LMFDB - Embedded newform 3744.2.c.b.3457.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3744,2,Mod(3457,3744)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3744, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("3744.3457");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3744 = 2^{5} \cdot 3^{2} \cdot 13$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3744.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:	$29.8959905168$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} + 1$ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 416)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			3457.2
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	3744.3457
Dual form			3744.2.c.b.3457.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.00000i q^{5} -1.00000i q^{7} -4.00000i q^{11} +(-2.00000 + 3.00000i) q^{13} +5.00000 q^{17} -6.00000i q^{19} -6.00000 q^{23} -4.00000 q^{25} +4.00000 q^{29} +3.00000 q^{35} +3.00000i q^{37} -12.0000i q^{41} +3.00000 q^{43} -7.00000i q^{47} +6.00000 q^{49} +2.00000 q^{53} +12.0000 q^{55} +2.00000i q^{59} -12.0000 q^{61} +(-9.00000 - 6.00000i) q^{65} -4.00000i q^{67} -11.0000i q^{71} -6.00000i q^{73} -4.00000 q^{77} -6.00000 q^{79} +10.0000i q^{83} +15.0000i q^{85} +6.00000i q^{89} +(3.00000 + 2.00000i) q^{91} +18.0000 q^{95} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q - 4 q^{13} + 10 q^{17} - 12 q^{23} - 8 q^{25} + 8 q^{29} + 6 q^{35} + 6 q^{43} + 12 q^{49} + 4 q^{53} + 24 q^{55} - 24 q^{61} - 18 q^{65} - 8 q^{77} - 12 q^{79} + 6 q^{91} + 36 q^{95}+O(q^{100})$ 2 * q - 4 * q^13 + 10 * q^17 - 12 * q^23 - 8 * q^25 + 8 * q^29 + 6 * q^35 + 6 * q^43 + 12 * q^49 + 4 * q^53 + 24 * q^55 - 24 * q^61 - 18 * q^65 - 8 * q^77 - 12 * q^79 + 6 * q^91 + 36 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3744\mathbb{Z}\right)^\times$ .

$n$	$703$	$2017$	$2081$	$2341$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$7$		−	1.00000i		−	0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$		−	1.00000i		−	0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	4.00000i		−	1.20605i	−0.797724	−	0.603023i	$-0.793963\pi$
$11$		−	4.00000i		−	1.20605i	0.797724	−	0.603023i	$-0.206037\pi$
$12$		0			0
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$13$	−2.00000	+	3.00000i	−0.554700	+	0.832050i
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.00000			1.21268			0.606339	−	0.795206i	$-0.292637\pi$
$17$	5.00000			1.21268			0.606339	+	0.795206i	$0.292637\pi$
$18$		0			0
$19$		−	6.00000i		−	1.37649i	−0.725476	−	0.688247i	$-0.758380\pi$
$19$		−	6.00000i		−	1.37649i	0.725476	−	0.688247i	$-0.241620\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−6.00000			−1.25109			−0.625543	−	0.780189i	$-0.715123\pi$
$23$	−6.00000			−1.25109			−0.625543	+	0.780189i	$0.715123\pi$
$24$		0			0
$25$	−4.00000			−0.800000
$26$		0			0
$27$		0			0
$28$		0			0
$29$	4.00000			0.742781			0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000			0.742781			0.371391	+	0.928477i	$0.378881\pi$
$30$		0			0
$31$		0			0		1.00000			$0$
$31$		0			0		−1.00000			$\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.00000			0.507093
$36$		0			0
$37$			3.00000i			0.493197i	0.969118	+	0.246598i	$0.0793129\pi$
$37$			3.00000i			0.493197i	−0.969118	+	0.246598i	$0.920687\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	12.0000i		−	1.87409i	−0.349215	−	0.937043i	$-0.613552\pi$
$41$		−	12.0000i		−	1.87409i	0.349215	−	0.937043i	$-0.386448\pi$
$42$		0			0
$43$	3.00000			0.457496			0.228748	−	0.973486i	$-0.426537\pi$
$43$	3.00000			0.457496			0.228748	+	0.973486i	$0.426537\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	7.00000i		−	1.02105i	−0.859861	−	0.510527i	$-0.829450\pi$
$47$		−	7.00000i		−	1.02105i	0.859861	−	0.510527i	$-0.170550\pi$
$48$		0			0
$49$	6.00000			0.857143
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.00000			0.274721			0.137361	−	0.990521i	$-0.456138\pi$
$53$	2.00000			0.274721			0.137361	+	0.990521i	$0.456138\pi$
$54$		0			0
$55$	12.0000			1.61808
$56$		0			0
$57$		0			0
$58$		0			0
$59$			2.00000i			0.260378i	0.991489	+	0.130189i	$0.0415584\pi$
$59$			2.00000i			0.260378i	−0.991489	+	0.130189i	$0.958442\pi$
$60$		0			0
$61$	−12.0000			−1.53644			−0.768221	−	0.640184i	$-0.778858\pi$
$61$	−12.0000			−1.53644			−0.768221	+	0.640184i	$0.778858\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−9.00000	−	6.00000i	−1.11631	−	0.744208i
$66$		0			0
$67$		−	4.00000i		−	0.488678i	−0.969690	−	0.244339i	$-0.921429\pi$
$67$		−	4.00000i		−	0.488678i	0.969690	−	0.244339i	$-0.0785709\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	11.0000i		−	1.30546i	−0.757591	−	0.652730i	$-0.773624\pi$
$71$		−	11.0000i		−	1.30546i	0.757591	−	0.652730i	$-0.226376\pi$
$72$		0			0
$73$		−	6.00000i		−	0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$		−	6.00000i		−	0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.00000			−0.455842
$78$		0			0
$79$	−6.00000			−0.675053			−0.337526	−	0.941316i	$-0.609590\pi$
$79$	−6.00000			−0.675053			−0.337526	+	0.941316i	$0.609590\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			10.0000i			1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$			10.0000i			1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$		0			0
$85$			15.0000i			1.62698i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			6.00000i			0.635999i	0.948091	+	0.317999i	$0.103011\pi$
$89$			6.00000i			0.635999i	−0.948091	+	0.317999i	$0.896989\pi$
$90$		0			0
$91$	3.00000	+	2.00000i	0.314485	+	0.209657i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	18.0000			1.84676
$96$		0			0
$97$		0			0		1.00000			$0$
$97$		0			0		−1.00000			$\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	4.00000			0.398015			0.199007	−	0.979998i	$-0.436228\pi$
$101$	4.00000			0.398015			0.199007	+	0.979998i	$0.436228\pi$
$102$		0			0
$103$	6.00000			0.591198			0.295599	−	0.955312i	$-0.404481\pi$
$103$	6.00000			0.591198			0.295599	+	0.955312i	$0.404481\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.0000			1.16008			0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		0			0
$109$		−	15.0000i		−	1.43674i	−0.695662	−	0.718370i	$-0.744889\pi$
$109$		−	15.0000i		−	1.43674i	0.695662	−	0.718370i	$-0.255111\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	14.0000			1.31701			0.658505	−	0.752577i	$-0.271189\pi$
$113$	14.0000			1.31701			0.658505	+	0.752577i	$0.271189\pi$
$114$		0			0
$115$		−	18.0000i		−	1.67851i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	5.00000i		−	0.458349i
$120$		0			0
$121$	−5.00000			−0.454545
$122$		0			0
$123$		0			0
$124$		0			0
$125$			3.00000i			0.268328i
$126$		0			0
$127$	18.0000			1.59724			0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000			1.59724			0.798621	+	0.601834i	$0.205563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−9.00000			−0.786334			−0.393167	−	0.919467i	$-0.628621\pi$
$131$	−9.00000			−0.786334			−0.393167	+	0.919467i	$0.628621\pi$
$132$		0			0
$133$	−6.00000			−0.520266
$134$		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
$139$	9.00000			0.763370			0.381685	−	0.924292i	$-0.375344\pi$
$139$	9.00000			0.763370			0.381685	+	0.924292i	$0.375344\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	12.0000	+	8.00000i	1.00349	+	0.668994i
$144$		0			0
$145$			12.0000i			0.996546i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			6.00000i			0.491539i	0.969328	+	0.245770i	$0.0790407\pi$
$149$			6.00000i			0.491539i	−0.969328	+	0.245770i	$0.920959\pi$
$150$		0			0
$151$		−	3.00000i		−	0.244137i	−0.992522	−	0.122068i	$-0.961047\pi$
$151$		−	3.00000i		−	0.244137i	0.992522	−	0.122068i	$-0.0389527\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.00000i			0.472866i
$162$		0			0
$163$			18.0000i			1.40987i	0.709273	+	0.704934i	$0.249024\pi$
$163$			18.0000i			1.40987i	−0.709273	+	0.704934i	$0.750976\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			4.00000i			0.309529i	0.987951	+	0.154765i	$0.0494619\pi$
$167$			4.00000i			0.309529i	−0.987951	+	0.154765i	$0.950538\pi$
$168$		0			0
$169$	−5.00000	−	12.0000i	−0.384615	−	0.923077i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	14.0000			1.06440			0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000			1.06440			0.532200	+	0.846619i	$0.321365\pi$
$174$		0			0
$175$			4.00000i			0.302372i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	21.0000			1.56961			0.784807	−	0.619740i	$-0.212762\pi$
$179$	21.0000			1.56961			0.784807	+	0.619740i	$0.212762\pi$
$180$		0			0
$181$	−6.00000			−0.445976			−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000			−0.445976			−0.222988	+	0.974821i	$0.571581\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−9.00000			−0.661693
$186$		0			0
$187$		−	20.0000i		−	1.46254i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−24.0000			−1.73658			−0.868290	−	0.496058i	$-0.834780\pi$
$191$	−24.0000			−1.73658			−0.868290	+	0.496058i	$0.834780\pi$
$192$		0			0
$193$		−	18.0000i		−	1.29567i	−0.761781	−	0.647834i	$-0.775675\pi$
$193$		−	18.0000i		−	1.29567i	0.761781	−	0.647834i	$-0.224325\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	3.00000i		−	0.213741i	−0.994273	−	0.106871i	$-0.965917\pi$
$197$		−	3.00000i		−	0.213741i	0.994273	−	0.106871i	$-0.0340831\pi$
$198$		0			0
$199$	12.0000			0.850657			0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000			0.850657			0.425329	+	0.905039i	$0.360158\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	4.00000i		−	0.280745i
$204$		0			0
$205$	36.0000			2.51435
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−24.0000			−1.66011
$210$		0			0
$211$	−3.00000			−0.206529			−0.103264	−	0.994654i	$-0.532929\pi$
$211$	−3.00000			−0.206529			−0.103264	+	0.994654i	$0.532929\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$			9.00000i			0.613795i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−10.0000	+	15.0000i	−0.672673	+	1.00901i
$222$		0			0
$223$		−	21.0000i		−	1.40626i	−0.711059	−	0.703132i	$-0.751784\pi$
$223$		−	21.0000i		−	1.40626i	0.711059	−	0.703132i	$-0.248216\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			8.00000i			0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$			8.00000i			0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$		0			0
$229$			9.00000i			0.594737i	0.954763	+	0.297368i	$0.0961089\pi$
$229$			9.00000i			0.594737i	−0.954763	+	0.297368i	$0.903891\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−19.0000			−1.24473			−0.622366	−	0.782727i	$-0.713828\pi$
$233$	−19.0000			−1.24473			−0.622366	+	0.782727i	$0.713828\pi$
$234$		0			0
$235$	21.0000			1.36989
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	5.00000i		−	0.323423i	−0.986838	−	0.161712i	$-0.948299\pi$
$239$		−	5.00000i		−	0.323423i	0.986838	−	0.161712i	$-0.0517014\pi$
$240$		0			0
$241$		−	6.00000i		−	0.386494i	−0.981150	−	0.193247i	$-0.938098\pi$
$241$		−	6.00000i		−	0.386494i	0.981150	−	0.193247i	$-0.0619019\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$			18.0000i			1.14998i
$246$		0			0
$247$	18.0000	+	12.0000i	1.14531	+	0.763542i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$			24.0000i			1.50887i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.00000			0.311891			0.155946	−	0.987766i	$-0.450158\pi$
$257$	5.00000			0.311891			0.155946	+	0.987766i	$0.450158\pi$
$258$		0			0
$259$	3.00000			0.186411
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.0000			0.739952			0.369976	−	0.929041i	$-0.379366\pi$
$263$	12.0000			0.739952			0.369976	+	0.929041i	$0.379366\pi$
$264$		0			0
$265$			6.00000i			0.368577i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.00000			0.243884			0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000			0.243884			0.121942	+	0.992537i	$0.461088\pi$
$270$		0			0
$271$			11.0000i			0.668202i	0.942537	+	0.334101i	$0.108433\pi$
$271$			11.0000i			0.668202i	−0.942537	+	0.334101i	$0.891567\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$			16.0000i			0.964836i
$276$		0			0
$277$	12.0000			0.721010			0.360505	−	0.932757i	$-0.382604\pi$
$277$	12.0000			0.721010			0.360505	+	0.932757i	$0.382604\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	18.0000i		−	1.07379i	−0.843649	−	0.536895i	$-0.819597\pi$
$281$		−	18.0000i		−	1.07379i	0.843649	−	0.536895i	$-0.180403\pi$
$282$		0			0
$283$	−12.0000			−0.713326			−0.356663	−	0.934233i	$-0.616086\pi$
$283$	−12.0000			−0.713326			−0.356663	+	0.934233i	$0.616086\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−12.0000			−0.708338
$288$		0			0
$289$	8.00000			0.470588
$290$		0			0
$291$		0			0
$292$		0			0
$293$			9.00000i			0.525786i	0.964825	+	0.262893i	$0.0846766\pi$
$293$			9.00000i			0.525786i	−0.964825	+	0.262893i	$0.915323\pi$
$294$		0			0
$295$	−6.00000			−0.349334
$296$		0			0
$297$		0			0
$298$		0			0
$299$	12.0000	−	18.0000i	0.693978	−	1.04097i
$300$		0			0
$301$		−	3.00000i		−	0.172917i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	36.0000i		−	2.06135i
$306$		0			0
$307$		−	2.00000i		−	0.114146i	−0.998370	−	0.0570730i	$-0.981823\pi$
$307$		−	2.00000i		−	0.114146i	0.998370	−	0.0570730i	$-0.0181768\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−6.00000			−0.340229			−0.170114	−	0.985424i	$-0.554414\pi$
$311$	−6.00000			−0.340229			−0.170114	+	0.985424i	$0.554414\pi$
$312$		0			0
$313$	−21.0000			−1.18699			−0.593495	−	0.804838i	$-0.702252\pi$
$313$	−21.0000			−1.18699			−0.593495	+	0.804838i	$0.702252\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			6.00000i			0.336994i	0.985702	+	0.168497i	$0.0538913\pi$
$317$			6.00000i			0.336994i	−0.985702	+	0.168497i	$0.946109\pi$
$318$		0			0
$319$		−	16.0000i		−	0.895828i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	30.0000i		−	1.66924i
$324$		0			0
$325$	8.00000	−	12.0000i	0.443760	−	0.665640i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−7.00000			−0.385922
$330$		0			0
$331$		−	26.0000i		−	1.42909i	−0.699590	−	0.714545i	$-0.746634\pi$
$331$		−	26.0000i		−	1.42909i	0.699590	−	0.714545i	$-0.253366\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	12.0000			0.655630
$336$		0			0
$337$	13.0000			0.708155			0.354078	−	0.935216i	$-0.384795\pi$
$337$	13.0000			0.708155			0.354078	+	0.935216i	$0.384795\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		−	13.0000i		−	0.701934i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−21.0000			−1.12734			−0.563670	−	0.826000i	$-0.690611\pi$
$347$	−21.0000			−1.12734			−0.563670	+	0.826000i	$0.690611\pi$
$348$		0			0
$349$		−	21.0000i		−	1.12410i	−0.827102	−	0.562052i	$-0.810012\pi$
$349$		−	21.0000i		−	1.12410i	0.827102	−	0.562052i	$-0.189988\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$			18.0000i			0.958043i	0.877803	+	0.479022i	$0.159008\pi$
$353$			18.0000i			0.958043i	−0.877803	+	0.479022i	$0.840992\pi$
$354$		0			0
$355$	33.0000			1.75146
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	32.0000i		−	1.68890i	−0.535638	−	0.844448i	$-0.679929\pi$
$359$		−	32.0000i		−	1.68890i	0.535638	−	0.844448i	$-0.320071\pi$
$360$		0			0
$361$	−17.0000			−0.894737
$362$		0			0
$363$		0			0
$364$		0			0
$365$	18.0000			0.942163
$366$		0			0
$367$		0			0			−	1.00000i	$-0.5\pi$
$367$		0			0				1.00000i	$0.5\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	2.00000i		−	0.103835i
$372$		0			0
$373$	12.0000			0.621336			0.310668	−	0.950518i	$-0.399447\pi$
$373$	12.0000			0.621336			0.310668	+	0.950518i	$0.399447\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−8.00000	+	12.0000i	−0.412021	+	0.618031i
$378$		0			0
$379$			30.0000i			1.54100i	0.637442	+	0.770498i	$0.279993\pi$
$379$			30.0000i			1.54100i	−0.637442	+	0.770498i	$0.720007\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	11.0000i		−	0.562074i	−0.959697	−	0.281037i	$-0.909322\pi$
$383$		−	11.0000i		−	0.562074i	0.959697	−	0.281037i	$-0.0906783\pi$
$384$		0			0
$385$		−	12.0000i		−	0.611577i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	2.00000			0.101404			0.0507020	−	0.998714i	$-0.483854\pi$
$389$	2.00000			0.101404			0.0507020	+	0.998714i	$0.483854\pi$
$390$		0			0
$391$	−30.0000			−1.51717
$392$		0			0
$393$		0			0
$394$		0			0
$395$		−	18.0000i		−	0.905678i
$396$		0			0
$397$		−	30.0000i		−	1.50566i	−0.658217	−	0.752828i	$-0.728689\pi$
$397$		−	30.0000i		−	1.50566i	0.658217	−	0.752828i	$-0.271311\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			6.00000i			0.299626i	0.988714	+	0.149813i	$0.0478671\pi$
$401$			6.00000i			0.299626i	−0.988714	+	0.149813i	$0.952133\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	12.0000			0.594818
$408$		0			0
$409$			6.00000i			0.296681i	0.988936	+	0.148340i	$0.0473931\pi$
$409$			6.00000i			0.296681i	−0.988936	+	0.148340i	$0.952607\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	2.00000			0.0984136
$414$		0			0
$415$	−30.0000			−1.47264
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−3.00000			−0.146560			−0.0732798	−	0.997311i	$-0.523347\pi$
$419$	−3.00000			−0.146560			−0.0732798	+	0.997311i	$0.523347\pi$
$420$		0			0
$421$			9.00000i			0.438633i	0.975654	+	0.219317i	$0.0703828\pi$
$421$			9.00000i			0.438633i	−0.975654	+	0.219317i	$0.929617\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−20.0000			−0.970143
$426$		0			0
$427$			12.0000i			0.580721i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			19.0000i			0.915198i	0.889159	+	0.457599i	$0.151290\pi$
$431$			19.0000i			0.915198i	−0.889159	+	0.457599i	$0.848710\pi$
$432$		0			0
$433$	−3.00000			−0.144171			−0.0720854	−	0.997398i	$-0.522965\pi$
$433$	−3.00000			−0.144171			−0.0720854	+	0.997398i	$0.522965\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			36.0000i			1.72211i
$438$		0			0
$439$	−6.00000			−0.286364			−0.143182	−	0.989696i	$-0.545733\pi$
$439$	−6.00000			−0.286364			−0.143182	+	0.989696i	$0.545733\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−39.0000			−1.85295			−0.926473	−	0.376361i	$-0.877175\pi$
$443$	−39.0000			−1.85295			−0.926473	+	0.376361i	$0.877175\pi$
$444$		0			0
$445$	−18.0000			−0.853282
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	12.0000i		−	0.566315i	−0.959073	−	0.283158i	$-0.908618\pi$
$449$		−	12.0000i		−	0.566315i	0.959073	−	0.283158i	$-0.0913819\pi$
$450$		0			0
$451$	−48.0000			−2.26023
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−6.00000	+	9.00000i	−0.281284	+	0.421927i
$456$		0			0
$457$			18.0000i			0.842004i	0.907060	+	0.421002i	$0.138322\pi$
$457$			18.0000i			0.842004i	−0.907060	+	0.421002i	$0.861678\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	33.0000i		−	1.53696i	−0.639872	−	0.768482i	$-0.721013\pi$
$461$		−	33.0000i		−	1.53696i	0.639872	−	0.768482i	$-0.278987\pi$
$462$		0			0
$463$		−	32.0000i		−	1.48717i	−0.668644	−	0.743583i	$-0.733125\pi$
$463$		−	32.0000i		−	1.48717i	0.668644	−	0.743583i	$-0.266875\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	−4.00000			−0.184703
$470$		0			0
$471$		0			0
$472$		0			0
$473$		−	12.0000i		−	0.551761i
$474$		0			0
$475$			24.0000i			1.10120i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			37.0000i			1.69057i	0.534313	+	0.845287i	$0.320570\pi$
$479$			37.0000i			1.69057i	−0.534313	+	0.845287i	$0.679430\pi$
$480$		0			0
$481$	−9.00000	−	6.00000i	−0.410365	−	0.273576i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$			12.0000i			0.543772i	0.962329	+	0.271886i	$0.0876473\pi$
$487$			12.0000i			0.543772i	−0.962329	+	0.271886i	$0.912353\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−15.0000			−0.676941			−0.338470	−	0.940977i	$-0.609909\pi$
$491$	−15.0000			−0.676941			−0.338470	+	0.940977i	$0.609909\pi$
$492$		0			0
$493$	20.0000			0.900755
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−11.0000			−0.493417
$498$		0			0
$499$		−	32.0000i		−	1.43252i	−0.697835	−	0.716258i	$-0.745853\pi$
$499$		−	32.0000i		−	1.43252i	0.697835	−	0.716258i	$-0.254147\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$			12.0000i			0.533993i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	18.0000i		−	0.797836i	−0.916987	−	0.398918i	$-0.869386\pi$
$509$		−	18.0000i		−	0.797836i	0.916987	−	0.398918i	$-0.130614\pi$
$510$		0			0
$511$	−6.00000			−0.265424
$512$		0			0
$513$		0			0
$514$		0			0
$515$			18.0000i			0.793175i
$516$		0			0
$517$	−28.0000			−1.23144
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−11.0000			−0.481919			−0.240959	−	0.970535i	$-0.577462\pi$
$521$	−11.0000			−0.481919			−0.240959	+	0.970535i	$0.577462\pi$
$522$		0			0
$523$	24.0000			1.04945			0.524723	−	0.851273i	$-0.324169\pi$
$523$	24.0000			1.04945			0.524723	+	0.851273i	$0.324169\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	13.0000			0.565217
$530$		0			0
$531$		0			0
$532$		0			0
$533$	36.0000	+	24.0000i	1.55933	+	1.03956i
$534$		0			0
$535$			36.0000i			1.55642i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	24.0000i		−	1.03375i
$540$		0			0
$541$			33.0000i			1.41878i	0.704816	+	0.709390i	$0.251030\pi$
$541$			33.0000i			1.41878i	−0.704816	+	0.709390i	$0.748970\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	45.0000			1.92759
$546$		0			0
$547$	−33.0000			−1.41098			−0.705489	−	0.708721i	$-0.749273\pi$
$547$	−33.0000			−1.41098			−0.705489	+	0.708721i	$0.749273\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	24.0000i		−	1.02243i
$552$		0			0
$553$			6.00000i			0.255146i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			3.00000i			0.127114i	0.997978	+	0.0635570i	$0.0202445\pi$
$557$			3.00000i			0.127114i	−0.997978	+	0.0635570i	$0.979756\pi$
$558$		0			0
$559$	−6.00000	+	9.00000i	−0.253773	+	0.380659i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−3.00000			−0.126435			−0.0632175	−	0.998000i	$-0.520136\pi$
$563$	−3.00000			−0.126435			−0.0632175	+	0.998000i	$0.520136\pi$
$564$		0			0
$565$			42.0000i			1.76695i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	11.0000			0.461144			0.230572	−	0.973055i	$-0.425940\pi$
$569$	11.0000			0.461144			0.230572	+	0.973055i	$0.425940\pi$
$570$		0			0
$571$	21.0000			0.878823			0.439411	−	0.898286i	$-0.355187\pi$
$571$	21.0000			0.878823			0.439411	+	0.898286i	$0.355187\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.0000			1.00087
$576$		0			0
$577$		−	6.00000i		−	0.249783i	−0.992170	−	0.124892i	$-0.960142\pi$
$577$		−	6.00000i		−	0.249783i	0.992170	−	0.124892i	$-0.0398583\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	10.0000			0.414870
$582$		0			0
$583$		−	8.00000i		−	0.331326i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			2.00000i			0.0825488i	0.999148	+	0.0412744i	$0.0131418\pi$
$587$			2.00000i			0.0825488i	−0.999148	+	0.0412744i	$0.986858\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$	15.0000			0.614940
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−30.0000			−1.22577			−0.612883	−	0.790173i	$-0.709990\pi$
$599$	−30.0000			−1.22577			−0.612883	+	0.790173i	$0.709990\pi$
$600$		0			0
$601$	−3.00000			−0.122373			−0.0611863	−	0.998126i	$-0.519488\pi$
$601$	−3.00000			−0.122373			−0.0611863	+	0.998126i	$0.519488\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$		−	15.0000i		−	0.609837i
$606$		0			0
$607$	30.0000			1.21766			0.608831	−	0.793300i	$-0.291639\pi$
$607$	30.0000			1.21766			0.608831	+	0.793300i	$0.291639\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	21.0000	+	14.0000i	0.849569	+	0.566379i
$612$		0			0
$613$			6.00000i			0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$			6.00000i			0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	36.0000i		−	1.44931i	−0.689114	−	0.724653i	$-0.742000\pi$
$617$		−	36.0000i		−	1.44931i	0.689114	−	0.724653i	$-0.258000\pi$
$618$		0			0
$619$			44.0000i			1.76851i	0.467005	+	0.884255i	$0.345333\pi$
$619$			44.0000i			1.76851i	−0.467005	+	0.884255i	$0.654667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000			0.240385
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			15.0000i			0.598089i
$630$		0			0
$631$			21.0000i			0.835997i	0.908448	+	0.417998i	$0.137268\pi$
$631$			21.0000i			0.835997i	−0.908448	+	0.417998i	$0.862732\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$			54.0000i			2.14292i
$636$		0			0
$637$	−12.0000	+	18.0000i	−0.475457	+	0.713186i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	10.0000			0.394976			0.197488	−	0.980305i	$-0.436722\pi$
$641$	10.0000			0.394976			0.197488	+	0.980305i	$0.436722\pi$
$642$		0			0
$643$			4.00000i			0.157745i	0.996885	+	0.0788723i	$0.0251319\pi$
$643$			4.00000i			0.157745i	−0.996885	+	0.0788723i	$0.974868\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0000			−0.707653			−0.353827	−	0.935311i	$-0.615120\pi$
$647$	−18.0000			−0.707653			−0.353827	+	0.935311i	$0.615120\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−28.0000			−1.09572			−0.547862	−	0.836569i	$-0.684558\pi$
$653$	−28.0000			−1.09572			−0.547862	+	0.836569i	$0.684558\pi$
$654$		0			0
$655$		−	27.0000i		−	1.05498i
$656$		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
$661$		−	42.0000i		−	1.63361i	−0.576913	−	0.816805i	$-0.695743\pi$
$661$		−	42.0000i		−	1.63361i	0.576913	−	0.816805i	$-0.304257\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	18.0000i		−	0.698010i
$666$		0			0
$667$	−24.0000			−0.929284
$668$		0			0
$669$		0			0
$670$		0			0
$671$			48.0000i			1.85302i
$672$		0			0
$673$	−1.00000			−0.0385472			−0.0192736	−	0.999814i	$-0.506135\pi$
$673$	−1.00000			−0.0385472			−0.0192736	+	0.999814i	$0.506135\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	14.0000			0.538064			0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000			0.538064			0.269032	+	0.963131i	$0.413296\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	14.0000i		−	0.535695i	−0.963461	−	0.267848i	$-0.913688\pi$
$683$		−	14.0000i		−	0.535695i	0.963461	−	0.267848i	$-0.0863124\pi$
$684$		0			0
$685$	18.0000			0.687745
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−4.00000	+	6.00000i	−0.152388	+	0.228582i
$690$		0			0
$691$			8.00000i			0.304334i	0.988355	+	0.152167i	$0.0486252\pi$
$691$			8.00000i			0.304334i	−0.988355	+	0.152167i	$0.951375\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			27.0000i			1.02417i
$696$		0			0
$697$		−	60.0000i		−	2.27266i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	26.0000			0.982006			0.491003	−	0.871158i	$-0.336630\pi$
$701$	26.0000			0.982006			0.491003	+	0.871158i	$0.336630\pi$
$702$		0			0
$703$	18.0000			0.678883
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	4.00000i		−	0.150435i
$708$		0			0
$709$			42.0000i			1.57734i	0.614815	+	0.788672i	$0.289231\pi$
$709$			42.0000i			1.57734i	−0.614815	+	0.788672i	$0.710769\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$	−24.0000	+	36.0000i	−0.897549	+	1.34632i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	48.0000			1.79010			0.895049	−	0.445968i	$-0.147140\pi$
$719$	48.0000			1.79010			0.895049	+	0.445968i	$0.147140\pi$
$720$		0			0
$721$		−	6.00000i		−	0.223452i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−16.0000			−0.594225
$726$		0			0
$727$	−24.0000			−0.890111			−0.445055	−	0.895503i	$-0.646816\pi$
$727$	−24.0000			−0.890111			−0.445055	+	0.895503i	$0.646816\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	15.0000			0.554795
$732$		0			0
$733$		−	33.0000i		−	1.21888i	−0.792831	−	0.609441i	$-0.791394\pi$
$733$		−	33.0000i		−	1.21888i	0.792831	−	0.609441i	$-0.208606\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−16.0000			−0.589368
$738$		0			0
$739$			16.0000i			0.588570i	0.955718	+	0.294285i	$0.0950814\pi$
$739$			16.0000i			0.588570i	−0.955718	+	0.294285i	$0.904919\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			11.0000i			0.403551i	0.979432	+	0.201775i	$0.0646711\pi$
$743$			11.0000i			0.403551i	−0.979432	+	0.201775i	$0.935329\pi$
$744$		0			0
$745$	−18.0000			−0.659469
$746$		0			0
$747$		0			0
$748$		0			0
$749$		−	12.0000i		−	0.438470i
$750$		0			0
$751$	−36.0000			−1.31366			−0.656829	−	0.754039i	$-0.728103\pi$
$751$	−36.0000			−1.31366			−0.656829	+	0.754039i	$0.728103\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.00000			0.327544
$756$		0			0
$757$	2.00000			0.0726912			0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000			0.0726912			0.0363456	+	0.999339i	$0.488428\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	30.0000i		−	1.08750i	−0.839248	−	0.543750i	$-0.817004\pi$
$761$		−	30.0000i		−	1.08750i	0.839248	−	0.543750i	$-0.182996\pi$
$762$		0			0
$763$	−15.0000			−0.543036
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−6.00000	−	4.00000i	−0.216647	−	0.144432i
$768$		0			0
$769$			48.0000i			1.73092i	0.500974	+	0.865462i	$0.332975\pi$
$769$			48.0000i			1.73092i	−0.500974	+	0.865462i	$0.667025\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$			27.0000i			0.971123i	0.874203	+	0.485561i	$0.161385\pi$
$773$			27.0000i			0.971123i	−0.874203	+	0.485561i	$0.838615\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−72.0000			−2.57967
$780$		0			0
$781$	−44.0000			−1.57444
$782$		0			0
$783$		0			0
$784$		0			0
$785$			42.0000i			1.49904i
$786$		0			0
$787$			32.0000i			1.14068i	0.821410	+	0.570338i	$0.193188\pi$
$787$			32.0000i			1.14068i	−0.821410	+	0.570338i	$0.806812\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	14.0000i		−	0.497783i
$792$		0			0
$793$	24.0000	−	36.0000i	0.852265	−	1.27840i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−50.0000			−1.77109			−0.885545	−	0.464553i	$-0.846215\pi$
$797$	−50.0000			−1.77109			−0.885545	+	0.464553i	$0.846215\pi$
$798$		0			0
$799$		−	35.0000i		−	1.23821i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−24.0000			−0.846942
$804$		0			0
$805$	−18.0000			−0.634417
$806$		0			0
$807$		0			0
$808$		0			0
$809$	25.0000			0.878953			0.439477	−	0.898254i	$-0.355164\pi$
$809$	25.0000			0.878953			0.439477	+	0.898254i	$0.355164\pi$
$810$		0			0
$811$			24.0000i			0.842754i	0.906886	+	0.421377i	$0.138453\pi$
$811$			24.0000i			0.842754i	−0.906886	+	0.421377i	$0.861547\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−54.0000			−1.89154
$816$		0			0
$817$		−	18.0000i		−	0.629740i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	45.0000i		−	1.57051i	−0.619172	−	0.785255i	$-0.712532\pi$
$821$		−	45.0000i		−	1.57051i	0.619172	−	0.785255i	$-0.287468\pi$
$822$		0			0
$823$	−30.0000			−1.04573			−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000			−1.04573			−0.522867	+	0.852414i	$0.675138\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			14.0000i			0.486828i	0.969923	+	0.243414i	$0.0782673\pi$
$827$			14.0000i			0.486828i	−0.969923	+	0.243414i	$0.921733\pi$
$828$		0			0
$829$	−52.0000			−1.80603			−0.903017	−	0.429604i	$-0.858653\pi$
$829$	−52.0000			−1.80603			−0.903017	+	0.429604i	$0.858653\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	30.0000			1.03944
$834$		0			0
$835$	−12.0000			−0.415277
$836$		0			0
$837$		0			0
$838$		0			0
$839$			56.0000i			1.93333i	0.256036	+	0.966667i	$0.417584\pi$
$839$			56.0000i			1.93333i	−0.256036	+	0.966667i	$0.582416\pi$
$840$		0			0
$841$	−13.0000			−0.448276
$842$		0			0
$843$		0			0
$844$		0			0
$845$	36.0000	−	15.0000i	1.23844	−	0.516016i
$846$		0			0
$847$			5.00000i			0.171802i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	18.0000i		−	0.617032i
$852$		0			0
$853$			33.0000i			1.12990i	0.825126	+	0.564949i	$0.191104\pi$
$853$			33.0000i			1.12990i	−0.825126	+	0.564949i	$0.808896\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−10.0000			−0.341593			−0.170797	−	0.985306i	$-0.554634\pi$
$857$	−10.0000			−0.341593			−0.170797	+	0.985306i	$0.554634\pi$
$858$		0			0
$859$	−48.0000			−1.63774			−0.818869	−	0.573980i	$-0.805399\pi$
$859$	−48.0000			−1.63774			−0.818869	+	0.573980i	$0.805399\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	29.0000i		−	0.987171i	−0.869697	−	0.493586i	$-0.835686\pi$
$863$		−	29.0000i		−	0.987171i	0.869697	−	0.493586i	$-0.164314\pi$
$864$		0			0
$865$			42.0000i			1.42804i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			24.0000i			0.814144i
$870$		0			0
$871$	12.0000	+	8.00000i	0.406604	+	0.271070i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.00000			0.101419
$876$		0			0
$877$			27.0000i			0.911725i	0.890050	+	0.455863i	$0.150669\pi$
$877$			27.0000i			0.911725i	−0.890050	+	0.455863i	$0.849331\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	41.0000			1.38133			0.690663	−	0.723177i	$-0.257319\pi$
$881$	41.0000			1.38133			0.690663	+	0.723177i	$0.257319\pi$
$882$		0			0
$883$	9.00000			0.302874			0.151437	−	0.988467i	$-0.451610\pi$
$883$	9.00000			0.302874			0.151437	+	0.988467i	$0.451610\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−24.0000			−0.805841			−0.402921	−	0.915235i	$-0.632005\pi$
$887$	−24.0000			−0.805841			−0.402921	+	0.915235i	$0.632005\pi$
$888$		0			0
$889$		−	18.0000i		−	0.603701i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−42.0000			−1.40548
$894$		0			0
$895$			63.0000i			2.10586i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$		0			0
$901$	10.0000			0.333148
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	18.0000i		−	0.598340i
$906$		0			0
$907$	−21.0000			−0.697294			−0.348647	−	0.937254i	$-0.613359\pi$
$907$	−21.0000			−0.697294			−0.348647	+	0.937254i	$0.613359\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0			−	1.00000i	$-0.5\pi$
$911$		0			0				1.00000i	$0.5\pi$
$912$		0			0
$913$	40.0000			1.32381
$914$		0			0
$915$		0			0
$916$		0			0
$917$			9.00000i			0.297206i
$918$		0			0
$919$	−36.0000			−1.18753			−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000			−1.18753			−0.593765	+	0.804638i	$0.702359\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	33.0000	+	22.0000i	1.08621	+	0.724139i
$924$		0			0
$925$		−	12.0000i		−	0.394558i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			36.0000i			1.18112i	0.806993	+	0.590561i	$0.201093\pi$
$929$			36.0000i			1.18112i	−0.806993	+	0.590561i	$0.798907\pi$
$930$		0			0
$931$		−	36.0000i		−	1.17985i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	60.0000			1.96221
$936$		0			0
$937$	54.0000			1.76410			0.882052	−	0.471153i	$-0.156162\pi$
$937$	54.0000			1.76410			0.882052	+	0.471153i	$0.156162\pi$
$938$		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
$943$			72.0000i			2.34464i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	20.0000i		−	0.649913i	−0.945729	−	0.324956i	$-0.894650\pi$
$947$		−	20.0000i		−	0.649913i	0.945729	−	0.324956i	$-0.105350\pi$
$948$		0			0
$949$	18.0000	+	12.0000i	0.584305	+	0.389536i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−41.0000			−1.32812			−0.664060	−	0.747679i	$-0.731168\pi$
$953$	−41.0000			−1.32812			−0.664060	+	0.747679i	$0.731168\pi$
$954$		0			0
$955$		−	72.0000i		−	2.32987i
$956$		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
$965$	54.0000			1.73832
$966$		0			0
$967$		−	33.0000i		−	1.06121i	−0.847620	−	0.530604i	$-0.821965\pi$
$967$		−	33.0000i		−	1.06121i	0.847620	−	0.530604i	$-0.178035\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−15.0000			−0.481373			−0.240686	−	0.970603i	$-0.577373\pi$
$971$	−15.0000			−0.481373			−0.240686	+	0.970603i	$0.577373\pi$
$972$		0			0
$973$		−	9.00000i		−	0.288527i
$974$		0			0
$975$		0			0
$976$		0			0
$977$		−	12.0000i		−	0.383914i	−0.981403	−	0.191957i	$-0.938517\pi$
$977$		−	12.0000i		−	0.383914i	0.981403	−	0.191957i	$-0.0614834\pi$
$978$		0			0
$979$	24.0000			0.767043
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	35.0000i		−	1.11633i	−0.829731	−	0.558163i	$-0.811506\pi$
$983$		−	35.0000i		−	1.11633i	0.829731	−	0.558163i	$-0.188494\pi$
$984$		0			0
$985$	9.00000			0.286764
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−18.0000			−0.572367
$990$		0			0
$991$	−54.0000			−1.71537			−0.857683	−	0.514178i	$-0.828097\pi$
$991$	−54.0000			−1.71537			−0.857683	+	0.514178i	$0.828097\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$			36.0000i			1.14128i
$996$		0			0
$997$	28.0000			0.886769			0.443384	−	0.896332i	$-0.353778\pi$
$997$	28.0000			0.886769			0.443384	+	0.896332i	$0.353778\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3744.2.c.b.3457.2		2
3.2	odd	2		416.2.f.c.129.1	yes	2
4.3	odd	2		3744.2.c.c.3457.2		2
12.11	even	2		416.2.f.a.129.1	✓	2
13.12	even	2	inner	3744.2.c.b.3457.1		2
24.5	odd	2		832.2.f.a.129.2		2
24.11	even	2		832.2.f.e.129.2		2
39.5	even	4		5408.2.a.l.1.1		1
39.8	even	4		5408.2.a.m.1.1		1
39.38	odd	2		416.2.f.c.129.2	yes	2
52.51	odd	2		3744.2.c.c.3457.1		2
156.47	odd	4		5408.2.a.b.1.1		1
156.83	odd	4		5408.2.a.a.1.1		1
156.155	even	2		416.2.f.a.129.2	yes	2
312.77	odd	2		832.2.f.a.129.1		2
312.155	even	2		832.2.f.e.129.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.2.f.a.129.1	✓	2	12.11	even	2
416.2.f.a.129.2	yes	2	156.155	even	2
416.2.f.c.129.1	yes	2	3.2	odd	2
416.2.f.c.129.2	yes	2	39.38	odd	2
832.2.f.a.129.1		2	312.77	odd	2
832.2.f.a.129.2		2	24.5	odd	2
832.2.f.e.129.1		2	312.155	even	2
832.2.f.e.129.2		2	24.11	even	2
3744.2.c.b.3457.1		2	13.12	even	2	inner
3744.2.c.b.3457.2		2	1.1	even	1	trivial
3744.2.c.c.3457.1		2	52.51	odd	2
3744.2.c.c.3457.2		2	4.3	odd	2
5408.2.a.a.1.1		1	156.83	odd	4
5408.2.a.b.1.1		1	156.47	odd	4
5408.2.a.l.1.1		1	39.5	even	4
5408.2.a.m.1.1		1	39.8	even	4

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Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3744.2.c.b.3457.2

Level	$3744$
Weight	$2$
Character	3744.3457
Analytic conductor	$29.896$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$