Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [120,6,Mod(1,120)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(120, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("120.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.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: | \(19.2460583776\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{1489}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 372 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| Twist minimal: | yes |
| 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 \(\beta = 4\sqrt{1489}\). We also show the integral \(q\)-expansion of the trace form.
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 | 9.00000 | 0 | −25.0000 | 0 | −146.350 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||
| 1.2 | 0 | 9.00000 | 0 | −25.0000 | 0 | 162.350 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( +1 \) |
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 | 120.6.a.g | ✓ | 2 |
| 3.b | odd | 2 | 1 | 360.6.a.n | 2 | ||
| 4.b | odd | 2 | 1 | 240.6.a.o | 2 | ||
| 5.b | even | 2 | 1 | 600.6.a.k | 2 | ||
| 5.c | odd | 4 | 2 | 600.6.f.k | 4 | ||
| 8.b | even | 2 | 1 | 960.6.a.bi | 2 | ||
| 8.d | odd | 2 | 1 | 960.6.a.bn | 2 | ||
| 12.b | even | 2 | 1 | 720.6.a.bf | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.6.a.g | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 240.6.a.o | 2 | 4.b | odd | 2 | 1 | ||
| 360.6.a.n | 2 | 3.b | odd | 2 | 1 | ||
| 600.6.a.k | 2 | 5.b | even | 2 | 1 | ||
| 600.6.f.k | 4 | 5.c | odd | 4 | 2 | ||
| 720.6.a.bf | 2 | 12.b | even | 2 | 1 | ||
| 960.6.a.bi | 2 | 8.b | even | 2 | 1 | ||
| 960.6.a.bn | 2 | 8.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} - 16T_{7} - 23760 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(120))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( (T - 9)^{2} \)
|
| $5$ |
\( (T + 25)^{2} \)
|
| $7$ |
\( T^{2} - 16T - 23760 \)
|
| $11$ |
\( T^{2} - 64T - 380160 \)
|
| $13$ |
\( T^{2} - 1164 T + 124308 \)
|
| $17$ |
\( T^{2} - 2164 T + 575124 \)
|
| $19$ |
\( T^{2} - 2640 T + 575024 \)
|
| $23$ |
\( T^{2} - 40 T - 22894464 \)
|
| $29$ |
\( T^{2} - 1940 T - 23454876 \)
|
| $31$ |
\( T^{2} - 6328 T + 5984640 \)
|
| $37$ |
\( T^{2} - 3532 T - 26065644 \)
|
| $41$ |
\( T^{2} - 2404 T - 266241660 \)
|
| $43$ |
\( T^{2} - 31928 T + 216730896 \)
|
| $47$ |
\( T^{2} - 38232 T + 352818560 \)
|
| $53$ |
\( T^{2} + 24572 T + 116543940 \)
|
| $59$ |
\( T^{2} - 5184 T - 24157440 \)
|
| $61$ |
\( T^{2} + 53956 T - 61333692 \)
|
| $67$ |
\( T^{2} + \cdots + 2079363344 \)
|
| $71$ |
\( T^{2} + 71552 T + 249200640 \)
|
| $73$ |
\( T^{2} + \cdots + 2299236900 \)
|
| $79$ |
\( T^{2} - 29288 T - 482190848 \)
|
| $83$ |
\( T^{2} + \cdots + 4235146128 \)
|
| $89$ |
\( T^{2} + \cdots + 4838673508 \)
|
| $97$ |
\( T^{2} + \cdots - 2349969404 \)
|
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