Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [520,2,Mod(61,520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(520, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 3, 0, 4]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("520.61");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 520 = 2^{3} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 520.by (of order \(6\), 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: | \(4.15222090511\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| 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: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| 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 primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(261\) | \(391\) | \(417\) |
| \(\chi(n)\) | \(-1 + \zeta_{12}^{2}\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 61.1 |
|
−0.366025 | − | 1.36603i | −1.73205 | − | 1.00000i | −1.73205 | + | 1.00000i | − | 1.00000i | −0.732051 | + | 2.73205i | 1.50000 | + | 2.59808i | 2.00000 | + | 2.00000i | 0.500000 | + | 0.866025i | −1.36603 | + | 0.366025i | |||||||||||||
| 61.2 | 1.36603 | − | 0.366025i | 1.73205 | + | 1.00000i | 1.73205 | − | 1.00000i | 1.00000i | 2.73205 | + | 0.732051i | 1.50000 | + | 2.59808i | 2.00000 | − | 2.00000i | 0.500000 | + | 0.866025i | 0.366025 | + | 1.36603i | |||||||||||||||
| 341.1 | −0.366025 | + | 1.36603i | −1.73205 | + | 1.00000i | −1.73205 | − | 1.00000i | 1.00000i | −0.732051 | − | 2.73205i | 1.50000 | − | 2.59808i | 2.00000 | − | 2.00000i | 0.500000 | − | 0.866025i | −1.36603 | − | 0.366025i | |||||||||||||||
| 341.2 | 1.36603 | + | 0.366025i | 1.73205 | − | 1.00000i | 1.73205 | + | 1.00000i | − | 1.00000i | 2.73205 | − | 0.732051i | 1.50000 | − | 2.59808i | 2.00000 | + | 2.00000i | 0.500000 | − | 0.866025i | 0.366025 | − | 1.36603i | ||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
| 13.c | even | 3 | 1 | inner |
| 104.r | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 520.2.by.b | ✓ | 4 |
| 8.b | even | 2 | 1 | inner | 520.2.by.b | ✓ | 4 |
| 13.c | even | 3 | 1 | inner | 520.2.by.b | ✓ | 4 |
| 104.r | even | 6 | 1 | inner | 520.2.by.b | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.by.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 520.2.by.b | ✓ | 4 | 8.b | even | 2 | 1 | inner |
| 520.2.by.b | ✓ | 4 | 13.c | even | 3 | 1 | inner |
| 520.2.by.b | ✓ | 4 | 104.r | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{4} - 4T_{3}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 2 T^{3} + \cdots + 4 \)
|
| $3$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $5$ |
\( (T^{2} + 1)^{2} \)
|
| $7$ |
\( (T^{2} - 3 T + 9)^{2} \)
|
| $11$ |
\( T^{4} - 9T^{2} + 81 \)
|
| $13$ |
\( T^{4} - T^{2} + 169 \)
|
| $17$ |
\( (T^{2} + 2 T + 4)^{2} \)
|
| $19$ |
\( T^{4} - 25T^{2} + 625 \)
|
| $23$ |
\( (T^{2} + 4 T + 16)^{2} \)
|
| $29$ |
\( T^{4} - 64T^{2} + 4096 \)
|
| $31$ |
\( (T - 4)^{4} \)
|
| $37$ |
\( T^{4} - 121 T^{2} + 14641 \)
|
| $41$ |
\( (T^{2} - 10 T + 100)^{2} \)
|
| $43$ |
\( T^{4} - 16T^{2} + 256 \)
|
| $47$ |
\( (T - 5)^{4} \)
|
| $53$ |
\( (T^{2} + 121)^{2} \)
|
| $59$ |
\( T^{4} - 16T^{2} + 256 \)
|
| $61$ |
\( T^{4} - 4T^{2} + 16 \)
|
| $67$ |
\( T^{4} - 100 T^{2} + 10000 \)
|
| $71$ |
\( (T^{2} - 12 T + 144)^{2} \)
|
| $73$ |
\( (T + 10)^{4} \)
|
| $79$ |
\( (T - 8)^{4} \)
|
| $83$ |
\( (T^{2} + 16)^{2} \)
|
| $89$ |
\( (T^{2} + 15 T + 225)^{2} \)
|
| $97$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
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