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: | \(1\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} - 120x^{4} + 70x^{3} + 2825x^{2} - 2101x - 2690 \)
|
| 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 a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - x^{5} - 120x^{4} + 70x^{3} + 2825x^{2} - 2101x - 2690 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 40 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} - 2\nu^{2} - 73\nu + 68 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} - \nu^{3} - 91\nu^{2} + 17\nu + 730 ) / 8 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} - 120\nu^{3} - 42\nu^{2} + 2703\nu - 190 ) / 16 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 40 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + 2\beta_{2} + 73\beta _1 + 12 \)
|
| \(\nu^{4}\) | \(=\) |
\( 8\beta_{4} + 2\beta_{3} + 93\beta_{2} + 56\beta _1 + 2922 \)
|
| \(\nu^{5}\) | \(=\) |
\( 16\beta_{5} + 240\beta_{3} + 282\beta_{2} + 6057\beta _1 + 3310 \)
|
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 |
|
−10.0376 | 0 | 68.7528 | 25.0000 | 0 | −84.1978 | −368.909 | 0 | −250.939 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −6.78580 | 0 | 14.0471 | 25.0000 | 0 | 47.2008 | 121.825 | 0 | −169.645 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −1.68043 | 0 | −29.1761 | 25.0000 | 0 | 193.803 | 102.802 | 0 | −42.0108 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 0.471685 | 0 | −31.7775 | 25.0000 | 0 | −149.335 | −30.0829 | 0 | 11.7921 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 4.25379 | 0 | −13.9052 | 25.0000 | 0 | 30.2542 | −195.271 | 0 | 106.345 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 8.77833 | 0 | 45.0590 | 25.0000 | 0 | −117.725 | 114.636 | 0 | 219.458 | |||||||||||||||||||||||||||||||||||||
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.k | ✓ | 6 |
| 3.b | odd | 2 | 1 | 495.6.a.m | yes | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 495.6.a.k | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.m | yes | 6 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 5T_{2}^{5} - 110T_{2}^{4} - 400T_{2}^{3} + 2320T_{2}^{2} + 3280T_{2} - 2016 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 5 T^{5} + \cdots - 2016 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T - 25)^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 409661768704 \)
|
| $11$ |
\( (T - 121)^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 44\!\cdots\!76 \)
|
| $17$ |
\( T^{6} + \cdots + 27\!\cdots\!44 \)
|
| $19$ |
\( T^{6} + \cdots + 46\!\cdots\!36 \)
|
| $23$ |
\( T^{6} + \cdots + 67\!\cdots\!24 \)
|
| $29$ |
\( T^{6} + \cdots - 11\!\cdots\!84 \)
|
| $31$ |
\( T^{6} + \cdots + 98\!\cdots\!36 \)
|
| $37$ |
\( T^{6} + \cdots - 23\!\cdots\!96 \)
|
| $41$ |
\( T^{6} + \cdots - 70\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots - 36\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots + 52\!\cdots\!84 \)
|
| $53$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $59$ |
\( T^{6} + \cdots + 64\!\cdots\!24 \)
|
| $61$ |
\( T^{6} + \cdots - 30\!\cdots\!24 \)
|
| $67$ |
\( T^{6} + \cdots - 50\!\cdots\!36 \)
|
| $71$ |
\( T^{6} + \cdots - 70\!\cdots\!00 \)
|
| $73$ |
\( T^{6} + \cdots - 14\!\cdots\!04 \)
|
| $79$ |
\( T^{6} + \cdots - 15\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $89$ |
\( T^{6} + \cdots + 25\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots + 62\!\cdots\!24 \)
|
| show more | |
| show less |