Newspace parameters
Level: | \( N \) | \(=\) | \( 309 = 3 \cdot 103 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 309.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(2.46737742246\) |
Analytic rank: | \(0\) |
Dimension: | \(3\) |
Coefficient field: | 3.3.148.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
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Defining polynomial: | \( x^{3} - x^{2} - 3x + 1 \) |
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 \( x^{3} - x^{2} - 3x + 1 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( \nu^{2} - \nu - 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{2} + \beta _1 + 2 \) |
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 |
|
−1.48119 | −1.00000 | 0.193937 | −1.48119 | 1.48119 | −4.15633 | 2.67513 | 1.00000 | 2.19394 | |||||||||||||||||||||||||||
1.2 | 0.311108 | −1.00000 | −1.90321 | 0.311108 | −0.311108 | 1.52543 | −1.21432 | 1.00000 | 0.0967881 | ||||||||||||||||||||||||||||
1.3 | 2.17009 | −1.00000 | 2.70928 | 2.17009 | −2.17009 | 0.630898 | 1.53919 | 1.00000 | 4.70928 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(3\) | \(1\) |
\(103\) | \(-1\) |
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 | 309.2.a.b | ✓ | 3 |
3.b | odd | 2 | 1 | 927.2.a.c | 3 | ||
4.b | odd | 2 | 1 | 4944.2.a.v | 3 | ||
5.b | even | 2 | 1 | 7725.2.a.p | 3 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
309.2.a.b | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
927.2.a.c | 3 | 3.b | odd | 2 | 1 | ||
4944.2.a.v | 3 | 4.b | odd | 2 | 1 | ||
7725.2.a.p | 3 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(309))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 3T + 1 \)
$3$
\( (T + 1)^{3} \)
$5$
\( T^{3} - T^{2} - 3T + 1 \)
$7$
\( T^{3} + 2 T^{2} - 8 T + 4 \)
$11$
\( T^{3} - 8 T^{2} + 16 T - 4 \)
$13$
\( T^{3} + 3 T^{2} - 13 T - 31 \)
$17$
\( T^{3} - 4 T^{2} - 8 T + 16 \)
$19$
\( T^{3} - 12 T^{2} + 32 T + 16 \)
$23$
\( T^{3} - 7 T^{2} + 13 T - 5 \)
$29$
\( T^{3} - 28T + 52 \)
$31$
\( T^{3} - 7 T^{2} + 3 T + 19 \)
$37$
\( T^{3} + 6 T^{2} - 28 T - 148 \)
$41$
\( T^{3} - 2 T^{2} - 32 T - 52 \)
$43$
\( T^{3} + T^{2} - 9T - 13 \)
$47$
\( T^{3} - 12 T^{2} + 8 T + 20 \)
$53$
\( T^{3} + 8 T^{2} + 12 T + 4 \)
$59$
\( T^{3} - 15 T^{2} + 41 T - 17 \)
$61$
\( T^{3} + 19 T^{2} + 31 T - 607 \)
$67$
\( T^{3} + 3 T^{2} - 61 T - 191 \)
$71$
\( T^{3} - 8 T^{2} - 72 T + 368 \)
$73$
\( T^{3} + 12 T^{2} + 8 T - 20 \)
$79$
\( T^{3} - 2 T^{2} - 44 T + 20 \)
$83$
\( T^{3} - 17 T^{2} + 83 T - 125 \)
$89$
\( T^{3} - 24 T^{2} + 92 T + 556 \)
$97$
\( T^{3} - 3 T^{2} - 157 T - 449 \)
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