# Properties

 Label 560.2.bd.a Level 560 Weight 2 Character orbit 560.bd Analytic conductor 4.472 Analytic rank 0 Dimension 44 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.bd (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$44$$ Relative dimension: $$22$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$44q + 12q^{4} + 12q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$44q + 12q^{4} + 12q^{6} - 4q^{10} + 12q^{11} - 16q^{12} + 4q^{14} + 8q^{15} - 8q^{16} - 20q^{18} + 8q^{19} - 36q^{22} + 12q^{24} + 44q^{26} - 24q^{27} - 4q^{28} + 12q^{29} - 40q^{32} - 16q^{34} + 4q^{36} + 28q^{37} - 16q^{38} + 4q^{42} - 44q^{43} - 32q^{44} - 16q^{46} - 32q^{48} - 44q^{49} + 4q^{50} - 8q^{51} + 16q^{52} - 12q^{53} - 80q^{54} + 8q^{56} + 4q^{58} + 24q^{59} - 16q^{61} + 28q^{63} + 72q^{64} - 40q^{65} - 20q^{66} - 28q^{67} + 56q^{68} + 40q^{69} + 12q^{72} + 24q^{74} - 12q^{77} + 84q^{78} - 16q^{79} + 20q^{81} + 48q^{82} + 16q^{85} - 64q^{86} + 28q^{88} - 36q^{90} - 16q^{92} + 88q^{93} + 96q^{94} - 32q^{95} + 48q^{96} + 28q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
141.1 −1.41203 0.0784738i 0.257753 + 0.257753i 1.98768 + 0.221615i −0.707107 + 0.707107i −0.343730 0.384184i 1.00000i −2.78929 0.468910i 2.86713i 1.05395 0.942970i
141.2 −1.40016 0.198865i 1.92081 + 1.92081i 1.92091 + 0.556887i 0.707107 0.707107i −2.30746 3.07142i 1.00000i −2.57883 1.16173i 4.37899i −1.13068 + 0.849445i
141.3 −1.39986 + 0.200956i −2.22143 2.22143i 1.91923 0.562622i −0.707107 + 0.707107i 3.55611 + 2.66329i 1.00000i −2.57360 + 1.17328i 6.86952i 0.847755 1.13195i
141.4 −1.37680 + 0.323129i −1.67958 1.67958i 1.79118 0.889770i 0.707107 0.707107i 2.85517 + 1.76973i 1.00000i −2.17859 + 1.80382i 2.64195i −0.745060 + 1.20203i
141.5 −1.16029 + 0.808533i 0.925827 + 0.925827i 0.692549 1.87627i −0.707107 + 0.707107i −1.82279 0.325667i 1.00000i 0.713464 + 2.73696i 1.28569i 0.248730 1.39217i
141.6 −1.04918 0.948274i 0.448521 + 0.448521i 0.201553 + 1.98982i 0.707107 0.707107i −0.0452579 0.895900i 1.00000i 1.67543 2.27880i 2.59766i −1.41241 + 0.0713505i
141.7 −0.914710 + 1.07857i −1.42818 1.42818i −0.326610 1.97315i 0.707107 0.707107i 2.84676 0.234016i 1.00000i 2.42693 + 1.45259i 1.07940i 0.115864 + 1.40946i
141.8 −0.630992 + 1.26564i 0.659301 + 0.659301i −1.20370 1.59722i 0.707107 0.707107i −1.25045 + 0.418425i 1.00000i 2.78103 0.515617i 2.13064i 0.448765 + 1.34112i
141.9 −0.590325 + 1.28511i −0.839605 0.839605i −1.30303 1.51727i −0.707107 + 0.707107i 1.57463 0.583348i 1.00000i 2.71908 0.778863i 1.59013i −0.491290 1.32614i
141.10 −0.560807 1.29827i 0.693100 + 0.693100i −1.37099 + 1.45615i −0.707107 + 0.707107i 0.511134 1.28852i 1.00000i 2.65934 + 0.963293i 2.03922i 1.31456 + 0.521463i
141.11 −0.551277 1.30234i −1.16279 1.16279i −1.39219 + 1.43590i 0.707107 0.707107i −0.873327 + 2.15536i 1.00000i 2.63752 + 1.02152i 0.295857i −1.31071 0.531083i
141.12 0.381101 + 1.36190i −0.605289 0.605289i −1.70952 + 1.03804i −0.707107 + 0.707107i 0.593665 1.05502i 1.00000i −2.06521 1.93260i 2.26725i −1.23249 0.693527i
141.13 0.504607 + 1.32113i 2.22016 + 2.22016i −1.49074 + 1.33330i 0.707107 0.707107i −1.81280 + 4.05342i 1.00000i −2.51369 1.29667i 6.85821i 1.29099 + 0.577366i
141.14 0.517296 1.31621i −0.296675 0.296675i −1.46481 1.36174i −0.707107 + 0.707107i −0.543956 + 0.237018i 1.00000i −2.55007 + 1.22357i 2.82397i 0.564917 + 1.29648i
141.15 0.817344 1.15410i 1.28029 + 1.28029i −0.663897 1.88660i 0.707107 0.707107i 2.52403 0.431148i 1.00000i −2.71995 0.775794i 0.278310i −0.238123 1.39402i
141.16 1.13939 + 0.837723i −1.78962 1.78962i 0.596439 + 1.90899i −0.707107 + 0.707107i −0.539875 3.53828i 1.00000i −0.919631 + 2.67475i 3.40546i −1.39803 + 0.213314i
141.17 1.17062 0.793500i −1.25769 1.25769i 0.740715 1.85778i 0.707107 0.707107i −2.47025 0.474301i 1.00000i −0.607051 2.76252i 0.163545i 0.266666 1.38884i
141.18 1.18248 + 0.775723i 1.63220 + 1.63220i 0.796507 + 1.83455i −0.707107 + 0.707107i 0.663904 + 3.19617i 1.00000i −0.481253 + 2.78718i 2.32815i −1.38466 + 0.287619i
141.19 1.20137 0.746130i 2.18833 + 2.18833i 0.886580 1.79276i −0.707107 + 0.707107i 4.26177 + 0.996215i 1.00000i −0.272519 2.81527i 6.57756i −0.321903 + 1.37709i
141.20 1.32415 + 0.496608i −0.576404 0.576404i 1.50676 + 1.31517i 0.707107 0.707107i −0.477000 1.04949i 1.00000i 1.34206 + 2.48976i 2.33552i 1.28747 0.585162i
See all 44 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.22 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
16.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.bd.a 44
4.b odd 2 1 2240.2.bd.a 44
16.e even 4 1 inner 560.2.bd.a 44
16.f odd 4 1 2240.2.bd.a 44

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.a 44 1.a even 1 1 trivial
560.2.bd.a 44 16.e even 4 1 inner
2240.2.bd.a 44 4.b odd 2 1
2240.2.bd.a 44 16.f odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{44} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database