↑

Properties

Label 2268.2.i.c
Level $2268$
Weight $2$
Character orbit 2268.i
Analytic conductor $18.110$
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] = [2268,2,Mod(865,2268)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2268, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.865");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.i (of order \(3\), degree \(2\), not 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: \(18.1100711784\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
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.

\(f(q)\) \(=\) \( q + (2 \zeta_{6} - 3) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \zeta_{6} - 3) q^{7} - 5 \zeta_{6} q^{13} + \zeta_{6} q^{19} + 5 \zeta_{6} q^{25} + 11 q^{31} - 11 \zeta_{6} q^{37} + ( - 13 \zeta_{6} + 13) q^{43} + ( - 8 \zeta_{6} + 5) q^{49} + 14 q^{61} + 5 q^{67} + (17 \zeta_{6} - 17) q^{73} + 17 q^{79} + (5 \zeta_{6} + 10) q^{91} + (14 \zeta_{6} - 14) q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{7} - 5 q^{13} + q^{19} + 5 q^{25} + 22 q^{31} - 11 q^{37} + 13 q^{43} + 2 q^{49} + 28 q^{61} + 10 q^{67} - 17 q^{73} + 34 q^{79} + 25 q^{91} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1541\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(-\zeta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
865.1
0.500000 + 0.866025i
0.500000 − 0.866025i
0 0 0 0 0 −2.00000 + 1.73205i 0 0 0
2053.1 0 0 0 0 0 −2.00000 − 1.73205i 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 \(\Q(\sqrt{-3}) \)
63.h even 3 1 inner
63.j odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.i.c 2
3.b odd 2 1 CM 2268.2.i.c 2
7.c even 3 1 2268.2.l.e 2
9.c even 3 1 252.2.k.b ✓ 2
9.c even 3 1 2268.2.l.e 2
9.d odd 6 1 252.2.k.b ✓ 2
9.d odd 6 1 2268.2.l.e 2
21.h odd 6 1 2268.2.l.e 2
36.f odd 6 1 1008.2.s.i 2
36.h even 6 1 1008.2.s.i 2
63.g even 3 1 252.2.k.b ✓ 2
63.h even 3 1 1764.2.a.f 1
63.h even 3 1 inner 2268.2.i.c 2
63.i even 6 1 1764.2.a.d 1
63.j odd 6 1 1764.2.a.f 1
63.j odd 6 1 inner 2268.2.i.c 2
63.k odd 6 1 1764.2.k.f 2
63.l odd 6 1 1764.2.k.f 2
63.n odd 6 1 252.2.k.b ✓ 2
63.o even 6 1 1764.2.k.f 2
63.s even 6 1 1764.2.k.f 2
63.t odd 6 1 1764.2.a.d 1
252.o even 6 1 1008.2.s.i 2
252.r odd 6 1 7056.2.a.z 1
252.u odd 6 1 7056.2.a.be 1
252.bb even 6 1 7056.2.a.be 1
252.bj even 6 1 7056.2.a.z 1
252.bl odd 6 1 1008.2.s.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.k.b ✓ 2 9.c even 3 1
252.2.k.b ✓ 2 9.d odd 6 1
252.2.k.b ✓ 2 63.g even 3 1
252.2.k.b ✓ 2 63.n odd 6 1
1008.2.s.i 2 36.f odd 6 1
1008.2.s.i 2 36.h even 6 1
1008.2.s.i 2 252.o even 6 1
1008.2.s.i 2 252.bl odd 6 1
1764.2.a.d 1 63.i even 6 1
1764.2.a.d 1 63.t odd 6 1
1764.2.a.f 1 63.h even 3 1
1764.2.a.f 1 63.j odd 6 1
1764.2.k.f 2 63.k odd 6 1
1764.2.k.f 2 63.l odd 6 1
1764.2.k.f 2 63.o even 6 1
1764.2.k.f 2 63.s even 6 1
2268.2.i.c 2 1.a even 1 1 trivial
2268.2.i.c 2 3.b odd 2 1 CM
2268.2.i.c 2 63.h even 3 1 inner
2268.2.i.c 2 63.j odd 6 1 inner
2268.2.l.e 2 7.c even 3 1
2268.2.l.e 2 9.c even 3 1
2268.2.l.e 2 9.d odd 6 1
2268.2.l.e 2 21.h odd 6 1
7056.2.a.z 1 252.r odd 6 1
7056.2.a.z 1 252.bj even 6 1
7056.2.a.be 1 252.u odd 6 1
7056.2.a.be 1 252.bb even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\):

\( T_{5} \) Copy content Toggle raw display
\( T_{13}^{2} + 5T_{13} + 25 \) Copy content Toggle raw display
\( T_{19}^{2} - T_{19} + 1 \) 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
$7$ \( T^{2} + 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} \) Copy content Toggle raw display
$31$ \( (T - 11)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( T^{2} \) Copy content Toggle raw display
$43$ \( T^{2} - 13T + 169 \) 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
$61$ \( (T - 14)^{2} \) Copy content Toggle raw display
$67$ \( (T - 5)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 17T + 289 \) Copy content Toggle raw display
$79$ \( (T - 17)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
show more
show less