Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [242,3,Mod(241,242)]
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([1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("242.241");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 242 = 2 \cdot 11^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 242.b (of order \(2\), degree \(1\), minimal) |
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: | no |
| Analytic conductor: | \(6.59402239752\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{97})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 2x^{3} - 43x^{2} + 44x + 678 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} - 2x^{3} - 43x^{2} + 44x + 678 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 35\nu - 18 ) / 105 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -2\nu^{3} + 3\nu^{2} + 140\nu - 18 ) / 105 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 51\nu^{2} - 70\nu - 1146 ) / 105 \)
|
| \(\nu\) | \(=\) |
\( \beta_{2} - \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 22 \)
|
| \(\nu^{3}\) | \(=\) |
\( 3\beta_{3} + 19\beta_{2} - 70\beta _1 + 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/242\mathbb{Z}\right)^\times\).
| \(n\) | \(123\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 241.1 |
|
− | 1.41421i | −5.42443 | −2.00000 | 5.42443 | 7.67130i | 2.01445i | 2.82843i | 20.4244 | − | 7.67130i | ||||||||||||||||||||||||||||
| 241.2 | − | 1.41421i | 4.42443 | −2.00000 | −4.42443 | − | 6.25709i | − | 11.9139i | 2.82843i | 10.5756 | 6.25709i | ||||||||||||||||||||||||||||
| 241.3 | 1.41421i | −5.42443 | −2.00000 | 5.42443 | − | 7.67130i | − | 2.01445i | − | 2.82843i | 20.4244 | 7.67130i | ||||||||||||||||||||||||||||
| 241.4 | 1.41421i | 4.42443 | −2.00000 | −4.42443 | 6.25709i | 11.9139i | − | 2.82843i | 10.5756 | − | 6.25709i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 11.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 242.3.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 2178.3.d.f | 4 | ||
| 11.b | odd | 2 | 1 | inner | 242.3.b.b | ✓ | 4 |
| 11.c | even | 5 | 4 | 242.3.d.g | 16 | ||
| 11.d | odd | 10 | 4 | 242.3.d.g | 16 | ||
| 33.d | even | 2 | 1 | 2178.3.d.f | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 242.3.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 242.3.b.b | ✓ | 4 | 11.b | odd | 2 | 1 | inner |
| 242.3.d.g | 16 | 11.c | even | 5 | 4 | ||
| 242.3.d.g | 16 | 11.d | odd | 10 | 4 | ||
| 2178.3.d.f | 4 | 3.b | odd | 2 | 1 | ||
| 2178.3.d.f | 4 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + T_{3} - 24 \)
acting on \(S_{3}^{\mathrm{new}}(242, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( (T^{2} + T - 24)^{2} \)
|
| $5$ |
\( (T^{2} - T - 24)^{2} \)
|
| $7$ |
\( T^{4} + 146T^{2} + 576 \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} + 178T^{2} + 64 \)
|
| $17$ |
\( T^{4} + 788T^{2} + 36 \)
|
| $19$ |
\( (T^{2} + 32)^{2} \)
|
| $23$ |
\( (T^{2} - 19 T - 516)^{2} \)
|
| $29$ |
\( T^{4} + 2546 T^{2} + 1327104 \)
|
| $31$ |
\( (T^{2} - 69 T + 972)^{2} \)
|
| $37$ |
\( (T^{2} - 15 T - 162)^{2} \)
|
| $41$ |
\( T^{4} + 1314 T^{2} + 46656 \)
|
| $43$ |
\( T^{4} + 5282 T^{2} + 4460544 \)
|
| $47$ |
\( (T^{2} - 28 T - 192)^{2} \)
|
| $53$ |
\( (T^{2} + 34 T + 192)^{2} \)
|
| $59$ |
\( (T^{2} - 89 T + 792)^{2} \)
|
| $61$ |
\( T^{4} + 10706 T^{2} + 8573184 \)
|
| $67$ |
\( (T^{2} + 151 T + 4512)^{2} \)
|
| $71$ |
\( (T^{2} - 171 T + 7092)^{2} \)
|
| $73$ |
\( T^{4} + 14338 T^{2} + 20866624 \)
|
| $79$ |
\( T^{4} + 15778 T^{2} + 1024 \)
|
| $83$ |
\( T^{4} + 8354 T^{2} + 3069504 \)
|
| $89$ |
\( (T^{2} - 169 T + 4206)^{2} \)
|
| $97$ |
\( (T^{2} + 113 T + 3168)^{2} \)
|
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