LMFDB - Embedded newform 21.41.h.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,41,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, 41, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 41);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 41 $
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:	$212.818798913$
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			21.41.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)

$f(q)$	$=$	$q+(-1.74339e9 + 3.01964e9i) q^{3} +(-5.49756e11 + 9.52205e11i) q^{4} +(1.96603e16 + 7.73323e16i) q^{7} +(-6.07883e18 - 1.05288e19i) q^{9} +O(q^{10})$	\(q+(-1.74339e9 + 3.01964e9i) q^{3} +(-5.49756e11 + 9.52205e11i) q^{4} +(1.96603e16 + 7.73323e16i) q^{7} +(-6.07883e18 - 1.05288e19i) q^{9} +(-1.91688e21 - 3.32013e21i) q^{12} -3.65529e22 q^{13} +(-6.04463e23 - 1.04696e24i) q^{16} +(-3.44798e25 - 5.97208e25i) q^{19} +(-2.67792e26 - 7.54533e25i) q^{21} +(-4.54747e27 + 7.87646e27i) q^{25} +4.23912e28 q^{27} +(-8.44445e28 - 2.37932e28i) q^{28} +(3.14672e29 - 5.45028e29i) q^{31} +1.33675e31 q^{36} +(2.28625e31 + 3.95990e31i) q^{37} +(6.37261e31 - 1.10377e32i) q^{39} -9.32306e32 q^{43} +4.21526e33 q^{48} +(-5.59375e33 + 3.04075e33i) q^{49} +(2.00952e34 - 3.48059e34i) q^{52} +2.40448e35 q^{57} +(-4.54120e35 - 7.86559e35i) q^{61} +(6.94708e35 - 6.77090e35i) q^{63} +1.32923e36 q^{64} +(1.20948e36 - 2.09487e36i) q^{67} +(7.12783e36 - 1.23458e37i) q^{73} +(-1.58561e37 - 2.74635e37i) q^{75} +7.58220e37 q^{76} +(-1.58083e37 - 2.73808e37i) q^{79} +(-7.39044e37 + 1.28006e38i) q^{81} +(2.19067e38 - 2.13512e38i) q^{84} +(-7.18642e38 - 2.82672e39i) q^{91} +(1.09719e39 + 1.90040e39i) q^{93} +1.06455e40 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots - 12\!\cdots\!01 q^{9}+O(q^{10}) $ 2 * q - 3486784401 * q^3 - 1099511627776 * q^4 + 39320628860422799 * q^7 - 12157665459056928801 * q^9	$ 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots + 21\!\cdots\!48 q^{97}+O(q^{100}) $ 2 * q - 3486784401 * q^3 - 1099511627776 * q^4 + 39320628860422799 * q^7 - 12157665459056928801 * q^9 - 3833759992447475122176 * q^12 - 73105816473271055020802 * q^13 - 1208925819614629174706176 * q^16 - 68959680500986052300849999 * q^19 - 535583022247728139930999326 * q^21 - 9094947017729282379150390625 * q^25 + 84782316550432407028588866402 * q^27 - 168889065934762161812669464576 * q^28 + 629344461662676553680973098673 * q^31 + 26734989077687468135677691953152 * q^36 + 45724987453244821079270923454401 * q^37 + 127452110250685174045672572054801 * q^39 - 1864612348242241782755125725507746 * q^43 + 8430527379596857675529996470321152 * q^48 - 11187499667838941626759933239453601 * q^49 + 40190347635209886650593432280498176 * q^52 + 480895076537564064562347871108131198 * q^57 - 908239954252096873493225827240445474 * q^61 + 1389415476089055164206164058938179727 * q^63 + 2658455991569831745807614120560689152 * q^64 + 2418952203220787909486714319608792401 * q^67 + 14255661140299481178564160453025016001 * q^73 - 31712119389339932240545749664306640625 * q^75 + 151643941117104123076759704937795944448 * q^76 - 31616623256762405475971963610201308399 * q^79 - 147808829414345923316083210206383297601 * q^81 + 438133906797824707239241321619077988352 * q^84 - 1437283338541837165376338775168730032399 * q^91 + 2194388451781163131283256130953650199873 * q^93 + 21291032132139656523090370242468502316548 * 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	$0.333333\pi$
$3$	−1.74339e9	+	3.01964e9i	−0.500000	+	0.866025i
$3$	−1.74339e9	+	3.01964e9i	−0.500000	+	0.866025i
$4$	−5.49756e11	+	9.52205e11i	−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
$7$	1.96603e16	+	7.73323e16i	0.246394	+	0.969170i
$7$	1.96603e16	+	7.73323e16i	0.246394	+	0.969170i
$8$		0			0
$9$	−6.07883e18	−	1.05288e19i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−1.91688e21	−	3.32013e21i	−0.500000	−	0.866025i
$13$	−3.65529e22			−1.92333			−0.961667	−	0.274219i	$-0.911581\pi$
$13$	−3.65529e22			−1.92333			−0.961667	+	0.274219i	$0.911581\pi$
$14$		0			0
$15$		0			0
$16$	−6.04463e23	−	1.04696e24i	−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
$19$	−3.44798e25	−	5.97208e25i	−0.917262	−	1.58874i	−0.803556	−	0.595229i	$-0.797061\pi$
$19$	−3.44798e25	−	5.97208e25i	−0.917262	−	1.58874i	−0.113706	−	0.993514i	$-0.536272\pi$
$20$		0			0
$21$	−2.67792e26	−	7.54533e25i	−0.962523	−	0.271202i
$22$		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
$25$	−4.54747e27	+	7.87646e27i	−0.500000	+	0.866025i
$26$		0			0
$27$	4.23912e28			1.00000
$28$	−8.44445e28	−	2.37932e28i	−0.962523	−	0.271202i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	3.14672e29	−	5.45028e29i	0.468408	−	0.811307i	−0.530940	−	0.847410i	$-0.678161\pi$
$31$	3.14672e29	−	5.45028e29i	0.468408	−	0.811307i	0.999348	+	0.0361026i	$0.0114943\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.33675e31			1.00000
$37$	2.28625e31	+	3.95990e31i	0.988756	+	1.71258i	0.623882	+	0.781519i	$0.285555\pi$
$37$	2.28625e31	+	3.95990e31i	0.988756	+	1.71258i	0.364874	+	0.931057i	$0.381112\pi$
$38$		0			0
$39$	6.37261e31	−	1.10377e32i	0.961667	−	1.66566i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−9.32306e32			−1.99613			−0.998066	−	0.0621561i	$-0.980202\pi$
$43$	−9.32306e32			−1.99613			−0.998066	+	0.0621561i	$0.980202\pi$
$44$		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	$0.333333\pi$
$48$	4.21526e33			1.00000
$49$	−5.59375e33	+	3.04075e33i	−0.878580	+	0.477595i
$50$		0			0
$51$		0			0
$52$	2.00952e34	−	3.48059e34i	0.961667	−	1.66566i
$53$		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
$57$	2.40448e35			1.83452
$58$		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
$61$	−4.54120e35	−	7.86559e35i	−0.892429	−	1.54573i	−0.836954	−	0.547273i	$-0.815666\pi$
$61$	−4.54120e35	−	7.86559e35i	−0.892429	−	1.54573i	−0.0554754	−	0.998460i	$-0.517667\pi$
$62$		0			0
$63$	6.94708e35	−	6.77090e35i	0.716129	−	0.697968i
$64$	1.32923e36			1.00000
$65$		0			0
$66$		0			0
$67$	1.20948e36	−	2.09487e36i	0.364000	−	0.630466i	−0.624615	−	0.780933i	$-0.714744\pi$
$67$	1.20948e36	−	2.09487e36i	0.364000	−	0.630466i	0.988615	+	0.150466i	$0.0480775\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	7.12783e36	−	1.23458e37i	0.385923	−	0.668437i	−0.605974	−	0.795484i	$-0.707217\pi$
$73$	7.12783e36	−	1.23458e37i	0.385923	−	0.668437i	0.991897	+	0.127047i	$0.0405499\pi$
$74$		0			0
$75$	−1.58561e37	−	2.74635e37i	−0.500000	−	0.866025i
$76$	7.58220e37			1.83452
$77$		0			0
$78$		0			0
$79$	−1.58083e37	−	2.73808e37i	−0.176337	−	0.305425i	0.764286	−	0.644877i	$-0.223092\pi$
$79$	−1.58083e37	−	2.73808e37i	−0.176337	−	0.305425i	−0.940623	+	0.339453i	$0.889758\pi$
$80$		0			0
$81$	−7.39044e37	+	1.28006e38i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	2.19067e38	−	2.13512e38i	0.716129	−	0.697968i
$85$		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
$91$	−7.18642e38	−	2.82672e39i	−0.473898	−	1.86404i
$92$		0			0
$93$	1.09719e39	+	1.90040e39i	0.468408	+	0.811307i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.06455e40			1.95764			0.978818	−	0.204731i	$-0.0656321\pi$
$97$	1.06455e40			1.95764			0.978818	+	0.204731i	$0.0656321\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e39	−	8.66025e39i	−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
$103$	−9.63249e38	−	1.66840e39i	−0.0533328	−	0.0923751i	0.838126	−	0.545476i	$-0.183651\pi$
$103$	−9.63249e38	−	1.66840e39i	−0.0533328	−	0.0923751i	−0.891459	+	0.453101i	$0.850318\pi$
$104$		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	$0.333333\pi$
$108$	−2.33048e40	+	4.03651e40i	−0.500000	+	0.866025i
$109$	−5.26154e40	+	9.11326e40i	−0.938822	+	1.62609i	−0.171150	+	0.985245i	$0.554748\pi$
$109$	−5.26154e40	+	9.11326e40i	−0.938822	+	1.62609i	−0.767672	+	0.640842i	$0.778585\pi$
$110$		0			0
$111$	−1.59433e41			−1.97751
$112$	6.90799e40	−	6.73281e40i	0.716129	−	0.697968i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.22199e41	+	3.84860e41i	0.961667	+	1.66566i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−2.26296e41	−	3.91957e41i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	3.45986e41	+	5.99265e41i	0.468408	+	0.811307i
$125$		0			0
$126$		0			0
$127$	−1.77869e42			−1.49288			−0.746441	−	0.665451i	$-0.768239\pi$
$127$	−1.77869e42			−1.49288			−0.746441	+	0.665451i	$0.768239\pi$
$128$		0			0
$129$	1.62538e42	−	2.81523e42i	0.998066	−	1.72870i
$130$		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
$133$	3.94046e42	−	3.84053e42i	1.31375	−	1.28044i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	1.26535e43			1.74548			0.872742	−	0.488181i	$-0.162340\pi$
$139$	1.26535e43			1.74548			0.872742	+	0.488181i	$0.162340\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−7.34886e42	+	1.27286e43i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	5.70109e41	−	2.21924e43i	0.0256810	−	0.999670i
$148$	−5.02752e43			−1.97751
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−3.67388e43	+	6.36335e43i	−0.967356	+	1.67551i	−0.264210	+	0.964465i	$0.585111\pi$
$151$	−3.67388e43	+	6.36335e43i	−0.967356	+	1.67551i	−0.703146	+	0.711045i	$0.748222\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	7.00675e43	+	1.21361e44i	0.961667	+	1.66566i
$157$	−1.22796e43	+	2.12689e43i	−0.148317	+	0.256892i	−0.930605	−	0.366024i	$-0.880719\pi$
$157$	−1.22796e43	+	2.12689e43i	−0.148317	+	0.256892i	0.782289	+	0.622916i	$0.214052\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.72442e44	−	2.98678e44i	−0.983764	−	1.70393i	−0.647306	−	0.762231i	$-0.724104\pi$
$163$	−1.72442e44	−	2.98678e44i	−0.983764	−	1.70393i	−0.336458	−	0.941698i	$-0.609229\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	9.74926e44			2.69922
$170$		0			0
$171$	−4.19194e44	+	7.26066e44i	−0.917262	+	1.58874i
$172$	5.12541e44	−	8.87747e44i	0.998066	−	1.72870i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−6.98509e44	−	1.96813e44i	−0.962523	−	0.271202i
$176$		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
$181$	−1.28581e45			−0.902829			−0.451415	−	0.892314i	$-0.649080\pi$
$181$	−1.28581e45			−0.902829			−0.451415	+	0.892314i	$0.649080\pi$
$182$		0			0
$183$	3.16684e45			1.78486
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.33424e44	+	3.27820e45i	0.246394	+	0.969170i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−2.31737e45	+	4.01380e45i	−0.500000	+	0.866025i
$193$	−3.62892e45	+	6.28547e45i	−0.705718	+	1.22234i	0.260714	+	0.965416i	$0.416042\pi$
$193$	−3.62892e45	+	6.28547e45i	−0.705718	+	1.22234i	−0.966432	+	0.256923i	$0.917291\pi$
$194$		0			0
$195$		0			0
$196$	1.79776e44	−	6.99807e45i	0.0256810	−	0.999670i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.98949e45	+	3.44590e45i	−0.209739	+	0.363279i	−0.951632	−	0.307239i	$-0.900595\pi$
$199$	−1.98949e45	+	3.44590e45i	−0.209739	+	0.363279i	0.741893	+	0.670518i	$0.233928\pi$
$200$		0			0
$201$	4.21718e45	+	7.30437e45i	0.364000	+	0.630466i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	2.20949e46	+	3.82695e46i	0.961667	+	1.66566i
$209$		0			0
$210$		0			0
$211$	−1.40567e46			−0.459446			−0.229723	−	0.973256i	$-0.573782\pi$
$211$	−1.40567e46			−0.459446			−0.229723	+	0.973256i	$0.573782\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	4.83348e46	+	1.36189e46i	0.901707	+	0.254066i
$218$		0			0
$219$	2.48532e46	+	4.30470e46i	0.385923	+	0.668437i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−1.45259e46			−0.157053			−0.0785263	−	0.996912i	$-0.525021\pi$
$223$	−1.45259e46			−0.157053			−0.0785263	+	0.996912i	$0.525021\pi$
$224$		0			0
$225$	1.10573e47			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	−1.32187e47	+	2.28955e47i	−0.917262	+	1.58874i
$229$	1.56674e47	+	2.71367e47i	0.996061	+	1.72523i	0.574819	+	0.818280i	$0.305072\pi$
$229$	1.56674e47	+	2.71367e47i	0.996061	+	1.72523i	0.421242	+	0.906948i	$0.361594\pi$
$230$		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
$237$	1.10240e47			0.352674
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−4.32694e47	+	7.49448e47i	−0.990467	+	1.71554i	−0.375940	+	0.926644i	$0.622680\pi$
$241$	−4.32694e47	+	7.49448e47i	−0.990467	+	1.71554i	−0.614527	+	0.788896i	$0.710653\pi$
$242$		0			0
$243$	−2.57689e47	−	4.46330e47i	−0.500000	−	0.866025i
$244$	9.98620e47			1.78486
$245$		0			0
$246$		0			0
$247$	1.26034e48	+	2.18297e48i	1.76420	+	3.05569i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	2.62809e47	+	1.03374e48i	0.246394	+	0.969170i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−7.30751e47	+	1.26570e48i	−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
$259$	−2.61280e48	+	2.54654e48i	−1.41615	+	1.38024i
$260$		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
$268$	1.32983e48	+	2.30334e48i	0.364000	+	0.630466i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	4.50023e48	+	7.79464e48i	0.985937	+	1.70769i	0.637695	+	0.770289i	$0.279888\pi$
$271$	4.50023e48	+	7.79464e48i	0.985937	+	1.70769i	0.348243	+	0.937404i	$0.386778\pi$
$272$		0			0
$273$	9.78856e48	+	2.75804e48i	1.85125	+	0.521612i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	4.95172e48	−	8.57663e48i	0.700101	−	1.21261i	−0.268330	−	0.963327i	$-0.586472\pi$
$277$	4.95172e48	−	8.57663e48i	0.700101	−	1.21261i	0.968431	−	0.249283i	$-0.0801951\pi$
$278$		0			0
$279$	−7.65136e48			−0.936817
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	5.48314e48	−	9.49707e48i	0.505010	−	0.874704i	−0.494973	−	0.868908i	$-0.664822\pi$
$283$	5.48314e48	−	9.49707e48i	0.505010	−	0.874704i	0.999983	−	0.00579524i	$-0.00184469\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.25899e48	−	1.43050e49i	−0.500000	−	0.866025i
$290$		0			0
$291$	−1.85593e49	+	3.21457e49i	−0.978818	+	1.69536i
$292$	7.83713e48	+	1.35743e49i	0.385923	+	0.668437i
$293$		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
$300$	3.48678e49			1.00000
$301$	−1.83294e49	−	7.20973e49i	−0.491835	−	1.93459i
$302$		0			0
$303$		0			0
$304$	−4.16836e49	+	7.21981e49i	−0.917262	+	1.58874i
$305$		0			0
$306$		0			0
$307$	−4.37938e49			−0.791853			−0.395927	−	0.918282i	$-0.629577\pi$
$307$	−4.37938e49			−0.791853			−0.395927	+	0.918282i	$0.629577\pi$
$308$		0			0
$309$	6.71728e48			0.106666
$310$		0			0
$311$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$311$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$312$		0			0
$313$	−7.54014e49	−	1.30599e50i	−0.925748	−	1.60344i	−0.790353	−	0.612651i	$-0.790103\pi$
$313$	−7.54014e49	−	1.30599e50i	−0.925748	−	1.60344i	−0.135395	−	0.990792i	$-0.543230\pi$
$314$		0			0
$315$		0			0
$316$	3.47628e49			0.352674
$317$		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
$324$	−8.12588e49	−	1.40744e50i	−0.500000	−	0.866025i
$325$	1.66223e50	−	2.87907e50i	0.961667	−	1.66566i
$326$		0			0
$327$	−1.83459e50	−	3.17760e50i	−0.938822	−	1.62609i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.13164e50	−	1.96006e50i	−0.454097	−	0.786519i	0.544539	−	0.838736i	$-0.316705\pi$
$331$	−1.13164e50	−	1.96006e50i	−0.454097	−	0.786519i	−0.998636	+	0.0522165i	$0.983371\pi$
$332$		0			0
$333$	2.77955e50	−	4.81431e50i	0.988756	−	1.71258i
$334$		0			0
$335$		0			0
$336$	8.28734e49	+	3.25976e50i	0.246394	+	0.969170i
$337$	4.87623e50			1.36611			0.683055	−	0.730367i	$-0.260651\pi$
$337$	4.87623e50			1.36611			0.683055	+	0.730367i	$0.260651\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−3.45123e50	−	3.72795e50i	−0.679347	−	0.733817i
$344$		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
$349$	−1.35259e51			−1.88217			−0.941084	−	0.338173i	$-0.890191\pi$
$349$	−1.35259e51			−1.88217			−0.941084	+	0.338173i	$0.890191\pi$
$350$		0			0
$351$	−1.54952e51			−1.92333
$352$		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
$361$	−1.67122e51	+	2.89463e51i	−1.18274	+	2.04856i
$362$		0			0
$363$	1.57809e51			1.00000
$364$	3.08669e51	+	8.69711e50i	1.85125	+	0.521612i
$365$		0			0
$366$		0			0
$367$	9.73544e50	−	1.68623e51i	0.495489	−	0.858212i	−0.504497	−	0.863413i	$-0.668322\pi$
$367$	9.73544e50	−	1.68623e51i	0.495489	−	0.858212i	0.999986	+	0.00520096i	$0.00165552\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.41276e51			−0.936817
$373$	−1.07533e51	−	1.86252e51i	−0.395699	−	0.685370i	0.597491	−	0.801875i	$-0.296164\pi$
$373$	−1.07533e51	−	1.86252e51i	−0.395699	−	0.685370i	−0.993190	+	0.116505i	$0.962831\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	7.44115e51			1.99001			0.995006	−	0.0998121i	$-0.0318242\pi$
$379$	7.44115e51			1.99001			0.995006	+	0.0998121i	$0.0318242\pi$
$380$		0			0
$381$	3.10095e51	−	5.37101e51i	0.746441	−	1.29287i
$382$		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
$387$	5.66733e51	+	9.81611e51i	0.998066	+	1.72870i
$388$	−5.85243e51	+	1.01367e52i	−0.978818	+	1.69536i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−4.36995e51	−	7.56897e51i	−0.462026	−	0.800252i	0.537036	−	0.843559i	$-0.319544\pi$
$397$	−4.36995e51	−	7.56897e51i	−0.462026	−	0.800252i	−0.999062	+	0.0433074i	$0.986211\pi$
$398$		0			0
$399$	4.72727e51	+	1.85944e52i	0.452015	+	1.77796i
$400$	1.09951e52			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
$403$	−1.15022e52	+	1.99224e52i	−0.900906	+	1.56041i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.34018e52	+	2.32126e52i	−0.781082	+	1.35287i	0.150230	+	0.988651i	$0.451999\pi$
$409$	−1.34018e52	+	2.32126e52i	−0.781082	+	1.35287i	−0.931312	+	0.364222i	$0.881335\pi$
$410$		0			0
$411$		0			0
$412$	2.11821e51			0.106666
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−2.20601e52	+	3.82092e52i	−0.872742	+	1.51163i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	5.62244e52			1.83774			0.918868	−	0.394564i	$-0.129104\pi$
$421$	5.62244e52			1.83774			0.918868	+	0.394564i	$0.129104\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.18982e52	−	5.05821e52i	1.27819	−	1.24577i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	−2.56239e52	−	4.43819e52i	−0.500000	−	0.866025i
$433$	−6.87412e52			−1.28073			−0.640367	−	0.768069i	$-0.721218\pi$
$433$	−6.87412e52			−1.28073			−0.640367	+	0.768069i	$0.721218\pi$
$434$		0			0
$435$		0			0
$436$	−5.78513e52	−	1.00201e53i	−0.938822	−	1.62609i
$437$		0			0
$438$		0			0
$439$	3.84107e52	+	6.65294e52i	0.543453	+	0.941289i	0.998702	+	0.0509247i	$0.0162168\pi$
$439$	3.84107e52	+	6.65294e52i	0.543453	+	0.941289i	−0.455249	+	0.890364i	$0.650450\pi$
$440$		0			0
$441$	6.60191e52	+	4.04115e52i	0.852899	+	0.522075i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	8.76493e52	−	1.51813e53i	0.988756	−	1.71258i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	2.61330e52	+	1.02792e53i	0.246394	+	0.969170i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.28100e53	−	2.21876e53i	−0.967356	−	1.67551i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−7.64246e52	−	1.32371e53i	−0.484071	−	0.838436i	0.515761	−	0.856732i	$-0.327509\pi$
$457$	−7.64246e52	−	1.32371e53i	−0.484071	−	0.838436i	−0.999833	+	0.0182964i	$0.994176\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	−3.46582e53			−1.69116			−0.845582	−	0.533846i	$-0.820746\pi$
$463$	−3.46582e53			−1.69116			−0.845582	+	0.533846i	$0.820746\pi$
$464$		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	$0.333333\pi$
$468$	−4.88621e53			−1.92333
$469$	1.85780e53	+	5.23456e52i	0.700716	+	0.197435i
$470$		0			0
$471$	−4.28163e52	−	7.41599e52i	−0.148317	−	0.256892i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	6.27185e53			1.83452
$476$		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
$481$	−8.35691e53	−	1.44746e54i	−1.90171	−	3.29386i
$482$		0			0
$483$		0			0
$484$	4.97631e53			1.00000
$485$		0			0
$486$		0			0
$487$	−2.82086e53	+	4.88588e53i	−0.500959	+	0.867687i	0.499040	+	0.866579i	$0.333686\pi$
$487$	−2.82086e53	+	4.88588e53i	−0.500959	+	0.867687i	−0.999999	+	0.00110815i	$0.999647\pi$
$488$		0			0
$489$	1.20254e54			1.96753
$490$		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
$496$	−7.60831e53			−0.936817
$497$		0			0
$498$		0			0
$499$	2.28853e51	+	3.96385e51i	0.00249773	+	0.00432620i	0.867272	−	0.497835i	$-0.165872\pi$
$499$	2.28853e51	+	3.96385e51i	0.00249773	+	0.00432620i	−0.864774	+	0.502162i	$0.832538\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.69968e54	+	2.94393e54i	−1.34961	+	2.33759i
$508$	9.77845e53	−	1.69368e54i	0.746441	−	1.29287i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	1.09486e54	+	3.08490e53i	0.742918	+	0.209326i
$512$		0			0
$513$	−1.46164e54	−	2.53164e54i	−0.917262	−	1.58874i
$514$		0			0
$515$		0			0
$516$	1.78712e54	+	3.09538e54i	0.998066	+	1.72870i
$517$		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
$523$	2.24115e54	+	3.88179e54i	0.955955	+	1.65576i	0.732166	+	0.681126i	$0.238509\pi$
$523$	2.24115e54	+	3.88179e54i	0.955955	+	1.65576i	0.223789	+	0.974638i	$0.428157\pi$
$524$		0			0
$525$	1.81208e54	−	1.76613e54i	0.716129	−	0.697968i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.47260e54	+	2.55061e54i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	1.49068e54	+	5.86348e54i	0.452015	+	1.77796i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.00658e54	+	1.74344e54i	0.218224	+	0.377975i	0.954265	−	0.298962i	$-0.0966404\pi$
$541$	1.00658e54	+	1.74344e54i	0.218224	+	0.377975i	−0.736041	+	0.676937i	$0.763307\pi$
$542$		0			0
$543$	2.24167e54	−	3.88269e54i	0.451415	−	0.781873i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−1.12907e55			−1.96325			−0.981626	−	0.190817i	$-0.938886\pi$
$547$	−1.12907e55			−1.96325			−0.981626	+	0.190817i	$0.938886\pi$
$548$		0			0
$549$	−5.52104e54	+	9.56272e54i	−0.892429	+	1.54573i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.80662e54	−	1.76081e54i	0.252560	−	0.246155i
$554$		0			0
$555$		0			0
$556$	−6.95636e54	+	1.20488e55i	−0.872742	+	1.51163i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	3.40785e55			3.83923
$560$		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
$567$	−1.13520e55	−	3.19855e54i	−0.962523	−	0.271202i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	1.35626e55	−	2.34912e55i	0.999130	−	1.73054i	0.463455	−	0.886120i	$-0.346610\pi$
$571$	1.35626e55	−	2.34912e55i	0.999130	−	1.73054i	0.535675	−	0.844424i	$-0.320057\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−8.08015e54	−	1.39952e55i	−0.500000	−	0.866025i
$577$	−5.57187e54	+	9.65076e54i	−0.333031	+	0.576826i	−0.983105	−	0.183045i	$-0.941405\pi$
$577$	−5.57187e54	+	9.65076e54i	−0.333031	+	0.576826i	0.650074	+	0.759871i	$0.274738\pi$
$578$		0			0
$579$	−1.26533e55	−	2.19161e55i	−0.705718	−	1.22234i
$580$		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			$\pi$
$588$	2.08183e55	+	1.27432e55i	0.852899	+	0.522075i
$589$	−4.33994e55			−1.71861
$590$		0			0
$591$		0			0
$592$	2.76391e55	−	4.78723e55i	0.988756	−	1.71258i
$593$		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
$597$	−6.93692e54	−	1.20151e55i	−0.209739	−	0.363279i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$	1.18552e55			0.313632			0.156816	−	0.987628i	$-0.449877\pi$
$601$	1.18552e55			0.313632			0.156816	+	0.987628i	$0.449877\pi$
$602$		0			0
$603$	−2.94088e55			−0.728000
$604$	−4.03948e55	−	6.99658e55i	−0.967356	−	1.67551i
$605$		0			0
$606$		0			0
$607$	2.15539e55	+	3.73325e55i	0.467469	+	0.809680i	0.999309	−	0.0371648i	$-0.0118326\pi$
$607$	2.15539e55	+	3.73325e55i	0.467469	+	0.809680i	−0.531840	+	0.846845i	$0.678499\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.55060e55	−	2.68572e55i	0.276242	−	0.478466i	−0.694205	−	0.719777i	$-0.744244\pi$
$613$	1.55060e55	−	2.68572e55i	0.276242	−	0.478466i	0.970448	+	0.241311i	$0.0775774\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−5.15388e55	+	8.92678e55i	−0.755651	+	1.30883i	0.189399	+	0.981900i	$0.439346\pi$
$619$	−5.15388e55	+	8.92678e55i	−0.755651	+	1.30883i	−0.945050	+	0.326926i	$0.893987\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−1.54080e56			−1.92333
$625$	−4.13590e55	−	7.16359e55i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−1.35015e55	−	2.33854e55i	−0.148317	−	0.256892i
$629$		0			0
$630$		0			0
$631$	5.39281e55			0.538552			0.269276	−	0.963063i	$-0.413216\pi$
$631$	5.39281e55			0.538552			0.269276	+	0.963063i	$0.413216\pi$
$632$		0			0
$633$	2.45063e55	−	4.24462e55i	0.229723	−	0.397892i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	2.04468e56	−	1.11148e56i	1.68980	−	0.918575i
$638$		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
$643$	2.14690e56			1.47093			0.735466	−	0.677561i	$-0.236963\pi$
$643$	2.14690e56			1.47093			0.735466	+	0.677561i	$0.236963\pi$
$644$		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
$651$	−1.25391e56	+	1.22211e56i	−0.670881	+	0.653868i
$652$	3.79204e56			1.96753
$653$		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
$657$	−1.73316e56			−0.771845
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	−2.45892e56	+	4.25898e56i	−0.969876	+	1.67987i	−0.273973	+	0.961737i	$0.588338\pi$
$661$	−2.45892e56	+	4.25898e56i	−0.969876	+	1.67987i	−0.695902	+	0.718136i	$0.744995\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.53243e55	−	4.38630e55i	0.0785263	−	0.136012i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.84632e56			1.88433			0.942163	−	0.335155i	$-0.108789\pi$
$673$	6.84632e56			1.88433			0.942163	+	0.335155i	$0.108789\pi$
$674$		0			0
$675$	−1.92773e56	+	3.33892e56i	−0.500000	+	0.866025i
$676$	−5.35971e56	+	9.28330e56i	−1.34961	+	2.33759i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	2.09294e56	+	8.23242e56i	0.482349	+	1.89728i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−4.60909e56	−	7.98318e56i	−0.917262	−	1.58874i
$685$		0			0
$686$		0			0
$687$	−1.09257e57			−1.99212
$688$	5.63545e56	+	9.76088e56i	0.998066	+	1.72870i
$689$		0			0
$690$		0			0
$691$	3.95102e56	+	6.84337e56i	0.641429	+	1.11099i	0.985114	+	0.171903i	$0.0549915\pi$
$691$	3.95102e56	+	6.84337e56i	0.641429	+	1.11099i	−0.343685	+	0.939085i	$0.611675\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	5.71415e56	−	5.56925e56i	0.716129	−	0.697968i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.57659e57	−	2.73073e57i	1.81390	−	3.14176i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.48824e56	+	4.30976e56i	0.241528	+	0.418339i	0.961150	−	0.276028i	$-0.0890181\pi$
$709$	2.48824e56	+	4.30976e56i	0.241528	+	0.418339i	−0.719622	+	0.694366i	$0.755685\pi$
$710$		0			0
$711$	−1.92192e56	+	3.32887e56i	−0.176337	+	0.305425i
$712$		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
$721$	1.10083e56	−	1.07291e56i	0.0763863	−	0.0744491i
$722$		0			0
$723$	−1.50871e57	−	2.61316e57i	−0.990467	−	1.71554i
$724$	7.06882e56	−	1.22436e57i	0.451415	−	0.781873i
$725$		0			0
$726$		0			0
$727$	−1.66414e56			−0.0978370			−0.0489185	−	0.998803i	$-0.515577\pi$
$727$	−1.66414e56			−0.0978370			−0.0489185	+	0.998803i	$0.515577\pi$
$728$		0			0
$729$	1.79701e57			1.00000
$730$		0			0
$731$		0			0
$732$	−1.74099e57	+	3.01548e57i	−0.892429	+	1.54573i
$733$	−9.13575e56	−	1.58236e57i	−0.455685	−	0.789269i	0.543042	−	0.839705i	$-0.317272\pi$
$733$	−9.13575e56	−	1.58236e57i	−0.455685	−	0.789269i	−0.998727	+	0.0504358i	$0.983939\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	2.14592e57	−	3.71684e57i	0.909337	−	1.57502i	0.0943493	−	0.995539i	$-0.469923\pi$
$739$	2.14592e57	−	3.71684e57i	0.909337	−	1.57502i	0.814988	−	0.579478i	$-0.196744\pi$
$740$		0			0
$741$	−8.78906e57			−3.52840
$742$		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
$751$	2.37761e57	+	4.11815e57i	0.730033	+	1.26445i	0.956868	+	0.290522i	$0.0938288\pi$
$751$	2.37761e57	+	4.11815e57i	0.730033	+	1.26445i	−0.226835	+	0.973933i	$0.572838\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−3.57970e57	−	1.00862e57i	−0.962523	−	0.271202i
$757$	5.12530e57			1.34215			0.671075	−	0.741389i	$-0.265833\pi$
$757$	5.12530e57			1.34215			0.671075	+	0.741389i	$0.265833\pi$
$758$		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
$763$	−8.08193e57	−	2.27717e57i	−1.80727	−	0.509220i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−2.54797e57	−	4.41321e57i	−0.500000	−	0.866025i
$769$	8.31709e57			1.59017			0.795087	−	0.606495i	$-0.207425\pi$
$769$	8.31709e57			1.59017			0.795087	+	0.606495i	$0.207425\pi$
$770$		0			0
$771$		0			0
$772$	−3.99004e57	−	6.91095e57i	−0.705718	−	1.22234i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	2.86193e57	+	4.95700e57i	0.468408	+	0.811307i
$776$		0			0
$777$	−3.13451e57	−	1.23293e58i	−0.487247	−	1.91654i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	6.56476e57	+	4.01841e57i	0.852899	+	0.522075i
$785$		0			0
$786$		0			0
$787$	6.50999e57	−	1.12756e58i	0.783584	−	1.35721i	−0.146258	−	0.989246i	$-0.546723\pi$
$787$	6.50999e57	−	1.12756e58i	0.783584	−	1.35721i	0.929842	−	0.367960i	$-0.119944\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.65994e58	+	2.87510e58i	1.71644	+	2.97296i
$794$		0			0
$795$		0			0
$796$	−2.18747e57	−	3.78880e57i	−0.209739	−	0.363279i
$797$		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
$804$	−9.27368e57			−0.728000
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$	1.49082e58			0.984029			0.492014	−	0.870587i	$-0.336261\pi$
$811$	1.49082e58			0.984029			0.492014	+	0.870587i	$0.336261\pi$
$812$		0			0
$813$	−3.13827e58			−1.97187
$814$		0			0
$815$		0			0
$816$		0			0
$817$	3.21458e58	+	5.56781e58i	1.83098	+	3.17134i
$818$		0			0
$819$	−2.53936e58	+	2.47496e58i	−1.37736	+	1.34243i
$820$		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
$823$	−3.53814e57	+	6.12824e57i	−0.174092	+	0.301536i	−0.939847	−	0.341597i	$-0.889032\pi$
$823$	−3.53814e57	+	6.12824e57i	−0.174092	+	0.301536i	0.765755	+	0.643133i	$0.222366\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−3.79213e57	+	6.56817e57i	−0.161359	+	0.279482i	−0.935356	−	0.353707i	$-0.884921\pi$
$829$	−3.79213e57	+	6.56817e57i	−0.161359	+	0.279482i	0.773997	+	0.633189i	$0.218254\pi$
$830$		0			0
$831$	1.72656e58	+	2.99049e58i	0.700101	+	1.21261i
$832$	−4.85871e58			−1.92333
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.33393e58	−	2.31044e58i	0.468408	−	0.811307i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	3.13271e58			1.00000
$842$		0			0
$843$		0			0
$844$	7.72775e57	−	1.33849e58i	0.229723	−	0.397892i
$845$		0			0
$846$		0			0
$847$	2.58618e58	−	2.52060e58i	0.716129	−	0.697968i
$848$		0			0
$849$	1.91185e58	+	3.31142e58i	0.505010	+	0.874704i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	7.01736e58			1.68730			0.843651	−	0.536891i	$-0.180401\pi$
$853$	7.01736e58			1.68730			0.843651	+	0.536891i	$0.180401\pi$
$854$		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
$859$	−4.03378e58	−	6.98671e58i	−0.843041	−	1.46019i	−0.887312	−	0.461170i	$-0.847430\pi$
$859$	−4.03378e58	−	6.98671e58i	−0.843041	−	1.46019i	0.0442710	−	0.999020i	$-0.485903\pi$
$860$		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
$867$	5.75946e58			1.00000
$868$	−3.95403e58	+	3.85376e58i	−0.670881	+	0.653868i
$869$		0			0
$870$		0			0
$871$	−4.42099e58	+	7.65737e58i	−0.700094	+	1.21260i
$872$		0			0
$873$	−6.47123e58	−	1.12085e59i	−0.978818	−	1.69536i
$874$		0			0
$875$		0			0
$876$	−5.46528e58			−0.771845
$877$	−5.65543e58	−	9.79549e58i	−0.780681	−	1.35218i	−0.931545	−	0.363625i	$-0.881539\pi$
$877$	−5.65543e58	−	9.79549e58i	−0.780681	−	1.35218i	0.150864	−	0.988555i	$-0.451794\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	5.83035e58			0.702232			0.351116	−	0.936332i	$-0.385802\pi$
$883$	5.83035e58			0.702232			0.351116	+	0.936332i	$0.385802\pi$
$884$		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
$889$	−3.49696e58	−	1.37550e59i	−0.367837	−	1.44686i
$890$		0			0
$891$		0			0
$892$	7.98568e57	−	1.38316e58i	0.0785263	−	0.136012i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−6.07883e58	+	1.05288e59i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	2.49664e59	+	7.03456e58i	1.92132	+	0.541355i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.55405e58	−	6.15580e58i	0.250368	−	0.433649i	−0.713259	−	0.700900i	$-0.752782\pi$
$907$	3.55405e58	−	6.15580e58i	0.250368	−	0.433649i	0.963627	+	0.267251i	$0.0861152\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−1.45342e59	−	2.51739e59i	−0.917262	−	1.58874i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−3.44529e59			−1.99212
$917$		0			0
$918$		0			0
$919$	−1.45885e59	−	2.52680e59i	−0.790134	−	1.36855i	−0.925884	−	0.377809i	$-0.876678\pi$
$919$	−1.45885e59	−	2.52680e59i	−0.790134	−	1.36855i	0.135750	−	0.990743i	$-0.456656\pi$
$920$		0			0
$921$	7.63497e58	−	1.32242e59i	0.395927	−	0.685765i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.15866e59			−1.97751
$926$		0			0
$927$	−1.17109e58	+	2.02838e58i	−0.0533328	+	0.0923751i
$928$		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
$931$	3.74468e59	+	2.29219e59i	1.56466	+	0.957760i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	3.83211e59			1.40814			0.704071	−	0.710130i	$-0.251364\pi$
$937$	3.83211e59			1.40814			0.704071	+	0.710130i	$0.251364\pi$
$938$		0			0
$939$	5.25817e59			1.85150
$940$		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	$0.333333\pi$
$948$	−6.06053e58	+	1.04971e59i	−0.176337	+	0.305425i
$949$	−2.60543e59	+	4.51274e59i	−0.742258	+	1.28563i
$950$		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
$961$	2.76140e58	+	4.78289e58i	0.0611874	+	0.105980i
$962$		0			0
$963$		0			0
$964$	−4.75752e59	−	8.24027e59i	−0.990467	−	1.71554i
$965$		0			0
$966$		0			0
$967$	5.99615e57			0.0117312			0.00586561	−	0.999983i	$-0.498133\pi$
$967$	5.99615e57			0.0117312			0.00586561	+	0.999983i	$0.498133\pi$
$968$		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	$0.333333\pi$
$972$	5.66664e59			1.00000
$973$	2.48773e59	+	9.78527e59i	0.430076	+	1.69167i
$974$		0			0
$975$	5.79585e59	+	1.00387e60i	0.961667	+	1.66566i
$976$	−5.48997e59	+	9.50891e59i	−0.892429	+	1.54573i
$977$		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
$981$	1.27936e60			1.87764
$982$		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
$988$	−2.77151e60			−3.52840
$989$		0			0
$990$		0			0
$991$	4.95284e58	−	8.57857e58i	0.0593446	−	0.102788i	−0.834827	−	0.550513i	$-0.814432\pi$
$991$	4.95284e58	−	8.57857e58i	0.0593446	−	0.102788i	0.894171	+	0.447725i	$0.147766\pi$
$992$		0			0
$993$	7.89157e59			0.908194
$994$		0			0
$995$		0			0
$996$		0			0
$997$	5.32685e59	−	9.22637e59i	0.565675	−	0.979778i	−0.431312	−	0.902203i	$-0.641949\pi$
$997$	5.32685e59	−	9.22637e59i	0.565675	−	0.979778i	0.996987	−	0.0775748i	$-0.0247177\pi$
$998$		0			0
$999$	9.69168e59	+	1.67865e60i	0.988756	+	1.71258i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.41.h.a.11.1	yes	2
3.2	odd	2	CM	21.41.h.a.11.1	yes	2
7.2	even	3	inner	21.41.h.a.2.1	✓	2
21.2	odd	6	inner	21.41.h.a.2.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.41.h.a.2.1	✓	2	7.2	even	3	inner
21.41.h.a.2.1	✓	2	21.2	odd	6	inner
21.41.h.a.11.1	yes	2	1.1	even	1	trivial
21.41.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:	\( k \)	\(=\)	\( 41 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.74339e9 + 3.01964e9i) q^{3} +(-5.49756e11 + 9.52205e11i) q^{4} +(1.96603e16 + 7.73323e16i) q^{7} +(-6.07883e18 - 1.05288e19i) q^{9} +O(q^{10})\)	\(q+(-1.74339e9 + 3.01964e9i) q^{3} +(-5.49756e11 + 9.52205e11i) q^{4} +(1.96603e16 + 7.73323e16i) q^{7} +(-6.07883e18 - 1.05288e19i) q^{9} +(-1.91688e21 - 3.32013e21i) q^{12} -3.65529e22 q^{13} +(-6.04463e23 - 1.04696e24i) q^{16} +(-3.44798e25 - 5.97208e25i) q^{19} +(-2.67792e26 - 7.54533e25i) q^{21} +(-4.54747e27 + 7.87646e27i) q^{25} +4.23912e28 q^{27} +(-8.44445e28 - 2.37932e28i) q^{28} +(3.14672e29 - 5.45028e29i) q^{31} +1.33675e31 q^{36} +(2.28625e31 + 3.95990e31i) q^{37} +(6.37261e31 - 1.10377e32i) q^{39} -9.32306e32 q^{43} +4.21526e33 q^{48} +(-5.59375e33 + 3.04075e33i) q^{49} +(2.00952e34 - 3.48059e34i) q^{52} +2.40448e35 q^{57} +(-4.54120e35 - 7.86559e35i) q^{61} +(6.94708e35 - 6.77090e35i) q^{63} +1.32923e36 q^{64} +(1.20948e36 - 2.09487e36i) q^{67} +(7.12783e36 - 1.23458e37i) q^{73} +(-1.58561e37 - 2.74635e37i) q^{75} +7.58220e37 q^{76} +(-1.58083e37 - 2.73808e37i) q^{79} +(-7.39044e37 + 1.28006e38i) q^{81} +(2.19067e38 - 2.13512e38i) q^{84} +(-7.18642e38 - 2.82672e39i) q^{91} +(1.09719e39 + 1.90040e39i) q^{93} +1.06455e40 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots - 12\!\cdots\!01 q^{9}+O(q^{10}) \) 2 * q - 3486784401 * q^3 - 1099511627776 * q^4 + 39320628860422799 * q^7 - 12157665459056928801 * q^9	\( 2 q - 3486784401 q^{3} - 1099511627776 q^{4} + 39\!\cdots\!99 q^{7}+ \cdots + 21\!\cdots\!48 q^{97}+O(q^{100}) \) 2 * q - 3486784401 * q^3 - 1099511627776 * q^4 + 39320628860422799 * q^7 - 12157665459056928801 * q^9 - 3833759992447475122176 * q^12 - 73105816473271055020802 * q^13 - 1208925819614629174706176 * q^16 - 68959680500986052300849999 * q^19 - 535583022247728139930999326 * q^21 - 9094947017729282379150390625 * q^25 + 84782316550432407028588866402 * q^27 - 168889065934762161812669464576 * q^28 + 629344461662676553680973098673 * q^31 + 26734989077687468135677691953152 * q^36 + 45724987453244821079270923454401 * q^37 + 127452110250685174045672572054801 * q^39 - 1864612348242241782755125725507746 * q^43 + 8430527379596857675529996470321152 * q^48 - 11187499667838941626759933239453601 * q^49 + 40190347635209886650593432280498176 * q^52 + 480895076537564064562347871108131198 * q^57 - 908239954252096873493225827240445474 * q^61 + 1389415476089055164206164058938179727 * q^63 + 2658455991569831745807614120560689152 * q^64 + 2418952203220787909486714319608792401 * q^67 + 14255661140299481178564160453025016001 * q^73 - 31712119389339932240545749664306640625 * q^75 + 151643941117104123076759704937795944448 * q^76 - 31616623256762405475971963610201308399 * q^79 - 147808829414345923316083210206383297601 * q^81 + 438133906797824707239241321619077988352 * q^84 - 1437283338541837165376338775168730032399 * q^91 + 2194388451781163131283256130953650199873 * q^93 + 21291032132139656523090370242468502316548 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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