Newspace parameters
| Level: | \( N \) | \(=\) | \( 120 = 2^{3} \cdot 3 \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 120.f (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(7.08022920069\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
0 | − | 3.00000i | 0 | 5.00000 | − | 10.0000i | 0 | − | 4.00000i | 0 | −9.00000 | 0 | ||||||||||||||||||||
| 49.2 | 0 | 3.00000i | 0 | 5.00000 | + | 10.0000i | 0 | 4.00000i | 0 | −9.00000 | 0 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 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 | 120.4.f.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 360.4.f.a | 2 | ||
| 4.b | odd | 2 | 1 | 240.4.f.e | 2 | ||
| 5.b | even | 2 | 1 | inner | 120.4.f.c | ✓ | 2 |
| 5.c | odd | 4 | 1 | 600.4.a.f | 1 | ||
| 5.c | odd | 4 | 1 | 600.4.a.k | 1 | ||
| 8.b | even | 2 | 1 | 960.4.f.b | 2 | ||
| 8.d | odd | 2 | 1 | 960.4.f.a | 2 | ||
| 12.b | even | 2 | 1 | 720.4.f.b | 2 | ||
| 15.d | odd | 2 | 1 | 360.4.f.a | 2 | ||
| 15.e | even | 4 | 1 | 1800.4.a.o | 1 | ||
| 15.e | even | 4 | 1 | 1800.4.a.u | 1 | ||
| 20.d | odd | 2 | 1 | 240.4.f.e | 2 | ||
| 20.e | even | 4 | 1 | 1200.4.a.l | 1 | ||
| 20.e | even | 4 | 1 | 1200.4.a.z | 1 | ||
| 40.e | odd | 2 | 1 | 960.4.f.a | 2 | ||
| 40.f | even | 2 | 1 | 960.4.f.b | 2 | ||
| 60.h | even | 2 | 1 | 720.4.f.b | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 120.4.f.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 120.4.f.c | ✓ | 2 | 5.b | even | 2 | 1 | inner |
| 240.4.f.e | 2 | 4.b | odd | 2 | 1 | ||
| 240.4.f.e | 2 | 20.d | odd | 2 | 1 | ||
| 360.4.f.a | 2 | 3.b | odd | 2 | 1 | ||
| 360.4.f.a | 2 | 15.d | odd | 2 | 1 | ||
| 600.4.a.f | 1 | 5.c | odd | 4 | 1 | ||
| 600.4.a.k | 1 | 5.c | odd | 4 | 1 | ||
| 720.4.f.b | 2 | 12.b | even | 2 | 1 | ||
| 720.4.f.b | 2 | 60.h | even | 2 | 1 | ||
| 960.4.f.a | 2 | 8.d | odd | 2 | 1 | ||
| 960.4.f.a | 2 | 40.e | odd | 2 | 1 | ||
| 960.4.f.b | 2 | 8.b | even | 2 | 1 | ||
| 960.4.f.b | 2 | 40.f | even | 2 | 1 | ||
| 1200.4.a.l | 1 | 20.e | even | 4 | 1 | ||
| 1200.4.a.z | 1 | 20.e | even | 4 | 1 | ||
| 1800.4.a.o | 1 | 15.e | even | 4 | 1 | ||
| 1800.4.a.u | 1 | 15.e | even | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(120, [\chi])\):
|
\( T_{7}^{2} + 16 \)
|
|
\( T_{11} + 28 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} \)
|
| $3$ |
\( T^{2} + 9 \)
|
| $5$ |
\( T^{2} - 10T + 125 \)
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| $7$ |
\( T^{2} + 16 \)
|
| $11$ |
\( (T + 28)^{2} \)
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| $13$ |
\( T^{2} + 256 \)
|
| $17$ |
\( T^{2} + 11664 \)
|
| $19$ |
\( (T + 32)^{2} \)
|
| $23$ |
\( T^{2} + 784 \)
|
| $29$ |
\( (T - 238)^{2} \)
|
| $31$ |
\( (T + 180)^{2} \)
|
| $37$ |
\( T^{2} + 1600 \)
|
| $41$ |
\( (T - 422)^{2} \)
|
| $43$ |
\( T^{2} + 76176 \)
|
| $47$ |
\( T^{2} + 3600 \)
|
| $53$ |
\( T^{2} + 48400 \)
|
| $59$ |
\( (T - 804)^{2} \)
|
| $61$ |
\( (T + 358)^{2} \)
|
| $67$ |
\( T^{2} + 781456 \)
|
| $71$ |
\( (T + 64)^{2} \)
|
| $73$ |
\( T^{2} + 23104 \)
|
| $79$ |
\( (T - 932)^{2} \)
|
| $83$ |
\( T^{2} + 1669264 \)
|
| $89$ |
\( (T - 1146)^{2} \)
|
| $97$ |
\( T^{2} + 678976 \)
|
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