LMFDB - Embedded newform 324.3.c.a.161.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [324,3,Mod(161,324)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(324, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("324.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$8.82836056527$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 4x^{2} + 1 $ x^4 + 4*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.2
Root			$-0.517638i$ of defining polynomial
Character	$\chi$	$=$	324.161
Dual form			324.3.c.a.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.55291i q^{5} +4.19615 q^{7} +O(q^{10})$	$q-1.55291i q^{5} +4.19615 q^{7} +15.8338i q^{11} +9.39230 q^{13} -23.9029i q^{17} +18.9808 q^{19} -25.1512i q^{23} +22.5885 q^{25} +55.2658i q^{29} +39.1769 q^{31} -6.51626i q^{35} +19.1962 q^{37} -2.80122i q^{41} +33.7654 q^{43} +16.1384i q^{47} -31.3923 q^{49} -53.7129i q^{53} +24.5885 q^{55} -9.92670i q^{59} -62.7654 q^{61} -14.5854i q^{65} +20.5885 q^{67} +113.942i q^{71} -110.765 q^{73} +66.4408i q^{77} +40.6269 q^{79} -164.549i q^{83} -37.1192 q^{85} -13.6716i q^{89} +39.4115 q^{91} -29.4755i q^{95} -154.315 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 - 4 * q^13 - 28 * q^19 + 28 * q^25 + 32 * q^31 + 56 * q^37 - 52 * q^43 - 84 * q^49 + 36 * q^55 - 64 * q^61 + 20 * q^67 - 256 * q^73 + 308 * q^79 + 288 * q^85 + 220 * q^91 - 160 * q^97

Display 100 coefficients 10 coefficients

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
$5$	− 1.55291i	− 0.310583i	−0.987869	−	0.155291i	$-0.950368\pi$
$5$	− 1.55291i	− 0.310583i	0.987869	−	0.155291i	$-0.0496317\pi$
$6$	0	0
$7$	4.19615	0.599450	0.299725	−	0.954026i	$-0.403105\pi$
$7$	4.19615	0.599450	0.299725	+	0.954026i	$0.403105\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	15.8338i	1.43943i	0.694269	+	0.719716i	$0.255728\pi$
$11$	15.8338i	1.43943i	−0.694269	+	0.719716i	$0.744272\pi$
$12$	0	0
$13$	9.39230	0.722485	0.361242	−	0.932472i	$-0.382353\pi$
$13$	9.39230	0.722485	0.361242	+	0.932472i	$0.382353\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 23.9029i	− 1.40605i	−0.711163	−	0.703027i	$-0.751831\pi$
$17$	− 23.9029i	− 1.40605i	0.711163	−	0.703027i	$-0.248169\pi$
$18$	0	0
$19$	18.9808	0.998987	0.499494	−	0.866317i	$-0.333519\pi$
$19$	18.9808	0.998987	0.499494	+	0.866317i	$0.333519\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 25.1512i	− 1.09353i	−0.837285	−	0.546766i	$-0.815859\pi$
$23$	− 25.1512i	− 1.09353i	0.837285	−	0.546766i	$-0.184141\pi$
$24$	0	0
$25$	22.5885	0.903538
$26$	0	0
$27$	0	0
$28$	0	0
$29$	55.2658i	1.90572i	0.303413	+	0.952859i	$0.401874\pi$
$29$	55.2658i	1.90572i	−0.303413	+	0.952859i	$0.598126\pi$
$30$	0	0
$31$	39.1769	1.26377	0.631886	−	0.775062i	$-0.282281\pi$
$31$	39.1769	1.26377	0.631886	+	0.775062i	$0.282281\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 6.51626i	− 0.186179i
$36$	0	0
$37$	19.1962	0.518815	0.259407	−	0.965768i	$-0.416473\pi$
$37$	19.1962	0.518815	0.259407	+	0.965768i	$0.416473\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 2.80122i	− 0.0683225i	−0.999416	−	0.0341612i	$-0.989124\pi$
$41$	− 2.80122i	− 0.0683225i	0.999416	−	0.0341612i	$-0.0108760\pi$
$42$	0	0
$43$	33.7654	0.785241	0.392621	−	0.919701i	$-0.371569\pi$
$43$	33.7654	0.785241	0.392621	+	0.919701i	$0.371569\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	16.1384i	0.343369i	0.985152	+	0.171685i	$0.0549210\pi$
$47$	16.1384i	0.343369i	−0.985152	+	0.171685i	$0.945079\pi$
$48$	0	0
$49$	−31.3923	−0.640659
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 53.7129i	− 1.01345i	−0.862107	−	0.506726i	$-0.830856\pi$
$53$	− 53.7129i	− 1.01345i	0.862107	−	0.506726i	$-0.169144\pi$
$54$	0	0
$55$	24.5885	0.447063
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.92670i	− 0.168249i	−0.996455	−	0.0841246i	$-0.973191\pi$
$59$	− 9.92670i	− 0.168249i	0.996455	−	0.0841246i	$-0.0268094\pi$
$60$	0	0
$61$	−62.7654	−1.02894	−0.514470	−	0.857508i	$-0.672011\pi$
$61$	−62.7654	−1.02894	−0.514470	+	0.857508i	$0.672011\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 14.5854i	− 0.224391i
$66$	0	0
$67$	20.5885	0.307290	0.153645	−	0.988126i	$-0.450899\pi$
$67$	20.5885	0.307290	0.153645	+	0.988126i	$0.450899\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	113.942i	1.60482i	0.596774	+	0.802409i	$0.296449\pi$
$71$	113.942i	1.60482i	−0.596774	+	0.802409i	$0.703551\pi$
$72$	0	0
$73$	−110.765	−1.51733	−0.758667	−	0.651479i	$-0.774149\pi$
$73$	−110.765	−1.51733	−0.758667	+	0.651479i	$0.774149\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	66.4408i	0.862868i
$78$	0	0
$79$	40.6269	0.514265	0.257132	−	0.966376i	$-0.417222\pi$
$79$	40.6269	0.514265	0.257132	+	0.966376i	$0.417222\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 164.549i	− 1.98252i	−0.131923	−	0.991260i	$-0.542115\pi$
$83$	− 164.549i	− 1.98252i	0.131923	−	0.991260i	$-0.457885\pi$
$84$	0	0
$85$	−37.1192	−0.436696
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 13.6716i	− 0.153614i	−0.997046	−	0.0768069i	$-0.975528\pi$
$89$	− 13.6716i	− 0.153614i	0.997046	−	0.0768069i	$-0.0244725\pi$
$90$	0	0
$91$	39.4115	0.433094
$92$	0	0
$93$	0	0
$94$	0	0
$95$	− 29.4755i	− 0.310268i
$96$	0	0
$97$	−154.315	−1.59088	−0.795440	−	0.606032i	$-0.792760\pi$
$97$	−154.315	−1.59088	−0.795440	+	0.606032i	$0.792760\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 76.0629i	− 0.753098i	−0.926397	−	0.376549i	$-0.877111\pi$
$101$	− 76.0629i	− 0.753098i	0.926397	−	0.376549i	$-0.122889\pi$
$102$	0	0
$103$	−112.000	−1.08738	−0.543689	−	0.839287i	$-0.682973\pi$
$103$	−112.000	−1.08738	−0.543689	+	0.839287i	$0.682973\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	155.841i	1.45646i	0.685334	+	0.728228i	$0.259656\pi$
$107$	155.841i	1.45646i	−0.685334	+	0.728228i	$0.740344\pi$
$108$	0	0
$109$	−77.7077	−0.712914	−0.356457	−	0.934312i	$-0.616015\pi$
$109$	−77.7077	−0.712914	−0.356457	+	0.934312i	$0.616015\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	159.890i	1.41496i	0.706734	+	0.707480i	$0.250168\pi$
$113$	159.890i	1.41496i	−0.706734	+	0.707480i	$0.749832\pi$
$114$	0	0
$115$	−39.0577	−0.339632
$116$	0	0
$117$	0	0
$118$	0	0
$119$	− 100.300i	− 0.842860i
$120$	0	0
$121$	−129.708	−1.07196
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 73.9008i	− 0.591206i
$126$	0	0
$127$	−161.765	−1.27374	−0.636872	−	0.770970i	$-0.719772\pi$
$127$	−161.765	−1.27374	−0.636872	+	0.770970i	$0.719772\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 7.12548i	− 0.0543930i	−0.999630	−	0.0271965i	$-0.991342\pi$
$131$	− 7.12548i	− 0.0543930i	0.999630	−	0.0271965i	$-0.00865798\pi$
$132$	0	0
$133$	79.6462	0.598843
$134$	0	0
$135$	0	0
$136$	0	0
$137$	4.35413i	0.0317820i	0.999874	+	0.0158910i	$0.00505848\pi$
$137$	4.35413i	0.0317820i	−0.999874	+	0.0158910i	$0.994942\pi$
$138$	0	0
$139$	197.962	1.42418	0.712092	−	0.702086i	$-0.247748\pi$
$139$	197.962	1.42418	0.712092	+	0.702086i	$0.247748\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	148.715i	1.03997i
$144$	0	0
$145$	85.8231	0.591883
$146$	0	0
$147$	0	0
$148$	0	0
$149$	30.4192i	0.204156i	0.994776	+	0.102078i	$0.0325491\pi$
$149$	30.4192i	0.204156i	−0.994776	+	0.102078i	$0.967451\pi$
$150$	0	0
$151$	−78.3538	−0.518900	−0.259450	−	0.965757i	$-0.583541\pi$
$151$	−78.3538	−0.518900	−0.259450	+	0.965757i	$0.583541\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 60.8384i	− 0.392506i
$156$	0	0
$157$	−138.412	−0.881602	−0.440801	−	0.897605i	$-0.645306\pi$
$157$	−138.412	−0.881602	−0.440801	+	0.897605i	$0.645306\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 105.538i	− 0.655518i
$162$	0	0
$163$	235.023	1.44186	0.720929	−	0.693008i	$-0.243715\pi$
$163$	235.023	1.44186	0.720929	+	0.693008i	$0.243715\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 240.003i	− 1.43714i	−0.695453	−	0.718571i	$-0.744796\pi$
$167$	− 240.003i	− 1.43714i	0.695453	−	0.718571i	$-0.255204\pi$
$168$	0	0
$169$	−80.7846	−0.478015
$170$	0	0
$171$	0	0
$172$	0	0
$173$	98.7474i	0.570794i	0.958409	+	0.285397i	$0.0921255\pi$
$173$	98.7474i	0.570794i	−0.958409	+	0.285397i	$0.907874\pi$
$174$	0	0
$175$	94.7846	0.541626
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 101.214i	− 0.565442i	−0.959202	−	0.282721i	$-0.908763\pi$
$179$	− 101.214i	− 0.565442i	0.959202	−	0.282721i	$-0.0912371\pi$
$180$	0	0
$181$	−248.392	−1.37233	−0.686167	−	0.727444i	$-0.740708\pi$
$181$	−248.392	−1.37233	−0.686167	+	0.727444i	$0.740708\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 29.8100i	− 0.161135i
$186$	0	0
$187$	378.473	2.02392
$188$	0	0
$189$	0	0
$190$	0	0
$191$	305.775i	1.60091i	0.599390	+	0.800457i	$0.295410\pi$
$191$	305.775i	1.60091i	−0.599390	+	0.800457i	$0.704590\pi$
$192$	0	0
$193$	286.885	1.48645	0.743224	−	0.669042i	$-0.233296\pi$
$193$	286.885	1.48645	0.743224	+	0.669042i	$0.233296\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 163.910i	− 0.832031i	−0.909358	−	0.416015i	$-0.863426\pi$
$197$	− 163.910i	− 0.832031i	0.909358	−	0.416015i	$-0.136574\pi$
$198$	0	0
$199$	205.569	1.03301	0.516506	−	0.856284i	$-0.327233\pi$
$199$	205.569	1.03301	0.516506	+	0.856284i	$0.327233\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	231.904i	1.14238i
$204$	0	0
$205$	−4.35006	−0.0212198
$206$	0	0
$207$	0	0
$208$	0	0
$209$	300.537i	1.43797i
$210$	0	0
$211$	−133.488	−0.632647	−0.316323	−	0.948651i	$-0.602448\pi$
$211$	−133.488	−0.632647	−0.316323	+	0.948651i	$0.602448\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	− 52.4347i	− 0.243882i
$216$	0	0
$217$	164.392	0.757568
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 224.504i	− 1.01585i
$222$	0	0
$223$	−101.881	−0.456865	−0.228432	−	0.973560i	$-0.573360\pi$
$223$	−101.881	−0.456865	−0.228432	+	0.973560i	$0.573360\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	117.048i	0.515630i	0.966194	+	0.257815i	$0.0830024\pi$
$227$	117.048i	0.515630i	−0.966194	+	0.257815i	$0.916998\pi$
$228$	0	0
$229$	69.7077	0.304400	0.152200	−	0.988350i	$-0.451364\pi$
$229$	69.7077	0.304400	0.152200	+	0.988350i	$0.451364\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 121.097i	− 0.519731i	−0.965645	−	0.259866i	$-0.916322\pi$
$233$	− 121.097i	− 0.519731i	0.965645	−	0.259866i	$-0.0836783\pi$
$234$	0	0
$235$	25.0615	0.106645
$236$	0	0
$237$	0	0
$238$	0	0
$239$	− 264.545i	− 1.10688i	−0.832888	−	0.553441i	$-0.813314\pi$
$239$	− 264.545i	− 1.10688i	0.832888	−	0.553441i	$-0.186686\pi$
$240$	0	0
$241$	363.550	1.50851	0.754253	−	0.656584i	$-0.227999\pi$
$241$	363.550	1.50851	0.754253	+	0.656584i	$0.227999\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	48.7496i	0.198978i
$246$	0	0
$247$	178.273	0.721753
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 164.854i	− 0.656788i	−0.944541	−	0.328394i	$-0.893493\pi$
$251$	− 164.854i	− 0.656788i	0.944541	−	0.328394i	$-0.106507\pi$
$252$	0	0
$253$	398.238	1.57406
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 216.709i	− 0.843226i	−0.906776	−	0.421613i	$-0.861464\pi$
$257$	− 216.709i	− 0.843226i	0.906776	−	0.421613i	$-0.138536\pi$
$258$	0	0
$259$	80.5500	0.311004
$260$	0	0
$261$	0	0
$262$	0	0
$263$	140.981i	0.536048i	0.963412	+	0.268024i	$0.0863707\pi$
$263$	140.981i	0.536048i	−0.963412	+	0.268024i	$0.913629\pi$
$264$	0	0
$265$	−83.4115	−0.314761
$266$	0	0
$267$	0	0
$268$	0	0
$269$	184.128i	0.684490i	0.939611	+	0.342245i	$0.111187\pi$
$269$	184.128i	0.684490i	−0.939611	+	0.342245i	$0.888813\pi$
$270$	0	0
$271$	−248.004	−0.915143	−0.457572	−	0.889173i	$-0.651281\pi$
$271$	−248.004	−0.915143	−0.457572	+	0.889173i	$0.651281\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	357.660i	1.30058i
$276$	0	0
$277$	−200.708	−0.724576	−0.362288	−	0.932066i	$-0.618004\pi$
$277$	−200.708	−0.724576	−0.362288	+	0.932066i	$0.618004\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	63.0005i	0.224201i	0.993697	+	0.112101i	$0.0357579\pi$
$281$	63.0005i	0.224201i	−0.993697	+	0.112101i	$0.964242\pi$
$282$	0	0
$283$	332.946	1.17649	0.588244	−	0.808684i	$-0.299819\pi$
$283$	332.946	1.17649	0.588244	+	0.808684i	$0.299819\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 11.7543i	− 0.0409559i
$288$	0	0
$289$	−282.350	−0.976990
$290$	0	0
$291$	0	0
$292$	0	0
$293$	300.871i	1.02686i	0.858130	+	0.513432i	$0.171626\pi$
$293$	300.871i	1.02686i	−0.858130	+	0.513432i	$0.828374\pi$
$294$	0	0
$295$	−15.4153	−0.0522553
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 236.228i	− 0.790060i
$300$	0	0
$301$	141.685	0.470713
$302$	0	0
$303$	0	0
$304$	0	0
$305$	97.4692i	0.319571i
$306$	0	0
$307$	−14.7077	−0.0479077	−0.0239538	−	0.999713i	$-0.507625\pi$
$307$	−14.7077	−0.0479077	−0.0239538	+	0.999713i	$0.507625\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	− 485.548i	− 1.56125i	−0.625000	−	0.780624i	$-0.714901\pi$
$311$	− 485.548i	− 1.56125i	0.625000	−	0.780624i	$-0.285099\pi$
$312$	0	0
$313$	−4.33082	−0.0138365	−0.00691824	−	0.999976i	$-0.502202\pi$
$313$	−4.33082	−0.0138365	−0.00691824	+	0.999976i	$0.502202\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 226.391i	− 0.714167i	−0.934072	−	0.357084i	$-0.883771\pi$
$317$	− 226.391i	− 0.714167i	0.934072	−	0.357084i	$-0.116229\pi$
$318$	0	0
$319$	−875.065	−2.74315
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 453.696i	− 1.40463i
$324$	0	0
$325$	212.158	0.652793
$326$	0	0
$327$	0	0
$328$	0	0
$329$	67.7190i	0.205833i
$330$	0	0
$331$	−421.258	−1.27268	−0.636341	−	0.771408i	$-0.719553\pi$
$331$	−421.258	−1.27268	−0.636341	+	0.771408i	$0.719553\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	− 31.9721i	− 0.0954391i
$336$	0	0
$337$	−164.277	−0.487468	−0.243734	−	0.969842i	$-0.578372\pi$
$337$	−164.277	−0.487468	−0.243734	+	0.969842i	$0.578372\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	620.317i	1.81911i
$342$	0	0
$343$	−337.338	−0.983494
$344$	0	0
$345$	0	0
$346$	0	0
$347$	485.244i	1.39840i	0.714927	+	0.699199i	$0.246460\pi$
$347$	485.244i	1.39840i	−0.714927	+	0.699199i	$0.753540\pi$
$348$	0	0
$349$	172.785	0.495085	0.247542	−	0.968877i	$-0.420377\pi$
$349$	172.785	0.495085	0.247542	+	0.968877i	$0.420377\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 633.045i	− 1.79333i	−0.442710	−	0.896665i	$-0.645983\pi$
$353$	− 633.045i	− 1.79333i	0.442710	−	0.896665i	$-0.354017\pi$
$354$	0	0
$355$	176.942	0.498429
$356$	0	0
$357$	0	0
$358$	0	0
$359$	300.232i	0.836301i	0.908378	+	0.418150i	$0.137322\pi$
$359$	300.232i	0.836301i	−0.908378	+	0.418150i	$0.862678\pi$
$360$	0	0
$361$	−0.730670	−0.00202402
$362$	0	0
$363$	0	0
$364$	0	0
$365$	172.009i	0.471258i
$366$	0	0
$367$	−430.946	−1.17424	−0.587120	−	0.809500i	$-0.699738\pi$
$367$	−430.946	−1.17424	−0.587120	+	0.809500i	$0.699738\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 225.388i	− 0.607514i
$372$	0	0
$373$	179.100	0.480161	0.240080	−	0.970753i	$-0.422826\pi$
$373$	179.100	0.480161	0.240080	+	0.970753i	$0.422826\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	519.073i	1.37685i
$378$	0	0
$379$	−201.454	−0.531540	−0.265770	−	0.964036i	$-0.585626\pi$
$379$	−201.454	−0.531540	−0.265770	+	0.964036i	$0.585626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	299.014i	0.780714i	0.920663	+	0.390357i	$0.127648\pi$
$383$	299.014i	0.780714i	−0.920663	+	0.390357i	$0.872352\pi$
$384$	0	0
$385$	103.177	0.267992
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 414.479i	− 1.06550i	−0.846273	−	0.532749i	$-0.821159\pi$
$389$	− 414.479i	− 1.06550i	0.846273	−	0.532749i	$-0.178841\pi$
$390$	0	0
$391$	−601.188	−1.53757
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 63.0901i	− 0.159722i
$396$	0	0
$397$	−395.119	−0.995262	−0.497631	−	0.867389i	$-0.665797\pi$
$397$	−395.119	−0.995262	−0.497631	+	0.867389i	$0.665797\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 576.806i	− 1.43842i	−0.694793	−	0.719210i	$-0.744504\pi$
$401$	− 576.806i	− 1.43842i	0.694793	−	0.719210i	$-0.255496\pi$
$402$	0	0
$403$	367.962	0.913056
$404$	0	0
$405$	0	0
$406$	0	0
$407$	303.947i	0.746799i
$408$	0	0
$409$	113.904	0.278493	0.139247	−	0.990258i	$-0.455532\pi$
$409$	113.904	0.278493	0.139247	+	0.990258i	$0.455532\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 41.6540i	− 0.100857i
$414$	0	0
$415$	−255.531	−0.615737
$416$	0	0
$417$	0	0
$418$	0	0
$419$	706.612i	1.68642i	0.537581	+	0.843212i	$0.319338\pi$
$419$	706.612i	1.68642i	−0.537581	+	0.843212i	$0.680662\pi$
$420$	0	0
$421$	−542.692	−1.28906	−0.644528	−	0.764581i	$-0.722946\pi$
$421$	−542.692	−1.28906	−0.644528	+	0.764581i	$0.722946\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 539.930i	− 1.27042i
$426$	0	0
$427$	−263.373	−0.616799
$428$	0	0
$429$	0	0
$430$	0	0
$431$	− 111.081i	− 0.257729i	−0.991662	−	0.128864i	$-0.958867\pi$
$431$	− 111.081i	− 0.257729i	0.991662	−	0.128864i	$-0.0411332\pi$
$432$	0	0
$433$	424.654	0.980725	0.490362	−	0.871519i	$-0.336864\pi$
$433$	424.654	0.980725	0.490362	+	0.871519i	$0.336864\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 477.390i	− 1.09242i
$438$	0	0
$439$	−733.300	−1.67039	−0.835193	−	0.549956i	$-0.814644\pi$
$439$	−733.300	−1.67039	−0.835193	+	0.549956i	$0.814644\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 538.897i	− 1.21647i	−0.793756	−	0.608236i	$-0.791877\pi$
$443$	− 538.897i	− 1.21647i	0.793756	−	0.608236i	$-0.208123\pi$
$444$	0	0
$445$	−21.2309	−0.0477098
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.0319i	0.0713404i	0.999364	+	0.0356702i	$0.0113566\pi$
$449$	32.0319i	0.0713404i	−0.999364	+	0.0356702i	$0.988643\pi$
$450$	0	0
$451$	44.3538	0.0983455
$452$	0	0
$453$	0	0
$454$	0	0
$455$	− 61.2027i	− 0.134512i
$456$	0	0
$457$	−24.4115	−0.0534169	−0.0267085	−	0.999643i	$-0.508503\pi$
$457$	−24.4115	−0.0534169	−0.0267085	+	0.999643i	$0.508503\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	529.944i	1.14955i	0.818310	+	0.574776i	$0.194911\pi$
$461$	529.944i	1.14955i	−0.818310	+	0.574776i	$0.805089\pi$
$462$	0	0
$463$	92.9770	0.200814	0.100407	−	0.994946i	$-0.467985\pi$
$463$	92.9770	0.200814	0.100407	+	0.994946i	$0.467985\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 309.854i	− 0.663499i	−0.943367	−	0.331750i	$-0.892361\pi$
$467$	− 309.854i	− 0.663499i	0.943367	−	0.331750i	$-0.107639\pi$
$468$	0	0
$469$	86.3923	0.184205
$470$	0	0
$471$	0	0
$472$	0	0
$473$	534.632i	1.13030i
$474$	0	0
$475$	428.746	0.902623
$476$	0	0
$477$	0	0
$478$	0	0
$479$	310.708i	0.648660i	0.945944	+	0.324330i	$0.105139\pi$
$479$	310.708i	0.648660i	−0.945944	+	0.324330i	$0.894861\pi$
$480$	0	0
$481$	180.296	0.374836
$482$	0	0
$483$	0	0
$484$	0	0
$485$	239.639i	0.494100i
$486$	0	0
$487$	−428.908	−0.880714	−0.440357	−	0.897823i	$-0.645148\pi$
$487$	−428.908	−0.880714	−0.440357	+	0.897823i	$0.645148\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	92.1415i	0.187661i	0.995588	+	0.0938305i	$0.0299112\pi$
$491$	92.1415i	0.187661i	−0.995588	+	0.0938305i	$0.970089\pi$
$492$	0	0
$493$	1321.02	2.67954
$494$	0	0
$495$	0	0
$496$	0	0
$497$	478.118i	0.962009i
$498$	0	0
$499$	517.296	1.03667	0.518333	−	0.855179i	$-0.326553\pi$
$499$	517.296	1.03667	0.518333	+	0.855179i	$0.326553\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 477.509i	− 0.949322i	−0.880169	−	0.474661i	$-0.842571\pi$
$503$	− 477.509i	− 0.949322i	0.880169	−	0.474661i	$-0.157429\pi$
$504$	0	0
$505$	−118.119	−0.233899
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 520.077i	− 1.02176i	−0.859651	−	0.510881i	$-0.829319\pi$
$509$	− 520.077i	− 1.02176i	0.859651	−	0.510881i	$-0.170681\pi$
$510$	0	0
$511$	−464.788	−0.909566
$512$	0	0
$513$	0	0
$514$	0	0
$515$	173.926i	0.337721i
$516$	0	0
$517$	−255.531	−0.494257
$518$	0	0
$519$	0	0
$520$	0	0
$521$	777.801i	1.49290i	0.665441	+	0.746450i	$0.268243\pi$
$521$	777.801i	1.49290i	−0.665441	+	0.746450i	$0.731757\pi$
$522$	0	0
$523$	808.481	1.54585	0.772926	−	0.634496i	$-0.218792\pi$
$523$	808.481	1.54585	0.772926	+	0.634496i	$0.218792\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 936.443i	− 1.77693i
$528$	0	0
$529$	−103.585	−0.195812
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 26.3099i	− 0.0493619i
$534$	0	0
$535$	242.008	0.452351
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 497.058i	− 0.922185i
$540$	0	0
$541$	510.492	0.943609	0.471804	−	0.881703i	$-0.343603\pi$
$541$	510.492	0.943609	0.471804	+	0.881703i	$0.343603\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	120.673i	0.221419i
$546$	0	0
$547$	−23.4078	−0.0427930	−0.0213965	−	0.999771i	$-0.506811\pi$
$547$	−23.4078	−0.0427930	−0.0213965	+	0.999771i	$0.506811\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	1048.99i	1.90379i
$552$	0	0
$553$	170.477	0.308276
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 501.657i	− 0.900641i	−0.892867	−	0.450320i	$-0.851310\pi$
$557$	− 501.657i	− 0.900641i	0.892867	−	0.450320i	$-0.148690\pi$
$558$	0	0
$559$	317.135	0.567325
$560$	0	0
$561$	0	0
$562$	0	0
$563$	742.054i	1.31804i	0.752127	+	0.659018i	$0.229028\pi$
$563$	742.054i	1.31804i	−0.752127	+	0.659018i	$0.770972\pi$
$564$	0	0
$565$	248.296	0.439462
$566$	0	0
$567$	0	0
$568$	0	0
$569$	172.254i	0.302731i	0.988478	+	0.151366i	$0.0483671\pi$
$569$	172.254i	0.302731i	−0.988478	+	0.151366i	$0.951633\pi$
$570$	0	0
$571$	−245.923	−0.430688	−0.215344	−	0.976538i	$-0.569087\pi$
$571$	−245.923	−0.430688	−0.215344	+	0.976538i	$0.569087\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	− 568.128i	− 0.988048i
$576$	0	0
$577$	313.008	0.542474	0.271237	−	0.962513i	$-0.412567\pi$
$577$	313.008	0.542474	0.271237	+	0.962513i	$0.412567\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	− 690.473i	− 1.18842i
$582$	0	0
$583$	850.477	1.45879
$584$	0	0
$585$	0	0
$586$	0	0
$587$	531.891i	0.906118i	0.891481	+	0.453059i	$0.149667\pi$
$587$	531.891i	0.906118i	−0.891481	+	0.453059i	$0.850333\pi$
$588$	0	0
$589$	743.608	1.26249
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 733.925i	− 1.23765i	−0.785530	−	0.618824i	$-0.787609\pi$
$593$	− 733.925i	− 1.23765i	0.785530	−	0.618824i	$-0.212391\pi$
$594$	0	0
$595$	−155.758	−0.261778
$596$	0	0
$597$	0	0
$598$	0	0
$599$	− 571.293i	− 0.953745i	−0.878973	−	0.476872i	$-0.841770\pi$
$599$	− 571.293i	− 0.953745i	0.878973	−	0.476872i	$-0.158230\pi$
$600$	0	0
$601$	−125.477	−0.208780	−0.104390	−	0.994536i	$-0.533289\pi$
$601$	−125.477	−0.208780	−0.104390	+	0.994536i	$0.533289\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	201.425i	0.332934i
$606$	0	0
$607$	73.7269	0.121461	0.0607306	−	0.998154i	$-0.480657\pi$
$607$	73.7269	0.121461	0.0607306	+	0.998154i	$0.480657\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	151.576i	0.248079i
$612$	0	0
$613$	106.008	0.172932	0.0864662	−	0.996255i	$-0.472443\pi$
$613$	106.008	0.172932	0.0864662	+	0.996255i	$0.472443\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	110.502i	0.179095i	0.995983	+	0.0895476i	$0.0285421\pi$
$617$	110.502i	0.179095i	−0.995983	+	0.0895476i	$0.971458\pi$
$618$	0	0
$619$	229.538	0.370821	0.185411	−	0.982661i	$-0.440638\pi$
$619$	229.538	0.370821	0.185411	+	0.982661i	$0.440638\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 57.3682i	− 0.0920838i
$624$	0	0
$625$	449.950	0.719920
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 458.844i	− 0.729482i
$630$	0	0
$631$	−1156.40	−1.83265	−0.916323	−	0.400440i	$-0.868857\pi$
$631$	−1156.40	−1.83265	−0.916323	+	0.400440i	$0.868857\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	251.208i	0.395603i
$636$	0	0
$637$	−294.846	−0.462867
$638$	0	0
$639$	0	0
$640$	0	0
$641$	984.828i	1.53639i	0.640214	+	0.768197i	$0.278846\pi$
$641$	984.828i	1.53639i	−0.640214	+	0.768197i	$0.721154\pi$
$642$	0	0
$643$	389.184	0.605264	0.302632	−	0.953108i	$-0.402135\pi$
$643$	389.184	0.605264	0.302632	+	0.953108i	$0.402135\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	1203.18i	1.85963i	0.368029	+	0.929814i	$0.380033\pi$
$647$	1203.18i	1.85963i	−0.368029	+	0.929814i	$0.619967\pi$
$648$	0	0
$649$	157.177	0.242183
$650$	0	0
$651$	0	0
$652$	0	0
$653$	620.562i	0.950325i	0.879898	+	0.475163i	$0.157611\pi$
$653$	620.562i	0.950325i	−0.879898	+	0.475163i	$0.842389\pi$
$654$	0	0
$655$	−11.0653	−0.0168935
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 93.3600i	− 0.141669i	−0.997488	−	0.0708346i	$-0.977434\pi$
$659$	− 93.3600i	− 0.141669i	0.997488	−	0.0708346i	$-0.0225662\pi$
$660$	0	0
$661$	−241.358	−0.365140	−0.182570	−	0.983193i	$-0.558442\pi$
$661$	−241.358	−0.365140	−0.182570	+	0.983193i	$0.558442\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 123.684i	− 0.185990i
$666$	0	0
$667$	1390.00	2.08396
$668$	0	0
$669$	0	0
$670$	0	0
$671$	− 993.811i	− 1.48109i
$672$	0	0
$673$	−337.546	−0.501555	−0.250777	−	0.968045i	$-0.580686\pi$
$673$	−337.546	−0.501555	−0.250777	+	0.968045i	$0.580686\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 399.009i	− 0.589379i	−0.955593	−	0.294689i	$-0.904784\pi$
$677$	− 399.009i	− 0.589379i	0.955593	−	0.294689i	$-0.0952162\pi$
$678$	0	0
$679$	−647.531	−0.953654
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 548.394i	− 0.802919i	−0.915877	−	0.401460i	$-0.868503\pi$
$683$	− 548.394i	− 0.802919i	0.915877	−	0.401460i	$-0.131497\pi$
$684$	0	0
$685$	6.76160	0.00987095
$686$	0	0
$687$	0	0
$688$	0	0
$689$	− 504.488i	− 0.732203i
$690$	0	0
$691$	476.358	0.689374	0.344687	−	0.938718i	$-0.387985\pi$
$691$	476.358	0.689374	0.344687	+	0.938718i	$0.387985\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	− 307.417i	− 0.442327i
$696$	0	0
$697$	−66.9574	−0.0960651
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 248.622i	− 0.354667i	−0.984151	−	0.177333i	$-0.943253\pi$
$701$	− 248.622i	− 0.354667i	0.984151	−	0.177333i	$-0.0567471\pi$
$702$	0	0
$703$	364.358	0.518290
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 319.172i	− 0.451445i
$708$	0	0
$709$	409.154	0.577086	0.288543	−	0.957467i	$-0.406829\pi$
$709$	409.154	0.577086	0.288543	+	0.957467i	$0.406829\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 985.348i	− 1.38197i
$714$	0	0
$715$	230.942	0.322996
$716$	0	0
$717$	0	0
$718$	0	0
$719$	880.723i	1.22493i	0.790498	+	0.612464i	$0.209822\pi$
$719$	880.723i	1.22493i	−0.790498	+	0.612464i	$0.790178\pi$
$720$	0	0
$721$	−469.969	−0.651830
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1248.37i	1.72189i
$726$	0	0
$727$	−628.358	−0.864316	−0.432158	−	0.901798i	$-0.642248\pi$
$727$	−628.358	−0.864316	−0.432158	+	0.901798i	$0.642248\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 807.091i	− 1.10409i
$732$	0	0
$733$	−387.054	−0.528041	−0.264020	−	0.964517i	$-0.585049\pi$
$733$	−387.054	−0.528041	−0.264020	+	0.964517i	$0.585049\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	325.992i	0.442324i
$738$	0	0
$739$	−838.831	−1.13509	−0.567544	−	0.823343i	$-0.692106\pi$
$739$	−838.831	−1.13509	−0.567544	+	0.823343i	$0.692106\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1335.03i	1.79681i	0.439170	+	0.898404i	$0.355273\pi$
$743$	1335.03i	1.79681i	−0.439170	+	0.898404i	$0.644727\pi$
$744$	0	0
$745$	47.2384	0.0634073
$746$	0	0
$747$	0	0
$748$	0	0
$749$	653.932i	0.873074i
$750$	0	0
$751$	−318.396	−0.423963	−0.211981	−	0.977274i	$-0.567992\pi$
$751$	−318.396	−0.423963	−0.211981	+	0.977274i	$0.567992\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	121.677i	0.161161i
$756$	0	0
$757$	−1356.72	−1.79224	−0.896118	−	0.443816i	$-0.853624\pi$
$757$	−1356.72	−1.79224	−0.896118	+	0.443816i	$0.853624\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 248.317i	− 0.326303i	−0.986601	−	0.163152i	$-0.947834\pi$
$761$	− 248.317i	− 0.326303i	0.986601	−	0.163152i	$-0.0521660\pi$
$762$	0	0
$763$	−326.073	−0.427357
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 93.2346i	− 0.121558i
$768$	0	0
$769$	−994.731	−1.29354	−0.646769	−	0.762686i	$-0.723880\pi$
$769$	−994.731	−1.29354	−0.646769	+	0.762686i	$0.723880\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	717.667i	0.928418i	0.885726	+	0.464209i	$0.153661\pi$
$773$	717.667i	0.928418i	−0.885726	+	0.464209i	$0.846339\pi$
$774$	0	0
$775$	884.946	1.14187
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 53.1693i	− 0.0682533i
$780$	0	0
$781$	−1804.13	−2.31003
$782$	0	0
$783$	0	0
$784$	0	0
$785$	214.941i	0.273811i
$786$	0	0
$787$	276.119	0.350850	0.175425	−	0.984493i	$-0.443870\pi$
$787$	276.119	0.350850	0.175425	+	0.984493i	$0.443870\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	670.925i	0.848198i
$792$	0	0
$793$	−589.512	−0.743394
$794$	0	0
$795$	0	0
$796$	0	0
$797$	253.979i	0.318669i	0.987225	+	0.159334i	$0.0509348\pi$
$797$	253.979i	0.318669i	−0.987225	+	0.159334i	$0.949065\pi$
$798$	0	0
$799$	385.754	0.482796
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 1753.83i	− 2.18410i
$804$	0	0
$805$	−163.892	−0.203593
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 730.939i	− 0.903509i	−0.892142	−	0.451754i	$-0.850798\pi$
$809$	− 730.939i	− 0.903509i	0.892142	−	0.451754i	$-0.149202\pi$
$810$	0	0
$811$	−1038.12	−1.28005	−0.640026	−	0.768353i	$-0.721077\pi$
$811$	−1038.12	−1.28005	−0.640026	+	0.768353i	$0.721077\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	− 364.971i	− 0.447817i
$816$	0	0
$817$	640.892	0.784446
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 874.601i	− 1.06529i	−0.846339	−	0.532644i	$-0.821199\pi$
$821$	− 874.601i	− 1.06529i	0.846339	−	0.532644i	$-0.178801\pi$
$822$	0	0
$823$	1005.81	1.22212	0.611062	−	0.791583i	$-0.290743\pi$
$823$	1005.81	1.22212	0.611062	+	0.791583i	$0.290743\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 140.616i	− 0.170032i	−0.996380	−	0.0850159i	$-0.972906\pi$
$827$	− 140.616i	− 0.170032i	0.996380	−	0.0850159i	$-0.0270941\pi$
$828$	0	0
$829$	704.400	0.849698	0.424849	−	0.905264i	$-0.360327\pi$
$829$	704.400	0.849698	0.424849	+	0.905264i	$0.360327\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	750.368i	0.900802i
$834$	0	0
$835$	−372.704	−0.446352
$836$	0	0
$837$	0	0
$838$	0	0
$839$	− 546.142i	− 0.650944i	−0.945552	−	0.325472i	$-0.894477\pi$
$839$	− 546.142i	− 0.650944i	0.945552	−	0.325472i	$-0.105523\pi$
$840$	0	0
$841$	−2213.31	−2.63176
$842$	0	0
$843$	0	0
$844$	0	0
$845$	125.452i	0.148463i
$846$	0	0
$847$	−544.273	−0.642589
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 482.807i	− 0.567341i
$852$	0	0
$853$	−1053.54	−1.23510	−0.617549	−	0.786533i	$-0.711874\pi$
$853$	−1053.54	−1.23510	−0.617549	+	0.786533i	$0.711874\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 1197.18i	− 1.39695i	−0.715636	−	0.698473i	$-0.753863\pi$
$857$	− 1197.18i	− 1.39695i	0.715636	−	0.698473i	$-0.246137\pi$
$858$	0	0
$859$	427.485	0.497654	0.248827	−	0.968548i	$-0.419955\pi$
$859$	427.485	0.497654	0.248827	+	0.968548i	$0.419955\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	1250.81i	1.44937i	0.689080	+	0.724685i	$0.258015\pi$
$863$	1250.81i	1.44937i	−0.689080	+	0.724685i	$0.741985\pi$
$864$	0	0
$865$	153.346	0.177279
$866$	0	0
$867$	0	0
$868$	0	0
$869$	643.277i	0.740249i
$870$	0	0
$871$	193.373	0.222013
$872$	0	0
$873$	0	0
$874$	0	0
$875$	− 310.099i	− 0.354399i
$876$	0	0
$877$	−517.389	−0.589953	−0.294976	−	0.955505i	$-0.595312\pi$
$877$	−517.389	−0.589953	−0.294976	+	0.955505i	$0.595312\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 881.942i	− 1.00107i	−0.865717	−	0.500534i	$-0.833137\pi$
$881$	− 881.942i	− 1.00107i	0.865717	−	0.500534i	$-0.166863\pi$
$882$	0	0
$883$	1557.81	1.76422	0.882111	−	0.471042i	$-0.156122\pi$
$883$	1557.81	1.76422	0.882111	+	0.471042i	$0.156122\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 551.685i	− 0.621967i	−0.950415	−	0.310983i	$-0.899342\pi$
$887$	− 551.685i	− 0.621967i	0.950415	−	0.310983i	$-0.100658\pi$
$888$	0	0
$889$	−678.792	−0.763546
$890$	0	0
$891$	0	0
$892$	0	0
$893$	306.318i	0.343022i
$894$	0	0
$895$	−157.177	−0.175617
$896$	0	0
$897$	0	0
$898$	0	0
$899$	2165.14i	2.40839i
$900$	0	0
$901$	−1283.90	−1.42497
$902$	0	0
$903$	0	0
$904$	0	0
$905$	385.732i	0.426223i
$906$	0	0
$907$	628.038	0.692435	0.346217	−	0.938154i	$-0.387466\pi$
$907$	628.038	0.692435	0.346217	+	0.938154i	$0.387466\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	236.168i	0.259241i	0.991564	+	0.129620i	$0.0413759\pi$
$911$	236.168i	0.259241i	−0.991564	+	0.129620i	$0.958624\pi$
$912$	0	0
$913$	2605.43	2.85370
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 29.8996i	− 0.0326059i
$918$	0	0
$919$	−434.350	−0.472633	−0.236317	−	0.971676i	$-0.575940\pi$
$919$	−434.350	−0.472633	−0.236317	+	0.971676i	$0.575940\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	1070.18i	1.15946i
$924$	0	0
$925$	433.611	0.468769
$926$	0	0
$927$	0	0
$928$	0	0
$929$	1194.93i	1.28626i	0.765759	+	0.643128i	$0.222364\pi$
$929$	1194.93i	1.28626i	−0.765759	+	0.643128i	$0.777636\pi$
$930$	0	0
$931$	−595.850	−0.640011
$932$	0	0
$933$	0	0
$934$	0	0
$935$	− 587.736i	− 0.628595i
$936$	0	0
$937$	−1528.24	−1.63099	−0.815495	−	0.578764i	$-0.803535\pi$
$937$	−1528.24	−1.63099	−0.815495	+	0.578764i	$0.803535\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 1317.09i	− 1.39967i	−0.714303	−	0.699836i	$-0.753256\pi$
$941$	− 1317.09i	− 1.39967i	0.714303	−	0.699836i	$-0.246744\pi$
$942$	0	0
$943$	−70.4542	−0.0747128
$944$	0	0
$945$	0	0
$946$	0	0
$947$	720.498i	0.760822i	0.924818	+	0.380411i	$0.124217\pi$
$947$	720.498i	0.760822i	−0.924818	+	0.380411i	$0.875783\pi$
$948$	0	0
$949$	−1040.34	−1.09625
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 1469.22i	− 1.54168i	−0.637031	−	0.770838i	$-0.719838\pi$
$953$	− 1469.22i	− 1.54168i	0.637031	−	0.770838i	$-0.280162\pi$
$954$	0	0
$955$	474.842	0.497217
$956$	0	0
$957$	0	0
$958$	0	0
$959$	18.2706i	0.0190517i
$960$	0	0
$961$	573.831	0.597118
$962$	0	0
$963$	0	0
$964$	0	0
$965$	− 445.507i	− 0.461665i
$966$	0	0
$967$	−871.058	−0.900784	−0.450392	−	0.892831i	$-0.648716\pi$
$967$	−871.058	−0.900784	−0.450392	+	0.892831i	$0.648716\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	720.009i	0.741512i	0.928730	+	0.370756i	$0.120901\pi$
$971$	720.009i	0.741512i	−0.928730	+	0.370756i	$0.879099\pi$
$972$	0	0
$973$	830.677	0.853727
$974$	0	0
$975$	0	0
$976$	0	0
$977$	191.468i	0.195976i	0.995188	+	0.0979879i	$0.0312406\pi$
$977$	191.468i	0.195976i	−0.995188	+	0.0979879i	$0.968759\pi$
$978$	0	0
$979$	216.473	0.221116
$980$	0	0
$981$	0	0
$982$	0	0
$983$	− 644.005i	− 0.655143i	−0.944826	−	0.327571i	$-0.893770\pi$
$983$	− 644.005i	− 0.655143i	0.944826	−	0.327571i	$-0.106230\pi$
$984$	0	0
$985$	−254.538	−0.258415
$986$	0	0
$987$	0	0
$988$	0	0
$989$	− 849.241i	− 0.858686i
$990$	0	0
$991$	1178.82	1.18953	0.594763	−	0.803901i	$-0.297246\pi$
$991$	1178.82	1.18953	0.594763	+	0.803901i	$0.297246\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 319.231i	− 0.320836i
$996$	0	0
$997$	−683.350	−0.685406	−0.342703	−	0.939444i	$-0.611342\pi$
$997$	−683.350	−0.685406	−0.342703	+	0.939444i	$0.611342\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	324.3.c.a.161.2	✓	4
3.2	odd	2	inner	324.3.c.a.161.3	yes	4
4.3	odd	2		1296.3.e.f.161.2		4
9.2	odd	6		324.3.g.d.53.2		8
9.4	even	3		324.3.g.d.269.2		8
9.5	odd	6		324.3.g.d.269.3		8
9.7	even	3		324.3.g.d.53.3		8
12.11	even	2		1296.3.e.f.161.3		4
36.7	odd	6		1296.3.q.n.1025.3		8
36.11	even	6		1296.3.q.n.1025.2		8
36.23	even	6		1296.3.q.n.593.3		8
36.31	odd	6		1296.3.q.n.593.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
324.3.c.a.161.2	✓	4	1.1	even	1	trivial
324.3.c.a.161.3	yes	4	3.2	odd	2	inner
324.3.g.d.53.2		8	9.2	odd	6
324.3.g.d.53.3		8	9.7	even	3
324.3.g.d.269.2		8	9.4	even	3
324.3.g.d.269.3		8	9.5	odd	6
1296.3.e.f.161.2		4	4.3	odd	2
1296.3.e.f.161.3		4	12.11	even	2
1296.3.q.n.593.2		8	36.31	odd	6
1296.3.q.n.593.3		8	36.23	even	6
1296.3.q.n.1025.2		8	36.11	even	6
1296.3.q.n.1025.3		8	36.7	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.55291i q^{5} +4.19615 q^{7} +O(q^{10})\)	\(q-1.55291i q^{5} +4.19615 q^{7} +15.8338i q^{11} +9.39230 q^{13} -23.9029i q^{17} +18.9808 q^{19} -25.1512i q^{23} +22.5885 q^{25} +55.2658i q^{29} +39.1769 q^{31} -6.51626i q^{35} +19.1962 q^{37} -2.80122i q^{41} +33.7654 q^{43} +16.1384i q^{47} -31.3923 q^{49} -53.7129i q^{53} +24.5885 q^{55} -9.92670i q^{59} -62.7654 q^{61} -14.5854i q^{65} +20.5885 q^{67} +113.942i q^{71} -110.765 q^{73} +66.4408i q^{77} +40.6269 q^{79} -164.549i q^{83} -37.1192 q^{85} -13.6716i q^{89} +39.4115 q^{91} -29.4755i q^{95} -154.315 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} - 4 q^{13} - 28 q^{19} + 28 q^{25} + 32 q^{31} + 56 q^{37} - 52 q^{43} - 84 q^{49} + 36 q^{55} - 64 q^{61} + 20 q^{67} - 256 q^{73} + 308 q^{79} + 288 q^{85} + 220 q^{91} - 160 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 - 4 * q^13 - 28 * q^19 + 28 * q^25 + 32 * q^31 + 56 * q^37 - 52 * q^43 - 84 * q^49 + 36 * q^55 - 64 * q^61 + 20 * q^67 - 256 * q^73 + 308 * q^79 + 288 * q^85 + 220 * q^91 - 160 * q^97

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