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 $8$

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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:

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 \(S_{3}^{\mathrm{new}}(684, [\chi])\).