Properties

Label 252.8.k.a
Level $252$
Weight $8$
Character orbit 252.k
Analytic conductor $78.721$
Analytic rank $0$
Dimension $2$
CM discriminant -3
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [252,8,Mod(37,252)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(252, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("252.37");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.k (of order \(3\), degree \(2\), minimal)

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: no
Analytic conductor: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{3}]$

$q$-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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 757 \zeta_{6} - 249) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 757 \zeta_{6} - 249) q^{7} + 12605 q^{13} + (43091 \zeta_{6} - 43091) q^{19} + 78125 \zeta_{6} q^{25} - 152471 \zeta_{6} q^{31} + ( - 615373 \zeta_{6} + 615373) q^{37} - 625729 q^{43} + (950035 \zeta_{6} - 511048) q^{49} + ( - 3535546 \zeta_{6} + 3535546) q^{61} + 4058455 \zeta_{6} q^{67} - 5038001 \zeta_{6} q^{73} + ( - 4245427 \zeta_{6} + 4245427) q^{79} + ( - 9541985 \zeta_{6} - 3138645) q^{91} + 12245198 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1255 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1255 q^{7} + 25210 q^{13} - 43091 q^{19} + 78125 q^{25} - 152471 q^{31} + 615373 q^{37} - 1251458 q^{43} - 72061 q^{49} + 3535546 q^{61} + 4058455 q^{67} - 5038001 q^{73} + 4245427 q^{79} - 15819275 q^{91} + 24490396 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(1\) \(-1 + \zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0 0 0 −627.500 655.581i 0 0 0
109.1 0 0 0 0 0 −627.500 + 655.581i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
7.c even 3 1 inner
21.h odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 252.8.k.a 2
3.b odd 2 1 CM 252.8.k.a 2
7.c even 3 1 inner 252.8.k.a 2
21.h odd 6 1 inner 252.8.k.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.8.k.a 2 1.a even 1 1 trivial
252.8.k.a 2 3.b odd 2 1 CM
252.8.k.a 2 7.c even 3 1 inner
252.8.k.a 2 21.h odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5} \) acting on \(S_{8}^{\mathrm{new}}(252, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1255 T + 823543 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( (T - 12605)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 1856834281 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + \cdots + 23247405841 \) Copy content Toggle raw display
$37$ \( T^{2} + \cdots + 378683929129 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( (T + 625729)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( T^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} + \cdots + 12500085518116 \) Copy content Toggle raw display
$67$ \( T^{2} + \cdots + 16471056987025 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + \cdots + 25381454076001 \) Copy content Toggle raw display
$79$ \( T^{2} + \cdots + 18023650412329 \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( (T - 12245198)^{2} \) Copy content Toggle raw display
show more
show less