Properties

Label 5376.2.a.bg
Level $5376$
Weight $2$
Character orbit 5376.a
Self dual yes
Analytic conductor $42.928$
Analytic rank $0$
Dimension $3$
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] = [5376,2,Mod(1,5376)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5376, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("5376.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5376.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: \(42.9275761266\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.1016.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 2688)
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)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_{2} q^{5} - q^{7} + q^{9} + ( - \beta_{2} - \beta_1 + 1) q^{11} + (\beta_1 - 1) q^{13} - \beta_{2} q^{15} + (\beta_{2} - \beta_1 - 1) q^{17} + ( - \beta_1 - 3) q^{19} + q^{21} + ( - \beta_{2} + \beta_1 - 1) q^{23} + ( - 2 \beta_{2} - \beta_1 + 4) q^{25} - q^{27} + ( - \beta_1 - 3) q^{29} + ( - \beta_1 + 1) q^{31} + (\beta_{2} + \beta_1 - 1) q^{33} - \beta_{2} q^{35} + (2 \beta_{2} + \beta_1 + 3) q^{37} + ( - \beta_1 + 1) q^{39} + (\beta_{2} + \beta_1 + 5) q^{41} + 2 \beta_{2} q^{43} + \beta_{2} q^{45} + (2 \beta_{2} + 2 \beta_1 - 2) q^{47} + q^{49} + ( - \beta_{2} + \beta_1 + 1) q^{51} + (2 \beta_{2} + \beta_1 + 3) q^{53} + (4 \beta_{2} - \beta_1 - 7) q^{55} + (\beta_1 + 3) q^{57} + ( - 2 \beta_{2} + 2 \beta_1 + 2) q^{59} + (2 \beta_{2} - \beta_1 - 7) q^{61} - q^{63} + ( - 2 \beta_{2} + 2 \beta_1 - 2) q^{65} - 8 q^{67} + (\beta_{2} - \beta_1 + 1) q^{69} + ( - 3 \beta_{2} - \beta_1 + 1) q^{71} + (2 \beta_{2} + 10) q^{73} + (2 \beta_{2} + \beta_1 - 4) q^{75} + (\beta_{2} + \beta_1 - 1) q^{77} + (2 \beta_{2} + 8) q^{79} + q^{81} + (4 \beta_{2} + 4) q^{83} + ( - 2 \beta_{2} - 3 \beta_1 + 11) q^{85} + (\beta_1 + 3) q^{87} + ( - \beta_{2} - \beta_1 + 11) q^{89} + ( - \beta_1 + 1) q^{91} + (\beta_1 - 1) q^{93} + ( - 2 \beta_{2} - 2 \beta_1 + 2) q^{95} + ( - 2 \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{2} - \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} - 3 q^{7} + 3 q^{9} + 4 q^{11} - 4 q^{13} - 2 q^{17} - 8 q^{19} + 3 q^{21} - 4 q^{23} + 13 q^{25} - 3 q^{27} - 8 q^{29} + 4 q^{31} - 4 q^{33} + 8 q^{37} + 4 q^{39} + 14 q^{41} - 8 q^{47} + 3 q^{49} + 2 q^{51} + 8 q^{53} - 20 q^{55} + 8 q^{57} + 4 q^{59} - 20 q^{61} - 3 q^{63} - 8 q^{65} - 24 q^{67} + 4 q^{69} + 4 q^{71} + 30 q^{73} - 13 q^{75} - 4 q^{77} + 24 q^{79} + 3 q^{81} + 12 q^{83} + 36 q^{85} + 8 q^{87} + 34 q^{89} + 4 q^{91} - 4 q^{93} + 8 q^{95} + 2 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 6x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{2} + \beta _1 + 9 ) / 2 \) 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
0.321637
2.85577
−2.17741
0 −1.00000 0 −4.21819 0 −1.00000 0 1.00000 0
1.2 0 −1.00000 0 1.29966 0 −1.00000 0 1.00000 0
1.3 0 −1.00000 0 2.91852 0 −1.00000 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5376.2.a.bg 3
4.b odd 2 1 5376.2.a.bj 3
8.b even 2 1 5376.2.a.bi 3
8.d odd 2 1 5376.2.a.bh 3
16.e even 4 2 2688.2.c.h yes 6
16.f odd 4 2 2688.2.c.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2688.2.c.g 6 16.f odd 4 2
2688.2.c.h yes 6 16.e even 4 2
5376.2.a.bg 3 1.a even 1 1 trivial
5376.2.a.bh 3 8.d odd 2 1
5376.2.a.bi 3 8.b even 2 1
5376.2.a.bj 3 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5376))\):

\( T_{5}^{3} - 14T_{5} + 16 \) Copy content Toggle raw display
\( T_{11}^{3} - 4T_{11}^{2} - 26T_{11} + 96 \) Copy content Toggle raw display
\( T_{13}^{3} + 4T_{13}^{2} - 20T_{13} - 32 \) Copy content Toggle raw display
\( T_{19}^{3} + 8T_{19}^{2} - 4T_{19} - 48 \) Copy content Toggle raw display
\( T_{29}^{3} + 8T_{29}^{2} - 4T_{29} - 48 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( (T + 1)^{3} \) Copy content Toggle raw display
$5$ \( T^{3} - 14T + 16 \) Copy content Toggle raw display
$7$ \( (T + 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} - 4 T^{2} - 26 T + 96 \) Copy content Toggle raw display
$13$ \( T^{3} + 4 T^{2} - 20 T - 32 \) Copy content Toggle raw display
$17$ \( T^{3} + 2 T^{2} - 46 T - 156 \) Copy content Toggle raw display
$19$ \( T^{3} + 8 T^{2} - 4 T - 48 \) Copy content Toggle raw display
$23$ \( T^{3} + 4 T^{2} - 42 T + 64 \) Copy content Toggle raw display
$29$ \( T^{3} + 8 T^{2} - 4 T - 48 \) Copy content Toggle raw display
$31$ \( T^{3} - 4 T^{2} - 20 T + 32 \) Copy content Toggle raw display
$37$ \( T^{3} - 8 T^{2} - 44 T + 208 \) Copy content Toggle raw display
$41$ \( T^{3} - 14 T^{2} + 34 T - 12 \) Copy content Toggle raw display
$43$ \( T^{3} - 56T + 128 \) Copy content Toggle raw display
$47$ \( T^{3} + 8 T^{2} - 104 T - 768 \) Copy content Toggle raw display
$53$ \( T^{3} - 8 T^{2} - 44 T + 208 \) Copy content Toggle raw display
$59$ \( T^{3} - 4 T^{2} - 184 T + 1248 \) Copy content Toggle raw display
$61$ \( T^{3} + 20 T^{2} + 36 T - 576 \) Copy content Toggle raw display
$67$ \( (T + 8)^{3} \) Copy content Toggle raw display
$71$ \( T^{3} - 4 T^{2} - 122 T - 256 \) Copy content Toggle raw display
$73$ \( T^{3} - 30 T^{2} + 244 T - 312 \) Copy content Toggle raw display
$79$ \( T^{3} - 24 T^{2} + 136 T + 64 \) Copy content Toggle raw display
$83$ \( T^{3} - 12 T^{2} - 176 T + 1856 \) Copy content Toggle raw display
$89$ \( T^{3} - 34 T^{2} + 354 T - 1044 \) Copy content Toggle raw display
$97$ \( T^{3} - 2 T^{2} - 124 T + 536 \) Copy content Toggle raw display
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