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Properties

Label 8008.2.a.o
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: \( x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{4} q^{5} + q^{7} + \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{4} q^{5} + q^{7} + \beta_{2} q^{9} - q^{11} + q^{13} - \beta_{3} q^{15} + (\beta_{8} + \beta_{7} - \beta_{6} + \cdots - 1) q^{17}+ \cdots - \beta_{2} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 2 q^{3} - 4 q^{5} + 9 q^{7} + q^{9} - 9 q^{11} + 9 q^{13} - 4 q^{15} - 9 q^{17} + 9 q^{19} - 2 q^{21} - 10 q^{23} - q^{25} - 8 q^{27} - 13 q^{29} + 7 q^{31} + 2 q^{33} - 4 q^{35} - 15 q^{37} - 2 q^{39} - 18 q^{41} + 7 q^{43} + 9 q^{45} - 11 q^{47} + 9 q^{49} + q^{51} - 16 q^{53} + 4 q^{55} - 26 q^{57} - 2 q^{59} + 2 q^{61} + q^{63} - 4 q^{65} + 13 q^{67} - 3 q^{69} - q^{71} - 6 q^{73} - 4 q^{75} - 9 q^{77} - 21 q^{79} - 23 q^{81} - 30 q^{83} + q^{85} + q^{87} - 36 q^{89} + 9 q^{91} - 22 q^{93} - 23 q^{95} - 5 q^{97} - q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 2x^{8} - 12x^{7} + 20x^{6} + 47x^{5} - 55x^{4} - 68x^{3} + 37x^{2} + 21x - 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{8} + 4\nu^{7} + 11\nu^{6} - 42\nu^{5} - 40\nu^{4} + 114\nu^{3} + 57\nu^{2} - 46\nu - 6 ) / 7 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 36\nu^{6} - 7\nu^{5} - 220\nu^{4} - 10\nu^{3} + 478\nu^{2} + 97\nu - 180 ) / 21 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{8} + \nu^{7} + 29\nu^{6} - 7\nu^{5} - 136\nu^{4} + 4\nu^{3} + 212\nu^{2} + 27\nu - 47 ) / 7 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{8} - \nu^{7} - 29\nu^{6} + 7\nu^{5} + 136\nu^{4} + 3\nu^{3} - 219\nu^{2} - 62\nu + 61 ) / 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 2\nu^{7} + 12\nu^{6} - 20\nu^{5} - 47\nu^{4} + 55\nu^{3} + 68\nu^{2} - 34\nu - 18 ) / 3 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -6\nu^{8} + 10\nu^{7} + 73\nu^{6} - 91\nu^{5} - 289\nu^{4} + 194\nu^{3} + 412\nu^{2} - 10\nu - 99 ) / 7 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 5\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{4} + \beta_{3} + 8\beta_{2} + \beta _1 + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{8} - 3\beta_{7} + 11\beta_{6} + 11\beta_{5} + \beta_{3} + 12\beta_{2} + 29\beta _1 + 11 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -12\beta_{7} + 14\beta_{6} + 25\beta_{5} - 9\beta_{4} + 12\beta_{3} + 60\beta_{2} + 12\beta _1 + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 11\beta_{8} - 37\beta_{7} + 95\beta_{6} + 97\beta_{5} - \beta_{4} + 17\beta_{3} + 110\beta_{2} + 180\beta _1 + 93 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 2 \beta_{8} - 114 \beta_{7} + 146 \beta_{6} + 235 \beta_{5} - 63 \beta_{4} + 111 \beta_{3} + 447 \beta_{2} + \cdots + 546 \) 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
2.76592
2.33191
1.93909
0.559729
0.390946
−0.706153
−1.07573
−1.79806
−2.40766
0 −2.76592 0 1.47621 0 1.00000 0 4.65032 0
1.2 0 −2.33191 0 −2.59697 0 1.00000 0 2.43780 0
1.3 0 −1.93909 0 2.25153 0 1.00000 0 0.760073 0
1.4 0 −0.559729 0 0.0677230 0 1.00000 0 −2.68670 0
1.5 0 −0.390946 0 −3.55681 0 1.00000 0 −2.84716 0
1.6 0 0.706153 0 −2.65916 0 1.00000 0 −2.50135 0
1.7 0 1.07573 0 2.25022 0 1.00000 0 −1.84282 0
1.8 0 1.79806 0 0.876486 0 1.00000 0 0.233011 0
1.9 0 2.40766 0 −2.10923 0 1.00000 0 2.79683 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.o ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.o ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{9} + 2T_{3}^{8} - 12T_{3}^{7} - 20T_{3}^{6} + 47T_{3}^{5} + 55T_{3}^{4} - 68T_{3}^{3} - 37T_{3}^{2} + 21T_{3} + 9 \) Copy content Toggle raw display
\( T_{5}^{9} + 4T_{5}^{8} - 14T_{5}^{7} - 56T_{5}^{6} + 79T_{5}^{5} + 259T_{5}^{4} - 244T_{5}^{3} - 389T_{5}^{2} + 367T_{5} - 23 \) Copy content Toggle raw display
\( T_{17}^{9} + 9 T_{17}^{8} - 33 T_{17}^{7} - 435 T_{17}^{6} - 212 T_{17}^{5} + 4349 T_{17}^{4} + \cdots + 11420 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 2 T^{8} + \cdots + 9 \) Copy content Toggle raw display
$5$ \( T^{9} + 4 T^{8} + \cdots - 23 \) 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
$17$ \( T^{9} + 9 T^{8} + \cdots + 11420 \) Copy content Toggle raw display
$19$ \( T^{9} - 9 T^{8} + \cdots - 2252 \) Copy content Toggle raw display
$23$ \( T^{9} + 10 T^{8} + \cdots - 2116 \) Copy content Toggle raw display
$29$ \( T^{9} + 13 T^{8} + \cdots + 710000 \) Copy content Toggle raw display
$31$ \( T^{9} - 7 T^{8} + \cdots + 10532 \) Copy content Toggle raw display
$37$ \( T^{9} + 15 T^{8} + \cdots - 5260 \) Copy content Toggle raw display
$41$ \( T^{9} + 18 T^{8} + \cdots + 2271520 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} + \cdots + 17561790 \) Copy content Toggle raw display
$47$ \( T^{9} + 11 T^{8} + \cdots - 5443136 \) Copy content Toggle raw display
$53$ \( T^{9} + 16 T^{8} + \cdots - 5926 \) Copy content Toggle raw display
$59$ \( T^{9} + 2 T^{8} + \cdots - 292676 \) Copy content Toggle raw display
$61$ \( T^{9} - 2 T^{8} + \cdots + 2218040 \) Copy content Toggle raw display
$67$ \( T^{9} - 13 T^{8} + \cdots - 40789 \) Copy content Toggle raw display
$71$ \( T^{9} + T^{8} + \cdots - 16205296 \) Copy content Toggle raw display
$73$ \( T^{9} + 6 T^{8} + \cdots - 285928 \) Copy content Toggle raw display
$79$ \( T^{9} + 21 T^{8} + \cdots + 655940 \) Copy content Toggle raw display
$83$ \( T^{9} + 30 T^{8} + \cdots - 1233038 \) Copy content Toggle raw display
$89$ \( T^{9} + 36 T^{8} + \cdots + 911190517 \) Copy content Toggle raw display
$97$ \( T^{9} + 5 T^{8} + \cdots + 20530000 \) Copy content Toggle raw display
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