LMFDB - Embedded newform 21.27.h.a.2.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,27,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, 27, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 27);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 27 $
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:	$89.9415133532$
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			2.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.2
Dual form			21.27.h.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(797162. + 1.38072e6i) q^{3} +(-3.35544e7 - 5.81180e7i) q^{4} +(-2.21712e10 + 9.43182e10i) q^{7} +(-1.27093e12 + 2.20132e12i) q^{9} +O(q^{10})$	\(q+(797162. + 1.38072e6i) q^{3} +(-3.35544e7 - 5.81180e7i) q^{4} +(-2.21712e10 + 9.43182e10i) q^{7} +(-1.27093e12 + 2.20132e12i) q^{9} +(5.34966e13 - 9.26588e13i) q^{12} +6.05583e14 q^{13} +(-2.25180e15 + 3.90023e15i) q^{16} +(2.63184e16 - 4.55848e16i) q^{19} +(-1.47901e17 + 4.45745e16i) q^{21} +(-7.45058e17 - 1.29048e18i) q^{25} -4.05256e18 q^{27} +(6.22552e18 - 1.87625e18i) q^{28} +(9.40525e18 + 1.62904e19i) q^{31} +1.70582e20 q^{36} +(-1.31193e20 + 2.27232e20i) q^{37} +(4.82748e20 + 8.36144e20i) q^{39} +2.52622e21 q^{43} -7.18019e21 q^{48} +(-8.40435e21 - 4.18230e21i) q^{49} +(-2.03200e22 - 3.51953e22i) q^{52} +8.39201e22 q^{57} +(-1.31928e23 + 2.28506e23i) q^{61} +(-1.79446e23 - 1.68678e23i) q^{63} +3.02231e23 q^{64} +(5.22507e23 + 9.05008e23i) q^{67} +(-3.66365e22 - 6.34563e22i) q^{73} +(1.18786e24 - 2.05744e24i) q^{75} -3.53240e24 q^{76} +(-3.67351e24 + 6.36270e24i) q^{79} +(-3.23054e24 - 5.59546e24i) q^{81} +(7.55333e24 + 7.10006e24i) q^{84} +(-1.34265e25 + 5.71175e25i) q^{91} +(-1.49950e25 + 2.59721e25i) q^{93} +1.79863e25 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1594323 q^{3} - 67108864 q^{4} - 44342429053 q^{7} - 2541865828329 q^{9}+O(q^{10}) $ 2 * q + 1594323 * q^3 - 67108864 * q^4 - 44342429053 * q^7 - 2541865828329 * q^9	$ 2 q + 1594323 q^{3} - 67108864 q^{4} - 44342429053 q^{7} - 2541865828329 q^{9} + 106993205379072 q^{12} + 12\!\cdots\!66 q^{13}+ \cdots + 35\!\cdots\!84 q^{97}+O(q^{100}) $ 2 * q + 1594323 * q^3 - 67108864 * q^4 - 44342429053 * q^7 - 2541865828329 * q^9 + 106993205379072 * q^12 + 1211166615597166 * q^13 - 4503599627370496 * q^16 + 52636841466889669 * q^19 - 295802846374395366 * q^21 - 1490116119384765625 * q^25 - 8105110306037952534 * q^27 + 12451048494033010688 * q^28 + 18810497092058970493 * q^31 + 341163456359156416512 * q^36 - 262385069961736743767 * q^37 + 965495396039360244309 * q^39 + 5052437468927806557478 * q^43 - 14360384937416422588416 * q^48 - 16808709660975170134489 * q^49 - 40640007843725245939712 * q^52 + 167840253996031875498174 * q^57 - 263855512891444028536874 * q^61 - 358892776285241383064781 * q^63 + 604462909807314587353088 * q^64 + 1045013466379120827629053 * q^67 - 73273034217751719361823 * q^73 + 2375726401805877685546875 * q^75 - 7064797270782118599852032 * q^76 - 7347014375367289352249531 * q^79 - 6461081889226673298932241 * q^81 + 15106654369477633568145408 * q^84 - 26853034861739728291431899 * q^91 - 29990008155302734013311239 * q^93 + 35972525076053703782561284 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$e\left(\frac{1}{3}\right)$

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	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	797162.	+	1.38072e6i	0.500000	+	0.866025i
$3$	797162.	+	1.38072e6i	0.500000	+	0.866025i
$4$	−3.35544e7	−	5.81180e7i	−0.500000	−	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	−2.21712e10	+	9.43182e10i	−0.228831	+	0.973466i
$7$	−2.21712e10	+	9.43182e10i	−0.228831	+	0.973466i
$8$		0			0
$9$	−1.27093e12	+	2.20132e12i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	5.34966e13	−	9.26588e13i	0.500000	−	0.866025i
$13$	6.05583e14			1.99945			0.999724	−	0.0234733i	$-0.00747246\pi$
$13$	6.05583e14			1.99945			0.999724	+	0.0234733i	$0.00747246\pi$
$14$		0			0
$15$		0			0
$16$	−2.25180e15	+	3.90023e15i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	2.63184e16	−	4.55848e16i	0.625840	−	1.08399i	−0.362538	−	0.931969i	$-0.618090\pi$
$19$	2.63184e16	−	4.55848e16i	0.625840	−	1.08399i	0.988378	−	0.152017i	$-0.0485769\pi$
$20$		0			0
$21$	−1.47901e17	+	4.45745e16i	−0.957462	+	0.288560i
$22$		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
$25$	−7.45058e17	−	1.29048e18i	−0.500000	−	0.866025i
$26$		0			0
$27$	−4.05256e18			−1.00000
$28$	6.22552e18	−	1.87625e18i	0.957462	−	0.288560i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	9.40525e18	+	1.62904e19i	0.385184	+	0.667158i	0.991795	−	0.127841i	$-0.0408046\pi$
$31$	9.40525e18	+	1.62904e19i	0.385184	+	0.667158i	−0.606611	+	0.794999i	$0.707471\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	1.70582e20			1.00000
$37$	−1.31193e20	+	2.27232e20i	−0.538625	+	0.932926i	0.460353	+	0.887736i	$0.347723\pi$
$37$	−1.31193e20	+	2.27232e20i	−0.538625	+	0.932926i	−0.998978	+	0.0451904i	$0.985611\pi$
$38$		0			0
$39$	4.82748e20	+	8.36144e20i	0.999724	+	1.73157i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	2.52622e21			1.47022			0.735108	−	0.677950i	$-0.237132\pi$
$43$	2.52622e21			1.47022			0.735108	+	0.677950i	$0.237132\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	−7.18019e21			−1.00000
$49$	−8.40435e21	−	4.18230e21i	−0.895273	−	0.445519i
$50$		0			0
$51$		0			0
$52$	−2.03200e22	−	3.51953e22i	−0.999724	−	1.73157i
$53$		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
$57$	8.39201e22			1.25168
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−1.31928e23	+	2.28506e23i	−0.814795	+	1.41127i	0.0946801	+	0.995508i	$0.469817\pi$
$61$	−1.31928e23	+	2.28506e23i	−0.814795	+	1.41127i	−0.909475	+	0.415758i	$0.863516\pi$
$62$		0			0
$63$	−1.79446e23	−	1.68678e23i	−0.728631	−	0.684907i
$64$	3.02231e23			1.00000
$65$		0			0
$66$		0			0
$67$	5.22507e23	+	9.05008e23i	0.953058	+	1.65075i	0.738750	+	0.673979i	$0.235416\pi$
$67$	5.22507e23	+	9.05008e23i	0.953058	+	1.65075i	0.214308	+	0.976766i	$0.431250\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−3.66365e22	−	6.34563e22i	−0.0219138	−	0.0379558i	0.854861	−	0.518858i	$-0.173643\pi$
$73$	−3.66365e22	−	6.34563e22i	−0.0219138	−	0.0379558i	−0.876774	+	0.480902i	$0.840309\pi$
$74$		0			0
$75$	1.18786e24	−	2.05744e24i	0.500000	−	0.866025i
$76$	−3.53240e24			−1.25168
$77$		0			0
$78$		0			0
$79$	−3.67351e24	+	6.36270e24i	−0.786917	+	1.36298i	0.140931	+	0.990019i	$0.454990\pi$
$79$	−3.67351e24	+	6.36270e24i	−0.786917	+	1.36298i	−0.927847	+	0.372960i	$0.878343\pi$
$80$		0			0
$81$	−3.23054e24	−	5.59546e24i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	7.55333e24	+	7.10006e24i	0.728631	+	0.684907i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−1.34265e25	+	5.71175e25i	−0.457536	+	1.94640i
$92$		0			0
$93$	−1.49950e25	+	2.59721e25i	−0.385184	+	0.667158i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	1.79863e25			0.267244			0.133622	−	0.991032i	$-0.457339\pi$
$97$	1.79863e25			0.267244			0.133622	+	0.991032i	$0.457339\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e25	+	8.66025e25i	−0.500000	+	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	−1.41521e26	+	2.45122e26i	−0.963690	+	1.66916i	−0.250597	+	0.968092i	$0.580627\pi$
$103$	−1.41521e26	+	2.45122e26i	−0.963690	+	1.66916i	−0.713094	+	0.701069i	$0.752707\pi$
$104$		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	$0.666667\pi$
$108$	1.35981e26	+	2.35526e26i	0.500000	+	0.866025i
$109$	2.82406e26	+	4.89141e26i	0.921148	+	1.59547i	0.797642	+	0.603131i	$0.206080\pi$
$109$	2.82406e26	+	4.89141e26i	0.921148	+	1.59547i	0.123506	+	0.992344i	$0.460586\pi$
$110$		0			0
$111$	−4.18327e26			−1.07725
$112$	−3.17938e26	−	2.98859e26i	−0.728631	−	0.684907i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	−7.69656e26	+	1.33308e27i	−0.999724	+	1.73157i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.95909e26	+	1.03214e27i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	6.31176e26	−	1.09323e27i	0.385184	−	0.667158i
$125$		0			0
$126$		0			0
$127$	−2.63575e27			−1.17884			−0.589422	−	0.807825i	$-0.700645\pi$
$127$	−2.63575e27			−1.17884			−0.589422	+	0.807825i	$0.700645\pi$
$128$		0			0
$129$	2.01380e27	+	3.48801e27i	0.735108	+	1.27324i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	3.71597e27	+	3.49298e27i	0.912012	+	0.857283i
$134$		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
$139$	−2.50809e26			−0.0346857			−0.0173429	−	0.999850i	$-0.505521\pi$
$139$	−2.50809e26			−0.0346857			−0.0173429	+	0.999850i	$0.505521\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−5.72377e27	−	9.91387e27i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−9.25030e26	−	1.49381e28i	−0.0618059	−	0.998088i
$148$	1.76084e28			1.07725
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−9.45342e27	−	1.63738e28i	−0.445542	−	0.771702i	0.552548	−	0.833481i	$-0.313656\pi$
$151$	−9.45342e27	−	1.63738e28i	−0.445542	−	0.771702i	−0.998090	+	0.0617797i	$0.980322\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	3.23966e28	−	5.61126e28i	0.999724	−	1.73157i
$157$	−3.36817e28	−	5.83384e28i	−0.956530	−	1.65676i	−0.730828	−	0.682561i	$-0.760866\pi$
$157$	−3.36817e28	−	5.83384e28i	−0.956530	−	1.65676i	−0.225701	−	0.974197i	$-0.572467\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.39778e28	+	4.15308e28i	−0.418187	+	0.724322i	−0.995757	−	0.0920196i	$-0.970668\pi$
$163$	−2.39778e28	+	4.15308e28i	−0.418187	+	0.724322i	0.577570	+	0.816341i	$0.304001\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	2.74998e29			2.99780
$170$		0			0
$171$	6.68979e28	+	1.15871e29i	0.625840	+	1.08399i
$172$	−8.47658e28	−	1.46819e29i	−0.735108	−	1.27324i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	1.38234e29	−	4.16610e28i	0.957462	−	0.288560i
$176$		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
$181$	−3.79314e29			−1.69503			−0.847515	−	0.530771i	$-0.821902\pi$
$181$	−3.79314e29			−1.69503			−0.847515	+	0.530771i	$0.821902\pi$
$182$		0			0
$183$	−4.20671e29			−1.62959
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	8.98501e28	−	3.82230e29i	0.228831	−	0.973466i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	2.40927e29	+	4.17298e29i	0.500000	+	0.866025i
$193$	−3.49452e29	−	6.05268e29i	−0.677864	−	1.17409i	−0.975623	−	0.219453i	$-0.929573\pi$
$193$	−3.49452e29	−	6.05268e29i	−0.677864	−	1.17409i	0.297759	−	0.954641i	$-0.403761\pi$
$194$		0			0
$195$		0			0
$196$	3.89367e28	+	6.28779e29i	0.0618059	+	0.998088i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	7.66194e29	+	1.32709e30i	0.998271	+	1.72906i	0.550038	+	0.835140i	$0.314613\pi$
$199$	7.66194e29	+	1.32709e30i	0.998271	+	1.72906i	0.448233	+	0.893917i	$0.352053\pi$
$200$		0			0
$201$	−8.33045e29	+	1.44288e30i	−0.953058	+	1.65075i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.36365e30	+	2.36192e30i	−0.999724	+	1.73157i
$209$		0			0
$210$		0			0
$211$	2.81295e30			1.71195			0.855974	−	0.517019i	$-0.172958\pi$
$211$	2.81295e30			1.71195			0.855974	+	0.517019i	$0.172958\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−1.74500e30	+	5.25909e29i	−0.737598	+	0.222297i
$218$		0			0
$219$	5.84104e28	−	1.01170e29i	0.0219138	−	0.0379558i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	−6.63067e30			−1.96605			−0.983024	−	0.183479i	$-0.941264\pi$
$223$	−6.63067e30			−1.96605			−0.983024	+	0.183479i	$0.941264\pi$
$224$		0			0
$225$	3.78768e30			1.00000
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−2.81589e30	−	4.87727e30i	−0.625840	−	1.08399i
$229$	4.65335e30	−	8.05983e30i	0.977021	−	1.69225i	0.303925	−	0.952696i	$-0.401703\pi$
$229$	4.65335e30	−	8.05983e30i	0.977021	−	1.69225i	0.673097	−	0.739554i	$-0.264964\pi$
$230$		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
$237$	−1.17135e31			−1.57383
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	8.50697e30	+	1.47345e31i	0.919503	+	1.59263i	0.800171	+	0.599772i	$0.204742\pi$
$241$	8.50697e30	+	1.47345e31i	0.919503	+	1.59263i	0.119332	+	0.992854i	$0.461925\pi$
$242$		0			0
$243$	5.15053e30	−	8.92097e30i	0.500000	−	0.866025i
$244$	1.77070e31			1.62959
$245$		0			0
$246$		0			0
$247$	1.59380e31	−	2.76054e31i	1.25133	−	2.16737i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	−3.78200e30	+	1.60890e31i	−0.228831	+	0.973466i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−1.01412e31	−	1.75651e31i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	−1.85234e31	−	1.74119e31i	−0.784918	−	0.737816i
$260$		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
$268$	3.50648e31	−	6.07341e31i	0.953058	−	1.65075i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	1.11865e31	−	1.93756e31i	0.263083	−	0.455673i	−0.703977	−	0.710223i	$-0.748594\pi$
$271$	1.11865e31	−	1.93756e31i	0.263083	−	0.455673i	0.967060	+	0.254550i	$0.0819273\pi$
$272$		0			0
$273$	−8.95666e31	+	2.69936e31i	−1.91440	+	0.576960i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	5.44995e31	+	9.43960e31i	0.964175	+	1.67000i	0.711816	+	0.702366i	$0.247873\pi$
$277$	5.44995e31	+	9.43960e31i	0.964175	+	1.67000i	0.252359	+	0.967634i	$0.418793\pi$
$278$		0			0
$279$	−4.78138e31			−0.770368
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	5.01816e31	+	8.69171e31i	0.671925	+	1.16381i	0.977357	+	0.211595i	$0.0678657\pi$
$283$	5.01816e31	+	8.69171e31i	0.671925	+	1.16381i	−0.305432	+	0.952214i	$0.598801\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−4.90503e31	+	8.49577e31i	−0.500000	+	0.866025i
$290$		0			0
$291$	1.43380e31	+	2.48341e31i	0.133622	+	0.231440i
$292$	−2.45864e30	+	4.25848e30i	−0.0219138	+	0.0379558i
$293$		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
$300$	−1.59432e32			−1.00000
$301$	−5.60093e31	+	2.38268e32i	−0.336431	+	1.43121i
$302$		0			0
$303$		0			0
$304$	1.18528e32	+	2.05296e32i	0.625840	+	1.08399i
$305$		0			0
$306$		0			0
$307$	1.21072e32			0.562658			0.281329	−	0.959611i	$-0.409225\pi$
$307$	1.21072e32			0.562658			0.281329	+	0.959611i	$0.409225\pi$
$308$		0			0
$309$	−4.51261e32			−1.92738
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	2.13363e32	−	3.69556e32i	0.770981	−	1.33538i	−0.166045	−	0.986118i	$-0.553100\pi$
$313$	2.13363e32	−	3.69556e32i	0.770981	−	1.33538i	0.937026	−	0.349260i	$-0.113567\pi$
$314$		0			0
$315$		0			0
$316$	4.93050e32			1.57383
$317$		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
$324$	−2.16798e32	+	3.75505e32i	−0.500000	+	0.866025i
$325$	−4.51195e32	−	7.81492e32i	−0.999724	−	1.73157i
$326$		0			0
$327$	−4.50246e32	+	7.79849e32i	−0.921148	+	1.59547i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.32416e32	−	7.48967e32i	0.755333	−	1.30827i	−0.189876	−	0.981808i	$-0.560809\pi$
$331$	4.32416e32	−	7.48967e32i	0.755333	−	1.30827i	0.945209	−	0.326467i	$-0.105858\pi$
$332$		0			0
$333$	−3.33474e32	−	5.77594e32i	−0.538625	−	0.932926i
$334$		0			0
$335$		0			0
$336$	1.59194e32	−	6.77223e32i	0.228831	−	0.973466i
$337$	1.32618e33			1.83407			0.917033	−	0.398811i	$-0.130577\pi$
$337$	1.32618e33			1.83407			0.917033	+	0.398811i	$0.130577\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	5.80801e32	−	6.99957e32i	0.638563	−	0.769569i
$344$		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
$349$	1.92956e33			1.69327			0.846636	−	0.532172i	$-0.178624\pi$
$349$	1.92956e33			1.69327			0.846636	+	0.532172i	$0.178624\pi$
$350$		0			0
$351$	−2.45416e33			−1.99945
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		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
$361$	−5.01092e32	−	8.67917e32i	−0.283350	−	0.490777i
$362$		0			0
$363$	−1.90014e33			−1.00000
$364$	3.77007e33	−	1.13622e33i	1.91440	−	0.576960i
$365$		0			0
$366$		0			0
$367$	2.21663e32	+	3.83931e32i	0.101166	+	0.175224i	0.912165	−	0.409823i	$-0.134409\pi$
$367$	2.21663e32	+	3.83931e32i	0.101166	+	0.175224i	−0.810999	+	0.585047i	$0.801076\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	2.01260e33			0.770368
$373$	−2.37405e33	+	4.11197e33i	−0.877555	+	1.51997i	−0.0235385	+	0.999723i	$0.507493\pi$
$373$	−2.37405e33	+	4.11197e33i	−0.877555	+	1.51997i	−0.854016	+	0.520246i	$0.825840\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−2.46428e33			−0.740253			−0.370127	−	0.928981i	$-0.620686\pi$
$379$	−2.46428e33			−0.740253			−0.370127	+	0.928981i	$0.620686\pi$
$380$		0			0
$381$	−2.10112e33	−	3.63925e33i	−0.589422	−	1.02091i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−3.21065e33	+	5.56102e33i	−0.735108	+	1.27324i
$388$	−6.03519e32	−	1.04533e33i	−0.133622	−	0.231440i
$389$		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
$397$	−5.61740e33	+	9.72961e33i	−0.923121	+	1.59889i	−0.128564	+	0.991701i	$0.541037\pi$
$397$	−5.61740e33	+	9.72961e33i	−0.923121	+	1.59889i	−0.794556	+	0.607190i	$0.792296\pi$
$398$		0			0
$399$	−1.86061e33	+	7.91519e33i	−0.286423	+	1.21847i
$400$	6.71089e33			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
$403$	5.69566e33	+	9.86518e33i	0.770156	+	1.33395i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.69218e33	−	2.93095e33i	−0.188818	−	0.327043i	0.756038	−	0.654527i	$-0.227132\pi$
$409$	−1.69218e33	−	2.93095e33i	−0.188818	−	0.327043i	−0.944857	+	0.327485i	$0.893799\pi$
$410$		0			0
$411$		0			0
$412$	1.89946e34			1.92738
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.99935e32	−	3.46298e32i	−0.0173429	−	0.0300387i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−2.60701e34			−1.99745			−0.998724	−	0.0505019i	$-0.983918\pi$
$421$	−2.60701e34			−1.99745			−0.998724	+	0.0505019i	$0.983918\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−1.86272e34	−	1.75094e34i	−1.18737	−	1.11612i
$428$		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	$0.333333\pi$
$432$	9.12554e33	−	1.58059e34i	0.500000	−	0.866025i
$433$	−2.49829e34			−1.32831			−0.664155	−	0.747595i	$-0.731209\pi$
$433$	−2.49829e34			−1.32831			−0.664155	+	0.747595i	$0.731209\pi$
$434$		0			0
$435$		0			0
$436$	1.89519e34	−	3.28257e34i	0.921148	−	1.59547i
$437$		0			0
$438$		0			0
$439$	2.21438e34	−	3.83542e34i	0.984495	−	1.70520i	0.340336	−	0.940304i	$-0.389459\pi$
$439$	2.21438e34	−	3.83542e34i	0.984495	−	1.70520i	0.644159	−	0.764892i	$-0.277208\pi$
$440$		0			0
$441$	1.98879e34	−	1.31853e34i	0.833467	−	0.552570i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	1.40367e34	+	2.43123e34i	0.538625	+	0.932926i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−6.70084e33	+	2.85059e34i	−0.228831	+	0.973466i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.50718e34	−	2.61051e34i	0.445542	−	0.771702i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.40107e34	+	2.42673e34i	−0.369446	+	0.639899i	−0.989479	−	0.144677i	$-0.953786\pi$
$457$	−1.40107e34	+	2.42673e34i	−0.369446	+	0.639899i	0.620033	+	0.784576i	$0.287119\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	4.48946e34			0.999174			0.499587	−	0.866264i	$-0.333485\pi$
$463$	4.48946e34			0.999174			0.499587	+	0.866264i	$0.333485\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	1.03301e35			1.99945
$469$	−9.69433e34	+	2.92167e34i	−1.82503	+	0.550028i
$470$		0			0
$471$	5.36995e34	−	9.30103e34i	0.956530	−	1.65676i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−7.84350e34			−1.25168
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−7.94480e34	+	1.37608e35i	−1.07695	+	1.86534i
$482$		0			0
$483$		0			0
$484$	7.99815e34			1.00000
$485$		0			0
$486$		0			0
$487$	−8.65560e34	−	1.49919e35i	−0.998667	−	1.72974i	−0.544035	−	0.839063i	$-0.683104\pi$
$487$	−8.65560e34	−	1.49919e35i	−0.998667	−	1.72974i	−0.454632	−	0.890679i	$-0.650229\pi$
$488$		0			0
$489$	−7.64568e34			−0.836375
$490$		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
$496$	−8.47149e34			−0.770368
$497$		0			0
$498$		0			0
$499$	−7.73884e34	+	1.34041e35i	−0.650682	+	1.12701i	0.332276	+	0.943182i	$0.392183\pi$
$499$	−7.73884e34	+	1.34041e35i	−0.650682	+	1.12701i	−0.982958	+	0.183832i	$0.941150\pi$
$500$		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
$507$	2.19218e35	+	3.79696e35i	1.49890	+	2.59617i
$508$	8.84412e34	+	1.53185e35i	0.589422	+	1.02091i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	6.79736e33	−	2.04859e33i	0.0419632	−	0.0126469i
$512$		0			0
$513$	−1.06657e35	+	1.84735e35i	−0.625840	+	1.08399i
$514$		0			0
$515$		0			0
$516$	1.35144e35	−	2.34076e35i	0.735108	−	1.27324i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	−1.72383e35	+	2.98576e35i	−0.786995	+	1.36312i	0.140805	+	0.990037i	$0.455031\pi$
$523$	−1.72383e35	+	2.98576e35i	−0.786995	+	1.36312i	−0.927800	+	0.373078i	$0.878302\pi$
$524$		0			0
$525$	1.67718e35	+	1.57653e35i	0.728631	+	0.684907i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−1.27026e35	−	2.20016e35i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	7.83176e34	−	3.33169e35i	0.286423	−	1.21847i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	7.47223e34	−	1.29423e35i	0.219728	−	0.380581i	−0.734997	−	0.678071i	$-0.762816\pi$
$541$	7.47223e34	−	1.29423e35i	0.219728	−	0.380581i	0.954725	+	0.297490i	$0.0961496\pi$
$542$		0			0
$543$	−3.02375e35	−	5.23728e35i	−0.847515	−	1.46794i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−2.53378e34			−0.0645557			−0.0322778	−	0.999479i	$-0.510276\pi$
$547$	−2.53378e34			−0.0645557			−0.0322778	+	0.999479i	$0.510276\pi$
$548$		0			0
$549$	−3.35343e35	−	5.80831e35i	−0.814795	−	1.41127i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−5.18672e35	−	4.87547e35i	−1.14674	−	1.07793i
$554$		0			0
$555$		0			0
$556$	8.41575e33	+	1.45765e34i	0.0173429	+	0.0300387i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	1.52984e36			2.93962
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	5.99379e35	−	1.80641e35i	0.957462	−	0.288560i
$568$		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
$571$	6.02488e35	+	1.04354e36i	0.878373	+	1.52139i	0.853126	+	0.521705i	$0.174704\pi$
$571$	6.02488e35	+	1.04354e36i	0.878373	+	1.52139i	0.0252467	+	0.999681i	$0.491963\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−3.84116e35	+	6.65308e35i	−0.500000	+	0.866025i
$577$	−5.63846e35	−	9.76610e35i	−0.717587	−	1.24290i	−0.961953	−	0.273215i	$-0.911913\pi$
$577$	−5.63846e35	−	9.76610e35i	−0.717587	−	1.24290i	0.244366	−	0.969683i	$-0.421420\pi$
$578$		0			0
$579$	5.57139e35	−	9.64993e35i	0.677864	−	1.17409i
$580$		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			$\pi$
$588$	−8.37131e35	+	5.54999e35i	−0.833467	+	0.552570i
$589$	9.90125e35			0.964254
$590$		0			0
$591$		0			0
$592$	−5.90839e35	−	1.02336e36i	−0.538625	−	0.932926i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.22156e36	+	2.11580e36i	−0.998271	+	1.72906i
$598$		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
$601$	1.61995e36			1.21376			0.606882	−	0.794792i	$-0.292420\pi$
$601$	1.61995e36			1.21376			0.606882	+	0.794792i	$0.292420\pi$
$602$		0			0
$603$	−2.65628e36			−1.90612
$604$	−6.34408e35	+	1.09883e36i	−0.445542	+	0.771702i
$605$		0			0
$606$		0			0
$607$	1.40186e36	−	2.42809e36i	0.923106	−	1.59887i	0.128526	−	0.991706i	$-0.458975\pi$
$607$	1.40186e36	−	2.42809e36i	0.923106	−	1.59887i	0.794580	−	0.607160i	$-0.207691\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	9.37994e35	+	1.62465e36i	0.543518	+	0.941401i	0.998699	+	0.0510016i	$0.0162414\pi$
$613$	9.37994e35	+	1.62465e36i	0.543518	+	0.941401i	−0.455181	+	0.890399i	$0.650425\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	3.20032e35	+	5.54312e35i	0.163386	+	0.282993i	0.936081	−	0.351784i	$-0.114425\pi$
$619$	3.20032e35	+	5.54312e35i	0.163386	+	0.282993i	−0.772695	+	0.634778i	$0.781092\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−4.34820e36			−1.99945
$625$	−1.11022e36	+	1.92296e36i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−2.26034e36	+	3.91502e36i	−0.956530	+	1.65676i
$629$		0			0
$630$		0			0
$631$	4.33927e36			1.72598			0.862988	−	0.505224i	$-0.168590\pi$
$631$	4.33927e36			1.72598			0.862988	+	0.505224i	$0.168590\pi$
$632$		0			0
$633$	2.24238e36	+	3.88391e36i	0.855974	+	1.48259i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−5.08954e36	−	2.53273e36i	−1.79005	−	0.890792i
$638$		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
$643$	5.75982e36			1.79335			0.896677	−	0.442685i	$-0.145974\pi$
$643$	5.75982e36			1.79335			0.896677	+	0.442685i	$0.145974\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−2.11718e36	−	1.99013e36i	−0.561314	−	0.527630i
$652$	3.21825e36			0.836375
$653$		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
$657$	1.86250e35			0.0438275
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	7.06329e35	+	1.22340e36i	0.153599	+	0.266041i	0.932548	−	0.361046i	$-0.117580\pi$
$661$	7.06329e35	+	1.22340e36i	0.153599	+	0.266041i	−0.778949	+	0.627087i	$0.784247\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−5.28571e36	−	9.15512e36i	−0.983024	−	1.70265i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	2.56032e36			0.440656			0.220328	−	0.975426i	$-0.429287\pi$
$673$	2.56032e36			0.440656			0.220328	+	0.975426i	$0.429287\pi$
$674$		0			0
$675$	3.01939e36	+	5.22973e36i	0.500000	+	0.866025i
$676$	−9.22740e36	−	1.59823e37i	−1.49890	−	2.59617i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−3.98777e35	+	1.69643e36i	−0.0611538	+	0.260153i
$680$		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	$0.333333\pi$
$684$	4.48944e36	−	7.77594e36i	0.625840	−	1.08399i
$685$		0			0
$686$		0			0
$687$	1.48379e37			1.95404
$688$	−5.68854e36	+	9.85284e36i	−0.735108	+	1.27324i
$689$		0			0
$690$		0			0
$691$	7.99662e36	−	1.38506e37i	0.976545	−	1.69142i	0.301804	−	0.953370i	$-0.402411\pi$
$691$	7.99662e36	−	1.38506e37i	0.976545	−	1.69142i	0.674741	−	0.738055i	$-0.264256\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−7.05963e36	−	6.63599e36i	−0.728631	−	0.684907i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	6.90556e36	+	1.19608e37i	0.674186	+	1.16772i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	8.80625e35	−	1.52529e36i	0.0769821	−	0.133337i	−0.824964	−	0.565185i	$-0.808805\pi$
$709$	8.80625e35	−	1.52529e36i	0.0769821	−	0.133337i	0.901946	+	0.431848i	$0.142138\pi$
$710$		0			0
$711$	−9.33756e36	−	1.61731e37i	−0.786917	−	1.36298i
$712$		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	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	−1.99817e37	−	1.87827e37i	−1.40435	−	1.32008i
$722$		0			0
$723$	−1.35629e37	+	2.34916e37i	−0.919503	+	1.59263i
$724$	1.27277e37	+	2.20450e37i	0.847515	+	1.46794i
$725$		0			0
$726$		0			0
$727$	6.41492e36			0.404803			0.202402	−	0.979303i	$-0.435125\pi$
$727$	6.41492e36			0.404803			0.202402	+	0.979303i	$0.435125\pi$
$728$		0			0
$729$	1.64232e37			1.00000
$730$		0			0
$731$		0			0
$732$	1.41154e37	+	2.44485e37i	0.814795	+	1.41127i
$733$	−1.71737e37	+	2.97457e37i	−0.973893	+	1.68683i	−0.290354	+	0.956919i	$0.593773\pi$
$733$	−1.71737e37	+	2.97457e37i	−0.973893	+	1.68683i	−0.683540	+	0.729913i	$0.739560\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.77714e37	−	3.07809e37i	−0.906448	−	1.57001i	−0.818962	−	0.573848i	$-0.805450\pi$
$739$	−1.77714e37	−	3.07809e37i	−0.906448	−	1.57001i	−0.0874864	−	0.996166i	$-0.527883\pi$
$740$		0			0
$741$	5.08206e37			2.50267
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−2.41685e37	+	4.18611e37i	−0.999841	+	1.73178i	−0.484473	+	0.874806i	$0.660989\pi$
$751$	−2.41685e37	+	4.18611e37i	−0.999841	+	1.73178i	−0.515368	+	0.856969i	$0.672345\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−2.52293e37	+	7.60359e36i	−0.957462	+	0.288560i
$757$	−5.22180e37			−1.94793			−0.973966	−	0.226692i	$-0.927209\pi$
$757$	−5.22180e37			−1.94793			−0.973966	+	0.226692i	$0.927209\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	−5.23962e37	+	1.57912e37i	−1.76393	+	0.531612i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	1.61684e37	−	2.80044e37i	0.500000	−	0.866025i
$769$	6.53834e37			1.98804			0.994021	−	0.109192i	$-0.0348264\pi$
$769$	6.53834e37			1.98804			0.994021	+	0.109192i	$0.0348264\pi$
$770$		0			0
$771$		0			0
$772$	−2.34513e37	+	4.06189e37i	−0.677864	+	1.17409i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	1.40149e37	−	2.42745e37i	0.385184	−	0.667158i
$776$		0			0
$777$	9.27481e36	−	3.94558e37i	0.246508	−	1.04867i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	3.52369e37	−	2.33612e37i	0.833467	−	0.552570i
$785$		0			0
$786$		0			0
$787$	1.57646e36	+	2.73052e36i	0.0354824	+	0.0614573i	0.883221	−	0.468957i	$-0.155370\pi$
$787$	1.57646e36	+	2.73052e36i	0.0354824	+	0.0614573i	−0.847739	+	0.530414i	$0.822037\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−7.98932e37	+	1.38379e38i	−1.62914	+	2.82175i
$794$		0			0
$795$		0			0
$796$	5.14184e37	−	8.90592e37i	0.998271	−	1.72906i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	1.11809e38			1.90612
$805$		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
$811$	7.65605e36			0.116609			0.0583047	−	0.998299i	$-0.481431\pi$
$811$	7.65605e36			0.116609			0.0583047	+	0.998299i	$0.481431\pi$
$812$		0			0
$813$	3.56697e37			0.526166
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.64861e37	−	1.15157e38i	0.920119	−	1.59369i
$818$		0			0
$819$	−1.08670e38	−	1.02149e38i	−1.45686	−	1.36944i
$820$		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
$823$	−2.72019e37	−	4.71150e37i	−0.342295	−	0.592873i	0.642563	−	0.766233i	$-0.277871\pi$
$823$	−2.72019e37	−	4.71150e37i	−0.342295	−	0.592873i	−0.984859	+	0.173360i	$0.944538\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−8.73078e37	−	1.51221e38i	−0.999641	−	1.73143i	−0.523031	−	0.852314i	$-0.675199\pi$
$829$	−8.73078e37	−	1.51221e38i	−0.999641	−	1.73143i	−0.476610	−	0.879115i	$-0.658135\pi$
$830$		0			0
$831$	−8.68899e37	+	1.50498e38i	−0.964175	+	1.67000i
$832$	1.83026e38			1.99945
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.81153e37	−	6.60176e37i	−0.385184	−	0.667158i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	1.05281e38			1.00000
$842$		0			0
$843$		0			0
$844$	−9.43870e37	−	1.63483e38i	−0.855974	−	1.48259i
$845$		0			0
$846$		0			0
$847$	−8.41379e37	−	7.90889e37i	−0.728631	−	0.684907i
$848$		0			0
$849$	−8.00057e37	+	1.38574e38i	−0.671925	+	1.16381i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	1.37475e38			1.08614			0.543072	−	0.839686i	$-0.317261\pi$
$853$	1.37475e38			1.08614			0.543072	+	0.839686i	$0.317261\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	−7.61698e36	+	1.31930e37i	−0.0549379	+	0.0951552i	−0.892186	−	0.451667i	$-0.850829\pi$
$859$	−7.61698e36	+	1.31930e37i	−0.0549379	+	0.0951552i	0.837249	+	0.546822i	$0.184163\pi$
$860$		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
$867$	−1.56404e38			−1.00000
$868$	8.91173e37	+	8.37695e37i	0.561314	+	0.527630i
$869$		0			0
$870$		0			0
$871$	3.16421e38	+	5.48058e38i	1.90559	+	3.30058i
$872$		0			0
$873$	−2.28593e37	+	3.95935e37i	−0.133622	+	0.231440i
$874$		0			0
$875$		0			0
$876$	−7.83972e36			−0.0438275
$877$	−1.78298e38	+	3.08821e38i	−0.982091	+	1.70103i	−0.327878	+	0.944720i	$0.606334\pi$
$877$	−1.78298e38	+	3.08821e38i	−0.982091	+	1.70103i	−0.654212	+	0.756311i	$0.727000\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−6.11158e37			−0.308080			−0.154040	−	0.988065i	$-0.549229\pi$
$883$	−6.11158e37			−0.308080			−0.154040	+	0.988065i	$0.549229\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	5.84379e37	−	2.48600e38i	0.269756	−	1.14757i
$890$		0			0
$891$		0			0
$892$	2.22488e38	+	3.85361e38i	0.983024	+	1.70265i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−1.27093e38	−	2.20132e38i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−3.73631e38	+	1.12605e38i	−1.40768	+	0.424245i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.48658e38	−	2.57483e38i	−0.528802	−	0.915911i	−0.999436	−	0.0335829i	$-0.989308\pi$
$907$	−1.48658e38	−	2.57483e38i	−0.528802	−	0.915911i	0.470634	−	0.882328i	$-0.344025\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−1.88971e38	+	3.27308e38i	−0.625840	+	1.08399i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	−6.24561e38			−1.95404
$917$		0			0
$918$		0			0
$919$	3.03304e38	−	5.25338e38i	0.909445	−	1.57520i	0.0946072	−	0.995515i	$-0.469840\pi$
$919$	3.03304e38	−	5.25338e38i	0.909445	−	1.57520i	0.814837	−	0.579690i	$-0.196826\pi$
$920$		0			0
$921$	9.65139e37	+	1.67167e38i	0.281329	+	0.487276i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.90984e38			1.07725
$926$		0			0
$927$	−3.59728e38	−	6.23067e38i	−0.963690	−	1.66916i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	−4.11839e38	+	2.73040e38i	−1.04323	+	0.691640i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	8.52639e38			1.98678			0.993392	−	0.114771i	$-0.0366135\pi$
$937$	8.52639e38			1.98678			0.993392	+	0.114771i	$0.0366135\pi$
$938$		0			0
$939$	6.80341e38			1.54196
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		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	$0.666667\pi$
$948$	3.93040e38	+	6.80766e38i	0.786917	+	1.36298i
$949$	−2.21865e37	−	3.84281e37i	−0.0438155	−	0.0758906i
$950$		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
$961$	1.21191e38	−	2.09909e38i	0.203267	−	0.352068i
$962$		0			0
$963$		0			0
$964$	5.70893e38	−	9.88816e38i	0.919503	−	1.59263i
$965$		0			0
$966$		0			0
$967$	4.90224e38			0.758316			0.379158	−	0.925332i	$-0.376214\pi$
$967$	4.90224e38			0.758316			0.379158	+	0.925332i	$0.376214\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−6.91292e38			−1.00000
$973$	5.56074e36	−	2.36558e37i	0.00793717	−	0.0337654i
$974$		0			0
$975$	7.19350e38	−	1.24595e39i	0.999724	−	1.73157i
$976$	−5.94150e38	−	1.02910e39i	−0.814795	−	1.41127i
$977$		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
$981$	−1.43568e39			−1.84230
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	−2.13916e39			−2.50267
$989$		0			0
$990$		0			0
$991$	6.03092e38	+	1.04459e39i	0.678306	+	1.17486i	0.975491	+	0.220041i	$0.0706190\pi$
$991$	6.03092e38	+	1.04459e39i	0.678306	+	1.17486i	−0.297184	+	0.954820i	$0.596048\pi$
$992$		0			0
$993$	1.37882e39			1.51067
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−4.34984e38	−	7.53414e38i	−0.452310	−	0.783423i	0.546219	−	0.837642i	$-0.316066\pi$
$997$	−4.34984e38	−	7.53414e38i	−0.452310	−	0.783423i	−0.998529	+	0.0542188i	$0.982733\pi$
$998$		0			0
$999$	5.31665e38	−	9.20871e38i	0.538625	−	0.932926i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.27.h.a.2.1	✓	2
3.2	odd	2	CM	21.27.h.a.2.1	✓	2
7.4	even	3	inner	21.27.h.a.11.1	yes	2
21.11	odd	6	inner	21.27.h.a.11.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.27.h.a.2.1	✓	2	1.1	even	1	trivial
21.27.h.a.2.1	✓	2	3.2	odd	2	CM
21.27.h.a.11.1	yes	2	7.4	even	3	inner
21.27.h.a.11.1	yes	2	21.11	odd	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:	\( k \)	\(=\)	\( 27 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(797162. + 1.38072e6i) q^{3} +(-3.35544e7 - 5.81180e7i) q^{4} +(-2.21712e10 + 9.43182e10i) q^{7} +(-1.27093e12 + 2.20132e12i) q^{9} +O(q^{10})\)	\(q+(797162. + 1.38072e6i) q^{3} +(-3.35544e7 - 5.81180e7i) q^{4} +(-2.21712e10 + 9.43182e10i) q^{7} +(-1.27093e12 + 2.20132e12i) q^{9} +(5.34966e13 - 9.26588e13i) q^{12} +6.05583e14 q^{13} +(-2.25180e15 + 3.90023e15i) q^{16} +(2.63184e16 - 4.55848e16i) q^{19} +(-1.47901e17 + 4.45745e16i) q^{21} +(-7.45058e17 - 1.29048e18i) q^{25} -4.05256e18 q^{27} +(6.22552e18 - 1.87625e18i) q^{28} +(9.40525e18 + 1.62904e19i) q^{31} +1.70582e20 q^{36} +(-1.31193e20 + 2.27232e20i) q^{37} +(4.82748e20 + 8.36144e20i) q^{39} +2.52622e21 q^{43} -7.18019e21 q^{48} +(-8.40435e21 - 4.18230e21i) q^{49} +(-2.03200e22 - 3.51953e22i) q^{52} +8.39201e22 q^{57} +(-1.31928e23 + 2.28506e23i) q^{61} +(-1.79446e23 - 1.68678e23i) q^{63} +3.02231e23 q^{64} +(5.22507e23 + 9.05008e23i) q^{67} +(-3.66365e22 - 6.34563e22i) q^{73} +(1.18786e24 - 2.05744e24i) q^{75} -3.53240e24 q^{76} +(-3.67351e24 + 6.36270e24i) q^{79} +(-3.23054e24 - 5.59546e24i) q^{81} +(7.55333e24 + 7.10006e24i) q^{84} +(-1.34265e25 + 5.71175e25i) q^{91} +(-1.49950e25 + 2.59721e25i) q^{93} +1.79863e25 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1594323 q^{3} - 67108864 q^{4} - 44342429053 q^{7} - 2541865828329 q^{9}+O(q^{10}) \) 2 * q + 1594323 * q^3 - 67108864 * q^4 - 44342429053 * q^7 - 2541865828329 * q^9	\( 2 q + 1594323 q^{3} - 67108864 q^{4} - 44342429053 q^{7} - 2541865828329 q^{9} + 106993205379072 q^{12} + 12\!\cdots\!66 q^{13}+ \cdots + 35\!\cdots\!84 q^{97}+O(q^{100}) \) 2 * q + 1594323 * q^3 - 67108864 * q^4 - 44342429053 * q^7 - 2541865828329 * q^9 + 106993205379072 * q^12 + 1211166615597166 * q^13 - 4503599627370496 * q^16 + 52636841466889669 * q^19 - 295802846374395366 * q^21 - 1490116119384765625 * q^25 - 8105110306037952534 * q^27 + 12451048494033010688 * q^28 + 18810497092058970493 * q^31 + 341163456359156416512 * q^36 - 262385069961736743767 * q^37 + 965495396039360244309 * q^39 + 5052437468927806557478 * q^43 - 14360384937416422588416 * q^48 - 16808709660975170134489 * q^49 - 40640007843725245939712 * q^52 + 167840253996031875498174 * q^57 - 263855512891444028536874 * q^61 - 358892776285241383064781 * q^63 + 604462909807314587353088 * q^64 + 1045013466379120827629053 * q^67 - 73273034217751719361823 * q^73 + 2375726401805877685546875 * q^75 - 7064797270782118599852032 * q^76 - 7347014375367289352249531 * q^79 - 6461081889226673298932241 * q^81 + 15106654369477633568145408 * q^84 - 26853034861739728291431899 * q^91 - 29990008155302734013311239 * q^93 + 35972525076053703782561284 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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