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Properties

Label 3.16.a.a
Level $3$
Weight $16$
Character orbit 3.a
Self dual yes
Analytic conductor $4.281$
Analytic rank $0$
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 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);
 
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")
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,-234] 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: \(0\)
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)
 
\(f(q)\) \(=\) \( q - 234 q^{2} - 2187 q^{3} + 21988 q^{4} + 280710 q^{5} + 511758 q^{6} - 1373344 q^{7} + 2522520 q^{8} + 4782969 q^{9} - 65686140 q^{10} + 34031052 q^{11} - 48087756 q^{12} + 384022262 q^{13} + 321362496 q^{14}+ \cdots + 162769466753388 q^{99}+O(q^{100}) \) 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
−234.000 −2187.00 21988.0 280710. 511758. −1.37334e6 2.52252e6 4.78297e6 −6.56861e7
\(n\): 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.a ✓ 1
3.b odd 2 1 9.16.a.d 1
4.b odd 2 1 48.16.a.g 1
5.b even 2 1 75.16.a.b 1
5.c odd 4 2 75.16.b.a 2
7.b odd 2 1 147.16.a.a 1
12.b even 2 1 144.16.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.16.a.a ✓ 1 1.a even 1 1 trivial
9.16.a.d 1 3.b odd 2 1
48.16.a.g 1 4.b odd 2 1
75.16.a.b 1 5.b even 2 1
75.16.b.a 2 5.c odd 4 2
144.16.a.b 1 12.b even 2 1
147.16.a.a 1 7.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 234 \) acting on \(S_{16}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 234 \) Copy content Toggle raw display
$3$ \( T + 2187 \) Copy content Toggle raw display
$5$ \( T - 280710 \) Copy content Toggle raw display
$7$ \( T + 1373344 \) Copy content Toggle raw display
$11$ \( T - 34031052 \) Copy content Toggle raw display
$13$ \( T - 384022262 \) Copy content Toggle raw display
$17$ \( T - 1259207586 \) Copy content Toggle raw display
$19$ \( T + 2499071020 \) Copy content Toggle raw display
$23$ \( T - 11284833672 \) Copy content Toggle raw display
$29$ \( T + 48413458530 \) Copy content Toggle raw display
$31$ \( T - 130547265752 \) Copy content Toggle raw display
$37$ \( T + 200223317554 \) Copy content Toggle raw display
$41$ \( T - 679141724202 \) Copy content Toggle raw display
$43$ \( T - 279482194892 \) Copy content Toggle raw display
$47$ \( T - 1520672832576 \) Copy content Toggle raw display
$53$ \( T - 2646053822502 \) Copy content Toggle raw display
$59$ \( T - 7399371294540 \) Copy content Toggle raw display
$61$ \( T + 42659617819498 \) Copy content Toggle raw display
$67$ \( T + 56408026065964 \) Copy content Toggle raw display
$71$ \( T + 133149677299848 \) Copy content Toggle raw display
$73$ \( T - 105603350884922 \) Copy content Toggle raw display
$79$ \( T + 55665674361880 \) Copy content Toggle raw display
$83$ \( T - 378077412997332 \) Copy content Toggle raw display
$89$ \( T - 219315065897610 \) Copy content Toggle raw display
$97$ \( T - 703322682162626 \) Copy content Toggle raw display
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