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Properties

Label 3.16.a.b
Level $3$
Weight $16$
Character orbit 3.a
Self dual yes
Analytic conductor $4.281$
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] = [3,16,Mod(1,3)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([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("3.1"); S:= CuspForms(chi, 16); N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-72] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(4.28080515300\)
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)
 
\(f(q)\) \(=\) \( q - 72 q^{2} + 2187 q^{3} - 27584 q^{4} - 221490 q^{5} - 157464 q^{6} - 2149000 q^{7} + 4345344 q^{8} + 4782969 q^{9} + 15947280 q^{10} + 37169316 q^{11} - 60326208 q^{12} - 279974266 q^{13} + 154728000 q^{14}+ \cdots + 177779686179204 q^{99}+O(q^{100}) \) 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. −2.14900e6 4.34534e6 4.78297e6 1.59473e7
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 3.16.a.b ✓ 1
3.b odd 2 1 9.16.a.c 1
4.b odd 2 1 48.16.a.a 1
5.b even 2 1 75.16.a.a 1
5.c odd 4 2 75.16.b.b 2
7.b odd 2 1 147.16.a.b 1
12.b even 2 1 144.16.a.l 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.b ✓ 1 1.a even 1 1 trivial
9.16.a.c 1 3.b odd 2 1
48.16.a.a 1 4.b odd 2 1
75.16.a.a 1 5.b even 2 1
75.16.b.b 2 5.c odd 4 2
144.16.a.l 1 12.b even 2 1
147.16.a.b 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 72 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\). 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
$7$ \( T + 2149000 \) 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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