Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [5,10,Mod(1,5)] 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("5.1"); S:= CuspForms(chi, 10); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 10, names="a")
 
Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 5.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1] 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: \(2.57517918082\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
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)
 
\(f(q)\) \(=\) \( q - 8 q^{2} - 114 q^{3} - 448 q^{4} - 625 q^{5} + 912 q^{6} + 4242 q^{7} + 7680 q^{8} - 6687 q^{9} + 5000 q^{10} - 46208 q^{11} + 51072 q^{12} - 115934 q^{13} - 33936 q^{14} + 71250 q^{15} + 167936 q^{16}+ \cdots + 308992896 q^{99}+O(q^{100}) \) 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
−8.00000 −114.000 −448.000 −625.000 912.000 4242.00 7680.00 −6687.00 5000.00
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)

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 5.10.a.a ✓ 1
3.b odd 2 1 45.10.a.c 1
4.b odd 2 1 80.10.a.d 1
5.b even 2 1 25.10.a.a 1
5.c odd 4 2 25.10.b.a 2
7.b odd 2 1 245.10.a.a 1
8.b even 2 1 320.10.a.h 1
8.d odd 2 1 320.10.a.c 1
15.d odd 2 1 225.10.a.b 1
15.e even 4 2 225.10.b.d 2
20.d odd 2 1 400.10.a.c 1
20.e even 4 2 400.10.c.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5.10.a.a ✓ 1 1.a even 1 1 trivial
25.10.a.a 1 5.b even 2 1
25.10.b.a 2 5.c odd 4 2
45.10.a.c 1 3.b odd 2 1
80.10.a.d 1 4.b odd 2 1
225.10.a.b 1 15.d odd 2 1
225.10.b.d 2 15.e even 4 2
245.10.a.a 1 7.b odd 2 1
320.10.a.c 1 8.d odd 2 1
320.10.a.h 1 8.b even 2 1
400.10.a.c 1 20.d odd 2 1
400.10.c.e 2 20.e even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 8 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(5))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 8 \) Copy content Toggle raw display
$3$ \( T + 114 \) Copy content Toggle raw display
$5$ \( T + 625 \) Copy content Toggle raw display
$7$ \( T - 4242 \) Copy content Toggle raw display
$11$ \( T + 46208 \) Copy content Toggle raw display
$13$ \( T + 115934 \) Copy content Toggle raw display
$17$ \( T - 494842 \) Copy content Toggle raw display
$19$ \( T + 1008740 \) Copy content Toggle raw display
$23$ \( T + 532554 \) Copy content Toggle raw display
$29$ \( T - 4196390 \) Copy content Toggle raw display
$31$ \( T + 3365028 \) Copy content Toggle raw display
$37$ \( T + 14931358 \) Copy content Toggle raw display
$41$ \( T - 11056262 \) Copy content Toggle raw display
$43$ \( T + 6396794 \) Copy content Toggle raw display
$47$ \( T + 35559158 \) Copy content Toggle raw display
$53$ \( T - 39738586 \) Copy content Toggle raw display
$59$ \( T + 85185620 \) Copy content Toggle raw display
$61$ \( T - 45748642 \) Copy content Toggle raw display
$67$ \( T + 45286158 \) Copy content Toggle raw display
$71$ \( T + 189967468 \) Copy content Toggle raw display
$73$ \( T - 412170946 \) Copy content Toggle raw display
$79$ \( T - 95040840 \) Copy content Toggle raw display
$83$ \( T - 261706326 \) Copy content Toggle raw display
$89$ \( T + 19938630 \) Copy content Toggle raw display
$97$ \( T + 19503358 \) Copy content Toggle raw display
show more
show less