Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [225,4,Mod(49,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.49");
S:= CuspForms(chi, 4);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 225 = 3^{2} \cdot 5^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 225.k (of order \(6\), degree \(2\), 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: | \(13.2754297513\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 45) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | −4.66357 | + | 2.69252i | 3.19791 | − | 4.09553i | 10.4993 | − | 18.1853i | 0 | −3.88642 | + | 27.7102i | −10.8861 | + | 6.28510i | 69.9976i | −6.54673 | − | 26.1943i | 0 | ||||||
49.2 | −4.60336 | + | 2.65775i | −0.153351 | + | 5.19389i | 10.1273 | − | 17.5410i | 0 | −13.0981 | − | 24.3169i | −11.6339 | + | 6.71686i | 65.1396i | −26.9530 | − | 1.59298i | 0 | ||||||
49.3 | −3.69081 | + | 2.13089i | 2.76977 | − | 4.39640i | 5.08138 | − | 8.80120i | 0 | −0.854448 | + | 22.1284i | 26.6423 | − | 15.3820i | 9.21718i | −11.6567 | − | 24.3541i | 0 | ||||||
49.4 | −2.63422 | + | 1.52087i | 3.29398 | + | 4.01867i | 0.626094 | − | 1.08443i | 0 | −14.7890 | − | 5.57636i | 11.8751 | − | 6.85611i | − | 20.5251i | −5.29939 | + | 26.4748i | 0 | |||||
49.5 | −1.90044 | + | 1.09722i | −5.19206 | − | 0.206141i | −1.59221 | + | 2.75778i | 0 | 10.0934 | − | 5.30509i | 2.39547 | − | 1.38302i | − | 24.5436i | 26.9150 | + | 2.14059i | 0 | |||||
49.6 | −1.35984 | + | 0.785104i | −3.43441 | − | 3.89934i | −2.76722 | + | 4.79297i | 0 | 7.73164 | + | 2.60611i | −29.6525 | + | 17.1199i | − | 21.2519i | −3.40971 | + | 26.7838i | 0 | |||||
49.7 | −0.195072 | + | 0.112625i | 4.76710 | − | 2.06755i | −3.97463 | + | 6.88426i | 0 | −0.697071 | + | 0.940215i | −27.0148 | + | 15.5970i | − | 3.59257i | 18.4505 | − | 19.7124i | 0 | |||||
49.8 | 0.195072 | − | 0.112625i | −4.76710 | + | 2.06755i | −3.97463 | + | 6.88426i | 0 | −0.697071 | + | 0.940215i | 27.0148 | − | 15.5970i | 3.59257i | 18.4505 | − | 19.7124i | 0 | ||||||
49.9 | 1.35984 | − | 0.785104i | 3.43441 | + | 3.89934i | −2.76722 | + | 4.79297i | 0 | 7.73164 | + | 2.60611i | 29.6525 | − | 17.1199i | 21.2519i | −3.40971 | + | 26.7838i | 0 | ||||||
49.10 | 1.90044 | − | 1.09722i | 5.19206 | + | 0.206141i | −1.59221 | + | 2.75778i | 0 | 10.0934 | − | 5.30509i | −2.39547 | + | 1.38302i | 24.5436i | 26.9150 | + | 2.14059i | 0 | ||||||
49.11 | 2.63422 | − | 1.52087i | −3.29398 | − | 4.01867i | 0.626094 | − | 1.08443i | 0 | −14.7890 | − | 5.57636i | −11.8751 | + | 6.85611i | 20.5251i | −5.29939 | + | 26.4748i | 0 | ||||||
49.12 | 3.69081 | − | 2.13089i | −2.76977 | + | 4.39640i | 5.08138 | − | 8.80120i | 0 | −0.854448 | + | 22.1284i | −26.6423 | + | 15.3820i | − | 9.21718i | −11.6567 | − | 24.3541i | 0 | |||||
49.13 | 4.60336 | − | 2.65775i | 0.153351 | − | 5.19389i | 10.1273 | − | 17.5410i | 0 | −13.0981 | − | 24.3169i | 11.6339 | − | 6.71686i | − | 65.1396i | −26.9530 | − | 1.59298i | 0 | |||||
49.14 | 4.66357 | − | 2.69252i | −3.19791 | + | 4.09553i | 10.4993 | − | 18.1853i | 0 | −3.88642 | + | 27.7102i | 10.8861 | − | 6.28510i | − | 69.9976i | −6.54673 | − | 26.1943i | 0 | |||||
124.1 | −4.66357 | − | 2.69252i | 3.19791 | + | 4.09553i | 10.4993 | + | 18.1853i | 0 | −3.88642 | − | 27.7102i | −10.8861 | − | 6.28510i | − | 69.9976i | −6.54673 | + | 26.1943i | 0 | |||||
124.2 | −4.60336 | − | 2.65775i | −0.153351 | − | 5.19389i | 10.1273 | + | 17.5410i | 0 | −13.0981 | + | 24.3169i | −11.6339 | − | 6.71686i | − | 65.1396i | −26.9530 | + | 1.59298i | 0 | |||||
124.3 | −3.69081 | − | 2.13089i | 2.76977 | + | 4.39640i | 5.08138 | + | 8.80120i | 0 | −0.854448 | − | 22.1284i | 26.6423 | + | 15.3820i | − | 9.21718i | −11.6567 | + | 24.3541i | 0 | |||||
124.4 | −2.63422 | − | 1.52087i | 3.29398 | − | 4.01867i | 0.626094 | + | 1.08443i | 0 | −14.7890 | + | 5.57636i | 11.8751 | + | 6.85611i | 20.5251i | −5.29939 | − | 26.4748i | 0 | ||||||
124.5 | −1.90044 | − | 1.09722i | −5.19206 | + | 0.206141i | −1.59221 | − | 2.75778i | 0 | 10.0934 | + | 5.30509i | 2.39547 | + | 1.38302i | 24.5436i | 26.9150 | − | 2.14059i | 0 | ||||||
124.6 | −1.35984 | − | 0.785104i | −3.43441 | + | 3.89934i | −2.76722 | − | 4.79297i | 0 | 7.73164 | − | 2.60611i | −29.6525 | − | 17.1199i | 21.2519i | −3.40971 | − | 26.7838i | 0 | ||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 225.4.k.d | 28 | |
5.b | even | 2 | 1 | inner | 225.4.k.d | 28 | |
5.c | odd | 4 | 1 | 45.4.e.c | ✓ | 14 | |
5.c | odd | 4 | 1 | 225.4.e.d | 14 | ||
9.c | even | 3 | 1 | inner | 225.4.k.d | 28 | |
15.e | even | 4 | 1 | 135.4.e.c | 14 | ||
45.j | even | 6 | 1 | inner | 225.4.k.d | 28 | |
45.k | odd | 12 | 1 | 45.4.e.c | ✓ | 14 | |
45.k | odd | 12 | 1 | 225.4.e.d | 14 | ||
45.k | odd | 12 | 1 | 405.4.a.m | 7 | ||
45.k | odd | 12 | 1 | 2025.4.a.bb | 7 | ||
45.l | even | 12 | 1 | 135.4.e.c | 14 | ||
45.l | even | 12 | 1 | 405.4.a.n | 7 | ||
45.l | even | 12 | 1 | 2025.4.a.ba | 7 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
45.4.e.c | ✓ | 14 | 5.c | odd | 4 | 1 | |
45.4.e.c | ✓ | 14 | 45.k | odd | 12 | 1 | |
135.4.e.c | 14 | 15.e | even | 4 | 1 | ||
135.4.e.c | 14 | 45.l | even | 12 | 1 | ||
225.4.e.d | 14 | 5.c | odd | 4 | 1 | ||
225.4.e.d | 14 | 45.k | odd | 12 | 1 | ||
225.4.k.d | 28 | 1.a | even | 1 | 1 | trivial | |
225.4.k.d | 28 | 5.b | even | 2 | 1 | inner | |
225.4.k.d | 28 | 9.c | even | 3 | 1 | inner | |
225.4.k.d | 28 | 45.j | even | 6 | 1 | inner | |
405.4.a.m | 7 | 45.k | odd | 12 | 1 | ||
405.4.a.n | 7 | 45.l | even | 12 | 1 | ||
2025.4.a.ba | 7 | 45.l | even | 12 | 1 | ||
2025.4.a.bb | 7 | 45.k | odd | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{28} - 92 T_{2}^{26} + 5274 T_{2}^{24} - 189744 T_{2}^{22} + 4999971 T_{2}^{20} + \cdots + 6879707136 \) acting on \(S_{4}^{\mathrm{new}}(225, [\chi])\).