Newspace parameters
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 6 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(26.4633302691\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.34253.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 52x + 48 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2^{2} \) |
| 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 \(1,\beta_1,\beta_2\) 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
\( x^{3} - x^{2} - 52x + 48 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 35 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + 35 \)
|
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 |
|
−9.25531 | 9.00000 | 53.6607 | 25.0000 | −83.2977 | 36.4478 | −200.476 | 81.0000 | −231.383 | |||||||||||||||||||||||||||
| 1.2 | −2.92180 | 9.00000 | −23.4631 | 25.0000 | −26.2962 | −85.0105 | 162.052 | 81.0000 | −73.0450 | ||||||||||||||||||||||||||||
| 1.3 | 5.17710 | 9.00000 | −5.19759 | 25.0000 | 46.5939 | −123.437 | −192.576 | 81.0000 | 129.428 | ||||||||||||||||||||||||||||
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 | 165.6.a.a | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.6.a.e | 3 | ||
| 5.b | even | 2 | 1 | 825.6.a.j | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.6.a.a | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 495.6.a.e | 3 | 3.b | odd | 2 | 1 | ||
| 825.6.a.j | 3 | 5.b | even | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 7T_{2}^{2} - 36T_{2} - 140 \)
acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{3} + 7 T^{2} + \cdots - 140 \)
|
| $3$ |
\( (T - 9)^{3} \)
|
| $5$ |
\( (T - 25)^{3} \)
|
| $7$ |
\( T^{3} + 172 T^{2} + \cdots - 382464 \)
|
| $11$ |
\( (T + 121)^{3} \)
|
| $13$ |
\( T^{3} + 654 T^{2} + \cdots - 317918392 \)
|
| $17$ |
\( T^{3} + 2366 T^{2} + \cdots - 137826264 \)
|
| $19$ |
\( T^{3} + 2872 T^{2} + \cdots + 11543616 \)
|
| $23$ |
\( T^{3} + \cdots + 3706904576 \)
|
| $29$ |
\( T^{3} + \cdots + 5220625848 \)
|
| $31$ |
\( T^{3} - 568 T^{2} + \cdots - 289787904 \)
|
| $37$ |
\( T^{3} + \cdots - 129972509048 \)
|
| $41$ |
\( T^{3} + \cdots - 1122652557432 \)
|
| $43$ |
\( T^{3} + \cdots - 518908872384 \)
|
| $47$ |
\( T^{3} + \cdots - 1508908531200 \)
|
| $53$ |
\( T^{3} + \cdots + 728896505288 \)
|
| $59$ |
\( T^{3} + \cdots - 6267836310080 \)
|
| $61$ |
\( T^{3} + \cdots - 13500896397400 \)
|
| $67$ |
\( T^{3} + \cdots - 8479952260160 \)
|
| $71$ |
\( T^{3} + \cdots - 17011990639616 \)
|
| $73$ |
\( T^{3} + \cdots - 9750515676328 \)
|
| $79$ |
\( T^{3} + \cdots + 9069370346752 \)
|
| $83$ |
\( T^{3} + \cdots - 739830059345664 \)
|
| $89$ |
\( T^{3} + \cdots - 158914472576552 \)
|
| $97$ |
\( T^{3} + \cdots + 352998320493112 \)
|
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