Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [350,3,Mod(251,350)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(350, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("350.251");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 350 = 2 \cdot 5^{2} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 350.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: | \(9.53680925261\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.0.7168.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 6x^{2} + 7 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 6x^{2} + 7 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 3\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} + 5\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -3\beta_{2} + 5\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 251.1 |
|
−1.41421 | − | 1.78089i | 2.00000 | 0 | 2.51856i | 1.58579 | − | 6.81801i | −2.82843 | 5.82843 | 0 | |||||||||||||||||||||||||||
| 251.2 | −1.41421 | 1.78089i | 2.00000 | 0 | − | 2.51856i | 1.58579 | + | 6.81801i | −2.82843 | 5.82843 | 0 | ||||||||||||||||||||||||||||
| 251.3 | 1.41421 | − | 2.97127i | 2.00000 | 0 | − | 4.20201i | 4.41421 | + | 5.43275i | 2.82843 | 0.171573 | 0 | |||||||||||||||||||||||||||
| 251.4 | 1.41421 | 2.97127i | 2.00000 | 0 | 4.20201i | 4.41421 | − | 5.43275i | 2.82843 | 0.171573 | 0 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 350.3.b.b | yes | 4 |
| 5.b | even | 2 | 1 | 350.3.b.a | ✓ | 4 | |
| 5.c | odd | 4 | 2 | 350.3.d.a | 8 | ||
| 7.b | odd | 2 | 1 | inner | 350.3.b.b | yes | 4 |
| 35.c | odd | 2 | 1 | 350.3.b.a | ✓ | 4 | |
| 35.f | even | 4 | 2 | 350.3.d.a | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 350.3.b.a | ✓ | 4 | 5.b | even | 2 | 1 | |
| 350.3.b.a | ✓ | 4 | 35.c | odd | 2 | 1 | |
| 350.3.b.b | yes | 4 | 1.a | even | 1 | 1 | trivial |
| 350.3.b.b | yes | 4 | 7.b | odd | 2 | 1 | inner |
| 350.3.d.a | 8 | 5.c | odd | 4 | 2 | ||
| 350.3.d.a | 8 | 35.f | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(350, [\chi])\):
|
\( T_{3}^{4} + 12T_{3}^{2} + 28 \)
|
|
\( T_{23}^{2} - 18T_{23} + 79 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2)^{2} \)
|
| $3$ |
\( T^{4} + 12T^{2} + 28 \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 12 T^{3} + \cdots + 2401 \)
|
| $11$ |
\( (T^{2} - 10 T + 7)^{2} \)
|
| $13$ |
\( T^{4} + 276T^{2} + 8092 \)
|
| $17$ |
\( T^{4} + 944 T^{2} + 129472 \)
|
| $19$ |
\( T^{4} + 768 T^{2} + 114688 \)
|
| $23$ |
\( (T^{2} - 18 T + 79)^{2} \)
|
| $29$ |
\( (T^{2} + 42 T - 71)^{2} \)
|
| $31$ |
\( T^{4} + 1356 T^{2} + 149212 \)
|
| $37$ |
\( (T^{2} + 54 T - 239)^{2} \)
|
| $41$ |
\( T^{4} + 5556 T^{2} + 3489052 \)
|
| $43$ |
\( (T^{2} - 38 T + 311)^{2} \)
|
| $47$ |
\( T^{4} + 1872 T^{2} + 430528 \)
|
| $53$ |
\( (T^{2} + 8 T - 2872)^{2} \)
|
| $59$ |
\( T^{4} + 8300 T^{2} + 16817500 \)
|
| $61$ |
\( T^{4} + 7668 T^{2} + 4502428 \)
|
| $67$ |
\( (T^{2} + 42 T - 6057)^{2} \)
|
| $71$ |
\( (T^{2} - 46 T - 2833)^{2} \)
|
| $73$ |
\( T^{4} + 12340 T^{2} + 14812 \)
|
| $79$ |
\( (T^{2} - 102 T - 3449)^{2} \)
|
| $83$ |
\( T^{4} + 36588 T^{2} + 285109468 \)
|
| $89$ |
\( T^{4} + 9684 T^{2} + 1849372 \)
|
| $97$ |
\( T^{4} + 34996 T^{2} + 276596572 \)
|
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