LMFDB - Embedded newform 3332.1.ce.c.1391.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,1,Mod(31,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(48))

chi = DirichletCharacter(H, H._module([24, 8, 27]))

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

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

chi := DirichletCharacter("3332.31");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	3332.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:	$1.66288462209$
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:	yes
Projective image:	$D_{16}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{16} - \cdots)$

Embedding invariants

Embedding label			1391.1
Root			$-0.130526 + 0.991445i$ of defining polynomial
Character	$\chi$	$=$	3332.1391
Dual form			3332.1.ce.c.2187.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.130526 + 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(1.95737 - 0.128293i) q^{5} +(0.382683 - 0.923880i) q^{8} +(-0.793353 + 0.608761i) q^{9} +O(q^{10})$	\(q+(-0.130526 + 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(1.95737 - 0.128293i) q^{5} +(0.382683 - 0.923880i) q^{8} +(-0.793353 + 0.608761i) q^{9} +(-0.128293 + 1.95737i) q^{10} +(1.00000 - 1.00000i) q^{13} +(0.866025 + 0.500000i) q^{16} +(0.608761 - 0.793353i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-1.92388 - 0.382683i) q^{20} +(2.82340 - 0.371707i) q^{25} +(0.860919 + 1.12197i) q^{26} +(-1.08979 - 1.63099i) q^{29} +(-0.608761 + 0.793353i) q^{32} +(0.707107 + 0.707107i) q^{34} +(0.923880 - 0.382683i) q^{36} +(-0.293353 - 0.257264i) q^{37} +(0.630526 - 1.85747i) q^{40} +(-0.617317 + 0.923880i) q^{41} +(-1.47479 + 1.29335i) q^{45} +2.84776i q^{50} +(-1.22474 + 0.707107i) q^{52} +(0.607206 + 0.465926i) q^{53} +(1.75928 - 0.867580i) q^{58} +(-1.49144 + 0.735499i) q^{61} +(-0.707107 - 0.707107i) q^{64} +(1.82908 - 2.08566i) q^{65} +(-0.793353 + 0.608761i) q^{68} +(0.258819 + 0.965926i) q^{72} +(-0.172572 + 0.349942i) q^{73} +(0.293353 - 0.257264i) q^{74} +(1.75928 + 0.867580i) q^{80} +(0.258819 - 0.965926i) q^{81} +(-0.835400 - 0.732626i) q^{82} +(1.08979 - 1.63099i) q^{85} +(-1.08979 - 1.63099i) q^{90} +(-0.324423 + 0.216773i) q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{13} - 8 q^{18} - 16 q^{20} + 8 q^{37} + 8 q^{40} - 16 q^{41} - 8 q^{61} - 8 q^{74}+O(q^{100}) $ 16 * q + 16 * q^13 - 8 * q^18 - 16 * q^20 + 8 * q^37 + 8 * q^40 - 16 * q^41 - 8 * q^61 - 8 * q^74

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3332\mathbb{Z}\right)^\times$.

$n$	$785$	$885$	$1667$
$\chi(n)$	$e\left(\frac{9}{16}\right)$	$e\left(\frac{5}{6}\right)$	$-1$

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 $

$2$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$2$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$3$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$3$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$4$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$5$	1.95737	−	0.128293i	1.95737	−	0.128293i	0.965926	−	0.258819i	$-0.0833333\pi$
$5$	1.95737	−	0.128293i	1.95737	−	0.128293i	0.991445	+	0.130526i	$0.0416667\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$	0.382683	−	0.923880i	0.382683	−	0.923880i
$9$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$10$	−0.128293	+	1.95737i	−0.128293	+	1.95737i
$11$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$11$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$12$		0			0
$13$	1.00000	−	1.00000i	1.00000	−	1.00000i		−	1.00000i	$-0.5\pi$
$13$	1.00000	−	1.00000i	1.00000	−	1.00000i	1.00000			$0$
$14$		0			0
$15$		0			0
$16$	0.866025	+	0.500000i	0.866025	+	0.500000i
$17$	0.608761	−	0.793353i	0.608761	−	0.793353i
$17$	0.608761	−	0.793353i	0.608761	−	0.793353i
$18$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$19$		0			0		−0.991445	−	0.130526i	$-0.958333\pi$
$19$		0			0		0.991445	+	0.130526i	$0.0416667\pi$
$20$	−1.92388	−	0.382683i	−1.92388	−	0.382683i
$21$		0			0
$22$		0			0
$23$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$23$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$24$		0			0
$25$	2.82340	−	0.371707i	2.82340	−	0.371707i
$26$	0.860919	+	1.12197i	0.860919	+	1.12197i
$27$		0			0
$28$		0			0
$29$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.707107	−	0.707107i	$-0.750000\pi$
$29$	−1.08979	−	1.63099i	−1.08979	−	1.63099i	−0.382683	−	0.923880i	$-0.625000\pi$
$30$		0			0
$31$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$31$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$32$	−0.608761	+	0.793353i	−0.608761	+	0.793353i
$33$		0			0
$34$	0.707107	+	0.707107i	0.707107	+	0.707107i
$35$		0			0
$36$	0.923880	−	0.382683i	0.923880	−	0.382683i
$37$	−0.293353	−	0.257264i	−0.293353	−	0.257264i	0.500000	−	0.866025i	$-0.333333\pi$
$37$	−0.293353	−	0.257264i	−0.293353	−	0.257264i	−0.793353	+	0.608761i	$0.791667\pi$
$38$		0			0
$39$		0			0
$40$	0.630526	−	1.85747i	0.630526	−	1.85747i
$41$	−0.617317	+	0.923880i	−0.617317	+	0.923880i	0.382683	+	0.923880i	$0.375000\pi$
$41$	−0.617317	+	0.923880i	−0.617317	+	0.923880i	−1.00000			$\pi$
$42$		0			0
$43$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$43$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$44$		0			0
$45$	−1.47479	+	1.29335i	−1.47479	+	1.29335i
$46$		0			0
$47$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$47$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$48$		0			0
$49$		0			0
$50$			2.84776i			2.84776i
$51$		0			0
$52$	−1.22474	+	0.707107i	−1.22474	+	0.707107i
$53$	0.607206	+	0.465926i	0.607206	+	0.465926i	0.866025	−	0.500000i	$-0.166667\pi$
$53$	0.607206	+	0.465926i	0.607206	+	0.465926i	−0.258819	+	0.965926i	$0.583333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$		0			0
$58$	1.75928	−	0.867580i	1.75928	−	0.867580i
$59$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$59$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$60$		0			0
$61$	−1.49144	+	0.735499i	−1.49144	+	0.735499i	−0.991445	−	0.130526i	$-0.958333\pi$
$61$	−1.49144	+	0.735499i	−1.49144	+	0.735499i	−0.500000	+	0.866025i	$0.666667\pi$
$62$		0			0
$63$		0			0
$64$	−0.707107	−	0.707107i	−0.707107	−	0.707107i
$65$	1.82908	−	2.08566i	1.82908	−	2.08566i
$66$		0			0
$67$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$67$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$68$	−0.793353	+	0.608761i	−0.793353	+	0.608761i
$69$		0			0
$70$		0			0
$71$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$71$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$72$	0.258819	+	0.965926i	0.258819	+	0.965926i
$73$	−0.172572	+	0.349942i	−0.172572	+	0.349942i	−0.965926	−	0.258819i	$-0.916667\pi$
$73$	−0.172572	+	0.349942i	−0.172572	+	0.349942i	0.793353	+	0.608761i	$0.208333\pi$
$74$	0.293353	−	0.257264i	0.293353	−	0.257264i
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$		0			0		0.321439	−	0.946930i	$-0.395833\pi$
$79$		0			0		−0.321439	+	0.946930i	$0.604167\pi$
$80$	1.75928	+	0.867580i	1.75928	+	0.867580i
$81$	0.258819	−	0.965926i	0.258819	−	0.965926i
$82$	−0.835400	−	0.732626i	−0.835400	−	0.732626i
$83$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$83$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$84$		0			0
$85$	1.08979	−	1.63099i	1.08979	−	1.63099i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$89$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$90$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	−0.707107	−	0.707107i	$-0.750000\pi$
$97$	−0.324423	+	0.216773i	−0.324423	+	0.216773i	0.382683	+	0.923880i	$0.375000\pi$
$98$		0			0
$99$		0			0
$100$	−2.82340	−	0.371707i	−2.82340	−	0.371707i
$101$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$101$	1.00000	+	1.73205i	1.00000	+	1.73205i	0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$103$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$104$	−0.541196	−	1.30656i	−0.541196	−	1.30656i
$105$		0			0
$106$	−0.541196	+	0.541196i	−0.541196	+	0.541196i
$107$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$107$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$108$		0			0
$109$	−0.0726721	+	1.10876i	−0.0726721	+	1.10876i	0.793353	+	0.608761i	$0.208333\pi$
$109$	−0.0726721	+	1.10876i	−0.0726721	+	1.10876i	−0.866025	+	0.500000i	$0.833333\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.707107	−	0.707107i	$-0.250000\pi$
$113$	1.63099	−	0.324423i	1.63099	−	0.324423i	0.923880	+	0.382683i	$0.125000\pi$
$114$		0			0
$115$		0			0
$116$	0.630526	+	1.85747i	0.630526	+	1.85747i
$117$	−0.184592	+	1.40211i	−0.184592	+	1.40211i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−0.130526	+	0.991445i	−0.130526	+	0.991445i
$122$	−0.534534	−	1.57469i	−0.534534	−	1.57469i
$123$		0			0
$124$		0			0
$125$	3.55487	−	0.707107i	3.55487	−	0.707107i
$126$		0			0
$127$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$127$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$128$	0.793353	−	0.608761i	0.793353	−	0.608761i
$129$		0			0
$130$	1.82908	+	2.08566i	1.82908	+	2.08566i
$131$		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			0
$135$		0			0
$136$	−0.500000	−	0.866025i	−0.500000	−	0.866025i
$137$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	0.608761	−	0.793353i	$-0.291667\pi$
$137$	−0.382683	−	0.662827i	−0.382683	−	0.662827i	−0.991445	+	0.130526i	$0.958333\pi$
$138$		0			0
$139$		0			0		−0.980785	−	0.195090i	$-0.937500\pi$
$139$		0			0		0.980785	+	0.195090i	$0.0625000\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−0.991445	+	0.130526i	−0.991445	+	0.130526i
$145$	−2.34237	−	3.05263i	−2.34237	−	3.05263i
$146$	−0.324423	−	0.216773i	−0.324423	−	0.216773i
$147$		0			0
$148$	0.216773	+	0.324423i	0.216773	+	0.324423i
$149$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.866025	−	0.500000i	$-0.833333\pi$
$149$	−1.36603	+	0.366025i	−1.36603	+	0.366025i	−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$151$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$152$		0			0
$153$			1.00000i			1.00000i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	0.198092	−	0.739288i	0.198092	−	0.739288i	−0.793353	−	0.608761i	$-0.791667\pi$
$157$	0.198092	−	0.739288i	0.198092	−	0.739288i	0.991445	−	0.130526i	$-0.0416667\pi$
$158$		0			0
$159$		0			0
$160$	−1.08979	+	1.63099i	−1.08979	+	1.63099i
$161$		0			0
$162$	0.923880	+	0.382683i	0.923880	+	0.382683i
$163$		0			0		−0.442289	−	0.896873i	$-0.645833\pi$
$163$		0			0		0.442289	+	0.896873i	$0.354167\pi$
$164$	0.835400	−	0.732626i	0.835400	−	0.732626i
$165$		0			0
$166$		0			0
$167$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$167$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$168$		0			0
$169$		−	1.00000i		−	1.00000i
$170$	1.47479	+	1.29335i	1.47479	+	1.29335i
$171$		0			0
$172$		0			0
$173$	−0.732626	+	0.835400i	−0.732626	+	0.835400i	−0.991445	−	0.130526i	$-0.958333\pi$
$173$	−0.732626	+	0.835400i	−0.732626	+	0.835400i	0.258819	+	0.965926i	$0.416667\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.130526	−	0.991445i	$-0.541667\pi$
$179$		0			0		0.130526	+	0.991445i	$0.458333\pi$
$180$	1.75928	−	0.867580i	1.75928	−	0.867580i
$181$	0.324423	+	1.63099i	0.324423	+	1.63099i	0.707107	+	0.707107i	$0.250000\pi$
$181$	0.324423	+	1.63099i	0.324423	+	1.63099i	−0.382683	+	0.923880i	$0.625000\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.607206	−	0.465926i	−0.607206	−	0.465926i
$186$		0			0
$187$		0			0
$188$		0			0
$189$		0			0
$190$		0			0
$191$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$191$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$192$		0			0
$193$	−0.293353	+	0.257264i	−0.293353	+	0.257264i	−0.793353	−	0.608761i	$-0.791667\pi$
$193$	−0.293353	+	0.257264i	−0.293353	+	0.257264i	0.500000	+	0.866025i	$0.333333\pi$
$194$	−0.172572	−	0.349942i	−0.172572	−	0.349942i
$195$		0			0
$196$		0			0
$197$	−0.923880	+	1.38268i	−0.923880	+	1.38268i			1.00000i	$0.5\pi$
$197$	−0.923880	+	1.38268i	−0.923880	+	1.38268i	−0.923880	+	0.382683i	$0.875000\pi$
$198$		0			0
$199$		0			0		−0.896873	−	0.442289i	$-0.854167\pi$
$199$		0			0		0.896873	+	0.442289i	$0.145833\pi$
$200$	0.737054	−	2.75072i	0.737054	−	2.75072i
$201$		0			0
$202$	−1.84776	+	0.765367i	−1.84776	+	0.765367i
$203$		0			0
$204$		0			0
$205$	−1.08979	+	1.88757i	−1.08979	+	1.88757i
$206$		0			0
$207$		0			0
$208$	1.36603	−	0.366025i	1.36603	−	0.366025i
$209$		0			0
$210$		0			0
$211$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$211$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$212$	−0.465926	−	0.607206i	−0.465926	−	0.607206i
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$		0			0
$218$	−1.08979	−	0.216773i	−1.08979	−	0.216773i
$219$		0			0
$220$		0			0
$221$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$222$		0			0
$223$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$223$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$224$		0			0
$225$	−2.01367	+	2.01367i	−2.01367	+	2.01367i
$226$	0.108761	+	1.65938i	0.108761	+	1.65938i
$227$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$227$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$228$		0			0
$229$	−1.46593	+	1.12484i	−1.46593	+	1.12484i	−0.500000	+	0.866025i	$0.666667\pi$
$229$	−1.46593	+	1.12484i	−1.46593	+	1.12484i	−0.965926	+	0.258819i	$0.916667\pi$
$230$		0			0
$231$		0			0
$232$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$233$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.608761	−	0.793353i	$-0.708333\pi$
$233$	−1.10876	+	0.0726721i	−1.10876	+	0.0726721i	−0.500000	+	0.866025i	$0.666667\pi$
$234$	−1.36603	−	0.366025i	−1.36603	−	0.366025i
$235$		0			0
$236$		0			0
$237$		0			0
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−0.125419	−	0.369474i	−0.125419	−	0.369474i	0.866025	−	0.500000i	$-0.166667\pi$
$241$	−0.125419	−	0.369474i	−0.125419	−	0.369474i	−0.991445	+	0.130526i	$0.958333\pi$
$242$	−0.965926	−	0.258819i	−0.965926	−	0.258819i
$243$		0			0
$244$	1.63099	−	0.324423i	1.63099	−	0.324423i
$245$		0			0
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$	0.237054	+	3.61675i	0.237054	+	3.61675i
$251$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$251$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$252$		0			0
$253$		0			0
$254$		0			0
$255$		0			0
$256$	0.500000	+	0.866025i	0.500000	+	0.866025i
$257$	1.83195	+	0.241181i	1.83195	+	0.241181i	0.965926	−	0.258819i	$-0.0833333\pi$
$257$	1.83195	+	0.241181i	1.83195	+	0.241181i	0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$		0			0
$260$	−2.30656	+	1.54120i	−2.30656	+	1.54120i
$261$	1.85747	+	0.630526i	1.85747	+	0.630526i
$262$		0			0
$263$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$263$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$264$		0			0
$265$	1.24830	+	0.834089i	1.24830	+	0.834089i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	−0.500000	−	0.866025i	$-0.666667\pi$
$269$	−0.369474	+	0.125419i	−0.369474	+	0.125419i	0.130526	+	0.991445i	$0.458333\pi$
$270$		0			0
$271$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$271$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$272$	0.923880	−	0.382683i	0.923880	−	0.382683i
$273$		0			0
$274$	0.707107	−	0.292893i	0.707107	−	0.292893i
$275$		0			0
$276$		0			0
$277$	−0.996552	−	0.491445i	−0.996552	−	0.491445i	−0.130526	−	0.991445i	$-0.541667\pi$
$277$	−0.996552	−	0.491445i	−0.996552	−	0.491445i	−0.866025	+	0.500000i	$0.833333\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$281$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$282$		0			0
$283$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$283$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		−	1.00000i		−	1.00000i
$289$	−0.258819	−	0.965926i	−0.258819	−	0.965926i
$290$	3.33226	−	1.92388i	3.33226	−	1.92388i
$291$		0			0
$292$	0.257264	−	0.293353i	0.257264	−	0.293353i
$293$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	0.382683	−	0.923880i	$-0.375000\pi$
$293$	−0.541196	−	0.541196i	−0.541196	−	0.541196i	−0.923880	+	0.382683i	$0.875000\pi$
$294$		0			0
$295$		0			0
$296$	−0.349942	+	0.172572i	−0.349942	+	0.172572i
$297$		0			0
$298$	−0.184592	−	1.40211i	−0.184592	−	1.40211i
$299$		0			0
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.82495	+	1.63099i	−2.82495	+	1.63099i
$306$	−0.991445	−	0.130526i	−0.991445	−	0.130526i
$307$		0			0			−	1.00000i	$-0.5\pi$
$307$		0			0				1.00000i	$0.5\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$311$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$312$		0			0
$313$	0.735499	+	1.49144i	0.735499	+	1.49144i	0.866025	+	0.500000i	$0.166667\pi$
$313$	0.735499	+	1.49144i	0.735499	+	1.49144i	−0.130526	+	0.991445i	$0.541667\pi$
$314$	0.707107	+	0.292893i	0.707107	+	0.292893i
$315$		0			0
$316$		0			0
$317$	0.630526	−	1.85747i	0.630526	−	1.85747i	0.130526	−	0.991445i	$-0.458333\pi$
$317$	0.630526	−	1.85747i	0.630526	−	1.85747i	0.500000	−	0.866025i	$-0.333333\pi$
$318$		0			0
$319$		0			0
$320$	−1.47479	−	1.29335i	−1.47479	−	1.29335i
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−0.500000	+	0.866025i	−0.500000	+	0.866025i
$325$	2.45169	−	3.19510i	2.45169	−	3.19510i
$326$		0			0
$327$		0			0
$328$	0.617317	+	0.923880i	0.617317	+	0.923880i
$329$		0			0
$330$		0			0
$331$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$331$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$332$		0			0
$333$	0.389345	+	0.0255190i	0.389345	+	0.0255190i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.923880	−	0.382683i	$-0.125000\pi$
$337$	1.63099	+	0.324423i	1.63099	+	0.324423i	0.707107	+	0.707107i	$0.250000\pi$
$338$	0.991445	+	0.130526i	0.991445	+	0.130526i
$339$		0			0
$340$	−1.47479	+	1.29335i	−1.47479	+	1.29335i
$341$		0			0
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$	−0.732626	−	0.835400i	−0.732626	−	0.835400i
$347$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$347$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$348$		0			0
$349$		0			0		−0.923880	−	0.382683i	$-0.875000\pi$
$349$		0			0		0.923880	+	0.382683i	$0.125000\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.793353	−	0.608761i	$-0.791667\pi$
$353$	−1.78480	−	0.478235i	−1.78480	−	0.478235i	−0.991445	+	0.130526i	$0.958333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$359$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$360$	0.630526	+	1.85747i	0.630526	+	1.85747i
$361$	0.965926	+	0.258819i	0.965926	+	0.258819i
$362$	−1.65938	+	0.108761i	−1.65938	+	0.108761i
$363$		0			0
$364$		0			0
$365$	−0.292893	+	0.707107i	−0.292893	+	0.707107i
$366$		0			0
$367$		0			0		0.0654031	−	0.997859i	$-0.479167\pi$
$367$		0			0		−0.0654031	+	0.997859i	$0.520833\pi$
$368$		0			0
$369$	−0.0726721	−	1.10876i	−0.0726721	−	1.10876i
$370$	0.541196	−	0.541196i	0.541196	−	0.541196i
$371$		0			0
$372$		0			0
$373$	−1.60021	−	0.923880i	−1.60021	−	0.923880i	−0.991445	−	0.130526i	$-0.958333\pi$
$373$	−1.60021	−	0.923880i	−1.60021	−	0.923880i	−0.608761	−	0.793353i	$-0.708333\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−2.72078	−	0.541196i	−2.72078	−	0.541196i
$378$		0			0
$379$		0			0		0.831470	−	0.555570i	$-0.187500\pi$
$379$		0			0		−0.831470	+	0.555570i	$0.812500\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		0			0		−0.608761	−	0.793353i	$-0.708333\pi$
$383$		0			0		0.608761	+	0.793353i	$0.291667\pi$
$384$		0			0
$385$		0			0
$386$	−0.216773	−	0.324423i	−0.216773	−	0.324423i
$387$		0			0
$388$	0.369474	−	0.125419i	0.369474	−	0.125419i
$389$	0.860919	−	1.12197i	0.860919	−	1.12197i	−0.130526	−	0.991445i	$-0.541667\pi$
$389$	0.860919	−	1.12197i	0.860919	−	1.12197i	0.991445	−	0.130526i	$-0.0416667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$	−1.25026	−	1.09645i	−1.25026	−	1.09645i
$395$		0			0
$396$		0			0
$397$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	0.608761	+	0.793353i	$0.291667\pi$
$397$	−0.357164	+	1.05217i	−0.357164	+	1.05217i	−0.965926	+	0.258819i	$0.916667\pi$
$398$		0			0
$399$		0			0
$400$	2.63099	+	1.08979i	2.63099	+	1.08979i
$401$	−0.172572	−	0.349942i	−0.172572	−	0.349942i	0.793353	−	0.608761i	$-0.208333\pi$
$401$	−0.172572	−	0.349942i	−0.172572	−	0.349942i	−0.965926	+	0.258819i	$0.916667\pi$
$402$		0			0
$403$		0			0
$404$	−0.517638	−	1.93185i	−0.517638	−	1.93185i
$405$	0.382683	−	1.92388i	0.382683	−	1.92388i
$406$		0			0
$407$		0			0
$408$		0			0
$409$	0.662827	−	0.382683i	0.662827	−	0.382683i	−0.130526	−	0.991445i	$-0.541667\pi$
$409$	0.662827	−	0.382683i	0.662827	−	0.382683i	0.793353	+	0.608761i	$0.208333\pi$
$410$	−1.72918	−	1.32684i	−1.72918	−	1.32684i
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$		0			0
$416$	0.184592	+	1.40211i	0.184592	+	1.40211i
$417$		0			0
$418$		0			0
$419$		0			0		−0.195090	−	0.980785i	$-0.562500\pi$
$419$		0			0		0.195090	+	0.980785i	$0.437500\pi$
$420$		0			0
$421$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.923880	−	0.382683i	$-0.875000\pi$
$421$	−1.30656	−	1.30656i	−1.30656	−	1.30656i	−0.382683	−	0.923880i	$-0.625000\pi$
$422$		0			0
$423$		0			0
$424$	0.662827	−	0.382683i	0.662827	−	0.382683i
$425$	1.42388	−	2.46623i	1.42388	−	2.46623i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.751840	−	0.659346i	$-0.229167\pi$
$431$		0			0		−0.751840	+	0.659346i	$0.770833\pi$
$432$		0			0
$433$	−0.707107	−	0.292893i	−0.707107	−	0.292893i		−	1.00000i	$-0.5\pi$
$433$	−0.707107	−	0.292893i	−0.707107	−	0.292893i	−0.707107	+	0.707107i	$0.750000\pi$
$434$		0			0
$435$		0			0
$436$	0.357164	−	1.05217i	0.357164	−	1.05217i
$437$		0			0
$438$		0			0
$439$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$439$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$440$		0			0
$441$		0			0
$442$	1.41421			1.41421
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.923880	−	0.382683i	$-0.875000\pi$
$449$	−1.63099	−	1.08979i	−1.63099	−	1.08979i	−0.707107	−	0.707107i	$-0.750000\pi$
$450$	−1.73361	−	2.25928i	−1.73361	−	2.25928i
$451$		0			0
$452$	−1.65938	−	0.108761i	−1.65938	−	0.108761i
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.40211	+	0.184592i	1.40211	+	0.184592i	0.793353	−	0.608761i	$-0.208333\pi$
$457$	1.40211	+	0.184592i	1.40211	+	0.184592i	0.608761	+	0.793353i	$0.291667\pi$
$458$	−0.923880	−	1.60021i	−0.923880	−	1.60021i
$459$		0			0
$460$		0			0
$461$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$461$	−0.765367	−	1.84776i	−0.765367	−	1.84776i	−0.382683	−	0.923880i	$-0.625000\pi$
$462$		0			0
$463$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$463$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$464$	−0.128293	−	1.95737i	−0.128293	−	1.95737i
$465$		0			0
$466$	0.0726721	−	1.10876i	0.0726721	−	1.10876i
$467$		0			0		0.793353	−	0.608761i	$-0.208333\pi$
$467$		0			0		−0.793353	+	0.608761i	$0.791667\pi$
$468$	0.541196	−	1.30656i	0.541196	−	1.30656i
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$		0			0
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−0.765367			−0.765367
$478$		0			0
$479$		0			0		−0.321439	−	0.946930i	$-0.604167\pi$
$479$		0			0		0.321439	+	0.946930i	$0.395833\pi$
$480$		0			0
$481$	−0.550617	+	0.0360894i	−0.550617	+	0.0360894i
$482$	0.382683	−	0.0761205i	0.382683	−	0.0761205i
$483$		0			0
$484$	0.382683	−	0.923880i	0.382683	−	0.923880i
$485$	−0.607206	+	0.465926i	−0.607206	+	0.465926i
$486$		0			0
$487$		0			0		−0.659346	−	0.751840i	$-0.729167\pi$
$487$		0			0		0.659346	+	0.751840i	$0.270833\pi$
$488$	0.108761	+	1.65938i	0.108761	+	1.65938i
$489$		0			0
$490$		0			0
$491$		0			0		−0.382683	−	0.923880i	$-0.625000\pi$
$491$		0			0		0.382683	+	0.923880i	$0.375000\pi$
$492$		0			0
$493$	−1.95737	−	0.128293i	−1.95737	−	0.128293i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$499$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$500$	−3.61675	−	0.237054i	−3.61675	−	0.237054i
$501$		0			0
$502$		0			0
$503$		0			0		−0.831470	−	0.555570i	$-0.812500\pi$
$503$		0			0		0.831470	+	0.555570i	$0.187500\pi$
$504$		0			0
$505$	2.17958	+	3.26197i	2.17958	+	3.26197i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.130526	−	0.991445i	$-0.458333\pi$
$509$	0.923880	−	1.60021i	0.923880	−	1.60021i	0.793353	−	0.608761i	$-0.208333\pi$
$510$		0			0
$511$		0			0
$512$	−0.923880	+	0.382683i	−0.923880	+	0.382683i
$513$		0			0
$514$	−0.478235	+	1.78480i	−0.478235	+	1.78480i
$515$		0			0
$516$		0			0
$517$		0			0
$518$		0			0
$519$		0			0
$520$	−1.22694	−	2.48800i	−1.22694	−	2.48800i
$521$	0.835400	−	0.732626i	0.835400	−	0.732626i	−0.130526	−	0.991445i	$-0.541667\pi$
$521$	0.835400	−	0.732626i	0.835400	−	0.732626i	0.965926	+	0.258819i	$0.0833333\pi$
$522$	−0.867580	+	1.75928i	−0.867580	+	1.75928i
$523$		0			0		−0.258819	−	0.965926i	$-0.583333\pi$
$523$		0			0		0.258819	+	0.965926i	$0.416667\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		0			0
$528$		0			0
$529$	0.793353	+	0.608761i	0.793353	+	0.608761i
$530$	−0.989890	+	1.12875i	−0.989890	+	1.12875i
$531$		0			0
$532$		0			0
$533$	0.306563	+	1.54120i	0.306563	+	1.54120i
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$539$		0			0
$540$		0			0
$541$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.130526	−	0.991445i	$-0.458333\pi$
$541$	1.09645	−	1.25026i	1.09645	−	1.25026i	0.965926	−	0.258819i	$-0.0833333\pi$
$542$		0			0
$543$		0			0
$544$	0.258819	+	0.965926i	0.258819	+	0.965926i
$545$			2.17958i			2.17958i
$546$		0			0
$547$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$547$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$548$	0.198092	+	0.739288i	0.198092	+	0.739288i
$549$	0.735499	−	1.49144i	0.735499	−	1.49144i
$550$		0			0
$551$		0			0
$552$		0			0
$553$		0			0
$554$	0.617317	−	0.923880i	0.617317	−	0.923880i
$555$		0			0
$556$		0			0
$557$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	0.130526	+	0.991445i	$0.458333\pi$
$557$	−0.478235	+	1.78480i	−0.478235	+	1.78480i	−0.608761	+	0.793353i	$0.708333\pi$
$558$		0			0
$559$		0			0
$560$		0			0
$561$		0			0
$562$	0.382683	−	0.662827i	0.382683	−	0.662827i
$563$		0			0		0.608761	−	0.793353i	$-0.291667\pi$
$563$		0			0		−0.608761	+	0.793353i	$0.708333\pi$
$564$		0			0
$565$	3.15082	−	0.844261i	3.15082	−	0.844261i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.12484	−	1.46593i	−1.12484	−	1.46593i	−0.866025	−	0.500000i	$-0.833333\pi$
$569$	−1.12484	−	1.46593i	−1.12484	−	1.46593i	−0.258819	−	0.965926i	$-0.583333\pi$
$570$		0			0
$571$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$571$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	0.991445	+	0.130526i	0.991445	+	0.130526i
$577$	0.707107	+	1.22474i	0.707107	+	1.22474i	0.965926	+	0.258819i	$0.0833333\pi$
$577$	0.707107	+	1.22474i	0.707107	+	1.22474i	−0.258819	+	0.965926i	$0.583333\pi$
$578$	0.991445	−	0.130526i	0.991445	−	0.130526i
$579$		0			0
$580$	1.47247	+	3.55487i	1.47247	+	3.55487i
$581$		0			0
$582$		0			0
$583$		0			0
$584$	0.257264	+	0.293353i	0.257264	+	0.293353i
$585$	−0.181433	+	2.76814i	−0.181433	+	2.76814i
$586$	0.607206	−	0.465926i	0.607206	−	0.465926i
$587$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$587$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$	−0.125419	−	0.369474i	−0.125419	−	0.369474i
$593$	−0.261052	+	1.98289i	−0.261052	+	1.98289i	−0.130526	+	0.991445i	$0.541667\pi$
$593$	−0.261052	+	1.98289i	−0.261052	+	1.98289i	−0.130526	+	0.991445i	$0.541667\pi$
$594$		0			0
$595$		0			0
$596$	1.41421			1.41421
$597$		0			0
$598$		0			0
$599$		0			0		−0.965926	−	0.258819i	$-0.916667\pi$
$599$		0			0		0.965926	+	0.258819i	$0.0833333\pi$
$600$		0			0
$601$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.923880	−	0.382683i	$-0.875000\pi$
$601$	−1.63099	+	0.324423i	−1.63099	+	0.324423i	−0.707107	+	0.707107i	$0.750000\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.128293	+	1.95737i	−0.128293	+	1.95737i
$606$		0			0
$607$		0			0		−0.0654031	−	0.997859i	$-0.520833\pi$
$607$		0			0		0.0654031	+	0.997859i	$0.479167\pi$
$608$		0			0
$609$		0			0
$610$	−1.24830	−	3.01367i	−1.24830	−	3.01367i
$611$		0			0
$612$	0.258819	−	0.965926i	0.258819	−	0.965926i
$613$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	−0.965926	−	0.258819i	$-0.916667\pi$
$613$	−0.707107	−	1.22474i	−0.707107	−	1.22474i	0.258819	−	0.965926i	$-0.416667\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.923880	−	0.617317i	0.923880	−	0.617317i		−	1.00000i	$-0.5\pi$
$617$	0.923880	−	0.617317i	0.923880	−	0.617317i	0.923880	+	0.382683i	$0.125000\pi$
$618$		0			0
$619$		0			0		−0.997859	−	0.0654031i	$-0.979167\pi$
$619$		0			0		0.997859	+	0.0654031i	$0.0208333\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	4.11675	−	1.10308i	4.11675	−	1.10308i
$626$	−1.57469	+	0.534534i	−1.57469	+	0.534534i
$627$		0			0
$628$	−0.382683	+	0.662827i	−0.382683	+	0.662827i
$629$	−0.382683	+	0.0761205i	−0.382683	+	0.0761205i
$630$		0			0
$631$		0			0		0.923880	−	0.382683i	$-0.125000\pi$
$631$		0			0		−0.923880	+	0.382683i	$0.875000\pi$
$632$		0			0
$633$		0			0
$634$	1.75928	+	0.867580i	1.75928	+	0.867580i
$635$		0			0
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$	1.47479	−	1.29335i	1.47479	−	1.29335i
$641$	0.491445	−	0.996552i	0.491445	−	0.996552i	−0.500000	−	0.866025i	$-0.666667\pi$
$641$	0.491445	−	0.996552i	0.491445	−	0.996552i	0.991445	−	0.130526i	$-0.0416667\pi$
$642$		0			0
$643$		0			0		0.195090	−	0.980785i	$-0.437500\pi$
$643$		0			0		−0.195090	+	0.980785i	$0.562500\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$	−0.793353	−	0.608761i	−0.793353	−	0.608761i
$649$		0			0
$650$	2.84776	+	2.84776i	2.84776	+	2.84776i
$651$		0			0
$652$		0			0
$653$	−0.349942	+	0.172572i	−0.349942	+	0.172572i	−0.608761	−	0.793353i	$-0.708333\pi$
$653$	−0.349942	+	0.172572i	−0.349942	+	0.172572i	0.258819	+	0.965926i	$0.416667\pi$
$654$		0			0
$655$		0			0
$656$	−0.996552	+	0.491445i	−0.996552	+	0.491445i
$657$	−0.0761205	−	0.382683i	−0.0761205	−	0.382683i
$658$		0			0
$659$		0			0		−0.707107	−	0.707107i	$-0.750000\pi$
$659$		0			0		0.707107	+	0.707107i	$0.250000\pi$
$660$		0			0
$661$	1.12197	+	0.860919i	1.12197	+	0.860919i	0.991445	−	0.130526i	$-0.0416667\pi$
$661$	1.12197	+	0.860919i	1.12197	+	0.860919i	0.130526	+	0.991445i	$0.458333\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$	−0.0761205	+	0.382683i	−0.0761205	+	0.382683i
$667$		0			0
$668$		0			0
$669$		0			0
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−0.923880	+	1.38268i	−0.923880	+	1.38268i			1.00000i	$0.5\pi$
$673$	−0.923880	+	1.38268i	−0.923880	+	1.38268i	−0.923880	+	0.382683i	$0.875000\pi$
$674$	−0.534534	+	1.57469i	−0.534534	+	1.57469i
$675$		0			0
$676$	−0.258819	+	0.965926i	−0.258819	+	0.965926i
$677$	−0.293353	−	0.257264i	−0.293353	−	0.257264i	0.500000	−	0.866025i	$-0.333333\pi$
$677$	−0.293353	−	0.257264i	−0.293353	−	0.257264i	−0.793353	+	0.608761i	$0.791667\pi$
$678$		0			0
$679$		0			0
$680$	−1.08979	−	1.63099i	−1.08979	−	1.63099i
$681$		0			0
$682$		0			0
$683$		0			0		0.946930	−	0.321439i	$-0.104167\pi$
$683$		0			0		−0.946930	+	0.321439i	$0.895833\pi$
$684$		0			0
$685$	−0.834089	−	1.24830i	−0.834089	−	1.24830i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.07313	−	0.141281i	1.07313	−	0.141281i
$690$		0			0
$691$		0			0		−0.946930	−	0.321439i	$-0.895833\pi$
$691$		0			0		0.946930	+	0.321439i	$0.104167\pi$
$692$	0.923880	−	0.617317i	0.923880	−	0.617317i
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$	0.357164	+	1.05217i	0.357164	+	1.05217i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.382683	−	0.923880i	$-0.375000\pi$
$701$	1.30656	−	1.30656i	1.30656	−	1.30656i	0.923880	−	0.382683i	$-0.125000\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$	0.707107	−	1.70711i	0.707107	−	1.70711i
$707$		0			0
$708$		0			0
$709$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.991445	−	0.130526i	$-0.958333\pi$
$709$	−1.95737	+	0.128293i	−1.95737	+	0.128293i	−0.965926	+	0.258819i	$0.916667\pi$
$710$		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		0.997859	−	0.0654031i	$-0.0208333\pi$
$719$		0			0		−0.997859	+	0.0654031i	$0.979167\pi$
$720$	−1.92388	+	0.382683i	−1.92388	+	0.382683i
$721$		0			0
$722$	−0.382683	+	0.923880i	−0.382683	+	0.923880i
$723$		0			0
$724$	0.108761	−	1.65938i	0.108761	−	1.65938i
$725$	−3.68316	−	4.19984i	−3.68316	−	4.19984i
$726$		0			0
$727$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$727$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$728$		0			0
$729$	0.382683	+	0.923880i	0.382683	+	0.923880i
$730$	−0.662827	−	0.382683i	−0.662827	−	0.382683i
$731$		0			0
$732$		0			0
$733$		0			0		0.130526	−	0.991445i	$-0.458333\pi$
$733$		0			0		−0.130526	+	0.991445i	$0.541667\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$	1.10876	+	0.0726721i	1.10876	+	0.0726721i
$739$		0			0		0.991445	−	0.130526i	$-0.0416667\pi$
$739$		0			0		−0.991445	+	0.130526i	$0.958333\pi$
$740$	0.465926	+	0.607206i	0.465926	+	0.607206i
$741$		0			0
$742$		0			0
$743$		0			0		−0.555570	−	0.831470i	$-0.687500\pi$
$743$		0			0		0.555570	+	0.831470i	$0.312500\pi$
$744$		0			0
$745$	−2.62686	+	0.891699i	−2.62686	+	0.891699i
$746$	1.12484	−	1.46593i	1.12484	−	1.46593i
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$		0			0		−0.751840	−	0.659346i	$-0.770833\pi$
$751$		0			0		0.751840	+	0.659346i	$0.229167\pi$
$752$		0			0
$753$		0			0
$754$	0.891699	−	2.62686i	0.891699	−	2.62686i
$755$		0			0
$756$		0			0
$757$		0			0		0.382683	−	0.923880i	$-0.375000\pi$
$757$		0			0		−0.382683	+	0.923880i	$0.625000\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	0.366025	+	1.36603i	0.366025	+	1.36603i	0.866025	+	0.500000i	$0.166667\pi$
$761$	0.366025	+	1.36603i	0.366025	+	1.36603i	−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	0.128293	+	1.95737i	0.128293	+	1.95737i
$766$		0			0
$767$		0			0
$768$		0			0
$769$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$769$	1.41421	+	1.41421i	1.41421	+	1.41421i	0.707107	+	0.707107i	$0.250000\pi$
$770$		0			0
$771$		0			0
$772$	0.349942	−	0.172572i	0.349942	−	0.172572i
$773$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	−0.500000	−	0.866025i	$-0.666667\pi$
$773$	−0.241181	−	1.83195i	−0.241181	−	1.83195i	0.258819	−	0.965926i	$-0.416667\pi$
$774$		0			0
$775$		0			0
$776$	0.0761205	+	0.382683i	0.0761205	+	0.382683i
$777$		0			0
$778$	1.00000	+	1.00000i	1.00000	+	1.00000i
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$		0			0
$785$	0.292893	−	1.47247i	0.292893	−	1.47247i
$786$		0			0
$787$		0			0		0.442289	−	0.896873i	$-0.354167\pi$
$787$		0			0		−0.442289	+	0.896873i	$0.645833\pi$
$788$	1.25026	−	1.09645i	1.25026	−	1.09645i
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.755946	+	2.22694i	−0.755946	+	2.22694i
$794$	−0.996552	−	0.491445i	−0.996552	−	0.491445i
$795$		0			0

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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	3332.ce (of order \(48\), degree \(16\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.130526 + 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(1.95737 - 0.128293i) q^{5} +(0.382683 - 0.923880i) q^{8} +(-0.793353 + 0.608761i) q^{9} +O(q^{10})\)	\(q+(-0.130526 + 0.991445i) q^{2} +(-0.965926 - 0.258819i) q^{4} +(1.95737 - 0.128293i) q^{5} +(0.382683 - 0.923880i) q^{8} +(-0.793353 + 0.608761i) q^{9} +(-0.128293 + 1.95737i) q^{10} +(1.00000 - 1.00000i) q^{13} +(0.866025 + 0.500000i) q^{16} +(0.608761 - 0.793353i) q^{17} +(-0.500000 - 0.866025i) q^{18} +(-1.92388 - 0.382683i) q^{20} +(2.82340 - 0.371707i) q^{25} +(0.860919 + 1.12197i) q^{26} +(-1.08979 - 1.63099i) q^{29} +(-0.608761 + 0.793353i) q^{32} +(0.707107 + 0.707107i) q^{34} +(0.923880 - 0.382683i) q^{36} +(-0.293353 - 0.257264i) q^{37} +(0.630526 - 1.85747i) q^{40} +(-0.617317 + 0.923880i) q^{41} +(-1.47479 + 1.29335i) q^{45} +2.84776i q^{50} +(-1.22474 + 0.707107i) q^{52} +(0.607206 + 0.465926i) q^{53} +(1.75928 - 0.867580i) q^{58} +(-1.49144 + 0.735499i) q^{61} +(-0.707107 - 0.707107i) q^{64} +(1.82908 - 2.08566i) q^{65} +(-0.793353 + 0.608761i) q^{68} +(0.258819 + 0.965926i) q^{72} +(-0.172572 + 0.349942i) q^{73} +(0.293353 - 0.257264i) q^{74} +(1.75928 + 0.867580i) q^{80} +(0.258819 - 0.965926i) q^{81} +(-0.835400 - 0.732626i) q^{82} +(1.08979 - 1.63099i) q^{85} +(-1.08979 - 1.63099i) q^{90} +(-0.324423 + 0.216773i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{13} - 8 q^{18} - 16 q^{20} + 8 q^{37} + 8 q^{40} - 16 q^{41} - 8 q^{61} - 8 q^{74}+O(q^{100}) \) 16 * q + 16 * q^13 - 8 * q^18 - 16 * q^20 + 8 * q^37 + 8 * q^40 - 16 * q^41 - 8 * q^61 - 8 * q^74

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more