Properties

Label 684.3.g.d
Level $684$
Weight $3$
Character orbit 684.g
Analytic conductor $18.638$
Analytic rank $0$
Dimension $36$
CM no
Inner twists $4$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(18.6376500822\)
Analytic rank: \(0\)
Dimension: \(36\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{4} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 36 q - 12 q^{4} + 8 q^{10} + 24 q^{13} - 92 q^{16} - 60 q^{22} + 44 q^{25} - 48 q^{28} - 148 q^{34} + 200 q^{37} + 180 q^{40} + 140 q^{46} - 332 q^{49} + 60 q^{52} - 64 q^{58} + 40 q^{61} + 60 q^{64} + 36 q^{70} - 200 q^{73} + 312 q^{82} + 16 q^{85} + 104 q^{88} + 184 q^{94} + 280 q^{97} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
343.1 −1.98942 0.205433i 0 3.91559 + 0.817384i −0.0956487 0 9.56723i −7.62185 2.43051i 0 0.190286 + 0.0196494i
343.2 −1.98942 + 0.205433i 0 3.91559 0.817384i −0.0956487 0 9.56723i −7.62185 + 2.43051i 0 0.190286 0.0196494i
343.3 −1.83969 0.784567i 0 2.76891 + 2.88672i −6.85656 0 3.48151i −2.82911 7.48306i 0 12.6139 + 5.37943i
343.4 −1.83969 + 0.784567i 0 2.76891 2.88672i −6.85656 0 3.48151i −2.82911 + 7.48306i 0 12.6139 5.37943i
343.5 −1.58671 1.21751i 0 1.03532 + 3.86369i 1.46825 0 6.38949i 3.06136 7.39108i 0 −2.32969 1.78762i
343.6 −1.58671 + 1.21751i 0 1.03532 3.86369i 1.46825 0 6.38949i 3.06136 + 7.39108i 0 −2.32969 + 1.78762i
343.7 −1.55174 1.26178i 0 0.815807 + 3.91592i 2.61138 0 10.9942i 3.67512 7.10587i 0 −4.05218 3.29499i
343.8 −1.55174 + 1.26178i 0 0.815807 3.91592i 2.61138 0 10.9942i 3.67512 + 7.10587i 0 −4.05218 + 3.29499i
343.9 −1.42379 1.40457i 0 0.0543607 + 3.99963i 5.95758 0 3.23592i 5.54037 5.77099i 0 −8.48235 8.36785i
343.10 −1.42379 + 1.40457i 0 0.0543607 3.99963i 5.95758 0 3.23592i 5.54037 + 5.77099i 0 −8.48235 + 8.36785i
343.11 −1.07034 1.68949i 0 −1.70873 + 3.61666i −6.66582 0 12.1510i 7.93923 0.984190i 0 7.13471 + 11.2618i
343.12 −1.07034 + 1.68949i 0 −1.70873 3.61666i −6.66582 0 12.1510i 7.93923 + 0.984190i 0 7.13471 11.2618i
343.13 −0.690218 1.87713i 0 −3.04720 + 2.59125i 8.91109 0 7.74115i 6.96733 + 3.93145i 0 −6.15059 16.7272i
343.14 −0.690218 + 1.87713i 0 −3.04720 2.59125i 8.91109 0 7.74115i 6.96733 3.93145i 0 −6.15059 + 16.7272i
343.15 −0.671560 1.88388i 0 −3.09801 + 2.53028i −4.54598 0 5.76325i 6.84725 + 4.13706i 0 3.05290 + 8.56408i
343.16 −0.671560 + 1.88388i 0 −3.09801 2.53028i −4.54598 0 5.76325i 6.84725 4.13706i 0 3.05290 8.56408i
343.17 −0.363289 1.96673i 0 −3.73604 + 1.42898i −0.0632854 0 2.71864i 4.16769 + 6.82864i 0 0.0229909 + 0.124465i
343.18 −0.363289 + 1.96673i 0 −3.73604 1.42898i −0.0632854 0 2.71864i 4.16769 6.82864i 0 0.0229909 0.124465i
343.19 0.363289 1.96673i 0 −3.73604 1.42898i 0.0632854 0 2.71864i −4.16769 + 6.82864i 0 0.0229909 0.124465i
343.20 0.363289 + 1.96673i 0 −3.73604 + 1.42898i 0.0632854 0 2.71864i −4.16769 6.82864i 0 0.0229909 + 0.124465i
See all 36 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 343.36
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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 684.3.g.d 36
3.b odd 2 1 inner 684.3.g.d 36
4.b odd 2 1 inner 684.3.g.d 36
12.b even 2 1 inner 684.3.g.d 36
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
684.3.g.d 36 1.a even 1 1 trivial
684.3.g.d 36 3.b odd 2 1 inner
684.3.g.d 36 4.b odd 2 1 inner
684.3.g.d 36 12.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{18} - \cdots\) acting on \(S_{3}^{\mathrm{new}}(684, [\chi])\).