LMFDB - Embedded newform 21.15.h.a.2.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,15,Mod(2,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([3, 2]))

N = Newforms(chi, 15, names="a")

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

chi := DirichletCharacter("21.2");

S:= CuspForms(chi, 15);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 15 $
Character orbit:	$[\chi]$	$=$	21.h (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$26.1090833119$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{4}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{6}]$

Embedding invariants

Embedding label			2.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.2
Dual form			21.15.h.a.11.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1093.50 + 1894.00i) q^{3} +(-8192.00 - 14189.0i) q^{4} +(730542. - 380174. i) q^{7} +(-2.39148e6 + 4.14217e6i) q^{9} +O(q^{10})$	\(q+(1093.50 + 1894.00i) q^{3} +(-8192.00 - 14189.0i) q^{4} +(730542. - 380174. i) q^{7} +(-2.39148e6 + 4.14217e6i) q^{9} +(1.79159e7 - 3.10313e7i) q^{12} -1.21461e8 q^{13} +(-1.34218e8 + 2.32472e8i) q^{16} +(7.90810e8 - 1.36972e9i) q^{19} +(1.51890e9 + 9.67924e8i) q^{21} +(-3.05176e9 - 5.28580e9i) q^{25} -1.04604e10 q^{27} +(-1.13789e10 - 7.25124e9i) q^{28} +(-2.73961e10 - 4.74514e10i) q^{31} +7.83642e10 q^{36} +(3.85971e10 - 6.68521e10i) q^{37} +(-1.32818e11 - 2.30047e11i) q^{39} -3.75951e11 q^{43} -5.87068e11 q^{48} +(3.89159e11 - 5.55466e11i) q^{49} +(9.95008e11 + 1.72340e12i) q^{52} +3.45900e12 q^{57} +(-3.10730e12 + 5.38200e12i) q^{61} +(-1.72333e11 + 3.93521e12i) q^{63} +4.39805e12 q^{64} +(-3.81175e12 - 6.60215e12i) q^{67} +(1.02826e13 + 1.78099e13i) q^{73} +(6.67419e12 - 1.15600e13i) q^{75} -2.59133e13 q^{76} +(8.99948e12 - 1.55876e13i) q^{79} +(-1.14384e13 - 1.98119e13i) q^{81} +(1.29104e12 - 2.94808e13i) q^{84} +(-8.87323e13 + 4.61763e13i) q^{91} +(5.99152e13 - 1.03776e14i) q^{93} -1.16517e13 q^{97} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2187 q^{3} - 16384 q^{4} + 1461083 q^{7} - 4782969 q^{9}+O(q^{10}) $ 2 * q + 2187 * q^3 - 16384 * q^4 + 1461083 * q^7 - 4782969 * q^9	$ 2 q + 2187 q^{3} - 16384 q^{4} + 1461083 q^{7} - 4782969 q^{9} + 35831808 q^{12} - 242921906 q^{13} - 268435456 q^{16} + 1581620029 q^{19} + 3037791114 q^{21} - 6103515625 q^{25} - 20920706406 q^{27} - 22757736448 q^{28} - 54792115547 q^{31} + 156728328192 q^{36} + 77194104697 q^{37} - 265635104211 q^{39} - 751902134378 q^{43} - 1174136684544 q^{48} + 778317387191 q^{49} + 1990016253952 q^{52} + 6918006006846 q^{57} - 6214599846074 q^{61} - 344665529109 q^{63} + 8796093022208 q^{64} - 7623508989083 q^{67} + 20565100535713 q^{73} + 13348388671875 q^{75} - 51826525110272 q^{76} + 17998954613629 q^{79} - 22876792454961 q^{81} + 2582075916288 q^{84} - 177464533592099 q^{91} + 119830356701289 q^{93} - 23303389794044 q^{97}+O(q^{100}) $ 2 * q + 2187 * q^3 - 16384 * q^4 + 1461083 * q^7 - 4782969 * q^9 + 35831808 * q^12 - 242921906 * q^13 - 268435456 * q^16 + 1581620029 * q^19 + 3037791114 * q^21 - 6103515625 * q^25 - 20920706406 * q^27 - 22757736448 * q^28 - 54792115547 * q^31 + 156728328192 * q^36 + 77194104697 * q^37 - 265635104211 * q^39 - 751902134378 * q^43 - 1174136684544 * q^48 + 778317387191 * q^49 + 1990016253952 * q^52 + 6918006006846 * q^57 - 6214599846074 * q^61 - 344665529109 * q^63 + 8796093022208 * q^64 - 7623508989083 * q^67 + 20565100535713 * q^73 + 13348388671875 * q^75 - 51826525110272 * q^76 + 17998954613629 * q^79 - 22876792454961 * q^81 + 2582075916288 * q^84 - 177464533592099 * q^91 + 119830356701289 * q^93 - 23303389794044 * q^97

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$2$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$3$	1093.50	+	1894.00i	0.500000	+	0.866025i
$3$	1093.50	+	1894.00i	0.500000	+	0.866025i
$4$	−8192.00	−	14189.0i	−0.500000	−	0.866025i
$5$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$5$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$6$		0			0
$7$	730542.	−	380174.i	0.887071	−	0.461632i
$7$	730542.	−	380174.i	0.887071	−	0.461632i
$8$		0			0
$9$	−2.39148e6	+	4.14217e6i	−0.500000	+	0.866025i
$10$		0			0
$11$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$11$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$12$	1.79159e7	−	3.10313e7i	0.500000	−	0.866025i
$13$	−1.21461e8			−1.93568			−0.967839	−	0.251570i	$-0.919053\pi$
$13$	−1.21461e8			−1.93568			−0.967839	+	0.251570i	$0.919053\pi$
$14$		0			0
$15$		0			0
$16$	−1.34218e8	+	2.32472e8i	−0.500000	+	0.866025i
$17$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$17$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$18$		0			0
$19$	7.90810e8	−	1.36972e9i	0.884702	−	1.53235i	0.0386469	−	0.999253i	$-0.487695\pi$
$19$	7.90810e8	−	1.36972e9i	0.884702	−	1.53235i	0.846055	−	0.533096i	$-0.178971\pi$
$20$		0			0
$21$	1.51890e9	+	9.67924e8i	0.843321	+	0.537410i
$22$		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
$25$	−3.05176e9	−	5.28580e9i	−0.500000	−	0.866025i
$26$		0			0
$27$	−1.04604e10			−1.00000
$28$	−1.13789e10	−	7.25124e9i	−0.843321	−	0.537410i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$	−2.73961e10	−	4.74514e10i	−0.995764	−	1.72471i	−0.577514	−	0.816381i	$-0.695977\pi$
$31$	−2.73961e10	−	4.74514e10i	−0.995764	−	1.72471i	−0.418250	−	0.908332i	$-0.637356\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	7.83642e10			1.00000
$37$	3.85971e10	−	6.68521e10i	0.406576	−	0.704211i	−0.587927	−	0.808914i	$-0.700056\pi$
$37$	3.85971e10	−	6.68521e10i	0.406576	−	0.704211i	0.994504	+	0.104703i	$0.0333892\pi$
$38$		0			0
$39$	−1.32818e11	−	2.30047e11i	−0.967839	−	1.67635i
$40$		0			0
$41$		0			0		1.00000			$0$
$41$		0			0		−1.00000			$\pi$
$42$		0			0
$43$	−3.75951e11			−1.38310			−0.691548	−	0.722331i	$-0.743071\pi$
$43$	−3.75951e11			−1.38310			−0.691548	+	0.722331i	$0.743071\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$47$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$48$	−5.87068e11			−1.00000
$49$	3.89159e11	−	5.55466e11i	0.573792	−	0.819001i
$50$		0			0
$51$		0			0
$52$	9.95008e11	+	1.72340e12i	0.967839	+	1.67635i
$53$		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
$57$	3.45900e12			1.76940
$58$		0			0
$59$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$59$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$60$		0			0
$61$	−3.10730e12	+	5.38200e12i	−0.988722	+	1.71252i	−0.364665	+	0.931139i	$0.618816\pi$
$61$	−3.10730e12	+	5.38200e12i	−0.988722	+	1.71252i	−0.624058	+	0.781378i	$0.714517\pi$
$62$		0			0
$63$	−1.72333e11	+	3.93521e12i	−0.0437506	+	0.999042i
$64$	4.39805e12			1.00000
$65$		0			0
$66$		0			0
$67$	−3.81175e12	−	6.60215e12i	−0.628929	−	1.08934i	−0.987767	−	0.155937i	$-0.950160\pi$
$67$	−3.81175e12	−	6.60215e12i	−0.628929	−	1.08934i	0.358839	−	0.933400i	$-0.383173\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$	1.02826e13	+	1.78099e13i	0.930767	+	1.61214i	0.782014	+	0.623261i	$0.214192\pi$
$73$	1.02826e13	+	1.78099e13i	0.930767	+	1.61214i	0.148753	+	0.988874i	$0.452474\pi$
$74$		0			0
$75$	6.67419e12	−	1.15600e13i	0.500000	−	0.866025i
$76$	−2.59133e13			−1.76940
$77$		0			0
$78$		0			0
$79$	8.99948e12	−	1.55876e13i	0.468627	−	0.811686i	−0.530730	−	0.847541i	$-0.678082\pi$
$79$	8.99948e12	−	1.55876e13i	0.468627	−	0.811686i	0.999357	+	0.0358548i	$0.0114154\pi$
$80$		0			0
$81$	−1.14384e13	−	1.98119e13i	−0.500000	−	0.866025i
$82$		0			0
$83$		0			0		1.00000			$0$
$83$		0			0		−1.00000			$\pi$
$84$	1.29104e12	−	2.94808e13i	0.0437506	−	0.999042i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$89$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$90$		0			0
$91$	−8.87323e13	+	4.61763e13i	−1.71709	+	0.893571i
$92$		0			0
$93$	5.99152e13	−	1.03776e14i	0.995764	−	1.72471i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−1.16517e13			−0.144207			−0.0721036	−	0.997397i	$-0.522971\pi$
$97$	−1.16517e13			−0.144207			−0.0721036	+	0.997397i	$0.522971\pi$
$98$		0			0
$99$		0			0
$100$	−5.00000e13	+	8.66025e13i	−0.500000	+	0.866025i
$101$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$101$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$102$		0			0
$103$	2.64726e13	−	4.58519e13i	0.215246	−	0.372818i	−0.738102	−	0.674689i	$-0.764278\pi$
$103$	2.64726e13	−	4.58519e13i	0.215246	−	0.372818i	0.953349	+	0.301871i	$0.0976112\pi$
$104$		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	$0.666667\pi$
$108$	8.56912e13	+	1.48422e14i	0.500000	+	0.866025i
$109$	1.33584e14	+	2.31374e14i	0.730748	+	1.26569i	0.956564	+	0.291523i	$0.0941618\pi$
$109$	1.33584e14	+	2.31374e14i	0.730748	+	1.26569i	−0.225816	+	0.974170i	$0.572505\pi$
$110$		0			0
$111$	1.68824e14			0.813153
$112$	−9.67186e12	+	2.20856e14i	−0.0437506	+	0.999042i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	2.90472e14	−	5.03112e14i	0.967839	−	1.67635i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−1.89875e14	+	3.28873e14i	−0.500000	+	0.866025i
$122$		0			0
$123$		0			0
$124$	−4.48857e14	+	7.77443e14i	−0.995764	+	1.72471i
$125$		0			0
$126$		0			0
$127$	−1.04016e14			−0.195198			−0.0975991	−	0.995226i	$-0.531116\pi$
$127$	−1.04016e14			−0.195198			−0.0975991	+	0.995226i	$0.531116\pi$
$128$		0			0
$129$	−4.11102e14	−	7.12050e14i	−0.691548	−	1.19780i
$130$		0			0
$131$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$131$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$132$		0			0
$133$	5.69866e13	−	1.30128e15i	0.0774125	−	1.76771i
$134$		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
$139$	9.48351e14			0.945944			0.472972	−	0.881077i	$-0.343181\pi$
$139$	9.48351e14			0.945944			0.472972	+	0.881077i	$0.343181\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−6.41959e14	−	1.11191e15i	−0.500000	−	0.866025i
$145$		0			0
$146$		0			0
$147$	1.47760e15	+	1.29664e14i	0.996172	+	0.0874174i
$148$	−1.26475e15			−0.813153
$149$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$149$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$150$		0			0
$151$	−1.17756e15	−	2.03959e15i	−0.657875	−	1.13947i	−0.981165	−	0.193172i	$-0.938122\pi$
$151$	−1.17756e15	−	2.03959e15i	−0.657875	−	1.13947i	0.323290	−	0.946300i	$-0.395211\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.17608e15	+	3.76909e15i	−0.967839	+	1.67635i
$157$	−1.01169e15	−	1.75230e15i	−0.430279	−	0.745266i	0.566618	−	0.823981i	$-0.308252\pi$
$157$	−1.01169e15	−	1.75230e15i	−0.430279	−	0.745266i	−0.996897	+	0.0787150i	$0.974918\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−7.59317e14	+	1.31517e15i	−0.248376	+	0.430200i	−0.963075	−	0.269232i	$-0.913230\pi$
$163$	−7.59317e14	+	1.31517e15i	−0.248376	+	0.430200i	0.714699	+	0.699432i	$0.246564\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		1.00000			$0$
$167$		0			0		−1.00000			$\pi$
$168$		0			0
$169$	1.08154e16			2.74685
$170$		0			0
$171$	3.78242e15	+	6.55134e15i	0.884702	+	1.53235i
$172$	3.07979e15	+	5.33435e15i	0.691548	+	1.19780i
$173$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$173$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$174$		0			0
$175$	−4.23896e15	−	2.70130e15i	−0.843321	−	0.537410i
$176$		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
$181$	1.08982e16			1.71239			0.856196	−	0.516652i	$-0.172822\pi$
$181$	1.08982e16			1.71239			0.856196	+	0.516652i	$0.172822\pi$
$182$		0			0
$183$	−1.35913e16			−1.97744
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−7.64172e15	+	3.97675e15i	−0.887071	+	0.461632i
$190$		0			0
$191$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$191$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$192$	4.80926e15	+	8.32989e15i	0.500000	+	0.866025i
$193$	−9.96731e15	−	1.72639e16i	−0.999256	−	1.73076i	−0.533034	−	0.846094i	$-0.678948\pi$
$193$	−9.96731e15	−	1.72639e16i	−0.999256	−	1.73076i	−0.466222	−	0.884668i	$-0.654385\pi$
$194$		0			0
$195$		0			0
$196$	−1.10695e16	−	9.71383e14i	−0.996172	−	0.0874174i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$	1.87737e15	+	3.25170e15i	0.151907	+	0.263111i	0.931929	−	0.362642i	$-0.118125\pi$
$199$	1.87737e15	+	3.25170e15i	0.151907	+	0.263111i	−0.780021	+	0.625753i	$0.784792\pi$
$200$		0			0
$201$	8.33631e15	−	1.44389e16i	0.628929	−	1.08934i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$	1.63022e16	−	2.82363e16i	0.967839	−	1.67635i
$209$		0			0
$210$		0			0
$211$	1.14160e16			0.613110			0.306555	−	0.951853i	$-0.400824\pi$
$211$	1.14160e16			0.613110			0.306555	+	0.951853i	$0.400824\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−3.80537e16	−	2.42499e16i	−1.67950	−	1.07027i
$218$		0			0
$219$	−2.24879e16	+	3.89503e16i	−0.930767	+	1.61214i
$220$		0			0
$221$		0			0
$222$		0			0
$223$	1.42664e16			0.520212			0.260106	−	0.965580i	$-0.416242\pi$
$223$	1.42664e16			0.520212			0.260106	+	0.965580i	$0.416242\pi$
$224$		0			0
$225$	2.91929e16			1.00000
$226$		0			0
$227$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$227$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$228$	−2.83362e16	−	4.90797e16i	−0.884702	−	1.53235i
$229$	−3.27636e16	+	5.67482e16i	−0.992072	+	1.71832i	−0.387201	+	0.921995i	$0.626558\pi$
$229$	−3.27636e16	+	5.67482e16i	−0.992072	+	1.71832i	−0.604871	+	0.796323i	$0.706775\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$233$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	3.93637e16			0.937255
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$	4.55611e16	+	7.89142e16i	0.964884	+	1.67123i	0.709925	+	0.704277i	$0.248729\pi$
$241$	4.55611e16	+	7.89142e16i	0.964884	+	1.67123i	0.254959	+	0.966952i	$0.417938\pi$
$242$		0			0
$243$	2.50158e16	−	4.33286e16i	0.500000	−	0.866025i
$244$	1.01820e17			1.97744
$245$		0			0
$246$		0			0
$247$	−9.60525e16	+	1.66368e17i	−1.71250	+	2.96613i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0		1.00000			$0$
$251$		0			0		−1.00000			$\pi$
$252$	5.72483e16	−	2.97920e16i	0.887071	−	0.461632i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	−3.60288e16	−	6.24037e16i	−0.500000	−	0.866025i
$257$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$257$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$258$		0			0
$259$	2.78134e15	−	6.35118e16i	0.0355759	−	0.812374i
$260$		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
$268$	−6.24518e16	+	1.08170e17i	−0.628929	+	1.08934i
$269$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$269$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$270$		0			0
$271$	−1.05366e16	+	1.82499e16i	−0.0981554	+	0.170010i	−0.910921	−	0.412581i	$-0.864628\pi$
$271$	−1.05366e16	+	1.82499e16i	−0.0981554	+	0.170010i	0.812766	+	0.582591i	$0.197961\pi$
$272$		0			0
$273$	−1.84487e17	−	1.17565e17i	−1.63240	−	1.04025i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−6.07637e16	−	1.05246e17i	−0.485608	−	0.841097i	0.514256	−	0.857637i	$-0.328068\pi$
$277$	−6.07637e16	−	1.05246e17i	−0.485608	−	0.841097i	−0.999863	+	0.0165399i	$0.994735\pi$
$278$		0			0
$279$	2.62069e17			1.99153
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$	−1.42264e17	−	2.46409e17i	−0.978567	−	1.69493i	−0.667622	−	0.744501i	$-0.732688\pi$
$283$	−1.42264e17	−	2.46409e17i	−0.978567	−	1.69493i	−0.310945	−	0.950428i	$-0.600646\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−8.41889e16	+	1.45819e17i	−0.500000	+	0.866025i
$290$		0			0
$291$	−1.27411e16	−	2.20683e16i	−0.0721036	−	0.124887i
$292$	1.68469e17	−	2.91797e17i	0.930767	−	1.61214i
$293$		0			0		1.00000			$0$
$293$		0			0		−1.00000			$\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$		0			0
$298$		0			0
$299$		0			0
$300$	−2.18700e17			−1.00000
$301$	−2.74648e17	+	1.42927e17i	−1.22690	+	0.638481i
$302$		0			0
$303$		0			0
$304$	2.12281e17	+	3.67682e17i	0.884702	+	1.53235i
$305$		0			0
$306$		0			0
$307$	3.95714e17			1.53962			0.769808	−	0.638275i	$-0.220352\pi$
$307$	3.95714e17			1.53962			0.769808	+	0.638275i	$0.220352\pi$
$308$		0			0
$309$	1.15791e17			0.430493
$310$		0			0
$311$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$311$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$312$		0			0
$313$	1.39185e17	−	2.41075e17i	0.472912	−	0.819108i	−0.526607	−	0.850109i	$-0.676536\pi$
$313$	1.39185e17	−	2.41075e17i	0.472912	−	0.819108i	0.999519	+	0.0310007i	$0.00986940\pi$
$314$		0			0
$315$		0			0
$316$	−2.94895e17			−0.937255
$317$		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
$324$	−1.87407e17	+	3.24598e17i	−0.500000	+	0.866025i
$325$	3.70669e17	+	6.42018e17i	0.967839	+	1.67635i
$326$		0			0
$327$	−2.92147e17	+	5.06014e17i	−0.730748	+	1.26569i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	4.03474e17	−	6.98838e17i	0.926872	−	1.60539i	0.138350	−	0.990383i	$-0.455820\pi$
$331$	4.03474e17	−	6.98838e17i	0.926872	−	1.60539i	0.788522	−	0.615007i	$-0.210847\pi$
$332$		0			0
$333$	1.84609e17	+	3.19751e17i	0.406576	+	0.704211i
$334$		0			0
$335$		0			0
$336$	−4.28878e17	+	2.23188e17i	−0.887071	+	0.461632i
$337$	3.30843e17			0.670213			0.335106	−	0.942180i	$-0.391228\pi$
$337$	3.30843e17			0.670213			0.335106	+	0.942180i	$0.391228\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	7.31231e16	−	5.53739e17i	0.130917	−	0.991393i
$344$		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
$349$	−7.77937e17			−1.23358			−0.616789	−	0.787129i	$-0.711567\pi$
$349$	−7.77937e17			−1.23358			−0.616789	+	0.787129i	$0.711567\pi$
$350$		0			0
$351$	1.27052e18			1.93568
$352$		0			0
$353$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$353$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$359$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$360$		0			0
$361$	−8.51258e17	−	1.47442e18i	−1.06539	−	1.84532i
$362$		0			0
$363$	−8.30513e17			−1.00000
$364$	1.38209e18	+	8.80742e17i	1.63240	+	1.04025i
$365$		0			0
$366$		0			0
$367$	−8.19868e17	−	1.42005e18i	−0.914284	−	1.58359i	−0.807946	−	0.589257i	$-0.799421\pi$
$367$	−8.19868e17	−	1.42005e18i	−0.914284	−	1.58359i	−0.106339	−	0.994330i	$-0.533913\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−1.96330e18			−1.99153
$373$	9.35159e16	−	1.61974e17i	0.0930944	−	0.161244i	−0.815717	−	0.578451i	$-0.803657\pi$
$373$	9.35159e16	−	1.61974e17i	0.0930944	−	0.161244i	0.908812	+	0.417206i	$0.136991\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−6.19031e17			−0.551110			−0.275555	−	0.961285i	$-0.588862\pi$
$379$	−6.19031e17			−0.551110			−0.275555	+	0.961285i	$0.588862\pi$
$380$		0			0
$381$	−1.13742e17	−	1.97007e17i	−0.0975991	−	0.169047i
$382$		0			0
$383$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$383$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	8.99081e17	−	1.55725e18i	0.691548	−	1.19780i
$388$	9.54507e16	+	1.65325e17i	0.0721036	+	0.124887i
$389$		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
$397$	−1.19295e18	+	2.06624e18i	−0.767515	+	1.32938i	0.171392	+	0.985203i	$0.445174\pi$
$397$	−1.19295e18	+	2.06624e18i	−0.767515	+	1.32938i	−0.938907	+	0.344172i	$0.888160\pi$
$398$		0			0
$399$	2.52695e18	−	1.31502e18i	1.56959	−	0.816814i
$400$	1.63840e18			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
$403$	3.32755e18	+	5.76349e18i	1.92748	+	3.33849i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−1.45635e18	−	2.52247e18i	−0.760679	−	1.31753i	−0.942501	−	0.334203i	$-0.891533\pi$
$409$	−1.45635e18	−	2.52247e18i	−0.760679	−	1.31753i	0.181822	−	0.983331i	$-0.441801\pi$
$410$		0			0
$411$		0			0
$412$	−8.67454e17			−0.430493
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.03702e18	+	1.79617e18i	0.472972	+	0.819212i
$418$		0			0
$419$		0			0		1.00000			$0$
$419$		0			0		−1.00000			$\pi$
$420$		0			0
$421$	−4.20891e18			−1.79554			−0.897770	−	0.440465i	$-0.854813\pi$
$421$	−4.20891e18			−1.79554			−0.897770	+	0.440465i	$0.854813\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	−2.23915e17	+	5.11309e18i	−0.0865144	+	1.97555i
$428$		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	$0.333333\pi$
$432$	1.40396e18	−	2.43174e18i	0.500000	−	0.866025i
$433$	−8.49855e17			−0.297803			−0.148902	−	0.988852i	$-0.547574\pi$
$433$	−8.49855e17			−0.297803			−0.148902	+	0.988852i	$0.547574\pi$
$434$		0			0
$435$		0			0
$436$	2.18863e18	−	3.79082e18i	0.730748	−	1.26569i
$437$		0			0
$438$		0			0
$439$	−2.14393e18	+	3.71340e18i	−0.682275	+	1.18173i	0.292010	+	0.956415i	$0.405676\pi$
$439$	−2.14393e18	+	3.71340e18i	−0.682275	+	1.18173i	−0.974285	+	0.225319i	$0.927657\pi$
$440$		0			0
$441$	1.37017e18	+	2.94035e18i	0.422380	+	0.906419i
$442$		0			0
$443$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$443$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$444$	−1.38300e18	−	2.39543e18i	−0.406576	−	0.704211i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	3.21296e18	−	1.67202e18i	0.887071	−	0.461632i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$	2.57532e18	−	4.46058e18i	0.657875	−	1.13947i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	2.64057e18	−	4.57360e18i	0.634284	−	1.09861i	−0.352382	−	0.935856i	$-0.614628\pi$
$457$	2.64057e18	−	4.57360e18i	0.634284	−	1.09861i	0.986666	−	0.162756i	$-0.0520385\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		1.00000			$0$
$461$		0			0		−1.00000			$\pi$
$462$		0			0
$463$	3.64971e17			0.0800187			0.0400094	−	0.999199i	$-0.487261\pi$
$463$	3.64971e17			0.0800187			0.0400094	+	0.999199i	$0.487261\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$467$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$468$	−9.51819e18			−1.93568
$469$	−5.29461e18	−	3.37402e18i	−1.06078	−	0.675985i
$470$		0			0
$471$	2.21257e18	−	3.83228e18i	0.430279	−	0.745266i
$472$		0			0
$473$		0			0
$474$		0			0
$475$	−9.65344e18			−1.76940
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$479$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$480$		0			0
$481$	−4.68803e18	+	8.11991e18i	−0.787001	+	1.36313i
$482$		0			0
$483$		0			0
$484$	6.22182e18			1.00000
$485$		0			0
$486$		0			0
$487$	−3.54055e18	−	6.13241e18i	−0.544964	−	0.943906i	−0.998609	−	0.0527233i	$-0.983210\pi$
$487$	−3.54055e18	−	6.13241e18i	−0.544964	−	0.943906i	0.453645	−	0.891183i	$-0.350123\pi$
$488$		0			0
$489$	−3.32125e18			−0.496752
$490$		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
$496$	1.47081e19			1.99153
$497$		0			0
$498$		0			0
$499$	5.86237e18	−	1.01539e19i	0.760973	−	1.31804i	−0.181376	−	0.983414i	$-0.558055\pi$
$499$	5.86237e18	−	1.01539e19i	0.760973	−	1.31804i	0.942349	−	0.334631i	$-0.108612\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		0			0		1.00000			$0$
$503$		0			0		−1.00000			$\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	1.18266e19	+	2.04843e19i	1.37343	+	2.37884i
$508$	8.52103e17	+	1.47589e18i	0.0975991	+	0.169047i
$509$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$509$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$510$		0			0
$511$	1.42827e19	+	9.10171e18i	1.56987	+	1.00041i
$512$		0			0
$513$	−8.27215e18	+	1.43278e19i	−0.884702	+	1.53235i
$514$		0			0
$515$		0			0
$516$	−6.73550e18	+	1.16662e19i	−0.691548	+	1.19780i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$521$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$522$		0			0
$523$	9.99051e18	−	1.73041e19i	0.933417	−	1.61673i	0.155985	−	0.987759i	$-0.450145\pi$
$523$	9.99051e18	−	1.73041e19i	0.933417	−	1.61673i	0.777432	−	0.628966i	$-0.216522\pi$
$524$		0			0
$525$	4.80949e17	−	1.09824e19i	0.0437506	−	0.999042i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	−5.79642e18	−	1.00397e19i	−0.500000	−	0.866025i
$530$		0			0
$531$		0			0
$532$	−1.89307e19	+	9.85155e18i	−1.56959	+	0.816814i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−1.35638e19	+	2.34931e19i	−0.999999	+	1.73205i	−0.498664	+	0.866795i	$0.666176\pi$
$541$	−1.35638e19	+	2.34931e19i	−0.999999	+	1.73205i	−0.501335	+	0.865253i	$0.667157\pi$
$542$		0			0
$543$	1.19171e19	+	2.06411e19i	0.856196	+	1.48297i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	6.51749e18			0.444804			0.222402	−	0.974955i	$-0.428610\pi$
$547$	6.51749e18			0.444804			0.222402	+	0.974955i	$0.428610\pi$
$548$		0			0
$549$	−1.48621e19	−	2.57419e19i	−0.988722	−	1.71252i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	6.48511e17	−	1.48087e19i	0.0410055	−	0.936357i
$554$		0			0
$555$		0			0
$556$	−7.76889e18	−	1.34561e19i	−0.472972	−	0.819212i
$557$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$557$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$558$		0			0
$559$	4.56634e19			2.67723
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$563$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.58882e19	−	1.01248e19i	−0.843321	−	0.537410i
$568$		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
$571$	−1.77626e19	−	3.07657e19i	−0.897539	−	1.55458i	−0.830631	−	0.556823i	$-0.812020\pi$
$571$	−1.77626e19	−	3.07657e19i	−0.897539	−	1.55458i	−0.0669078	−	0.997759i	$-0.521313\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.05179e19	+	1.82175e19i	−0.500000	+	0.866025i
$577$	8.89465e18	+	1.54060e19i	0.417732	+	0.723534i	0.995711	−	0.0925182i	$-0.0294916\pi$
$577$	8.89465e18	+	1.54060e19i	0.417732	+	0.723534i	−0.577979	+	0.816052i	$0.696158\pi$
$578$		0			0
$579$	2.17985e19	−	3.77561e19i	0.999256	−	1.73076i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0		1.00000			$0$
$587$		0			0		−1.00000			$\pi$
$588$	−1.02647e19	−	2.20278e19i	−0.422380	−	0.906419i
$589$	−8.66603e19			−3.52382
$590$		0			0
$591$		0			0
$592$	1.03608e19	+	1.79455e19i	0.406576	+	0.704211i
$593$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$593$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−4.10581e18	+	7.11148e18i	−0.151907	+	0.263111i
$598$		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
$601$	4.98536e19			1.76025			0.880127	−	0.474737i	$-0.157457\pi$
$601$	4.98536e19			1.76025			0.880127	+	0.474737i	$0.157457\pi$
$602$		0			0
$603$	3.64630e19			1.25786
$604$	−1.92931e19	+	3.34166e19i	−0.657875	+	1.13947i
$605$		0			0
$606$		0			0
$607$	−2.78051e18	+	4.81598e18i	−0.0915806	+	0.158622i	−0.908176	−	0.418588i	$-0.862525\pi$
$607$	−2.78051e18	+	4.81598e18i	−0.0915806	+	0.158622i	0.816596	+	0.577210i	$0.195859\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−3.11229e19	−	5.39064e19i	−0.956878	−	1.65736i	−0.730012	−	0.683435i	$-0.760485\pi$
$613$	−3.11229e19	−	5.39064e19i	−0.956878	−	1.65736i	−0.226866	−	0.973926i	$-0.572848\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$	2.67991e19	+	4.64174e19i	0.769636	+	1.33305i	0.937761	+	0.347283i	$0.112896\pi$
$619$	2.67991e19	+	4.64174e19i	0.769636	+	1.33305i	−0.168125	+	0.985766i	$0.553771\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	7.13059e19			1.93568
$625$	−1.86265e19	+	3.22620e19i	−0.500000	+	0.866025i
$626$		0			0
$627$		0			0
$628$	−1.65756e19	+	2.87097e19i	−0.430279	+	0.745266i
$629$		0			0
$630$		0			0
$631$	5.72705e19			1.43789			0.718945	−	0.695067i	$-0.244625\pi$
$631$	5.72705e19			1.43789			0.718945	+	0.695067i	$0.244625\pi$
$632$		0			0
$633$	1.24834e19	+	2.16220e19i	0.306555	+	0.530969i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−4.72676e19	+	6.74674e19i	−1.11068	+	1.58532i
$638$		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
$643$	3.26770e19			0.719060			0.359530	−	0.933134i	$-0.382937\pi$
$643$	3.26770e19			0.719060			0.359530	+	0.933134i	$0.382937\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$647$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	4.31755e18	−	9.85910e19i	0.0871305	−	1.98962i
$652$	2.48813e19			0.496752
$653$		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
$657$	−9.83622e19			−1.86153
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$	2.14843e19	+	3.72118e19i	0.389682	+	0.674949i	0.992407	−	0.123000i	$-0.0392516\pi$
$661$	2.14843e19	+	3.72118e19i	0.389682	+	0.674949i	−0.602725	+	0.797949i	$0.705918\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	1.56003e19	+	2.70205e19i	0.260106	+	0.450517i
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.49245e19			−0.238669			−0.119334	−	0.992854i	$-0.538076\pi$
$673$	−1.49245e19			−0.238669			−0.119334	+	0.992854i	$0.538076\pi$
$674$		0			0
$675$	3.19225e19	+	5.52913e19i	0.500000	+	0.866025i
$676$	−8.85996e19	−	1.53459e20i	−1.37343	−	2.37884i
$677$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$677$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$678$		0			0
$679$	−8.51205e18	+	4.42967e18i	−0.127922	+	0.0665707i
$680$		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	$0.333333\pi$
$684$	6.19712e19	−	1.07337e20i	0.884702	−	1.53235i
$685$		0			0
$686$		0			0
$687$	−1.43308e20			−1.98414
$688$	5.04593e19	−	8.73981e19i	0.691548	−	1.19780i
$689$		0			0
$690$		0			0
$691$	−3.46924e19	+	6.00890e19i	−0.461198	+	0.798819i	−0.999021	−	0.0442390i	$-0.985914\pi$
$691$	−3.46924e19	+	6.00890e19i	−0.461198	+	0.798819i	0.537823	+	0.843058i	$0.319247\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−3.60305e18	+	8.22754e19i	−0.0437506	+	0.999042i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$	−6.10459e19	−	1.05735e20i	−0.719398	−	1.24603i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	8.54582e19	−	1.48018e20i	0.948921	−	1.64358i	0.201217	−	0.979547i	$-0.435510\pi$
$709$	8.54582e19	−	1.48018e20i	0.948921	−	1.64358i	0.747704	−	0.664032i	$-0.231156\pi$
$710$		0			0
$711$	4.30442e19	+	7.45548e19i	0.468627	+	0.811686i
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$719$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$720$		0			0
$721$	1.90764e18	−	4.35609e19i	0.0188343	−	0.430081i
$722$		0			0
$723$	−9.96422e19	+	1.72585e20i	−0.964884	+	1.67123i
$724$	−8.92777e19	−	1.54633e20i	−0.856196	−	1.48297i
$725$		0			0
$726$		0			0
$727$	2.07849e20			1.93646			0.968229	−	0.250065i	$-0.0804521\pi$
$727$	2.07849e20			1.93646			0.968229	+	0.250065i	$0.0804521\pi$
$728$		0			0
$729$	1.09419e20			1.00000
$730$		0			0
$731$		0			0
$732$	1.11340e20	+	1.92847e20i	0.988722	+	1.71252i
$733$	−2.74816e19	+	4.75996e19i	−0.241721	+	0.418674i	−0.961205	−	0.275836i	$-0.911045\pi$
$733$	−2.74816e19	+	4.75996e19i	−0.241721	+	0.418674i	0.719483	+	0.694510i	$0.244379\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.20365e20	+	2.08479e20i	0.999976	+	1.73201i	0.505956	+	0.862559i	$0.331140\pi$
$739$	1.20365e20	+	2.08479e20i	0.999976	+	1.73201i	0.494020	+	0.869451i	$0.335527\pi$
$740$		0			0
$741$	−4.20134e20			−3.42500
$742$		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
$751$	9.99876e19	−	1.73184e20i	0.742107	−	1.28537i	−0.209427	−	0.977824i	$-0.567160\pi$
$751$	9.99876e19	−	1.73184e20i	0.742107	−	1.28537i	0.951534	−	0.307543i	$-0.0995067\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	1.19027e20	+	7.58505e19i	0.843321	+	0.537410i
$757$	−2.34099e20			−1.64334			−0.821672	−	0.569960i	$-0.806959\pi$
$757$	−2.34099e20			−1.64334			−0.821672	+	0.569960i	$0.806959\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$761$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$762$		0			0
$763$	1.85551e20	+	1.18243e20i	1.23251	+	0.785423i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$	7.87950e19	−	1.36477e20i	0.500000	−	0.866025i
$769$	2.72440e20			1.71311			0.856557	−	0.516052i	$-0.172599\pi$
$769$	2.72440e20			1.71311			0.856557	+	0.516052i	$0.172599\pi$
$770$		0			0
$771$		0			0
$772$	−1.63304e20	+	2.82851e20i	−0.999256	+	1.73076i
$773$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$773$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$774$		0			0
$775$	−1.67212e20	+	2.89620e20i	−0.995764	+	1.72471i
$776$		0			0
$777$	1.23333e20	−	6.41823e19i	0.721324	−	0.375377i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	7.68982e19	+	1.65022e20i	0.422380	+	0.906419i
$785$		0			0
$786$		0			0
$787$	−1.21301e20	−	2.10100e20i	−0.648698	−	1.12358i	−0.983434	−	0.181267i	$-0.941980\pi$
$787$	−1.21301e20	−	2.10100e20i	−0.648698	−	1.12358i	0.334736	−	0.942312i	$-0.391353\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	3.77416e20	−	6.53703e20i	1.91385	−	3.31488i
$794$		0			0
$795$		0			0
$796$	3.07589e19	−	5.32759e19i	0.151907	−	0.263111i
$797$		0			0		1.00000			$0$
$797$		0			0		−1.00000			$\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$	−2.73164e20			−1.25786
$805$		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
$811$	−3.87857e20			−1.68084			−0.840419	−	0.541937i	$-0.817691\pi$
$811$	−3.87857e20			−1.68084			−0.840419	+	0.541937i	$0.817691\pi$
$812$		0			0
$813$	−4.60869e19			−0.196311
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−2.97306e20	+	5.14949e20i	−1.22363	+	2.11938i
$818$		0			0
$819$	2.09317e19	−	4.77974e20i	0.0846871	−	1.93383i
$820$		0			0
$821$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$821$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$822$		0			0
$823$	−2.51862e20	−	4.36238e20i	−0.984835	−	1.70578i	−0.642667	−	0.766146i	$-0.722172\pi$
$823$	−2.51862e20	−	4.36238e20i	−0.984835	−	1.70578i	−0.342168	−	0.939639i	$-0.611161\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$	−3.62858e19	−	6.28489e19i	−0.134851	−	0.233569i	0.790689	−	0.612218i	$-0.209722\pi$
$829$	−3.62858e19	−	6.28489e19i	−0.134851	−	0.233569i	−0.925541	+	0.378648i	$0.876389\pi$
$830$		0			0
$831$	1.32890e20	−	2.30172e20i	0.485608	−	0.841097i
$832$	−5.34191e20			−1.93568
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	2.86572e20	+	4.96358e20i	0.995764	+	1.72471i
$838$		0			0
$839$		0			0		1.00000			$0$
$839$		0			0		−1.00000			$\pi$
$840$		0			0
$841$	2.97558e20			1.00000
$842$		0			0
$843$		0			0
$844$	−9.35202e19	−	1.61982e20i	−0.306555	−	0.530969i
$845$		0			0
$846$		0			0
$847$	−1.36826e19	+	3.12441e20i	−0.0437506	+	0.999042i
$848$		0			0
$849$	3.11132e20	−	5.38896e20i	0.978567	−	1.69493i
$850$		0			0
$851$		0			0
$852$		0			0
$853$	5.97351e20			1.81797			0.908984	−	0.416831i	$-0.136859\pi$
$853$	5.97351e20			1.81797			0.908984	+	0.416831i	$0.136859\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$857$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$858$		0			0
$859$	1.69351e20	−	2.93325e20i	0.490723	−	0.849956i	−0.509220	−	0.860636i	$-0.670066\pi$
$859$	1.69351e20	−	2.93325e20i	0.490723	−	0.849956i	0.999943	+	0.0106798i	$0.00339955\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$863$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−3.68242e20			−1.00000
$868$	−3.23451e19	+	7.38598e20i	−0.0871305	+	1.98962i
$869$		0			0
$870$		0			0
$871$	4.62979e20	+	8.01904e20i	1.21740	+	2.10860i
$872$		0			0
$873$	2.78648e19	−	4.82633e19i	0.0721036	−	0.124887i
$874$		0			0
$875$		0			0
$876$	7.36885e20			1.86153
$877$	−2.58949e20	+	4.48512e20i	−0.648958	+	1.12403i	0.334415	+	0.942426i	$0.391461\pi$
$877$	−2.58949e20	+	4.48512e20i	−0.648958	+	1.12403i	−0.983372	+	0.181601i	$0.941872\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0		1.00000			$0$
$881$		0			0		−1.00000			$\pi$
$882$		0			0
$883$	−8.19682e20			−1.95848			−0.979242	−	0.202693i	$-0.935031\pi$
$883$	−8.19682e20			−1.95848			−0.979242	+	0.202693i	$0.935031\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$887$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$888$		0			0
$889$	−7.59883e19	+	3.95443e19i	−0.173155	+	0.0901098i
$890$		0			0
$891$		0			0
$892$	−1.16870e20	−	2.02425e20i	−0.260106	−	0.450517i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	−2.39148e20	−	4.14217e20i	−0.500000	−	0.866025i
$901$		0			0
$902$		0			0
$903$	−5.71030e20	−	3.63892e20i	−1.16639	−	0.743290i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.81780e20	+	8.34467e20i	0.954108	+	1.65256i	0.736395	+	0.676551i	$0.236526\pi$
$907$	4.81780e20	+	8.34467e20i	0.954108	+	1.65256i	0.217713	+	0.976013i	$0.430140\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	−4.64260e20	+	8.04121e20i	−0.884702	+	1.53235i
$913$		0			0
$914$		0			0
$915$		0			0
$916$	1.07360e21			1.98414
$917$		0			0
$918$		0			0
$919$	5.38921e20	−	9.33439e20i	0.973456	−	1.68608i	0.288519	−	0.957474i	$-0.406837\pi$
$919$	5.38921e20	−	9.33439e20i	0.973456	−	1.68608i	0.684938	−	0.728602i	$-0.259829\pi$
$920$		0			0
$921$	4.32713e20	+	7.49481e20i	0.769808	+	1.33335i
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−4.71155e20			−0.813153
$926$		0			0
$927$	1.26618e20	+	2.19308e20i	0.215246	+	0.372818i
$928$		0			0
$929$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$929$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$930$		0			0
$931$	−4.53084e20	−	9.72307e20i	−0.747361	−	1.60382i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$	7.83676e20			1.23583			0.617916	−	0.786244i	$-0.287977\pi$
$937$	7.83676e20			1.23583			0.617916	+	0.786244i	$0.287977\pi$
$938$		0			0
$939$	6.08793e20			0.945825
$940$		0			0
$941$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$941$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$947$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$948$	−3.22468e20	−	5.58530e20i	−0.468627	−	0.811686i
$949$	−1.24893e21	−	2.16321e21i	−1.80166	−	3.12058i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		0			0		1.00000			$0$
$953$		0			0		−1.00000			$\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−1.12262e21	+	1.94443e21i	−1.48309	+	2.56879i
$962$		0			0
$963$		0			0
$964$	7.46474e20	−	1.29293e21i	0.964884	−	1.67123i
$965$		0			0
$966$		0			0
$967$	−1.45969e21			−1.84618			−0.923092	−	0.384579i	$-0.874347\pi$
$967$	−1.45969e21			−1.84618			−0.923092	+	0.384579i	$0.874347\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0		0.500000	−	0.866025i	$-0.333333\pi$
$971$		0			0		−0.500000	+	0.866025i	$0.666667\pi$
$972$	−8.19717e20			−1.00000
$973$	6.92810e20	−	3.60538e20i	0.839120	−	0.436678i
$974$		0			0
$975$	−8.10654e20	+	1.40409e21i	−0.967839	+	1.67635i
$976$	−8.34109e20	−	1.44472e21i	−0.988722	−	1.71252i
$977$		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
$981$	−1.27785e21			−1.46150
$982$		0			0
$983$		0			0		−0.500000	−	0.866025i	$-0.666667\pi$
$983$		0			0		0.500000	+	0.866025i	$0.333333\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	3.14745e21			3.42500
$989$		0			0
$990$		0			0
$991$	−7.26289e20	−	1.25797e21i	−0.773738	−	1.34015i	−0.935501	−	0.353324i	$-0.885051\pi$
$991$	−7.26289e20	−	1.25797e21i	−0.773738	−	1.34015i	0.161763	−	0.986830i	$-0.448282\pi$
$992$		0			0
$993$	1.76480e21			1.85374
$994$		0			0
$995$		0			0
$996$		0			0
$997$	9.70683e20	+	1.68127e21i	0.991314	+	1.71701i	0.609553	+	0.792745i	$0.291349\pi$
$997$	9.70683e20	+	1.68127e21i	0.991314	+	1.71701i	0.381761	+	0.924261i	$0.375318\pi$
$998$		0			0
$999$	−4.03739e20	+	6.99296e20i	−0.406576	+	0.704211i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.15.h.a.2.1	✓	2
3.2	odd	2	CM	21.15.h.a.2.1	✓	2
7.4	even	3	inner	21.15.h.a.11.1	yes	2
21.11	odd	6	inner	21.15.h.a.11.1	yes	2

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 15 \)
Character orbit:	\([\chi]\)	\(=\)	21.h (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1093.50 + 1894.00i) q^{3} +(-8192.00 - 14189.0i) q^{4} +(730542. - 380174. i) q^{7} +(-2.39148e6 + 4.14217e6i) q^{9} +O(q^{10})\)	\(q+(1093.50 + 1894.00i) q^{3} +(-8192.00 - 14189.0i) q^{4} +(730542. - 380174. i) q^{7} +(-2.39148e6 + 4.14217e6i) q^{9} +(1.79159e7 - 3.10313e7i) q^{12} -1.21461e8 q^{13} +(-1.34218e8 + 2.32472e8i) q^{16} +(7.90810e8 - 1.36972e9i) q^{19} +(1.51890e9 + 9.67924e8i) q^{21} +(-3.05176e9 - 5.28580e9i) q^{25} -1.04604e10 q^{27} +(-1.13789e10 - 7.25124e9i) q^{28} +(-2.73961e10 - 4.74514e10i) q^{31} +7.83642e10 q^{36} +(3.85971e10 - 6.68521e10i) q^{37} +(-1.32818e11 - 2.30047e11i) q^{39} -3.75951e11 q^{43} -5.87068e11 q^{48} +(3.89159e11 - 5.55466e11i) q^{49} +(9.95008e11 + 1.72340e12i) q^{52} +3.45900e12 q^{57} +(-3.10730e12 + 5.38200e12i) q^{61} +(-1.72333e11 + 3.93521e12i) q^{63} +4.39805e12 q^{64} +(-3.81175e12 - 6.60215e12i) q^{67} +(1.02826e13 + 1.78099e13i) q^{73} +(6.67419e12 - 1.15600e13i) q^{75} -2.59133e13 q^{76} +(8.99948e12 - 1.55876e13i) q^{79} +(-1.14384e13 - 1.98119e13i) q^{81} +(1.29104e12 - 2.94808e13i) q^{84} +(-8.87323e13 + 4.61763e13i) q^{91} +(5.99152e13 - 1.03776e14i) q^{93} -1.16517e13 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2187 q^{3} - 16384 q^{4} + 1461083 q^{7} - 4782969 q^{9}+O(q^{10}) \) 2 * q + 2187 * q^3 - 16384 * q^4 + 1461083 * q^7 - 4782969 * q^9	\( 2 q + 2187 q^{3} - 16384 q^{4} + 1461083 q^{7} - 4782969 q^{9} + 35831808 q^{12} - 242921906 q^{13} - 268435456 q^{16} + 1581620029 q^{19} + 3037791114 q^{21} - 6103515625 q^{25} - 20920706406 q^{27} - 22757736448 q^{28} - 54792115547 q^{31} + 156728328192 q^{36} + 77194104697 q^{37} - 265635104211 q^{39} - 751902134378 q^{43} - 1174136684544 q^{48} + 778317387191 q^{49} + 1990016253952 q^{52} + 6918006006846 q^{57} - 6214599846074 q^{61} - 344665529109 q^{63} + 8796093022208 q^{64} - 7623508989083 q^{67} + 20565100535713 q^{73} + 13348388671875 q^{75} - 51826525110272 q^{76} + 17998954613629 q^{79} - 22876792454961 q^{81} + 2582075916288 q^{84} - 177464533592099 q^{91} + 119830356701289 q^{93} - 23303389794044 q^{97}+O(q^{100}) \) 2 * q + 2187 * q^3 - 16384 * q^4 + 1461083 * q^7 - 4782969 * q^9 + 35831808 * q^12 - 242921906 * q^13 - 268435456 * q^16 + 1581620029 * q^19 + 3037791114 * q^21 - 6103515625 * q^25 - 20920706406 * q^27 - 22757736448 * q^28 - 54792115547 * q^31 + 156728328192 * q^36 + 77194104697 * q^37 - 265635104211 * q^39 - 751902134378 * q^43 - 1174136684544 * q^48 + 778317387191 * q^49 + 1990016253952 * q^52 + 6918006006846 * q^57 - 6214599846074 * q^61 - 344665529109 * q^63 + 8796093022208 * q^64 - 7623508989083 * q^67 + 20565100535713 * q^73 + 13348388671875 * q^75 - 51826525110272 * q^76 + 17998954613629 * q^79 - 22876792454961 * q^81 + 2582075916288 * q^84 - 177464533592099 * q^91 + 119830356701289 * q^93 - 23303389794044 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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