# Properties

 Label 900.3.y.a Level $900$ Weight $3$ Character orbit 900.y Analytic conductor $24.523$ Analytic rank $0$ Dimension $80$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.y (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$20$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 60q^{19} + 56q^{25} - 120q^{31} + 20q^{37} - 680q^{49} - 56q^{55} - 80q^{61} - 280q^{67} - 360q^{73} + 40q^{79} + 192q^{85} + 140q^{91} - 40q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
89.1 0 0 0 −4.98948 + 0.324198i 0 11.3476i 0 0 0
89.2 0 0 0 −4.98426 0.396382i 0 2.05133i 0 0 0
89.3 0 0 0 −4.98271 0.415486i 0 3.91927i 0 0 0
89.4 0 0 0 −4.08889 2.87767i 0 4.24698i 0 0 0
89.5 0 0 0 −3.94179 + 3.07609i 0 7.41454i 0 0 0
89.6 0 0 0 −2.56782 4.29026i 0 2.96738i 0 0 0
89.7 0 0 0 −2.55393 + 4.29854i 0 5.96546i 0 0 0
89.8 0 0 0 −1.41055 + 4.79691i 0 7.51506i 0 0 0
89.9 0 0 0 −0.741250 4.94475i 0 10.0354i 0 0 0
89.10 0 0 0 −0.657760 + 4.95655i 0 12.7771i 0 0 0
89.11 0 0 0 0.657760 4.95655i 0 12.7771i 0 0 0
89.12 0 0 0 0.741250 + 4.94475i 0 10.0354i 0 0 0
89.13 0 0 0 1.41055 4.79691i 0 7.51506i 0 0 0
89.14 0 0 0 2.55393 4.29854i 0 5.96546i 0 0 0
89.15 0 0 0 2.56782 + 4.29026i 0 2.96738i 0 0 0
89.16 0 0 0 3.94179 3.07609i 0 7.41454i 0 0 0
89.17 0 0 0 4.08889 + 2.87767i 0 4.24698i 0 0 0
89.18 0 0 0 4.98271 + 0.415486i 0 3.91927i 0 0 0
89.19 0 0 0 4.98426 + 0.396382i 0 2.05133i 0 0 0
89.20 0 0 0 4.98948 0.324198i 0 11.3476i 0 0 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 809.20 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
25.e even 10 1 inner
75.h odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.y.a 80
3.b odd 2 1 inner 900.3.y.a 80
25.e even 10 1 inner 900.3.y.a 80
75.h odd 10 1 inner 900.3.y.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.y.a 80 1.a even 1 1 trivial
900.3.y.a 80 3.b odd 2 1 inner
900.3.y.a 80 25.e even 10 1 inner
900.3.y.a 80 75.h odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(900, [\chi])$$.