# Properties

 Label 690.3.f.a Level $690$ Weight $3$ Character orbit 690.f Analytic conductor $18.801$ Analytic rank $0$ Dimension $48$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 690.f (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.8011382409$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$48q - 96q^{4} - 144q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q - 96q^{4} - 144q^{9} + 192q^{16} + 96q^{25} + 64q^{26} - 152q^{29} - 8q^{31} + 56q^{35} + 288q^{36} - 48q^{39} + 40q^{41} - 160q^{46} + 424q^{49} + 96q^{50} + 32q^{55} + 360q^{59} - 384q^{64} + 192q^{69} - 496q^{70} - 152q^{71} + 144q^{75} + 432q^{81} - 136q^{85} + 256q^{94} + 496q^{95} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
229.1 1.41421i 1.73205i −2.00000 −1.71908 + 4.69518i 2.44949 −12.9397 2.82843i −3.00000 6.63999 + 2.43115i
229.2 1.41421i 1.73205i −2.00000 1.71908 4.69518i 2.44949 12.9397 2.82843i −3.00000 −6.63999 2.43115i
229.3 1.41421i 1.73205i −2.00000 −1.71908 4.69518i 2.44949 −12.9397 2.82843i −3.00000 6.63999 2.43115i
229.4 1.41421i 1.73205i −2.00000 1.71908 + 4.69518i 2.44949 12.9397 2.82843i −3.00000 −6.63999 + 2.43115i
229.5 1.41421i 1.73205i −2.00000 −3.10921 3.91571i −2.44949 3.09410 2.82843i −3.00000 −5.53765 + 4.39709i
229.6 1.41421i 1.73205i −2.00000 3.10921 + 3.91571i −2.44949 −3.09410 2.82843i −3.00000 5.53765 4.39709i
229.7 1.41421i 1.73205i −2.00000 −3.10921 + 3.91571i −2.44949 3.09410 2.82843i −3.00000 −5.53765 4.39709i
229.8 1.41421i 1.73205i −2.00000 3.10921 3.91571i −2.44949 −3.09410 2.82843i −3.00000 5.53765 + 4.39709i
229.9 1.41421i 1.73205i −2.00000 −4.91938 + 0.894267i −2.44949 6.65471 2.82843i −3.00000 1.26468 + 6.95705i
229.10 1.41421i 1.73205i −2.00000 4.91938 0.894267i −2.44949 −6.65471 2.82843i −3.00000 −1.26468 6.95705i
229.11 1.41421i 1.73205i −2.00000 −4.91938 0.894267i −2.44949 6.65471 2.82843i −3.00000 1.26468 6.95705i
229.12 1.41421i 1.73205i −2.00000 4.91938 + 0.894267i −2.44949 −6.65471 2.82843i −3.00000 −1.26468 + 6.95705i
229.13 1.41421i 1.73205i −2.00000 −4.27075 2.60014i 2.44949 −3.19570 2.82843i −3.00000 −3.67715 + 6.03975i
229.14 1.41421i 1.73205i −2.00000 4.27075 + 2.60014i 2.44949 3.19570 2.82843i −3.00000 3.67715 6.03975i
229.15 1.41421i 1.73205i −2.00000 −4.27075 + 2.60014i 2.44949 −3.19570 2.82843i −3.00000 −3.67715 6.03975i
229.16 1.41421i 1.73205i −2.00000 4.27075 2.60014i 2.44949 3.19570 2.82843i −3.00000 3.67715 + 6.03975i
229.17 1.41421i 1.73205i −2.00000 −1.15789 + 4.86408i 2.44949 1.79343 2.82843i −3.00000 6.87885 + 1.63751i
229.18 1.41421i 1.73205i −2.00000 1.15789 4.86408i 2.44949 −1.79343 2.82843i −3.00000 −6.87885 1.63751i
229.19 1.41421i 1.73205i −2.00000 −1.15789 4.86408i 2.44949 1.79343 2.82843i −3.00000 6.87885 1.63751i
229.20 1.41421i 1.73205i −2.00000 1.15789 + 4.86408i 2.44949 −1.79343 2.82843i −3.00000 −6.87885 + 1.63751i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 229.48 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.f.a 48
5.b even 2 1 inner 690.3.f.a 48
23.b odd 2 1 inner 690.3.f.a 48
115.c odd 2 1 inner 690.3.f.a 48

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(690, [\chi])$$.