Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [1110,2,Mod(889,1110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(1110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1, 0]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("1110.889");
S:= CuspForms(chi, 2);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 1110 = 2 \cdot 3 \cdot 5 \cdot 37 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1110.d (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: | \(8.86339462436\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.5161984.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| 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 \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 4x^{3} + 25x^{2} - 20x + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -5\nu^{5} - 2\nu^{4} - 25\nu^{3} + 10\nu^{2} - 121\nu + 100 ) / 121 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 7\nu^{5} + 27\nu^{4} + 35\nu^{3} - 14\nu^{2} + 223 ) / 121 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -25\nu^{5} - 10\nu^{4} - 4\nu^{3} + 50\nu^{2} - 605\nu + 258 ) / 242 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -65\nu^{5} - 26\nu^{4} + 38\nu^{3} + 372\nu^{2} - 1331\nu + 574 ) / 242 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{5} - 3\beta_{4} + \beta_{2} - \beta_1 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} - 5\beta_{2} + 2 \)
|
| \(\nu^{4}\) | \(=\) |
\( 5\beta_{3} + 7\beta_{2} + 7\beta _1 - 15 \)
|
| \(\nu^{5}\) | \(=\) |
\( 2\beta_{5} - 16\beta_{4} - 2\beta_{3} - 29\beta _1 + 16 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1110\mathbb{Z}\right)^\times\).
| \(n\) | \(371\) | \(631\) | \(667\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 889.1 |
|
− | 1.00000i | − | 1.00000i | −1.00000 | −2.19388 | + | 0.432320i | −1.00000 | 1.86464i | 1.00000i | −1.00000 | 0.432320 | + | 2.19388i | ||||||||||||||||||||||||||||||
| 889.2 | − | 1.00000i | − | 1.00000i | −1.00000 | 1.38900 | − | 1.75233i | −1.00000 | − | 2.50466i | 1.00000i | −1.00000 | −1.75233 | − | 1.38900i | ||||||||||||||||||||||||||||||
| 889.3 | − | 1.00000i | − | 1.00000i | −1.00000 | 1.80487 | + | 1.32001i | −1.00000 | 3.64002i | 1.00000i | −1.00000 | 1.32001 | − | 1.80487i | |||||||||||||||||||||||||||||||
| 889.4 | 1.00000i | 1.00000i | −1.00000 | −2.19388 | − | 0.432320i | −1.00000 | − | 1.86464i | − | 1.00000i | −1.00000 | 0.432320 | − | 2.19388i | |||||||||||||||||||||||||||||||
| 889.5 | 1.00000i | 1.00000i | −1.00000 | 1.38900 | + | 1.75233i | −1.00000 | 2.50466i | − | 1.00000i | −1.00000 | −1.75233 | + | 1.38900i | ||||||||||||||||||||||||||||||||
| 889.6 | 1.00000i | 1.00000i | −1.00000 | 1.80487 | − | 1.32001i | −1.00000 | − | 3.64002i | − | 1.00000i | −1.00000 | 1.32001 | + | 1.80487i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1110.2.d.i | ✓ | 6 |
| 3.b | odd | 2 | 1 | 3330.2.d.n | 6 | ||
| 5.b | even | 2 | 1 | inner | 1110.2.d.i | ✓ | 6 |
| 5.c | odd | 4 | 1 | 5550.2.a.cf | 3 | ||
| 5.c | odd | 4 | 1 | 5550.2.a.cg | 3 | ||
| 15.d | odd | 2 | 1 | 3330.2.d.n | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 1110.2.d.i | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 1110.2.d.i | ✓ | 6 | 5.b | even | 2 | 1 | inner |
| 3330.2.d.n | 6 | 3.b | odd | 2 | 1 | ||
| 3330.2.d.n | 6 | 15.d | odd | 2 | 1 | ||
| 5550.2.a.cf | 3 | 5.c | odd | 4 | 1 | ||
| 5550.2.a.cg | 3 | 5.c | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1110, [\chi])\):
|
\( T_{7}^{6} + 23T_{7}^{4} + 151T_{7}^{2} + 289 \)
|
|
\( T_{11}^{3} + 5T_{11}^{2} - 11T_{11} - 59 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{3} \)
|
| $3$ |
\( (T^{2} + 1)^{3} \)
|
| $5$ |
\( T^{6} - 2 T^{5} + \cdots + 125 \)
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| $7$ |
\( T^{6} + 23 T^{4} + \cdots + 289 \)
|
| $11$ |
\( (T^{3} + 5 T^{2} - 11 T - 59)^{2} \)
|
| $13$ |
\( T^{6} + 14 T^{4} + \cdots + 64 \)
|
| $17$ |
\( T^{6} + 55 T^{4} + \cdots + 1 \)
|
| $19$ |
\( (T^{3} + 10 T^{2} + 25 T + 8)^{2} \)
|
| $23$ |
\( T^{6} + 82 T^{4} + \cdots + 256 \)
|
| $29$ |
\( (T^{3} - 17 T^{2} + \cdots + 50)^{2} \)
|
| $31$ |
\( (T^{3} + T^{2} - 24 T + 20)^{2} \)
|
| $37$ |
\( (T^{2} + 1)^{3} \)
|
| $41$ |
\( (T^{3} + 9 T^{2} - 30 T - 34)^{2} \)
|
| $43$ |
\( T^{6} + 185 T^{4} + \cdots + 59536 \)
|
| $47$ |
\( T^{6} + 44 T^{4} + \cdots + 64 \)
|
| $53$ |
\( T^{6} + 91 T^{4} + \cdots + 121 \)
|
| $59$ |
\( (T^{3} - 4 T^{2} + \cdots + 232)^{2} \)
|
| $61$ |
\( (T^{3} - 3 T^{2} + \cdots + 108)^{2} \)
|
| $67$ |
\( T^{6} + 312 T^{4} + \cdots + 295936 \)
|
| $71$ |
\( (T^{3} - 2 T^{2} + \cdots - 148)^{2} \)
|
| $73$ |
\( T^{6} + 54 T^{4} + \cdots + 4 \)
|
| $79$ |
\( (T^{3} - 12 T^{2} + \cdots - 16)^{2} \)
|
| $83$ |
\( T^{6} + 206 T^{4} + \cdots + 1936 \)
|
| $89$ |
\( (T^{3} + 6 T^{2} - 45 T + 4)^{2} \)
|
| $97$ |
\( T^{6} + 493 T^{4} + \cdots + 1085764 \)
|
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