Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [240,3,Mod(31,240)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(240, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 0, 0]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("240.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 240 = 2^{4} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 240.e (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(6.53952634465\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - x^{3} + 2x^{2} + x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 2\nu^{2} - 2\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{3} - 2\nu^{2} + 6\nu + 1 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} - \beta _1 + 1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} + 3\beta_{2} + \beta _1 - 3 ) / 4 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta _1 - 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(97\) | \(161\) | \(181\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0 | − | 1.73205i | 0 | −2.23607 | 0 | 11.2101i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 31.2 | 0 | − | 1.73205i | 0 | 2.23607 | 0 | − | 4.28187i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||
| 31.3 | 0 | 1.73205i | 0 | −2.23607 | 0 | − | 11.2101i | 0 | −3.00000 | 0 | ||||||||||||||||||||||||||||||
| 31.4 | 0 | 1.73205i | 0 | 2.23607 | 0 | 4.28187i | 0 | −3.00000 | 0 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 4.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 240.3.e.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 720.3.e.a | 4 | ||
| 4.b | odd | 2 | 1 | inner | 240.3.e.a | ✓ | 4 |
| 5.b | even | 2 | 1 | 1200.3.e.n | 4 | ||
| 5.c | odd | 4 | 2 | 1200.3.j.f | 8 | ||
| 8.b | even | 2 | 1 | 960.3.e.b | 4 | ||
| 8.d | odd | 2 | 1 | 960.3.e.b | 4 | ||
| 12.b | even | 2 | 1 | 720.3.e.a | 4 | ||
| 15.d | odd | 2 | 1 | 3600.3.e.bh | 4 | ||
| 15.e | even | 4 | 2 | 3600.3.j.l | 8 | ||
| 20.d | odd | 2 | 1 | 1200.3.e.n | 4 | ||
| 20.e | even | 4 | 2 | 1200.3.j.f | 8 | ||
| 24.f | even | 2 | 1 | 2880.3.e.f | 4 | ||
| 24.h | odd | 2 | 1 | 2880.3.e.f | 4 | ||
| 60.h | even | 2 | 1 | 3600.3.e.bh | 4 | ||
| 60.l | odd | 4 | 2 | 3600.3.j.l | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 240.3.e.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 240.3.e.a | ✓ | 4 | 4.b | odd | 2 | 1 | inner |
| 720.3.e.a | 4 | 3.b | odd | 2 | 1 | ||
| 720.3.e.a | 4 | 12.b | even | 2 | 1 | ||
| 960.3.e.b | 4 | 8.b | even | 2 | 1 | ||
| 960.3.e.b | 4 | 8.d | odd | 2 | 1 | ||
| 1200.3.e.n | 4 | 5.b | even | 2 | 1 | ||
| 1200.3.e.n | 4 | 20.d | odd | 2 | 1 | ||
| 1200.3.j.f | 8 | 5.c | odd | 4 | 2 | ||
| 1200.3.j.f | 8 | 20.e | even | 4 | 2 | ||
| 2880.3.e.f | 4 | 24.f | even | 2 | 1 | ||
| 2880.3.e.f | 4 | 24.h | odd | 2 | 1 | ||
| 3600.3.e.bh | 4 | 15.d | odd | 2 | 1 | ||
| 3600.3.e.bh | 4 | 60.h | even | 2 | 1 | ||
| 3600.3.j.l | 8 | 15.e | even | 4 | 2 | ||
| 3600.3.j.l | 8 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{13}^{2} + 16T_{13} - 116 \)
acting on \(S_{3}^{\mathrm{new}}(240, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} \)
|
| $3$ |
\( (T^{2} + 3)^{2} \)
|
| $5$ |
\( (T^{2} - 5)^{2} \)
|
| $7$ |
\( T^{4} + 144T^{2} + 2304 \)
|
| $11$ |
\( T^{4} + 144T^{2} + 2304 \)
|
| $13$ |
\( (T^{2} + 16 T - 116)^{2} \)
|
| $17$ |
\( (T^{2} - 24 T - 36)^{2} \)
|
| $19$ |
\( T^{4} + 576 T^{2} + 36864 \)
|
| $23$ |
\( T^{4} + 2016 T^{2} + 831744 \)
|
| $29$ |
\( (T^{2} + 24 T - 36)^{2} \)
|
| $31$ |
\( T^{4} + 2880 T^{2} + 921600 \)
|
| $37$ |
\( (T^{2} - 40 T - 1220)^{2} \)
|
| $41$ |
\( (T + 30)^{4} \)
|
| $43$ |
\( T^{4} + 2016 T^{2} + 831744 \)
|
| $47$ |
\( T^{4} + 6624 T^{2} + 8020224 \)
|
| $53$ |
\( (T^{2} - 48 T - 1044)^{2} \)
|
| $59$ |
\( T^{4} + 14544 T^{2} + 52533504 \)
|
| $61$ |
\( (T^{2} - 28 T - 6284)^{2} \)
|
| $67$ |
\( T^{4} + 18144 T^{2} + 22581504 \)
|
| $71$ |
\( T^{4} + 28224 T^{2} + 185831424 \)
|
| $73$ |
\( (T^{2} + 4 T - 716)^{2} \)
|
| $79$ |
\( T^{4} + 12096 T^{2} + 31002624 \)
|
| $83$ |
\( T^{4} + 7776 T^{2} + 186624 \)
|
| $89$ |
\( (T^{2} - 12 T - 684)^{2} \)
|
| $97$ |
\( (T + 14)^{4} \)
|
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