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: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 143x^{3} + 71x^{2} + 4216x - 3740 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} - 3\nu^{2} - 81\nu + 145 ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - \nu - 57 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - 2\nu^{3} - 109\nu^{2} + 74\nu + 1465 ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + \beta _1 + 57 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 5\beta_{2} + 84\beta _1 + 26 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{4} + 115\beta_{3} + 10\beta_{2} + 203\beta _1 + 4800 \)
|
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 |
|
−9.46271 | 0 | 57.5429 | 25.0000 | 0 | −112.299 | −241.706 | 0 | −236.568 | |||||||||||||||||||||||||||||||||
| 1.2 | −6.98466 | 0 | 16.7854 | 25.0000 | 0 | 180.375 | 106.269 | 0 | −174.616 | ||||||||||||||||||||||||||||||||||
| 1.3 | 0.898099 | 0 | −31.1934 | 25.0000 | 0 | 120.732 | −56.7540 | 0 | 22.4525 | ||||||||||||||||||||||||||||||||||
| 1.4 | 5.93734 | 0 | 3.25206 | 25.0000 | 0 | −105.553 | −170.686 | 0 | 148.434 | ||||||||||||||||||||||||||||||||||
| 1.5 | 10.6119 | 0 | 80.6130 | 25.0000 | 0 | 32.7446 | 515.877 | 0 | 265.298 | ||||||||||||||||||||||||||||||||||
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.i | 5 | |
| 3.b | odd | 2 | 1 | 165.6.a.g | ✓ | 5 | |
| 15.d | odd | 2 | 1 | 825.6.a.k | 5 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.g | ✓ | 5 | 3.b | odd | 2 | 1 | |
| 495.6.a.i | 5 | 1.a | even | 1 | 1 | trivial | |
| 825.6.a.k | 5 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - T_{2}^{4} - 143T_{2}^{3} + 71T_{2}^{2} + 4216T_{2} - 3740 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - T^{4} + \cdots - 3740 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( (T - 25)^{5} \)
|
| $7$ |
\( T^{5} + \cdots - 8452488256 \)
|
| $11$ |
\( (T - 121)^{5} \)
|
| $13$ |
\( T^{5} + \cdots + 13445413206016 \)
|
| $17$ |
\( T^{5} + \cdots - 475773584554112 \)
|
| $19$ |
\( T^{5} + \cdots - 25\!\cdots\!92 \)
|
| $23$ |
\( T^{5} + \cdots - 23\!\cdots\!04 \)
|
| $29$ |
\( T^{5} + \cdots - 18\!\cdots\!88 \)
|
| $31$ |
\( T^{5} + \cdots + 55\!\cdots\!00 \)
|
| $37$ |
\( T^{5} + \cdots - 11\!\cdots\!48 \)
|
| $41$ |
\( T^{5} + \cdots + 86\!\cdots\!12 \)
|
| $43$ |
\( T^{5} + \cdots - 69\!\cdots\!08 \)
|
| $47$ |
\( T^{5} + \cdots + 59\!\cdots\!00 \)
|
| $53$ |
\( T^{5} + \cdots + 27\!\cdots\!48 \)
|
| $59$ |
\( T^{5} + \cdots + 17\!\cdots\!20 \)
|
| $61$ |
\( T^{5} + \cdots + 27\!\cdots\!80 \)
|
| $67$ |
\( T^{5} + \cdots - 76\!\cdots\!00 \)
|
| $71$ |
\( T^{5} + \cdots + 45\!\cdots\!12 \)
|
| $73$ |
\( T^{5} + \cdots + 70\!\cdots\!96 \)
|
| $79$ |
\( T^{5} + \cdots - 19\!\cdots\!84 \)
|
| $83$ |
\( T^{5} + \cdots - 58\!\cdots\!00 \)
|
| $89$ |
\( T^{5} + \cdots - 35\!\cdots\!96 \)
|
| $97$ |
\( T^{5} + \cdots + 11\!\cdots\!12 \)
|
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