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Properties

Label 7865.2.a.z
Level $7865$
Weight $2$
Character orbit 7865.a
Self dual yes
Analytic conductor $62.802$
Analytic rank $0$
Dimension $9$
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] = [7865,2,Mod(1,7865)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(7865, 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("7865.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 7865 = 5 \cdot 11^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7865.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [9,1,-2,5,9,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(62.8023411897\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 11x^{7} + 7x^{6} + 39x^{5} - 6x^{4} - 50x^{3} - 14x^{2} + 8x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{5} q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + (\beta_{7} + \beta_{4} + \beta_1) q^{6} + (\beta_{8} - \beta_1 + 1) q^{7} + (\beta_{3} + 1) q^{8} + (\beta_{8} - \beta_{6} - \beta_{4} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 7) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} - 2 q^{3} + 5 q^{4} + 9 q^{5} + q^{6} + 10 q^{7} + 9 q^{8} + 11 q^{9} + q^{10} + 8 q^{12} - 9 q^{13} - 16 q^{14} - 2 q^{15} + q^{16} + 3 q^{17} + 15 q^{18} + 13 q^{19} + 5 q^{20} + 2 q^{21}+ \cdots - 48 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 11x^{7} + 7x^{6} + 39x^{5} - 6x^{4} - 50x^{3} - 14x^{2} + 8x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 6\nu^{3} + 14\nu^{2} - 4\nu - 3 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{8} + 2\nu^{7} + 9\nu^{6} - 15\nu^{5} - 24\nu^{4} + 22\nu^{3} + 27\nu^{2} + \nu - 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 23\nu^{4} - 29\nu^{3} - 21\nu^{2} + 8\nu + 1 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{8} - 2\nu^{7} - 9\nu^{6} + 16\nu^{5} + 24\nu^{4} - 29\nu^{3} - 27\nu^{2} + 7\nu + 5 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( \nu^{8} - \nu^{7} - 11\nu^{6} + 8\nu^{5} + 38\nu^{4} - 14\nu^{3} - 44\nu^{2} + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + 6\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + \beta_{5} + 7\beta_{3} + 20\beta _1 + 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} - 8\beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 34\beta_{2} + 8\beta _1 + 73 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} + 11\beta_{7} - 2\beta_{6} + 10\beta_{5} + 2\beta_{4} + 43\beta_{3} + \beta_{2} + 109\beta _1 + 34 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2\beta_{8} + 64\beta_{7} - 52\beta_{6} + 13\beta_{5} + 13\beta_{4} + 12\beta_{3} + 191\beta_{2} + 55\beta _1 + 398 \) 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
−2.34372
−1.32130
−1.20260
−0.643977
−0.228262
0.400677
1.70614
2.16813
2.46492
−2.34372 −0.308136 3.49302 1.00000 0.722185 3.44492 −3.49923 −2.90505 −2.34372
1.2 −1.32130 2.86458 −0.254156 1.00000 −3.78499 4.19083 2.97842 5.20584 −1.32130
1.3 −1.20260 −1.31130 −0.553749 1.00000 1.57697 2.01380 3.07114 −1.28049 −1.20260
1.4 −0.643977 −1.26807 −1.58529 1.00000 0.816608 −2.92401 2.30885 −1.39200 −0.643977
1.5 −0.228262 −3.13778 −1.94790 1.00000 0.716235 4.19891 0.901153 6.84566 −0.228262
1.6 0.400677 1.41688 −1.83946 1.00000 0.567710 −1.34956 −1.53838 −0.992452 0.400677
1.7 1.70614 −0.297397 0.910902 1.00000 −0.507400 2.72512 −1.85815 −2.91155 1.70614
1.8 2.16813 −2.66539 2.70078 1.00000 −5.77891 −2.25474 1.51937 4.10432 2.16813
1.9 2.46492 2.70661 4.07585 1.00000 6.67159 −0.0452711 5.11681 4.32574 2.46492
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( -1 \)
\(11\) \( +1 \)
\(13\) \( +1 \)

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 7865.2.a.z yes 9
11.b odd 2 1 7865.2.a.v ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7865.2.a.v ✓ 9 11.b odd 2 1
7865.2.a.z yes 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7865))\):

\( T_{2}^{9} - T_{2}^{8} - 11T_{2}^{7} + 7T_{2}^{6} + 39T_{2}^{5} - 6T_{2}^{4} - 50T_{2}^{3} - 14T_{2}^{2} + 8T_{2} + 2 \) Copy content Toggle raw display
\( T_{3}^{9} + 2T_{3}^{8} - 17T_{3}^{7} - 34T_{3}^{6} + 79T_{3}^{5} + 172T_{3}^{4} - 46T_{3}^{3} - 218T_{3}^{2} - 104T_{3} - 14 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} - T^{8} - 11 T^{7} + \cdots + 2 \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots - 14 \) Copy content Toggle raw display
$5$ \( (T - 1)^{9} \) Copy content Toggle raw display
$7$ \( T^{9} - 10 T^{8} + \cdots - 134 \) Copy content Toggle raw display
$11$ \( T^{9} \) Copy content Toggle raw display
$13$ \( (T + 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} - 3 T^{8} + \cdots - 9472 \) Copy content Toggle raw display
$19$ \( T^{9} - 13 T^{8} + \cdots + 2773 \) Copy content Toggle raw display
$23$ \( T^{9} + 10 T^{8} + \cdots - 9440 \) Copy content Toggle raw display
$29$ \( T^{9} - 16 T^{8} + \cdots - 58016 \) Copy content Toggle raw display
$31$ \( T^{9} - 8 T^{8} + \cdots - 304000 \) Copy content Toggle raw display
$37$ \( T^{9} - T^{8} + \cdots - 11920 \) Copy content Toggle raw display
$41$ \( T^{9} - 15 T^{8} + \cdots - 13231381 \) Copy content Toggle raw display
$43$ \( T^{9} - 13 T^{8} + \cdots - 115472 \) Copy content Toggle raw display
$47$ \( T^{9} - T^{8} + \cdots - 3248 \) Copy content Toggle raw display
$53$ \( T^{9} - 14 T^{8} + \cdots + 8330240 \) Copy content Toggle raw display
$59$ \( T^{9} - 10 T^{8} + \cdots + 2383360 \) Copy content Toggle raw display
$61$ \( T^{9} - 16 T^{8} + \cdots + 116000 \) Copy content Toggle raw display
$67$ \( T^{9} - 3 T^{8} + \cdots + 153040 \) Copy content Toggle raw display
$71$ \( T^{9} + 10 T^{8} + \cdots + 2595040 \) Copy content Toggle raw display
$73$ \( T^{9} - 32 T^{8} + \cdots - 15761170 \) Copy content Toggle raw display
$79$ \( T^{9} - 28 T^{8} + \cdots - 100519040 \) Copy content Toggle raw display
$83$ \( T^{9} - 22 T^{8} + \cdots + 23888776 \) Copy content Toggle raw display
$89$ \( T^{9} + 8 T^{8} + \cdots + 5920 \) Copy content Toggle raw display
$97$ \( T^{9} - 11 T^{8} + \cdots - 5098240 \) Copy content Toggle raw display
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