Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1520,2,Mod(881,1520)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1520, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([0, 0, 0, 2]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1520.881");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1520 = 2^{4} \cdot 5 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1520.q (of order \(3\), 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: | \(12.1372611072\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 8.0.4601315889.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 760) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{8} - x^{7} + 6x^{6} - 3x^{5} + 26x^{4} - 14x^{3} + 31x^{2} + 12x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -26\nu^{7} - 189\nu^{6} + 729\nu^{5} - 911\nu^{4} + 3051\nu^{3} - 3618\nu^{2} + 14317\nu - 1215 ) / 4243 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 115\nu^{7} + 20\nu^{6} + 529\nu^{5} + 276\nu^{4} + 3314\nu^{3} + 989\nu^{2} + 483\nu + 1947 ) / 4243 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -135\nu^{7} + 161\nu^{6} - 621\nu^{5} - 324\nu^{4} - 2599\nu^{3} - 1161\nu^{2} - 567\nu - 11694 ) / 4243 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -649\nu^{7} + 994\nu^{6} - 3834\nu^{5} + 3534\nu^{4} - 16046\nu^{3} + 19028\nu^{2} - 17152\nu + 6390 ) / 12729 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 434\nu^{7} - 109\nu^{6} + 2845\nu^{5} + 193\nu^{4} + 12064\nu^{3} + 338\nu^{2} + 16249\nu + 6573 ) / 4243 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 514\nu^{7} - 833\nu^{6} + 3213\nu^{5} - 3858\nu^{4} + 13447\nu^{3} - 15946\nu^{2} + 16585\nu - 5355 ) / 4243 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + 3\beta_{5} - \beta_{4} - 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{6} + 4\beta_{3} + \beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( -5\beta_{7} - 12\beta_{5} + \beta_{2} \)
|
| \(\nu^{5}\) | \(=\) |
\( -\beta_{7} + 6\beta_{6} + \beta_{4} - 17\beta_{3} - 17\beta_1 \)
|
| \(\nu^{6}\) | \(=\) |
\( 7\beta_{6} + 23\beta_{4} - \beta_{3} - 7\beta_{2} + 51 \)
|
| \(\nu^{7}\) | \(=\) |
\( 8\beta_{7} + 3\beta_{5} - 30\beta_{2} + 74\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1520\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(401\) | \(1141\) | \(1217\) |
| \(\chi(n)\) | \(1\) | \(-1 + \beta_{5}\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 881.1 |
|
0 | −1.37901 | − | 2.38851i | 0 | −0.500000 | − | 0.866025i | 0 | −4.11474 | 0 | −2.30333 | + | 3.98948i | 0 | ||||||||||||||||||||||||||||||||||||
| 881.2 | 0 | −0.548719 | − | 0.950409i | 0 | −0.500000 | − | 0.866025i | 0 | 0.416295 | 0 | 0.897815 | − | 1.55506i | 0 | |||||||||||||||||||||||||||||||||||||
| 881.3 | 0 | 0.595455 | + | 1.03136i | 0 | −0.500000 | − | 0.866025i | 0 | 3.62891 | 0 | 0.790867 | − | 1.36982i | 0 | |||||||||||||||||||||||||||||||||||||
| 881.4 | 0 | 0.832272 | + | 1.44154i | 0 | −0.500000 | − | 0.866025i | 0 | −1.93047 | 0 | 0.114645 | − | 0.198571i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.1 | 0 | −1.37901 | + | 2.38851i | 0 | −0.500000 | + | 0.866025i | 0 | −4.11474 | 0 | −2.30333 | − | 3.98948i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.2 | 0 | −0.548719 | + | 0.950409i | 0 | −0.500000 | + | 0.866025i | 0 | 0.416295 | 0 | 0.897815 | + | 1.55506i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.3 | 0 | 0.595455 | − | 1.03136i | 0 | −0.500000 | + | 0.866025i | 0 | 3.62891 | 0 | 0.790867 | + | 1.36982i | 0 | |||||||||||||||||||||||||||||||||||||
| 961.4 | 0 | 0.832272 | − | 1.44154i | 0 | −0.500000 | + | 0.866025i | 0 | −1.93047 | 0 | 0.114645 | + | 0.198571i | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 19.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1520.2.q.k | 8 | |
| 4.b | odd | 2 | 1 | 760.2.q.f | ✓ | 8 | |
| 19.c | even | 3 | 1 | inner | 1520.2.q.k | 8 | |
| 76.g | odd | 6 | 1 | 760.2.q.f | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 760.2.q.f | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 760.2.q.f | ✓ | 8 | 76.g | odd | 6 | 1 | |
| 1520.2.q.k | 8 | 1.a | even | 1 | 1 | trivial | |
| 1520.2.q.k | 8 | 19.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1520, [\chi])\):
|
\( T_{3}^{8} + T_{3}^{7} + 7T_{3}^{6} - 4T_{3}^{5} + 31T_{3}^{4} - 6T_{3}^{3} + 37T_{3}^{2} - 6T_{3} + 36 \)
|
|
\( T_{7}^{4} + 2T_{7}^{3} - 15T_{7}^{2} - 23T_{7} + 12 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} + T^{7} + \cdots + 36 \)
|
| $5$ |
\( (T^{2} + T + 1)^{4} \)
|
| $7$ |
\( (T^{4} + 2 T^{3} - 15 T^{2} + \cdots + 12)^{2} \)
|
| $11$ |
\( (T^{4} - 4 T^{3} - 15 T^{2} + \cdots + 27)^{2} \)
|
| $13$ |
\( T^{8} - T^{7} + \cdots + 144 \)
|
| $17$ |
\( T^{8} - 13 T^{7} + \cdots + 1296 \)
|
| $19$ |
\( T^{8} + T^{7} + \cdots + 130321 \)
|
| $23$ |
\( T^{8} + 8 T^{7} + \cdots + 27556 \)
|
| $29$ |
\( T^{8} + 3 T^{7} + \cdots + 7921 \)
|
| $31$ |
\( (T^{4} + 2 T^{3} - 39 T^{2} + \cdots - 69)^{2} \)
|
| $37$ |
\( (T^{4} - 10 T^{3} + \cdots - 346)^{2} \)
|
| $41$ |
\( T^{8} + 8 T^{7} + \cdots + 27556 \)
|
| $43$ |
\( T^{8} - 3 T^{7} + \cdots + 1444 \)
|
| $47$ |
\( T^{8} + 10 T^{7} + \cdots + 1024 \)
|
| $53$ |
\( T^{8} + 11 T^{7} + \cdots + 381924 \)
|
| $59$ |
\( T^{8} + T^{7} + \cdots + 410881 \)
|
| $61$ |
\( T^{8} + 74 T^{6} + \cdots + 961 \)
|
| $67$ |
\( T^{8} - 8 T^{7} + \cdots + 1024 \)
|
| $71$ |
\( T^{8} - 4 T^{7} + \cdots + 5958481 \)
|
| $73$ |
\( T^{8} + 20 T^{7} + \cdots + 68644 \)
|
| $79$ |
\( T^{8} - 3 T^{7} + \cdots + 20720704 \)
|
| $83$ |
\( (T^{4} - 15 T^{3} + \cdots - 3634)^{2} \)
|
| $89$ |
\( T^{8} - 17 T^{7} + \cdots + 76457536 \)
|
| $97$ |
\( T^{8} - 11 T^{7} + \cdots + 68644 \)
|
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