Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [520,2,Mod(19,520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(520, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([6, 6, 6, 5]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("520.19");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 520 = 2^{3} \cdot 5 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 520.cz (of order \(12\), degree \(4\), 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: | \(8\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{12})\) |
| Coefficient field: | 8.0.3317760000.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 25x^{4} + 625 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{12}]$ |
$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) 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
\( x^{8} - 25x^{4} + 625 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 5 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 5 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( \nu^{4} ) / 25 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{5} ) / 25 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( \nu^{6} ) / 125 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( \nu^{7} ) / 125 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 5\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 5\beta_{3} \)
|
| \(\nu^{4}\) | \(=\) |
\( 25\beta_{4} \)
|
| \(\nu^{5}\) | \(=\) |
\( 25\beta_{5} \)
|
| \(\nu^{6}\) | \(=\) |
\( 125\beta_{6} \)
|
| \(\nu^{7}\) | \(=\) |
\( 125\beta_{7} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/520\mathbb{Z}\right)^\times\).
| \(n\) | \(41\) | \(261\) | \(391\) | \(417\) |
| \(\chi(n)\) | \(-\beta_{2} + \beta_{6}\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 19.1 |
|
0.366025 | + | 1.36603i | 0 | −1.73205 | + | 1.00000i | −1.58114 | + | 1.58114i | 0 | 1.21271 | − | 4.52590i | −2.00000 | − | 2.00000i | −1.50000 | − | 2.59808i | −2.73861 | − | 1.58114i | ||||||||||||||||||||||||||||
| 19.2 | 0.366025 | + | 1.36603i | 0 | −1.73205 | + | 1.00000i | 1.58114 | − | 1.58114i | 0 | 0.0552376 | − | 0.206150i | −2.00000 | − | 2.00000i | −1.50000 | − | 2.59808i | 2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||
| 59.1 | −1.36603 | + | 0.366025i | 0 | 1.73205 | − | 1.00000i | −1.58114 | − | 1.58114i | 0 | 0.206150 | + | 0.0552376i | −2.00000 | + | 2.00000i | −1.50000 | − | 2.59808i | 2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||
| 59.2 | −1.36603 | + | 0.366025i | 0 | 1.73205 | − | 1.00000i | 1.58114 | + | 1.58114i | 0 | 4.52590 | + | 1.21271i | −2.00000 | + | 2.00000i | −1.50000 | − | 2.59808i | −2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||
| 219.1 | 0.366025 | − | 1.36603i | 0 | −1.73205 | − | 1.00000i | −1.58114 | − | 1.58114i | 0 | 1.21271 | + | 4.52590i | −2.00000 | + | 2.00000i | −1.50000 | + | 2.59808i | −2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||
| 219.2 | 0.366025 | − | 1.36603i | 0 | −1.73205 | − | 1.00000i | 1.58114 | + | 1.58114i | 0 | 0.0552376 | + | 0.206150i | −2.00000 | + | 2.00000i | −1.50000 | + | 2.59808i | 2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||
| 379.1 | −1.36603 | − | 0.366025i | 0 | 1.73205 | + | 1.00000i | −1.58114 | + | 1.58114i | 0 | 0.206150 | − | 0.0552376i | −2.00000 | − | 2.00000i | −1.50000 | + | 2.59808i | 2.73861 | − | 1.58114i | |||||||||||||||||||||||||||||
| 379.2 | −1.36603 | − | 0.366025i | 0 | 1.73205 | + | 1.00000i | 1.58114 | − | 1.58114i | 0 | 4.52590 | − | 1.21271i | −2.00000 | − | 2.00000i | −1.50000 | + | 2.59808i | −2.73861 | + | 1.58114i | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 40.e | odd | 2 | 1 | CM by \(\Q(\sqrt{-10}) \) |
| 13.f | odd | 12 | 1 | inner |
| 520.cz | even | 12 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 520.2.cz.a | ✓ | 8 |
| 5.b | even | 2 | 1 | 520.2.cz.b | yes | 8 | |
| 8.d | odd | 2 | 1 | 520.2.cz.b | yes | 8 | |
| 13.f | odd | 12 | 1 | inner | 520.2.cz.a | ✓ | 8 |
| 40.e | odd | 2 | 1 | CM | 520.2.cz.a | ✓ | 8 |
| 65.s | odd | 12 | 1 | 520.2.cz.b | yes | 8 | |
| 104.u | even | 12 | 1 | 520.2.cz.b | yes | 8 | |
| 520.cz | even | 12 | 1 | inner | 520.2.cz.a | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 520.2.cz.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 520.2.cz.a | ✓ | 8 | 13.f | odd | 12 | 1 | inner |
| 520.2.cz.a | ✓ | 8 | 40.e | odd | 2 | 1 | CM |
| 520.2.cz.a | ✓ | 8 | 520.cz | even | 12 | 1 | inner |
| 520.2.cz.b | yes | 8 | 5.b | even | 2 | 1 | |
| 520.2.cz.b | yes | 8 | 8.d | odd | 2 | 1 | |
| 520.2.cz.b | yes | 8 | 65.s | odd | 12 | 1 | |
| 520.2.cz.b | yes | 8 | 104.u | even | 12 | 1 | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(520, [\chi])\):
|
\( T_{3} \)
|
|
\( T_{7}^{8} - 12T_{7}^{7} + 72T_{7}^{6} - 288T_{7}^{5} + 623T_{7}^{4} - 288T_{7}^{3} + 72T_{7}^{2} - 12T_{7} + 1 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} + 2 T^{3} + 2 T^{2} + \cdots + 4)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 25)^{2} \)
|
| $7$ |
\( T^{8} - 12 T^{7} + \cdots + 1 \)
|
| $11$ |
\( T^{8} - 4 T^{7} + \cdots + 1521 \)
|
| $13$ |
\( T^{8} - 8 T^{7} + \cdots + 28561 \)
|
| $17$ |
\( T^{8} \)
|
| $19$ |
\( T^{8} + 24 T^{7} + \cdots + 1164241 \)
|
| $23$ |
\( T^{8} - 92 T^{6} + \cdots + 456976 \)
|
| $29$ |
\( T^{8} \)
|
| $31$ |
\( T^{8} \)
|
| $37$ |
\( T^{8} + 16 T^{7} + \cdots + 83521 \)
|
| $41$ |
\( T^{8} + 4 T^{7} + \cdots + 37015056 \)
|
| $43$ |
\( T^{8} \)
|
| $47$ |
\( (T^{4} + 4 T^{3} + \cdots + 17689)^{2} \)
|
| $53$ |
\( (T^{4} + 8 T^{3} + \cdots - 2951)^{2} \)
|
| $59$ |
\( T^{8} + 28 T^{7} + \cdots + 37015056 \)
|
| $61$ |
\( T^{8} \)
|
| $67$ |
\( T^{8} \)
|
| $71$ |
\( T^{8} \)
|
| $73$ |
\( T^{8} \)
|
| $79$ |
\( T^{8} \)
|
| $83$ |
\( T^{8} \)
|
| $89$ |
\( T^{8} + 56 T^{7} + \cdots + 549386721 \)
|
| $97$ |
\( T^{8} \)
|
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