Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [600,6,Mod(1,600)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(600, 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("600.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 600.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: | \(96.2302918878\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.20073.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 30x - 24 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 120) |
| 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
\( x^{3} - x^{2} - 30x - 24 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + 9\nu + 17 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} - 7\nu - 59 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 8 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 9\beta_{2} + 7\beta _1 + 412 ) / 20 \)
|
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 | 0 | 0 | −85.8595 | 0 | 81.0000 | 0 | |||||||||||||||||||||||||||
| 1.2 | 0 | −9.00000 | 0 | 0 | 0 | 21.3742 | 0 | 81.0000 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 0 | −9.00000 | 0 | 0 | 0 | 168.485 | 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 | 600.6.a.r | 3 | |
| 5.b | even | 2 | 1 | 600.6.a.s | 3 | ||
| 5.c | odd | 4 | 2 | 120.6.f.a | ✓ | 6 | |
| 15.e | even | 4 | 2 | 360.6.f.a | 6 | ||
| 20.e | even | 4 | 2 | 240.6.f.e | 6 | ||
| 60.l | odd | 4 | 2 | 720.6.f.l | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.6.f.a | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 240.6.f.e | 6 | 20.e | even | 4 | 2 | ||
| 360.6.f.a | 6 | 15.e | even | 4 | 2 | ||
| 600.6.a.r | 3 | 1.a | even | 1 | 1 | trivial | |
| 600.6.a.s | 3 | 5.b | even | 2 | 1 | ||
| 720.6.f.l | 6 | 60.l | odd | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 104T_{7}^{2} - 12700T_{7} + 309200 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(600))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} \)
|
| $3$ |
\( (T + 9)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} - 104 T^{2} + \cdots + 309200 \)
|
| $11$ |
\( T^{3} + 332 T^{2} + \cdots + 751600 \)
|
| $13$ |
\( T^{3} - 88 T^{2} + \cdots - 862016 \)
|
| $17$ |
\( T^{3} + 1116 T^{2} + \cdots - 563332752 \)
|
| $19$ |
\( T^{3} + \cdots - 2584131584 \)
|
| $23$ |
\( T^{3} + \cdots + 14809498432 \)
|
| $29$ |
\( T^{3} + \cdots - 37303134464 \)
|
| $31$ |
\( T^{3} + \cdots - 71486323200 \)
|
| $37$ |
\( T^{3} + \cdots - 23317590944 \)
|
| $41$ |
\( T^{3} + \cdots + 289547160040 \)
|
| $43$ |
\( T^{3} + \cdots + 202238316864 \)
|
| $47$ |
\( T^{3} + \cdots + 5434818969600 \)
|
| $53$ |
\( T^{3} + \cdots - 1233623106640 \)
|
| $59$ |
\( T^{3} + \cdots + 6990217203600 \)
|
| $61$ |
\( T^{3} + \cdots - 30927698302664 \)
|
| $67$ |
\( T^{3} + \cdots - 186009692416 \)
|
| $71$ |
\( T^{3} + \cdots - 37186025753600 \)
|
| $73$ |
\( T^{3} + \cdots + 144372500393600 \)
|
| $79$ |
\( T^{3} + \cdots - 4303737800704 \)
|
| $83$ |
\( T^{3} + \cdots - 58214767068352 \)
|
| $89$ |
\( T^{3} + \cdots - 11306161037592 \)
|
| $97$ |
\( T^{3} + \cdots + 591535197394432 \)
|
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