Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(143,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([27, 34]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.143");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.bb (of order \(36\), degree \(12\), 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: | \(7.58578819202\) |
Analytic rank: | \(0\) |
Dimension: | \(72\) |
Relative dimension: | \(6\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
143.1 | −0.573576 | + | 0.819152i | −0.181419 | + | 2.07363i | −0.342020 | − | 0.939693i | 0 | −1.59456 | − | 1.33799i | 1.21655 | + | 4.54023i | 0.965926 | + | 0.258819i | −1.31260 | − | 0.231446i | 0 | ||||
143.2 | −0.573576 | + | 0.819152i | 0.00886944 | − | 0.101378i | −0.342020 | − | 0.939693i | 0 | 0.0779568 | + | 0.0654135i | −0.0402147 | − | 0.150083i | 0.965926 | + | 0.258819i | 2.94422 | + | 0.519146i | 0 | ||||
143.3 | −0.573576 | + | 0.819152i | 0.218924 | − | 2.50231i | −0.342020 | − | 0.939693i | 0 | 1.92421 | + | 1.61460i | 0.801272 | + | 2.99039i | 0.965926 | + | 0.258819i | −3.25922 | − | 0.574689i | 0 | ||||
143.4 | 0.573576 | − | 0.819152i | −0.218924 | + | 2.50231i | −0.342020 | − | 0.939693i | 0 | 1.92421 | + | 1.61460i | −0.801272 | − | 2.99039i | −0.965926 | − | 0.258819i | −3.25922 | − | 0.574689i | 0 | ||||
143.5 | 0.573576 | − | 0.819152i | −0.00886944 | + | 0.101378i | −0.342020 | − | 0.939693i | 0 | 0.0779568 | + | 0.0654135i | 0.0402147 | + | 0.150083i | −0.965926 | − | 0.258819i | 2.94422 | + | 0.519146i | 0 | ||||
143.6 | 0.573576 | − | 0.819152i | 0.181419 | − | 2.07363i | −0.342020 | − | 0.939693i | 0 | −1.59456 | − | 1.33799i | −1.21655 | − | 4.54023i | −0.965926 | − | 0.258819i | −1.31260 | − | 0.231446i | 0 | ||||
193.1 | −0.0871557 | − | 0.996195i | −1.14235 | + | 2.44977i | −0.984808 | + | 0.173648i | 0 | 2.54001 | + | 0.924487i | −1.86007 | − | 0.498404i | 0.258819 | + | 0.965926i | −2.76804 | − | 3.29882i | 0 | ||||
193.2 | −0.0871557 | − | 0.996195i | −0.475045 | + | 1.01874i | −0.984808 | + | 0.173648i | 0 | 1.05626 | + | 0.384448i | 3.66894 | + | 0.983089i | 0.258819 | + | 0.965926i | 1.11621 | + | 1.33024i | 0 | ||||
193.3 | −0.0871557 | − | 0.996195i | 0.400509 | − | 0.858895i | −0.984808 | + | 0.173648i | 0 | −0.890533 | − | 0.324128i | 0.243409 | + | 0.0652212i | 0.258819 | + | 0.965926i | 1.35107 | + | 1.61014i | 0 | ||||
193.4 | 0.0871557 | + | 0.996195i | −0.400509 | + | 0.858895i | −0.984808 | + | 0.173648i | 0 | −0.890533 | − | 0.324128i | −0.243409 | − | 0.0652212i | −0.258819 | − | 0.965926i | 1.35107 | + | 1.61014i | 0 | ||||
193.5 | 0.0871557 | + | 0.996195i | 0.475045 | − | 1.01874i | −0.984808 | + | 0.173648i | 0 | 1.05626 | + | 0.384448i | −3.66894 | − | 0.983089i | −0.258819 | − | 0.965926i | 1.11621 | + | 1.33024i | 0 | ||||
193.6 | 0.0871557 | + | 0.996195i | 1.14235 | − | 2.44977i | −0.984808 | + | 0.173648i | 0 | 2.54001 | + | 0.924487i | 1.86007 | + | 0.498404i | −0.258819 | − | 0.965926i | −2.76804 | − | 3.29882i | 0 | ||||
243.1 | −0.422618 | − | 0.906308i | −2.29045 | − | 1.60379i | −0.642788 | + | 0.766044i | 0 | −0.485541 | + | 2.75364i | 0.367125 | + | 1.37013i | 0.965926 | + | 0.258819i | 1.64795 | + | 4.52771i | 0 | ||||
243.2 | −0.422618 | − | 0.906308i | −0.0850875 | − | 0.0595789i | −0.642788 | + | 0.766044i | 0 | −0.0180373 | + | 0.102295i | 0.497281 | + | 1.85588i | 0.965926 | + | 0.258819i | −1.02237 | − | 2.80894i | 0 | ||||
243.3 | −0.422618 | − | 0.906308i | 1.84087 | + | 1.28899i | −0.642788 | + | 0.766044i | 0 | 0.390238 | − | 2.21315i | −0.947246 | − | 3.53517i | 0.965926 | + | 0.258819i | 0.701247 | + | 1.92666i | 0 | ||||
243.4 | 0.422618 | + | 0.906308i | −1.84087 | − | 1.28899i | −0.642788 | + | 0.766044i | 0 | 0.390238 | − | 2.21315i | 0.947246 | + | 3.53517i | −0.965926 | − | 0.258819i | 0.701247 | + | 1.92666i | 0 | ||||
243.5 | 0.422618 | + | 0.906308i | 0.0850875 | + | 0.0595789i | −0.642788 | + | 0.766044i | 0 | −0.0180373 | + | 0.102295i | −0.497281 | − | 1.85588i | −0.965926 | − | 0.258819i | −1.02237 | − | 2.80894i | 0 | ||||
243.6 | 0.422618 | + | 0.906308i | 2.29045 | + | 1.60379i | −0.642788 | + | 0.766044i | 0 | −0.485541 | + | 2.75364i | −0.367125 | − | 1.37013i | −0.965926 | − | 0.258819i | 1.64795 | + | 4.52771i | 0 | ||||
257.1 | −0.819152 | − | 0.573576i | −2.50231 | − | 0.218924i | 0.342020 | + | 0.939693i | 0 | 1.92421 | + | 1.61460i | −2.99039 | + | 0.801272i | 0.258819 | − | 0.965926i | 3.25922 | + | 0.574689i | 0 | ||||
257.2 | −0.819152 | − | 0.573576i | −0.101378 | − | 0.00886944i | 0.342020 | + | 0.939693i | 0 | 0.0779568 | + | 0.0654135i | 0.150083 | − | 0.0402147i | 0.258819 | − | 0.965926i | −2.94422 | − | 0.519146i | 0 | ||||
See all 72 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
5.c | odd | 4 | 2 | inner |
19.f | odd | 18 | 1 | inner |
95.o | odd | 18 | 1 | inner |
95.r | even | 36 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.2.bb.c | ✓ | 72 |
5.b | even | 2 | 1 | inner | 950.2.bb.c | ✓ | 72 |
5.c | odd | 4 | 2 | inner | 950.2.bb.c | ✓ | 72 |
19.f | odd | 18 | 1 | inner | 950.2.bb.c | ✓ | 72 |
95.o | odd | 18 | 1 | inner | 950.2.bb.c | ✓ | 72 |
95.r | even | 36 | 2 | inner | 950.2.bb.c | ✓ | 72 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
950.2.bb.c | ✓ | 72 | 1.a | even | 1 | 1 | trivial |
950.2.bb.c | ✓ | 72 | 5.b | even | 2 | 1 | inner |
950.2.bb.c | ✓ | 72 | 5.c | odd | 4 | 2 | inner |
950.2.bb.c | ✓ | 72 | 19.f | odd | 18 | 1 | inner |
950.2.bb.c | ✓ | 72 | 95.o | odd | 18 | 1 | inner |
950.2.bb.c | ✓ | 72 | 95.r | even | 36 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{72} + 42 T_{3}^{68} + 1317 T_{3}^{64} - 124986 T_{3}^{60} - 2508630 T_{3}^{56} - 235773723 T_{3}^{52} + 17508117579 T_{3}^{48} + 164300946747 T_{3}^{44} - 4034387282280 T_{3}^{40} + \cdots + 1 \)
acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).