LMFDB - Embedded newform 324.8.e.m.217.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,8,Mod(109,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("324.109");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	324.e (of order $3$, 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:	$101.212748257$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 3205 x^{14} + 7140274 x^{12} + 8220484645 x^{10} + 6820694102626 x^{8} + \cdots + 57\!\cdots\!00 $ x^16 + 3205x^14 + 7140274x^12 + 8220484645x^10 + 6820694102626x^8 + 2359067894603905x^6 + 587831501492949925x^4 + 68841856420333385500*x^2 + 5743303187209972810000
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{32}\cdot 3^{44} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.3
Root			$-6.08630 - 10.5418i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.8.e.m.109.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-82.6178 - 143.098i) q^{5} +(122.405 - 212.011i) q^{7} +O(q^{10})$	$q+(-82.6178 - 143.098i) q^{5} +(122.405 - 212.011i) q^{7} +(-3966.92 + 6870.90i) q^{11} +(6470.24 + 11206.8i) q^{13} +8572.76 q^{17} +14729.6 q^{19} +(-25623.9 - 44381.9i) q^{23} +(25411.1 - 44013.3i) q^{25} +(35350.2 - 61228.4i) q^{29} +(-131050. - 226985. i) q^{31} -40451.2 q^{35} -108156. q^{37} +(57801.9 + 100116. i) q^{41} +(189803. - 328749. i) q^{43} +(-465181. + 805716. i) q^{47} +(381806. + 661307. i) q^{49} +372981. q^{53} +1.31095e6 q^{55} +(-378164. - 654999. i) q^{59} +(-1.33564e6 + 2.31340e6i) q^{61} +(1.06911e6 - 1.85176e6i) q^{65} +(-887141. - 1.53657e6i) q^{67} -3.44236e6 q^{71} -2.75839e6 q^{73} +(971138. + 1.68206e6i) q^{77} +(831088. - 1.43949e6i) q^{79} +(-1.46060e6 + 2.52984e6i) q^{83} +(-708262. - 1.22675e6i) q^{85} -8.29306e6 q^{89} +3.16795e6 q^{91} +(-1.21692e6 - 2.10777e6i) q^{95} +(-3.53367e6 + 6.12049e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 560 q^{7}+O(q^{10}) $ 16 * q + 560 * q^7	$ 16 q + 560 q^{7} - 1480 q^{13} - 55264 q^{19} - 77936 q^{25} + 247424 q^{31} + 220400 q^{37} - 897040 q^{43} - 1329672 q^{49} + 1910880 q^{55} - 494968 q^{61} - 4698160 q^{67} + 21452240 q^{73} - 8887312 q^{79} - 15973992 q^{85} + 30743008 q^{91} - 33664240 q^{97}+O(q^{100}) $ 16 * q + 560 * q^7 - 1480 * q^13 - 55264 * q^19 - 77936 * q^25 + 247424 * q^31 + 220400 * q^37 - 897040 * q^43 - 1329672 * q^49 + 1910880 * q^55 - 494968 * q^61 - 4698160 * q^67 + 21452240 * q^73 - 8887312 * q^79 - 15973992 * q^85 + 30743008 * q^91 - 33664240 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

$n$	$163$	$245$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−82.6178	−	143.098i	−0.295582	−	0.511964i	0.679538	−	0.733640i	$-0.262180\pi$
$5$	−82.6178	−	143.098i	−0.295582	−	0.511964i	−0.975120	+	0.221677i	$0.928847\pi$
$6$		0			0
$7$	122.405	−	212.011i	0.134882	−	0.233623i	−0.790670	−	0.612242i	$-0.790268\pi$
$7$	122.405	−	212.011i	0.134882	−	0.233623i	0.925552	+	0.378619i	$0.123601\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3966.92	+	6870.90i	−0.898626	+	1.55647i	−0.0693739	+	0.997591i	$0.522100\pi$
$11$	−3966.92	+	6870.90i	−0.898626	+	1.55647i	−0.829252	+	0.558875i	$0.811233\pi$
$12$		0			0
$13$	6470.24	+	11206.8i	0.816806	+	1.41475i	0.908024	+	0.418918i	$0.137591\pi$
$13$	6470.24	+	11206.8i	0.816806	+	1.41475i	−0.0912185	+	0.995831i	$0.529076\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	8572.76			0.423204			0.211602	−	0.977356i	$-0.432132\pi$
$17$	8572.76			0.423204			0.211602	+	0.977356i	$0.432132\pi$
$18$		0			0
$19$	14729.6			0.492665			0.246333	−	0.969185i	$-0.420774\pi$
$19$	14729.6			0.492665			0.246333	+	0.969185i	$0.420774\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−25623.9	−	44381.9i	−0.439134	−	0.760603i	0.558489	−	0.829512i	$-0.311381\pi$
$23$	−25623.9	−	44381.9i	−0.439134	−	0.760603i	−0.997623	+	0.0689092i	$0.978048\pi$
$24$		0			0
$25$	25411.1	−	44013.3i	0.325262	−	0.563371i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	35350.2	−	61228.4i	0.269153	−	0.466187i	−0.699490	−	0.714642i	$-0.746590\pi$
$29$	35350.2	−	61228.4i	0.269153	−	0.466187i	0.968643	+	0.248455i	$0.0799229\pi$
$30$		0			0
$31$	−131050.	−	226985.i	−0.790079	−	1.36846i	−0.925917	−	0.377726i	$-0.876706\pi$
$31$	−131050.	−	226985.i	−0.790079	−	1.36846i	0.135838	−	0.990731i	$-0.456627\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−40451.2			−0.159475
$36$		0			0
$37$	−108156.			−0.351031			−0.175516	−	0.984477i	$-0.556159\pi$
$37$	−108156.			−0.351031			−0.175516	+	0.984477i	$0.556159\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	57801.9	+	100116.i	0.130978	+	0.226861i	0.924054	−	0.382262i	$-0.124855\pi$
$41$	57801.9	+	100116.i	0.130978	+	0.226861i	−0.793076	+	0.609123i	$0.791522\pi$
$42$		0			0
$43$	189803.	−	328749.i	0.364052	−	0.630557i	−0.624571	−	0.780968i	$-0.714726\pi$
$43$	189803.	−	328749.i	0.364052	−	0.630557i	0.988624	+	0.150411i	$0.0480597\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−465181.	+	805716.i	−0.653550	+	1.13198i	0.328705	+	0.944433i	$0.393388\pi$
$47$	−465181.	+	805716.i	−0.653550	+	1.13198i	−0.982255	+	0.187550i	$0.939945\pi$
$48$		0			0
$49$	381806.	+	661307.i	0.463614	+	0.803002i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	372981.			0.344129			0.172064	−	0.985086i	$-0.444956\pi$
$53$	372981.			0.344129			0.172064	+	0.985086i	$0.444956\pi$
$54$		0			0
$55$	1.31095e6			1.06247
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−378164.	−	654999.i	−0.239716	−	0.415201i	0.720916	−	0.693022i	$-0.243721\pi$
$59$	−378164.	−	654999.i	−0.239716	−	0.415201i	−0.960633	+	0.277821i	$0.910388\pi$
$60$		0			0
$61$	−1.33564e6	+	2.31340e6i	−0.753416	+	1.30496i	0.192742	+	0.981250i	$0.438262\pi$
$61$	−1.33564e6	+	2.31340e6i	−0.753416	+	1.30496i	−0.946158	+	0.323706i	$0.895071\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.06911e6	−	1.85176e6i	0.482867	−	0.836349i
$66$		0			0
$67$	−887141.	−	1.53657e6i	−0.360355	−	0.624154i	0.627664	−	0.778484i	$-0.284011\pi$
$67$	−887141.	−	1.53657e6i	−0.360355	−	0.624154i	−0.988019	+	0.154331i	$0.950678\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.44236e6			−1.14144			−0.570718	−	0.821146i	$-0.693335\pi$
$71$	−3.44236e6			−1.14144			−0.570718	+	0.821146i	$0.693335\pi$
$72$		0			0
$73$	−2.75839e6			−0.829898			−0.414949	−	0.909845i	$-0.636201\pi$
$73$	−2.75839e6			−0.829898			−0.414949	+	0.909845i	$0.636201\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	971138.	+	1.68206e6i	0.242417	+	0.419879i
$78$		0			0
$79$	831088.	−	1.43949e6i	0.189650	−	0.328483i	−0.755484	−	0.655167i	$-0.772598\pi$
$79$	831088.	−	1.43949e6i	0.189650	−	0.328483i	0.945133	+	0.326685i	$0.105931\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.46060e6	+	2.52984e6i	−0.280387	+	0.485645i	−0.971480	−	0.237121i	$-0.923796\pi$
$83$	−1.46060e6	+	2.52984e6i	−0.280387	+	0.485645i	0.691093	+	0.722766i	$0.257130\pi$
$84$		0			0
$85$	−708262.	−	1.22675e6i	−0.125092	−	0.216665i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−8.29306e6			−1.24695			−0.623476	−	0.781843i	$-0.714280\pi$
$89$	−8.29306e6			−1.24695			−0.623476	+	0.781843i	$0.714280\pi$
$90$		0			0
$91$	3.16795e6			0.440690
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.21692e6	−	2.10777e6i	−0.145623	−	0.252227i
$96$		0			0
$97$	−3.53367e6	+	6.12049e6i	−0.393119	+	0.680903i	−0.992859	−	0.119292i	$-0.961937\pi$
$97$	−3.53367e6	+	6.12049e6i	−0.393119	+	0.680903i	0.599740	+	0.800195i	$0.295271\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.73370e6	+	1.33952e7i	−0.746900	+	1.29367i	0.202401	+	0.979303i	$0.435125\pi$
$101$	−7.73370e6	+	1.33952e7i	−0.746900	+	1.29367i	−0.949302	+	0.314367i	$0.898208\pi$
$102$		0			0
$103$	2.47805e6	+	4.29212e6i	0.223450	+	0.387027i	0.955853	−	0.293844i	$-0.0949347\pi$
$103$	2.47805e6	+	4.29212e6i	0.223450	+	0.387027i	−0.732403	+	0.680871i	$0.761601\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−62339.7			−0.00491951			−0.00245975	−	0.999997i	$-0.500783\pi$
$107$	−62339.7			−0.00491951			−0.00245975	+	0.999997i	$0.500783\pi$
$108$		0			0
$109$	2.09948e7			1.55281			0.776407	−	0.630232i	$-0.217040\pi$
$109$	2.09948e7			1.55281			0.776407	+	0.630232i	$0.217040\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.41549e7	−	2.45170e7i	−0.922851	−	1.59843i	−0.794981	−	0.606635i	$-0.792519\pi$
$113$	−1.41549e7	−	2.45170e7i	−0.922851	−	1.59843i	−0.127871	−	0.991791i	$-0.540814\pi$
$114$		0			0
$115$	−4.23397e6	+	7.33346e6i	−0.259601	+	0.449642i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.04935e6	−	1.81752e6i	0.0570827	−	0.0988701i
$120$		0			0
$121$	−2.17293e7	−	3.76363e7i	−1.11506	−	1.93134i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.13067e7			−0.975732
$126$		0			0
$127$	8.74447e6			0.378809			0.189405	−	0.981899i	$-0.439344\pi$
$127$	8.74447e6			0.378809			0.189405	+	0.981899i	$0.439344\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.40318e7	+	2.43037e7i	0.545334	+	0.944547i	0.998586	+	0.0531642i	$0.0169307\pi$
$131$	1.40318e7	+	2.43037e7i	0.545334	+	0.944547i	−0.453251	+	0.891383i	$0.649736\pi$
$132$		0			0
$133$	1.80297e6	−	3.12283e6i	0.0664518	−	0.115098i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−2.34506e7	+	4.06177e7i	−0.779171	+	1.34956i	0.153250	+	0.988188i	$0.451026\pi$
$137$	−2.34506e7	+	4.06177e7i	−0.779171	+	1.34956i	−0.932420	+	0.361376i	$0.882307\pi$
$138$		0			0
$139$	−9.37514e6	−	1.62382e7i	−0.296092	−	0.512846i	0.679147	−	0.734003i	$-0.262350\pi$
$139$	−9.37514e6	−	1.62382e7i	−0.296092	−	0.512846i	−0.975238	+	0.221157i	$0.929017\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−1.02668e8			−2.93601
$144$		0			0
$145$	−1.16822e7			−0.318228
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.01572e7	−	1.75927e7i	−0.251548	−	0.435694i	0.712404	−	0.701769i	$-0.247606\pi$
$149$	−1.01572e7	−	1.75927e7i	−0.251548	−	0.435694i	−0.963952	+	0.266075i	$0.914273\pi$
$150$		0			0
$151$	−2.03320e7	+	3.52160e7i	−0.480573	+	0.832378i	−0.999752	−	0.0222884i	$-0.992905\pi$
$151$	−2.03320e7	+	3.52160e7i	−0.480573	+	0.832378i	0.519178	+	0.854666i	$0.326238\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.16541e7	+	3.75060e7i	−0.467067	+	0.808984i
$156$		0			0
$157$	2.67373e6	+	4.63103e6i	0.0551402	+	0.0955057i	0.892278	−	0.451486i	$-0.149106\pi$
$157$	2.67373e6	+	4.63103e6i	0.0551402	+	0.0955057i	−0.837138	+	0.546992i	$0.815773\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−1.25459e7			−0.236926
$162$		0			0
$163$	9.01917e7			1.63121			0.815605	−	0.578608i	$-0.196404\pi$
$163$	9.01917e7			1.63121			0.815605	+	0.578608i	$0.196404\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−4.86150e7	−	8.42037e7i	−0.807724	−	1.39902i	−0.914437	−	0.404729i	$-0.867366\pi$
$167$	−4.86150e7	−	8.42037e7i	−0.807724	−	1.39902i	0.106713	−	0.994290i	$-0.465967\pi$
$168$		0			0
$169$	−5.23538e7	+	9.06794e7i	−0.834343	+	1.44512i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.80454e6	+	1.00538e7i	−0.0852327	+	0.147627i	−0.905490	−	0.424367i	$-0.860497\pi$
$173$	−5.80454e6	+	1.00538e7i	−0.0852327	+	0.147627i	0.820258	+	0.571994i	$0.193830\pi$
$174$		0			0
$175$	−6.22088e6	−	1.07749e7i	−0.0877442	−	0.151977i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−9.38751e7			−1.22339			−0.611695	−	0.791094i	$-0.709512\pi$
$179$	−9.38751e7			−1.22339			−0.611695	+	0.791094i	$0.709512\pi$
$180$		0			0
$181$	5.68871e7			0.713080			0.356540	−	0.934280i	$-0.383956\pi$
$181$	5.68871e7			0.713080			0.356540	+	0.934280i	$0.383956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	8.93563e6	+	1.54770e7i	0.103759	+	0.179715i
$186$		0			0
$187$	−3.40074e7	+	5.89026e7i	−0.380302	+	0.658702i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−3.07932e7	+	5.33355e7i	−0.319771	+	0.553859i	−0.980440	−	0.196818i	$-0.936939\pi$
$191$	−3.07932e7	+	5.33355e7i	−0.319771	+	0.553859i	0.660669	+	0.750677i	$0.270273\pi$
$192$		0			0
$193$	−2.95198e7	−	5.11297e7i	−0.295571	−	0.511944i	0.679546	−	0.733633i	$-0.262177\pi$
$193$	−2.95198e7	−	5.11297e7i	−0.295571	−	0.511944i	−0.975118	+	0.221688i	$0.928843\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.05205e8			−1.91230			−0.956150	−	0.292877i	$-0.905387\pi$
$197$	−2.05205e8			−1.91230			−0.956150	+	0.292877i	$0.905387\pi$
$198$		0			0
$199$	2.01123e7			0.180915			0.0904577	−	0.995900i	$-0.471167\pi$
$199$	2.01123e7			0.180915			0.0904577	+	0.995900i	$0.471167\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−8.65406e6	−	1.49893e7i	−0.0726079	−	0.125761i
$204$		0			0
$205$	9.55093e6	−	1.65427e7i	0.0774296	−	0.134112i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−5.84309e7	+	1.01205e8i	−0.442722	+	0.766816i
$210$		0			0
$211$	−1.29415e8	−	2.24153e8i	−0.948409	−	1.64269i	−0.748778	−	0.662821i	$-0.769359\pi$
$211$	−1.29415e8	−	2.24153e8i	−0.948409	−	1.64269i	−0.199630	−	0.979871i	$-0.563974\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.27244e7			−0.430429
$216$		0			0
$217$	−6.41644e7			−0.426271
$218$		0			0
$219$		0			0
$220$		0			0
$221$	5.54678e7	+	9.60731e7i	0.345675	+	0.598727i
$222$		0			0
$223$	7.93753e7	−	1.37482e8i	0.479312	−	0.830192i	−0.520407	−	0.853919i	$-0.674220\pi$
$223$	7.93753e7	−	1.37482e8i	0.479312	−	0.830192i	0.999718	+	0.0237262i	$0.00755298\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.01321e8	+	1.75492e8i	−0.574919	+	0.995790i	0.421131	+	0.907000i	$0.361633\pi$
$227$	−1.01321e8	+	1.75492e8i	−0.574919	+	0.995790i	−0.996050	+	0.0887898i	$0.971700\pi$
$228$		0			0
$229$	1.00345e7	+	1.73803e7i	0.0552171	+	0.0956388i	0.892313	−	0.451418i	$-0.149082\pi$
$229$	1.00345e7	+	1.73803e7i	0.0552171	+	0.0956388i	−0.837096	+	0.547057i	$0.815748\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.28702e7			0.0666561			0.0333281	−	0.999444i	$-0.489389\pi$
$233$	1.28702e7			0.0666561			0.0333281	+	0.999444i	$0.489389\pi$
$234$		0			0
$235$	1.53729e8			0.772712
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−1.34495e8	−	2.32953e8i	−0.637258	−	1.10376i	−0.986032	−	0.166556i	$-0.946735\pi$
$239$	−1.34495e8	−	2.32953e8i	−0.637258	−	1.10376i	0.348774	−	0.937207i	$-0.386598\pi$
$240$		0			0
$241$	−1.46299e8	+	2.53397e8i	−0.673258	+	1.16612i	0.303716	+	0.952762i	$0.401772\pi$
$241$	−1.46299e8	+	2.53397e8i	−0.673258	+	1.16612i	−0.976975	+	0.213355i	$0.931561\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	6.30879e7	−	1.09271e8i	0.274072	−	0.474706i
$246$		0			0
$247$	9.53037e7	+	1.65071e8i	0.402412	+	0.696998i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	1.85208e8			0.739269			0.369635	−	0.929177i	$-0.379483\pi$
$251$	1.85208e8			0.739269			0.369635	+	0.929177i	$0.379483\pi$
$252$		0			0
$253$	4.06591e8			1.57847
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.65195e8	−	2.86127e8i	−0.607060	−	1.05146i	−0.991722	−	0.128402i	$-0.959015\pi$
$257$	−1.65195e8	−	2.86127e8i	−0.607060	−	1.05146i	0.384662	−	0.923058i	$-0.374318\pi$
$258$		0			0
$259$	−1.32388e7	+	2.29303e7i	−0.0473479	+	0.0820089i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	1.25644e8	−	2.17621e8i	0.425888	−	0.737659i	−0.570615	−	0.821218i	$-0.693295\pi$
$263$	1.25644e8	−	2.17621e8i	0.425888	−	0.737659i	0.996503	+	0.0835582i	$0.0266285\pi$
$264$		0			0
$265$	−3.08148e7	−	5.33728e7i	−0.101718	−	0.176181i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.60063e8			−0.501369			−0.250685	−	0.968069i	$-0.580656\pi$
$269$	−1.60063e8			−0.501369			−0.250685	+	0.968069i	$0.580656\pi$
$270$		0			0
$271$	−4.46626e8			−1.36318			−0.681588	−	0.731736i	$-0.738710\pi$
$271$	−4.46626e8			−1.36318			−0.681588	+	0.731736i	$0.738710\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2.01608e8	+	3.49195e8i	0.584578	+	1.01252i
$276$		0			0
$277$	1.89550e8	−	3.28310e8i	0.535851	−	0.928121i	−0.463271	−	0.886217i	$-0.653324\pi$
$277$	1.89550e8	−	3.28310e8i	0.535851	−	0.928121i	0.999122	−	0.0419039i	$-0.0133423\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−2.16761e8	+	3.75440e8i	−0.582785	+	1.00941i	0.412363	+	0.911020i	$0.364704\pi$
$281$	−2.16761e8	+	3.75440e8i	−0.582785	+	1.00941i	−0.995148	+	0.0983932i	$0.968630\pi$
$282$		0			0
$283$	9.60881e7	+	1.66429e8i	0.252010	+	0.436493i	0.964079	−	0.265616i	$-0.0855752\pi$
$283$	9.60881e7	+	1.66429e8i	0.252010	+	0.436493i	−0.712069	+	0.702109i	$0.752242\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	2.83009e7			0.0706665
$288$		0			0
$289$	−3.36846e8			−0.820899
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.39360e8	+	5.87790e8i	0.788178	+	1.36517i	0.927082	+	0.374859i	$0.122309\pi$
$293$	3.39360e8	+	5.87790e8i	0.788178	+	1.36517i	−0.138903	+	0.990306i	$0.544358\pi$
$294$		0			0
$295$	−6.24861e7	+	1.08229e8i	−0.141712	+	0.245452i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	3.31585e8	−	5.74323e8i	0.717375	−	1.24253i
$300$		0			0
$301$	−4.64655e7	−	8.04807e7i	−0.0982083	−	0.170102i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.41390e8			0.890786
$306$		0			0
$307$	−8.55694e8			−1.68785			−0.843926	−	0.536460i	$-0.819761\pi$
$307$	−8.55694e8			−1.68785			−0.843926	+	0.536460i	$0.819761\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.92731e7	−	8.53436e7i	−0.0928857	−	0.160883i	0.815839	−	0.578280i	$-0.196276\pi$
$311$	−4.92731e7	−	8.53436e7i	−0.0928857	−	0.160883i	−0.908724	+	0.417397i	$0.862942\pi$
$312$		0			0
$313$	1.56596e8	−	2.71231e8i	0.288652	−	0.499960i	−0.684836	−	0.728697i	$-0.740126\pi$
$313$	1.56596e8	−	2.71231e8i	0.288652	−	0.499960i	0.973488	+	0.228737i	$0.0734597\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.13670e8	−	7.16497e8i	0.729367	−	1.26330i	−0.227783	−	0.973712i	$-0.573148\pi$
$317$	4.13670e8	−	7.16497e8i	0.729367	−	1.26330i	0.957151	−	0.289590i	$-0.0935189\pi$
$318$		0			0
$319$	2.80463e8	+	4.85776e8i	0.483736	+	0.837855i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	1.26273e8			0.208498
$324$		0			0
$325$	6.57664e8			1.06270
$326$		0			0
$327$		0			0
$328$		0			0
$329$	1.13880e8	+	1.97247e8i	0.176305	+	0.305369i
$330$		0			0
$331$	−1.46120e8	+	2.53087e8i	−0.221468	+	0.383593i	−0.955254	−	0.295787i	$-0.904418\pi$
$331$	−1.46120e8	+	2.53087e8i	−0.221468	+	0.383593i	0.733786	+	0.679381i	$0.237751\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.46587e8	+	2.53896e8i	−0.213029	+	0.368977i
$336$		0			0
$337$	1.42540e8	+	2.46886e8i	0.202876	+	0.351392i	0.949454	−	0.313906i	$-0.101638\pi$
$337$	1.42540e8	+	2.46886e8i	0.202876	+	0.351392i	−0.746578	+	0.665298i	$0.768304\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.07946e9			2.83994
$342$		0			0
$343$	3.88550e8			0.519897
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−2.93656e8	−	5.08627e8i	−0.377299	−	0.653501i	0.613369	−	0.789796i	$-0.289814\pi$
$347$	−2.93656e8	−	5.08627e8i	−0.377299	−	0.653501i	−0.990668	+	0.136295i	$0.956480\pi$
$348$		0			0
$349$	3.53318e8	−	6.11965e8i	0.444915	−	0.770615i	−0.553132	−	0.833094i	$-0.686567\pi$
$349$	3.53318e8	−	6.11965e8i	0.444915	−	0.770615i	0.998046	+	0.0624791i	$0.0199007\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	3.83333e8	−	6.63953e8i	0.463837	−	0.803388i	−0.535312	−	0.844655i	$-0.679806\pi$
$353$	3.83333e8	−	6.63953e8i	0.463837	−	0.803388i	0.999148	+	0.0412662i	$0.0131392\pi$
$354$		0			0
$355$	2.84400e8	+	4.92595e8i	0.337388	+	0.584374i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	6.18455e8			0.705468			0.352734	−	0.935724i	$-0.385252\pi$
$359$	6.18455e8			0.705468			0.352734	+	0.935724i	$0.385252\pi$
$360$		0			0
$361$	−6.76912e8			−0.757281
$362$		0			0
$363$		0			0
$364$		0			0
$365$	2.27892e8	+	3.94720e8i	0.245303	+	0.424878i
$366$		0			0
$367$	4.76399e8	−	8.25147e8i	0.503083	−	0.871365i	−0.496911	−	0.867801i	$-0.665532\pi$
$367$	4.76399e8	−	8.25147e8i	0.503083	−	0.871365i	0.999994	−	0.00356318i	$-0.00113420\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	4.56546e7	−	7.90760e7i	0.0464168	−	0.0803963i
$372$		0			0
$373$	3.73085e8	+	6.46202e8i	0.372243	+	0.644744i	0.989910	−	0.141696i	$-0.0452555\pi$
$373$	3.73085e8	+	6.46202e8i	0.372243	+	0.644744i	−0.617667	+	0.786440i	$0.711922\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	9.14898e8			0.879383
$378$		0			0
$379$	−1.09849e9			−1.03647			−0.518236	−	0.855238i	$-0.673411\pi$
$379$	−1.09849e9			−1.03647			−0.518236	+	0.855238i	$0.673411\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	2.02739e8	+	3.51155e8i	0.184392	+	0.319376i	0.943371	−	0.331738i	$-0.107635\pi$
$383$	2.02739e8	+	3.51155e8i	0.184392	+	0.319376i	−0.758979	+	0.651115i	$0.774302\pi$
$384$		0			0
$385$	1.60467e8	−	2.77936e8i	0.143308	−	0.248218i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	1.51925e8	−	2.63141e8i	0.130859	−	0.226655i	−0.793149	−	0.609028i	$-0.791560\pi$
$389$	1.51925e8	−	2.63141e8i	0.130859	−	0.226655i	0.924008	+	0.382373i	$0.124893\pi$
$390$		0			0
$391$	−2.19667e8	−	3.80475e8i	−0.185843	−	0.321890i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.74650e8			−0.224228
$396$		0			0
$397$	−2.05580e9			−1.64897			−0.824487	−	0.565881i	$-0.808536\pi$
$397$	−2.05580e9			−1.64897			−0.824487	+	0.565881i	$0.808536\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	9.36535e8	+	1.62213e9i	0.725302	+	1.25626i	0.958850	+	0.283914i	$0.0916331\pi$
$401$	9.36535e8	+	1.62213e9i	0.725302	+	1.25626i	−0.233548	+	0.972345i	$0.575034\pi$
$402$		0			0
$403$	1.69585e9	−	2.93730e9i	1.29068	−	2.23553i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	4.29047e8	−	7.43131e8i	0.315446	−	0.546368i
$408$		0			0
$409$	−3.88034e8	−	6.72095e8i	−0.280439	−	0.485735i	0.691054	−	0.722803i	$-0.257147\pi$
$409$	−3.88034e8	−	6.72095e8i	−0.280439	−	0.485735i	−0.971493	+	0.237068i	$0.923813\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.85156e8			−0.129334
$414$		0			0
$415$	4.82686e8			0.331510
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−5.90284e8	−	1.02240e9i	−0.392023	−	0.679005i	0.600693	−	0.799480i	$-0.294891\pi$
$419$	−5.90284e8	−	1.02240e9i	−0.392023	−	0.679005i	−0.992716	+	0.120475i	$0.961558\pi$
$420$		0			0
$421$	−1.32679e9	+	2.29806e9i	−0.866590	+	1.50098i	−0.00113024	+	0.999999i	$0.500360\pi$
$421$	−1.32679e9	+	2.29806e9i	−0.866590	+	1.50098i	−0.865460	+	0.500978i	$0.832974\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	2.17843e8	−	3.77316e8i	0.137652	−	0.238421i
$426$		0			0
$427$	3.26977e8	+	5.66341e8i	0.203245	+	0.352030i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	2.85665e9			1.71865			0.859323	−	0.511433i	$-0.170885\pi$
$431$	2.85665e9			1.71865			0.859323	+	0.511433i	$0.170885\pi$
$432$		0			0
$433$	2.34183e9			1.38627			0.693134	−	0.720809i	$-0.256229\pi$
$433$	2.34183e9			1.38627			0.693134	+	0.720809i	$0.256229\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−3.77428e8	−	6.53725e8i	−0.216346	−	0.374723i
$438$		0			0
$439$	−3.32702e8	+	5.76256e8i	−0.187685	+	0.325080i	−0.944478	−	0.328575i	$-0.893432\pi$
$439$	−3.32702e8	+	5.76256e8i	−0.187685	+	0.325080i	0.756793	+	0.653655i	$0.226765\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.68045e9	−	2.91062e9i	0.918358	−	1.59064i	0.116448	−	0.993197i	$-0.462849\pi$
$443$	1.68045e9	−	2.91062e9i	0.918358	−	1.59064i	0.801909	−	0.597446i	$-0.203818\pi$
$444$		0			0
$445$	6.85154e8	+	1.18672e9i	0.368577	+	0.638394i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.04062e8			0.106390			0.0531949	−	0.998584i	$-0.483060\pi$
$449$	2.04062e8			0.106390			0.0531949	+	0.998584i	$0.483060\pi$
$450$		0			0
$451$	−9.17182e8			−0.470801
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−2.61729e8	−	4.53328e8i	−0.130260	−	0.225617i
$456$		0			0
$457$	−1.06852e9	+	1.85072e9i	−0.523690	+	0.907057i	0.475930	+	0.879483i	$0.342112\pi$
$457$	−1.06852e9	+	1.85072e9i	−0.523690	+	0.907057i	−0.999620	+	0.0275740i	$0.991222\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−1.66342e9	+	2.88112e9i	−0.790765	+	1.36965i	0.134728	+	0.990883i	$0.456984\pi$
$461$	−1.66342e9	+	2.88112e9i	−0.790765	+	1.36965i	−0.925494	+	0.378763i	$0.876349\pi$
$462$		0			0
$463$	1.39859e9	+	2.42243e9i	0.654873	+	1.13427i	0.981926	+	0.189267i	$0.0606113\pi$
$463$	1.39859e9	+	2.42243e9i	0.654873	+	1.13427i	−0.327052	+	0.945006i	$0.606055\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−2.25354e9			−1.02390			−0.511948	−	0.859017i	$-0.671076\pi$
$467$	−2.25354e9			−1.02390			−0.511948	+	0.859017i	$0.671076\pi$
$468$		0			0
$469$	−4.34361e8			−0.194422
$470$		0			0
$471$		0			0
$472$		0			0
$473$	1.50587e9	+	2.60824e9i	0.654293	+	1.13327i
$474$		0			0
$475$	3.74294e8	−	6.48297e8i	0.160245	−	0.277553i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−1.05515e9	+	1.82757e9i	−0.438672	+	0.759802i	−0.997587	−	0.0694230i	$-0.977884\pi$
$479$	−1.05515e9	+	1.82757e9i	−0.438672	+	0.759802i	0.558916	+	0.829224i	$0.311218\pi$
$480$		0			0
$481$	−6.99797e8	−	1.21208e9i	−0.286724	−	0.496621i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	1.16777e9			0.464796
$486$		0			0
$487$	−7.90646e8			−0.310192			−0.155096	−	0.987899i	$-0.549569\pi$
$487$	−7.90646e8			−0.310192			−0.155096	+	0.987899i	$0.549569\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−8.40717e8	−	1.45616e9i	−0.320527	−	0.555169i	0.660070	−	0.751204i	$-0.270527\pi$
$491$	−8.40717e8	−	1.45616e9i	−0.320527	−	0.555169i	−0.980597	+	0.196035i	$0.937193\pi$
$492$		0			0
$493$	3.03049e8	−	5.24896e8i	0.113907	−	0.197292i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.21360e8	+	7.29818e8i	−0.153959	+	0.266666i
$498$		0			0
$499$	−4.71012e8	−	8.15816e8i	−0.169699	−	0.293928i	0.768615	−	0.639712i	$-0.220946\pi$
$499$	−4.71012e8	−	8.15816e8i	−0.169699	−	0.293928i	−0.938314	+	0.345784i	$0.887613\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.96913e9			−1.74098			−0.870488	−	0.492190i	$-0.836197\pi$
$503$	−4.96913e9			−1.74098			−0.870488	+	0.492190i	$0.836197\pi$
$504$		0			0
$505$	2.55576e9			0.883082
$506$		0			0
$507$		0			0
$508$		0			0
$509$	9.45851e8	+	1.63826e9i	0.317915	+	0.550644i	0.980053	−	0.198737i	$-0.0636841\pi$
$509$	9.45851e8	+	1.63826e9i	0.317915	+	0.550644i	−0.662138	+	0.749382i	$0.730351\pi$
$510$		0			0
$511$	−3.37639e8	+	5.84808e8i	−0.111939	+	0.193883i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	4.09463e8	−	7.09210e8i	0.132096	−	0.228797i
$516$		0			0
$517$	−3.69067e9	−	6.39242e9i	−1.17459	−	2.03446i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−1.91165e9			−0.592210			−0.296105	−	0.955155i	$-0.595688\pi$
$521$	−1.91165e9			−0.592210			−0.296105	+	0.955155i	$0.595688\pi$
$522$		0			0
$523$	−4.77755e8			−0.146032			−0.0730162	−	0.997331i	$-0.523262\pi$
$523$	−4.77755e8			−0.146032			−0.0730162	+	0.997331i	$0.523262\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.12346e9	−	1.94589e9i	−0.334364	−	0.579136i
$528$		0			0
$529$	3.89247e8	−	6.74195e8i	0.114322	−	0.198012i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−7.47984e8	+	1.29555e9i	−0.213967	+	0.370602i
$534$		0			0
$535$	5.15037e6	+	8.92070e6i	0.00145412	+	0.00251861i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.05837e9			−1.66646
$540$		0			0
$541$	−3.09043e8			−0.0839128			−0.0419564	−	0.999119i	$-0.513359\pi$
$541$	−3.09043e8			−0.0839128			−0.0419564	+	0.999119i	$0.513359\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−1.73454e9	−	3.00432e9i	−0.458984	−	0.794984i
$546$		0			0
$547$	−5.29968e8	+	9.17932e8i	−0.138450	+	0.239803i	−0.926910	−	0.375283i	$-0.877545\pi$
$547$	−5.29968e8	+	9.17932e8i	−0.138450	+	0.239803i	0.788460	+	0.615086i	$0.210879\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.20693e8	−	9.01867e8i	0.132602	−	0.229674i
$552$		0			0
$553$	−2.03458e8	−	3.52399e8i	−0.0511607	−	0.0886130i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	4.43145e9			1.08656			0.543278	−	0.839553i	$-0.317183\pi$
$557$	4.43145e9			1.08656			0.543278	+	0.839553i	$0.317183\pi$
$558$		0			0
$559$	4.91229e9			1.18944
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.13365e9	−	5.42763e9i	−0.740066	−	1.28183i	−0.952465	−	0.304649i	$-0.901461\pi$
$563$	−3.13365e9	−	5.42763e9i	−0.740066	−	1.28183i	0.212398	−	0.977183i	$-0.431873\pi$
$564$		0			0
$565$	−2.33889e9	+	4.05108e9i	−0.545557	+	0.944932i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.75145e9	+	4.76565e9i	−0.626136	+	1.08450i	0.362184	+	0.932106i	$0.382031\pi$
$569$	−2.75145e9	+	4.76565e9i	−0.626136	+	1.08450i	−0.988320	+	0.152392i	$0.951302\pi$
$570$		0			0
$571$	−2.14316e7	−	3.71206e7i	−0.00481757	−	0.00834428i	0.863607	−	0.504166i	$-0.168200\pi$
$571$	−2.14316e7	−	3.71206e7i	−0.00481757	−	0.00834428i	−0.868424	+	0.495822i	$0.834867\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.60452e9			−0.571335
$576$		0			0
$577$	7.95093e9			1.72307			0.861534	−	0.507700i	$-0.169504\pi$
$577$	7.95093e9			1.72307			0.861534	+	0.507700i	$0.169504\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	3.57569e8	+	6.19327e8i	0.0756385	+	0.131010i
$582$		0			0
$583$	−1.47958e9	+	2.56271e9i	−0.309243	+	0.535624i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−3.34732e9	+	5.79773e9i	−0.683068	+	1.18311i	0.290972	+	0.956732i	$0.406021\pi$
$587$	−3.34732e9	+	5.79773e9i	−0.683068	+	1.18311i	−0.974040	+	0.226377i	$0.927312\pi$
$588$		0			0
$589$	−1.93031e9	−	3.34339e9i	−0.389245	−	0.674191i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−3.30233e9			−0.650323			−0.325162	−	0.945658i	$-0.605419\pi$
$593$	−3.30233e9			−0.650323			−0.325162	+	0.945658i	$0.605419\pi$
$594$		0			0
$595$	−3.46778e8			−0.0674905
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1.59658e9	−	2.76536e9i	−0.303526	−	0.525723i	0.673406	−	0.739273i	$-0.264831\pi$
$599$	−1.59658e9	−	2.76536e9i	−0.303526	−	0.525723i	−0.976932	+	0.213550i	$0.931497\pi$
$600$		0			0
$601$	1.44439e9	−	2.50176e9i	0.271409	−	0.470094i	−0.697814	−	0.716279i	$-0.745844\pi$
$601$	1.44439e9	−	2.50176e9i	0.271409	−	0.470094i	0.969223	+	0.246185i	$0.0791771\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.59045e9	+	6.21885e9i	−0.659182	+	1.14174i
$606$		0			0
$607$	2.57337e9	+	4.45720e9i	0.467026	+	0.808913i	0.999290	−	0.0376651i	$-0.0119920\pi$
$607$	2.57337e9	+	4.45720e9i	0.467026	+	0.808913i	−0.532264	+	0.846578i	$0.678659\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.20393e10			−2.13529
$612$		0			0
$613$	3.95231e8			0.0693009			0.0346505	−	0.999399i	$-0.488968\pi$
$613$	3.95231e8			0.0693009			0.0346505	+	0.999399i	$0.488968\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	4.21163e9	+	7.29475e9i	0.721858	+	1.25029i	0.960254	+	0.279127i	$0.0900450\pi$
$617$	4.21163e9	+	7.29475e9i	0.721858	+	1.25029i	−0.238396	+	0.971168i	$0.576622\pi$
$618$		0			0
$619$	−6.62889e8	+	1.14816e9i	−0.112337	+	0.194574i	−0.916712	−	0.399548i	$-0.869167\pi$
$619$	−6.62889e8	+	1.14816e9i	−0.112337	+	0.194574i	0.804375	+	0.594122i	$0.202500\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.01511e9	+	1.75822e9i	−0.168192	+	0.291316i
$624$		0			0
$625$	−2.24935e8	−	3.89598e8i	−0.0368533	−	0.0638318i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−9.27198e8			−0.148558
$630$		0			0
$631$	−6.91872e9			−1.09628			−0.548142	−	0.836385i	$-0.684665\pi$
$631$	−6.91872e9			−1.09628			−0.548142	+	0.836385i	$0.684665\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−7.22448e8	−	1.25132e9i	−0.111969	−	0.193936i
$636$		0			0
$637$	−4.94075e9	+	8.55763e9i	−0.757364	+	1.31179i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	4.35056e9	−	7.53538e9i	0.652442	−	1.13006i	−0.330087	−	0.943950i	$-0.607078\pi$
$641$	4.35056e9	−	7.53538e9i	0.652442	−	1.13006i	0.982529	−	0.186111i	$-0.0595885\pi$
$642$		0			0
$643$	−6.48016e9	−	1.12240e10i	−0.961275	−	1.66498i	−0.719307	−	0.694693i	$-0.755540\pi$
$643$	−6.48016e9	−	1.12240e10i	−0.961275	−	1.66498i	−0.241968	−	0.970284i	$-0.577793\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−8.42694e9			−1.22322			−0.611611	−	0.791159i	$-0.709478\pi$
$647$	−8.42694e9			−1.22322			−0.611611	+	0.791159i	$0.709478\pi$
$648$		0			0
$649$	6.00058e9			0.861661
$650$		0			0
$651$		0			0
$652$		0			0
$653$	5.06846e9	+	8.77883e9i	0.712328	+	1.23379i	0.963981	+	0.265971i	$0.0856925\pi$
$653$	5.06846e9	+	8.77883e9i	0.712328	+	1.23379i	−0.251653	+	0.967818i	$0.580974\pi$
$654$		0			0
$655$	2.31855e9	−	4.01584e9i	0.322382	−	0.558383i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.95202e9	−	3.38100e9i	0.265696	−	0.460199i	−0.702050	−	0.712128i	$-0.747732\pi$
$659$	1.95202e9	−	3.38100e9i	0.265696	−	0.460199i	0.967746	+	0.251929i	$0.0810649\pi$
$660$		0			0
$661$	−1.69568e9	−	2.93700e9i	−0.228370	−	0.395548i	0.728955	−	0.684561i	$-0.240006\pi$
$661$	−1.69568e9	−	2.93700e9i	−0.228370	−	0.395548i	−0.957325	+	0.289013i	$0.906673\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.95828e8			−0.0785679
$666$		0			0
$667$	−3.62324e9			−0.472777
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.05967e10	−	1.83541e10i	−1.35408	−	2.34533i
$672$		0			0
$673$	−5.94162e9	+	1.02912e10i	−0.751368	+	1.30141i	0.195792	+	0.980645i	$0.437272\pi$
$673$	−5.94162e9	+	1.02912e10i	−0.751368	+	1.30141i	−0.947160	+	0.320762i	$0.896061\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	1.56500e9	−	2.71067e9i	0.193845	−	0.335750i	−0.752676	−	0.658391i	$-0.771237\pi$
$677$	1.56500e9	−	2.71067e9i	0.193845	−	0.335750i	0.946521	+	0.322641i	$0.104571\pi$
$678$		0			0
$679$	8.65074e8	+	1.49835e9i	0.106050	+	0.183683i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1.46773e10			1.76268			0.881342	−	0.472478i	$-0.156640\pi$
$683$	1.46773e10			1.76268			0.881342	+	0.472478i	$0.156640\pi$
$684$		0			0
$685$	7.74976e9			0.921236
$686$		0			0
$687$		0			0
$688$		0			0
$689$	2.41327e9	+	4.17991e9i	0.281086	+	0.486856i
$690$		0			0
$691$	3.11687e9	−	5.39858e9i	0.359373	−	0.622452i	−0.628483	−	0.777823i	$-0.716324\pi$
$691$	3.11687e9	−	5.39858e9i	0.359373	−	0.622452i	0.987856	+	0.155371i	$0.0496573\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.54911e9	+	2.68313e9i	−0.175039	+	0.303176i
$696$		0			0
$697$	4.95522e8	+	8.58269e8i	0.0554304	+	0.0960083i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.11118e10			−1.21835			−0.609176	−	0.793035i	$-0.708500\pi$
$701$	−1.11118e10			−1.21835			−0.609176	+	0.793035i	$0.708500\pi$
$702$		0			0
$703$	−1.59309e9			−0.172941
$704$		0			0
$705$		0			0
$706$		0			0
$707$	1.89328e9	+	3.27926e9i	0.201487	+	0.348986i
$708$		0			0
$709$	−2.43745e9	+	4.22178e9i	−0.256846	+	0.444871i	−0.965395	−	0.260791i	$-0.916017\pi$
$709$	−2.43745e9	+	4.22178e9i	−0.256846	+	0.444871i	0.708549	+	0.705662i	$0.249350\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−6.71601e9	+	1.16325e10i	−0.693902	+	1.20187i
$714$		0			0
$715$	8.48217e9	+	1.46916e10i	0.867833	+	1.50313i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−1.51665e9			−0.152172			−0.0760858	−	0.997101i	$-0.524242\pi$
$719$	−1.51665e9			−0.152172			−0.0760858	+	0.997101i	$0.524242\pi$
$720$		0			0
$721$	1.21330e9			0.120558
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.79658e9	−	3.11176e9i	−0.175091	−	0.303266i
$726$		0			0
$727$	8.78805e9	−	1.52213e10i	0.848246	−	1.46921i	−0.0345261	−	0.999404i	$-0.510992\pi$
$727$	8.78805e9	−	1.52213e10i	0.848246	−	1.46921i	0.882772	−	0.469801i	$-0.155674\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.62714e9	−	2.81828e9i	0.154068	−	0.266854i
$732$		0			0
$733$	1.24920e9	+	2.16368e9i	0.117157	+	0.202922i	0.918640	−	0.395096i	$-0.129289\pi$
$733$	1.24920e9	+	2.16368e9i	0.117157	+	0.202922i	−0.801483	+	0.598018i	$0.795955\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.40769e10			1.29530
$738$		0			0
$739$	−1.16341e10			−1.06042			−0.530211	−	0.847866i	$-0.677887\pi$
$739$	−1.16341e10			−1.06042			−0.530211	+	0.847866i	$0.677887\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−2.99558e9	−	5.18849e9i	−0.267929	−	0.464067i	0.700398	−	0.713753i	$-0.253006\pi$
$743$	−2.99558e9	−	5.18849e9i	−0.267929	−	0.464067i	−0.968327	+	0.249686i	$0.919673\pi$
$744$		0			0
$745$	−1.67833e9	+	2.90695e9i	−0.148706	+	0.257567i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−7.63067e6	+	1.32167e7i	−0.000663554	+	0.00114931i
$750$		0			0
$751$	7.51759e9	+	1.30208e10i	0.647647	+	1.12176i	0.983683	+	0.179909i	$0.0575804\pi$
$751$	7.51759e9	+	1.30208e10i	0.647647	+	1.12176i	−0.336036	+	0.941849i	$0.609086\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	6.71912e9			0.568196
$756$		0			0
$757$	1.63730e10			1.37180			0.685902	−	0.727694i	$-0.259408\pi$
$757$	1.63730e10			1.37180			0.685902	+	0.727694i	$0.259408\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7.25561e9	+	1.25671e10i	0.596799	+	1.03369i	0.993290	+	0.115647i	$0.0368942\pi$
$761$	7.25561e9	+	1.25671e10i	0.596799	+	1.03369i	−0.396492	+	0.918038i	$0.629772\pi$
$762$		0			0
$763$	2.56986e9	−	4.45113e9i	0.209447	−	0.362773i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.89362e9	−	8.47600e9i	0.391603	−	0.678277i
$768$		0			0
$769$	−9.97611e9	−	1.72791e10i	−0.791077	−	1.37019i	−0.925300	−	0.379235i	$-0.876187\pi$
$769$	−9.97611e9	−	1.72791e10i	−0.791077	−	1.37019i	0.134223	−	0.990951i	$-0.457146\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.79071e10			1.39443			0.697217	−	0.716861i	$-0.254422\pi$
$773$	1.79071e10			1.39443			0.697217	+	0.716861i	$0.254422\pi$
$774$		0			0
$775$	−1.33205e10			−1.02793
$776$		0			0
$777$		0			0
$778$		0			0
$779$	8.51396e8	+	1.47466e9i	0.0645283	+	0.111766i
$780$		0			0
$781$	1.36556e10	−	2.36521e10i	1.02572	−	1.77661i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.41795e8	−	7.65211e8i	0.0325970	−	0.0564596i
$786$		0			0
$787$	−7.69528e9	−	1.33286e10i	−0.562747	−	0.974706i	−0.997255	−	0.0740383i	$-0.976411\pi$
$787$	−7.69528e9	−	1.33286e10i	−0.562747	−	0.974706i	0.434509	−	0.900668i	$-0.356922\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−6.93049e9			−0.497905
$792$		0			0
$793$	−3.45676e10			−2.46158
$794$		0			0
$795$		0			0
$796$		0			0
$797$	7.39691e9	+	1.28118e10i	0.517543	+	0.896411i	0.999792	+	0.0203768i	$0.00648660\pi$
$797$	7.39691e9	+	1.28118e10i	0.517543	+	0.896411i	−0.482249	+	0.876034i	$0.660180\pi$
$798$		0			0
$799$	−3.98788e9	+	6.90721e9i	−0.276585	+	0.479059i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	1.09423e10	−	1.89526e10i	0.745768	−	1.29171i
$804$		0			0
$805$	1.03652e9	+	1.79530e9i	0.0700310	+	0.121297i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	2.86671e10			1.90355			0.951774	−	0.306801i	$-0.0992587\pi$
$809$	2.86671e10			1.90355			0.951774	+	0.306801i	$0.0992587\pi$
$810$		0			0
$811$	1.63843e10			1.07858			0.539292	−	0.842119i	$-0.318692\pi$
$811$	1.63843e10			1.07858			0.539292	+	0.842119i	$0.318692\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.45144e9	−	1.29063e10i	−0.482157	−	0.835120i
$816$		0			0
$817$	2.79571e9	−	4.84232e9i	0.179356	−	0.310653i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.17149e10	−	2.02908e10i	0.738819	−	1.27967i	−0.214209	−	0.976788i	$-0.568717\pi$
$821$	1.17149e10	−	2.02908e10i	0.738819	−	1.27967i	0.953027	−	0.302884i	$-0.0979494\pi$
$822$		0			0
$823$	3.73062e9	+	6.46162e9i	0.233282	+	0.404056i	0.958772	−	0.284177i	$-0.0917202\pi$
$823$	3.73062e9	+	6.46162e9i	0.233282	+	0.404056i	−0.725490	+	0.688233i	$0.758387\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−6.02982e9			−0.370711			−0.185355	−	0.982672i	$-0.559344\pi$
$827$	−6.02982e9			−0.370711			−0.185355	+	0.982672i	$0.559344\pi$
$828$		0			0
$829$	4.54823e9			0.277269			0.138635	−	0.990344i	$-0.455729\pi$
$829$	4.54823e9			0.277269			0.138635	+	0.990344i	$0.455729\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	3.27313e9	+	5.66923e9i	0.196203	+	0.339834i
$834$		0			0
$835$	−8.03293e9	+	1.39134e10i	−0.477498	+	0.827050i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	4.57920e9	−	7.93140e9i	0.267684	−	0.463642i	−0.700579	−	0.713575i	$-0.747075\pi$
$839$	4.57920e9	−	7.93140e9i	0.267684	−	0.463642i	0.968263	+	0.249932i	$0.0804083\pi$
$840$		0			0
$841$	6.12566e9	+	1.06100e10i	0.355113	+	0.615074i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	1.73014e10			0.986468
$846$		0			0
$847$	−1.06391e10			−0.601605
$848$		0			0
$849$		0			0
$850$		0			0
$851$	2.77138e9	+	4.80018e9i	0.154150	+	0.266995i
$852$		0			0
$853$	5.32049e8	−	9.21535e8i	0.0293515	−	0.0508382i	−0.850977	−	0.525204i	$-0.823989\pi$
$853$	5.32049e8	−	9.21535e8i	0.0293515	−	0.0508382i	0.880328	+	0.474365i	$0.157322\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−5.13931e9	+	8.90155e9i	−0.278915	+	0.483095i	−0.971115	−	0.238610i	$-0.923308\pi$
$857$	−5.13931e9	+	8.90155e9i	−0.278915	+	0.483095i	0.692200	+	0.721705i	$0.256641\pi$
$858$		0			0
$859$	1.65770e10	+	2.87121e10i	0.892337	+	1.54557i	0.837066	+	0.547101i	$0.184269\pi$
$859$	1.65770e10	+	2.87121e10i	0.892337	+	1.54557i	0.0552705	+	0.998471i	$0.482398\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.28480e10			0.680452			0.340226	−	0.940344i	$-0.389496\pi$
$863$	1.28480e10			0.680452			0.340226	+	0.940344i	$0.389496\pi$
$864$		0			0
$865$	1.91823e9			0.100773
$866$		0			0
$867$		0			0
$868$		0			0
$869$	6.59371e9	+	1.14206e10i	0.340848	+	0.590366i
$870$		0			0
$871$	1.14800e10	−	1.98840e10i	0.588680	−	1.01962i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−2.60803e9	+	4.51725e9i	−0.131609	+	0.227953i
$876$		0			0
$877$	1.60122e10	+	2.77340e10i	0.801591	+	1.38840i	0.918569	+	0.395261i	$0.129346\pi$
$877$	1.60122e10	+	2.77340e10i	0.801591	+	1.38840i	−0.116978	+	0.993134i	$0.537321\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	3.34497e10			1.64807			0.824036	−	0.566537i	$-0.191717\pi$
$881$	3.34497e10			1.64807			0.824036	+	0.566537i	$0.191717\pi$
$882$		0			0
$883$	−2.66332e10			−1.30185			−0.650926	−	0.759141i	$-0.725619\pi$
$883$	−2.66332e10			−1.30185			−0.650926	+	0.759141i	$0.725619\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.34168e10	+	2.32385e10i	0.645529	+	1.11809i	0.984179	+	0.177176i	$0.0566962\pi$
$887$	1.34168e10	+	2.32385e10i	0.645529	+	1.11809i	−0.338651	+	0.940912i	$0.609970\pi$
$888$		0			0
$889$	1.07036e9	−	1.85392e9i	0.0510946	−	0.0884985i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.85190e9	+	1.18678e10i	−0.321982	+	0.557688i
$894$		0			0
$895$	7.75575e9	+	1.34333e10i	0.361612	+	0.626331i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−1.85306e10			−0.850609
$900$		0			0
$901$	3.19747e9			0.145636
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.69988e9	−	8.14043e9i	−0.210774	−	0.365071i
$906$		0			0
$907$	4.36155e9	−	7.55442e9i	0.194095	−	0.336183i	−0.752508	−	0.658583i	$-0.771156\pi$
$907$	4.36155e9	−	7.55442e9i	0.194095	−	0.336183i	0.946604	+	0.322400i	$0.104490\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	6.49806e9	−	1.12550e10i	0.284754	−	0.493208i	−0.687796	−	0.725904i	$-0.741421\pi$
$911$	6.49806e9	−	1.12550e10i	0.284754	−	0.493208i	0.972549	+	0.232696i	$0.0747548\pi$
$912$		0			0
$913$	−1.15882e10	−	2.00713e10i	−0.503927	−	0.872826i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	6.87021e9			0.294224
$918$		0			0
$919$	−8.23373e9			−0.349939			−0.174970	−	0.984574i	$-0.555983\pi$
$919$	−8.23373e9			−0.349939			−0.174970	+	0.984574i	$0.555983\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−2.22729e10	−	3.85778e10i	−0.932332	−	1.61485i
$924$		0			0
$925$	−2.74837e9	+	4.76032e9i	−0.114177	+	0.197761i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1.68192e10	−	2.91318e10i	0.688258	−	1.19210i	−0.284143	−	0.958782i	$-0.591709\pi$
$929$	1.68192e10	−	2.91318e10i	0.688258	−	1.19210i	0.972401	−	0.233316i	$-0.0749577\pi$
$930$		0			0
$931$	5.62383e9	+	9.74075e9i	0.228406	+	0.395611i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	1.12385e10			0.449642
$936$		0			0
$937$	−1.72301e10			−0.684224			−0.342112	−	0.939659i	$-0.611142\pi$
$937$	−1.72301e10			−0.684224			−0.342112	+	0.939659i	$0.611142\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.65184e10	+	2.86106e10i	0.646254	+	1.11934i	0.984010	+	0.178111i	$0.0569986\pi$
$941$	1.65184e10	+	2.86106e10i	0.646254	+	1.11934i	−0.337757	+	0.941233i	$0.609668\pi$
$942$		0			0
$943$	2.96222e9	−	5.13071e9i	0.115034	−	0.199245i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	2.02843e10	−	3.51334e10i	0.776131	−	1.34430i	−0.158026	−	0.987435i	$-0.550513\pi$
$947$	2.02843e10	−	3.51334e10i	0.776131	−	1.34430i	0.934157	−	0.356863i	$-0.116154\pi$
$948$		0			0
$949$	−1.78474e10	−	3.09126e10i	−0.677866	−	1.17410i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.23972e10			−0.838240			−0.419120	−	0.907931i	$-0.637661\pi$
$953$	−2.23972e10			−0.838240			−0.419120	+	0.907931i	$0.637661\pi$
$954$		0			0
$955$	1.01763e10			0.378074
$956$		0			0
$957$		0			0
$958$		0			0
$959$	5.74093e9	+	9.94359e9i	0.210193	+	0.364064i
$960$		0			0
$961$	−2.05918e10	+	3.56661e10i	−0.748451	+	1.29635i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−4.87771e9	+	8.44845e9i	−0.174731	+	0.302643i
$966$		0			0
$967$	−2.50274e10	−	4.33488e10i	−0.890068	−	1.54164i	−0.839792	−	0.542908i	$-0.817323\pi$
$967$	−2.50274e10	−	4.33488e10i	−0.890068	−	1.54164i	−0.0502766	−	0.998735i	$-0.516010\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.56566e10			0.899358			0.449679	−	0.893190i	$-0.351538\pi$
$971$	2.56566e10			0.899358			0.449679	+	0.893190i	$0.351538\pi$
$972$		0			0
$973$	−4.59024e9			−0.159750
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−5.81779e9	−	1.00767e10i	−0.199585	−	0.345691i	0.748809	−	0.662786i	$-0.230626\pi$
$977$	−5.81779e9	−	1.00767e10i	−0.199585	−	0.345691i	−0.948394	+	0.317095i	$0.897293\pi$
$978$		0			0
$979$	3.28979e10	−	5.69808e10i	1.12054	−	1.94084i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−5.59306e9	+	9.68746e9i	−0.187807	+	0.325291i	−0.944519	−	0.328457i	$-0.893471\pi$
$983$	−5.59306e9	+	9.68746e9i	−0.187807	+	0.325291i	0.756712	+	0.653749i	$0.226805\pi$
$984$		0			0
$985$	1.69536e10	+	2.93645e10i	0.565242	+	0.979028i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−1.94540e10			−0.639471
$990$		0			0
$991$	2.68238e10			0.875513			0.437756	−	0.899094i	$-0.355773\pi$
$991$	2.68238e10			0.875513			0.437756	+	0.899094i	$0.355773\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.66163e9	−	2.87803e9i	−0.0534754	−	0.0926221i
$996$		0			0
$997$	−1.11028e9	+	1.92306e9i	−0.0354813	+	0.0614554i	−0.883221	−	0.468957i	$-0.844630\pi$
$997$	−1.11028e9	+	1.92306e9i	−0.0354813	+	0.0614554i	0.847739	+	0.530413i	$0.177963\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.8.e.m.217.3		16
3.2	odd	2	inner	324.8.e.m.217.6		16
9.2	odd	6		324.8.a.e.1.3	✓	8
9.4	even	3	inner	324.8.e.m.109.3		16
9.5	odd	6	inner	324.8.e.m.109.6		16
9.7	even	3		324.8.a.e.1.6	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.8.a.e.1.3	✓	8	9.2	odd	6
324.8.a.e.1.6	yes	8	9.7	even	3
324.8.e.m.109.3		16	9.4	even	3	inner
324.8.e.m.109.6		16	9.5	odd	6	inner
324.8.e.m.217.3		16	1.1	even	1	trivial
324.8.e.m.217.6		16	3.2	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:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	324.e (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-82.6178 - 143.098i) q^{5} +(122.405 - 212.011i) q^{7} +O(q^{10})\)	\(q+(-82.6178 - 143.098i) q^{5} +(122.405 - 212.011i) q^{7} +(-3966.92 + 6870.90i) q^{11} +(6470.24 + 11206.8i) q^{13} +8572.76 q^{17} +14729.6 q^{19} +(-25623.9 - 44381.9i) q^{23} +(25411.1 - 44013.3i) q^{25} +(35350.2 - 61228.4i) q^{29} +(-131050. - 226985. i) q^{31} -40451.2 q^{35} -108156. q^{37} +(57801.9 + 100116. i) q^{41} +(189803. - 328749. i) q^{43} +(-465181. + 805716. i) q^{47} +(381806. + 661307. i) q^{49} +372981. q^{53} +1.31095e6 q^{55} +(-378164. - 654999. i) q^{59} +(-1.33564e6 + 2.31340e6i) q^{61} +(1.06911e6 - 1.85176e6i) q^{65} +(-887141. - 1.53657e6i) q^{67} -3.44236e6 q^{71} -2.75839e6 q^{73} +(971138. + 1.68206e6i) q^{77} +(831088. - 1.43949e6i) q^{79} +(-1.46060e6 + 2.52984e6i) q^{83} +(-708262. - 1.22675e6i) q^{85} -8.29306e6 q^{89} +3.16795e6 q^{91} +(-1.21692e6 - 2.10777e6i) q^{95} +(-3.53367e6 + 6.12049e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 560 q^{7}+O(q^{10}) \) 16 * q + 560 * q^7	\( 16 q + 560 q^{7} - 1480 q^{13} - 55264 q^{19} - 77936 q^{25} + 247424 q^{31} + 220400 q^{37} - 897040 q^{43} - 1329672 q^{49} + 1910880 q^{55} - 494968 q^{61} - 4698160 q^{67} + 21452240 q^{73} - 8887312 q^{79} - 15973992 q^{85} + 30743008 q^{91} - 33664240 q^{97}+O(q^{100}) \) 16 * q + 560 * q^7 - 1480 * q^13 - 55264 * q^19 - 77936 * q^25 + 247424 * q^31 + 220400 * q^37 - 897040 * q^43 - 1329672 * q^49 + 1910880 * q^55 - 494968 * q^61 - 4698160 * q^67 + 21452240 * q^73 - 8887312 * q^79 - 15973992 * q^85 + 30743008 * q^91 - 33664240 * q^97

Introduction

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