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Properties

Label 6288.2.a.bh
Level $6288$
Weight $2$
Character orbit 6288.a
Self dual yes
Analytic conductor $50.210$
Analytic rank $0$
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: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.254064397.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 14x^{3} + 29x^{2} - 46x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 1572)
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.

\(f(q)\) \(=\) \( q - q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{3} + \beta_{3} q^{5} + \beta_{2} q^{7} + q^{9} + (\beta_1 - 2) q^{11} + (\beta_{5} + 1) q^{13} - \beta_{3} q^{15} + (\beta_1 + 2) q^{17} + (\beta_{5} - \beta_{3} - \beta_{2} - 1) q^{19} - \beta_{2} q^{21} + (2 \beta_{3} + \beta_{2}) q^{23} + ( - \beta_{5} + \beta_{4} + \beta_{3} + \cdots + 1) q^{25}+ \cdots + (\beta_1 - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{3} + 2 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{3} + 2 q^{5} + 6 q^{9} - 11 q^{11} + 6 q^{13} - 2 q^{15} + 13 q^{17} - 8 q^{19} + 4 q^{23} + 6 q^{25} - 6 q^{27} + q^{29} + 9 q^{31} + 11 q^{33} - 13 q^{35} - 2 q^{37} - 6 q^{39} + 21 q^{41} - 9 q^{43} + 2 q^{45} - 15 q^{47} + 16 q^{49} - 13 q^{51} + 16 q^{53} - 9 q^{55} + 8 q^{57} - 10 q^{59} - q^{61} + 17 q^{65} + 2 q^{67} - 4 q^{69} - 25 q^{71} + 12 q^{73} - 6 q^{75} + 13 q^{77} + 28 q^{79} + 6 q^{81} - 11 q^{83} - q^{85} - q^{87} + 10 q^{89} + 17 q^{91} - 9 q^{93} - 10 q^{95} + 6 q^{97} - 11 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 12x^{4} + 14x^{3} + 29x^{2} - 46x + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} - \nu^{4} - 12\nu^{3} + 12\nu^{2} + 31\nu - 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\nu^{5} + 12\nu^{3} - 2\nu^{2} - 30\nu + 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - 12\nu^{3} + 3\nu^{2} + 31\nu - 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3\nu^{5} - \nu^{4} - 38\nu^{3} + 18\nu^{2} + 109\nu - 76 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 5\nu^{5} - \nu^{4} - 60\nu^{3} + 22\nu^{2} + 159\nu - 104 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} + 4\beta_{3} + 2\beta_{2} + \beta _1 + 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{5} - 3\beta_{4} - 5\beta_{3} + 8\beta_{2} - 5\beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{5} + 13\beta_{3} + 6\beta_{2} + \beta _1 + 23 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 68\beta_{5} - 36\beta_{4} - 38\beta_{3} + 59\beta_{2} - 32\beta _1 - 16 ) / 3 \) 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
1.70569
0.653797
−2.13409
0.817992
2.84525
−2.88864
0 −1.00000 0 −3.50767 0 4.12274 0 1.00000 0
1.2 0 −1.00000 0 −1.68407 0 −1.23468 0 1.00000 0
1.3 0 −1.00000 0 −0.126583 0 −1.45316 0 1.00000 0
1.4 0 −1.00000 0 1.16338 0 −3.67628 0 1.00000 0
1.5 0 −1.00000 0 2.55353 0 4.38718 0 1.00000 0
1.6 0 −1.00000 0 3.60141 0 −2.14580 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.bh 6
4.b odd 2 1 1572.2.a.f ✓ 6
12.b even 2 1 4716.2.a.j 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1572.2.a.f ✓ 6 4.b odd 2 1
4716.2.a.j 6 12.b even 2 1
6288.2.a.bh 6 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6288))\):

\( T_{5}^{6} - 2T_{5}^{5} - 16T_{5}^{4} + 29T_{5}^{3} + 45T_{5}^{2} - 58T_{5} - 8 \) Copy content Toggle raw display
\( T_{7}^{6} - 29T_{7}^{4} - 30T_{7}^{3} + 203T_{7}^{2} + 452T_{7} + 256 \) Copy content Toggle raw display
\( T_{17}^{6} - 13T_{17}^{5} + 46T_{17}^{4} - 7T_{17}^{3} - 186T_{17}^{2} + 241T_{17} - 64 \) 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
$5$ \( T^{6} - 2 T^{5} + \cdots - 8 \) Copy content Toggle raw display
$7$ \( T^{6} - 29 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( T^{6} + 11 T^{5} + \cdots + 36 \) Copy content Toggle raw display
$13$ \( T^{6} - 6 T^{5} + \cdots + 2 \) Copy content Toggle raw display
$17$ \( T^{6} - 13 T^{5} + \cdots - 64 \) Copy content Toggle raw display
$19$ \( T^{6} + 8 T^{5} + \cdots - 202 \) Copy content Toggle raw display
$23$ \( T^{6} - 4 T^{5} + \cdots + 1472 \) Copy content Toggle raw display
$29$ \( T^{6} - T^{5} + \cdots + 36 \) Copy content Toggle raw display
$31$ \( T^{6} - 9 T^{5} + \cdots - 35642 \) Copy content Toggle raw display
$37$ \( T^{6} + 2 T^{5} + \cdots - 232 \) Copy content Toggle raw display
$41$ \( T^{6} - 21 T^{5} + \cdots + 72 \) Copy content Toggle raw display
$43$ \( T^{6} + 9 T^{5} + \cdots - 632 \) Copy content Toggle raw display
$47$ \( T^{6} + 15 T^{5} + \cdots + 38824 \) Copy content Toggle raw display
$53$ \( T^{6} - 16 T^{5} + \cdots + 2368 \) Copy content Toggle raw display
$59$ \( T^{6} + 10 T^{5} + \cdots + 184284 \) Copy content Toggle raw display
$61$ \( T^{6} + T^{5} - 24 T^{4} + \cdots + 2 \) Copy content Toggle raw display
$67$ \( T^{6} - 2 T^{5} + \cdots - 6712 \) Copy content Toggle raw display
$71$ \( T^{6} + 25 T^{5} + \cdots - 1696 \) Copy content Toggle raw display
$73$ \( T^{6} - 12 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$79$ \( T^{6} - 28 T^{5} + \cdots - 18072 \) Copy content Toggle raw display
$83$ \( T^{6} + 11 T^{5} + \cdots - 681048 \) Copy content Toggle raw display
$89$ \( T^{6} - 10 T^{5} + \cdots - 1635836 \) Copy content Toggle raw display
$97$ \( T^{6} - 6 T^{5} + \cdots - 304884 \) Copy content Toggle raw display
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