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Properties

Label 6468.2.a.bf
Level $6468$
Weight $2$
Character orbit 6468.a
Self dual yes
Analytic conductor $51.647$
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] = [6468,2,Mod(1,6468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6468, 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("6468.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6468 = 2^{2} \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6468.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: \(51.6472400274\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.152932864.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 15x^{4} + 20x^{3} + 58x^{2} - 16x - 28 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_1 + 1) q^{5} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_1 + 1) q^{5} + q^{9} + q^{11} + (\beta_{5} + 1) q^{13} + ( - \beta_1 + 1) q^{15} + (\beta_{4} + 1) q^{17} + ( - \beta_{5} - \beta_{3} + 1) q^{19} + ( - \beta_{5} - \beta_{3} - \beta_{2} + \cdots + 2) q^{23}+ \cdots + q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 6 q^{3} + 4 q^{5} + 6 q^{9} + 6 q^{11} + 8 q^{13} + 4 q^{15} + 8 q^{17} + 4 q^{19} + 8 q^{23} + 6 q^{25} + 6 q^{27} + 4 q^{29} + 8 q^{31} + 6 q^{33} + 4 q^{37} + 8 q^{39} + 4 q^{41} + 8 q^{43} + 4 q^{45} + 4 q^{47} + 8 q^{51} - 4 q^{53} + 4 q^{55} + 4 q^{57} + 16 q^{59} + 20 q^{65} - 8 q^{67} + 8 q^{69} + 12 q^{71} + 20 q^{73} + 6 q^{75} - 16 q^{79} + 6 q^{81} + 8 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} + 8 q^{93} - 4 q^{95} + 24 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - 2x^{5} - 15x^{4} + 20x^{3} + 58x^{2} - 16x - 28 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{5} + 2\nu^{4} - 7\nu^{3} - 26\nu^{2} - 28\nu + 16 ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -2\nu^{5} - 4\nu^{4} + 23\nu^{3} + 43\nu^{2} - 16\nu - 14 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{5} - 2\nu^{4} + 13\nu^{3} + 26\nu^{2} - 26\nu - 34 ) / 6 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{5} + 5\nu^{4} + 23\nu^{3} - 47\nu^{2} - 52\nu + 40 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{4} - \beta_{3} - \beta_{2} + \beta _1 + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{2} + 9\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{5} + 10\beta_{4} - 11\beta_{3} - 10\beta_{2} + 14\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{5} + 13\beta_{4} - 4\beta_{3} + 33\beta_{2} + 89\beta _1 + 47 \) 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
3.31746
3.19601
0.799723
−0.668473
−1.64899
−2.99573
0 1.00000 0 −2.31746 0 0 0 1.00000 0
1.2 0 1.00000 0 −2.19601 0 0 0 1.00000 0
1.3 0 1.00000 0 0.200277 0 0 0 1.00000 0
1.4 0 1.00000 0 1.66847 0 0 0 1.00000 0
1.5 0 1.00000 0 2.64899 0 0 0 1.00000 0
1.6 0 1.00000 0 3.99573 0 0 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
\(2\) \(-1\)
\(3\) \(-1\)
\(7\) \(1\)
\(11\) \(-1\)

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 6468.2.a.bf yes 6
7.b odd 2 1 6468.2.a.bc ✓ 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6468.2.a.bc ✓ 6 7.b odd 2 1
6468.2.a.bf yes 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(6468))\):

\( T_{5}^{6} - 4T_{5}^{5} - 10T_{5}^{4} + 40T_{5}^{3} + 23T_{5}^{2} - 96T_{5} + 18 \) Copy content Toggle raw display
\( T_{13}^{6} - 8T_{13}^{5} - 12T_{13}^{4} + 144T_{13}^{3} + 13T_{13}^{2} - 136T_{13} + 14 \) Copy content Toggle raw display
\( T_{17}^{6} - 8T_{17}^{5} - 8T_{17}^{4} + 160T_{17}^{3} - 240T_{17}^{2} - 64T_{17} + 56 \) Copy content Toggle raw display
\( T_{23}^{6} - 8T_{23}^{5} - 30T_{23}^{4} + 344T_{23}^{3} - 132T_{23}^{2} - 3584T_{23} + 6272 \) 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
$5$ \( T^{6} - 4 T^{5} + \cdots + 18 \) Copy content Toggle raw display
$7$ \( T^{6} \) Copy content Toggle raw display
$11$ \( (T - 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{6} - 8 T^{5} + \cdots + 14 \) Copy content Toggle raw display
$17$ \( T^{6} - 8 T^{5} + \cdots + 56 \) Copy content Toggle raw display
$19$ \( T^{6} - 4 T^{5} + \cdots + 568 \) Copy content Toggle raw display
$23$ \( T^{6} - 8 T^{5} + \cdots + 6272 \) Copy content Toggle raw display
$29$ \( T^{6} - 4 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$31$ \( T^{6} - 8 T^{5} + \cdots - 12832 \) Copy content Toggle raw display
$37$ \( T^{6} - 4 T^{5} + \cdots - 1792 \) Copy content Toggle raw display
$41$ \( T^{6} - 4 T^{5} + \cdots - 3896 \) Copy content Toggle raw display
$43$ \( T^{6} - 8 T^{5} + \cdots - 6272 \) Copy content Toggle raw display
$47$ \( T^{6} - 4 T^{5} + \cdots + 568 \) Copy content Toggle raw display
$53$ \( T^{6} + 4 T^{5} + \cdots + 24944 \) Copy content Toggle raw display
$59$ \( T^{6} - 16 T^{5} + \cdots + 4336 \) Copy content Toggle raw display
$61$ \( T^{6} - 138 T^{4} + \cdots + 3296 \) Copy content Toggle raw display
$67$ \( T^{6} + 8 T^{5} + \cdots - 45712 \) Copy content Toggle raw display
$71$ \( T^{6} - 12 T^{5} + \cdots + 10304 \) Copy content Toggle raw display
$73$ \( T^{6} - 20 T^{5} + \cdots + 351166 \) Copy content Toggle raw display
$79$ \( T^{6} + 16 T^{5} + \cdots + 50176 \) Copy content Toggle raw display
$83$ \( T^{6} - 8 T^{5} + \cdots - 800 \) Copy content Toggle raw display
$89$ \( T^{6} - 4 T^{5} + \cdots + 224 \) Copy content Toggle raw display
$97$ \( T^{6} - 24 T^{5} + \cdots + 88776 \) Copy content Toggle raw display
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