Newspace parameters
| Level: | \( N \) | \(=\) | \( 231 = 3 \cdot 7 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 231.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(1.84454428669\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.837.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 6x - 1 \)
|
| 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
\( x^{3} - 6x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
−2.36147 | −1.00000 | 3.57653 | −3.93800 | 2.36147 | −1.00000 | −3.72294 | 1.00000 | 9.29947 | |||||||||||||||||||||||||||
| 1.2 | −0.167449 | −1.00000 | −1.97196 | 3.80451 | 0.167449 | −1.00000 | 0.665102 | 1.00000 | −0.637062 | ||||||||||||||||||||||||||||
| 1.3 | 2.52892 | −1.00000 | 4.39543 | 0.133492 | −2.52892 | −1.00000 | 6.05784 | 1.00000 | 0.337590 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(1\) |
| \(7\) | \(1\) |
| \(11\) | \(-1\) |
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 | 231.2.a.d | ✓ | 3 |
| 3.b | odd | 2 | 1 | 693.2.a.m | 3 | ||
| 4.b | odd | 2 | 1 | 3696.2.a.bp | 3 | ||
| 5.b | even | 2 | 1 | 5775.2.a.bw | 3 | ||
| 7.b | odd | 2 | 1 | 1617.2.a.s | 3 | ||
| 11.b | odd | 2 | 1 | 2541.2.a.bi | 3 | ||
| 21.c | even | 2 | 1 | 4851.2.a.bp | 3 | ||
| 33.d | even | 2 | 1 | 7623.2.a.cb | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 231.2.a.d | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 693.2.a.m | 3 | 3.b | odd | 2 | 1 | ||
| 1617.2.a.s | 3 | 7.b | odd | 2 | 1 | ||
| 2541.2.a.bi | 3 | 11.b | odd | 2 | 1 | ||
| 3696.2.a.bp | 3 | 4.b | odd | 2 | 1 | ||
| 4851.2.a.bp | 3 | 21.c | even | 2 | 1 | ||
| 5775.2.a.bw | 3 | 5.b | even | 2 | 1 | ||
| 7623.2.a.cb | 3 | 33.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 6T_{2} - 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(231))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 6T - 1 \)
$3$
\( (T + 1)^{3} \)
$5$
\( T^{3} - 15T + 2 \)
$7$
\( (T + 1)^{3} \)
$11$
\( (T - 1)^{3} \)
$13$
\( T^{3} - 15T + 2 \)
$17$
\( T^{3} - 24T + 8 \)
$19$
\( T^{3} - 12 T^{2} + 27 T + 36 \)
$23$
\( T^{3} + 6 T^{2} - 12 T - 32 \)
$29$
\( T^{3} - 12 T^{2} + 33 T - 6 \)
$31$
\( T^{3} + 6 T^{2} - 36 T + 32 \)
$37$
\( T^{3} - 75T - 246 \)
$41$
\( T^{3} - 6 T^{2} - 72 T + 32 \)
$43$
\( T^{3} - 6 T^{2} - 12 T + 48 \)
$47$
\( T^{3} + 24 T^{2} + 171 T + 328 \)
$53$
\( T^{3} - 48T - 120 \)
$59$
\( T^{3} + 24 T^{2} + 87 T - 716 \)
$61$
\( (T - 6)^{3} \)
$67$
\( T^{3} - 12 T^{2} + 27 T + 4 \)
$71$
\( T^{3} - 12 T^{2} - 48 T + 384 \)
$73$
\( T^{3} - 24 T^{2} + 177 T - 394 \)
$79$
\( T^{3} - 12 T^{2} - 48 T + 256 \)
$83$
\( T^{3} + 18 T^{2} + 60 T + 48 \)
$89$
\( T^{3} - 18 T^{2} - 84 T + 1896 \)
$97$
\( T^{3} - 24 T^{2} + 144 T - 8 \)
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