Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1150,3,Mod(1149,1150)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1150, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1150.1149");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1150 = 2 \cdot 5^{2} \cdot 23 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 1150.c (of order \(2\), degree \(1\), 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: | \(31.3352304014\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | no (minimal twist has level 230) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1149.1 | − | 1.41421i | 1.43837i | −2.00000 | 0 | 2.03417 | −10.1866 | 2.82843i | 6.93108 | 0 | |||||||||||||||||
1149.2 | 1.41421i | − | 1.43837i | −2.00000 | 0 | 2.03417 | −10.1866 | − | 2.82843i | 6.93108 | 0 | ||||||||||||||||
1149.3 | − | 1.41421i | 5.41949i | −2.00000 | 0 | 7.66432 | 8.24199 | 2.82843i | −20.3709 | 0 | |||||||||||||||||
1149.4 | 1.41421i | − | 5.41949i | −2.00000 | 0 | 7.66432 | 8.24199 | − | 2.82843i | −20.3709 | 0 | ||||||||||||||||
1149.5 | − | 1.41421i | − | 4.76369i | −2.00000 | 0 | −6.73687 | −7.05858 | 2.82843i | −13.6927 | 0 | ||||||||||||||||
1149.6 | 1.41421i | 4.76369i | −2.00000 | 0 | −6.73687 | −7.05858 | − | 2.82843i | −13.6927 | 0 | |||||||||||||||||
1149.7 | − | 1.41421i | 0.278523i | −2.00000 | 0 | 0.393890 | 8.51262 | 2.82843i | 8.92243 | 0 | |||||||||||||||||
1149.8 | 1.41421i | − | 0.278523i | −2.00000 | 0 | 0.393890 | 8.51262 | − | 2.82843i | 8.92243 | 0 | ||||||||||||||||
1149.9 | − | 1.41421i | − | 3.79379i | −2.00000 | 0 | −5.36524 | −7.10180 | 2.82843i | −5.39287 | 0 | ||||||||||||||||
1149.10 | 1.41421i | 3.79379i | −2.00000 | 0 | −5.36524 | −7.10180 | − | 2.82843i | −5.39287 | 0 | |||||||||||||||||
1149.11 | − | 1.41421i | − | 2.34854i | −2.00000 | 0 | −3.32134 | 7.61815 | 2.82843i | 3.48436 | 0 | ||||||||||||||||
1149.12 | 1.41421i | 2.34854i | −2.00000 | 0 | −3.32134 | 7.61815 | − | 2.82843i | 3.48436 | 0 | |||||||||||||||||
1149.13 | − | 1.41421i | 4.30716i | −2.00000 | 0 | 6.09125 | −1.47532 | 2.82843i | −9.55167 | 0 | |||||||||||||||||
1149.14 | 1.41421i | − | 4.30716i | −2.00000 | 0 | 6.09125 | −1.47532 | − | 2.82843i | −9.55167 | 0 | ||||||||||||||||
1149.15 | − | 1.41421i | − | 3.36596i | −2.00000 | 0 | −4.76019 | 1.16919 | 2.82843i | −2.32968 | 0 | ||||||||||||||||
1149.16 | 1.41421i | 3.36596i | −2.00000 | 0 | −4.76019 | 1.16919 | − | 2.82843i | −2.32968 | 0 | |||||||||||||||||
1149.17 | − | 1.41421i | − | 3.36596i | −2.00000 | 0 | −4.76019 | −1.16919 | 2.82843i | −2.32968 | 0 | ||||||||||||||||
1149.18 | 1.41421i | 3.36596i | −2.00000 | 0 | −4.76019 | −1.16919 | − | 2.82843i | −2.32968 | 0 | |||||||||||||||||
1149.19 | − | 1.41421i | 4.30716i | −2.00000 | 0 | 6.09125 | 1.47532 | 2.82843i | −9.55167 | 0 | |||||||||||||||||
1149.20 | 1.41421i | − | 4.30716i | −2.00000 | 0 | 6.09125 | 1.47532 | − | 2.82843i | −9.55167 | 0 | ||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
23.b | odd | 2 | 1 | inner |
115.c | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1150.3.c.c | 32 | |
5.b | even | 2 | 1 | inner | 1150.3.c.c | 32 | |
5.c | odd | 4 | 1 | 230.3.d.a | ✓ | 16 | |
5.c | odd | 4 | 1 | 1150.3.d.b | 16 | ||
15.e | even | 4 | 1 | 2070.3.c.a | 16 | ||
20.e | even | 4 | 1 | 1840.3.k.d | 16 | ||
23.b | odd | 2 | 1 | inner | 1150.3.c.c | 32 | |
115.c | odd | 2 | 1 | inner | 1150.3.c.c | 32 | |
115.e | even | 4 | 1 | 230.3.d.a | ✓ | 16 | |
115.e | even | 4 | 1 | 1150.3.d.b | 16 | ||
345.l | odd | 4 | 1 | 2070.3.c.a | 16 | ||
460.k | odd | 4 | 1 | 1840.3.k.d | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
230.3.d.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
230.3.d.a | ✓ | 16 | 115.e | even | 4 | 1 | |
1150.3.c.c | 32 | 1.a | even | 1 | 1 | trivial | |
1150.3.c.c | 32 | 5.b | even | 2 | 1 | inner | |
1150.3.c.c | 32 | 23.b | odd | 2 | 1 | inner | |
1150.3.c.c | 32 | 115.c | odd | 2 | 1 | inner | |
1150.3.d.b | 16 | 5.c | odd | 4 | 1 | ||
1150.3.d.b | 16 | 115.e | even | 4 | 1 | ||
1840.3.k.d | 16 | 20.e | even | 4 | 1 | ||
1840.3.k.d | 16 | 460.k | odd | 4 | 1 | ||
2070.3.c.a | 16 | 15.e | even | 4 | 1 | ||
2070.3.c.a | 16 | 345.l | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 104 T_{3}^{14} + 4362 T_{3}^{12} + 94700 T_{3}^{10} + 1132361 T_{3}^{8} + 7286420 T_{3}^{6} + \cdots + 1784896 \) acting on \(S_{3}^{\mathrm{new}}(1150, [\chi])\).