Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1760,1,Mod(1649,1760)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1760, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 1, 1]))
N = Newforms(chi, 1, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1760.1649");
S:= CuspForms(chi, 1);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1760 = 2^{5} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 1 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1760.o (of order \(2\), degree \(1\), 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: | yes |
| Analytic conductor: | \(0.878354422234\) |
| Analytic rank: | \(0\) |
| Dimension: | \(1\) |
| Coefficient field: | \(\mathbb{Q}\) |
| Coefficient ring: | \(\mathbb{Z}\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 440) |
| Projective image: | \(D_{3}\) |
| Projective field: | Galois closure of 3.1.440.1 |
| Artin image: | $D_6$ |
| Artin field: | Galois closure of 6.0.3097600.1 |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1760\mathbb{Z}\right)^\times\).
| \(n\) | \(321\) | \(991\) | \(1057\) | \(1541\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1649.1 |
|
0 | 1.00000 | 0 | 1.00000 | 0 | 1.00000 | 0 | 0 | 0 | |||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 440.o | odd | 2 | 1 | CM by \(\Q(\sqrt{-110}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1760.1.o.d | 1 | |
| 4.b | odd | 2 | 1 | 440.1.o.c | yes | 1 | |
| 5.b | even | 2 | 1 | 1760.1.o.a | 1 | ||
| 8.b | even | 2 | 1 | 1760.1.o.b | 1 | ||
| 8.d | odd | 2 | 1 | 440.1.o.d | yes | 1 | |
| 11.b | odd | 2 | 1 | 1760.1.o.c | 1 | ||
| 12.b | even | 2 | 1 | 3960.1.x.a | 1 | ||
| 20.d | odd | 2 | 1 | 440.1.o.b | yes | 1 | |
| 20.e | even | 4 | 2 | 2200.1.d.e | 2 | ||
| 24.f | even | 2 | 1 | 3960.1.x.b | 1 | ||
| 40.e | odd | 2 | 1 | 440.1.o.a | ✓ | 1 | |
| 40.f | even | 2 | 1 | 1760.1.o.c | 1 | ||
| 40.k | even | 4 | 2 | 2200.1.d.f | 2 | ||
| 44.c | even | 2 | 1 | 440.1.o.a | ✓ | 1 | |
| 55.d | odd | 2 | 1 | 1760.1.o.b | 1 | ||
| 60.h | even | 2 | 1 | 3960.1.x.d | 1 | ||
| 88.b | odd | 2 | 1 | 1760.1.o.a | 1 | ||
| 88.g | even | 2 | 1 | 440.1.o.b | yes | 1 | |
| 120.m | even | 2 | 1 | 3960.1.x.c | 1 | ||
| 132.d | odd | 2 | 1 | 3960.1.x.c | 1 | ||
| 220.g | even | 2 | 1 | 440.1.o.d | yes | 1 | |
| 220.i | odd | 4 | 2 | 2200.1.d.f | 2 | ||
| 264.p | odd | 2 | 1 | 3960.1.x.d | 1 | ||
| 440.c | even | 2 | 1 | 440.1.o.c | yes | 1 | |
| 440.o | odd | 2 | 1 | CM | 1760.1.o.d | 1 | |
| 440.w | odd | 4 | 2 | 2200.1.d.e | 2 | ||
| 660.g | odd | 2 | 1 | 3960.1.x.b | 1 | ||
| 1320.b | odd | 2 | 1 | 3960.1.x.a | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 440.1.o.a | ✓ | 1 | 40.e | odd | 2 | 1 | |
| 440.1.o.a | ✓ | 1 | 44.c | even | 2 | 1 | |
| 440.1.o.b | yes | 1 | 20.d | odd | 2 | 1 | |
| 440.1.o.b | yes | 1 | 88.g | even | 2 | 1 | |
| 440.1.o.c | yes | 1 | 4.b | odd | 2 | 1 | |
| 440.1.o.c | yes | 1 | 440.c | even | 2 | 1 | |
| 440.1.o.d | yes | 1 | 8.d | odd | 2 | 1 | |
| 440.1.o.d | yes | 1 | 220.g | even | 2 | 1 | |
| 1760.1.o.a | 1 | 5.b | even | 2 | 1 | ||
| 1760.1.o.a | 1 | 88.b | odd | 2 | 1 | ||
| 1760.1.o.b | 1 | 8.b | even | 2 | 1 | ||
| 1760.1.o.b | 1 | 55.d | odd | 2 | 1 | ||
| 1760.1.o.c | 1 | 11.b | odd | 2 | 1 | ||
| 1760.1.o.c | 1 | 40.f | even | 2 | 1 | ||
| 1760.1.o.d | 1 | 1.a | even | 1 | 1 | trivial | |
| 1760.1.o.d | 1 | 440.o | odd | 2 | 1 | CM | |
| 2200.1.d.e | 2 | 20.e | even | 4 | 2 | ||
| 2200.1.d.e | 2 | 440.w | odd | 4 | 2 | ||
| 2200.1.d.f | 2 | 40.k | even | 4 | 2 | ||
| 2200.1.d.f | 2 | 220.i | odd | 4 | 2 | ||
| 3960.1.x.a | 1 | 12.b | even | 2 | 1 | ||
| 3960.1.x.a | 1 | 1320.b | odd | 2 | 1 | ||
| 3960.1.x.b | 1 | 24.f | even | 2 | 1 | ||
| 3960.1.x.b | 1 | 660.g | odd | 2 | 1 | ||
| 3960.1.x.c | 1 | 120.m | even | 2 | 1 | ||
| 3960.1.x.c | 1 | 132.d | odd | 2 | 1 | ||
| 3960.1.x.d | 1 | 60.h | even | 2 | 1 | ||
| 3960.1.x.d | 1 | 264.p | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1760, [\chi])\):
|
\( T_{3} - 1 \)
|
|
\( T_{7} - 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T \)
|
| $3$ |
\( T - 1 \)
|
| $5$ |
\( T - 1 \)
|
| $7$ |
\( T - 1 \)
|
| $11$ |
\( T + 1 \)
|
| $13$ |
\( T \)
|
| $17$ |
\( T + 1 \)
|
| $19$ |
\( T - 1 \)
|
| $23$ |
\( T \)
|
| $29$ |
\( T + 1 \)
|
| $31$ |
\( T - 1 \)
|
| $37$ |
\( T + 1 \)
|
| $41$ |
\( T \)
|
| $43$ |
\( T \)
|
| $47$ |
\( T \)
|
| $53$ |
\( T + 1 \)
|
| $59$ |
\( T \)
|
| $61$ |
\( T + 1 \)
|
| $67$ |
\( T + 2 \)
|
| $71$ |
\( T - 1 \)
|
| $73$ |
\( T - 2 \)
|
| $79$ |
\( T \)
|
| $83$ |
\( T \)
|
| $89$ |
\( T + 1 \)
|
| $97$ |
\( T \)
|
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