Properties

Label 12.24.a.b
Level $12$
Weight $24$
Character orbit 12.a
Self dual yes
Analytic conductor $40.224$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [12,24,Mod(1,12)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(12, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.1");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 12.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(40.2244844814\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\mathbb{Q}[x]/(x^{2} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4674852 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{3}\cdot 5 \)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = 8640\sqrt{18699409}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 177147 q^{3} + ( - \beta + 36953550) q^{5} + ( - 231 \beta - 765614248) q^{7} + 31381059609 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 177147 q^{3} + ( - \beta + 36953550) q^{5} + ( - 231 \beta - 765614248) q^{7} + 31381059609 q^{9} + (12194 \beta + 428001419556) q^{11} + ( - 79806 \beta + 770193809510) q^{13} + ( - 177147 \beta + 6546210521850) q^{15} + (3586318 \beta + 93895276928562) q^{17} + ( - 6507462 \beta + 177778332794780) q^{19} + ( - 40920957 \beta - 135626267190456) q^{21} + (158242162 \beta - 569328282916632) q^{23} + ( - 73907100 \beta - 91\!\cdots\!25) q^{25}+ \cdots + (382660640872146 \beta + 13\!\cdots\!04) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 354294 q^{3} + 73907100 q^{5} - 1531228496 q^{7} + 62762119218 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 354294 q^{3} + 73907100 q^{5} - 1531228496 q^{7} + 62762119218 q^{9} + 856002839112 q^{11} + 1540387619020 q^{13} + 13092421043700 q^{15} + 187790553857124 q^{17} + 355556665589560 q^{19} - 271252534380912 q^{21} - 11\!\cdots\!64 q^{23}+ \cdots + 26\!\cdots\!08 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
2162.64
−2161.64
0 177147. 0 −408241. 0 −9.39619e9 0 3.13811e10 0
1.2 0 177147. 0 7.43153e7 0 7.86496e9 0 3.13811e10 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)

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 12.24.a.b 2
3.b odd 2 1 36.24.a.b 2
4.b odd 2 1 48.24.a.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.24.a.b 2 1.a even 1 1 trivial
36.24.a.b 2 3.b odd 2 1
48.24.a.f 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 73907100T_{5} - 30338544483900 \) acting on \(S_{24}^{\mathrm{new}}(\Gamma_0(12))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T - 177147)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} - 73907100 T - 30338544483900 \) Copy content Toggle raw display
$7$ \( T^{2} + 1531228496 T - 73\!\cdots\!96 \) Copy content Toggle raw display
$11$ \( T^{2} - 856002839112 T - 24\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( T^{2} - 1540387619020 T - 82\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{2} - 187790553857124 T - 91\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{2} - 355556665589560 T - 27\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} + \cdots - 34\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 39\!\cdots\!64 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots - 17\!\cdots\!24 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 53\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 22\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 39\!\cdots\!24 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 87\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 48\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 36\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 19\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 15\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 46\!\cdots\!64 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots - 30\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots - 31\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots - 59\!\cdots\!84 \) Copy content Toggle raw display
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