# Properties

 Label 1344.4.p.d Level $1344$ Weight $4$ Character orbit 1344.p Analytic conductor $79.299$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1344 = 2^{6} \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 1344.p (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$79.2985670477$$ Analytic rank: $$0$$ Dimension: $$32$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 288q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 288q^{9} + 224q^{13} + 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} + 544q^{61} + 1536q^{65} + 144q^{69} + 1632q^{77} + 2592q^{81} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
223.1 0 3.00000i 0 12.5168 0 −18.2936 2.88837i 0 −9.00000 0
223.2 0 3.00000i 0 12.5168 0 −18.2936 + 2.88837i 0 −9.00000 0
223.3 0 3.00000i 0 3.15485 0 6.11026 + 17.4833i 0 −9.00000 0
223.4 0 3.00000i 0 3.15485 0 6.11026 17.4833i 0 −9.00000 0
223.5 0 3.00000i 0 12.8225 0 −6.32908 + 17.4053i 0 −9.00000 0
223.6 0 3.00000i 0 12.8225 0 −6.32908 17.4053i 0 −9.00000 0
223.7 0 3.00000i 0 −20.7003 0 11.8166 + 14.2607i 0 −9.00000 0
223.8 0 3.00000i 0 −20.7003 0 11.8166 14.2607i 0 −9.00000 0
223.9 0 3.00000i 0 15.4018 0 6.29285 17.4184i 0 −9.00000 0
223.10 0 3.00000i 0 15.4018 0 6.29285 + 17.4184i 0 −9.00000 0
223.11 0 3.00000i 0 15.4018 0 −6.29285 17.4184i 0 −9.00000 0
223.12 0 3.00000i 0 15.4018 0 −6.29285 + 17.4184i 0 −9.00000 0
223.13 0 3.00000i 0 −20.7003 0 −11.8166 + 14.2607i 0 −9.00000 0
223.14 0 3.00000i 0 −20.7003 0 −11.8166 14.2607i 0 −9.00000 0
223.15 0 3.00000i 0 12.8225 0 6.32908 + 17.4053i 0 −9.00000 0
223.16 0 3.00000i 0 12.8225 0 6.32908 17.4053i 0 −9.00000 0
223.17 0 3.00000i 0 −1.64166 0 15.2747 10.4729i 0 −9.00000 0
223.18 0 3.00000i 0 −1.64166 0 15.2747 + 10.4729i 0 −9.00000 0
223.19 0 3.00000i 0 −1.64166 0 −15.2747 10.4729i 0 −9.00000 0
223.20 0 3.00000i 0 −1.64166 0 −15.2747 + 10.4729i 0 −9.00000 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 223.32 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 1344.4.p.d yes 32
4.b odd 2 1 inner 1344.4.p.d yes 32
7.b odd 2 1 1344.4.p.c 32
8.b even 2 1 1344.4.p.c 32
8.d odd 2 1 1344.4.p.c 32
28.d even 2 1 1344.4.p.c 32
56.e even 2 1 inner 1344.4.p.d yes 32
56.h odd 2 1 inner 1344.4.p.d yes 32

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1344.4.p.c 32 7.b odd 2 1
1344.4.p.c 32 8.b even 2 1
1344.4.p.c 32 8.d odd 2 1
1344.4.p.c 32 28.d even 2 1
1344.4.p.d yes 32 1.a even 1 1 trivial
1344.4.p.d yes 32 4.b odd 2 1 inner
1344.4.p.d yes 32 56.e even 2 1 inner
1344.4.p.d yes 32 56.h odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{5}^{8} - 640 T_{5}^{6} + 1728 T_{5}^{5} + 116464 T_{5}^{4} - 458400 T_{5}^{3} - 4875024 T_{5}^{2} + 8581632 T_{5} + 24385536$$ acting on $$S_{4}^{\mathrm{new}}(1344, [\chi])$$.