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: | 8.0.81622204416.6 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 14x^{6} + 165x^{4} + 434x^{2} + 961 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{4} \) |
| 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
\( x^{8} + 14x^{6} + 165x^{4} + 434x^{2} + 961 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{6} + 110\nu^{4} - 1870\nu^{2} - 2139 ) / 5115 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 14\nu^{7} + 165\nu^{5} + 2310\nu^{3} + 961\nu ) / 5115 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 26\nu^{7} + 550\nu^{5} + 5995\nu^{3} + 28334\nu ) / 5115 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 14\nu^{7} + 165\nu^{5} + 2310\nu^{3} + 11191\nu ) / 1705 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( \nu^{6} - 721 ) / 165 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -28\nu^{7} - 330\nu^{5} - 3597\nu^{3} - 1922\nu ) / 1023 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( 83\nu^{6} + 1100\nu^{4} + 11990\nu^{2} + 18972 ) / 5115 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{4} - 3\beta_{2} ) / 6 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{7} - 3\beta_{5} - 10\beta _1 - 21 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( \beta_{6} + 10\beta_{2} \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 17\beta_{7} - 42\beta_{5} + 109\beta _1 - 201 ) / 6 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -42\beta_{6} - 109\beta_{4} + 126\beta_{3} - 327\beta_{2} ) / 6 \)
|
| \(\nu^{6}\) | \(=\) |
\( 165\beta_{5} + 721 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -495\beta_{6} + 1216\beta_{4} - 1485\beta_{3} - 3648\beta_{2} ) / 6 \)
|
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 |
|
−3.35300 | 0 | 7.24264 | − | 2.40558i | 0 | 2.24264 | + | 6.63103i | −10.8726 | 0 | 8.06591i | |||||||||||||||||||||||||||||||||||||||
| 55.2 | −3.35300 | 0 | 7.24264 | 2.40558i | 0 | 2.24264 | − | 6.63103i | −10.8726 | 0 | − | 8.06591i | ||||||||||||||||||||||||||||||||||||||||
| 55.3 | −1.66053 | 0 | −1.24264 | − | 6.94357i | 0 | −6.24264 | + | 3.16693i | 8.70556 | 0 | 11.5300i | ||||||||||||||||||||||||||||||||||||||||
| 55.4 | −1.66053 | 0 | −1.24264 | 6.94357i | 0 | −6.24264 | − | 3.16693i | 8.70556 | 0 | − | 11.5300i | ||||||||||||||||||||||||||||||||||||||||
| 55.5 | 1.66053 | 0 | −1.24264 | − | 6.94357i | 0 | −6.24264 | − | 3.16693i | −8.70556 | 0 | − | 11.5300i | |||||||||||||||||||||||||||||||||||||||
| 55.6 | 1.66053 | 0 | −1.24264 | 6.94357i | 0 | −6.24264 | + | 3.16693i | −8.70556 | 0 | 11.5300i | |||||||||||||||||||||||||||||||||||||||||
| 55.7 | 3.35300 | 0 | 7.24264 | − | 2.40558i | 0 | 2.24264 | − | 6.63103i | 10.8726 | 0 | − | 8.06591i | |||||||||||||||||||||||||||||||||||||||
| 55.8 | 3.35300 | 0 | 7.24264 | 2.40558i | 0 | 2.24264 | + | 6.63103i | 10.8726 | 0 | 8.06591i | |||||||||||||||||||||||||||||||||||||||||
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.d | ✓ | 8 |
| 3.b | odd | 2 | 1 | inner | 189.3.d.d | ✓ | 8 |
| 7.b | odd | 2 | 1 | inner | 189.3.d.d | ✓ | 8 |
| 21.c | even | 2 | 1 | inner | 189.3.d.d | ✓ | 8 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.3.d.d | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 189.3.d.d | ✓ | 8 | 3.b | odd | 2 | 1 | inner |
| 189.3.d.d | ✓ | 8 | 7.b | odd | 2 | 1 | inner |
| 189.3.d.d | ✓ | 8 | 21.c | even | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - 14T_{2}^{2} + 31 \)
acting on \(S_{3}^{\mathrm{new}}(189, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{4} - 14 T^{2} + 31)^{2} \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( (T^{4} + 54 T^{2} + 279)^{2} \)
|
| $7$ |
\( (T^{4} + 8 T^{3} + \cdots + 2401)^{2} \)
|
| $11$ |
\( (T^{4} - 14 T^{2} + 31)^{2} \)
|
| $13$ |
\( (T^{4} + 396 T^{2} + 34596)^{2} \)
|
| $17$ |
\( (T^{4} + 756 T^{2} + 90396)^{2} \)
|
| $19$ |
\( (T^{4} + 162 T^{2} + 729)^{2} \)
|
| $23$ |
\( (T^{4} - 2702 T^{2} + 1770751)^{2} \)
|
| $29$ |
\( (T^{4} - 1724 T^{2} + 65596)^{2} \)
|
| $31$ |
\( (T^{4} + 3618 T^{2} + 2752281)^{2} \)
|
| $37$ |
\( (T^{2} - 58 T + 769)^{4} \)
|
| $41$ |
\( (T^{4} + 8118 T^{2} + 15936759)^{2} \)
|
| $43$ |
\( (T^{2} + 44 T - 398)^{4} \)
|
| $47$ |
\( (T^{4} + 9324 T^{2} + 10500444)^{2} \)
|
| $53$ |
\( (T^{4} - 1736 T^{2} + 476656)^{2} \)
|
| $59$ |
\( (T^{4} + 9972 T^{2} + 1875996)^{2} \)
|
| $61$ |
\( (T^{4} + 5976 T^{2} + 3283344)^{2} \)
|
| $67$ |
\( (T^{2} - 88 T - 1106)^{4} \)
|
| $71$ |
\( (T^{4} - 21266 T^{2} + 71528191)^{2} \)
|
| $73$ |
\( (T^{4} + 11556 T^{2} + 23097636)^{2} \)
|
| $79$ |
\( (T^{2} - 28 T + 178)^{4} \)
|
| $83$ |
\( (T^{4} + 13356 T^{2} + 41569884)^{2} \)
|
| $89$ |
\( (T^{4} + 12150 T^{2} + 14124375)^{2} \)
|
| $97$ |
\( (T^{4} + 23256 T^{2} + 57335184)^{2} \)
|
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