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Properties

Label 147.16.a.b
Level $147$
Weight $16$
Character orbit 147.a
Self dual yes
Analytic conductor $209.759$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [147,16,Mod(1,147)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(147, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 16, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("147.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-72,-2187] 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: \(209.759452497\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
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)
 
\(f(q)\) \(=\) \( q - 72 q^{2} - 2187 q^{3} - 27584 q^{4} + 221490 q^{5} + 157464 q^{6} + 4345344 q^{8} + 4782969 q^{9} - 15947280 q^{10} + 37169316 q^{11} + 60326208 q^{12} + 279974266 q^{13} - 484398630 q^{15} + 591007744 q^{16}+ \cdots + 177779686179204 q^{99}+O(q^{100}) \) 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
−72.0000 −2187.00 −27584.0 221490. 157464. 0 4.34534e6 4.78297e6 −1.59473e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(7\) \( -1 \)

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 147.16.a.b 1
7.b odd 2 1 3.16.a.b ✓ 1
21.c even 2 1 9.16.a.c 1
28.d even 2 1 48.16.a.a 1
35.c odd 2 1 75.16.a.a 1
35.f even 4 2 75.16.b.b 2
84.h odd 2 1 144.16.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.b ✓ 1 7.b odd 2 1
9.16.a.c 1 21.c even 2 1
48.16.a.a 1 28.d even 2 1
75.16.a.a 1 35.c odd 2 1
75.16.b.b 2 35.f even 4 2
144.16.a.l 1 84.h odd 2 1
147.16.a.b 1 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(147))\):

\( T_{2} + 72 \) Copy content Toggle raw display
\( T_{5} - 221490 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 72 \) Copy content Toggle raw display
$3$ \( T + 2187 \) Copy content Toggle raw display
$5$ \( T - 221490 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 37169316 \) Copy content Toggle raw display
$13$ \( T - 279974266 \) Copy content Toggle raw display
$17$ \( T + 2492912754 \) Copy content Toggle raw display
$19$ \( T - 4669782244 \) Copy content Toggle raw display
$23$ \( T + 18467933400 \) Copy content Toggle raw display
$29$ \( T + 115953449418 \) Copy content Toggle raw display
$31$ \( T - 56187023200 \) Copy content Toggle raw display
$37$ \( T - 614764926830 \) Copy content Toggle raw display
$41$ \( T + 549859792410 \) Copy content Toggle raw display
$43$ \( T + 982884444028 \) Copy content Toggle raw display
$47$ \( T + 2076144322896 \) Copy content Toggle raw display
$53$ \( T + 12048378188130 \) Copy content Toggle raw display
$59$ \( T + 23087905758324 \) Copy content Toggle raw display
$61$ \( T - 8505809142442 \) Copy content Toggle raw display
$67$ \( T + 12331010771476 \) Copy content Toggle raw display
$71$ \( T - 58989192692472 \) Copy content Toggle raw display
$73$ \( T - 5609828808070 \) Copy content Toggle raw display
$79$ \( T - 159918683826800 \) Copy content Toggle raw display
$83$ \( T + 57675894342876 \) Copy content Toggle raw display
$89$ \( T - 362287610413974 \) Copy content Toggle raw display
$97$ \( T - 539786645144926 \) Copy content Toggle raw display
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