# Properties

 Label 280.2.ba.b Level $280$ Weight $2$ Character orbit 280.ba Analytic conductor $2.236$ Analytic rank $0$ Dimension $80$ 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.ba (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$2.23581125660$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 6q^{4} - 52q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 6q^{4} - 52q^{9} - 12q^{10} - 6q^{14} + 10q^{16} - 48q^{19} + 12q^{24} - 22q^{25} + 54q^{26} - 20q^{30} - 22q^{35} + 28q^{36} + 12q^{40} + 38q^{44} + 14q^{46} + 16q^{49} - 40q^{50} + 20q^{51} + 60q^{54} - 104q^{56} - 60q^{59} - 42q^{60} - 60q^{64} - 12q^{65} + 84q^{66} - 22q^{70} + 34q^{74} - 6q^{75} - 96q^{80} - 100q^{84} - 8q^{86} - 36q^{89} + 72q^{91} - 66q^{94} + 96q^{96} + 104q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
19.1 −1.41178 + 0.0828556i 0.653375 + 1.13168i 1.98627 0.233948i 1.67181 + 1.48495i −1.01619 1.54355i 1.03631 + 2.43435i −2.78480 + 0.494858i 0.646203 1.11926i −2.48327 1.95791i
19.2 −1.39533 + 0.230335i −0.929162 1.60936i 1.89389 0.642786i 2.18875 0.457553i 1.66718 + 2.03156i 1.10610 2.40345i −2.49455 + 1.33313i −0.226682 + 0.392626i −2.94864 + 1.14258i
19.3 −1.38731 0.274535i 1.60233 + 2.77531i 1.84926 + 0.761731i −2.12896 0.683763i −1.46101 4.29012i −2.23600 + 1.41432i −2.35638 1.56444i −3.63491 + 6.29586i 2.76581 + 1.53307i
19.4 −1.36769 + 0.359747i −1.46528 2.53794i 1.74116 0.984045i −1.36405 + 1.77183i 2.91706 + 2.94399i 1.45426 + 2.21023i −2.02737 + 1.97225i −2.79408 + 4.83948i 1.22819 2.91403i
19.5 −1.32026 0.506870i 0.360276 + 0.624015i 1.48617 + 1.33840i −1.56432 + 1.59778i −0.159362 1.00647i 1.35637 2.27162i −1.28373 2.52033i 1.24040 2.14844i 2.87518 1.31657i
19.6 −1.30291 + 0.549923i 0.833565 + 1.44378i 1.39517 1.43300i −0.660510 2.13629i −1.88003 1.42272i 2.56833 0.635367i −1.02974 + 2.63432i 0.110340 0.191114i 2.03538 + 2.42017i
19.7 −1.24532 0.670206i −0.818911 1.41839i 1.10165 + 1.66924i 0.951297 + 2.02362i 0.0691894 + 2.31520i −2.64158 0.148518i −0.253167 2.81707i 0.158771 0.274999i 0.171572 3.15762i
19.8 −1.21328 + 0.726608i 0.0769760 + 0.133326i 0.944081 1.76315i −1.85193 + 1.25314i −0.190269 0.105830i −2.33880 1.23694i 0.135691 + 2.82517i 1.48815 2.57755i 1.33636 2.86603i
19.9 −1.20308 0.743376i −0.818911 1.41839i 0.894783 + 1.78868i −0.951297 2.02362i −0.0691894 + 2.31520i 2.64158 + 0.148518i 0.253167 2.81707i 0.158771 0.274999i −0.359828 + 3.14174i
19.10 −1.09909 0.889943i 0.360276 + 0.624015i 0.416004 + 1.95626i 1.56432 1.59778i 0.159362 1.00647i −1.35637 + 2.27162i 1.28373 2.52033i 1.24040 2.14844i −3.14126 + 0.363945i
19.11 −1.02807 + 0.971124i 1.39824 + 2.42182i 0.113837 1.99676i 2.07888 0.823569i −3.78937 1.13193i −2.32367 1.26513i 1.82207 + 2.16335i −2.41015 + 4.17451i −1.33744 + 2.86553i
19.12 −0.931410 1.06418i 1.60233 + 2.77531i −0.264952 + 1.98237i 2.12896 + 0.683763i 1.46101 4.29012i 2.23600 1.41432i 2.35638 1.56444i −3.63491 + 6.29586i −1.25529 2.90246i
19.13 −0.881474 + 1.10590i −1.13535 1.96648i −0.446008 1.94964i 1.70095 1.45147i 3.17551 + 0.477826i −0.551936 + 2.58754i 2.54924 + 1.22531i −1.07804 + 1.86722i 0.105836 + 3.16051i
19.14 −0.634137 1.26407i 0.653375 + 1.13168i −1.19574 + 1.60319i −1.67181 1.48495i 1.01619 1.54355i −1.03631 2.43435i 2.78480 + 0.494858i 0.646203 1.11926i −0.816920 + 3.05494i
19.15 −0.516996 + 1.31633i 1.13535 + 1.96648i −1.46543 1.36107i −0.406539 + 2.19880i −3.17551 + 0.477826i −0.551936 + 2.58754i 2.54924 1.22531i −1.07804 + 1.86722i −2.68416 1.67191i
19.16 −0.498189 1.32356i −0.929162 1.60936i −1.50361 + 1.31877i −2.18875 + 0.457553i −1.66718 + 2.03156i −1.10610 + 2.40345i 2.49455 + 1.33313i −0.226682 + 0.392626i 1.69601 + 2.66900i
19.17 −0.372297 1.36433i −1.46528 2.53794i −1.72279 + 1.01587i 1.36405 1.77183i −2.91706 + 2.94399i −1.45426 2.21023i 2.02737 + 1.97225i −2.79408 + 4.83948i −2.92519 1.20136i
19.18 −0.326985 + 1.37589i −1.39824 2.42182i −1.78616 0.899793i 0.326208 + 2.21215i 3.78937 1.13193i −2.32367 1.26513i 1.82207 2.16335i −2.41015 + 4.17451i −3.15034 0.274512i
19.19 −0.175210 1.40332i 0.833565 + 1.44378i −1.93860 + 0.491749i 0.660510 + 2.13629i 1.88003 1.42272i −2.56833 + 0.635367i 1.02974 + 2.63432i 0.110340 0.191114i 2.88216 1.30120i
19.20 −0.0226228 + 1.41403i −0.0769760 0.133326i −1.99898 0.0639789i 0.159286 2.23039i 0.190269 0.105830i −2.33880 1.23694i 0.135691 2.82517i 1.48815 2.57755i 3.15024 + 0.275693i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 59.40 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.d odd 6 1 inner
8.d odd 2 1 inner
35.i odd 6 1 inner
40.e odd 2 1 inner
56.m even 6 1 inner
280.ba 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.ba.b 80
4.b odd 2 1 1120.2.bq.b 80
5.b even 2 1 inner 280.2.ba.b 80
7.d odd 6 1 inner 280.2.ba.b 80
8.b even 2 1 1120.2.bq.b 80
8.d odd 2 1 inner 280.2.ba.b 80
20.d odd 2 1 1120.2.bq.b 80
28.f even 6 1 1120.2.bq.b 80
35.i odd 6 1 inner 280.2.ba.b 80
40.e odd 2 1 inner 280.2.ba.b 80
40.f even 2 1 1120.2.bq.b 80
56.j odd 6 1 1120.2.bq.b 80
56.m even 6 1 inner 280.2.ba.b 80
140.s even 6 1 1120.2.bq.b 80
280.ba even 6 1 inner 280.2.ba.b 80
280.bk odd 6 1 1120.2.bq.b 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.2.ba.b 80 1.a even 1 1 trivial
280.2.ba.b 80 5.b even 2 1 inner
280.2.ba.b 80 7.d odd 6 1 inner
280.2.ba.b 80 8.d odd 2 1 inner
280.2.ba.b 80 35.i odd 6 1 inner
280.2.ba.b 80 40.e odd 2 1 inner
280.2.ba.b 80 56.m even 6 1 inner
280.2.ba.b 80 280.ba even 6 1 inner
1120.2.bq.b 80 4.b odd 2 1
1120.2.bq.b 80 8.b even 2 1
1120.2.bq.b 80 20.d odd 2 1
1120.2.bq.b 80 28.f even 6 1
1120.2.bq.b 80 40.f even 2 1
1120.2.bq.b 80 56.j odd 6 1
1120.2.bq.b 80 140.s even 6 1
1120.2.bq.b 80 280.bk odd 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.