# Properties

 Label 280.2.n.b Level $280$ Weight $2$ Character orbit 280.n Analytic conductor $2.236$ Analytic rank $0$ Dimension $40$ 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.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

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

 $$\operatorname{Tr}(f)(q) =$$ $$40q - 12q^{4} + 40q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q - 12q^{4} + 40q^{9} - 12q^{14} - 28q^{16} + 16q^{25} - 28q^{30} + 16q^{35} - 28q^{36} - 8q^{44} - 32q^{46} + 8q^{49} + 4q^{50} - 32q^{51} - 4q^{56} + 12q^{60} - 84q^{64} - 24q^{65} + 40q^{70} + 80q^{74} - 72q^{81} - 8q^{84} + 80q^{86} - 128q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
139.1 −1.27891 0.603655i −1.64952 1.27120 + 1.54404i −1.78913 1.34127i 2.10957 + 0.995738i −0.157160 + 2.64108i −0.693682 2.74204i −0.279099 1.47847 + 2.79538i
139.2 −1.27891 0.603655i 1.64952 1.27120 + 1.54404i 1.78913 + 1.34127i −2.10957 0.995738i −0.157160 2.64108i −0.693682 2.74204i −0.279099 −1.47847 2.79538i
139.3 −1.27891 + 0.603655i −1.64952 1.27120 1.54404i −1.78913 + 1.34127i 2.10957 0.995738i −0.157160 2.64108i −0.693682 + 2.74204i −0.279099 1.47847 2.79538i
139.4 −1.27891 + 0.603655i 1.64952 1.27120 1.54404i 1.78913 1.34127i −2.10957 + 0.995738i −0.157160 + 2.64108i −0.693682 + 2.74204i −0.279099 −1.47847 + 2.79538i
139.5 −1.20452 0.741038i −2.57969 0.901725 + 1.78519i −0.460102 + 2.18822i 3.10728 + 1.91165i 2.31487 1.28116i 0.236748 2.81850i 3.65479 2.17576 2.29480i
139.6 −1.20452 0.741038i 2.57969 0.901725 + 1.78519i 0.460102 2.18822i −3.10728 1.91165i 2.31487 + 1.28116i 0.236748 2.81850i 3.65479 −2.17576 + 2.29480i
139.7 −1.20452 + 0.741038i −2.57969 0.901725 1.78519i −0.460102 2.18822i 3.10728 1.91165i 2.31487 + 1.28116i 0.236748 + 2.81850i 3.65479 2.17576 + 2.29480i
139.8 −1.20452 + 0.741038i 2.57969 0.901725 1.78519i 0.460102 + 2.18822i −3.10728 + 1.91165i 2.31487 1.28116i 0.236748 + 2.81850i 3.65479 −2.17576 2.29480i
139.9 −0.918230 1.07557i −1.19038 −0.313707 + 1.97524i 2.22369 0.234978i 1.09304 + 1.28034i −2.11797 + 1.58562i 2.41257 1.47631i −1.58299 −2.29459 2.17597i
139.10 −0.918230 1.07557i 1.19038 −0.313707 + 1.97524i −2.22369 + 0.234978i −1.09304 1.28034i −2.11797 1.58562i 2.41257 1.47631i −1.58299 2.29459 + 2.17597i
139.11 −0.918230 + 1.07557i −1.19038 −0.313707 1.97524i 2.22369 + 0.234978i 1.09304 1.28034i −2.11797 1.58562i 2.41257 + 1.47631i −1.58299 −2.29459 + 2.17597i
139.12 −0.918230 + 1.07557i 1.19038 −0.313707 1.97524i −2.22369 0.234978i −1.09304 + 1.28034i −2.11797 + 1.58562i 2.41257 + 1.47631i −1.58299 2.29459 2.17597i
139.13 −0.510308 1.31893i −0.857983 −1.47917 + 1.34612i 1.33029 1.79731i 0.437835 + 1.13162i 2.41097 1.08959i 2.53028 + 1.26399i −2.26387 −3.04939 0.837376i
139.14 −0.510308 1.31893i 0.857983 −1.47917 + 1.34612i −1.33029 + 1.79731i −0.437835 1.13162i 2.41097 + 1.08959i 2.53028 + 1.26399i −2.26387 3.04939 + 0.837376i
139.15 −0.510308 + 1.31893i −0.857983 −1.47917 1.34612i 1.33029 + 1.79731i 0.437835 1.13162i 2.41097 + 1.08959i 2.53028 1.26399i −2.26387 −3.04939 + 0.837376i
139.16 −0.510308 + 1.31893i 0.857983 −1.47917 1.34612i −1.33029 1.79731i −0.437835 + 1.13162i 2.41097 1.08959i 2.53028 1.26399i −2.26387 3.04939 0.837376i
139.17 −0.244900 1.39285i −2.91053 −1.88005 + 0.682218i −1.83654 1.27559i 0.712789 + 4.05392i −1.52252 2.16378i 1.41065 + 2.45154i 5.47117 −1.32694 + 2.87041i
139.18 −0.244900 1.39285i 2.91053 −1.88005 + 0.682218i 1.83654 + 1.27559i −0.712789 4.05392i −1.52252 + 2.16378i 1.41065 + 2.45154i 5.47117 1.32694 2.87041i
139.19 −0.244900 + 1.39285i −2.91053 −1.88005 0.682218i −1.83654 + 1.27559i 0.712789 4.05392i −1.52252 + 2.16378i 1.41065 2.45154i 5.47117 −1.32694 2.87041i
139.20 −0.244900 + 1.39285i 2.91053 −1.88005 0.682218i 1.83654 1.27559i −0.712789 + 4.05392i −1.52252 2.16378i 1.41065 2.45154i 5.47117 1.32694 + 2.87041i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 139.40 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.b odd 2 1 inner
8.d odd 2 1 inner
35.c odd 2 1 inner
40.e odd 2 1 inner
56.e even 2 1 inner
280.n 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.n.b 40
4.b odd 2 1 1120.2.n.b 40
5.b even 2 1 inner 280.2.n.b 40
7.b odd 2 1 inner 280.2.n.b 40
8.b even 2 1 1120.2.n.b 40
8.d odd 2 1 inner 280.2.n.b 40
20.d odd 2 1 1120.2.n.b 40
28.d even 2 1 1120.2.n.b 40
35.c odd 2 1 inner 280.2.n.b 40
40.e odd 2 1 inner 280.2.n.b 40
40.f even 2 1 1120.2.n.b 40
56.e even 2 1 inner 280.2.n.b 40
56.h odd 2 1 1120.2.n.b 40
140.c even 2 1 1120.2.n.b 40
280.c odd 2 1 1120.2.n.b 40
280.n even 2 1 inner 280.2.n.b 40

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{10} - 20 T_{3}^{8} + 137 T_{3}^{6} - 382 T_{3}^{4} + 432 T_{3}^{2} - 160$$ acting on $$S_{2}^{\mathrm{new}}(280, [\chi])$$.