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Properties

Label 1350.2.e.b
Level $1350$
Weight $2$
Character orbit 1350.e
Analytic conductor $10.780$
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] = [1350,2,Mod(451,1350)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1350, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1350.451");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1350.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: \(10.7798042729\)
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 90)
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.

\(f(q)\) \(=\) \( q + (\zeta_{6} - 1) q^{2} - \zeta_{6} q^{4} + (\zeta_{6} - 1) q^{7} + q^{8} + ( - 6 \zeta_{6} + 6) q^{11} + 2 \zeta_{6} q^{13} - \zeta_{6} q^{14} + (\zeta_{6} - 1) q^{16} - 4 q^{19} + 6 \zeta_{6} q^{22} + \cdots - 6 q^{98} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{2} - q^{4} - q^{7} + 2 q^{8} + 6 q^{11} + 2 q^{13} - q^{14} - q^{16} - 8 q^{19} + 6 q^{22} - 9 q^{23} - 4 q^{26} + 2 q^{28} + 3 q^{29} + 4 q^{31} - q^{32} - 16 q^{37} + 4 q^{38} - 3 q^{41}+ \cdots - 12 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 − 0.866025i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0 0 −0.500000 + 0.866025i 1.00000 0 0
901.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0 0 −0.500000 − 0.866025i 1.00000 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
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 1350.2.e.b 2
3.b odd 2 1 450.2.e.e 2
5.b even 2 1 270.2.e.b 2
5.c odd 4 2 1350.2.j.e 4
9.c even 3 1 inner 1350.2.e.b 2
9.c even 3 1 4050.2.a.ba 1
9.d odd 6 1 450.2.e.e 2
9.d odd 6 1 4050.2.a.n 1
15.d odd 2 1 90.2.e.a ✓ 2
15.e even 4 2 450.2.j.c 4
20.d odd 2 1 2160.2.q.b 2
45.h odd 6 1 90.2.e.a ✓ 2
45.h odd 6 1 810.2.a.g 1
45.j even 6 1 270.2.e.b 2
45.j even 6 1 810.2.a.b 1
45.k odd 12 2 1350.2.j.e 4
45.k odd 12 2 4050.2.c.a 2
45.l even 12 2 450.2.j.c 4
45.l even 12 2 4050.2.c.t 2
60.h even 2 1 720.2.q.b 2
180.n even 6 1 720.2.q.b 2
180.n even 6 1 6480.2.a.g 1
180.p odd 6 1 2160.2.q.b 2
180.p odd 6 1 6480.2.a.v 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
90.2.e.a ✓ 2 15.d odd 2 1
90.2.e.a ✓ 2 45.h odd 6 1
270.2.e.b 2 5.b even 2 1
270.2.e.b 2 45.j even 6 1
450.2.e.e 2 3.b odd 2 1
450.2.e.e 2 9.d odd 6 1
450.2.j.c 4 15.e even 4 2
450.2.j.c 4 45.l even 12 2
720.2.q.b 2 60.h even 2 1
720.2.q.b 2 180.n even 6 1
810.2.a.b 1 45.j even 6 1
810.2.a.g 1 45.h odd 6 1
1350.2.e.b 2 1.a even 1 1 trivial
1350.2.e.b 2 9.c even 3 1 inner
1350.2.j.e 4 5.c odd 4 2
1350.2.j.e 4 45.k odd 12 2
2160.2.q.b 2 20.d odd 2 1
2160.2.q.b 2 180.p odd 6 1
4050.2.a.n 1 9.d odd 6 1
4050.2.a.ba 1 9.c even 3 1
4050.2.c.a 2 45.k odd 12 2
4050.2.c.t 2 45.l even 12 2
6480.2.a.g 1 180.n even 6 1
6480.2.a.v 1 180.p odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{2} + T_{7} + 1 \) Copy content Toggle raw display
\( T_{11}^{2} - 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{17} \) 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
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$11$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$17$ \( T^{2} \) Copy content Toggle raw display
$19$ \( (T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 9T + 81 \) Copy content Toggle raw display
$29$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$31$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$37$ \( (T + 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} - 8T + 64 \) Copy content Toggle raw display
$47$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$53$ \( (T - 6)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} - 6T + 36 \) Copy content Toggle raw display
$61$ \( T^{2} - 13T + 169 \) Copy content Toggle raw display
$67$ \( T^{2} + 13T + 169 \) Copy content Toggle raw display
$71$ \( (T - 6)^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} - 10T + 100 \) Copy content Toggle raw display
$83$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$89$ \( (T + 9)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
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