# Properties

 Label 560.2.x.b Level 560 Weight 2 Character orbit 560.x Analytic conductor 4.472 Analytic rank 0 Dimension 24 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$560 = 2^{4} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 560.x (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$24$$ Relative dimension: $$12$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 8q^{13} + 8q^{17} - 8q^{21} + 32q^{25} + 24q^{33} - 16q^{37} + 32q^{41} - 24q^{45} + 8q^{53} + 40q^{57} + 16q^{61} - 16q^{73} + 16q^{77} - 104q^{81} - 8q^{85} - 8q^{93} - 24q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
127.1 0 −2.31224 2.31224i 0 2.03291 + 0.931271i 0 0.707107 0.707107i 0 7.69286i 0
127.2 0 −1.77851 1.77851i 0 −2.23308 + 0.115477i 0 −0.707107 + 0.707107i 0 3.32619i 0
127.3 0 −1.38994 1.38994i 0 −0.674859 + 2.13180i 0 0.707107 0.707107i 0 0.863866i 0
127.4 0 −0.636375 0.636375i 0 1.68320 1.47202i 0 −0.707107 + 0.707107i 0 2.19005i 0
127.5 0 −0.333196 0.333196i 0 −2.17403 + 0.523047i 0 0.707107 0.707107i 0 2.77796i 0
127.6 0 −0.206273 0.206273i 0 1.36587 + 1.77043i 0 −0.707107 + 0.707107i 0 2.91490i 0
127.7 0 0.206273 + 0.206273i 0 1.36587 + 1.77043i 0 0.707107 0.707107i 0 2.91490i 0
127.8 0 0.333196 + 0.333196i 0 −2.17403 + 0.523047i 0 −0.707107 + 0.707107i 0 2.77796i 0
127.9 0 0.636375 + 0.636375i 0 1.68320 1.47202i 0 0.707107 0.707107i 0 2.19005i 0
127.10 0 1.38994 + 1.38994i 0 −0.674859 + 2.13180i 0 −0.707107 + 0.707107i 0 0.863866i 0
127.11 0 1.77851 + 1.77851i 0 −2.23308 + 0.115477i 0 0.707107 0.707107i 0 3.32619i 0
127.12 0 2.31224 + 2.31224i 0 2.03291 + 0.931271i 0 −0.707107 + 0.707107i 0 7.69286i 0
463.1 0 −2.31224 + 2.31224i 0 2.03291 0.931271i 0 0.707107 + 0.707107i 0 7.69286i 0
463.2 0 −1.77851 + 1.77851i 0 −2.23308 0.115477i 0 −0.707107 0.707107i 0 3.32619i 0
463.3 0 −1.38994 + 1.38994i 0 −0.674859 2.13180i 0 0.707107 + 0.707107i 0 0.863866i 0
463.4 0 −0.636375 + 0.636375i 0 1.68320 + 1.47202i 0 −0.707107 0.707107i 0 2.19005i 0
463.5 0 −0.333196 + 0.333196i 0 −2.17403 0.523047i 0 0.707107 + 0.707107i 0 2.77796i 0
463.6 0 −0.206273 + 0.206273i 0 1.36587 1.77043i 0 −0.707107 0.707107i 0 2.91490i 0
463.7 0 0.206273 0.206273i 0 1.36587 1.77043i 0 0.707107 + 0.707107i 0 2.91490i 0
463.8 0 0.333196 0.333196i 0 −2.17403 0.523047i 0 −0.707107 0.707107i 0 2.77796i 0
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 463.12 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.c odd 4 1 inner
20.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.x.b 24
4.b odd 2 1 inner 560.2.x.b 24
5.c odd 4 1 inner 560.2.x.b 24
20.e even 4 1 inner 560.2.x.b 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.x.b 24 1.a even 1 1 trivial
560.2.x.b 24 4.b odd 2 1 inner
560.2.x.b 24 5.c odd 4 1 inner
560.2.x.b 24 20.e even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{24} + 170 T_{3}^{20} + 7001 T_{3}^{16} + 73224 T_{3}^{12} + 48936 T_{3}^{8} + 2560 T_{3}^{4} + 16$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database