Newspace parameters
| Level: | \( N \) | \(=\) | \( 93 = 3 \cdot 31 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 93.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(0.742608738798\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.229.1 |
|
|
|
| Defining polynomial: |
\( x^{3} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| 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
\( x^{3} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 3 \)
|
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 |
|
−2.11491 | 1.00000 | 2.47283 | −0.357926 | −2.11491 | 1.64207 | −1.00000 | 1.00000 | 0.756981 | |||||||||||||||||||||||||||
| 1.2 | 0.254102 | 1.00000 | −1.93543 | 1.68133 | 0.254102 | 3.68133 | −1.00000 | 1.00000 | 0.427229 | ||||||||||||||||||||||||||||
| 1.3 | 1.86081 | 1.00000 | 1.46260 | −3.32340 | 1.86081 | −1.32340 | −1.00000 | 1.00000 | −6.18421 | ||||||||||||||||||||||||||||
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 | 93.2.a.b | ✓ | 3 |
| 3.b | odd | 2 | 1 | 279.2.a.c | 3 | ||
| 4.b | odd | 2 | 1 | 1488.2.a.t | 3 | ||
| 5.b | even | 2 | 1 | 2325.2.a.s | 3 | ||
| 5.c | odd | 4 | 2 | 2325.2.c.n | 6 | ||
| 7.b | odd | 2 | 1 | 4557.2.a.v | 3 | ||
| 8.b | even | 2 | 1 | 5952.2.a.bz | 3 | ||
| 8.d | odd | 2 | 1 | 5952.2.a.cf | 3 | ||
| 12.b | even | 2 | 1 | 4464.2.a.bq | 3 | ||
| 15.d | odd | 2 | 1 | 6975.2.a.bb | 3 | ||
| 31.b | odd | 2 | 1 | 2883.2.a.f | 3 | ||
| 93.c | even | 2 | 1 | 8649.2.a.p | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 93.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 279.2.a.c | 3 | 3.b | odd | 2 | 1 | ||
| 1488.2.a.t | 3 | 4.b | odd | 2 | 1 | ||
| 2325.2.a.s | 3 | 5.b | even | 2 | 1 | ||
| 2325.2.c.n | 6 | 5.c | odd | 4 | 2 | ||
| 2883.2.a.f | 3 | 31.b | odd | 2 | 1 | ||
| 4464.2.a.bq | 3 | 12.b | even | 2 | 1 | ||
| 4557.2.a.v | 3 | 7.b | odd | 2 | 1 | ||
| 5952.2.a.bz | 3 | 8.b | even | 2 | 1 | ||
| 5952.2.a.cf | 3 | 8.d | odd | 2 | 1 | ||
| 6975.2.a.bb | 3 | 15.d | odd | 2 | 1 | ||
| 8649.2.a.p | 3 | 93.c | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 4T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(93))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} - 4T + 1 \)
|
| $3$ |
\( (T - 1)^{3} \)
|
| $5$ |
\( T^{3} + 2 T^{2} + \cdots - 2 \)
|
| $7$ |
\( T^{3} - 4T^{2} - T + 8 \)
|
| $11$ |
\( T^{3} + 2 T^{2} + \cdots + 16 \)
|
| $13$ |
\( T^{3} - 4 T^{2} + \cdots + 56 \)
|
| $17$ |
\( T^{3} + 2 T^{2} + \cdots - 32 \)
|
| $19$ |
\( T^{3} - 4 T^{2} + \cdots + 196 \)
|
| $23$ |
\( T^{3} + 6 T^{2} + \cdots - 32 \)
|
| $29$ |
\( T^{3} + 8 T^{2} + \cdots - 392 \)
|
| $31$ |
\( (T + 1)^{3} \)
|
| $37$ |
\( T^{3} - 16T + 8 \)
|
| $41$ |
\( T^{3} + 10 T^{2} + \cdots - 262 \)
|
| $43$ |
\( T^{3} - 14 T^{2} + \cdots + 368 \)
|
| $47$ |
\( T^{3} - 12 T^{2} + \cdots + 256 \)
|
| $53$ |
\( T^{3} + 10 T^{2} + \cdots - 32 \)
|
| $59$ |
\( T^{3} - 26 T^{2} + \cdots - 556 \)
|
| $61$ |
\( T^{3} + 2 T^{2} + \cdots - 512 \)
|
| $67$ |
\( (T - 4)^{3} \)
|
| $71$ |
\( T^{3} + 10 T^{2} + \cdots - 712 \)
|
| $73$ |
\( T^{3} + 12 T^{2} + \cdots - 728 \)
|
| $79$ |
\( T^{3} - 8 T^{2} + \cdots + 64 \)
|
| $83$ |
\( T^{3} - 20 T^{2} + \cdots - 112 \)
|
| $89$ |
\( (T + 6)^{3} \)
|
| $97$ |
\( T^{3} - 4 T^{2} + \cdots + 94 \)
|
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