Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [360,2,Mod(49,360)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(360, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 2, 3]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("360.49");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 360.bi (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: | \(2.87461447277\) |
Analytic rank: | \(0\) |
Dimension: | \(32\) |
Relative dimension: | \(16\) 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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
49.1 | 0 | −1.72704 | + | 0.131617i | 0 | 2.23514 | − | 0.0643013i | 0 | −1.19102 | + | 0.687633i | 0 | 2.96535 | − | 0.454616i | 0 | ||||||||||
49.2 | 0 | −1.56081 | + | 0.750911i | 0 | −2.18001 | − | 0.497547i | 0 | 2.51643 | − | 1.45286i | 0 | 1.87227 | − | 2.34406i | 0 | ||||||||||
49.3 | 0 | −1.54437 | − | 0.784174i | 0 | 0.554267 | − | 2.16628i | 0 | −0.608912 | + | 0.351555i | 0 | 1.77014 | + | 2.42211i | 0 | ||||||||||
49.4 | 0 | −1.27766 | + | 1.16944i | 0 | 0.307984 | + | 2.21476i | 0 | −3.28422 | + | 1.89614i | 0 | 0.264813 | − | 2.98829i | 0 | ||||||||||
49.5 | 0 | −1.19594 | − | 1.25289i | 0 | −2.14796 | − | 0.621508i | 0 | −1.98012 | + | 1.14322i | 0 | −0.139458 | + | 2.99676i | 0 | ||||||||||
49.6 | 0 | −1.11432 | − | 1.32601i | 0 | −0.355011 | + | 2.20771i | 0 | 2.19623 | − | 1.26799i | 0 | −0.516592 | + | 2.95519i | 0 | ||||||||||
49.7 | 0 | −0.557359 | + | 1.63992i | 0 | 0.668629 | + | 2.13376i | 0 | 3.90215 | − | 2.25291i | 0 | −2.37870 | − | 1.82805i | 0 | ||||||||||
49.8 | 0 | −0.284761 | − | 1.70848i | 0 | 1.16287 | − | 1.90990i | 0 | 3.55262 | − | 2.05111i | 0 | −2.83782 | + | 0.973019i | 0 | ||||||||||
49.9 | 0 | 0.284761 | + | 1.70848i | 0 | 1.07259 | − | 1.96203i | 0 | −3.55262 | + | 2.05111i | 0 | −2.83782 | + | 0.973019i | 0 | ||||||||||
49.10 | 0 | 0.557359 | − | 1.63992i | 0 | −2.18221 | + | 0.487830i | 0 | −3.90215 | + | 2.25291i | 0 | −2.37870 | − | 1.82805i | 0 | ||||||||||
49.11 | 0 | 1.11432 | + | 1.32601i | 0 | −1.73442 | + | 1.41130i | 0 | −2.19623 | + | 1.26799i | 0 | −0.516592 | + | 2.95519i | 0 | ||||||||||
49.12 | 0 | 1.19594 | + | 1.25289i | 0 | 1.61222 | + | 1.54943i | 0 | 1.98012 | − | 1.14322i | 0 | −0.139458 | + | 2.99676i | 0 | ||||||||||
49.13 | 0 | 1.27766 | − | 1.16944i | 0 | −2.07203 | + | 0.840657i | 0 | 3.28422 | − | 1.89614i | 0 | 0.264813 | − | 2.98829i | 0 | ||||||||||
49.14 | 0 | 1.54437 | + | 0.784174i | 0 | 1.59892 | − | 1.56315i | 0 | 0.608912 | − | 0.351555i | 0 | 1.77014 | + | 2.42211i | 0 | ||||||||||
49.15 | 0 | 1.56081 | − | 0.750911i | 0 | 1.52089 | + | 1.63917i | 0 | −2.51643 | + | 1.45286i | 0 | 1.87227 | − | 2.34406i | 0 | ||||||||||
49.16 | 0 | 1.72704 | − | 0.131617i | 0 | −1.06189 | − | 1.96784i | 0 | 1.19102 | − | 0.687633i | 0 | 2.96535 | − | 0.454616i | 0 | ||||||||||
169.1 | 0 | −1.72704 | − | 0.131617i | 0 | 2.23514 | + | 0.0643013i | 0 | −1.19102 | − | 0.687633i | 0 | 2.96535 | + | 0.454616i | 0 | ||||||||||
169.2 | 0 | −1.56081 | − | 0.750911i | 0 | −2.18001 | + | 0.497547i | 0 | 2.51643 | + | 1.45286i | 0 | 1.87227 | + | 2.34406i | 0 | ||||||||||
169.3 | 0 | −1.54437 | + | 0.784174i | 0 | 0.554267 | + | 2.16628i | 0 | −0.608912 | − | 0.351555i | 0 | 1.77014 | − | 2.42211i | 0 | ||||||||||
169.4 | 0 | −1.27766 | − | 1.16944i | 0 | 0.307984 | − | 2.21476i | 0 | −3.28422 | − | 1.89614i | 0 | 0.264813 | + | 2.98829i | 0 | ||||||||||
See all 32 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
9.c | even | 3 | 1 | inner |
45.j | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 360.2.bi.b | ✓ | 32 |
3.b | odd | 2 | 1 | 1080.2.bi.b | 32 | ||
4.b | odd | 2 | 1 | 720.2.by.f | 32 | ||
5.b | even | 2 | 1 | inner | 360.2.bi.b | ✓ | 32 |
9.c | even | 3 | 1 | inner | 360.2.bi.b | ✓ | 32 |
9.c | even | 3 | 1 | 3240.2.f.k | 16 | ||
9.d | odd | 6 | 1 | 1080.2.bi.b | 32 | ||
9.d | odd | 6 | 1 | 3240.2.f.i | 16 | ||
12.b | even | 2 | 1 | 2160.2.by.f | 32 | ||
15.d | odd | 2 | 1 | 1080.2.bi.b | 32 | ||
20.d | odd | 2 | 1 | 720.2.by.f | 32 | ||
36.f | odd | 6 | 1 | 720.2.by.f | 32 | ||
36.h | even | 6 | 1 | 2160.2.by.f | 32 | ||
45.h | odd | 6 | 1 | 1080.2.bi.b | 32 | ||
45.h | odd | 6 | 1 | 3240.2.f.i | 16 | ||
45.j | even | 6 | 1 | inner | 360.2.bi.b | ✓ | 32 |
45.j | even | 6 | 1 | 3240.2.f.k | 16 | ||
60.h | even | 2 | 1 | 2160.2.by.f | 32 | ||
180.n | even | 6 | 1 | 2160.2.by.f | 32 | ||
180.p | odd | 6 | 1 | 720.2.by.f | 32 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
360.2.bi.b | ✓ | 32 | 1.a | even | 1 | 1 | trivial |
360.2.bi.b | ✓ | 32 | 5.b | even | 2 | 1 | inner |
360.2.bi.b | ✓ | 32 | 9.c | even | 3 | 1 | inner |
360.2.bi.b | ✓ | 32 | 45.j | even | 6 | 1 | inner |
720.2.by.f | 32 | 4.b | odd | 2 | 1 | ||
720.2.by.f | 32 | 20.d | odd | 2 | 1 | ||
720.2.by.f | 32 | 36.f | odd | 6 | 1 | ||
720.2.by.f | 32 | 180.p | odd | 6 | 1 | ||
1080.2.bi.b | 32 | 3.b | odd | 2 | 1 | ||
1080.2.bi.b | 32 | 9.d | odd | 6 | 1 | ||
1080.2.bi.b | 32 | 15.d | odd | 2 | 1 | ||
1080.2.bi.b | 32 | 45.h | odd | 6 | 1 | ||
2160.2.by.f | 32 | 12.b | even | 2 | 1 | ||
2160.2.by.f | 32 | 36.h | even | 6 | 1 | ||
2160.2.by.f | 32 | 60.h | even | 2 | 1 | ||
2160.2.by.f | 32 | 180.n | even | 6 | 1 | ||
3240.2.f.i | 16 | 9.d | odd | 6 | 1 | ||
3240.2.f.i | 16 | 45.h | odd | 6 | 1 | ||
3240.2.f.k | 16 | 9.c | even | 3 | 1 | ||
3240.2.f.k | 16 | 45.j | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{32} - 74 T_{7}^{30} + 3261 T_{7}^{28} - 94822 T_{7}^{26} + 2048402 T_{7}^{24} + \cdots + 1700843738896 \) acting on \(S_{2}^{\mathrm{new}}(360, [\chi])\).