Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [441,3,Mod(31,441)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(441, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([2, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("441.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 441 = 3^{2} \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 441.k (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: | \(12.0163796583\) |
Analytic rank: | \(0\) |
Dimension: | \(28\) |
Relative dimension: | \(14\) over \(\Q(\zeta_{6})\) |
Twist minimal: | no (minimal twist has level 63) |
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 | −1.85100 | + | 3.20603i | −2.30657 | − | 1.91826i | −4.85242 | − | 8.40464i | − | 5.48291i | 10.4195 | − | 3.84424i | 0 | 21.1194 | 1.64056 | + | 8.84921i | 17.5784 | + | 10.1489i | |||||
31.2 | −1.85100 | + | 3.20603i | 2.30657 | + | 1.91826i | −4.85242 | − | 8.40464i | 5.48291i | −10.4195 | + | 3.84424i | 0 | 21.1194 | 1.64056 | + | 8.84921i | −17.5784 | − | 10.1489i | ||||||
31.3 | −1.06935 | + | 1.85216i | −2.45962 | + | 1.71764i | −0.287009 | − | 0.497114i | − | 1.60844i | −0.551161 | − | 6.39237i | 0 | −7.32713 | 3.09944 | − | 8.44947i | 2.97910 | + | 1.71998i | |||||
31.4 | −1.06935 | + | 1.85216i | 2.45962 | − | 1.71764i | −0.287009 | − | 0.497114i | 1.60844i | 0.551161 | + | 6.39237i | 0 | −7.32713 | 3.09944 | − | 8.44947i | −2.97910 | − | 1.71998i | ||||||
31.5 | −0.952438 | + | 1.64967i | −1.69506 | − | 2.47523i | 0.185723 | + | 0.321682i | 8.23162i | 5.69776 | − | 0.438798i | 0 | −8.32706 | −3.25351 | + | 8.39135i | −13.5795 | − | 7.84010i | ||||||
31.6 | −0.952438 | + | 1.64967i | 1.69506 | + | 2.47523i | 0.185723 | + | 0.321682i | − | 8.23162i | −5.69776 | + | 0.438798i | 0 | −8.32706 | −3.25351 | + | 8.39135i | 13.5795 | + | 7.84010i | |||||
31.7 | −0.0644125 | + | 0.111566i | −0.0616869 | + | 2.99937i | 1.99170 | + | 3.44973i | 6.63133i | −0.330653 | − | 0.200079i | 0 | −1.02846 | −8.99239 | − | 0.370043i | −0.739829 | − | 0.427141i | ||||||
31.8 | −0.0644125 | + | 0.111566i | 0.0616869 | − | 2.99937i | 1.99170 | + | 3.44973i | − | 6.63133i | 0.330653 | + | 0.200079i | 0 | −1.02846 | −8.99239 | − | 0.370043i | 0.739829 | + | 0.427141i | |||||
31.9 | 0.435704 | − | 0.754662i | −2.98422 | + | 0.307314i | 1.62032 | + | 2.80648i | − | 2.19279i | −1.06832 | + | 2.38597i | 0 | 6.30956 | 8.81112 | − | 1.83419i | −1.65481 | − | 0.955407i | |||||
31.10 | 0.435704 | − | 0.754662i | 2.98422 | − | 0.307314i | 1.62032 | + | 2.80648i | 2.19279i | 1.06832 | − | 2.38597i | 0 | 6.30956 | 8.81112 | − | 1.83419i | 1.65481 | + | 0.955407i | ||||||
31.11 | 1.30753 | − | 2.26471i | −2.30352 | − | 1.92193i | −1.41927 | − | 2.45825i | 3.68512i | −7.36452 | + | 2.70382i | 0 | 3.03728 | 1.61239 | + | 8.85439i | 8.34571 | + | 4.81840i | ||||||
31.12 | 1.30753 | − | 2.26471i | 2.30352 | + | 1.92193i | −1.41927 | − | 2.45825i | − | 3.68512i | 7.36452 | − | 2.70382i | 0 | 3.03728 | 1.61239 | + | 8.85439i | −8.34571 | − | 4.81840i | |||||
31.13 | 1.69397 | − | 2.93404i | −1.24145 | + | 2.73108i | −3.73905 | − | 6.47622i | 5.12135i | 5.91012 | + | 8.26882i | 0 | −11.7836 | −5.91761 | − | 6.78100i | 15.0262 | + | 8.67539i | ||||||
31.14 | 1.69397 | − | 2.93404i | 1.24145 | − | 2.73108i | −3.73905 | − | 6.47622i | − | 5.12135i | −5.91012 | − | 8.26882i | 0 | −11.7836 | −5.91761 | − | 6.78100i | −15.0262 | − | 8.67539i | |||||
313.1 | −1.85100 | − | 3.20603i | −2.30657 | + | 1.91826i | −4.85242 | + | 8.40464i | 5.48291i | 10.4195 | + | 3.84424i | 0 | 21.1194 | 1.64056 | − | 8.84921i | 17.5784 | − | 10.1489i | ||||||
313.2 | −1.85100 | − | 3.20603i | 2.30657 | − | 1.91826i | −4.85242 | + | 8.40464i | − | 5.48291i | −10.4195 | − | 3.84424i | 0 | 21.1194 | 1.64056 | − | 8.84921i | −17.5784 | + | 10.1489i | |||||
313.3 | −1.06935 | − | 1.85216i | −2.45962 | − | 1.71764i | −0.287009 | + | 0.497114i | 1.60844i | −0.551161 | + | 6.39237i | 0 | −7.32713 | 3.09944 | + | 8.44947i | 2.97910 | − | 1.71998i | ||||||
313.4 | −1.06935 | − | 1.85216i | 2.45962 | + | 1.71764i | −0.287009 | + | 0.497114i | − | 1.60844i | 0.551161 | − | 6.39237i | 0 | −7.32713 | 3.09944 | + | 8.44947i | −2.97910 | + | 1.71998i | |||||
313.5 | −0.952438 | − | 1.64967i | −1.69506 | + | 2.47523i | 0.185723 | − | 0.321682i | − | 8.23162i | 5.69776 | + | 0.438798i | 0 | −8.32706 | −3.25351 | − | 8.39135i | −13.5795 | + | 7.84010i | |||||
313.6 | −0.952438 | − | 1.64967i | 1.69506 | − | 2.47523i | 0.185723 | − | 0.321682i | 8.23162i | −5.69776 | − | 0.438798i | 0 | −8.32706 | −3.25351 | − | 8.39135i | 13.5795 | − | 7.84010i | ||||||
See all 28 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.b | odd | 2 | 1 | inner |
63.g | even | 3 | 1 | inner |
63.k | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 441.3.k.a | 28 | |
7.b | odd | 2 | 1 | inner | 441.3.k.a | 28 | |
7.c | even | 3 | 1 | 63.3.l.a | ✓ | 28 | |
7.c | even | 3 | 1 | 441.3.t.b | 28 | ||
7.d | odd | 6 | 1 | 63.3.l.a | ✓ | 28 | |
7.d | odd | 6 | 1 | 441.3.t.b | 28 | ||
9.c | even | 3 | 1 | 441.3.t.b | 28 | ||
21.g | even | 6 | 1 | 189.3.l.a | 28 | ||
21.h | odd | 6 | 1 | 189.3.l.a | 28 | ||
63.g | even | 3 | 1 | inner | 441.3.k.a | 28 | |
63.g | even | 3 | 1 | 567.3.d.h | 14 | ||
63.h | even | 3 | 1 | 63.3.l.a | ✓ | 28 | |
63.i | even | 6 | 1 | 189.3.l.a | 28 | ||
63.j | odd | 6 | 1 | 189.3.l.a | 28 | ||
63.k | odd | 6 | 1 | inner | 441.3.k.a | 28 | |
63.k | odd | 6 | 1 | 567.3.d.h | 14 | ||
63.l | odd | 6 | 1 | 441.3.t.b | 28 | ||
63.n | odd | 6 | 1 | 567.3.d.g | 14 | ||
63.s | even | 6 | 1 | 567.3.d.g | 14 | ||
63.t | odd | 6 | 1 | 63.3.l.a | ✓ | 28 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
63.3.l.a | ✓ | 28 | 7.c | even | 3 | 1 | |
63.3.l.a | ✓ | 28 | 7.d | odd | 6 | 1 | |
63.3.l.a | ✓ | 28 | 63.h | even | 3 | 1 | |
63.3.l.a | ✓ | 28 | 63.t | odd | 6 | 1 | |
189.3.l.a | 28 | 21.g | even | 6 | 1 | ||
189.3.l.a | 28 | 21.h | odd | 6 | 1 | ||
189.3.l.a | 28 | 63.i | even | 6 | 1 | ||
189.3.l.a | 28 | 63.j | odd | 6 | 1 | ||
441.3.k.a | 28 | 1.a | even | 1 | 1 | trivial | |
441.3.k.a | 28 | 7.b | odd | 2 | 1 | inner | |
441.3.k.a | 28 | 63.g | even | 3 | 1 | inner | |
441.3.k.a | 28 | 63.k | odd | 6 | 1 | inner | |
441.3.t.b | 28 | 7.c | even | 3 | 1 | ||
441.3.t.b | 28 | 7.d | odd | 6 | 1 | ||
441.3.t.b | 28 | 9.c | even | 3 | 1 | ||
441.3.t.b | 28 | 63.l | odd | 6 | 1 | ||
567.3.d.g | 14 | 63.n | odd | 6 | 1 | ||
567.3.d.g | 14 | 63.s | even | 6 | 1 | ||
567.3.d.h | 14 | 63.g | even | 3 | 1 | ||
567.3.d.h | 14 | 63.k | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{14} + T_{2}^{13} + 21 T_{2}^{12} + 14 T_{2}^{11} + 314 T_{2}^{10} + 213 T_{2}^{9} + 2054 T_{2}^{8} + \cdots + 225 \) acting on \(S_{3}^{\mathrm{new}}(441, [\chi])\).