Newspace parameters
| Level: | \( N \) | \(=\) | \( 273 = 3 \cdot 7 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 273.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(2.17991597518\) |
| Analytic rank: | \(0\) |
| Dimension: | \(4\) |
| Coefficient field: | 4.4.17428.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{4} - x^{3} - 6x^{2} + 4x + 6 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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 \(1,\beta_1,\beta_2,\beta_3\) 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
\( x^{4} - x^{3} - 6x^{2} + 4x + 6 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 3 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{3} - 4\nu - 1 \)
|
| \(\beta_{3}\) | \(=\) |
\( -\nu^{2} + 2\nu + 3 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{3} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 3 \)
|
| \(\nu^{3}\) | \(=\) |
\( 2\beta_{3} + \beta_{2} + 2\beta _1 + 1 \)
|
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 |
|
−2.37951 | 1.00000 | 3.66208 | −2.66208 | −2.37951 | 1.00000 | −3.95493 | 1.00000 | 6.33445 | ||||||||||||||||||||||||||||||
| 1.2 | −0.670843 | 1.00000 | −1.54997 | 2.54997 | −0.670843 | 1.00000 | 2.38147 | 1.00000 | −1.71063 | |||||||||||||||||||||||||||||||
| 1.3 | 1.43986 | 1.00000 | 0.0731828 | 0.926817 | 1.43986 | 1.00000 | −2.77434 | 1.00000 | 1.33448 | |||||||||||||||||||||||||||||||
| 1.4 | 2.61050 | 1.00000 | 4.81471 | −3.81471 | 2.61050 | 1.00000 | 7.34780 | 1.00000 | −9.95830 | |||||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(7\) | \(-1\) |
| \(13\) | \(-1\) |
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 | 273.2.a.e | ✓ | 4 |
| 3.b | odd | 2 | 1 | 819.2.a.k | 4 | ||
| 4.b | odd | 2 | 1 | 4368.2.a.br | 4 | ||
| 5.b | even | 2 | 1 | 6825.2.a.bg | 4 | ||
| 7.b | odd | 2 | 1 | 1911.2.a.s | 4 | ||
| 13.b | even | 2 | 1 | 3549.2.a.w | 4 | ||
| 21.c | even | 2 | 1 | 5733.2.a.bf | 4 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 273.2.a.e | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
| 819.2.a.k | 4 | 3.b | odd | 2 | 1 | ||
| 1911.2.a.s | 4 | 7.b | odd | 2 | 1 | ||
| 3549.2.a.w | 4 | 13.b | even | 2 | 1 | ||
| 4368.2.a.br | 4 | 4.b | odd | 2 | 1 | ||
| 5733.2.a.bf | 4 | 21.c | even | 2 | 1 | ||
| 6825.2.a.bg | 4 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} - T_{2}^{3} - 7T_{2}^{2} + 5T_{2} + 6 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(273))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} - T^{3} - 7 T^{2} + 5 T + 6 \)
$3$
\( (T - 1)^{4} \)
$5$
\( T^{4} + 3 T^{3} - 10 T^{2} - 20 T + 24 \)
$7$
\( (T - 1)^{4} \)
$11$
\( T^{4} + 2 T^{3} - 24 T^{2} - 32 T + 96 \)
$13$
\( (T - 1)^{4} \)
$17$
\( T^{4} + 2 T^{3} - 28 T^{2} - 40 T + 96 \)
$19$
\( T^{4} - 7 T^{3} - 12 T^{2} + 48 T + 64 \)
$23$
\( T^{4} - 3 T^{3} - 52 T^{2} + 256 T - 288 \)
$29$
\( T^{4} - T^{3} - 30 T^{2} + 52 T + 72 \)
$31$
\( T^{4} - 3 T^{3} - 128 T^{2} + \cdots + 3968 \)
$37$
\( T^{4} - 10 T^{3} - 84 T^{2} + \cdots - 128 \)
$41$
\( T^{4} + 16 T^{3} - 688 T - 1392 \)
$43$
\( T^{4} - 3 T^{3} - 44 T^{2} + 112 T - 64 \)
$47$
\( T^{4} - 5 T^{3} - 40 T^{2} + 16 T + 144 \)
$53$
\( T^{4} - 5 T^{3} - 38 T^{2} + 68 T - 24 \)
$59$
\( T^{4} + 20 T^{3} + 80 T^{2} + \cdots - 1536 \)
$61$
\( T^{4} - 12 T^{3} - 64 T^{2} + \cdots + 496 \)
$67$
\( T^{4} + 22 T^{3} - 40 T^{2} + \cdots - 15488 \)
$71$
\( T^{4} - 232 T^{2} + 304 T + 10176 \)
$73$
\( T^{4} + 13 T^{3} - 166 T^{2} + \cdots - 11672 \)
$79$
\( T^{4} - 11 T^{3} - 120 T^{2} + \cdots - 3456 \)
$83$
\( T^{4} - T^{3} - 36 T^{2} - 80 T - 48 \)
$89$
\( T^{4} + 5 T^{3} - 162 T^{2} + \cdots - 1704 \)
$97$
\( T^{4} + 17 T^{3} - 14 T^{2} + \cdots - 1528 \)
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