LMFDB - Embedded newform 39.11.i.a.35.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [39,11,Mod(29,39)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(39, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 11, names="a")

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

chi := DirichletCharacter("39.29");

S:= CuspForms(chi, 11);

N := Newforms(S);

Level:	$ N $	$=$	$ 39 = 3 \cdot 13 $
Weight:	$ k $	$=$	$ 11 $
Character orbit:	$[\chi]$	$=$	39.i (of order $6$, degree $2$, 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:	$24.7789328543$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			35.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	39.35
Dual form			39.11.i.a.29.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(121.500 - 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-5453.50 - 9445.74i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})$	$q+(121.500 - 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-5453.50 - 9445.74i) q^{7} +(-29524.5 - 51137.9i) q^{9} -248832. q^{12} +(-70980.5 - 364445. i) q^{13} +(-524288. + 908093. i) q^{16} +(1.45079e6 + 2.51284e6i) q^{19} -2.65040e6 q^{21} +9.76562e6 q^{25} -1.43489e7 q^{27} +(-5.58438e6 + 9.67244e6i) q^{28} -4.98436e7 q^{31} +(-3.02331e7 + 5.23652e7i) q^{36} +(-6.76073e7 + 1.17099e8i) q^{37} +(-8.53195e7 - 2.93427e7i) q^{39} +(-3.58361e7 - 6.20700e7i) q^{43} +(1.27402e8 + 2.20667e8i) q^{48} +(8.17563e7 - 1.41606e8i) q^{49} +(-2.86852e8 + 2.49542e8i) q^{52} +7.05082e8 q^{57} +(-7.75745e8 - 1.34363e9i) q^{61} +(-3.22024e8 + 5.57761e8i) q^{63} +1.07374e9 q^{64} +(6.30441e8 - 1.09196e9i) q^{67} +2.18636e9 q^{73} +(1.18652e9 - 2.05512e9i) q^{75} +(1.48561e9 - 2.57314e9i) q^{76} +2.10088e9 q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +(1.35701e9 + 2.35040e9i) q^{84} +(-3.05536e9 + 2.65796e9i) q^{91} +(-6.05599e9 + 1.04893e10i) q^{93} +(-7.20575e9 - 1.24807e10i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 243 q^{3} - 1024 q^{4} - 10907 q^{7} - 59049 q^{9}+O(q^{10}) $ 2 * q + 243 * q^3 - 1024 * q^4 - 10907 * q^7 - 59049 * q^9	$ 2 q + 243 q^{3} - 1024 q^{4} - 10907 q^{7} - 59049 q^{9} - 497664 q^{12} - 141961 q^{13} - 1048576 q^{16} + 2901574 q^{19} - 5300802 q^{21} + 19531250 q^{25} - 28697814 q^{27} - 11168768 q^{28} - 99687146 q^{31} - 60466176 q^{36} - 135214586 q^{37} - 170638974 q^{39} - 71672243 q^{43} + 254803968 q^{48} + 163512600 q^{49} - 573703168 q^{52} + 1410164964 q^{57} - 1551490727 q^{61} - 644047443 q^{63} + 2147483648 q^{64} + 1260882493 q^{67} + 4372711486 q^{73} + 2373046875 q^{75} + 2971211776 q^{76} + 4201763302 q^{79} - 3486784401 q^{81} + 2714010624 q^{84} - 6110723099 q^{91} - 12111988239 q^{93} - 14411495111 q^{97}+O(q^{100}) $ 2 * q + 243 * q^3 - 1024 * q^4 - 10907 * q^7 - 59049 * q^9 - 497664 * q^12 - 141961 * q^13 - 1048576 * q^16 + 2901574 * q^19 - 5300802 * q^21 + 19531250 * q^25 - 28697814 * q^27 - 11168768 * q^28 - 99687146 * q^31 - 60466176 * q^36 - 135214586 * q^37 - 170638974 * q^39 - 71672243 * q^43 + 254803968 * q^48 + 163512600 * q^49 - 573703168 * q^52 + 1410164964 * q^57 - 1551490727 * q^61 - 644047443 * q^63 + 2147483648 * q^64 + 1260882493 * q^67 + 4372711486 * q^73 + 2373046875 * q^75 + 2971211776 * q^76 + 4201763302 * q^79 - 3486784401 * q^81 + 2714010624 * q^84 - 6110723099 * q^91 - 12111988239 * q^93 - 14411495111 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/39\mathbb{Z}\right)^\times$.

$n$	$14$	$28$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

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			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	121.500	−	210.444i	0.500000	−	0.866025i
$3$	121.500	−	210.444i	0.500000	−	0.866025i
$4$	−512.000	−	886.810i	−0.500000	−	0.866025i
$5$		0			0		1.00000			$0$
$5$		0			0		−1.00000			$\pi$
$6$		0			0
$7$	−5453.50	−	9445.74i	−0.324478	−	0.562012i	0.656929	−	0.753953i	$-0.271855\pi$
$7$	−5453.50	−	9445.74i	−0.324478	−	0.562012i	−0.981407	+	0.191941i	$0.938522\pi$
$8$		0			0
$9$	−29524.5	−	51137.9i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−248832.			−1.00000
$13$	−70980.5	−	364445.i	−0.191171	−	0.981557i
$13$	−70980.5	−	364445.i	−0.191171	−	0.981557i
$14$		0			0
$15$		0			0
$16$	−524288.	+	908093.i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	1.45079e6	+	2.51284e6i	0.585916	+	1.01484i	0.994761	+	0.102233i	$0.0325986\pi$
$19$	1.45079e6	+	2.51284e6i	0.585916	+	1.01484i	−0.408844	+	0.912604i	$0.634068\pi$
$20$		0			0
$21$	−2.65040e6			−0.648956
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	9.76562e6			1.00000
$26$		0			0
$27$	−1.43489e7			−1.00000
$28$	−5.58438e6	+	9.67244e6i	−0.324478	+	0.562012i
$29$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$29$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$30$		0			0
$31$	−4.98436e7			−1.74101			−0.870504	−	0.492162i	$-0.836207\pi$
$31$	−4.98436e7			−1.74101			−0.870504	+	0.492162i	$0.836207\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.02331e7	+	5.23652e7i	−0.500000	+	0.866025i
$37$	−6.76073e7	+	1.17099e8i	−0.974956	+	1.68867i	−0.294875	+	0.955536i	$0.595278\pi$
$37$	−6.76073e7	+	1.17099e8i	−0.974956	+	1.68867i	−0.680081	+	0.733137i	$0.738055\pi$
$38$		0			0
$39$	−8.53195e7	−	2.93427e7i	−0.945639	−	0.325219i
$40$		0			0
$41$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$41$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$42$		0			0
$43$	−3.58361e7	−	6.20700e7i	−0.243769	−	0.422221i	0.718016	−	0.696027i	$-0.245051\pi$
$43$	−3.58361e7	−	6.20700e7i	−0.243769	−	0.422221i	−0.961785	+	0.273806i	$0.911717\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		1.00000			$0$
$47$		0			0		−1.00000			$\pi$
$48$	1.27402e8	+	2.20667e8i	0.500000	+	0.866025i
$49$	8.17563e7	−	1.41606e8i	0.289428	−	0.501304i
$50$		0			0
$51$		0			0
$52$	−2.86852e8	+	2.49542e8i	−0.754467	+	0.656337i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.05082e8			1.17183
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−7.75745e8	−	1.34363e9i	−0.918481	−	1.59086i	−0.801724	−	0.597695i	$-0.796084\pi$
$61$	−7.75745e8	−	1.34363e9i	−0.918481	−	1.59086i	−0.116757	−	0.993161i	$-0.537250\pi$
$62$		0			0
$63$	−3.22024e8	+	5.57761e8i	−0.324478	+	0.562012i
$64$	1.07374e9			1.00000
$65$		0			0
$66$		0			0
$67$	6.30441e8	−	1.09196e9i	0.466950	−	0.808782i	−0.532337	−	0.846533i	$-0.678686\pi$
$67$	6.30441e8	−	1.09196e9i	0.466950	−	0.808782i	0.999287	+	0.0377510i	$0.0120194\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$71$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$72$		0			0
$73$	2.18636e9			1.05465			0.527323	−	0.849665i	$-0.323196\pi$
$73$	2.18636e9			1.05465			0.527323	+	0.849665i	$0.323196\pi$
$74$		0			0
$75$	1.18652e9	−	2.05512e9i	0.500000	−	0.866025i
$76$	1.48561e9	−	2.57314e9i	0.585916	−	1.01484i
$77$		0			0
$78$		0			0
$79$	2.10088e9			0.682757			0.341378	−	0.939926i	$-0.389106\pi$
$79$	2.10088e9			0.682757			0.341378	+	0.939926i	$0.389106\pi$
$80$		0			0
$81$	−1.74339e9	+	3.01964e9i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	1.35701e9	+	2.35040e9i	0.324478	+	0.562012i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−3.05536e9	+	2.65796e9i	−0.489616	+	0.425934i
$92$		0			0
$93$	−6.05599e9	+	1.04893e10i	−0.870504	+	1.50776i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−7.20575e9	−	1.24807e10i	−0.839113	−	1.45339i	−0.890637	−	0.454714i	$-0.849741\pi$
$97$	−7.20575e9	−	1.24807e10i	−0.839113	−	1.45339i	0.0515245	−	0.998672i	$-0.483592\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e9	−	8.66025e9i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	−1.82044e10			−1.57032			−0.785162	−	0.619290i	$-0.787421\pi$
$103$	−1.82044e10			−1.57032			−0.785162	+	0.619290i	$0.787421\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	7.34664e9	+	1.27248e10i	0.500000	+	0.866025i
$109$	−1.29873e10			−0.844087			−0.422043	−	0.906576i	$-0.638687\pi$
$109$	−1.29873e10			−0.844087			−0.422043	+	0.906576i	$0.638687\pi$
$110$		0			0
$111$	1.64286e10	+	2.84551e10i	0.974956	+	1.68867i
$112$	1.14368e10			0.648956
$113$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$113$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.65413e10	+	1.43899e10i	−0.754467	+	0.656337i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.29687e10	−	2.24625e10i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	2.55199e10	+	4.42018e10i	0.870504	+	1.50776i
$125$		0			0
$126$		0			0
$127$	2.98375e10	−	5.16800e10i	0.903116	−	1.56424i	0.0796902	−	0.996820i	$-0.474607\pi$
$127$	2.98375e10	−	5.16800e10i	0.903116	−	1.56424i	0.823426	−	0.567424i	$-0.192060\pi$
$128$		0			0
$129$	−1.74164e10			−0.487538
$130$		0			0
$131$		0			0		1.00000			$0$
$131$		0			0		−1.00000			$\pi$
$132$		0			0
$133$	1.58237e10	−	2.74075e10i	0.380234	−	0.658584i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$	−5.14676e10	−	8.91446e10i	−0.991883	−	1.71799i	−0.606062	−	0.795418i	$-0.707252\pi$
$139$	−5.14676e10	−	8.91446e10i	−0.991883	−	1.71799i	−0.385821	−	0.922574i	$-0.626082\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	6.19174e10			1.00000
$145$		0			0
$146$		0			0
$147$	−1.98668e10	−	3.44103e10i	−0.289428	−	0.501304i
$148$	1.38460e11			1.94991
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	1.00499e10			0.128019			0.0640096	−	0.997949i	$-0.479611\pi$
$151$	1.00499e10			0.128019			0.0640096	+	0.997949i	$0.479611\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	1.76622e10	+	9.06856e10i	0.191171	+	0.981557i
$157$	3.19220e10			0.334651			0.167325	−	0.985902i	$-0.446487\pi$
$157$	3.19220e10			0.334651			0.167325	+	0.985902i	$0.446487\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	5.29795e10	+	9.17633e10i	0.460437	+	0.797500i	0.998983	−	0.0450961i	$-0.0143594\pi$
$163$	5.29795e10	+	9.17633e10i	0.460437	+	0.797500i	−0.538546	+	0.842596i	$0.681026\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$167$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$168$		0			0
$169$	−1.27782e11	+	5.17370e10i	−0.926907	+	0.375291i
$170$		0			0
$171$	8.56675e10	−	1.48380e11i	0.585916	−	1.01484i
$172$	−3.66962e10	+	6.35597e10i	−0.243769	+	0.422221i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−5.32568e10	−	9.22435e10i	−0.324478	−	0.562012i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	−3.33760e11			−1.71807			−0.859037	−	0.511914i	$-0.828937\pi$
$181$	−3.33760e11			−1.71807			−0.859037	+	0.511914i	$0.828937\pi$
$182$		0			0
$183$	−3.77012e11			−1.83696
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	7.82518e10	+	1.35536e11i	0.324478	+	0.562012i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	1.30460e11	−	2.25963e11i	0.500000	−	0.866025i
$193$	−2.53616e11	+	4.39276e11i	−0.947088	+	1.64040i	−0.195571	+	0.980689i	$0.562656\pi$
$193$	−2.53616e11	+	4.39276e11i	−0.947088	+	1.64040i	−0.751516	+	0.659715i	$0.770677\pi$
$194$		0			0
$195$		0			0
$196$	−1.67437e11			−0.578856
$197$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$197$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$198$		0			0
$199$	3.06827e11	+	5.31440e11i	0.983170	+	1.70290i	0.649803	+	0.760102i	$0.274851\pi$
$199$	3.06827e11	+	5.31440e11i	0.983170	+	1.70290i	0.333366	+	0.942797i	$0.391815\pi$
$200$		0			0
$201$	−1.53197e11	−	2.65345e11i	−0.466950	−	0.808782i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	3.68164e11	+	1.26617e11i	0.945639	+	0.325219i
$209$		0			0
$210$		0			0
$211$	−3.04832e10	+	5.27985e10i	−0.0728868	+	0.126244i	−0.900165	−	0.435548i	$-0.856555\pi$
$211$	−3.04832e10	+	5.27985e10i	−0.0728868	+	0.126244i	0.827279	+	0.561792i	$0.189888\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.71822e11	+	4.70809e11i	0.564919	+	0.978468i
$218$		0			0
$219$	2.65642e11	−	4.60106e11i	0.527323	−	0.913350i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	5.05249e11	−	8.75117e11i	0.916181	−	1.58687i	0.111019	−	0.993818i	$-0.464589\pi$
$223$	5.05249e11	−	8.75117e11i	0.916181	−	1.58687i	0.805163	−	0.593054i	$-0.202078\pi$
$224$		0			0
$225$	−2.88325e11	−	4.99394e11i	−0.500000	−	0.866025i
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−3.61002e11	−	6.25274e11i	−0.585916	−	1.01484i
$229$	3.52911e11			0.560387			0.280193	−	0.959944i	$-0.409601\pi$
$229$	3.52911e11			0.560387			0.280193	+	0.959944i	$0.409601\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	2.55257e11	−	4.42118e11i	0.341378	−	0.591285i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−2.80808e10	−	4.86373e10i	−0.0345401	−	0.0598252i	0.848239	−	0.529614i	$-0.177663\pi$
$241$	−2.80808e10	−	4.86373e10i	−0.0345401	−	0.0598252i	−0.882779	+	0.469789i	$0.844330\pi$
$242$		0			0
$243$	4.23644e11	+	7.33773e11i	0.500000	+	0.866025i
$244$	−7.94363e11	+	1.37588e12i	−0.918481	+	1.59086i
$245$		0			0
$246$		0			0
$247$	8.12814e11	−	7.07095e11i	0.884110	−	0.769118i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$251$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$252$	6.59505e11			0.648956
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−5.49756e11	−	9.52205e11i	−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
$259$	1.47479e12			1.26541
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−1.29114e12			−0.933900
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−3.18726e11	+	5.52050e11i	−0.218058	+	0.377687i	−0.954214	−	0.299125i	$-0.903305\pi$
$271$	−3.18726e11	+	5.52050e11i	−0.218058	+	0.377687i	0.736157	+	0.676811i	$0.236639\pi$
$272$		0			0
$273$	1.88127e11	+	9.65926e11i	0.124062	+	0.636987i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.46083e12	−	2.53023e12i	−0.895780	−	1.55154i	−0.832836	−	0.553519i	$-0.813285\pi$
$277$	−1.46083e12	−	2.53023e12i	−0.895780	−	1.55154i	−0.0629434	−	0.998017i	$-0.520049\pi$
$278$		0			0
$279$	1.47161e12	+	2.54890e12i	0.870504	+	1.50776i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−9.05338e11	+	1.56809e12i	−0.498745	+	0.863851i	−0.999999	−	0.00144882i	$-0.999539\pi$
$283$	−9.05338e11	+	1.56809e12i	−0.498745	+	0.863851i	0.501254	+	0.865300i	$0.332872\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.00800e12	+	1.74590e12i	−0.500000	+	0.866025i
$290$		0			0
$291$	−3.50199e12			−1.67823
$292$	−1.11941e12	−	1.93888e12i	−0.527323	−	0.913350i
$293$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$293$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−2.43000e12			−1.00000
$301$	−3.90865e11	+	6.76997e11i	−0.158195	+	0.274002i
$302$		0			0
$303$		0			0
$304$	−3.04252e12			−1.17183
$305$		0			0
$306$		0			0
$307$	4.80080e12			1.76044			0.880221	−	0.474564i	$-0.157394\pi$
$307$	4.80080e12			1.76044			0.880221	+	0.474564i	$0.157394\pi$
$308$		0			0
$309$	−2.21183e12	+	3.83100e12i	−0.785162	+	1.35994i
$310$		0			0
$311$		0			0		1.00000			$0$
$311$		0			0		−1.00000			$\pi$
$312$		0			0
$313$	4.88812e12			1.62712			0.813561	−	0.581479i	$-0.197526\pi$
$313$	4.88812e12			1.62712			0.813561	+	0.581479i	$0.197526\pi$
$314$		0			0
$315$		0			0
$316$	−1.07565e12	−	1.86308e12i	−0.341378	−	0.591285i
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	3.57047e12			1.00000
$325$	−6.93169e11	−	3.55903e12i	−0.191171	−	0.981557i
$326$		0			0
$327$	−1.57796e12	+	2.73311e12i	−0.422043	+	0.731001i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.84592e12	−	4.92929e12i	−0.716281	−	1.24064i	−0.962463	−	0.271412i	$-0.912510\pi$
$331$	−2.84592e12	−	4.92929e12i	−0.716281	−	1.24064i	0.246182	−	0.969224i	$-0.420824\pi$
$332$		0			0
$333$	7.98429e12			1.94991
$334$		0			0
$335$		0			0
$336$	1.38957e12	−	2.40681e12i	0.324478	−	0.562012i
$337$	8.25494e12			1.89917			0.949586	−	0.313507i	$-0.101504\pi$
$337$	8.25494e12			1.89917			0.949586	+	0.313507i	$0.101504\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−4.86439e12			−1.02461
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$	4.86715e12	−	8.43014e12i	0.940042	−	1.62820i	0.174655	−	0.984630i	$-0.444119\pi$
$349$	4.86715e12	−	8.43014e12i	0.940042	−	1.62820i	0.765387	−	0.643571i	$-0.222548\pi$
$350$		0			0
$351$	1.01849e12	+	5.22939e12i	0.191171	+	0.981557i
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	−1.14403e12	+	1.98152e12i	−0.186596	+	0.323194i
$362$		0			0
$363$	−6.30279e12			−1.00000
$364$	3.92145e12	+	1.34865e12i	0.613678	+	0.211053i
$365$		0			0
$366$		0			0
$367$	−6.65779e12	+	1.15316e13i	−0.999999	+	1.73205i	−0.498874	+	0.866675i	$0.666253\pi$
$367$	−6.65779e12	+	1.15316e13i	−0.999999	+	1.73205i	−0.501126	+	0.865375i	$0.667081\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.24027e13			1.74101
$373$	−6.91658e12	−	1.19799e13i	−0.957960	−	1.65923i	−0.727445	−	0.686166i	$-0.759292\pi$
$373$	−6.91658e12	−	1.19799e13i	−0.957960	−	1.65923i	−0.230515	−	0.973069i	$-0.574041\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.81736e12	−	1.35401e13i	0.999688	−	1.73151i	0.478200	−	0.878251i	$-0.341289\pi$
$379$	7.81736e12	−	1.35401e13i	0.999688	−	1.73151i	0.521488	−	0.853259i	$-0.325377\pi$
$380$		0			0
$381$	−7.25051e12	−	1.25583e13i	−0.903116	−	1.56424i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.11609e12	+	3.66517e12i	−0.243769	+	0.422221i
$388$	−7.37869e12	+	1.27803e13i	−0.839113	+	1.45339i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	9.55277e12	+	1.65459e13i	0.968672	+	1.67779i	0.699407	+	0.714723i	$0.253447\pi$
$397$	9.55277e12	+	1.65459e13i	0.968672	+	1.67779i	0.269265	+	0.963066i	$0.413219\pi$
$398$		0			0
$399$	−3.84517e12	−	6.66003e12i	−0.380234	−	0.658584i
$400$	−5.12000e12	+	8.86810e12i	−0.500000	+	0.866025i
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	3.53792e12	+	1.81652e13i	0.332830	+	1.70890i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.81089e12	−	1.52609e13i	−0.769845	−	1.33341i	−0.937647	−	0.347589i	$-0.887000\pi$
$409$	−8.81089e12	−	1.52609e13i	−0.769845	−	1.33341i	0.167802	−	0.985821i	$-0.446333\pi$
$410$		0			0
$411$		0			0
$412$	9.32064e12	+	1.61438e13i	0.785162	+	1.35994i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.50133e13			−1.98377
$418$		0			0
$419$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$419$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$420$		0			0
$421$	−2.67749e12			−0.202449			−0.101225	−	0.994864i	$-0.532276\pi$
$421$	−2.67749e12			−0.202449			−0.101225	+	0.994864i	$0.532276\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−8.46105e12	+	1.46550e13i	−0.596053	+	1.03239i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	7.52296e12	−	1.30301e13i	0.500000	−	0.866025i
$433$	−1.28436e13	−	2.22458e13i	−0.843815	−	1.46153i	−0.886646	−	0.462448i	$-0.846971\pi$
$433$	−1.28436e13	−	2.22458e13i	−0.843815	−	1.46153i	0.0428312	−	0.999082i	$-0.486362\pi$
$434$		0			0
$435$		0			0
$436$	6.64951e12	+	1.15173e13i	0.422043	+	0.731001i
$437$		0			0
$438$		0			0
$439$	−1.63037e13	+	2.82389e13i	−0.999919	+	1.73191i	−0.488929	+	0.872323i	$0.662612\pi$
$439$	−1.63037e13	+	2.82389e13i	−0.999919	+	1.73191i	−0.510989	+	0.859587i	$0.670721\pi$
$440$		0			0
$441$	−9.65526e12			−0.578856
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$	1.68229e13	−	2.91380e13i	0.974956	−	1.68867i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−5.85565e12	−	1.01423e13i	−0.324478	−	0.562012i
$449$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$449$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.22106e12	−	2.11493e12i	0.0640096	−	0.110868i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.92304e13	−	3.33080e13i	0.964732	−	1.67096i	0.254398	−	0.967100i	$-0.418123\pi$
$457$	1.92304e13	−	3.33080e13i	0.964732	−	1.67096i	0.710334	−	0.703865i	$-0.248544\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$461$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$462$		0			0
$463$	3.91514e13			1.84010			0.920051	−	0.391798i	$-0.128147\pi$
$463$	3.91514e13			1.84010			0.920051	+	0.391798i	$0.128147\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		1.00000			$0$
$467$		0			0		−1.00000			$\pi$
$468$	2.12302e13	+	7.30139e12i	0.945639	+	0.325219i
$469$	−1.37524e13			−0.606060
$470$		0			0
$471$	3.87852e12	−	6.71780e12i	0.167325	−	0.289816i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.41678e13	+	2.45394e13i	0.585916	+	1.01484i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	4.74751e13	+	1.63274e13i	1.84391	+	0.634149i
$482$		0			0
$483$		0			0
$484$	−1.32800e13	+	2.30016e13i	−0.500000	+	0.866025i
$485$		0			0
$486$		0			0
$487$	−1.93646e13	−	3.35405e13i	−0.706911	−	1.22441i	−0.965997	−	0.258551i	$-0.916755\pi$
$487$	−1.93646e13	−	3.35405e13i	−0.706911	−	1.22441i	0.259087	−	0.965854i	$-0.416578\pi$
$488$		0			0
$489$	2.57481e13			0.920874
$490$		0			0
$491$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$491$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	2.61324e13	−	4.52626e13i	0.870504	−	1.50776i
$497$		0			0
$498$		0			0
$499$	8.03832e12			0.259814			0.129907	−	0.991526i	$-0.458532\pi$
$499$	8.03832e12			0.259814			0.129907	+	0.991526i	$0.458532\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$503$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−4.63777e12	+	3.31770e13i	−0.138442	+	0.990370i
$508$	−6.11072e13			−1.80623
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	−1.19233e13	−	2.06517e13i	−0.342209	−	0.592724i
$512$		0			0
$513$	−2.08172e13	−	3.60565e13i	−0.585916	−	1.01484i
$514$		0			0
$515$		0			0
$516$	8.91717e12	+	1.54450e13i	0.243769	+	0.422221i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$	−7.28720e12	+	1.26218e13i	−0.186231	+	0.322562i	−0.943991	−	0.329972i	$-0.892961\pi$
$523$	−7.28720e12	+	1.26218e13i	−0.186231	+	0.322562i	0.757760	+	0.652534i	$0.226294\pi$
$524$		0			0
$525$	−2.58828e13			−0.648956
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.07133e13	−	3.58764e13i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−3.24070e13			−0.760468
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.65236e13			1.65124			0.825618	−	0.564230i	$-0.190827\pi$
$541$	7.65236e13			1.65124			0.825618	+	0.564230i	$0.190827\pi$
$542$		0			0
$543$	−4.05519e13	+	7.02379e13i	−0.859037	+	1.48790i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.98590e13			0.609732			0.304866	−	0.952395i	$-0.401388\pi$
$547$	2.98590e13			0.609732			0.304866	+	0.952395i	$0.401388\pi$
$548$		0			0
$549$	−4.58070e13	+	7.93400e13i	−0.918481	+	1.59086i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.14572e13	−	1.98444e13i	−0.221540	−	0.383718i
$554$		0			0
$555$		0			0
$556$	−5.27029e13	+	9.12841e13i	−0.991883	+	1.71799i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	−2.00774e13	+	1.74661e13i	−0.367832	+	0.319990i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	3.80304e13			0.648956
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	1.16927e14			1.92635			0.963175	−	0.268877i	$-0.0866524\pi$
$571$	1.16927e14			1.92635			0.963175	+	0.268877i	$0.0866524\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.17017e13	−	5.49089e13i	−0.500000	−	0.866025i
$577$	−1.24945e14			−1.95362			−0.976810	−	0.214108i	$-0.931316\pi$
$577$	−1.24945e14			−1.95362			−0.976810	+	0.214108i	$0.931316\pi$
$578$		0			0
$579$	6.16287e13	+	1.06744e14i	0.947088	+	1.64040i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$587$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$588$	−2.03436e13	+	3.52361e13i	−0.289428	+	0.501304i
$589$	−7.23124e13	−	1.25249e14i	−1.02008	−	1.76684i
$590$		0			0
$591$		0			0
$592$	−7.08914e13	−	1.22787e14i	−0.974956	−	1.68867i
$593$		0			0		1.00000			$0$
$593$		0			0		−1.00000			$\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.49118e14			1.96634
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$	6.02802e13	−	1.04408e14i	0.768780	−	1.33157i	−0.169445	−	0.985540i	$-0.554197\pi$
$601$	6.02802e13	−	1.04408e14i	0.768780	−	1.33157i	0.938225	−	0.346026i	$-0.112469\pi$
$602$		0			0
$603$	−7.44539e13			−0.933900
$604$	−5.14553e12	−	8.91231e12i	−0.0640096	−	0.110868i
$605$		0			0
$606$		0			0
$607$	−4.18630e13	−	7.25088e13i	−0.508027	−	0.879928i	−0.999957	−	0.00929320i	$-0.997042\pi$
$607$	−4.18630e13	−	7.25088e13i	−0.508027	−	0.879928i	0.491930	−	0.870635i	$-0.336291\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.77627e13	+	8.27274e13i	−0.551806	+	0.955756i	0.446339	+	0.894864i	$0.352728\pi$
$613$	−4.77627e13	+	8.27274e13i	−0.551806	+	0.955756i	−0.998144	+	0.0608915i	$0.980606\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$617$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$618$		0			0
$619$	1.71040e14			1.88211			0.941055	−	0.338255i	$-0.109837\pi$
$619$	1.71040e14			1.88211			0.941055	+	0.338255i	$0.109837\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	7.13779e13	−	6.20941e13i	0.754467	−	0.656337i
$625$	9.53674e13			1.00000
$626$		0			0
$627$		0			0
$628$	−1.63441e13	−	2.83088e13i	−0.167325	−	0.289816i
$629$		0			0
$630$		0			0
$631$	9.80584e13	+	1.69842e14i	0.980253	+	1.69785i	0.661381	+	0.750050i	$0.269971\pi$
$631$	9.80584e13	+	1.69842e14i	0.980253	+	1.69785i	0.318872	+	0.947798i	$0.396696\pi$
$632$		0			0
$633$	7.40743e12	+	1.28300e13i	0.0728868	+	0.126244i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.74107e13	−	1.97444e13i	−0.547389	−	0.188255i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$	8.33758e13	+	1.44411e14i	0.758551	+	1.31385i	0.943589	+	0.331118i	$0.107426\pi$
$643$	8.33758e13	+	1.44411e14i	0.758551	+	1.31385i	−0.185038	+	0.982731i	$0.559241\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.32105e14			1.12984
$652$	5.42511e13	−	9.39656e13i	0.460437	−	0.797500i
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−6.45511e13	−	1.11806e14i	−0.527323	−	0.913350i
$658$		0			0
$659$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$659$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$660$		0			0
$661$	1.25946e14	−	2.18145e14i	0.998108	−	1.72877i	0.445811	−	0.895127i	$-0.352916\pi$
$661$	1.25946e14	−	2.18145e14i	0.998108	−	1.72877i	0.552298	−	0.833647i	$-0.313751\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.22776e14	−	2.12654e14i	−0.916181	−	1.58687i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.13238e14	+	1.96135e14i	−0.820197	+	1.42062i	0.0853386	+	0.996352i	$0.472803\pi$
$673$	−1.13238e14	+	1.96135e14i	−0.820197	+	1.42062i	−0.905535	+	0.424271i	$0.860531\pi$
$674$		0			0
$675$	−1.40126e14			−1.00000
$676$	1.11305e14	+	8.68290e13i	0.788465	+	0.615080i
$677$		0			0		1.00000			$0$
$677$		0			0		−1.00000			$\pi$
$678$		0			0
$679$	−7.85931e13	+	1.36127e14i	−0.544547	+	0.943183i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	−1.75447e14			−1.17183
$685$		0			0
$686$		0			0
$687$	4.28787e13	−	7.42681e13i	0.280193	−	0.485309i
$688$	7.51538e13			0.487538
$689$		0			0
$690$		0			0
$691$	−1.39707e14	+	2.41980e14i	−0.886806	+	1.53599i	−0.0431768	+	0.999067i	$0.513748\pi$
$691$	−1.39707e14	+	2.41980e14i	−0.886806	+	1.53599i	−0.843629	+	0.536926i	$0.819585\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−5.45350e13	+	9.44574e13i	−0.324478	+	0.562012i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−3.92335e14			−2.28497
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.00198e14	−	1.73548e14i	−0.559277	−	0.968696i	−0.997557	−	0.0698577i	$-0.977745\pi$
$709$	−1.00198e14	−	1.73548e14i	−0.559277	−	0.968696i	0.438280	−	0.898839i	$-0.355588\pi$
$710$		0			0
$711$	−6.20275e13	−	1.07435e14i	−0.341378	−	0.591285i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	9.92775e13	+	1.71954e14i	0.509536	+	0.882542i
$722$		0			0
$723$	−1.36472e13			−0.0690802
$724$	1.70885e14	+	2.95982e14i	0.859037	+	1.48790i
$725$		0			0
$726$		0			0
$727$	−1.60012e14			−0.787915			−0.393958	−	0.919129i	$-0.628894\pi$
$727$	−1.60012e14			−0.787915			−0.393958	+	0.919129i	$0.628894\pi$
$728$		0			0
$729$	2.05891e14			1.00000
$730$		0			0
$731$		0			0
$732$	1.93030e14	+	3.34338e14i	0.918481	+	1.59086i
$733$	−1.14681e14			−0.541965			−0.270983	−	0.962584i	$-0.587349\pi$
$733$	−1.14681e14			−0.541965			−0.270983	+	0.962584i	$0.587349\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.17779e14	+	3.77205e14i	−0.988085	+	1.71141i	−0.360754	+	0.932661i	$0.617481\pi$
$739$	−2.17779e14	+	3.77205e14i	−0.988085	+	1.71141i	−0.627331	+	0.778753i	$0.715853\pi$
$740$		0			0
$741$	−5.00471e13	−	2.56964e14i	−0.224021	−	1.15022i
$742$		0			0
$743$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$743$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.38243e14	−	4.12648e14i	0.997286	−	1.72735i	0.434881	−	0.900488i	$-0.356791\pi$
$751$	2.38243e14	−	4.12648e14i	0.997286	−	1.72735i	0.562405	−	0.826862i	$-0.309876\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	8.01298e13	−	1.38789e14i	0.324478	−	0.562012i
$757$	5.15250e13	−	8.92439e13i	0.207271	−	0.359004i	−0.743583	−	0.668644i	$-0.766875\pi$
$757$	5.15250e13	−	8.92439e13i	0.207271	−	0.359004i	0.950854	+	0.309640i	$0.100208\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	7.08264e13	+	1.22675e14i	0.273888	+	0.474387i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−2.67181e14			−1.00000
$769$	2.51621e14	−	4.35821e14i	0.935654	−	1.62060i	0.162192	−	0.986759i	$-0.448144\pi$
$769$	2.51621e14	−	4.35821e14i	0.935654	−	1.62060i	0.773463	−	0.633842i	$-0.218523\pi$
$770$		0			0
$771$		0			0
$772$	5.19406e14			1.89418
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−4.86754e14			−1.74101
$776$		0			0
$777$	1.79186e14	−	3.10360e14i	0.632703	−	1.09587i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	8.57277e13	+	1.48485e14i	0.289428	+	0.501304i
$785$		0			0
$786$		0			0
$787$	−2.83987e13	−	4.91880e13i	−0.0940643	−	0.162924i	0.815153	−	0.579245i	$-0.196653\pi$
$787$	−2.83987e13	−	4.91880e13i	−0.0940643	−	0.162924i	−0.909218	+	0.416321i	$0.863319\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.34617e14	+	3.78088e14i	−1.38593	+	1.20567i
$794$		0			0
$795$		0			0
$796$	3.14191e14	−	5.44195e14i	0.983170	−	1.70290i
$797$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$797$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−1.56874e14	+	2.71714e14i	−0.466950	+	0.808782i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	6.78179e14			1.93304			0.966518	−	0.256598i	$-0.0826015\pi$
$811$	6.78179e14			1.93304			0.966518	+	0.256598i	$0.0826015\pi$
$812$		0			0
$813$	7.74504e13	+	1.34148e14i	0.218058	+	0.377687i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.03981e14	−	1.80101e14i	0.285657	−	0.494772i
$818$		0			0
$819$	2.26131e14	+	7.77698e13i	0.613678	+	0.211053i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	1.29097e14	+	2.23603e14i	0.341914	+	0.592213i	0.984788	−	0.173759i	$-0.0555914\pi$
$823$	1.29097e14	+	2.23603e14i	0.341914	+	0.592213i	−0.642874	+	0.765972i	$0.722258\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	2.50739e14	−	4.34293e14i	0.640397	−	1.10920i	−0.344947	−	0.938622i	$-0.612103\pi$
$829$	2.50739e14	−	4.34293e14i	0.640397	−	1.10920i	0.985344	−	0.170578i	$-0.0545636\pi$
$830$		0			0
$831$	−7.09964e14			−1.79156
$832$	−7.62147e13	−	3.91320e14i	−0.191171	−	0.981557i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	7.15201e14			1.74101
$838$		0			0
$839$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$839$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$840$		0			0
$841$	−2.10354e14	−	3.64343e14i	−0.500000	−	0.866025i
$842$		0			0
$843$		0			0
$844$	6.24297e13			0.145774
$845$		0			0
$846$		0			0
$847$	−1.41450e14	+	2.44998e14i	−0.324478	+	0.562012i
$848$		0			0
$849$	2.19997e14	+	3.81046e14i	0.498745	+	0.863851i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−8.94131e14			−1.97996			−0.989979	−	0.141213i	$-0.954900\pi$
$853$	−8.94131e14			−1.97996			−0.989979	+	0.141213i	$0.954900\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		1.00000			$0$
$857$		0			0		−1.00000			$\pi$
$858$		0			0
$859$	5.77614e14			1.23501			0.617507	−	0.786565i	$-0.288143\pi$
$859$	5.77614e14			1.23501			0.617507	+	0.786565i	$0.288143\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	2.44943e14	+	4.24254e14i	0.500000	+	0.866025i
$868$	2.78346e14	−	4.82109e14i	0.564919	−	0.978468i
$869$		0			0
$870$		0			0
$871$	−4.42707e14	−	1.52254e14i	−0.883132	−	0.303722i
$872$		0			0
$873$	−4.25492e14	+	7.36974e14i	−0.839113	+	1.45339i
$874$		0			0
$875$		0			0
$876$	−5.44035e14			−1.05465
$877$	5.11427e14	+	8.85818e14i	0.985794	+	1.70744i	0.638355	+	0.769742i	$0.279615\pi$
$877$	5.11427e14	+	8.85818e14i	0.985794	+	1.70744i	0.347439	+	0.937703i	$0.387051\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$881$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$882$		0			0
$883$	−7.89764e14			−1.47127			−0.735637	−	0.677376i	$-0.763117\pi$
$883$	−7.89764e14			−1.47127			−0.735637	+	0.677376i	$0.763117\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−6.50875e14			−1.17217
$890$		0			0
$891$		0			0
$892$	−1.03475e15			−1.83236
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.95245e14	+	5.11379e14i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	9.49801e13	+	1.64510e14i	0.158195	+	0.274002i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.42131e14	−	5.92588e14i	0.557386	−	0.965420i	−0.440328	−	0.897837i	$-0.645138\pi$
$907$	3.42131e14	−	5.92588e14i	0.557386	−	0.965420i	0.997714	−	0.0675832i	$-0.0215288\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−3.69666e14	+	6.40281e14i	−0.585916	+	1.01484i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−1.80690e14	−	3.12965e14i	−0.280193	−	0.485309i
$917$		0			0
$918$		0			0
$919$	−4.16517e14	−	7.21428e14i	−0.635411	−	1.10056i	−0.986428	−	0.164195i	$-0.947497\pi$
$919$	−4.16517e14	−	7.21428e14i	−0.635411	−	1.10056i	0.351017	−	0.936369i	$-0.385836\pi$
$920$		0			0
$921$	5.83297e14	−	1.01030e15i	0.880221	−	1.52459i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−6.60227e14	+	1.14355e15i	−0.974956	+	1.68867i
$926$		0			0
$927$	5.37475e14	+	9.30934e14i	0.785162	+	1.35994i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	4.74444e14			0.678323
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	4.62865e14			0.640850			0.320425	−	0.947274i	$-0.396174\pi$
$937$	4.62865e14			0.640850			0.320425	+	0.947274i	$0.396174\pi$
$938$		0			0
$939$	5.93907e14	−	1.02868e15i	0.813561	−	1.40913i
$940$		0			0
$941$		0			0		1.00000			$0$
$941$		0			0		−1.00000			$\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−5.22767e14			−0.682757
$949$	−1.55189e14	−	7.96807e14i	−0.201618	−	1.03519i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$953$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	1.66475e15			2.03111
$962$		0			0
$963$		0			0
$964$	−2.87547e13	+	4.98046e13i	−0.0345401	+	0.0598252i
$965$		0			0
$966$		0			0
$967$	1.67628e15			1.98250			0.991252	−	0.131980i	$-0.0421334\pi$
$967$	1.67628e15			1.98250			0.991252	+	0.131980i	$0.0421334\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	4.33812e14	−	7.51384e14i	0.500000	−	0.866025i
$973$	−5.61358e14	+	9.72300e14i	−0.643688	+	1.11490i
$974$		0			0
$975$	−8.33198e14	−	2.86549e14i	−0.945639	−	0.325219i
$976$	1.62686e15			1.83696
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	3.83444e14	+	6.64145e14i	0.422043	+	0.731001i
$982$		0			0
$983$		0			0		1.00000			$0$
$983$		0			0		−1.00000			$\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−1.04322e15	−	3.58779e14i	−1.10813	−	0.381103i
$989$		0			0
$990$		0			0
$991$	1.10677e14	−	1.91698e14i	0.115794	−	0.200562i	−0.802303	−	0.596918i	$-0.796392\pi$
$991$	1.10677e14	−	1.91698e14i	0.115794	−	0.200562i	0.918097	+	0.396356i	$0.129725\pi$
$992$		0			0
$993$	−1.38312e15			−1.43256
$994$		0			0
$995$		0			0
$996$		0			0
$997$	8.62513e14	+	1.49392e15i	0.875568	+	1.51653i	0.856156	+	0.516717i	$0.172846\pi$
$997$	8.62513e14	+	1.49392e15i	0.875568	+	1.51653i	0.0194120	+	0.999812i	$0.493821\pi$
$998$		0			0
$999$	9.70091e14	−	1.68025e15i	0.974956	−	1.68867i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	39.11.i.a.35.1	yes	2
3.2	odd	2	CM	39.11.i.a.35.1	yes	2
13.3	even	3	inner	39.11.i.a.29.1	✓	2
39.29	odd	6	inner	39.11.i.a.29.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.11.i.a.29.1	✓	2	13.3	even	3	inner
39.11.i.a.29.1	✓	2	39.29	odd	6	inner
39.11.i.a.35.1	yes	2	1.1	even	1	trivial
39.11.i.a.35.1	yes	2	3.2	odd	2	CM

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:	\( N \)	\(=\)	\( 39 = 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 11 \)
Character orbit:	\([\chi]\)	\(=\)	39.i (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(121.500 - 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-5453.50 - 9445.74i) q^{7} +(-29524.5 - 51137.9i) q^{9} +O(q^{10})\)	\(q+(121.500 - 210.444i) q^{3} +(-512.000 - 886.810i) q^{4} +(-5453.50 - 9445.74i) q^{7} +(-29524.5 - 51137.9i) q^{9} -248832. q^{12} +(-70980.5 - 364445. i) q^{13} +(-524288. + 908093. i) q^{16} +(1.45079e6 + 2.51284e6i) q^{19} -2.65040e6 q^{21} +9.76562e6 q^{25} -1.43489e7 q^{27} +(-5.58438e6 + 9.67244e6i) q^{28} -4.98436e7 q^{31} +(-3.02331e7 + 5.23652e7i) q^{36} +(-6.76073e7 + 1.17099e8i) q^{37} +(-8.53195e7 - 2.93427e7i) q^{39} +(-3.58361e7 - 6.20700e7i) q^{43} +(1.27402e8 + 2.20667e8i) q^{48} +(8.17563e7 - 1.41606e8i) q^{49} +(-2.86852e8 + 2.49542e8i) q^{52} +7.05082e8 q^{57} +(-7.75745e8 - 1.34363e9i) q^{61} +(-3.22024e8 + 5.57761e8i) q^{63} +1.07374e9 q^{64} +(6.30441e8 - 1.09196e9i) q^{67} +2.18636e9 q^{73} +(1.18652e9 - 2.05512e9i) q^{75} +(1.48561e9 - 2.57314e9i) q^{76} +2.10088e9 q^{79} +(-1.74339e9 + 3.01964e9i) q^{81} +(1.35701e9 + 2.35040e9i) q^{84} +(-3.05536e9 + 2.65796e9i) q^{91} +(-6.05599e9 + 1.04893e10i) q^{93} +(-7.20575e9 - 1.24807e10i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 243 q^{3} - 1024 q^{4} - 10907 q^{7} - 59049 q^{9}+O(q^{10}) \) 2 * q + 243 * q^3 - 1024 * q^4 - 10907 * q^7 - 59049 * q^9	\( 2 q + 243 q^{3} - 1024 q^{4} - 10907 q^{7} - 59049 q^{9} - 497664 q^{12} - 141961 q^{13} - 1048576 q^{16} + 2901574 q^{19} - 5300802 q^{21} + 19531250 q^{25} - 28697814 q^{27} - 11168768 q^{28} - 99687146 q^{31} - 60466176 q^{36} - 135214586 q^{37} - 170638974 q^{39} - 71672243 q^{43} + 254803968 q^{48} + 163512600 q^{49} - 573703168 q^{52} + 1410164964 q^{57} - 1551490727 q^{61} - 644047443 q^{63} + 2147483648 q^{64} + 1260882493 q^{67} + 4372711486 q^{73} + 2373046875 q^{75} + 2971211776 q^{76} + 4201763302 q^{79} - 3486784401 q^{81} + 2714010624 q^{84} - 6110723099 q^{91} - 12111988239 q^{93} - 14411495111 q^{97}+O(q^{100}) \) 2 * q + 243 * q^3 - 1024 * q^4 - 10907 * q^7 - 59049 * q^9 - 497664 * q^12 - 141961 * q^13 - 1048576 * q^16 + 2901574 * q^19 - 5300802 * q^21 + 19531250 * q^25 - 28697814 * q^27 - 11168768 * q^28 - 99687146 * q^31 - 60466176 * q^36 - 135214586 * q^37 - 170638974 * q^39 - 71672243 * q^43 + 254803968 * q^48 + 163512600 * q^49 - 573703168 * q^52 + 1410164964 * q^57 - 1551490727 * q^61 - 644047443 * q^63 + 2147483648 * q^64 + 1260882493 * q^67 + 4372711486 * q^73 + 2373046875 * q^75 + 2971211776 * q^76 + 4201763302 * q^79 - 3486784401 * q^81 + 2714010624 * q^84 - 6110723099 * q^91 - 12111988239 * q^93 - 14411495111 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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