Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,3,Mod(949,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, 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("950.949");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.d (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: | \(25.8856251142\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Twist minimal: | no (minimal twist has level 190) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
949.1 | −1.41421 | 3.15928 | 2.00000 | 0 | −4.46790 | − | 13.6136i | −2.82843 | 0.981053 | 0 | |||||||||||||||||
949.2 | −1.41421 | −5.26410 | 2.00000 | 0 | 7.44456 | 12.5505i | −2.82843 | 18.7107 | 0 | ||||||||||||||||||
949.3 | −1.41421 | 1.23540 | 2.00000 | 0 | −1.74712 | − | 8.54633i | −2.82843 | −7.47378 | 0 | |||||||||||||||||
949.4 | −1.41421 | −5.62107 | 2.00000 | 0 | 7.94939 | 6.51823i | −2.82843 | 22.5964 | 0 | ||||||||||||||||||
949.5 | −1.41421 | −1.95901 | 2.00000 | 0 | 2.77046 | 5.01111i | −2.82843 | −5.16227 | 0 | ||||||||||||||||||
949.6 | −1.41421 | 5.15315 | 2.00000 | 0 | −7.28765 | 0.0751428i | −2.82843 | 17.5549 | 0 | ||||||||||||||||||
949.7 | −1.41421 | −1.12134 | 2.00000 | 0 | 1.58581 | 6.17053i | −2.82843 | −7.74260 | 0 | ||||||||||||||||||
949.8 | −1.41421 | −1.23917 | 2.00000 | 0 | 1.75244 | 6.19957i | −2.82843 | −7.46447 | 0 | ||||||||||||||||||
949.9 | −1.41421 | −1.23917 | 2.00000 | 0 | 1.75244 | − | 6.19957i | −2.82843 | −7.46447 | 0 | |||||||||||||||||
949.10 | −1.41421 | −1.12134 | 2.00000 | 0 | 1.58581 | − | 6.17053i | −2.82843 | −7.74260 | 0 | |||||||||||||||||
949.11 | −1.41421 | 5.15315 | 2.00000 | 0 | −7.28765 | − | 0.0751428i | −2.82843 | 17.5549 | 0 | |||||||||||||||||
949.12 | −1.41421 | −1.95901 | 2.00000 | 0 | 2.77046 | − | 5.01111i | −2.82843 | −5.16227 | 0 | |||||||||||||||||
949.13 | −1.41421 | −5.62107 | 2.00000 | 0 | 7.94939 | − | 6.51823i | −2.82843 | 22.5964 | 0 | |||||||||||||||||
949.14 | −1.41421 | 1.23540 | 2.00000 | 0 | −1.74712 | 8.54633i | −2.82843 | −7.47378 | 0 | ||||||||||||||||||
949.15 | −1.41421 | −5.26410 | 2.00000 | 0 | 7.44456 | − | 12.5505i | −2.82843 | 18.7107 | 0 | |||||||||||||||||
949.16 | −1.41421 | 3.15928 | 2.00000 | 0 | −4.46790 | 13.6136i | −2.82843 | 0.981053 | 0 | ||||||||||||||||||
949.17 | 1.41421 | −3.15928 | 2.00000 | 0 | −4.46790 | − | 13.6136i | 2.82843 | 0.981053 | 0 | |||||||||||||||||
949.18 | 1.41421 | 5.26410 | 2.00000 | 0 | 7.44456 | 12.5505i | 2.82843 | 18.7107 | 0 | ||||||||||||||||||
949.19 | 1.41421 | −1.23540 | 2.00000 | 0 | −1.74712 | − | 8.54633i | 2.82843 | −7.47378 | 0 | |||||||||||||||||
949.20 | 1.41421 | 5.62107 | 2.00000 | 0 | 7.94939 | 6.51823i | 2.82843 | 22.5964 | 0 | ||||||||||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
19.b | odd | 2 | 1 | inner |
95.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.3.d.c | 32 | |
5.b | even | 2 | 1 | inner | 950.3.d.c | 32 | |
5.c | odd | 4 | 1 | 190.3.c.a | ✓ | 16 | |
5.c | odd | 4 | 1 | 950.3.c.d | 16 | ||
15.e | even | 4 | 1 | 1710.3.h.a | 16 | ||
19.b | odd | 2 | 1 | inner | 950.3.d.c | 32 | |
20.e | even | 4 | 1 | 1520.3.h.c | 16 | ||
95.d | odd | 2 | 1 | inner | 950.3.d.c | 32 | |
95.g | even | 4 | 1 | 190.3.c.a | ✓ | 16 | |
95.g | even | 4 | 1 | 950.3.c.d | 16 | ||
285.j | odd | 4 | 1 | 1710.3.h.a | 16 | ||
380.j | odd | 4 | 1 | 1520.3.h.c | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
190.3.c.a | ✓ | 16 | 5.c | odd | 4 | 1 | |
190.3.c.a | ✓ | 16 | 95.g | even | 4 | 1 | |
950.3.c.d | 16 | 5.c | odd | 4 | 1 | ||
950.3.c.d | 16 | 95.g | even | 4 | 1 | ||
950.3.d.c | 32 | 1.a | even | 1 | 1 | trivial | |
950.3.d.c | 32 | 5.b | even | 2 | 1 | inner | |
950.3.d.c | 32 | 19.b | odd | 2 | 1 | inner | |
950.3.d.c | 32 | 95.d | odd | 2 | 1 | inner | |
1520.3.h.c | 16 | 20.e | even | 4 | 1 | ||
1520.3.h.c | 16 | 380.j | odd | 4 | 1 | ||
1710.3.h.a | 16 | 15.e | even | 4 | 1 | ||
1710.3.h.a | 16 | 285.j | odd | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{16} - 104 T_{3}^{14} + 4112 T_{3}^{12} - 76896 T_{3}^{10} + 699096 T_{3}^{8} - 3068640 T_{3}^{6} + 6595840 T_{3}^{4} - 6739200 T_{3}^{2} + 2624400 \)
acting on \(S_{3}^{\mathrm{new}}(950, [\chi])\).