Newspace parameters
| Level: | \( N \) | \(=\) | \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 1350.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(79.6525785077\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.29021904.1 |
|
|
|
| Defining polynomial: |
\( x^{4} - 50x^{2} + 109 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 270) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{4} - 50x^{2} + 109 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} + 63\nu ) / 4 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} - 195\nu ) / 4 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( 15\nu^{2} - 377 ) / 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 5\beta_1 ) / 30 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( 4\beta_{3} + 377 ) / 15 \)
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| \(\nu^{3}\) | \(=\) |
\( ( 21\beta_{2} + 65\beta_1 ) / 10 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.00000 | 0 | 4.00000 | 0 | 0 | −26.3948 | −8.00000 | 0 | 0 | ||||||||||||||||||||||||||||||
| 1.2 | −2.00000 | 0 | 4.00000 | 0 | 0 | −22.9416 | −8.00000 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 0 | 4.00000 | 0 | 0 | 22.9416 | −8.00000 | 0 | 0 | |||||||||||||||||||||||||||||||
| 1.4 | −2.00000 | 0 | 4.00000 | 0 | 0 | 26.3948 | −8.00000 | 0 | 0 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(3\) | \( -1 \) |
| \(5\) | \( -1 \) |
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 15.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 1350.4.a.bu | 4 | |
| 3.b | odd | 2 | 1 | 1350.4.a.bv | 4 | ||
| 5.b | even | 2 | 1 | 1350.4.a.bv | 4 | ||
| 5.c | odd | 4 | 2 | 270.4.c.e | ✓ | 8 | |
| 15.d | odd | 2 | 1 | inner | 1350.4.a.bu | 4 | |
| 15.e | even | 4 | 2 | 270.4.c.e | ✓ | 8 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 270.4.c.e | ✓ | 8 | 5.c | odd | 4 | 2 | |
| 270.4.c.e | ✓ | 8 | 15.e | even | 4 | 2 | |
| 1350.4.a.bu | 4 | 1.a | even | 1 | 1 | trivial | |
| 1350.4.a.bu | 4 | 15.d | odd | 2 | 1 | inner | |
| 1350.4.a.bv | 4 | 3.b | odd | 2 | 1 | ||
| 1350.4.a.bv | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1350))\):
|
\( T_{7}^{4} - 1223T_{7}^{2} + 366676 \)
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|
\( T_{11}^{4} - 1223T_{11}^{2} + 366676 \)
|
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\( T_{17}^{2} + 55T_{17} - 6500 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} - 1223 T^{2} + 366676 \)
|
| $11$ |
\( T^{4} - 1223 T^{2} + 366676 \)
|
| $13$ |
\( T^{4} - 10908 T^{2} + 20342016 \)
|
| $17$ |
\( (T^{2} + 55 T - 6500)^{2} \)
|
| $19$ |
\( (T^{2} - 21 T - 7146)^{2} \)
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| $23$ |
\( (T^{2} + 61 T - 6326)^{2} \)
|
| $29$ |
\( T^{4} + \cdots + 1647703296 \)
|
| $31$ |
\( (T^{2} - 200 T - 19025)^{2} \)
|
| $37$ |
\( T^{4} - 132812 T^{2} + 262548736 \)
|
| $41$ |
\( T^{4} + \cdots + 9169688656 \)
|
| $43$ |
\( T^{4} - 113000 T^{2} + 916690000 \)
|
| $47$ |
\( (T^{2} + 386 T + 8224)^{2} \)
|
| $53$ |
\( (T^{2} + 186 T - 107451)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 67407499216 \)
|
| $61$ |
\( (T^{2} - 781 T - 28916)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 110836055296 \)
|
| $71$ |
\( T^{4} + \cdots + 12713760000 \)
|
| $73$ |
\( T^{4} + \cdots + 106363578196 \)
|
| $79$ |
\( (T^{2} - 1629 T + 598104)^{2} \)
|
| $83$ |
\( (T^{2} - 632 T - 161369)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 1839447376 \)
|
| $97$ |
\( T^{4} - 556883 T^{2} + 365828416 \)
|
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