Newspace parameters
Level: | \( N \) | \(=\) | \( 1161 = 3^{3} \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1161.f (of order \(3\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(9.27063167467\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) over \(\Q(\zeta_{3})\) |
Twist minimal: | no (minimal twist has level 387) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
388.1 | −1.38785 | − | 2.40383i | 0 | −2.85226 | + | 4.94026i | −1.14436 | + | 1.98209i | 0 | 2.15274 | + | 3.72866i | 10.2826 | 0 | 6.35282 | ||||||||||
388.2 | −1.28103 | − | 2.21881i | 0 | −2.28209 | + | 3.95269i | 1.16256 | − | 2.01361i | 0 | −0.00975820 | − | 0.0169017i | 6.56959 | 0 | −5.95711 | ||||||||||
388.3 | −1.18355 | − | 2.04996i | 0 | −1.80157 | + | 3.12041i | 0.867534 | − | 1.50261i | 0 | −2.31405 | − | 4.00804i | 3.79478 | 0 | −4.10707 | ||||||||||
388.4 | −1.16730 | − | 2.02182i | 0 | −1.72517 | + | 2.98809i | −1.89009 | + | 3.27373i | 0 | −0.823076 | − | 1.42561i | 3.38597 | 0 | 8.82518 | ||||||||||
388.5 | −1.16214 | − | 2.01289i | 0 | −1.70115 | + | 2.94647i | −1.73647 | + | 3.00766i | 0 | −0.946105 | − | 1.63870i | 3.25933 | 0 | 8.07210 | ||||||||||
388.6 | −0.856434 | − | 1.48339i | 0 | −0.466959 | + | 0.808796i | −0.461242 | + | 0.798894i | 0 | −1.56362 | − | 2.70827i | −1.82606 | 0 | 1.58009 | ||||||||||
388.7 | −0.774348 | − | 1.34121i | 0 | −0.199230 | + | 0.345076i | 0.0851621 | − | 0.147505i | 0 | 0.931330 | + | 1.61311i | −2.48030 | 0 | −0.263781 | ||||||||||
388.8 | −0.535034 | − | 0.926706i | 0 | 0.427478 | − | 0.740413i | 1.71565 | − | 2.97160i | 0 | 0.0278005 | + | 0.0481519i | −3.05500 | 0 | −3.67173 | ||||||||||
388.9 | −0.352536 | − | 0.610611i | 0 | 0.751436 | − | 1.30153i | 0.464078 | − | 0.803807i | 0 | 2.55550 | + | 4.42626i | −2.46978 | 0 | −0.654418 | ||||||||||
388.10 | −0.342086 | − | 0.592511i | 0 | 0.765954 | − | 1.32667i | −0.741129 | + | 1.28367i | 0 | −1.90727 | − | 3.30349i | −2.41643 | 0 | 1.01412 | ||||||||||
388.11 | −0.254302 | − | 0.440465i | 0 | 0.870661 | − | 1.50803i | −1.90300 | + | 3.29610i | 0 | 0.340171 | + | 0.589193i | −1.90285 | 0 | 1.93575 | ||||||||||
388.12 | 0.0942911 | + | 0.163317i | 0 | 0.982218 | − | 1.70125i | −1.89854 | + | 3.28836i | 0 | 2.00490 | + | 3.47259i | 0.747622 | 0 | −0.716061 | ||||||||||
388.13 | 0.110277 | + | 0.191005i | 0 | 0.975678 | − | 1.68992i | 0.797629 | − | 1.38153i | 0 | −0.709781 | − | 1.22938i | 0.871488 | 0 | 0.351841 | ||||||||||
388.14 | 0.391509 | + | 0.678114i | 0 | 0.693441 | − | 1.20108i | −1.47053 | + | 2.54704i | 0 | −1.24831 | − | 2.16213i | 2.65199 | 0 | −2.30291 | ||||||||||
388.15 | 0.562737 | + | 0.974689i | 0 | 0.366654 | − | 0.635063i | 1.19873 | − | 2.07626i | 0 | 0.174845 | + | 0.302841i | 3.07627 | 0 | 2.69827 | ||||||||||
388.16 | 0.875101 | + | 1.51572i | 0 | −0.531605 | + | 0.920767i | −1.34854 | + | 2.33575i | 0 | −1.33599 | − | 2.31400i | 1.63957 | 0 | −4.72045 | ||||||||||
388.17 | 0.911404 | + | 1.57860i | 0 | −0.661315 | + | 1.14543i | −0.216442 | + | 0.374888i | 0 | 0.345529 | + | 0.598473i | 1.23472 | 0 | −0.789063 | ||||||||||
388.18 | 0.988037 | + | 1.71133i | 0 | −0.952435 | + | 1.64967i | −0.717842 | + | 1.24334i | 0 | −2.37609 | − | 4.11551i | 0.187983 | 0 | −2.83702 | ||||||||||
388.19 | 1.04491 | + | 1.80984i | 0 | −1.18368 | + | 2.05019i | 0.136674 | − | 0.236726i | 0 | 1.21012 | + | 2.09599i | −0.767710 | 0 | 0.571248 | ||||||||||
388.20 | 1.31834 | + | 2.28344i | 0 | −2.47606 | + | 4.28866i | −1.39983 | + | 2.42457i | 0 | 1.99111 | + | 3.44870i | −7.78382 | 0 | −7.38182 | ||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1161.2.f.d | 40 | |
3.b | odd | 2 | 1 | 387.2.f.d | ✓ | 40 | |
9.c | even | 3 | 1 | inner | 1161.2.f.d | 40 | |
9.c | even | 3 | 1 | 3483.2.a.u | 20 | ||
9.d | odd | 6 | 1 | 387.2.f.d | ✓ | 40 | |
9.d | odd | 6 | 1 | 3483.2.a.t | 20 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
387.2.f.d | ✓ | 40 | 3.b | odd | 2 | 1 | |
387.2.f.d | ✓ | 40 | 9.d | odd | 6 | 1 | |
1161.2.f.d | 40 | 1.a | even | 1 | 1 | trivial | |
1161.2.f.d | 40 | 9.c | even | 3 | 1 | inner | |
3483.2.a.t | 20 | 9.d | odd | 6 | 1 | ||
3483.2.a.u | 20 | 9.c | even | 3 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{40} + 6 T_{2}^{39} + 49 T_{2}^{38} + 196 T_{2}^{37} + 1015 T_{2}^{36} + 3316 T_{2}^{35} + 13489 T_{2}^{34} + 37496 T_{2}^{33} + 127462 T_{2}^{32} + 308874 T_{2}^{31} + 912872 T_{2}^{30} + 1954589 T_{2}^{29} + \cdots + 6561 \)
acting on \(S_{2}^{\mathrm{new}}(1161, [\chi])\).