Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [81,10,Mod(28,81)]
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([2]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("81.28");
S:= CuspForms(chi, 10);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 81 = 3^{4} \) |
| Weight: | \( k \) | \(=\) | \( 10 \) |
| Character orbit: | \([\chi]\) | \(=\) | 81.c (of order \(3\), 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: | \(41.7179027293\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{6} - \cdots)\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - x^{5} + 119x^{4} - 154x^{3} + 14060x^{2} - 16048x + 18496 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2}\cdot 3^{8} \) |
| Twist minimal: | no (minimal twist has level 27) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{6} - x^{5} + 119x^{4} - 154x^{3} + 14060x^{2} - 16048x + 18496 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{5} + 119\nu^{4} - 14161\nu^{3} + 14060\nu^{2} - 16048\nu + 1357348 ) / 552364 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 413\nu^{5} - 409\nu^{4} + 48671\nu^{3} - 6958\nu^{2} + 5750540\nu - 6563632 ) / 6628368 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -413\nu^{5} + 409\nu^{4} - 48671\nu^{3} + 6958\nu^{2} + 14134564\nu - 64736 ) / 6628368 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -88\nu^{5} + 10472\nu^{4} - 3349\nu^{3} + 1237280\nu^{2} - 1412224\nu + 97490155 ) / 138091 \)
|
| \(\beta_{5}\) | \(=\) |
\( ( -294445\nu^{5} + 288761\nu^{4} - 34362559\nu^{3} + 64278926\nu^{2} - 4059979660\nu + 4634036528 ) / 6628368 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_{2} + 1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{5} - 2\beta_{3} + 711\beta_{2} + 2\beta_1 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( \beta_{4} - 352\beta _1 + 159 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( -119\beta_{5} + 119\beta_{4} + 292\beta_{3} - 83331\beta_{2} - 83331 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -101\beta_{5} - 41516\beta_{3} + 32127\beta_{2} + 41516\beta_1 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).
| \(n\) | \(2\) |
| \(\chi(n)\) | \(\beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 28.1 |
|
−16.3980 | − | 28.4022i | 0 | −281.788 | + | 488.072i | −657.691 | + | 1139.16i | 0 | −2579.27 | − | 4467.43i | 1691.52 | 0 | 43139.3 | ||||||||||||||||||||||||||||
| 28.2 | 1.73143 | + | 2.99892i | 0 | 250.004 | − | 433.020i | 702.004 | − | 1215.91i | 0 | −832.785 | − | 1442.43i | 3504.44 | 0 | 4861.88 | |||||||||||||||||||||||||||||
| 28.3 | 16.1666 | + | 28.0013i | 0 | −266.716 | + | 461.965i | −1035.81 | + | 1794.08i | 0 | 5258.56 | + | 9108.09i | −692.954 | 0 | −66982.2 | |||||||||||||||||||||||||||||
| 55.1 | −16.3980 | + | 28.4022i | 0 | −281.788 | − | 488.072i | −657.691 | − | 1139.16i | 0 | −2579.27 | + | 4467.43i | 1691.52 | 0 | 43139.3 | |||||||||||||||||||||||||||||
| 55.2 | 1.73143 | − | 2.99892i | 0 | 250.004 | + | 433.020i | 702.004 | + | 1215.91i | 0 | −832.785 | + | 1442.43i | 3504.44 | 0 | 4861.88 | |||||||||||||||||||||||||||||
| 55.3 | 16.1666 | − | 28.0013i | 0 | −266.716 | − | 461.965i | −1035.81 | − | 1794.08i | 0 | 5258.56 | − | 9108.09i | −692.954 | 0 | −66982.2 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 9.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 81.10.c.h | 6 | |
| 3.b | odd | 2 | 1 | 81.10.c.g | 6 | ||
| 9.c | even | 3 | 1 | 27.10.a.b | ✓ | 3 | |
| 9.c | even | 3 | 1 | inner | 81.10.c.h | 6 | |
| 9.d | odd | 6 | 1 | 27.10.a.c | yes | 3 | |
| 9.d | odd | 6 | 1 | 81.10.c.g | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 27.10.a.b | ✓ | 3 | 9.c | even | 3 | 1 | |
| 27.10.a.c | yes | 3 | 9.d | odd | 6 | 1 | |
| 81.10.c.g | 6 | 3.b | odd | 2 | 1 | ||
| 81.10.c.g | 6 | 9.d | odd | 6 | 1 | ||
| 81.10.c.h | 6 | 1.a | even | 1 | 1 | trivial | |
| 81.10.c.h | 6 | 9.c | even | 3 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} - 3T_{2}^{5} + 1071T_{2}^{4} - 4158T_{2}^{3} + 1138860T_{2}^{2} - 3899664T_{2} + 13483584 \)
acting on \(S_{10}^{\mathrm{new}}(81, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} - 3 T^{5} + \cdots + 13483584 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( T^{6} + \cdots + 14\!\cdots\!25 \)
|
| $7$ |
\( T^{6} + \cdots + 81\!\cdots\!25 \)
|
| $11$ |
\( T^{6} + \cdots + 92\!\cdots\!25 \)
|
| $13$ |
\( T^{6} + \cdots + 91\!\cdots\!00 \)
|
| $17$ |
\( (T^{3} + \cdots - 78\!\cdots\!84)^{2} \)
|
| $19$ |
\( (T^{3} + \cdots + 44\!\cdots\!64)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 20\!\cdots\!84 \)
|
| $29$ |
\( T^{6} + \cdots + 18\!\cdots\!00 \)
|
| $31$ |
\( T^{6} + \cdots + 39\!\cdots\!61 \)
|
| $37$ |
\( (T^{3} + \cdots - 10\!\cdots\!80)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 13\!\cdots\!00 \)
|
| $43$ |
\( T^{6} + \cdots + 20\!\cdots\!00 \)
|
| $47$ |
\( T^{6} + \cdots + 10\!\cdots\!00 \)
|
| $53$ |
\( (T^{3} + \cdots + 13\!\cdots\!97)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 27\!\cdots\!00 \)
|
| $61$ |
\( T^{6} + \cdots + 13\!\cdots\!04 \)
|
| $67$ |
\( T^{6} + \cdots + 36\!\cdots\!00 \)
|
| $71$ |
\( (T^{3} + \cdots + 98\!\cdots\!40)^{2} \)
|
| $73$ |
\( (T^{3} + \cdots - 64\!\cdots\!45)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 69\!\cdots\!64 \)
|
| $83$ |
\( T^{6} + \cdots + 45\!\cdots\!49 \)
|
| $89$ |
\( (T^{3} + \cdots - 10\!\cdots\!00)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 67\!\cdots\!25 \)
|
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