LMFDB - Embedded newform 21.14.g.a.5.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,14,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, 14, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 14);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 14 $
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:	$22.5184950799$
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			21.14.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)

$f(q)$	$=$	$q+(-1093.50 + 631.333i) q^{3} +(-4096.00 - 7094.48i) q^{4} +(193284. - 243988. i) q^{7} +(797162. - 1.38072e6i) q^{9} +O(q^{10})$	\(q+(-1093.50 + 631.333i) q^{3} +(-4096.00 - 7094.48i) q^{4} +(193284. - 243988. i) q^{7} +(797162. - 1.38072e6i) q^{9} +(8.95795e6 + 5.17188e6i) q^{12} +408541. i q^{13} +(-3.35544e7 + 5.81180e7i) q^{16} +(-3.20245e8 - 1.84894e8i) q^{19} +(-5.73191e7 + 3.88828e8i) q^{21} +(6.10352e8 + 1.05716e9i) q^{25} +2.01310e9i q^{27} +(-2.52266e9 - 3.71878e8i) q^{28} +(-7.12279e9 + 4.11235e9i) q^{31} -1.30607e10 q^{36} +(-1.36887e10 + 2.37095e10i) q^{37} +(-2.57925e8 - 4.46739e8i) q^{39} +7.72188e10 q^{43} -8.47360e10i q^{48} +(-2.21712e10 - 9.43182e10i) q^{49} +(2.89838e9 - 1.67338e9i) q^{52} +4.66918e11 q^{57} +(-2.12088e11 - 1.22449e11i) q^{61} +(-1.82801e11 - 4.61370e11i) q^{63} +5.49756e11 q^{64} +(1.13436e11 + 1.96477e11i) q^{67} +(-1.56615e12 + 9.04216e11i) q^{73} +(-1.33484e12 - 7.70670e11i) q^{75} +3.02930e12i q^{76} +(-2.04227e12 + 3.53732e12i) q^{79} +(-1.27093e12 - 2.20132e12i) q^{81} +(2.99331e12 - 1.18599e12i) q^{84} +(9.96790e10 + 7.89646e10i) q^{91} +(5.19252e12 - 8.99370e12i) q^{93} +1.07990e13i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2187 q^{3} - 8192 q^{4} + 386569 q^{7} + 1594323 q^{9}+O(q^{10}) $ 2 * q - 2187 * q^3 - 8192 * q^4 + 386569 * q^7 + 1594323 * q^9	$ 2 q - 2187 q^{3} - 8192 q^{4} + 386569 q^{7} + 1594323 q^{9} + 17915904 q^{12} - 67108864 q^{16} - 640490769 q^{19} - 114638166 q^{21} + 1220703125 q^{25} - 5045321728 q^{28} - 14245587705 q^{31} - 26121388032 q^{36} - 27377427169 q^{37} - 515849877 q^{39} + 154437650630 q^{43} - 44342429053 q^{49} + 5796765696 q^{52} + 933835541202 q^{57} - 424175890308 q^{61} - 365602178745 q^{63} + 1099511627776 q^{64} + 226871917261 q^{67} - 3132295953723 q^{73} - 2669677734375 q^{75} - 4084540747547 q^{79} - 2541865828329 q^{81} + 5986617237504 q^{84} + 199357933329 q^{91} + 10385033436945 q^{93}+O(q^{100}) $ 2 * q - 2187 * q^3 - 8192 * q^4 + 386569 * q^7 + 1594323 * q^9 + 17915904 * q^12 - 67108864 * q^16 - 640490769 * q^19 - 114638166 * q^21 + 1220703125 * q^25 - 5045321728 * q^28 - 14245587705 * q^31 - 26121388032 * q^36 - 27377427169 * q^37 - 515849877 * q^39 + 154437650630 * q^43 - 44342429053 * q^49 + 5796765696 * q^52 + 933835541202 * q^57 - 424175890308 * q^61 - 365602178745 * q^63 + 1099511627776 * q^64 + 226871917261 * q^67 - 3132295953723 * q^73 - 2669677734375 * q^75 - 4084540747547 * q^79 - 2541865828329 * q^81 + 5986617237504 * q^84 + 199357933329 * q^91 + 10385033436945 * 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	$0.666667\pi$
$3$	−1093.50	+	631.333i	−0.866025	+	0.500000i
$3$	−1093.50	+	631.333i	−0.866025	+	0.500000i
$4$	−4096.00	−	7094.48i	−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
$7$	193284.	−	243988.i	0.620954	−	0.783847i
$7$	193284.	−	243988.i	0.620954	−	0.783847i
$8$		0			0
$9$	797162.	−	1.38072e6i	0.500000	−	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	8.95795e6	+	5.17188e6i	0.866025	+	0.500000i
$13$			408541.i			0.0234749i	0.999931	+	0.0117374i	$0.00373623\pi$
$13$			408541.i			0.0234749i	−0.999931	+	0.0117374i	$0.996264\pi$
$14$		0			0
$15$		0			0
$16$	−3.35544e7	+	5.81180e7i	−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
$19$	−3.20245e8	−	1.84894e8i	−1.56165	−	0.901621i	−0.997090	−	0.0762303i	$-0.975712\pi$
$19$	−3.20245e8	−	1.84894e8i	−1.56165	−	0.901621i	−0.564563	−	0.825390i	$-0.690955\pi$
$20$		0			0
$21$	−5.73191e7	+	3.88828e8i	−0.145839	+	0.989308i
$22$		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
$25$	6.10352e8	+	1.05716e9i	0.500000	+	0.866025i
$26$		0			0
$27$			2.01310e9i			1.00000i
$28$	−2.52266e9	−	3.71878e8i	−0.989308	−	0.145839i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−7.12279e9	+	4.11235e9i	−1.44145	+	0.832221i	−0.997947	−	0.0640519i	$-0.979598\pi$
$31$	−7.12279e9	+	4.11235e9i	−1.44145	+	0.832221i	−0.443503	+	0.896273i	$0.646264\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.30607e10			−1.00000
$37$	−1.36887e10	+	2.37095e10i	−0.877105	+	1.51919i	−0.0226010	+	0.999745i	$0.507195\pi$
$37$	−1.36887e10	+	2.37095e10i	−0.877105	+	1.51919i	−0.854504	+	0.519445i	$0.826139\pi$
$38$		0			0
$39$	−2.57925e8	−	4.46739e8i	−0.0117374	−	0.0203298i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	7.72188e10			1.86285			0.931426	−	0.363931i	$-0.118566\pi$
$43$	7.72188e10			1.86285			0.931426	+	0.363931i	$0.118566\pi$
$44$		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	$0.166667\pi$
$48$		−	8.47360e10i		−	1.00000i
$49$	−2.21712e10	−	9.43182e10i	−0.228831	−	0.973466i
$50$		0			0
$51$		0			0
$52$	2.89838e9	−	1.67338e9i	0.0203298	−	0.0117374i
$53$		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
$57$	4.66918e11			1.80324
$58$		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
$61$	−2.12088e11	−	1.22449e11i	−0.527075	−	0.304307i	0.212750	−	0.977107i	$-0.431758\pi$
$61$	−2.12088e11	−	1.22449e11i	−0.527075	−	0.304307i	−0.739824	+	0.672800i	$0.765091\pi$
$62$		0			0
$63$	−1.82801e11	−	4.61370e11i	−0.368354	−	0.929686i
$64$	5.49756e11			1.00000
$65$		0			0
$66$		0			0
$67$	1.13436e11	+	1.96477e11i	0.153202	+	0.265354i	0.932403	−	0.361421i	$-0.117708\pi$
$67$	1.13436e11	+	1.96477e11i	0.153202	+	0.265354i	−0.779201	+	0.626774i	$0.784375\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.56615e12	+	9.04216e11i	−1.21125	+	0.699316i	−0.963032	−	0.269388i	$-0.913179\pi$
$73$	−1.56615e12	+	9.04216e11i	−1.21125	+	0.699316i	−0.248219	+	0.968704i	$0.579845\pi$
$74$		0			0
$75$	−1.33484e12	−	7.70670e11i	−0.866025	−	0.500000i
$76$			3.02930e12i			1.80324i
$77$		0			0
$78$		0			0
$79$	−2.04227e12	+	3.53732e12i	−0.945229	+	1.63719i	−0.189938	+	0.981796i	$0.560829\pi$
$79$	−2.04227e12	+	3.53732e12i	−0.945229	+	1.63719i	−0.755292	+	0.655389i	$0.772505\pi$
$80$		0			0
$81$	−1.27093e12	−	2.20132e12i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	2.99331e12	−	1.18599e12i	0.929686	−	0.368354i
$85$		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
$91$	9.96790e10	+	7.89646e10i	0.0184007	+	0.0145768i
$92$		0			0
$93$	5.19252e12	−	8.99370e12i	0.832221	−	1.44145i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.07990e13i			1.31634i	0.752869	+	0.658171i	$0.228670\pi$
$97$			1.07990e13i			1.31634i	−0.752869	+	0.658171i	$0.771330\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e12	−	8.66025e12i	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
$103$	−2.82813e12	−	1.63282e12i	−0.233377	−	0.134740i	0.378752	−	0.925498i	$-0.376353\pi$
$103$	−2.82813e12	−	1.63282e12i	−0.233377	−	0.134740i	−0.612129	+	0.790758i	$0.709687\pi$
$104$		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	$0.666667\pi$
$108$	1.42819e13	−	8.24564e12i	0.866025	−	0.500000i
$109$	−3.47667e12	−	6.02178e12i	−0.198560	−	0.343916i	0.749502	−	0.662002i	$-0.230293\pi$
$109$	−3.47667e12	−	6.02178e12i	−0.198560	−	0.343916i	−0.948062	+	0.318086i	$0.896960\pi$
$110$		0			0
$111$		−	3.45685e13i		−	1.75421i
$112$	7.69453e12	+	1.94202e13i	0.368354	+	0.929686i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	5.64082e11	+	3.25673e11i	0.0203298	+	0.0117374i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.72614e13	+	2.98975e13i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	5.83499e13	+	3.36883e13i	1.44145	+	0.832221i
$125$		0			0
$126$		0			0
$127$	−4.28486e13			−0.906176			−0.453088	−	0.891466i	$-0.649678\pi$
$127$	−4.28486e13			−0.906176			−0.453088	+	0.891466i	$0.649678\pi$
$128$		0			0
$129$	−8.44388e13	+	4.87508e13i	−1.61328	+	0.931426i
$130$		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
$133$	−1.07010e14	+	4.23989e13i	−1.67645	+	0.664231i
$134$		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
$139$		−	1.21296e14i		−	1.42642i	−0.700948	−	0.713212i	$-0.747240\pi$
$139$		−	1.21296e14i		−	1.42642i	0.700948	−	0.713212i	$-0.252760\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	5.34966e13	+	9.26588e13i	0.500000	+	0.866025i
$145$		0			0
$146$		0			0
$147$	8.37904e13	+	8.91395e13i	0.684907	+	0.728631i
$148$	2.24276e14			1.75421
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−1.23837e14	−	2.14492e14i	−0.850159	−	1.47252i	−0.881064	−	0.472997i	$-0.843172\pi$
$151$	−1.23837e14	−	2.14492e14i	−0.850159	−	1.47252i	0.0309046	−	0.999522i	$-0.490161\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.11292e12	+	3.65969e12i	−0.0117374	+	0.0203298i
$157$	4.79171e13	−	2.76649e13i	0.255354	−	0.147429i	−0.366859	−	0.930276i	$-0.619567\pi$
$157$	4.79171e13	−	2.76649e13i	0.255354	−	0.147429i	0.622213	+	0.782848i	$0.286234\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	2.01637e14	−	3.49246e14i	0.842077	−	1.45852i	−0.0460587	−	0.998939i	$-0.514666\pi$
$163$	2.01637e14	−	3.49246e14i	0.842077	−	1.45852i	0.888136	−	0.459581i	$-0.152001\pi$
$164$		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
$169$	3.02708e14			0.999449
$170$		0			0
$171$	−5.10575e14	+	2.94780e14i	−1.56165	+	0.901621i
$172$	−3.16288e14	−	5.47827e14i	−0.931426	−	1.61328i
$173$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$173$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$174$		0			0
$175$	3.75906e14	+	5.54142e13i	0.989308	+	0.145839i
$176$		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
$181$			9.09326e14i			1.92225i	0.276120	+	0.961123i	$0.410951\pi$
$181$			9.09326e14i			1.92225i	−0.276120	+	0.961123i	$0.589049\pi$
$182$		0			0
$183$	3.09224e14			0.608613
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	4.91171e14	+	3.89100e14i	0.783847	+	0.620954i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	−6.01158e14	+	3.47079e14i	−0.866025	+	0.500000i
$193$	−6.57636e14	−	1.13906e15i	−0.915932	−	1.58644i	−0.805531	−	0.592554i	$-0.798120\pi$
$193$	−6.57636e14	−	1.13906e15i	−0.915932	−	1.58644i	−0.110402	−	0.993887i	$-0.535214\pi$
$194$		0			0
$195$		0			0
$196$	−5.78325e14	+	5.43620e14i	−0.728631	+	0.684907i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−1.51676e15	+	8.75704e14i	−1.73130	+	0.999568i	−0.850950	+	0.525246i	$0.823973\pi$
$199$	−1.51676e15	+	8.75704e14i	−1.73130	+	0.999568i	−0.880352	+	0.474322i	$0.842693\pi$
$200$		0			0
$201$	−2.48084e14	−	1.43232e14i	−0.265354	−	0.153202i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−2.37436e13	−	1.37083e13i	−0.0203298	−	0.0117374i
$209$		0			0
$210$		0			0
$211$	2.46966e15			1.92664			0.963321	−	0.268352i	$-0.0864791\pi$
$211$	2.46966e15			1.92664			0.963321	+	0.268352i	$0.0864791\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.73363e14	+	2.53273e15i	−0.242741	+	1.64665i
$218$		0			0
$219$	1.14172e15	−	1.97752e15i	0.699316	−	1.21125i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	3.65730e15i		−	1.99149i	−0.0921315	−	0.995747i	$-0.529368\pi$
$223$		−	3.65730e15i		−	1.99149i	0.0921315	−	0.995747i	$-0.470632\pi$
$224$		0			0
$225$	1.94620e15			1.00000
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$	−1.91250e15	−	3.31254e15i	−0.901621	−	1.56165i
$229$	3.75822e15	+	2.16981e15i	1.72207	+	0.994239i	0.914630	+	0.404291i	$0.132482\pi$
$229$	3.75822e15	+	2.16981e15i	1.72207	+	0.994239i	0.807442	+	0.589947i	$0.200851\pi$
$230$		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
$237$		−	5.15741e15i		−	1.89046i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−5.16120e15	+	2.97982e15i	−1.69684	+	0.979669i	−0.748108	+	0.663577i	$0.769038\pi$
$241$	−5.16120e15	+	2.97982e15i	−1.69684	+	0.979669i	−0.948728	+	0.316093i	$0.897629\pi$
$242$		0			0
$243$	2.77953e15	+	1.60476e15i	0.866025	+	0.500000i
$244$			2.00620e15i			0.608613i
$245$		0			0
$246$		0			0
$247$	7.55366e13	−	1.30833e14i	0.0211654	−	0.0366596i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−2.52443e15	+	3.18665e15i	−0.620954	+	0.783847i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−2.25180e15	−	3.90023e15i	−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
$259$	3.13903e15	+	7.92257e15i	0.646170	+	1.63086i
$260$		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
$268$	9.29267e14	−	1.60954e15i	0.153202	−	0.265354i
$269$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$269$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$270$		0			0
$271$	−8.97557e15	−	5.18205e15i	−1.37645	−	0.794696i	−0.384723	−	0.923032i	$-0.625703\pi$
$271$	−8.97557e15	−	5.18205e15i	−1.37645	−	0.794696i	−0.991731	+	0.128336i	$0.959036\pi$
$272$		0			0
$273$	−1.58852e14	−	2.34172e13i	−0.0232239	−	0.00342355i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.00623e15	+	1.74284e15i	0.133838	+	0.231814i	0.925153	−	0.379595i	$-0.123937\pi$
$277$	1.00623e15	+	1.74284e15i	0.133838	+	0.231814i	−0.791315	+	0.611408i	$0.790603\pi$
$278$		0			0
$279$			1.31128e16i			1.66444i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.36857e16	+	7.90142e15i	−1.58363	+	0.914310i	−0.589308	+	0.807908i	$0.700600\pi$
$283$	−1.36857e16	+	7.90142e15i	−1.58363	+	0.914310i	−0.994323	+	0.106401i	$0.966067\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	4.95229e15	−	8.57762e15i	0.500000	−	0.866025i
$290$		0			0
$291$	−6.81778e15	−	1.18087e16i	−0.658171	−	1.13999i
$292$	1.28299e16	+	7.40734e15i	1.21125	+	0.699316i
$293$		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
$300$			1.26267e16i			1.00000i
$301$	1.49252e16	−	1.88405e16i	1.15675	−	1.46019i
$302$		0			0
$303$		0			0
$304$	2.14913e16	−	1.24080e16i	1.56165	−	0.901621i
$305$		0			0
$306$		0			0
$307$		−	1.75865e16i		−	1.19889i	−0.800415	−	0.599446i	$-0.795388\pi$
$307$		−	1.75865e16i		−	1.19889i	0.800415	−	0.599446i	$-0.204612\pi$
$308$		0			0
$309$	4.12341e15			0.269480
$310$		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
$313$	−2.71138e16	−	1.56542e16i	−1.62987	−	0.941005i	−0.984131	−	0.177446i	$-0.943217\pi$
$313$	−2.71138e16	−	1.56542e16i	−1.62987	−	0.941005i	−0.645738	−	0.763559i	$-0.723450\pi$
$314$		0			0
$315$		0			0
$316$	3.34606e16			1.89046
$317$		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
$324$	−1.04115e16	+	1.80332e16i	−0.500000	+	0.866025i
$325$	−4.31893e14	+	2.49353e14i	−0.0203298	+	0.0117374i
$326$		0			0
$327$	7.60349e15	+	4.38988e15i	0.343916	+	0.198560i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−8.36864e15	+	1.44949e16i	−0.349762	+	0.605806i	−0.986207	−	0.165516i	$-0.947071\pi$
$331$	−8.36864e15	+	1.44949e16i	−0.349762	+	0.605806i	0.636445	+	0.771322i	$0.280404\pi$
$332$		0			0
$333$	2.18242e16	+	3.78007e16i	0.877105	+	1.51919i
$334$		0			0
$335$		0			0
$336$	−2.06746e16	−	1.63782e16i	−0.783847	−	0.620954i
$337$	−5.26531e16			−1.95808			−0.979039	−	0.203675i	$-0.934711\pi$
$337$	−5.26531e16			−1.95808			−0.979039	+	0.203675i	$0.934711\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−2.72978e16	−	1.28207e16i	−0.905142	−	0.425110i
$344$		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
$349$			1.86957e16i			0.553830i	0.960894	+	0.276915i	$0.0893121\pi$
$349$			1.86957e16i			0.553830i	−0.960894	+	0.276915i	$0.910688\pi$
$350$		0			0
$351$	−8.22431e14			−0.0234749
$352$		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
$361$	4.73449e16	+	8.20038e16i	1.12584	+	1.95001i
$362$		0			0
$363$		−	4.35906e16i		−	1.00000i
$364$	1.51927e14	−	1.03061e15i	0.00342355	−	0.0232239i
$365$		0			0
$366$		0			0
$367$	6.01591e16	−	3.47329e16i	1.28520	−	0.742013i	0.307409	−	0.951577i	$-0.400538\pi$
$367$	6.01591e16	−	3.47329e16i	1.28520	−	0.742013i	0.977795	+	0.209565i	$0.0672047\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−8.50742e16			−1.66444
$373$	−5.03951e16	+	8.72870e16i	−0.968905	+	1.67819i	−0.270170	+	0.962813i	$0.587080\pi$
$373$	−5.03951e16	+	8.72870e16i	−0.968905	+	1.67819i	−0.698735	+	0.715381i	$0.746253\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.47585e16			−1.12238			−0.561192	−	0.827686i	$-0.689657\pi$
$379$	−6.47585e16			−1.12238			−0.561192	+	0.827686i	$0.689657\pi$
$380$		0			0
$381$	4.68550e16	−	2.70517e16i	0.784772	−	0.453088i
$382$		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
$387$	6.15559e16	−	1.06618e17i	0.931426	−	1.61328i
$388$	7.66135e16	−	4.42328e16i	1.13999	−	0.658171i
$389$		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
$397$	2.64904e16	+	1.52943e16i	0.339587	+	0.196061i	0.660089	−	0.751187i	$-0.270518\pi$
$397$	2.64904e16	+	1.52943e16i	0.339587	+	0.196061i	−0.320502	+	0.947248i	$0.603852\pi$
$398$		0			0
$399$	9.02480e16	−	1.13922e17i	1.11973	−	1.41346i
$400$	−8.19200e16			−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
$403$	−1.68006e15	−	2.90995e15i	−0.0195363	−	0.0338378i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	1.04425e17	−	6.02900e16i	1.10307	−	0.636860i	0.166047	−	0.986118i	$-0.446900\pi$
$409$	1.04425e17	−	6.02900e16i	1.10307	−	0.636860i	0.937027	+	0.349258i	$0.113566\pi$
$410$		0			0
$411$		0			0
$412$			2.67521e16i			0.269480i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	7.65778e16	+	1.32637e17i	0.713212	+	1.23532i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	5.77139e15			0.0505181			0.0252590	−	0.999681i	$-0.491959\pi$
$421$	5.77139e15			0.0505181			0.0252590	+	0.999681i	$0.491959\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−7.08694e16	+	2.80794e16i	−0.565819	+	0.224185i
$428$		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	$0.333333\pi$
$432$	−1.16997e17	−	6.75483e16i	−0.866025	−	0.500000i
$433$			2.50198e17i			1.82437i	0.409783	+	0.912183i	$0.365604\pi$
$433$			2.50198e17i			1.82437i	−0.409783	+	0.912183i	$0.634396\pi$
$434$		0			0
$435$		0			0
$436$	−2.84809e16	+	4.93304e16i	−0.198560	+	0.343916i
$437$		0			0
$438$		0			0
$439$	−2.58756e17	−	1.49393e17i	−1.72532	−	0.996116i	−0.906686	−	0.421806i	$-0.861396\pi$
$439$	−2.58756e17	−	1.49393e17i	−1.72532	−	0.996116i	−0.818638	−	0.574310i	$-0.805270\pi$
$440$		0			0
$441$	−1.47901e17	−	4.45745e16i	−0.957462	−	0.288560i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−2.45246e17	+	1.41593e17i	−1.51919	+	0.877105i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	1.06259e17	−	1.34134e17i	0.620954	−	0.783847i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.70832e17	+	1.56365e17i	1.47252	+	0.850159i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.61143e17	+	2.79109e17i	−0.827480	+	1.43324i	0.0725294	+	0.997366i	$0.476893\pi$
$457$	−1.61143e17	+	2.79109e17i	−0.827480	+	1.43324i	−0.900009	+	0.435871i	$0.856440\pi$
$458$		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
$463$	3.67094e17			1.73181			0.865906	−	0.500206i	$-0.166743\pi$
$463$	3.67094e17			1.73181			0.865906	+	0.500206i	$0.166743\pi$
$464$		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	$0.166667\pi$
$468$		−	5.33582e15i		−	0.0234749i
$469$	6.98634e16	+	1.02989e16i	0.303128	+	0.0446857i
$470$		0			0
$471$	−3.49316e16	+	6.05032e16i	−0.147429	+	0.255354i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	4.51401e17i		−	1.80324i
$476$		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
$481$	−9.68631e15	−	5.59239e15i	−0.0356628	−	0.0205899i
$482$		0			0
$483$		0			0
$484$	2.82810e17			1.00000
$485$		0			0
$486$		0			0
$487$	−2.94302e17	−	5.09746e17i	−0.999667	−	1.73147i	−0.522191	−	0.852828i	$-0.674885\pi$
$487$	−2.94302e17	−	5.09746e17i	−0.999667	−	1.73147i	−0.477475	−	0.878645i	$-0.658448\pi$
$488$		0			0
$489$			5.09201e17i			1.68415i
$490$		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
$496$		−	5.51950e17i		−	1.66444i
$497$		0			0
$498$		0			0
$499$	3.13307e17	−	5.42664e17i	0.908483	−	1.57354i	0.0923099	−	0.995730i	$-0.470575\pi$
$499$	3.13307e17	−	5.42664e17i	0.908483	−	1.57354i	0.816173	−	0.577808i	$-0.196092\pi$
$500$		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
$507$	−3.31011e17	+	1.91110e17i	−0.865548	+	0.499724i
$508$	1.75508e17	+	3.03989e17i	0.453088	+	0.784772i
$509$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$509$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$510$		0			0
$511$	−8.20943e16	+	5.56892e17i	−0.203975	+	1.38368i
$512$		0			0
$513$	3.72209e17	−	6.44685e17i	0.901621	−	1.56165i
$514$		0			0
$515$		0			0
$516$	6.91723e17	+	3.99366e17i	1.61328	+	0.931426i
$517$		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
$523$	−2.64546e17	−	1.52736e17i	−0.565250	−	0.326347i	0.190000	−	0.981784i	$-0.439151\pi$
$523$	−2.64546e17	−	1.52736e17i	−0.565250	−	0.326347i	−0.755250	+	0.655437i	$0.772484\pi$
$524$		0			0
$525$	−4.46038e17	+	1.76726e17i	−0.929686	+	0.368354i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−2.52018e17	−	4.36508e17i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	7.39112e17	+	5.85517e17i	1.41346	+	1.11973i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.64242e17	+	6.30886e17i	−0.624609	+	1.08185i	0.364008	+	0.931396i	$0.381408\pi$
$541$	−3.64242e17	+	6.30886e17i	−0.624609	+	1.08185i	−0.988616	+	0.150458i	$0.951925\pi$
$542$		0			0
$543$	−5.74087e17	−	9.94348e17i	−0.961123	−	1.66471i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	−8.71581e17			−1.39120			−0.695601	−	0.718428i	$-0.744862\pi$
$547$	−8.71581e17			−1.39120			−0.695601	+	0.718428i	$0.744862\pi$
$548$		0			0
$549$	−3.38137e17	+	1.95223e17i	−0.527075	+	0.304307i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	4.68323e17	+	1.18200e18i	0.696358	+	1.75753i
$554$		0			0
$555$		0			0
$556$	−8.60529e17	+	4.96827e17i	−1.23532	+	0.713212i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$			3.15470e16i			0.0437302i
$560$		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
$567$	−7.82747e17	−	1.15389e17i	−0.989308	−	0.145839i
$568$		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
$571$	2.04237e17	+	3.53750e17i	0.246604	+	0.427131i	0.962581	−	0.270993i	$-0.0873519\pi$
$571$	2.04237e17	+	3.53750e17i	0.246604	+	0.427131i	−0.715977	+	0.698124i	$0.754019\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	4.38244e17	−	7.59061e17i	0.500000	−	0.866025i
$577$	−5.76940e17	+	3.33096e17i	−0.650860	+	0.375774i	−0.788786	−	0.614668i	$-0.789290\pi$
$577$	−5.76940e17	+	3.33096e17i	−0.650860	+	0.375774i	0.137926	+	0.990443i	$0.455957\pi$
$578$		0			0
$579$	1.43825e18	+	8.30375e17i	1.58644	+	0.915932i
$580$		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	$0.5\pi$
$588$	2.89193e17	−	9.59564e17i	0.288560	−	0.957462i
$589$	3.04139e18			3.00139
$590$		0			0
$591$		0			0
$592$	−9.18634e17	−	1.59112e18i	−0.877105	−	1.51919i
$593$		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
$597$	1.10572e18	−	1.91516e18i	0.999568	−	1.73130i
$598$		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
$601$		−	1.02438e18i		−	0.886700i	−0.896349	−	0.443350i	$-0.853790\pi$
$601$		−	1.02438e18i		−	0.886700i	0.896349	−	0.443350i	$-0.146210\pi$
$602$		0			0
$603$	3.61707e17			0.306404
$604$	−1.01447e18	+	1.75712e18i	−0.850159	+	1.47252i
$605$		0			0
$606$		0			0
$607$	−2.09302e18	−	1.20841e18i	−1.69843	−	0.980588i	−0.947254	−	0.320484i	$-0.896154\pi$
$607$	−2.09302e18	−	1.20841e18i	−1.69843	−	0.980588i	−0.751174	−	0.660104i	$-0.770512\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	6.27610e17	+	1.08705e18i	0.477746	+	0.827480i	0.999675	−	0.0255091i	$-0.00812068\pi$
$613$	6.27610e17	+	1.08705e18i	0.477746	+	0.827480i	−0.521929	+	0.852989i	$0.674787\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	1.84883e18	−	1.06742e18i	1.32101	−	0.762688i	0.337124	−	0.941460i	$-0.390546\pi$
$619$	1.84883e18	−	1.06742e18i	1.32101	−	0.762688i	0.983890	+	0.178772i	$0.0572125\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	3.46181e16			0.0234749
$625$	−7.45058e17	+	1.29048e18i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−3.92537e17	−	2.26631e17i	−0.255354	−	0.147429i
$629$		0			0
$630$		0			0
$631$	3.06063e18			1.93028			0.965139	−	0.261736i	$-0.0842951\pi$
$631$	3.06063e18			1.93028			0.965139	+	0.261736i	$0.0842951\pi$
$632$		0			0
$633$	−2.70057e18	+	1.55918e18i	−1.66852	+	0.963321i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	3.85328e16	−	9.05784e15i	0.0228520	−	0.00537178i
$638$		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
$643$			8.14675e17i			0.454583i	0.973827	+	0.227291i	$0.0729870\pi$
$643$			8.14675e17i			0.454583i	−0.973827	+	0.227291i	$0.927013\pi$
$644$		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
$651$	−1.19072e18	−	3.00525e18i	−0.613104	−	1.54741i
$652$	−3.30363e18			−1.68415
$653$		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
$657$			2.88322e18i			1.39863i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	2.82087e18	−	1.62863e18i	1.31545	−	0.759473i	0.332454	−	0.943120i	$-0.392123\pi$
$661$	2.82087e18	−	1.62863e18i	1.31545	−	0.759473i	0.982992	+	0.183646i	$0.0587902\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.30897e18	+	3.99926e18i	0.995747	+	1.72468i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−3.76575e18			−1.56226			−0.781130	−	0.624369i	$-0.785356\pi$
$673$	−3.76575e18			−1.56226			−0.781130	+	0.624369i	$0.785356\pi$
$674$		0			0
$675$	−2.12816e18	+	1.22870e18i	−0.866025	+	0.500000i
$676$	−1.23989e18	−	2.14756e18i	−0.499724	−	0.865548i
$677$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$677$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$678$		0			0
$679$	2.63483e18	+	2.08729e18i	1.03181	+	0.817388i
$680$		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	$0.333333\pi$
$684$	4.18263e18	+	2.41484e18i	1.56165	+	0.901621i
$685$		0			0
$686$		0			0
$687$	−5.47948e18			−1.98848
$688$	−2.59103e18	+	4.48780e18i	−0.931426	+	1.61328i
$689$		0			0
$690$		0			0
$691$	−4.92727e18	−	2.84476e18i	−1.72186	−	0.994119i	−0.915079	−	0.403274i	$-0.867872\pi$
$691$	−4.92727e18	−	2.84476e18i	−1.72186	−	0.994119i	−0.806785	−	0.590845i	$-0.798794\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−1.14658e18	−	2.89383e18i	−0.368354	−	0.929686i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	8.76749e18	−	5.06192e18i	2.73947	−	1.58163i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	2.29769e18	−	3.97971e18i	0.679345	−	1.17666i	−0.295834	−	0.955239i	$-0.595598\pi$
$709$	2.29769e18	−	3.97971e18i	0.679345	−	1.17666i	0.975179	−	0.221420i	$-0.0710691\pi$
$710$		0			0
$711$	3.25604e18	+	5.63962e18i	0.945229	+	1.63719i
$712$		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
$721$	−9.45022e17	+	3.74430e17i	−0.250532	+	0.0992640i
$722$		0			0
$723$	3.76252e18	−	6.51687e18i	0.979669	−	1.69684i
$724$	6.45120e18	−	3.72460e18i	1.66471	−	0.961123i
$725$		0			0
$726$		0			0
$727$		−	5.02783e18i		−	1.26301i	−0.775371	−	0.631505i	$-0.782437\pi$
$727$		−	5.02783e18i		−	1.26301i	0.775371	−	0.631505i	$-0.217563\pi$
$728$		0			0
$729$	−4.05256e18			−1.00000
$730$		0			0
$731$		0			0
$732$	−1.26658e18	−	2.19379e18i	−0.304307	−	0.527075i
$733$	8.30992e17	+	4.79773e17i	0.197889	+	0.114251i	0.595670	−	0.803229i	$-0.296887\pi$
$733$	8.30992e17	+	4.79773e17i	0.197889	+	0.114251i	−0.397782	+	0.917480i	$0.630220\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	4.32301e18	+	7.48767e18i	0.976332	+	1.69106i	0.675468	+	0.737389i	$0.263942\pi$
$739$	4.32301e18	+	7.48767e18i	0.976332	+	1.69106i	0.300864	+	0.953667i	$0.402725\pi$
$740$		0			0
$741$			1.90755e17i			0.0423309i
$742$		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
$751$	−4.91634e18	+	8.51536e18i	−0.999960	+	1.73198i	−0.492256	+	0.870450i	$0.663828\pi$
$751$	−4.91634e18	+	8.51536e18i	−0.999960	+	1.73198i	−0.507704	+	0.861532i	$0.669506\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	7.48627e17	−	5.07836e18i	0.145839	−	0.989308i
$757$	−1.18142e18			−0.228182			−0.114091	−	0.993470i	$-0.536396\pi$
$757$	−1.18142e18			−0.228182			−0.114091	+	0.993470i	$0.536396\pi$
$758$		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
$763$	−2.14123e18	−	3.15649e17i	−0.392874	−	0.0579156i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	4.92469e18	+	2.84327e18i	0.866025	+	0.500000i
$769$		−	6.27137e17i		−	0.109356i	−0.998504	−	0.0546778i	$-0.982587\pi$
$769$		−	6.27137e17i		−	0.109356i	0.998504	−	0.0546778i	$-0.0174132\pi$
$770$		0			0
$771$		0			0
$772$	−5.38736e18	+	9.33118e18i	−0.915932	+	1.58644i
$773$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$773$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$774$		0			0
$775$	−8.69482e18	−	5.01995e18i	−1.44145	−	0.832221i
$776$		0			0
$777$	−8.43430e18	−	6.68156e18i	−1.37503	−	1.08928i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	6.22552e18	+	1.87625e18i	0.957462	+	0.288560i
$785$		0			0
$786$		0			0
$787$	−8.30716e18	+	4.79614e18i	−1.24628	+	0.719542i	−0.970366	−	0.241639i	$-0.922315\pi$
$787$	−8.30716e18	+	4.79614e18i	−1.24628	+	0.719542i	−0.275918	+	0.961181i	$0.588982\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	5.00254e16	−	8.66465e16i	0.00714356	−	0.0123730i
$794$		0			0
$795$		0			0
$796$	1.24253e19	+	7.17377e18i	1.73130	+	0.999568i
$797$		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
$804$			2.34671e18i			0.306404i
$805$		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
$811$			1.11200e19i			1.37237i	0.727429	+	0.686183i	$0.240715\pi$
$811$			1.11200e19i			1.37237i	−0.727429	+	0.686183i	$0.759285\pi$
$812$		0			0
$813$	1.30864e19			1.58939
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.47290e19	−	1.42773e19i	−2.90913	−	1.67959i
$818$		0			0
$819$	1.88489e17	−	7.46817e16i	0.0218243	−	0.00864706i
$820$		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
$823$	7.30311e18	+	1.26494e19i	0.819236	+	1.41896i	0.906246	+	0.422751i	$0.138935\pi$
$823$	7.30311e18	+	1.26494e19i	0.819236	+	1.41896i	−0.0870099	+	0.996207i	$0.527731\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−2.16931e17	+	1.25245e17i	−0.0232123	+	0.0134016i	−0.511561	−	0.859247i	$-0.670933\pi$
$829$	−2.16931e17	+	1.25245e17i	−0.0232123	+	0.0134016i	0.488349	+	0.872648i	$0.337599\pi$
$830$		0			0
$831$	−2.20062e18	−	1.27053e18i	−0.231814	−	0.133838i
$832$			2.24598e17i			0.0234749i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−8.27855e18	−	1.43389e19i	−0.832221	−	1.44145i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.02606e19			1.00000
$842$		0			0
$843$		0			0
$844$	−1.01157e19	−	1.75209e19i	−0.963321	−	1.66852i
$845$		0			0
$846$		0			0
$847$	3.95829e18	+	9.99029e18i	0.368354	+	0.929686i
$848$		0			0
$849$	9.97685e18	−	1.72804e19i	0.914310	−	1.58363i
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	1.07549e19i		−	0.955958i	−0.878371	−	0.477979i	$-0.841370\pi$
$853$		−	1.07549e19i		−	0.955958i	0.878371	−	0.477979i	$-0.158630\pi$
$854$		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
$859$	1.40195e19	+	8.09414e18i	1.19063	+	0.687409i	0.958449	−	0.285264i	$-0.0920814\pi$
$859$	1.40195e19	+	8.09414e18i	1.19063	+	0.687409i	0.232178	+	0.972673i	$0.425415\pi$
$860$		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
$867$			1.25062e19i			1.00000i
$868$	1.94977e19	−	7.72524e18i	1.54741	−	0.613104i
$869$		0			0
$870$		0			0
$871$	−8.02688e16	+	4.63432e16i	−0.00622915	+	0.00359640i
$872$		0			0
$873$	1.49105e19	+	8.60857e18i	1.13999	+	0.658171i
$874$		0			0
$875$		0			0
$876$	−1.87060e19			−1.39863
$877$	1.34136e19	−	2.32330e19i	0.995513	−	1.72428i	0.415805	−	0.909454i	$-0.363500\pi$
$877$	1.34136e19	−	2.32330e19i	0.995513	−	1.72428i	0.579708	−	0.814825i	$-0.303167\pi$
$878$		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
$883$	1.83204e19			1.30074			0.650369	−	0.759618i	$-0.274614\pi$
$883$	1.83204e19			1.30074			0.650369	+	0.759618i	$0.274614\pi$
$884$		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
$889$	−8.28197e18	+	1.04545e19i	−0.562694	+	0.710303i
$890$		0			0
$891$		0			0
$892$	−2.59466e19	+	1.49803e19i	−1.72468	+	0.995747i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−7.97162e18	−	1.38072e19i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−4.42611e18	+	3.00248e19i	−0.271676	+	1.84293i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.46591e19	−	2.53903e19i	−0.874300	−	1.51433i	−0.857506	−	0.514473i	$-0.827987\pi$
$907$	−1.46591e19	−	2.53903e19i	−0.874300	−	1.51433i	−0.0167938	−	0.999859i	$-0.505346\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−1.56672e19	+	2.71363e19i	−0.901621	+	1.56165i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	3.55501e19i		−	1.98848i
$917$		0			0
$918$		0			0
$919$	−3.88591e18	+	6.73060e18i	−0.212786	+	0.368555i	−0.952585	−	0.304272i	$-0.901587\pi$
$919$	−3.88591e18	+	6.73060e18i	−0.212786	+	0.368555i	0.739800	+	0.672827i	$0.234920\pi$
$920$		0			0
$921$	1.11029e19	+	1.92308e19i	0.599446	+	1.03827i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−3.34197e19			−1.75421
$926$		0			0
$927$	−4.50895e18	+	2.60324e18i	−0.233377	+	0.134740i
$928$		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
$931$	−1.03386e19	+	3.43043e19i	−0.520343	+	1.72653i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			2.38155e18i			0.114961i	0.998347	+	0.0574807i	$0.0183068\pi$
$937$			2.38155e18i			0.114961i	−0.998347	+	0.0574807i	$0.981693\pi$
$938$		0			0
$939$	3.95320e19			1.88201
$940$		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	$0.666667\pi$
$948$	−3.65891e19	+	2.11247e19i	−1.63719	+	0.945229i
$949$	−3.69409e17	−	6.39835e17i	−0.0164164	−	0.0284340i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	2.16140e19	−	3.74366e19i	0.885184	−	1.53318i
$962$		0			0
$963$		0			0
$964$	4.22806e19	+	2.44107e19i	1.69684	+	0.979669i
$965$		0			0
$966$		0			0
$967$	4.22274e19			1.66082			0.830409	−	0.557154i	$-0.188107\pi$
$967$	4.22274e19			1.66082			0.830409	+	0.557154i	$0.188107\pi$
$968$		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	$0.166667\pi$
$972$		−	2.62924e19i		−	1.00000i
$973$	−2.95947e19	−	2.34445e19i	−1.11810	−	0.885744i
$974$		0			0
$975$	3.14850e17	−	5.45336e17i	0.0117374	−	0.0203298i
$976$	1.42330e19	−	8.21742e18i	0.527075	−	0.304307i
$977$		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
$981$	−1.10859e19			−0.397120
$982$		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
$988$	−1.23759e18			−0.0423309
$989$		0			0
$990$		0			0
$991$	−1.19587e19	−	2.07131e19i	−0.401057	−	0.694651i	0.592797	−	0.805352i	$-0.298024\pi$
$991$	−1.19587e19	−	2.07131e19i	−0.401057	−	0.694651i	−0.993854	+	0.110701i	$0.964690\pi$
$992$		0			0
$993$		−	2.11336e19i		−	0.699524i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.81081e19	+	1.62282e19i	−0.906386	+	0.523302i	−0.879267	−	0.476330i	$-0.841967\pi$
$997$	−2.81081e19	+	1.62282e19i	−0.906386	+	0.523302i	−0.0271194	+	0.999632i	$0.508633\pi$
$998$		0			0
$999$	−4.77296e19	−	2.75567e19i	−1.51919	−	0.877105i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.14.g.a.5.1	✓	2
3.2	odd	2	CM	21.14.g.a.5.1	✓	2
7.3	odd	6	inner	21.14.g.a.17.1	yes	2
21.17	even	6	inner	21.14.g.a.17.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.14.g.a.5.1	✓	2	1.1	even	1	trivial
21.14.g.a.5.1	✓	2	3.2	odd	2	CM
21.14.g.a.17.1	yes	2	7.3	odd	6	inner
21.14.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:	\( k \)	\(=\)	\( 14 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-1093.50 + 631.333i) q^{3} +(-4096.00 - 7094.48i) q^{4} +(193284. - 243988. i) q^{7} +(797162. - 1.38072e6i) q^{9} +O(q^{10})\)	\(q+(-1093.50 + 631.333i) q^{3} +(-4096.00 - 7094.48i) q^{4} +(193284. - 243988. i) q^{7} +(797162. - 1.38072e6i) q^{9} +(8.95795e6 + 5.17188e6i) q^{12} +408541. i q^{13} +(-3.35544e7 + 5.81180e7i) q^{16} +(-3.20245e8 - 1.84894e8i) q^{19} +(-5.73191e7 + 3.88828e8i) q^{21} +(6.10352e8 + 1.05716e9i) q^{25} +2.01310e9i q^{27} +(-2.52266e9 - 3.71878e8i) q^{28} +(-7.12279e9 + 4.11235e9i) q^{31} -1.30607e10 q^{36} +(-1.36887e10 + 2.37095e10i) q^{37} +(-2.57925e8 - 4.46739e8i) q^{39} +7.72188e10 q^{43} -8.47360e10i q^{48} +(-2.21712e10 - 9.43182e10i) q^{49} +(2.89838e9 - 1.67338e9i) q^{52} +4.66918e11 q^{57} +(-2.12088e11 - 1.22449e11i) q^{61} +(-1.82801e11 - 4.61370e11i) q^{63} +5.49756e11 q^{64} +(1.13436e11 + 1.96477e11i) q^{67} +(-1.56615e12 + 9.04216e11i) q^{73} +(-1.33484e12 - 7.70670e11i) q^{75} +3.02930e12i q^{76} +(-2.04227e12 + 3.53732e12i) q^{79} +(-1.27093e12 - 2.20132e12i) q^{81} +(2.99331e12 - 1.18599e12i) q^{84} +(9.96790e10 + 7.89646e10i) q^{91} +(5.19252e12 - 8.99370e12i) q^{93} +1.07990e13i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2187 q^{3} - 8192 q^{4} + 386569 q^{7} + 1594323 q^{9}+O(q^{10}) \) 2 * q - 2187 * q^3 - 8192 * q^4 + 386569 * q^7 + 1594323 * q^9	\( 2 q - 2187 q^{3} - 8192 q^{4} + 386569 q^{7} + 1594323 q^{9} + 17915904 q^{12} - 67108864 q^{16} - 640490769 q^{19} - 114638166 q^{21} + 1220703125 q^{25} - 5045321728 q^{28} - 14245587705 q^{31} - 26121388032 q^{36} - 27377427169 q^{37} - 515849877 q^{39} + 154437650630 q^{43} - 44342429053 q^{49} + 5796765696 q^{52} + 933835541202 q^{57} - 424175890308 q^{61} - 365602178745 q^{63} + 1099511627776 q^{64} + 226871917261 q^{67} - 3132295953723 q^{73} - 2669677734375 q^{75} - 4084540747547 q^{79} - 2541865828329 q^{81} + 5986617237504 q^{84} + 199357933329 q^{91} + 10385033436945 q^{93}+O(q^{100}) \) 2 * q - 2187 * q^3 - 8192 * q^4 + 386569 * q^7 + 1594323 * q^9 + 17915904 * q^12 - 67108864 * q^16 - 640490769 * q^19 - 114638166 * q^21 + 1220703125 * q^25 - 5045321728 * q^28 - 14245587705 * q^31 - 26121388032 * q^36 - 27377427169 * q^37 - 515849877 * q^39 + 154437650630 * q^43 - 44342429053 * q^49 + 5796765696 * q^52 + 933835541202 * q^57 - 424175890308 * q^61 - 365602178745 * q^63 + 1099511627776 * q^64 + 226871917261 * q^67 - 3132295953723 * q^73 - 2669677734375 * q^75 - 4084540747547 * q^79 - 2541865828329 * q^81 + 5986617237504 * q^84 + 199357933329 * q^91 + 10385033436945 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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