Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [121,8,Mod(1,121)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(121, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 8, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("121.1");
S:= CuspForms(chi, 8);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 121 = 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 8 \) |
| Character orbit: | \([\chi]\) | \(=\) | 121.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: | \(37.7985880836\) |
| Analytic rank: | \(1\) |
| 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: |
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2\cdot 3 \) |
| Twist minimal: | no (minimal twist has level 11) |
| 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
\( x^{4} - x^{3} - 341x^{2} + 1417x - 1412 \)
:
| \(\beta_{1}\) | \(=\) |
\( 3\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 4\nu^{2} - 321\nu + 204 ) / 28 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -9\nu^{3} - 8\nu^{2} + 3029\nu - 6652 ) / 28 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 3\beta_{3} + 27\beta_{2} - 5\beta _1 + 511 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -12\beta_{3} - 24\beta_{2} + 341\beta _1 - 2335 ) / 3 \)
|
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 |
|
−20.7673 | −29.4867 | 303.281 | 223.515 | 612.360 | 1410.66 | −3640.12 | −1317.53 | −4641.80 | ||||||||||||||||||||||||||||||
| 1.2 | −11.0598 | 59.6211 | −5.68000 | −60.1766 | −659.400 | −698.069 | 1478.48 | 1367.68 | 665.544 | |||||||||||||||||||||||||||||||
| 1.3 | 10.6261 | 12.2971 | −15.0863 | 512.130 | 130.670 | −973.904 | −1520.45 | −2035.78 | 5441.94 | |||||||||||||||||||||||||||||||
| 1.4 | 21.2011 | −77.4315 | 321.485 | −138.468 | −1641.63 | 91.3115 | 4102.09 | 3808.63 | −2935.68 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(11\) | \(-1\) |
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 | 121.8.a.c | 4 | |
| 11.b | odd | 2 | 1 | 11.8.a.b | ✓ | 4 | |
| 33.d | even | 2 | 1 | 99.8.a.g | 4 | ||
| 44.c | even | 2 | 1 | 176.8.a.j | 4 | ||
| 55.d | odd | 2 | 1 | 275.8.a.b | 4 | ||
| 77.b | even | 2 | 1 | 539.8.a.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 11.8.a.b | ✓ | 4 | 11.b | odd | 2 | 1 | |
| 99.8.a.g | 4 | 33.d | even | 2 | 1 | ||
| 121.8.a.c | 4 | 1.a | even | 1 | 1 | trivial | |
| 176.8.a.j | 4 | 44.c | even | 2 | 1 | ||
| 275.8.a.b | 4 | 55.d | odd | 2 | 1 | ||
| 539.8.a.b | 4 | 77.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 558T_{2}^{2} - 140T_{2} + 51744 \)
acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(121))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 558 T^{2} + \cdots + 51744 \)
|
| $3$ |
\( T^{4} + 35 T^{3} + \cdots + 1673964 \)
|
| $5$ |
\( T^{4} - 537 T^{3} + \cdots + 953818350 \)
|
| $7$ |
\( T^{4} + \cdots + 87571440704 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + \cdots - 518474088880000 \)
|
| $17$ |
\( T^{4} + \cdots - 58\!\cdots\!56 \)
|
| $19$ |
\( T^{4} + \cdots - 31\!\cdots\!84 \)
|
| $23$ |
\( T^{4} + \cdots + 10\!\cdots\!44 \)
|
| $29$ |
\( T^{4} + \cdots - 41\!\cdots\!64 \)
|
| $31$ |
\( T^{4} + \cdots - 61\!\cdots\!36 \)
|
| $37$ |
\( T^{4} + \cdots + 41\!\cdots\!94 \)
|
| $41$ |
\( T^{4} + \cdots - 66\!\cdots\!84 \)
|
| $43$ |
\( T^{4} + \cdots + 28\!\cdots\!64 \)
|
| $47$ |
\( T^{4} + \cdots - 74\!\cdots\!04 \)
|
| $53$ |
\( T^{4} + \cdots + 18\!\cdots\!84 \)
|
| $59$ |
\( T^{4} + \cdots + 29\!\cdots\!64 \)
|
| $61$ |
\( T^{4} + \cdots + 10\!\cdots\!84 \)
|
| $67$ |
\( T^{4} + \cdots - 84\!\cdots\!64 \)
|
| $71$ |
\( T^{4} + \cdots + 59\!\cdots\!44 \)
|
| $73$ |
\( T^{4} + \cdots - 60\!\cdots\!76 \)
|
| $79$ |
\( T^{4} + \cdots + 25\!\cdots\!16 \)
|
| $83$ |
\( T^{4} + \cdots + 70\!\cdots\!96 \)
|
| $89$ |
\( T^{4} + \cdots - 24\!\cdots\!34 \)
|
| $97$ |
\( T^{4} + \cdots + 46\!\cdots\!46 \)
|
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