Properties

Label 2240.2.e.g
Level $2240$
Weight $2$
Character orbit 2240.e
Analytic conductor $17.886$
Analytic rank $0$
Dimension $48$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: no (minimal twist has level 1120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 48q - 48q^{9} - 8q^{21} - 16q^{25} - 16q^{29} + 24q^{49} - 16q^{65} - 32q^{81} - 32q^{85} + O(q^{100}) \)

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.

Label \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
2239.1 0 2.21376i 0 −1.81856 1.30110i 0 1.09155 2.41009i 0 −1.90072 0
2239.2 0 2.21376i 0 −1.81856 + 1.30110i 0 1.09155 + 2.41009i 0 −1.90072 0
2239.3 0 3.06386i 0 −1.39003 + 1.75152i 0 2.46063 0.972270i 0 −6.38723 0
2239.4 0 3.06386i 0 −1.39003 1.75152i 0 2.46063 + 0.972270i 0 −6.38723 0
2239.5 0 1.86733i 0 2.07366 0.836614i 0 2.64040 + 0.168201i 0 −0.486908 0
2239.6 0 1.86733i 0 2.07366 + 0.836614i 0 2.64040 0.168201i 0 −0.486908 0
2239.7 0 3.06386i 0 −1.39003 1.75152i 0 −2.46063 0.972270i 0 −6.38723 0
2239.8 0 3.06386i 0 −1.39003 + 1.75152i 0 −2.46063 + 0.972270i 0 −6.38723 0
2239.9 0 1.86733i 0 −2.07366 + 0.836614i 0 −2.64040 + 0.168201i 0 −0.486908 0
2239.10 0 1.86733i 0 −2.07366 0.836614i 0 −2.64040 0.168201i 0 −0.486908 0
2239.11 0 0.326439i 0 −0.278349 2.21868i 0 −2.25431 1.38495i 0 2.89344 0
2239.12 0 0.326439i 0 −0.278349 + 2.21868i 0 −2.25431 + 1.38495i 0 2.89344 0
2239.13 0 0.326439i 0 0.278349 2.21868i 0 −2.25431 1.38495i 0 2.89344 0
2239.14 0 0.326439i 0 0.278349 + 2.21868i 0 −2.25431 + 1.38495i 0 2.89344 0
2239.15 0 3.06386i 0 1.39003 + 1.75152i 0 2.46063 0.972270i 0 −6.38723 0
2239.16 0 3.06386i 0 1.39003 1.75152i 0 2.46063 + 0.972270i 0 −6.38723 0
2239.17 0 2.40740i 0 0.475379 + 2.18495i 0 −0.283298 + 2.63054i 0 −2.79558 0
2239.18 0 2.40740i 0 0.475379 2.18495i 0 −0.283298 2.63054i 0 −2.79558 0
2239.19 0 3.06386i 0 1.39003 1.75152i 0 −2.46063 0.972270i 0 −6.38723 0
2239.20 0 3.06386i 0 1.39003 + 1.75152i 0 −2.46063 + 0.972270i 0 −6.38723 0
See all 48 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2239.48
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
5.b even 2 1 inner
7.b odd 2 1 inner
20.d odd 2 1 inner
28.d even 2 1 inner
35.c odd 2 1 inner
140.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.e.g 48
4.b odd 2 1 inner 2240.2.e.g 48
5.b even 2 1 inner 2240.2.e.g 48
7.b odd 2 1 inner 2240.2.e.g 48
8.b even 2 1 1120.2.e.a 48
8.d odd 2 1 1120.2.e.a 48
20.d odd 2 1 inner 2240.2.e.g 48
28.d even 2 1 inner 2240.2.e.g 48
35.c odd 2 1 inner 2240.2.e.g 48
40.e odd 2 1 1120.2.e.a 48
40.f even 2 1 1120.2.e.a 48
56.e even 2 1 1120.2.e.a 48
56.h odd 2 1 1120.2.e.a 48
140.c even 2 1 inner 2240.2.e.g 48
280.c odd 2 1 1120.2.e.a 48
280.n even 2 1 1120.2.e.a 48
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1120.2.e.a 48 8.b even 2 1
1120.2.e.a 48 8.d odd 2 1
1120.2.e.a 48 40.e odd 2 1
1120.2.e.a 48 40.f even 2 1
1120.2.e.a 48 56.e even 2 1
1120.2.e.a 48 56.h odd 2 1
1120.2.e.a 48 280.c odd 2 1
1120.2.e.a 48 280.n even 2 1
2240.2.e.g 48 1.a even 1 1 trivial
2240.2.e.g 48 4.b odd 2 1 inner
2240.2.e.g 48 5.b even 2 1 inner
2240.2.e.g 48 7.b odd 2 1 inner
2240.2.e.g 48 20.d odd 2 1 inner
2240.2.e.g 48 28.d even 2 1 inner
2240.2.e.g 48 35.c odd 2 1 inner
2240.2.e.g 48 140.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2240, [\chi])\):

\( T_{3}^{12} + 24 T_{3}^{10} + 209 T_{3}^{8} + 802 T_{3}^{6} + 1244 T_{3}^{4} + 424 T_{3}^{2} + 32 \)
\( T_{11}^{12} + 82 T_{11}^{10} + 2417 T_{11}^{8} + 30236 T_{11}^{6} + 138480 T_{11}^{4} + 76096 T_{11}^{2} + 9216 \)