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Properties

Label 6288.2.a.bi
Level $6288$
Weight $2$
Character orbit 6288.a
Self dual yes
Analytic conductor $50.210$
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] = [6288,2,Mod(1,6288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6288, 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("6288.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6288 = 2^{4} \cdot 3 \cdot 131 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6288.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: \(50.2099327910\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 15x^{4} + 28x^{3} + 39x^{2} - 44x - 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3144)
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.

\(f(q)\) \(=\) \( q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_1 - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + ( - \beta_{2} + 1) q^{5} + ( - \beta_1 - 1) q^{7} + q^{9} + \beta_{3} q^{11} + ( - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{13} + (\beta_{2} - 1) q^{15} - \beta_{3} q^{17} + (\beta_{4} - 3) q^{19} + (\beta_1 + 1) q^{21} + ( - \beta_{5} + \beta_1 - 1) q^{23} + ( - \beta_{2} + \beta_1 + 2) q^{25} - q^{27} + (\beta_{5} + \beta_1 - 2) q^{29} + (\beta_{5} - \beta_{3} + \beta_1 - 1) q^{31} - \beta_{3} q^{33} + (\beta_{5} - \beta_{4} - \beta_{3} + \cdots - 1) q^{35}+ \cdots + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 3 q^{5} - 8 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 3 q^{5} - 8 q^{7} + 6 q^{9} + q^{11} + 9 q^{13} - 3 q^{15} - q^{17} - 20 q^{19} + 8 q^{21} - 5 q^{23} + 11 q^{25} - 6 q^{27} - 9 q^{29} - 4 q^{31} - q^{33} + 2 q^{35} + 12 q^{37} - 9 q^{39} + 20 q^{41} - 9 q^{43} + 3 q^{45} + 2 q^{47} + 2 q^{49} + q^{51} - 5 q^{53} + 20 q^{57} - 24 q^{59} + 4 q^{61} - 8 q^{63} - 23 q^{65} - 26 q^{67} + 5 q^{69} - 2 q^{71} + 20 q^{73} - 11 q^{75} + 2 q^{77} - 52 q^{79} + 6 q^{81} - 10 q^{83} + 9 q^{87} - 7 q^{89} + 3 q^{91} + 4 q^{93} - 16 q^{95} - 12 q^{97} + q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 15x^{4} + 28x^{3} + 39x^{2} - 44x - 36 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} - 15\nu^{3} + 2\nu^{2} + 39\nu + 6 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{5} + 15\nu^{3} + 6\nu^{2} - 39\nu - 46 ) / 8 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 11\nu^{3} - 54\nu^{2} + 9\nu + 66 ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{5} + 4\nu^{4} + 15\nu^{3} - 50\nu^{2} - 27\nu + 50 ) / 8 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{5} - 2\beta_{4} - \beta_{3} - \beta_{2} + 9\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{5} + 12\beta_{3} + 14\beta_{2} - 3\beta _1 + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 30\beta_{5} - 30\beta_{4} - 17\beta_{3} - 9\beta_{2} + 96\beta _1 - 31 \) 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
3.41157
1.49057
−3.47507
−1.22623
−0.672231
2.47139
0 −1.00000 0 −2.60831 0 −4.41157 0 1.00000 0
1.2 0 −1.00000 0 −2.28219 0 −2.49057 0 1.00000 0
1.3 0 −1.00000 0 −1.16581 0 2.47507 0 1.00000 0
1.4 0 −1.00000 0 2.74138 0 0.226229 0 1.00000 0
1.5 0 −1.00000 0 2.86173 0 −0.327769 0 1.00000 0
1.6 0 −1.00000 0 3.45320 0 −3.47139 0 1.00000 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
\(2\) \( +1 \)
\(3\) \( +1 \)
\(131\) \( +1 \)

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 6288.2.a.bi 6
4.b odd 2 1 3144.2.a.k ✓ 6
12.b even 2 1 9432.2.a.q 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3144.2.a.k ✓ 6 4.b odd 2 1
6288.2.a.bi 6 1.a even 1 1 trivial
9432.2.a.q 6 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6288))\):

\( T_{5}^{6} - 3T_{5}^{5} - 16T_{5}^{4} + 39T_{5}^{3} + 90T_{5}^{2} - 127T_{5} - 188 \) Copy content Toggle raw display
\( T_{7}^{6} + 8T_{7}^{5} + 10T_{7}^{4} - 48T_{7}^{3} - 100T_{7}^{2} - 6T_{7} + 7 \) Copy content Toggle raw display
\( T_{17}^{6} + T_{17}^{5} - 46T_{17}^{4} - 65T_{17}^{3} + 480T_{17}^{2} + 443T_{17} - 1514 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( (T + 1)^{6} \) Copy content Toggle raw display
$5$ \( T^{6} - 3 T^{5} + \cdots - 188 \) Copy content Toggle raw display
$7$ \( T^{6} + 8 T^{5} + \cdots + 7 \) Copy content Toggle raw display
$11$ \( T^{6} - T^{5} + \cdots - 1514 \) Copy content Toggle raw display
$13$ \( T^{6} - 9 T^{5} + \cdots + 968 \) Copy content Toggle raw display
$17$ \( T^{6} + T^{5} + \cdots - 1514 \) Copy content Toggle raw display
$19$ \( T^{6} + 20 T^{5} + \cdots - 696 \) Copy content Toggle raw display
$23$ \( T^{6} + 5 T^{5} + \cdots + 5302 \) Copy content Toggle raw display
$29$ \( T^{6} + 9 T^{5} + \cdots + 10808 \) Copy content Toggle raw display
$31$ \( T^{6} + 4 T^{5} + \cdots - 5596 \) Copy content Toggle raw display
$37$ \( T^{6} - 12 T^{5} + \cdots - 2725 \) Copy content Toggle raw display
$41$ \( T^{6} - 20 T^{5} + \cdots + 82438 \) Copy content Toggle raw display
$43$ \( T^{6} + 9 T^{5} + \cdots - 144 \) Copy content Toggle raw display
$47$ \( T^{6} - 2 T^{5} + \cdots - 1316 \) Copy content Toggle raw display
$53$ \( T^{6} + 5 T^{5} + \cdots + 58 \) Copy content Toggle raw display
$59$ \( T^{6} + 24 T^{5} + \cdots - 5972 \) Copy content Toggle raw display
$61$ \( T^{6} - 4 T^{5} + \cdots - 45464 \) Copy content Toggle raw display
$67$ \( T^{6} + 26 T^{5} + \cdots + 5713 \) Copy content Toggle raw display
$71$ \( T^{6} + 2 T^{5} + \cdots - 36762 \) Copy content Toggle raw display
$73$ \( T^{6} - 20 T^{5} + \cdots - 392 \) Copy content Toggle raw display
$79$ \( T^{6} + 52 T^{5} + \cdots + 43144 \) Copy content Toggle raw display
$83$ \( T^{6} + 10 T^{5} + \cdots + 5082 \) Copy content Toggle raw display
$89$ \( T^{6} + 7 T^{5} + \cdots - 17400 \) Copy content Toggle raw display
$97$ \( T^{6} + 12 T^{5} + \cdots - 76696 \) Copy content Toggle raw display
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