Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [135,4,Mod(46,135)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(135, base_ring=CyclotomicField(6))
chi = DirichletCharacter(H, H._module([4, 0]))
N = Newforms(chi, 4, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("135.46");
S:= CuspForms(chi, 4);
N := Newforms(S);
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.e (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: | \(7.96525785077\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Relative dimension: | \(3\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | 6.0.15759792.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
| Coefficient ring index: | \( 3^{3} \) |
| Twist minimal: | no (minimal twist has level 45) |
| 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
\( x^{6} - 3x^{5} + 16x^{4} - 27x^{3} + 52x^{2} - 39x + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{2} + \nu - 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{4} - 2\nu^{3} + 10\nu^{2} - 9\nu + 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 2\nu^{5} - 5\nu^{4} + 30\nu^{3} - 40\nu^{2} + 88\nu - 39 ) / 3 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -8\nu^{5} + 20\nu^{4} - 117\nu^{3} + 157\nu^{2} - 334\nu + 147 ) / 3 \)
|
| \(\beta_{5}\) | \(=\) |
\( 5\nu^{5} - 12\nu^{4} + 72\nu^{3} - 92\nu^{2} + 198\nu - 81 \)
|
| \(\nu\) | \(=\) |
\( ( -2\beta_{5} - 4\beta_{4} - \beta_{3} + \beta_{2} - 2\beta _1 + 4 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -2\beta_{5} - 4\beta_{4} - \beta_{3} + \beta_{2} - 11\beta _1 - 32 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 10\beta_{5} + 29\beta_{4} + 41\beta_{3} - 5\beta_{2} + \beta _1 - 29 ) / 9 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 22\beta_{5} + 62\beta_{4} + 83\beta_{3} - 2\beta_{2} + 94\beta _1 + 217 ) / 9 \)
|
| \(\nu^{5}\) | \(=\) |
\( ( -47\beta_{5} - 184\beta_{4} - 370\beta_{3} + 46\beta_{2} + 88\beta _1 + 337 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(\beta_{3}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−2.28679 | + | 3.96084i | 0 | −6.45882 | − | 11.1870i | 2.50000 | + | 4.33013i | 0 | 10.0573 | − | 17.4197i | 22.4912 | 0 | −22.8679 | ||||||||||||||||||||||||||||
| 46.2 | −0.0874923 | + | 0.151541i | 0 | 3.98469 | + | 6.90169i | 2.50000 | + | 4.33013i | 0 | −4.23186 | + | 7.32979i | −2.79440 | 0 | −0.874923 | |||||||||||||||||||||||||||||
| 46.3 | 1.87428 | − | 3.24635i | 0 | −3.02587 | − | 5.24096i | 2.50000 | + | 4.33013i | 0 | 15.6746 | − | 27.1492i | 7.30318 | 0 | 18.7428 | |||||||||||||||||||||||||||||
| 91.1 | −2.28679 | − | 3.96084i | 0 | −6.45882 | + | 11.1870i | 2.50000 | − | 4.33013i | 0 | 10.0573 | + | 17.4197i | 22.4912 | 0 | −22.8679 | |||||||||||||||||||||||||||||
| 91.2 | −0.0874923 | − | 0.151541i | 0 | 3.98469 | − | 6.90169i | 2.50000 | − | 4.33013i | 0 | −4.23186 | − | 7.32979i | −2.79440 | 0 | −0.874923 | |||||||||||||||||||||||||||||
| 91.3 | 1.87428 | + | 3.24635i | 0 | −3.02587 | + | 5.24096i | 2.50000 | − | 4.33013i | 0 | 15.6746 | + | 27.1492i | 7.30318 | 0 | 18.7428 | |||||||||||||||||||||||||||||
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 | 135.4.e.b | 6 | |
| 3.b | odd | 2 | 1 | 45.4.e.b | ✓ | 6 | |
| 9.c | even | 3 | 1 | inner | 135.4.e.b | 6 | |
| 9.c | even | 3 | 1 | 405.4.a.j | 3 | ||
| 9.d | odd | 6 | 1 | 45.4.e.b | ✓ | 6 | |
| 9.d | odd | 6 | 1 | 405.4.a.h | 3 | ||
| 15.d | odd | 2 | 1 | 225.4.e.c | 6 | ||
| 15.e | even | 4 | 2 | 225.4.k.c | 12 | ||
| 45.h | odd | 6 | 1 | 225.4.e.c | 6 | ||
| 45.h | odd | 6 | 1 | 2025.4.a.s | 3 | ||
| 45.j | even | 6 | 1 | 2025.4.a.q | 3 | ||
| 45.l | even | 12 | 2 | 225.4.k.c | 12 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 45.4.e.b | ✓ | 6 | 3.b | odd | 2 | 1 | |
| 45.4.e.b | ✓ | 6 | 9.d | odd | 6 | 1 | |
| 135.4.e.b | 6 | 1.a | even | 1 | 1 | trivial | |
| 135.4.e.b | 6 | 9.c | even | 3 | 1 | inner | |
| 225.4.e.c | 6 | 15.d | odd | 2 | 1 | ||
| 225.4.e.c | 6 | 45.h | odd | 6 | 1 | ||
| 225.4.k.c | 12 | 15.e | even | 4 | 2 | ||
| 225.4.k.c | 12 | 45.l | even | 12 | 2 | ||
| 405.4.a.h | 3 | 9.d | odd | 6 | 1 | ||
| 405.4.a.j | 3 | 9.c | even | 3 | 1 | ||
| 2025.4.a.q | 3 | 45.j | even | 6 | 1 | ||
| 2025.4.a.s | 3 | 45.h | odd | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{6} + T_{2}^{5} + 18T_{2}^{4} - 11T_{2}^{3} + 292T_{2}^{2} + 51T_{2} + 9 \)
acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} + T^{5} + 18 T^{4} + \cdots + 9 \)
|
| $3$ |
\( T^{6} \)
|
| $5$ |
\( (T^{2} - 5 T + 25)^{3} \)
|
| $7$ |
\( T^{6} - 43 T^{5} + \cdots + 28483569 \)
|
| $11$ |
\( T^{6} + \cdots + 1896428304 \)
|
| $13$ |
\( T^{6} + \cdots + 5679732496 \)
|
| $17$ |
\( (T^{3} - 166 T^{2} + \cdots - 156324)^{2} \)
|
| $19$ |
\( (T^{3} + 164 T^{2} + \cdots + 57316)^{2} \)
|
| $23$ |
\( T^{6} + \cdots + 3746541681 \)
|
| $29$ |
\( T^{6} + \cdots + 11463342489 \)
|
| $31$ |
\( T^{6} + \cdots + 97238137683600 \)
|
| $37$ |
\( (T^{3} - 402 T^{2} + \cdots + 3335284)^{2} \)
|
| $41$ |
\( T^{6} + \cdots + 52247959475625 \)
|
| $43$ |
\( T^{6} + \cdots + 238533321586576 \)
|
| $47$ |
\( T^{6} + \cdots + 6187571675289 \)
|
| $53$ |
\( (T^{3} - 730 T^{2} + \cdots - 3250536)^{2} \)
|
| $59$ |
\( T^{6} + \cdots + 16\!\cdots\!16 \)
|
| $61$ |
\( T^{6} + \cdots + 30\!\cdots\!09 \)
|
| $67$ |
\( T^{6} + \cdots + 43\!\cdots\!81 \)
|
| $71$ |
\( (T^{3} + 70 T^{2} + \cdots - 223775052)^{2} \)
|
| $73$ |
\( (T^{3} + 368 T^{2} + \cdots - 134927744)^{2} \)
|
| $79$ |
\( T^{6} + \cdots + 19\!\cdots\!56 \)
|
| $83$ |
\( T^{6} + \cdots + 91\!\cdots\!41 \)
|
| $89$ |
\( (T^{3} + 1719 T^{2} + \cdots - 125506395)^{2} \)
|
| $97$ |
\( T^{6} + \cdots + 285551326213696 \)
|
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