LMFDB - Embedded newform 432.2.c.c.431.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [432,2,Mod(431,432)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(432, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("432.431");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 432 = 2^{4} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	432.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:	$3.44953736732$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			431.4
Root			$0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	432.431
Dual form			432.2.c.c.431.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{5} +1.73205i q^{7} +O(q^{10})$	$q+3.00000i q^{5} +1.73205i q^{7} -5.19615 q^{11} -2.00000 q^{13} +6.00000i q^{17} -6.92820i q^{19} -4.00000 q^{25} +6.00000i q^{29} +5.19615i q^{31} -5.19615 q^{35} +8.00000 q^{37} +10.3923i q^{43} +10.3923 q^{47} +4.00000 q^{49} -9.00000i q^{53} -15.5885i q^{55} -10.3923 q^{59} -4.00000 q^{61} -6.00000i q^{65} +3.46410i q^{67} +10.3923 q^{71} +1.00000 q^{73} -9.00000i q^{77} +3.46410i q^{79} +5.19615 q^{83} -18.0000 q^{85} -6.00000i q^{89} -3.46410i q^{91} +20.7846 q^{95} -5.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 8 q^{13} - 16 q^{25} + 32 q^{37} + 16 q^{49} - 16 q^{61} + 4 q^{73} - 72 q^{85} - 20 q^{97}+O(q^{100}) $ 4 * q - 8 * q^13 - 16 * q^25 + 32 * q^37 + 16 * q^49 - 16 * q^61 + 4 * q^73 - 72 * q^85 - 20 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/432\mathbb{Z}\right)^\times$.

$n$	$271$	$325$	$353$
$\chi(n)$	$-1$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	0	0
$3$	0	0
$4$	0	0
$5$	3.00000i	1.34164i	0.741620	+	0.670820i	$0.234058\pi$
$5$	3.00000i	1.34164i	−0.741620	+	0.670820i	$0.765942\pi$
$6$	0	0
$7$	1.73205i	0.654654i	0.944911	+	0.327327i	$0.106148\pi$
$7$	1.73205i	0.654654i	−0.944911	+	0.327327i	$0.893852\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.00000i	1.45521i	0.685994	+	0.727607i	$0.259367\pi$
$17$	6.00000i	1.45521i	−0.685994	+	0.727607i	$0.740633\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	0	0
$25$	−4.00000	−0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	6.00000i	1.11417i	0.830455	+	0.557086i	$0.188081\pi$
$29$	6.00000i	1.11417i	−0.830455	+	0.557086i	$0.811919\pi$
$30$	0	0
$31$	5.19615i	0.933257i	0.884454	+	0.466628i	$0.154531\pi$
$31$	5.19615i	0.933257i	−0.884454	+	0.466628i	$0.845469\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−5.19615	−0.878310
$36$	0	0
$37$	8.00000	1.31519	0.657596	−	0.753371i	$-0.271573\pi$
$37$	8.00000	1.31519	0.657596	+	0.753371i	$0.271573\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0	0	1.00000			$0$
$41$	0	0	−1.00000			$\pi$
$42$	0	0
$43$	10.3923i	1.58481i	0.609994	+	0.792406i	$0.291172\pi$
$43$	10.3923i	1.58481i	−0.609994	+	0.792406i	$0.708828\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	10.3923	1.51587	0.757937	−	0.652328i	$-0.226208\pi$
$47$	10.3923	1.51587	0.757937	+	0.652328i	$0.226208\pi$
$48$	0	0
$49$	4.00000	0.571429
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 9.00000i	− 1.23625i	−0.786082	−	0.618123i	$-0.787894\pi$
$53$	− 9.00000i	− 1.23625i	0.786082	−	0.618123i	$-0.212106\pi$
$54$	0	0
$55$	− 15.5885i	− 2.10195i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.3923	−1.35296	−0.676481	−	0.736460i	$-0.736496\pi$
$59$	−10.3923	−1.35296	−0.676481	+	0.736460i	$0.736496\pi$
$60$	0	0
$61$	−4.00000	−0.512148	−0.256074	−	0.966657i	$-0.582429\pi$
$61$	−4.00000	−0.512148	−0.256074	+	0.966657i	$0.582429\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 6.00000i	− 0.744208i
$66$	0	0
$67$	3.46410i	0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$	3.46410i	0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 9.00000i	− 1.02565i
$78$	0	0
$79$	3.46410i	0.389742i	0.980829	+	0.194871i	$0.0624288\pi$
$79$	3.46410i	0.389742i	−0.980829	+	0.194871i	$0.937571\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	−18.0000	−1.95237
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 6.00000i	− 0.635999i	−0.948091	−	0.317999i	$-0.896989\pi$
$89$	− 6.00000i	− 0.635999i	0.948091	−	0.317999i	$-0.103011\pi$
$90$	0	0
$91$	− 3.46410i	− 0.363137i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	20.7846	2.13246
$96$	0	0
$97$	−5.00000	−0.507673	−0.253837	−	0.967247i	$-0.581693\pi$
$97$	−5.00000	−0.507673	−0.253837	+	0.967247i	$0.581693\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 9.00000i	− 0.895533i	−0.894150	−	0.447767i	$-0.852219\pi$
$101$	− 9.00000i	− 0.895533i	0.894150	−	0.447767i	$-0.147781\pi$
$102$	0	0
$103$	− 3.46410i	− 0.341328i	−0.985329	−	0.170664i	$-0.945409\pi$
$103$	− 3.46410i	− 0.341328i	0.985329	−	0.170664i	$-0.0545913\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.19615	−0.502331	−0.251166	−	0.967944i	$-0.580814\pi$
$107$	−5.19615	−0.502331	−0.251166	+	0.967944i	$0.580814\pi$
$108$	0	0
$109$	16.0000	1.53252	0.766261	−	0.642529i	$-0.222115\pi$
$109$	16.0000	1.53252	0.766261	+	0.642529i	$0.222115\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.0000i	− 1.12887i	−0.825479	−	0.564433i	$-0.809095\pi$
$113$	− 12.0000i	− 1.12887i	0.825479	−	0.564433i	$-0.190905\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.3923	−0.952661
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	3.00000i	0.268328i
$126$	0	0
$127$	5.19615i	0.461084i	0.973062	+	0.230542i	$0.0740499\pi$
$127$	5.19615i	0.461084i	−0.973062	+	0.230542i	$0.925950\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	5.19615	0.453990	0.226995	−	0.973896i	$-0.427110\pi$
$131$	5.19615	0.453990	0.226995	+	0.973896i	$0.427110\pi$
$132$	0	0
$133$	12.0000	1.04053
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 6.00000i	− 0.512615i	−0.966595	−	0.256307i	$-0.917494\pi$
$137$	− 6.00000i	− 0.512615i	0.966595	−	0.256307i	$-0.0825059\pi$
$138$	0	0
$139$	6.92820i	0.587643i	0.955860	+	0.293821i	$0.0949270\pi$
$139$	6.92820i	0.587643i	−0.955860	+	0.293821i	$0.905073\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	10.3923	0.869048
$144$	0	0
$145$	−18.0000	−1.49482
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0000i	1.22885i	0.788976	+	0.614424i	$0.210612\pi$
$149$	15.0000i	1.22885i	−0.788976	+	0.614424i	$0.789388\pi$
$150$	0	0
$151$	8.66025i	0.704761i	0.935857	+	0.352381i	$0.114628\pi$
$151$	8.66025i	0.704761i	−0.935857	+	0.352381i	$0.885372\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−15.5885	−1.25210
$156$	0	0
$157$	−4.00000	−0.319235	−0.159617	−	0.987179i	$-0.551026\pi$
$157$	−4.00000	−0.319235	−0.159617	+	0.987179i	$0.551026\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	− 3.46410i	− 0.271329i	−0.990755	−	0.135665i	$-0.956683\pi$
$163$	− 3.46410i	− 0.271329i	0.990755	−	0.135665i	$-0.0433170\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−10.3923	−0.804181	−0.402090	−	0.915600i	$-0.631716\pi$
$167$	−10.3923	−0.804181	−0.402090	+	0.915600i	$0.631716\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 3.00000i	− 0.228086i	−0.993476	−	0.114043i	$-0.963620\pi$
$173$	− 3.00000i	− 0.228086i	0.993476	−	0.114043i	$-0.0363801\pi$
$174$	0	0
$175$	− 6.92820i	− 0.523723i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	5.19615	0.388379	0.194189	−	0.980964i	$-0.437792\pi$
$179$	5.19615	0.388379	0.194189	+	0.980964i	$0.437792\pi$
$180$	0	0
$181$	−8.00000	−0.594635	−0.297318	−	0.954779i	$-0.596092\pi$
$181$	−8.00000	−0.594635	−0.297318	+	0.954779i	$0.596092\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	24.0000i	1.76452i
$186$	0	0
$187$	− 31.1769i	− 2.27988i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	10.3923	0.751961	0.375980	−	0.926628i	$-0.377306\pi$
$191$	10.3923	0.751961	0.375980	+	0.926628i	$0.377306\pi$
$192$	0	0
$193$	11.0000	0.791797	0.395899	−	0.918294i	$-0.370433\pi$
$193$	11.0000	0.791797	0.395899	+	0.918294i	$0.370433\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.0000i	1.06871i	0.845262	+	0.534353i	$0.179445\pi$
$197$	15.0000i	1.06871i	−0.845262	+	0.534353i	$0.820555\pi$
$198$	0	0
$199$	22.5167i	1.59616i	0.602549	+	0.798082i	$0.294152\pi$
$199$	22.5167i	1.59616i	−0.602549	+	0.798082i	$0.705848\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.3923	−0.729397
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	36.0000i	2.49017i
$210$	0	0
$211$	− 13.8564i	− 0.953914i	−0.878927	−	0.476957i	$-0.841740\pi$
$211$	− 13.8564i	− 0.953914i	0.878927	−	0.476957i	$-0.158260\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−31.1769	−2.12625
$216$	0	0
$217$	−9.00000	−0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 12.0000i	− 0.807207i
$222$	0	0
$223$	− 10.3923i	− 0.695920i	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	− 10.3923i	− 0.695920i	0.937509	−	0.347960i	$-0.113126\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	10.3923	0.689761	0.344881	−	0.938647i	$-0.387919\pi$
$227$	10.3923	0.689761	0.344881	+	0.938647i	$0.387919\pi$
$228$	0	0
$229$	−14.0000	−0.925146	−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000	−0.925146	−0.462573	+	0.886581i	$0.653074\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	31.1769i	2.03376i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.7846	−1.34444	−0.672222	−	0.740349i	$-0.734660\pi$
$239$	−20.7846	−1.34444	−0.672222	+	0.740349i	$0.734660\pi$
$240$	0	0
$241$	22.0000	1.41714	0.708572	−	0.705638i	$-0.249340\pi$
$241$	22.0000	1.41714	0.708572	+	0.705638i	$0.249340\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	12.0000i	0.766652i
$246$	0	0
$247$	13.8564i	0.881662i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−10.3923	−0.655956	−0.327978	−	0.944685i	$-0.606367\pi$
$251$	−10.3923	−0.655956	−0.327978	+	0.944685i	$0.606367\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 6.00000i	− 0.374270i	−0.982334	−	0.187135i	$-0.940080\pi$
$257$	− 6.00000i	− 0.374270i	0.982334	−	0.187135i	$-0.0599201\pi$
$258$	0	0
$259$	13.8564i	0.860995i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	27.0000	1.65860
$266$	0	0
$267$	0	0
$268$	0	0
$269$	30.0000i	1.82913i	0.404436	+	0.914566i	$0.367468\pi$
$269$	30.0000i	1.82913i	−0.404436	+	0.914566i	$0.632532\pi$
$270$	0	0
$271$	− 1.73205i	− 0.105215i	−0.998615	−	0.0526073i	$-0.983247\pi$
$271$	− 1.73205i	− 0.105215i	0.998615	−	0.0526073i	$-0.0167532\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	20.7846	1.25336
$276$	0	0
$277$	28.0000	1.68236	0.841178	−	0.540758i	$-0.181862\pi$
$277$	28.0000	1.68236	0.841178	+	0.540758i	$0.181862\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.00000i	0.357930i	0.983855	+	0.178965i	$0.0572749\pi$
$281$	6.00000i	0.357930i	−0.983855	+	0.178965i	$0.942725\pi$
$282$	0	0
$283$	− 20.7846i	− 1.23552i	−0.786368	−	0.617758i	$-0.788041\pi$
$283$	− 20.7846i	− 1.23552i	0.786368	−	0.617758i	$-0.211959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 6.00000i	− 0.350524i	−0.984522	−	0.175262i	$-0.943923\pi$
$293$	− 6.00000i	− 0.350524i	0.984522	−	0.175262i	$-0.0560772\pi$
$294$	0	0
$295$	− 31.1769i	− 1.81519i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	0	0
$301$	−18.0000	−1.03750
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.0000i	− 0.687118i
$306$	0	0
$307$	31.1769i	1.77936i	0.456584	+	0.889680i	$0.349073\pi$
$307$	31.1769i	1.77936i	−0.456584	+	0.889680i	$0.650927\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−20.7846	−1.17859	−0.589294	−	0.807919i	$-0.700594\pi$
$311$	−20.7846	−1.17859	−0.589294	+	0.807919i	$0.700594\pi$
$312$	0	0
$313$	−31.0000	−1.75222	−0.876112	−	0.482108i	$-0.839871\pi$
$313$	−31.0000	−1.75222	−0.876112	+	0.482108i	$0.839871\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 15.0000i	− 0.842484i	−0.906948	−	0.421242i	$-0.861594\pi$
$317$	− 15.0000i	− 0.842484i	0.906948	−	0.421242i	$-0.138406\pi$
$318$	0	0
$319$	− 31.1769i	− 1.74557i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	41.5692	2.31297
$324$	0	0
$325$	8.00000	0.443760
$326$	0	0
$327$	0	0
$328$	0	0
$329$	18.0000i	0.992372i
$330$	0	0
$331$	− 3.46410i	− 0.190404i	−0.995458	−	0.0952021i	$-0.969650\pi$
$331$	− 3.46410i	− 0.190404i	0.995458	−	0.0952021i	$-0.0303497\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−10.3923	−0.567792
$336$	0	0
$337$	−2.00000	−0.108947	−0.0544735	−	0.998515i	$-0.517348\pi$
$337$	−2.00000	−0.108947	−0.0544735	+	0.998515i	$0.517348\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	− 27.0000i	− 1.46213i
$342$	0	0
$343$	19.0526i	1.02874i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	15.5885	0.836832	0.418416	−	0.908255i	$-0.362585\pi$
$347$	15.5885	0.836832	0.418416	+	0.908255i	$0.362585\pi$
$348$	0	0
$349$	−4.00000	−0.214115	−0.107058	−	0.994253i	$-0.534143\pi$
$349$	−4.00000	−0.214115	−0.107058	+	0.994253i	$0.534143\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 36.0000i	− 1.91609i	−0.286623	−	0.958043i	$-0.592533\pi$
$353$	− 36.0000i	− 1.91609i	0.286623	−	0.958043i	$-0.407467\pi$
$354$	0	0
$355$	31.1769i	1.65470i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−10.3923	−0.548485	−0.274242	−	0.961661i	$-0.588427\pi$
$359$	−10.3923	−0.548485	−0.274242	+	0.961661i	$0.588427\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	3.00000i	0.157027i
$366$	0	0
$367$	− 8.66025i	− 0.452062i	−0.974120	−	0.226031i	$-0.927425\pi$
$367$	− 8.66025i	− 0.452062i	0.974120	−	0.226031i	$-0.0725750\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	15.5885	0.809312
$372$	0	0
$373$	2.00000	0.103556	0.0517780	−	0.998659i	$-0.483511\pi$
$373$	2.00000	0.103556	0.0517780	+	0.998659i	$0.483511\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 12.0000i	− 0.618031i
$378$	0	0
$379$	− 27.7128i	− 1.42351i	−0.702427	−	0.711756i	$-0.747900\pi$
$379$	− 27.7128i	− 1.42351i	0.702427	−	0.711756i	$-0.252100\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	31.1769	1.59307	0.796533	−	0.604595i	$-0.206665\pi$
$383$	31.1769	1.59307	0.796533	+	0.604595i	$0.206665\pi$
$384$	0	0
$385$	27.0000	1.37605
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 9.00000i	− 0.456318i	−0.973624	−	0.228159i	$-0.926729\pi$
$389$	− 9.00000i	− 0.456318i	0.973624	−	0.228159i	$-0.0732706\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−10.3923	−0.522894
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 24.0000i	− 1.19850i	−0.800561	−	0.599251i	$-0.795465\pi$
$401$	− 24.0000i	− 1.19850i	0.800561	−	0.599251i	$-0.204535\pi$
$402$	0	0
$403$	− 10.3923i	− 0.517678i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−41.5692	−2.06051
$408$	0	0
$409$	25.0000	1.23617	0.618085	−	0.786111i	$-0.287909\pi$
$409$	25.0000	1.23617	0.618085	+	0.786111i	$0.287909\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 18.0000i	− 0.885722i
$414$	0	0
$415$	15.5885i	0.765207i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	10.3923	0.507697	0.253849	−	0.967244i	$-0.418303\pi$
$419$	10.3923	0.507697	0.253849	+	0.967244i	$0.418303\pi$
$420$	0	0
$421$	4.00000	0.194948	0.0974740	−	0.995238i	$-0.468924\pi$
$421$	4.00000	0.194948	0.0974740	+	0.995238i	$0.468924\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 24.0000i	− 1.16417i
$426$	0	0
$427$	− 6.92820i	− 0.335279i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.1769	1.50174	0.750870	−	0.660451i	$-0.229635\pi$
$431$	31.1769	1.50174	0.750870	+	0.660451i	$0.229635\pi$
$432$	0	0
$433$	11.0000	0.528626	0.264313	−	0.964437i	$-0.414855\pi$
$433$	11.0000	0.528626	0.264313	+	0.964437i	$0.414855\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	15.5885i	0.743996i	0.928233	+	0.371998i	$0.121327\pi$
$439$	15.5885i	0.743996i	−0.928233	+	0.371998i	$0.878673\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	31.1769	1.48126	0.740630	−	0.671913i	$-0.234527\pi$
$443$	31.1769	1.48126	0.740630	+	0.671913i	$0.234527\pi$
$444$	0	0
$445$	18.0000	0.853282
$446$	0	0
$447$	0	0
$448$	0	0
$449$	36.0000i	1.69895i	0.527633	+	0.849473i	$0.323080\pi$
$449$	36.0000i	1.69895i	−0.527633	+	0.849473i	$0.676920\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	10.3923	0.487199
$456$	0	0
$457$	1.00000	0.0467780	0.0233890	−	0.999726i	$-0.492554\pi$
$457$	1.00000	0.0467780	0.0233890	+	0.999726i	$0.492554\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.00000i	0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$	9.00000i	0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$	0	0
$463$	32.9090i	1.52941i	0.644381	+	0.764705i	$0.277115\pi$
$463$	32.9090i	1.52941i	−0.644381	+	0.764705i	$0.722885\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	25.9808	1.20225	0.601123	−	0.799156i	$-0.294720\pi$
$467$	25.9808	1.20225	0.601123	+	0.799156i	$0.294720\pi$
$468$	0	0
$469$	−6.00000	−0.277054
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 54.0000i	− 2.48292i
$474$	0	0
$475$	27.7128i	1.27155i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−10.3923	−0.474837	−0.237418	−	0.971408i	$-0.576301\pi$
$479$	−10.3923	−0.474837	−0.237418	+	0.971408i	$0.576301\pi$
$480$	0	0
$481$	−16.0000	−0.729537
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 15.0000i	− 0.681115i
$486$	0	0
$487$	− 24.2487i	− 1.09881i	−0.835555	−	0.549407i	$-0.814854\pi$
$487$	− 24.2487i	− 1.09881i	0.835555	−	0.549407i	$-0.185146\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−5.19615	−0.234499	−0.117250	−	0.993102i	$-0.537408\pi$
$491$	−5.19615	−0.234499	−0.117250	+	0.993102i	$0.537408\pi$
$492$	0	0
$493$	−36.0000	−1.62136
$494$	0	0
$495$	0	0
$496$	0	0
$497$	18.0000i	0.807410i
$498$	0	0
$499$	− 31.1769i	− 1.39567i	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	− 31.1769i	− 1.39567i	0.716258	−	0.697835i	$-0.245853\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−41.5692	−1.85348	−0.926740	−	0.375703i	$-0.877401\pi$
$503$	−41.5692	−1.85348	−0.926740	+	0.375703i	$0.877401\pi$
$504$	0	0
$505$	27.0000	1.20148
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.00000i	0.398918i	0.979906	+	0.199459i	$0.0639185\pi$
$509$	9.00000i	0.398918i	−0.979906	+	0.199459i	$0.936082\pi$
$510$	0	0
$511$	1.73205i	0.0766214i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	10.3923	0.457940
$516$	0	0
$517$	−54.0000	−2.37492
$518$	0	0
$519$	0	0
$520$	0	0
$521$	18.0000i	0.788594i	0.918983	+	0.394297i	$0.129012\pi$
$521$	18.0000i	0.788594i	−0.918983	+	0.394297i	$0.870988\pi$
$522$	0	0
$523$	− 17.3205i	− 0.757373i	−0.925525	−	0.378686i	$-0.876376\pi$
$523$	− 17.3205i	− 0.757373i	0.925525	−	0.378686i	$-0.123624\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−31.1769	−1.35809
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	0	0
$534$	0	0
$535$	− 15.5885i	− 0.673948i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−20.7846	−0.895257
$540$	0	0
$541$	28.0000	1.20381	0.601907	−	0.798566i	$-0.294408\pi$
$541$	28.0000	1.20381	0.601907	+	0.798566i	$0.294408\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	48.0000i	2.05609i
$546$	0	0
$547$	− 24.2487i	− 1.03680i	−0.855138	−	0.518400i	$-0.826528\pi$
$547$	− 24.2487i	− 1.03680i	0.855138	−	0.518400i	$-0.173472\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	41.5692	1.77091
$552$	0	0
$553$	−6.00000	−0.255146
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.0000i	− 1.65248i	−0.563316	−	0.826242i	$-0.690475\pi$
$557$	− 39.0000i	− 1.65248i	0.563316	−	0.826242i	$-0.309525\pi$
$558$	0	0
$559$	− 20.7846i	− 0.879095i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−15.5885	−0.656975	−0.328488	−	0.944508i	$-0.606539\pi$
$563$	−15.5885	−0.656975	−0.328488	+	0.944508i	$0.606539\pi$
$564$	0	0
$565$	36.0000	1.51453
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24.0000i	1.00613i	0.864248	+	0.503066i	$0.167795\pi$
$569$	24.0000i	1.00613i	−0.864248	+	0.503066i	$0.832205\pi$
$570$	0	0
$571$	24.2487i	1.01478i	0.861717	+	0.507388i	$0.169389\pi$
$571$	24.2487i	1.01478i	−0.861717	+	0.507388i	$0.830611\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−10.0000	−0.416305	−0.208153	−	0.978096i	$-0.566745\pi$
$577$	−10.0000	−0.416305	−0.208153	+	0.978096i	$0.566745\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	9.00000i	0.373383i
$582$	0	0
$583$	46.7654i	1.93682i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−5.19615	−0.214468	−0.107234	−	0.994234i	$-0.534199\pi$
$587$	−5.19615	−0.214468	−0.107234	+	0.994234i	$0.534199\pi$
$588$	0	0
$589$	36.0000	1.48335
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.00000i	− 0.246390i	−0.992382	−	0.123195i	$-0.960686\pi$
$593$	− 6.00000i	− 0.246390i	0.992382	−	0.123195i	$-0.0393141\pi$
$594$	0	0
$595$	− 31.1769i	− 1.27813i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−31.1769	−1.27385	−0.636927	−	0.770924i	$-0.719795\pi$
$599$	−31.1769	−1.27385	−0.636927	+	0.770924i	$0.719795\pi$
$600$	0	0
$601$	−19.0000	−0.775026	−0.387513	−	0.921864i	$-0.626666\pi$
$601$	−19.0000	−0.775026	−0.387513	+	0.921864i	$0.626666\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	48.0000i	1.95148i
$606$	0	0
$607$	38.1051i	1.54664i	0.634017	+	0.773320i	$0.281405\pi$
$607$	38.1051i	1.54664i	−0.634017	+	0.773320i	$0.718595\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.7846	−0.840855
$612$	0	0
$613$	−10.0000	−0.403896	−0.201948	−	0.979396i	$-0.564727\pi$
$613$	−10.0000	−0.403896	−0.201948	+	0.979396i	$0.564727\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	12.0000i	0.483102i	0.970388	+	0.241551i	$0.0776561\pi$
$617$	12.0000i	0.483102i	−0.970388	+	0.241551i	$0.922344\pi$
$618$	0	0
$619$	24.2487i	0.974638i	0.873224	+	0.487319i	$0.162025\pi$
$619$	24.2487i	0.974638i	−0.873224	+	0.487319i	$0.837975\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.3923	0.416359
$624$	0	0
$625$	−29.0000	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	48.0000i	1.91389i
$630$	0	0
$631$	− 12.1244i	− 0.482663i	−0.970443	−	0.241331i	$-0.922416\pi$
$631$	− 12.1244i	− 0.482663i	0.970443	−	0.241331i	$-0.0775841\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−15.5885	−0.618609
$636$	0	0
$637$	−8.00000	−0.316972
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 24.0000i	− 0.947943i	−0.880540	−	0.473972i	$-0.842820\pi$
$641$	− 24.0000i	− 0.947943i	0.880540	−	0.473972i	$-0.157180\pi$
$642$	0	0
$643$	0	0	1.00000			$0$
$643$	0	0	−1.00000			$\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	54.0000	2.11969
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9.00000i	0.352197i	0.984373	+	0.176099i	$0.0563478\pi$
$653$	9.00000i	0.352197i	−0.984373	+	0.176099i	$0.943652\pi$
$654$	0	0
$655$	15.5885i	0.609091i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−36.3731	−1.41689	−0.708447	−	0.705764i	$-0.750604\pi$
$659$	−36.3731	−1.41689	−0.708447	+	0.705764i	$0.750604\pi$
$660$	0	0
$661$	20.0000	0.777910	0.388955	−	0.921257i	$-0.372836\pi$
$661$	20.0000	0.777910	0.388955	+	0.921257i	$0.372836\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	36.0000i	1.39602i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	20.7846	0.802381
$672$	0	0
$673$	23.0000	0.886585	0.443292	−	0.896377i	$-0.353810\pi$
$673$	23.0000	0.886585	0.443292	+	0.896377i	$0.353810\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	18.0000i	0.691796i	0.938272	+	0.345898i	$0.112426\pi$
$677$	18.0000i	0.691796i	−0.938272	+	0.345898i	$0.887574\pi$
$678$	0	0
$679$	− 8.66025i	− 0.332350i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	10.3923	0.397650	0.198825	−	0.980035i	$-0.436287\pi$
$683$	10.3923	0.397650	0.198825	+	0.980035i	$0.436287\pi$
$684$	0	0
$685$	18.0000	0.687745
$686$	0	0
$687$	0	0
$688$	0	0
$689$	18.0000i	0.685745i
$690$	0	0
$691$	20.7846i	0.790684i	0.918534	+	0.395342i	$0.129374\pi$
$691$	20.7846i	0.790684i	−0.918534	+	0.395342i	$0.870626\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−20.7846	−0.788405
$696$	0	0
$697$	0	0
$698$	0	0
$699$	0	0
$700$	0	0
$701$	21.0000i	0.793159i	0.918000	+	0.396580i	$0.129803\pi$
$701$	21.0000i	0.793159i	−0.918000	+	0.396580i	$0.870197\pi$
$702$	0	0
$703$	− 55.4256i	− 2.09042i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.5885	0.586264
$708$	0	0
$709$	−32.0000	−1.20179	−0.600893	−	0.799330i	$-0.705188\pi$
$709$	−32.0000	−1.20179	−0.600893	+	0.799330i	$0.705188\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	31.1769i	1.16595i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.7846	−0.775135	−0.387568	−	0.921841i	$-0.626685\pi$
$719$	−20.7846	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$720$	0	0
$721$	6.00000	0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.0000i	− 0.891338i
$726$	0	0
$727$	− 46.7654i	− 1.73443i	−0.497932	−	0.867216i	$-0.665907\pi$
$727$	− 46.7654i	− 1.73443i	0.497932	−	0.867216i	$-0.334093\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−62.3538	−2.30624
$732$	0	0
$733$	4.00000	0.147743	0.0738717	−	0.997268i	$-0.476464\pi$
$733$	4.00000	0.147743	0.0738717	+	0.997268i	$0.476464\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 18.0000i	− 0.663039i
$738$	0	0
$739$	− 10.3923i	− 0.382287i	−0.981562	−	0.191144i	$-0.938780\pi$
$739$	− 10.3923i	− 0.382287i	0.981562	−	0.191144i	$-0.0612196\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	31.1769	1.14377	0.571885	−	0.820334i	$-0.306212\pi$
$743$	31.1769	1.14377	0.571885	+	0.820334i	$0.306212\pi$
$744$	0	0
$745$	−45.0000	−1.64867
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 9.00000i	− 0.328853i
$750$	0	0
$751$	− 22.5167i	− 0.821645i	−0.911715	−	0.410822i	$-0.865242\pi$
$751$	− 22.5167i	− 0.821645i	0.911715	−	0.410822i	$-0.134758\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−25.9808	−0.945537
$756$	0	0
$757$	−26.0000	−0.944986	−0.472493	−	0.881334i	$-0.656646\pi$
$757$	−26.0000	−0.944986	−0.472493	+	0.881334i	$0.656646\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 24.0000i	− 0.869999i	−0.900431	−	0.435000i	$-0.856748\pi$
$761$	− 24.0000i	− 0.869999i	0.900431	−	0.435000i	$-0.143252\pi$
$762$	0	0
$763$	27.7128i	1.00327i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	20.7846	0.750489
$768$	0	0
$769$	−25.0000	−0.901523	−0.450762	−	0.892644i	$-0.648848\pi$
$769$	−25.0000	−0.901523	−0.450762	+	0.892644i	$0.648848\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.00000i	0.215805i	0.994161	+	0.107903i	$0.0344134\pi$
$773$	6.00000i	0.215805i	−0.994161	+	0.107903i	$0.965587\pi$
$774$	0	0
$775$	− 20.7846i	− 0.746605i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−54.0000	−1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 12.0000i	− 0.428298i
$786$	0	0
$787$	34.6410i	1.23482i	0.786642	+	0.617409i	$0.211818\pi$
$787$	34.6410i	1.23482i	−0.786642	+	0.617409i	$0.788182\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	20.7846	0.739016
$792$	0	0
$793$	8.00000	0.284088
$794$	0	0
$795$	0	0
$796$	0	0
$797$	33.0000i	1.16892i	0.811423	+	0.584460i	$0.198694\pi$
$797$	33.0000i	1.16892i	−0.811423	+	0.584460i	$0.801306\pi$
$798$	0	0
$799$	62.3538i	2.20592i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.19615	−0.183368
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	− 6.00000i	− 0.210949i	−0.994422	−	0.105474i	$-0.966364\pi$
$809$	− 6.00000i	− 0.210949i	0.994422	−	0.105474i	$-0.0336361\pi$
$810$	0	0
$811$	− 20.7846i	− 0.729846i	−0.931038	−	0.364923i	$-0.881095\pi$
$811$	− 20.7846i	− 0.729846i	0.931038	−	0.364923i	$-0.118905\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.3923	0.364027
$816$	0	0
$817$	72.0000	2.51896
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 42.0000i	− 1.46581i	−0.680331	−	0.732905i	$-0.738164\pi$
$821$	− 42.0000i	− 1.46581i	0.680331	−	0.732905i	$-0.261836\pi$
$822$	0	0
$823$	− 32.9090i	− 1.14713i	−0.819159	−	0.573567i	$-0.805559\pi$
$823$	− 32.9090i	− 1.14713i	0.819159	−	0.573567i	$-0.194441\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	31.1769	1.08413	0.542064	−	0.840337i	$-0.317643\pi$
$827$	31.1769	1.08413	0.542064	+	0.840337i	$0.317643\pi$
$828$	0	0
$829$	−26.0000	−0.903017	−0.451509	−	0.892267i	$-0.649114\pi$
$829$	−26.0000	−0.903017	−0.451509	+	0.892267i	$0.649114\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	24.0000i	0.831551i
$834$	0	0
$835$	− 31.1769i	− 1.07892i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.1769	1.07635	0.538173	−	0.842834i	$-0.319115\pi$
$839$	31.1769	1.07635	0.538173	+	0.842834i	$0.319115\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	− 27.0000i	− 0.928828i
$846$	0	0
$847$	27.7128i	0.952224i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	50.0000	1.71197	0.855984	−	0.517003i	$-0.172952\pi$
$853$	50.0000	1.71197	0.855984	+	0.517003i	$0.172952\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	30.0000i	1.02478i	0.858753	+	0.512390i	$0.171240\pi$
$857$	30.0000i	1.02478i	−0.858753	+	0.512390i	$0.828760\pi$
$858$	0	0
$859$	6.92820i	0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$	6.92820i	0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	20.7846	0.707516	0.353758	−	0.935337i	$-0.384904\pi$
$863$	20.7846	0.707516	0.353758	+	0.935337i	$0.384904\pi$
$864$	0	0
$865$	9.00000	0.306009
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 18.0000i	− 0.610608i
$870$	0	0
$871$	− 6.92820i	− 0.234753i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−5.19615	−0.175662
$876$	0	0
$877$	38.0000	1.28317	0.641584	−	0.767052i	$-0.278277\pi$
$877$	38.0000	1.28317	0.641584	+	0.767052i	$0.278277\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 54.0000i	− 1.81931i	−0.415369	−	0.909653i	$-0.636347\pi$
$881$	− 54.0000i	− 1.81931i	0.415369	−	0.909653i	$-0.363653\pi$
$882$	0	0
$883$	6.92820i	0.233153i	0.993182	+	0.116576i	$0.0371920\pi$
$883$	6.92820i	0.233153i	−0.993182	+	0.116576i	$0.962808\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	10.3923	0.348939	0.174470	−	0.984663i	$-0.444179\pi$
$887$	10.3923	0.348939	0.174470	+	0.984663i	$0.444179\pi$
$888$	0	0
$889$	−9.00000	−0.301850
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 72.0000i	− 2.40939i
$894$	0	0
$895$	15.5885i	0.521065i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−31.1769	−1.03981
$900$	0	0
$901$	54.0000	1.79900
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 24.0000i	− 0.797787i
$906$	0	0
$907$	− 27.7128i	− 0.920189i	−0.887870	−	0.460094i	$-0.847816\pi$
$907$	− 27.7128i	− 0.920189i	0.887870	−	0.460094i	$-0.152184\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	10.3923	0.344312	0.172156	−	0.985070i	$-0.444927\pi$
$911$	10.3923	0.344312	0.172156	+	0.985070i	$0.444927\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	0	0
$915$	0	0
$916$	0	0
$917$	9.00000i	0.297206i
$918$	0	0
$919$	15.5885i	0.514216i	0.966383	+	0.257108i	$0.0827696\pi$
$919$	15.5885i	0.514216i	−0.966383	+	0.257108i	$0.917230\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−20.7846	−0.684134
$924$	0	0
$925$	−32.0000	−1.05215
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 12.0000i	− 0.393707i	−0.980433	−	0.196854i	$-0.936928\pi$
$929$	− 12.0000i	− 0.393707i	0.980433	−	0.196854i	$-0.0630724\pi$
$930$	0	0
$931$	− 27.7128i	− 0.908251i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	93.5307	3.05878
$936$	0	0
$937$	17.0000	0.555366	0.277683	−	0.960673i	$-0.410434\pi$
$937$	17.0000	0.555366	0.277683	+	0.960673i	$0.410434\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 3.00000i	− 0.0977972i	−0.998804	−	0.0488986i	$-0.984429\pi$
$941$	− 3.00000i	− 0.0977972i	0.998804	−	0.0488986i	$-0.0155711\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	46.7654	1.51967	0.759835	−	0.650116i	$-0.225280\pi$
$947$	46.7654	1.51967	0.759835	+	0.650116i	$0.225280\pi$
$948$	0	0
$949$	−2.00000	−0.0649227
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.0000i	− 0.777436i	−0.921357	−	0.388718i	$-0.872918\pi$
$953$	− 24.0000i	− 0.777436i	0.921357	−	0.388718i	$-0.127082\pi$
$954$	0	0
$955$	31.1769i	1.00886i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	10.3923	0.335585
$960$	0	0
$961$	4.00000	0.129032
$962$	0	0
$963$	0	0
$964$	0	0
$965$	33.0000i	1.06231i
$966$	0	0
$967$	50.2295i	1.61527i	0.589682	+	0.807635i	$0.299253\pi$
$967$	50.2295i	1.61527i	−0.589682	+	0.807635i	$0.700747\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.9808	−0.833762	−0.416881	−	0.908961i	$-0.636877\pi$
$971$	−25.9808	−0.833762	−0.416881	+	0.908961i	$0.636877\pi$
$972$	0	0
$973$	−12.0000	−0.384702
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.0000i	− 1.72761i	−0.503824	−	0.863807i	$-0.668074\pi$
$977$	− 54.0000i	− 1.72761i	0.503824	−	0.863807i	$-0.331926\pi$
$978$	0	0
$979$	31.1769i	0.996419i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−10.3923	−0.331463	−0.165732	−	0.986171i	$-0.552999\pi$
$983$	−10.3923	−0.331463	−0.165732	+	0.986171i	$0.552999\pi$
$984$	0	0
$985$	−45.0000	−1.43382
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	32.9090i	1.04539i	0.852520	+	0.522694i	$0.175073\pi$
$991$	32.9090i	1.04539i	−0.852520	+	0.522694i	$0.824927\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−67.5500	−2.14148
$996$	0	0
$997$	26.0000	0.823428	0.411714	−	0.911313i	$-0.364930\pi$
$997$	26.0000	0.823428	0.411714	+	0.911313i	$0.364930\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	432.2.c.c.431.4	yes	4
3.2	odd	2	inner	432.2.c.c.431.2	yes	4
4.3	odd	2	inner	432.2.c.c.431.3	yes	4
8.3	odd	2		1728.2.c.e.1727.1		4
8.5	even	2		1728.2.c.e.1727.2		4
9.2	odd	6		1296.2.s.i.863.2		4
9.4	even	3		1296.2.s.g.431.2		4
9.5	odd	6		1296.2.s.g.431.1		4
9.7	even	3		1296.2.s.i.863.1		4
12.11	even	2	inner	432.2.c.c.431.1	✓	4
24.5	odd	2		1728.2.c.e.1727.4		4
24.11	even	2		1728.2.c.e.1727.3		4
36.7	odd	6		1296.2.s.g.863.1		4
36.11	even	6		1296.2.s.g.863.2		4
36.23	even	6		1296.2.s.i.431.1		4
36.31	odd	6		1296.2.s.i.431.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
432.2.c.c.431.1	✓	4	12.11	even	2	inner
432.2.c.c.431.2	yes	4	3.2	odd	2	inner
432.2.c.c.431.3	yes	4	4.3	odd	2	inner
432.2.c.c.431.4	yes	4	1.1	even	1	trivial
1296.2.s.g.431.1		4	9.5	odd	6
1296.2.s.g.431.2		4	9.4	even	3
1296.2.s.g.863.1		4	36.7	odd	6
1296.2.s.g.863.2		4	36.11	even	6
1296.2.s.i.431.1		4	36.23	even	6
1296.2.s.i.431.2		4	36.31	odd	6
1296.2.s.i.863.1		4	9.7	even	3
1296.2.s.i.863.2		4	9.2	odd	6
1728.2.c.e.1727.1		4	8.3	odd	2
1728.2.c.e.1727.2		4	8.5	even	2
1728.2.c.e.1727.3		4	24.11	even	2
1728.2.c.e.1727.4		4	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 432 = 2^{4} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	432.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.00000i q^{5} +1.73205i q^{7} +O(q^{10})\)	\(q+3.00000i q^{5} +1.73205i q^{7} -5.19615 q^{11} -2.00000 q^{13} +6.00000i q^{17} -6.92820i q^{19} -4.00000 q^{25} +6.00000i q^{29} +5.19615i q^{31} -5.19615 q^{35} +8.00000 q^{37} +10.3923i q^{43} +10.3923 q^{47} +4.00000 q^{49} -9.00000i q^{53} -15.5885i q^{55} -10.3923 q^{59} -4.00000 q^{61} -6.00000i q^{65} +3.46410i q^{67} +10.3923 q^{71} +1.00000 q^{73} -9.00000i q^{77} +3.46410i q^{79} +5.19615 q^{83} -18.0000 q^{85} -6.00000i q^{89} -3.46410i q^{91} +20.7846 q^{95} -5.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 8 q^{13} - 16 q^{25} + 32 q^{37} + 16 q^{49} - 16 q^{61} + 4 q^{73} - 72 q^{85} - 20 q^{97}+O(q^{100}) \) 4 * q - 8 * q^13 - 16 * q^25 + 32 * q^37 + 16 * q^49 - 16 * q^61 + 4 * q^73 - 72 * q^85 - 20 * q^97

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