Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [585,2,Mod(196,585)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(585, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("585.196");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 585 = 3^{2} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 585.i (of order \(3\), degree \(2\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(4.67124851824\) |
Analytic rank: | \(0\) |
Dimension: | \(26\) |
Relative dimension: | \(13\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
196.1 | −1.23671 | + | 2.14204i | −1.15563 | − | 1.29016i | −2.05890 | − | 3.56612i | −0.500000 | − | 0.866025i | 4.19276 | − | 0.879859i | −2.17152 | + | 3.76119i | 5.23822 | −0.329032 | + | 2.98190i | 2.47342 | ||||
196.2 | −1.20515 | + | 2.08737i | 1.64592 | − | 0.539392i | −1.90475 | − | 3.29913i | −0.500000 | − | 0.866025i | −0.857662 | + | 4.08570i | −0.497502 | + | 0.861700i | 4.36143 | 2.41811 | − | 1.77559i | 2.41029 | ||||
196.3 | −0.877561 | + | 1.51998i | 0.0726124 | − | 1.73053i | −0.540227 | − | 0.935700i | −0.500000 | − | 0.866025i | 2.56665 | + | 1.62901i | 1.83503 | − | 3.17836i | −1.61392 | −2.98945 | − | 0.251316i | 1.75512 | ||||
196.4 | −0.780853 | + | 1.35248i | 0.0458444 | + | 1.73144i | −0.219464 | − | 0.380123i | −0.500000 | − | 0.866025i | −2.37754 | − | 1.29000i | −2.47307 | + | 4.28349i | −2.43794 | −2.99580 | + | 0.158754i | 1.56171 | ||||
196.5 | −0.643841 | + | 1.11517i | −1.73180 | − | 0.0293427i | 0.170938 | + | 0.296073i | −0.500000 | − | 0.866025i | 1.14773 | − | 1.91235i | 0.391685 | − | 0.678419i | −3.01559 | 2.99828 | + | 0.101632i | 1.28768 | ||||
196.6 | −0.0377835 | + | 0.0654430i | −1.16946 | + | 1.27764i | 0.997145 | + | 1.72711i | −0.500000 | − | 0.866025i | −0.0394262 | − | 0.124807i | −0.0159135 | + | 0.0275629i | −0.301837 | −0.264725 | − | 2.98830i | 0.0755671 | ||||
196.7 | 0.0514441 | − | 0.0891038i | 1.71492 | + | 0.243008i | 0.994707 | + | 1.72288i | −0.500000 | − | 0.866025i | 0.109875 | − | 0.140305i | −2.41664 | + | 4.18575i | 0.410464 | 2.88189 | + | 0.833478i | −0.102888 | ||||
196.8 | 0.238506 | − | 0.413105i | 1.59029 | + | 0.686267i | 0.886230 | + | 1.53499i | −0.500000 | − | 0.866025i | 0.662795 | − | 0.493279i | 0.335964 | − | 0.581906i | 1.79951 | 2.05807 | + | 2.18273i | −0.477012 | ||||
196.9 | 0.669930 | − | 1.16035i | 0.478101 | − | 1.66476i | 0.102389 | + | 0.177343i | −0.500000 | − | 0.866025i | −1.61141 | − | 1.67004i | −0.186958 | + | 0.323821i | 2.95409 | −2.54284 | − | 1.59184i | −1.33986 | ||||
196.10 | 0.768621 | − | 1.33129i | −1.52975 | − | 0.812318i | −0.181557 | − | 0.314466i | −0.500000 | − | 0.866025i | −2.25723 | + | 1.41218i | −1.69912 | + | 2.94296i | 2.51629 | 1.68028 | + | 2.48529i | −1.53724 | ||||
196.11 | 0.853509 | − | 1.47832i | 0.733344 | + | 1.56914i | −0.456955 | − | 0.791470i | −0.500000 | − | 0.866025i | 2.94561 | + | 0.255158i | 1.35209 | − | 2.34188i | 1.85397 | −1.92441 | + | 2.30144i | −1.70702 | ||||
196.12 | 1.34334 | − | 2.32674i | −1.55252 | + | 0.767915i | −2.60914 | − | 4.51916i | −0.500000 | − | 0.866025i | −0.298825 | + | 4.64387i | 1.78091 | − | 3.08463i | −8.64648 | 1.82061 | − | 2.38440i | −2.68668 | ||||
196.13 | 1.35654 | − | 2.34960i | 1.35813 | − | 1.07494i | −2.68041 | − | 4.64261i | −0.500000 | − | 0.866025i | −0.683325 | − | 4.64925i | −1.23495 | + | 2.13900i | −9.11822 | 0.689008 | − | 2.91981i | −2.71308 | ||||
391.1 | −1.23671 | − | 2.14204i | −1.15563 | + | 1.29016i | −2.05890 | + | 3.56612i | −0.500000 | + | 0.866025i | 4.19276 | + | 0.879859i | −2.17152 | − | 3.76119i | 5.23822 | −0.329032 | − | 2.98190i | 2.47342 | ||||
391.2 | −1.20515 | − | 2.08737i | 1.64592 | + | 0.539392i | −1.90475 | + | 3.29913i | −0.500000 | + | 0.866025i | −0.857662 | − | 4.08570i | −0.497502 | − | 0.861700i | 4.36143 | 2.41811 | + | 1.77559i | 2.41029 | ||||
391.3 | −0.877561 | − | 1.51998i | 0.0726124 | + | 1.73053i | −0.540227 | + | 0.935700i | −0.500000 | + | 0.866025i | 2.56665 | − | 1.62901i | 1.83503 | + | 3.17836i | −1.61392 | −2.98945 | + | 0.251316i | 1.75512 | ||||
391.4 | −0.780853 | − | 1.35248i | 0.0458444 | − | 1.73144i | −0.219464 | + | 0.380123i | −0.500000 | + | 0.866025i | −2.37754 | + | 1.29000i | −2.47307 | − | 4.28349i | −2.43794 | −2.99580 | − | 0.158754i | 1.56171 | ||||
391.5 | −0.643841 | − | 1.11517i | −1.73180 | + | 0.0293427i | 0.170938 | − | 0.296073i | −0.500000 | + | 0.866025i | 1.14773 | + | 1.91235i | 0.391685 | + | 0.678419i | −3.01559 | 2.99828 | − | 0.101632i | 1.28768 | ||||
391.6 | −0.0377835 | − | 0.0654430i | −1.16946 | − | 1.27764i | 0.997145 | − | 1.72711i | −0.500000 | + | 0.866025i | −0.0394262 | + | 0.124807i | −0.0159135 | − | 0.0275629i | −0.301837 | −0.264725 | + | 2.98830i | 0.0755671 | ||||
391.7 | 0.0514441 | + | 0.0891038i | 1.71492 | − | 0.243008i | 0.994707 | − | 1.72288i | −0.500000 | + | 0.866025i | 0.109875 | + | 0.140305i | −2.41664 | − | 4.18575i | 0.410464 | 2.88189 | − | 0.833478i | −0.102888 | ||||
See all 26 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 585.2.i.g | ✓ | 26 |
3.b | odd | 2 | 1 | 1755.2.i.g | 26 | ||
9.c | even | 3 | 1 | inner | 585.2.i.g | ✓ | 26 |
9.c | even | 3 | 1 | 5265.2.a.bg | 13 | ||
9.d | odd | 6 | 1 | 1755.2.i.g | 26 | ||
9.d | odd | 6 | 1 | 5265.2.a.bh | 13 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
585.2.i.g | ✓ | 26 | 1.a | even | 1 | 1 | trivial |
585.2.i.g | ✓ | 26 | 9.c | even | 3 | 1 | inner |
1755.2.i.g | 26 | 3.b | odd | 2 | 1 | ||
1755.2.i.g | 26 | 9.d | odd | 6 | 1 | ||
5265.2.a.bg | 13 | 9.c | even | 3 | 1 | ||
5265.2.a.bh | 13 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(585, [\chi])\):
\( T_{2}^{26} - T_{2}^{25} + 21 T_{2}^{24} - 14 T_{2}^{23} + 267 T_{2}^{22} - 149 T_{2}^{21} + 2136 T_{2}^{20} + \cdots + 4 \) |
\( T_{7}^{26} + 10 T_{7}^{25} + 110 T_{7}^{24} + 596 T_{7}^{23} + 4027 T_{7}^{22} + 16491 T_{7}^{21} + \cdots + 36864 \) |