Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [495,6,Mod(1,495)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(495, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("495.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.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: | \(79.3899908074\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 55) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - 2x^{3} - 33x^{2} - 8x + 116 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 4\nu^{2} - 17\nu + 22 ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 41\nu + 38 ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 8\nu^{2} + 17\nu - 90 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 17 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 41\beta_{3} + 17\beta_{2} + 82\beta _1 + 310 ) / 6 \)
|
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 |
|
−7.82466 | 0 | 29.2253 | 25.0000 | 0 | −125.436 | 21.7111 | 0 | −195.616 | ||||||||||||||||||||||||||||||
| 1.2 | −2.64192 | 0 | −25.0202 | 25.0000 | 0 | −12.8802 | 150.643 | 0 | −66.0481 | |||||||||||||||||||||||||||||||
| 1.3 | 6.96278 | 0 | 16.4803 | 25.0000 | 0 | −169.118 | −108.060 | 0 | 174.070 | |||||||||||||||||||||||||||||||
| 1.4 | 8.50380 | 0 | 40.3146 | 25.0000 | 0 | 217.433 | 70.7058 | 0 | 212.595 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
| \(11\) | \(-1\) |
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 | 495.6.a.g | 4 | |
| 3.b | odd | 2 | 1 | 55.6.a.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 880.6.a.n | 4 | ||
| 15.d | odd | 2 | 1 | 275.6.a.d | 4 | ||
| 15.e | even | 4 | 2 | 275.6.b.d | 8 | ||
| 33.d | even | 2 | 1 | 605.6.a.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 275.6.a.d | 4 | 15.d | odd | 2 | 1 | ||
| 275.6.b.d | 8 | 15.e | even | 4 | 2 | ||
| 495.6.a.g | 4 | 1.a | even | 1 | 1 | trivial | |
| 605.6.a.c | 4 | 33.d | even | 2 | 1 | ||
| 880.6.a.n | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 5T_{2}^{3} - 82T_{2}^{2} + 300T_{2} + 1224 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 5 T^{3} + \cdots + 1224 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T - 25)^{4} \)
|
| $7$ |
\( T^{4} + 90 T^{3} + \cdots - 59409676 \)
|
| $11$ |
\( (T - 121)^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 4887860000 \)
|
| $17$ |
\( T^{4} + \cdots - 500025407256 \)
|
| $19$ |
\( T^{4} + \cdots + 123062234816 \)
|
| $23$ |
\( T^{4} + \cdots - 26010252218016 \)
|
| $29$ |
\( T^{4} + \cdots + 563529236434956 \)
|
| $31$ |
\( T^{4} + \cdots - 2285702081856 \)
|
| $37$ |
\( T^{4} + \cdots + 21956626606324 \)
|
| $41$ |
\( T^{4} + \cdots + 17\!\cdots\!96 \)
|
| $43$ |
\( T^{4} + \cdots + 559080671259904 \)
|
| $47$ |
\( T^{4} + \cdots + 11\!\cdots\!16 \)
|
| $53$ |
\( T^{4} + \cdots + 21\!\cdots\!44 \)
|
| $59$ |
\( T^{4} + \cdots - 60\!\cdots\!76 \)
|
| $61$ |
\( T^{4} + \cdots - 81\!\cdots\!16 \)
|
| $67$ |
\( T^{4} + \cdots - 10\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots - 30\!\cdots\!76 \)
|
| $73$ |
\( T^{4} + \cdots + 48\!\cdots\!44 \)
|
| $79$ |
\( T^{4} + \cdots - 14\!\cdots\!44 \)
|
| $83$ |
\( T^{4} + \cdots - 77\!\cdots\!04 \)
|
| $89$ |
\( T^{4} + \cdots + 56\!\cdots\!36 \)
|
| $97$ |
\( T^{4} + \cdots - 54\!\cdots\!24 \)
|
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