Newspace parameters
| Level: | \( N \) | \(=\) | \( 338 = 2 \cdot 13^{2} \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 338.b (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(19.9426455819\) |
| Analytic rank: | \(0\) |
| Dimension: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-1}) \) |
|
|
|
| Defining polynomial: |
\( x^{2} + 1 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 26) |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 337.1 |
|
− | 2.00000i | −1.00000 | −4.00000 | − | 17.0000i | 2.00000i | − | 35.0000i | 8.00000i | −26.0000 | −34.0000 | |||||||||||||||||||||
| 337.2 | 2.00000i | −1.00000 | −4.00000 | 17.0000i | − | 2.00000i | 35.0000i | − | 8.00000i | −26.0000 | −34.0000 | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 13.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 338.4.b.b | 2 | |
| 13.b | even | 2 | 1 | inner | 338.4.b.b | 2 | |
| 13.c | even | 3 | 2 | 338.4.e.c | 4 | ||
| 13.d | odd | 4 | 1 | 26.4.a.b | ✓ | 1 | |
| 13.d | odd | 4 | 1 | 338.4.a.b | 1 | ||
| 13.e | even | 6 | 2 | 338.4.e.c | 4 | ||
| 13.f | odd | 12 | 2 | 338.4.c.c | 2 | ||
| 13.f | odd | 12 | 2 | 338.4.c.g | 2 | ||
| 39.f | even | 4 | 1 | 234.4.a.a | 1 | ||
| 52.f | even | 4 | 1 | 208.4.a.e | 1 | ||
| 65.f | even | 4 | 1 | 650.4.b.d | 2 | ||
| 65.g | odd | 4 | 1 | 650.4.a.c | 1 | ||
| 65.k | even | 4 | 1 | 650.4.b.d | 2 | ||
| 91.i | even | 4 | 1 | 1274.4.a.f | 1 | ||
| 104.j | odd | 4 | 1 | 832.4.a.j | 1 | ||
| 104.m | even | 4 | 1 | 832.4.a.g | 1 | ||
| 156.l | odd | 4 | 1 | 1872.4.a.b | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.4.a.b | ✓ | 1 | 13.d | odd | 4 | 1 | |
| 208.4.a.e | 1 | 52.f | even | 4 | 1 | ||
| 234.4.a.a | 1 | 39.f | even | 4 | 1 | ||
| 338.4.a.b | 1 | 13.d | odd | 4 | 1 | ||
| 338.4.b.b | 2 | 1.a | even | 1 | 1 | trivial | |
| 338.4.b.b | 2 | 13.b | even | 2 | 1 | inner | |
| 338.4.c.c | 2 | 13.f | odd | 12 | 2 | ||
| 338.4.c.g | 2 | 13.f | odd | 12 | 2 | ||
| 338.4.e.c | 4 | 13.c | even | 3 | 2 | ||
| 338.4.e.c | 4 | 13.e | even | 6 | 2 | ||
| 650.4.a.c | 1 | 65.g | odd | 4 | 1 | ||
| 650.4.b.d | 2 | 65.f | even | 4 | 1 | ||
| 650.4.b.d | 2 | 65.k | even | 4 | 1 | ||
| 832.4.a.g | 1 | 104.m | even | 4 | 1 | ||
| 832.4.a.j | 1 | 104.j | odd | 4 | 1 | ||
| 1274.4.a.f | 1 | 91.i | even | 4 | 1 | ||
| 1872.4.a.b | 1 | 156.l | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3} + 1 \)
acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} + 4 \)
|
| $3$ |
\( (T + 1)^{2} \)
|
| $5$ |
\( T^{2} + 289 \)
|
| $7$ |
\( T^{2} + 1225 \)
|
| $11$ |
\( T^{2} + 4 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( (T - 19)^{2} \)
|
| $19$ |
\( T^{2} + 8836 \)
|
| $23$ |
\( (T - 72)^{2} \)
|
| $29$ |
\( (T - 246)^{2} \)
|
| $31$ |
\( T^{2} + 10000 \)
|
| $37$ |
\( T^{2} + 121 \)
|
| $41$ |
\( T^{2} + 78400 \)
|
| $43$ |
\( (T + 241)^{2} \)
|
| $47$ |
\( T^{2} + 18769 \)
|
| $53$ |
\( (T + 232)^{2} \)
|
| $59$ |
\( T^{2} + 148996 \)
|
| $61$ |
\( (T - 64)^{2} \)
|
| $67$ |
\( T^{2} + 448900 \)
|
| $71$ |
\( T^{2} + 3025 \)
|
| $73$ |
\( T^{2} + 702244 \)
|
| $79$ |
\( (T - 1016)^{2} \)
|
| $83$ |
\( T^{2} + 176400 \)
|
| $89$ |
\( T^{2} + 872356 \)
|
| $97$ |
\( T^{2} + 1331716 \)
|
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