Properties

Label 5824.2.a.ck
Level $5824$
Weight $2$
Character orbit 5824.a
Self dual yes
Analytic conductor $46.505$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.6329476.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 13x^{3} - 2x^{2} + 40x + 18 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + (\beta_{3} - 1) q^{5} + q^{7} + (\beta_{2} + 2) q^{9} + (\beta_{3} + 2) q^{11} - q^{13} + (\beta_{4} + \beta_{2}) q^{15} + ( - \beta_{3} + \beta_{2} + 1) q^{17} + ( - \beta_{3} + 2 \beta_{2} - 1) q^{19} + \beta_1 q^{21} + ( - \beta_{4} - \beta_1) q^{23} + ( - \beta_{2} + 3) q^{25} + (\beta_{3} - \beta_{2} + \beta_1 + 1) q^{27} + (\beta_{4} + \beta_{3} + \beta_{2} - 1) q^{29} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{31} + (\beta_{4} + \beta_{2} + 3 \beta_1) q^{33} + (\beta_{3} - 1) q^{35} + ( - \beta_{4} - \beta_{3} + \beta_1 - 3) q^{37} - \beta_1 q^{39} + ( - \beta_{3} - \beta_{2} + \beta_1 + 5) q^{41} + \beta_{2} q^{43} + (\beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{45} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 - 1) q^{47} + q^{49} + ( - \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{51} + (\beta_{2} + 2 \beta_1 - 2) q^{53} + (3 \beta_{3} - \beta_{2} + 5) q^{55} + ( - \beta_{4} + 2 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 2) q^{57} + (\beta_{4} - \beta_{3} + 3) q^{59} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 2 \beta_1 + 2) q^{61} + (\beta_{2} + 2) q^{63} + ( - \beta_{3} + 1) q^{65} + (\beta_{4} + \beta_{3} + \beta_{2} + 2 \beta_1) q^{67} + ( - \beta_{4} - 4 \beta_{3} - 1) q^{69} + ( - \beta_{4} + \beta_{2} - 2 \beta_1 + 2) q^{71} + ( - \beta_{4} + \beta_{3} - 3 \beta_{2} - \beta_1 + 1) q^{73} + ( - \beta_{3} + \beta_{2} + \beta_1 - 1) q^{75} + (\beta_{3} + 2) q^{77} + (\beta_{3} - 2 \beta_{2} + \beta_1 - 1) q^{79} + (\beta_{4} - \beta_{3} - 2) q^{81} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{83} + (\beta_{4} - \beta_{2} + 2 \beta_1 - 6) q^{85} + (2 \beta_{4} + 5 \beta_{3} - \beta_{2} + 2 \beta_1 - 3) q^{87} + ( - \beta_{4} + 2 \beta_{2} + 2) q^{89} - q^{91} + (\beta_{4} + \beta_{3} - 2 \beta_{2} + 5 \beta_1 - 9) q^{93} + (2 \beta_{4} - 2 \beta_{3} - 3 \beta_{2} + 4 \beta_1 - 2) q^{95} + ( - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 + 5) q^{97} + (\beta_{4} + 2 \beta_{3} + \beta_{2} + 2 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 3 q^{5} + 5 q^{7} + 11 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 3 q^{5} + 5 q^{7} + 11 q^{9} + 12 q^{11} - 5 q^{13} + 2 q^{15} + 4 q^{17} - 5 q^{19} - q^{23} + 14 q^{25} + 6 q^{27} - q^{29} + 13 q^{31} + 2 q^{33} - 3 q^{35} - 18 q^{37} + 22 q^{41} + q^{43} + 3 q^{45} - 7 q^{47} + 5 q^{49} + 4 q^{51} - 9 q^{53} + 30 q^{55} + 10 q^{57} + 14 q^{59} + 12 q^{61} + 11 q^{63} + 3 q^{65} + 4 q^{67} - 14 q^{69} + 10 q^{71} + 3 q^{73} - 6 q^{75} + 12 q^{77} - 5 q^{79} - 11 q^{81} + 11 q^{83} - 30 q^{85} - 4 q^{87} + 11 q^{89} - 5 q^{91} - 44 q^{93} - 15 q^{95} + 27 q^{97} + 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - 13x^{3} - 2x^{2} + 40x + 18 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + \nu^{2} - 7\nu - 6 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + \nu^{3} - 8\nu^{2} - 7\nu + 5 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - \beta_{2} + 7\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - \beta_{3} + 9\beta_{2} + 34 \) 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
−2.86011
−1.92896
−0.472539
2.50170
2.75990
0 −2.86011 0 −2.19539 0 1.00000 0 5.18025 0
1.2 0 −1.92896 0 3.04617 0 1.00000 0 0.720870 0
1.3 0 −0.472539 0 −3.57445 0 1.00000 0 −2.77671 0
1.4 0 2.50170 0 −2.59644 0 1.00000 0 3.25852 0
1.5 0 2.75990 0 2.32011 0 1.00000 0 4.61707 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.5
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.ck 5
4.b odd 2 1 5824.2.a.cj 5
8.b even 2 1 2912.2.a.s yes 5
8.d odd 2 1 2912.2.a.r 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2912.2.a.r 5 8.d odd 2 1
2912.2.a.s yes 5 8.b even 2 1
5824.2.a.cj 5 4.b odd 2 1
5824.2.a.ck 5 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5824))\):

\( T_{3}^{5} - 13T_{3}^{3} - 2T_{3}^{2} + 40T_{3} + 18 \) Copy content Toggle raw display
\( T_{5}^{5} + 3T_{5}^{4} - 15T_{5}^{3} - 43T_{5}^{2} + 52T_{5} + 144 \) Copy content Toggle raw display
\( T_{11}^{5} - 12T_{11}^{4} + 39T_{11}^{3} - 16T_{11}^{2} - 14T_{11} + 6 \) Copy content Toggle raw display
\( T_{17}^{5} - 4T_{17}^{4} - 24T_{17}^{3} + 74T_{17}^{2} + 88T_{17} - 32 \) Copy content Toggle raw display
\( T_{19}^{5} + 5T_{19}^{4} - 75T_{19}^{3} - 205T_{19}^{2} + 1484T_{19} - 1132 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( T^{5} - 13 T^{3} - 2 T^{2} + 40 T + 18 \) Copy content Toggle raw display
$5$ \( T^{5} + 3 T^{4} - 15 T^{3} - 43 T^{2} + \cdots + 144 \) Copy content Toggle raw display
$7$ \( (T - 1)^{5} \) Copy content Toggle raw display
$11$ \( T^{5} - 12 T^{4} + 39 T^{3} - 16 T^{2} + \cdots + 6 \) Copy content Toggle raw display
$13$ \( (T + 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 4 T^{4} - 24 T^{3} + 74 T^{2} + \cdots - 32 \) Copy content Toggle raw display
$19$ \( T^{5} + 5 T^{4} - 75 T^{3} + \cdots - 1132 \) Copy content Toggle raw display
$23$ \( T^{5} + T^{4} - 74 T^{3} - 50 T^{2} + \cdots + 321 \) Copy content Toggle raw display
$29$ \( T^{5} + T^{4} - 93 T^{3} - 77 T^{2} + \cdots + 1728 \) Copy content Toggle raw display
$31$ \( T^{5} - 13 T^{4} - 18 T^{3} + \cdots - 2979 \) Copy content Toggle raw display
$37$ \( T^{5} + 18 T^{4} + 29 T^{3} + \cdots - 16278 \) Copy content Toggle raw display
$41$ \( T^{5} - 22 T^{4} + 139 T^{3} - 230 T^{2} + \cdots - 8 \) Copy content Toggle raw display
$43$ \( T^{5} - T^{4} - 21 T^{3} + 41 T^{2} + \cdots - 64 \) Copy content Toggle raw display
$47$ \( T^{5} + 7 T^{4} - 100 T^{3} + \cdots + 151 \) Copy content Toggle raw display
$53$ \( T^{5} + 9 T^{4} - 53 T^{3} + \cdots + 6544 \) Copy content Toggle raw display
$59$ \( T^{5} - 14 T^{4} - 28 T^{3} + \cdots - 5424 \) Copy content Toggle raw display
$61$ \( T^{5} - 12 T^{4} - 107 T^{3} + \cdots - 5736 \) Copy content Toggle raw display
$67$ \( T^{5} - 4 T^{4} - 131 T^{3} + \cdots - 312 \) Copy content Toggle raw display
$71$ \( T^{5} - 10 T^{4} - 86 T^{3} + \cdots - 16736 \) Copy content Toggle raw display
$73$ \( T^{5} - 3 T^{4} - 238 T^{3} + \cdots - 9221 \) Copy content Toggle raw display
$79$ \( T^{5} + 5 T^{4} - 78 T^{3} - 2 T^{2} + \cdots + 93 \) Copy content Toggle raw display
$83$ \( T^{5} - 11 T^{4} - 145 T^{3} + \cdots + 16 \) Copy content Toggle raw display
$89$ \( T^{5} - 11 T^{4} - 141 T^{3} + \cdots + 12604 \) Copy content Toggle raw display
$97$ \( T^{5} - 27 T^{4} + 106 T^{3} + \cdots + 14703 \) Copy content Toggle raw display
show more
show less