Properties

Label 684.3.m.a
Level $684$
Weight $3$
Character orbit 684.m
Analytic conductor $18.638$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 684 = 2^{2} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 684.m (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(80\)
Relative dimension: \(40\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80q - 2q^{3} + q^{7} - 2q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 80q - 2q^{3} + q^{7} - 2q^{9} + 18q^{11} - 5q^{13} - 2q^{15} - 9q^{17} + 20q^{19} - 30q^{21} + 72q^{23} - 400q^{25} + 25q^{27} - 8q^{31} - 64q^{33} + 22q^{37} + 39q^{39} - 44q^{43} - 196q^{45} - 267q^{49} - 47q^{51} - 36q^{53} + 84q^{57} - 14q^{61} - 260q^{63} - 144q^{65} - 77q^{67} + 44q^{69} - 135q^{71} + 43q^{73} + 69q^{75} + 216q^{77} - 17q^{79} - 254q^{81} - 171q^{83} - 244q^{87} + 216q^{89} + 122q^{91} + 292q^{93} - 288q^{95} - 8q^{97} + 172q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
353.1 0 −2.99548 0.164631i 0 2.57048i 0 1.88687 3.26816i 0 8.94579 + 0.986299i 0
353.2 0 −2.99460 0.179992i 0 0.944063i 0 −5.01764 + 8.69081i 0 8.93521 + 1.07801i 0
353.3 0 −2.88193 + 0.833354i 0 7.26979i 0 3.75776 6.50863i 0 7.61104 4.80334i 0
353.4 0 −2.84492 0.952067i 0 6.24143i 0 4.14502 7.17938i 0 7.18714 + 5.41711i 0
353.5 0 −2.80552 + 1.06257i 0 5.14812i 0 2.38165 4.12514i 0 6.74191 5.96210i 0
353.6 0 −2.74110 1.21916i 0 1.38038i 0 −1.29258 + 2.23881i 0 6.02728 + 6.68370i 0
353.7 0 −2.64568 + 1.41435i 0 6.34957i 0 −3.73312 + 6.46596i 0 4.99921 7.48384i 0
353.8 0 −2.55262 1.57611i 0 9.08186i 0 3.71777 6.43936i 0 4.03172 + 8.04644i 0
353.9 0 −2.38029 + 1.82598i 0 5.89395i 0 −0.351537 + 0.608880i 0 2.33159 8.69274i 0
353.10 0 −2.11779 2.12485i 0 8.55202i 0 −5.68709 + 9.85033i 0 −0.0299448 + 8.99995i 0
353.11 0 −1.79577 + 2.40317i 0 4.55087i 0 5.85150 10.1351i 0 −2.55043 8.63107i 0
353.12 0 −1.70356 2.46939i 0 9.28650i 0 −2.28067 + 3.95023i 0 −3.19575 + 8.41351i 0
353.13 0 −1.61882 2.52575i 0 0.497329i 0 5.73504 9.93338i 0 −3.75887 + 8.17746i 0
353.14 0 −1.60029 2.53754i 0 1.56237i 0 −4.59594 + 7.96040i 0 −3.87817 + 8.12156i 0
353.15 0 −1.55115 + 2.56787i 0 5.60337i 0 −5.05578 + 8.75686i 0 −4.18789 7.96628i 0
353.16 0 −1.21870 + 2.74131i 0 2.92593i 0 −2.71227 + 4.69779i 0 −6.02956 6.68165i 0
353.17 0 −1.06149 + 2.80593i 0 1.84834i 0 2.46982 4.27785i 0 −6.74649 5.95692i 0
353.18 0 −1.03515 2.81575i 0 0.329507i 0 2.81673 4.87871i 0 −6.85692 + 5.82947i 0
353.19 0 −0.358884 2.97846i 0 0.408400i 0 −0.674294 + 1.16791i 0 −8.74240 + 2.13784i 0
353.20 0 −0.0106609 + 2.99998i 0 4.51007i 0 1.48435 2.57096i 0 −8.99977 0.0639652i 0
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 653.40
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
171.n odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.m.a 80
3.b odd 2 1 2052.3.m.a 80
9.c even 3 1 2052.3.be.a 80
9.d odd 6 1 684.3.be.a yes 80
19.c even 3 1 684.3.be.a yes 80
57.h odd 6 1 2052.3.be.a 80
171.g even 3 1 2052.3.m.a 80
171.n odd 6 1 inner 684.3.m.a 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.m.a 80 1.a even 1 1 trivial
684.3.m.a 80 171.n odd 6 1 inner
684.3.be.a yes 80 9.d odd 6 1
684.3.be.a yes 80 19.c even 3 1
2052.3.m.a 80 3.b odd 2 1
2052.3.m.a 80 171.g even 3 1
2052.3.be.a 80 9.c even 3 1
2052.3.be.a 80 57.h odd 6 1

Hecke kernels

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