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Properties

Label 8008.2.a.r
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: \( x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 \) 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.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 1) q^{3} + \beta_{6} q^{5} - q^{7} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 1) q^{3} + \beta_{6} q^{5} - q^{7} + (\beta_{7} - \beta_{6} + \beta_{5} + \cdots + 1) q^{9}+ \cdots + ( - \beta_{7} + \beta_{6} - \beta_{5} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 5 q^{3} + 3 q^{5} - 9 q^{7} + 8 q^{9} - 9 q^{11} - 9 q^{13} - 3 q^{15} - 7 q^{17} + 13 q^{19} - 5 q^{21} + 9 q^{23} - 2 q^{25} + 5 q^{27} + q^{29} + 10 q^{31} - 5 q^{33} - 3 q^{35} + 14 q^{37} - 5 q^{39} - 2 q^{41} + 5 q^{43} - 15 q^{45} + 13 q^{47} + 9 q^{49} - 3 q^{51} + 22 q^{53} - 3 q^{55} + 16 q^{57} + 43 q^{59} - 10 q^{61} - 8 q^{63} - 3 q^{65} + 26 q^{67} - 30 q^{69} + 18 q^{71} - 8 q^{73} + 28 q^{75} + 9 q^{77} - 9 q^{79} + 33 q^{81} + 4 q^{83} + 5 q^{85} + 33 q^{87} + 7 q^{89} + 9 q^{91} + 13 q^{93} + 7 q^{95} + 2 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 4x^{8} - 9x^{7} + 36x^{6} + 23x^{5} - 89x^{4} - 20x^{3} + 51x^{2} + 18x + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -2\nu^{8} + 59\nu^{7} - 166\nu^{6} - 565\nu^{5} + 1504\nu^{4} + 1441\nu^{3} - 3276\nu^{2} - 520\nu + 992 ) / 139 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{8} - 95\nu^{7} - 6\nu^{6} + 832\nu^{5} - 577\nu^{4} - 1981\nu^{3} + 1079\nu^{2} + 981\nu + 205 ) / 139 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -41\nu^{8} + 167\nu^{7} + 350\nu^{6} - 1505\nu^{5} - 721\nu^{4} + 3895\nu^{3} + 118\nu^{2} - 2876\nu - 375 ) / 139 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -44\nu^{8} + 186\nu^{7} + 379\nu^{6} - 1727\nu^{5} - 967\nu^{4} + 4597\nu^{3} + 1042\nu^{2} - 3378\nu - 833 ) / 139 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 66\nu^{8} - 279\nu^{7} - 499\nu^{6} + 2382\nu^{5} + 686\nu^{4} - 5297\nu^{3} + 661\nu^{2} + 2287\nu + 207 ) / 139 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 95\nu^{8} - 370\nu^{7} - 872\nu^{6} + 3277\nu^{5} + 2230\nu^{4} - 7913\nu^{3} - 1321\nu^{2} + 4406\nu + 418 ) / 139 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 141 \nu^{8} - 615 \nu^{7} - 1085 \nu^{6} + 5569 \nu^{5} + 1693 \nu^{4} - 13812 \nu^{3} + 496 \nu^{2} + \cdots + 1232 ) / 139 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 2\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} + 2\beta_{7} - 3\beta_{6} + 4\beta_{5} + 3\beta_{3} + \beta_{2} + 11\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{8} + 13\beta_{7} - 17\beta_{6} + 20\beta_{5} - 2\beta_{4} + 18\beta_{3} + 4\beta_{2} + 36\beta _1 + 19 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18\beta_{8} + 41\beta_{7} - 63\beta_{6} + 81\beta_{5} - 9\beta_{4} + 64\beta_{3} + 20\beta_{2} + 157\beta _1 + 35 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 64 \beta_{8} + 188 \beta_{7} - 273 \beta_{6} + 342 \beta_{5} - 49 \beta_{4} + 289 \beta_{3} + 81 \beta_{2} + \cdots + 187 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 289 \beta_{8} + 695 \beta_{7} - 1068 \beta_{6} + 1375 \beta_{5} - 203 \beta_{4} + 1105 \beta_{3} + \cdots + 561 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1105 \beta_{8} + 2895 \beta_{7} - 4357 \beta_{6} + 5578 \beta_{5} - 883 \beta_{4} + 4590 \beta_{3} + \cdots + 2452 \) 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
3.98888
2.28175
1.66280
1.14890
−0.0694951
−0.302339
−0.688949
−1.74591
−2.27563
0 −2.98888 0 −0.559279 0 −1.00000 0 5.93339 0
1.2 0 −1.28175 0 0.660545 0 −1.00000 0 −1.35712 0
1.3 0 −0.662800 0 3.68243 0 −1.00000 0 −2.56070 0
1.4 0 −0.148900 0 −0.337922 0 −1.00000 0 −2.97783 0
1.5 0 1.06950 0 0.381633 0 −1.00000 0 −1.85618 0
1.6 0 1.30234 0 −2.00169 0 −1.00000 0 −1.30391 0
1.7 0 1.68895 0 3.11266 0 −1.00000 0 −0.147453 0
1.8 0 2.74591 0 −3.50460 0 −1.00000 0 4.54004 0
1.9 0 3.27563 0 1.56622 0 −1.00000 0 7.72976 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.r ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.r ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{9} - 5T_{3}^{8} - 5T_{3}^{7} + 55T_{3}^{6} - 48T_{3}^{5} - 97T_{3}^{4} + 119T_{3}^{3} + 38T_{3}^{2} - 51T_{3} - 8 \) Copy content Toggle raw display
\( T_{5}^{9} - 3T_{5}^{8} - 17T_{5}^{7} + 49T_{5}^{6} + 58T_{5}^{5} - 157T_{5}^{4} - 9T_{5}^{3} + 66T_{5}^{2} - T_{5} - 6 \) Copy content Toggle raw display
\( T_{17}^{9} + 7 T_{17}^{8} - 71 T_{17}^{7} - 693 T_{17}^{6} + 90 T_{17}^{5} + 15915 T_{17}^{4} + \cdots - 1958 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} - 5 T^{8} + \cdots - 8 \) Copy content Toggle raw display
$5$ \( T^{9} - 3 T^{8} + \cdots - 6 \) 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
$17$ \( T^{9} + 7 T^{8} + \cdots - 1958 \) Copy content Toggle raw display
$19$ \( T^{9} - 13 T^{8} + \cdots - 704 \) Copy content Toggle raw display
$23$ \( T^{9} - 9 T^{8} + \cdots - 2784 \) Copy content Toggle raw display
$29$ \( T^{9} - T^{8} + \cdots + 9944 \) Copy content Toggle raw display
$31$ \( T^{9} - 10 T^{8} + \cdots - 115232 \) Copy content Toggle raw display
$37$ \( T^{9} - 14 T^{8} + \cdots - 9496 \) Copy content Toggle raw display
$41$ \( T^{9} + 2 T^{8} + \cdots + 116888 \) Copy content Toggle raw display
$43$ \( T^{9} - 5 T^{8} + \cdots + 3186868 \) Copy content Toggle raw display
$47$ \( T^{9} - 13 T^{8} + \cdots - 192896 \) Copy content Toggle raw display
$53$ \( T^{9} - 22 T^{8} + \cdots - 872034 \) Copy content Toggle raw display
$59$ \( T^{9} - 43 T^{8} + \cdots - 1779216 \) Copy content Toggle raw display
$61$ \( T^{9} + 10 T^{8} + \cdots + 78447650 \) Copy content Toggle raw display
$67$ \( T^{9} - 26 T^{8} + \cdots + 17253076 \) Copy content Toggle raw display
$71$ \( T^{9} - 18 T^{8} + \cdots - 145388736 \) Copy content Toggle raw display
$73$ \( T^{9} + 8 T^{8} + \cdots + 51814712 \) Copy content Toggle raw display
$79$ \( T^{9} + 9 T^{8} + \cdots + 224396 \) Copy content Toggle raw display
$83$ \( T^{9} - 4 T^{8} + \cdots + 1621376 \) Copy content Toggle raw display
$89$ \( T^{9} - 7 T^{8} + \cdots + 337886 \) Copy content Toggle raw display
$97$ \( T^{9} - 2 T^{8} + \cdots + 2808608 \) Copy content Toggle raw display
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