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Properties

Label 8008.2.a.p
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
Dimension $9$
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: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{3} q^{5} - q^{7} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{3} q^{5} - q^{7} + (\beta_{7} + \beta_{5} - \beta_{3}) q^{9} + q^{11} + q^{13} + ( - \beta_{8} - \beta_{6} + \beta_{3} + \cdots - 1) q^{15}+ \cdots + (\beta_{7} + \beta_{5} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{3} - 4 q^{5} - 9 q^{7} + 4 q^{9} + 9 q^{11} + 9 q^{13} - 9 q^{15} - 11 q^{17} + 10 q^{19} - q^{21} - 14 q^{23} - q^{25} - 5 q^{27} - 10 q^{29} + 5 q^{31} + q^{33} + 4 q^{35} - 16 q^{37} + q^{39} + 2 q^{41} + 4 q^{43} - 30 q^{45} + 9 q^{49} + 3 q^{51} - 23 q^{53} - 4 q^{55} + 14 q^{57} + 9 q^{59} - 14 q^{61} - 4 q^{63} - 4 q^{65} + 8 q^{67} - 26 q^{69} - 20 q^{71} - 23 q^{73} + 32 q^{75} - 9 q^{77} + 2 q^{79} - 11 q^{81} - 9 q^{83} - 3 q^{85} - 7 q^{87} - 6 q^{89} - 9 q^{91} - 19 q^{93} - 4 q^{95} - 3 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 15x^{7} + 15x^{6} + 66x^{5} - 59x^{4} - 77x^{3} + 34x^{2} + 11x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 2\nu^{7} + 2\nu^{6} - 25\nu^{5} - 19\nu^{4} + 77\nu^{3} + 40\nu^{2} - 13\nu - 10 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{8} - 15\nu^{6} + 66\nu^{4} + 7\nu^{3} - 79\nu^{2} - 45\nu + 11 ) / 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2\nu^{8} - 3\nu^{7} - 27\nu^{6} + 45\nu^{5} + 93\nu^{4} - 181\nu^{3} - 26\nu^{2} + 129\nu - 14 ) / 9 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 31\nu^{6} - 17\nu^{5} - 146\nu^{4} + 74\nu^{3} + 205\nu^{2} - 47\nu - 45 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3\nu^{8} - 2\nu^{7} - 44\nu^{6} + 31\nu^{5} + 190\nu^{4} - 116\nu^{3} - 217\nu^{2} + 25\nu + 13 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{8} - \nu^{7} - 46\nu^{6} + 17\nu^{5} + 212\nu^{4} - 67\nu^{3} - 275\nu^{2} + 2\nu + 29 ) / 9 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -3\nu^{8} + 4\nu^{7} + 46\nu^{6} - 56\nu^{5} - 209\nu^{4} + 202\nu^{3} + 257\nu^{2} - 83\nu - 23 ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + \beta_{5} - \beta_{3} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + \beta_{6} - \beta_{2} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} + 5\beta_{7} + \beta_{6} + 6\beta_{5} - \beta_{4} - 7\beta_{3} + \beta_{2} - \beta _1 + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 12\beta_{8} + \beta_{7} + 10\beta_{6} - 2\beta_{5} + \beta_{4} - 3\beta_{3} - 11\beta_{2} + 27\beta _1 - 4 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{8} + 23\beta_{7} + 12\beta_{6} + 38\beta_{5} - 11\beta_{4} - 46\beta_{3} + 14\beta_{2} - 12\beta _1 + 120 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 115\beta_{8} + 17\beta_{7} + 84\beta_{6} - 26\beta_{5} + 14\beta_{4} - 38\beta_{3} - 99\beta_{2} + 154\beta _1 - 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 136 \beta_{8} + 94 \beta_{7} + 107 \beta_{6} + 253 \beta_{5} - 99 \beta_{4} - 298 \beta_{3} + \cdots + 838 \) 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
−2.82075
−2.37534
−1.00974
−0.333725
0.141545
0.536705
1.94061
2.24883
2.67186
0 −2.82075 0 −1.92109 0 −1.00000 0 4.95665 0
1.2 0 −2.37534 0 −0.155697 0 −1.00000 0 2.64224 0
1.3 0 −1.00974 0 2.49743 0 −1.00000 0 −1.98043 0
1.4 0 −0.333725 0 1.97301 0 −1.00000 0 −2.88863 0
1.5 0 0.141545 0 0.343768 0 −1.00000 0 −2.97996 0
1.6 0 0.536705 0 −3.30011 0 −1.00000 0 −2.71195 0
1.7 0 1.94061 0 1.48281 0 −1.00000 0 0.765982 0
1.8 0 2.24883 0 −0.903831 0 −1.00000 0 2.05724 0
1.9 0 2.67186 0 −4.01631 0 −1.00000 0 4.13885 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
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.p ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.p ✓ 9 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{9} - T_{3}^{8} - 15T_{3}^{7} + 15T_{3}^{6} + 66T_{3}^{5} - 59T_{3}^{4} - 77T_{3}^{3} + 34T_{3}^{2} + 11T_{3} - 2 \) Copy content Toggle raw display
\( T_{5}^{9} + 4T_{5}^{8} - 14T_{5}^{7} - 50T_{5}^{6} + 73T_{5}^{5} + 173T_{5}^{4} - 138T_{5}^{3} - 159T_{5}^{2} + 37T_{5} + 9 \) Copy content Toggle raw display
\( T_{17}^{9} + 11 T_{17}^{8} - 25 T_{17}^{7} - 619 T_{17}^{6} - 1068 T_{17}^{5} + 8059 T_{17}^{4} + \cdots - 70614 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - T^{8} - 15 T^{7} + \cdots - 2 \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( (T + 1)^{9} \) Copy content Toggle raw display
$11$ \( (T - 1)^{9} \) Copy content Toggle raw display
$13$ \( (T - 1)^{9} \) Copy content Toggle raw display
$17$ \( T^{9} + 11 T^{8} + \cdots - 70614 \) Copy content Toggle raw display
$19$ \( T^{9} - 10 T^{8} + \cdots + 2349 \) Copy content Toggle raw display
$23$ \( T^{9} + 14 T^{8} + \cdots - 7236 \) Copy content Toggle raw display
$29$ \( T^{9} + 10 T^{8} + \cdots + 1388 \) Copy content Toggle raw display
$31$ \( T^{9} - 5 T^{8} + \cdots - 3336612 \) Copy content Toggle raw display
$37$ \( T^{9} + 16 T^{8} + \cdots + 7045848 \) Copy content Toggle raw display
$41$ \( T^{9} - 2 T^{8} + \cdots + 832472 \) Copy content Toggle raw display
$43$ \( T^{9} - 4 T^{8} + \cdots + 15441 \) Copy content Toggle raw display
$47$ \( T^{9} - 201 T^{7} + \cdots + 311652 \) Copy content Toggle raw display
$53$ \( T^{9} + 23 T^{8} + \cdots - 6408999 \) Copy content Toggle raw display
$59$ \( T^{9} - 9 T^{8} + \cdots - 373248 \) Copy content Toggle raw display
$61$ \( T^{9} + 14 T^{8} + \cdots - 5885662 \) Copy content Toggle raw display
$67$ \( T^{9} - 8 T^{8} + \cdots + 378174 \) Copy content Toggle raw display
$71$ \( T^{9} + 20 T^{8} + \cdots - 1319904 \) Copy content Toggle raw display
$73$ \( T^{9} + 23 T^{8} + \cdots - 512844 \) Copy content Toggle raw display
$79$ \( T^{9} - 2 T^{8} + \cdots + 79277529 \) Copy content Toggle raw display
$83$ \( T^{9} + 9 T^{8} + \cdots + 2314287 \) Copy content Toggle raw display
$89$ \( T^{9} + 6 T^{8} + \cdots + 103041 \) Copy content Toggle raw display
$97$ \( T^{9} + 3 T^{8} + \cdots + 28521072 \) Copy content Toggle raw display
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