Newspace parameters
| Level: | \( N \) | \(=\) | \( 850 = 2 \cdot 5^{2} \cdot 17 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 850.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(50.1516235049\) |
| Analytic rank: | \(1\) |
| Dimension: | \(5\) |
| Coefficient field: | \(\mathbb{Q}[x]/(x^{5} - \cdots)\) |
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|
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| Defining polynomial: |
\( x^{5} - x^{4} - 42x^{3} + 50x^{2} + 349x - 317 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{7}]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | no (minimal twist has level 170) |
| Fricke sign: | \(-1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-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
\( x^{5} - x^{4} - 42x^{3} + 50x^{2} + 349x - 317 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - 17 ) / 2 \)
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| \(\beta_{3}\) | \(=\) |
\( ( \nu^{4} + 4\nu^{3} - 28\nu^{2} - 84\nu + 91 ) / 12 \)
|
| \(\beta_{4}\) | \(=\) |
\( ( -\nu^{4} + 2\nu^{3} + 34\nu^{2} - 66\nu - 133 ) / 12 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + 17 \)
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| \(\nu^{3}\) | \(=\) |
\( 2\beta_{4} + 2\beta_{3} - 2\beta_{2} + 25\beta _1 - 10 \)
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| \(\nu^{4}\) | \(=\) |
\( -8\beta_{4} + 4\beta_{3} + 64\beta_{2} - 16\beta _1 + 425 \)
|
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
2.00000 | −8.82384 | 4.00000 | 0 | −17.6477 | 18.3177 | 8.00000 | 50.8601 | 0 | |||||||||||||||||||||||||||||||||
| 1.2 | 2.00000 | −5.87390 | 4.00000 | 0 | −11.7478 | −15.1251 | 8.00000 | 7.50273 | 0 | ||||||||||||||||||||||||||||||||||
| 1.3 | 2.00000 | 0.130417 | 4.00000 | 0 | 0.260834 | 5.61659 | 8.00000 | −26.9830 | 0 | ||||||||||||||||||||||||||||||||||
| 1.4 | 2.00000 | 2.45415 | 4.00000 | 0 | 4.90829 | −12.8054 | 8.00000 | −20.9772 | 0 | ||||||||||||||||||||||||||||||||||
| 1.5 | 2.00000 | 7.11318 | 4.00000 | 0 | 14.2264 | −22.0038 | 8.00000 | 23.5973 | 0 | ||||||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( -1 \) |
| \(5\) | \( -1 \) |
| \(17\) | \( -1 \) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 850.4.a.s | 5 | |
| 5.b | even | 2 | 1 | 850.4.a.r | 5 | ||
| 5.c | odd | 4 | 2 | 170.4.c.a | ✓ | 10 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 170.4.c.a | ✓ | 10 | 5.c | odd | 4 | 2 | |
| 850.4.a.r | 5 | 5.b | even | 2 | 1 | ||
| 850.4.a.s | 5 | 1.a | even | 1 | 1 | trivial | |
Hecke kernels
This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(850))\):
|
\( T_{3}^{5} + 5T_{3}^{4} - 72T_{3}^{3} - 230T_{3}^{2} + 936T_{3} - 118 \)
|
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\( T_{7}^{5} + 26T_{7}^{4} - 284T_{7}^{3} - 9946T_{7}^{2} - 18846T_{7} + 438464 \)
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Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T - 2)^{5} \)
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| $3$ |
\( T^{5} + 5 T^{4} + \cdots - 118 \)
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| $5$ |
\( T^{5} \)
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| $7$ |
\( T^{5} + 26 T^{4} + \cdots + 438464 \)
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| $11$ |
\( T^{5} + 104 T^{4} + \cdots - 6730100 \)
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| $13$ |
\( T^{5} + 79 T^{4} + \cdots - 54676180 \)
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| $17$ |
\( (T - 17)^{5} \)
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| $19$ |
\( T^{5} + 9 T^{4} + \cdots - 323725936 \)
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| $23$ |
\( T^{5} - 28 T^{4} + \cdots - 244176896 \)
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| $29$ |
\( T^{5} + 429 T^{4} + \cdots - 79797328 \)
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| $31$ |
\( T^{5} + \cdots + 183983556422 \)
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| $37$ |
\( T^{5} + \cdots + 81652362112 \)
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| $41$ |
\( T^{5} + \cdots - 473106238784 \)
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| $43$ |
\( T^{5} + 138 T^{4} + \cdots - 912218896 \)
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| $47$ |
\( T^{5} + \cdots + 101704831756 \)
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| $53$ |
\( T^{5} + \cdots - 2420910879056 \)
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| $59$ |
\( T^{5} + \cdots - 678267250832 \)
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| $61$ |
\( T^{5} + \cdots + 80879412314024 \)
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| $67$ |
\( T^{5} + \cdots - 28652020483688 \)
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| $71$ |
\( T^{5} + \cdots + 54066860182 \)
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| $73$ |
\( T^{5} + \cdots + 392981834872 \)
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| $79$ |
\( T^{5} + \cdots - 56359197318400 \)
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| $83$ |
\( T^{5} + \cdots + 6713457927440 \)
|
| $89$ |
\( T^{5} + \cdots + 123503446265116 \)
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| $97$ |
\( T^{5} + \cdots - 5118591823600 \)
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