Properties

Label 8004.2.a.d
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
Analytic rank $1$
Dimension $8$
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] = [8004,2,Mod(1,8004)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8004.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8004 = 2^{2} \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8004.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: \(63.9122617778\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{3} + ( - \beta_{7} - 1) q^{5} + (\beta_{7} + \beta_{6}) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} + ( - \beta_{7} - 1) q^{5} + (\beta_{7} + \beta_{6}) q^{7} + q^{9} + (\beta_{7} - \beta_{3}) q^{11} + (\beta_{4} - 1) q^{13} + ( - \beta_{7} - 1) q^{15} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{17} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{19} + (\beta_{7} + \beta_{6}) q^{21} + q^{23} + (\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{25} + q^{27} + q^{29} + ( - \beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2}) q^{31} + (\beta_{7} - \beta_{3}) q^{33} + (\beta_{5} + \beta_{4} - \beta_{2} - \beta_1 - 3) q^{35} + (\beta_{7} + \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} - 2) q^{37} + (\beta_{4} - 1) q^{39} + ( - \beta_{7} + \beta_{5} + \beta_{3} - 4 \beta_1 - 1) q^{41} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - \beta_1 - 1) q^{43} + ( - \beta_{7} - 1) q^{45} + ( - \beta_{7} - 3 \beta_{6} + 2 \beta_{5} - \beta_{4} + 2 \beta_{2} - 2) q^{47} + ( - 2 \beta_{7} - 3 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1 - 3) q^{49} + ( - \beta_{6} - \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{51} + (\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 4 \beta_1 - 1) q^{53} + ( - 2 \beta_{7} - \beta_{6} + 2 \beta_{3} + \beta_{2} - 3) q^{55} + ( - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_1 - 1) q^{57} + (2 \beta_{7} + 4 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 2) q^{59} + ( - \beta_{7} - \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{61} + (\beta_{7} + \beta_{6}) q^{63} + (\beta_{7} - \beta_{6} - \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_{2} + 2) q^{65} + (3 \beta_{6} + 2 \beta_{5} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_1 - 1) q^{67} + q^{69} + (\beta_{7} - 3 \beta_{6} - 3 \beta_{4} + 3 \beta_{3} + \beta_{2} + 3 \beta_1) q^{71} + (3 \beta_{7} + \beta_{6} - \beta_{3} - 4 \beta_{2} + 5 \beta_1 - 5) q^{73} + (\beta_{7} - \beta_{4} - \beta_{3} + \beta_{2} + 1) q^{75} + ( - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{2} + \beta_1) q^{77} + (\beta_{7} - 3 \beta_{6} + \beta_{5} - 3 \beta_{4} - \beta_{2} + 4 \beta_1 - 2) q^{79} + q^{81} + (4 \beta_{7} + 7 \beta_{6} - 2 \beta_{5} + 3 \beta_{4} - 3 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{83}+ \cdots + (\beta_{7} - \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{3} - 5 q^{5} - 4 q^{7} + 8 q^{9} - 5 q^{11} - 4 q^{13} - 5 q^{15} - 3 q^{17} - 5 q^{19} - 4 q^{21} + 8 q^{23} - 5 q^{25} + 8 q^{27} + 8 q^{29} - 2 q^{31} - 5 q^{33} - 15 q^{35} - 10 q^{37} - 4 q^{39} - 11 q^{41} - 7 q^{43} - 5 q^{45} - 14 q^{47} - 18 q^{49} - 3 q^{51} - 15 q^{53} - 17 q^{55} - 5 q^{57} + 4 q^{59} + q^{61} - 4 q^{63} - 5 q^{67} + 8 q^{69} - q^{71} - 21 q^{73} - 5 q^{75} - 8 q^{79} + 8 q^{81} + 3 q^{83} + 8 q^{87} - 20 q^{89} - 7 q^{91} - 2 q^{93} - 3 q^{95} - 7 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} - 8x^{6} + 19x^{5} + 19x^{4} - 35x^{3} - 10x^{2} + 18x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 5\nu^{7} - 13\nu^{6} - 46\nu^{5} + 79\nu^{4} + 131\nu^{3} - 131\nu^{2} - 108\nu + 50 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} + 13\nu^{6} + 46\nu^{5} - 79\nu^{4} - 131\nu^{3} + 133\nu^{2} + 106\nu - 56 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( 5\nu^{7} - 13\nu^{6} - 45\nu^{5} + 76\nu^{4} + 126\nu^{3} - 122\nu^{2} - 102\nu + 50 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11\nu^{7} + 29\nu^{6} + 98\nu^{5} - 171\nu^{4} - 269\nu^{3} + 273\nu^{2} + 210\nu - 104 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{7} + 33\nu^{6} + 120\nu^{5} - 195\nu^{4} - 339\nu^{3} + 317\nu^{2} + 272\nu - 132 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 7\nu^{7} - 18\nu^{6} - 64\nu^{5} + 107\nu^{4} + 179\nu^{3} - 175\nu^{2} - 143\nu + 71 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 2\beta_{3} + 2\beta_{2} + 6\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 5\beta_{7} + 4\beta_{6} + 3\beta_{5} + 2\beta_{4} + 10\beta_{3} + 9\beta_{2} + 16\beta _1 + 20 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 20\beta_{7} + 17\beta_{6} + 14\beta_{5} + 12\beta_{4} + 31\beta_{3} + 26\beta_{2} + 63\beta _1 + 48 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 77\beta_{7} + 60\beta_{6} + 48\beta_{5} + 34\beta_{4} + 120\beta_{3} + 98\beta_{2} + 211\beta _1 + 199 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 279\beta_{7} + 223\beta_{6} + 180\beta_{5} + 141\beta_{4} + 413\beta_{3} + 326\beta_{2} + 766\beta _1 + 633 \) Copy content Toggle raw display

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
−1.85152
0.386133
−1.03502
3.55127
−1.58311
1.24887
0.411238
1.87214
0 1.00000 0 −3.41642 0 2.05098 0 1.00000 0
1.2 0 1.00000 0 −2.77471 0 0.556399 0 1.00000 0
1.3 0 1.00000 0 −1.84132 0 −2.79101 0 1.00000 0
1.4 0 1.00000 0 −1.29538 0 2.07218 0 1.00000 0
1.5 0 1.00000 0 −0.270840 0 0.409136 0 1.00000 0
1.6 0 1.00000 0 0.138622 0 −3.33662 0 1.00000 0
1.7 0 1.00000 0 1.71910 0 0.210007 0 1.00000 0
1.8 0 1.00000 0 2.74094 0 −3.17108 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(23\) \(-1\)
\(29\) \(-1\)

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 8004.2.a.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.d 8 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{8} + 5T_{5}^{7} - 5T_{5}^{6} - 52T_{5}^{5} - 35T_{5}^{4} + 107T_{5}^{3} + 122T_{5}^{2} + 10T_{5} - 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 4T_{7}^{7} - 11T_{7}^{6} - 41T_{7}^{5} + 56T_{7}^{4} + 104T_{7}^{3} - 145T_{7}^{2} + 54T_{7} - 6 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 5 T^{7} - 5 T^{6} - 52 T^{5} + \cdots - 4 \) Copy content Toggle raw display
$7$ \( T^{8} + 4 T^{7} - 11 T^{6} - 41 T^{5} + \cdots - 6 \) Copy content Toggle raw display
$11$ \( T^{8} + 5 T^{7} - 12 T^{6} - 91 T^{5} + \cdots - 96 \) Copy content Toggle raw display
$13$ \( T^{8} + 4 T^{7} - 14 T^{6} - 45 T^{5} + \cdots - 1 \) Copy content Toggle raw display
$17$ \( T^{8} + 3 T^{7} - 34 T^{6} - 127 T^{5} + \cdots - 461 \) Copy content Toggle raw display
$19$ \( T^{8} + 5 T^{7} - 45 T^{6} + \cdots - 3418 \) Copy content Toggle raw display
$23$ \( (T - 1)^{8} \) Copy content Toggle raw display
$29$ \( (T - 1)^{8} \) Copy content Toggle raw display
$31$ \( T^{8} + 2 T^{7} - 144 T^{6} + \cdots + 67344 \) Copy content Toggle raw display
$37$ \( T^{8} + 10 T^{7} - 45 T^{6} + \cdots - 2944 \) Copy content Toggle raw display
$41$ \( T^{8} + 11 T^{7} - 111 T^{6} + \cdots + 969324 \) Copy content Toggle raw display
$43$ \( T^{8} + 7 T^{7} - 86 T^{6} + \cdots + 11216 \) Copy content Toggle raw display
$47$ \( T^{8} + 14 T^{7} - 46 T^{6} - 1292 T^{5} + \cdots - 54 \) Copy content Toggle raw display
$53$ \( T^{8} + 15 T^{7} - 127 T^{6} + \cdots + 43448 \) Copy content Toggle raw display
$59$ \( T^{8} - 4 T^{7} - 227 T^{6} + \cdots + 2324416 \) Copy content Toggle raw display
$61$ \( T^{8} - T^{7} - 206 T^{6} + \cdots - 712512 \) Copy content Toggle raw display
$67$ \( T^{8} + 5 T^{7} - 169 T^{6} - 278 T^{5} + \cdots - 8 \) Copy content Toggle raw display
$71$ \( T^{8} + T^{7} - 259 T^{6} + \cdots - 5344884 \) Copy content Toggle raw display
$73$ \( T^{8} + 21 T^{7} - 105 T^{6} + \cdots + 7960224 \) Copy content Toggle raw display
$79$ \( T^{8} + 8 T^{7} - 191 T^{6} + \cdots + 15902 \) Copy content Toggle raw display
$83$ \( T^{8} - 3 T^{7} - 325 T^{6} + \cdots - 3977348 \) Copy content Toggle raw display
$89$ \( T^{8} + 20 T^{7} + \cdots + 194617089 \) Copy content Toggle raw display
$97$ \( T^{8} + 7 T^{7} - 184 T^{6} + \cdots - 85056 \) Copy content Toggle raw display
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