Newspace parameters
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 12 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(115.251477084\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{3} - \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{3} - 175039x - 13178910 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
| Coefficient ring index: | \( 2^{3}\cdot 3\cdot 5^{3} \) |
| Twist minimal: | no (minimal twist has level 30) |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 175039x - 13178910 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( 5\nu^{2} + 305\nu - 583481 ) / 53 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -20\nu^{2} + 6730\nu + 2333871 ) / 53 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + 4\beta _1 + 1 ) / 150 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( -61\beta_{2} + 1346\beta _1 + 17504369 ) / 150 \)
|
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 |
|
32.0000 | −243.000 | 1024.00 | 0 | −7776.00 | −52907.3 | 32768.0 | 59049.0 | 0 | |||||||||||||||||||||||||||
| 1.2 | 32.0000 | −243.000 | 1024.00 | 0 | −7776.00 | 41527.6 | 32768.0 | 59049.0 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 32.0000 | −243.000 | 1024.00 | 0 | −7776.00 | 78763.7 | 32768.0 | 59049.0 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(3\) | \( +1 \) |
| \(5\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 150.12.a.u | 3 | |
| 5.b | even | 2 | 1 | 150.12.a.t | 3 | ||
| 5.c | odd | 4 | 2 | 30.12.c.b | ✓ | 6 | |
| 15.e | even | 4 | 2 | 90.12.c.c | 6 | ||
| 20.e | even | 4 | 2 | 240.12.f.b | 6 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.12.c.b | ✓ | 6 | 5.c | odd | 4 | 2 | |
| 90.12.c.c | 6 | 15.e | even | 4 | 2 | ||
| 150.12.a.t | 3 | 5.b | even | 2 | 1 | ||
| 150.12.a.u | 3 | 1.a | even | 1 | 1 | trivial | |
| 240.12.f.b | 6 | 20.e | even | 4 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{3} - 67384T_{7}^{2} - 3093414748T_{7} + 173052509440048 \)
acting on \(S_{12}^{\mathrm{new}}(\Gamma_0(150))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 32)^{3} \)
|
| $3$ |
\( (T + 243)^{3} \)
|
| $5$ |
\( T^{3} \)
|
| $7$ |
\( T^{3} + \cdots + 173052509440048 \)
|
| $11$ |
\( T^{3} + \cdots + 22\!\cdots\!00 \)
|
| $13$ |
\( T^{3} + \cdots - 14\!\cdots\!44 \)
|
| $17$ |
\( T^{3} + \cdots + 69\!\cdots\!88 \)
|
| $19$ |
\( T^{3} + \cdots + 29\!\cdots\!36 \)
|
| $23$ |
\( T^{3} + \cdots + 52\!\cdots\!00 \)
|
| $29$ |
\( T^{3} + \cdots - 12\!\cdots\!16 \)
|
| $31$ |
\( T^{3} + \cdots + 26\!\cdots\!88 \)
|
| $37$ |
\( T^{3} + \cdots + 21\!\cdots\!44 \)
|
| $41$ |
\( T^{3} + \cdots + 12\!\cdots\!16 \)
|
| $43$ |
\( T^{3} + \cdots - 24\!\cdots\!48 \)
|
| $47$ |
\( T^{3} + \cdots - 69\!\cdots\!00 \)
|
| $53$ |
\( T^{3} + \cdots - 58\!\cdots\!12 \)
|
| $59$ |
\( T^{3} + \cdots + 58\!\cdots\!00 \)
|
| $61$ |
\( T^{3} + \cdots - 34\!\cdots\!00 \)
|
| $67$ |
\( T^{3} + \cdots + 50\!\cdots\!48 \)
|
| $71$ |
\( T^{3} + \cdots + 78\!\cdots\!00 \)
|
| $73$ |
\( T^{3} + \cdots - 61\!\cdots\!44 \)
|
| $79$ |
\( T^{3} + \cdots + 29\!\cdots\!12 \)
|
| $83$ |
\( T^{3} + \cdots + 24\!\cdots\!96 \)
|
| $89$ |
\( T^{3} + \cdots + 39\!\cdots\!32 \)
|
| $97$ |
\( T^{3} + \cdots - 77\!\cdots\!56 \)
|
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