Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [950,2,Mod(107,950)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(950, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("950.107");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 950 = 2 \cdot 5^{2} \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 950.q (of order \(12\), degree \(4\), 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: | \(32\) |
Relative dimension: | \(8\) over \(\Q(\zeta_{12})\) |
Twist minimal: | no (minimal twist has level 190) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
107.1 | −0.965926 | − | 0.258819i | −0.570807 | + | 2.13028i | 0.866025 | + | 0.500000i | 0 | 1.10271 | − | 1.90996i | −0.526345 | − | 0.526345i | −0.707107 | − | 0.707107i | −1.61420 | − | 0.931958i | 0 | ||||
107.2 | −0.965926 | − | 0.258819i | 0.186170 | − | 0.694795i | 0.866025 | + | 0.500000i | 0 | −0.359652 | + | 0.622936i | −1.01416 | − | 1.01416i | −0.707107 | − | 0.707107i | 2.15000 | + | 1.24130i | 0 | ||||
107.3 | −0.965926 | − | 0.258819i | 0.338196 | − | 1.26216i | 0.866025 | + | 0.500000i | 0 | −0.653344 | + | 1.13162i | −1.48691 | − | 1.48691i | −0.707107 | − | 0.707107i | 1.11940 | + | 0.646284i | 0 | ||||
107.4 | −0.965926 | − | 0.258819i | 0.680416 | − | 2.53935i | 0.866025 | + | 0.500000i | 0 | −1.31446 | + | 2.27672i | 2.47691 | + | 2.47691i | −0.707107 | − | 0.707107i | −3.38724 | − | 1.95563i | 0 | ||||
107.5 | 0.965926 | + | 0.258819i | −0.407975 | + | 1.52258i | 0.866025 | + | 0.500000i | 0 | −0.788148 | + | 1.36511i | −2.78714 | − | 2.78714i | 0.707107 | + | 0.707107i | 0.446257 | + | 0.257647i | 0 | ||||
107.6 | 0.965926 | + | 0.258819i | 0.0641036 | − | 0.239238i | 0.866025 | + | 0.500000i | 0 | 0.123839 | − | 0.214495i | 2.17543 | + | 2.17543i | 0.707107 | + | 0.707107i | 2.54495 | + | 1.46933i | 0 | ||||
107.7 | 0.965926 | + | 0.258819i | 0.108675 | − | 0.405580i | 0.866025 | + | 0.500000i | 0 | 0.209944 | − | 0.363633i | −2.14890 | − | 2.14890i | 0.707107 | + | 0.707107i | 2.44539 | + | 1.41185i | 0 | ||||
107.8 | 0.965926 | + | 0.258819i | 0.869171 | − | 3.24379i | 0.866025 | + | 0.500000i | 0 | 1.67911 | − | 2.90830i | −2.68888 | − | 2.68888i | 0.707107 | + | 0.707107i | −7.16865 | − | 4.13882i | 0 | ||||
293.1 | −0.965926 | + | 0.258819i | −0.570807 | − | 2.13028i | 0.866025 | − | 0.500000i | 0 | 1.10271 | + | 1.90996i | −0.526345 | + | 0.526345i | −0.707107 | + | 0.707107i | −1.61420 | + | 0.931958i | 0 | ||||
293.2 | −0.965926 | + | 0.258819i | 0.186170 | + | 0.694795i | 0.866025 | − | 0.500000i | 0 | −0.359652 | − | 0.622936i | −1.01416 | + | 1.01416i | −0.707107 | + | 0.707107i | 2.15000 | − | 1.24130i | 0 | ||||
293.3 | −0.965926 | + | 0.258819i | 0.338196 | + | 1.26216i | 0.866025 | − | 0.500000i | 0 | −0.653344 | − | 1.13162i | −1.48691 | + | 1.48691i | −0.707107 | + | 0.707107i | 1.11940 | − | 0.646284i | 0 | ||||
293.4 | −0.965926 | + | 0.258819i | 0.680416 | + | 2.53935i | 0.866025 | − | 0.500000i | 0 | −1.31446 | − | 2.27672i | 2.47691 | − | 2.47691i | −0.707107 | + | 0.707107i | −3.38724 | + | 1.95563i | 0 | ||||
293.5 | 0.965926 | − | 0.258819i | −0.407975 | − | 1.52258i | 0.866025 | − | 0.500000i | 0 | −0.788148 | − | 1.36511i | −2.78714 | + | 2.78714i | 0.707107 | − | 0.707107i | 0.446257 | − | 0.257647i | 0 | ||||
293.6 | 0.965926 | − | 0.258819i | 0.0641036 | + | 0.239238i | 0.866025 | − | 0.500000i | 0 | 0.123839 | + | 0.214495i | 2.17543 | − | 2.17543i | 0.707107 | − | 0.707107i | 2.54495 | − | 1.46933i | 0 | ||||
293.7 | 0.965926 | − | 0.258819i | 0.108675 | + | 0.405580i | 0.866025 | − | 0.500000i | 0 | 0.209944 | + | 0.363633i | −2.14890 | + | 2.14890i | 0.707107 | − | 0.707107i | 2.44539 | − | 1.41185i | 0 | ||||
293.8 | 0.965926 | − | 0.258819i | 0.869171 | + | 3.24379i | 0.866025 | − | 0.500000i | 0 | 1.67911 | + | 2.90830i | −2.68888 | + | 2.68888i | 0.707107 | − | 0.707107i | −7.16865 | + | 4.13882i | 0 | ||||
407.1 | −0.258819 | − | 0.965926i | −2.13028 | + | 0.570807i | −0.866025 | + | 0.500000i | 0 | 1.10271 | + | 1.90996i | −0.526345 | − | 0.526345i | 0.707107 | + | 0.707107i | 1.61420 | − | 0.931958i | 0 | ||||
407.2 | −0.258819 | − | 0.965926i | 0.694795 | − | 0.186170i | −0.866025 | + | 0.500000i | 0 | −0.359652 | − | 0.622936i | −1.01416 | − | 1.01416i | 0.707107 | + | 0.707107i | −2.15000 | + | 1.24130i | 0 | ||||
407.3 | −0.258819 | − | 0.965926i | 1.26216 | − | 0.338196i | −0.866025 | + | 0.500000i | 0 | −0.653344 | − | 1.13162i | −1.48691 | − | 1.48691i | 0.707107 | + | 0.707107i | −1.11940 | + | 0.646284i | 0 | ||||
407.4 | −0.258819 | − | 0.965926i | 2.53935 | − | 0.680416i | −0.866025 | + | 0.500000i | 0 | −1.31446 | − | 2.27672i | 2.47691 | + | 2.47691i | 0.707107 | + | 0.707107i | 3.38724 | − | 1.95563i | 0 | ||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
19.d | odd | 6 | 1 | inner |
95.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 950.2.q.g | 32 | |
5.b | even | 2 | 1 | 190.2.m.b | ✓ | 32 | |
5.c | odd | 4 | 1 | 190.2.m.b | ✓ | 32 | |
5.c | odd | 4 | 1 | inner | 950.2.q.g | 32 | |
19.d | odd | 6 | 1 | inner | 950.2.q.g | 32 | |
95.h | odd | 6 | 1 | 190.2.m.b | ✓ | 32 | |
95.l | even | 12 | 1 | 190.2.m.b | ✓ | 32 | |
95.l | even | 12 | 1 | inner | 950.2.q.g | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
190.2.m.b | ✓ | 32 | 5.b | even | 2 | 1 | |
190.2.m.b | ✓ | 32 | 5.c | odd | 4 | 1 | |
190.2.m.b | ✓ | 32 | 95.h | odd | 6 | 1 | |
190.2.m.b | ✓ | 32 | 95.l | even | 12 | 1 | |
950.2.q.g | 32 | 1.a | even | 1 | 1 | trivial | |
950.2.q.g | 32 | 5.c | odd | 4 | 1 | inner | |
950.2.q.g | 32 | 19.d | odd | 6 | 1 | inner | |
950.2.q.g | 32 | 95.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):
\( T_{3}^{32} - 12 T_{3}^{31} + 72 T_{3}^{30} - 288 T_{3}^{29} + 760 T_{3}^{28} - 1068 T_{3}^{27} + \cdots + 81 \) |
\( T_{7}^{16} + 12 T_{7}^{15} + 72 T_{7}^{14} + 248 T_{7}^{13} + 784 T_{7}^{12} + 3640 T_{7}^{11} + \cdots + 1214404 \) |