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.
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 |
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 \(S_{3}^{\mathrm{new}}(690, [\chi])\).