Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [945,2,Mod(13,945)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(945, base_ring=CyclotomicField(36))
chi = DirichletCharacter(H, H._module([16, 27, 18]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("945.13");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 945 = 3^{3} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 945.di (of order \(36\), degree \(12\), 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: | \(1680\) |
Relative dimension: | \(140\) over \(\Q(\zeta_{36})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{36}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
13.1 | −2.23186 | − | 1.56277i | −0.188802 | − | 1.72173i | 1.85493 | + | 5.09638i | −1.68115 | − | 1.47436i | −2.26928 | + | 4.13772i | 1.95113 | + | 1.78692i | 2.41415 | − | 9.00972i | −2.92871 | + | 0.650134i | 1.44802 | + | 5.91781i |
13.2 | −2.23186 | − | 1.56277i | 0.188802 | + | 1.72173i | 1.85493 | + | 5.09638i | 1.68115 | + | 1.47436i | 2.26928 | − | 4.13772i | −0.114699 | − | 2.64326i | 2.41415 | − | 9.00972i | −2.92871 | + | 0.650134i | −1.44802 | − | 5.91781i |
13.3 | −2.20304 | − | 1.54258i | −1.55032 | − | 0.772338i | 1.78977 | + | 4.91734i | −1.29639 | + | 1.82191i | 2.22402 | + | 4.09299i | 1.40635 | − | 2.24102i | 2.25035 | − | 8.39841i | 1.80699 | + | 2.39474i | 5.66645 | − | 2.01395i |
13.4 | −2.20304 | − | 1.54258i | 1.55032 | + | 0.772338i | 1.78977 | + | 4.91734i | 1.29639 | − | 1.82191i | −2.22402 | − | 4.09299i | 2.62071 | + | 0.363171i | 2.25035 | − | 8.39841i | 1.80699 | + | 2.39474i | −5.66645 | + | 2.01395i |
13.5 | −2.18942 | − | 1.53305i | −1.17801 | + | 1.26976i | 1.75928 | + | 4.83359i | −1.27192 | − | 1.83908i | 4.52576 | − | 0.974103i | −1.60301 | − | 2.10484i | 2.17478 | − | 8.11640i | −0.224598 | − | 2.99158i | −0.0346433 | + | 5.97644i |
13.6 | −2.18942 | − | 1.53305i | 1.17801 | − | 1.26976i | 1.75928 | + | 4.83359i | 1.27192 | + | 1.83908i | −4.52576 | + | 0.974103i | 0.582007 | + | 2.58094i | 2.17478 | − | 8.11640i | −0.224598 | − | 2.99158i | 0.0346433 | − | 5.97644i |
13.7 | −2.16435 | − | 1.51550i | −0.918543 | − | 1.46843i | 1.70366 | + | 4.68076i | 2.23443 | + | 0.0855556i | −0.237345 | + | 4.57025i | −2.64242 | − | 0.132811i | 2.03866 | − | 7.60839i | −1.31256 | + | 2.69763i | −4.70644 | − | 3.57145i |
13.8 | −2.16435 | − | 1.51550i | 0.918543 | + | 1.46843i | 1.70366 | + | 4.68076i | −2.23443 | − | 0.0855556i | 0.237345 | − | 4.57025i | −1.59677 | + | 2.10958i | 2.03866 | − | 7.60839i | −1.31256 | + | 2.69763i | 4.70644 | + | 3.57145i |
13.9 | −2.06973 | − | 1.44924i | −1.62734 | + | 0.593088i | 1.49944 | + | 4.11967i | −0.292912 | + | 2.21680i | 4.22768 | + | 1.13088i | −2.50221 | + | 0.859606i | 1.55906 | − | 5.81847i | 2.29649 | − | 1.93032i | 3.81892 | − | 4.16367i |
13.10 | −2.06973 | − | 1.44924i | 1.62734 | − | 0.593088i | 1.49944 | + | 4.11967i | 0.292912 | − | 2.21680i | −4.22768 | − | 1.13088i | −2.26689 | + | 1.36426i | 1.55906 | − | 5.81847i | 2.29649 | − | 1.93032i | −3.81892 | + | 4.16367i |
13.11 | −2.01289 | − | 1.40944i | −1.37448 | + | 1.05394i | 1.38115 | + | 3.79469i | −2.12880 | + | 0.684275i | 4.25214 | − | 0.184224i | 2.24970 | + | 1.39243i | 1.29629 | − | 4.83782i | 0.778401 | − | 2.89726i | 5.24946 | + | 1.62304i |
13.12 | −2.01289 | − | 1.40944i | 1.37448 | − | 1.05394i | 1.38115 | + | 3.79469i | 2.12880 | − | 0.684275i | −4.25214 | + | 0.184224i | 0.379418 | − | 2.61840i | 1.29629 | − | 4.83782i | 0.778401 | − | 2.89726i | −5.24946 | − | 1.62304i |
13.13 | −1.97645 | − | 1.38392i | −1.68367 | − | 0.406506i | 1.30706 | + | 3.59111i | −0.176115 | − | 2.22912i | 2.76511 | + | 3.13351i | −0.353046 | + | 2.62209i | 1.13753 | − | 4.24534i | 2.66951 | + | 1.36885i | −2.73685 | + | 4.64947i |
13.14 | −1.97645 | − | 1.38392i | 1.68367 | + | 0.406506i | 1.30706 | + | 3.59111i | 0.176115 | + | 2.22912i | −2.76511 | − | 3.13351i | −2.23557 | − | 1.41500i | 1.13753 | − | 4.24534i | 2.66951 | + | 1.36885i | 2.73685 | − | 4.64947i |
13.15 | −1.86385 | − | 1.30508i | −0.269430 | + | 1.71097i | 1.08665 | + | 2.98555i | 0.200151 | − | 2.22709i | 2.73512 | − | 2.83735i | 2.64495 | + | 0.0652897i | 0.693225 | − | 2.58715i | −2.85481 | − | 0.921972i | −3.27958 | + | 3.88974i |
13.16 | −1.86385 | − | 1.30508i | 0.269430 | − | 1.71097i | 1.08665 | + | 2.98555i | −0.200151 | + | 2.22709i | −2.73512 | + | 2.83735i | 1.65012 | − | 2.06811i | 0.693225 | − | 2.58715i | −2.85481 | − | 0.921972i | 3.27958 | − | 3.88974i |
13.17 | −1.85841 | − | 1.30127i | −1.69310 | − | 0.365267i | 1.07633 | + | 2.95721i | 1.71668 | + | 1.43284i | 2.67116 | + | 2.88200i | 1.95976 | + | 1.77746i | 0.673494 | − | 2.51351i | 2.73316 | + | 1.23686i | −1.32578 | − | 4.89666i |
13.18 | −1.85841 | − | 1.30127i | 1.69310 | + | 0.365267i | 1.07633 | + | 2.95721i | −1.71668 | − | 1.43284i | −2.67116 | − | 2.88200i | −0.101903 | − | 2.64379i | 0.673494 | − | 2.51351i | 2.73316 | + | 1.23686i | 1.32578 | + | 4.89666i |
13.19 | −1.77405 | − | 1.24221i | −0.0211915 | − | 1.73192i | 0.920150 | + | 2.52809i | −1.98532 | + | 1.02884i | −2.11381 | + | 3.09884i | −2.60042 | + | 0.487680i | 0.386955 | − | 1.44414i | −2.99910 | + | 0.0734040i | 4.80009 | + | 0.640953i |
13.20 | −1.77405 | − | 1.24221i | 0.0211915 | + | 1.73192i | 0.920150 | + | 2.52809i | 1.98532 | − | 1.02884i | 2.11381 | − | 3.09884i | −2.04510 | + | 1.67856i | 0.386955 | − | 1.44414i | −2.99910 | + | 0.0734040i | −4.80009 | − | 0.640953i |
See next 80 embeddings (of 1680 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
7.b | odd | 2 | 1 | inner |
27.e | even | 9 | 1 | inner |
35.f | even | 4 | 1 | inner |
135.r | odd | 36 | 1 | inner |
189.y | odd | 18 | 1 | inner |
945.di | even | 36 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 945.2.di.a | ✓ | 1680 |
5.c | odd | 4 | 1 | inner | 945.2.di.a | ✓ | 1680 |
7.b | odd | 2 | 1 | inner | 945.2.di.a | ✓ | 1680 |
27.e | even | 9 | 1 | inner | 945.2.di.a | ✓ | 1680 |
35.f | even | 4 | 1 | inner | 945.2.di.a | ✓ | 1680 |
135.r | odd | 36 | 1 | inner | 945.2.di.a | ✓ | 1680 |
189.y | odd | 18 | 1 | inner | 945.2.di.a | ✓ | 1680 |
945.di | even | 36 | 1 | inner | 945.2.di.a | ✓ | 1680 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
945.2.di.a | ✓ | 1680 | 1.a | even | 1 | 1 | trivial |
945.2.di.a | ✓ | 1680 | 5.c | odd | 4 | 1 | inner |
945.2.di.a | ✓ | 1680 | 7.b | odd | 2 | 1 | inner |
945.2.di.a | ✓ | 1680 | 27.e | even | 9 | 1 | inner |
945.2.di.a | ✓ | 1680 | 35.f | even | 4 | 1 | inner |
945.2.di.a | ✓ | 1680 | 135.r | odd | 36 | 1 | inner |
945.2.di.a | ✓ | 1680 | 189.y | odd | 18 | 1 | inner |
945.2.di.a | ✓ | 1680 | 945.di | even | 36 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(945, [\chi])\).