# Properties

 Label 784.4.f.j Level $784$ Weight $4$ Character orbit 784.f Analytic conductor $46.257$ Analytic rank $0$ Dimension $24$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$784 = 2^{4} \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 784.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$46.2574974445$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q + 88q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q + 88q^{9} - 856q^{25} + 896q^{29} - 2496q^{53} + 4416q^{57} - 4416q^{65} - 4904q^{81} - 2432q^{85} - 11968q^{93} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
783.1 0 −8.86009 0 7.47612i 0 0 0 51.5012 0
783.2 0 −8.86009 0 7.47612i 0 0 0 51.5012 0
783.3 0 −6.34240 0 1.20378i 0 0 0 13.2260 0
783.4 0 −6.34240 0 1.20378i 0 0 0 13.2260 0
783.5 0 −5.68961 0 20.4139i 0 0 0 5.37166 0
783.6 0 −5.68961 0 20.4139i 0 0 0 5.37166 0
783.7 0 −4.30860 0 7.29492i 0 0 0 −8.43597 0
783.8 0 −4.30860 0 7.29492i 0 0 0 −8.43597 0
783.9 0 −3.30410 0 13.4612i 0 0 0 −16.0830 0
783.10 0 −3.30410 0 13.4612i 0 0 0 −16.0830 0
783.11 0 −1.84933 0 15.9847i 0 0 0 −23.5800 0
783.12 0 −1.84933 0 15.9847i 0 0 0 −23.5800 0
783.13 0 1.84933 0 15.9847i 0 0 0 −23.5800 0
783.14 0 1.84933 0 15.9847i 0 0 0 −23.5800 0
783.15 0 3.30410 0 13.4612i 0 0 0 −16.0830 0
783.16 0 3.30410 0 13.4612i 0 0 0 −16.0830 0
783.17 0 4.30860 0 7.29492i 0 0 0 −8.43597 0
783.18 0 4.30860 0 7.29492i 0 0 0 −8.43597 0
783.19 0 5.68961 0 20.4139i 0 0 0 5.37166 0
783.20 0 5.68961 0 20.4139i 0 0 0 5.37166 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 783.24 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
7.b odd 2 1 inner
28.d even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.4.f.j 24
4.b odd 2 1 inner 784.4.f.j 24
7.b odd 2 1 inner 784.4.f.j 24
28.d even 2 1 inner 784.4.f.j 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.4.f.j 24 1.a even 1 1 trivial
784.4.f.j 24 4.b odd 2 1 inner
784.4.f.j 24 7.b odd 2 1 inner
784.4.f.j 24 28.d even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{12} - 184 T_{3}^{10} + 12276 T_{3}^{8} - 379120 T_{3}^{6} + 5592772 T_{3}^{4} - 35876352 T_{3}^{2} + 70852608$$ acting on $$S_{4}^{\mathrm{new}}(784, [\chi])$$.