Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [429,2,Mod(298,429)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(429, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 0, 1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("429.298");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 429 = 3 \cdot 11 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 429.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: | \(3.42558224671\) |
| Analytic rank: | \(0\) |
| Dimension: | \(10\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{10} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{10} + 13x^{8} + 54x^{6} + 74x^{4} + 21x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\ldots,\beta_{9}\) 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
\( x^{10} + 13x^{8} + 54x^{6} + 74x^{4} + 21x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{6} + 7\nu^{4} + 9\nu^{2} + 1 ) / 2 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{8} - 12\nu^{6} - 44\nu^{4} - 46\nu^{2} - 3 ) / 2 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{9} + 13\nu^{7} + 53\nu^{5} + 67\nu^{3} + 12\nu ) / 2 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{8} + 13\nu^{6} + 53\nu^{4} + 67\nu^{2} + 10 ) / 2 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -\nu^{9} - 13\nu^{7} - 53\nu^{5} - 65\nu^{3} - 2\nu ) / 2 \)
|
| \(\beta_{8}\) | \(=\) |
\( ( \nu^{9} + 13\nu^{7} + 55\nu^{5} + 81\nu^{3} + 30\nu ) / 2 \)
|
| \(\beta_{9}\) | \(=\) |
\( ( -3\nu^{9} - 38\nu^{7} - 152\nu^{5} - 192\nu^{3} - 35\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{7} + \beta_{5} - 5\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{6} + \beta_{4} - \beta_{3} - 6\beta_{2} + 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{8} - 7\beta_{7} - 8\beta_{5} + 26\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( -7\beta_{6} - 7\beta_{4} + 9\beta_{3} + 33\beta_{2} - 79 \)
|
| \(\nu^{7}\) | \(=\) |
\( 2\beta_{9} - 7\beta_{8} + 40\beta_{7} + 53\beta_{5} - 138\beta_1 \)
|
| \(\nu^{8}\) | \(=\) |
\( 40\beta_{6} + 38\beta_{4} - 64\beta_{3} - 178\beta_{2} + 423 \)
|
| \(\nu^{9}\) | \(=\) |
\( -26\beta_{9} + 38\beta_{8} - 216\beta_{7} - 330\beta_{5} + 739\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/429\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(79\) | \(287\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 298.1 |
|
− | 2.35969i | −1.00000 | −3.56813 | − | 1.80373i | 2.35969i | − | 4.78347i | 3.70030i | 1.00000 | −4.25625 | |||||||||||||||||||||||||||||||||||||||||||||
| 298.2 | − | 2.25835i | −1.00000 | −3.10015 | 3.76912i | 2.25835i | − | 0.701152i | 2.48452i | 1.00000 | 8.51198 | |||||||||||||||||||||||||||||||||||||||||||||||
| 298.3 | − | 1.40449i | −1.00000 | 0.0273977 | − | 3.56736i | 1.40449i | − | 0.116494i | − | 2.84747i | 1.00000 | −5.01033 | |||||||||||||||||||||||||||||||||||||||||||||
| 298.4 | − | 0.547285i | −1.00000 | 1.70048 | 0.955178i | 0.547285i | − | 4.37449i | − | 2.02522i | 1.00000 | 0.522755 | ||||||||||||||||||||||||||||||||||||||||||||||
| 298.5 | − | 0.244130i | −1.00000 | 1.94040 | 0.949685i | 0.244130i | − | 2.34031i | − | 0.961969i | 1.00000 | 0.231846 | ||||||||||||||||||||||||||||||||||||||||||||||
| 298.6 | 0.244130i | −1.00000 | 1.94040 | − | 0.949685i | − | 0.244130i | 2.34031i | 0.961969i | 1.00000 | 0.231846 | |||||||||||||||||||||||||||||||||||||||||||||||
| 298.7 | 0.547285i | −1.00000 | 1.70048 | − | 0.955178i | − | 0.547285i | 4.37449i | 2.02522i | 1.00000 | 0.522755 | |||||||||||||||||||||||||||||||||||||||||||||||
| 298.8 | 1.40449i | −1.00000 | 0.0273977 | 3.56736i | − | 1.40449i | 0.116494i | 2.84747i | 1.00000 | −5.01033 | ||||||||||||||||||||||||||||||||||||||||||||||||
| 298.9 | 2.25835i | −1.00000 | −3.10015 | − | 3.76912i | − | 2.25835i | 0.701152i | − | 2.48452i | 1.00000 | 8.51198 | ||||||||||||||||||||||||||||||||||||||||||||||
| 298.10 | 2.35969i | −1.00000 | −3.56813 | 1.80373i | − | 2.35969i | 4.78347i | − | 3.70030i | 1.00000 | −4.25625 | |||||||||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 429.2.b.a | ✓ | 10 |
| 3.b | odd | 2 | 1 | 1287.2.b.a | 10 | ||
| 13.b | even | 2 | 1 | inner | 429.2.b.a | ✓ | 10 |
| 13.d | odd | 4 | 1 | 5577.2.a.q | 5 | ||
| 13.d | odd | 4 | 1 | 5577.2.a.t | 5 | ||
| 39.d | odd | 2 | 1 | 1287.2.b.a | 10 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 429.2.b.a | ✓ | 10 | 1.a | even | 1 | 1 | trivial |
| 429.2.b.a | ✓ | 10 | 13.b | even | 2 | 1 | inner |
| 1287.2.b.a | 10 | 3.b | odd | 2 | 1 | ||
| 1287.2.b.a | 10 | 39.d | odd | 2 | 1 | ||
| 5577.2.a.q | 5 | 13.d | odd | 4 | 1 | ||
| 5577.2.a.t | 5 | 13.d | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{10} + 13T_{2}^{8} + 54T_{2}^{6} + 74T_{2}^{4} + 21T_{2}^{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(429, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{10} + 13 T^{8} + \cdots + 1 \)
|
| $3$ |
\( (T + 1)^{10} \)
|
| $5$ |
\( T^{10} + 32 T^{8} + \cdots + 484 \)
|
| $7$ |
\( T^{10} + 48 T^{8} + \cdots + 16 \)
|
| $11$ |
\( (T^{2} + 1)^{5} \)
|
| $13$ |
\( T^{10} - 12 T^{9} + \cdots + 371293 \)
|
| $17$ |
\( (T^{5} - 10 T^{4} + \cdots - 186)^{2} \)
|
| $19$ |
\( T^{10} + 148 T^{8} + \cdots + 4443664 \)
|
| $23$ |
\( (T^{5} - 4 T^{4} + \cdots + 1276)^{2} \)
|
| $29$ |
\( (T^{5} - 4 T^{4} + \cdots + 2214)^{2} \)
|
| $31$ |
\( T^{10} + 208 T^{8} + \cdots + 10797796 \)
|
| $37$ |
\( T^{10} + 196 T^{8} + \cdots + 1827904 \)
|
| $41$ |
\( T^{10} + 208 T^{8} + \cdots + 4963984 \)
|
| $43$ |
\( (T^{5} + 12 T^{4} + \cdots + 2194)^{2} \)
|
| $47$ |
\( T^{10} + 124 T^{8} + \cdots + 107584 \)
|
| $53$ |
\( (T^{5} + 2 T^{4} + \cdots + 848)^{2} \)
|
| $59$ |
\( T^{10} + 196 T^{8} + \cdots + 1024 \)
|
| $61$ |
\( (T^{5} - 18 T^{4} + \cdots + 6024)^{2} \)
|
| $67$ |
\( T^{10} + \cdots + 1150159396 \)
|
| $71$ |
\( T^{10} + 272 T^{8} + \cdots + 2585664 \)
|
| $73$ |
\( T^{10} + 148 T^{8} + \cdots + 5438224 \)
|
| $79$ |
\( (T^{5} - 18 T^{4} + \cdots - 6014)^{2} \)
|
| $83$ |
\( T^{10} + \cdots + 1673791744 \)
|
| $89$ |
\( T^{10} + \cdots + 3978834084 \)
|
| $97$ |
\( T^{10} + \cdots + 7929546304 \)
|
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