Newspace parameters
Level: | \( N \) | \(=\) | \( 165 = 3 \cdot 5 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 165.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(4.49592436194\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-11}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{2} - x + 3 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/165\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) | \(67\) |
\(\chi(n)\) | \(1\) | \(-1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
89.1 |
|
−1.00000 | −2.50000 | − | 1.65831i | −3.00000 | −5.00000 | 2.50000 | + | 1.65831i | − | 9.94987i | 7.00000 | 3.50000 | + | 8.29156i | 5.00000 | |||||||||||||||||
89.2 | −1.00000 | −2.50000 | + | 1.65831i | −3.00000 | −5.00000 | 2.50000 | − | 1.65831i | 9.94987i | 7.00000 | 3.50000 | − | 8.29156i | 5.00000 | |||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
15.d | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 165.3.g.a | ✓ | 2 |
3.b | odd | 2 | 1 | 165.3.g.b | yes | 2 | |
5.b | even | 2 | 1 | 165.3.g.b | yes | 2 | |
15.d | odd | 2 | 1 | inner | 165.3.g.a | ✓ | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
165.3.g.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
165.3.g.a | ✓ | 2 | 15.d | odd | 2 | 1 | inner |
165.3.g.b | yes | 2 | 3.b | odd | 2 | 1 | |
165.3.g.b | yes | 2 | 5.b | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} + 1 \)
acting on \(S_{3}^{\mathrm{new}}(165, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( (T + 1)^{2} \)
$3$
\( T^{2} + 5T + 9 \)
$5$
\( (T + 5)^{2} \)
$7$
\( T^{2} + 99 \)
$11$
\( T^{2} + 11 \)
$13$
\( T^{2} + 396 \)
$17$
\( (T + 19)^{2} \)
$19$
\( (T - 11)^{2} \)
$23$
\( (T - 32)^{2} \)
$29$
\( T^{2} + 99 \)
$31$
\( (T + 31)^{2} \)
$37$
\( T^{2} + 2475 \)
$41$
\( T^{2} + 396 \)
$43$
\( T^{2} + 3564 \)
$47$
\( (T - 14)^{2} \)
$53$
\( (T + 13)^{2} \)
$59$
\( T^{2} \)
$61$
\( (T - 47)^{2} \)
$67$
\( T^{2} + 1584 \)
$71$
\( T^{2} + 8019 \)
$73$
\( T^{2} + 6336 \)
$79$
\( (T + 100)^{2} \)
$83$
\( (T + 22)^{2} \)
$89$
\( T^{2} + 28611 \)
$97$
\( T^{2} + 25344 \)
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