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: | \(38\) |
Relative dimension: | \(19\) 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 | −1.27673 | − | 2.21136i | 0 | −2.26008 | + | 3.91458i | 1.39974 | − | 2.42442i | 0 | 0.0769910 | + | 0.133352i | 6.43513 | 0 | −7.14837 | ||||||||||
388.2 | −1.12806 | − | 1.95386i | 0 | −1.54504 | + | 2.67608i | 0.722147 | − | 1.25080i | 0 | −0.444392 | − | 0.769710i | 2.45933 | 0 | −3.25850 | ||||||||||
388.3 | −1.09981 | − | 1.90493i | 0 | −1.41918 | + | 2.45809i | −0.829860 | + | 1.43736i | 0 | 1.66958 | + | 2.89180i | 1.84408 | 0 | 3.65077 | ||||||||||
388.4 | −0.887526 | − | 1.53724i | 0 | −0.575406 | + | 0.996632i | 1.84963 | − | 3.20365i | 0 | 0.0448503 | + | 0.0776831i | −1.50735 | 0 | −6.56638 | ||||||||||
388.5 | −0.880687 | − | 1.52540i | 0 | −0.551220 | + | 0.954742i | −0.398628 | + | 0.690444i | 0 | 0.958880 | + | 1.66083i | −1.58094 | 0 | 1.40427 | ||||||||||
388.6 | −0.649321 | − | 1.12466i | 0 | 0.156764 | − | 0.271524i | −1.38166 | + | 2.39311i | 0 | −0.516159 | − | 0.894013i | −3.00445 | 0 | 3.58856 | ||||||||||
388.7 | −0.326572 | − | 0.565640i | 0 | 0.786701 | − | 1.36261i | 1.96622 | − | 3.40559i | 0 | −2.10690 | − | 3.64925i | −2.33395 | 0 | −2.56845 | ||||||||||
388.8 | −0.152464 | − | 0.264076i | 0 | 0.953509 | − | 1.65153i | 1.13542 | − | 1.96661i | 0 | 1.25241 | + | 2.16924i | −1.19136 | 0 | −0.692446 | ||||||||||
388.9 | −0.0180090 | − | 0.0311926i | 0 | 0.999351 | − | 1.73093i | −0.638135 | + | 1.10528i | 0 | −1.94045 | − | 3.36095i | −0.144026 | 0 | 0.0459688 | ||||||||||
388.10 | 0.100298 | + | 0.173722i | 0 | 0.979881 | − | 1.69720i | 0.316846 | − | 0.548794i | 0 | 0.192248 | + | 0.332984i | 0.794314 | 0 | 0.127116 | ||||||||||
388.11 | 0.449049 | + | 0.777775i | 0 | 0.596711 | − | 1.03353i | −0.818891 | + | 1.41836i | 0 | 0.930316 | + | 1.61135i | 2.86800 | 0 | −1.47089 | ||||||||||
388.12 | 0.451326 | + | 0.781720i | 0 | 0.592609 | − | 1.02643i | −0.0415173 | + | 0.0719100i | 0 | −1.41989 | − | 2.45931i | 2.87515 | 0 | −0.0749514 | ||||||||||
388.13 | 0.651400 | + | 1.12826i | 0 | 0.151357 | − | 0.262158i | 1.84461 | − | 3.19495i | 0 | −1.71786 | − | 2.97543i | 2.99997 | 0 | 4.80630 | ||||||||||
388.14 | 0.724508 | + | 1.25489i | 0 | −0.0498244 | + | 0.0862984i | −1.28186 | + | 2.22025i | 0 | 0.350586 | + | 0.607232i | 2.75364 | 0 | −3.71487 | ||||||||||
388.15 | 0.987453 | + | 1.71032i | 0 | −0.950127 | + | 1.64567i | −1.66142 | + | 2.87766i | 0 | 2.32616 | + | 4.02902i | 0.196989 | 0 | −6.56230 | ||||||||||
388.16 | 1.09093 | + | 1.88955i | 0 | −1.38028 | + | 2.39071i | 0.845931 | − | 1.46520i | 0 | −1.08430 | − | 1.87806i | −1.65943 | 0 | 3.69142 | ||||||||||
388.17 | 1.26912 | + | 2.19819i | 0 | −2.22135 | + | 3.84749i | 0.946835 | − | 1.63997i | 0 | 0.412403 | + | 0.714304i | −6.20016 | 0 | 4.80660 | ||||||||||
388.18 | 1.33284 | + | 2.30855i | 0 | −2.55293 | + | 4.42180i | −0.659842 | + | 1.14288i | 0 | −0.548434 | − | 0.949916i | −8.27921 | 0 | −3.51785 | ||||||||||
388.19 | 1.36225 | + | 2.35949i | 0 | −2.71146 | + | 4.69638i | 1.18444 | − | 2.05150i | 0 | −1.93605 | − | 3.35333i | −9.32575 | 0 | 6.45400 | ||||||||||
775.1 | −1.27673 | + | 2.21136i | 0 | −2.26008 | − | 3.91458i | 1.39974 | + | 2.42442i | 0 | 0.0769910 | − | 0.133352i | 6.43513 | 0 | −7.14837 | ||||||||||
See all 38 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.c | 38 | |
3.b | odd | 2 | 1 | 387.2.f.c | ✓ | 38 | |
9.c | even | 3 | 1 | inner | 1161.2.f.c | 38 | |
9.c | even | 3 | 1 | 3483.2.a.r | 19 | ||
9.d | odd | 6 | 1 | 387.2.f.c | ✓ | 38 | |
9.d | odd | 6 | 1 | 3483.2.a.s | 19 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
387.2.f.c | ✓ | 38 | 3.b | odd | 2 | 1 | |
387.2.f.c | ✓ | 38 | 9.d | odd | 6 | 1 | |
1161.2.f.c | 38 | 1.a | even | 1 | 1 | trivial | |
1161.2.f.c | 38 | 9.c | even | 3 | 1 | inner | |
3483.2.a.r | 19 | 9.c | even | 3 | 1 | ||
3483.2.a.s | 19 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{38} - 4 T_{2}^{37} + 38 T_{2}^{36} - 112 T_{2}^{35} + 703 T_{2}^{34} - 1772 T_{2}^{33} + 8664 T_{2}^{32} - 18846 T_{2}^{31} + 77334 T_{2}^{30} - 148750 T_{2}^{29} + 529864 T_{2}^{28} - 903317 T_{2}^{27} + 2841518 T_{2}^{26} + \cdots + 81 \)
acting on \(S_{2}^{\mathrm{new}}(1161, [\chi])\).