# Properties

 Label 560.2.bb.d Level 560 Weight 2 Character orbit 560.bb Analytic conductor 4.472 Analytic rank 0 Dimension 70 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.bb (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.47162251319$$ Analytic rank: $$0$$ Dimension: $$70$$ Relative dimension: $$35$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$70q + 2q^{2} + 2q^{3} + 2q^{5} + 70q^{7} + 8q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$70q + 2q^{2} + 2q^{3} + 2q^{5} + 70q^{7} + 8q^{8} - 18q^{10} - 2q^{11} - 4q^{12} + 6q^{13} + 2q^{14} - 6q^{15} + 4q^{16} - 18q^{18} + 14q^{19} + 12q^{20} + 2q^{21} - 12q^{22} + 20q^{24} + 6q^{25} - 36q^{26} + 8q^{27} + 2q^{29} + 8q^{30} + 16q^{31} - 8q^{32} + 4q^{34} + 2q^{35} - 40q^{36} + 10q^{37} - 12q^{38} - 24q^{40} + 2q^{43} - 24q^{44} - 24q^{45} - 16q^{46} - 44q^{48} + 70q^{49} - 10q^{50} + 8q^{51} + 28q^{52} - 30q^{53} - 32q^{54} + 6q^{55} + 8q^{56} - 76q^{57} + 56q^{58} + 2q^{59} - 8q^{60} + 30q^{61} + 48q^{62} + 12q^{64} - 10q^{65} + 80q^{66} + 6q^{67} - 36q^{68} - 16q^{69} - 18q^{70} + 4q^{72} - 36q^{73} - 32q^{74} - 2q^{75} + 44q^{76} - 2q^{77} - 84q^{78} - 40q^{79} + 12q^{80} - 82q^{81} + 24q^{82} + 10q^{83} - 4q^{84} + 32q^{85} + 32q^{86} - 4q^{87} + 32q^{88} + 18q^{90} + 6q^{91} - 92q^{92} - 56q^{93} - 20q^{94} + 6q^{95} + 16q^{96} + 2q^{98} - 34q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.41250 + 0.0695215i 0.861811 0.861811i 1.99033 0.196399i −1.76836 1.36854i −1.15740 + 1.27722i 1.00000 −2.79770 + 0.415785i 1.51457i 2.59295 + 1.81014i
29.2 −1.40611 0.151214i −1.45780 + 1.45780i 1.95427 + 0.425246i −2.23574 + 0.0381166i 2.27026 1.82938i 1.00000 −2.68361 0.893453i 1.25035i 3.14946 + 0.284479i
29.3 −1.34972 + 0.422212i 1.56336 1.56336i 1.64347 1.13973i 2.04190 + 0.911403i −1.45003 + 2.77016i 1.00000 −1.73702 + 2.23221i 1.88819i −3.14079 0.368023i
29.4 −1.33379 0.470109i 0.760413 0.760413i 1.55800 + 1.25405i 0.975010 2.01230i −1.37171 + 0.656755i 1.00000 −1.48850 2.40507i 1.84355i −2.24646 + 2.22563i
29.5 −1.31401 + 0.522843i −1.20613 + 1.20613i 1.45327 1.37405i 2.07792 0.825973i 0.954258 2.21549i 1.00000 −1.19121 + 2.56535i 0.0904875i −2.29857 + 2.17177i
29.6 −1.20051 0.747507i −1.99217 + 1.99217i 0.882467 + 1.79478i 0.520478 2.17465i 3.88079 0.902467i 1.00000 0.282200 2.81431i 4.93747i −2.25041 + 2.22164i
29.7 −1.17664 0.784556i −0.0419138 + 0.0419138i 0.768944 + 1.84627i 0.673232 + 2.13231i 0.0822009 0.0164335i 1.00000 0.543737 2.77567i 2.99649i 0.880770 3.03714i
29.8 −1.08190 + 0.910767i 1.99977 1.99977i 0.341008 1.97071i −2.16539 + 0.557749i −0.342224 + 3.98487i 1.00000 1.42592 + 2.44269i 4.99816i 1.83475 2.57559i
29.9 −0.868292 1.11627i 2.38527 2.38527i −0.492138 + 1.93850i −0.763308 2.10175i −4.73373 0.591506i 1.00000 2.59122 1.13383i 8.37902i −1.68336 + 2.67700i
29.10 −0.866818 + 1.11742i 0.969878 0.969878i −0.497254 1.93720i 1.42081 + 1.72664i 0.243053 + 1.92447i 1.00000 2.59569 + 1.12356i 1.11867i −3.16097 + 0.0909610i
29.11 −0.744791 + 1.20220i −2.31279 + 2.31279i −0.890574 1.79078i −1.10225 + 1.94552i −1.05789 4.50298i 1.00000 2.81616 + 0.263104i 7.69797i −1.51795 2.77413i
29.12 −0.735407 1.20796i −1.40023 + 1.40023i −0.918355 + 1.77669i 2.12747 + 0.688383i 2.72116 + 0.661689i 1.00000 2.82154 0.197250i 0.921283i −0.733014 3.07615i
29.13 −0.639532 1.26135i −0.819103 + 0.819103i −1.18200 + 1.61335i −2.21530 0.304027i 1.55702 + 0.509331i 1.00000 2.79091 + 0.459123i 1.65814i 1.03327 + 2.98870i
29.14 −0.332249 + 1.37463i −1.50237 + 1.50237i −1.77922 0.913441i 2.16429 + 0.562013i −1.56605 2.56437i 1.00000 1.84679 2.14228i 1.51426i −1.49164 + 2.78837i
29.15 −0.289368 1.38429i 1.70369 1.70369i −1.83253 + 0.801140i 0.0668319 + 2.23507i −2.85141 1.86542i 1.00000 1.63929 + 2.30494i 2.80515i 3.07465 0.739272i
29.16 −0.239753 + 1.39374i 0.826272 0.826272i −1.88504 0.668309i −1.16974 + 1.90570i 0.953509 + 1.34971i 1.00000 1.38339 2.46703i 1.63455i −2.37561 2.08722i
29.17 −0.171434 1.40378i 0.104638 0.104638i −1.94122 + 0.481312i 2.23517 + 0.0634380i −0.164828 0.128951i 1.00000 1.00845 + 2.64254i 2.97810i −0.294130 3.14857i
29.18 −0.168286 + 1.40417i −0.0271728 + 0.0271728i −1.94336 0.472603i −1.76632 1.37117i −0.0335823 0.0427279i 1.00000 0.990653 2.64927i 2.99852i 2.22260 2.24946i
29.19 0.125099 + 1.40867i 1.73140 1.73140i −1.96870 + 0.352445i 1.71263 1.43768i 2.65556 + 2.22237i 1.00000 −0.742760 2.72916i 2.99546i 2.23946 + 2.23267i
29.20 0.435695 1.34543i −2.28818 + 2.28818i −1.62034 1.17239i −1.81745 1.30264i 2.08163 + 4.07553i 1.00000 −2.28334 + 1.66924i 7.47157i −2.54446 + 1.87769i
See all 70 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 309.35 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.q 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.bb.d yes 70
5.b even 2 1 560.2.bb.c 70
16.e even 4 1 560.2.bb.c 70
80.q even 4 1 inner 560.2.bb.d yes 70

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
560.2.bb.c 70 5.b even 2 1
560.2.bb.c 70 16.e even 4 1
560.2.bb.d yes 70 1.a even 1 1 trivial
560.2.bb.d yes 70 80.q even 4 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{70} - \cdots$$ acting on $$S_{2}^{\mathrm{new}}(560, [\chi])$$.

## Hecke characteristic polynomials

There are no characteristic polynomials of Hecke operators in the database