Properties

Label 1638.2.a.f
Level $1638$
Weight $2$
Character orbit 1638.a
Self dual yes
Analytic conductor $13.079$
Analytic rank $0$
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] = [1638,2,Mod(1,1638)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1638, 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("1638.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1638.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: \(13.0794958511\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 182)
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)
 
\(f(q)\) \(=\) \( q - q^{2} + q^{4} + q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + q^{4} + q^{7} - q^{8} + 3 q^{11} + q^{13} - q^{14} + q^{16} + 2 q^{19} - 3 q^{22} + 3 q^{23} - 5 q^{25} - q^{26} + q^{28} + 5 q^{31} - q^{32} - 7 q^{37} - 2 q^{38} - 3 q^{41} + 8 q^{43} + 3 q^{44} - 3 q^{46} + 3 q^{47} + q^{49} + 5 q^{50} + q^{52} + 12 q^{53} - q^{56} - 6 q^{59} - q^{61} - 5 q^{62} + q^{64} + 5 q^{67} - 12 q^{71} + 11 q^{73} + 7 q^{74} + 2 q^{76} + 3 q^{77} - q^{79} + 3 q^{82} - 12 q^{83} - 8 q^{86} - 3 q^{88} + 18 q^{89} + q^{91} + 3 q^{92} - 3 q^{94} + 17 q^{97} - q^{98}+O(q^{100}) \) 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 0 1.00000 0 0 1.00000 −1.00000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(3\) \( -1 \)
\(7\) \( -1 \)
\(13\) \( -1 \)

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 1638.2.a.f 1
3.b odd 2 1 182.2.a.d 1
12.b even 2 1 1456.2.a.d 1
15.d odd 2 1 4550.2.a.c 1
21.c even 2 1 1274.2.a.j 1
21.g even 6 2 1274.2.f.i 2
21.h odd 6 2 1274.2.f.d 2
24.f even 2 1 5824.2.a.x 1
24.h odd 2 1 5824.2.a.k 1
39.d odd 2 1 2366.2.a.e 1
39.f even 4 2 2366.2.d.e 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.2.a.d 1 3.b odd 2 1
1274.2.a.j 1 21.c even 2 1
1274.2.f.d 2 21.h odd 6 2
1274.2.f.i 2 21.g even 6 2
1456.2.a.d 1 12.b even 2 1
1638.2.a.f 1 1.a even 1 1 trivial
2366.2.a.e 1 39.d odd 2 1
2366.2.d.e 2 39.f even 4 2
4550.2.a.c 1 15.d odd 2 1
5824.2.a.k 1 24.h odd 2 1
5824.2.a.x 1 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1638))\):

\( T_{5} \) Copy content Toggle raw display
\( T_{11} - 3 \) Copy content Toggle raw display
\( T_{17} \) Copy content Toggle raw display
\( T_{19} - 2 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 1 \) Copy content Toggle raw display
$3$ \( T \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 1 \) Copy content Toggle raw display
$11$ \( T - 3 \) Copy content Toggle raw display
$13$ \( T - 1 \) Copy content Toggle raw display
$17$ \( T \) Copy content Toggle raw display
$19$ \( T - 2 \) Copy content Toggle raw display
$23$ \( T - 3 \) Copy content Toggle raw display
$29$ \( T \) Copy content Toggle raw display
$31$ \( T - 5 \) Copy content Toggle raw display
$37$ \( T + 7 \) Copy content Toggle raw display
$41$ \( T + 3 \) Copy content Toggle raw display
$43$ \( T - 8 \) Copy content Toggle raw display
$47$ \( T - 3 \) Copy content Toggle raw display
$53$ \( T - 12 \) Copy content Toggle raw display
$59$ \( T + 6 \) Copy content Toggle raw display
$61$ \( T + 1 \) Copy content Toggle raw display
$67$ \( T - 5 \) Copy content Toggle raw display
$71$ \( T + 12 \) Copy content Toggle raw display
$73$ \( T - 11 \) Copy content Toggle raw display
$79$ \( T + 1 \) Copy content Toggle raw display
$83$ \( T + 12 \) Copy content Toggle raw display
$89$ \( T - 18 \) Copy content Toggle raw display
$97$ \( T - 17 \) Copy content Toggle raw display
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