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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [796,2,Mod(1,796)] 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("796.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(796, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 796 = 2^{2} \cdot 199 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 796.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(6.35609200089\)
Analytic rank: \(0\)
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: \( x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 47x^{4} - 82x^{3} - 28x^{2} + 64x - 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{6} - \beta_{4}) q^{5} + (\beta_{3} + 1) q^{7} + (\beta_{6} - \beta_{5} - \beta_{2} + \cdots + 1) q^{9} + (\beta_{5} - \beta_{4} - \beta_1 + 2) q^{11} + (\beta_{5} + \beta_{2}) q^{13}+ \cdots + (\beta_{7} + \beta_{6} - 2 \beta_{5} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{3} + 2 q^{5} + 4 q^{7} + 6 q^{9} + 15 q^{11} + 3 q^{13} + 8 q^{15} - 3 q^{17} + 4 q^{19} + 6 q^{21} + 15 q^{23} + 8 q^{25} + 2 q^{27} + 18 q^{29} + 15 q^{31} - q^{33} + 26 q^{35} + 3 q^{37}+ \cdots + 13 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 13x^{6} + 24x^{5} + 47x^{4} - 82x^{3} - 28x^{2} + 64x - 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} - \nu^{2} - 5\nu + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} - 17\nu^{5} + 2\nu^{4} + 79\nu^{3} - 12\nu^{2} - 76\nu + 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 13\nu^{5} + 20\nu^{4} + 51\nu^{3} - 54\nu^{2} - 52\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 13\nu^{5} + 24\nu^{4} + 47\nu^{3} - 82\nu^{2} - 28\nu + 56 ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} - 2\nu^{6} - 13\nu^{5} + 24\nu^{4} + 51\nu^{3} - 78\nu^{2} - 56\nu + 40 ) / 8 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} - 4\nu^{6} - 39\nu^{5} + 42\nu^{4} + 145\nu^{3} - 120\nu^{2} - 120\nu + 72 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} - \beta_{5} - \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} - \beta_{5} + \beta_{2} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 8\beta_{6} - 6\beta_{5} - 2\beta_{4} - 6\beta_{2} + 7\beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} + 9\beta_{6} - 9\beta_{5} - 4\beta_{4} - 2\beta_{3} + 9\beta_{2} + 39\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 4\beta_{7} + 55\beta_{6} - 37\beta_{5} - 30\beta_{4} - 29\beta_{2} + 48\beta _1 + 156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 34\beta_{7} + 70\beta_{6} - 74\beta_{5} - 64\beta_{4} - 26\beta_{3} + 74\beta_{2} + 263\beta _1 + 60 \) 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
−2.51609
−2.28208
−1.08730
0.339347
0.736242
1.42093
2.49136
2.89759
0 −2.51609 0 1.87996 0 3.61687 0 3.33072 0
1.2 0 −2.28208 0 −2.46642 0 −3.48032 0 2.20789 0
1.3 0 −1.08730 0 −0.133273 0 1.21649 0 −1.81777 0
1.4 0 0.339347 0 −4.07465 0 −1.01682 0 −2.88484 0
1.5 0 0.736242 0 2.93220 0 −2.23802 0 −2.45795 0
1.6 0 1.42093 0 3.51416 0 3.97523 0 −0.980953 0
1.7 0 2.49136 0 −0.440123 0 1.86463 0 3.20690 0
1.8 0 2.89759 0 0.788142 0 0.0619435 0 5.39601 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(199\) \( +1 \)

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 796.2.a.c ✓ 8
3.b odd 2 1 7164.2.a.l 8
4.b odd 2 1 3184.2.a.o 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
796.2.a.c ✓ 8 1.a even 1 1 trivial
3184.2.a.o 8 4.b odd 2 1
7164.2.a.l 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 2T_{3}^{7} - 13T_{3}^{6} + 24T_{3}^{5} + 47T_{3}^{4} - 82T_{3}^{3} - 28T_{3}^{2} + 64T_{3} - 16 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(796))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 2 T^{7} + \cdots - 16 \) Copy content Toggle raw display
$5$ \( T^{8} - 2 T^{7} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots - 16 \) Copy content Toggle raw display
$11$ \( T^{8} - 15 T^{7} + \cdots + 80 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + \cdots - 31 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} + \cdots - 128 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + \cdots - 3392 \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + \cdots - 95573 \) Copy content Toggle raw display
$29$ \( T^{8} - 18 T^{7} + \cdots - 2878 \) Copy content Toggle raw display
$31$ \( T^{8} - 15 T^{7} + \cdots - 210659 \) Copy content Toggle raw display
$37$ \( T^{8} - 3 T^{7} + \cdots + 640 \) Copy content Toggle raw display
$41$ \( T^{8} - 2 T^{7} + \cdots + 1280 \) Copy content Toggle raw display
$43$ \( T^{8} - 12 T^{7} + \cdots - 39715 \) Copy content Toggle raw display
$47$ \( T^{8} - 13 T^{7} + \cdots - 27344 \) Copy content Toggle raw display
$53$ \( T^{8} - 16 T^{7} + \cdots - 2216765 \) Copy content Toggle raw display
$59$ \( T^{8} - 26 T^{7} + \cdots + 5186448 \) Copy content Toggle raw display
$61$ \( T^{8} + 14 T^{7} + \cdots + 309703 \) Copy content Toggle raw display
$67$ \( T^{8} + 11 T^{7} + \cdots + 3207376 \) Copy content Toggle raw display
$71$ \( T^{8} - 20 T^{7} + \cdots - 244336 \) Copy content Toggle raw display
$73$ \( T^{8} + 16 T^{7} + \cdots - 846368 \) Copy content Toggle raw display
$79$ \( T^{8} - 16 T^{7} + \cdots - 6204173 \) Copy content Toggle raw display
$83$ \( T^{8} - 19 T^{7} + \cdots - 6669088 \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + \cdots - 5289593 \) Copy content Toggle raw display
$97$ \( T^{8} + 36 T^{7} + \cdots + 4608 \) Copy content Toggle raw display
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