Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [110,4,Mod(89,110)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(110, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("110.89");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 110 = 2 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 110.b (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: | \(6.49021010063\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{89})\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} + 45x^{2} + 484 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{3} \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 45x^{2} + 484 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 23\nu ) / 11 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 45 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 45 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 22\beta_{2} - 23\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/110\mathbb{Z}\right)^\times\).
| \(n\) | \(67\) | \(101\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 89.1 |
|
− | 2.00000i | 2.00000i | −4.00000 | −9.43398 | − | 6.00000i | 4.00000 | 26.4340i | 8.00000i | 23.0000 | −12.0000 | + | 18.8680i | |||||||||||||||||||||||||
| 89.2 | − | 2.00000i | 2.00000i | −4.00000 | 9.43398 | − | 6.00000i | 4.00000 | 7.56602i | 8.00000i | 23.0000 | −12.0000 | − | 18.8680i | ||||||||||||||||||||||||||
| 89.3 | 2.00000i | − | 2.00000i | −4.00000 | −9.43398 | + | 6.00000i | 4.00000 | − | 26.4340i | − | 8.00000i | 23.0000 | −12.0000 | − | 18.8680i | ||||||||||||||||||||||||
| 89.4 | 2.00000i | − | 2.00000i | −4.00000 | 9.43398 | + | 6.00000i | 4.00000 | − | 7.56602i | − | 8.00000i | 23.0000 | −12.0000 | + | 18.8680i | ||||||||||||||||||||||||
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 | 110.4.b.b | ✓ | 4 |
| 3.b | odd | 2 | 1 | 990.4.c.e | 4 | ||
| 4.b | odd | 2 | 1 | 880.4.b.d | 4 | ||
| 5.b | even | 2 | 1 | inner | 110.4.b.b | ✓ | 4 |
| 5.c | odd | 4 | 1 | 550.4.a.p | 2 | ||
| 5.c | odd | 4 | 1 | 550.4.a.x | 2 | ||
| 15.d | odd | 2 | 1 | 990.4.c.e | 4 | ||
| 20.d | odd | 2 | 1 | 880.4.b.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 110.4.b.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 110.4.b.b | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 550.4.a.p | 2 | 5.c | odd | 4 | 1 | ||
| 550.4.a.x | 2 | 5.c | odd | 4 | 1 | ||
| 880.4.b.d | 4 | 4.b | odd | 2 | 1 | ||
| 880.4.b.d | 4 | 20.d | odd | 2 | 1 | ||
| 990.4.c.e | 4 | 3.b | odd | 2 | 1 | ||
| 990.4.c.e | 4 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{2} + 4 \)
acting on \(S_{4}^{\mathrm{new}}(110, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4)^{2} \)
|
| $3$ |
\( (T^{2} + 4)^{2} \)
|
| $5$ |
\( T^{4} - 106 T^{2} + 15625 \)
|
| $7$ |
\( T^{4} + 756 T^{2} + 40000 \)
|
| $11$ |
\( (T - 11)^{4} \)
|
| $13$ |
\( T^{4} + 8820 T^{2} + 18593344 \)
|
| $17$ |
\( T^{4} + 12772 T^{2} + 5456896 \)
|
| $19$ |
\( (T^{2} - 48 T - 12240)^{2} \)
|
| $23$ |
\( T^{4} + 12648 T^{2} + 12082576 \)
|
| $29$ |
\( (T^{2} - 112 T - 39940)^{2} \)
|
| $31$ |
\( (T^{2} + 280 T + 18176)^{2} \)
|
| $37$ |
\( T^{4} + 81872 T^{2} + 280093696 \)
|
| $41$ |
\( (T^{2} + 472 T - 4468)^{2} \)
|
| $43$ |
\( T^{4} + 98100 T^{2} + 867184704 \)
|
| $47$ |
\( T^{4} + \cdots + 5124841744 \)
|
| $53$ |
\( T^{4} + \cdots + 16433188864 \)
|
| $59$ |
\( (T^{2} + 896 T + 199280)^{2} \)
|
| $61$ |
\( (T^{2} + 84 T - 459612)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 1571646736 \)
|
| $71$ |
\( (T^{2} - 1680 T + 682816)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 230968670464 \)
|
| $79$ |
\( (T^{2} + 1100 T + 222400)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 1099763300416 \)
|
| $89$ |
\( (T^{2} + 856 T - 603220)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 377723326464 \)
|
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