Newspace parameters
| Level: | \( N \) | \(=\) | \( 738 = 2 \cdot 3^{2} \cdot 41 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 738.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(5.89295966917\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | \(\Q(i, \sqrt{57})\) |
|
|
|
| Defining polynomial: |
\( x^{4} + 29x^{2} + 196 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{13}]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 246) |
| 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
\( x^{4} + 29x^{2} + 196 \)
:
| \(\beta_{1}\) | \(=\) |
\( ( -\nu^{3} - 29\nu ) / 14 \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{3} + 15\nu ) / 7 \)
|
| \(\beta_{3}\) | \(=\) |
\( \nu^{2} + 15 \)
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| \(\nu\) | \(=\) |
\( ( -\beta_{2} - 2\beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{3} - 15 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 29\beta_{2} + 30\beta_1 ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/738\mathbb{Z}\right)^\times\).
| \(n\) | \(83\) | \(703\) |
| \(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 163.1 |
|
1.00000 | 0 | 1.00000 | −4.27492 | 0 | − | 2.00000i | 1.00000 | 0 | −4.27492 | |||||||||||||||||||||||||||||
| 163.2 | 1.00000 | 0 | 1.00000 | −4.27492 | 0 | 2.00000i | 1.00000 | 0 | −4.27492 | |||||||||||||||||||||||||||||||
| 163.3 | 1.00000 | 0 | 1.00000 | 3.27492 | 0 | − | 2.00000i | 1.00000 | 0 | 3.27492 | ||||||||||||||||||||||||||||||
| 163.4 | 1.00000 | 0 | 1.00000 | 3.27492 | 0 | 2.00000i | 1.00000 | 0 | 3.27492 | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 41.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 738.2.d.h | 4 | |
| 3.b | odd | 2 | 1 | 246.2.d.b | ✓ | 4 | |
| 12.b | even | 2 | 1 | 1968.2.j.c | 4 | ||
| 41.b | even | 2 | 1 | inner | 738.2.d.h | 4 | |
| 123.b | odd | 2 | 1 | 246.2.d.b | ✓ | 4 | |
| 492.d | even | 2 | 1 | 1968.2.j.c | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 246.2.d.b | ✓ | 4 | 3.b | odd | 2 | 1 | |
| 246.2.d.b | ✓ | 4 | 123.b | odd | 2 | 1 | |
| 738.2.d.h | 4 | 1.a | even | 1 | 1 | trivial | |
| 738.2.d.h | 4 | 41.b | even | 2 | 1 | inner | |
| 1968.2.j.c | 4 | 12.b | even | 2 | 1 | ||
| 1968.2.j.c | 4 | 492.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(738, [\chi])\):
|
\( T_{5}^{2} + T_{5} - 14 \)
|
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\( T_{7}^{2} + 4 \)
|
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 1)^{4} \)
|
| $3$ |
\( T^{4} \)
|
| $5$ |
\( (T^{2} + T - 14)^{2} \)
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| $7$ |
\( (T^{2} + 4)^{2} \)
|
| $11$ |
\( (T^{2} + 16)^{2} \)
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| $13$ |
\( T^{4} + 29T^{2} + 196 \)
|
| $17$ |
\( T^{4} + 41T^{2} + 64 \)
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| $19$ |
\( T^{4} + 41T^{2} + 64 \)
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| $23$ |
\( T^{4} \)
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| $29$ |
\( T^{4} + 116T^{2} + 3136 \)
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| $31$ |
\( (T^{2} + 11 T + 16)^{2} \)
|
| $37$ |
\( (T + 6)^{4} \)
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| $41$ |
\( T^{4} - 10 T^{3} + \cdots + 1681 \)
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| $43$ |
\( (T^{2} + 10 T - 32)^{2} \)
|
| $47$ |
\( T^{4} + 164T^{2} + 1024 \)
|
| $53$ |
\( (T^{2} + 4)^{2} \)
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| $59$ |
\( (T^{2} + 3 T - 12)^{2} \)
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| $61$ |
\( (T^{2} - 2 T - 56)^{2} \)
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| $67$ |
\( T^{4} + 89T^{2} + 256 \)
|
| $71$ |
\( T^{4} + 141T^{2} + 1764 \)
|
| $73$ |
\( (T^{2} + 7 T - 2)^{2} \)
|
| $79$ |
\( T^{4} + 116T^{2} + 3136 \)
|
| $83$ |
\( (T^{2} - 5 T - 8)^{2} \)
|
| $89$ |
\( T^{4} + 281 T^{2} + 13456 \)
|
| $97$ |
\( T^{4} + 212T^{2} + 64 \)
|
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