Properties

Label 2268.2.k.a
Level $2268$
Weight $2$
Character orbit 2268.k
Analytic conductor $18.110$
Analytic rank $1$
Dimension $2$
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] = [2268,2,Mod(1297,2268)]
 
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, 0, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2268.1297");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2268.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: \(18.1100711784\)
Analytic rank: \(1\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 252)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q - 2 \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 \zeta_{6} q^{5} + (3 \zeta_{6} - 1) q^{7} + (4 \zeta_{6} - 4) q^{11} + 3 q^{13} + (7 \zeta_{6} - 7) q^{17} - 5 \zeta_{6} q^{19} - 4 \zeta_{6} q^{23} + ( - \zeta_{6} + 1) q^{25} - q^{29} + ( - 3 \zeta_{6} + 3) q^{31} + ( - 4 \zeta_{6} + 6) q^{35} - 11 \zeta_{6} q^{37} - 9 q^{41} + 5 q^{43} - 3 \zeta_{6} q^{47} + (3 \zeta_{6} - 8) q^{49} + (3 \zeta_{6} - 3) q^{53} + 8 q^{55} + ( - 7 \zeta_{6} + 7) q^{59} - 3 \zeta_{6} q^{61} - 6 \zeta_{6} q^{65} + (13 \zeta_{6} - 13) q^{67} - 8 q^{71} + (7 \zeta_{6} - 7) q^{73} + ( - 4 \zeta_{6} - 8) q^{77} + 9 \zeta_{6} q^{79} + q^{83} + 14 q^{85} - 15 \zeta_{6} q^{89} + (9 \zeta_{6} - 3) q^{91} + (10 \zeta_{6} - 10) q^{95} - 17 q^{97} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + q^{7} - 4 q^{11} + 6 q^{13} - 7 q^{17} - 5 q^{19} - 4 q^{23} + q^{25} - 2 q^{29} + 3 q^{31} + 8 q^{35} - 11 q^{37} - 18 q^{41} + 10 q^{43} - 3 q^{47} - 13 q^{49} - 3 q^{53} + 16 q^{55} + 7 q^{59} - 3 q^{61} - 6 q^{65} - 13 q^{67} - 16 q^{71} - 7 q^{73} - 20 q^{77} + 9 q^{79} + 2 q^{83} + 28 q^{85} - 15 q^{89} + 3 q^{91} - 10 q^{95} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1297.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 0 −1.00000 + 1.73205i 0 0.500000 2.59808i 0 0 0
1621.1 0 0 0 −1.00000 1.73205i 0 0.500000 + 2.59808i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2268.2.k.a 2
3.b odd 2 1 2268.2.k.b 2
7.c even 3 1 inner 2268.2.k.a 2
9.c even 3 1 252.2.i.a 2
9.c even 3 1 252.2.l.a yes 2
9.d odd 6 1 756.2.i.a 2
9.d odd 6 1 756.2.l.a 2
21.h odd 6 1 2268.2.k.b 2
36.f odd 6 1 1008.2.q.f 2
36.f odd 6 1 1008.2.t.b 2
36.h even 6 1 3024.2.q.e 2
36.h even 6 1 3024.2.t.b 2
63.g even 3 1 252.2.i.a 2
63.g even 3 1 1764.2.j.c 2
63.h even 3 1 252.2.l.a yes 2
63.h even 3 1 1764.2.j.c 2
63.i even 6 1 5292.2.j.b 2
63.i even 6 1 5292.2.l.b 2
63.j odd 6 1 756.2.l.a 2
63.j odd 6 1 5292.2.j.c 2
63.k odd 6 1 1764.2.i.b 2
63.k odd 6 1 1764.2.j.a 2
63.l odd 6 1 1764.2.i.b 2
63.l odd 6 1 1764.2.l.b 2
63.n odd 6 1 756.2.i.a 2
63.n odd 6 1 5292.2.j.c 2
63.o even 6 1 5292.2.i.b 2
63.o even 6 1 5292.2.l.b 2
63.s even 6 1 5292.2.i.b 2
63.s even 6 1 5292.2.j.b 2
63.t odd 6 1 1764.2.j.a 2
63.t odd 6 1 1764.2.l.b 2
252.o even 6 1 3024.2.q.e 2
252.u odd 6 1 1008.2.t.b 2
252.bb even 6 1 3024.2.t.b 2
252.bl odd 6 1 1008.2.q.f 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.a 2 9.c even 3 1
252.2.i.a 2 63.g even 3 1
252.2.l.a yes 2 9.c even 3 1
252.2.l.a yes 2 63.h even 3 1
756.2.i.a 2 9.d odd 6 1
756.2.i.a 2 63.n odd 6 1
756.2.l.a 2 9.d odd 6 1
756.2.l.a 2 63.j odd 6 1
1008.2.q.f 2 36.f odd 6 1
1008.2.q.f 2 252.bl odd 6 1
1008.2.t.b 2 36.f odd 6 1
1008.2.t.b 2 252.u odd 6 1
1764.2.i.b 2 63.k odd 6 1
1764.2.i.b 2 63.l odd 6 1
1764.2.j.a 2 63.k odd 6 1
1764.2.j.a 2 63.t odd 6 1
1764.2.j.c 2 63.g even 3 1
1764.2.j.c 2 63.h even 3 1
1764.2.l.b 2 63.l odd 6 1
1764.2.l.b 2 63.t odd 6 1
2268.2.k.a 2 1.a even 1 1 trivial
2268.2.k.a 2 7.c even 3 1 inner
2268.2.k.b 2 3.b odd 2 1
2268.2.k.b 2 21.h odd 6 1
3024.2.q.e 2 36.h even 6 1
3024.2.q.e 2 252.o even 6 1
3024.2.t.b 2 36.h even 6 1
3024.2.t.b 2 252.bb even 6 1
5292.2.i.b 2 63.o even 6 1
5292.2.i.b 2 63.s even 6 1
5292.2.j.b 2 63.i even 6 1
5292.2.j.b 2 63.s even 6 1
5292.2.j.c 2 63.j odd 6 1
5292.2.j.c 2 63.n odd 6 1
5292.2.l.b 2 63.i even 6 1
5292.2.l.b 2 63.o even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{2} + 2T_{5} + 4 \) acting on \(S_{2}^{\mathrm{new}}(2268, [\chi])\). 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
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( (T - 3)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$19$ \( T^{2} + 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$29$ \( (T + 1)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$37$ \( T^{2} + 11T + 121 \) Copy content Toggle raw display
$41$ \( (T + 9)^{2} \) Copy content Toggle raw display
$43$ \( (T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$53$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$59$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$61$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$79$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$83$ \( (T - 1)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 15T + 225 \) Copy content Toggle raw display
$97$ \( (T + 17)^{2} \) Copy content Toggle raw display
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