Newspace parameters
Level: | \( N \) | \(=\) | \( 546 = 2 \cdot 3 \cdot 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 546.u (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.35983195036\) |
Analytic rank: | \(0\) |
Dimension: | \(76\) |
Relative dimension: | \(38\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
185.1 | −0.866025 | − | 0.500000i | −1.67595 | + | 0.437249i | 0.500000 | + | 0.866025i | 1.40279 | + | 2.42970i | 1.67004 | + | 0.459307i | 1.42569 | − | 2.22877i | − | 1.00000i | 2.61763 | − | 1.46562i | − | 2.80557i | ||
185.2 | −0.866025 | − | 0.500000i | −1.59603 | + | 0.672816i | 0.500000 | + | 0.866025i | 1.07511 | + | 1.86215i | 1.71861 | + | 0.215340i | 0.447408 | + | 2.60765i | − | 1.00000i | 2.09464 | − | 2.14767i | − | 2.15023i | ||
185.3 | −0.866025 | − | 0.500000i | −1.56696 | − | 0.737995i | 0.500000 | + | 0.866025i | −1.41378 | − | 2.44875i | 0.988029 | + | 1.42260i | −2.45899 | + | 0.976416i | − | 1.00000i | 1.91073 | + | 2.31282i | 2.82757i | |||
185.4 | −0.866025 | − | 0.500000i | −1.51699 | − | 0.835899i | 0.500000 | + | 0.866025i | −0.699535 | − | 1.21163i | 0.895807 | + | 1.48241i | 1.97618 | + | 1.75918i | − | 1.00000i | 1.60255 | + | 2.53611i | 1.39907i | |||
185.5 | −0.866025 | − | 0.500000i | −1.38893 | + | 1.03484i | 0.500000 | + | 0.866025i | −1.75377 | − | 3.03762i | 1.72026 | − | 0.201730i | 1.37341 | − | 2.26136i | − | 1.00000i | 0.858232 | − | 2.87462i | 3.50754i | |||
185.6 | −0.866025 | − | 0.500000i | −1.18075 | − | 1.26722i | 0.500000 | + | 0.866025i | 0.386492 | + | 0.669424i | 0.388948 | + | 1.68782i | 0.662023 | − | 2.56159i | − | 1.00000i | −0.211678 | + | 2.99252i | − | 0.772984i | ||
185.7 | −0.866025 | − | 0.500000i | −0.685609 | + | 1.59058i | 0.500000 | + | 0.866025i | −0.587127 | − | 1.01693i | 1.38904 | − | 1.03468i | −1.73688 | + | 1.99580i | − | 1.00000i | −2.05988 | − | 2.18103i | 1.17425i | |||
185.8 | −0.866025 | − | 0.500000i | −0.266485 | + | 1.71143i | 0.500000 | + | 0.866025i | 0.279399 | + | 0.483934i | 1.08650 | − | 1.34890i | 0.253211 | − | 2.63361i | − | 1.00000i | −2.85797 | − | 0.912140i | − | 0.558798i | ||
185.9 | −0.866025 | − | 0.500000i | −0.219501 | − | 1.71809i | 0.500000 | + | 0.866025i | 0.356048 | + | 0.616693i | −0.668950 | + | 1.59766i | −2.23074 | − | 1.42260i | − | 1.00000i | −2.90364 | + | 0.754242i | − | 0.712096i | ||
185.10 | −0.866025 | − | 0.500000i | −0.0864870 | − | 1.72989i | 0.500000 | + | 0.866025i | −0.886718 | − | 1.53584i | −0.790045 | + | 1.54137i | 2.61154 | + | 0.424103i | − | 1.00000i | −2.98504 | + | 0.299226i | 1.77344i | |||
185.11 | −0.866025 | − | 0.500000i | 0.323515 | + | 1.70157i | 0.500000 | + | 0.866025i | 1.99252 | + | 3.45115i | 0.570612 | − | 1.63536i | −2.44529 | − | 1.01022i | − | 1.00000i | −2.79068 | + | 1.10097i | − | 3.98504i | ||
185.12 | −0.866025 | − | 0.500000i | 0.696855 | + | 1.58568i | 0.500000 | + | 0.866025i | −1.42622 | − | 2.47029i | 0.189348 | − | 1.72167i | 2.54641 | + | 0.718201i | − | 1.00000i | −2.02879 | + | 2.20998i | 2.85245i | |||
185.13 | −0.866025 | − | 0.500000i | 0.804592 | − | 1.53383i | 0.500000 | + | 0.866025i | 2.14723 | + | 3.71911i | −1.46371 | + | 0.926039i | 2.58099 | − | 0.581819i | − | 1.00000i | −1.70526 | − | 2.46821i | − | 4.29446i | ||
185.14 | −0.866025 | − | 0.500000i | 0.896713 | − | 1.48186i | 0.500000 | + | 0.866025i | −0.277881 | − | 0.481304i | −1.51751 | + | 0.834971i | −0.887101 | + | 2.49260i | − | 1.00000i | −1.39181 | − | 2.65760i | 0.555761i | |||
185.15 | −0.866025 | − | 0.500000i | 1.11365 | − | 1.32657i | 0.500000 | + | 0.866025i | −2.12998 | − | 3.68924i | −1.62773 | + | 0.592023i | −1.07162 | − | 2.41901i | − | 1.00000i | −0.519589 | − | 2.95466i | 4.25996i | |||
185.16 | −0.866025 | − | 0.500000i | 1.30626 | + | 1.13741i | 0.500000 | + | 0.866025i | −0.604752 | − | 1.04746i | −0.562548 | − | 1.63815i | −2.45540 | + | 0.985399i | − | 1.00000i | 0.412611 | + | 2.97149i | 1.20950i | |||
185.17 | −0.866025 | − | 0.500000i | 1.62717 | + | 0.593575i | 0.500000 | + | 0.866025i | 1.76347 | + | 3.05441i | −1.11238 | − | 1.32763i | 1.45770 | + | 2.20797i | − | 1.00000i | 2.29534 | + | 1.93169i | − | 3.52693i | ||
185.18 | −0.866025 | − | 0.500000i | 1.69393 | + | 0.361382i | 0.500000 | + | 0.866025i | 0.0272832 | + | 0.0472559i | −1.28630 | − | 1.15993i | 2.23165 | − | 1.42118i | − | 1.00000i | 2.73881 | + | 1.22431i | − | 0.0545664i | ||
185.19 | −0.866025 | − | 0.500000i | 1.72102 | − | 0.195177i | 0.500000 | + | 0.866025i | 0.349437 | + | 0.605242i | −1.58803 | − | 0.691481i | −1.78017 | − | 1.95729i | − | 1.00000i | 2.92381 | − | 0.671806i | − | 0.698873i | ||
185.20 | 0.866025 | + | 0.500000i | −1.72102 | − | 0.195177i | 0.500000 | + | 0.866025i | −0.349437 | − | 0.605242i | −1.39286 | − | 1.02954i | −1.78017 | − | 1.95729i | 1.00000i | 2.92381 | + | 0.671806i | − | 0.698873i | |||
See all 76 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
91.m | odd | 6 | 1 | inner |
273.bf | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 546.2.u.a | ✓ | 76 |
3.b | odd | 2 | 1 | inner | 546.2.u.a | ✓ | 76 |
7.d | odd | 6 | 1 | 546.2.bb.a | yes | 76 | |
13.c | even | 3 | 1 | 546.2.bb.a | yes | 76 | |
21.g | even | 6 | 1 | 546.2.bb.a | yes | 76 | |
39.i | odd | 6 | 1 | 546.2.bb.a | yes | 76 | |
91.m | odd | 6 | 1 | inner | 546.2.u.a | ✓ | 76 |
273.bf | even | 6 | 1 | inner | 546.2.u.a | ✓ | 76 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
546.2.u.a | ✓ | 76 | 1.a | even | 1 | 1 | trivial |
546.2.u.a | ✓ | 76 | 3.b | odd | 2 | 1 | inner |
546.2.u.a | ✓ | 76 | 91.m | odd | 6 | 1 | inner |
546.2.u.a | ✓ | 76 | 273.bf | even | 6 | 1 | inner |
546.2.bb.a | yes | 76 | 7.d | odd | 6 | 1 | |
546.2.bb.a | yes | 76 | 13.c | even | 3 | 1 | |
546.2.bb.a | yes | 76 | 21.g | even | 6 | 1 | |
546.2.bb.a | yes | 76 | 39.i | odd | 6 | 1 |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(546, [\chi])\).