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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8664,2,Mod(1,8664)] 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("8664.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8664, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8664 = 2^{3} \cdot 3 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8664.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [6,0,6,0,1,0,2] 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: \(69.1823883112\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.142368125.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 16x^{4} + 13x^{3} + 51x^{2} - 60x + 16 \) 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.

\(f(q)\) \(=\) \( q + q^{3} + \beta_{5} q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_{2} + \beta_1) q^{11} + (\beta_{5} - \beta_{2} - \beta_1 + 1) q^{13} + \beta_{5} q^{15} + (\beta_{3} + \beta_{2} - 1) q^{17} + \beta_{2} q^{21}+ \cdots + (\beta_{2} + \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + q^{5} + 2 q^{7} + 6 q^{9} + 3 q^{11} + 4 q^{13} + q^{15} - q^{17} + 2 q^{21} + 4 q^{23} + 19 q^{25} + 6 q^{27} - 2 q^{29} - 11 q^{31} + 3 q^{33} + 4 q^{35} + 5 q^{37} + 4 q^{39} + 20 q^{41}+ \cdots + 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 16x^{4} + 13x^{3} + 51x^{2} - 60x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{5} + 2\nu^{4} + 14\nu^{3} - 17\nu^{2} - 44\nu + 24 ) / 10 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{5} + \nu^{4} + 52\nu^{3} - 11\nu^{2} - 187\nu + 112 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 16\nu^{3} + 3\nu^{2} - 48\nu + 14 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{5} - \nu^{4} - 48\nu^{3} + 7\nu^{2} + 151\nu - 80 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} + \beta_{4} + 6\beta_{3} + \beta_{2} + 9\beta _1 - 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{5} + 10\beta_{4} + 16\beta_{3} + 16\beta_{2} + 7\beta _1 + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 35\beta_{5} + 17\beta_{4} + 99\beta_{3} + 19\beta_{2} + 96\beta _1 - 19 \) 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
0.437886
1.73281
−3.09636
−2.48937
3.66952
0.745515
0 1.00000 0 −4.13892 0 0.270628 0 1.00000 0
1.2 0 1.00000 0 −2.30478 0 −2.80375 0 1.00000 0
1.3 0 1.00000 0 −0.316286 0 5.01002 0 1.00000 0
1.4 0 1.00000 0 0.690627 0 −1.53852 0 1.00000 0
1.5 0 1.00000 0 2.83026 0 2.26789 0 1.00000 0
1.6 0 1.00000 0 4.23910 0 −1.20627 0 1.00000 0
\(n\): e.g. 2-40 or 80-90
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 8664.2.a.bj yes 6
19.b odd 2 1 8664.2.a.bg ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8664.2.a.bg ✓ 6 19.b odd 2 1
8664.2.a.bj yes 6 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8664))\):

\( T_{5}^{6} - T_{5}^{5} - 24T_{5}^{4} + 19T_{5}^{3} + 116T_{5}^{2} - 45T_{5} - 25 \) Copy content Toggle raw display
\( T_{7}^{6} - 2T_{7}^{5} - 19T_{7}^{4} + 4T_{7}^{3} + 71T_{7}^{2} + 40T_{7} - 16 \) Copy content Toggle raw display
\( T_{13}^{6} - 4T_{13}^{5} - 35T_{13}^{4} + 60T_{13}^{3} + 405T_{13}^{2} + 416T_{13} + 76 \) Copy content Toggle raw display
\( T_{29}^{6} + 2T_{29}^{5} - 85T_{29}^{4} + 1835T_{29}^{2} - 1598T_{29} - 6676 \) 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
$5$ \( T^{6} - T^{5} + \cdots - 25 \) Copy content Toggle raw display
$7$ \( T^{6} - 2 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{6} - 3 T^{5} + \cdots - 16 \) Copy content Toggle raw display
$13$ \( T^{6} - 4 T^{5} + \cdots + 76 \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} + \cdots - 100 \) Copy content Toggle raw display
$19$ \( T^{6} \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( T^{6} + 2 T^{5} + \cdots - 6676 \) Copy content Toggle raw display
$31$ \( T^{6} + 11 T^{5} + \cdots - 1024 \) Copy content Toggle raw display
$37$ \( T^{6} - 5 T^{5} + \cdots - 496 \) Copy content Toggle raw display
$41$ \( T^{6} - 20 T^{5} + \cdots - 716 \) Copy content Toggle raw display
$43$ \( T^{6} - 20 T^{5} + \cdots - 3136 \) Copy content Toggle raw display
$47$ \( T^{6} - 11 T^{5} + \cdots - 47296 \) Copy content Toggle raw display
$53$ \( T^{6} + 4 T^{5} + \cdots + 35081 \) Copy content Toggle raw display
$59$ \( T^{6} - 9 T^{5} + \cdots - 29504 \) Copy content Toggle raw display
$61$ \( T^{6} + 27 T^{5} + \cdots - 179600 \) Copy content Toggle raw display
$67$ \( T^{6} + 33 T^{5} + \cdots - 102656 \) Copy content Toggle raw display
$71$ \( T^{6} + 5 T^{5} + \cdots + 8000 \) Copy content Toggle raw display
$73$ \( T^{6} - 124 T^{4} + \cdots - 589 \) Copy content Toggle raw display
$79$ \( T^{6} - 29 T^{5} + \cdots + 1280320 \) Copy content Toggle raw display
$83$ \( T^{6} - 11 T^{5} + \cdots + 67520 \) Copy content Toggle raw display
$89$ \( T^{6} - 11 T^{5} + \cdots - 72256 \) Copy content Toggle raw display
$97$ \( T^{6} - 11 T^{5} + \cdots + 147376 \) Copy content Toggle raw display
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