Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1104,4,Mod(1,1104)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1104, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1104.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1104 = 2^{4} \cdot 3 \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1104.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: | \(65.1381086463\) |
| Analytic rank: | \(1\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.1140200.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 20x^{2} + 25x + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 69) |
| 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
\( x^{4} - x^{3} - 20x^{2} + 25x + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 10\nu - 35 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -2\nu^{3} + 2\nu^{2} + 30\nu - 35 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} - \beta _1 + 21 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{3} + \beta_{2} + 7\beta _1 - 7 \)
|
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 | 3.00000 | 0 | −19.5058 | 0 | 20.6068 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | 3.00000 | 0 | −6.33125 | 0 | −17.5951 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.00000 | 0 | 2.99280 | 0 | −20.4098 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 3.00000 | 0 | 20.8442 | 0 | −14.6019 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( -1 \) |
| \(23\) | \( -1 \) |
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 | 1104.4.a.v | 4 | |
| 4.b | odd | 2 | 1 | 69.4.a.c | ✓ | 4 | |
| 12.b | even | 2 | 1 | 207.4.a.f | 4 | ||
| 20.d | odd | 2 | 1 | 1725.4.a.s | 4 | ||
| 92.b | even | 2 | 1 | 1587.4.a.e | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 69.4.a.c | ✓ | 4 | 4.b | odd | 2 | 1 | |
| 207.4.a.f | 4 | 12.b | even | 2 | 1 | ||
| 1104.4.a.v | 4 | 1.a | even | 1 | 1 | trivial | |
| 1587.4.a.e | 4 | 92.b | even | 2 | 1 | ||
| 1725.4.a.s | 4 | 20.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1104))\):
|
\( T_{5}^{4} + 2T_{5}^{3} - 430T_{5}^{2} - 1332T_{5} + 7704 \)
|
|
\( T_{7}^{4} + 32T_{7}^{3} - 170T_{7}^{2} - 13592T_{7} - 108056 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T - 3)^{4} \)
|
| $5$ |
\( T^{4} + 2 T^{3} + \cdots + 7704 \)
|
| $7$ |
\( T^{4} + 32 T^{3} + \cdots - 108056 \)
|
| $11$ |
\( T^{4} + 62 T^{3} + \cdots + 1835520 \)
|
| $13$ |
\( T^{4} - 32 T^{3} + \cdots - 5184 \)
|
| $17$ |
\( T^{4} + 28 T^{3} + \cdots + 3442464 \)
|
| $19$ |
\( T^{4} + 42 T^{3} + \cdots - 3033320 \)
|
| $23$ |
\( (T - 23)^{4} \)
|
| $29$ |
\( T^{4} + 20 T^{3} + \cdots + 329043600 \)
|
| $31$ |
\( T^{4} + 444 T^{3} + \cdots - 13145472 \)
|
| $37$ |
\( T^{4} + \cdots + 1193345264 \)
|
| $41$ |
\( T^{4} + \cdots + 1853697360 \)
|
| $43$ |
\( T^{4} + \cdots - 7030549368 \)
|
| $47$ |
\( T^{4} + 36 T^{3} + \cdots + 431257536 \)
|
| $53$ |
\( T^{4} - 358 T^{3} + \cdots + 25456680 \)
|
| $59$ |
\( T^{4} + \cdots - 14547689280 \)
|
| $61$ |
\( T^{4} - 294 T^{3} + \cdots - 56561712 \)
|
| $67$ |
\( T^{4} + \cdots + 26618219240 \)
|
| $71$ |
\( T^{4} + \cdots - 16923156480 \)
|
| $73$ |
\( T^{4} + \cdots + 5089871152 \)
|
| $79$ |
\( T^{4} + \cdots + 6874959400 \)
|
| $83$ |
\( T^{4} + \cdots + 15308700480 \)
|
| $89$ |
\( T^{4} + \cdots - 1279749600 \)
|
| $97$ |
\( T^{4} + \cdots - 285081598064 \)
|
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