Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1152,3,Mod(1025,1152)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1152, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1152.1025");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1152 = 2^{7} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1152.e (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: | \(31.3897264543\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{41}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
comment: q-expansion
sage: f.q_expansion() # note that sage often uses an isomorphic number field
gp: mfcoefs(f, 20)
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{2} - 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{3} + 4\nu \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} + 2 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1152\mathbb{Z}\right)^\times\).
| \(n\) | \(127\) | \(641\) | \(901\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1025.1 |
|
0 | 0 | 0 | − | 4.87832i | 0 | −11.7980 | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 1025.2 | 0 | 0 | 0 | − | 2.04989i | 0 | 7.79796 | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1025.3 | 0 | 0 | 0 | 2.04989i | 0 | 7.79796 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1025.4 | 0 | 0 | 0 | 4.87832i | 0 | −11.7980 | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1152.3.e.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 1152.3.e.b | ✓ | 4 |
| 4.b | odd | 2 | 1 | 1152.3.e.f | yes | 4 | |
| 8.b | even | 2 | 1 | 1152.3.e.d | yes | 4 | |
| 8.d | odd | 2 | 1 | 1152.3.e.h | yes | 4 | |
| 12.b | even | 2 | 1 | 1152.3.e.f | yes | 4 | |
| 16.e | even | 4 | 2 | 2304.3.h.k | 8 | ||
| 16.f | odd | 4 | 2 | 2304.3.h.i | 8 | ||
| 24.f | even | 2 | 1 | 1152.3.e.h | yes | 4 | |
| 24.h | odd | 2 | 1 | 1152.3.e.d | yes | 4 | |
| 48.i | odd | 4 | 2 | 2304.3.h.k | 8 | ||
| 48.k | even | 4 | 2 | 2304.3.h.i | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1152.3.e.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 1152.3.e.b | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 1152.3.e.d | yes | 4 | 8.b | even | 2 | 1 | |
| 1152.3.e.d | yes | 4 | 24.h | odd | 2 | 1 | |
| 1152.3.e.f | yes | 4 | 4.b | odd | 2 | 1 | |
| 1152.3.e.f | yes | 4 | 12.b | even | 2 | 1 | |
| 1152.3.e.h | yes | 4 | 8.d | odd | 2 | 1 | |
| 1152.3.e.h | yes | 4 | 24.f | even | 2 | 1 | |
| 2304.3.h.i | 8 | 16.f | odd | 4 | 2 | ||
| 2304.3.h.i | 8 | 48.k | even | 4 | 2 | ||
| 2304.3.h.k | 8 | 16.e | even | 4 | 2 | ||
| 2304.3.h.k | 8 | 48.i | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(1152, [\chi])\):
|
\( T_{5}^{4} + 28T_{5}^{2} + 100 \)
|
|
\( T_{7}^{2} + 4T_{7} - 92 \)
|
|
\( T_{13}^{2} + 16T_{13} - 152 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 28T^{2} + 100 \)
|
| $7$ |
\( (T^{2} + 4 T - 92)^{2} \)
|
| $11$ |
\( T^{4} + 112T^{2} + 1600 \)
|
| $13$ |
\( (T^{2} + 16 T - 152)^{2} \)
|
| $17$ |
\( T^{4} + 868 T^{2} + 184900 \)
|
| $19$ |
\( (T^{2} - 16 T - 32)^{2} \)
|
| $23$ |
\( (T^{2} + 200)^{2} \)
|
| $29$ |
\( T^{4} + 1212 T^{2} + 324900 \)
|
| $31$ |
\( (T^{2} - 60 T + 804)^{2} \)
|
| $37$ |
\( (T^{2} + 44 T + 100)^{2} \)
|
| $41$ |
\( T^{4} + 4164 T^{2} + 101124 \)
|
| $43$ |
\( (T^{2} - 72 T - 240)^{2} \)
|
| $47$ |
\( T^{4} + 1936 T^{2} + 322624 \)
|
| $53$ |
\( T^{4} + 3516 T^{2} + 1340964 \)
|
| $59$ |
\( T^{4} + 15424 T^{2} + 37356544 \)
|
| $61$ |
\( (T^{2} + 100 T + 964)^{2} \)
|
| $67$ |
\( (T^{2} - 56 T + 688)^{2} \)
|
| $71$ |
\( T^{4} + 15120 T^{2} + 16842816 \)
|
| $73$ |
\( (T^{2} - 136 T + 4240)^{2} \)
|
| $79$ |
\( (T^{2} - 244 T + 14020)^{2} \)
|
| $83$ |
\( T^{4} + 29040 T^{2} + 96353856 \)
|
| $89$ |
\( T^{4} + 19972 T^{2} + 77968900 \)
|
| $97$ |
\( (T^{2} - 80 T - 14624)^{2} \)
|
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