Newspace parameters
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.i (of order \(4\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.50967773947\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
47.1 | −0.707107 | + | 0.707107i | −1.42268 | − | 0.987925i | − | 1.00000i | 2.16582 | + | 0.556076i | 1.70455 | − | 0.307415i | 3.32202 | + | 3.32202i | 0.707107 | + | 0.707107i | 1.04801 | + | 2.81099i | −1.92467 | + | 1.13826i | |
47.2 | −0.707107 | + | 0.707107i | −1.35428 | + | 1.07978i | − | 1.00000i | −1.85580 | + | 1.24740i | 0.194102 | − | 1.72114i | −2.75926 | − | 2.75926i | 0.707107 | + | 0.707107i | 0.668152 | − | 2.92465i | 0.430209 | − | 2.19429i | |
47.3 | −0.707107 | + | 0.707107i | −0.639676 | − | 1.60960i | − | 1.00000i | −1.63199 | + | 1.52859i | 1.59048 | + | 0.685841i | 0.772441 | + | 0.772441i | 0.707107 | + | 0.707107i | −2.18163 | + | 2.05924i | 0.0731134 | − | 2.23487i | |
47.4 | −0.707107 | + | 0.707107i | 0.451971 | + | 1.67204i | − | 1.00000i | 0.664484 | + | 2.13506i | −1.50190 | − | 0.862720i | 2.34139 | + | 2.34139i | 0.707107 | + | 0.707107i | −2.59144 | + | 1.51143i | −1.97957 | − | 1.03985i | |
47.5 | −0.707107 | + | 0.707107i | 0.471935 | − | 1.66652i | − | 1.00000i | −1.55923 | − | 1.60274i | 0.844697 | + | 1.51211i | −3.44975 | − | 3.44975i | 0.707107 | + | 0.707107i | −2.55455 | − | 1.57298i | 2.23586 | + | 0.0307665i | |
47.6 | −0.707107 | + | 0.707107i | 1.31856 | + | 1.12312i | − | 1.00000i | −1.11299 | − | 1.93940i | −1.72653 | + | 0.138191i | 1.56700 | + | 1.56700i | 0.707107 | + | 0.707107i | 0.477181 | + | 2.96181i | 2.15836 | + | 0.584357i | |
47.7 | −0.707107 | + | 0.707107i | 1.40859 | + | 1.00790i | − | 1.00000i | −0.318132 | + | 2.21332i | −1.70872 | + | 0.283328i | −2.49652 | − | 2.49652i | 0.707107 | + | 0.707107i | 0.968258 | + | 2.83945i | −1.34010 | − | 1.79001i | |
47.8 | −0.707107 | + | 0.707107i | 1.47268 | − | 0.911701i | − | 1.00000i | 2.23363 | + | 0.104337i | −0.396675 | + | 1.68602i | −1.29732 | − | 1.29732i | 0.707107 | + | 0.707107i | 1.33760 | − | 2.68530i | −1.65319 | + | 1.50564i | |
47.9 | 0.707107 | − | 0.707107i | −1.67204 | − | 0.451971i | − | 1.00000i | −0.664484 | − | 2.13506i | −1.50190 | + | 0.862720i | 2.34139 | + | 2.34139i | −0.707107 | − | 0.707107i | 2.59144 | + | 1.51143i | −1.97957 | − | 1.03985i | |
47.10 | 0.707107 | − | 0.707107i | −1.12312 | − | 1.31856i | − | 1.00000i | 1.11299 | + | 1.93940i | −1.72653 | − | 0.138191i | 1.56700 | + | 1.56700i | −0.707107 | − | 0.707107i | −0.477181 | + | 2.96181i | 2.15836 | + | 0.584357i | |
47.11 | 0.707107 | − | 0.707107i | −1.07978 | + | 1.35428i | − | 1.00000i | 1.85580 | − | 1.24740i | 0.194102 | + | 1.72114i | −2.75926 | − | 2.75926i | −0.707107 | − | 0.707107i | −0.668152 | − | 2.92465i | 0.430209 | − | 2.19429i | |
47.12 | 0.707107 | − | 0.707107i | −1.00790 | − | 1.40859i | − | 1.00000i | 0.318132 | − | 2.21332i | −1.70872 | − | 0.283328i | −2.49652 | − | 2.49652i | −0.707107 | − | 0.707107i | −0.968258 | + | 2.83945i | −1.34010 | − | 1.79001i | |
47.13 | 0.707107 | − | 0.707107i | 0.911701 | − | 1.47268i | − | 1.00000i | −2.23363 | − | 0.104337i | −0.396675 | − | 1.68602i | −1.29732 | − | 1.29732i | −0.707107 | − | 0.707107i | −1.33760 | − | 2.68530i | −1.65319 | + | 1.50564i | |
47.14 | 0.707107 | − | 0.707107i | 0.987925 | + | 1.42268i | − | 1.00000i | −2.16582 | − | 0.556076i | 1.70455 | + | 0.307415i | 3.32202 | + | 3.32202i | −0.707107 | − | 0.707107i | −1.04801 | + | 2.81099i | −1.92467 | + | 1.13826i | |
47.15 | 0.707107 | − | 0.707107i | 1.60960 | + | 0.639676i | − | 1.00000i | 1.63199 | − | 1.52859i | 1.59048 | − | 0.685841i | 0.772441 | + | 0.772441i | −0.707107 | − | 0.707107i | 2.18163 | + | 2.05924i | 0.0731134 | − | 2.23487i | |
47.16 | 0.707107 | − | 0.707107i | 1.66652 | − | 0.471935i | − | 1.00000i | 1.55923 | + | 1.60274i | 0.844697 | − | 1.51211i | −3.44975 | − | 3.44975i | −0.707107 | − | 0.707107i | 2.55455 | − | 1.57298i | 2.23586 | + | 0.0307665i | |
323.1 | −0.707107 | − | 0.707107i | −1.42268 | + | 0.987925i | 1.00000i | 2.16582 | − | 0.556076i | 1.70455 | + | 0.307415i | 3.32202 | − | 3.32202i | 0.707107 | − | 0.707107i | 1.04801 | − | 2.81099i | −1.92467 | − | 1.13826i | ||
323.2 | −0.707107 | − | 0.707107i | −1.35428 | − | 1.07978i | 1.00000i | −1.85580 | − | 1.24740i | 0.194102 | + | 1.72114i | −2.75926 | + | 2.75926i | 0.707107 | − | 0.707107i | 0.668152 | + | 2.92465i | 0.430209 | + | 2.19429i | ||
323.3 | −0.707107 | − | 0.707107i | −0.639676 | + | 1.60960i | 1.00000i | −1.63199 | − | 1.52859i | 1.59048 | − | 0.685841i | 0.772441 | − | 0.772441i | 0.707107 | − | 0.707107i | −2.18163 | − | 2.05924i | 0.0731134 | + | 2.23487i | ||
323.4 | −0.707107 | − | 0.707107i | 0.451971 | − | 1.67204i | 1.00000i | 0.664484 | − | 2.13506i | −1.50190 | + | 0.862720i | 2.34139 | − | 2.34139i | 0.707107 | − | 0.707107i | −2.59144 | − | 1.51143i | −1.97957 | + | 1.03985i | ||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
5.c | odd | 4 | 1 | inner |
15.e | even | 4 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.2.i.e | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 690.2.i.e | ✓ | 32 |
5.c | odd | 4 | 1 | inner | 690.2.i.e | ✓ | 32 |
15.e | even | 4 | 1 | inner | 690.2.i.e | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.i.e | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
690.2.i.e | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
690.2.i.e | ✓ | 32 | 5.c | odd | 4 | 1 | inner |
690.2.i.e | ✓ | 32 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \(T_{7}^{16} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).