LMFDB - Embedded newform 21.16.c.a.20.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,16,Mod(20,21)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(21, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1]))

N = Newforms(chi, 16, names="a")

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

chi := DirichletCharacter("21.20");

S:= CuspForms(chi, 16);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 16 $
Character orbit:	$[\chi]$	$=$	21.c (of order $2$, degree $1$, 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:	$29.9656360710$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label			20.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.20
Dual form			21.16.c.a.20.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3788.00i q^{3} +32768.0 q^{4} +(-622450. - 2.08809e6i) q^{7} -1.43489e7 q^{9} +O(q^{10})$	$q-3788.00i q^{3} +32768.0 q^{4} +(-622450. - 2.08809e6i) q^{7} -1.43489e7 q^{9} -1.24125e8i q^{12} -2.15688e8i q^{13} +1.07374e9 q^{16} +1.19238e9i q^{19} +(-7.90967e9 + 2.35784e9i) q^{21} -3.05176e10 q^{25} +5.43536e10i q^{27} +(-2.03964e10 - 6.84225e10i) q^{28} -2.19548e11i q^{31} -4.70185e11 q^{36} -1.09016e12 q^{37} -8.17023e11 q^{39} -1.44065e12 q^{43} -4.06733e12i q^{48} +(-3.97267e12 + 2.59946e12i) q^{49} -7.06765e12i q^{52} +4.51671e12 q^{57} +2.81172e13i q^{61} +(8.93148e12 + 2.99618e13i) q^{63} +3.51844e13 q^{64} +9.90590e13 q^{67} -1.88563e14i q^{73} +1.15600e14i q^{75} +3.90718e13i q^{76} -8.86923e13 q^{79} +2.05891e14 q^{81} +(-2.59184e14 + 7.72616e13i) q^{84} +(-4.50375e14 + 1.34255e14i) q^{91} -8.31647e14 q^{93} -1.20896e15i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 65536 q^{4} - 1244900 q^{7} - 28697814 q^{9}+O(q^{10}) $ 2 * q + 65536 * q^4 - 1244900 * q^7 - 28697814 * q^9	$ 2 q + 65536 q^{4} - 1244900 q^{7} - 28697814 q^{9} + 2147483648 q^{16} - 15819345198 q^{21} - 61035156250 q^{25} - 40792883200 q^{28} - 940369969152 q^{36} - 2180317819900 q^{37} - 1634046915672 q^{39} - 2881308305200 q^{43} - 7945347009886 q^{49} + 9033428213100 q^{57} + 17862954324300 q^{63} + 70368744177664 q^{64} + 198118034672800 q^{67} - 177384618158072 q^{79} + 411782264189298 q^{81} - 518368303448064 q^{84} - 900749869963704 q^{91} - 16\!\cdots\!00 q^{93}+O(q^{100}) $ 2 * q + 65536 * q^4 - 1244900 * q^7 - 28697814 * q^9 + 2147483648 * q^16 - 15819345198 * q^21 - 61035156250 * q^25 - 40792883200 * q^28 - 940369969152 * q^36 - 2180317819900 * q^37 - 1634046915672 * q^39 - 2881308305200 * q^43 - 7945347009886 * q^49 + 9033428213100 * q^57 + 17862954324300 * q^63 + 70368744177664 * q^64 + 198118034672800 * q^67 - 177384618158072 * q^79 + 411782264189298 * q^81 - 518368303448064 * q^84 - 900749869963704 * q^91 - 1663294481766900 * q^93

Display 100 coefficients 10 coefficients

Character values

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

$n$	$8$	$10$
$\chi(n)$	$-1$	$-1$

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		1.00000			$0$
$2$		0			0		−1.00000			$\pi$
$3$		−	3788.00i		−	1.00000i
$3$		−	3788.00i		−	1.00000i
$4$	32768.0			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−622450.	−	2.08809e6i	−0.285673	−	0.958327i
$7$	−622450.	−	2.08809e6i	−0.285673	−	0.958327i
$8$		0			0
$9$	−1.43489e7			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	1.24125e8i		−	1.00000i
$13$		−	2.15688e8i		−	0.953345i	−0.879081	−	0.476672i	$-0.841843\pi$
$13$		−	2.15688e8i		−	0.953345i	0.879081	−	0.476672i	$-0.158157\pi$
$14$		0			0
$15$		0			0
$16$	1.07374e9			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			1.19238e9i			0.306028i	0.988224	+	0.153014i	$0.0488980\pi$
$19$			1.19238e9i			0.306028i	−0.988224	+	0.153014i	$0.951102\pi$
$20$		0			0
$21$	−7.90967e9	+	2.35784e9i	−0.958327	+	0.285673i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−3.05176e10			−1.00000
$26$		0			0
$27$			5.43536e10i			1.00000i
$28$	−2.03964e10	−	6.84225e10i	−0.285673	−	0.958327i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	2.19548e11i		−	1.43323i	−0.697467	−	0.716617i	$-0.745690\pi$
$31$		−	2.19548e11i		−	1.43323i	0.697467	−	0.716617i	$-0.254310\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−4.70185e11			−1.00000
$37$	−1.09016e12			−1.88789			−0.943945	−	0.330102i	$-0.892917\pi$
$37$	−1.09016e12			−1.88789			−0.943945	+	0.330102i	$0.892917\pi$
$38$		0			0
$39$	−8.17023e11			−0.953345
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−1.44065e12			−0.808251			−0.404126	−	0.914704i	$-0.632424\pi$
$43$	−1.44065e12			−0.808251			−0.404126	+	0.914704i	$0.632424\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		−	4.06733e12i		−	1.00000i
$49$	−3.97267e12	+	2.59946e12i	−0.836782	+	0.547536i
$50$		0			0
$51$		0			0
$52$		−	7.06765e12i		−	0.953345i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	4.51671e12			0.306028
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			2.81172e13i			1.14551i	0.819727	+	0.572755i	$0.194125\pi$
$61$			2.81172e13i			1.14551i	−0.819727	+	0.572755i	$0.805875\pi$
$62$		0			0
$63$	8.93148e12	+	2.99618e13i	0.285673	+	0.958327i
$64$	3.51844e13			1.00000
$65$		0			0
$66$		0			0
$67$	9.90590e13			1.99679			0.998396	−	0.0566097i	$-0.0180291\pi$
$67$	9.90590e13			1.99679			0.998396	+	0.0566097i	$0.0180291\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	1.88563e14i		−	1.99772i	−0.0477535	−	0.998859i	$-0.515206\pi$
$73$		−	1.88563e14i		−	1.99772i	0.0477535	−	0.998859i	$-0.484794\pi$
$74$		0			0
$75$			1.15600e14i			1.00000i
$76$			3.90718e13i			0.306028i
$77$		0			0
$78$		0			0
$79$	−8.86923e13			−0.519616			−0.259808	−	0.965660i	$-0.583659\pi$
$79$	−8.86923e13			−0.519616			−0.259808	+	0.965660i	$0.583659\pi$
$80$		0			0
$81$	2.05891e14			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	−2.59184e14	+	7.72616e13i	−0.958327	+	0.285673i
$85$		0			0
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	−4.50375e14	+	1.34255e14i	−0.913616	+	0.272345i
$92$		0			0
$93$	−8.31647e14			−1.43323
$94$		0			0
$95$		0			0
$96$		0			0
$97$		−	1.20896e15i		−	1.51923i	−0.650374	−	0.759614i	$-0.725388\pi$
$97$		−	1.20896e15i		−	1.51923i	0.650374	−	0.759614i	$-0.274612\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000e15			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$		−	1.34510e15i		−	1.07765i	−0.842419	−	0.538823i	$-0.818869\pi$
$103$		−	1.34510e15i		−	1.07765i	0.842419	−	0.538823i	$-0.181131\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			1.78106e15i			1.00000i
$109$	3.78393e15			1.98264			0.991320	−	0.131469i	$-0.0419695\pi$
$109$	3.78393e15			1.98264			0.991320	+	0.131469i	$0.0419695\pi$
$110$		0			0
$111$			4.12952e15i			1.88789i
$112$	−6.68351e14	−	2.24207e15i	−0.285673	−	0.958327i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$			3.09488e15i			0.953345i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.17725e15			1.00000
$122$		0			0
$123$		0			0
$124$		−	7.19415e15i		−	1.43323i
$125$		0			0
$126$		0			0
$127$	7.41809e15			1.23528			0.617638	−	0.786462i	$-0.288090\pi$
$127$	7.41809e15			1.23528			0.617638	+	0.786462i	$0.288090\pi$
$128$		0			0
$129$			5.45719e15i			0.808251i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	2.48979e15	−	7.42194e14i	0.293275	−	0.0874239i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	4.49361e15i		−	0.380176i	−0.981767	−	0.190088i	$-0.939123\pi$
$139$		−	4.49361e15i		−	0.380176i	0.981767	−	0.190088i	$-0.0608773\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.54070e16			−1.00000
$145$		0			0
$146$		0			0
$147$	9.84675e15	+	1.50485e16i	0.547536	+	0.836782i
$148$	−3.57223e16			−1.88789
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−2.79783e16			−1.27202			−0.636011	−	0.771680i	$-0.719417\pi$
$151$	−2.79783e16			−1.27202			−0.636011	+	0.771680i	$0.719417\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−2.67722e16			−0.953345
$157$		−	5.07420e16i		−	1.72235i	−0.508312	−	0.861173i	$-0.669730\pi$
$157$		−	5.07420e16i		−	1.72235i	0.508312	−	0.861173i	$-0.330270\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.78829e16			1.99542			0.997711	−	0.0676154i	$-0.0215391\pi$
$163$	7.78829e16			1.99542			0.997711	+	0.0676154i	$0.0215391\pi$
$164$		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
$169$	4.66477e15			0.0911338
$170$		0			0
$171$		−	1.71093e16i		−	0.306028i
$172$	−4.72074e16			−0.808251
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	1.89957e16	+	6.37234e16i	0.285673	+	0.958327i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$			1.18554e17i			1.38461i	0.721604	+	0.692306i	$0.243405\pi$
$181$			1.18554e17i			1.38461i	−0.721604	+	0.692306i	$0.756595\pi$
$182$		0			0
$183$	1.06508e17			1.14551
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	1.13495e17	−	3.38324e16i	0.958327	−	0.285673i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	1.33278e17i		−	1.00000i
$193$	2.44518e17			1.76454			0.882268	−	0.470748i	$-0.156016\pi$
$193$	2.44518e17			1.76454			0.882268	+	0.470748i	$0.156016\pi$
$194$		0			0
$195$		0			0
$196$	−1.30177e17	+	8.51792e16i	−0.836782	+	0.547536i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			3.35523e17i			1.92453i	0.272115	+	0.962265i	$0.412277\pi$
$199$			3.35523e17i			1.92453i	−0.272115	+	0.962265i	$0.587723\pi$
$200$		0			0
$201$		−	3.75235e17i		−	1.99679i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		−	2.31593e17i		−	0.953345i
$209$		0			0
$210$		0			0
$211$	−3.38441e17			−1.25131			−0.625655	−	0.780100i	$-0.715168\pi$
$211$	−3.38441e17			−1.25131			−0.625655	+	0.780100i	$0.715168\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−4.58436e17	+	1.36658e17i	−1.37351	+	0.409436i
$218$		0			0
$219$	−7.14274e17			−1.99772
$220$		0			0
$221$		0			0
$222$		0			0
$223$			6.67317e17i			1.62947i	0.579835	+	0.814734i	$0.303117\pi$
$223$			6.67317e17i			1.62947i	−0.579835	+	0.814734i	$0.696883\pi$
$224$		0			0
$225$	4.37894e17			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	1.48004e17			0.306028
$229$		−	9.35667e17i		−	1.87221i	−0.351717	−	0.936106i	$-0.614402\pi$
$229$		−	9.35667e17i		−	1.87221i	0.351717	−	0.936106i	$-0.385598\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		0			0		1.00000			$0$
$233$		0			0		−1.00000			$\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$			3.35966e17i			0.519616i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	1.08898e18i		−	1.48557i	−0.669531	−	0.742784i	$-0.733505\pi$
$241$		−	1.08898e18i		−	1.48557i	0.669531	−	0.742784i	$-0.266495\pi$
$242$		0			0
$243$		−	7.79915e17i		−	1.00000i
$244$			9.21345e17i			1.14551i
$245$		0			0
$246$		0			0
$247$	2.57181e17			0.291750
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	2.92667e17	+	9.81788e17i	0.285673	+	0.958327i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	1.15292e18			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	6.78569e17	+	2.27635e18i	0.539319	+	1.80922i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		1.00000			$0$
$263$		0			0		−1.00000			$\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$	3.24597e18			1.99679
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		−	3.17367e18i		−	1.79594i	−0.440057	−	0.897970i	$-0.645042\pi$
$271$		−	3.17367e18i		−	1.79594i	0.440057	−	0.897970i	$-0.354958\pi$
$272$		0			0
$273$	5.08556e17	+	1.70602e18i	0.272345	+	0.913616i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	−3.20990e18			−1.54132			−0.770660	−	0.637247i	$-0.780073\pi$
$277$	−3.20990e18			−1.54132			−0.770660	+	0.637247i	$0.780073\pi$
$278$		0			0
$279$			3.15028e18i			1.43323i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$		−	4.89133e18i		−	2.00000i	−0.00217322	−	0.999998i	$-0.500692\pi$
$283$		−	4.89133e18i		−	2.00000i	0.00217322	−	0.999998i	$-0.499308\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−2.86242e18			−1.00000
$290$		0			0
$291$	−4.57952e18			−1.51923
$292$		−	6.17882e18i		−	1.99772i
$293$		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
$300$			3.78800e18i			1.00000i
$301$	8.96735e17	+	3.00821e18i	0.230896	+	0.774569i
$302$		0			0
$303$		0			0
$304$			1.28030e18i			0.306028i
$305$		0			0
$306$		0			0
$307$			6.08445e18i			1.35109i	0.737320	+	0.675544i	$0.236091\pi$
$307$			6.08445e18i			1.35109i	−0.737320	+	0.675544i	$0.763909\pi$
$308$		0			0
$309$	−5.09524e18			−1.07765
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	8.35306e18i		−	1.60422i	−0.597178	−	0.802109i	$-0.703711\pi$
$313$		−	8.35306e18i		−	1.60422i	0.597178	−	0.802109i	$-0.296289\pi$
$314$		0			0
$315$		0			0
$316$	−2.90627e18			−0.519616
$317$		0			0		1.00000			$0$
$317$		0			0		−1.00000			$\pi$
$318$		0			0
$319$		0			0
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$	6.74664e18			1.00000
$325$			6.58226e18i			0.953345i
$326$		0			0
$327$		−	1.43335e19i		−	1.98264i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−6.34699e18			−0.801415			−0.400708	−	0.916206i	$-0.631236\pi$
$331$	−6.34699e18			−0.801415			−0.400708	+	0.916206i	$0.631236\pi$
$332$		0			0
$333$	1.56426e19			1.88789
$334$		0			0
$335$		0			0
$336$	−8.49295e18	+	2.53171e18i	−0.958327	+	0.285673i
$337$	1.60899e19			1.77553			0.887767	−	0.460294i	$-0.152256\pi$
$337$	1.60899e19			1.77553			0.887767	+	0.460294i	$0.152256\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	7.90070e18	+	6.67726e18i	0.763765	+	0.645494i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			2.04237e19i			1.73358i	0.498671	+	0.866791i	$0.333822\pi$
$349$			2.04237e19i			1.73358i	−0.498671	+	0.866791i	$0.666178\pi$
$350$		0			0
$351$	1.17234e19			0.953345
$352$		0			0
$353$		0			0			−	1.00000i	$-0.5\pi$
$353$		0			0				1.00000i	$0.5\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$		0			0
$358$		0			0
$359$		0			0		1.00000			$0$
$359$		0			0		−1.00000			$\pi$
$360$		0			0
$361$	1.37594e19			0.906347
$362$		0			0
$363$		−	1.58234e19i		−	1.00000i
$364$	−1.47579e19	+	4.39926e18i	−0.913616	+	0.272345i
$365$		0			0
$366$		0			0
$367$			6.70103e16i			0.00390073i	0.999998	+	0.00195037i	$0.000620821\pi$
$367$			6.70103e16i			0.00390073i	−0.999998	+	0.00195037i	$0.999379\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	−2.72514e19			−1.43323
$373$	−3.51634e19			−1.81248			−0.906242	−	0.422759i	$-0.861062\pi$
$373$	−3.51634e19			−1.81248			−0.906242	+	0.422759i	$0.861062\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	1.63931e18			0.0749664			0.0374832	−	0.999297i	$-0.488066\pi$
$379$	1.63931e18			0.0749664			0.0374832	+	0.999297i	$0.488066\pi$
$380$		0			0
$381$		−	2.80997e19i		−	1.23528i
$382$		0			0
$383$		0			0			−	1.00000i	$-0.5\pi$
$383$		0			0				1.00000i	$0.5\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	2.06718e19			0.808251
$388$		−	3.96151e19i		−	1.51923i
$389$		0			0		1.00000			$0$
$389$		0			0		−1.00000			$\pi$
$390$		0			0
$391$		0			0
$392$		0			0
$393$		0			0
$394$		0			0
$395$		0			0
$396$		0			0
$397$			5.76010e19i			1.85995i	0.367625	+	0.929974i	$0.380171\pi$
$397$			5.76010e19i			1.85995i	−0.367625	+	0.929974i	$0.619829\pi$
$398$		0			0
$399$	−2.81143e18	−	9.43130e18i	−0.0874239	−	0.293275i
$400$	−3.27680e19			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−4.73538e19			−1.36637
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			6.67161e19i			1.72308i	0.507687	+	0.861541i	$0.330501\pi$
$409$			6.67161e19i			1.72308i	−0.507687	+	0.861541i	$0.669499\pi$
$410$		0			0
$411$		0			0
$412$		−	4.40763e19i		−	1.07765i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.70218e19			−0.380176
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	7.75395e19			1.61216			0.806079	−	0.591808i	$-0.201586\pi$
$421$	7.75395e19			1.61216			0.806079	+	0.591808i	$0.201586\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	5.87113e19	−	1.75016e19i	1.09777	−	0.327241i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			5.83617e19i			1.00000i
$433$		−	8.91997e19i		−	1.50212i	−0.660235	−	0.751059i	$-0.729543\pi$
$433$		−	8.91997e19i		−	1.50212i	0.660235	−	0.751059i	$-0.270457\pi$
$434$		0			0
$435$		0			0
$436$	1.23992e20			1.98264
$437$		0			0
$438$		0			0
$439$			2.51556e18i			0.0382077i	0.999818	+	0.0191039i	$0.00608132\pi$
$439$			2.51556e18i			0.0382077i	−0.999818	+	0.0191039i	$0.993919\pi$
$440$		0			0
$441$	5.70035e19	−	3.72994e19i	0.836782	−	0.547536i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$			1.35316e20i			1.88789i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−2.19005e19	−	7.34681e19i	−0.285673	−	0.958327i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$			1.05982e20i			1.27202i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	6.92415e19			0.778027			0.389014	−	0.921232i	$-0.372816\pi$
$457$	6.92415e19			0.778027			0.389014	+	0.921232i	$0.372816\pi$
$458$		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
$463$	−1.61570e20			−1.64628			−0.823140	−	0.567838i	$-0.807780\pi$
$463$	−1.61570e20			−1.64628			−0.823140	+	0.567838i	$0.807780\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$			1.01413e20i			0.953345i
$469$	−6.16593e19	−	2.06844e20i	−0.570430	−	1.91358i
$470$		0			0
$471$	−1.92211e20			−1.72235
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	3.63884e19i		−	0.306028i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		0			0			−	1.00000i	$-0.5\pi$
$479$		0			0				1.00000i	$0.5\pi$
$480$		0			0
$481$			2.35134e20i			1.79981i
$482$		0			0
$483$		0			0
$484$	1.36880e20			1.00000
$485$		0			0
$486$		0			0
$487$	−1.09623e20			−0.764599			−0.382299	−	0.924038i	$-0.624868\pi$
$487$	−1.09623e20			−0.764599			−0.382299	+	0.924038i	$0.624868\pi$
$488$		0			0
$489$		−	2.95020e20i		−	1.99542i
$490$		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
$496$		−	2.35738e20i		−	1.43323i
$497$		0			0
$498$		0			0
$499$	−1.91483e20			−1.11270			−0.556349	−	0.830949i	$-0.687798\pi$
$499$	−1.91483e20			−1.11270			−0.556349	+	0.830949i	$0.687798\pi$
$500$		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
$507$		−	1.76701e19i		−	0.0911338i
$508$	2.43076e20			1.23528
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−3.93736e20	+	1.17371e20i	−1.91447	+	0.570694i
$512$		0			0
$513$	−6.48099e19			−0.306028
$514$		0			0
$515$		0			0
$516$			1.78821e20i			0.808251i
$517$		0			0
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0			−	1.00000i	$-0.5\pi$
$521$		0			0				1.00000i	$0.5\pi$
$522$		0			0
$523$			4.28577e20i			1.75092i	0.483292	+	0.875459i	$0.339441\pi$
$523$			4.28577e20i			1.75092i	−0.483292	+	0.875459i	$0.660559\pi$
$524$		0			0
$525$	2.41384e20	−	7.19555e19i	0.958327	−	0.285673i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	2.66635e20			1.00000
$530$		0			0
$531$		0			0
$532$	8.15854e19	−	2.43202e19i	0.293275	−	0.0874239i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	3.92589e20			1.24439			0.622197	−	0.782860i	$-0.286240\pi$
$541$	3.92589e20			1.24439			0.622197	+	0.782860i	$0.286240\pi$
$542$		0			0
$543$	4.49083e20			1.38461
$544$		0			0
$545$		0			0
$546$		0			0
$547$	2.16056e20			0.630464			0.315232	−	0.949015i	$-0.397918\pi$
$547$	2.16056e20			0.630464			0.315232	+	0.949015i	$0.397918\pi$
$548$		0			0
$549$		−	4.03452e20i		−	1.14551i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	5.52065e19	+	1.85197e20i	0.148440	+	0.497962i
$554$		0			0
$555$		0			0
$556$		−	1.47247e20i		−	0.380176i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			3.10731e20i			0.770542i
$560$		0			0
$561$		0			0
$562$		0			0
$563$		0			0			−	1.00000i	$-0.5\pi$
$563$		0			0				1.00000i	$0.5\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−1.28157e20	−	4.29919e20i	−0.285673	−	0.958327i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	−8.68041e20			−1.83556			−0.917781	−	0.397086i	$-0.870021\pi$
$571$	−8.68041e20			−1.83556			−0.917781	+	0.397086i	$0.870021\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−5.04857e20			−1.00000
$577$			9.69821e20i			1.89615i	0.318046	+	0.948075i	$0.396973\pi$
$577$			9.69821e20i			1.89615i	−0.318046	+	0.948075i	$0.603027\pi$
$578$		0			0
$579$		−	9.26232e20i		−	1.76454i
$580$		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	$0.5\pi$
$588$	3.22658e20	+	4.93108e20i	0.547536	+	0.836782i
$589$	2.61784e20			0.438610
$590$		0			0
$591$		0			0
$592$	−1.17055e21			−1.88789
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	1.27096e21			1.92453
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			7.02001e20i			1.01106i	0.862808	+	0.505532i	$0.168704\pi$
$601$			7.02001e20i			1.01106i	−0.862808	+	0.505532i	$0.831296\pi$
$602$		0			0
$603$	−1.42139e21			−1.99679
$604$	−9.16794e20			−1.27202
$605$		0			0
$606$		0			0
$607$		−	1.49590e21i		−	1.99981i	−0.0139395	−	0.999903i	$-0.504437\pi$
$607$		−	1.49590e21i		−	1.99981i	0.0139395	−	0.999903i	$-0.495563\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	1.46993e20			0.182533			0.0912666	−	0.995826i	$-0.470908\pi$
$613$	1.46993e20			0.182533			0.0912666	+	0.995826i	$0.470908\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$		−	8.57295e20i		−	0.989579i	−0.869013	−	0.494789i	$-0.835245\pi$
$619$		−	8.57295e20i		−	0.989579i	0.869013	−	0.494789i	$-0.164755\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−8.77272e20			−0.953345
$625$	9.31323e20			1.00000
$626$		0			0
$627$		0			0
$628$		−	1.66271e21i		−	1.72235i
$629$		0			0
$630$		0			0
$631$	5.88644e20			0.588346			0.294173	−	0.955752i	$-0.404956\pi$
$631$	5.88644e20			0.588346			0.294173	+	0.955752i	$0.404956\pi$
$632$		0			0
$633$			1.28201e21i			1.25131i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	5.60672e20	+	8.56856e20i	0.521991	+	0.797742i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			1.11751e21i			0.969772i	0.874577	+	0.484886i	$0.161139\pi$
$643$			1.11751e21i			0.969772i	−0.874577	+	0.484886i	$0.838861\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$		0			0			−	1.00000i	$-0.5\pi$
$647$		0			0				1.00000i	$0.5\pi$
$648$		0			0
$649$		0			0
$650$		0			0
$651$	5.17659e20	+	1.73655e21i	0.409436	+	1.37351i
$652$	2.55207e21			1.99542
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			2.70567e21i			1.99772i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$		−	2.82286e21i		−	1.99149i	−0.0921479	−	0.995745i	$-0.529373\pi$
$661$		−	2.82286e21i		−	1.99149i	0.0921479	−	0.995745i	$-0.470627\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	2.52779e21			1.62947
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.98214e21			1.22186			0.610930	−	0.791685i	$-0.290796\pi$
$673$	1.98214e21			1.22186			0.610930	+	0.791685i	$0.290796\pi$
$674$		0			0
$675$		−	1.65874e21i		−	1.00000i
$676$	1.52855e20			0.0911338
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−2.52441e21	+	7.52515e20i	−1.45592	+	0.434002i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		−	5.60637e20i		−	0.306028i
$685$		0			0
$686$		0			0
$687$	−3.54430e21			−1.87221
$688$	−1.54689e21			−0.808251
$689$		0			0
$690$		0			0
$691$		−	2.60950e21i		−	1.31969i	−0.751401	−	0.659845i	$-0.770622\pi$
$691$		−	2.60950e21i		−	1.31969i	0.751401	−	0.659845i	$-0.229378\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	6.22450e20	+	2.08809e21i	0.285673	+	0.958327i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	1.29988e21i		−	0.577747i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−5.02042e20			−0.209360			−0.104680	−	0.994506i	$-0.533382\pi$
$709$	−5.02042e20			−0.209360			−0.104680	+	0.994506i	$0.533382\pi$
$710$		0			0
$711$	1.27264e21			0.519616
$712$		0			0
$713$		0			0
$714$		0			0
$715$		0			0
$716$		0			0
$717$		0			0
$718$		0			0
$719$		0			0			−	1.00000i	$-0.5\pi$
$719$		0			0				1.00000i	$0.5\pi$
$720$		0			0
$721$	−2.80869e21	+	8.37259e20i	−1.03274	+	0.307854i
$722$		0			0
$723$	−4.12505e21			−1.48557
$724$			3.88479e21i			1.38461i
$725$		0			0
$726$		0			0
$727$		−	1.02484e21i		−	0.354119i	−0.984200	−	0.177059i	$-0.943342\pi$
$727$		−	1.02484e21i		−	0.354119i	0.984200	−	0.177059i	$-0.0566585\pi$
$728$		0			0
$729$	−2.95431e21			−1.00000
$730$		0			0
$731$		0			0
$732$	3.49005e21			1.14551
$733$		−	2.24783e21i		−	0.730270i	−0.930955	−	0.365135i	$-0.881023\pi$
$733$		−	2.24783e21i		−	0.730270i	0.930955	−	0.365135i	$-0.118977\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−6.36930e21			−1.94652			−0.973258	−	0.229713i	$-0.926221\pi$
$739$	−6.36930e21			−1.94652			−0.973258	+	0.229713i	$0.926221\pi$
$740$		0			0
$741$		−	9.74199e20i		−	0.291750i
$742$		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
$751$	8.14633e20			0.220629			0.110314	−	0.993897i	$-0.464814\pi$
$751$	8.14633e20			0.220629			0.110314	+	0.993897i	$0.464814\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	3.71901e21	−	1.10862e21i	0.958327	−	0.285673i
$757$	6.98091e21			1.78112			0.890560	−	0.454866i	$-0.150313\pi$
$757$	6.98091e21			1.78112			0.890560	+	0.454866i	$0.150313\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		0			0			−	1.00000i	$-0.5\pi$
$761$		0			0				1.00000i	$0.5\pi$
$762$		0			0
$763$	−2.35531e21	−	7.90118e21i	−0.566387	−	1.90002i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	4.36726e21i		−	1.00000i
$769$			7.56047e21i			1.71436i	0.515020	+	0.857178i	$0.327784\pi$
$769$			7.56047e21i			1.71436i	−0.515020	+	0.857178i	$0.672216\pi$
$770$		0			0
$771$		0			0
$772$	8.01236e21			1.76454
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$			6.70008e21i			1.43323i
$776$		0			0
$777$	8.62280e21	−	2.57042e21i	1.80922	−	0.539319i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	−4.26563e21	+	2.79115e21i	−0.836782	+	0.547536i
$785$		0			0
$786$		0			0
$787$			8.38953e21i			1.59929i	0.600473	+	0.799645i	$0.294979\pi$
$787$			8.38953e21i			1.59929i	−0.600473	+	0.799645i	$0.705021\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	6.06454e21			1.09207
$794$		0			0
$795$		0			0
$796$			1.09944e22i			1.92453i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$		−	1.22957e22i		−	1.99679i
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$		0			0		1.00000			$0$
$809$		0			0		−1.00000			$\pi$
$810$		0			0
$811$		−	4.98761e21i		−	0.758990i	−0.925194	−	0.379495i	$-0.876098\pi$
$811$		−	4.98761e21i		−	0.758990i	0.925194	−	0.379495i	$-0.123902\pi$
$812$		0			0
$813$	−1.20218e22			−1.79594
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	1.71780e21i		−	0.247347i
$818$		0			0
$819$	6.46239e21	−	1.92641e21i	0.913616	−	0.272345i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−1.41727e22			−1.93176			−0.965882	−	0.258982i	$-0.916613\pi$
$823$	−1.41727e22			−1.93176			−0.965882	+	0.258982i	$0.916613\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	3.94037e21i		−	0.508602i	−0.967125	−	0.254301i	$-0.918155\pi$
$829$		−	3.94037e21i		−	0.508602i	0.967125	−	0.254301i	$-0.0818454\pi$
$830$		0			0
$831$			1.21591e22i			1.54132i
$832$		−	7.58883e21i		−	0.953345i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	1.19332e22			1.43323
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	8.62919e21			1.00000
$842$		0			0
$843$		0			0
$844$	−1.10900e22			−1.25131
$845$		0			0
$846$		0			0
$847$	−2.60013e21	−	8.72247e21i	−0.285673	−	0.958327i
$848$		0			0
$849$	−1.85283e22			−2.00000
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	1.87614e22i		−	1.95500i	−0.210934	−	0.977500i	$-0.567651\pi$
$853$		−	1.87614e22i		−	1.95500i	0.210934	−	0.977500i	$-0.432349\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		0			0			−	1.00000i	$-0.5\pi$
$857$		0			0				1.00000i	$0.5\pi$
$858$		0			0
$859$			1.97722e22i			1.95482i	0.211351	+	0.977410i	$0.432214\pi$
$859$			1.97722e22i			1.95482i	−0.211351	+	0.977410i	$0.567786\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0		1.00000			$0$
$863$		0			0		−1.00000			$\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$			1.08428e22i			1.00000i
$868$	−1.50220e22	+	4.47800e21i	−1.37351	+	0.409436i
$869$		0			0
$870$		0			0
$871$		−	2.13658e22i		−	1.90363i
$872$		0			0
$873$			1.73472e22i			1.51923i
$874$		0			0
$875$		0			0
$876$	−2.34053e22			−1.99772
$877$	5.91889e21			0.500891			0.250445	−	0.968131i	$-0.419423\pi$
$877$	5.91889e21			0.500891			0.250445	+	0.968131i	$0.419423\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		0			0			−	1.00000i	$-0.5\pi$
$881$		0			0				1.00000i	$0.5\pi$
$882$		0			0
$883$	2.23481e22			1.79695			0.898475	−	0.439024i	$-0.144676\pi$
$883$	2.23481e22			1.79695			0.898475	+	0.439024i	$0.144676\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		0			0			−	1.00000i	$-0.5\pi$
$887$		0			0				1.00000i	$0.5\pi$
$888$		0			0
$889$	−4.61739e21	−	1.54896e22i	−0.352885	−	1.18380i
$890$		0			0
$891$		0			0
$892$			2.18666e22i			1.62947i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	1.43489e22			1.00000
$901$		0			0
$902$		0			0
$903$	1.13951e22	−	3.39683e21i	0.774569	−	0.230896i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	3.02584e22			1.98972			0.994858	−	0.101278i	$-0.0322932\pi$
$907$	3.02584e22			1.98972			0.994858	+	0.101278i	$0.0322932\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	4.84978e21			0.306028
$913$		0			0
$914$		0			0
$915$		0			0
$916$		−	3.06599e22i		−	1.87221i
$917$		0			0
$918$		0			0
$919$	−8.22015e21			−0.489794			−0.244897	−	0.969549i	$-0.578754\pi$
$919$	−8.22015e21			−0.489794			−0.244897	+	0.969549i	$0.578754\pi$
$920$		0			0
$921$	2.30479e22			1.35109
$922$		0			0
$923$		0			0
$924$		0			0
$925$	3.32690e22			1.88789
$926$		0			0
$927$			1.93007e22i			1.07765i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−3.09954e21	−	4.73692e21i	−0.167561	−	0.256079i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			3.71321e22i			1.91295i	0.291822	+	0.956473i	$0.405739\pi$
$937$			3.71321e22i			1.91295i	−0.291822	+	0.956473i	$0.594261\pi$
$938$		0			0
$939$	−3.16414e22			−1.60422
$940$		0			0
$941$		0			0			−	1.00000i	$-0.5\pi$
$941$		0			0				1.00000i	$0.5\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$		0			0		1.00000			$0$
$947$		0			0		−1.00000			$\pi$
$948$			1.10089e22i			0.519616i
$949$	−4.06706e22			−1.90451
$950$		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
$961$	−2.47361e22			−1.05416
$962$		0			0
$963$		0			0
$964$		−	3.56837e22i		−	1.48557i
$965$		0			0
$966$		0			0
$967$	4.82071e22			1.96071			0.980354	−	0.197247i	$-0.0632002\pi$
$967$	4.82071e22			1.96071			0.980354	+	0.197247i	$0.0632002\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	2.55562e22i		−	1.00000i
$973$	−9.38306e21	+	2.79705e21i	−0.364333	+	0.108606i
$974$		0			0
$975$	2.49336e22			0.953345
$976$			3.01906e22i			1.14551i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−5.42952e22			−1.98264
$982$		0			0
$983$		0			0			−	1.00000i	$-0.5\pi$
$983$		0			0				1.00000i	$0.5\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$		0			0
$988$	8.42730e21			0.291750
$989$		0			0
$990$		0			0
$991$	−4.83960e22			−1.63779			−0.818893	−	0.573946i	$-0.805412\pi$
$991$	−4.83960e22			−1.63779			−0.818893	+	0.573946i	$0.805412\pi$
$992$		0			0
$993$			2.40424e22i			0.801415i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	4.49794e22i		−	1.45479i	−0.686219	−	0.727395i	$-0.740731\pi$
$997$		−	4.49794e22i		−	1.45479i	0.686219	−	0.727395i	$-0.259269\pi$
$998$		0			0
$999$		−	5.92541e22i		−	1.88789i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.16.c.a.20.1	✓	2
3.2	odd	2	CM	21.16.c.a.20.1	✓	2
7.6	odd	2	inner	21.16.c.a.20.2	yes	2
21.20	even	2	inner	21.16.c.a.20.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.16.c.a.20.1	✓	2	1.1	even	1	trivial
21.16.c.a.20.1	✓	2	3.2	odd	2	CM
21.16.c.a.20.2	yes	2	7.6	odd	2	inner
21.16.c.a.20.2	yes	2	21.20	even	2	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:	\( k \)	\(=\)	\( 16 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3788.00i q^{3} +32768.0 q^{4} +(-622450. - 2.08809e6i) q^{7} -1.43489e7 q^{9} +O(q^{10})\)	\(q-3788.00i q^{3} +32768.0 q^{4} +(-622450. - 2.08809e6i) q^{7} -1.43489e7 q^{9} -1.24125e8i q^{12} -2.15688e8i q^{13} +1.07374e9 q^{16} +1.19238e9i q^{19} +(-7.90967e9 + 2.35784e9i) q^{21} -3.05176e10 q^{25} +5.43536e10i q^{27} +(-2.03964e10 - 6.84225e10i) q^{28} -2.19548e11i q^{31} -4.70185e11 q^{36} -1.09016e12 q^{37} -8.17023e11 q^{39} -1.44065e12 q^{43} -4.06733e12i q^{48} +(-3.97267e12 + 2.59946e12i) q^{49} -7.06765e12i q^{52} +4.51671e12 q^{57} +2.81172e13i q^{61} +(8.93148e12 + 2.99618e13i) q^{63} +3.51844e13 q^{64} +9.90590e13 q^{67} -1.88563e14i q^{73} +1.15600e14i q^{75} +3.90718e13i q^{76} -8.86923e13 q^{79} +2.05891e14 q^{81} +(-2.59184e14 + 7.72616e13i) q^{84} +(-4.50375e14 + 1.34255e14i) q^{91} -8.31647e14 q^{93} -1.20896e15i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 65536 q^{4} - 1244900 q^{7} - 28697814 q^{9}+O(q^{10}) \) 2 * q + 65536 * q^4 - 1244900 * q^7 - 28697814 * q^9	\( 2 q + 65536 q^{4} - 1244900 q^{7} - 28697814 q^{9} + 2147483648 q^{16} - 15819345198 q^{21} - 61035156250 q^{25} - 40792883200 q^{28} - 940369969152 q^{36} - 2180317819900 q^{37} - 1634046915672 q^{39} - 2881308305200 q^{43} - 7945347009886 q^{49} + 9033428213100 q^{57} + 17862954324300 q^{63} + 70368744177664 q^{64} + 198118034672800 q^{67} - 177384618158072 q^{79} + 411782264189298 q^{81} - 518368303448064 q^{84} - 900749869963704 q^{91} - 16\!\cdots\!00 q^{93}+O(q^{100}) \) 2 * q + 65536 * q^4 - 1244900 * q^7 - 28697814 * q^9 + 2147483648 * q^16 - 15819345198 * q^21 - 61035156250 * q^25 - 40792883200 * q^28 - 940369969152 * q^36 - 2180317819900 * q^37 - 1634046915672 * q^39 - 2881308305200 * q^43 - 7945347009886 * q^49 + 9033428213100 * q^57 + 17862954324300 * q^63 + 70368744177664 * q^64 + 198118034672800 * q^67 - 177384618158072 * q^79 + 411782264189298 * q^81 - 518368303448064 * q^84 - 900749869963704 * q^91 - 1663294481766900 * q^93

Introduction

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