LMFDB - Newform orbit 42.12.a.c

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [42,12,Mod(1,42)] 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("42.1"); S:= CuspForms(chi, 12); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 12, names="a")

Level:	$ N $	$=$	$ 42 = 2 \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 12 $
Character orbit:	$[\chi]$	$=$	42.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,-32,243,1024,1080] 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:	$32.2704135835$
Analytic rank:	$0$
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 - 32 q^{2} + 243 q^{3} + 1024 q^{4} + 1080 q^{5} - 7776 q^{6} + 16807 q^{7} - 32768 q^{8} + 59049 q^{9} - 34560 q^{10} + 454950 q^{11} + 248832 q^{12} + 333386 q^{13} - 537824 q^{14} + 262440 q^{15}+ \cdots + 26864342550 q^{99}+O(q^{100}) $

q - 32 * q^2 + 243 * q^3 + 1024 * q^4 + 1080 * q^5 - 7776 * q^6 + 16807 * q^7 - 32768 * q^8 + 59049 * q^9 - 34560 * q^10 + 454950 * q^11 + 248832 * q^12 + 333386 * q^13 - 537824 * q^14 + 262440 * q^15 + 1048576 * q^16 + 329448 * q^17 - 1889568 * q^18 - 4876828 * q^19 + 1105920 * q^20 + 4084101 * q^21 - 14558400 * q^22 - 4054050 * q^23 - 7962624 * q^24 - 47661725 * q^25 - 10668352 * q^26 + 14348907 * q^27 + 17210368 * q^28 + 64835934 * q^29 - 8398080 * q^30 + 64240136 * q^31 - 33554432 * q^32 + 110552850 * q^33 - 10542336 * q^34 + 18151560 * q^35 + 60466176 * q^36 + 439691690 * q^37 + 156058496 * q^38 + 81012798 * q^39 - 35389440 * q^40 + 1147482084 * q^41 - 130691232 * q^42 + 682653668 * q^43 + 465868800 * q^44 + 63772920 * q^45 + 129729600 * q^46 + 2242598340 * q^47 + 254803968 * q^48 + 282475249 * q^49 + 1525175200 * q^50 + 80055864 * q^51 + 341387264 * q^52 + 1692909282 * q^53 - 459165024 * q^54 + 491346000 * q^55 - 550731776 * q^56 - 1185069204 * q^57 - 2074749888 * q^58 - 856900848 * q^59 + 268738560 * q^60 + 328162910 * q^61 - 2055684352 * q^62 + 992436543 * q^63 + 1073741824 * q^64 + 360056880 * q^65 - 3537691200 * q^66 - 6834378088 * q^67 + 337354752 * q^68 - 985134150 * q^69 - 580849920 * q^70 + 18096946722 * q^71 - 1934917632 * q^72 + 4621386710 * q^73 - 14070134080 * q^74 - 11581799175 * q^75 - 4993871872 * q^76 + 7646344650 * q^77 - 2592409536 * q^78 - 24901224268 * q^79 + 1132462080 * q^80 + 3486784401 * q^81 - 36719426688 * q^82 + 24895768956 * q^83 + 4182119424 * q^84 + 355803840 * q^85 - 21844917376 * q^86 + 15755131962 * q^87 - 14907801600 * q^88 + 25356289956 * q^89 - 2040733440 * q^90 + 5603218502 * q^91 - 4151347200 * q^92 + 15610353048 * q^93 - 71763146880 * q^94 - 5266974240 * q^95 - 8153726976 * q^96 - 40168146010 * q^97 - 9039207968 * q^98 + 26864342550 * 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

−32.0000

243.000

1024.00

1080.00

−7776.00

16807.0

−32768.0

59049.0

−34560.0

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	42.12.a.c	✓	1
3.b	odd	2	1		126.12.a.f		1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} - 1080 $ T5 - 1080 acting on $S_{12}^{\mathrm{new}}(\Gamma_0(42))$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T + 32 $ T + 32
$3$	$ T - 243 $ T - 243
$5$	$ T - 1080 $ T - 1080
$7$	$ T - 16807 $ T - 16807
$11$	$ T - 454950 $ T - 454950
$13$	$ T - 333386 $ T - 333386
$17$	$ T - 329448 $ T - 329448
$19$	$ T + 4876828 $ T + 4876828
$23$	$ T + 4054050 $ T + 4054050
$29$	$ T - 64835934 $ T - 64835934
$31$	$ T - 64240136 $ T - 64240136
$37$	$ T - 439691690 $ T - 439691690
$41$	$ T - 1147482084 $ T - 1147482084
$43$	$ T - 682653668 $ T - 682653668
$47$	$ T - 2242598340 $ T - 2242598340
$53$	$ T - 1692909282 $ T - 1692909282
$59$	$ T + 856900848 $ T + 856900848
$61$	$ T - 328162910 $ T - 328162910
$67$	$ T + 6834378088 $ T + 6834378088
$71$	$ T - 18096946722 $ T - 18096946722
$73$	$ T - 4621386710 $ T - 4621386710
$79$	$ T + 24901224268 $ T + 24901224268
$83$	$ T - 24895768956 $ T - 24895768956
$89$	$ T - 25356289956 $ T - 25356289956
$97$	$ T + 40168146010 $ T + 40168146010
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Elliptic curves over $\Q$
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Level:	\( N \)	\(=\)	\( 42 = 2 \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 12 \)
Character orbit:	\([\chi]\)	\(=\)	42.a (trivial)

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