Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [338,2,Mod(337,338)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(338, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("338.337");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.b (of order \(2\), degree \(1\), not 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: | \(2.69894358832\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.153664.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
|
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} + 5x^{4} + 6x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} + 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{4}\) | \(=\) |
\( \nu^{4} + 3\nu^{2} + 1 \)
|
| \(\beta_{5}\) | \(=\) |
\( \nu^{5} + 4\nu^{3} + 3\nu \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} - 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{3} - 3\beta_1 \)
|
| \(\nu^{4}\) | \(=\) |
\( \beta_{4} - 3\beta_{2} + 5 \)
|
| \(\nu^{5}\) | \(=\) |
\( \beta_{5} - 4\beta_{3} + 9\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 1.00000i | −2.04892 | −1.00000 | − | 3.60388i | 2.04892i | − | 1.10992i | 1.00000i | 1.19806 | −3.60388 | |||||||||||||||||||||||||||||||||
| 337.2 | − | 1.00000i | 2.35690 | −1.00000 | − | 0.890084i | − | 2.35690i | − | 4.49396i | 1.00000i | 2.55496 | −0.890084 | |||||||||||||||||||||||||||||||||
| 337.3 | − | 1.00000i | 2.69202 | −1.00000 | 2.49396i | − | 2.69202i | 1.60388i | 1.00000i | 4.24698 | 2.49396 | |||||||||||||||||||||||||||||||||||
| 337.4 | 1.00000i | −2.04892 | −1.00000 | 3.60388i | − | 2.04892i | 1.10992i | − | 1.00000i | 1.19806 | −3.60388 | |||||||||||||||||||||||||||||||||||
| 337.5 | 1.00000i | 2.35690 | −1.00000 | 0.890084i | 2.35690i | 4.49396i | − | 1.00000i | 2.55496 | −0.890084 | ||||||||||||||||||||||||||||||||||||
| 337.6 | 1.00000i | 2.69202 | −1.00000 | − | 2.49396i | 2.69202i | − | 1.60388i | − | 1.00000i | 4.24698 | 2.49396 | ||||||||||||||||||||||||||||||||||
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 | 338.2.b.d | 6 | |
| 3.b | odd | 2 | 1 | 3042.2.b.n | 6 | ||
| 4.b | odd | 2 | 1 | 2704.2.f.m | 6 | ||
| 13.b | even | 2 | 1 | inner | 338.2.b.d | 6 | |
| 13.c | even | 3 | 2 | 338.2.e.e | 12 | ||
| 13.d | odd | 4 | 1 | 338.2.a.g | ✓ | 3 | |
| 13.d | odd | 4 | 1 | 338.2.a.h | yes | 3 | |
| 13.e | even | 6 | 2 | 338.2.e.e | 12 | ||
| 13.f | odd | 12 | 2 | 338.2.c.h | 6 | ||
| 13.f | odd | 12 | 2 | 338.2.c.i | 6 | ||
| 39.d | odd | 2 | 1 | 3042.2.b.n | 6 | ||
| 39.f | even | 4 | 1 | 3042.2.a.z | 3 | ||
| 39.f | even | 4 | 1 | 3042.2.a.bi | 3 | ||
| 52.b | odd | 2 | 1 | 2704.2.f.m | 6 | ||
| 52.f | even | 4 | 1 | 2704.2.a.v | 3 | ||
| 52.f | even | 4 | 1 | 2704.2.a.w | 3 | ||
| 65.g | odd | 4 | 1 | 8450.2.a.bn | 3 | ||
| 65.g | odd | 4 | 1 | 8450.2.a.bx | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 338.2.a.g | ✓ | 3 | 13.d | odd | 4 | 1 | |
| 338.2.a.h | yes | 3 | 13.d | odd | 4 | 1 | |
| 338.2.b.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 338.2.b.d | 6 | 13.b | even | 2 | 1 | inner | |
| 338.2.c.h | 6 | 13.f | odd | 12 | 2 | ||
| 338.2.c.i | 6 | 13.f | odd | 12 | 2 | ||
| 338.2.e.e | 12 | 13.c | even | 3 | 2 | ||
| 338.2.e.e | 12 | 13.e | even | 6 | 2 | ||
| 2704.2.a.v | 3 | 52.f | even | 4 | 1 | ||
| 2704.2.a.w | 3 | 52.f | even | 4 | 1 | ||
| 2704.2.f.m | 6 | 4.b | odd | 2 | 1 | ||
| 2704.2.f.m | 6 | 52.b | odd | 2 | 1 | ||
| 3042.2.a.z | 3 | 39.f | even | 4 | 1 | ||
| 3042.2.a.bi | 3 | 39.f | even | 4 | 1 | ||
| 3042.2.b.n | 6 | 3.b | odd | 2 | 1 | ||
| 3042.2.b.n | 6 | 39.d | odd | 2 | 1 | ||
| 8450.2.a.bn | 3 | 65.g | odd | 4 | 1 | ||
| 8450.2.a.bx | 3 | 65.g | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - 3T_{3}^{2} - 4T_{3} + 13 \)
acting on \(S_{2}^{\mathrm{new}}(338, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( (T^{3} - 3 T^{2} - 4 T + 13)^{2} \)
|
| $5$ |
\( T^{6} + 20 T^{4} + \cdots + 64 \)
|
| $7$ |
\( T^{6} + 24 T^{4} + \cdots + 64 \)
|
| $11$ |
\( T^{6} + 17 T^{4} + \cdots + 169 \)
|
| $13$ |
\( T^{6} \)
|
| $17$ |
\( (T^{3} + 5 T^{2} - 22 T - 97)^{2} \)
|
| $19$ |
\( T^{6} + 33 T^{4} + \cdots + 169 \)
|
| $23$ |
\( (T^{3} - 28 T - 56)^{2} \)
|
| $29$ |
\( (T^{3} + 10 T^{2} + \cdots - 104)^{2} \)
|
| $31$ |
\( T^{6} + 104 T^{4} + \cdots + 10816 \)
|
| $37$ |
\( T^{6} + 84 T^{4} + \cdots + 3136 \)
|
| $41$ |
\( T^{6} + 77 T^{4} + \cdots + 8281 \)
|
| $43$ |
\( (T^{3} + 11 T^{2} - 4 T - 1)^{2} \)
|
| $47$ |
\( T^{6} + 76 T^{4} + \cdots + 64 \)
|
| $53$ |
\( (T^{3} - 28 T - 56)^{2} \)
|
| $59$ |
\( T^{6} + 129 T^{4} + \cdots + 16129 \)
|
| $61$ |
\( (T^{3} + 4 T^{2} - 144 T - 64)^{2} \)
|
| $67$ |
\( T^{6} + 273 T^{4} + \cdots + 82369 \)
|
| $71$ |
\( T^{6} + 104 T^{4} + \cdots + 64 \)
|
| $73$ |
\( T^{6} + 173 T^{4} + \cdots + 63001 \)
|
| $79$ |
\( (T^{3} + 18 T^{2} + \cdots - 232)^{2} \)
|
| $83$ |
\( T^{6} + 33 T^{4} + \cdots + 169 \)
|
| $89$ |
\( T^{6} + 437 T^{4} + \cdots + 573049 \)
|
| $97$ |
\( T^{6} + 405 T^{4} + \cdots + 779689 \)
|
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