Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [490,6,Mod(1,490)] 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("490.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(490, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 490 = 2 \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 490.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-4,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.5880717084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
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)
 
\(f(q)\) \(=\) \( q - 4 q^{2} + 5 q^{3} + 16 q^{4} + 25 q^{5} - 20 q^{6} - 64 q^{8} - 218 q^{9} - 100 q^{10} + 198 q^{11} + 80 q^{12} + 340 q^{13} + 125 q^{15} + 256 q^{16} - 1848 q^{17} + 872 q^{18} + 1210 q^{19} + 400 q^{20}+ \cdots - 43164 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
−4.00000 5.00000 16.0000 25.0000 −20.0000 0 −64.0000 −218.000 −100.000
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(5\) \( -1 \)
\(7\) \( -1 \)

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 490.6.a.f 1
7.b odd 2 1 490.6.a.d 1
7.d odd 6 2 70.6.e.a ✓ 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.6.e.a ✓ 2 7.d odd 6 2
490.6.a.d 1 7.b odd 2 1
490.6.a.f 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} - 5 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(490))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 4 \) Copy content Toggle raw display
$3$ \( T - 5 \) Copy content Toggle raw display
$5$ \( T - 25 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 198 \) Copy content Toggle raw display
$13$ \( T - 340 \) Copy content Toggle raw display
$17$ \( T + 1848 \) Copy content Toggle raw display
$19$ \( T - 1210 \) Copy content Toggle raw display
$23$ \( T - 2823 \) Copy content Toggle raw display
$29$ \( T + 4539 \) Copy content Toggle raw display
$31$ \( T - 712 \) Copy content Toggle raw display
$37$ \( T + 7324 \) Copy content Toggle raw display
$41$ \( T + 15633 \) Copy content Toggle raw display
$43$ \( T - 15827 \) Copy content Toggle raw display
$47$ \( T - 3192 \) Copy content Toggle raw display
$53$ \( T + 20046 \) Copy content Toggle raw display
$59$ \( T + 23046 \) Copy content Toggle raw display
$61$ \( T - 379 \) Copy content Toggle raw display
$67$ \( T + 35473 \) Copy content Toggle raw display
$71$ \( T - 71814 \) Copy content Toggle raw display
$73$ \( T + 31664 \) Copy content Toggle raw display
$79$ \( T - 8534 \) Copy content Toggle raw display
$83$ \( T - 106551 \) Copy content Toggle raw display
$89$ \( T - 12303 \) Copy content Toggle raw display
$97$ \( T - 102802 \) Copy content Toggle raw display
show more
show less