LMFDB - Embedded newform 324.8.e.m.109.2

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			109.2
Root			$8.11082 - 14.0484i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.8.e.m.217.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-103.256 + 178.845i) q^{5} +(-503.506 - 872.097i) q^{7} +O(q^{10})$	$q+(-103.256 + 178.845i) q^{5} +(-503.506 - 872.097i) q^{7} +(1651.73 + 2860.88i) q^{11} +(-6641.32 + 11503.1i) q^{13} +18005.2 q^{17} +21732.3 q^{19} +(32417.0 - 56147.9i) q^{23} +(17738.8 + 30724.5i) q^{25} +(-105660. - 183009. i) q^{29} +(-42368.6 + 73384.6i) q^{31} +207960. q^{35} +163466. q^{37} +(-270687. + 468844. i) q^{41} +(152718. + 264516. i) q^{43} +(405133. + 701710. i) q^{47} +(-95264.1 + 165002. i) q^{49} -1.40849e6 q^{53} -682206. q^{55} +(-69842.7 + 120971. i) q^{59} +(-145701. - 252362. i) q^{61} +(-1.37152e6 - 2.37554e6i) q^{65} +(808071. - 1.39962e6i) q^{67} -962505. q^{71} +211780. q^{73} +(1.66331e6 - 2.88094e6i) q^{77} +(-3.76039e6 - 6.51319e6i) q^{79} +(1.74040e6 + 3.01445e6i) q^{83} +(-1.85915e6 + 3.22013e6i) q^{85} -964328. q^{89} +1.33758e7 q^{91} +(-2.24399e6 + 3.88671e6i) q^{95} +(-491976. - 852128. i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

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

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
$5$	−103.256	+	178.845i	−0.369421	+	0.639856i	−0.989475	−	0.144703i	$-0.953777\pi$
$5$	−103.256	+	178.845i	−0.369421	+	0.639856i	0.620054	+	0.784559i	$0.287111\pi$
$6$		0			0
$7$	−503.506	−	872.097i	−0.554831	−	0.960996i	−0.997917	−	0.0645168i	$-0.979449\pi$
$7$	−503.506	−	872.097i	−0.554831	−	0.960996i	0.443085	−	0.896480i	$-0.353884\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1651.73	+	2860.88i	0.374166	+	0.648075i	0.990202	−	0.139643i	$-0.0445955\pi$
$11$	1651.73	+	2860.88i	0.374166	+	0.648075i	−0.616035	+	0.787718i	$0.711262\pi$
$12$		0			0
$13$	−6641.32	+	11503.1i	−0.838403	+	1.45216i	0.0528259	+	0.998604i	$0.483177\pi$
$13$	−6641.32	+	11503.1i	−0.838403	+	1.45216i	−0.891229	+	0.453553i	$0.850156\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	18005.2			0.888845			0.444422	−	0.895817i	$-0.353409\pi$
$17$	18005.2			0.888845			0.444422	+	0.895817i	$0.353409\pi$
$18$		0			0
$19$	21732.3			0.726888			0.363444	−	0.931616i	$-0.381601\pi$
$19$	21732.3			0.726888			0.363444	+	0.931616i	$0.381601\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	32417.0	−	56147.9i	0.555553	−	0.962246i	−0.442308	−	0.896863i	$-0.645840\pi$
$23$	32417.0	−	56147.9i	0.555553	−	0.962246i	0.997860	−	0.0653822i	$-0.0208267\pi$
$24$		0			0
$25$	17738.8	+	30724.5i	0.227057	+	0.393274i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−105660.	−	183009.i	−0.804486	−	1.39341i	−0.916638	−	0.399719i	$-0.869108\pi$
$29$	−105660.	−	183009.i	−0.804486	−	1.39341i	0.112152	−	0.993691i	$-0.464226\pi$
$30$		0			0
$31$	−42368.6	+	73384.6i	−0.255434	+	0.442424i	−0.965013	−	0.262201i	$-0.915552\pi$
$31$	−42368.6	+	73384.6i	−0.255434	+	0.442424i	0.709579	+	0.704625i	$0.248885\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	207960.			0.819865
$36$		0			0
$37$	163466.			0.530545			0.265273	−	0.964173i	$-0.414538\pi$
$37$	163466.			0.530545			0.265273	+	0.964173i	$0.414538\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−270687.	+	468844.i	−0.613373	+	1.06239i	0.377295	+	0.926093i	$0.376854\pi$
$41$	−270687.	+	468844.i	−0.613373	+	1.06239i	−0.990668	+	0.136300i	$0.956479\pi$
$42$		0			0
$43$	152718.	+	264516.i	0.292922	+	0.507355i	0.974499	−	0.224391i	$-0.0720391\pi$
$43$	152718.	+	264516.i	0.292922	+	0.507355i	−0.681578	+	0.731746i	$0.738706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	405133.	+	701710.i	0.569187	+	0.985860i	0.996647	+	0.0818260i	$0.0260752\pi$
$47$	405133.	+	701710.i	0.569187	+	0.985860i	−0.427460	+	0.904034i	$0.640591\pi$
$48$		0			0
$49$	−95264.1	+	165002.i	−0.115676	+	0.200357i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−1.40849e6			−1.29953			−0.649766	−	0.760135i	$-0.725133\pi$
$53$	−1.40849e6			−1.29953			−0.649766	+	0.760135i	$0.725133\pi$
$54$		0			0
$55$	−682206.			−0.552899
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−69842.7	+	120971.i	−0.0442730	+	0.0766831i	−0.887313	−	0.461168i	$-0.847430\pi$
$59$	−69842.7	+	120971.i	−0.0442730	+	0.0766831i	0.843040	+	0.537851i	$0.180764\pi$
$60$		0			0
$61$	−145701.	−	252362.i	−0.0821882	−	0.142354i	0.822001	−	0.569485i	$-0.192858\pi$
$61$	−145701.	−	252362.i	−0.0821882	−	0.142354i	−0.904190	+	0.427131i	$0.859524\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−1.37152e6	−	2.37554e6i	−0.619447	−	1.07291i
$66$		0			0
$67$	808071.	−	1.39962e6i	0.328237	−	0.568524i	−0.653925	−	0.756560i	$-0.726879\pi$
$67$	808071.	−	1.39962e6i	0.328237	−	0.568524i	0.982162	+	0.188036i	$0.0602121\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−962505.			−0.319153			−0.159576	−	0.987186i	$-0.551013\pi$
$71$	−962505.			−0.319153			−0.159576	+	0.987186i	$0.551013\pi$
$72$		0			0
$73$	211780.			0.0637170			0.0318585	−	0.999492i	$-0.489857\pi$
$73$	211780.			0.0637170			0.0318585	+	0.999492i	$0.489857\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.66331e6	−	2.88094e6i	0.415199	−	0.719145i
$78$		0			0
$79$	−3.76039e6	−	6.51319e6i	−0.858100	−	1.48627i	−0.873739	−	0.486396i	$-0.838311\pi$
$79$	−3.76039e6	−	6.51319e6i	−0.858100	−	1.48627i	0.0156383	−	0.999878i	$-0.495022\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.74040e6	+	3.01445e6i	0.334099	+	0.578676i	0.983311	−	0.181931i	$-0.0582348\pi$
$83$	1.74040e6	+	3.01445e6i	0.334099	+	0.578676i	−0.649213	+	0.760607i	$0.724901\pi$
$84$		0			0
$85$	−1.85915e6	+	3.22013e6i	−0.328358	+	0.568732i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−964328.			−0.144997			−0.0724986	−	0.997369i	$-0.523097\pi$
$89$	−964328.			−0.144997			−0.0724986	+	0.997369i	$0.523097\pi$
$90$		0			0
$91$	1.33758e7			1.86069
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.24399e6	+	3.88671e6i	−0.268527	+	0.465103i
$96$		0			0
$97$	−491976.	−	852128.i	−0.0547322	−	0.0947990i	0.837361	−	0.546650i	$-0.184097\pi$
$97$	−491976.	−	852128.i	−0.0547322	−	0.0947990i	−0.892093	+	0.451851i	$0.850764\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−7.20543e6	−	1.24802e7i	−0.695881	−	1.20530i	−0.969883	−	0.243572i	$-0.921681\pi$
$101$	−7.20543e6	−	1.24802e7i	−0.695881	−	1.20530i	0.274002	−	0.961729i	$-0.411653\pi$
$102$		0			0
$103$	−7.60168e6	+	1.31665e7i	−0.685455	+	1.18724i	0.287838	+	0.957679i	$0.407063\pi$
$103$	−7.60168e6	+	1.31665e7i	−0.685455	+	1.18724i	−0.973293	+	0.229564i	$0.926270\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−2.16120e7			−1.70550			−0.852751	−	0.522317i	$-0.825068\pi$
$107$	−2.16120e7			−1.70550			−0.852751	+	0.522317i	$0.825068\pi$
$108$		0			0
$109$	−1.81069e7			−1.33922			−0.669611	−	0.742712i	$-0.733539\pi$
$109$	−1.81069e7			−1.33922			−0.669611	+	0.742712i	$0.733539\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.38216e6	+	9.32218e6i	−0.350899	+	0.607775i	−0.986407	−	0.164319i	$-0.947457\pi$
$113$	−5.38216e6	+	9.32218e6i	−0.350899	+	0.607775i	0.635508	+	0.772094i	$0.280791\pi$
$114$		0			0
$115$	6.69451e6	+	1.15952e7i	0.410465	+	0.710947i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−9.06570e6	−	1.57023e7i	−0.493159	−	0.854176i
$120$		0			0
$121$	4.28716e6	−	7.42557e6i	0.219999	−	0.381049i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−2.34604e7			−1.07436
$126$		0			0
$127$	3.81170e7			1.65122			0.825612	−	0.564239i	$-0.190830\pi$
$127$	3.81170e7			1.65122			0.825612	+	0.564239i	$0.190830\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.75683e7	+	3.04291e7i	−0.682778	+	1.18261i	0.291351	+	0.956616i	$0.405895\pi$
$131$	−1.75683e7	+	3.04291e7i	−0.682778	+	1.18261i	−0.974130	+	0.225990i	$0.927438\pi$
$132$		0			0
$133$	−1.09423e7	−	1.89526e7i	−0.403300	−	0.698536i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−9.38202e6	−	1.62501e7i	−0.311727	−	0.539927i	0.667009	−	0.745049i	$-0.267574\pi$
$137$	−9.38202e6	−	1.62501e7i	−0.311727	−	0.539927i	−0.978736	+	0.205122i	$0.934241\pi$
$138$		0			0
$139$	−1.74869e7	+	3.02881e7i	−0.552281	+	0.956579i	0.445829	+	0.895118i	$0.352909\pi$
$139$	−1.74869e7	+	3.02881e7i	−0.552281	+	0.956579i	−0.998110	+	0.0614603i	$0.980424\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−4.38787e7			−1.25481
$144$		0			0
$145$	4.36403e7			1.18878
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−8.98086e6	+	1.55553e7i	−0.222416	+	0.385236i	−0.955541	−	0.294858i	$-0.904728\pi$
$149$	−8.98086e6	+	1.55553e7i	−0.222416	+	0.385236i	0.733125	+	0.680094i	$0.238061\pi$
$150$		0			0
$151$	−2.96554e7	−	5.13647e7i	−0.700947	−	1.21408i	−0.968134	−	0.250431i	$-0.919428\pi$
$151$	−2.96554e7	−	5.13647e7i	−0.700947	−	1.21408i	0.267187	−	0.963645i	$-0.413906\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−8.74965e6	−	1.51548e7i	−0.188725	−	0.326881i
$156$		0			0
$157$	−3.17469e7	+	5.49872e7i	−0.654715	+	1.13400i	0.327250	+	0.944938i	$0.393878\pi$
$157$	−3.17469e7	+	5.49872e7i	−0.654715	+	1.13400i	−0.981965	+	0.189062i	$0.939455\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−6.52885e7			−1.23295
$162$		0			0
$163$	−6.83086e6			−0.123543			−0.0617716	−	0.998090i	$-0.519675\pi$
$163$	−6.83086e6			−0.123543			−0.0617716	+	0.998090i	$0.519675\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.67873e7	−	8.10379e7i	0.777356	−	1.34642i	−0.156105	−	0.987740i	$-0.549894\pi$
$167$	4.67873e7	−	8.10379e7i	0.777356	−	1.34642i	0.933461	−	0.358679i	$-0.116773\pi$
$168$		0			0
$169$	−5.68401e7	−	9.84500e7i	−0.905840	−	1.56896i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−5.74229e7	−	9.94594e7i	−0.843187	−	1.46044i	−0.887186	−	0.461412i	$-0.847343\pi$
$173$	−5.74229e7	−	9.94594e7i	−0.843187	−	1.46044i	0.0439990	−	0.999032i	$-0.485990\pi$
$174$		0			0
$175$	1.78632e7	−	3.09399e7i	0.251956	−	0.436401i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.12778e8			−1.46973			−0.734864	−	0.678214i	$-0.762754\pi$
$179$	−1.12778e8			−1.46973			−0.734864	+	0.678214i	$0.762754\pi$
$180$		0			0
$181$	4.01383e7			0.503134			0.251567	−	0.967840i	$-0.419054\pi$
$181$	4.01383e7			0.503134			0.251567	+	0.967840i	$0.419054\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.68789e7	+	2.92351e7i	−0.195994	+	0.339472i
$186$		0			0
$187$	2.97397e7	+	5.15106e7i	0.332576	+	0.576038i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.76221e7	−	4.78429e7i	−0.286840	−	0.496822i	0.686214	−	0.727400i	$-0.259272\pi$
$191$	−2.76221e7	−	4.78429e7i	−0.286840	−	0.496822i	−0.973054	+	0.230578i	$0.925938\pi$
$192$		0			0
$193$	−6.89104e7	+	1.19356e8i	−0.689977	+	1.19507i	0.281868	+	0.959453i	$0.409046\pi$
$193$	−6.89104e7	+	1.19356e8i	−0.689977	+	1.19507i	−0.971845	+	0.235621i	$0.924287\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.64184e8			−1.53003			−0.765015	−	0.644012i	$-0.777269\pi$
$197$	−1.64184e8			−1.53003			−0.765015	+	0.644012i	$0.777269\pi$
$198$		0			0
$199$	1.09221e8			0.982468			0.491234	−	0.871028i	$-0.336546\pi$
$199$	1.09221e8			0.982468			0.491234	+	0.871028i	$0.336546\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.06401e8	+	1.84292e8i	−0.892708	+	1.54622i
$204$		0			0
$205$	−5.59003e7	−	9.68222e7i	−0.453185	−	0.784940i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	3.58958e7	+	6.21734e7i	0.271977	+	0.471078i
$210$		0			0
$211$	3.35228e7	−	5.80632e7i	0.245670	−	0.425513i	−0.716650	−	0.697433i	$-0.754325\pi$
$211$	3.35228e7	−	5.80632e7i	0.245670	−	0.425513i	0.962320	+	0.271920i	$0.0876588\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−6.30765e7			−0.432845
$216$		0			0
$217$	8.53313e7			0.566891
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.19578e8	+	2.07115e8i	−0.745210	+	1.29074i
$222$		0			0
$223$	−3.58056e7	−	6.20171e7i	−0.216214	−	0.374494i	0.737433	−	0.675420i	$-0.236038\pi$
$223$	−3.58056e7	−	6.20171e7i	−0.216214	−	0.374494i	−0.953647	+	0.300926i	$0.902704\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.09860e8	−	1.90283e8i	−0.623375	−	1.07972i	−0.988853	−	0.148897i	$-0.952428\pi$
$227$	−1.09860e8	−	1.90283e8i	−0.623375	−	1.07972i	0.365478	−	0.930820i	$-0.380906\pi$
$228$		0			0
$229$	−5.33939e7	+	9.24809e7i	−0.293810	+	0.508895i	−0.974707	−	0.223485i	$-0.928257\pi$
$229$	−5.33939e7	+	9.24809e7i	−0.293810	+	0.508895i	0.680897	+	0.732379i	$0.261590\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	1.00966e8			0.522914			0.261457	−	0.965215i	$-0.415797\pi$
$233$	1.00966e8			0.522914			0.261457	+	0.965215i	$0.415797\pi$
$234$		0			0
$235$	−1.67330e8			−0.841078
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.54545e7	−	1.65332e8i	0.452276	−	0.783366i	−0.546251	−	0.837622i	$-0.683945\pi$
$239$	9.54545e7	−	1.65332e8i	0.452276	−	0.783366i	0.998527	+	0.0542561i	$0.0172787\pi$
$240$		0			0
$241$	2.04331e8	+	3.53911e8i	0.940316	+	1.62867i	0.764869	+	0.644186i	$0.222804\pi$
$241$	2.04331e8	+	3.53911e8i	0.940316	+	1.62867i	0.175446	+	0.984489i	$0.443863\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.96732e7	−	3.40750e7i	−0.0854662	−	0.148032i
$246$		0			0
$247$	−1.44331e8	+	2.49989e8i	−0.609425	+	1.05556i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.17519e8			−1.26739			−0.633697	−	0.773582i	$-0.718463\pi$
$251$	−3.17519e8			−1.26739			−0.633697	+	0.773582i	$0.718463\pi$
$252$		0			0
$253$	2.14177e8			0.831477
$254$		0			0
$255$		0			0
$256$		0			0
$257$	1.76938e8	−	3.06465e8i	0.650211	−	1.12620i	−0.332860	−	0.942976i	$-0.608014\pi$
$257$	1.76938e8	−	3.06465e8i	0.650211	−	1.12620i	0.983071	−	0.183223i	$-0.0586529\pi$
$258$		0			0
$259$	−8.23062e7	−	1.42559e8i	−0.294363	−	0.509852i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−8.92703e7	−	1.54621e8i	−0.302595	−	0.524110i	0.674128	−	0.738615i	$-0.264520\pi$
$263$	−8.92703e7	−	1.54621e8i	−0.302595	−	0.524110i	−0.976723	+	0.214505i	$0.931186\pi$
$264$		0			0
$265$	1.45435e8	−	2.51901e8i	0.480074	−	0.831512i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2.99987e8			0.939657			0.469828	−	0.882758i	$-0.344316\pi$
$269$	2.99987e8			0.939657			0.469828	+	0.882758i	$0.344316\pi$
$270$		0			0
$271$	−1.50818e7			−0.0460322			−0.0230161	−	0.999735i	$-0.507327\pi$
$271$	−1.50818e7			−0.0460322			−0.0230161	+	0.999735i	$0.507327\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−5.85994e7	+	1.01497e8i	−0.169914	+	0.294300i
$276$		0			0
$277$	−2.20864e8	−	3.82548e8i	−0.624376	−	1.08145i	−0.988661	−	0.150164i	$-0.952020\pi$
$277$	−2.20864e8	−	3.82548e8i	−0.624376	−	1.08145i	0.364285	−	0.931287i	$-0.381313\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1.60639e8	+	2.78235e8i	0.431896	+	0.748066i	0.997037	−	0.0769286i	$-0.0245114\pi$
$281$	1.60639e8	+	2.78235e8i	0.431896	+	0.748066i	−0.565140	+	0.824995i	$0.691178\pi$
$282$		0			0
$283$	−5.17705e6	+	8.96692e6i	−0.0135778	+	0.0235175i	−0.872734	−	0.488195i	$-0.837655\pi$
$283$	−5.17705e6	+	8.96692e6i	−0.0135778	+	0.0235175i	0.859157	+	0.511713i	$0.170989\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.45170e8			1.36127
$288$		0			0
$289$	−8.61527e7			−0.209955
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2.20904e8	−	3.82617e8i	0.513058	−	0.888643i	−0.486827	−	0.873499i	$-0.661846\pi$
$293$	2.20904e8	−	3.82617e8i	0.513058	−	0.888643i	0.999885	−	0.0151449i	$-0.00482094\pi$
$294$		0			0
$295$	−1.44234e7	−	2.49821e7i	−0.0327108	−	0.0566567i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	4.30584e8	+	7.45793e8i	0.931555	+	1.61350i
$300$		0			0
$301$	1.53789e8	−	2.66370e8i	0.325044	−	0.562993i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	6.01783e7			0.121448
$306$		0			0
$307$	−4.06829e8			−0.802468			−0.401234	−	0.915976i	$-0.631419\pi$
$307$	−4.06829e8			−0.802468			−0.401234	+	0.915976i	$0.631419\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−4.51114e8	+	7.81352e8i	−0.850403	+	1.47294i	0.0304427	+	0.999537i	$0.490308\pi$
$311$	−4.51114e8	+	7.81352e8i	−0.850403	+	1.47294i	−0.880845	+	0.473404i	$0.843025\pi$
$312$		0			0
$313$	1.71584e8	+	2.97192e8i	0.316279	+	0.547812i	0.979709	−	0.200427i	$-0.0642329\pi$
$313$	1.71584e8	+	2.97192e8i	0.316279	+	0.547812i	−0.663429	+	0.748239i	$0.730900\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−6.30969e7	−	1.09287e8i	−0.111250	−	0.192691i	0.805024	−	0.593242i	$-0.202152\pi$
$317$	−6.30969e7	−	1.09287e8i	−0.111250	−	0.192691i	−0.916275	+	0.400551i	$0.868819\pi$
$318$		0			0
$319$	3.49044e8	−	6.04563e8i	0.602023	−	1.04273i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.91293e8			0.646090
$324$		0			0
$325$	−4.71236e8			−0.761460
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.07973e8	−	7.06630e8i	0.631605	−	1.09397i
$330$		0			0
$331$	5.25373e8	+	9.09972e8i	0.796287	+	1.37921i	0.922019	+	0.387145i	$0.126539\pi$
$331$	5.25373e8	+	9.09972e8i	0.796287	+	1.37921i	−0.125732	+	0.992064i	$0.540128\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.66877e8	+	2.89039e8i	0.242515	+	0.420049i
$336$		0			0
$337$	2.88024e8	−	4.98873e8i	0.409944	−	0.710044i	−0.584939	−	0.811077i	$-0.698882\pi$
$337$	2.88024e8	−	4.98873e8i	0.409944	−	0.710044i	0.994883	+	0.101033i	$0.0322149\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−2.79926e8			−0.382299
$342$		0			0
$343$	−6.37453e8			−0.852940
$344$		0			0
$345$		0			0
$346$		0			0
$347$	2.73198e8	−	4.73192e8i	0.351014	−	0.607973i	−0.635414	−	0.772172i	$-0.719170\pi$
$347$	2.73198e8	−	4.73192e8i	0.351014	−	0.607973i	0.986427	+	0.164199i	$0.0525038\pi$
$348$		0			0
$349$	2.01757e8	+	3.49453e8i	0.254062	+	0.440048i	0.964640	−	0.263570i	$-0.0849001\pi$
$349$	2.01757e8	+	3.49453e8i	0.254062	+	0.440048i	−0.710579	+	0.703618i	$0.751567\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	1.14865e8	+	1.98953e8i	0.138988	+	0.240735i	0.927114	−	0.374780i	$-0.122282\pi$
$353$	1.14865e8	+	1.98953e8i	0.138988	+	0.240735i	−0.788126	+	0.615514i	$0.788948\pi$
$354$		0			0
$355$	9.93846e7	−	1.72139e8i	0.117902	−	0.204212i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.31991e9			1.50561			0.752805	−	0.658244i	$-0.228700\pi$
$359$	1.31991e9			1.50561			0.752805	+	0.658244i	$0.228700\pi$
$360$		0			0
$361$	−4.21581e8			−0.471634
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−2.18676e7	+	3.78759e7i	−0.0235384	+	0.0407697i
$366$		0			0
$367$	3.21314e8	+	5.56532e8i	0.339311	+	0.587705i	0.984303	−	0.176485i	$-0.0564726\pi$
$367$	3.21314e8	+	5.56532e8i	0.339311	+	0.587705i	−0.644992	+	0.764189i	$0.723139\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	7.09180e8	+	1.22834e9i	0.721021	+	1.24884i
$372$		0			0
$373$	−7.76157e8	+	1.34434e9i	−0.774406	+	1.34131i	0.160721	+	0.987000i	$0.448618\pi$
$373$	−7.76157e8	+	1.34434e9i	−0.774406	+	1.34131i	−0.935128	+	0.354311i	$0.884715\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	2.80689e9			2.69793
$378$		0			0
$379$	−1.39252e8			−0.131391			−0.0656955	−	0.997840i	$-0.520927\pi$
$379$	−1.39252e8			−0.131391			−0.0656955	+	0.997840i	$0.520927\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−3.13820e8	+	5.43552e8i	−0.285420	+	0.494362i	−0.972711	−	0.232020i	$-0.925466\pi$
$383$	−3.13820e8	+	5.43552e8i	−0.285420	+	0.494362i	0.687291	+	0.726382i	$0.258800\pi$
$384$		0			0
$385$	3.43494e8	+	5.94950e8i	0.306766	+	0.531334i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.63593e8	−	4.56557e8i	−0.227044	−	0.393252i	0.729887	−	0.683568i	$-0.239573\pi$
$389$	−2.63593e8	−	4.56557e8i	−0.227044	−	0.393252i	−0.956931	+	0.290316i	$0.906240\pi$
$390$		0			0
$391$	5.83673e8	−	1.01095e9i	0.493800	−	0.855287i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.55313e9			1.26800
$396$		0			0
$397$	1.58499e9			1.27134			0.635668	−	0.771963i	$-0.280725\pi$
$397$	1.58499e9			1.27134			0.635668	+	0.771963i	$0.280725\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	5.05776e8	−	8.76030e8i	0.391700	−	0.678444i	−0.600974	−	0.799268i	$-0.705221\pi$
$401$	5.05776e8	−	8.76030e8i	0.391700	−	0.678444i	0.992674	+	0.120825i	$0.0385539\pi$
$402$		0			0
$403$	−5.62767e8	−	9.74742e8i	−0.428313	−	0.741860i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	2.70002e8	+	4.67658e8i	0.198512	+	0.343833i
$408$		0			0
$409$	6.49180e8	−	1.12441e9i	0.469173	−	0.812632i	−0.530206	−	0.847869i	$-0.677885\pi$
$409$	6.49180e8	−	1.12441e9i	0.469173	−	0.812632i	0.999379	+	0.0352370i	$0.0112186\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	1.40665e8			0.0982563
$414$		0			0
$415$	−7.18827e8			−0.493692
$416$		0			0
$417$		0			0
$418$		0			0
$419$	7.52027e8	−	1.30255e9i	0.499441	−	0.865058i	−0.500558	−	0.865703i	$-0.666872\pi$
$419$	7.52027e8	−	1.30255e9i	0.499441	−	0.865058i	1.00000		0.000644995i	$0.000205308\pi$
$420$		0			0
$421$	1.07441e8	+	1.86093e8i	0.0701750	+	0.121547i	0.898978	−	0.437994i	$-0.144311\pi$
$421$	1.07441e8	+	1.86093e8i	0.0701750	+	0.121547i	−0.828803	+	0.559541i	$0.810978\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.19390e8	+	5.53200e8i	0.201818	+	0.349559i
$426$		0			0
$427$	−1.46723e8	+	2.54132e8i	−0.0912012	+	0.157965i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−3.06745e9			−1.84547			−0.922735	−	0.385436i	$-0.874051\pi$
$431$	−3.06745e9			−1.84547			−0.922735	+	0.385436i	$0.874051\pi$
$432$		0			0
$433$	−1.51503e9			−0.896837			−0.448418	−	0.893824i	$-0.648013\pi$
$433$	−1.51503e9			−0.896837			−0.448418	+	0.893824i	$0.648013\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	7.04495e8	−	1.22022e9i	0.403824	−	0.699444i
$438$		0			0
$439$	1.36864e9	+	2.37056e9i	0.772084	+	1.33729i	0.936419	+	0.350884i	$0.114119\pi$
$439$	1.36864e9	+	2.37056e9i	0.772084	+	1.33729i	−0.164335	+	0.986405i	$0.552548\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−3.92701e8	−	6.80178e8i	−0.214609	−	0.371714i	0.738542	−	0.674207i	$-0.235515\pi$
$443$	−3.92701e8	−	6.80178e8i	−0.214609	−	0.371714i	−0.953152	+	0.302493i	$0.902181\pi$
$444$		0			0
$445$	9.95729e7	−	1.72465e8i	0.0535650	−	0.0927773i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	3.05948e9			1.59509			0.797545	−	0.603259i	$-0.206132\pi$
$449$	3.05948e9			1.59509			0.797545	+	0.603259i	$0.206132\pi$
$450$		0			0
$451$	−1.78841e9			−0.918014
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1.38113e9	+	2.39219e9i	−0.687378	+	1.19057i
$456$		0			0
$457$	−1.25749e9	−	2.17804e9i	−0.616310	−	1.06748i	−0.990153	−	0.139988i	$-0.955294\pi$
$457$	−1.25749e9	−	2.17804e9i	−0.616310	−	1.06748i	0.373844	−	0.927492i	$-0.378040\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.83330e8	+	8.37152e8i	0.229769	+	0.397971i	0.957739	−	0.287637i	$-0.0928698\pi$
$461$	4.83330e8	+	8.37152e8i	0.229769	+	0.397971i	−0.727971	+	0.685608i	$0.759536\pi$
$462$		0			0
$463$	1.48512e9	−	2.57231e9i	0.695390	−	1.20445i	−0.274659	−	0.961542i	$-0.588565\pi$
$463$	1.48512e9	−	2.57231e9i	0.695390	−	1.20445i	0.970049	−	0.242909i	$-0.0781018\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−3.56135e9			−1.61810			−0.809050	−	0.587739i	$-0.800018\pi$
$467$	−3.56135e9			−1.61810			−0.809050	+	0.587739i	$0.800018\pi$
$468$		0			0
$469$	−1.62747e9			−0.728466
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−5.04499e8	+	8.73818e8i	−0.219203	+	0.379671i
$474$		0			0
$475$	3.85504e8	+	6.67713e8i	0.165045	+	0.285866i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	8.33828e8	+	1.44423e9i	0.346659	+	0.600431i	0.985654	−	0.168780i	$-0.0539829\pi$
$479$	8.33828e8	+	1.44423e9i	0.346659	+	0.600431i	−0.638995	+	0.769211i	$0.720650\pi$
$480$		0			0
$481$	−1.08563e9	+	1.88037e9i	−0.444811	+	0.770435i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.03198e8			0.0808769
$486$		0			0
$487$	3.24188e9			1.27188			0.635940	−	0.771739i	$-0.280613\pi$
$487$	3.24188e9			1.27188			0.635940	+	0.771739i	$0.280613\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.09192e8	+	8.81946e8i	−0.194132	+	0.336246i	−0.946615	−	0.322365i	$-0.895522\pi$
$491$	−5.09192e8	+	8.81946e8i	−0.194132	+	0.336246i	0.752484	+	0.658611i	$0.228856\pi$
$492$		0			0
$493$	−1.90243e9	−	3.29510e9i	−0.715063	−	1.23853i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	4.84626e8	+	8.39397e8i	0.177076	+	0.306705i
$498$		0			0
$499$	−8.88409e8	+	1.53877e9i	−0.320082	+	0.554398i	−0.980505	−	0.196496i	$-0.937044\pi$
$499$	−8.88409e8	+	1.53877e9i	−0.320082	+	0.554398i	0.660423	+	0.750894i	$0.270377\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.92852e9			−1.37639			−0.688194	−	0.725527i	$-0.741596\pi$
$503$	−3.92852e9			−1.37639			−0.688194	+	0.725527i	$0.741596\pi$
$504$		0			0
$505$	2.97602e9			1.02829
$506$		0			0
$507$		0			0
$508$		0			0
$509$	2.33018e8	−	4.03599e8i	0.0783208	−	0.135656i	−0.824205	−	0.566292i	$-0.808378\pi$
$509$	2.33018e8	−	4.03599e8i	0.0783208	−	0.135656i	0.902526	+	0.430636i	$0.141711\pi$
$510$		0			0
$511$	−1.06633e8	−	1.84693e8i	−0.0353522	−	0.0612318i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.56984e9	−	2.71904e9i	−0.506443	−	0.877185i
$516$		0			0
$517$	−1.33834e9	+	2.31807e9i	−0.425941	+	0.737752i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	1.86744e9			0.578515			0.289258	−	0.957251i	$-0.406592\pi$
$521$	1.86744e9			0.578515			0.289258	+	0.957251i	$0.406592\pi$
$522$		0			0
$523$	−6.56935e7			−0.0200801			−0.0100401	−	0.999950i	$-0.503196\pi$
$523$	−6.56935e7			−0.0200801			−0.0100401	+	0.999950i	$0.503196\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−7.62854e8	+	1.32130e9i	−0.227041	+	0.393246i
$528$		0			0
$529$	−3.99310e8	−	6.91626e8i	−0.117278	−	0.203131i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−3.59544e9	−	6.22749e9i	−1.02851	−	1.78143i
$534$		0			0
$535$	2.23158e9	−	3.86520e9i	0.630048	−	1.09128i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.29403e8			−0.173128
$540$		0			0
$541$	7.05261e9			1.91496			0.957479	−	0.288503i	$-0.0931576\pi$
$541$	7.05261e9			1.91496			0.957479	+	0.288503i	$0.0931576\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.86965e9	−	3.23834e9i	0.494736	−	0.856908i
$546$		0			0
$547$	1.97436e9	+	3.41969e9i	0.515787	+	0.893369i	0.999832	+	0.0183263i	$0.00583377\pi$
$547$	1.97436e9	+	3.41969e9i	0.515787	+	0.893369i	−0.484045	+	0.875043i	$0.660833\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−2.29623e9	−	3.97720e9i	−0.584771	−	1.01285i
$552$		0			0
$553$	−3.78675e9	+	6.55885e9i	−0.952202	+	1.64926i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−5.88371e9			−1.44264			−0.721320	−	0.692602i	$-0.756464\pi$
$557$	−5.88371e9			−1.44264			−0.721320	+	0.692602i	$0.756464\pi$
$558$		0			0
$559$	−4.05701e9			−0.982346
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.62408e9	+	2.81299e9i	−0.383555	+	0.664336i	−0.991568	−	0.129591i	$-0.958634\pi$
$563$	−1.62408e9	+	2.81299e9i	−0.383555	+	0.664336i	0.608013	+	0.793927i	$0.291967\pi$
$564$		0			0
$565$	−1.11148e9	−	1.92515e9i	−0.259259	−	0.449050i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.89886e9	−	3.28892e9i	−0.432116	−	0.748447i	0.564939	−	0.825132i	$-0.308900\pi$
$569$	−1.89886e9	−	3.28892e9i	−0.432116	−	0.748447i	−0.997055	+	0.0766857i	$0.975566\pi$
$570$		0			0
$571$	−3.20931e9	+	5.55869e9i	−0.721415	+	1.24953i	0.239017	+	0.971015i	$0.423175\pi$
$571$	−3.20931e9	+	5.55869e9i	−0.721415	+	1.24953i	−0.960433	+	0.278513i	$0.910159\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.30015e9			0.504568
$576$		0			0
$577$	−3.61207e8			−0.0782781			−0.0391390	−	0.999234i	$-0.512462\pi$
$577$	−3.61207e8			−0.0782781			−0.0391390	+	0.999234i	$0.512462\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.75260e9	−	3.03559e9i	0.370737	−	0.642135i
$582$		0			0
$583$	−2.32644e9	−	4.02951e9i	−0.486241	−	0.842194i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.97028e9	+	3.41262e9i	0.402063	+	0.696394i	0.993975	−	0.109609i	$-0.0349599\pi$
$587$	1.97028e9	+	3.41262e9i	0.402063	+	0.696394i	−0.591912	+	0.806003i	$0.701627\pi$
$588$		0			0
$589$	−9.20766e8	+	1.59481e9i	−0.185672	+	0.321593i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−9.82926e9			−1.93566			−0.967831	−	0.251603i	$-0.919042\pi$
$593$	−9.82926e9			−1.93566			−0.967831	+	0.251603i	$0.919042\pi$
$594$		0			0
$595$	3.74436e9			0.728733
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−5.16478e9	+	8.94566e9i	−0.981879	+	1.70066i	−0.326820	+	0.945087i	$0.605977\pi$
$599$	−5.16478e9	+	8.94566e9i	−0.981879	+	1.70066i	−0.655059	+	0.755577i	$0.727356\pi$
$600$		0			0
$601$	−3.30732e9	−	5.72844e9i	−0.621463	−	1.07640i	−0.989214	−	0.146481i	$-0.953205\pi$
$601$	−3.30732e9	−	5.72844e9i	−0.621463	−	1.07640i	0.367751	−	0.929924i	$-0.380128\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	8.85352e8	+	1.53347e9i	0.162544	+	0.281535i
$606$		0			0
$607$	5.39174e8	−	9.33878e8i	0.0978518	−	0.169484i	−0.812943	−	0.582343i	$-0.802136\pi$
$607$	5.39174e8	−	9.33878e8i	0.0978518	−	0.169484i	0.910795	+	0.412858i	$0.135470\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−1.07625e10			−1.90883
$612$		0			0
$613$	7.04475e9			1.23525			0.617624	−	0.786474i	$-0.288096\pi$
$613$	7.04475e9			1.23525			0.617624	+	0.786474i	$0.288096\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.33992e9	+	2.32080e9i	−0.229657	+	0.397778i	−0.957706	−	0.287747i	$-0.907094\pi$
$617$	−1.33992e9	+	2.32080e9i	−0.229657	+	0.397778i	0.728049	+	0.685525i	$0.240427\pi$
$618$		0			0
$619$	2.79134e9	+	4.83474e9i	0.473037	+	0.819324i	0.999524	−	0.0308594i	$-0.00982442\pi$
$619$	2.79134e9	+	4.83474e9i	0.473037	+	0.819324i	−0.526487	+	0.850183i	$0.676491\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	4.85545e8	+	8.40988e8i	0.0804490	+	0.139342i
$624$		0			0
$625$	1.03658e9	−	1.79542e9i	0.169834	−	0.294161i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	2.94324e9			0.471572
$630$		0			0
$631$	6.79334e8			0.107642			0.0538208	−	0.998551i	$-0.482860\pi$
$631$	6.79334e8			0.107642			0.0538208	+	0.998551i	$0.482860\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−3.93582e9	+	6.81704e9i	−0.609996	+	1.05654i
$636$		0			0
$637$	−1.26536e9	−	2.19167e9i	−0.193966	−	0.335959i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.66790e9	−	9.81709e9i	−0.850000	−	1.47224i	−0.881207	−	0.472730i	$-0.843269\pi$
$641$	−5.66790e9	−	9.81709e9i	−0.850000	−	1.47224i	0.0312072	−	0.999513i	$-0.490065\pi$
$642$		0			0
$643$	7.49800e8	−	1.29869e9i	0.111226	−	0.192649i	−0.805039	−	0.593222i	$-0.797856\pi$
$643$	7.49800e8	−	1.29869e9i	0.111226	−	0.192649i	0.916265	+	0.400573i	$0.131189\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−7.57854e9			−1.10007			−0.550035	−	0.835141i	$-0.685386\pi$
$647$	−7.57854e9			−1.10007			−0.550035	+	0.835141i	$0.685386\pi$
$648$		0			0
$649$	−4.61446e8			−0.0662619
$650$		0			0
$651$		0			0
$652$		0			0
$653$	3.29135e9	−	5.70078e9i	0.462570	−	0.801195i	−0.536518	−	0.843889i	$-0.680261\pi$
$653$	3.29135e9	−	5.70078e9i	0.462570	−	0.801195i	0.999088	+	0.0426941i	$0.0135941\pi$
$654$		0			0
$655$	−3.62807e9	−	6.28400e9i	−0.504465	−	0.873759i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	4.49002e9	+	7.77694e9i	0.611152	+	1.05855i	0.991047	+	0.133516i	$0.0426268\pi$
$659$	4.49002e9	+	7.77694e9i	0.611152	+	1.05855i	−0.379895	+	0.925030i	$0.624040\pi$
$660$		0			0
$661$	3.15432e9	−	5.46343e9i	0.424815	−	0.735801i	−0.571588	−	0.820541i	$-0.693672\pi$
$661$	3.15432e9	−	5.46343e9i	0.424815	−	0.735801i	0.996403	+	0.0847394i	$0.0270058\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	4.51945e9			0.595950
$666$		0			0
$667$	−1.37007e10			−1.78774
$668$		0			0
$669$		0			0
$670$		0			0
$671$	4.81319e8	−	8.33669e8i	0.0615041	−	0.106528i
$672$		0			0
$673$	1.09644e9	+	1.89909e9i	0.138654	+	0.240156i	0.926987	−	0.375093i	$-0.122389\pi$
$673$	1.09644e9	+	1.89909e9i	0.138654	+	0.240156i	−0.788333	+	0.615248i	$0.789056\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−4.54133e9	−	7.86582e9i	−0.562500	−	0.974279i	−0.997277	−	0.0737411i	$-0.976506\pi$
$677$	−4.54133e9	−	7.86582e9i	−0.562500	−	0.974279i	0.434777	−	0.900538i	$-0.356827\pi$
$678$		0			0
$679$	−4.95425e8	+	8.58102e8i	−0.0607343	+	0.105195i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−1.44563e10			−1.73614			−0.868068	−	0.496444i	$-0.834639\pi$
$683$	−1.44563e10			−1.73614			−0.868068	+	0.496444i	$0.834639\pi$
$684$		0			0
$685$	3.87501e9			0.460634
$686$		0			0
$687$		0			0
$688$		0			0
$689$	9.35421e9	−	1.62020e10i	1.08953	−	1.88712i
$690$		0			0
$691$	−5.85120e9	−	1.01346e10i	−0.674640	−	1.16851i	−0.976574	−	0.215182i	$-0.930966\pi$
$691$	−5.85120e9	−	1.01346e10i	−0.674640	−	1.16851i	0.301934	−	0.953329i	$-0.402368\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−3.61125e9	−	6.25488e9i	−0.408048	−	0.706760i
$696$		0			0
$697$	−4.87377e9	+	8.44162e9i	−0.545193	+	0.944302i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.39696e10			1.53169			0.765847	−	0.643022i	$-0.222320\pi$
$701$	1.39696e10			1.53169			0.765847	+	0.643022i	$0.222320\pi$
$702$		0			0
$703$	3.55249e9			0.385647
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−7.25595e9	+	1.25677e10i	−0.772193	+	1.33748i
$708$		0			0
$709$	4.32835e9	+	7.49692e9i	0.456101	+	0.789989i	0.998751	−	0.0499695i	$-0.0159124\pi$
$709$	4.32835e9	+	7.49692e9i	0.456101	+	0.789989i	−0.542650	+	0.839959i	$0.682579\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	2.74693e9	+	4.75782e9i	0.283814	+	0.491580i
$714$		0			0
$715$	4.53075e9	−	7.84749e9i	0.463553	−	0.802897i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	5.45100e9			0.546922			0.273461	−	0.961883i	$-0.411832\pi$
$719$	5.45100e9			0.546922			0.273461	+	0.961883i	$0.411832\pi$
$720$		0			0
$721$	1.53099e10			1.52125
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.74857e9	−	6.49271e9i	0.365328	−	0.632766i
$726$		0			0
$727$	3.94770e9	+	6.83762e9i	0.381043	+	0.659985i	0.991212	−	0.132286i	$-0.0422317\pi$
$727$	3.94770e9	+	6.83762e9i	0.381043	+	0.659985i	−0.610169	+	0.792271i	$0.708898\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	2.74972e9	+	4.76265e9i	0.260362	+	0.450960i
$732$		0			0
$733$	−1.02353e10	+	1.77281e10i	−0.959928	+	1.66264i	−0.237264	+	0.971445i	$0.576251\pi$
$733$	−1.02353e10	+	1.77281e10i	−0.959928	+	1.66264i	−0.722665	+	0.691199i	$0.757083\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	5.33887e9			0.491262
$738$		0			0
$739$	−2.86408e9			−0.261054			−0.130527	−	0.991445i	$-0.541667\pi$
$739$	−2.86408e9			−0.261054			−0.130527	+	0.991445i	$0.541667\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.24008e9	+	2.14789e9i	−0.110915	+	0.192110i	−0.916139	−	0.400860i	$-0.868711\pi$
$743$	−1.24008e9	+	2.14789e9i	−0.110915	+	0.192110i	0.805224	+	0.592970i	$0.202045\pi$
$744$		0			0
$745$	−1.85466e9	−	3.21236e9i	−0.164330	−	0.284628i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1.08818e10	+	1.88478e10i	0.946266	+	1.63898i
$750$		0			0
$751$	−4.78442e9	+	8.28687e9i	−0.412183	+	0.713922i	−0.995128	−	0.0985897i	$-0.968567\pi$
$751$	−4.78442e9	+	8.28687e9i	−0.412183	+	0.713922i	0.582945	+	0.812511i	$0.301900\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.22484e10			1.03578
$756$		0			0
$757$	−1.37910e10			−1.15547			−0.577737	−	0.816223i	$-0.696064\pi$
$757$	−1.37910e10			−1.15547			−0.577737	+	0.816223i	$0.696064\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.54558e9	−	6.14113e9i	0.291636	−	0.505129i	−0.682561	−	0.730829i	$-0.739134\pi$
$761$	3.54558e9	−	6.14113e9i	0.291636	−	0.505129i	0.974197	+	0.225700i	$0.0724670\pi$
$762$		0			0
$763$	9.11694e9	+	1.57910e10i	0.743042	+	1.28699i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−9.27697e8	−	1.60682e9i	−0.0742373	−	0.128583i
$768$		0			0
$769$	1.09654e10	−	1.89927e10i	0.869527	−	1.50607i	0.00704673	−	0.999975i	$-0.497757\pi$
$769$	1.09654e10	−	1.89927e10i	0.869527	−	1.50607i	0.862481	−	0.506090i	$-0.168910\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1.26416e10			0.984407			0.492204	−	0.870480i	$-0.336192\pi$
$773$	1.26416e10			0.984407			0.492204	+	0.870480i	$0.336192\pi$
$774$		0			0
$775$	−3.00627e9			−0.231992
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.88265e9	+	1.01890e10i	−0.445853	+	0.772240i
$780$		0			0
$781$	−1.58980e9	−	2.75361e9i	−0.119416	−	0.206835i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−6.55613e9	−	1.13555e10i	−0.483731	−	0.837846i
$786$		0			0
$787$	8.05398e9	−	1.39499e10i	0.588978	−	1.02014i	−0.405389	−	0.914144i	$-0.632864\pi$
$787$	8.05398e9	−	1.39499e10i	0.588978	−	1.02014i	0.994367	−	0.105995i	$-0.0338028\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	1.08398e10			0.778760
$792$		0			0
$793$	3.87060e9			0.275627
$794$		0			0
$795$		0			0
$796$		0			0
$797$	3.38094e8	−	5.85597e8i	0.0236556	−	0.0409727i	−0.853955	−	0.520346i	$-0.825803\pi$
$797$	3.38094e8	−	5.85597e8i	0.0236556	−	0.0409727i	0.877611	+	0.479374i	$0.159136\pi$
$798$		0			0
$799$	7.29448e9	+	1.26344e10i	0.505919	+	0.876277i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	3.49804e8	+	6.05878e8i	0.0238408	+	0.0412934i
$804$		0			0
$805$	6.74145e9	−	1.16765e10i	0.455478	−	0.788912i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.50279e9			−0.232592			−0.116296	−	0.993215i	$-0.537102\pi$
$809$	−3.50279e9			−0.232592			−0.116296	+	0.993215i	$0.537102\pi$
$810$		0			0
$811$	−2.59913e10			−1.71102			−0.855510	−	0.517786i	$-0.826756\pi$
$811$	−2.59913e10			−1.71102			−0.855510	+	0.517786i	$0.826756\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	7.05329e8	−	1.22167e9i	0.0456394	−	0.0790498i
$816$		0			0
$817$	3.31891e9	+	5.74853e9i	0.212921	+	0.368790i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	1.37619e9	+	2.38363e9i	0.0867914	+	0.150327i	0.906153	−	0.422950i	$-0.139005\pi$
$821$	1.37619e9	+	2.38363e9i	0.0867914	+	0.150327i	−0.819362	+	0.573277i	$0.805672\pi$
$822$		0			0
$823$	1.32706e10	−	2.29853e10i	0.829831	−	1.43731i	−0.0683391	−	0.997662i	$-0.521770\pi$
$823$	1.32706e10	−	2.29853e10i	0.829831	−	1.43731i	0.898170	−	0.439648i	$-0.144897\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−3.46034e9			−0.212740			−0.106370	−	0.994327i	$-0.533923\pi$
$827$	−3.46034e9			−0.212740			−0.106370	+	0.994327i	$0.533923\pi$
$828$		0			0
$829$	−1.43017e10			−0.871857			−0.435929	−	0.899981i	$-0.643580\pi$
$829$	−1.43017e10			−0.871857			−0.435929	+	0.899981i	$0.643580\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.71525e9	+	2.97089e9i	−0.102818	+	0.178086i
$834$		0			0
$835$	9.66215e9	+	1.67353e10i	0.574343	+	0.994791i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.17565e10	+	2.03629e10i	0.687245	+	1.19034i	0.972726	+	0.231959i	$0.0745134\pi$
$839$	1.17565e10	+	2.03629e10i	0.687245	+	1.19034i	−0.285481	+	0.958384i	$0.592153\pi$
$840$		0			0
$841$	−1.37032e10	+	2.37347e10i	−0.794395	+	1.37593i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.34764e10			1.33854
$846$		0			0
$847$	−8.63443e9			−0.488249
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.29909e9	−	9.17829e9i	0.294746	−	0.510515i
$852$		0			0
$853$	−7.42212e9	−	1.28555e10i	−0.409455	−	0.709198i	0.585373	−	0.810764i	$-0.300948\pi$
$853$	−7.42212e9	−	1.28555e10i	−0.409455	−	0.709198i	−0.994829	+	0.101566i	$0.967615\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.10920e10	+	1.92120e10i	0.601975	+	1.04265i	0.992522	+	0.122068i	$0.0389527\pi$
$857$	1.10920e10	+	1.92120e10i	0.601975	+	1.04265i	−0.390547	+	0.920583i	$0.627714\pi$
$858$		0			0
$859$	2.51392e9	−	4.35423e9i	0.135324	−	0.234388i	−0.790397	−	0.612595i	$-0.790126\pi$
$859$	2.51392e9	−	4.35423e9i	0.135324	−	0.234388i	0.925721	+	0.378207i	$0.123459\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	2.58042e9			0.136664			0.0683318	−	0.997663i	$-0.478232\pi$
$863$	2.58042e9			0.136664			0.0683318	+	0.997663i	$0.478232\pi$
$864$		0			0
$865$	2.37171e10			1.24596
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1.24223e10	−	2.15161e10i	0.642145	−	1.11223i
$870$		0			0
$871$	1.07333e10	+	1.85907e10i	0.550390	+	0.953304i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.18124e10	+	2.04597e10i	0.596088	+	1.03246i
$876$		0			0
$877$	4.00303e9	−	6.93345e9i	0.200397	−	0.347097i	−0.748260	−	0.663406i	$-0.769110\pi$
$877$	4.00303e9	−	6.93345e9i	0.200397	−	0.347097i	0.948656	+	0.316309i	$0.102444\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.77186e10			0.872998			0.436499	−	0.899705i	$-0.356218\pi$
$881$	1.77186e10			0.872998			0.436499	+	0.899705i	$0.356218\pi$
$882$		0			0
$883$	−6.14136e9			−0.300194			−0.150097	−	0.988671i	$-0.547959\pi$
$883$	−6.14136e9			−0.300194			−0.150097	+	0.988671i	$0.547959\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	8.39938e9	−	1.45481e10i	0.404124	−	0.699963i	−0.590095	−	0.807334i	$-0.700910\pi$
$887$	8.39938e9	−	1.45481e10i	0.404124	−	0.699963i	0.994219	+	0.107371i	$0.0342432\pi$
$888$		0			0
$889$	−1.91921e10	−	3.32417e10i	−0.916151	−	1.58682i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	8.80445e9	+	1.52498e10i	0.413735	+	0.716610i
$894$		0			0
$895$	1.16450e10	−	2.01697e10i	0.542948	−	0.940414i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	1.79067e10			0.821971
$900$		0			0
$901$	−2.53600e10			−1.15508
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.14453e9	+	7.17853e9i	−0.185868	+	0.321933i
$906$		0			0
$907$	−8.97823e9	−	1.55508e10i	−0.399545	−	0.692032i	0.594125	−	0.804373i	$-0.297498\pi$
$907$	−8.97823e9	−	1.55508e10i	−0.399545	−	0.692032i	−0.993670	+	0.112341i	$0.964165\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.42336e9	+	2.46533e9i	0.0623734	+	0.108034i	0.895526	−	0.445009i	$-0.146800\pi$
$911$	1.42336e9	+	2.46533e9i	0.0623734	+	0.108034i	−0.833152	+	0.553043i	$0.813466\pi$
$912$		0			0
$913$	−5.74933e9	+	9.95813e9i	−0.250017	+	0.433042i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	3.53829e10			1.51531
$918$		0			0
$919$	−4.85382e8			−0.0206290			−0.0103145	−	0.999947i	$-0.503283\pi$
$919$	−4.85382e8			−0.0206290			−0.0103145	+	0.999947i	$0.503283\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	6.39230e9	−	1.10718e10i	0.267579	−	0.463460i
$924$		0			0
$925$	2.89970e9	+	5.02242e9i	0.120464	+	0.208649i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.53087e10	−	2.65154e10i	−0.626444	−	1.08503i	−0.988260	−	0.152783i	$-0.951176\pi$
$929$	−1.53087e10	−	2.65154e10i	−0.626444	−	1.08503i	0.361816	−	0.932250i	$-0.382157\pi$
$930$		0			0
$931$	−2.07030e9	+	3.58587e9i	−0.0840834	+	0.145637i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1.22832e10			−0.491442
$936$		0			0
$937$	−5.42261e9			−0.215338			−0.107669	−	0.994187i	$-0.534339\pi$
$937$	−5.42261e9			−0.215338			−0.107669	+	0.994187i	$0.534339\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	6.51113e9	−	1.12776e10i	0.254738	−	0.441218i	−0.710087	−	0.704114i	$-0.751344\pi$
$941$	6.51113e9	−	1.12776e10i	0.254738	−	0.441218i	0.964824	+	0.262896i	$0.0846776\pi$
$942$		0			0
$943$	1.75497e10	+	3.03970e10i	0.681522	+	1.18043i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−4.74133e9	−	8.21222e9i	−0.181416	−	0.314221i	0.760947	−	0.648814i	$-0.224735\pi$
$947$	−4.74133e9	−	8.21222e9i	−0.181416	−	0.314221i	−0.942363	+	0.334593i	$0.891401\pi$
$948$		0			0
$949$	−1.40650e9	+	2.43613e9i	−0.0534206	+	0.0925271i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−3.04803e9			−0.114076			−0.0570379	−	0.998372i	$-0.518166\pi$
$953$	−3.04803e9			−0.114076			−0.0570379	+	0.998372i	$0.518166\pi$
$954$		0			0
$955$	1.14086e10			0.423859
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−9.44780e9	+	1.63641e10i	−0.345912	+	0.599137i
$960$		0			0
$961$	1.01661e10	+	1.76082e10i	0.369507	+	0.640005i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−1.42309e10	−	2.46486e10i	−0.509783	−	0.882971i
$966$		0			0
$967$	5.89632e8	−	1.02127e9i	0.0209695	−	0.0363203i	−0.855350	−	0.518050i	$-0.826658\pi$
$967$	5.89632e8	−	1.02127e9i	0.0209695	−	0.0363203i	0.876320	+	0.481730i	$0.159991\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.58426e10			−0.555340			−0.277670	−	0.960676i	$-0.589562\pi$
$971$	−1.58426e10			−0.555340			−0.277670	+	0.960676i	$0.589562\pi$
$972$		0			0
$973$	3.52189e10			1.22569
$974$		0			0
$975$		0			0
$976$		0			0
$977$	7.77746e9	−	1.34709e10i	0.266813	−	0.462133i	−0.701224	−	0.712941i	$-0.747363\pi$
$977$	7.77746e9	−	1.34709e10i	0.266813	−	0.462133i	0.968037	+	0.250808i	$0.0806961\pi$
$978$		0			0
$979$	−1.59281e9	−	2.75883e9i	−0.0542531	−	0.0939691i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−6.75768e9	−	1.17046e10i	−0.226913	−	0.393025i	0.729978	−	0.683470i	$-0.239530\pi$
$983$	−6.75768e9	−	1.17046e10i	−0.226913	−	0.393025i	−0.956892	+	0.290445i	$0.906197\pi$
$984$		0			0
$985$	1.69531e10	−	2.93636e10i	0.565225	−	0.978998i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	1.98027e10			0.650934
$990$		0			0
$991$	−2.44751e10			−0.798854			−0.399427	−	0.916765i	$-0.630791\pi$
$991$	−2.44751e10			−0.798854			−0.399427	+	0.916765i	$0.630791\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.12777e10	+	1.95335e10i	−0.362944	+	0.628638i
$996$		0			0
$997$	1.45998e10	+	2.52876e10i	0.466566	+	0.808116i	0.999271	−	0.0381850i	$-0.0121576\pi$
$997$	1.45998e10	+	2.52876e10i	0.466566	+	0.808116i	−0.532705	+	0.846301i	$0.678824\pi$
$998$		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.109.2		16
3.2	odd	2	inner	324.8.e.m.109.7		16
9.2	odd	6	inner	324.8.e.m.217.7		16
9.4	even	3		324.8.a.e.1.7	yes	8
9.5	odd	6		324.8.a.e.1.2	✓	8
9.7	even	3	inner	324.8.e.m.217.2		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.8.a.e.1.2	✓	8	9.5	odd	6
324.8.a.e.1.7	yes	8	9.4	even	3
324.8.e.m.109.2		16	1.1	even	1	trivial
324.8.e.m.109.7		16	3.2	odd	2	inner
324.8.e.m.217.2		16	9.7	even	3	inner
324.8.e.m.217.7		16	9.2	odd	6	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)

\(f(q)\)	\(=\)	\(q+(-103.256 + 178.845i) q^{5} +(-503.506 - 872.097i) q^{7} +O(q^{10})\)	\(q+(-103.256 + 178.845i) q^{5} +(-503.506 - 872.097i) q^{7} +(1651.73 + 2860.88i) q^{11} +(-6641.32 + 11503.1i) q^{13} +18005.2 q^{17} +21732.3 q^{19} +(32417.0 - 56147.9i) q^{23} +(17738.8 + 30724.5i) q^{25} +(-105660. - 183009. i) q^{29} +(-42368.6 + 73384.6i) q^{31} +207960. q^{35} +163466. q^{37} +(-270687. + 468844. i) q^{41} +(152718. + 264516. i) q^{43} +(405133. + 701710. i) q^{47} +(-95264.1 + 165002. i) q^{49} -1.40849e6 q^{53} -682206. q^{55} +(-69842.7 + 120971. i) q^{59} +(-145701. - 252362. i) q^{61} +(-1.37152e6 - 2.37554e6i) q^{65} +(808071. - 1.39962e6i) q^{67} -962505. q^{71} +211780. q^{73} +(1.66331e6 - 2.88094e6i) q^{77} +(-3.76039e6 - 6.51319e6i) q^{79} +(1.74040e6 + 3.01445e6i) q^{83} +(-1.85915e6 + 3.22013e6i) q^{85} -964328. q^{89} +1.33758e7 q^{91} +(-2.24399e6 + 3.88671e6i) q^{95} +(-491976. - 852128. i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 560 q^{7}+O(q^{10}) \) 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

L-functions

Modular forms

Varieties

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Representations

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Database

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