Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [245,2,Mod(48,245)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(245, base_ring=CyclotomicField(4))
chi = DirichletCharacter(H, H._module([3, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("245.48");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 245 = 5 \cdot 7^{2} \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 245.f (of order \(4\), degree \(2\), 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: | \(1.95633484952\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(i)\) |
| Coefficient field: | \(\Q(\zeta_{12})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 35) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{4}]$ |
$q$-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
| \(\beta_{1}\) | \(=\) |
\( \zeta_{12}^{2} + \zeta_{12} \)
|
| \(\beta_{2}\) | \(=\) |
\( \zeta_{12}^{3} \)
|
| \(\beta_{3}\) | \(=\) |
\( -\zeta_{12}^{2} + \zeta_{12} \)
|
| \(\zeta_{12}\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{2}\) | \(=\) |
\( ( -\beta_{3} + \beta_1 ) / 2 \)
|
| \(\zeta_{12}^{3}\) | \(=\) |
\( \beta_{2} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(197\) |
| \(\chi(n)\) | \(-1\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 48.1 |
|
−0.366025 | + | 0.366025i | 0.366025 | − | 0.366025i | 1.73205i | 2.00000 | + | 1.00000i | 0.267949i | 0 | −1.36603 | − | 1.36603i | 2.73205i | −1.09808 | + | 0.366025i | ||||||||||||||||||||
| 48.2 | 1.36603 | − | 1.36603i | −1.36603 | + | 1.36603i | − | 1.73205i | 2.00000 | + | 1.00000i | 3.73205i | 0 | 0.366025 | + | 0.366025i | − | 0.732051i | 4.09808 | − | 1.36603i | |||||||||||||||||||
| 97.1 | −0.366025 | − | 0.366025i | 0.366025 | + | 0.366025i | − | 1.73205i | 2.00000 | − | 1.00000i | − | 0.267949i | 0 | −1.36603 | + | 1.36603i | − | 2.73205i | −1.09808 | − | 0.366025i | ||||||||||||||||||
| 97.2 | 1.36603 | + | 1.36603i | −1.36603 | − | 1.36603i | 1.73205i | 2.00000 | − | 1.00000i | − | 3.73205i | 0 | 0.366025 | − | 0.366025i | 0.732051i | 4.09808 | + | 1.36603i | ||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 35.f | even | 4 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 245.2.f.a | 4 | |
| 5.c | odd | 4 | 1 | 245.2.f.b | 4 | ||
| 7.b | odd | 2 | 1 | 245.2.f.b | 4 | ||
| 7.c | even | 3 | 1 | 35.2.k.a | ✓ | 4 | |
| 7.c | even | 3 | 1 | 245.2.l.b | 4 | ||
| 7.d | odd | 6 | 1 | 35.2.k.b | yes | 4 | |
| 7.d | odd | 6 | 1 | 245.2.l.a | 4 | ||
| 21.g | even | 6 | 1 | 315.2.bz.a | 4 | ||
| 21.h | odd | 6 | 1 | 315.2.bz.b | 4 | ||
| 28.f | even | 6 | 1 | 560.2.ci.b | 4 | ||
| 28.g | odd | 6 | 1 | 560.2.ci.a | 4 | ||
| 35.f | even | 4 | 1 | inner | 245.2.f.a | 4 | |
| 35.i | odd | 6 | 1 | 175.2.o.a | 4 | ||
| 35.j | even | 6 | 1 | 175.2.o.b | 4 | ||
| 35.k | even | 12 | 1 | 35.2.k.a | ✓ | 4 | |
| 35.k | even | 12 | 1 | 175.2.o.b | 4 | ||
| 35.k | even | 12 | 1 | 245.2.l.b | 4 | ||
| 35.l | odd | 12 | 1 | 35.2.k.b | yes | 4 | |
| 35.l | odd | 12 | 1 | 175.2.o.a | 4 | ||
| 35.l | odd | 12 | 1 | 245.2.l.a | 4 | ||
| 105.w | odd | 12 | 1 | 315.2.bz.b | 4 | ||
| 105.x | even | 12 | 1 | 315.2.bz.a | 4 | ||
| 140.w | even | 12 | 1 | 560.2.ci.b | 4 | ||
| 140.x | odd | 12 | 1 | 560.2.ci.a | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 35.2.k.a | ✓ | 4 | 7.c | even | 3 | 1 | |
| 35.2.k.a | ✓ | 4 | 35.k | even | 12 | 1 | |
| 35.2.k.b | yes | 4 | 7.d | odd | 6 | 1 | |
| 35.2.k.b | yes | 4 | 35.l | odd | 12 | 1 | |
| 175.2.o.a | 4 | 35.i | odd | 6 | 1 | ||
| 175.2.o.a | 4 | 35.l | odd | 12 | 1 | ||
| 175.2.o.b | 4 | 35.j | even | 6 | 1 | ||
| 175.2.o.b | 4 | 35.k | even | 12 | 1 | ||
| 245.2.f.a | 4 | 1.a | even | 1 | 1 | trivial | |
| 245.2.f.a | 4 | 35.f | even | 4 | 1 | inner | |
| 245.2.f.b | 4 | 5.c | odd | 4 | 1 | ||
| 245.2.f.b | 4 | 7.b | odd | 2 | 1 | ||
| 245.2.l.a | 4 | 7.d | odd | 6 | 1 | ||
| 245.2.l.a | 4 | 35.l | odd | 12 | 1 | ||
| 245.2.l.b | 4 | 7.c | even | 3 | 1 | ||
| 245.2.l.b | 4 | 35.k | even | 12 | 1 | ||
| 315.2.bz.a | 4 | 21.g | even | 6 | 1 | ||
| 315.2.bz.a | 4 | 105.x | even | 12 | 1 | ||
| 315.2.bz.b | 4 | 21.h | odd | 6 | 1 | ||
| 315.2.bz.b | 4 | 105.w | odd | 12 | 1 | ||
| 560.2.ci.a | 4 | 28.g | odd | 6 | 1 | ||
| 560.2.ci.a | 4 | 140.x | odd | 12 | 1 | ||
| 560.2.ci.b | 4 | 28.f | even | 6 | 1 | ||
| 560.2.ci.b | 4 | 140.w | even | 12 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(245, [\chi])\):
|
\( T_{2}^{4} - 2T_{2}^{3} + 2T_{2}^{2} + 2T_{2} + 1 \)
|
|
\( T_{3}^{4} + 2T_{3}^{3} + 2T_{3}^{2} - 2T_{3} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 1 \)
|
| $3$ |
\( T^{4} + 2 T^{3} + \cdots + 1 \)
|
| $5$ |
\( (T^{2} - 4 T + 5)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( (T^{2} + 2 T - 2)^{2} \)
|
| $13$ |
\( (T^{2} - 4 T + 8)^{2} \)
|
| $17$ |
\( T^{4} - 4 T^{3} + \cdots + 16 \)
|
| $19$ |
\( (T^{2} + 2 T - 2)^{2} \)
|
| $23$ |
\( T^{4} + 10 T^{3} + \cdots + 1 \)
|
| $29$ |
\( (T^{2} + 9)^{2} \)
|
| $31$ |
\( T^{4} + 56T^{2} + 16 \)
|
| $37$ |
\( T^{4} + 576 \)
|
| $41$ |
\( T^{4} + 42T^{2} + 9 \)
|
| $43$ |
\( T^{4} + 6 T^{3} + \cdots + 1089 \)
|
| $47$ |
\( T^{4} + 12 T^{3} + \cdots + 36 \)
|
| $53$ |
\( (T^{2} + 10 T + 50)^{2} \)
|
| $59$ |
\( (T^{2} + 6 T - 18)^{2} \)
|
| $61$ |
\( T^{4} + 74T^{2} + 169 \)
|
| $67$ |
\( T^{4} - 14 T^{3} + \cdots + 169 \)
|
| $71$ |
\( (T^{2} - 6 T + 6)^{2} \)
|
| $73$ |
\( T^{4} - 24 T^{3} + \cdots + 2304 \)
|
| $79$ |
\( T^{4} + 56T^{2} + 484 \)
|
| $83$ |
\( T^{4} + 2 T^{3} + \cdots + 169 \)
|
| $89$ |
\( (T^{2} - 16 T - 11)^{2} \)
|
| $97$ |
\( T^{4} + 4 T^{3} + \cdots + 8836 \)
|
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