Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,11,Mod(26,81)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(81, base_ring=CyclotomicField(6))
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("81.26");
S:= CuspForms(chi, 11);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 11 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.d (of order \(6\), degree \(2\), not 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: | \(51.4639374666\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Relative dimension: | \(4\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{8} + \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{8} + 86x^{6} + 6052x^{4} + 115584x^{2} + 1806336 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{4}\cdot 3^{19} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{8} + 86x^{6} + 6052x^{4} + 115584x^{2} + 1806336 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -43\nu^{6} - 3026\nu^{4} - 260236\nu^{2} - 4970112 ) / 4066944 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 43\nu^{7} + 3026\nu^{5} + 260236\nu^{3} + 9037056\nu ) / 1355648 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 43\nu^{7} + 3026\nu^{5} + 260236\nu^{3} - 3163776\nu ) / 1355648 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -13633\nu^{7} - 2236214\nu^{5} - 137410660\nu^{3} - 4581228288\nu ) / 2711296 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( 27\nu^{6} - 3905604 ) / 6052 \)
|
| \(\beta_{6}\) | \(=\) |
\( ( -10593\nu^{6} - 1171062\nu^{4} - 64108836\nu^{2} - 1224381312 ) / 1355648 \)
|
| \(\beta_{7}\) | \(=\) |
\( ( -40849\nu^{7} - 2236214\nu^{5} - 137410660\nu^{3} + 1289452416\nu ) / 2711296 \)
|
| \(\nu\) | \(=\) |
\( ( -\beta_{3} + \beta_{2} ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{6} - \beta_{5} - 1161\beta _1 - 1161 ) / 27 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 8\beta_{7} + 4\beta_{4} + 2956\beta_{3} + 1478\beta_{2} ) / 243 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -86\beta_{6} + 63558\beta_1 ) / 27 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -344\beta_{7} - 688\beta_{4} - 90820\beta_{3} - 181640\beta_{2} ) / 243 \)
|
| \(\nu^{6}\) | \(=\) |
\( ( 6052\beta_{5} + 3905604 ) / 27 \)
|
| \(\nu^{7}\) | \(=\) |
\( ( -24208\beta_{7} + 24208\beta_{4} - 5824088\beta_{3} + 5824088\beta_{2} ) / 243 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(-\beta_{1}\) |
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 |
|
−36.4117 | − | 21.0223i | 0 | 371.875 | + | 644.106i | 4203.27 | − | 2426.76i | 0 | −15913.0 | + | 27562.1i | 11783.0i | 0 | −204064. | ||||||||||||||||||||||||||||||||||
| 26.2 | −20.3884 | − | 11.7713i | 0 | −234.875 | − | 406.815i | −3831.75 | + | 2212.26i | 0 | 10784.0 | − | 18678.4i | 35166.6i | 0 | 104164. | |||||||||||||||||||||||||||||||||||
| 26.3 | 20.3884 | + | 11.7713i | 0 | −234.875 | − | 406.815i | 3831.75 | − | 2212.26i | 0 | 10784.0 | − | 18678.4i | − | 35166.6i | 0 | 104164. | ||||||||||||||||||||||||||||||||||
| 26.4 | 36.4117 | + | 21.0223i | 0 | 371.875 | + | 644.106i | −4203.27 | + | 2426.76i | 0 | −15913.0 | + | 27562.1i | − | 11783.0i | 0 | −204064. | ||||||||||||||||||||||||||||||||||
| 53.1 | −36.4117 | + | 21.0223i | 0 | 371.875 | − | 644.106i | 4203.27 | + | 2426.76i | 0 | −15913.0 | − | 27562.1i | − | 11783.0i | 0 | −204064. | ||||||||||||||||||||||||||||||||||
| 53.2 | −20.3884 | + | 11.7713i | 0 | −234.875 | + | 406.815i | −3831.75 | − | 2212.26i | 0 | 10784.0 | + | 18678.4i | − | 35166.6i | 0 | 104164. | ||||||||||||||||||||||||||||||||||
| 53.3 | 20.3884 | − | 11.7713i | 0 | −234.875 | + | 406.815i | 3831.75 | + | 2212.26i | 0 | 10784.0 | + | 18678.4i | 35166.6i | 0 | 104164. | |||||||||||||||||||||||||||||||||||
| 53.4 | 36.4117 | − | 21.0223i | 0 | 371.875 | − | 644.106i | −4203.27 | − | 2426.76i | 0 | −15913.0 | − | 27562.1i | 11783.0i | 0 | −204064. | |||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 9.c | even | 3 | 1 | inner |
| 9.d | odd | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.11.d.e | 8 | |
| 3.b | odd | 2 | 1 | inner | 81.11.d.e | 8 | |
| 9.c | even | 3 | 1 | 27.11.b.c | ✓ | 4 | |
| 9.c | even | 3 | 1 | inner | 81.11.d.e | 8 | |
| 9.d | odd | 6 | 1 | 27.11.b.c | ✓ | 4 | |
| 9.d | odd | 6 | 1 | inner | 81.11.d.e | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.11.b.c | ✓ | 4 | 9.c | even | 3 | 1 | |
| 27.11.b.c | ✓ | 4 | 9.d | odd | 6 | 1 | |
| 81.11.d.e | 8 | 1.a | even | 1 | 1 | trivial | |
| 81.11.d.e | 8 | 3.b | odd | 2 | 1 | inner | |
| 81.11.d.e | 8 | 9.c | even | 3 | 1 | inner | |
| 81.11.d.e | 8 | 9.d | odd | 6 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{8} - 2322T_{2}^{6} + 4411908T_{2}^{4} - 2275039872T_{2}^{2} + 959961010176 \)
acting on \(S_{11}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} + \cdots + 959961010176 \)
|
| $3$ |
\( T^{8} \)
|
| $5$ |
\( T^{8} + \cdots + 21\!\cdots\!00 \)
|
| $7$ |
\( (T^{4} + \cdots + 47\!\cdots\!41)^{2} \)
|
| $11$ |
\( T^{8} + \cdots + 60\!\cdots\!96 \)
|
| $13$ |
\( (T^{4} + \cdots + 11\!\cdots\!25)^{2} \)
|
| $17$ |
\( (T^{4} + \cdots + 15\!\cdots\!36)^{2} \)
|
| $19$ |
\( (T^{2} + \cdots + 3727788408121)^{4} \)
|
| $23$ |
\( T^{8} + \cdots + 25\!\cdots\!16 \)
|
| $29$ |
\( T^{8} + \cdots + 68\!\cdots\!36 \)
|
| $31$ |
\( (T^{4} + \cdots + 94\!\cdots\!00)^{2} \)
|
| $37$ |
\( (T^{2} + \cdots + 487162727648041)^{4} \)
|
| $41$ |
\( T^{8} + \cdots + 30\!\cdots\!36 \)
|
| $43$ |
\( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \)
|
| $47$ |
\( T^{8} + \cdots + 57\!\cdots\!36 \)
|
| $53$ |
\( (T^{4} + \cdots + 17\!\cdots\!56)^{2} \)
|
| $59$ |
\( T^{8} + \cdots + 73\!\cdots\!00 \)
|
| $61$ |
\( (T^{4} + \cdots + 54\!\cdots\!21)^{2} \)
|
| $67$ |
\( (T^{4} + \cdots + 44\!\cdots\!21)^{2} \)
|
| $71$ |
\( (T^{4} + \cdots + 80\!\cdots\!16)^{2} \)
|
| $73$ |
\( (T^{2} + \cdots - 51\!\cdots\!55)^{4} \)
|
| $79$ |
\( (T^{4} + \cdots + 15\!\cdots\!21)^{2} \)
|
| $83$ |
\( T^{8} + \cdots + 85\!\cdots\!16 \)
|
| $89$ |
\( (T^{4} + \cdots + 15\!\cdots\!96)^{2} \)
|
| $97$ |
\( (T^{4} + \cdots + 47\!\cdots\!21)^{2} \)
|
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