Newspace parameters
| Level: | \( N \) | \(=\) | \( 306 = 2 \cdot 3^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 3 \) |
| Character orbit: | \([\chi]\) | \(=\) | 306.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(8.33789608943\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-17})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 16x^{2} + 81 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | yes |
| 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
\( x^{4} - 16x^{2} + 81 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} - 7\nu ) / 9 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 25\nu ) / 9 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} - 8 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 8 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 7\beta_{2} + 25\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/306\mathbb{Z}\right)^\times\).
| \(n\) | \(37\) | \(137\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 35.1 |
|
− | 1.41421i | 0 | −2.00000 | − | 6.95153i | 0 | −7.83095 | 2.82843i | 0 | −9.83095 | ||||||||||||||||||||||||||||
| 35.2 | − | 1.41421i | 0 | −2.00000 | 1.29468i | 0 | 3.83095 | 2.82843i | 0 | 1.83095 | ||||||||||||||||||||||||||||||
| 35.3 | 1.41421i | 0 | −2.00000 | − | 1.29468i | 0 | 3.83095 | − | 2.82843i | 0 | 1.83095 | |||||||||||||||||||||||||||||
| 35.4 | 1.41421i | 0 | −2.00000 | 6.95153i | 0 | −7.83095 | − | 2.82843i | 0 | −9.83095 | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 306.3.c.a | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 306.3.c.a | ✓ | 4 |
| 4.b | odd | 2 | 1 | 2448.3.g.b | 4 | ||
| 12.b | even | 2 | 1 | 2448.3.g.b | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 306.3.c.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 306.3.c.a | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 2448.3.g.b | 4 | 4.b | odd | 2 | 1 | ||
| 2448.3.g.b | 4 | 12.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{7}^{2} + 4T_{7} - 30 \)
acting on \(S_{3}^{\mathrm{new}}(306, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 2)^{2} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} + 50T^{2} + 81 \)
|
| $7$ |
\( (T^{2} + 4 T - 30)^{2} \)
|
| $11$ |
\( T^{4} + 50T^{2} + 81 \)
|
| $13$ |
\( (T^{2} - 6 T - 127)^{2} \)
|
| $17$ |
\( (T^{2} + 17)^{2} \)
|
| $19$ |
\( (T^{2} - 18 T - 55)^{2} \)
|
| $23$ |
\( (T^{2} + 153)^{2} \)
|
| $29$ |
\( T^{4} + 944T^{2} + 5184 \)
|
| $31$ |
\( (T^{2} + 32 T - 50)^{2} \)
|
| $37$ |
\( (T^{2} + 12 T - 814)^{2} \)
|
| $41$ |
\( T^{4} + 4370 T^{2} + 269361 \)
|
| $43$ |
\( (T^{2} + 34 T - 255)^{2} \)
|
| $47$ |
\( T^{4} + 7812 T^{2} + 980100 \)
|
| $53$ |
\( T^{4} + 5220 T^{2} + 5225796 \)
|
| $59$ |
\( T^{4} + 3528 T^{2} + 291600 \)
|
| $61$ |
\( (T^{2} + 44 T - 2270)^{2} \)
|
| $67$ |
\( (T^{2} - 56 T - 5880)^{2} \)
|
| $71$ |
\( (T^{2} + 2448)^{2} \)
|
| $73$ |
\( (T^{2} - 168 T + 5390)^{2} \)
|
| $79$ |
\( (T^{2} + 36 T - 1852)^{2} \)
|
| $83$ |
\( T^{4} + 9188 T^{2} + 16892100 \)
|
| $89$ |
\( T^{4} + 13932 T^{2} + 16402500 \)
|
| $97$ |
\( (T^{2} + 72 T - 12304)^{2} \)
|
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