Newspace parameters
Level: | \( N \) | \(=\) | \( 378 = 2 \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 378.u (of order \(9\), degree \(6\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.01834519640\) |
Analytic rank: | \(0\) |
Dimension: | \(30\) |
Relative dimension: | \(5\) over \(\Q(\zeta_{9})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{9}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
43.1 | 0.766044 | + | 0.642788i | −1.62854 | − | 0.589807i | 0.173648 | + | 0.984808i | −1.36620 | − | 0.497255i | −0.868410 | − | 1.49862i | −0.173648 | + | 0.984808i | −0.500000 | + | 0.866025i | 2.30426 | + | 1.92104i | −0.726937 | − | 1.25909i |
43.2 | 0.766044 | + | 0.642788i | −0.764934 | − | 1.55399i | 0.173648 | + | 0.984808i | 3.01809 | + | 1.09849i | 0.412911 | − | 1.68211i | −0.173648 | + | 0.984808i | −0.500000 | + | 0.866025i | −1.82975 | + | 2.37739i | 1.60589 | + | 2.78148i |
43.3 | 0.766044 | + | 0.642788i | −0.167318 | + | 1.72395i | 0.173648 | + | 0.984808i | 2.66798 | + | 0.971065i | −1.23631 | + | 1.21307i | −0.173648 | + | 0.984808i | −0.500000 | + | 0.866025i | −2.94401 | − | 0.576895i | 1.41960 | + | 2.45882i |
43.4 | 0.766044 | + | 0.642788i | 0.503821 | + | 1.65716i | 0.173648 | + | 0.984808i | −4.01520 | − | 1.46142i | −0.679250 | + | 1.59330i | −0.173648 | + | 0.984808i | −0.500000 | + | 0.866025i | −2.49233 | + | 1.66982i | −2.13645 | − | 3.70043i |
43.5 | 0.766044 | + | 0.642788i | 1.73061 | − | 0.0705192i | 0.173648 | + | 0.984808i | −0.918005 | − | 0.334126i | 1.37106 | + | 1.05840i | −0.173648 | + | 0.984808i | −0.500000 | + | 0.866025i | 2.99005 | − | 0.244083i | −0.488460 | − | 0.846038i |
85.1 | −0.939693 | − | 0.342020i | −1.73005 | + | 0.0833302i | 0.766044 | + | 0.642788i | 0.111050 | − | 0.629795i | 1.65421 | + | 0.513405i | −0.766044 | + | 0.642788i | −0.500000 | − | 0.866025i | 2.98611 | − | 0.288330i | −0.319755 | + | 0.553832i |
85.2 | −0.939693 | − | 0.342020i | −1.21448 | − | 1.23493i | 0.766044 | + | 0.642788i | −0.386963 | + | 2.19458i | 0.718864 | + | 1.57583i | −0.766044 | + | 0.642788i | −0.500000 | − | 0.866025i | −0.0500961 | + | 2.99958i | 1.11422 | − | 1.92988i |
85.3 | −0.939693 | − | 0.342020i | 0.344942 | − | 1.69736i | 0.766044 | + | 0.642788i | −0.445239 | + | 2.52508i | −0.904669 | + | 1.47702i | −0.766044 | + | 0.642788i | −0.500000 | − | 0.866025i | −2.76203 | − | 1.17098i | 1.28201 | − | 2.22051i |
85.4 | −0.939693 | − | 0.342020i | 1.22833 | + | 1.22115i | 0.766044 | + | 0.642788i | −0.0262315 | + | 0.148766i | −0.736599 | − | 1.56762i | −0.766044 | + | 0.642788i | −0.500000 | − | 0.866025i | 0.0176026 | + | 2.99995i | 0.0755306 | − | 0.130823i |
85.5 | −0.939693 | − | 0.342020i | 1.63729 | − | 0.565046i | 0.766044 | + | 0.642788i | 0.654987 | − | 3.71462i | −1.73181 | − | 0.0290168i | −0.766044 | + | 0.642788i | −0.500000 | − | 0.866025i | 2.36145 | − | 1.85029i | −1.88596 | + | 3.26658i |
169.1 | −0.939693 | + | 0.342020i | −1.73005 | − | 0.0833302i | 0.766044 | − | 0.642788i | 0.111050 | + | 0.629795i | 1.65421 | − | 0.513405i | −0.766044 | − | 0.642788i | −0.500000 | + | 0.866025i | 2.98611 | + | 0.288330i | −0.319755 | − | 0.553832i |
169.2 | −0.939693 | + | 0.342020i | −1.21448 | + | 1.23493i | 0.766044 | − | 0.642788i | −0.386963 | − | 2.19458i | 0.718864 | − | 1.57583i | −0.766044 | − | 0.642788i | −0.500000 | + | 0.866025i | −0.0500961 | − | 2.99958i | 1.11422 | + | 1.92988i |
169.3 | −0.939693 | + | 0.342020i | 0.344942 | + | 1.69736i | 0.766044 | − | 0.642788i | −0.445239 | − | 2.52508i | −0.904669 | − | 1.47702i | −0.766044 | − | 0.642788i | −0.500000 | + | 0.866025i | −2.76203 | + | 1.17098i | 1.28201 | + | 2.22051i |
169.4 | −0.939693 | + | 0.342020i | 1.22833 | − | 1.22115i | 0.766044 | − | 0.642788i | −0.0262315 | − | 0.148766i | −0.736599 | + | 1.56762i | −0.766044 | − | 0.642788i | −0.500000 | + | 0.866025i | 0.0176026 | − | 2.99995i | 0.0755306 | + | 0.130823i |
169.5 | −0.939693 | + | 0.342020i | 1.63729 | + | 0.565046i | 0.766044 | − | 0.642788i | 0.654987 | + | 3.71462i | −1.73181 | + | 0.0290168i | −0.766044 | − | 0.642788i | −0.500000 | + | 0.866025i | 2.36145 | + | 1.85029i | −1.88596 | − | 3.26658i |
211.1 | 0.766044 | − | 0.642788i | −1.62854 | + | 0.589807i | 0.173648 | − | 0.984808i | −1.36620 | + | 0.497255i | −0.868410 | + | 1.49862i | −0.173648 | − | 0.984808i | −0.500000 | − | 0.866025i | 2.30426 | − | 1.92104i | −0.726937 | + | 1.25909i |
211.2 | 0.766044 | − | 0.642788i | −0.764934 | + | 1.55399i | 0.173648 | − | 0.984808i | 3.01809 | − | 1.09849i | 0.412911 | + | 1.68211i | −0.173648 | − | 0.984808i | −0.500000 | − | 0.866025i | −1.82975 | − | 2.37739i | 1.60589 | − | 2.78148i |
211.3 | 0.766044 | − | 0.642788i | −0.167318 | − | 1.72395i | 0.173648 | − | 0.984808i | 2.66798 | − | 0.971065i | −1.23631 | − | 1.21307i | −0.173648 | − | 0.984808i | −0.500000 | − | 0.866025i | −2.94401 | + | 0.576895i | 1.41960 | − | 2.45882i |
211.4 | 0.766044 | − | 0.642788i | 0.503821 | − | 1.65716i | 0.173648 | − | 0.984808i | −4.01520 | + | 1.46142i | −0.679250 | − | 1.59330i | −0.173648 | − | 0.984808i | −0.500000 | − | 0.866025i | −2.49233 | − | 1.66982i | −2.13645 | + | 3.70043i |
211.5 | 0.766044 | − | 0.642788i | 1.73061 | + | 0.0705192i | 0.173648 | − | 0.984808i | −0.918005 | + | 0.334126i | 1.37106 | − | 1.05840i | −0.173648 | − | 0.984808i | −0.500000 | − | 0.866025i | 2.99005 | + | 0.244083i | −0.488460 | + | 0.846038i |
See all 30 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
27.e | even | 9 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 378.2.u.d | ✓ | 30 |
27.e | even | 9 | 1 | inner | 378.2.u.d | ✓ | 30 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
378.2.u.d | ✓ | 30 | 1.a | even | 1 | 1 | trivial |
378.2.u.d | ✓ | 30 | 27.e | even | 9 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{30} - 3 T_{5}^{29} - 6 T_{5}^{28} + 53 T_{5}^{27} - 81 T_{5}^{26} - 333 T_{5}^{25} + 3834 T_{5}^{24} - 9945 T_{5}^{23} + 2925 T_{5}^{22} - 44418 T_{5}^{21} + 338166 T_{5}^{20} - 587880 T_{5}^{19} + 2858250 T_{5}^{18} + \cdots + 185761 \)
acting on \(S_{2}^{\mathrm{new}}(378, [\chi])\).