Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [46,3,Mod(45,46)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(46, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("46.45");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 46 = 2 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 46.b (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: | \(1.25340921606\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.613376.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 58x^{2} + 599 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{4} + 58x^{2} + 599 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 29\nu ) / 11 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + 29 ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 51\nu ) / 11 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( 11\beta_{2} - 29 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -29\beta_{3} + 51\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/46\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 45.1 |
|
−1.41421 | −2.41421 | 2.00000 | − | 9.43995i | 3.41421 | − | 3.91016i | −2.82843 | −3.17157 | 13.3501i | ||||||||||||||||||||||||||||
| 45.2 | −1.41421 | −2.41421 | 2.00000 | 9.43995i | 3.41421 | 3.91016i | −2.82843 | −3.17157 | − | 13.3501i | ||||||||||||||||||||||||||||||
| 45.3 | 1.41421 | 0.414214 | 2.00000 | − | 5.18530i | 0.585786 | 12.5184i | 2.82843 | −8.82843 | − | 7.33312i | |||||||||||||||||||||||||||||
| 45.4 | 1.41421 | 0.414214 | 2.00000 | 5.18530i | 0.585786 | − | 12.5184i | 2.82843 | −8.82843 | 7.33312i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 46.3.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 414.3.b.a | 4 | ||
| 4.b | odd | 2 | 1 | 368.3.f.c | 4 | ||
| 5.b | even | 2 | 1 | 1150.3.d.a | 4 | ||
| 5.c | odd | 4 | 2 | 1150.3.c.a | 8 | ||
| 8.b | even | 2 | 1 | 1472.3.f.f | 4 | ||
| 8.d | odd | 2 | 1 | 1472.3.f.c | 4 | ||
| 23.b | odd | 2 | 1 | inner | 46.3.b.a | ✓ | 4 |
| 69.c | even | 2 | 1 | 414.3.b.a | 4 | ||
| 92.b | even | 2 | 1 | 368.3.f.c | 4 | ||
| 115.c | odd | 2 | 1 | 1150.3.d.a | 4 | ||
| 115.e | even | 4 | 2 | 1150.3.c.a | 8 | ||
| 184.e | odd | 2 | 1 | 1472.3.f.f | 4 | ||
| 184.h | even | 2 | 1 | 1472.3.f.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 46.3.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 46.3.b.a | ✓ | 4 | 23.b | odd | 2 | 1 | inner |
| 368.3.f.c | 4 | 4.b | odd | 2 | 1 | ||
| 368.3.f.c | 4 | 92.b | even | 2 | 1 | ||
| 414.3.b.a | 4 | 3.b | odd | 2 | 1 | ||
| 414.3.b.a | 4 | 69.c | even | 2 | 1 | ||
| 1150.3.c.a | 8 | 5.c | odd | 4 | 2 | ||
| 1150.3.c.a | 8 | 115.e | even | 4 | 2 | ||
| 1150.3.d.a | 4 | 5.b | even | 2 | 1 | ||
| 1150.3.d.a | 4 | 115.c | odd | 2 | 1 | ||
| 1472.3.f.c | 4 | 8.d | odd | 2 | 1 | ||
| 1472.3.f.c | 4 | 184.h | even | 2 | 1 | ||
| 1472.3.f.f | 4 | 8.b | even | 2 | 1 | ||
| 1472.3.f.f | 4 | 184.e | odd | 2 | 1 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{3}^{\mathrm{new}}(46, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( (T^{2} + 2 T - 1)^{2} \)
|
| $5$ |
\( T^{4} + 116T^{2} + 2396 \)
|
| $7$ |
\( T^{4} + 172T^{2} + 2396 \)
|
| $11$ |
\( T^{4} + 172T^{2} + 2396 \)
|
| $13$ |
\( (T^{2} + 2 T - 127)^{2} \)
|
| $17$ |
\( T^{4} + 692 T^{2} + 117404 \)
|
| $19$ |
\( T^{4} + 928 T^{2} + 153344 \)
|
| $23$ |
\( T^{4} - 52 T^{3} + \cdots + 279841 \)
|
| $29$ |
\( (T^{2} - 42 T + 241)^{2} \)
|
| $31$ |
\( (T^{2} - 58 T + 823)^{2} \)
|
| $37$ |
\( T^{4} + 4148 T^{2} + 4027676 \)
|
| $41$ |
\( (T^{2} - 10 T - 47)^{2} \)
|
| $43$ |
\( T^{4} + 2992 T^{2} + 1878464 \)
|
| $47$ |
\( (T^{2} + 14 T - 529)^{2} \)
|
| $53$ |
\( T^{4} + 6704 T^{2} + 11079104 \)
|
| $59$ |
\( (T^{2} + 92 T + 1724)^{2} \)
|
| $61$ |
\( T^{4} + 2752 T^{2} + 613376 \)
|
| $67$ |
\( T^{4} + 4716 T^{2} + 194076 \)
|
| $71$ |
\( (T^{2} + 50 T - 6817)^{2} \)
|
| $73$ |
\( (T^{2} + 74 T - 5831)^{2} \)
|
| $79$ |
\( T^{4} + 14432 T^{2} + 7513856 \)
|
| $83$ |
\( T^{4} + 18700 T^{2} + 73377500 \)
|
| $89$ |
\( T^{4} + 17200 T^{2} + 23960000 \)
|
| $97$ |
\( T^{4} + 5684 T^{2} + 5752796 \)
|
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