Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [144,13,Mod(127,144)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(144, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0]))
N = Newforms(chi, 13, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("144.127");
S:= CuspForms(chi, 13);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 144 = 2^{4} \cdot 3^{2} \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 144.g (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: | \(131.615109688\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-2803})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 700x^{2} - 701x + 491401 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{12}\cdot 3^{12} \) |
| Twist minimal: | no (minimal twist has level 48) |
| 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
\( x^{4} - x^{3} - 700x^{2} - 701x + 491401 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -1458\nu^{3} - 1020600\nu^{2} + 1020600\nu + 358742358 ) / 122675 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -432\nu^{3} + 432\nu^{2} + 605232\nu + 151416 ) / 701 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 76086\nu^{3} + 264600\nu^{2} - 264600\nu - 172576386 ) / 122675 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} + 3\beta_{2} + \beta _1 + 648 ) / 2592 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -27\beta_{3} + 27\beta_{2} - 2809\beta _1 + 8170632 ) / 23328 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 4725\beta_{3} + 1225\beta _1 + 3064716 ) / 2916 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/144\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(65\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 127.1 |
|
0 | 0 | 0 | −25405.3 | 0 | − | 30429.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||
| 127.2 | 0 | 0 | 0 | −25405.3 | 0 | 30429.0i | 0 | 0 | 0 | |||||||||||||||||||||||||||||||
| 127.3 | 0 | 0 | 0 | 14209.3 | 0 | − | 83928.6i | 0 | 0 | 0 | ||||||||||||||||||||||||||||||
| 127.4 | 0 | 0 | 0 | 14209.3 | 0 | 83928.6i | 0 | 0 | 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 | 144.13.g.f | 4 | |
| 3.b | odd | 2 | 1 | 48.13.g.b | ✓ | 4 | |
| 4.b | odd | 2 | 1 | inner | 144.13.g.f | 4 | |
| 12.b | even | 2 | 1 | 48.13.g.b | ✓ | 4 | |
| 24.f | even | 2 | 1 | 192.13.g.b | 4 | ||
| 24.h | odd | 2 | 1 | 192.13.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 48.13.g.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 48.13.g.b | ✓ | 4 | 12.b | even | 2 | 1 | |
| 144.13.g.f | 4 | 1.a | even | 1 | 1 | trivial | |
| 144.13.g.f | 4 | 4.b | odd | 2 | 1 | inner | |
| 192.13.g.b | 4 | 24.f | even | 2 | 1 | ||
| 192.13.g.b | 4 | 24.h | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} + 11196T_{5} - 360992700 \)
acting on \(S_{13}^{\mathrm{new}}(144, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 11196 T - 360992700)^{2} \)
|
| $7$ |
\( T^{4} + \cdots + 65\!\cdots\!64 \)
|
| $11$ |
\( T^{4} + \cdots + 16\!\cdots\!84 \)
|
| $13$ |
\( (T^{2} + \cdots + 4830142701604)^{2} \)
|
| $17$ |
\( (T^{2} + \cdots + 316516185520164)^{2} \)
|
| $19$ |
\( T^{4} + \cdots + 51\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 87\!\cdots\!64 \)
|
| $29$ |
\( (T^{2} + \cdots - 10\!\cdots\!16)^{2} \)
|
| $31$ |
\( T^{4} + \cdots + 57\!\cdots\!00 \)
|
| $37$ |
\( (T^{2} + \cdots - 14\!\cdots\!00)^{2} \)
|
| $41$ |
\( (T^{2} + \cdots - 54\!\cdots\!00)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 38\!\cdots\!04 \)
|
| $47$ |
\( T^{4} + \cdots + 64\!\cdots\!44 \)
|
| $53$ |
\( (T^{2} + \cdots + 59\!\cdots\!36)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 10\!\cdots\!24 \)
|
| $61$ |
\( (T^{2} + \cdots - 13\!\cdots\!84)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 22\!\cdots\!04 \)
|
| $71$ |
\( T^{4} + \cdots + 31\!\cdots\!44 \)
|
| $73$ |
\( (T^{2} + \cdots - 12\!\cdots\!76)^{2} \)
|
| $79$ |
\( T^{4} + \cdots + 59\!\cdots\!64 \)
|
| $83$ |
\( T^{4} + \cdots + 99\!\cdots\!24 \)
|
| $89$ |
\( (T^{2} + \cdots + 23\!\cdots\!76)^{2} \)
|
| $97$ |
\( (T^{2} + \cdots - 12\!\cdots\!04)^{2} \)
|
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