LMFDB - Newform orbit 126.12.a.c

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,-32,0,1024,11880] 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:	$0$
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} + 11880 q^{5} + 16807 q^{7} - 32768 q^{8} - 380160 q^{10} - 727110 q^{11} - 1895734 q^{13} - 537824 q^{14} + 1048576 q^{16} + 10233912 q^{17} - 12792796 q^{19} + 12165120 q^{20}+ \cdots - 9039207968 q^{98}+O(q^{100}) $

q - 32 * q^2 + 1024 * q^4 + 11880 * q^5 + 16807 * q^7 - 32768 * q^8 - 380160 * q^10 - 727110 * q^11 - 1895734 * q^13 - 537824 * q^14 + 1048576 * q^16 + 10233912 * q^17 - 12792796 * q^19 + 12165120 * q^20 + 23267520 * q^22 + 27412290 * q^23 + 92306275 * q^25 + 60663488 * q^26 + 17210368 * q^28 + 108082914 * q^29 - 202243384 * q^31 - 33554432 * q^32 - 327485184 * q^34 + 199667160 * q^35 + 412454954 * q^37 + 409369472 * q^38 - 389283840 * q^40 + 245108604 * q^41 + 509839844 * q^43 - 744560640 * q^44 - 877193280 * q^46 - 699876996 * q^47 + 282475249 * q^49 - 2953800800 * q^50 - 1941231616 * q^52 + 4836690462 * q^53 - 8638066800 * q^55 - 550731776 * q^56 - 3458653248 * q^58 - 2462248272 * q^59 + 9054716702 * q^61 + 6471788288 * q^62 + 1073741824 * q^64 - 22521319920 * q^65 - 2923148584 * q^67 + 10479525888 * q^68 - 6389349120 * q^70 + 11268869310 * q^71 + 2809231382 * q^73 - 13198558528 * q^74 - 13099823104 * q^76 - 12220537770 * q^77 + 25573574516 * q^79 + 12457082880 * q^80 - 7843475328 * q^82 + 38718767364 * q^83 + 121578874560 * q^85 - 16314875008 * q^86 + 23825940480 * q^88 - 90783707844 * q^89 - 31861601338 * q^91 + 28070184960 * q^92 + 22396063872 * q^94 - 151978416480 * q^95 + 130559880038 * 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

11880.0

16807.0

−32768.0

−380160.

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.c		1
3.b	odd	2	1		42.12.a.f	✓	1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 11880 $ T5 - 11880 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 - 11880 $ T - 11880
$7$	$ T - 16807 $ T - 16807
$11$	$ T + 727110 $ T + 727110
$13$	$ T + 1895734 $ T + 1895734
$17$	$ T - 10233912 $ T - 10233912
$19$	$ T + 12792796 $ T + 12792796
$23$	$ T - 27412290 $ T - 27412290
$29$	$ T - 108082914 $ T - 108082914
$31$	$ T + 202243384 $ T + 202243384
$37$	$ T - 412454954 $ T - 412454954
$41$	$ T - 245108604 $ T - 245108604
$43$	$ T - 509839844 $ T - 509839844
$47$	$ T + 699876996 $ T + 699876996
$53$	$ T - 4836690462 $ T - 4836690462
$59$	$ T + 2462248272 $ T + 2462248272
$61$	$ T - 9054716702 $ T - 9054716702
$67$	$ T + 2923148584 $ T + 2923148584
$71$	$ T - 11268869310 $ T - 11268869310
$73$	$ T - 2809231382 $ T - 2809231382
$79$	$ T - 25573574516 $ T - 25573574516
$83$	$ T - 38718767364 $ T - 38718767364
$89$	$ T + 90783707844 $ T + 90783707844
$97$	$ T - 130559880038 $ T - 130559880038
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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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