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Properties

Label 6642.2.a.bu
Level $6642$
Weight $2$
Character orbit 6642.a
Self dual yes
Analytic conductor $53.037$
Analytic rank $0$
Dimension $7$
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] = [6642,2,Mod(1,6642)] 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("6642.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(6642, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 6642 = 2 \cdot 3^{4} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6642.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,-7,0,7,5,0,1] 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: \(53.0366370225\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 15x^{5} + 11x^{4} + 58x^{3} - 12x^{2} - 60x - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 738)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + ( - \beta_{6} + 1) q^{5} - \beta_{3} q^{7} - q^{8} + (\beta_{6} - 1) q^{10} + (\beta_{2} - \beta_1 + 2) q^{11} + ( - \beta_{6} + \beta_{3} - \beta_1) q^{13} + \beta_{3} q^{14} + q^{16}+ \cdots + (\beta_{6} + \beta_{5} - \beta_{4} + \cdots + 3) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} + 7 q^{4} + 5 q^{5} + q^{7} - 7 q^{8} - 5 q^{10} + 10 q^{11} - 4 q^{13} - q^{14} + 7 q^{16} + 13 q^{17} - q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 14 q^{25} + 4 q^{26} + q^{28} + 30 q^{29}+ \cdots + 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 15x^{5} + 11x^{4} + 58x^{3} - 12x^{2} - 60x - 6 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 3\nu^{6} - 7\nu^{5} - 36\nu^{4} + 80\nu^{3} + 70\nu^{2} - 120\nu - 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -9\nu^{6} + 22\nu^{5} + 105\nu^{4} - 252\nu^{3} - 178\nu^{2} + 382\nu + 22 ) / 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{6} - 12\nu^{5} - 59\nu^{4} + 138\nu^{3} + 106\nu^{2} - 212\nu - 20 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{6} - 12\nu^{5} - 59\nu^{4} + 138\nu^{3} + 108\nu^{2} - 214\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\nu^{6} - 44\nu^{5} - 225\nu^{4} + 504\nu^{3} + 414\nu^{2} - 762\nu - 90 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - \beta_{4} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} + 2\beta_{3} - 2\beta_{2} + 7\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + 10\beta_{5} - 11\beta_{4} - \beta_{3} - 3\beta_{2} + 10\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{6} - 2\beta_{5} + 35\beta_{4} + 25\beta_{3} - 27\beta_{2} + 60\beta _1 - 20 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{6} + 92\beta_{5} - 107\beta_{4} - 7\beta_{3} - 45\beta_{2} + 90\beta _1 + 326 \) 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
−1.34981
1.29672
−3.10593
3.08538
−0.103166
−1.36443
2.54123
−1.00000 0 1.00000 −3.62831 0 0.660484 −1.00000 0 3.62831
1.2 −1.00000 0 1.00000 −1.45288 0 −0.831424 −1.00000 0 1.45288
1.3 −1.00000 0 1.00000 −0.411382 0 −0.435560 −1.00000 0 0.411382
1.4 −1.00000 0 1.00000 0.571825 0 −1.74317 −1.00000 0 −0.571825
1.5 −1.00000 0 1.00000 2.88980 0 4.75394 −1.00000 0 −2.88980
1.6 −1.00000 0 1.00000 3.23241 0 −2.83031 −1.00000 0 −3.23241
1.7 −1.00000 0 1.00000 3.79853 0 1.42604 −1.00000 0 −3.79853
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(41\) \( +1 \)

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 6642.2.a.bu 7
3.b odd 2 1 6642.2.a.bv 7
9.c even 3 2 2214.2.e.j 14
9.d odd 6 2 738.2.e.j ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
738.2.e.j ✓ 14 9.d odd 6 2
2214.2.e.j 14 9.c even 3 2
6642.2.a.bu 7 1.a even 1 1 trivial
6642.2.a.bv 7 3.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6642))\):

\( T_{5}^{7} - 5T_{5}^{6} - 12T_{5}^{5} + 81T_{5}^{4} - 18T_{5}^{3} - 204T_{5}^{2} + 32T_{5} + 44 \) Copy content Toggle raw display
\( T_{7}^{7} - T_{7}^{6} - 18T_{7}^{5} - 9T_{7}^{4} + 42T_{7}^{3} + 24T_{7}^{2} - 16T_{7} - 8 \) Copy content Toggle raw display
\( T_{11}^{7} - 10T_{11}^{6} + 15T_{11}^{5} + 139T_{11}^{4} - 538T_{11}^{3} + 336T_{11}^{2} + 940T_{11} - 1108 \) Copy content Toggle raw display
\( T_{17}^{7} - 13T_{17}^{6} - 6T_{17}^{5} + 559T_{17}^{4} - 1114T_{17}^{3} - 3828T_{17}^{2} + 3760T_{17} + 7688 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 5 T^{6} + \cdots + 44 \) Copy content Toggle raw display
$7$ \( T^{7} - T^{6} - 18 T^{5} + \cdots - 8 \) Copy content Toggle raw display
$11$ \( T^{7} - 10 T^{6} + \cdots - 1108 \) Copy content Toggle raw display
$13$ \( T^{7} + 4 T^{6} + \cdots + 899 \) Copy content Toggle raw display
$17$ \( T^{7} - 13 T^{6} + \cdots + 7688 \) Copy content Toggle raw display
$19$ \( T^{7} + T^{6} + \cdots - 4335 \) Copy content Toggle raw display
$23$ \( T^{7} - 15 T^{6} + \cdots - 143517 \) Copy content Toggle raw display
$29$ \( T^{7} - 30 T^{6} + \cdots + 2025 \) Copy content Toggle raw display
$31$ \( T^{7} - 13 T^{6} + \cdots + 42215 \) Copy content Toggle raw display
$37$ \( T^{7} - 7 T^{6} + \cdots + 147880 \) Copy content Toggle raw display
$41$ \( (T + 1)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} + 5 T^{6} + \cdots + 1044 \) Copy content Toggle raw display
$47$ \( T^{7} - 3 T^{6} + \cdots - 28164 \) Copy content Toggle raw display
$53$ \( T^{7} - 12 T^{6} + \cdots + 311796 \) Copy content Toggle raw display
$59$ \( T^{7} - 13 T^{6} + \cdots + 393544 \) Copy content Toggle raw display
$61$ \( T^{7} + 55 T^{6} + \cdots + 99908 \) Copy content Toggle raw display
$67$ \( T^{7} + 15 T^{6} + \cdots - 9261 \) Copy content Toggle raw display
$71$ \( T^{7} - 12 T^{6} + \cdots - 325368 \) Copy content Toggle raw display
$73$ \( T^{7} - 270 T^{5} + \cdots + 93192 \) Copy content Toggle raw display
$79$ \( T^{7} - 12 T^{6} + \cdots + 42684 \) Copy content Toggle raw display
$83$ \( T^{7} - 36 T^{6} + \cdots + 152856 \) Copy content Toggle raw display
$89$ \( T^{7} - 16 T^{6} + \cdots + 9980 \) Copy content Toggle raw display
$97$ \( T^{7} - T^{6} + \cdots - 15141444 \) Copy content Toggle raw display
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