Newspace parameters
Level: | \( N \) | \(=\) | \( 690 = 2 \cdot 3 \cdot 5 \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 690.m (of order \(11\), degree \(10\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.50967773947\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{11})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{11}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | −0.959493 | + | 0.281733i | −2.43922 | − | 1.56759i | 0.415415 | − | 0.909632i | −0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i |
31.2 | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | −0.959493 | + | 0.281733i | −1.19214 | − | 0.766143i | 0.415415 | − | 0.909632i | −0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i |
31.3 | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | −0.959493 | + | 0.281733i | 4.15900 | + | 2.67283i | 0.415415 | − | 0.909632i | −0.654861 | + | 0.755750i | 0.841254 | − | 0.540641i |
121.1 | 0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | −0.654861 | + | 0.755750i | −0.498068 | + | 3.46414i | −0.959493 | + | 0.281733i | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i |
121.2 | 0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | −0.654861 | + | 0.755750i | −0.188781 | + | 1.31300i | −0.959493 | + | 0.281733i | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i |
121.3 | 0.415415 | − | 0.909632i | −0.959493 | − | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | −0.654861 | + | 0.755750i | 0.743473 | − | 5.17097i | −0.959493 | + | 0.281733i | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i |
151.1 | −0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | −0.142315 | − | 0.989821i | −2.83591 | − | 0.832699i | 0.841254 | − | 0.540641i | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i |
151.2 | −0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | −0.142315 | − | 0.989821i | −1.69063 | − | 0.496413i | 0.841254 | − | 0.540641i | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i |
151.3 | −0.654861 | − | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.415415 | − | 0.909632i | −0.142315 | − | 0.989821i | 2.52297 | + | 0.740810i | 0.841254 | − | 0.540641i | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i |
211.1 | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.654861 | − | 0.755750i | −0.498068 | − | 3.46414i | −0.959493 | − | 0.281733i | 0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i |
211.2 | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.654861 | − | 0.755750i | −0.188781 | − | 1.31300i | −0.959493 | − | 0.281733i | 0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i |
211.3 | 0.415415 | + | 0.909632i | −0.959493 | + | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.654861 | − | 0.755750i | 0.743473 | + | 5.17097i | −0.959493 | − | 0.281733i | 0.841254 | − | 0.540641i | −0.142315 | + | 0.989821i |
271.1 | −0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.841254 | − | 0.540641i | −1.37681 | − | 3.01480i | −0.654861 | − | 0.755750i | −0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i |
271.2 | −0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.841254 | − | 0.540641i | −0.117211 | − | 0.256656i | −0.654861 | − | 0.755750i | −0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i |
271.3 | −0.959493 | − | 0.281733i | −0.654861 | + | 0.755750i | 0.841254 | + | 0.540641i | −0.142315 | + | 0.989821i | 0.841254 | − | 0.540641i | 1.68254 | + | 3.68425i | −0.654861 | − | 0.755750i | −0.142315 | − | 0.989821i | 0.415415 | − | 0.909632i |
301.1 | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | 0.415415 | − | 0.909632i | −2.82773 | + | 3.26337i | −0.142315 | + | 0.989821i | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i |
301.2 | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | 0.415415 | − | 0.909632i | −1.70152 | + | 1.96366i | −0.142315 | + | 0.989821i | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i |
301.3 | 0.841254 | + | 0.540641i | −0.142315 | − | 0.989821i | 0.415415 | + | 0.909632i | −0.959493 | − | 0.281733i | 0.415415 | − | 0.909632i | 1.76004 | − | 2.03119i | −0.142315 | + | 0.989821i | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i |
331.1 | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | 0.841254 | + | 0.540641i | −1.37681 | + | 3.01480i | −0.654861 | + | 0.755750i | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i |
331.2 | −0.959493 | + | 0.281733i | −0.654861 | − | 0.755750i | 0.841254 | − | 0.540641i | −0.142315 | − | 0.989821i | 0.841254 | + | 0.540641i | −0.117211 | + | 0.256656i | −0.654861 | + | 0.755750i | −0.142315 | + | 0.989821i | 0.415415 | + | 0.909632i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
23.c | even | 11 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 690.2.m.g | ✓ | 30 |
23.c | even | 11 | 1 | inner | 690.2.m.g | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
690.2.m.g | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
690.2.m.g | ✓ | 30 | 23.c | even | 11 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \(43\!\cdots\!84\)\( T_{7}^{8} + \)\(13\!\cdots\!88\)\( T_{7}^{7} + \)\(28\!\cdots\!88\)\( T_{7}^{6} + \)\(42\!\cdots\!88\)\( T_{7}^{5} + \)\(45\!\cdots\!76\)\( T_{7}^{4} + \)\(33\!\cdots\!36\)\( T_{7}^{3} + \)\(15\!\cdots\!16\)\( T_{7}^{2} + \)\(35\!\cdots\!64\)\( T_{7} + 593713398784 \)">\(T_{7}^{30} + \cdots\) acting on \(S_{2}^{\mathrm{new}}(690, [\chi])\).