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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,16,Mod(1,144)] 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("144.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 16, names="a")
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,251890,0,-1374072] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(205.478647344\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
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)
 
\(f(q)\) \(=\) \( q + 251890 q^{5} - 1374072 q^{7} - 43286716 q^{11} - 323161466 q^{13} + 191653646 q^{17} + 6515456644 q^{19} + 23880801512 q^{23} + 32930993975 q^{25} - 176820596982 q^{29} + 152007193888 q^{31} - 346114996080 q^{35}+ \cdots + 309183128990882 q^{97}+O(q^{100}) \) 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
0 0 0 251890. 0 −1.37407e6 0 0 0
\(n\): 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 144.16.a.m 1
3.b odd 2 1 16.16.a.b 1
4.b odd 2 1 72.16.a.c 1
12.b even 2 1 8.16.a.b ✓ 1
24.f even 2 1 64.16.a.d 1
24.h odd 2 1 64.16.a.h 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.16.a.b ✓ 1 12.b even 2 1
16.16.a.b 1 3.b odd 2 1
64.16.a.d 1 24.f even 2 1
64.16.a.h 1 24.h odd 2 1
72.16.a.c 1 4.b odd 2 1
144.16.a.m 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} - 251890 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(144))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T - 251890 \) Copy content Toggle raw display
$7$ \( T + 1374072 \) Copy content Toggle raw display
$11$ \( T + 43286716 \) Copy content Toggle raw display
$13$ \( T + 323161466 \) Copy content Toggle raw display
$17$ \( T - 191653646 \) Copy content Toggle raw display
$19$ \( T - 6515456644 \) Copy content Toggle raw display
$23$ \( T - 23880801512 \) Copy content Toggle raw display
$29$ \( T + 176820596982 \) Copy content Toggle raw display
$31$ \( T - 152007193888 \) Copy content Toggle raw display
$37$ \( T - 21581233902 \) Copy content Toggle raw display
$41$ \( T - 245334499686 \) Copy content Toggle raw display
$43$ \( T + 2769961534756 \) Copy content Toggle raw display
$47$ \( T - 2811771943248 \) Copy content Toggle raw display
$53$ \( T - 3491413730722 \) Copy content Toggle raw display
$59$ \( T + 15827800893676 \) Copy content Toggle raw display
$61$ \( T + 24609047974442 \) Copy content Toggle raw display
$67$ \( T - 20706233653684 \) Copy content Toggle raw display
$71$ \( T + 719982528200 \) Copy content Toggle raw display
$73$ \( T - 29883036220282 \) Copy content Toggle raw display
$79$ \( T - 148100908648400 \) Copy content Toggle raw display
$83$ \( T + 302806756982468 \) Copy content Toggle raw display
$89$ \( T - 496150966996374 \) Copy content Toggle raw display
$97$ \( T - 309183128990882 \) Copy content Toggle raw display
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