Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [2,16,Mod(1,2)] 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("2.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2, base_ring=CyclotomicField(1)) chi = DirichletCharacter(H, H._module([])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(2.85387010200\)
Analytic rank: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 128 q^{2} + 6252 q^{3} + 16384 q^{4} + 90510 q^{5} - 800256 q^{6} + 56 q^{7} - 2097152 q^{8} + 24738597 q^{9} - 11585280 q^{10} - 95889948 q^{11} + 102432768 q^{12} - 59782138 q^{13} - 7168 q^{14}+ \cdots - 23\!\cdots\!56 q^{99}+O(q^{100}) \) 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
−128.000 6252.00 16384.0 90510.0 −800256. 56.0000 −2.09715e6 2.47386e7 −1.15853e7
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

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

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 2.16.a.a ✓ 1
3.b odd 2 1 18.16.a.e 1
4.b odd 2 1 16.16.a.a 1
5.b even 2 1 50.16.a.b 1
5.c odd 4 2 50.16.b.a 2
7.b odd 2 1 98.16.a.a 1
7.c even 3 2 98.16.c.a 2
7.d odd 6 2 98.16.c.d 2
8.b even 2 1 64.16.a.a 1
8.d odd 2 1 64.16.a.k 1
12.b even 2 1 144.16.a.d 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.16.a.a ✓ 1 1.a even 1 1 trivial
16.16.a.a 1 4.b odd 2 1
18.16.a.e 1 3.b odd 2 1
50.16.a.b 1 5.b even 2 1
50.16.b.a 2 5.c odd 4 2
64.16.a.a 1 8.b even 2 1
64.16.a.k 1 8.d odd 2 1
98.16.a.a 1 7.b odd 2 1
98.16.c.a 2 7.c even 3 2
98.16.c.d 2 7.d odd 6 2
144.16.a.d 1 12.b even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(\Gamma_0(2))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 128 \) Copy content Toggle raw display
$3$ \( T - 6252 \) Copy content Toggle raw display
$5$ \( T - 90510 \) Copy content Toggle raw display
$7$ \( T - 56 \) Copy content Toggle raw display
$11$ \( T + 95889948 \) Copy content Toggle raw display
$13$ \( T + 59782138 \) Copy content Toggle raw display
$17$ \( T + 1355814414 \) Copy content Toggle raw display
$19$ \( T - 3783593180 \) Copy content Toggle raw display
$23$ \( T + 11608845528 \) Copy content Toggle raw display
$29$ \( T + 28959105930 \) Copy content Toggle raw display
$31$ \( T - 253685353952 \) Copy content Toggle raw display
$37$ \( T - 817641294446 \) Copy content Toggle raw display
$41$ \( T + 682333284198 \) Copy content Toggle raw display
$43$ \( T - 366945604292 \) Copy content Toggle raw display
$47$ \( T - 695741581776 \) Copy content Toggle raw display
$53$ \( T - 12993372468702 \) Copy content Toggle raw display
$59$ \( T - 9209035340340 \) Copy content Toggle raw display
$61$ \( T + 42338641200298 \) Copy content Toggle raw display
$67$ \( T - 30029787950636 \) Copy content Toggle raw display
$71$ \( T - 115328696975352 \) Copy content Toggle raw display
$73$ \( T - 43787346432122 \) Copy content Toggle raw display
$79$ \( T - 79603813043120 \) Copy content Toggle raw display
$83$ \( T + 3417068864868 \) Copy content Toggle raw display
$89$ \( T + 377306179184790 \) Copy content Toggle raw display
$97$ \( T + 166982186657374 \) Copy content Toggle raw display
show more
show less