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Properties

Label 8008.2.a.q
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
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: \(0\)
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: \( x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 5 \)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + ( - \beta_{7} + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + ( - \beta_{7} + 1) q^{5} + q^{7} + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{9} + q^{11} - q^{13} + ( - \beta_{7} - \beta_{4} + \beta_1 - 1) q^{15} + (\beta_{4} + \beta_{2} + \beta_1) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \cdots + 2) q^{19}+ \cdots + (\beta_{3} + \beta_{2} + \beta_1 + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 3 q^{3} + 8 q^{5} + 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} - 9 q^{15} + q^{17} + 16 q^{19} + 3 q^{21} + 6 q^{23} + 11 q^{25} + 9 q^{27} - 2 q^{29} + 3 q^{31} + 3 q^{33} + 8 q^{35} - 4 q^{37} - 3 q^{39} + 20 q^{41} + 20 q^{43} + 8 q^{45} - 6 q^{47} + 9 q^{49} + 15 q^{51} + 15 q^{53} + 8 q^{55} + 16 q^{57} + 21 q^{59} + 12 q^{63} - 8 q^{65} + 18 q^{67} + 10 q^{69} + 12 q^{71} + 29 q^{73} + 12 q^{75} + 9 q^{77} - 6 q^{79} + 5 q^{81} + 25 q^{83} + 13 q^{85} + 17 q^{87} + 32 q^{89} - 9 q^{91} - 13 q^{93} + 38 q^{95} + 17 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 53x^{3} + 252x^{2} - 69x - 26 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 25\nu^{8} - 54\nu^{7} - 419\nu^{6} + 771\nu^{5} + 2232\nu^{4} - 3127\nu^{3} - 3917\nu^{2} + 2939\nu + 773 ) / 17 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -25\nu^{8} + 54\nu^{7} + 419\nu^{6} - 771\nu^{5} - 2232\nu^{4} + 3127\nu^{3} + 3934\nu^{2} - 2956\nu - 841 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -50\nu^{8} + 108\nu^{7} + 838\nu^{6} - 1542\nu^{5} - 4464\nu^{4} + 6271\nu^{3} + 7834\nu^{2} - 6014\nu - 1563 ) / 17 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 110 \nu^{8} + 241 \nu^{7} + 1847 \nu^{6} - 3457 \nu^{5} - 9865 \nu^{4} + 14126 \nu^{3} + 17347 \nu^{2} + \cdots - 3374 ) / 17 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 123 \nu^{8} + 265 \nu^{7} + 2071 \nu^{6} - 3794 \nu^{5} - 11095 \nu^{4} + 15461 \nu^{3} + \cdots - 3965 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 287 \nu^{8} - 624 \nu^{7} - 4821 \nu^{6} + 8932 \nu^{5} + 25758 \nu^{4} - 36393 \nu^{3} - 45333 \nu^{2} + \cdots + 9008 ) / 17 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 383 \nu^{8} + 830 \nu^{7} + 6432 \nu^{6} - 11877 \nu^{5} - 34335 \nu^{4} + 48383 \nu^{3} + \cdots - 11960 ) / 17 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 2\beta_{2} + 8\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} - 2\beta_{4} + 8\beta_{3} + 10\beta_{2} + 11\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{7} - 2\beta_{6} - 4\beta_{5} + 8\beta_{4} + 23\beta_{2} + 68\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 12\beta_{8} - 4\beta_{7} - 13\beta_{6} - 17\beta_{5} - 28\beta_{4} + 62\beta_{3} + 97\beta_{2} + 116\beta _1 + 238 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{8} - 53\beta_{7} - 38\beta_{6} - 67\beta_{5} + 46\beta_{4} + 9\beta_{3} + 243\beta_{2} + 608\beta _1 + 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 114 \beta_{8} - 89 \beta_{7} - 149 \beta_{6} - 217 \beta_{5} - 313 \beta_{4} + 501 \beta_{3} + \cdots + 2079 \) 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
−2.78620
−2.31438
−1.45026
−0.211829
0.827072
0.829765
2.27011
2.61891
3.21682
0 −2.78620 0 0.423993 0 1.00000 0 4.76292 0
1.2 0 −2.31438 0 3.17142 0 1.00000 0 2.35637 0
1.3 0 −1.45026 0 1.74448 0 1.00000 0 −0.896739 0
1.4 0 −0.211829 0 1.56322 0 1.00000 0 −2.95513 0
1.5 0 0.827072 0 −3.42555 0 1.00000 0 −2.31595 0
1.6 0 0.829765 0 4.12811 0 1.00000 0 −2.31149 0
1.7 0 2.27011 0 1.08525 0 1.00000 0 2.15339 0
1.8 0 2.61891 0 −2.59083 0 1.00000 0 3.85871 0
1.9 0 3.21682 0 1.89991 0 1.00000 0 7.34791 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.q ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.q ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{9} - 3T_{3}^{8} - 15T_{3}^{7} + 45T_{3}^{6} + 64T_{3}^{5} - 201T_{3}^{4} - 53T_{3}^{3} + 252T_{3}^{2} - 69T_{3} - 26 \) Copy content Toggle raw display
\( T_{5}^{9} - 8T_{5}^{8} + 4T_{5}^{7} + 118T_{5}^{6} - 319T_{5}^{5} - 91T_{5}^{4} + 1458T_{5}^{3} - 2241T_{5}^{2} + 1357T_{5} - 277 \) Copy content Toggle raw display
\( T_{17}^{9} - T_{17}^{8} - 49 T_{17}^{7} + 37 T_{17}^{6} + 820 T_{17}^{5} - 451 T_{17}^{4} - 5625 T_{17}^{3} + \cdots + 374 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 3 T^{8} + \cdots - 26 \) Copy content Toggle raw display
$5$ \( T^{9} - 8 T^{8} + \cdots - 277 \) 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
$17$ \( T^{9} - T^{8} + \cdots + 374 \) Copy content Toggle raw display
$19$ \( T^{9} - 16 T^{8} + \cdots - 6227 \) Copy content Toggle raw display
$23$ \( T^{9} - 6 T^{8} + \cdots + 56300 \) Copy content Toggle raw display
$29$ \( T^{9} + 2 T^{8} + \cdots - 52 \) Copy content Toggle raw display
$31$ \( T^{9} - 3 T^{8} + \cdots + 86476 \) Copy content Toggle raw display
$37$ \( T^{9} + 4 T^{8} + \cdots - 2008 \) Copy content Toggle raw display
$41$ \( T^{9} - 20 T^{8} + \cdots - 4456 \) Copy content Toggle raw display
$43$ \( T^{9} - 20 T^{8} + \cdots + 2194673 \) Copy content Toggle raw display
$47$ \( T^{9} + 6 T^{8} + \cdots - 56300 \) Copy content Toggle raw display
$53$ \( T^{9} - 15 T^{8} + \cdots - 15125 \) Copy content Toggle raw display
$59$ \( T^{9} - 21 T^{8} + \cdots - 2333888 \) Copy content Toggle raw display
$61$ \( T^{9} - 171 T^{7} + \cdots - 82946 \) Copy content Toggle raw display
$67$ \( T^{9} - 18 T^{8} + \cdots - 63586 \) Copy content Toggle raw display
$71$ \( T^{9} - 12 T^{8} + \cdots - 840800 \) Copy content Toggle raw display
$73$ \( T^{9} - 29 T^{8} + \cdots + 8113636 \) Copy content Toggle raw display
$79$ \( T^{9} + 6 T^{8} + \cdots + 5853775 \) Copy content Toggle raw display
$83$ \( T^{9} - 25 T^{8} + \cdots + 23491 \) Copy content Toggle raw display
$89$ \( T^{9} - 32 T^{8} + \cdots + 30937 \) Copy content Toggle raw display
$97$ \( T^{9} - 17 T^{8} + \cdots - 252472208 \) Copy content Toggle raw display
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