Newspace parameters
| Level: | \( N \) | \(=\) | \( 495 = 3^{2} \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 495.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.95259490005\) |
| Analytic rank: | \(0\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.148.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 3x + 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| Twist minimal: | no (minimal twist has level 165) |
| 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
\( x^{3} - x^{2} - 3x + 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu^{2} - 2 \)
|
| \(\beta_{2}\) | \(=\) |
\( -\nu^{2} + 2\nu + 2 \)
|
| \(\nu\) | \(=\) |
\( ( \beta_{2} + \beta_1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta _1 + 2 \)
|
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 |
|
−1.90321 | 0 | 1.62222 | −1.00000 | 0 | −4.42864 | 0.719004 | 0 | 1.90321 | |||||||||||||||||||||||||||
| 1.2 | 0.193937 | 0 | −1.96239 | −1.00000 | 0 | 3.35026 | −0.768452 | 0 | −0.193937 | ||||||||||||||||||||||||||||
| 1.3 | 2.70928 | 0 | 5.34017 | −1.00000 | 0 | 1.07838 | 9.04945 | 0 | −2.70928 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(3\) | \(-1\) |
| \(5\) | \(1\) |
| \(11\) | \(1\) |
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 | 495.2.a.e | 3 | |
| 3.b | odd | 2 | 1 | 165.2.a.c | ✓ | 3 | |
| 4.b | odd | 2 | 1 | 7920.2.a.cj | 3 | ||
| 5.b | even | 2 | 1 | 2475.2.a.bb | 3 | ||
| 5.c | odd | 4 | 2 | 2475.2.c.r | 6 | ||
| 11.b | odd | 2 | 1 | 5445.2.a.z | 3 | ||
| 12.b | even | 2 | 1 | 2640.2.a.be | 3 | ||
| 15.d | odd | 2 | 1 | 825.2.a.k | 3 | ||
| 15.e | even | 4 | 2 | 825.2.c.g | 6 | ||
| 21.c | even | 2 | 1 | 8085.2.a.bk | 3 | ||
| 33.d | even | 2 | 1 | 1815.2.a.m | 3 | ||
| 165.d | even | 2 | 1 | 9075.2.a.cf | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.2.a.c | ✓ | 3 | 3.b | odd | 2 | 1 | |
| 495.2.a.e | 3 | 1.a | even | 1 | 1 | trivial | |
| 825.2.a.k | 3 | 15.d | odd | 2 | 1 | ||
| 825.2.c.g | 6 | 15.e | even | 4 | 2 | ||
| 1815.2.a.m | 3 | 33.d | even | 2 | 1 | ||
| 2475.2.a.bb | 3 | 5.b | even | 2 | 1 | ||
| 2475.2.c.r | 6 | 5.c | odd | 4 | 2 | ||
| 2640.2.a.be | 3 | 12.b | even | 2 | 1 | ||
| 5445.2.a.z | 3 | 11.b | odd | 2 | 1 | ||
| 7920.2.a.cj | 3 | 4.b | odd | 2 | 1 | ||
| 8085.2.a.bk | 3 | 21.c | even | 2 | 1 | ||
| 9075.2.a.cf | 3 | 165.d | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 5T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(495))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 5T + 1 \)
$3$
\( T^{3} \)
$5$
\( (T + 1)^{3} \)
$7$
\( T^{3} - 16T + 16 \)
$11$
\( (T + 1)^{3} \)
$13$
\( T^{3} + 2 T^{2} - 12 T - 8 \)
$17$
\( T^{3} - 2 T^{2} - 52 T + 184 \)
$19$
\( T^{3} - 8 T^{2} - 16 T + 160 \)
$23$
\( T^{3} - 64T + 128 \)
$29$
\( T^{3} - 10 T^{2} + 12 T + 40 \)
$31$
\( T^{3} - 8 T^{2} - 32 T + 128 \)
$37$
\( (T + 2)^{3} \)
$41$
\( T^{3} - 14 T^{2} + 44 T - 8 \)
$43$
\( T^{3} - 4 T^{2} - 80 T + 400 \)
$47$
\( T^{3} - 8 T^{2} - 32 T + 128 \)
$53$
\( T^{3} - 6 T^{2} - 52 T - 8 \)
$59$
\( T^{3} + 12 T^{2} - 16 T - 320 \)
$61$
\( T^{3} + 6 T^{2} - 52 T - 248 \)
$67$
\( T^{3} + 4 T^{2} - 48 T - 64 \)
$71$
\( T^{3} + 8 T^{2} - 32 T - 128 \)
$73$
\( T^{3} + 14 T^{2} + 4 T - 344 \)
$79$
\( T^{3} - 12 T^{2} - 64 T + 800 \)
$83$
\( T^{3} - 120T - 16 \)
$89$
\( T^{3} - 10 T^{2} - 52 T + 200 \)
$97$
\( T^{3} - 22 T^{2} + 108 T - 8 \)
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