Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,10,Mod(1,242)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(242, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.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: | \(124.638672352\) |
| Analytic rank: | \(0\) |
| 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: |
\( x^{3} - x^{2} - 1503x + 9963 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 11 \) |
| 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
\( x^{3} - x^{2} - 1503x + 9963 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 2\nu^{2} + 34\nu - 2016 ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{2} + 98\nu + 1971 ) / 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 3\beta _1 + 15 ) / 44 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -17\beta_{2} + 147\beta _1 + 44097 ) / 44 \)
|
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 |
|
−16.0000 | −215.411 | 256.000 | −2564.71 | 3446.58 | 7292.27 | −4096.00 | 26719.0 | 41035.4 | |||||||||||||||||||||||||||
| 1.2 | −16.0000 | −24.5710 | 256.000 | 1002.27 | 393.136 | 4370.69 | −4096.00 | −19079.3 | −16036.4 | ||||||||||||||||||||||||||||
| 1.3 | −16.0000 | 161.982 | 256.000 | −926.562 | −2591.72 | −3900.96 | −4096.00 | 6555.26 | 14825.0 | ||||||||||||||||||||||||||||
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 | 242.10.a.g | ✓ | 3 |
| 11.b | odd | 2 | 1 | 242.10.a.h | yes | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 242.10.a.g | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 242.10.a.h | yes | 3 | 11.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{3} + 78T_{3}^{2} - 33580T_{3} - 857352 \)
|
|
\( T_{7}^{3} - 7762T_{7}^{2} - 13624492T_{7} + 124332345592 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 16)^{3} \)
|
| $3$ |
\( T^{3} + 78 T^{2} + \cdots - 857352 \)
|
| $5$ |
\( T^{3} + \cdots - 2381766525 \)
|
| $7$ |
\( T^{3} + \cdots + 124332345592 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 31491061593435 \)
|
| $17$ |
\( T^{3} + \cdots + 27\!\cdots\!11 \)
|
| $19$ |
\( T^{3} + \cdots + 24\!\cdots\!32 \)
|
| $23$ |
\( T^{3} + \cdots + 10\!\cdots\!60 \)
|
| $29$ |
\( T^{3} + \cdots - 10\!\cdots\!83 \)
|
| $31$ |
\( T^{3} + \cdots + 28\!\cdots\!24 \)
|
| $37$ |
\( T^{3} + \cdots + 15\!\cdots\!95 \)
|
| $41$ |
\( T^{3} + \cdots + 10\!\cdots\!75 \)
|
| $43$ |
\( T^{3} + \cdots - 21\!\cdots\!76 \)
|
| $47$ |
\( T^{3} + \cdots + 72\!\cdots\!40 \)
|
| $53$ |
\( T^{3} + \cdots + 33\!\cdots\!63 \)
|
| $59$ |
\( T^{3} + \cdots + 57\!\cdots\!80 \)
|
| $61$ |
\( T^{3} + \cdots + 11\!\cdots\!72 \)
|
| $67$ |
\( T^{3} + \cdots - 11\!\cdots\!24 \)
|
| $71$ |
\( T^{3} + \cdots + 39\!\cdots\!48 \)
|
| $73$ |
\( T^{3} + \cdots + 14\!\cdots\!72 \)
|
| $79$ |
\( T^{3} + \cdots + 30\!\cdots\!20 \)
|
| $83$ |
\( T^{3} + \cdots + 31\!\cdots\!76 \)
|
| $89$ |
\( T^{3} + \cdots - 47\!\cdots\!83 \)
|
| $97$ |
\( T^{3} + \cdots + 55\!\cdots\!81 \)
|
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