LMFDB - Embedded newform 3104.1.et.a.1327.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3104,1,Mod(335,3104)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3104, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 24, 29]))

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("3104.335");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$N$	$=$	$3104 = 2^{5} \cdot 97$
Weight:	$k$	$=$	$1$
Character orbit:	$[\chi]$	$=$	3104.et (of order $48$ , degree $16$ , not 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:	$1.54909779921$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\Q(\zeta_{48})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{16} - x^{8} + 1$ x^16 - x^8 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 776)
Projective image:	$D_{48}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{48} + \cdots)$

Embedding invariants

Embedding label			1327.1
Root			$0.991445 - 0.130526i$ of defining polynomial
Character	$\chi$	$=$	3104.1327
Dual form			3104.1.et.a.2159.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.207107 - 1.57313i) q^{3} +(-1.46593 - 0.392794i) q^{9} +(1.53264 + 1.17604i) q^{11} +(1.75928 + 0.867580i) q^{17} +(-0.923880 - 1.38268i) q^{19} +(0.991445 - 0.130526i) q^{25} +(-0.314313 + 0.758819i) q^{27} +(2.16748 - 2.16748i) q^{33} +(-0.835400 - 0.732626i) q^{41} +(0.252157 - 0.0675653i) q^{43} +(-0.608761 + 0.793353i) q^{49} +(1.72918 - 2.58790i) q^{51} +(-2.36649 + 1.16702i) q^{57} +(0.125419 - 0.369474i) q^{59} +(-1.25026 + 0.835400i) q^{67} +(-0.366025 - 1.36603i) q^{73} -1.58671i q^{75} +(-0.185687 - 0.107206i) q^{81} +(-0.837633 + 1.69855i) q^{83} +(1.83195 - 0.758819i) q^{89} +(-0.130526 + 0.991445i) q^{97} +(-1.78480 - 2.32599i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$16 q - 8 q^{3} - 8 q^{9} + 16 q^{27} + 8 q^{73} - 24 q^{81}+O(q^{100})$ 16 * q - 8 * q^3 - 8 * q^9 + 16 * q^27 + 8 * q^73 - 24 * q^81

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3104\mathbb{Z}\right)^\times$ .

$n$	$389$	$2721$	$2911$
$\chi(n)$	$-1$	$e\left(\frac{47}{48}\right)$	$-1$

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
$3$	0.207107	−	1.57313i	0.207107	−	1.57313i	−0.500000	−	0.866025i	$-0.666667\pi$
$3$	0.207107	−	1.57313i	0.207107	−	1.57313i	0.707107	−	0.707107i	$-0.250000\pi$
$4$		0			0
$5$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$5$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$6$		0			0
$7$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$7$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$8$		0			0
$9$	−1.46593	−	0.392794i	−1.46593	−	0.392794i
$10$		0			0
$11$	1.53264	+	1.17604i	1.53264	+	1.17604i	0.923880	+	0.382683i	$0.125000\pi$
$11$	1.53264	+	1.17604i	1.53264	+	1.17604i	0.608761	+	0.793353i	$0.291667\pi$
$12$		0			0
$13$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$13$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.75928	+	0.867580i	1.75928	+	0.867580i	0.965926	+	0.258819i	$0.0833333\pi$
$17$	1.75928	+	0.867580i	1.75928	+	0.867580i	0.793353	+	0.608761i	$0.208333\pi$
$18$		0			0
$19$	−0.923880	−	1.38268i	−0.923880	−	1.38268i	−0.923880	−	0.382683i	$-0.875000\pi$
$19$	−0.923880	−	1.38268i	−0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$23$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$24$		0			0
$25$	0.991445	−	0.130526i	0.991445	−	0.130526i
$26$		0			0
$27$	−0.314313	+	0.758819i	−0.314313	+	0.758819i
$28$		0			0
$29$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$29$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$30$		0			0
$31$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$31$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$32$		0			0
$33$	2.16748	−	2.16748i	2.16748	−	2.16748i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$37$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	0.130526	−	0.991445i	$-0.458333\pi$
$41$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	−0.965926	+	0.258819i	$0.916667\pi$
$42$		0			0
$43$	0.252157	−	0.0675653i	0.252157	−	0.0675653i	−0.130526	−	0.991445i	$-0.541667\pi$
$43$	0.252157	−	0.0675653i	0.252157	−	0.0675653i	0.382683	+	0.923880i	$0.375000\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$47$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$48$		0			0
$49$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$50$		0			0
$51$	1.72918	−	2.58790i	1.72918	−	2.58790i
$52$		0			0
$53$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$53$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.36649	+	1.16702i	−2.36649	+	1.16702i
$58$		0			0
$59$	0.125419	−	0.369474i	0.125419	−	0.369474i	−0.866025	−	0.500000i	$-0.833333\pi$
$59$	0.125419	−	0.369474i	0.125419	−	0.369474i	0.991445	+	0.130526i	$0.0416667\pi$
$60$		0			0
$61$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$61$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		0			0
$66$		0			0
$67$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.991445	−	0.130526i	$-0.958333\pi$
$67$	−1.25026	+	0.835400i	−1.25026	+	0.835400i	−0.258819	+	0.965926i	$0.583333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$71$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$72$		0			0
$73$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	−0.866025	−	0.500000i	$-0.833333\pi$
$73$	−0.366025	−	1.36603i	−0.366025	−	1.36603i	0.500000	−	0.866025i	$-0.333333\pi$
$74$		0			0
$75$		−	1.58671i		−	1.58671i
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$79$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$80$		0			0
$81$	−0.185687	−	0.107206i	−0.185687	−	0.107206i
$82$		0			0
$83$	−0.837633	+	1.69855i	−0.837633	+	1.69855i	−0.130526	+	0.991445i	$0.541667\pi$
$83$	−0.837633	+	1.69855i	−0.837633	+	1.69855i	−0.707107	+	0.707107i	$0.750000\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.866025	−	0.500000i	$-0.166667\pi$
$89$	1.83195	−	0.758819i	1.83195	−	0.758819i	0.965926	−	0.258819i	$-0.0833333\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$97$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$98$		0			0
$99$	−1.78480	−	2.32599i	−1.78480	−	2.32599i
$100$		0			0
$101$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$101$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$102$		0			0
$103$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$103$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−0.583242	−	0.665060i	−0.583242	−	0.665060i	0.382683	−	0.923880i	$-0.375000\pi$
$107$	−0.583242	−	0.665060i	−0.583242	−	0.665060i	−0.965926	+	0.258819i	$0.916667\pi$
$108$		0			0
$109$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$109$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.923880	−	0.382683i	$-0.875000\pi$
$113$	−1.71723	−	0.991445i	−1.71723	−	0.991445i	−0.793353	−	0.608761i	$-0.791667\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	0.707107	+	2.63896i	0.707107	+	2.63896i
$122$		0			0
$123$	−1.32553	+	1.16246i	−1.32553	+	1.16246i
$124$		0			0
$125$		0			0
$126$		0			0
$127$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$127$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$128$		0			0
$129$	−0.0540657	−	0.410670i	−0.0540657	−	0.410670i
$130$		0			0
$131$	0.172572	+	0.867580i	0.172572	+	0.867580i	0.965926	+	0.258819i	$0.0833333\pi$
$131$	0.172572	+	0.867580i	0.172572	+	0.867580i	−0.793353	+	0.608761i	$0.791667\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.117317	+	0.0578541i	−0.117317	+	0.0578541i	−0.500000	−	0.866025i	$-0.666667\pi$
$137$	−0.117317	+	0.0578541i	−0.117317	+	0.0578541i	0.382683	+	0.923880i	$0.375000\pi$
$138$		0			0
$139$	−0.108761	−	0.0726721i	−0.108761	−	0.0726721i	0.500000	−	0.866025i	$-0.333333\pi$
$139$	−0.108761	−	0.0726721i	−0.108761	−	0.0726721i	−0.608761	+	0.793353i	$0.708333\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$		0			0
$145$		0			0
$146$		0			0
$147$	1.12197	+	1.12197i	1.12197	+	1.12197i
$148$		0			0
$149$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$149$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$150$		0			0
$151$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$151$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$152$		0			0
$153$	−2.23819	−	1.96284i	−2.23819	−	1.96284i
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$157$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−0.991445	−	0.130526i	−0.991445	−	0.130526i	−0.382683	−	0.923880i	$-0.625000\pi$
$163$	−0.991445	−	0.130526i	−0.991445	−	0.130526i	−0.608761	+	0.793353i	$0.708333\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$167$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$168$		0			0
$169$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$170$		0			0
$171$	0.811230	+	2.38981i	0.811230	+	2.38981i
$172$		0			0
$173$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$173$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−0.555256	−	0.273822i	−0.555256	−	0.273822i
$178$		0			0
$179$	0.0255190	−	0.389345i	0.0255190	−	0.389345i	−0.965926	−	0.258819i	$-0.916667\pi$
$179$	0.0255190	−	0.389345i	0.0255190	−	0.389345i	0.991445	−	0.130526i	$-0.0416667\pi$
$180$		0			0
$181$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$181$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	1.67604	+	3.39867i	1.67604	+	3.39867i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$191$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$192$		0			0
$193$	1.00000			1.00000			0.500000	−	0.866025i	$-0.333333\pi$
$193$	1.00000			1.00000			0.500000	+	0.866025i	$0.333333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$197$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$198$		0			0
$199$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$199$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$200$		0			0
$201$	1.05526	+	2.13985i	1.05526	+	2.13985i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.210111	−	3.20567i	0.210111	−	3.20567i
$210$		0			0
$211$	1.18270	+	0.583242i	1.18270	+	0.583242i	0.923880	−	0.382683i	$-0.125000\pi$
$211$	1.18270	+	0.583242i	1.18270	+	0.583242i	0.258819	+	0.965926i	$0.416667\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−2.22474	+	0.292893i	−2.22474	+	0.292893i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$223$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$224$		0			0
$225$	−1.50465	−	0.198092i	−1.50465	−	0.198092i
$226$		0			0
$227$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$227$	1.22474	−	1.22474i	1.22474	−	1.22474i	0.965926	−	0.258819i	$-0.0833333\pi$
$228$		0			0
$229$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$229$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−0.641502	−	0.0420463i	−0.641502	−	0.0420463i	−0.258819	−	0.965926i	$-0.583333\pi$
$233$	−0.641502	−	0.0420463i	−0.641502	−	0.0420463i	−0.382683	+	0.923880i	$0.625000\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$239$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$240$		0			0
$241$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$241$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$242$		0			0
$243$	−0.707107	+	0.921519i	−0.707107	+	0.921519i
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$	2.49857	+	1.66949i	2.49857	+	1.66949i
$250$		0			0
$251$	0.996552	−	0.491445i	0.996552	−	0.491445i	0.130526	−	0.991445i	$-0.458333\pi$
$251$	0.996552	−	0.491445i	0.996552	−	0.491445i	0.866025	+	0.500000i	$0.166667\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.349942	−	1.75928i	−0.349942	−	1.75928i	−0.608761	−	0.793353i	$-0.708333\pi$
$257$	−0.349942	−	1.75928i	−0.349942	−	1.75928i	0.258819	−	0.965926i	$-0.416667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$263$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−0.814313	−	3.03906i	−0.814313	−	3.03906i
$268$		0			0
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$271$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	1.67303	+	0.965926i	1.67303	+	0.965926i
$276$		0			0
$277$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$277$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.991445	−	1.13053i	−0.991445	−	1.13053i	−0.991445	−	0.130526i	$-0.958333\pi$
$281$	−0.991445	−	1.13053i	−0.991445	−	1.13053i		−	1.00000i	$-0.5\pi$
$282$		0			0
$283$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$283$	−1.30656	+	0.541196i	−1.30656	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.73361	+	2.25928i	1.73361	+	2.25928i
$290$		0			0
$291$	1.53264	+	0.410670i	1.53264	+	0.410670i
$292$		0			0
$293$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$293$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−1.37413	+	0.793353i	−1.37413	+	0.793353i
$298$		0			0
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.923880	−	0.382683i	$-0.125000\pi$
$307$	1.05441	+	0.608761i	1.05441	+	0.608761i	0.130526	+	0.991445i	$0.458333\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$311$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$312$		0			0
$313$		−	0.261052i		−	0.261052i	−0.991445	−	0.130526i	$-0.958333\pi$
$313$		−	0.261052i		−	0.261052i	0.991445	−	0.130526i	$-0.0416667\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$317$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$	−1.16702	+	0.779779i	−1.16702	+	0.779779i
$322$		0			0
$323$	−0.425773	−	3.23407i	−0.425773	−	3.23407i
$324$		0			0
$325$		0			0
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.78990	+	0.882683i	−1.78990	+	0.882683i	−0.866025	+	0.500000i	$0.833333\pi$
$331$	−1.78990	+	0.882683i	−1.78990	+	0.882683i	−0.923880	+	0.382683i	$0.875000\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−0.357164	+	0.534534i	−0.357164	+	0.534534i	−0.965926	−	0.258819i	$-0.916667\pi$
$337$	−0.357164	+	0.534534i	−0.357164	+	0.534534i	0.608761	+	0.793353i	$0.291667\pi$
$338$		0			0
$339$	−1.91532	+	2.49610i	−1.91532	+	2.49610i
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	0.483342	+	0.423880i	0.483342	+	0.423880i	0.866025	−	0.500000i	$-0.166667\pi$
$347$	0.483342	+	0.423880i	0.483342	+	0.423880i	−0.382683	+	0.923880i	$0.625000\pi$
$348$		0			0
$349$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$349$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$359$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$360$		0			0
$361$	−0.675577	+	1.63099i	−0.675577	+	1.63099i
$362$		0			0
$363$	4.29788	−	0.565826i	4.29788	−	0.565826i
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$367$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$368$		0			0
$369$	0.936863	+	1.40211i	0.936863	+	1.40211i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$373$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.965926	−	0.258819i	$-0.0833333\pi$
$379$	1.67303	+	0.448288i	1.67303	+	0.448288i	0.707107	+	0.707107i	$0.250000\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$383$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−0.396183			−0.396183
$388$		0			0
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$	1.40056	−	0.0917975i	1.40056	−	0.0917975i
$394$		0			0
$395$		0			0
$396$		0			0
$397$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$397$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	0.130526	+	1.99144i	0.130526	+	1.99144i	0.130526	+	0.991445i	$0.458333\pi$
$401$	0.130526	+	1.99144i	0.130526	+	1.99144i			1.00000i	$0.5\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.79335	−	0.608761i	1.79335	−	0.608761i	0.793353	−	0.608761i	$-0.208333\pi$
$409$	1.79335	−	0.608761i	1.79335	−	0.608761i	1.00000			$0$
$410$		0			0
$411$	0.0667151	+	0.196536i	0.0667151	+	0.196536i
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−0.136848	+	0.156045i	−0.136848	+	0.156045i
$418$		0			0
$419$	1.20711	+	0.158919i	1.20711	+	0.158919i	0.707107	−	0.707107i	$-0.250000\pi$
$419$	1.20711	+	0.158919i	1.20711	+	0.158919i	0.500000	+	0.866025i	$0.333333\pi$
$420$		0			0
$421$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$421$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.85747	+	0.630526i	1.85747	+	0.630526i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$431$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$432$		0			0
$433$	−1.47479	−	0.293353i	−1.47479	−	0.293353i	−0.608761	−	0.793353i	$-0.708333\pi$
$433$	−1.47479	−	0.293353i	−1.47479	−	0.293353i	−0.866025	+	0.500000i	$0.833333\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$439$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$440$		0			0
$441$	1.20402	−	0.923880i	1.20402	−	0.923880i
$442$		0			0
$443$	1.65938	+	1.10876i	1.65938	+	1.10876i	0.866025	+	0.500000i	$0.166667\pi$
$443$	1.65938	+	1.10876i	1.65938	+	1.10876i	0.793353	+	0.608761i	$0.208333\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	0.382683	+	0.923880i	$0.375000\pi$
$449$	−0.608761	+	1.05441i	−0.608761	+	1.05441i	−0.991445	+	0.130526i	$0.958333\pi$
$450$		0			0
$451$	−0.418773	−	2.10531i	−0.418773	−	2.10531i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.75928	+	0.349942i	−1.75928	+	0.349942i	−0.965926	−	0.258819i	$-0.916667\pi$
$457$	−1.75928	+	0.349942i	−1.75928	+	0.349942i	−0.793353	+	0.608761i	$0.791667\pi$
$458$		0			0
$459$	−1.21130	+	1.06228i	−1.21130	+	1.06228i
$460$		0			0
$461$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$461$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$462$		0			0
$463$		0			0			−	1.00000i	$-0.5\pi$
$463$		0			0				1.00000i	$0.5\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	0.292893	+	0.707107i	0.292893	+	0.707107i	1.00000			$0$
$467$	0.292893	+	0.707107i	0.292893	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.465926	+	0.192993i	0.465926	+	0.192993i
$474$		0			0
$475$	−1.09645	−	1.25026i	−1.09645	−	1.25026i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$		0			0
$486$		0			0
$487$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$487$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$488$		0			0
$489$	−0.410670	+	1.53264i	−0.410670	+	1.53264i
$490$		0			0
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i	−0.866025	−	0.500000i	$-0.833333\pi$
$491$	−0.866025	+	0.500000i	−0.866025	+	0.500000i			1.00000i	$0.5\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−0.735499	+	1.49144i	−0.735499	+	1.49144i	0.130526	+	0.991445i	$0.458333\pi$
$499$	−0.735499	+	1.49144i	−0.735499	+	1.49144i	−0.866025	+	0.500000i	$0.833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$503$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$			1.58671i			1.58671i
$508$		0			0
$509$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$509$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	1.33959	−	0.266462i	1.33959	−	0.266462i
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	0.500000	+	0.866025i	$0.333333\pi$
$521$	−0.500000	+	0.866025i	−0.500000	+	0.866025i	−1.00000			$\pi$
$522$		0			0
$523$	0.206647	−	0.608761i	0.206647	−	0.608761i	−0.793353	−	0.608761i	$-0.791667\pi$
$523$	0.206647	−	0.608761i	0.206647	−	0.608761i	1.00000			$0$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$530$		0			0
$531$	−0.328983	+	0.492357i	−0.328983	+	0.492357i
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−0.607206	−	0.120781i	−0.607206	−	0.120781i
$538$		0			0
$539$	−1.86603	+	0.500000i	−1.86603	+	0.500000i
$540$		0			0
$541$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$541$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	0.793353	−	0.608761i	$-0.208333\pi$
$547$	−0.130526	−	0.226078i	−0.130526	−	0.226078i	−0.923880	+	0.382683i	$0.875000\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$557$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$	5.69367	−	1.93274i	5.69367	−	1.93274i
$562$		0			0
$563$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$563$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$		0			0
$569$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	0.793353	+	0.608761i	$0.208333\pi$
$569$	0.0862466	+	1.31587i	0.0862466	+	1.31587i	−0.707107	+	0.707107i	$0.750000\pi$
$570$		0			0
$571$	−0.965926	−	0.741181i	−0.965926	−	0.741181i		−	1.00000i	$-0.5\pi$
$571$	−0.965926	−	0.741181i	−0.965926	−	0.741181i	−0.965926	+	0.258819i	$0.916667\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−1.31587	+	0.0862466i	−1.31587	+	0.0862466i	−0.707107	−	0.707107i	$-0.750000\pi$
$577$	−1.31587	+	0.0862466i	−1.31587	+	0.0862466i	−0.608761	+	0.793353i	$0.708333\pi$
$578$		0			0
$579$	0.207107	−	1.57313i	0.207107	−	1.57313i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−1.99144	+	0.130526i	−1.99144	+	0.130526i	−0.991445	+	0.130526i	$0.958333\pi$
$587$	−1.99144	+	0.130526i	−1.99144	+	0.130526i	−1.00000			$1.00000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$	0.793353	+	0.608761i	0.793353	+	0.608761i	0.923880	−	0.382683i	$-0.125000\pi$
$593$	0.793353	+	0.608761i	0.793353	+	0.608761i	−0.130526	+	0.991445i	$0.541667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$599$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$600$		0			0
$601$	0.216773	+	0.324423i	0.216773	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$601$	0.216773	+	0.324423i	0.216773	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$	2.16093	−	0.733538i	2.16093	−	0.733538i
$604$		0			0
$605$		0			0
$606$		0			0
$607$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$607$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$613$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.793353	−	0.608761i	$-0.791667\pi$
$617$	−0.923880	−	1.60021i	−0.923880	−	1.60021i	−0.130526	−	0.991445i	$-0.541667\pi$
$618$		0			0
$619$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.793353	−	0.608761i	$-0.208333\pi$
$619$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.258819	+	0.965926i	$0.416667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.965926	−	0.258819i	0.965926	−	0.258819i
$626$		0			0
$627$	−4.99943	−	0.994449i	−4.99943	−	0.994449i
$628$		0			0
$629$		0			0
$630$		0			0
$631$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$631$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$632$		0			0
$633$	1.16246	−	1.73975i	1.16246	−	1.73975i
$634$		0			0
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i	−0.965926	−	0.258819i	$-0.916667\pi$
$641$	−0.0420463	+	0.123864i	−0.0420463	+	0.123864i	0.923880	+	0.382683i	$0.125000\pi$
$642$		0			0
$643$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$643$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$647$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$648$		0			0
$649$	0.626738	−	0.418773i	0.626738	−	0.418773i
$650$		0			0
$651$		0			0
$652$		0			0
$653$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$653$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			2.14626i			2.14626i
$658$		0			0
$659$	0.349942	−	1.75928i	0.349942	−	1.75928i	−0.258819	−	0.965926i	$-0.583333\pi$
$659$	0.349942	−	1.75928i	0.349942	−	1.75928i	0.608761	−	0.793353i	$-0.291667\pi$
$660$		0			0
$661$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$661$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.923880	−	0.382683i	$-0.875000\pi$
$673$	−1.05441	+	0.608761i	−1.05441	+	0.608761i	−0.130526	+	0.991445i	$0.541667\pi$
$674$		0			0
$675$	−0.212578	+	0.793353i	−0.212578	+	0.793353i
$676$		0			0
$677$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$677$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.67303	−	2.18034i	−1.67303	−	2.18034i
$682$		0			0
$683$	−0.315118	+	1.17604i	−0.315118	+	1.17604i	0.608761	+	0.793353i	$0.291667\pi$
$683$	−0.315118	+	1.17604i	−0.315118	+	1.17604i	−0.923880	+	0.382683i	$0.875000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	0.707107	+	0.292893i	0.707107	+	0.292893i	0.707107	−	0.707107i	$-0.250000\pi$
$691$	0.707107	+	0.292893i	0.707107	+	0.292893i			1.00000i	$0.5\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−0.834089	−	2.01367i	−0.834089	−	2.01367i
$698$		0			0
$699$	−0.199004	+	1.00046i	−0.199004	+	1.00046i
$700$		0			0
$701$		0			0			−	1.00000i	$-0.5\pi$
$701$		0			0				1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$709$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.896873	−	0.442289i	$-0.145833\pi$
$719$		0			0		−0.896873	+	0.442289i	$0.854167\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$		0			0
$726$		0			0
$727$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$727$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$728$		0			0
$729$	1.15161	+	1.15161i	1.15161	+	1.15161i
$730$		0			0
$731$	0.502233	+	0.0999004i	0.502233	+	0.0999004i
$732$		0			0
$733$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$733$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−2.89867	−	0.189989i	−2.89867	−	0.189989i
$738$		0			0
$739$	0.123864	+	0.0420463i	0.123864	+	0.0420463i	0.382683	−	0.923880i	$-0.375000\pi$
$739$	0.123864	+	0.0420463i	0.123864	+	0.0420463i	−0.258819	+	0.965926i	$0.583333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$743$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	1.89509	−	2.16093i	1.89509	−	2.16093i
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$751$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$752$		0			0
$753$	−0.566715	−	1.66949i	−0.566715	−	1.66949i
$754$		0			0
$755$		0			0
$756$		0			0
$757$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$757$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−0.128293	+	1.95737i	−0.128293	+	1.95737i	0.130526	+	0.991445i	$0.458333\pi$
$761$	−0.128293	+	1.95737i	−0.128293	+	1.95737i	−0.258819	+	0.965926i	$0.583333\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	−0.866025	−	0.500000i	$-0.833333\pi$
$769$	−0.735499	−	1.49144i	−0.735499	−	1.49144i	0.130526	−	0.991445i	$-0.458333\pi$
$770$		0			0
$771$	−2.84005	+	0.186147i	−2.84005	+	0.186147i
$772$		0			0
$773$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$773$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−0.241181	+	1.83195i	−0.241181	+	1.83195i
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.965926	−	0.258819i	$-0.916667\pi$
$787$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.500000	−	0.866025i	$-0.666667\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$797$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$	−2.98356	+	0.392794i	−2.98356	+	0.392794i
$802$		0			0
$803$	1.04551	−	2.52409i	1.04551	−	2.52409i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$809$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$810$		0			0
$811$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.608761	−	0.793353i	$-0.708333\pi$
$811$	−0.991445	−	1.71723i	−0.991445	−	1.71723i	−0.382683	−	0.923880i	$-0.625000\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−0.326384	−	0.286231i	−0.326384	−	0.286231i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$821$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$822$		0			0
$823$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$823$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$824$		0			0
$825$	1.86603	−	2.43185i	1.86603	−	2.43185i
$826$		0			0
$827$	−0.732626	+	1.09645i	−0.732626	+	1.09645i	0.258819	+	0.965926i	$0.416667\pi$
$827$	−0.732626	+	1.09645i	−0.732626	+	1.09645i	−0.991445	+	0.130526i	$0.958333\pi$
$828$		0			0
$829$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$829$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−1.75928	+	0.867580i	−1.75928	+	0.867580i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$839$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$840$		0			0
$841$	−0.130526	−	0.991445i	−0.130526	−	0.991445i
$842$		0			0
$843$	−1.98380	+	1.32553i	−1.98380	+	1.32553i
$844$		0			0
$845$		0			0
$846$		0			0
$847$		0			0
$848$		0			0
$849$	0.580775	+	2.16748i	0.580775	+	2.16748i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$853$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.965926	+	0.258819i	$0.0833333\pi$
$857$	1.67303	+	0.965926i	1.67303	+	0.965926i	0.707107	+	0.707107i	$0.250000\pi$
$858$		0			0
$859$	0.391239	−	0.793353i	0.391239	−	0.793353i	−0.608761	−	0.793353i	$-0.708333\pi$
$859$	0.391239	−	0.793353i	0.391239	−	0.793353i	1.00000			$0$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$863$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	3.91319	−	2.25928i	3.91319	−	2.25928i
$868$		0			0
$869$		0			0
$870$		0			0
$871$		0			0
$872$		0			0
$873$	0.580775	−	1.40211i	0.580775	−	1.40211i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$877$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	1.60021	−	0.662827i	1.60021	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$881$	1.60021	−	0.662827i	1.60021	−	0.662827i	0.991445	+	0.130526i	$0.0416667\pi$
$882$		0			0
$883$	0.583242	+	0.665060i	0.583242	+	0.665060i	0.965926	−	0.258819i	$-0.0833333\pi$
$883$	0.583242	+	0.665060i	0.583242	+	0.665060i	−0.382683	+	0.923880i	$0.625000\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$887$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−0.158513	−	0.382683i	−0.158513	−	0.382683i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		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
$905$		0			0
$906$		0			0
$907$	0.369474	+	1.85747i	0.369474	+	1.85747i	0.500000	+	0.866025i	$0.333333\pi$
$907$	0.369474	+	1.85747i	0.369474	+	1.85747i	−0.130526	+	0.991445i	$0.541667\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$911$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$912$		0			0
$913$	−3.28135	+	1.61818i	−3.28135	+	1.61818i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$919$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$920$		0			0
$921$	1.17604	−	1.53264i	1.17604	−	1.53264i
$922$		0			0
$923$		0			0
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−1.34861	−	1.18270i	−1.34861	−	1.18270i	−0.965926	−	0.258819i	$-0.916667\pi$
$929$	−1.34861	−	1.18270i	−1.34861	−	1.18270i	−0.382683	−	0.923880i	$-0.625000\pi$
$930$		0			0
$931$	1.65938	+	0.108761i	1.65938	+	0.108761i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$937$	−1.22474	+	1.22474i	−1.22474	+	1.22474i	−0.965926	+	0.258819i	$0.916667\pi$
$938$		0			0
$939$	−0.410670	−	0.0540657i	−0.410670	−	0.0540657i
$940$		0			0
$941$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$941$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.608761	−	1.79335i	−0.608761	−	1.79335i	−0.608761	−	0.793353i	$-0.708333\pi$
$947$	−0.608761	−	1.79335i	−0.608761	−	1.79335i		−	1.00000i	$-0.5\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	0.349942	+	0.172572i	0.349942	+	0.172572i	0.608761	−	0.793353i	$-0.291667\pi$
$953$	0.349942	+	0.172572i	0.349942	+	0.172572i	−0.258819	+	0.965926i	$0.583333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.965926	+	0.258819i	0.965926	+	0.258819i
$962$		0			0
$963$	0.593759	+	1.20402i	0.593759	+	1.20402i
$964$		0			0
$965$		0			0
$966$		0			0
$967$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$967$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$968$		0			0
$969$	−5.17579			−5.17579
$970$		0			0
$971$	1.84776			1.84776			0.923880	−	0.382683i	$-0.125000\pi$
$971$	1.84776			1.84776			0.923880	+	0.382683i	$0.125000\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.608761	+	0.793353i	$0.291667\pi$
$977$	0.867580	+	1.75928i	0.867580	+	1.75928i	0.258819	+	0.965926i	$0.416667\pi$
$978$		0			0
$979$	3.70012	+	0.991445i	3.70012	+	0.991445i
$980$		0			0
$981$		0			0
$982$		0			0
$983$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$983$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$		0			0
$990$		0			0
$991$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$991$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$992$		0			0
$993$	1.01788	+	2.99857i	1.01788	+	2.99857i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$997$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$998$		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	3104.1.et.a.1327.1		16
4.3	odd	2		776.1.bp.a.163.1	✓	16
8.3	odd	2	CM	3104.1.et.a.1327.1		16
8.5	even	2		776.1.bp.a.163.1	✓	16
97.25	even	48	inner	3104.1.et.a.2159.1		16
388.219	odd	48		776.1.bp.a.219.1	yes	16
776.219	odd	48	inner	3104.1.et.a.2159.1		16
776.413	even	48		776.1.bp.a.219.1	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
776.1.bp.a.163.1	✓	16	4.3	odd	2
776.1.bp.a.163.1	✓	16	8.5	even	2
776.1.bp.a.219.1	yes	16	388.219	odd	48
776.1.bp.a.219.1	yes	16	776.413	even	48
3104.1.et.a.1327.1		16	1.1	even	1	trivial
3104.1.et.a.1327.1		16	8.3	odd	2	CM
3104.1.et.a.2159.1		16	97.25	even	48	inner
3104.1.et.a.2159.1		16	776.219	odd	48	inner

Overview	Random
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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

Related objects

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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	3104.1.et.a.1327.1

Level	$3104$
Weight	$1$
Character	3104.1327
Analytic conductor	$1.549$
Analytic rank	$0$
Dimension	$16$
Projective image	$D_{48}$
CM discriminant	-8
Inner twists	$4$