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: | \(48\) |
Relative dimension: | \(4\) 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 | −0.133530 | + | 1.52626i | −0.342020 | − | 0.939693i | 0 | −1.17365 | − | 0.984808i | −1.06982 | − | 3.99264i | 0.965926 | + | 0.258819i | 0.642788 | + | 0.113341i | 0 | ||||
143.2 | −0.573576 | + | 0.819152i | −0.133530 | + | 1.52626i | −0.342020 | − | 0.939693i | 0 | −1.17365 | − | 0.984808i | 0.227319 | + | 0.848367i | 0.965926 | + | 0.258819i | 0.642788 | + | 0.113341i | 0 | ||||
143.3 | 0.573576 | − | 0.819152i | 0.133530 | − | 1.52626i | −0.342020 | − | 0.939693i | 0 | −1.17365 | − | 0.984808i | −0.227319 | − | 0.848367i | −0.965926 | − | 0.258819i | 0.642788 | + | 0.113341i | 0 | ||||
143.4 | 0.573576 | − | 0.819152i | 0.133530 | − | 1.52626i | −0.342020 | − | 0.939693i | 0 | −1.17365 | − | 0.984808i | 1.06982 | + | 3.99264i | −0.965926 | − | 0.258819i | 0.642788 | + | 0.113341i | 0 | ||||
193.1 | −0.0871557 | − | 0.996195i | 0.794263 | − | 1.70330i | −0.984808 | + | 0.173648i | 0 | −1.76604 | − | 0.642788i | −3.76160 | − | 1.00792i | 0.258819 | + | 0.965926i | −0.342020 | − | 0.407604i | 0 | ||||
193.2 | −0.0871557 | − | 0.996195i | 0.794263 | − | 1.70330i | −0.984808 | + | 0.173648i | 0 | −1.76604 | − | 0.642788i | 3.18056 | + | 0.852228i | 0.258819 | + | 0.965926i | −0.342020 | − | 0.407604i | 0 | ||||
193.3 | 0.0871557 | + | 0.996195i | −0.794263 | + | 1.70330i | −0.984808 | + | 0.173648i | 0 | −1.76604 | − | 0.642788i | −3.18056 | − | 0.852228i | −0.258819 | − | 0.965926i | −0.342020 | − | 0.407604i | 0 | ||||
193.4 | 0.0871557 | + | 0.996195i | −0.794263 | + | 1.70330i | −0.984808 | + | 0.173648i | 0 | −1.76604 | − | 0.642788i | 3.76160 | + | 1.00792i | −0.258819 | − | 0.965926i | −0.342020 | − | 0.407604i | 0 | ||||
243.1 | −0.422618 | − | 0.906308i | −0.284489 | − | 0.199201i | −0.642788 | + | 0.766044i | 0 | −0.0603074 | + | 0.342020i | −0.127266 | − | 0.474963i | 0.965926 | + | 0.258819i | −0.984808 | − | 2.70574i | 0 | ||||
243.2 | −0.422618 | − | 0.906308i | −0.284489 | − | 0.199201i | −0.642788 | + | 0.766044i | 0 | −0.0603074 | + | 0.342020i | 0.814083 | + | 3.03820i | 0.965926 | + | 0.258819i | −0.984808 | − | 2.70574i | 0 | ||||
243.3 | 0.422618 | + | 0.906308i | 0.284489 | + | 0.199201i | −0.642788 | + | 0.766044i | 0 | −0.0603074 | + | 0.342020i | −0.814083 | − | 3.03820i | −0.965926 | − | 0.258819i | −0.984808 | − | 2.70574i | 0 | ||||
243.4 | 0.422618 | + | 0.906308i | 0.284489 | + | 0.199201i | −0.642788 | + | 0.766044i | 0 | −0.0603074 | + | 0.342020i | 0.127266 | + | 0.474963i | −0.965926 | − | 0.258819i | −0.984808 | − | 2.70574i | 0 | ||||
257.1 | −0.819152 | − | 0.573576i | 1.52626 | + | 0.133530i | 0.342020 | + | 0.939693i | 0 | −1.17365 | − | 0.984808i | −0.848367 | + | 0.227319i | 0.258819 | − | 0.965926i | −0.642788 | − | 0.113341i | 0 | ||||
257.2 | −0.819152 | − | 0.573576i | 1.52626 | + | 0.133530i | 0.342020 | + | 0.939693i | 0 | −1.17365 | − | 0.984808i | 3.99264 | − | 1.06982i | 0.258819 | − | 0.965926i | −0.642788 | − | 0.113341i | 0 | ||||
257.3 | 0.819152 | + | 0.573576i | −1.52626 | − | 0.133530i | 0.342020 | + | 0.939693i | 0 | −1.17365 | − | 0.984808i | −3.99264 | + | 1.06982i | −0.258819 | + | 0.965926i | −0.642788 | − | 0.113341i | 0 | ||||
257.4 | 0.819152 | + | 0.573576i | −1.52626 | − | 0.133530i | 0.342020 | + | 0.939693i | 0 | −1.17365 | − | 0.984808i | 0.848367 | − | 0.227319i | −0.258819 | + | 0.965926i | −0.642788 | − | 0.113341i | 0 | ||||
307.1 | −0.996195 | + | 0.0871557i | 1.70330 | + | 0.794263i | 0.984808 | − | 0.173648i | 0 | −1.76604 | − | 0.642788i | −1.00792 | + | 3.76160i | −0.965926 | + | 0.258819i | 0.342020 | + | 0.407604i | 0 | ||||
307.2 | −0.996195 | + | 0.0871557i | 1.70330 | + | 0.794263i | 0.984808 | − | 0.173648i | 0 | −1.76604 | − | 0.642788i | 0.852228 | − | 3.18056i | −0.965926 | + | 0.258819i | 0.342020 | + | 0.407604i | 0 | ||||
307.3 | 0.996195 | − | 0.0871557i | −1.70330 | − | 0.794263i | 0.984808 | − | 0.173648i | 0 | −1.76604 | − | 0.642788i | −0.852228 | + | 3.18056i | 0.965926 | − | 0.258819i | 0.342020 | + | 0.407604i | 0 | ||||
307.4 | 0.996195 | − | 0.0871557i | −1.70330 | − | 0.794263i | 0.984808 | − | 0.173648i | 0 | −1.76604 | − | 0.642788i | 1.00792 | − | 3.76160i | 0.965926 | − | 0.258819i | 0.342020 | + | 0.407604i | 0 | ||||
See all 48 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.b | ✓ | 48 |
5.b | even | 2 | 1 | inner | 950.2.bb.b | ✓ | 48 |
5.c | odd | 4 | 2 | inner | 950.2.bb.b | ✓ | 48 |
19.f | odd | 18 | 1 | inner | 950.2.bb.b | ✓ | 48 |
95.o | odd | 18 | 1 | inner | 950.2.bb.b | ✓ | 48 |
95.r | even | 36 | 2 | inner | 950.2.bb.b | ✓ | 48 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
950.2.bb.b | ✓ | 48 | 1.a | even | 1 | 1 | trivial |
950.2.bb.b | ✓ | 48 | 5.b | even | 2 | 1 | inner |
950.2.bb.b | ✓ | 48 | 5.c | odd | 4 | 2 | inner |
950.2.bb.b | ✓ | 48 | 19.f | odd | 18 | 1 | inner |
950.2.bb.b | ✓ | 48 | 95.o | odd | 18 | 1 | inner |
950.2.bb.b | ✓ | 48 | 95.r | even | 36 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{24} - 6T_{3}^{20} + 141T_{3}^{16} - 1477T_{3}^{12} + 4692T_{3}^{8} + 105T_{3}^{4} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).