# Properties

 Label 560.2.bc.a Level 560 Weight 2 Character orbit 560.bc Analytic conductor 4.472 Analytic rank 0 Dimension 128 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.bc (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$128$$ Relative dimension: $$64$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$128q - 4q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$128q - 4q^{4} + 8q^{11} - 8q^{14} + 20q^{16} + 20q^{18} - 52q^{22} - 16q^{23} + 44q^{28} + 16q^{29} + 40q^{32} + 16q^{37} + 60q^{42} - 8q^{43} - 108q^{44} - 40q^{46} - 4q^{50} + 80q^{51} + 16q^{53} + 12q^{56} - 76q^{58} - 56q^{60} - 52q^{64} + 40q^{67} - 8q^{70} - 64q^{71} - 36q^{72} + 60q^{74} - 48q^{78} - 128q^{81} - 32q^{84} + 76q^{86} + 20q^{88} - 32q^{91} + 64q^{92} - 76q^{98} - 40q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
251.1 −1.40740 + 0.138663i −0.176544 + 0.176544i 1.96155 0.390307i −0.707107 + 0.707107i 0.223988 0.272948i −2.53089 + 0.771085i −2.70656 + 0.821311i 2.93766i 0.897132 1.09323i
251.2 −1.40740 + 0.138663i 0.176544 0.176544i 1.96155 0.390307i 0.707107 0.707107i −0.223988 + 0.272948i −2.53089 0.771085i −2.70656 + 0.821311i 2.93766i −0.897132 + 1.09323i
251.3 −1.40603 0.151956i −0.503741 + 0.503741i 1.95382 + 0.427308i 0.707107 0.707107i 0.784820 0.631727i 1.62955 + 2.08436i −2.68219 0.897701i 2.49249i −1.10166 + 0.886762i
251.4 −1.40603 0.151956i 0.503741 0.503741i 1.95382 + 0.427308i −0.707107 + 0.707107i −0.784820 + 0.631727i 1.62955 2.08436i −2.68219 0.897701i 2.49249i 1.10166 0.886762i
251.5 −1.34988 + 0.421701i −1.91918 + 1.91918i 1.64434 1.13849i −0.707107 + 0.707107i 1.78134 3.39997i 2.54641 + 0.718206i −1.73955 + 2.23024i 4.36648i 0.656320 1.25269i
251.6 −1.34988 + 0.421701i 1.91918 1.91918i 1.64434 1.13849i 0.707107 0.707107i −1.78134 + 3.39997i 2.54641 0.718206i −1.73955 + 2.23024i 4.36648i −0.656320 + 1.25269i
251.7 −1.30816 0.537317i −1.40559 + 1.40559i 1.42258 + 1.40580i −0.707107 + 0.707107i 2.59399 1.08349i −0.814506 + 2.51726i −1.10561 2.60339i 0.951378i 1.30495 0.545070i
251.8 −1.30816 0.537317i 1.40559 1.40559i 1.42258 + 1.40580i 0.707107 0.707107i −2.59399 + 1.08349i −0.814506 2.51726i −1.10561 2.60339i 0.951378i −1.30495 + 0.545070i
251.9 −1.30320 + 0.549239i −0.985188 + 0.985188i 1.39667 1.43154i 0.707107 0.707107i 0.742796 1.82500i 1.76064 1.97488i −1.03389 + 2.63269i 1.05881i −0.533133 + 1.30987i
251.10 −1.30320 + 0.549239i 0.985188 0.985188i 1.39667 1.43154i −0.707107 + 0.707107i −0.742796 + 1.82500i 1.76064 + 1.97488i −1.03389 + 2.63269i 1.05881i 0.533133 1.30987i
251.11 −1.16313 + 0.804438i −1.78548 + 1.78548i 0.705760 1.87134i 0.707107 0.707107i 0.640444 3.51306i −0.850060 + 2.50547i 0.684482 + 2.74436i 3.37588i −0.253636 + 1.39128i
251.12 −1.16313 + 0.804438i 1.78548 1.78548i 0.705760 1.87134i −0.707107 + 0.707107i −0.640444 + 3.51306i −0.850060 2.50547i 0.684482 + 2.74436i 3.37588i 0.253636 1.39128i
251.13 −1.13815 0.839407i −1.07700 + 1.07700i 0.590792 + 1.91075i −0.707107 + 0.707107i 2.12984 0.321752i −0.292665 2.62951i 0.931484 2.67064i 0.680136i 1.39835 0.211247i
251.14 −1.13815 0.839407i 1.07700 1.07700i 0.590792 + 1.91075i 0.707107 0.707107i −2.12984 + 0.321752i −0.292665 + 2.62951i 0.931484 2.67064i 0.680136i −1.39835 + 0.211247i
251.15 −1.04480 0.953102i −1.26143 + 1.26143i 0.183195 + 1.99159i 0.707107 0.707107i 2.52022 0.115666i −1.76061 1.97491i 1.70679 2.25541i 0.182435i −1.41273 + 0.0648373i
251.16 −1.04480 0.953102i 1.26143 1.26143i 0.183195 + 1.99159i −0.707107 + 0.707107i −2.52022 + 0.115666i −1.76061 + 1.97491i 1.70679 2.25541i 0.182435i 1.41273 0.0648373i
251.17 −1.02851 0.970651i −2.22970 + 2.22970i 0.115673 + 1.99665i 0.707107 0.707107i 4.45753 0.129012i 2.24619 + 1.39808i 1.81908 2.16586i 6.94309i −1.41362 + 0.0409138i
251.18 −1.02851 0.970651i 2.22970 2.22970i 0.115673 + 1.99665i −0.707107 + 0.707107i −4.45753 + 0.129012i 2.24619 1.39808i 1.81908 2.16586i 6.94309i 1.41362 0.0409138i
251.19 −0.883798 + 1.10404i −1.16174 + 1.16174i −0.437803 1.95149i −0.707107 + 0.707107i −0.255863 2.30935i 2.48811 0.899608i 2.54145 + 1.24137i 0.300718i −0.155734 1.40561i
251.20 −0.883798 + 1.10404i 1.16174 1.16174i −0.437803 1.95149i 0.707107 0.707107i 0.255863 + 2.30935i 2.48811 + 0.899608i 2.54145 + 1.24137i 0.300718i 0.155734 + 1.40561i
See next 80 embeddings (of 128 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 531.64 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
16.f odd 4 1 inner
112.j 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.bc.a 128
7.b odd 2 1 inner 560.2.bc.a 128
16.f odd 4 1 inner 560.2.bc.a 128
112.j even 4 1 inner 560.2.bc.a 128

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bc.a 128 1.a even 1 1 trivial
560.2.bc.a 128 7.b odd 2 1 inner
560.2.bc.a 128 16.f odd 4 1 inner
560.2.bc.a 128 112.j 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