Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,12,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, 12, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.1");
S:= CuspForms(chi, 12);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| 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: | \(185.939049695\) |
| 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: |
\( x^{3} - x^{2} - 206434x + 34594984 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 5 \) |
| Twist minimal: | no (minimal twist has level 22) |
| 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
\( x^{3} - x^{2} - 206434x + 34594984 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{2} + 261\nu - 137728 ) / 54 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{2} + 279\nu + 137539 ) / 27 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 2\beta _1 + 7 ) / 20 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -261\beta_{2} + 558\beta _1 + 2752733 ) / 20 \)
|
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 |
|
−32.0000 | −542.722 | 1024.00 | −8758.18 | 17367.1 | −21611.7 | −32768.0 | 117400. | 280262. | |||||||||||||||||||||||||||
| 1.2 | −32.0000 | 154.206 | 1024.00 | 9891.53 | −4934.59 | −9234.32 | −32768.0 | −153368. | −316529. | ||||||||||||||||||||||||||||
| 1.3 | −32.0000 | 758.516 | 1024.00 | −3061.35 | −24272.5 | −27626.0 | −32768.0 | 398200. | 97963.4 | ||||||||||||||||||||||||||||
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 | 242.12.a.b | 3 | |
| 11.b | odd | 2 | 1 | 22.12.a.d | ✓ | 3 | |
| 33.d | even | 2 | 1 | 198.12.a.j | 3 | ||
| 44.c | even | 2 | 1 | 176.12.a.b | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 22.12.a.d | ✓ | 3 | 11.b | odd | 2 | 1 | |
| 176.12.a.b | 3 | 44.c | even | 2 | 1 | ||
| 198.12.a.j | 3 | 33.d | even | 2 | 1 | ||
| 242.12.a.b | 3 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(242))\):
|
\( T_{3}^{3} - 370T_{3}^{2} - 378387T_{3} + 63480996 \)
|
|
\( T_{7}^{3} + 58472T_{7}^{2} + 1051720796T_{7} + 5513298935200 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 32)^{3} \)
|
| $3$ |
\( T^{3} - 370 T^{2} + \cdots + 63480996 \)
|
| $5$ |
\( T^{3} + \cdots - 265210541550 \)
|
| $7$ |
\( T^{3} + \cdots + 5513298935200 \)
|
| $11$ |
\( T^{3} \)
|
| $13$ |
\( T^{3} + \cdots - 16\!\cdots\!84 \)
|
| $17$ |
\( T^{3} + \cdots - 12\!\cdots\!28 \)
|
| $19$ |
\( T^{3} + \cdots - 10\!\cdots\!00 \)
|
| $23$ |
\( T^{3} + \cdots - 17\!\cdots\!36 \)
|
| $29$ |
\( T^{3} + \cdots - 18\!\cdots\!00 \)
|
| $31$ |
\( T^{3} + \cdots - 14\!\cdots\!28 \)
|
| $37$ |
\( T^{3} + \cdots - 44\!\cdots\!66 \)
|
| $41$ |
\( T^{3} + \cdots + 12\!\cdots\!92 \)
|
| $43$ |
\( T^{3} + \cdots - 11\!\cdots\!00 \)
|
| $47$ |
\( T^{3} + \cdots - 18\!\cdots\!04 \)
|
| $53$ |
\( T^{3} + \cdots - 27\!\cdots\!00 \)
|
| $59$ |
\( T^{3} + \cdots + 13\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 88\!\cdots\!40 \)
|
| $67$ |
\( T^{3} + \cdots - 39\!\cdots\!28 \)
|
| $71$ |
\( T^{3} + \cdots - 63\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 96\!\cdots\!48 \)
|
| $79$ |
\( T^{3} + \cdots - 27\!\cdots\!00 \)
|
| $83$ |
\( T^{3} + \cdots - 52\!\cdots\!48 \)
|
| $89$ |
\( T^{3} + \cdots - 62\!\cdots\!50 \)
|
| $97$ |
\( T^{3} + \cdots + 10\!\cdots\!94 \)
|
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