# Properties

 Label 2052.3.bl.a Level $2052$ Weight $3$ Character orbit 2052.bl Analytic conductor $55.913$ Analytic rank $0$ Dimension $80$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2052 = 2^{2} \cdot 3^{3} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 2052.bl (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$55.9129502467$$ Analytic rank: $$0$$ Dimension: $$80$$ Relative dimension: $$40$$ over $$\Q(\zeta_{6})$$ Twist minimal: no (minimal twist has level 684) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q - q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80 q - q^{7} + 6 q^{11} - 15 q^{13} + 21 q^{17} - 20 q^{19} - 24 q^{23} + 400 q^{25} + 24 q^{31} + 54 q^{35} + 76 q^{43} - 24 q^{47} - 267 q^{49} + 36 q^{53} + 14 q^{61} - 288 q^{65} - 21 q^{67} + 81 q^{71} + 55 q^{73} - 30 q^{77} - 51 q^{79} + 93 q^{83} - 216 q^{89} + 96 q^{91} + 432 q^{95} + 90 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1 0 0 0 −8.90430 0 −2.28143 3.95155i 0 0 0
145.2 0 0 0 −8.37687 0 −4.50755 7.80731i 0 0 0
145.3 0 0 0 −8.35221 0 0.352343 + 0.610277i 0 0 0
145.4 0 0 0 −7.87469 0 2.97107 + 5.14604i 0 0 0
145.5 0 0 0 −6.99525 0 0.133800 + 0.231748i 0 0 0
145.6 0 0 0 −6.81841 0 3.70019 + 6.40891i 0 0 0
145.7 0 0 0 −6.03262 0 −3.98752 6.90658i 0 0 0
145.8 0 0 0 −5.96221 0 6.83689 + 11.8418i 0 0 0
145.9 0 0 0 −5.52048 0 0.838417 + 1.45218i 0 0 0
145.10 0 0 0 −4.76547 0 3.51868 + 6.09454i 0 0 0
145.11 0 0 0 −4.14557 0 −5.99339 10.3809i 0 0 0
145.12 0 0 0 −4.10591 0 −6.48935 11.2399i 0 0 0
145.13 0 0 0 −3.96127 0 2.79341 + 4.83834i 0 0 0
145.14 0 0 0 −3.77174 0 −0.501515 0.868649i 0 0 0
145.15 0 0 0 −3.57346 0 2.11933 + 3.67079i 0 0 0
145.16 0 0 0 −3.05736 0 −5.58622 9.67561i 0 0 0
145.17 0 0 0 −1.84061 0 3.31490 + 5.74157i 0 0 0
145.18 0 0 0 −0.638534 0 −1.11619 1.93329i 0 0 0
145.19 0 0 0 −0.387901 0 5.33508 + 9.24062i 0 0 0
145.20 0 0 0 −0.356865 0 −3.52002 6.09686i 0 0 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1585.40 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.s odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2052.3.bl.a 80
3.b odd 2 1 684.3.bl.a yes 80
9.c even 3 1 2052.3.s.a 80
9.d odd 6 1 684.3.s.a 80
19.d odd 6 1 2052.3.s.a 80
57.f even 6 1 684.3.s.a 80
171.k even 6 1 684.3.bl.a yes 80
171.s odd 6 1 inner 2052.3.bl.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.s.a 80 9.d odd 6 1
684.3.s.a 80 57.f even 6 1
684.3.bl.a yes 80 3.b odd 2 1
684.3.bl.a yes 80 171.k even 6 1
2052.3.s.a 80 9.c even 3 1
2052.3.s.a 80 19.d odd 6 1
2052.3.bl.a 80 1.a even 1 1 trivial
2052.3.bl.a 80 171.s odd 6 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(2052, [\chi])$$.