# Properties

 Label 2052.3.m.a Level $2052$ Weight $3$ Character orbit 2052.m 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.m (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.

 $$\operatorname{Tr}(f)(q) =$$ $$80 q + q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80 q + q^{7} - 18 q^{11} - 5 q^{13} + 9 q^{17} + 20 q^{19} - 72 q^{23} - 400 q^{25} - 8 q^{31} + 22 q^{37} - 44 q^{43} - 267 q^{49} + 36 q^{53} - 14 q^{61} + 144 q^{65} - 77 q^{67} + 135 q^{71} + 43 q^{73} - 216 q^{77} - 17 q^{79} + 171 q^{83} - 216 q^{89} + 122 q^{91} + 288 q^{95} - 8 q^{97} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1 0 0 0 9.28650i 0 −2.28067 3.95023i 0 0 0
881.2 0 0 0 8.16148i 0 −1.62444 2.81362i 0 0 0
881.3 0 0 0 7.26979i 0 3.75776 + 6.50863i 0 0 0
881.4 0 0 0 7.03696i 0 2.95479 + 5.11784i 0 0 0
881.5 0 0 0 6.78017i 0 −0.841737 1.45793i 0 0 0
881.6 0 0 0 6.34957i 0 −3.73312 6.46596i 0 0 0
881.7 0 0 0 6.32167i 0 2.00114 + 3.46607i 0 0 0
881.8 0 0 0 6.24143i 0 4.14502 + 7.17938i 0 0 0
881.9 0 0 0 6.14879i 0 −6.67945 11.5692i 0 0 0
881.10 0 0 0 5.73236i 0 5.68826 + 9.85235i 0 0 0
881.11 0 0 0 5.68625i 0 −1.27053 2.20061i 0 0 0
881.12 0 0 0 4.55087i 0 5.85150 + 10.1351i 0 0 0
881.13 0 0 0 3.96092i 0 −6.03683 10.4561i 0 0 0
881.14 0 0 0 2.92593i 0 −2.71227 4.69779i 0 0 0
881.15 0 0 0 2.23654i 0 1.30590 + 2.26189i 0 0 0
881.16 0 0 0 1.84834i 0 2.46982 + 4.27785i 0 0 0
881.17 0 0 0 1.38038i 0 −1.29258 2.23881i 0 0 0
881.18 0 0 0 0.944063i 0 −5.01764 8.69081i 0 0 0
881.19 0 0 0 0.497329i 0 5.73504 + 9.93338i 0 0 0
881.20 0 0 0 0.434480i 0 −4.09970 7.10089i 0 0 0
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1493.40 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.n 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.m.a 80
3.b odd 2 1 684.3.m.a 80
9.c even 3 1 684.3.be.a yes 80
9.d odd 6 1 2052.3.be.a 80
19.c even 3 1 2052.3.be.a 80
57.h odd 6 1 684.3.be.a yes 80
171.g even 3 1 684.3.m.a 80
171.n odd 6 1 inner 2052.3.m.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.m.a 80 3.b odd 2 1
684.3.m.a 80 171.g even 3 1
684.3.be.a yes 80 9.c even 3 1
684.3.be.a yes 80 57.h odd 6 1
2052.3.m.a 80 1.a even 1 1 trivial
2052.3.m.a 80 171.n odd 6 1 inner
2052.3.be.a 80 9.d odd 6 1
2052.3.be.a 80 19.c even 3 1

## Hecke kernels

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