# Properties

 Label 1120.3.c.g Level $1120$ Weight $3$ Character orbit 1120.c Analytic conductor $30.518$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1120 = 2^{5} \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 1120.c (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5177896084$$ Analytic rank: $$0$$ Dimension: $$80$$ Twist minimal: no (minimal twist has level 280) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 224q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 224q^{9} + 72q^{15} - 104q^{25} + 112q^{39} + 192q^{49} + 472q^{65} - 800q^{71} - 480q^{79} - 896q^{81} - 1176q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
209.1 0 3.68915i 0 −0.693644 4.95165i 0 4.51720 5.34742i 0 −4.60986 0
209.2 0 3.68915i 0 −0.693644 + 4.95165i 0 4.51720 + 5.34742i 0 −4.60986 0
209.3 0 1.25300i 0 3.15168 3.88161i 0 −5.63086 + 4.15854i 0 7.43000 0
209.4 0 1.25300i 0 3.15168 + 3.88161i 0 −5.63086 4.15854i 0 7.43000 0
209.5 0 3.68915i 0 0.693644 4.95165i 0 −4.51720 + 5.34742i 0 −4.60986 0
209.6 0 3.68915i 0 0.693644 + 4.95165i 0 −4.51720 5.34742i 0 −4.60986 0
209.7 0 3.23212i 0 −3.82711 3.21764i 0 6.97525 + 0.588096i 0 −1.44659 0
209.8 0 3.23212i 0 −3.82711 + 3.21764i 0 6.97525 0.588096i 0 −1.44659 0
209.9 0 3.23212i 0 3.82711 3.21764i 0 6.97525 0.588096i 0 −1.44659 0
209.10 0 3.23212i 0 3.82711 + 3.21764i 0 6.97525 + 0.588096i 0 −1.44659 0
209.11 0 2.74983i 0 −4.72637 + 1.63138i 0 1.41172 6.85617i 0 1.43843 0
209.12 0 2.74983i 0 −4.72637 1.63138i 0 1.41172 + 6.85617i 0 1.43843 0
209.13 0 3.68915i 0 0.693644 4.95165i 0 4.51720 + 5.34742i 0 −4.60986 0
209.14 0 3.68915i 0 0.693644 + 4.95165i 0 4.51720 5.34742i 0 −4.60986 0
209.15 0 2.74983i 0 4.72637 + 1.63138i 0 −1.41172 + 6.85617i 0 1.43843 0
209.16 0 2.74983i 0 4.72637 1.63138i 0 −1.41172 6.85617i 0 1.43843 0
209.17 0 1.25300i 0 3.15168 3.88161i 0 5.63086 + 4.15854i 0 7.43000 0
209.18 0 1.25300i 0 3.15168 + 3.88161i 0 5.63086 4.15854i 0 7.43000 0
209.19 0 5.26463i 0 4.99643 0.188883i 0 6.12587 + 3.38729i 0 −18.7163 0
209.20 0 5.26463i 0 4.99643 + 0.188883i 0 6.12587 3.38729i 0 −18.7163 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 209.80 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.b even 2 1 inner
35.c odd 2 1 inner
40.f even 2 1 inner
56.h odd 2 1 inner
280.c odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1120.3.c.g 80
4.b odd 2 1 280.3.c.g 80
5.b even 2 1 inner 1120.3.c.g 80
7.b odd 2 1 inner 1120.3.c.g 80
8.b even 2 1 inner 1120.3.c.g 80
8.d odd 2 1 280.3.c.g 80
20.d odd 2 1 280.3.c.g 80
28.d even 2 1 280.3.c.g 80
35.c odd 2 1 inner 1120.3.c.g 80
40.e odd 2 1 280.3.c.g 80
40.f even 2 1 inner 1120.3.c.g 80
56.e even 2 1 280.3.c.g 80
56.h odd 2 1 inner 1120.3.c.g 80
140.c even 2 1 280.3.c.g 80
280.c odd 2 1 inner 1120.3.c.g 80
280.n even 2 1 280.3.c.g 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
280.3.c.g 80 4.b odd 2 1
280.3.c.g 80 8.d odd 2 1
280.3.c.g 80 20.d odd 2 1
280.3.c.g 80 28.d even 2 1
280.3.c.g 80 40.e odd 2 1
280.3.c.g 80 56.e even 2 1
280.3.c.g 80 140.c even 2 1
280.3.c.g 80 280.n even 2 1
1120.3.c.g 80 1.a even 1 1 trivial
1120.3.c.g 80 5.b even 2 1 inner
1120.3.c.g 80 7.b odd 2 1 inner
1120.3.c.g 80 8.b even 2 1 inner
1120.3.c.g 80 35.c odd 2 1 inner
1120.3.c.g 80 40.f even 2 1 inner
1120.3.c.g 80 56.h odd 2 1 inner
1120.3.c.g 80 280.c odd 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{3}^{\mathrm{new}}(1120, [\chi])$$:

 $$T_{3}^{20} + \cdots$$ $$19\!\cdots\!29$$$$T_{17}^{12} -$$$$64\!\cdots\!40$$$$T_{17}^{10} +$$$$13\!\cdots\!28$$$$T_{17}^{8} -$$$$17\!\cdots\!20$$$$T_{17}^{6} +$$$$12\!\cdots\!84$$$$T_{17}^{4} -$$$$36\!\cdots\!40$$$$T_{17}^{2} +$$$$11\!\cdots\!00$$">$$T_{17}^{20} - \cdots$$ $$11\!\cdots\!52$$$$T_{19}^{12} -$$$$25\!\cdots\!04$$$$T_{19}^{10} +$$$$32\!\cdots\!20$$$$T_{19}^{8} -$$$$22\!\cdots\!24$$$$T_{19}^{6} +$$$$85\!\cdots\!08$$$$T_{19}^{4} -$$$$16\!\cdots\!12$$$$T_{19}^{2} +$$$$10\!\cdots\!80$$">$$T_{19}^{20} - \cdots$$