# Properties

 Label 720.2.u.a Level $720$ Weight $2$ Character orbit 720.u Analytic conductor $5.749$ Analytic rank $0$ Dimension $96$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.u (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$5.74922894553$$ Analytic rank: $$0$$ Dimension: $$96$$ Relative dimension: $$48$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$96q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$96q - 8q^{16} - 16q^{19} + 72q^{34} + 8q^{40} + 8q^{46} - 96q^{49} + 64q^{55} - 32q^{61} + 48q^{64} + 24q^{70} + 40q^{76} - 88q^{94} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
179.1 −1.41373 + 0.0371259i 0 1.99724 0.104972i −1.81012 1.31281i 0 1.40695i −2.81966 + 0.222551i 0 2.60775 + 1.78875i
179.2 −1.41373 + 0.0371259i 0 1.99724 0.104972i 1.31281 + 1.81012i 0 1.40695i −2.81966 + 0.222551i 0 −1.92316 2.51027i
179.3 −1.39867 + 0.209116i 0 1.91254 0.584968i 1.41522 1.73123i 0 3.80565i −2.55268 + 1.21812i 0 −1.61739 + 2.71736i
179.4 −1.39867 + 0.209116i 0 1.91254 0.584968i 1.73123 1.41522i 0 3.80565i −2.55268 + 1.21812i 0 −2.12547 + 2.34145i
179.5 −1.31018 0.532382i 0 1.43314 + 1.39503i −1.50698 1.65197i 0 4.30751i −1.13498 2.59072i 0 1.09494 + 2.96667i
179.6 −1.31018 0.532382i 0 1.43314 + 1.39503i 1.65197 + 1.50698i 0 4.30751i −1.13498 2.59072i 0 −1.36208 2.85390i
179.7 −1.23687 + 0.685679i 0 1.05969 1.69619i −2.19631 + 0.419806i 0 0.263783i −0.147653 + 2.82457i 0 2.42869 2.02521i
179.8 −1.23687 + 0.685679i 0 1.05969 1.69619i −0.419806 + 2.19631i 0 0.263783i −0.147653 + 2.82457i 0 −0.986717 3.00440i
179.9 −1.22941 0.698958i 0 1.02291 + 1.71862i −1.91479 + 1.15480i 0 3.02955i −0.0563436 2.82787i 0 3.16123 0.0813664i
179.10 −1.22941 0.698958i 0 1.02291 + 1.71862i −1.15480 + 1.91479i 0 3.02955i −0.0563436 2.82787i 0 2.75809 1.54692i
179.11 −1.04776 0.949842i 0 0.195601 + 1.99041i 0.469629 2.18619i 0 2.94937i 1.68563 2.27126i 0 −2.56860 + 1.84453i
179.12 −1.04776 0.949842i 0 0.195601 + 1.99041i 2.18619 0.469629i 0 2.94937i 1.68563 2.27126i 0 −2.73668 1.58448i
179.13 −0.956592 + 1.04160i 0 −0.169864 1.99277i 1.11222 1.93984i 0 1.78786i 2.23816 + 1.72934i 0 0.956594 + 3.01412i
179.14 −0.956592 + 1.04160i 0 −0.169864 1.99277i 1.93984 1.11222i 0 1.78786i 2.23816 + 1.72934i 0 −0.697143 + 3.08448i
179.15 −0.638599 1.26182i 0 −1.18438 + 1.61160i −2.14049 0.646750i 0 0.594230i 2.78989 + 0.465314i 0 0.550835 + 3.11393i
179.16 −0.638599 1.26182i 0 −1.18438 + 1.61160i 0.646750 + 2.14049i 0 0.594230i 2.78989 + 0.465314i 0 2.28790 2.18300i
179.17 −0.612434 1.27473i 0 −1.24985 + 1.56137i −0.263678 2.22047i 0 2.95946i 2.75577 + 0.636981i 0 −2.66900 + 1.69601i
179.18 −0.612434 1.27473i 0 −1.24985 + 1.56137i 2.22047 + 0.263678i 0 2.95946i 2.75577 + 0.636981i 0 −1.02377 2.99197i
179.19 −0.581659 + 1.28906i 0 −1.32335 1.49958i −2.23581 0.0342493i 0 4.97879i 2.70279 0.833625i 0 1.34463 2.86216i
179.20 −0.581659 + 1.28906i 0 −1.32335 1.49958i 0.0342493 + 2.23581i 0 4.97879i 2.70279 0.833625i 0 −2.90201 1.25633i
See all 96 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 539.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner
16.f odd 4 1 inner
48.k even 4 1 inner
80.k odd 4 1 inner
240.t even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 720.2.u.a 96
3.b odd 2 1 inner 720.2.u.a 96
4.b odd 2 1 2880.2.u.a 96
5.b even 2 1 inner 720.2.u.a 96
12.b even 2 1 2880.2.u.a 96
15.d odd 2 1 inner 720.2.u.a 96
16.e even 4 1 2880.2.u.a 96
16.f odd 4 1 inner 720.2.u.a 96
20.d odd 2 1 2880.2.u.a 96
48.i odd 4 1 2880.2.u.a 96
48.k even 4 1 inner 720.2.u.a 96
60.h even 2 1 2880.2.u.a 96
80.k odd 4 1 inner 720.2.u.a 96
80.q even 4 1 2880.2.u.a 96
240.t even 4 1 inner 720.2.u.a 96
240.bm odd 4 1 2880.2.u.a 96

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
720.2.u.a 96 1.a even 1 1 trivial
720.2.u.a 96 3.b odd 2 1 inner
720.2.u.a 96 5.b even 2 1 inner
720.2.u.a 96 15.d odd 2 1 inner
720.2.u.a 96 16.f odd 4 1 inner
720.2.u.a 96 48.k even 4 1 inner
720.2.u.a 96 80.k odd 4 1 inner
720.2.u.a 96 240.t even 4 1 inner
2880.2.u.a 96 4.b odd 2 1
2880.2.u.a 96 12.b even 2 1
2880.2.u.a 96 16.e even 4 1
2880.2.u.a 96 20.d odd 2 1
2880.2.u.a 96 48.i odd 4 1
2880.2.u.a 96 60.h even 2 1
2880.2.u.a 96 80.q even 4 1
2880.2.u.a 96 240.bm odd 4 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(720, [\chi])$$.