LMFDB - Embedded newform 324.9.c.a.161.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [324,9,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, 9); 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, 9, names="a")

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:traces = [16,0,0,0,0,0,-1846] 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:	$131.990669660$
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} - 3 x^{15} + 1777 x^{14} - 2036 x^{13} - 1362401 x^{12} - 229443563 x^{11} + \cdots + 41\!\cdots\!00 $ x^16 - 3x^15 + 1777x^14 - 2036x^13 - 1362401x^12 - 229443563x^11 + 12565251279x^10 - 924267732266x^9 + 53903003620983x^8 - 669407211102857x^7 + 37488371175442929x^6 - 415198738773859682x^5 + 80523268287397086568x^4 - 10099200795405062595224x^3 + 445357305762676329633376x^2 - 15926706368268795333431520*x + 417889304743672757045502400
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{28}\cdot 3^{88} $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.6
Root			$39.4192 + 13.8761i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.9.c.a.161.11

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-318.248i q^{5} -3671.20 q^{7} +21286.9i q^{11} -29706.2 q^{13} +135818. i q^{17} -156912. q^{19} -78505.2i q^{23} +289344. q^{25} +888666. i q^{29} +745846. q^{31} +1.16835e6i q^{35} +1.59748e6 q^{37} -122830. i q^{41} -2.96183e6 q^{43} +224405. i q^{47} +7.71290e6 q^{49} -2.65410e6i q^{53} +6.77450e6 q^{55} -2.22710e7i q^{59} -2.21083e7 q^{61} +9.45394e6i q^{65} +1.04604e7 q^{67} +8.33031e6i q^{71} -4.29505e7 q^{73} -7.81483e7i q^{77} -1.02405e7 q^{79} +1.26098e7i q^{83} +4.32237e7 q^{85} +5.84973e7i q^{89} +1.09057e8 q^{91} +4.99369e7i q^{95} -7.32199e7 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 1846 q^{7} - 3370 q^{13} + 84518 q^{19} - 911606 q^{25} + 756614 q^{31} - 1671664 q^{37} - 679024 q^{43} + 6052890 q^{49} - 17497638 q^{55} - 34901410 q^{61} + 70053452 q^{67} + 28071218 q^{73} - 61378918 q^{79}+ \cdots + 66842156 q^{97}+O(q^{100}) $ 16 * q - 1846 * q^7 - 3370 * q^13 + 84518 * q^19 - 911606 * q^25 + 756614 * q^31 - 1671664 * q^37 - 679024 * q^43 + 6052890 * q^49 - 17497638 * q^55 - 34901410 * q^61 + 70053452 * q^67 + 28071218 * q^73 - 61378918 * q^79 + 79786044 * q^85 - 55534214 * q^91 + 66842156 * 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
$5$	− 318.248i	− 0.509196i	−0.967047	−	0.254598i	$-0.918057\pi$
$5$	− 318.248i	− 0.509196i	0.967047	−	0.254598i	$-0.0819432\pi$
$6$	0	0
$7$	−3671.20	−1.52903	−0.764514	−	0.644607i	$-0.777021\pi$
$7$	−3671.20	−1.52903	−0.764514	+	0.644607i	$0.777021\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	21286.9i	1.45392i	0.686679	+	0.726961i	$0.259068\pi$
$11$	21286.9i	1.45392i	−0.686679	+	0.726961i	$0.740932\pi$
$12$	0	0
$13$	−29706.2	−1.04010	−0.520049	−	0.854136i	$-0.674086\pi$
$13$	−29706.2	−1.04010	−0.520049	+	0.854136i	$0.674086\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	135818.i	1.62615i	0.582157	+	0.813077i	$0.302209\pi$
$17$	135818.i	1.62615i	−0.582157	+	0.813077i	$0.697791\pi$
$18$	0	0
$19$	−156912.	−1.20404	−0.602022	−	0.798480i	$-0.705638\pi$
$19$	−156912.	−1.20404	−0.602022	+	0.798480i	$0.705638\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 78505.2i	− 0.280535i	−0.990114	−	0.140268i	$-0.955204\pi$
$23$	− 78505.2i	− 0.280535i	0.990114	−	0.140268i	$-0.0447963\pi$
$24$	0	0
$25$	289344.	0.740719
$26$	0	0
$27$	0	0
$28$	0	0
$29$	888666.i	1.25645i	0.778030	+	0.628227i	$0.216219\pi$
$29$	888666.i	1.25645i	−0.778030	+	0.628227i	$0.783781\pi$
$30$	0	0
$31$	745846.	0.807612	0.403806	−	0.914845i	$-0.367687\pi$
$31$	745846.	0.807612	0.403806	+	0.914845i	$0.367687\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.16835e6i	0.778575i
$36$	0	0
$37$	1.59748e6	0.852370	0.426185	−	0.904636i	$-0.359857\pi$
$37$	1.59748e6	0.852370	0.426185	+	0.904636i	$0.359857\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 122830.i	− 0.0434678i	−0.999764	−	0.0217339i	$-0.993081\pi$
$41$	− 122830.i	− 0.0434678i	0.999764	−	0.0217339i	$-0.00691866\pi$
$42$	0	0
$43$	−2.96183e6	−0.866337	−0.433169	−	0.901313i	$-0.642605\pi$
$43$	−2.96183e6	−0.866337	−0.433169	+	0.901313i	$0.642605\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	224405.i	0.0459877i	0.999736	+	0.0229939i	$0.00731982\pi$
$47$	224405.i	0.0459877i	−0.999736	+	0.0229939i	$0.992680\pi$
$48$	0	0
$49$	7.71290e6	1.33793
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 2.65410e6i	− 0.336367i	−0.985756	−	0.168183i	$-0.946210\pi$
$53$	− 2.65410e6i	− 0.336367i	0.985756	−	0.168183i	$-0.0537901\pi$
$54$	0	0
$55$	6.77450e6	0.740331
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 2.22710e7i	− 1.83794i	−0.394323	−	0.918972i	$-0.629021\pi$
$59$	− 2.22710e7i	− 1.83794i	0.394323	−	0.918972i	$-0.370979\pi$
$60$	0	0
$61$	−2.21083e7	−1.59675	−0.798374	−	0.602161i	$-0.794306\pi$
$61$	−2.21083e7	−1.59675	−0.798374	+	0.602161i	$0.794306\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	9.45394e6i	0.529614i
$66$	0	0
$67$	1.04604e7	0.519098	0.259549	−	0.965730i	$-0.416426\pi$
$67$	1.04604e7	0.519098	0.259549	+	0.965730i	$0.416426\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.33031e6i	0.327814i	0.986476	+	0.163907i	$0.0524098\pi$
$71$	8.33031e6i	0.327814i	−0.986476	+	0.163907i	$0.947590\pi$
$72$	0	0
$73$	−4.29505e7	−1.51243	−0.756217	−	0.654320i	$-0.772955\pi$
$73$	−4.29505e7	−1.51243	−0.756217	+	0.654320i	$0.772955\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 7.81483e7i	− 2.22309i
$78$	0	0
$79$	−1.02405e7	−0.262913	−0.131457	−	0.991322i	$-0.541965\pi$
$79$	−1.02405e7	−0.262913	−0.131457	+	0.991322i	$0.541965\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	1.26098e7i	0.265703i	0.991136	+	0.132852i	$0.0424134\pi$
$83$	1.26098e7i	0.265703i	−0.991136	+	0.132852i	$0.957587\pi$
$84$	0	0
$85$	4.32237e7	0.828031
$86$	0	0
$87$	0	0
$88$	0	0
$89$	5.84973e7i	0.932344i	0.884694	+	0.466172i	$0.154367\pi$
$89$	5.84973e7i	0.932344i	−0.884694	+	0.466172i	$0.845633\pi$
$90$	0	0
$91$	1.09057e8	1.59034
$92$	0	0
$93$	0	0
$94$	0	0
$95$	4.99369e7i	0.613094i
$96$	0	0
$97$	−7.32199e7	−0.827070	−0.413535	−	0.910488i	$-0.635706\pi$
$97$	−7.32199e7	−0.827070	−0.413535	+	0.910488i	$0.635706\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1.23285e8i	1.18474i	0.805665	+	0.592372i	$0.201808\pi$
$101$	1.23285e8i	1.18474i	−0.805665	+	0.592372i	$0.798192\pi$
$102$	0	0
$103$	1.14434e7	0.101674	0.0508368	−	0.998707i	$-0.483811\pi$
$103$	1.14434e7	0.101674	0.0508368	+	0.998707i	$0.483811\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1.05908e8i	− 0.807968i	−0.914766	−	0.403984i	$-0.867625\pi$
$107$	− 1.05908e8i	− 0.807968i	0.914766	−	0.403984i	$-0.132375\pi$
$108$	0	0
$109$	−1.32205e8	−0.936577	−0.468289	−	0.883576i	$-0.655129\pi$
$109$	−1.32205e8	−0.936577	−0.468289	+	0.883576i	$0.655129\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1.88976e8i	1.15903i	0.814962	+	0.579514i	$0.196758\pi$
$113$	1.88976e8i	1.15903i	−0.814962	+	0.579514i	$0.803242\pi$
$114$	0	0
$115$	−2.49841e7	−0.142847
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 4.98615e8i	− 2.48644i
$120$	0	0
$121$	−2.38772e8	−1.11389
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 2.16398e8i	− 0.886367i
$126$	0	0
$127$	2.28100e6	0.00876821	0.00438411	−	0.999990i	$-0.498604\pi$
$127$	2.28100e6	0.00876821	0.00438411	+	0.999990i	$0.498604\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 5.56662e8i	− 1.89019i	−0.326791	−	0.945097i	$-0.605967\pi$
$131$	− 5.56662e8i	− 1.89019i	0.326791	−	0.945097i	$-0.394033\pi$
$132$	0	0
$133$	5.76056e8	1.84102
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 2.37091e8i	− 0.673027i	−0.941679	−	0.336513i	$-0.890752\pi$
$137$	− 2.37091e8i	− 0.673027i	0.941679	−	0.336513i	$-0.109248\pi$
$138$	0	0
$139$	−1.13143e8	−0.303089	−0.151544	−	0.988450i	$-0.548425\pi$
$139$	−1.13143e8	−0.303089	−0.151544	+	0.988450i	$0.548425\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 6.32353e8i	− 1.51222i
$144$	0	0
$145$	2.82816e8	0.639782
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.08402e8i	0.625708i	0.949801	+	0.312854i	$0.101285\pi$
$149$	3.08402e8i	0.625708i	−0.949801	+	0.312854i	$0.898715\pi$
$150$	0	0
$151$	6.89761e8	1.32675	0.663377	−	0.748285i	$-0.269122\pi$
$151$	6.89761e8	1.32675	0.663377	+	0.748285i	$0.269122\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 2.37364e8i	− 0.411233i
$156$	0	0
$157$	1.27441e8	0.209754	0.104877	−	0.994485i	$-0.466555\pi$
$157$	1.27441e8	0.209754	0.104877	+	0.994485i	$0.466555\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2.88208e8i	0.428946i
$162$	0	0
$163$	−4.34390e8	−0.615360	−0.307680	−	0.951490i	$-0.599553\pi$
$163$	−4.34390e8	−0.615360	−0.307680	+	0.951490i	$0.599553\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	4.33615e8i	0.557491i	0.960365	+	0.278746i	$0.0899186\pi$
$167$	4.33615e8i	0.557491i	−0.960365	+	0.278746i	$0.910081\pi$
$168$	0	0
$169$	6.67299e7	0.0818038
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 1.71571e9i	− 1.91540i	−0.287772	−	0.957699i	$-0.592914\pi$
$173$	− 1.71571e9i	− 1.91540i	0.287772	−	0.957699i	$-0.407086\pi$
$174$	0	0
$175$	−1.06224e9	−1.13258
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 1.56008e9i	− 1.51962i	−0.650148	−	0.759808i	$-0.725293\pi$
$179$	− 1.56008e9i	− 1.51962i	0.650148	−	0.759808i	$-0.274707\pi$
$180$	0	0
$181$	−5.47223e8	−0.509859	−0.254929	−	0.966960i	$-0.582052\pi$
$181$	−5.47223e8	−0.509859	−0.254929	+	0.966960i	$0.582052\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.08393e8i	− 0.434023i
$186$	0	0
$187$	−2.89114e9	−2.36430
$188$	0	0
$189$	0	0
$190$	0	0
$191$	2.06569e9i	1.55214i	0.630645	+	0.776071i	$0.282790\pi$
$191$	2.06569e9i	1.55214i	−0.630645	+	0.776071i	$0.717210\pi$
$192$	0	0
$193$	1.20554e8	0.0868863	0.0434431	−	0.999056i	$-0.486167\pi$
$193$	1.20554e8	0.0868863	0.0434431	+	0.999056i	$0.486167\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 4.55329e8i	− 0.302316i	−0.988510	−	0.151158i	$-0.951700\pi$
$197$	− 4.55329e8i	− 0.302316i	0.988510	−	0.151158i	$-0.0483002\pi$
$198$	0	0
$199$	2.31537e9	1.47642	0.738208	−	0.674573i	$-0.235672\pi$
$199$	2.31537e9	1.47642	0.738208	+	0.674573i	$0.235672\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.26247e9i	− 1.92116i
$204$	0	0
$205$	−3.90902e7	−0.0221336
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 3.34017e9i	− 1.75059i
$210$	0	0
$211$	3.51493e9	1.77332	0.886659	−	0.462424i	$-0.153020\pi$
$211$	3.51493e9	1.77332	0.886659	+	0.462424i	$0.153020\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	9.42597e8i	0.441136i
$216$	0	0
$217$	−2.73815e9	−1.23486
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 4.03464e9i	− 1.69136i
$222$	0	0
$223$	2.65136e9	1.07213	0.536067	−	0.844175i	$-0.319909\pi$
$223$	2.65136e9	1.07213	0.536067	+	0.844175i	$0.319909\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	1.33978e9i	0.504581i	0.967652	+	0.252290i	$0.0811838\pi$
$227$	1.33978e9i	0.504581i	−0.967652	+	0.252290i	$0.918816\pi$
$228$	0	0
$229$	−1.02094e9	−0.371244	−0.185622	−	0.982621i	$-0.559430\pi$
$229$	−1.02094e9	−0.371244	−0.185622	+	0.982621i	$0.559430\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 2.38627e9i	− 0.809647i	−0.914395	−	0.404824i	$-0.867333\pi$
$233$	− 2.38627e9i	− 0.809647i	0.914395	−	0.404824i	$-0.132667\pi$
$234$	0	0
$235$	7.14165e7	0.0234168
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 5.12645e9i	− 1.57118i	−0.618748	−	0.785589i	$-0.712360\pi$
$239$	− 5.12645e9i	− 1.57118i	0.618748	−	0.785589i	$-0.287640\pi$
$240$	0	0
$241$	−3.69617e9	−1.09568	−0.547840	−	0.836583i	$-0.684550\pi$
$241$	−3.69617e9	−1.09568	−0.547840	+	0.836583i	$0.684550\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 2.45461e9i	− 0.681268i
$246$	0	0
$247$	4.66127e9	1.25232
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 3.98463e9i	− 1.00391i	−0.864895	−	0.501953i	$-0.832615\pi$
$251$	− 3.98463e9i	− 1.00391i	0.864895	−	0.501953i	$-0.167385\pi$
$252$	0	0
$253$	1.67113e9	0.407876
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 4.51379e9i	− 1.03469i	−0.855778	−	0.517343i	$-0.826921\pi$
$257$	− 4.51379e9i	− 1.03469i	0.855778	−	0.517343i	$-0.173079\pi$
$258$	0	0
$259$	−5.86466e9	−1.30330
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 1.56751e9i	− 0.327632i	−0.986491	−	0.163816i	$-0.947620\pi$
$263$	− 1.56751e9i	− 0.327632i	0.986491	−	0.163816i	$-0.0523804\pi$
$264$	0	0
$265$	−8.44659e8	−0.171277
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 1.36738e9i	− 0.261145i	−0.991439	−	0.130572i	$-0.958319\pi$
$269$	− 1.36738e9i	− 0.261145i	0.991439	−	0.130572i	$-0.0416815\pi$
$270$	0	0
$271$	6.66201e9	1.23517	0.617587	−	0.786503i	$-0.288110\pi$
$271$	6.66201e9	1.23517	0.617587	+	0.786503i	$0.288110\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	6.15922e9i	1.07695i
$276$	0	0
$277$	1.05859e10	1.79809	0.899044	−	0.437859i	$-0.144263\pi$
$277$	1.05859e10	1.79809	0.899044	+	0.437859i	$0.144263\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 5.11945e9i	− 0.821103i	−0.911837	−	0.410552i	$-0.865336\pi$
$281$	− 5.11945e9i	− 0.821103i	0.911837	−	0.410552i	$-0.134664\pi$
$282$	0	0
$283$	−1.02978e10	−1.60545	−0.802727	−	0.596346i	$-0.796619\pi$
$283$	−1.02978e10	−1.60545	−0.802727	+	0.596346i	$0.796619\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	4.50932e8i	0.0664636i
$288$	0	0
$289$	−1.14708e10	−1.64437
$290$	0	0
$291$	0	0
$292$	0	0
$293$	1.41363e10i	1.91808i	0.283273	+	0.959039i	$0.408580\pi$
$293$	1.41363e10i	1.91808i	−0.283273	+	0.959039i	$0.591420\pi$
$294$	0	0
$295$	−7.08770e9	−0.935874
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.33210e9i	0.291784i
$300$	0	0
$301$	1.08735e10	1.32465
$302$	0	0
$303$	0	0
$304$	0	0
$305$	7.03592e9i	0.813058i
$306$	0	0
$307$	8.25753e9	0.929601	0.464800	−	0.885415i	$-0.346126\pi$
$307$	8.25753e9	0.929601	0.464800	+	0.885415i	$0.346126\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 3.03154e7i	− 0.00324057i	−0.999999	−	0.00162029i	$-0.999484\pi$
$311$	− 3.03154e7i	− 0.00324057i	0.999999	−	0.00162029i	$-0.000515753\pi$
$312$	0	0
$313$	1.49534e10	1.55798	0.778990	−	0.627037i	$-0.215732\pi$
$313$	1.49534e10	1.55798	0.778990	+	0.627037i	$0.215732\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.01943e9i	0.199982i	0.994988	+	0.0999911i	$0.0318814\pi$
$317$	2.01943e9i	0.199982i	−0.994988	+	0.0999911i	$0.968119\pi$
$318$	0	0
$319$	−1.89169e10	−1.82679
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 2.13115e10i	− 1.95796i
$324$	0	0
$325$	−8.59531e9	−0.770421
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 8.23837e8i	− 0.0703166i
$330$	0	0
$331$	1.02463e10	0.853602	0.426801	−	0.904346i	$-0.359641\pi$
$331$	1.02463e10	0.853602	0.426801	+	0.904346i	$0.359641\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 3.32900e9i	− 0.264323i
$336$	0	0
$337$	−6.61910e9	−0.513191	−0.256596	−	0.966519i	$-0.582601\pi$
$337$	−6.61910e9	−0.513191	−0.256596	+	0.966519i	$0.582601\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	1.58767e10i	1.17420i
$342$	0	0
$343$	−7.15185e9	−0.516704
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.47363e8i	− 0.0515482i	−0.999668	−	0.0257741i	$-0.991795\pi$
$347$	− 7.47363e8i	− 0.0515482i	0.999668	−	0.0257741i	$-0.00820507\pi$
$348$	0	0
$349$	7.88701e9	0.531632	0.265816	−	0.964024i	$-0.414359\pi$
$349$	7.88701e9	0.531632	0.265816	+	0.964024i	$0.414359\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 1.70679e10i	− 1.09921i	−0.835424	−	0.549605i	$-0.814778\pi$
$353$	− 1.70679e10i	− 1.09921i	0.835424	−	0.549605i	$-0.185222\pi$
$354$	0	0
$355$	2.65110e9	0.166922
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.32530e10i	0.797876i	0.916978	+	0.398938i	$0.130621\pi$
$359$	1.32530e10i	0.797876i	−0.916978	+	0.398938i	$0.869379\pi$
$360$	0	0
$361$	7.63787e9	0.449721
$362$	0	0
$363$	0	0
$364$	0	0
$365$	1.36689e10i	0.770126i
$366$	0	0
$367$	−1.55907e10	−0.859414	−0.429707	−	0.902968i	$-0.641383\pi$
$367$	−1.55907e10	−0.859414	−0.429707	+	0.902968i	$0.641383\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	9.74371e9i	0.514314i
$372$	0	0
$373$	3.23735e9	0.167245	0.0836227	−	0.996497i	$-0.473351\pi$
$373$	3.23735e9	0.167245	0.0836227	+	0.996497i	$0.473351\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 2.63989e10i	− 1.30684i
$378$	0	0
$379$	−3.34929e9	−0.162329	−0.0811644	−	0.996701i	$-0.525864\pi$
$379$	−3.34929e9	−0.162329	−0.0811644	+	0.996701i	$0.525864\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 7.65092e9i	− 0.355564i	−0.984070	−	0.177782i	$-0.943108\pi$
$383$	− 7.65092e9i	− 0.355564i	0.984070	−	0.177782i	$-0.0568923\pi$
$384$	0	0
$385$	−2.48705e10	−1.13199
$386$	0	0
$387$	0	0
$388$	0	0
$389$	2.73660e10i	1.19513i	0.801822	+	0.597563i	$0.203864\pi$
$389$	2.73660e10i	1.19513i	−0.801822	+	0.597563i	$0.796136\pi$
$390$	0	0
$391$	1.06624e10	0.456193
$392$	0	0
$393$	0	0
$394$	0	0
$395$	3.25901e9i	0.133874i
$396$	0	0
$397$	1.96729e10	0.791965	0.395982	−	0.918258i	$-0.370404\pi$
$397$	1.96729e10	0.791965	0.395982	+	0.918258i	$0.370404\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	3.41991e10i	1.32263i	0.750110	+	0.661314i	$0.230001\pi$
$401$	3.41991e10i	1.32263i	−0.750110	+	0.661314i	$0.769999\pi$
$402$	0	0
$403$	−2.21563e10	−0.839995
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.40053e10i	1.23928i
$408$	0	0
$409$	−3.94278e9	−0.140899	−0.0704497	−	0.997515i	$-0.522443\pi$
$409$	−3.94278e9	−0.140899	−0.0704497	+	0.997515i	$0.522443\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.17614e10i	2.81027i
$414$	0	0
$415$	4.01305e9	0.135295
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 1.01113e10i	− 0.328058i	−0.986456	−	0.164029i	$-0.947551\pi$
$419$	− 1.01113e10i	− 0.328058i	0.986456	−	0.164029i	$-0.0524491\pi$
$420$	0	0
$421$	1.10666e10	0.352277	0.176139	−	0.984365i	$-0.443639\pi$
$421$	1.10666e10	0.352277	0.176139	+	0.984365i	$0.443639\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.92980e10i	1.20452i
$426$	0	0
$427$	8.11641e10	2.44148
$428$	0	0
$429$	0	0
$430$	0	0
$431$	4.00671e10i	1.16113i	0.814216	+	0.580563i	$0.197167\pi$
$431$	4.00671e10i	1.16113i	−0.814216	+	0.580563i	$0.802833\pi$
$432$	0	0
$433$	4.32592e9	0.123063	0.0615315	−	0.998105i	$-0.480402\pi$
$433$	4.32592e9	0.123063	0.0615315	+	0.998105i	$0.480402\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.23184e10i	0.337777i
$438$	0	0
$439$	2.90106e10	0.781087	0.390543	−	0.920584i	$-0.372287\pi$
$439$	2.90106e10	0.781087	0.390543	+	0.920584i	$0.372287\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	2.44097e10i	0.633793i	0.948460	+	0.316896i	$0.102641\pi$
$443$	2.44097e10i	0.633793i	−0.948460	+	0.316896i	$0.897359\pi$
$444$	0	0
$445$	1.86166e10	0.474746
$446$	0	0
$447$	0	0
$448$	0	0
$449$	3.47585e10i	0.855215i	0.903964	+	0.427608i	$0.140643\pi$
$449$	3.47585e10i	0.855215i	−0.903964	+	0.427608i	$0.859357\pi$
$450$	0	0
$451$	2.61466e9	0.0631988
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 3.47073e10i	− 0.809795i
$456$	0	0
$457$	−6.50854e9	−0.149217	−0.0746086	−	0.997213i	$-0.523771\pi$
$457$	−6.50854e9	−0.149217	−0.0746086	+	0.997213i	$0.523771\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.18104e10i	1.36854i	0.729228	+	0.684271i	$0.239879\pi$
$461$	6.18104e10i	1.36854i	−0.729228	+	0.684271i	$0.760121\pi$
$462$	0	0
$463$	−6.00567e10	−1.30689	−0.653443	−	0.756976i	$-0.726676\pi$
$463$	−6.00567e10	−1.30689	−0.653443	+	0.756976i	$0.726676\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 4.35906e10i	− 0.916485i	−0.888827	−	0.458242i	$-0.848479\pi$
$467$	− 4.35906e10i	− 0.916485i	0.888827	−	0.458242i	$-0.151521\pi$
$468$	0	0
$469$	−3.84022e10	−0.793716
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 6.30482e10i	− 1.25959i
$474$	0	0
$475$	−4.54015e10	−0.891859
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 6.31123e10i	− 1.19887i	−0.800423	−	0.599435i	$-0.795392\pi$
$479$	− 6.31123e10i	− 1.19887i	0.800423	−	0.599435i	$-0.204608\pi$
$480$	0	0
$481$	−4.74551e10	−0.886548
$482$	0	0
$483$	0	0
$484$	0	0
$485$	2.33020e10i	0.421141i
$486$	0	0
$487$	−3.21573e10	−0.571694	−0.285847	−	0.958275i	$-0.592275\pi$
$487$	−3.21573e10	−0.571694	−0.285847	+	0.958275i	$0.592275\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	7.65396e10i	1.31692i	0.752615	+	0.658461i	$0.228792\pi$
$491$	7.65396e10i	1.31692i	−0.752615	+	0.658461i	$0.771208\pi$
$492$	0	0
$493$	−1.20697e11	−2.04319
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 3.05822e10i	− 0.501238i
$498$	0	0
$499$	6.86455e9	0.110716	0.0553580	−	0.998467i	$-0.482370\pi$
$499$	6.86455e9	0.110716	0.0553580	+	0.998467i	$0.482370\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 3.21431e10i	− 0.502129i	−0.967970	−	0.251064i	$-0.919219\pi$
$503$	− 3.21431e10i	− 0.502129i	0.967970	−	0.251064i	$-0.0807806\pi$
$504$	0	0
$505$	3.92351e10	0.603267
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.41545e10i	1.40272i	0.712809	+	0.701359i	$0.247423\pi$
$509$	9.41545e10i	1.40272i	−0.712809	+	0.701359i	$0.752577\pi$
$510$	0	0
$511$	1.57680e11	2.31256
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 3.64185e9i	− 0.0517718i
$516$	0	0
$517$	−4.77689e9	−0.0668626
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 1.19731e11i	− 1.62501i	−0.582957	−	0.812503i	$-0.698104\pi$
$521$	− 1.19731e11i	− 1.62501i	0.582957	−	0.812503i	$-0.301896\pi$
$522$	0	0
$523$	−4.58622e10	−0.612983	−0.306491	−	0.951873i	$-0.599155\pi$
$523$	−4.58622e10	−0.612983	−0.306491	+	0.951873i	$0.599155\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	1.01299e11i	1.31330i
$528$	0	0
$529$	7.21479e10	0.921300
$530$	0	0
$531$	0	0
$532$	0	0
$533$	3.64881e9i	0.0452108i
$534$	0	0
$535$	−3.37050e10	−0.411414
$536$	0	0
$537$	0	0
$538$	0	0
$539$	1.64183e11i	1.94525i
$540$	0	0
$541$	9.06414e10	1.05813	0.529063	−	0.848583i	$-0.322544\pi$
$541$	9.06414e10	1.05813	0.529063	+	0.848583i	$0.322544\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	4.20741e10i	0.476901i
$546$	0	0
$547$	5.98296e9	0.0668292	0.0334146	−	0.999442i	$-0.489362\pi$
$547$	5.98296e9	0.0668292	0.0334146	+	0.999442i	$0.489362\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 1.39443e11i	− 1.51283i
$552$	0	0
$553$	3.75949e10	0.402002
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8.85076e10i	− 0.919517i	−0.888044	−	0.459759i	$-0.847936\pi$
$557$	− 8.85076e10i	− 0.919517i	0.888044	−	0.459759i	$-0.152064\pi$
$558$	0	0
$559$	8.79850e10	0.901076
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 7.99638e10i	− 0.795903i	−0.917406	−	0.397951i	$-0.869721\pi$
$563$	− 7.99638e10i	− 0.795903i	0.917406	−	0.397951i	$-0.130279\pi$
$564$	0	0
$565$	6.01413e10	0.590172
$566$	0	0
$567$	0	0
$568$	0	0
$569$	1.22580e11i	1.16942i	0.811244	+	0.584708i	$0.198791\pi$
$569$	1.22580e11i	1.16942i	−0.811244	+	0.584708i	$0.801209\pi$
$570$	0	0
$571$	−1.28527e11	−1.20907	−0.604533	−	0.796580i	$-0.706640\pi$
$571$	−1.28527e11	−1.20907	−0.604533	+	0.796580i	$0.706640\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 2.27150e10i	− 0.207798i
$576$	0	0
$577$	−6.45300e10	−0.582181	−0.291091	−	0.956696i	$-0.594018\pi$
$577$	−6.45300e10	−0.582181	−0.291091	+	0.956696i	$0.594018\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 4.62932e10i	− 0.406268i
$582$	0	0
$583$	5.64974e10	0.489051
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 8.15148e10i	− 0.686569i	−0.939232	−	0.343284i	$-0.888461\pi$
$587$	− 8.15148e10i	− 0.686569i	0.939232	−	0.343284i	$-0.111539\pi$
$588$	0	0
$589$	−1.17032e11	−0.972400
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.35630e10i	− 0.514027i	−0.966408	−	0.257013i	$-0.917262\pi$
$593$	− 6.35630e10i	− 0.514027i	0.966408	−	0.257013i	$-0.0827384\pi$
$594$	0	0
$595$	−1.58683e11	−1.26608
$596$	0	0
$597$	0	0
$598$	0	0
$599$	4.87502e10i	0.378677i	0.981912	+	0.189339i	$0.0606344\pi$
$599$	4.87502e10i	0.378677i	−0.981912	+	0.189339i	$0.939366\pi$
$600$	0	0
$601$	−1.04167e10	−0.0798422	−0.0399211	−	0.999203i	$-0.512711\pi$
$601$	−1.04167e10	−0.0798422	−0.0399211	+	0.999203i	$0.512711\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	7.59887e10i	0.567188i
$606$	0	0
$607$	2.14841e11	1.58257	0.791283	−	0.611450i	$-0.209413\pi$
$607$	2.14841e11	1.58257	0.791283	+	0.611450i	$0.209413\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 6.66624e9i	− 0.0478317i
$612$	0	0
$613$	−1.37183e11	−0.971534	−0.485767	−	0.874088i	$-0.661460\pi$
$613$	−1.37183e11	−0.971534	−0.485767	+	0.874088i	$0.661460\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 9.43805e10i	− 0.651241i	−0.945501	−	0.325620i	$-0.894427\pi$
$617$	− 9.43805e10i	− 0.651241i	0.945501	−	0.325620i	$-0.105573\pi$
$618$	0	0
$619$	1.17365e11	0.799421	0.399710	−	0.916642i	$-0.369111\pi$
$619$	1.17365e11	0.799421	0.399710	+	0.916642i	$0.369111\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 2.14755e11i	− 1.42558i
$624$	0	0
$625$	4.41566e10	0.289385
$626$	0	0
$627$	0	0
$628$	0	0
$629$	2.16966e11i	1.38608i
$630$	0	0
$631$	3.62435e9	0.0228619	0.0114310	−	0.999935i	$-0.496361\pi$
$631$	3.62435e9	0.0228619	0.0114310	+	0.999935i	$0.496361\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.25924e8i	− 0.00446474i
$636$	0	0
$637$	−2.29121e11	−1.39158
$638$	0	0
$639$	0	0
$640$	0	0
$641$	8.09388e10i	0.479429i	0.970843	+	0.239715i	$0.0770539\pi$
$641$	8.09388e10i	0.479429i	−0.970843	+	0.239715i	$0.922946\pi$
$642$	0	0
$643$	−1.44459e11	−0.845083	−0.422542	−	0.906344i	$-0.638862\pi$
$643$	−1.44459e11	−0.845083	−0.422542	+	0.906344i	$0.638862\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 3.11720e11i	− 1.77889i	−0.457046	−	0.889443i	$-0.651093\pi$
$647$	− 3.11720e11i	− 1.77889i	0.457046	−	0.889443i	$-0.348907\pi$
$648$	0	0
$649$	4.74081e11	2.67223
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 2.89314e11i	− 1.59117i	−0.605842	−	0.795585i	$-0.707164\pi$
$653$	− 2.89314e11i	− 1.59117i	0.605842	−	0.795585i	$-0.292836\pi$
$654$	0	0
$655$	−1.77156e11	−0.962479
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 1.51054e11i	− 0.800925i	−0.916313	−	0.400463i	$-0.868849\pi$
$659$	− 1.51054e11i	− 0.800925i	0.916313	−	0.400463i	$-0.131151\pi$
$660$	0	0
$661$	2.34913e11	1.23056	0.615278	−	0.788310i	$-0.289044\pi$
$661$	2.34913e11	1.23056	0.615278	+	0.788310i	$0.289044\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 1.83328e11i	− 0.937439i
$666$	0	0
$667$	6.97650e10	0.352480
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 4.70617e11i	− 2.32155i
$672$	0	0
$673$	−1.86487e11	−0.909053	−0.454526	−	0.890733i	$-0.650191\pi$
$673$	−1.86487e11	−0.909053	−0.454526	+	0.890733i	$0.650191\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 8.13139e10i	− 0.387088i	−0.981092	−	0.193544i	$-0.938002\pi$
$677$	− 8.13139e10i	− 0.387088i	0.981092	−	0.193544i	$-0.0619983\pi$
$678$	0	0
$679$	2.68805e11	1.26461
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 2.90702e11i	− 1.33587i	−0.744218	−	0.667937i	$-0.767178\pi$
$683$	− 2.90702e11i	− 1.33587i	0.744218	−	0.667937i	$-0.232822\pi$
$684$	0	0
$685$	−7.54535e10	−0.342702
$686$	0	0
$687$	0	0
$688$	0	0
$689$	7.88432e10i	0.349854i
$690$	0	0
$691$	−1.77614e11	−0.779049	−0.389525	−	0.921016i	$-0.627361\pi$
$691$	−1.77614e11	−0.779049	−0.389525	+	0.921016i	$0.627361\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	3.60076e10i	0.154332i
$696$	0	0
$697$	1.66825e10	0.0706853
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.59619e10i	0.273163i	0.990629	+	0.136581i	$0.0436115\pi$
$701$	6.59619e10i	0.273163i	−0.990629	+	0.136581i	$0.956389\pi$
$702$	0	0
$703$	−2.50664e11	−1.02629
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 4.52603e11i	− 1.81151i
$708$	0	0
$709$	1.72962e11	0.684489	0.342244	−	0.939611i	$-0.388813\pi$
$709$	1.72962e11	0.684489	0.342244	+	0.939611i	$0.388813\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 5.85528e10i	− 0.226563i
$714$	0	0
$715$	−2.01245e11	−0.770017
$716$	0	0
$717$	0	0
$718$	0	0
$719$	2.12925e11i	0.796731i	0.917227	+	0.398365i	$0.130422\pi$
$719$	2.12925e11i	0.796731i	−0.917227	+	0.398365i	$0.869578\pi$
$720$	0	0
$721$	−4.20112e10	−0.155462
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.57130e11i	0.930680i
$726$	0	0
$727$	−1.70085e11	−0.608874	−0.304437	−	0.952533i	$-0.598468\pi$
$727$	−1.70085e11	−0.608874	−0.304437	+	0.952533i	$0.598468\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 4.02270e11i	− 1.40880i
$732$	0	0
$733$	−6.90724e10	−0.239270	−0.119635	−	0.992818i	$-0.538172\pi$
$733$	−6.90724e10	−0.239270	−0.119635	+	0.992818i	$0.538172\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2.22669e11i	0.754728i
$738$	0	0
$739$	−1.39045e11	−0.466206	−0.233103	−	0.972452i	$-0.574888\pi$
$739$	−1.39045e11	−0.466206	−0.233103	+	0.972452i	$0.574888\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 2.50291e11i	− 0.821277i	−0.911798	−	0.410639i	$-0.865306\pi$
$743$	− 2.50291e11i	− 0.821277i	0.911798	−	0.410639i	$-0.134694\pi$
$744$	0	0
$745$	9.81481e10	0.318608
$746$	0	0
$747$	0	0
$748$	0	0
$749$	3.88810e11i	1.23541i
$750$	0	0
$751$	2.64592e11	0.831797	0.415899	−	0.909411i	$-0.363467\pi$
$751$	2.64592e11	0.831797	0.415899	+	0.909411i	$0.363467\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 2.19515e11i	− 0.675578i
$756$	0	0
$757$	−2.80272e11	−0.853487	−0.426743	−	0.904373i	$-0.640339\pi$
$757$	−2.80272e11	−0.853487	−0.426743	+	0.904373i	$0.640339\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	2.68619e11i	0.800935i	0.916311	+	0.400468i	$0.131152\pi$
$761$	2.68619e11i	0.800935i	−0.916311	+	0.400468i	$0.868848\pi$
$762$	0	0
$763$	4.85353e11	1.43205
$764$	0	0
$765$	0	0
$766$	0	0
$767$	6.61589e11i	1.91164i
$768$	0	0
$769$	−4.73505e11	−1.35400	−0.677001	−	0.735983i	$-0.736721\pi$
$769$	−4.73505e11	−1.35400	−0.677001	+	0.735983i	$0.736721\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	3.66187e11i	1.02562i	0.858503	+	0.512809i	$0.171395\pi$
$773$	3.66187e11i	1.02562i	−0.858503	+	0.512809i	$0.828605\pi$
$774$	0	0
$775$	2.15806e11	0.598214
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.92735e10i	0.0523372i
$780$	0	0
$781$	−1.77326e11	−0.476617
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 4.05578e10i	− 0.106806i
$786$	0	0
$787$	6.82820e10	0.177995	0.0889974	−	0.996032i	$-0.471634\pi$
$787$	6.82820e10	0.177995	0.0889974	+	0.996032i	$0.471634\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 6.93770e11i	− 1.77219i
$792$	0	0
$793$	6.56755e11	1.66078
$794$	0	0
$795$	0	0
$796$	0	0
$797$	3.33352e11i	0.826172i	0.910692	+	0.413086i	$0.135549\pi$
$797$	3.33352e11i	0.826172i	−0.910692	+	0.413086i	$0.864451\pi$
$798$	0	0
$799$	−3.04783e10	−0.0747831
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 9.14282e11i	− 2.19896i
$804$	0	0
$805$	9.17216e10	0.218418
$806$	0	0
$807$	0	0
$808$	0	0
$809$	2.70933e11i	0.632510i	0.948674	+	0.316255i	$0.102425\pi$
$809$	2.70933e11i	0.632510i	−0.948674	+	0.316255i	$0.897575\pi$
$810$	0	0
$811$	2.12542e10	0.0491316	0.0245658	−	0.999698i	$-0.492180\pi$
$811$	2.12542e10	0.0491316	0.0245658	+	0.999698i	$0.492180\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	1.38243e11i	0.313339i
$816$	0	0
$817$	4.64748e11	1.04311
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 4.41120e11i	− 0.970921i	−0.874259	−	0.485460i	$-0.838652\pi$
$821$	− 4.41120e11i	− 0.970921i	0.874259	−	0.485460i	$-0.161348\pi$
$822$	0	0
$823$	−3.38323e11	−0.737449	−0.368724	−	0.929539i	$-0.620205\pi$
$823$	−3.38323e11	−0.737449	−0.368724	+	0.929539i	$0.620205\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 2.01813e11i	− 0.431446i	−0.976455	−	0.215723i	$-0.930789\pi$
$827$	− 2.01813e11i	− 0.431446i	0.976455	−	0.215723i	$-0.0692108\pi$
$828$	0	0
$829$	−7.92241e11	−1.67741	−0.838705	−	0.544586i	$-0.816687\pi$
$829$	−7.92241e11	−1.67741	−0.838705	+	0.544586i	$0.816687\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1.04755e12i	2.17568i
$834$	0	0
$835$	1.37997e11	0.283872
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 3.29121e11i	− 0.664214i	−0.943242	−	0.332107i	$-0.892241\pi$
$839$	− 3.29121e11i	− 0.664214i	0.943242	−	0.332107i	$-0.107759\pi$
$840$	0	0
$841$	−2.89482e11	−0.578678
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 2.12366e10i	− 0.0416542i
$846$	0	0
$847$	8.76580e11	1.70317
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.25410e11i	− 0.239120i
$852$	0	0
$853$	6.85129e11	1.29413	0.647063	−	0.762437i	$-0.275997\pi$
$853$	6.85129e11	1.29413	0.647063	+	0.762437i	$0.275997\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	4.96288e10i	0.0920049i	0.998941	+	0.0460024i	$0.0146482\pi$
$857$	4.96288e10i	0.0920049i	−0.998941	+	0.0460024i	$0.985352\pi$
$858$	0	0
$859$	−1.71095e11	−0.314242	−0.157121	−	0.987579i	$-0.550221\pi$
$859$	−1.71095e11	−0.314242	−0.157121	+	0.987579i	$0.550221\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	− 6.12227e11i	− 1.10375i	−0.833928	−	0.551873i	$-0.813913\pi$
$863$	− 6.12227e11i	− 1.10375i	0.833928	−	0.551873i	$-0.186087\pi$
$864$	0	0
$865$	−5.46020e11	−0.975313
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 2.17988e11i	− 0.382255i
$870$	0	0
$871$	−3.10739e11	−0.539913
$872$	0	0
$873$	0	0
$874$	0	0
$875$	7.94441e11i	1.35528i
$876$	0	0
$877$	9.23259e11	1.56072	0.780360	−	0.625331i	$-0.215036\pi$
$877$	9.23259e11	1.56072	0.780360	+	0.625331i	$0.215036\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 4.63714e11i	− 0.769745i	−0.922970	−	0.384872i	$-0.874245\pi$
$881$	− 4.63714e11i	− 0.769745i	0.922970	−	0.384872i	$-0.125755\pi$
$882$	0	0
$883$	4.40769e11	0.725050	0.362525	−	0.931974i	$-0.381915\pi$
$883$	4.40769e11	0.725050	0.362525	+	0.931974i	$0.381915\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	4.09560e11i	0.661642i	0.943694	+	0.330821i	$0.107326\pi$
$887$	4.09560e11i	0.661642i	−0.943694	+	0.330821i	$0.892674\pi$
$888$	0	0
$889$	−8.37402e9	−0.0134069
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 3.52119e10i	− 0.0553712i
$894$	0	0
$895$	−4.96491e11	−0.773782
$896$	0	0
$897$	0	0
$898$	0	0
$899$	6.62809e11i	1.01473i
$900$	0	0
$901$	3.60474e11	0.546984
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.74152e11i	0.259618i
$906$	0	0
$907$	−8.35824e11	−1.23505	−0.617526	−	0.786550i	$-0.711865\pi$
$907$	−8.35824e11	−1.23505	−0.617526	+	0.786550i	$0.711865\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	4.39938e11i	0.638731i	0.947632	+	0.319365i	$0.103470\pi$
$911$	4.39938e11i	0.638731i	−0.947632	+	0.319365i	$0.896530\pi$
$912$	0	0
$913$	−2.68424e11	−0.386312
$914$	0	0
$915$	0	0
$916$	0	0
$917$	2.04362e12i	2.89016i
$918$	0	0
$919$	−1.16424e12	−1.63222	−0.816111	−	0.577896i	$-0.803874\pi$
$919$	−1.16424e12	−1.63222	−0.816111	+	0.577896i	$0.803874\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 2.47462e11i	− 0.340959i
$924$	0	0
$925$	4.62220e11	0.631367
$926$	0	0
$927$	0	0
$928$	0	0
$929$	9.70594e11i	1.30309i	0.758610	+	0.651545i	$0.225879\pi$
$929$	9.70594e11i	1.30309i	−0.758610	+	0.651545i	$0.774121\pi$
$930$	0	0
$931$	−1.21025e12	−1.61093
$932$	0	0
$933$	0	0
$934$	0	0
$935$	9.20098e11i	1.20389i
$936$	0	0
$937$	−1.40587e12	−1.82384	−0.911920	−	0.410369i	$-0.865400\pi$
$937$	−1.40587e12	−1.82384	−0.911920	+	0.410369i	$0.865400\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 7.86048e11i	− 1.00251i	−0.865298	−	0.501257i	$-0.832871\pi$
$941$	− 7.86048e11i	− 1.00251i	0.865298	−	0.501257i	$-0.167129\pi$
$942$	0	0
$943$	−9.64278e9	−0.0121943
$944$	0	0
$945$	0	0
$946$	0	0
$947$	8.16138e11i	1.01476i	0.861722	+	0.507380i	$0.169386\pi$
$947$	8.16138e11i	1.01476i	−0.861722	+	0.507380i	$0.830614\pi$
$948$	0	0
$949$	1.27590e12	1.57308
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.20108e11i	0.145613i	0.997346	+	0.0728065i	$0.0231956\pi$
$953$	1.20108e11i	0.145613i	−0.997346	+	0.0728065i	$0.976804\pi$
$954$	0	0
$955$	6.57401e11	0.790345
$956$	0	0
$957$	0	0
$958$	0	0
$959$	8.70407e11i	1.02908i
$960$	0	0
$961$	−2.96604e11	−0.347763
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 3.83659e10i	− 0.0442421i
$966$	0	0
$967$	4.21798e11	0.482390	0.241195	−	0.970477i	$-0.422461\pi$
$967$	4.21798e11	0.482390	0.241195	+	0.970477i	$0.422461\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.66843e10i	0.0862640i	0.999069	+	0.0431320i	$0.0137336\pi$
$971$	7.66843e10i	0.0862640i	−0.999069	+	0.0431320i	$0.986266\pi$
$972$	0	0
$973$	4.15372e11	0.463432
$974$	0	0
$975$	0	0
$976$	0	0
$977$	1.06340e12i	1.16713i	0.812068	+	0.583563i	$0.198342\pi$
$977$	1.06340e12i	1.16713i	−0.812068	+	0.583563i	$0.801658\pi$
$978$	0	0
$979$	−1.24523e12	−1.35556
$980$	0	0
$981$	0	0
$982$	0	0
$983$	1.82502e11i	0.195458i	0.995213	+	0.0977292i	$0.0311579\pi$
$983$	1.82502e11i	0.195458i	−0.995213	+	0.0977292i	$0.968842\pi$
$984$	0	0
$985$	−1.44907e11	−0.153938
$986$	0	0
$987$	0	0
$988$	0	0
$989$	2.32520e11i	0.243038i
$990$	0	0
$991$	8.82094e11	0.914577	0.457289	−	0.889318i	$-0.348821\pi$
$991$	8.82094e11	0.914577	0.457289	+	0.889318i	$0.348821\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 7.36862e11i	− 0.751785i
$996$	0	0
$997$	−3.89945e11	−0.394660	−0.197330	−	0.980337i	$-0.563227\pi$
$997$	−3.89945e11	−0.394660	−0.197330	+	0.980337i	$0.563227\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.9.c.a.161.6		16
3.2	odd	2	inner	324.9.c.a.161.11		16
9.2	odd	6		36.9.g.a.5.4	✓	16
9.4	even	3		36.9.g.a.29.4	yes	16
9.5	odd	6		108.9.g.a.89.6		16
9.7	even	3		108.9.g.a.17.6		16
36.7	odd	6		432.9.q.b.17.6		16
36.11	even	6		144.9.q.c.113.5		16
36.23	even	6		432.9.q.b.305.6		16
36.31	odd	6		144.9.q.c.65.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.9.g.a.5.4	✓	16	9.2	odd	6
36.9.g.a.29.4	yes	16	9.4	even	3
108.9.g.a.17.6		16	9.7	even	3
108.9.g.a.89.6		16	9.5	odd	6
144.9.q.c.65.5		16	36.31	odd	6
144.9.q.c.113.5		16	36.11	even	6
324.9.c.a.161.6		16	1.1	even	1	trivial
324.9.c.a.161.11		16	3.2	odd	2	inner
432.9.q.b.17.6		16	36.7	odd	6
432.9.q.b.305.6		16	36.23	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q-318.248i q^{5} -3671.20 q^{7} +21286.9i q^{11} -29706.2 q^{13} +135818. i q^{17} -156912. q^{19} -78505.2i q^{23} +289344. q^{25} +888666. i q^{29} +745846. q^{31} +1.16835e6i q^{35} +1.59748e6 q^{37} -122830. i q^{41} -2.96183e6 q^{43} +224405. i q^{47} +7.71290e6 q^{49} -2.65410e6i q^{53} +6.77450e6 q^{55} -2.22710e7i q^{59} -2.21083e7 q^{61} +9.45394e6i q^{65} +1.04604e7 q^{67} +8.33031e6i q^{71} -4.29505e7 q^{73} -7.81483e7i q^{77} -1.02405e7 q^{79} +1.26098e7i q^{83} +4.32237e7 q^{85} +5.84973e7i q^{89} +1.09057e8 q^{91} +4.99369e7i q^{95} -7.32199e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 1846 q^{7} - 3370 q^{13} + 84518 q^{19} - 911606 q^{25} + 756614 q^{31} - 1671664 q^{37} - 679024 q^{43} + 6052890 q^{49} - 17497638 q^{55} - 34901410 q^{61} + 70053452 q^{67} + 28071218 q^{73} - 61378918 q^{79}+ \cdots + 66842156 q^{97}+O(q^{100}) \) 16 * q - 1846 * q^7 - 3370 * q^13 + 84518 * q^19 - 911606 * q^25 + 756614 * q^31 - 1671664 * q^37 - 679024 * q^43 + 6052890 * q^49 - 17497638 * q^55 - 34901410 * q^61 + 70053452 * q^67 + 28071218 * q^73 - 61378918 * q^79 + 79786044 * q^85 - 55534214 * q^91 + 66842156 * q^97

Introduction

L-functions

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Varieties

Fields

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Groups

Database

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