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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [8325,2,Mod(1,8325)] 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("8325.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(8325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 8325 = 3^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8325.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,1,0,7,0,0,0,0,0,0,-16,0,1] 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: \(66.4754596827\)
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: \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 925)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 1) q^{4} + \beta_{6} q^{7} + (\beta_{5} + \beta_{4} + \beta_{2} + \beta_1) q^{8} + (\beta_{5} - 2) q^{11} + (\beta_{5} - \beta_{3} - \beta_{2} + \cdots + 1) q^{13} + ( - \beta_{5} + \beta_{4}) q^{14}+ \cdots + (3 \beta_{6} - 2 \beta_{5} + \cdots - 3 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + q^{2} + 7 q^{4} - 16 q^{11} + q^{13} + 3 q^{14} + 3 q^{16} + 4 q^{17} + 9 q^{19} - q^{22} - q^{23} - 24 q^{26} - 7 q^{28} - 3 q^{29} + 11 q^{31} + 21 q^{32} - 23 q^{34} - 7 q^{37} - 32 q^{38}+ \cdots + 6 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - x^{6} - 10x^{5} + 9x^{4} + 26x^{3} - 23x^{2} - 9x + 5 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{4} - \nu^{3} - 6\nu^{2} + 4\nu + 4 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 5\nu^{3} + 9\nu^{2} - 1 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{5} + 2\nu^{4} + 6\nu^{3} - 10\nu^{2} - 5\nu + 4 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - \nu^{5} - 8\nu^{4} + 5\nu^{3} + 15\nu^{2} - 4\nu - 4 \) 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
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + \beta_{4} + \beta_{3} + 7\beta_{2} + \beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 7\beta_{5} + 8\beta_{4} + 2\beta_{3} + 10\beta_{2} + 27\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 10\beta_{5} + 11\beta_{4} + 10\beta_{3} + 46\beta_{2} + 14\beta _1 + 73 \) 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
−2.20055
−2.13289
−0.548343
0.365590
1.16970
1.74907
2.59742
−2.20055 0 2.84240 0 0 1.71679 −1.85374 0 0
1.2 −2.13289 0 2.54923 0 0 −3.01925 −1.17144 0 0
1.3 −0.548343 0 −1.69932 0 0 1.23269 2.02850 0 0
1.4 0.365590 0 −1.86634 0 0 −3.36026 −1.41350 0 0
1.5 1.16970 0 −0.631803 0 0 5.24193 −3.07842 0 0
1.6 1.74907 0 1.05925 0 0 −0.963428 −1.64544 0 0
1.7 2.59742 0 4.74659 0 0 −0.848469 7.13404 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(37\) \( +1 \)

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 8325.2.a.cn 7
3.b odd 2 1 925.2.a.j ✓ 7
5.b even 2 1 8325.2.a.cm 7
15.d odd 2 1 925.2.a.k yes 7
15.e even 4 2 925.2.b.i 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
925.2.a.j ✓ 7 3.b odd 2 1
925.2.a.k yes 7 15.d odd 2 1
925.2.b.i 14 15.e even 4 2
8325.2.a.cm 7 5.b even 2 1
8325.2.a.cn 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8325))\):

\( T_{2}^{7} - T_{2}^{6} - 10T_{2}^{5} + 9T_{2}^{4} + 26T_{2}^{3} - 23T_{2}^{2} - 9T_{2} + 5 \) Copy content Toggle raw display
\( T_{7}^{7} - 27T_{7}^{5} - 28T_{7}^{4} + 120T_{7}^{3} + 97T_{7}^{2} - 116T_{7} - 92 \) Copy content Toggle raw display
\( T_{11}^{7} + 16T_{11}^{6} + 87T_{11}^{5} + 177T_{11}^{4} + 67T_{11}^{3} - 145T_{11}^{2} - 144T_{11} - 36 \) Copy content Toggle raw display
\( T_{13}^{7} - T_{13}^{6} - 51T_{13}^{5} + 13T_{13}^{4} + 424T_{13}^{3} - 355T_{13}^{2} - 264T_{13} + 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - T^{6} - 10 T^{5} + \cdots + 5 \) Copy content Toggle raw display
$3$ \( T^{7} \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( T^{7} - 27 T^{5} + \cdots - 92 \) Copy content Toggle raw display
$11$ \( T^{7} + 16 T^{6} + \cdots - 36 \) Copy content Toggle raw display
$13$ \( T^{7} - T^{6} - 51 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$17$ \( T^{7} - 4 T^{6} + \cdots + 1616 \) Copy content Toggle raw display
$19$ \( T^{7} - 9 T^{6} + \cdots + 5245 \) Copy content Toggle raw display
$23$ \( T^{7} + T^{6} - 66 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$29$ \( T^{7} + 3 T^{6} + \cdots - 920 \) Copy content Toggle raw display
$31$ \( T^{7} - 11 T^{6} + \cdots - 2628 \) Copy content Toggle raw display
$37$ \( (T + 1)^{7} \) Copy content Toggle raw display
$41$ \( T^{7} + 29 T^{6} + \cdots - 83 \) Copy content Toggle raw display
$43$ \( T^{7} + 3 T^{6} + \cdots + 74561 \) Copy content Toggle raw display
$47$ \( T^{7} + 2 T^{6} + \cdots - 9668 \) Copy content Toggle raw display
$53$ \( T^{7} + 6 T^{6} + \cdots - 7353 \) Copy content Toggle raw display
$59$ \( T^{7} + 42 T^{6} + \cdots + 1068435 \) Copy content Toggle raw display
$61$ \( T^{7} + 11 T^{6} + \cdots + 250612 \) Copy content Toggle raw display
$67$ \( T^{7} + 3 T^{6} + \cdots - 2784632 \) Copy content Toggle raw display
$71$ \( T^{7} + 10 T^{6} + \cdots + 180 \) Copy content Toggle raw display
$73$ \( T^{7} - 14 T^{6} + \cdots - 14944 \) Copy content Toggle raw display
$79$ \( T^{7} + 9 T^{6} + \cdots + 321075 \) Copy content Toggle raw display
$83$ \( T^{7} + 6 T^{6} + \cdots - 78048 \) Copy content Toggle raw display
$89$ \( T^{7} + 12 T^{6} + \cdots - 115400 \) Copy content Toggle raw display
$97$ \( T^{7} - T^{6} + \cdots + 687692 \) Copy content Toggle raw display
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