Newspace parameters
| Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
| Weight: | \( k \) | \(=\) | \( 4 \) |
| Character orbit: | \([\chi]\) | \(=\) | 165.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(9.73531515095\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.788.1 |
| Defining polynomial: |
\( x^{3} - x^{2} - 7x - 3 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 2 \) |
| 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
\( x^{3} - x^{2} - 7x - 3 \)
:
| \(\beta_{1}\) | \(=\) |
\( 2\nu - 1 \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - 2\nu - 4 \)
|
| \(\nu\) | \(=\) |
\( ( \beta _1 + 1 ) / 2 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 5 \)
|
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 |
|
−2.82009 | −3.00000 | −0.0470959 | 5.00000 | 8.46027 | −7.12434 | 22.6935 | 9.00000 | −14.1004 | |||||||||||||||||||||||||||
| 1.2 | 0.540637 | −3.00000 | −7.70771 | 5.00000 | −1.62191 | 24.4573 | −8.49217 | 9.00000 | 2.70319 | ||||||||||||||||||||||||||||
| 1.3 | 3.27945 | −3.00000 | 2.75481 | 5.00000 | −9.83836 | −33.3329 | −17.2014 | 9.00000 | 16.3973 | ||||||||||||||||||||||||||||
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 | 165.4.a.f | ✓ | 3 |
| 3.b | odd | 2 | 1 | 495.4.a.g | 3 | ||
| 5.b | even | 2 | 1 | 825.4.a.n | 3 | ||
| 5.c | odd | 4 | 2 | 825.4.c.o | 6 | ||
| 11.b | odd | 2 | 1 | 1815.4.a.p | 3 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.w | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.f | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 495.4.a.g | 3 | 3.b | odd | 2 | 1 | ||
| 825.4.a.n | 3 | 5.b | even | 2 | 1 | ||
| 825.4.c.o | 6 | 5.c | odd | 4 | 2 | ||
| 1815.4.a.p | 3 | 11.b | odd | 2 | 1 | ||
| 2475.4.a.w | 3 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} - T_{2}^{2} - 9T_{2} + 5 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} - T^{2} - 9T + 5 \)
$3$
\( (T + 3)^{3} \)
$5$
\( (T - 5)^{3} \)
$7$
\( T^{3} + 16 T^{2} - 752 T - 5808 \)
$11$
\( (T + 11)^{3} \)
$13$
\( T^{3} + 42 T^{2} - 5228 T - 137416 \)
$17$
\( T^{3} + 34 T^{2} - 4660 T - 179064 \)
$19$
\( T^{3} + 280 T^{2} + 18624 T - 97056 \)
$23$
\( T^{3} + 112 T^{2} - 17024 T - 1916288 \)
$29$
\( T^{3} + 290 T^{2} - 26500 T - 9251496 \)
$31$
\( T^{3} + 392 T^{2} + 32160 T - 316800 \)
$37$
\( T^{3} - 570 T^{2} + 60940 T + 1039624 \)
$41$
\( T^{3} + 662 T^{2} + \cdots - 68561784 \)
$43$
\( T^{3} + 68 T^{2} - 227376 T - 20491056 \)
$47$
\( T^{3} - 264 T^{2} + \cdots - 14121600 \)
$53$
\( T^{3} + 94 T^{2} - 57812 T - 2403992 \)
$59$
\( T^{3} + 612 T^{2} + \cdots - 162128320 \)
$61$
\( T^{3} + 582 T^{2} + \cdots - 21355000 \)
$67$
\( T^{3} - 940 T^{2} + \cdots + 394498240 \)
$71$
\( T^{3} + 1616 T^{2} + \cdots + 40198784 \)
$73$
\( T^{3} - 738 T^{2} + \cdots - 12046264 \)
$79$
\( T^{3} - 124 T^{2} + \cdots + 26871328 \)
$83$
\( T^{3} + 1232 T^{2} + \cdots - 15656400 \)
$89$
\( T^{3} - 838 T^{2} + \cdots + 831946232 \)
$97$
\( T^{3} + 90 T^{2} + \cdots - 924170696 \)
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