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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [3936,2,Mod(1,3936)] 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("3936.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3936, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 3936 = 2^{5} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3936.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [7,0,7,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(31.4291182356\)
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: \( x^{7} - 3x^{6} - 14x^{5} + 29x^{4} + 81x^{3} - 58x^{2} - 174x - 58 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 3x^{6} - 14x^{5} + 29x^{4} + 81x^{3} - 58x^{2} - 174x - 58 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 4\nu^{5} - 8\nu^{4} + 35\nu^{3} + 22\nu^{2} - 74\nu - 32 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 8\nu^{4} - 33\nu^{3} - 26\nu^{2} + 66\nu + 38 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 4\nu^{5} + 8\nu^{4} - 35\nu^{3} - 24\nu^{2} + 80\nu + 40 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{6} + 5\nu^{5} + 6\nu^{4} - 44\nu^{3} - 15\nu^{2} + 98\nu + 41 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{6} + 14\nu^{5} + 18\nu^{4} - 117\nu^{3} - 42\nu^{2} + 250\nu + 100 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} + 3\beta _1 + 11 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{4} + \beta_{3} + 4\beta_{2} + 5\beta _1 + 10 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{6} + 2\beta_{5} + 13\beta_{4} + 6\beta_{3} + 17\beta_{2} + 35\beta _1 + 91 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{6} + 6\beta_{5} + 49\beta_{4} + 30\beta_{3} + 79\beta_{2} + 115\beta _1 + 243 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -16\beta_{6} + 20\beta_{5} + 71\beta_{4} + 49\beta_{3} + 114\beta_{2} + 199\beta _1 + 448 \) 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
−0.420924
2.16207
−1.57602
−1.73722
−2.09604
3.23137
3.43676
0 1.00000 0 −3.40190 0 1.12166 0 1.00000 0
1.2 0 1.00000 0 −1.48751 0 2.46937 0 1.00000 0
1.3 0 1.00000 0 0.0598794 0 4.56019 0 1.00000 0
1.4 0 1.00000 0 0.755167 0 −0.318138 0 1.00000 0
1.5 0 1.00000 0 2.48940 0 −4.16412 0 1.00000 0
1.6 0 1.00000 0 3.21040 0 −0.752408 0 1.00000 0
1.7 0 1.00000 0 4.37455 0 5.08346 0 1.00000 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
\(2\) \( -1 \)
\(3\) \( -1 \)
\(41\) \( -1 \)

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 3936.2.a.v yes 7
4.b odd 2 1 3936.2.a.u ✓ 7
8.b even 2 1 7872.2.a.cq 7
8.d odd 2 1 7872.2.a.cr 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3936.2.a.u ✓ 7 4.b odd 2 1
3936.2.a.v yes 7 1.a even 1 1 trivial
7872.2.a.cq 7 8.b even 2 1
7872.2.a.cr 7 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3936))\):

\( T_{5}^{7} - 6T_{5}^{6} - 7T_{5}^{5} + 84T_{5}^{4} - 66T_{5}^{3} - 170T_{5}^{2} + 144T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{7} - 8T_{7}^{6} - 4T_{7}^{5} + 146T_{7}^{4} - 240T_{7}^{3} - 120T_{7}^{2} + 192T_{7} + 64 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 1)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} - 6 T^{6} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{7} - 8 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{7} + T^{6} + \cdots - 7744 \) Copy content Toggle raw display
$13$ \( T^{7} - 6 T^{6} + \cdots - 632 \) Copy content Toggle raw display
$17$ \( T^{7} - 7 T^{6} + \cdots + 13852 \) Copy content Toggle raw display
$19$ \( T^{7} + 2 T^{6} + \cdots - 23648 \) Copy content Toggle raw display
$23$ \( T^{7} - 8 T^{6} + \cdots - 256 \) Copy content Toggle raw display
$29$ \( T^{7} - 13 T^{6} + \cdots - 201472 \) Copy content Toggle raw display
$31$ \( T^{7} - 9 T^{6} + \cdots + 6352 \) Copy content Toggle raw display
$37$ \( T^{7} - 3 T^{6} + \cdots + 186256 \) Copy content Toggle raw display
$41$ \( (T - 1)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} - 9 T^{6} + \cdots + 2816 \) Copy content Toggle raw display
$47$ \( T^{7} - 15 T^{6} + \cdots + 1609216 \) Copy content Toggle raw display
$53$ \( T^{7} - 12 T^{6} + \cdots + 443968 \) Copy content Toggle raw display
$59$ \( T^{7} + 8 T^{6} + \cdots + 128 \) Copy content Toggle raw display
$61$ \( T^{7} - 23 T^{6} + \cdots + 5488 \) Copy content Toggle raw display
$67$ \( T^{7} + 6 T^{6} + \cdots - 917504 \) Copy content Toggle raw display
$71$ \( T^{7} - T^{6} + \cdots - 87752 \) Copy content Toggle raw display
$73$ \( T^{7} - 29 T^{6} + \cdots - 2296148 \) Copy content Toggle raw display
$79$ \( T^{7} - 32 T^{6} + \cdots + 354304 \) Copy content Toggle raw display
$83$ \( T^{7} + 16 T^{6} + \cdots - 1211744 \) Copy content Toggle raw display
$89$ \( T^{7} - 8 T^{6} + \cdots + 118496 \) Copy content Toggle raw display
$97$ \( T^{7} + 8 T^{6} + \cdots - 1124128 \) Copy content Toggle raw display
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