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Properties

Label 1372.2.a.c
Level $1372$
Weight $2$
Character orbit 1372.a
Self dual yes
Analytic conductor $10.955$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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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] = [1372,2,Mod(1,1372)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1372.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1372, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 1372 = 2^{2} \cdot 7^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1372.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,0,0,0,0,2,0,10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(10.9554751573\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.1229312.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 24x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - 2 \beta_{5} + \beta_{3} + \beta_1) q^{5} + (2 \beta_{2} + 1) q^{9} + (\beta_{4} - \beta_{2} + 1) q^{11} + (\beta_{5} - \beta_{3}) q^{13} + ( - 2 \beta_{4} + 2 \beta_{2} + 2) q^{15}+ \cdots + ( - \beta_{4} + 5 \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{9} + 10 q^{11} + 4 q^{15} + 18 q^{23} + 10 q^{25} + 6 q^{29} + 22 q^{37} + 12 q^{39} + 18 q^{43} + 32 q^{51} + 18 q^{53} + 8 q^{57} - 20 q^{65} + 42 q^{67} + 70 q^{71} + 6 q^{79} - 22 q^{81}+ \cdots - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} + 24x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} - 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} - 6\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{4} - 6\nu^{2} + 4 ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 12\nu ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{4} + 12\beta_{2} + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4\beta_{5} + 16\beta_{3} + 36\beta_1 \) 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
−2.54832
−1.76350
−0.629384
0.629384
1.76350
2.54832
0 −2.54832 0 −0.349282 0 0 0 3.49396 0
1.2 0 −1.76350 0 −2.04360 0 0 0 0.109916 0
1.3 0 −0.629384 0 3.96254 0 0 0 −2.60388 0
1.4 0 0.629384 0 −3.96254 0 0 0 −2.60388 0
1.5 0 1.76350 0 2.04360 0 0 0 0.109916 0
1.6 0 2.54832 0 0.349282 0 0 0 3.49396 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
\(2\) \( -1 \)
\(7\) \( +1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1372.2.a.c ✓ 6
4.b odd 2 1 5488.2.a.j 6
7.b odd 2 1 inner 1372.2.a.c ✓ 6
7.c even 3 2 1372.2.e.b 12
7.d odd 6 2 1372.2.e.b 12
28.d even 2 1 5488.2.a.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1372.2.a.c ✓ 6 1.a even 1 1 trivial
1372.2.a.c ✓ 6 7.b odd 2 1 inner
1372.2.e.b 12 7.c even 3 2
1372.2.e.b 12 7.d odd 6 2
5488.2.a.j 6 4.b odd 2 1
5488.2.a.j 6 28.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1372))\):

\( T_{3}^{6} - 10T_{3}^{4} + 24T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{11}^{3} - 5T_{11}^{2} + 6T_{11} - 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 10 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{6} - 20 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T^{3} - 5 T^{2} + 6 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} - 12 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$17$ \( T^{6} - 34 T^{4} + \cdots - 8 \) Copy content Toggle raw display
$19$ \( T^{6} - 52 T^{4} + \cdots - 1352 \) Copy content Toggle raw display
$23$ \( (T^{3} - 9 T^{2} + 20 T + 1)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 3 T^{2} - 46 T + 97)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} - 84 T^{4} + \cdots - 392 \) Copy content Toggle raw display
$37$ \( (T^{3} - 11 T^{2} + \cdots + 29)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 178 T^{4} + \cdots - 129032 \) Copy content Toggle raw display
$43$ \( (T^{3} - 9 T^{2} - 22 T + 71)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 180 T^{4} + \cdots - 129032 \) Copy content Toggle raw display
$53$ \( (T^{3} - 9 T^{2} + \cdots + 169)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 250 T^{4} + \cdots - 228488 \) Copy content Toggle raw display
$61$ \( (T^{2} - 162)^{3} \) Copy content Toggle raw display
$67$ \( (T^{3} - 21 T^{2} + \cdots - 189)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} - 35 T^{2} + \cdots - 1379)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} - 598 T^{4} + \cdots - 6237512 \) Copy content Toggle raw display
$79$ \( (T^{3} - 3 T^{2} - 46 T + 97)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} - 404 T^{4} + \cdots - 631688 \) Copy content Toggle raw display
$89$ \( T^{6} - 262 T^{4} + \cdots - 40328 \) Copy content Toggle raw display
$97$ \( T^{6} - 426 T^{4} + \cdots - 262088 \) Copy content Toggle raw display
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