Newspace parameters
| Level: | \( N \) | \(=\) | \( 736 = 2^{5} \cdot 23 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 736.e (of order \(2\), degree \(1\), not minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(76.0802928297\) |
| Analytic rank: | \(0\) |
| Dimension: | \(6\) |
| Coefficient field: | 6.0.8869743.1 |
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|
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| Defining polynomial: |
\( x^{6} - 3x^{3} + 8 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{29}]\) |
| Coefficient ring index: | \( 2^{17} \) |
| Twist minimal: | no (minimal twist has level 184) |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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
\( x^{6} - 3x^{3} + 8 \)
:
| \(\beta_{1}\) | \(=\) |
\( -\nu^{5} - 2\nu^{4} + 7\nu^{2} + 10\nu \)
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| \(\beta_{2}\) | \(=\) |
\( ( 3\nu^{5} + 10\nu^{4} + 11\nu^{2} - 18\nu ) / 4 \)
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| \(\beta_{3}\) | \(=\) |
\( ( 5\nu^{5} - 6\nu^{4} - 3\nu^{2} - 2\nu ) / 4 \)
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| \(\beta_{4}\) | \(=\) |
\( ( 7\nu^{5} + 2\nu^{4} - 17\nu^{2} + 22\nu ) / 4 \)
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| \(\beta_{5}\) | \(=\) |
\( 32\nu^{3} - 48 \)
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| \(\nu\) | \(=\) |
\( ( 5\beta_{4} - 4\beta_{3} - \beta_{2} + 3\beta_1 ) / 64 \)
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| \(\nu^{2}\) | \(=\) |
\( ( -3\beta_{4} + 4\beta_{3} + 7\beta_{2} + 5\beta_1 ) / 64 \)
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| \(\nu^{3}\) | \(=\) |
\( ( \beta_{5} + 48 ) / 32 \)
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| \(\nu^{4}\) | \(=\) |
\( ( 9\beta_{4} - 20\beta_{3} + 11\beta_{2} - \beta_1 ) / 64 \)
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| \(\nu^{5}\) | \(=\) |
\( ( 11\beta_{4} + 28\beta_{3} + 17\beta_{2} + 3\beta_1 ) / 64 \)
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Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/736\mathbb{Z}\right)^\times\).
| \(n\) | \(97\) | \(415\) | \(645\) |
| \(\chi(n)\) | \(-1\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 689.1 |
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0 | − | 17.9999i | 0 | 0 | 0 | 0 | 0 | −242.997 | 0 | |||||||||||||||||||||||||||||||||||
| 689.2 | 0 | − | 9.04985i | 0 | 0 | 0 | 0 | 0 | −0.899768 | 0 | ||||||||||||||||||||||||||||||||||||
| 689.3 | 0 | − | 8.95006i | 0 | 0 | 0 | 0 | 0 | 0.896448 | 0 | ||||||||||||||||||||||||||||||||||||
| 689.4 | 0 | 8.95006i | 0 | 0 | 0 | 0 | 0 | 0.896448 | 0 | |||||||||||||||||||||||||||||||||||||
| 689.5 | 0 | 9.04985i | 0 | 0 | 0 | 0 | 0 | −0.899768 | 0 | |||||||||||||||||||||||||||||||||||||
| 689.6 | 0 | 17.9999i | 0 | 0 | 0 | 0 | 0 | −242.997 | 0 | |||||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 23.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-23}) \) |
| 8.b | even | 2 | 1 | inner |
| 184.e | odd | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 736.5.e.d | 6 | |
| 4.b | odd | 2 | 1 | 184.5.e.d | ✓ | 6 | |
| 8.b | even | 2 | 1 | inner | 736.5.e.d | 6 | |
| 8.d | odd | 2 | 1 | 184.5.e.d | ✓ | 6 | |
| 23.b | odd | 2 | 1 | CM | 736.5.e.d | 6 | |
| 92.b | even | 2 | 1 | 184.5.e.d | ✓ | 6 | |
| 184.e | odd | 2 | 1 | inner | 736.5.e.d | 6 | |
| 184.h | even | 2 | 1 | 184.5.e.d | ✓ | 6 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 184.5.e.d | ✓ | 6 | 4.b | odd | 2 | 1 | |
| 184.5.e.d | ✓ | 6 | 8.d | odd | 2 | 1 | |
| 184.5.e.d | ✓ | 6 | 92.b | even | 2 | 1 | |
| 184.5.e.d | ✓ | 6 | 184.h | even | 2 | 1 | |
| 736.5.e.d | 6 | 1.a | even | 1 | 1 | trivial | |
| 736.5.e.d | 6 | 8.b | even | 2 | 1 | inner | |
| 736.5.e.d | 6 | 23.b | odd | 2 | 1 | CM | |
| 736.5.e.d | 6 | 184.e | odd | 2 | 1 | inner | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{5}^{\mathrm{new}}(736, [\chi])\):
|
\( T_{3}^{6} + 486T_{3}^{4} + 59049T_{3}^{2} + 2125568 \)
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\( T_{5} \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{6} \)
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| $3$ |
\( T^{6} + 486 T^{4} + \cdots + 2125568 \)
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| $5$ |
\( T^{6} \)
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| $7$ |
\( T^{6} \)
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| $11$ |
\( T^{6} \)
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| $13$ |
\( T^{6} + \cdots + 21232849874688 \)
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| $17$ |
\( T^{6} \)
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| $19$ |
\( T^{6} \)
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| $23$ |
\( (T - 529)^{6} \)
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| $29$ |
\( T^{6} + \cdots + 13\!\cdots\!88 \)
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| $31$ |
\( (T^{3} - 2770563 T - 1677025154)^{2} \)
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| $37$ |
\( T^{6} \)
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| $41$ |
\( (T^{3} - 8477283 T - 7596282526)^{2} \)
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| $43$ |
\( T^{6} \)
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| $47$ |
\( (T^{3} - 14639043 T + 20606906306)^{2} \)
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| $53$ |
\( T^{6} \)
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| $59$ |
\( (T^{2} + 8955648)^{3} \)
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| $61$ |
\( T^{6} \)
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| $67$ |
\( T^{6} \)
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| $71$ |
\( (T^{3} - 76235043 T + 188893891874)^{2} \)
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| $73$ |
\( (T^{3} - 85194723 T + 223017449186)^{2} \)
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| $79$ |
\( T^{6} \)
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| $83$ |
\( T^{6} \)
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| $89$ |
\( T^{6} \)
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| $97$ |
\( T^{6} \)
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