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 | −1.68613 | − | 0.396187i | − | 1.00000i | 2.10894 | − | 0.743206i | 1.47242 | − | 0.912127i | −2.17056 | − | 2.17056i | 0.707107 | + | 0.707107i | 2.68607 | + | 1.33605i | −0.965722 | + | 2.01677i | |
47.2 | −0.707107 | + | 0.707107i | −1.51580 | + | 0.838057i | − | 1.00000i | −2.23509 | − | 0.0661068i | 0.479239 | − | 1.66443i | 2.39575 | + | 2.39575i | 0.707107 | + | 0.707107i | 1.59532 | − | 2.54066i | 1.62719 | − | 1.53370i | |
47.3 | −0.707107 | + | 0.707107i | −1.16628 | − | 1.28054i | − | 1.00000i | −1.31868 | − | 1.80585i | 1.73017 | + | 0.0807905i | 1.43599 | + | 1.43599i | 0.707107 | + | 0.707107i | −0.279562 | + | 2.98695i | 2.20937 | + | 0.344476i | |
47.4 | −0.707107 | + | 0.707107i | −0.144577 | − | 1.72601i | − | 1.00000i | 1.25275 | + | 1.85219i | 1.32270 | + | 1.11824i | −2.31531 | − | 2.31531i | 0.707107 | + | 0.707107i | −2.95819 | + | 0.499082i | −2.19553 | − | 0.423868i | |
47.5 | −0.707107 | + | 0.707107i | −0.0456124 | − | 1.73145i | − | 1.00000i | 1.54063 | − | 1.62063i | 1.25657 | + | 1.19207i | 0.528026 | + | 0.528026i | 0.707107 | + | 0.707107i | −2.99584 | + | 0.157951i | 0.0565714 | + | 2.23535i | |
47.6 | −0.707107 | + | 0.707107i | 0.906765 | + | 1.47573i | − | 1.00000i | −1.96911 | − | 1.05954i | −1.68468 | − | 0.402318i | −0.621187 | − | 0.621187i | 0.707107 | + | 0.707107i | −1.35555 | + | 2.67628i | 2.14157 | − | 0.643162i | |
47.7 | −0.707107 | + | 0.707107i | 1.12660 | − | 1.31559i | − | 1.00000i | −0.192443 | + | 2.22777i | 0.133641 | + | 1.72689i | 1.96010 | + | 1.96010i | 0.707107 | + | 0.707107i | −0.461565 | − | 2.96428i | −1.43919 | − | 1.71135i | |
47.8 | −0.707107 | + | 0.707107i | 1.40373 | + | 1.01467i | − | 1.00000i | 2.22721 | − | 0.198845i | −1.71006 | + | 0.275105i | 0.787190 | + | 0.787190i | 0.707107 | + | 0.707107i | 0.940895 | + | 2.84863i | −1.43427 | + | 1.71548i | |
47.9 | 0.707107 | − | 0.707107i | −1.47573 | − | 0.906765i | − | 1.00000i | 1.96911 | + | 1.05954i | −1.68468 | + | 0.402318i | −0.621187 | − | 0.621187i | −0.707107 | − | 0.707107i | 1.35555 | + | 2.67628i | 2.14157 | − | 0.643162i | |
47.10 | 0.707107 | − | 0.707107i | −1.01467 | − | 1.40373i | − | 1.00000i | −2.22721 | + | 0.198845i | −1.71006 | − | 0.275105i | 0.787190 | + | 0.787190i | −0.707107 | − | 0.707107i | −0.940895 | + | 2.84863i | −1.43427 | + | 1.71548i | |
47.11 | 0.707107 | − | 0.707107i | −0.838057 | + | 1.51580i | − | 1.00000i | 2.23509 | + | 0.0661068i | 0.479239 | + | 1.66443i | 2.39575 | + | 2.39575i | −0.707107 | − | 0.707107i | −1.59532 | − | 2.54066i | 1.62719 | − | 1.53370i | |
47.12 | 0.707107 | − | 0.707107i | 0.396187 | + | 1.68613i | − | 1.00000i | −2.10894 | + | 0.743206i | 1.47242 | + | 0.912127i | −2.17056 | − | 2.17056i | −0.707107 | − | 0.707107i | −2.68607 | + | 1.33605i | −0.965722 | + | 2.01677i | |
47.13 | 0.707107 | − | 0.707107i | 1.28054 | + | 1.16628i | − | 1.00000i | 1.31868 | + | 1.80585i | 1.73017 | − | 0.0807905i | 1.43599 | + | 1.43599i | −0.707107 | − | 0.707107i | 0.279562 | + | 2.98695i | 2.20937 | + | 0.344476i | |
47.14 | 0.707107 | − | 0.707107i | 1.31559 | − | 1.12660i | − | 1.00000i | 0.192443 | − | 2.22777i | 0.133641 | − | 1.72689i | 1.96010 | + | 1.96010i | −0.707107 | − | 0.707107i | 0.461565 | − | 2.96428i | −1.43919 | − | 1.71135i | |
47.15 | 0.707107 | − | 0.707107i | 1.72601 | + | 0.144577i | − | 1.00000i | −1.25275 | − | 1.85219i | 1.32270 | − | 1.11824i | −2.31531 | − | 2.31531i | −0.707107 | − | 0.707107i | 2.95819 | + | 0.499082i | −2.19553 | − | 0.423868i | |
47.16 | 0.707107 | − | 0.707107i | 1.73145 | + | 0.0456124i | − | 1.00000i | −1.54063 | + | 1.62063i | 1.25657 | − | 1.19207i | 0.528026 | + | 0.528026i | −0.707107 | − | 0.707107i | 2.99584 | + | 0.157951i | 0.0565714 | + | 2.23535i | |
323.1 | −0.707107 | − | 0.707107i | −1.68613 | + | 0.396187i | 1.00000i | 2.10894 | + | 0.743206i | 1.47242 | + | 0.912127i | −2.17056 | + | 2.17056i | 0.707107 | − | 0.707107i | 2.68607 | − | 1.33605i | −0.965722 | − | 2.01677i | ||
323.2 | −0.707107 | − | 0.707107i | −1.51580 | − | 0.838057i | 1.00000i | −2.23509 | + | 0.0661068i | 0.479239 | + | 1.66443i | 2.39575 | − | 2.39575i | 0.707107 | − | 0.707107i | 1.59532 | + | 2.54066i | 1.62719 | + | 1.53370i | ||
323.3 | −0.707107 | − | 0.707107i | −1.16628 | + | 1.28054i | 1.00000i | −1.31868 | + | 1.80585i | 1.73017 | − | 0.0807905i | 1.43599 | − | 1.43599i | 0.707107 | − | 0.707107i | −0.279562 | − | 2.98695i | 2.20937 | − | 0.344476i | ||
323.4 | −0.707107 | − | 0.707107i | −0.144577 | + | 1.72601i | 1.00000i | 1.25275 | − | 1.85219i | 1.32270 | − | 1.11824i | −2.31531 | + | 2.31531i | 0.707107 | − | 0.707107i | −2.95819 | − | 0.499082i | −2.19553 | + | 0.423868i | ||
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.f | ✓ | 32 |
3.b | odd | 2 | 1 | inner | 690.2.i.f | ✓ | 32 |
5.c | odd | 4 | 1 | inner | 690.2.i.f | ✓ | 32 |
15.e | even | 4 | 1 | inner | 690.2.i.f | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.i.f | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
690.2.i.f | ✓ | 32 | 3.b | odd | 2 | 1 | inner |
690.2.i.f | ✓ | 32 | 5.c | odd | 4 | 1 | inner |
690.2.i.f | ✓ | 32 | 15.e | even | 4 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{16} - 4 T_{7}^{15} + 8 T_{7}^{14} - 8 T_{7}^{13} + 207 T_{7}^{12} - 816 T_{7}^{11} + 1640 T_{7}^{10} - 1636 T_{7}^{9} + 10817 T_{7}^{8} - 41128 T_{7}^{7} + 84512 T_{7}^{6} - 77072 T_{7}^{5} + 40056 T_{7}^{4} + \cdots + 19600 \)
acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).