Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,4,Mod(99,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.99");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.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: | \(14.4554679514\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\zeta_{8})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
| \(\beta_{1}\) | \(=\) |
\( 3\zeta_{8}^{2} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{8}^{3} + 2\zeta_{8} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{8}^{3} + \zeta_{8} \)
|
| \(\zeta_{8}\) | \(=\) |
\( ( \beta_{3} + \beta_{2} ) / 3 \)
|
| \(\zeta_{8}^{2}\) | \(=\) |
\( ( \beta_1 ) / 3 \)
|
| \(\zeta_{8}^{3}\) | \(=\) |
\( ( -2\beta_{3} + \beta_{2} ) / 3 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 99.1 |
|
− | 3.00000i | − | 8.48528i | −1.00000 | −3.53553 | + | 10.6066i | −25.4558 | 0 | − | 21.0000i | −45.0000 | 31.8198 | + | 10.6066i | |||||||||||||||||||||||
| 99.2 | − | 3.00000i | 8.48528i | −1.00000 | 3.53553 | − | 10.6066i | 25.4558 | 0 | − | 21.0000i | −45.0000 | −31.8198 | − | 10.6066i | |||||||||||||||||||||||||
| 99.3 | 3.00000i | − | 8.48528i | −1.00000 | 3.53553 | + | 10.6066i | 25.4558 | 0 | 21.0000i | −45.0000 | −31.8198 | + | 10.6066i | ||||||||||||||||||||||||||
| 99.4 | 3.00000i | 8.48528i | −1.00000 | −3.53553 | − | 10.6066i | −25.4558 | 0 | 21.0000i | −45.0000 | 31.8198 | − | 10.6066i | |||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 35.c | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.4.b.b | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 245.4.b.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | 1225.4.a.n | 2 | ||
| 5.c | odd | 4 | 1 | 1225.4.a.w | 2 | ||
| 7.b | odd | 2 | 1 | inner | 245.4.b.b | ✓ | 4 |
| 7.c | even | 3 | 2 | 245.4.j.b | 8 | ||
| 7.d | odd | 6 | 2 | 245.4.j.b | 8 | ||
| 35.c | odd | 2 | 1 | inner | 245.4.b.b | ✓ | 4 |
| 35.f | even | 4 | 1 | 1225.4.a.n | 2 | ||
| 35.f | even | 4 | 1 | 1225.4.a.w | 2 | ||
| 35.i | odd | 6 | 2 | 245.4.j.b | 8 | ||
| 35.j | even | 6 | 2 | 245.4.j.b | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 245.4.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 245.4.b.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 245.4.b.b | ✓ | 4 | 7.b | odd | 2 | 1 | inner |
| 245.4.b.b | ✓ | 4 | 35.c | odd | 2 | 1 | inner |
| 245.4.j.b | 8 | 7.c | even | 3 | 2 | ||
| 245.4.j.b | 8 | 7.d | odd | 6 | 2 | ||
| 245.4.j.b | 8 | 35.i | odd | 6 | 2 | ||
| 245.4.j.b | 8 | 35.j | even | 6 | 2 | ||
| 1225.4.a.n | 2 | 5.c | odd | 4 | 1 | ||
| 1225.4.a.n | 2 | 35.f | even | 4 | 1 | ||
| 1225.4.a.w | 2 | 5.c | odd | 4 | 1 | ||
| 1225.4.a.w | 2 | 35.f | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{2} + 9 \)
|
|
\( T_{19}^{2} - 19208 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 9)^{2} \)
|
| $3$ |
\( (T^{2} + 72)^{2} \)
|
| $5$ |
\( T^{4} + 200 T^{2} + 15625 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T + 40)^{4} \)
|
| $13$ |
\( (T^{2} + 5202)^{2} \)
|
| $17$ |
\( (T^{2} + 882)^{2} \)
|
| $19$ |
\( (T^{2} - 19208)^{2} \)
|
| $23$ |
\( (T^{2} + 144)^{2} \)
|
| $29$ |
\( (T - 40)^{4} \)
|
| $31$ |
\( (T^{2} - 3872)^{2} \)
|
| $37$ |
\( (T^{2} + 101124)^{2} \)
|
| $41$ |
\( (T^{2} - 49298)^{2} \)
|
| $43$ |
\( (T^{2} + 129600)^{2} \)
|
| $47$ |
\( (T^{2} + 115200)^{2} \)
|
| $53$ |
\( (T^{2} + 20736)^{2} \)
|
| $59$ |
\( (T^{2} - 17672)^{2} \)
|
| $61$ |
\( (T^{2} - 837218)^{2} \)
|
| $67$ |
\( (T^{2} + 1296)^{2} \)
|
| $71$ |
\( (T + 148)^{4} \)
|
| $73$ |
\( (T^{2} + 399618)^{2} \)
|
| $79$ |
\( (T - 612)^{4} \)
|
| $83$ |
\( (T^{2} + 677448)^{2} \)
|
| $89$ |
\( (T^{2} - 960498)^{2} \)
|
| $97$ |
\( (T^{2} + 882)^{2} \)
|
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