Properties

Label 8008.2.a.k
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
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] = [8008,2,Mod(1,8008)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, 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("8008.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8008.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: \(63.9442019386\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.244558277.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 29x^{2} + 5x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{5} q^{5} + q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{5} q^{5} + q^{7} + (\beta_{2} + 2) q^{9} - q^{11} - q^{13} + (\beta_{5} - \beta_{4} - \beta_{3} - \beta_1) q^{15} + (\beta_{5} + \beta_{4} + \beta_1) q^{17} + (\beta_{5} - \beta_{4} + \beta_{2} + 1) q^{19} - \beta_1 q^{21} + ( - \beta_{4} - \beta_{3} - 2 \beta_1 + 2) q^{23} + ( - \beta_{5} - \beta_{3} + \beta_1 + 1) q^{25} + (\beta_{5} - \beta_{4} + \beta_{2} - 3 \beta_1 + 1) q^{27} + ( - \beta_{5} - \beta_{3} + 2) q^{29} + (\beta_{5} - \beta_{4} - 2 \beta_1 + 4) q^{31} + \beta_1 q^{33} - \beta_{5} q^{35} + (\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{37} + \beta_1 q^{39} + ( - \beta_{5} - \beta_{3} - 2 \beta_1) q^{41} + (\beta_{4} + \beta_{3} + \beta_1 + 2) q^{43} + ( - \beta_{5} + 2 \beta_{4} + 2 \beta_{3} + 3 \beta_1) q^{45} + ( - \beta_{3} + \beta_{2} - 2 \beta_1 + 1) q^{47} + q^{49} + (\beta_{4} + \beta_{3} - \beta_{2} - 3) q^{51} + ( - \beta_{4} + \beta_{3} + 3 \beta_1) q^{53} + \beta_{5} q^{55} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 3 \beta_1 - 1) q^{57} + ( - 2 \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1) q^{59} + ( - 2 \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 - 3) q^{61} + (\beta_{2} + 2) q^{63} + \beta_{5} q^{65} + ( - 2 \beta_{5} - \beta_{3} - \beta_{2} - \beta_1 - 5) q^{67} + ( - 3 \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} + 5) q^{69} + (\beta_{2} - 2 \beta_1 - 1) q^{71} + ( - 2 \beta_{5} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 1) q^{73} + ( - \beta_{5} - 2 \beta_{2} - \beta_1 - 8) q^{75} - q^{77} + ( - 2 \beta_{5} + \beta_{4} - 2 \beta_1 + 4) q^{79} + ( - \beta_{5} + \beta_{3} + \beta_{2} - 3 \beta_1 + 8) q^{81} + (\beta_{5} + \beta_{4} - \beta_{2} + 4 \beta_1 + 3) q^{83} + (2 \beta_{3} + \beta_{2} + \beta_1 - 7) q^{85} + ( - \beta_{5} - \beta_{2} - 2 \beta_1 - 3) q^{87} + (\beta_{5} - \beta_{2} - \beta_1 + 9) q^{89} - q^{91} + ( - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} - 2 \beta_1 + 8) q^{93} + (\beta_{5} + 3 \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 5) q^{95} + ( - 2 \beta_{5} - \beta_{4} - \beta_{2} + 2 \beta_1 + 5) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - q^{3} + q^{5} + 6 q^{7} + 9 q^{9} - 6 q^{11} - 6 q^{13} - q^{15} + 2 q^{19} - q^{21} + 11 q^{23} + 9 q^{25} - q^{27} + 14 q^{29} + 21 q^{31} + q^{33} + q^{35} - 5 q^{37} + q^{39} + 12 q^{43} + 2 q^{45} + 2 q^{47} + 6 q^{49} - 16 q^{51} + 2 q^{53} - q^{55} - 12 q^{57} + 3 q^{59} - 18 q^{61} + 9 q^{63} - q^{65} - 25 q^{67} + 29 q^{69} - 11 q^{71} + 14 q^{73} - 42 q^{75} - 6 q^{77} + 24 q^{79} + 42 q^{81} + 24 q^{83} - 46 q^{85} - 16 q^{87} + 55 q^{89} - 6 q^{91} + 41 q^{93} - 30 q^{95} + 37 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} - x^{5} - 13x^{4} + 11x^{3} + 29x^{2} + 5x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{5} - \nu^{4} - 12\nu^{3} + 12\nu^{2} + 19\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 2\nu^{4} - 11\nu^{3} + 23\nu^{2} + 7\nu - 10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 2\nu^{4} - 12\nu^{3} + 22\nu^{2} + 16\nu - 6 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{5} + \beta_{4} - \beta_{2} + 9\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{5} + \beta_{3} + 10\beta_{2} - 3\beta _1 + 44 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -13\beta_{5} + 12\beta_{4} + 2\beta_{3} - 14\beta_{2} + 86\beta _1 - 28 \) 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
2.98317
2.50454
0.287253
−0.676795
−0.847024
−3.25115
0 −2.98317 0 3.19751 0 1.00000 0 5.89931 0
1.2 0 −2.50454 0 −3.40126 0 1.00000 0 3.27272 0
1.3 0 −0.287253 0 −0.115270 0 1.00000 0 −2.91749 0
1.4 0 0.676795 0 3.59312 0 1.00000 0 −2.54195 0
1.5 0 0.847024 0 −2.05840 0 1.00000 0 −2.28255 0
1.6 0 3.25115 0 −0.215704 0 1.00000 0 7.56995 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
\(2\) \(1\)
\(7\) \(-1\)
\(11\) \(1\)
\(13\) \(1\)

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 8008.2.a.k 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.k 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{6} + T_{3}^{5} - 13T_{3}^{4} - 11T_{3}^{3} + 29T_{3}^{2} - 5T_{3} - 4 \) Copy content Toggle raw display
\( T_{5}^{6} - T_{5}^{5} - 19T_{5}^{4} + 9T_{5}^{3} + 85T_{5}^{2} + 27T_{5} + 2 \) Copy content Toggle raw display
\( T_{17}^{6} - 71T_{17}^{4} - 20T_{17}^{3} + 1225T_{17}^{2} + 200T_{17} - 1792 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} + T^{5} - 13 T^{4} - 11 T^{3} + \cdots - 4 \) Copy content Toggle raw display
$5$ \( T^{6} - T^{5} - 19 T^{4} + 9 T^{3} + \cdots + 2 \) Copy content Toggle raw display
$7$ \( (T - 1)^{6} \) Copy content Toggle raw display
$11$ \( (T + 1)^{6} \) Copy content Toggle raw display
$13$ \( (T + 1)^{6} \) Copy content Toggle raw display
$17$ \( T^{6} - 71 T^{4} - 20 T^{3} + \cdots - 1792 \) Copy content Toggle raw display
$19$ \( T^{6} - 2 T^{5} - 79 T^{4} + \cdots + 1088 \) Copy content Toggle raw display
$23$ \( T^{6} - 11 T^{5} - 26 T^{4} + \cdots + 1984 \) Copy content Toggle raw display
$29$ \( T^{6} - 14 T^{5} + 24 T^{4} + \cdots + 512 \) Copy content Toggle raw display
$31$ \( T^{6} - 21 T^{5} + 96 T^{4} + \cdots - 12736 \) Copy content Toggle raw display
$37$ \( T^{6} + 5 T^{5} - 94 T^{4} + \cdots + 13888 \) Copy content Toggle raw display
$41$ \( T^{6} - 84 T^{4} - 112 T^{3} + \cdots - 1664 \) Copy content Toggle raw display
$43$ \( T^{6} - 12 T^{5} + T^{4} + 326 T^{3} + \cdots + 3688 \) Copy content Toggle raw display
$47$ \( T^{6} - 2 T^{5} - 124 T^{4} + \cdots - 1024 \) Copy content Toggle raw display
$53$ \( T^{6} - 2 T^{5} - 231 T^{4} + \cdots - 111076 \) Copy content Toggle raw display
$59$ \( T^{6} - 3 T^{5} - 292 T^{4} + \cdots - 294304 \) Copy content Toggle raw display
$61$ \( T^{6} + 18 T^{5} - 93 T^{4} + \cdots + 7576 \) Copy content Toggle raw display
$67$ \( T^{6} + 25 T^{5} + 99 T^{4} + \cdots - 152236 \) Copy content Toggle raw display
$71$ \( T^{6} + 11 T^{5} - 53 T^{4} + \cdots - 15694 \) Copy content Toggle raw display
$73$ \( T^{6} - 14 T^{5} - 248 T^{4} + \cdots + 445504 \) Copy content Toggle raw display
$79$ \( T^{6} - 24 T^{5} + 67 T^{4} + \cdots + 160496 \) Copy content Toggle raw display
$83$ \( T^{6} - 24 T^{5} - 19 T^{4} + \cdots + 6568 \) Copy content Toggle raw display
$89$ \( T^{6} - 55 T^{5} + 1169 T^{4} + \cdots - 151876 \) Copy content Toggle raw display
$97$ \( T^{6} - 37 T^{5} + 284 T^{4} + \cdots + 220864 \) Copy content Toggle raw display
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