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Properties

Label 7935.2.a.bk
Level $7935$
Weight $2$
Character orbit 7935.a
Self dual yes
Analytic conductor $63.361$
Analytic rank $0$
Dimension $10$
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] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7935, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [10,0,-10,16,10,0,6] 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: \(63.3612940039\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 18x^{8} + 111x^{6} - 4x^{5} - 270x^{4} + 32x^{3} + 218x^{2} - 60x - 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - q^{3} + (\beta_{2} + 2) q^{4} + q^{5} - \beta_1 q^{6} + ( - \beta_{3} + \beta_{2} + 1) q^{7} + (\beta_{4} + \beta_{3} + 2 \beta_1) q^{8} + q^{9} + \beta_1 q^{10} - \beta_{9} q^{11}+ \cdots - \beta_{9} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 10 q^{3} + 16 q^{4} + 10 q^{5} + 6 q^{7} + 10 q^{9} - 4 q^{11} - 16 q^{12} - 4 q^{13} - 10 q^{15} + 28 q^{16} + 10 q^{17} + 8 q^{19} + 16 q^{20} - 6 q^{21} - 4 q^{22} + 10 q^{25} - 8 q^{26} - 10 q^{27}+ \cdots - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 18x^{8} + 111x^{6} - 4x^{5} - 270x^{4} + 32x^{3} + 218x^{2} - 60x - 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{9} + 12\nu^{8} + 69\nu^{7} - 150\nu^{6} - 312\nu^{5} + 566\nu^{4} + 522\nu^{3} - 664\nu^{2} - 214\nu + 96 ) / 78 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 5\nu^{9} - 12\nu^{8} - 69\nu^{7} + 150\nu^{6} + 312\nu^{5} - 566\nu^{4} - 444\nu^{3} + 664\nu^{2} - 254\nu - 96 ) / 78 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{9} + 18 \nu^{8} - 33 \nu^{7} - 186 \nu^{6} + 468 \nu^{5} + 472 \nu^{4} - 1596 \nu^{3} + \cdots - 246 ) / 78 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{9} + 7\nu^{8} + 50\nu^{7} - 81\nu^{6} - 182\nu^{5} + 276\nu^{4} + 168\nu^{3} - 292\nu^{2} + 68\nu + 30 ) / 26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{9} + 3\nu^{8} - 38\nu^{7} - 44\nu^{6} + 234\nu^{5} + 200\nu^{4} - 513\nu^{3} - 283\nu^{2} + 291\nu + 37 ) / 13 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} - 2\nu^{8} - 44\nu^{7} + 12\nu^{6} + 221\nu^{5} + 27\nu^{4} - 451\nu^{3} - 145\nu^{2} + 287\nu + 36 ) / 13 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2\nu^{9} + 3\nu^{8} - 38\nu^{7} - 44\nu^{6} + 234\nu^{5} + 213\nu^{4} - 513\nu^{3} - 374\nu^{2} + 291\nu + 102 ) / 13 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - \beta_{7} + 7\beta_{2} + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{5} + 9\beta_{4} + 10\beta_{3} + 38\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 10\beta_{9} - 11\beta_{7} - 2\beta_{6} + 2\beta_{5} + \beta_{4} + 3\beta_{3} + 47\beta_{2} + \beta _1 + 147 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{9} + 12 \beta_{8} - 14 \beta_{7} - 4 \beta_{6} + 13 \beta_{5} + 68 \beta_{4} + 87 \beta_{3} + \cdots + 29 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 80 \beta_{9} + 2 \beta_{8} - 95 \beta_{7} - 32 \beta_{6} + 30 \beta_{5} + 17 \beta_{4} + 59 \beta_{3} + \cdots + 985 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 19 \beta_{9} + 108 \beta_{8} - 142 \beta_{7} - 72 \beta_{6} + 129 \beta_{5} + 492 \beta_{4} + 717 \beta_{3} + \cdots + 321 \) 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
−2.57634
−2.55225
−1.80133
−1.32647
−0.184637
0.580737
0.899827
1.77882
2.39815
2.78349
−2.57634 −1.00000 4.63755 1.00000 2.57634 2.22333 −6.79523 1.00000 −2.57634
1.2 −2.55225 −1.00000 4.51396 1.00000 2.55225 4.92817 −6.41625 1.00000 −2.55225
1.3 −1.80133 −1.00000 1.24478 1.00000 1.80133 −1.16943 1.36039 1.00000 −1.80133
1.4 −1.32647 −1.00000 −0.240473 1.00000 1.32647 0.173740 2.97192 1.00000 −1.32647
1.5 −0.184637 −1.00000 −1.96591 1.00000 0.184637 −4.38012 0.732252 1.00000 −0.184637
1.6 0.580737 −1.00000 −1.66274 1.00000 −0.580737 −1.24853 −2.12709 1.00000 0.580737
1.7 0.899827 −1.00000 −1.19031 1.00000 −0.899827 −0.776098 −2.87073 1.00000 0.899827
1.8 1.77882 −1.00000 1.16421 1.00000 −1.77882 −1.25001 −1.48673 1.00000 1.77882
1.9 2.39815 −1.00000 3.75114 1.00000 −2.39815 4.16535 4.19950 1.00000 2.39815
1.10 2.78349 −1.00000 5.74780 1.00000 −2.78349 3.33359 10.4320 1.00000 2.78349
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(23\) \( +1 \)

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 7935.2.a.bk yes 10
23.b odd 2 1 7935.2.a.bj ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7935.2.a.bj ✓ 10 23.b odd 2 1
7935.2.a.bk yes 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{10} - 18T_{2}^{8} + 111T_{2}^{6} - 4T_{2}^{5} - 270T_{2}^{4} + 32T_{2}^{3} + 218T_{2}^{2} - 60T_{2} - 18 \) Copy content Toggle raw display
\( T_{7}^{10} - 6 T_{7}^{9} - 23 T_{7}^{8} + 160 T_{7}^{7} + 103 T_{7}^{6} - 962 T_{7}^{5} - 705 T_{7}^{4} + \cdots - 164 \) Copy content Toggle raw display
\( T_{11}^{10} + 4 T_{11}^{9} - 62 T_{11}^{8} - 248 T_{11}^{7} + 1050 T_{11}^{6} + 4476 T_{11}^{5} + \cdots + 1152 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 18 T^{8} + \cdots - 18 \) Copy content Toggle raw display
$3$ \( (T + 1)^{10} \) Copy content Toggle raw display
$5$ \( (T - 1)^{10} \) Copy content Toggle raw display
$7$ \( T^{10} - 6 T^{9} + \cdots - 164 \) Copy content Toggle raw display
$11$ \( T^{10} + 4 T^{9} + \cdots + 1152 \) Copy content Toggle raw display
$13$ \( T^{10} + 4 T^{9} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( T^{10} - 10 T^{9} + \cdots - 157968 \) Copy content Toggle raw display
$19$ \( T^{10} - 8 T^{9} + \cdots + 2949704 \) Copy content Toggle raw display
$23$ \( T^{10} \) Copy content Toggle raw display
$29$ \( T^{10} - 2 T^{9} + \cdots + 51662736 \) Copy content Toggle raw display
$31$ \( T^{10} - 2 T^{9} + \cdots + 2101392 \) Copy content Toggle raw display
$37$ \( T^{10} - 2 T^{9} + \cdots - 2689252 \) Copy content Toggle raw display
$41$ \( T^{10} - 6 T^{9} + \cdots - 222192 \) Copy content Toggle raw display
$43$ \( T^{10} - 44 T^{9} + \cdots + 916336 \) Copy content Toggle raw display
$47$ \( T^{10} - 8 T^{9} + \cdots + 5703552 \) Copy content Toggle raw display
$53$ \( T^{10} - 2 T^{9} + \cdots - 13895568 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 1336781376 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 160982152 \) Copy content Toggle raw display
$67$ \( T^{10} - 2 T^{9} + \cdots + 3231868 \) Copy content Toggle raw display
$71$ \( T^{10} - 6 T^{9} + \cdots - 19584 \) Copy content Toggle raw display
$73$ \( T^{10} + 12 T^{9} + \cdots + 19459584 \) Copy content Toggle raw display
$79$ \( T^{10} - 444 T^{8} + \cdots + 47492992 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots - 875635776 \) Copy content Toggle raw display
$89$ \( T^{10} - 56 T^{9} + \cdots - 30218112 \) Copy content Toggle raw display
$97$ \( T^{10} - 56 T^{9} + \cdots + 7857136 \) Copy content Toggle raw display
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