Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2352,3,Mod(1471,2352)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2352.1471");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2352.m (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: | \(64.0873581775\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-7})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{8} \) |
| Twist minimal: | no (minimal twist has level 336) |
| 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
\( x^{4} - x^{3} - x^{2} - 2x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + \nu^{2} - \nu - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( -4\nu^{3} + 4\nu^{2} + 12\nu + 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( 8\nu^{3} - 20 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - 4\beta _1 + 4 ) / 16 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} + 12\beta _1 + 12 ) / 16 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{3} + 20 ) / 8 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).
| \(n\) | \(785\) | \(1471\) | \(1765\) | \(2257\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1471.1 |
|
0 | − | 1.73205i | 0 | 2.00000 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 1471.2 | 0 | − | 1.73205i | 0 | 2.00000 | 0 | 0 | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 1471.3 | 0 | 1.73205i | 0 | 2.00000 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
| 1471.4 | 0 | 1.73205i | 0 | 2.00000 | 0 | 0 | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2352.3.m.i | 4 | |
| 4.b | odd | 2 | 1 | inner | 2352.3.m.i | 4 | |
| 7.b | odd | 2 | 1 | 336.3.m.b | ✓ | 4 | |
| 21.c | even | 2 | 1 | 1008.3.m.e | 4 | ||
| 28.d | even | 2 | 1 | 336.3.m.b | ✓ | 4 | |
| 56.e | even | 2 | 1 | 1344.3.m.b | 4 | ||
| 56.h | odd | 2 | 1 | 1344.3.m.b | 4 | ||
| 84.h | odd | 2 | 1 | 1008.3.m.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 336.3.m.b | ✓ | 4 | 7.b | odd | 2 | 1 | |
| 336.3.m.b | ✓ | 4 | 28.d | even | 2 | 1 | |
| 1008.3.m.e | 4 | 21.c | even | 2 | 1 | ||
| 1008.3.m.e | 4 | 84.h | odd | 2 | 1 | ||
| 1344.3.m.b | 4 | 56.e | even | 2 | 1 | ||
| 1344.3.m.b | 4 | 56.h | odd | 2 | 1 | ||
| 2352.3.m.i | 4 | 1.a | even | 1 | 1 | trivial | |
| 2352.3.m.i | 4 | 4.b | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5} - 2 \)
acting on \(S_{3}^{\mathrm{new}}(2352, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( (T - 2)^{4} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} + 320T^{2} + 4096 \)
|
| $13$ |
\( (T^{2} + 4 T - 332)^{2} \)
|
| $17$ |
\( (T^{2} + 28 T - 140)^{2} \)
|
| $19$ |
\( T^{4} + 992 T^{2} + 160000 \)
|
| $23$ |
\( T^{4} + 1088 T^{2} + 102400 \)
|
| $29$ |
\( (T^{2} + 28 T - 1148)^{2} \)
|
| $31$ |
\( (T^{2} + 768)^{2} \)
|
| $37$ |
\( (T^{2} - 52 T - 668)^{2} \)
|
| $41$ |
\( (T^{2} - 20 T - 236)^{2} \)
|
| $43$ |
\( T^{4} + 2432 T^{2} + 102400 \)
|
| $47$ |
\( T^{4} + 9600 T^{2} + 10653696 \)
|
| $53$ |
\( (T^{2} - 20 T - 1244)^{2} \)
|
| $59$ |
\( T^{4} + 1760T^{2} + 256 \)
|
| $61$ |
\( (T^{2} - 12 T - 300)^{2} \)
|
| $67$ |
\( (T^{2} + 3072)^{2} \)
|
| $71$ |
\( T^{4} + 11840 T^{2} + 25563136 \)
|
| $73$ |
\( (T^{2} + 116 T + 2020)^{2} \)
|
| $79$ |
\( (T^{2} + 9408)^{2} \)
|
| $83$ |
\( T^{4} + 15200 T^{2} + 45373696 \)
|
| $89$ |
\( (T^{2} + 76 T - 1580)^{2} \)
|
| $97$ |
\( (T^{2} - 44 T - 4892)^{2} \)
|
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