# Properties

 Label 1380.2.t.a Level $1380$ Weight $2$ Character orbit 1380.t Analytic conductor $11.019$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.t (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$24$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 8q^{13} - 16q^{23} - 8q^{25} + 8q^{31} + 8q^{35} - 24q^{41} + 8q^{47} - 32q^{55} - 24q^{71} + 8q^{73} + 32q^{75} + 40q^{77} - 48q^{81} + 24q^{85} - 40q^{87} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1057.1 0 −0.707107 + 0.707107i 0 −0.811448 2.08364i 0 3.03238 3.03238i 0 1.00000i 0
1057.2 0 −0.707107 + 0.707107i 0 0.811448 + 2.08364i 0 −3.03238 + 3.03238i 0 1.00000i 0
1057.3 0 −0.707107 + 0.707107i 0 −1.62167 + 1.53954i 0 2.42328 2.42328i 0 1.00000i 0
1057.4 0 −0.707107 + 0.707107i 0 1.62167 1.53954i 0 −2.42328 + 2.42328i 0 1.00000i 0
1057.5 0 −0.707107 + 0.707107i 0 −1.68477 1.47022i 0 −2.29608 + 2.29608i 0 1.00000i 0
1057.6 0 −0.707107 + 0.707107i 0 1.68477 + 1.47022i 0 2.29608 2.29608i 0 1.00000i 0
1057.7 0 −0.707107 + 0.707107i 0 −1.35578 + 1.77816i 0 0.243134 0.243134i 0 1.00000i 0
1057.8 0 −0.707107 + 0.707107i 0 1.35578 1.77816i 0 −0.243134 + 0.243134i 0 1.00000i 0
1057.9 0 −0.707107 + 0.707107i 0 −2.09989 + 0.768407i 0 −0.0117285 + 0.0117285i 0 1.00000i 0
1057.10 0 −0.707107 + 0.707107i 0 2.09989 0.768407i 0 0.0117285 0.0117285i 0 1.00000i 0
1057.11 0 −0.707107 + 0.707107i 0 −0.0653842 2.23511i 0 −0.522235 + 0.522235i 0 1.00000i 0
1057.12 0 −0.707107 + 0.707107i 0 0.0653842 + 2.23511i 0 0.522235 0.522235i 0 1.00000i 0
1057.13 0 0.707107 0.707107i 0 −0.951596 + 2.02348i 0 3.48236 3.48236i 0 1.00000i 0
1057.14 0 0.707107 0.707107i 0 0.951596 2.02348i 0 −3.48236 + 3.48236i 0 1.00000i 0
1057.15 0 0.707107 0.707107i 0 −2.23126 + 0.146576i 0 2.93317 2.93317i 0 1.00000i 0
1057.16 0 0.707107 0.707107i 0 2.23126 0.146576i 0 −2.93317 + 2.93317i 0 1.00000i 0
1057.17 0 0.707107 0.707107i 0 −1.86909 1.22740i 0 0.329148 0.329148i 0 1.00000i 0
1057.18 0 0.707107 0.707107i 0 1.86909 + 1.22740i 0 −0.329148 + 0.329148i 0 1.00000i 0
1057.19 0 0.707107 0.707107i 0 −0.577284 + 2.16026i 0 −0.575518 + 0.575518i 0 1.00000i 0
1057.20 0 0.707107 0.707107i 0 0.577284 2.16026i 0 0.575518 0.575518i 0 1.00000i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1333.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.t.a 48
5.c odd 4 1 inner 1380.2.t.a 48
23.b odd 2 1 inner 1380.2.t.a 48
115.e even 4 1 inner 1380.2.t.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.t.a 48 1.a even 1 1 trivial
1380.2.t.a 48 5.c odd 4 1 inner
1380.2.t.a 48 23.b odd 2 1 inner
1380.2.t.a 48 115.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{2}^{\mathrm{new}}(1380, [\chi])$$.