Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [2640,2,Mod(1871,2640)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(2640, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0, 1, 0, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("2640.1871");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 2640 = 2^{4} \cdot 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 2640.k (of order \(2\), degree \(1\), 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: | \(21.0805061336\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.195105024.2 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{8} - 4x^{7} + 5x^{6} + 4x^{5} - 20x^{4} + 12x^{3} + 45x^{2} - 108x + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{7} - 4\nu^{6} + 5\nu^{5} + 4\nu^{4} - 20\nu^{3} + 12\nu^{2} + 45\nu - 81 ) / 27 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{7} - 13\nu^{6} + 2\nu^{5} + 22\nu^{4} - 26\nu^{3} - 54\nu^{2} + 225\nu - 243 ) / 108 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 5\nu^{7} - 17\nu^{6} + 22\nu^{5} + 26\nu^{4} - 70\nu^{3} - 18\nu^{2} + 243\nu - 351 ) / 108 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 5\nu^{7} - 8\nu^{6} + 4\nu^{5} + 26\nu^{4} - 52\nu^{3} - 18\nu^{2} + 153\nu - 216 ) / 54 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -3\nu^{7} + 8\nu^{6} - 2\nu^{5} - 20\nu^{4} + 38\nu^{3} + 32\nu^{2} - 141\nu + 144 ) / 36 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 10\nu^{7} - 25\nu^{6} + 8\nu^{5} + 70\nu^{4} - 104\nu^{3} - 54\nu^{2} + 378\nu - 459 ) / 108 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{7} + \beta_{6} - \beta_{4} + \beta_{2} + \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{7} + 2\beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta _1 - 1 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{7} + 3\beta_{6} + 2\beta_{4} - 3\beta_{3} - 3\beta_{2} + \beta_1 \)
|
| \(\nu^{5}\) | \(=\) |
\( 4\beta_{6} + 2\beta_{5} + 10\beta_{4} - 2\beta_{3} - 5\beta_{2} + 5 \)
|
| \(\nu^{6}\) | \(=\) |
\( -4\beta_{7} + 4\beta_{6} + 12\beta_{5} + 6\beta_{4} - 6\beta_{3} - 8\beta_{2} + 8\beta _1 - 3 \)
|
| \(\nu^{7}\) | \(=\) |
\( -8\beta_{7} + 12\beta_{6} + 18\beta_{5} - 2\beta_{4} + 18\beta_{3} - 9\beta _1 + 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2640\mathbb{Z}\right)^\times\).
| \(n\) | \(661\) | \(881\) | \(991\) | \(1057\) | \(1201\) |
| \(\chi(n)\) | \(1\) | \(-1\) | \(-1\) | \(1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1871.1 |
|
0 | −1.58726 | − | 0.693255i | 0 | − | 1.00000i | 0 | 0.613491i | 0 | 2.03880 | + | 2.20075i | 0 | |||||||||||||||||||||||||||||||||||||
| 1871.2 | 0 | −1.58726 | + | 0.693255i | 0 | 1.00000i | 0 | − | 0.613491i | 0 | 2.03880 | − | 2.20075i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1871.3 | 0 | 0.560908 | − | 1.63871i | 0 | − | 1.00000i | 0 | − | 1.27743i | 0 | −2.37076 | − | 1.83834i | 0 | |||||||||||||||||||||||||||||||||||||
| 1871.4 | 0 | 0.560908 | + | 1.63871i | 0 | 1.00000i | 0 | 1.27743i | 0 | −2.37076 | + | 1.83834i | 0 | |||||||||||||||||||||||||||||||||||||||
| 1871.5 | 0 | 1.30512 | − | 1.13871i | 0 | 1.00000i | 0 | − | 4.27743i | 0 | 0.406663 | − | 2.97231i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1871.6 | 0 | 1.30512 | + | 1.13871i | 0 | − | 1.00000i | 0 | 4.27743i | 0 | 0.406663 | + | 2.97231i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1871.7 | 0 | 1.72124 | − | 0.193255i | 0 | 1.00000i | 0 | − | 2.38651i | 0 | 2.92531 | − | 0.665273i | 0 | ||||||||||||||||||||||||||||||||||||||
| 1871.8 | 0 | 1.72124 | + | 0.193255i | 0 | − | 1.00000i | 0 | 2.38651i | 0 | 2.92531 | + | 0.665273i | 0 | ||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 12.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 2640.2.k.f | yes | 8 |
| 3.b | odd | 2 | 1 | 2640.2.k.d | ✓ | 8 | |
| 4.b | odd | 2 | 1 | 2640.2.k.d | ✓ | 8 | |
| 12.b | even | 2 | 1 | inner | 2640.2.k.f | yes | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 2640.2.k.d | ✓ | 8 | 3.b | odd | 2 | 1 | |
| 2640.2.k.d | ✓ | 8 | 4.b | odd | 2 | 1 | |
| 2640.2.k.f | yes | 8 | 1.a | even | 1 | 1 | trivial |
| 2640.2.k.f | yes | 8 | 12.b | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2640, [\chi])\):
|
\( T_{7}^{8} + 26T_{7}^{6} + 153T_{7}^{4} + 224T_{7}^{2} + 64 \)
|
|
\( T_{23}^{4} + 4T_{23}^{3} - 40T_{23}^{2} - 112T_{23} + 208 \)
|
|
\( T_{47}^{4} - 12T_{47}^{3} - 4T_{47}^{2} + 432T_{47} - 1184 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} \)
|
| $3$ |
\( T^{8} - 4 T^{7} + \cdots + 81 \)
|
| $5$ |
\( (T^{2} + 1)^{4} \)
|
| $7$ |
\( T^{8} + 26 T^{6} + \cdots + 64 \)
|
| $11$ |
\( (T - 1)^{8} \)
|
| $13$ |
\( (T^{2} - 4 T - 8)^{4} \)
|
| $17$ |
\( T^{8} + 66 T^{6} + \cdots + 256 \)
|
| $19$ |
\( T^{8} + 102 T^{6} + \cdots + 16 \)
|
| $23$ |
\( (T^{4} + 4 T^{3} + \cdots + 208)^{2} \)
|
| $29$ |
\( T^{8} + 134 T^{6} + \cdots + 952576 \)
|
| $31$ |
\( T^{8} + 54 T^{6} + \cdots + 16 \)
|
| $37$ |
\( (T^{4} - 12 T^{3} + \cdots - 92)^{2} \)
|
| $41$ |
\( T^{8} + 192 T^{6} + \cdots + 16384 \)
|
| $43$ |
\( T^{8} + 152 T^{6} + \cdots + 1024 \)
|
| $47$ |
\( (T^{4} - 12 T^{3} + \cdots - 1184)^{2} \)
|
| $53$ |
\( T^{8} + 282 T^{6} + \cdots + 10061584 \)
|
| $59$ |
\( (T^{4} - 108 T^{2} + \cdots + 576)^{2} \)
|
| $61$ |
\( (T^{4} - 14 T^{3} + \cdots - 488)^{2} \)
|
| $67$ |
\( T^{8} + 248 T^{6} + \cdots + 2166784 \)
|
| $71$ |
\( (T^{4} + 22 T^{3} + \cdots - 908)^{2} \)
|
| $73$ |
\( (T^{4} + 12 T^{3} + \cdots - 4608)^{2} \)
|
| $79$ |
\( T^{8} + 232 T^{6} + \cdots + 173056 \)
|
| $83$ |
\( (T^{4} + 28 T^{3} + \cdots - 128)^{2} \)
|
| $89$ |
\( T^{8} + 454 T^{6} + \cdots + 59228416 \)
|
| $97$ |
\( (T^{4} + 36 T^{3} + \cdots - 6656)^{2} \)
|
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