# Properties

 Label 2240.2.h.f Level $2240$ Weight $2$ Character orbit 2240.h Analytic conductor $17.886$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$2240 = 2^{6} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2240.h (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$17.8864900528$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{5} - 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 24q^{5} - 36q^{9} + 4q^{13} + 8q^{21} + 24q^{25} - 36q^{45} + 24q^{57} + 56q^{61} + 4q^{65} + 96q^{69} - 12q^{77} + 24q^{81} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
671.1 0 2.84405i 0 1.00000 0 −0.712435 + 2.54803i 0 −5.08861 0
671.2 0 2.84405i 0 1.00000 0 −0.712435 2.54803i 0 −5.08861 0
671.3 0 0.237691i 0 1.00000 0 −2.63997 + 0.174795i 0 2.94350 0
671.4 0 0.237691i 0 1.00000 0 −2.63997 0.174795i 0 2.94350 0
671.5 0 2.59614i 0 1.00000 0 −2.28275 1.33755i 0 −3.73996 0
671.6 0 2.59614i 0 1.00000 0 −2.28275 + 1.33755i 0 −3.73996 0
671.7 0 2.91632i 0 1.00000 0 −2.36081 + 1.19440i 0 −5.50491 0
671.8 0 2.91632i 0 1.00000 0 −2.36081 1.19440i 0 −5.50491 0
671.9 0 1.85258i 0 1.00000 0 1.29892 2.30495i 0 −0.432061 0
671.10 0 1.85258i 0 1.00000 0 1.29892 + 2.30495i 0 −0.432061 0
671.11 0 0.421861i 0 1.00000 0 −1.02538 2.43897i 0 2.82203 0
671.12 0 0.421861i 0 1.00000 0 −1.02538 + 2.43897i 0 2.82203 0
671.13 0 0.421861i 0 1.00000 0 1.02538 2.43897i 0 2.82203 0
671.14 0 0.421861i 0 1.00000 0 1.02538 + 2.43897i 0 2.82203 0
671.15 0 1.85258i 0 1.00000 0 −1.29892 2.30495i 0 −0.432061 0
671.16 0 1.85258i 0 1.00000 0 −1.29892 + 2.30495i 0 −0.432061 0
671.17 0 2.91632i 0 1.00000 0 2.36081 + 1.19440i 0 −5.50491 0
671.18 0 2.91632i 0 1.00000 0 2.36081 1.19440i 0 −5.50491 0
671.19 0 2.59614i 0 1.00000 0 2.28275 1.33755i 0 −3.73996 0
671.20 0 2.59614i 0 1.00000 0 2.28275 + 1.33755i 0 −3.73996 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 671.24 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
56.e even 2 1 inner
56.h odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2240.2.h.f yes 24
4.b odd 2 1 inner 2240.2.h.f yes 24
7.b odd 2 1 2240.2.h.e 24
8.b even 2 1 2240.2.h.e 24
8.d odd 2 1 2240.2.h.e 24
28.d even 2 1 2240.2.h.e 24
56.e even 2 1 inner 2240.2.h.f yes 24
56.h odd 2 1 inner 2240.2.h.f yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2240.2.h.e 24 7.b odd 2 1
2240.2.h.e 24 8.b even 2 1
2240.2.h.e 24 8.d odd 2 1
2240.2.h.e 24 28.d even 2 1
2240.2.h.f yes 24 1.a even 1 1 trivial
2240.2.h.f yes 24 4.b odd 2 1 inner
2240.2.h.f yes 24 56.e even 2 1 inner
2240.2.h.f yes 24 56.h odd 2 1 inner

## Hecke kernels

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

 $$T_{3}^{12} + 27 T_{3}^{10} + 267 T_{3}^{8} + 1145 T_{3}^{6} + 1848 T_{3}^{4} + 384 T_{3}^{2} + 16$$ $$T_{13}^{6} - T_{13}^{5} - 43 T_{13}^{4} + 69 T_{13}^{3} + 346 T_{13}^{2} - 404 T_{13} - 376$$