LMFDB - Newform orbit 126.12.a.h

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [126,12,Mod(1,126)] 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("126.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(126, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 126 = 2 \cdot 3^{2} \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	126.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,32,0,1024,8010] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$96.8112407505$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
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 + 32 q^{2} + 1024 q^{4} + 8010 q^{5} + 16807 q^{7} + 32768 q^{8} + 256320 q^{10} - 452052 q^{11} - 1091458 q^{13} + 537824 q^{14} + 1048576 q^{16} - 9665922 q^{17} - 3348844 q^{19} + 8202240 q^{20} - 14465664 q^{22}+ \cdots + 9039207968 q^{98}+O(q^{100}) $

q + 32 * q^2 + 1024 * q^4 + 8010 * q^5 + 16807 * q^7 + 32768 * q^8 + 256320 * q^10 - 452052 * q^11 - 1091458 * q^13 + 537824 * q^14 + 1048576 * q^16 - 9665922 * q^17 - 3348844 * q^19 + 8202240 * q^20 - 14465664 * q^22 - 59956344 * q^23 + 15331975 * q^25 - 34926656 * q^26 + 17210368 * q^28 + 55940106 * q^29 - 128343952 * q^31 + 33554432 * q^32 - 309309504 * q^34 + 134624070 * q^35 - 578037778 * q^37 - 107163008 * q^38 + 262471680 * q^40 + 840886998 * q^41 + 333428 * q^43 - 462901248 * q^44 - 1918603008 * q^46 - 333958944 * q^47 + 282475249 * q^49 + 490623200 * q^50 - 1117652992 * q^52 + 3918222258 * q^53 - 3620936520 * q^55 + 550731776 * q^56 + 1790083392 * q^58 + 7463827092 * q^59 - 1084935010 * q^61 - 4107006464 * q^62 + 1073741824 * q^64 - 8742578580 * q^65 + 5115031244 * q^67 - 9897904128 * q^68 + 4307970240 * q^70 - 29627533944 * q^71 - 23084922598 * q^73 - 18497208896 * q^74 - 3429216256 * q^76 - 7597637964 * q^77 + 28623995312 * q^79 + 8399093760 * q^80 + 26908383936 * q^82 + 18819334956 * q^83 - 77424035220 * q^85 + 10669696 * q^86 - 14812839936 * q^88 - 45381862410 * q^89 - 18344134606 * q^91 - 61395296256 * q^92 - 10686686208 * q^94 - 26824240440 * q^95 + 30119202770 * q^97 + 9039207968 * q^98

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

32.0000

1024.00

8010.00

16807.0

32768.0

256320.

Atkin-Lehner signs

$ p $	Sign
$2$	$ -1 $
$3$	$ -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	126.12.a.h		1
3.b	odd	2	1		42.12.a.a	✓	1

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
42.12.a.a	✓	1	3.b	odd	2	1
126.12.a.h		1	1.a	even	1	1	trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 8010 $ T5 - 8010 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(126))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T - 32 $ T - 32
$3$	$ T $ T
$5$	$ T - 8010 $ T - 8010
$7$	$ T - 16807 $ T - 16807
$11$	$ T + 452052 $ T + 452052
$13$	$ T + 1091458 $ T + 1091458
$17$	$ T + 9665922 $ T + 9665922
$19$	$ T + 3348844 $ T + 3348844
$23$	$ T + 59956344 $ T + 59956344
$29$	$ T - 55940106 $ T - 55940106
$31$	$ T + 128343952 $ T + 128343952
$37$	$ T + 578037778 $ T + 578037778
$41$	$ T - 840886998 $ T - 840886998
$43$	$ T - 333428 $ T - 333428
$47$	$ T + 333958944 $ T + 333958944
$53$	$ T - 3918222258 $ T - 3918222258
$59$	$ T - 7463827092 $ T - 7463827092
$61$	$ T + 1084935010 $ T + 1084935010
$67$	$ T - 5115031244 $ T - 5115031244
$71$	$ T + 29627533944 $ T + 29627533944
$73$	$ T + 23084922598 $ T + 23084922598
$79$	$ T - 28623995312 $ T - 28623995312
$83$	$ T - 18819334956 $ T - 18819334956
$89$	$ T + 45381862410 $ T + 45381862410
$97$	$ T - 30119202770 $ T - 30119202770
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Level:	\( N \)	\(=\)	\( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	126.a (trivial)

\(n\):		e.g. 2-40 or 990-1000
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