# 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

## 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: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 $$S_{2}^{\mathrm{new}}(2240, [\chi])$$:

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