LMFDB - Embedded newform 324.5.c.b.161.6

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			161.6
Root			$5.23219 - 0.517638i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.5.c.b.161.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+23.7512i q^{5} +86.6692 q^{7} -178.986i q^{11} +126.043 q^{13} -40.6840i q^{17} -157.085 q^{19} -784.115i q^{23} +60.8803 q^{25} +1377.20i q^{29} -376.079 q^{31} +2058.50i q^{35} +1828.71 q^{37} -734.617i q^{41} +359.163 q^{43} -3001.03i q^{47} +5110.56 q^{49} +2955.02i q^{53} +4251.14 q^{55} -4921.52i q^{59} +6775.98 q^{61} +2993.67i q^{65} -523.022 q^{67} +5104.50i q^{71} -6301.79 q^{73} -15512.6i q^{77} +4305.58 q^{79} +4808.46i q^{83} +966.293 q^{85} +9205.60i q^{89} +10924.0 q^{91} -3730.96i q^{95} -8935.86 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	23.7512i	0.950048i	0.879973	+	0.475024i	$0.157561\pi$
$5$	23.7512i	0.950048i	−0.879973	+	0.475024i	$0.842439\pi$
$6$	0	0
$7$	86.6692	1.76876	0.884380	−	0.466768i	$-0.154582\pi$
$7$	86.6692	1.76876	0.884380	+	0.466768i	$0.154582\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	− 178.986i	− 1.47923i	−0.673032	−	0.739613i	$-0.735008\pi$
$11$	− 178.986i	− 1.47923i	0.673032	−	0.739613i	$-0.264992\pi$
$12$	0	0
$13$	126.043	0.745815	0.372907	−	0.927869i	$-0.378361\pi$
$13$	126.043	0.745815	0.372907	+	0.927869i	$0.378361\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 40.6840i	− 0.140775i	−0.997520	−	0.0703875i	$-0.977576\pi$
$17$	− 40.6840i	− 0.140775i	0.997520	−	0.0703875i	$-0.0224236\pi$
$18$	0	0
$19$	−157.085	−0.435139	−0.217569	−	0.976045i	$-0.569813\pi$
$19$	−157.085	−0.435139	−0.217569	+	0.976045i	$0.569813\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 784.115i	− 1.48226i	−0.671362	−	0.741130i	$-0.734290\pi$
$23$	− 784.115i	− 1.48226i	0.671362	−	0.741130i	$-0.265710\pi$
$24$	0	0
$25$	60.8803	0.0974085
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1377.20i	1.63757i	0.574099	+	0.818786i	$0.305352\pi$
$29$	1377.20i	1.63757i	−0.574099	+	0.818786i	$0.694648\pi$
$30$	0	0
$31$	−376.079	−0.391341	−0.195671	−	0.980670i	$-0.562688\pi$
$31$	−376.079	−0.391341	−0.195671	+	0.980670i	$0.562688\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2058.50i	1.68041i
$36$	0	0
$37$	1828.71	1.33580	0.667900	−	0.744251i	$-0.267194\pi$
$37$	1828.71	1.33580	0.667900	+	0.744251i	$0.267194\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 734.617i	− 0.437012i	−0.975836	−	0.218506i	$-0.929882\pi$
$41$	− 734.617i	− 0.437012i	0.975836	−	0.218506i	$-0.0701183\pi$
$42$	0	0
$43$	359.163	0.194247	0.0971237	−	0.995272i	$-0.469036\pi$
$43$	359.163	0.194247	0.0971237	+	0.995272i	$0.469036\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 3001.03i	− 1.35855i	−0.733886	−	0.679273i	$-0.762295\pi$
$47$	− 3001.03i	− 1.35855i	0.733886	−	0.679273i	$-0.237705\pi$
$48$	0	0
$49$	5110.56	2.12851
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2955.02i	1.05198i	0.850490	+	0.525992i	$0.176306\pi$
$53$	2955.02i	1.05198i	−0.850490	+	0.525992i	$0.823694\pi$
$54$	0	0
$55$	4251.14	1.40534
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4921.52i	− 1.41382i	−0.707301	−	0.706912i	$-0.750087\pi$
$59$	− 4921.52i	− 1.41382i	0.707301	−	0.706912i	$-0.249913\pi$
$60$	0	0
$61$	6775.98	1.82101	0.910505	−	0.413498i	$-0.135693\pi$
$61$	6775.98	1.82101	0.910505	+	0.413498i	$0.135693\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	2993.67i	0.708560i
$66$	0	0
$67$	−523.022	−0.116512	−0.0582559	−	0.998302i	$-0.518554\pi$
$67$	−523.022	−0.116512	−0.0582559	+	0.998302i	$0.518554\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5104.50i	1.01260i	0.862359	+	0.506298i	$0.168986\pi$
$71$	5104.50i	1.01260i	−0.862359	+	0.506298i	$0.831014\pi$
$72$	0	0
$73$	−6301.79	−1.18255	−0.591273	−	0.806472i	$-0.701374\pi$
$73$	−6301.79	−1.18255	−0.591273	+	0.806472i	$0.701374\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15512.6i	− 2.61640i
$78$	0	0
$79$	4305.58	0.689885	0.344943	−	0.938624i	$-0.387898\pi$
$79$	4305.58	0.689885	0.344943	+	0.938624i	$0.387898\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4808.46i	0.697992i	0.937124	+	0.348996i	$0.113477\pi$
$83$	4808.46i	0.697992i	−0.937124	+	0.348996i	$0.886523\pi$
$84$	0	0
$85$	966.293	0.133743
$86$	0	0
$87$	0	0
$88$	0	0
$89$	9205.60i	1.16218i	0.813841	+	0.581088i	$0.197373\pi$
$89$	9205.60i	1.16218i	−0.813841	+	0.581088i	$0.802627\pi$
$90$	0	0
$91$	10924.0	1.31917
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 3730.96i	− 0.413403i
$96$	0	0
$97$	−8935.86	−0.949714	−0.474857	−	0.880063i	$-0.657500\pi$
$97$	−8935.86	−0.949714	−0.474857	+	0.880063i	$0.657500\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 6535.65i	− 0.640687i	−0.947301	−	0.320343i	$-0.896202\pi$
$101$	− 6535.65i	− 0.640687i	0.947301	−	0.320343i	$-0.103798\pi$
$102$	0	0
$103$	19772.1	1.86371	0.931856	−	0.362829i	$-0.118189\pi$
$103$	19772.1	1.86371	0.931856	+	0.362829i	$0.118189\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 884.646i	− 0.0772684i	−0.999253	−	0.0386342i	$-0.987699\pi$
$107$	− 884.646i	− 0.0772684i	0.999253	−	0.0386342i	$-0.0123007\pi$
$108$	0	0
$109$	15820.7	1.33160	0.665798	−	0.746132i	$-0.268091\pi$
$109$	15820.7	1.33160	0.665798	+	0.746132i	$0.268091\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15078.7i	1.18088i	0.807082	+	0.590440i	$0.201046\pi$
$113$	15078.7i	1.18088i	−0.807082	+	0.590440i	$0.798954\pi$
$114$	0	0
$115$	18623.7	1.40822
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 3526.05i	− 0.248997i
$120$	0	0
$121$	−17395.2	−1.18811
$122$	0	0
$123$	0	0
$124$	0	0
$125$	16290.5i	1.04259i
$126$	0	0
$127$	−2433.13	−0.150854	−0.0754271	−	0.997151i	$-0.524032\pi$
$127$	−2433.13	−0.150854	−0.0754271	+	0.997151i	$0.524032\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	13786.9i	0.803388i	0.915774	+	0.401694i	$0.131578\pi$
$131$	13786.9i	0.803388i	−0.915774	+	0.401694i	$0.868422\pi$
$132$	0	0
$133$	−13614.4	−0.769656
$134$	0	0
$135$	0	0
$136$	0	0
$137$	12641.9i	0.673554i	0.941584	+	0.336777i	$0.109337\pi$
$137$	12641.9i	0.673554i	−0.941584	+	0.336777i	$0.890663\pi$
$138$	0	0
$139$	−3110.48	−0.160990	−0.0804949	−	0.996755i	$-0.525650\pi$
$139$	−3110.48	−0.160990	−0.0804949	+	0.996755i	$0.525650\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 22559.9i	− 1.10323i
$144$	0	0
$145$	−32710.1	−1.55577
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 41742.7i	− 1.88022i	−0.340877	−	0.940108i	$-0.610724\pi$
$149$	− 41742.7i	− 1.88022i	0.340877	−	0.940108i	$-0.389276\pi$
$150$	0	0
$151$	−13882.7	−0.608862	−0.304431	−	0.952534i	$-0.598466\pi$
$151$	−13882.7	−0.608862	−0.304431	+	0.952534i	$0.598466\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 8932.33i	− 0.371793i
$156$	0	0
$157$	−19248.5	−0.780903	−0.390452	−	0.920623i	$-0.627681\pi$
$157$	−19248.5	−0.780903	−0.390452	+	0.920623i	$0.627681\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 67958.7i	− 2.62176i
$162$	0	0
$163$	−38351.4	−1.44346	−0.721732	−	0.692173i	$-0.756654\pi$
$163$	−38351.4	−1.44346	−0.721732	+	0.692173i	$0.756654\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	46059.7i	1.65153i	0.564011	+	0.825767i	$0.309258\pi$
$167$	46059.7i	1.65153i	−0.564011	+	0.825767i	$0.690742\pi$
$168$	0	0
$169$	−12674.2	−0.443761
$170$	0	0
$171$	0	0
$172$	0	0
$173$	18815.9i	0.628686i	0.949310	+	0.314343i	$0.101784\pi$
$173$	18815.9i	0.628686i	−0.949310	+	0.314343i	$0.898216\pi$
$174$	0	0
$175$	5276.45	0.172292
$176$	0	0
$177$	0	0
$178$	0	0
$179$	15586.9i	0.486467i	0.969968	+	0.243233i	$0.0782081\pi$
$179$	15586.9i	0.486467i	−0.969968	+	0.243233i	$0.921792\pi$
$180$	0	0
$181$	12059.0	0.368089	0.184044	−	0.982918i	$-0.441081\pi$
$181$	12059.0	0.368089	0.184044	+	0.982918i	$0.441081\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	43434.0i	1.26907i
$186$	0	0
$187$	−7281.88	−0.208238
$188$	0	0
$189$	0	0
$190$	0	0
$191$	− 65282.1i	− 1.78948i	−0.446585	−	0.894741i	$-0.647360\pi$
$191$	− 65282.1i	− 1.78948i	0.446585	−	0.894741i	$-0.352640\pi$
$192$	0	0
$193$	−40646.2	−1.09120	−0.545601	−	0.838045i	$-0.683698\pi$
$193$	−40646.2	−1.09120	−0.545601	+	0.838045i	$0.683698\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 29562.3i	− 0.761739i	−0.924629	−	0.380869i	$-0.875625\pi$
$197$	− 29562.3i	− 0.761739i	0.924629	−	0.380869i	$-0.124375\pi$
$198$	0	0
$199$	12760.8	0.322233	0.161117	−	0.986935i	$-0.448490\pi$
$199$	12760.8	0.322233	0.161117	+	0.986935i	$0.448490\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	119361.i	2.89647i
$204$	0	0
$205$	17448.0	0.415182
$206$	0	0
$207$	0	0
$208$	0	0
$209$	28116.1i	0.643669i
$210$	0	0
$211$	−20715.9	−0.465307	−0.232654	−	0.972560i	$-0.574741\pi$
$211$	−20715.9	−0.465307	−0.232654	+	0.972560i	$0.574741\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	8530.56i	0.184544i
$216$	0	0
$217$	−32594.5	−0.692189
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 5127.91i	− 0.104992i
$222$	0	0
$223$	−83790.3	−1.68494	−0.842469	−	0.538745i	$-0.818899\pi$
$223$	−83790.3	−1.68494	−0.842469	+	0.538745i	$0.818899\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 23432.3i	− 0.454739i	−0.973809	−	0.227370i	$-0.926987\pi$
$227$	− 23432.3i	− 0.454739i	0.973809	−	0.227370i	$-0.0730126\pi$
$228$	0	0
$229$	28876.8	0.550653	0.275326	−	0.961351i	$-0.411214\pi$
$229$	28876.8	0.550653	0.275326	+	0.961351i	$0.411214\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 79153.1i	− 1.45800i	−0.684516	−	0.728998i	$-0.739986\pi$
$233$	− 79153.1i	− 1.45800i	0.684516	−	0.728998i	$-0.260014\pi$
$234$	0	0
$235$	71278.0	1.29068
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 44604.9i	− 0.780884i	−0.920627	−	0.390442i	$-0.872322\pi$
$239$	− 44604.9i	− 0.780884i	0.920627	−	0.390442i	$-0.127678\pi$
$240$	0	0
$241$	−110358.	−1.90007	−0.950036	−	0.312141i	$-0.898954\pi$
$241$	−110358.	−1.90007	−0.950036	+	0.312141i	$0.898954\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	121382.i	2.02219i
$246$	0	0
$247$	−19799.4	−0.324533
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 24428.5i	− 0.387748i	−0.981026	−	0.193874i	$-0.937895\pi$
$251$	− 24428.5i	− 0.387748i	0.981026	−	0.193874i	$-0.0621053\pi$
$252$	0	0
$253$	−140346.	−2.19260
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 63139.9i	− 0.955956i	−0.878372	−	0.477978i	$-0.841370\pi$
$257$	− 63139.9i	− 0.955956i	0.878372	−	0.477978i	$-0.158630\pi$
$258$	0	0
$259$	158493.	2.36271
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 60485.0i	− 0.874452i	−0.899352	−	0.437226i	$-0.855961\pi$
$263$	− 60485.0i	− 0.874452i	0.899352	−	0.437226i	$-0.144039\pi$
$264$	0	0
$265$	−70185.3	−0.999435
$266$	0	0
$267$	0	0
$268$	0	0
$269$	109374.i	1.51150i	0.654861	+	0.755749i	$0.272727\pi$
$269$	109374.i	1.51150i	−0.654861	+	0.755749i	$0.727273\pi$
$270$	0	0
$271$	−37620.6	−0.512257	−0.256128	−	0.966643i	$-0.582447\pi$
$271$	−37620.6	−0.512257	−0.256128	+	0.966643i	$0.582447\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 10896.8i	− 0.144089i
$276$	0	0
$277$	8871.32	0.115619	0.0578094	−	0.998328i	$-0.481588\pi$
$277$	8871.32	0.115619	0.0578094	+	0.998328i	$0.481588\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 9467.56i	− 0.119902i	−0.998201	−	0.0599509i	$-0.980906\pi$
$281$	− 9467.56i	− 0.119902i	0.998201	−	0.0599509i	$-0.0190944\pi$
$282$	0	0
$283$	53409.2	0.666873	0.333436	−	0.942773i	$-0.391792\pi$
$283$	53409.2	0.666873	0.333436	+	0.942773i	$0.391792\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 63668.7i	− 0.772969i
$288$	0	0
$289$	81865.8	0.980182
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 5225.87i	− 0.0608728i	−0.999537	−	0.0304364i	$-0.990310\pi$
$293$	− 5225.87i	− 0.0608728i	0.999537	−	0.0304364i	$-0.00968970\pi$
$294$	0	0
$295$	116892.	1.34320
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 98832.0i	− 1.10549i
$300$	0	0
$301$	31128.4	0.343577
$302$	0	0
$303$	0	0
$304$	0	0
$305$	160938.i	1.73005i
$306$	0	0
$307$	117913.	1.25108	0.625542	−	0.780191i	$-0.284878\pi$
$307$	117913.	1.25108	0.625542	+	0.780191i	$0.284878\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	57436.2i	0.593834i	0.954903	+	0.296917i	$0.0959585\pi$
$311$	57436.2i	0.593834i	−0.954903	+	0.296917i	$0.904042\pi$
$312$	0	0
$313$	5741.93	0.0586097	0.0293048	−	0.999571i	$-0.490671\pi$
$313$	5741.93	0.0586097	0.0293048	+	0.999571i	$0.490671\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8429.42i	0.0838840i	0.999120	+	0.0419420i	$0.0133545\pi$
$317$	8429.42i	0.0838840i	−0.999120	+	0.0419420i	$0.986646\pi$
$318$	0	0
$319$	246500.	2.42234
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6390.84i	0.0612566i
$324$	0	0
$325$	7673.52	0.0726487
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 260097.i	− 2.40294i
$330$	0	0
$331$	85349.2	0.779011	0.389505	−	0.921024i	$-0.372646\pi$
$331$	85349.2	0.779011	0.389505	+	0.921024i	$0.372646\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 12422.4i	− 0.110692i
$336$	0	0
$337$	20182.7	0.177713	0.0888565	−	0.996044i	$-0.471679\pi$
$337$	20182.7	0.177713	0.0888565	+	0.996044i	$0.471679\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	67313.0i	0.578883i
$342$	0	0
$343$	234835.	1.99607
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 14500.2i	− 0.120425i	−0.998186	−	0.0602123i	$-0.980822\pi$
$347$	− 14500.2i	− 0.120425i	0.998186	−	0.0602123i	$-0.0191778\pi$
$348$	0	0
$349$	−199023.	−1.63400	−0.817001	−	0.576636i	$-0.804365\pi$
$349$	−199023.	−1.63400	−0.817001	+	0.576636i	$0.804365\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	112370.i	0.901781i	0.892579	+	0.450890i	$0.148893\pi$
$353$	112370.i	0.901781i	−0.892579	+	0.450890i	$0.851107\pi$
$354$	0	0
$355$	−121238.	−0.962015
$356$	0	0
$357$	0	0
$358$	0	0
$359$	196146.i	1.52191i	0.648802	+	0.760957i	$0.275270\pi$
$359$	196146.i	1.52191i	−0.648802	+	0.760957i	$0.724730\pi$
$360$	0	0
$361$	−105645.	−0.810654
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 149675.i	− 1.12348i
$366$	0	0
$367$	−52979.7	−0.393348	−0.196674	−	0.980469i	$-0.563014\pi$
$367$	−52979.7	−0.393348	−0.196674	+	0.980469i	$0.563014\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	256109.i	1.86071i
$372$	0	0
$373$	−27297.1	−0.196200	−0.0980998	−	0.995177i	$-0.531276\pi$
$373$	−27297.1	−0.196200	−0.0980998	+	0.995177i	$0.531276\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	173586.i	1.22133i
$378$	0	0
$379$	44799.6	0.311886	0.155943	−	0.987766i	$-0.450158\pi$
$379$	44799.6	0.311886	0.155943	+	0.987766i	$0.450158\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 97981.4i	− 0.667953i	−0.942581	−	0.333977i	$-0.891609\pi$
$383$	− 97981.4i	− 0.667953i	0.942581	−	0.333977i	$-0.108391\pi$
$384$	0	0
$385$	368443.	2.48570
$386$	0	0
$387$	0	0
$388$	0	0
$389$	69789.7i	0.461203i	0.973048	+	0.230602i	$0.0740694\pi$
$389$	69789.7i	0.461203i	−0.973048	+	0.230602i	$0.925931\pi$
$390$	0	0
$391$	−31900.9	−0.208665
$392$	0	0
$393$	0	0
$394$	0	0
$395$	102263.i	0.655424i
$396$	0	0
$397$	−1193.74	−0.00757408	−0.00378704	−	0.999993i	$-0.501205\pi$
$397$	−1193.74	−0.00757408	−0.00378704	+	0.999993i	$0.501205\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 137748.i	− 0.856637i	−0.903628	−	0.428319i	$-0.859106\pi$
$401$	− 137748.i	− 0.856637i	0.903628	−	0.428319i	$-0.140894\pi$
$402$	0	0
$403$	−47402.0	−0.291868
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 327314.i	− 1.97595i
$408$	0	0
$409$	26316.5	0.157319	0.0786597	−	0.996902i	$-0.474936\pi$
$409$	26316.5	0.157319	0.0786597	+	0.996902i	$0.474936\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 426545.i	− 2.50072i
$414$	0	0
$415$	−114207.	−0.663126
$416$	0	0
$417$	0	0
$418$	0	0
$419$	30153.9i	0.171758i	0.996306	+	0.0858788i	$0.0273698\pi$
$419$	30153.9i	0.171758i	−0.996306	+	0.0858788i	$0.972630\pi$
$420$	0	0
$421$	−5937.01	−0.0334968	−0.0167484	−	0.999860i	$-0.505331\pi$
$421$	−5937.01	−0.0334968	−0.0167484	+	0.999860i	$0.505331\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2476.85i	− 0.0137127i
$426$	0	0
$427$	587269.	3.22093
$428$	0	0
$429$	0	0
$430$	0	0
$431$	166431.i	0.895942i	0.894048	+	0.447971i	$0.147853\pi$
$431$	166431.i	0.895942i	−0.894048	+	0.447971i	$0.852147\pi$
$432$	0	0
$433$	219214.	1.16921	0.584605	−	0.811318i	$-0.301250\pi$
$433$	219214.	1.16921	0.584605	+	0.811318i	$0.301250\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	123173.i	0.644988i
$438$	0	0
$439$	201027.	1.04310	0.521550	−	0.853220i	$-0.325354\pi$
$439$	201027.	1.04310	0.521550	+	0.853220i	$0.325354\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 263344.i	− 1.34189i	−0.741509	−	0.670943i	$-0.765890\pi$
$443$	− 263344.i	− 1.34189i	0.741509	−	0.670943i	$-0.234110\pi$
$444$	0	0
$445$	−218644.	−1.10412
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 70689.4i	− 0.350640i	−0.984511	−	0.175320i	$-0.943904\pi$
$449$	− 70689.4i	− 0.350640i	0.984511	−	0.175320i	$-0.0560960\pi$
$450$	0	0
$451$	−131486.	−0.646440
$452$	0	0
$453$	0	0
$454$	0	0
$455$	259459.i	1.25327i
$456$	0	0
$457$	−166296.	−0.796248	−0.398124	−	0.917332i	$-0.630339\pi$
$457$	−166296.	−0.796248	−0.398124	+	0.917332i	$0.630339\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	− 109252.i	− 0.514074i	−0.966402	−	0.257037i	$-0.917254\pi$
$461$	− 109252.i	− 0.514074i	0.966402	−	0.257037i	$-0.0827462\pi$
$462$	0	0
$463$	−379106.	−1.76847	−0.884237	−	0.467039i	$-0.845321\pi$
$463$	−379106.	−1.76847	−0.884237	+	0.467039i	$0.845321\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 20498.9i	− 0.0939932i	−0.998895	−	0.0469966i	$-0.985035\pi$
$467$	− 20498.9i	− 0.0939932i	0.998895	−	0.0469966i	$-0.0149650\pi$
$468$	0	0
$469$	−45329.9	−0.206082
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 64285.4i	− 0.287336i
$474$	0	0
$475$	−9563.39	−0.0423862
$476$	0	0
$477$	0	0
$478$	0	0
$479$	32431.7i	0.141351i	0.997499	+	0.0706754i	$0.0225154\pi$
$479$	32431.7i	0.141351i	−0.997499	+	0.0706754i	$0.977485\pi$
$480$	0	0
$481$	230495.	0.996259
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 212237.i	− 0.902274i
$486$	0	0
$487$	−320004.	−1.34927	−0.674633	−	0.738153i	$-0.735698\pi$
$487$	−320004.	−1.34927	−0.674633	+	0.738153i	$0.735698\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	98170.4i	0.407209i	0.979053	+	0.203605i	$0.0652657\pi$
$491$	98170.4i	0.407209i	−0.979053	+	0.203605i	$0.934734\pi$
$492$	0	0
$493$	56029.9	0.230529
$494$	0	0
$495$	0	0
$496$	0	0
$497$	442403.i	1.79104i
$498$	0	0
$499$	−334328.	−1.34268	−0.671340	−	0.741150i	$-0.734281\pi$
$499$	−334328.	−1.34268	−0.671340	+	0.741150i	$0.734281\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	71289.4i	0.281766i	0.990026	+	0.140883i	$0.0449942\pi$
$503$	71289.4i	0.281766i	−0.990026	+	0.140883i	$0.955006\pi$
$504$	0	0
$505$	155229.	0.608683
$506$	0	0
$507$	0	0
$508$	0	0
$509$	306381.i	1.18257i	0.806463	+	0.591285i	$0.201379\pi$
$509$	306381.i	1.18257i	−0.806463	+	0.591285i	$0.798621\pi$
$510$	0	0
$511$	−546171.	−2.09164
$512$	0	0
$513$	0	0
$514$	0	0
$515$	469611.i	1.77062i
$516$	0	0
$517$	−537143.	−2.00960
$518$	0	0
$519$	0	0
$520$	0	0
$521$	77716.8i	0.286312i	0.989700	+	0.143156i	$0.0457251\pi$
$521$	77716.8i	0.286312i	−0.989700	+	0.143156i	$0.954275\pi$
$522$	0	0
$523$	−47075.7	−0.172105	−0.0860525	−	0.996291i	$-0.527425\pi$
$523$	−47075.7	−0.172105	−0.0860525	+	0.996291i	$0.527425\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	15300.4i	0.0550910i
$528$	0	0
$529$	−334996.	−1.19709
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 92593.1i	− 0.325930i
$534$	0	0
$535$	21011.4	0.0734087
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 914720.i	− 3.14855i
$540$	0	0
$541$	−94797.1	−0.323892	−0.161946	−	0.986800i	$-0.551777\pi$
$541$	−94797.1	−0.323892	−0.161946	+	0.986800i	$0.551777\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	375760.i	1.26508i
$546$	0	0
$547$	−133762.	−0.447050	−0.223525	−	0.974698i	$-0.571756\pi$
$547$	−133762.	−0.447050	−0.223525	+	0.974698i	$0.571756\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	− 216337.i	− 0.712571i
$552$	0	0
$553$	373161.	1.22024
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 8093.05i	− 0.0260857i	−0.999915	−	0.0130428i	$-0.995848\pi$
$557$	− 8093.05i	− 0.0260857i	0.999915	−	0.0130428i	$-0.00415178\pi$
$558$	0	0
$559$	45269.9	0.144873
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 135086.i	− 0.426179i	−0.977033	−	0.213090i	$-0.931647\pi$
$563$	− 135086.i	− 0.426179i	0.977033	−	0.213090i	$-0.0683527\pi$
$564$	0	0
$565$	−358136.	−1.12189
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 456758.i	− 1.41079i	−0.708815	−	0.705394i	$-0.750770\pi$
$569$	− 456758.i	− 1.41079i	0.708815	−	0.705394i	$-0.249230\pi$
$570$	0	0
$571$	−69964.5	−0.214588	−0.107294	−	0.994227i	$-0.534219\pi$
$571$	−69964.5	−0.214588	−0.107294	+	0.994227i	$0.534219\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 47737.2i	− 0.144385i
$576$	0	0
$577$	−553043.	−1.66114	−0.830571	−	0.556912i	$-0.811986\pi$
$577$	−553043.	−1.66114	−0.830571	+	0.556912i	$0.811986\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	416746.i	1.23458i
$582$	0	0
$583$	528909.	1.55612
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 277782.i	− 0.806171i	−0.915162	−	0.403085i	$-0.867938\pi$
$587$	− 277782.i	− 0.806171i	0.915162	−	0.403085i	$-0.132062\pi$
$588$	0	0
$589$	59076.4	0.170288
$590$	0	0
$591$	0	0
$592$	0	0
$593$	147873.i	0.420514i	0.977646	+	0.210257i	$0.0674301\pi$
$593$	147873.i	0.420514i	−0.977646	+	0.210257i	$0.932570\pi$
$594$	0	0
$595$	83747.9	0.236559
$596$	0	0
$597$	0	0
$598$	0	0
$599$	596299.i	1.66192i	0.556331	+	0.830961i	$0.312209\pi$
$599$	596299.i	1.66192i	−0.556331	+	0.830961i	$0.687791\pi$
$600$	0	0
$601$	414549.	1.14770	0.573848	−	0.818962i	$-0.305450\pi$
$601$	414549.	1.14770	0.573848	+	0.818962i	$0.305450\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 413156.i	− 1.12876i
$606$	0	0
$607$	−393182.	−1.06713	−0.533563	−	0.845760i	$-0.679147\pi$
$607$	−393182.	−1.06713	−0.533563	+	0.845760i	$0.679147\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 378258.i	− 1.01322i
$612$	0	0
$613$	324774.	0.864292	0.432146	−	0.901804i	$-0.357757\pi$
$613$	324774.	0.864292	0.432146	+	0.901804i	$0.357757\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	139692.i	0.366946i	0.983025	+	0.183473i	$0.0587339\pi$
$617$	139692.i	0.366946i	−0.983025	+	0.183473i	$0.941266\pi$
$618$	0	0
$619$	−227426.	−0.593553	−0.296776	−	0.954947i	$-0.595912\pi$
$619$	−227426.	−0.593553	−0.296776	+	0.954947i	$0.595912\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	797842.i	2.05561i
$624$	0	0
$625$	−348868.	−0.893103
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 74399.1i	− 0.188047i
$630$	0	0
$631$	−363707.	−0.913468	−0.456734	−	0.889603i	$-0.650981\pi$
$631$	−363707.	−0.913468	−0.456734	+	0.889603i	$0.650981\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 57789.7i	− 0.143319i
$636$	0	0
$637$	644148.	1.58747
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 662932.i	− 1.61344i	−0.590934	−	0.806720i	$-0.701241\pi$
$641$	− 662932.i	− 1.61344i	0.590934	−	0.806720i	$-0.298759\pi$
$642$	0	0
$643$	419706.	1.01513	0.507567	−	0.861612i	$-0.330545\pi$
$643$	419706.	1.01513	0.507567	+	0.861612i	$0.330545\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	103936.i	0.248289i	0.992264	+	0.124145i	$0.0396186\pi$
$647$	103936.i	0.248289i	−0.992264	+	0.124145i	$0.960381\pi$
$648$	0	0
$649$	−880886.	−2.09137
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 326907.i	− 0.766650i	−0.923613	−	0.383325i	$-0.874779\pi$
$653$	− 326907.i	− 0.766650i	0.923613	−	0.383325i	$-0.125221\pi$
$654$	0	0
$655$	−327456.	−0.763257
$656$	0	0
$657$	0	0
$658$	0	0
$659$	709809.i	1.63445i	0.576322	+	0.817223i	$0.304487\pi$
$659$	709809.i	1.63445i	−0.576322	+	0.817223i	$0.695513\pi$
$660$	0	0
$661$	476388.	1.09033	0.545165	−	0.838328i	$-0.316467\pi$
$661$	476388.	1.09033	0.545165	+	0.838328i	$0.316467\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 323359.i	− 0.731210i
$666$	0	0
$667$	1.07988e6	2.42731
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 1.21281e6i	− 2.69369i
$672$	0	0
$673$	−489359.	−1.08043	−0.540216	−	0.841527i	$-0.681657\pi$
$673$	−489359.	−1.08043	−0.540216	+	0.841527i	$0.681657\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	313807.i	0.684676i	0.939577	+	0.342338i	$0.111219\pi$
$677$	313807.i	0.684676i	−0.939577	+	0.342338i	$0.888781\pi$
$678$	0	0
$679$	−774464.	−1.67982
$680$	0	0
$681$	0	0
$682$	0	0
$683$	7855.89i	0.0168405i	0.999965	+	0.00842023i	$0.00268027\pi$
$683$	7855.89i	0.0168405i	−0.999965	+	0.00842023i	$0.997320\pi$
$684$	0	0
$685$	−300261.	−0.639909
$686$	0	0
$687$	0	0
$688$	0	0
$689$	372459.i	0.784585i
$690$	0	0
$691$	−231267.	−0.484348	−0.242174	−	0.970233i	$-0.577860\pi$
$691$	−231267.	−0.484348	−0.242174	+	0.970233i	$0.577860\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 73877.7i	− 0.152948i
$696$	0	0
$697$	−29887.1	−0.0615203
$698$	0	0
$699$	0	0
$700$	0	0
$701$	872510.i	1.77556i	0.460272	+	0.887778i	$0.347752\pi$
$701$	872510.i	1.77556i	−0.460272	+	0.887778i	$0.652248\pi$
$702$	0	0
$703$	−287263.	−0.581258
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 566439.i	− 1.13322i
$708$	0	0
$709$	−312389.	−0.621445	−0.310722	−	0.950501i	$-0.600571\pi$
$709$	−312389.	−0.621445	−0.310722	+	0.950501i	$0.600571\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	294889.i	0.580069i
$714$	0	0
$715$	535826.	1.04812
$716$	0	0
$717$	0	0
$718$	0	0
$719$	870129.i	1.68316i	0.540132	+	0.841581i	$0.318374\pi$
$719$	870129.i	1.68316i	−0.540132	+	0.841581i	$0.681626\pi$
$720$	0	0
$721$	1.71363e6	3.29646
$722$	0	0
$723$	0	0
$724$	0	0
$725$	83844.3i	0.159513i
$726$	0	0
$727$	−445948.	−0.843752	−0.421876	−	0.906653i	$-0.638628\pi$
$727$	−445948.	−0.843752	−0.421876	+	0.906653i	$0.638628\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 14612.2i	− 0.0273452i
$732$	0	0
$733$	1.04537e6	1.94563	0.972816	−	0.231581i	$-0.0743899\pi$
$733$	1.04537e6	1.94563	0.972816	+	0.231581i	$0.0743899\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	93613.8i	0.172348i
$738$	0	0
$739$	−614455.	−1.12513	−0.562564	−	0.826754i	$-0.690185\pi$
$739$	−614455.	−1.12513	−0.562564	+	0.826754i	$0.690185\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	− 329504.i	− 0.596874i	−0.954429	−	0.298437i	$-0.903535\pi$
$743$	− 329504.i	− 0.596874i	0.954429	−	0.298437i	$-0.0964652\pi$
$744$	0	0
$745$	991438.	1.78630
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 76671.5i	− 0.136669i
$750$	0	0
$751$	−764579.	−1.35563	−0.677817	−	0.735231i	$-0.737074\pi$
$751$	−764579.	−1.35563	−0.677817	+	0.735231i	$0.737074\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	− 329730.i	− 0.578448i
$756$	0	0
$757$	794136.	1.38581	0.692904	−	0.721030i	$-0.256331\pi$
$757$	794136.	1.38581	0.692904	+	0.721030i	$0.256331\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	587558.i	1.01457i	0.861779	+	0.507284i	$0.169351\pi$
$761$	587558.i	1.01457i	−0.861779	+	0.507284i	$0.830649\pi$
$762$	0	0
$763$	1.37117e6	2.35527
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 620322.i	− 1.05445i
$768$	0	0
$769$	−81625.4	−0.138030	−0.0690149	−	0.997616i	$-0.521986\pi$
$769$	−81625.4	−0.138030	−0.0690149	+	0.997616i	$0.521986\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	771118.i	1.29051i	0.763967	+	0.645256i	$0.223249\pi$
$773$	771118.i	1.29051i	−0.763967	+	0.645256i	$0.776751\pi$
$774$	0	0
$775$	−22895.8	−0.0381200
$776$	0	0
$777$	0	0
$778$	0	0
$779$	115397.i	0.190161i
$780$	0	0
$781$	913636.	1.49786
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 457175.i	− 0.741896i
$786$	0	0
$787$	314632.	0.507989	0.253994	−	0.967206i	$-0.418256\pi$
$787$	314632.	0.507989	0.253994	+	0.967206i	$0.418256\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	1.30686e6i	2.08869i
$792$	0	0
$793$	854063.	1.35814
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 926192.i	− 1.45809i	−0.684466	−	0.729045i	$-0.739964\pi$
$797$	− 926192.i	− 1.45809i	0.684466	−	0.729045i	$-0.260036\pi$
$798$	0	0
$799$	−122094.	−0.191249
$800$	0	0
$801$	0	0
$802$	0	0
$803$	1.12793e6i	1.74925i
$804$	0	0
$805$	1.61410e6	2.49080
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 814711.i	− 1.24482i	−0.782691	−	0.622410i	$-0.786154\pi$
$809$	− 814711.i	− 1.24482i	0.782691	−	0.622410i	$-0.213846\pi$
$810$	0	0
$811$	−464849.	−0.706758	−0.353379	−	0.935480i	$-0.614967\pi$
$811$	−464849.	−0.706758	−0.353379	+	0.935480i	$0.614967\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 910892.i	− 1.37136i
$816$	0	0
$817$	−56419.2	−0.0845245
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 568912.i	− 0.844032i	−0.906588	−	0.422016i	$-0.861323\pi$
$821$	− 568912.i	− 0.844032i	0.906588	−	0.422016i	$-0.138677\pi$
$822$	0	0
$823$	517711.	0.764341	0.382171	−	0.924092i	$-0.375177\pi$
$823$	517711.	0.764341	0.382171	+	0.924092i	$0.375177\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	108209.i	0.158216i	0.996866	+	0.0791080i	$0.0252072\pi$
$827$	108209.i	0.158216i	−0.996866	+	0.0791080i	$0.974793\pi$
$828$	0	0
$829$	526645.	0.766318	0.383159	−	0.923682i	$-0.374836\pi$
$829$	526645.	0.766318	0.383159	+	0.923682i	$0.374836\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 207918.i	− 0.299641i
$834$	0	0
$835$	−1.09397e6	−1.56904
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 831834.i	− 1.18172i	−0.806776	−	0.590858i	$-0.798789\pi$
$839$	− 831834.i	− 1.18172i	0.806776	−	0.590858i	$-0.201211\pi$
$840$	0	0
$841$	−1.18939e6	−1.68164
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 301029.i	− 0.421594i
$846$	0	0
$847$	−1.50762e6	−2.10149
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 1.43392e6i	− 1.98000i
$852$	0	0
$853$	908743.	1.24894	0.624472	−	0.781047i	$-0.285314\pi$
$853$	908743.	1.24894	0.624472	+	0.781047i	$0.285314\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15049.8i	0.0204912i	0.999948	+	0.0102456i	$0.00326134\pi$
$857$	15049.8i	0.0204912i	−0.999948	+	0.0102456i	$0.996739\pi$
$858$	0	0
$859$	384604.	0.521228	0.260614	−	0.965443i	$-0.416075\pi$
$859$	384604.	0.521228	0.260614	+	0.965443i	$0.416075\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	577369.i	0.775232i	0.921821	+	0.387616i	$0.126701\pi$
$863$	577369.i	0.775232i	−0.921821	+	0.387616i	$0.873299\pi$
$864$	0	0
$865$	−446901.	−0.597282
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 770640.i	− 1.02050i
$870$	0	0
$871$	−65923.1	−0.0868963
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.41188e6i	1.84409i
$876$	0	0
$877$	−648747.	−0.843482	−0.421741	−	0.906716i	$-0.638581\pi$
$877$	−648747.	−0.843482	−0.421741	+	0.906716i	$0.638581\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	19219.1i	0.0247617i	0.999923	+	0.0123808i	$0.00394104\pi$
$881$	19219.1i	0.0247617i	−0.999923	+	0.0123808i	$0.996059\pi$
$882$	0	0
$883$	166919.	0.214085	0.107042	−	0.994254i	$-0.465862\pi$
$883$	166919.	0.214085	0.107042	+	0.994254i	$0.465862\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	281186.i	0.357393i	0.983904	+	0.178697i	$0.0571881\pi$
$887$	281186.i	0.357393i	−0.983904	+	0.178697i	$0.942812\pi$
$888$	0	0
$889$	−210877.	−0.266825
$890$	0	0
$891$	0	0
$892$	0	0
$893$	471417.i	0.591156i
$894$	0	0
$895$	−370207.	−0.462167
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 517935.i	− 0.640849i
$900$	0	0
$901$	120222.	0.148093
$902$	0	0
$903$	0	0
$904$	0	0
$905$	286415.i	0.349702i
$906$	0	0
$907$	169309.	0.205810	0.102905	−	0.994691i	$-0.467186\pi$
$907$	169309.	0.205810	0.102905	+	0.994691i	$0.467186\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	711986.i	0.857896i	0.903329	+	0.428948i	$0.141116\pi$
$911$	711986.i	0.857896i	−0.903329	+	0.428948i	$0.858884\pi$
$912$	0	0
$913$	860650.	1.03249
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.19490e6i	1.42100i
$918$	0	0
$919$	1.30687e6	1.54739	0.773696	−	0.633556i	$-0.218406\pi$
$919$	1.30687e6	1.54739	0.773696	+	0.633556i	$0.218406\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	643384.i	0.755209i
$924$	0	0
$925$	111332.	0.130118
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 671466.i	− 0.778023i	−0.921233	−	0.389012i	$-0.872817\pi$
$929$	− 671466.i	− 0.778023i	0.921233	−	0.389012i	$-0.127183\pi$
$930$	0	0
$931$	−802792.	−0.926197
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 172953.i	− 0.197836i
$936$	0	0
$937$	−485798.	−0.553320	−0.276660	−	0.960968i	$-0.589228\pi$
$937$	−485798.	−0.553320	−0.276660	+	0.960968i	$0.589228\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	1.44727e6i	1.63444i	0.576323	+	0.817222i	$0.304487\pi$
$941$	1.44727e6i	1.63444i	−0.576323	+	0.817222i	$0.695513\pi$
$942$	0	0
$943$	−576024.	−0.647765
$944$	0	0
$945$	0	0
$946$	0	0
$947$	300354.i	0.334914i	0.985879	+	0.167457i	$0.0535556\pi$
$947$	300354.i	0.334914i	−0.985879	+	0.167457i	$0.946444\pi$
$948$	0	0
$949$	−794294.	−0.881960
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 922083.i	− 1.01528i	−0.861570	−	0.507638i	$-0.830519\pi$
$953$	− 922083.i	− 1.01528i	0.861570	−	0.507638i	$-0.169481\pi$
$954$	0	0
$955$	1.55053e6	1.70009
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.09567e6i	1.19136i
$960$	0	0
$961$	−782086.	−0.846852
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 965395.i	− 1.03669i
$966$	0	0
$967$	844788.	0.903430	0.451715	−	0.892162i	$-0.350812\pi$
$967$	844788.	0.903430	0.451715	+	0.892162i	$0.350812\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 83930.8i	− 0.0890190i	−0.999009	−	0.0445095i	$-0.985828\pi$
$971$	− 83930.8i	− 0.0890190i	0.999009	−	0.0445095i	$-0.0141725\pi$
$972$	0	0
$973$	−269583.	−0.284752
$974$	0	0
$975$	0	0
$976$	0	0
$977$	784388.i	0.821753i	0.911691	+	0.410877i	$0.134777\pi$
$977$	784388.i	0.821753i	−0.911691	+	0.410877i	$0.865223\pi$
$978$	0	0
$979$	1.64768e6	1.71912
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 694563.i	− 0.718794i	−0.933185	−	0.359397i	$-0.882982\pi$
$983$	− 694563.i	− 0.718794i	0.933185	−	0.359397i	$-0.117018\pi$
$984$	0	0
$985$	702141.	0.723689
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 281625.i	− 0.287925i
$990$	0	0
$991$	−1.54902e6	−1.57728	−0.788641	−	0.614854i	$-0.789215\pi$
$991$	−1.54902e6	−1.57728	−0.788641	+	0.614854i	$0.789215\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	303083.i	0.306137i
$996$	0	0
$997$	−951773.	−0.957510	−0.478755	−	0.877949i	$-0.658912\pi$
$997$	−951773.	−0.957510	−0.478755	+	0.877949i	$0.658912\pi$
$998$	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.161.6	yes	8
3.2	odd	2	inner	324.5.c.b.161.3	✓	8
4.3	odd	2		1296.5.e.f.161.6		8
9.2	odd	6		324.5.g.f.53.6		16
9.4	even	3		324.5.g.f.269.6		16
9.5	odd	6		324.5.g.f.269.3		16
9.7	even	3		324.5.g.f.53.3		16
12.11	even	2		1296.5.e.f.161.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.5.c.b.161.3	✓	8	3.2	odd	2	inner
324.5.c.b.161.6	yes	8	1.1	even	1	trivial
324.5.g.f.53.3		16	9.7	even	3
324.5.g.f.53.6		16	9.2	odd	6
324.5.g.f.269.3		16	9.5	odd	6
324.5.g.f.269.6		16	9.4	even	3
1296.5.e.f.161.3		8	12.11	even	2
1296.5.e.f.161.6		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)

\(f(q)\)	\(=\)	\(q+23.7512i q^{5} +86.6692 q^{7} -178.986i q^{11} +126.043 q^{13} -40.6840i q^{17} -157.085 q^{19} -784.115i q^{23} +60.8803 q^{25} +1377.20i q^{29} -376.079 q^{31} +2058.50i q^{35} +1828.71 q^{37} -734.617i q^{41} +359.163 q^{43} -3001.03i q^{47} +5110.56 q^{49} +2955.02i q^{53} +4251.14 q^{55} -4921.52i q^{59} +6775.98 q^{61} +2993.67i q^{65} -523.022 q^{67} +5104.50i q^{71} -6301.79 q^{73} -15512.6i q^{77} +4305.58 q^{79} +4808.46i q^{83} +966.293 q^{85} +9205.60i q^{89} +10924.0 q^{91} -3730.96i q^{95} -8935.86 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 52 q^{7} + 220 q^{13} - 308 q^{19} - 328 q^{25} - 1472 q^{31} + 1228 q^{37} + 388 q^{43} + 3432 q^{49} - 36 q^{55} + 1804 q^{61} - 2180 q^{67} - 4544 q^{73} - 5012 q^{79} - 5364 q^{85} - 1756 q^{91}+ \cdots - 2528 q^{97}+O(q^{100}) \) 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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