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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{-3}) \) |
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|
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| Defining polynomial: |
\( x^{2} - x + 1 \)
|
| 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 primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/338\mathbb{Z}\right)^\times\).
| \(n\) | \(171\) |
| \(\chi(n)\) | \(-\zeta_{6}\) |
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 | −2.00000 | + | 3.46410i | −2.00000 | − | 3.46410i | 18.0000 | 4.00000 | + | 6.92820i | 10.0000 | + | 17.3205i | −8.00000 | 5.50000 | + | 9.52628i | 18.0000 | − | 31.1769i | ||||||||||
| 315.1 | 1.00000 | + | 1.73205i | −2.00000 | − | 3.46410i | −2.00000 | + | 3.46410i | 18.0000 | 4.00000 | − | 6.92820i | 10.0000 | − | 17.3205i | −8.00000 | 5.50000 | − | 9.52628i | 18.0000 | + | 31.1769i | |||||||||||
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.e | 2 | |
| 13.b | even | 2 | 1 | 338.4.c.a | 2 | ||
| 13.c | even | 3 | 1 | 338.4.a.c | 1 | ||
| 13.c | even | 3 | 1 | inner | 338.4.c.e | 2 | |
| 13.d | odd | 4 | 2 | 338.4.e.a | 4 | ||
| 13.e | even | 6 | 1 | 26.4.a.c | ✓ | 1 | |
| 13.e | even | 6 | 1 | 338.4.c.a | 2 | ||
| 13.f | odd | 12 | 2 | 338.4.b.d | 2 | ||
| 13.f | odd | 12 | 2 | 338.4.e.a | 4 | ||
| 39.h | odd | 6 | 1 | 234.4.a.e | 1 | ||
| 52.i | odd | 6 | 1 | 208.4.a.b | 1 | ||
| 65.l | even | 6 | 1 | 650.4.a.b | 1 | ||
| 65.r | odd | 12 | 2 | 650.4.b.f | 2 | ||
| 91.t | odd | 6 | 1 | 1274.4.a.d | 1 | ||
| 104.p | odd | 6 | 1 | 832.4.a.o | 1 | ||
| 104.s | even | 6 | 1 | 832.4.a.d | 1 | ||
| 156.r | even | 6 | 1 | 1872.4.a.q | 1 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 26.4.a.c | ✓ | 1 | 13.e | even | 6 | 1 | |
| 208.4.a.b | 1 | 52.i | odd | 6 | 1 | ||
| 234.4.a.e | 1 | 39.h | odd | 6 | 1 | ||
| 338.4.a.c | 1 | 13.c | even | 3 | 1 | ||
| 338.4.b.d | 2 | 13.f | odd | 12 | 2 | ||
| 338.4.c.a | 2 | 13.b | even | 2 | 1 | ||
| 338.4.c.a | 2 | 13.e | even | 6 | 1 | ||
| 338.4.c.e | 2 | 1.a | even | 1 | 1 | trivial | |
| 338.4.c.e | 2 | 13.c | even | 3 | 1 | inner | |
| 338.4.e.a | 4 | 13.d | odd | 4 | 2 | ||
| 338.4.e.a | 4 | 13.f | odd | 12 | 2 | ||
| 650.4.a.b | 1 | 65.l | even | 6 | 1 | ||
| 650.4.b.f | 2 | 65.r | odd | 12 | 2 | ||
| 832.4.a.d | 1 | 104.s | even | 6 | 1 | ||
| 832.4.a.o | 1 | 104.p | odd | 6 | 1 | ||
| 1274.4.a.d | 1 | 91.t | odd | 6 | 1 | ||
| 1872.4.a.q | 1 | 156.r | even | 6 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(338, [\chi])\):
|
\( T_{3}^{2} + 4T_{3} + 16 \)
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\( T_{5} - 18 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{2} - 2T + 4 \)
|
| $3$ |
\( T^{2} + 4T + 16 \)
|
| $5$ |
\( (T - 18)^{2} \)
|
| $7$ |
\( T^{2} - 20T + 400 \)
|
| $11$ |
\( T^{2} + 48T + 2304 \)
|
| $13$ |
\( T^{2} \)
|
| $17$ |
\( T^{2} + 66T + 4356 \)
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| $19$ |
\( T^{2} + 16T + 256 \)
|
| $23$ |
\( T^{2} + 168T + 28224 \)
|
| $29$ |
\( T^{2} + 6T + 36 \)
|
| $31$ |
\( (T + 20)^{2} \)
|
| $37$ |
\( T^{2} - 254T + 64516 \)
|
| $41$ |
\( T^{2} + 390T + 152100 \)
|
| $43$ |
\( T^{2} - 124T + 15376 \)
|
| $47$ |
\( (T - 468)^{2} \)
|
| $53$ |
\( (T - 558)^{2} \)
|
| $59$ |
\( T^{2} + 96T + 9216 \)
|
| $61$ |
\( T^{2} - 826T + 682276 \)
|
| $67$ |
\( T^{2} + 160T + 25600 \)
|
| $71$ |
\( T^{2} + 420T + 176400 \)
|
| $73$ |
\( (T + 362)^{2} \)
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| $79$ |
\( (T - 776)^{2} \)
|
| $83$ |
\( T^{2} \)
|
| $89$ |
\( T^{2} - 1626 T + 2643876 \)
|
| $97$ |
\( T^{2} + 1294 T + 1674436 \)
|
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