Newspace parameters
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.m (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.08022920069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{3}, \sqrt{-5})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 3^{2} \) |
| 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
\( x^{4} + x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 3\nu ) / 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( 3\nu^{2} + 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( -\nu^{3} + 3\nu ) / 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 3 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{2} - 1 ) / 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( -\beta_{3} + \beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/120\mathbb{Z}\right)^\times\).
| \(n\) | \(31\) | \(41\) | \(61\) | \(97\) |
| \(\chi(n)\) | \(-1\) | \(-1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 59.1 |
|
−2.59808 | − | 1.11803i | −5.19615 | 5.50000 | + | 5.80948i | 11.1803i | 13.5000 | + | 5.80948i | 0 | −7.79423 | − | 21.2426i | 27.0000 | 12.5000 | − | 29.0474i | ||||||||||||||||||||
| 59.2 | −2.59808 | + | 1.11803i | −5.19615 | 5.50000 | − | 5.80948i | − | 11.1803i | 13.5000 | − | 5.80948i | 0 | −7.79423 | + | 21.2426i | 27.0000 | 12.5000 | + | 29.0474i | ||||||||||||||||||||
| 59.3 | 2.59808 | − | 1.11803i | 5.19615 | 5.50000 | − | 5.80948i | 11.1803i | 13.5000 | − | 5.80948i | 0 | 7.79423 | − | 21.2426i | 27.0000 | 12.5000 | + | 29.0474i | |||||||||||||||||||||
| 59.4 | 2.59808 | + | 1.11803i | 5.19615 | 5.50000 | + | 5.80948i | − | 11.1803i | 13.5000 | + | 5.80948i | 0 | 7.79423 | + | 21.2426i | 27.0000 | 12.5000 | − | 29.0474i | ||||||||||||||||||||
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 |
| 8.d | odd | 2 | 1 | inner |
| 24.f | even | 2 | 1 | inner |
| 40.e | odd | 2 | 1 | inner |
| 120.m | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 120.4.m.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 480.4.m.a | 4 | ||
| 5.b | even | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| 8.b | even | 2 | 1 | 480.4.m.a | 4 | ||
| 8.d | odd | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| 12.b | even | 2 | 1 | 480.4.m.a | 4 | ||
| 15.d | odd | 2 | 1 | CM | 120.4.m.a | ✓ | 4 |
| 20.d | odd | 2 | 1 | 480.4.m.a | 4 | ||
| 24.f | even | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| 24.h | odd | 2 | 1 | 480.4.m.a | 4 | ||
| 40.e | odd | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| 40.f | even | 2 | 1 | 480.4.m.a | 4 | ||
| 60.h | even | 2 | 1 | 480.4.m.a | 4 | ||
| 120.i | odd | 2 | 1 | 480.4.m.a | 4 | ||
| 120.m | even | 2 | 1 | inner | 120.4.m.a | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.4.m.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 120.4.m.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 120.4.m.a | ✓ | 4 | 5.b | even | 2 | 1 | inner |
| 120.4.m.a | ✓ | 4 | 8.d | odd | 2 | 1 | inner |
| 120.4.m.a | ✓ | 4 | 15.d | odd | 2 | 1 | CM |
| 120.4.m.a | ✓ | 4 | 24.f | even | 2 | 1 | inner |
| 120.4.m.a | ✓ | 4 | 40.e | odd | 2 | 1 | inner |
| 120.4.m.a | ✓ | 4 | 120.m | even | 2 | 1 | inner |
| 480.4.m.a | 4 | 4.b | odd | 2 | 1 | ||
| 480.4.m.a | 4 | 8.b | even | 2 | 1 | ||
| 480.4.m.a | 4 | 12.b | even | 2 | 1 | ||
| 480.4.m.a | 4 | 20.d | odd | 2 | 1 | ||
| 480.4.m.a | 4 | 24.h | odd | 2 | 1 | ||
| 480.4.m.a | 4 | 40.f | even | 2 | 1 | ||
| 480.4.m.a | 4 | 60.h | even | 2 | 1 | ||
| 480.4.m.a | 4 | 120.i | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7} \)
acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} - 11T^{2} + 64 \)
|
| $3$ |
\( (T^{2} - 27)^{2} \)
|
| $5$ |
\( (T^{2} + 125)^{2} \)
|
| $7$ |
\( T^{4} \)
|
| $11$ |
\( T^{4} \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( (T^{2} - 432)^{2} \)
|
| $19$ |
\( (T + 164)^{4} \)
|
| $23$ |
\( (T^{2} + 9680)^{2} \)
|
| $29$ |
\( T^{4} \)
|
| $31$ |
\( (T^{2} + 65340)^{2} \)
|
| $37$ |
\( T^{4} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( T^{4} \)
|
| $47$ |
\( (T^{2} + 297680)^{2} \)
|
| $53$ |
\( (T^{2} + 386420)^{2} \)
|
| $59$ |
\( T^{4} \)
|
| $61$ |
\( (T^{2} + 779760)^{2} \)
|
| $67$ |
\( T^{4} \)
|
| $71$ |
\( T^{4} \)
|
| $73$ |
\( T^{4} \)
|
| $79$ |
\( (T^{2} + 1879740)^{2} \)
|
| $83$ |
\( (T^{2} - 674028)^{2} \)
|
| $89$ |
\( T^{4} \)
|
| $97$ |
\( T^{4} \)
|
| show more | |
| show less |