Properties

Label 5390.2.a.bf
Level $5390$
Weight $2$
Character orbit 5390.a
Self dual yes
Analytic conductor $43.039$
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] = [5390,2,Mod(1,5390)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5390, 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("5390.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5390 = 2 \cdot 5 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5390.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: \(43.0393666895\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
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)
 
\(f(q)\) \(=\) \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} - 2 q^{9} - q^{10} + q^{11} + q^{12} + 6 q^{13} - q^{15} + q^{16} + 7 q^{17} - 2 q^{18} - 5 q^{19} - q^{20} + q^{22} - 6 q^{23} + q^{24} + q^{25} + 6 q^{26} - 5 q^{27} + 5 q^{29} - q^{30} + 3 q^{31} + q^{32} + q^{33} + 7 q^{34} - 2 q^{36} + 3 q^{37} - 5 q^{38} + 6 q^{39} - q^{40} - 2 q^{41} + 4 q^{43} + q^{44} + 2 q^{45} - 6 q^{46} + 2 q^{47} + q^{48} + q^{50} + 7 q^{51} + 6 q^{52} - q^{53} - 5 q^{54} - q^{55} - 5 q^{57} + 5 q^{58} + 10 q^{59} - q^{60} - 7 q^{61} + 3 q^{62} + q^{64} - 6 q^{65} + q^{66} + 8 q^{67} + 7 q^{68} - 6 q^{69} + 7 q^{71} - 2 q^{72} - 14 q^{73} + 3 q^{74} + q^{75} - 5 q^{76} + 6 q^{78} + 10 q^{79} - q^{80} + q^{81} - 2 q^{82} + 6 q^{83} - 7 q^{85} + 4 q^{86} + 5 q^{87} + q^{88} + 15 q^{89} + 2 q^{90} - 6 q^{92} + 3 q^{93} + 2 q^{94} + 5 q^{95} + q^{96} + 12 q^{97} - 2 q^{99}+O(q^{100}) \) 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 −1.00000 1.00000 0 1.00000 −2.00000 −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(1\)
\(7\) \(-1\)
\(11\) \(-1\)

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 5390.2.a.bf 1
7.b odd 2 1 110.2.a.b 1
21.c even 2 1 990.2.a.d 1
28.d even 2 1 880.2.a.i 1
35.c odd 2 1 550.2.a.f 1
35.f even 4 2 550.2.b.a 2
56.e even 2 1 3520.2.a.h 1
56.h odd 2 1 3520.2.a.y 1
77.b even 2 1 1210.2.a.b 1
84.h odd 2 1 7920.2.a.d 1
105.g even 2 1 4950.2.a.bc 1
105.k odd 4 2 4950.2.c.m 2
140.c even 2 1 4400.2.a.l 1
140.j odd 4 2 4400.2.b.i 2
308.g odd 2 1 9680.2.a.x 1
385.h even 2 1 6050.2.a.bj 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
110.2.a.b 1 7.b odd 2 1
550.2.a.f 1 35.c odd 2 1
550.2.b.a 2 35.f even 4 2
880.2.a.i 1 28.d even 2 1
990.2.a.d 1 21.c even 2 1
1210.2.a.b 1 77.b even 2 1
3520.2.a.h 1 56.e even 2 1
3520.2.a.y 1 56.h odd 2 1
4400.2.a.l 1 140.c even 2 1
4400.2.b.i 2 140.j odd 4 2
4950.2.a.bc 1 105.g even 2 1
4950.2.c.m 2 105.k odd 4 2
5390.2.a.bf 1 1.a even 1 1 trivial
6050.2.a.bj 1 385.h even 2 1
7920.2.a.d 1 84.h odd 2 1
9680.2.a.x 1 308.g odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5390))\):

\( T_{3} - 1 \) Copy content Toggle raw display
\( T_{13} - 6 \) Copy content Toggle raw display
\( T_{17} - 7 \) Copy content Toggle raw display
\( T_{19} + 5 \) Copy content Toggle raw display
\( T_{31} - 3 \) 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
$5$ \( T + 1 \) Copy content Toggle raw display
$7$ \( T \) Copy content Toggle raw display
$11$ \( T - 1 \) Copy content Toggle raw display
$13$ \( T - 6 \) Copy content Toggle raw display
$17$ \( T - 7 \) Copy content Toggle raw display
$19$ \( T + 5 \) Copy content Toggle raw display
$23$ \( T + 6 \) Copy content Toggle raw display
$29$ \( T - 5 \) Copy content Toggle raw display
$31$ \( T - 3 \) Copy content Toggle raw display
$37$ \( T - 3 \) Copy content Toggle raw display
$41$ \( T + 2 \) Copy content Toggle raw display
$43$ \( T - 4 \) Copy content Toggle raw display
$47$ \( T - 2 \) Copy content Toggle raw display
$53$ \( T + 1 \) Copy content Toggle raw display
$59$ \( T - 10 \) Copy content Toggle raw display
$61$ \( T + 7 \) Copy content Toggle raw display
$67$ \( T - 8 \) Copy content Toggle raw display
$71$ \( T - 7 \) Copy content Toggle raw display
$73$ \( T + 14 \) Copy content Toggle raw display
$79$ \( T - 10 \) Copy content Toggle raw display
$83$ \( T - 6 \) Copy content Toggle raw display
$89$ \( T - 15 \) Copy content Toggle raw display
$97$ \( T - 12 \) Copy content Toggle raw display
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