LMFDB - Embedded newform 1296.2.c.b.1295.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1296,2,Mod(1295,1296)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1296, 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("1296.1295");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1296 = 2^{4} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1296.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:	$10.3486121020$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1295.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1296.1295
Dual form			1296.2.c.b.1295.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.46410i q^{5} +3.46410i q^{7} -3.00000 q^{11} +4.00000 q^{13} +1.73205i q^{17} +1.73205i q^{19} -7.00000 q^{25} -3.46410i q^{29} -12.0000 q^{35} +2.00000 q^{37} +5.19615i q^{41} -5.19615i q^{43} -12.0000 q^{47} -5.00000 q^{49} -10.3923i q^{55} -15.0000 q^{59} +8.00000 q^{61} +13.8564i q^{65} -8.66025i q^{67} +6.00000 q^{71} -11.0000 q^{73} -10.3923i q^{77} -3.46410i q^{79} +12.0000 q^{83} -6.00000 q^{85} +13.8564i q^{89} +13.8564i q^{91} -6.00000 q^{95} +13.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{11} + 8 q^{13} - 14 q^{25} - 24 q^{35} + 4 q^{37} - 24 q^{47} - 10 q^{49} - 30 q^{59} + 16 q^{61} + 12 q^{71} - 22 q^{73} + 24 q^{83} - 12 q^{85} - 12 q^{95} + 26 q^{97}+O(q^{100}) $ 2 * q - 6 * q^11 + 8 * q^13 - 14 * q^25 - 24 * q^35 + 4 * q^37 - 24 * q^47 - 10 * q^49 - 30 * q^59 + 16 * q^61 + 12 * q^71 - 22 * q^73 + 24 * q^83 - 12 * q^85 - 12 * q^95 + 26 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1217$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	3.46410i	1.54919i	0.632456	+	0.774597i	$0.282047\pi$
$5$	3.46410i	1.54919i	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	3.46410i	1.30931i	0.755929	+	0.654654i	$0.227186\pi$
$7$	3.46410i	1.30931i	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	4.00000	1.10940	0.554700	−	0.832050i	$-0.312833\pi$
$13$	4.00000	1.10940	0.554700	+	0.832050i	$0.312833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	1.73205i	0.420084i	0.977692	+	0.210042i	$0.0673601\pi$
$17$	1.73205i	0.420084i	−0.977692	+	0.210042i	$0.932640\pi$
$18$	0	0
$19$	1.73205i	0.397360i	0.980064	+	0.198680i	$0.0636654\pi$
$19$	1.73205i	0.397360i	−0.980064	+	0.198680i	$0.936335\pi$
$20$	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
$25$	−7.00000	−1.40000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 3.46410i	− 0.643268i	−0.946864	−	0.321634i	$-0.895768\pi$
$29$	− 3.46410i	− 0.643268i	0.946864	−	0.321634i	$-0.104232\pi$
$30$	0	0
$31$	0	0	1.00000			$0$
$31$	0	0	−1.00000			$\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−12.0000	−2.02837
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.19615i	0.811503i	0.913984	+	0.405751i	$0.132990\pi$
$41$	5.19615i	0.811503i	−0.913984	+	0.405751i	$0.867010\pi$
$42$	0	0
$43$	− 5.19615i	− 0.792406i	−0.918163	−	0.396203i	$-0.870328\pi$
$43$	− 5.19615i	− 0.792406i	0.918163	−	0.396203i	$-0.129672\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	0	0
$49$	−5.00000	−0.714286
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	− 10.3923i	− 1.40130i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−15.0000	−1.95283	−0.976417	−	0.215894i	$-0.930733\pi$
$59$	−15.0000	−1.95283	−0.976417	+	0.215894i	$0.930733\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	13.8564i	1.71868i
$66$	0	0
$67$	− 8.66025i	− 1.05802i	−0.848616	−	0.529009i	$-0.822564\pi$
$67$	− 8.66025i	− 1.05802i	0.848616	−	0.529009i	$-0.177436\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.00000	0.712069	0.356034	−	0.934473i	$-0.384129\pi$
$71$	6.00000	0.712069	0.356034	+	0.934473i	$0.384129\pi$
$72$	0	0
$73$	−11.0000	−1.28745	−0.643726	−	0.765256i	$-0.722612\pi$
$73$	−11.0000	−1.28745	−0.643726	+	0.765256i	$0.722612\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 10.3923i	− 1.18431i
$78$	0	0
$79$	− 3.46410i	− 0.389742i	−0.980829	−	0.194871i	$-0.937571\pi$
$79$	− 3.46410i	− 0.389742i	0.980829	−	0.194871i	$-0.0624288\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	0	0
$85$	−6.00000	−0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.8564i	1.46878i	0.678730	+	0.734388i	$0.262531\pi$
$89$	13.8564i	1.46878i	−0.678730	+	0.734388i	$0.737469\pi$
$90$	0	0
$91$	13.8564i	1.45255i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.00000	−0.615587
$96$	0	0
$97$	13.0000	1.31995	0.659975	−	0.751288i	$-0.270567\pi$
$97$	13.0000	1.31995	0.659975	+	0.751288i	$0.270567\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	10.3923i	1.03407i	0.855963	+	0.517036i	$0.172965\pi$
$101$	10.3923i	1.03407i	−0.855963	+	0.517036i	$0.827035\pi$
$102$	0	0
$103$	13.8564i	1.36531i	0.730740	+	0.682656i	$0.239175\pi$
$103$	13.8564i	1.36531i	−0.730740	+	0.682656i	$0.760825\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	0	0
$109$	4.00000	0.383131	0.191565	−	0.981480i	$-0.438644\pi$
$109$	4.00000	0.383131	0.191565	+	0.981480i	$0.438644\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.92820i	0.651751i	0.945413	+	0.325875i	$0.105659\pi$
$113$	6.92820i	0.651751i	−0.945413	+	0.325875i	$0.894341\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 6.92820i	− 0.619677i
$126$	0	0
$127$	0	0	1.00000			$0$
$127$	0	0	−1.00000			$\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	0	0
$133$	−6.00000	−0.520266
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 1.73205i	− 0.147979i	−0.997259	−	0.0739895i	$-0.976427\pi$
$137$	− 1.73205i	− 0.147979i	0.997259	−	0.0739895i	$-0.0235731\pi$
$138$	0	0
$139$	19.0526i	1.61602i	0.589171	+	0.808008i	$0.299454\pi$
$139$	19.0526i	1.61602i	−0.589171	+	0.808008i	$0.700546\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.0000	−1.00349
$144$	0	0
$145$	12.0000	0.996546
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 13.8564i	− 1.13516i	−0.823318	−	0.567581i	$-0.807880\pi$
$149$	− 13.8564i	− 1.13516i	0.823318	−	0.567581i	$-0.192120\pi$
$150$	0	0
$151$	6.92820i	0.563809i	0.959442	+	0.281905i	$0.0909662\pi$
$151$	6.92820i	0.563809i	−0.959442	+	0.281905i	$0.909034\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	8.00000	0.638470	0.319235	−	0.947676i	$-0.396574\pi$
$157$	8.00000	0.638470	0.319235	+	0.947676i	$0.396574\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	3.46410i	0.271329i	0.990755	+	0.135665i	$0.0433170\pi$
$163$	3.46410i	0.271329i	−0.990755	+	0.135665i	$0.956683\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−6.00000	−0.464294	−0.232147	−	0.972681i	$-0.574575\pi$
$167$	−6.00000	−0.464294	−0.232147	+	0.972681i	$0.574575\pi$
$168$	0	0
$169$	3.00000	0.230769
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 24.2487i	− 1.84360i	−0.387671	−	0.921798i	$-0.626720\pi$
$173$	− 24.2487i	− 1.84360i	0.387671	−	0.921798i	$-0.373280\pi$
$174$	0	0
$175$	− 24.2487i	− 1.83303i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	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
$185$	6.92820i	0.509372i
$186$	0	0
$187$	− 5.19615i	− 0.379980i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	6.00000	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$191$	6.00000	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$192$	0	0
$193$	23.0000	1.65558	0.827788	−	0.561041i	$-0.189599\pi$
$193$	23.0000	1.65558	0.827788	+	0.561041i	$0.189599\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 13.8564i	− 0.987228i	−0.869681	−	0.493614i	$-0.835676\pi$
$197$	− 13.8564i	− 0.987228i	0.869681	−	0.493614i	$-0.164324\pi$
$198$	0	0
$199$	3.46410i	0.245564i	0.992434	+	0.122782i	$0.0391815\pi$
$199$	3.46410i	0.245564i	−0.992434	+	0.122782i	$0.960818\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	12.0000	0.842235
$204$	0	0
$205$	−18.0000	−1.25717
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 5.19615i	− 0.359425i
$210$	0	0
$211$	− 17.3205i	− 1.19239i	−0.802839	−	0.596196i	$-0.796678\pi$
$211$	− 17.3205i	− 1.19239i	0.802839	−	0.596196i	$-0.203322\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	18.0000	1.22759
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.92820i	0.466041i
$222$	0	0
$223$	20.7846i	1.39184i	0.718119	+	0.695920i	$0.245003\pi$
$223$	20.7846i	1.39184i	−0.718119	+	0.695920i	$0.754997\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−3.00000	−0.199117	−0.0995585	−	0.995032i	$-0.531743\pi$
$227$	−3.00000	−0.199117	−0.0995585	+	0.995032i	$0.531743\pi$
$228$	0	0
$229$	−26.0000	−1.71813	−0.859064	−	0.511868i	$-0.828954\pi$
$229$	−26.0000	−1.71813	−0.859064	+	0.511868i	$0.828954\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 12.1244i	− 0.794293i	−0.917755	−	0.397146i	$-0.870000\pi$
$233$	− 12.1244i	− 0.794293i	0.917755	−	0.397146i	$-0.130000\pi$
$234$	0	0
$235$	− 41.5692i	− 2.71168i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	24.0000	1.55243	0.776215	−	0.630468i	$-0.217137\pi$
$239$	24.0000	1.55243	0.776215	+	0.630468i	$0.217137\pi$
$240$	0	0
$241$	−17.0000	−1.09507	−0.547533	−	0.836784i	$-0.684433\pi$
$241$	−17.0000	−1.09507	−0.547533	+	0.836784i	$0.684433\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	− 17.3205i	− 1.10657i
$246$	0	0
$247$	6.92820i	0.440831i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	21.0000	1.32551	0.662754	−	0.748837i	$-0.269387\pi$
$251$	21.0000	1.32551	0.662754	+	0.748837i	$0.269387\pi$
$252$	0	0
$253$	0	0
$254$	0	0
$255$	0	0
$256$	0	0
$257$	19.0526i	1.18847i	0.804293	+	0.594233i	$0.202544\pi$
$257$	19.0526i	1.18847i	−0.804293	+	0.594233i	$0.797456\pi$
$258$	0	0
$259$	6.92820i	0.430498i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	18.0000	1.10993	0.554964	−	0.831875i	$-0.312732\pi$
$263$	18.0000	1.10993	0.554964	+	0.831875i	$0.312732\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 6.92820i	− 0.422420i	−0.977441	−	0.211210i	$-0.932260\pi$
$269$	− 6.92820i	− 0.422420i	0.977441	−	0.211210i	$-0.0677404\pi$
$270$	0	0
$271$	6.92820i	0.420858i	0.977609	+	0.210429i	$0.0674861\pi$
$271$	6.92820i	0.420858i	−0.977609	+	0.210429i	$0.932514\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	21.0000	1.26635
$276$	0	0
$277$	−8.00000	−0.480673	−0.240337	−	0.970690i	$-0.577258\pi$
$277$	−8.00000	−0.480673	−0.240337	+	0.970690i	$0.577258\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	6.92820i	0.413302i	0.978415	+	0.206651i	$0.0662565\pi$
$281$	6.92820i	0.413302i	−0.978415	+	0.206651i	$0.933744\pi$
$282$	0	0
$283$	− 10.3923i	− 0.617758i	−0.951101	−	0.308879i	$-0.900046\pi$
$283$	− 10.3923i	− 0.617758i	0.951101	−	0.308879i	$-0.0999539\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.0000	−1.06251
$288$	0	0
$289$	14.0000	0.823529
$290$	0	0
$291$	0	0
$292$	0	0
$293$	3.46410i	0.202375i	0.994867	+	0.101187i	$0.0322642\pi$
$293$	3.46410i	0.202375i	−0.994867	+	0.101187i	$0.967736\pi$
$294$	0	0
$295$	− 51.9615i	− 3.02532i
$296$	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
$305$	27.7128i	1.58683i
$306$	0	0
$307$	25.9808i	1.48280i	0.671063	+	0.741400i	$0.265838\pi$
$307$	25.9808i	1.48280i	−0.671063	+	0.741400i	$0.734162\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	6.00000	0.340229	0.170114	−	0.985424i	$-0.445586\pi$
$311$	6.00000	0.340229	0.170114	+	0.985424i	$0.445586\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 6.92820i	− 0.389127i	−0.980890	−	0.194563i	$-0.937671\pi$
$317$	− 6.92820i	− 0.389127i	0.980890	−	0.194563i	$-0.0623290\pi$
$318$	0	0
$319$	10.3923i	0.581857i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.00000	−0.166924
$324$	0	0
$325$	−28.0000	−1.55316
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 41.5692i	− 2.29179i
$330$	0	0
$331$	24.2487i	1.33283i	0.745581	+	0.666415i	$0.232172\pi$
$331$	24.2487i	1.33283i	−0.745581	+	0.666415i	$0.767828\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	30.0000	1.63908
$336$	0	0
$337$	−11.0000	−0.599208	−0.299604	−	0.954064i	$-0.596855\pi$
$337$	−11.0000	−0.599208	−0.299604	+	0.954064i	$0.596855\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	6.92820i	0.374088i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−27.0000	−1.44944	−0.724718	−	0.689046i	$-0.758030\pi$
$347$	−27.0000	−1.44944	−0.724718	+	0.689046i	$0.758030\pi$
$348$	0	0
$349$	32.0000	1.71292	0.856460	−	0.516213i	$-0.172659\pi$
$349$	32.0000	1.71292	0.856460	+	0.516213i	$0.172659\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.9808i	1.38282i	0.722464	+	0.691408i	$0.243009\pi$
$353$	25.9808i	1.38282i	−0.722464	+	0.691408i	$0.756991\pi$
$354$	0	0
$355$	20.7846i	1.10313i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	0	0
$361$	16.0000	0.842105
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 38.1051i	− 1.99451i
$366$	0	0
$367$	3.46410i	0.180825i	0.995904	+	0.0904123i	$0.0288185\pi$
$367$	3.46410i	0.180825i	−0.995904	+	0.0904123i	$0.971182\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	0	0
$372$	0	0
$373$	20.0000	1.03556	0.517780	−	0.855514i	$-0.326758\pi$
$373$	20.0000	1.03556	0.517780	+	0.855514i	$0.326758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	− 13.8564i	− 0.713641i
$378$	0	0
$379$	− 19.0526i	− 0.978664i	−0.872098	−	0.489332i	$-0.837241\pi$
$379$	− 19.0526i	− 0.978664i	0.872098	−	0.489332i	$-0.162759\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	18.0000	0.919757	0.459879	−	0.887982i	$-0.347893\pi$
$383$	18.0000	0.919757	0.459879	+	0.887982i	$0.347893\pi$
$384$	0	0
$385$	36.0000	1.83473
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 10.3923i	− 0.526911i	−0.964672	−	0.263455i	$-0.915138\pi$
$389$	− 10.3923i	− 0.526911i	0.964672	−	0.263455i	$-0.0848622\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.0000	0.603786
$396$	0	0
$397$	14.0000	0.702640	0.351320	−	0.936255i	$-0.385733\pi$
$397$	14.0000	0.702640	0.351320	+	0.936255i	$0.385733\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	8.66025i	0.432472i	0.976341	+	0.216236i	$0.0693781\pi$
$401$	8.66025i	0.432472i	−0.976341	+	0.216236i	$0.930622\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6.00000	−0.297409
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 51.9615i	− 2.55686i
$414$	0	0
$415$	41.5692i	2.04055i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.0000	−0.586238	−0.293119	−	0.956076i	$-0.594693\pi$
$419$	−12.0000	−0.586238	−0.293119	+	0.956076i	$0.594693\pi$
$420$	0	0
$421$	10.0000	0.487370	0.243685	−	0.969854i	$-0.421644\pi$
$421$	10.0000	0.487370	0.243685	+	0.969854i	$0.421644\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 12.1244i	− 0.588118i
$426$	0	0
$427$	27.7128i	1.34112i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−18.0000	−0.867029	−0.433515	−	0.901146i	$-0.642727\pi$
$431$	−18.0000	−0.867029	−0.433515	+	0.901146i	$0.642727\pi$
$432$	0	0
$433$	−31.0000	−1.48976	−0.744882	−	0.667196i	$-0.767494\pi$
$433$	−31.0000	−1.48976	−0.744882	+	0.667196i	$0.767494\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	20.7846i	0.991995i	0.868324	+	0.495998i	$0.165198\pi$
$439$	20.7846i	0.991995i	−0.868324	+	0.495998i	$0.834802\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−9.00000	−0.427603	−0.213801	−	0.976877i	$-0.568585\pi$
$443$	−9.00000	−0.427603	−0.213801	+	0.976877i	$0.568585\pi$
$444$	0	0
$445$	−48.0000	−2.27542
$446$	0	0
$447$	0	0
$448$	0	0
$449$	25.9808i	1.22611i	0.790041	+	0.613054i	$0.210059\pi$
$449$	25.9808i	1.22611i	−0.790041	+	0.613054i	$0.789941\pi$
$450$	0	0
$451$	− 15.5885i	− 0.734032i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−48.0000	−2.25027
$456$	0	0
$457$	−11.0000	−0.514558	−0.257279	−	0.966337i	$-0.582826\pi$
$457$	−11.0000	−0.514558	−0.257279	+	0.966337i	$0.582826\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.7846i	0.968036i	0.875058	+	0.484018i	$0.160823\pi$
$461$	20.7846i	0.968036i	−0.875058	+	0.484018i	$0.839177\pi$
$462$	0	0
$463$	− 17.3205i	− 0.804952i	−0.915430	−	0.402476i	$-0.868150\pi$
$463$	− 17.3205i	− 0.804952i	0.915430	−	0.402476i	$-0.131850\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−3.00000	−0.138823	−0.0694117	−	0.997588i	$-0.522112\pi$
$467$	−3.00000	−0.138823	−0.0694117	+	0.997588i	$0.522112\pi$
$468$	0	0
$469$	30.0000	1.38527
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.5885i	0.716758i
$474$	0	0
$475$	− 12.1244i	− 0.556304i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	30.0000	1.37073	0.685367	−	0.728197i	$-0.259642\pi$
$479$	30.0000	1.37073	0.685367	+	0.728197i	$0.259642\pi$
$480$	0	0
$481$	8.00000	0.364769
$482$	0	0
$483$	0	0
$484$	0	0
$485$	45.0333i	2.04486i
$486$	0	0
$487$	− 17.3205i	− 0.784867i	−0.919780	−	0.392434i	$-0.871633\pi$
$487$	− 17.3205i	− 0.784867i	0.919780	−	0.392434i	$-0.128367\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.0000	0.676941	0.338470	−	0.940977i	$-0.390091\pi$
$491$	15.0000	0.676941	0.338470	+	0.940977i	$0.390091\pi$
$492$	0	0
$493$	6.00000	0.270226
$494$	0	0
$495$	0	0
$496$	0	0
$497$	20.7846i	0.932317i
$498$	0	0
$499$	− 15.5885i	− 0.697835i	−0.937153	−	0.348918i	$-0.886549\pi$
$499$	− 15.5885i	− 0.697835i	0.937153	−	0.348918i	$-0.113451\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.0000	−1.07011	−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000	−1.07011	−0.535054	+	0.844818i	$0.679709\pi$
$504$	0	0
$505$	−36.0000	−1.60198
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 20.7846i	− 0.921262i	−0.887592	−	0.460631i	$-0.847623\pi$
$509$	− 20.7846i	− 0.921262i	0.887592	−	0.460631i	$-0.152377\pi$
$510$	0	0
$511$	− 38.1051i	− 1.68567i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−48.0000	−2.11513
$516$	0	0
$517$	36.0000	1.58328
$518$	0	0
$519$	0	0
$520$	0	0
$521$	− 15.5885i	− 0.682943i	−0.939892	−	0.341471i	$-0.889075\pi$
$521$	− 15.5885i	− 0.682943i	0.939892	−	0.341471i	$-0.110925\pi$
$522$	0	0
$523$	17.3205i	0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$	17.3205i	0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−23.0000	−1.00000
$530$	0	0
$531$	0	0
$532$	0	0
$533$	20.7846i	0.900281i
$534$	0	0
$535$	− 10.3923i	− 0.449299i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	15.0000	0.646096
$540$	0	0
$541$	−2.00000	−0.0859867	−0.0429934	−	0.999075i	$-0.513689\pi$
$541$	−2.00000	−0.0859867	−0.0429934	+	0.999075i	$0.513689\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	13.8564i	0.593543i
$546$	0	0
$547$	8.66025i	0.370286i	0.982712	+	0.185143i	$0.0592748\pi$
$547$	8.66025i	0.370286i	−0.982712	+	0.185143i	$0.940725\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.00000	0.255609
$552$	0	0
$553$	12.0000	0.510292
$554$	0	0
$555$	0	0
$556$	0	0
$557$	38.1051i	1.61457i	0.590165	+	0.807283i	$0.299063\pi$
$557$	38.1051i	1.61457i	−0.590165	+	0.807283i	$0.700937\pi$
$558$	0	0
$559$	− 20.7846i	− 0.879095i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.00000	0.379305	0.189652	−	0.981851i	$-0.439264\pi$
$563$	9.00000	0.379305	0.189652	+	0.981851i	$0.439264\pi$
$564$	0	0
$565$	−24.0000	−1.00969
$566$	0	0
$567$	0	0
$568$	0	0
$569$	− 19.0526i	− 0.798725i	−0.916793	−	0.399362i	$-0.869232\pi$
$569$	− 19.0526i	− 0.798725i	0.916793	−	0.399362i	$-0.130768\pi$
$570$	0	0
$571$	12.1244i	0.507388i	0.967284	+	0.253694i	$0.0816457\pi$
$571$	12.1244i	0.507388i	−0.967284	+	0.253694i	$0.918354\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−7.00000	−0.291414	−0.145707	−	0.989328i	$-0.546546\pi$
$577$	−7.00000	−0.291414	−0.145707	+	0.989328i	$0.546546\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	41.5692i	1.72458i
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−3.00000	−0.123823	−0.0619116	−	0.998082i	$-0.519720\pi$
$587$	−3.00000	−0.123823	−0.0619116	+	0.998082i	$0.519720\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 6.92820i	− 0.284507i	−0.989830	−	0.142254i	$-0.954565\pi$
$593$	− 6.92820i	− 0.284507i	0.989830	−	0.142254i	$-0.0454349\pi$
$594$	0	0
$595$	− 20.7846i	− 0.852086i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	18.0000	0.735460	0.367730	−	0.929933i	$-0.380135\pi$
$599$	18.0000	0.735460	0.367730	+	0.929933i	$0.380135\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 6.92820i	− 0.281672i
$606$	0	0
$607$	− 27.7128i	− 1.12483i	−0.826856	−	0.562414i	$-0.809873\pi$
$607$	− 27.7128i	− 1.12483i	0.826856	−	0.562414i	$-0.190127\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−48.0000	−1.94187
$612$	0	0
$613$	26.0000	1.05013	0.525065	−	0.851062i	$-0.324041\pi$
$613$	26.0000	1.05013	0.525065	+	0.851062i	$0.324041\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	29.4449i	1.18541i	0.805421	+	0.592703i	$0.201939\pi$
$617$	29.4449i	1.18541i	−0.805421	+	0.592703i	$0.798061\pi$
$618$	0	0
$619$	− 29.4449i	− 1.18349i	−0.806126	−	0.591744i	$-0.798439\pi$
$619$	− 29.4449i	− 1.18349i	0.806126	−	0.591744i	$-0.201561\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−48.0000	−1.92308
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	3.46410i	0.138123i
$630$	0	0
$631$	− 3.46410i	− 0.137904i	−0.997620	−	0.0689519i	$-0.978035\pi$
$631$	− 3.46410i	− 0.137904i	0.997620	−	0.0689519i	$-0.0219655\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−20.0000	−0.792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	− 43.3013i	− 1.71030i	−0.518383	−	0.855149i	$-0.673466\pi$
$641$	− 43.3013i	− 1.71030i	0.518383	−	0.855149i	$-0.326534\pi$
$642$	0	0
$643$	− 25.9808i	− 1.02458i	−0.858812	−	0.512291i	$-0.828797\pi$
$643$	− 25.9808i	− 1.02458i	0.858812	−	0.512291i	$-0.171203\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−42.0000	−1.65119	−0.825595	−	0.564263i	$-0.809160\pi$
$647$	−42.0000	−1.65119	−0.825595	+	0.564263i	$0.809160\pi$
$648$	0	0
$649$	45.0000	1.76640
$650$	0	0
$651$	0	0
$652$	0	0
$653$	31.1769i	1.22005i	0.792383	+	0.610023i	$0.208840\pi$
$653$	31.1769i	1.22005i	−0.792383	+	0.610023i	$0.791160\pi$
$654$	0	0
$655$	41.5692i	1.62424i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	24.0000	0.934907	0.467454	−	0.884018i	$-0.345171\pi$
$659$	24.0000	0.934907	0.467454	+	0.884018i	$0.345171\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	− 20.7846i	− 0.805993i
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−24.0000	−0.926510
$672$	0	0
$673$	−46.0000	−1.77317	−0.886585	−	0.462566i	$-0.846929\pi$
$673$	−46.0000	−1.77317	−0.886585	+	0.462566i	$0.846929\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	20.7846i	0.798817i	0.916773	+	0.399409i	$0.130785\pi$
$677$	20.7846i	0.798817i	−0.916773	+	0.399409i	$0.869215\pi$
$678$	0	0
$679$	45.0333i	1.72822i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	15.0000	0.573959	0.286980	−	0.957937i	$-0.407349\pi$
$683$	15.0000	0.573959	0.286980	+	0.957937i	$0.407349\pi$
$684$	0	0
$685$	6.00000	0.229248
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	31.1769i	1.18603i	0.805193	+	0.593013i	$0.202062\pi$
$691$	31.1769i	1.18603i	−0.805193	+	0.593013i	$0.797938\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−66.0000	−2.50352
$696$	0	0
$697$	−9.00000	−0.340899
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 27.7128i	− 1.04670i	−0.852118	−	0.523349i	$-0.824682\pi$
$701$	− 27.7128i	− 1.04670i	0.852118	−	0.523349i	$-0.175318\pi$
$702$	0	0
$703$	3.46410i	0.130651i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−36.0000	−1.35392
$708$	0	0
$709$	28.0000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$709$	28.0000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	− 41.5692i	− 1.55460i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	42.0000	1.56634	0.783168	−	0.621810i	$-0.213603\pi$
$719$	42.0000	1.56634	0.783168	+	0.621810i	$0.213603\pi$
$720$	0	0
$721$	−48.0000	−1.78761
$722$	0	0
$723$	0	0
$724$	0	0
$725$	24.2487i	0.900575i
$726$	0	0
$727$	41.5692i	1.54172i	0.637006	+	0.770859i	$0.280172\pi$
$727$	41.5692i	1.54172i	−0.637006	+	0.770859i	$0.719828\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	9.00000	0.332877
$732$	0	0
$733$	22.0000	0.812589	0.406294	−	0.913742i	$-0.366821\pi$
$733$	22.0000	0.812589	0.406294	+	0.913742i	$0.366821\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	25.9808i	0.957014i
$738$	0	0
$739$	25.9808i	0.955718i	0.878437	+	0.477859i	$0.158587\pi$
$739$	25.9808i	0.955718i	−0.878437	+	0.477859i	$0.841413\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−18.0000	−0.660356	−0.330178	−	0.943919i	$-0.607109\pi$
$743$	−18.0000	−0.660356	−0.330178	+	0.943919i	$0.607109\pi$
$744$	0	0
$745$	48.0000	1.75858
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 10.3923i	− 0.379727i
$750$	0	0
$751$	38.1051i	1.39048i	0.718780	+	0.695238i	$0.244701\pi$
$751$	38.1051i	1.39048i	−0.718780	+	0.695238i	$0.755299\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−24.0000	−0.873449
$756$	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
$761$	− 48.4974i	− 1.75803i	−0.476794	−	0.879015i	$-0.658201\pi$
$761$	− 48.4974i	− 1.75803i	0.476794	−	0.879015i	$-0.341799\pi$
$762$	0	0
$763$	13.8564i	0.501636i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−60.0000	−2.16647
$768$	0	0
$769$	14.0000	0.504853	0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000	0.504853	0.252426	+	0.967616i	$0.418771\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	6.92820i	0.249190i	0.992208	+	0.124595i	$0.0397632\pi$
$773$	6.92820i	0.249190i	−0.992208	+	0.124595i	$0.960237\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−9.00000	−0.322458
$780$	0	0
$781$	−18.0000	−0.644091
$782$	0	0
$783$	0	0
$784$	0	0
$785$	27.7128i	0.989113i
$786$	0	0
$787$	17.3205i	0.617409i	0.951158	+	0.308705i	$0.0998955\pi$
$787$	17.3205i	0.617409i	−0.951158	+	0.308705i	$0.900105\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−24.0000	−0.853342
$792$	0	0
$793$	32.0000	1.13635
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 3.46410i	− 0.122705i	−0.998116	−	0.0613524i	$-0.980459\pi$
$797$	− 3.46410i	− 0.122705i	0.998116	−	0.0613524i	$-0.0195413\pi$
$798$	0	0
$799$	− 20.7846i	− 0.735307i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	33.0000	1.16454
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	19.0526i	0.669852i	0.942244	+	0.334926i	$0.108711\pi$
$809$	19.0526i	0.669852i	−0.942244	+	0.334926i	$0.891289\pi$
$810$	0	0
$811$	36.3731i	1.27723i	0.769526	+	0.638616i	$0.220493\pi$
$811$	36.3731i	1.27723i	−0.769526	+	0.638616i	$0.779507\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.0000	−0.420342
$816$	0	0
$817$	9.00000	0.314870
$818$	0	0
$819$	0	0
$820$	0	0
$821$	3.46410i	0.120898i	0.998171	+	0.0604490i	$0.0192532\pi$
$821$	3.46410i	0.120898i	−0.998171	+	0.0604490i	$0.980747\pi$
$822$	0	0
$823$	− 24.2487i	− 0.845257i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	− 24.2487i	− 0.845257i	0.906303	−	0.422628i	$-0.138892\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−36.0000	−1.25184	−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000	−1.25184	−0.625921	+	0.779886i	$0.715277\pi$
$828$	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
$833$	− 8.66025i	− 0.300060i
$834$	0	0
$835$	− 20.7846i	− 0.719281i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	36.0000	1.24286	0.621429	−	0.783470i	$-0.286552\pi$
$839$	36.0000	1.24286	0.621429	+	0.783470i	$0.286552\pi$
$840$	0	0
$841$	17.0000	0.586207
$842$	0	0
$843$	0	0
$844$	0	0
$845$	10.3923i	0.357506i
$846$	0	0
$847$	− 6.92820i	− 0.238056i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	0	0
$852$	0	0
$853$	32.0000	1.09566	0.547830	−	0.836590i	$-0.315454\pi$
$853$	32.0000	1.09566	0.547830	+	0.836590i	$0.315454\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	34.6410i	1.18331i	0.806190	+	0.591657i	$0.201526\pi$
$857$	34.6410i	1.18331i	−0.806190	+	0.591657i	$0.798474\pi$
$858$	0	0
$859$	29.4449i	1.00465i	0.864680	+	0.502323i	$0.167521\pi$
$859$	29.4449i	1.00465i	−0.864680	+	0.502323i	$0.832479\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	30.0000	1.02121	0.510606	−	0.859815i	$-0.329421\pi$
$863$	30.0000	1.02121	0.510606	+	0.859815i	$0.329421\pi$
$864$	0	0
$865$	84.0000	2.85609
$866$	0	0
$867$	0	0
$868$	0	0
$869$	10.3923i	0.352535i
$870$	0	0
$871$	− 34.6410i	− 1.17377i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	24.0000	0.811348
$876$	0	0
$877$	−40.0000	−1.35070	−0.675352	−	0.737496i	$-0.736008\pi$
$877$	−40.0000	−1.35070	−0.675352	+	0.737496i	$0.736008\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	− 20.7846i	− 0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$	− 20.7846i	− 0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$	0	0
$883$	8.66025i	0.291441i	0.989326	+	0.145720i	$0.0465500\pi$
$883$	8.66025i	0.291441i	−0.989326	+	0.145720i	$0.953450\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	42.0000	1.41022	0.705111	−	0.709097i	$-0.250897\pi$
$887$	42.0000	1.41022	0.705111	+	0.709097i	$0.250897\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 20.7846i	− 0.695530i
$894$	0	0
$895$	41.5692i	1.38951i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 27.7128i	− 0.921205i
$906$	0	0
$907$	− 39.8372i	− 1.32277i	−0.750046	−	0.661386i	$-0.769969\pi$
$907$	− 39.8372i	− 1.32277i	0.750046	−	0.661386i	$-0.230031\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	0	0
$913$	−36.0000	−1.19143
$914$	0	0
$915$	0	0
$916$	0	0
$917$	41.5692i	1.37274i
$918$	0	0
$919$	− 41.5692i	− 1.37124i	−0.727959	−	0.685621i	$-0.759531\pi$
$919$	− 41.5692i	− 1.37124i	0.727959	−	0.685621i	$-0.240469\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.0000	0.789970
$924$	0	0
$925$	−14.0000	−0.460317
$926$	0	0
$927$	0	0
$928$	0	0
$929$	6.92820i	0.227307i	0.993520	+	0.113653i	$0.0362554\pi$
$929$	6.92820i	0.227307i	−0.993520	+	0.113653i	$0.963745\pi$
$930$	0	0
$931$	− 8.66025i	− 0.283828i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	18.0000	0.588663
$936$	0	0
$937$	−34.0000	−1.11073	−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000	−1.11073	−0.555366	+	0.831606i	$0.687422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	− 13.8564i	− 0.451706i	−0.974161	−	0.225853i	$-0.927483\pi$
$941$	− 13.8564i	− 0.451706i	0.974161	−	0.225853i	$-0.0725169\pi$
$942$	0	0
$943$	0	0
$944$	0	0
$945$	0	0
$946$	0	0
$947$	9.00000	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$947$	9.00000	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$948$	0	0
$949$	−44.0000	−1.42830
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 12.1244i	− 0.392746i	−0.980529	−	0.196373i	$-0.937084\pi$
$953$	− 12.1244i	− 0.392746i	0.980529	−	0.196373i	$-0.0629164\pi$
$954$	0	0
$955$	20.7846i	0.672574i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	6.00000	0.193750
$960$	0	0
$961$	31.0000	1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	79.6743i	2.56481i
$966$	0	0
$967$	− 24.2487i	− 0.779786i	−0.920860	−	0.389893i	$-0.872512\pi$
$967$	− 24.2487i	− 0.779786i	0.920860	−	0.389893i	$-0.127488\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.0000	0.385098	0.192549	−	0.981287i	$-0.438325\pi$
$971$	12.0000	0.385098	0.192549	+	0.981287i	$0.438325\pi$
$972$	0	0
$973$	−66.0000	−2.11586
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.5885i	0.498719i	0.968411	+	0.249359i	$0.0802201\pi$
$977$	15.5885i	0.498719i	−0.968411	+	0.249359i	$0.919780\pi$
$978$	0	0
$979$	− 41.5692i	− 1.32856i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−6.00000	−0.191370	−0.0956851	−	0.995412i	$-0.530504\pi$
$983$	−6.00000	−0.191370	−0.0956851	+	0.995412i	$0.530504\pi$
$984$	0	0
$985$	48.0000	1.52941
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 48.4974i	− 1.54057i	−0.637699	−	0.770286i	$-0.720114\pi$
$991$	− 48.4974i	− 1.54057i	0.637699	−	0.770286i	$-0.279886\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−12.0000	−0.380426
$996$	0	0
$997$	20.0000	0.633406	0.316703	−	0.948525i	$-0.397424\pi$
$997$	20.0000	0.633406	0.316703	+	0.948525i	$0.397424\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1296.2.c.b.1295.2		2
3.2	odd	2		1296.2.c.d.1295.1		2
4.3	odd	2		1296.2.c.d.1295.2		2
8.3	odd	2		5184.2.c.a.5183.1		2
8.5	even	2		5184.2.c.c.5183.1		2
9.2	odd	6		432.2.s.d.287.1		2
9.4	even	3		432.2.s.c.143.1		2
9.5	odd	6		144.2.s.a.47.1	✓	2
9.7	even	3		144.2.s.d.95.1	yes	2
12.11	even	2	inner	1296.2.c.b.1295.1		2
24.5	odd	2		5184.2.c.a.5183.2		2
24.11	even	2		5184.2.c.c.5183.2		2
36.7	odd	6		144.2.s.a.95.1	yes	2
36.11	even	6		432.2.s.c.287.1		2
36.23	even	6		144.2.s.d.47.1	yes	2
36.31	odd	6		432.2.s.d.143.1		2
72.5	odd	6		576.2.s.d.191.1		2
72.11	even	6		1728.2.s.a.1151.1		2
72.13	even	6		1728.2.s.a.575.1		2
72.29	odd	6		1728.2.s.b.1151.1		2
72.43	odd	6		576.2.s.d.383.1		2
72.59	even	6		576.2.s.a.191.1		2
72.61	even	6		576.2.s.a.383.1		2
72.67	odd	6		1728.2.s.b.575.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.a.47.1	✓	2	9.5	odd	6
144.2.s.a.95.1	yes	2	36.7	odd	6
144.2.s.d.47.1	yes	2	36.23	even	6
144.2.s.d.95.1	yes	2	9.7	even	3
432.2.s.c.143.1		2	9.4	even	3
432.2.s.c.287.1		2	36.11	even	6
432.2.s.d.143.1		2	36.31	odd	6
432.2.s.d.287.1		2	9.2	odd	6
576.2.s.a.191.1		2	72.59	even	6
576.2.s.a.383.1		2	72.61	even	6
576.2.s.d.191.1		2	72.5	odd	6
576.2.s.d.383.1		2	72.43	odd	6
1296.2.c.b.1295.1		2	12.11	even	2	inner
1296.2.c.b.1295.2		2	1.1	even	1	trivial
1296.2.c.d.1295.1		2	3.2	odd	2
1296.2.c.d.1295.2		2	4.3	odd	2
1728.2.s.a.575.1		2	72.13	even	6
1728.2.s.a.1151.1		2	72.11	even	6
1728.2.s.b.575.1		2	72.67	odd	6
1728.2.s.b.1151.1		2	72.29	odd	6
5184.2.c.a.5183.1		2	8.3	odd	2
5184.2.c.a.5183.2		2	24.5	odd	2
5184.2.c.c.5183.1		2	8.5	even	2
5184.2.c.c.5183.2		2	24.11	even	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:	\( N \)	\(=\)	\( 1296 = 2^{4} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1296.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.46410i q^{5} +3.46410i q^{7} -3.00000 q^{11} +4.00000 q^{13} +1.73205i q^{17} +1.73205i q^{19} -7.00000 q^{25} -3.46410i q^{29} -12.0000 q^{35} +2.00000 q^{37} +5.19615i q^{41} -5.19615i q^{43} -12.0000 q^{47} -5.00000 q^{49} -10.3923i q^{55} -15.0000 q^{59} +8.00000 q^{61} +13.8564i q^{65} -8.66025i q^{67} +6.00000 q^{71} -11.0000 q^{73} -10.3923i q^{77} -3.46410i q^{79} +12.0000 q^{83} -6.00000 q^{85} +13.8564i q^{89} +13.8564i q^{91} -6.00000 q^{95} +13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{11} + 8 q^{13} - 14 q^{25} - 24 q^{35} + 4 q^{37} - 24 q^{47} - 10 q^{49} - 30 q^{59} + 16 q^{61} + 12 q^{71} - 22 q^{73} + 24 q^{83} - 12 q^{85} - 12 q^{95} + 26 q^{97}+O(q^{100}) \) 2 * q - 6 * q^11 + 8 * q^13 - 14 * q^25 - 24 * q^35 + 4 * q^37 - 24 * q^47 - 10 * q^49 - 30 * q^59 + 16 * q^61 + 12 * q^71 - 22 * q^73 + 24 * q^83 - 12 * q^85 - 12 * q^95 + 26 * q^97

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