LMFDB - Embedded newform 21.21.h.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 21);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 21 $
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:	$53.2378906716$
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			21.21.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)

$f(q)$	$=$	$q+(-29524.5 + 51137.9i) q^{3} +(-524288. + 908093. i) q^{4} +(-2.22994e8 + 1.73395e8i) q^{7} +(-1.74339e9 - 3.01964e9i) q^{9} +O(q^{10})$	\(q+(-29524.5 + 51137.9i) q^{3} +(-524288. + 908093. i) q^{4} +(-2.22994e8 + 1.73395e8i) q^{7} +(-1.74339e9 - 3.01964e9i) q^{9} +(-3.09587e10 - 5.36220e10i) q^{12} +3.81709e10 q^{13} +(-5.49756e11 - 9.52205e11i) q^{16} +(-6.00291e12 - 1.03973e13i) q^{19} +(-2.28330e12 - 1.65229e13i) q^{21} +(-4.76837e13 + 8.25906e13i) q^{25} +2.05891e14 q^{27} +(-4.05462e13 - 2.93408e14i) q^{28} +(-4.22563e14 + 7.31900e14i) q^{31} +3.65616e15 q^{36} +(3.60548e14 + 6.24488e14i) q^{37} +(-1.12698e15 + 1.95198e15i) q^{39} +1.34394e15 q^{43} +6.49251e16 q^{48} +(1.96603e16 - 7.73323e16i) q^{49} +(-2.00126e16 + 3.46628e16i) q^{52} +7.08931e17 q^{57} +(6.93894e17 + 1.20186e18i) q^{61} +(9.12358e17 + 3.71066e17i) q^{63} +1.15292e18 q^{64} +(1.02793e18 - 1.78042e18i) q^{67} +(2.38136e18 - 4.12464e18i) q^{73} +(-2.81568e18 - 4.87689e18i) q^{75} +1.25890e19 q^{76} +(7.26142e18 + 1.25772e19i) q^{79} +(-6.07883e18 + 1.05288e19i) q^{81} +(1.62014e19 + 6.58929e18i) q^{84} +(-8.51188e18 + 6.61866e18i) q^{91} +(-2.49519e19 - 4.32180e19i) q^{93} -1.46702e20 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 59049 q^{3} - 1048576 q^{4} - 445987849 q^{7} - 3486784401 q^{9}+O(q^{10}) $ 2 * q - 59049 * q^3 - 1048576 * q^4 - 445987849 * q^7 - 3486784401 * q^9	$ 2 q - 59049 q^{3} - 1048576 q^{4} - 445987849 q^{7} - 3486784401 q^{9} - 61917364224 q^{12} + 76341844702 q^{13} - 1099511627776 q^{16} - 12005816399399 q^{19} - 4566600118926 q^{21} - 95367431640625 q^{25} + 411782264189298 q^{27} - 81092436557824 q^{28} - 845125195444727 q^{31} + 73\!\cdots\!52 q^{36}+ \cdots - 29\!\cdots\!48 q^{97}+O(q^{100}) $ 2 * q - 59049 * q^3 - 1048576 * q^4 - 445987849 * q^7 - 3486784401 * q^9 - 61917364224 * q^12 + 76341844702 * q^13 - 1099511627776 * q^16 - 12005816399399 * q^19 - 4566600118926 * q^21 - 95367431640625 * q^25 + 411782264189298 * q^27 - 81092436557824 * q^28 - 845125195444727 * q^31 + 7312316880125952 * q^36 + 721096304301649 * q^37 - 2253954793904199 * q^39 + 2687870111203246 * q^43 + 129850124217090048 * q^48 + 39320628860422799 * q^49 - 40025113075122176 * q^52 + 1417862905136223102 * q^57 + 1387788230779990126 * q^61 + 1824716645351204823 * q^63 + 2305843009213693952 * q^64 + 2055850947949627849 * q^67 + 4762721323358202049 * q^73 - 5631351470947265625 * q^75 + 25178021873632411648 * q^76 + 14522848453745208601 * q^79 - 12157665459056928801 * q^81 + 32402819372314263552 * q^84 - 17023767553668512999 * q^91 - 49903797665815684623 * q^93 - 293403496397740791548 * 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	$0.333333\pi$
$3$	−29524.5	+	51137.9i	−0.500000	+	0.866025i
$3$	−29524.5	+	51137.9i	−0.500000	+	0.866025i
$4$	−524288.	+	908093.i	−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
$7$	−2.22994e8	+	1.73395e8i	−0.789428	+	0.613843i
$7$	−2.22994e8	+	1.73395e8i	−0.789428	+	0.613843i
$8$		0			0
$9$	−1.74339e9	−	3.01964e9i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−3.09587e10	−	5.36220e10i	−0.500000	−	0.866025i
$13$	3.81709e10			0.276885			0.138442	−	0.990370i	$-0.455790\pi$
$13$	3.81709e10			0.276885			0.138442	+	0.990370i	$0.455790\pi$
$14$		0			0
$15$		0			0
$16$	−5.49756e11	−	9.52205e11i	−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
$19$	−6.00291e12	−	1.03973e13i	−0.979097	−	1.69585i	−0.665693	−	0.746226i	$-0.731864\pi$
$19$	−6.00291e12	−	1.03973e13i	−0.979097	−	1.69585i	−0.313404	−	0.949620i	$-0.601469\pi$
$20$		0			0
$21$	−2.28330e12	−	1.65229e13i	−0.136889	−	0.990586i
$22$		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
$25$	−4.76837e13	+	8.25906e13i	−0.500000	+	0.866025i
$26$		0			0
$27$	2.05891e14			1.00000
$28$	−4.05462e13	−	2.93408e14i	−0.136889	−	0.990586i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−4.22563e14	+	7.31900e14i	−0.515554	+	0.892966i	0.484283	+	0.874911i	$0.339080\pi$
$31$	−4.22563e14	+	7.31900e14i	−0.515554	+	0.892966i	−0.999837	+	0.0180543i	$0.994253\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	3.65616e15			1.00000
$37$	3.60548e14	+	6.24488e14i	0.0749801	+	0.129869i	0.901078	−	0.433658i	$-0.142777\pi$
$37$	3.60548e14	+	6.24488e14i	0.0749801	+	0.129869i	−0.826098	+	0.563527i	$0.809444\pi$
$38$		0			0
$39$	−1.12698e15	+	1.95198e15i	−0.138442	+	0.239789i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	1.34394e15			0.0621862			0.0310931	−	0.999516i	$-0.490101\pi$
$43$	1.34394e15			0.0621862			0.0310931	+	0.999516i	$0.490101\pi$
$44$		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	$0.333333\pi$
$48$	6.49251e16			1.00000
$49$	1.96603e16	−	7.73323e16i	0.246394	−	0.969170i
$50$		0			0
$51$		0			0
$52$	−2.00126e16	+	3.46628e16i	−0.138442	+	0.239789i
$53$		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
$57$	7.08931e17			1.95819
$58$		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
$61$	6.93894e17	+	1.20186e18i	0.972736	+	1.68483i	0.687213	+	0.726456i	$0.258834\pi$
$61$	6.93894e17	+	1.20186e18i	0.972736	+	1.68483i	0.285522	+	0.958372i	$0.407833\pi$
$62$		0			0
$63$	9.12358e17	+	3.71066e17i	0.926318	+	0.376743i
$64$	1.15292e18			1.00000
$65$		0			0
$66$		0			0
$67$	1.02793e18	−	1.78042e18i	0.563915	−	0.976729i	−0.433235	−	0.901281i	$-0.642628\pi$
$67$	1.02793e18	−	1.78042e18i	0.563915	−	0.976729i	0.997150	−	0.0754482i	$-0.0240388\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	2.38136e18	−	4.12464e18i	0.554111	−	0.959748i	−0.443861	−	0.896095i	$-0.646392\pi$
$73$	2.38136e18	−	4.12464e18i	0.554111	−	0.959748i	0.997972	−	0.0636525i	$-0.0202749\pi$
$74$		0			0
$75$	−2.81568e18	−	4.87689e18i	−0.500000	−	0.866025i
$76$	1.25890e19			1.95819
$77$		0			0
$78$		0			0
$79$	7.26142e18	+	1.25772e19i	0.766921	+	1.32835i	0.939225	+	0.343303i	$0.111546\pi$
$79$	7.26142e18	+	1.25772e19i	0.766921	+	1.32835i	−0.172303	+	0.985044i	$0.555121\pi$
$80$		0			0
$81$	−6.07883e18	+	1.05288e19i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	1.62014e19	+	6.58929e18i	0.926318	+	0.376743i
$85$		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
$91$	−8.51188e18	+	6.61866e18i	−0.218581	+	0.169964i
$92$		0			0
$93$	−2.49519e19	−	4.32180e19i	−0.515554	−	0.892966i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.46702e20			−1.98938			−0.994690	−	0.102912i	$-0.967184\pi$
$97$	−1.46702e20			−1.98938			−0.994690	+	0.102912i	$0.967184\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e19	−	8.66025e19i	−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
$103$	−9.75304e19	−	1.68928e20i	−0.725718	−	1.25698i	−0.958678	−	0.284494i	$-0.908174\pi$
$103$	−9.75304e19	−	1.68928e20i	−0.725718	−	1.25698i	0.232960	−	0.972486i	$-0.425159\pi$
$104$		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	$0.333333\pi$
$108$	−1.07946e20	+	1.86968e20i	−0.500000	+	0.866025i
$109$	−2.33087e20	+	4.03719e20i	−0.984587	+	1.70535i	−0.340828	+	0.940126i	$0.610707\pi$
$109$	−2.33087e20	+	4.03719e20i	−0.984587	+	1.70535i	−0.643759	+	0.765229i	$0.722626\pi$
$110$		0			0
$111$	−4.25800e19			−0.149960
$112$	2.87700e20	+	1.17011e20i	0.926318	+	0.376743i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−6.65469e19	−	1.15263e20i	−0.138442	−	0.239789i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.36375e20	−	5.82619e20i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−4.43089e20	−	7.67453e20i	−0.515554	−	0.892966i
$125$		0			0
$126$		0			0
$127$	7.77305e20			0.712122			0.356061	−	0.934463i	$-0.384120\pi$
$127$	7.77305e20			0.712122			0.356061	+	0.934463i	$0.384120\pi$
$128$		0			0
$129$	−3.96790e19	+	6.87261e19i	−0.0310931	+	0.0538548i
$130$		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
$133$	3.14146e21	+	1.27767e21i	1.81391	+	0.737737i
$134$		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
$139$	5.21077e21			1.93533			0.967663	−	0.252248i	$-0.0811698\pi$
$139$	5.21077e21			1.93533			0.967663	+	0.252248i	$0.0811698\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.91688e21	+	3.32013e21i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	3.37415e21	+	3.28858e21i	0.716129	+	0.697968i
$148$	−7.56124e20			−0.149960
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	6.11218e21	−	1.05866e22i	0.991806	−	1.71786i	0.385262	−	0.922807i	$-0.374111\pi$
$151$	6.11218e21	−	1.05866e22i	0.991806	−	1.71786i	0.606543	−	0.795050i	$-0.292556\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.18172e21	−	2.04680e21i	−0.138442	−	0.239789i
$157$	−6.89465e21	+	1.19419e22i	−0.757732	+	1.31243i	0.186272	+	0.982498i	$0.440359\pi$
$157$	−6.89465e21	+	1.19419e22i	−0.757732	+	1.31243i	−0.944004	+	0.329933i	$0.892974\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.31858e22	−	2.28385e22i	−0.995933	−	1.72501i	−0.575996	−	0.817453i	$-0.695385\pi$
$163$	−1.31858e22	−	2.28385e22i	−0.995933	−	1.72501i	−0.419937	−	0.907553i	$-0.637948\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−1.75479e22			−0.923335
$170$		0			0
$171$	−2.09308e22	+	3.62533e22i	−0.979097	+	1.69585i
$172$	−7.04609e20	+	1.22042e21i	−0.0310931	+	0.0538548i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−3.68766e21	−	2.66853e22i	−0.136889	−	0.990586i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$	3.95296e22			1.04746			0.523730	−	0.851885i	$-0.324540\pi$
$181$	3.95296e22			1.04746			0.523730	+	0.851885i	$0.324540\pi$
$182$		0			0
$183$	−8.19475e22			−1.94547
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−4.59125e22	+	3.57006e22i	−0.789428	+	0.613843i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−3.40394e22	+	5.89580e22i	−0.500000	+	0.866025i
$193$	6.62234e22	−	1.14702e23i	0.923504	−	1.59956i	0.129554	−	0.991572i	$-0.458646\pi$
$193$	6.62234e22	−	1.14702e23i	0.923504	−	1.59956i	0.793950	−	0.607983i	$-0.208021\pi$
$194$		0			0
$195$		0			0
$196$	5.99172e22	+	5.83978e22i	0.716129	+	0.697968i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	7.57463e22	−	1.31197e23i	0.777734	−	1.34707i	−0.155511	−	0.987834i	$-0.549703\pi$
$199$	7.57463e22	−	1.31197e23i	0.777734	−	1.34707i	0.933245	−	0.359240i	$-0.116964\pi$
$200$		0			0
$201$	6.06980e22	+	1.05132e23i	0.563915	+	0.976729i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2.09847e22	−	3.63465e22i	−0.138442	−	0.239789i
$209$		0			0
$210$		0			0
$211$	2.17102e23			1.24119			0.620595	−	0.784131i	$-0.286891\pi$
$211$	2.17102e23			1.24119			0.620595	+	0.784131i	$0.286891\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.26792e22	−	2.36480e23i	−0.141148	−	1.02140i
$218$		0			0
$219$	1.40617e23	+	2.43556e23i	0.554111	+	0.959748i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	4.12862e23			1.35755			0.678776	−	0.734345i	$-0.262511\pi$
$223$	4.12862e23			1.35755			0.678776	+	0.734345i	$0.262511\pi$
$224$		0			0
$225$	3.32526e23			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−3.71684e23	+	6.43776e23i	−0.979097	+	1.69585i
$229$	1.76002e22	+	3.04845e22i	0.0443776	+	0.0768642i	0.887361	−	0.461075i	$-0.152536\pi$
$229$	1.76002e22	+	3.04845e22i	0.0443776	+	0.0768642i	−0.842983	+	0.537940i	$0.819203\pi$
$230$		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
$237$	−8.57560e23			−1.53384
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	6.59376e23	−	1.14207e24i	0.997614	−	1.72792i	0.439017	−	0.898479i	$-0.355327\pi$
$241$	6.59376e23	−	1.14207e24i	0.997614	−	1.72792i	0.558597	−	0.829440i	$-0.311340\pi$
$242$		0			0
$243$	−3.58949e23	−	6.21718e23i	−0.500000	−	0.866025i
$244$	−1.45520e24			−1.94547
$245$		0			0
$246$		0			0
$247$	−2.29137e23	−	3.96876e23i	−0.271097	−	0.469554i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−8.15301e23	+	6.33961e23i	−0.789428	+	0.613843i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−6.04463e23	+	1.04696e24i	−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
$259$	−1.88683e23	−	7.67396e22i	−0.138911	−	0.0564965i
$260$		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
$268$	1.07786e24	+	1.86690e24i	0.563915	+	0.976729i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−1.79147e23	−	3.10292e23i	−0.0838528	−	0.145237i	0.821049	−	0.570858i	$-0.193389\pi$
$271$	−1.79147e23	−	3.10292e23i	−0.0838528	−	0.145237i	−0.904902	+	0.425620i	$0.860056\pi$
$272$		0			0
$273$	−8.71557e22	−	6.30693e23i	−0.0379026	−	0.274278i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−1.02984e24	+	1.78374e24i	−0.387233	+	0.670707i	−0.992076	−	0.125637i	$-0.959902\pi$
$277$	−1.02984e24	+	1.78374e24i	−0.387233	+	0.670707i	0.604843	+	0.796345i	$0.293236\pi$
$278$		0			0
$279$	2.94677e24			1.03111
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	1.63926e24	−	2.83928e24i	0.497488	−	0.861675i	−0.502507	−	0.864573i	$-0.667589\pi$
$283$	1.63926e24	−	2.83928e24i	0.497488	−	0.861675i	0.999996	+	0.00289763i	$0.000922346\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.03212e24	−	3.51973e24i	−0.500000	−	0.866025i
$290$		0			0
$291$	4.33130e24	−	7.50202e24i	0.994690	−	1.72285i
$292$	2.49704e24	+	4.32500e24i	0.554111	+	0.959748i
$293$		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
$300$	5.90490e24			1.00000
$301$	−2.99689e23	+	2.33032e23i	−0.0490915	+	0.0381725i
$302$		0			0
$303$		0			0
$304$	−6.60027e24	+	1.14320e25i	−0.979097	+	1.69585i
$305$		0			0
$306$		0			0
$307$	8.17417e24			1.09916			0.549579	−	0.835442i	$-0.314788\pi$
$307$	8.17417e24			1.09916			0.549579	+	0.835442i	$0.314788\pi$
$308$		0			0
$309$	1.15181e25			1.45144
$310$		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
$313$	8.85581e24	+	1.53387e25i	0.981261	+	1.69959i	0.657497	+	0.753457i	$0.271615\pi$
$313$	8.85581e24	+	1.53387e25i	0.981261	+	1.69959i	0.323764	+	0.946138i	$0.395052\pi$
$314$		0			0
$315$		0			0
$316$	−1.52283e25			−1.53384
$317$		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
$324$	−6.37412e24	−	1.10403e25i	−0.500000	−	0.866025i
$325$	−1.82013e24	+	3.15256e24i	−0.138442	+	0.239789i
$326$		0			0
$327$	−1.37636e25	−	2.38392e25i	−0.984587	−	1.70535i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.34605e25	−	2.33143e25i	−0.852671	−	1.47687i	−0.878789	−	0.477211i	$-0.841648\pi$
$331$	−1.34605e25	−	2.33143e25i	−0.852671	−	1.47687i	0.0261172	−	0.999659i	$-0.491686\pi$
$332$		0			0
$333$	1.25715e24	−	2.17745e24i	0.0749801	−	0.129869i
$334$		0			0
$335$		0			0
$336$	−1.44779e25	+	1.12577e25i	−0.789428	+	0.613843i
$337$	−3.46628e25			−1.83470			−0.917348	−	0.398086i	$-0.869675\pi$
$337$	−3.46628e25			−1.83470			−0.917348	+	0.398086i	$0.869675\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	9.02493e24	+	2.06536e25i	0.400408	+	0.916337i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	9.20209e24			0.343267			0.171634	−	0.985161i	$-0.445095\pi$
$349$	9.20209e24			0.343267			0.171634	+	0.985161i	$0.445095\pi$
$350$		0			0
$351$	7.85905e24			0.276885
$352$		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
$361$	−5.32748e25	+	9.22747e25i	−1.41726	+	2.45477i
$362$		0			0
$363$	3.97252e25			1.00000
$364$	−1.54769e24	−	1.11997e25i	−0.0379026	−	0.274278i
$365$		0			0
$366$		0			0
$367$	2.22629e25	−	3.85604e25i	0.502250	−	0.869923i	−0.497746	−	0.867323i	$-0.665839\pi$
$367$	2.22629e25	−	3.85604e25i	0.502250	−	0.869923i	0.999997	−	0.00260049i	$-0.000827762\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	5.23279e25			1.03111
$373$	−4.35481e25	−	7.54275e25i	−0.835374	−	1.44691i	−0.893726	−	0.448614i	$-0.851918\pi$
$373$	−4.35481e25	−	7.54275e25i	−0.835374	−	1.44691i	0.0583519	−	0.998296i	$-0.481415\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.22146e26			1.99750			0.998751	−	0.0499685i	$-0.0159121\pi$
$379$	1.22146e26			1.99750			0.998751	+	0.0499685i	$0.0159121\pi$
$380$		0			0
$381$	−2.29495e25	+	3.97498e25i	−0.356061	+	0.616716i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.34301e24	−	4.05821e24i	−0.0310931	−	0.0538548i
$388$	7.69140e25	−	1.33219e26i	0.994690	−	1.72285i
$389$		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
$397$	8.31510e25	+	1.44022e26i	0.854993	+	1.48089i	0.876652	+	0.481126i	$0.159772\pi$
$397$	8.31510e25	+	1.44022e26i	0.854993	+	1.48089i	−0.0216588	+	0.999765i	$0.506895\pi$
$398$		0			0
$399$	−1.58087e26	+	1.22925e26i	−1.54585	+	1.20202i
$400$	1.04858e26			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
$403$	−1.61296e25	+	2.79373e25i	−0.142749	+	0.247249i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.23612e26	−	2.14102e26i	0.943685	−	1.63451i	0.185321	−	0.982678i	$-0.440667\pi$
$409$	1.23612e26	−	2.14102e26i	0.943685	−	1.63451i	0.758363	−	0.651832i	$-0.225999\pi$
$410$		0			0
$411$		0			0
$412$	2.04536e26			1.45144
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.53845e26	+	2.66468e26i	−0.967663	+	1.67604i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−3.42656e26			−1.95901			−0.979507	−	0.201410i	$-0.935448\pi$
$421$	−3.42656e26			−1.95901			−0.979507	+	0.201410i	$0.935448\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−3.63131e26	−	1.47689e26i	−1.80212	−	0.732944i
$428$		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	$0.666667\pi$
$432$	−1.13190e26	−	1.96051e26i	−0.500000	−	0.866025i
$433$	1.96482e26			0.848096			0.424048	−	0.905640i	$-0.360609\pi$
$433$	1.96482e26			0.848096			0.424048	+	0.905640i	$0.360609\pi$
$434$		0			0
$435$		0			0
$436$	−2.44410e26	−	4.23330e26i	−0.984587	−	1.70535i
$437$		0			0
$438$		0			0
$439$	1.27020e26	+	2.20005e26i	0.477780	+	0.827539i	0.999676	−	0.0254706i	$-0.00810842\pi$
$439$	1.27020e26	+	2.20005e26i	0.477780	+	0.827539i	−0.521896	+	0.853009i	$0.674775\pi$
$440$		0			0
$441$	−2.67792e26	+	7.54533e25i	−0.962523	+	0.271202i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	2.23242e25	−	3.86666e25i	0.0749801	−	0.129869i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2.57094e26	+	1.99911e26i	−0.789428	+	0.613843i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	3.60918e26	+	6.25128e26i	0.991806	+	1.71786i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.42274e26	−	5.92837e26i	−0.861415	−	1.49201i	−0.870563	−	0.492056i	$-0.836245\pi$
$457$	−3.42274e26	−	5.92837e26i	−0.861415	−	1.49201i	0.00914856	−	0.999958i	$-0.497088\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	2.51579e26			0.555730			0.277865	−	0.960620i	$-0.410373\pi$
$463$	2.51579e26			0.555730			0.277865	+	0.960620i	$0.410373\pi$
$464$		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	$0.333333\pi$
$468$	1.39559e26			0.276885
$469$	7.94954e25	+	5.75260e26i	0.154388	+	1.11721i
$470$		0			0
$471$	−4.07122e26	−	7.05157e26i	−0.757732	−	1.31243i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	1.14496e27			1.95819
$476$		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
$481$	1.37625e25	+	2.38373e25i	0.0207609	+	0.0359588i
$482$		0			0
$483$		0			0
$484$	7.05429e26			1.00000
$485$		0			0
$486$		0			0
$487$	−6.50068e26	+	1.12595e27i	−0.866302	+	1.50048i	−0.000554074		1.00000i	$0.500176\pi$
$487$	−6.50068e26	+	1.12595e27i	−0.866302	+	1.50048i	−0.865748	+	0.500480i	$0.833157\pi$
$488$		0			0
$489$	1.55721e27			1.99187
$490$		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
$496$	9.29225e26			1.03111
$497$		0			0
$498$		0			0
$499$	−6.76001e26	−	1.17087e27i	−0.706223	−	1.22321i	−0.966248	−	0.257613i	$-0.917064\pi$
$499$	−6.76001e26	−	1.17087e27i	−0.706223	−	1.22321i	0.260025	−	0.965602i	$-0.416269\pi$
$500$		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
$507$	5.18094e26	−	8.97366e26i	0.461667	−	0.799631i
$508$	−4.07532e26	+	7.05866e26i	−0.356061	+	0.616716i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	1.84164e26	+	1.33269e27i	0.151704	+	1.09779i
$512$		0			0
$513$	−1.23595e27	−	2.14072e27i	−0.979097	−	1.69585i
$514$		0			0
$515$		0			0
$516$	−4.16065e25	−	7.20645e25i	−0.0310931	−	0.0538548i
$517$		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
$523$	−2.27221e26	−	3.93558e26i	−0.148399	−	0.257035i	0.782237	−	0.622981i	$-0.214079\pi$
$523$	−2.27221e26	−	3.93558e26i	−0.148399	−	0.257035i	−0.930636	+	0.365946i	$0.880745\pi$
$524$		0			0
$525$	1.47351e27	+	5.99292e26i	0.926318	+	0.376743i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−8.58078e26	+	1.48623e27i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−2.80727e27	+	2.18288e27i	−1.54585	+	1.20202i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.34276e27	−	2.32573e27i	−0.625210	−	1.08290i	−0.988500	−	0.151220i	$-0.951680\pi$
$541$	−1.34276e27	−	2.32573e27i	−0.625210	−	1.08290i	0.363290	−	0.931676i	$-0.381654\pi$
$542$		0			0
$543$	−1.16709e27	+	2.02146e27i	−0.523730	+	0.907126i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−4.59721e26			−0.191699			−0.0958497	−	0.995396i	$-0.530557\pi$
$547$	−4.59721e26			−0.191699			−0.0958497	+	0.995396i	$0.530557\pi$
$548$		0			0
$549$	2.41946e27	−	4.19063e27i	0.972736	−	1.68483i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.80007e27	−	1.54553e27i	−1.42083	−	0.577865i
$554$		0			0
$555$		0			0
$556$	−2.73194e27	+	4.73187e27i	−0.967663	+	1.67604i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	5.12992e25			0.0172184
$560$		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
$567$	−4.70111e26	−	3.40191e27i	−0.136889	−	0.990586i
$568$		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
$571$	−7.68287e25	+	1.33071e26i	−0.0208527	+	0.0361180i	−0.876263	−	0.481832i	$-0.839971\pi$
$571$	−7.68287e25	+	1.33071e26i	−0.0208527	+	0.0361180i	0.855411	+	0.517950i	$0.173305\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−2.00999e27	−	3.48141e27i	−0.500000	−	0.866025i
$577$	3.33936e27	−	5.78394e27i	0.816404	−	1.41405i	−0.0919114	−	0.995767i	$-0.529298\pi$
$577$	3.33936e27	−	5.78394e27i	0.816404	−	1.41405i	0.908315	−	0.418286i	$-0.137369\pi$
$578$		0			0
$579$	3.91043e27	+	6.77306e27i	0.923504	+	1.59956i
$580$		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			$\pi$
$588$	−4.75537e27	+	1.33988e27i	−0.962523	+	0.271202i
$589$	1.01464e28			2.01911
$590$		0			0
$591$		0			0
$592$	3.96427e26	−	6.86632e26i	0.0749801	−	0.129869i
$593$		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
$597$	4.47275e27	+	7.74702e27i	0.777734	+	1.34707i
$598$		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
$601$	9.35173e27			1.52106			0.760531	−	0.649301i	$-0.224939\pi$
$601$	9.35173e27			1.52106			0.760531	+	0.649301i	$0.224939\pi$
$602$		0			0
$603$	−7.16831e27			−1.12783
$604$	6.40908e27	+	1.11009e28i	0.991806	+	1.71786i
$605$		0			0
$606$		0			0
$607$	3.50384e27	+	6.06883e27i	0.516009	+	0.893754i	0.999827	+	0.0185856i	$0.00591632\pi$
$607$	3.50384e27	+	6.06883e27i	0.516009	+	0.893754i	−0.483818	+	0.875169i	$0.660750\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.50699e27	−	7.80633e27i	0.601564	−	1.04194i	−0.391021	−	0.920382i	$-0.627878\pi$
$613$	4.50699e27	−	7.80633e27i	0.601564	−	1.04194i	0.992584	−	0.121557i	$-0.0387887\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	7.73768e27	−	1.34021e28i	0.936923	−	1.62280i	0.165756	−	0.986167i	$-0.446994\pi$
$619$	7.73768e27	−	1.34021e28i	0.936923	−	1.62280i	0.771168	−	0.636632i	$-0.219673\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	2.47825e27			0.276885
$625$	−4.54747e27	−	7.87646e27i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−7.22957e27	−	1.25220e28i	−0.757732	−	1.31243i
$629$		0			0
$630$		0			0
$631$	−1.59436e28			−1.59328			−0.796642	−	0.604452i	$-0.793392\pi$
$631$	−1.59436e28			−1.59328			−0.796642	+	0.604452i	$0.793392\pi$
$632$		0			0
$633$	−6.40982e27	+	1.11021e28i	−0.620595	+	1.07490i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	7.50452e26	−	2.95184e27i	0.0682227	−	0.268348i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−2.25078e28			−1.86304			−0.931522	−	0.363685i	$-0.881518\pi$
$643$	−2.25078e28			−1.86304			−0.931522	+	0.363685i	$0.881518\pi$
$644$		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
$651$	1.30579e28	+	5.31080e27i	0.955133	+	0.388463i
$652$	2.76526e28			1.99187
$653$		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
$657$	−1.66066e28			−1.10822
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−1.58023e28	+	2.73703e28i	−0.992440	+	1.71896i	−0.389934	+	0.920843i	$0.627502\pi$
$661$	−1.58023e28	+	2.73703e28i	−0.992440	+	1.71896i	−0.602506	+	0.798114i	$0.705831\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−1.21895e28	+	2.11129e28i	−0.678776	+	1.17567i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−3.75672e28			−1.97087			−0.985435	−	0.170055i	$-0.945606\pi$
$673$	−3.75672e28			−1.97087			−0.985435	+	0.170055i	$0.945606\pi$
$674$		0			0
$675$	−9.81765e27	+	1.70047e28i	−0.500000	+	0.866025i
$676$	9.20018e27	−	1.59352e28i	0.461667	−	0.799631i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	3.27136e28	−	2.54374e28i	1.57047	−	1.22117i
$680$		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	$0.666667\pi$
$684$	−2.19476e28	−	3.80143e28i	−0.979097	−	1.69585i
$685$		0			0
$686$		0			0
$687$	−2.07855e27			−0.0887552
$688$	−7.38836e26	−	1.27970e27i	−0.0310931	−	0.0538548i
$689$		0			0
$690$		0			0
$691$	−1.05088e28	−	1.82018e28i	−0.423421	−	0.733387i	0.572850	−	0.819660i	$-0.305838\pi$
$691$	−1.05088e28	−	1.82018e28i	−0.423421	−	0.733387i	−0.996272	+	0.0862731i	$0.972504\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.61662e28	+	1.06421e28i	0.926318	+	0.376743i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	4.32867e27	−	7.49748e27i	0.146826	−	0.254309i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.97659e28	+	3.42356e28i	0.615821	+	1.06663i	0.990240	+	0.139374i	$0.0445090\pi$
$709$	1.97659e28	+	3.42356e28i	0.615821	+	1.06663i	−0.374418	+	0.927260i	$0.622158\pi$
$710$		0			0
$711$	2.53190e28	−	4.38538e28i	0.766921	−	1.32835i
$712$		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
$721$	5.10400e28	+	2.07585e28i	1.34449	+	0.546819i
$722$		0			0
$723$	3.89355e28	+	6.74382e28i	0.997614	+	1.72792i
$724$	−2.07249e28	+	3.58966e28i	−0.523730	+	0.907126i
$725$		0			0
$726$		0			0
$727$	−5.68811e28			−1.37919			−0.689595	−	0.724196i	$-0.742211\pi$
$727$	−5.68811e28			−1.37919			−0.689595	+	0.724196i	$0.742211\pi$
$728$		0			0
$729$	4.23912e28			1.00000
$730$		0			0
$731$		0			0
$732$	4.29641e28	−	7.44160e28i	0.972736	−	1.68483i
$733$	3.81996e28	+	6.61636e28i	0.853137	+	1.47768i	0.878363	+	0.477995i	$0.158636\pi$
$733$	3.81996e28	+	6.61636e28i	0.853137	+	1.47768i	−0.0252259	+	0.999682i	$0.508031\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.03430e28	−	1.79145e28i	0.212912	−	0.368775i	−0.739713	−	0.672923i	$-0.765039\pi$
$739$	1.03430e28	−	1.79145e28i	0.212912	−	0.368775i	0.952625	+	0.304148i	$0.0983719\pi$
$740$		0			0
$741$	2.70606e28			0.542194
$742$		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
$751$	2.09672e28	+	3.63162e28i	0.367401	+	0.636357i	0.989158	−	0.146853i	$-0.0469144\pi$
$751$	2.09672e28	+	3.63162e28i	0.367401	+	0.636357i	−0.621758	+	0.783210i	$0.713581\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−8.34811e27	−	6.04102e28i	−0.136889	−	0.990586i
$757$	−1.12972e29			−1.82816			−0.914078	−	0.405540i	$-0.867084\pi$
$757$	−1.12972e29			−1.82816			−0.914078	+	0.405540i	$0.867084\pi$
$758$		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
$763$	−1.80260e28	−	1.30443e29i	−0.269559	−	1.95064i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−3.56929e28	−	6.18220e28i	−0.500000	−	0.866025i
$769$	−1.37032e29			−1.89478			−0.947388	−	0.320088i	$-0.896287\pi$
$769$	−1.37032e29			−1.89478			−0.947388	+	0.320088i	$0.896287\pi$
$770$		0			0
$771$		0			0
$772$	6.94403e28	+	1.20274e29i	0.923504	+	1.59956i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	−4.02987e28	−	6.97994e28i	−0.515554	−	0.892966i
$776$		0			0
$777$	9.49508e27	−	7.38318e27i	0.118383	−	0.0920520i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−8.44445e28	+	2.37932e28i	−0.962523	+	0.271202i
$785$		0			0
$786$		0			0
$787$	−2.99832e28	+	5.19324e28i	−0.328950	+	0.569758i	−0.982304	−	0.187294i	$-0.940028\pi$
$787$	−2.99832e28	+	5.19324e28i	−0.328950	+	0.569758i	0.653354	+	0.757053i	$0.273361\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.64866e28	+	4.58761e28i	0.269336	+	0.466503i
$794$		0			0
$795$		0			0
$796$	7.94258e28	+	1.37570e29i	0.777734	+	1.34707i
$797$		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
$804$	−1.27293e29			−1.12783
$805$		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
$811$	−2.12623e29			−1.72743			−0.863717	−	0.503977i	$-0.831870\pi$
$811$	−2.12623e29			−1.72743			−0.863717	+	0.503977i	$0.831870\pi$
$812$		0			0
$813$	2.11570e28			0.167706
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−8.06752e27	−	1.39734e28i	−0.0608863	−	0.105458i
$818$		0			0
$819$	3.48256e28	+	1.41639e28i	0.256483	+	0.104315i
$820$		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
$823$	1.09228e29	−	1.89189e29i	0.766189	−	1.32708i	−0.173426	−	0.984847i	$-0.555484\pi$
$823$	1.09228e29	−	1.89189e29i	0.766189	−	1.32708i	0.939616	−	0.342232i	$-0.111183\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	1.16819e29	−	2.02336e29i	0.762023	−	1.31986i	−0.179783	−	0.983706i	$-0.557539\pi$
$829$	1.16819e29	−	2.02336e29i	0.762023	−	1.31986i	0.941806	−	0.336157i	$-0.109127\pi$
$830$		0			0
$831$	−6.08111e28	−	1.05328e29i	−0.387233	−	0.670707i
$832$	4.40081e28			0.276885
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−8.70019e28	+	1.50692e29i	−0.515554	+	0.892966i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.76995e29			1.00000
$842$		0			0
$843$		0			0
$844$	−1.13824e29	+	1.97149e29i	−0.620595	+	1.07490i
$845$		0			0
$846$		0			0
$847$	1.76033e29	+	7.15945e28i	0.926318	+	0.376743i
$848$		0			0
$849$	9.67966e28	+	1.67657e29i	0.497488	+	0.861675i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	3.91602e29			1.92024			0.960118	−	0.279597i	$-0.0902007\pi$
$853$	3.91602e29			1.92024			0.960118	+	0.279597i	$0.0902007\pi$
$854$		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
$859$	−2.09983e29	−	3.63701e29i	−0.959959	−	1.66270i	−0.722589	−	0.691277i	$-0.757048\pi$
$859$	−2.09983e29	−	3.63701e29i	−0.959959	−	1.66270i	−0.237369	−	0.971420i	$-0.576285\pi$
$860$		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
$867$	2.39989e29			1.00000
$868$	2.31879e29	+	9.43077e28i	0.955133	+	0.388463i
$869$		0			0
$870$		0			0
$871$	3.92369e28	−	6.79602e28i	0.156139	−	0.270442i
$872$		0			0
$873$	2.55759e29	+	4.42987e29i	0.994690	+	1.72285i
$874$		0			0
$875$		0			0
$876$	−2.94895e29			−1.10822
$877$	−2.53965e29	−	4.39880e29i	−0.943579	−	1.63433i	−0.758572	−	0.651589i	$-0.774103\pi$
$877$	−2.53965e29	−	4.39880e29i	−0.943579	−	1.63433i	−0.185006	−	0.982737i	$-0.559231\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	4.73662e29			1.64385			0.821923	−	0.569598i	$-0.192901\pi$
$883$	4.73662e29			1.64385			0.821923	+	0.569598i	$0.192901\pi$
$884$		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
$889$	−1.73334e29	+	1.34781e29i	−0.562169	+	0.437131i
$890$		0			0
$891$		0			0
$892$	−2.16459e29	+	3.74917e29i	−0.678776	+	1.17567i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.74339e29	+	3.01964e29i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−3.06861e27	−	2.22057e28i	−0.00851263	−	0.0616008i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	2.30665e29	−	3.99524e29i	0.612222	−	1.06040i	−0.378643	−	0.925543i	$-0.623609\pi$
$907$	2.30665e29	−	3.99524e29i	0.612222	−	1.06040i	0.990865	−	0.134857i	$-0.0430576\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−3.89739e29	−	6.75048e29i	−0.979097	−	1.69585i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.69104e28			−0.0887552
$917$		0			0
$918$		0			0
$919$	−4.06521e29	−	7.04115e29i	−0.946080	−	1.63866i	−0.753575	−	0.657362i	$-0.771672\pi$
$919$	−4.06521e29	−	7.04115e29i	−0.946080	−	1.63866i	−0.192505	−	0.981296i	$-0.561661\pi$
$920$		0			0
$921$	−2.41338e29	+	4.18010e29i	−0.549579	+	0.951898i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−6.87691e28			−0.149960
$926$		0			0
$927$	−3.40068e29	+	5.89014e29i	−0.725718	+	1.25698i
$928$		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
$931$	−9.22069e29	+	2.59803e29i	−1.88481	+	0.531065i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	9.63063e29			1.84612			0.923058	−	0.384662i	$-0.125682\pi$
$937$	9.63063e29			1.84612			0.923058	+	0.384662i	$0.125682\pi$
$938$		0			0
$939$	−1.04585e30			−1.96252
$940$		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	$0.333333\pi$
$948$	4.49608e29	−	7.78744e29i	0.766921	−	1.32835i
$949$	9.08987e28	−	1.57441e29i	0.153425	−	0.265740i
$950$		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
$961$	−2.12230e28	−	3.67594e28i	−0.0315917	−	0.0547185i
$962$		0			0
$963$		0			0
$964$	6.91406e29	+	1.19755e30i	0.997614	+	1.72792i
$965$		0			0
$966$		0			0
$967$	−1.01403e30			−1.41836			−0.709178	−	0.705030i	$-0.750934\pi$
$967$	−1.01403e30			−1.41836			−0.709178	+	0.705030i	$0.750934\pi$
$968$		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	$0.333333\pi$
$972$	7.52771e29			1.00000
$973$	−1.16197e30	+	9.03524e29i	−1.52780	+	1.18799i
$974$		0			0
$975$	−1.07477e29	−	1.86156e29i	−0.138442	−	0.239789i
$976$	7.62945e29	−	1.32146e30i	0.972736	−	1.68483i
$977$		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
$981$	1.62545e30			1.96917
$982$		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
$988$	4.80534e29			0.542194
$989$		0			0
$990$		0			0
$991$	−6.26523e29	+	1.08517e30i	−0.685804	+	1.18785i	0.287379	+	0.957817i	$0.407216\pi$
$991$	−6.26523e29	+	1.08517e30i	−0.685804	+	1.18785i	−0.973183	+	0.230031i	$0.926117\pi$
$992$		0			0
$993$	1.58966e30			1.70534
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−4.52214e29	+	7.83258e29i	−0.466007	+	0.807148i	−0.999246	−	0.0388167i	$-0.987641\pi$
$997$	−4.52214e29	+	7.83258e29i	−0.466007	+	0.807148i	0.533239	+	0.845964i	$0.320975\pi$
$998$		0			0
$999$	7.42337e28	+	1.28576e29i	0.0749801	+	0.129869i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.21.h.a.11.1	yes	2
3.2	odd	2	CM	21.21.h.a.11.1	yes	2
7.2	even	3	inner	21.21.h.a.2.1	✓	2
21.2	odd	6	inner	21.21.h.a.2.1	✓	2

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

\(f(q)\)	\(=\)	\(q+(-29524.5 + 51137.9i) q^{3} +(-524288. + 908093. i) q^{4} +(-2.22994e8 + 1.73395e8i) q^{7} +(-1.74339e9 - 3.01964e9i) q^{9} +O(q^{10})\)	\(q+(-29524.5 + 51137.9i) q^{3} +(-524288. + 908093. i) q^{4} +(-2.22994e8 + 1.73395e8i) q^{7} +(-1.74339e9 - 3.01964e9i) q^{9} +(-3.09587e10 - 5.36220e10i) q^{12} +3.81709e10 q^{13} +(-5.49756e11 - 9.52205e11i) q^{16} +(-6.00291e12 - 1.03973e13i) q^{19} +(-2.28330e12 - 1.65229e13i) q^{21} +(-4.76837e13 + 8.25906e13i) q^{25} +2.05891e14 q^{27} +(-4.05462e13 - 2.93408e14i) q^{28} +(-4.22563e14 + 7.31900e14i) q^{31} +3.65616e15 q^{36} +(3.60548e14 + 6.24488e14i) q^{37} +(-1.12698e15 + 1.95198e15i) q^{39} +1.34394e15 q^{43} +6.49251e16 q^{48} +(1.96603e16 - 7.73323e16i) q^{49} +(-2.00126e16 + 3.46628e16i) q^{52} +7.08931e17 q^{57} +(6.93894e17 + 1.20186e18i) q^{61} +(9.12358e17 + 3.71066e17i) q^{63} +1.15292e18 q^{64} +(1.02793e18 - 1.78042e18i) q^{67} +(2.38136e18 - 4.12464e18i) q^{73} +(-2.81568e18 - 4.87689e18i) q^{75} +1.25890e19 q^{76} +(7.26142e18 + 1.25772e19i) q^{79} +(-6.07883e18 + 1.05288e19i) q^{81} +(1.62014e19 + 6.58929e18i) q^{84} +(-8.51188e18 + 6.61866e18i) q^{91} +(-2.49519e19 - 4.32180e19i) q^{93} -1.46702e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 59049 q^{3} - 1048576 q^{4} - 445987849 q^{7} - 3486784401 q^{9}+O(q^{10}) \) 2 * q - 59049 * q^3 - 1048576 * q^4 - 445987849 * q^7 - 3486784401 * q^9	\( 2 q - 59049 q^{3} - 1048576 q^{4} - 445987849 q^{7} - 3486784401 q^{9} - 61917364224 q^{12} + 76341844702 q^{13} - 1099511627776 q^{16} - 12005816399399 q^{19} - 4566600118926 q^{21} - 95367431640625 q^{25} + 411782264189298 q^{27} - 81092436557824 q^{28} - 845125195444727 q^{31} + 73\!\cdots\!52 q^{36}+ \cdots - 29\!\cdots\!48 q^{97}+O(q^{100}) \) 2 * q - 59049 * q^3 - 1048576 * q^4 - 445987849 * q^7 - 3486784401 * q^9 - 61917364224 * q^12 + 76341844702 * q^13 - 1099511627776 * q^16 - 12005816399399 * q^19 - 4566600118926 * q^21 - 95367431640625 * q^25 + 411782264189298 * q^27 - 81092436557824 * q^28 - 845125195444727 * q^31 + 7312316880125952 * q^36 + 721096304301649 * q^37 - 2253954793904199 * q^39 + 2687870111203246 * q^43 + 129850124217090048 * q^48 + 39320628860422799 * q^49 - 40025113075122176 * q^52 + 1417862905136223102 * q^57 + 1387788230779990126 * q^61 + 1824716645351204823 * q^63 + 2305843009213693952 * q^64 + 2055850947949627849 * q^67 + 4762721323358202049 * q^73 - 5631351470947265625 * q^75 + 25178021873632411648 * q^76 + 14522848453745208601 * q^79 - 12157665459056928801 * q^81 + 32402819372314263552 * q^84 - 17023767553668512999 * q^91 - 49903797665815684623 * q^93 - 293403496397740791548 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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