Properties

 Label 560.2.bd.b Level 560 Weight 2 Character orbit 560.bd Analytic conductor 4.472 Analytic rank 0 Dimension 52 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: $$52$$ Relative dimension: $$26$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$52q - 8q^{4} + 12q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$52q - 8q^{4} + 12q^{6} - 4q^{10} - 4q^{11} + 40q^{12} + 8q^{15} - 4q^{16} + 8q^{19} + 32q^{22} + 12q^{24} - 12q^{26} - 24q^{27} + 4q^{28} + 4q^{29} - 16q^{34} + 4q^{36} - 12q^{37} + 16q^{38} - 4q^{42} + 36q^{43} + 28q^{44} - 16q^{46} + 32q^{48} - 52q^{49} - 8q^{51} - 80q^{52} - 4q^{53} + 48q^{54} - 12q^{56} - 56q^{58} + 24q^{59} - 16q^{61} - 72q^{62} - 68q^{63} + 4q^{64} + 40q^{65} - 12q^{66} - 12q^{67} - 72q^{69} - 8q^{72} + 68q^{74} - 4q^{77} - 4q^{78} - 16q^{79} - 32q^{80} - 116q^{81} + 104q^{82} + 16q^{85} - 68q^{86} - 48q^{88} + 36q^{90} + 64q^{92} + 8q^{93} - 72q^{94} - 32q^{95} - 52q^{99} + O(q^{100})$$

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 −1.36236 + 0.379431i 0.0811559 + 0.0811559i 1.71206 1.03385i 0.707107 0.707107i −0.141357 0.0797707i 1.00000i −1.94018 + 2.05808i 2.98683i −0.695038 + 1.23163i
141.2 −1.32495 + 0.494478i −1.01907 1.01907i 1.51098 1.31032i −0.707107 + 0.707107i 1.85412 + 0.846307i 1.00000i −1.35405 + 2.48325i 0.922998i 0.587232 1.28653i
141.3 −1.30578 + 0.543081i 2.37790 + 2.37790i 1.41013 1.41829i −0.707107 + 0.707107i −4.39640 1.81362i 1.00000i −1.07107 + 2.61779i 8.30879i 0.539310 1.30734i
141.4 −1.30541 0.543961i −2.02124 2.02124i 1.40821 + 1.42019i 0.707107 0.707107i 1.53908 + 3.73803i 1.00000i −1.06577 2.61995i 5.17082i −1.30771 + 0.538429i
141.5 −1.30240 0.551150i −1.24266 1.24266i 1.39247 + 1.43563i −0.707107 + 0.707107i 0.933546 + 2.30334i 1.00000i −1.02229 2.63722i 0.0884307i 1.31065 0.531211i
141.6 −0.961413 1.03715i 2.07558 + 2.07558i −0.151369 + 1.99426i 0.707107 0.707107i 0.157202 4.14819i 1.00000i 2.21388 1.76032i 5.61608i −1.41320 0.0535555i
141.7 −0.908304 + 1.08397i 0.523494 + 0.523494i −0.349967 1.96914i 0.707107 0.707107i −1.04294 + 0.0919583i 1.00000i 2.45236 + 1.40923i 2.45191i 0.124212 + 1.40875i
141.8 −0.807148 + 1.16125i 2.00539 + 2.00539i −0.697026 1.87461i 0.707107 0.707107i −3.94742 + 0.710123i 1.00000i 2.73950 + 0.703661i 5.04319i 0.250392 + 1.39187i
141.9 −0.557588 + 1.29965i −2.40733 2.40733i −1.37819 1.44934i −0.707107 + 0.707107i 4.47099 1.78639i 1.00000i 2.65210 0.983032i 8.59045i −0.524718 1.31327i
141.10 −0.513830 1.31757i 0.363076 + 0.363076i −1.47196 + 1.35401i −0.707107 + 0.707107i 0.291817 0.664935i 1.00000i 2.54033 + 1.24367i 2.73635i 1.29499 + 0.568327i
141.11 −0.488835 1.32704i −1.83016 1.83016i −1.52208 + 1.29741i 0.707107 0.707107i −1.53405 + 3.32334i 1.00000i 2.46576 + 1.38565i 3.69896i −1.28402 0.592702i
141.12 −0.171805 + 1.40374i −0.303944 0.303944i −1.94097 0.482338i −0.707107 + 0.707107i 0.478877 0.374439i 1.00000i 1.01054 2.64174i 2.81524i −0.871109 1.11408i
141.13 −0.0789169 + 1.41201i −0.991372 0.991372i −1.98754 0.222863i 0.707107 0.707107i 1.47806 1.32159i 1.00000i 0.471536 2.78884i 1.03436i 0.942639 + 1.05424i
141.14 −0.0328336 1.41383i 0.359066 + 0.359066i −1.99784 + 0.0928423i 0.707107 0.707107i 0.495870 0.519449i 1.00000i 0.196860 + 2.82157i 2.74214i −1.02295 0.976514i
141.15 0.252525 1.39149i 1.86516 + 1.86516i −1.87246 0.702770i −0.707107 + 0.707107i 3.06634 2.12434i 1.00000i −1.45074 + 2.42804i 3.95763i 0.805366 + 1.16249i
141.16 0.328623 1.37550i −1.80727 1.80727i −1.78401 0.904043i −0.707107 + 0.707107i −3.07981 + 1.89199i 1.00000i −1.82978 + 2.15683i 3.53242i 0.740256 + 1.20500i
141.17 0.642045 + 1.26007i 0.975138 + 0.975138i −1.17556 + 1.61804i −0.707107 + 0.707107i −0.602660 + 1.85482i 1.00000i −2.79361 0.442425i 1.09821i −1.34500 0.437010i
141.18 0.668656 1.24615i −1.21993 1.21993i −1.10580 1.66650i 0.707107 0.707107i −2.33593 + 0.704506i 1.00000i −2.81611 + 0.263684i 0.0235503i −0.408353 1.35397i
141.19 0.865210 + 1.11867i −1.52217 1.52217i −0.502824 + 1.93576i 0.707107 0.707107i 0.385801 3.01979i 1.00000i −2.60052 + 1.11235i 1.63398i 1.40281 + 0.179220i
141.20 0.912154 + 1.08073i −1.81546 1.81546i −0.335949 + 1.97158i −0.707107 + 0.707107i 0.306040 3.61800i 1.00000i −2.43718 + 1.43532i 3.59179i −1.40918 0.119200i
See all 52 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 421.26 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.b 52
4.b odd 2 1 2240.2.bd.b 52
16.e even 4 1 inner 560.2.bd.b 52
16.f odd 4 1 2240.2.bd.b 52

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bd.b 52 1.a even 1 1 trivial
560.2.bd.b 52 16.e even 4 1 inner
2240.2.bd.b 52 4.b odd 2 1
2240.2.bd.b 52 16.f odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{52} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database