Newspace parameters
Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 231.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(6.29429410672\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
166.1 | −1.77117 | + | 3.06776i | 1.50000 | − | 0.866025i | −4.27410 | − | 7.40296i | −7.27902 | − | 4.20254i | 6.13552i | 6.51694 | + | 2.55529i | 16.1113 | 1.50000 | − | 2.59808i | 25.7848 | − | 14.8869i | ||||
166.2 | −1.58858 | + | 2.75151i | 1.50000 | − | 0.866025i | −3.04719 | − | 5.27789i | −0.961730 | − | 0.555255i | 5.50301i | −6.99783 | − | 0.174356i | 6.65421 | 1.50000 | − | 2.59808i | 3.05558 | − | 1.76414i | ||||
166.3 | −1.56641 | + | 2.71310i | 1.50000 | − | 0.866025i | −2.90728 | − | 5.03555i | 1.45988 | + | 0.842865i | 5.42620i | 3.76448 | − | 5.90159i | 5.68466 | 1.50000 | − | 2.59808i | −4.57355 | + | 2.64054i | ||||
166.4 | −1.19231 | + | 2.06513i | 1.50000 | − | 0.866025i | −0.843183 | − | 1.46044i | 2.80248 | + | 1.61801i | 4.13027i | −6.38656 | + | 2.86564i | −5.51712 | 1.50000 | − | 2.59808i | −6.68282 | + | 3.85833i | ||||
166.5 | −0.960419 | + | 1.66349i | 1.50000 | − | 0.866025i | 0.155192 | + | 0.268801i | 5.90040 | + | 3.40660i | 3.32699i | 5.01823 | + | 4.88030i | −8.27955 | 1.50000 | − | 2.59808i | −11.3337 | + | 6.54352i | ||||
166.6 | −0.702448 | + | 1.21667i | 1.50000 | − | 0.866025i | 1.01313 | + | 1.75480i | −7.60842 | − | 4.39272i | 2.43335i | −0.216994 | + | 6.99664i | −8.46628 | 1.50000 | − | 2.59808i | 10.6890 | − | 6.17131i | ||||
166.7 | −0.0824997 | + | 0.142894i | 1.50000 | − | 0.866025i | 1.98639 | + | 3.44052i | 6.61692 | + | 3.82028i | 0.285787i | 2.84959 | − | 6.39373i | −1.31550 | 1.50000 | − | 2.59808i | −1.09179 | + | 0.630344i | ||||
166.8 | 0.235283 | − | 0.407522i | 1.50000 | − | 0.866025i | 1.88928 | + | 3.27234i | 1.22197 | + | 0.705503i | − | 0.815044i | −1.82511 | + | 6.75788i | 3.66033 | 1.50000 | − | 2.59808i | 0.575016 | − | 0.331985i | |||
166.9 | 0.482489 | − | 0.835695i | 1.50000 | − | 0.866025i | 1.53441 | + | 2.65767i | −3.16765 | − | 1.82885i | − | 1.67139i | 6.98831 | − | 0.404391i | 6.82125 | 1.50000 | − | 2.59808i | −3.05672 | + | 1.76480i | |||
166.10 | 1.06901 | − | 1.85157i | 1.50000 | − | 0.866025i | −0.285549 | − | 0.494586i | −5.40395 | − | 3.11997i | − | 3.70315i | 0.976308 | − | 6.93158i | 7.33104 | 1.50000 | − | 2.59808i | −11.5537 | + | 6.67054i | |||
166.11 | 1.10214 | − | 1.90897i | 1.50000 | − | 0.866025i | −0.429434 | − | 0.743802i | 5.58603 | + | 3.22509i | − | 3.81793i | −6.22295 | − | 3.20545i | 6.92395 | 1.50000 | − | 2.59808i | 12.3132 | − | 7.10903i | |||
166.12 | 1.37561 | − | 2.38263i | 1.50000 | − | 0.866025i | −1.78461 | − | 3.09104i | 3.80869 | + | 2.19895i | − | 4.76526i | 5.12759 | + | 4.76527i | 1.18516 | 1.50000 | − | 2.59808i | 10.4786 | − | 6.04979i | |||
166.13 | 1.68556 | − | 2.91948i | 1.50000 | − | 0.866025i | −3.68224 | − | 6.37782i | −3.91191 | − | 2.25854i | − | 5.83896i | −6.46810 | + | 2.67650i | −11.3421 | 1.50000 | − | 2.59808i | −13.1875 | + | 7.61382i | |||
166.14 | 1.91374 | − | 3.31470i | 1.50000 | − | 0.866025i | −5.32482 | − | 9.22286i | 0.936300 | + | 0.540573i | − | 6.62940i | 4.87609 | − | 5.02232i | −25.4514 | 1.50000 | − | 2.59808i | 3.58367 | − | 2.06904i | |||
199.1 | −1.77117 | − | 3.06776i | 1.50000 | + | 0.866025i | −4.27410 | + | 7.40296i | −7.27902 | + | 4.20254i | − | 6.13552i | 6.51694 | − | 2.55529i | 16.1113 | 1.50000 | + | 2.59808i | 25.7848 | + | 14.8869i | |||
199.2 | −1.58858 | − | 2.75151i | 1.50000 | + | 0.866025i | −3.04719 | + | 5.27789i | −0.961730 | + | 0.555255i | − | 5.50301i | −6.99783 | + | 0.174356i | 6.65421 | 1.50000 | + | 2.59808i | 3.05558 | + | 1.76414i | |||
199.3 | −1.56641 | − | 2.71310i | 1.50000 | + | 0.866025i | −2.90728 | + | 5.03555i | 1.45988 | − | 0.842865i | − | 5.42620i | 3.76448 | + | 5.90159i | 5.68466 | 1.50000 | + | 2.59808i | −4.57355 | − | 2.64054i | |||
199.4 | −1.19231 | − | 2.06513i | 1.50000 | + | 0.866025i | −0.843183 | + | 1.46044i | 2.80248 | − | 1.61801i | − | 4.13027i | −6.38656 | − | 2.86564i | −5.51712 | 1.50000 | + | 2.59808i | −6.68282 | − | 3.85833i | |||
199.5 | −0.960419 | − | 1.66349i | 1.50000 | + | 0.866025i | 0.155192 | − | 0.268801i | 5.90040 | − | 3.40660i | − | 3.32699i | 5.01823 | − | 4.88030i | −8.27955 | 1.50000 | + | 2.59808i | −11.3337 | − | 6.54352i | |||
199.6 | −0.702448 | − | 1.21667i | 1.50000 | + | 0.866025i | 1.01313 | − | 1.75480i | −7.60842 | + | 4.39272i | − | 2.43335i | −0.216994 | − | 6.99664i | −8.46628 | 1.50000 | + | 2.59808i | 10.6890 | + | 6.17131i | |||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 231.3.m.b | ✓ | 28 |
7.d | odd | 6 | 1 | inner | 231.3.m.b | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
231.3.m.b | ✓ | 28 | 1.a | even | 1 | 1 | trivial |
231.3.m.b | ✓ | 28 | 7.d | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{28} + 44 T_{2}^{26} + 4 T_{2}^{25} + 1181 T_{2}^{24} + 132 T_{2}^{23} + 20428 T_{2}^{22} + 2094 T_{2}^{21} + 260734 T_{2}^{20} + 19404 T_{2}^{19} + 2437672 T_{2}^{18} + 71432 T_{2}^{17} + 17439685 T_{2}^{16} + \cdots + 8088336 \)
acting on \(S_{3}^{\mathrm{new}}(231, [\chi])\).