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: | \(2\) |
| Coefficient field: | \(\Q(\sqrt{17}) \) |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{2} - x - 4 \)
|
| 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 \(\beta = \frac{1}{2}(1 + \sqrt{17})\). We also show the integral \(q\)-expansion of the trace form.
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 |
|
−2.56155 | 3.00000 | −1.43845 | −5.00000 | −7.68466 | 6.24621 | 24.1771 | 9.00000 | 12.8078 | ||||||||||||||||||||||||
| 1.2 | 1.56155 | 3.00000 | −5.56155 | −5.00000 | 4.68466 | −10.2462 | −21.1771 | 9.00000 | −7.80776 | |||||||||||||||||||||||||
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.c | ✓ | 2 |
| 3.b | odd | 2 | 1 | 495.4.a.d | 2 | ||
| 5.b | even | 2 | 1 | 825.4.a.m | 2 | ||
| 5.c | odd | 4 | 2 | 825.4.c.j | 4 | ||
| 11.b | odd | 2 | 1 | 1815.4.a.n | 2 | ||
| 15.d | odd | 2 | 1 | 2475.4.a.n | 2 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 165.4.a.c | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
| 495.4.a.d | 2 | 3.b | odd | 2 | 1 | ||
| 825.4.a.m | 2 | 5.b | even | 2 | 1 | ||
| 825.4.c.j | 4 | 5.c | odd | 4 | 2 | ||
| 1815.4.a.n | 2 | 11.b | odd | 2 | 1 | ||
| 2475.4.a.n | 2 | 15.d | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} - 4 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(165))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T - 4 \)
$3$
\( (T - 3)^{2} \)
$5$
\( (T + 5)^{2} \)
$7$
\( T^{2} + 4T - 64 \)
$11$
\( (T + 11)^{2} \)
$13$
\( T^{2} + 90T + 2008 \)
$17$
\( T^{2} + 16T - 8164 \)
$19$
\( T^{2} + 170T + 5168 \)
$23$
\( T^{2} + 124T - 11456 \)
$29$
\( T^{2} + 158T + 1328 \)
$31$
\( T^{2} - 60T + 288 \)
$37$
\( T^{2} + 372T - 18716 \)
$41$
\( T^{2} - 38T - 100432 \)
$43$
\( T^{2} + 516T + 1216 \)
$47$
\( T^{2} - 224T - 185744 \)
$53$
\( T^{2} - 472T - 107572 \)
$59$
\( T^{2} - 248T - 229424 \)
$61$
\( T^{2} - 72T - 561812 \)
$67$
\( T^{2} + 744T + 137296 \)
$71$
\( T^{2} - 2060 T + 1052672 \)
$73$
\( T^{2} + 486T + 44752 \)
$79$
\( T^{2} - 642T - 294912 \)
$83$
\( T^{2} + 286T - 750824 \)
$89$
\( T^{2} - 244T - 54748 \)
$97$
\( T^{2} + 168T - 771476 \)
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