# Properties

 Label 280.2.l.a Level $280$ Weight $2$ Character orbit 280.l Analytic conductor $2.236$ Analytic rank $0$ Dimension $36$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$280 = 2^{3} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 280.l (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$36$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{4} - 4q^{6} + 36q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36q + 4q^{4} - 4q^{6} + 36q^{9} - 8q^{10} + 20q^{16} - 24q^{20} - 48q^{24} + 4q^{25} - 4q^{26} + 4q^{30} - 16q^{31} + 12q^{34} - 20q^{36} - 32q^{39} + 16q^{40} - 8q^{41} + 56q^{44} - 36q^{49} - 12q^{50} - 52q^{54} - 32q^{55} + 12q^{56} - 20q^{60} - 20q^{64} - 24q^{65} - 28q^{66} - 12q^{70} + 56q^{71} - 24q^{74} + 48q^{76} + 24q^{79} + 64q^{80} + 36q^{81} + 24q^{86} - 40q^{89} - 52q^{90} - 92q^{94} + 40q^{95} + 48q^{96} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
29.1 −1.38397 0.290898i −1.83171 1.83076 + 0.805189i 1.41945 1.72776i 2.53504 + 0.532841i 1.00000i −2.29949 1.64692i 0.355171 −2.46708 + 1.97826i
29.2 −1.38397 + 0.290898i −1.83171 1.83076 0.805189i 1.41945 + 1.72776i 2.53504 0.532841i 1.00000i −2.29949 + 1.64692i 0.355171 −2.46708 1.97826i
29.3 −1.37343 0.337187i 3.10393 1.77261 + 0.926204i −0.0152300 2.23602i −4.26302 1.04660i 1.00000i −2.12225 1.86978i 6.63436 −0.733038 + 3.07614i
29.4 −1.37343 + 0.337187i 3.10393 1.77261 0.926204i −0.0152300 + 2.23602i −4.26302 + 1.04660i 1.00000i −2.12225 + 1.86978i 6.63436 −0.733038 3.07614i
29.5 −1.36669 0.363546i 0.460762 1.73567 + 0.993708i −2.19557 + 0.423658i −0.629718 0.167508i 1.00000i −2.01086 1.98908i −2.78770 3.15467 + 0.219182i
29.6 −1.36669 + 0.363546i 0.460762 1.73567 0.993708i −2.19557 0.423658i −0.629718 + 0.167508i 1.00000i −2.01086 + 1.98908i −2.78770 3.15467 0.219182i
29.7 −1.23615 0.686976i −0.359051 1.05613 + 1.69841i 0.565870 + 2.16328i 0.443841 + 0.246660i 1.00000i −0.138765 2.82502i −2.87108 0.786624 3.06288i
29.8 −1.23615 + 0.686976i −0.359051 1.05613 1.69841i 0.565870 2.16328i 0.443841 0.246660i 1.00000i −0.138765 + 2.82502i −2.87108 0.786624 + 3.06288i
29.9 −1.12571 0.856025i 1.65329 0.534444 + 1.92727i 2.21195 + 0.327549i −1.86112 1.41526i 1.00000i 1.04816 2.62704i −0.266637 −2.20962 2.26221i
29.10 −1.12571 + 0.856025i 1.65329 0.534444 1.92727i 2.21195 0.327549i −1.86112 + 1.41526i 1.00000i 1.04816 + 2.62704i −0.266637 −2.20962 + 2.26221i
29.11 −0.752500 1.19739i −2.12140 −0.867487 + 1.80207i −2.02293 + 0.952769i 1.59636 + 2.54015i 1.00000i 2.81057 0.317339i 1.50036 2.66309 + 1.70527i
29.12 −0.752500 + 1.19739i −2.12140 −0.867487 1.80207i −2.02293 0.952769i 1.59636 2.54015i 1.00000i 2.81057 + 0.317339i 1.50036 2.66309 1.70527i
29.13 −0.614166 1.27389i −2.96656 −1.24560 + 1.56476i 2.18846 + 0.458941i 1.82196 + 3.77907i 1.00000i 2.75834 + 0.625738i 5.80046 −0.759437 3.06973i
29.14 −0.614166 + 1.27389i −2.96656 −1.24560 1.56476i 2.18846 0.458941i 1.82196 3.77907i 1.00000i 2.75834 0.625738i 5.80046 −0.759437 + 3.06973i
29.15 −0.269098 1.38838i 2.55593 −1.85517 + 0.747219i 0.790907 + 2.09152i −0.687797 3.54859i 1.00000i 1.53664 + 2.37460i 3.53279 2.69099 1.66090i
29.16 −0.269098 + 1.38838i 2.55593 −1.85517 0.747219i 0.790907 2.09152i −0.687797 + 3.54859i 1.00000i 1.53664 2.37460i 3.53279 2.69099 + 1.66090i
29.17 −0.139020 1.40736i −0.319826 −1.96135 + 0.391304i −1.20181 + 1.88564i 0.0444622 + 0.450112i 1.00000i 0.823373 + 2.70593i −2.89771 2.82086 + 1.42925i
29.18 −0.139020 + 1.40736i −0.319826 −1.96135 0.391304i −1.20181 1.88564i 0.0444622 0.450112i 1.00000i 0.823373 2.70593i −2.89771 2.82086 1.42925i
29.19 0.139020 1.40736i 0.319826 −1.96135 0.391304i 1.20181 1.88564i 0.0444622 0.450112i 1.00000i −0.823373 + 2.70593i −2.89771 −2.48671 1.95353i
29.20 0.139020 + 1.40736i 0.319826 −1.96135 + 0.391304i 1.20181 + 1.88564i 0.0444622 + 0.450112i 1.00000i −0.823373 2.70593i −2.89771 −2.48671 + 1.95353i
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 29.36 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
8.b even 2 1 inner
40.f even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.l.a 36
4.b odd 2 1 1120.2.l.a 36
5.b even 2 1 inner 280.2.l.a 36
8.b even 2 1 inner 280.2.l.a 36
8.d odd 2 1 1120.2.l.a 36
20.d odd 2 1 1120.2.l.a 36
40.e odd 2 1 1120.2.l.a 36
40.f even 2 1 inner 280.2.l.a 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.l.a 36 1.a even 1 1 trivial
280.2.l.a 36 5.b even 2 1 inner
280.2.l.a 36 8.b even 2 1 inner
280.2.l.a 36 40.f even 2 1 inner
1120.2.l.a 36 4.b odd 2 1
1120.2.l.a 36 8.d odd 2 1
1120.2.l.a 36 20.d odd 2 1
1120.2.l.a 36 40.e odd 2 1

## Hecke kernels

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