# Properties

 Label 280.2.bf.a Level $280$ Weight $2$ Character orbit 280.bf Analytic conductor $2.236$ Analytic rank $0$ Dimension $88$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.bf (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$88$$ Relative dimension: $$44$$ over $$\Q(\zeta_{6})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} - 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$88q - 2q^{4} - 40q^{9} + 6q^{10} - 6q^{14} + 4q^{15} + 2q^{16} - 24q^{20} - 8q^{24} - 2q^{25} - 14q^{26} + 14q^{30} - 28q^{31} + 24q^{34} + 60q^{36} - 16q^{39} - 32q^{40} - 16q^{41} - 2q^{44} - 42q^{46} - 8q^{49} - 12q^{50} + 8q^{54} - 28q^{55} + 36q^{56} + 10q^{60} + 52q^{64} - 12q^{65} - 44q^{66} - 48q^{70} + 16q^{71} - 42q^{74} - 36q^{76} - 4q^{79} - 40q^{80} - 20q^{81} + 12q^{84} - 24q^{86} + 4q^{89} - 96q^{90} - 22q^{94} - 34q^{95} + 80q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
109.1 −1.41234 0.0727260i −1.37008 + 2.37305i 1.98942 + 0.205428i −2.18145 + 0.491206i 2.10761 3.25192i −2.02537 1.70232i −2.79480 0.434817i −2.25425 3.90447i 3.11668 0.535104i
109.2 −1.39743 + 0.217250i 0.284947 0.493542i 1.90560 0.607183i 0.172645 + 2.22939i −0.290970 + 0.751594i −2.45778 + 0.979438i −2.53103 + 1.26249i 1.33761 + 2.31681i −0.725596 3.07791i
109.3 −1.38464 0.287716i 1.62181 2.80906i 1.83444 + 0.796764i −0.194592 + 2.22758i −3.05383 + 3.42291i 1.16173 2.37705i −2.31079 1.63103i −3.76055 6.51347i 0.910350 3.02841i
109.4 −1.38305 0.295272i −0.496594 + 0.860127i 1.82563 + 0.816749i 1.99653 1.00690i 0.940784 1.04296i 2.24690 1.39695i −2.28377 1.66866i 1.00679 + 1.74381i −3.05861 + 0.803068i
109.5 −1.34506 + 0.436831i 0.866705 1.50118i 1.61836 1.17513i 1.99503 1.00988i −0.510007 + 2.39777i 1.01592 + 2.44293i −1.66345 + 2.28756i −0.00235378 0.00407687i −2.24228 + 2.22984i
109.6 −1.32820 0.485693i 0.390744 0.676788i 1.52820 + 1.29019i −2.21752 0.287435i −0.847695 + 0.709125i 1.02493 + 2.43916i −1.40312 2.45586i 1.19464 + 2.06917i 2.80569 + 1.45880i
109.7 −1.29160 + 0.576005i 0.755963 1.30937i 1.33644 1.48793i −1.54124 1.62006i −0.222196 + 2.12661i 0.727432 2.54379i −0.869078 + 2.69160i 0.357040 + 0.618411i 2.92382 + 1.20470i
109.8 −1.19702 0.753089i −1.40092 + 2.42646i 0.865713 + 1.80293i 0.791109 + 2.09145i 3.50426 1.84950i 2.13463 + 1.56312i 0.321488 2.81010i −2.42513 4.20044i 0.628073 3.09928i
109.9 −1.14463 + 0.830552i −0.755963 + 1.30937i 0.620368 1.90135i 1.54124 + 1.62006i −0.222196 2.12661i 0.727432 2.54379i 0.869078 + 2.69160i 0.357040 + 0.618411i −3.10969 0.574295i
109.10 −1.05084 + 0.946438i −0.866705 + 1.50118i 0.208510 1.98910i −1.99503 + 1.00988i −0.510007 2.39777i 1.01592 + 2.44293i 1.66345 + 2.28756i −0.00235378 0.00407687i 1.14066 2.94939i
109.11 −1.02388 0.975536i 0.299892 0.519428i 0.0966599 + 1.99766i 2.04027 + 0.915040i −0.813774 + 0.239277i −2.46748 0.954749i 1.84982 2.13966i 1.32013 + 2.28653i −1.19634 2.92725i
109.12 −0.912095 1.08078i −0.945048 + 1.63687i −0.336164 + 1.97155i −0.0512873 2.23548i 2.63107 0.471595i −1.18371 + 2.36619i 2.43742 1.43492i −0.286230 0.495764i −2.36928 + 2.09440i
109.13 −0.886858 + 1.10158i −0.284947 + 0.493542i −0.426967 1.95389i −0.172645 2.22939i −0.290970 0.751594i −2.45778 + 0.979438i 2.53103 + 1.26249i 1.33761 + 2.31681i 2.60897 + 1.78697i
109.14 −0.805587 1.16234i 1.22223 2.11697i −0.702060 + 1.87273i 1.10428 1.94437i −3.44524 + 0.284754i 2.60788 0.446057i 2.74231 0.692614i −1.48770 2.57677i −3.14961 + 0.282810i
109.15 −0.643189 + 1.25949i 1.37008 2.37305i −1.17262 1.62018i 2.18145 0.491206i 2.10761 + 3.25192i −2.02537 1.70232i 2.79480 0.434817i −2.25425 3.90447i −0.784414 + 3.06344i
109.16 −0.603821 1.27883i 1.22223 2.11697i −1.27080 + 1.54437i −2.23601 0.0158492i −3.44524 0.284754i −2.60788 + 0.446057i 2.74231 + 0.692614i −1.48770 2.57677i 1.32988 + 2.86904i
109.17 −0.479934 1.33029i −0.945048 + 1.63687i −1.53933 + 1.27690i −1.91034 1.16216i 2.63107 + 0.471595i 1.18371 2.36619i 2.43742 + 1.43492i −0.286230 0.495764i −0.629165 + 3.09906i
109.18 −0.443149 + 1.34299i −1.62181 + 2.80906i −1.60724 1.19029i 0.194592 2.22758i −3.05383 3.42291i 1.16173 2.37705i 2.31079 1.63103i −3.76055 6.51347i 2.90539 + 1.24849i
109.19 −0.435810 + 1.34539i 0.496594 0.860127i −1.62014 1.17267i −1.99653 + 1.00690i 0.940784 + 1.04296i 2.24690 1.39695i 2.28377 1.66866i 1.00679 + 1.74381i −0.484562 3.12493i
109.20 −0.332899 1.37447i 0.299892 0.519428i −1.77836 + 0.915122i −0.227688 + 2.22445i −0.813774 0.239277i 2.46748 + 0.954749i 1.84982 + 2.13966i 1.32013 + 2.28653i 3.13324 0.427565i
See all 88 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 149.44 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.c even 3 1 inner
8.b even 2 1 inner
35.j even 6 1 inner
40.f even 2 1 inner
56.p even 6 1 inner
280.bf even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.bf.a 88
4.b odd 2 1 1120.2.bv.a 88
5.b even 2 1 inner 280.2.bf.a 88
7.c even 3 1 inner 280.2.bf.a 88
8.b even 2 1 inner 280.2.bf.a 88
8.d odd 2 1 1120.2.bv.a 88
20.d odd 2 1 1120.2.bv.a 88
28.g odd 6 1 1120.2.bv.a 88
35.j even 6 1 inner 280.2.bf.a 88
40.e odd 2 1 1120.2.bv.a 88
40.f even 2 1 inner 280.2.bf.a 88
56.k odd 6 1 1120.2.bv.a 88
56.p even 6 1 inner 280.2.bf.a 88
140.p odd 6 1 1120.2.bv.a 88
280.bf even 6 1 inner 280.2.bf.a 88
280.bi odd 6 1 1120.2.bv.a 88

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.bf.a 88 1.a even 1 1 trivial
280.2.bf.a 88 5.b even 2 1 inner
280.2.bf.a 88 7.c even 3 1 inner
280.2.bf.a 88 8.b even 2 1 inner
280.2.bf.a 88 35.j even 6 1 inner
280.2.bf.a 88 40.f even 2 1 inner
280.2.bf.a 88 56.p even 6 1 inner
280.2.bf.a 88 280.bf even 6 1 inner
1120.2.bv.a 88 4.b odd 2 1
1120.2.bv.a 88 8.d odd 2 1
1120.2.bv.a 88 20.d odd 2 1
1120.2.bv.a 88 28.g odd 6 1
1120.2.bv.a 88 40.e odd 2 1
1120.2.bv.a 88 56.k odd 6 1
1120.2.bv.a 88 140.p odd 6 1
1120.2.bv.a 88 280.bi odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(280, [\chi])$$.