Properties

Label 105.8.a.f
Level $105$
Weight $8$
Character orbit 105.a
Self dual yes
Analytic conductor $32.800$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [105,8,Mod(1,105)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(105, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("105.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 105.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: \(32.8004276758\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 466x^{2} + 520x + 23440 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + 27 q^{3} + (\beta_{2} + 105) q^{4} + 125 q^{5} - 27 \beta_1 q^{6} - 343 q^{7} + ( - 2 \beta_{3} + \beta_{2} + \cdots + 131) q^{8} + 729 q^{9} - 125 \beta_1 q^{10} + (3 \beta_{3} + 10 \beta_{2} + \cdots + 1965) q^{11}+ \cdots + (2187 \beta_{3} + 7290 \beta_{2} + \cdots + 1432485) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 108 q^{3} + 421 q^{4} + 500 q^{5} - 27 q^{6} - 1372 q^{7} + 417 q^{8} + 2916 q^{9} - 125 q^{10} + 7852 q^{11} + 11367 q^{12} + 18532 q^{13} + 343 q^{14} + 13500 q^{15} + 47601 q^{16} + 33976 q^{17}+ \cdots + 5724108 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} - 466x^{2} + 520x + 23440 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 233 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + \nu^{2} - 364\nu - 102 ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 233 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{3} - \beta_{2} + 364\beta _1 - 131 \) 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
20.0439
8.26635
−6.94744
−20.3628
−20.0439 27.0000 273.757 125.000 −541.185 −343.000 −2921.54 729.000 −2505.48
1.2 −8.26635 27.0000 −59.6674 125.000 −223.192 −343.000 1551.33 729.000 −1033.29
1.3 6.94744 27.0000 −79.7331 125.000 187.581 −343.000 −1443.21 729.000 868.429
1.4 20.3628 27.0000 286.643 125.000 549.796 −343.000 3230.42 729.000 2545.35
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( -1 \)
\(5\) \( -1 \)
\(7\) \( +1 \)

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 105.8.a.f 4
3.b odd 2 1 315.8.a.i 4
5.b even 2 1 525.8.a.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.8.a.f 4 1.a even 1 1 trivial
315.8.a.i 4 3.b odd 2 1
525.8.a.k 4 5.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + T_{2}^{3} - 466T_{2}^{2} - 520T_{2} + 23440 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(105))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + T^{3} + \cdots + 23440 \) Copy content Toggle raw display
$3$ \( (T - 27)^{4} \) Copy content Toggle raw display
$5$ \( (T - 125)^{4} \) Copy content Toggle raw display
$7$ \( (T + 343)^{4} \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots - 183343152835136 \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 28607321995840 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots - 69\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots - 66\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{4} + \cdots + 11\!\cdots\!56 \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 47\!\cdots\!60 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots - 97\!\cdots\!76 \) Copy content Toggle raw display
$41$ \( T^{4} + \cdots - 44\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots - 44\!\cdots\!60 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots - 35\!\cdots\!60 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots - 39\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots - 20\!\cdots\!40 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 17\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 45\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( T^{4} + \cdots - 23\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots - 27\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 69\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 18\!\cdots\!24 \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
show more
show less