Properties

Label 12.22.a.b
Level $12$
Weight $22$
Character orbit 12.a
Self dual yes
Analytic conductor $33.537$
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] = [12,22,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, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("12.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 12 = 2^{2} \cdot 3 \)
Weight: \( k \) \(=\) \( 22 \)
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: \(33.5372813144\)
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: \( x^{2} - x - 797544 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{2}\cdot 5 \)
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 \(\beta = 2880\sqrt{3190177}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 59049 q^{3} + ( - 5 \beta + 14413950) q^{5} + ( - 207 \beta + 254834864) q^{7} + 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 59049 q^{3} + ( - 5 \beta + 14413950) q^{5} + ( - 207 \beta + 254834864) q^{7} + 3486784401 q^{9} + ( - 3446 \beta - 38025427644) q^{11} + (33282 \beta - 404663021650) q^{13} + (295245 \beta - 851129333550) q^{15} + ( - 1929874 \beta + 415692948834) q^{17} + ( - 143838 \beta + 14908784326460) q^{19} + (12223143 \beta - 15047743884336) q^{21} + (22074626 \beta + 31340320703064) q^{23} + ( - 144139500 \beta + 392439899119375) q^{25} - 205891132094649 q^{27} + (547532665 \beta - 10\!\cdots\!26) q^{29}+ \cdots + ( - 12015459045846 \beta - 13\!\cdots\!44) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 118098 q^{3} + 28827900 q^{5} + 509669728 q^{7} + 6973568802 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 118098 q^{3} + 28827900 q^{5} + 509669728 q^{7} + 6973568802 q^{9} - 76050855288 q^{11} - 809326043300 q^{13} - 1702258667100 q^{15} + 831385897668 q^{17} + 29817568652920 q^{19} - 30095487768672 q^{21} + 62680641406128 q^{23} + 784879798238750 q^{25} - 411782264189298 q^{27} - 21\!\cdots\!52 q^{29}+ \cdots - 26\!\cdots\!88 q^{99}+O(q^{100}) \) 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
893.553
−892.553
0 −59049.0 0 −1.13060e7 0 −8.09970e8 0 3.48678e9 0
1.2 0 −59049.0 0 4.01339e7 0 1.31964e9 0 3.48678e9 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 12.22.a.b 2
3.b odd 2 1 36.22.a.b 2
4.b odd 2 1 48.22.a.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
12.22.a.b 2 1.a even 1 1 trivial
36.22.a.b 2 3.b odd 2 1
48.22.a.i 2 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} - 28827900T_{5} - 453753148117500 \) acting on \(S_{22}^{\mathrm{new}}(\Gamma_0(12))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( (T + 59049)^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + \cdots - 453753148117500 \) Copy content Toggle raw display
$7$ \( T^{2} - 509669728 T - 10\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( T^{2} + 76050855288 T + 11\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( T^{2} + 809326043300 T + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{2} - 831385897668 T - 98\!\cdots\!44 \) Copy content Toggle raw display
$19$ \( T^{2} - 29817568652920 T + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{2} - 62680641406128 T - 11\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( T^{2} + \cdots - 68\!\cdots\!24 \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 85\!\cdots\!76 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots - 10\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{2} + \cdots + 30\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{2} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{2} + \cdots + 81\!\cdots\!44 \) Copy content Toggle raw display
$59$ \( T^{2} + \cdots - 88\!\cdots\!64 \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 75\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots - 37\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{2} + \cdots - 77\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots - 77\!\cdots\!16 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots - 86\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{2} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( T^{2} + \cdots + 96\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{2} + \cdots + 34\!\cdots\!64 \) Copy content Toggle raw display
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