Properties

Label 429.2.a.a
Level $429$
Weight $2$
Character orbit 429.a
Self dual yes
Analytic conductor $3.426$
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] = [429,2,Mod(1,429)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(429, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("429.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 429 = 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 429.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: \(3.42558224671\)
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)
 
\(f(q)\) \(=\) \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} - q^{3} - q^{4} + q^{6} + 3 q^{8} + q^{9} + q^{11} + q^{12} + q^{13} - q^{16} - 4 q^{17} - q^{18} - 8 q^{19} - q^{22} - 3 q^{24} - 5 q^{25} - q^{26} - q^{27} + 4 q^{29} - 6 q^{31} - 5 q^{32} - q^{33} + 4 q^{34} - q^{36} - 6 q^{37} + 8 q^{38} - q^{39} + 6 q^{41} - 2 q^{43} - q^{44} - 8 q^{47} + q^{48} - 7 q^{49} + 5 q^{50} + 4 q^{51} - q^{52} + 6 q^{53} + q^{54} + 8 q^{57} - 4 q^{58} - 14 q^{61} + 6 q^{62} + 7 q^{64} + q^{66} + 14 q^{67} + 4 q^{68} - 4 q^{71} + 3 q^{72} + 6 q^{73} + 6 q^{74} + 5 q^{75} + 8 q^{76} + q^{78} - 10 q^{79} + q^{81} - 6 q^{82} - 12 q^{83} + 2 q^{86} - 4 q^{87} + 3 q^{88} + 12 q^{89} + 6 q^{93} + 8 q^{94} + 5 q^{96} - 2 q^{97} + 7 q^{98} + q^{99}+O(q^{100}) \) 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
−1.00000 −1.00000 −1.00000 0 1.00000 0 3.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(11\) \(-1\)
\(13\) \(-1\)

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 429.2.a.a 1
3.b odd 2 1 1287.2.a.d 1
4.b odd 2 1 6864.2.a.w 1
11.b odd 2 1 4719.2.a.j 1
13.b even 2 1 5577.2.a.f 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
429.2.a.a 1 1.a even 1 1 trivial
1287.2.a.d 1 3.b odd 2 1
4719.2.a.j 1 11.b odd 2 1
5577.2.a.f 1 13.b even 2 1
6864.2.a.w 1 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(429))\):

\( T_{2} + 1 \) Copy content Toggle raw display
\( T_{5} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T + 1 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T + 4 \) Copy content Toggle raw display
$19$ \( T + 8 \) Copy content Toggle raw display
$23$ \( T \) Copy content Toggle raw display
$29$ \( T - 4 \) Copy content Toggle raw display
$31$ \( T + 6 \) Copy content Toggle raw display
$37$ \( T + 6 \) Copy content Toggle raw display
$41$ \( T - 6 \) Copy content Toggle raw display
$43$ \( T + 2 \) Copy content Toggle raw display
$47$ \( T + 8 \) Copy content Toggle raw display
$53$ \( T - 6 \) Copy content Toggle raw display
$59$ \( T \) Copy content Toggle raw display
$61$ \( T + 14 \) Copy content Toggle raw display
$67$ \( T - 14 \) Copy content Toggle raw display
$71$ \( T + 4 \) Copy content Toggle raw display
$73$ \( T - 6 \) Copy content Toggle raw display
$79$ \( T + 10 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 12 \) Copy content Toggle raw display
$97$ \( T + 2 \) Copy content Toggle raw display
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