LMFDB - Embedded newform 21.28.c.a.20.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [21,28,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, 28, names="a")

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

chi := DirichletCharacter("21.20");

S:= CuspForms(chi, 28);

N := Newforms(S);

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 28 $
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:	$96.9896707160$
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^{4} $
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			21.28.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)

$f(q)$	$=$	$q-2.76145e6i q^{3} +1.34218e8 q^{4} +(2.33960e11 + 1.04762e11i) q^{7} -7.62560e12 q^{9} +O(q^{10})$	$q-2.76145e6i q^{3} +1.34218e8 q^{4} +(2.33960e11 + 1.04762e11i) q^{7} -7.62560e12 q^{9} -3.70635e14i q^{12} +5.74217e14i q^{13} +1.80144e16 q^{16} +3.66350e17i q^{19} +(2.89294e17 - 6.46069e17i) q^{21} -7.45058e18 q^{25} +2.10577e19i q^{27} +(3.14016e19 + 1.40609e19i) q^{28} +2.39186e20i q^{31} -1.02349e21 q^{36} -2.35695e21 q^{37} +1.58567e21 q^{39} -2.04392e22 q^{43} -4.97458e22i q^{48} +(4.37624e22 + 4.90201e22i) q^{49} +7.70701e22i q^{52} +1.01166e24 q^{57} -4.00161e23i q^{61} +(-1.78409e24 - 7.98870e23i) q^{63} +2.41785e24 q^{64} -7.76022e24 q^{67} -1.11290e25i q^{73} +2.05744e25i q^{75} +4.91706e25i q^{76} -1.28305e25 q^{79} +5.81497e25 q^{81} +(3.88283e25 - 8.67139e25i) q^{84} +(-6.01559e25 + 1.34344e26i) q^{91} +6.60500e26 q^{93} +1.05840e27i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 268435456 q^{4} + 467920383820 q^{7} - 15251194969974 q^{9}+O(q^{10}) $ 2 * q + 268435456 * q^4 + 467920383820 * q^7 - 15251194969974 * q^9	$ 2 q + 268435456 q^{4} + 467920383820 q^{7} - 15251194969974 q^{9} + 36\!\cdots\!68 q^{16}+ \cdots + 13\!\cdots\!20 q^{93}+O(q^{100}) $ 2 * q + 268435456 * q^4 + 467920383820 * q^7 - 15251194969974 * q^9 + 36028797018963968 * q^16 + 578587464434659842 * q^21 - 14901161193847656250 * q^25 + 62803210801208360960 * q^28 - 2046980738154938499072 * q^36 - 4713890406868719236380 * q^37 + 3171340163073136542408 * q^39 - 40878358325687530326640 * q^43 + 87524760867187557513314 * q^49 + 2023311244175931564165420 * q^57 - 3568172502031943727710340 * q^63 + 4835703278458516698824704 * q^64 - 15520447866648287687606240 * q^67 - 25660963406174729383556312 * q^79 + 116299474006080119380780338 * q^81 + 77656694925700848446078976 * q^84 - 120311730813537349841403864 * q^91 + 1321000852240421053599117420 * 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			$\pi$
$3$		−	2.76145e6i		−	1.00000i
$3$		−	2.76145e6i		−	1.00000i
$4$	1.34218e8			1.00000
$5$		0			0			−	1.00000i	$-0.5\pi$
$5$		0			0				1.00000i	$0.5\pi$
$6$		0			0
$7$	2.33960e11	+	1.04762e11i	0.912680	+	0.408675i
$7$	2.33960e11	+	1.04762e11i	0.912680	+	0.408675i
$8$		0			0
$9$	−7.62560e12			−1.00000
$10$		0			0
$11$		0			0		1.00000			$0$
$11$		0			0		−1.00000			$\pi$
$12$		−	3.70635e14i		−	1.00000i
$13$			5.74217e14i			0.525824i	0.964820	+	0.262912i	$0.0846829\pi$
$13$			5.74217e14i			0.525824i	−0.964820	+	0.262912i	$0.915317\pi$
$14$		0			0
$15$		0			0
$16$	1.80144e16			1.00000
$17$		0			0			−	1.00000i	$-0.5\pi$
$17$		0			0				1.00000i	$0.5\pi$
$18$		0			0
$19$			3.66350e17i			1.99858i	0.0376389	+	0.999291i	$0.488016\pi$
$19$			3.66350e17i			1.99858i	−0.0376389	+	0.999291i	$0.511984\pi$
$20$		0			0
$21$	2.89294e17	−	6.46069e17i	0.408675	−	0.912680i
$22$		0			0
$23$		0			0		1.00000			$0$
$23$		0			0		−1.00000			$\pi$
$24$		0			0
$25$	−7.45058e18			−1.00000
$26$		0			0
$27$			2.10577e19i			1.00000i
$28$	3.14016e19	+	1.40609e19i	0.912680	+	0.408675i
$29$		0			0		1.00000			$0$
$29$		0			0		−1.00000			$\pi$
$30$		0			0
$31$			2.39186e20i			1.75935i	0.475571	+	0.879677i	$0.342241\pi$
$31$			2.39186e20i			1.75935i	−0.475571	+	0.879677i	$0.657759\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$	−1.02349e21			−1.00000
$37$	−2.35695e21			−1.59084			−0.795419	−	0.606059i	$-0.792749\pi$
$37$	−2.35695e21			−1.59084			−0.795419	+	0.606059i	$0.792749\pi$
$38$		0			0
$39$	1.58567e21			0.525824
$40$		0			0
$41$		0			0			−	1.00000i	$-0.5\pi$
$41$		0			0				1.00000i	$0.5\pi$
$42$		0			0
$43$	−2.04392e22			−1.81401			−0.907004	−	0.421122i	$-0.861637\pi$
$43$	−2.04392e22			−1.81401			−0.907004	+	0.421122i	$0.861637\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		0			0			−	1.00000i	$-0.5\pi$
$47$		0			0				1.00000i	$0.5\pi$
$48$		−	4.97458e22i		−	1.00000i
$49$	4.37624e22	+	4.90201e22i	0.665969	+	0.745980i
$50$		0			0
$51$		0			0
$52$			7.70701e22i			0.525824i
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	1.01166e24			1.99858
$58$		0			0
$59$		0			0			−	1.00000i	$-0.5\pi$
$59$		0			0				1.00000i	$0.5\pi$
$60$		0			0
$61$		−	4.00161e23i		−	0.316434i	−0.987404	−	0.158217i	$-0.949426\pi$
$61$		−	4.00161e23i		−	0.316434i	0.987404	−	0.158217i	$-0.0505745\pi$
$62$		0			0
$63$	−1.78409e24	−	7.98870e23i	−0.912680	−	0.408675i
$64$	2.41785e24			1.00000
$65$		0			0
$66$		0			0
$67$	−7.76022e24			−1.72928			−0.864638	−	0.502395i	$-0.832452\pi$
$67$	−7.76022e24			−1.72928			−0.864638	+	0.502395i	$0.832452\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		0			0		1.00000			$0$
$71$		0			0		−1.00000			$\pi$
$72$		0			0
$73$		−	1.11290e25i		−	0.779110i	−0.921003	−	0.389555i	$-0.872629\pi$
$73$		−	1.11290e25i		−	0.779110i	0.921003	−	0.389555i	$-0.127371\pi$
$74$		0			0
$75$			2.05744e25i			1.00000i
$76$			4.91706e25i			1.99858i
$77$		0			0
$78$		0			0
$79$	−1.28305e25			−0.309227			−0.154613	−	0.987975i	$-0.549413\pi$
$79$	−1.28305e25			−0.309227			−0.154613	+	0.987975i	$0.549413\pi$
$80$		0			0
$81$	5.81497e25			1.00000
$82$		0			0
$83$		0			0			−	1.00000i	$-0.5\pi$
$83$		0			0				1.00000i	$0.5\pi$
$84$	3.88283e25	−	8.67139e25i	0.408675	−	0.912680i
$85$		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
$91$	−6.01559e25	+	1.34344e26i	−0.214891	+	0.479909i
$92$		0			0
$93$	6.60500e26			1.75935
$94$		0			0
$95$		0			0
$96$		0			0
$97$			1.05840e27i			1.59673i	0.602173	+	0.798365i	$0.294302\pi$
$97$			1.05840e27i			1.59673i	−0.602173	+	0.798365i	$0.705698\pi$
$98$		0			0
$99$		0			0
$100$	−1.00000e27			−1.00000
$101$		0			0			−	1.00000i	$-0.5\pi$
$101$		0			0				1.00000i	$0.5\pi$
$102$		0			0
$103$			2.97083e27i			1.99331i	0.0817019	+	0.996657i	$0.473964\pi$
$103$			2.97083e27i			1.99331i	−0.0817019	+	0.996657i	$0.526036\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$		0			0		1.00000			$0$
$107$		0			0		−1.00000			$\pi$
$108$			2.82632e27i			1.00000i
$109$	3.35416e27			1.04792			0.523958	−	0.851744i	$-0.324455\pi$
$109$	3.35416e27			1.04792			0.523958	+	0.851744i	$0.324455\pi$
$110$		0			0
$111$			6.50858e27i			1.59084i
$112$	4.21465e27	+	1.88722e27i	0.912680	+	0.408675i
$113$		0			0		1.00000			$0$
$113$		0			0		−1.00000			$\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$		−	4.37875e27i		−	0.525824i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	1.31100e28			1.00000
$122$		0			0
$123$		0			0
$124$			3.21030e28i			1.75935i
$125$		0			0
$126$		0			0
$127$	4.69387e28			1.86286			0.931431	−	0.363919i	$-0.118561\pi$
$127$	4.69387e28			1.86286			0.931431	+	0.363919i	$0.118561\pi$
$128$		0			0
$129$			5.64417e28i			1.81401i
$130$		0			0
$131$		0			0			−	1.00000i	$-0.5\pi$
$131$		0			0				1.00000i	$0.5\pi$
$132$		0			0
$133$	−3.83794e28	+	8.57112e28i	−0.816772	+	1.82407i
$134$		0			0
$135$		0			0
$136$		0			0
$137$		0			0		1.00000			$0$
$137$		0			0		−1.00000			$\pi$
$138$		0			0
$139$		−	4.77850e28i		−	0.560521i	−0.959924	−	0.280260i	$-0.909579\pi$
$139$		−	4.77850e28i		−	0.560521i	0.959924	−	0.280260i	$-0.0904208\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$		0			0
$144$	−1.37371e29			−1.00000
$145$		0			0
$146$		0			0
$147$	1.35366e29	−	1.20848e29i	0.745980	−	0.665969i
$148$	−3.16344e29			−1.59084
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	4.98429e29			1.91168			0.955839	−	0.293891i	$-0.0949502\pi$
$151$	4.98429e29			1.91168			0.955839	+	0.293891i	$0.0949502\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$		0			0
$156$	2.12825e29			0.525824
$157$		−	8.43829e29i		−	1.91253i	−0.292494	−	0.956267i	$-0.594485\pi$
$157$		−	8.43829e29i		−	1.91253i	0.292494	−	0.956267i	$-0.405515\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−1.07113e30			−1.46322			−0.731608	−	0.681726i	$-0.761230\pi$
$163$	−1.07113e30			−1.46322			−0.731608	+	0.681726i	$0.761230\pi$
$164$		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
$169$	8.62808e29			0.723509
$170$		0			0
$171$		−	2.79363e30i		−	1.99858i
$172$	−2.74330e30			−1.81401
$173$		0			0			−	1.00000i	$-0.5\pi$
$173$		0			0				1.00000i	$0.5\pi$
$174$		0			0
$175$	−1.74314e30	−	7.80535e29i	−0.912680	−	0.408675i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		0			0		1.00000			$0$
$179$		0			0		−1.00000			$\pi$
$180$		0			0
$181$		−	4.09600e30i		−	1.36050i	−0.732979	−	0.680251i	$-0.761871\pi$
$181$		−	4.09600e30i		−	1.36050i	0.732979	−	0.680251i	$-0.238129\pi$
$182$		0			0
$183$	−1.10503e30			−0.316434
$184$		0			0
$185$		0			0
$186$		0			0
$187$		0			0
$188$		0			0
$189$	−2.20604e30	+	4.92666e30i	−0.408675	+	0.912680i
$190$		0			0
$191$		0			0		1.00000			$0$
$191$		0			0		−1.00000			$\pi$
$192$		−	6.67677e30i		−	1.00000i
$193$	1.33315e31			1.86146			0.930731	−	0.365703i	$-0.119172\pi$
$193$	1.33315e31			1.86146			0.930731	+	0.365703i	$0.119172\pi$
$194$		0			0
$195$		0			0
$196$	5.87369e30	+	6.57936e30i	0.665969	+	0.745980i
$197$		0			0		1.00000			$0$
$197$		0			0		−1.00000			$\pi$
$198$		0			0
$199$		−	3.91732e30i		−	0.361803i	−0.983501	−	0.180902i	$-0.942098\pi$
$199$		−	3.91732e30i		−	0.361803i	0.983501	−	0.180902i	$-0.0579016\pi$
$200$		0			0
$201$			2.14295e31i			1.72928i
$202$		0			0
$203$		0			0
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$			1.03442e31i			0.525824i
$209$		0			0
$210$		0			0
$211$	1.90364e30			0.0797575			0.0398787	−	0.999205i	$-0.487303\pi$
$211$	1.90364e30			0.0797575			0.0398787	+	0.999205i	$0.487303\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		0			0
$216$		0			0
$217$	−2.50575e31	+	5.59601e31i	−0.719005	+	1.60573i
$218$		0			0
$219$	−3.07322e31			−0.779110
$220$		0			0
$221$		0			0
$222$		0			0
$223$			1.71275e31i			0.340079i	0.985437	+	0.170039i	$0.0543895\pi$
$223$			1.71275e31i			0.340079i	−0.985437	+	0.170039i	$0.945611\pi$
$224$		0			0
$225$	5.68151e31			1.00000
$226$		0			0
$227$		0			0			−	1.00000i	$-0.5\pi$
$227$		0			0				1.00000i	$0.5\pi$
$228$	1.35782e32			1.99858
$229$			1.44123e32i			1.99965i	0.0186093	+	0.999827i	$0.494076\pi$
$229$			1.44123e32i			1.99965i	−0.0186093	+	0.999827i	$0.505924\pi$
$230$		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
$237$			3.54307e31i			0.309227i
$238$		0			0
$239$		0			0		1.00000			$0$
$239$		0			0		−1.00000			$\pi$
$240$		0			0
$241$		−	2.86666e32i		−	1.99593i	−0.0637595	−	0.997965i	$-0.520309\pi$
$241$		−	2.86666e32i		−	1.99593i	0.0637595	−	0.997965i	$-0.479691\pi$
$242$		0			0
$243$		−	1.60578e32i		−	1.00000i
$244$		−	5.37088e31i		−	0.316434i
$245$		0			0
$246$		0			0
$247$	−2.10364e32			−1.05090
$248$		0			0
$249$		0			0
$250$		0			0
$251$		0			0			−	1.00000i	$-0.5\pi$
$251$		0			0				1.00000i	$0.5\pi$
$252$	−2.39456e32	−	1.07222e32i	−0.912680	−	0.408675i
$253$		0			0
$254$		0			0
$255$		0			0
$256$	3.24519e32			1.00000
$257$		0			0			−	1.00000i	$-0.5\pi$
$257$		0			0				1.00000i	$0.5\pi$
$258$		0			0
$259$	−5.51431e32	−	2.46917e32i	−1.45193	−	0.650137i
$260$		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
$268$	−1.04156e33			−1.72928
$269$		0			0			−	1.00000i	$-0.5\pi$
$269$		0			0				1.00000i	$0.5\pi$
$270$		0			0
$271$		−	9.54885e32i		−	1.36416i	−0.731277	−	0.682080i	$-0.761075\pi$
$271$		−	9.54885e32i		−	1.36416i	0.731277	−	0.682080i	$-0.238925\pi$
$272$		0			0
$273$	3.70984e32	+	1.66117e32i	0.479909	+	0.214891i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	1.50769e33			1.60264			0.801320	−	0.598237i	$-0.204132\pi$
$277$	1.50769e33			1.60264			0.801320	+	0.598237i	$0.204132\pi$
$278$		0			0
$279$		−	1.82394e33i		−	1.75935i
$280$		0			0
$281$		0			0		1.00000			$0$
$281$		0			0		−1.00000			$\pi$
$282$		0			0
$283$			2.03861e33i			1.62262i	0.584616	+	0.811310i	$0.301245\pi$
$283$			2.03861e33i			1.62262i	−0.584616	+	0.811310i	$0.698755\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−1.66771e33			−1.00000
$290$		0			0
$291$	2.92272e33			1.59673
$292$		−	1.49371e33i		−	0.779110i
$293$		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
$300$			2.76145e33i			1.00000i
$301$	−4.78195e33	−	2.14124e33i	−1.65561	−	0.741341i
$302$		0			0
$303$		0			0
$304$			6.59957e33i			1.99858i
$305$		0			0
$306$		0			0
$307$			7.33192e33i			1.94469i	0.233555	+	0.972344i	$0.424964\pi$
$307$			7.33192e33i			1.94469i	−0.233555	+	0.972344i	$0.575036\pi$
$308$		0			0
$309$	8.20380e33			1.99331
$310$		0			0
$311$		0			0			−	1.00000i	$-0.5\pi$
$311$		0			0				1.00000i	$0.5\pi$
$312$		0			0
$313$			7.27728e33i			1.48635i	0.669098	+	0.743174i	$0.266680\pi$
$313$			7.27728e33i			1.48635i	−0.669098	+	0.743174i	$0.733320\pi$
$314$		0			0
$315$		0			0
$316$	−1.72208e33			−0.309227
$317$		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
$324$	7.80473e33			1.00000
$325$		−	4.27825e33i		−	0.525824i
$326$		0			0
$327$		−	9.26234e33i		−	1.04792i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−1.40786e34			−1.35170			−0.675852	−	0.737037i	$-0.736224\pi$
$331$	−1.40786e34			−1.35170			−0.675852	+	0.737037i	$0.736224\pi$
$332$		0			0
$333$	1.79731e34			1.59084
$334$		0			0
$335$		0			0
$336$	5.21145e33	−	1.16385e34i	0.408675	−	0.912680i
$337$	−1.38045e34			−1.03996			−0.519980	−	0.854178i	$-0.674061\pi$
$337$	−1.38045e34			−1.03996			−0.519980	+	0.854178i	$0.674061\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		0			0
$342$		0			0
$343$	5.10323e33	+	1.60534e34i	0.302953	+	0.953006i
$344$		0			0
$345$		0			0
$346$		0			0
$347$		0			0		1.00000			$0$
$347$		0			0		−1.00000			$\pi$
$348$		0			0
$349$		−	2.49308e34i		−	1.17110i	−0.810637	−	0.585549i	$-0.800879\pi$
$349$		−	2.49308e34i		−	1.17110i	0.810637	−	0.585549i	$-0.199121\pi$
$350$		0			0
$351$	−1.20917e34			−0.525824
$352$		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
$361$	−1.00611e35			−2.99433
$362$		0			0
$363$		−	3.62026e34i		−	1.00000i
$364$	−8.07398e33	+	1.80313e34i	−0.214891	+	0.479909i
$365$		0			0
$366$		0			0
$367$		−	7.97498e34i		−	1.89993i	−0.312354	−	0.949966i	$-0.601117\pi$
$367$		−	7.97498e34i		−	1.89993i	0.312354	−	0.949966i	$-0.398883\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$	8.86509e34			1.75935
$373$	1.63179e34			0.312316			0.156158	−	0.987732i	$-0.450089\pi$
$373$	1.63179e34			0.312316			0.156158	+	0.987732i	$0.450089\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		0			0
$378$		0			0
$379$	−8.30844e34			−1.28201			−0.641003	−	0.767539i	$-0.721481\pi$
$379$	−8.30844e34			−1.28201			−0.641003	+	0.767539i	$0.721481\pi$
$380$		0			0
$381$		−	1.29619e35i		−	1.86286i
$382$		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
$387$	1.55861e35			1.81401
$388$			1.42056e35i			1.59673i
$389$		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
$397$			1.88920e35i			1.55814i	0.626938	+	0.779069i	$0.284308\pi$
$397$			1.88920e35i			1.55814i	−0.626938	+	0.779069i	$0.715692\pi$
$398$		0			0
$399$	2.36687e35	+	1.05983e35i	1.82407	+	0.816772i
$400$	−1.34218e35			−1.00000
$401$		0			0		1.00000			$0$
$401$		0			0		−1.00000			$\pi$
$402$		0			0
$403$	−1.37345e35			−0.925111
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$			5.80649e33i			0.0320368i	0.999872	+	0.0160184i	$0.00509903\pi$
$409$			5.80649e33i			0.0320368i	−0.999872	+	0.0160184i	$0.994901\pi$
$410$		0			0
$411$		0			0
$412$			3.98738e35i			1.99331i
$413$		0			0
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−1.31956e35			−0.560521
$418$		0			0
$419$		0			0			−	1.00000i	$-0.5\pi$
$419$		0			0				1.00000i	$0.5\pi$
$420$		0			0
$421$	2.23691e35			0.835296			0.417648	−	0.908609i	$-0.362854\pi$
$421$	2.23691e35			0.835296			0.417648	+	0.908609i	$0.362854\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		0			0
$426$		0			0
$427$	4.19216e34	−	9.36219e34i	0.129319	−	0.288802i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0		1.00000			$0$
$431$		0			0		−1.00000			$\pi$
$432$			3.79342e35i			1.00000i
$433$		−	2.10862e35i		−	0.538779i	−0.963031	−	0.269390i	$-0.913178\pi$
$433$		−	2.10862e35i		−	0.538779i	0.963031	−	0.269390i	$-0.0868219\pi$
$434$		0			0
$435$		0			0
$436$	4.50188e35			1.04792
$437$		0			0
$438$		0			0
$439$		−	5.79903e35i		−	1.23051i	−0.788329	−	0.615254i	$-0.789054\pi$
$439$		−	5.79903e35i		−	1.23051i	0.788329	−	0.615254i	$-0.210946\pi$
$440$		0			0
$441$	−3.33714e35	−	3.73807e35i	−0.665969	−	0.745980i
$442$		0			0
$443$		0			0		1.00000			$0$
$443$		0			0		−1.00000			$\pi$
$444$			8.73567e35i			1.59084i
$445$		0			0
$446$		0			0
$447$		0			0
$448$	5.65681e35	+	2.53298e35i	0.912680	+	0.408675i
$449$		0			0		1.00000			$0$
$449$		0			0		−1.00000			$\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		−	1.37639e36i		−	1.91168i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.47170e35			−0.181531			−0.0907655	−	0.995872i	$-0.528931\pi$
$457$	−1.47170e35			−0.181531			−0.0907655	+	0.995872i	$0.528931\pi$
$458$		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
$463$	−8.99450e35			−0.930323			−0.465162	−	0.885226i	$-0.654004\pi$
$463$	−8.99450e35			−0.930323			−0.465162	+	0.885226i	$0.654004\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		0			0			−	1.00000i	$-0.5\pi$
$467$		0			0				1.00000i	$0.5\pi$
$468$		−	5.87705e35i		−	0.525824i
$469$	−1.81558e36	−	8.12973e35i	−1.57828	−	0.706713i
$470$		0			0
$471$	−2.33019e36			−1.91253
$472$		0			0
$473$		0			0
$474$		0			0
$475$		−	2.72952e36i		−	1.99858i
$476$		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
$481$		−	1.35340e36i		−	0.836502i
$482$		0			0
$483$		0			0
$484$	1.75959e36			1.00000
$485$		0			0
$486$		0			0
$487$	2.00118e36			1.04627			0.523136	−	0.852249i	$-0.324762\pi$
$487$	2.00118e36			1.04627			0.523136	+	0.852249i	$0.324762\pi$
$488$		0			0
$489$			2.95786e36i			1.46322i
$490$		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
$496$			4.30880e36i			1.75935i
$497$		0			0
$498$		0			0
$499$	2.23272e36			0.840380			0.420190	−	0.907436i	$-0.361963\pi$
$499$	2.23272e36			0.840380			0.420190	+	0.907436i	$0.361963\pi$
$500$		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
$507$		−	2.38260e36i		−	0.723509i
$508$	6.30000e36			1.86286
$509$		0			0			−	1.00000i	$-0.5\pi$
$509$		0			0				1.00000i	$0.5\pi$
$510$		0			0
$511$	1.16589e36	−	2.60375e36i	0.318403	−	0.711078i
$512$		0			0
$513$	−7.71448e36			−1.99858
$514$		0			0
$515$		0			0
$516$			7.57548e36i			1.81401i
$517$		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
$523$		−	3.45951e35i		−	0.0690623i	−0.999404	−	0.0345312i	$-0.989006\pi$
$523$		−	3.45951e35i		−	0.0690623i	0.999404	−	0.0345312i	$-0.0109938\pi$
$524$		0			0
$525$	−2.15541e36	+	4.81359e36i	−0.408675	+	0.912680i
$526$		0			0
$527$		0			0
$528$		0			0
$529$	5.84321e36			1.00000
$530$		0			0
$531$		0			0
$532$	−5.15119e36	+	1.15040e37i	−0.816772	+	1.82407i
$533$		0			0
$534$		0			0
$535$		0			0
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	9.90004e36			1.25162			0.625812	−	0.779974i	$-0.284768\pi$
$541$	9.90004e36			1.25162			0.625812	+	0.779974i	$0.284768\pi$
$542$		0			0
$543$	−1.13109e37			−1.36050
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.00193e37			1.09146			0.545732	−	0.837960i	$-0.316252\pi$
$547$	1.00193e37			1.09146			0.545732	+	0.837960i	$0.316252\pi$
$548$		0			0
$549$			3.05147e36i			0.316434i
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−3.00182e36	−	1.34414e36i	−0.282225	−	0.126373i
$554$		0			0
$555$		0			0
$556$		−	6.41359e36i		−	0.560521i
$557$		0			0		1.00000			$0$
$557$		0			0		−1.00000			$\pi$
$558$		0			0
$559$		−	1.17365e37i		−	0.953850i
$560$		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
$567$	1.36047e37	+	6.09186e36i	0.912680	+	0.408675i
$568$		0			0
$569$		0			0		1.00000			$0$
$569$		0			0		−1.00000			$\pi$
$570$		0			0
$571$	6.75219e36			0.411962			0.205981	−	0.978556i	$-0.433961\pi$
$571$	6.75219e36			0.411962			0.205981	+	0.978556i	$0.433961\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		0			0
$576$	−1.84376e37			−1.00000
$577$		−	3.77094e37i		−	1.99791i	−0.0457001	−	0.998955i	$-0.514552\pi$
$577$		−	3.77094e37i		−	1.99791i	0.0457001	−	0.998955i	$-0.485448\pi$
$578$		0			0
$579$		−	3.68142e37i		−	1.86146i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$		0			0			−	1.00000i	$-0.5\pi$
$587$		0			0				1.00000i	$0.5\pi$
$588$	1.81686e37	−	1.62199e37i	0.745980	−	0.665969i
$589$	−8.76258e37			−3.51622
$590$		0			0
$591$		0			0
$592$	−4.24590e37			−1.59084
$593$		0			0			−	1.00000i	$-0.5\pi$
$593$		0			0				1.00000i	$0.5\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−1.08175e37			−0.361803
$598$		0			0
$599$		0			0		1.00000			$0$
$599$		0			0		−1.00000			$\pi$
$600$		0			0
$601$			5.33807e37i			1.63147i	0.578426	+	0.815735i	$0.303667\pi$
$601$			5.33807e37i			1.63147i	−0.578426	+	0.815735i	$0.696333\pi$
$602$		0			0
$603$	5.91763e37			1.72928
$604$	6.68980e37			1.91168
$605$		0			0
$606$		0			0
$607$		−	2.13310e37i		−	0.570117i	−0.958510	−	0.285058i	$-0.907987\pi$
$607$		−	2.13310e37i		−	0.570117i	0.958510	−	0.285058i	$-0.0920130\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		0			0
$612$		0			0
$613$	−6.08743e37			−1.42468			−0.712340	−	0.701834i	$-0.752365\pi$
$613$	−6.08743e37			−1.42468			−0.712340	+	0.701834i	$0.752365\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		0			0		1.00000			$0$
$617$		0			0		−1.00000			$\pi$
$618$		0			0
$619$			9.74604e37i			1.99988i	0.0108111	+	0.999942i	$0.496559\pi$
$619$			9.74604e37i			1.99988i	−0.0108111	+	0.999942i	$0.503441\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$	2.85649e37			0.525824
$625$	5.55112e37			1.00000
$626$		0			0
$627$		0			0
$628$		−	1.13257e38i		−	1.91253i
$629$		0			0
$630$		0			0
$631$	6.46637e37			1.02391			0.511957	−	0.859011i	$-0.328921\pi$
$631$	6.46637e37			1.02391			0.511957	+	0.859011i	$0.328921\pi$
$632$		0			0
$633$		−	5.25680e36i		−	0.0797575i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−2.81482e37	+	2.51291e37i	−0.392254	+	0.350183i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		0			0		1.00000			$0$
$641$		0			0		−1.00000			$\pi$
$642$		0			0
$643$			1.34704e38i			1.65398i	0.562213	+	0.826992i	$0.309950\pi$
$643$			1.34704e38i			1.65398i	−0.562213	+	0.826992i	$0.690050\pi$
$644$		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
$651$	1.54531e38	+	6.91951e37i	1.60573	+	0.719005i
$652$	−1.43764e38			−1.46322
$653$		0			0		1.00000			$0$
$653$		0			0		−1.00000			$\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$			8.48655e37i			0.779110i
$658$		0			0
$659$		0			0		1.00000			$0$
$659$		0			0		−1.00000			$\pi$
$660$		0			0
$661$			2.11643e38i			1.79013i	0.445933	+	0.895066i	$0.352872\pi$
$661$			2.11643e38i			1.79013i	−0.445933	+	0.895066i	$0.647128\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$		0			0
$668$		0			0
$669$	4.72968e37			0.340079
$670$		0			0
$671$		0			0
$672$		0			0
$673$	−1.58778e38			−1.05338			−0.526691	−	0.850057i	$-0.676568\pi$
$673$	−1.58778e38			−1.05338			−0.526691	+	0.850057i	$0.676568\pi$
$674$		0			0
$675$		−	1.56892e38i		−	1.00000i
$676$	1.15804e38			0.723509
$677$		0			0			−	1.00000i	$-0.5\pi$
$677$		0			0				1.00000i	$0.5\pi$
$678$		0			0
$679$	−1.10880e38	+	2.47624e38i	−0.652545	+	1.45730i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		0			0		1.00000			$0$
$683$		0			0		−1.00000			$\pi$
$684$		−	3.74955e38i		−	1.99858i
$685$		0			0
$686$		0			0
$687$	3.97989e38			1.99965
$688$	−3.68200e38			−1.81401
$689$		0			0
$690$		0			0
$691$		−	2.38750e38i		−	1.10915i	−0.832134	−	0.554575i	$-0.812881\pi$
$691$		−	2.38750e38i		−	1.10915i	0.832134	−	0.554575i	$-0.187119\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		0			0
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$	−2.33960e38	−	1.04762e38i	−0.912680	−	0.408675i
$701$		0			0		1.00000			$0$
$701$		0			0		−1.00000			$\pi$
$702$		0			0
$703$		−	8.63466e38i		−	3.17942i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	5.33759e38			1.75235			0.876175	−	0.481994i	$-0.160087\pi$
$709$	5.33759e38			1.75235			0.876175	+	0.481994i	$0.160087\pi$
$710$		0			0
$711$	9.78401e37			0.309227
$712$		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
$721$	−3.11229e38	+	6.95057e38i	−0.814618	+	1.81926i
$722$		0			0
$723$	−7.91612e38			−1.99593
$724$		−	5.49756e38i		−	1.36050i
$725$		0			0
$726$		0			0
$727$			6.88211e38i			1.61067i	0.592819	+	0.805335i	$0.298015\pi$
$727$			6.88211e38i			1.61067i	−0.592819	+	0.805335i	$0.701985\pi$
$728$		0			0
$729$	−4.43426e38			−1.00000
$730$		0			0
$731$		0			0
$732$	−1.48314e38			−0.316434
$733$		−	8.94094e38i		−	1.87275i	−0.351002	−	0.936375i	$-0.614159\pi$
$733$		−	8.94094e38i		−	1.87275i	0.351002	−	0.936375i	$-0.385841\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	1.04227e39			1.95560			0.977799	−	0.209544i	$-0.0671980\pi$
$739$	1.04227e39			1.95560			0.977799	+	0.209544i	$0.0671980\pi$
$740$		0			0
$741$			5.80910e38i			1.05090i
$742$		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
$751$	−9.75227e38			−1.47220			−0.736099	−	0.676873i	$-0.763334\pi$
$751$	−9.75227e38			−1.47220			−0.736099	+	0.676873i	$0.763334\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		0			0
$756$	−2.96089e38	+	6.61245e38i	−0.408675	+	0.912680i
$757$	9.73549e38			1.31997			0.659984	−	0.751279i	$-0.270563\pi$
$757$	9.73549e38			1.31997			0.659984	+	0.751279i	$0.270563\pi$
$758$		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
$763$	7.84740e38	+	3.51387e38i	0.956411	+	0.428257i
$764$		0			0
$765$		0			0
$766$		0			0
$767$		0			0
$768$		−	8.96141e38i		−	1.00000i
$769$			1.02532e39i			1.12423i	0.827059	+	0.562115i	$0.190012\pi$
$769$			1.02532e39i			1.12423i	−0.827059	+	0.562115i	$0.809988\pi$
$770$		0			0
$771$		0			0
$772$	1.78932e39			1.86146
$773$		0			0			−	1.00000i	$-0.5\pi$
$773$		0			0				1.00000i	$0.5\pi$
$774$		0			0
$775$		−	1.78208e39i		−	1.75935i
$776$		0			0
$777$	−6.81849e38	+	1.52275e39i	−0.650137	+	1.45193i
$778$		0			0
$779$		0			0
$780$		0			0
$781$		0			0
$782$		0			0
$783$		0			0
$784$	7.88353e38	+	8.83067e38i	0.665969	+	0.745980i
$785$		0			0
$786$		0			0
$787$			2.48102e39i			1.99055i	0.0971167	+	0.995273i	$0.469038\pi$
$787$			2.48102e39i			1.99055i	−0.0971167	+	0.995273i	$0.530962\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	2.29779e38			0.166388
$794$		0			0
$795$		0			0
$796$		−	5.25774e38i		−	0.361803i
$797$		0			0			−	1.00000i	$-0.5\pi$
$797$		0			0				1.00000i	$0.5\pi$
$798$		0			0
$799$		0			0
$800$		0			0
$801$		0			0
$802$		0			0
$803$		0			0
$804$			2.87621e39i			1.72928i
$805$		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
$811$			3.63067e39i			1.94180i	0.239477	+	0.970902i	$0.423024\pi$
$811$			3.63067e39i			1.94180i	−0.239477	+	0.970902i	$0.576976\pi$
$812$		0			0
$813$	−2.63686e39			−1.36416
$814$		0			0
$815$		0			0
$816$		0			0
$817$		−	7.48789e39i		−	3.62545i
$818$		0			0
$819$	4.58724e38	−	1.02445e39i	0.214891	−	0.479909i
$820$		0			0
$821$		0			0		1.00000			$0$
$821$		0			0		−1.00000			$\pi$
$822$		0			0
$823$	−3.22511e39			−1.41464			−0.707322	−	0.706892i	$-0.750097\pi$
$823$	−3.22511e39			−1.41464			−0.707322	+	0.706892i	$0.750097\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0		1.00000			$0$
$827$		0			0		−1.00000			$\pi$
$828$		0			0
$829$		−	4.97392e39i		−	1.97794i	−0.148118	−	0.988970i	$-0.547322\pi$
$829$		−	4.97392e39i		−	1.97794i	0.148118	−	0.988970i	$-0.452678\pi$
$830$		0			0
$831$		−	4.16341e39i		−	1.60264i
$832$			1.38837e39i			0.525824i
$833$		0			0
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−5.03671e39			−1.75935
$838$		0			0
$839$		0			0			−	1.00000i	$-0.5\pi$
$839$		0			0				1.00000i	$0.5\pi$
$840$		0			0
$841$	3.05313e39			1.00000
$842$		0			0
$843$		0			0
$844$	2.55502e38			0.0797575
$845$		0			0
$846$		0			0
$847$	3.06722e39	+	1.37342e39i	0.912680	+	0.408675i
$848$		0			0
$849$	5.62951e39			1.62262
$850$		0			0
$851$		0			0
$852$		0			0
$853$			5.05279e38i			0.136685i	0.997662	+	0.0683423i	$0.0217710\pi$
$853$			5.05279e38i			0.136685i	−0.997662	+	0.0683423i	$0.978229\pi$
$854$		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
$859$		−	7.88445e39i		−	1.94028i	−0.242553	−	0.970138i	$-0.577985\pi$
$859$		−	7.88445e39i		−	1.94028i	0.242553	−	0.970138i	$-0.422015\pi$
$860$		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
$867$			4.60530e39i			1.00000i
$868$	−3.36316e39	+	7.51083e39i	−0.719005	+	1.60573i
$869$		0			0
$870$		0			0
$871$		−	4.45605e39i		−	0.909296i
$872$		0			0
$873$		−	8.07094e39i		−	1.59673i
$874$		0			0
$875$		0			0
$876$	−4.12481e39			−0.779110
$877$	−7.72101e39			−1.43608			−0.718042	−	0.696000i	$-0.754961\pi$
$877$	−7.72101e39			−1.43608			−0.718042	+	0.696000i	$0.754961\pi$
$878$		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
$883$	−5.69320e39			−0.965800			−0.482900	−	0.875676i	$-0.660417\pi$
$883$	−5.69320e39			−0.965800			−0.482900	+	0.875676i	$0.660417\pi$
$884$		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
$889$	1.09818e40	+	4.91737e39i	1.70020	+	0.761306i
$890$		0			0
$891$		0			0
$892$			2.29882e39i			0.340079i
$893$		0			0
$894$		0			0
$895$		0			0
$896$		0			0
$897$		0			0
$898$		0			0
$899$		0			0
$900$	7.62560e39			1.00000
$901$		0			0
$902$		0			0
$903$	−5.91293e39	+	1.32051e40i	−0.741341	+	1.65561i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−1.16637e40			−1.37765			−0.688823	−	0.724930i	$-0.741872\pi$
$907$	−1.16637e40			−1.37765			−0.688823	+	0.724930i	$0.741872\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		0			0		1.00000			$0$
$911$		0			0		−1.00000			$\pi$
$912$	1.82244e40			1.99858
$913$		0			0
$914$		0			0
$915$		0			0
$916$			1.93439e40i			1.99965i
$917$		0			0
$918$		0			0
$919$	−8.71030e39			−0.861537			−0.430768	−	0.902462i	$-0.641757\pi$
$919$	−8.71030e39			−0.861537			−0.430768	+	0.902462i	$0.641757\pi$
$920$		0			0
$921$	2.02467e40			1.94469
$922$		0			0
$923$		0			0
$924$		0			0
$925$	1.75606e40			1.59084
$926$		0			0
$927$		−	2.26544e40i		−	1.99331i
$928$		0			0
$929$		0			0			−	1.00000i	$-0.5\pi$
$929$		0			0				1.00000i	$0.5\pi$
$930$		0			0
$931$	−1.79585e40	+	1.60323e40i	−1.49090	+	1.33099i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		0			0
$936$		0			0
$937$			1.96899e40i			1.49885i	0.662088	+	0.749426i	$0.269671\pi$
$937$			1.96899e40i			1.49885i	−0.662088	+	0.749426i	$0.730329\pi$
$938$		0			0
$939$	2.00958e40			1.48635
$940$		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			$\pi$
$948$			4.75543e39i			0.309227i
$949$	6.39048e39			0.409675
$950$		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
$961$	−3.87273e40			−2.09533
$962$		0			0
$963$		0			0
$964$		−	3.84756e40i		−	1.99593i
$965$		0			0
$966$		0			0
$967$	−2.22045e40			−1.10455			−0.552274	−	0.833663i	$-0.686240\pi$
$967$	−2.22045e40			−1.10455			−0.552274	+	0.833663i	$0.686240\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		0			0			−	1.00000i	$-0.5\pi$
$971$		0			0				1.00000i	$0.5\pi$
$972$		−	2.15523e40i		−	1.00000i
$973$	5.00603e39	−	1.11798e40i	0.229071	−	0.511576i
$974$		0			0
$975$	−1.18142e40			−0.525824
$976$		−	7.20867e39i		−	0.316434i
$977$		0			0		1.00000			$0$
$977$		0			0		−1.00000			$\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$	−2.55775e40			−1.04792
$982$		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
$988$	−2.82346e40			−1.05090
$989$		0			0
$990$		0			0
$991$	4.98569e40			1.78128			0.890638	−	0.454713i	$-0.150258\pi$
$991$	4.98569e40			1.78128			0.890638	+	0.454713i	$0.150258\pi$
$992$		0			0
$993$			3.88774e40i			1.35170i
$994$		0			0
$995$		0			0
$996$		0			0
$997$		−	2.46987e40i		−	0.813371i	−0.913568	−	0.406686i	$-0.866684\pi$
$997$		−	2.46987e40i		−	0.813371i	0.913568	−	0.406686i	$-0.133316\pi$
$998$		0			0
$999$		−	4.96318e40i		−	1.59084i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	21.28.c.a.20.1	✓	2
3.2	odd	2	CM	21.28.c.a.20.1	✓	2
7.6	odd	2	inner	21.28.c.a.20.2	yes	2
21.20	even	2	inner	21.28.c.a.20.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.28.c.a.20.1	✓	2	1.1	even	1	trivial
21.28.c.a.20.1	✓	2	3.2	odd	2	CM
21.28.c.a.20.2	yes	2	7.6	odd	2	inner
21.28.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:	\( k \)	\(=\)	\( 28 \)
Character orbit:	\([\chi]\)	\(=\)	21.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.76145e6i q^{3} +1.34218e8 q^{4} +(2.33960e11 + 1.04762e11i) q^{7} -7.62560e12 q^{9} +O(q^{10})\)	\(q-2.76145e6i q^{3} +1.34218e8 q^{4} +(2.33960e11 + 1.04762e11i) q^{7} -7.62560e12 q^{9} -3.70635e14i q^{12} +5.74217e14i q^{13} +1.80144e16 q^{16} +3.66350e17i q^{19} +(2.89294e17 - 6.46069e17i) q^{21} -7.45058e18 q^{25} +2.10577e19i q^{27} +(3.14016e19 + 1.40609e19i) q^{28} +2.39186e20i q^{31} -1.02349e21 q^{36} -2.35695e21 q^{37} +1.58567e21 q^{39} -2.04392e22 q^{43} -4.97458e22i q^{48} +(4.37624e22 + 4.90201e22i) q^{49} +7.70701e22i q^{52} +1.01166e24 q^{57} -4.00161e23i q^{61} +(-1.78409e24 - 7.98870e23i) q^{63} +2.41785e24 q^{64} -7.76022e24 q^{67} -1.11290e25i q^{73} +2.05744e25i q^{75} +4.91706e25i q^{76} -1.28305e25 q^{79} +5.81497e25 q^{81} +(3.88283e25 - 8.67139e25i) q^{84} +(-6.01559e25 + 1.34344e26i) q^{91} +6.60500e26 q^{93} +1.05840e27i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 268435456 q^{4} + 467920383820 q^{7} - 15251194969974 q^{9}+O(q^{10}) \) 2 * q + 268435456 * q^4 + 467920383820 * q^7 - 15251194969974 * q^9	\( 2 q + 268435456 q^{4} + 467920383820 q^{7} - 15251194969974 q^{9} + 36\!\cdots\!68 q^{16}+ \cdots + 13\!\cdots\!20 q^{93}+O(q^{100}) \) 2 * q + 268435456 * q^4 + 467920383820 * q^7 - 15251194969974 * q^9 + 36028797018963968 * q^16 + 578587464434659842 * q^21 - 14901161193847656250 * q^25 + 62803210801208360960 * q^28 - 2046980738154938499072 * q^36 - 4713890406868719236380 * q^37 + 3171340163073136542408 * q^39 - 40878358325687530326640 * q^43 + 87524760867187557513314 * q^49 + 2023311244175931564165420 * q^57 - 3568172502031943727710340 * q^63 + 4835703278458516698824704 * q^64 - 15520447866648287687606240 * q^67 - 25660963406174729383556312 * q^79 + 116299474006080119380780338 * q^81 + 77656694925700848446078976 * q^84 - 120311730813537349841403864 * q^91 + 1321000852240421053599117420 * q^93

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more