Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,26,Mod(1,48)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(48, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0]))
N = Newforms(chi, 26, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.1");
S:= CuspForms(chi, 26);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 26 \) |
| Character orbit: | \([\chi]\) | \(=\) | 48.a (trivial) |
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: | yes |
| Analytic conductor: | \(190.078454377\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 1647414391x^{2} - 16173027965094x + 235708219135253151 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{42}\cdot 3^{8}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 24) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 1647414391x^{2} - 16173027965094x + 235708219135253151 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 1207296\nu^{3} - 16208524416\nu^{2} - 1200570932736000\nu - 1293145787712959040 ) / 15788235953 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -738524160\nu^{3} + 18800802781824\nu^{2} + 736596216180486144\nu - 6528227613131678661312 ) / 15788235953 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 1183942656\nu^{3} - 33122495920896\nu^{2} - 1067557716188651520\nu + 12922284959544684769920 ) / 2255462279 \)
|
| \(\nu\) | \(=\) |
\( ( 7\beta_{3} + 95\beta_{2} + 10061\beta_1 ) / 353894400 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -287\beta_{3} + 100905\beta_{2} + 63695499\beta _1 + 48584227287859200 ) / 58982400 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 27751555377\beta_{3} + 410396010745\beta_{2} + 79055253221371\beta _1 + 17170632083670486220800 ) / 1415577600 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
0 | 531441. | 0 | −6.59716e8 | 0 | 5.77845e9 | 0 | 2.82430e11 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 531441. | 0 | −1.16924e8 | 0 | 3.09827e10 | 0 | 2.82430e11 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 531441. | 0 | 7.35582e6 | 0 | −7.04630e10 | 0 | 2.82430e11 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 531441. | 0 | 1.03572e9 | 0 | 3.32173e10 | 0 | 2.82430e11 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(3\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 48.26.a.k | 4 | |
| 4.b | odd | 2 | 1 | 24.26.a.d | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.26.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 48.26.a.k | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 266436088 T_{5}^{3} + \cdots + 58\!\cdots\!00 \)
acting on \(S_{26}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 531441)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots - 41\!\cdots\!00 \)
|
| $11$ |
\( T^{4} + \cdots + 25\!\cdots\!40 \)
|
| $13$ |
\( T^{4} + \cdots - 89\!\cdots\!88 \)
|
| $17$ |
\( T^{4} + \cdots + 21\!\cdots\!76 \)
|
| $19$ |
\( T^{4} + \cdots + 80\!\cdots\!64 \)
|
| $23$ |
\( T^{4} + \cdots + 35\!\cdots\!64 \)
|
| $29$ |
\( T^{4} + \cdots - 20\!\cdots\!36 \)
|
| $31$ |
\( T^{4} + \cdots + 71\!\cdots\!60 \)
|
| $37$ |
\( T^{4} + \cdots - 46\!\cdots\!04 \)
|
| $41$ |
\( T^{4} + \cdots - 73\!\cdots\!60 \)
|
| $43$ |
\( T^{4} + \cdots - 18\!\cdots\!24 \)
|
| $47$ |
\( T^{4} + \cdots + 16\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots - 37\!\cdots\!40 \)
|
| $59$ |
\( T^{4} + \cdots - 51\!\cdots\!40 \)
|
| $61$ |
\( T^{4} + \cdots - 11\!\cdots\!88 \)
|
| $67$ |
\( T^{4} + \cdots - 27\!\cdots\!96 \)
|
| $71$ |
\( T^{4} + \cdots - 27\!\cdots\!40 \)
|
| $73$ |
\( T^{4} + \cdots + 18\!\cdots\!00 \)
|
| $79$ |
\( T^{4} + \cdots - 59\!\cdots\!72 \)
|
| $83$ |
\( T^{4} + \cdots + 19\!\cdots\!28 \)
|
| $89$ |
\( T^{4} + \cdots - 47\!\cdots\!72 \)
|
| $97$ |
\( T^{4} + \cdots - 93\!\cdots\!36 \)
|
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