Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [27,11,Mod(26,27)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(27, base_ring=CyclotomicField(2))
chi = DirichletCharacter(H, H._module([1]))
N = Newforms(chi, 11, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("27.26");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 27 = 3^{3} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 27.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: | \(17.1546458222\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 196x^{3} + 11881x^{2} - 21364x + 19208 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{5}\cdot 3^{21} \) |
| 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
\( x^{6} - 196x^{3} + 11881x^{2} - 21364x + 19208 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -39349\nu^{5} - 35378\nu^{4} - 124264\nu^{3} + 8895068\nu^{2} - 474116157\nu + 432503890 ) / 8398110 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -1092\nu^{5} - 22209\nu^{4} - 119028\nu^{3} + 107016\nu^{2} - 144804102 ) / 142825 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -4128\nu^{5} + 144879\nu^{4} - 449952\nu^{3} + 404544\nu^{2} + 1265125722 ) / 142825 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( - 3642017 \nu^{5} - 3274474 \nu^{4} - 1326038 \nu^{3} + 268737511 \nu^{2} - 42773545215 \nu + 39033977318 ) / 20995275 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( - 2429719 \nu^{5} - 2184518 \nu^{4} + 29008784 \nu^{3} + 817582052 \nu^{2} - 25277368455 \nu + 23111397526 ) / 13996850 \)
|
| \(\nu\) | \(=\) |
\( ( 27\beta_{5} + 16\beta_{4} + 9\beta_{3} + 63\beta_{2} - 1751\beta_1 ) / 17496 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 27\beta_{5} - 146\beta_{4} + 4405\beta_1 ) / 4374 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 2943\beta_{5} + 2528\beta_{4} - 981\beta_{3} - 6867\beta_{2} - 185371\beta _1 + 1714608 ) / 17496 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 455\beta_{3} - 1720\beta_{2} - 5774166 ) / 729 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -103401\beta_{5} - 110928\beta_{4} - 38387\beta_{3} - 233429\beta_{2} + 7310733\beta _1 + 103829040 ) / 5832 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 26.1 |
|
− | 49.7909i | 0 | −1455.14 | 4680.95i | 0 | 10645.1 | 21466.7i | 0 | 233069. | |||||||||||||||||||||||||||||||||||
| 26.2 | − | 43.0534i | 0 | −829.595 | − | 2154.41i | 0 | −11290.6 | − | 8369.79i | 0 | −92754.7 | ||||||||||||||||||||||||||||||||||
| 26.3 | − | 8.26247i | 0 | 955.732 | − | 2842.36i | 0 | 2964.51 | − | 16357.5i | 0 | −23484.9 | ||||||||||||||||||||||||||||||||||
| 26.4 | 8.26247i | 0 | 955.732 | 2842.36i | 0 | 2964.51 | 16357.5i | 0 | −23484.9 | |||||||||||||||||||||||||||||||||||||
| 26.5 | 43.0534i | 0 | −829.595 | 2154.41i | 0 | −11290.6 | 8369.79i | 0 | −92754.7 | |||||||||||||||||||||||||||||||||||||
| 26.6 | 49.7909i | 0 | −1455.14 | − | 4680.95i | 0 | 10645.1 | − | 21466.7i | 0 | 233069. | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 27.11.b.d | ✓ | 6 |
| 3.b | odd | 2 | 1 | inner | 27.11.b.d | ✓ | 6 |
| 9.c | even | 3 | 2 | 81.11.d.g | 12 | ||
| 9.d | odd | 6 | 2 | 81.11.d.g | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.11.b.d | ✓ | 6 | 1.a | even | 1 | 1 | trivial |
| 27.11.b.d | ✓ | 6 | 3.b | odd | 2 | 1 | inner |
| 81.11.d.g | 12 | 9.c | even | 3 | 2 | ||
| 81.11.d.g | 12 | 9.d | odd | 6 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + 4401T_{2}^{4} + 4891104T_{2}^{2} + 313714944 \)
acting on \(S_{11}^{\mathrm{new}}(27, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + 4401 T^{4} + \cdots + 313714944 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 82\!\cdots\!25 \)
|
| $7$ |
\( (T^{3} - 2319 T^{2} + \cdots + 356301122275)^{2} \)
|
| $11$ |
\( T^{6} + \cdots + 13\!\cdots\!25 \)
|
| $13$ |
\( (T^{3} + \cdots + 50\!\cdots\!00)^{2} \)
|
| $17$ |
\( T^{6} + \cdots + 36\!\cdots\!04 \)
|
| $19$ |
\( (T^{3} + \cdots + 30\!\cdots\!36)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 23\!\cdots\!00 \)
|
| $29$ |
\( T^{6} + \cdots + 51\!\cdots\!00 \)
|
| $31$ |
\( (T^{3} + \cdots - 10\!\cdots\!79)^{2} \)
|
| $37$ |
\( (T^{3} + \cdots + 40\!\cdots\!00)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $43$ |
\( (T^{3} + \cdots + 15\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{6} + \cdots + 89\!\cdots\!16 \)
|
| $53$ |
\( T^{6} + \cdots + 28\!\cdots\!89 \)
|
| $59$ |
\( T^{6} + \cdots + 17\!\cdots\!00 \)
|
| $61$ |
\( (T^{3} + \cdots + 61\!\cdots\!16)^{2} \)
|
| $67$ |
\( (T^{3} + \cdots - 53\!\cdots\!00)^{2} \)
|
| $71$ |
\( T^{6} + \cdots + 63\!\cdots\!00 \)
|
| $73$ |
\( (T^{3} + \cdots + 49\!\cdots\!75)^{2} \)
|
| $79$ |
\( (T^{3} + \cdots + 76\!\cdots\!20)^{2} \)
|
| $83$ |
\( T^{6} + \cdots + 23\!\cdots\!89 \)
|
| $89$ |
\( T^{6} + \cdots + 35\!\cdots\!00 \)
|
| $97$ |
\( (T^{3} + \cdots - 23\!\cdots\!75)^{2} \)
|
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