Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [810,2,Mod(17,810)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(810, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([22, 9]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("810.17");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 810 = 2 \cdot 3^{4} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 810.s (of order \(36\), degree \(12\), not 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: | \(6.46788256372\) |
Analytic rank: | \(0\) |
Dimension: | \(216\) |
Relative dimension: | \(18\) over \(\Q(\zeta_{36})\) |
Twist minimal: | no (minimal twist has level 270) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
17.1 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.23440 | − | 0.0864530i | 0 | −3.15874 | − | 0.276354i | −0.258819 | + | 0.965926i | 0 | 2.06159 | − | 0.865944i | ||||||
17.2 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.00343 | + | 0.993102i | 0 | 0.000512095 | 0 | 4.48025e-5i | −0.258819 | + | 0.965926i | 0 | 1.39602 | − | 1.74674i | ||||||
17.3 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | −0.993387 | − | 2.00329i | 0 | 1.47317 | + | 0.128886i | −0.258819 | + | 0.965926i | 0 | 1.74694 | + | 1.39578i | ||||||
17.4 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | −0.842674 | + | 2.07121i | 0 | 1.73135 | + | 0.151473i | −0.258819 | + | 0.965926i | 0 | −0.111608 | − | 2.23328i | ||||||
17.5 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 0.460985 | − | 2.18803i | 0 | −1.84790 | − | 0.161670i | −0.258819 | + | 0.965926i | 0 | 0.506909 | + | 2.17785i | ||||||
17.6 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 0.893056 | + | 2.04999i | 0 | −3.58615 | − | 0.313747i | −0.258819 | + | 0.965926i | 0 | −1.67575 | − | 1.48050i | ||||||
17.7 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 0.975338 | − | 2.01214i | 0 | 5.12168 | + | 0.448089i | −0.258819 | + | 0.965926i | 0 | −0.0335884 | + | 2.23582i | ||||||
17.8 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 1.39590 | + | 1.74684i | 0 | 1.62157 | + | 0.141869i | −0.258819 | + | 0.965926i | 0 | −2.00336 | − | 0.993246i | ||||||
17.9 | −0.906308 | + | 0.422618i | 0 | 0.642788 | − | 0.766044i | 2.12538 | + | 0.694824i | 0 | −3.31761 | − | 0.290253i | −0.258819 | + | 0.965926i | 0 | −2.21989 | + | 0.268498i | ||||||
17.10 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.23426 | − | 0.0899855i | 0 | −1.89299 | − | 0.165615i | 0.258819 | − | 0.965926i | 0 | −2.06295 | + | 0.862683i | ||||||
17.11 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.23011 | + | 0.163105i | 0 | 3.73281 | + | 0.326578i | 0.258819 | − | 0.965926i | 0 | −1.95224 | + | 1.09031i | ||||||
17.12 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −2.04220 | + | 0.910725i | 0 | −1.36563 | − | 0.119477i | 0.258819 | − | 0.965926i | 0 | −1.46597 | + | 1.68847i | ||||||
17.13 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | −0.639439 | − | 2.14269i | 0 | 3.64169 | + | 0.318607i | 0.258819 | − | 0.965926i | 0 | −1.48507 | − | 1.67170i | ||||||
17.14 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 0.0174118 | + | 2.23600i | 0 | −4.56637 | − | 0.399506i | 0.258819 | − | 0.965926i | 0 | 0.960755 | + | 2.01915i | ||||||
17.15 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 1.28469 | + | 1.83018i | 0 | 0.618317 | + | 0.0540957i | 0.258819 | − | 0.965926i | 0 | 1.93780 | + | 1.11577i | ||||||
17.16 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 1.58821 | − | 1.57404i | 0 | 1.23570 | + | 0.108110i | 0.258819 | − | 0.965926i | 0 | 0.774190 | − | 2.09777i | ||||||
17.17 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 2.00011 | − | 0.999773i | 0 | −3.41173 | − | 0.298488i | 0.258819 | − | 0.965926i | 0 | 1.39020 | − | 1.75139i | ||||||
17.18 | 0.906308 | − | 0.422618i | 0 | 0.642788 | − | 0.766044i | 2.03234 | + | 0.932518i | 0 | 3.97033 | + | 0.347359i | 0.258819 | − | 0.965926i | 0 | 2.23603 | − | 0.0137560i | ||||||
143.1 | −0.906308 | − | 0.422618i | 0 | 0.642788 | + | 0.766044i | −2.23440 | + | 0.0864530i | 0 | −3.15874 | + | 0.276354i | −0.258819 | − | 0.965926i | 0 | 2.06159 | + | 0.865944i | ||||||
143.2 | −0.906308 | − | 0.422618i | 0 | 0.642788 | + | 0.766044i | −2.00343 | − | 0.993102i | 0 | 0.000512095 | 0 | 4.48025e-5i | −0.258819 | − | 0.965926i | 0 | 1.39602 | + | 1.74674i | ||||||
See next 80 embeddings (of 216 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
27.f | odd | 18 | 1 | inner |
135.q | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 810.2.s.a | 216 | |
3.b | odd | 2 | 1 | 270.2.r.a | ✓ | 216 | |
5.c | odd | 4 | 1 | inner | 810.2.s.a | 216 | |
15.e | even | 4 | 1 | 270.2.r.a | ✓ | 216 | |
27.e | even | 9 | 1 | 270.2.r.a | ✓ | 216 | |
27.f | odd | 18 | 1 | inner | 810.2.s.a | 216 | |
135.q | even | 36 | 1 | inner | 810.2.s.a | 216 | |
135.r | odd | 36 | 1 | 270.2.r.a | ✓ | 216 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
270.2.r.a | ✓ | 216 | 3.b | odd | 2 | 1 | |
270.2.r.a | ✓ | 216 | 15.e | even | 4 | 1 | |
270.2.r.a | ✓ | 216 | 27.e | even | 9 | 1 | |
270.2.r.a | ✓ | 216 | 135.r | odd | 36 | 1 | |
810.2.s.a | 216 | 1.a | even | 1 | 1 | trivial | |
810.2.s.a | 216 | 5.c | odd | 4 | 1 | inner | |
810.2.s.a | 216 | 27.f | odd | 18 | 1 | inner | |
810.2.s.a | 216 | 135.q | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(810, [\chi])\).