Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [845,2,Mod(654,845)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(845, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([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("845.654");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 845 = 5 \cdot 13^{2} \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 845.l (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: | \(6.74735897080\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Relative dimension: | \(12\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 65) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
654.1 | −1.27287 | + | 2.20467i | −1.86449 | − | 1.07646i | −2.24039 | − | 3.88048i | 2.08125 | − | 0.817544i | 4.74650 | − | 2.74039i | 1.46928 | + | 2.54486i | 6.31544 | 0.817544 | + | 1.41603i | −0.846746 | + | 5.62912i | ||
654.2 | −1.27287 | + | 2.20467i | 1.86449 | + | 1.07646i | −2.24039 | − | 3.88048i | 2.08125 | + | 0.817544i | −4.74650 | + | 2.74039i | 1.46928 | + | 2.54486i | 6.31544 | 0.817544 | + | 1.41603i | −4.45158 | + | 3.54786i | ||
654.3 | −0.593667 | + | 1.02826i | −0.298874 | − | 0.172555i | 0.295120 | + | 0.511162i | −1.71029 | + | 1.44045i | 0.354863 | − | 0.204880i | 1.01478 | + | 1.75765i | −3.07548 | −1.44045 | − | 2.49493i | −0.465813 | − | 2.61378i | ||
654.4 | −0.593667 | + | 1.02826i | 0.298874 | + | 0.172555i | 0.295120 | + | 0.511162i | −1.71029 | − | 1.44045i | −0.354863 | + | 0.204880i | 1.01478 | + | 1.75765i | −3.07548 | −1.44045 | − | 2.49493i | 2.49650 | − | 0.903481i | ||
654.5 | −0.165418 | + | 0.286513i | −2.33117 | − | 1.34590i | 0.945274 | + | 1.63726i | 0.702335 | + | 2.12291i | 0.771236 | − | 0.445274i | 1.67674 | + | 2.90420i | −1.28714 | 2.12291 | + | 3.67698i | −0.724419 | − | 0.149939i | ||
654.6 | −0.165418 | + | 0.286513i | 2.33117 | + | 1.34590i | 0.945274 | + | 1.63726i | 0.702335 | − | 2.12291i | −0.771236 | + | 0.445274i | 1.67674 | + | 2.90420i | −1.28714 | 2.12291 | + | 3.67698i | 0.492061 | + | 0.552395i | ||
654.7 | 0.165418 | − | 0.286513i | −2.33117 | − | 1.34590i | 0.945274 | + | 1.63726i | −0.702335 | − | 2.12291i | −0.771236 | + | 0.445274i | −1.67674 | − | 2.90420i | 1.28714 | 2.12291 | + | 3.67698i | −0.724419 | − | 0.149939i | ||
654.8 | 0.165418 | − | 0.286513i | 2.33117 | + | 1.34590i | 0.945274 | + | 1.63726i | −0.702335 | + | 2.12291i | 0.771236 | − | 0.445274i | −1.67674 | − | 2.90420i | 1.28714 | 2.12291 | + | 3.67698i | 0.492061 | + | 0.552395i | ||
654.9 | 0.593667 | − | 1.02826i | −0.298874 | − | 0.172555i | 0.295120 | + | 0.511162i | 1.71029 | − | 1.44045i | −0.354863 | + | 0.204880i | −1.01478 | − | 1.75765i | 3.07548 | −1.44045 | − | 2.49493i | −0.465813 | − | 2.61378i | ||
654.10 | 0.593667 | − | 1.02826i | 0.298874 | + | 0.172555i | 0.295120 | + | 0.511162i | 1.71029 | + | 1.44045i | 0.354863 | − | 0.204880i | −1.01478 | − | 1.75765i | 3.07548 | −1.44045 | − | 2.49493i | 2.49650 | − | 0.903481i | ||
654.11 | 1.27287 | − | 2.20467i | −1.86449 | − | 1.07646i | −2.24039 | − | 3.88048i | −2.08125 | + | 0.817544i | −4.74650 | + | 2.74039i | −1.46928 | − | 2.54486i | −6.31544 | 0.817544 | + | 1.41603i | −0.846746 | + | 5.62912i | ||
654.12 | 1.27287 | − | 2.20467i | 1.86449 | + | 1.07646i | −2.24039 | − | 3.88048i | −2.08125 | − | 0.817544i | 4.74650 | − | 2.74039i | −1.46928 | − | 2.54486i | −6.31544 | 0.817544 | + | 1.41603i | −4.45158 | + | 3.54786i | ||
699.1 | −1.27287 | − | 2.20467i | −1.86449 | + | 1.07646i | −2.24039 | + | 3.88048i | 2.08125 | + | 0.817544i | 4.74650 | + | 2.74039i | 1.46928 | − | 2.54486i | 6.31544 | 0.817544 | − | 1.41603i | −0.846746 | − | 5.62912i | ||
699.2 | −1.27287 | − | 2.20467i | 1.86449 | − | 1.07646i | −2.24039 | + | 3.88048i | 2.08125 | − | 0.817544i | −4.74650 | − | 2.74039i | 1.46928 | − | 2.54486i | 6.31544 | 0.817544 | − | 1.41603i | −4.45158 | − | 3.54786i | ||
699.3 | −0.593667 | − | 1.02826i | −0.298874 | + | 0.172555i | 0.295120 | − | 0.511162i | −1.71029 | − | 1.44045i | 0.354863 | + | 0.204880i | 1.01478 | − | 1.75765i | −3.07548 | −1.44045 | + | 2.49493i | −0.465813 | + | 2.61378i | ||
699.4 | −0.593667 | − | 1.02826i | 0.298874 | − | 0.172555i | 0.295120 | − | 0.511162i | −1.71029 | + | 1.44045i | −0.354863 | − | 0.204880i | 1.01478 | − | 1.75765i | −3.07548 | −1.44045 | + | 2.49493i | 2.49650 | + | 0.903481i | ||
699.5 | −0.165418 | − | 0.286513i | −2.33117 | + | 1.34590i | 0.945274 | − | 1.63726i | 0.702335 | − | 2.12291i | 0.771236 | + | 0.445274i | 1.67674 | − | 2.90420i | −1.28714 | 2.12291 | − | 3.67698i | −0.724419 | + | 0.149939i | ||
699.6 | −0.165418 | − | 0.286513i | 2.33117 | − | 1.34590i | 0.945274 | − | 1.63726i | 0.702335 | + | 2.12291i | −0.771236 | − | 0.445274i | 1.67674 | − | 2.90420i | −1.28714 | 2.12291 | − | 3.67698i | 0.492061 | − | 0.552395i | ||
699.7 | 0.165418 | + | 0.286513i | −2.33117 | + | 1.34590i | 0.945274 | − | 1.63726i | −0.702335 | + | 2.12291i | −0.771236 | − | 0.445274i | −1.67674 | + | 2.90420i | 1.28714 | 2.12291 | − | 3.67698i | −0.724419 | + | 0.149939i | ||
699.8 | 0.165418 | + | 0.286513i | 2.33117 | − | 1.34590i | 0.945274 | − | 1.63726i | −0.702335 | − | 2.12291i | 0.771236 | + | 0.445274i | −1.67674 | + | 2.90420i | 1.28714 | 2.12291 | − | 3.67698i | 0.492061 | − | 0.552395i | ||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
13.b | even | 2 | 1 | inner |
13.c | even | 3 | 1 | inner |
13.e | even | 6 | 1 | inner |
65.d | even | 2 | 1 | inner |
65.l | even | 6 | 1 | inner |
65.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 845.2.l.f | 24 | |
5.b | even | 2 | 1 | inner | 845.2.l.f | 24 | |
13.b | even | 2 | 1 | inner | 845.2.l.f | 24 | |
13.c | even | 3 | 1 | 845.2.d.d | 12 | ||
13.c | even | 3 | 1 | inner | 845.2.l.f | 24 | |
13.d | odd | 4 | 1 | 65.2.n.a | ✓ | 12 | |
13.d | odd | 4 | 1 | 845.2.n.e | 12 | ||
13.e | even | 6 | 1 | 845.2.d.d | 12 | ||
13.e | even | 6 | 1 | inner | 845.2.l.f | 24 | |
13.f | odd | 12 | 1 | 65.2.n.a | ✓ | 12 | |
13.f | odd | 12 | 1 | 845.2.b.d | 6 | ||
13.f | odd | 12 | 1 | 845.2.b.e | 6 | ||
13.f | odd | 12 | 1 | 845.2.n.e | 12 | ||
39.f | even | 4 | 1 | 585.2.bs.a | 12 | ||
39.k | even | 12 | 1 | 585.2.bs.a | 12 | ||
52.f | even | 4 | 1 | 1040.2.dh.a | 12 | ||
52.l | even | 12 | 1 | 1040.2.dh.a | 12 | ||
65.d | even | 2 | 1 | inner | 845.2.l.f | 24 | |
65.f | even | 4 | 1 | 325.2.e.e | 12 | ||
65.g | odd | 4 | 1 | 65.2.n.a | ✓ | 12 | |
65.g | odd | 4 | 1 | 845.2.n.e | 12 | ||
65.k | even | 4 | 1 | 325.2.e.e | 12 | ||
65.l | even | 6 | 1 | 845.2.d.d | 12 | ||
65.l | even | 6 | 1 | inner | 845.2.l.f | 24 | |
65.n | even | 6 | 1 | 845.2.d.d | 12 | ||
65.n | even | 6 | 1 | inner | 845.2.l.f | 24 | |
65.o | even | 12 | 1 | 325.2.e.e | 12 | ||
65.o | even | 12 | 1 | 4225.2.a.bq | 6 | ||
65.o | even | 12 | 1 | 4225.2.a.br | 6 | ||
65.s | odd | 12 | 1 | 65.2.n.a | ✓ | 12 | |
65.s | odd | 12 | 1 | 845.2.b.d | 6 | ||
65.s | odd | 12 | 1 | 845.2.b.e | 6 | ||
65.s | odd | 12 | 1 | 845.2.n.e | 12 | ||
65.t | even | 12 | 1 | 325.2.e.e | 12 | ||
65.t | even | 12 | 1 | 4225.2.a.bq | 6 | ||
65.t | even | 12 | 1 | 4225.2.a.br | 6 | ||
195.n | even | 4 | 1 | 585.2.bs.a | 12 | ||
195.bh | even | 12 | 1 | 585.2.bs.a | 12 | ||
260.u | even | 4 | 1 | 1040.2.dh.a | 12 | ||
260.bc | even | 12 | 1 | 1040.2.dh.a | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
65.2.n.a | ✓ | 12 | 13.d | odd | 4 | 1 | |
65.2.n.a | ✓ | 12 | 13.f | odd | 12 | 1 | |
65.2.n.a | ✓ | 12 | 65.g | odd | 4 | 1 | |
65.2.n.a | ✓ | 12 | 65.s | odd | 12 | 1 | |
325.2.e.e | 12 | 65.f | even | 4 | 1 | ||
325.2.e.e | 12 | 65.k | even | 4 | 1 | ||
325.2.e.e | 12 | 65.o | even | 12 | 1 | ||
325.2.e.e | 12 | 65.t | even | 12 | 1 | ||
585.2.bs.a | 12 | 39.f | even | 4 | 1 | ||
585.2.bs.a | 12 | 39.k | even | 12 | 1 | ||
585.2.bs.a | 12 | 195.n | even | 4 | 1 | ||
585.2.bs.a | 12 | 195.bh | even | 12 | 1 | ||
845.2.b.d | 6 | 13.f | odd | 12 | 1 | ||
845.2.b.d | 6 | 65.s | odd | 12 | 1 | ||
845.2.b.e | 6 | 13.f | odd | 12 | 1 | ||
845.2.b.e | 6 | 65.s | odd | 12 | 1 | ||
845.2.d.d | 12 | 13.c | even | 3 | 1 | ||
845.2.d.d | 12 | 13.e | even | 6 | 1 | ||
845.2.d.d | 12 | 65.l | even | 6 | 1 | ||
845.2.d.d | 12 | 65.n | even | 6 | 1 | ||
845.2.l.f | 24 | 1.a | even | 1 | 1 | trivial | |
845.2.l.f | 24 | 5.b | even | 2 | 1 | inner | |
845.2.l.f | 24 | 13.b | even | 2 | 1 | inner | |
845.2.l.f | 24 | 13.c | even | 3 | 1 | inner | |
845.2.l.f | 24 | 13.e | even | 6 | 1 | inner | |
845.2.l.f | 24 | 65.d | even | 2 | 1 | inner | |
845.2.l.f | 24 | 65.l | even | 6 | 1 | inner | |
845.2.l.f | 24 | 65.n | even | 6 | 1 | inner | |
845.2.n.e | 12 | 13.d | odd | 4 | 1 | ||
845.2.n.e | 12 | 13.f | odd | 12 | 1 | ||
845.2.n.e | 12 | 65.g | odd | 4 | 1 | ||
845.2.n.e | 12 | 65.s | odd | 12 | 1 | ||
1040.2.dh.a | 12 | 52.f | even | 4 | 1 | ||
1040.2.dh.a | 12 | 52.l | even | 12 | 1 | ||
1040.2.dh.a | 12 | 260.u | even | 4 | 1 | ||
1040.2.dh.a | 12 | 260.bc | even | 12 | 1 | ||
4225.2.a.bq | 6 | 65.o | even | 12 | 1 | ||
4225.2.a.bq | 6 | 65.t | even | 12 | 1 | ||
4225.2.a.br | 6 | 65.o | even | 12 | 1 | ||
4225.2.a.br | 6 | 65.t | even | 12 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 8T_{2}^{10} + 54T_{2}^{8} + 78T_{2}^{6} + 92T_{2}^{4} + 10T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(845, [\chi])\).