LMFDB - Newform orbit 12.57.c.a

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [12,57,Mod(5,12)] 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("12.5"); S:= CuspForms(chi, 57); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(12, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 57, names="a")

Level:	$ N $	$=$	$ 12 = 2^{2} \cdot 3 $
Weight:	$ k $	$=$	$ 57 $
Character orbit:	$[\chi]$	$=$	12.c (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$238.332749918$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$ q + 22876792454961 q^{3} - 59\!\cdots\!06 q^{7} + 52\!\cdots\!21 q^{9} - 30\!\cdots\!86 q^{13} - 80\!\cdots\!26 q^{19} - 13\!\cdots\!66 q^{21} + 13\!\cdots\!25 q^{25} + 11\!\cdots\!81 q^{27} - 17\!\cdots\!86 q^{31}+ \cdots + 81\!\cdots\!94 q^{97}+O(q^{100}) $

q + 22876792454961 * q^3 - 591717446468983676495806 * q^7 + 523347633027360537213511521 * q^9 - 30991922180269535381092636301086 * q^13 - 801094118290345631094986560492669726 * q^19 - 13536597214850435181565125176560393566 * q^21 + 1387778780781445675529539585113525390625 * q^25 + 11972515182562019788602740026717047105681 * q^27 - 177219193906377821405866631209929992114686 * q^31 - 131479260864817427088935383416643128485153758 * q^37 - 708995771498328571907505059908880700290387646 * q^39 + 6260644778529028190982080116975624058745258914 * q^43 + 138541922653349171125793439094796140881475396035 * q^49 - 18326463881018213762162678455792403394704503210686 * q^57 + 160882166479384547464784483706890952741807228139874 * q^61 - 309673925030536522223716443758186314500613489180926 * q^63 + 2186336628514884999406009485093209387409497271945314 * q^67 - 26824669443334398767307847312585979941233078135570238 * q^73 + 31747927141335952061496072929003275930881500244140625 * q^75 + 269218387719281559373178978274179197128698492212533122 * q^79 + 273892744995340833777347939263771534786080723599733441 * q^81 + 18338461053674546673081493783833097674528370706032245316 * q^91 - 4054206718031694572554718506564765166070270909801657246 * q^93 + 81712258162086248351688385708540726414381613455170984194 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/12\mathbb{Z}\right)^\times$.

$n$	$5$	$7$
$\chi(n)$	$-1$	$1$

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} $

5.1

2.28768e13

−5.91717e23

5.23348e26

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
3.b	odd	2	1	CM by $\Q(\sqrt{-3}) $

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	12.57.c.a	✓	1
3.b	odd	2	1	CM	12.57.c.a	✓	1

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

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $ T_{5} $ T5 acting on $S_{57}^{\mathrm{new}}(12, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T $ T
$3$	$ T - 22876792454961 $ T - 22876792454961
$5$	$ T $ T
$7$	$ T + 59\!\cdots\!06 $ T + 591717446468983676495806
$11$	$ T $ T
$13$	$ T + 30\!\cdots\!86 $ T + 30991922180269535381092636301086
$17$	$ T $ T
$19$	$ T + 80\!\cdots\!26 $ T + 801094118290345631094986560492669726
$23$	$ T $ T
$29$	$ T $ T
$31$	$ T + 17\!\cdots\!86 $ T + 177219193906377821405866631209929992114686
$37$	$ T + 13\!\cdots\!58 $ T + 131479260864817427088935383416643128485153758
$41$	$ T $ T
$43$	$ T - 62\!\cdots\!14 $ T - 6260644778529028190982080116975624058745258914
$47$	$ T $ T
$53$	$ T $ T
$59$	$ T $ T
$61$	$ T - 16\!\cdots\!74 $ T - 160882166479384547464784483706890952741807228139874
$67$	$ T - 21\!\cdots\!14 $ T - 2186336628514884999406009485093209387409497271945314
$71$	$ T $ T
$73$	$ T + 26\!\cdots\!38 $ T + 26824669443334398767307847312585979941233078135570238
$79$	$ T - 26\!\cdots\!22 $ T - 269218387719281559373178978274179197128698492212533122
$83$	$ T $ T
$89$	$ T $ T
$97$	$ T - 81\!\cdots\!94 $ T - 81712258162086248351688385708540726414381613455170984194
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Level:	\( N \)	\(=\)	\( 12 = 2^{2} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 57 \)
Character orbit:	\([\chi]\)	\(=\)	12.c (of order \(2\), degree \(1\), minimal)

\(n\)	\(5\)	\(7\)
\(\chi(n)\)	\(-1\)	\(1\)

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