Newspace parameters
Level: | \( N \) | \(=\) | \( 99 = 3^{2} \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 99.g (of order \(6\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.790518980011\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Relative dimension: | \(2\) over \(\Q(\zeta_{6})\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-11})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{11}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{U}(1)[D_{6}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) 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 \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 2\nu^{2} - 2\nu - 3 ) / 6 \) |
\(\beta_{3}\) | \(=\) | \( ( -\nu^{3} + 2\nu + 3 ) / 2 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( \beta_{3} + 3\beta_{2} \) |
\(\nu^{3}\) | \(=\) | \( -2\beta_{3} + 2\beta _1 + 3 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).
\(n\) | \(46\) | \(56\) |
\(\chi(n)\) | \(-1\) | \(1 - \beta_{2}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
32.1 |
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0 | −1.68614 | − | 0.396143i | 1.00000 | − | 1.73205i | −3.68614 | − | 2.12819i | 0 | 0 | 0 | 2.68614 | + | 1.33591i | 0 | ||||||||||||||||||||||
32.2 | 0 | 1.18614 | + | 1.26217i | 1.00000 | − | 1.73205i | −0.813859 | − | 0.469882i | 0 | 0 | 0 | −0.186141 | + | 2.99422i | 0 | |||||||||||||||||||||||
65.1 | 0 | −1.68614 | + | 0.396143i | 1.00000 | + | 1.73205i | −3.68614 | + | 2.12819i | 0 | 0 | 0 | 2.68614 | − | 1.33591i | 0 | |||||||||||||||||||||||
65.2 | 0 | 1.18614 | − | 1.26217i | 1.00000 | + | 1.73205i | −0.813859 | + | 0.469882i | 0 | 0 | 0 | −0.186141 | − | 2.99422i | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
11.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-11}) \) |
9.d | odd | 6 | 1 | inner |
99.g | even | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 99.2.g.a | ✓ | 4 |
3.b | odd | 2 | 1 | 297.2.g.a | 4 | ||
9.c | even | 3 | 1 | 297.2.g.a | 4 | ||
9.c | even | 3 | 1 | 891.2.d.a | 4 | ||
9.d | odd | 6 | 1 | inner | 99.2.g.a | ✓ | 4 |
9.d | odd | 6 | 1 | 891.2.d.a | 4 | ||
11.b | odd | 2 | 1 | CM | 99.2.g.a | ✓ | 4 |
33.d | even | 2 | 1 | 297.2.g.a | 4 | ||
99.g | even | 6 | 1 | inner | 99.2.g.a | ✓ | 4 |
99.g | even | 6 | 1 | 891.2.d.a | 4 | ||
99.h | odd | 6 | 1 | 297.2.g.a | 4 | ||
99.h | odd | 6 | 1 | 891.2.d.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
99.2.g.a | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
99.2.g.a | ✓ | 4 | 9.d | odd | 6 | 1 | inner |
99.2.g.a | ✓ | 4 | 11.b | odd | 2 | 1 | CM |
99.2.g.a | ✓ | 4 | 99.g | even | 6 | 1 | inner |
297.2.g.a | 4 | 3.b | odd | 2 | 1 | ||
297.2.g.a | 4 | 9.c | even | 3 | 1 | ||
297.2.g.a | 4 | 33.d | even | 2 | 1 | ||
297.2.g.a | 4 | 99.h | odd | 6 | 1 | ||
891.2.d.a | 4 | 9.c | even | 3 | 1 | ||
891.2.d.a | 4 | 9.d | odd | 6 | 1 | ||
891.2.d.a | 4 | 99.g | even | 6 | 1 | ||
891.2.d.a | 4 | 99.h | odd | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} + T^{3} - 2 T^{2} + 3 T + 9 \)
$5$
\( T^{4} + 9 T^{3} + 31 T^{2} + 36 T + 16 \)
$7$
\( T^{4} \)
$11$
\( T^{4} - 11T^{2} + 121 \)
$13$
\( T^{4} \)
$17$
\( T^{4} \)
$19$
\( T^{4} \)
$23$
\( T^{4} - 11T^{2} + 121 \)
$29$
\( T^{4} \)
$31$
\( T^{4} - 5 T^{3} + 93 T^{2} + \cdots + 4624 \)
$37$
\( (T^{2} - 7 T - 62)^{2} \)
$41$
\( T^{4} \)
$43$
\( T^{4} \)
$47$
\( T^{4} - 36 T^{3} + 529 T^{2} + \cdots + 9409 \)
$53$
\( T^{4} + 142T^{2} + 289 \)
$59$
\( T^{4} - 45 T^{3} + 841 T^{2} + \cdots + 27556 \)
$61$
\( T^{4} \)
$67$
\( T^{4} + 13 T^{3} + 201 T^{2} + \cdots + 1024 \)
$71$
\( T^{4} + 151T^{2} + 3844 \)
$73$
\( T^{4} \)
$79$
\( T^{4} \)
$83$
\( T^{4} \)
$89$
\( (T^{2} + 275)^{2} \)
$97$
\( T^{4} - 17 T^{3} + 291 T^{2} + 34 T + 4 \)
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