Newspace parameters
| Level: | \( N \) | \(=\) | \( 513 = 3^{3} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 513.m (of order \(6\), degree \(2\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.09632562369\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
| Coefficient field: | \(\Q(\sqrt{-2}, \sqrt{-3})\) |
|
|
|
| Defining polynomial: |
\( x^{4} - 2x^{2} + 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-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
\( x^{4} - 2x^{2} + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} ) / 2 \)
|
| \(\beta_{3}\) | \(=\) |
\( ( \nu^{3} ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/513\mathbb{Z}\right)^\times\).
| \(n\) | \(191\) | \(325\) |
| \(\chi(n)\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 107.1 |
|
−1.22474 | − | 2.12132i | 0 | −2.00000 | + | 3.46410i | −1.22474 | + | 0.707107i | 0 | 2.00000 | 4.89898 | 0 | 3.00000 | + | 1.73205i | ||||||||||||||||||||||
| 107.2 | 1.22474 | + | 2.12132i | 0 | −2.00000 | + | 3.46410i | 1.22474 | − | 0.707107i | 0 | 2.00000 | −4.89898 | 0 | 3.00000 | + | 1.73205i | |||||||||||||||||||||||
| 350.1 | −1.22474 | + | 2.12132i | 0 | −2.00000 | − | 3.46410i | −1.22474 | − | 0.707107i | 0 | 2.00000 | 4.89898 | 0 | 3.00000 | − | 1.73205i | |||||||||||||||||||||||
| 350.2 | 1.22474 | − | 2.12132i | 0 | −2.00000 | − | 3.46410i | 1.22474 | + | 0.707107i | 0 | 2.00000 | −4.89898 | 0 | 3.00000 | − | 1.73205i | |||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 19.d | odd | 6 | 1 | inner |
| 57.f | even | 6 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 513.2.m.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 513.2.m.c | ✓ | 4 |
| 19.d | odd | 6 | 1 | inner | 513.2.m.c | ✓ | 4 |
| 57.f | even | 6 | 1 | inner | 513.2.m.c | ✓ | 4 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 513.2.m.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 513.2.m.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 513.2.m.c | ✓ | 4 | 19.d | odd | 6 | 1 | inner |
| 513.2.m.c | ✓ | 4 | 57.f | even | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(513, [\chi])\):
|
\( T_{2}^{4} + 6T_{2}^{2} + 36 \)
|
|
\( T_{5}^{4} - 2T_{5}^{2} + 4 \)
|
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\( T_{7} - 2 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{4} + 6T^{2} + 36 \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( T^{4} - 2T^{2} + 4 \)
|
| $7$ |
\( (T - 2)^{4} \)
|
| $11$ |
\( (T^{2} + 32)^{2} \)
|
| $13$ |
\( (T^{2} + 3 T + 3)^{2} \)
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| $17$ |
\( T^{4} - 2T^{2} + 4 \)
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| $19$ |
\( (T^{2} + 7 T + 19)^{2} \)
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| $23$ |
\( T^{4} - 8T^{2} + 64 \)
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| $29$ |
\( T^{4} + 54T^{2} + 2916 \)
|
| $31$ |
\( (T^{2} + 3)^{2} \)
|
| $37$ |
\( (T^{2} + 27)^{2} \)
|
| $41$ |
\( T^{4} \)
|
| $43$ |
\( (T^{2} + 11 T + 121)^{2} \)
|
| $47$ |
\( T^{4} - 128 T^{2} + 16384 \)
|
| $53$ |
\( T^{4} + 6T^{2} + 36 \)
|
| $59$ |
\( T^{4} + 54T^{2} + 2916 \)
|
| $61$ |
\( (T^{2} - 13 T + 169)^{2} \)
|
| $67$ |
\( (T^{2} - 24 T + 192)^{2} \)
|
| $71$ |
\( T^{4} + 24T^{2} + 576 \)
|
| $73$ |
\( (T^{2} - 4 T + 16)^{2} \)
|
| $79$ |
\( (T^{2} + 21 T + 147)^{2} \)
|
| $83$ |
\( (T^{2} + 2)^{2} \)
|
| $89$ |
\( T^{4} + 96T^{2} + 9216 \)
|
| $97$ |
\( (T^{2} + 3 T + 3)^{2} \)
|
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