Newspace parameters
| Level: | \( N \) | \(=\) | \( 418 = 2 \cdot 11 \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 418.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.33774680449\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.469.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 5x + 4 \)
|
| Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
| 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
\( x^{3} - x^{2} - 5x + 4 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 4 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 4 \)
|
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 |
|
1.00000 | −2.16425 | 1.00000 | 4.16425 | −2.16425 | −0.683969 | 1.00000 | 1.68397 | 4.16425 | |||||||||||||||||||||||||||
| 1.2 | 1.00000 | 0.772866 | 1.00000 | 1.22713 | 0.772866 | 3.40268 | 1.00000 | −2.40268 | 1.22713 | ||||||||||||||||||||||||||||
| 1.3 | 1.00000 | 2.39138 | 1.00000 | −0.391382 | 2.39138 | −1.71871 | 1.00000 | 2.71871 | −0.391382 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(2\) | \(-1\) |
| \(11\) | \(1\) |
| \(19\) | \(1\) |
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 | 418.2.a.h | ✓ | 3 |
| 3.b | odd | 2 | 1 | 3762.2.a.bd | 3 | ||
| 4.b | odd | 2 | 1 | 3344.2.a.p | 3 | ||
| 11.b | odd | 2 | 1 | 4598.2.a.bm | 3 | ||
| 19.b | odd | 2 | 1 | 7942.2.a.bc | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 418.2.a.h | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 3344.2.a.p | 3 | 4.b | odd | 2 | 1 | ||
| 3762.2.a.bd | 3 | 3.b | odd | 2 | 1 | ||
| 4598.2.a.bm | 3 | 11.b | odd | 2 | 1 | ||
| 7942.2.a.bc | 3 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{3}^{3} - T_{3}^{2} - 5T_{3} + 4 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(418))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T - 1)^{3} \)
$3$
\( T^{3} - T^{2} - 5T + 4 \)
$5$
\( T^{3} - 5 T^{2} + 3 T + 2 \)
$7$
\( T^{3} - T^{2} - 7T - 4 \)
$11$
\( (T + 1)^{3} \)
$13$
\( T^{3} - 5T^{2} + T + 14 \)
$17$
\( T^{3} - 4 T^{2} - 24 T + 88 \)
$19$
\( (T + 1)^{3} \)
$23$
\( T^{3} - 2 T^{2} - 28 T - 32 \)
$29$
\( T^{3} - 7 T^{2} - 19 T + 86 \)
$31$
\( T^{3} + 3 T^{2} - 25 T - 76 \)
$37$
\( T^{3} + 14 T^{2} + 16 T - 128 \)
$41$
\( T^{3} + 3 T^{2} - 25 T + 22 \)
$43$
\( T^{3} + 5 T^{2} - 67 T - 268 \)
$47$
\( T^{3} - 52T + 128 \)
$53$
\( T^{3} + 16 T^{2} + 64 T + 56 \)
$59$
\( T^{3} + 12 T^{2} - 160 T - 1792 \)
$61$
\( (T - 2)^{3} \)
$67$
\( T^{3} - T^{2} - 7T - 4 \)
$71$
\( T^{3} + 3 T^{2} - 31 T + 28 \)
$73$
\( T^{3} - 4 T^{2} - 136 T + 56 \)
$79$
\( T^{3} - 2 T^{2} - 116 T - 352 \)
$83$
\( T^{3} - 7 T^{2} - 143 T + 1108 \)
$89$
\( T^{3} - 16 T^{2} - 280 T + 4424 \)
$97$
\( T^{3} + 14 T^{2} + 16 T - 128 \)
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