↑

Properties

Label 8619.2.a.bf
Level $8619$
Weight $2$
Character orbit 8619.a
Self dual yes
Analytic conductor $68.823$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8619,2,Mod(1,8619)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8619, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("8619.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8619.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: \(68.8230615021\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.59052888064.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 10x^{6} + 28x^{4} - 24x^{2} + 1 \) 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 663)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_1) q^{2} - q^{3} + (\beta_{5} + 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{7} - \beta_1) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{7} - \beta_{3} + 2 \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_1) q^{2} - q^{3} + (\beta_{5} + 2) q^{4} + (\beta_{4} - \beta_1) q^{5} + ( - \beta_{7} - \beta_1) q^{6} + ( - \beta_{3} + \beta_1) q^{7} + (\beta_{7} - \beta_{3} + 2 \beta_1) q^{8} + q^{9} + ( - \beta_{6} - \beta_{5} - \beta_{2} - 1) q^{10} + ( - \beta_{7} + \beta_{4} + \beta_{3}) q^{11} + ( - \beta_{5} - 2) q^{12} + ( - \beta_{6} + 2 \beta_{5} - \beta_{2}) q^{14} + ( - \beta_{4} + \beta_1) q^{15} + ( - \beta_{6} + \beta_{5} - \beta_{2}) q^{16} - q^{17} + (\beta_{7} + \beta_1) q^{18} + ( - \beta_{7} - \beta_{3} - \beta_1) q^{19} + ( - \beta_{7} + \beta_{4} - \beta_{3}) q^{20} + (\beta_{3} - \beta_1) q^{21} + ( - \beta_{5} + 3 \beta_{2} + 1) q^{22} + (\beta_{6} + \beta_{5} - \beta_{2} - 3) q^{23} + ( - \beta_{7} + \beta_{3} - 2 \beta_1) q^{24} + ( - 2 \beta_{5} + 1) q^{25} - q^{27} + (3 \beta_{7} + 3 \beta_{4} + \cdots + 3 \beta_1) q^{28}+ \cdots + ( - \beta_{7} + \beta_{4} + \beta_{3}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{3} + 16 q^{4} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{3} + 16 q^{4} + 8 q^{9} - 8 q^{10} - 16 q^{12} - 8 q^{17} - 4 q^{22} - 16 q^{23} + 8 q^{25} - 8 q^{27} + 8 q^{30} - 4 q^{35} + 16 q^{36} - 44 q^{38} - 12 q^{40} + 36 q^{43} + 28 q^{49} + 8 q^{51} + 4 q^{53} + 84 q^{56} + 28 q^{61} - 4 q^{62} - 12 q^{64} + 4 q^{66} - 16 q^{68} + 16 q^{69} + 36 q^{74} - 8 q^{75} - 52 q^{77} + 64 q^{79} + 8 q^{81} + 120 q^{82} - 56 q^{88} - 8 q^{90} - 4 q^{92} + 52 q^{94} + 24 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 10x^{6} + 28x^{4} - 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 4\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{5} - 8\nu^{3} + 12\nu \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{6} - 8\nu^{4} + 12\nu^{2} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -\nu^{6} + 9\nu^{4} - 19\nu^{2} + 7 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} - 9\nu^{5} + 20\nu^{3} - 12\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} + \beta_{5} + 7\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{4} + 8\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 8\beta_{6} + 9\beta_{5} + 44\beta_{2} + 76 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} + 9\beta_{4} + 52\beta_{3} + 112\beta_1 \) 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
−1.27474
0.209470
−2.44735
1.53025
−1.53025
2.44735
−0.209470
1.27474
−2.58192 −1.00000 4.66632 −0.816926 2.58192 −4.30230 −6.88422 1.00000 2.10924
1.2 −2.12396 −1.00000 2.51122 2.23105 2.12396 1.03816 −1.08580 1.00000 −4.73866
1.3 −1.93632 −1.00000 1.74934 2.54977 1.93632 2.42168 0.485353 1.00000 −4.93717
1.4 −1.03592 −1.00000 −0.926874 −3.44293 1.03592 4.06792 3.03200 1.00000 3.56659
1.5 1.03592 −1.00000 −0.926874 3.44293 −1.03592 −4.06792 −3.03200 1.00000 3.56659
1.6 1.93632 −1.00000 1.74934 −2.54977 −1.93632 −2.42168 −0.485353 1.00000 −4.93717
1.7 2.12396 −1.00000 2.51122 −2.23105 −2.12396 −1.03816 1.08580 1.00000 −4.73866
1.8 2.58192 −1.00000 4.66632 0.816926 −2.58192 4.30230 6.88422 1.00000 2.10924
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(13\) \(-1\)
\(17\) \(1\)

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8619.2.a.bf 8
13.b even 2 1 inner 8619.2.a.bf 8
13.d odd 4 2 663.2.b.d ✓ 8
39.f even 4 2 1989.2.b.g 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
663.2.b.d ✓ 8 13.d odd 4 2
1989.2.b.g 8 39.f even 4 2
8619.2.a.bf 8 1.a even 1 1 trivial
8619.2.a.bf 8 13.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8619))\):

\( T_{2}^{8} - 16T_{2}^{6} + 88T_{2}^{4} - 190T_{2}^{2} + 121 \) Copy content Toggle raw display
\( T_{5}^{8} - 24T_{5}^{6} + 184T_{5}^{4} - 496T_{5}^{2} + 256 \) Copy content Toggle raw display
\( T_{7}^{8} - 42T_{7}^{6} + 556T_{7}^{4} - 2348T_{7}^{2} + 1936 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 16 T^{6} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} - 24 T^{6} + \cdots + 256 \) Copy content Toggle raw display
$7$ \( T^{8} - 42 T^{6} + \cdots + 1936 \) Copy content Toggle raw display
$11$ \( T^{8} - 52 T^{6} + \cdots + 7744 \) Copy content Toggle raw display
$13$ \( T^{8} \) Copy content Toggle raw display
$17$ \( (T + 1)^{8} \) Copy content Toggle raw display
$19$ \( T^{8} - 68 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$23$ \( (T^{4} + 8 T^{3} + \cdots - 704)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 52 T^{2} + \cdots - 152)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} - 134 T^{6} + \cdots + 110224 \) Copy content Toggle raw display
$37$ \( T^{8} - 262 T^{6} + \cdots + 2483776 \) Copy content Toggle raw display
$41$ \( T^{8} - 248 T^{6} + \cdots + 7573504 \) Copy content Toggle raw display
$43$ \( (T^{4} - 18 T^{3} + \cdots - 2288)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 418 T^{6} + \cdots + 45751696 \) Copy content Toggle raw display
$53$ \( (T^{4} - 2 T^{3} + \cdots - 416)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 322 T^{6} + \cdots + 190096 \) Copy content Toggle raw display
$61$ \( (T^{4} - 14 T^{3} + \cdots - 5696)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} - 148 T^{6} + \cdots + 1024 \) Copy content Toggle raw display
$71$ \( T^{8} - 68 T^{6} + \cdots + 64 \) Copy content Toggle raw display
$73$ \( T^{8} - 278 T^{6} + \cdots + 1478656 \) Copy content Toggle raw display
$79$ \( (T - 8)^{8} \) Copy content Toggle raw display
$83$ \( T^{8} - 310 T^{6} + \cdots + 21418384 \) Copy content Toggle raw display
$89$ \( T^{8} - 782 T^{6} + \cdots + 368025856 \) Copy content Toggle raw display
$97$ \( T^{8} - 306 T^{6} + \cdots + 7139584 \) Copy content Toggle raw display
show more
show less