Newspace parameters
| Level: | \( N \) | \(=\) | \( 522 = 2 \cdot 3^{2} \cdot 29 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 522.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(4.16819098551\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{13})\) |
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|
|
| Defining polynomial: |
\( x^{4} + 7x^{2} + 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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
\( x^{4} + 7x^{2} + 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( \nu^{3} + 4\nu ) / 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 10\nu ) / 3 \)
|
| \(\beta_{3}\) | \(=\) |
\( 2\nu^{2} + 7 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} - \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{3} - 7 ) / 2 \)
|
| \(\nu^{3}\) | \(=\) |
\( -2\beta_{2} + 5\beta_1 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/522\mathbb{Z}\right)^\times\).
| \(n\) | \(379\) | \(407\) |
| \(\chi(n)\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 289.1 |
|
− | 1.00000i | 0 | −1.00000 | −3.60555 | 0 | 1.00000 | 1.00000i | 0 | 3.60555i | |||||||||||||||||||||||||||||
| 289.2 | − | 1.00000i | 0 | −1.00000 | 3.60555 | 0 | 1.00000 | 1.00000i | 0 | − | 3.60555i | |||||||||||||||||||||||||||||
| 289.3 | 1.00000i | 0 | −1.00000 | −3.60555 | 0 | 1.00000 | − | 1.00000i | 0 | − | 3.60555i | |||||||||||||||||||||||||||||
| 289.4 | 1.00000i | 0 | −1.00000 | 3.60555 | 0 | 1.00000 | − | 1.00000i | 0 | 3.60555i | ||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 3.b | odd | 2 | 1 | inner |
| 29.b | even | 2 | 1 | inner |
| 87.d | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 522.2.d.c | ✓ | 4 |
| 3.b | odd | 2 | 1 | inner | 522.2.d.c | ✓ | 4 |
| 4.b | odd | 2 | 1 | 4176.2.o.m | 4 | ||
| 12.b | even | 2 | 1 | 4176.2.o.m | 4 | ||
| 29.b | even | 2 | 1 | inner | 522.2.d.c | ✓ | 4 |
| 87.d | odd | 2 | 1 | inner | 522.2.d.c | ✓ | 4 |
| 116.d | odd | 2 | 1 | 4176.2.o.m | 4 | ||
| 348.b | even | 2 | 1 | 4176.2.o.m | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 522.2.d.c | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 522.2.d.c | ✓ | 4 | 3.b | odd | 2 | 1 | inner |
| 522.2.d.c | ✓ | 4 | 29.b | even | 2 | 1 | inner |
| 522.2.d.c | ✓ | 4 | 87.d | odd | 2 | 1 | inner |
| 4176.2.o.m | 4 | 4.b | odd | 2 | 1 | ||
| 4176.2.o.m | 4 | 12.b | even | 2 | 1 | ||
| 4176.2.o.m | 4 | 116.d | odd | 2 | 1 | ||
| 4176.2.o.m | 4 | 348.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(522, [\chi])\):
|
\( T_{5}^{2} - 13 \)
|
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\( T_{7} - 1 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T^{2} + 1)^{2} \)
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| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} - 13)^{2} \)
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| $7$ |
\( (T - 1)^{4} \)
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| $11$ |
\( (T^{2} + 4)^{2} \)
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| $13$ |
\( (T - 2)^{4} \)
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| $17$ |
\( (T^{2} + 25)^{2} \)
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| $19$ |
\( (T^{2} + 13)^{2} \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( T^{4} + 6T^{2} + 841 \)
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| $31$ |
\( (T^{2} + 52)^{2} \)
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| $37$ |
\( (T^{2} + 13)^{2} \)
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| $41$ |
\( (T^{2} + 25)^{2} \)
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| $43$ |
\( (T^{2} + 117)^{2} \)
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| $47$ |
\( (T^{2} + 9)^{2} \)
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| $53$ |
\( (T^{2} - 52)^{2} \)
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| $59$ |
\( (T^{2} - 13)^{2} \)
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| $61$ |
\( (T^{2} + 52)^{2} \)
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| $67$ |
\( (T + 10)^{4} \)
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| $71$ |
\( (T^{2} - 52)^{2} \)
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| $73$ |
\( (T^{2} + 52)^{2} \)
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| $79$ |
\( (T^{2} + 208)^{2} \)
|
| $83$ |
\( (T^{2} - 208)^{2} \)
|
| $89$ |
\( (T^{2} + 100)^{2} \)
|
| $97$ |
\( (T^{2} + 208)^{2} \)
|
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