# Properties

 Label 690.3.g.a Level $690$ Weight $3$ Character orbit 690.g Analytic conductor $18.801$ Analytic rank $0$ Dimension $56$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 690.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$18.8011382409$$ Analytic rank: $$0$$ Dimension: $$56$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$56q - 8q^{3} - 112q^{4} + 16q^{6} - 16q^{7} + 16q^{12} + 80q^{13} - 40q^{15} + 224q^{16} - 32q^{18} - 64q^{19} + 56q^{21} - 96q^{22} - 32q^{24} - 280q^{25} + 40q^{27} + 32q^{28} - 80q^{31} + 32q^{33} + 192q^{34} + 240q^{37} - 56q^{39} - 144q^{43} - 32q^{48} + 72q^{49} - 24q^{51} - 160q^{52} + 16q^{54} - 16q^{57} + 80q^{60} + 112q^{61} - 64q^{63} - 448q^{64} + 160q^{66} + 832q^{67} + 64q^{72} - 608q^{73} + 40q^{75} + 128q^{76} - 320q^{78} + 48q^{79} - 32q^{81} - 448q^{82} - 112q^{84} + 240q^{85} + 200q^{87} + 192q^{88} + 80q^{91} - 232q^{93} + 160q^{94} + 64q^{96} - 448q^{97} + 464q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
461.1 1.41421i 2.78199 1.12272i −2.00000 2.23607i −1.58777 3.93433i 9.71289 2.82843i 6.47899 6.24682i −3.16228
461.2 1.41421i 2.78199 + 1.12272i −2.00000 2.23607i −1.58777 + 3.93433i 9.71289 2.82843i 6.47899 + 6.24682i −3.16228
461.3 1.41421i −2.87483 + 0.857518i −2.00000 2.23607i 1.21271 + 4.06563i 5.06417 2.82843i 7.52932 4.93044i 3.16228
461.4 1.41421i −2.87483 0.857518i −2.00000 2.23607i 1.21271 4.06563i 5.06417 2.82843i 7.52932 + 4.93044i 3.16228
461.5 1.41421i −2.73248 1.23837i −2.00000 2.23607i −1.75132 + 3.86431i 4.14117 2.82843i 5.93289 + 6.76763i 3.16228
461.6 1.41421i −2.73248 + 1.23837i −2.00000 2.23607i −1.75132 3.86431i 4.14117 2.82843i 5.93289 6.76763i 3.16228
461.7 1.41421i 1.56849 + 2.55731i −2.00000 2.23607i 3.61658 2.21819i −1.76788 2.82843i −4.07965 + 8.02225i 3.16228
461.8 1.41421i 1.56849 2.55731i −2.00000 2.23607i 3.61658 + 2.21819i −1.76788 2.82843i −4.07965 8.02225i 3.16228
461.9 1.41421i 2.16025 + 2.08166i −2.00000 2.23607i 2.94392 3.05506i 12.7530 2.82843i 0.333364 + 8.99382i 3.16228
461.10 1.41421i 2.16025 2.08166i −2.00000 2.23607i 2.94392 + 3.05506i 12.7530 2.82843i 0.333364 8.99382i 3.16228
461.11 1.41421i −1.59356 2.54176i −2.00000 2.23607i −3.59459 + 2.25364i 10.3156 2.82843i −3.92110 + 8.10092i −3.16228
461.12 1.41421i −1.59356 + 2.54176i −2.00000 2.23607i −3.59459 2.25364i 10.3156 2.82843i −3.92110 8.10092i −3.16228
461.13 1.41421i −1.34299 2.68261i −2.00000 2.23607i −3.79378 + 1.89928i −8.50482 2.82843i −5.39274 + 7.20544i −3.16228
461.14 1.41421i −1.34299 + 2.68261i −2.00000 2.23607i −3.79378 1.89928i −8.50482 2.82843i −5.39274 7.20544i −3.16228
461.15 1.41421i 0.880203 2.86797i −2.00000 2.23607i −4.05592 1.24480i −3.20877 2.82843i −7.45049 5.04879i 3.16228
461.16 1.41421i 0.880203 + 2.86797i −2.00000 2.23607i −4.05592 + 1.24480i −3.20877 2.82843i −7.45049 + 5.04879i 3.16228
461.17 1.41421i −1.05752 + 2.80743i −2.00000 2.23607i 3.97030 + 1.49556i 11.9645 2.82843i −6.76329 5.93784i 3.16228
461.18 1.41421i −1.05752 2.80743i −2.00000 2.23607i 3.97030 1.49556i 11.9645 2.82843i −6.76329 + 5.93784i 3.16228
461.19 1.41421i −2.91669 0.702099i −2.00000 2.23607i −0.992918 + 4.12482i −8.52828 2.82843i 8.01411 + 4.09561i −3.16228
461.20 1.41421i −2.91669 + 0.702099i −2.00000 2.23607i −0.992918 4.12482i −8.52828 2.82843i 8.01411 4.09561i −3.16228
See all 56 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 461.56 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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 690.3.g.a 56
3.b odd 2 1 inner 690.3.g.a 56

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
690.3.g.a 56 1.a even 1 1 trivial
690.3.g.a 56 3.b odd 2 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{3}^{\mathrm{new}}(690, [\chi])$$.