Newspace parameters
| Level: | \( N \) | \(=\) | \( 86 = 2 \cdot 43 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 86.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.07416426049\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.85685.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 32x - 41 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(+1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) 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
\( x^{3} - 32x - 41 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 21 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2\beta _1 + 21 \)
|
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 | −9.21283 | 4.00000 | −9.25207 | 18.4257 | −24.7728 | −8.00000 | 57.8762 | 18.5041 | |||||||||||||||||||||||||||
| 1.2 | −2.00000 | −1.64017 | 4.00000 | −15.7116 | 3.28034 | 33.5821 | −8.00000 | −24.3098 | 31.4231 | ||||||||||||||||||||||||||||
| 1.3 | −2.00000 | 1.85300 | 4.00000 | 19.9636 | −3.70600 | −16.8092 | −8.00000 | −23.5664 | −39.9272 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \( +1 \) |
| \(43\) | \( +1 \) |
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 | 86.4.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 774.4.a.j | 3 | ||
| 4.b | odd | 2 | 1 | 688.4.a.d | 3 | ||
| 5.b | even | 2 | 1 | 2150.4.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 86.4.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 688.4.a.d | 3 | 4.b | odd | 2 | 1 | ||
| 774.4.a.j | 3 | 3.b | odd | 2 | 1 | ||
| 2150.4.a.j | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} + 9T_{3}^{2} - 5T_{3} - 28 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(86))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( (T + 2)^{3} \)
|
| $3$ |
\( T^{3} + 9 T^{2} + \cdots - 28 \)
|
| $5$ |
\( T^{3} + 5 T^{2} + \cdots - 2902 \)
|
| $7$ |
\( T^{3} + 8 T^{2} + \cdots - 13984 \)
|
| $11$ |
\( T^{3} - 96 T^{2} + \cdots - 13760 \)
|
| $13$ |
\( T^{3} - 174 T^{2} + \cdots - 162792 \)
|
| $17$ |
\( T^{3} - 113 T^{2} + \cdots - 8430 \)
|
| $19$ |
\( T^{3} - 133 T^{2} + \cdots + 443180 \)
|
| $23$ |
\( T^{3} + 175 T^{2} + \cdots - 927648 \)
|
| $29$ |
\( T^{3} + 97 T^{2} + \cdots - 178686 \)
|
| $31$ |
\( T^{3} - 215 T^{2} + \cdots + 7121032 \)
|
| $37$ |
\( T^{3} + 183 T^{2} + \cdots - 21099454 \)
|
| $41$ |
\( T^{3} - 21 T^{2} + \cdots + 7650850 \)
|
| $43$ |
\( (T + 43)^{3} \)
|
| $47$ |
\( T^{3} + 551 T^{2} + \cdots - 19705240 \)
|
| $53$ |
\( T^{3} - 812 T^{2} + \cdots + 336355560 \)
|
| $59$ |
\( T^{3} + 524 T^{2} + \cdots - 89985168 \)
|
| $61$ |
\( T^{3} - 1106 T^{2} + \cdots + 101721640 \)
|
| $67$ |
\( T^{3} - 952 T^{2} + \cdots + 48477360 \)
|
| $71$ |
\( T^{3} + 312 T^{2} + \cdots - 58882208 \)
|
| $73$ |
\( T^{3} + 1226 T^{2} + \cdots + 23640984 \)
|
| $79$ |
\( T^{3} - 709 T^{2} + \cdots - 94472200 \)
|
| $83$ |
\( T^{3} + 106 T^{2} + \cdots - 244974000 \)
|
| $89$ |
\( T^{3} - 372 T^{2} + \cdots - 365455720 \)
|
| $97$ |
\( T^{3} + \cdots - 1491830494 \)
|
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