Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1120,2,Mod(289,1120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1120, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1120.289");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1120 = 2^{5} \cdot 5 \cdot 7 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1120.bw (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: | \(8.94324502638\) |
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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
289.1 | 0 | −2.73849 | + | 1.58107i | 0 | 2.12487 | + | 0.696361i | 0 | −2.30440 | − | 1.29991i | 0 | 3.49953 | − | 6.06137i | 0 | ||||||||||
289.2 | 0 | −2.73849 | + | 1.58107i | 0 | −1.66550 | − | 1.49201i | 0 | −2.30440 | − | 1.29991i | 0 | 3.49953 | − | 6.06137i | 0 | ||||||||||
289.3 | 0 | −2.25275 | + | 1.30063i | 0 | −1.24022 | + | 1.86061i | 0 | 2.63098 | − | 0.279192i | 0 | 1.88326 | − | 3.26190i | 0 | ||||||||||
289.4 | 0 | −2.25275 | + | 1.30063i | 0 | −0.991222 | + | 2.00437i | 0 | 2.63098 | − | 0.279192i | 0 | 1.88326 | − | 3.26190i | 0 | ||||||||||
289.5 | 0 | −2.04145 | + | 1.17863i | 0 | −0.800879 | − | 2.08772i | 0 | 0.745750 | + | 2.53848i | 0 | 1.27835 | − | 2.21416i | 0 | ||||||||||
289.6 | 0 | −2.04145 | + | 1.17863i | 0 | 2.20846 | − | 0.350280i | 0 | 0.745750 | + | 2.53848i | 0 | 1.27835 | − | 2.21416i | 0 | ||||||||||
289.7 | 0 | −1.21235 | + | 0.699953i | 0 | −1.99091 | − | 1.01797i | 0 | 0.374974 | − | 2.61904i | 0 | −0.520132 | + | 0.900895i | 0 | ||||||||||
289.8 | 0 | −1.21235 | + | 0.699953i | 0 | 1.87704 | + | 1.21520i | 0 | 0.374974 | − | 2.61904i | 0 | −0.520132 | + | 0.900895i | 0 | ||||||||||
289.9 | 0 | −0.733819 | + | 0.423671i | 0 | −2.17978 | + | 0.498541i | 0 | −1.14933 | + | 2.38308i | 0 | −1.14101 | + | 1.97628i | 0 | ||||||||||
289.10 | 0 | −0.733819 | + | 0.423671i | 0 | 0.658143 | + | 2.13702i | 0 | −1.14933 | + | 2.38308i | 0 | −1.14101 | + | 1.97628i | 0 | ||||||||||
289.11 | 0 | 0.733819 | − | 0.423671i | 0 | 0.658143 | + | 2.13702i | 0 | 1.14933 | − | 2.38308i | 0 | −1.14101 | + | 1.97628i | 0 | ||||||||||
289.12 | 0 | 0.733819 | − | 0.423671i | 0 | −2.17978 | + | 0.498541i | 0 | 1.14933 | − | 2.38308i | 0 | −1.14101 | + | 1.97628i | 0 | ||||||||||
289.13 | 0 | 1.21235 | − | 0.699953i | 0 | 1.87704 | + | 1.21520i | 0 | −0.374974 | + | 2.61904i | 0 | −0.520132 | + | 0.900895i | 0 | ||||||||||
289.14 | 0 | 1.21235 | − | 0.699953i | 0 | −1.99091 | − | 1.01797i | 0 | −0.374974 | + | 2.61904i | 0 | −0.520132 | + | 0.900895i | 0 | ||||||||||
289.15 | 0 | 2.04145 | − | 1.17863i | 0 | 2.20846 | − | 0.350280i | 0 | −0.745750 | − | 2.53848i | 0 | 1.27835 | − | 2.21416i | 0 | ||||||||||
289.16 | 0 | 2.04145 | − | 1.17863i | 0 | −0.800879 | − | 2.08772i | 0 | −0.745750 | − | 2.53848i | 0 | 1.27835 | − | 2.21416i | 0 | ||||||||||
289.17 | 0 | 2.25275 | − | 1.30063i | 0 | −0.991222 | + | 2.00437i | 0 | −2.63098 | + | 0.279192i | 0 | 1.88326 | − | 3.26190i | 0 | ||||||||||
289.18 | 0 | 2.25275 | − | 1.30063i | 0 | −1.24022 | + | 1.86061i | 0 | −2.63098 | + | 0.279192i | 0 | 1.88326 | − | 3.26190i | 0 | ||||||||||
289.19 | 0 | 2.73849 | − | 1.58107i | 0 | −1.66550 | − | 1.49201i | 0 | 2.30440 | + | 1.29991i | 0 | 3.49953 | − | 6.06137i | 0 | ||||||||||
289.20 | 0 | 2.73849 | − | 1.58107i | 0 | 2.12487 | + | 0.696361i | 0 | 2.30440 | + | 1.29991i | 0 | 3.49953 | − | 6.06137i | 0 | ||||||||||
See all 40 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
4.b | odd | 2 | 1 | inner |
5.b | even | 2 | 1 | inner |
7.c | even | 3 | 1 | inner |
20.d | odd | 2 | 1 | inner |
28.g | odd | 6 | 1 | inner |
35.j | even | 6 | 1 | inner |
140.p | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1120.2.bw.h | ✓ | 40 |
4.b | odd | 2 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
5.b | even | 2 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
7.c | even | 3 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
20.d | odd | 2 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
28.g | odd | 6 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
35.j | even | 6 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
140.p | odd | 6 | 1 | inner | 1120.2.bw.h | ✓ | 40 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1120.2.bw.h | ✓ | 40 | 1.a | even | 1 | 1 | trivial |
1120.2.bw.h | ✓ | 40 | 4.b | odd | 2 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 5.b | even | 2 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 7.c | even | 3 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 20.d | odd | 2 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 28.g | odd | 6 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 35.j | even | 6 | 1 | inner |
1120.2.bw.h | ✓ | 40 | 140.p | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1120, [\chi])\):
\( T_{3}^{20} - 25 T_{3}^{18} + 403 T_{3}^{16} - 3874 T_{3}^{14} + 27101 T_{3}^{12} - 124915 T_{3}^{10} + 415293 T_{3}^{8} - 798378 T_{3}^{6} + 1076987 T_{3}^{4} - 652257 T_{3}^{2} + 279841 \) |
\( T_{11}^{20} + 66 T_{11}^{18} + 2927 T_{11}^{16} + 72554 T_{11}^{14} + 1308857 T_{11}^{12} + 13557888 T_{11}^{10} + 96520448 T_{11}^{8} + 152625152 T_{11}^{6} + 183566336 T_{11}^{4} + 61865984 T_{11}^{2} + \cdots + 16777216 \) |