Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [975,2,Mod(443,975)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(975, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([2, 3, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("975.443");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 975 = 3 \cdot 5^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 975.m (of order \(4\), 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: | \(7.78541419707\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(i)\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
443.1 | −1.87465 | − | 1.87465i | −1.24513 | + | 1.20401i | 5.02865i | 0 | 4.59130 | + | 0.0770814i | 3.00921 | − | 3.00921i | 5.67767 | − | 5.67767i | 0.100703 | − | 2.99831i | 0 | ||||||
443.2 | −1.63741 | − | 1.63741i | −1.36197 | − | 1.07007i | 3.36220i | 0 | 0.477964 | + | 3.98223i | 1.97947 | − | 1.97947i | 2.23048 | − | 2.23048i | 0.709918 | + | 2.91479i | 0 | ||||||
443.3 | −1.51844 | − | 1.51844i | 1.72589 | + | 0.145999i | 2.61134i | 0 | −2.39897 | − | 2.84235i | −0.988598 | + | 0.988598i | 0.928279 | − | 0.928279i | 2.95737 | + | 0.503955i | 0 | ||||||
443.4 | −1.47851 | − | 1.47851i | 0.625353 | + | 1.61522i | 2.37199i | 0 | 1.46353 | − | 3.31271i | 2.32423 | − | 2.32423i | 0.549992 | − | 0.549992i | −2.21787 | + | 2.02017i | 0 | ||||||
443.5 | −1.08900 | − | 1.08900i | 1.68742 | − | 0.390653i | 0.371844i | 0 | −2.26302 | − | 1.41218i | 2.02596 | − | 2.02596i | −1.77306 | + | 1.77306i | 2.69478 | − | 1.31839i | 0 | ||||||
443.6 | −0.727164 | − | 0.727164i | −1.65767 | − | 0.502116i | − | 0.942464i | 0 | 0.840280 | + | 1.57052i | −1.99337 | + | 1.99337i | −2.13965 | + | 2.13965i | 2.49576 | + | 1.66469i | 0 | |||||
443.7 | −0.572109 | − | 0.572109i | −0.867167 | + | 1.49934i | − | 1.34538i | 0 | 1.35390 | − | 0.361672i | 0.327985 | − | 0.327985i | −1.91392 | + | 1.91392i | −1.49604 | − | 2.60036i | 0 | |||||
443.8 | −0.520491 | − | 0.520491i | 1.69638 | − | 0.349729i | − | 1.45818i | 0 | −1.06498 | − | 0.700917i | −3.38546 | + | 3.38546i | −1.79995 | + | 1.79995i | 2.75538 | − | 1.18654i | 0 | |||||
443.9 | 0.520491 | + | 0.520491i | −1.69638 | + | 0.349729i | − | 1.45818i | 0 | −1.06498 | − | 0.700917i | 3.38546 | − | 3.38546i | 1.79995 | − | 1.79995i | 2.75538 | − | 1.18654i | 0 | |||||
443.10 | 0.572109 | + | 0.572109i | 0.867167 | − | 1.49934i | − | 1.34538i | 0 | 1.35390 | − | 0.361672i | −0.327985 | + | 0.327985i | 1.91392 | − | 1.91392i | −1.49604 | − | 2.60036i | 0 | |||||
443.11 | 0.727164 | + | 0.727164i | 1.65767 | + | 0.502116i | − | 0.942464i | 0 | 0.840280 | + | 1.57052i | 1.99337 | − | 1.99337i | 2.13965 | − | 2.13965i | 2.49576 | + | 1.66469i | 0 | |||||
443.12 | 1.08900 | + | 1.08900i | −1.68742 | + | 0.390653i | 0.371844i | 0 | −2.26302 | − | 1.41218i | −2.02596 | + | 2.02596i | 1.77306 | − | 1.77306i | 2.69478 | − | 1.31839i | 0 | ||||||
443.13 | 1.47851 | + | 1.47851i | −0.625353 | − | 1.61522i | 2.37199i | 0 | 1.46353 | − | 3.31271i | −2.32423 | + | 2.32423i | −0.549992 | + | 0.549992i | −2.21787 | + | 2.02017i | 0 | ||||||
443.14 | 1.51844 | + | 1.51844i | −1.72589 | − | 0.145999i | 2.61134i | 0 | −2.39897 | − | 2.84235i | 0.988598 | − | 0.988598i | −0.928279 | + | 0.928279i | 2.95737 | + | 0.503955i | 0 | ||||||
443.15 | 1.63741 | + | 1.63741i | 1.36197 | + | 1.07007i | 3.36220i | 0 | 0.477964 | + | 3.98223i | −1.97947 | + | 1.97947i | −2.23048 | + | 2.23048i | 0.709918 | + | 2.91479i | 0 | ||||||
443.16 | 1.87465 | + | 1.87465i | 1.24513 | − | 1.20401i | 5.02865i | 0 | 4.59130 | + | 0.0770814i | −3.00921 | + | 3.00921i | −5.67767 | + | 5.67767i | 0.100703 | − | 2.99831i | 0 | ||||||
482.1 | −1.87465 | + | 1.87465i | −1.24513 | − | 1.20401i | − | 5.02865i | 0 | 4.59130 | − | 0.0770814i | 3.00921 | + | 3.00921i | 5.67767 | + | 5.67767i | 0.100703 | + | 2.99831i | 0 | |||||
482.2 | −1.63741 | + | 1.63741i | −1.36197 | + | 1.07007i | − | 3.36220i | 0 | 0.477964 | − | 3.98223i | 1.97947 | + | 1.97947i | 2.23048 | + | 2.23048i | 0.709918 | − | 2.91479i | 0 | |||||
482.3 | −1.51844 | + | 1.51844i | 1.72589 | − | 0.145999i | − | 2.61134i | 0 | −2.39897 | + | 2.84235i | −0.988598 | − | 0.988598i | 0.928279 | + | 0.928279i | 2.95737 | − | 0.503955i | 0 | |||||
482.4 | −1.47851 | + | 1.47851i | 0.625353 | − | 1.61522i | − | 2.37199i | 0 | 1.46353 | + | 3.31271i | 2.32423 | + | 2.32423i | 0.549992 | + | 0.549992i | −2.21787 | − | 2.02017i | 0 | |||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
15.e | even | 4 | 2 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 975.2.m.c | yes | 32 |
3.b | odd | 2 | 1 | 975.2.m.b | ✓ | 32 | |
5.b | even | 2 | 1 | inner | 975.2.m.c | yes | 32 |
5.c | odd | 4 | 2 | 975.2.m.b | ✓ | 32 | |
15.d | odd | 2 | 1 | 975.2.m.b | ✓ | 32 | |
15.e | even | 4 | 2 | inner | 975.2.m.c | yes | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
975.2.m.b | ✓ | 32 | 3.b | odd | 2 | 1 | |
975.2.m.b | ✓ | 32 | 5.c | odd | 4 | 2 | |
975.2.m.b | ✓ | 32 | 15.d | odd | 2 | 1 | |
975.2.m.c | yes | 32 | 1.a | even | 1 | 1 | trivial |
975.2.m.c | yes | 32 | 5.b | even | 2 | 1 | inner |
975.2.m.c | yes | 32 | 15.e | even | 4 | 2 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(975, [\chi])\):
\( T_{2}^{32} + 126 T_{2}^{28} + 5879 T_{2}^{24} + 127668 T_{2}^{20} + 1299639 T_{2}^{16} + 5343510 T_{2}^{12} + \cdots + 456976 \) |
\( T_{29}^{8} - 100T_{29}^{6} - 72T_{29}^{5} + 2086T_{29}^{4} - 432T_{29}^{3} - 8964T_{29}^{2} + 6552T_{29} + 1521 \) |