LMFDB - Embedded newform 21.33.h.a.11.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,33,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, 33, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 33);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 33 $
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:	$136.219975799$
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			21.33.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)

$f(q)$	$=$	$q+(-2.15234e7 + 3.72796e7i) q^{3} +(-2.14748e9 + 3.71955e9i) q^{4} +(3.25760e13 + 6.57520e12i) q^{7} +(-9.26510e14 - 1.60476e15i) q^{9} +O(q^{10})$	\(q+(-2.15234e7 + 3.72796e7i) q^{3} +(-2.14748e9 + 3.71955e9i) q^{4} +(3.25760e13 + 6.57520e12i) q^{7} +(-9.26510e14 - 1.60476e15i) q^{9} +(-9.24421e16 - 1.60114e17i) q^{12} +1.22181e17 q^{13} +(-9.22337e18 - 1.59753e19i) q^{16} +(3.91058e18 + 6.77332e18i) q^{19} +(-9.46265e20 + 1.07290e21i) q^{21} +(-1.16415e22 + 2.01637e22i) q^{25} +7.97664e22 q^{27} +(-9.44132e22 + 1.07048e23i) q^{28} +(7.06862e23 - 1.22432e24i) q^{31} +7.95866e24 q^{36} +(1.07781e25 + 1.86682e25i) q^{37} +(-2.62975e24 + 4.55487e24i) q^{39} +2.72890e26 q^{43} +7.94072e26 q^{48} +(1.01796e27 + 4.28387e26i) q^{49} +(-2.62383e26 + 4.54460e26i) q^{52} -3.36675e26 q^{57} +(1.97705e28 + 3.42436e28i) q^{61} +(-1.96303e28 - 5.83687e28i) q^{63} +7.92282e28 q^{64} +(-2.65665e27 + 4.60145e27i) q^{67} +(-3.83758e29 + 6.64688e29i) q^{73} +(-5.01130e29 - 8.67982e29i) q^{75} -3.35916e28 q^{76} +(-1.01348e30 - 1.75540e30i) q^{79} +(-1.71684e30 + 2.97366e30i) q^{81} +(-1.95861e30 - 5.82371e30i) q^{84} +(3.98018e30 + 8.03367e29i) q^{91} +(3.04281e31 + 5.27030e31i) q^{93} +1.21183e32 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 43046721 q^{3} - 4294967296 q^{4} + 65151959156159 q^{7} - 18\!\cdots\!41 q^{9}+O(q^{10}) $ 2 * q - 43046721 * q^3 - 4294967296 * q^4 + 65151959156159 * q^7 - 1853020188851841 * q^9	$ 2 q - 43046721 q^{3} - 4294967296 q^{4} + 65151959156159 q^{7} - 18\!\cdots\!41 q^{9}+ \cdots + 24\!\cdots\!28 q^{97}+O(q^{100}) $ 2 * q - 43046721 * q^3 - 4294967296 * q^4 + 65151959156159 * q^7 - 1853020188851841 * q^9 - 184884258895036416 * q^12 + 244362804407950078 * q^13 - 18446744073709551616 * q^16 + 7821153231347348161 * q^19 - 1892530001344659310206 * q^21 - 23283064365386962890625 * q^25 + 159532886153745019726722 * q^27 - 188826332729829706694656 * q^28 + 1413723680265064241474113 * q^31 + 15917322219892801768783872 * q^36 + 21556186719627646822930561 * q^37 - 5259508732063298594797119 * q^39 + 545779797112520941017987454 * q^43 + 1588143690998757006898102272 * q^48 + 2035922433397969262737034879 * q^49 - 524765126645495113705324544 * q^52 - 673350002096115500752860162 * q^57 + 39541057150634567761997420926 * q^61 - 39260684707600007410943504193 * q^63 + 158456325028528675187087900672 * q^64 - 5313292745304228970622516159 * q^67 - 767515778351719658853870420479 * q^73 - 1002259575761854648590087890625 * q^75 - 67183194691283164735641485312 * q^76 - 2026961502021606539826450603839 * q^79 - 3433683820292512484657849089281 * q^81 - 3917217221671991101932246663168 * q^84 + 7960357726035616964721939115201 * q^91 + 60856168835463426449812771033473 * q^93 + 242366393497161848998074563303428 * 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	$0.333333\pi$
$3$	−2.15234e7	+	3.72796e7i	−0.500000	+	0.866025i
$3$	−2.15234e7	+	3.72796e7i	−0.500000	+	0.866025i
$4$	−2.14748e9	+	3.71955e9i	−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
$7$	3.25760e13	+	6.57520e12i	0.980232	+	0.197852i
$7$	3.25760e13	+	6.57520e12i	0.980232	+	0.197852i
$8$		0			0
$9$	−9.26510e14	−	1.60476e15i	−0.500000	−	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−9.24421e16	−	1.60114e17i	−0.500000	−	0.866025i
$13$	1.22181e17			0.183616			0.0918082	−	0.995777i	$-0.470735\pi$
$13$	1.22181e17			0.183616			0.0918082	+	0.995777i	$0.470735\pi$
$14$		0			0
$15$		0			0
$16$	−9.22337e18	−	1.59753e19i	−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
$19$	3.91058e18	+	6.77332e18i	0.0135576	+	0.0234825i	0.872725	−	0.488213i	$-0.162351\pi$
$19$	3.91058e18	+	6.77332e18i	0.0135576	+	0.0234825i	−0.859167	+	0.511695i	$0.829018\pi$
$20$		0			0
$21$	−9.46265e20	+	1.07290e21i	−0.661461	+	0.749980i
$22$		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
$25$	−1.16415e22	+	2.01637e22i	−0.500000	+	0.866025i
$26$		0			0
$27$	7.97664e22			1.00000
$28$	−9.44132e22	+	1.07048e23i	−0.661461	+	0.749980i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	7.06862e23	−	1.22432e24i	0.971734	−	1.68309i	0.281417	−	0.959585i	$-0.409195\pi$
$31$	7.06862e23	−	1.22432e24i	0.971734	−	1.68309i	0.690317	−	0.723507i	$-0.257471\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	7.95866e24			1.00000
$37$	1.07781e25	+	1.86682e25i	0.873603	+	1.51313i	0.858243	+	0.513244i	$0.171556\pi$
$37$	1.07781e25	+	1.86682e25i	0.873603	+	1.51313i	0.0153605	+	0.999882i	$0.495110\pi$
$38$		0			0
$39$	−2.62975e24	+	4.55487e24i	−0.0918082	+	0.159016i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	2.72890e26			1.99752			0.998762	−	0.0497364i	$-0.0158381\pi$
$43$	2.72890e26			1.99752			0.998762	+	0.0497364i	$0.0158381\pi$
$44$		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	$0.333333\pi$
$48$	7.94072e26			1.00000
$49$	1.01796e27	+	4.28387e26i	0.921709	+	0.387881i
$50$		0			0
$51$		0			0
$52$	−2.62383e26	+	4.54460e26i	−0.0918082	+	0.159016i
$53$		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
$57$	−3.36675e26			−0.0271152
$58$		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
$61$	1.97705e28	+	3.42436e28i	0.537949	+	0.931755i	0.999014	+	0.0443885i	$0.0141340\pi$
$61$	1.97705e28	+	3.42436e28i	0.537949	+	0.931755i	−0.461066	+	0.887366i	$0.652533\pi$
$62$		0			0
$63$	−1.96303e28	−	5.83687e28i	−0.318771	−	0.947832i
$64$	7.92282e28			1.00000
$65$		0			0
$66$		0			0
$67$	−2.65665e27	+	4.60145e27i	−0.0161115	+	0.0279060i	−0.873969	−	0.485982i	$-0.838462\pi$
$67$	−2.65665e27	+	4.60145e27i	−0.0161115	+	0.0279060i	0.857857	+	0.513888i	$0.171795\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−3.83758e29	+	6.64688e29i	−0.590054	+	1.02200i	0.404171	+	0.914684i	$0.367560\pi$
$73$	−3.83758e29	+	6.64688e29i	−0.590054	+	1.02200i	−0.994225	+	0.107320i	$0.965773\pi$
$74$		0			0
$75$	−5.01130e29	−	8.67982e29i	−0.500000	−	0.866025i
$76$	−3.35916e28			−0.0271152
$77$		0			0
$78$		0			0
$79$	−1.01348e30	−	1.75540e30i	−0.440334	−	0.762681i	0.557380	−	0.830257i	$-0.311807\pi$
$79$	−1.01348e30	−	1.75540e30i	−0.440334	−	0.762681i	−0.997714	+	0.0675768i	$0.978473\pi$
$80$		0			0
$81$	−1.71684e30	+	2.97366e30i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	−1.95861e30	−	5.82371e30i	−0.318771	−	0.947832i
$85$		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
$91$	3.98018e30	+	8.03367e29i	0.179987	+	0.0363289i
$92$		0			0
$93$	3.04281e31	+	5.27030e31i	0.971734	+	1.68309i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.21183e32			1.97285			0.986426	−	0.164204i	$-0.0525057\pi$
$97$	1.21183e32			1.97285			0.986426	+	0.164204i	$0.0525057\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e31	−	8.66025e31i	−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
$103$	−4.30303e31	−	7.45307e31i	−0.268151	−	0.464450i	0.700234	−	0.713914i	$-0.253079\pi$
$103$	−4.30303e31	−	7.45307e31i	−0.268151	−	0.464450i	−0.968384	+	0.249463i	$0.919746\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−1.71297e32	+	2.96695e32i	−0.500000	+	0.866025i
$109$	−2.22684e32	+	3.85700e32i	−0.560874	+	0.971462i	0.436546	+	0.899682i	$0.356202\pi$
$109$	−2.22684e32	+	3.85700e32i	−0.560874	+	0.971462i	−0.997420	+	0.0717807i	$0.977132\pi$
$110$		0			0
$111$	−9.27923e32			−1.74721
$112$	−1.95419e32	−	5.81058e32i	−0.318771	−	0.947832i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.13202e32	−	1.96072e32i	−0.0918082	−	0.159016i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.05569e33	−	1.82851e33i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	3.03595e33	+	5.25842e33i	0.971734	+	1.68309i
$125$		0			0
$126$		0			0
$127$	7.15599e33			1.56247			0.781233	−	0.624240i	$-0.214591\pi$
$127$	7.15599e33			1.56247			0.781233	+	0.624240i	$0.214591\pi$
$128$		0			0
$129$	−5.87351e33	+	1.01732e34i	−0.998762	+	1.72991i
$130$		0			0
$131$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$131$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$132$		0			0
$133$	8.28550e31	+	2.46360e32i	0.00864355	+	0.0257007i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$	−3.79001e34			−1.95166			−0.975829	−	0.218537i	$-0.929872\pi$
$139$	−3.79001e34			−1.95166			−0.975829	+	0.218537i	$0.929872\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.70911e34	+	2.96026e34i	−0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−3.78800e34	+	2.87288e34i	−0.796770	+	0.604283i
$148$	−9.25831e34			−1.74721
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	−3.62428e34	+	6.27745e34i	−0.496125	+	0.859314i	−0.999990	−	0.00446863i	$-0.998578\pi$
$151$	−3.62428e34	+	6.27745e34i	−0.496125	+	0.859314i	0.503865	+	0.863782i	$0.331911\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−1.12947e34	−	1.95630e34i	−0.0918082	−	0.159016i
$157$	−1.35303e35	+	2.34352e35i	−0.992917	+	1.71978i	−0.393565	+	0.919297i	$0.628758\pi$
$157$	−1.35303e35	+	2.34352e35i	−0.992917	+	1.71978i	−0.599352	+	0.800485i	$0.704575\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	1.77804e35	+	3.07966e35i	0.716046	+	1.24023i	0.962555	+	0.271088i	$0.0873833\pi$
$163$	1.77804e35	+	3.07966e35i	0.716046	+	1.24023i	−0.246509	+	0.969141i	$0.579283\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	−4.27851e35			−0.966285
$170$		0			0
$171$	7.24638e33	−	1.25511e34i	0.0135576	−	0.0234825i
$172$	−5.86027e35	+	1.01503e36i	−0.998762	+	1.72991i
$173$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$173$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$174$		0			0
$175$	−5.11815e35	+	5.80308e35i	−0.661461	+	0.749980i
$176$		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
$181$	−1.42721e36			−1.07555			−0.537777	−	0.843087i	$-0.680736\pi$
$181$	−1.42721e36			−1.07555			−0.537777	+	0.843087i	$0.680736\pi$
$182$		0			0
$183$	−1.70211e36			−1.07590
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	2.59847e36	+	5.24480e35i	0.980232	+	0.197852i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	−1.70526e36	+	2.95359e36i	−0.500000	+	0.866025i
$193$	1.13971e36	−	1.97404e36i	0.307524	−	0.532647i	−0.670296	−	0.742094i	$-0.733833\pi$
$193$	1.13971e36	−	1.97404e36i	0.307524	−	0.532647i	0.977820	+	0.209447i	$0.0671663\pi$
$194$		0			0
$195$		0			0
$196$	−3.77946e36	+	2.86640e36i	−0.796770	+	0.604283i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	4.22546e36	−	7.31872e36i	0.698594	−	1.21000i	−0.270360	−	0.962759i	$-0.587143\pi$
$199$	4.22546e36	−	7.31872e36i	0.698594	−	1.21000i	0.968954	−	0.247241i	$-0.0795241\pi$
$200$		0			0
$201$	−1.14360e35	−	1.98077e35i	−0.0161115	−	0.0279060i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.12692e36	−	1.95189e36i	−0.0918082	−	0.159016i
$209$		0			0
$210$		0			0
$211$	3.03417e37			1.96571			0.982856	−	0.184373i	$-0.0590256\pi$
$211$	3.03417e37			1.96571			0.982856	+	0.184373i	$0.0590256\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	3.10769e37	−	3.52357e37i	1.28553	−	1.45756i
$218$		0			0
$219$	−1.65195e37	−	2.86126e37i	−0.590054	−	1.02200i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	2.75399e37			0.736349			0.368175	−	0.929757i	$-0.379983\pi$
$223$	2.75399e37			0.736349			0.368175	+	0.929757i	$0.379983\pi$
$224$		0			0
$225$	4.31440e37			1.00000
$226$		0			0
$227$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$227$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$228$	7.23004e35	−	1.25228e36i	0.0135576	−	0.0234825i
$229$	−2.14905e37	−	3.72226e37i	−0.375732	−	0.650787i	0.614704	−	0.788758i	$-0.289275\pi$
$229$	−2.14905e37	−	3.72226e37i	−0.375732	−	0.650787i	−0.990436	+	0.137971i	$0.955942\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	8.72540e37			0.880668
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	9.57310e37	−	1.65811e38i	0.739231	−	1.28039i	−0.213611	−	0.976919i	$-0.568523\pi$
$241$	9.57310e37	−	1.65811e38i	0.739231	−	1.28039i	0.952842	−	0.303467i	$-0.0981441\pi$
$242$		0			0
$243$	−7.39044e37	−	1.28006e38i	−0.500000	−	0.866025i
$244$	−1.69828e38			−1.07590
$245$		0			0
$246$		0			0
$247$	4.77800e35	+	8.27573e35i	0.00248940	+	0.00431177i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	2.59261e38	+	5.23298e37i	0.980232	+	0.197852i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.70141e38	+	2.94693e38i	−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
$259$	2.28360e38	+	6.79003e38i	0.556959	+	1.65606i
$260$		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
$268$	−1.14102e37	−	1.97631e37i	−0.0161115	−	0.0279060i
$269$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$269$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$270$		0			0
$271$	−3.66940e38	−	6.35560e38i	−0.433598	−	0.751013i	0.563582	−	0.826060i	$-0.309423\pi$
$271$	−3.66940e38	−	6.35560e38i	−0.433598	−	0.751013i	−0.997180	+	0.0750466i	$0.976089\pi$
$272$		0			0
$273$	−1.15616e38	+	1.31088e38i	−0.121455	+	0.137709i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−9.66328e38	+	1.67373e39i	−0.804356	+	1.39319i	0.112369	+	0.993667i	$0.464156\pi$
$277$	−9.66328e38	+	1.67373e39i	−0.804356	+	1.39319i	−0.916725	+	0.399519i	$0.869177\pi$
$278$		0			0
$279$	−2.61966e39			−1.94347
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.54316e39	+	2.67283e39i	−0.911650	+	1.57902i	−0.0999165	+	0.994996i	$0.531858\pi$
$283$	−1.54316e39	+	2.67283e39i	−0.911650	+	1.57902i	−0.811733	+	0.584028i	$0.801476\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.18396e39	−	2.05067e39i	−0.500000	−	0.866025i
$290$		0			0
$291$	−2.60827e39	+	4.51766e39i	−0.986426	+	1.70854i
$292$	−1.64823e39	−	2.85481e39i	−0.590054	−	1.02200i
$293$		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
$300$	4.30467e39			1.00000
$301$	8.88966e39	+	1.79431e39i	1.95804	+	0.395214i
$302$		0			0
$303$		0			0
$304$	7.21374e37	−	1.24946e38i	0.0135576	−	0.0234825i
$305$		0			0
$306$		0			0
$307$	−1.43223e38			−0.0230038			−0.0115019	−	0.999934i	$-0.503661\pi$
$307$	−1.43223e38			−0.0230038			−0.0115019	+	0.999934i	$0.503661\pi$
$308$		0			0
$309$	3.70463e39			0.536301
$310$		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
$313$	−3.33486e37	−	5.77614e37i	−0.00392978	−	0.00680657i	0.864054	−	0.503399i	$-0.167918\pi$
$313$	−3.33486e37	−	5.77614e37i	−0.00392978	−	0.00680657i	−0.867984	+	0.496593i	$0.834584\pi$
$314$		0			0
$315$		0			0
$316$	8.70573e39			0.880668
$317$		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
$324$	−7.37378e39	−	1.27718e40i	−0.500000	−	0.866025i
$325$	−1.42238e39	+	2.46363e39i	−0.0918082	+	0.159016i
$326$		0			0
$327$	−9.58582e39	−	1.66031e40i	−0.560874	−	0.971462i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.32351e40	−	2.29238e40i	−0.637498	−	1.10418i	−0.985980	−	0.166863i	$-0.946636\pi$
$331$	−1.32351e40	−	2.29238e40i	−0.637498	−	1.10418i	0.348482	−	0.937315i	$-0.386697\pi$
$332$		0			0
$333$	1.99720e40	−	3.45926e40i	0.873603	−	1.51313i
$334$		0			0
$335$		0			0
$336$	2.58677e40	+	5.22118e39i	0.980232	+	0.197852i
$337$	−1.85065e40			−0.668723			−0.334361	−	0.942445i	$-0.608521\pi$
$337$	−1.85065e40			−0.668723			−0.334361	+	0.942445i	$0.608521\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	3.03444e40	+	2.06484e40i	0.826746	+	0.562576i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$	9.32145e40			1.92432			0.962159	−	0.272490i	$-0.0878472\pi$
$349$	9.32145e40			1.92432			0.962159	+	0.272490i	$0.0878472\pi$
$350$		0			0
$351$	9.74598e39			0.183616
$352$		0			0
$353$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$353$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	4.15686e40	−	7.19990e40i	0.499632	−	0.865389i
$362$		0			0
$363$	9.08879e40			1.00000
$364$	−1.15355e40	+	1.30793e40i	−0.121455	+	0.137709i
$365$		0			0
$366$		0			0
$367$	−7.21361e40	+	1.24943e41i	−0.666033	+	1.15360i	0.312971	+	0.949763i	$0.398676\pi$
$367$	−7.21361e40	+	1.24943e41i	−0.666033	+	1.15360i	−0.979004	+	0.203840i	$0.934658\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.61375e41			−1.94347
$373$	1.58690e39	+	2.74860e39i	0.0113034	+	0.0195781i	0.871622	−	0.490179i	$-0.163069\pi$
$373$	1.58690e39	+	2.74860e39i	0.0113034	+	0.0195781i	−0.860318	+	0.509757i	$0.829735\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.75275e41			−1.51894			−0.759468	−	0.650545i	$-0.774541\pi$
$379$	−2.75275e41			−1.51894			−0.759468	+	0.650545i	$0.774541\pi$
$380$		0			0
$381$	−1.54021e41	+	2.66772e41i	−0.781233	+	1.35313i
$382$		0			0
$383$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$383$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−2.52835e41	−	4.37923e41i	−0.998762	−	1.72991i
$388$	−2.60239e41	+	4.50747e41i	−0.986426	+	1.70854i
$389$		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
$397$	3.69470e41	+	6.39941e41i	0.970356	+	1.68071i	0.694479	+	0.719513i	$0.255635\pi$
$397$	3.69470e41	+	6.39941e41i	0.970356	+	1.68071i	0.275877	+	0.961193i	$0.411032\pi$
$398$		0			0
$399$	−1.09675e40	−	2.21370e39i	−0.0265792	−	0.00536480i
$400$	4.29497e41			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
$403$	8.63654e40	−	1.49589e41i	0.178426	−	0.309043i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−4.62140e41	+	8.00450e41i	−0.753703	+	1.30545i	0.192313	+	0.981334i	$0.438401\pi$
$409$	−4.62140e41	+	8.00450e41i	−0.753703	+	1.30545i	−0.946016	+	0.324119i	$0.894932\pi$
$410$		0			0
$411$		0			0
$412$	3.69627e41			0.536301
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	8.15738e41	−	1.41290e42i	0.975829	−	1.69018i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−1.85857e42			−1.90839			−0.954194	−	0.299190i	$-0.903284\pi$
$421$	−1.85857e42			−1.90839			−0.954194	+	0.299190i	$0.903284\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.18886e41	+	1.24551e42i	0.342965	+	1.01977i
$428$		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	$0.666667\pi$
$432$	−7.35716e41	−	1.27430e42i	−0.500000	−	0.866025i
$433$	2.59382e42			1.69877			0.849383	−	0.527777i	$-0.176974\pi$
$433$	2.59382e42			1.69877			0.849383	+	0.527777i	$0.176974\pi$
$434$		0			0
$435$		0			0
$436$	−9.56421e41	−	1.65657e42i	−0.560874	−	0.971462i
$437$		0			0
$438$		0			0
$439$	2.75863e41	+	4.77809e41i	0.144965	+	0.251086i	0.929360	−	0.369175i	$-0.120360\pi$
$439$	2.75863e41	+	4.77809e41i	0.144965	+	0.251086i	−0.784395	+	0.620261i	$0.787027\pi$
$440$		0			0
$441$	−2.55692e41	−	2.03049e42i	−0.124939	−	0.992164i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	1.99270e42	−	3.45146e42i	0.873603	−	1.51313i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	2.58094e42	+	5.20941e41i	0.980232	+	0.197852i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−1.56014e42	−	2.70223e42i	−0.496125	−	0.859314i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	3.52910e42	+	6.11258e42i	0.975000	+	1.68875i	0.679937	+	0.733271i	$0.262007\pi$
$457$	3.52910e42	+	6.11258e42i	0.975000	+	1.68875i	0.295063	+	0.955478i	$0.404659\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	6.17477e42			1.38460			0.692301	−	0.721609i	$-0.256597\pi$
$463$	6.17477e42			1.38460			0.692301	+	0.721609i	$0.256597\pi$
$464$		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	$0.333333\pi$
$468$	9.72400e41			0.183616
$469$	−1.16798e41	+	1.32429e41i	−0.0213143	+	0.0241667i
$470$		0			0
$471$	−5.82435e42	−	1.00881e43i	−0.992917	−	1.71978i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−1.82100e41			−0.0271152
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$479$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$480$		0			0
$481$	1.31688e42	+	2.28091e42i	0.160408	+	0.277835i
$482$		0			0
$483$		0			0
$484$	9.06830e42			1.00000
$485$		0			0
$486$		0			0
$487$	−9.14159e42	+	1.58337e43i	−0.913185	+	1.58168i	−0.103647	+	0.994614i	$0.533051\pi$
$487$	−9.14159e42	+	1.58337e43i	−0.913185	+	1.58168i	−0.809538	+	0.587068i	$0.800282\pi$
$488$		0			0
$489$	−1.53078e43			−1.43209
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$	−2.60786e43			−1.94347
$497$		0			0
$498$		0			0
$499$	1.19381e43	+	2.06774e43i	0.807841	+	1.39922i	0.914356	+	0.404910i	$0.132697\pi$
$499$	1.19381e43	+	2.06774e43i	0.807841	+	1.39922i	−0.106515	+	0.994311i	$0.533969\pi$
$500$		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
$507$	9.20879e42	−	1.59501e43i	0.483143	−	0.836827i
$508$	−1.53674e43	+	2.66171e43i	−0.781233	+	1.35313i
$509$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$509$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$510$		0			0
$511$	−1.68717e43	+	1.91296e43i	−0.780595	+	0.885057i
$512$		0			0
$513$	3.11933e41	+	5.40283e41i	0.0135576	+	0.0234825i
$514$		0			0
$515$		0			0
$516$	−2.52265e43	−	4.36936e43i	−0.998762	−	1.72991i
$517$		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
$523$	2.89818e43	+	5.01980e43i	0.924909	+	1.60199i	0.791707	+	0.610901i	$0.209193\pi$
$523$	2.89818e43	+	5.01980e43i	0.924909	+	1.60199i	0.133202	+	0.991089i	$0.457474\pi$
$524$		0			0
$525$	−1.06176e43	−	3.15704e43i	−0.318771	−	0.947832i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.88045e43	+	3.25703e43i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	−1.09428e42	−	2.20871e41i	−0.0265792	−	0.00536480i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	4.84312e43	+	8.38854e43i	0.899434	+	1.55787i	0.828219	+	0.560405i	$0.189354\pi$
$541$	4.84312e43	+	8.38854e43i	0.899434	+	1.55787i	0.0712152	+	0.997461i	$0.477312\pi$
$542$		0			0
$543$	3.07184e43	−	5.32059e43i	0.537777	−	0.931456i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.05403e43			0.319750			0.159875	−	0.987137i	$-0.448891\pi$
$547$	2.05403e43			0.319750			0.159875	+	0.987137i	$0.448891\pi$
$548$		0			0
$549$	3.66352e43	−	6.34540e43i	0.537949	−	0.931755i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.14730e43	−	6.38477e43i	−0.280731	−	0.834725i
$554$		0			0
$555$		0			0
$556$	8.13899e43	−	1.40971e44i	0.975829	−	1.69018i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$	3.33421e43			0.366778
$560$		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
$567$	−7.54802e43	+	8.55812e43i	−0.661461	+	0.749980i
$568$		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
$571$	1.05755e44	−	1.83173e44i	0.828176	−	1.43444i	−0.0712925	−	0.997455i	$-0.522712\pi$
$571$	1.05755e44	−	1.83173e44i	0.828176	−	1.43444i	0.899468	−	0.436987i	$-0.143954\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−7.34057e43	−	1.27142e44i	−0.500000	−	0.866025i
$577$	−8.34478e43	+	1.44536e44i	−0.552843	+	0.957552i	0.445225	+	0.895419i	$0.353124\pi$
$577$	−8.34478e43	+	1.44536e44i	−0.552843	+	0.957552i	−0.998068	+	0.0621333i	$0.980210\pi$
$578$		0			0
$579$	4.90609e43	+	8.49760e43i	0.307524	+	0.532647i
$580$		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			$\pi$
$588$	−2.55115e43	−	2.02591e44i	−0.124939	−	0.992164i
$589$	1.10569e43			0.0526976
$590$		0			0
$591$		0			0
$592$	1.98821e44	−	3.44368e44i	0.873603	−	1.51313i
$593$		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
$597$	1.81892e44	+	3.15047e44i	0.698594	+	1.21000i
$598$		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
$601$	2.46883e44			0.852118			0.426059	−	0.904695i	$-0.359902\pi$
$601$	2.46883e44			0.852118			0.426059	+	0.904695i	$0.359902\pi$
$602$		0			0
$603$	9.84564e42			0.0322231
$604$	−1.55662e44	−	2.69614e44i	−0.496125	−	0.859314i
$605$		0			0
$606$		0			0
$607$	3.29972e44	+	5.71528e44i	0.971533	+	1.68274i	0.690932	+	0.722920i	$0.257201\pi$
$607$	3.29972e44	+	5.71528e44i	0.971533	+	1.68274i	0.280601	+	0.959824i	$0.409466\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.87605e44	+	6.71351e44i	−0.975038	+	1.68881i	−0.295227	+	0.955427i	$0.595395\pi$
$613$	−3.87605e44	+	6.71351e44i	−0.975038	+	1.68881i	−0.679811	+	0.733388i	$0.737938\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−3.59678e44	+	6.22980e44i	−0.774218	+	1.34098i	0.161016	+	0.986952i	$0.448523\pi$
$619$	−3.59678e44	+	6.22980e44i	−0.774218	+	1.34098i	−0.935233	+	0.354032i	$0.884810\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	9.70208e43			0.183616
$625$	−2.71051e44	−	4.69473e44i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	−5.81122e44	−	1.00653e45i	−0.992917	−	1.71978i
$629$		0			0
$630$		0			0
$631$	−1.15848e45			−1.83408			−0.917039	−	0.398798i	$-0.869428\pi$
$631$	−1.15848e45			−1.83408			−0.917039	+	0.398798i	$0.869428\pi$
$632$		0			0
$633$	−6.53054e44	+	1.13112e45i	−0.982856	+	1.70236i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	1.24376e44	+	5.23409e43i	0.169241	+	0.0712214i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$	−4.74213e44			−0.555390			−0.277695	−	0.960669i	$-0.589571\pi$
$643$	−4.74213e44			−0.555390			−0.277695	+	0.960669i	$0.589571\pi$
$644$		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
$651$	6.44692e44	+	1.91692e45i	0.619522	+	1.84208i
$652$	−1.52733e45			−1.43209
$653$		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
$657$	1.42222e45			1.18011
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1.20637e45	−	2.08949e45i	0.908357	−	1.57332i	0.0920114	−	0.995758i	$-0.470670\pi$
$661$	1.20637e45	−	2.08949e45i	0.908357	−	1.57332i	0.816346	−	0.577563i	$-0.195996\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−5.92750e44	+	1.02667e45i	−0.368175	+	0.637697i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	3.41061e45			1.92571			0.962854	−	0.270022i	$-0.0870310\pi$
$673$	3.41061e45			1.92571			0.962854	+	0.270022i	$0.0870310\pi$
$674$		0			0
$675$	−9.28604e44	+	1.60839e45i	−0.500000	+	0.866025i
$676$	9.18803e44	−	1.59141e45i	0.483143	−	0.836827i
$677$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$677$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$678$		0			0
$679$	3.94766e45	+	7.96804e44i	1.93385	+	0.390333i
$680$		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	$0.666667\pi$
$684$	3.11230e43	+	5.39065e43i	0.0135576	+	0.0234825i
$685$		0			0
$686$		0			0
$687$	1.85019e45			0.751464
$688$	−2.51697e45	−	4.35951e45i	−0.998762	−	1.72991i
$689$		0			0
$690$		0			0
$691$	−2.06721e45	−	3.58052e45i	−0.765132	−	1.32525i	−0.940177	−	0.340688i	$-0.889340\pi$
$691$	−2.06721e45	−	3.58052e45i	−0.765132	−	1.32525i	0.175044	−	0.984561i	$-0.443993\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.05937e45	−	3.14992e45i	−0.318771	−	0.947832i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−8.42971e43	+	1.46007e44i	−0.0236880	+	0.0410287i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−1.98784e45	−	3.44305e45i	−0.487576	−	0.844507i	0.512322	−	0.858794i	$-0.328786\pi$
$709$	−1.98784e45	−	3.44305e45i	−0.487576	−	0.844507i	−0.999898	+	0.0142869i	$0.995452\pi$
$710$		0			0
$711$	−1.87800e45	+	3.25279e45i	−0.440334	+	0.762681i
$712$		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
$721$	−9.11700e44	−	2.71084e45i	−0.170957	−	0.508323i
$722$		0			0
$723$	4.12091e45	+	7.13762e45i	0.739231	+	1.28039i
$724$	3.06492e45	−	5.30860e45i	0.537777	−	0.931456i
$725$		0			0
$726$		0			0
$727$	4.21369e45			0.692010			0.346005	−	0.938233i	$-0.387538\pi$
$727$	4.21369e45			0.692010			0.346005	+	0.938233i	$0.387538\pi$
$728$		0			0
$729$	6.36269e45			1.00000
$730$		0			0
$731$		0			0
$732$	3.65526e45	−	6.33110e45i	0.537949	−	0.931755i
$733$	3.71231e45	+	6.42992e45i	0.534541	+	0.925852i	0.999185	+	0.0403548i	$0.0128488\pi$
$733$	3.71231e45	+	6.42992e45i	0.534541	+	0.925852i	−0.464644	+	0.885497i	$0.653818\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−4.83528e45	+	8.37495e45i	−0.611098	+	1.05845i	0.379958	+	0.925004i	$0.375938\pi$
$739$	−4.83528e45	+	8.37495e45i	−0.611098	+	1.05845i	−0.991056	+	0.133448i	$0.957395\pi$
$740$		0			0
$741$	−4.11354e43			−0.00497880
$742$		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
$751$	3.41961e45	+	5.92294e45i	0.333994	+	0.578494i	0.983291	−	0.182041i	$-0.0582703\pi$
$751$	3.41961e45	+	5.92294e45i	0.333994	+	0.578494i	−0.649297	+	0.760535i	$0.724937\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−7.53100e45	+	8.53883e45i	−0.661461	+	0.749980i
$757$	−2.32391e46			−1.99842			−0.999208	−	0.0397813i	$-0.987334\pi$
$757$	−2.32391e46			−1.99842			−0.999208	+	0.0397813i	$0.987334\pi$
$758$		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
$763$	−9.79021e45	+	1.11004e46i	−0.741992	+	0.841289i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−7.32402e45	−	1.26856e46i	−0.500000	−	0.866025i
$769$	−2.97413e46			−1.98856			−0.994281	−	0.106798i	$-0.965940\pi$
$769$	−2.97413e46			−1.98856			−0.994281	+	0.106798i	$0.965940\pi$
$770$		0			0
$771$		0			0
$772$	4.89503e45	+	8.47845e45i	0.307524	+	0.532647i
$773$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$773$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$774$		0			0
$775$	1.64579e46	+	2.85059e46i	0.971734	+	1.68309i
$776$		0			0
$777$	−3.02280e46	−	6.10128e45i	−1.71267	−	0.345688i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−2.54540e45	−	2.02135e46i	−0.124939	−	0.992164i
$785$		0			0
$786$		0			0
$787$	8.55652e45	−	1.48203e46i	0.395095	−	0.684325i	−0.598018	−	0.801482i	$-0.704045\pi$
$787$	8.55652e45	−	1.48203e46i	0.395095	−	0.684325i	0.993113	+	0.117158i	$0.0373783\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.41559e45	+	4.18393e45i	0.0987762	+	0.171085i
$794$		0			0
$795$		0			0
$796$	1.81482e46	+	3.14336e46i	0.698594	+	1.21000i
$797$		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
$804$	9.82344e44			0.0322231
$805$		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
$811$	6.41952e46			1.83303			0.916513	−	0.400004i	$-0.130991\pi$
$811$	6.41952e46			1.83303			0.916513	+	0.400004i	$0.130991\pi$
$812$		0			0
$813$	3.15912e46			0.867195
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.06716e45	+	1.84837e45i	0.0270817	+	0.0469068i
$818$		0			0
$819$	−2.39846e45	−	7.13157e45i	−0.0585316	−	0.174037i
$820$		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
$823$	−7.67678e45	+	1.32966e46i	−0.173293	+	0.300153i	−0.939569	−	0.342359i	$-0.888774\pi$
$823$	−7.67678e45	+	1.32966e46i	−0.173293	+	0.300153i	0.766276	+	0.642511i	$0.222107\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	3.61366e46	−	6.25904e46i	0.726231	−	1.25787i	−0.232234	−	0.972660i	$-0.574603\pi$
$829$	3.61366e46	−	6.25904e46i	0.726231	−	1.25787i	0.958465	−	0.285210i	$-0.0920632\pi$
$830$		0			0
$831$	−4.15972e46	−	7.20485e46i	−0.804356	−	1.39319i
$832$	9.68021e45			0.183616
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5.63839e46	−	9.76597e46i	0.971734	−	1.68309i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	6.26233e46			1.00000
$842$		0			0
$843$		0			0
$844$	−6.51582e46	+	1.12857e47i	−0.982856	+	1.70236i
$845$		0			0
$846$		0			0
$847$	−2.23673e46	−	6.65068e46i	−0.318771	−	0.947832i
$848$		0			0
$849$	−6.64280e46	−	1.15057e47i	−0.911650	−	1.57902i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	−2.18058e46			−0.277579			−0.138790	−	0.990322i	$-0.544321\pi$
$853$	−2.18058e46			−0.277579			−0.138790	+	0.990322i	$0.544321\pi$
$854$		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
$859$	8.65530e46	+	1.49914e47i	0.984898	+	1.70589i	0.642388	+	0.766380i	$0.277944\pi$
$859$	8.65530e46	+	1.49914e47i	0.984898	+	1.70589i	0.342510	+	0.939514i	$0.388723\pi$
$860$		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
$867$	1.01931e47			1.00000
$868$	6.43239e46	+	1.91260e47i	0.619522	+	1.84208i
$869$		0			0
$870$		0			0
$871$	−3.24593e44	+	5.62211e44i	−0.00295834	+	0.00512400i
$872$		0			0
$873$	−1.12277e47	−	1.94470e47i	−0.986426	−	1.70854i
$874$		0			0
$875$		0			0
$876$	1.41902e47			1.18011
$877$	−9.23396e46	−	1.59937e47i	−0.754040	−	1.30604i	−0.945850	−	0.324604i	$-0.894769\pi$
$877$	−9.23396e46	−	1.59937e47i	−0.754040	−	1.30604i	0.191809	−	0.981432i	$-0.438564\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−1.66493e47			−1.21906			−0.609530	−	0.792763i	$-0.708642\pi$
$883$	−1.66493e47			−1.21906			−0.609530	+	0.792763i	$0.708642\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$887$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$888$		0			0
$889$	2.33113e47	+	4.70521e46i	1.53158	+	0.309137i
$890$		0			0
$891$		0			0
$892$	−5.91414e46	+	1.02436e47i	−0.368175	+	0.637697i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−9.26510e46	+	1.60476e47i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	−2.58226e47	+	2.92783e47i	−1.32128	+	1.49810i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−2.05473e47	+	3.55890e47i	−0.979577	+	1.69668i	−0.315659	+	0.948873i	$0.602226\pi$
$907$	−2.05473e47	+	3.55890e47i	−0.979577	+	1.69668i	−0.663919	+	0.747805i	$0.731108\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	3.10528e45	+	5.37850e45i	0.0135576	+	0.0234825i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.84602e47			0.751464
$917$		0			0
$918$		0			0
$919$	−2.23623e47	−	3.87326e47i	−0.863909	−	1.49634i	−0.868126	−	0.496344i	$-0.834675\pi$
$919$	−2.23623e47	−	3.87326e47i	−0.863909	−	1.49634i	0.00421644	−	0.999991i	$-0.498658\pi$
$920$		0			0
$921$	3.08265e45	−	5.33930e45i	0.0115019	−	0.0199219i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−5.01894e47			−1.74721
$926$		0			0
$927$	−7.97360e46	+	1.38107e47i	−0.268151	+	0.464450i
$928$		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
$931$	1.07921e45	+	8.57021e45i	0.00338776	+	0.0269028i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	5.72662e47			1.62205			0.811027	−	0.585009i	$-0.198909\pi$
$937$	5.72662e47			1.62205			0.811027	+	0.585009i	$0.198909\pi$
$938$		0			0
$939$	2.87109e45			0.00785955
$940$		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	$0.333333\pi$
$948$	−1.87377e47	+	3.24546e47i	−0.440334	+	0.762681i
$949$	−4.68881e46	+	8.12125e46i	−0.108344	+	0.187657i
$950$		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
$961$	−7.34735e47	−	1.27260e48i	−1.38853	−	2.40501i
$962$		0			0
$963$		0			0
$964$	4.11162e47	+	7.12153e47i	0.739231	+	1.28039i
$965$		0			0
$966$		0			0
$967$	1.16909e48			1.99998			0.999989	−	0.00469250i	$-0.00149367\pi$
$967$	1.16909e48			1.99998			0.999989	+	0.00469250i	$0.00149367\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$971$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$972$	6.34834e47			1.00000
$973$	−1.23463e48	−	2.49201e47i	−1.91308	−	0.386139i
$974$		0			0
$975$	−6.12287e46	−	1.06051e47i	−0.0918082	−	0.159016i
$976$	3.64702e47	−	6.31682e47i	0.537949	−	0.931755i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	8.25277e47			1.12175
$982$		0			0
$983$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$983$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−4.10427e45			−0.00497880
$989$		0			0
$990$		0			0
$991$	6.75119e47	−	1.16934e48i	0.780193	−	1.35133i	−0.151636	−	0.988436i	$-0.548454\pi$
$991$	6.75119e47	−	1.16934e48i	0.780193	−	1.35133i	0.931829	−	0.362897i	$-0.118212\pi$
$992$		0			0
$993$	1.13945e48			1.27500
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.42742e47	−	1.63288e48i	0.989168	−	1.71329i	0.367465	−	0.930037i	$-0.380226\pi$
$997$	9.42742e47	−	1.63288e48i	0.989168	−	1.71329i	0.621704	−	0.783252i	$-0.286441\pi$
$998$		0			0
$999$	8.59730e47	+	1.48910e48i	0.873603	+	1.51313i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.33.h.a.11.1	yes	2
3.2	odd	2	CM	21.33.h.a.11.1	yes	2
7.2	even	3	inner	21.33.h.a.2.1	✓	2
21.2	odd	6	inner	21.33.h.a.2.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.33.h.a.2.1	✓	2	7.2	even	3	inner
21.33.h.a.2.1	✓	2	21.2	odd	6	inner
21.33.h.a.11.1	yes	2	1.1	even	1	trivial
21.33.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:	\( k \)	\(=\)	\( 33 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.15234e7 + 3.72796e7i) q^{3} +(-2.14748e9 + 3.71955e9i) q^{4} +(3.25760e13 + 6.57520e12i) q^{7} +(-9.26510e14 - 1.60476e15i) q^{9} +O(q^{10})\)	\(q+(-2.15234e7 + 3.72796e7i) q^{3} +(-2.14748e9 + 3.71955e9i) q^{4} +(3.25760e13 + 6.57520e12i) q^{7} +(-9.26510e14 - 1.60476e15i) q^{9} +(-9.24421e16 - 1.60114e17i) q^{12} +1.22181e17 q^{13} +(-9.22337e18 - 1.59753e19i) q^{16} +(3.91058e18 + 6.77332e18i) q^{19} +(-9.46265e20 + 1.07290e21i) q^{21} +(-1.16415e22 + 2.01637e22i) q^{25} +7.97664e22 q^{27} +(-9.44132e22 + 1.07048e23i) q^{28} +(7.06862e23 - 1.22432e24i) q^{31} +7.95866e24 q^{36} +(1.07781e25 + 1.86682e25i) q^{37} +(-2.62975e24 + 4.55487e24i) q^{39} +2.72890e26 q^{43} +7.94072e26 q^{48} +(1.01796e27 + 4.28387e26i) q^{49} +(-2.62383e26 + 4.54460e26i) q^{52} -3.36675e26 q^{57} +(1.97705e28 + 3.42436e28i) q^{61} +(-1.96303e28 - 5.83687e28i) q^{63} +7.92282e28 q^{64} +(-2.65665e27 + 4.60145e27i) q^{67} +(-3.83758e29 + 6.64688e29i) q^{73} +(-5.01130e29 - 8.67982e29i) q^{75} -3.35916e28 q^{76} +(-1.01348e30 - 1.75540e30i) q^{79} +(-1.71684e30 + 2.97366e30i) q^{81} +(-1.95861e30 - 5.82371e30i) q^{84} +(3.98018e30 + 8.03367e29i) q^{91} +(3.04281e31 + 5.27030e31i) q^{93} +1.21183e32 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 43046721 q^{3} - 4294967296 q^{4} + 65151959156159 q^{7} - 18\!\cdots\!41 q^{9}+O(q^{10}) \) 2 * q - 43046721 * q^3 - 4294967296 * q^4 + 65151959156159 * q^7 - 1853020188851841 * q^9	\( 2 q - 43046721 q^{3} - 4294967296 q^{4} + 65151959156159 q^{7} - 18\!\cdots\!41 q^{9}+ \cdots + 24\!\cdots\!28 q^{97}+O(q^{100}) \) 2 * q - 43046721 * q^3 - 4294967296 * q^4 + 65151959156159 * q^7 - 1853020188851841 * q^9 - 184884258895036416 * q^12 + 244362804407950078 * q^13 - 18446744073709551616 * q^16 + 7821153231347348161 * q^19 - 1892530001344659310206 * q^21 - 23283064365386962890625 * q^25 + 159532886153745019726722 * q^27 - 188826332729829706694656 * q^28 + 1413723680265064241474113 * q^31 + 15917322219892801768783872 * q^36 + 21556186719627646822930561 * q^37 - 5259508732063298594797119 * q^39 + 545779797112520941017987454 * q^43 + 1588143690998757006898102272 * q^48 + 2035922433397969262737034879 * q^49 - 524765126645495113705324544 * q^52 - 673350002096115500752860162 * q^57 + 39541057150634567761997420926 * q^61 - 39260684707600007410943504193 * q^63 + 158456325028528675187087900672 * q^64 - 5313292745304228970622516159 * q^67 - 767515778351719658853870420479 * q^73 - 1002259575761854648590087890625 * q^75 - 67183194691283164735641485312 * q^76 - 2026961502021606539826450603839 * q^79 - 3433683820292512484657849089281 * q^81 - 3917217221671991101932246663168 * q^84 + 7960357726035616964721939115201 * q^91 + 60856168835463426449812771033473 * q^93 + 242366393497161848998074563303428 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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