Properties

Label 6864.2.a.ce
Level $6864$
Weight $2$
Character orbit 6864.a
Self dual yes
Analytic conductor $54.809$
Analytic rank $0$
Dimension $5$
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] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.46437524.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 3432)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_1 q^{5} + (\beta_{4} + 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_1 q^{5} + (\beta_{4} + 1) q^{7} + q^{9} - q^{11} + q^{13} - \beta_1 q^{15} + ( - \beta_{4} + \beta_1 + 1) q^{17} + ( - \beta_{2} - 2) q^{19} + ( - \beta_{4} - 1) q^{21} + ( - \beta_{4} + 1) q^{23} + (\beta_{3} + \beta_1 + 3) q^{25} - q^{27} + ( - \beta_{4} - \beta_{3} - \beta_{2} - 1) q^{29} + (\beta_{4} + \beta_{2} + \beta_1 + 1) q^{31} + q^{33} + ( - \beta_{4} - \beta_{2} - 1) q^{35} + (2 \beta_{4} + 2) q^{37} - q^{39} + (\beta_{4} + 3) q^{41} + (2 \beta_{4} + \beta_{2} + \beta_1) q^{43} + \beta_1 q^{45} + ( - \beta_{4} + \beta_{3} + \beta_1 - 1) q^{47} + (2 \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 + 3) q^{49} + (\beta_{4} - \beta_1 - 1) q^{51} + (\beta_{4} + \beta_{3} + \beta_1 + 1) q^{53} - \beta_1 q^{55} + (\beta_{2} + 2) q^{57} + ( - \beta_{3} + \beta_{2} - \beta_1 - 4) q^{59} + ( - \beta_{4} - \beta_{2} - 2 \beta_1 - 3) q^{61} + (\beta_{4} + 1) q^{63} + \beta_1 q^{65} + (\beta_1 - 2) q^{67} + (\beta_{4} - 1) q^{69} + (\beta_{4} - \beta_{3} + 2 \beta_{2} + \beta_1 + 1) q^{71} + ( - \beta_{3} - \beta_1 + 4) q^{73} + ( - \beta_{3} - \beta_1 - 3) q^{75} + ( - \beta_{4} - 1) q^{77} + (\beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{79} + q^{81} + 2 \beta_1 q^{83} + (\beta_{4} + \beta_{3} + \beta_{2} + 3 \beta_1 + 9) q^{85} + (\beta_{4} + \beta_{3} + \beta_{2} + 1) q^{87} + (3 \beta_{4} + \beta_{2} + 3 \beta_1 + 3) q^{89} + (\beta_{4} + 1) q^{91} + ( - \beta_{4} - \beta_{2} - \beta_1 - 1) q^{93} + (\beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 1) q^{95} + (\beta_{4} + \beta_{3} - \beta_1 + 7) q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 4\nu^{3} - 17\nu^{2} - 60\nu + 8 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} - \nu - 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{4} + 5\nu^{3} + 51\nu^{2} - 58\nu - 109 ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{2} + 14\beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -4\beta_{4} + 17\beta_{3} + 5\beta_{2} + 21\beta _1 + 108 \) 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
−3.73149
−1.57692
−0.447720
2.49314
4.26299
0 −1.00000 0 −3.73149 0 −0.404219 0 1.00000 0
1.2 0 −1.00000 0 −1.57692 0 5.18377 0 1.00000 0
1.3 0 −1.00000 0 −0.447720 0 −3.31638 0 1.00000 0
1.4 0 −1.00000 0 2.49314 0 2.46927 0 1.00000 0
1.5 0 −1.00000 0 4.26299 0 −0.932447 0 1.00000 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
\(2\) \(1\)
\(3\) \(1\)
\(11\) \(1\)
\(13\) \(-1\)

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 6864.2.a.ce 5
4.b odd 2 1 3432.2.a.x 5
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3432.2.a.x 5 4.b odd 2 1
6864.2.a.ce 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6864))\):

\( T_{5}^{5} - T_{5}^{4} - 20T_{5}^{3} + 8T_{5}^{2} + 70T_{5} + 28 \) Copy content Toggle raw display
\( T_{7}^{5} - 3T_{7}^{4} - 18T_{7}^{3} + 24T_{7}^{2} + 52T_{7} + 16 \) Copy content Toggle raw display
\( T_{17}^{5} - 8T_{17}^{4} - 20T_{17}^{3} + 202T_{17}^{2} - 56T_{17} - 448 \) Copy content Toggle raw display
\( T_{19}^{5} + 10T_{19}^{4} - 16T_{19}^{3} - 420T_{19}^{2} - 1232T_{19} - 784 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{5} \) Copy content Toggle raw display
$3$ \( (T + 1)^{5} \) Copy content Toggle raw display
$5$ \( T^{5} - T^{4} - 20 T^{3} + 8 T^{2} + \cdots + 28 \) Copy content Toggle raw display
$7$ \( T^{5} - 3 T^{4} - 18 T^{3} + 24 T^{2} + \cdots + 16 \) Copy content Toggle raw display
$11$ \( (T + 1)^{5} \) Copy content Toggle raw display
$13$ \( (T - 1)^{5} \) Copy content Toggle raw display
$17$ \( T^{5} - 8 T^{4} - 20 T^{3} + 202 T^{2} + \cdots - 448 \) Copy content Toggle raw display
$19$ \( T^{5} + 10 T^{4} - 16 T^{3} + \cdots - 784 \) Copy content Toggle raw display
$23$ \( T^{5} - 7 T^{4} - 2 T^{3} + 76 T^{2} + \cdots - 56 \) Copy content Toggle raw display
$29$ \( T^{5} + 3 T^{4} - 180 T^{3} + \cdots + 31468 \) Copy content Toggle raw display
$31$ \( T^{5} - 4 T^{4} - 70 T^{3} + \cdots + 2016 \) Copy content Toggle raw display
$37$ \( T^{5} - 6 T^{4} - 72 T^{3} + 192 T^{2} + \cdots + 512 \) Copy content Toggle raw display
$41$ \( T^{5} - 13 T^{4} + 46 T^{3} - 20 T^{2} + \cdots + 72 \) Copy content Toggle raw display
$43$ \( T^{5} + 3 T^{4} - 118 T^{3} + \cdots + 9184 \) Copy content Toggle raw display
$47$ \( T^{5} + 2 T^{4} - 156 T^{3} + \cdots + 14336 \) Copy content Toggle raw display
$53$ \( T^{5} - 4 T^{4} - 112 T^{3} + \cdots - 128 \) Copy content Toggle raw display
$59$ \( T^{5} + 21 T^{4} + 32 T^{3} + \cdots + 28096 \) Copy content Toggle raw display
$61$ \( T^{5} + 15 T^{4} - 42 T^{3} + \cdots + 224 \) Copy content Toggle raw display
$67$ \( T^{5} + 9 T^{4} + 12 T^{3} - 56 T^{2} + \cdots + 56 \) Copy content Toggle raw display
$71$ \( T^{5} - 4 T^{4} - 296 T^{3} + \cdots - 90944 \) Copy content Toggle raw display
$73$ \( T^{5} - 19 T^{4} + 28 T^{3} + \cdots - 7712 \) Copy content Toggle raw display
$79$ \( T^{5} + 8 T^{4} - 312 T^{3} + \cdots - 21056 \) Copy content Toggle raw display
$83$ \( T^{5} - 2 T^{4} - 80 T^{3} + 64 T^{2} + \cdots + 896 \) Copy content Toggle raw display
$89$ \( T^{5} - 12 T^{4} - 290 T^{3} + \cdots - 164704 \) Copy content Toggle raw display
$97$ \( T^{5} - 32 T^{4} + 260 T^{3} + \cdots + 8064 \) Copy content Toggle raw display
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