# Properties

 Label 1120.2.bq.b Level $1120$ Weight $2$ Character orbit 1120.bq Analytic conductor $8.943$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1120.bq (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$8.94324502638$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 280) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 52q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 52q^{9} + 48q^{19} - 22q^{25} + 22q^{35} + 16q^{49} - 20q^{51} + 60q^{59} - 12q^{65} + 6q^{75} - 36q^{89} - 72q^{91} - 104q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
719.1 0 −1.60233 2.77531i 0 2.12896 + 0.683763i 0 −2.23600 + 1.41432i 0 −3.63491 + 6.29586i 0
719.2 0 −1.60233 2.77531i 0 −2.12896 0.683763i 0 2.23600 1.41432i 0 −3.63491 + 6.29586i 0
719.3 0 −1.46528 2.53794i 0 −0.852425 + 2.06721i 0 1.45426 + 2.21023i 0 −2.79408 + 4.83948i 0
719.4 0 −1.46528 2.53794i 0 0.852425 2.06721i 0 −1.45426 2.21023i 0 −2.79408 + 4.83948i 0
719.5 0 −1.39824 2.42182i 0 −2.07888 + 0.823569i 0 −2.32367 1.26513i 0 −2.41015 + 4.17451i 0
719.6 0 −1.39824 2.42182i 0 2.07888 0.823569i 0 2.32367 + 1.26513i 0 −2.41015 + 4.17451i 0
719.7 0 −1.13535 1.96648i 0 0.406539 2.19880i 0 −0.551936 + 2.58754i 0 −1.07804 + 1.86722i 0
719.8 0 −1.13535 1.96648i 0 −0.406539 + 2.19880i 0 0.551936 2.58754i 0 −1.07804 + 1.86722i 0
719.9 0 −0.929162 1.60936i 0 −0.698124 2.12429i 0 1.10610 2.40345i 0 −0.226682 + 0.392626i 0
719.10 0 −0.929162 1.60936i 0 0.698124 + 2.12429i 0 −1.10610 + 2.40345i 0 −0.226682 + 0.392626i 0
719.11 0 −0.833565 1.44378i 0 0.660510 + 2.13629i 0 2.56833 0.635367i 0 0.110340 0.191114i 0
719.12 0 −0.833565 1.44378i 0 −0.660510 2.13629i 0 −2.56833 + 0.635367i 0 0.110340 0.191114i 0
719.13 0 −0.818911 1.41839i 0 −2.22815 + 0.187962i 0 −2.64158 0.148518i 0 0.158771 0.274999i 0
719.14 0 −0.818911 1.41839i 0 2.22815 0.187962i 0 2.64158 + 0.148518i 0 0.158771 0.274999i 0
719.15 0 −0.653375 1.13168i 0 1.67181 + 1.48495i 0 −1.03631 2.43435i 0 0.646203 1.11926i 0
719.16 0 −0.653375 1.13168i 0 −1.67181 1.48495i 0 1.03631 + 2.43435i 0 0.646203 1.11926i 0
719.17 0 −0.360276 0.624015i 0 −1.56432 + 1.59778i 0 −1.35637 + 2.27162i 0 1.24040 2.14844i 0
719.18 0 −0.360276 0.624015i 0 1.56432 1.59778i 0 1.35637 2.27162i 0 1.24040 2.14844i 0
719.19 0 −0.0769760 0.133326i 0 −1.85193 + 1.25314i 0 2.33880 + 1.23694i 0 1.48815 2.57755i 0
719.20 0 −0.0769760 0.133326i 0 1.85193 1.25314i 0 −2.33880 1.23694i 0 1.48815 2.57755i 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1039.40 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 1120.2.bq.b 80
4.b odd 2 1 280.2.ba.b 80
5.b even 2 1 inner 1120.2.bq.b 80
7.d odd 6 1 inner 1120.2.bq.b 80
8.b even 2 1 280.2.ba.b 80
8.d odd 2 1 inner 1120.2.bq.b 80
20.d odd 2 1 280.2.ba.b 80
28.f even 6 1 280.2.ba.b 80
35.i odd 6 1 inner 1120.2.bq.b 80
40.e odd 2 1 inner 1120.2.bq.b 80
40.f even 2 1 280.2.ba.b 80
56.j odd 6 1 280.2.ba.b 80
56.m even 6 1 inner 1120.2.bq.b 80
140.s even 6 1 280.2.ba.b 80
280.ba even 6 1 inner 1120.2.bq.b 80
280.bk odd 6 1 280.2.ba.b 80

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{3}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(1120, [\chi])$$.