LMFDB - Embedded newform 324.5.c.b.161.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,5,Mod(161,324)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("324.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(324, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 324 = 2^{2} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	324.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,52] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$33.4918680392$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 130x^{6} + 404x^{5} + 7007x^{4} - 14692x^{3} - 164750x^{2} + 172164x + 1445046 $ x^8 - 4x^7 - 130x^6 + 404x^5 + 7007x^4 - 14692x^3 - 164750x^2 + 172164*x + 1445046
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{20} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$7.57859 - 1.93185i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.5.c.b.161.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-26.3577i q^{5} +20.4355 q^{7} -54.1285i q^{11} -238.301 q^{13} -195.038i q^{17} +59.1338 q^{19} -269.521i q^{23} -69.7268 q^{25} +1193.70i q^{29} +373.847 q^{31} -538.632i q^{35} -1793.37 q^{37} +361.849i q^{41} -3241.91 q^{43} +3519.76i q^{47} -1983.39 q^{49} -1522.31i q^{53} -1426.70 q^{55} -6068.46i q^{59} +899.781 q^{61} +6281.06i q^{65} -6954.12 q^{67} -9017.89i q^{71} -3063.58 q^{73} -1106.14i q^{77} -5579.53 q^{79} +4248.36i q^{83} -5140.73 q^{85} -6532.00i q^{89} -4869.80 q^{91} -1558.63i q^{95} -4567.53 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) $ 8 * q + 52 * q^7 + 220 * q^13 - 308 * q^19 - 328 * q^25 - 1472 * q^31 + 1228 * q^37 + 388 * q^43 + 3432 * q^49 - 36 * q^55 + 1804 * q^61 - 2180 * q^67 - 4544 * q^73 - 5012 * q^79 - 5364 * q^85 - 1756 * q^91 - 2528 * 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
$5$	− 26.3577i	− 1.05431i	−0.849770	−	0.527153i	$-0.823259\pi$
$5$	− 26.3577i	− 1.05431i	0.849770	−	0.527153i	$-0.176741\pi$
$6$	0	0
$7$	20.4355	0.417051	0.208525	−	0.978017i	$-0.433134\pi$
$7$	20.4355	0.417051	0.208525	+	0.978017i	$0.433134\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 54.1285i	− 0.447343i	−0.974665	−	0.223672i	$-0.928196\pi$
$11$	− 54.1285i	− 0.447343i	0.974665	−	0.223672i	$-0.0718043\pi$
$12$	0	0
$13$	−238.301	−1.41006	−0.705032	−	0.709175i	$-0.749068\pi$
$13$	−238.301	−1.41006	−0.705032	+	0.709175i	$0.749068\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 195.038i	− 0.674870i	−0.941349	−	0.337435i	$-0.890441\pi$
$17$	− 195.038i	− 0.674870i	0.941349	−	0.337435i	$-0.109559\pi$
$18$	0	0
$19$	59.1338	0.163806	0.0819028	−	0.996640i	$-0.473900\pi$
$19$	59.1338	0.163806	0.0819028	+	0.996640i	$0.473900\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 269.521i	− 0.509492i	−0.967008	−	0.254746i	$-0.918008\pi$
$23$	− 269.521i	− 0.509492i	0.967008	−	0.254746i	$-0.0819919\pi$
$24$	0	0
$25$	−69.7268	−0.111563
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1193.70i	1.41938i	0.704514	+	0.709690i	$0.251165\pi$
$29$	1193.70i	1.41938i	−0.704514	+	0.709690i	$0.748835\pi$
$30$	0	0
$31$	373.847	0.389018	0.194509	−	0.980901i	$-0.437689\pi$
$31$	373.847	0.389018	0.194509	+	0.980901i	$0.437689\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 538.632i	− 0.439699i
$36$	0	0
$37$	−1793.37	−1.30999	−0.654994	−	0.755634i	$-0.727329\pi$
$37$	−1793.37	−1.30999	−0.654994	+	0.755634i	$0.727329\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	361.849i	0.215258i	0.994191	+	0.107629i	$0.0343259\pi$
$41$	361.849i	0.215258i	−0.994191	+	0.107629i	$0.965674\pi$
$42$	0	0
$43$	−3241.91	−1.75333	−0.876666	−	0.481099i	$-0.840238\pi$
$43$	−3241.91	−1.75333	−0.876666	+	0.481099i	$0.840238\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3519.76i	1.59337i	0.604394	+	0.796686i	$0.293415\pi$
$47$	3519.76i	1.59337i	−0.604394	+	0.796686i	$0.706585\pi$
$48$	0	0
$49$	−1983.39	−0.826069
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 1522.31i	− 0.541940i	−0.962588	−	0.270970i	$-0.912656\pi$
$53$	− 1522.31i	− 0.541940i	0.962588	−	0.270970i	$-0.0873443\pi$
$54$	0	0
$55$	−1426.70	−0.471637
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 6068.46i	− 1.74331i	−0.490121	−	0.871655i	$-0.663047\pi$
$59$	− 6068.46i	− 1.74331i	0.490121	−	0.871655i	$-0.336953\pi$
$60$	0	0
$61$	899.781	0.241812	0.120906	−	0.992664i	$-0.461420\pi$
$61$	899.781	0.241812	0.120906	+	0.992664i	$0.461420\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	6281.06i	1.48664i
$66$	0	0
$67$	−6954.12	−1.54915	−0.774574	−	0.632483i	$-0.782036\pi$
$67$	−6954.12	−1.54915	−0.774574	+	0.632483i	$0.782036\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	− 9017.89i	− 1.78891i	−0.447160	−	0.894454i	$-0.647564\pi$
$71$	− 9017.89i	− 1.78891i	0.447160	−	0.894454i	$-0.352436\pi$
$72$	0	0
$73$	−3063.58	−0.574888	−0.287444	−	0.957797i	$-0.592806\pi$
$73$	−3063.58	−0.574888	−0.287444	+	0.957797i	$0.592806\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 1106.14i	− 0.186565i
$78$	0	0
$79$	−5579.53	−0.894012	−0.447006	−	0.894531i	$-0.647510\pi$
$79$	−5579.53	−0.894012	−0.447006	+	0.894531i	$0.647510\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4248.36i	0.616688i	0.951275	+	0.308344i	$0.0997748\pi$
$83$	4248.36i	0.616688i	−0.951275	+	0.308344i	$0.900225\pi$
$84$	0	0
$85$	−5140.73	−0.711520
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6532.00i	− 0.824644i	−0.911038	−	0.412322i	$-0.864718\pi$
$89$	− 6532.00i	− 0.824644i	0.911038	−	0.412322i	$-0.135282\pi$
$90$	0	0
$91$	−4869.80	−0.588069
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 1558.63i	− 0.172701i
$96$	0	0
$97$	−4567.53	−0.485443	−0.242722	−	0.970096i	$-0.578040\pi$
$97$	−4567.53	−0.485443	−0.242722	+	0.970096i	$0.578040\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	18560.7i	1.81950i	0.415157	+	0.909750i	$0.363727\pi$
$101$	18560.7i	1.81950i	−0.415157	+	0.909750i	$0.636273\pi$
$102$	0	0
$103$	5070.63	0.477955	0.238978	−	0.971025i	$-0.423188\pi$
$103$	5070.63	0.477955	0.238978	+	0.971025i	$0.423188\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	16248.3i	1.41919i	0.704609	+	0.709595i	$0.251122\pi$
$107$	16248.3i	1.41919i	−0.704609	+	0.709595i	$0.748878\pi$
$108$	0	0
$109$	−2690.46	−0.226451	−0.113225	−	0.993569i	$-0.536118\pi$
$109$	−2690.46	−0.226451	−0.113225	+	0.993569i	$0.536118\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 5097.49i	− 0.399208i	−0.979877	−	0.199604i	$-0.936034\pi$
$113$	− 5097.49i	− 0.399208i	0.979877	−	0.199604i	$-0.0639656\pi$
$114$	0	0
$115$	−7103.96	−0.537161
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3985.69i	− 0.281455i
$120$	0	0
$121$	11711.1	0.799884
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 14635.7i	− 0.936685i
$126$	0	0
$127$	−11705.0	−0.725714	−0.362857	−	0.931845i	$-0.618199\pi$
$127$	−11705.0	−0.725714	−0.362857	+	0.931845i	$0.618199\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12122.3i	0.706389i	0.935550	+	0.353194i	$0.114905\pi$
$131$	12122.3i	0.706389i	−0.935550	+	0.353194i	$0.885095\pi$
$132$	0	0
$133$	1208.43	0.0683153
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 23958.5i	− 1.27649i	−0.769833	−	0.638246i	$-0.779660\pi$
$137$	− 23958.5i	− 1.27649i	0.769833	−	0.638246i	$-0.220340\pi$
$138$	0	0
$139$	−19547.4	−1.01172	−0.505858	−	0.862617i	$-0.668824\pi$
$139$	−19547.4	−1.01172	−0.505858	+	0.862617i	$0.668824\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	12898.9i	0.630783i
$144$	0	0
$145$	31463.1	1.49646
$146$	0	0
$147$	0	0
$148$	0	0
$149$	9267.88i	0.417453i	0.977974	+	0.208727i	$0.0669319\pi$
$149$	9267.88i	0.417453i	−0.977974	+	0.208727i	$0.933068\pi$
$150$	0	0
$151$	31783.3	1.39395	0.696973	−	0.717098i	$-0.254530\pi$
$151$	31783.3	1.39395	0.696973	+	0.717098i	$0.254530\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 9853.73i	− 0.410145i
$156$	0	0
$157$	17103.5	0.693883	0.346941	−	0.937887i	$-0.387220\pi$
$157$	17103.5	0.693883	0.346941	+	0.937887i	$0.387220\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 5507.80i	− 0.212484i
$162$	0	0
$163$	14463.6	0.544379	0.272189	−	0.962244i	$-0.412252\pi$
$163$	14463.6	0.544379	0.272189	+	0.962244i	$0.412252\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 25314.2i	− 0.907678i	−0.891084	−	0.453839i	$-0.850054\pi$
$167$	− 25314.2i	− 0.907678i	0.891084	−	0.453839i	$-0.149946\pi$
$168$	0	0
$169$	28226.3	0.988282
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 40904.9i	− 1.36673i	−0.730077	−	0.683365i	$-0.760516\pi$
$173$	− 40904.9i	− 1.36673i	0.730077	−	0.683365i	$-0.239484\pi$
$174$	0	0
$175$	−1424.90	−0.0465274
$176$	0	0
$177$	0	0
$178$	0	0
$179$	45433.1i	1.41797i	0.705225	+	0.708984i	$0.250846\pi$
$179$	45433.1i	1.41797i	−0.705225	+	0.708984i	$0.749154\pi$
$180$	0	0
$181$	41107.8	1.25478	0.627389	−	0.778706i	$-0.284124\pi$
$181$	41107.8	1.25478	0.627389	+	0.778706i	$0.284124\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	47269.1i	1.38113i
$186$	0	0
$187$	−10557.1	−0.301899
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 60676.3i	− 1.66323i	−0.555353	−	0.831615i	$-0.687417\pi$
$191$	− 60676.3i	− 1.66323i	0.555353	−	0.831615i	$-0.312583\pi$
$192$	0	0
$193$	44731.3	1.20087	0.600436	−	0.799673i	$-0.294994\pi$
$193$	44731.3	1.20087	0.600436	+	0.799673i	$0.294994\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	464.194i	0.0119610i	0.999982	+	0.00598050i	$0.00190366\pi$
$197$	464.194i	0.0119610i	−0.999982	+	0.00598050i	$0.998096\pi$
$198$	0	0
$199$	−48133.7	−1.21547	−0.607733	−	0.794142i	$-0.707921\pi$
$199$	−48133.7	−1.21547	−0.607733	+	0.794142i	$0.707921\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	24393.8i	0.591954i
$204$	0	0
$205$	9537.49	0.226948
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3200.83i	− 0.0732774i
$210$	0	0
$211$	32194.8	0.723138	0.361569	−	0.932345i	$-0.382241\pi$
$211$	32194.8	0.723138	0.361569	+	0.932345i	$0.382241\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	85449.2i	1.84855i
$216$	0	0
$217$	7639.74	0.162240
$218$	0	0
$219$	0	0
$220$	0	0
$221$	46477.6i	0.951611i
$222$	0	0
$223$	2292.15	0.0460927	0.0230464	−	0.999734i	$-0.492663\pi$
$223$	2292.15	0.0460927	0.0230464	+	0.999734i	$0.492663\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26794.9i	− 0.519997i	−0.965609	−	0.259998i	$-0.916278\pi$
$227$	− 26794.9i	− 0.519997i	0.965609	−	0.259998i	$-0.0837221\pi$
$228$	0	0
$229$	48060.1	0.916460	0.458230	−	0.888834i	$-0.348484\pi$
$229$	48060.1	0.916460	0.458230	+	0.888834i	$0.348484\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 27107.9i	− 0.499326i	−0.968333	−	0.249663i	$-0.919680\pi$
$233$	− 27107.9i	− 0.499326i	0.968333	−	0.249663i	$-0.0803199\pi$
$234$	0	0
$235$	92772.6	1.67990
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 61272.5i	− 1.07268i	−0.844003	−	0.536339i	$-0.819807\pi$
$239$	− 61272.5i	− 1.07268i	0.844003	−	0.536339i	$-0.180193\pi$
$240$	0	0
$241$	37849.9	0.651675	0.325837	−	0.945426i	$-0.394354\pi$
$241$	37849.9	0.651675	0.325837	+	0.945426i	$0.394354\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	52277.6i	0.870930i
$246$	0	0
$247$	−14091.6	−0.230977
$248$	0	0
$249$	0	0
$250$	0	0
$251$	25694.6i	0.407844i	0.978987	+	0.203922i	$0.0653688\pi$
$251$	25694.6i	0.407844i	−0.978987	+	0.203922i	$0.934631\pi$
$252$	0	0
$253$	−14588.8	−0.227918
$254$	0	0
$255$	0	0
$256$	0	0
$257$	25717.4i	0.389369i	0.980866	+	0.194685i	$0.0623683\pi$
$257$	25717.4i	0.389369i	−0.980866	+	0.194685i	$0.937632\pi$
$258$	0	0
$259$	−36648.4	−0.546331
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 99701.3i	− 1.44142i	−0.693239	−	0.720708i	$-0.743817\pi$
$263$	− 99701.3i	− 1.44142i	0.693239	−	0.720708i	$-0.256183\pi$
$264$	0	0
$265$	−40124.5	−0.571371
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 126040.i	− 1.74182i	−0.491445	−	0.870909i	$-0.663531\pi$
$269$	− 126040.i	− 1.74182i	0.491445	−	0.870909i	$-0.336469\pi$
$270$	0	0
$271$	−73237.4	−0.997228	−0.498614	−	0.866824i	$-0.666157\pi$
$271$	−73237.4	−0.997228	−0.498614	+	0.866824i	$0.666157\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	3774.21i	0.0499069i
$276$	0	0
$277$	104435.	1.36109	0.680547	−	0.732704i	$-0.261742\pi$
$277$	104435.	1.36109	0.680547	+	0.732704i	$0.261742\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	54894.5i	0.695210i	0.937641	+	0.347605i	$0.113005\pi$
$281$	54894.5i	0.695210i	−0.937641	+	0.347605i	$0.886995\pi$
$282$	0	0
$283$	−115201.	−1.43841	−0.719204	−	0.694799i	$-0.755493\pi$
$283$	−115201.	−1.43841	−0.719204	+	0.694799i	$0.755493\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7394.56i	0.0897735i
$288$	0	0
$289$	45481.4	0.544550
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 36524.3i	− 0.425448i	−0.977112	−	0.212724i	$-0.931766\pi$
$293$	− 36524.3i	− 0.425448i	0.977112	−	0.212724i	$-0.0682335\pi$
$294$	0	0
$295$	−159950.	−1.83798
$296$	0	0
$297$	0	0
$298$	0	0
$299$	64227.2i	0.718417i
$300$	0	0
$301$	−66250.1	−0.731229
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 23716.1i	− 0.254944i
$306$	0	0
$307$	−95335.4	−1.01153	−0.505763	−	0.862672i	$-0.668789\pi$
$307$	−95335.4	−1.01153	−0.505763	+	0.862672i	$0.668789\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 181809.i	− 1.87973i	−0.341545	−	0.939865i	$-0.610950\pi$
$311$	− 181809.i	− 1.87973i	0.341545	−	0.939865i	$-0.389050\pi$
$312$	0	0
$313$	60391.5	0.616434	0.308217	−	0.951316i	$-0.400268\pi$
$313$	60391.5	0.616434	0.308217	+	0.951316i	$0.400268\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	94986.4i	0.945242i	0.881266	+	0.472621i	$0.156692\pi$
$317$	94986.4i	0.945242i	−0.881266	+	0.472621i	$0.843308\pi$
$318$	0	0
$319$	64613.2	0.634950
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 11533.3i	− 0.110548i
$324$	0	0
$325$	16616.0	0.157311
$326$	0	0
$327$	0	0
$328$	0	0
$329$	71928.0i	0.664517i
$330$	0	0
$331$	−138129.	−1.26075	−0.630377	−	0.776289i	$-0.717100\pi$
$331$	−138129.	−1.26075	−0.630377	+	0.776289i	$0.717100\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	183295.i	1.63328i
$336$	0	0
$337$	80799.9	0.711461	0.355731	−	0.934589i	$-0.384232\pi$
$337$	80799.9	0.711461	0.355731	+	0.934589i	$0.384232\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 20235.8i	− 0.174025i
$342$	0	0
$343$	−89597.2	−0.761563
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 11663.2i	− 0.0968630i	−0.998827	−	0.0484315i	$-0.984578\pi$
$347$	− 11663.2i	− 0.0968630i	0.998827	−	0.0484315i	$-0.0154223\pi$
$348$	0	0
$349$	−156501.	−1.28489	−0.642447	−	0.766330i	$-0.722081\pi$
$349$	−156501.	−1.28489	−0.642447	+	0.766330i	$0.722081\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	196742.i	1.57887i	0.613831	+	0.789437i	$0.289628\pi$
$353$	196742.i	1.57887i	−0.613831	+	0.789437i	$0.710372\pi$
$354$	0	0
$355$	−237690.	−1.88606
$356$	0	0
$357$	0	0
$358$	0	0
$359$	− 36095.4i	− 0.280068i	−0.990147	−	0.140034i	$-0.955279\pi$
$359$	− 36095.4i	− 0.280068i	0.990147	−	0.140034i	$-0.0447211\pi$
$360$	0	0
$361$	−126824.	−0.973168
$362$	0	0
$363$	0	0
$364$	0	0
$365$	80748.8i	0.606108i
$366$	0	0
$367$	41547.9	0.308473	0.154236	−	0.988034i	$-0.450708\pi$
$367$	41547.9	0.308473	0.154236	+	0.988034i	$0.450708\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 31109.1i	− 0.226016i
$372$	0	0
$373$	174472.	1.25403	0.627017	−	0.779006i	$-0.284276\pi$
$373$	174472.	1.25403	0.627017	+	0.779006i	$0.284276\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 284460.i	− 2.00142i
$378$	0	0
$379$	−164788.	−1.14722	−0.573612	−	0.819127i	$-0.694458\pi$
$379$	−164788.	−1.14722	−0.573612	+	0.819127i	$0.694458\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 12680.8i	− 0.0864468i	−0.999065	−	0.0432234i	$-0.986237\pi$
$383$	− 12680.8i	− 0.0864468i	0.999065	−	0.0432234i	$-0.0137627\pi$
$384$	0	0
$385$	−29155.4	−0.196697
$386$	0	0
$387$	0	0
$388$	0	0
$389$	64430.6i	0.425787i	0.977075	+	0.212894i	$0.0682888\pi$
$389$	64430.6i	0.425787i	−0.977075	+	0.212894i	$0.931711\pi$
$390$	0	0
$391$	−52566.8	−0.343841
$392$	0	0
$393$	0	0
$394$	0	0
$395$	147063.i	0.942563i
$396$	0	0
$397$	143446.	0.910135	0.455068	−	0.890457i	$-0.349615\pi$
$397$	143446.	0.910135	0.455068	+	0.890457i	$0.349615\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 212374.i	− 1.32073i	−0.750946	−	0.660364i	$-0.770402\pi$
$401$	− 212374.i	− 1.32073i	0.750946	−	0.660364i	$-0.229598\pi$
$402$	0	0
$403$	−89088.0	−0.548541
$404$	0	0
$405$	0	0
$406$	0	0
$407$	97072.6i	0.586014i
$408$	0	0
$409$	−212213.	−1.26860	−0.634301	−	0.773086i	$-0.718712\pi$
$409$	−212213.	−1.26860	−0.634301	+	0.773086i	$0.718712\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 124012.i	− 0.727048i
$414$	0	0
$415$	111977.	0.650178
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 121549.i	− 0.692347i	−0.938170	−	0.346174i	$-0.887481\pi$
$419$	− 121549.i	− 0.692347i	0.938170	−	0.346174i	$-0.112519\pi$
$420$	0	0
$421$	63011.6	0.355514	0.177757	−	0.984074i	$-0.443116\pi$
$421$	63011.6	0.355514	0.177757	+	0.984074i	$0.443116\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	13599.3i	0.0752905i
$426$	0	0
$427$	18387.5	0.100848
$428$	0	0
$429$	0	0
$430$	0	0
$431$	128883.i	0.693812i	0.937900	+	0.346906i	$0.112768\pi$
$431$	128883.i	0.693812i	−0.937900	+	0.346906i	$0.887232\pi$
$432$	0	0
$433$	6365.28	0.0339502	0.0169751	−	0.999856i	$-0.494596\pi$
$433$	6365.28	0.0339502	0.0169751	+	0.999856i	$0.494596\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 15937.8i	− 0.0834577i
$438$	0	0
$439$	350295.	1.81763	0.908815	−	0.417200i	$-0.136988\pi$
$439$	350295.	1.81763	0.908815	+	0.417200i	$0.136988\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 186346.i	− 0.949537i	−0.880111	−	0.474769i	$-0.842532\pi$
$443$	− 186346.i	− 0.949537i	0.880111	−	0.474769i	$-0.157468\pi$
$444$	0	0
$445$	−172168.	−0.869427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 46711.0i	− 0.231700i	−0.993267	−	0.115850i	$-0.963041\pi$
$449$	− 46711.0i	− 0.231700i	0.993267	−	0.115850i	$-0.0369593\pi$
$450$	0	0
$451$	19586.3	0.0962943
$452$	0	0
$453$	0	0
$454$	0	0
$455$	128356.i	0.620005i
$456$	0	0
$457$	−290223.	−1.38963	−0.694816	−	0.719188i	$-0.744514\pi$
$457$	−290223.	−1.38963	−0.694816	+	0.719188i	$0.744514\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 237263.i	− 1.11642i	−0.829699	−	0.558211i	$-0.811488\pi$
$461$	− 237263.i	− 1.11642i	0.829699	−	0.558211i	$-0.188512\pi$
$462$	0	0
$463$	−126164.	−0.588538	−0.294269	−	0.955723i	$-0.595076\pi$
$463$	−126164.	−0.588538	−0.294269	+	0.955723i	$0.595076\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20073.4i	0.0920421i	0.998940	+	0.0460210i	$0.0146541\pi$
$467$	20073.4i	0.0920421i	−0.998940	+	0.0460210i	$0.985346\pi$
$468$	0	0
$469$	−142111.	−0.646073
$470$	0	0
$471$	0	0
$472$	0	0
$473$	175480.i	0.784342i
$474$	0	0
$475$	−4123.21	−0.0182746
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1426.46i	0.00621712i	0.999995	+	0.00310856i	$0.000989486\pi$
$479$	1426.46i	0.00621712i	−0.999995	+	0.00310856i	$0.999011\pi$
$480$	0	0
$481$	427362.	1.84717
$482$	0	0
$483$	0	0
$484$	0	0
$485$	120390.i	0.511806i
$486$	0	0
$487$	74085.9	0.312376	0.156188	−	0.987727i	$-0.450079\pi$
$487$	74085.9	0.312376	0.156188	+	0.987727i	$0.450079\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	113556.i	0.471029i	0.971871	+	0.235514i	$0.0756775\pi$
$491$	113556.i	0.471029i	−0.971871	+	0.235514i	$0.924322\pi$
$492$	0	0
$493$	232816.	0.957898
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 184285.i	− 0.746066i
$498$	0	0
$499$	280796.	1.12769	0.563845	−	0.825881i	$-0.309322\pi$
$499$	280796.	1.12769	0.563845	+	0.825881i	$0.309322\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	89107.8i	0.352192i	0.984373	+	0.176096i	$0.0563470\pi$
$503$	89107.8i	0.352192i	−0.984373	+	0.176096i	$0.943653\pi$
$504$	0	0
$505$	489217.	1.91831
$506$	0	0
$507$	0	0
$508$	0	0
$509$	117960.i	0.455302i	0.973743	+	0.227651i	$0.0731046\pi$
$509$	117960.i	0.455302i	−0.973743	+	0.227651i	$0.926895\pi$
$510$	0	0
$511$	−62605.7	−0.239757
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 133650.i	− 0.503911i
$516$	0	0
$517$	190519.	0.712784
$518$	0	0
$519$	0	0
$520$	0	0
$521$	316710.i	1.16677i	0.812194	+	0.583387i	$0.198273\pi$
$521$	316710.i	1.16677i	−0.812194	+	0.583387i	$0.801727\pi$
$522$	0	0
$523$	−390205.	−1.42656	−0.713280	−	0.700880i	$-0.752791\pi$
$523$	−390205.	−1.42656	−0.713280	+	0.700880i	$0.752791\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 72914.1i	− 0.262537i
$528$	0	0
$529$	207199.	0.740418
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 86228.9i	− 0.303528i
$534$	0	0
$535$	428268.	1.49626
$536$	0	0
$537$	0	0
$538$	0	0
$539$	107358.i	0.369536i
$540$	0	0
$541$	99617.9	0.340363	0.170182	−	0.985413i	$-0.445565\pi$
$541$	99617.9	0.340363	0.170182	+	0.985413i	$0.445565\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	70914.3i	0.238749i
$546$	0	0
$547$	248048.	0.829012	0.414506	−	0.910047i	$-0.363954\pi$
$547$	248048.	0.829012	0.414506	+	0.910047i	$0.363954\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	70588.0i	0.232503i
$552$	0	0
$553$	−114020.	−0.372848
$554$	0	0
$555$	0	0
$556$	0	0
$557$	242800.i	0.782598i	0.920264	+	0.391299i	$0.127974\pi$
$557$	242800.i	0.782598i	−0.920264	+	0.391299i	$0.872026\pi$
$558$	0	0
$559$	772551.	2.47231
$560$	0	0
$561$	0	0
$562$	0	0
$563$	566496.i	1.78723i	0.448837	+	0.893613i	$0.351838\pi$
$563$	566496.i	1.78723i	−0.448837	+	0.893613i	$0.648162\pi$
$564$	0	0
$565$	−134358.	−0.420888
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 562499.i	− 1.73739i	−0.495347	−	0.868695i	$-0.664959\pi$
$569$	− 562499.i	− 1.73739i	0.495347	−	0.868695i	$-0.335041\pi$
$570$	0	0
$571$	−155999.	−0.478463	−0.239231	−	0.970963i	$-0.576895\pi$
$571$	−155999.	−0.478463	−0.239231	+	0.970963i	$0.576895\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	18792.9i	0.0568404i
$576$	0	0
$577$	−374817.	−1.12582	−0.562908	−	0.826519i	$-0.690318\pi$
$577$	−374817.	−1.12582	−0.562908	+	0.826519i	$0.690318\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	86817.4i	0.257190i
$582$	0	0
$583$	−82400.3	−0.242433
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 527481.i	− 1.53084i	−0.643530	−	0.765421i	$-0.722531\pi$
$587$	− 527481.i	− 1.53084i	0.643530	−	0.765421i	$-0.277469\pi$
$588$	0	0
$589$	22107.0	0.0637234
$590$	0	0
$591$	0	0
$592$	0	0
$593$	316522.i	0.900108i	0.893001	+	0.450054i	$0.148595\pi$
$593$	316522.i	0.900108i	−0.893001	+	0.450054i	$0.851405\pi$
$594$	0	0
$595$	−105053.	−0.296740
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 379683.i	− 1.05820i	−0.848559	−	0.529100i	$-0.822529\pi$
$599$	− 379683.i	− 1.05820i	0.848559	−	0.529100i	$-0.177471\pi$
$600$	0	0
$601$	441678.	1.22280	0.611402	−	0.791320i	$-0.290606\pi$
$601$	441678.	1.22280	0.611402	+	0.791320i	$0.290606\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 308677.i	− 0.843323i
$606$	0	0
$607$	−82363.1	−0.223540	−0.111770	−	0.993734i	$-0.535652\pi$
$607$	−82363.1	−0.223540	−0.111770	+	0.993734i	$0.535652\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 838762.i	− 2.24676i
$612$	0	0
$613$	154087.	0.410057	0.205029	−	0.978756i	$-0.434271\pi$
$613$	154087.	0.410057	0.205029	+	0.978756i	$0.434271\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 339303.i	− 0.891287i	−0.895210	−	0.445644i	$-0.852975\pi$
$617$	− 339303.i	− 0.891287i	0.895210	−	0.445644i	$-0.147025\pi$
$618$	0	0
$619$	−122692.	−0.320209	−0.160105	−	0.987100i	$-0.551183\pi$
$619$	−122692.	−0.320209	−0.160105	+	0.987100i	$0.551183\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 133485.i	− 0.343918i
$624$	0	0
$625$	−429342.	−1.09912
$626$	0	0
$627$	0	0
$628$	0	0
$629$	349775.i	0.884071i
$630$	0	0
$631$	517548.	1.29985	0.649923	−	0.760000i	$-0.274801\pi$
$631$	517548.	1.29985	0.649923	+	0.760000i	$0.274801\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	308517.i	0.765125i
$636$	0	0
$637$	472644.	1.16481
$638$	0	0
$639$	0	0
$640$	0	0
$641$	663334.i	1.61442i	0.590265	+	0.807209i	$0.299023\pi$
$641$	663334.i	1.61442i	−0.590265	+	0.807209i	$0.700977\pi$
$642$	0	0
$643$	388977.	0.940810	0.470405	−	0.882451i	$-0.344108\pi$
$643$	388977.	0.940810	0.470405	+	0.882451i	$0.344108\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	464654.i	1.11000i	0.831852	+	0.554998i	$0.187281\pi$
$647$	464654.i	1.11000i	−0.831852	+	0.554998i	$0.812719\pi$
$648$	0	0
$649$	−328477.	−0.779858
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 339224.i	− 0.795537i	−0.917486	−	0.397768i	$-0.869785\pi$
$653$	− 339224.i	− 0.795537i	0.917486	−	0.397768i	$-0.130215\pi$
$654$	0	0
$655$	319517.	0.744750
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 215923.i	− 0.497196i	−0.968607	−	0.248598i	$-0.920030\pi$
$659$	− 215923.i	− 0.497196i	0.968607	−	0.248598i	$-0.0799698\pi$
$660$	0	0
$661$	39052.6	0.0893813	0.0446906	−	0.999001i	$-0.485770\pi$
$661$	39052.6	0.0893813	0.0446906	+	0.999001i	$0.485770\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 31851.4i	− 0.0720253i
$666$	0	0
$667$	321727.	0.723163
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 48703.8i	− 0.108173i
$672$	0	0
$673$	−832865.	−1.83884	−0.919421	−	0.393276i	$-0.871342\pi$
$673$	−832865.	−1.83884	−0.919421	+	0.393276i	$0.871342\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	338988.i	0.739617i	0.929108	+	0.369809i	$0.120577\pi$
$677$	338988.i	0.739617i	−0.929108	+	0.369809i	$0.879423\pi$
$678$	0	0
$679$	−93339.8	−0.202454
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 928607.i	− 1.99063i	−0.0966825	−	0.995315i	$-0.530823\pi$
$683$	− 928607.i	− 1.99063i	0.0966825	−	0.995315i	$-0.469177\pi$
$684$	0	0
$685$	−631489.	−1.34581
$686$	0	0
$687$	0	0
$688$	0	0
$689$	362767.i	0.764170i
$690$	0	0
$691$	219271.	0.459224	0.229612	−	0.973282i	$-0.426254\pi$
$691$	219271.	0.459224	0.229612	+	0.973282i	$0.426254\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	515223.i	1.06666i
$696$	0	0
$697$	70574.1	0.145271
$698$	0	0
$699$	0	0
$700$	0	0
$701$	118463.i	0.241071i	0.992709	+	0.120536i	$0.0384612\pi$
$701$	118463.i	0.241071i	−0.992709	+	0.120536i	$0.961539\pi$
$702$	0	0
$703$	−106049.	−0.214583
$704$	0	0
$705$	0	0
$706$	0	0
$707$	379297.i	0.758824i
$708$	0	0
$709$	−814890.	−1.62109	−0.810544	−	0.585678i	$-0.800828\pi$
$709$	−814890.	−1.62109	−0.810544	+	0.585678i	$0.800828\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 100760.i	− 0.198202i
$714$	0	0
$715$	339984.	0.665039
$716$	0	0
$717$	0	0
$718$	0	0
$719$	789.716i	0.00152761i	1.00000		0.000763806i	$0.000243127\pi$
$719$	789.716i	0.00152761i	−1.00000		0.000763806i	$0.999757\pi$
$720$	0	0
$721$	103621.	0.199332
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 83232.8i	− 0.158350i
$726$	0	0
$727$	−601293.	−1.13767	−0.568836	−	0.822451i	$-0.692606\pi$
$727$	−601293.	−1.13767	−0.568836	+	0.822451i	$0.692606\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	632294.i	1.18327i
$732$	0	0
$733$	−1.00924e6	−1.87838	−0.939192	−	0.343391i	$-0.888424\pi$
$733$	−1.00924e6	−1.87838	−0.939192	+	0.343391i	$0.888424\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	376417.i	0.693001i
$738$	0	0
$739$	−753999.	−1.38065	−0.690323	−	0.723502i	$-0.742531\pi$
$739$	−753999.	−1.38065	−0.690323	+	0.723502i	$0.742531\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 197835.i	− 0.358365i	−0.983816	−	0.179182i	$-0.942655\pi$
$743$	− 197835.i	− 0.358365i	0.983816	−	0.179182i	$-0.0573453\pi$
$744$	0	0
$745$	244280.	0.440124
$746$	0	0
$747$	0	0
$748$	0	0
$749$	332042.i	0.591875i
$750$	0	0
$751$	772087.	1.36895	0.684473	−	0.729038i	$-0.260032\pi$
$751$	772087.	1.36895	0.684473	+	0.729038i	$0.260032\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 837735.i	− 1.46965i
$756$	0	0
$757$	−459844.	−0.802452	−0.401226	−	0.915979i	$-0.631416\pi$
$757$	−459844.	−0.802452	−0.401226	+	0.915979i	$0.631416\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 859603.i	− 1.48432i	−0.670221	−	0.742161i	$-0.733801\pi$
$761$	− 859603.i	− 1.48432i	0.670221	−	0.742161i	$-0.266199\pi$
$762$	0	0
$763$	−54980.9	−0.0944415
$764$	0	0
$765$	0	0
$766$	0	0
$767$	1.44612e6i	2.45818i
$768$	0	0
$769$	5919.28	0.0100096	0.00500479	−	0.999987i	$-0.498407\pi$
$769$	5919.28	0.0100096	0.00500479	+	0.999987i	$0.498407\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	80749.0i	0.135138i	0.997715	+	0.0675691i	$0.0215243\pi$
$773$	80749.0i	0.135138i	−0.997715	+	0.0675691i	$0.978476\pi$
$774$	0	0
$775$	−26067.1	−0.0434000
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21397.5i	0.0352605i
$780$	0	0
$781$	−488125.	−0.800256
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 450809.i	− 0.731565i
$786$	0	0
$787$	−1.06105e6	−1.71312	−0.856558	−	0.516050i	$-0.827402\pi$
$787$	−1.06105e6	−1.71312	−0.856558	+	0.516050i	$0.827402\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 104170.i	− 0.166490i
$792$	0	0
$793$	−214419.	−0.340970
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 929793.i	− 1.46376i	−0.681434	−	0.731879i	$-0.738643\pi$
$797$	− 929793.i	− 1.46376i	0.681434	−	0.731879i	$-0.261357\pi$
$798$	0	0
$799$	686485.	1.07532
$800$	0	0
$801$	0	0
$802$	0	0
$803$	165827.i	0.257172i
$804$	0	0
$805$	−145173.	−0.224024
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 181072.i	− 0.276665i	−0.990386	−	0.138333i	$-0.955826\pi$
$809$	− 181072.i	− 0.276665i	0.990386	−	0.138333i	$-0.0441743\pi$
$810$	0	0
$811$	49745.9	0.0756338	0.0378169	−	0.999285i	$-0.487960\pi$
$811$	49745.9	0.0756338	0.0378169	+	0.999285i	$0.487960\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 381227.i	− 0.573942i
$816$	0	0
$817$	−191707.	−0.287206
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 6952.22i	− 0.0103142i	−0.999987	−	0.00515712i	$-0.998358\pi$
$821$	− 6952.22i	− 0.0103142i	0.999987	−	0.00515712i	$-0.00164157\pi$
$822$	0	0
$823$	−758252.	−1.11947	−0.559737	−	0.828670i	$-0.689098\pi$
$823$	−758252.	−1.11947	−0.559737	+	0.828670i	$0.689098\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 641019.i	− 0.937260i	−0.883394	−	0.468630i	$-0.844748\pi$
$827$	− 641019.i	− 0.937260i	0.883394	−	0.468630i	$-0.155252\pi$
$828$	0	0
$829$	549710.	0.799879	0.399940	−	0.916541i	$-0.369031\pi$
$829$	549710.	0.799879	0.399940	+	0.916541i	$0.369031\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	386836.i	0.557489i
$834$	0	0
$835$	−667224.	−0.956971
$836$	0	0
$837$	0	0
$838$	0	0
$839$	917081.i	1.30282i	0.758727	+	0.651409i	$0.225822\pi$
$839$	917081.i	1.30282i	−0.758727	+	0.651409i	$0.774178\pi$
$840$	0	0
$841$	−717636.	−1.01464
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 743980.i	− 1.04195i
$846$	0	0
$847$	239322.	0.333592
$848$	0	0
$849$	0	0
$850$	0	0
$851$	483352.i	0.667428i
$852$	0	0
$853$	−1.00142e6	−1.37632	−0.688160	−	0.725559i	$-0.741581\pi$
$853$	−1.00142e6	−1.37632	−0.688160	+	0.725559i	$0.741581\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 256079.i	− 0.348668i	−0.984687	−	0.174334i	$-0.944223\pi$
$857$	− 256079.i	− 0.348668i	0.984687	−	0.174334i	$-0.0557772\pi$
$858$	0	0
$859$	−326298.	−0.442209	−0.221105	−	0.975250i	$-0.570966\pi$
$859$	−326298.	−0.442209	−0.221105	+	0.975250i	$0.570966\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 952268.i	− 1.27861i	−0.768954	−	0.639304i	$-0.779222\pi$
$863$	− 952268.i	− 1.27861i	0.768954	−	0.639304i	$-0.220778\pi$
$864$	0	0
$865$	−1.07816e6	−1.44095
$866$	0	0
$867$	0	0
$868$	0	0
$869$	302012.i	0.399930i
$870$	0	0
$871$	1.65717e6	2.18440
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 299088.i	− 0.390645i
$876$	0	0
$877$	−21518.1	−0.0279773	−0.0139886	−	0.999902i	$-0.504453\pi$
$877$	−21518.1	−0.0279773	−0.0139886	+	0.999902i	$0.504453\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	784125.i	1.01026i	0.863043	+	0.505130i	$0.168556\pi$
$881$	784125.i	1.01026i	−0.863043	+	0.505130i	$0.831444\pi$
$882$	0	0
$883$	−67879.2	−0.0870593	−0.0435296	−	0.999052i	$-0.513860\pi$
$883$	−67879.2	−0.0870593	−0.0435296	+	0.999052i	$0.513860\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 605558.i	− 0.769678i	−0.922984	−	0.384839i	$-0.874257\pi$
$887$	− 605558.i	− 0.769678i	0.922984	−	0.384839i	$-0.125743\pi$
$888$	0	0
$889$	−239198.	−0.302659
$890$	0	0
$891$	0	0
$892$	0	0
$893$	208137.i	0.261003i
$894$	0	0
$895$	1.19751e6	1.49497
$896$	0	0
$897$	0	0
$898$	0	0
$899$	446260.i	0.552165i
$900$	0	0
$901$	−296907.	−0.365739
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 1.08350e6i	− 1.32292i
$906$	0	0
$907$	−51373.7	−0.0624491	−0.0312246	−	0.999512i	$-0.509941\pi$
$907$	−51373.7	−0.0624491	−0.0312246	+	0.999512i	$0.509941\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	370894.i	0.446902i	0.974715	+	0.223451i	$0.0717323\pi$
$911$	370894.i	0.446902i	−0.974715	+	0.223451i	$0.928268\pi$
$912$	0	0
$913$	229958.	0.275871
$914$	0	0
$915$	0	0
$916$	0	0
$917$	247726.i	0.294600i
$918$	0	0
$919$	891724.	1.05584	0.527922	−	0.849293i	$-0.322971\pi$
$919$	891724.	1.05584	0.527922	+	0.849293i	$0.322971\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	2.14897e6i	2.52248i
$924$	0	0
$925$	125046.	0.146146
$926$	0	0
$927$	0	0
$928$	0	0
$929$	682726.i	0.791071i	0.918451	+	0.395535i	$0.129441\pi$
$929$	682726.i	0.791071i	−0.918451	+	0.395535i	$0.870559\pi$
$930$	0	0
$931$	−117286.	−0.135315
$932$	0	0
$933$	0	0
$934$	0	0
$935$	278260.i	0.318294i
$936$	0	0
$937$	1.02178e6	1.16380	0.581899	−	0.813261i	$-0.302310\pi$
$937$	1.02178e6	1.16380	0.581899	+	0.813261i	$0.302310\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1.20924e6i	− 1.36563i	−0.730593	−	0.682813i	$-0.760756\pi$
$941$	− 1.20924e6i	− 1.36563i	0.730593	−	0.682813i	$-0.239244\pi$
$942$	0	0
$943$	97526.0	0.109672
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 547109.i	− 0.610062i	−0.952342	−	0.305031i	$-0.901333\pi$
$947$	− 547109.i	− 0.610062i	0.952342	−	0.305031i	$-0.0986668\pi$
$948$	0	0
$949$	730054.	0.810629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	251720.i	0.277161i	0.990351	+	0.138581i	$0.0442540\pi$
$953$	251720.i	0.277161i	−0.990351	+	0.138581i	$0.955746\pi$
$954$	0	0
$955$	−1.59929e6	−1.75355
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 489603.i	− 0.532362i
$960$	0	0
$961$	−783760.	−0.848665
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 1.17901e6i	− 1.26609i
$966$	0	0
$967$	1.81734e6	1.94349	0.971746	−	0.236028i	$-0.0758458\pi$
$967$	1.81734e6	1.94349	0.971746	+	0.236028i	$0.0758458\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	870954.i	0.923755i	0.886944	+	0.461878i	$0.152824\pi$
$971$	870954.i	0.923755i	−0.886944	+	0.461878i	$0.847176\pi$
$972$	0	0
$973$	−399460.	−0.421937
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.16437e6i	1.21983i	0.792465	+	0.609917i	$0.208797\pi$
$977$	1.16437e6i	1.21983i	−0.792465	+	0.609917i	$0.791203\pi$
$978$	0	0
$979$	−353568.	−0.368899
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 1.01777e6i	− 1.05328i	−0.850088	−	0.526641i	$-0.823451\pi$
$983$	− 1.01777e6i	− 1.05328i	0.850088	−	0.526641i	$-0.176549\pi$
$984$	0	0
$985$	12235.1	0.0126106
$986$	0	0
$987$	0	0
$988$	0	0
$989$	873765.i	0.893309i
$990$	0	0
$991$	758495.	0.772334	0.386167	−	0.922429i	$-0.373799\pi$
$991$	758495.	0.772334	0.386167	+	0.922429i	$0.373799\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.26869e6i	1.28147i
$996$	0	0
$997$	483994.	0.486911	0.243456	−	0.969912i	$-0.421719\pi$
$997$	483994.	0.486911	0.243456	+	0.969912i	$0.421719\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.5.c.b.161.2	✓	8
3.2	odd	2	inner	324.5.c.b.161.7	yes	8
4.3	odd	2		1296.5.e.f.161.2		8
9.2	odd	6		324.5.g.f.53.2		16
9.4	even	3		324.5.g.f.269.2		16
9.5	odd	6		324.5.g.f.269.7		16
9.7	even	3		324.5.g.f.53.7		16
12.11	even	2		1296.5.e.f.161.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.2	✓	8	1.1	even	1	trivial
324.5.c.b.161.7	yes	8	3.2	odd	2	inner
324.5.g.f.53.2		16	9.2	odd	6
324.5.g.f.53.7		16	9.7	even	3
324.5.g.f.269.2		16	9.4	even	3
324.5.g.f.269.7		16	9.5	odd	6
1296.5.e.f.161.2		8	4.3	odd	2
1296.5.e.f.161.7		8	12.11	even	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 324 = 2^{2} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	324.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-26.3577i q^{5} +20.4355 q^{7} -54.1285i q^{11} -238.301 q^{13} -195.038i q^{17} +59.1338 q^{19} -269.521i q^{23} -69.7268 q^{25} +1193.70i q^{29} +373.847 q^{31} -538.632i q^{35} -1793.37 q^{37} +361.849i q^{41} -3241.91 q^{43} +3519.76i q^{47} -1983.39 q^{49} -1522.31i q^{53} -1426.70 q^{55} -6068.46i q^{59} +899.781 q^{61} +6281.06i q^{65} -6954.12 q^{67} -9017.89i q^{71} -3063.58 q^{73} -1106.14i q^{77} -5579.53 q^{79} +4248.36i q^{83} -5140.73 q^{85} -6532.00i q^{89} -4869.80 q^{91} -1558.63i q^{95} -4567.53 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) \) 8 * q + 52 * q^7 + 220 * q^13 - 308 * q^19 - 328 * q^25 - 1472 * q^31 + 1228 * q^37 + 388 * q^43 + 3432 * q^49 - 36 * q^55 + 1804 * q^61 - 2180 * q^67 - 4544 * q^73 - 5012 * q^79 - 5364 * q^85 - 1756 * q^91 - 2528 * q^97

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