# Properties

 Label 560.2.t.a Level 560 Weight 2 Character orbit 560.t Analytic conductor 4.472 Analytic rank 0 Dimension 144 CM no Inner twists 2

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$144$$ Relative dimension: $$72$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$144q + 8q^{4} - 144q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$144q + 8q^{4} - 144q^{9} - 16q^{12} + 8q^{16} - 28q^{18} - 16q^{19} + 4q^{22} + 32q^{26} - 36q^{30} + 32q^{34} - 32q^{36} - 56q^{38} + 8q^{40} - 20q^{42} + 64q^{43} - 56q^{44} - 32q^{46} - 80q^{48} - 48q^{50} - 16q^{51} - 16q^{52} + 32q^{54} - 24q^{56} + 52q^{58} + 72q^{60} - 32q^{61} - 40q^{62} + 32q^{64} + 32q^{66} - 24q^{68} - 32q^{69} + 80q^{72} - 16q^{73} - 32q^{74} - 88q^{75} + 76q^{78} + 128q^{80} + 144q^{81} + 56q^{82} + 32q^{86} - 112q^{87} + 88q^{88} + 88q^{90} - 136q^{92} - 8q^{94} + 80q^{95} + 32q^{96} + 64q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
43.1 −1.41270 0.0654069i 1.49919i 1.99144 + 0.184801i 1.45710 1.69613i 0.0980575 2.11791i −0.707107 + 0.707107i −2.80123 0.391322i 0.752424 −2.16939 + 2.30082i
43.2 −1.41154 0.0869980i 1.58286i 1.98486 + 0.245601i 1.89869 1.18110i −0.137706 + 2.23427i 0.707107 0.707107i −2.78034 0.519354i 0.494538 −2.78282 + 1.50198i
43.3 −1.40839 + 0.128258i 0.748637i 1.96710 0.361275i 0.813660 + 2.08278i 0.0960190 + 1.05437i −0.707107 + 0.707107i −2.72410 + 0.761111i 2.43954 −1.41308 2.82899i
43.4 −1.40199 0.185504i 1.80332i 1.93118 + 0.520150i −0.550864 + 2.16715i −0.334523 + 2.52824i 0.707107 0.707107i −2.61101 1.08749i −0.251963 1.17432 2.93615i
43.5 −1.39923 + 0.205299i 0.670195i 1.91570 0.574522i −1.47177 1.68342i 0.137590 + 0.937758i −0.707107 + 0.707107i −2.56257 + 1.19718i 2.55084 2.40495 + 2.05334i
43.6 −1.39015 0.259757i 2.75657i 1.86505 + 0.722204i −1.95957 + 1.07707i 0.716039 3.83206i −0.707107 + 0.707107i −2.40511 1.48843i −4.59869 3.00388 0.988279i
43.7 −1.37092 + 0.347225i 2.77381i 1.75887 0.952039i 1.05275 1.97274i −0.963136 3.80268i 0.707107 0.707107i −2.08071 + 1.91590i −4.69401 −0.758256 + 3.07002i
43.8 −1.36754 0.360316i 3.13969i 1.74034 + 0.985496i 1.64497 + 1.51462i 1.13128 4.29365i 0.707107 0.707107i −2.02490 1.97478i −6.85762 −1.70383 2.66401i
43.9 −1.36029 + 0.386778i 1.62010i 1.70080 1.05227i −2.19508 0.426188i −0.626621 2.20382i 0.707107 0.707107i −1.90660 + 2.08923i 0.375266 3.15079 0.269268i
43.10 −1.33923 + 0.454387i 3.28908i 1.58707 1.21706i 2.05100 0.890731i 1.49451 + 4.40483i −0.707107 + 0.707107i −1.57243 + 2.35106i −7.81803 −2.34202 + 2.12484i
43.11 −1.29547 0.567247i 0.154841i 1.35646 + 1.46970i −0.497515 2.18002i 0.0878329 0.200591i 0.707107 0.707107i −0.923567 2.67339i 2.97602 −0.592095 + 3.10635i
43.12 −1.22105 0.713475i 2.06291i 0.981908 + 1.74237i 1.83957 + 1.27121i −1.47183 + 2.51891i −0.707107 + 0.707107i 0.0441824 2.82808i −1.25559 −1.33923 2.86469i
43.13 −1.18735 + 0.768249i 0.0697127i 0.819588 1.82436i 2.20646 + 0.362656i 0.0535567 + 0.0827732i 0.707107 0.707107i 0.428424 + 2.79579i 2.99514 −2.89845 + 1.26451i
43.14 −1.16689 0.798984i 1.17950i 0.723250 + 1.86465i 0.0146410 + 2.23602i 0.942403 1.37635i 0.707107 0.707107i 0.645873 2.75370i 1.60878 1.76946 2.62088i
43.15 −1.11562 + 0.869134i 2.40982i 0.489211 1.93925i −2.07668 + 0.829085i 2.09446 + 2.68844i −0.707107 + 0.707107i 1.13969 + 2.58865i −2.80723 1.59620 2.72986i
43.16 −1.09978 0.889090i 2.37538i 0.419039 + 1.95561i −0.533053 2.17160i 2.11192 2.61239i −0.707107 + 0.707107i 1.27786 2.52331i −2.64241 −1.34451 + 2.86222i
43.17 −1.09013 + 0.900895i 2.66085i 0.376777 1.96419i 1.75709 + 1.38298i −2.39714 2.90067i −0.707107 + 0.707107i 1.35879 + 2.48066i −4.08010 −3.16138 + 0.0753290i
43.18 −1.07328 + 0.920909i 1.27333i 0.303853 1.97678i −1.92329 + 1.14059i 1.17262 + 1.36664i 0.707107 0.707107i 1.49432 + 2.40146i 1.37862 1.01385 2.99535i
43.19 −0.975664 1.02376i 0.864258i −0.0961601 + 1.99769i −2.07047 0.844473i −0.884791 + 0.843225i −0.707107 + 0.707107i 2.13897 1.85063i 2.25306 1.15555 + 2.94359i
43.20 −0.918365 + 1.07546i 1.03076i −0.313212 1.97532i 1.39292 1.74922i 1.10853 + 0.946610i −0.707107 + 0.707107i 2.41202 + 1.47722i 1.93754 0.601999 + 3.10445i
See next 80 embeddings (of 144 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 547.72 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
80.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.t.a 144
5.c odd 4 1 560.2.bl.a yes 144
16.f odd 4 1 560.2.bl.a yes 144
80.j even 4 1 inner 560.2.t.a 144

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.t.a 144 1.a even 1 1 trivial
560.2.t.a 144 80.j even 4 1 inner
560.2.bl.a yes 144 5.c odd 4 1
560.2.bl.a yes 144 16.f odd 4 1

## 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