Properties

Label 5824.2.a.bv
Level $5824$
Weight $2$
Character orbit 5824.a
Self dual yes
Analytic conductor $46.505$
Analytic rank $1$
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] = [5824,2,Mod(1,5824)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, 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("5824.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5824 = 2^{6} \cdot 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5824.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: \(46.5048741372\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 2912)
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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_1 q^{5} + q^{7} + ( - \beta_{2} - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_1 q^{5} + q^{7} + ( - \beta_{2} - \beta_1) q^{9} + (2 \beta_{2} + \beta_1 - 1) q^{11} + q^{13} + (\beta_1 - 1) q^{15} + (\beta_{2} - 2 \beta_1 - 2) q^{17} + (\beta_1 + 2) q^{19} - \beta_{2} q^{21} + (2 \beta_1 - 3) q^{23} + (\beta_{2} + \beta_1 - 3) q^{25} + (2 \beta_{2} + 2) q^{27} + ( - 2 \beta_{2} - 2 \beta_1 + 3) q^{29} + ( - \beta_{2} + 4 \beta_1 - 3) q^{31} + (3 \beta_{2} + \beta_1 - 5) q^{33} - \beta_1 q^{35} + (2 \beta_{2} + 3 \beta_1 + 1) q^{37} - \beta_{2} q^{39} + ( - 2 \beta_{2} - 2 \beta_1) q^{41} + (\beta_{2} + 3 \beta_1 - 4) q^{43} + (\beta_{2} + 2 \beta_1 + 1) q^{45} + ( - 2 \beta_{2} - 5 \beta_1) q^{47} + q^{49} + (3 \beta_{2} + 3 \beta_1 - 5) q^{51} + (5 \beta_{2} + 3 \beta_1) q^{53} + ( - \beta_{2} - 2 \beta_1) q^{55} + ( - 2 \beta_{2} - \beta_1 + 1) q^{57} + ( - \beta_{2} - 3 \beta_1 - 5) q^{59} + ( - 2 \beta_{2} + 2 \beta_1 + 4) q^{61} + ( - \beta_{2} - \beta_1) q^{63} - \beta_1 q^{65} + (4 \beta_{2} + 2 \beta_1 - 4) q^{67} + (3 \beta_{2} - 2 \beta_1 + 2) q^{69} + (4 \beta_{2} + \beta_1 - 1) q^{71} + (\beta_{2} + 4 \beta_1 - 9) q^{73} + (4 \beta_{2} - 2) q^{75} + (2 \beta_{2} + \beta_1 - 1) q^{77} + ( - 3 \beta_{2} + 5 \beta_1 - 4) q^{79} + (3 \beta_{2} + 5 \beta_1 - 6) q^{81} + ( - 7 \beta_{2} - 2 \beta_1 + 3) q^{83} + (2 \beta_{2} + 3 \beta_1 + 5) q^{85} + ( - 5 \beta_{2} + 4) q^{87} + (5 \beta_{2} - 1) q^{89} + q^{91} + (2 \beta_{2} - 5 \beta_1 + 7) q^{93} + ( - \beta_{2} - 3 \beta_1 - 2) q^{95} + (3 \beta_{2} - 4 \beta_1 - 1) q^{97} + (2 \beta_{2} - \beta_1 - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{5} + 3 q^{7} - q^{9} - 2 q^{11} + 3 q^{13} - 2 q^{15} - 8 q^{17} + 7 q^{19} - 7 q^{23} - 8 q^{25} + 6 q^{27} + 7 q^{29} - 5 q^{31} - 14 q^{33} - q^{35} + 6 q^{37} - 2 q^{41} - 9 q^{43} + 5 q^{45} - 5 q^{47} + 3 q^{49} - 12 q^{51} + 3 q^{53} - 2 q^{55} + 2 q^{57} - 18 q^{59} + 14 q^{61} - q^{63} - q^{65} - 10 q^{67} + 4 q^{69} - 2 q^{71} - 23 q^{73} - 6 q^{75} - 2 q^{77} - 7 q^{79} - 13 q^{81} + 7 q^{83} + 18 q^{85} + 12 q^{87} - 3 q^{89} + 3 q^{91} + 16 q^{93} - 9 q^{95} - 7 q^{97} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{3} - x^{2} - 3x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 2 \) 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
−1.48119
2.17009
0.311108
0 −1.67513 0 1.48119 0 1.00000 0 −0.193937 0
1.2 0 −0.539189 0 −2.17009 0 1.00000 0 −2.70928 0
1.3 0 2.21432 0 −0.311108 0 1.00000 0 1.90321 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

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

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 5824.2.a.bv 3
4.b odd 2 1 5824.2.a.bu 3
8.b even 2 1 2912.2.a.k yes 3
8.d odd 2 1 2912.2.a.j 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2912.2.a.j 3 8.d odd 2 1
2912.2.a.k yes 3 8.b even 2 1
5824.2.a.bu 3 4.b odd 2 1
5824.2.a.bv 3 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(5824))\):

\( T_{3}^{3} - 4T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{3} + T_{5}^{2} - 3T_{5} - 1 \) Copy content Toggle raw display
\( T_{11}^{3} + 2T_{11}^{2} - 14T_{11} + 10 \) Copy content Toggle raw display
\( T_{17}^{3} + 8T_{17}^{2} - 74 \) Copy content Toggle raw display
\( T_{19}^{3} - 7T_{19}^{2} + 13T_{19} - 5 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{3} \) Copy content Toggle raw display
$3$ \( T^{3} - 4T - 2 \) Copy content Toggle raw display
$5$ \( T^{3} + T^{2} - 3T - 1 \) Copy content Toggle raw display
$7$ \( (T - 1)^{3} \) Copy content Toggle raw display
$11$ \( T^{3} + 2 T^{2} - 14 T + 10 \) Copy content Toggle raw display
$13$ \( (T - 1)^{3} \) Copy content Toggle raw display
$17$ \( T^{3} + 8T^{2} - 74 \) Copy content Toggle raw display
$19$ \( T^{3} - 7 T^{2} + 13 T - 5 \) Copy content Toggle raw display
$23$ \( T^{3} + 7 T^{2} + 3 T - 19 \) Copy content Toggle raw display
$29$ \( T^{3} - 7 T^{2} - 5 T + 43 \) Copy content Toggle raw display
$31$ \( T^{3} + 5 T^{2} - 57 T + 25 \) Copy content Toggle raw display
$37$ \( T^{3} - 6 T^{2} - 22 T - 2 \) Copy content Toggle raw display
$41$ \( T^{3} + 2 T^{2} - 20 T - 8 \) Copy content Toggle raw display
$43$ \( T^{3} + 9 T^{2} - T - 109 \) Copy content Toggle raw display
$47$ \( T^{3} + 5 T^{2} - 71 T + 139 \) Copy content Toggle raw display
$53$ \( T^{3} - 3 T^{2} - 97 T + 367 \) Copy content Toggle raw display
$59$ \( T^{3} + 18 T^{2} + 80 T + 100 \) Copy content Toggle raw display
$61$ \( T^{3} - 14 T^{2} + 28 T + 152 \) Copy content Toggle raw display
$67$ \( T^{3} + 10 T^{2} - 28 T - 8 \) Copy content Toggle raw display
$71$ \( T^{3} + 2 T^{2} - 58 T + 134 \) Copy content Toggle raw display
$73$ \( T^{3} + 23 T^{2} + 127 T - 29 \) Copy content Toggle raw display
$79$ \( T^{3} + 7 T^{2} - 133 T + 361 \) Copy content Toggle raw display
$83$ \( T^{3} - 7 T^{2} - 165 T - 527 \) Copy content Toggle raw display
$89$ \( T^{3} + 3 T^{2} - 97 T + 151 \) Copy content Toggle raw display
$97$ \( T^{3} + 7 T^{2} - 97 T - 713 \) Copy content Toggle raw display
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