LMFDB - Embedded newform 21.32.g.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,32,Mod(5,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

N = Newforms(chi, 32, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 32);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 32 $
Character orbit:	$[\chi]$	$=$	21.g (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$127.841978920$
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			5.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.5
Dual form			21.32.g.a.17.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.15234e7 - 1.24265e7i) q^{3} +(-1.07374e9 - 1.85978e9i) q^{4} +(4.76727e12 - 1.16210e13i) q^{7} +(3.08837e14 - 5.34921e14i) q^{9} +O(q^{10})$	\(q+(2.15234e7 - 1.24265e7i) q^{3} +(-1.07374e9 - 1.85978e9i) q^{4} +(4.76727e12 - 1.16210e13i) q^{7} +(3.08837e14 - 5.34921e14i) q^{9} +(-4.62211e16 - 2.66857e16i) q^{12} -3.48649e17i q^{13} +(-2.30584e18 + 3.99384e18i) q^{16} +(-9.29484e19 - 5.36638e19i) q^{19} +(-4.18015e19 - 3.09364e20i) q^{21} +(2.32831e21 + 4.03275e21i) q^{25} -1.53511e22i q^{27} +(-2.67313e22 + 3.61195e21i) q^{28} +(1.37433e23 - 7.93468e22i) q^{31} -1.32644e24 q^{36} +(1.12935e24 - 1.95608e24i) q^{37} +(-4.33249e24 - 7.50409e24i) q^{39} -1.77815e25 q^{43} +1.14614e26i q^{48} +(-1.12322e26 - 1.10801e26i) q^{49} +(-6.48408e26 + 3.74359e26i) q^{52} -2.66742e27 q^{57} +(8.10239e27 + 4.67792e27i) q^{61} +(-4.74403e27 - 6.13911e27i) q^{63} +9.90352e27 q^{64} +(-1.51355e28 - 2.62155e28i) q^{67} +(1.14574e29 - 6.61496e28i) q^{73} +(1.00226e29 + 5.78655e28i) q^{75} +2.30484e29i q^{76} +(-7.51860e28 + 1.30226e29i) q^{79} +(-1.90760e29 - 3.30406e29i) q^{81} +(-5.30464e29 + 4.09919e29i) q^{84} +(-4.05166e30 - 1.66210e30i) q^{91} +(1.97201e30 - 3.41562e30i) q^{93} +1.01112e31i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 43046721 q^{3} - 2147483648 q^{4} + 9534530086957 q^{7} + 617673396283947 q^{9}+O(q^{10}) $ 2 * q + 43046721 * q^3 - 2147483648 * q^4 + 9534530086957 * q^7 + 617673396283947 * q^9	$ 2 q + 43046721 q^{3} - 2147483648 q^{4} + 9534530086957 q^{7} + 617673396283947 q^{9} - 92\!\cdots\!08 q^{12}+ \cdots + 39\!\cdots\!05 q^{93}+O(q^{100}) $ 2 * q + 43046721 * q^3 - 2147483648 * q^4 + 9534530086957 * q^7 + 617673396283947 * q^9 - 92442129447518208 * q^12 - 4611686018427387904 * q^16 - 185896844889165280341 * q^19 - 83602991755761479406 * q^21 + 4656612873077392578125 * q^25 - 53462671458706502189056 * q^28 + 274865422539577564919715 * q^31 - 2652887036648800294797312 * q^36 + 2258691937157207814386543 * q^37 - 8664977206989331233197553 * q^39 - 35563070302396847726017330 * q^43 - 224643500090603355248565637 * q^49 - 1296816326309892463150497792 * q^52 - 5334833077816115830483874574 * q^57 + 16204788248998670740089934212 * q^61 - 9488060241657771257110507005 * q^63 + 19807040628566084398385987584 * q^64 - 30271048416507274043668039127 * q^67 + 229148784216062228941507200789 * q^73 + 200451915152370929718017578125 * q^75 - 150372080821436019808890048863 * q^79 - 381520424476945831628649898809 * q^81 - 1060928322239112616902754566144 * q^84 - 8103319462859891108445297803691 * q^91 + 3944018385536102298319453001505 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	2.15234e7	−	1.24265e7i	0.866025	−	0.500000i
$3$	2.15234e7	−	1.24265e7i	0.866025	−	0.500000i
$4$	−1.07374e9	−	1.85978e9i	−0.500000	−	0.866025i
$5$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$5$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$6$		0			0
$7$	4.76727e12	−	1.16210e13i	0.379533	−	0.925178i
$7$	4.76727e12	−	1.16210e13i	0.379533	−	0.925178i
$8$		0			0
$9$	3.08837e14	−	5.34921e14i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	−4.62211e16	−	2.66857e16i	−0.866025	−	0.500000i
$13$		−	3.48649e17i		−	1.88915i	−0.328298	−	0.944574i	$-0.606475\pi$
$13$		−	3.48649e17i		−	1.88915i	0.328298	−	0.944574i	$-0.393525\pi$
$14$		0			0
$15$		0			0
$16$	−2.30584e18	+	3.99384e18i	−0.500000	+	0.866025i
$17$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$17$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$18$		0			0
$19$	−9.29484e19	−	5.36638e19i	−1.40463	−	0.810962i	−0.409765	−	0.912191i	$-0.634389\pi$
$19$	−9.29484e19	−	5.36638e19i	−1.40463	−	0.810962i	−0.994863	+	0.101229i	$0.967722\pi$
$20$		0			0
$21$	−4.18015e19	−	3.09364e20i	−0.133904	−	0.990994i
$22$		0			0
$23$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$23$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$24$		0			0
$25$	2.32831e21	+	4.03275e21i	0.500000	+	0.866025i
$26$		0			0
$27$		−	1.53511e22i		−	1.00000i
$28$	−2.67313e22	+	3.61195e21i	−0.990994	+	0.133904i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	1.37433e23	−	7.93468e22i	1.05192	−	0.607328i	0.128736	−	0.991679i	$-0.458908\pi$
$31$	1.37433e23	−	7.93468e22i	1.05192	−	0.607328i	0.923187	+	0.384351i	$0.125575\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.32644e24			−1.00000
$37$	1.12935e24	−	1.95608e24i	0.556801	−	0.964408i	−0.440960	−	0.897527i	$-0.645362\pi$
$37$	1.12935e24	−	1.95608e24i	0.556801	−	0.964408i	0.997761	−	0.0668813i	$-0.0213049\pi$
$38$		0			0
$39$	−4.33249e24	−	7.50409e24i	−0.944574	−	1.63605i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−1.77815e25			−0.853509			−0.426755	−	0.904367i	$-0.640343\pi$
$43$	−1.77815e25			−0.853509			−0.426755	+	0.904367i	$0.640343\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$47$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$48$			1.14614e26i			1.00000i
$49$	−1.12322e26	−	1.10801e26i	−0.711909	−	0.702271i
$50$		0			0
$51$		0			0
$52$	−6.48408e26	+	3.74359e26i	−1.63605	+	0.944574i
$53$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$53$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−2.66742e27			−1.62192
$58$		0			0
$59$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$59$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$60$		0			0
$61$	8.10239e27	+	4.67792e27i	1.72187	+	0.994123i	0.915063	+	0.403312i	$0.132141\pi$
$61$	8.10239e27	+	4.67792e27i	1.72187	+	0.994123i	0.806810	+	0.590811i	$0.201192\pi$
$62$		0			0
$63$	−4.74403e27	−	6.13911e27i	−0.611461	−	0.791274i
$64$	9.90352e27			1.00000
$65$		0			0
$66$		0			0
$67$	−1.51355e28	−	2.62155e28i	−0.751343	−	1.30136i	−0.947172	−	0.320726i	$-0.896073\pi$
$67$	−1.51355e28	−	2.62155e28i	−0.751343	−	1.30136i	0.195830	−	0.980638i	$-0.437260\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1.14574e29	−	6.61496e28i	1.50516	−	0.869006i	0.505180	−	0.863014i	$-0.331426\pi$
$73$	1.14574e29	−	6.61496e28i	1.50516	−	0.869006i	0.999982	−	0.00599159i	$-0.00190719\pi$
$74$		0			0
$75$	1.00226e29	+	5.78655e28i	0.866025	+	0.500000i
$76$			2.30484e29i			1.62192i
$77$		0			0
$78$		0			0
$79$	−7.51860e28	+	1.30226e29i	−0.290347	+	0.502896i	−0.973892	−	0.227013i	$-0.927104\pi$
$79$	−7.51860e28	+	1.30226e29i	−0.290347	+	0.502896i	0.683545	+	0.729909i	$0.260437\pi$
$80$		0			0
$81$	−1.90760e29	−	3.30406e29i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−5.30464e29	+	4.09919e29i	−0.791274	+	0.611461i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$89$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$90$		0			0
$91$	−4.05166e30	−	1.66210e30i	−1.74780	−	0.716994i
$92$		0			0
$93$	1.97201e30	−	3.41562e30i	0.607328	−	1.05192i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.01112e31i			1.62122i	0.585585	+	0.810611i	$0.300865\pi$
$97$			1.01112e31i			1.62122i	−0.585585	+	0.810611i	$0.699135\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e30	−	8.66025e30i	0.500000	−	0.866025i
$101$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$101$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$102$		0			0
$103$	2.20558e30	+	1.27339e30i	0.139491	+	0.0805351i	0.568121	−	0.822945i	$-0.307670\pi$
$103$	2.20558e30	+	1.27339e30i	0.139491	+	0.0805351i	−0.428630	+	0.903480i	$0.641004\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$107$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$108$	−2.85495e31	+	1.64831e31i	−0.866025	+	0.500000i
$109$	3.56465e31	+	6.17415e31i	0.937359	+	1.62355i	0.770373	+	0.637594i	$0.220070\pi$
$109$	3.56465e31	+	6.17415e31i	0.937359	+	1.62355i	0.166986	+	0.985959i	$0.446596\pi$
$110$		0			0
$111$		−	5.61354e31i		−	1.11360i
$112$	3.54200e31	+	4.58360e31i	0.611461	+	0.791274i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−1.86499e32	−	1.07675e32i	−1.63605	−	0.944574i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−9.59717e31	+	1.66228e32i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−2.95135e32	−	1.70396e32i	−1.05192	−	0.607328i
$125$		0			0
$126$		0			0
$127$	−2.62351e32			−0.645544			−0.322772	−	0.946477i	$-0.604615\pi$
$127$	−2.62351e32			−0.645544			−0.322772	+	0.946477i	$0.604615\pi$
$128$		0			0
$129$	−3.82718e32	+	2.20963e32i	−0.739161	+	0.426755i
$130$		0			0
$131$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$131$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$132$		0			0
$133$	−1.06674e33	+	8.24328e32i	−1.28339	+	0.991744i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$137$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$138$		0			0
$139$			3.28345e33i			1.99343i	0.0810055	+	0.996714i	$0.474187\pi$
$139$			3.28345e33i			1.99343i	−0.0810055	+	0.996714i	$0.525813\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.42426e33	+	2.46689e33i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	−3.79441e33	−	9.89045e32i	−0.967667	−	0.252230i
$148$	−4.85050e33			−1.11360
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−5.57286e33	−	9.65247e33i	−0.937422	−	1.62366i	−0.770257	−	0.637734i	$-0.779872\pi$
$151$	−5.57286e33	−	9.65247e33i	−0.937422	−	1.62366i	−0.167165	−	0.985929i	$-0.553461\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−9.30395e33	+	1.61149e34i	−0.944574	+	1.63605i
$157$	−1.67627e34	+	9.67797e33i	−1.54135	+	0.889897i	−0.542592	+	0.839996i	$0.682557\pi$
$157$	−1.67627e34	+	9.67797e33i	−1.54135	+	0.889897i	−0.998754	+	0.0499006i	$0.984110\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−8.52991e33	+	1.47742e34i	−0.438568	+	0.759622i	−0.997579	−	0.0695382i	$-0.977847\pi$
$163$	−8.52991e33	+	1.47742e34i	−0.438568	+	0.759622i	0.559011	+	0.829160i	$0.311181\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0			−	1.00000i	$-0.5\pi$
$167$		0			0				1.00000i	$0.5\pi$
$168$		0			0
$169$	−8.74959e34			−2.56888
$170$		0			0
$171$	−5.74118e34	+	3.31467e34i	−1.40463	+	0.810962i
$172$	1.90928e34	+	3.30697e34i	0.426755	+	0.739161i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	5.79643e34	−	7.83218e33i	0.990994	−	0.133904i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$179$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$180$		0			0
$181$		−	5.91650e34i		−	0.599856i	−0.953962	−	0.299928i	$-0.903037\pi$
$181$		−	5.91650e34i		−	0.599856i	0.953962	−	0.299928i	$-0.0969626\pi$
$182$		0			0
$183$	2.32521e35			1.98825
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.78395e35	−	7.31826e34i	−0.925178	−	0.379533i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	2.13157e35	−	1.23066e35i	0.866025	−	0.500000i
$193$	−3.69527e34	−	6.40039e34i	−0.138518	−	0.239921i	0.788418	−	0.615140i	$-0.210901\pi$
$193$	−3.69527e34	−	6.40039e34i	−0.138518	−	0.239921i	−0.926936	+	0.375220i	$0.877567\pi$
$194$		0			0
$195$		0			0
$196$	−8.54607e34	+	3.27865e35i	−0.252230	+	0.967667i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	4.63657e35	−	2.67693e35i	1.08137	−	0.624329i	0.150105	−	0.988670i	$-0.452039\pi$
$199$	4.63657e35	−	2.67693e35i	1.08137	−	0.624329i	0.931266	+	0.364341i	$0.118706\pi$
$200$		0			0
$201$	−6.51535e35	−	3.76164e35i	−1.30136	−	0.751343i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.39245e36	+	8.03929e35i	1.63605	+	0.944574i
$209$		0			0
$210$		0			0
$211$	1.47745e36			1.39039			0.695193	−	0.718823i	$-0.255319\pi$
$211$	1.47745e36			1.39039			0.695193	+	0.718823i	$0.255319\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.66914e35	−	1.97538e36i	−0.162647	−	1.20372i
$218$		0			0
$219$	1.64402e36	−	2.84752e36i	0.869006	−	1.50516i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	5.00788e36i		−	1.99953i	−0.0216490	−	0.999766i	$-0.506892\pi$
$223$		−	5.00788e36i		−	1.99953i	0.0216490	−	0.999766i	$-0.493108\pi$
$224$		0			0
$225$	2.87627e36			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	2.86412e36	+	4.96080e36i	0.810962	+	1.40463i
$229$	3.79622e36	+	2.19175e36i	1.00439	+	0.579884i	0.909544	−	0.415608i	$-0.136431\pi$
$229$	3.79622e36	+	2.19175e36i	1.00439	+	0.579884i	0.0948450	+	0.995492i	$0.469764\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			3.73720e36i			0.580694i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−5.14584e36	+	2.97095e36i	−0.616868	+	0.356149i	−0.775649	−	0.631165i	$-0.782577\pi$
$241$	−5.14584e36	+	2.97095e36i	−0.616868	+	0.356149i	0.158781	+	0.987314i	$0.449244\pi$
$242$		0			0
$243$	−8.21160e36	−	4.74097e36i	−0.866025	−	0.500000i
$244$		−	2.00915e37i		−	1.98825i
$245$		0			0
$246$		0			0
$247$	−1.87098e37	+	3.24063e37i	−1.53203	+	2.65355i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−6.32351e36	+	1.54147e37i	−0.379533	+	0.925178i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.06338e37	−	1.84183e37i	−0.500000	−	0.866025i
$257$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$257$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$258$		0			0
$259$	−1.73478e37	−	2.24493e37i	−0.680925	−	0.881165i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$263$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	−3.25033e37	+	5.62974e37i	−0.751343	+	1.30136i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	8.85449e37	+	5.11214e37i	1.72242	+	0.994441i	0.913860	+	0.406029i	$0.133087\pi$
$271$	8.85449e37	+	5.11214e37i	1.72242	+	0.994441i	0.808561	+	0.588412i	$0.200247\pi$
$272$		0			0
$273$	−1.07859e38	+	1.45740e37i	−1.87214	+	0.252964i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.84511e37	+	6.65993e37i	0.532687	+	0.922642i	0.999271	+	0.0381648i	$0.0121512\pi$
$277$	3.84511e37	+	6.65993e37i	0.532687	+	0.922642i	−0.466584	+	0.884477i	$0.654515\pi$
$278$		0			0
$279$		−	9.80208e37i		−	1.21466i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.70308e38	+	9.83275e37i	−1.69257	+	0.977204i	−0.740132	+	0.672462i	$0.765237\pi$
$283$	−1.70308e38	+	9.83275e37i	−1.69257	+	0.977204i	−0.952435	+	0.304742i	$0.901430\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	6.96445e37	−	1.20628e38i	0.500000	−	0.866025i
$290$		0			0
$291$	1.25647e38	+	2.17628e38i	0.810611	+	1.40402i
$292$	−2.46047e38	−	1.42055e38i	−1.50516	−	0.869006i
$293$		0			0			−	1.00000i	$-0.5\pi$
$293$		0			0				1.00000i	$0.5\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$		−	2.48530e38i		−	1.00000i
$301$	−8.47693e37	+	2.06640e38i	−0.323935	+	0.789648i
$302$		0			0
$303$		0			0
$304$	4.28649e38	−	2.47481e38i	1.40463	−	0.810962i
$305$		0			0
$306$		0			0
$307$		−	7.10170e38i		−	1.99856i	−0.0379356	−	0.999280i	$-0.512078\pi$
$307$		−	7.10170e38i		−	1.99856i	0.0379356	−	0.999280i	$-0.487922\pi$
$308$		0			0
$309$	6.32953e37			0.161070
$310$		0			0
$311$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$311$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$312$		0			0
$313$	8.22269e38	+	4.74737e38i	1.71426	+	0.989728i	0.928611	+	0.371054i	$0.121003\pi$
$313$	8.22269e38	+	4.74737e38i	1.71426	+	0.989728i	0.785648	+	0.618674i	$0.212330\pi$
$314$		0			0
$315$		0			0
$316$	3.22922e38			0.580694
$317$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$317$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−4.09654e38	+	7.09542e38i	−0.500000	+	0.866025i
$325$	1.40601e39	−	8.11761e38i	1.63605	−	0.944574i
$326$		0			0
$327$	1.53446e39	+	8.85923e38i	1.62355	+	0.937359i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.58859e38	−	1.48759e39i	0.752641	−	1.30361i	−0.193897	−	0.981022i	$-0.562113\pi$
$331$	8.58859e38	−	1.48759e39i	0.752641	−	1.30361i	0.946538	−	0.322591i	$-0.104554\pi$
$332$		0			0
$333$	−6.97567e38	−	1.20822e39i	−0.556801	−	0.964408i
$334$		0			0
$335$		0			0
$336$	1.33194e39	+	5.46397e38i	0.925178	+	0.379533i
$337$	−2.67775e39			−1.77626			−0.888130	−	0.459592i	$-0.847996\pi$
$337$	−2.67775e39			−1.77626			−0.888130	+	0.459592i	$0.847996\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.82309e39	+	7.77077e38i	−0.919919	+	0.392108i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$347$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$348$		0			0
$349$		−	4.09665e39i		−	1.57992i	−0.613160	−	0.789959i	$-0.710102\pi$
$349$		−	4.09665e39i		−	1.57992i	0.613160	−	0.789959i	$-0.289898\pi$
$350$		0			0
$351$	−5.35213e39			−1.88915
$352$		0			0
$353$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$353$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	3.57017e39	+	6.18372e39i	0.815319	+	1.41217i
$362$		0			0
$363$			4.77038e39i			1.00000i
$364$	1.25930e39	+	9.31984e39i	0.252964	+	1.87214i
$365$		0			0
$366$		0			0
$367$	−4.01891e39	+	2.32032e39i	−0.710861	+	0.410416i	−0.811380	−	0.584520i	$-0.801283\pi$
$367$	−4.01891e39	+	2.32032e39i	−0.710861	+	0.410416i	0.100519	+	0.994935i	$0.467950\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−8.46971e39			−1.21466
$373$	6.81705e39	−	1.18075e40i	0.937799	−	1.62431i	0.168233	−	0.985747i	$-0.446194\pi$
$373$	6.81705e39	−	1.18075e40i	0.937799	−	1.62431i	0.769566	−	0.638568i	$-0.220473\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−1.72085e40			−1.84857			−0.924283	−	0.381707i	$-0.875336\pi$
$379$	−1.72085e40			−1.84857			−0.924283	+	0.381707i	$0.875336\pi$
$380$		0			0
$381$	−5.64668e39	+	3.26011e39i	−0.559057	+	0.322772i
$382$		0			0
$383$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$383$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−5.49159e39	+	9.51171e39i	−0.426755	+	0.739161i
$388$	1.88046e40	−	1.08569e40i	1.40402	−	0.810611i
$389$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$389$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	8.33000e39	+	4.80933e39i	0.435906	+	0.251670i	0.701859	−	0.712316i	$-0.252354\pi$
$397$	8.33000e39	+	4.80933e39i	0.435906	+	0.251670i	−0.265954	+	0.963986i	$0.585687\pi$
$398$		0			0
$399$	−1.27163e40	+	3.09982e40i	−0.615574	+	1.50057i
$400$	−2.14748e40			−1.00000
$401$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$401$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$402$		0			0
$403$	−2.76642e40	−	4.79157e40i	−1.14733	−	1.98724i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	3.30096e40	−	1.90581e40i	1.08875	−	0.628590i	0.155507	−	0.987835i	$-0.450299\pi$
$409$	3.30096e40	−	1.90581e40i	1.08875	−	0.628590i	0.933244	+	0.359244i	$0.116966\pi$
$410$		0			0
$411$		0			0
$412$		−	5.46918e39i		−	0.161070i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	4.08018e40	+	7.06708e40i	0.996714	+	1.72636i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	6.71137e40			1.41396			0.706982	−	0.707231i	$-0.250056\pi$
$421$	6.71137e40			1.41396			0.706982	+	0.707231i	$0.250056\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	9.29885e40	−	7.18574e40i	1.57325	−	1.21574i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$431$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$432$	6.13096e40	+	3.53971e40i	0.866025	+	0.500000i
$433$		−	1.32580e41i		−	1.80682i	−0.428780	−	0.903409i	$-0.641056\pi$
$433$		−	1.32580e41i		−	1.80682i	0.428780	−	0.903409i	$-0.358944\pi$
$434$		0			0
$435$		0			0
$436$	7.65502e40	−	1.32589e41i	0.937359	−	1.62355i
$437$		0			0
$438$		0			0
$439$	1.20969e41	+	6.98416e40i	1.33191	+	0.768979i	0.985592	−	0.169139i	$-0.0540986\pi$
$439$	1.20969e41	+	6.98416e40i	1.33191	+	0.768979i	0.346318	+	0.938117i	$0.387432\pi$
$440$		0			0
$441$	−9.39589e40	+	2.58638e40i	−0.964140	+	0.265396i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.04399e41	+	6.02749e40i	−0.964408	+	0.556801i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	4.72127e40	−	1.15089e41i	0.379533	−	0.925178i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	−2.39893e41	−	1.38502e41i	−1.62366	−	0.937422i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.47638e41	+	2.55717e41i	−0.871963	+	1.51028i	−0.0119992	+	0.999928i	$0.503820\pi$
$457$	−1.47638e41	+	2.55717e41i	−0.871963	+	1.51028i	−0.859963	+	0.510356i	$0.829514\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	4.06180e41			1.95981			0.979904	−	0.199469i	$-0.0639216\pi$
$463$	4.06180e41			1.95981			0.979904	+	0.199469i	$0.0639216\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$467$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$468$			4.62463e41i			1.88915i
$469$	−3.76806e41	+	5.09143e40i	−1.48915	+	0.201215i
$470$		0			0
$471$	−2.40527e41	+	4.16605e41i	−0.889897	+	1.54135i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	4.99783e41i		−	1.62192i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$479$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$480$		0			0
$481$	−6.81986e41	−	3.93745e41i	−1.82191	−	1.05188i
$482$		0			0
$483$		0			0
$484$	4.12195e41			1.00000
$485$		0			0
$486$		0			0
$487$	−8.22839e40	−	1.42520e41i	−0.181391	−	0.314178i	0.760963	−	0.648795i	$-0.224727\pi$
$487$	−8.22839e40	−	1.42520e41i	−0.181391	−	0.314178i	−0.942354	+	0.334616i	$0.891393\pi$
$488$		0			0
$489$			4.23989e41i			0.877136i
$490$		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
$496$			7.31845e41i			1.21466i
$497$		0			0
$498$		0			0
$499$	4.08551e41	−	7.07631e41i	0.617573	−	1.06967i	−0.372355	−	0.928091i	$-0.621449\pi$
$499$	4.08551e41	−	7.07631e41i	0.617573	−	1.06967i	0.989927	−	0.141577i	$-0.0452172\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.88321e42	+	1.08727e42i	−2.22472	+	1.28444i
$508$	2.81698e41	+	4.87915e41i	0.322772	+	0.559057i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−2.22520e41	−	1.64683e42i	−0.232726	−	1.72236i
$512$		0			0
$513$	−8.23796e41	+	1.42686e42i	−0.810962	+	1.40463i
$514$		0			0
$515$		0			0
$516$	8.21881e41	+	4.74513e41i	0.739161	+	0.426755i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$521$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$522$		0			0
$523$	−7.22638e41	−	4.17215e41i	−0.527406	−	0.304498i	0.212554	−	0.977149i	$-0.431822\pi$
$523$	−7.22638e41	−	4.17215e41i	−0.527406	−	0.304498i	−0.739959	+	0.672652i	$0.765155\pi$
$524$		0			0
$525$	1.15026e42	−	8.88870e41i	0.791274	−	0.611461i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−8.17585e41	−	1.41610e42i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	2.67847e42	+	1.09878e42i	1.50057	+	0.615574i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.08919e42	−	3.61859e42i	0.902445	−	1.56308i	0.0781343	−	0.996943i	$-0.475104\pi$
$541$	2.08919e42	−	3.61859e42i	0.902445	−	1.56308i	0.824311	−	0.566138i	$-0.191563\pi$
$542$		0			0
$543$	−7.35215e41	−	1.27343e42i	−0.299928	−	0.519490i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.06008e42			0.750034			0.375017	−	0.927018i	$-0.377637\pi$
$547$	2.06008e42			0.750034			0.375017	+	0.927018i	$0.377637\pi$
$548$		0			0
$549$	5.00463e42	−	2.88943e42i	1.72187	−	0.994123i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	1.15493e42	+	1.49456e42i	0.355072	+	0.459488i
$554$		0			0
$555$		0			0
$556$	6.10647e42	−	3.52557e42i	1.72636	−	0.996714i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			6.19951e42i			1.61241i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$563$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−4.74907e42	+	6.41697e41i	−0.990994	+	0.133904i
$568$		0			0
$569$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$569$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$570$		0			0
$571$	−5.91234e41	−	1.02405e42i	−0.110637	−	0.191629i	0.805390	−	0.592745i	$-0.201956\pi$
$571$	−5.91234e41	−	1.02405e42i	−0.110637	−	0.191629i	−0.916027	+	0.401116i	$0.868622\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	3.05857e42	−	5.29760e42i	0.500000	−	0.866025i
$577$	−1.07292e43	+	6.19448e42i	−1.70742	+	0.985780i	−0.769689	+	0.638419i	$0.779589\pi$
$577$	−1.07292e43	+	6.19448e42i	−1.70742	+	0.985780i	−0.937732	+	0.347360i	$0.887078\pi$
$578$		0			0
$579$	−1.59069e42	−	9.18386e41i	−0.239921	−	0.138518i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	2.23482e42	+	8.11874e42i	0.265396	+	0.964140i
$589$	−1.70322e43			−1.97008
$590$		0			0
$591$		0			0
$592$	5.20819e42	+	9.02085e42i	0.556801	+	0.964408i
$593$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$593$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.65298e42	−	1.15233e43i	0.624329	−	1.08137i
$598$		0			0
$599$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$599$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$600$		0			0
$601$		−	2.10148e43i		−	1.77816i	−0.457752	−	0.889080i	$-0.651345\pi$
$601$		−	2.10148e43i		−	1.77816i	0.457752	−	0.889080i	$-0.348655\pi$
$602$		0			0
$603$	−1.86976e43			−1.50269
$604$	−1.19676e43	+	2.07285e43i	−0.937422	+	1.62366i
$605$		0			0
$606$		0			0
$607$	3.17885e42	+	1.83531e42i	0.230592	+	0.133133i	0.610845	−	0.791750i	$-0.290830\pi$
$607$	3.17885e42	+	1.83531e42i	0.230592	+	0.133133i	−0.380253	+	0.924883i	$0.624163\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.30074e41	−	5.71705e41i	−0.0205577	−	0.0356070i	0.855564	−	0.517698i	$-0.173211\pi$
$613$	−3.30074e41	−	5.71705e41i	−0.0205577	−	0.0356070i	−0.876121	+	0.482091i	$0.839878\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−2.45154e43	+	1.41540e43i	−1.31291	+	0.758007i	−0.982576	−	0.185859i	$-0.940493\pi$
$619$	−2.45154e43	+	1.41540e43i	−1.31291	+	0.758007i	−0.330330	+	0.943866i	$0.607160\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	3.99602e43			1.88915
$625$	−1.08420e43	+	1.87789e43i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	3.59977e43	+	2.07833e43i	1.54135	+	0.889897i
$629$		0			0
$630$		0			0
$631$	−5.00708e43			−1.99127			−0.995633	−	0.0933511i	$-0.970242\pi$
$631$	−5.00708e43			−1.99127			−0.995633	+	0.0933511i	$0.970242\pi$
$632$		0			0
$633$	3.17997e43	−	1.83596e43i	1.20411	−	0.695193i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−3.86307e43	+	3.91608e43i	−1.32669	+	1.34490i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$641$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$642$		0			0
$643$		−	2.74518e43i		−	0.815270i	−0.913145	−	0.407635i	$-0.866354\pi$
$643$		−	2.74518e43i		−	0.815270i	0.913145	−	0.407635i	$-0.133646\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$647$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−3.02920e43	−	3.92000e43i	−0.742715	−	0.961126i
$652$	3.66357e43			0.877136
$653$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$653$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	8.17176e43i		−	1.73801i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	1.10172e43	−	6.36076e42i	0.213278	−	0.123136i	−0.389556	−	0.921003i	$-0.627371\pi$
$661$	1.10172e43	−	6.36076e42i	0.213278	−	0.123136i	0.602834	+	0.797867i	$0.294038\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−6.22305e43	−	1.07786e44i	−0.999766	−	1.73164i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	6.79069e43			0.994674			0.497337	−	0.867557i	$-0.334311\pi$
$673$	6.79069e43			0.994674			0.497337	+	0.867557i	$0.334311\pi$
$674$		0			0
$675$	6.19069e43	−	3.57420e43i	0.866025	−	0.500000i
$676$	9.39480e43	+	1.62723e44i	1.28444	+	2.22472i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	1.17503e44	+	4.82030e43i	1.49992	+	0.615307i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$683$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$684$	1.23291e44	+	7.11820e43i	1.40463	+	0.810962i
$685$		0			0
$686$		0			0
$687$	1.08943e44			1.15977
$688$	4.10014e43	−	7.10166e43i	0.426755	−	0.739161i
$689$		0			0
$690$		0			0
$691$	−1.56193e44	−	9.01780e43i	−1.51968	−	0.877386i	−0.999731	−	0.0231879i	$-0.992618\pi$
$691$	−1.56193e44	−	9.01780e43i	−1.51968	−	0.877386i	−0.519947	−	0.854199i	$-0.674048\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−7.68048e43	−	9.93909e43i	−0.611461	−	0.791274i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−2.09942e44	+	1.21210e44i	−1.56420	+	0.903090i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	9.08999e43	−	1.57443e44i	0.593672	−	1.02827i	−0.400061	−	0.916489i	$-0.631011\pi$
$709$	9.08999e43	−	1.57443e44i	0.593672	−	1.02827i	0.993733	−	0.111782i	$-0.0356557\pi$
$710$		0			0
$711$	4.64404e43	+	8.04372e43i	0.290347	+	0.502896i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$719$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$720$		0			0
$721$	2.53127e43	−	1.95605e43i	0.127451	−	0.0984882i
$722$		0			0
$723$	−7.38372e43	+	1.27890e44i	−0.356149	+	0.616868i
$724$	−1.10034e44	+	6.35279e43i	−0.519490	+	0.299928i
$725$		0			0
$726$		0			0
$727$			3.19802e44i			1.41611i	0.706157	+	0.708056i	$0.250427\pi$
$727$			3.19802e44i			1.41611i	−0.706157	+	0.708056i	$0.749573\pi$
$728$		0			0
$729$	−2.35655e44			−1.00000
$730$		0			0
$731$		0			0
$732$	−2.49668e44	−	4.32437e44i	−0.994123	−	1.72187i
$733$	−2.84044e44	−	1.63993e44i	−1.10732	−	0.639312i	−0.169186	−	0.985584i	$-0.554114\pi$
$733$	−2.84044e44	−	1.63993e44i	−1.10732	−	0.639312i	−0.938134	+	0.346272i	$0.887447\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−2.24767e44	−	3.89309e44i	−0.772227	−	1.33754i	−0.936340	−	0.351094i	$-0.885810\pi$
$739$	−2.24767e44	−	3.89309e44i	−0.772227	−	1.33754i	0.164114	−	0.986441i	$-0.447524\pi$
$740$		0			0
$741$			9.29991e44i			3.06406i
$742$		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
$751$	−3.27048e44	+	5.66464e44i	−0.875373	+	1.51619i	−0.0190076	+	0.999819i	$0.506051\pi$
$751$	−3.27048e44	+	5.66464e44i	−0.875373	+	1.51619i	−0.856365	+	0.516371i	$0.827283\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	5.54473e43	+	4.10354e44i	0.133904	+	0.990994i
$757$	3.69450e44			0.874117			0.437058	−	0.899433i	$-0.356020\pi$
$757$	3.69450e44			0.874117			0.437058	+	0.899433i	$0.356020\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$761$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$762$		0			0
$763$	8.87437e44	−	1.19911e44i	1.85783	−	0.251032i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−4.57751e44	−	2.64283e44i	−0.866025	−	0.500000i
$769$		−	6.04186e44i		−	1.12024i	−0.828411	−	0.560121i	$-0.810755\pi$
$769$		−	6.04186e44i		−	1.12024i	0.828411	−	0.560121i	$-0.189245\pi$
$770$		0			0
$771$		0			0
$772$	−7.93552e43	+	1.37447e44i	−0.138518	+	0.239921i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	6.39971e44	+	3.69487e44i	1.05192	+	0.607328i
$776$		0			0
$777$	−6.52351e44	−	2.67612e44i	−1.03028	−	0.422649i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	7.01518e44	−	1.93105e44i	0.964140	−	0.265396i
$785$		0			0
$786$		0			0
$787$	−1.25860e45	+	7.26654e44i	−1.63035	+	0.941281i	−0.646362	+	0.763031i	$0.723710\pi$
$787$	−1.25860e45	+	7.26654e44i	−1.63035	+	0.941281i	−0.983985	+	0.178250i	$0.942956\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	1.63095e45	−	2.82489e45i	1.87805	−	3.25287i
$794$		0			0
$795$		0			0
$796$	−9.95697e44	−	5.74866e44i	−1.08137	−	0.624329i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			1.61561e45i			1.50269i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$809$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$810$		0			0
$811$		−	1.37861e45i		−	1.12104i	−0.828142	−	0.560518i	$-0.810602\pi$
$811$		−	1.37861e45i		−	1.12104i	0.828142	−	0.560518i	$-0.189398\pi$
$812$		0			0
$813$	2.54104e45			1.98888
$814$		0			0
$815$		0			0
$816$		0			0
$817$	1.65277e45	+	9.54225e44i	1.19886	+	0.692164i
$818$		0			0
$819$	−2.14039e45	+	1.65400e45i	−1.49483	+	1.15514i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	8.85263e44	+	1.53332e45i	0.573290	+	0.992968i	0.996225	+	0.0868079i	$0.0276666\pi$
$823$	8.85263e44	+	1.53332e45i	0.573290	+	0.992968i	−0.422935	+	0.906160i	$0.639000\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	2.40788e45	−	1.39019e45i	1.39328	−	0.804412i	0.399605	−	0.916687i	$-0.369147\pi$
$829$	2.40788e45	−	1.39019e45i	1.39328	−	0.804412i	0.993677	+	0.112275i	$0.0358138\pi$
$830$		0			0
$831$	1.65519e45	+	9.55627e44i	0.922642	+	0.532687i
$832$		−	3.45285e45i		−	1.88915i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.21806e45	−	2.10974e45i	−0.607328	−	1.05192i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	2.15942e45			1.00000
$842$		0			0
$843$		0			0
$844$	−1.58640e45	−	2.74773e45i	−0.695193	−	1.20411i
$845$		0			0
$846$		0			0
$847$	1.47422e45	+	1.90774e45i	0.611461	+	0.791274i
$848$		0			0
$849$	−2.44374e45	+	4.23268e45i	−0.977204	+	1.69257i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	2.59438e45i		−	0.964547i	−0.876021	−	0.482274i	$-0.839811\pi$
$853$		−	2.59438e45i		−	0.964547i	0.876021	−	0.482274i	$-0.160189\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$857$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$858$		0			0
$859$	−6.52391e44	−	3.76658e44i	−0.217577	−	0.125618i	0.387251	−	0.921974i	$-0.373425\pi$
$859$	−6.52391e44	−	3.76658e44i	−0.217577	−	0.125618i	−0.604828	+	0.796356i	$0.706758\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		−	3.46175e45i		−	1.00000i
$868$	−3.38716e45	+	2.61745e45i	−0.961126	+	0.742715i
$869$		0			0
$870$		0			0
$871$	−9.14000e45	+	5.27698e45i	−2.45847	+	1.41940i
$872$		0			0
$873$	5.40871e45	+	3.12272e45i	1.40402	+	0.810611i
$874$		0			0
$875$		0			0
$876$	−7.06100e45			−1.73801
$877$	−4.34061e44	+	7.51816e44i	−0.104968	+	0.181810i	−0.913725	−	0.406333i	$-0.866807\pi$
$877$	−4.34061e44	+	7.51816e44i	−0.104968	+	0.181810i	0.808757	+	0.588143i	$0.200141\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	−2.89719e43			−0.00630357			−0.00315178	−	0.999995i	$-0.501003\pi$
$883$	−2.89719e43			−0.00630357			−0.00315178	+	0.999995i	$0.501003\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$887$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$888$		0			0
$889$	−1.25070e45	+	3.04880e45i	−0.245005	+	0.597243i
$890$		0			0
$891$		0			0
$892$	−9.31352e45	+	5.37717e45i	−1.73164	+	0.999766i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−3.08837e45	−	5.34921e45i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	7.43295e44	+	5.50097e45i	0.114288	+	0.845823i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	6.96487e45	+	1.20635e46i	1.00000	+	1.73205i	0.500198	+	0.865911i	$0.333261\pi$
$907$	6.96487e45	+	1.20635e46i	1.00000	+	1.73205i	0.499802	+	0.866139i	$0.333406\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	6.15064e45	−	1.06532e46i	0.810962	−	1.40463i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	9.41350e45i		−	1.15977i
$917$		0			0
$918$		0			0
$919$	4.17823e45	−	7.23690e45i	0.489330	−	0.847544i	−0.510595	−	0.859821i	$-0.670575\pi$
$919$	4.17823e45	−	7.23690e45i	0.489330	−	0.847544i	0.999925	+	0.0122775i	$0.00390814\pi$
$920$		0			0
$921$	−8.82494e45	−	1.52852e46i	−0.999280	−	1.73080i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.05179e46			1.11360
$926$		0			0
$927$	1.36233e45	−	7.86541e44i	0.139491	−	0.0805351i
$928$		0			0
$929$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$929$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$930$		0			0
$931$	4.49412e45	+	1.63264e46i	0.430452	+	1.56376i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			2.30641e46i			1.99973i	0.0163137	+	0.999867i	$0.494807\pi$
$937$			2.30641e46i			1.99973i	−0.0163137	+	0.999867i	$0.505193\pi$
$938$		0			0
$939$	2.35973e46			1.97946
$940$		0			0
$941$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$941$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	6.95036e45	−	4.01279e45i	0.502896	−	0.290347i
$949$	−2.30630e46	−	3.99462e46i	−1.64168	−	2.84347i
$950$		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
$961$	4.05725e45	−	7.02736e45i	0.237694	−	0.411699i
$962$		0			0
$963$		0			0
$964$	1.10506e46	+	6.38007e45i	0.616868	+	0.356149i
$965$		0			0
$966$		0			0
$967$	−3.68398e46			−1.95978			−0.979888	−	0.199547i	$-0.936053\pi$
$967$	−3.68398e46			−1.95978			−0.979888	+	0.199547i	$0.936053\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$971$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$972$			2.03623e46i			1.00000i
$973$	3.81571e46	+	1.56531e46i	1.84428	+	0.756572i
$974$		0			0
$975$	2.01747e46	−	3.49436e46i	0.944574	−	1.63605i
$976$	−3.73657e46	+	2.15731e46i	−1.72187	+	0.994123i
$977$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$977$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	4.40358e46			1.87472
$982$		0			0
$983$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$983$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	8.03580e46			3.06406
$989$		0			0
$990$		0			0
$991$	−2.23467e46	−	3.87057e46i	−0.812966	−	1.40810i	−0.910780	−	0.412893i	$-0.864518\pi$
$991$	−2.23467e46	−	3.87057e46i	−0.812966	−	1.40810i	0.0978139	−	0.995205i	$-0.468815\pi$
$992$		0			0
$993$		−	4.26905e46i		−	1.50528i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−3.85768e46	+	2.22723e46i	−1.27806	+	0.737889i	−0.976492	−	0.215555i	$-0.930844\pi$
$997$	−3.85768e46	+	2.22723e46i	−1.27806	+	0.737889i	−0.301569	+	0.953444i	$0.597510\pi$
$998$		0			0
$999$	−3.00280e46	−	1.73367e46i	−0.964408	−	0.556801i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.32.g.a.5.1	✓	2
3.2	odd	2	CM	21.32.g.a.5.1	✓	2
7.3	odd	6	inner	21.32.g.a.17.1	yes	2
21.17	even	6	inner	21.32.g.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.32.g.a.5.1	✓	2	1.1	even	1	trivial
21.32.g.a.5.1	✓	2	3.2	odd	2	CM
21.32.g.a.17.1	yes	2	7.3	odd	6	inner
21.32.g.a.17.1	yes	2	21.17	even	6	inner

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:	\( k \)	\(=\)	\( 32 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.15234e7 - 1.24265e7i) q^{3} +(-1.07374e9 - 1.85978e9i) q^{4} +(4.76727e12 - 1.16210e13i) q^{7} +(3.08837e14 - 5.34921e14i) q^{9} +O(q^{10})\)	\(q+(2.15234e7 - 1.24265e7i) q^{3} +(-1.07374e9 - 1.85978e9i) q^{4} +(4.76727e12 - 1.16210e13i) q^{7} +(3.08837e14 - 5.34921e14i) q^{9} +(-4.62211e16 - 2.66857e16i) q^{12} -3.48649e17i q^{13} +(-2.30584e18 + 3.99384e18i) q^{16} +(-9.29484e19 - 5.36638e19i) q^{19} +(-4.18015e19 - 3.09364e20i) q^{21} +(2.32831e21 + 4.03275e21i) q^{25} -1.53511e22i q^{27} +(-2.67313e22 + 3.61195e21i) q^{28} +(1.37433e23 - 7.93468e22i) q^{31} -1.32644e24 q^{36} +(1.12935e24 - 1.95608e24i) q^{37} +(-4.33249e24 - 7.50409e24i) q^{39} -1.77815e25 q^{43} +1.14614e26i q^{48} +(-1.12322e26 - 1.10801e26i) q^{49} +(-6.48408e26 + 3.74359e26i) q^{52} -2.66742e27 q^{57} +(8.10239e27 + 4.67792e27i) q^{61} +(-4.74403e27 - 6.13911e27i) q^{63} +9.90352e27 q^{64} +(-1.51355e28 - 2.62155e28i) q^{67} +(1.14574e29 - 6.61496e28i) q^{73} +(1.00226e29 + 5.78655e28i) q^{75} +2.30484e29i q^{76} +(-7.51860e28 + 1.30226e29i) q^{79} +(-1.90760e29 - 3.30406e29i) q^{81} +(-5.30464e29 + 4.09919e29i) q^{84} +(-4.05166e30 - 1.66210e30i) q^{91} +(1.97201e30 - 3.41562e30i) q^{93} +1.01112e31i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 43046721 q^{3} - 2147483648 q^{4} + 9534530086957 q^{7} + 617673396283947 q^{9}+O(q^{10}) \) 2 * q + 43046721 * q^3 - 2147483648 * q^4 + 9534530086957 * q^7 + 617673396283947 * q^9	\( 2 q + 43046721 q^{3} - 2147483648 q^{4} + 9534530086957 q^{7} + 617673396283947 q^{9} - 92\!\cdots\!08 q^{12}+ \cdots + 39\!\cdots\!05 q^{93}+O(q^{100}) \) 2 * q + 43046721 * q^3 - 2147483648 * q^4 + 9534530086957 * q^7 + 617673396283947 * q^9 - 92442129447518208 * q^12 - 4611686018427387904 * q^16 - 185896844889165280341 * q^19 - 83602991755761479406 * q^21 + 4656612873077392578125 * q^25 - 53462671458706502189056 * q^28 + 274865422539577564919715 * q^31 - 2652887036648800294797312 * q^36 + 2258691937157207814386543 * q^37 - 8664977206989331233197553 * q^39 - 35563070302396847726017330 * q^43 - 224643500090603355248565637 * q^49 - 1296816326309892463150497792 * q^52 - 5334833077816115830483874574 * q^57 + 16204788248998670740089934212 * q^61 - 9488060241657771257110507005 * q^63 + 19807040628566084398385987584 * q^64 - 30271048416507274043668039127 * q^67 + 229148784216062228941507200789 * q^73 + 200451915152370929718017578125 * q^75 - 150372080821436019808890048863 * q^79 - 381520424476945831628649898809 * q^81 - 1060928322239112616902754566144 * q^84 - 8103319462859891108445297803691 * q^91 + 3944018385536102298319453001505 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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