Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(269,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 3, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.269");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.bb (of order \(6\), 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: | \(7.54586299101\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) over \(\Q(\zeta_{6})\) |
Twist minimal: | yes |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
269.1 | −1.23319 | − | 2.13595i | 0 | −2.04153 | + | 3.53604i | −2.20106 | − | 0.394102i | 0 | 0.465281 | + | 2.60452i | 5.13765 | 0 | 1.87255 | + | 5.18738i | ||||||||
269.2 | −1.23319 | − | 2.13595i | 0 | −2.04153 | + | 3.53604i | −0.759230 | + | 2.10323i | 0 | −0.465281 | − | 2.60452i | 5.13765 | 0 | 5.42868 | − | 0.972008i | ||||||||
269.3 | −1.04316 | − | 1.80680i | 0 | −1.17634 | + | 2.03749i | 1.42338 | − | 1.72453i | 0 | 2.64510 | − | 0.0586776i | 0.735821 | 0 | −4.60068 | − | 0.772800i | ||||||||
269.4 | −1.04316 | − | 1.80680i | 0 | −1.17634 | + | 2.03749i | 2.20517 | − | 0.370415i | 0 | −2.64510 | + | 0.0586776i | 0.735821 | 0 | −2.96960 | − | 3.59790i | ||||||||
269.5 | −0.640286 | − | 1.10901i | 0 | 0.180069 | − | 0.311888i | −1.97382 | − | 1.05073i | 0 | −2.58281 | − | 0.573647i | −3.02232 | 0 | 0.0985445 | + | 2.86175i | ||||||||
269.6 | −0.640286 | − | 1.10901i | 0 | 0.180069 | − | 0.311888i | −0.0769535 | + | 2.23474i | 0 | 2.58281 | + | 0.573647i | −3.02232 | 0 | 2.52762 | − | 1.34553i | ||||||||
269.7 | −0.104913 | − | 0.181715i | 0 | 0.977986 | − | 1.69392i | 0.346458 | − | 2.20906i | 0 | 0.694926 | − | 2.55286i | −0.830069 | 0 | −0.437769 | + | 0.168804i | ||||||||
269.8 | −0.104913 | − | 0.181715i | 0 | 0.977986 | − | 1.69392i | 2.08634 | + | 0.804491i | 0 | −0.694926 | + | 2.55286i | −0.830069 | 0 | −0.0726961 | − | 0.463521i | ||||||||
269.9 | 0.187042 | + | 0.323967i | 0 | 0.930030 | − | 1.61086i | −2.23199 | + | 0.135023i | 0 | 2.46237 | − | 0.967860i | 1.44399 | 0 | −0.461219 | − | 0.697834i | ||||||||
269.10 | 0.187042 | + | 0.323967i | 0 | 0.930030 | − | 1.61086i | −1.23293 | + | 1.86545i | 0 | −2.46237 | + | 0.967860i | 1.44399 | 0 | −0.834952 | − | 0.0505098i | ||||||||
269.11 | 0.598512 | + | 1.03665i | 0 | 0.283567 | − | 0.491153i | 1.06736 | − | 1.96488i | 0 | −1.89415 | − | 1.84722i | 3.07292 | 0 | 2.67572 | − | 0.0695206i | ||||||||
269.12 | 0.598512 | + | 1.03665i | 0 | 0.283567 | − | 0.491153i | 2.23531 | + | 0.0580779i | 0 | 1.89415 | + | 1.84722i | 3.07292 | 0 | 1.27766 | + | 2.35200i | ||||||||
269.13 | 0.921715 | + | 1.59646i | 0 | −0.699118 | + | 1.21091i | −1.28751 | − | 1.82820i | 0 | −0.918398 | + | 2.48124i | 1.10931 | 0 | 1.73193 | − | 3.74054i | ||||||||
269.14 | 0.921715 | + | 1.59646i | 0 | −0.699118 | + | 1.21091i | 0.939513 | + | 2.02912i | 0 | 0.918398 | − | 2.48124i | 1.10931 | 0 | −2.37344 | + | 3.37016i | ||||||||
269.15 | 1.31428 | + | 2.27640i | 0 | −2.45466 | + | 4.25159i | −1.97038 | − | 1.05717i | 0 | 1.84082 | − | 1.90036i | −7.64730 | 0 | −0.183092 | − | 5.87478i | ||||||||
269.16 | 1.31428 | + | 2.27640i | 0 | −2.45466 | + | 4.25159i | −0.0696551 | + | 2.23498i | 0 | −1.84082 | + | 1.90036i | −7.64730 | 0 | −5.17926 | + | 2.77883i | ||||||||
404.1 | −1.23319 | + | 2.13595i | 0 | −2.04153 | − | 3.53604i | −2.20106 | + | 0.394102i | 0 | 0.465281 | − | 2.60452i | 5.13765 | 0 | 1.87255 | − | 5.18738i | ||||||||
404.2 | −1.23319 | + | 2.13595i | 0 | −2.04153 | − | 3.53604i | −0.759230 | − | 2.10323i | 0 | −0.465281 | + | 2.60452i | 5.13765 | 0 | 5.42868 | + | 0.972008i | ||||||||
404.3 | −1.04316 | + | 1.80680i | 0 | −1.17634 | − | 2.03749i | 1.42338 | + | 1.72453i | 0 | 2.64510 | + | 0.0586776i | 0.735821 | 0 | −4.60068 | + | 0.772800i | ||||||||
404.4 | −1.04316 | + | 1.80680i | 0 | −1.17634 | − | 2.03749i | 2.20517 | + | 0.370415i | 0 | −2.64510 | − | 0.0586776i | 0.735821 | 0 | −2.96960 | + | 3.59790i | ||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
15.d | odd | 2 | 1 | inner |
105.p | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.bb.a | ✓ | 32 |
3.b | odd | 2 | 1 | 945.2.bb.b | yes | 32 | |
5.b | even | 2 | 1 | 945.2.bb.b | yes | 32 | |
7.d | odd | 6 | 1 | inner | 945.2.bb.a | ✓ | 32 |
15.d | odd | 2 | 1 | inner | 945.2.bb.a | ✓ | 32 |
21.g | even | 6 | 1 | 945.2.bb.b | yes | 32 | |
35.i | odd | 6 | 1 | 945.2.bb.b | yes | 32 | |
105.p | even | 6 | 1 | inner | 945.2.bb.a | ✓ | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
945.2.bb.a | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
945.2.bb.a | ✓ | 32 | 7.d | odd | 6 | 1 | inner |
945.2.bb.a | ✓ | 32 | 15.d | odd | 2 | 1 | inner |
945.2.bb.a | ✓ | 32 | 105.p | even | 6 | 1 | inner |
945.2.bb.b | yes | 32 | 3.b | odd | 2 | 1 | |
945.2.bb.b | yes | 32 | 5.b | even | 2 | 1 | |
945.2.bb.b | yes | 32 | 21.g | even | 6 | 1 | |
945.2.bb.b | yes | 32 | 35.i | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{16} + 12 T_{2}^{14} + 102 T_{2}^{12} - 3 T_{2}^{11} + 420 T_{2}^{10} - 66 T_{2}^{9} + 1257 T_{2}^{8} - 198 T_{2}^{7} + 1701 T_{2}^{6} - 378 T_{2}^{5} + 1656 T_{2}^{4} - 234 T_{2}^{3} + 162 T_{2}^{2} + 18 T_{2} + 9 \)
acting on \(S_{2}^{\mathrm{new}}(945, [\chi])\).