# Properties

 Label 560.2.ci.e Level $560$ Weight $2$ Character orbit 560.ci Analytic conductor $4.472$ Analytic rank $0$ Dimension $48$ 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.ci (of order $$12$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{12})$$ Twist minimal: no (minimal twist has level 280) Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 4q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 4q^{7} - 4q^{11} - 8q^{15} - 4q^{21} + 4q^{23} - 8q^{25} - 36q^{33} - 24q^{35} + 8q^{37} + 16q^{43} + 48q^{45} - 24q^{51} + 16q^{53} - 96q^{57} - 36q^{61} + 68q^{63} + 12q^{65} + 16q^{67} + 64q^{71} - 48q^{73} + 48q^{75} + 4q^{77} - 40q^{85} + 12q^{87} + 80q^{91} - 24q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
17.1 0 −3.19510 + 0.856125i 0 0.672461 + 2.13256i 0 2.52104 + 0.802724i 0 6.87765 3.97081i 0
17.2 0 −2.20364 + 0.590462i 0 −0.352053 2.20818i 0 1.22435 + 2.34541i 0 1.90929 1.10233i 0
17.3 0 −2.19810 + 0.588978i 0 −1.19880 1.88756i 0 −1.40409 2.24244i 0 1.88665 1.08926i 0
17.4 0 −1.66856 + 0.447090i 0 2.21600 + 0.298941i 0 −0.900705 2.48772i 0 −0.0138674 + 0.00800633i 0
17.5 0 −0.280752 + 0.0752271i 0 2.21121 0.332503i 0 −1.13104 + 2.39181i 0 −2.52491 + 1.45776i 0
17.6 0 −0.120281 + 0.0322291i 0 −2.04377 + 0.907205i 0 2.46167 0.969635i 0 −2.58465 + 1.49225i 0
17.7 0 0.0749389 0.0200798i 0 −0.965007 + 2.01712i 0 −1.45892 2.20716i 0 −2.59286 + 1.49699i 0
17.8 0 1.16710 0.312724i 0 −0.121837 2.23275i 0 2.59977 0.491095i 0 −1.33374 + 0.770036i 0
17.9 0 1.35944 0.364260i 0 −2.13431 0.666887i 0 −2.59782 + 0.501338i 0 −0.882692 + 0.509622i 0
17.10 0 2.00047 0.536025i 0 1.38383 + 1.75642i 0 −1.24986 + 2.33192i 0 1.11649 0.644605i 0
17.11 0 2.30560 0.617784i 0 1.37859 1.76054i 0 0.755351 2.53563i 0 2.33607 1.34873i 0
17.12 0 2.75887 0.739238i 0 −1.04631 + 1.97617i 0 1.91229 + 1.82843i 0 4.46683 2.57893i 0
33.1 0 −3.19510 0.856125i 0 0.672461 2.13256i 0 2.52104 0.802724i 0 6.87765 + 3.97081i 0
33.2 0 −2.20364 0.590462i 0 −0.352053 + 2.20818i 0 1.22435 2.34541i 0 1.90929 + 1.10233i 0
33.3 0 −2.19810 0.588978i 0 −1.19880 + 1.88756i 0 −1.40409 + 2.24244i 0 1.88665 + 1.08926i 0
33.4 0 −1.66856 0.447090i 0 2.21600 0.298941i 0 −0.900705 + 2.48772i 0 −0.0138674 0.00800633i 0
33.5 0 −0.280752 0.0752271i 0 2.21121 + 0.332503i 0 −1.13104 2.39181i 0 −2.52491 1.45776i 0
33.6 0 −0.120281 0.0322291i 0 −2.04377 0.907205i 0 2.46167 + 0.969635i 0 −2.58465 1.49225i 0
33.7 0 0.0749389 + 0.0200798i 0 −0.965007 2.01712i 0 −1.45892 + 2.20716i 0 −2.59286 1.49699i 0
33.8 0 1.16710 + 0.312724i 0 −0.121837 + 2.23275i 0 2.59977 + 0.491095i 0 −1.33374 0.770036i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 353.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
7.d odd 6 1 inner
35.k even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 560.2.ci.e 48
4.b odd 2 1 280.2.bo.a 48
5.c odd 4 1 inner 560.2.ci.e 48
7.d odd 6 1 inner 560.2.ci.e 48
20.e even 4 1 280.2.bo.a 48
28.f even 6 1 280.2.bo.a 48
35.k even 12 1 inner 560.2.ci.e 48
140.x odd 12 1 280.2.bo.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bo.a 48 4.b odd 2 1
280.2.bo.a 48 20.e even 4 1
280.2.bo.a 48 28.f even 6 1
280.2.bo.a 48 140.x odd 12 1
560.2.ci.e 48 1.a even 1 1 trivial
560.2.ci.e 48 5.c odd 4 1 inner
560.2.ci.e 48 7.d odd 6 1 inner
560.2.ci.e 48 35.k even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{48} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.