Newspace parameters
| Level: | \( N \) | \(=\) | \( 150 = 2 \cdot 3 \cdot 5^{2} \) |
| Weight: | \( k \) | \(=\) | \( 14 \) |
| Character orbit: | \([\chi]\) | \(=\) | 150.c (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(160.846393428\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{4} + \cdots)\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 11557705x^{2} + 33395130437904 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 2^{6}\cdot 3^{2}\cdot 5^{2} \) |
| Twist minimal: | no (minimal twist has level 30) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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
\( x^{4} + 11557705x^{2} + 33395130437904 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 5778853\nu ) / 5778852 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( 5\nu^{3} + 86682785\nu ) / 481571 \)
|
| \(\beta_{3}\) | \(=\) |
\( 120\nu^{2} + 693462300 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - 60\beta_1 ) / 120 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 693462300 ) / 120 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( -5778853\beta_{2} + 1040193420\beta_1 ) / 120 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/150\mathbb{Z}\right)^\times\).
| \(n\) | \(101\) | \(127\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 49.1 |
|
− | 64.0000i | − | 729.000i | −4096.00 | 0 | −46656.0 | − | 406927.i | 262144.i | −531441. | 0 | |||||||||||||||||||||||||||
| 49.2 | − | 64.0000i | − | 729.000i | −4096.00 | 0 | −46656.0 | 170015.i | 262144.i | −531441. | 0 | |||||||||||||||||||||||||||||
| 49.3 | 64.0000i | 729.000i | −4096.00 | 0 | −46656.0 | − | 170015.i | − | 262144.i | −531441. | 0 | |||||||||||||||||||||||||||||
| 49.4 | 64.0000i | 729.000i | −4096.00 | 0 | −46656.0 | 406927.i | − | 262144.i | −531441. | 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 | 150.14.c.j | 4 | |
| 5.b | even | 2 | 1 | inner | 150.14.c.j | 4 | |
| 5.c | odd | 4 | 1 | 30.14.a.g | ✓ | 2 | |
| 5.c | odd | 4 | 1 | 150.14.a.n | 2 | ||
| 15.e | even | 4 | 1 | 90.14.a.m | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 30.14.a.g | ✓ | 2 | 5.c | odd | 4 | 1 | |
| 90.14.a.m | 2 | 15.e | even | 4 | 1 | ||
| 150.14.a.n | 2 | 5.c | odd | 4 | 1 | ||
| 150.14.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 150.14.c.j | 4 | 5.b | even | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{4} + 194494592672T_{7}^{2} + 4786377214790329559296 \)
acting on \(S_{14}^{\mathrm{new}}(150, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 4096)^{2} \)
|
| $3$ |
\( (T^{2} + 531441)^{2} \)
|
| $5$ |
\( T^{4} \)
|
| $7$ |
\( T^{4} + \cdots + 47\!\cdots\!96 \)
|
| $11$ |
\( (T^{2} + \cdots - 61203724243200)^{2} \)
|
| $13$ |
\( T^{4} + \cdots + 41\!\cdots\!96 \)
|
| $17$ |
\( T^{4} + \cdots + 25\!\cdots\!76 \)
|
| $19$ |
\( (T^{2} + \cdots - 83\!\cdots\!24)^{2} \)
|
| $23$ |
\( T^{4} + \cdots + 55\!\cdots\!00 \)
|
| $29$ |
\( (T^{2} + \cdots - 21\!\cdots\!04)^{2} \)
|
| $31$ |
\( (T^{2} + \cdots + 30\!\cdots\!44)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 30\!\cdots\!96 \)
|
| $41$ |
\( (T^{2} + \cdots + 10\!\cdots\!96)^{2} \)
|
| $43$ |
\( T^{4} + \cdots + 22\!\cdots\!16 \)
|
| $47$ |
\( T^{4} + \cdots + 81\!\cdots\!00 \)
|
| $53$ |
\( T^{4} + \cdots + 77\!\cdots\!36 \)
|
| $59$ |
\( (T^{2} + \cdots - 79\!\cdots\!00)^{2} \)
|
| $61$ |
\( (T^{2} + \cdots + 18\!\cdots\!00)^{2} \)
|
| $67$ |
\( T^{4} + \cdots + 12\!\cdots\!96 \)
|
| $71$ |
\( (T^{2} + \cdots + 13\!\cdots\!00)^{2} \)
|
| $73$ |
\( T^{4} + \cdots + 96\!\cdots\!56 \)
|
| $79$ |
\( (T^{2} + \cdots + 38\!\cdots\!84)^{2} \)
|
| $83$ |
\( T^{4} + \cdots + 34\!\cdots\!56 \)
|
| $89$ |
\( (T^{2} + \cdots + 12\!\cdots\!64)^{2} \)
|
| $97$ |
\( T^{4} + \cdots + 36\!\cdots\!96 \)
|
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