Newspace parameters
| Level: | \( N \) | \(=\) | \( 119 = 7 \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 5 \) |
| Character orbit: | \([\chi]\) | \(=\) | 119.d (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(12.3010256070\) |
| Analytic rank: | \(0\) |
| Dimension: | \(5\) |
| Coefficient field: | 5.5.44253125.1 |
|
|
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| Defining polynomial: |
\( x^{5} - 10x^{3} + 20x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 1 \) |
| 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
\( x^{5} - 10x^{3} + 20x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
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| \(\beta_{3}\) | \(=\) |
\( \nu^{3} - 6\nu \)
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| \(\beta_{4}\) | \(=\) |
\( \nu^{4} - 8\nu^{2} - \nu + 8 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
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| \(\nu^{3}\) | \(=\) |
\( \beta_{3} + 6\beta_1 \)
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| \(\nu^{4}\) | \(=\) |
\( \beta_{4} + 8\beta_{2} + \beta _1 + 24 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/119\mathbb{Z}\right)^\times\).
| \(n\) | \(52\) | \(71\) |
| \(\chi(n)\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 118.1 |
|
−7.32538 | 16.2398 | 37.6612 | −46.0925 | −118.962 | 49.0000 | −158.676 | 182.730 | 337.645 | |||||||||||||||||||||||||||||||||
| 118.2 | −5.32170 | −2.36506 | 12.3204 | 48.6794 | 12.5861 | 49.0000 | 19.5815 | −75.4065 | −259.057 | ||||||||||||||||||||||||||||||||||
| 118.3 | 0.794362 | 12.4018 | −15.3690 | 25.8998 | 9.85151 | 49.0000 | −24.9183 | 72.8042 | 20.5738 | ||||||||||||||||||||||||||||||||||
| 118.4 | 4.03639 | −17.7014 | 0.292450 | −32.6724 | −71.4500 | 49.0000 | −63.4018 | 232.341 | −131.879 | ||||||||||||||||||||||||||||||||||
| 118.5 | 7.81632 | −8.57504 | 45.0949 | 4.18570 | −67.0253 | 49.0000 | 227.415 | −7.46873 | 32.7168 | ||||||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 119.d | odd | 2 | 1 | CM by \(\Q(\sqrt{-119}) \) |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 119.5.d.a | ✓ | 5 |
| 7.b | odd | 2 | 1 | 119.5.d.b | yes | 5 | |
| 17.b | even | 2 | 1 | 119.5.d.b | yes | 5 | |
| 119.d | odd | 2 | 1 | CM | 119.5.d.a | ✓ | 5 |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 119.5.d.a | ✓ | 5 | 1.a | even | 1 | 1 | trivial |
| 119.5.d.a | ✓ | 5 | 119.d | odd | 2 | 1 | CM |
| 119.5.d.b | yes | 5 | 7.b | odd | 2 | 1 | |
| 119.5.d.b | yes | 5 | 17.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 \(S_{5}^{\mathrm{new}}(119, [\chi])\):
|
\( T_{2}^{5} - 80T_{2}^{3} + 1280T_{2} - 977 \)
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\( T_{3}^{5} - 405T_{3}^{3} + 32805T_{3} + 72302 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{5} - 80 T^{3} + \cdots - 977 \)
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| $3$ |
\( T^{5} - 405 T^{3} + \cdots + 72302 \)
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| $5$ |
\( T^{5} - 3125 T^{3} + \cdots - 7947314 \)
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| $7$ |
\( (T - 49)^{5} \)
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| $11$ |
\( T^{5} \)
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| $13$ |
\( T^{5} \)
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| $17$ |
\( (T - 289)^{5} \)
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| $19$ |
\( T^{5} \)
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| $23$ |
\( T^{5} \)
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| $29$ |
\( T^{5} \)
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| $31$ |
\( T^{5} + \cdots - 347038462375298 \)
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| $37$ |
\( T^{5} \)
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| $41$ |
\( T^{5} + \cdots - 46\!\cdots\!98 \)
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| $43$ |
\( T^{5} + \cdots - 20\!\cdots\!02 \)
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| $47$ |
\( T^{5} \)
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| $53$ |
\( T^{5} + \cdots + 32\!\cdots\!98 \)
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| $59$ |
\( T^{5} \)
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| $61$ |
\( T^{5} + \cdots + 76\!\cdots\!02 \)
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| $67$ |
\( T^{5} + \cdots - 12\!\cdots\!02 \)
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| $71$ |
\( T^{5} \)
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| $73$ |
\( T^{5} + \cdots + 17\!\cdots\!02 \)
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| $79$ |
\( T^{5} \)
|
| $83$ |
\( T^{5} \)
|
| $89$ |
\( T^{5} \)
|
| $97$ |
\( T^{5} + \cdots + 15\!\cdots\!02 \)
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