LMFDB - Embedded newform 324.9.c.b.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,9,Mod(161,324)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 9, names="a")

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

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 9);

N := Newforms(S);

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 9 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, 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:	$131.990669660$
Analytic rank:	$0$
Dimension:	$16$
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} + 456016 x^{14} + 81688865380 x^{12} + \cdots + 60\!\cdots\!25 $ x^16 + 456016x^14 + 81688865380x^12 + 7460070919828720x^10 + 382449673662084695494x^8 + 11367182393998798416041200x^6 + 192579026012424892778300582500x^4 + 1706442887222936191415620923250000*x^2 + 6056656355062095659654264284612890625
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{88} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$-167.188i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.9.c.b.161.12

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-501.563i q^{5} -3090.68 q^{7} -11241.7i q^{11} +3930.44 q^{13} +46705.5i q^{17} +131459. q^{19} -384942. i q^{23} +139059. q^{25} -817545. i q^{29} +1.59414e6 q^{31} +1.55017e6i q^{35} -645124. q^{37} -1.32509e6i q^{41} -1.20800e6 q^{43} -453328. i q^{47} +3.78749e6 q^{49} -1.24042e7i q^{53} -5.63843e6 q^{55} +2.58079e6i q^{59} -1.28437e7 q^{61} -1.97136e6i q^{65} -2.98465e7 q^{67} +2.77799e6i q^{71} +7.42060e6 q^{73} +3.47445e7i q^{77} +2.32710e7 q^{79} +1.96704e7i q^{83} +2.34257e7 q^{85} +9.34649e7i q^{89} -1.21477e7 q^{91} -6.59349e7i q^{95} +6.07429e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 3692 q^{7} - 50380 q^{13} + 185108 q^{19} - 1958288 q^{25} - 285568 q^{31} - 2910052 q^{37} - 8122852 q^{43} + 14602560 q^{49} + 38320164 q^{55} - 5700244 q^{61} - 6328540 q^{67} - 51728704 q^{73}+ \cdots - 181751968 q^{97}+O(q^{100}) $ 16 * q + 3692 * q^7 - 50380 * q^13 + 185108 * q^19 - 1958288 * q^25 - 285568 * q^31 - 2910052 * q^37 - 8122852 * q^43 + 14602560 * q^49 + 38320164 * q^55 - 5700244 * q^61 - 6328540 * q^67 - 51728704 * q^73 + 168603092 * q^79 - 96322644 * q^85 - 199743716 * q^91 - 181751968 * q^97

Character values

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

$n$	$163$	$245$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	− 501.563i	− 0.802501i	−0.915968	−	0.401250i	$-0.868576\pi$
$5$	− 501.563i	− 0.802501i	0.915968	−	0.401250i	$-0.131424\pi$
$6$	0	0
$7$	−3090.68	−1.28725	−0.643623	−	0.765343i	$-0.722570\pi$
$7$	−3090.68	−1.28725	−0.643623	+	0.765343i	$0.722570\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 11241.7i	− 0.767825i	−0.923370	−	0.383912i	$-0.874576\pi$
$11$	− 11241.7i	− 0.767825i	0.923370	−	0.383912i	$-0.125424\pi$
$12$	0	0
$13$	3930.44	0.137616	0.0688078	−	0.997630i	$-0.478080\pi$
$13$	3930.44	0.137616	0.0688078	+	0.997630i	$0.478080\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	46705.5i	0.559206i	0.960116	+	0.279603i	$0.0902029\pi$
$17$	46705.5i	0.559206i	−0.960116	+	0.279603i	$0.909797\pi$
$18$	0	0
$19$	131459.	1.00873	0.504365	−	0.863490i	$-0.331727\pi$
$19$	131459.	1.00873	0.504365	+	0.863490i	$0.331727\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 384942.i	− 1.37557i	−0.725913	−	0.687787i	$-0.758582\pi$
$23$	− 384942.i	− 1.37557i	0.725913	−	0.687787i	$-0.241418\pi$
$24$	0	0
$25$	139059.	0.355992
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 817545.i	− 1.15590i	−0.816073	−	0.577949i	$-0.803853\pi$
$29$	− 817545.i	− 1.15590i	0.816073	−	0.577949i	$-0.196147\pi$
$30$	0	0
$31$	1.59414e6	1.72615	0.863077	−	0.505073i	$-0.168534\pi$
$31$	1.59414e6	1.72615	0.863077	+	0.505073i	$0.168534\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.55017e6i	1.03302i
$36$	0	0
$37$	−645124.	−0.344220	−0.172110	−	0.985078i	$-0.555058\pi$
$37$	−645124.	−0.344220	−0.172110	+	0.985078i	$0.555058\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 1.32509e6i	− 0.468931i	−0.972124	−	0.234466i	$-0.924666\pi$
$41$	− 1.32509e6i	− 0.468931i	0.972124	−	0.234466i	$-0.0753340\pi$
$42$	0	0
$43$	−1.20800e6	−0.353341	−0.176670	−	0.984270i	$-0.556533\pi$
$43$	−1.20800e6	−0.353341	−0.176670	+	0.984270i	$0.556533\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 453328.i	− 0.0929011i	−0.998921	−	0.0464505i	$-0.985209\pi$
$47$	− 453328.i	− 0.0929011i	0.998921	−	0.0464505i	$-0.0147910\pi$
$48$	0	0
$49$	3.78749e6	0.657002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1.24042e7i	− 1.57204i	−0.618199	−	0.786022i	$-0.712137\pi$
$53$	− 1.24042e7i	− 1.57204i	0.618199	−	0.786022i	$-0.287863\pi$
$54$	0	0
$55$	−5.63843e6	−0.616180
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.58079e6i	0.212983i	0.994314	+	0.106492i	$0.0339617\pi$
$59$	2.58079e6i	0.212983i	−0.994314	+	0.106492i	$0.966038\pi$
$60$	0	0
$61$	−1.28437e7	−0.927623	−0.463811	−	0.885934i	$-0.653518\pi$
$61$	−1.28437e7	−0.927623	−0.463811	+	0.885934i	$0.653518\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.97136e6i	− 0.110437i
$66$	0	0
$67$	−2.98465e7	−1.48114	−0.740568	−	0.671982i	$-0.765443\pi$
$67$	−2.98465e7	−1.48114	−0.740568	+	0.671982i	$0.765443\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.77799e6i	0.109319i	0.998505	+	0.0546596i	$0.0174074\pi$
$71$	2.77799e6i	0.109319i	−0.998505	+	0.0546596i	$0.982593\pi$
$72$	0	0
$73$	7.42060e6	0.261305	0.130652	−	0.991428i	$-0.458293\pi$
$73$	7.42060e6	0.261305	0.130652	+	0.991428i	$0.458293\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.47445e7i	0.988379i
$78$	0	0
$79$	2.32710e7	0.597457	0.298729	−	0.954338i	$-0.403437\pi$
$79$	2.32710e7	0.597457	0.298729	+	0.954338i	$0.403437\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.96704e7i	0.414477i	0.978290	+	0.207239i	$0.0664477\pi$
$83$	1.96704e7i	0.414477i	−0.978290	+	0.207239i	$0.933552\pi$
$84$	0	0
$85$	2.34257e7	0.448764
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9.34649e7i	1.48966i	0.667252	+	0.744832i	$0.267471\pi$
$89$	9.34649e7i	1.48966i	−0.667252	+	0.744832i	$0.732529\pi$
$90$	0	0
$91$	−1.21477e7	−0.177145
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 6.59349e7i	− 0.809507i
$96$	0	0
$97$	6.07429e7	0.686133	0.343067	−	0.939311i	$-0.388534\pi$
$97$	6.07429e7	0.686133	0.343067	+	0.939311i	$0.388534\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1.32284e8i	− 1.27122i	−0.772011	−	0.635610i	$-0.780749\pi$
$101$	− 1.32284e8i	− 1.27122i	0.772011	−	0.635610i	$-0.219251\pi$
$102$	0	0
$103$	−1.15239e8	−1.02388	−0.511942	−	0.859020i	$-0.671074\pi$
$103$	−1.15239e8	−1.02388	−0.511942	+	0.859020i	$0.671074\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.24901e8i	0.952865i	0.879211	+	0.476432i	$0.158070\pi$
$107$	1.24901e8i	0.952865i	−0.879211	+	0.476432i	$0.841930\pi$
$108$	0	0
$109$	7.75512e7	0.549392	0.274696	−	0.961531i	$-0.411423\pi$
$109$	7.75512e7	0.549392	0.274696	+	0.961531i	$0.411423\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.54869e8i	0.949839i	0.880029	+	0.474919i	$0.157523\pi$
$113$	1.54869e8i	0.949839i	−0.880029	+	0.474919i	$0.842477\pi$
$114$	0	0
$115$	−1.93073e8	−1.10390
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1.44352e8i	− 0.719836i
$120$	0	0
$121$	8.79825e7	0.410445
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.65670e8i	− 1.08819i
$126$	0	0
$127$	−3.72217e8	−1.43081	−0.715404	−	0.698711i	$-0.753757\pi$
$127$	−3.72217e8	−1.43081	−0.715404	+	0.698711i	$0.753757\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.54228e8i	− 0.523695i	−0.965109	−	0.261848i	$-0.915668\pi$
$131$	− 1.54228e8i	− 0.523695i	0.965109	−	0.261848i	$-0.0843318\pi$
$132$	0	0
$133$	−4.06297e8	−1.29848
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.74558e8i	− 0.779385i	−0.920945	−	0.389692i	$-0.872581\pi$
$137$	− 2.74558e8i	− 0.779385i	0.920945	−	0.389692i	$-0.127419\pi$
$138$	0	0
$139$	−1.67621e8	−0.449023	−0.224511	−	0.974471i	$-0.572079\pi$
$139$	−1.67621e8	−0.449023	−0.224511	+	0.974471i	$0.572079\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 4.41849e7i	− 0.105665i
$144$	0	0
$145$	−4.10050e8	−0.927609
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9.25112e8i	1.87693i	0.345368	+	0.938467i	$0.387754\pi$
$149$	9.25112e8i	1.87693i	−0.345368	+	0.938467i	$0.612246\pi$
$150$	0	0
$151$	−6.59348e7	−0.126826	−0.0634128	−	0.997987i	$-0.520198\pi$
$151$	−6.59348e7	−0.126826	−0.0634128	+	0.997987i	$0.520198\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 7.99561e8i	− 1.38524i
$156$	0	0
$157$	−4.25762e8	−0.700758	−0.350379	−	0.936608i	$-0.613947\pi$
$157$	−4.25762e8	−0.700758	−0.350379	+	0.936608i	$0.613947\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.18973e9i	1.77070i
$162$	0	0
$163$	−8.70098e8	−1.23259	−0.616294	−	0.787516i	$-0.711367\pi$
$163$	−8.70098e8	−1.23259	−0.616294	+	0.787516i	$0.711367\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 3.97837e8i	− 0.511492i	−0.966744	−	0.255746i	$-0.917679\pi$
$167$	− 3.97837e8i	− 0.511492i	0.966744	−	0.255746i	$-0.0823211\pi$
$168$	0	0
$169$	−8.00282e8	−0.981062
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.06311e9i	− 1.18685i	−0.804891	−	0.593423i	$-0.797776\pi$
$173$	− 1.06311e9i	− 1.18685i	0.804891	−	0.593423i	$-0.202224\pi$
$174$	0	0
$175$	−4.29788e8	−0.458249
$176$	0	0
$177$	0	0
$178$	0	0
$179$	9.73761e8i	0.948507i	0.880388	+	0.474253i	$0.157282\pi$
$179$	9.73761e8i	0.948507i	−0.880388	+	0.474253i	$0.842718\pi$
$180$	0	0
$181$	1.59538e9	1.48644	0.743222	−	0.669045i	$-0.233297\pi$
$181$	1.59538e9	1.48644	0.743222	+	0.669045i	$0.233297\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3.23570e8i	0.276237i
$186$	0	0
$187$	5.25050e8	0.429373
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 2.28553e9i	− 1.71733i	−0.512538	−	0.858665i	$-0.671295\pi$
$191$	− 2.28553e9i	− 1.71733i	0.512538	−	0.858665i	$-0.328705\pi$
$192$	0	0
$193$	−1.44122e9	−1.03873	−0.519363	−	0.854554i	$-0.673831\pi$
$193$	−1.44122e9	−1.03873	−0.519363	+	0.854554i	$0.673831\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2.62973e9i	1.74601i	0.487710	+	0.873006i	$0.337832\pi$
$197$	2.62973e9i	1.74601i	−0.487710	+	0.873006i	$0.662168\pi$
$198$	0	0
$199$	−9.92440e8	−0.632837	−0.316419	−	0.948620i	$-0.602480\pi$
$199$	−9.92440e8	−0.632837	−0.316419	+	0.948620i	$0.602480\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	2.52677e9i	1.48793i
$204$	0	0
$205$	−6.64615e8	−0.376318
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 1.47782e9i	− 0.774528i
$210$	0	0
$211$	−3.49959e9	−1.76558	−0.882790	−	0.469769i	$-0.844337\pi$
$211$	−3.49959e9	−1.76558	−0.882790	+	0.469769i	$0.844337\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	6.05890e8i	0.283557i
$216$	0	0
$217$	−4.92697e9	−2.22198
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1.83573e8i	0.0769555i
$222$	0	0
$223$	−3.13526e9	−1.26781	−0.633904	−	0.773411i	$-0.718549\pi$
$223$	−3.13526e9	−1.26781	−0.633904	+	0.773411i	$0.718549\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 3.28088e9i	− 1.23562i	−0.786325	−	0.617812i	$-0.788019\pi$
$227$	− 3.28088e9i	− 1.23562i	0.786325	−	0.617812i	$-0.211981\pi$
$228$	0	0
$229$	1.06424e9	0.386989	0.193494	−	0.981101i	$-0.438018\pi$
$229$	1.06424e9	0.386989	0.193494	+	0.981101i	$0.438018\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 1.01877e9i	− 0.345663i	−0.984951	−	0.172832i	$-0.944708\pi$
$233$	− 1.01877e9i	− 0.345663i	0.984951	−	0.172832i	$-0.0552917\pi$
$234$	0	0
$235$	−2.27372e8	−0.0745532
$236$	0	0
$237$	0	0
$238$	0	0
$239$	4.66078e9i	1.42846i	0.699913	+	0.714228i	$0.253222\pi$
$239$	4.66078e9i	1.42846i	−0.699913	+	0.714228i	$0.746778\pi$
$240$	0	0
$241$	−5.17328e9	−1.53355	−0.766775	−	0.641916i	$-0.778140\pi$
$241$	−5.17328e9	−1.53355	−0.766775	+	0.641916i	$0.778140\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 1.89966e9i	− 0.527245i
$246$	0	0
$247$	5.16690e8	0.138817
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 5.93933e8i	− 0.149638i	−0.997197	−	0.0748191i	$-0.976162\pi$
$251$	− 5.93933e8i	− 0.149638i	0.997197	−	0.0748191i	$-0.0238379\pi$
$252$	0	0
$253$	−4.32741e9	−1.05620
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 2.14652e9i	− 0.492043i	−0.969264	−	0.246022i	$-0.920877\pi$
$257$	− 2.14652e9i	− 0.492043i	0.969264	−	0.246022i	$-0.0791234\pi$
$258$	0	0
$259$	1.99387e9	0.443096
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 5.23736e9i	− 1.09469i	−0.836908	−	0.547343i	$-0.815639\pi$
$263$	− 5.23736e9i	− 1.09469i	0.836908	−	0.547343i	$-0.184361\pi$
$264$	0	0
$265$	−6.22148e9	−1.26157
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4.63768e9i	0.885710i	0.896593	+	0.442855i	$0.146034\pi$
$269$	4.63768e9i	0.885710i	−0.896593	+	0.442855i	$0.853966\pi$
$270$	0	0
$271$	8.80767e7	0.0163299	0.00816496	−	0.999967i	$-0.497401\pi$
$271$	8.80767e7	0.0163299	0.00816496	+	0.999967i	$0.497401\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 1.56327e9i	− 0.273340i
$276$	0	0
$277$	−5.67482e9	−0.963903	−0.481951	−	0.876198i	$-0.660072\pi$
$277$	−5.67482e9	−0.963903	−0.481951	+	0.876198i	$0.660072\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 1.05405e10i	− 1.69059i	−0.534303	−	0.845293i	$-0.679426\pi$
$281$	− 1.05405e10i	− 1.69059i	0.534303	−	0.845293i	$-0.320574\pi$
$282$	0	0
$283$	5.52106e9	0.860749	0.430375	−	0.902650i	$-0.358381\pi$
$283$	5.52106e9	0.860749	0.430375	+	0.902650i	$0.358381\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.09542e9i	0.603630i
$288$	0	0
$289$	4.79436e9	0.687288
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 1.28590e9i	− 0.174476i	−0.996188	−	0.0872379i	$-0.972196\pi$
$293$	− 1.28590e9i	− 0.174476i	0.996188	−	0.0872379i	$-0.0278040\pi$
$294$	0	0
$295$	1.29443e9	0.170919
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 1.51299e9i	− 0.189300i
$300$	0	0
$301$	3.73355e9	0.454837
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.44193e9i	0.744418i
$306$	0	0
$307$	−2.84077e9	−0.319804	−0.159902	−	0.987133i	$-0.551118\pi$
$307$	−2.84077e9	−0.319804	−0.159902	+	0.987133i	$0.551118\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	3.67624e9i	0.392973i	0.980507	+	0.196486i	$0.0629531\pi$
$311$	3.67624e9i	0.392973i	−0.980507	+	0.196486i	$0.937047\pi$
$312$	0	0
$313$	1.68833e10	1.75906	0.879530	−	0.475844i	$-0.157857\pi$
$313$	1.68833e10	1.75906	0.879530	+	0.475844i	$0.157857\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1.82921e10i	1.81145i	0.423866	+	0.905725i	$0.360673\pi$
$317$	1.82921e10i	1.81145i	−0.423866	+	0.905725i	$0.639327\pi$
$318$	0	0
$319$	−9.19061e9	−0.887527
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.13984e9i	0.564089i
$324$	0	0
$325$	5.46565e8	0.0489901
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.40109e9i	0.119587i
$330$	0	0
$331$	−1.40053e10	−1.16676	−0.583379	−	0.812200i	$-0.698270\pi$
$331$	−1.40053e10	−1.16676	−0.583379	+	0.812200i	$0.698270\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.49699e10i	1.18861i
$336$	0	0
$337$	−8.30186e9	−0.643659	−0.321829	−	0.946798i	$-0.604298\pi$
$337$	−8.30186e9	−0.643659	−0.321829	+	0.946798i	$0.604298\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 1.79209e10i	− 1.32538i
$342$	0	0
$343$	6.11124e9	0.441522
$344$	0	0
$345$	0	0
$346$	0	0
$347$	8.96363e9i	0.618252i	0.951021	+	0.309126i	$0.100037\pi$
$347$	8.96363e9i	0.618252i	−0.951021	+	0.309126i	$0.899963\pi$
$348$	0	0
$349$	1.57508e10	1.06170	0.530848	−	0.847467i	$-0.321873\pi$
$349$	1.57508e10	1.06170	0.530848	+	0.847467i	$0.321873\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 2.26808e10i	− 1.46069i	−0.683077	−	0.730347i	$-0.739358\pi$
$353$	− 2.26808e10i	− 1.46069i	0.683077	−	0.730347i	$-0.260642\pi$
$354$	0	0
$355$	1.39334e9	0.0877288
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 2.70853e10i	− 1.63063i	−0.579015	−	0.815317i	$-0.696563\pi$
$359$	− 2.70853e10i	− 1.63063i	0.579015	−	0.815317i	$-0.303437\pi$
$360$	0	0
$361$	2.97840e8	0.0175369
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.72190e9i	− 0.209697i
$366$	0	0
$367$	−1.46499e10	−0.807550	−0.403775	−	0.914858i	$-0.632302\pi$
$367$	−1.46499e10	−0.807550	−0.403775	+	0.914858i	$0.632302\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	3.83373e10i	2.02361i
$372$	0	0
$373$	1.16112e10	0.599851	0.299925	−	0.953963i	$-0.403038\pi$
$373$	1.16112e10	0.599851	0.299925	+	0.953963i	$0.403038\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 3.21331e9i	− 0.159070i
$378$	0	0
$379$	2.42458e9	0.117511	0.0587556	−	0.998272i	$-0.481287\pi$
$379$	2.42458e9	0.117511	0.0587556	+	0.998272i	$0.481287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 3.58404e10i	− 1.66562i	−0.553555	−	0.832812i	$-0.686729\pi$
$383$	− 3.58404e10i	− 1.66562i	0.553555	−	0.832812i	$-0.313271\pi$
$384$	0	0
$385$	1.74266e10	0.793175
$386$	0	0
$387$	0	0
$388$	0	0
$389$	3.02212e10i	1.31982i	0.751346	+	0.659909i	$0.229405\pi$
$389$	3.02212e10i	1.31982i	−0.751346	+	0.659909i	$0.770595\pi$
$390$	0	0
$391$	1.79789e10	0.769230
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 1.16719e10i	− 0.479460i
$396$	0	0
$397$	3.21443e10	1.29402	0.647011	−	0.762481i	$-0.276019\pi$
$397$	3.21443e10	1.29402	0.647011	+	0.762481i	$0.276019\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.58108e9i	0.215844i	0.994159	+	0.107922i	$0.0344197\pi$
$401$	5.58108e9i	0.215844i	−0.994159	+	0.107922i	$0.965580\pi$
$402$	0	0
$403$	6.26566e9	0.237546
$404$	0	0
$405$	0	0
$406$	0	0
$407$	7.25230e9i	0.264301i
$408$	0	0
$409$	4.02008e10	1.43662	0.718310	−	0.695724i	$-0.244916\pi$
$409$	4.02008e10	1.43662	0.718310	+	0.695724i	$0.244916\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 7.97640e9i	− 0.274162i
$414$	0	0
$415$	9.86595e9	0.332618
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 2.71804e10i	− 0.881860i	−0.897541	−	0.440930i	$-0.854649\pi$
$419$	− 2.71804e10i	− 0.881860i	0.897541	−	0.440930i	$-0.145351\pi$
$420$	0	0
$421$	2.37614e10	0.756386	0.378193	−	0.925727i	$-0.376546\pi$
$421$	2.37614e10	0.756386	0.378193	+	0.925727i	$0.376546\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.49484e9i	0.199073i
$426$	0	0
$427$	3.96958e10	1.19408
$428$	0	0
$429$	0	0
$430$	0	0
$431$	1.31683e10i	0.381610i	0.981628	+	0.190805i	$0.0611099\pi$
$431$	1.31683e10i	0.381610i	−0.981628	+	0.190805i	$0.938890\pi$
$432$	0	0
$433$	−4.82332e9	−0.137213	−0.0686064	−	0.997644i	$-0.521855\pi$
$433$	−4.82332e9	−0.137213	−0.0686064	+	0.997644i	$0.521855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 5.06040e10i	− 1.38758i
$438$	0	0
$439$	4.36207e10	1.17445	0.587225	−	0.809424i	$-0.300220\pi$
$439$	4.36207e10	1.17445	0.587225	+	0.809424i	$0.300220\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1.56683e10i	0.406824i	0.979093	+	0.203412i	$0.0652031\pi$
$443$	1.56683e10i	0.406824i	−0.979093	+	0.203412i	$0.934797\pi$
$444$	0	0
$445$	4.68785e10	1.19546
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 1.28411e10i	− 0.315948i	−0.987443	−	0.157974i	$-0.949504\pi$
$449$	− 1.28411e10i	− 0.315948i	0.987443	−	0.157974i	$-0.0504962\pi$
$450$	0	0
$451$	−1.48963e10	−0.360057
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.09285e9i	0.142159i
$456$	0	0
$457$	−1.43273e10	−0.328473	−0.164236	−	0.986421i	$-0.552516\pi$
$457$	−1.43273e10	−0.328473	−0.164236	+	0.986421i	$0.552516\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.25362e10i	1.38461i	0.721604	+	0.692306i	$0.243405\pi$
$461$	6.25362e10i	1.38461i	−0.721604	+	0.692306i	$0.756595\pi$
$462$	0	0
$463$	−6.77591e10	−1.47450	−0.737248	−	0.675622i	$-0.763875\pi$
$463$	−6.77591e10	−1.47450	−0.737248	+	0.675622i	$0.763875\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.69195e10i	− 0.986474i	−0.869895	−	0.493237i	$-0.835814\pi$
$467$	− 4.69195e10i	− 0.986474i	0.869895	−	0.493237i	$-0.164186\pi$
$468$	0	0
$469$	9.22460e10	1.90659
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1.35800e10i	0.271304i
$474$	0	0
$475$	1.82806e10	0.359100
$476$	0	0
$477$	0	0
$478$	0	0
$479$	8.68411e10i	1.64962i	0.565411	+	0.824809i	$0.308717\pi$
$479$	8.68411e10i	1.64962i	−0.565411	+	0.824809i	$0.691283\pi$
$480$	0	0
$481$	−2.53562e9	−0.0473700
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 3.04664e10i	− 0.550622i
$486$	0	0
$487$	6.69006e10	1.18936	0.594681	−	0.803962i	$-0.297278\pi$
$487$	6.69006e10	1.18936	0.594681	+	0.803962i	$0.297278\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	5.31276e10i	0.914100i	0.889441	+	0.457050i	$0.151094\pi$
$491$	5.31276e10i	0.914100i	−0.889441	+	0.457050i	$0.848906\pi$
$492$	0	0
$493$	3.81838e10	0.646386
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 8.58586e9i	− 0.140721i
$498$	0	0
$499$	−4.38611e10	−0.707420	−0.353710	−	0.935355i	$-0.615080\pi$
$499$	−4.38611e10	−0.707420	−0.353710	+	0.935355i	$0.615080\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 1.12677e11i	− 1.76021i	−0.474780	−	0.880105i	$-0.657472\pi$
$503$	− 1.12677e11i	− 1.76021i	0.474780	−	0.880105i	$-0.342528\pi$
$504$	0	0
$505$	−6.63486e10	−1.02015
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 6.63733e10i	− 0.988832i	−0.869225	−	0.494416i	$-0.835382\pi$
$509$	− 6.63733e10i	− 0.988832i	0.869225	−	0.494416i	$-0.164618\pi$
$510$	0	0
$511$	−2.29347e10	−0.336364
$512$	0	0
$513$	0	0
$514$	0	0
$515$	5.77996e10i	0.821668i
$516$	0	0
$517$	−5.09618e9	−0.0713317
$518$	0	0
$519$	0	0
$520$	0	0
$521$	1.35296e11i	1.83625i	0.396286	+	0.918127i	$0.370299\pi$
$521$	1.35296e11i	1.83625i	−0.396286	+	0.918127i	$0.629701\pi$
$522$	0	0
$523$	−8.95612e10	−1.19705	−0.598526	−	0.801103i	$-0.704247\pi$
$523$	−8.95612e10	−1.19705	−0.598526	+	0.801103i	$0.704247\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	7.44550e10i	0.965276i
$528$	0	0
$529$	−6.98693e10	−0.892203
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.20818e9i	− 0.0645322i
$534$	0	0
$535$	6.26458e10	0.764675
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 4.25779e10i	− 0.504463i
$540$	0	0
$541$	4.88612e10	0.570395	0.285197	−	0.958469i	$-0.407941\pi$
$541$	4.88612e10	0.570395	0.285197	+	0.958469i	$0.407941\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 3.88968e10i	− 0.440888i
$546$	0	0
$547$	1.05883e11	1.18270	0.591352	−	0.806413i	$-0.298594\pi$
$547$	1.05883e11	1.18270	0.591352	+	0.806413i	$0.298594\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.07473e11i	− 1.16599i
$552$	0	0
$553$	−7.19232e10	−0.769075
$554$	0	0
$555$	0	0
$556$	0	0
$557$	9.35207e10i	0.971599i	0.874070	+	0.485799i	$0.161471\pi$
$557$	9.35207e10i	0.971599i	−0.874070	+	0.485799i	$0.838529\pi$
$558$	0	0
$559$	−4.74798e9	−0.0486252
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2.79666e10i	0.278359i	0.990267	+	0.139180i	$0.0444466\pi$
$563$	2.79666e10i	0.278359i	−0.990267	+	0.139180i	$0.955553\pi$
$564$	0	0
$565$	7.76764e10	0.762246
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.12009e10i	0.106858i	0.998572	+	0.0534288i	$0.0170150\pi$
$569$	1.12009e10i	0.106858i	−0.998572	+	0.0534288i	$0.982985\pi$
$570$	0	0
$571$	5.88813e10	0.553902	0.276951	−	0.960884i	$-0.410676\pi$
$571$	5.88813e10	0.553902	0.276951	+	0.960884i	$0.410676\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 5.35298e10i	− 0.489693i
$576$	0	0
$577$	6.31803e10	0.570005	0.285003	−	0.958527i	$-0.408006\pi$
$577$	6.31803e10	0.570005	0.285003	+	0.958527i	$0.408006\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 6.07949e10i	− 0.533534i
$582$	0	0
$583$	−1.39444e11	−1.20705
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 1.68634e11i	− 1.42034i	−0.704031	−	0.710169i	$-0.748618\pi$
$587$	− 1.68634e11i	− 1.42034i	0.704031	−	0.710169i	$-0.251382\pi$
$588$	0	0
$589$	2.09563e11	1.74122
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.23251e11i	1.80541i	0.430263	+	0.902703i	$0.358421\pi$
$593$	2.23251e11i	1.80541i	−0.430263	+	0.902703i	$0.641579\pi$
$594$	0	0
$595$	−7.24014e10	−0.577669
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 8.61787e10i	− 0.669411i	−0.942323	−	0.334705i	$-0.891363\pi$
$599$	− 8.61787e10i	− 0.669411i	0.942323	−	0.334705i	$-0.108637\pi$
$600$	0	0
$601$	−1.48549e11	−1.13860	−0.569302	−	0.822128i	$-0.692787\pi$
$601$	−1.48549e11	−1.13860	−0.569302	+	0.822128i	$0.692787\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 4.41288e10i	− 0.329383i
$606$	0	0
$607$	−1.30829e11	−0.963719	−0.481860	−	0.876248i	$-0.660038\pi$
$607$	−1.30829e11	−0.963719	−0.481860	+	0.876248i	$0.660038\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 1.78178e9i	− 0.0127846i
$612$	0	0
$613$	−9.29664e10	−0.658391	−0.329196	−	0.944262i	$-0.606778\pi$
$613$	−9.29664e10	−0.658391	−0.329196	+	0.944262i	$0.606778\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.07823e10i	0.0743997i	0.999308	+	0.0371999i	$0.0118438\pi$
$617$	1.07823e10i	0.0743997i	−0.999308	+	0.0371999i	$0.988156\pi$
$618$	0	0
$619$	−1.00598e11	−0.685216	−0.342608	−	0.939478i	$-0.611310\pi$
$619$	−1.00598e11	−0.685216	−0.342608	+	0.939478i	$0.611310\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.88870e11i	− 1.91756i
$624$	0	0
$625$	−7.89303e10	−0.517277
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 3.01308e10i	− 0.192490i
$630$	0	0
$631$	−9.04320e10	−0.570433	−0.285216	−	0.958463i	$-0.592065\pi$
$631$	−9.04320e10	−0.570433	−0.285216	+	0.958463i	$0.592065\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	1.86690e11i	1.14822i
$636$	0	0
$637$	1.48865e10	0.0904137
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 2.96556e11i	− 1.75661i	−0.478101	−	0.878305i	$-0.658675\pi$
$641$	− 2.96556e11i	− 1.75661i	0.478101	−	0.878305i	$-0.341325\pi$
$642$	0	0
$643$	−2.26677e11	−1.32606	−0.663031	−	0.748592i	$-0.730730\pi$
$643$	−2.26677e11	−1.32606	−0.663031	+	0.748592i	$0.730730\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 1.26102e11i	− 0.719620i	−0.933026	−	0.359810i	$-0.882841\pi$
$647$	− 1.26102e11i	− 0.719620i	0.933026	−	0.359810i	$-0.117159\pi$
$648$	0	0
$649$	2.90126e10	0.163534
$650$	0	0
$651$	0	0
$652$	0	0
$653$	2.12723e11i	1.16993i	0.811057	+	0.584967i	$0.198892\pi$
$653$	2.12723e11i	1.16993i	−0.811057	+	0.584967i	$0.801108\pi$
$654$	0	0
$655$	−7.73552e10	−0.420266
$656$	0	0
$657$	0	0
$658$	0	0
$659$	3.53272e11i	1.87313i	0.350498	+	0.936564i	$0.386012\pi$
$659$	3.53272e11i	1.87313i	−0.350498	+	0.936564i	$0.613988\pi$
$660$	0	0
$661$	−1.75821e11	−0.921011	−0.460505	−	0.887657i	$-0.652332\pi$
$661$	−1.75821e11	−0.921011	−0.460505	+	0.887657i	$0.652332\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.03783e11i	1.04203i
$666$	0	0
$667$	−3.14707e11	−1.59002
$668$	0	0
$669$	0	0
$670$	0	0
$671$	1.44385e11i	0.712252i
$672$	0	0
$673$	−3.39934e10	−0.165704	−0.0828522	−	0.996562i	$-0.526403\pi$
$673$	−3.39934e10	−0.165704	−0.0828522	+	0.996562i	$0.526403\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 1.27996e11i	− 0.609313i	−0.952462	−	0.304657i	$-0.901458\pi$
$677$	− 1.27996e11i	− 0.609313i	0.952462	−	0.304657i	$-0.0985417\pi$
$678$	0	0
$679$	−1.87737e11	−0.883222
$680$	0	0
$681$	0	0
$682$	0	0
$683$	1.35606e11i	0.623154i	0.950221	+	0.311577i	$0.100857\pi$
$683$	1.35606e11i	0.623154i	−0.950221	+	0.311577i	$0.899143\pi$
$684$	0	0
$685$	−1.37708e11	−0.625457
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 4.87539e10i	− 0.216338i
$690$	0	0
$691$	−2.39232e11	−1.04932	−0.524660	−	0.851312i	$-0.675807\pi$
$691$	−2.39232e11	−1.04932	−0.524660	+	0.851312i	$0.675807\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	8.40724e10i	0.360341i
$696$	0	0
$697$	6.18889e10	0.262229
$698$	0	0
$699$	0	0
$700$	0	0
$701$	1.15359e11i	0.477728i	0.971053	+	0.238864i	$0.0767751\pi$
$701$	1.15359e11i	0.477728i	−0.971053	+	0.238864i	$0.923225\pi$
$702$	0	0
$703$	−8.48072e10	−0.347225
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.08846e11i	1.63637i
$708$	0	0
$709$	2.69218e11	1.06542	0.532708	−	0.846299i	$-0.321174\pi$
$709$	2.69218e11	1.06542	0.532708	+	0.846299i	$0.321174\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 6.13651e11i	− 2.37445i
$714$	0	0
$715$	−2.21615e10	−0.0847960
$716$	0	0
$717$	0	0
$718$	0	0
$719$	3.59958e10i	0.134690i	0.997730	+	0.0673452i	$0.0214529\pi$
$719$	3.59958e10i	0.134690i	−0.997730	+	0.0673452i	$0.978547\pi$
$720$	0	0
$721$	3.56167e11	1.31799
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 1.13687e11i	− 0.411491i
$726$	0	0
$727$	−4.48988e11	−1.60730	−0.803651	−	0.595101i	$-0.797112\pi$
$727$	−4.48988e11	−1.60730	−0.803651	+	0.595101i	$0.797112\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 5.64203e10i	− 0.197591i
$732$	0	0
$733$	−1.82966e11	−0.633804	−0.316902	−	0.948458i	$-0.602643\pi$
$733$	−1.82966e11	−0.633804	−0.316902	+	0.948458i	$0.602643\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	3.35527e11i	1.13725i
$738$	0	0
$739$	−1.01363e11	−0.339860	−0.169930	−	0.985456i	$-0.554354\pi$
$739$	−1.01363e11	−0.339860	−0.169930	+	0.985456i	$0.554354\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 1.38026e11i	− 0.452903i	−0.974022	−	0.226451i	$-0.927288\pi$
$743$	− 1.38026e11i	− 0.452903i	0.974022	−	0.226451i	$-0.0727124\pi$
$744$	0	0
$745$	4.64002e11	1.50624
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.86029e11i	− 1.22657i
$750$	0	0
$751$	−5.88823e11	−1.85108	−0.925540	−	0.378650i	$-0.876388\pi$
$751$	−5.88823e11	−1.85108	−0.925540	+	0.378650i	$0.876388\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.30705e10i	0.101778i
$756$	0	0
$757$	−7.69827e10	−0.234428	−0.117214	−	0.993107i	$-0.537396\pi$
$757$	−7.69827e10	−0.234428	−0.117214	+	0.993107i	$0.537396\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 5.08457e10i	− 0.151606i	−0.997123	−	0.0758029i	$-0.975848\pi$
$761$	− 5.08457e10i	− 0.151606i	0.997123	−	0.0758029i	$-0.0241520\pi$
$762$	0	0
$763$	−2.39686e11	−0.707203
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.01436e10i	0.0293098i
$768$	0	0
$769$	6.47421e11	1.85132	0.925660	−	0.378356i	$-0.123510\pi$
$769$	6.47421e11	1.85132	0.925660	+	0.378356i	$0.123510\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.58110e11i	1.84324i	0.388099	+	0.921618i	$0.373132\pi$
$773$	6.58110e11i	1.84324i	−0.388099	+	0.921618i	$0.626868\pi$
$774$	0	0
$775$	2.21680e11	0.614497
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 1.74194e11i	− 0.473025i
$780$	0	0
$781$	3.12293e10	0.0839380
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.13546e11i	0.562359i
$786$	0	0
$787$	1.27466e11	0.332273	0.166137	−	0.986103i	$-0.446871\pi$
$787$	1.27466e11	0.332273	0.166137	+	0.986103i	$0.446871\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 4.78649e11i	− 1.22268i
$792$	0	0
$793$	−5.04814e10	−0.127655
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 4.88186e11i	− 1.20991i	−0.796261	−	0.604953i	$-0.793192\pi$
$797$	− 4.88186e11i	− 1.20991i	0.796261	−	0.604953i	$-0.206808\pi$
$798$	0	0
$799$	2.11729e10	0.0519509
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 8.34203e10i	− 0.200636i
$804$	0	0
$805$	5.96725e11	1.42099
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 6.80814e11i	− 1.58940i	−0.606999	−	0.794702i	$-0.707627\pi$
$809$	− 6.80814e11i	− 1.58940i	0.606999	−	0.794702i	$-0.292373\pi$
$810$	0	0
$811$	−5.81532e11	−1.34428	−0.672141	−	0.740423i	$-0.734625\pi$
$811$	−5.81532e11	−1.34428	−0.672141	+	0.740423i	$0.734625\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	4.36409e11i	0.989153i
$816$	0	0
$817$	−1.58803e11	−0.356426
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 3.35193e11i	− 0.737773i	−0.929475	−	0.368886i	$-0.879739\pi$
$821$	− 3.35193e11i	− 0.737773i	0.929475	−	0.368886i	$-0.120261\pi$
$822$	0	0
$823$	1.19433e11	0.260330	0.130165	−	0.991492i	$-0.458449\pi$
$823$	1.19433e11	0.260330	0.130165	+	0.991492i	$0.458449\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.40606e11i	− 0.514380i	−0.966361	−	0.257190i	$-0.917203\pi$
$827$	− 2.40606e11i	− 0.514380i	0.966361	−	0.257190i	$-0.0827966\pi$
$828$	0	0
$829$	4.85810e11	1.02860	0.514302	−	0.857609i	$-0.328051\pi$
$829$	4.85810e11	1.02860	0.514302	+	0.857609i	$0.328051\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.76896e11i	0.367400i
$834$	0	0
$835$	−1.99540e11	−0.410473
$836$	0	0
$837$	0	0
$838$	0	0
$839$	6.68559e11i	1.34925i	0.738161	+	0.674625i	$0.235694\pi$
$839$	6.68559e11i	1.34925i	−0.738161	+	0.674625i	$0.764306\pi$
$840$	0	0
$841$	−1.68133e11	−0.336101
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.01392e11i	0.787303i
$846$	0	0
$847$	−2.71926e11	−0.528344
$848$	0	0
$849$	0	0
$850$	0	0
$851$	2.48335e11i	0.473500i
$852$	0	0
$853$	−4.93071e10	−0.0931350	−0.0465675	−	0.998915i	$-0.514828\pi$
$853$	−4.93071e10	−0.0931350	−0.0465675	+	0.998915i	$0.514828\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	5.49130e11i	1.01801i	0.860764	+	0.509005i	$0.169986\pi$
$857$	5.49130e11i	1.01801i	−0.860764	+	0.509005i	$0.830014\pi$
$858$	0	0
$859$	−1.91345e11	−0.351434	−0.175717	−	0.984441i	$-0.556224\pi$
$859$	−1.91345e11	−0.351434	−0.175717	+	0.984441i	$0.556224\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 7.76668e11i	− 1.40021i	−0.714042	−	0.700103i	$-0.753137\pi$
$863$	− 7.76668e11i	− 1.40021i	0.714042	−	0.700103i	$-0.246863\pi$
$864$	0	0
$865$	−5.33217e11	−0.952445
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.61606e11i	− 0.458743i
$870$	0	0
$871$	−1.17310e11	−0.203827
$872$	0	0
$873$	0	0
$874$	0	0
$875$	8.21101e11i	1.40076i
$876$	0	0
$877$	−4.31657e11	−0.729694	−0.364847	−	0.931068i	$-0.618879\pi$
$877$	−4.31657e11	−0.729694	−0.364847	+	0.931068i	$0.618879\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	6.30763e10i	0.104704i	0.998629	+	0.0523519i	$0.0166718\pi$
$881$	6.30763e10i	0.104704i	−0.998629	+	0.0523519i	$0.983328\pi$
$882$	0	0
$883$	5.31035e11	0.873535	0.436767	−	0.899575i	$-0.356123\pi$
$883$	5.31035e11	0.873535	0.436767	+	0.899575i	$0.356123\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.66644e9i	0.00753860i	0.999993	+	0.00376930i	$0.00119981\pi$
$887$	4.66644e9i	0.00753860i	−0.999993	+	0.00376930i	$0.998800\pi$
$888$	0	0
$889$	1.15040e12	1.84180
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 5.95939e10i	− 0.0937121i
$894$	0	0
$895$	4.88403e11	0.761178
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.30328e12i	− 1.99526i
$900$	0	0
$901$	5.79343e11	0.879097
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 8.00181e11i	− 1.19287i
$906$	0	0
$907$	5.94481e11	0.878433	0.439216	−	0.898381i	$-0.355256\pi$
$907$	5.94481e11	0.878433	0.439216	+	0.898381i	$0.355256\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	− 6.84082e10i	− 0.0993195i	−0.998766	−	0.0496597i	$-0.984186\pi$
$911$	− 6.84082e10i	− 0.0993195i	0.998766	−	0.0496597i	$-0.0158137\pi$
$912$	0	0
$913$	2.21129e11	0.318246
$914$	0	0
$915$	0	0
$916$	0	0
$917$	4.76670e11i	0.674125i
$918$	0	0
$919$	1.12042e12	1.57080	0.785400	−	0.618989i	$-0.212457\pi$
$919$	1.12042e12	1.57080	0.785400	+	0.618989i	$0.212457\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.09187e10i	0.0150440i
$924$	0	0
$925$	−8.97106e10	−0.122540
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1.44778e12i	1.94374i	0.235508	+	0.971872i	$0.424325\pi$
$929$	1.44778e12i	1.94374i	−0.235508	+	0.971872i	$0.575675\pi$
$930$	0	0
$931$	4.97898e11	0.662738
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 2.63346e11i	− 0.344572i
$936$	0	0
$937$	1.47839e12	1.91793	0.958963	−	0.283531i	$-0.0915060\pi$
$937$	1.47839e12	1.91793	0.958963	+	0.283531i	$0.0915060\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 9.07930e11i	− 1.15796i	−0.815341	−	0.578981i	$-0.803451\pi$
$941$	− 9.07930e11i	− 1.15796i	0.815341	−	0.578981i	$-0.196549\pi$
$942$	0	0
$943$	−5.10082e11	−0.645049
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.50346e11i	0.559946i	0.960008	+	0.279973i	$0.0903256\pi$
$947$	4.50346e11i	0.559946i	−0.960008	+	0.279973i	$0.909674\pi$
$948$	0	0
$949$	2.91662e10	0.0359596
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.18835e12i	1.44069i	0.693614	+	0.720347i	$0.256017\pi$
$953$	1.18835e12i	1.44069i	−0.693614	+	0.720347i	$0.743983\pi$
$954$	0	0
$955$	−1.14634e12	−1.37816
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.48571e11i	1.00326i
$960$	0	0
$961$	1.68839e12	1.97960
$962$	0	0
$963$	0	0
$964$	0	0
$965$	7.22863e11i	0.833579i
$966$	0	0
$967$	−4.85818e11	−0.555607	−0.277804	−	0.960638i	$-0.589606\pi$
$967$	−4.85818e11	−0.555607	−0.277804	+	0.960638i	$0.589606\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.02451e11i	0.115249i	0.998338	+	0.0576246i	$0.0183526\pi$
$971$	1.02451e11i	0.115249i	−0.998338	+	0.0576246i	$0.981647\pi$
$972$	0	0
$973$	5.18062e11	0.578003
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.39842e12i	− 1.53483i	−0.641153	−	0.767413i	$-0.721544\pi$
$977$	− 1.39842e12i	− 1.53483i	0.641153	−	0.767413i	$-0.278456\pi$
$978$	0	0
$979$	1.05071e12	1.14380
$980$	0	0
$981$	0	0
$982$	0	0
$983$	2.77559e11i	0.297264i	0.988893	+	0.148632i	$0.0474869\pi$
$983$	2.77559e11i	0.297264i	−0.988893	+	0.148632i	$0.952513\pi$
$984$	0	0
$985$	1.31898e12	1.40118
$986$	0	0
$987$	0	0
$988$	0	0
$989$	4.65011e11i	0.486047i
$990$	0	0
$991$	−2.27286e11	−0.235656	−0.117828	−	0.993034i	$-0.537593\pi$
$991$	−2.27286e11	−0.235656	−0.117828	+	0.993034i	$0.537593\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	4.97772e11i	0.507853i
$996$	0	0
$997$	−5.71256e11	−0.578163	−0.289082	−	0.957304i	$-0.593350\pi$
$997$	−5.71256e11	−0.578163	−0.289082	+	0.957304i	$0.593350\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.9.c.b.161.5	✓	16
3.2	odd	2	inner	324.9.c.b.161.12	yes	16
9.2	odd	6		324.9.g.h.53.5		32
9.4	even	3		324.9.g.h.269.5		32
9.5	odd	6		324.9.g.h.269.12		32
9.7	even	3		324.9.g.h.53.12		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.9.c.b.161.5	✓	16	1.1	even	1	trivial
324.9.c.b.161.12	yes	16	3.2	odd	2	inner
324.9.g.h.53.5		32	9.2	odd	6
324.9.g.h.53.12		32	9.7	even	3
324.9.g.h.269.5		32	9.4	even	3
324.9.g.h.269.12		32	9.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 9 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-501.563i q^{5} -3090.68 q^{7} -11241.7i q^{11} +3930.44 q^{13} +46705.5i q^{17} +131459. q^{19} -384942. i q^{23} +139059. q^{25} -817545. i q^{29} +1.59414e6 q^{31} +1.55017e6i q^{35} -645124. q^{37} -1.32509e6i q^{41} -1.20800e6 q^{43} -453328. i q^{47} +3.78749e6 q^{49} -1.24042e7i q^{53} -5.63843e6 q^{55} +2.58079e6i q^{59} -1.28437e7 q^{61} -1.97136e6i q^{65} -2.98465e7 q^{67} +2.77799e6i q^{71} +7.42060e6 q^{73} +3.47445e7i q^{77} +2.32710e7 q^{79} +1.96704e7i q^{83} +2.34257e7 q^{85} +9.34649e7i q^{89} -1.21477e7 q^{91} -6.59349e7i q^{95} +6.07429e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 3692 q^{7} - 50380 q^{13} + 185108 q^{19} - 1958288 q^{25} - 285568 q^{31} - 2910052 q^{37} - 8122852 q^{43} + 14602560 q^{49} + 38320164 q^{55} - 5700244 q^{61} - 6328540 q^{67} - 51728704 q^{73}+ \cdots - 181751968 q^{97}+O(q^{100}) \) 16 * q + 3692 * q^7 - 50380 * q^13 + 185108 * q^19 - 1958288 * q^25 - 285568 * q^31 - 2910052 * q^37 - 8122852 * q^43 + 14602560 * q^49 + 38320164 * q^55 - 5700244 * q^61 - 6328540 * q^67 - 51728704 * q^73 + 168603092 * q^79 - 96322644 * q^85 - 199743716 * q^91 - 181751968 * q^97

Introduction

L-functions

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Database

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