LMFDB - Embedded newform 3136.1.ce.a.1293.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3136,1,Mod(117,3136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3136, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([0, 15, 40]))

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

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

chi := DirichletCharacter("3136.117");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3136 = 2^{6} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3136.ce (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:	$1.56506787962$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 448)
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			1293.1
Root			$-0.991445 + 0.130526i$ of defining polynomial
Character	$\chi$	$=$	3136.1293
Dual form			3136.1.ce.a.325.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.130526 - 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(-0.382683 + 0.923880i) q^{8} +(0.608761 + 0.793353i) q^{9} +(-0.389345 - 0.0255190i) q^{11} +(0.866025 + 0.500000i) q^{16} +(0.866025 - 0.500000i) q^{18} +(-0.0761205 + 0.382683i) q^{22} +(0.607206 - 0.465926i) q^{23} +(-0.793353 - 0.608761i) q^{25} +(1.08979 + 1.63099i) q^{29} +(0.608761 - 0.793353i) q^{32} +(-0.382683 - 0.923880i) q^{36} +(1.05217 + 0.357164i) q^{37} +(0.923880 + 0.617317i) q^{43} +(0.369474 + 0.125419i) q^{44} +(-0.382683 - 0.662827i) q^{46} +(-0.707107 + 0.707107i) q^{50} +(1.65938 + 0.108761i) q^{53} +(1.75928 - 0.867580i) q^{58} +(-0.707107 - 0.707107i) q^{64} +(1.49144 + 0.735499i) q^{67} +(-0.541196 - 1.30656i) q^{71} +(-0.965926 + 0.258819i) q^{72} +(0.491445 - 0.996552i) q^{74} +(-0.478235 - 1.78480i) q^{79} +(-0.258819 + 0.965926i) q^{81} +(0.732626 - 0.835400i) q^{86} +(0.172572 - 0.349942i) q^{88} +(-0.707107 + 0.292893i) q^{92} +(-0.216773 - 0.324423i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{22} + 8 q^{44} + 8 q^{67} - 8 q^{74}+O(q^{100}) $ 16 * q - 16 * q^22 + 8 * q^44 + 8 * q^67 - 8 * q^74

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3136\mathbb{Z}\right)^\times$.

$n$	$197$	$1471$	$1473$
$\chi(n)$	$e\left(\frac{15}{16}\right)$	$1$	$e\left(\frac{5}{6}\right)$

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 $

$2$	0.130526	−	0.991445i	0.130526	−	0.991445i
$2$	0.130526	−	0.991445i	0.130526	−	0.991445i
$3$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$3$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$4$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$5$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$5$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$9$	0.608761	+	0.793353i	0.608761	+	0.793353i
$10$		0			0
$11$	−0.389345	−	0.0255190i	−0.389345	−	0.0255190i	−0.130526	−	0.991445i	$-0.541667\pi$
$11$	−0.389345	−	0.0255190i	−0.389345	−	0.0255190i	−0.258819	+	0.965926i	$0.583333\pi$
$12$		0			0
$13$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$13$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$14$		0			0
$15$		0			0
$16$	0.866025	+	0.500000i	0.866025	+	0.500000i
$17$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$17$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$18$	0.866025	−	0.500000i	0.866025	−	0.500000i
$19$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$19$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$20$		0			0
$21$		0			0
$22$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$23$	0.607206	−	0.465926i	0.607206	−	0.465926i	−0.258819	−	0.965926i	$-0.583333\pi$
$23$	0.607206	−	0.465926i	0.607206	−	0.465926i	0.866025	+	0.500000i	$0.166667\pi$
$24$		0			0
$25$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.08979	+	1.63099i	1.08979	+	1.63099i	0.707107	+	0.707107i	$0.250000\pi$
$29$	1.08979	+	1.63099i	1.08979	+	1.63099i	0.382683	+	0.923880i	$0.375000\pi$
$30$		0			0
$31$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$31$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$32$	0.608761	−	0.793353i	0.608761	−	0.793353i
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$37$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.793353	−	0.608761i	$-0.208333\pi$
$37$	1.05217	+	0.357164i	1.05217	+	0.357164i	0.258819	+	0.965926i	$0.416667\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$41$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$42$		0			0
$43$	0.923880	+	0.617317i	0.923880	+	0.617317i	0.923880	−	0.382683i	$-0.125000\pi$
$43$	0.923880	+	0.617317i	0.923880	+	0.617317i			1.00000i	$0.5\pi$
$44$	0.369474	+	0.125419i	0.369474	+	0.125419i
$45$		0			0
$46$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$47$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$47$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$48$		0			0
$49$		0			0
$50$	−0.707107	+	0.707107i	−0.707107	+	0.707107i
$51$		0			0
$52$		0			0
$53$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.866025	−	0.500000i	$-0.166667\pi$
$53$	1.65938	+	0.108761i	1.65938	+	0.108761i	0.793353	+	0.608761i	$0.208333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.75928	−	0.867580i	1.75928	−	0.867580i
$59$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$59$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$60$		0			0
$61$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$61$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$		0			0
$66$		0			0
$67$	1.49144	+	0.735499i	1.49144	+	0.735499i	0.991445	−	0.130526i	$-0.0416667\pi$
$67$	1.49144	+	0.735499i	1.49144	+	0.735499i	0.500000	+	0.866025i	$0.333333\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$71$	−0.541196	−	1.30656i	−0.541196	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$72$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$73$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$73$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$74$	0.491445	−	0.996552i	0.491445	−	0.996552i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	−0.608761	−	0.793353i	$-0.708333\pi$
$79$	−0.478235	−	1.78480i	−0.478235	−	1.78480i	0.130526	−	0.991445i	$-0.458333\pi$
$80$		0			0
$81$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$82$		0			0
$83$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$83$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$84$		0			0
$85$		0			0
$86$	0.732626	−	0.835400i	0.732626	−	0.835400i
$87$		0			0
$88$	0.172572	−	0.349942i	0.172572	−	0.349942i
$89$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$89$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$90$		0			0
$91$		0			0
$92$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$		0			0			−	1.00000i	$-0.5\pi$
$97$		0			0				1.00000i	$0.5\pi$
$98$		0			0
$99$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$100$	0.608761	+	0.793353i	0.608761	+	0.793353i
$101$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$101$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$102$		0			0
$103$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$103$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$104$		0			0
$105$		0			0
$106$	0.324423	−	1.63099i	0.324423	−	1.63099i
$107$	−1.49144	+	0.735499i	−1.49144	+	0.735499i	−0.991445	−	0.130526i	$-0.958333\pi$
$107$	−1.49144	+	0.735499i	−1.49144	+	0.735499i	−0.500000	+	0.866025i	$0.666667\pi$
$108$		0			0
$109$	0.369474	−	0.125419i	0.369474	−	0.125419i	−0.130526	−	0.991445i	$-0.541667\pi$
$109$	0.369474	−	0.125419i	0.369474	−	0.125419i	0.500000	+	0.866025i	$0.333333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	−0.923880	−	0.382683i	$-0.875000\pi$
$113$	−0.541196	+	0.541196i	−0.541196	+	0.541196i	0.382683	+	0.923880i	$0.375000\pi$
$114$		0			0
$115$		0			0
$116$	−0.630526	−	1.85747i	−0.630526	−	1.85747i
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.840506	−	0.110655i	−0.840506	−	0.110655i
$122$		0			0
$123$		0			0
$124$		0			0
$125$		0			0
$126$		0			0
$127$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$127$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$128$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$129$		0			0
$130$		0			0
$131$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$131$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$132$		0			0
$133$		0			0
$134$	0.923880	−	1.38268i	0.923880	−	1.38268i
$135$		0			0
$136$		0			0
$137$	−1.83195	+	0.241181i	−1.83195	+	0.241181i	−0.965926	−	0.258819i	$-0.916667\pi$
$137$	−1.83195	+	0.241181i	−1.83195	+	0.241181i	−0.866025	+	0.500000i	$0.833333\pi$
$138$		0			0
$139$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$139$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$140$		0			0
$141$		0			0
$142$	−1.36603	+	0.366025i	−1.36603	+	0.366025i
$143$		0			0
$144$	0.130526	+	0.991445i	0.130526	+	0.991445i
$145$		0			0
$146$		0			0
$147$		0			0
$148$	−0.923880	−	0.617317i	−0.923880	−	0.617317i
$149$	1.49144	−	0.735499i	1.49144	−	0.735499i	0.500000	−	0.866025i	$-0.333333\pi$
$149$	1.49144	−	0.735499i	1.49144	−	0.735499i	0.991445	+	0.130526i	$0.0416667\pi$
$150$		0			0
$151$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$151$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	0.258819	−	0.965926i	$-0.416667\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$		0			0
$157$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$157$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$158$	−1.83195	+	0.241181i	−1.83195	+	0.241181i
$159$		0			0
$160$		0			0
$161$		0			0
$162$	0.923880	+	0.382683i	0.923880	+	0.382683i
$163$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$163$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$167$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$168$		0			0
$169$	0.923880	+	0.382683i	0.923880	+	0.382683i
$170$		0			0
$171$		0			0
$172$	−0.732626	−	0.835400i	−0.732626	−	0.835400i
$173$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$173$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$174$		0			0
$175$		0			0
$176$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$177$		0			0
$178$		0			0
$179$	0.357164	+	1.05217i	0.357164	+	1.05217i	0.965926	+	0.258819i	$0.0833333\pi$
$179$	0.357164	+	1.05217i	0.357164	+	1.05217i	−0.608761	+	0.793353i	$0.708333\pi$
$180$		0			0
$181$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$181$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$182$		0			0
$183$		0			0
$184$	0.198092	+	0.739288i	0.198092	+	0.739288i
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.130526	+	0.991445i	$0.541667\pi$
$191$	−0.923880	+	1.60021i	−0.923880	+	1.60021i	−0.793353	+	0.608761i	$0.791667\pi$
$192$		0			0
$193$	0.382683	+	0.662827i	0.382683	+	0.662827i	0.991445	−	0.130526i	$-0.0416667\pi$
$193$	0.382683	+	0.662827i	0.382683	+	0.662827i	−0.608761	+	0.793353i	$0.708333\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$197$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$198$	−0.349942	+	0.172572i	−0.349942	+	0.172572i
$199$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$199$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$200$	0.866025	−	0.500000i	0.866025	−	0.500000i
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$	0.739288	+	0.198092i	0.739288	+	0.198092i
$208$		0			0
$209$		0			0
$210$		0			0
$211$	0.382683	−	1.92388i	0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$211$	0.382683	−	1.92388i	0.382683	−	1.92388i	0.382683	−	0.923880i	$-0.375000\pi$
$212$	−1.57469	−	0.534534i	−1.57469	−	0.534534i
$213$		0			0
$214$	0.534534	+	1.57469i	0.534534	+	1.57469i
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$219$		0			0
$220$		0			0
$221$		0			0
$222$		0			0
$223$		0			0			−	1.00000i	$-0.5\pi$
$223$		0			0				1.00000i	$0.5\pi$
$224$		0			0
$225$		−	1.00000i		−	1.00000i
$226$	0.465926	+	0.607206i	0.465926	+	0.607206i
$227$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$227$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$228$		0			0
$229$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$229$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$230$		0			0
$231$		0			0
$232$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$233$	1.46593	−	1.12484i	1.46593	−	1.12484i	0.500000	−	0.866025i	$-0.333333\pi$
$233$	1.46593	−	1.12484i	1.46593	−	1.12484i	0.965926	−	0.258819i	$-0.0833333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$	1.00000	+	1.00000i	1.00000	+	1.00000i	1.00000			$0$
$239$	1.00000	+	1.00000i	1.00000	+	1.00000i			1.00000i	$0.5\pi$
$240$		0			0
$241$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$241$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$242$	−0.219416	+	0.818872i	−0.219416	+	0.818872i
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		0.980785	−	0.195090i	$-0.0625000\pi$
$251$		0			0		−0.980785	+	0.195090i	$0.937500\pi$
$252$		0			0
$253$	−0.248303	+	0.165911i	−0.248303	+	0.165911i
$254$	0.184592	−	1.40211i	0.184592	−	1.40211i
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−0.630526	+	1.85747i	−0.630526	+	1.85747i
$262$		0			0
$263$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$263$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.25026	−	1.09645i	−1.25026	−	1.09645i
$269$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$269$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$270$		0			0
$271$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$271$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$272$		0			0
$273$		0			0
$274$			1.84776i			1.84776i
$275$	0.293353	+	0.257264i	0.293353	+	0.257264i
$276$		0			0
$277$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i	0.965926	+	0.258819i	$0.0833333\pi$
$277$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i	−0.991445	+	0.130526i	$0.958333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$281$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$282$		0			0
$283$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$283$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$284$	0.184592	+	1.40211i	0.184592	+	1.40211i
$285$		0			0
$286$		0			0
$287$		0			0
$288$	1.00000			1.00000
$289$	−0.866025	−	0.500000i	−0.866025	−	0.500000i
$290$		0			0
$291$		0			0
$292$		0			0
$293$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$293$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$294$		0			0
$295$		0			0
$296$	−0.732626	+	0.835400i	−0.732626	+	0.835400i
$297$		0			0
$298$	−0.534534	−	1.57469i	−0.534534	−	1.57469i
$299$		0			0
$300$		0			0
$301$		0			0
$302$	−1.84776			−1.84776
$303$		0			0
$304$		0			0
$305$		0			0
$306$		0			0
$307$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$307$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$311$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$312$		0			0
$313$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$313$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$314$		0			0
$315$		0			0
$316$			1.84776i			1.84776i
$317$	0.128293	+	1.95737i	0.128293	+	1.95737i	0.258819	+	0.965926i	$0.416667\pi$
$317$	0.128293	+	1.95737i	0.128293	+	1.95737i	−0.130526	+	0.991445i	$0.541667\pi$
$318$		0			0
$319$	−0.382683	−	0.662827i	−0.382683	−	0.662827i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	0.500000	−	0.866025i	0.500000	−	0.866025i
$325$		0			0
$326$	−0.128293	+	1.95737i	−0.128293	+	1.95737i
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.608761	−	0.793353i	$-0.708333\pi$
$331$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.258819	−	0.965926i	$-0.583333\pi$
$332$		0			0
$333$	0.357164	+	1.05217i	0.357164	+	1.05217i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	0.541196	+	0.541196i	0.541196	+	0.541196i	0.923880	−	0.382683i	$-0.125000\pi$
$337$	0.541196	+	0.541196i	0.541196	+	0.541196i	−0.382683	+	0.923880i	$0.625000\pi$
$338$	0.500000	−	0.866025i	0.500000	−	0.866025i
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$		0			0
$344$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$345$		0			0
$346$		0			0
$347$	−1.47479	+	1.29335i	−1.47479	+	1.29335i	−0.608761	+	0.793353i	$0.708333\pi$
$347$	−1.47479	+	1.29335i	−1.47479	+	1.29335i	−0.866025	+	0.500000i	$0.833333\pi$
$348$		0			0
$349$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$349$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$350$		0			0
$351$		0			0
$352$	−0.257264	+	0.293353i	−0.257264	+	0.293353i
$353$		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
$358$	1.08979	−	0.216773i	1.08979	−	0.216773i
$359$	0.0999004	−	0.758819i	0.0999004	−	0.758819i	−0.866025	−	0.500000i	$-0.833333\pi$
$359$	0.0999004	−	0.758819i	0.0999004	−	0.758819i	0.965926	−	0.258819i	$-0.0833333\pi$
$360$		0			0
$361$	0.130526	+	0.991445i	0.130526	+	0.991445i
$362$		0			0
$363$		0			0
$364$		0			0
$365$		0			0
$366$		0			0
$367$		0			0		0.258819	−	0.965926i	$-0.416667\pi$
$367$		0			0		−0.258819	+	0.965926i	$0.583333\pi$
$368$	0.758819	−	0.0999004i	0.758819	−	0.0999004i
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	0.172572	+	0.349942i	0.172572	+	0.349942i	0.965926	−	0.258819i	$-0.0833333\pi$
$373$	0.172572	+	0.349942i	0.172572	+	0.349942i	−0.793353	+	0.608761i	$0.791667\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$379$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$380$		0			0
$381$		0			0
$382$	1.46593	+	1.12484i	1.46593	+	1.12484i
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$	0.707107	−	0.292893i	0.707107	−	0.292893i
$387$	0.0726721	+	1.10876i	0.0726721	+	1.10876i
$388$		0			0
$389$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	0.258819	−	0.965926i	$-0.416667\pi$
$389$	−0.732626	−	0.835400i	−0.732626	−	0.835400i	−0.991445	+	0.130526i	$0.958333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	0.0726721	+	1.10876i	0.0726721	+	1.10876i
$395$		0			0
$396$	0.125419	+	0.369474i	0.125419	+	0.369474i
$397$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$397$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$398$		0			0
$399$		0			0
$400$	−0.382683	−	0.923880i	−0.382683	−	0.923880i
$401$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.130526	−	0.991445i	$-0.458333\pi$
$401$	0.739288	−	0.198092i	0.739288	−	0.198092i	0.608761	+	0.793353i	$0.291667\pi$
$402$		0			0
$403$		0			0
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−0.400544	−	0.165911i	−0.400544	−	0.165911i
$408$		0			0
$409$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$409$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$	0.292893	−	0.707107i	0.292893	−	0.707107i
$415$		0			0
$416$		0			0
$417$		0			0
$418$		0			0
$419$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$419$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$420$		0			0
$421$	−0.382683	−	1.92388i	−0.382683	−	1.92388i	−0.382683	−	0.923880i	$-0.625000\pi$
$421$	−0.382683	−	1.92388i	−0.382683	−	1.92388i		−	1.00000i	$-0.5\pi$
$422$	−1.85747	−	0.630526i	−1.85747	−	0.630526i
$423$		0			0
$424$	−0.735499	+	1.49144i	−0.735499	+	1.49144i
$425$		0			0
$426$		0			0
$427$		0			0
$428$	1.63099	−	0.324423i	1.63099	−	0.324423i
$429$		0			0
$430$		0			0
$431$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$431$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$432$		0			0
$433$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$433$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	−0.389345	+	0.0255190i	−0.389345	+	0.0255190i
$437$		0			0
$438$		0			0
$439$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$439$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$443$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	1.41421			1.41421			0.707107	−	0.707107i	$-0.250000\pi$
$449$	1.41421			1.41421			0.707107	+	0.707107i	$0.250000\pi$
$450$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$451$		0			0
$452$	0.662827	−	0.382683i	0.662827	−	0.382683i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−0.860919	−	1.12197i	−0.860919	−	1.12197i	−0.991445	−	0.130526i	$-0.958333\pi$
$457$	−0.860919	−	1.12197i	−0.860919	−	1.12197i	0.130526	−	0.991445i	$-0.458333\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$461$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$462$		0			0
$463$	0.541196	−	0.541196i	0.541196	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$463$	0.541196	−	0.541196i	0.541196	−	0.541196i	0.923880	+	0.382683i	$0.125000\pi$
$464$	0.128293	+	1.95737i	0.128293	+	1.95737i
$465$		0			0
$466$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$467$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$467$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.343955	−	0.263926i	−0.343955	−	0.263926i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	0.923880	+	1.38268i	0.923880	+	1.38268i
$478$	1.12197	−	0.860919i	1.12197	−	0.860919i
$479$		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
$484$	0.783227	+	0.324423i	0.783227	+	0.324423i
$485$		0			0
$486$		0			0
$487$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.965926	−	0.258819i	$-0.916667\pi$
$487$	−1.46593	−	1.12484i	−1.46593	−	1.12484i	−0.500000	−	0.866025i	$-0.666667\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−0.382683	−	0.923880i	$-0.625000\pi$
$491$	−1.38268	−	0.923880i	−1.38268	−	0.923880i	−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.130526	−	0.991445i	$-0.458333\pi$
$499$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$503$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$504$		0			0
$505$		0			0
$506$	0.132081	+	0.267834i	0.132081	+	0.267834i
$507$		0			0
$508$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$509$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$509$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$510$		0			0
$511$		0			0
$512$	0.923880	−	0.382683i	0.923880	−	0.382683i
$513$		0			0
$514$		0			0
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$521$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$522$	1.75928	+	0.867580i	1.75928	+	0.867580i
$523$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$523$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−0.107206	+	0.400100i	−0.107206	+	0.400100i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$		0			0
$536$	−1.25026	+	1.09645i	−1.25026	+	1.09645i
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.793353	−	0.608761i	$-0.208333\pi$
$541$	1.65938	−	0.108761i	1.65938	−	0.108761i	0.866025	+	0.500000i	$0.166667\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$547$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$548$	1.83195	+	0.241181i	1.83195	+	0.241181i
$549$		0			0
$550$	0.293353	−	0.257264i	0.293353	−	0.257264i
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.382683	+	0.0761205i	0.382683	+	0.0761205i
$555$		0			0
$556$		0			0
$557$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	−0.500000	−	0.866025i	$-0.666667\pi$
$557$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	0.130526	+	0.991445i	$0.458333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.478235	+	1.78480i	0.478235	+	1.78480i
$563$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$563$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$		0			0
$568$	1.41421			1.41421
$569$	−1.40211	−	0.184592i	−1.40211	−	0.184592i	−0.608761	−	0.793353i	$-0.708333\pi$
$569$	−1.40211	−	0.184592i	−1.40211	−	0.184592i	−0.793353	+	0.608761i	$0.791667\pi$
$570$		0			0
$571$	1.29335	+	1.47479i	1.29335	+	1.47479i	0.793353	+	0.608761i	$0.208333\pi$
$571$	1.29335	+	1.47479i	1.29335	+	1.47479i	0.500000	+	0.866025i	$0.333333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.765367			−0.765367
$576$	0.130526	−	0.991445i	0.130526	−	0.991445i
$577$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$577$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$578$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−0.643296	−	0.0846915i	−0.643296	−	0.0846915i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.555570	−	0.831470i	$-0.312500\pi$
$587$		0			0		−0.555570	+	0.831470i	$0.687500\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	0.732626	+	0.835400i	0.732626	+	0.835400i
$593$		0			0		0.965926	−	0.258819i	$-0.0833333\pi$
$593$		0			0		−0.965926	+	0.258819i	$0.916667\pi$
$594$		0			0
$595$		0			0
$596$	−1.63099	+	0.324423i	−1.63099	+	0.324423i
$597$		0			0
$598$		0			0
$599$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$599$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$600$		0			0
$601$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$601$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$602$		0			0
$603$	0.324423	+	1.63099i	0.324423	+	1.63099i
$604$	−0.241181	+	1.83195i	−0.241181	+	1.83195i
$605$		0			0
$606$		0			0
$607$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$607$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.47479	−	1.29335i	1.47479	−	1.29335i	0.608761	−	0.793353i	$-0.291667\pi$
$613$	1.47479	−	1.29335i	1.47479	−	1.29335i	0.866025	−	0.500000i	$-0.166667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.382683	−	0.923880i	$-0.625000\pi$
$617$	−1.30656	−	0.541196i	−1.30656	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$618$		0			0
$619$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$619$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	0.258819	+	0.965926i	0.258819	+	0.965926i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	0.382683	+	0.923880i	$0.375000\pi$
$631$	−0.541196	+	1.30656i	−0.541196	+	1.30656i	−0.923880	+	0.382683i	$0.875000\pi$
$632$	1.83195	+	0.241181i	1.83195	+	0.241181i
$633$		0			0
$634$	1.95737	+	0.128293i	1.95737	+	0.128293i
$635$		0			0
$636$		0			0
$637$		0			0
$638$	−0.707107	+	0.292893i	−0.707107	+	0.292893i
$639$	0.707107	−	1.22474i	0.707107	−	1.22474i
$640$		0			0
$641$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$641$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$642$		0			0
$643$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$643$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$647$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$648$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$649$		0			0
$650$		0			0
$651$		0			0
$652$	1.92388	+	0.382683i	1.92388	+	0.382683i
$653$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	−0.866025	−	0.500000i	$-0.833333\pi$
$653$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	0.608761	+	0.793353i	$0.291667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$659$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i	−1.00000			$\pi$
$660$		0			0
$661$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$661$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$662$	−1.85747	+	0.630526i	−1.85747	+	0.630526i
$663$		0			0
$664$		0			0
$665$		0			0
$666$	1.08979	−	0.216773i	1.08979	−	0.216773i
$667$	1.42165	+	0.482584i	1.42165	+	0.482584i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	1.41421i		−	1.41421i	−0.707107	−	0.707107i	$-0.750000\pi$
$673$		−	1.41421i		−	1.41421i	0.707107	−	0.707107i	$-0.250000\pi$
$674$	0.607206	−	0.465926i	0.607206	−	0.465926i
$675$		0			0
$676$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$677$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$677$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−0.0726721	+	1.10876i	−0.0726721	+	1.10876i	0.793353	+	0.608761i	$0.208333\pi$
$683$	−0.0726721	+	1.10876i	−0.0726721	+	1.10876i	−0.866025	+	0.500000i	$0.833333\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$		0			0
$688$	0.491445	+	0.996552i	0.491445	+	0.996552i
$689$		0			0
$690$		0			0
$691$		0			0		0.659346	−	0.751840i	$-0.270833\pi$
$691$		0			0		−0.659346	+	0.751840i	$0.729167\pi$
$692$		0			0
$693$		0			0
$694$	1.08979	+	1.63099i	1.08979	+	1.63099i
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−0.923880	+	0.617317i	−0.923880	+	0.617317i	−0.923880	−	0.382683i	$-0.875000\pi$
$701$	−0.923880	+	0.617317i	−0.923880	+	0.617317i			1.00000i	$0.5\pi$
$702$		0			0
$703$		0			0
$704$	0.257264	+	0.293353i	0.257264	+	0.293353i
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	−0.991445	−	0.130526i	$-0.958333\pi$
$709$	−0.125419	+	0.369474i	−0.125419	+	0.369474i	0.866025	+	0.500000i	$0.166667\pi$
$710$		0			0
$711$	1.12484	−	1.46593i	1.12484	−	1.46593i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i
$717$		0			0
$718$	−0.739288	−	0.198092i	−0.739288	−	0.198092i
$719$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$719$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$720$		0			0
$721$		0			0
$722$	1.00000			1.00000
$723$		0			0
$724$		0			0
$725$	0.128293	−	1.95737i	0.128293	−	1.95737i
$726$		0			0
$727$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$727$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$728$		0			0
$729$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$730$		0			0
$731$		0			0
$732$		0			0
$733$		0			0		0.997859	−	0.0654031i	$-0.0208333\pi$
$733$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$734$		0			0
$735$		0			0
$736$		−	0.765367i		−	0.765367i
$737$	−0.561918	−	0.324423i	−0.561918	−	0.324423i
$738$		0			0
$739$	0.172572	−	0.349942i	0.172572	−	0.349942i	−0.793353	−	0.608761i	$-0.791667\pi$
$739$	0.172572	−	0.349942i	0.172572	−	0.349942i	0.965926	+	0.258819i	$0.0833333\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−0.707107	+	0.707107i	$0.750000\pi$
$743$	−1.70711	+	0.707107i	−1.70711	+	0.707107i	−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$	0.369474	−	0.125419i	0.369474	−	0.125419i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.500000	−	0.866025i	$-0.333333\pi$
$751$	1.36603	−	0.366025i	1.36603	−	0.366025i	0.866025	+	0.500000i	$0.166667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$		0			0
$757$	0.923880	−	1.38268i	0.923880	−	1.38268i		−	1.00000i	$-0.5\pi$
$757$	0.923880	−	1.38268i	0.923880	−	1.38268i	0.923880	−	0.382683i	$-0.125000\pi$
$758$	0.108761	+	1.65938i	0.108761	+	1.65938i
$759$		0			0
$760$		0			0
$761$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$761$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$762$		0			0
$763$		0			0
$764$	1.30656	−	1.30656i	1.30656	−	1.30656i
$765$		0			0
$766$		0			0
$767$		0			0
$768$		0			0
$769$		0			0		1.00000			$0$
$769$		0			0		−1.00000			$\pi$
$770$		0			0
$771$		0			0
$772$	−0.198092	−	0.739288i	−0.198092	−	0.739288i
$773$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$773$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$774$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$775$		0			0
$776$		0			0
$777$		0			0
$778$	−0.923880	+	0.617317i	−0.923880	+	0.617317i
$779$		0			0
$780$		0			0
$781$	0.177370	+	0.522515i	0.177370	+	0.522515i
$782$		0			0
$783$		0			0
$784$		0			0
$785$		0			0
$786$		0			0
$787$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$787$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$788$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$789$		0			0
$790$		0			0
$791$		0			0
$792$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$793$		0			0
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$797$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$798$		0			0
$799$		0			0
$800$	−0.965926	+	0.258819i	−0.965926	+	0.258819i
$801$		0			0
$802$	−0.0999004	−	0.758819i	−0.0999004	−	0.758819i
$803$		0			0
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−0.184592	−	1.40211i	−0.184592	−	1.40211i	−0.793353	−	0.608761i	$-0.791667\pi$
$809$	−0.184592	−	1.40211i	−0.184592	−	1.40211i	0.608761	−	0.793353i	$-0.291667\pi$
$810$		0			0
$811$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$811$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$812$		0			0
$813$		0			0
$814$	−0.216773	+	0.375461i	−0.216773	+	0.375461i
$815$		0			0
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.608761	−	0.793353i	$-0.708333\pi$
$821$	−0.867580	−	1.75928i	−0.867580	−	1.75928i	−0.258819	−	0.965926i	$-0.583333\pi$
$822$		0			0
$823$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$823$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$827$	−1.08979	+	0.216773i	−1.08979	+	0.216773i	−0.382683	+	0.923880i	$0.625000\pi$
$828$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$829$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$829$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$839$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$840$		0			0
$841$	−1.08979	+	2.63099i	−1.08979	+	2.63099i
$842$	−1.95737	+	0.128293i	−1.95737	+	0.128293i
$843$		0			0
$844$	−0.867580	+	1.75928i	−0.867580	+	1.75928i
$845$		0			0
$846$		0			0
$847$		0			0
$848$	1.38268	+	0.923880i	1.38268	+	0.923880i
$849$		0			0
$850$		0			0
$851$	0.805298	−	0.273362i	0.805298	−	0.273362i
$852$		0			0
$853$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$853$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$854$		0			0
$855$		0			0
$856$	−0.108761	−	1.65938i	−0.108761	−	1.65938i
$857$		0			0		−0.793353	−	0.608761i	$-0.791667\pi$
$857$		0			0		0.793353	+	0.608761i	$0.208333\pi$
$858$		0			0
$859$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$859$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.991445	+	0.130526i	$0.0416667\pi$
$863$	1.60021	+	0.923880i	1.60021	+	0.923880i	0.608761	+	0.793353i	$0.291667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	0.140652	+	0.707107i	0.140652	+	0.707107i
$870$		0			0
$871$		0			0
$872$	−0.0255190	+	0.389345i	−0.0255190	+	0.389345i
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	0.130526	−	0.991445i	$-0.458333\pi$
$877$	−0.835400	−	0.732626i	−0.835400	−	0.732626i	−0.965926	+	0.258819i	$0.916667\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$881$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$882$		0			0
$883$	1.92388	+	0.382683i	1.92388	+	0.382683i	1.00000			$0$
$883$	1.92388	+	0.382683i	1.92388	+	0.382683i	0.923880	+	0.382683i	$0.125000\pi$
$884$		0			0
$885$		0			0
$886$	0.996552	+	0.491445i	0.996552	+	0.491445i
$887$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$887$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	0.125419	−	0.369474i	0.125419	−	0.369474i
$892$		0			0
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$	0.184592	−	1.40211i	0.184592	−	1.40211i
$899$		0			0
$900$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$901$		0			0
$902$		0			0
$903$		0			0
$904$	−0.292893	−	0.707107i	−0.292893	−	0.707107i
$905$		0			0
$906$		0			0
$907$	0.389345	+	0.0255190i	0.389345	+	0.0255190i	0.258819	−	0.965926i	$-0.416667\pi$
$907$	0.389345	+	0.0255190i	0.389345	+	0.0255190i	0.130526	+	0.991445i	$0.458333\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$911$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$912$		0			0
$913$		0			0
$914$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	1.12197	−	0.860919i	1.12197	−	0.860919i	0.130526	−	0.991445i	$-0.458333\pi$
$919$	1.12197	−	0.860919i	1.12197	−	0.860919i	0.991445	+	0.130526i	$0.0416667\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−0.617317	−	0.923880i	−0.617317	−	0.923880i
$926$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$927$		0			0
$928$	1.95737	+	0.128293i	1.95737	+	0.128293i
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$		0			0
$932$	−1.70711	+	0.707107i	−1.70711	+	0.707107i
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$937$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$941$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$	−0.306563	+	0.306563i	−0.306563	+	0.306563i
$947$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	−0.866025	−	0.500000i	$-0.833333\pi$
$947$	−0.257264	+	0.293353i	−0.257264	+	0.293353i	0.608761	+	0.793353i	$0.291667\pi$
$948$		0			0
$949$		0			0
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$953$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$954$	1.49144	−	0.735499i	1.49144	−	0.735499i
$955$		0			0
$956$	−0.707107	−	1.22474i	−0.707107	−	1.22474i
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	0.500000	−	0.866025i	0.500000	−	0.866025i
$962$		0			0
$963$	−1.49144	−	0.735499i	−1.49144	−	0.735499i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−0.707107	−	1.70711i	−0.707107	−	1.70711i	−0.707107	−	0.707107i	$-0.750000\pi$
$967$	−0.707107	−	1.70711i	−0.707107	−	1.70711i		−	1.00000i	$-0.5\pi$
$968$	0.423880	−	0.734181i	0.423880	−	0.734181i
$969$		0			0
$970$		0			0
$971$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$971$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$972$		0			0
$973$		0			0
$974$	−1.30656	+	1.30656i	−1.30656	+	1.30656i
$975$		0			0
$976$		0			0
$977$	0.366025	−	1.36603i	0.366025	−	1.36603i	−0.500000	−	0.866025i	$-0.666667\pi$
$977$	0.366025	−	1.36603i	0.366025	−	1.36603i	0.866025	−	0.500000i	$-0.166667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	0.324423	+	0.216773i	0.324423	+	0.216773i
$982$	−1.09645	+	1.25026i	−1.09645	+	1.25026i
$983$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$983$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.848609	−	0.0556208i	0.848609	−	0.0556208i
$990$		0			0
$991$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	−0.793353	−	0.608761i	$-0.791667\pi$
$991$	−0.662827	+	0.382683i	−0.662827	+	0.382683i	0.130526	+	0.991445i	$0.458333\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		0			0
$996$		0			0
$997$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$997$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$998$	−1.09645	−	1.25026i	−1.09645	−	1.25026i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3136.1.ce.a.1293.1		16
7.2	even	3		448.1.bf.a.13.1	✓	8
7.3	odd	6	inner	3136.1.ce.a.717.1		16
7.4	even	3	inner	3136.1.ce.a.717.1		16
7.5	odd	6		448.1.bf.a.13.1	✓	8
7.6	odd	2	CM	3136.1.ce.a.1293.1		16
28.19	even	6		1792.1.bf.a.657.1		8
28.23	odd	6		1792.1.bf.a.657.1		8
56.5	odd	6		3584.1.bf.a.545.1		8
56.19	even	6		3584.1.bf.b.545.1		8
56.37	even	6		3584.1.bf.a.545.1		8
56.51	odd	6		3584.1.bf.b.545.1		8
64.5	even	16	inner	3136.1.ce.a.901.1		16
448.5	odd	48		448.1.bf.a.69.1	yes	8
448.37	even	48		3584.1.bf.a.993.1		8
448.69	odd	16	inner	3136.1.ce.a.901.1		16
448.187	even	48		1792.1.bf.a.881.1		8
448.219	odd	48		3584.1.bf.b.993.1		8
448.229	odd	48		3584.1.bf.a.993.1		8
448.261	even	48		448.1.bf.a.69.1	yes	8
448.325	odd	48	inner	3136.1.ce.a.325.1		16
448.389	even	48	inner	3136.1.ce.a.325.1		16
448.411	even	48		3584.1.bf.b.993.1		8
448.443	odd	48		1792.1.bf.a.881.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
448.1.bf.a.13.1	✓	8	7.2	even	3
448.1.bf.a.13.1	✓	8	7.5	odd	6
448.1.bf.a.69.1	yes	8	448.5	odd	48
448.1.bf.a.69.1	yes	8	448.261	even	48
1792.1.bf.a.657.1		8	28.19	even	6
1792.1.bf.a.657.1		8	28.23	odd	6
1792.1.bf.a.881.1		8	448.187	even	48
1792.1.bf.a.881.1		8	448.443	odd	48
3136.1.ce.a.325.1		16	448.325	odd	48	inner
3136.1.ce.a.325.1		16	448.389	even	48	inner
3136.1.ce.a.717.1		16	7.3	odd	6	inner
3136.1.ce.a.717.1		16	7.4	even	3	inner
3136.1.ce.a.901.1		16	64.5	even	16	inner
3136.1.ce.a.901.1		16	448.69	odd	16	inner
3136.1.ce.a.1293.1		16	1.1	even	1	trivial
3136.1.ce.a.1293.1		16	7.6	odd	2	CM
3584.1.bf.a.545.1		8	56.5	odd	6
3584.1.bf.a.545.1		8	56.37	even	6
3584.1.bf.a.993.1		8	448.37	even	48
3584.1.bf.a.993.1		8	448.229	odd	48
3584.1.bf.b.545.1		8	56.19	even	6
3584.1.bf.b.545.1		8	56.51	odd	6
3584.1.bf.b.993.1		8	448.219	odd	48
3584.1.bf.b.993.1		8	448.411	even	48

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 3136 = 2^{6} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3136.ce (of order \(48\), degree \(16\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.130526 - 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(-0.382683 + 0.923880i) q^{8} +(0.608761 + 0.793353i) q^{9} +(-0.389345 - 0.0255190i) q^{11} +(0.866025 + 0.500000i) q^{16} +(0.866025 - 0.500000i) q^{18} +(-0.0761205 + 0.382683i) q^{22} +(0.607206 - 0.465926i) q^{23} +(-0.793353 - 0.608761i) q^{25} +(1.08979 + 1.63099i) q^{29} +(0.608761 - 0.793353i) q^{32} +(-0.382683 - 0.923880i) q^{36} +(1.05217 + 0.357164i) q^{37} +(0.923880 + 0.617317i) q^{43} +(0.369474 + 0.125419i) q^{44} +(-0.382683 - 0.662827i) q^{46} +(-0.707107 + 0.707107i) q^{50} +(1.65938 + 0.108761i) q^{53} +(1.75928 - 0.867580i) q^{58} +(-0.707107 - 0.707107i) q^{64} +(1.49144 + 0.735499i) q^{67} +(-0.541196 - 1.30656i) q^{71} +(-0.965926 + 0.258819i) q^{72} +(0.491445 - 0.996552i) q^{74} +(-0.478235 - 1.78480i) q^{79} +(-0.258819 + 0.965926i) q^{81} +(0.732626 - 0.835400i) q^{86} +(0.172572 - 0.349942i) q^{88} +(-0.707107 + 0.292893i) q^{92} +(-0.216773 - 0.324423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{22} + 8 q^{44} + 8 q^{67} - 8 q^{74}+O(q^{100}) \) 16 * q - 16 * q^22 + 8 * q^44 + 8 * q^67 - 8 * q^74

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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