Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,3,Mod(55,189)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(189, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([0, 1]))
N = Newforms(chi, 3, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.55");
S:= CuspForms(chi, 3);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 189.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: | \(5.14987699641\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | 8.0.2355463701504.14 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 9x^{6} + 79x^{4} + 18x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{4} \) |
| 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
\( x^{8} + 9x^{6} + 79x^{4} + 18x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{6} - 377 ) / 79 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 44\nu^{6} + 395\nu^{4} + 3397\nu^{2} + 397 ) / 79 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 34\nu^{7} + 316\nu^{5} + 2765\nu^{3} + 1244\nu ) / 79 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 36\nu^{7} + 316\nu^{5} + 2765\nu^{3} + 16\nu ) / 79 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -81\nu^{6} - 711\nu^{4} - 6399\nu^{2} - 747 ) / 79 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( 95\nu^{7} + 869\nu^{5} + 7663\nu^{3} + 3448\nu ) / 79 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -99\nu^{7} - 869\nu^{5} - 7663\nu^{3} - 44\nu ) / 79 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{7} + \beta_{6} + 2\beta_{4} - 2\beta_{3} ) / 12 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -5\beta_{5} - 9\beta_{2} - 9\beta _1 - 45 ) / 18 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{7} - 11\beta_{4} ) / 3 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 43\beta_{5} + 81\beta_{2} - 81\beta _1 - 387 ) / 18 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( 35\beta_{7} - 35\beta_{6} + 97\beta_{4} + 97\beta_{3} ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 79\beta _1 + 377 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( 307\beta_{7} + 307\beta_{6} + 851\beta_{4} - 851\beta_{3} ) / 6 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/189\mathbb{Z}\right)^\times\).
| \(n\) | \(29\) | \(136\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 55.1 |
|
−3.71106 | 0 | 9.77200 | − | 8.26535i | 0 | −2.77200 | − | 6.42775i | −21.4203 | 0 | 30.6732i | |||||||||||||||||||||||||||||||||||||||
| 55.2 | −3.71106 | 0 | 9.77200 | 8.26535i | 0 | −2.77200 | + | 6.42775i | −21.4203 | 0 | − | 30.6732i | ||||||||||||||||||||||||||||||||||||||||
| 55.3 | −2.28648 | 0 | 1.22800 | − | 6.53330i | 0 | 5.77200 | + | 3.96030i | 6.33813 | 0 | 14.9383i | ||||||||||||||||||||||||||||||||||||||||
| 55.4 | −2.28648 | 0 | 1.22800 | 6.53330i | 0 | 5.77200 | − | 3.96030i | 6.33813 | 0 | − | 14.9383i | ||||||||||||||||||||||||||||||||||||||||
| 55.5 | 2.28648 | 0 | 1.22800 | − | 6.53330i | 0 | 5.77200 | − | 3.96030i | −6.33813 | 0 | − | 14.9383i | |||||||||||||||||||||||||||||||||||||||
| 55.6 | 2.28648 | 0 | 1.22800 | 6.53330i | 0 | 5.77200 | + | 3.96030i | −6.33813 | 0 | 14.9383i | |||||||||||||||||||||||||||||||||||||||||
| 55.7 | 3.71106 | 0 | 9.77200 | − | 8.26535i | 0 | −2.77200 | + | 6.42775i | 21.4203 | 0 | − | 30.6732i | |||||||||||||||||||||||||||||||||||||||
| 55.8 | 3.71106 | 0 | 9.77200 | 8.26535i | 0 | −2.77200 | − | 6.42775i | 21.4203 | 0 | 30.6732i | |||||||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 7.b | odd | 2 | 1 | inner |
| 21.c | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 189.3.d.e | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 189.3.d.e | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 189.3.d.e | ✓ | 8 |
| 21.c | even | 2 | 1 | inner | 189.3.d.e | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.3.d.e | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 189.3.d.e | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 189.3.d.e | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 189.3.d.e | ✓ | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 19T_{2}^{2} + 72 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 19 T^{2} + 72)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 111 T^{2} + 2916)^{2} \)
|
| $7$ |
\( (T^{4} - 6 T^{3} + \cdots + 2401)^{2} \)
|
| $11$ |
\( (T^{4} - 280 T^{2} + 18432)^{2} \)
|
| $13$ |
\( (T^{4} + 57 T^{2} + 648)^{2} \)
|
| $17$ |
\( (T^{2} + 243)^{4} \)
|
| $19$ |
\( (T^{4} + 972 T^{2} + 23328)^{2} \)
|
| $23$ |
\( (T^{4} - 835 T^{2} + 2592)^{2} \)
|
| $29$ |
\( (T^{4} - 1435 T^{2} + 5832)^{2} \)
|
| $31$ |
\( (T^{4} + 3585 T^{2} + 887112)^{2} \)
|
| $37$ |
\( (T^{2} + 75 T + 1388)^{4} \)
|
| $41$ |
\( (T^{4} + 2163 T^{2} + 944784)^{2} \)
|
| $43$ |
\( (T^{2} + 42 T + 149)^{4} \)
|
| $47$ |
\( (T^{4} + 2775 T^{2} + 1822500)^{2} \)
|
| $53$ |
\( (T^{4} - 1963 T^{2} + 103968)^{2} \)
|
| $59$ |
\( (T^{4} + 4494 T^{2} + 3272481)^{2} \)
|
| $61$ |
\( (T^{4} + 12684 T^{2} + 12340512)^{2} \)
|
| $67$ |
\( (T^{2} - 55 T - 3350)^{4} \)
|
| $71$ |
\( (T^{4} - 6355 T^{2} + 609408)^{2} \)
|
| $73$ |
\( (T^{4} + 2052 T^{2} + 839808)^{2} \)
|
| $79$ |
\( (T^{2} - 175 T + 7492)^{4} \)
|
| $83$ |
\( (T^{4} + 18651 T^{2} + 43401744)^{2} \)
|
| $89$ |
\( (T^{4} + 5859 T^{2} + 3779136)^{2} \)
|
| $97$ |
\( (T^{4} + 20172 T^{2} + 41806368)^{2} \)
|
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