Properties

Label 8004.2.a.e
Level $8004$
Weight $2$
Character orbit 8004.a
Self dual yes
Analytic conductor $63.912$
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] = [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: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - 3x^{8} - 13x^{7} + 32x^{6} + 40x^{5} - 79x^{4} - 39x^{3} + 58x^{2} + 9x - 11 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 93 \nu^{8} - 7 \nu^{7} - 1649 \nu^{6} - 1742 \nu^{5} + 8484 \nu^{4} + 13848 \nu^{3} - 21492 \nu^{2} - 13414 \nu + 16720 ) / 4877 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 676 \nu^{8} - 54 \nu^{7} + 14451 \nu^{6} + 5373 \nu^{5} - 80862 \nu^{4} - 26245 \nu^{3} + 118464 \nu^{2} + 29593 \nu - 29868 ) / 4877 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1248 \nu^{8} + 4027 \nu^{7} + 15049 \nu^{6} - 42227 \nu^{5} - 37863 \nu^{4} + 93731 \nu^{3} + 26624 \nu^{2} - 41031 \nu - 4120 ) / 4877 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1613 \nu^{8} - 4474 \nu^{7} - 21416 \nu^{6} + 45249 \nu^{5} + 67542 \nu^{4} - 97748 \nu^{3} - 66924 \nu^{2} + 53254 \nu + 16986 ) / 4877 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1706 \nu^{8} - 4481 \nu^{7} - 23065 \nu^{6} + 43507 \nu^{5} + 76026 \nu^{4} - 83900 \nu^{3} - 83539 \nu^{2} + 34963 \nu + 19075 ) / 4877 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2082 \nu^{8} + 5663 \nu^{7} + 27005 \nu^{6} - 53822 \nu^{5} - 79649 \nu^{4} + 92100 \nu^{3} + 68801 \nu^{2} - 18907 \nu - 12313 ) / 4877 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2159 \nu^{8} - 5931 \nu^{7} - 29524 \nu^{6} + 60980 \nu^{5} + 102091 \nu^{4} - 136012 \nu^{3} - 122465 \nu^{2} + 64804 \nu + 35858 ) / 4877 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 2535 \nu^{8} - 7113 \nu^{7} - 33464 \nu^{6} + 71295 \nu^{5} + 105714 \nu^{4} - 144212 \nu^{3} - 107727 \nu^{2} + 58502 \nu + 24219 ) / 4877 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + \beta_{6} + 3\beta_{5} - 2\beta_{4} - 2\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{8} - 5\beta_{7} + 5\beta_{6} + 6\beta_{5} - 3\beta_{4} - 4\beta_{3} - \beta_{2} + 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\beta_{8} - 19\beta_{7} + 23\beta_{6} + 53\beta_{5} - 32\beta_{4} - 14\beta_{3} - 2\beta_{2} - 18\beta _1 + 75 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 83 \beta_{8} - 129 \beta_{7} + 135 \beta_{6} + 193 \beta_{5} - 100 \beta_{4} - 130 \beta_{3} - 30 \beta_{2} - 14 \beta _1 + 225 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 96 \beta_{8} - 174 \beta_{7} + 201 \beta_{6} + 413 \beta_{5} - 231 \beta_{4} - 167 \beta_{3} - 29 \beta_{2} - 99 \beta _1 + 520 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1033 \beta_{8} - 1823 \beta_{7} + 1943 \beta_{6} + 3085 \beta_{5} - 1588 \beta_{4} - 1928 \beta_{3} - 424 \beta_{2} - 342 \beta _1 + 3591 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 3035 \beta_{8} - 5877 \beta_{7} + 6591 \beta_{6} + 12709 \beta_{5} - 6834 \beta_{4} - 6034 \beta_{3} - 1142 \beta_{2} - 2514 \beta _1 + 15401 ) / 2 \) 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
−2.86352
−1.47135
3.93556
2.25739
0.761439
−0.975163
1.39711
0.511914
−0.553378
0 −1.00000 0 −3.70748 0 −1.59469 0 1.00000 0
1.2 0 −1.00000 0 −3.48372 0 2.18462 0 1.00000 0
1.3 0 −1.00000 0 −2.36120 0 2.82157 0 1.00000 0
1.4 0 −1.00000 0 −0.650765 0 −1.33403 0 1.00000 0
1.5 0 −1.00000 0 0.461945 0 0.732054 0 1.00000 0
1.6 0 −1.00000 0 0.900888 0 3.90501 0 1.00000 0
1.7 0 −1.00000 0 1.20115 0 −1.61263 0 1.00000 0
1.8 0 −1.00000 0 1.34732 0 2.91839 0 1.00000 0
1.9 0 −1.00000 0 3.29187 0 −1.02028 0 1.00000 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
\(2\) \(-1\)
\(3\) \(1\)
\(23\) \(1\)
\(29\) \(-1\)

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.e 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.e 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{9} + 3T_{5}^{8} - 19T_{5}^{7} - 44T_{5}^{6} + 117T_{5}^{5} + 117T_{5}^{4} - 314T_{5}^{3} + 66T_{5}^{2} + 116T_{5} - 44 \) Copy content Toggle raw display
\( T_{7}^{9} - 7T_{7}^{8} + 2T_{7}^{7} + 65T_{7}^{6} - 53T_{7}^{5} - 245T_{7}^{4} + 138T_{7}^{3} + 418T_{7}^{2} - 36T_{7} - 180 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( (T + 1)^{9} \) Copy content Toggle raw display
$5$ \( T^{9} + 3 T^{8} - 19 T^{7} - 44 T^{6} + \cdots - 44 \) Copy content Toggle raw display
$7$ \( T^{9} - 7 T^{8} + 2 T^{7} + 65 T^{6} + \cdots - 180 \) Copy content Toggle raw display
$11$ \( T^{9} + 2 T^{8} - 50 T^{7} - 81 T^{6} + \cdots - 496 \) Copy content Toggle raw display
$13$ \( T^{9} + 7 T^{8} - 30 T^{7} + \cdots + 11717 \) Copy content Toggle raw display
$17$ \( T^{9} - 55 T^{7} - 115 T^{6} + \cdots + 107 \) Copy content Toggle raw display
$19$ \( T^{9} + T^{8} - 74 T^{7} - 38 T^{6} + \cdots + 2047 \) Copy content Toggle raw display
$23$ \( (T + 1)^{9} \) Copy content Toggle raw display
$29$ \( (T - 1)^{9} \) Copy content Toggle raw display
$31$ \( T^{9} - 8 T^{8} - 126 T^{7} + \cdots + 333252 \) Copy content Toggle raw display
$37$ \( T^{9} + 8 T^{8} - 80 T^{7} + \cdots + 236939 \) Copy content Toggle raw display
$41$ \( T^{9} + 19 T^{8} + 31 T^{7} + \cdots - 47916 \) Copy content Toggle raw display
$43$ \( T^{9} + 3 T^{8} - 267 T^{7} + \cdots - 2401487 \) Copy content Toggle raw display
$47$ \( T^{9} + 3 T^{8} - 145 T^{7} + \cdots - 24692 \) Copy content Toggle raw display
$53$ \( T^{9} + 17 T^{8} - 79 T^{7} + \cdots + 3308752 \) Copy content Toggle raw display
$59$ \( T^{9} + 10 T^{8} - 104 T^{7} + \cdots + 1772000 \) Copy content Toggle raw display
$61$ \( T^{9} - T^{8} - 134 T^{7} + 207 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$67$ \( T^{9} - 12 T^{8} - 329 T^{7} + \cdots + 99286848 \) Copy content Toggle raw display
$71$ \( T^{9} + 7 T^{8} - 106 T^{7} + \cdots - 3875 \) Copy content Toggle raw display
$73$ \( T^{9} - 13 T^{8} - 151 T^{7} + \cdots - 18180 \) Copy content Toggle raw display
$79$ \( T^{9} + 10 T^{8} - 286 T^{7} + \cdots + 21425 \) Copy content Toggle raw display
$83$ \( T^{9} - 9 T^{8} - 325 T^{7} + \cdots + 69516 \) Copy content Toggle raw display
$89$ \( T^{9} + 5 T^{8} - 510 T^{7} + \cdots - 61987357 \) Copy content Toggle raw display
$97$ \( T^{9} + 7 T^{8} - 604 T^{7} + \cdots - 15236672 \) Copy content Toggle raw display
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