# Properties

 Label 684.3.b.a Level $684$ Weight $3$ Character orbit 684.b Analytic conductor $18.638$ Analytic rank $0$ Dimension $80$ CM no Inner twists $8$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.6376500822$$ Analytic rank: $$0$$ Dimension: $$80$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$80q - 8q^{4} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$80q - 8q^{4} - 56q^{16} - 400q^{25} - 464q^{49} - 272q^{58} - 352q^{61} - 200q^{64} + 480q^{73} + 152q^{76} + 32q^{82} + 704q^{85} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
683.1 −1.98202 0.267543i 0 3.85684 + 1.06055i 4.56953i 0 9.67098i −7.36061 3.13391i 0 1.22255 9.05693i
683.2 −1.98202 0.267543i 0 3.85684 + 1.06055i 4.56953i 0 9.67098i −7.36061 3.13391i 0 −1.22255 + 9.05693i
683.3 −1.98202 + 0.267543i 0 3.85684 1.06055i 4.56953i 0 9.67098i −7.36061 + 3.13391i 0 1.22255 + 9.05693i
683.4 −1.98202 + 0.267543i 0 3.85684 1.06055i 4.56953i 0 9.67098i −7.36061 + 3.13391i 0 −1.22255 9.05693i
683.5 −1.90361 0.613402i 0 3.24748 + 2.33536i 6.64226i 0 0.470842i −4.74942 6.43762i 0 −4.07438 + 12.6443i
683.6 −1.90361 0.613402i 0 3.24748 + 2.33536i 6.64226i 0 0.470842i −4.74942 6.43762i 0 4.07438 12.6443i
683.7 −1.90361 + 0.613402i 0 3.24748 2.33536i 6.64226i 0 0.470842i −4.74942 + 6.43762i 0 −4.07438 12.6443i
683.8 −1.90361 + 0.613402i 0 3.24748 2.33536i 6.64226i 0 0.470842i −4.74942 + 6.43762i 0 4.07438 + 12.6443i
683.9 −1.88680 0.663313i 0 3.12003 + 2.50308i 0.647124i 0 4.63116i −4.22656 6.79236i 0 0.429246 1.22099i
683.10 −1.88680 0.663313i 0 3.12003 + 2.50308i 0.647124i 0 4.63116i −4.22656 6.79236i 0 −0.429246 + 1.22099i
683.11 −1.88680 + 0.663313i 0 3.12003 2.50308i 0.647124i 0 4.63116i −4.22656 + 6.79236i 0 0.429246 + 1.22099i
683.12 −1.88680 + 0.663313i 0 3.12003 2.50308i 0.647124i 0 4.63116i −4.22656 + 6.79236i 0 −0.429246 1.22099i
683.13 −1.58380 1.22130i 0 1.01686 + 3.86859i 8.58011i 0 1.97492i 3.11420 7.36897i 0 −10.4789 + 13.5892i
683.14 −1.58380 1.22130i 0 1.01686 + 3.86859i 8.58011i 0 1.97492i 3.11420 7.36897i 0 10.4789 13.5892i
683.15 −1.58380 + 1.22130i 0 1.01686 3.86859i 8.58011i 0 1.97492i 3.11420 + 7.36897i 0 −10.4789 13.5892i
683.16 −1.58380 + 1.22130i 0 1.01686 3.86859i 8.58011i 0 1.97492i 3.11420 + 7.36897i 0 10.4789 + 13.5892i
683.17 −1.45112 1.37632i 0 0.211495 + 3.99440i 4.00161i 0 12.0281i 5.19067 6.08744i 0 5.50748 5.80681i
683.18 −1.45112 1.37632i 0 0.211495 + 3.99440i 4.00161i 0 12.0281i 5.19067 6.08744i 0 −5.50748 + 5.80681i
683.19 −1.45112 + 1.37632i 0 0.211495 3.99440i 4.00161i 0 12.0281i 5.19067 + 6.08744i 0 5.50748 + 5.80681i
683.20 −1.45112 + 1.37632i 0 0.211495 3.99440i 4.00161i 0 12.0281i 5.19067 + 6.08744i 0 −5.50748 5.80681i
See all 80 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 683.80 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
4.b odd 2 1 inner
12.b even 2 1 inner
19.b odd 2 1 inner
57.d even 2 1 inner
76.d even 2 1 inner
228.b odd 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.b.a 80
3.b odd 2 1 inner 684.3.b.a 80
4.b odd 2 1 inner 684.3.b.a 80
12.b even 2 1 inner 684.3.b.a 80
19.b odd 2 1 inner 684.3.b.a 80
57.d even 2 1 inner 684.3.b.a 80
76.d even 2 1 inner 684.3.b.a 80
228.b odd 2 1 inner 684.3.b.a 80

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.b.a 80 1.a even 1 1 trivial
684.3.b.a 80 3.b odd 2 1 inner
684.3.b.a 80 4.b odd 2 1 inner
684.3.b.a 80 12.b even 2 1 inner
684.3.b.a 80 19.b odd 2 1 inner
684.3.b.a 80 57.d even 2 1 inner
684.3.b.a 80 76.d even 2 1 inner
684.3.b.a 80 228.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(684, [\chi])$$.