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Properties

Label 9520.2.a.ca
Level $9520$
Weight $2$
Character orbit 9520.a
Self dual yes
Analytic conductor $76.018$
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] = [9520,2,Mod(1,9520)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(9520, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 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("9520.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 9520 = 2^{4} \cdot 5 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9520.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,3,0,7,0,7,0,16,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(76.0175827243\)
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: \( x^{7} - 17x^{5} - 8x^{4} + 38x^{3} + 21x^{2} - 17x - 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4760)
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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + q^{5} + q^{7} + (\beta_1 + 2) q^{9} + ( - \beta_{2} - 1) q^{11} - \beta_{4} q^{13} - \beta_{5} q^{15} + q^{17} + ( - \beta_{2} - 1) q^{19} - \beta_{5} q^{21} + ( - \beta_{3} + 2) q^{23}+ \cdots + (\beta_{6} + \beta_{4} - 2 \beta_{2} - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 3 q^{3} + 7 q^{5} + 7 q^{7} + 16 q^{9} - 4 q^{11} + q^{13} + 3 q^{15} + 7 q^{17} - 4 q^{19} + 3 q^{21} + 14 q^{23} + 7 q^{25} - 3 q^{27} + 12 q^{29} - 6 q^{31} + 8 q^{33} + 7 q^{35} + 10 q^{37} + 11 q^{39}+ \cdots - 29 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 17x^{5} - 8x^{4} + 38x^{3} + 21x^{2} - 17x - 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{6} - \nu^{5} - 16\nu^{4} + 8\nu^{3} + 30\nu^{2} - 10\nu - 9 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{6} - 6\nu^{5} - 116\nu^{4} + 42\nu^{3} + 265\nu^{2} - 48\nu - 125 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{6} - 4\nu^{5} - 134\nu^{4} + 3\nu^{3} + 300\nu^{2} + 13\nu - 120 ) / 5 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -13\nu^{6} + 9\nu^{5} + 214\nu^{4} - 43\nu^{3} - 455\nu^{2} + 32\nu + 190 ) / 5 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\nu^{6} - 17\nu^{5} - 397\nu^{4} + 89\nu^{3} + 865\nu^{2} - 96\nu - 360 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -34\nu^{6} + 22\nu^{5} + 562\nu^{4} - 94\nu^{3} - 1205\nu^{2} + 111\nu + 495 ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + 2\beta_{5} - \beta_{4} - \beta_{3} - 2\beta_{2} - \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{6} - 2\beta_{5} - 3\beta_{4} - 3\beta_{3} + 2\beta_{2} - 3\beta _1 + 20 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 7\beta_{6} + 22\beta_{5} - 7\beta_{4} - 19\beta_{3} - 22\beta_{2} - 15\beta _1 + 16 ) / 4 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{6} - 22\beta_{5} - 51\beta_{4} - 55\beta_{3} + 14\beta_{2} - 47\beta _1 + 248 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 79\beta_{6} + 286\beta_{5} - 115\beta_{4} - 299\beta_{3} - 298\beta_{2} - 239\beta _1 + 364 ) / 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -145\beta_{6} - 162\beta_{5} - 795\beta_{4} - 947\beta_{3} + 22\beta_{2} - 787\beta _1 + 3640 ) / 4 \) 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
−0.709116
1.40011
−1.37628
4.05169
0.783603
−3.48942
−0.660583
0 −3.40363 0 1.00000 0 1.00000 0 8.58471 0
1.2 0 −1.85289 0 1.00000 0 1.00000 0 0.433187 0
1.3 0 −0.247178 0 1.00000 0 1.00000 0 −2.93890 0
1.4 0 0.586698 0 1.00000 0 1.00000 0 −2.65579 0
1.5 0 2.08272 0 1.00000 0 1.00000 0 1.33772 0
1.6 0 2.58574 0 1.00000 0 1.00000 0 3.68603 0
1.7 0 3.24854 0 1.00000 0 1.00000 0 7.55304 0
\(n\): 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 9520.2.a.ca 7
4.b odd 2 1 4760.2.a.p ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4760.2.a.p ✓ 7 4.b odd 2 1
9520.2.a.ca 7 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(9520))\):

\( T_{3}^{7} - 3T_{3}^{6} - 14T_{3}^{5} + 46T_{3}^{4} + 26T_{3}^{3} - 129T_{3}^{2} + 32T_{3} + 16 \) Copy content Toggle raw display
\( T_{11}^{7} + 4T_{11}^{6} - 35T_{11}^{5} - 69T_{11}^{4} + 366T_{11}^{3} + 9T_{11}^{2} - 348T_{11} + 32 \) Copy content Toggle raw display
\( T_{13}^{7} - T_{13}^{6} - 60T_{13}^{5} + 49T_{13}^{4} + 836T_{13}^{3} - 404T_{13}^{2} - 688T_{13} + 320 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( T^{7} - 3 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( (T - 1)^{7} \) Copy content Toggle raw display
$7$ \( (T - 1)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} + 4 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} + \cdots + 320 \) Copy content Toggle raw display
$17$ \( (T - 1)^{7} \) Copy content Toggle raw display
$19$ \( T^{7} + 4 T^{6} + \cdots + 32 \) Copy content Toggle raw display
$23$ \( T^{7} - 14 T^{6} + \cdots + 1600 \) Copy content Toggle raw display
$29$ \( T^{7} - 12 T^{6} + \cdots - 460 \) Copy content Toggle raw display
$31$ \( T^{7} + 6 T^{6} + \cdots - 640 \) Copy content Toggle raw display
$37$ \( T^{7} - 10 T^{6} + \cdots + 33392 \) Copy content Toggle raw display
$41$ \( T^{7} - 12 T^{6} + \cdots - 5228 \) Copy content Toggle raw display
$43$ \( T^{7} + 10 T^{6} + \cdots + 65840 \) Copy content Toggle raw display
$47$ \( T^{7} - 6 T^{6} + \cdots - 1600 \) Copy content Toggle raw display
$53$ \( T^{7} - 12 T^{6} + \cdots + 47356 \) Copy content Toggle raw display
$59$ \( T^{7} + 10 T^{6} + \cdots + 354976 \) Copy content Toggle raw display
$61$ \( T^{7} - 25 T^{6} + \cdots + 2000 \) Copy content Toggle raw display
$67$ \( T^{7} - 8 T^{6} + \cdots - 2224 \) Copy content Toggle raw display
$71$ \( T^{7} + 3 T^{6} + \cdots - 6400 \) Copy content Toggle raw display
$73$ \( T^{7} - 3 T^{6} + \cdots + 16 \) Copy content Toggle raw display
$79$ \( T^{7} - 4 T^{6} + \cdots - 1024 \) Copy content Toggle raw display
$83$ \( T^{7} - 4 T^{6} + \cdots - 14720 \) Copy content Toggle raw display
$89$ \( T^{7} + 3 T^{6} + \cdots - 505600 \) Copy content Toggle raw display
$97$ \( T^{7} - 32 T^{6} + \cdots + 195632 \) Copy content Toggle raw display
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