Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [22,12,Mod(1,22)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(22, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("22.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 22 = 2 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 22.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: | \(16.9035499723\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 331687x - 40657734 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - 331687x - 40657734 \)
:
| \(\beta_{1}\) | \(=\) |
\( 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 213\nu - 221198 ) / 220 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 4 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 880\beta_{2} + 213\beta _1 + 884792 ) / 4 \)
|
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 |
|
−32.0000 | −827.781 | 1024.00 | −553.980 | 26489.0 | 22407.2 | −32768.0 | 508075. | 17727.4 | |||||||||||||||||||||||||||
| 1.2 | −32.0000 | 163.327 | 1024.00 | −4579.34 | −5226.46 | 32606.6 | −32768.0 | −150471. | 146539. | ||||||||||||||||||||||||||||
| 1.3 | −32.0000 | 594.455 | 1024.00 | −490.681 | −19022.5 | −85589.8 | −32768.0 | 176229. | 15701.8 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(1\) |
| \(11\) | \(-1\) |
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 | 22.12.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 198.12.a.l | 3 | ||
| 4.b | odd | 2 | 1 | 176.12.a.d | 3 | ||
| 11.b | odd | 2 | 1 | 242.12.a.c | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.12.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 176.12.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 198.12.a.l | 3 | 3.b | odd | 2 | 1 | ||
| 242.12.a.c | 3 | 11.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 70T_{3}^{2} - 530187T_{3} + 80369604 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(22))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{3} \)
|
| $3$ |
\( T^{3} + 70 T^{2} + \cdots + 80369604 \)
|
| $5$ |
\( T^{3} + \cdots + 1244790450 \)
|
| $7$ |
\( T^{3} + \cdots + 62533813132000 \)
|
| $11$ |
\( (T - 161051)^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 32\!\cdots\!88 \)
|
| $17$ |
\( T^{3} + \cdots - 15\!\cdots\!72 \)
|
| $19$ |
\( T^{3} + \cdots - 78\!\cdots\!60 \)
|
| $23$ |
\( T^{3} + \cdots - 27\!\cdots\!16 \)
|
| $29$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots + 54\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots + 28\!\cdots\!94 \)
|
| $41$ |
\( T^{3} + \cdots + 73\!\cdots\!64 \)
|
| $43$ |
\( T^{3} + \cdots + 70\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots + 87\!\cdots\!12 \)
|
| $53$ |
\( T^{3} + \cdots - 94\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots - 26\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 59\!\cdots\!20 \)
|
| $67$ |
\( T^{3} + \cdots + 47\!\cdots\!72 \)
|
| $71$ |
\( T^{3} + \cdots + 25\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 42\!\cdots\!16 \)
|
| $79$ |
\( T^{3} + \cdots - 96\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots + 54\!\cdots\!28 \)
|
| $89$ |
\( T^{3} + \cdots - 92\!\cdots\!30 \)
|
| $97$ |
\( T^{3} + \cdots - 13\!\cdots\!02 \)
|
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