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Properties

Label 1053.2.e.b
Level $1053$
Weight $2$
Character orbit 1053.e
Analytic conductor $8.408$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1053,2,Mod(352,1053)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1053, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1053.352");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1053 = 3^{4} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1053.e (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: \(8.40824733284\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} + \zeta_{6} q^{4} - 2 \zeta_{6} q^{5} + ( - 4 \zeta_{6} + 4) q^{7} - 3 q^{8} + 2 q^{10} + (4 \zeta_{6} - 4) q^{11} - \zeta_{6} q^{13} + 4 \zeta_{6} q^{14} + ( - \zeta_{6} + 1) q^{16} + \cdots + 9 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} + q^{4} - 2 q^{5} + 4 q^{7} - 6 q^{8} + 4 q^{10} - 4 q^{11} - q^{13} + 4 q^{14} + q^{16} + 4 q^{17} + 2 q^{20} - 4 q^{22} + q^{25} + 2 q^{26} + 8 q^{28} + 10 q^{29} - 4 q^{31} - 5 q^{32}+ \cdots + 18 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(730\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
352.1
0.500000 − 0.866025i
0.500000 + 0.866025i
−0.500000 − 0.866025i 0 0.500000 − 0.866025i −1.00000 + 1.73205i 0 2.00000 + 3.46410i −3.00000 0 2.00000
703.1 −0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00000 − 1.73205i 0 2.00000 − 3.46410i −3.00000 0 2.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1053.2.e.b 2
3.b odd 2 1 1053.2.e.d 2
9.c even 3 1 39.2.a.a ✓ 1
9.c even 3 1 inner 1053.2.e.b 2
9.d odd 6 1 117.2.a.a 1
9.d odd 6 1 1053.2.e.d 2
36.f odd 6 1 624.2.a.i 1
36.h even 6 1 1872.2.a.h 1
45.h odd 6 1 2925.2.a.p 1
45.j even 6 1 975.2.a.f 1
45.k odd 12 2 975.2.c.f 2
45.l even 12 2 2925.2.c.e 2
63.l odd 6 1 1911.2.a.f 1
63.o even 6 1 5733.2.a.e 1
72.j odd 6 1 7488.2.a.bl 1
72.l even 6 1 7488.2.a.by 1
72.n even 6 1 2496.2.a.q 1
72.p odd 6 1 2496.2.a.e 1
99.h odd 6 1 4719.2.a.c 1
117.f even 3 1 507.2.e.a 2
117.h even 3 1 507.2.e.a 2
117.l even 6 1 507.2.e.b 2
117.n odd 6 1 1521.2.a.e 1
117.r even 6 1 507.2.e.b 2
117.t even 6 1 507.2.a.a 1
117.w odd 12 2 507.2.j.e 4
117.y odd 12 2 507.2.b.a 2
117.z even 12 2 1521.2.b.b 2
117.bb odd 12 2 507.2.j.e 4
468.bg odd 6 1 8112.2.a.s 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
39.2.a.a ✓ 1 9.c even 3 1
117.2.a.a 1 9.d odd 6 1
507.2.a.a 1 117.t even 6 1
507.2.b.a 2 117.y odd 12 2
507.2.e.a 2 117.f even 3 1
507.2.e.a 2 117.h even 3 1
507.2.e.b 2 117.l even 6 1
507.2.e.b 2 117.r even 6 1
507.2.j.e 4 117.w odd 12 2
507.2.j.e 4 117.bb odd 12 2
624.2.a.i 1 36.f odd 6 1
975.2.a.f 1 45.j even 6 1
975.2.c.f 2 45.k odd 12 2
1053.2.e.b 2 1.a even 1 1 trivial
1053.2.e.b 2 9.c even 3 1 inner
1053.2.e.d 2 3.b odd 2 1
1053.2.e.d 2 9.d odd 6 1
1521.2.a.e 1 117.n odd 6 1
1521.2.b.b 2 117.z even 12 2
1872.2.a.h 1 36.h even 6 1
1911.2.a.f 1 63.l odd 6 1
2496.2.a.e 1 72.p odd 6 1
2496.2.a.q 1 72.n even 6 1
2925.2.a.p 1 45.h odd 6 1
2925.2.c.e 2 45.l even 12 2
4719.2.a.c 1 99.h odd 6 1
5733.2.a.e 1 63.o even 6 1
7488.2.a.bl 1 72.j odd 6 1
7488.2.a.by 1 72.l even 6 1
8112.2.a.s 1 468.bg odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1053, [\chi])\):

\( T_{2}^{2} + T_{2} + 1 \) Copy content Toggle raw display
\( T_{5}^{2} + 2T_{5} + 4 \) Copy content Toggle raw display
\( T_{7}^{2} - 4T_{7} + 16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$11$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$13$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$17$ \( (T - 2)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$31$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 2)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$43$ \( T^{2} - 12T + 144 \) Copy content Toggle raw display
$47$ \( T^{2} \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} + 12T + 144 \) Copy content Toggle raw display
$61$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$67$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 2)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} + 8T + 64 \) Copy content Toggle raw display
$83$ \( T^{2} + 4T + 16 \) Copy content Toggle raw display
$89$ \( (T + 2)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
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