Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.c (of order \(3\), degree \(2\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(19.9426455819\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{3})\) |
| Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{217})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{9}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-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
\( x^{4} - x^{3} + 55x^{2} + 54x + 2916 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( -\nu^{3} + 55\nu^{2} - 55\nu + 2916 ) / 2970 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} + 109 ) / 55 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} + 54\beta_{2} + \beta _1 - 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( 55\beta_{3} - 109 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-1 + \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 191.1 |
|
1.00000 | − | 1.73205i | −2.93273 | + | 5.07964i | −2.00000 | − | 3.46410i | 10.8655 | 5.86546 | + | 10.1593i | −14.9327 | − | 25.8642i | −8.00000 | −3.70181 | − | 6.41172i | 10.8655 | − | 18.8195i | ||||||||||||||||
| 191.2 | 1.00000 | − | 1.73205i | 4.43273 | − | 7.67771i | −2.00000 | − | 3.46410i | −3.86546 | −8.86546 | − | 15.3554i | −7.56727 | − | 13.1069i | −8.00000 | −25.7982 | − | 44.6838i | −3.86546 | + | 6.69517i | |||||||||||||||||
| 315.1 | 1.00000 | + | 1.73205i | −2.93273 | − | 5.07964i | −2.00000 | + | 3.46410i | 10.8655 | 5.86546 | − | 10.1593i | −14.9327 | + | 25.8642i | −8.00000 | −3.70181 | + | 6.41172i | 10.8655 | + | 18.8195i | |||||||||||||||||
| 315.2 | 1.00000 | + | 1.73205i | 4.43273 | + | 7.67771i | −2.00000 | + | 3.46410i | −3.86546 | −8.86546 | + | 15.3554i | −7.56727 | + | 13.1069i | −8.00000 | −25.7982 | + | 44.6838i | −3.86546 | − | 6.69517i | |||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.c | even | 3 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 338.4.c.j | 4 | |
| 13.b | even | 2 | 1 | 26.4.c.b | ✓ | 4 | |
| 13.c | even | 3 | 1 | 338.4.a.g | 2 | ||
| 13.c | even | 3 | 1 | inner | 338.4.c.j | 4 | |
| 13.d | odd | 4 | 2 | 338.4.e.f | 8 | ||
| 13.e | even | 6 | 1 | 26.4.c.b | ✓ | 4 | |
| 13.e | even | 6 | 1 | 338.4.a.h | 2 | ||
| 13.f | odd | 12 | 2 | 338.4.b.e | 4 | ||
| 13.f | odd | 12 | 2 | 338.4.e.f | 8 | ||
| 39.d | odd | 2 | 1 | 234.4.h.h | 4 | ||
| 39.h | odd | 6 | 1 | 234.4.h.h | 4 | ||
| 52.b | odd | 2 | 1 | 208.4.i.d | 4 | ||
| 52.i | odd | 6 | 1 | 208.4.i.d | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.4.c.b | ✓ | 4 | 13.b | even | 2 | 1 | |
| 26.4.c.b | ✓ | 4 | 13.e | even | 6 | 1 | |
| 208.4.i.d | 4 | 52.b | odd | 2 | 1 | ||
| 208.4.i.d | 4 | 52.i | odd | 6 | 1 | ||
| 234.4.h.h | 4 | 39.d | odd | 2 | 1 | ||
| 234.4.h.h | 4 | 39.h | odd | 6 | 1 | ||
| 338.4.a.g | 2 | 13.c | even | 3 | 1 | ||
| 338.4.a.h | 2 | 13.e | even | 6 | 1 | ||
| 338.4.b.e | 4 | 13.f | odd | 12 | 2 | ||
| 338.4.c.j | 4 | 1.a | even | 1 | 1 | trivial | |
| 338.4.c.j | 4 | 13.c | even | 3 | 1 | inner | |
| 338.4.e.f | 8 | 13.d | odd | 4 | 2 | ||
| 338.4.e.f | 8 | 13.f | odd | 12 | 2 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):
|
\( T_{3}^{4} - 3T_{3}^{3} + 61T_{3}^{2} + 156T_{3} + 2704 \)
|
|
\( T_{5}^{2} - 7T_{5} - 42 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} - 2 T + 4)^{2} \)
|
| $3$ |
\( T^{4} - 3 T^{3} + \cdots + 2704 \)
|
| $5$ |
\( (T^{2} - 7 T - 42)^{2} \)
|
| $7$ |
\( T^{4} + 45 T^{3} + \cdots + 204304 \)
|
| $11$ |
\( T^{4} - 5 T^{3} + \cdots + 7033104 \)
|
| $13$ |
\( T^{4} \)
|
| $17$ |
\( T^{4} - 130 T^{3} + \cdots + 567009 \)
|
| $19$ |
\( T^{4} - 3 T^{3} + \cdots + 2704 \)
|
| $23$ |
\( T^{4} + 33 T^{3} + \cdots + 46656 \)
|
| $29$ |
\( T^{4} - 198 T^{3} + \cdots + 3956121 \)
|
| $31$ |
\( (T^{2} + 280 T - 11648)^{2} \)
|
| $37$ |
\( T^{4} + \cdots + 14965362889 \)
|
| $41$ |
\( T^{4} - 242 T^{3} + \cdots + 49829481 \)
|
| $43$ |
\( T^{4} + \cdots + 1007681536 \)
|
| $47$ |
\( (T^{2} + 224 T - 157584)^{2} \)
|
| $53$ |
\( (T^{2} + 535 T + 71502)^{2} \)
|
| $59$ |
\( T^{4} + \cdots + 1237069584 \)
|
| $61$ |
\( T^{4} + \cdots + 2639596129 \)
|
| $67$ |
\( T^{4} + \cdots + 45137551936 \)
|
| $71$ |
\( T^{4} + \cdots + 3764558736 \)
|
| $73$ |
\( (T^{2} - 623 T + 84826)^{2} \)
|
| $79$ |
\( (T^{2} - 760 T + 19408)^{2} \)
|
| $83$ |
\( (T^{2} - 1764 T + 707616)^{2} \)
|
| $89$ |
\( T^{4} + \cdots + 18913400676 \)
|
| $97$ |
\( T^{4} + \cdots + 10397065156 \)
|
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