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: |
\( x^{4} - 2x^{3} - 40x^{2} - 6x + 72 \)
|
| 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
\( x^{4} - 2x^{3} - 40x^{2} - 6x + 72 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 40\nu - 6 ) / 6 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 16\nu - 18 ) / 6 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 8\nu^{2} + 16\nu - 108 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta _1 + 2 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 42 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 12\beta_{2} - 4\beta _1 + 68 \)
|
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 | −8.56201 | 0 | 5.00000 | 0 | −7.00000 | 0 | 46.3080 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | 0 | −0.629788 | 0 | 5.00000 | 0 | −7.00000 | 0 | −26.6034 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | 0 | 3.35953 | 0 | 5.00000 | 0 | −7.00000 | 0 | −15.7136 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | 0 | 8.83227 | 0 | 5.00000 | 0 | −7.00000 | 0 | 51.0090 | 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.ch | 4 | |
| 4.b | odd | 2 | 1 | 2240.4.a.ce | 4 | ||
| 8.b | even | 2 | 1 | 1120.4.a.i | ✓ | 4 | |
| 8.d | odd | 2 | 1 | 1120.4.a.l | yes | 4 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1120.4.a.i | ✓ | 4 | 8.b | even | 2 | 1 | |
| 1120.4.a.l | yes | 4 | 8.d | odd | 2 | 1 | |
| 2240.4.a.ce | 4 | 4.b | odd | 2 | 1 | ||
| 2240.4.a.ch | 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 \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):
|
\( T_{3}^{4} - 3T_{3}^{3} - 77T_{3}^{2} + 207T_{3} + 160 \)
|
|
\( T_{11}^{4} - 77T_{11}^{3} - 617T_{11}^{2} + 64617T_{11} + 804420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 160 \)
|
| $5$ |
\( (T - 5)^{4} \)
|
| $7$ |
\( (T + 7)^{4} \)
|
| $11$ |
\( T^{4} - 77 T^{3} + \cdots + 804420 \)
|
| $13$ |
\( T^{4} + 103 T^{3} + \cdots + 95254 \)
|
| $17$ |
\( T^{4} - 101 T^{3} + \cdots - 1814762 \)
|
| $19$ |
\( T^{4} - 96 T^{3} + \cdots - 5567616 \)
|
| $23$ |
\( T^{4} - 12928 T^{2} + \cdots + 9997568 \)
|
| $29$ |
\( T^{4} - 105 T^{3} + \cdots - 250382326 \)
|
| $31$ |
\( T^{4} - 218 T^{3} + \cdots - 54144640 \)
|
| $37$ |
\( T^{4} + \cdots + 9565696064 \)
|
| $41$ |
\( T^{4} - 428 T^{3} + \cdots - 978820048 \)
|
| $43$ |
\( T^{4} + \cdots + 3788745856 \)
|
| $47$ |
\( T^{4} + \cdots - 10377464464 \)
|
| $53$ |
\( T^{4} + 342 T^{3} + \cdots - 165640064 \)
|
| $59$ |
\( T^{4} + \cdots - 245828212736 \)
|
| $61$ |
\( T^{4} - 60 T^{3} + \cdots + 281659760 \)
|
| $67$ |
\( T^{4} + \cdots + 228730988288 \)
|
| $71$ |
\( T^{4} + \cdots + 30595472640 \)
|
| $73$ |
\( T^{4} + \cdots + 2631346320 \)
|
| $79$ |
\( T^{4} + \cdots - 7900808020 \)
|
| $83$ |
\( T^{4} + \cdots + 83579334656 \)
|
| $89$ |
\( T^{4} + \cdots - 108379876560 \)
|
| $97$ |
\( T^{4} + \cdots + 1395955346374 \)
|
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