Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [55,6,Mod(1,55)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(55, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([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("55.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 55 = 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 55.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: | \(8.82111008971\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{6} - 3x^{5} - 149x^{4} + 297x^{3} + 4052x^{2} - 6522x - 20692 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{6} - 3x^{5} - 149x^{4} + 297x^{3} + 4052x^{2} - 6522x - 20692 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{5} - 135\nu^{3} - 120\nu^{2} + 1946\nu + 804 ) / 78 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{5} + 8\nu^{4} - 153\nu^{3} - 1218\nu^{2} + 2930\nu + 17668 ) / 312 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{5} - 135\nu^{3} - 198\nu^{2} + 2024\nu + 4782 ) / 78 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -\nu^{5} - \nu^{4} + 147\nu^{3} + 267\nu^{2} - 3044\nu - 3692 ) / 78 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( -\beta_{4} + \beta_{2} + \beta _1 + 51 \)
|
| \(\nu^{3}\) | \(=\) |
\( 8\beta_{5} + \beta_{4} + 4\beta_{3} + 6\beta_{2} + 99\beta _1 + 29 \)
|
| \(\nu^{4}\) | \(=\) |
\( 18\beta_{5} - 135\beta_{4} + 48\beta_{3} + 141\beta_{2} + 237\beta _1 + 4957 \)
|
| \(\nu^{5}\) | \(=\) |
\( 1080\beta_{5} + 15\beta_{4} + 540\beta_{3} + 1008\beta_{2} + 11539\beta _1 + 9231 \)
|
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 |
|
−10.5183 | −10.1483 | 78.6353 | −25.0000 | 106.743 | −228.119 | −490.525 | −140.013 | 262.958 | ||||||||||||||||||||||||||||||||||||
| 1.2 | −3.74230 | 13.5338 | −17.9952 | −25.0000 | −50.6477 | 134.766 | 187.097 | −59.8351 | 93.5576 | |||||||||||||||||||||||||||||||||||||
| 1.3 | −3.01414 | −13.7145 | −22.9149 | −25.0000 | 41.3374 | −131.565 | 165.521 | −54.9135 | 75.3536 | |||||||||||||||||||||||||||||||||||||
| 1.4 | 2.73538 | −29.5841 | −24.5177 | −25.0000 | −80.9238 | 175.856 | −154.598 | 632.218 | −68.3846 | |||||||||||||||||||||||||||||||||||||
| 1.5 | 6.32252 | 28.4424 | 7.97424 | −25.0000 | 179.828 | 115.579 | −151.903 | 565.969 | −158.063 | |||||||||||||||||||||||||||||||||||||
| 1.6 | 11.2169 | 11.4706 | 93.8183 | −25.0000 | 128.664 | −132.518 | 693.408 | −111.426 | −280.422 | |||||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(1\) |
| \(11\) | \(-1\) |
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 | 55.6.a.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | 495.6.a.l | 6 | ||
| 4.b | odd | 2 | 1 | 880.6.a.u | 6 | ||
| 5.b | even | 2 | 1 | 275.6.a.f | 6 | ||
| 5.c | odd | 4 | 2 | 275.6.b.f | 12 | ||
| 11.b | odd | 2 | 1 | 605.6.a.e | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 275.6.a.f | 6 | 5.b | even | 2 | 1 | ||
| 275.6.b.f | 12 | 5.c | odd | 4 | 2 | ||
| 495.6.a.l | 6 | 3.b | odd | 2 | 1 | ||
| 605.6.a.e | 6 | 11.b | odd | 2 | 1 | ||
| 880.6.a.u | 6 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} - 149T_{2}^{4} + 309T_{2}^{3} + 4034T_{2}^{2} - 1868T_{2} - 23016 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(55))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots - 23016 \)
|
| $3$ |
\( T^{6} - 1145 T^{4} + \cdots - 18180288 \)
|
| $5$ |
\( (T + 25)^{6} \)
|
| $7$ |
\( T^{6} + \cdots - 10894071216816 \)
|
| $11$ |
\( (T - 121)^{6} \)
|
| $13$ |
\( T^{6} + \cdots - 24\!\cdots\!32 \)
|
| $17$ |
\( T^{6} + \cdots - 20\!\cdots\!92 \)
|
| $19$ |
\( T^{6} + \cdots - 13\!\cdots\!00 \)
|
| $23$ |
\( T^{6} + \cdots - 28\!\cdots\!68 \)
|
| $29$ |
\( T^{6} + \cdots + 94\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 11\!\cdots\!00 \)
|
| $37$ |
\( T^{6} + \cdots + 22\!\cdots\!12 \)
|
| $41$ |
\( T^{6} + \cdots + 80\!\cdots\!56 \)
|
| $43$ |
\( T^{6} + \cdots + 11\!\cdots\!76 \)
|
| $47$ |
\( T^{6} + \cdots - 70\!\cdots\!68 \)
|
| $53$ |
\( T^{6} + \cdots - 18\!\cdots\!56 \)
|
| $59$ |
\( T^{6} + \cdots - 84\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots - 24\!\cdots\!64 \)
|
| $67$ |
\( T^{6} + \cdots - 26\!\cdots\!16 \)
|
| $71$ |
\( T^{6} + \cdots - 10\!\cdots\!28 \)
|
| $73$ |
\( T^{6} + \cdots - 23\!\cdots\!72 \)
|
| $79$ |
\( T^{6} + \cdots + 29\!\cdots\!00 \)
|
| $83$ |
\( T^{6} + \cdots - 30\!\cdots\!84 \)
|
| $89$ |
\( T^{6} + \cdots + 30\!\cdots\!00 \)
|
| $97$ |
\( T^{6} + \cdots - 41\!\cdots\!28 \)
|
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