LMFDB - Embedded newform 324.8.e.m.109.8

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			109.8
Root			$18.0990 - 31.3483i$ of defining polynomial
Character	$\chi$	$=$	324.109
Dual form			324.8.e.m.217.8

$q$-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 $\left(\mathbb{Z}/324\mathbb{Z}\right)^\times$.

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

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
$5$	265.209	−	459.356i	0.948842	−	1.64344i	0.200973	−	0.979597i	$-0.435590\pi$
$5$	265.209	−	459.356i	0.948842	−	1.64344i	0.747869	−	0.663846i	$-0.231077\pi$
$6$		0			0
$7$	−280.503	−	485.846i	−0.309097	−	0.535372i	0.669068	−	0.743201i	$-0.266693\pi$
$7$	−280.503	−	485.846i	−0.309097	−	0.535372i	−0.978165	+	0.207829i	$0.933360\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−56.0795	−	97.1325i	−0.0127037	−	0.0220034i	0.859604	−	0.510961i	$-0.170710\pi$
$11$	−56.0795	−	97.1325i	−0.0127037	−	0.0220034i	−0.872307	+	0.488958i	$0.837377\pi$
$12$		0			0
$13$	1899.11	−	3289.35i	0.239744	−	0.415249i	−0.720897	−	0.693043i	$-0.756270\pi$
$13$	1899.11	−	3289.35i	0.239744	−	0.415249i	0.960641	+	0.277793i	$0.0896031\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−6165.82			−0.304382			−0.152191	−	0.988351i	$-0.548633\pi$
$17$	−6165.82			−0.304382			−0.152191	+	0.988351i	$0.548633\pi$
$18$		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	−0.110708	−	0.993853i	$-0.535312\pi$
$23$	46992.8	−	81393.9i	0.805348	−	1.39490i	0.916056	−	0.401051i	$-0.131355\pi$
$24$		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	0.992364	+	0.123345i	$0.0393622\pi$
$29$	79197.6	+	137174.i	0.603002	+	1.04443i	−0.389362	+	0.921085i	$0.627304\pi$
$30$		0			0
$31$	108296.	−	187574.i	0.652898	−	1.13085i	−0.329519	−	0.944149i	$-0.606886\pi$
$31$	108296.	−	187574.i	0.652898	−	1.13085i	0.982416	−	0.186703i	$-0.0597802\pi$
$32$		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	$-0.704597\pi$
$41$	173656.	−	300781.i	0.393502	−	0.681565i	0.992909	+	0.118879i	$0.0379302\pi$
$42$		0			0
$43$	−367099.	−	635834.i	−0.704115	−	1.21956i	−0.967010	−	0.254738i	$-0.918011\pi$
$43$	−367099.	−	635834.i	−0.704115	−	1.21956i	0.262895	−	0.964824i	$-0.415323\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	84548.3	+	146442.i	0.118785	+	0.205742i	0.919287	−	0.393589i	$-0.128767\pi$
$47$	84548.3	+	146442.i	0.118785	+	0.205742i	−0.800501	+	0.599331i	$0.795433\pi$
$48$		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	$-0.582610\pi$
$53$	−556281.			−0.513249			−0.256625	+	0.966511i	$0.582610\pi$
$54$		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	0.222502	+	0.974932i	$0.428577\pi$
$59$	−1.15644e6	+	2.00302e6i	−0.733065	+	1.26971i	−0.955567	+	0.294773i	$0.904756\pi$
$60$		0			0
$61$	1.54385e6	+	2.67402e6i	0.870863	+	1.50838i	0.861105	+	0.508427i	$0.169773\pi$
$61$	1.54385e6	+	2.67402e6i	0.870863	+	1.50838i	0.00975860	+	0.999952i	$0.496894\pi$
$62$		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	−0.133346	+	0.991070i	$0.542572\pi$
$67$	−2.27712e6	+	3.94410e6i	−0.924964	+	1.60209i	−0.791619	+	0.611016i	$0.790761\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	2.84155e6			0.942217			0.471109	−	0.882075i	$-0.343854\pi$
$71$	2.84155e6			0.942217			0.471109	+	0.882075i	$0.343854\pi$
$72$		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	0.923719	−	0.383070i	$-0.125133\pi$
$79$	570175.	+	987572.i	0.130111	+	0.225359i	−0.793608	+	0.608429i	$0.791800\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−3.54815e6	−	6.14558e6i	−0.681129	−	1.17975i	−0.974637	−	0.223793i	$-0.928156\pi$
$83$	−3.54815e6	−	6.14558e6i	−0.681129	−	1.17975i	0.293508	−	0.955957i	$-0.405177\pi$
$84$		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	$-0.660584\pi$
$89$	−6.42932e6			−0.966719			−0.483359	+	0.875422i	$0.660584\pi$
$90$		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	0.726462	−	0.687206i	$-0.241163\pi$
$97$	−2.08456e6	−	3.61056e6i	−0.231907	−	0.401674i	−0.958369	+	0.285532i	$0.907830\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	630068.	+	1.09131e6i	0.0608503	+	0.105396i	0.894846	−	0.446376i	$-0.147285\pi$
$101$	630068.	+	1.09131e6i	0.0608503	+	0.105396i	−0.833995	+	0.551771i	$0.813952\pi$
$102$		0			0
$103$	−345346.	+	598157.i	−0.0311404	+	0.0539368i	−0.881176	−	0.472789i	$-0.843247\pi$
$103$	−345346.	+	598157.i	−0.0311404	+	0.0539368i	0.850035	+	0.526726i	$0.176581\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−1.62238e7			−1.28029			−0.640147	−	0.768253i	$-0.721126\pi$
$107$	−1.62238e7			−1.28029			−0.640147	+	0.768253i	$0.721126\pi$
$108$		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	$-0.714934\pi$
$113$	5.57458e6	−	9.65546e6i	0.363444	−	0.629504i	0.988525	+	0.151056i	$0.0482674\pi$
$114$		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	0.242157	+	0.970237i	$0.422145\pi$
$131$	−1.85047e7	+	3.20511e7i	−0.719172	+	1.24564i	−0.961328	+	0.275405i	$0.911188\pi$
$132$		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	$-0.296359\pi$
$137$	−1.19262e7	−	2.06568e7i	−0.396260	−	0.686342i	−0.993261	+	0.115898i	$0.963025\pi$
$138$		0			0
$139$	−2.13929e7	+	3.70536e7i	−0.675644	+	1.17025i	0.300636	+	0.953739i	$0.402801\pi$
$139$	−2.13929e7	+	3.70536e7i	−0.675644	+	1.17025i	−0.976280	+	0.216511i	$0.930532\pi$
$140$		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	$-0.749228\pi$
$149$	1.05453e7	−	1.82650e7i	0.261161	−	0.452344i	0.966551	+	0.256476i	$0.0825614\pi$
$150$		0			0
$151$	−1.22534e7	−	2.12235e7i	−0.289625	−	0.501646i	0.684095	−	0.729393i	$-0.260197\pi$
$151$	−1.22534e7	−	2.12235e7i	−0.289625	−	0.501646i	−0.973720	+	0.227747i	$0.926864\pi$
$152$		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	0.297776	+	0.954636i	$0.403755\pi$
$157$	−3.28687e7	+	5.69303e7i	−0.677851	+	1.17407i	−0.975627	+	0.219436i	$0.929578\pi$
$158$		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	0.273102	+	0.961985i	$0.411950\pi$
$167$	−4.19239e7	+	7.26143e7i	−0.696552	+	1.20646i	−0.969655	+	0.244479i	$0.921383\pi$
$168$		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	$-0.192672\pi$
$173$	−5.55767e6	−	9.62617e6i	−0.0816078	−	0.141349i	−0.903941	+	0.427658i	$0.859339\pi$
$174$		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	$-0.654485\pi$
$179$	−7.15923e7			−0.932998			−0.466499	+	0.884522i	$0.654485\pi$
$180$		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	−0.847531	−	0.530745i	$-0.821912\pi$
$191$	−8.50700e7	−	1.47346e8i	−0.883405	−	1.53010i	−0.0358732	−	0.999356i	$-0.511421\pi$
$192$		0			0
$193$	−3.91925e7	+	6.78833e7i	−0.392421	+	0.679692i	−0.992768	−	0.120047i	$-0.961696\pi$
$193$	−3.91925e7	+	6.78833e7i	−0.392421	+	0.679692i	0.600348	+	0.799739i	$0.295029\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−3.26452e7			−0.304220			−0.152110	−	0.988364i	$-0.548607\pi$
$197$	−3.26452e7			−0.304220			−0.152110	+	0.988364i	$0.548607\pi$
$198$		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	−0.472740	−	0.881202i	$-0.656735\pi$
$211$	7.18807e7	−	1.24501e8i	0.526773	−	0.912398i	0.999513	−	0.0311959i	$-0.00993156\pi$
$212$		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	−0.999993	−	0.00374059i	$-0.998809\pi$
$223$	−8.33372e7	−	1.44344e8i	−0.503236	−	0.871630i	0.496757	−	0.867890i	$-0.334524\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.35156e8	+	2.34097e8i	0.766912	+	1.32833i	0.939230	+	0.343289i	$0.111541\pi$
$227$	1.35156e8	+	2.34097e8i	0.766912	+	1.32833i	−0.172318	+	0.985041i	$0.555126\pi$
$228$		0			0
$229$	5.57020e7	−	9.64786e7i	0.306511	−	0.530893i	−0.671085	−	0.741380i	$-0.734172\pi$
$229$	5.57020e7	−	9.64786e7i	0.306511	−	0.530893i	0.977597	+	0.210487i	$0.0675050\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−2.21831e8			−1.14888			−0.574442	−	0.818545i	$-0.694781\pi$
$233$	−2.21831e8			−1.14888			−0.574442	+	0.818545i	$0.694781\pi$
$234$		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	$-0.880736\pi$
$239$	−3.13141e7	+	5.42376e7i	−0.148371	+	0.256985i	0.782255	+	0.622959i	$0.214069\pi$
$240$		0			0
$241$	−1.48123e8	−	2.56557e8i	−0.681653	−	1.18066i	−0.974476	−	0.224491i	$-0.927928\pi$
$241$	−1.48123e8	−	2.56557e8i	−0.681653	−	1.18066i	0.292824	−	0.956166i	$-0.405405\pi$
$242$		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	$-0.274306\pi$
$251$	3.26241e8			1.30221			0.651104	+	0.758989i	$0.274306\pi$
$252$		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	−0.321590	−	0.946879i	$-0.604217\pi$
$257$	1.79391e8	−	3.10714e8i	0.659226	−	1.14181i	0.980816	−	0.194934i	$-0.0624493\pi$
$258$		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	$-0.124382\pi$
$263$	3.90755e7	+	6.76808e7i	0.132452	+	0.229414i	−0.792169	+	0.610302i	$0.791048\pi$
$264$		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	$-0.398351\pi$
$269$	2.00452e8			0.627882			0.313941	+	0.949443i	$0.398351\pi$
$270$		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	0.937056	−	0.349180i	$-0.113540\pi$
$277$	5.87657e7	+	1.01785e8i	0.166129	+	0.287743i	−0.770927	+	0.636924i	$0.780207\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	7.43884e7	+	1.28844e8i	0.200001	+	0.346412i	0.948529	−	0.316692i	$-0.102572\pi$
$281$	7.43884e7	+	1.28844e8i	0.200001	+	0.346412i	−0.748527	+	0.663104i	$0.769239\pi$
$282$		0			0
$283$	1.38403e8	−	2.39721e8i	0.362989	−	0.628715i	−0.625463	−	0.780254i	$-0.715090\pi$
$283$	1.38403e8	−	2.39721e8i	0.362989	−	0.628715i	0.988451	+	0.151540i	$0.0484231\pi$
$284$		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	−0.0661027	+	0.997813i	$0.521056\pi$
$293$	−3.86294e8	+	6.69080e8i	−0.897183	+	1.55397i	−0.831080	+	0.556153i	$0.812277\pi$
$294$		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	−0.382532	−	0.923942i	$-0.624948\pi$
$311$	3.22999e8	−	5.59451e8i	0.608891	−	1.05463i	0.991423	−	0.130689i	$-0.0417188\pi$
$312$		0			0
$313$	1.97858e8	+	3.42700e8i	0.364711	+	0.631697i	0.988730	−	0.149712i	$-0.0478346\pi$
$313$	1.97858e8	+	3.42700e8i	0.364711	+	0.631697i	−0.624019	+	0.781409i	$0.714501\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.96290e8	−	3.39985e8i	−0.346092	−	0.599449i	0.639460	−	0.768825i	$-0.279158\pi$
$317$	−1.96290e8	−	3.39985e8i	−0.346092	−	0.599449i	−0.985551	+	0.169376i	$0.945825\pi$
$318$		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	0.935326	−	0.353787i	$-0.115106\pi$
$331$	1.06406e8	+	1.84300e8i	0.161275	+	0.279336i	−0.774051	+	0.633123i	$0.781773\pi$
$332$		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	−0.439539	−	0.898223i	$-0.644858\pi$
$337$	3.92128e8	−	6.79186e8i	0.558115	−	0.966683i	0.997654	−	0.0684596i	$-0.0218084\pi$
$338$		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	0.289468	+	0.957188i	$0.406521\pi$
$347$	−5.32532e8	+	9.22372e8i	−0.684215	+	1.18509i	−0.973683	+	0.227907i	$0.926812\pi$
$348$		0			0
$349$	−3.00319e8	−	5.20168e8i	−0.378176	−	0.655020i	0.612621	−	0.790377i	$-0.290115\pi$
$349$	−3.00319e8	−	5.20168e8i	−0.378176	−	0.655020i	−0.990797	+	0.135357i	$0.956782\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	2.32650e7	+	4.02962e7i	0.0281509	+	0.0487588i	0.879758	−	0.475422i	$-0.157705\pi$
$353$	2.32650e7	+	4.02962e7i	0.0281509	+	0.0487588i	−0.851607	+	0.524181i	$0.824371\pi$
$354$		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	$-0.334115\pi$
$359$	8.72929e8			0.995745			0.497873	+	0.867250i	$0.334115\pi$
$360$		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	0.736939	−	0.675959i	$-0.236270\pi$
$367$	−2.05422e8	−	3.55801e8i	−0.216928	−	0.375730i	−0.953867	+	0.300229i	$0.902937\pi$
$368$		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	−0.537953	−	0.842975i	$-0.680802\pi$
$373$	4.62104e8	−	8.00387e8i	0.461061	−	0.798582i	0.999014	+	0.0443933i	$0.0141355\pi$
$374$		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	$-0.696156\pi$
$383$	4.59303e8	−	7.95537e8i	0.417738	−	0.723543i	0.995712	+	0.0925122i	$0.0294897\pi$
$384$		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	0.987063	+	0.160331i	$0.0512562\pi$
$389$	7.34181e8	+	1.27164e9i	0.632382	+	1.09532i	−0.354681	+	0.934987i	$0.615410\pi$
$390$		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	$-0.680939\pi$
$401$	5.94845e8	−	1.03030e9i	0.460679	−	0.797920i	0.998995	+	0.0448234i	$0.0142725\pi$
$402$		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	−0.666349	−	0.745640i	$-0.732144\pi$
$409$	4.32491e8	−	7.49096e8i	0.312569	−	0.541385i	0.978918	+	0.204255i	$0.0654772\pi$
$410$		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	0.365100	+	0.930968i	$0.381035\pi$
$419$	−9.39117e8	+	1.62660e9i	−0.623692	+	1.08027i	−0.988792	+	0.149298i	$0.952298\pi$
$420$		0			0
$421$	4.91277e8	+	8.50917e8i	0.320877	+	0.555776i	0.980669	−	0.195672i	$-0.0626889\pi$
$421$	4.91277e8	+	8.50917e8i	0.320877	+	0.555776i	−0.659792	+	0.751448i	$0.729356\pi$
$422$		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	$-0.488611\pi$
$431$	1.18914e8			0.0715424			0.0357712	+	0.999360i	$0.488611\pi$
$432$		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	0.877731	+	0.479154i	$0.159056\pi$
$439$	1.51354e9	+	2.62153e9i	0.853825	+	1.47887i	−0.0239061	+	0.999714i	$0.507610\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−1.12348e9	−	1.94593e9i	−0.613978	−	1.06344i	−0.990563	−	0.137059i	$-0.956235\pi$
$443$	−1.12348e9	−	1.94593e9i	−0.613978	−	1.06344i	0.376585	−	0.926382i	$-0.377098\pi$
$444$		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	$-0.354725\pi$
$449$	1.69064e9			0.881431			0.440716	+	0.897647i	$0.354725\pi$
$450$		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	0.999330	+	0.0366030i	$0.0116537\pi$
$457$	1.08417e9	+	1.87784e9i	0.531364	+	0.920350i	−0.467966	+	0.883747i	$0.655013\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−4.28840e8	−	7.42773e8i	−0.203865	−	0.353104i	0.745906	−	0.666052i	$-0.232017\pi$
$461$	−4.28840e8	−	7.42773e8i	−0.203865	−	0.353104i	−0.949770	+	0.312947i	$0.898684\pi$
$462$		0			0
$463$	−1.26464e9	+	2.19042e9i	−0.592153	+	1.02564i	0.401788	+	0.915733i	$0.368389\pi$
$463$	−1.26464e9	+	2.19042e9i	−0.592153	+	1.02564i	−0.993942	+	0.109907i	$0.964945\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.13072e9			0.513745			0.256872	−	0.966445i	$-0.417308\pi$
$467$	1.13072e9			0.513745			0.256872	+	0.966445i	$0.417308\pi$
$468$		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	$-0.189267\pi$
$479$	−1.70634e8	−	2.95546e8i	−0.0709399	−	0.122872i	−0.899314	+	0.437304i	$0.855933\pi$
$480$		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	0.0476396	+	0.998865i	$0.484830\pi$
$491$	−2.20646e9	+	3.82170e9i	−0.841222	+	1.45704i	−0.888862	+	0.458175i	$0.848503\pi$
$492$		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	0.0938023	−	0.995591i	$-0.470098\pi$
$499$	2.52329e9	−	4.37047e9i	0.909108	−	1.57462i	0.815306	−	0.579031i	$-0.196569\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−3.84853e9			−1.34836			−0.674182	−	0.738565i	$-0.735504\pi$
$503$	−3.84853e9			−1.34836			−0.674182	+	0.738565i	$0.735504\pi$
$504$		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	−0.260712	−	0.965417i	$-0.583957\pi$
$509$	2.09964e9	−	3.63668e9i	0.705720	−	1.22234i	0.966431	−	0.256925i	$-0.0827095\pi$
$510$		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	$-0.658487\pi$
$521$	−3.08327e9			−0.955168			−0.477584	+	0.878586i	$0.658487\pi$
$522$		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	−0.913527	−	0.406779i	$-0.866652\pi$
$547$	−3.09691e9	−	5.36400e9i	−0.809044	−	1.40131i	0.104482	−	0.994527i	$-0.466681\pi$
$548$		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	$-0.527856\pi$
$557$	−7.12914e8			−0.174801			−0.0874005	+	0.996173i	$0.527856\pi$
$558$		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	$-0.967677\pi$
$563$	−1.73450e9	+	3.00425e9i	−0.409634	+	0.709507i	0.585215	+	0.810879i	$0.301010\pi$
$564$		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	$-0.0840213\pi$
$569$	1.12816e9	+	1.95403e9i	0.256731	+	0.444671i	−0.708633	+	0.705577i	$0.750688\pi$
$570$		0			0
$571$	4.19749e9	−	7.27027e9i	0.943547	−	1.63427i	0.184913	−	0.982755i	$-0.440800\pi$
$571$	4.19749e9	−	7.27027e9i	0.943547	−	1.63427i	0.758634	−	0.651517i	$-0.225867\pi$
$572$		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	$-0.248082\pi$
$587$	−1.23978e9	−	2.14736e9i	−0.252995	−	0.438200i	−0.964349	+	0.264634i	$0.914749\pi$
$588$		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	$-0.443358\pi$
$593$	1.79770e9			0.354018			0.177009	+	0.984209i	$0.443358\pi$
$594$		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	$-0.869780\pi$
$599$	−6.00964e8	+	1.04090e9i	−0.114250	+	0.197886i	0.803230	+	0.595669i	$0.203113\pi$
$600$		0			0
$601$	−5.77215e8	−	9.99766e8i	−0.108462	−	0.187861i	0.806685	−	0.590981i	$-0.201259\pi$
$601$	−5.77215e8	−	9.99766e8i	−0.108462	−	0.187861i	−0.915147	+	0.403120i	$0.867926\pi$
$602$		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	−0.976273	−	0.216544i	$-0.930522\pi$
$607$	−1.65636e9	+	2.86890e9i	−0.300604	+	0.520661i	0.675669	+	0.737205i	$0.263855\pi$
$608$		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	$-0.757070\pi$
$617$	1.38453e9	−	2.39807e9i	0.237303	−	0.411022i	0.959940	+	0.280207i	$0.0904031\pi$
$618$		0			0
$619$	3.20312e9	+	5.54797e9i	0.542821	+	0.940193i	0.998741	+	0.0501720i	$0.0159769\pi$
$619$	3.20312e9	+	5.54797e9i	0.542821	+	0.940193i	−0.455920	+	0.890021i	$0.650690\pi$
$620$		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	−0.916852	−	0.399228i	$-0.869278\pi$
$641$	−5.36228e9	−	9.28774e9i	−0.804168	−	1.39286i	0.112684	−	0.993631i	$-0.464055\pi$
$642$		0			0
$643$	2.52124e9	−	4.36691e9i	0.374003	−	0.647792i	−0.616174	−	0.787610i	$-0.711318\pi$
$643$	2.52124e9	−	4.36691e9i	0.374003	−	0.647792i	0.990177	+	0.139818i	$0.0446516\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−3.76257e9			−0.546160			−0.273080	−	0.961991i	$-0.588042\pi$
$647$	−3.76257e9			−0.546160			−0.273080	+	0.961991i	$0.588042\pi$
$648$		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	$-0.930633\pi$
$653$	−2.14128e9	+	3.70881e9i	−0.300939	+	0.521241i	0.675410	+	0.737442i	$0.263967\pi$
$654$		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	$-0.293533\pi$
$659$	−2.85125e9	−	4.93851e9i	−0.388093	−	0.672197i	−0.992193	+	0.124711i	$0.960199\pi$
$660$		0			0
$661$	−5.54124e8	+	9.59771e8i	−0.0746280	+	0.129260i	−0.900924	−	0.433976i	$-0.857110\pi$
$661$	−5.54124e8	+	9.59771e8i	−0.0746280	+	0.129260i	0.826296	+	0.563235i	$0.190444\pi$
$662$		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	0.627809	−	0.778367i	$-0.283952\pi$
$673$	−2.84822e9	−	4.93326e9i	−0.360181	−	0.623852i	−0.987990	+	0.154515i	$0.950618\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−7.88175e8	−	1.36516e9i	−0.0976253	−	0.169092i	0.813076	−	0.582158i	$-0.197791\pi$
$677$	−7.88175e8	−	1.36516e9i	−0.0976253	−	0.169092i	−0.910701	+	0.413066i	$0.864458\pi$
$678$		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	$-0.696493\pi$
$683$	−9.63957e9			−1.15767			−0.578836	+	0.815444i	$0.696493\pi$
$684$		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	0.659484	−	0.751719i	$-0.270775\pi$
$691$	−2.78636e9	−	4.82612e9i	−0.321265	−	0.556448i	−0.980749	+	0.195271i	$0.937441\pi$
$692$		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	$-0.248888\pi$
$701$	1.29431e10			1.41914			0.709572	+	0.704633i	$0.248888\pi$
$702$		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	0.614289	−	0.789081i	$-0.289443\pi$
$709$	−3.57029e9	−	6.18393e9i	−0.376220	−	0.651633i	−0.990509	+	0.137449i	$0.956110\pi$
$710$		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	$-0.368454\pi$
$719$	8.00524e9			0.803200			0.401600	+	0.915815i	$0.368454\pi$
$720$		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	0.967308	−	0.253604i	$-0.0816159\pi$
$727$	2.73539e9	+	4.73783e9i	0.264027	+	0.457308i	−0.703281	+	0.710912i	$0.748283\pi$
$728$		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	−0.841948	−	0.539559i	$-0.818591\pi$
$733$	4.93662e8	−	8.55048e8i	0.0462984	−	0.0801912i	0.888246	+	0.459368i	$0.151924\pi$
$734$		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	$-0.916433\pi$
$743$	−2.88579e9	+	4.99833e9i	−0.258109	+	0.447058i	0.707626	+	0.706587i	$0.249766\pi$
$744$		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	−0.0885361	−	0.996073i	$-0.528219\pi$
$751$	9.49910e9	−	1.64529e10i	0.818356	−	1.41743i	0.906893	−	0.421362i	$-0.138448\pi$
$752$		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	$-0.836240\pi$
$761$	−1.11029e8	+	1.92307e8i	−0.00913247	+	0.0158179i	0.861423	+	0.507888i	$0.169574\pi$
$762$		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	0.270084	−	0.962837i	$-0.412949\pi$
$769$	1.22184e10	−	2.11629e10i	0.968883	−	1.67815i	0.698799	−	0.715318i	$-0.253718\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.73556e9			0.602370			0.301185	−	0.953566i	$-0.402618\pi$
$773$	7.73556e9			0.602370			0.301185	+	0.953566i	$0.402618\pi$
$774$		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	−0.484496	−	0.874793i	$-0.660997\pi$
$787$	7.04709e9	−	1.22059e10i	0.515345	−	0.892604i	0.999841	−	0.0178108i	$-0.00566964\pi$
$788$		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	$-0.732860\pi$
$797$	4.43678e9	−	7.68472e9i	0.310430	−	0.537680i	0.978455	+	0.206458i	$0.0661937\pi$
$798$		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	$-0.206061\pi$
$809$	2.40258e10			1.59536			0.797678	+	0.603084i	$0.206061\pi$
$810$		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	$-0.0653737\pi$
$821$	4.96107e9	+	8.59283e9i	0.312878	+	0.541920i	−0.666106	+	0.745857i	$0.732040\pi$
$822$		0			0
$823$	9.64630e9	−	1.67079e10i	0.603200	−	1.04477i	−0.389133	−	0.921182i	$-0.627225\pi$
$823$	9.64630e9	−	1.67079e10i	0.603200	−	1.04477i	0.992333	−	0.123592i	$-0.0394413\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	2.36031e10			1.45111			0.725555	−	0.688164i	$-0.241583\pi$
$827$	2.36031e10			1.45111			0.725555	+	0.688164i	$0.241583\pi$
$828$		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	−0.992192	−	0.124723i	$-0.960196\pi$
$839$	−1.03343e10	−	1.78996e10i	−0.604109	−	1.04635i	0.388083	−	0.921625i	$-0.373137\pi$
$840$		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	0.999997	+	0.00242465i	$0.000771789\pi$
$853$	9.10145e9	+	1.57642e10i	0.502098	+	0.869660i	−0.497899	+	0.867235i	$0.665895\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.42653e9	−	1.11311e10i	−0.348774	−	0.604094i	0.637258	−	0.770650i	$-0.280068\pi$
$857$	−6.42653e9	−	1.11311e10i	−0.348774	−	0.604094i	−0.986032	+	0.166557i	$0.946735\pi$
$858$		0			0
$859$	−9.90150e8	+	1.71499e9i	−0.0532997	+	0.0923179i	−0.891444	−	0.453130i	$-0.850307\pi$
$859$	−9.90150e8	+	1.71499e9i	−0.0532997	+	0.0923179i	0.838145	+	0.545448i	$0.183641\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.28817e10			0.682238			0.341119	−	0.940020i	$-0.389194\pi$
$863$	1.28817e10			0.682238			0.341119	+	0.940020i	$0.389194\pi$
$864$		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	−0.358284	−	0.933613i	$-0.616638\pi$
$877$	1.25724e10	−	2.17761e10i	0.629390	−	1.09014i	0.987674	−	0.156523i	$-0.0500287\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−4.62474e9			−0.227862			−0.113931	−	0.993489i	$-0.536344\pi$
$881$	−4.62474e9			−0.227862			−0.113931	+	0.993489i	$0.536344\pi$
$882$		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	0.0901620	+	0.995927i	$0.471262\pi$
$887$	−1.69893e10	+	2.94264e10i	−0.817417	+	1.41581i	−0.907579	+	0.419881i	$0.862072\pi$
$888$		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	0.629534	−	0.776973i	$-0.283246\pi$
$907$	−8.04718e9	−	1.39381e10i	−0.358111	−	0.620267i	−0.987645	+	0.156706i	$0.949913\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−1.17471e10	−	2.03466e10i	−0.514773	−	0.891614i	−0.999853	−	0.0171438i	$-0.994543\pi$
$911$	−1.17471e10	−	2.03466e10i	−0.514773	−	0.891614i	0.485080	−	0.874470i	$-0.338791\pi$
$912$		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	$-0.117277\pi$
$929$	3.77657e9	+	6.54121e9i	0.154540	+	0.267672i	−0.778351	+	0.627829i	$0.783944\pi$
$930$		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	$-0.855864\pi$
$941$	−1.80772e9	+	3.13106e9i	−0.0707241	+	0.122498i	0.828495	+	0.559997i	$0.189198\pi$
$942$		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	$-0.219106\pi$
$947$	−4.28609e9	−	7.42373e9i	−0.163997	−	0.284052i	−0.936299	+	0.351205i	$0.885772\pi$
$948$		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	$-0.375736\pi$
$953$	2.03359e10			0.761093			0.380546	+	0.924762i	$0.375736\pi$
$954$		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	−0.235697	−	0.971827i	$-0.575737\pi$
$967$	2.03516e10	−	3.52500e10i	0.723778	−	1.25362i	0.959475	−	0.281794i	$-0.0909293\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.44101e10			−0.855662			−0.427831	−	0.903859i	$-0.640722\pi$
$971$	−2.44101e10			−0.855662			−0.427831	+	0.903859i	$0.640722\pi$
$972$		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	$-0.954560\pi$
$977$	−1.08350e10	+	1.87668e10i	−0.371705	+	0.643813i	0.618123	+	0.786082i	$0.287893\pi$
$978$		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	$-0.0331053\pi$
$983$	1.21325e10	+	2.10140e10i	0.407391	+	0.705622i	−0.587206	+	0.809438i	$0.699772\pi$
$984$		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	−0.866361	−	0.499418i	$-0.833547\pi$
$997$	−2.70891e10	−	4.69197e10i	−0.865689	−	1.49942i	0.000672287	−	1.00000i	$-0.499786\pi$
$998$		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.109.8		16
3.2	odd	2	inner	324.8.e.m.109.1		16
9.2	odd	6	inner	324.8.e.m.217.1		16
9.4	even	3		324.8.a.e.1.1	✓	8
9.5	odd	6		324.8.a.e.1.8	yes	8
9.7	even	3	inner	324.8.e.m.217.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.8.a.e.1.1	✓	8	9.4	even	3
324.8.a.e.1.8	yes	8	9.5	odd	6
324.8.e.m.109.1		16	3.2	odd	2	inner
324.8.e.m.109.8		16	1.1	even	1	trivial
324.8.e.m.217.1		16	9.2	odd	6	inner
324.8.e.m.217.8		16	9.7	even	3	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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