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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [18,16,Mod(1,18)] 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("18.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(18, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,128,0,16384,-90510] 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: \(25.6848309180\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2)
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)
 
\(f(q)\) \(=\) \( q + 128 q^{2} + 16384 q^{4} - 90510 q^{5} + 56 q^{7} + 2097152 q^{8} - 11585280 q^{10} + 95889948 q^{11} - 59782138 q^{13} + 7168 q^{14} + 268435456 q^{16} + 1355814414 q^{17} + 3783593180 q^{19} - 1482915840 q^{20}+ \cdots - 607687872871296 q^{98}+O(q^{100}) \) 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 0 16384.0 −90510.0 0 56.0000 2.09715e6 0 −1.15853e7
\(n\): e.g. 2-40 or 80-90
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( -1 \)
\(3\) \( -1 \)

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 18.16.a.e 1
3.b odd 2 1 2.16.a.a ✓ 1
4.b odd 2 1 144.16.a.d 1
12.b even 2 1 16.16.a.a 1
15.d odd 2 1 50.16.a.b 1
15.e even 4 2 50.16.b.a 2
21.c even 2 1 98.16.a.a 1
21.g even 6 2 98.16.c.d 2
21.h odd 6 2 98.16.c.a 2
24.f even 2 1 64.16.a.k 1
24.h odd 2 1 64.16.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2.16.a.a ✓ 1 3.b odd 2 1
16.16.a.a 1 12.b even 2 1
18.16.a.e 1 1.a even 1 1 trivial
50.16.a.b 1 15.d odd 2 1
50.16.b.a 2 15.e even 4 2
64.16.a.a 1 24.h odd 2 1
64.16.a.k 1 24.f even 2 1
98.16.a.a 1 21.c even 2 1
98.16.c.a 2 21.h odd 6 2
98.16.c.d 2 21.g even 6 2
144.16.a.d 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} + 90510 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(18))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T - 128 \) Copy content Toggle raw display
$3$ \( T \) 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
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