# Properties

 Label 280.2.s.c Level $280$ Weight $2$ Character orbit 280.s Analytic conductor $2.236$ Analytic rank $0$ Dimension $72$ 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.s (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 4q^{2} - 4q^{7} - 16q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 4q^{2} - 4q^{7} - 16q^{8} - 8q^{15} + 40q^{16} - 40q^{18} + 4q^{22} + 40q^{23} - 8q^{25} - 4q^{28} - 44q^{30} - 24q^{32} - 72q^{36} + 40q^{42} + 16q^{46} - 60q^{50} + 32q^{56} + 32q^{57} - 68q^{58} + 64q^{60} - 100q^{63} - 56q^{65} + 16q^{70} - 80q^{71} + 16q^{72} - 4q^{78} + 72q^{81} - 72q^{86} + 48q^{88} + 72q^{92} + 80q^{95} + 40q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −1.39086 0.255964i −1.92461 1.92461i 1.86896 + 0.712018i 2.22709 + 0.200181i 2.18423 + 3.16949i −0.516406 + 2.59487i −2.41721 1.46870i 4.40827i −3.04632 0.848477i
13.2 −1.39086 0.255964i 1.92461 + 1.92461i 1.86896 + 0.712018i −2.22709 0.200181i −2.18423 3.16949i −2.59487 + 0.516406i −2.41721 1.46870i 4.40827i 3.04632 + 0.848477i
13.3 −1.39041 + 0.258373i −1.88603 1.88603i 1.86649 0.718491i −1.79374 1.33510i 3.10966 + 2.13506i −2.30054 1.30671i −2.40954 + 1.48125i 4.11423i 2.83899 + 1.39289i
13.4 −1.39041 + 0.258373i 1.88603 + 1.88603i 1.86649 0.718491i 1.79374 + 1.33510i −3.10966 2.13506i 1.30671 + 2.30054i −2.40954 + 1.48125i 4.11423i −2.83899 1.39289i
13.5 −1.37024 + 0.349904i −0.152811 0.152811i 1.75513 0.958907i −2.05351 + 0.884920i 0.262858 + 0.155920i 2.63870 + 0.192988i −2.06944 + 1.92806i 2.95330i 2.50418 1.93109i
13.6 −1.37024 + 0.349904i 0.152811 + 0.152811i 1.75513 0.958907i 2.05351 0.884920i −0.262858 0.155920i −0.192988 2.63870i −2.06944 + 1.92806i 2.95330i −2.50418 + 1.93109i
13.7 −1.25643 0.649134i −0.639680 0.639680i 1.15725 + 1.63119i −1.45663 + 1.69653i 0.388478 + 1.21895i 0.0280541 + 2.64560i −0.395151 2.80069i 2.18162i 2.93144 1.18603i
13.8 −1.25643 0.649134i 0.639680 + 0.639680i 1.15725 + 1.63119i 1.45663 1.69653i −0.388478 1.21895i −2.64560 0.0280541i −0.395151 2.80069i 2.18162i −2.93144 + 1.18603i
13.9 −1.19418 0.757577i −1.10062 1.10062i 0.852154 + 1.80937i −0.557471 2.16546i 0.480540 + 2.14815i 2.20653 1.45987i 0.353110 2.80630i 0.577263i −0.974781 + 3.00829i
13.10 −1.19418 0.757577i 1.10062 + 1.10062i 0.852154 + 1.80937i 0.557471 + 2.16546i −0.480540 2.14815i 1.45987 2.20653i 0.353110 2.80630i 0.577263i 0.974781 3.00829i
13.11 −0.788440 1.17404i −1.96562 1.96562i −0.756725 + 1.85131i −0.686018 + 2.12823i −0.757936 + 3.85748i −1.91879 1.82161i 2.77014 0.571227i 4.72730i 3.03951 0.872574i
13.12 −0.788440 1.17404i 1.96562 + 1.96562i −0.756725 + 1.85131i 0.686018 2.12823i 0.757936 3.85748i 1.82161 + 1.91879i 2.77014 0.571227i 4.72730i −3.03951 + 0.872574i
13.13 −0.482209 1.32946i −0.851048 0.851048i −1.53495 + 1.28216i 2.18999 + 0.451591i −0.721055 + 1.54182i 2.50752 + 0.844017i 2.44475 + 1.42239i 1.55144i −0.455660 3.12928i
13.14 −0.482209 1.32946i 0.851048 + 0.851048i −1.53495 + 1.28216i −2.18999 0.451591i 0.721055 1.54182i −0.844017 2.50752i 2.44475 + 1.42239i 1.55144i 0.455660 + 3.12928i
13.15 −0.349904 + 1.37024i −0.152811 0.152811i −1.75513 0.958907i −2.05351 + 0.884920i 0.262858 0.155920i −0.192988 2.63870i 1.92806 2.06944i 2.95330i −0.494024 3.12345i
13.16 −0.349904 + 1.37024i 0.152811 + 0.152811i −1.75513 0.958907i 2.05351 0.884920i −0.262858 + 0.155920i 2.63870 + 0.192988i 1.92806 2.06944i 2.95330i 0.494024 + 3.12345i
13.17 −0.258373 + 1.39041i −1.88603 1.88603i −1.86649 0.718491i −1.79374 1.33510i 3.10966 2.13506i 1.30671 + 2.30054i 1.48125 2.40954i 4.11423i 2.31980 2.14908i
13.18 −0.258373 + 1.39041i 1.88603 + 1.88603i −1.86649 0.718491i 1.79374 + 1.33510i −3.10966 + 2.13506i −2.30054 1.30671i 1.48125 2.40954i 4.11423i −2.31980 + 2.14908i
13.19 0.00402547 1.41421i −1.09655 1.09655i −1.99997 0.0113857i 0.721255 2.11655i −1.55517 + 1.54634i −2.63247 + 0.264726i −0.0241525 + 2.82832i 0.595144i −2.99034 1.02852i
13.20 0.00402547 1.41421i 1.09655 + 1.09655i −1.99997 0.0113857i −0.721255 + 2.11655i 1.55517 1.54634i −0.264726 + 2.63247i −0.0241525 + 2.82832i 0.595144i 2.99034 + 1.02852i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 237.36 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.c odd 4 1 inner
7.b odd 2 1 inner
8.b even 2 1 inner
35.f even 4 1 inner
40.i odd 4 1 inner
56.h odd 2 1 inner
280.s even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 280.2.s.c 72
4.b odd 2 1 1120.2.w.c 72
5.c odd 4 1 inner 280.2.s.c 72
7.b odd 2 1 inner 280.2.s.c 72
8.b even 2 1 inner 280.2.s.c 72
8.d odd 2 1 1120.2.w.c 72
20.e even 4 1 1120.2.w.c 72
28.d even 2 1 1120.2.w.c 72
35.f even 4 1 inner 280.2.s.c 72
40.i odd 4 1 inner 280.2.s.c 72
40.k even 4 1 1120.2.w.c 72
56.e even 2 1 1120.2.w.c 72
56.h odd 2 1 inner 280.2.s.c 72
140.j odd 4 1 1120.2.w.c 72
280.s even 4 1 inner 280.2.s.c 72
280.y odd 4 1 1120.2.w.c 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.s.c 72 1.a even 1 1 trivial
280.2.s.c 72 5.c odd 4 1 inner
280.2.s.c 72 7.b odd 2 1 inner
280.2.s.c 72 8.b even 2 1 inner
280.2.s.c 72 35.f even 4 1 inner
280.2.s.c 72 40.i odd 4 1 inner
280.2.s.c 72 56.h odd 2 1 inner
280.2.s.c 72 280.s even 4 1 inner
1120.2.w.c 72 4.b odd 2 1
1120.2.w.c 72 8.d odd 2 1
1120.2.w.c 72 20.e even 4 1
1120.2.w.c 72 28.d even 2 1
1120.2.w.c 72 40.k even 4 1
1120.2.w.c 72 56.e even 2 1
1120.2.w.c 72 140.j odd 4 1
1120.2.w.c 72 280.y odd 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{36} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.