LMFDB - Embedded newform 21.26.g.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,26,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, 26, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 26);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 26 $
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:	$83.1593237900$
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			5.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.5
Dual form			21.26.g.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-797161. + 460241. i) q^{3} +(-1.67772e7 - 2.90590e7i) q^{4} +(-3.65902e10 + 1.49192e9i) q^{7} +(4.23644e11 - 7.33773e11i) q^{9} +O(q^{10})$	\(q+(-797161. + 460241. i) q^{3} +(-1.67772e7 - 2.90590e7i) q^{4} +(-3.65902e10 + 1.49192e9i) q^{7} +(4.23644e11 - 7.33773e11i) q^{9} +(2.67483e13 + 1.54431e13i) q^{12} +3.65142e13i q^{13} +(-5.62950e14 + 9.75058e14i) q^{16} +(4.23170e15 + 2.44317e15i) q^{19} +(2.84817e16 - 1.80296e16i) q^{21} +(1.49012e17 + 2.58096e17i) q^{25} +7.79915e17i q^{27} +(6.57235e17 + 1.03824e18i) q^{28} +(-2.13105e18 + 1.23036e18i) q^{31} -2.84303e19 q^{36} +(-3.53964e19 + 6.13083e19i) q^{37} +(-1.68053e19 - 2.91077e19i) q^{39} +1.81583e20 q^{43} -1.03637e21i q^{48} +(1.33662e21 - 1.09179e20i) q^{49} +(1.06107e21 - 6.12606e20i) q^{52} -4.49780e21 q^{57} +(-3.16341e22 - 1.82640e22i) q^{61} +(-1.44065e22 + 2.74810e22i) q^{63} +3.77789e22 q^{64} +(2.78726e22 + 4.82767e22i) q^{67} +(1.45578e23 - 8.40496e22i) q^{73} +(-2.37573e23 - 1.37163e23i) q^{75} -1.63959e23i q^{76} +(5.23296e23 - 9.06375e23i) q^{79} +(-3.58949e23 - 6.21718e23i) q^{81} +(-1.00177e24 - 5.25161e23i) q^{84} +(-5.44761e22 - 1.33606e24i) q^{91} +(1.13253e24 - 1.96160e24i) q^{93} -8.34202e24i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 1594323 q^{3} - 33554432 q^{4} - 73180401839 q^{7} + 847288609443 q^{9}+O(q^{10}) $ 2 * q - 1594323 * q^3 - 33554432 * q^4 - 73180401839 * q^7 + 847288609443 * q^9	$ 2 q - 1594323 q^{3} - 33554432 q^{4} - 73180401839 q^{7} + 847288609443 q^{9} + 53496602689536 q^{12} - 11\!\cdots\!24 q^{16}+ \cdots + 22\!\cdots\!25 q^{93}+O(q^{100}) $ 2 * q - 1594323 * q^3 - 33554432 * q^4 - 73180401839 * q^7 + 847288609443 * q^9 + 53496602689536 * q^12 - 1125899906842624 * q^16 + 8463400225141671 * q^19 + 56963314943414874 * q^21 + 298023223876953125 * q^25 + 1314470612537704448 * q^28 - 4262104410474914625 * q^31 - 56860576059859402752 * q^36 - 70792710091786388161 * q^37 - 33610669399525960557 * q^39 + 363166767001127849750 * q^43 + 2673233973989584780307 * q^49 + 2122130059293269950464 * q^52 - 8995595758099029555822 * q^57 - 63268297182424491524628 * q^61 - 28813002257883762194625 * q^63 + 75557863725914323419136 * q^64 + 55745137266360974031589 * q^67 + 291156389317363370673573 * q^73 - 475145280361175537109375 * q^75 + 1046591204768194897309693 * q^79 - 717897987691852588770249 * q^81 - 2003531204078174403035136 * q^84 - 108952233186028470179751 * q^91 + 2265057030007199103224625 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	−797161.	+	460241.i	−0.866025	+	0.500000i
$3$	−797161.	+	460241.i	−0.866025	+	0.500000i
$4$	−1.67772e7	−	2.90590e7i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	−3.65902e10	+	1.49192e9i	−0.999170	+	0.0407398i
$7$	−3.65902e10	+	1.49192e9i	−0.999170	+	0.0407398i
$8$		0			0
$9$	4.23644e11	−	7.33773e11i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	2.67483e13	+	1.54431e13i	0.866025	+	0.500000i
$13$			3.65142e13i			0.434680i	0.976096	+	0.217340i	$0.0697380\pi$
$13$			3.65142e13i			0.434680i	−0.976096	+	0.217340i	$0.930262\pi$
$14$		0			0
$15$		0			0
$16$	−5.62950e14	+	9.75058e14i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	4.23170e15	+	2.44317e15i	0.438627	+	0.253241i	0.703015	−	0.711175i	$-0.251837\pi$
$19$	4.23170e15	+	2.44317e15i	0.438627	+	0.253241i	−0.264388	+	0.964416i	$0.585170\pi$
$20$		0			0
$21$	2.84817e16	−	1.80296e16i	0.844937	−	0.534867i
$22$		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
$25$	1.49012e17	+	2.58096e17i	0.500000	+	0.866025i
$26$		0			0
$27$			7.79915e17i			1.00000i
$28$	6.57235e17	+	1.03824e18i	0.534867	+	0.844937i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−2.13105e18	+	1.23036e18i	−0.485929	+	0.280551i	−0.722884	−	0.690969i	$-0.757184\pi$
$31$	−2.13105e18	+	1.23036e18i	−0.485929	+	0.280551i	0.236955	+	0.971521i	$0.423851\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.84303e19			−1.00000
$37$	−3.53964e19	+	6.13083e19i	−0.883969	+	1.53108i	−0.0370780	+	0.999312i	$0.511805\pi$
$37$	−3.53964e19	+	6.13083e19i	−0.883969	+	1.53108i	−0.846891	+	0.531767i	$0.821528\pi$
$38$		0			0
$39$	−1.68053e19	−	2.91077e19i	−0.217340	−	0.376444i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	1.81583e20			0.692980			0.346490	−	0.938054i	$-0.387374\pi$
$43$	1.81583e20			0.692980			0.346490	+	0.938054i	$0.387374\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$		−	1.03637e21i		−	1.00000i
$49$	1.33662e21	−	1.09179e20i	0.996681	−	0.0814120i
$50$		0			0
$51$		0			0
$52$	1.06107e21	−	6.12606e20i	0.376444	−	0.217340i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−4.49780e21			−0.506482
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	−3.16341e22	−	1.82640e22i	−1.52593	−	0.880993i	−0.999527	−	0.0307551i	$-0.990209\pi$
$61$	−3.16341e22	−	1.82640e22i	−1.52593	−	0.880993i	−0.526398	−	0.850238i	$-0.676458\pi$
$62$		0			0
$63$	−1.44065e22	+	2.74810e22i	−0.464303	+	0.885676i
$64$	3.77789e22			1.00000
$65$		0			0
$66$		0			0
$67$	2.78726e22	+	4.82767e22i	0.416142	+	0.720780i	0.995548	−	0.0942598i	$-0.0300484\pi$
$67$	2.78726e22	+	4.82767e22i	0.416142	+	0.720780i	−0.579405	+	0.815040i	$0.696715\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1.45578e23	−	8.40496e22i	0.743979	−	0.429536i	−0.0795354	−	0.996832i	$-0.525344\pi$
$73$	1.45578e23	−	8.40496e22i	0.743979	−	0.429536i	0.823514	+	0.567296i	$0.192010\pi$
$74$		0			0
$75$	−2.37573e23	−	1.37163e23i	−0.866025	−	0.500000i
$76$		−	1.63959e23i		−	0.506482i
$77$		0			0
$78$		0			0
$79$	5.23296e23	−	9.06375e23i	0.996342	−	1.72571i	0.424165	−	0.905585i	$-0.360568\pi$
$79$	5.23296e23	−	9.06375e23i	0.996342	−	1.72571i	0.572177	−	0.820130i	$-0.306099\pi$
$80$		0			0
$81$	−3.58949e23	−	6.21718e23i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−1.00177e24	−	5.25161e23i	−0.885676	−	0.464303i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−5.44761e22	−	1.33606e24i	−0.0177088	−	0.434319i
$92$		0			0
$93$	1.13253e24	−	1.96160e24i	0.280551	−	0.485929i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	8.34202e24i		−	1.22074i	−0.792115	−	0.610372i	$-0.791020\pi$
$97$		−	8.34202e24i		−	1.22074i	0.792115	−	0.610372i	$-0.208980\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e24	−	8.66025e24i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	−2.03629e25	−	1.17565e25i	−1.40726	−	0.812483i	−0.412139	−	0.911121i	$-0.635218\pi$
$103$	−2.03629e25	−	1.17565e25i	−1.40726	−	0.812483i	−0.995123	+	0.0986380i	$0.968551\pi$
$104$		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	$0.666667\pi$
$108$	2.26635e25	−	1.30848e25i	0.866025	−	0.500000i
$109$	−2.86589e25	−	4.96387e25i	−0.975952	−	1.69040i	−0.676758	−	0.736205i	$-0.736616\pi$
$109$	−2.86589e25	−	4.96387e25i	−0.975952	−	1.69040i	−0.299193	−	0.954192i	$-0.596718\pi$
$110$		0			0
$111$		−	6.51635e25i		−	1.76794i
$112$	1.91437e25	−	3.65174e25i	0.464303	−	0.885676i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.67931e25	+	1.54690e25i	0.376444	+	0.217340i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.41735e25	+	9.38313e25i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	7.15062e25	+	4.12842e25i	0.485929	+	0.280551i
$125$		0			0
$126$		0			0
$127$	2.47296e25			0.124644			0.0623219	−	0.998056i	$-0.480149\pi$
$127$	2.47296e25			0.124644			0.0623219	+	0.998056i	$0.480149\pi$
$128$		0			0
$129$	−1.44751e26	+	8.35722e25i	−0.600138	+	0.346490i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−1.58484e26	−	8.30829e25i	−0.448579	−	0.235161i
$134$		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
$139$			3.84406e26i			0.626765i	0.949627	+	0.313383i	$0.101462\pi$
$139$			3.84406e26i			0.626765i	−0.949627	+	0.313383i	$0.898538\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	4.76981e26	+	8.26155e26i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−1.01525e27	+	7.02200e26i	−0.822445	+	0.568845i
$148$	2.37541e27			1.76794
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	1.40001e27	+	2.42488e27i	0.810808	+	1.40436i	0.912299	+	0.409525i	$0.134305\pi$
$151$	1.40001e27	+	2.42488e27i	0.810808	+	1.40436i	−0.101491	+	0.994836i	$0.532361\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−5.63893e26	+	9.76692e26i	−0.217340	+	0.376444i
$157$	−3.06856e27	+	1.77163e27i	−1.09191	+	0.630417i	−0.934085	−	0.357050i	$-0.883783\pi$
$157$	−3.06856e27	+	1.77163e27i	−1.09191	+	0.630417i	−0.157828	+	0.987467i	$0.550449\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−4.46249e27	+	7.72926e27i	−0.993647	+	1.72105i	−0.399362	+	0.916793i	$0.630768\pi$
$163$	−4.46249e27	+	7.72926e27i	−0.993647	+	1.72105i	−0.594286	+	0.804254i	$0.702565\pi$
$164$		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
$169$	5.72313e27			0.811053
$170$		0			0
$171$	3.58547e27	−	2.07007e27i	0.438627	−	0.253241i
$172$	−3.04646e27	−	5.27663e27i	−0.346490	−	0.600138i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	−5.83742e27	−	9.22146e27i	−0.534867	−	0.844937i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	3.18169e28i		−	1.91283i	−0.292012	−	0.956415i	$-0.594325\pi$
$181$		−	3.18169e28i		−	1.91283i	0.292012	−	0.956415i	$-0.405675\pi$
$182$		0			0
$183$	3.36234e28			1.76199
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.16357e27	−	2.85372e28i	−0.0407398	−	0.999170i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−3.01159e28	+	1.73874e28i	−0.866025	+	0.500000i
$193$	−1.07283e28	−	1.85820e28i	−0.289111	−	0.500755i	0.684487	−	0.729025i	$-0.260026\pi$
$193$	−1.07283e28	−	1.85820e28i	−0.289111	−	0.500755i	−0.973598	+	0.228270i	$0.926693\pi$
$194$		0			0
$195$		0			0
$196$	−2.55973e28	−	3.70090e28i	−0.568845	−	0.822445i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−6.05359e27	+	3.49504e27i	−0.111263	+	0.0642375i	−0.554599	−	0.832118i	$-0.687128\pi$
$199$	−6.05359e27	+	3.49504e27i	−0.111263	+	0.0642375i	0.443336	+	0.896356i	$0.353795\pi$
$200$		0			0
$201$	−4.44379e28	−	2.56562e28i	−0.720780	−	0.416142i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−3.56034e28	−	2.05557e28i	−0.376444	−	0.217340i
$209$		0			0
$210$		0			0
$211$	2.04283e29			1.80593			0.902964	−	0.429716i	$-0.141386\pi$
$211$	2.04283e29			1.80593			0.902964	+	0.429716i	$0.141386\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	7.61400e28	−	4.81986e28i	0.474096	−	0.300115i
$218$		0			0
$219$	−7.73662e28	+	1.34002e29i	−0.429536	+	0.743979i
$220$		0			0
$221$		0			0
$222$		0			0
$223$			2.32199e29i			1.02813i	0.857750	+	0.514067i	$0.171862\pi$
$223$			2.32199e29i			1.02813i	−0.857750	+	0.514067i	$0.828138\pi$
$224$		0			0
$225$	2.52512e29			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	7.54605e28	+	1.30701e29i	0.253241	+	0.438627i
$229$	−4.91294e29	−	2.83649e29i	−1.56098	−	0.901233i	−0.997158	−	0.0753380i	$-0.975996\pi$
$229$	−4.91294e29	−	2.83649e29i	−1.56098	−	0.901233i	−0.563824	−	0.825895i	$-0.690670\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			9.63369e29i			1.99268i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−6.64211e29	+	3.83482e29i	−1.11453	+	0.643476i	−0.940000	−	0.341175i	$-0.889175\pi$
$241$	−6.64211e29	+	3.83482e29i	−1.11453	+	0.643476i	−0.174533	+	0.984651i	$0.555842\pi$
$242$		0			0
$243$	5.72281e29	+	3.30406e29i	0.866025	+	0.500000i
$244$			1.22568e30i			1.76199i
$245$		0			0
$246$		0			0
$247$	−8.92105e28	+	1.54517e29i	−0.110079	+	0.190662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	1.04027e30	−	4.24156e28i	0.999170	−	0.0407398i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−6.33825e29	−	1.09782e30i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	1.20369e30	−	2.29609e30i	0.820859	−	1.56582i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	9.35248e29	−	1.61990e30i	0.416142	−	0.720780i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−4.28985e30	−	2.47675e30i	−1.66083	−	0.958882i	−0.972319	−	0.233659i	$-0.924930\pi$
$271$	−4.28985e30	−	2.47675e30i	−1.66083	−	0.958882i	−0.688514	−	0.725223i	$-0.741737\pi$
$272$		0			0
$273$	6.58337e29	+	1.03998e30i	0.232496	+	0.367277i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.37776e30	−	5.85045e30i	−0.994562	−	1.72263i	−0.587475	−	0.809242i	$-0.699878\pi$
$277$	−3.37776e30	−	5.85045e30i	−0.994562	−	1.72263i	−0.407087	−	0.913390i	$-0.633455\pi$
$278$		0			0
$279$			2.08495e30i			0.561103i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	3.86878e30	−	2.23364e30i	0.871453	−	0.503133i	0.00362204	−	0.999993i	$-0.498847\pi$
$283$	3.86878e30	−	2.23364e30i	0.871453	−	0.503133i	0.867831	+	0.496860i	$0.165514\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	2.88531e30	−	4.99751e30i	0.500000	−	0.866025i
$290$		0			0
$291$	3.83934e30	+	6.64994e30i	0.610372	+	1.05719i
$292$	−4.88479e30	−	2.82024e30i	−0.743979	−	0.429536i
$293$		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
$300$			9.20483e30i			1.00000i
$301$	−6.64417e30	+	2.70907e29i	−0.692404	+	0.0282319i
$302$		0			0
$303$		0			0
$304$	−4.76447e30	+	2.75077e30i	−0.438627	+	0.253241i
$305$		0			0
$306$		0			0
$307$		−	2.31994e31i		−	1.88907i	−0.328416	−	0.944533i	$-0.606515\pi$
$307$		−	2.31994e31i		−	1.88907i	0.328416	−	0.944533i	$-0.393485\pi$
$308$		0			0
$309$	2.16434e31			1.62497
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	1.40001e31	+	8.08297e30i	0.895009	+	0.516734i	0.875578	−	0.483078i	$-0.160481\pi$
$313$	1.40001e31	+	8.08297e30i	0.895009	+	0.516734i	0.0194314	+	0.999811i	$0.493814\pi$
$314$		0			0
$315$		0			0
$316$	−3.51178e31			−1.99268
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−1.20443e31	+	2.08614e31i	−0.500000	+	0.866025i
$325$	−9.42415e30	+	5.44104e30i	−0.376444	+	0.217340i
$326$		0			0
$327$	4.56915e31	+	2.63800e31i	1.69040	+	0.975952i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.99604e31	+	3.45724e31i	−0.634336	+	1.09870i	0.352320	+	0.935880i	$0.385393\pi$
$331$	−1.99604e31	+	3.45724e31i	−0.634336	+	1.09870i	−0.986656	+	0.162822i	$0.947940\pi$
$332$		0			0
$333$	2.99909e31	+	5.19458e31i	0.883969	+	1.53108i
$334$		0			0
$335$		0			0
$336$	1.54618e30	+	3.79210e31i	0.0407398	+	0.999170i
$337$	2.12866e31			0.540421			0.270211	−	0.962801i	$-0.412907\pi$
$337$	2.12866e31			0.540421			0.270211	+	0.962801i	$0.412907\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−4.87442e31	+	5.98901e30i	−0.992536	+	0.121949i
$344$		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
$349$		−	1.20089e32i		−	1.96872i	−0.176165	−	0.984361i	$-0.556369\pi$
$349$		−	1.20089e32i		−	1.96872i	0.176165	−	0.984361i	$-0.443631\pi$
$350$		0			0
$351$	−2.84779e31			−0.434680
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−3.46001e31	−	5.99291e31i	−0.371738	−	0.643869i
$362$		0			0
$363$		−	9.97316e31i		−	1.00000i
$364$	−3.79106e31	+	2.39984e31i	−0.367277	+	0.232496i
$365$		0			0
$366$		0			0
$367$	1.71881e32	−	9.92358e31i	1.50281	−	0.867646i	0.502812	−	0.864396i	$-0.332299\pi$
$367$	1.71881e32	−	9.92358e31i	1.50281	−	0.867646i	0.999995	−	0.00325061i	$-0.00103470\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−7.60027e31			−0.561103
$373$	1.04615e32	−	1.81199e32i	0.746853	−	1.29359i	−0.202471	−	0.979288i	$-0.564897\pi$
$373$	1.04615e32	−	1.81199e32i	0.746853	−	1.29359i	0.949324	−	0.314299i	$-0.101769\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.13562e31			−0.124892			−0.0624460	−	0.998048i	$-0.519890\pi$
$379$	−2.13562e31			−0.124892			−0.0624460	+	0.998048i	$0.519890\pi$
$380$		0			0
$381$	−1.97135e31	+	1.13816e31i	−0.107945	+	0.0623219i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.69268e31	−	1.33241e32i	0.346490	−	0.600138i
$388$	−2.42411e32	+	1.39956e32i	−1.05719	+	0.610372i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−5.48226e31	−	3.16518e31i	−0.179506	−	0.103638i	0.407555	−	0.913181i	$-0.366382\pi$
$397$	−5.48226e31	−	3.16518e31i	−0.179506	−	0.103638i	−0.587060	+	0.809543i	$0.699715\pi$
$398$		0			0
$399$	1.64575e32	−	6.71034e30i	0.506062	−	0.0206340i
$400$	−3.35544e32			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−4.49257e31	−	7.78136e31i	−0.121950	−	0.211224i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.73183e32	−	1.57723e32i	0.616471	−	0.355920i	−0.159022	−	0.987275i	$-0.550834\pi$
$409$	2.73183e32	−	1.57723e32i	0.616471	−	0.355920i	0.775494	+	0.631355i	$0.217501\pi$
$410$		0			0
$411$		0			0
$412$			7.88968e32i			1.62497i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.76919e32	−	3.06433e32i	−0.313383	−	0.542794i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6.45227e32			1.01435			0.507174	−	0.861844i	$-0.330690\pi$
$421$	6.45227e32			1.01435			0.507174	+	0.861844i	$0.330690\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.18475e33	+	6.21087e32i	1.56055	+	0.818096i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	−7.60462e32	−	4.39053e32i	−0.866025	−	0.500000i
$433$			1.78612e33i			1.97612i	0.154073	+	0.988060i	$0.450761\pi$
$433$			1.78612e33i			1.97612i	−0.154073	+	0.988060i	$0.549239\pi$
$434$		0			0
$435$		0			0
$436$	−9.61633e32	+	1.66560e33i	−0.975952	+	1.69040i
$437$		0			0
$438$		0			0
$439$	1.57985e33	+	9.12128e32i	1.47167	+	0.849668i	0.999493	−	0.0318363i	$-0.0101355\pi$
$439$	1.57985e33	+	9.12128e32i	1.47167	+	0.849668i	0.472175	+	0.881505i	$0.343469\pi$
$440$		0			0
$441$	4.86137e32	−	1.02703e33i	0.427835	−	0.903857i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.89358e33	+	1.09326e33i	−1.53108	+	0.883969i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.38234e33	+	5.63630e31i	−0.999170	+	0.0407398i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−2.23206e33	−	1.28868e33i	−1.40436	−	0.810808i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.09596e33	−	1.89827e33i	0.617794	−	1.07005i	−0.372093	−	0.928195i	$-0.621360\pi$
$457$	1.09596e33	−	1.89827e33i	0.617794	−	1.07005i	0.989887	−	0.141855i	$-0.0453068\pi$
$458$		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
$463$	4.17284e33			1.99834			0.999170	−	0.0407241i	$-0.0129665\pi$
$463$	4.17284e33			1.99834			0.999170	+	0.0407241i	$0.0129665\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$		−	1.03811e33i		−	0.434680i
$469$	−1.09189e33	−	1.72487e33i	−0.445161	−	0.703228i
$470$		0			0
$471$	1.63076e33	−	2.82455e33i	0.630417	−	1.09191i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			1.45624e33i			0.506482i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−2.23862e33	−	1.29247e33i	−0.665529	−	0.384244i
$482$		0			0
$483$		0			0
$484$	3.63552e33			1.00000
$485$		0			0
$486$		0			0
$487$	−2.38873e33	−	4.13740e33i	−0.608212	−	1.05345i	−0.991535	−	0.129839i	$-0.958554\pi$
$487$	−2.38873e33	−	4.13740e33i	−0.608212	−	1.05345i	0.383323	−	0.923614i	$-0.374780\pi$
$488$		0			0
$489$		−	8.21529e33i		−	1.98729i
$490$		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
$496$		−	2.77053e33i		−	0.561103i
$497$		0			0
$498$		0			0
$499$	−4.43155e33	+	7.67567e33i	−0.832336	+	1.44165i	0.0638453	+	0.997960i	$0.479664\pi$
$499$	−4.43155e33	+	7.67567e33i	−0.832336	+	1.44165i	−0.896181	+	0.443688i	$0.853670\pi$
$500$		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
$507$	−4.56226e33	+	2.63402e33i	−0.702393	+	0.405527i
$508$	−4.14894e32	−	7.18617e32i	−0.0623219	−	0.107945i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−5.20134e33	+	3.29258e33i	−0.725862	+	0.459489i
$512$		0			0
$513$	−1.90547e33	+	3.30036e33i	−0.253241	+	0.438627i
$514$		0			0
$515$		0			0
$516$	4.85705e33	+	2.80422e33i	0.600138	+	0.346490i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	1.62369e34	+	9.37441e33i	1.69525	+	0.978752i	0.950148	+	0.311799i	$0.100932\pi$
$523$	1.62369e34	+	9.37441e33i	1.69525	+	0.978752i	0.745100	+	0.666953i	$0.232402\pi$
$524$		0			0
$525$	8.89747e33	+	4.66437e33i	0.885676	+	0.464303i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−5.52288e33	−	9.56592e33i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	2.44613e32	+	5.99928e33i	0.0206340	+	0.506062i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.31552e34	−	2.27855e34i	0.899771	−	1.55845i	0.0719842	−	0.997406i	$-0.477067\pi$
$541$	1.31552e34	−	2.27855e34i	0.899771	−	1.55845i	0.827787	−	0.561043i	$-0.189600\pi$
$542$		0			0
$543$	1.46435e34	+	2.53632e34i	0.956415	+	1.65656i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.64640e34			−0.981056			−0.490528	−	0.871425i	$-0.663196\pi$
$547$	−1.64640e34			−0.981056			−0.490528	+	0.871425i	$0.663196\pi$
$548$		0			0
$549$	−2.68033e34	+	1.54749e34i	−1.52593	+	0.880993i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−1.77953e34	+	3.39451e34i	−0.925210	+	1.76487i
$554$		0			0
$555$		0			0
$556$	1.11704e34	−	6.44926e33i	0.542794	−	0.313383i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			6.63037e33i			0.301224i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.40616e34	+	2.22133e34i	0.534867	+	0.844937i
$568$		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
$571$	−2.67969e34	−	4.64136e34i	−0.933540	−	1.61694i	−0.777217	−	0.629233i	$-0.783369\pi$
$571$	−2.67969e34	−	4.64136e34i	−0.933540	−	1.61694i	−0.156324	−	0.987706i	$-0.549964\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.60048e34	−	2.77212e34i	0.500000	−	0.866025i
$577$	8.97873e32	−	5.18387e32i	0.0274484	−	0.0158474i	−0.486213	−	0.873840i	$-0.661622\pi$
$577$	8.97873e32	−	5.18387e32i	0.0274484	−	0.0158474i	0.513661	+	0.857993i	$0.328289\pi$
$578$		0			0
$579$	1.71044e34	+	9.87522e33i	0.500755	+	0.289111i
$580$		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	$0.5\pi$
$588$	3.74383e34	+	1.77212e34i	0.903857	+	0.427835i
$589$	−1.20240e34			−0.284189
$590$		0			0
$591$		0			0
$592$	−3.98528e34	−	6.90270e34i	−0.883969	−	1.53108i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.21712e33	−	5.57222e33i	0.0642375	−	0.111263i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		−	8.41512e34i		−	1.54571i	−0.634580	−	0.772857i	$-0.718827\pi$
$601$		−	8.41512e34i		−	1.54571i	0.634580	−	0.772857i	$-0.281173\pi$
$602$		0			0
$603$	4.72322e34			0.832285
$604$	4.69764e34	−	8.13655e34i	0.810808	−	1.40436i
$605$		0			0
$606$		0			0
$607$	−5.12190e34	−	2.95713e34i	−0.830945	−	0.479746i	0.0232312	−	0.999730i	$-0.492605\pi$
$607$	−5.12190e34	−	2.95713e34i	−0.830945	−	0.479746i	−0.854176	+	0.519984i	$0.825938\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.49545e34	−	1.12504e35i	−0.931865	−	1.61404i	−0.780131	−	0.625617i	$-0.784847\pi$
$613$	−6.49545e34	−	1.12504e35i	−0.931865	−	1.61404i	−0.151735	−	0.988421i	$-0.548486\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−1.28599e35	+	7.42466e34i	−1.63344	+	0.943069i	−0.650424	+	0.759572i	$0.725409\pi$
$619$	−1.28599e35	+	7.42466e34i	−1.63344	+	0.943069i	−0.983020	+	0.183498i	$0.941258\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	3.78422e34			0.434680
$625$	−4.44089e34	+	7.69185e34i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	1.02964e35	+	5.94461e34i	1.09191	+	0.630417i
$629$		0			0
$630$		0			0
$631$	2.00102e35			1.99933			0.999663	−	0.0259459i	$-0.00825975\pi$
$631$	2.00102e35			1.99933			0.999663	+	0.0259459i	$0.00825975\pi$
$632$		0			0
$633$	−1.62846e35	+	9.40193e34i	−1.56398	+	0.902964i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.98658e33	+	4.88055e34i	0.0353882	+	0.433237i
$638$		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
$643$			2.40449e35i			1.89839i	0.314682	+	0.949197i	$0.398102\pi$
$643$			2.40449e35i			1.89839i	−0.314682	+	0.949197i	$0.601898\pi$
$644$		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			0
$649$		0			0
$650$		0			0
$651$	−3.85129e34	+	7.34649e34i	−0.260522	+	0.496955i
$652$	2.99473e35			1.98729
$653$		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
$657$		−	1.42429e35i		−	0.859073i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−3.06141e35	+	1.76751e35i	−1.71161	+	0.988196i	−0.779207	+	0.626767i	$0.784378\pi$
$661$	−3.06141e35	+	1.76751e35i	−1.71161	+	0.988196i	−0.932399	+	0.361430i	$0.882289\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.06868e35	−	1.85100e35i	−0.514067	−	0.890389i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	3.76686e35			1.68187			0.840933	−	0.541140i	$-0.182007\pi$
$673$	3.76686e35			1.68187			0.840933	+	0.541140i	$0.182007\pi$
$674$		0			0
$675$	−2.01293e35	+	1.16216e35i	−0.866025	+	0.500000i
$676$	−9.60181e34	−	1.66308e35i	−0.405527	−	0.702393i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	1.24456e34	+	3.05236e35i	0.0497329	+	1.21973i
$680$		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	$0.333333\pi$
$684$	−1.20308e35	−	6.94601e34i	−0.438627	−	0.253241i
$685$		0			0
$686$		0			0
$687$	5.22187e35			1.80247
$688$	−1.02222e35	+	1.77054e35i	−0.346490	+	0.600138i
$689$		0			0
$690$		0			0
$691$	8.26696e34	+	4.77293e34i	0.265381	+	0.153218i	0.626787	−	0.779191i	$-0.284370\pi$
$691$	8.26696e34	+	4.77293e34i	0.265381	+	0.153218i	−0.361406	+	0.932409i	$0.617703\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.70031e35	+	3.24340e35i	−0.464303	+	0.885676i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−2.99574e35	+	1.72959e35i	−0.775464	+	0.447715i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	4.03742e35	−	6.99302e35i	0.939779	−	1.62775i	0.173898	−	0.984764i	$-0.444364\pi$
$709$	4.03742e35	−	6.99302e35i	0.939779	−	1.62775i	0.765881	−	0.642982i	$-0.222303\pi$
$710$		0			0
$711$	−4.43382e35	−	7.67961e35i	−0.996342	−	1.72571i
$712$		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	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	7.62624e35	+	3.99794e35i	1.43919	+	0.754477i
$722$		0			0
$723$	3.52989e35	−	6.11395e35i	0.643476	−	1.11453i
$724$	−9.24568e35	+	5.33800e35i	−1.65656	+	0.956415i
$725$		0			0
$726$		0			0
$727$		−	1.14533e36i		−	1.94872i	−0.224993	−	0.974360i	$-0.572236\pi$
$727$		−	1.14533e36i		−	1.94872i	0.224993	−	0.974360i	$-0.427764\pi$
$728$		0			0
$729$	−6.08267e35			−1.00000
$730$		0			0
$731$		0			0
$732$	−5.64106e35	−	9.77061e35i	−0.880993	−	1.52593i
$733$	1.12257e36	+	6.48115e35i	1.72351	+	0.995067i	0.911354	+	0.411624i	$0.135038\pi$
$733$	1.12257e36	+	6.48115e35i	1.72351	+	0.995067i	0.812153	+	0.583444i	$0.198295\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−5.69347e35	−	9.86139e35i	−0.789445	−	1.36736i	−0.926307	−	0.376769i	$-0.877035\pi$
$739$	−5.69347e35	−	9.86139e35i	−0.789445	−	1.36736i	0.136862	−	0.990590i	$-0.456298\pi$
$740$		0			0
$741$		−	1.64233e35i		−	0.220158i
$742$		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
$751$	6.70168e35	−	1.16076e36i	0.759774	−	1.31597i	−0.183192	−	0.983077i	$-0.558643\pi$
$751$	6.70168e35	−	1.16076e36i	0.759774	−	1.31597i	0.942966	−	0.332890i	$-0.108024\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−8.09742e35	+	5.12587e35i	−0.844937	+	0.534867i
$757$	5.07423e35			0.520801			0.260400	−	0.965501i	$-0.416145\pi$
$757$	5.07423e35			0.520801			0.260400	+	0.965501i	$0.416145\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	1.12269e36	+	1.77353e36i	1.04401	+	1.64923i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.01052e36	+	5.83425e35i	0.866025	+	0.500000i
$769$			8.75669e35i			0.738347i	0.929360	+	0.369174i	$0.120359\pi$
$769$			8.75669e35i			0.738347i	−0.929360	+	0.369174i	$0.879641\pi$
$770$		0			0
$771$		0			0
$772$	−3.59982e35	+	6.23507e35i	−0.289111	+	0.500755i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−6.35103e35	−	3.66677e35i	−0.485929	−	0.280551i
$776$		0			0
$777$	9.72185e34	+	2.38434e36i	0.0720255	+	1.76647i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−6.45993e35	+	1.36474e36i	−0.427835	+	0.903857i
$785$		0			0
$786$		0			0
$787$	−2.74217e36	+	1.58319e36i	−1.73145	+	0.999654i	−0.852582	+	0.522593i	$0.824965\pi$
$787$	−2.74217e36	+	1.58319e36i	−1.73145	+	0.999654i	−0.878870	+	0.477062i	$0.841702\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.66894e35	−	1.15509e36i	0.382950	−	0.663289i
$794$		0			0
$795$		0			0
$796$	2.03125e35	+	1.17274e35i	0.111263	+	0.0642375i
$797$		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
$804$			1.72176e36i			0.832285i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	4.57489e36i		−	1.98436i	−0.124825	−	0.992179i	$-0.539837\pi$
$811$		−	4.57489e36i		−	1.98436i	0.124825	−	0.992179i	$-0.460163\pi$
$812$		0			0
$813$	4.55960e36			1.91776
$814$		0			0
$815$		0			0
$816$		0			0
$817$	7.68406e35	+	4.43640e35i	0.303959	+	0.175491i
$818$		0			0
$819$	−1.00344e36	−	5.26041e35i	−0.384986	−	0.201823i
$820$		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
$823$	−2.40654e35	−	4.16826e35i	−0.0868752	−	0.150472i	0.819314	−	0.573346i	$-0.194355\pi$
$823$	−2.40654e35	−	4.16826e35i	−0.0868752	−	0.150472i	−0.906189	+	0.422873i	$0.861021\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−5.23031e36	+	3.01972e36i	−1.72423	+	0.995485i	−0.814659	+	0.579941i	$0.803076\pi$
$829$	−5.23031e36	+	3.01972e36i	−1.72423	+	0.995485i	−0.909573	+	0.415545i	$0.863591\pi$
$830$		0			0
$831$	5.38524e36	+	3.10917e36i	1.72263	+	0.994562i
$832$			1.37947e36i			0.434680i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−9.59579e35	−	1.66204e36i	−0.280551	−	0.485929i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	3.63036e36			1.00000
$842$		0			0
$843$		0			0
$844$	−3.42729e36	−	5.93624e36i	−0.902964	−	1.56398i
$845$		0			0
$846$		0			0
$847$	1.84223e36	−	3.51413e36i	0.464303	−	0.885676i
$848$		0			0
$849$	−2.05603e36	+	3.56115e36i	−0.503133	+	0.871453i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	8.12939e36i		−	1.87584i	−0.346857	−	0.937918i	$-0.612751\pi$
$853$		−	8.12939e36i		−	1.87584i	0.346857	−	0.937918i	$-0.387249\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−7.68291e36	−	4.43573e36i	−1.62409	−	0.937672i	−0.985809	−	0.167872i	$-0.946310\pi$
$859$	−7.68291e36	−	4.43573e36i	−1.62409	−	0.937672i	−0.638286	−	0.769799i	$-0.720356\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			5.31176e36i			1.00000i
$868$	−2.67802e36	−	1.40391e36i	−0.496955	−	0.260522i
$869$		0			0
$870$		0			0
$871$	−1.76278e36	+	1.01774e36i	−0.313308	+	0.180889i
$872$		0			0
$873$	−6.12115e36	−	3.53405e36i	−1.05719	−	0.610372i
$874$		0			0
$875$		0			0
$876$	5.19196e36			0.859073
$877$	2.51040e36	−	4.34815e36i	0.409495	−	0.709266i	−0.585338	−	0.810789i	$-0.699038\pi$
$877$	2.51040e36	−	4.34815e36i	0.409495	−	0.709266i	0.994833	+	0.101523i	$0.0323715\pi$
$878$		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
$883$	9.70062e36			1.45308			0.726542	−	0.687122i	$-0.241126\pi$
$883$	9.70062e36			1.45308			0.726542	+	0.687122i	$0.241126\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−9.04861e35	+	3.68945e34i	−0.124540	+	0.00507797i
$890$		0			0
$891$		0			0
$892$	6.74747e36	−	3.89565e36i	0.890389	−	0.514067i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−4.23644e36	−	7.33773e36i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	5.17180e36	−	3.27388e36i	0.585524	−	0.370652i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.24031e36	+	5.61238e36i	0.347133	+	0.601251i	0.985739	−	0.168282i	$-0.0538220\pi$
$907$	3.24031e36	+	5.61238e36i	0.347133	+	0.601251i	−0.638606	+	0.769534i	$0.720489\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	2.53204e36	−	4.38561e36i	0.253241	−	0.438627i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			1.90353e37i			1.80247i
$917$		0			0
$918$		0			0
$919$	−4.40897e36	+	7.63657e36i	−0.400769	+	0.694152i	−0.993819	−	0.111014i	$-0.964590\pi$
$919$	−4.40897e36	+	7.63657e36i	−0.400769	+	0.694152i	0.593050	+	0.805165i	$0.297924\pi$
$920$		0			0
$921$	1.06773e37	+	1.84937e37i	0.944533	+	1.63598i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−2.10979e37			−1.76794
$926$		0			0
$927$	−1.72533e37	+	9.96119e36i	−1.40726	+	0.812483i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	5.92291e36	+	2.80357e36i	0.457787	+	0.216691i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			2.46934e37i			1.76131i	0.473757	+	0.880656i	$0.342898\pi$
$937$			2.46934e37i			1.76131i	−0.473757	+	0.880656i	$0.657102\pi$
$938$		0			0
$939$	−1.48805e37			−1.03347
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		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	$0.666667\pi$
$948$	2.79945e37	−	1.61627e37i	1.72571	−	0.996342i
$949$	3.06900e36	+	5.31567e36i	0.186711	+	0.323393i
$950$		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
$961$	−6.58881e36	+	1.14121e37i	−0.342582	+	0.593369i
$962$		0			0
$963$		0			0
$964$	2.22872e37	+	1.28675e37i	1.11453	+	0.643476i
$965$		0			0
$966$		0			0
$967$	−7.96186e36			−0.382987			−0.191493	−	0.981494i	$-0.561333\pi$
$967$	−7.96186e36			−0.382987			−0.191493	+	0.981494i	$0.561333\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$		−	2.21732e37i		−	1.00000i
$973$	−5.73501e35	−	1.40655e37i	−0.0255343	−	0.626245i
$974$		0			0
$975$	5.00838e36	−	8.67477e36i	0.217340	−	0.376444i
$976$	3.56169e37	−	2.05634e37i	1.52593	−	0.880993i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−4.85647e37			−1.95190
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	5.98681e36			0.220158
$989$		0			0
$990$		0			0
$991$	1.48092e37	+	2.56503e37i	0.524337	+	0.908179i	0.999599	+	0.0283339i	$0.00902016\pi$
$991$	1.48092e37	+	2.56503e37i	0.524337	+	0.908179i	−0.475261	+	0.879845i	$0.657647\pi$
$992$		0			0
$993$		−	3.67464e37i		−	1.26867i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.15582e37	−	2.97672e37i	1.69281	−	0.977345i	0.740579	−	0.671970i	$-0.234551\pi$
$997$	5.15582e37	−	2.97672e37i	1.69281	−	0.977345i	0.952232	−	0.305375i	$-0.0987819\pi$
$998$		0			0
$999$	−4.78152e37	−	2.76061e37i	−1.53108	−	0.883969i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.26.g.a.5.1	✓	2
3.2	odd	2	CM	21.26.g.a.5.1	✓	2
7.3	odd	6	inner	21.26.g.a.17.1	yes	2
21.17	even	6	inner	21.26.g.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.26.g.a.5.1	✓	2	1.1	even	1	trivial
21.26.g.a.5.1	✓	2	3.2	odd	2	CM
21.26.g.a.17.1	yes	2	7.3	odd	6	inner
21.26.g.a.17.1	yes	2	21.17	even	6	inner

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:	\( k \)	\(=\)	\( 26 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-797161. + 460241. i) q^{3} +(-1.67772e7 - 2.90590e7i) q^{4} +(-3.65902e10 + 1.49192e9i) q^{7} +(4.23644e11 - 7.33773e11i) q^{9} +O(q^{10})\)	\(q+(-797161. + 460241. i) q^{3} +(-1.67772e7 - 2.90590e7i) q^{4} +(-3.65902e10 + 1.49192e9i) q^{7} +(4.23644e11 - 7.33773e11i) q^{9} +(2.67483e13 + 1.54431e13i) q^{12} +3.65142e13i q^{13} +(-5.62950e14 + 9.75058e14i) q^{16} +(4.23170e15 + 2.44317e15i) q^{19} +(2.84817e16 - 1.80296e16i) q^{21} +(1.49012e17 + 2.58096e17i) q^{25} +7.79915e17i q^{27} +(6.57235e17 + 1.03824e18i) q^{28} +(-2.13105e18 + 1.23036e18i) q^{31} -2.84303e19 q^{36} +(-3.53964e19 + 6.13083e19i) q^{37} +(-1.68053e19 - 2.91077e19i) q^{39} +1.81583e20 q^{43} -1.03637e21i q^{48} +(1.33662e21 - 1.09179e20i) q^{49} +(1.06107e21 - 6.12606e20i) q^{52} -4.49780e21 q^{57} +(-3.16341e22 - 1.82640e22i) q^{61} +(-1.44065e22 + 2.74810e22i) q^{63} +3.77789e22 q^{64} +(2.78726e22 + 4.82767e22i) q^{67} +(1.45578e23 - 8.40496e22i) q^{73} +(-2.37573e23 - 1.37163e23i) q^{75} -1.63959e23i q^{76} +(5.23296e23 - 9.06375e23i) q^{79} +(-3.58949e23 - 6.21718e23i) q^{81} +(-1.00177e24 - 5.25161e23i) q^{84} +(-5.44761e22 - 1.33606e24i) q^{91} +(1.13253e24 - 1.96160e24i) q^{93} -8.34202e24i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 1594323 q^{3} - 33554432 q^{4} - 73180401839 q^{7} + 847288609443 q^{9}+O(q^{10}) \) 2 * q - 1594323 * q^3 - 33554432 * q^4 - 73180401839 * q^7 + 847288609443 * q^9	\( 2 q - 1594323 q^{3} - 33554432 q^{4} - 73180401839 q^{7} + 847288609443 q^{9} + 53496602689536 q^{12} - 11\!\cdots\!24 q^{16}+ \cdots + 22\!\cdots\!25 q^{93}+O(q^{100}) \) 2 * q - 1594323 * q^3 - 33554432 * q^4 - 73180401839 * q^7 + 847288609443 * q^9 + 53496602689536 * q^12 - 1125899906842624 * q^16 + 8463400225141671 * q^19 + 56963314943414874 * q^21 + 298023223876953125 * q^25 + 1314470612537704448 * q^28 - 4262104410474914625 * q^31 - 56860576059859402752 * q^36 - 70792710091786388161 * q^37 - 33610669399525960557 * q^39 + 363166767001127849750 * q^43 + 2673233973989584780307 * q^49 + 2122130059293269950464 * q^52 - 8995595758099029555822 * q^57 - 63268297182424491524628 * q^61 - 28813002257883762194625 * q^63 + 75557863725914323419136 * q^64 + 55745137266360974031589 * q^67 + 291156389317363370673573 * q^73 - 475145280361175537109375 * q^75 + 1046591204768194897309693 * q^79 - 717897987691852588770249 * q^81 - 2003531204078174403035136 * q^84 - 108952233186028470179751 * q^91 + 2265057030007199103224625 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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