LMFDB - Embedded newform 21.40.c.a.20.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,40,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, 40, names="a")

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

chi := DirichletCharacter("21.20");

S:= CuspForms(chi, 40);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 40 $
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:	$202.313057918$
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			20.2
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	21.20
Dual form			21.40.c.a.20.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.01310e9i q^{3} +5.49756e11 q^{4} +(-2.72978e16 - 1.28207e16i) q^{7} -4.05256e18 q^{9} +O(q^{10})$	$q+2.01310e9i q^{3} +5.49756e11 q^{4} +(-2.72978e16 - 1.28207e16i) q^{7} -4.05256e18 q^{9} +1.10671e21i q^{12} +3.71142e20i q^{13} +3.02231e23 q^{16} -3.91378e24i q^{19} +(2.58094e25 - 5.49532e25i) q^{21} -1.81899e27 q^{25} -8.15818e27i q^{27} +(-1.50072e28 - 7.04827e27i) q^{28} -4.61163e28i q^{31} -2.22792e30 q^{36} +5.15129e29 q^{37} -7.47145e29 q^{39} +6.23893e31 q^{43} +6.08421e32i q^{48} +(5.80801e32 + 6.99957e32i) q^{49} +2.04038e32i q^{52} +7.87881e33 q^{57} +1.04270e35i q^{61} +(1.10626e35 + 5.19567e34i) q^{63} +1.66153e35 q^{64} +3.61465e35 q^{67} -3.15594e36i q^{73} -3.66180e36i q^{75} -2.15162e36i q^{76} +1.09416e37 q^{79} +1.64232e37 q^{81} +(1.41888e37 - 3.02108e37i) q^{84} +(4.75831e36 - 1.01314e37i) q^{91} +9.28365e37 q^{93} +9.21039e38i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 1099511627776 q^{4} - 54\!\cdots\!40 q^{7}+ \cdots - 81\!\cdots\!34 q^{9}+O(q^{10}) $ 2 * q + 1099511627776 * q^4 - 54595696320612740 * q^7 - 8105110306037952534 * q^9	$ 2 q + 1099511627776 q^{4} - 54\!\cdots\!40 q^{7}+ \cdots + 18\!\cdots\!60 q^{93}+O(q^{100}) $ 2 * q + 1099511627776 * q^4 - 54595696320612740 * q^7 - 8105110306037952534 * q^9 + 604462909807314587353088 * q^16 + 51618725887612315581087282 * q^21 - 3637978807091712951660156250 * q^25 - 30014301465520543869561733120 * q^28 - 4455831512947911355946281992192 * q^36 + 1030258893049640524286886149060 * q^37 - 1494289312090346104866844889112 * q^39 + 124778611812125711120578395302480 * q^43 + 1161602696472845194686199010529314 * q^49 + 15757623329045360684807265247182060 * q^57 + 221252070456758322114880084957841580 * q^63 + 332306998946228968225951765070086144 * q^64 + 722929920286442150829066081726002080 * q^67 + 21883213004409135930743906637420415048 * q^79 + 32846406536521316292462935601418510578 * q^81 + 28377694662205883769292942375875772416 * q^84 + 9516626854556119484632475271509421576 * q^91 + 185672922604118864405863993995056634060 * 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			$\pi$
$3$			2.01310e9i			1.00000i
$3$			2.01310e9i			1.00000i
$4$	5.49756e11			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	−2.72978e16	−	1.28207e16i	−0.905142	−	0.425110i
$7$	−2.72978e16	−	1.28207e16i	−0.905142	−	0.425110i
$8$		0			0
$9$	−4.05256e18			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$			1.10671e21i			1.00000i
$13$			3.71142e20i			0.0704117i	0.999380	+	0.0352059i	$0.0112087\pi$
$13$			3.71142e20i			0.0704117i	−0.999380	+	0.0352059i	$0.988791\pi$
$14$		0			0
$15$		0			0
$16$	3.02231e23			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$		−	3.91378e24i		−	0.453838i	−0.973914	−	0.226919i	$-0.927135\pi$
$19$		−	3.91378e24i		−	0.453838i	0.973914	−	0.226919i	$-0.0728653\pi$
$20$		0			0
$21$	2.58094e25	−	5.49532e25i	0.425110	−	0.905142i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−1.81899e27			−1.00000
$26$		0			0
$27$		−	8.15818e27i		−	1.00000i
$28$	−1.50072e28	−	7.04827e27i	−0.905142	−	0.425110i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$		−	4.61163e28i		−	0.382209i	−0.981570	−	0.191105i	$-0.938793\pi$
$31$		−	4.61163e28i		−	0.382209i	0.981570	−	0.191105i	$-0.0612070\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−2.22792e30			−1.00000
$37$	5.15129e29			0.135514			0.0677568	−	0.997702i	$-0.478416\pi$
$37$	5.15129e29			0.135514			0.0677568	+	0.997702i	$0.478416\pi$
$38$		0			0
$39$	−7.47145e29			−0.0704117
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	6.23893e31			0.875942			0.437971	−	0.898989i	$-0.355697\pi$
$43$	6.23893e31			0.875942			0.437971	+	0.898989i	$0.355697\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$			6.08421e32i			1.00000i
$49$	5.80801e32	+	6.99957e32i	0.638563	+	0.769569i
$50$		0			0
$51$		0			0
$52$			2.04038e32i			0.0704117i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	7.87881e33			0.453838
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$			1.04270e35i			1.60040i	0.599731	+	0.800202i	$0.295274\pi$
$61$			1.04270e35i			1.60040i	−0.599731	+	0.800202i	$0.704726\pi$
$62$		0			0
$63$	1.10626e35	+	5.19567e34i	0.905142	+	0.425110i
$64$	1.66153e35			1.00000
$65$		0			0
$66$		0			0
$67$	3.61465e35			0.890446			0.445223	−	0.895420i	$-0.353124\pi$
$67$	3.61465e35			0.890446			0.445223	+	0.895420i	$0.353124\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	3.15594e36i		−	1.45993i	−0.683484	−	0.729965i	$-0.739536\pi$
$73$		−	3.15594e36i		−	1.45993i	0.683484	−	0.729965i	$-0.260464\pi$
$74$		0			0
$75$		−	3.66180e36i		−	1.00000i
$76$		−	2.15162e36i		−	0.453838i
$77$		0			0
$78$		0			0
$79$	1.09416e37			1.08481			0.542404	−	0.840118i	$-0.317514\pi$
$79$	1.09416e37			1.08481			0.542404	+	0.840118i	$0.317514\pi$
$80$		0			0
$81$	1.64232e37			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	1.41888e37	−	3.02108e37i	0.425110	−	0.905142i
$85$		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
$91$	4.75831e36	−	1.01314e37i	0.0299327	−	0.0637326i
$92$		0			0
$93$	9.28365e37			0.382209
$94$		0			0
$95$		0			0
$96$		0			0
$97$			9.21039e38i			1.66813i	0.551669	+	0.834063i	$0.313991\pi$
$97$			9.21039e38i			1.66813i	−0.551669	+	0.834063i	$0.686009\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000e39			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			1.40389e39i			0.788871i	0.918924	+	0.394435i	$0.129060\pi$
$103$			1.40389e39i			0.788871i	−0.918924	+	0.394435i	$0.870940\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$		−	4.48501e39i		−	1.00000i
$109$	−6.05909e39			−1.12873			−0.564366	−	0.825525i	$-0.690879\pi$
$109$	−6.05909e39			−1.12873			−0.564366	+	0.825525i	$0.690879\pi$
$110$		0			0
$111$			1.03700e39i			0.135514i
$112$	−8.25027e39	−	3.87483e39i	−0.905142	−	0.425110i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	1.50407e39i		−	0.0704117i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	4.11448e40			1.00000
$122$		0			0
$123$		0			0
$124$		−	2.53527e40i		−	0.382209i
$125$		0			0
$126$		0			0
$127$	2.08743e41			1.97442			0.987209	−	0.159434i	$-0.0509668\pi$
$127$	2.08743e41			1.97442			0.987209	+	0.159434i	$0.0509668\pi$
$128$		0			0
$129$			1.25596e41i			0.875942i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−5.01775e40	+	1.06838e41i	−0.192931	+	0.410788i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	8.46654e41i		−	1.37695i	−0.725261	−	0.688474i	$-0.758281\pi$
$139$		−	8.46654e41i		−	1.37695i	0.725261	−	0.688474i	$-0.241719\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.22481e42			−1.00000
$145$		0			0
$146$		0			0
$147$	−1.40908e42	+	1.16921e42i	−0.769569	+	0.638563i
$148$	2.83195e41			0.135514
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−5.72363e41			−0.185192			−0.0925958	−	0.995704i	$-0.529516\pi$
$151$	−5.72363e41			−0.185192			−0.0925958	+	0.995704i	$0.529516\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	−4.10747e41			−0.0704117
$157$			5.67551e42i			0.858937i	0.903082	+	0.429469i	$0.141299\pi$
$157$			5.67551e42i			0.858937i	−0.903082	+	0.429469i	$0.858701\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	3.78348e42			0.275571			0.137785	−	0.990462i	$-0.456002\pi$
$163$	3.78348e42			0.275571			0.137785	+	0.990462i	$0.456002\pi$
$164$		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
$169$	2.76460e43			0.995042
$170$		0			0
$171$			1.58608e43i			0.453838i
$172$	3.42989e43			0.875942
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	4.96545e43	+	2.33208e43i	0.905142	+	0.425110i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	1.41431e44i		−	1.33602i	−0.744153	−	0.668009i	$-0.767147\pi$
$181$		−	1.41431e44i		−	1.33602i	0.744153	−	0.668009i	$-0.232853\pi$
$182$		0			0
$183$	−2.09906e44			−1.60040
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−1.04594e44	+	2.22701e44i	−0.425110	+	0.905142i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$			3.34483e44i			1.00000i
$193$	2.41201e44			0.651644			0.325822	−	0.945431i	$-0.394359\pi$
$193$	2.41201e44			0.651644			0.325822	+	0.945431i	$0.394359\pi$
$194$		0			0
$195$		0			0
$196$	3.19299e44	+	3.84805e44i	0.638563	+	0.769569i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$			1.33959e45i			1.99222i	0.0881016	+	0.996111i	$0.471920\pi$
$199$			1.33959e45i			1.99222i	−0.0881016	+	0.996111i	$0.528080\pi$
$200$		0			0
$201$			7.27664e44i			0.890446i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			1.12171e44i			0.0704117i
$209$		0			0
$210$		0			0
$211$	2.88906e45			1.37167			0.685834	−	0.727758i	$-0.259437\pi$
$211$	2.88906e45			1.37167			0.685834	+	0.727758i	$0.259437\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−5.91244e44	+	1.25887e45i	−0.162481	+	0.345953i
$218$		0			0
$219$	6.35320e45			1.45993
$220$		0			0
$221$		0			0
$222$		0			0
$223$			1.19158e46i			1.92388i	0.273266	+	0.961938i	$0.411896\pi$
$223$			1.19158e46i			1.92388i	−0.273266	+	0.961938i	$0.588104\pi$
$224$		0			0
$225$	7.37155e45			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	4.33142e45			0.453838
$229$			1.97188e46i			1.89709i	0.316639	+	0.948546i	$0.397446\pi$
$229$			1.97188e46i			1.89709i	−0.316639	+	0.948546i	$0.602554\pi$
$230$		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
$237$			2.20265e46i			1.08481i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$			4.62602e46i			1.64390i	0.569561	+	0.821949i	$0.307113\pi$
$241$			4.62602e46i			1.64390i	−0.569561	+	0.821949i	$0.692887\pi$
$242$		0			0
$243$			3.30615e46i			1.00000i
$244$			5.73233e46i			1.60040i
$245$		0			0
$246$		0			0
$247$	1.45257e45			0.0319555
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	6.08173e46	+	2.85635e46i	0.905142	+	0.425110i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	9.13439e46			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−1.40619e46	−	6.60434e45i	−0.122659	−	0.0576081i
$260$		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
$268$	1.98717e47			0.890446
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$			2.08813e47i			0.753108i	0.926395	+	0.376554i	$0.122891\pi$
$271$			2.08813e47i			0.753108i	−0.926395	+	0.376554i	$0.877109\pi$
$272$		0			0
$273$	2.03954e46	+	9.57894e45i	0.0637326	+	0.0299327i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	3.33109e47			0.783847			0.391923	−	0.919998i	$-0.371810\pi$
$277$	3.33109e47			0.783847			0.391923	+	0.919998i	$0.371810\pi$
$278$		0			0
$279$			1.86889e47i			0.382209i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			4.05815e47i			0.628772i	0.949295	+	0.314386i	$0.101799\pi$
$283$			4.05815e47i			0.628772i	−0.949295	+	0.314386i	$0.898201\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−9.71646e47			−1.00000
$290$		0			0
$291$	−1.85414e48			−1.66813
$292$		−	1.73499e48i		−	1.45993i
$293$		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
$300$		−	2.01310e48i		−	1.00000i
$301$	−1.70309e48	−	7.99877e47i	−0.792852	−	0.372371i
$302$		0			0
$303$		0			0
$304$		−	1.18287e48i		−	0.453838i
$305$		0			0
$306$		0			0
$307$		−	5.91347e48i		−	1.87346i	−0.350055	−	0.936729i	$-0.613837\pi$
$307$		−	5.91347e48i		−	1.87346i	0.350055	−	0.936729i	$-0.386163\pi$
$308$		0			0
$309$	−2.82616e48			−0.788871
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$		−	4.69575e48i		−	1.01998i	−0.860181	−	0.509989i	$-0.829650\pi$
$313$		−	4.69575e48i		−	1.01998i	0.860181	−	0.509989i	$-0.170350\pi$
$314$		0			0
$315$		0			0
$316$	6.01521e48			1.08481
$317$		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
$324$	9.02875e48			1.00000
$325$		−	6.75104e47i		−	0.0704117i
$326$		0			0
$327$		−	1.21975e49i		−	1.12873i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−2.40568e49			−1.75627			−0.878136	−	0.478411i	$-0.841213\pi$
$331$	−2.40568e49			−1.75627			−0.878136	+	0.478411i	$0.841213\pi$
$332$		0			0
$333$	−2.08759e48			−0.135514
$334$		0			0
$335$		0			0
$336$	7.80040e48	−	1.66086e49i	0.425110	−	0.905142i
$337$	−3.17551e49			−1.63317			−0.816583	−	0.577228i	$-0.804134\pi$
$337$	−3.17551e49			−1.63317			−0.816583	+	0.577228i	$0.804134\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	−6.88067e48	−	2.65536e49i	−0.250839	−	0.968029i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$			5.73790e49i			1.49162i	0.666161	+	0.745808i	$0.267936\pi$
$349$			5.73790e49i			1.49162i	−0.666161	+	0.745808i	$0.732064\pi$
$350$		0			0
$351$	3.02784e48			0.0704117
$352$		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
$361$	5.90511e49			0.794031
$362$		0			0
$363$			8.28284e49i			1.00000i
$364$	2.61591e48	−	5.56979e48i	0.0299327	−	0.0637326i
$365$		0			0
$366$		0			0
$367$			1.21409e50i			1.18376i	0.806026	+	0.591880i	$0.201614\pi$
$367$			1.21409e50i			1.18376i	−0.806026	+	0.591880i	$0.798386\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	5.10374e49			0.382209
$373$	2.05894e50			1.46326			0.731630	−	0.681702i	$-0.238760\pi$
$373$	2.05894e50			1.46326			0.731630	+	0.681702i	$0.238760\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	3.75162e50			1.95323			0.976617	−	0.214989i	$-0.0689714\pi$
$379$	3.75162e50			1.95323			0.976617	+	0.214989i	$0.0689714\pi$
$380$		0			0
$381$			4.20219e50i			1.97442i
$382$		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
$387$	−2.52836e50			−0.875942
$388$			5.06346e50i			1.66813i
$389$		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
$397$		−	5.29794e50i		−	1.11607i	−0.829817	−	0.558036i	$-0.811555\pi$
$397$		−	5.29794e50i		−	1.11607i	0.829817	−	0.558036i	$-0.188445\pi$
$398$		0			0
$399$	−2.15075e50	−	1.01012e50i	−0.410788	−	0.192931i
$400$	−5.49756e50			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	1.71157e49			0.0269120
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			1.48872e51i			1.75472i	0.479829	+	0.877362i	$0.340699\pi$
$409$			1.48872e51i			1.75472i	−0.479829	+	0.877362i	$0.659301\pi$
$410$		0			0
$411$		0			0
$412$			7.71794e50i			0.788871i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	1.70440e51			1.37695
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	−2.25787e50			−0.151425			−0.0757126	−	0.997130i	$-0.524123\pi$
$421$	−2.25787e50			−0.151425			−0.0757126	+	0.997130i	$0.524123\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	1.33682e51	−	2.84636e51i	0.680347	−	1.44859i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$		−	2.46566e51i		−	1.00000i
$433$		−	1.54493e51i		−	0.598957i	−0.954103	−	0.299479i	$-0.903187\pi$
$433$		−	1.54493e51i		−	0.598957i	0.954103	−	0.299479i	$-0.0968128\pi$
$434$		0			0
$435$		0			0
$436$	−3.33102e51			−1.12873
$437$		0			0
$438$		0			0
$439$		−	6.51205e51i		−	1.93045i	−0.261413	−	0.965227i	$-0.584189\pi$
$439$		−	6.51205e51i		−	1.93045i	0.261413	−	0.965227i	$-0.415811\pi$
$440$		0			0
$441$	−2.35373e51	−	2.83661e51i	−0.638563	−	0.769569i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$			5.70100e50i			0.135514i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	−4.53563e51	−	2.13021e51i	−0.905142	−	0.425110i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	1.15222e51i		−	0.185192i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.19135e51			−0.432124			−0.216062	−	0.976380i	$-0.569321\pi$
$457$	−3.19135e51			−0.432124			−0.216062	+	0.976380i	$0.569321\pi$
$458$		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
$463$	−1.36255e49			−0.00143062			−0.000715309	−	1.00000i	$-0.500228\pi$
$463$	−1.36255e49			−0.00143062			−0.000715309		1.00000i	$0.500228\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		−	8.26873e50i		−	0.0704117i
$469$	−9.86722e51	−	4.63425e51i	−0.805980	−	0.378537i
$470$		0			0
$471$	−1.14253e52			−0.858937
$472$		0			0
$473$		0			0
$474$		0			0
$475$			7.11912e51i			0.453838i
$476$		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
$481$			1.91186e50i			0.00954174i
$482$		0			0
$483$		0			0
$484$	2.26196e52			1.00000
$485$		0			0
$486$		0			0
$487$	5.08792e52			1.99400			0.997002	−	0.0773819i	$-0.0246561\pi$
$487$	5.08792e52			1.99400			0.997002	+	0.0773819i	$0.0246561\pi$
$488$		0			0
$489$			7.61650e51i			0.275571i
$490$		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
$496$		−	1.39378e52i		−	0.382209i
$497$		0			0
$498$		0			0
$499$	−2.24594e52			−0.547567			−0.273784	−	0.961791i	$-0.588275\pi$
$499$	−2.24594e52			−0.547567			−0.273784	+	0.961791i	$0.588275\pi$
$500$		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
$507$			5.56540e52i			0.995042i
$508$	1.14758e53			1.97442
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	−4.04614e52	+	8.61502e52i	−0.620631	+	1.32144i
$512$		0			0
$513$	−3.19293e52			−0.453838
$514$		0			0
$515$		0			0
$516$			6.90469e52i			0.875942i
$517$		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
$523$			1.72226e53i			1.68003i	0.542565	+	0.840014i	$0.317453\pi$
$523$			1.72226e53i			1.68003i	−0.542565	+	0.840014i	$0.682547\pi$
$524$		0			0
$525$	−4.69470e52	+	9.99593e52i	−0.425110	+	0.905142i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	1.28052e53			1.00000
$530$		0			0
$531$		0			0
$532$	−2.75854e52	+	5.87347e52i	−0.192931	+	0.410788i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−3.56602e53			−1.79819			−0.899097	−	0.437750i	$-0.855776\pi$
$541$	−3.56602e53			−1.79819			−0.899097	+	0.437750i	$0.855776\pi$
$542$		0			0
$543$	2.84714e53			1.33602
$544$		0			0
$545$		0			0
$546$		0			0
$547$	3.64176e53			1.48101			0.740506	−	0.672049i	$-0.234586\pi$
$547$	3.64176e53			1.48101			0.740506	+	0.672049i	$0.234586\pi$
$548$		0			0
$549$		−	4.22562e53i		−	1.60040i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−2.98682e53	−	1.40279e53i	−0.981905	−	0.461162i
$554$		0			0
$555$		0			0
$556$		−	4.65453e53i		−	1.37695i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$			2.31553e52i			0.0616766i
$560$		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
$567$	−4.48318e53	−	2.10557e53i	−0.905142	−	0.425110i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	7.72380e53			1.35965			0.679825	−	0.733374i	$-0.262056\pi$
$571$	7.72380e53			1.35965			0.679825	+	0.733374i	$0.262056\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−6.73346e53			−1.00000
$577$		−	1.27472e54i		−	1.83015i	−0.403282	−	0.915076i	$-0.632130\pi$
$577$		−	1.27472e54i		−	1.83015i	0.403282	−	0.915076i	$-0.367870\pi$
$578$		0			0
$579$			4.85560e53i			0.651644i
$580$		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	$0.5\pi$
$588$	−7.74650e53	+	6.42779e53i	−0.769569	+	0.638563i
$589$	−1.80489e53			−0.173461
$590$		0			0
$591$		0			0
$592$	1.55688e53			0.135514
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−2.69673e54			−1.99222
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$		−	3.02663e54i		−	1.96294i	−0.191607	−	0.981472i	$-0.561370\pi$
$601$		−	3.02663e54i		−	1.96294i	0.191607	−	0.981472i	$-0.438630\pi$
$602$		0			0
$603$	−1.46486e54			−0.890446
$604$	−3.14660e53			−0.185192
$605$		0			0
$606$		0			0
$607$		−	3.10581e54i		−	1.65957i	−0.558084	−	0.829785i	$-0.688463\pi$
$607$		−	3.10581e54i		−	1.65957i	0.558084	−	0.829785i	$-0.311537\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	4.52102e54			1.99415			0.997073	−	0.0764609i	$-0.0243620\pi$
$613$	4.52102e54			1.99415			0.997073	+	0.0764609i	$0.0243620\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			2.81518e54i			1.02693i	0.858112	+	0.513463i	$0.171638\pi$
$619$			2.81518e54i			1.02693i	−0.858112	+	0.513463i	$0.828362\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	−2.25811e53			−0.0704117
$625$	3.30872e54			1.00000
$626$		0			0
$627$		0			0
$628$			3.12014e54i			0.858937i
$629$		0			0
$630$		0			0
$631$	5.58618e54			1.40134			0.700668	−	0.713487i	$-0.252885\pi$
$631$	5.58618e54			1.40134			0.700668	+	0.713487i	$0.252885\pi$
$632$		0			0
$633$			5.81596e54i			1.37167i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.59783e53	+	2.15560e53i	−0.0541867	+	0.0449623i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			7.30892e54i			1.26981i	0.772590	+	0.634905i	$0.218961\pi$
$643$			7.30892e54i			1.26981i	−0.772590	+	0.634905i	$0.781039\pi$
$644$		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
$651$	−2.53424e54	−	1.19023e54i	−0.345953	−	0.162481i
$652$	2.07999e54			0.275571
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			1.27896e55i			1.45993i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			1.03772e55i			1.05233i	0.850383	+	0.526165i	$0.176370\pi$
$661$			1.03772e55i			1.05233i	−0.850383	+	0.526165i	$0.823630\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	−2.39876e55			−1.92388
$670$		0			0
$671$		0			0
$672$		0			0
$673$	1.22384e55			0.873841			0.436921	−	0.899500i	$-0.356069\pi$
$673$	1.22384e55			0.873841			0.436921	+	0.899500i	$0.356069\pi$
$674$		0			0
$675$			1.48396e55i			1.00000i
$676$	1.51985e55			0.995042
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	1.18084e55	−	2.51424e55i	0.709137	−	1.50989i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$			8.71957e54i			0.453838i
$685$		0			0
$686$		0			0
$687$	−3.96958e55			−1.89709
$688$	1.88560e55			0.875942
$689$		0			0
$690$		0			0
$691$		−	4.44042e55i		−	1.89497i	−0.319803	−	0.947484i	$-0.603617\pi$
$691$		−	4.44042e55i		−	1.89497i	0.319803	−	0.947484i	$-0.396383\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	2.72978e55	+	1.28207e55i	0.905142	+	0.425110i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	2.01610e54i		−	0.0615012i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.06616e55			1.56788			0.783939	−	0.620838i	$-0.213207\pi$
$709$	6.06616e55			1.56788			0.783939	+	0.620838i	$0.213207\pi$
$710$		0			0
$711$	−4.43415e55			−1.08481
$712$		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
$721$	1.79988e55	−	3.83231e55i	0.335357	−	0.714040i
$722$		0			0
$723$	−9.31261e55			−1.64390
$724$		−	7.77526e55i		−	1.33602i
$725$		0			0
$726$		0			0
$727$		−	1.11929e56i		−	1.77428i	−0.461498	−	0.887141i	$-0.652688\pi$
$727$		−	1.11929e56i		−	1.77428i	0.461498	−	0.887141i	$-0.347312\pi$
$728$		0			0
$729$	−6.65559e55			−1.00000
$730$		0			0
$731$		0			0
$732$	−1.15397e56			−1.60040
$733$		−	4.98785e55i		−	0.673576i	−0.941581	−	0.336788i	$-0.890660\pi$
$733$		−	4.98785e55i		−	0.673576i	0.941581	−	0.336788i	$-0.109340\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−1.37794e56			−1.58731			−0.793656	−	0.608366i	$-0.791825\pi$
$739$	−1.37794e56			−1.58731			−0.793656	+	0.608366i	$0.791825\pi$
$740$		0			0
$741$			2.92416e54i			0.0319555i
$742$		0			0
$743$		0			0		1.00000			$0$
$743$		0			0		−1.00000			$\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	2.37604e56			1.99928			0.999642	−	0.0267535i	$-0.00851692\pi$
$751$	2.37604e56			1.99928			0.999642	+	0.0267535i	$0.00851692\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−5.75011e55	+	1.22431e56i	−0.425110	+	0.905142i
$757$	9.33619e55			0.672667			0.336333	−	0.941743i	$-0.390813\pi$
$757$	9.33619e55			0.672667			0.336333	+	0.941743i	$0.390813\pi$
$758$		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
$763$	1.65400e56	+	7.76820e55i	1.02166	+	0.479835i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$			1.83884e56i			1.00000i
$769$		−	6.16299e55i		−	0.326759i	−0.986563	−	0.163380i	$-0.947760\pi$
$769$		−	6.16299e55i		−	0.326759i	0.986563	−	0.163380i	$-0.0522395\pi$
$770$		0			0
$771$		0			0
$772$	1.32601e56			0.651644
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$			8.38850e55i			0.382209i
$776$		0			0
$777$	1.32952e55	−	2.83080e55i	0.0576081	−	0.122659i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	1.75536e56	+	2.11549e56i	0.638563	+	0.769569i
$785$		0			0
$786$		0			0
$787$		−	3.95937e56i		−	1.33696i	−0.743730	−	0.668480i	$-0.766945\pi$
$787$		−	3.95937e56i		−	1.33696i	0.743730	−	0.668480i	$-0.233055\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−3.86992e55			−0.112687
$794$		0			0
$795$		0			0
$796$			7.36449e56i			1.99222i
$797$		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
$804$			4.00037e56i			0.890446i
$805$		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
$811$			8.15226e56i			1.53240i	0.642604	+	0.766199i	$0.277854\pi$
$811$			8.15226e56i			1.53240i	−0.642604	+	0.766199i	$0.722146\pi$
$812$		0			0
$813$	−4.20361e56			−0.753108
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	2.44178e56i		−	0.397536i
$818$		0			0
$819$	−1.92833e55	+	4.10580e55i	−0.0299327	+	0.0637326i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	3.66109e56			0.516789			0.258395	−	0.966039i	$-0.416806\pi$
$823$	3.66109e56			0.516789			0.258395	+	0.966039i	$0.416806\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	6.56173e55i		−	0.0803905i	−0.999192	−	0.0401952i	$-0.987202\pi$
$829$		−	6.56173e55i		−	0.0803905i	0.999192	−	0.0401952i	$-0.0127980\pi$
$830$		0			0
$831$			6.70581e56i			0.783847i
$832$			6.16666e55i			0.0704117i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−3.76225e56			−0.382209
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	1.08024e57			1.00000
$842$		0			0
$843$		0			0
$844$	1.58828e57			1.37167
$845$		0			0
$846$		0			0
$847$	−1.12316e57	−	5.27506e56i	−0.905142	−	0.425110i
$848$		0			0
$849$	−8.16945e56			−0.628772
$850$		0			0
$851$		0			0
$852$		0			0
$853$		−	2.83981e57i		−	1.99427i	−0.0756669	−	0.997133i	$-0.524109\pi$
$853$		−	2.83981e57i		−	1.99427i	0.0756669	−	0.997133i	$-0.475891\pi$
$854$		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
$859$		−	2.49107e57i		−	1.52588i	−0.646471	−	0.762939i	$-0.723756\pi$
$859$		−	2.49107e57i		−	1.52588i	0.646471	−	0.762939i	$-0.276244\pi$
$860$		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
$867$		−	1.95602e57i		−	1.00000i
$868$	−3.25040e56	+	6.92074e56i	−0.162481	+	0.345953i
$869$		0			0
$870$		0			0
$871$			1.34155e56i			0.0626978i
$872$		0			0
$873$		−	3.73256e57i		−	1.66813i
$874$		0			0
$875$		0			0
$876$	3.49271e57			1.45993
$877$	−4.69599e57			−1.91971			−0.959855	−	0.280498i	$-0.909500\pi$
$877$	−4.69599e57			−1.91971			−0.959855	+	0.280498i	$0.909500\pi$
$878$		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
$883$	−4.75399e57			−1.70147			−0.850735	−	0.525595i	$-0.823843\pi$
$883$	−4.75399e57			−1.70147			−0.850735	+	0.525595i	$0.823843\pi$
$884$		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
$889$	−5.69823e57	−	2.67624e57i	−1.78713	−	0.839344i
$890$		0			0
$891$		0			0
$892$			6.55076e57i			1.92388i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	4.05256e57			1.00000
$901$		0			0
$902$		0			0
$903$	1.61023e57	−	3.42849e57i	0.372371	−	0.792852i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	4.74766e56			0.100725			0.0503625	−	0.998731i	$-0.483962\pi$
$907$	4.74766e56			0.100725			0.0503625	+	0.998731i	$0.483962\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	2.38122e57			0.453838
$913$		0			0
$914$		0			0
$915$		0			0
$916$			1.08405e58i			1.89709i
$917$		0			0
$918$		0			0
$919$	−7.30639e57			−1.19964			−0.599819	−	0.800136i	$-0.704761\pi$
$919$	−7.30639e57			−1.19964			−0.599819	+	0.800136i	$0.704761\pi$
$920$		0			0
$921$	1.19044e58			1.87346
$922$		0			0
$923$		0			0
$924$		0			0
$925$	−9.37015e56			−0.135514
$926$		0			0
$927$		−	5.68932e57i		−	0.788871i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	2.73948e57	−	2.27313e57i	0.349260	−	0.289804i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			3.05265e57i			0.343365i	0.985152	+	0.171682i	$0.0549203\pi$
$937$			3.05265e57i			0.343365i	−0.985152	+	0.171682i	$0.945080\pi$
$938$		0			0
$939$	9.45300e57			1.01998
$940$		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			$\pi$
$948$			1.21092e58i			1.08481i
$949$	1.17130e57			0.102796
$950$		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
$961$	1.24314e58			0.853916
$962$		0			0
$963$		0			0
$964$			2.54318e58i			1.64390i
$965$		0			0
$966$		0			0
$967$	−6.59759e57			−0.401392			−0.200696	−	0.979654i	$-0.564320\pi$
$967$	−6.59759e57			−0.401392			−0.200696	+	0.979654i	$0.564320\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$			1.81757e58i			1.00000i
$973$	−1.08547e58	+	2.31118e58i	−0.585354	+	1.24633i
$974$		0			0
$975$	1.35905e57			0.0704117
$976$			3.15138e58i			1.60040i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	2.45548e58			1.12873
$982$		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
$988$	7.98558e56			0.0319555
$989$		0			0
$990$		0			0
$991$	−5.01142e58			−1.89027			−0.945136	−	0.326676i	$-0.894071\pi$
$991$	−5.01142e58			−1.89027			−0.945136	+	0.326676i	$0.894071\pi$
$992$		0			0
$993$		−	4.84286e58i		−	1.75627i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	5.94492e58i		−	1.99338i	−0.0812784	−	0.996691i	$-0.525900\pi$
$997$		−	5.94492e58i		−	1.99338i	0.0812784	−	0.996691i	$-0.474100\pi$
$998$		0			0
$999$		−	4.20252e57i		−	0.135514i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.40.c.a.20.2	yes	2
3.2	odd	2	CM	21.40.c.a.20.2	yes	2
7.6	odd	2	inner	21.40.c.a.20.1	✓	2
21.20	even	2	inner	21.40.c.a.20.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.40.c.a.20.1	✓	2	7.6	odd	2	inner
21.40.c.a.20.1	✓	2	21.20	even	2	inner
21.40.c.a.20.2	yes	2	1.1	even	1	trivial
21.40.c.a.20.2	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:	\( k \)	\(=\)	\( 40 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.01310e9i q^{3} +5.49756e11 q^{4} +(-2.72978e16 - 1.28207e16i) q^{7} -4.05256e18 q^{9} +O(q^{10})\)	\(q+2.01310e9i q^{3} +5.49756e11 q^{4} +(-2.72978e16 - 1.28207e16i) q^{7} -4.05256e18 q^{9} +1.10671e21i q^{12} +3.71142e20i q^{13} +3.02231e23 q^{16} -3.91378e24i q^{19} +(2.58094e25 - 5.49532e25i) q^{21} -1.81899e27 q^{25} -8.15818e27i q^{27} +(-1.50072e28 - 7.04827e27i) q^{28} -4.61163e28i q^{31} -2.22792e30 q^{36} +5.15129e29 q^{37} -7.47145e29 q^{39} +6.23893e31 q^{43} +6.08421e32i q^{48} +(5.80801e32 + 6.99957e32i) q^{49} +2.04038e32i q^{52} +7.87881e33 q^{57} +1.04270e35i q^{61} +(1.10626e35 + 5.19567e34i) q^{63} +1.66153e35 q^{64} +3.61465e35 q^{67} -3.15594e36i q^{73} -3.66180e36i q^{75} -2.15162e36i q^{76} +1.09416e37 q^{79} +1.64232e37 q^{81} +(1.41888e37 - 3.02108e37i) q^{84} +(4.75831e36 - 1.01314e37i) q^{91} +9.28365e37 q^{93} +9.21039e38i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 1099511627776 q^{4} - 54\!\cdots\!40 q^{7}+ \cdots - 81\!\cdots\!34 q^{9}+O(q^{10}) \) 2 * q + 1099511627776 * q^4 - 54595696320612740 * q^7 - 8105110306037952534 * q^9	\( 2 q + 1099511627776 q^{4} - 54\!\cdots\!40 q^{7}+ \cdots + 18\!\cdots\!60 q^{93}+O(q^{100}) \) 2 * q + 1099511627776 * q^4 - 54595696320612740 * q^7 - 8105110306037952534 * q^9 + 604462909807314587353088 * q^16 + 51618725887612315581087282 * q^21 - 3637978807091712951660156250 * q^25 - 30014301465520543869561733120 * q^28 - 4455831512947911355946281992192 * q^36 + 1030258893049640524286886149060 * q^37 - 1494289312090346104866844889112 * q^39 + 124778611812125711120578395302480 * q^43 + 1161602696472845194686199010529314 * q^49 + 15757623329045360684807265247182060 * q^57 + 221252070456758322114880084957841580 * q^63 + 332306998946228968225951765070086144 * q^64 + 722929920286442150829066081726002080 * q^67 + 21883213004409135930743906637420415048 * q^79 + 32846406536521316292462935601418510578 * q^81 + 28377694662205883769292942375875772416 * q^84 + 9516626854556119484632475271509421576 * q^91 + 185672922604118864405863993995056634060 * q^93

Introduction

L-functions

Modular forms

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