LMFDB - Embedded newform 324.5.c.b.161.8

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			161.8
Root			$-6.57859 - 1.93185i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.5.c.b.161.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+33.7061i q^{5} -33.4163 q^{7} +137.845i q^{11} +256.928 q^{13} +56.9994i q^{17} -328.391 q^{19} -65.7667i q^{23} -511.104 q^{25} +462.307i q^{29} -1833.04 q^{31} -1126.33i q^{35} +359.661 q^{37} -2650.81i q^{41} +860.347 q^{43} +3168.85i q^{47} -1284.35 q^{49} -5281.61i q^{53} -4646.21 q^{55} +1532.62i q^{59} -7229.76 q^{61} +8660.05i q^{65} -590.092 q^{67} +4706.23i q^{71} -2748.96 q^{73} -4606.25i q^{77} -5967.05 q^{79} +11938.8i q^{83} -1921.23 q^{85} -9339.75i q^{89} -8585.57 q^{91} -11068.8i q^{95} +3197.68 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	33.7061i	1.34825i	0.738619	+	0.674123i	$0.235478\pi$
$5$	33.7061i	1.34825i	−0.738619	+	0.674123i	$0.764522\pi$
$6$	0	0
$7$	−33.4163	−0.681964	−0.340982	−	0.940070i	$-0.610760\pi$
$7$	−33.4163	−0.681964	−0.340982	+	0.940070i	$0.610760\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	137.845i	1.13921i	0.821918	+	0.569606i	$0.192904\pi$
$11$	137.845i	1.13921i	−0.821918	+	0.569606i	$0.807096\pi$
$12$	0	0
$13$	256.928	1.52028	0.760142	−	0.649757i	$-0.225130\pi$
$13$	256.928	1.52028	0.760142	+	0.649757i	$0.225130\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	56.9994i	0.197230i	0.995126	+	0.0986149i	$0.0314412\pi$
$17$	56.9994i	0.197230i	−0.995126	+	0.0986149i	$0.968559\pi$
$18$	0	0
$19$	−328.391	−0.909672	−0.454836	−	0.890575i	$-0.650302\pi$
$19$	−328.391	−0.909672	−0.454836	+	0.890575i	$0.650302\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 65.7667i	− 0.124323i	−0.998066	−	0.0621613i	$-0.980201\pi$
$23$	− 65.7667i	− 0.124323i	0.998066	−	0.0621613i	$-0.0197993\pi$
$24$	0	0
$25$	−511.104	−0.817766
$26$	0	0
$27$	0	0
$28$	0	0
$29$	462.307i	0.549711i	0.961486	+	0.274855i	$0.0886300\pi$
$29$	462.307i	0.549711i	−0.961486	+	0.274855i	$0.911370\pi$
$30$	0	0
$31$	−1833.04	−1.90743	−0.953714	−	0.300715i	$-0.902775\pi$
$31$	−1833.04	−1.90743	−0.953714	+	0.300715i	$0.902775\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 1126.33i	− 0.919455i
$36$	0	0
$37$	359.661	0.262718	0.131359	−	0.991335i	$-0.458066\pi$
$37$	359.661	0.262718	0.131359	+	0.991335i	$0.458066\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2650.81i	− 1.57693i	−0.615082	−	0.788463i	$-0.710877\pi$
$41$	− 2650.81i	− 1.57693i	0.615082	−	0.788463i	$-0.289123\pi$
$42$	0	0
$43$	860.347	0.465304	0.232652	−	0.972560i	$-0.425260\pi$
$43$	860.347	0.465304	0.232652	+	0.972560i	$0.425260\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	3168.85i	1.43452i	0.696808	+	0.717258i	$0.254603\pi$
$47$	3168.85i	1.43452i	−0.696808	+	0.717258i	$0.745397\pi$
$48$	0	0
$49$	−1284.35	−0.534925
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 5281.61i	− 1.88025i	−0.340836	−	0.940123i	$-0.610710\pi$
$53$	− 5281.61i	− 1.88025i	0.340836	−	0.940123i	$-0.389290\pi$
$54$	0	0
$55$	−4646.21	−1.53594
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1532.62i	0.440280i	0.975468	+	0.220140i	$0.0706515\pi$
$59$	1532.62i	0.440280i	−0.975468	+	0.220140i	$0.929349\pi$
$60$	0	0
$61$	−7229.76	−1.94296	−0.971481	−	0.237118i	$-0.923797\pi$
$61$	−7229.76	−1.94296	−0.971481	+	0.237118i	$0.923797\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	8660.05i	2.04972i
$66$	0	0
$67$	−590.092	−0.131453	−0.0657265	−	0.997838i	$-0.520936\pi$
$67$	−590.092	−0.131453	−0.0657265	+	0.997838i	$0.520936\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	4706.23i	0.933591i	0.884365	+	0.466795i	$0.154592\pi$
$71$	4706.23i	0.933591i	−0.884365	+	0.466795i	$0.845408\pi$
$72$	0	0
$73$	−2748.96	−0.515849	−0.257924	−	0.966165i	$-0.583039\pi$
$73$	−2748.96	−0.515849	−0.257924	+	0.966165i	$0.583039\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 4606.25i	− 0.776901i
$78$	0	0
$79$	−5967.05	−0.956105	−0.478053	−	0.878331i	$-0.658657\pi$
$79$	−5967.05	−0.956105	−0.478053	+	0.878331i	$0.658657\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	11938.8i	1.73303i	0.499151	+	0.866515i	$0.333645\pi$
$83$	11938.8i	1.73303i	−0.499151	+	0.866515i	$0.666355\pi$
$84$	0	0
$85$	−1921.23	−0.265914
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 9339.75i	− 1.17911i	−0.807727	−	0.589556i	$-0.799303\pi$
$89$	− 9339.75i	− 1.17911i	0.807727	−	0.589556i	$-0.200697\pi$
$90$	0	0
$91$	−8585.57	−1.03678
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 11068.8i	− 1.22646i
$96$	0	0
$97$	3197.68	0.339853	0.169927	−	0.985457i	$-0.445647\pi$
$97$	3197.68	0.339853	0.169927	+	0.985457i	$0.445647\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 4937.17i	− 0.483989i	−0.970278	−	0.241994i	$-0.922198\pi$
$101$	− 4937.17i	− 0.483989i	0.970278	−	0.241994i	$-0.0778015\pi$
$102$	0	0
$103$	−15625.4	−1.47285	−0.736423	−	0.676521i	$-0.763487\pi$
$103$	−15625.4	−1.47285	−0.736423	+	0.676521i	$0.763487\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 15974.3i	− 1.39526i	−0.716460	−	0.697629i	$-0.754239\pi$
$107$	− 15974.3i	− 1.39526i	0.716460	−	0.697629i	$-0.245761\pi$
$108$	0	0
$109$	4163.62	0.350444	0.175222	−	0.984529i	$-0.443936\pi$
$109$	4163.62	0.350444	0.175222	+	0.984529i	$0.443936\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	13823.5i	1.08259i	0.840834	+	0.541293i	$0.182065\pi$
$113$	13823.5i	1.08259i	−0.840834	+	0.541293i	$0.817935\pi$
$114$	0	0
$115$	2216.74	0.167617
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 1904.71i	− 0.134504i
$120$	0	0
$121$	−4360.12	−0.297802
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3839.00i	0.245696i
$126$	0	0
$127$	19659.0	1.21886	0.609432	−	0.792839i	$-0.291398\pi$
$127$	19659.0	1.21886	0.609432	+	0.792839i	$0.291398\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 20817.6i	− 1.21307i	−0.795055	−	0.606537i	$-0.792558\pi$
$131$	− 20817.6i	− 1.21307i	0.795055	−	0.606537i	$-0.207442\pi$
$132$	0	0
$133$	10973.6	0.620364
$134$	0	0
$135$	0	0
$136$	0	0
$137$	17149.0i	0.913686i	0.889547	+	0.456843i	$0.151020\pi$
$137$	17149.0i	0.913686i	−0.889547	+	0.456843i	$0.848980\pi$
$138$	0	0
$139$	15916.9	0.823813	0.411906	−	0.911226i	$-0.364863\pi$
$139$	15916.9	0.823813	0.411906	+	0.911226i	$0.364863\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	35416.1i	1.73192i
$144$	0	0
$145$	−15582.6	−0.741145
$146$	0	0
$147$	0	0
$148$	0	0
$149$	26457.7i	1.19173i	0.803084	+	0.595866i	$0.203191\pi$
$149$	26457.7i	1.19173i	−0.803084	+	0.595866i	$0.796809\pi$
$150$	0	0
$151$	−26761.4	−1.17369	−0.586846	−	0.809698i	$-0.699631\pi$
$151$	−26761.4	−1.17369	−0.586846	+	0.809698i	$0.699631\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 61784.7i	− 2.57168i
$156$	0	0
$157$	41145.8	1.66927	0.834635	−	0.550804i	$-0.185679\pi$
$157$	41145.8	1.66927	0.834635	+	0.550804i	$0.185679\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2197.68i	0.0847836i
$162$	0	0
$163$	−18225.9	−0.685982	−0.342991	−	0.939339i	$-0.611440\pi$
$163$	−18225.9	−0.685982	−0.342991	+	0.939339i	$0.611440\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	28664.7i	1.02781i	0.857846	+	0.513906i	$0.171802\pi$
$167$	28664.7i	1.02781i	−0.857846	+	0.513906i	$0.828198\pi$
$168$	0	0
$169$	37450.9	1.31126
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 5456.36i	− 0.182310i	−0.995837	−	0.0911551i	$-0.970944\pi$
$173$	− 5456.36i	− 0.182310i	0.995837	−	0.0911551i	$-0.0290559\pi$
$174$	0	0
$175$	17079.2	0.557687
$176$	0	0
$177$	0	0
$178$	0	0
$179$	41680.0i	1.30083i	0.759577	+	0.650417i	$0.225406\pi$
$179$	41680.0i	1.30083i	−0.759577	+	0.650417i	$0.774594\pi$
$180$	0	0
$181$	49304.0	1.50496	0.752480	−	0.658615i	$-0.228857\pi$
$181$	49304.0	1.50496	0.752480	+	0.658615i	$0.228857\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	12122.8i	0.354209i
$186$	0	0
$187$	−7857.05	−0.224686
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 30621.3i	− 0.839376i	−0.907668	−	0.419688i	$-0.862139\pi$
$191$	− 30621.3i	− 0.839376i	0.907668	−	0.419688i	$-0.137861\pi$
$192$	0	0
$193$	−9769.13	−0.262266	−0.131133	−	0.991365i	$-0.541861\pi$
$193$	−9769.13	−0.262266	−0.131133	+	0.991365i	$0.541861\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 19789.3i	− 0.509916i	−0.966952	−	0.254958i	$-0.917938\pi$
$197$	− 19789.3i	− 0.509916i	0.966952	−	0.254958i	$-0.0820616\pi$
$198$	0	0
$199$	53235.2	1.34429	0.672144	−	0.740420i	$-0.265373\pi$
$199$	53235.2	1.34429	0.672144	+	0.740420i	$0.265373\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 15448.6i	− 0.374883i
$204$	0	0
$205$	89348.7	2.12608
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 45267.0i	− 1.03631i
$210$	0	0
$211$	−2613.91	−0.0587118	−0.0293559	−	0.999569i	$-0.509346\pi$
$211$	−2613.91	−0.0587118	−0.0293559	+	0.999569i	$0.509346\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	28999.0i	0.627344i
$216$	0	0
$217$	61253.3	1.30080
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14644.7i	0.299845i
$222$	0	0
$223$	24129.2	0.485215	0.242607	−	0.970125i	$-0.421997\pi$
$223$	24129.2	0.485215	0.242607	+	0.970125i	$0.421997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 80003.8i	− 1.55260i	−0.630365	−	0.776299i	$-0.717095\pi$
$227$	− 80003.8i	− 1.55260i	0.630365	−	0.776299i	$-0.282905\pi$
$228$	0	0
$229$	15864.2	0.302515	0.151258	−	0.988494i	$-0.451668\pi$
$229$	15864.2	0.302515	0.151258	+	0.988494i	$0.451668\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 44937.9i	− 0.827753i	−0.910333	−	0.413876i	$-0.864175\pi$
$233$	− 44937.9i	− 0.827753i	0.910333	−	0.413876i	$-0.135825\pi$
$234$	0	0
$235$	−106810.	−1.93408
$236$	0	0
$237$	0	0
$238$	0	0
$239$	5577.13i	0.0976371i	0.998808	+	0.0488186i	$0.0155456\pi$
$239$	5577.13i	0.0976371i	−0.998808	+	0.0488186i	$0.984454\pi$
$240$	0	0
$241$	−47914.5	−0.824961	−0.412480	−	0.910967i	$-0.635337\pi$
$241$	−47914.5	−0.824961	−0.412480	+	0.910967i	$0.635337\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 43290.6i	− 0.721210i
$246$	0	0
$247$	−84372.9	−1.38296
$248$	0	0
$249$	0	0
$250$	0	0
$251$	79435.9i	1.26087i	0.776243	+	0.630434i	$0.217123\pi$
$251$	79435.9i	1.26087i	−0.776243	+	0.630434i	$0.782877\pi$
$252$	0	0
$253$	9065.58	0.141630
$254$	0	0
$255$	0	0
$256$	0	0
$257$	20964.3i	0.317405i	0.987326	+	0.158703i	$0.0507311\pi$
$257$	20964.3i	0.317405i	−0.987326	+	0.158703i	$0.949269\pi$
$258$	0	0
$259$	−12018.5	−0.179164
$260$	0	0
$261$	0	0
$262$	0	0
$263$	126060.i	1.82250i	0.411858	+	0.911248i	$0.364880\pi$
$263$	126060.i	1.82250i	−0.411858	+	0.911248i	$0.635120\pi$
$264$	0	0
$265$	178023.	2.53503
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4162.75i	0.0575276i	0.999586	+	0.0287638i	$0.00915706\pi$
$269$	4162.75i	0.0575276i	−0.999586	+	0.0287638i	$0.990843\pi$
$270$	0	0
$271$	−1002.99	−0.0136571	−0.00682853	−	0.999977i	$-0.502174\pi$
$271$	−1002.99	−0.0136571	−0.00682853	+	0.999977i	$0.502174\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 70452.9i	− 0.931608i
$276$	0	0
$277$	−40285.3	−0.525033	−0.262517	−	0.964927i	$-0.584552\pi$
$277$	−40285.3	−0.525033	−0.262517	+	0.964927i	$0.584552\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	105670.i	1.33825i	0.743148	+	0.669127i	$0.233332\pi$
$281$	105670.i	1.33825i	−0.743148	+	0.669127i	$0.766668\pi$
$282$	0	0
$283$	−29884.2	−0.373137	−0.186568	−	0.982442i	$-0.559737\pi$
$283$	−29884.2	−0.373137	−0.186568	+	0.982442i	$0.559737\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	88580.2i	1.07541i
$288$	0	0
$289$	80272.1	0.961100
$290$	0	0
$291$	0	0
$292$	0	0
$293$	110213.i	1.28380i	0.766787	+	0.641902i	$0.221854\pi$
$293$	110213.i	1.28380i	−0.766787	+	0.641902i	$0.778146\pi$
$294$	0	0
$295$	−51658.5	−0.593606
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 16897.3i	− 0.189006i
$300$	0	0
$301$	−28749.6	−0.317321
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 243687.i	− 2.61959i
$306$	0	0
$307$	−172215.	−1.82723	−0.913616	−	0.406579i	$-0.866722\pi$
$307$	−172215.	−1.82723	−0.913616	+	0.406579i	$0.866722\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 43943.4i	− 0.454331i	−0.973856	−	0.227166i	$-0.927054\pi$
$311$	− 43943.4i	− 0.454331i	0.973856	−	0.227166i	$-0.0729459\pi$
$312$	0	0
$313$	−54550.3	−0.556812	−0.278406	−	0.960464i	$-0.589806\pi$
$313$	−54550.3	−0.556812	−0.278406	+	0.960464i	$0.589806\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	169211.i	1.68388i	0.539571	+	0.841940i	$0.318586\pi$
$317$	169211.i	1.68388i	−0.539571	+	0.841940i	$0.681414\pi$
$318$	0	0
$319$	−63726.5	−0.626237
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 18718.1i	− 0.179414i
$324$	0	0
$325$	−131317.	−1.24324
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 105891.i	− 0.978289i
$330$	0	0
$331$	82532.0	0.753297	0.376648	−	0.926356i	$-0.377076\pi$
$331$	82532.0	0.753297	0.376648	+	0.926356i	$0.377076\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 19889.7i	− 0.177231i
$336$	0	0
$337$	−144015.	−1.26808	−0.634041	−	0.773300i	$-0.718605\pi$
$337$	−144015.	−1.26808	−0.634041	+	0.773300i	$0.718605\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 252674.i	− 2.17296i
$342$	0	0
$343$	123151.	1.04676
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 106957.i	− 0.888284i	−0.895956	−	0.444142i	$-0.853509\pi$
$347$	− 106957.i	− 0.888284i	0.895956	−	0.444142i	$-0.146491\pi$
$348$	0	0
$349$	−60877.4	−0.499811	−0.249905	−	0.968270i	$-0.580399\pi$
$349$	−60877.4	−0.499811	−0.249905	+	0.968270i	$0.580399\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20696.8i	0.166094i	0.996546	+	0.0830471i	$0.0264652\pi$
$353$	20696.8i	0.166094i	−0.996546	+	0.0830471i	$0.973535\pi$
$354$	0	0
$355$	−158629.	−1.25871
$356$	0	0
$357$	0	0
$358$	0	0
$359$	123377.i	0.957290i	0.878008	+	0.478645i	$0.158872\pi$
$359$	123377.i	0.957290i	−0.878008	+	0.478645i	$0.841128\pi$
$360$	0	0
$361$	−22480.0	−0.172497
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 92656.8i	− 0.695491i
$366$	0	0
$367$	−97980.5	−0.727457	−0.363729	−	0.931505i	$-0.618496\pi$
$367$	−97980.5	−0.727457	−0.363729	+	0.931505i	$0.618496\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	176492.i	1.28226i
$372$	0	0
$373$	−91031.9	−0.654299	−0.327149	−	0.944973i	$-0.606088\pi$
$373$	−91031.9	−0.654299	−0.327149	+	0.944973i	$0.606088\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	118780.i	0.835716i
$378$	0	0
$379$	−85165.4	−0.592905	−0.296452	−	0.955048i	$-0.595804\pi$
$379$	−85165.4	−0.592905	−0.296452	+	0.955048i	$0.595804\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 200100.i	− 1.36411i	−0.731301	−	0.682055i	$-0.761086\pi$
$383$	− 200100.i	− 1.36411i	0.731301	−	0.682055i	$-0.238914\pi$
$384$	0	0
$385$	155259.	1.04745
$386$	0	0
$387$	0	0
$388$	0	0
$389$	154487.i	1.02092i	0.859901	+	0.510460i	$0.170525\pi$
$389$	154487.i	1.02092i	−0.859901	+	0.510460i	$0.829475\pi$
$390$	0	0
$391$	3748.66	0.0245201
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 201126.i	− 1.28906i
$396$	0	0
$397$	154482.	0.980162	0.490081	−	0.871677i	$-0.336967\pi$
$397$	154482.	0.980162	0.490081	+	0.871677i	$0.336967\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	199641.i	1.24154i	0.783992	+	0.620771i	$0.213180\pi$
$401$	199641.i	1.24154i	−0.783992	+	0.620771i	$0.786820\pi$
$402$	0	0
$403$	−470959.	−2.89983
$404$	0	0
$405$	0	0
$406$	0	0
$407$	49577.3i	0.299292i
$408$	0	0
$409$	10076.5	0.0602369	0.0301184	−	0.999546i	$-0.490412\pi$
$409$	10076.5	0.0602369	0.0301184	+	0.999546i	$0.490412\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 51214.3i	− 0.300255i
$414$	0	0
$415$	−402412.	−2.33655
$416$	0	0
$417$	0	0
$418$	0	0
$419$	273275.i	1.55658i	0.627903	+	0.778292i	$0.283913\pi$
$419$	273275.i	1.55658i	−0.627903	+	0.778292i	$0.716087\pi$
$420$	0	0
$421$	201765.	1.13837	0.569183	−	0.822211i	$-0.307260\pi$
$421$	201765.	1.13837	0.569183	+	0.822211i	$0.307260\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 29132.6i	− 0.161288i
$426$	0	0
$427$	241591.	1.32503
$428$	0	0
$429$	0	0
$430$	0	0
$431$	44036.2i	0.237058i	0.992951	+	0.118529i	$0.0378179\pi$
$431$	44036.2i	0.237058i	−0.992951	+	0.118529i	$0.962182\pi$
$432$	0	0
$433$	−139788.	−0.745580	−0.372790	−	0.927916i	$-0.621599\pi$
$433$	−139788.	−0.745580	−0.372790	+	0.927916i	$0.621599\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	21597.2i	0.113093i
$438$	0	0
$439$	−47650.0	−0.247249	−0.123624	−	0.992329i	$-0.539452\pi$
$439$	−47650.0	−0.247249	−0.123624	+	0.992329i	$0.539452\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 127856.i	− 0.651499i	−0.945456	−	0.325749i	$-0.894383\pi$
$443$	− 127856.i	− 0.651499i	0.945456	−	0.325749i	$-0.105617\pi$
$444$	0	0
$445$	314807.	1.58973
$446$	0	0
$447$	0	0
$448$	0	0
$449$	77490.5i	0.384376i	0.981358	+	0.192188i	$0.0615583\pi$
$449$	77490.5i	0.384376i	−0.981358	+	0.192188i	$0.938442\pi$
$450$	0	0
$451$	365400.	1.79645
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 289386.i	− 1.39783i
$456$	0	0
$457$	97774.5	0.468159	0.234079	−	0.972218i	$-0.424792\pi$
$457$	97774.5	0.468159	0.234079	+	0.972218i	$0.424792\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	197650.i	0.930027i	0.885303	+	0.465014i	$0.153951\pi$
$461$	197650.i	0.930027i	−0.885303	+	0.465014i	$0.846049\pi$
$462$	0	0
$463$	−76082.4	−0.354913	−0.177457	−	0.984129i	$-0.556787\pi$
$463$	−76082.4	−0.354913	−0.177457	+	0.984129i	$0.556787\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	78375.5i	0.359374i	0.983724	+	0.179687i	$0.0575085\pi$
$467$	78375.5i	0.359374i	−0.983724	+	0.179687i	$0.942492\pi$
$468$	0	0
$469$	19718.7	0.0896462
$470$	0	0
$471$	0	0
$472$	0	0
$473$	118594.i	0.530080i
$474$	0	0
$475$	167842.	0.743899
$476$	0	0
$477$	0	0
$478$	0	0
$479$	− 206044.i	− 0.898027i	−0.893525	−	0.449014i	$-0.851775\pi$
$479$	− 206044.i	− 0.898027i	0.893525	−	0.449014i	$-0.148225\pi$
$480$	0	0
$481$	92407.0	0.399406
$482$	0	0
$483$	0	0
$484$	0	0
$485$	107781.i	0.458206i
$486$	0	0
$487$	200708.	0.846264	0.423132	−	0.906068i	$-0.360931\pi$
$487$	200708.	0.846264	0.423132	+	0.906068i	$0.360931\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	209263.i	0.868018i	0.900908	+	0.434009i	$0.142901\pi$
$491$	209263.i	0.868018i	−0.900908	+	0.434009i	$0.857099\pi$
$492$	0	0
$493$	−26351.2	−0.108419
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 157265.i	− 0.636675i
$498$	0	0
$499$	−354112.	−1.42213	−0.711066	−	0.703125i	$-0.751787\pi$
$499$	−354112.	−1.42213	−0.711066	+	0.703125i	$0.751787\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	99176.2i	0.391987i	0.980605	+	0.195993i	$0.0627931\pi$
$503$	99176.2i	0.391987i	−0.980605	+	0.195993i	$0.937207\pi$
$504$	0	0
$505$	166413.	0.652535
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 55928.5i	− 0.215872i	−0.994158	−	0.107936i	$-0.965576\pi$
$509$	− 55928.5i	− 0.215872i	0.994158	−	0.107936i	$-0.0344242\pi$
$510$	0	0
$511$	91859.9	0.351791
$512$	0	0
$513$	0	0
$514$	0	0
$515$	− 526673.i	− 1.98576i
$516$	0	0
$517$	−436808.	−1.63422
$518$	0	0
$519$	0	0
$520$	0	0
$521$	217215.i	0.800229i	0.916465	+	0.400114i	$0.131030\pi$
$521$	217215.i	0.800229i	−0.916465	+	0.400114i	$0.868970\pi$
$522$	0	0
$523$	104089.	0.380542	0.190271	−	0.981732i	$-0.439063\pi$
$523$	104089.	0.380542	0.190271	+	0.981732i	$0.439063\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 104482.i	− 0.376202i
$528$	0	0
$529$	275516.	0.984544
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 681068.i	− 2.39737i
$534$	0	0
$535$	538432.	1.88115
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 177041.i	− 0.609392i
$540$	0	0
$541$	−116397.	−0.397693	−0.198846	−	0.980031i	$-0.563720\pi$
$541$	−116397.	−0.397693	−0.198846	+	0.980031i	$0.563720\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	140340.i	0.472484i
$546$	0	0
$547$	398649.	1.33234	0.666171	−	0.745799i	$-0.267932\pi$
$547$	398649.	1.33234	0.666171	+	0.745799i	$0.267932\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 151818.i	− 0.500056i
$552$	0	0
$553$	199396.	0.652030
$554$	0	0
$555$	0	0
$556$	0	0
$557$	203998.i	0.657530i	0.944412	+	0.328765i	$0.106632\pi$
$557$	203998.i	0.657530i	−0.944412	+	0.328765i	$0.893368\pi$
$558$	0	0
$559$	221047.	0.707394
$560$	0	0
$561$	0	0
$562$	0	0
$563$	379444.i	1.19710i	0.801084	+	0.598551i	$0.204257\pi$
$563$	379444.i	1.19710i	−0.801084	+	0.598551i	$0.795743\pi$
$564$	0	0
$565$	−465938.	−1.45959
$566$	0	0
$567$	0	0
$568$	0	0
$569$	91989.2i	0.284127i	0.989858	+	0.142063i	$0.0453737\pi$
$569$	91989.2i	0.284127i	−0.989858	+	0.142063i	$0.954626\pi$
$570$	0	0
$571$	−79434.0	−0.243632	−0.121816	−	0.992553i	$-0.538872\pi$
$571$	−79434.0	−0.243632	−0.121816	+	0.992553i	$0.538872\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	33613.6i	0.101667i
$576$	0	0
$577$	41340.4	0.124172	0.0620860	−	0.998071i	$-0.480225\pi$
$577$	41340.4	0.124172	0.0620860	+	0.998071i	$0.480225\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 398951.i	− 1.18186i
$582$	0	0
$583$	728041.	2.14200
$584$	0	0
$585$	0	0
$586$	0	0
$587$	249186.i	0.723183i	0.932337	+	0.361592i	$0.117766\pi$
$587$	249186.i	0.723183i	−0.932337	+	0.361592i	$0.882234\pi$
$588$	0	0
$589$	601954.	1.73513
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 37868.4i	− 0.107688i	−0.998549	−	0.0538440i	$-0.982853\pi$
$593$	− 37868.4i	− 0.107688i	0.998549	−	0.0538440i	$-0.0171474\pi$
$594$	0	0
$595$	64200.3	0.181344
$596$	0	0
$597$	0	0
$598$	0	0
$599$	290093.i	0.808505i	0.914647	+	0.404253i	$0.132468\pi$
$599$	290093.i	0.808505i	−0.914647	+	0.404253i	$0.867532\pi$
$600$	0	0
$601$	8522.48	0.0235949	0.0117974	−	0.999930i	$-0.496245\pi$
$601$	8522.48	0.0235949	0.0117974	+	0.999930i	$0.496245\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 146963.i	− 0.401510i
$606$	0	0
$607$	695218.	1.88688	0.943439	−	0.331547i	$-0.107571\pi$
$607$	695218.	1.88688	0.943439	+	0.331547i	$0.107571\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	814165.i	2.18087i
$612$	0	0
$613$	457316.	1.21701	0.608506	−	0.793549i	$-0.291769\pi$
$613$	457316.	1.21701	0.608506	+	0.793549i	$0.291769\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	198593.i	0.521668i	0.965384	+	0.260834i	$0.0839975\pi$
$617$	198593.i	0.521668i	−0.965384	+	0.260834i	$0.916002\pi$
$618$	0	0
$619$	−692654.	−1.80774	−0.903868	−	0.427812i	$-0.859284\pi$
$619$	−692654.	−1.80774	−0.903868	+	0.427812i	$0.859284\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	312100.i	0.804113i
$624$	0	0
$625$	−448838.	−1.14902
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20500.5i	0.0518158i
$630$	0	0
$631$	−314958.	−0.791032	−0.395516	−	0.918459i	$-0.629434\pi$
$631$	−314958.	−0.791032	−0.395516	+	0.918459i	$0.629434\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	662630.i	1.64333i
$636$	0	0
$637$	−329986.	−0.813237
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 690670.i	− 1.68095i	−0.541850	−	0.840475i	$-0.682276\pi$
$641$	− 690670.i	− 1.68095i	0.541850	−	0.840475i	$-0.317724\pi$
$642$	0	0
$643$	63240.7	0.152959	0.0764794	−	0.997071i	$-0.475632\pi$
$643$	63240.7	0.152959	0.0764794	+	0.997071i	$0.475632\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 597558.i	− 1.42748i	−0.700409	−	0.713742i	$-0.746999\pi$
$647$	− 597558.i	− 1.42748i	0.700409	−	0.713742i	$-0.253001\pi$
$648$	0	0
$649$	−211263.	−0.501572
$650$	0	0
$651$	0	0
$652$	0	0
$653$	568444.i	1.33310i	0.745462	+	0.666548i	$0.232229\pi$
$653$	568444.i	1.33310i	−0.745462	+	0.666548i	$0.767771\pi$
$654$	0	0
$655$	701680.	1.63552
$656$	0	0
$657$	0	0
$658$	0	0
$659$	418151.i	0.962858i	0.876485	+	0.481429i	$0.159882\pi$
$659$	418151.i	0.962858i	−0.876485	+	0.481429i	$0.840118\pi$
$660$	0	0
$661$	−270806.	−0.619804	−0.309902	−	0.950768i	$-0.600296\pi$
$661$	−270806.	−0.619804	−0.309902	+	0.950768i	$0.600296\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	369878.i	0.836403i
$666$	0	0
$667$	30404.4	0.0683415
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 996583.i	− 2.21344i
$672$	0	0
$673$	−536876.	−1.18534	−0.592672	−	0.805444i	$-0.701927\pi$
$673$	−536876.	−1.18534	−0.592672	+	0.805444i	$0.701927\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 260528.i	− 0.568430i	−0.958761	−	0.284215i	$-0.908267\pi$
$677$	− 260528.i	− 0.568430i	0.958761	−	0.284215i	$-0.0917329\pi$
$678$	0	0
$679$	−106854.	−0.231768
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 427175.i	− 0.915723i	−0.889023	−	0.457862i	$-0.848615\pi$
$683$	− 427175.i	− 0.915723i	0.889023	−	0.457862i	$-0.151385\pi$
$684$	0	0
$685$	−578026.	−1.23187
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 1.35699e6i	− 2.85851i
$690$	0	0
$691$	−205589.	−0.430571	−0.215285	−	0.976551i	$-0.569068\pi$
$691$	−205589.	−0.430571	−0.215285	+	0.976551i	$0.569068\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	536497.i	1.11070i
$696$	0	0
$697$	151095.	0.311017
$698$	0	0
$699$	0	0
$700$	0	0
$701$	176791.i	0.359769i	0.983688	+	0.179884i	$0.0575724\pi$
$701$	176791.i	0.359769i	−0.983688	+	0.179884i	$0.942428\pi$
$702$	0	0
$703$	−118110.	−0.238987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	164982.i	0.330063i
$708$	0	0
$709$	753167.	1.49830	0.749150	−	0.662400i	$-0.230462\pi$
$709$	753167.	1.49830	0.749150	+	0.662400i	$0.230462\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	120553.i	0.237137i
$714$	0	0
$715$	−1.19374e6	−2.33506
$716$	0	0
$717$	0	0
$718$	0	0
$719$	− 653122.i	− 1.26339i	−0.775218	−	0.631694i	$-0.782360\pi$
$719$	− 653122.i	− 1.26339i	0.775218	−	0.631694i	$-0.217640\pi$
$720$	0	0
$721$	522143.	1.00443
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 236287.i	− 0.449535i
$726$	0	0
$727$	374569.	0.708701	0.354351	−	0.935113i	$-0.384702\pi$
$727$	374569.	0.708701	0.354351	+	0.935113i	$0.384702\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	49039.3i	0.0917718i
$732$	0	0
$733$	−698725.	−1.30046	−0.650232	−	0.759736i	$-0.725328\pi$
$733$	−698725.	−1.30046	−0.650232	+	0.759736i	$0.725328\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 81341.0i	− 0.149753i
$738$	0	0
$739$	4561.08	0.00835177	0.00417588	−	0.999991i	$-0.498671\pi$
$739$	4561.08	0.00835177	0.00417588	+	0.999991i	$0.498671\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	305605.i	0.553583i	0.960930	+	0.276792i	$0.0892712\pi$
$743$	305605.i	0.553583i	−0.960930	+	0.276792i	$0.910729\pi$
$744$	0	0
$745$	−891786.	−1.60675
$746$	0	0
$747$	0	0
$748$	0	0
$749$	533801.i	0.951516i
$750$	0	0
$751$	725165.	1.28575	0.642875	−	0.765971i	$-0.277741\pi$
$751$	725165.	1.28575	0.642875	+	0.765971i	$0.277741\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 902022.i	− 1.58243i
$756$	0	0
$757$	−273369.	−0.477043	−0.238521	−	0.971137i	$-0.576663\pi$
$757$	−273369.	−0.477043	−0.238521	+	0.971137i	$0.576663\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 490874.i	− 0.847619i	−0.905751	−	0.423810i	$-0.860693\pi$
$761$	− 490874.i	− 0.847619i	0.905751	−	0.423810i	$-0.139307\pi$
$762$	0	0
$763$	−139133.	−0.238990
$764$	0	0
$765$	0	0
$766$	0	0
$767$	393772.i	0.669351i
$768$	0	0
$769$	−660652.	−1.11717	−0.558586	−	0.829447i	$-0.688656\pi$
$769$	−660652.	−1.11717	−0.558586	+	0.829447i	$0.688656\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	69649.7i	0.116563i	0.998300	+	0.0582814i	$0.0185621\pi$
$773$	69649.7i	0.116563i	−0.998300	+	0.0582814i	$0.981438\pi$
$774$	0	0
$775$	936873.	1.55983
$776$	0	0
$777$	0	0
$778$	0	0
$779$	870504.i	1.43448i
$780$	0	0
$781$	−648728.	−1.06356
$782$	0	0
$783$	0	0
$784$	0	0
$785$	1.38687e6i	2.25059i
$786$	0	0
$787$	−277046.	−0.447303	−0.223652	−	0.974669i	$-0.571798\pi$
$787$	−277046.	−0.447303	−0.223652	+	0.974669i	$0.571798\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	− 461931.i	− 0.738285i
$792$	0	0
$793$	−1.85753e6	−2.95385
$794$	0	0
$795$	0	0
$796$	0	0
$797$	15292.4i	0.0240746i	0.999928	+	0.0120373i	$0.00383168\pi$
$797$	15292.4i	0.0240746i	−0.999928	+	0.0120373i	$0.996168\pi$
$798$	0	0
$799$	−180622.	−0.282929
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 378929.i	− 0.587661i
$804$	0	0
$805$	−74075.2	−0.114309
$806$	0	0
$807$	0	0
$808$	0	0
$809$	264170.i	0.403632i	0.979423	+	0.201816i	$0.0646843\pi$
$809$	264170.i	0.403632i	−0.979423	+	0.201816i	$0.935316\pi$
$810$	0	0
$811$	−303869.	−0.462002	−0.231001	−	0.972953i	$-0.574200\pi$
$811$	−303869.	−0.462002	−0.231001	+	0.972953i	$0.574200\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 614323.i	− 0.924872i
$816$	0	0
$817$	−282531.	−0.423274
$818$	0	0
$819$	0	0
$820$	0	0
$821$	827669.i	1.22792i	0.789337	+	0.613961i	$0.210425\pi$
$821$	827669.i	1.22792i	−0.789337	+	0.613961i	$0.789575\pi$
$822$	0	0
$823$	674989.	0.996545	0.498272	−	0.867021i	$-0.333968\pi$
$823$	674989.	0.996545	0.498272	+	0.867021i	$0.333968\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 1.05229e6i	− 1.53860i	−0.638887	−	0.769301i	$-0.720605\pi$
$827$	− 1.05229e6i	− 1.53860i	0.638887	−	0.769301i	$-0.279395\pi$
$828$	0	0
$829$	−622368.	−0.905603	−0.452802	−	0.891611i	$-0.649575\pi$
$829$	−622368.	−0.905603	−0.452802	+	0.891611i	$0.649575\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 73207.4i	− 0.105503i
$834$	0	0
$835$	−966175.	−1.38574
$836$	0	0
$837$	0	0
$838$	0	0
$839$	653970.i	0.929038i	0.885563	+	0.464519i	$0.153773\pi$
$839$	653970.i	0.929038i	−0.885563	+	0.464519i	$0.846227\pi$
$840$	0	0
$841$	493553.	0.697818
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1.26233e6i	1.76790i
$846$	0	0
$847$	145699.	0.203090
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 23653.7i	− 0.0326618i
$852$	0	0
$853$	1.05227e6	1.44621	0.723104	−	0.690739i	$-0.242715\pi$
$853$	1.05227e6	1.44621	0.723104	+	0.690739i	$0.242715\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	336092.i	0.457611i	0.973472	+	0.228806i	$0.0734821\pi$
$857$	336092.i	0.457611i	−0.973472	+	0.228806i	$0.926518\pi$
$858$	0	0
$859$	929784.	1.26007	0.630036	−	0.776566i	$-0.283040\pi$
$859$	929784.	1.26007	0.630036	+	0.776566i	$0.283040\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	760939.i	1.02171i	0.859666	+	0.510856i	$0.170671\pi$
$863$	760939.i	1.02171i	−0.859666	+	0.510856i	$0.829329\pi$
$864$	0	0
$865$	183913.	0.245799
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 822526.i	− 1.08921i
$870$	0	0
$871$	−151611.	−0.199846
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 128285.i	− 0.167556i
$876$	0	0
$877$	−50193.9	−0.0652607	−0.0326304	−	0.999467i	$-0.510388\pi$
$877$	−50193.9	−0.0652607	−0.0326304	+	0.999467i	$0.510388\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 421710.i	− 0.543328i	−0.962392	−	0.271664i	$-0.912426\pi$
$881$	− 421710.i	− 0.543328i	0.962392	−	0.271664i	$-0.0875740\pi$
$882$	0	0
$883$	−712187.	−0.913424	−0.456712	−	0.889615i	$-0.650973\pi$
$883$	−712187.	−0.913424	−0.456712	+	0.889615i	$0.650973\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 1.08271e6i	− 1.37615i	−0.725639	−	0.688075i	$-0.758456\pi$
$887$	− 1.08271e6i	− 1.37615i	0.725639	−	0.688075i	$-0.241544\pi$
$888$	0	0
$889$	−656932.	−0.831221
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 1.04062e6i	− 1.30494i
$894$	0	0
$895$	−1.40487e6	−1.75384
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 847426.i	− 1.04853i
$900$	0	0
$901$	301048.	0.370840
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.66185e6i	2.02906i
$906$	0	0
$907$	−73511.1	−0.0893590	−0.0446795	−	0.999001i	$-0.514227\pi$
$907$	−73511.1	−0.0893590	−0.0446795	+	0.999001i	$0.514227\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	1.20000e6i	1.44592i	0.690892	+	0.722958i	$0.257218\pi$
$911$	1.20000e6i	1.44592i	−0.690892	+	0.722958i	$0.742782\pi$
$912$	0	0
$913$	−1.64570e6	−1.97429
$914$	0	0
$915$	0	0
$916$	0	0
$917$	695645.i	0.827274i
$918$	0	0
$919$	311383.	0.368692	0.184346	−	0.982861i	$-0.440983\pi$
$919$	311383.	0.368692	0.184346	+	0.982861i	$0.440983\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1.20916e6i	1.41932i
$924$	0	0
$925$	−183824.	−0.214842
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 1.30768e6i	− 1.51520i	−0.652716	−	0.757602i	$-0.726371\pi$
$929$	− 1.30768e6i	− 1.51520i	0.652716	−	0.757602i	$-0.273629\pi$
$930$	0	0
$931$	421771.	0.486606
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 264831.i	− 0.302932i
$936$	0	0
$937$	860787.	0.980430	0.490215	−	0.871601i	$-0.336918\pi$
$937$	860787.	0.980430	0.490215	+	0.871601i	$0.336918\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 260061.i	− 0.293695i	−0.989159	−	0.146848i	$-0.953087\pi$
$941$	− 260061.i	− 0.293695i	0.989159	−	0.146848i	$-0.0469127\pi$
$942$	0	0
$943$	−174335.	−0.196048
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 630525.i	− 0.703077i	−0.936174	−	0.351538i	$-0.885659\pi$
$947$	− 630525.i	− 0.703077i	0.936174	−	0.351538i	$-0.114341\pi$
$948$	0	0
$949$	−706284.	−0.784236
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 619620.i	− 0.682244i	−0.940019	−	0.341122i	$-0.889193\pi$
$953$	− 619620.i	− 0.682244i	0.940019	−	0.341122i	$-0.110807\pi$
$954$	0	0
$955$	1.03213e6	1.13169
$956$	0	0
$957$	0	0
$958$	0	0
$959$	− 573054.i	− 0.623101i
$960$	0	0
$961$	2.43651e6	2.63828
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 329280.i	− 0.353598i
$966$	0	0
$967$	1.13400e6	1.21272	0.606359	−	0.795191i	$-0.292629\pi$
$967$	1.13400e6	1.21272	0.606359	+	0.795191i	$0.292629\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 510281.i	− 0.541216i	−0.962690	−	0.270608i	$-0.912775\pi$
$971$	− 510281.i	− 0.541216i	0.962690	−	0.270608i	$-0.0872247\pi$
$972$	0	0
$973$	−531883.	−0.561811
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 1.65002e6i	− 1.72863i	−0.502954	−	0.864313i	$-0.667753\pi$
$977$	− 1.65002e6i	− 1.72863i	0.502954	−	0.864313i	$-0.332247\pi$
$978$	0	0
$979$	1.28743e6	1.34326
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 483351.i	− 0.500214i	−0.968218	−	0.250107i	$-0.919534\pi$
$983$	− 483351.i	− 0.500214i	0.968218	−	0.250107i	$-0.0804658\pi$
$984$	0	0
$985$	667022.	0.687492
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 56582.2i	− 0.0578478i
$990$	0	0
$991$	1.47602e6	1.50295	0.751477	−	0.659759i	$-0.229342\pi$
$991$	1.47602e6	1.50295	0.751477	+	0.659759i	$0.229342\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	1.79435e6i	1.81243i
$996$	0	0
$997$	22655.9	0.0227924	0.0113962	−	0.999935i	$-0.496372\pi$
$997$	22655.9	0.0227924	0.0113962	+	0.999935i	$0.496372\pi$
$998$	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.161.8	yes	8
3.2	odd	2	inner	324.5.c.b.161.1	✓	8
4.3	odd	2		1296.5.e.f.161.8		8
9.2	odd	6		324.5.g.f.53.8		16
9.4	even	3		324.5.g.f.269.8		16
9.5	odd	6		324.5.g.f.269.1		16
9.7	even	3		324.5.g.f.53.1		16
12.11	even	2		1296.5.e.f.161.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.1	✓	8	3.2	odd	2	inner
324.5.c.b.161.8	yes	8	1.1	even	1	trivial
324.5.g.f.53.1		16	9.7	even	3
324.5.g.f.53.8		16	9.2	odd	6
324.5.g.f.269.1		16	9.5	odd	6
324.5.g.f.269.8		16	9.4	even	3
1296.5.e.f.161.1		8	12.11	even	2
1296.5.e.f.161.8		8	4.3	odd	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)

\(f(q)\)	\(=\)	\(q+33.7061i q^{5} -33.4163 q^{7} +137.845i q^{11} +256.928 q^{13} +56.9994i q^{17} -328.391 q^{19} -65.7667i q^{23} -511.104 q^{25} +462.307i q^{29} -1833.04 q^{31} -1126.33i q^{35} +359.661 q^{37} -2650.81i q^{41} +860.347 q^{43} +3168.85i q^{47} -1284.35 q^{49} -5281.61i q^{53} -4646.21 q^{55} +1532.62i q^{59} -7229.76 q^{61} +8660.05i q^{65} -590.092 q^{67} +4706.23i q^{71} -2748.96 q^{73} -4606.25i q^{77} -5967.05 q^{79} +11938.8i q^{83} -1921.23 q^{85} -9339.75i q^{89} -8585.57 q^{91} -11068.8i q^{95} +3197.68 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) \) 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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