Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2240,4,Mod(1,2240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2240, 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("2240.1");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2240 = 2^{6} \cdot 5 \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2240.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: | \(132.164278413\) |
| 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: |
\( x^{4} - 2x^{3} - 68x^{2} - 168x - 99 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | no (minimal twist has level 1120) |
| 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
\( x^{4} - 2x^{3} - 68x^{2} - 168x - 99 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 5\nu^{2} - 53\nu - 27 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{3} - 23\nu^{2} - 443\nu - 585 ) / 6 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 7\beta_{2} + 3\beta _1 + 72 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{3} - 46\beta_{2} + 83\beta _1 + 934 ) / 4 \)
|
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 | −5.92058 | 0 | −5.00000 | 0 | 7.00000 | 0 | 8.05323 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.0969232 | 0 | −5.00000 | 0 | 7.00000 | 0 | −26.9906 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 4.75423 | 0 | −5.00000 | 0 | 7.00000 | 0 | −4.39727 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 10.2633 | 0 | −5.00000 | 0 | 7.00000 | 0 | 78.3346 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(5\) | \(1\) |
| \(7\) | \(-1\) |
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 | 2240.4.a.cl | 4 | |
| 4.b | odd | 2 | 1 | 2240.4.a.ca | 4 | ||
| 8.b | even | 2 | 1 | 1120.4.a.e | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 1120.4.a.p | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1120.4.a.e | ✓ | 4 | 8.b | even | 2 | 1 | |
| 1120.4.a.p | yes | 4 | 8.d | odd | 2 | 1 | |
| 2240.4.a.ca | 4 | 4.b | odd | 2 | 1 | ||
| 2240.4.a.cl | 4 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):
|
\( T_{3}^{4} - 9T_{3}^{3} - 41T_{3}^{2} + 285T_{3} + 28 \)
|
|
\( T_{11}^{4} - 33T_{11}^{3} - 1769T_{11}^{2} + 45525T_{11} + 255388 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 9 T^{3} + \cdots + 28 \)
|
| $5$ |
\( (T + 5)^{4} \)
|
| $7$ |
\( (T - 7)^{4} \)
|
| $11$ |
\( T^{4} - 33 T^{3} + \cdots + 255388 \)
|
| $13$ |
\( T^{4} - 83 T^{3} + \cdots - 811146 \)
|
| $17$ |
\( T^{4} + 13 T^{3} + \cdots + 4921962 \)
|
| $19$ |
\( T^{4} - 132 T^{3} + \cdots - 5605632 \)
|
| $23$ |
\( T^{4} + 92 T^{3} + \cdots - 2700800 \)
|
| $29$ |
\( T^{4} - 113 T^{3} + \cdots + 337345458 \)
|
| $31$ |
\( T^{4} + 94 T^{3} + \cdots - 130638144 \)
|
| $37$ |
\( T^{4} - 54 T^{3} + \cdots + 12778432 \)
|
| $41$ |
\( T^{4} + \cdots - 2804834384 \)
|
| $43$ |
\( T^{4} + \cdots - 3522396800 \)
|
| $47$ |
\( T^{4} + \cdots - 1181540696 \)
|
| $53$ |
\( T^{4} + 554 T^{3} + \cdots + 69798912 \)
|
| $59$ |
\( T^{4} + \cdots + 2209334784 \)
|
| $61$ |
\( T^{4} + \cdots + 10917583408 \)
|
| $67$ |
\( T^{4} - 484 T^{3} + \cdots - 461318144 \)
|
| $71$ |
\( T^{4} + \cdots + 29925186816 \)
|
| $73$ |
\( T^{4} + \cdots - 155433416208 \)
|
| $79$ |
\( T^{4} + \cdots + 1424762644 \)
|
| $83$ |
\( T^{4} + \cdots + 39267818496 \)
|
| $89$ |
\( T^{4} + \cdots - 104478749424 \)
|
| $97$ |
\( T^{4} + \cdots - 98500524374 \)
|
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