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Properties

Label 8008.2.a.s
Level $8008$
Weight $2$
Character orbit 8008.a
Self dual yes
Analytic conductor $63.944$
Analytic rank $1$
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: \(1\)
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: \( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + \beta_{9} q^{5} + q^{7} + ( - \beta_{6} + \beta_{5} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + \beta_{9} q^{5} + q^{7} + ( - \beta_{6} + \beta_{5} + 1) q^{9} + q^{11} - q^{13} + ( - \beta_{9} + \beta_{6} + \beta_1 - 1) q^{15} + ( - \beta_{9} - \beta_{4} - \beta_{2} - 1) q^{17} + ( - \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 1) q^{19}+ \cdots + ( - \beta_{6} + \beta_{5} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 3 q^{3} - 4 q^{5} + 10 q^{7} + 9 q^{9} + 10 q^{11} - 10 q^{13} - 5 q^{15} - 11 q^{17} + 2 q^{19} - 3 q^{21} - 8 q^{23} + 2 q^{25} - 15 q^{27} - 8 q^{29} - 23 q^{31} - 3 q^{33} - 4 q^{35} + 10 q^{37} + 3 q^{39} - 18 q^{41} + 12 q^{43} - 10 q^{45} - 36 q^{47} + 10 q^{49} + 9 q^{51} - 21 q^{53} - 4 q^{55} - 30 q^{57} - 13 q^{59} - 2 q^{61} + 9 q^{63} + 4 q^{65} - 4 q^{67} - 26 q^{69} - 24 q^{71} - 23 q^{73} - 28 q^{75} + 10 q^{77} + 14 q^{79} + 30 q^{81} - 9 q^{83} - 17 q^{85} + 7 q^{87} - 18 q^{89} - 10 q^{91} + q^{93} - 4 q^{95} - 9 q^{97} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 3x^{9} - 15x^{8} + 43x^{7} + 66x^{6} - 173x^{5} - 127x^{4} + 246x^{3} + 99x^{2} - 82x + 8 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{9} + 41\nu^{7} + 7\nu^{6} - 255\nu^{5} - 64\nu^{4} + 501\nu^{3} + 157\nu^{2} - 173\nu + 16 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -5\nu^{9} + 15\nu^{8} + 62\nu^{7} - 180\nu^{6} - 159\nu^{5} + 452\nu^{4} + 108\nu^{3} - 245\nu^{2} + 42\nu + 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{9} + 29\nu^{8} + 147\nu^{7} - 343\nu^{6} - 474\nu^{5} + 827\nu^{4} + 525\nu^{3} - 356\nu^{2} - \nu + 10 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} + \cdots + 18 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{9} + 41 \nu^{8} + 277 \nu^{7} - 469 \nu^{6} - 1090 \nu^{5} + 1015 \nu^{4} + 1547 \nu^{3} + \cdots + 26 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 19 \nu^{9} + 33 \nu^{8} + 299 \nu^{7} - 365 \nu^{6} - 1350 \nu^{5} + 705 \nu^{4} + 2169 \nu^{3} + \cdots + 62 ) / 2 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13\nu^{9} - 9\nu^{8} - 241\nu^{7} + 71\nu^{6} + 1352\nu^{5} + 71\nu^{4} - 2493\nu^{3} - 684\nu^{2} + 735\nu - 76 ) / 2 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 15\nu^{9} - 11\nu^{8} - 277\nu^{7} + 91\nu^{6} + 1550\nu^{5} + 45\nu^{4} - 2875\nu^{3} - 732\nu^{2} + 895\nu - 96 ) / 2 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} - \beta_{8} - \beta_{6} + \beta_{5} - 2\beta_{4} + 2\beta_{3} + \beta_{2} + 8\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} + 4\beta_{7} - 12\beta_{6} + 10\beta_{5} - 2\beta_{4} + \beta_{3} + \beta_{2} + 4\beta _1 + 29 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 18 \beta_{9} - 14 \beta_{8} + 6 \beta_{7} - 17 \beta_{6} + 14 \beta_{5} - 27 \beta_{4} + 26 \beta_{3} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 42 \beta_{9} - 6 \beta_{8} + 63 \beta_{7} - 132 \beta_{6} + 102 \beta_{5} - 39 \beta_{4} + 23 \beta_{3} + \cdots + 263 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 245 \beta_{9} - 159 \beta_{8} + 123 \beta_{7} - 233 \beta_{6} + 174 \beta_{5} - 314 \beta_{4} + \cdots + 321 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 635 \beta_{9} - 138 \beta_{8} + 807 \beta_{7} - 1454 \beta_{6} + 1077 \beta_{5} - 571 \beta_{4} + \cdots + 2642 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3061 \beta_{9} - 1746 \beta_{8} + 1849 \beta_{7} - 3016 \beta_{6} + 2148 \beta_{5} - 3568 \beta_{4} + \cdots + 4081 \) 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
3.44977
2.46329
1.79551
1.79314
0.364324
0.119595
−1.01146
−1.48379
−1.49066
−2.99972
0 −3.44977 0 0.464152 0 1.00000 0 8.90091 0
1.2 0 −2.46329 0 −1.35093 0 1.00000 0 3.06778 0
1.3 0 −1.79551 0 3.68086 0 1.00000 0 0.223871 0
1.4 0 −1.79314 0 −4.09989 0 1.00000 0 0.215350 0
1.5 0 −0.364324 0 2.29944 0 1.00000 0 −2.86727 0
1.6 0 −0.119595 0 −2.15143 0 1.00000 0 −2.98570 0
1.7 0 1.01146 0 −0.0780254 0 1.00000 0 −1.97695 0
1.8 0 1.48379 0 1.24336 0 1.00000 0 −0.798370 0
1.9 0 1.49066 0 −2.23108 0 1.00000 0 −0.777924 0
1.10 0 2.99972 0 −1.77646 0 1.00000 0 5.99829 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.s ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.s ✓ 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(8008))\):

\( T_{3}^{10} + 3T_{3}^{9} - 15T_{3}^{8} - 43T_{3}^{7} + 66T_{3}^{6} + 173T_{3}^{5} - 127T_{3}^{4} - 246T_{3}^{3} + 99T_{3}^{2} + 82T_{3} + 8 \) Copy content Toggle raw display
\( T_{5}^{10} + 4 T_{5}^{9} - 18 T_{5}^{8} - 82 T_{5}^{7} + 47 T_{5}^{6} + 431 T_{5}^{5} + 168 T_{5}^{4} + \cdots + 18 \) Copy content Toggle raw display
\( T_{17}^{10} + 11 T_{17}^{9} - 5 T_{17}^{8} - 349 T_{17}^{7} - 380 T_{17}^{6} + 4001 T_{17}^{5} + \cdots + 6316 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 3 T^{9} + \cdots + 8 \) Copy content Toggle raw display
$5$ \( T^{10} + 4 T^{9} + \cdots + 18 \) 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
$17$ \( T^{10} + 11 T^{9} + \cdots + 6316 \) Copy content Toggle raw display
$19$ \( T^{10} - 2 T^{9} + \cdots + 12148 \) Copy content Toggle raw display
$23$ \( T^{10} + 8 T^{9} + \cdots + 186592 \) Copy content Toggle raw display
$29$ \( T^{10} + 8 T^{9} + \cdots + 2254648 \) Copy content Toggle raw display
$31$ \( T^{10} + 23 T^{9} + \cdots + 321552 \) Copy content Toggle raw display
$37$ \( T^{10} - 10 T^{9} + \cdots - 3555312 \) Copy content Toggle raw display
$41$ \( T^{10} + 18 T^{9} + \cdots - 28325264 \) Copy content Toggle raw display
$43$ \( T^{10} - 12 T^{9} + \cdots + 87736 \) Copy content Toggle raw display
$47$ \( T^{10} + 36 T^{9} + \cdots - 40392432 \) Copy content Toggle raw display
$53$ \( T^{10} + 21 T^{9} + \cdots + 2179822 \) Copy content Toggle raw display
$59$ \( T^{10} + 13 T^{9} + \cdots - 99072 \) Copy content Toggle raw display
$61$ \( T^{10} + 2 T^{9} + \cdots - 12429956 \) Copy content Toggle raw display
$67$ \( T^{10} + 4 T^{9} + \cdots + 12248 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 119759488 \) Copy content Toggle raw display
$73$ \( T^{10} + 23 T^{9} + \cdots + 2875592 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots - 2657567288 \) Copy content Toggle raw display
$83$ \( T^{10} + 9 T^{9} + \cdots + 12897556 \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 579216766 \) Copy content Toggle raw display
$97$ \( T^{10} + 9 T^{9} + \cdots + 58810528 \) Copy content Toggle raw display
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