Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [756,2,Mod(19,756)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(756, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([3, 4, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("756.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 756.n (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: | \(6.03669039281\) |
Analytic rank: | \(0\) |
Dimension: | \(84\) |
Relative dimension: | \(42\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 252) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
19.1 | −1.41039 | + | 0.103929i | 0 | 1.97840 | − | 0.293161i | − | 1.91789i | 0 | −1.85299 | + | 1.88850i | −2.75984 | + | 0.619084i | 0 | 0.199325 | + | 2.70497i | |||||||
19.2 | −1.37782 | + | 0.318759i | 0 | 1.79679 | − | 0.878385i | − | 0.594537i | 0 | 2.59555 | + | 0.512964i | −2.19566 | + | 1.78300i | 0 | 0.189514 | + | 0.819166i | |||||||
19.3 | −1.36968 | + | 0.352122i | 0 | 1.75202 | − | 0.964586i | 2.71053i | 0 | 1.53391 | − | 2.15572i | −2.06005 | + | 1.93809i | 0 | −0.954438 | − | 3.71255i | ||||||||
19.4 | −1.34403 | − | 0.439974i | 0 | 1.61285 | + | 1.18268i | 2.88398i | 0 | −2.24777 | + | 1.39554i | −1.64737 | − | 2.29917i | 0 | 1.26887 | − | 3.87616i | ||||||||
19.5 | −1.30137 | + | 0.553558i | 0 | 1.38715 | − | 1.44077i | − | 4.05112i | 0 | −2.21649 | − | 1.44470i | −1.00765 | + | 2.64285i | 0 | 2.24253 | + | 5.27202i | |||||||
19.6 | −1.28099 | − | 0.599230i | 0 | 1.28185 | + | 1.53521i | − | 2.56839i | 0 | 0.477445 | − | 2.60232i | −0.722083 | − | 2.73470i | 0 | −1.53906 | + | 3.29008i | |||||||
19.7 | −1.27769 | − | 0.606229i | 0 | 1.26497 | + | 1.54914i | − | 0.139892i | 0 | −1.99326 | − | 1.73980i | −0.677105 | − | 2.74618i | 0 | −0.0848066 | + | 0.178738i | |||||||
19.8 | −1.17926 | + | 0.780599i | 0 | 0.781329 | − | 1.84107i | 2.00240i | 0 | −2.64523 | + | 0.0525529i | 0.515742 | + | 2.78101i | 0 | −1.56307 | − | 2.36135i | ||||||||
19.9 | −1.08144 | + | 0.911308i | 0 | 0.339034 | − | 1.97105i | − | 0.815110i | 0 | 0.448419 | − | 2.60747i | 1.42959 | + | 2.44055i | 0 | 0.742817 | + | 0.881495i | |||||||
19.10 | −1.07941 | − | 0.913714i | 0 | 0.330255 | + | 1.97254i | − | 3.48216i | 0 | −0.638585 | + | 2.56753i | 1.44586 | − | 2.43094i | 0 | −3.18169 | + | 3.75868i | |||||||
19.11 | −0.883516 | + | 1.10426i | 0 | −0.438800 | − | 1.95127i | − | 2.93241i | 0 | 2.24678 | + | 1.39713i | 2.54241 | + | 1.23943i | 0 | 3.23816 | + | 2.59083i | |||||||
19.12 | −0.825493 | − | 1.14829i | 0 | −0.637123 | + | 1.89580i | − | 0.299382i | 0 | 2.33495 | − | 1.24419i | 2.70287 | − | 0.833374i | 0 | −0.343777 | + | 0.247138i | |||||||
19.13 | −0.746337 | + | 1.20124i | 0 | −0.885963 | − | 1.79306i | 0.724862i | 0 | 1.25991 | + | 2.32651i | 2.81513 | + | 0.273973i | 0 | −0.870735 | − | 0.540991i | ||||||||
19.14 | −0.726200 | − | 1.21352i | 0 | −0.945266 | + | 1.76252i | 2.80831i | 0 | −0.514387 | − | 2.59527i | 2.82531 | − | 0.132842i | 0 | 3.40794 | − | 2.03940i | ||||||||
19.15 | −0.687840 | − | 1.23567i | 0 | −1.05375 | + | 1.69988i | 2.80831i | 0 | 0.514387 | + | 2.59527i | 2.82531 | + | 0.132842i | 0 | 3.47014 | − | 1.93167i | ||||||||
19.16 | −0.640648 | + | 1.26078i | 0 | −1.17914 | − | 1.61543i | 1.05993i | 0 | −0.349222 | − | 2.62260i | 2.79212 | − | 0.451713i | 0 | −1.33634 | − | 0.679041i | ||||||||
19.17 | −0.581699 | − | 1.28904i | 0 | −1.32325 | + | 1.49967i | − | 0.299382i | 0 | −2.33495 | + | 1.24419i | 2.70287 | + | 0.833374i | 0 | −0.385916 | + | 0.174150i | |||||||
19.18 | −0.447581 | + | 1.34152i | 0 | −1.59934 | − | 1.20088i | 3.94623i | 0 | −2.37341 | + | 1.16916i | 2.32683 | − | 1.60806i | 0 | −5.29393 | − | 1.76626i | ||||||||
19.19 | −0.270834 | + | 1.38804i | 0 | −1.85330 | − | 0.751856i | − | 2.69249i | 0 | −1.51663 | + | 2.16791i | 1.54554 | − | 2.36882i | 0 | 3.73728 | + | 0.729218i | |||||||
19.20 | −0.251594 | − | 1.39165i | 0 | −1.87340 | + | 0.700263i | − | 3.48216i | 0 | 0.638585 | − | 2.56753i | 1.44586 | + | 2.43094i | 0 | −4.84596 | + | 0.876089i | |||||||
See all 84 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
63.k | odd | 6 | 1 | inner |
252.n | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 756.2.n.b | 84 | |
3.b | odd | 2 | 1 | 252.2.n.b | ✓ | 84 | |
4.b | odd | 2 | 1 | inner | 756.2.n.b | 84 | |
7.d | odd | 6 | 1 | 756.2.bj.b | 84 | ||
9.c | even | 3 | 1 | 756.2.bj.b | 84 | ||
9.d | odd | 6 | 1 | 252.2.bj.b | yes | 84 | |
12.b | even | 2 | 1 | 252.2.n.b | ✓ | 84 | |
21.g | even | 6 | 1 | 252.2.bj.b | yes | 84 | |
28.f | even | 6 | 1 | 756.2.bj.b | 84 | ||
36.f | odd | 6 | 1 | 756.2.bj.b | 84 | ||
36.h | even | 6 | 1 | 252.2.bj.b | yes | 84 | |
63.k | odd | 6 | 1 | inner | 756.2.n.b | 84 | |
63.s | even | 6 | 1 | 252.2.n.b | ✓ | 84 | |
84.j | odd | 6 | 1 | 252.2.bj.b | yes | 84 | |
252.n | even | 6 | 1 | inner | 756.2.n.b | 84 | |
252.bn | odd | 6 | 1 | 252.2.n.b | ✓ | 84 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
252.2.n.b | ✓ | 84 | 3.b | odd | 2 | 1 | |
252.2.n.b | ✓ | 84 | 12.b | even | 2 | 1 | |
252.2.n.b | ✓ | 84 | 63.s | even | 6 | 1 | |
252.2.n.b | ✓ | 84 | 252.bn | odd | 6 | 1 | |
252.2.bj.b | yes | 84 | 9.d | odd | 6 | 1 | |
252.2.bj.b | yes | 84 | 21.g | even | 6 | 1 | |
252.2.bj.b | yes | 84 | 36.h | even | 6 | 1 | |
252.2.bj.b | yes | 84 | 84.j | odd | 6 | 1 | |
756.2.n.b | 84 | 1.a | even | 1 | 1 | trivial | |
756.2.n.b | 84 | 4.b | odd | 2 | 1 | inner | |
756.2.n.b | 84 | 63.k | odd | 6 | 1 | inner | |
756.2.n.b | 84 | 252.n | even | 6 | 1 | inner | |
756.2.bj.b | 84 | 7.d | odd | 6 | 1 | ||
756.2.bj.b | 84 | 9.c | even | 3 | 1 | ||
756.2.bj.b | 84 | 28.f | even | 6 | 1 | ||
756.2.bj.b | 84 | 36.f | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{42} + 113 T_{5}^{40} + 5811 T_{5}^{38} + 180273 T_{5}^{36} + 3769914 T_{5}^{34} + 56237622 T_{5}^{32} + 617515849 T_{5}^{30} + 5076052073 T_{5}^{28} + 31455323817 T_{5}^{26} + 146814741693 T_{5}^{24} + \cdots + 2187 \)
acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\).