# 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 $4$

# Related objects

## 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: 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

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