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Properties

Label 8925.2.a.co
Level $8925$
Weight $2$
Character orbit 8925.a
Self dual yes
Analytic conductor $71.266$
Analytic rank $1$
Dimension $8$
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] = [8925,2,Mod(1,8925)] 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("8925.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8925 = 3 \cdot 5^{2} \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8925.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-1,8,7,0,-1,-8,-3,8,0,-12,7,1,1,0,5,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(71.2664838040\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 35x^{4} - 24x^{3} - 37x^{2} + 17x + 10 \) 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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + q^{3} + (\beta_{2} + 1) q^{4} - \beta_1 q^{6} - q^{7} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots - 1) q^{8} + q^{9} + (\beta_{3} - \beta_{2} - 2) q^{11} + (\beta_{2} + 1) q^{12}+ \cdots + (\beta_{3} - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - q^{2} + 8 q^{3} + 7 q^{4} - q^{6} - 8 q^{7} - 3 q^{8} + 8 q^{9} - 12 q^{11} + 7 q^{12} + q^{13} + q^{14} + 5 q^{16} - 8 q^{17} - q^{18} - 17 q^{19} - 8 q^{21} + 5 q^{22} + 6 q^{23} - 3 q^{24}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} - 11x^{6} + 9x^{5} + 35x^{4} - 24x^{3} - 37x^{2} + 17x + 10 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{7} - 10\nu^{5} - 2\nu^{4} + 24\nu^{3} + 7\nu^{2} - 13\nu - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{7} + \nu^{6} - 10\nu^{5} - 11\nu^{4} + 22\nu^{3} + 23\nu^{2} - 9\nu - 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 2\nu^{7} + \nu^{6} - 20\nu^{5} - 13\nu^{4} + 47\nu^{3} + 29\nu^{2} - 27\nu - 13 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -3\nu^{7} - \nu^{6} + 31\nu^{5} + 14\nu^{4} - 79\nu^{3} - 30\nu^{2} + 51\nu + 13 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -3\nu^{7} - 2\nu^{6} + 31\nu^{5} + 24\nu^{4} - 78\nu^{3} - 54\nu^{2} + 51\nu + 26 \) 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
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} - 2\beta_{3} + 9\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{7} + 10\beta_{5} - 8\beta_{4} - 9\beta_{3} + 11\beta_{2} + 30\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9\beta_{7} - 9\beta_{6} + 11\beta_{5} - \beta_{4} - 21\beta_{3} + 67\beta_{2} + 15\beta _1 + 92 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 12\beta_{7} - 2\beta_{6} + 78\beta_{5} - 56\beta_{4} - 69\beta_{3} + 97\beta_{2} + 195\beta _1 + 101 \) 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
2.72652
1.52490
1.47540
0.914764
−0.384966
−1.10555
−1.72832
−2.42274
−2.72652 1.00000 5.43389 0 −2.72652 −1.00000 −9.36256 1.00000 0
1.2 −1.52490 1.00000 0.325311 0 −1.52490 −1.00000 2.55373 1.00000 0
1.3 −1.47540 1.00000 0.176805 0 −1.47540 −1.00000 2.68994 1.00000 0
1.4 −0.914764 1.00000 −1.16321 0 −0.914764 −1.00000 2.89359 1.00000 0
1.5 0.384966 1.00000 −1.85180 0 0.384966 −1.00000 −1.48281 1.00000 0
1.6 1.10555 1.00000 −0.777755 0 1.10555 −1.00000 −3.07095 1.00000 0
1.7 1.72832 1.00000 0.987097 0 1.72832 −1.00000 −1.75062 1.00000 0
1.8 2.42274 1.00000 3.86966 0 2.42274 −1.00000 4.52969 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)
\(17\) \( +1 \)

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 8925.2.a.co ✓ 8
5.b even 2 1 8925.2.a.cp yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8925.2.a.co ✓ 8 1.a even 1 1 trivial
8925.2.a.cp yes 8 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8925))\):

\( T_{2}^{8} + T_{2}^{7} - 11T_{2}^{6} - 9T_{2}^{5} + 35T_{2}^{4} + 24T_{2}^{3} - 37T_{2}^{2} - 17T_{2} + 10 \) Copy content Toggle raw display
\( T_{11}^{8} + 12T_{11}^{7} + 31T_{11}^{6} - 71T_{11}^{5} - 249T_{11}^{4} + 151T_{11}^{3} + 332T_{11}^{2} + 64T_{11} - 15 \) Copy content Toggle raw display
\( T_{13}^{8} - T_{13}^{7} - 52T_{13}^{6} + 59T_{13}^{5} + 709T_{13}^{4} - 911T_{13}^{3} - 2226T_{13}^{2} + 1684T_{13} + 2400 \) Copy content Toggle raw display
\( T_{23}^{8} - 6T_{23}^{7} - 74T_{23}^{6} + 235T_{23}^{5} + 1835T_{23}^{4} - 1207T_{23}^{3} - 11777T_{23}^{2} - 626T_{23} + 13059 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + T^{7} + \cdots + 10 \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T + 1)^{8} \) Copy content Toggle raw display
$11$ \( T^{8} + 12 T^{7} + \cdots - 15 \) Copy content Toggle raw display
$13$ \( T^{8} - T^{7} + \cdots + 2400 \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} + 17 T^{7} + \cdots - 640 \) Copy content Toggle raw display
$23$ \( T^{8} - 6 T^{7} + \cdots + 13059 \) Copy content Toggle raw display
$29$ \( T^{8} - 2 T^{7} + \cdots - 39186 \) Copy content Toggle raw display
$31$ \( T^{8} + 5 T^{7} + \cdots - 1828 \) Copy content Toggle raw display
$37$ \( T^{8} - 18 T^{7} + \cdots - 75656 \) Copy content Toggle raw display
$41$ \( T^{8} + 7 T^{7} + \cdots - 457296 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + \cdots - 2431584 \) Copy content Toggle raw display
$47$ \( T^{8} + 23 T^{7} + \cdots + 272268 \) Copy content Toggle raw display
$53$ \( T^{8} - 9 T^{7} + \cdots + 904 \) Copy content Toggle raw display
$59$ \( T^{8} + 13 T^{7} + \cdots - 1556 \) Copy content Toggle raw display
$61$ \( T^{8} + 6 T^{7} + \cdots + 297900 \) Copy content Toggle raw display
$67$ \( T^{8} - 8 T^{7} + \cdots + 103399 \) Copy content Toggle raw display
$71$ \( T^{8} + 27 T^{7} + \cdots + 551043 \) Copy content Toggle raw display
$73$ \( T^{8} + 2 T^{7} + \cdots + 1030076 \) Copy content Toggle raw display
$79$ \( T^{8} + 17 T^{7} + \cdots + 13951164 \) Copy content Toggle raw display
$83$ \( T^{8} + 31 T^{7} + \cdots - 182578356 \) Copy content Toggle raw display
$89$ \( T^{8} + 2 T^{7} + \cdots - 1488224 \) Copy content Toggle raw display
$97$ \( T^{8} - 10 T^{7} + \cdots + 149400 \) Copy content Toggle raw display
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