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: | \(3\) |
| Coefficient field: | 3.3.11853.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 30x - 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 280) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 30x - 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + \nu - 20 ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 11\nu + 20 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{2} + 11\beta _1 + 80 ) / 4 \)
|
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 | −8.72637 | 0 | −5.00000 | 0 | 7.00000 | 0 | 49.1496 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −0.887801 | 0 | −5.00000 | 0 | 7.00000 | 0 | −26.2118 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | 3.61417 | 0 | −5.00000 | 0 | 7.00000 | 0 | −13.9378 | 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.bq | 3 | |
| 4.b | odd | 2 | 1 | 2240.4.a.by | 3 | ||
| 8.b | even | 2 | 1 | 560.4.a.x | 3 | ||
| 8.d | odd | 2 | 1 | 280.4.a.g | ✓ | 3 | |
| 40.e | odd | 2 | 1 | 1400.4.a.n | 3 | ||
| 40.k | even | 4 | 2 | 1400.4.g.m | 6 | ||
| 56.e | even | 2 | 1 | 1960.4.a.q | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 280.4.a.g | ✓ | 3 | 8.d | odd | 2 | 1 | |
| 560.4.a.x | 3 | 8.b | even | 2 | 1 | ||
| 1400.4.a.n | 3 | 40.e | odd | 2 | 1 | ||
| 1400.4.g.m | 6 | 40.k | even | 4 | 2 | ||
| 1960.4.a.q | 3 | 56.e | even | 2 | 1 | ||
| 2240.4.a.bq | 3 | 1.a | even | 1 | 1 | trivial | |
| 2240.4.a.by | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2240))\):
|
\( T_{3}^{3} + 6T_{3}^{2} - 27T_{3} - 28 \)
|
|
\( T_{11}^{3} + 54T_{11}^{2} - 1023T_{11} - 2420 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( T^{3} + 6 T^{2} + \cdots - 28 \)
|
| $5$ |
\( (T + 5)^{3} \)
|
| $7$ |
\( (T - 7)^{3} \)
|
| $11$ |
\( T^{3} + 54 T^{2} + \cdots - 2420 \)
|
| $13$ |
\( T^{3} - 24 T^{2} + \cdots + 79518 \)
|
| $17$ |
\( T^{3} - 48 T^{2} + \cdots + 959126 \)
|
| $19$ |
\( T^{3} + 132 T^{2} + \cdots - 325648 \)
|
| $23$ |
\( T^{3} - 144 T^{2} + \cdots + 866464 \)
|
| $29$ |
\( T^{3} - 12 T^{2} + \cdots + 282534 \)
|
| $31$ |
\( T^{3} - 180 T^{2} + \cdots + 286720 \)
|
| $37$ |
\( T^{3} - 402 T^{2} + \cdots + 4841128 \)
|
| $41$ |
\( T^{3} + 66 T^{2} + \cdots + 11863712 \)
|
| $43$ |
\( T^{3} + 684 T^{2} + \cdots - 71835312 \)
|
| $47$ |
\( T^{3} - 1074 T^{2} + \cdots - 22845312 \)
|
| $53$ |
\( T^{3} + 126 T^{2} + \cdots + 16752512 \)
|
| $59$ |
\( T^{3} + 36 T^{2} + \cdots + 5320384 \)
|
| $61$ |
\( T^{3} - 750 T^{2} + \cdots + 5580192 \)
|
| $67$ |
\( T^{3} + 1236 T^{2} + \cdots + 6943744 \)
|
| $71$ |
\( T^{3} - 696 T^{2} + \cdots - 26919424 \)
|
| $73$ |
\( T^{3} - 654 T^{2} + \cdots + 2941560 \)
|
| $79$ |
\( T^{3} + 762 T^{2} + \cdots - 417847024 \)
|
| $83$ |
\( T^{3} - 444 T^{2} + \cdots + 7436992 \)
|
| $89$ |
\( T^{3} - 1782 T^{2} + \cdots + 836292576 \)
|
| $97$ |
\( T^{3} - 1560 T^{2} + \cdots + 3281334 \)
|
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