Newspace parameters
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(5.48717763053\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.2089.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 13x - 4 \)
|
| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| 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
\( x^{3} - 13x - 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( ( \nu^{2} - \nu - 8 ) / 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( 2\beta_{2} + \beta _1 + 8 \)
|
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 |
|
−2.44055 | −3.00000 | −2.04371 | −7.52008 | 7.32165 | −13.6354 | 24.5122 | 9.00000 | 18.3531 | |||||||||||||||||||||||||||
| 1.2 | 0.690016 | −3.00000 | −7.52388 | 6.17700 | −2.07005 | 26.6748 | −10.7117 | 9.00000 | 4.26223 | ||||||||||||||||||||||||||||
| 1.3 | 4.75054 | −3.00000 | 14.5676 | 9.34308 | −14.2516 | 5.96059 | 31.1995 | 9.00000 | 44.3846 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \( +1 \) |
| \(31\) | \( +1 \) |
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 | 93.4.a.e | ✓ | 3 |
| 3.b | odd | 2 | 1 | 279.4.a.f | 3 | ||
| 4.b | odd | 2 | 1 | 1488.4.a.q | 3 | ||
| 5.b | even | 2 | 1 | 2325.4.a.o | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.4.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 279.4.a.f | 3 | 3.b | odd | 2 | 1 | ||
| 1488.4.a.q | 3 | 4.b | odd | 2 | 1 | ||
| 2325.4.a.o | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2}^{2} - 10T_{2} + 8 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(93))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 3 T^{2} + \cdots + 8 \)
|
| $3$ |
\( (T + 3)^{3} \)
|
| $5$ |
\( T^{3} - 8 T^{2} + \cdots + 434 \)
|
| $7$ |
\( T^{3} - 19 T^{2} + \cdots + 2168 \)
|
| $11$ |
\( T^{3} - 59 T^{2} + \cdots - 3184 \)
|
| $13$ |
\( T^{3} - 25 T^{2} + \cdots - 3136 \)
|
| $17$ |
\( T^{3} - 71 T^{2} + \cdots + 93584 \)
|
| $19$ |
\( T^{3} - 52 T^{2} + \cdots + 61516 \)
|
| $23$ |
\( T^{3} - 120 T^{2} + \cdots - 12544 \)
|
| $29$ |
\( T^{3} + 70 T^{2} + \cdots - 124864 \)
|
| $31$ |
\( (T + 31)^{3} \)
|
| $37$ |
\( T^{3} + 198 T^{2} + \cdots - 506488 \)
|
| $41$ |
\( T^{3} - 403 T^{2} + \cdots + 534388 \)
|
| $43$ |
\( T^{3} - 32 T^{2} + \cdots + 2465552 \)
|
| $47$ |
\( T^{3} - 675 T^{2} + \cdots + 9589304 \)
|
| $53$ |
\( T^{3} - 454 T^{2} + \cdots + 103574624 \)
|
| $59$ |
\( T^{3} - 1159 T^{2} + \cdots + 156232684 \)
|
| $61$ |
\( T^{3} + 563 T^{2} + \cdots + 4561216 \)
|
| $67$ |
\( T^{3} - 801 T^{2} + \cdots + 450212528 \)
|
| $71$ |
\( T^{3} - 1306 T^{2} + \cdots + 471768226 \)
|
| $73$ |
\( T^{3} - 282 T^{2} + \cdots + 338525032 \)
|
| $79$ |
\( T^{3} - 1025 T^{2} + \cdots - 25506944 \)
|
| $83$ |
\( T^{3} - 1393 T^{2} + \cdots - 69563504 \)
|
| $89$ |
\( T^{3} + 1452 T^{2} + \cdots - 421613856 \)
|
| $97$ |
\( T^{3} - 148 T^{2} + \cdots - 252557894 \)
|
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