Properties

Label 3.14.a.a
Level $3$
Weight $14$
Character orbit 3.a
Self dual yes
Analytic conductor $3.217$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3,14,Mod(1,3)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(3.21692786856\)
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

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 12 q^{2} - 729 q^{3} - 8048 q^{4} - 30210 q^{5} + 8748 q^{6} + 235088 q^{7} + 194880 q^{8} + 531441 q^{9} + 362520 q^{10} - 11182908 q^{11} + 5866992 q^{12} + 8049614 q^{13} - 2821056 q^{14} + 22023090 q^{15}+ \cdots - 5943055810428 q^{99}+O(q^{100}) \) 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.

comment: embeddings in the coefficient field
 
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
−12.0000 −729.000 −8048.00 −30210.0 8748.00 235088. 194880. 531441. 362520.
\(n\): 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 3.14.a.a 1
3.b odd 2 1 9.14.a.a 1
4.b odd 2 1 48.14.a.c 1
5.b even 2 1 75.14.a.a 1
5.c odd 4 2 75.14.b.b 2
7.b odd 2 1 147.14.a.a 1
8.b even 2 1 192.14.a.j 1
8.d odd 2 1 192.14.a.e 1
12.b even 2 1 144.14.a.k 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.14.a.a 1 1.a even 1 1 trivial
9.14.a.a 1 3.b odd 2 1
48.14.a.c 1 4.b odd 2 1
75.14.a.a 1 5.b even 2 1
75.14.b.b 2 5.c odd 4 2
144.14.a.k 1 12.b even 2 1
147.14.a.a 1 7.b odd 2 1
192.14.a.e 1 8.d odd 2 1
192.14.a.j 1 8.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 12 \) acting on \(S_{14}^{\mathrm{new}}(\Gamma_0(3))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 12 \) Copy content Toggle raw display
$3$ \( T + 729 \) Copy content Toggle raw display
$5$ \( T + 30210 \) Copy content Toggle raw display
$7$ \( T - 235088 \) Copy content Toggle raw display
$11$ \( T + 11182908 \) Copy content Toggle raw display
$13$ \( T - 8049614 \) Copy content Toggle raw display
$17$ \( T + 117494622 \) Copy content Toggle raw display
$19$ \( T + 214061380 \) Copy content Toggle raw display
$23$ \( T - 830555544 \) Copy content Toggle raw display
$29$ \( T + 1252400250 \) Copy content Toggle raw display
$31$ \( T - 6159350552 \) Copy content Toggle raw display
$37$ \( T + 5498191402 \) Copy content Toggle raw display
$41$ \( T + 4678687878 \) Copy content Toggle raw display
$43$ \( T - 7115013764 \) Copy content Toggle raw display
$47$ \( T + 29528776992 \) Copy content Toggle raw display
$53$ \( T + 204125042466 \) Copy content Toggle raw display
$59$ \( T + 29909821020 \) Copy content Toggle raw display
$61$ \( T + 134392006738 \) Copy content Toggle raw display
$67$ \( T - 348518801948 \) Copy content Toggle raw display
$71$ \( T - 1314335409192 \) Copy content Toggle raw display
$73$ \( T + 1178875922326 \) Copy content Toggle raw display
$79$ \( T + 1072420659640 \) Copy content Toggle raw display
$83$ \( T - 1124025139644 \) Copy content Toggle raw display
$89$ \( T - 2235610909530 \) Copy content Toggle raw display
$97$ \( T + 14215257165502 \) Copy content Toggle raw display
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