Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [495,2,Mod(166,495)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("495.166");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 495.i (of order \(3\), 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: | \(3.95259490005\) |
Analytic rank: | \(0\) |
Dimension: | \(22\) |
Relative dimension: | \(11\) over \(\Q(\zeta_{3})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
166.1 | −1.26605 | − | 2.19286i | 0.415004 | + | 1.68160i | −2.20577 | + | 3.82050i | 0.500000 | − | 0.866025i | 3.16210 | − | 3.03904i | 1.22474 | + | 2.12131i | 6.10626 | −2.65554 | + | 1.39574i | −2.53210 | ||||
166.2 | −1.00630 | − | 1.74296i | −0.598177 | − | 1.62548i | −1.02527 | + | 1.77582i | 0.500000 | − | 0.866025i | −2.23120 | + | 2.67831i | −1.47315 | − | 2.55156i | 0.101717 | −2.28437 | + | 1.94465i | −2.01260 | ||||
166.3 | −0.949086 | − | 1.64387i | −1.54093 | + | 0.790898i | −0.801529 | + | 1.38829i | 0.500000 | − | 0.866025i | 2.76261 | + | 1.78246i | 0.751191 | + | 1.30110i | −0.753463 | 1.74896 | − | 2.43745i | −1.89817 | ||||
166.4 | −0.350077 | − | 0.606351i | 1.65559 | − | 0.508929i | 0.754892 | − | 1.30751i | 0.500000 | − | 0.866025i | −0.888175 | − | 0.825707i | −0.588842 | − | 1.01990i | −2.45739 | 2.48198 | − | 1.68516i | −0.700154 | ||||
166.5 | 0.151059 | + | 0.261642i | −1.73181 | − | 0.0286900i | 0.954362 | − | 1.65300i | 0.500000 | − | 0.866025i | −0.254099 | − | 0.457448i | −2.35692 | − | 4.08230i | 1.18090 | 2.99835 | + | 0.0993714i | 0.302118 | ||||
166.6 | 0.167960 | + | 0.290915i | −0.721147 | + | 1.57478i | 0.943579 | − | 1.63433i | 0.500000 | − | 0.866025i | −0.579253 | + | 0.0547084i | −0.114478 | − | 0.198282i | 1.30577 | −1.95990 | − | 2.27130i | 0.335920 | ||||
166.7 | 0.461676 | + | 0.799647i | 0.817694 | − | 1.52688i | 0.573710 | − | 0.993694i | 0.500000 | − | 0.866025i | 1.59848 | − | 0.0510597i | 2.41373 | + | 4.18070i | 2.90618 | −1.66275 | − | 2.49705i | 0.923353 | ||||
166.8 | 0.726237 | + | 1.25788i | 0.843757 | + | 1.51264i | −0.0548416 | + | 0.0949884i | 0.500000 | − | 0.866025i | −1.28995 | + | 2.15988i | 0.128378 | + | 0.222358i | 2.74564 | −1.57615 | + | 2.55260i | 1.45247 | ||||
166.9 | 0.979421 | + | 1.69641i | 0.695938 | − | 1.58609i | −0.918533 | + | 1.59095i | 0.500000 | − | 0.866025i | 3.37227 | − | 0.372853i | −1.92591 | − | 3.33577i | 0.319163 | −2.03134 | − | 2.20764i | 1.95884 | ||||
166.10 | 1.19567 | + | 2.07097i | −1.41076 | − | 1.00487i | −1.85927 | + | 3.22035i | 0.500000 | − | 0.866025i | 0.394240 | − | 4.12313i | 1.20845 | + | 2.09310i | −4.10964 | 0.980483 | + | 2.83525i | 2.39135 | ||||
166.11 | 1.38948 | + | 2.40666i | 1.57484 | + | 0.721017i | −2.86133 | + | 4.95596i | 0.500000 | − | 0.866025i | 0.452979 | + | 4.79195i | −0.767199 | − | 1.32883i | −10.3451 | 1.96027 | + | 2.27098i | 2.77897 | ||||
331.1 | −1.26605 | + | 2.19286i | 0.415004 | − | 1.68160i | −2.20577 | − | 3.82050i | 0.500000 | + | 0.866025i | 3.16210 | + | 3.03904i | 1.22474 | − | 2.12131i | 6.10626 | −2.65554 | − | 1.39574i | −2.53210 | ||||
331.2 | −1.00630 | + | 1.74296i | −0.598177 | + | 1.62548i | −1.02527 | − | 1.77582i | 0.500000 | + | 0.866025i | −2.23120 | − | 2.67831i | −1.47315 | + | 2.55156i | 0.101717 | −2.28437 | − | 1.94465i | −2.01260 | ||||
331.3 | −0.949086 | + | 1.64387i | −1.54093 | − | 0.790898i | −0.801529 | − | 1.38829i | 0.500000 | + | 0.866025i | 2.76261 | − | 1.78246i | 0.751191 | − | 1.30110i | −0.753463 | 1.74896 | + | 2.43745i | −1.89817 | ||||
331.4 | −0.350077 | + | 0.606351i | 1.65559 | + | 0.508929i | 0.754892 | + | 1.30751i | 0.500000 | + | 0.866025i | −0.888175 | + | 0.825707i | −0.588842 | + | 1.01990i | −2.45739 | 2.48198 | + | 1.68516i | −0.700154 | ||||
331.5 | 0.151059 | − | 0.261642i | −1.73181 | + | 0.0286900i | 0.954362 | + | 1.65300i | 0.500000 | + | 0.866025i | −0.254099 | + | 0.457448i | −2.35692 | + | 4.08230i | 1.18090 | 2.99835 | − | 0.0993714i | 0.302118 | ||||
331.6 | 0.167960 | − | 0.290915i | −0.721147 | − | 1.57478i | 0.943579 | + | 1.63433i | 0.500000 | + | 0.866025i | −0.579253 | − | 0.0547084i | −0.114478 | + | 0.198282i | 1.30577 | −1.95990 | + | 2.27130i | 0.335920 | ||||
331.7 | 0.461676 | − | 0.799647i | 0.817694 | + | 1.52688i | 0.573710 | + | 0.993694i | 0.500000 | + | 0.866025i | 1.59848 | + | 0.0510597i | 2.41373 | − | 4.18070i | 2.90618 | −1.66275 | + | 2.49705i | 0.923353 | ||||
331.8 | 0.726237 | − | 1.25788i | 0.843757 | − | 1.51264i | −0.0548416 | − | 0.0949884i | 0.500000 | + | 0.866025i | −1.28995 | − | 2.15988i | 0.128378 | − | 0.222358i | 2.74564 | −1.57615 | − | 2.55260i | 1.45247 | ||||
331.9 | 0.979421 | − | 1.69641i | 0.695938 | + | 1.58609i | −0.918533 | − | 1.59095i | 0.500000 | + | 0.866025i | 3.37227 | + | 0.372853i | −1.92591 | + | 3.33577i | 0.319163 | −2.03134 | + | 2.20764i | 1.95884 | ||||
See all 22 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
9.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 495.2.i.f | ✓ | 22 |
3.b | odd | 2 | 1 | 1485.2.i.f | 22 | ||
9.c | even | 3 | 1 | inner | 495.2.i.f | ✓ | 22 |
9.c | even | 3 | 1 | 4455.2.a.u | 11 | ||
9.d | odd | 6 | 1 | 1485.2.i.f | 22 | ||
9.d | odd | 6 | 1 | 4455.2.a.v | 11 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
495.2.i.f | ✓ | 22 | 1.a | even | 1 | 1 | trivial |
495.2.i.f | ✓ | 22 | 9.c | even | 3 | 1 | inner |
1485.2.i.f | 22 | 3.b | odd | 2 | 1 | ||
1485.2.i.f | 22 | 9.d | odd | 6 | 1 | ||
4455.2.a.u | 11 | 9.c | even | 3 | 1 | ||
4455.2.a.v | 11 | 9.d | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(495, [\chi])\):
\( T_{2}^{22} - 3 T_{2}^{21} + 22 T_{2}^{20} - 47 T_{2}^{19} + 251 T_{2}^{18} - 481 T_{2}^{17} + 1836 T_{2}^{16} + \cdots + 144 \) |
\( T_{13}^{22} + 9 T_{13}^{21} + 92 T_{13}^{20} + 505 T_{13}^{19} + 3466 T_{13}^{18} + 16561 T_{13}^{17} + \cdots + 4096 \) |