LMFDB - Embedded newform 21.20.g.a.17.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,20,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, 20, names="a")

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

chi := DirichletCharacter("21.5");

S:= CuspForms(chi, 20);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 20 $
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:	$48.0515062768$
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			17.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.17
Dual form			21.20.g.a.5.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(29524.5 + 17046.0i) q^{3} +(-262144. + 454047. i) q^{4} +(4.84282e7 - 9.51504e7i) q^{7} +(5.81131e8 + 1.00655e9i) q^{9} +O(q^{10})$	\(q+(29524.5 + 17046.0i) q^{3} +(-262144. + 454047. i) q^{4} +(4.84282e7 - 9.51504e7i) q^{7} +(5.81131e8 + 1.00655e9i) q^{9} +(-1.54793e10 + 8.93700e9i) q^{12} -7.60595e10i q^{13} +(-1.37439e11 - 2.38051e11i) q^{16} +(1.02386e12 - 5.91128e11i) q^{19} +(3.05175e12 - 1.98376e12i) q^{21} +(9.53674e12 - 1.65181e13i) q^{25} +3.96238e13i q^{27} +(3.05076e13 + 4.69318e13i) q^{28} +(-2.39571e14 - 1.38317e14i) q^{31} -6.09360e14 q^{36} +(7.90506e14 + 1.36920e15i) q^{37} +(1.29651e15 - 2.24562e15i) q^{39} +6.01253e15 q^{43} -9.37113e15i q^{48} +(-6.70831e15 - 9.21593e15i) q^{49} +(3.45346e16 + 1.99385e16i) q^{52} +4.03054e16 q^{57} +(1.01134e17 - 5.83897e16i) q^{61} +(1.23917e17 - 6.54950e15i) q^{63} +1.44115e17 q^{64} +(2.20587e17 - 3.82068e17i) q^{67} +(-1.43298e17 - 8.27332e16i) q^{73} +(5.63135e17 - 3.25126e17i) q^{75} +6.19843e17i q^{76} +(-1.31195e17 - 2.27236e17i) q^{79} +(-6.75426e17 + 1.16987e18i) q^{81} +(1.00722e17 + 1.90567e18i) q^{84} +(-7.23710e18 - 3.68343e18i) q^{91} +(-4.71548e18 - 8.16746e18i) q^{93} +1.15735e19i q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9}+O(q^{10}) $ 2 * q + 59049 * q^3 - 524288 * q^4 + 96856453 * q^7 + 1162261467 * q^9	$ 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9} - 30958682112 q^{12} - 274877906944 q^{16} + 2047727487699 q^{19} + 6103502217954 q^{21} + 19073486328125 q^{25} + 61015133782016 q^{28} - 479142790563765 q^{31} - 12\!\cdots\!92 q^{36}+ \cdots - 94\!\cdots\!95 q^{93}+O(q^{100}) $ 2 * q + 59049 * q^3 - 524288 * q^4 + 96856453 * q^7 + 1162261467 * q^9 - 30958682112 * q^12 - 274877906944 * q^16 + 2047727487699 * q^19 + 6103502217954 * q^21 + 19073486328125 * q^25 + 61015133782016 * q^28 - 479142790563765 * q^31 - 1218719480020992 * q^36 + 1581011344413647 * q^37 + 2593018074647367 * q^39 + 12025059029572670 * q^43 - 13416617883005077 * q^49 + 69069159189184512 * q^52 + 80610840280758834 * q^57 + 202267850921264532 * q^61 + 247833179315769195 * q^63 + 288230376151711744 * q^64 + 441174519368848177 * q^67 - 286596050169531339 * q^73 + 1126270294189453125 * q^75 - 262389124703851223 * q^79 - 1350851717672992089 * q^81 + 201444831923798016 * q^84 - 14474191718377587411 * q^91 - 9430967546666586495 * q^93

Display 100 coefficients 10 coefficients

Character values

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

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

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	$-0.666667\pi$
$2$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$3$	29524.5	+	17046.0i	0.866025	+	0.500000i
$3$	29524.5	+	17046.0i	0.866025	+	0.500000i
$4$	−262144.	+	454047.i	−0.500000	+	0.866025i
$5$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$5$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$6$		0			0
$7$	4.84282e7	−	9.51504e7i	0.453594	−	0.891209i
$7$	4.84282e7	−	9.51504e7i	0.453594	−	0.891209i
$8$		0			0
$9$	5.81131e8	+	1.00655e9i	0.500000	+	0.866025i
$10$		0			0
$11$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$11$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$12$	−1.54793e10	+	8.93700e9i	−0.866025	+	0.500000i
$13$		−	7.60595e10i		−	1.98926i	−0.103490	−	0.994631i	$-0.533001\pi$
$13$		−	7.60595e10i		−	1.98926i	0.103490	−	0.994631i	$-0.466999\pi$
$14$		0			0
$15$		0			0
$16$	−1.37439e11	−	2.38051e11i	−0.500000	−	0.866025i
$17$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$17$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$18$		0			0
$19$	1.02386e12	−	5.91128e11i	0.727919	−	0.420264i	−0.0897415	−	0.995965i	$-0.528604\pi$
$19$	1.02386e12	−	5.91128e11i	0.727919	−	0.420264i	0.817660	+	0.575701i	$0.195271\pi$
$20$		0			0
$21$	3.05175e12	−	1.98376e12i	0.838428	−	0.545012i
$22$		0			0
$23$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$23$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$24$		0			0
$25$	9.53674e12	−	1.65181e13i	0.500000	−	0.866025i
$26$		0			0
$27$			3.96238e13i			1.00000i
$28$	3.05076e13	+	4.69318e13i	0.545012	+	0.838428i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−2.39571e14	−	1.38317e14i	−1.62742	−	0.939590i	−0.984860	−	0.173352i	$-0.944540\pi$
$31$	−2.39571e14	−	1.38317e14i	−1.62742	−	0.939590i	−0.642557	−	0.766238i	$-0.722126\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−6.09360e14			−1.00000
$37$	7.90506e14	+	1.36920e15i	0.999974	+	1.73201i	0.506258	+	0.862382i	$0.331028\pi$
$37$	7.90506e14	+	1.36920e15i	0.999974	+	1.73201i	0.493715	+	0.869624i	$0.335638\pi$
$38$		0			0
$39$	1.29651e15	−	2.24562e15i	0.994631	−	1.72275i
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.01253e15			1.82434			0.912172	−	0.409807i	$-0.134404\pi$
$43$	6.01253e15			1.82434			0.912172	+	0.409807i	$0.134404\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$47$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$48$		−	9.37113e15i		−	1.00000i
$49$	−6.70831e15	−	9.21593e15i	−0.588505	−	0.808493i
$50$		0			0
$51$		0			0
$52$	3.45346e16	+	1.99385e16i	1.72275	+	0.994631i
$53$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$53$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.03054e16			0.840528
$58$		0			0
$59$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$59$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$60$		0			0
$61$	1.01134e17	−	5.83897e16i	1.10730	−	0.639297i	0.169168	−	0.985587i	$-0.445892\pi$
$61$	1.01134e17	−	5.83897e16i	1.10730	−	0.639297i	0.938127	+	0.346290i	$0.112559\pi$
$62$		0			0
$63$	1.23917e17	−	6.54950e15i	0.998606	−	0.0527804i
$64$	1.44115e17			1.00000
$65$		0			0
$66$		0			0
$67$	2.20587e17	−	3.82068e17i	0.990535	−	1.71566i	0.376396	−	0.926459i	$-0.377163\pi$
$67$	2.20587e17	−	3.82068e17i	0.990535	−	1.71566i	0.614139	−	0.789198i	$-0.289503\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	−1.43298e17	−	8.27332e16i	−0.284887	−	0.164480i	0.350747	−	0.936470i	$-0.385928\pi$
$73$	−1.43298e17	−	8.27332e16i	−0.284887	−	0.164480i	−0.635634	+	0.771991i	$0.719261\pi$
$74$		0			0
$75$	5.63135e17	−	3.25126e17i	0.866025	−	0.500000i
$76$			6.19843e17i			0.840528i
$77$		0			0
$78$		0			0
$79$	−1.31195e17	−	2.27236e17i	−0.123157	−	0.213314i	0.797854	−	0.602851i	$-0.205968\pi$
$79$	−1.31195e17	−	2.27236e17i	−0.123157	−	0.213314i	−0.921011	+	0.389537i	$0.872635\pi$
$80$		0			0
$81$	−6.75426e17	+	1.16987e18i	−0.500000	+	0.866025i
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	1.00722e17	+	1.90567e18i	0.0527804	+	0.998606i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$89$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$90$		0			0
$91$	−7.23710e18	−	3.68343e18i	−1.77285	−	0.902317i
$92$		0			0
$93$	−4.71548e18	−	8.16746e18i	−0.939590	−	1.62742i
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.15735e19i			1.54572i	0.634575	+	0.772861i	$0.281175\pi$
$97$			1.15735e19i			1.54572i	−0.634575	+	0.772861i	$0.718825\pi$
$98$		0			0
$99$		0			0
$100$	5.00000e18	+	8.66025e18i	0.500000	+	0.866025i
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	−2.10657e19	+	1.21623e19i	−1.59082	+	0.918461i	−0.597655	+	0.801753i	$0.703901\pi$
$103$	−2.10657e19	+	1.21623e19i	−1.59082	+	0.918461i	−0.993166	+	0.116708i	$0.962766\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$107$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$108$	−1.79910e19	−	1.03871e19i	−0.866025	−	0.500000i
$109$	1.58738e19	−	2.74942e19i	0.700049	−	1.21252i	−0.268399	−	0.963308i	$-0.586495\pi$
$109$	1.58738e19	−	2.74942e19i	0.700049	−	1.21252i	0.968449	−	0.249213i	$-0.0801720\pi$
$110$		0			0
$111$			5.38998e19i			1.99995i
$112$	−2.93066e19	+	1.54897e18i	−0.998606	+	0.0527804i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	7.65576e19	−	4.42005e19i	1.72275	−	0.994631i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−3.05795e19	−	5.29653e19i	−0.500000	−	0.866025i
$122$		0			0
$123$		0			0
$124$	1.25604e20	−	7.25177e19i	1.62742	−	0.939590i
$125$		0			0
$126$		0			0
$127$	3.92316e19			0.405043			0.202521	−	0.979278i	$-0.435086\pi$
$127$	3.92316e19			0.405043			0.202521	+	0.979278i	$0.435086\pi$
$128$		0			0
$129$	1.77517e20	+	1.02489e20i	1.57993	+	0.912172i
$130$		0			0
$131$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$131$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$132$		0			0
$133$	−6.66217e18	−	1.26048e20i	−0.0443634	−	0.839357i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$137$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$138$		0			0
$139$			3.87845e20i			1.69831i	0.528142	+	0.849156i	$0.322889\pi$
$139$			3.87845e20i			1.69831i	−0.528142	+	0.849156i	$0.677111\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	1.59740e20	−	2.76678e20i	0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	−4.09649e19	−	3.86445e20i	−0.105414	−	0.994428i
$148$	−8.28905e20			−1.99995
$149$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$149$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$150$		0			0
$151$	1.77389e19	−	3.07246e19i	0.0353708	−	0.0612640i	−0.847798	−	0.530319i	$-0.822072\pi$
$151$	1.77389e19	−	3.07246e19i	0.0353708	−	0.0612640i	0.883169	+	0.469055i	$0.155405\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	6.79744e20	+	1.17735e21i	0.994631	+	1.72275i
$157$	−3.31927e20	−	1.91638e20i	−0.457084	−	0.263898i	0.253733	−	0.967274i	$-0.418341\pi$
$157$	−3.31927e20	−	1.91638e20i	−0.457084	−	0.263898i	−0.710817	+	0.703377i	$0.751675\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−2.34847e20	−	4.06767e20i	−0.226466	−	0.392250i	0.730292	−	0.683135i	$-0.239384\pi$
$163$	−2.34847e20	−	4.06767e20i	−0.226466	−	0.392250i	−0.956758	+	0.290884i	$0.906050\pi$
$164$		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
$169$	−4.32313e21			−2.95716
$170$		0			0
$171$	1.19000e21	+	6.87045e20i	0.727919	+	0.420264i
$172$	−1.57615e21	+	2.72997e21i	−0.912172	+	1.57993i
$173$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$173$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$174$		0			0
$175$	−1.10986e21	−	1.70737e21i	−0.545012	−	0.838428i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$179$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$180$		0			0
$181$		−	1.59372e21i		−	0.568152i	−0.958802	−	0.284076i	$-0.908313\pi$
$181$		−	1.59372e21i		−	0.568152i	0.958802	−	0.284076i	$-0.0916868\pi$
$182$		0			0
$183$	3.98124e21			1.27859
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	3.77022e21	+	1.91891e21i	0.891209	+	0.453594i
$190$		0			0
$191$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$191$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$192$	4.25493e21	+	2.45658e21i	0.866025	+	0.500000i
$193$	3.42509e21	−	5.93243e21i	0.663556	−	1.14931i	−0.316119	−	0.948719i	$-0.602380\pi$
$193$	3.42509e21	−	5.93243e21i	0.663556	−	1.14931i	0.979675	−	0.200592i	$-0.0642867\pi$
$194$		0			0
$195$		0			0
$196$	5.94301e21	−	6.29984e20i	0.994428	−	0.105414i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	−8.60187e21	−	4.96629e21i	−1.24591	−	0.719329i	−0.275623	−	0.961266i	$-0.588884\pi$
$199$	−8.60187e21	−	4.96629e21i	−1.24591	−	0.719329i	−0.970292	+	0.241937i	$0.922217\pi$
$200$		0			0
$201$	1.30255e22	−	7.52025e21i	1.71566	−	0.990535i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	−1.81061e22	+	1.04535e22i	−1.72275	+	0.994631i
$209$		0			0
$210$		0			0
$211$	2.39936e22			1.99257			0.996283	−	0.0861410i	$-0.0274536\pi$
$211$	2.39936e22			1.99257			0.996283	+	0.0861410i	$0.0274536\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.47629e22	+	1.60969e22i	−1.57556	+	1.02418i
$218$		0			0
$219$	−2.82054e21	−	4.88531e21i	−0.164480	−	0.284887i
$220$		0			0
$221$		0			0
$222$		0			0
$223$		−	1.84214e22i		−	0.904537i	−0.891882	−	0.452269i	$-0.850615\pi$
$223$		−	1.84214e22i		−	0.904537i	0.891882	−	0.452269i	$-0.149385\pi$
$224$		0			0
$225$	2.21684e22			1.00000
$226$		0			0
$227$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$227$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$228$	−1.05658e22	+	1.83005e22i	−0.420264	+	0.727919i
$229$	−1.24485e22	+	7.18715e21i	−0.474985	+	0.274233i	−0.718324	−	0.695708i	$-0.755091\pi$
$229$	−1.24485e22	+	7.18715e21i	−0.474985	+	0.274233i	0.243339	+	0.969941i	$0.421757\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$233$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$		−	8.94536e21i		−	0.246314i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	−6.33683e22	−	3.65857e22i	−1.48837	−	0.859310i	−0.488456	−	0.872589i	$-0.662440\pi$
$241$	−6.33683e22	−	3.65857e22i	−1.48837	−	0.859310i	−0.999912	+	0.0132790i	$0.995773\pi$
$242$		0			0
$243$	−3.98832e22	+	2.30266e22i	−0.866025	+	0.500000i
$244$			6.12260e22i			1.27859i
$245$		0			0
$246$		0			0
$247$	−4.49609e22	−	7.78746e22i	−0.836015	−	1.44802i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−2.95102e22	+	5.79808e22i	−0.453594	+	0.891209i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−3.77789e22	+	6.54350e22i	−0.500000	+	0.866025i
$257$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$257$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$258$		0			0
$259$	1.68562e23	−	8.90921e21i	1.99716	−	0.105558i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$263$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	1.15651e23	+	2.00314e23i	0.990535	+	1.71566i
$269$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$269$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$270$		0			0
$271$	2.00409e23	−	1.15706e23i	1.54422	−	0.891557i	0.545657	−	0.838009i	$-0.316280\pi$
$271$	2.00409e23	−	1.15706e23i	1.54422	−	0.891557i	0.998565	−	0.0535483i	$-0.0170531\pi$
$272$		0			0
$273$	−1.50884e23	−	2.32115e23i	−1.08417	−	1.66785i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.11289e23	−	1.92758e23i	0.696456	−	1.20630i	−0.273232	−	0.961948i	$-0.588093\pi$
$277$	1.11289e23	−	1.92758e23i	0.696456	−	1.20630i	0.969688	−	0.244348i	$-0.0785740\pi$
$278$		0			0
$279$		−	3.21520e23i		−	1.87918i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	2.26314e23	+	1.30662e23i	1.15542	+	0.667082i	0.950202	−	0.311634i	$-0.100876\pi$
$283$	2.26314e23	+	1.30662e23i	1.15542	+	0.667082i	0.205218	+	0.978716i	$0.434210\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	1.19536e23	+	2.07043e23i	0.500000	+	0.866025i
$290$		0			0
$291$	−1.97281e23	+	3.41701e23i	−0.772861	+	1.33864i
$292$	7.51294e22	−	4.33760e22i	0.284887	−	0.164480i
$293$		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
$300$			3.40920e23i			1.00000i
$301$	2.91176e23	−	5.72095e23i	0.827512	−	1.62587i
$302$		0			0
$303$		0			0
$304$	−2.81438e23	−	1.62488e23i	−0.727919	−	0.420264i
$305$		0			0
$306$		0			0
$307$			6.85262e23i			1.61451i	0.590200	+	0.807257i	$0.299049\pi$
$307$			6.85262e23i			1.61451i	−0.590200	+	0.807257i	$0.700951\pi$
$308$		0			0
$309$	−8.29271e23			−1.83692
$310$		0			0
$311$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$311$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$312$		0			0
$313$	−2.95718e23	+	1.70733e23i	−0.579704	+	0.334692i	−0.761016	−	0.648734i	$-0.775299\pi$
$313$	−2.95718e23	+	1.70733e23i	−0.579704	+	0.334692i	0.181312	+	0.983426i	$0.441966\pi$
$314$		0			0
$315$		0			0
$316$	1.37567e23			0.246314
$317$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$317$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	−3.54118e23	−	6.13350e23i	−0.500000	−	0.866025i
$325$	−1.25636e24	−	7.25360e23i	−1.72275	−	0.994631i
$326$		0			0
$327$	9.37331e23	−	5.41168e23i	1.21252	−	0.700049i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.32820e23	+	7.49666e23i	0.498817	+	0.863976i	0.999999	−	0.00136580i	$-0.000434747\pi$
$331$	4.32820e23	+	7.49666e23i	0.498817	+	0.863976i	−0.501182	+	0.865342i	$0.667101\pi$
$332$		0			0
$333$	−9.18774e23	+	1.59136e24i	−0.999974	+	1.73201i
$334$		0			0
$335$		0			0
$336$	−8.91667e23	−	4.53827e23i	−0.891209	−	0.453594i
$337$	1.06890e24			1.03861			0.519305	−	0.854589i	$-0.326191\pi$
$337$	1.06890e24			1.03861			0.519305	+	0.854589i	$0.326191\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−1.20177e24	+	1.91987e23i	−0.987479	+	0.157753i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$347$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$348$		0			0
$349$			1.51610e24i			1.05654i	0.849076	+	0.528270i	$0.177159\pi$
$349$			1.51610e24i			1.05654i	−0.849076	+	0.528270i	$0.822841\pi$
$350$		0			0
$351$	3.01376e24			1.98926
$352$		0			0
$353$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$353$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$359$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$360$		0			0
$361$	−2.90345e23	+	5.02893e23i	−0.146756	+	0.254189i
$362$		0			0
$363$		−	2.08503e24i		−	1.00000i
$364$	3.56961e24	−	2.32039e24i	1.66785	−	1.08417i
$365$		0			0
$366$		0			0
$367$	−3.98463e24	−	2.30053e24i	−1.72211	−	0.994261i	−0.914547	−	0.404479i	$-0.867453\pi$
$367$	−3.98463e24	−	2.30053e24i	−1.72211	−	0.994261i	−0.807562	−	0.589782i	$-0.799214\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	4.94454e24			1.87918
$373$	−6.38263e23	−	1.10550e24i	−0.236464	−	0.409568i	0.723233	−	0.690604i	$-0.242655\pi$
$373$	−6.38263e23	−	1.10550e24i	−0.236464	−	0.409568i	−0.959697	+	0.281036i	$0.909322\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	6.06002e24			1.92931			0.964654	−	0.263520i	$-0.0848835\pi$
$379$	6.06002e24			1.92931			0.964654	+	0.263520i	$0.0848835\pi$
$380$		0			0
$381$	1.15829e24	+	6.68742e23i	0.350777	+	0.202521i
$382$		0			0
$383$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$383$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	3.49407e24	+	6.05190e24i	0.912172	+	1.57993i
$388$	−5.25489e24	−	3.03391e24i	−1.33864	−	0.772861i
$389$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$389$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$	−3.95150e23	+	2.28140e23i	−0.0809567	+	0.0467404i	−0.539932	−	0.841709i	$-0.681550\pi$
$397$	−3.95150e23	+	2.28140e23i	−0.0809567	+	0.0467404i	0.458975	+	0.888449i	$0.348217\pi$
$398$		0			0
$399$	1.95192e24	−	3.83508e24i	0.381258	−	0.749086i
$400$	−5.24288e24			−1.00000
$401$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$401$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$402$		0			0
$403$	−1.05203e25	+	1.82217e25i	−1.86909	+	3.23736i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.10693e25	−	6.39084e24i	−1.70902	−	0.986703i	−0.935778	−	0.352591i	$-0.885301\pi$
$409$	−1.10693e25	−	6.39084e24i	−1.70902	−	0.986703i	−0.773242	−	0.634112i	$-0.781366\pi$
$410$		0			0
$411$		0			0
$412$		−	1.27531e25i		−	1.83692i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−6.61119e24	+	1.14509e25i	−0.849156	+	1.47078i
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	4.68847e24			0.549985			0.274993	−	0.961446i	$-0.411325\pi$
$421$	4.68847e24			0.549985			0.274993	+	0.961446i	$0.411325\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−6.58068e23	−	1.24506e25i	−0.0674848	−	1.27681i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$431$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$432$	9.43249e24	−	5.44585e24i	0.866025	−	0.500000i
$433$			2.20683e25i			1.98213i	0.133365	+	0.991067i	$0.457422\pi$
$433$			2.20683e25i			1.98213i	−0.133365	+	0.991067i	$0.542578\pi$
$434$		0			0
$435$		0			0
$436$	8.32243e24	+	1.44149e25i	0.700049	+	1.21252i
$437$		0			0
$438$		0			0
$439$	−5.08816e24	+	2.93765e24i	−0.401003	+	0.231519i	−0.686917	−	0.726736i	$-0.741036\pi$
$439$	−5.08816e24	+	2.93765e24i	−0.401003	+	0.231519i	0.285913	+	0.958255i	$0.407703\pi$
$440$		0			0
$441$	5.37787e24	−	1.21079e25i	0.405923	−	0.913907i
$442$		0			0
$443$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$443$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$444$	−2.44730e25	−	1.41295e25i	−1.73201	−	0.999974i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	6.97924e24	−	1.37126e25i	0.453594	−	0.891209i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	1.04746e24	−	6.04753e23i	0.0612640	−	0.0353708i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	1.26766e25	+	2.19566e25i	0.682024	+	1.18130i	0.974362	+	0.224986i	$0.0722335\pi$
$457$	1.26766e25	+	2.19566e25i	0.682024	+	1.18130i	−0.292338	+	0.956315i	$0.594433\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0			−	1.00000i	$-0.5\pi$
$461$		0			0				1.00000i	$0.5\pi$
$462$		0			0
$463$	4.04121e25			1.92084			0.960421	−	0.278552i	$-0.0898545\pi$
$463$	4.04121e25			1.92084			0.960421	+	0.278552i	$0.0898545\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$467$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$468$			4.63476e25i			1.98926i
$469$	−2.56713e25	−	3.94919e25i	−1.07971	−	1.66098i
$470$		0			0
$471$	−6.53332e24	−	1.13160e25i	−0.263898	−	0.457084i
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	2.25497e25i		−	0.840528i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$479$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$480$		0			0
$481$	1.04140e26	−	6.01255e25i	3.44541	−	1.98921i
$482$		0			0
$483$		0			0
$484$	3.20650e25			1.00000
$485$		0			0
$486$		0			0
$487$	−3.06835e25	+	5.31453e25i	−0.902359	+	1.56293i	−0.0779343	+	0.996958i	$0.524832\pi$
$487$	−3.06835e25	+	5.31453e25i	−0.902359	+	1.56293i	−0.824424	+	0.565972i	$0.808501\pi$
$488$		0			0
$489$		−	1.60128e25i		−	0.452932i
$490$		0			0
$491$		0			0		1.00000			$0$
$491$		0			0		−1.00000			$\pi$
$492$		0			0
$493$		0			0
$494$		0			0
$495$		0			0
$496$			7.60404e25i			1.87918i
$497$		0			0
$498$		0			0
$499$	−2.08933e25	−	3.61882e25i	−0.487586	−	0.844523i	0.512312	−	0.858799i	$-0.328789\pi$
$499$	−2.08933e25	−	3.61882e25i	−0.487586	−	0.844523i	−0.999898	+	0.0142759i	$0.995456\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0			−	1.00000i	$-0.5\pi$
$503$		0			0				1.00000i	$0.5\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−1.27638e26	−	7.36920e25i	−2.56098	−	1.47858i
$508$	−1.02843e25	+	1.78130e25i	−0.202521	+	0.350777i
$509$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$509$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$510$		0			0
$511$	−1.48118e25	+	9.62825e24i	−0.275809	+	0.179287i
$512$		0			0
$513$	2.34227e25	+	4.05693e25i	0.420264	+	0.727919i
$514$		0			0
$515$		0			0
$516$	−9.30700e25	+	5.37340e25i	−1.57993	+	0.912172i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$521$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$522$		0			0
$523$	−7.66777e25	+	4.42699e25i	−1.14526	+	0.661214i	−0.947727	−	0.319083i	$-0.896625\pi$
$523$	−7.66777e25	+	4.42699e25i	−1.14526	+	0.661214i	−0.197530	+	0.980297i	$0.563292\pi$
$524$		0			0
$525$	−3.66426e24	−	6.93278e25i	−0.0527804	−	0.998606i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−3.73077e25	+	6.46189e25i	−0.500000	+	0.866025i
$530$		0			0
$531$		0			0
$532$	5.89783e25	+	3.00179e25i	0.749086	+	0.381258i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	2.03584e25	+	3.52618e25i	0.220480	+	0.381883i	0.954954	−	0.296754i	$-0.0959042\pi$
$541$	2.03584e25	+	3.52618e25i	0.220480	+	0.381883i	−0.734474	+	0.678637i	$0.762571\pi$
$542$		0			0
$543$	2.71664e25	−	4.70537e25i	0.284076	−	0.492034i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.22629e26			1.19596			0.597978	−	0.801513i	$-0.295971\pi$
$547$	1.22629e26			1.19596			0.597978	+	0.801513i	$0.295971\pi$
$548$		0			0
$549$	1.17544e26	+	6.78641e25i	1.10730	+	0.639297i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.79751e25	+	1.47860e24i	−0.245970	+	0.0130005i
$554$		0			0
$555$		0			0
$556$	−1.76100e26	−	1.01671e26i	−1.47078	−	0.849156i
$557$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$557$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$558$		0			0
$559$		−	4.57310e26i		−	3.62910i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$563$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	7.86041e25	+	1.20922e26i	0.545012	+	0.838428i
$568$		0			0
$569$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$569$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$570$		0			0
$571$	−1.15237e26	+	1.99596e26i	−0.747391	+	1.29452i	0.201678	+	0.979452i	$0.435360\pi$
$571$	−1.15237e26	+	1.99596e26i	−0.747391	+	1.29452i	−0.949069	+	0.315067i	$0.897973\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	8.37498e25	+	1.45059e26i	0.500000	+	0.866025i
$577$	2.89027e26	+	1.66870e26i	1.69734	+	0.979959i	0.948269	+	0.317469i	$0.102833\pi$
$577$	2.89027e26	+	1.66870e26i	1.69734	+	0.979959i	0.749070	+	0.662491i	$0.230501\pi$
$578$		0			0
$579$	2.02248e26	−	1.16768e26i	1.14931	−	0.663556i
$580$		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	$0.5\pi$
$588$	1.86203e26	+	8.27044e25i	0.913907	+	0.405923i
$589$	−3.27051e26			−1.57950
$590$		0			0
$591$		0			0
$592$	2.17293e26	−	3.76362e26i	0.999974	−	1.73201i
$593$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$593$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.69311e26	−	2.93254e26i	−0.719329	−	1.24591i
$598$		0			0
$599$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$599$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$600$		0			0
$601$			3.11996e26i			1.24406i	0.782994	+	0.622030i	$0.213692\pi$
$601$			3.11996e26i			1.24406i	−0.782994	+	0.622030i	$0.786308\pi$
$602$		0			0
$603$	5.12760e26			1.98107
$604$	9.30028e24	+	1.61086e25i	0.0353708	+	0.0612640i
$605$		0			0
$606$		0			0
$607$	−4.32601e26	+	2.49762e26i	−1.56962	+	0.906222i	−0.573409	+	0.819269i	$0.694379\pi$
$607$	−4.32601e26	+	2.49762e26i	−1.56962	+	0.906222i	−0.996212	+	0.0869525i	$0.972287\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.99985e26	−	3.46385e26i	0.660881	−	1.14468i	−0.319503	−	0.947585i	$-0.603516\pi$
$613$	1.99985e26	−	3.46385e26i	0.660881	−	1.14468i	0.980384	−	0.197095i	$-0.0631507\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	−1.04142e26	−	6.01263e25i	−0.313736	−	0.181135i	0.334861	−	0.942267i	$-0.391311\pi$
$619$	−1.04142e26	−	6.01263e25i	−0.313736	−	0.181135i	−0.648597	+	0.761132i	$0.724644\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−7.12763e26			−1.98926
$625$	−1.81899e26	−	3.15058e26i	−0.500000	−	0.866025i
$626$		0			0
$627$		0			0
$628$	1.74025e26	−	1.00474e26i	0.457084	−	0.263898i
$629$		0			0
$630$		0			0
$631$	−7.88321e26			−1.97890			−0.989450	−	0.144873i	$-0.953723\pi$
$631$	−7.88321e26			−1.97890			−0.989450	+	0.144873i	$0.953723\pi$
$632$		0			0
$633$	7.08400e26	+	4.08995e26i	1.72561	+	0.996283i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−7.00959e26	+	5.10231e26i	−1.60830	+	1.17069i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$641$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$642$		0			0
$643$			8.31292e26i			1.74482i	0.488779	+	0.872408i	$0.337443\pi$
$643$			8.31292e26i			1.74482i	−0.488779	+	0.872408i	$0.662557\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$647$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	−1.00550e27	+	5.31448e25i	−1.87656	+	0.0991839i
$652$	2.46255e26			0.452932
$653$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$653$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$		−	1.92315e26i		−	0.328959i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	4.48193e26	+	2.58764e26i	0.723687	+	0.417821i	0.816108	−	0.577899i	$-0.196127\pi$
$661$	4.48193e26	+	2.58764e26i	0.723687	+	0.417821i	−0.0924209	+	0.995720i	$0.529461\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	3.14011e26	−	5.43883e26i	0.452269	−	0.783352i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−8.57491e26			−1.16704			−0.583521	−	0.812098i	$-0.698325\pi$
$673$	−8.57491e26			−1.16704			−0.583521	+	0.812098i	$0.698325\pi$
$674$		0			0
$675$	6.54510e26	+	3.77882e26i	0.866025	+	0.500000i
$676$	1.13328e27	−	1.96290e27i	1.47858	−	2.56098i
$677$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$677$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$678$		0			0
$679$	1.10122e27	+	5.60482e26i	1.37756	+	0.701130i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$683$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$684$	−6.23901e26	+	3.60210e26i	−0.727919	+	0.420264i
$685$		0			0
$686$		0			0
$687$	−4.90048e26			−0.548466
$688$	−8.26356e26	−	1.43129e27i	−0.912172	−	1.57993i
$689$		0			0
$690$		0			0
$691$	1.12985e27	−	6.52318e26i	1.19668	−	0.690905i	0.236868	−	0.971542i	$-0.423879\pi$
$691$	1.12985e27	−	6.52318e26i	1.19668	−	0.690905i	0.959814	+	0.280637i	$0.0905457\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	1.06617e27	−	5.63513e25i	0.998606	−	0.0527804i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	1.61874e27	+	9.34580e26i	1.45580	+	0.840506i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	1.17157e27	+	2.02922e27i	0.971916	+	1.68341i	0.689759	+	0.724039i	$0.257716\pi$
$709$	1.17157e27	+	2.02922e27i	0.971916	+	1.68341i	0.282156	+	0.959368i	$0.408950\pi$
$710$		0			0
$711$	1.52482e26	−	2.64107e26i	0.123157	−	0.213314i
$712$		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	$-0.166667\pi$
$719$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$720$		0			0
$721$	1.37072e26	+	2.59340e27i	0.0969536	+	1.83436i
$722$		0			0
$723$	−1.24728e27	−	2.16035e27i	−0.859310	−	1.48837i
$724$	7.23621e26	+	4.17783e26i	0.492034	+	0.284076i
$725$		0			0
$726$		0			0
$727$		−	2.89394e27i		−	1.89197i	−0.324217	−	0.945983i	$-0.605101\pi$
$727$		−	2.89394e27i		−	1.89197i	0.324217	−	0.945983i	$-0.394899\pi$
$728$		0			0
$729$	−1.57004e27			−1.00000
$730$		0			0
$731$		0			0
$732$	−1.04366e27	+	1.80767e27i	−0.639297	+	1.10730i
$733$	−2.23012e27	+	1.28756e27i	−1.34846	+	0.778537i	−0.988032	−	0.154249i	$-0.950704\pi$
$733$	−2.23012e27	+	1.28756e27i	−1.34846	+	0.778537i	−0.360433	+	0.932785i	$0.617371\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	7.60254e26	−	1.31680e27i	0.425438	−	0.736881i	−0.571023	−	0.820934i	$-0.693453\pi$
$739$	7.60254e26	−	1.31680e27i	0.425438	−	0.736881i	0.996461	+	0.0840533i	$0.0267866\pi$
$740$		0			0
$741$		−	3.06561e27i		−	1.67203i
$742$		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
$751$	7.13011e26	+	1.23497e27i	0.342387	+	0.593031i	0.984875	−	0.173264i	$-0.0554313\pi$
$751$	7.13011e26	+	1.23497e27i	0.342387	+	0.593031i	−0.642489	+	0.766295i	$0.722098\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−1.85961e27	+	1.20882e27i	−0.838428	+	0.545012i
$757$	3.34562e26			0.148959			0.0744793	−	0.997223i	$-0.476271\pi$
$757$	3.34562e26			0.148959			0.0744793	+	0.997223i	$0.476271\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$761$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$762$		0			0
$763$	−1.84735e27	−	2.84189e27i	−0.763071	−	1.17388i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	−2.23081e27	+	1.28796e27i	−0.866025	+	0.500000i
$769$			8.39780e26i			0.322007i	0.986954	+	0.161003i	$0.0514730\pi$
$769$			8.39780e26i			0.322007i	−0.986954	+	0.161003i	$0.948527\pi$
$770$		0			0
$771$		0			0
$772$	1.79573e27	+	3.11030e27i	0.663556	+	1.14931i
$773$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$773$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$774$		0			0
$775$	−4.56946e27	+	2.63818e27i	−1.62742	+	0.939590i
$776$		0			0
$777$	5.12859e27	+	2.61027e27i	1.78237	+	0.907164i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−1.27188e27	+	2.86355e27i	−0.405923	+	0.913907i
$785$		0			0
$786$		0			0
$787$	−5.18994e27	−	2.99642e27i	−1.59736	−	0.922236i	−0.991994	−	0.126289i	$-0.959693\pi$
$787$	−5.18994e27	−	2.99642e27i	−1.59736	−	0.922236i	−0.605366	−	0.795947i	$-0.706973\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−4.44109e27	−	7.69220e27i	−1.27173	−	2.20270i
$794$		0			0
$795$		0			0
$796$	4.50985e27	−	2.60377e27i	1.24591	−	0.719329i
$797$		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
$804$			7.88555e27i			1.98107i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$809$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$810$		0			0
$811$			2.92108e27i			0.675843i	0.941174	+	0.337922i	$0.109724\pi$
$811$			2.92108e27i			0.675843i	−0.941174	+	0.337922i	$0.890276\pi$
$812$		0			0
$813$	7.88932e27			1.78311
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.15601e27	−	3.55417e27i	1.32797	−	0.766707i
$818$		0			0
$819$	−4.98152e26	−	9.42504e27i	−0.104994	−	1.98649i
$820$		0			0
$821$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$821$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$822$		0			0
$823$	−2.10110e27	+	3.63921e27i	−0.422812	+	0.732332i	−0.996213	−	0.0869425i	$-0.972290\pi$
$823$	−2.10110e27	+	3.63921e27i	−0.422812	+	0.732332i	0.573401	+	0.819275i	$0.305624\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	6.24939e27	+	3.60809e27i	1.17373	+	0.677656i	0.954557	−	0.298029i	$-0.0963292\pi$
$829$	6.24939e27	+	3.60809e27i	1.17373	+	0.677656i	0.219178	+	0.975685i	$0.429663\pi$
$830$		0			0
$831$	6.57149e27	−	3.79405e27i	1.20630	−	0.696456i
$832$		−	1.09613e28i		−	1.98926i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	5.48063e27	−	9.49272e27i	0.939590	−	1.62742i
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	6.10326e27			1.00000
$842$		0			0
$843$		0			0
$844$	−6.28979e27	+	1.08942e28i	−0.996283	+	1.72561i
$845$		0			0
$846$		0			0
$847$	−6.52059e27	+	3.44640e26i	−0.998606	+	0.0527804i
$848$		0			0
$849$	4.45454e27	+	7.71548e27i	0.667082	+	1.15542i
$850$		0			0
$851$		0			0
$852$		0			0
$853$			1.30743e28i			1.87242i	0.351444	+	0.936209i	$0.385691\pi$
$853$			1.30743e28i			1.87242i	−0.351444	+	0.936209i	$0.614309\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$857$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$858$		0			0
$859$	7.12643e27	−	4.11445e27i	0.954855	−	0.551286i	0.0602690	−	0.998182i	$-0.480804\pi$
$859$	7.12643e27	−	4.11445e27i	0.954855	−	0.551286i	0.894586	+	0.446897i	$0.147471\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$863$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			8.15045e27i			1.00000i
$868$	−8.17294e26	−	1.54632e28i	−0.0991839	−	1.87656i
$869$		0			0
$870$		0			0
$871$	−2.90599e28	−	1.67778e28i	−3.41289	−	1.97043i
$872$		0			0
$873$	−1.16492e28	+	6.72569e27i	−1.33864	+	0.772861i
$874$		0			0
$875$		0			0
$876$	2.95755e27			0.328959
$877$	7.38731e27	+	1.27952e28i	0.812812	+	1.40783i	0.910889	+	0.412653i	$0.135398\pi$
$877$	7.38731e27	+	1.27952e28i	0.812812	+	1.40783i	−0.0980766	+	0.995179i	$0.531269\pi$
$878$		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
$883$	−1.93667e28			−1.99723			−0.998617	−	0.0525783i	$-0.983256\pi$
$883$	−1.93667e28			−1.99723			−0.998617	+	0.0525783i	$0.983256\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$887$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$888$		0			0
$889$	1.89992e27	−	3.73291e27i	0.183725	−	0.360978i
$890$		0			0
$891$		0			0
$892$	8.36418e27	+	4.82906e27i	0.783352	+	0.452269i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−5.81131e27	+	1.00655e28i	−0.500000	+	0.866025i
$901$		0			0
$902$		0			0
$903$	1.83487e28	−	1.19274e28i	1.52958	−	0.994290i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−9.49346e27	+	1.64432e28i	−0.758849	+	1.31437i	0.184589	+	0.982816i	$0.440905\pi$
$907$	−9.49346e27	+	1.64432e28i	−0.758849	+	1.31437i	−0.943438	+	0.331549i	$0.892429\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−5.53953e27	−	9.59476e27i	−0.420264	−	0.727919i
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	7.53627e27i		−	0.548466i
$917$		0			0
$918$		0			0
$919$	4.36969e27	+	7.56852e27i	0.308286	+	0.533966i	0.977987	−	0.208664i	$-0.0669114\pi$
$919$	4.36969e27	+	7.56852e27i	0.308286	+	0.533966i	−0.669702	+	0.742630i	$0.733578\pi$
$920$		0			0
$921$	−1.16810e28	+	2.02320e28i	−0.807257	+	1.39821i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.01554e28			1.99995
$926$		0			0
$927$	−2.44838e28	−	1.41357e28i	−1.59082	−	0.918461i
$928$		0			0
$929$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$929$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$930$		0			0
$931$	−1.23162e28	−	5.47039e27i	−0.768165	−	0.341190i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$		−	1.83178e28i		−	1.07485i	−0.843313	−	0.537423i	$-0.819398\pi$
$937$		−	1.83178e28i		−	1.07485i	0.843313	−	0.537423i	$-0.180602\pi$
$938$		0			0
$939$	−1.16412e28			−0.669384
$940$		0			0
$941$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$941$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$947$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$948$	4.06161e27	+	2.34497e27i	0.213314	+	0.123157i
$949$	−6.29264e27	+	1.08992e28i	−0.327193	+	0.566715i
$950$		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
$961$	2.74276e28	+	4.75061e28i	1.26566	+	2.19218i
$962$		0			0
$963$		0			0
$964$	3.32232e28	−	1.91815e28i	1.48837	−	0.859310i
$965$		0			0
$966$		0			0
$967$	3.83031e28			1.66603			0.833014	−	0.553251i	$-0.186613\pi$
$967$	3.83031e28			1.66603			0.833014	+	0.553251i	$0.186613\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$971$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$972$		−	2.41451e28i		−	1.00000i
$973$	3.69036e28	+	1.87826e28i	1.51355	+	0.770344i
$974$		0			0
$975$	−2.47289e28	−	4.28318e28i	−0.994631	−	1.72275i
$976$	−2.77995e28	−	1.60500e28i	−1.10730	−	0.639297i
$977$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$977$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	3.68990e28			1.40010
$982$		0			0
$983$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$983$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	4.71449e28			1.67203
$989$		0			0
$990$		0			0
$991$	2.57098e28	−	4.45307e28i	0.885929	−	1.53447i	0.0412835	−	0.999147i	$-0.486855\pi$
$991$	2.57098e28	−	4.45307e28i	0.885929	−	1.53447i	0.844645	−	0.535326i	$-0.179811\pi$
$992$		0			0
$993$			2.95113e28i			0.997633i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	−2.08959e28	−	1.20643e28i	−0.679919	−	0.392552i	0.119905	−	0.992785i	$-0.461741\pi$
$997$	−2.08959e28	−	1.20643e28i	−0.679919	−	0.392552i	−0.799825	+	0.600234i	$0.795074\pi$
$998$		0			0
$999$	−5.42527e28	+	3.13228e28i	−1.73201	+	0.999974i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.20.g.a.17.1	yes	2
3.2	odd	2	CM	21.20.g.a.17.1	yes	2
7.5	odd	6	inner	21.20.g.a.5.1	✓	2
21.5	even	6	inner	21.20.g.a.5.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.20.g.a.5.1	✓	2	7.5	odd	6	inner
21.20.g.a.5.1	✓	2	21.5	even	6	inner
21.20.g.a.17.1	yes	2	1.1	even	1	trivial
21.20.g.a.17.1	yes	2	3.2	odd	2	CM

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 20 \)
Character orbit:	\([\chi]\)	\(=\)	21.g (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(29524.5 + 17046.0i) q^{3} +(-262144. + 454047. i) q^{4} +(4.84282e7 - 9.51504e7i) q^{7} +(5.81131e8 + 1.00655e9i) q^{9} +O(q^{10})\)	\(q+(29524.5 + 17046.0i) q^{3} +(-262144. + 454047. i) q^{4} +(4.84282e7 - 9.51504e7i) q^{7} +(5.81131e8 + 1.00655e9i) q^{9} +(-1.54793e10 + 8.93700e9i) q^{12} -7.60595e10i q^{13} +(-1.37439e11 - 2.38051e11i) q^{16} +(1.02386e12 - 5.91128e11i) q^{19} +(3.05175e12 - 1.98376e12i) q^{21} +(9.53674e12 - 1.65181e13i) q^{25} +3.96238e13i q^{27} +(3.05076e13 + 4.69318e13i) q^{28} +(-2.39571e14 - 1.38317e14i) q^{31} -6.09360e14 q^{36} +(7.90506e14 + 1.36920e15i) q^{37} +(1.29651e15 - 2.24562e15i) q^{39} +6.01253e15 q^{43} -9.37113e15i q^{48} +(-6.70831e15 - 9.21593e15i) q^{49} +(3.45346e16 + 1.99385e16i) q^{52} +4.03054e16 q^{57} +(1.01134e17 - 5.83897e16i) q^{61} +(1.23917e17 - 6.54950e15i) q^{63} +1.44115e17 q^{64} +(2.20587e17 - 3.82068e17i) q^{67} +(-1.43298e17 - 8.27332e16i) q^{73} +(5.63135e17 - 3.25126e17i) q^{75} +6.19843e17i q^{76} +(-1.31195e17 - 2.27236e17i) q^{79} +(-6.75426e17 + 1.16987e18i) q^{81} +(1.00722e17 + 1.90567e18i) q^{84} +(-7.23710e18 - 3.68343e18i) q^{91} +(-4.71548e18 - 8.16746e18i) q^{93} +1.15735e19i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9}+O(q^{10}) \) 2 * q + 59049 * q^3 - 524288 * q^4 + 96856453 * q^7 + 1162261467 * q^9	\( 2 q + 59049 q^{3} - 524288 q^{4} + 96856453 q^{7} + 1162261467 q^{9} - 30958682112 q^{12} - 274877906944 q^{16} + 2047727487699 q^{19} + 6103502217954 q^{21} + 19073486328125 q^{25} + 61015133782016 q^{28} - 479142790563765 q^{31} - 12\!\cdots\!92 q^{36}+ \cdots - 94\!\cdots\!95 q^{93}+O(q^{100}) \) 2 * q + 59049 * q^3 - 524288 * q^4 + 96856453 * q^7 + 1162261467 * q^9 - 30958682112 * q^12 - 274877906944 * q^16 + 2047727487699 * q^19 + 6103502217954 * q^21 + 19073486328125 * q^25 + 61015133782016 * q^28 - 479142790563765 * q^31 - 1218719480020992 * q^36 + 1581011344413647 * q^37 + 2593018074647367 * q^39 + 12025059029572670 * q^43 - 13416617883005077 * q^49 + 69069159189184512 * q^52 + 80610840280758834 * q^57 + 202267850921264532 * q^61 + 247833179315769195 * q^63 + 288230376151711744 * q^64 + 441174519368848177 * q^67 - 286596050169531339 * q^73 + 1126270294189453125 * q^75 - 262389124703851223 * q^79 - 1350851717672992089 * q^81 + 201444831923798016 * q^84 - 14474191718377587411 * q^91 - 9430967546666586495 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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