Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,4,Mod(53,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([0, 2, 2, 3]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("120.53");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.w (of order \(4\), degree \(2\), 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: | \(7.08022920069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(i, \sqrt{6})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{4}]$ |
$q$-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
\( x^{4} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( 3\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(-\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 53.1 |
|
−2.00000 | + | 2.00000i | −3.67423 | − | 3.67423i | − | 8.00000i | −3.32577 | − | 10.6742i | 14.6969 | 9.65153 | − | 9.65153i | 16.0000 | + | 16.0000i | 27.0000i | 28.0000 | + | 14.6969i | |||||||||||||||||
| 53.2 | −2.00000 | + | 2.00000i | 3.67423 | + | 3.67423i | − | 8.00000i | −10.6742 | − | 3.32577i | −14.6969 | 24.3485 | − | 24.3485i | 16.0000 | + | 16.0000i | 27.0000i | 28.0000 | − | 14.6969i | ||||||||||||||||||
| 77.1 | −2.00000 | − | 2.00000i | −3.67423 | + | 3.67423i | 8.00000i | −3.32577 | + | 10.6742i | 14.6969 | 9.65153 | + | 9.65153i | 16.0000 | − | 16.0000i | − | 27.0000i | 28.0000 | − | 14.6969i | ||||||||||||||||||
| 77.2 | −2.00000 | − | 2.00000i | 3.67423 | − | 3.67423i | 8.00000i | −10.6742 | + | 3.32577i | −14.6969 | 24.3485 | + | 24.3485i | 16.0000 | − | 16.0000i | − | 27.0000i | 28.0000 | + | 14.6969i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 24.h | odd | 2 | 1 | CM by \(\Q(\sqrt{-6}) \) |
| 5.c | odd | 4 | 1 | inner |
| 120.w | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 120.4.w.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | 120.4.w.b | yes | 4 | |
| 5.c | odd | 4 | 1 | inner | 120.4.w.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 120.4.w.b | yes | 4 | |
| 15.e | even | 4 | 1 | 120.4.w.b | yes | 4 | |
| 24.h | odd | 2 | 1 | CM | 120.4.w.a | ✓ | 4 |
| 40.i | odd | 4 | 1 | 120.4.w.b | yes | 4 | |
| 120.w | even | 4 | 1 | inner | 120.4.w.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.4.w.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 120.4.w.a | ✓ | 4 | 5.c | odd | 4 | 1 | inner |
| 120.4.w.a | ✓ | 4 | 24.h | odd | 2 | 1 | CM |
| 120.4.w.a | ✓ | 4 | 120.w | even | 4 | 1 | inner |
| 120.4.w.b | yes | 4 | 3.b | odd | 2 | 1 | |
| 120.4.w.b | yes | 4 | 8.b | even | 2 | 1 | |
| 120.4.w.b | yes | 4 | 15.e | even | 4 | 1 | |
| 120.4.w.b | yes | 4 | 40.i | odd | 4 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\):
|
\( T_{7}^{4} - 68T_{7}^{3} + 2312T_{7}^{2} - 31960T_{7} + 220900 \)
|
|
\( T_{11}^{2} - 8T_{11} - 2630 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4 T + 8)^{2} \)
|
| $3$ |
\( T^{4} + 729 \)
|
| $5$ |
\( T^{4} + 28 T^{3} + \cdots + 15625 \)
|
| $7$ |
\( T^{4} - 68 T^{3} + \cdots + 220900 \)
|
| $11$ |
\( (T^{2} - 8 T - 2630)^{2} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} \)
|
| $19$ |
\( T^{4} \)
|
| $23$ |
\( T^{4} \)
|
| $29$ |
\( T^{4} + 97556 T^{2} + 1322500 \)
|
| $31$ |
\( (T^{2} - 114264)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( T^{4} \)
|
| $53$ |
\( T^{4} + 67240638864 \)
|
| $59$ |
\( T^{4} + \cdots + 10697764900 \)
|
| $61$ |
\( T^{4} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} + \cdots + 454747922500 \)
|
| $79$ |
\( (T^{2} + 95256)^{2} \)
|
| $83$ |
\( (T^{2} + 1736 T + 1506848)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} + \cdots + 2237627056900 \)
|
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