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Properties

Label 8330.2.a.cr
Level $8330$
Weight $2$
Character orbit 8330.a
Self dual yes
Analytic conductor $66.515$
Analytic rank $1$
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] = [8330,2,Mod(1,8330)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8330, 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("8330.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 8330 = 2 \cdot 5 \cdot 7^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8330.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,7,-1,7,-7,-1,0,7,8,-7,0,-1,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.5153848837\)
Analytic rank: \(1\)
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: \( x^{7} - x^{6} - 14x^{5} + 6x^{4} + 55x^{3} + 7x^{2} - 36x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1190)
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.

\(f(q)\) \(=\) \( q + q^{2} - \beta_1 q^{3} + q^{4} - q^{5} - \beta_1 q^{6} + q^{8} + (\beta_{2} + \beta_1 + 1) q^{9} - q^{10} - \beta_{6} q^{11} - \beta_1 q^{12} + ( - \beta_{3} - 1) q^{13} + \beta_1 q^{15} + q^{16}+ \cdots + (\beta_{6} + 2 \beta_{5} + 2 \beta_{4} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} - q^{3} + 7 q^{4} - 7 q^{5} - q^{6} + 7 q^{8} + 8 q^{9} - 7 q^{10} - q^{12} - 6 q^{13} + q^{15} + 7 q^{16} - 7 q^{17} + 8 q^{18} - 10 q^{19} - 7 q^{20} - q^{23} - q^{24} + 7 q^{25} - 6 q^{26}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 14x^{5} + 6x^{4} + 55x^{3} + 7x^{2} - 36x - 14 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 16\nu^{3} + 41\nu^{2} - 20\nu - 22 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} - 2\nu^{5} - 12\nu^{4} + 18\nu^{3} + 39\nu^{2} - 32\nu - 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{6} - 3\nu^{5} - 26\nu^{4} + 25\nu^{3} + 93\nu^{2} - 35\nu - 48 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 5\nu^{6} - 8\nu^{5} - 66\nu^{4} + 70\nu^{3} + 241\nu^{2} - 110\nu - 128 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} - \beta_{3} + \beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} + 9\beta_{2} + 11\beta _1 + 27 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{6} + 3\beta_{5} + 11\beta_{4} - 13\beta_{3} + 14\beta_{2} + 55\beta _1 + 28 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} + 18\beta_{5} + 30\beta_{4} - 20\beta_{3} + 79\beta_{2} + 109\beta _1 + 206 \) 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
3.12679
2.65487
0.957805
−0.556702
−0.613936
−2.02610
−2.54274
1.00000 −3.12679 1.00000 −1.00000 −3.12679 0 1.00000 6.77683 −1.00000
1.2 1.00000 −2.65487 1.00000 −1.00000 −2.65487 0 1.00000 4.04836 −1.00000
1.3 1.00000 −0.957805 1.00000 −1.00000 −0.957805 0 1.00000 −2.08261 −1.00000
1.4 1.00000 0.556702 1.00000 −1.00000 0.556702 0 1.00000 −2.69008 −1.00000
1.5 1.00000 0.613936 1.00000 −1.00000 0.613936 0 1.00000 −2.62308 −1.00000
1.6 1.00000 2.02610 1.00000 −1.00000 2.02610 0 1.00000 1.10507 −1.00000
1.7 1.00000 2.54274 1.00000 −1.00000 2.54274 0 1.00000 3.46552 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(5\) \( +1 \)
\(7\) \( -1 \)
\(17\) \( +1 \)

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 8330.2.a.cr 7
7.b odd 2 1 8330.2.a.cs 7
7.d odd 6 2 1190.2.i.o ✓ 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1190.2.i.o ✓ 14 7.d odd 6 2
8330.2.a.cr 7 1.a even 1 1 trivial
8330.2.a.cs 7 7.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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8330))\):

\( T_{3}^{7} + T_{3}^{6} - 14T_{3}^{5} - 6T_{3}^{4} + 55T_{3}^{3} - 7T_{3}^{2} - 36T_{3} + 14 \) Copy content Toggle raw display
\( T_{11}^{7} - 41T_{11}^{5} - 28T_{11}^{4} + 467T_{11}^{3} + 598T_{11}^{2} - 911T_{11} - 982 \) Copy content Toggle raw display
\( T_{13}^{7} + 6T_{13}^{6} - 23T_{13}^{5} - 116T_{13}^{4} - 21T_{13}^{3} + 202T_{13}^{2} + 131T_{13} + 4 \) Copy content Toggle raw display
\( T_{19}^{7} + 10T_{19}^{6} - 55T_{19}^{5} - 806T_{19}^{4} - 1536T_{19}^{3} + 4240T_{19}^{2} + 9280T_{19} + 3392 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{7} \) Copy content Toggle raw display
$3$ \( T^{7} + T^{6} + \cdots + 14 \) Copy content Toggle raw display
$5$ \( (T + 1)^{7} \) Copy content Toggle raw display
$7$ \( T^{7} \) Copy content Toggle raw display
$11$ \( T^{7} - 41 T^{5} + \cdots - 982 \) Copy content Toggle raw display
$13$ \( T^{7} + 6 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$17$ \( (T + 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} + 10 T^{6} + \cdots + 3392 \) Copy content Toggle raw display
$23$ \( T^{7} + T^{6} + \cdots + 92 \) Copy content Toggle raw display
$29$ \( T^{7} + 9 T^{6} + \cdots - 153464 \) Copy content Toggle raw display
$31$ \( T^{7} + 18 T^{6} + \cdots - 784 \) Copy content Toggle raw display
$37$ \( T^{7} - 8 T^{6} + \cdots + 1544 \) Copy content Toggle raw display
$41$ \( T^{7} + 15 T^{6} + \cdots + 2192 \) Copy content Toggle raw display
$43$ \( T^{7} + 9 T^{6} + \cdots - 223024 \) Copy content Toggle raw display
$47$ \( T^{7} + 10 T^{6} + \cdots - 74432 \) Copy content Toggle raw display
$53$ \( T^{7} - 10 T^{6} + \cdots - 190768 \) Copy content Toggle raw display
$59$ \( T^{7} - 6 T^{6} + \cdots + 25216 \) Copy content Toggle raw display
$61$ \( T^{7} + 23 T^{6} + \cdots + 299488 \) Copy content Toggle raw display
$67$ \( T^{7} - 13 T^{6} + \cdots - 317952 \) Copy content Toggle raw display
$71$ \( T^{7} + 20 T^{6} + \cdots - 75312 \) Copy content Toggle raw display
$73$ \( T^{7} + 22 T^{6} + \cdots + 156064 \) Copy content Toggle raw display
$79$ \( T^{7} + 8 T^{6} + \cdots + 617724 \) Copy content Toggle raw display
$83$ \( T^{7} + 3 T^{6} + \cdots - 502456 \) Copy content Toggle raw display
$89$ \( T^{7} + 21 T^{6} + \cdots - 3905432 \) Copy content Toggle raw display
$97$ \( T^{7} + 28 T^{6} + \cdots + 216576 \) Copy content Toggle raw display
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