Newspace parameters
Level: | \( N \) | \(=\) | \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 468.bj (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(3.73699881460\) |
Analytic rank: | \(0\) |
Dimension: | \(6\) |
Relative dimension: | \(3\) over \(\Q(\zeta_{6})\) |
Coefficient field: | 6.0.954288.1 |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 3 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) 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 \( x^{6} - x^{5} - 2x^{4} + 3x^{3} - 6x^{2} - 9x + 27 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( -2\nu^{5} - \nu^{4} - 2\nu^{3} + 12\nu + 9 ) / 27 \) |
\(\beta_{3}\) | \(=\) | \( ( 2\nu^{5} + \nu^{4} + 2\nu^{3} + 27\nu^{2} - 12\nu - 36 ) / 27 \) |
\(\beta_{4}\) | \(=\) | \( ( -\nu^{5} - 2\nu^{4} + 2\nu^{3} + 6\nu + 18 ) / 9 \) |
\(\beta_{5}\) | \(=\) | \( ( -7\nu^{5} + \nu^{4} + 11\nu^{3} + 15\nu + 72 ) / 27 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + \beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( \beta_{5} + \beta_{4} - 5\beta_{2} + \beta _1 - 3 \) |
\(\nu^{4}\) | \(=\) | \( 2\beta_{5} - 4\beta_{4} - \beta_{2} + 2\beta _1 + 3 \) |
\(\nu^{5}\) | \(=\) | \( -2\beta_{5} + \beta_{4} - 8\beta_{2} + 4\beta _1 + 6 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).
\(n\) | \(145\) | \(209\) | \(235\) |
\(\chi(n)\) | \(-\beta_{2}\) | \(-1 - \beta_{2}\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
121.1 |
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0 | −1.66044 | + | 0.492881i | 0 | 1.50000 | + | 0.866025i | 0 | −3.77121 | − | 2.17731i | 0 | 2.51414 | − | 1.63680i | 0 | ||||||||||||||||||||||||||||
121.2 | 0 | −0.675970 | − | 1.59470i | 0 | 1.50000 | + | 0.866025i | 0 | 3.12920 | + | 1.80664i | 0 | −2.08613 | + | 2.15594i | 0 | |||||||||||||||||||||||||||||
121.3 | 0 | 1.33641 | + | 1.10182i | 0 | 1.50000 | + | 0.866025i | 0 | −0.857990 | − | 0.495361i | 0 | 0.571993 | + | 2.94497i | 0 | |||||||||||||||||||||||||||||
205.1 | 0 | −1.66044 | − | 0.492881i | 0 | 1.50000 | − | 0.866025i | 0 | −3.77121 | + | 2.17731i | 0 | 2.51414 | + | 1.63680i | 0 | |||||||||||||||||||||||||||||
205.2 | 0 | −0.675970 | + | 1.59470i | 0 | 1.50000 | − | 0.866025i | 0 | 3.12920 | − | 1.80664i | 0 | −2.08613 | − | 2.15594i | 0 | |||||||||||||||||||||||||||||
205.3 | 0 | 1.33641 | − | 1.10182i | 0 | 1.50000 | − | 0.866025i | 0 | −0.857990 | + | 0.495361i | 0 | 0.571993 | − | 2.94497i | 0 | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
117.l | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 468.2.bj.b | yes | 6 |
3.b | odd | 2 | 1 | 1404.2.bj.b | 6 | ||
9.c | even | 3 | 1 | 468.2.be.b | ✓ | 6 | |
9.d | odd | 6 | 1 | 1404.2.be.b | 6 | ||
13.e | even | 6 | 1 | 468.2.be.b | ✓ | 6 | |
39.h | odd | 6 | 1 | 1404.2.be.b | 6 | ||
117.l | even | 6 | 1 | inner | 468.2.bj.b | yes | 6 |
117.v | odd | 6 | 1 | 1404.2.bj.b | 6 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
468.2.be.b | ✓ | 6 | 9.c | even | 3 | 1 | |
468.2.be.b | ✓ | 6 | 13.e | even | 6 | 1 | |
468.2.bj.b | yes | 6 | 1.a | even | 1 | 1 | trivial |
468.2.bj.b | yes | 6 | 117.l | even | 6 | 1 | inner |
1404.2.be.b | 6 | 9.d | odd | 6 | 1 | ||
1404.2.be.b | 6 | 39.h | odd | 6 | 1 | ||
1404.2.bj.b | 6 | 3.b | odd | 2 | 1 | ||
1404.2.bj.b | 6 | 117.v | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 3T_{5} + 3 \)
acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{6} \)
$3$
\( T^{6} + 2 T^{5} + T^{4} + 3 T^{2} + \cdots + 27 \)
$5$
\( (T^{2} - 3 T + 3)^{3} \)
$7$
\( T^{6} + 3 T^{5} - 12 T^{4} - 45 T^{3} + \cdots + 243 \)
$11$
\( (T^{2} + 12)^{3} \)
$13$
\( (T^{2} + 2 T + 13)^{3} \)
$17$
\( T^{6} + 9 T^{5} + 78 T^{4} + \cdots + 6561 \)
$19$
\( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 49923 \)
$23$
\( T^{6} + 3 T^{5} + 42 T^{4} + \cdots + 13689 \)
$29$
\( (T^{3} + 6 T^{2} - 72 T - 324)^{2} \)
$31$
\( T^{6} - 3 T^{5} - 30 T^{4} + \cdots + 2187 \)
$37$
\( T^{6} + 3 T^{5} - 24 T^{4} - 81 T^{3} + \cdots + 243 \)
$41$
\( (T^{2} - 3 T + 3)^{3} \)
$43$
\( T^{6} + 3 T^{5} + 108 T^{4} + \cdots + 1681 \)
$47$
\( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 49923 \)
$53$
\( (T^{3} + 6 T^{2} - 132 T - 936)^{2} \)
$59$
\( (T^{2} + 12)^{3} \)
$61$
\( T^{6} + 9 T^{5} + 138 T^{4} + \cdots + 151321 \)
$67$
\( T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 9747 \)
$71$
\( T^{6} - 9 T^{5} - 18 T^{4} + \cdots + 2187 \)
$73$
\( T^{6} + 288 T^{4} + 25920 T^{2} + \cdots + 726192 \)
$79$
\( T^{6} - 9 T^{5} + 102 T^{4} + \cdots + 22801 \)
$83$
\( T^{6} + 9 T^{5} - 108 T^{4} + \cdots + 4563 \)
$89$
\( T^{6} + 27 T^{5} + 108 T^{4} + \cdots + 4255443 \)
$97$
\( T^{6} + 27 T^{5} + 36 T^{4} + \cdots + 8137827 \)
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