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Properties

Label 5824.2.a.cm
Level $5824$
Weight $2$
Character orbit 5824.a
Self dual yes
Analytic conductor $46.505$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.153499364.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 7x^{4} + 18x^{3} + 19x^{2} - 25x - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{3} + \beta_{5} q^{5} + q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1) q^{3} + \beta_{5} q^{5} + q^{7} + ( - \beta_{5} + \beta_{3} - \beta_1 + 1) q^{9} + ( - \beta_{4} + \beta_{3} + \beta_1 - 2) q^{11} + q^{13} + ( - \beta_{4} - \beta_{3} + \cdots - 2 \beta_1) q^{15}+ \cdots + (2 \beta_{5} + 3 \beta_{4} - \beta_{3} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{3} - 3 q^{5} + 6 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{3} - 3 q^{5} + 6 q^{7} + 8 q^{9} - 8 q^{11} + 6 q^{13} - 2 q^{15} - 2 q^{17} - 9 q^{19} - 4 q^{21} - 11 q^{23} + 21 q^{25} - 22 q^{27} - 7 q^{29} + q^{31} + 18 q^{33} - 3 q^{35} - 16 q^{37} - 4 q^{39} - q^{43} - 29 q^{45} - 11 q^{47} + 6 q^{49} + 8 q^{51} - 23 q^{53} + 18 q^{55} + 10 q^{57} - 10 q^{59} - 10 q^{61} + 8 q^{63} - 3 q^{65} + 8 q^{67} - 6 q^{69} - 14 q^{71} + 11 q^{73} - 14 q^{75} - 8 q^{77} - 3 q^{79} + 30 q^{81} - 17 q^{83} - 6 q^{85} - 36 q^{87} - 5 q^{89} + 6 q^{91} + 16 q^{93} - 45 q^{95} + 27 q^{97} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 3x^{5} - 7x^{4} + 18x^{3} + 19x^{2} - 25x - 19 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu^{2} - \nu - 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{3} - 2\nu^{2} - 4\nu + 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{5} - 2\nu^{4} - 7\nu^{3} + 7\nu^{2} + 14\nu + 1 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - \nu^{3} + 15\nu^{2} - 4\nu - 11 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{5} - 4\nu^{4} - 3\nu^{3} + 19\nu^{2} - 17 ) / 2 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + \beta_{4} - \beta_{2} + 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{5} + 3\beta_{4} - 2\beta_{2} + 2\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -13\beta_{5} + 11\beta_{4} + 2\beta_{3} - 7\beta_{2} + 20\beta _1 + 37 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -47\beta_{5} + 43\beta_{4} + 8\beta_{3} - 21\beta_{2} + 54\beta _1 + 79 ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
1.49402
−0.672304
2.37463
−1.44305
−1.80744
3.05415
0 −3.26192 0 1.46001 0 1.00000 0 7.64015 0
1.2 0 −2.87570 0 −4.22752 0 1.00000 0 5.26967 0
1.3 0 −0.735766 0 −0.856892 0 1.00000 0 −2.45865 0
1.4 0 −0.474556 0 3.98872 0 1.00000 0 −2.77480 0
1.5 0 1.07428 0 0.402677 0 1.00000 0 −1.84591 0
1.6 0 2.27366 0 −3.76699 0 1.00000 0 2.16955 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.6
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.cm 6
4.b odd 2 1 5824.2.a.cn 6
8.b even 2 1 2912.2.a.v yes 6
8.d odd 2 1 2912.2.a.u ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2912.2.a.u ✓ 6 8.d odd 2 1
2912.2.a.v yes 6 8.b even 2 1
5824.2.a.cm 6 1.a even 1 1 trivial
5824.2.a.cn 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(5824))\):

\( T_{3}^{6} + 4T_{3}^{5} - 5T_{3}^{4} - 26T_{3}^{3} + 22T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{6} + 3T_{5}^{5} - 21T_{5}^{4} - 51T_{5}^{3} + 82T_{5}^{2} + 56T_{5} - 32 \) Copy content Toggle raw display
\( T_{11}^{6} + 8T_{11}^{5} - 13T_{11}^{4} - 224T_{11}^{3} - 370T_{11}^{2} + 366T_{11} + 536 \) Copy content Toggle raw display
\( T_{17}^{6} + 2T_{17}^{5} - 100T_{17}^{4} - 118T_{17}^{3} + 2900T_{17}^{2} + 1040T_{17} - 21952 \) Copy content Toggle raw display
\( T_{19}^{6} + 9T_{19}^{5} + 9T_{19}^{4} - 77T_{19}^{3} - 116T_{19}^{2} + 164T_{19} + 224 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + 4 T^{5} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{6} + 3 T^{5} + \cdots - 32 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( T^{6} + 8 T^{5} + \cdots + 536 \) Copy content Toggle raw display
$13$ \( (T - 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} + 2 T^{5} + \cdots - 21952 \) Copy content Toggle raw display
$19$ \( T^{6} + 9 T^{5} + \cdots + 224 \) Copy content Toggle raw display
$23$ \( T^{6} + 11 T^{5} + \cdots - 1204 \) Copy content Toggle raw display
$29$ \( T^{6} + 7 T^{5} + \cdots + 8800 \) Copy content Toggle raw display
$31$ \( T^{6} - T^{5} + \cdots - 72104 \) Copy content Toggle raw display
$37$ \( T^{6} + 16 T^{5} + \cdots - 6404 \) Copy content Toggle raw display
$41$ \( T^{6} - 137 T^{4} + \cdots + 3344 \) Copy content Toggle raw display
$43$ \( T^{6} + T^{5} + \cdots - 54592 \) Copy content Toggle raw display
$47$ \( T^{6} + 11 T^{5} + \cdots - 304 \) Copy content Toggle raw display
$53$ \( T^{6} + 23 T^{5} + \cdots - 103712 \) Copy content Toggle raw display
$59$ \( T^{6} + 10 T^{5} + \cdots + 110848 \) Copy content Toggle raw display
$61$ \( T^{6} + 10 T^{5} + \cdots + 112 \) Copy content Toggle raw display
$67$ \( T^{6} - 8 T^{5} + \cdots - 148832 \) Copy content Toggle raw display
$71$ \( T^{6} + 14 T^{5} + \cdots + 4096 \) Copy content Toggle raw display
$73$ \( T^{6} - 11 T^{5} + \cdots - 27542 \) Copy content Toggle raw display
$79$ \( T^{6} + 3 T^{5} + \cdots + 20108 \) Copy content Toggle raw display
$83$ \( T^{6} + 17 T^{5} + \cdots + 1408 \) Copy content Toggle raw display
$89$ \( T^{6} + 5 T^{5} + \cdots - 30968 \) Copy content Toggle raw display
$97$ \( T^{6} - 27 T^{5} + \cdots - 662438 \) Copy content Toggle raw display
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