LMFDB - Embedded newform 21.12.g.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,12,Mod(5,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5]))

N = Newforms(chi, 12, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 12);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	21.g (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:	$16.1352067918$
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			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.17
Dual form			21.12.g.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-364.500 - 210.444i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(-38576.5 + 22117.4i) q^{7} +(88573.5 + 153414. i) q^{9} +O(q^{10})$	$q+(-364.500 - 210.444i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(-38576.5 + 22117.4i) q^{7} +(88573.5 + 153414. i) q^{9} +(746496. - 430990. i) q^{12} -1.22066e6i q^{13} +(-2.09715e6 - 3.63237e6i) q^{16} +(1.26186e7 - 7.28537e6i) q^{19} +(1.87156e7 + 56399.0i) q^{21} +(2.44141e7 - 4.22864e7i) q^{25} -7.45591e7i q^{27} +(274432. - 9.10683e7i) q^{28} +(1.01496e8 + 5.85989e7i) q^{31} -3.62797e8 q^{36} +(3.31773e8 + 5.74647e8i) q^{37} +(-2.56880e8 + 4.44930e8i) q^{39} -1.76836e9 q^{43} +1.76533e9i q^{48} +(9.98966e8 - 1.70643e9i) q^{49} +(2.16498e9 + 1.24995e9i) q^{52} -6.13266e9 q^{57} +(1.07618e10 - 6.21334e9i) q^{61} +(-6.80997e9 - 3.95915e9i) q^{63} +8.58993e9 q^{64} +(1.07050e10 - 1.85416e10i) q^{67} +(2.13618e9 + 1.23332e9i) q^{73} +(-1.77979e10 + 1.02756e10i) q^{75} +2.98409e10i q^{76} +(-1.07052e10 - 1.85419e10i) q^{79} +(-1.56905e10 + 2.71768e10i) q^{81} +(-1.92648e10 + 3.31366e10i) q^{84} +(2.69978e10 + 4.70887e10i) q^{91} +(-2.46636e10 - 4.27186e10i) q^{93} -1.26194e11i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 729 q^{3} - 2048 q^{4} - 77153 q^{7} + 177147 q^{9}+O(q^{10}) $ 2 * q - 729 * q^3 - 2048 * q^4 - 77153 * q^7 + 177147 * q^9	$ 2 q - 729 q^{3} - 2048 q^{4} - 77153 q^{7} + 177147 q^{9} + 1492992 q^{12} - 4194304 q^{16} + 25237269 q^{19} + 37431234 q^{21} + 48828125 q^{25} + 548864 q^{28} + 202992585 q^{31} - 725594112 q^{36} + 663545123 q^{37} - 513760563 q^{39} - 3536716270 q^{43} + 1997931923 q^{49} + 4329965568 q^{52} - 12265312734 q^{57} + 21523645452 q^{61} - 13619947095 q^{63} + 17179869184 q^{64} + 21410042863 q^{67} + 4272359229 q^{73} - 35595703125 q^{75} - 21410392133 q^{79} - 31381059609 q^{81} - 38529644544 q^{84} + 53995600899 q^{91} - 49327198155 q^{93}+O(q^{100}) $ 2 * q - 729 * q^3 - 2048 * q^4 - 77153 * q^7 + 177147 * q^9 + 1492992 * q^12 - 4194304 * q^16 + 25237269 * q^19 + 37431234 * q^21 + 48828125 * q^25 + 548864 * q^28 + 202992585 * q^31 - 725594112 * q^36 + 663545123 * q^37 - 513760563 * q^39 - 3536716270 * q^43 + 1997931923 * q^49 + 4329965568 * q^52 - 12265312734 * q^57 + 21523645452 * q^61 - 13619947095 * q^63 + 17179869184 * q^64 + 21410042863 * q^67 + 4272359229 * q^73 - 35595703125 * q^75 - 21410392133 * q^79 - 31381059609 * q^81 - 38529644544 * q^84 + 53995600899 * q^91 - 49327198155 * q^93

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/21\mathbb{Z}\right)^\times$.

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{1}{6}\right)$

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			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	−364.500	−	210.444i	−0.866025	−	0.500000i
$3$	−364.500	−	210.444i	−0.866025	−	0.500000i
$4$	−1024.00	+	1773.62i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−38576.5	+	22117.4i	−0.867528	+	0.497388i
$7$	−38576.5	+	22117.4i	−0.867528	+	0.497388i
$8$		0			0
$9$	88573.5	+	153414.i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	746496.	−	430990.i	0.866025	−	0.500000i
$13$		−	1.22066e6i		−	0.911812i	−0.890028	−	0.455906i	$-0.849315\pi$
$13$		−	1.22066e6i		−	0.911812i	0.890028	−	0.455906i	$-0.150685\pi$
$14$		0			0
$15$		0			0
$16$	−2.09715e6	−	3.63237e6i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	1.26186e7	−	7.28537e6i	1.16914	−	0.675005i	0.215665	−	0.976467i	$-0.430808\pi$
$19$	1.26186e7	−	7.28537e6i	1.16914	−	0.675005i	0.953478	+	0.301463i	$0.0974748\pi$
$20$		0			0
$21$	1.87156e7	+	56399.0i	0.999995	+	0.00301346i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	2.44141e7	−	4.22864e7i	0.500000	−	0.866025i
$26$		0			0
$27$		−	7.45591e7i		−	1.00000i
$28$	274432.	−	9.10683e7i	0.00301346	−	0.999995i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	1.01496e8	+	5.85989e7i	0.636738	+	0.367621i	0.783357	−	0.621572i	$-0.213506\pi$
$31$	1.01496e8	+	5.85989e7i	0.636738	+	0.367621i	−0.146619	+	0.989193i	$0.546839\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−3.62797e8			−1.00000
$37$	3.31773e8	+	5.74647e8i	0.786558	+	1.36236i	0.928064	+	0.372422i	$0.121472\pi$
$37$	3.31773e8	+	5.74647e8i	0.786558	+	1.36236i	−0.141505	+	0.989938i	$0.545194\pi$
$38$		0			0
$39$	−2.56880e8	+	4.44930e8i	−0.455906	+	0.789653i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−1.76836e9			−1.83440			−0.917199	−	0.398429i	$-0.869556\pi$
$43$	−1.76836e9			−1.83440			−0.917199	+	0.398429i	$0.869556\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$			1.76533e9i			1.00000i
$49$	9.98966e8	−	1.70643e9i	0.505210	−	0.862996i
$50$		0			0
$51$		0			0
$52$	2.16498e9	+	1.24995e9i	0.789653	+	0.455906i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−6.13266e9			−1.35001
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	1.07618e10	−	6.21334e9i	1.63144	−	0.941914i	0.647794	−	0.761816i	$-0.275692\pi$
$61$	1.07618e10	−	6.21334e9i	1.63144	−	0.941914i	0.983649	−	0.180098i	$-0.0576415\pi$
$62$		0			0
$63$	−6.80997e9	−	3.95915e9i	−0.864515	−	0.502607i
$64$	8.58993e9			1.00000
$65$		0			0
$66$		0			0
$67$	1.07050e10	−	1.85416e10i	0.968671	−	1.67779i	0.269259	−	0.963068i	$-0.413221\pi$
$67$	1.07050e10	−	1.85416e10i	0.968671	−	1.67779i	0.699412	−	0.714719i	$-0.253445\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	2.13618e9	+	1.23332e9i	0.120604	+	0.0696308i	0.559088	−	0.829108i	$-0.311151\pi$
$73$	2.13618e9	+	1.23332e9i	0.120604	+	0.0696308i	−0.438484	+	0.898739i	$0.644485\pi$
$74$		0			0
$75$	−1.77979e10	+	1.02756e10i	−0.866025	+	0.500000i
$76$			2.98409e10i			1.35001i
$77$		0			0
$78$		0			0
$79$	−1.07052e10	−	1.85419e10i	−0.391422	−	0.677963i	0.601215	−	0.799087i	$-0.294684\pi$
$79$	−1.07052e10	−	1.85419e10i	−0.391422	−	0.677963i	−0.992637	+	0.121124i	$0.961350\pi$
$80$		0			0
$81$	−1.56905e10	+	2.71768e10i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−1.92648e10	+	3.31366e10i	−0.502607	+	0.864515i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	2.69978e10	+	4.70887e10i	0.453524	+	0.791023i
$92$		0			0
$93$	−2.46636e10	−	4.27186e10i	−0.367621	−	0.636738i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.26194e11i		−	1.49209i	−0.665897	−	0.746043i	$-0.731951\pi$
$97$		−	1.26194e11i		−	1.49209i	0.665897	−	0.746043i	$-0.268049\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e10	+	8.66025e10i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−1.51655e11	+	8.75581e10i	−1.28900	+	0.744203i	−0.978476	−	0.206363i	$-0.933837\pi$
$103$	−1.51655e11	+	8.75581e10i	−1.28900	+	0.744203i	−0.310523	+	0.950566i	$0.600504\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	1.32240e11	+	7.63485e10i	0.866025	+	0.500000i
$109$	1.38466e11	−	2.39831e11i	0.861983	−	1.49300i	−0.00802899	−	0.999968i	$-0.502556\pi$
$109$	1.38466e11	−	2.39831e11i	0.861983	−	1.49300i	0.870012	−	0.493031i	$-0.164111\pi$
$110$		0			0
$111$		−	2.79278e11i		−	1.57312i
$112$	1.61240e11	+	9.37407e10i	0.864515	+	0.502607i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	1.87266e11	−	1.08118e11i	0.789653	−	0.455906i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.42656e11	−	2.47087e11i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−2.07864e11	+	1.20011e11i	−0.636738	+	0.367621i
$125$		0			0
$126$		0			0
$127$	−7.44181e11			−1.99875			−0.999374	−	0.0353850i	$-0.988734\pi$
$127$	−7.44181e11			−1.99875			−0.999374	+	0.0353850i	$0.988734\pi$
$128$		0			0
$129$	6.44567e11	+	3.72141e11i	1.58864	+	0.917199i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−3.25649e11	+	5.60136e11i	−0.678525	+	1.16710i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$			1.20507e12i			1.96983i	0.173033	+	0.984916i	$0.444643\pi$
$139$			1.20507e12i			1.96983i	−0.173033	+	0.984916i	$0.555357\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	3.71504e11	−	6.43464e11i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−7.23230e11	+	4.11766e11i	−0.869023	+	0.494771i
$148$	−1.35894e12			−1.57312
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	9.39793e11	−	1.62777e12i	0.974225	−	1.68741i	0.291754	−	0.956493i	$-0.405761\pi$
$151$	9.39793e11	−	1.62777e12i	0.974225	−	1.68741i	0.682470	−	0.730913i	$-0.260906\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−5.26091e11	−	9.11216e11i	−0.455906	−	0.789653i
$157$	2.01217e12	+	1.16173e12i	1.68352	+	0.971979i	0.959292	+	0.282415i	$0.0911356\pi$
$157$	2.01217e12	+	1.16173e12i	1.68352	+	0.971979i	0.724225	+	0.689564i	$0.242198\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.22698e11	−	9.05340e11i	−0.355811	−	0.616283i	0.631445	−	0.775420i	$-0.282462\pi$
$163$	−5.22698e11	−	9.05340e11i	−0.355811	−	0.616283i	−0.987256	+	0.159138i	$0.949129\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	3.02155e11			0.168598
$170$		0			0
$171$	2.23535e12	+	1.29058e12i	1.16914	+	0.675005i
$172$	1.81080e12	−	3.13640e12i	0.917199	−	1.58864i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−6.54297e9	+	2.17124e12i	−0.00301346	+	0.999995i
$176$		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
$181$		−	5.06898e12i		−	1.93949i	−0.244114	−	0.969747i	$-0.578497\pi$
$181$		−	5.06898e12i		−	1.93949i	0.244114	−	0.969747i	$-0.421503\pi$
$182$		0			0
$183$	−5.23025e12			−1.88383
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.64906e12	+	2.87623e12i	0.497388	+	0.867528i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−3.13103e12	−	1.80770e12i	−0.866025	−	0.500000i
$193$	−9.37375e11	+	1.62358e12i	−0.251969	+	0.436424i	−0.964068	−	0.265656i	$-0.914412\pi$
$193$	−9.37375e11	+	1.62358e12i	−0.251969	+	0.436424i	0.712099	+	0.702080i	$0.247745\pi$
$194$		0			0
$195$		0			0
$196$	2.00361e12	+	3.51917e12i	0.494771	+	0.869023i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	7.44676e12	+	4.29939e12i	1.69151	+	0.976595i	0.953299	+	0.302028i	$0.0976636\pi$
$199$	7.44676e12	+	4.29939e12i	1.69151	+	0.976595i	0.738213	+	0.674567i	$0.235670\pi$
$200$		0			0
$201$	−7.80396e12	+	4.50562e12i	−1.67779	+	0.968671i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−4.43388e12	+	2.55990e12i	−0.789653	+	0.455906i
$209$		0			0
$210$		0			0
$211$	−1.60648e12			−0.264436			−0.132218	−	0.991221i	$-0.542210\pi$
$211$	−1.60648e12			−0.264436			−0.132218	+	0.991221i	$0.542210\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.21143e12	−	1.57045e10i	−0.735239	−	0.00221562i
$218$		0			0
$219$	−5.19092e11	−	8.99093e11i	−0.0696308	−	0.120604i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.14390e13i		−	1.38903i	−0.719478	−	0.694515i	$-0.755619\pi$
$223$		−	1.14390e13i		−	1.38903i	0.719478	−	0.694515i	$-0.244381\pi$
$224$		0			0
$225$	8.64976e12			1.00000
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$	6.27984e12	−	1.08770e13i	0.675005	−	1.16914i
$229$	1.38594e13	−	8.00172e12i	1.45428	−	0.839630i	0.455562	−	0.890204i	$-0.349438\pi$
$229$	1.38594e13	−	8.00172e12i	1.45428	−	0.839630i	0.998720	+	0.0505740i	$0.0161051\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			9.01138e12i			0.782845i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−9.17716e12	−	5.29844e12i	−0.727134	−	0.419811i	0.0902386	−	0.995920i	$-0.471237\pi$
$241$	−9.17716e12	−	5.29844e12i	−0.727134	−	0.419811i	−0.817373	+	0.576109i	$0.804570\pi$
$242$		0			0
$243$	1.14384e13	−	6.60396e12i	0.866025	−	0.500000i
$244$			2.54498e13i			1.88383i
$245$		0			0
$246$		0			0
$247$	−8.89294e12	−	1.54030e13i	−0.615478	−	1.06604i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	1.39954e13	−	8.02414e12i	0.867528	−	0.497388i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−8.79609e12	+	1.52353e13i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	−2.55083e13	−	1.48299e13i	−1.35998	−	0.790660i
$260$		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
$268$	2.19239e13	+	3.79733e13i	0.968671	+	1.67779i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	−7.84832e12	+	4.53123e12i	−0.326171	+	0.188315i	−0.654140	−	0.756374i	$-0.726969\pi$
$271$	−7.84832e12	+	4.53123e12i	−0.326171	+	0.188315i	0.327969	+	0.944689i	$0.393636\pi$
$272$		0			0
$273$	6.88439e10	−	2.28454e13i	0.00274771	−	0.911808i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.55797e13	+	2.69848e13i	−0.574010	+	0.994215i	0.422138	+	0.906531i	$0.361280\pi$
$277$	−1.55797e13	+	2.69848e13i	−0.574010	+	0.994215i	−0.996148	+	0.0876833i	$0.972054\pi$
$278$		0			0
$279$			2.07612e13i			0.735242i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−3.54539e13	−	2.04693e13i	−1.16102	−	0.670314i	−0.209470	−	0.977815i	$-0.567174\pi$
$283$	−3.54539e13	−	2.04693e13i	−1.16102	−	0.670314i	−0.951548	+	0.307501i	$0.900507\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.71359e13	+	2.96803e13i	0.500000	+	0.866025i
$290$		0			0
$291$	−2.65568e13	+	4.59977e13i	−0.746043	+	1.29218i
$292$	−4.37490e12	+	2.52585e12i	−0.120604	+	0.0696308i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	4.20888e13i		−	1.00000i
$301$	6.82171e13	−	3.91115e13i	1.59139	−	0.912408i
$302$		0			0
$303$		0			0
$304$	−5.29264e13	−	3.05571e13i	−1.16914	−	0.675005i
$305$		0			0
$306$		0			0
$307$		−	4.94530e13i		−	1.03498i	−0.855689	−	0.517490i	$-0.826866\pi$
$307$		−	4.94530e13i		−	1.03498i	0.855689	−	0.517490i	$-0.173134\pi$
$308$		0			0
$309$	7.37044e13			1.48841
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	9.19394e13	−	5.30813e13i	1.72985	−	0.998728i	0.839716	−	0.543026i	$-0.182722\pi$
$313$	9.19394e13	−	5.30813e13i	1.72985	−	0.998728i	0.890132	−	0.455702i	$-0.150612\pi$
$314$		0			0
$315$		0			0
$316$	4.38485e13			0.782845
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−3.21342e13	−	5.56581e13i	−0.500000	−	0.866025i
$325$	−5.16172e13	−	2.98012e13i	−0.789653	−	0.455906i
$326$		0			0
$327$	−1.00942e14	+	5.82789e13i	−1.49300	+	0.861983i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.15239e13	+	3.72804e13i	0.297760	+	0.515736i	0.975623	−	0.219453i	$-0.0704271\pi$
$331$	2.15239e13	+	3.72804e13i	0.297760	+	0.515736i	−0.677863	+	0.735188i	$0.737094\pi$
$332$		0			0
$333$	−5.87725e13	+	1.01797e14i	−0.786558	+	1.36236i
$334$		0			0
$335$		0			0
$336$	−3.90446e13	−	6.81004e13i	−0.497388	−	0.867528i
$337$	1.59216e14			1.99536			0.997682	−	0.0680538i	$-0.0216789\pi$
$337$	1.59216e14			1.99536			0.997682	+	0.0680538i	$0.0216789\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−7.94876e11	+	8.79225e13i	−0.00904027	+	0.999959i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$			9.06273e13i			0.936956i	0.883475	+	0.468478i	$0.155197\pi$
$349$			9.06273e13i			0.936956i	−0.883475	+	0.468478i	$0.844803\pi$
$350$		0			0
$351$	−9.10111e13			−0.911812
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	4.79082e13	−	8.29794e13i	0.411263	−	0.712329i
$362$		0			0
$363$			1.20084e14i			1.00000i
$364$	−1.11163e14	−	3.34988e11i	−0.911808	−	0.00274771i
$365$		0			0
$366$		0			0
$367$	−2.19737e14	−	1.26865e14i	−1.72282	−	0.994670i	−0.912963	−	0.408043i	$-0.866211\pi$
$367$	−2.19737e14	−	1.26865e14i	−1.72282	−	0.994670i	−0.809857	−	0.586628i	$-0.800455\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	1.01022e14			0.735242
$373$	8.26306e13	+	1.43120e14i	0.592574	+	1.02637i	0.993884	+	0.110426i	$0.0352215\pi$
$373$	8.26306e13	+	1.43120e14i	0.592574	+	1.02637i	−0.401311	+	0.915942i	$0.631445\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.86873e14			−1.88440			−0.942202	−	0.335045i	$-0.891248\pi$
$379$	−2.86873e14			−1.88440			−0.942202	+	0.335045i	$0.891248\pi$
$380$		0			0
$381$	2.71254e14	+	1.56608e14i	1.73097	+	0.999374i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−1.56630e14	−	2.71291e14i	−0.917199	−	1.58864i
$388$	2.23820e14	+	1.29223e14i	1.29218	+	0.746043i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.44063e14	−	1.40910e14i	1.24209	−	0.717123i	0.272573	−	0.962135i	$-0.412125\pi$
$397$	2.44063e14	−	1.40910e14i	1.24209	−	0.717123i	0.969520	+	0.245013i	$0.0787921\pi$
$398$		0			0
$399$	2.36576e14	−	1.35639e14i	1.17117	−	0.671479i
$400$	−2.04800e14			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	7.15292e13	−	1.23892e14i	0.335201	−	0.580586i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.34655e14	+	7.77432e13i	0.581762	+	0.335880i	0.761833	−	0.647773i	$-0.224300\pi$
$409$	1.34655e14	+	7.77432e13i	0.581762	+	0.335880i	−0.180072	+	0.983654i	$0.557633\pi$
$410$		0			0
$411$		0			0
$412$		−	3.58638e14i		−	1.48841i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.53599e14	−	4.39246e14i	0.984916	−	1.70592i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−4.53143e14			−1.66987			−0.834937	−	0.550345i	$-0.814496\pi$
$421$	−4.53143e14			−1.66987			−0.834937	+	0.550345i	$0.814496\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.77730e14	+	4.77713e14i	−0.946826	+	1.62860i
$428$		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	$0.666667\pi$
$432$	−2.70827e14	+	1.56362e14i	−0.866025	+	0.500000i
$433$			3.31368e12i			0.0104623i	0.999986	+	0.00523115i	$0.00166513\pi$
$433$			3.31368e12i			0.0104623i	−0.999986	+	0.00523115i	$0.998335\pi$
$434$		0			0
$435$		0			0
$436$	2.83579e14	+	4.91174e14i	0.861983	+	1.49300i
$437$		0			0
$438$		0			0
$439$	−1.14904e14	+	6.63401e13i	−0.336343	+	0.194188i	−0.658654	−	0.752446i	$-0.728874\pi$
$439$	−1.14904e14	+	6.63401e13i	−0.336343	+	0.194188i	0.322311	+	0.946634i	$0.395540\pi$
$440$		0			0
$441$	3.50271e14	+	2.11109e12i	0.999982	+	0.00602689i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	4.95334e14	+	2.85981e14i	1.36236	+	0.786558i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−3.31370e14	+	1.89987e14i	−0.867528	+	0.497388i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−6.85109e14	+	3.95548e14i	−1.68741	+	0.974225i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−4.26032e14	−	7.37909e14i	−0.999776	−	1.73166i	−0.518202	−	0.855258i	$-0.673398\pi$
$457$	−4.26032e14	−	7.37909e14i	−0.999776	−	1.73166i	−0.481575	−	0.876405i	$-0.659935\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	7.38627e13			0.161335			0.0806677	−	0.996741i	$-0.474295\pi$
$463$	7.38627e13			0.161335			0.0806677	+	0.996741i	$0.474295\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$			4.42851e14i			0.911812i
$469$	−2.86895e12	+	9.52039e14i	−0.00583810	+	1.93733i
$470$		0			0
$471$	−4.88958e14	−	8.46901e14i	−0.971979	−	1.68352i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	7.11462e14i		−	1.35001i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	7.01447e14	−	4.04981e14i	1.24222	−	0.717194i
$482$		0			0
$483$		0			0
$484$	5.84318e14			1.00000
$485$		0			0
$486$		0			0
$487$	2.01966e14	−	3.49815e14i	0.334094	−	0.578668i	−0.649216	−	0.760604i	$-0.724903\pi$
$487$	2.01966e14	−	3.49815e14i	0.334094	−	0.578668i	0.983310	+	0.181936i	$0.0582363\pi$
$488$		0			0
$489$			4.39995e14i			0.711622i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$		−	4.91563e14i		−	0.735242i
$497$		0			0
$498$		0			0
$499$	6.86273e14	+	1.18866e15i	0.992987	+	1.71990i	0.598875	+	0.800843i	$0.295615\pi$
$499$	6.86273e14	+	1.18866e15i	0.992987	+	1.71990i	0.394113	+	0.919062i	$0.371052\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.10136e14	−	6.35868e13i	−0.146011	−	0.0842992i
$508$	7.62041e14	−	1.31989e15i	0.999374	−	1.73097i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−1.09684e14	−	3.30531e11i	−0.139261	−	0.000419659i
$512$		0			0
$513$	−5.43191e14	−	9.40834e14i	−0.675005	−	1.16914i
$514$		0			0
$515$		0			0
$516$	−1.32007e15	+	7.62144e14i	−1.58864	+	0.917199i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	1.53005e15	−	8.83373e14i	1.70980	−	0.987154i	0.775013	−	0.631945i	$-0.217743\pi$
$523$	1.53005e15	−	8.83373e14i	1.70980	−	0.987154i	0.934787	−	0.355208i	$-0.115590\pi$
$524$		0			0
$525$	4.59309e14	−	7.90039e14i	0.502607	−	0.864515i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−4.76405e14	+	8.25157e14i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−6.60003e14	−	1.15116e15i	−0.671479	−	1.17117i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.70602e14	+	1.68113e15i	0.900443	+	1.55961i	0.826921	+	0.562319i	$0.190091\pi$
$541$	9.70602e14	+	1.68113e15i	0.900443	+	1.55961i	0.0735219	+	0.997294i	$0.476576\pi$
$542$		0			0
$543$	−1.06674e15	+	1.84764e15i	−0.969747	+	1.67965i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.19604e15			−1.91739			−0.958695	−	0.284436i	$-0.908194\pi$
$547$	−2.19604e15			−1.91739			−0.958695	+	0.284436i	$0.908194\pi$
$548$		0			0
$549$	1.90642e15	+	1.10067e15i	1.63144	+	0.941914i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	8.23069e14	+	4.78512e14i	0.676781	+	0.393464i
$554$		0			0
$555$		0			0
$556$	−2.13733e15	−	1.23399e15i	−1.70592	−	0.984916i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$			2.15856e15i			1.67263i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.20506e12	−	1.39542e15i	0.00301346	−	0.999995i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−1.44456e15	+	2.50206e15i	−0.995952	+	1.72504i	−0.420131	+	0.907464i	$0.638016\pi$
$571$	−1.44456e15	+	2.50206e15i	−0.995952	+	1.72504i	−0.575821	+	0.817576i	$0.695318\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	7.60841e14	+	1.31781e15i	0.500000	+	0.866025i
$577$	−2.39969e15	−	1.38546e15i	−1.56203	−	0.901836i	−0.997052	−	0.0767272i	$-0.975553\pi$
$577$	−2.39969e15	−	1.38546e15i	−1.56203	−	0.901836i	−0.564974	−	0.825109i	$-0.691114\pi$
$578$		0			0
$579$	6.83346e14	−	3.94530e14i	0.436424	−	0.251969i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	1.02723e13	−	1.70438e15i	0.00602689	−	0.999982i
$589$	1.70766e15			0.992584
$590$		0			0
$591$		0			0
$592$	1.39155e15	−	2.41024e15i	0.786558	−	1.36236i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.80956e15	−	3.13425e15i	−0.976595	−	1.69151i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$		−	1.45689e15i		−	0.757908i	−0.925415	−	0.378954i	$-0.876284\pi$
$601$		−	1.45689e15i		−	0.757908i	0.925415	−	0.378954i	$-0.123716\pi$
$602$		0			0
$603$	3.79272e15			1.93734
$604$	1.92470e15	+	3.33367e15i	0.974225	+	1.68741i
$605$		0			0
$606$		0			0
$607$	3.22685e15	−	1.86302e15i	1.58943	−	0.917656i	0.596025	−	0.802966i	$-0.296746\pi$
$607$	3.22685e15	−	1.86302e15i	1.58943	−	0.917656i	0.993401	−	0.114689i	$-0.0365873\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−1.68506e15	+	2.91861e15i	−0.786289	+	1.36189i	0.141936	+	0.989876i	$0.454667\pi$
$613$	−1.68506e15	+	2.91861e15i	−0.786289	+	1.36189i	−0.928226	+	0.372017i	$0.878666\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−3.67404e15	−	2.12121e15i	−1.62497	−	0.938177i	−0.985563	−	0.169309i	$-0.945846\pi$
$619$	−3.67404e15	−	2.12121e15i	−1.62497	−	0.938177i	−0.639407	−	0.768868i	$-0.720820\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	2.15487e15			0.911812
$625$	−1.19209e15	−	2.06477e15i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−4.12093e15	+	2.37922e15i	−1.68352	+	0.971979i
$629$		0			0
$630$		0			0
$631$	3.70257e15			1.47347			0.736737	−	0.676180i	$-0.236366\pi$
$631$	3.70257e15			1.47347			0.736737	+	0.676180i	$0.236366\pi$
$632$		0			0
$633$	5.85561e14	+	3.38074e14i	0.229009	+	0.132218i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.08296e15	−	1.21940e15i	−0.786891	−	0.460657i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			2.65469e14i			0.0952476i	0.998865	+	0.0476238i	$0.0151649\pi$
$643$			2.65469e14i			0.0952476i	−0.998865	+	0.0476238i	$0.984835\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	1.89626e15	+	1.10244e15i	0.635627	+	0.369538i
$652$	2.14097e15			0.711622
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			4.36959e14i			0.139262i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.37106e15	−	7.91585e14i	−0.422620	−	0.244000i	0.273578	−	0.961850i	$-0.411793\pi$
$661$	−1.37106e15	−	7.91585e14i	−0.422620	−	0.244000i	−0.696198	+	0.717850i	$0.745126\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.40727e15	+	4.16952e15i	−0.694515	+	1.20293i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−4.56750e15			−1.27525			−0.637625	−	0.770347i	$-0.720083\pi$
$673$	−4.56750e15			−1.27525			−0.637625	+	0.770347i	$0.720083\pi$
$674$		0			0
$675$	−3.15284e15	−	1.82029e15i	−0.866025	−	0.500000i
$676$	−3.09407e14	+	5.35909e14i	−0.0842992	+	0.146011i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	2.79109e15	+	4.86812e15i	0.742146	+	1.29443i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−4.57800e15	+	2.64311e15i	−1.16914	+	0.675005i
$685$		0			0
$686$		0			0
$687$	−6.73566e15			−1.67926
$688$	3.70852e15	+	6.42334e15i	0.917199	+	1.58864i
$689$		0			0
$690$		0			0
$691$	5.78241e15	−	3.33848e15i	1.39630	−	0.806156i	0.402301	−	0.915508i	$-0.368211\pi$
$691$	5.78241e15	−	3.33848e15i	1.39630	−	0.806156i	0.994003	+	0.109351i	$0.0348773\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−3.84425e15	−	2.23495e15i	−0.864515	−	0.502607i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	8.37303e15	+	4.83417e15i	1.83920	+	1.06186i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−4.76855e15	−	8.25937e15i	−0.999613	−	1.73138i	−0.523891	−	0.851785i	$-0.675520\pi$
$709$	−4.76855e15	−	8.25937e15i	−0.999613	−	1.73138i	−0.475722	−	0.879595i	$-0.657813\pi$
$710$		0			0
$711$	1.89639e15	−	3.28465e15i	0.391422	−	0.677963i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	3.91376e15	−	6.73191e15i	0.748084	−	1.28675i
$722$		0			0
$723$	2.23005e15	+	3.86256e15i	0.419811	+	0.727134i
$724$	8.99044e15	+	5.19063e15i	1.67965	+	0.969747i
$725$		0			0
$726$		0			0
$727$		−	1.09477e16i		−	1.99933i	−0.0258091	−	0.999667i	$-0.508216\pi$
$727$		−	1.09477e16i		−	1.99933i	0.0258091	−	0.999667i	$-0.491784\pi$
$728$		0			0
$729$	−5.55906e15			−1.00000
$730$		0			0
$731$		0			0
$732$	5.35577e15	−	9.27647e15i	0.941914	−	1.63144i
$733$	−1.43156e15	+	8.26511e14i	−0.249883	+	0.144270i	−0.619711	−	0.784830i	$-0.712750\pi$
$733$	−1.43156e15	+	8.26511e14i	−0.249883	+	0.144270i	0.369827	+	0.929100i	$0.379417\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.70460e15	−	8.14860e15i	0.785196	−	1.36000i	−0.143686	−	0.989623i	$-0.545895\pi$
$739$	4.70460e15	−	8.14860e15i	0.785196	−	1.36000i	0.928882	−	0.370376i	$-0.120771\pi$
$740$		0			0
$741$			7.48587e15i			1.23096i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−8.38135e14	−	1.45169e15i	−0.128025	−	0.221746i	0.794886	−	0.606758i	$-0.207530\pi$
$751$	−8.38135e14	−	1.45169e15i	−0.128025	−	0.221746i	−0.922911	+	0.385013i	$0.874197\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−6.78997e15	−	2.04614e13i	−0.999995	−	0.00301346i
$757$	−1.04544e16			−1.52852			−0.764261	−	0.644907i	$-0.776896\pi$
$757$	−1.04544e16			−1.52852			−0.764261	+	0.644907i	$0.776896\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−3.71090e13	+	1.23144e16i	−0.00519510	+	1.72396i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	6.41235e15	−	3.70217e15i	0.866025	−	0.500000i
$769$			1.18696e16i			1.59163i	0.605538	+	0.795817i	$0.292958\pi$
$769$			1.18696e16i			1.59163i	−0.605538	+	0.795817i	$0.707042\pi$
$770$		0			0
$771$		0			0
$772$	−1.91974e15	−	3.32509e15i	−0.251969	−	0.436424i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	4.95587e15	−	2.86127e15i	0.636738	−	0.367621i
$776$		0			0
$777$	6.17692e15	+	1.07736e16i	0.782449	+	1.36472i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−8.29336e15	−	4.99841e13i	−0.999982	−	0.00602689i
$785$		0			0
$786$		0			0
$787$	9.21945e15	+	5.32285e15i	1.08854	+	0.628468i	0.933187	−	0.359391i	$-0.117015\pi$
$787$	9.21945e15	+	5.32285e15i	1.08854	+	0.628468i	0.155352	+	0.987859i	$0.450349\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.58436e15	−	1.31365e16i	−0.858848	−	1.48757i
$794$		0			0
$795$		0			0
$796$	−1.52510e16	+	8.80514e15i	−1.69151	+	0.976595i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	1.84550e16i		−	1.93734i
$805$		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
$811$			1.80139e16i			1.80299i	0.432792	+	0.901494i	$0.357528\pi$
$811$			1.80139e16i			1.80299i	−0.432792	+	0.901494i	$0.642472\pi$
$812$		0			0
$813$	3.81428e15			0.376630
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.23143e16	+	1.28831e16i	−2.14467	+	1.23823i
$818$		0			0
$819$	−4.83277e15	+	8.31265e15i	−0.458284	+	0.788275i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	9.45387e14	−	1.63746e15i	0.0872792	−	0.151172i	−0.819081	−	0.573678i	$-0.805516\pi$
$823$	9.45387e14	−	1.63746e15i	0.0872792	−	0.151172i	0.906360	+	0.422506i	$0.138849\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	4.45756e15	+	2.57357e15i	0.395410	+	0.228290i	0.684501	−	0.729012i	$-0.260020\pi$
$829$	4.45756e15	+	2.57357e15i	0.395410	+	0.228290i	−0.289092	+	0.957301i	$0.593353\pi$
$830$		0			0
$831$	1.13576e16	−	6.55730e15i	0.994215	−	0.574010i
$832$		−	1.04854e16i		−	0.911812i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	4.36908e15	−	7.56747e15i	0.367621	−	0.636738i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.22005e16			1.00000
$842$		0			0
$843$		0			0
$844$	1.64503e15	−	2.84928e15i	0.132218	−	0.229009i
$845$		0			0
$846$		0			0
$847$	1.09681e16	+	6.37658e15i	0.864515	+	0.502607i
$848$		0			0
$849$	8.61530e15	+	1.49221e16i	0.670314	+	1.16102i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	1.86752e16i		−	1.41594i	−0.706241	−	0.707972i	$-0.749610\pi$
$853$		−	1.86752e16i		−	1.41594i	0.706241	−	0.707972i	$-0.250390\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−3.73549e15	+	2.15669e15i	−0.272512	+	0.157335i	−0.630028	−	0.776572i	$-0.716957\pi$
$859$	−3.73549e15	+	2.15669e15i	−0.272512	+	0.157335i	0.357517	+	0.933907i	$0.383623\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	1.44246e16i		−	1.00000i
$868$	5.36436e15	−	9.22701e15i	0.369538	−	0.635627i
$869$		0			0
$870$		0			0
$871$	−2.26330e16	−	1.30672e16i	−1.52983	−	0.883246i
$872$		0			0
$873$	1.93599e16	−	1.11774e16i	1.29218	−	0.746043i
$874$		0			0
$875$		0			0
$876$	2.12620e15			0.139262
$877$	−6.58130e15	−	1.13991e16i	−0.428365	−	0.741949i	0.568363	−	0.822778i	$-0.307577\pi$
$877$	−6.58130e15	−	1.13991e16i	−0.428365	−	0.741949i	−0.996728	+	0.0808282i	$0.974243\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−1.82537e16			−1.14437			−0.572186	−	0.820124i	$-0.693904\pi$
$883$	−1.82537e16			−1.14437			−0.572186	+	0.820124i	$0.693904\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	2.87079e16	−	1.64594e16i	1.73397	−	0.994153i
$890$		0			0
$891$		0			0
$892$	2.02884e16	+	1.17135e16i	1.20293	+	0.694515i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−8.85735e15	+	1.53414e16i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−3.30959e16	−	9.97337e13i	−1.83439	−	0.00552789i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.13141e16	+	1.95967e16i	−0.612042	+	1.06009i	0.378854	+	0.925457i	$0.376318\pi$
$907$	−1.13141e16	+	1.95967e16i	−0.612042	+	1.06009i	−0.990896	+	0.134631i	$0.957015\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	1.28611e16	+	2.22761e16i	0.675005	+	1.16914i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			3.27750e16i			1.67926i
$917$		0			0
$918$		0			0
$919$	3.58570e15	+	6.21062e15i	0.180443	+	0.312536i	0.942031	−	0.335525i	$-0.108914\pi$
$919$	3.58570e15	+	6.21062e15i	0.180443	+	0.312536i	−0.761589	+	0.648061i	$0.775580\pi$
$920$		0			0
$921$	−1.04071e16	+	1.80256e16i	−0.517490	+	0.896319i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.23997e16			1.57312
$926$		0			0
$927$	−2.68652e16	−	1.55107e16i	−1.28900	−	0.744203i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	1.73642e14	−	2.88106e16i	0.00813637	−	1.34999i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			4.05327e16i			1.83332i	0.399669	+	0.916660i	$0.369125\pi$
$937$			4.05327e16i			1.83332i	−0.399669	+	0.916660i	$0.630875\pi$
$938$		0			0
$939$	−4.46826e16			−1.99746
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	−1.59828e16	−	9.22766e15i	−0.677963	−	0.391422i
$949$	1.50547e15	−	2.60754e15i	0.0634902	−	0.109968i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−5.83657e15	−	1.01092e16i	−0.229710	−	0.397869i
$962$		0			0
$963$		0			0
$964$	1.87948e16	−	1.08512e16i	0.727134	−	0.419811i
$965$		0			0
$966$		0			0
$967$	2.91450e16			1.10846			0.554228	−	0.832365i	$-0.313013\pi$
$967$	2.91450e16			1.10846			0.554228	+	0.832365i	$0.313013\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$			2.70498e16i			1.00000i
$973$	−2.66529e16	−	4.64872e16i	−0.979771	−	1.70888i
$974$		0			0
$975$	1.25430e16	+	2.17251e16i	0.455906	+	0.789653i
$976$	−4.51384e16	−	2.60606e16i	−1.63144	−	0.941914i
$977$		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
$981$	4.90578e16			1.72397
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	3.64255e16			1.23096
$989$		0			0
$990$		0			0
$991$	−4.02399e15	+	6.96976e15i	−0.133737	+	0.231640i	−0.925114	−	0.379689i	$-0.876031\pi$
$991$	−4.02399e15	+	6.96976e15i	−0.133737	+	0.231640i	0.791377	+	0.611328i	$0.209364\pi$
$992$		0			0
$993$		−	1.81183e16i		−	0.595520i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.06971e16	+	2.92700e16i	1.62989	+	0.941020i	0.984123	+	0.177489i	$0.0567974\pi$
$997$	5.06971e16	+	2.92700e16i	1.62989	+	0.941020i	0.645771	+	0.763531i	$0.276536\pi$
$998$		0			0
$999$	4.28452e16	−	2.47367e16i	1.36236	−	0.786558i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.12.g.a.17.1	yes	2
3.2	odd	2	CM	21.12.g.a.17.1	yes	2
7.5	odd	6	inner	21.12.g.a.5.1	✓	2
21.5	even	6	inner	21.12.g.a.5.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.12.g.a.5.1	✓	2	7.5	odd	6	inner
21.12.g.a.5.1	✓	2	21.5	even	6	inner
21.12.g.a.17.1	yes	2	1.1	even	1	trivial
21.12.g.a.17.1	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:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-364.500 - 210.444i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(-38576.5 + 22117.4i) q^{7} +(88573.5 + 153414. i) q^{9} +O(q^{10})\)	\(q+(-364.500 - 210.444i) q^{3} +(-1024.00 + 1773.62i) q^{4} +(-38576.5 + 22117.4i) q^{7} +(88573.5 + 153414. i) q^{9} +(746496. - 430990. i) q^{12} -1.22066e6i q^{13} +(-2.09715e6 - 3.63237e6i) q^{16} +(1.26186e7 - 7.28537e6i) q^{19} +(1.87156e7 + 56399.0i) q^{21} +(2.44141e7 - 4.22864e7i) q^{25} -7.45591e7i q^{27} +(274432. - 9.10683e7i) q^{28} +(1.01496e8 + 5.85989e7i) q^{31} -3.62797e8 q^{36} +(3.31773e8 + 5.74647e8i) q^{37} +(-2.56880e8 + 4.44930e8i) q^{39} -1.76836e9 q^{43} +1.76533e9i q^{48} +(9.98966e8 - 1.70643e9i) q^{49} +(2.16498e9 + 1.24995e9i) q^{52} -6.13266e9 q^{57} +(1.07618e10 - 6.21334e9i) q^{61} +(-6.80997e9 - 3.95915e9i) q^{63} +8.58993e9 q^{64} +(1.07050e10 - 1.85416e10i) q^{67} +(2.13618e9 + 1.23332e9i) q^{73} +(-1.77979e10 + 1.02756e10i) q^{75} +2.98409e10i q^{76} +(-1.07052e10 - 1.85419e10i) q^{79} +(-1.56905e10 + 2.71768e10i) q^{81} +(-1.92648e10 + 3.31366e10i) q^{84} +(2.69978e10 + 4.70887e10i) q^{91} +(-2.46636e10 - 4.27186e10i) q^{93} -1.26194e11i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 729 q^{3} - 2048 q^{4} - 77153 q^{7} + 177147 q^{9}+O(q^{10}) \) 2 * q - 729 * q^3 - 2048 * q^4 - 77153 * q^7 + 177147 * q^9	\( 2 q - 729 q^{3} - 2048 q^{4} - 77153 q^{7} + 177147 q^{9} + 1492992 q^{12} - 4194304 q^{16} + 25237269 q^{19} + 37431234 q^{21} + 48828125 q^{25} + 548864 q^{28} + 202992585 q^{31} - 725594112 q^{36} + 663545123 q^{37} - 513760563 q^{39} - 3536716270 q^{43} + 1997931923 q^{49} + 4329965568 q^{52} - 12265312734 q^{57} + 21523645452 q^{61} - 13619947095 q^{63} + 17179869184 q^{64} + 21410042863 q^{67} + 4272359229 q^{73} - 35595703125 q^{75} - 21410392133 q^{79} - 31381059609 q^{81} - 38529644544 q^{84} + 53995600899 q^{91} - 49327198155 q^{93}+O(q^{100}) \) 2 * q - 729 * q^3 - 2048 * q^4 - 77153 * q^7 + 177147 * q^9 + 1492992 * q^12 - 4194304 * q^16 + 25237269 * q^19 + 37431234 * q^21 + 48828125 * q^25 + 548864 * q^28 + 202992585 * q^31 - 725594112 * q^36 + 663545123 * q^37 - 513760563 * q^39 - 3536716270 * q^43 + 1997931923 * q^49 + 4329965568 * q^52 - 12265312734 * q^57 + 21523645452 * q^61 - 13619947095 * q^63 + 17179869184 * q^64 + 21410042863 * q^67 + 4272359229 * q^73 - 35595703125 * q^75 - 21410392133 * q^79 - 31381059609 * q^81 - 38529644544 * q^84 + 53995600899 * q^91 - 49327198155 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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