Properties

Label 8004.2.a.f
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: \( x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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.

\(f(q)\) \(=\) \( q + q^{3} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + q^{3} - \beta_1 q^{5} + ( - \beta_{6} - 1) q^{7} + q^{9} + ( - \beta_{3} - 1) q^{11} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{13} - \beta_1 q^{15} + (\beta_{7} + \beta_1) q^{17} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{19} + ( - \beta_{6} - 1) q^{21} - q^{23} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{25} + q^{27} - q^{29} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{31} + ( - \beta_{3} - 1) q^{33} + ( - \beta_{8} + 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1 + 2) q^{35} + (\beta_{6} - \beta_{4} - 2 \beta_{2} + 1) q^{37} + ( - \beta_{7} + \beta_{6} + \beta_{3} + \beta_1) q^{39} + (\beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 - 3) q^{41} + ( - \beta_{8} + \beta_{6} + \beta_{5} - \beta_1 - 1) q^{43} - \beta_1 q^{45} + (2 \beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} + 2 \beta_1 - 2) q^{47} + (\beta_{8} + 2 \beta_{5} - \beta_{3} - 1) q^{49} + (\beta_{7} + \beta_1) q^{51} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 + 1) q^{53} + (\beta_{8} - 2 \beta_{6} - \beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_1 - 3) q^{55} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_1 - 1) q^{57} + (\beta_{8} - 2 \beta_{7} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 2 \beta_1 - 1) q^{59} + ( - 2 \beta_{8} + \beta_{7} + 2 \beta_{6} + 2 \beta_{4} - \beta_{2}) q^{61} + ( - \beta_{6} - 1) q^{63} + (\beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - 4 \beta_1 - 2) q^{65} + (\beta_{8} + \beta_{7} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 - 3) q^{67} - q^{69} + ( - 3 \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} + 3 \beta_{3} + 2 \beta_1 + 3) q^{71} + ( - 4 \beta_{6} - \beta_{5} - 2 \beta_{4} - 4 \beta_{3} + 3 \beta_{2} - \beta_1 - 5) q^{73} + ( - \beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2 \beta_1 - 1) q^{75} + ( - \beta_{8} + \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 1) q^{77} + (2 \beta_{8} + \beta_{6} + \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} + 1) q^{79} + q^{81} + ( - 2 \beta_{8} + 4 \beta_{6} - \beta_{5} + \beta_{4} + 4 \beta_{3} - 2 \beta_{2} + 3) q^{83} + (\beta_{7} - \beta_{5} - \beta_{4} - \beta_{2} - 5) q^{85} - q^{87} + (\beta_{8} - \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 3) q^{89} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} - \beta_{5} + \beta_{4} - 2 \beta_1 - 2) q^{91} + (\beta_{7} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{93} + (\beta_{8} - \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} + 5 \beta_1 + 2) q^{95} + (\beta_{8} + \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 4) q^{97} + ( - \beta_{3} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + 9 q^{3} - q^{5} - 5 q^{7} + 9 q^{9} - 8 q^{11} - q^{13} - q^{15} - 2 q^{17} - 11 q^{19} - 5 q^{21} - 9 q^{23} - 10 q^{25} + 9 q^{27} - 9 q^{29} - 8 q^{33} + q^{35} - 2 q^{37} - q^{39} - 3 q^{41} - 19 q^{43} - q^{45} - 3 q^{47} - 6 q^{49} - 2 q^{51} - 9 q^{53} - 7 q^{55} - 11 q^{57} - 2 q^{59} - 25 q^{61} - 5 q^{63} - 12 q^{65} - 20 q^{67} - 9 q^{69} + 9 q^{71} - 11 q^{73} - 10 q^{75} - 19 q^{77} + 4 q^{79} + 9 q^{81} - 9 q^{83} - 50 q^{85} - 9 q^{87} - 29 q^{89} - 38 q^{91} + 23 q^{95} - 43 q^{97} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 17x^{7} + 4x^{6} + 75x^{5} + x^{4} - 118x^{3} - 26x^{2} + 60x + 24 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -11\nu^{8} - 2\nu^{7} + 180\nu^{6} + 161\nu^{5} - 559\nu^{4} - 480\nu^{3} + 553\nu^{2} + 258\nu - 182 ) / 34 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{8} - 4\nu^{7} + 88\nu^{6} + 118\nu^{5} - 302\nu^{4} - 382\nu^{3} + 307\nu^{2} + 278\nu - 41 ) / 17 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -21\nu^{8} + 24\nu^{7} + 322\nu^{6} - 113\nu^{5} - 1061\nu^{4} + 252\nu^{3} + 895\nu^{2} - 104\nu - 162 ) / 34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\nu^{8} - 13\nu^{7} - 343\nu^{6} - 67\nu^{5} + 1050\nu^{4} + 229\nu^{3} - 749\nu^{2} - 74\nu + 24 ) / 34 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4\nu^{8} + 10\nu^{7} - 67\nu^{6} - 210\nu^{5} + 109\nu^{4} + 683\nu^{3} + 193\nu^{2} - 559\nu - 297 ) / 17 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -23\nu^{8} + 36\nu^{7} + 364\nu^{6} - 297\nu^{5} - 1447\nu^{4} + 854\nu^{3} + 1861\nu^{2} - 598\nu - 702 ) / 34 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{8} + 28\nu^{7} + 251\nu^{6} - 248\nu^{5} - 1014\nu^{4} + 685\nu^{3} + 1319\nu^{2} - 484\nu - 495 ) / 17 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3} + 2\beta _1 + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{8} - 3\beta_{7} + 2\beta_{6} - 2\beta_{5} + 2\beta_{3} - \beta_{2} + 12\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - 16\beta_{7} + 15\beta_{6} - 4\beta_{5} + 12\beta_{4} + 16\beta_{3} - 4\beta_{2} + 37\beta _1 + 38 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 17\beta_{8} - 58\beta_{7} + 39\beta_{6} - 28\beta_{5} + 12\beta_{4} + 41\beta_{3} - 16\beta_{2} + 173\beta _1 + 69 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30 \beta_{8} - 246 \beta_{7} + 215 \beta_{6} - 82 \beta_{5} + 147 \beta_{4} + 233 \beta_{3} - 73 \beta_{2} + 612 \beta _1 + 480 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 238 \beta_{8} - 944 \beta_{7} + 675 \beta_{6} - 402 \beta_{5} + 300 \beta_{4} + 719 \beta_{3} - 258 \beta_{2} + 2611 \beta _1 + 1298 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 602 \beta_{8} - 3809 \beta_{7} + 3167 \beta_{6} - 1388 \beta_{5} + 1967 \beta_{4} + 3432 \beta_{3} - 1138 \beta_{2} + 9792 \beta _1 + 6751 \) 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
3.95129
1.58421
1.51482
1.05365
−0.509166
−0.808455
−1.09926
−1.91513
−2.77196
0 1.00000 0 −3.95129 0 −1.61455 0 1.00000 0
1.2 0 1.00000 0 −1.58421 0 1.44549 0 1.00000 0
1.3 0 1.00000 0 −1.51482 0 −4.32415 0 1.00000 0
1.4 0 1.00000 0 −1.05365 0 3.84396 0 1.00000 0
1.5 0 1.00000 0 0.509166 0 1.30726 0 1.00000 0
1.6 0 1.00000 0 0.808455 0 −2.11918 0 1.00000 0
1.7 0 1.00000 0 1.09926 0 −1.62400 0 1.00000 0
1.8 0 1.00000 0 1.91513 0 −2.97719 0 1.00000 0
1.9 0 1.00000 0 2.77196 0 1.06236 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.f 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8004.2.a.f 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8004))\):

\( T_{5}^{9} + T_{5}^{8} - 17T_{5}^{7} - 4T_{5}^{6} + 75T_{5}^{5} - T_{5}^{4} - 118T_{5}^{3} + 26T_{5}^{2} + 60T_{5} - 24 \) Copy content Toggle raw display
\( T_{7}^{9} + 5T_{7}^{8} - 16T_{7}^{7} - 101T_{7}^{6} + 7T_{7}^{5} + 463T_{7}^{4} + 146T_{7}^{3} - 834T_{7}^{2} - 212T_{7} + 552 \) 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
$5$ \( T^{9} + T^{8} - 17 T^{7} - 4 T^{6} + \cdots - 24 \) Copy content Toggle raw display
$7$ \( T^{9} + 5 T^{8} - 16 T^{7} - 101 T^{6} + \cdots + 552 \) Copy content Toggle raw display
$11$ \( T^{9} + 8 T^{8} - 4 T^{7} - 147 T^{6} + \cdots - 192 \) Copy content Toggle raw display
$13$ \( T^{9} + T^{8} - 53 T^{7} - 82 T^{6} + \cdots - 2904 \) Copy content Toggle raw display
$17$ \( T^{9} + 2 T^{8} - 64 T^{7} + \cdots - 60174 \) Copy content Toggle raw display
$19$ \( T^{9} + 11 T^{8} - 19 T^{7} + \cdots - 20138 \) Copy content Toggle raw display
$23$ \( (T + 1)^{9} \) Copy content Toggle raw display
$29$ \( (T + 1)^{9} \) Copy content Toggle raw display
$31$ \( T^{9} - 80 T^{7} + 101 T^{6} + \cdots + 576 \) Copy content Toggle raw display
$37$ \( T^{9} + 2 T^{8} - 177 T^{7} + \cdots - 940008 \) Copy content Toggle raw display
$41$ \( T^{9} + 3 T^{8} - 121 T^{7} + \cdots + 319824 \) Copy content Toggle raw display
$43$ \( T^{9} + 19 T^{8} + 38 T^{7} + \cdots + 15258 \) Copy content Toggle raw display
$47$ \( T^{9} + 3 T^{8} - 187 T^{7} + \cdots - 1714992 \) Copy content Toggle raw display
$53$ \( T^{9} + 9 T^{8} - 139 T^{7} + \cdots + 410472 \) Copy content Toggle raw display
$59$ \( T^{9} + 2 T^{8} - 173 T^{7} + \cdots + 2844288 \) Copy content Toggle raw display
$61$ \( T^{9} + 25 T^{8} + 102 T^{7} + \cdots - 226976 \) Copy content Toggle raw display
$67$ \( T^{9} + 20 T^{8} - 61 T^{7} + \cdots - 226816 \) Copy content Toggle raw display
$71$ \( T^{9} - 9 T^{8} - 477 T^{7} + \cdots - 54048 \) Copy content Toggle raw display
$73$ \( T^{9} + 11 T^{8} + \cdots - 287789584 \) Copy content Toggle raw display
$79$ \( T^{9} - 4 T^{8} - 569 T^{7} + \cdots - 120367822 \) Copy content Toggle raw display
$83$ \( T^{9} + 9 T^{8} - 283 T^{7} + \cdots + 2865024 \) Copy content Toggle raw display
$89$ \( T^{9} + 29 T^{8} + 19 T^{7} + \cdots - 65904558 \) Copy content Toggle raw display
$97$ \( T^{9} + 43 T^{8} + 546 T^{7} + \cdots - 5813856 \) Copy content Toggle raw display
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