Newspace parameters
| Level: | \( N \) | \(=\) | \( 135 = 3^{3} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 135.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.96525785077\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 3x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 3^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} + 3x^{2} + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{3} + 4\nu \)
|
| \(\beta_{2}\) | \(=\) |
\( -5\nu^{3} - 11\nu \)
|
| \(\beta_{3}\) | \(=\) |
\( 9\nu^{2} + 14 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 9 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 14 ) / 9 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -4\beta_{2} - 11\beta_1 ) / 9 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/135\mathbb{Z}\right)^\times\).
| \(n\) | \(56\) | \(82\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 109.1 |
|
− | 5.61803i | 0 | −23.5623 | 11.1803i | 0 | 0 | 87.4296i | 0 | 62.8115 | |||||||||||||||||||||||||||||
| 109.2 | − | 3.38197i | 0 | −3.43769 | − | 11.1803i | 0 | 0 | − | 15.4296i | 0 | −37.8115 | ||||||||||||||||||||||||||||
| 109.3 | 3.38197i | 0 | −3.43769 | 11.1803i | 0 | 0 | 15.4296i | 0 | −37.8115 | |||||||||||||||||||||||||||||||
| 109.4 | 5.61803i | 0 | −23.5623 | − | 11.1803i | 0 | 0 | − | 87.4296i | 0 | 62.8115 | |||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-15}) \) |
| 3.b | odd | 2 | 1 | inner |
| 5.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 135.4.b.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 135.4.b.a | ✓ | 4 |
| 5.b | even | 2 | 1 | inner | 135.4.b.a | ✓ | 4 |
| 5.c | odd | 4 | 1 | 675.4.a.k | 2 | ||
| 5.c | odd | 4 | 1 | 675.4.a.o | 2 | ||
| 15.d | odd | 2 | 1 | CM | 135.4.b.a | ✓ | 4 |
| 15.e | even | 4 | 1 | 675.4.a.k | 2 | ||
| 15.e | even | 4 | 1 | 675.4.a.o | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 135.4.b.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 135.4.b.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 135.4.b.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 135.4.b.a | ✓ | 4 | 15.d | odd | 2 | 1 | CM |
| 675.4.a.k | 2 | 5.c | odd | 4 | 1 | ||
| 675.4.a.k | 2 | 15.e | even | 4 | 1 | ||
| 675.4.a.o | 2 | 5.c | odd | 4 | 1 | ||
| 675.4.a.o | 2 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 43T_{2}^{2} + 361 \)
acting on \(S_{4}^{\mathrm{new}}(135, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 43T^{2} + 361 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + 125)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} + 10258 T^{2} + 20079361 \)
|
| $19$ |
\( (T^{2} + 164 T + 6319)^{2} \)
|
| $23$ |
\( T^{4} + 63322 T^{2} + 719366041 \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} - 232 T - 35549)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 297680)^{2} \)
|
| $53$ |
\( T^{4} + \cdots + 3625364521 \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} - 358 T - 552779)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} - 304 T - 1386701)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 10453222081 \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
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