LMFDB - Newform orbit 21.10.a.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [21,10,Mod(1,21)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("21.1"); S:= CuspForms(chi, 10); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(21, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 10, names="a")

Level:	$ N $	$=$	$ 21 = 3 \cdot 7 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	21.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-24] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.8157525594$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q - 24 q^{2} + 81 q^{3} + 64 q^{4} - 144 q^{5} - 1944 q^{6} + 2401 q^{7} + 10752 q^{8} + 6561 q^{9} + 3456 q^{10} - 15030 q^{11} + 5184 q^{12} - 151486 q^{13} - 57624 q^{14} - 11664 q^{15} - 290816 q^{16}+ \cdots - 98611830 q^{99}+O(q^{100}) $

q - 24 * q^2 + 81 * q^3 + 64 * q^4 - 144 * q^5 - 1944 * q^6 + 2401 * q^7 + 10752 * q^8 + 6561 * q^9 + 3456 * q^10 - 15030 * q^11 + 5184 * q^12 - 151486 * q^13 - 57624 * q^14 - 11664 * q^15 - 290816 * q^16 - 350448 * q^17 - 157464 * q^18 - 691108 * q^19 - 9216 * q^20 + 194481 * q^21 + 360720 * q^22 + 892458 * q^23 + 870912 * q^24 - 1932389 * q^25 + 3635664 * q^26 + 531441 * q^27 + 153664 * q^28 + 1648518 * q^29 + 279936 * q^30 - 3734296 * q^31 + 1474560 * q^32 - 1217430 * q^33 + 8410752 * q^34 - 345744 * q^35 + 419904 * q^36 - 11471902 * q^37 + 16586592 * q^38 - 12270366 * q^39 - 1548288 * q^40 + 13985724 * q^41 - 4667544 * q^42 + 16794524 * q^43 - 961920 * q^44 - 944784 * q^45 - 21418992 * q^46 - 14012052 * q^47 - 23556096 * q^48 + 5764801 * q^49 + 46377336 * q^50 - 28386288 * q^51 - 9695104 * q^52 - 97439910 * q^53 - 12754584 * q^54 + 2164320 * q^55 + 25815552 * q^56 - 55979748 * q^57 - 39564432 * q^58 + 110798304 * q^59 - 746496 * q^60 - 93816682 * q^61 + 89623104 * q^62 + 15752961 * q^63 + 113508352 * q^64 + 21813984 * q^65 + 29218320 * q^66 - 122446456 * q^67 - 22428672 * q^68 + 72289098 * q^69 + 8297856 * q^70 + 206197398 * q^71 + 70543872 * q^72 + 250337558 * q^73 + 275325648 * q^74 - 156523509 * q^75 - 44230912 * q^76 - 36087030 * q^77 + 294488784 * q^78 - 38314852 * q^79 + 41877504 * q^80 + 43046721 * q^81 - 335657376 * q^82 - 514086924 * q^83 + 12446784 * q^84 + 50464512 * q^85 - 403068576 * q^86 + 133529958 * q^87 - 161602560 * q^88 - 1061294916 * q^89 + 22674816 * q^90 - 363717886 * q^91 + 57117312 * q^92 - 302477976 * q^93 + 336289248 * q^94 + 99519552 * q^95 + 119439360 * q^96 - 73841578 * q^97 - 138355224 * q^98 - 98611830 * q^99

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment:embeddings in the coefficient field

gp:mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

1.1

−24.0000

81.0000

64.0000

−144.000

−1944.00

2401.00

10752.0

6561.00

3456.00

Atkin-Lehner signs

$ p $	Sign
$3$	$ -1 $
$7$	$ -1 $

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	21.10.a.a	✓	1
3.b	odd	2	1		63.10.a.a		1
4.b	odd	2	1		336.10.a.d		1
7.b	odd	2	1		147.10.a.b		1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
21.10.a.a	✓	1	1.a	even	1	1	trivial
63.10.a.a		1	3.b	odd	2	1
147.10.a.b		1	7.b	odd	2	1
336.10.a.d		1	4.b	odd	2	1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{2} + 24 $ T2 + 24 acting on $S_{10}^{\mathrm{new}}(\Gamma_0(21))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 24 $ T + 24
$3$	$ T - 81 $ T - 81
$5$	$ T + 144 $ T + 144
$7$	$ T - 2401 $ T - 2401
$11$	$ T + 15030 $ T + 15030
$13$	$ T + 151486 $ T + 151486
$17$	$ T + 350448 $ T + 350448
$19$	$ T + 691108 $ T + 691108
$23$	$ T - 892458 $ T - 892458
$29$	$ T - 1648518 $ T - 1648518
$31$	$ T + 3734296 $ T + 3734296
$37$	$ T + 11471902 $ T + 11471902
$41$	$ T - 13985724 $ T - 13985724
$43$	$ T - 16794524 $ T - 16794524
$47$	$ T + 14012052 $ T + 14012052
$53$	$ T + 97439910 $ T + 97439910
$59$	$ T - 110798304 $ T - 110798304
$61$	$ T + 93816682 $ T + 93816682
$67$	$ T + 122446456 $ T + 122446456
$71$	$ T - 206197398 $ T - 206197398
$73$	$ T - 250337558 $ T - 250337558
$79$	$ T + 38314852 $ T + 38314852
$83$	$ T + 514086924 $ T + 514086924
$89$	$ T + 1061294916 $ T + 1061294916
$97$	$ T + 73841578 $ T + 73841578
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Classical	Maass
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 21 = 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	21.a (trivial)

\(n\):		e.g. 2-40 or 80-90
Significant digits:
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