LMFDB - Embedded newform 324.8.e.m.217.1

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			217.1
Root			$-18.0990 - 31.3483i$ of defining polynomial
Character	$\chi$	$=$	324.217
Dual form			324.8.e.m.109.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-265.209 - 459.356i) q^{5} +(-280.503 + 485.846i) q^{7} +O(q^{10})$	$q+(-265.209 - 459.356i) q^{5} +(-280.503 + 485.846i) q^{7} +(56.0795 - 97.1325i) q^{11} +(1899.11 + 3289.35i) q^{13} +6165.82 q^{17} -6799.85 q^{19} +(-46992.8 - 81393.9i) q^{23} +(-101610. + 175993. i) q^{25} +(-79197.6 + 137174. i) q^{29} +(108296. + 187574. i) q^{31} +297568. q^{35} -210516. q^{37} +(-173656. - 300781. i) q^{41} +(-367099. + 635834. i) q^{43} +(-84548.3 + 146442. i) q^{47} +(254407. + 440646. i) q^{49} +556281. q^{53} -59491.2 q^{55} +(1.15644e6 + 2.00302e6i) q^{59} +(1.54385e6 - 2.67402e6i) q^{61} +(1.00732e6 - 1.74474e6i) q^{65} +(-2.27712e6 - 3.94410e6i) q^{67} -2.84155e6 q^{71} +5.10749e6 q^{73} +(31461.0 + 54492.0i) q^{77} +(570175. - 987572. i) q^{79} +(3.54815e6 - 6.14558e6i) q^{83} +(-1.63523e6 - 2.83231e6i) q^{85} +6.42932e6 q^{89} -2.13083e6 q^{91} +(1.80338e6 + 3.12355e6i) q^{95} +(-2.08456e6 + 3.61056e6i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−265.209	−	459.356i	−0.948842	−	1.64344i	−0.747869	−	0.663846i	$-0.768923\pi$
$5$	−265.209	−	459.356i	−0.948842	−	1.64344i	−0.200973	−	0.979597i	$-0.564410\pi$
$6$		0			0
$7$	−280.503	+	485.846i	−0.309097	+	0.535372i	−0.978165	−	0.207829i	$-0.933360\pi$
$7$	−280.503	+	485.846i	−0.309097	+	0.535372i	0.669068	+	0.743201i	$0.266693\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	56.0795	−	97.1325i	0.0127037	−	0.0220034i	−0.859604	−	0.510961i	$-0.829290\pi$
$11$	56.0795	−	97.1325i	0.0127037	−	0.0220034i	0.872307	+	0.488958i	$0.162623\pi$
$12$		0			0
$13$	1899.11	+	3289.35i	0.239744	+	0.415249i	0.960641	−	0.277793i	$-0.0896031\pi$
$13$	1899.11	+	3289.35i	0.239744	+	0.415249i	−0.720897	+	0.693043i	$0.756270\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	6165.82			0.304382			0.152191	−	0.988351i	$-0.451367\pi$
$17$	6165.82			0.304382			0.152191	+	0.988351i	$0.451367\pi$
$18$		0			0
$19$	−6799.85			−0.227437			−0.113719	−	0.993513i	$-0.536276\pi$
$19$	−6799.85			−0.227437			−0.113719	+	0.993513i	$0.536276\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−46992.8	−	81393.9i	−0.805348	−	1.39490i	−0.916056	−	0.401051i	$-0.868645\pi$
$23$	−46992.8	−	81393.9i	−0.805348	−	1.39490i	0.110708	−	0.993853i	$-0.464688\pi$
$24$		0			0
$25$	−101610.	+	175993.i	−1.30060	+	2.25271i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−79197.6	+	137174.i	−0.603002	+	1.04443i	0.389362	+	0.921085i	$0.372696\pi$
$29$	−79197.6	+	137174.i	−0.603002	+	1.04443i	−0.992364	+	0.123345i	$0.960638\pi$
$30$		0			0
$31$	108296.	+	187574.i	0.652898	+	1.13085i	0.982416	+	0.186703i	$0.0597802\pi$
$31$	108296.	+	187574.i	0.652898	+	1.13085i	−0.329519	+	0.944149i	$0.606886\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	297568.			1.17314
$36$		0			0
$37$	−210516.			−0.683248			−0.341624	−	0.939837i	$-0.610977\pi$
$37$	−210516.			−0.683248			−0.341624	+	0.939837i	$0.610977\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−173656.	−	300781.i	−0.393502	−	0.681565i	0.599407	−	0.800444i	$-0.295403\pi$
$41$	−173656.	−	300781.i	−0.393502	−	0.681565i	−0.992909	+	0.118879i	$0.962070\pi$
$42$		0			0
$43$	−367099.	+	635834.i	−0.704115	+	1.21956i	0.262895	+	0.964824i	$0.415323\pi$
$43$	−367099.	+	635834.i	−0.704115	+	1.21956i	−0.967010	+	0.254738i	$0.918011\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−84548.3	+	146442.i	−0.118785	+	0.205742i	−0.919287	−	0.393589i	$-0.871233\pi$
$47$	−84548.3	+	146442.i	−0.118785	+	0.205742i	0.800501	+	0.599331i	$0.204567\pi$
$48$		0			0
$49$	254407.	+	440646.i	0.308918	+	0.535062i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	556281.			0.513249			0.256625	−	0.966511i	$-0.417390\pi$
$53$	556281.			0.513249			0.256625	+	0.966511i	$0.417390\pi$
$54$		0			0
$55$	−59491.2			−0.0482152
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.15644e6	+	2.00302e6i	0.733065	+	1.26971i	0.955567	+	0.294773i	$0.0952441\pi$
$59$	1.15644e6	+	2.00302e6i	0.733065	+	1.26971i	−0.222502	+	0.974932i	$0.571423\pi$
$60$		0			0
$61$	1.54385e6	−	2.67402e6i	0.870863	−	1.50838i	0.00975860	−	0.999952i	$-0.496894\pi$
$61$	1.54385e6	−	2.67402e6i	0.870863	−	1.50838i	0.861105	−	0.508427i	$-0.169773\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	1.00732e6	−	1.74474e6i	0.454959	−	0.788012i
$66$		0			0
$67$	−2.27712e6	−	3.94410e6i	−0.924964	−	1.60209i	−0.791619	−	0.611016i	$-0.790761\pi$
$67$	−2.27712e6	−	3.94410e6i	−0.924964	−	1.60209i	−0.133346	−	0.991070i	$-0.542572\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−2.84155e6			−0.942217			−0.471109	−	0.882075i	$-0.656146\pi$
$71$	−2.84155e6			−0.942217			−0.471109	+	0.882075i	$0.656146\pi$
$72$		0			0
$73$	5.10749e6			1.53666			0.768330	−	0.640055i	$-0.221088\pi$
$73$	5.10749e6			1.53666			0.768330	+	0.640055i	$0.221088\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	31461.0	+	54492.0i	0.00785334	+	0.0136024i
$78$		0			0
$79$	570175.	−	987572.i	0.130111	−	0.225359i	−0.793608	−	0.608429i	$-0.791800\pi$
$79$	570175.	−	987572.i	0.130111	−	0.225359i	0.923719	+	0.383070i	$0.125133\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.54815e6	−	6.14558e6i	0.681129	−	1.17975i	−0.293508	−	0.955957i	$-0.594823\pi$
$83$	3.54815e6	−	6.14558e6i	0.681129	−	1.17975i	0.974637	−	0.223793i	$-0.0718439\pi$
$84$		0			0
$85$	−1.63523e6	−	2.83231e6i	−0.288811	−	0.500235i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.42932e6			0.966719			0.483359	−	0.875422i	$-0.339416\pi$
$89$	6.42932e6			0.966719			0.483359	+	0.875422i	$0.339416\pi$
$90$		0			0
$91$	−2.13083e6			−0.296417
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.80338e6	+	3.12355e6i	0.215802	+	0.373780i
$96$		0			0
$97$	−2.08456e6	+	3.61056e6i	−0.231907	+	0.401674i	−0.958369	−	0.285532i	$-0.907830\pi$
$97$	−2.08456e6	+	3.61056e6i	−0.231907	+	0.401674i	0.726462	+	0.687206i	$0.241163\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−630068.	+	1.09131e6i	−0.0608503	+	0.105396i	−0.894846	−	0.446376i	$-0.852715\pi$
$101$	−630068.	+	1.09131e6i	−0.0608503	+	0.105396i	0.833995	+	0.551771i	$0.186048\pi$
$102$		0			0
$103$	−345346.	−	598157.i	−0.0311404	−	0.0539368i	0.850035	−	0.526726i	$-0.176581\pi$
$103$	−345346.	−	598157.i	−0.0311404	−	0.0539368i	−0.881176	+	0.472789i	$0.843247\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	1.62238e7			1.28029			0.640147	−	0.768253i	$-0.278874\pi$
$107$	1.62238e7			1.28029			0.640147	+	0.768253i	$0.278874\pi$
$108$		0			0
$109$	1.34204e7			0.992599			0.496300	−	0.868151i	$-0.334692\pi$
$109$	1.34204e7			0.992599			0.496300	+	0.868151i	$0.334692\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.57458e6	−	9.65546e6i	−0.363444	−	0.629504i	0.625081	−	0.780560i	$-0.285066\pi$
$113$	−5.57458e6	−	9.65546e6i	−0.363444	−	0.629504i	−0.988525	+	0.151056i	$0.951733\pi$
$114$		0			0
$115$	−2.49259e7	+	4.31728e7i	−1.52830	+	2.64709i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−1.72953e6	+	2.99564e6i	−0.0940836	+	0.162958i
$120$		0			0
$121$	9.73730e6	+	1.68655e7i	0.499677	+	0.865466i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	6.63523e7			3.03858
$126$		0			0
$127$	−2.69340e7			−1.16678			−0.583389	−	0.812193i	$-0.698274\pi$
$127$	−2.69340e7			−1.16678			−0.583389	+	0.812193i	$0.698274\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	1.85047e7	+	3.20511e7i	0.719172	+	1.24564i	0.961328	+	0.275405i	$0.0888119\pi$
$131$	1.85047e7	+	3.20511e7i	0.719172	+	1.24564i	−0.242157	+	0.970237i	$0.577855\pi$
$132$		0			0
$133$	1.90738e6	−	3.30368e6i	0.0703002	−	0.121764i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.19262e7	−	2.06568e7i	0.396260	−	0.686342i	−0.597001	−	0.802240i	$-0.703641\pi$
$137$	1.19262e7	−	2.06568e7i	0.396260	−	0.686342i	0.993261	+	0.115898i	$0.0369746\pi$
$138$		0			0
$139$	−2.13929e7	−	3.70536e7i	−0.675644	−	1.17025i	−0.976280	−	0.216511i	$-0.930532\pi$
$139$	−2.13929e7	−	3.70536e7i	−0.675644	−	1.17025i	0.300636	−	0.953739i	$-0.402801\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	426004.			0.0121825
$144$		0			0
$145$	8.40157e7			2.28861
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−1.05453e7	−	1.82650e7i	−0.261161	−	0.452344i	0.705390	−	0.708820i	$-0.250772\pi$
$149$	−1.05453e7	−	1.82650e7i	−0.261161	−	0.452344i	−0.966551	+	0.256476i	$0.917439\pi$
$150$		0			0
$151$	−1.22534e7	+	2.12235e7i	−0.289625	+	0.501646i	−0.973720	−	0.227747i	$-0.926864\pi$
$151$	−1.22534e7	+	2.12235e7i	−0.289625	+	0.501646i	0.684095	+	0.729393i	$0.260197\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	5.74421e7	−	9.94926e7i	1.23899	−	2.14600i
$156$		0			0
$157$	−3.28687e7	−	5.69303e7i	−0.677851	−	1.17407i	−0.975627	−	0.219436i	$-0.929578\pi$
$157$	−3.28687e7	−	5.69303e7i	−0.677851	−	1.17407i	0.297776	−	0.954636i	$-0.403755\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	5.27265e7			0.995723
$162$		0			0
$163$	2.18202e7			0.394642			0.197321	−	0.980339i	$-0.436776\pi$
$163$	2.18202e7			0.394642			0.197321	+	0.980339i	$0.436776\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	4.19239e7	+	7.26143e7i	0.696552	+	1.20646i	0.969655	+	0.244479i	$0.0786170\pi$
$167$	4.19239e7	+	7.26143e7i	0.696552	+	1.20646i	−0.273102	+	0.961985i	$0.588050\pi$
$168$		0			0
$169$	2.41610e7	−	4.18481e7i	0.385045	−	0.666918i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	5.55767e6	−	9.62617e6i	0.0816078	−	0.141349i	−0.822333	−	0.569007i	$-0.807328\pi$
$173$	5.55767e6	−	9.62617e6i	0.0816078	−	0.141349i	0.903941	+	0.427658i	$0.140661\pi$
$174$		0			0
$175$	−5.70036e7	−	9.87332e7i	−0.804025	−	1.39261i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	7.15923e7			0.932998			0.466499	−	0.884522i	$-0.345515\pi$
$179$	7.15923e7			0.932998			0.466499	+	0.884522i	$0.345515\pi$
$180$		0			0
$181$	1.29435e8			1.62248			0.811238	−	0.584716i	$-0.198794\pi$
$181$	1.29435e8			1.62248			0.811238	+	0.584716i	$0.198794\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	5.58308e7	+	9.67017e7i	0.648295	+	1.12288i
$186$		0			0
$187$	345776.	−	598901.i	0.00386678	−	0.00669745i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	8.50700e7	−	1.47346e8i	0.883405	−	1.53010i	0.0358732	−	0.999356i	$-0.488579\pi$
$191$	8.50700e7	−	1.47346e8i	0.883405	−	1.53010i	0.847531	−	0.530745i	$-0.178088\pi$
$192$		0			0
$193$	−3.91925e7	−	6.78833e7i	−0.392421	−	0.679692i	0.600348	−	0.799739i	$-0.295029\pi$
$193$	−3.91925e7	−	6.78833e7i	−0.392421	−	0.679692i	−0.992768	+	0.120047i	$0.961696\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	3.26452e7			0.304220			0.152110	−	0.988364i	$-0.451393\pi$
$197$	3.26452e7			0.304220			0.152110	+	0.988364i	$0.451393\pi$
$198$		0			0
$199$	4.01408e6			0.0361077			0.0180539	−	0.999837i	$-0.494253\pi$
$199$	4.01408e6			0.0361077			0.0180539	+	0.999837i	$0.494253\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−4.44303e7	−	7.69556e7i	−0.372772	−	0.645660i
$204$		0			0
$205$	−9.21105e7	+	1.59540e8i	−0.746742	+	1.29340i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−381332.	+	660487.i	−0.00288929	+	0.00500440i
$210$		0			0
$211$	7.18807e7	+	1.24501e8i	0.526773	+	0.912398i	0.999513	+	0.0311959i	$0.00993156\pi$
$211$	7.18807e7	+	1.24501e8i	0.526773	+	0.912398i	−0.472740	+	0.881202i	$0.656735\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	3.89432e8			2.67237
$216$		0			0
$217$	−1.21509e8			−0.807235
$218$		0			0
$219$		0			0
$220$		0			0
$221$	1.17096e7	+	2.02816e7i	0.0729739	+	0.126395i
$222$		0			0
$223$	−8.33372e7	+	1.44344e8i	−0.503236	+	0.871630i	0.496757	+	0.867890i	$0.334524\pi$
$223$	−8.33372e7	+	1.44344e8i	−0.503236	+	0.871630i	−0.999993	+	0.00374059i	$0.998809\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−1.35156e8	+	2.34097e8i	−0.766912	+	1.32833i	0.172318	+	0.985041i	$0.444874\pi$
$227$	−1.35156e8	+	2.34097e8i	−0.766912	+	1.32833i	−0.939230	+	0.343289i	$0.888459\pi$
$228$		0			0
$229$	5.57020e7	+	9.64786e7i	0.306511	+	0.530893i	0.977597	−	0.210487i	$-0.0675050\pi$
$229$	5.57020e7	+	9.64786e7i	0.306511	+	0.530893i	−0.671085	+	0.741380i	$0.734172\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	2.21831e8			1.14888			0.574442	−	0.818545i	$-0.305219\pi$
$233$	2.21831e8			1.14888			0.574442	+	0.818545i	$0.305219\pi$
$234$		0			0
$235$	8.96920e7			0.450834
$236$		0			0
$237$		0			0
$238$		0			0
$239$	3.13141e7	+	5.42376e7i	0.148371	+	0.256985i	0.930625	−	0.365973i	$-0.119264\pi$
$239$	3.13141e7	+	5.42376e7i	0.148371	+	0.256985i	−0.782255	+	0.622959i	$0.785931\pi$
$240$		0			0
$241$	−1.48123e8	+	2.56557e8i	−0.681653	+	1.18066i	0.292824	+	0.956166i	$0.405405\pi$
$241$	−1.48123e8	+	2.56557e8i	−0.681653	+	1.18066i	−0.974476	+	0.224491i	$0.927928\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.34942e8	−	2.33727e8i	0.586229	−	1.01538i
$246$		0			0
$247$	−1.29137e7	−	2.23671e7i	−0.0545268	−	0.0944432i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−3.26241e8			−1.30221			−0.651104	−	0.758989i	$-0.725694\pi$
$251$	−3.26241e8			−1.30221			−0.651104	+	0.758989i	$0.725694\pi$
$252$		0			0
$253$	−1.05413e7			−0.0409235
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.79391e8	−	3.10714e8i	−0.659226	−	1.14181i	−0.980816	−	0.194934i	$-0.937551\pi$
$257$	−1.79391e8	−	3.10714e8i	−0.659226	−	1.14181i	0.321590	−	0.946879i	$-0.395783\pi$
$258$		0			0
$259$	5.90504e7	−	1.02278e8i	0.211190	−	0.365792i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.90755e7	+	6.76808e7i	−0.132452	+	0.229414i	−0.924621	−	0.380888i	$-0.875618\pi$
$263$	−3.90755e7	+	6.76808e7i	−0.132452	+	0.229414i	0.792169	+	0.610302i	$0.208952\pi$
$264$		0			0
$265$	−1.47531e8	−	2.55531e8i	−0.486993	−	0.843496i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.00452e8			−0.627882			−0.313941	−	0.949443i	$-0.601649\pi$
$269$	−2.00452e8			−0.627882			−0.313941	+	0.949443i	$0.601649\pi$
$270$		0			0
$271$	1.58275e8			0.483082			0.241541	−	0.970391i	$-0.422347\pi$
$271$	1.58275e8			0.483082			0.241541	+	0.970391i	$0.422347\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.13964e7	+	1.97392e7i	0.0330449	+	0.0572354i
$276$		0			0
$277$	5.87657e7	−	1.01785e8i	0.166129	−	0.287743i	−0.770927	−	0.636924i	$-0.780207\pi$
$277$	5.87657e7	−	1.01785e8i	0.166129	−	0.287743i	0.937056	+	0.349180i	$0.113540\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−7.43884e7	+	1.28844e8i	−0.200001	+	0.346412i	−0.948529	−	0.316692i	$-0.897428\pi$
$281$	−7.43884e7	+	1.28844e8i	−0.200001	+	0.346412i	0.748527	+	0.663104i	$0.230761\pi$
$282$		0			0
$283$	1.38403e8	+	2.39721e8i	0.362989	+	0.628715i	0.988451	−	0.151540i	$-0.0484231\pi$
$283$	1.38403e8	+	2.39721e8i	0.362989	+	0.628715i	−0.625463	+	0.780254i	$0.715090\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	1.94844e8			0.486521
$288$		0			0
$289$	−3.72321e8			−0.907351
$290$		0			0
$291$		0			0
$292$		0			0
$293$	3.86294e8	+	6.69080e8i	0.897183	+	1.55397i	0.831080	+	0.556153i	$0.187723\pi$
$293$	3.86294e8	+	6.69080e8i	0.897183	+	1.55397i	0.0661027	+	0.997813i	$0.478944\pi$
$294$		0			0
$295$	6.13399e8	−	1.06244e9i	1.39113	−	2.40950i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.78489e8	−	3.09152e8i	0.386155	−	0.668840i
$300$		0			0
$301$	−2.05945e8	−	3.56707e8i	−0.435279	−	0.753926i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−1.63777e9			−3.30525
$306$		0			0
$307$	5.70352e8			1.12502			0.562508	−	0.826792i	$-0.309836\pi$
$307$	5.70352e8			1.12502			0.562508	+	0.826792i	$0.309836\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−3.22999e8	−	5.59451e8i	−0.608891	−	1.05463i	−0.991423	−	0.130689i	$-0.958281\pi$
$311$	−3.22999e8	−	5.59451e8i	−0.608891	−	1.05463i	0.382532	−	0.923942i	$-0.375052\pi$
$312$		0			0
$313$	1.97858e8	−	3.42700e8i	0.364711	−	0.631697i	−0.624019	−	0.781409i	$-0.714501\pi$
$313$	1.97858e8	−	3.42700e8i	0.364711	−	0.631697i	0.988730	+	0.149712i	$0.0478346\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.96290e8	−	3.39985e8i	0.346092	−	0.599449i	−0.639460	−	0.768825i	$-0.720842\pi$
$317$	1.96290e8	−	3.39985e8i	0.346092	−	0.599449i	0.985551	+	0.169376i	$0.0541752\pi$
$318$		0			0
$319$	8.88272e6	+	1.53853e7i	0.0153207	+	0.0265362i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−4.19266e7			−0.0692279
$324$		0			0
$325$	−7.71871e8			−1.24725
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.74321e7	−	8.21549e7i	−0.0734323	−	0.127188i
$330$		0			0
$331$	1.06406e8	−	1.84300e8i	0.161275	−	0.279336i	−0.774051	−	0.633123i	$-0.781773\pi$
$331$	1.06406e8	−	1.84300e8i	0.161275	−	0.279336i	0.935326	+	0.353787i	$0.115106\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.20783e9	+	2.09202e9i	−1.75529	+	3.04025i
$336$		0			0
$337$	3.92128e8	+	6.79186e8i	0.558115	+	0.966683i	0.997654	+	0.0684596i	$0.0218084\pi$
$337$	3.92128e8	+	6.79186e8i	0.558115	+	0.966683i	−0.439539	+	0.898223i	$0.644858\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	2.42927e7			0.0331768
$342$		0			0
$343$	−7.47461e8			−1.00014
$344$		0			0
$345$		0			0
$346$		0			0
$347$	5.32532e8	+	9.22372e8i	0.684215	+	1.18509i	0.973683	+	0.227907i	$0.0731882\pi$
$347$	5.32532e8	+	9.22372e8i	0.684215	+	1.18509i	−0.289468	+	0.957188i	$0.593479\pi$
$348$		0			0
$349$	−3.00319e8	+	5.20168e8i	−0.378176	+	0.655020i	−0.990797	−	0.135357i	$-0.956782\pi$
$349$	−3.00319e8	+	5.20168e8i	−0.378176	+	0.655020i	0.612621	+	0.790377i	$0.290115\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.32650e7	+	4.02962e7i	−0.0281509	+	0.0487588i	−0.879758	−	0.475422i	$-0.842295\pi$
$353$	−2.32650e7	+	4.02962e7i	−0.0281509	+	0.0487588i	0.851607	+	0.524181i	$0.175629\pi$
$354$		0			0
$355$	7.53606e8	+	1.30528e9i	0.894015	+	1.54848i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−8.72929e8			−0.995745			−0.497873	−	0.867250i	$-0.665885\pi$
$359$	−8.72929e8			−0.995745			−0.497873	+	0.867250i	$0.665885\pi$
$360$		0			0
$361$	−8.47634e8			−0.948272
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.35455e9	−	2.34616e9i	−1.45805	−	2.52541i
$366$		0			0
$367$	−2.05422e8	+	3.55801e8i	−0.216928	+	0.375730i	−0.953867	−	0.300229i	$-0.902937\pi$
$367$	−2.05422e8	+	3.55801e8i	−0.216928	+	0.375730i	0.736939	+	0.675959i	$0.236270\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−1.56039e8	+	2.70267e8i	−0.158644	+	0.274779i
$372$		0			0
$373$	4.62104e8	+	8.00387e8i	0.461061	+	0.798582i	0.999014	−	0.0443933i	$-0.0141355\pi$
$373$	4.62104e8	+	8.00387e8i	0.461061	+	0.798582i	−0.537953	+	0.842975i	$0.680802\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−6.01619e8			−0.578265
$378$		0			0
$379$	−1.61945e8			−0.152802			−0.0764011	−	0.997077i	$-0.524343\pi$
$379$	−1.61945e8			−0.152802			−0.0764011	+	0.997077i	$0.524343\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.59303e8	−	7.95537e8i	−0.417738	−	0.723543i	0.577974	−	0.816055i	$-0.303844\pi$
$383$	−4.59303e8	−	7.95537e8i	−0.417738	−	0.723543i	−0.995712	+	0.0925122i	$0.970510\pi$
$384$		0			0
$385$	1.66875e7	−	2.89036e7i	0.0149032	−	0.0258130i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−7.34181e8	+	1.27164e9i	−0.632382	+	1.09532i	0.354681	+	0.934987i	$0.384590\pi$
$389$	−7.34181e8	+	1.27164e9i	−0.632382	+	1.09532i	−0.987063	+	0.160331i	$0.948744\pi$
$390$		0			0
$391$	−2.89749e8	−	5.01860e8i	−0.245134	−	0.424584i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−6.04863e8			−0.493818
$396$		0			0
$397$	7.03759e8			0.564491			0.282246	−	0.959342i	$-0.408921\pi$
$397$	7.03759e8			0.564491			0.282246	+	0.959342i	$0.408921\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−5.94845e8	−	1.03030e9i	−0.460679	−	0.797920i	0.538316	−	0.842743i	$-0.319061\pi$
$401$	−5.94845e8	−	1.03030e9i	−0.460679	−	0.797920i	−0.998995	+	0.0448234i	$0.985727\pi$
$402$		0			0
$403$	−4.11331e8	+	7.12446e8i	−0.313057	+	0.542231i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−1.18056e7	+	2.04479e7i	−0.00867977	+	0.0150338i
$408$		0			0
$409$	4.32491e8	+	7.49096e8i	0.312569	+	0.541385i	0.978918	−	0.204255i	$-0.0654772\pi$
$409$	4.32491e8	+	7.49096e8i	0.312569	+	0.541385i	−0.666349	+	0.745640i	$0.732144\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−1.29754e9			−0.906352
$414$		0			0
$415$	−3.76402e9			−2.58513
$416$		0			0
$417$		0			0
$418$		0			0
$419$	9.39117e8	+	1.62660e9i	0.623692	+	1.08027i	0.988792	+	0.149298i	$0.0477015\pi$
$419$	9.39117e8	+	1.62660e9i	0.623692	+	1.08027i	−0.365100	+	0.930968i	$0.618965\pi$
$420$		0			0
$421$	4.91277e8	−	8.50917e8i	0.320877	−	0.555776i	−0.659792	−	0.751448i	$-0.729356\pi$
$421$	4.91277e8	−	8.50917e8i	0.320877	−	0.555776i	0.980669	+	0.195672i	$0.0626889\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−6.26506e8	+	1.08514e9i	−0.395880	+	0.685685i
$426$		0			0
$427$	8.66109e8	+	1.50014e9i	0.538363	+	0.932471i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.18914e8			−0.0715424			−0.0357712	−	0.999360i	$-0.511389\pi$
$431$	−1.18914e8			−0.0715424			−0.0357712	+	0.999360i	$0.511389\pi$
$432$		0			0
$433$	3.20548e9			1.89751			0.948757	−	0.316008i	$-0.102343\pi$
$433$	3.20548e9			1.89751			0.948757	+	0.316008i	$0.102343\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	3.19544e8	+	5.53466e8i	0.183166	+	0.317253i
$438$		0			0
$439$	1.51354e9	−	2.62153e9i	0.853825	−	1.47887i	−0.0239061	−	0.999714i	$-0.507610\pi$
$439$	1.51354e9	−	2.62153e9i	0.853825	−	1.47887i	0.877731	−	0.479154i	$-0.159056\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	1.12348e9	−	1.94593e9i	0.613978	−	1.06344i	−0.376585	−	0.926382i	$-0.622902\pi$
$443$	1.12348e9	−	1.94593e9i	0.613978	−	1.06344i	0.990563	−	0.137059i	$-0.0437649\pi$
$444$		0			0
$445$	−1.70512e9	−	2.95335e9i	−0.917263	−	1.58875i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.69064e9			−0.881431			−0.440716	−	0.897647i	$-0.645275\pi$
$449$	−1.69064e9			−0.881431			−0.440716	+	0.897647i	$0.645275\pi$
$450$		0			0
$451$	−3.89542e7			−0.0199957
$452$		0			0
$453$		0			0
$454$		0			0
$455$	5.65115e8	+	9.78808e8i	0.281253	+	0.487144i
$456$		0			0
$457$	1.08417e9	−	1.87784e9i	0.531364	−	0.920350i	−0.467966	−	0.883747i	$-0.655013\pi$
$457$	1.08417e9	−	1.87784e9i	0.531364	−	0.920350i	0.999330	−	0.0366030i	$-0.0116537\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	4.28840e8	−	7.42773e8i	0.203865	−	0.353104i	−0.745906	−	0.666052i	$-0.767983\pi$
$461$	4.28840e8	−	7.42773e8i	0.203865	−	0.353104i	0.949770	+	0.312947i	$0.101316\pi$
$462$		0			0
$463$	−1.26464e9	−	2.19042e9i	−0.592153	−	1.02564i	−0.993942	−	0.109907i	$-0.964945\pi$
$463$	−1.26464e9	−	2.19042e9i	−0.592153	−	1.02564i	0.401788	−	0.915733i	$-0.368389\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−1.13072e9			−0.513745			−0.256872	−	0.966445i	$-0.582692\pi$
$467$	−1.13072e9			−0.513745			−0.256872	+	0.966445i	$0.582692\pi$
$468$		0			0
$469$	2.55496e9			1.14361
$470$		0			0
$471$		0			0
$472$		0			0
$473$	4.11734e7	+	7.13145e7i	0.0178897	+	0.0309859i
$474$		0			0
$475$	6.90930e8	−	1.19673e9i	0.295806	−	0.512350i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	1.70634e8	−	2.95546e8i	0.0709399	−	0.122872i	−0.828374	−	0.560176i	$-0.810733\pi$
$479$	1.70634e8	−	2.95546e8i	0.0709399	−	0.122872i	0.899314	+	0.437304i	$0.144067\pi$
$480$		0			0
$481$	−3.99793e8	−	6.92461e8i	−0.163805	−	0.283718i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	2.21138e9			0.880171
$486$		0			0
$487$	−3.05349e9			−1.19797			−0.598984	−	0.800761i	$-0.704429\pi$
$487$	−3.05349e9			−1.19797			−0.598984	+	0.800761i	$0.704429\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	2.20646e9	+	3.82170e9i	0.841222	+	1.45704i	0.888862	+	0.458175i	$0.151497\pi$
$491$	2.20646e9	+	3.82170e9i	0.841222	+	1.45704i	−0.0476396	+	0.998865i	$0.515170\pi$
$492$		0			0
$493$	−4.88317e8	+	8.45791e8i	−0.183543	+	0.317906i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	7.97064e8	−	1.38056e9i	0.291237	−	0.504436i
$498$		0			0
$499$	2.52329e9	+	4.37047e9i	0.909108	+	1.57462i	0.815306	+	0.579031i	$0.196569\pi$
$499$	2.52329e9	+	4.37047e9i	0.909108	+	1.57462i	0.0938023	+	0.995591i	$0.470098\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	3.84853e9			1.34836			0.674182	−	0.738565i	$-0.264496\pi$
$503$	3.84853e9			1.34836			0.674182	+	0.738565i	$0.264496\pi$
$504$		0			0
$505$	6.68399e8			0.230949
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−2.09964e9	−	3.63668e9i	−0.705720	−	1.22234i	−0.966431	−	0.256925i	$-0.917291\pi$
$509$	−2.09964e9	−	3.63668e9i	−0.705720	−	1.22234i	0.260712	−	0.965417i	$-0.416043\pi$
$510$		0			0
$511$	−1.43267e9	+	2.48145e9i	−0.474977	+	0.822684i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−1.83178e8	+	3.17274e8i	−0.0590947	+	0.102355i
$516$		0			0
$517$	9.48285e6	+	1.64248e7i	0.00301802	+	0.00522736i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	3.08327e9			0.955168			0.477584	−	0.878586i	$-0.341513\pi$
$521$	3.08327e9			0.955168			0.477584	+	0.878586i	$0.341513\pi$
$522$		0			0
$523$	1.03577e9			0.316597			0.158298	−	0.987391i	$-0.449399\pi$
$523$	1.03577e9			0.316597			0.158298	+	0.987391i	$0.449399\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	6.67731e8	+	1.15654e9i	0.198730	+	0.344211i
$528$		0			0
$529$	−2.71423e9	+	4.70118e9i	−0.797171	+	1.38074i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.59584e8	−	1.14243e9i	0.188680	−	0.326803i
$534$		0			0
$535$	−4.30270e9	−	7.45250e9i	−1.21480	−	2.10409i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	5.70681e7			0.0156976
$540$		0			0
$541$	5.48956e9			1.49055			0.745276	−	0.666756i	$-0.232318\pi$
$541$	5.48956e9			1.49055			0.745276	+	0.666756i	$0.232318\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−3.55923e9	−	6.16476e9i	−0.941820	−	1.63128i
$546$		0			0
$547$	−3.09691e9	+	5.36400e9i	−0.809044	+	1.40131i	0.104482	+	0.994527i	$0.466681\pi$
$547$	−3.09691e9	+	5.36400e9i	−0.809044	+	1.40131i	−0.913527	+	0.406779i	$0.866652\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	5.38532e8	−	9.32764e8i	0.137145	−	0.237542i
$552$		0			0
$553$	3.19872e8	+	5.54034e8i	0.0804337	+	0.139315i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	7.12914e8			0.174801			0.0874005	−	0.996173i	$-0.472144\pi$
$557$	7.12914e8			0.174801			0.0874005	+	0.996173i	$0.472144\pi$
$558$		0			0
$559$	−2.78864e9			−0.675230
$560$		0			0
$561$		0			0
$562$		0			0
$563$	1.73450e9	+	3.00425e9i	0.409634	+	0.709507i	0.994849	−	0.101371i	$-0.0323230\pi$
$563$	1.73450e9	+	3.00425e9i	0.409634	+	0.709507i	−0.585215	+	0.810879i	$0.698990\pi$
$564$		0			0
$565$	−2.95686e9	+	5.12144e9i	−0.689702	+	1.19460i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.12816e9	+	1.95403e9i	−0.256731	+	0.444671i	−0.965364	−	0.260906i	$-0.915979\pi$
$569$	−1.12816e9	+	1.95403e9i	−0.256731	+	0.444671i	0.708633	+	0.705577i	$0.249312\pi$
$570$		0			0
$571$	4.19749e9	+	7.27027e9i	0.943547	+	1.63427i	0.758634	+	0.651517i	$0.225867\pi$
$571$	4.19749e9	+	7.27027e9i	0.943547	+	1.63427i	0.184913	+	0.982755i	$0.440800\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	1.90997e10			4.18975
$576$		0			0
$577$	8.82809e8			0.191316			0.0956581	−	0.995414i	$-0.469504\pi$
$577$	8.82809e8			0.191316			0.0956581	+	0.995414i	$0.469504\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	1.99054e9	+	3.44771e9i	0.421070	+	0.729314i
$582$		0			0
$583$	3.11959e7	−	5.40330e7i	0.00652016	−	0.0112932i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	1.23978e9	−	2.14736e9i	0.252995	−	0.438200i	−0.711354	−	0.702834i	$-0.751918\pi$
$587$	1.23978e9	−	2.14736e9i	0.252995	−	0.438200i	0.964349	+	0.264634i	$0.0852511\pi$
$588$		0			0
$589$	−7.36395e8	−	1.27547e9i	−0.148493	−	0.257198i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−1.79770e9			−0.354018			−0.177009	−	0.984209i	$-0.556642\pi$
$593$	−1.79770e9			−0.354018			−0.177009	+	0.984209i	$0.556642\pi$
$594$		0			0
$595$	1.83475e9			0.357082
$596$		0			0
$597$		0			0
$598$		0			0
$599$	6.00964e8	+	1.04090e9i	0.114250	+	0.197886i	0.917480	−	0.397783i	$-0.130220\pi$
$599$	6.00964e8	+	1.04090e9i	0.114250	+	0.197886i	−0.803230	+	0.595669i	$0.796887\pi$
$600$		0			0
$601$	−5.77215e8	+	9.99766e8i	−0.108462	+	0.187861i	−0.915147	−	0.403120i	$-0.867926\pi$
$601$	−5.77215e8	+	9.99766e8i	−0.108462	+	0.187861i	0.806685	+	0.590981i	$0.201259\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	5.16485e9	−	8.94577e9i	0.948230	−	1.64238i
$606$		0			0
$607$	−1.65636e9	−	2.86890e9i	−0.300604	−	0.520661i	0.675669	−	0.737205i	$-0.263855\pi$
$607$	−1.65636e9	−	2.86890e9i	−0.300604	−	0.520661i	−0.976273	+	0.216544i	$0.930522\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−6.42266e8			−0.113912
$612$		0			0
$613$	2.19911e9			0.385598			0.192799	−	0.981238i	$-0.438244\pi$
$613$	2.19911e9			0.385598			0.192799	+	0.981238i	$0.438244\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.38453e9	−	2.39807e9i	−0.237303	−	0.411022i	0.722636	−	0.691229i	$-0.242930\pi$
$617$	−1.38453e9	−	2.39807e9i	−0.237303	−	0.411022i	−0.959940	+	0.280207i	$0.909597\pi$
$618$		0			0
$619$	3.20312e9	−	5.54797e9i	0.542821	−	0.940193i	−0.455920	−	0.890021i	$-0.650690\pi$
$619$	3.20312e9	−	5.54797e9i	0.542821	−	0.940193i	0.998741	−	0.0501720i	$-0.0159769\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−1.80345e9	+	3.12366e9i	−0.298810	+	0.517554i
$624$		0			0
$625$	−9.65900e9	−	1.67299e10i	−1.58253	−	2.74102i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−1.29800e9			−0.207969
$630$		0			0
$631$	3.86848e9			0.612968			0.306484	−	0.951876i	$-0.400847\pi$
$631$	3.86848e9			0.612968			0.306484	+	0.951876i	$0.400847\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	7.14316e9	+	1.23723e10i	1.10709	+	1.91753i
$636$		0			0
$637$	−9.66295e8	+	1.67367e9i	−0.148123	+	0.256556i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	5.36228e9	−	9.28774e9i	0.804168	−	1.39286i	−0.112684	−	0.993631i	$-0.535945\pi$
$641$	5.36228e9	−	9.28774e9i	0.804168	−	1.39286i	0.916852	−	0.399228i	$-0.130722\pi$
$642$		0			0
$643$	2.52124e9	+	4.36691e9i	0.374003	+	0.647792i	0.990177	−	0.139818i	$-0.0446516\pi$
$643$	2.52124e9	+	4.36691e9i	0.374003	+	0.647792i	−0.616174	+	0.787610i	$0.711318\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	3.76257e9			0.546160			0.273080	−	0.961991i	$-0.411958\pi$
$647$	3.76257e9			0.546160			0.273080	+	0.961991i	$0.411958\pi$
$648$		0			0
$649$	2.59411e8			0.0372505
$650$		0			0
$651$		0			0
$652$		0			0
$653$	2.14128e9	+	3.70881e9i	0.300939	+	0.521241i	0.976349	−	0.216201i	$-0.0693667\pi$
$653$	2.14128e9	+	3.70881e9i	0.300939	+	0.521241i	−0.675410	+	0.737442i	$0.736033\pi$
$654$		0			0
$655$	9.81524e9	−	1.70005e10i	1.36476	−	2.36384i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	2.85125e9	−	4.93851e9i	0.388093	−	0.672197i	−0.604100	−	0.796909i	$-0.706467\pi$
$659$	2.85125e9	−	4.93851e9i	0.388093	−	0.672197i	0.992193	+	0.124711i	$0.0398005\pi$
$660$		0			0
$661$	−5.54124e8	−	9.59771e8i	−0.0746280	−	0.129260i	0.826296	−	0.563235i	$-0.190444\pi$
$661$	−5.54124e8	−	9.59771e8i	−0.0746280	−	0.129260i	−0.900924	+	0.433976i	$0.857110\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−2.02342e9			−0.266815
$666$		0			0
$667$	1.48868e10			1.94251
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.73156e8	−	2.99916e8i	−0.0221263	−	0.0383240i
$672$		0			0
$673$	−2.84822e9	+	4.93326e9i	−0.360181	+	0.623852i	−0.987990	−	0.154515i	$-0.950618\pi$
$673$	−2.84822e9	+	4.93326e9i	−0.360181	+	0.623852i	0.627809	+	0.778367i	$0.283952\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.88175e8	−	1.36516e9i	0.0976253	−	0.169092i	−0.813076	−	0.582158i	$-0.802209\pi$
$677$	7.88175e8	−	1.36516e9i	0.0976253	−	0.169092i	0.910701	+	0.413066i	$0.135542\pi$
$678$		0			0
$679$	−1.16945e9	−	2.02555e9i	−0.143363	−	0.248313i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	9.63957e9			1.15767			0.578836	−	0.815444i	$-0.303507\pi$
$683$	9.63957e9			1.15767			0.578836	+	0.815444i	$0.303507\pi$
$684$		0			0
$685$	−1.26518e10			−1.50395
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.05644e9	+	1.82980e9i	0.123049	+	0.213126i
$690$		0			0
$691$	−2.78636e9	+	4.82612e9i	−0.321265	+	0.556448i	−0.980749	−	0.195271i	$-0.937441\pi$
$691$	−2.78636e9	+	4.82612e9i	−0.321265	+	0.556448i	0.659484	+	0.751719i	$0.270775\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.13472e10	+	1.96539e10i	−1.28216	+	2.22076i
$696$		0			0
$697$	−1.07073e9	−	1.85456e9i	−0.119775	−	0.207456i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−1.29431e10			−1.41914			−0.709572	−	0.704633i	$-0.751112\pi$
$701$	−1.29431e10			−1.41914			−0.709572	+	0.704633i	$0.751112\pi$
$702$		0			0
$703$	1.43148e9			0.155396
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−3.53472e8	−	6.12232e8i	−0.0376173	−	0.0651550i
$708$		0			0
$709$	−3.57029e9	+	6.18393e9i	−0.376220	+	0.651633i	−0.990509	−	0.137449i	$-0.956110\pi$
$709$	−3.57029e9	+	6.18393e9i	−0.376220	+	0.651633i	0.614289	+	0.789081i	$0.289443\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.01782e10	−	1.76292e10i	1.05162	−	1.82146i
$714$		0			0
$715$	−1.12980e8	−	1.95688e8i	−0.0115593	−	0.0200213i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−8.00524e9			−0.803200			−0.401600	−	0.915815i	$-0.631546\pi$
$719$	−8.00524e9			−0.803200			−0.401600	+	0.915815i	$0.631546\pi$
$720$		0			0
$721$	3.87483e8			0.0385016
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−1.60945e10	−	2.78764e10i	−1.56853	−	2.71678i
$726$		0			0
$727$	2.73539e9	−	4.73783e9i	0.264027	−	0.457308i	−0.703281	−	0.710912i	$-0.748283\pi$
$727$	2.73539e9	−	4.73783e9i	0.264027	−	0.457308i	0.967308	+	0.253604i	$0.0816159\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−2.26346e9	+	3.92043e9i	−0.214320	+	0.371213i
$732$		0			0
$733$	4.93662e8	+	8.55048e8i	0.0462984	+	0.0801912i	0.888246	−	0.459368i	$-0.151924\pi$
$733$	4.93662e8	+	8.55048e8i	0.0462984	+	0.0801912i	−0.841948	+	0.539559i	$0.818591\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−5.10800e8			−0.0470018
$738$		0			0
$739$	−2.25070e9			−0.205145			−0.102573	−	0.994726i	$-0.532707\pi$
$739$	−2.25070e9			−0.205145			−0.102573	+	0.994726i	$0.532707\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	2.88579e9	+	4.99833e9i	0.258109	+	0.447058i	0.965735	−	0.259529i	$-0.0835672\pi$
$743$	2.88579e9	+	4.99833e9i	0.258109	+	0.447058i	−0.707626	+	0.706587i	$0.750234\pi$
$744$		0			0
$745$	−5.59344e9	+	9.68812e9i	−0.495601	+	0.858405i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−4.55083e9	+	7.88227e9i	−0.395735	+	0.685433i
$750$		0			0
$751$	9.49910e9	+	1.64529e10i	0.818356	+	1.41743i	0.906893	+	0.421362i	$0.138448\pi$
$751$	9.49910e9	+	1.64529e10i	0.818356	+	1.41743i	−0.0885361	+	0.996073i	$0.528219\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	1.29988e10			1.09924
$756$		0			0
$757$	1.01006e9			0.0846274			0.0423137	−	0.999104i	$-0.486527\pi$
$757$	1.01006e9			0.0846274			0.0423137	+	0.999104i	$0.486527\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	1.11029e8	+	1.92307e8i	0.00913247	+	0.0158179i	0.870556	−	0.492070i	$-0.163760\pi$
$761$	1.11029e8	+	1.92307e8i	0.00913247	+	0.0158179i	−0.861423	+	0.507888i	$0.830426\pi$
$762$		0			0
$763$	−3.76448e9	+	6.52026e9i	−0.306809	+	0.531409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−4.39243e9	+	7.60790e9i	−0.351496	+	0.608809i
$768$		0			0
$769$	1.22184e10	+	2.11629e10i	0.968883	+	1.67815i	0.698799	+	0.715318i	$0.253718\pi$
$769$	1.22184e10	+	2.11629e10i	0.968883	+	1.67815i	0.270084	+	0.962837i	$0.412949\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.73556e9			−0.602370			−0.301185	−	0.953566i	$-0.597382\pi$
$773$	−7.73556e9			−0.602370			−0.301185	+	0.953566i	$0.597382\pi$
$774$		0			0
$775$	−4.40155e10			−3.39664
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.18084e9	+	2.04527e9i	0.0894970	+	0.155013i
$780$		0			0
$781$	−1.59353e8	+	2.76007e8i	−0.0119696	+	0.0207320i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.74342e10	+	3.01969e10i	−1.28635	+	2.22802i
$786$		0			0
$787$	7.04709e9	+	1.22059e10i	0.515345	+	0.892604i	0.999841	+	0.0178108i	$0.00566964\pi$
$787$	7.04709e9	+	1.22059e10i	0.515345	+	0.892604i	−0.484496	+	0.874793i	$0.660997\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.25475e9			0.449358
$792$		0			0
$793$	1.17277e10			0.835138
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−4.43678e9	−	7.68472e9i	−0.310430	−	0.537680i	0.668026	−	0.744138i	$-0.267140\pi$
$797$	−4.43678e9	−	7.68472e9i	−0.310430	−	0.537680i	−0.978455	+	0.206458i	$0.933806\pi$
$798$		0			0
$799$	−5.21309e8	+	9.02934e8i	−0.0361561	+	0.0626242i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	2.86425e8	−	4.96103e8i	0.0195212	−	0.0338118i
$804$		0			0
$805$	−1.39836e10	−	2.42202e10i	−0.944784	−	1.63641i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−2.40258e10			−1.59536			−0.797678	−	0.603084i	$-0.793939\pi$
$809$	−2.40258e10			−1.59536			−0.797678	+	0.603084i	$0.793939\pi$
$810$		0			0
$811$	1.48835e9			0.0979789			0.0489895	−	0.998799i	$-0.484400\pi$
$811$	1.48835e9			0.0979789			0.0489895	+	0.998799i	$0.484400\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.78694e9	−	1.00233e10i	−0.374453	−	0.648571i
$816$		0			0
$817$	2.49622e9	−	4.32358e9i	0.160142	−	0.277374i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−4.96107e9	+	8.59283e9i	−0.312878	+	0.541920i	−0.978984	−	0.203937i	$-0.934626\pi$
$821$	−4.96107e9	+	8.59283e9i	−0.312878	+	0.541920i	0.666106	+	0.745857i	$0.267960\pi$
$822$		0			0
$823$	9.64630e9	+	1.67079e10i	0.603200	+	1.04477i	0.992333	+	0.123592i	$0.0394413\pi$
$823$	9.64630e9	+	1.67079e10i	0.603200	+	1.04477i	−0.389133	+	0.921182i	$0.627225\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−2.36031e10			−1.45111			−0.725555	−	0.688164i	$-0.758417\pi$
$827$	−2.36031e10			−1.45111			−0.725555	+	0.688164i	$0.758417\pi$
$828$		0			0
$829$	5.81223e9			0.354325			0.177162	−	0.984182i	$-0.443308\pi$
$829$	5.81223e9			0.354325			0.177162	+	0.984182i	$0.443308\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	1.56863e9	+	2.71694e9i	0.0940292	+	0.162863i
$834$		0			0
$835$	2.22372e10	−	3.85160e10i	1.32184	−	2.28949i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.03343e10	−	1.78996e10i	0.604109	−	1.04635i	−0.388083	−	0.921625i	$-0.626863\pi$
$839$	1.03343e10	−	1.78996e10i	0.604109	−	1.04635i	0.992192	−	0.124723i	$-0.0398041\pi$
$840$		0			0
$841$	−3.91957e9	−	6.78889e9i	−0.227223	−	0.393562i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.56309e10			−1.46139
$846$		0			0
$847$	−1.09254e10			−0.617795
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.89272e9	+	1.71347e10i	0.550253	+	0.953066i
$852$		0			0
$853$	9.10145e9	−	1.57642e10i	0.502098	−	0.869660i	−0.497899	−	0.867235i	$-0.665895\pi$
$853$	9.10145e9	−	1.57642e10i	0.502098	−	0.869660i	0.999997	−	0.00242465i	$-0.000771789\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	6.42653e9	−	1.11311e10i	0.348774	−	0.604094i	−0.637258	−	0.770650i	$-0.719932\pi$
$857$	6.42653e9	−	1.11311e10i	0.348774	−	0.604094i	0.986032	+	0.166557i	$0.0532649\pi$
$858$		0			0
$859$	−9.90150e8	−	1.71499e9i	−0.0532997	−	0.0923179i	0.838145	−	0.545448i	$-0.183641\pi$
$859$	−9.90150e8	−	1.71499e9i	−0.0532997	−	0.0923179i	−0.891444	+	0.453130i	$0.850307\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.28817e10			−0.682238			−0.341119	−	0.940020i	$-0.610806\pi$
$863$	−1.28817e10			−0.682238			−0.341119	+	0.940020i	$0.610806\pi$
$864$		0			0
$865$	−5.89579e9			−0.309732
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−6.39502e7	−	1.10765e8i	−0.00330577	−	0.00572577i
$870$		0			0
$871$	8.64902e9	−	1.49805e10i	0.443510	−	0.768182i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−1.86120e10	+	3.22370e10i	−0.939216	+	1.62677i
$876$		0			0
$877$	1.25724e10	+	2.17761e10i	0.629390	+	1.09014i	0.987674	+	0.156523i	$0.0500287\pi$
$877$	1.25724e10	+	2.17761e10i	0.629390	+	1.09014i	−0.358284	+	0.933613i	$0.616638\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	4.62474e9			0.227862			0.113931	−	0.993489i	$-0.463656\pi$
$881$	4.62474e9			0.227862			0.113931	+	0.993489i	$0.463656\pi$
$882$		0			0
$883$	−1.69814e10			−0.830062			−0.415031	−	0.909807i	$-0.636229\pi$
$883$	−1.69814e10			−0.830062			−0.415031	+	0.909807i	$0.636229\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	1.69893e10	+	2.94264e10i	0.817417	+	1.41581i	0.907579	+	0.419881i	$0.137928\pi$
$887$	1.69893e10	+	2.94264e10i	0.817417	+	1.41581i	−0.0901620	+	0.995927i	$0.528738\pi$
$888$		0			0
$889$	7.55508e9	−	1.30858e10i	0.360648	−	0.624660i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.74916e8	−	9.95783e8i	0.0270162	−	0.0467934i
$894$		0			0
$895$	−1.89870e10	−	3.28864e10i	−0.885268	−	1.53333i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−3.43070e10			−1.57479
$900$		0			0
$901$	3.42992e9			0.156224
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.43275e10	−	5.94570e10i	−1.53947	−	2.66645i
$906$		0			0
$907$	−8.04718e9	+	1.39381e10i	−0.358111	+	0.620267i	−0.987645	−	0.156706i	$-0.949913\pi$
$907$	−8.04718e9	+	1.39381e10i	−0.358111	+	0.620267i	0.629534	+	0.776973i	$0.283246\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.17471e10	−	2.03466e10i	0.514773	−	0.891614i	−0.485080	−	0.874470i	$-0.661209\pi$
$911$	1.17471e10	−	2.03466e10i	0.514773	−	0.891614i	0.999853	−	0.0171438i	$-0.00545731\pi$
$912$		0			0
$913$	−3.97957e8	−	6.89282e8i	−0.0173057	−	0.0299743i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.07625e10			−0.889175
$918$		0			0
$919$	2.84798e10			1.21041			0.605206	−	0.796069i	$-0.293091\pi$
$919$	2.84798e10			1.21041			0.605206	+	0.796069i	$0.293091\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−5.39641e9	−	9.34686e9i	−0.225891	−	0.391255i
$924$		0			0
$925$	2.13904e10	−	3.70493e10i	0.888635	−	1.53916i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−3.77657e9	+	6.54121e9i	−0.154540	+	0.267672i	−0.932892	−	0.360157i	$-0.882723\pi$
$929$	−3.77657e9	+	6.54121e9i	−0.154540	+	0.267672i	0.778351	+	0.627829i	$0.216056\pi$
$930$		0			0
$931$	−1.72993e9	−	2.99633e9i	−0.0702595	−	0.121693i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−3.66812e8			−0.0146758
$936$		0			0
$937$	3.71519e10			1.47534			0.737670	−	0.675161i	$-0.235926\pi$
$937$	3.71519e10			1.47534			0.737670	+	0.675161i	$0.235926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1.80772e9	+	3.13106e9i	0.0707241	+	0.122498i	0.899219	−	0.437499i	$-0.144136\pi$
$941$	1.80772e9	+	3.13106e9i	0.0707241	+	0.122498i	−0.828495	+	0.559997i	$0.810802\pi$
$942$		0			0
$943$	−1.63212e10	+	2.82691e10i	−0.633812	+	1.09779i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4.28609e9	−	7.42373e9i	0.163997	−	0.284052i	−0.772301	−	0.635256i	$-0.780894\pi$
$947$	4.28609e9	−	7.42373e9i	0.163997	−	0.284052i	0.936299	+	0.351205i	$0.114228\pi$
$948$		0			0
$949$	9.69968e9	+	1.68003e10i	0.368405	+	0.638097i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−2.03359e10			−0.761093			−0.380546	−	0.924762i	$-0.624264\pi$
$953$	−2.03359e10			−0.761093			−0.380546	+	0.924762i	$0.624264\pi$
$954$		0			0
$955$	−9.02455e10			−3.35285
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.69068e9	+	1.15886e10i	0.244965	+	0.424293i
$960$		0			0
$961$	−9.69960e9	+	1.68002e10i	−0.352551	+	0.610636i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.07884e10	+	3.60066e10i	−0.744690	+	1.28984i
$966$		0			0
$967$	2.03516e10	+	3.52500e10i	0.723778	+	1.25362i	0.959475	+	0.281794i	$0.0909293\pi$
$967$	2.03516e10	+	3.52500e10i	0.723778	+	1.25362i	−0.235697	+	0.971827i	$0.575737\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	2.44101e10			0.855662			0.427831	−	0.903859i	$-0.359278\pi$
$971$	2.44101e10			0.855662			0.427831	+	0.903859i	$0.359278\pi$
$972$		0			0
$973$	2.40031e10			0.835358
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.08350e10	+	1.87668e10i	0.371705	+	0.643813i	0.989828	−	0.142269i	$-0.0454398\pi$
$977$	1.08350e10	+	1.87668e10i	0.371705	+	0.643813i	−0.618123	+	0.786082i	$0.712107\pi$
$978$		0			0
$979$	3.60553e8	−	6.24496e8i	0.0122809	−	0.0212711i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−1.21325e10	+	2.10140e10i	−0.407391	+	0.705622i	−0.994597	−	0.103816i	$-0.966895\pi$
$983$	−1.21325e10	+	2.10140e10i	−0.407391	+	0.705622i	0.587206	+	0.809438i	$0.300228\pi$
$984$		0			0
$985$	−8.65782e9	−	1.49958e10i	−0.288657	−	0.499968i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	6.90040e10			2.26823
$990$		0			0
$991$	−1.19297e10			−0.389378			−0.194689	−	0.980865i	$-0.562370\pi$
$991$	−1.19297e10			−0.389378			−0.194689	+	0.980865i	$0.562370\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−1.06457e9	−	1.84389e9i	−0.0342605	−	0.0593410i
$996$		0			0
$997$	−2.70891e10	+	4.69197e10i	−0.865689	+	1.49942i	0.000672287		1.00000i	$0.499786\pi$
$997$	−2.70891e10	+	4.69197e10i	−0.865689	+	1.49942i	−0.866361	+	0.499418i	$0.833547\pi$
$998$		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.217.1		16
3.2	odd	2	inner	324.8.e.m.217.8		16
9.2	odd	6		324.8.a.e.1.1	✓	8
9.4	even	3	inner	324.8.e.m.109.1		16
9.5	odd	6	inner	324.8.e.m.109.8		16
9.7	even	3		324.8.a.e.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.8.a.e.1.1	✓	8	9.2	odd	6
324.8.a.e.1.8	yes	8	9.7	even	3
324.8.e.m.109.1		16	9.4	even	3	inner
324.8.e.m.109.8		16	9.5	odd	6	inner
324.8.e.m.217.1		16	1.1	even	1	trivial
324.8.e.m.217.8		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)

\(f(q)\)	\(=\)	\(q+(-265.209 - 459.356i) q^{5} +(-280.503 + 485.846i) q^{7} +O(q^{10})\)	\(q+(-265.209 - 459.356i) q^{5} +(-280.503 + 485.846i) q^{7} +(56.0795 - 97.1325i) q^{11} +(1899.11 + 3289.35i) q^{13} +6165.82 q^{17} -6799.85 q^{19} +(-46992.8 - 81393.9i) q^{23} +(-101610. + 175993. i) q^{25} +(-79197.6 + 137174. i) q^{29} +(108296. + 187574. i) q^{31} +297568. q^{35} -210516. q^{37} +(-173656. - 300781. i) q^{41} +(-367099. + 635834. i) q^{43} +(-84548.3 + 146442. i) q^{47} +(254407. + 440646. i) q^{49} +556281. q^{53} -59491.2 q^{55} +(1.15644e6 + 2.00302e6i) q^{59} +(1.54385e6 - 2.67402e6i) q^{61} +(1.00732e6 - 1.74474e6i) q^{65} +(-2.27712e6 - 3.94410e6i) q^{67} -2.84155e6 q^{71} +5.10749e6 q^{73} +(31461.0 + 54492.0i) q^{77} +(570175. - 987572. i) q^{79} +(3.54815e6 - 6.14558e6i) q^{83} +(-1.63523e6 - 2.83231e6i) q^{85} +6.42932e6 q^{89} -2.13083e6 q^{91} +(1.80338e6 + 3.12355e6i) q^{95} +(-2.08456e6 + 3.61056e6i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 560 q^{7}+O(q^{10}) \) 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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Modular forms

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