# Properties

 Label 684.3.g.c Level $684$ Weight $3$ Character orbit 684.g Analytic conductor $18.638$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# 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: no (minimal twist has level 228) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 20 q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$36 q - 4 q^{2} + 12 q^{4} - 8 q^{5} + 20 q^{8} + 8 q^{10} - 24 q^{13} + 12 q^{14} + 4 q^{16} + 40 q^{17} + 80 q^{20} + 12 q^{22} + 284 q^{25} + 112 q^{26} - 48 q^{28} - 104 q^{29} - 44 q^{32} + 140 q^{34} - 184 q^{37} + 180 q^{40} + 200 q^{41} - 96 q^{44} - 28 q^{46} - 332 q^{49} - 176 q^{50} + 276 q^{52} - 264 q^{53} + 192 q^{56} - 184 q^{58} + 40 q^{61} + 240 q^{62} - 372 q^{64} - 176 q^{65} + 104 q^{68} - 60 q^{70} + 424 q^{73} + 104 q^{74} + 400 q^{77} - 704 q^{80} + 528 q^{82} - 128 q^{85} - 668 q^{86} - 496 q^{88} + 520 q^{89} + 456 q^{92} - 32 q^{94} - 440 q^{97} + 472 q^{98} + O(q^{100})$$

## 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 −1.99932 0.0520666i 0 3.99458 + 0.208196i −6.84194 0 12.2496i −7.97561 0.624234i 0 13.6792 + 0.356236i
343.2 −1.99932 + 0.0520666i 0 3.99458 0.208196i −6.84194 0 12.2496i −7.97561 + 0.624234i 0 13.6792 0.356236i
343.3 −1.95612 0.416662i 0 3.65279 + 1.63008i −1.84551 0 7.45193i −6.46608 4.71060i 0 3.61003 + 0.768954i
343.4 −1.95612 + 0.416662i 0 3.65279 1.63008i −1.84551 0 7.45193i −6.46608 + 4.71060i 0 3.61003 0.768954i
343.5 −1.79512 0.881776i 0 2.44494 + 3.16579i 9.55892 0 12.8591i −1.59745 7.83889i 0 −17.1594 8.42882i
343.6 −1.79512 + 0.881776i 0 2.44494 3.16579i 9.55892 0 12.8591i −1.59745 + 7.83889i 0 −17.1594 + 8.42882i
343.7 −1.78242 0.907175i 0 2.35407 + 3.23394i 3.42194 0 1.09734i −1.26219 7.89980i 0 −6.09935 3.10430i
343.8 −1.78242 + 0.907175i 0 2.35407 3.23394i 3.42194 0 1.09734i −1.26219 + 7.89980i 0 −6.09935 + 3.10430i
343.9 −1.54492 1.27013i 0 0.773559 + 3.92449i −7.21023 0 4.48213i 3.78951 7.04554i 0 11.1392 + 9.15790i
343.10 −1.54492 + 1.27013i 0 0.773559 3.92449i −7.21023 0 4.48213i 3.78951 + 7.04554i 0 11.1392 9.15790i
343.11 −1.28823 1.52986i 0 −0.680940 + 3.94161i −3.40651 0 4.01056i 6.90732 4.03595i 0 4.38835 + 5.21148i
343.12 −1.28823 + 1.52986i 0 −0.680940 3.94161i −3.40651 0 4.01056i 6.90732 + 4.03595i 0 4.38835 5.21148i
343.13 −1.26893 1.54590i 0 −0.779622 + 3.92329i 6.46378 0 11.6492i 7.05430 3.77317i 0 −8.20210 9.99237i
343.14 −1.26893 + 1.54590i 0 −0.779622 3.92329i 6.46378 0 11.6492i 7.05430 + 3.77317i 0 −8.20210 + 9.99237i
343.15 −0.895709 1.78821i 0 −2.39541 + 3.20344i 5.60753 0 0.452718i 7.87402 + 1.41416i 0 −5.02271 10.0275i
343.16 −0.895709 + 1.78821i 0 −2.39541 3.20344i 5.60753 0 0.452718i 7.87402 1.41416i 0 −5.02271 + 10.0275i
343.17 −0.234708 1.98618i 0 −3.88982 + 0.932347i 0.816108 0 3.29246i 2.76478 + 7.50706i 0 −0.191547 1.62094i
343.18 −0.234708 + 1.98618i 0 −3.88982 0.932347i 0.816108 0 3.29246i 2.76478 7.50706i 0 −0.191547 + 1.62094i
343.19 −0.157997 1.99375i 0 −3.95007 + 0.630012i −9.57494 0 3.50228i 1.88018 + 7.77592i 0 1.51281 + 19.0900i
343.20 −0.157997 + 1.99375i 0 −3.95007 0.630012i −9.57494 0 3.50228i 1.88018 7.77592i 0 1.51281 19.0900i
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
4.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.g.c 36
3.b odd 2 1 228.3.g.a 36
4.b odd 2 1 inner 684.3.g.c 36
12.b even 2 1 228.3.g.a 36

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

## Hecke kernels

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