# Properties

 Label 560.2.be.a Level 560 Weight 2 Character orbit 560.be Analytic conductor 4.472 Analytic rank 0 Dimension 184 CM no Inner twists 8

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$184$$ Relative dimension: $$92$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$184q - 8q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$184q - 8q^{4} - 8q^{11} - 4q^{14} - 8q^{16} - 16q^{21} - 8q^{29} + 8q^{30} - 24q^{35} - 16q^{36} - 16q^{39} - 24q^{44} + 8q^{46} - 8q^{49} - 24q^{50} - 32q^{51} + 24q^{56} + 16q^{60} - 104q^{64} - 8q^{65} - 28q^{70} + 48q^{71} + 104q^{74} - 136q^{81} + 16q^{84} - 24q^{85} - 64q^{86} - 48q^{91} + 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.41383 0.0328882i −1.31817 1.31817i 1.99784 + 0.0929966i 1.05420 1.97197i 1.82031 + 1.90702i 2.24452 1.40075i −2.82155 0.197187i 0.475122i −1.55532 + 2.75336i
139.2 −1.41383 0.0328882i 1.31817 + 1.31817i 1.99784 + 0.0929966i −1.05420 + 1.97197i −1.82031 1.90702i −2.24452 1.40075i −2.82155 0.197187i 0.475122i 1.55532 2.75336i
139.3 −1.40363 + 0.172701i −0.710544 0.710544i 1.94035 0.484816i 1.39309 + 1.74909i 1.12005 + 0.874629i −2.62345 + 0.342784i −2.63980 + 1.01560i 1.99025i −2.25745 2.21448i
139.4 −1.40363 + 0.172701i 0.710544 + 0.710544i 1.94035 0.484816i −1.39309 1.74909i −1.12005 0.874629i 2.62345 + 0.342784i −2.63980 + 1.01560i 1.99025i 2.25745 + 2.21448i
139.5 −1.40049 0.196510i −1.64523 1.64523i 1.92277 + 0.550422i −1.65460 + 1.50409i 1.98083 + 2.62744i −0.240324 + 2.63481i −2.58466 1.14871i 2.41357i 2.61283 1.78133i
139.6 −1.40049 0.196510i 1.64523 + 1.64523i 1.92277 + 0.550422i 1.65460 1.50409i −1.98083 2.62744i 0.240324 + 2.63481i −2.58466 1.14871i 2.41357i −2.61283 + 1.78133i
139.7 −1.33517 0.466181i −1.96435 1.96435i 1.56535 + 1.24486i 0.776425 + 2.09694i 1.70699 + 3.53848i 1.99873 1.73352i −1.50968 2.39184i 4.71733i −0.0591043 3.16173i
139.8 −1.33517 0.466181i 1.96435 + 1.96435i 1.56535 + 1.24486i −0.776425 2.09694i −1.70699 3.53848i −1.99873 1.73352i −1.50968 2.39184i 4.71733i 0.0591043 + 3.16173i
139.9 −1.30971 + 0.533547i −1.34417 1.34417i 1.43065 1.39758i −2.08996 + 0.795024i 2.47764 + 1.04329i 0.486119 2.60071i −1.12806 + 2.59374i 0.613574i 2.31305 2.15634i
139.10 −1.30971 + 0.533547i 1.34417 + 1.34417i 1.43065 1.39758i 2.08996 0.795024i −2.47764 1.04329i −0.486119 2.60071i −1.12806 + 2.59374i 0.613574i −2.31305 + 2.15634i
139.11 −1.30750 0.538918i −0.289321 0.289321i 1.41913 + 1.40928i −2.02485 0.948664i 0.222368 + 0.534210i −1.73658 + 1.99607i −1.09604 2.60743i 2.83259i 2.13625 + 2.33161i
139.12 −1.30750 0.538918i 0.289321 + 0.289321i 1.41913 + 1.40928i 2.02485 + 0.948664i −0.222368 0.534210i 1.73658 + 1.99607i −1.09604 2.60743i 2.83259i −2.13625 2.33161i
139.13 −1.27264 + 0.616746i −0.0442516 0.0442516i 1.23925 1.56980i −0.0460683 + 2.23559i 0.0836085 + 0.0290246i 2.16475 + 1.52114i −0.608959 + 2.76210i 2.99608i −1.32016 2.87353i
139.14 −1.27264 + 0.616746i 0.0442516 + 0.0442516i 1.23925 1.56980i 0.0460683 2.23559i −0.0836085 0.0290246i −2.16475 + 1.52114i −0.608959 + 2.76210i 2.99608i 1.32016 + 2.87353i
139.15 −1.22124 0.713144i −0.422516 0.422516i 0.982851 + 1.74184i 1.94457 1.10393i 0.214678 + 0.817308i −1.83593 1.90509i 0.0418853 2.82812i 2.64296i −3.16204 0.0385979i
139.16 −1.22124 0.713144i 0.422516 + 0.422516i 0.982851 + 1.74184i −1.94457 + 1.10393i −0.214678 0.817308i 1.83593 1.90509i 0.0418853 2.82812i 2.64296i 3.16204 + 0.0385979i
139.17 −1.16263 + 0.805161i −2.03615 2.03615i 0.703430 1.87221i −1.56280 1.59927i 4.00673 + 0.727866i 0.889133 + 2.49188i 0.689603 + 2.74307i 5.29182i 3.10463 + 0.601051i
139.18 −1.16263 + 0.805161i 2.03615 + 2.03615i 0.703430 1.87221i 1.56280 + 1.59927i −4.00673 0.727866i −0.889133 + 2.49188i 0.689603 + 2.74307i 5.29182i −3.10463 0.601051i
139.19 −1.15591 + 0.814782i −2.22344 2.22344i 0.672261 1.88363i 2.21099 + 0.333957i 4.38172 + 0.758481i −1.99535 1.73741i 0.757674 + 2.72506i 6.88739i −2.82781 + 1.41545i
139.20 −1.15591 + 0.814782i 2.22344 + 2.22344i 0.672261 1.88363i −2.21099 0.333957i −4.38172 0.758481i 1.99535 1.73741i 0.757674 + 2.72506i 6.88739i 2.82781 1.41545i
See next 80 embeddings (of 184 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 419.92 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.b even 2 1 inner
7.b odd 2 1 inner
16.f odd 4 1 inner
35.c odd 2 1 inner
80.k odd 4 1 inner
112.j even 4 1 inner
560.be 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.be.a 184
5.b even 2 1 inner 560.2.be.a 184
7.b odd 2 1 inner 560.2.be.a 184
16.f odd 4 1 inner 560.2.be.a 184
35.c odd 2 1 inner 560.2.be.a 184
80.k odd 4 1 inner 560.2.be.a 184
112.j even 4 1 inner 560.2.be.a 184
560.be even 4 1 inner 560.2.be.a 184

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.be.a 184 1.a even 1 1 trivial
560.2.be.a 184 5.b even 2 1 inner
560.2.be.a 184 7.b odd 2 1 inner
560.2.be.a 184 16.f odd 4 1 inner
560.2.be.a 184 35.c odd 2 1 inner
560.2.be.a 184 80.k odd 4 1 inner
560.2.be.a 184 112.j even 4 1 inner
560.2.be.a 184 560.be even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database