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Properties

Label 4332.2.a.t
Level $4332$
Weight $2$
Character orbit 4332.a
Self dual yes
Analytic conductor $34.591$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [4332,2,Mod(1,4332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4332, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4332.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 4332 = 2^{2} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4332.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,-6,0,3,0,9] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(34.5911941556\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.73227321.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 18x^{4} + 27x^{3} + 96x^{2} - 48x - 127 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 228)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_1 + 1) q^{5} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{7} + q^{9} + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{11} + (\beta_{4} - \beta_1 - 1) q^{13} + (\beta_1 - 1) q^{15} + ( - \beta_{4} + \beta_{2} + 1) q^{17}+ \cdots + (\beta_{5} + \beta_{4} + \beta_{3} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 3 q^{5} + 9 q^{7} + 6 q^{9} + 9 q^{11} - 9 q^{13} - 3 q^{15} + 9 q^{17} - 9 q^{21} + 12 q^{23} + 15 q^{25} - 6 q^{27} - 9 q^{29} + 6 q^{31} - 9 q^{33} - 3 q^{35} + 6 q^{37} + 9 q^{39} + 18 q^{41}+ \cdots + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 18x^{4} + 27x^{3} + 96x^{2} - 48x - 127 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 5\nu^{4} - 5\nu^{3} + 28\nu^{2} + \nu - 8 ) / 6 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{5} + 28\nu^{4} + 22\nu^{3} - 209\nu^{2} + 13\nu + 283 ) / 12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 8\nu^{5} - 43\nu^{4} - 43\nu^{3} + 323\nu^{2} + 14\nu - 445 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{5} + 58\nu^{4} + 64\nu^{3} - 433\nu^{2} - 49\nu + 579 ) / 4 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 3\beta_{4} + 3\beta_{3} + 2\beta _1 + 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 5\beta_{5} + 14\beta_{4} + 13\beta_{3} + 3\beta_{2} + 14\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 28\beta_{5} + 83\beta_{4} + 86\beta_{3} + 13\beta_{2} + 54\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 137\beta_{5} + 401\beta_{4} + 411\beta_{3} + 86\beta_{2} + 283\beta _1 + 332 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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
4.85683
2.60676
1.46277
−1.25946
−2.32474
−2.34215
0 −1.00000 0 −3.85683 0 3.68676 0 1.00000 0
1.2 0 −1.00000 0 −1.60676 0 −2.89911 0 1.00000 0
1.3 0 −1.00000 0 −0.462767 0 4.24109 0 1.00000 0
1.4 0 −1.00000 0 2.25946 0 4.36702 0 1.00000 0
1.5 0 −1.00000 0 3.32474 0 1.19263 0 1.00000 0
1.6 0 −1.00000 0 3.34215 0 −1.58839 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( +1 \)
\(19\) \( +1 \)

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 4332.2.a.t 6
19.b odd 2 1 4332.2.a.u 6
19.e even 9 2 228.2.q.b ✓ 12
57.l odd 18 2 684.2.bo.e 12
76.l odd 18 2 912.2.bo.g 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
228.2.q.b ✓ 12 19.e even 9 2
684.2.bo.e 12 57.l odd 18 2
912.2.bo.g 12 76.l odd 18 2
4332.2.a.t 6 1.a even 1 1 trivial
4332.2.a.u 6 19.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4332))\):

\( T_{5}^{6} - 3T_{5}^{5} - 18T_{5}^{4} + 55T_{5}^{3} + 54T_{5}^{2} - 144T_{5} - 72 \) Copy content Toggle raw display
\( T_{7}^{6} - 9T_{7}^{5} + 9T_{7}^{4} + 101T_{7}^{3} - 195T_{7}^{2} - 225T_{7} + 375 \) Copy content Toggle raw display
\( T_{13}^{6} + 9T_{13}^{5} + 12T_{13}^{4} - 77T_{13}^{3} - 174T_{13}^{2} + 36T_{13} + 57 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots - 72 \) Copy content Toggle raw display
$7$ \( T^{6} - 9 T^{5} + \cdots + 375 \) Copy content Toggle raw display
$11$ \( T^{6} - 9 T^{5} + \cdots + 152 \) Copy content Toggle raw display
$13$ \( T^{6} + 9 T^{5} + \cdots + 57 \) Copy content Toggle raw display
$17$ \( T^{6} - 9 T^{5} + \cdots + 24 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 12 T^{5} + \cdots + 584 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots - 2136 \) Copy content Toggle raw display
$31$ \( T^{6} - 6 T^{5} + \cdots - 4833 \) Copy content Toggle raw display
$37$ \( T^{6} - 6 T^{5} + \cdots - 7757 \) Copy content Toggle raw display
$41$ \( T^{6} - 18 T^{5} + \cdots + 257032 \) Copy content Toggle raw display
$43$ \( T^{6} - 15 T^{5} + \cdots - 3831 \) Copy content Toggle raw display
$47$ \( T^{6} - 21 T^{5} + \cdots + 150344 \) Copy content Toggle raw display
$53$ \( T^{6} + 6 T^{5} + \cdots - 8 \) Copy content Toggle raw display
$59$ \( T^{6} + 15 T^{5} + \cdots + 318744 \) Copy content Toggle raw display
$61$ \( T^{6} + 21 T^{5} + \cdots - 127871 \) Copy content Toggle raw display
$67$ \( T^{6} + 18 T^{5} + \cdots + 44136 \) Copy content Toggle raw display
$71$ \( T^{6} - 15 T^{5} + \cdots + 32104 \) Copy content Toggle raw display
$73$ \( T^{6} - 18 T^{5} + \cdots + 813 \) Copy content Toggle raw display
$79$ \( T^{6} - 9 T^{5} + \cdots - 750367 \) Copy content Toggle raw display
$83$ \( T^{6} + 3 T^{5} + \cdots + 24 \) Copy content Toggle raw display
$89$ \( T^{6} + 36 T^{5} + \cdots + 78216 \) Copy content Toggle raw display
$97$ \( T^{6} + 39 T^{5} + \cdots + 8873 \) Copy content Toggle raw display
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