Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1755,2,Mod(1054,1755)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1755, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1755.1054");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 1755 = 3^{3} \cdot 5 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 1755.c (of order \(2\), degree \(1\), 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: | \(14.0137455547\) |
Analytic rank: | \(0\) |
Dimension: | \(24\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
1054.1 | − | 2.74548i | 0 | −5.53764 | 2.17097 | + | 0.535606i | 0 | − | 3.22219i | 9.71252i | 0 | 1.47049 | − | 5.96036i | ||||||||||||
1054.2 | − | 2.41139i | 0 | −3.81481 | −2.23354 | − | 0.106317i | 0 | − | 3.03216i | 4.37623i | 0 | −0.256373 | + | 5.38594i | ||||||||||||
1054.3 | − | 2.16130i | 0 | −2.67123 | −0.878285 | − | 2.05636i | 0 | 3.06336i | 1.45074i | 0 | −4.44442 | + | 1.89824i | |||||||||||||
1054.4 | − | 2.15187i | 0 | −2.63053 | −1.22772 | + | 1.86888i | 0 | − | 1.91033i | 1.35682i | 0 | 4.02158 | + | 2.64188i | ||||||||||||
1054.5 | − | 2.02678i | 0 | −2.10785 | 2.15690 | + | 0.589721i | 0 | − | 3.38762i | 0.218585i | 0 | 1.19524 | − | 4.37157i | ||||||||||||
1054.6 | − | 1.61293i | 0 | −0.601541 | 1.84270 | − | 1.26667i | 0 | 2.78276i | − | 2.25562i | 0 | −2.04305 | − | 2.97214i | ||||||||||||
1054.7 | − | 1.55410i | 0 | −0.415232 | 0.130512 | + | 2.23226i | 0 | 3.05332i | − | 2.46289i | 0 | 3.46915 | − | 0.202829i | ||||||||||||
1054.8 | − | 1.26785i | 0 | 0.392563 | 1.19142 | − | 1.89222i | 0 | 1.78234i | − | 3.03340i | 0 | −2.39905 | − | 1.51054i | ||||||||||||
1054.9 | − | 0.872948i | 0 | 1.23796 | 1.62865 | − | 1.53215i | 0 | − | 4.38491i | − | 2.82657i | 0 | −1.33749 | − | 1.42173i | |||||||||||
1054.10 | − | 0.819591i | 0 | 1.32827 | −2.12060 | + | 0.709260i | 0 | − | 0.824970i | − | 2.72782i | 0 | 0.581303 | + | 1.73803i | |||||||||||
1054.11 | − | 0.347415i | 0 | 1.87930 | −2.15064 | + | 0.612168i | 0 | 0.968821i | − | 1.34773i | 0 | 0.212676 | + | 0.747163i | ||||||||||||
1054.12 | − | 0.243423i | 0 | 1.94075 | −0.510383 | + | 2.17704i | 0 | 4.88099i | − | 0.959268i | 0 | 0.529942 | + | 0.124239i | ||||||||||||
1054.13 | 0.243423i | 0 | 1.94075 | −0.510383 | − | 2.17704i | 0 | − | 4.88099i | 0.959268i | 0 | 0.529942 | − | 0.124239i | |||||||||||||
1054.14 | 0.347415i | 0 | 1.87930 | −2.15064 | − | 0.612168i | 0 | − | 0.968821i | 1.34773i | 0 | 0.212676 | − | 0.747163i | |||||||||||||
1054.15 | 0.819591i | 0 | 1.32827 | −2.12060 | − | 0.709260i | 0 | 0.824970i | 2.72782i | 0 | 0.581303 | − | 1.73803i | ||||||||||||||
1054.16 | 0.872948i | 0 | 1.23796 | 1.62865 | + | 1.53215i | 0 | 4.38491i | 2.82657i | 0 | −1.33749 | + | 1.42173i | ||||||||||||||
1054.17 | 1.26785i | 0 | 0.392563 | 1.19142 | + | 1.89222i | 0 | − | 1.78234i | 3.03340i | 0 | −2.39905 | + | 1.51054i | |||||||||||||
1054.18 | 1.55410i | 0 | −0.415232 | 0.130512 | − | 2.23226i | 0 | − | 3.05332i | 2.46289i | 0 | 3.46915 | + | 0.202829i | |||||||||||||
1054.19 | 1.61293i | 0 | −0.601541 | 1.84270 | + | 1.26667i | 0 | − | 2.78276i | 2.25562i | 0 | −2.04305 | + | 2.97214i | |||||||||||||
1054.20 | 2.02678i | 0 | −2.10785 | 2.15690 | − | 0.589721i | 0 | 3.38762i | − | 0.218585i | 0 | 1.19524 | + | 4.37157i | |||||||||||||
See all 24 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.b | even | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 1755.2.c.c | yes | 24 |
3.b | odd | 2 | 1 | 1755.2.c.b | ✓ | 24 | |
5.b | even | 2 | 1 | inner | 1755.2.c.c | yes | 24 |
5.c | odd | 4 | 1 | 8775.2.a.cl | 12 | ||
5.c | odd | 4 | 1 | 8775.2.a.cm | 12 | ||
15.d | odd | 2 | 1 | 1755.2.c.b | ✓ | 24 | |
15.e | even | 4 | 1 | 8775.2.a.ck | 12 | ||
15.e | even | 4 | 1 | 8775.2.a.cn | 12 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
1755.2.c.b | ✓ | 24 | 3.b | odd | 2 | 1 | |
1755.2.c.b | ✓ | 24 | 15.d | odd | 2 | 1 | |
1755.2.c.c | yes | 24 | 1.a | even | 1 | 1 | trivial |
1755.2.c.c | yes | 24 | 5.b | even | 2 | 1 | inner |
8775.2.a.ck | 12 | 15.e | even | 4 | 1 | ||
8775.2.a.cl | 12 | 5.c | odd | 4 | 1 | ||
8775.2.a.cm | 12 | 5.c | odd | 4 | 1 | ||
8775.2.a.cn | 12 | 15.e | even | 4 | 1 |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1755, [\chi])\):
\( T_{2}^{24} + 35 T_{2}^{22} + 529 T_{2}^{20} + 4534 T_{2}^{18} + 24322 T_{2}^{16} + 85014 T_{2}^{14} + 195433 T_{2}^{12} + 291375 T_{2}^{10} + 271189 T_{2}^{8} + 146744 T_{2}^{6} + 40572 T_{2}^{4} + 4372 T_{2}^{2} + \cdots + 144 \) |
\( T_{11}^{12} - 9 T_{11}^{11} - 36 T_{11}^{10} + 550 T_{11}^{9} - 646 T_{11}^{8} - 8896 T_{11}^{7} + 30071 T_{11}^{6} + 7763 T_{11}^{5} - 163498 T_{11}^{4} + 217476 T_{11}^{3} - 13872 T_{11}^{2} - 89152 T_{11} + 12544 \) |