Newspace parameters
| Level: | \( N \) | \(=\) | \( 47 \) |
| Weight: | \( k \) | \(=\) | \( 13 \) |
| Character orbit: | \([\chi]\) | \(=\) | 47.b (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(42.9577094120\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.6903125.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{5} - 10x^{3} + 20x - 9 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 5^{2}\cdot 11^{2} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{U}(1)[D_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3,\beta_4\) 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
\( x^{5} - 10x^{3} + 20x - 9 \)
:
| \(\beta_{1}\) | \(=\) |
\( -3\nu^{4} + 24\nu^{2} + \nu - 24 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{3} + 9\nu^{2} + 6\nu - 36 \)
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| \(\beta_{3}\) | \(=\) |
\( -5\nu^{4} + 40\nu^{2} + 20\nu - 40 \)
|
| \(\beta_{4}\) | \(=\) |
\( 7\nu^{3} - 8\nu^{2} - 42\nu + 32 \)
|
| \(\nu\) | \(=\) |
\( ( 3\beta_{3} - 5\beta_1 ) / 55 \)
|
| \(\nu^{2}\) | \(=\) |
\( ( \beta_{4} + 7\beta_{2} + 220 ) / 55 \)
|
| \(\nu^{3}\) | \(=\) |
\( ( 9\beta_{4} + 18\beta_{3} + 8\beta_{2} - 30\beta_1 ) / 55 \)
|
| \(\nu^{4}\) | \(=\) |
\( ( 8\beta_{4} + \beta_{3} + 56\beta_{2} - 20\beta _1 + 1320 ) / 55 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/47\mathbb{Z}\right)^\times\).
| \(n\) | \(5\) |
| \(\chi(n)\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 46.1 |
|
−121.396 | 670.582 | 10641.0 | 0 | −81406.1 | −125010. | −794540. | −81760.7 | 0 | |||||||||||||||||||||||||||||||||
| 46.2 | −76.1102 | −1303.48 | 1696.77 | 0 | 99208.2 | 150956. | 182606. | 1.16762e6 | 0 | ||||||||||||||||||||||||||||||||||
| 46.3 | 1.08329 | 218.456 | −4094.83 | 0 | 236.652 | −228217. | −8873.06 | −483718. | 0 | ||||||||||||||||||||||||||||||||||
| 46.4 | 74.3574 | 1438.49 | 1433.03 | 0 | 106963. | 218306. | −198012. | 1.53782e6 | 0 | ||||||||||||||||||||||||||||||||||
| 46.5 | 122.066 | −1024.05 | 10804.0 | 0 | −125001. | −16035.6 | 818818. | 517240. | 0 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 47.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-47}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 47.13.b.a | ✓ | 5 |
| 47.b | odd | 2 | 1 | CM | 47.13.b.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 47.13.b.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 47.13.b.a | ✓ | 5 | 47.b | odd | 2 | 1 | CM |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{5} - 20480T_{2}^{3} + 83886080T_{2} - 90847073 \)
acting on \(S_{13}^{\mathrm{new}}(47, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 20480 T^{3} + \cdots - 90847073 \)
|
| $3$ |
\( T^{5} + \cdots - 281287279384498 \)
|
| $5$ |
\( T^{5} \)
|
| $7$ |
\( T^{5} + \cdots + 15\!\cdots\!02 \)
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| $11$ |
\( T^{5} \)
|
| $13$ |
\( T^{5} \)
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| $17$ |
\( T^{5} + \cdots + 14\!\cdots\!02 \)
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| $19$ |
\( T^{5} \)
|
| $23$ |
\( T^{5} \)
|
| $29$ |
\( T^{5} \)
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| $31$ |
\( T^{5} \)
|
| $37$ |
\( T^{5} + \cdots - 17\!\cdots\!98 \)
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| $41$ |
\( T^{5} \)
|
| $43$ |
\( T^{5} \)
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| $47$ |
\( (T - 10779215329)^{5} \)
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| $53$ |
\( T^{5} + \cdots - 19\!\cdots\!98 \)
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| $59$ |
\( T^{5} + \cdots + 18\!\cdots\!98 \)
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| $61$ |
\( T^{5} + \cdots - 91\!\cdots\!02 \)
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| $67$ |
\( T^{5} \)
|
| $71$ |
\( T^{5} + \cdots + 67\!\cdots\!98 \)
|
| $73$ |
\( T^{5} \)
|
| $79$ |
\( T^{5} + \cdots + 12\!\cdots\!98 \)
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| $83$ |
\( (T + 456721658062)^{5} \)
|
| $89$ |
\( T^{5} + \cdots + 36\!\cdots\!98 \)
|
| $97$ |
\( T^{5} + \cdots + 59\!\cdots\!02 \)
|
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