Newspace parameters
Level: | \( N \) | \(=\) | \( 666 = 2 \cdot 3^{2} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 666.bj (of order \(18\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(5.31803677462\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(4\) over \(\Q(\zeta_{18})\) |
Twist minimal: | no (minimal twist has level 222) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{18}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | −3.98902 | + | 0.703371i | 0 | −0.673690 | − | 3.82069i | 0.866025 | + | 0.500000i | 0 | 2.02528 | + | 3.50788i | ||||||
289.2 | −0.342020 | − | 0.939693i | 0 | −0.766044 | + | 0.642788i | 3.64700 | − | 0.643064i | 0 | −0.469097 | − | 2.66038i | 0.866025 | + | 0.500000i | 0 | −1.85163 | − | 3.20712i | ||||||
289.3 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | −1.76204 | + | 0.310695i | 0 | −0.343122 | − | 1.94594i | −0.866025 | − | 0.500000i | 0 | −0.894610 | − | 1.54951i | ||||||
289.4 | 0.342020 | + | 0.939693i | 0 | −0.766044 | + | 0.642788i | 2.10406 | − | 0.371002i | 0 | 0.485910 | + | 2.75573i | −0.866025 | − | 0.500000i | 0 | 1.06826 | + | 1.85028i | ||||||
361.1 | −0.984808 | + | 0.173648i | 0 | 0.939693 | − | 0.342020i | −0.980233 | + | 1.16820i | 0 | −2.89806 | − | 2.43176i | −0.866025 | + | 0.500000i | 0 | 0.762486 | − | 1.32066i | ||||||
361.2 | −0.984808 | + | 0.173648i | 0 | 0.939693 | − | 0.342020i | −0.00457474 | + | 0.00545196i | 0 | 2.74008 | + | 2.29920i | −0.866025 | + | 0.500000i | 0 | 0.00355851 | − | 0.00616353i | ||||||
361.3 | 0.984808 | − | 0.173648i | 0 | 0.939693 | − | 0.342020i | −1.34200 | + | 1.59933i | 0 | 1.56789 | + | 1.31561i | 0.866025 | − | 0.500000i | 0 | −1.04389 | + | 1.80807i | ||||||
361.4 | 0.984808 | − | 0.173648i | 0 | 0.939693 | − | 0.342020i | 2.32681 | − | 2.77298i | 0 | −2.40991 | − | 2.02215i | 0.866025 | − | 0.500000i | 0 | 1.80993 | − | 3.13490i | ||||||
469.1 | −0.642788 | − | 0.766044i | 0 | −0.173648 | + | 0.984808i | −0.743263 | − | 2.04210i | 0 | −2.00421 | + | 0.729473i | 0.866025 | − | 0.500000i | 0 | −1.08658 | + | 1.88201i | ||||||
469.2 | −0.642788 | − | 0.766044i | 0 | −0.173648 | + | 0.984808i | 0.100476 | + | 0.276055i | 0 | 2.48902 | − | 0.905929i | 0.866025 | − | 0.500000i | 0 | 0.146886 | − | 0.254414i | ||||||
469.3 | 0.642788 | + | 0.766044i | 0 | −0.173648 | + | 0.984808i | −0.753389 | − | 2.06992i | 0 | −2.61124 | + | 0.950413i | −0.866025 | + | 0.500000i | 0 | 1.10138 | − | 1.90765i | ||||||
469.4 | 0.642788 | + | 0.766044i | 0 | −0.173648 | + | 0.984808i | 1.39618 | + | 3.83596i | 0 | 1.12643 | − | 0.409987i | −0.866025 | + | 0.500000i | 0 | −2.04107 | + | 3.53524i | ||||||
559.1 | −0.984808 | − | 0.173648i | 0 | 0.939693 | + | 0.342020i | −0.980233 | − | 1.16820i | 0 | −2.89806 | + | 2.43176i | −0.866025 | − | 0.500000i | 0 | 0.762486 | + | 1.32066i | ||||||
559.2 | −0.984808 | − | 0.173648i | 0 | 0.939693 | + | 0.342020i | −0.00457474 | − | 0.00545196i | 0 | 2.74008 | − | 2.29920i | −0.866025 | − | 0.500000i | 0 | 0.00355851 | + | 0.00616353i | ||||||
559.3 | 0.984808 | + | 0.173648i | 0 | 0.939693 | + | 0.342020i | −1.34200 | − | 1.59933i | 0 | 1.56789 | − | 1.31561i | 0.866025 | + | 0.500000i | 0 | −1.04389 | − | 1.80807i | ||||||
559.4 | 0.984808 | + | 0.173648i | 0 | 0.939693 | + | 0.342020i | 2.32681 | + | 2.77298i | 0 | −2.40991 | + | 2.02215i | 0.866025 | + | 0.500000i | 0 | 1.80993 | + | 3.13490i | ||||||
595.1 | −0.642788 | + | 0.766044i | 0 | −0.173648 | − | 0.984808i | −0.743263 | + | 2.04210i | 0 | −2.00421 | − | 0.729473i | 0.866025 | + | 0.500000i | 0 | −1.08658 | − | 1.88201i | ||||||
595.2 | −0.642788 | + | 0.766044i | 0 | −0.173648 | − | 0.984808i | 0.100476 | − | 0.276055i | 0 | 2.48902 | + | 0.905929i | 0.866025 | + | 0.500000i | 0 | 0.146886 | + | 0.254414i | ||||||
595.3 | 0.642788 | − | 0.766044i | 0 | −0.173648 | − | 0.984808i | −0.753389 | + | 2.06992i | 0 | −2.61124 | − | 0.950413i | −0.866025 | − | 0.500000i | 0 | 1.10138 | + | 1.90765i | ||||||
595.4 | 0.642788 | − | 0.766044i | 0 | −0.173648 | − | 0.984808i | 1.39618 | − | 3.83596i | 0 | 1.12643 | + | 0.409987i | −0.866025 | − | 0.500000i | 0 | −2.04107 | − | 3.53524i | ||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
37.h | even | 18 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 666.2.bj.e | 24 | |
3.b | odd | 2 | 1 | 222.2.n.a | ✓ | 24 | |
37.h | even | 18 | 1 | inner | 666.2.bj.e | 24 | |
111.n | odd | 18 | 1 | 222.2.n.a | ✓ | 24 | |
111.q | even | 36 | 1 | 8214.2.a.bh | 12 | ||
111.q | even | 36 | 1 | 8214.2.a.bj | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
222.2.n.a | ✓ | 24 | 3.b | odd | 2 | 1 | |
222.2.n.a | ✓ | 24 | 111.n | odd | 18 | 1 | |
666.2.bj.e | 24 | 1.a | even | 1 | 1 | trivial | |
666.2.bj.e | 24 | 37.h | even | 18 | 1 | inner | |
8214.2.a.bh | 12 | 111.q | even | 36 | 1 | ||
8214.2.a.bj | 12 | 111.q | even | 36 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{24} - 12 T_{5}^{22} + 54 T_{5}^{21} - 51 T_{5}^{20} - 882 T_{5}^{19} + 260 T_{5}^{18} + 1998 T_{5}^{17} + 6009 T_{5}^{16} - 49842 T_{5}^{15} + 53868 T_{5}^{14} + 927216 T_{5}^{13} + 2835415 T_{5}^{12} + 1361700 T_{5}^{11} + \cdots + 729 \)
acting on \(S_{2}^{\mathrm{new}}(666, [\chi])\).