Newspace parameters
Level: | \( N \) | \(=\) | \( 574 = 2 \cdot 7 \cdot 41 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 574.r (of order \(12\), degree \(4\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.58341307602\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(10\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
9.1 | 0.866025 | + | 0.500000i | −0.855646 | + | 3.19332i | 0.500000 | + | 0.866025i | 0.112148 | + | 0.0647489i | −2.33767 | + | 2.33767i | 1.76095 | + | 1.97460i | 1.00000i | −6.86706 | − | 3.96470i | 0.0647489 | + | 0.112148i | ||
9.2 | 0.866025 | + | 0.500000i | −0.593109 | + | 2.21351i | 0.500000 | + | 0.866025i | −2.00711 | − | 1.15881i | −1.62040 | + | 1.62040i | −2.64158 | − | 0.148454i | 1.00000i | −1.94978 | − | 1.12571i | −1.15881 | − | 2.00711i | ||
9.3 | 0.866025 | + | 0.500000i | −0.552179 | + | 2.06076i | 0.500000 | + | 0.866025i | 1.83592 | + | 1.05997i | −1.50858 | + | 1.50858i | 0.869194 | − | 2.49890i | 1.00000i | −1.34375 | − | 0.775813i | 1.05997 | + | 1.83592i | ||
9.4 | 0.866025 | + | 0.500000i | −0.321998 | + | 1.20171i | 0.500000 | + | 0.866025i | 2.36946 | + | 1.36801i | −0.879715 | + | 0.879715i | 2.54753 | + | 0.714193i | 1.00000i | 1.25765 | + | 0.726102i | 1.36801 | + | 2.36946i | ||
9.5 | 0.866025 | + | 0.500000i | −0.111957 | + | 0.417828i | 0.500000 | + | 0.866025i | 0.681525 | + | 0.393479i | −0.305871 | + | 0.305871i | −2.17302 | − | 1.50929i | 1.00000i | 2.43603 | + | 1.40644i | 0.393479 | + | 0.681525i | ||
9.6 | 0.866025 | + | 0.500000i | −0.0822802 | + | 0.307074i | 0.500000 | + | 0.866025i | −0.999657 | − | 0.577152i | −0.224794 | + | 0.224794i | 2.64547 | − | 0.0385485i | 1.00000i | 2.51055 | + | 1.44947i | −0.577152 | − | 0.999657i | ||
9.7 | 0.866025 | + | 0.500000i | 0.200396 | − | 0.747888i | 0.500000 | + | 0.866025i | 1.99851 | + | 1.15384i | 0.547492 | − | 0.547492i | −1.57271 | + | 2.12757i | 1.00000i | 2.07890 | + | 1.20025i | 1.15384 | + | 1.99851i | ||
9.8 | 0.866025 | + | 0.500000i | 0.320619 | − | 1.19657i | 0.500000 | + | 0.866025i | −1.97710 | − | 1.14148i | 0.875948 | − | 0.875948i | −1.58481 | − | 2.11858i | 1.00000i | 1.26910 | + | 0.732715i | −1.14148 | − | 1.97710i | ||
9.9 | 0.866025 | + | 0.500000i | 0.417822 | − | 1.55933i | 0.500000 | + | 0.866025i | 0.463562 | + | 0.267638i | 1.14151 | − | 1.14151i | 1.34738 | − | 2.27697i | 1.00000i | 0.341135 | + | 0.196954i | 0.267638 | + | 0.463562i | ||
9.10 | 0.866025 | + | 0.500000i | 0.846280 | − | 3.15836i | 0.500000 | + | 0.866025i | 2.71889 | + | 1.56975i | 2.31208 | − | 2.31208i | −2.46634 | − | 0.957682i | 1.00000i | −6.66098 | − | 3.84572i | 1.56975 | + | 2.71889i | ||
319.1 | 0.866025 | − | 0.500000i | −0.855646 | − | 3.19332i | 0.500000 | − | 0.866025i | 0.112148 | − | 0.0647489i | −2.33767 | − | 2.33767i | 1.76095 | − | 1.97460i | − | 1.00000i | −6.86706 | + | 3.96470i | 0.0647489 | − | 0.112148i | |
319.2 | 0.866025 | − | 0.500000i | −0.593109 | − | 2.21351i | 0.500000 | − | 0.866025i | −2.00711 | + | 1.15881i | −1.62040 | − | 1.62040i | −2.64158 | + | 0.148454i | − | 1.00000i | −1.94978 | + | 1.12571i | −1.15881 | + | 2.00711i | |
319.3 | 0.866025 | − | 0.500000i | −0.552179 | − | 2.06076i | 0.500000 | − | 0.866025i | 1.83592 | − | 1.05997i | −1.50858 | − | 1.50858i | 0.869194 | + | 2.49890i | − | 1.00000i | −1.34375 | + | 0.775813i | 1.05997 | − | 1.83592i | |
319.4 | 0.866025 | − | 0.500000i | −0.321998 | − | 1.20171i | 0.500000 | − | 0.866025i | 2.36946 | − | 1.36801i | −0.879715 | − | 0.879715i | 2.54753 | − | 0.714193i | − | 1.00000i | 1.25765 | − | 0.726102i | 1.36801 | − | 2.36946i | |
319.5 | 0.866025 | − | 0.500000i | −0.111957 | − | 0.417828i | 0.500000 | − | 0.866025i | 0.681525 | − | 0.393479i | −0.305871 | − | 0.305871i | −2.17302 | + | 1.50929i | − | 1.00000i | 2.43603 | − | 1.40644i | 0.393479 | − | 0.681525i | |
319.6 | 0.866025 | − | 0.500000i | −0.0822802 | − | 0.307074i | 0.500000 | − | 0.866025i | −0.999657 | + | 0.577152i | −0.224794 | − | 0.224794i | 2.64547 | + | 0.0385485i | − | 1.00000i | 2.51055 | − | 1.44947i | −0.577152 | + | 0.999657i | |
319.7 | 0.866025 | − | 0.500000i | 0.200396 | + | 0.747888i | 0.500000 | − | 0.866025i | 1.99851 | − | 1.15384i | 0.547492 | + | 0.547492i | −1.57271 | − | 2.12757i | − | 1.00000i | 2.07890 | − | 1.20025i | 1.15384 | − | 1.99851i | |
319.8 | 0.866025 | − | 0.500000i | 0.320619 | + | 1.19657i | 0.500000 | − | 0.866025i | −1.97710 | + | 1.14148i | 0.875948 | + | 0.875948i | −1.58481 | + | 2.11858i | − | 1.00000i | 1.26910 | − | 0.732715i | −1.14148 | + | 1.97710i | |
319.9 | 0.866025 | − | 0.500000i | 0.417822 | + | 1.55933i | 0.500000 | − | 0.866025i | 0.463562 | − | 0.267638i | 1.14151 | + | 1.14151i | 1.34738 | + | 2.27697i | − | 1.00000i | 0.341135 | − | 0.196954i | 0.267638 | − | 0.463562i | |
319.10 | 0.866025 | − | 0.500000i | 0.846280 | + | 3.15836i | 0.500000 | − | 0.866025i | 2.71889 | − | 1.56975i | 2.31208 | + | 2.31208i | −2.46634 | + | 0.957682i | − | 1.00000i | −6.66098 | + | 3.84572i | 1.56975 | − | 2.71889i | |
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.c | even | 3 | 1 | inner |
41.c | even | 4 | 1 | inner |
287.r | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 574.2.r.c | ✓ | 40 |
7.c | even | 3 | 1 | inner | 574.2.r.c | ✓ | 40 |
41.c | even | 4 | 1 | inner | 574.2.r.c | ✓ | 40 |
287.r | even | 12 | 1 | inner | 574.2.r.c | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
574.2.r.c | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
574.2.r.c | ✓ | 40 | 7.c | even | 3 | 1 | inner |
574.2.r.c | ✓ | 40 | 41.c | even | 4 | 1 | inner |
574.2.r.c | ✓ | 40 | 287.r | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{40} - 4 T_{3}^{39} + 8 T_{3}^{38} - 28 T_{3}^{37} - 67 T_{3}^{36} + 474 T_{3}^{35} - 968 T_{3}^{34} + 3426 T_{3}^{33} + 7278 T_{3}^{32} - 54078 T_{3}^{31} + 109642 T_{3}^{30} - 386712 T_{3}^{29} + 684529 T_{3}^{28} + \cdots + 38416 \)
acting on \(S_{2}^{\mathrm{new}}(574, [\chi])\).