# Properties

 Label 900.3.ba.a Level $900$ Weight $3$ Character orbit 900.ba 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.ba (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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q + 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q + 16q^{7} - 8q^{13} + 60q^{19} - 120q^{25} + 120q^{31} + 116q^{37} - 80q^{43} + 440q^{49} + 120q^{55} + 80q^{61} + 24q^{67} + 128q^{73} + 40q^{79} + 40q^{85} - 140q^{91} + 384q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
161.1 0 0 0 −4.89744 + 1.00750i 0 0.208186 0 0 0
161.2 0 0 0 −4.41393 + 2.34888i 0 −11.9145 0 0 0
161.3 0 0 0 −4.19489 + 2.72083i 0 5.95985 0 0 0
161.4 0 0 0 −4.15240 2.78525i 0 −1.13968 0 0 0
161.5 0 0 0 −3.66894 3.39689i 0 −6.21282 0 0 0
161.6 0 0 0 −3.45827 + 3.61115i 0 7.44862 0 0 0
161.7 0 0 0 −2.65177 4.23888i 0 6.40455 0 0 0
161.8 0 0 0 −1.42516 4.79259i 0 −1.71989 0 0 0
161.9 0 0 0 −0.922718 + 4.91412i 0 9.63452 0 0 0
161.10 0 0 0 −0.709639 + 4.94939i 0 −11.1410 0 0 0
161.11 0 0 0 0.709639 4.94939i 0 −11.1410 0 0 0
161.12 0 0 0 0.922718 4.91412i 0 9.63452 0 0 0
161.13 0 0 0 1.42516 + 4.79259i 0 −1.71989 0 0 0
161.14 0 0 0 2.65177 + 4.23888i 0 6.40455 0 0 0
161.15 0 0 0 3.45827 3.61115i 0 7.44862 0 0 0
161.16 0 0 0 3.66894 + 3.39689i 0 −6.21282 0 0 0
161.17 0 0 0 4.15240 + 2.78525i 0 −1.13968 0 0 0
161.18 0 0 0 4.19489 2.72083i 0 5.95985 0 0 0
161.19 0 0 0 4.41393 2.34888i 0 −11.9145 0 0 0
161.20 0 0 0 4.89744 1.00750i 0 0.208186 0 0 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 881.20 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.d even 5 1 inner
75.j 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.ba.a 80
3.b odd 2 1 inner 900.3.ba.a 80
25.d even 5 1 inner 900.3.ba.a 80
75.j odd 10 1 inner 900.3.ba.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.ba.a 80 1.a even 1 1 trivial
900.3.ba.a 80 3.b odd 2 1 inner
900.3.ba.a 80 25.d even 5 1 inner
900.3.ba.a 80 75.j odd 10 1 inner

## Hecke kernels

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