LMFDB - Embedded newform 21.35.h.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,35,Mod(2,21)]

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, 2]))

N = Newforms(chi, 35, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 35);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 35 $
Character orbit:	$[\chi]$	$=$	21.h (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:	$153.773873490$
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			11.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.11
Dual form			21.35.h.a.2.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.45701e7 - 1.11839e8i) q^{3} +(-8.58993e9 + 1.48782e10i) q^{4} +(-2.28827e14 - 4.18961e13i) q^{7} +(-8.33859e15 - 1.44429e16i) q^{9} +O(q^{10})$	\(q+(6.45701e7 - 1.11839e8i) q^{3} +(-8.58993e9 + 1.48782e10i) q^{4} +(-2.28827e14 - 4.18961e13i) q^{7} +(-8.33859e15 - 1.44429e16i) q^{9} +(1.10931e18 + 1.92137e18i) q^{12} -6.62859e18 q^{13} +(-1.47574e20 - 2.55606e20i) q^{16} +(5.26678e21 + 9.12232e21i) q^{19} +(-1.94610e22 + 2.28864e22i) q^{21} +(-2.91038e23 + 5.04093e23i) q^{25} -2.15369e24 q^{27} +(2.58895e24 - 3.04465e24i) q^{28} +(-9.79985e24 + 1.69738e25i) q^{31} +2.86512e26 q^{36} +(-4.29797e26 - 7.44431e26i) q^{37} +(-4.28009e26 + 7.41333e26i) q^{39} -8.73505e27 q^{43} -3.81154e28 q^{48} +(5.06064e28 + 1.91739e28i) q^{49} +(5.69391e28 - 9.86215e28i) q^{52} +1.36030e30 q^{57} +(2.23400e30 + 3.86940e30i) q^{61} +(1.30299e30 + 3.65427e30i) q^{63} +5.07060e30 q^{64} +(1.10114e31 - 1.90723e31i) q^{67} +(-1.00365e31 + 1.73838e31i) q^{73} +(3.75847e31 + 6.50987e31i) q^{75} -1.80965e32 q^{76} +(-1.68429e32 - 2.91728e32i) q^{79} +(-1.39064e32 + 2.40866e32i) q^{81} +(-1.73341e32 - 4.86137e32i) q^{84} +(1.51680e33 + 2.77712e32i) q^{91} +(1.26555e33 + 2.19200e33i) q^{93} +2.07782e33 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 129140163 q^{3} - 17179869184 q^{4} - 457653465545653 q^{7} - 16\!\cdots\!69 q^{9}+O(q^{10}) $ 2 * q + 129140163 * q^3 - 17179869184 * q^4 - 457653465545653 * q^7 - 16677181699666569 * q^9	$ 2 q + 129140163 q^{3} - 17179869184 q^{4} - 457653465545653 q^{7} - 16\!\cdots\!69 q^{9}+ \cdots + 41\!\cdots\!04 q^{97}+O(q^{100}) $ 2 * q + 129140163 * q^3 - 17179869184 * q^4 - 457653465545653 * q^7 - 16677181699666569 * q^9 + 2218611106740436992 * q^12 - 13257178113610601474 * q^13 - 295147905179352825856 * q^16 + 10533550932831642194149 * q^19 - 38921926981395836058246 * q^21 - 582076609134674072265625 * q^25 - 4307387926151115532621494 * q^27 + 5177890428476387049340928 * q^28 - 19599701438028304062305147 * q^31 + 573023599916140863676219392 * q^36 - 859594323581411974757802287 * q^37 - 856017071255852796440200131 * q^39 - 17470096463040699199417500122 * q^43 - 76230897167940336331108909056 * q^48 + 101212782450041974235107874711 * q^49 + 113878292870409011619438788608 * q^52 + 2720608968869360649020159012574 * q^57 + 4468002075713925041487910419766 * q^61 + 2605986005735392637396581440459 * q^63 + 10141204801825835211973625643008 * q^64 + 22022844874719432605769728964013 * q^67 - 20073068910160542409412259436943 * q^73 + 75169468182139098644256591796875 * q^75 - 361930054138097567182749100408832 * q^76 - 336858662185559380928110634527811 * q^79 - 278128389443693511257285776231761 * q^81 - 346681447768257186026240461504512 * q^84 + 3033596753524937217268086468046261 * q^91 + 2531108638458309585219648839318961 * q^93 + 4155647952096798288265379422464004 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{2}{3}\right)$

Coefficient data

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

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

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

$2$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	6.45701e7	−	1.11839e8i	0.500000	−	0.866025i
$3$	6.45701e7	−	1.11839e8i	0.500000	−	0.866025i
$4$	−8.58993e9	+	1.48782e10i	−0.500000	+	0.866025i
$5$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$5$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$6$		0			0
$7$	−2.28827e14	−	4.18961e13i	−0.983649	−	0.180097i
$7$	−2.28827e14	−	4.18961e13i	−0.983649	−	0.180097i
$8$		0			0
$9$	−8.33859e15	−	1.44429e16i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	1.10931e18	+	1.92137e18i	0.500000	+	0.866025i
$13$	−6.62859e18			−0.766274			−0.383137	−	0.923692i	$-0.625156\pi$
$13$	−6.62859e18			−0.766274			−0.383137	+	0.923692i	$0.625156\pi$
$14$		0			0
$15$		0			0
$16$	−1.47574e20	−	2.55606e20i	−0.500000	−	0.866025i
$17$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$17$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$18$		0			0
$19$	5.26678e21	+	9.12232e21i	0.961023	+	1.66454i	0.719941	+	0.694035i	$0.244169\pi$
$19$	5.26678e21	+	9.12232e21i	0.961023	+	1.66454i	0.241081	+	0.970505i	$0.422498\pi$
$20$		0			0
$21$	−1.94610e22	+	2.28864e22i	−0.647793	+	0.761816i
$22$		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
$25$	−2.91038e23	+	5.04093e23i	−0.500000	+	0.866025i
$26$		0			0
$27$	−2.15369e24			−1.00000
$28$	2.58895e24	−	3.04465e24i	0.647793	−	0.761816i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−9.79985e24	+	1.69738e25i	−0.434581	+	0.752716i	−0.997261	−	0.0739583i	$-0.976437\pi$
$31$	−9.79985e24	+	1.69738e25i	−0.434581	+	0.752716i	0.562680	+	0.826675i	$0.309770\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	2.86512e26			1.00000
$37$	−4.29797e26	−	7.44431e26i	−0.941530	−	1.63078i	−0.762554	−	0.646925i	$-0.776055\pi$
$37$	−4.29797e26	−	7.44431e26i	−0.941530	−	1.63078i	−0.178976	−	0.983853i	$-0.557279\pi$
$38$		0			0
$39$	−4.28009e26	+	7.41333e26i	−0.383137	+	0.663613i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−8.73505e27			−1.48697			−0.743484	−	0.668754i	$-0.766828\pi$
$43$	−8.73505e27			−1.48697			−0.743484	+	0.668754i	$0.766828\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$47$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$48$	−3.81154e28			−1.00000
$49$	5.06064e28	+	1.91739e28i	0.935130	+	0.354305i
$50$		0			0
$51$		0			0
$52$	5.69391e28	−	9.86215e28i	0.383137	−	0.663613i
$53$		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
$57$	1.36030e30			1.92205
$58$		0			0
$59$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$59$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$60$		0			0
$61$	2.23400e30	+	3.86940e30i	0.996497	+	1.72598i	0.570669	+	0.821180i	$0.306684\pi$
$61$	2.23400e30	+	3.86940e30i	0.996497	+	1.72598i	0.425829	+	0.904804i	$0.359983\pi$
$62$		0			0
$63$	1.30299e30	+	3.65427e30i	0.335856	+	0.941913i
$64$	5.07060e30			1.00000
$65$		0			0
$66$		0			0
$67$	1.10114e31	−	1.90723e31i	0.996717	−	1.72636i	0.428239	−	0.903665i	$-0.359134\pi$
$67$	1.10114e31	−	1.90723e31i	0.996717	−	1.72636i	0.568478	−	0.822699i	$-0.307533\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.00365e31	+	1.73838e31i	−0.211395	+	0.366147i	−0.952151	−	0.305627i	$-0.901134\pi$
$73$	−1.00365e31	+	1.73838e31i	−0.211395	+	0.366147i	0.740756	+	0.671774i	$0.234467\pi$
$74$		0			0
$75$	3.75847e31	+	6.50987e31i	0.500000	+	0.866025i
$76$	−1.80965e32			−1.92205
$77$		0			0
$78$		0			0
$79$	−1.68429e32	−	2.91728e32i	−0.926312	−	1.60442i	−0.789438	−	0.613831i	$-0.789628\pi$
$79$	−1.68429e32	−	2.91728e32i	−0.926312	−	1.60442i	−0.136874	−	0.990588i	$-0.543706\pi$
$80$		0			0
$81$	−1.39064e32	+	2.40866e32i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−1.73341e32	−	4.86137e32i	−0.335856	−	0.941913i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$89$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$90$		0			0
$91$	1.51680e33	+	2.77712e32i	0.753745	+	0.138004i
$92$		0			0
$93$	1.26555e33	+	2.19200e33i	0.434581	+	0.752716i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	2.07782e33			0.348730			0.174365	−	0.984681i	$-0.444213\pi$
$97$	2.07782e33			0.348730			0.174365	+	0.984681i	$0.444213\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e33	−	8.66025e33i	−0.500000	−	0.866025i
$101$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$101$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$102$		0			0
$103$	1.36868e34	+	2.37062e34i	0.828074	+	1.43427i	0.899547	+	0.436824i	$0.143897\pi$
$103$	1.36868e34	+	2.37062e34i	0.828074	+	1.43427i	−0.0714733	+	0.997443i	$0.522770\pi$
$104$		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	$0.333333\pi$
$108$	1.85001e34	−	3.20431e34i	0.500000	−	0.866025i
$109$	2.59698e34	−	4.49810e34i	0.600092	−	1.03939i	−0.392715	−	0.919660i	$-0.628464\pi$
$109$	2.59698e34	−	4.49810e34i	0.600092	−	1.03939i	0.992807	−	0.119729i	$-0.0382026\pi$
$110$		0			0
$111$	−1.11008e35			−1.88306
$112$	2.30600e34	+	6.46722e34i	0.335856	+	0.941913i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5.52731e34	+	9.57358e34i	0.383137	+	0.663613i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.27738e35	−	2.21249e35i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.68360e35	−	2.91608e35i	−0.434581	−	0.752716i
$125$		0			0
$126$		0			0
$127$	1.04145e36			1.79050			0.895250	−	0.445564i	$-0.146997\pi$
$127$	1.04145e36			1.79050			0.895250	+	0.445564i	$0.146997\pi$
$128$		0			0
$129$	−5.64023e35	+	9.76916e35i	−0.743484	+	1.28775i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	−8.22989e35	−	2.30809e36i	−0.645530	−	1.81040i
$134$		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
$139$	3.67066e36			1.35985			0.679927	−	0.733280i	$-0.262011\pi$
$139$	3.67066e36			1.35985			0.679927	+	0.733280i	$0.262011\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−2.46112e36	+	4.26278e36i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	5.41204e36	−	4.42169e36i	0.774402	−	0.632694i
$148$	1.47677e37			1.88306
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	8.25879e36	−	1.43046e37i	0.748701	−	1.29679i	−0.199745	−	0.979848i	$-0.564011\pi$
$151$	8.25879e36	−	1.43046e37i	0.748701	−	1.29679i	0.948446	−	0.316940i	$-0.102655\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−7.35313e36	−	1.27360e37i	−0.383137	−	0.663613i
$157$	5.62732e36	−	9.74681e36i	0.263031	−	0.455584i	−0.704015	−	0.710186i	$-0.748611\pi$
$157$	5.62732e36	−	9.74681e36i	0.263031	−	0.455584i	0.967046	+	0.254602i	$0.0819444\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−6.47963e36	−	1.12231e37i	−0.160089	−	0.277282i	0.774812	−	0.632192i	$-0.217845\pi$
$163$	−6.47963e36	−	1.12231e37i	−0.160089	−	0.277282i	−0.934900	+	0.354910i	$0.884511\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−3.08915e37			−0.412824
$170$		0			0
$171$	8.78350e37	−	1.52135e38i	0.961023	−	1.66454i
$172$	7.50335e37	−	1.29962e38i	0.743484	−	1.28775i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	8.77169e37	−	1.03157e38i	0.647793	−	0.761816i
$176$		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
$181$	4.26819e38			1.77709			0.888543	−	0.458794i	$-0.151718\pi$
$181$	4.26819e38			1.77709			0.888543	+	0.458794i	$0.151718\pi$
$182$		0			0
$183$	5.76999e38			1.99299
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.92823e38	+	9.02314e37i	0.983649	+	0.180097i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	3.27409e38	−	5.67089e38i	0.500000	−	0.866025i
$193$	6.70653e38	−	1.16160e39i	0.937613	−	1.62399i	0.167706	−	0.985837i	$-0.446364\pi$
$193$	6.70653e38	−	1.16160e39i	0.937613	−	1.62399i	0.769907	−	0.638156i	$-0.220302\pi$
$194$		0			0
$195$		0			0
$196$	−7.19979e38	+	5.88230e38i	−0.774402	+	0.632694i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	6.05626e38	−	1.04897e39i	0.503155	−	0.871491i	−0.496838	−	0.867843i	$-0.665506\pi$
$199$	6.05626e38	−	1.04897e39i	0.503155	−	0.871491i	0.999993	−	0.00364740i	$-0.00116101\pi$
$200$		0			0
$201$	−1.42202e39	−	2.46301e39i	−0.996717	−	1.72636i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	9.78207e38	+	1.69430e39i	0.383137	+	0.663613i
$209$		0			0
$210$		0			0
$211$	−3.62250e39			−1.11226			−0.556131	−	0.831095i	$-0.687715\pi$
$211$	−3.62250e39			−1.11226			−0.556131	+	0.831095i	$0.687715\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	2.95361e39	−	3.47349e39i	0.563037	−	0.662142i
$218$		0			0
$219$	1.29612e39	+	2.24495e39i	0.211395	+	0.366147i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.47728e40			1.77125			0.885623	−	0.464405i	$-0.153732\pi$
$223$	1.47728e40			1.77125			0.885623	+	0.464405i	$0.153732\pi$
$224$		0			0
$225$	9.70740e39			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−1.16849e40	+	2.02389e40i	−0.961023	+	1.66454i
$229$	−1.06778e39	−	1.84944e39i	−0.0815224	−	0.141201i	0.822382	−	0.568936i	$-0.192645\pi$
$229$	−1.06778e39	−	1.84944e39i	−0.0815224	−	0.141201i	−0.903904	+	0.427735i	$0.859312\pi$
$230$		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
$237$	−4.35020e40			−1.85262
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−3.06079e40	+	5.30144e40i	−0.980717	+	1.69865i	−0.321108	+	0.947043i	$0.604055\pi$
$241$	−3.06079e40	+	5.30144e40i	−0.980717	+	1.69865i	−0.659609	+	0.751609i	$0.729278\pi$
$242$		0			0
$243$	1.79588e40	+	3.11055e40i	0.500000	+	0.866025i
$244$	−7.67597e40			−1.99299
$245$		0			0
$246$		0			0
$247$	−3.49113e40	−	6.04681e40i	−0.736407	−	1.27549i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−6.55616e40	−	1.20037e40i	−0.983649	−	0.180097i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−4.35561e40	+	7.54415e40i	−0.500000	+	0.866025i
$257$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$257$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$258$		0			0
$259$	6.71604e40	+	1.88352e41i	0.632437	+	1.77368i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.89175e41	+	3.27660e41i	0.996717	+	1.72636i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−2.29315e41	−	3.97186e41i	−0.999897	−	1.73187i	−0.512386	−	0.858755i	$-0.671238\pi$
$271$	−2.29315e41	−	3.97186e41i	−0.999897	−	1.73187i	−0.487511	−	0.873117i	$-0.662095\pi$
$272$		0			0
$273$	1.28999e41	−	1.51705e41i	0.496387	−	0.583760i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.30789e41	+	5.72943e41i	−0.994019	+	1.72169i	−0.402430	+	0.915451i	$0.631834\pi$
$277$	−3.30789e41	+	5.72943e41i	−0.994019	+	1.72169i	−0.591589	+	0.806240i	$0.701499\pi$
$278$		0			0
$279$	3.26868e41			0.869162
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	3.18781e41	−	5.52145e41i	0.665462	−	1.15261i	−0.313698	−	0.949523i	$-0.601568\pi$
$283$	3.18781e41	−	5.52145e41i	0.665462	−	1.15261i	0.979160	−	0.203091i	$-0.0650987\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−3.42163e41	−	5.92644e41i	−0.500000	−	0.866025i
$290$		0			0
$291$	1.34165e41	−	2.32381e41i	0.174365	−	0.302009i
$292$	−1.72426e41	−	2.98651e41i	−0.211395	−	0.366147i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−1.29140e42			−1.00000
$301$	1.99881e42	+	3.65964e41i	1.46265	+	0.267799i
$302$		0			0
$303$		0			0
$304$	1.55448e42	−	2.69243e42i	0.961023	−	1.66454i
$305$		0			0
$306$		0			0
$307$	−4.21165e41			−0.220343			−0.110172	−	0.993913i	$-0.535140\pi$
$307$	−4.21165e41			−0.220343			−0.110172	+	0.993913i	$0.535140\pi$
$308$		0			0
$309$	3.53503e42			1.65615
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−7.81648e41	−	1.35385e42i	−0.294278	−	0.509704i	0.680539	−	0.732712i	$-0.261746\pi$
$313$	−7.81648e41	−	1.35385e42i	−0.294278	−	0.509704i	−0.974817	+	0.223008i	$0.928412\pi$
$314$		0			0
$315$		0			0
$316$	5.78719e42			1.85262
$317$		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
$324$	−2.38910e42	−	4.13805e42i	−0.500000	−	0.866025i
$325$	1.92917e42	−	3.34143e42i	0.383137	−	0.663613i
$326$		0			0
$327$	−3.35374e42	−	5.80885e42i	−0.600092	−	1.03939i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.08329e42	+	7.07247e42i	0.594203	+	1.02919i	0.993659	+	0.112437i	$0.0358657\pi$
$331$	4.08329e42	+	7.07247e42i	0.594203	+	1.02919i	−0.399456	+	0.916752i	$0.630801\pi$
$332$		0			0
$333$	−7.16781e42	+	1.24150e43i	−0.941530	+	1.63078i
$334$		0			0
$335$		0			0
$336$	8.72183e42	+	1.59689e42i	0.983649	+	0.180097i
$337$	1.26202e42			0.135319			0.0676593	−	0.997708i	$-0.478447\pi$
$337$	1.26202e42			0.135319			0.0676593	+	0.997708i	$0.478447\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.07768e43	−	6.50771e42i	−0.856030	−	0.516926i
$344$		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
$349$	−2.97983e43			−1.76262			−0.881310	−	0.472539i	$-0.843338\pi$
$349$	−2.97983e43			−1.76262			−0.881310	+	0.472539i	$0.843338\pi$
$350$		0			0
$351$	1.42760e43			0.766274
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−4.04605e43	+	7.00797e43i	−1.34713	+	2.33330i
$362$		0			0
$363$	−3.29923e43			−1.00000
$364$	−1.71611e43	+	2.01817e43i	−0.496387	+	0.583760i
$365$		0			0
$366$		0			0
$367$	4.32959e42	−	7.49907e42i	0.108924	−	0.188662i	−0.806411	−	0.591356i	$-0.798593\pi$
$367$	4.32959e42	−	7.49907e42i	0.108924	−	0.188662i	0.915335	+	0.402694i	$0.131926\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−4.34841e43			−0.869162
$373$	3.27321e43	+	5.66937e43i	0.625064	+	1.08264i	0.988529	+	0.151034i	$0.0482604\pi$
$373$	3.27321e43	+	5.66937e43i	0.625064	+	1.08264i	−0.363465	+	0.931608i	$0.618406\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−3.12076e43			−0.454353			−0.227177	−	0.973854i	$-0.572949\pi$
$379$	−3.12076e43			−0.454353			−0.227177	+	0.973854i	$0.572949\pi$
$380$		0			0
$381$	6.72464e43	−	1.16474e44i	0.895250	−	1.55062i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	7.28380e43	+	1.26159e44i	0.743484	+	1.28775i
$388$	−1.78484e43	+	3.09143e43i	−0.174365	+	0.302009i
$389$		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
$397$	4.89391e43	+	8.47650e43i	0.323756	+	0.560761i	0.981260	−	0.192690i	$-0.0617211\pi$
$397$	4.89391e43	+	8.47650e43i	0.323756	+	0.560761i	−0.657504	+	0.753451i	$0.728388\pi$
$398$		0			0
$399$	−3.11274e44	−	5.69914e43i	−1.89062	−	0.346155i
$400$	1.71799e44			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
$403$	6.49592e43	−	1.12513e44i	0.333008	−	0.576787i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	2.33880e44	−	4.05092e44i	0.932603	−	1.61532i	0.153748	−	0.988110i	$-0.450865\pi$
$409$	2.33880e44	−	4.05092e44i	0.932603	−	1.61532i	0.778854	−	0.627205i	$-0.215801\pi$
$410$		0			0
$411$		0			0
$412$	−4.70275e44			−1.65615
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	2.37015e44	−	4.10521e44i	0.679927	−	1.17767i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	6.48357e44			1.58132			0.790658	−	0.612258i	$-0.209739\pi$
$421$	6.48357e44			1.58132			0.790658	+	0.612258i	$0.209739\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.49086e44	−	9.79019e44i	−0.669359	−	1.87723i
$428$		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	$0.666667\pi$
$432$	3.17829e44	+	5.50496e44i	0.500000	+	0.866025i
$433$	1.10067e45			1.66480			0.832402	−	0.554172i	$-0.186965\pi$
$433$	1.10067e45			1.66480			0.832402	+	0.554172i	$0.186965\pi$
$434$		0			0
$435$		0			0
$436$	4.46157e44	+	7.72767e44i	0.600092	+	1.03939i
$437$		0			0
$438$		0			0
$439$	8.08864e44	+	1.40099e45i	0.968230	+	1.67702i	0.700675	+	0.713481i	$0.252882\pi$
$439$	8.08864e44	+	1.40099e45i	0.968230	+	1.67702i	0.267555	+	0.963543i	$0.413784\pi$
$440$		0			0
$441$	−1.45060e44	−	8.90784e44i	−0.160728	−	0.986999i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	9.53553e44	−	1.65160e45i	0.941530	−	1.63078i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−1.16029e45	−	2.12438e44i	−0.983649	−	0.180097i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.06654e45	−	1.84730e45i	−0.748701	−	1.29679i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.55332e45	+	2.69042e45i	0.939041	+	1.62647i	0.767265	+	0.641331i	$0.221617\pi$
$457$	1.55332e45	+	2.69042e45i	0.939041	+	1.62647i	0.171776	+	0.985136i	$0.445049\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−3.91788e45			−1.89747			−0.948734	−	0.316076i	$-0.897634\pi$
$463$	−3.91788e45			−1.89747			−0.948734	+	0.316076i	$0.897634\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$467$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$468$	−1.89917e45			−0.766274
$469$	−3.31876e45	+	3.90293e45i	−1.29133	+	1.51863i
$470$		0			0
$471$	−7.26713e44	−	1.25870e45i	−0.263031	−	0.455584i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−6.13133e45			−1.92205
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	2.84895e45	+	4.93452e45i	0.721470	+	1.24962i
$482$		0			0
$483$		0			0
$484$	4.38906e45			1.00000
$485$		0			0
$486$		0			0
$487$	−2.59734e44	+	4.49872e44i	−0.0532766	+	0.0922777i	−0.891434	−	0.453151i	$-0.850300\pi$
$487$	−2.59734e44	+	4.49872e44i	−0.0532766	+	0.0922777i	0.838157	+	0.545429i	$0.183633\pi$
$488$		0			0
$489$	−1.67356e45			−0.320178
$490$		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
$496$	5.78481e45			0.869162
$497$		0			0
$498$		0			0
$499$	7.35013e45	+	1.27308e46i	0.996749	+	1.72642i	0.568155	+	0.822922i	$0.307658\pi$
$499$	7.35013e45	+	1.27308e46i	0.996749	+	1.72642i	0.428594	+	0.903497i	$0.359009\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.99467e45	+	3.45486e45i	−0.206412	+	0.357516i
$508$	−8.94597e45	+	1.54949e46i	−0.895250	+	1.55062i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	3.02494e45	−	3.55738e45i	0.273881	−	0.322089i
$512$		0			0
$513$	−1.13430e46	−	1.96467e46i	−0.961023	−	1.66454i
$514$		0			0
$515$		0			0
$516$	−9.68984e45	−	1.67833e46i	−0.743484	−	1.28775i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$521$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$522$		0			0
$523$	−3.54478e45	−	6.13974e45i	−0.216302	−	0.374646i	0.737373	−	0.675486i	$-0.236066\pi$
$523$	−3.54478e45	−	6.13974e45i	−0.216302	−	0.374646i	−0.953675	+	0.300840i	$0.902733\pi$
$524$		0			0
$525$	−5.87301e45	−	1.64710e46i	−0.335856	−	0.941913i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−9.94756e45	+	1.72297e46i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	4.14096e46	+	7.58173e45i	1.89062	+	0.346155i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.82480e46	−	3.16064e46i	−0.626413	−	1.08498i	−0.988266	−	0.152744i	$-0.951189\pi$
$541$	−1.82480e46	−	3.16064e46i	−0.626413	−	1.08498i	0.361853	−	0.932235i	$-0.382144\pi$
$542$		0			0
$543$	2.75598e46	−	4.77349e46i	0.888543	−	1.53900i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.67035e46			1.89830			0.949149	−	0.314827i	$-0.101947\pi$
$547$	6.67035e46			1.89830			0.949149	+	0.314827i	$0.101947\pi$
$548$		0			0
$549$	3.72568e46	−	6.45307e46i	0.996497	−	1.72598i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	2.63189e46	+	7.38117e46i	0.622214	+	1.74501i
$554$		0			0
$555$		0			0
$556$	−3.15307e46	+	5.46128e46i	−0.679927	+	1.17767i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	5.79010e46			1.13942
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$563$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	4.19130e46	−	4.92904e46i	0.647793	−	0.761816i
$568$		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
$571$	3.07789e46	−	5.33107e46i	0.422124	−	0.731140i	−0.574023	−	0.818839i	$-0.694618\pi$
$571$	3.07789e46	−	5.33107e46i	0.422124	−	0.731140i	0.996147	+	0.0876992i	$0.0279514\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−4.22817e46	−	7.32340e46i	−0.500000	−	0.866025i
$577$	−7.53992e46	+	1.30595e47i	−0.865721	+	1.49947i	0.000609350		1.00000i	$0.499806\pi$
$577$	−7.53992e46	+	1.30595e47i	−0.865721	+	1.49947i	−0.866330	+	0.499472i	$0.833527\pi$
$578$		0			0
$579$	−8.66082e46	−	1.50010e47i	−0.937613	−	1.62399i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	1.92977e46	+	1.18503e47i	0.160728	+	0.986999i
$589$	−2.06454e47			−1.67057
$590$		0			0
$591$		0			0
$592$	−1.26854e47	+	2.19717e47i	−0.941530	+	1.63078i
$593$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$593$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−7.82106e46	−	1.35465e47i	−0.503155	−	0.871491i
$598$		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
$601$	1.25760e47			0.722234			0.361117	−	0.932521i	$-0.382396\pi$
$601$	1.25760e47			0.722234			0.361117	+	0.932521i	$0.382396\pi$
$602$		0			0
$603$	−3.67279e47			−1.99343
$604$	1.41885e47	+	2.45752e47i	0.748701	+	1.29679i
$605$		0			0
$606$		0			0
$607$	1.85595e47	+	3.21460e47i	0.900240	+	1.55926i	0.827183	+	0.561933i	$0.189942\pi$
$607$	1.85595e47	+	3.21460e47i	0.900240	+	1.55926i	0.0730573	+	0.997328i	$0.476724\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.96818e47	−	3.40898e47i	0.807675	−	1.39893i	−0.106796	−	0.994281i	$-0.534059\pi$
$613$	1.96818e47	−	3.40898e47i	0.807675	−	1.39893i	0.914471	−	0.404652i	$-0.132607\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−2.71540e47	+	4.70321e47i	−0.944262	+	1.63551i	−0.187039	+	0.982352i	$0.559889\pi$
$619$	−2.71540e47	+	4.70321e47i	−0.944262	+	1.63551i	−0.757223	+	0.653157i	$0.773444\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	2.52652e47			0.766274
$625$	−1.69407e47	−	2.93421e47i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	9.66766e46	+	1.67449e47i	0.263031	+	0.455584i
$629$		0			0
$630$		0			0
$631$	−1.21647e47			−0.305213			−0.152607	−	0.988287i	$-0.548767\pi$
$631$	−1.21647e47			−0.305213			−0.152607	+	0.988287i	$0.548767\pi$
$632$		0			0
$633$	−2.33905e47	+	4.05136e47i	−0.556131	+	0.963247i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.35449e47	−	1.27096e47i	−0.716566	−	0.271494i
$638$		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
$643$	4.89380e45			0.00891373			0.00445686	−	0.999990i	$-0.498581\pi$
$643$	4.89380e45			0.00891373			0.00445686	+	0.999990i	$0.498581\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$647$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.97756e47	−	5.54611e47i	−0.291913	−	0.818675i
$652$	2.22638e47			0.320178
$653$		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
$657$	3.34762e47			0.422790
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	4.38721e47	−	7.59887e47i	0.499763	−	0.865614i	−0.500237	−	0.865888i	$-0.666754\pi$
$661$	4.38721e47	−	7.59887e47i	0.499763	−	0.865614i	1.00000		0.000274157i	$8.72669e-5\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	9.53878e47	−	1.65217e48i	0.885623	−	1.53394i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.82831e47			0.572871			0.286436	−	0.958099i	$-0.407530\pi$
$673$	6.82831e47			0.572871			0.286436	+	0.958099i	$0.407530\pi$
$674$		0			0
$675$	6.26807e47	−	1.08566e48i	0.500000	−	0.866025i
$676$	2.65356e47	−	4.59610e47i	0.206412	−	0.357516i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	−4.75462e47	−	8.70527e46i	−0.343028	−	0.0628053i
$680$		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	$0.666667\pi$
$684$	1.50899e48	+	2.61365e48i	0.961023	+	1.66454i
$685$		0			0
$686$		0			0
$687$	−2.75785e47			−0.163045
$688$	1.28907e48	+	2.23273e48i	0.743484	+	1.28775i
$689$		0			0
$690$		0			0
$691$	−1.86527e48	−	3.23074e48i	−0.999113	−	1.73051i	−0.536029	−	0.844200i	$-0.680076\pi$
$691$	−1.86527e48	−	3.23074e48i	−0.999113	−	1.73051i	−0.463084	−	0.886314i	$-0.653257\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	7.81303e47	+	2.19118e48i	0.335856	+	0.941913i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	4.52729e48	−	7.84150e48i	1.80966	−	3.13443i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−9.70591e47	−	1.68111e48i	−0.335777	−	0.581582i	0.647857	−	0.761762i	$-0.275665\pi$
$709$	−9.70591e47	−	1.68111e48i	−0.335777	−	0.581582i	−0.983634	+	0.180180i	$0.942332\pi$
$710$		0			0
$711$	−2.80893e48	+	4.86520e48i	−0.926312	+	1.60442i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$719$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$720$		0			0
$721$	−2.13871e48	−	5.99804e48i	−0.556227	−	1.55995i
$722$		0			0
$723$	3.95271e48	+	6.84629e48i	0.980717	+	1.69865i
$724$	−3.66635e48	+	6.35030e48i	−0.888543	+	1.53900i
$725$		0			0
$726$		0			0
$727$	−4.30802e48			−0.973179			−0.486589	−	0.873631i	$-0.661759\pi$
$727$	−4.30802e48			−0.973179			−0.486589	+	0.873631i	$0.661759\pi$
$728$		0			0
$729$	4.63840e48			1.00000
$730$		0			0
$731$		0			0
$732$	−4.95638e48	+	8.58470e48i	−0.996497	+	1.72598i
$733$	−1.52715e48	−	2.64511e48i	−0.299996	−	0.519608i	0.676139	−	0.736774i	$-0.263652\pi$
$733$	−1.52715e48	−	2.64511e48i	−0.299996	−	0.519608i	−0.976135	+	0.217166i	$0.930319\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.20556e48	+	2.08809e48i	−0.206174	+	0.357103i	−0.950506	−	0.310706i	$-0.899435\pi$
$739$	−1.20556e48	+	2.08809e48i	−0.206174	+	0.357103i	0.744332	+	0.667809i	$0.232768\pi$
$740$		0			0
$741$	−9.01690e48			−1.47281
$742$		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
$751$	−6.88991e48	−	1.19337e49i	−0.896054	−	1.55201i	−0.832494	−	0.554034i	$-0.813088\pi$
$751$	−6.88991e48	−	1.19337e49i	−0.896054	−	1.55201i	−0.0635603	−	0.997978i	$-0.520246\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−5.57580e48	+	6.55724e48i	−0.647793	+	0.761816i
$757$	9.15388e48			1.03986			0.519931	−	0.854208i	$-0.325958\pi$
$757$	9.15388e48			1.03986			0.519931	+	0.854208i	$0.325958\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$761$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$762$		0			0
$763$	−7.82710e48	+	9.20481e48i	−0.777470	+	0.914319i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	5.62485e48	+	9.74252e48i	0.500000	+	0.866025i
$769$	−1.48668e49			−1.29262			−0.646310	−	0.763075i	$-0.723689\pi$
$769$	−1.48668e49			−1.29262			−0.646310	+	0.763075i	$0.723689\pi$
$770$		0			0
$771$		0			0
$772$	1.15217e49	+	1.99562e49i	0.937613	+	1.62399i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−5.70426e48	−	9.88007e48i	−0.434581	−	0.752716i
$776$		0			0
$777$	2.54016e49	+	4.65081e48i	1.85227	+	0.339134i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.56723e48	−	1.57648e49i	−0.160728	−	0.986999i
$785$		0			0
$786$		0			0
$787$	8.61234e48	−	1.49170e49i	0.505302	−	0.875208i	−0.494679	−	0.869076i	$-0.664714\pi$
$787$	8.61234e48	−	1.49170e49i	0.505302	−	0.875208i	0.999981	−	0.00613289i	$-0.00195217\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−1.48083e49	−	2.56487e49i	−0.763590	−	1.32258i
$794$		0			0
$795$		0			0
$796$	1.04046e49	+	1.80213e49i	0.503155	+	0.871491i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	4.88601e49			1.99343
$805$		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
$811$	5.67484e49			1.99802			0.999008	−	0.0445279i	$-0.0141784\pi$
$811$	5.67484e49			1.99802			0.999008	+	0.0445279i	$0.0141784\pi$
$812$		0			0
$813$	−5.92276e49			−1.99979
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−4.60055e49	−	7.96839e49i	−1.42901	−	2.47512i
$818$		0			0
$819$	−8.63701e48	−	2.42226e49i	−0.257358	−	0.721764i
$820$		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
$823$	2.34457e49	−	4.06091e49i	0.643081	−	1.11385i	−0.341660	−	0.939824i	$-0.610989\pi$
$823$	2.34457e49	−	4.06091e49i	0.643081	−	1.11385i	0.984741	−	0.174025i	$-0.0556775\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−2.22287e49	+	3.85013e49i	−0.538875	+	0.933359i	0.460090	+	0.887872i	$0.347817\pi$
$829$	−2.22287e49	+	3.85013e49i	−0.538875	+	0.933359i	−0.998965	+	0.0454865i	$0.985516\pi$
$830$		0			0
$831$	4.27181e49	+	7.39899e49i	0.994019	+	1.72169i
$832$	−3.36109e49			−0.766274
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.11059e49	−	3.65565e49i	0.434581	−	0.752716i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	5.26662e49			1.00000
$842$		0			0
$843$		0			0
$844$	3.11171e49	−	5.38963e49i	0.556131	−	0.963247i
$845$		0			0
$846$		0			0
$847$	1.99605e49	+	5.59795e49i	0.335856	+	0.941913i
$848$		0			0
$849$	−4.11674e49	−	7.13041e49i	−0.665462	−	1.15261i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.32581e50			1.97855			0.989276	−	0.146058i	$-0.0466588\pi$
$853$	1.32581e50			1.97855			0.989276	+	0.146058i	$0.0466588\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$857$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$858$		0			0
$859$	5.39871e49	+	9.35085e49i	0.715166	+	1.23870i	0.962896	+	0.269874i	$0.0869820\pi$
$859$	5.39871e49	+	9.35085e49i	0.715166	+	1.23870i	−0.247730	+	0.968829i	$0.579685\pi$
$860$		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
$867$	−8.83740e49			−1.00000
$868$	2.63081e49	+	7.37814e49i	0.291913	+	0.818675i
$869$		0			0
$870$		0			0
$871$	−7.29902e49	+	1.26423e50i	−0.763758	+	1.32287i
$872$		0			0
$873$	−1.73261e49	−	3.00097e49i	−0.174365	−	0.302009i
$874$		0			0
$875$		0			0
$876$	−4.45343e49			−0.422790
$877$	−3.87056e49	−	6.70401e49i	−0.360397	−	0.624225i	0.627630	−	0.778512i	$-0.284025\pi$
$877$	−3.87056e49	−	6.70401e49i	−0.360397	−	0.624225i	−0.988026	+	0.154287i	$0.950692\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−2.21972e50			−1.84063			−0.920316	−	0.391177i	$-0.872068\pi$
$883$	−2.21972e50			−1.84063			−0.920316	+	0.391177i	$0.872068\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	−2.38311e50	−	4.36326e49i	−1.76122	−	0.322464i
$890$		0			0
$891$		0			0
$892$	−1.26897e50	+	2.19792e50i	−0.885623	+	1.53394i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−8.33859e49	+	1.44429e50i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	1.69992e50	−	1.99914e50i	0.963247	−	1.13280i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.56177e50	+	2.70507e50i	−0.820906	+	1.42185i	0.0841017	+	0.996457i	$0.473198\pi$
$907$	−1.56177e50	+	2.70507e50i	−0.820906	+	1.42185i	−0.905008	+	0.425394i	$0.860135\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−2.00746e50	−	3.47701e50i	−0.961023	−	1.66454i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	3.66885e49			0.163045
$917$		0			0
$918$		0			0
$919$	2.01446e50	+	3.48915e50i	0.846828	+	1.46675i	0.884024	+	0.467441i	$0.154824\pi$
$919$	2.01446e50	+	3.48915e50i	0.846828	+	1.46675i	−0.0371963	+	0.999308i	$0.511843\pi$
$920$		0			0
$921$	−2.71947e49	+	4.71026e49i	−0.110172	+	0.190823i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	5.00350e50			1.88306
$926$		0			0
$927$	2.28257e50	−	3.95353e50i	0.828074	−	1.43427i
$928$		0			0
$929$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$929$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$930$		0			0
$931$	9.16221e49	+	5.62632e50i	0.308927	+	1.89706i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	1.77268e50			0.535867			0.267934	−	0.963437i	$-0.413659\pi$
$937$	1.77268e50			0.535867			0.267934	+	0.963437i	$0.413659\pi$
$938$		0			0
$939$	−2.01884e50			−0.588555
$940$		0			0
$941$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$941$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$942$		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	$0.333333\pi$
$948$	3.73679e50	−	6.47231e50i	0.926312	−	1.60442i
$949$	6.65281e49	−	1.15230e50i	0.161987	−	0.280569i
$950$		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
$961$	6.21797e49	+	1.07698e50i	0.122279	+	0.211793i
$962$		0			0
$963$		0			0
$964$	−5.25840e50	−	9.10781e50i	−0.980717	−	1.69865i
$965$		0			0
$966$		0			0
$967$	1.04661e51			1.85155			0.925776	−	0.378072i	$-0.123413\pi$
$967$	1.04661e51			1.85155			0.925776	+	0.378072i	$0.123413\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	−6.17059e50			−1.00000
$973$	−8.39945e50	−	1.53786e50i	−1.33762	−	0.244906i
$974$		0			0
$975$	−2.49134e50	−	4.31512e50i	−0.383137	−	0.663613i
$976$	6.59361e50	−	1.14205e51i	0.996497	−	1.72598i
$977$		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
$981$	−8.66205e50			−1.20018
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	1.19954e51			1.47281
$989$		0			0
$990$		0			0
$991$	−8.50409e50	+	1.47295e51i	−0.991689	+	1.71766i	−0.384426	+	0.923156i	$0.625601\pi$
$991$	−8.50409e50	+	1.47295e51i	−0.991689	+	1.71766i	−0.607263	+	0.794501i	$0.707733\pi$
$992$		0			0
$993$	1.05463e51			1.18841
$994$		0			0
$995$		0			0
$996$		0			0
$997$	3.28388e50	−	5.68784e50i	0.345596	−	0.598590i	−0.639866	−	0.768487i	$-0.721010\pi$
$997$	3.28388e50	−	5.68784e50i	0.345596	−	0.598590i	0.985462	+	0.169897i	$0.0543433\pi$
$998$		0			0
$999$	9.25652e50	+	1.60328e51i	0.941530	+	1.63078i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.35.h.a.11.1	yes	2
3.2	odd	2	CM	21.35.h.a.11.1	yes	2
7.2	even	3	inner	21.35.h.a.2.1	✓	2
21.2	odd	6	inner	21.35.h.a.2.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.35.h.a.2.1	✓	2	7.2	even	3	inner
21.35.h.a.2.1	✓	2	21.2	odd	6	inner
21.35.h.a.11.1	yes	2	1.1	even	1	trivial
21.35.h.a.11.1	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:	\( k \)	\(=\)	\( 35 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(6.45701e7 - 1.11839e8i) q^{3} +(-8.58993e9 + 1.48782e10i) q^{4} +(-2.28827e14 - 4.18961e13i) q^{7} +(-8.33859e15 - 1.44429e16i) q^{9} +O(q^{10})\)	\(q+(6.45701e7 - 1.11839e8i) q^{3} +(-8.58993e9 + 1.48782e10i) q^{4} +(-2.28827e14 - 4.18961e13i) q^{7} +(-8.33859e15 - 1.44429e16i) q^{9} +(1.10931e18 + 1.92137e18i) q^{12} -6.62859e18 q^{13} +(-1.47574e20 - 2.55606e20i) q^{16} +(5.26678e21 + 9.12232e21i) q^{19} +(-1.94610e22 + 2.28864e22i) q^{21} +(-2.91038e23 + 5.04093e23i) q^{25} -2.15369e24 q^{27} +(2.58895e24 - 3.04465e24i) q^{28} +(-9.79985e24 + 1.69738e25i) q^{31} +2.86512e26 q^{36} +(-4.29797e26 - 7.44431e26i) q^{37} +(-4.28009e26 + 7.41333e26i) q^{39} -8.73505e27 q^{43} -3.81154e28 q^{48} +(5.06064e28 + 1.91739e28i) q^{49} +(5.69391e28 - 9.86215e28i) q^{52} +1.36030e30 q^{57} +(2.23400e30 + 3.86940e30i) q^{61} +(1.30299e30 + 3.65427e30i) q^{63} +5.07060e30 q^{64} +(1.10114e31 - 1.90723e31i) q^{67} +(-1.00365e31 + 1.73838e31i) q^{73} +(3.75847e31 + 6.50987e31i) q^{75} -1.80965e32 q^{76} +(-1.68429e32 - 2.91728e32i) q^{79} +(-1.39064e32 + 2.40866e32i) q^{81} +(-1.73341e32 - 4.86137e32i) q^{84} +(1.51680e33 + 2.77712e32i) q^{91} +(1.26555e33 + 2.19200e33i) q^{93} +2.07782e33 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 129140163 q^{3} - 17179869184 q^{4} - 457653465545653 q^{7} - 16\!\cdots\!69 q^{9}+O(q^{10}) \) 2 * q + 129140163 * q^3 - 17179869184 * q^4 - 457653465545653 * q^7 - 16677181699666569 * q^9	\( 2 q + 129140163 q^{3} - 17179869184 q^{4} - 457653465545653 q^{7} - 16\!\cdots\!69 q^{9}+ \cdots + 41\!\cdots\!04 q^{97}+O(q^{100}) \) 2 * q + 129140163 * q^3 - 17179869184 * q^4 - 457653465545653 * q^7 - 16677181699666569 * q^9 + 2218611106740436992 * q^12 - 13257178113610601474 * q^13 - 295147905179352825856 * q^16 + 10533550932831642194149 * q^19 - 38921926981395836058246 * q^21 - 582076609134674072265625 * q^25 - 4307387926151115532621494 * q^27 + 5177890428476387049340928 * q^28 - 19599701438028304062305147 * q^31 + 573023599916140863676219392 * q^36 - 859594323581411974757802287 * q^37 - 856017071255852796440200131 * q^39 - 17470096463040699199417500122 * q^43 - 76230897167940336331108909056 * q^48 + 101212782450041974235107874711 * q^49 + 113878292870409011619438788608 * q^52 + 2720608968869360649020159012574 * q^57 + 4468002075713925041487910419766 * q^61 + 2605986005735392637396581440459 * q^63 + 10141204801825835211973625643008 * q^64 + 22022844874719432605769728964013 * q^67 - 20073068910160542409412259436943 * q^73 + 75169468182139098644256591796875 * q^75 - 361930054138097567182749100408832 * q^76 - 336858662185559380928110634527811 * q^79 - 278128389443693511257285776231761 * q^81 - 346681447768257186026240461504512 * q^84 + 3033596753524937217268086468046261 * q^91 + 2531108638458309585219648839318961 * q^93 + 4155647952096798288265379422464004 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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