# Properties

 Label 1380.2.n.a Level $1380$ Weight $2$ Character orbit 1380.n Analytic conductor $11.019$ Analytic rank $0$ Dimension $48$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1380.n (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$11.0193554789$$ Analytic rank: $$0$$ Dimension: $$48$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 16q^{31} + 24q^{39} + 112q^{49} + 8q^{55} - 16q^{69} + 20q^{75} - 8q^{81} + 32q^{85} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
689.1 0 −1.71333 0.253996i 0 −0.923440 2.03648i 0 −2.63922 0 2.87097 + 0.870355i 0
689.2 0 −1.71333 0.253996i 0 0.923440 + 2.03648i 0 2.63922 0 2.87097 + 0.870355i 0
689.3 0 −1.71333 + 0.253996i 0 −0.923440 + 2.03648i 0 −2.63922 0 2.87097 0.870355i 0
689.4 0 −1.71333 + 0.253996i 0 0.923440 2.03648i 0 2.63922 0 2.87097 0.870355i 0
689.5 0 −1.63297 0.577419i 0 −2.15297 + 0.603919i 0 1.63712 0 2.33317 + 1.88581i 0
689.6 0 −1.63297 0.577419i 0 2.15297 0.603919i 0 −1.63712 0 2.33317 + 1.88581i 0
689.7 0 −1.63297 + 0.577419i 0 −2.15297 0.603919i 0 1.63712 0 2.33317 1.88581i 0
689.8 0 −1.63297 + 0.577419i 0 2.15297 + 0.603919i 0 −1.63712 0 2.33317 1.88581i 0
689.9 0 −1.35581 1.07786i 0 −1.70581 1.44575i 0 −5.00303 0 0.676420 + 2.92275i 0
689.10 0 −1.35581 1.07786i 0 1.70581 + 1.44575i 0 5.00303 0 0.676420 + 2.92275i 0
689.11 0 −1.35581 + 1.07786i 0 −1.70581 + 1.44575i 0 −5.00303 0 0.676420 2.92275i 0
689.12 0 −1.35581 + 1.07786i 0 1.70581 1.44575i 0 5.00303 0 0.676420 2.92275i 0
689.13 0 −0.888021 1.48708i 0 −1.29815 + 1.82066i 0 0.470527 0 −1.42284 + 2.64112i 0
689.14 0 −0.888021 1.48708i 0 1.29815 1.82066i 0 −0.470527 0 −1.42284 + 2.64112i 0
689.15 0 −0.888021 + 1.48708i 0 −1.29815 1.82066i 0 0.470527 0 −1.42284 2.64112i 0
689.16 0 −0.888021 + 1.48708i 0 1.29815 + 1.82066i 0 −0.470527 0 −1.42284 2.64112i 0
689.17 0 −0.779801 1.54658i 0 −0.268100 2.21994i 0 2.49920 0 −1.78382 + 2.41205i 0
689.18 0 −0.779801 1.54658i 0 0.268100 + 2.21994i 0 −2.49920 0 −1.78382 + 2.41205i 0
689.19 0 −0.779801 + 1.54658i 0 −0.268100 + 2.21994i 0 2.49920 0 −1.78382 2.41205i 0
689.20 0 −0.779801 + 1.54658i 0 0.268100 2.21994i 0 −2.49920 0 −1.78382 2.41205i 0
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 689.48 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
5.b even 2 1 inner
15.d odd 2 1 inner
23.b odd 2 1 inner
69.c even 2 1 inner
115.c odd 2 1 inner
345.h even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1380.2.n.a 48
3.b odd 2 1 inner 1380.2.n.a 48
5.b even 2 1 inner 1380.2.n.a 48
15.d odd 2 1 inner 1380.2.n.a 48
23.b odd 2 1 inner 1380.2.n.a 48
69.c even 2 1 inner 1380.2.n.a 48
115.c odd 2 1 inner 1380.2.n.a 48
345.h even 2 1 inner 1380.2.n.a 48

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1380.2.n.a 48 1.a even 1 1 trivial
1380.2.n.a 48 3.b odd 2 1 inner
1380.2.n.a 48 5.b even 2 1 inner
1380.2.n.a 48 15.d odd 2 1 inner
1380.2.n.a 48 23.b odd 2 1 inner
1380.2.n.a 48 69.c even 2 1 inner
1380.2.n.a 48 115.c odd 2 1 inner
1380.2.n.a 48 345.h even 2 1 inner

## Hecke kernels

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