Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [189,6,Mod(1,189)]
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, 0]))
N = Newforms(chi, 6, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("189.1");
S:= CuspForms(chi, 6);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 189 = 3^{3} \cdot 7 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 189.a (trivial) |
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: | yes |
| Analytic conductor: | \(30.3125419447\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - x^{4} - 121x^{3} + 265x^{2} + 3580x - 11620 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2\cdot 3^{4} \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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 \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - x^{4} - 121x^{3} + 265x^{2} + 3580x - 11620 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{4} + 2\nu^{3} - 95\nu^{2} - 100\nu + 1940 ) / 40 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{2} + 9\nu - 148 ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( -3\nu^{2} + 9\nu + 144 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{4} + 44\nu^{3} - 665\nu^{2} - 2470\nu + 16480 ) / 40 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + 2\beta_{2} + 4 ) / 18 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -\beta_{3} + 2\beta_{2} + 292 ) / 6 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 24\beta_{4} + 59\beta_{3} + 118\beta_{2} - 168\beta _1 - 1504 ) / 18 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -16\beta_{4} - 101\beta_{3} + 178\beta_{2} + 352\beta _1 + 17236 ) / 6 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−10.3071 | 0 | 74.2371 | −107.925 | 0 | 49.0000 | −435.344 | 0 | 1112.40 | |||||||||||||||||||||||||||||||||
| 1.2 | −8.75089 | 0 | 44.5781 | 53.4758 | 0 | 49.0000 | −110.069 | 0 | −467.960 | ||||||||||||||||||||||||||||||||||
| 1.3 | −2.60945 | 0 | −25.1908 | −44.5138 | 0 | 49.0000 | 149.236 | 0 | 116.156 | ||||||||||||||||||||||||||||||||||
| 1.4 | 5.08011 | 0 | −6.19245 | 32.4109 | 0 | 49.0000 | −194.022 | 0 | 164.651 | ||||||||||||||||||||||||||||||||||
| 1.5 | 7.58736 | 0 | 25.5681 | −44.4480 | 0 | 49.0000 | −48.8014 | 0 | −337.243 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( -1 \) |
| \(7\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 189.6.a.m | ✓ | 5 |
| 3.b | odd | 2 | 1 | 189.6.a.n | yes | 5 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 189.6.a.m | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 189.6.a.n | yes | 5 | 3.b | odd | 2 | 1 | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} + 9T_{2}^{4} - 96T_{2}^{3} - 702T_{2}^{2} + 2412T_{2} + 9072 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(189))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} + 9 T^{4} + \cdots + 9072 \)
|
| $3$ |
\( T^{5} \)
|
| $5$ |
\( T^{5} + 111 T^{4} + \cdots + 370097856 \)
|
| $7$ |
\( (T - 49)^{5} \)
|
| $11$ |
\( T^{5} + \cdots + 212686718976 \)
|
| $13$ |
\( T^{5} + \cdots - 20384169483488 \)
|
| $17$ |
\( T^{5} + \cdots - 19\!\cdots\!57 \)
|
| $19$ |
\( T^{5} + \cdots - 35990315233856 \)
|
| $23$ |
\( T^{5} + \cdots + 57\!\cdots\!36 \)
|
| $29$ |
\( T^{5} + \cdots - 93\!\cdots\!56 \)
|
| $31$ |
\( T^{5} + \cdots - 23\!\cdots\!56 \)
|
| $37$ |
\( T^{5} + \cdots - 39\!\cdots\!24 \)
|
| $41$ |
\( T^{5} + \cdots + 98\!\cdots\!68 \)
|
| $43$ |
\( T^{5} + \cdots - 27\!\cdots\!19 \)
|
| $47$ |
\( T^{5} + \cdots - 60\!\cdots\!16 \)
|
| $53$ |
\( T^{5} + \cdots - 59\!\cdots\!68 \)
|
| $59$ |
\( T^{5} + \cdots - 26\!\cdots\!55 \)
|
| $61$ |
\( T^{5} + \cdots + 62\!\cdots\!48 \)
|
| $67$ |
\( T^{5} + \cdots + 73\!\cdots\!04 \)
|
| $71$ |
\( T^{5} + \cdots - 56\!\cdots\!88 \)
|
| $73$ |
\( T^{5} + \cdots - 50\!\cdots\!00 \)
|
| $79$ |
\( T^{5} + \cdots - 67\!\cdots\!84 \)
|
| $83$ |
\( T^{5} + \cdots + 36\!\cdots\!16 \)
|
| $89$ |
\( T^{5} + \cdots - 17\!\cdots\!96 \)
|
| $97$ |
\( T^{5} + \cdots - 19\!\cdots\!00 \)
|
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