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Properties

Label 8008.2.a.n
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: \( x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} - \beta_{6} q^{5} - q^{7} + ( - \beta_{5} + \beta_{4} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} - \beta_{6} q^{5} - q^{7} + ( - \beta_{5} + \beta_{4} + 1) q^{9} + q^{11} - q^{13} + (\beta_{8} + 2 \beta_{6} + \beta_{4} + \cdots + 2) q^{15}+ \cdots + ( - \beta_{5} + \beta_{4} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} + q^{5} - 9 q^{7} + 12 q^{9} + 9 q^{11} - 9 q^{13} + 11 q^{15} + 7 q^{17} - 17 q^{19} + 3 q^{21} + 11 q^{23} + 18 q^{25} - 9 q^{27} + 9 q^{29} - 8 q^{31} - 3 q^{33} - q^{35} + 2 q^{37} + 3 q^{39} + 18 q^{41} + 7 q^{43} + 5 q^{45} + 15 q^{47} + 9 q^{49} - 7 q^{51} - 4 q^{53} + q^{55} + 22 q^{57} - 23 q^{59} + 12 q^{61} - 12 q^{63} - q^{65} - 16 q^{67} - 32 q^{69} - 6 q^{71} + 4 q^{73} - 14 q^{75} - 9 q^{77} + 21 q^{79} + 5 q^{81} - 16 q^{83} + 53 q^{85} + 41 q^{87} + 5 q^{89} + 9 q^{91} + 29 q^{93} + 19 q^{95} + 18 q^{97} + 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 15x^{7} + 45x^{6} + 64x^{5} - 201x^{4} - 63x^{3} + 282x^{2} + 3x - 116 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{8} - 14\nu^{7} - 53\nu^{6} + 159\nu^{5} + 129\nu^{4} - 346\nu^{3} + 491\nu^{2} - 20\nu - 738 ) / 95 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8\nu^{8} - 28\nu^{7} - 106\nu^{6} + 413\nu^{5} + 353\nu^{4} - 1737\nu^{3} - 158\nu^{2} + 1670\nu + 139 ) / 95 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 29\nu^{8} - 54\nu^{7} - 503\nu^{6} + 749\nu^{5} + 2764\nu^{4} - 2841\nu^{3} - 5014\nu^{2} + 2705\nu + 2677 ) / 95 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 29\nu^{8} - 54\nu^{7} - 503\nu^{6} + 749\nu^{5} + 2764\nu^{4} - 2841\nu^{3} - 5109\nu^{2} + 2705\nu + 3057 ) / 95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 56\nu^{8} - 101\nu^{7} - 932\nu^{6} + 1371\nu^{5} + 4846\nu^{4} - 5034\nu^{3} - 8326\nu^{2} + 4375\nu + 4678 ) / 95 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -10\nu^{8} + 16\nu^{7} + 180\nu^{6} - 217\nu^{5} - 1054\nu^{4} + 808\nu^{3} + 2164\nu^{2} - 824\nu - 1328 ) / 19 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 62 \nu^{8} + 122 \nu^{7} + 1059 \nu^{6} - 1657 \nu^{5} - 5752 \nu^{4} + 6028 \nu^{3} + 10582 \nu^{2} + \cdots - 6326 ) / 95 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{5} + \beta_{4} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 6\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{8} - 11\beta_{5} + 9\beta_{4} - \beta_{2} + 2\beta _1 + 26 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{8} + 11\beta_{7} - 12\beta_{5} + 14\beta_{4} - 10\beta_{3} - 12\beta_{2} + 46\beta _1 + 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -13\beta_{8} + \beta_{7} + 2\beta_{6} - 104\beta_{5} + 76\beta_{4} - 14\beta_{2} + 34\beta _1 + 202 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 86 \beta_{8} + 103 \beta_{7} + 5 \beta_{6} - 126 \beta_{5} + 152 \beta_{4} - 92 \beta_{3} + \cdots + 200 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 130 \beta_{8} + 23 \beta_{7} + 44 \beta_{6} - 951 \beta_{5} + 656 \beta_{4} - 11 \beta_{3} + \cdots + 1682 \) 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
3.07040
3.00639
1.63046
1.21838
1.10969
−0.763644
−1.15169
−2.29224
−2.82774
0 −3.07040 0 −3.53994 0 −1.00000 0 6.42735 0
1.2 0 −3.00639 0 2.62191 0 −1.00000 0 6.03838 0
1.3 0 −1.63046 0 −1.01052 0 −1.00000 0 −0.341611 0
1.4 0 −1.21838 0 2.90030 0 −1.00000 0 −1.51556 0
1.5 0 −1.10969 0 −2.48203 0 −1.00000 0 −1.76859 0
1.6 0 0.763644 0 1.55421 0 −1.00000 0 −2.41685 0
1.7 0 1.15169 0 −3.19313 0 −1.00000 0 −1.67362 0
1.8 0 2.29224 0 3.91599 0 −1.00000 0 2.25436 0
1.9 0 2.82774 0 0.233208 0 −1.00000 0 4.99612 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.n ✓ 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.n ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{9} + 3T_{3}^{8} - 15T_{3}^{7} - 45T_{3}^{6} + 64T_{3}^{5} + 201T_{3}^{4} - 63T_{3}^{3} - 282T_{3}^{2} + 3T_{3} + 116 \) Copy content Toggle raw display
\( T_{5}^{9} - T_{5}^{8} - 31T_{5}^{7} + 27T_{5}^{6} + 320T_{5}^{5} - 247T_{5}^{4} - 1227T_{5}^{3} + 772T_{5}^{2} + 1201T_{5} - 306 \) Copy content Toggle raw display
\( T_{17}^{9} - 7 T_{17}^{8} - 59 T_{17}^{7} + 483 T_{17}^{6} + 552 T_{17}^{5} - 8979 T_{17}^{4} + \cdots - 21678 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + 3 T^{8} + \cdots + 116 \) Copy content Toggle raw display
$5$ \( T^{9} - T^{8} + \cdots - 306 \) 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
$17$ \( T^{9} - 7 T^{8} + \cdots - 21678 \) Copy content Toggle raw display
$19$ \( T^{9} + 17 T^{8} + \cdots - 156 \) Copy content Toggle raw display
$23$ \( T^{9} - 11 T^{8} + \cdots + 194944 \) Copy content Toggle raw display
$29$ \( T^{9} - 9 T^{8} + \cdots - 220704 \) Copy content Toggle raw display
$31$ \( T^{9} + 8 T^{8} + \cdots - 74176 \) Copy content Toggle raw display
$37$ \( T^{9} - 2 T^{8} + \cdots + 33056 \) Copy content Toggle raw display
$41$ \( T^{9} - 18 T^{8} + \cdots + 21536 \) Copy content Toggle raw display
$43$ \( T^{9} - 7 T^{8} + \cdots - 3003304 \) Copy content Toggle raw display
$47$ \( T^{9} - 15 T^{8} + \cdots + 2478144 \) Copy content Toggle raw display
$53$ \( T^{9} + 4 T^{8} + \cdots - 1334 \) Copy content Toggle raw display
$59$ \( T^{9} + 23 T^{8} + \cdots + 485952 \) Copy content Toggle raw display
$61$ \( T^{9} - 12 T^{8} + \cdots + 20177298 \) Copy content Toggle raw display
$67$ \( T^{9} + 16 T^{8} + \cdots + 2451556 \) Copy content Toggle raw display
$71$ \( T^{9} + 6 T^{8} + \cdots - 697408 \) Copy content Toggle raw display
$73$ \( T^{9} - 4 T^{8} + \cdots + 1146336 \) Copy content Toggle raw display
$79$ \( T^{9} - 21 T^{8} + \cdots + 939704 \) Copy content Toggle raw display
$83$ \( T^{9} + 16 T^{8} + \cdots + 28341132 \) Copy content Toggle raw display
$89$ \( T^{9} + \cdots - 1035802622 \) Copy content Toggle raw display
$97$ \( T^{9} - 18 T^{8} + \cdots + 2313824 \) Copy content Toggle raw display
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