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Properties

Label 3328.2.a.bo
Level $3328$
Weight $2$
Character orbit 3328.a
Self dual yes
Analytic conductor $26.574$
Analytic rank $1$
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] = [3328,2,Mod(1,3328)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3328, 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("3328.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3328 = 2^{8} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3328.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,0,0,-8,0,0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.5742137927\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.10323968.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 10x^{4} + 25x^{2} - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1664)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{2} - 1) q^{5} + \beta_{5} q^{7} + \beta_{2} q^{9} + ( - \beta_{5} + \beta_{3} + \beta_1) q^{11} - q^{13} + ( - \beta_{3} - 3 \beta_1) q^{15} + ( - \beta_{4} - 1) q^{17}+ \cdots + (3 \beta_{5} - \beta_{3} + 3 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 8 q^{5} + 2 q^{9} - 6 q^{13} - 4 q^{17} + 4 q^{21} + 14 q^{25} - 12 q^{29} + 16 q^{33} - 24 q^{37} + 20 q^{41} - 36 q^{45} - 2 q^{49} - 32 q^{53} - 24 q^{57} - 8 q^{61} + 8 q^{65} - 40 q^{69} - 12 q^{73}+ \cdots + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 10x^{4} + 25x^{2} - 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 5\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 8\nu^{3} + 13\nu ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{5} + 8\beta_{3} + 27\beta_1 \) 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
−2.47817
−1.86678
−0.611393
0.611393
1.86678
2.47817
0 −2.47817 0 −4.14134 0 −1.96435 0 3.14134 0
1.2 0 −1.86678 0 −1.48486 0 2.55248 0 0.484862 0
1.3 0 −0.611393 0 1.62620 0 −3.10261 0 −2.62620 0
1.4 0 0.611393 0 1.62620 0 3.10261 0 −2.62620 0
1.5 0 1.86678 0 −1.48486 0 −2.55248 0 0.484862 0
1.6 0 2.47817 0 −4.14134 0 1.96435 0 3.14134 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
\(2\) \( +1 \)
\(13\) \( +1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3328.2.a.bo 6
4.b odd 2 1 inner 3328.2.a.bo 6
8.b even 2 1 3328.2.a.bp 6
8.d odd 2 1 3328.2.a.bp 6
16.e even 4 2 1664.2.b.k ✓ 12
16.f odd 4 2 1664.2.b.k ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1664.2.b.k ✓ 12 16.e even 4 2
1664.2.b.k ✓ 12 16.f odd 4 2
3328.2.a.bo 6 1.a even 1 1 trivial
3328.2.a.bo 6 4.b odd 2 1 inner
3328.2.a.bp 6 8.b even 2 1
3328.2.a.bp 6 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3328))\):

\( T_{3}^{6} - 10T_{3}^{4} + 25T_{3}^{2} - 8 \) Copy content Toggle raw display
\( T_{5}^{3} + 4T_{5}^{2} - 3T_{5} - 10 \) Copy content Toggle raw display
\( T_{7}^{6} - 20T_{7}^{4} + 125T_{7}^{2} - 242 \) 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
$5$ \( (T^{3} + 4 T^{2} - 3 T - 10)^{2} \) Copy content Toggle raw display
$7$ \( T^{6} - 20 T^{4} + \cdots - 242 \) Copy content Toggle raw display
$11$ \( T^{6} - 42 T^{4} + \cdots - 800 \) Copy content Toggle raw display
$13$ \( (T + 1)^{6} \) Copy content Toggle raw display
$17$ \( (T^{3} + 2 T^{2} - 23 T - 44)^{2} \) Copy content Toggle raw display
$19$ \( T^{6} - 66 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$23$ \( T^{6} - 40 T^{4} + \cdots - 512 \) Copy content Toggle raw display
$29$ \( (T + 2)^{6} \) Copy content Toggle raw display
$31$ \( T^{6} - 34 T^{4} + \cdots - 128 \) Copy content Toggle raw display
$37$ \( (T^{3} + 12 T^{2} + \cdots - 50)^{2} \) Copy content Toggle raw display
$41$ \( (T^{3} - 10 T^{2} + \cdots + 512)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} - 34 T^{4} + \cdots - 32 \) Copy content Toggle raw display
$47$ \( T^{6} - 52 T^{4} + \cdots - 50 \) Copy content Toggle raw display
$53$ \( (T^{3} + 16 T^{2} + \cdots - 592)^{2} \) Copy content Toggle raw display
$59$ \( T^{6} - 130 T^{4} + \cdots - 2048 \) Copy content Toggle raw display
$61$ \( (T^{3} + 4 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} - 202 T^{4} + \cdots - 43808 \) Copy content Toggle raw display
$71$ \( T^{6} - 196 T^{4} + \cdots - 7442 \) Copy content Toggle raw display
$73$ \( (T^{3} + 6 T^{2} + \cdots + 128)^{2} \) Copy content Toggle raw display
$79$ \( T^{6} - 272 T^{4} + \cdots - 574592 \) Copy content Toggle raw display
$83$ \( T^{6} - 218 T^{4} + \cdots - 161312 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} - 32 T - 32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} - 10 T^{2} + \cdots + 488)^{2} \) Copy content Toggle raw display
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