Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [3,15,Mod(2,3)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(3, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 15, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("3.2");
S:= CuspForms(chi, 15);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 3 \) |
| Weight: | \( k \) | \(=\) | \( 15 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3.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: | \(3.72986904456\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.1929141760.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 364x^{2} + 3640 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{10}\cdot 3^{7} \) |
| 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
\( x^{4} + 364x^{2} + 3640 \)
:
| \(\beta_{1}\) | \(=\) |
\( 12\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( 2\nu^{3} + 8\nu^{2} + 688\nu + 1456 \)
|
| \(\beta_{3}\) | \(=\) |
\( -16\nu^{3} + 80\nu^{2} - 5504\nu + 14560 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 8\beta_{2} - 26208 ) / 144 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -\beta_{3} + 10\beta_{2} - 1032\beta_1 ) / 36 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 2.1 |
|
− | 225.686i | 1922.67 | − | 1042.26i | −34550.1 | 23159.5i | −235223. | − | 433920.i | 478389. | 4.09983e6i | 2.61037e6 | − | 4.00784e6i | 5.22678e6 | |||||||||||||||||||||||
| 2.2 | − | 38.4954i | −824.672 | + | 2025.56i | 14902.1 | 122931.i | 77974.7 | + | 31746.1i | −65585.1 | − | 1.20437e6i | −3.42280e6 | − | 3.34084e6i | 4.73226e6 | |||||||||||||||||||||||
| 2.3 | 38.4954i | −824.672 | − | 2025.56i | 14902.1 | − | 122931.i | 77974.7 | − | 31746.1i | −65585.1 | 1.20437e6i | −3.42280e6 | + | 3.34084e6i | 4.73226e6 | ||||||||||||||||||||||||
| 2.4 | 225.686i | 1922.67 | + | 1042.26i | −34550.1 | − | 23159.5i | −235223. | + | 433920.i | 478389. | − | 4.09983e6i | 2.61037e6 | + | 4.00784e6i | 5.22678e6 | |||||||||||||||||||||||
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 | 3.15.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 3.15.b.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 48.15.e.b | 4 | ||
| 5.b | even | 2 | 1 | 75.15.c.d | 4 | ||
| 5.c | odd | 4 | 2 | 75.15.d.b | 8 | ||
| 12.b | even | 2 | 1 | 48.15.e.b | 4 | ||
| 15.d | odd | 2 | 1 | 75.15.c.d | 4 | ||
| 15.e | even | 4 | 2 | 75.15.d.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3.15.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 3.15.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 48.15.e.b | 4 | 4.b | odd | 2 | 1 | ||
| 48.15.e.b | 4 | 12.b | even | 2 | 1 | ||
| 75.15.c.d | 4 | 5.b | even | 2 | 1 | ||
| 75.15.c.d | 4 | 15.d | odd | 2 | 1 | ||
| 75.15.d.b | 8 | 5.c | odd | 4 | 2 | ||
| 75.15.d.b | 8 | 15.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace is the entire newspace \(S_{15}^{\mathrm{new}}(3, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 52416 T^{2} + 75479040 \)
|
| $3$ |
\( T^{4} + \cdots + 22876792454961 \)
|
| $5$ |
\( T^{4} + \cdots + 81\!\cdots\!00 \)
|
| $7$ |
\( (T^{2} - 412804 T - 31375221500)^{2} \)
|
| $11$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $13$ |
\( (T^{2} + \cdots + 22\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{4} + \cdots + 43\!\cdots\!60 \)
|
| $19$ |
\( (T^{2} + \cdots + 10\!\cdots\!96)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 14\!\cdots\!40 \)
|
| $29$ |
\( T^{4} + \cdots + 32\!\cdots\!00 \)
|
| $31$ |
\( (T^{2} + \cdots - 45\!\cdots\!76)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots - 31\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{4} + \cdots + 12\!\cdots\!00 \)
|
| $43$ |
\( (T^{2} + \cdots - 26\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!60 \)
|
| $53$ |
\( T^{4} + \cdots + 17\!\cdots\!60 \)
|
| $59$ |
\( T^{4} + \cdots + 34\!\cdots\!00 \)
|
| $61$ |
\( (T^{2} + \cdots - 90\!\cdots\!96)^{2} \)
|
| $67$ |
\( (T^{2} + \cdots + 73\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{4} + \cdots + 40\!\cdots\!00 \)
|
| $73$ |
\( (T^{2} + \cdots + 12\!\cdots\!00)^{2} \)
|
| $79$ |
\( (T^{2} + \cdots - 37\!\cdots\!24)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 44\!\cdots\!60 \)
|
| $89$ |
\( T^{4} + \cdots + 81\!\cdots\!00 \)
|
| $97$ |
\( (T^{2} + \cdots - 46\!\cdots\!00)^{2} \)
|
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