LMFDB - Embedded newform 864.3.q.a.737.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,3,Mod(449,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

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

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

chi := DirichletCharacter("864.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	864.q (of order $6$, degree $2$, 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:	$23.5422948407$
Analytic rank:	$0$
Dimension:	$24$
Relative dimension:	$12$ over $\Q(\zeta_{6})$
Twist minimal:	no (minimal twist has level 288)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			737.5
Character	$\chi$	$=$	864.737
Dual form			864.3.q.a.449.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.721152 - 0.416357i) q^{5} +(1.26051 - 2.18326i) q^{7} +O(q^{10})$	$q+(0.721152 - 0.416357i) q^{5} +(1.26051 - 2.18326i) q^{7} +(-9.47693 - 5.47151i) q^{11} +(4.36453 + 7.55959i) q^{13} -20.8637i q^{17} +1.50150 q^{19} +(1.00209 - 0.578558i) q^{23} +(-12.1533 + 21.0501i) q^{25} +(-15.7347 - 9.08444i) q^{29} +(-25.6909 - 44.4980i) q^{31} -2.09929i q^{35} -7.93951 q^{37} +(21.8881 - 12.6371i) q^{41} +(19.3418 - 33.5010i) q^{43} +(-59.6559 - 34.4423i) q^{47} +(21.3222 + 36.9312i) q^{49} +46.5195i q^{53} -9.11241 q^{55} +(-89.1306 + 51.4596i) q^{59} +(44.1651 - 76.4962i) q^{61} +(6.29498 + 3.63441i) q^{65} +(-11.3192 - 19.6054i) q^{67} -104.256i q^{71} -75.2115 q^{73} +(-23.8915 + 13.7938i) q^{77} +(51.8676 - 89.8374i) q^{79} +(-53.7499 - 31.0325i) q^{83} +(-8.68677 - 15.0459i) q^{85} -1.95722i q^{89} +22.0061 q^{91} +(1.08281 - 0.625162i) q^{95} +(59.2171 - 102.567i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q+O(q^{10}) $ 24 * q	$ 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) $ 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	0.721152	−	0.416357i	0.144230	−	0.0832715i	−0.426148	−	0.904653i	$-0.640130\pi$
$5$	0.721152	−	0.416357i	0.144230	−	0.0832715i	0.570379	+	0.821382i	$0.306796\pi$
$6$		0			0
$7$	1.26051	−	2.18326i	0.180072	−	0.311895i	−0.761833	−	0.647774i	$-0.775700\pi$
$7$	1.26051	−	2.18326i	0.180072	−	0.311895i	0.941905	+	0.335879i	$0.109033\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−9.47693	−	5.47151i	−0.861540	−	0.497410i	0.00298800	−	0.999996i	$-0.499049\pi$
$11$	−9.47693	−	5.47151i	−0.861540	−	0.497410i	−0.864528	+	0.502585i	$0.832382\pi$
$12$		0			0
$13$	4.36453	+	7.55959i	0.335733	+	0.581507i	0.983625	−	0.180225i	$-0.0576826\pi$
$13$	4.36453	+	7.55959i	0.335733	+	0.581507i	−0.647892	+	0.761732i	$0.724349\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	20.8637i		−	1.22728i	−0.789586	−	0.613639i	$-0.789705\pi$
$17$		−	20.8637i		−	1.22728i	0.789586	−	0.613639i	$-0.210295\pi$
$18$		0			0
$19$	1.50150			0.0790265			0.0395132	−	0.999219i	$-0.487419\pi$
$19$	1.50150			0.0790265			0.0395132	+	0.999219i	$0.487419\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.00209	−	0.578558i	0.0435692	−	0.0251547i	−0.478057	−	0.878329i	$-0.658659\pi$
$23$	1.00209	−	0.578558i	0.0435692	−	0.0251547i	0.521626	+	0.853174i	$0.325325\pi$
$24$		0			0
$25$	−12.1533	+	21.0501i	−0.486132	+	0.842005i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−15.7347	−	9.08444i	−0.542576	−	0.313257i	0.203546	−	0.979065i	$-0.434753\pi$
$29$	−15.7347	−	9.08444i	−0.542576	−	0.313257i	−0.746122	+	0.665809i	$0.768087\pi$
$30$		0			0
$31$	−25.6909	−	44.4980i	−0.828739	−	1.43542i	−0.899028	−	0.437891i	$-0.855726\pi$
$31$	−25.6909	−	44.4980i	−0.828739	−	1.43542i	0.0702892	−	0.997527i	$-0.477608\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	2.09929i		−	0.0599796i
$36$		0			0
$37$	−7.93951			−0.214581			−0.107291	−	0.994228i	$-0.534218\pi$
$37$	−7.93951			−0.214581			−0.107291	+	0.994228i	$0.534218\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	21.8881	−	12.6371i	0.533857	−	0.308222i	−0.208729	−	0.977974i	$-0.566933\pi$
$41$	21.8881	−	12.6371i	0.533857	−	0.308222i	0.742586	+	0.669751i	$0.233599\pi$
$42$		0			0
$43$	19.3418	−	33.5010i	0.449810	−	0.779094i	−0.548563	−	0.836109i	$-0.684825\pi$
$43$	19.3418	−	33.5010i	0.449810	−	0.779094i	0.998373	+	0.0570153i	$0.0181584\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−59.6559	−	34.4423i	−1.26927	−	0.732815i	−0.294423	−	0.955675i	$-0.595127\pi$
$47$	−59.6559	−	34.4423i	−1.26927	−	0.732815i	−0.974851	+	0.222860i	$0.928461\pi$
$48$		0			0
$49$	21.3222	+	36.9312i	0.435148	+	0.753698i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			46.5195i			0.877727i	0.898554	+	0.438864i	$0.144619\pi$
$53$			46.5195i			0.877727i	−0.898554	+	0.438864i	$0.855381\pi$
$54$		0			0
$55$	−9.11241			−0.165680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−89.1306	+	51.4596i	−1.51069	+	0.872196i	−0.510766	+	0.859720i	$0.670638\pi$
$59$	−89.1306	+	51.4596i	−1.51069	+	0.872196i	−0.999922	+	0.0124764i	$0.996029\pi$
$60$		0			0
$61$	44.1651	−	76.4962i	0.724018	−	1.25404i	−0.235359	−	0.971909i	$-0.575627\pi$
$61$	44.1651	−	76.4962i	0.724018	−	1.25404i	0.959377	−	0.282127i	$-0.0910401\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.29498	+	3.63441i	0.0968459	+	0.0559140i
$66$		0			0
$67$	−11.3192	−	19.6054i	−0.168943	−	0.292618i	0.769105	−	0.639122i	$-0.220702\pi$
$67$	−11.3192	−	19.6054i	−0.168943	−	0.292618i	−0.938049	+	0.346504i	$0.887369\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	104.256i		−	1.46840i	−0.678935	−	0.734198i	$-0.737558\pi$
$71$		−	104.256i		−	1.46840i	0.678935	−	0.734198i	$-0.262442\pi$
$72$		0			0
$73$	−75.2115			−1.03030			−0.515148	−	0.857101i	$-0.672263\pi$
$73$	−75.2115			−1.03030			−0.515148	+	0.857101i	$0.672263\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−23.8915	+	13.7938i	−0.310279	+	0.179140i
$78$		0			0
$79$	51.8676	−	89.8374i	0.656552	−	1.13718i	−0.324950	−	0.945731i	$-0.605347\pi$
$79$	51.8676	−	89.8374i	0.656552	−	1.13718i	0.981502	−	0.191451i	$-0.0613192\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−53.7499	−	31.0325i	−0.647589	−	0.373886i	0.139943	−	0.990160i	$-0.455308\pi$
$83$	−53.7499	−	31.0325i	−0.647589	−	0.373886i	−0.787532	+	0.616274i	$0.788641\pi$
$84$		0			0
$85$	−8.68677	−	15.0459i	−0.102197	−	0.177011i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	1.95722i		−	0.0219912i	−0.999940	−	0.0109956i	$-0.996500\pi$
$89$		−	1.95722i		−	0.0219912i	0.999940	−	0.0109956i	$-0.00350008\pi$
$90$		0			0
$91$	22.0061			0.241825
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.08281	−	0.625162i	0.0113980	−	0.00658065i
$96$		0			0
$97$	59.2171	−	102.567i	0.610486	−	1.05739i	−0.380673	−	0.924710i	$-0.624308\pi$
$97$	59.2171	−	102.567i	0.610486	−	1.05739i	0.991159	−	0.132682i	$-0.0423590\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	48.2410	+	27.8519i	0.477633	+	0.275762i	0.719430	−	0.694565i	$-0.244403\pi$
$101$	48.2410	+	27.8519i	0.477633	+	0.275762i	−0.241796	+	0.970327i	$0.577737\pi$
$102$		0			0
$103$	−100.974	−	174.892i	−0.980328	−	1.69798i	−0.661098	−	0.750300i	$-0.729909\pi$
$103$	−100.974	−	174.892i	−0.980328	−	1.69798i	−0.319230	−	0.947677i	$-0.603424\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			55.1900i			0.515794i	0.966172	+	0.257897i	$0.0830295\pi$
$107$			55.1900i			0.515794i	−0.966172	+	0.257897i	$0.916970\pi$
$108$		0			0
$109$	−44.6887			−0.409988			−0.204994	−	0.978763i	$-0.565717\pi$
$109$	−44.6887			−0.409988			−0.204994	+	0.978763i	$0.565717\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−93.6725	+	54.0818i	−0.828960	+	0.478600i	−0.853497	−	0.521099i	$-0.825522\pi$
$113$	−93.6725	+	54.0818i	−0.828960	+	0.478600i	0.0245363	+	0.999699i	$0.492189\pi$
$114$		0			0
$115$	0.481774	−	0.834457i	0.00418934	−	0.00725615i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−45.5510	−	26.2989i	−0.382782	−	0.220999i
$120$		0			0
$121$	−0.625137	−	1.08277i	−0.00516642	−	0.00894850i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			41.0583i			0.328467i
$126$		0			0
$127$	−172.177			−1.35573			−0.677863	−	0.735188i	$-0.737094\pi$
$127$	−172.177			−1.35573			−0.677863	+	0.735188i	$0.737094\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−76.8188	+	44.3514i	−0.586403	+	0.338560i	−0.763674	−	0.645602i	$-0.776606\pi$
$131$	−76.8188	+	44.3514i	−0.586403	+	0.338560i	0.177271	+	0.984162i	$0.443273\pi$
$132$		0			0
$133$	1.89265	−	3.27817i	0.0142305	−	0.0246479i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	77.6156	+	44.8114i	0.566537	+	0.327090i	0.755765	−	0.654843i	$-0.227265\pi$
$137$	77.6156	+	44.8114i	0.566537	+	0.327090i	−0.189228	+	0.981933i	$0.560599\pi$
$138$		0			0
$139$	74.7919	+	129.543i	0.538071	+	0.931966i	0.999008	+	0.0445333i	$0.0141801\pi$
$139$	74.7919	+	129.543i	0.538071	+	0.931966i	−0.460937	+	0.887433i	$0.652487\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		−	95.5223i		−	0.667988i
$144$		0			0
$145$	−15.1295			−0.104341
$146$		0			0
$147$		0			0
$148$		0			0
$149$	121.312	−	70.0397i	0.814177	−	0.470065i	−0.0342275	−	0.999414i	$-0.510897\pi$
$149$	121.312	−	70.0397i	0.814177	−	0.470065i	0.848404	+	0.529349i	$0.177564\pi$
$150$		0			0
$151$	72.5660	−	125.688i	0.480570	−	0.832371i	−0.519182	−	0.854664i	$-0.673763\pi$
$151$	72.5660	−	125.688i	0.480570	−	0.832371i	0.999751	+	0.0222926i	$0.00709654\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−37.0541	−	21.3932i	−0.239059	−	0.138021i
$156$		0			0
$157$	123.540	+	213.977i	0.786876	+	1.36291i	0.927872	+	0.372899i	$0.121636\pi$
$157$	123.540	+	213.977i	0.786876	+	1.36291i	−0.140996	+	0.990010i	$0.545030\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	2.91711i		−	0.0181187i
$162$		0			0
$163$	189.342			1.16161			0.580805	−	0.814043i	$-0.302738\pi$
$163$	189.342			1.16161			0.580805	+	0.814043i	$0.302738\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	200.563	−	115.795i	1.20098	−	0.693384i	0.240204	−	0.970722i	$-0.422786\pi$
$167$	200.563	−	115.795i	1.20098	−	0.693384i	0.960772	+	0.277339i	$0.0894524\pi$
$168$		0			0
$169$	46.4017	−	80.3701i	0.274566	−	0.475563i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−224.906	−	129.849i	−1.30003	−	0.750574i	−0.319623	−	0.947545i	$-0.603556\pi$
$173$	−224.906	−	129.849i	−1.30003	−	0.750574i	−0.980409	+	0.196971i	$0.936890\pi$
$174$		0			0
$175$	30.6386	+	53.0677i	0.175078	+	0.303244i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			220.358i			1.23105i	0.788118	+	0.615524i	$0.211056\pi$
$179$			220.358i			1.23105i	−0.788118	+	0.615524i	$0.788944\pi$
$180$		0			0
$181$	−286.189			−1.58116			−0.790578	−	0.612362i	$-0.790220\pi$
$181$	−286.189			−1.58116			−0.790578	+	0.612362i	$0.790220\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−5.72560	+	3.30567i	−0.0309492	+	0.0178685i
$186$		0			0
$187$	−114.156	+	197.724i	−0.610461	+	1.05735i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.36792	+	0.789768i	0.00716188	+	0.00413491i	0.503577	−	0.863951i	$-0.332017\pi$
$191$	1.36792	+	0.789768i	0.00716188	+	0.00413491i	−0.496415	+	0.868085i	$0.665350\pi$
$192$		0			0
$193$	116.067	+	201.033i	0.601381	+	1.04162i	0.992612	+	0.121330i	$0.0387161\pi$
$193$	116.067	+	201.033i	0.601381	+	1.04162i	−0.391231	+	0.920293i	$0.627951\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	214.012i		−	1.08636i	−0.839618	−	0.543178i	$-0.817221\pi$
$197$		−	214.012i		−	1.08636i	0.839618	−	0.543178i	$-0.182779\pi$
$198$		0			0
$199$	25.8912			0.130106			0.0650532	−	0.997882i	$-0.479278\pi$
$199$	25.8912			0.130106			0.0650532	+	0.997882i	$0.479278\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−39.6674	+	22.9020i	−0.195406	+	0.112818i
$204$		0			0
$205$	10.5231	−	18.2266i	0.0513322	−	0.0889100i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−14.2296	−	8.21549i	−0.0680844	−	0.0393086i
$210$		0			0
$211$	46.9752	+	81.3635i	0.222631	+	0.385609i	0.955606	−	0.294647i	$-0.0952020\pi$
$211$	46.9752	+	81.3635i	0.222631	+	0.385609i	−0.732975	+	0.680256i	$0.761869\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	32.2124i		−	0.149825i
$216$		0			0
$217$	−129.534			−0.596932
$218$		0			0
$219$		0			0
$220$		0			0
$221$	157.721	−	91.0604i	0.713671	−	0.412038i
$222$		0			0
$223$	−149.686	+	259.264i	−0.671237	+	1.16262i	0.306316	+	0.951930i	$0.400904\pi$
$223$	−149.686	+	259.264i	−0.671237	+	1.16262i	−0.977554	+	0.210687i	$0.932430\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	315.904	+	182.387i	1.39165	+	0.803467i	0.993498	−	0.113853i	$-0.0363194\pi$
$227$	315.904	+	182.387i	1.39165	+	0.803467i	0.398149	+	0.917321i	$0.369653\pi$
$228$		0			0
$229$	140.979	+	244.183i	0.615629	+	1.06630i	0.990274	+	0.139132i	$0.0444311\pi$
$229$	140.979	+	244.183i	0.615629	+	1.06630i	−0.374645	+	0.927168i	$0.622236\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	208.568i		−	0.895140i	−0.894249	−	0.447570i	$-0.852290\pi$
$233$		−	208.568i		−	0.895140i	0.894249	−	0.447570i	$-0.147710\pi$
$234$		0			0
$235$	−57.3613			−0.244090
$236$		0			0
$237$		0			0
$238$		0			0
$239$	25.1232	−	14.5049i	0.105118	−	0.0606899i	−0.446519	−	0.894774i	$-0.647337\pi$
$239$	25.1232	−	14.5049i	0.105118	−	0.0606899i	0.551637	+	0.834084i	$0.314003\pi$
$240$		0			0
$241$	116.456	−	201.707i	0.483218	−	0.836959i	−0.516596	−	0.856229i	$-0.672801\pi$
$241$	116.456	−	201.707i	0.483218	−	0.836959i	0.999814	+	0.0192706i	$0.00613439\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	30.7532	+	17.7553i	0.125523	+	0.0724708i
$246$		0			0
$247$	6.55336	+	11.3507i	0.0265318	+	0.0459544i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			112.869i			0.449676i	0.974396	+	0.224838i	$0.0721853\pi$
$251$			112.869i			0.449676i	−0.974396	+	0.224838i	$0.927815\pi$
$252$		0			0
$253$	−12.6624			−0.0500488
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−292.209	+	168.707i	−1.13700	+	0.656448i	−0.945686	−	0.325081i	$-0.894608\pi$
$257$	−292.209	+	168.707i	−1.13700	+	0.656448i	−0.191315	+	0.981529i	$0.561275\pi$
$258$		0			0
$259$	−10.0078	+	17.3340i	−0.0386402	+	0.0669268i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	437.975	+	252.865i	1.66530	+	0.961464i	0.970118	+	0.242635i	$0.0780119\pi$
$263$	437.975	+	252.865i	1.66530	+	0.961464i	0.695187	+	0.718829i	$0.255321\pi$
$264$		0			0
$265$	19.3688	+	33.5477i	0.0730896	+	0.126595i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			337.414i			1.25433i	0.778887	+	0.627164i	$0.215784\pi$
$269$			337.414i			1.25433i	−0.778887	+	0.627164i	$0.784216\pi$
$270$		0			0
$271$	−12.0273			−0.0443813			−0.0221907	−	0.999754i	$-0.507064\pi$
$271$	−12.0273			−0.0443813			−0.0221907	+	0.999754i	$0.507064\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	230.352	−	132.994i	0.837643	−	0.483614i
$276$		0			0
$277$	99.9431	−	173.107i	0.360805	−	0.624933i	−0.627288	−	0.778787i	$-0.715835\pi$
$277$	99.9431	−	173.107i	0.360805	−	0.624933i	0.988094	+	0.153854i	$0.0491685\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	290.037	+	167.453i	1.03216	+	0.595918i	0.917603	−	0.397499i	$-0.130122\pi$
$281$	290.037	+	167.453i	1.03216	+	0.595918i	0.114557	+	0.993417i	$0.463455\pi$
$282$		0			0
$283$	235.076	+	407.164i	0.830659	+	1.43874i	0.897517	+	0.440980i	$0.145369\pi$
$283$	235.076	+	407.164i	0.830659	+	1.43874i	−0.0668583	+	0.997762i	$0.521298\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	63.7167i		−	0.222009i
$288$		0			0
$289$	−146.296			−0.506213
$290$		0			0
$291$		0			0
$292$		0			0
$293$	131.252	−	75.7785i	0.447960	−	0.258630i	−0.259008	−	0.965875i	$-0.583396\pi$
$293$	131.252	−	75.7785i	0.447960	−	0.258630i	0.706968	+	0.707245i	$0.250062\pi$
$294$		0			0
$295$	−42.8511	+	74.2204i	−0.145258	+	0.251594i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	8.74733	+	5.05027i	0.0292553	+	0.0168905i
$300$		0			0
$301$	−48.7610	−	84.4566i	−0.161997	−	0.280587i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	73.5538i		−	0.241160i
$306$		0			0
$307$	123.852			0.403426			0.201713	−	0.979445i	$-0.435349\pi$
$307$	123.852			0.403426			0.201713	+	0.979445i	$0.435349\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	83.9131	−	48.4472i	0.269817	−	0.155779i	−0.358987	−	0.933342i	$-0.616878\pi$
$311$	83.9131	−	48.4472i	0.269817	−	0.155779i	0.628804	+	0.777563i	$0.283545\pi$
$312$		0			0
$313$	48.2500	−	83.5714i	0.154153	−	0.267001i	−0.778597	−	0.627524i	$-0.784068\pi$
$313$	48.2500	−	83.5714i	0.154153	−	0.267001i	0.932750	+	0.360523i	$0.117402\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−119.837	−	69.1878i	−0.378034	−	0.218258i	0.298928	−	0.954276i	$-0.403371\pi$
$317$	−119.837	−	69.1878i	−0.378034	−	0.218258i	−0.676963	+	0.736017i	$0.736704\pi$
$318$		0			0
$319$	99.4112	+	172.185i	0.311634	+	0.539766i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	31.3270i		−	0.0969875i
$324$		0			0
$325$	−212.174			−0.652842
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−150.393	+	86.8296i	−0.457122	+	0.263920i
$330$		0			0
$331$	−149.493	+	258.929i	−0.451641	+	0.782264i	−0.998488	−	0.0549678i	$-0.982494\pi$
$331$	−149.493	+	258.929i	−0.451641	+	0.782264i	0.546848	+	0.837232i	$0.315828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−16.3257	−	9.42565i	−0.0487334	−	0.0281363i
$336$		0			0
$337$	23.8541	+	41.3165i	0.0707837	+	0.122601i	0.899245	−	0.437445i	$-0.144117\pi$
$337$	23.8541	+	41.3165i	0.0707837	+	0.122601i	−0.828461	+	0.560046i	$0.810783\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			562.272i			1.64889i
$342$		0			0
$343$	231.037			0.673577
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−104.774	+	60.4914i	−0.301943	+	0.174327i	−0.643315	−	0.765601i	$-0.722442\pi$
$347$	−104.774	+	60.4914i	−0.301943	+	0.174327i	0.341372	+	0.939928i	$0.389108\pi$
$348$		0			0
$349$	193.756	−	335.595i	0.555175	−	0.961592i	−0.442715	−	0.896663i	$-0.645985\pi$
$349$	193.756	−	335.595i	0.555175	−	0.961592i	0.997890	−	0.0649291i	$-0.0206821\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	6.85970	+	3.96045i	0.0194326	+	0.0112194i	0.509685	−	0.860361i	$-0.329762\pi$
$353$	6.85970	+	3.96045i	0.0194326	+	0.0112194i	−0.490252	+	0.871581i	$0.663095\pi$
$354$		0			0
$355$	−43.4078	−	75.1845i	−0.122276	−	0.211787i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			210.527i			0.586425i	0.956047	+	0.293212i	$0.0947243\pi$
$359$			210.527i			0.586425i	−0.956047	+	0.293212i	$0.905276\pi$
$360$		0			0
$361$	−358.745			−0.993755
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−54.2390	+	31.3149i	−0.148600	+	0.0857942i
$366$		0			0
$367$	−171.940	+	297.809i	−0.468502	+	0.811469i	−0.999352	−	0.0359967i	$-0.988539\pi$
$367$	−171.940	+	297.809i	−0.468502	+	0.811469i	0.530850	+	0.847466i	$0.321873\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	101.564	+	58.6382i	0.273758	+	0.158054i
$372$		0			0
$373$	−177.354	−	307.186i	−0.475480	−	0.823556i	0.524125	−	0.851641i	$-0.324392\pi$
$373$	−177.354	−	307.186i	−0.475480	−	0.823556i	−0.999606	+	0.0280853i	$0.991059\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	158.597i		−	0.420682i
$378$		0			0
$379$	−269.497			−0.711073			−0.355536	−	0.934662i	$-0.615702\pi$
$379$	−269.497			−0.711073			−0.355536	+	0.934662i	$0.615702\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−72.6913	+	41.9684i	−0.189795	+	0.109578i	−0.591886	−	0.806021i	$-0.701617\pi$
$383$	−72.6913	+	41.9684i	−0.189795	+	0.109578i	0.402092	+	0.915599i	$0.368283\pi$
$384$		0			0
$385$	−11.4863	+	19.8948i	−0.0298344	+	0.0516748i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−269.017	−	155.317i	−0.691559	−	0.399272i	0.112637	−	0.993636i	$-0.464070\pi$
$389$	−269.017	−	155.317i	−0.691559	−	0.399272i	−0.804196	+	0.594364i	$0.797404\pi$
$390$		0			0
$391$	−12.0709	−	20.9074i	−0.0308718	−	0.0534716i
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	86.3819i		−	0.218688i
$396$		0			0
$397$	436.104			1.09850			0.549249	−	0.835658i	$-0.314914\pi$
$397$	436.104			1.09850			0.549249	+	0.835658i	$0.314914\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	354.774	−	204.829i	0.884723	−	0.510795i	0.0125102	−	0.999922i	$-0.496018\pi$
$401$	354.774	−	204.829i	0.884723	−	0.510795i	0.872213	+	0.489127i	$0.162684\pi$
$402$		0			0
$403$	224.258	−	388.425i	0.556470	−	0.963835i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	75.2423	+	43.4411i	0.184870	+	0.106735i
$408$		0			0
$409$	211.082	+	365.604i	0.516092	+	0.893897i	0.999825	+	0.0186821i	$0.00594703\pi$
$409$	211.082	+	365.604i	0.516092	+	0.893897i	−0.483734	+	0.875215i	$0.660720\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			259.461i			0.628234i
$414$		0			0
$415$	−51.6825			−0.124536
$416$		0			0
$417$		0			0
$418$		0			0
$419$	549.980	−	317.531i	1.31260	−	0.757830i	0.330075	−	0.943955i	$-0.392926\pi$
$419$	549.980	−	317.531i	1.31260	−	0.757830i	0.982526	+	0.186124i	$0.0595927\pi$
$420$		0			0
$421$	23.7781	−	41.1849i	0.0564800	−	0.0978263i	−0.836403	−	0.548115i	$-0.815346\pi$
$421$	23.7781	−	41.1849i	0.0564800	−	0.0978263i	0.892883	+	0.450289i	$0.148679\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	439.184	+	253.563i	1.03337	+	0.596619i
$426$		0			0
$427$	−111.341	−	192.848i	−0.260751	−	0.451635i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			614.503i			1.42576i	0.701286	+	0.712880i	$0.252610\pi$
$431$			614.503i			1.42576i	−0.701286	+	0.712880i	$0.747390\pi$
$432$		0			0
$433$	118.672			0.274070			0.137035	−	0.990566i	$-0.456243\pi$
$433$	118.672			0.274070			0.137035	+	0.990566i	$0.456243\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	1.50464	−	0.868707i	0.00344312	−	0.00198789i
$438$		0			0
$439$	245.073	−	424.479i	0.558254	−	0.966924i	−0.439389	−	0.898297i	$-0.644805\pi$
$439$	245.073	−	424.479i	0.558254	−	0.966924i	0.997642	−	0.0686266i	$-0.0218617\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−167.219	−	96.5440i	−0.377470	−	0.217932i	0.299247	−	0.954176i	$-0.403264\pi$
$443$	−167.219	−	96.5440i	−0.377470	−	0.217932i	−0.676717	+	0.736243i	$0.736598\pi$
$444$		0			0
$445$	−0.814903	−	1.41145i	−0.00183124	−	0.00317180i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	449.191i		−	1.00043i	−0.865902	−	0.500213i	$-0.833255\pi$
$449$		−	449.191i		−	1.00043i	0.865902	−	0.500213i	$-0.166745\pi$
$450$		0			0
$451$	−276.576			−0.613252
$452$		0			0
$453$		0			0
$454$		0			0
$455$	15.8697	−	9.16240i	0.0348785	−	0.0201371i
$456$		0			0
$457$	−47.0282	+	81.4553i	−0.102906	+	0.178239i	−0.912881	−	0.408226i	$-0.866148\pi$
$457$	−47.0282	+	81.4553i	−0.102906	+	0.178239i	0.809975	+	0.586465i	$0.199481\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−295.285	−	170.483i	−0.640532	−	0.369811i	0.144288	−	0.989536i	$-0.453911\pi$
$461$	−295.285	−	170.483i	−0.640532	−	0.369811i	−0.784819	+	0.619725i	$0.787244\pi$
$462$		0			0
$463$	285.649	+	494.759i	0.616953	+	1.06859i	0.990038	+	0.140798i	$0.0449666\pi$
$463$	285.649	+	494.759i	0.616953	+	1.06859i	−0.373085	+	0.927797i	$0.621700\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	421.650i		−	0.902891i	−0.892299	−	0.451446i	$-0.850909\pi$
$467$		−	421.650i		−	0.902891i	0.892299	−	0.451446i	$-0.149091\pi$
$468$		0			0
$469$	−57.0716			−0.121688
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−366.602	+	211.658i	−0.775058	+	0.447480i
$474$		0			0
$475$	−18.2482	+	31.6068i	−0.0384173	+	0.0665407i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−700.337	−	404.340i	−1.46208	−	0.844133i	−0.462973	−	0.886372i	$-0.653217\pi$
$479$	−700.337	−	404.340i	−1.46208	−	0.844133i	−0.999108	+	0.0422395i	$0.986551\pi$
$480$		0			0
$481$	−34.6523	−	60.0195i	−0.0720421	−	0.124781i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	98.6219i		−	0.203344i
$486$		0			0
$487$	155.199			0.318684			0.159342	−	0.987223i	$-0.449063\pi$
$487$	155.199			0.318684			0.159342	+	0.987223i	$0.449063\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−546.817	+	315.705i	−1.11368	+	0.642984i	−0.939780	−	0.341779i	$-0.888970\pi$
$491$	−546.817	+	315.705i	−1.11368	+	0.642984i	−0.173901	+	0.984763i	$0.555637\pi$
$492$		0			0
$493$	−189.535	+	328.285i	−0.384453	+	0.665892i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−227.619	−	131.416i	−0.457985	−	0.264418i
$498$		0			0
$499$	−445.820	−	772.183i	−0.893427	−	1.54746i	−0.835739	−	0.549126i	$-0.814961\pi$
$499$	−445.820	−	772.183i	−0.893427	−	1.54746i	−0.0576878	−	0.998335i	$-0.518373\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	281.433i		−	0.559509i	−0.960072	−	0.279754i	$-0.909747\pi$
$503$		−	281.433i		−	0.559509i	0.960072	−	0.279754i	$-0.0902531\pi$
$504$		0			0
$505$	46.3854			0.0918523
$506$		0			0
$507$		0			0
$508$		0			0
$509$	426.209	−	246.072i	0.837346	−	0.483442i	−0.0190149	−	0.999819i	$-0.506053\pi$
$509$	426.209	−	246.072i	0.837346	−	0.483442i	0.856361	+	0.516377i	$0.172720\pi$
$510$		0			0
$511$	−94.8047	+	164.207i	−0.185528	+	0.321344i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−145.635	−	84.0823i	−0.282786	−	0.163267i
$516$		0			0
$517$	376.903	+	652.815i	0.729020	+	1.26270i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	633.968i		−	1.21683i	−0.793619	−	0.608415i	$-0.791806\pi$
$521$		−	633.968i		−	1.21683i	0.793619	−	0.608415i	$-0.208194\pi$
$522$		0			0
$523$	−42.6893			−0.0816240			−0.0408120	−	0.999167i	$-0.512994\pi$
$523$	−42.6893			−0.0816240			−0.0408120	+	0.999167i	$0.512994\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−928.394	+	536.008i	−1.76166	+	1.01709i
$528$		0			0
$529$	−263.831	+	456.968i	−0.498734	+	0.863833i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	191.063	+	110.310i	0.358467	+	0.206961i
$534$		0			0
$535$	22.9787	+	39.8004i	0.0429509	+	0.0743932i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	466.660i		−	0.865788i
$540$		0			0
$541$	−585.520			−1.08229			−0.541146	−	0.840929i	$-0.682009\pi$
$541$	−585.520			−1.08229			−0.541146	+	0.840929i	$0.682009\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−32.2273	+	18.6065i	−0.0591327	+	0.0341403i
$546$		0			0
$547$	429.811	−	744.455i	0.785761	−	1.36098i	−0.142783	−	0.989754i	$-0.545605\pi$
$547$	429.811	−	744.455i	0.785761	−	1.36098i	0.928544	−	0.371223i	$-0.121062\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−23.6257	−	13.6403i	−0.0428779	−	0.0247556i
$552$		0			0
$553$	−130.759	−	226.481i	−0.236454	−	0.409550i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	394.616i		−	0.708467i	−0.935157	−	0.354234i	$-0.884742\pi$
$557$		−	394.616i		−	0.708467i	0.935157	−	0.354234i	$-0.115258\pi$
$558$		0			0
$559$	337.672			0.604065
$560$		0			0
$561$		0			0
$562$		0			0
$563$	375.650	−	216.882i	0.667229	−	0.385225i	−0.127797	−	0.991800i	$-0.540791\pi$
$563$	375.650	−	216.882i	0.667229	−	0.385225i	0.795026	+	0.606576i	$0.207457\pi$
$564$		0			0
$565$	−45.0347	+	78.0025i	−0.0797075	+	0.138057i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−187.251	−	108.110i	−0.329088	−	0.189999i	0.326348	−	0.945250i	$-0.394182\pi$
$569$	−187.251	−	108.110i	−0.329088	−	0.189999i	−0.655436	+	0.755250i	$0.727515\pi$
$570$		0			0
$571$	−525.111	−	909.518i	−0.919633	−	1.59285i	−0.799972	−	0.600037i	$-0.795152\pi$
$571$	−525.111	−	909.518i	−0.919633	−	1.59285i	−0.119661	−	0.992815i	$-0.538181\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			28.1256i			0.0489140i
$576$		0			0
$577$	235.000			0.407279			0.203640	−	0.979046i	$-0.434723\pi$
$577$	235.000			0.407279			0.203640	+	0.979046i	$0.434723\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−135.504	+	78.2334i	−0.233226	+	0.134653i
$582$		0			0
$583$	254.532	−	440.863i	0.436590	−	0.756197i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−669.449	−	386.506i	−1.14046	−	0.658443i	−0.193914	−	0.981019i	$-0.562118\pi$
$587$	−669.449	−	386.506i	−1.14046	−	0.658443i	−0.946544	+	0.322575i	$0.895452\pi$
$588$		0			0
$589$	−38.5750	−	66.8138i	−0.0654923	−	0.113436i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			504.381i			0.850557i	0.905062	+	0.425279i	$0.139824\pi$
$593$			504.381i			0.850557i	−0.905062	+	0.425279i	$0.860176\pi$
$594$		0			0
$595$	−43.7989			−0.0736117
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−37.9297	+	21.8987i	−0.0633217	+	0.0365588i	−0.531327	−	0.847167i	$-0.678306\pi$
$599$	−37.9297	+	21.8987i	−0.0633217	+	0.0365588i	0.468005	+	0.883726i	$0.344973\pi$
$600$		0			0
$601$	4.89587	−	8.47990i	0.00814621	−	0.0141096i	−0.861924	−	0.507038i	$-0.830740\pi$
$601$	4.89587	−	8.47990i	0.00814621	−	0.0141096i	0.870070	+	0.492929i	$0.164074\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.901637	−	0.520561i	−0.00149031	−	0.000860431i
$606$		0			0
$607$	−175.724	−	304.363i	−0.289496	−	0.501421i	0.684194	−	0.729300i	$-0.260154\pi$
$607$	−175.724	−	304.363i	−0.289496	−	0.501421i	−0.973689	+	0.227879i	$0.926821\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	601.298i		−	0.984122i
$612$		0			0
$613$	733.118			1.19595			0.597976	−	0.801514i	$-0.295972\pi$
$613$	733.118			1.19595			0.597976	+	0.801514i	$0.295972\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−557.311	+	321.764i	−0.903260	+	0.521497i	−0.878256	−	0.478190i	$-0.841293\pi$
$617$	−557.311	+	321.764i	−0.903260	+	0.521497i	−0.0250035	+	0.999687i	$0.507960\pi$
$618$		0			0
$619$	−124.319	+	215.327i	−0.200839	+	0.347863i	−0.948799	−	0.315880i	$-0.897700\pi$
$619$	−124.319	+	215.327i	−0.200839	+	0.347863i	0.747960	+	0.663744i	$0.231033\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.27312	−	2.46709i	−0.00685895	−	0.00396001i
$624$		0			0
$625$	−286.737	−	496.644i	−0.458780	−	0.794630i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			165.648i			0.263351i
$630$		0			0
$631$	−479.055			−0.759200			−0.379600	−	0.925151i	$-0.623938\pi$
$631$	−479.055			−0.759200			−0.379600	+	0.925151i	$0.623938\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−124.166	+	71.6873i	−0.195537	+	0.112893i
$636$		0			0
$637$	−186.123	+	322.375i	−0.292187	+	0.506083i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1019.97	+	588.880i	1.59122	+	0.918689i	0.993099	+	0.117281i	$0.0374178\pi$
$641$	1019.97	+	588.880i	1.59122	+	0.918689i	0.598118	+	0.801408i	$0.295916\pi$
$642$		0			0
$643$	−88.0856	−	152.569i	−0.136992	−	0.237276i	0.789365	−	0.613924i	$-0.210410\pi$
$643$	−88.0856	−	152.569i	−0.136992	−	0.237276i	−0.926356	+	0.376648i	$0.877077\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			317.529i			0.490771i	0.969426	+	0.245386i	$0.0789146\pi$
$647$			317.529i			0.490771i	−0.969426	+	0.245386i	$0.921085\pi$
$648$		0			0
$649$	1126.25			1.73536
$650$		0			0
$651$		0			0
$652$		0			0
$653$	883.885	−	510.311i	1.35358	−	0.781488i	0.364828	−	0.931075i	$-0.381128\pi$
$653$	883.885	−	510.311i	1.35358	−	0.781488i	0.988749	+	0.149587i	$0.0477945\pi$
$654$		0			0
$655$	−36.9320	+	63.9681i	−0.0563848	+	0.0976613i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	127.595	+	73.6669i	0.193619	+	0.111786i	0.593676	−	0.804704i	$-0.297676\pi$
$659$	127.595	+	73.6669i	0.193619	+	0.111786i	−0.400057	+	0.916490i	$0.631010\pi$
$660$		0			0
$661$	−386.978	−	670.266i	−0.585444	−	1.01402i	−0.994820	−	0.101653i	$-0.967587\pi$
$661$	−386.978	−	670.266i	−0.585444	−	1.01402i	0.409376	−	0.912366i	$-0.365746\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	3.15208i		−	0.00473997i
$666$		0			0
$667$	−21.0235			−0.0315195
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−837.100	+	483.300i	−1.24754	+	0.720268i
$672$		0			0
$673$	62.7363	−	108.663i	0.0932189	−	0.161460i	−0.815645	−	0.578553i	$-0.803618\pi$
$673$	62.7363	−	108.663i	0.0932189	−	0.161460i	0.908864	+	0.417093i	$0.136951\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−953.760	−	550.654i	−1.40880	−	0.813374i	−0.413531	−	0.910490i	$-0.635705\pi$
$677$	−953.760	−	550.654i	−1.40880	−	0.813374i	−0.995273	+	0.0971164i	$0.969038\pi$
$678$		0			0
$679$	−149.287	−	258.573i	−0.219863	−	0.380814i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	469.992i		−	0.688128i	−0.938946	−	0.344064i	$-0.888196\pi$
$683$		−	469.992i		−	0.688128i	0.938946	−	0.344064i	$-0.111804\pi$
$684$		0			0
$685$	74.6302			0.108949
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−351.669	+	203.036i	−0.510404	+	0.294682i
$690$		0			0
$691$	263.870	−	457.036i	0.381866	−	0.661412i	−0.609463	−	0.792815i	$-0.708615\pi$
$691$	263.870	−	457.036i	0.381866	−	0.661412i	0.991329	+	0.131403i	$0.0419482\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	107.873	+	62.2803i	0.155212	+	0.0896119i
$696$		0			0
$697$	−263.657	−	456.668i	−0.378275	−	0.655191i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1162.38i			1.65818i	0.559116	+	0.829089i	$0.311141\pi$
$701$			1162.38i			1.65818i	−0.559116	+	0.829089i	$0.688859\pi$
$702$		0			0
$703$	−11.9212			−0.0169576
$704$		0			0
$705$		0			0
$706$		0			0
$707$	121.616	−	70.2151i	0.172017	−	0.0993142i
$708$		0			0
$709$	−100.029	+	173.255i	−0.141085	+	0.244366i	−0.927905	−	0.372816i	$-0.878392\pi$
$709$	−100.029	+	173.255i	−0.141085	+	0.244366i	0.786821	+	0.617182i	$0.211726\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−51.4893	−	29.7274i	−0.0722150	−	0.0416934i
$714$		0			0
$715$	−39.7714	−	68.8861i	−0.0556244	−	0.0963442i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			447.907i			0.622958i	0.950253	+	0.311479i	$0.100824\pi$
$719$			447.907i			0.622958i	−0.950253	+	0.311479i	$0.899176\pi$
$720$		0			0
$721$	−509.112			−0.706120
$722$		0			0
$723$		0			0
$724$		0			0
$725$	382.457	−	220.812i	0.527527	−	0.304568i
$726$		0			0
$727$	253.737	−	439.486i	0.349020	−	0.604520i	−0.637056	−	0.770818i	$-0.719848\pi$
$727$	253.737	−	439.486i	0.349020	−	0.604520i	0.986076	+	0.166298i	$0.0531813\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−698.957	−	403.543i	−0.956165	−	0.552042i
$732$		0			0
$733$	−545.542	−	944.907i	−0.744260	−	1.28910i	−0.950540	−	0.310603i	$-0.899469\pi$
$733$	−545.542	−	944.907i	−0.744260	−	1.28910i	0.206280	−	0.978493i	$-0.433864\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			247.732i			0.336136i
$738$		0			0
$739$	−114.753			−0.155281			−0.0776406	−	0.996981i	$-0.524739\pi$
$739$	−114.753			−0.155281			−0.0776406	+	0.996981i	$0.524739\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	531.028	−	306.589i	0.714708	−	0.412637i	−0.0980939	−	0.995177i	$-0.531275\pi$
$743$	531.028	−	306.589i	0.714708	−	0.412637i	0.812802	+	0.582540i	$0.197941\pi$
$744$		0			0
$745$	58.3231	−	101.019i	0.0782860	−	0.135595i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	120.494	+	69.5673i	0.160873	+	0.0928803i
$750$		0			0
$751$	193.945	+	335.922i	0.258248	+	0.447299i	0.965773	−	0.259390i	$-0.0835214\pi$
$751$	193.945	+	335.922i	0.258248	+	0.447299i	−0.707524	+	0.706689i	$0.750188\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	120.854i		−	0.160071i
$756$		0			0
$757$	732.340			0.967424			0.483712	−	0.875227i	$-0.339288\pi$
$757$	732.340			0.967424			0.483712	+	0.875227i	$0.339288\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−771.356	+	445.342i	−1.01361	+	0.585207i	−0.912246	−	0.409643i	$-0.865653\pi$
$761$	−771.356	+	445.342i	−1.01361	+	0.585207i	−0.101362	+	0.994850i	$0.532320\pi$
$762$		0			0
$763$	−56.3304	+	97.5671i	−0.0738275	+	0.127873i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−778.027	−	449.194i	−1.01438	−	0.585650i
$768$		0			0
$769$	−244.356	−	423.237i	−0.317758	−	0.550373i	0.662262	−	0.749272i	$-0.269597\pi$
$769$	−244.356	−	423.237i	−0.317758	−	0.550373i	−0.980020	+	0.198899i	$0.936263\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	579.450i		−	0.749612i	−0.927103	−	0.374806i	$-0.877709\pi$
$773$		−	579.450i		−	0.749612i	0.927103	−	0.374806i	$-0.122291\pi$
$774$		0			0
$775$	1248.92			1.61150
$776$		0			0
$777$		0			0
$778$		0			0
$779$	32.8651	−	18.9747i	0.0421888	−	0.0243577i
$780$		0			0
$781$	−570.439	+	988.029i	−0.730395	+	1.26508i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	178.182	+	102.873i	0.226983	+	0.131049i
$786$		0			0
$787$	−87.9405	−	152.317i	−0.111741	−	0.193542i	0.804731	−	0.593640i	$-0.202310\pi$
$787$	−87.9405	−	152.317i	−0.111741	−	0.193542i	−0.916472	+	0.400098i	$0.868976\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			272.682i			0.344731i
$792$		0			0
$793$	771.040			0.972308
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−397.577	+	229.541i	−0.498841	+	0.288006i	−0.728235	−	0.685328i	$-0.759659\pi$
$797$	−397.577	+	229.541i	−0.498841	+	0.288006i	0.229394	+	0.973334i	$0.426326\pi$
$798$		0			0
$799$	−718.596	+	1244.64i	−0.899369	+	1.55775i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	712.775	+	411.521i	0.887640	+	0.512479i
$804$		0			0
$805$	−1.21456	−	2.10368i	−0.00150877	−	0.00261326i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	89.1339i		−	0.110178i	−0.998481	−	0.0550890i	$-0.982456\pi$
$809$		−	89.1339i		−	0.110178i	0.998481	−	0.0550890i	$-0.0175442\pi$
$810$		0			0
$811$	−625.256			−0.770969			−0.385484	−	0.922714i	$-0.625966\pi$
$811$	−625.256			−0.770969			−0.385484	+	0.922714i	$0.625966\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	136.545	−	78.8341i	0.167539	−	0.0967290i
$816$		0			0
$817$	29.0418	−	50.3019i	0.0355469	−	0.0615690i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	625.391	+	361.070i	0.761743	+	0.439793i	0.829921	−	0.557880i	$-0.188385\pi$
$821$	625.391	+	361.070i	0.761743	+	0.439793i	−0.0681779	+	0.997673i	$0.521719\pi$
$822$		0			0
$823$	149.534	+	259.000i	0.181693	+	0.314702i	0.942457	−	0.334327i	$-0.108509\pi$
$823$	149.534	+	259.000i	0.181693	+	0.314702i	−0.760764	+	0.649029i	$0.775176\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1178.17i			1.42463i	0.701858	+	0.712317i	$0.252354\pi$
$827$			1178.17i			1.42463i	−0.701858	+	0.712317i	$0.747646\pi$
$828$		0			0
$829$	−451.760			−0.544946			−0.272473	−	0.962163i	$-0.587841\pi$
$829$	−451.760			−0.544946			−0.272473	+	0.962163i	$0.587841\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	770.523	−	444.862i	0.924998	−	0.534048i
$834$		0			0
$835$	96.4243	−	167.012i	0.115478	−	0.200014i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	274.743	+	158.623i	0.327465	+	0.189062i	0.654715	−	0.755876i	$-0.272789\pi$
$839$	274.743	+	158.623i	0.327465	+	0.189062i	−0.327250	+	0.944938i	$0.606122\pi$
$840$		0			0
$841$	−255.446	−	442.445i	−0.303741	−	0.526094i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	77.2788i		−	0.0914542i
$846$		0			0
$847$	−3.15196			−0.00372132
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−7.95613	+	4.59347i	−0.00934915	+	0.00539773i
$852$		0			0
$853$	55.9234	−	96.8622i	0.0655609	−	0.113555i	−0.831382	−	0.555702i	$-0.812450\pi$
$853$	55.9234	−	96.8622i	0.0655609	−	0.113555i	0.896943	+	0.442147i	$0.145783\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	522.040	+	301.400i	0.609148	+	0.351692i	0.772632	−	0.634854i	$-0.218940\pi$
$857$	522.040	+	301.400i	0.609148	+	0.351692i	−0.163484	+	0.986546i	$0.552273\pi$
$858$		0			0
$859$	649.175	+	1124.40i	0.755733	+	1.30897i	0.945009	+	0.327044i	$0.106053\pi$
$859$	649.175	+	1124.40i	0.755733	+	1.30897i	−0.189276	+	0.981924i	$0.560614\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	1332.38i		−	1.54390i	−0.635685	−	0.771949i	$-0.719282\pi$
$863$		−	1332.38i		−	1.54390i	0.635685	−	0.771949i	$-0.280718\pi$
$864$		0			0
$865$	−216.255			−0.250006
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−983.092	+	567.589i	−1.13129	+	0.653151i
$870$		0			0
$871$	98.8059	−	171.137i	0.113440	−	0.196483i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	89.6411	+	51.7543i	0.102447	+	0.0591478i
$876$		0			0
$877$	407.412	+	705.658i	0.464551	+	0.804627i	0.999181	−	0.0404597i	$-0.0128822\pi$
$877$	407.412	+	705.658i	0.464551	+	0.804627i	−0.534630	+	0.845086i	$0.679549\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			86.4877i			0.0981700i	0.998795	+	0.0490850i	$0.0156305\pi$
$881$			86.4877i			0.0981700i	−0.998795	+	0.0490850i	$0.984369\pi$
$882$		0			0
$883$	367.217			0.415875			0.207937	−	0.978142i	$-0.433325\pi$
$883$	367.217			0.415875			0.207937	+	0.978142i	$0.433325\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	587.456	−	339.168i	0.662295	−	0.382376i	−0.130856	−	0.991401i	$-0.541773\pi$
$887$	587.456	−	339.168i	0.662295	−	0.382376i	0.793151	+	0.609025i	$0.208439\pi$
$888$		0			0
$889$	−217.031	+	375.908i	−0.244129	+	0.422844i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−89.5734	−	51.7152i	−0.100306	−	0.0579118i
$894$		0			0
$895$	91.7475	+	158.911i	0.102511	+	0.177555i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			933.550i			1.03843i
$900$		0			0
$901$	970.571			1.07722
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−206.386	+	119.157i	−0.228051	+	0.131665i
$906$		0			0
$907$	−431.371	+	747.156i	−0.475602	+	0.823767i	−0.999609	−	0.0279469i	$-0.991103\pi$
$907$	−431.371	+	747.156i	−0.475602	+	0.823767i	0.524007	+	0.851714i	$0.324436\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1007.57	+	581.718i	1.10600	+	0.638549i	0.937790	−	0.347202i	$-0.112868\pi$
$911$	1007.57	+	581.718i	1.10600	+	0.638549i	0.168209	+	0.985751i	$0.446202\pi$
$912$		0			0
$913$	339.590	+	588.186i	0.371949	+	0.644235i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			223.621i			0.243861i
$918$		0			0
$919$	−832.360			−0.905724			−0.452862	−	0.891581i	$-0.649597\pi$
$919$	−832.360			−0.905724			−0.452862	+	0.891581i	$0.649597\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	788.134	−	455.029i	0.853883	−	0.492990i
$924$		0			0
$925$	96.4912	−	167.128i	0.104315	−	0.180679i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1540.10	−	889.178i	−1.65781	−	0.957135i	−0.973724	−	0.227730i	$-0.926870\pi$
$929$	−1540.10	−	889.178i	−1.65781	−	0.957135i	−0.684082	−	0.729405i	$-0.739797\pi$
$930$		0			0
$931$	32.0154	+	55.4523i	0.0343882	+	0.0595621i
$932$		0			0
$933$		0			0
$934$		0			0
$935$			190.119i			0.203336i
$936$		0			0
$937$	432.361			0.461431			0.230715	−	0.973021i	$-0.425893\pi$
$937$	432.361			0.461431			0.230715	+	0.973021i	$0.425893\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1521.40	−	878.378i	1.61679	−	0.933452i	0.629041	−	0.777372i	$-0.283448\pi$
$941$	1521.40	−	878.378i	1.61679	−	0.933452i	0.987744	−	0.156080i	$-0.0498857\pi$
$942$		0			0
$943$	14.6226	−	25.3271i	0.0155065	−	0.0268580i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	980.755	+	566.239i	1.03564	+	0.597929i	0.918596	−	0.395197i	$-0.129324\pi$
$947$	980.755	+	566.239i	1.03564	+	0.597929i	0.117048	+	0.993126i	$0.462657\pi$
$948$		0			0
$949$	−328.263	−	568.568i	−0.345904	−	0.599124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	1509.90i		−	1.58436i	−0.610286	−	0.792181i	$-0.708945\pi$
$953$		−	1509.90i		−	1.58436i	0.610286	−	0.792181i	$-0.291055\pi$
$954$		0			0
$955$	1.31530			0.00137728
$956$		0			0
$957$		0			0
$958$		0			0
$959$	195.670	−	112.970i	0.204035	−	0.117800i
$960$		0			0
$961$	−839.545	+	1454.13i	−0.873616	+	1.51315i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	167.403	+	96.6503i	0.173475	+	0.100156i
$966$		0			0
$967$	−244.990	−	424.335i	−0.253350	−	0.438816i	0.711096	−	0.703095i	$-0.248199\pi$
$967$	−244.990	−	424.335i	−0.253350	−	0.438816i	−0.964446	+	0.264279i	$0.914866\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			253.051i			0.260608i	0.991474	+	0.130304i	$0.0415954\pi$
$971$			253.051i			0.260608i	−0.991474	+	0.130304i	$0.958405\pi$
$972$		0			0
$973$	377.103			0.387567
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1407.52	−	812.635i	1.44066	−	0.831765i	0.442766	−	0.896637i	$-0.353997\pi$
$977$	1407.52	−	812.635i	1.44066	−	0.831765i	0.997894	+	0.0648717i	$0.0206638\pi$
$978$		0			0
$979$	−10.7089	+	18.5484i	−0.0109387	+	0.0189463i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1007.16	+	581.483i	1.02458	+	0.591540i	0.915426	−	0.402486i	$-0.131854\pi$
$983$	1007.16	+	581.483i	1.02458	+	0.591540i	0.109150	+	0.994025i	$0.465187\pi$
$984$		0			0
$985$	−89.1055	−	154.335i	−0.0904624	−	0.156686i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	44.7615i		−	0.0452594i
$990$		0			0
$991$	−1411.24			−1.42405			−0.712027	−	0.702152i	$-0.752223\pi$
$991$	−1411.24			−1.42405			−0.712027	+	0.702152i	$0.752223\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	18.6715	−	10.7800i	0.0187653	−	0.0108341i
$996$		0			0
$997$	−561.614	+	972.744i	−0.563304	+	0.975671i	0.433902	+	0.900960i	$0.357137\pi$
$997$	−561.614	+	972.744i	−0.563304	+	0.975671i	−0.997205	+	0.0747104i	$0.976197\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.3.q.a.737.5		24
3.2	odd	2		288.3.q.b.65.8	yes	24
4.3	odd	2	inner	864.3.q.a.737.6		24
8.3	odd	2		1728.3.q.k.1601.8		24
8.5	even	2		1728.3.q.k.1601.7		24
9.2	odd	6		2592.3.e.i.161.11		24
9.4	even	3		288.3.q.b.257.8	yes	24
9.5	odd	6	inner	864.3.q.a.449.5		24
9.7	even	3		2592.3.e.i.161.12		24
12.11	even	2		288.3.q.b.65.5	✓	24
24.5	odd	2		576.3.q.l.65.5		24
24.11	even	2		576.3.q.l.65.8		24
36.7	odd	6		2592.3.e.i.161.14		24
36.11	even	6		2592.3.e.i.161.13		24
36.23	even	6	inner	864.3.q.a.449.6		24
36.31	odd	6		288.3.q.b.257.5	yes	24
72.5	odd	6		1728.3.q.k.449.7		24
72.13	even	6		576.3.q.l.257.5		24
72.59	even	6		1728.3.q.k.449.8		24
72.67	odd	6		576.3.q.l.257.8		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
288.3.q.b.65.5	✓	24	12.11	even	2
288.3.q.b.65.8	yes	24	3.2	odd	2
288.3.q.b.257.5	yes	24	36.31	odd	6
288.3.q.b.257.8	yes	24	9.4	even	3
576.3.q.l.65.5		24	24.5	odd	2
576.3.q.l.65.8		24	24.11	even	2
576.3.q.l.257.5		24	72.13	even	6
576.3.q.l.257.8		24	72.67	odd	6
864.3.q.a.449.5		24	9.5	odd	6	inner
864.3.q.a.449.6		24	36.23	even	6	inner
864.3.q.a.737.5		24	1.1	even	1	trivial
864.3.q.a.737.6		24	4.3	odd	2	inner
1728.3.q.k.449.7		24	72.5	odd	6
1728.3.q.k.449.8		24	72.59	even	6
1728.3.q.k.1601.7		24	8.5	even	2
1728.3.q.k.1601.8		24	8.3	odd	2
2592.3.e.i.161.11		24	9.2	odd	6
2592.3.e.i.161.12		24	9.7	even	3
2592.3.e.i.161.13		24	36.11	even	6
2592.3.e.i.161.14		24	36.7	odd	6

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.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.721152 - 0.416357i) q^{5} +(1.26051 - 2.18326i) q^{7} +O(q^{10})\)	\(q+(0.721152 - 0.416357i) q^{5} +(1.26051 - 2.18326i) q^{7} +(-9.47693 - 5.47151i) q^{11} +(4.36453 + 7.55959i) q^{13} -20.8637i q^{17} +1.50150 q^{19} +(1.00209 - 0.578558i) q^{23} +(-12.1533 + 21.0501i) q^{25} +(-15.7347 - 9.08444i) q^{29} +(-25.6909 - 44.4980i) q^{31} -2.09929i q^{35} -7.93951 q^{37} +(21.8881 - 12.6371i) q^{41} +(19.3418 - 33.5010i) q^{43} +(-59.6559 - 34.4423i) q^{47} +(21.3222 + 36.9312i) q^{49} +46.5195i q^{53} -9.11241 q^{55} +(-89.1306 + 51.4596i) q^{59} +(44.1651 - 76.4962i) q^{61} +(6.29498 + 3.63441i) q^{65} +(-11.3192 - 19.6054i) q^{67} -104.256i q^{71} -75.2115 q^{73} +(-23.8915 + 13.7938i) q^{77} +(51.8676 - 89.8374i) q^{79} +(-53.7499 - 31.0325i) q^{83} +(-8.68677 - 15.0459i) q^{85} -1.95722i q^{89} +22.0061 q^{91} +(1.08281 - 0.625162i) q^{95} +(59.2171 - 102.567i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q+O(q^{10}) \) 24 * q	\( 24 q + 60 q^{25} - 72 q^{29} - 252 q^{41} - 36 q^{49} - 96 q^{61} + 288 q^{65} + 24 q^{73} + 720 q^{77} + 96 q^{85} - 132 q^{97}+O(q^{100}) \) 24 * q + 60 * q^25 - 72 * q^29 - 252 * q^41 - 36 * q^49 - 96 * q^61 + 288 * q^65 + 24 * q^73 + 720 * q^77 + 96 * q^85 - 132 * q^97

Introduction

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