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: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.21865.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 30x + 40 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 30x + 40 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} + \nu - 20 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} - \beta _1 + 20 \)
|
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 |
|
−8.45512 | −26.7755 | 39.4891 | 25.0000 | 226.390 | 22.2927 | −63.3212 | 473.926 | −211.378 | |||||||||||||||||||||||||||
| 1.2 | −4.94924 | 5.84068 | −7.50499 | 25.0000 | −28.9069 | −1.89401 | 195.520 | −208.886 | −123.731 | ||||||||||||||||||||||||||||
| 1.3 | 6.40437 | −15.0652 | 9.01590 | 25.0000 | −96.4830 | −122.399 | −147.199 | −16.0398 | 160.109 | ||||||||||||||||||||||||||||
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.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.6.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 880.6.a.l | 3 | ||
| 5.b | even | 2 | 1 | 275.6.a.c | 3 | ||
| 5.c | odd | 4 | 2 | 275.6.b.c | 6 | ||
| 11.b | odd | 2 | 1 | 605.6.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 55.6.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 275.6.a.c | 3 | 5.b | even | 2 | 1 | ||
| 275.6.b.c | 6 | 5.c | odd | 4 | 2 | ||
| 495.6.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 605.6.a.b | 3 | 11.b | odd | 2 | 1 | ||
| 880.6.a.l | 3 | 4.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 7T_{2}^{2} - 44T_{2} - 268 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(55))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 7 T^{2} + \cdots - 268 \)
|
| $3$ |
\( T^{3} + 36 T^{2} + \cdots - 2356 \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} + 102 T^{2} + \cdots - 5168 \)
|
| $11$ |
\( (T - 121)^{3} \)
|
| $13$ |
\( T^{3} + 1646 T^{2} + \cdots + 88934344 \)
|
| $17$ |
\( T^{3} + 1742 T^{2} + \cdots - 460035598 \)
|
| $19$ |
\( T^{3} + \cdots + 4578226300 \)
|
| $23$ |
\( T^{3} + 3876 T^{2} + \cdots - 81411776 \)
|
| $29$ |
\( T^{3} + \cdots + 146901534950 \)
|
| $31$ |
\( T^{3} + \cdots + 96581326832 \)
|
| $37$ |
\( T^{3} + \cdots + 153366614642 \)
|
| $41$ |
\( T^{3} + \cdots - 888835182328 \)
|
| $43$ |
\( T^{3} + \cdots - 313648586816 \)
|
| $47$ |
\( T^{3} + \cdots - 8738911157408 \)
|
| $53$ |
\( T^{3} + \cdots - 12910279016686 \)
|
| $59$ |
\( T^{3} + \cdots - 28840231836400 \)
|
| $61$ |
\( T^{3} + \cdots - 10687427094938 \)
|
| $67$ |
\( T^{3} + \cdots + 47800113135872 \)
|
| $71$ |
\( T^{3} + \cdots - 41054108151208 \)
|
| $73$ |
\( T^{3} + \cdots + 152396837064904 \)
|
| $79$ |
\( T^{3} + \cdots + 36194232363200 \)
|
| $83$ |
\( T^{3} + \cdots + 54540116863984 \)
|
| $89$ |
\( T^{3} + \cdots + 572172535345750 \)
|
| $97$ |
\( T^{3} + \cdots - 878543504051648 \)
|
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