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Properties

Label 8624.2.a.de
Level $8624$
Weight $2$
Character orbit 8624.a
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.98988261376.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 12x^{6} + 37x^{4} - 28x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4312)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{5} + (\beta_{7} - \beta_{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{3} + (\beta_{5} - \beta_{4} - \beta_{3}) q^{5} + (\beta_{7} - \beta_{2} + 1) q^{9} + q^{11} + (\beta_{6} + \beta_{4} - \beta_{3}) q^{13} + (\beta_{7} - 2 \beta_{2} - \beta_1 + 2) q^{15} + ( - \beta_{6} + \beta_{4} - \beta_{3}) q^{17} + (2 \beta_{5} - 2 \beta_{3}) q^{19} + ( - \beta_{2} + 2 \beta_1 + 3) q^{23} + ( - \beta_{7} - 3 \beta_{2} + 3) q^{25} + (\beta_{6} + \beta_{5} + \cdots - 2 \beta_{3}) q^{27}+ \cdots + (\beta_{7} - \beta_{2} + 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{9} + 8 q^{11} + 20 q^{15} + 16 q^{23} + 24 q^{25} - 24 q^{29} - 36 q^{37} + 32 q^{39} - 8 q^{43} + 48 q^{51} + 56 q^{57} - 32 q^{65} + 48 q^{67} + 64 q^{71} - 8 q^{79} + 20 q^{81} - 8 q^{85} - 48 q^{93} + 48 q^{95} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 12x^{6} + 37x^{4} - 28x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{6} - 10\nu^{4} + 17\nu^{2} + 2 ) / 4 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 14\nu^{4} + 49\nu^{2} - 22 ) / 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{7} + 34\nu^{5} - 83\nu^{3} + 2\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{7} + 14\nu^{5} - 57\nu^{3} + 70\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 34\nu^{5} + 91\nu^{3} - 42\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{7} + 62\nu^{5} - 197\nu^{3} + 110\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{6} - 34\nu^{4} + 91\nu^{2} - 38 ) / 4 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{6} + \beta_{4} + \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} - 2\beta_{2} - 2\beta _1 + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -5\beta_{6} + 2\beta_{5} + 5\beta_{4} + 9\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 4\beta_{7} - 10\beta_{2} - 7\beta _1 + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -29\beta_{6} + 22\beta_{5} + 35\beta_{4} + 69\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 63\beta_{7} - 166\beta_{2} - 98\beta _1 + 191 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -191\beta_{6} + 194\beta_{5} + 259\beta_{4} + 523\beta_{3} ) / 2 \) 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
2.75952
−0.917853
−1.83218
−0.430979
0.430979
1.83218
0.917853
−2.75952
0 −2.90087 0 −3.62563 0 0 0 5.41503 0
1.2 0 −2.61511 0 −0.436116 0 0 0 3.83882 0
1.3 0 −0.798145 0 0.293453 0 0 0 −2.36297 0
1.4 0 −0.330316 0 4.31028 0 0 0 −2.89089 0
1.5 0 0.330316 0 −4.31028 0 0 0 −2.89089 0
1.6 0 0.798145 0 −0.293453 0 0 0 −2.36297 0
1.7 0 2.61511 0 0.436116 0 0 0 3.83882 0
1.8 0 2.90087 0 3.62563 0 0 0 5.41503 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
\(2\) \(1\)
\(7\) \(1\)
\(11\) \(-1\)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8624.2.a.de 8
4.b odd 2 1 4312.2.a.bi ✓ 8
7.b odd 2 1 inner 8624.2.a.de 8
28.d even 2 1 4312.2.a.bi ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4312.2.a.bi ✓ 8 4.b odd 2 1
4312.2.a.bi ✓ 8 28.d even 2 1
8624.2.a.de 8 1.a even 1 1 trivial
8624.2.a.de 8 7.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8624))\):

\( T_{3}^{8} - 16T_{3}^{6} + 69T_{3}^{4} - 44T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{5}^{8} - 32T_{5}^{6} + 253T_{5}^{4} - 68T_{5}^{2} + 4 \) Copy content Toggle raw display
\( T_{13}^{8} - 80T_{13}^{6} + 2128T_{13}^{4} - 19712T_{13}^{2} + 25600 \) Copy content Toggle raw display
\( T_{17}^{8} - 80T_{17}^{6} + 1568T_{17}^{4} - 3328T_{17}^{2} + 256 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 16 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} - 32 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} - 80 T^{6} + \cdots + 25600 \) Copy content Toggle raw display
$17$ \( T^{8} - 80 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 80 T^{6} + \cdots + 4096 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} + \cdots - 716)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots - 112)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 160 T^{6} + \cdots + 1012036 \) Copy content Toggle raw display
$37$ \( (T^{4} + 18 T^{3} + \cdots - 284)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - 272 T^{6} + \cdots + 13897984 \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots + 7936)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 344 T^{6} + \cdots + 50410000 \) Copy content Toggle raw display
$53$ \( (T^{4} - 144 T^{2} + \cdots + 3584)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 32 T^{6} + \cdots + 2500 \) Copy content Toggle raw display
$61$ \( T^{8} - 240 T^{6} + \cdots + 952576 \) Copy content Toggle raw display
$67$ \( (T^{4} - 24 T^{3} + \cdots - 10828)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} - 32 T^{3} + \cdots + 820)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} - 112 T^{6} + \cdots + 215296 \) Copy content Toggle raw display
$79$ \( (T^{4} + 4 T^{3} + \cdots + 8864)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 512 T^{6} + \cdots + 163840000 \) Copy content Toggle raw display
$89$ \( T^{8} - 336 T^{6} + \cdots + 624100 \) Copy content Toggle raw display
$97$ \( T^{8} - 608 T^{6} + \cdots + 3161284 \) Copy content Toggle raw display
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