LMFDB - Embedded newform 21.36.g.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,36,Mod(5,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 36, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 36);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 36 $
Character orbit:	$[\chi]$	$=$	21.g (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$162.949774331$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.17
Dual form			21.36.g.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.93710e8 - 1.11839e8i) q^{3} +(-1.71799e10 + 2.97564e10i) q^{4} +(-2.23263e14 - 5.73561e14i) q^{7} +(2.50158e16 + 4.33286e16i) q^{9} +O(q^{10})$	\(q+(-1.93710e8 - 1.11839e8i) q^{3} +(-1.71799e10 + 2.97564e10i) q^{4} +(-2.23263e14 - 5.73561e14i) q^{7} +(2.50158e16 + 4.33286e16i) q^{9} +(6.65583e18 - 3.84275e18i) q^{12} -5.01917e19i q^{13} +(-5.90296e20 - 1.02242e21i) q^{16} +(-1.45153e22 + 8.38042e21i) q^{19} +(-2.08981e22 + 1.36074e23i) q^{21} +(1.45519e24 - 2.52047e24i) q^{25} -1.11909e25i q^{27} +(2.09027e25 + 3.21021e24i) q^{28} +(4.96202e25 + 2.86482e25i) q^{31} -1.71907e27 q^{36} +(7.17488e26 + 1.24273e27i) q^{37} +(-5.61337e27 + 9.72264e27i) q^{39} +6.94685e28 q^{43} +2.64072e29i q^{48} +(-2.79126e29 + 2.56110e29i) q^{49} +(1.49352e30 + 8.62287e29i) q^{52} +3.74902e30 q^{57} +(1.11310e31 - 6.42650e30i) q^{61} +(1.92665e31 - 2.40217e31i) q^{63} +4.05648e31 q^{64} +(5.51402e31 - 9.55057e31i) q^{67} +(4.16806e32 + 2.40643e32i) q^{73} +(-5.63771e32 + 3.25493e32i) q^{75} -5.75898e32i q^{76} +(6.79500e32 + 1.17693e33i) q^{79} +(-1.25158e33 + 2.16780e33i) q^{81} +(-3.69005e33 - 2.95959e33i) q^{84} +(-2.87880e34 + 1.12059e34i) q^{91} +(-6.40796e33 - 1.10989e34i) q^{93} -9.65334e34i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 387420489 q^{3} - 34359738368 q^{4} - 446525205377873 q^{7} + 50\!\cdots\!07 q^{9}+O(q^{10}) $ 2 * q - 387420489 * q^3 - 34359738368 * q^4 - 446525205377873 * q^7 + 50031545098999707 * q^9	$ 2 q - 387420489 q^{3} - 34359738368 q^{4} - 446525205377873 q^{7} + 50\!\cdots\!07 q^{9}+ \cdots - 12\!\cdots\!75 q^{93}+O(q^{100}) $ 2 * q - 387420489 * q^3 - 34359738368 * q^4 - 446525205377873 * q^7 + 50031545098999707 * q^9 + 13311666640442621952 * q^12 - 1180591620717411303424 * q^16 - 29030614593056510079771 * q^19 - 41796137645342869154526 * q^21 + 2910383045673370361328125 * q^25 + 41805487535667052978110464 * q^28 + 99240318692296271463039225 * q^31 - 3438141599496845182057316352 * q^36 + 1434976029760582590688944323 * q^37 - 11226741639714620351964140643 * q^39 + 138937021705263726870548760050 * q^43 - 558252625493577900644717247757 * q^49 + 2987048308632889626118374555648 * q^52 + 7498036601074992759825539885346 * q^57 + 22262061668307550275762391952172 * q^61 + 38533026035573202444818730766425 * q^63 + 81129638414606681695789005144064 * q^64 + 110280454033501822232789985372223 * q^67 + 833612588618178474625817730556989 * q^73 - 1127542022732086479663848876953125 * q^75 + 1359000989429499836036390683764907 * q^79 - 2503155504993241601315571986085849 * q^81 - 7380099034832319024082245767921664 * q^84 - 57576010078993069344020247768783501 * q^91 - 12815910932095087341029133991893675 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	−1.93710e8	−	1.11839e8i	−0.866025	−	0.500000i
$3$	−1.93710e8	−	1.11839e8i	−0.866025	−	0.500000i
$4$	−1.71799e10	+	2.97564e10i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	−2.23263e14	−	5.73561e14i	−0.362744	−	0.931889i
$7$	−2.23263e14	−	5.73561e14i	−0.362744	−	0.931889i
$8$		0			0
$9$	2.50158e16	+	4.33286e16i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	6.65583e18	−	3.84275e18i	0.866025	−	0.500000i
$13$		−	5.01917e19i		−	1.60925i	−0.593784	−	0.804624i	$-0.702367\pi$
$13$		−	5.01917e19i		−	1.60925i	0.593784	−	0.804624i	$-0.297633\pi$
$14$		0			0
$15$		0			0
$16$	−5.90296e20	−	1.02242e21i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	−1.45153e22	+	8.38042e21i	−0.607629	+	0.350815i	−0.772037	−	0.635578i	$-0.780762\pi$
$19$	−1.45153e22	+	8.38042e21i	−0.607629	+	0.350815i	0.164408	+	0.986392i	$0.447429\pi$
$20$		0			0
$21$	−2.08981e22	+	1.36074e23i	−0.151799	+	0.988411i
$22$		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
$25$	1.45519e24	−	2.52047e24i	0.500000	−	0.866025i
$26$		0			0
$27$		−	1.11909e25i		−	1.00000i
$28$	2.09027e25	+	3.21021e24i	0.988411	+	0.151799i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	4.96202e25	+	2.86482e25i	0.395211	+	0.228175i	0.684415	−	0.729092i	$-0.260058\pi$
$31$	4.96202e25	+	2.86482e25i	0.395211	+	0.228175i	−0.289205	+	0.957267i	$0.593391\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.71907e27			−1.00000
$37$	7.17488e26	+	1.24273e27i	0.258395	+	0.447554i	0.965812	−	0.259243i	$-0.0834730\pi$
$37$	7.17488e26	+	1.24273e27i	0.258395	+	0.447554i	−0.707417	+	0.706796i	$0.750140\pi$
$38$		0			0
$39$	−5.61337e27	+	9.72264e27i	−0.804624	+	1.39365i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.94685e28			1.80339			0.901696	−	0.432371i	$-0.142323\pi$
$43$	6.94685e28			1.80339			0.901696	+	0.432371i	$0.142323\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$			2.64072e29i			1.00000i
$49$	−2.79126e29	+	2.56110e29i	−0.736834	+	0.676074i
$50$		0			0
$51$		0			0
$52$	1.49352e30	+	8.62287e29i	1.39365	+	0.804624i
$53$		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
$57$	3.74902e30			0.701629
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	1.11310e31	−	6.42650e30i	0.635716	−	0.367031i	−0.147246	−	0.989100i	$-0.547041\pi$
$61$	1.11310e31	−	6.42650e30i	0.635716	−	0.367031i	0.782962	+	0.622069i	$0.213708\pi$
$62$		0			0
$63$	1.92665e31	−	2.40217e31i	0.625667	−	0.780090i
$64$	4.05648e31			1.00000
$65$		0			0
$66$		0			0
$67$	5.51402e31	−	9.55057e31i	0.609761	−	1.05614i	−0.381519	−	0.924361i	$-0.624599\pi$
$67$	5.51402e31	−	9.55057e31i	0.609761	−	1.05614i	0.991280	−	0.131776i	$-0.0420678\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	4.16806e32	+	2.40643e32i	1.02751	+	0.593231i	0.916269	−	0.400563i	$-0.131186\pi$
$73$	4.16806e32	+	2.40643e32i	1.02751	+	0.593231i	0.111237	+	0.993794i	$0.464519\pi$
$74$		0			0
$75$	−5.63771e32	+	3.25493e32i	−0.866025	+	0.500000i
$76$		−	5.75898e32i		−	0.701629i
$77$		0			0
$78$		0			0
$79$	6.79500e32	+	1.17693e33i	0.420451	+	0.728243i	0.995984	−	0.0895361i	$-0.0285384\pi$
$79$	6.79500e32	+	1.17693e33i	0.420451	+	0.728243i	−0.575532	+	0.817779i	$0.695205\pi$
$80$		0			0
$81$	−1.25158e33	+	2.16780e33i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−3.69005e33	−	2.95959e33i	−0.780090	−	0.625667i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−2.87880e34	+	1.12059e34i	−1.49964	+	0.583745i
$92$		0			0
$93$	−6.40796e33	−	1.10989e34i	−0.228175	−	0.395211i
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	9.65334e34i		−	1.64502i	−0.568748	−	0.822512i	$-0.692572\pi$
$97$		−	9.65334e34i		−	1.64502i	0.568748	−	0.822512i	$-0.307428\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e34	+	8.66025e34i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	2.82005e35	−	1.62816e35i	1.68115	−	0.970611i	0.720248	−	0.693717i	$-0.244028\pi$
$103$	2.82005e35	−	1.62816e35i	1.68115	−	0.970611i	0.960900	−	0.276895i	$-0.0893053\pi$
$104$		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	$0.333333\pi$
$108$	3.33002e35	+	1.92259e35i	0.866025	+	0.500000i
$109$	−4.30604e35	+	7.45828e35i	−0.953047	+	1.65073i	−0.214272	+	0.976774i	$0.568738\pi$
$109$	−4.30604e35	+	7.45828e35i	−0.953047	+	1.65073i	−0.738775	+	0.673952i	$0.764595\pi$
$110$		0			0
$111$		−	3.20972e35i		−	0.516790i
$112$	−4.54631e35	+	5.66839e35i	−0.625667	+	0.780090i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.17473e36	−	1.25558e36i	1.39365	−	0.804624i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.40512e36	−	2.43374e36i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	−1.70494e36	+	9.84345e35i	−0.395211	+	0.228175i
$125$		0			0
$126$		0			0
$127$	−6.75157e36			−1.03001			−0.515003	−	0.857188i	$-0.672209\pi$
$127$	−6.75157e36			−1.03001			−0.515003	+	0.857188i	$0.672209\pi$
$128$		0			0
$129$	−1.34568e37	−	7.76927e36i	−1.56178	−	0.901696i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	8.04741e36	+	6.45438e36i	0.547334	+	0.438986i
$134$		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
$139$			2.74701e37i			0.863179i	0.902070	+	0.431589i	$0.142047\pi$
$139$			2.74701e37i			0.863179i	−0.902070	+	0.431589i	$0.857953\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	2.95334e37	−	5.11534e37i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	8.27126e37	−	1.83939e37i	0.976154	−	0.217081i
$148$	−4.93054e37			−0.516790
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−7.21409e37	+	1.24952e38i	−0.532212	+	0.921818i	0.467081	+	0.884215i	$0.345306\pi$
$151$	−7.21409e37	+	1.24952e38i	−0.532212	+	0.921818i	−0.999293	+	0.0376036i	$0.988028\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.92874e38	−	3.34067e38i	−0.804624	−	1.39365i
$157$	−1.48964e38	−	8.60045e37i	−0.555697	−	0.320832i	0.195720	−	0.980660i	$-0.437296\pi$
$157$	−1.48964e38	−	8.60045e37i	−0.555697	−	0.320832i	−0.751416	+	0.659828i	$0.770629\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−5.10325e38	−	8.83909e38i	−0.987561	−	1.71051i	−0.629950	−	0.776636i	$-0.716925\pi$
$163$	−5.10325e38	−	8.83909e38i	−0.987561	−	1.71051i	−0.357612	−	0.933870i	$-0.616409\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−1.54642e39			−1.58968
$170$		0			0
$171$	−7.26223e38	−	4.19285e38i	−0.607629	−	0.350815i
$172$	−1.19346e39	+	2.06713e39i	−0.901696	+	1.56178i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−1.77053e39	−	2.71916e38i	−0.988411	−	0.151799i
$176$		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
$181$		−	6.31590e39i		−	1.95461i	−0.211839	−	0.977304i	$-0.567945\pi$
$181$		−	6.31590e39i		−	1.95461i	0.211839	−	0.977304i	$-0.432055\pi$
$182$		0			0
$183$	−2.87493e39			−0.734062
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−6.41868e39	+	2.49851e39i	−0.931889	+	0.362744i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−7.85782e39	−	4.53672e39i	−0.866025	−	0.500000i
$193$	9.89075e39	−	1.71313e40i	0.995352	−	1.72400i	0.414270	−	0.910154i	$-0.364037\pi$
$193$	9.89075e39	−	1.71313e40i	0.995352	−	1.72400i	0.581081	−	0.813845i	$-0.302630\pi$
$194$		0			0
$195$		0			0
$196$	−2.82555e39	−	1.27057e40i	−0.217081	−	0.976154i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−2.70403e40	−	1.56117e40i	−1.59251	−	0.919436i	−0.992875	−	0.119161i	$-0.961979\pi$
$199$	−2.70403e40	−	1.56117e40i	−1.59251	−	0.919436i	−0.599634	−	0.800274i	$-0.704687\pi$
$200$		0			0
$201$	−2.13625e40	+	1.23336e40i	−1.05614	+	0.609761i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−5.13171e40	+	2.96279e40i	−1.39365	+	0.804624i
$209$		0			0
$210$		0			0
$211$	−9.46158e40			−1.99996			−0.999979	−	0.00646922i	$-0.997941\pi$
$211$	−9.46158e40			−1.99996			−0.999979	+	0.00646922i	$0.997941\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5.35318e39	−	3.48563e40i	0.0692734	−	0.451061i
$218$		0			0
$219$	−5.38264e40	−	9.32301e40i	−0.593231	−	1.02751i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	3.16905e40i		−	0.254445i	−0.991874	−	0.127223i	$-0.959394\pi$
$223$		−	3.16905e40i		−	0.254445i	0.991874	−	0.127223i	$-0.0406062\pi$
$224$		0			0
$225$	1.45611e41			1.00000
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$	−6.44076e40	+	1.11557e41i	−0.350815	+	0.607629i
$229$	1.06366e41	−	6.14105e40i	0.536639	−	0.309829i	−0.207077	−	0.978325i	$-0.566395\pi$
$229$	1.06366e41	−	6.14105e40i	0.536639	−	0.309829i	0.743716	+	0.668496i	$0.233062\pi$
$230$		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
$237$		−	3.03978e41i		−	0.840903i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	6.60638e41	+	3.81419e41i	1.36353	+	0.787235i	0.990092	−	0.140419i	$-0.0448451\pi$
$241$	6.60638e41	+	3.81419e41i	1.36353	+	0.787235i	0.373439	+	0.927655i	$0.378178\pi$
$242$		0			0
$243$	4.84887e41	−	2.79950e41i	0.866025	−	0.500000i
$244$			4.41626e41i			0.734062i
$245$		0			0
$246$		0			0
$247$	4.20627e41	+	7.28548e41i	0.564548	+	0.977825i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	3.83804e41	+	9.85992e41i	0.362744	+	0.931889i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−6.96898e41	+	1.20706e42i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	5.52591e41	−	6.88977e41i	0.323339	−	0.403143i
$260$		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
$268$	1.89460e42	+	3.28155e42i	0.609761	+	1.05614i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	3.35986e42	−	1.93982e42i	0.889936	−	0.513805i	0.0160146	−	0.999872i	$-0.494902\pi$
$271$	3.35986e42	−	1.93982e42i	0.889936	−	0.513805i	0.873922	+	0.486067i	$0.161569\pi$
$272$		0			0
$273$	6.82979e42	+	1.04891e42i	1.59060	+	0.244282i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.90690e42	−	8.49901e42i	0.885955	−	1.53452i	0.0413399	−	0.999145i	$-0.486837\pi$
$277$	4.90690e42	−	8.49901e42i	0.885955	−	1.53452i	0.844615	−	0.535374i	$-0.179829\pi$
$278$		0			0
$279$			2.86663e42i			0.456350i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	6.91761e42	+	3.99389e42i	0.858408	+	0.495602i	0.863479	−	0.504385i	$-0.168281\pi$
$283$	6.91761e42	+	3.99389e42i	0.858408	+	0.495602i	−0.00507084	+	0.999987i	$0.501614\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	5.81677e42	+	1.00749e43i	0.500000	+	0.866025i
$290$		0			0
$291$	−1.07962e43	+	1.86995e43i	−0.822512	+	1.42463i
$292$	−1.43214e43	+	8.26844e42i	−1.02751	+	0.593231i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	2.23677e43i		−	1.00000i
$301$	−1.55097e43	−	3.98444e43i	−0.654170	−	1.68056i
$302$		0			0
$303$		0			0
$304$	1.71367e43	+	9.89385e42i	0.607629	+	0.350815i
$305$		0			0
$306$		0			0
$307$		−	6.61271e43i		−	1.97450i	−0.159163	−	0.987252i	$-0.550879\pi$
$307$		−	6.61271e43i		−	1.97450i	0.159163	−	0.987252i	$-0.449121\pi$
$308$		0			0
$309$	−7.28364e43			−1.94122
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−7.34281e43	+	4.23937e43i	−1.56256	+	0.902143i	−0.565561	+	0.824707i	$0.691340\pi$
$313$	−7.34281e43	+	4.23937e43i	−1.56256	+	0.902143i	−0.996997	+	0.0774365i	$0.975326\pi$
$314$		0			0
$315$		0			0
$316$	−4.66949e43			−0.840903
$317$		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
$324$	−4.30039e43	−	7.44849e43i	−0.500000	−	0.866025i
$325$	−1.26506e44	−	7.30385e43i	−1.39365	−	0.804624i
$326$		0			0
$327$	1.66825e44	−	9.63164e43i	1.65073	−	0.953047i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	1.02562e44	+	1.77642e44i	0.820342	+	1.42087i	0.905428	+	0.424500i	$0.139550\pi$
$331$	1.02562e44	+	1.77642e44i	0.820342	+	1.42087i	−0.0850864	+	0.996374i	$0.527117\pi$
$332$		0			0
$333$	−3.58970e43	+	6.21755e43i	−0.258395	+	0.447554i
$334$		0			0
$335$		0			0
$336$	1.51461e44	−	5.89573e43i	0.931889	−	0.362744i
$337$	−3.41603e44			−1.99526			−0.997628	−	0.0688426i	$-0.978069\pi$
$337$	−3.41603e44			−1.99526			−0.997628	+	0.0688426i	$0.978069\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	2.09213e44	+	1.02916e44i	0.897308	+	0.441405i
$344$		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
$349$		−	4.04281e44i		−	1.28008i	−0.768340	−	0.640042i	$-0.778917\pi$
$349$		−	4.04281e44i		−	1.28008i	0.768340	−	0.640042i	$-0.221083\pi$
$350$		0			0
$351$	−5.61691e44			−1.60925
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−1.44866e44	+	2.50916e44i	−0.253858	+	0.439695i
$362$		0			0
$363$			6.28588e44i			1.00000i
$364$	1.61126e44	−	1.04914e45i	0.244282	−	1.59060i
$365$		0			0
$366$		0			0
$367$	1.13920e45	+	6.57718e44i	1.49604	+	0.863741i	0.999990	−	0.00455084i	$-0.00144858\pi$
$367$	1.13920e45	+	6.57718e44i	1.49604	+	0.863741i	0.496054	+	0.868292i	$0.334782\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	4.40351e44			0.456350
$373$	5.30626e44	+	9.19072e44i	0.524668	+	0.908751i	0.999587	+	0.0287219i	$0.00914371\pi$
$373$	5.30626e44	+	9.19072e44i	0.524668	+	0.908751i	−0.474920	+	0.880029i	$0.657523\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	2.33656e44			0.174739			0.0873697	−	0.996176i	$-0.472154\pi$
$379$	2.33656e44			0.174739			0.0873697	+	0.996176i	$0.472154\pi$
$380$		0			0
$381$	1.30785e45	+	7.55087e44i	0.892012	+	0.515003i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	1.73781e45	+	3.00997e45i	0.901696	+	1.56178i
$388$	2.87249e45	+	1.65843e45i	1.42463	+	0.822512i
$389$		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
$397$	5.14253e45	−	2.96904e45i	1.70743	−	0.985786i	0.769713	−	0.638390i	$-0.220399\pi$
$397$	5.14253e45	−	2.96904e45i	1.70743	−	0.985786i	0.937718	−	0.347397i	$-0.112934\pi$
$398$		0			0
$399$	−8.37016e44	−	2.15029e45i	−0.254512	−	0.653840i
$400$	−3.43597e45			−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
$403$	1.43790e45	−	2.49052e45i	0.367190	−	0.635992i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−8.61753e45	−	4.97534e45i	−1.69912	−	0.980990i	−0.946592	−	0.322435i	$-0.895499\pi$
$409$	−8.61753e45	−	4.97534e45i	−1.69912	−	0.980990i	−0.752532	−	0.658555i	$-0.771168\pi$
$410$		0			0
$411$		0			0
$412$			1.11886e46i			1.94122i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	3.07222e45	−	5.32123e45i	0.431589	−	0.747535i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−1.52902e46			−1.81751			−0.908756	−	0.417329i	$-0.862966\pi$
$421$	−1.52902e46			−1.81751			−0.908756	+	0.417329i	$0.862966\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.17114e45	−	4.94953e45i	−0.572634	−	0.459279i
$428$		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	$0.666667\pi$
$432$	−1.14418e46	+	6.60595e45i	−0.866025	+	0.500000i
$433$			2.52152e46i			1.83284i	0.400218	+	0.916420i	$0.368935\pi$
$433$			2.52152e46i			1.83284i	−0.400218	+	0.916420i	$0.631065\pi$
$434$		0			0
$435$		0			0
$436$	−1.47954e46	−	2.56265e46i	−0.953047	−	1.65073i
$437$		0			0
$438$		0			0
$439$	2.69049e46	−	1.55336e46i	1.53710	−	0.887447i	0.538097	−	0.842883i	$-0.319144\pi$
$439$	2.69049e46	−	1.55336e46i	1.53710	−	0.887447i	0.999007	−	0.0445636i	$-0.0141897\pi$
$440$		0			0
$441$	−1.80794e46	−	5.68737e45i	−0.953914	−	0.300079i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	9.55096e45	+	5.51425e45i	0.447554	+	0.258395i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−9.05661e45	−	2.32664e46i	−0.362744	−	0.931889i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.79488e46	−	1.61363e46i	0.921818	−	0.532212i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−2.83954e46	−	4.91823e46i	−0.802998	−	1.39083i	−0.917635	−	0.397425i	$-0.869904\pi$
$457$	−2.83954e46	−	4.91823e46i	−0.802998	−	1.39083i	0.114637	−	0.993407i	$-0.463430\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	−6.87939e46			−1.54840			−0.774199	−	0.632942i	$-0.781847\pi$
$463$	−6.87939e46			−1.54840			−0.774199	+	0.632942i	$0.781847\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$			8.62831e46i			1.60925i
$469$	−6.70891e46	−	1.03035e46i	−1.20539	−	0.185122i
$470$		0			0
$471$	1.92373e46	+	3.33199e46i	0.320832	+	0.555697i
$472$		0			0
$473$		0			0
$474$		0			0
$475$			4.87804e46i			0.701629i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	6.23745e46	−	3.60119e46i	0.720225	−	0.415822i
$482$		0			0
$483$		0			0
$484$	9.65592e46			1.00000
$485$		0			0
$486$		0			0
$487$	8.52904e46	−	1.47727e47i	0.792763	−	1.37311i	−0.131487	−	0.991318i	$-0.541975\pi$
$487$	8.52904e46	−	1.47727e47i	0.792763	−	1.37311i	0.924250	−	0.381787i	$-0.124691\pi$
$488$		0			0
$489$			2.28296e47i			1.97512i
$490$		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
$496$		−	6.76437e46i		−	0.456350i
$497$		0			0
$498$		0			0
$499$	1.61630e47	+	2.79951e47i	0.981209	+	1.69950i	0.657701	+	0.753279i	$0.271529\pi$
$499$	1.61630e47	+	2.79951e47i	0.981209	+	1.69950i	0.323508	+	0.946225i	$0.395138\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	2.99557e47	+	1.72949e47i	1.37670	+	0.794840i
$508$	1.15991e47	−	2.00903e47i	0.515003	−	0.892012i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	4.49664e46	−	2.92791e47i	0.180103	−	1.17271i
$512$		0			0
$513$	9.37846e46	+	1.62440e47i	0.350815	+	0.607629i
$514$		0			0
$515$		0			0
$516$	4.62371e47	−	2.66950e47i	1.56178	−	0.901696i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−3.94587e47	+	2.27815e47i	−1.05284	+	0.607858i	−0.923443	−	0.383735i	$-0.874638\pi$
$523$	−3.94587e47	+	2.27815e47i	−1.05284	+	0.607858i	−0.129398	+	0.991593i	$0.541304\pi$
$524$		0			0
$525$	3.12559e47	+	2.50687e47i	0.780090	+	0.625667i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.28794e47	+	3.96282e47i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−3.30313e47	+	1.28576e47i	−0.653840	+	0.254512i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−6.77562e47	−	1.17357e48i	−0.999993	−	1.73204i	−0.503335	−	0.864092i	$-0.667894\pi$
$541$	−6.77562e47	−	1.17357e48i	−0.999993	−	1.73204i	−0.496658	−	0.867946i	$-0.665440\pi$
$542$		0			0
$543$	−7.06361e47	+	1.22345e48i	−0.977304	+	1.69274i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.60232e48			−1.94972			−0.974860	−	0.222818i	$-0.928474\pi$
$547$	−1.60232e48			−1.94972			−0.974860	+	0.222818i	$0.928474\pi$
$548$		0			0
$549$	5.56903e47	+	3.21528e47i	0.635716	+	0.367031i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.23334e47	−	6.52499e47i	0.526125	−	0.655980i
$554$		0			0
$555$		0			0
$556$	−8.17411e47	−	4.71932e47i	−0.747535	−	0.431589i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	3.48674e48i		−	2.90211i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	1.52279e48	+	2.33869e47i	0.988411	+	0.151799i
$568$		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
$571$	7.25052e47	−	1.25583e48i	0.416138	−	0.720772i	−0.579409	−	0.815037i	$-0.696717\pi$
$571$	7.25052e47	−	1.25583e48i	0.416138	−	0.720772i	0.995547	+	0.0942646i	$0.0300500\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	1.01476e48	+	1.75762e48i	0.500000	+	0.866025i
$577$	−3.47022e48	−	2.00353e48i	−1.65875	−	0.957678i	−0.973294	−	0.229561i	$-0.926271\pi$
$577$	−3.47022e48	−	2.00353e48i	−1.65875	−	0.957678i	−0.685453	−	0.728117i	$-0.740396\pi$
$578$		0			0
$579$	−3.83188e48	+	2.21234e48i	−1.72400	+	0.995352i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	−8.73654e47	+	2.77723e48i	−0.300079	+	0.953914i
$589$	−9.60336e47			−0.320188
$590$		0			0
$591$		0			0
$592$	8.47060e47	−	1.46715e48i	0.258395	−	0.447554i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	3.49198e48	+	6.04829e48i	0.919436	+	1.59251i
$598$		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
$601$		−	8.06538e48i		−	1.88939i	−0.327949	−	0.944695i	$-0.606358\pi$
$601$		−	8.06538e48i		−	1.88939i	0.327949	−	0.944695i	$-0.393642\pi$
$602$		0			0
$603$	5.51750e48			1.21952
$604$	−2.47874e48	−	4.29331e48i	−0.532212	−	0.921818i
$605$		0			0
$606$		0			0
$607$	−4.64423e48	+	2.68135e48i	−0.914349	+	0.527899i	−0.881828	−	0.471572i	$-0.843687\pi$
$607$	−4.64423e48	+	2.68135e48i	−0.914349	+	0.527899i	−0.0325209	+	0.999471i	$0.510354\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−4.46823e48	+	7.73920e48i	−0.740588	+	1.28274i	0.211639	+	0.977348i	$0.432120\pi$
$613$	−4.46823e48	+	7.73920e48i	−0.740588	+	1.28274i	−0.952228	+	0.305389i	$0.901214\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	1.98064e48	+	1.14352e48i	0.276833	+	0.159830i	0.631989	−	0.774977i	$-0.282239\pi$
$619$	1.98064e48	+	1.14352e48i	0.276833	+	0.159830i	−0.355156	+	0.934807i	$0.615572\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	1.32542e49			1.60925
$625$	−4.23516e48	−	7.33552e48i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	5.11837e48	−	2.95509e48i	0.555697	−	0.320832i
$629$		0			0
$630$		0			0
$631$	6.87524e48			0.686711			0.343356	−	0.939205i	$-0.388436\pi$
$631$	6.87524e48			0.686711			0.343356	+	0.939205i	$0.388436\pi$
$632$		0			0
$633$	1.83281e49	+	1.05817e49i	1.73201	+	0.999979i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.28546e49	+	1.40098e49i	1.08797	+	1.18575i
$638$		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
$643$			3.72018e48i			0.267222i	0.991034	+	0.133611i	$0.0426573\pi$
$643$			3.72018e48i			0.267222i	−0.991034	+	0.133611i	$0.957343\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−4.93524e48	+	6.15333e48i	−0.285523	+	0.355994i
$652$	3.50693e49			1.97512
$653$		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
$657$			2.40795e49i			1.18646i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−3.81892e49	−	2.20485e49i	−1.69206	−	0.976909i	−0.952856	−	0.303422i	$-0.901871\pi$
$661$	−3.81892e49	−	2.20485e49i	−1.69206	−	0.976909i	−0.739199	−	0.673487i	$-0.764796\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−3.54422e48	+	6.13877e48i	−0.127223	+	0.220356i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−2.89057e49			−0.934803			−0.467402	−	0.884045i	$-0.654810\pi$
$673$	−2.89057e49			−0.934803			−0.467402	+	0.884045i	$0.654810\pi$
$674$		0			0
$675$	−2.82063e49	−	1.62849e49i	−0.866025	−	0.500000i
$676$	2.65673e49	−	4.60159e49i	0.794840	−	1.37670i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	−5.53678e49	+	2.15523e49i	−1.53298	+	0.596723i
$680$		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	$0.666667\pi$
$684$	2.49528e49	−	1.44065e49i	0.607629	−	0.350815i
$685$		0			0
$686$		0			0
$687$	−2.74723e49			−0.619657
$688$	−4.10070e49	−	7.10262e49i	−0.901696	−	1.56178i
$689$		0			0
$690$		0			0
$691$	−3.41134e49	+	1.96954e49i	−0.695119	+	0.401327i	−0.805527	−	0.592559i	$-0.798118\pi$
$691$	−3.41134e49	+	1.96954e49i	−0.695119	+	0.401327i	0.110408	+	0.993886i	$0.464784\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	3.85087e49	−	4.80132e49i	0.625667	−	0.780090i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−2.08291e49	−	1.20257e49i	−0.314017	−	0.181298i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.53470e49	+	9.58638e49i	0.719093	+	1.24551i	0.961360	+	0.275296i	$0.0887758\pi$
$709$	5.53470e49	+	9.58638e49i	0.719093	+	1.24551i	−0.242267	+	0.970210i	$0.577891\pi$
$710$		0			0
$711$	−3.39965e49	+	5.88836e49i	−0.420451	+	0.728243i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	−1.56346e50	−	1.25397e50i	−1.51433	−	1.21456i
$722$		0			0
$723$	−8.53149e49	−	1.47770e50i	−0.787235	−	1.36353i
$724$	1.87938e50	+	1.08506e50i	1.69274	+	0.977304i
$725$		0			0
$726$		0			0
$727$			1.40999e50i			1.18131i	0.806925	+	0.590654i	$0.201130\pi$
$727$			1.40999e50i			1.18131i	−0.806925	+	0.590654i	$0.798870\pi$
$728$		0			0
$729$	−1.25237e50			−1.00000
$730$		0			0
$731$		0			0
$732$	4.93909e49	−	8.55475e49i	0.367031	−	0.635716i
$733$	4.15740e49	−	2.40027e49i	0.301649	−	0.174157i	−0.341534	−	0.939869i	$-0.610947\pi$
$733$	4.15740e49	−	2.40027e49i	0.301649	−	0.174157i	0.643183	+	0.765712i	$0.277613\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	2.73849e48	−	4.74320e48i	0.0172279	−	0.0298397i	−0.857283	−	0.514846i	$-0.827849\pi$
$739$	2.73849e48	−	4.74320e48i	0.0172279	−	0.0298397i	0.874511	+	0.485006i	$0.161183\pi$
$740$		0			0
$741$		−	1.88170e50i		−	1.12910i
$742$		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
$751$	2.03324e50	+	3.52167e50i	0.964916	+	1.67128i	0.709841	+	0.704362i	$0.248767\pi$
$751$	2.03324e50	+	3.52167e50i	0.964916	+	1.67128i	0.255075	+	0.966921i	$0.417900\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	3.59253e49	−	2.33921e50i	0.151799	−	0.988411i
$757$	−4.83681e50			−1.99701			−0.998507	−	0.0546252i	$-0.982604\pi$
$757$	−4.83681e50			−1.99701			−0.998507	+	0.0546252i	$0.982604\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	5.23916e50	+	8.04623e49i	1.88401	+	0.289343i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	2.69993e50	−	1.55880e50i	0.866025	−	0.500000i
$769$			6.23168e50i			1.95386i	0.213551	+	0.976932i	$0.431497\pi$
$769$			6.23168e50i			1.95386i	−0.213551	+	0.976932i	$0.568503\pi$
$770$		0			0
$771$		0			0
$772$	3.39844e50	+	5.88627e50i	0.995352	+	1.72400i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	1.44414e50	−	8.33773e49i	0.395211	−	0.228175i
$776$		0			0
$777$	−1.84097e50	+	7.16610e49i	−0.481591	+	0.187463i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	4.26619e50	+	1.34205e50i	0.953914	+	0.300079i
$785$		0			0
$786$		0			0
$787$	6.87461e50	+	3.96906e50i	1.43777	+	0.830099i	0.997695	−	0.0678610i	$-0.0216174\pi$
$787$	6.87461e50	+	3.96906e50i	1.43777	+	0.830099i	0.440078	+	0.897960i	$0.354951\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.22557e50	−	5.58685e50i	−0.590644	−	1.02303i
$794$		0			0
$795$		0			0
$796$	9.29096e50	−	5.36414e50i	1.59251	−	0.919436i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	8.47560e50i		−	1.21952i
$805$		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
$811$			2.24050e50i			0.277000i	0.990362	+	0.138500i	$0.0442282\pi$
$811$			2.24050e50i			0.277000i	−0.990362	+	0.138500i	$0.955772\pi$
$812$		0			0
$813$	−8.67786e50			−1.02761
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−1.00836e51	+	5.82175e50i	−1.09579	+	0.632656i
$818$		0			0
$819$	−1.20569e51	−	9.67019e50i	−1.25536	−	1.00685i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−9.48097e50	+	1.64215e51i	−0.906474	+	1.57006i	−0.0875485	+	0.996160i	$0.527903\pi$
$823$	−9.48097e50	+	1.64215e51i	−0.906474	+	1.57006i	−0.818926	+	0.573899i	$0.805430\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−8.89670e50	−	5.13652e50i	−0.749074	−	0.432478i	0.0762850	−	0.997086i	$-0.475694\pi$
$829$	−8.89670e50	−	5.13652e50i	−0.749074	−	0.432478i	−0.825359	+	0.564608i	$0.809027\pi$
$830$		0			0
$831$	−1.90104e51	+	1.09756e51i	−1.53452	+	0.885955i
$832$		−	2.03602e51i		−	1.60925i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	3.20600e50	−	5.55295e50i	0.228175	−	0.395211i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.52732e51			1.00000
$842$		0			0
$843$		0			0
$844$	1.62549e51	−	2.81543e51i	0.999979	−	1.73201i
$845$		0			0
$846$		0			0
$847$	−1.08219e51	+	1.34929e51i	−0.625667	+	0.780090i
$848$		0			0
$849$	−8.93342e50	−	1.54731e51i	−0.495602	−	0.858408i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			3.44265e51i			1.75908i	0.475827	+	0.879539i	$0.342149\pi$
$853$			3.44265e51i			1.75908i	−0.475827	+	0.879539i	$0.657851\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	−3.36870e51	+	1.94492e51i	−1.52259	+	0.879065i	−0.522942	+	0.852368i	$0.675166\pi$
$859$	−3.36870e51	+	1.94492e51i	−1.52259	+	0.879065i	−0.999644	+	0.0266968i	$0.991501\pi$
$860$		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
$867$		−	2.60216e51i		−	1.00000i
$868$	9.45231e50	+	7.58118e50i	0.355994	+	0.285523i
$869$		0			0
$870$		0			0
$871$	−4.79359e51	−	2.76758e51i	−1.69959	−	0.981257i
$872$		0			0
$873$	4.18266e51	−	2.41486e51i	1.42463	−	0.822512i
$874$		0			0
$875$		0			0
$876$	3.69892e51			1.18646
$877$	−1.77124e51	−	3.06788e51i	−0.556910	−	0.964596i	−0.997752	−	0.0670118i	$-0.978653\pi$
$877$	−1.77124e51	−	3.06788e51i	−0.556910	−	0.964596i	0.440842	−	0.897585i	$-0.354680\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	3.50048e51			0.976820			0.488410	−	0.872614i	$-0.337577\pi$
$883$	3.50048e51			0.976820			0.488410	+	0.872614i	$0.337577\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	1.50737e51	+	3.87244e51i	0.373629	+	0.959851i
$890$		0			0
$891$		0			0
$892$	9.42995e50	+	5.44439e50i	0.220356	+	0.127223i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.50158e51	+	4.33286e51i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−1.45176e51	+	9.45286e51i	−0.273753	+	1.78249i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.94860e51	−	6.83918e51i	0.689154	−	1.19365i	−0.282958	−	0.959132i	$-0.591316\pi$
$907$	3.94860e51	−	6.83918e51i	0.689154	−	1.19365i	0.972112	−	0.234517i	$-0.0753508\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−2.21303e51	−	3.83308e51i	−0.350815	−	0.607629i
$913$		0			0
$914$		0			0
$915$		0			0
$916$			4.22010e51i			0.619657i
$917$		0			0
$918$		0			0
$919$	3.93572e51	+	6.81687e51i	0.545762	+	0.945287i	0.998559	+	0.0536733i	$0.0170930\pi$
$919$	3.93572e51	+	6.81687e51i	0.545762	+	0.945287i	−0.452797	+	0.891614i	$0.649574\pi$
$920$		0			0
$921$	−7.39557e51	+	1.28095e52i	−0.987252	+	1.70997i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	4.17633e51			0.516790
$926$		0			0
$927$	1.41092e52	+	8.14593e51i	1.68115	+	0.970611i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	1.90530e51	−	6.05670e51i	0.210544	−	0.669294i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1.56069e52i		−	1.54125i	−0.637287	−	0.770627i	$-0.719943\pi$
$937$		−	1.56069e52i		−	1.54125i	0.637287	−	0.770627i	$-0.280057\pi$
$938$		0			0
$939$	1.89650e52			1.80429
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	9.04528e51	+	5.22230e51i	0.728243	+	0.420451i
$949$	1.20783e52	−	2.09202e52i	0.954656	−	1.65351i
$950$		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
$961$	−6.24043e51	−	1.08087e52i	−0.395872	−	0.685671i
$962$		0			0
$963$		0			0
$964$	−2.26993e52	+	1.31055e52i	−1.36353	+	0.787235i
$965$		0			0
$966$		0			0
$967$	−2.93306e52			−1.66862			−0.834310	−	0.551295i	$-0.814134\pi$
$967$	−2.93306e52			−1.66862			−0.834310	+	0.551295i	$0.814134\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$			1.92380e52i			1.00000i
$973$	1.57558e52	−	6.13304e51i	0.804387	−	0.313113i
$974$		0			0
$975$	1.63371e52	+	2.82966e52i	0.804624	+	1.39365i
$976$	−1.31412e52	−	7.58708e51i	−0.635716	−	0.367031i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−4.30876e52			−1.90609
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.89053e52			−1.12910
$989$		0			0
$990$		0			0
$991$	3.88439e51	−	6.72797e51i	0.143891	−	0.249227i	−0.785067	−	0.619410i	$-0.787372\pi$
$991$	3.88439e51	−	6.72797e51i	0.143891	−	0.249227i	0.928959	+	0.370183i	$0.120705\pi$
$992$		0			0
$993$		−	4.58814e52i		−	1.64068i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−4.91671e52	−	2.83866e52i	−1.63874	−	0.946125i	−0.981269	−	0.192641i	$-0.938295\pi$
$997$	−4.91671e52	−	2.83866e52i	−1.63874	−	0.946125i	−0.657467	−	0.753483i	$-0.728372\pi$
$998$		0			0
$999$	1.39072e52	−	8.02935e51i	0.447554	−	0.258395i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.36.g.a.17.1	yes	2
3.2	odd	2	CM	21.36.g.a.17.1	yes	2
7.5	odd	6	inner	21.36.g.a.5.1	✓	2
21.5	even	6	inner	21.36.g.a.5.1	✓	2

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 36 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.93710e8 - 1.11839e8i) q^{3} +(-1.71799e10 + 2.97564e10i) q^{4} +(-2.23263e14 - 5.73561e14i) q^{7} +(2.50158e16 + 4.33286e16i) q^{9} +O(q^{10})\)	\(q+(-1.93710e8 - 1.11839e8i) q^{3} +(-1.71799e10 + 2.97564e10i) q^{4} +(-2.23263e14 - 5.73561e14i) q^{7} +(2.50158e16 + 4.33286e16i) q^{9} +(6.65583e18 - 3.84275e18i) q^{12} -5.01917e19i q^{13} +(-5.90296e20 - 1.02242e21i) q^{16} +(-1.45153e22 + 8.38042e21i) q^{19} +(-2.08981e22 + 1.36074e23i) q^{21} +(1.45519e24 - 2.52047e24i) q^{25} -1.11909e25i q^{27} +(2.09027e25 + 3.21021e24i) q^{28} +(4.96202e25 + 2.86482e25i) q^{31} -1.71907e27 q^{36} +(7.17488e26 + 1.24273e27i) q^{37} +(-5.61337e27 + 9.72264e27i) q^{39} +6.94685e28 q^{43} +2.64072e29i q^{48} +(-2.79126e29 + 2.56110e29i) q^{49} +(1.49352e30 + 8.62287e29i) q^{52} +3.74902e30 q^{57} +(1.11310e31 - 6.42650e30i) q^{61} +(1.92665e31 - 2.40217e31i) q^{63} +4.05648e31 q^{64} +(5.51402e31 - 9.55057e31i) q^{67} +(4.16806e32 + 2.40643e32i) q^{73} +(-5.63771e32 + 3.25493e32i) q^{75} -5.75898e32i q^{76} +(6.79500e32 + 1.17693e33i) q^{79} +(-1.25158e33 + 2.16780e33i) q^{81} +(-3.69005e33 - 2.95959e33i) q^{84} +(-2.87880e34 + 1.12059e34i) q^{91} +(-6.40796e33 - 1.10989e34i) q^{93} -9.65334e34i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 387420489 q^{3} - 34359738368 q^{4} - 446525205377873 q^{7} + 50\!\cdots\!07 q^{9}+O(q^{10}) \) 2 * q - 387420489 * q^3 - 34359738368 * q^4 - 446525205377873 * q^7 + 50031545098999707 * q^9	\( 2 q - 387420489 q^{3} - 34359738368 q^{4} - 446525205377873 q^{7} + 50\!\cdots\!07 q^{9}+ \cdots - 12\!\cdots\!75 q^{93}+O(q^{100}) \) 2 * q - 387420489 * q^3 - 34359738368 * q^4 - 446525205377873 * q^7 + 50031545098999707 * q^9 + 13311666640442621952 * q^12 - 1180591620717411303424 * q^16 - 29030614593056510079771 * q^19 - 41796137645342869154526 * q^21 + 2910383045673370361328125 * q^25 + 41805487535667052978110464 * q^28 + 99240318692296271463039225 * q^31 - 3438141599496845182057316352 * q^36 + 1434976029760582590688944323 * q^37 - 11226741639714620351964140643 * q^39 + 138937021705263726870548760050 * q^43 - 558252625493577900644717247757 * q^49 + 2987048308632889626118374555648 * q^52 + 7498036601074992759825539885346 * q^57 + 22262061668307550275762391952172 * q^61 + 38533026035573202444818730766425 * q^63 + 81129638414606681695789005144064 * q^64 + 110280454033501822232789985372223 * q^67 + 833612588618178474625817730556989 * q^73 - 1127542022732086479663848876953125 * q^75 + 1359000989429499836036390683764907 * q^79 - 2503155504993241601315571986085849 * q^81 - 7380099034832319024082245767921664 * q^84 - 57576010078993069344020247768783501 * q^91 - 12815910932095087341029133991893675 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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