Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [48,28,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, 28, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("48.1");
S:= CuspForms(chi, 28);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 48 = 2^{4} \cdot 3 \) |
| Weight: | \( k \) | \(=\) | \( 28 \) |
| 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: | \(221.690675922\) |
| 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: |
\( x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{42}\cdot 3^{8}\cdot 5\cdot 7 \) |
| 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
\( x^{4} - 2x^{3} - 360714331909x^{2} - 43287560841177118x + 8819337660421091919513 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 45950336 \nu^{3} - 24972080483328 \nu^{2} + \cdots + 30\!\cdots\!28 ) / 784193203840095 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 39230848 \nu^{3} + 75313256469504 \nu^{2} + \cdots - 14\!\cdots\!80 ) / 156838640768019 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 3643222147456 \nu^{3} + \cdots - 70\!\cdots\!92 ) / 784193203840095 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} - 3549\beta_{2} - 64136\beta _1 + 1486356480 ) / 2972712960 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 147449\beta_{3} - 40817973\beta_{2} - 11516408584\beta _1 + 53615008466478563328 ) / 297271296 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 1036344307171 \beta_{3} + \cdots + 96\!\cdots\!40 ) / 2972712960 \)
|
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 | −1.59432e6 | 0 | −2.94911e9 | 0 | 2.52860e11 | 0 | 2.54187e12 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −1.59432e6 | 0 | −8.07135e8 | 0 | −1.51306e11 | 0 | 2.54187e12 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | −1.59432e6 | 0 | 3.92930e9 | 0 | −4.56279e11 | 0 | 2.54187e12 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | −1.59432e6 | 0 | 4.77599e9 | 0 | 4.42974e11 | 0 | 2.54187e12 | 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 | 48.28.a.l | 4 | |
| 4.b | odd | 2 | 1 | 24.28.a.d | ✓ | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 24.28.a.d | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 48.28.a.l | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} - 4949050424 T_{5}^{3} + \cdots + 44\!\cdots\!00 \)
acting on \(S_{28}^{\mathrm{new}}(\Gamma_0(48))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T + 1594323)^{4} \)
|
| $5$ |
\( T^{4} + \cdots + 44\!\cdots\!00 \)
|
| $7$ |
\( T^{4} + \cdots + 77\!\cdots\!80 \)
|
| $11$ |
\( T^{4} + \cdots + 41\!\cdots\!48 \)
|
| $13$ |
\( T^{4} + \cdots + 97\!\cdots\!48 \)
|
| $17$ |
\( T^{4} + \cdots + 28\!\cdots\!44 \)
|
| $19$ |
\( T^{4} + \cdots + 16\!\cdots\!32 \)
|
| $23$ |
\( T^{4} + \cdots + 48\!\cdots\!60 \)
|
| $29$ |
\( T^{4} + \cdots - 99\!\cdots\!84 \)
|
| $31$ |
\( T^{4} + \cdots + 10\!\cdots\!00 \)
|
| $37$ |
\( T^{4} + \cdots - 30\!\cdots\!00 \)
|
| $41$ |
\( T^{4} + \cdots + 26\!\cdots\!92 \)
|
| $43$ |
\( T^{4} + \cdots - 42\!\cdots\!76 \)
|
| $47$ |
\( T^{4} + \cdots + 88\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 58\!\cdots\!00 \)
|
| $59$ |
\( T^{4} + \cdots - 27\!\cdots\!52 \)
|
| $61$ |
\( T^{4} + \cdots + 11\!\cdots\!92 \)
|
| $67$ |
\( T^{4} + \cdots + 40\!\cdots\!52 \)
|
| $71$ |
\( T^{4} + \cdots + 59\!\cdots\!48 \)
|
| $73$ |
\( T^{4} + \cdots - 70\!\cdots\!64 \)
|
| $79$ |
\( T^{4} + \cdots - 66\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots + 61\!\cdots\!52 \)
|
| $89$ |
\( T^{4} + \cdots + 31\!\cdots\!60 \)
|
| $97$ |
\( T^{4} + \cdots + 93\!\cdots\!24 \)
|
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