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Properties

Label 3267.2.a.bh
Level $3267$
Weight $2$
Character orbit 3267.a
Self dual yes
Analytic conductor $26.087$
Analytic rank $0$
Dimension $8$
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] = [3267,2,Mod(1,3267)] 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("3267.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3267, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3267 = 3^{3} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3267.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,0,0,6,0,0,6,0,0,6,0,0,16,0,0,-6,0,0,10,0,0,0,0,0,22,0,0,10, 0,0,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(31)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(26.0871263404\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.92296160000.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 11x^{6} + 37x^{4} - 39x^{2} + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 297)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + (\beta_{7} + \beta_{6} + \beta_1) q^{5} + ( - \beta_{4} + 1) q^{7} + (\beta_{6} + \beta_{3} + \beta_1) q^{8} + (\beta_{5} + 2 \beta_{4} + \beta_{2} + 1) q^{10}+ \cdots + ( - 2 \beta_{7} - 6 \beta_{6} + \cdots - 2 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 6 q^{4} + 6 q^{7} + 6 q^{10} + 16 q^{13} - 6 q^{16} + 10 q^{19} + 22 q^{25} + 10 q^{28} + 2 q^{31} + 16 q^{34} - 24 q^{37} + 20 q^{40} + 40 q^{43} + 16 q^{46} - 10 q^{49} + 74 q^{52} - 30 q^{58} + 38 q^{61}+ \cdots + 10 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 11x^{6} + 37x^{4} - 39x^{2} + 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 9\nu^{5} - 19\nu^{3} - 2\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - 6\nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} - 9\nu^{4} + 22\nu^{2} - 13 ) / 3 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 9\nu^{5} + 22\nu^{3} - 13\nu ) / 3 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -2\nu^{7} + 21\nu^{5} - 62\nu^{3} + 38\nu ) / 3 \) 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
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + 6\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 8\beta_{6} + 6\beta_{3} + 26\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3\beta_{5} + 9\beta_{4} + 32\beta_{2} + 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 9\beta_{7} + 53\beta_{6} + 32\beta_{3} + 137\beta_1 \) 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
−2.34096
−1.98331
−1.06675
−0.669653
0.669653
1.06675
1.98331
2.34096
−2.34096 0 3.48008 −3.58816 0 −1.15081 −3.46481 0 8.39974
1.2 −1.98331 0 1.93353 2.99577 0 4.12852 0.131828 0 −5.94154
1.3 −1.06675 0 −0.862049 1.22762 0 1.53278 3.05308 0 −1.30957
1.4 −0.669653 0 −1.55157 −2.76467 0 −1.51049 2.37832 0 1.85137
1.5 0.669653 0 −1.55157 2.76467 0 −1.51049 −2.37832 0 1.85137
1.6 1.06675 0 −0.862049 −1.22762 0 1.53278 −3.05308 0 −1.30957
1.7 1.98331 0 1.93353 −2.99577 0 4.12852 −0.131828 0 −5.94154
1.8 2.34096 0 3.48008 3.58816 0 −1.15081 3.46481 0 8.39974
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(11\) \( -1 \)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3267.2.a.bh 8
3.b odd 2 1 inner 3267.2.a.bh 8
11.b odd 2 1 3267.2.a.bg 8
11.d odd 10 2 297.2.f.c ✓ 16
33.d even 2 1 3267.2.a.bg 8
33.f even 10 2 297.2.f.c ✓ 16
99.o odd 30 4 891.2.n.g 32
99.p even 30 4 891.2.n.g 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
297.2.f.c ✓ 16 11.d odd 10 2
297.2.f.c ✓ 16 33.f even 10 2
891.2.n.g 32 99.o odd 30 4
891.2.n.g 32 99.p even 30 4
3267.2.a.bg 8 11.b odd 2 1
3267.2.a.bg 8 33.d even 2 1
3267.2.a.bh 8 1.a even 1 1 trivial
3267.2.a.bh 8 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3267))\):

\( T_{2}^{8} - 11T_{2}^{6} + 37T_{2}^{4} - 39T_{2}^{2} + 11 \) Copy content Toggle raw display
\( T_{5}^{8} - 31T_{5}^{6} + 327T_{5}^{4} - 1309T_{5}^{2} + 1331 \) Copy content Toggle raw display
\( T_{7}^{4} - 3T_{7}^{3} - 7T_{7}^{2} + 7T_{7} + 11 \) Copy content Toggle raw display
\( T_{23}^{8} - 36T_{23}^{6} + 316T_{23}^{4} - 981T_{23}^{2} + 891 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 11 T^{6} + \cdots + 11 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 31 T^{6} + \cdots + 1331 \) Copy content Toggle raw display
$7$ \( (T^{4} - 3 T^{3} - 7 T^{2} + \cdots + 11)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + \cdots + 121)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} - 67 T^{6} + \cdots + 72171 \) Copy content Toggle raw display
$19$ \( (T^{4} - 5 T^{3} + T^{2} + \cdots - 1)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 36 T^{6} + \cdots + 891 \) Copy content Toggle raw display
$29$ \( T^{8} - 163 T^{6} + \cdots + 107811 \) Copy content Toggle raw display
$31$ \( (T^{4} - T^{3} - 88 T^{2} + \cdots + 1111)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 12 T^{3} + \cdots - 176)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 79 T^{6} + \cdots + 10571 \) Copy content Toggle raw display
$43$ \( (T^{4} - 20 T^{3} + \cdots - 1251)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 143 T^{6} + \cdots + 1331 \) Copy content Toggle raw display
$53$ \( T^{8} - 263 T^{6} + \cdots + 2739011 \) Copy content Toggle raw display
$59$ \( T^{8} - 208 T^{6} + \cdots + 161051 \) Copy content Toggle raw display
$61$ \( (T^{4} - 19 T^{3} + \cdots + 99)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 16 T^{3} + \cdots - 13761)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} - 418 T^{6} + \cdots + 23159411 \) Copy content Toggle raw display
$73$ \( (T^{4} - 8 T^{3} + \cdots + 1424)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 21 T^{3} + \cdots + 59)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 275 T^{6} + \cdots + 4296875 \) Copy content Toggle raw display
$89$ \( T^{8} - 409 T^{6} + \cdots + 56133891 \) Copy content Toggle raw display
$97$ \( (T^{4} - 5 T^{3} + \cdots - 261)^{2} \) Copy content Toggle raw display
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