Properties

Label 8008.2.a.t
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $0$
Dimension $10$
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: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{6} q^{5} + q^{7} + (\beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{6} q^{5} + q^{7} + (\beta_{2} + 1) q^{9} - q^{11} + q^{13} + ( - \beta_{9} - \beta_{8} + \beta_{6} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{15} + ( - \beta_{9} + \beta_{8} - \beta_{6} + 1) q^{17} + (\beta_{8} + \beta_{2} + 2) q^{19} + \beta_1 q^{21} + ( - \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 + 2) q^{23} + (\beta_{6} - \beta_{4} + \beta_{2} + \beta_1 + 1) q^{25} + (\beta_{9} - \beta_{7} + \beta_{6} + \beta_{3}) q^{27} + (\beta_{9} - \beta_{6} - \beta_{4} - \beta_{3}) q^{29} + ( - \beta_{9} + \beta_{7} - \beta_{4} - 1) q^{31} - \beta_1 q^{33} + \beta_{6} q^{35} + (\beta_{7} - \beta_{5} - \beta_{3} - \beta_1 - 1) q^{37} + \beta_1 q^{39} + ( - \beta_{9} + \beta_{8} + \beta_{6} + 2 \beta_{4} - \beta_1 + 1) q^{41} + (\beta_{8} + \beta_{4} - \beta_1 + 2) q^{43} + ( - \beta_{9} - 2 \beta_{8} + \beta_{6} - \beta_{5} - \beta_{4} - 2 \beta_{3} + \beta_1 - 1) q^{45} + (\beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{47} + q^{49} + (\beta_{8} - \beta_{7} + \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{51} + ( - \beta_{9} - 2 \beta_{8} + \beta_{5} + \beta_{4} + \beta_1 + 3) q^{53} - \beta_{6} q^{55} + (\beta_{9} - 2 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} + 4 \beta_1) q^{57} + ( - \beta_{9} + \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{59} + (\beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_1 + 1) q^{61} + (\beta_{2} + 1) q^{63} + \beta_{6} q^{65} + ( - \beta_{9} + \beta_{8} + 2 \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 + 2) q^{67} + (\beta_{9} - 2 \beta_{8} - \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + 2 \beta_1 + 3) q^{69} + (2 \beta_{9} + \beta_{8} + \beta_{7} + 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + \beta_{2}) q^{71} + (\beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 3 \beta_1 - 2) q^{73} + ( - \beta_{9} - \beta_{8} - \beta_{7} + 2 \beta_{6} - \beta_{5} - \beta_{3} + 2 \beta_{2} + \beta_1 + 6) q^{75} - q^{77} + (\beta_{9} - \beta_{7} + \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 1) q^{79} + (\beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{81} + (2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 + 4) q^{83} + ( - \beta_{9} + \beta_{6} + 2 \beta_{5} + 3 \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{85} + (\beta_{9} - \beta_{6} - \beta_{5} - \beta_{2} - 3) q^{87} + (2 \beta_{9} - \beta_{8} + \beta_{6} + \beta_{5} + \beta_{4} + 3 \beta_{3} - \beta_1 - 3) q^{89} + q^{91} + ( - 2 \beta_{9} - \beta_{8} + \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - 3 \beta_1 + 1) q^{93} + ( - \beta_{9} - 2 \beta_{8} + 2 \beta_{6} - \beta_{4} - 2 \beta_{3} - \beta_{2} + 1) q^{95} + ( - \beta_{9} + 2 \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} - 5 \beta_{3} + \beta_{2} - \beta_1) q^{97} + ( - \beta_{2} - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + q^{3} + 3 q^{5} + 10 q^{7} + 5 q^{9} - 10 q^{11} + 10 q^{13} + 9 q^{15} + 7 q^{17} + 17 q^{19} + q^{21} + 15 q^{23} + 13 q^{25} + 7 q^{27} - q^{29} - 6 q^{31} - q^{33} + 3 q^{35} - 12 q^{37} + q^{39} + 4 q^{41} + 17 q^{43} - 15 q^{45} + 21 q^{47} + 10 q^{49} + 3 q^{51} + 20 q^{53} - 3 q^{55} + 12 q^{57} + 9 q^{59} + 6 q^{61} + 5 q^{63} + 3 q^{65} + 26 q^{67} + 30 q^{69} + 18 q^{71} - 4 q^{73} + 48 q^{75} - 10 q^{77} + 19 q^{79} - 14 q^{81} + 40 q^{83} - 25 q^{85} - 25 q^{87} - 19 q^{89} + 10 q^{91} + q^{93} + 11 q^{95} - 22 q^{97} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - x^{9} - 17x^{8} + 13x^{7} + 96x^{6} - 49x^{5} - 207x^{4} + 58x^{3} + 169x^{2} - 12x - 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 15\nu^{9} + \nu^{8} - 247\nu^{7} - 65\nu^{6} + 1258\nu^{5} + 541\nu^{4} - 1949\nu^{3} - 864\nu^{2} + 641\nu + 138 ) / 26 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25 \nu^{9} + 7 \nu^{8} + 403 \nu^{7} - 13 \nu^{6} - 1984 \nu^{5} - 425 \nu^{4} + 2841 \nu^{3} + 1050 \nu^{2} - 739 \nu - 308 ) / 26 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{9} - 16\nu^{7} - 4\nu^{6} + 77\nu^{5} + 39\nu^{4} - 101\nu^{3} - 72\nu^{2} + 16\nu + 18 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23 \nu^{9} + 5 \nu^{8} - 377 \nu^{7} - 169 \nu^{6} + 1870 \nu^{5} + 1249 \nu^{4} - 2569 \nu^{3} - 2058 \nu^{2} + 449 \nu + 456 ) / 26 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21 \nu^{9} - 17 \nu^{8} + 351 \nu^{7} + 351 \nu^{6} - 1782 \nu^{5} - 2099 \nu^{4} + 2531 \nu^{3} + 3300 \nu^{2} - 445 \nu - 812 ) / 26 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 3\nu^{9} + \nu^{8} - 49\nu^{7} - 27\nu^{6} + 242\nu^{5} + 185\nu^{4} - 331\nu^{3} - 304\nu^{2} + 57\nu + 72 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 59 \nu^{9} - 23 \nu^{8} + 975 \nu^{7} + 585 \nu^{6} - 4910 \nu^{5} - 3889 \nu^{4} + 7075 \nu^{3} + 6222 \nu^{2} - 1691 \nu - 1406 ) / 26 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{7} + \beta_{6} + \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{8} - \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + 8\beta_{2} + \beta _1 + 25 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9\beta_{9} - \beta_{8} - 10\beta_{7} + 8\beta_{6} + \beta_{5} + 10\beta_{3} + \beta_{2} + 42\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{9} + 13 \beta_{8} + \beta_{7} - 10 \beta_{6} - 12 \beta_{5} - 10 \beta_{4} - 9 \beta_{3} + 59 \beta_{2} + 14 \beta _1 + 174 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 72 \beta_{9} - 12 \beta_{8} - 84 \beta_{7} + 57 \beta_{6} + 16 \beta_{5} + 2 \beta_{4} + 85 \beta_{3} + 15 \beta_{2} + 308 \beta _1 + 47 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{9} + 128 \beta_{8} + 13 \beta_{7} - 85 \beta_{6} - 110 \beta_{5} - 80 \beta_{4} - 64 \beta_{3} + 433 \beta_{2} + 147 \beta _1 + 1258 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 564 \beta_{9} - 102 \beta_{8} - 671 \beta_{7} + 396 \beta_{6} + 171 \beta_{5} + 31 \beta_{4} + 694 \beta_{3} + 159 \beta_{2} + 2301 \beta _1 + 512 \) 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
−2.72305
−2.25881
−1.29966
−1.02605
−0.516933
0.514400
1.37400
1.38780
2.73324
2.81506
0 −2.72305 0 −1.62788 0 1.00000 0 4.41500 0
1.2 0 −2.25881 0 −2.11934 0 1.00000 0 2.10224 0
1.3 0 −1.29966 0 0.367835 0 1.00000 0 −1.31089 0
1.4 0 −1.02605 0 3.56808 0 1.00000 0 −1.94722 0
1.5 0 −0.516933 0 1.90177 0 1.00000 0 −2.73278 0
1.6 0 0.514400 0 −2.27351 0 1.00000 0 −2.73539 0
1.7 0 1.37400 0 −0.949904 0 1.00000 0 −1.11212 0
1.8 0 1.38780 0 2.74127 0 1.00000 0 −1.07400 0
1.9 0 2.73324 0 4.21870 0 1.00000 0 4.47060 0
1.10 0 2.81506 0 −2.82702 0 1.00000 0 4.92456 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.10
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.t 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.t 10 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{10} - T_{3}^{9} - 17T_{3}^{8} + 13T_{3}^{7} + 96T_{3}^{6} - 49T_{3}^{5} - 207T_{3}^{4} + 58T_{3}^{3} + 169T_{3}^{2} - 12T_{3} - 32 \) Copy content Toggle raw display
\( T_{5}^{10} - 3 T_{5}^{9} - 27 T_{5}^{8} + 57 T_{5}^{7} + 290 T_{5}^{6} - 317 T_{5}^{5} - 1415 T_{5}^{4} + 344 T_{5}^{3} + 2595 T_{5}^{2} + 726 T_{5} - 608 \) Copy content Toggle raw display
\( T_{17}^{10} - 7 T_{17}^{9} - 61 T_{17}^{8} + 419 T_{17}^{7} + 1344 T_{17}^{6} - 8543 T_{17}^{5} - 14261 T_{17}^{4} + 71372 T_{17}^{3} + 76659 T_{17}^{2} - 204614 T_{17} - 162592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} - 17 T^{8} + 13 T^{7} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{10} - 3 T^{9} - 27 T^{8} + 57 T^{7} + \cdots - 608 \) Copy content Toggle raw display
$7$ \( (T - 1)^{10} \) Copy content Toggle raw display
$11$ \( (T + 1)^{10} \) Copy content Toggle raw display
$13$ \( (T - 1)^{10} \) Copy content Toggle raw display
$17$ \( T^{10} - 7 T^{9} - 61 T^{8} + \cdots - 162592 \) Copy content Toggle raw display
$19$ \( T^{10} - 17 T^{9} + 45 T^{8} + \cdots + 62312 \) Copy content Toggle raw display
$23$ \( T^{10} - 15 T^{9} - 16 T^{8} + \cdots + 1206272 \) Copy content Toggle raw display
$29$ \( T^{10} + T^{9} - 108 T^{8} + 174 T^{7} + \cdots - 304 \) Copy content Toggle raw display
$31$ \( T^{10} + 6 T^{9} - 140 T^{8} + \cdots - 141952 \) Copy content Toggle raw display
$37$ \( T^{10} + 12 T^{9} - 54 T^{8} + \cdots - 886384 \) Copy content Toggle raw display
$41$ \( T^{10} - 4 T^{9} - 232 T^{8} + \cdots + 1129888 \) Copy content Toggle raw display
$43$ \( T^{10} - 17 T^{9} + 30 T^{8} + \cdots + 33776 \) Copy content Toggle raw display
$47$ \( T^{10} - 21 T^{9} - 46 T^{8} + \cdots - 9531136 \) Copy content Toggle raw display
$53$ \( T^{10} - 20 T^{9} - 96 T^{8} + \cdots + 70797668 \) Copy content Toggle raw display
$59$ \( T^{10} - 9 T^{9} - 127 T^{8} + \cdots - 90176 \) Copy content Toggle raw display
$61$ \( T^{10} - 6 T^{9} - 183 T^{8} + \cdots + 5379244 \) Copy content Toggle raw display
$67$ \( T^{10} - 26 T^{9} - 67 T^{8} + \cdots + 1216912 \) Copy content Toggle raw display
$71$ \( T^{10} - 18 T^{9} + \cdots + 3319494656 \) Copy content Toggle raw display
$73$ \( T^{10} + 4 T^{9} - 372 T^{8} + \cdots + 26582944 \) Copy content Toggle raw display
$79$ \( T^{10} - 19 T^{9} + \cdots + 352678912 \) Copy content Toggle raw display
$83$ \( T^{10} - 40 T^{9} + \cdots + 655432936 \) Copy content Toggle raw display
$89$ \( T^{10} + 19 T^{9} + \cdots - 519184124 \) Copy content Toggle raw display
$97$ \( T^{10} + 22 T^{9} + \cdots + 1930390976 \) Copy content Toggle raw display
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