Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,3,Mod(31,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(6))
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("245.31");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.h (of order \(6\), degree \(2\), 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: | \(6.67576647683\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-5})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - 5x^{2} + 25 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 5x^{2} + 25 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(\beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 31.1 |
|
0.500000 | + | 0.866025i | −3.87298 | − | 2.23607i | 1.50000 | − | 2.59808i | 1.93649 | − | 1.11803i | − | 4.47214i | 0 | 7.00000 | 5.50000 | + | 9.52628i | 1.93649 | + | 1.11803i | |||||||||||||||||
| 31.2 | 0.500000 | + | 0.866025i | 3.87298 | + | 2.23607i | 1.50000 | − | 2.59808i | −1.93649 | + | 1.11803i | 4.47214i | 0 | 7.00000 | 5.50000 | + | 9.52628i | −1.93649 | − | 1.11803i | |||||||||||||||||||
| 166.1 | 0.500000 | − | 0.866025i | −3.87298 | + | 2.23607i | 1.50000 | + | 2.59808i | 1.93649 | + | 1.11803i | 4.47214i | 0 | 7.00000 | 5.50000 | − | 9.52628i | 1.93649 | − | 1.11803i | |||||||||||||||||||
| 166.2 | 0.500000 | − | 0.866025i | 3.87298 | − | 2.23607i | 1.50000 | + | 2.59808i | −1.93649 | − | 1.11803i | − | 4.47214i | 0 | 7.00000 | 5.50000 | − | 9.52628i | −1.93649 | + | 1.11803i | ||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 7.b | odd | 2 | 1 | inner |
| 7.c | even | 3 | 1 | inner |
| 7.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.3.h.b | 4 | |
| 7.b | odd | 2 | 1 | inner | 245.3.h.b | 4 | |
| 7.c | even | 3 | 1 | 35.3.d.a | ✓ | 2 | |
| 7.c | even | 3 | 1 | inner | 245.3.h.b | 4 | |
| 7.d | odd | 6 | 1 | 35.3.d.a | ✓ | 2 | |
| 7.d | odd | 6 | 1 | inner | 245.3.h.b | 4 | |
| 21.g | even | 6 | 1 | 315.3.h.b | 2 | ||
| 21.h | odd | 6 | 1 | 315.3.h.b | 2 | ||
| 28.f | even | 6 | 1 | 560.3.f.a | 2 | ||
| 28.g | odd | 6 | 1 | 560.3.f.a | 2 | ||
| 35.i | odd | 6 | 1 | 175.3.d.f | 2 | ||
| 35.j | even | 6 | 1 | 175.3.d.f | 2 | ||
| 35.k | even | 12 | 2 | 175.3.c.d | 4 | ||
| 35.l | odd | 12 | 2 | 175.3.c.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.3.d.a | ✓ | 2 | 7.c | even | 3 | 1 | |
| 35.3.d.a | ✓ | 2 | 7.d | odd | 6 | 1 | |
| 175.3.c.d | 4 | 35.k | even | 12 | 2 | ||
| 175.3.c.d | 4 | 35.l | odd | 12 | 2 | ||
| 175.3.d.f | 2 | 35.i | odd | 6 | 1 | ||
| 175.3.d.f | 2 | 35.j | even | 6 | 1 | ||
| 245.3.h.b | 4 | 1.a | even | 1 | 1 | trivial | |
| 245.3.h.b | 4 | 7.b | odd | 2 | 1 | inner | |
| 245.3.h.b | 4 | 7.c | even | 3 | 1 | inner | |
| 245.3.h.b | 4 | 7.d | odd | 6 | 1 | inner | |
| 315.3.h.b | 2 | 21.g | even | 6 | 1 | ||
| 315.3.h.b | 2 | 21.h | odd | 6 | 1 | ||
| 560.3.f.a | 2 | 28.f | even | 6 | 1 | ||
| 560.3.f.a | 2 | 28.g | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} - T_{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(245, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - T + 1)^{2} \)
|
| $3$ |
\( T^{4} - 20T^{2} + 400 \)
|
| $5$ |
\( T^{4} - 5T^{2} + 25 \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $13$ |
\( (T^{2} + 180)^{2} \)
|
| $17$ |
\( T^{4} - 720 T^{2} + 518400 \)
|
| $19$ |
\( T^{4} - 180 T^{2} + 32400 \)
|
| $23$ |
\( (T^{2} + 26 T + 676)^{2} \)
|
| $29$ |
\( (T + 22)^{4} \)
|
| $31$ |
\( T^{4} - 2880 T^{2} + 8294400 \)
|
| $37$ |
\( (T^{2} + 14 T + 196)^{2} \)
|
| $41$ |
\( (T^{2} + 720)^{2} \)
|
| $43$ |
\( (T + 34)^{4} \)
|
| $47$ |
\( T^{4} - 720 T^{2} + 518400 \)
|
| $53$ |
\( (T^{2} - 34 T + 1156)^{2} \)
|
| $59$ |
\( T^{4} - 1620 T^{2} + 2624400 \)
|
| $61$ |
\( T^{4} - 8820 T^{2} + 77792400 \)
|
| $67$ |
\( (T^{2} + 14 T + 196)^{2} \)
|
| $71$ |
\( (T - 62)^{4} \)
|
| $73$ |
\( T^{4} - 2880 T^{2} + 8294400 \)
|
| $79$ |
\( (T^{2} + 38 T + 1444)^{2} \)
|
| $83$ |
\( (T^{2} + 1620)^{2} \)
|
| $89$ |
\( T^{4} - 720 T^{2} + 518400 \)
|
| $97$ |
\( (T^{2} + 720)^{2} \)
|
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