Newspace parameters
| Level: | \( N \) | \(=\) | \( 3645 = 3^{6} \cdot 5 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 3645.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(29.1054715368\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | \(\Q(\zeta_{18})^+\) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - 3x - 1 \)
|
| 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 \(\nu = \zeta_{18} + \zeta_{18}^{-1}\):
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 2 \)
|
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 |
|
−1.87939 | 0 | 1.53209 | 1.00000 | 0 | −5.06418 | 0.879385 | 0 | −1.87939 | |||||||||||||||||||||||||||
| 1.2 | 0.347296 | 0 | −1.87939 | 1.00000 | 0 | 1.75877 | −1.34730 | 0 | 0.347296 | ||||||||||||||||||||||||||||
| 1.3 | 1.53209 | 0 | 0.347296 | 1.00000 | 0 | −2.69459 | −2.53209 | 0 | 1.53209 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(-1\) |
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 | 3645.2.a.b | yes | 3 |
| 3.b | odd | 2 | 1 | 3645.2.a.a | ✓ | 3 | |
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 3645.2.a.a | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 3645.2.a.b | yes | 3 | 1.a | even | 1 | 1 | trivial |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - 3T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3645))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - 3T + 1 \)
$3$
\( T^{3} \)
$5$
\( (T - 1)^{3} \)
$7$
\( T^{3} + 6T^{2} - 24 \)
$11$
\( T^{3} - 21T + 37 \)
$13$
\( T^{3} - 3T - 1 \)
$17$
\( T^{3} + 12 T^{2} + 39 T + 19 \)
$19$
\( T^{3} - 15 T^{2} + 54 T + 17 \)
$23$
\( T^{3} - 63T - 9 \)
$29$
\( T^{3} + 3 T^{2} - 18 T - 37 \)
$31$
\( T^{3} - 18 T^{2} + 72 T + 72 \)
$37$
\( T^{3} + 27 T^{2} + 240 T + 703 \)
$41$
\( T^{3} + 3 T^{2} - 36 T - 127 \)
$43$
\( T^{3} + 27 T^{2} + 222 T + 503 \)
$47$
\( T^{3} + 3 T^{2} - 33 T + 37 \)
$53$
\( T^{3} - 12 T^{2} - 27 T + 361 \)
$59$
\( T^{3} + 18 T^{2} + 87 T + 73 \)
$61$
\( T^{3} - 3 T^{2} - 6 T - 1 \)
$67$
\( T^{3} - 63T + 171 \)
$71$
\( T^{3} + 3 T^{2} - 45 T + 17 \)
$73$
\( T^{3} - 9 T^{2} + 15 T + 17 \)
$79$
\( T^{3} + 9 T^{2} - 36 T - 153 \)
$83$
\( T^{3} + 30 T^{2} + 261 T + 699 \)
$89$
\( T^{3} + 24 T^{2} + 63 T - 969 \)
$97$
\( T^{3} - 21T + 37 \)
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