LMFDB - Embedded newform 864.5.e.g.161.9

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [864,5,Mod(161,864)] 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("864.161"); S:= CuspForms(chi, 5); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(864, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 5, names="a")

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	864.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,0,0,0,0,0,0,0,0,0,-32] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$89.3116481044$
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} - 22 x^{14} - 60 x^{13} + 313 x^{12} + 1368 x^{11} + 1844 x^{10} - 4788 x^{9} - 11779 x^{8} + \cdots + 16848900 $ x^16 - 22x^14 - 60x^13 + 313x^12 + 1368x^11 + 1844x^10 - 4788x^9 - 11779x^8 - 45888x^7 + 8308x^6 + 466392x^5 + 2503364x^4 + 5337264x^3 + 10353832x^2 + 14664480x + 16848900
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{56}\cdot 3^{24} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.9
Root			$3.05115 + 1.25061i$ of defining polynomial
Character	$\chi$	$=$	864.161
Dual form			864.5.e.g.161.7

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+9.20720i q^{5} -58.8505 q^{7} +203.037i q^{11} -40.3208 q^{13} +411.924i q^{17} +674.335 q^{19} +295.738i q^{23} +540.227 q^{25} -1300.52i q^{29} +1271.23 q^{31} -541.849i q^{35} -2491.60 q^{37} +1607.08i q^{41} -995.060 q^{43} -2490.91i q^{47} +1062.38 q^{49} +3803.65i q^{53} -1869.40 q^{55} -1069.17i q^{59} +1422.99 q^{61} -371.242i q^{65} -6081.89 q^{67} +2585.90i q^{71} -9007.18 q^{73} -11948.8i q^{77} +5113.86 q^{79} +8679.67i q^{83} -3792.67 q^{85} +4627.16i q^{89} +2372.90 q^{91} +6208.74i q^{95} -8859.15 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) $ 16 * q - 32 * q^13 + 704 * q^25 - 2624 * q^37 + 1728 * q^49 + 3264 * q^61 - 5424 * q^73 + 4704 * q^85 - 80 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$1$	$-1$	$1$

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
$5$	9.20720i	0.368288i	0.982899	+	0.184144i	$0.0589513\pi$
$5$	9.20720i	0.368288i	−0.982899	+	0.184144i	$0.941049\pi$
$6$	0	0
$7$	−58.8505	−1.20103	−0.600516	−	0.799613i	$-0.705038\pi$
$7$	−58.8505	−1.20103	−0.600516	+	0.799613i	$0.705038\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	203.037i	1.67799i	0.544141	+	0.838994i	$0.316856\pi$
$11$	203.037i	1.67799i	−0.544141	+	0.838994i	$0.683144\pi$
$12$	0	0
$13$	−40.3208	−0.238585	−0.119292	−	0.992859i	$-0.538063\pi$
$13$	−40.3208	−0.238585	−0.119292	+	0.992859i	$0.538063\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	411.924i	1.42534i	0.701497	+	0.712672i	$0.252515\pi$
$17$	411.924i	1.42534i	−0.701497	+	0.712672i	$0.747485\pi$
$18$	0	0
$19$	674.335	1.86796	0.933982	−	0.357320i	$-0.116309\pi$
$19$	674.335	1.86796	0.933982	+	0.357320i	$0.116309\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	295.738i	0.559051i	0.960138	+	0.279526i	$0.0901772\pi$
$23$	295.738i	0.559051i	−0.960138	+	0.279526i	$0.909823\pi$
$24$	0	0
$25$	540.227	0.864364
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 1300.52i	− 1.54640i	−0.634162	−	0.773200i	$-0.718655\pi$
$29$	− 1300.52i	− 1.54640i	0.634162	−	0.773200i	$-0.281345\pi$
$30$	0	0
$31$	1271.23	1.32282	0.661409	−	0.750025i	$-0.269959\pi$
$31$	1271.23	1.32282	0.661409	+	0.750025i	$0.269959\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 541.849i	− 0.442325i
$36$	0	0
$37$	−2491.60	−1.82002	−0.910009	−	0.414589i	$-0.863925\pi$
$37$	−2491.60	−1.82002	−0.910009	+	0.414589i	$0.863925\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	1607.08i	0.956026i	0.878353	+	0.478013i	$0.158643\pi$
$41$	1607.08i	0.956026i	−0.878353	+	0.478013i	$0.841357\pi$
$42$	0	0
$43$	−995.060	−0.538161	−0.269081	−	0.963118i	$-0.586720\pi$
$43$	−995.060	−0.538161	−0.269081	+	0.963118i	$0.586720\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 2490.91i	− 1.12762i	−0.825906	−	0.563808i	$-0.809336\pi$
$47$	− 2490.91i	− 1.12762i	0.825906	−	0.563808i	$-0.190664\pi$
$48$	0	0
$49$	1062.38	0.442476
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3803.65i	1.35410i	0.735939	+	0.677048i	$0.236741\pi$
$53$	3803.65i	1.35410i	−0.735939	+	0.677048i	$0.763259\pi$
$54$	0	0
$55$	−1869.40	−0.617983
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 1069.17i	− 0.307144i	−0.988137	−	0.153572i	$-0.950922\pi$
$59$	− 1069.17i	− 0.307144i	0.988137	−	0.153572i	$-0.0490777\pi$
$60$	0	0
$61$	1422.99	0.382422	0.191211	−	0.981549i	$-0.438759\pi$
$61$	1422.99	0.382422	0.191211	+	0.981549i	$0.438759\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 371.242i	− 0.0878680i
$66$	0	0
$67$	−6081.89	−1.35484	−0.677421	−	0.735595i	$-0.736903\pi$
$67$	−6081.89	−1.35484	−0.677421	+	0.735595i	$0.736903\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2585.90i	0.512973i	0.966548	+	0.256486i	$0.0825649\pi$
$71$	2585.90i	0.512973i	−0.966548	+	0.256486i	$0.917435\pi$
$72$	0	0
$73$	−9007.18	−1.69022	−0.845110	−	0.534593i	$-0.820465\pi$
$73$	−9007.18	−1.69022	−0.845110	+	0.534593i	$0.820465\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 11948.8i	− 2.01532i
$78$	0	0
$79$	5113.86	0.819397	0.409699	−	0.912221i	$-0.365634\pi$
$79$	5113.86	0.819397	0.409699	+	0.912221i	$0.365634\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8679.67i	1.25993i	0.776623	+	0.629966i	$0.216931\pi$
$83$	8679.67i	1.25993i	−0.776623	+	0.629966i	$0.783069\pi$
$84$	0	0
$85$	−3792.67	−0.524937
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4627.16i	0.584163i	0.956393	+	0.292082i	$0.0943479\pi$
$89$	4627.16i	0.584163i	−0.956393	+	0.292082i	$0.905652\pi$
$90$	0	0
$91$	2372.90	0.286548
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6208.74i	0.687949i
$96$	0	0
$97$	−8859.15	−0.941561	−0.470781	−	0.882250i	$-0.656028\pi$
$97$	−8859.15	−0.941561	−0.470781	+	0.882250i	$0.656028\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 1192.36i	− 0.116886i	−0.998291	−	0.0584432i	$-0.981386\pi$
$101$	− 1192.36i	− 0.116886i	0.998291	−	0.0584432i	$-0.0186137\pi$
$102$	0	0
$103$	−11981.1	−1.12934	−0.564669	−	0.825318i	$-0.690996\pi$
$103$	−11981.1	−1.12934	−0.564669	+	0.825318i	$0.690996\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 1281.31i	− 0.111915i	−0.998433	−	0.0559575i	$-0.982179\pi$
$107$	− 1281.31i	− 0.111915i	0.998433	−	0.0559575i	$-0.0178211\pi$
$108$	0	0
$109$	3738.13	0.314631	0.157315	−	0.987548i	$-0.449716\pi$
$109$	3738.13	0.314631	0.157315	+	0.987548i	$0.449716\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 1663.91i	− 0.130308i	−0.997875	−	0.0651541i	$-0.979246\pi$
$113$	− 1663.91i	− 0.130308i	0.997875	−	0.0651541i	$-0.0207539\pi$
$114$	0	0
$115$	−2722.92	−0.205892
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 24242.0i	− 1.71188i
$120$	0	0
$121$	−26582.8	−1.81564
$122$	0	0
$123$	0	0
$124$	0	0
$125$	10728.5i	0.686623i
$126$	0	0
$127$	10655.7	0.660653	0.330326	−	0.943867i	$-0.392841\pi$
$127$	10655.7	0.660653	0.330326	+	0.943867i	$0.392841\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 7161.08i	− 0.417288i	−0.977992	−	0.208644i	$-0.933095\pi$
$131$	− 7161.08i	− 0.417288i	0.977992	−	0.208644i	$-0.0669050\pi$
$132$	0	0
$133$	−39685.0	−2.24348
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 25280.8i	− 1.34695i	−0.739212	−	0.673473i	$-0.764802\pi$
$137$	− 25280.8i	− 1.34695i	0.739212	−	0.673473i	$-0.235198\pi$
$138$	0	0
$139$	31985.5	1.65548	0.827738	−	0.561114i	$-0.189627\pi$
$139$	31985.5	1.65548	0.827738	+	0.561114i	$0.189627\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 8186.61i	− 0.400343i
$144$	0	0
$145$	11974.2	0.569521
$146$	0	0
$147$	0	0
$148$	0	0
$149$	39856.0i	1.79524i	0.440774	+	0.897618i	$0.354704\pi$
$149$	39856.0i	1.79524i	−0.440774	+	0.897618i	$0.645296\pi$
$150$	0	0
$151$	30792.7	1.35050	0.675249	−	0.737590i	$-0.264036\pi$
$151$	30792.7	1.35050	0.675249	+	0.737590i	$0.264036\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	11704.5i	0.487178i
$156$	0	0
$157$	−28412.4	−1.15268	−0.576339	−	0.817211i	$-0.695519\pi$
$157$	−28412.4	−1.15268	−0.576339	+	0.817211i	$0.695519\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 17404.3i	− 0.671438i
$162$	0	0
$163$	6548.84	0.246484	0.123242	−	0.992377i	$-0.460671\pi$
$163$	6548.84	0.246484	0.123242	+	0.992377i	$0.460671\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 1506.75i	− 0.0540267i	−0.999635	−	0.0270134i	$-0.991400\pi$
$167$	− 1506.75i	− 0.0540267i	0.999635	−	0.0270134i	$-0.00859967\pi$
$168$	0	0
$169$	−26935.2	−0.943077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 4460.99i	− 0.149053i	−0.997219	−	0.0745263i	$-0.976256\pi$
$173$	− 4460.99i	− 0.149053i	0.997219	−	0.0745263i	$-0.0237445\pi$
$174$	0	0
$175$	−31792.7	−1.03813
$176$	0	0
$177$	0	0
$178$	0	0
$179$	41140.6i	1.28400i	0.766705	+	0.641999i	$0.221895\pi$
$179$	41140.6i	1.28400i	−0.766705	+	0.641999i	$0.778105\pi$
$180$	0	0
$181$	−55119.0	−1.68246	−0.841230	−	0.540678i	$-0.818168\pi$
$181$	−55119.0	−1.68246	−0.841230	+	0.540678i	$0.818168\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 22940.7i	− 0.670291i
$186$	0	0
$187$	−83635.7	−2.39171
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 17927.6i	− 0.491423i	−0.969343	−	0.245712i	$-0.920978\pi$
$191$	− 17927.6i	− 0.491423i	0.969343	−	0.245712i	$-0.0790217\pi$
$192$	0	0
$193$	−17375.0	−0.466455	−0.233227	−	0.972422i	$-0.574929\pi$
$193$	−17375.0	−0.466455	−0.233227	+	0.972422i	$0.574929\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 20244.5i	− 0.521644i	−0.965387	−	0.260822i	$-0.916006\pi$
$197$	− 20244.5i	− 0.521644i	0.965387	−	0.260822i	$-0.0839935\pi$
$198$	0	0
$199$	−28268.6	−0.713835	−0.356918	−	0.934136i	$-0.616172\pi$
$199$	−28268.6	−0.713835	−0.356918	+	0.934136i	$0.616172\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	76536.5i	1.85728i
$204$	0	0
$205$	−14796.7	−0.352093
$206$	0	0
$207$	0	0
$208$	0	0
$209$	136915.i	3.13442i
$210$	0	0
$211$	11904.9	0.267400	0.133700	−	0.991022i	$-0.457314\pi$
$211$	11904.9	0.267400	0.133700	+	0.991022i	$0.457314\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 9161.72i	− 0.198198i
$216$	0	0
$217$	−74812.5	−1.58875
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 16609.1i	− 0.340066i
$222$	0	0
$223$	−4305.32	−0.0865757	−0.0432878	−	0.999063i	$-0.513783\pi$
$223$	−4305.32	−0.0865757	−0.0432878	+	0.999063i	$0.513783\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 60754.4i	− 1.17903i	−0.807756	−	0.589516i	$-0.799318\pi$
$227$	− 60754.4i	− 1.17903i	0.807756	−	0.589516i	$-0.200682\pi$
$228$	0	0
$229$	−30008.8	−0.572239	−0.286119	−	0.958194i	$-0.592365\pi$
$229$	−30008.8	−0.572239	−0.286119	+	0.958194i	$0.592365\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 47860.2i	− 0.881583i	−0.897610	−	0.440791i	$-0.854698\pi$
$233$	− 47860.2i	− 0.881583i	0.897610	−	0.440791i	$-0.145302\pi$
$234$	0	0
$235$	22934.3	0.415288
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 40605.1i	− 0.710861i	−0.934703	−	0.355430i	$-0.884334\pi$
$239$	− 40605.1i	− 0.710861i	0.934703	−	0.355430i	$-0.115666\pi$
$240$	0	0
$241$	−44909.7	−0.773226	−0.386613	−	0.922242i	$-0.626355\pi$
$241$	−44909.7	−0.773226	−0.386613	+	0.922242i	$0.626355\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	9781.58i	0.162958i
$246$	0	0
$247$	−27189.8	−0.445668
$248$	0	0
$249$	0	0
$250$	0	0
$251$	1832.88i	0.0290928i	0.999894	+	0.0145464i	$0.00463043\pi$
$251$	1832.88i	0.0290928i	−0.999894	+	0.0145464i	$0.995370\pi$
$252$	0	0
$253$	−60045.6	−0.938081
$254$	0	0
$255$	0	0
$256$	0	0
$257$	48645.0i	0.736499i	0.929727	+	0.368250i	$0.120043\pi$
$257$	48645.0i	0.736499i	−0.929727	+	0.368250i	$0.879957\pi$
$258$	0	0
$259$	146632.	2.18590
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 83740.8i	− 1.21067i	−0.795971	−	0.605335i	$-0.793039\pi$
$263$	− 83740.8i	− 1.21067i	0.795971	−	0.605335i	$-0.206961\pi$
$264$	0	0
$265$	−35021.0	−0.498697
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 87184.6i	− 1.20486i	−0.798173	−	0.602428i	$-0.794200\pi$
$269$	− 87184.6i	− 1.20486i	0.798173	−	0.602428i	$-0.205800\pi$
$270$	0	0
$271$	−35905.5	−0.488902	−0.244451	−	0.969662i	$-0.578608\pi$
$271$	−35905.5	−0.488902	−0.244451	+	0.969662i	$0.578608\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	109686.i	1.45039i
$276$	0	0
$277$	74382.8	0.969422	0.484711	−	0.874674i	$-0.338925\pi$
$277$	74382.8	0.969422	0.484711	+	0.874674i	$0.338925\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 105797.i	− 1.33987i	−0.742422	−	0.669933i	$-0.766323\pi$
$281$	− 105797.i	− 1.33987i	0.742422	−	0.669933i	$-0.233677\pi$
$282$	0	0
$283$	−36059.7	−0.450246	−0.225123	−	0.974330i	$-0.572278\pi$
$283$	−36059.7	−0.450246	−0.225123	+	0.974330i	$0.572278\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 94577.4i	− 1.14822i
$288$	0	0
$289$	−86160.8	−1.03161
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 66937.1i	− 0.779708i	−0.920877	−	0.389854i	$-0.872526\pi$
$293$	− 66937.1i	− 0.779708i	0.920877	−	0.389854i	$-0.127474\pi$
$294$	0	0
$295$	9844.04	0.113117
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 11924.4i	− 0.133381i
$300$	0	0
$301$	58559.8	0.646348
$302$	0	0
$303$	0	0
$304$	0	0
$305$	13101.8i	0.140841i
$306$	0	0
$307$	−71055.1	−0.753909	−0.376954	−	0.926232i	$-0.623029\pi$
$307$	−71055.1	−0.753909	−0.376954	+	0.926232i	$0.623029\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 79416.4i	− 0.821087i	−0.911841	−	0.410543i	$-0.865339\pi$
$311$	− 79416.4i	− 0.821087i	0.911841	−	0.410543i	$-0.134661\pi$
$312$	0	0
$313$	46492.0	0.474558	0.237279	−	0.971441i	$-0.423744\pi$
$313$	46492.0	0.474558	0.237279	+	0.971441i	$0.423744\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8615.98i	0.0857406i	0.999081	+	0.0428703i	$0.0136502\pi$
$317$	8615.98i	0.0857406i	−0.999081	+	0.0428703i	$0.986350\pi$
$318$	0	0
$319$	264054.	2.59484
$320$	0	0
$321$	0	0
$322$	0	0
$323$	277775.i	2.66249i
$324$	0	0
$325$	−21782.4	−0.206224
$326$	0	0
$327$	0	0
$328$	0	0
$329$	146591.i	1.35430i
$330$	0	0
$331$	−88930.9	−0.811703	−0.405851	−	0.913939i	$-0.633025\pi$
$331$	−88930.9	−0.811703	−0.405851	+	0.913939i	$0.633025\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 55997.2i	− 0.498972i
$336$	0	0
$337$	−119129.	−1.04896	−0.524478	−	0.851424i	$-0.675740\pi$
$337$	−119129.	−1.04896	−0.524478	+	0.851424i	$0.675740\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	258106.i	2.21967i
$342$	0	0
$343$	78778.3	0.669604
$344$	0	0
$345$	0	0
$346$	0	0
$347$	230105.i	1.91103i	0.294946	+	0.955514i	$0.404698\pi$
$347$	230105.i	1.91103i	−0.294946	+	0.955514i	$0.595302\pi$
$348$	0	0
$349$	−215366.	−1.76818	−0.884089	−	0.467318i	$-0.845220\pi$
$349$	−215366.	−1.76818	−0.884089	+	0.467318i	$0.845220\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 136919.i	− 1.09879i	−0.835564	−	0.549394i	$-0.814859\pi$
$353$	− 136919.i	− 1.09879i	0.835564	−	0.549394i	$-0.185141\pi$
$354$	0	0
$355$	−23808.9	−0.188922
$356$	0	0
$357$	0	0
$358$	0	0
$359$	54063.8i	0.419486i	0.977757	+	0.209743i	$0.0672627\pi$
$359$	54063.8i	0.419486i	−0.977757	+	0.209743i	$0.932737\pi$
$360$	0	0
$361$	324407.	2.48929
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 82930.9i	− 0.622488i
$366$	0	0
$367$	−150034.	−1.11393	−0.556965	−	0.830536i	$-0.688034\pi$
$367$	−150034.	−1.11393	−0.556965	+	0.830536i	$0.688034\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 223847.i	− 1.62631i
$372$	0	0
$373$	123883.	0.890415	0.445207	−	0.895427i	$-0.353130\pi$
$373$	123883.	0.890415	0.445207	+	0.895427i	$0.353130\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	52438.2i	0.368948i
$378$	0	0
$379$	116175.	0.808788	0.404394	−	0.914585i	$-0.367482\pi$
$379$	116175.	0.808788	0.404394	+	0.914585i	$0.367482\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	99372.9i	0.677439i	0.940887	+	0.338720i	$0.109994\pi$
$383$	99372.9i	0.677439i	−0.940887	+	0.338720i	$0.890006\pi$
$384$	0	0
$385$	110015.	0.742217
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 163660.i	− 1.08154i	−0.841169	−	0.540772i	$-0.818132\pi$
$389$	− 163660.i	− 1.08154i	0.841169	−	0.540772i	$-0.181868\pi$
$390$	0	0
$391$	−121822.	−0.796840
$392$	0	0
$393$	0	0
$394$	0	0
$395$	47084.3i	0.301774i
$396$	0	0
$397$	104776.	0.664782	0.332391	−	0.943142i	$-0.392145\pi$
$397$	104776.	0.664782	0.332391	+	0.943142i	$0.392145\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 238660.i	− 1.48419i	−0.670293	−	0.742097i	$-0.733832\pi$
$401$	− 238660.i	− 1.48419i	0.670293	−	0.742097i	$-0.266168\pi$
$402$	0	0
$403$	−51257.0	−0.315604
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 505887.i	− 3.05397i
$408$	0	0
$409$	135660.	0.810968	0.405484	−	0.914102i	$-0.367103\pi$
$409$	135660.	0.810968	0.405484	+	0.914102i	$0.367103\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	62921.1i	0.368889i
$414$	0	0
$415$	−79915.5	−0.464018
$416$	0	0
$417$	0	0
$418$	0	0
$419$	65750.8i	0.374518i	0.982311	+	0.187259i	$0.0599604\pi$
$419$	65750.8i	0.374518i	−0.982311	+	0.187259i	$0.940040\pi$
$420$	0	0
$421$	9953.15	0.0561560	0.0280780	−	0.999606i	$-0.491061\pi$
$421$	9953.15	0.0561560	0.0280780	+	0.999606i	$0.491061\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	222533.i	1.23202i
$426$	0	0
$427$	−83743.8	−0.459301
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 257673.i	− 1.38712i	−0.720399	−	0.693559i	$-0.756041\pi$
$431$	− 257673.i	− 1.38712i	0.720399	−	0.693559i	$-0.243959\pi$
$432$	0	0
$433$	128908.	0.687552	0.343776	−	0.939052i	$-0.388294\pi$
$433$	128908.	0.687552	0.343776	+	0.939052i	$0.388294\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	199427.i	1.04429i
$438$	0	0
$439$	−167706.	−0.870199	−0.435100	−	0.900382i	$-0.643287\pi$
$439$	−167706.	−0.870199	−0.435100	+	0.900382i	$0.643287\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	227763.i	1.16058i	0.814409	+	0.580292i	$0.197062\pi$
$443$	227763.i	1.16058i	−0.814409	+	0.580292i	$0.802938\pi$
$444$	0	0
$445$	−42603.1	−0.215140
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40005.1i	0.198437i	0.995066	+	0.0992185i	$0.0316343\pi$
$449$	40005.1i	0.198437i	−0.995066	+	0.0992185i	$0.968366\pi$
$450$	0	0
$451$	−326296.	−1.60420
$452$	0	0
$453$	0	0
$454$	0	0
$455$	21847.8i	0.105532i
$456$	0	0
$457$	29809.1	0.142730	0.0713651	−	0.997450i	$-0.477264\pi$
$457$	29809.1	0.142730	0.0713651	+	0.997450i	$0.477264\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	220157.i	1.03593i	0.855401	+	0.517966i	$0.173310\pi$
$461$	220157.i	1.03593i	−0.855401	+	0.517966i	$0.826690\pi$
$462$	0	0
$463$	−76372.8	−0.356268	−0.178134	−	0.984006i	$-0.557006\pi$
$463$	−76372.8	−0.356268	−0.178134	+	0.984006i	$0.557006\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	91142.2i	0.417913i	0.977925	+	0.208956i	$0.0670067\pi$
$467$	91142.2i	0.417913i	−0.977925	+	0.208956i	$0.932993\pi$
$468$	0	0
$469$	357922.	1.62721
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 202034.i	− 0.903028i
$474$	0	0
$475$	364294.	1.61460
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 335475.i	− 1.46214i	−0.682302	−	0.731070i	$-0.739021\pi$
$479$	− 335475.i	− 1.46214i	0.682302	−	0.731070i	$-0.260979\pi$
$480$	0	0
$481$	100464.	0.434229
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 81568.0i	− 0.346766i
$486$	0	0
$487$	275667.	1.16232	0.581162	−	0.813788i	$-0.302598\pi$
$487$	275667.	1.16232	0.581162	+	0.813788i	$0.302598\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	296740.i	1.23087i	0.788186	+	0.615437i	$0.211021\pi$
$491$	296740.i	1.23087i	−0.788186	+	0.615437i	$0.788979\pi$
$492$	0	0
$493$	535717.	2.20415
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 152181.i	− 0.616096i
$498$	0	0
$499$	192046.	0.771267	0.385633	−	0.922652i	$-0.373983\pi$
$499$	192046.	0.771267	0.385633	+	0.922652i	$0.373983\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	393624.i	1.55577i	0.628407	+	0.777885i	$0.283707\pi$
$503$	393624.i	1.55577i	−0.628407	+	0.777885i	$0.716293\pi$
$504$	0	0
$505$	10978.3	0.0430479
$506$	0	0
$507$	0	0
$508$	0	0
$509$	199010.i	0.768139i	0.923304	+	0.384070i	$0.125478\pi$
$509$	199010.i	0.768139i	−0.923304	+	0.384070i	$0.874522\pi$
$510$	0	0
$511$	530077.	2.03001
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 110313.i	− 0.415921i
$516$	0	0
$517$	505745.	1.89213
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 5744.00i	− 0.0211612i	−0.999944	−	0.0105806i	$-0.996632\pi$
$521$	− 5744.00i	− 0.0211612i	0.999944	−	0.0105806i	$-0.00336797\pi$
$522$	0	0
$523$	46438.8	0.169777	0.0848883	−	0.996390i	$-0.472947\pi$
$523$	46438.8	0.169777	0.0848883	+	0.996390i	$0.472947\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	523650.i	1.88547i
$528$	0	0
$529$	192380.	0.687462
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 64798.8i	− 0.228093i
$534$	0	0
$535$	11797.3	0.0412169
$536$	0	0
$537$	0	0
$538$	0	0
$539$	215703.i	0.742469i
$540$	0	0
$541$	−318888.	−1.08954	−0.544771	−	0.838585i	$-0.683383\pi$
$541$	−318888.	−1.08954	−0.544771	+	0.838585i	$0.683383\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	34417.7i	0.115875i
$546$	0	0
$547$	243528.	0.813906	0.406953	−	0.913449i	$-0.366591\pi$
$547$	243528.	0.813906	0.406953	+	0.913449i	$0.366591\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 876988.i	− 2.88862i
$552$	0	0
$553$	−300953.	−0.984121
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 279752.i	− 0.901701i	−0.892599	−	0.450851i	$-0.851121\pi$
$557$	− 279752.i	− 0.901701i	0.892599	−	0.450851i	$-0.148879\pi$
$558$	0	0
$559$	40121.7	0.128397
$560$	0	0
$561$	0	0
$562$	0	0
$563$	260930.i	0.823204i	0.911364	+	0.411602i	$0.135031\pi$
$563$	260930.i	0.823204i	−0.911364	+	0.411602i	$0.864969\pi$
$564$	0	0
$565$	15319.9	0.0479910
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 90744.7i	− 0.280283i	−0.990131	−	0.140142i	$-0.955244\pi$
$569$	− 90744.7i	− 0.280283i	0.990131	−	0.140142i	$-0.0447558\pi$
$570$	0	0
$571$	−409322.	−1.25543	−0.627715	−	0.778443i	$-0.716010\pi$
$571$	−409322.	−1.25543	−0.627715	+	0.778443i	$0.716010\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	159766.i	0.483224i
$576$	0	0
$577$	262977.	0.789889	0.394944	−	0.918705i	$-0.370764\pi$
$577$	262977.	0.789889	0.394944	+	0.918705i	$0.370764\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 510803.i	− 1.51322i
$582$	0	0
$583$	−772281.	−2.27216
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 292409.i	− 0.848622i	−0.905517	−	0.424311i	$-0.860516\pi$
$587$	− 292409.i	− 0.848622i	0.905517	−	0.424311i	$-0.139484\pi$
$588$	0	0
$589$	857234.	2.47098
$590$	0	0
$591$	0	0
$592$	0	0
$593$	439268.i	1.24916i	0.780959	+	0.624582i	$0.214731\pi$
$593$	439268.i	1.24916i	−0.780959	+	0.624582i	$0.785269\pi$
$594$	0	0
$595$	223201.	0.630466
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 184467.i	− 0.514120i	−0.966395	−	0.257060i	$-0.917246\pi$
$599$	− 184467.i	− 0.514120i	0.966395	−	0.257060i	$-0.0827539\pi$
$600$	0	0
$601$	607715.	1.68249	0.841243	−	0.540658i	$-0.181825\pi$
$601$	607715.	1.68249	0.841243	+	0.540658i	$0.181825\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 244754.i	− 0.668680i
$606$	0	0
$607$	−136724.	−0.371079	−0.185540	−	0.982637i	$-0.559403\pi$
$607$	−136724.	−0.371079	−0.185540	+	0.982637i	$0.559403\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	100435.i	0.269032i
$612$	0	0
$613$	−600964.	−1.59929	−0.799646	−	0.600472i	$-0.794979\pi$
$613$	−600964.	−1.59929	−0.799646	+	0.600472i	$0.794979\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	661738.i	1.73826i	0.494580	+	0.869132i	$0.335322\pi$
$617$	661738.i	1.73826i	−0.494580	+	0.869132i	$0.664678\pi$
$618$	0	0
$619$	−345696.	−0.902220	−0.451110	−	0.892468i	$-0.648972\pi$
$619$	−345696.	−0.902220	−0.451110	+	0.892468i	$0.648972\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 272310.i	− 0.701598i
$624$	0	0
$625$	238863.	0.611489
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 1.02635e6i	− 2.59415i
$630$	0	0
$631$	232800.	0.584689	0.292345	−	0.956313i	$-0.405565\pi$
$631$	232800.	0.584689	0.292345	+	0.956313i	$0.405565\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	98108.8i	0.243310i
$636$	0	0
$637$	−42836.2	−0.105568
$638$	0	0
$639$	0	0
$640$	0	0
$641$	200039.i	0.486855i	0.969919	+	0.243427i	$0.0782717\pi$
$641$	200039.i	0.486855i	−0.969919	+	0.243427i	$0.921728\pi$
$642$	0	0
$643$	64240.0	0.155376	0.0776879	−	0.996978i	$-0.475246\pi$
$643$	64240.0	0.155376	0.0776879	+	0.996978i	$0.475246\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 390522.i	− 0.932904i	−0.884547	−	0.466452i	$-0.845532\pi$
$647$	− 390522.i	− 0.932904i	0.884547	−	0.466452i	$-0.154468\pi$
$648$	0	0
$649$	217080.	0.515384
$650$	0	0
$651$	0	0
$652$	0	0
$653$	763743.i	1.79110i	0.444956	+	0.895552i	$0.353219\pi$
$653$	763743.i	1.79110i	−0.444956	+	0.895552i	$0.646781\pi$
$654$	0	0
$655$	65933.5	0.153682
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 556015.i	− 1.28031i	−0.768246	−	0.640155i	$-0.778870\pi$
$659$	− 556015.i	− 1.28031i	0.768246	−	0.640155i	$-0.221130\pi$
$660$	0	0
$661$	−137355.	−0.314371	−0.157185	−	0.987569i	$-0.550242\pi$
$661$	−137355.	−0.314371	−0.157185	+	0.987569i	$0.550242\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 365388.i	− 0.826248i
$666$	0	0
$667$	384614.	0.864517
$668$	0	0
$669$	0	0
$670$	0	0
$671$	288919.i	0.641700i
$672$	0	0
$673$	169016.	0.373163	0.186581	−	0.982440i	$-0.440259\pi$
$673$	169016.	0.373163	0.186581	+	0.982440i	$0.440259\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	717089.i	1.56457i	0.622919	+	0.782286i	$0.285947\pi$
$677$	717089.i	1.56457i	−0.622919	+	0.782286i	$0.714053\pi$
$678$	0	0
$679$	521366.	1.13084
$680$	0	0
$681$	0	0
$682$	0	0
$683$	74770.9i	0.160284i	0.996783	+	0.0801422i	$0.0255374\pi$
$683$	74770.9i	0.160284i	−0.996783	+	0.0801422i	$0.974463\pi$
$684$	0	0
$685$	232766.	0.496064
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 153367.i	− 0.323067i
$690$	0	0
$691$	−345313.	−0.723197	−0.361598	−	0.932334i	$-0.617769\pi$
$691$	−345313.	−0.723197	−0.361598	+	0.932334i	$0.617769\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	294497.i	0.609692i
$696$	0	0
$697$	−661995.	−1.36267
$698$	0	0
$699$	0	0
$700$	0	0
$701$	788179.i	1.60394i	0.597362	+	0.801971i	$0.296215\pi$
$701$	788179.i	1.60394i	−0.597362	+	0.801971i	$0.703785\pi$
$702$	0	0
$703$	−1.68018e6	−3.39973
$704$	0	0
$705$	0	0
$706$	0	0
$707$	70170.9i	0.140384i
$708$	0	0
$709$	−667638.	−1.32815	−0.664077	−	0.747664i	$-0.731175\pi$
$709$	−667638.	−1.32815	−0.664077	+	0.747664i	$0.731175\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	375951.i	0.739523i
$714$	0	0
$715$	75375.7	0.147441
$716$	0	0
$717$	0	0
$718$	0	0
$719$	591375.i	1.14394i	0.820273	+	0.571972i	$0.193821\pi$
$719$	591375.i	1.14394i	−0.820273	+	0.571972i	$0.806179\pi$
$720$	0	0
$721$	705096.	1.35637
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 702578.i	− 1.33665i
$726$	0	0
$727$	484117.	0.915970	0.457985	−	0.888960i	$-0.348571\pi$
$727$	484117.	0.915970	0.457985	+	0.888960i	$0.348571\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 409890.i	− 0.767065i
$732$	0	0
$733$	−36938.6	−0.0687499	−0.0343750	−	0.999409i	$-0.510944\pi$
$733$	−36938.6	−0.0687499	−0.0343750	+	0.999409i	$0.510944\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 1.23485e6i	− 2.27341i
$738$	0	0
$739$	2546.44	0.00466277	0.00233139	−	0.999997i	$-0.499258\pi$
$739$	2546.44	0.00466277	0.00233139	+	0.999997i	$0.499258\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	637751.i	1.15524i	0.816305	+	0.577622i	$0.196019\pi$
$743$	637751.i	1.15524i	−0.816305	+	0.577622i	$0.803981\pi$
$744$	0	0
$745$	−366963.	−0.661164
$746$	0	0
$747$	0	0
$748$	0	0
$749$	75406.0i	0.134413i
$750$	0	0
$751$	−506237.	−0.897582	−0.448791	−	0.893637i	$-0.648145\pi$
$751$	−506237.	−0.897582	−0.448791	+	0.893637i	$0.648145\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	283515.i	0.497372i
$756$	0	0
$757$	−41453.6	−0.0723386	−0.0361693	−	0.999346i	$-0.511516\pi$
$757$	−41453.6	−0.0723386	−0.0361693	+	0.999346i	$0.511516\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 550019.i	− 0.949747i	−0.880054	−	0.474874i	$-0.842494\pi$
$761$	− 550019.i	− 0.949747i	0.880054	−	0.474874i	$-0.157506\pi$
$762$	0	0
$763$	−219991.	−0.377881
$764$	0	0
$765$	0	0
$766$	0	0
$767$	43109.8i	0.0732799i
$768$	0	0
$769$	−785730.	−1.32868	−0.664341	−	0.747430i	$-0.731288\pi$
$769$	−785730.	−1.32868	−0.664341	+	0.747430i	$0.731288\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	790023.i	1.32215i	0.750319	+	0.661075i	$0.229900\pi$
$773$	790023.i	1.32215i	−0.750319	+	0.661075i	$0.770100\pi$
$774$	0	0
$775$	686752.	1.14340
$776$	0	0
$777$	0	0
$778$	0	0
$779$	1.08371e6i	1.78582i
$780$	0	0
$781$	−525032.	−0.860763
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 261598.i	− 0.424518i
$786$	0	0
$787$	52115.7	0.0841431	0.0420716	−	0.999115i	$-0.486604\pi$
$787$	52115.7	0.0841431	0.0420716	+	0.999115i	$0.486604\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	97921.7i	0.156504i
$792$	0	0
$793$	−57376.3	−0.0912401
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 332982.i	− 0.524208i	−0.965040	−	0.262104i	$-0.915584\pi$
$797$	− 332982.i	− 0.524208i	0.965040	−	0.262104i	$-0.0844164\pi$
$798$	0	0
$799$	1.02606e6	1.60724
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1.82879e6i	− 2.83617i
$804$	0	0
$805$	160245.	0.247282
$806$	0	0
$807$	0	0
$808$	0	0
$809$	1.09286e6i	1.66981i	0.550392	+	0.834906i	$0.314478\pi$
$809$	1.09286e6i	1.66981i	−0.550392	+	0.834906i	$0.685522\pi$
$810$	0	0
$811$	42430.5	0.0645113	0.0322557	−	0.999480i	$-0.489731\pi$
$811$	42430.5	0.0645113	0.0322557	+	0.999480i	$0.489731\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	60296.5i	0.0907772i
$816$	0	0
$817$	−671004.	−1.00527
$818$	0	0
$819$	0	0
$820$	0	0
$821$	323954.i	0.480615i	0.970697	+	0.240307i	$0.0772482\pi$
$821$	323954.i	0.480615i	−0.970697	+	0.240307i	$0.922752\pi$
$822$	0	0
$823$	23844.8	0.0352041	0.0176021	−	0.999845i	$-0.494397\pi$
$823$	23844.8	0.0352041	0.0176021	+	0.999845i	$0.494397\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	272909.i	0.399031i	0.979895	+	0.199515i	$0.0639368\pi$
$827$	272909.i	0.399031i	−0.979895	+	0.199515i	$0.936063\pi$
$828$	0	0
$829$	−186691.	−0.271652	−0.135826	−	0.990733i	$-0.543369\pi$
$829$	−186691.	−0.271652	−0.135826	+	0.990733i	$0.543369\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	437622.i	0.630680i
$834$	0	0
$835$	13873.0	0.0198974
$836$	0	0
$837$	0	0
$838$	0	0
$839$	5003.86i	0.00710856i	0.999994	+	0.00355428i	$0.00113136\pi$
$839$	5003.86i	0.00710856i	−0.999994	+	0.00355428i	$0.998869\pi$
$840$	0	0
$841$	−984079.	−1.39136
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 247998.i	− 0.347324i
$846$	0	0
$847$	1.56441e6	2.18065
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 736862.i	− 1.01748i
$852$	0	0
$853$	132546.	0.182167	0.0910834	−	0.995843i	$-0.470967\pi$
$853$	132546.	0.182167	0.0910834	+	0.995843i	$0.470967\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 492132.i	− 0.670070i	−0.942206	−	0.335035i	$-0.891252\pi$
$857$	− 492132.i	− 0.670070i	0.942206	−	0.335035i	$-0.108748\pi$
$858$	0	0
$859$	839031.	1.13708	0.568541	−	0.822655i	$-0.307508\pi$
$859$	839031.	1.13708	0.568541	+	0.822655i	$0.307508\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	508087.i	0.682208i	0.940026	+	0.341104i	$0.110801\pi$
$863$	508087.i	0.682208i	−0.940026	+	0.341104i	$0.889199\pi$
$864$	0	0
$865$	41073.3	0.0548943
$866$	0	0
$867$	0	0
$868$	0	0
$869$	1.03830e6i	1.37494i
$870$	0	0
$871$	245227.	0.323245
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 631377.i	− 0.824655i
$876$	0	0
$877$	828867.	1.07767	0.538835	−	0.842412i	$-0.318865\pi$
$877$	828867.	1.07767	0.538835	+	0.842412i	$0.318865\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.06305e6i	1.36963i	0.728718	+	0.684814i	$0.240117\pi$
$881$	1.06305e6i	1.36963i	−0.728718	+	0.684814i	$0.759883\pi$
$882$	0	0
$883$	−1.17415e6	−1.50593	−0.752963	−	0.658063i	$-0.771376\pi$
$883$	−1.17415e6	−1.50593	−0.752963	+	0.658063i	$0.771376\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 756633.i	− 0.961696i	−0.876804	−	0.480848i	$-0.840329\pi$
$887$	− 756633.i	− 0.961696i	0.876804	−	0.480848i	$-0.159671\pi$
$888$	0	0
$889$	−627091.	−0.793464
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.67971e6i	− 2.10635i
$894$	0	0
$895$	−378790.	−0.472881
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 1.65326e6i	− 2.04561i
$900$	0	0
$901$	−1.56682e6	−1.93005
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 507492.i	− 0.619630i
$906$	0	0
$907$	−365049.	−0.443748	−0.221874	−	0.975075i	$-0.571217\pi$
$907$	−365049.	−0.443748	−0.221874	+	0.975075i	$0.571217\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.23447e6i	1.48745i	0.668486	+	0.743725i	$0.266943\pi$
$911$	1.23447e6i	1.48745i	−0.668486	+	0.743725i	$0.733057\pi$
$912$	0	0
$913$	−1.76229e6	−2.11415
$914$	0	0
$915$	0	0
$916$	0	0
$917$	421433.i	0.501176i
$918$	0	0
$919$	−957839.	−1.13413	−0.567063	−	0.823674i	$-0.691921\pi$
$919$	−957839.	−1.13413	−0.567063	+	0.823674i	$0.691921\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 104266.i	− 0.122388i
$924$	0	0
$925$	−1.34603e6	−1.57316
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.53514e6i	− 1.77876i	−0.457167	−	0.889381i	$-0.651136\pi$
$929$	− 1.53514e6i	− 1.77876i	0.457167	−	0.889381i	$-0.348864\pi$
$930$	0	0
$931$	716403.	0.826528
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 770051.i	− 0.880838i
$936$	0	0
$937$	−275585.	−0.313889	−0.156944	−	0.987607i	$-0.550164\pi$
$937$	−275585.	−0.313889	−0.156944	+	0.987607i	$0.550164\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 237709.i	− 0.268452i	−0.990951	−	0.134226i	$-0.957145\pi$
$941$	− 237709.i	− 0.268452i	0.990951	−	0.134226i	$-0.0428547\pi$
$942$	0	0
$943$	−475274.	−0.534467
$944$	0	0
$945$	0	0
$946$	0	0
$947$	412084.i	0.459500i	0.973250	+	0.229750i	$0.0737908\pi$
$947$	412084.i	0.459500i	−0.973250	+	0.229750i	$0.926209\pi$
$948$	0	0
$949$	363177.	0.403261
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 716408.i	− 0.788814i	−0.918936	−	0.394407i	$-0.870950\pi$
$953$	− 716408.i	− 0.788814i	0.918936	−	0.394407i	$-0.129050\pi$
$954$	0	0
$955$	165063.	0.180985
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.48779e6i	1.61772i
$960$	0	0
$961$	692501.	0.749848
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 159975.i	− 0.171790i
$966$	0	0
$967$	−1.42980e6	−1.52905	−0.764524	−	0.644595i	$-0.777026\pi$
$967$	−1.42980e6	−1.52905	−0.764524	+	0.644595i	$0.777026\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.10577e6i	1.17280i	0.810021	+	0.586401i	$0.199456\pi$
$971$	1.10577e6i	1.17280i	−0.810021	+	0.586401i	$0.800544\pi$
$972$	0	0
$973$	−1.88236e6	−1.98828
$974$	0	0
$975$	0	0
$976$	0	0
$977$	143494.i	0.150329i	0.997171	+	0.0751646i	$0.0239482\pi$
$977$	143494.i	0.150329i	−0.997171	+	0.0751646i	$0.976052\pi$
$978$	0	0
$979$	−939482.	−0.980219
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 331189.i	− 0.342744i	−0.985206	−	0.171372i	$-0.945180\pi$
$983$	− 331189.i	− 0.342744i	0.985206	−	0.171372i	$-0.0548199\pi$
$984$	0	0
$985$	186395.	0.192115
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 294277.i	− 0.300860i
$990$	0	0
$991$	−1.31457e6	−1.33855	−0.669277	−	0.743013i	$-0.733396\pi$
$991$	−1.31457e6	−1.33855	−0.669277	+	0.743013i	$0.733396\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 260275.i	− 0.262897i
$996$	0	0
$997$	608752.	0.612421	0.306210	−	0.951964i	$-0.400939\pi$
$997$	608752.	0.612421	0.306210	+	0.951964i	$0.400939\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.5.e.g.161.9	yes	16
3.2	odd	2	inner	864.5.e.g.161.7	✓	16
4.3	odd	2	inner	864.5.e.g.161.10	yes	16
12.11	even	2	inner	864.5.e.g.161.8	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
864.5.e.g.161.7	✓	16	3.2	odd	2	inner
864.5.e.g.161.8	yes	16	12.11	even	2	inner
864.5.e.g.161.9	yes	16	1.1	even	1	trivial
864.5.e.g.161.10	yes	16	4.3	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	864.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+9.20720i q^{5} -58.8505 q^{7} +203.037i q^{11} -40.3208 q^{13} +411.924i q^{17} +674.335 q^{19} +295.738i q^{23} +540.227 q^{25} -1300.52i q^{29} +1271.23 q^{31} -541.849i q^{35} -2491.60 q^{37} +1607.08i q^{41} -995.060 q^{43} -2490.91i q^{47} +1062.38 q^{49} +3803.65i q^{53} -1869.40 q^{55} -1069.17i q^{59} +1422.99 q^{61} -371.242i q^{65} -6081.89 q^{67} +2585.90i q^{71} -9007.18 q^{73} -11948.8i q^{77} +5113.86 q^{79} +8679.67i q^{83} -3792.67 q^{85} +4627.16i q^{89} +2372.90 q^{91} +6208.74i q^{95} -8859.15 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{13} + 704 q^{25} - 2624 q^{37} + 1728 q^{49} + 3264 q^{61} - 5424 q^{73} + 4704 q^{85} - 80 q^{97}+O(q^{100}) \) 16 * q - 32 * q^13 + 704 * q^25 - 2624 * q^37 + 1728 * q^49 + 3264 * q^61 - 5424 * q^73 + 4704 * q^85 - 80 * q^97

Introduction

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