Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [380,2,Mod(31,380)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(380, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 0, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("380.31");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 380 = 2^{2} \cdot 5 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 380.n (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: | \(3.03431527681\) |
Analytic rank: | \(0\) |
Dimension: | \(40\) |
Relative dimension: | \(20\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
31.1 | −1.41224 | − | 0.0746014i | −1.09575 | − | 1.89790i | 1.98887 | + | 0.210711i | 0.500000 | + | 0.866025i | 1.40588 | + | 2.76204i | − | 2.65350i | −2.79305 | − | 0.445948i | −0.901338 | + | 1.56116i | −0.641516 | − | 1.26034i | |
31.2 | −1.39096 | + | 0.255414i | 1.57351 | + | 2.72539i | 1.86953 | − | 0.710539i | 0.500000 | + | 0.866025i | −2.88478 | − | 3.38901i | − | 3.91622i | −2.41895 | + | 1.46583i | −3.45184 | + | 5.97877i | −0.916674 | − | 1.07690i | |
31.3 | −1.38644 | + | 0.278901i | −0.418744 | − | 0.725286i | 1.84443 | − | 0.773359i | 0.500000 | + | 0.866025i | 0.782847 | + | 0.888777i | 3.40884i | −2.34150 | + | 1.58663i | 1.14931 | − | 1.99066i | −0.934755 | − | 1.06124i | ||
31.4 | −1.27347 | − | 0.615033i | 0.481867 | + | 0.834618i | 1.24347 | + | 1.56646i | 0.500000 | + | 0.866025i | −0.100327 | − | 1.35923i | 1.00869i | −0.620104 | − | 2.75961i | 1.03561 | − | 1.79373i | −0.104103 | − | 1.41038i | ||
31.5 | −0.934755 | + | 1.06124i | 0.418744 | + | 0.725286i | −0.252465 | − | 1.98400i | 0.500000 | + | 0.866025i | −1.16113 | − | 0.233577i | − | 3.40884i | 2.34150 | + | 1.58663i | 1.14931 | − | 1.99066i | −1.38644 | − | 0.278901i | |
31.6 | −0.916674 | + | 1.07690i | −1.57351 | − | 2.72539i | −0.319419 | − | 1.97433i | 0.500000 | + | 0.866025i | 4.37736 | + | 0.803789i | 3.91622i | 2.41895 | + | 1.46583i | −3.45184 | + | 5.97877i | −1.39096 | − | 0.255414i | ||
31.7 | −0.840559 | − | 1.13730i | 1.52787 | + | 2.64635i | −0.586922 | + | 1.91194i | 0.500000 | + | 0.866025i | 1.72544 | − | 3.96206i | 2.58057i | 2.66780 | − | 0.939591i | −3.16877 | + | 5.48848i | 0.564655 | − | 1.29660i | ||
31.8 | −0.776263 | − | 1.18212i | −1.45074 | − | 2.51275i | −0.794830 | + | 1.83528i | 0.500000 | + | 0.866025i | −1.84423 | + | 3.66550i | 1.19935i | 2.78652 | − | 0.485072i | −2.70927 | + | 4.69260i | 0.635617 | − | 1.26333i | ||
31.9 | −0.641516 | + | 1.26034i | 1.09575 | + | 1.89790i | −1.17692 | − | 1.61706i | 0.500000 | + | 0.866025i | −3.09494 | + | 0.163489i | 2.65350i | 2.79305 | − | 0.445948i | −0.901338 | + | 1.56116i | −1.41224 | + | 0.0746014i | ||
31.10 | −0.293307 | − | 1.38346i | −0.256977 | − | 0.445098i | −1.82794 | + | 0.811560i | 0.500000 | + | 0.866025i | −0.540403 | + | 0.486069i | − | 1.09058i | 1.65891 | + | 2.29085i | 1.36793 | − | 2.36932i | 1.05146 | − | 0.945743i | |
31.11 | −0.104103 | + | 1.41038i | −0.481867 | − | 0.834618i | −1.97833 | − | 0.293648i | 0.500000 | + | 0.866025i | 1.22729 | − | 0.592728i | − | 1.00869i | 0.620104 | − | 2.75961i | 1.03561 | − | 1.79373i | −1.27347 | + | 0.615033i | |
31.12 | 0.270280 | − | 1.38815i | 0.435984 | + | 0.755146i | −1.85390 | − | 0.750377i | 0.500000 | + | 0.866025i | 1.16609 | − | 0.401108i | 4.89965i | −1.54270 | + | 2.37067i | 1.11984 | − | 1.93961i | 1.33731 | − | 0.460003i | ||
31.13 | 0.564655 | + | 1.29660i | −1.52787 | − | 2.64635i | −1.36233 | + | 1.46426i | 0.500000 | + | 0.866025i | 2.56853 | − | 3.47531i | − | 2.58057i | −2.66780 | − | 0.939591i | −3.16877 | + | 5.48848i | −0.840559 | + | 1.13730i | |
31.14 | 0.635617 | + | 1.26333i | 1.45074 | + | 2.51275i | −1.19198 | + | 1.60598i | 0.500000 | + | 0.866025i | −2.25231 | + | 3.42990i | − | 1.19935i | −2.78652 | − | 0.485072i | −2.70927 | + | 4.69260i | −0.776263 | + | 1.18212i | |
31.15 | 0.759006 | − | 1.19328i | 0.600896 | + | 1.04078i | −0.847818 | − | 1.81141i | 0.500000 | + | 0.866025i | 1.69802 | + | 0.0729253i | − | 4.07590i | −2.80501 | − | 0.363190i | 0.777849 | − | 1.34727i | 1.41291 | + | 0.0606805i | |
31.16 | 1.05146 | + | 0.945743i | 0.256977 | + | 0.445098i | 0.211139 | + | 1.98882i | 0.500000 | + | 0.866025i | −0.150747 | + | 0.711038i | 1.09058i | −1.65891 | + | 2.29085i | 1.36793 | − | 2.36932i | −0.293307 | + | 1.38346i | ||
31.17 | 1.15432 | − | 0.817038i | −1.05340 | − | 1.82454i | 0.664897 | − | 1.88624i | 0.500000 | + | 0.866025i | −2.70668 | − | 1.24543i | − | 0.279006i | −0.773631 | − | 2.72057i | −0.719298 | + | 1.24586i | 1.28473 | + | 0.591149i | |
31.18 | 1.28473 | − | 0.591149i | 1.05340 | + | 1.82454i | 1.30109 | − | 1.51894i | 0.500000 | + | 0.866025i | 2.43191 | + | 1.72133i | 0.279006i | 0.773631 | − | 2.72057i | −0.719298 | + | 1.24586i | 1.15432 | + | 0.817038i | ||
31.19 | 1.33731 | + | 0.460003i | −0.435984 | − | 0.755146i | 1.57679 | + | 1.23033i | 0.500000 | + | 0.866025i | −0.235676 | − | 1.21042i | − | 4.89965i | 1.54270 | + | 2.37067i | 1.11984 | − | 1.93961i | 0.270280 | + | 1.38815i | |
31.20 | 1.41291 | − | 0.0606805i | −0.600896 | − | 1.04078i | 1.99264 | − | 0.171472i | 0.500000 | + | 0.866025i | −0.912167 | − | 1.43407i | 4.07590i | 2.80501 | − | 0.363190i | 0.777849 | − | 1.34727i | 0.759006 | + | 1.19328i | ||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
19.d | odd | 6 | 1 | inner |
76.f | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 380.2.n.a | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 380.2.n.a | ✓ | 40 |
19.d | odd | 6 | 1 | inner | 380.2.n.a | ✓ | 40 |
76.f | even | 6 | 1 | inner | 380.2.n.a | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
380.2.n.a | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
380.2.n.a | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
380.2.n.a | ✓ | 40 | 19.d | odd | 6 | 1 | inner |
380.2.n.a | ✓ | 40 | 76.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{40} + 41 T_{3}^{38} + 992 T_{3}^{36} + 15975 T_{3}^{34} + 191688 T_{3}^{32} + 1746767 T_{3}^{30} + \cdots + 9834496 \) acting on \(S_{2}^{\mathrm{new}}(380, [\chi])\).