Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3150,2,Mod(1151,3150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3150, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3150.1151");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 3150.bf (of order \(6\), 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: | \(25.1528766367\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1151.1 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −1.52781 | + | 2.16005i | 1.00000i | 0 | 0 | ||||||||||||
1151.2 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 2.34325 | + | 1.22849i | 1.00000i | 0 | 0 | ||||||||||||
1151.3 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0.295801 | − | 2.62916i | 1.00000i | 0 | 0 | ||||||||||||
1151.4 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −0.397202 | − | 2.61577i | 1.00000i | 0 | 0 | ||||||||||||
1151.5 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −2.43194 | + | 1.04195i | 1.00000i | 0 | 0 | ||||||||||||
1151.6 | −0.866025 | + | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0.717905 | + | 2.54649i | 1.00000i | 0 | 0 | ||||||||||||
1151.7 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0.717905 | + | 2.54649i | − | 1.00000i | 0 | 0 | |||||||||||
1151.8 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −2.43194 | + | 1.04195i | − | 1.00000i | 0 | 0 | |||||||||||
1151.9 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −0.397202 | − | 2.61577i | − | 1.00000i | 0 | 0 | |||||||||||
1151.10 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 0.295801 | − | 2.62916i | − | 1.00000i | 0 | 0 | |||||||||||
1151.11 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | 2.34325 | + | 1.22849i | − | 1.00000i | 0 | 0 | |||||||||||
1151.12 | 0.866025 | − | 0.500000i | 0 | 0.500000 | − | 0.866025i | 0 | 0 | −1.52781 | + | 2.16005i | − | 1.00000i | 0 | 0 | |||||||||||
1601.1 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −1.52781 | − | 2.16005i | − | 1.00000i | 0 | 0 | |||||||||||
1601.2 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 2.34325 | − | 1.22849i | − | 1.00000i | 0 | 0 | |||||||||||
1601.3 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0.295801 | + | 2.62916i | − | 1.00000i | 0 | 0 | |||||||||||
1601.4 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −0.397202 | + | 2.61577i | − | 1.00000i | 0 | 0 | |||||||||||
1601.5 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −2.43194 | − | 1.04195i | − | 1.00000i | 0 | 0 | |||||||||||
1601.6 | −0.866025 | − | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0.717905 | − | 2.54649i | − | 1.00000i | 0 | 0 | |||||||||||
1601.7 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | 0.717905 | − | 2.54649i | 1.00000i | 0 | 0 | ||||||||||||
1601.8 | 0.866025 | + | 0.500000i | 0 | 0.500000 | + | 0.866025i | 0 | 0 | −2.43194 | − | 1.04195i | 1.00000i | 0 | 0 | ||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
7.d | odd | 6 | 1 | inner |
21.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 3150.2.bf.d | ✓ | 24 |
3.b | odd | 2 | 1 | inner | 3150.2.bf.d | ✓ | 24 |
5.b | even | 2 | 1 | 3150.2.bf.e | yes | 24 | |
5.c | odd | 4 | 1 | 3150.2.bp.g | 24 | ||
5.c | odd | 4 | 1 | 3150.2.bp.h | 24 | ||
7.d | odd | 6 | 1 | inner | 3150.2.bf.d | ✓ | 24 |
15.d | odd | 2 | 1 | 3150.2.bf.e | yes | 24 | |
15.e | even | 4 | 1 | 3150.2.bp.g | 24 | ||
15.e | even | 4 | 1 | 3150.2.bp.h | 24 | ||
21.g | even | 6 | 1 | inner | 3150.2.bf.d | ✓ | 24 |
35.i | odd | 6 | 1 | 3150.2.bf.e | yes | 24 | |
35.k | even | 12 | 1 | 3150.2.bp.g | 24 | ||
35.k | even | 12 | 1 | 3150.2.bp.h | 24 | ||
105.p | even | 6 | 1 | 3150.2.bf.e | yes | 24 | |
105.w | odd | 12 | 1 | 3150.2.bp.g | 24 | ||
105.w | odd | 12 | 1 | 3150.2.bp.h | 24 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
3150.2.bf.d | ✓ | 24 | 1.a | even | 1 | 1 | trivial |
3150.2.bf.d | ✓ | 24 | 3.b | odd | 2 | 1 | inner |
3150.2.bf.d | ✓ | 24 | 7.d | odd | 6 | 1 | inner |
3150.2.bf.d | ✓ | 24 | 21.g | even | 6 | 1 | inner |
3150.2.bf.e | yes | 24 | 5.b | even | 2 | 1 | |
3150.2.bf.e | yes | 24 | 15.d | odd | 2 | 1 | |
3150.2.bf.e | yes | 24 | 35.i | odd | 6 | 1 | |
3150.2.bf.e | yes | 24 | 105.p | even | 6 | 1 | |
3150.2.bp.g | 24 | 5.c | odd | 4 | 1 | ||
3150.2.bp.g | 24 | 15.e | even | 4 | 1 | ||
3150.2.bp.g | 24 | 35.k | even | 12 | 1 | ||
3150.2.bp.g | 24 | 105.w | odd | 12 | 1 | ||
3150.2.bp.h | 24 | 5.c | odd | 4 | 1 | ||
3150.2.bp.h | 24 | 15.e | even | 4 | 1 | ||
3150.2.bp.h | 24 | 35.k | even | 12 | 1 | ||
3150.2.bp.h | 24 | 105.w | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3150, [\chi])\):
\( T_{11}^{24} - 68 T_{11}^{22} + 3232 T_{11}^{20} - 77312 T_{11}^{18} + 1330444 T_{11}^{16} - 9697232 T_{11}^{14} + 50195872 T_{11}^{12} - 126799808 T_{11}^{10} + 226614160 T_{11}^{8} - 136499904 T_{11}^{6} + \cdots + 1679616 \) |
\( T_{37}^{12} - 14 T_{37}^{11} + 253 T_{37}^{10} - 1642 T_{37}^{9} + 22759 T_{37}^{8} - 138392 T_{37}^{7} + 1343030 T_{37}^{6} - 3088724 T_{37}^{5} + 7024018 T_{37}^{4} - 3526600 T_{37}^{3} + 6136564 T_{37}^{2} + \cdots + 5080516 \) |