−1.08524
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 92.9103i	− 0.228281i
$408$	0	0
$409$	−127.263	−0.311157	−0.155579	−	0.987824i	$-0.549724\pi$
$409$	−127.263	−0.311157	−0.155579	+	0.987824i	$0.549724\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.7781i	0.0963151i
$414$	0	0
$415$	423.579	1.02067
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 47.5065i	− 0.113381i	−0.998392	−	0.0566903i	$-0.981945\pi$
$419$	− 47.5065i	− 0.113381i	0.998392	−	0.0566903i	$-0.0180548\pi$
$420$	0	0
$421$	−535.658	−1.27235	−0.636173	−	0.771546i	$-0.719484\pi$
$421$	−535.658	−1.27235	−0.636173	+	0.771546i	$0.719484\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	17.7708i	0.0418136i
$426$	0	0
$427$	−132.895	−0.311229
$428$	0	0
$429$	0	0
$430$	0	0
$431$	253.311i	0.587729i	0.955847	+	0.293865i	$0.0949415\pi$
$431$	253.311i	0.587729i	−0.955847	+	0.293865i	$0.905059\pi$
$432$	0	0
$433$	−369.157	−0.852557	−0.426279	−	0.904592i	$-0.640176\pi$
$433$	−369.157	−0.852557	−0.426279	+	0.904592i	$0.640176\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	48.9580i	0.112032i
$438$	0	0
$439$	202.877	0.462134	0.231067	−	0.972938i	$-0.425778\pi$
$439$	202.877	0.462134	0.231067	+	0.972938i	$0.425778\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 340.118i	− 0.767760i	−0.923383	−	0.383880i	$-0.874588\pi$
$443$	− 340.118i	− 0.767760i	0.923383	−	0.383880i	$-0.125412\pi$
$444$	0	0
$445$	−239.473	−0.538142
$446$	0	0
$447$	0	0
$448$	0	0
$449$	96.1977i	0.214249i	0.994246	+	0.107124i	$0.0341643\pi$
$449$	96.1977i	0.214249i	−0.994246	+	0.107124i	$0.965836\pi$
$450$	0	0
$451$	−610.386	−1.35341
$452$	0	0
$453$	0	0
$454$	0	0
$455$	207.759i	0.456612i
$456$	0	0
$457$	438.211	0.958885	0.479443	−	0.877573i	$-0.340839\pi$
$457$	438.211	0.958885	0.479443	+	0.877573i	$0.340839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 254.912i	− 0.552954i	−0.961021	−	0.276477i	$-0.910833\pi$
$461$	− 254.912i	− 0.552954i	0.961021	−	0.276477i	$-0.0891669\pi$
$462$	0	0
$463$	822.570	1.77661	0.888305	−	0.459255i	$-0.151883\pi$
$463$	822.570	1.77661	0.888305	+	0.459255i	$0.151883\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	497.859i	1.06608i	0.846091	+	0.533039i	$0.178950\pi$
$467$	497.859i	1.06608i	−0.846091	+	0.533039i	$0.821050\pi$
$468$	0	0
$469$	65.4997	0.139658
$470$	0	0
$471$	0	0
$472$	0	0
$473$	532.278i	1.12532i
$474$	0	0
$475$	−18.2456	−0.0384117
$476$	0	0
$477$	0	0
$478$	0	0
$479$	258.913i	0.540528i	0.962786	+	0.270264i	$0.0871109\pi$
$479$	258.913i	0.540528i	−0.962786	+	0.270264i	$0.912889\pi$
$480$	0	0
$481$	−167.316	−0.347850
$482$	0	0
$483$	0	0
$484$	0	0
$485$	362.007i	0.746407i
$486$	0	0
$487$	287.114	0.589556	0.294778	−	0.955566i	$-0.404754\pi$
$487$	287.114	0.589556	0.294778	+	0.955566i	$0.404754\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	93.0593i	0.189530i	0.995500	+	0.0947650i	$0.0302100\pi$
$491$	93.0593i	0.189530i	−0.995500	+	0.0947650i	$0.969790\pi$
$492$	0	0
$493$	−88.8427	−0.180208
$494$	0	0
$495$	0	0
$496$	0	0
$497$	366.480i	0.737384i
$498$	0	0
$499$	−362.421	−0.726295	−0.363148	−	0.931732i	$-0.618298\pi$
$499$	−362.421	−0.726295	−0.363148	+	0.931732i	$0.618298\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 839.726i	− 1.66944i	−0.550678	−	0.834718i	$-0.685631\pi$
$503$	− 839.726i	− 1.66944i	0.550678	−	0.834718i	$-0.314369\pi$
$504$	0	0
$505$	−631.789	−1.25107
$506$	0	0
$507$	0	0
$508$	0	0
$509$	451.450i	0.886936i	0.896290	+	0.443468i	$0.146252\pi$
$509$	451.450i	0.886936i	−0.896290	+	0.443468i	$0.853748\pi$
$510$	0	0
$511$	16.4477	0.0321872
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 465.220i	− 0.903340i
$516$	0	0
$517$	400.105	0.773898
$518$	0	0
$519$	0	0
$520$	0	0
$521$	106.978i	0.205332i	0.994716	+	0.102666i	$0.0327373\pi$
$521$	106.978i	0.205332i	−0.994716	+	0.102666i	$0.967263\pi$
$522$	0	0
$523$	−447.999	−0.856595	−0.428298	−	0.903638i	$-0.640887\pi$
$523$	−447.999	−0.856595	−0.428298	+	0.903638i	$0.640887\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 111.239i	− 0.211079i
$528$	0	0
$529$	349.000	0.659735
$530$	0	0
$531$	0	0
$532$	0	0
$533$	1099.20i	2.06230i
$534$	0	0
$535$	−572.456	−1.07001
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 294.454i	− 0.546298i
$540$	0	0
$541$	685.658	1.26739	0.633695	−	0.773583i	$-0.281538\pi$
$541$	685.658	1.26739	0.633695	+	0.773583i	$0.281538\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	402.021i	0.737654i
$546$	0	0
$547$	−685.053	−1.25238	−0.626191	−	0.779670i	$-0.715387\pi$
$547$	−685.053	−1.25238	−0.626191	+	0.779670i	$0.715387\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 91.2163i	− 0.165547i
$552$	0	0
$553$	−150.684	−0.272485
$554$	0	0
$555$	0	0
$556$	0	0
$557$	750.283i	1.34701i	0.739184	+	0.673504i	$0.235211\pi$
$557$	750.283i	1.34701i	−0.739184	+	0.673504i	$0.764789\pi$
$558$	0	0
$559$	958.544	1.71475
$560$	0	0
$561$	0	0
$562$	0	0
$563$	554.780i	0.985400i	0.870199	+	0.492700i	$0.163990\pi$
$563$	554.780i	0.985400i	−0.870199	+	0.492700i	$0.836010\pi$
$564$	0	0
$565$	−399.473	−0.707032
$566$	0	0
$567$	0	0
$568$	0	0
$569$	178.532i	0.313765i	0.987617	+	0.156882i	$0.0501444\pi$
$569$	178.532i	0.313765i	−0.987617	+	0.156882i	$0.949856\pi$
$570$	0	0
$571$	641.456	1.12339	0.561695	−	0.827344i	$-0.310150\pi$
$571$	641.456	1.12339	0.561695	+	0.827344i	$0.310150\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 67.0820i	− 0.116664i
$576$	0	0
$577$	935.789	1.62182	0.810909	−	0.585173i	$-0.198973\pi$
$577$	935.789	1.62182	0.810909	+	0.585173i	$0.198973\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 314.885i	− 0.541972i
$582$	0	0
$583$	−596.140	−1.02254
$584$	0	0
$585$	0	0
$586$	0	0
$587$	852.789i	1.45279i	0.687276	+	0.726396i	$0.258806\pi$
$587$	852.789i	1.45279i	−0.687276	+	0.726396i	$0.741194\pi$
$588$	0	0
$589$	114.211	0.193906
$590$	0	0
$591$	0	0
$592$	0	0
$593$	329.996i	0.556486i	0.960511	+	0.278243i	$0.0897520\pi$
$593$	329.996i	0.556486i	−0.960511	+	0.278243i	$0.910248\pi$
$594$	0	0
$595$	−52.8427	−0.0888112
$596$	0	0
$597$	0	0
$598$	0	0
$599$	100.819i	0.168312i	0.996453	+	0.0841559i	$0.0268194\pi$
$599$	100.819i	0.168312i	−0.996453	+	0.0841559i	$0.973181\pi$
$600$	0	0
$601$	1076.74	1.79157	0.895787	−	0.444484i	$-0.146613\pi$
$601$	1076.74	1.79157	0.895787	+	0.444484i	$0.146613\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	271.858i	0.449352i
$606$	0	0
$607$	989.622	1.63035	0.815175	−	0.579215i	$-0.196641\pi$
$607$	989.622	1.63035	0.815175	+	0.579215i	$0.196641\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 720.523i	− 1.17925i
$612$	0	0
$613$	−221.605	−0.361509	−0.180754	−	0.983528i	$-0.557854\pi$
$613$	−221.605	−0.361509	−0.180754	+	0.983528i	$0.557854\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	670.467i	1.08666i	0.839520	+	0.543328i	$0.182836\pi$
$617$	670.467i	1.08666i	−0.839520	+	0.543328i	$0.817164\pi$
$618$	0	0
$619$	−354.579	−0.572825	−0.286412	−	0.958106i	$-0.592463\pi$
$619$	−354.579	−0.572825	−0.286412	+	0.958106i	$0.592463\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	178.023i	0.285751i
$624$	0	0
$625$	−475.000	−0.760000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 42.5563i	− 0.0676570i
$630$	0	0
$631$	−165.957	−0.263006	−0.131503	−	0.991316i	$-0.541980\pi$
$631$	−165.957	−0.263006	−0.131503	+	0.991316i	$0.541980\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 176.218i	− 0.277509i
$636$	0	0
$637$	−530.263	−0.832439
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 277.508i	− 0.432930i	−0.976290	−	0.216465i	$-0.930547\pi$
$641$	− 277.508i	− 0.432930i	0.976290	−	0.216465i	$-0.0694527\pi$
$642$	0	0
$643$	−478.211	−0.743718	−0.371859	−	0.928289i	$-0.621280\pi$
$643$	−478.211	−0.743718	−0.371859	+	0.928289i	$0.621280\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 866.652i	− 1.33949i	−0.742590	−	0.669747i	$-0.766403\pi$
$647$	− 866.652i	− 1.33949i	0.742590	−	0.669747i	$-0.233597\pi$
$648$	0	0
$649$	−92.8427	−0.143055
$650$	0	0
$651$	0	0
$652$	0	0
$653$	571.727i	0.875539i	0.899087	+	0.437769i	$0.144231\pi$
$653$	571.727i	0.875539i	−0.899087	+	0.437769i	$0.855769\pi$
$654$	0	0
$655$	941.193	1.43694
$656$	0	0
$657$	0	0
$658$	0	0
$659$	954.259i	1.44804i	0.689778	+	0.724020i	$0.257708\pi$
$659$	954.259i	1.44804i	−0.689778	+	0.724020i	$0.742292\pi$
$660$	0	0
$661$	374.079	0.565929	0.282964	−	0.959130i	$-0.408682\pi$
$661$	374.079	0.565929	0.282964	+	0.959130i	$0.408682\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 54.2545i	− 0.0815857i
$666$	0	0
$667$	335.368	0.502801
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 310.178i	− 0.462262i
$672$	0	0
$673$	−386.315	−0.574020	−0.287010	−	0.957928i	$-0.592661\pi$
$673$	−386.315	−0.574020	−0.287010	+	0.957928i	$0.592661\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 278.308i	− 0.411090i	−0.978648	−	0.205545i	$-0.934103\pi$
$677$	− 278.308i	− 0.411090i	0.978648	−	0.205545i	$-0.0658968\pi$
$678$	0	0
$679$	269.114	0.396339
$680$	0	0
$681$	0	0
$682$	0	0
$683$	245.403i	0.359301i	0.983730	+	0.179651i	$0.0574967\pi$
$683$	245.403i	0.359301i	−0.983730	+	0.179651i	$0.942503\pi$
$684$	0	0
$685$	104.105	0.151979
$686$	0	0
$687$	0	0
$688$	0	0
$689$	1073.55i	1.55813i
$690$	0	0
$691$	−592.386	−0.857288	−0.428644	−	0.903474i	$-0.641008\pi$
$691$	−592.386	−0.857288	−0.428644	+	0.903474i	$0.641008\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	1142.59i	1.64402i
$696$	0	0
$697$	−279.579	−0.401117
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 112.039i	− 0.159827i	−0.996802	−	0.0799137i	$-0.974536\pi$
$701$	− 112.039i	− 0.159827i	0.996802	−	0.0799137i	$-0.0254645\pi$
$702$	0	0
$703$	43.6932	0.0621525
$704$	0	0
$705$	0	0
$706$	0	0
$707$	469.668i	0.664311i
$708$	0	0
$709$	−67.4470	−0.0951297	−0.0475649	−	0.998868i	$-0.515146\pi$
$709$	−67.4470	−0.0951297	−0.0475649	+	0.998868i	$0.515146\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	419.910i	0.588934i
$714$	0	0
$715$	−484.911	−0.678197
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 1124.26i	− 1.56365i	−0.623500	−	0.781824i	$-0.714290\pi$
$719$	− 1124.26i	− 1.56365i	0.623500	−	0.781824i	$-0.285710\pi$
$720$	0	0
$721$	−345.841	−0.479669
$722$	0	0
$723$	0	0
$724$	0	0
$725$	124.984i	0.172392i
$726$	0	0
$727$	−393.895	−0.541808	−0.270904	−	0.962606i	$-0.587323\pi$
$727$	−393.895	−0.541808	−0.270904	+	0.962606i	$0.587323\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	243.802i	0.333519i
$732$	0	0
$733$	464.841	0.634163	0.317081	−	0.948398i	$-0.397297\pi$
$733$	464.841	0.634163	0.317081	+	0.948398i	$0.397297\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	152.877i	0.207431i
$738$	0	0
$739$	−847.964	−1.14745	−0.573724	−	0.819049i	$-0.694502\pi$
$739$	−847.964	−1.14745	−0.573724	+	0.819049i	$0.694502\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	558.781i	0.752061i	0.926607	+	0.376031i	$0.122711\pi$
$743$	558.781i	0.752061i	−0.926607	+	0.376031i	$0.877289\pi$
$744$	0	0
$745$	568.211	0.762699
$746$	0	0
$747$	0	0
$748$	0	0
$749$	425.560i	0.568170i
$750$	0	0
$751$	574.465	0.764933	0.382467	−	0.923969i	$-0.375075\pi$
$751$	574.465	0.764933	0.382467	+	0.923969i	$0.375075\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 1305.04i	− 1.72853i
$756$	0	0
$757$	−1090.81	−1.44097	−0.720485	−	0.693470i	$-0.756081\pi$
$757$	−1090.81	−1.44097	−0.720485	+	0.693470i	$0.756081\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 533.996i	− 0.701703i	−0.936431	−	0.350851i	$-0.885892\pi$
$761$	− 533.996i	− 0.701703i	0.936431	−	0.350851i	$-0.114108\pi$
$762$	0	0
$763$	298.860	0.391690
$764$	0	0
$765$	0	0
$766$	0	0
$767$	167.194i	0.217985i
$768$	0	0
$769$	−703.947	−0.915405	−0.457703	−	0.889105i	$-0.651328\pi$
$769$	−703.947	−0.915405	−0.457703	+	0.889105i	$0.651328\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	753.813i	0.975179i	0.873073	+	0.487589i	$0.162124\pi$
$773$	753.813i	0.975179i	−0.873073	+	0.487589i	$0.837876\pi$
$774$	0	0
$775$	−156.491	−0.201924
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 287.048i	− 0.368483i
$780$	0	0
$781$	−855.368	−1.09522
$782$	0	0
$783$	0	0
$784$	0	0
$785$	509.352i	0.648857i
$786$	0	0
$787$	−277.369	−0.352438	−0.176219	−	0.984351i	$-0.556387\pi$
$787$	−277.369	−0.352438	−0.176219	+	0.984351i	$0.556387\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	296.966i	0.375431i
$792$	0	0
$793$	−558.579	−0.704387
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 1206.77i	− 1.51414i	−0.653333	−	0.757070i	$-0.726630\pi$
$797$	− 1206.77i	− 1.51414i	0.653333	−	0.757070i	$-0.273370\pi$
$798$	0	0
$799$	183.263	0.229365
$800$	0	0
$801$	0	0
$802$	0	0
$803$	38.3891i	0.0478071i
$804$	0	0
$805$	199.473	0.247793
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 921.496i	− 1.13906i	−0.821972	−	0.569528i	$-0.807126\pi$
$809$	− 921.496i	− 1.13906i	0.821972	−	0.569528i	$-0.192874\pi$
$810$	0	0
$811$	726.982	0.896402	0.448201	−	0.893933i	$-0.352065\pi$
$811$	726.982	0.896402	0.448201	+	0.893933i	$0.352065\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	894.583i	1.09765i
$816$	0	0
$817$	−250.316	−0.306384
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 564.712i	− 0.687834i	−0.939000	−	0.343917i	$-0.888246\pi$
$821$	− 564.712i	− 0.687834i	0.939000	−	0.343917i	$-0.111754\pi$
$822$	0	0
$823$	−586.623	−0.712786	−0.356393	−	0.934336i	$-0.615994\pi$
$823$	−586.623	−0.712786	−0.356393	+	0.934336i	$0.615994\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1364.81i	− 1.65031i	−0.564904	−	0.825156i	$-0.691087\pi$
$827$	− 1364.81i	− 1.65031i	0.564904	−	0.825156i	$-0.308913\pi$
$828$	0	0
$829$	1043.82	1.25913	0.629563	−	0.776949i	$-0.283234\pi$
$829$	1043.82	1.25913	0.629563	+	0.776949i	$0.283234\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 134.871i	− 0.161910i
$834$	0	0
$835$	262.386	0.314234
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 319.797i	− 0.381165i	−0.981671	−	0.190583i	$-0.938962\pi$
$839$	− 319.797i	− 0.381165i	0.981671	−	0.190583i	$-0.0610377\pi$
$840$	0	0
$841$	216.157	0.257024
$842$	0	0
$843$	0	0
$844$	0	0
$845$	117.453i	0.138998i
$846$	0	0
$847$	202.097	0.238604
$848$	0	0
$849$	0	0
$850$	0	0
$851$	160.644i	0.188770i
$852$	0	0
$853$	475.078	0.556950	0.278475	−	0.960443i	$-0.410171\pi$
$853$	475.078	0.556950	0.278475	+	0.960443i	$0.410171\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1314.57i	− 1.53392i	−0.641693	−	0.766962i	$-0.721768\pi$
$857$	− 1314.57i	− 1.53392i	0.641693	−	0.766962i	$-0.278232\pi$
$858$	0	0
$859$	492.912	0.573820	0.286910	−	0.957957i	$-0.407372\pi$
$859$	492.912	0.573820	0.286910	+	0.957957i	$0.407372\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 479.475i	− 0.555591i	−0.960640	−	0.277795i	$-0.910396\pi$
$863$	− 479.475i	− 0.555591i	0.960640	−	0.277795i	$-0.0896037\pi$
$864$	0	0
$865$	600.000	0.693642
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 351.698i	− 0.404716i
$870$	0	0
$871$	275.306	0.316080
$872$	0	0
$873$	0	0
$874$	0	0
$875$	446.036i	0.509755i
$876$	0	0
$877$	125.552	0.143161	0.0715806	−	0.997435i	$-0.477196\pi$
$877$	125.552	0.143161	0.0715806	+	0.997435i	$0.477196\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 1448.81i	− 1.64450i	−0.569125	−	0.822251i	$-0.692718\pi$
$881$	− 1448.81i	− 1.64450i	0.569125	−	0.822251i	$-0.307282\pi$
$882$	0	0
$883$	−102.562	−0.116151	−0.0580756	−	0.998312i	$-0.518496\pi$
$883$	−102.562	−0.116151	−0.0580756	+	0.998312i	$0.518496\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 49.7582i	− 0.0560971i	−0.999607	−	0.0280486i	$-0.991071\pi$
$887$	− 49.7582i	− 0.0560971i	0.999607	−	0.0280486i	$-0.00892931\pi$
$888$	0	0
$889$	−130.999	−0.147356
$890$	0	0
$891$	0	0
$892$	0	0
$893$	188.159i	0.210704i
$894$	0	0
$895$	−1478.95	−1.65245
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 782.357i	− 0.870253i
$900$	0	0
$901$	−273.053	−0.303056
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 98.0337i	− 0.108325i
$906$	0	0
$907$	1190.56	1.31264	0.656318	−	0.754485i	$-0.272113\pi$
$907$	1190.56	1.31264	0.656318	+	0.754485i	$0.272113\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	519.426i	0.570171i	0.958502	+	0.285086i	$0.0920220\pi$
$911$	519.426i	0.570171i	−0.958502	+	0.285086i	$0.907978\pi$
$912$	0	0
$913$	734.947	0.804980
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 699.676i	− 0.763006i
$918$	0	0
$919$	−953.684	−1.03774	−0.518871	−	0.854853i	$-0.673647\pi$
$919$	−953.684	−1.03774	−0.518871	+	0.854853i	$0.673647\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1540.38i	1.66888i
$924$	0	0
$925$	−59.8683	−0.0647225
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 910.903i	− 0.980519i	−0.871576	−	0.490260i	$-0.836902\pi$
$929$	− 910.903i	− 0.980519i	0.871576	−	0.490260i	$-0.163098\pi$
$930$	0	0
$931$	138.474	0.148737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 123.335i	− 0.131910i
$936$	0	0
$937$	−725.211	−0.773971	−0.386985	−	0.922086i	$-0.626484\pi$
$937$	−725.211	−0.773971	−0.386985	+	0.922086i	$0.626484\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 765.394i	− 0.813383i	−0.913566	−	0.406692i	$-0.866682\pi$
$941$	− 765.394i	− 0.813383i	0.913566	−	0.406692i	$-0.133318\pi$
$942$	0	0
$943$	1055.37	1.11916
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 505.563i	− 0.533857i	−0.963716	−	0.266929i	$-0.913991\pi$
$947$	− 505.563i	− 0.533857i	0.963716	−	0.266929i	$-0.0860088\pi$
$948$	0	0
$949$	69.1324	0.0728476
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1675.90i	1.75855i	0.476315	+	0.879275i	$0.341972\pi$
$953$	1675.90i	1.75855i	−0.476315	+	0.879275i	$0.658028\pi$
$954$	0	0
$955$	1210.74	1.26779
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 77.3912i	− 0.0806999i
$960$	0	0
$961$	18.5787	0.0193326
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 862.651i	− 0.893939i
$966$	0	0
$967$	−373.236	−0.385973	−0.192987	−	0.981201i	$-0.561817\pi$
$967$	−373.236	−0.385973	−0.192987	+	0.981201i	$0.561817\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 377.502i	− 0.388777i	−0.980925	−	0.194388i	$-0.937728\pi$
$971$	− 377.502i	− 0.388777i	0.980925	−	0.194388i	$-0.0622722\pi$
$972$	0	0
$973$	849.394	0.872964
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1314.34i	− 1.34528i	−0.739971	−	0.672639i	$-0.765161\pi$
$977$	− 1314.34i	− 1.34528i	0.739971	−	0.672639i	$-0.234839\pi$
$978$	0	0
$979$	−415.508	−0.424420
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1251.63i	1.27328i	0.771161	+	0.636640i	$0.219676\pi$
$983$	1251.63i	1.27328i	−0.771161	+	0.636640i	$0.780324\pi$
$984$	0	0
$985$	1003.58	1.01886
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 920.318i	− 0.930554i
$990$	0	0
$991$	−1646.53	−1.66149	−0.830744	−	0.556655i	$-0.812085\pi$
$991$	−1646.53	−1.66149	−0.830744	+	0.556655i	$0.812085\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 1257.18i	− 1.26350i
$996$	0	0
$997$	1768.42	1.77374	0.886871	−	0.462018i	$-0.152874\pi$
$997$	1768.42	1.77374	0.886871	+	0.462018i	$0.152874\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.e.a.161.4	yes	4
3.2	odd	2	inner	864.3.e.a.161.2	✓	4
4.3	odd	2		864.3.e.c.161.3	yes	4
8.3	odd	2		1728.3.e.t.1025.1		4
8.5	even	2		1728.3.e.q.1025.2		4
12.11	even	2		864.3.e.c.161.1	yes	4
24.5	odd	2		1728.3.e.q.1025.4		4
24.11	even	2		1728.3.e.t.1025.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.3.e.a.161.2	✓	4	3.2	odd	2	inner
864.3.e.a.161.4	yes	4	1.1	even	1	trivial
864.3.e.c.161.1	yes	4	12.11	even	2
864.3.e.c.161.3	yes	4	4.3	odd	2
1728.3.e.q.1025.2		4	8.5	even	2
1728.3.e.q.1025.4		4	24.5	odd	2
1728.3.e.t.1025.1		4	8.3	odd	2
1728.3.e.t.1025.3		4	24.11	even	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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	864.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+4.47214i q^{5} +3.32456 q^{7} +O(q^{10})\)	\(q+4.47214i q^{5} +3.32456 q^{7} +7.75955i q^{11} +13.9737 q^{13} +3.55415i q^{17} -3.64911 q^{19} -13.4164i q^{23} +5.00000 q^{25} +24.9969i q^{29} -31.2982 q^{31} +14.8679i q^{35} -11.9737 q^{37} +78.6625i q^{41} +68.5964 q^{43} -51.5629i q^{47} -37.9473 q^{49} +76.8265i q^{53} -34.7018 q^{55} +11.9650i q^{59} -39.9737 q^{61} +62.4921i q^{65} +19.7018 q^{67} +110.234i q^{71} +4.94733 q^{73} +25.7971i q^{77} -45.3246 q^{79} -94.7151i q^{83} -15.8947 q^{85} +53.5479i q^{89} +46.4562 q^{91} -16.3193i q^{95} +80.9473 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{7}+O(q^{10}) \) 4 * q - 12 * q^7	\( 4 q - 12 q^{7} - 20 q^{13} + 36 q^{19} + 20 q^{25} - 24 q^{31} + 28 q^{37} + 72 q^{43} - 240 q^{55} - 84 q^{61} + 180 q^{67} - 132 q^{73} - 156 q^{79} + 240 q^{85} + 540 q^{91} + 172 q^{97}+O(q^{100}) \) 4 * q - 12 * q^7 - 20 * q^13 + 36 * q^19 + 20 * q^25 - 24 * q^31 + 28 * q^37 + 72 * q^43 - 240 * q^55 - 84 * q^61 + 180 * q^67 - 132 * q^73 - 156 * q^79 + 240 * q^85 + 540 * q^91 + 172 * q^97

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