Newspace parameters
Level: | \( N \) | \(=\) | \( 528 = 2^{4} \cdot 3 \cdot 11 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 528.i (of order \(2\), degree \(1\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(14.3869579582\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
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: | \( 2 \) |
Twist minimal: | no (minimal twist has level 33) |
Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-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 \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) :
\(\beta_{1}\) | \(=\) | \( \nu \) |
\(\beta_{2}\) | \(=\) | \( ( \nu^{3} + 2\nu^{2} - 2\nu - 6 ) / 3 \) |
\(\beta_{3}\) | \(=\) | \( ( \nu^{3} - \nu^{2} - 2\nu - 3 ) / 3 \) |
\(\nu\) | \(=\) | \( \beta_1 \) |
\(\nu^{2}\) | \(=\) | \( -\beta_{3} + \beta_{2} + 1 \) |
\(\nu^{3}\) | \(=\) | \( 2\beta_{3} + \beta_{2} + 2\beta _1 + 4 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/528\mathbb{Z}\right)^\times\).
\(n\) | \(133\) | \(145\) | \(353\) | \(463\) |
\(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
353.1 |
|
0 | −0.186141 | − | 2.99422i | 0 | − | 2.52434i | 0 | 4.74456 | 0 | −8.93070 | + | 1.11469i | 0 | |||||||||||||||||||||||||
353.2 | 0 | −0.186141 | + | 2.99422i | 0 | 2.52434i | 0 | 4.74456 | 0 | −8.93070 | − | 1.11469i | 0 | |||||||||||||||||||||||||||
353.3 | 0 | 2.68614 | − | 1.33591i | 0 | 0.792287i | 0 | −6.74456 | 0 | 5.43070 | − | 7.17687i | 0 | |||||||||||||||||||||||||||
353.4 | 0 | 2.68614 | + | 1.33591i | 0 | − | 0.792287i | 0 | −6.74456 | 0 | 5.43070 | + | 7.17687i | 0 | ||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 528.3.i.d | 4 | |
3.b | odd | 2 | 1 | inner | 528.3.i.d | 4 | |
4.b | odd | 2 | 1 | 33.3.b.b | ✓ | 4 | |
12.b | even | 2 | 1 | 33.3.b.b | ✓ | 4 | |
44.c | even | 2 | 1 | 363.3.b.h | 4 | ||
44.g | even | 10 | 4 | 363.3.h.l | 16 | ||
44.h | odd | 10 | 4 | 363.3.h.m | 16 | ||
132.d | odd | 2 | 1 | 363.3.b.h | 4 | ||
132.n | odd | 10 | 4 | 363.3.h.l | 16 | ||
132.o | even | 10 | 4 | 363.3.h.m | 16 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
33.3.b.b | ✓ | 4 | 4.b | odd | 2 | 1 | |
33.3.b.b | ✓ | 4 | 12.b | even | 2 | 1 | |
363.3.b.h | 4 | 44.c | even | 2 | 1 | ||
363.3.b.h | 4 | 132.d | odd | 2 | 1 | ||
363.3.h.l | 16 | 44.g | even | 10 | 4 | ||
363.3.h.l | 16 | 132.n | odd | 10 | 4 | ||
363.3.h.m | 16 | 44.h | odd | 10 | 4 | ||
363.3.h.m | 16 | 132.o | even | 10 | 4 | ||
528.3.i.d | 4 | 1.a | even | 1 | 1 | trivial | |
528.3.i.d | 4 | 3.b | odd | 2 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{4} + 7T_{5}^{2} + 4 \)
acting on \(S_{3}^{\mathrm{new}}(528, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} \)
$3$
\( T^{4} - 5 T^{3} + 16 T^{2} - 45 T + 81 \)
$5$
\( T^{4} + 7T^{2} + 4 \)
$7$
\( (T^{2} + 2 T - 32)^{2} \)
$11$
\( (T^{2} + 11)^{2} \)
$13$
\( (T^{2} + 4 T - 128)^{2} \)
$17$
\( T^{4} + 1372 T^{2} + 440896 \)
$19$
\( (T^{2} - 18 T - 216)^{2} \)
$23$
\( T^{4} + 1639 T^{2} + 662596 \)
$29$
\( T^{4} + 3552 T^{2} + \cdots + 1937664 \)
$31$
\( (T^{2} + 23 T - 74)^{2} \)
$37$
\( (T^{2} + 45 T + 102)^{2} \)
$41$
\( T^{4} + 1264 T^{2} + 295936 \)
$43$
\( (T^{2} + 48 T + 444)^{2} \)
$47$
\( T^{4} + 2716 T^{2} + \cdots + 1700416 \)
$53$
\( T^{4} + 8124 T^{2} + 788544 \)
$59$
\( T^{4} + 4003 T^{2} + 824464 \)
$61$
\( (T^{2} + 12 T - 3264)^{2} \)
$67$
\( (T^{2} - 29 T - 2174)^{2} \)
$71$
\( T^{4} + 231T^{2} + 4356 \)
$73$
\( (T^{2} + 142 T + 4744)^{2} \)
$79$
\( (T^{2} - 38 T - 5216)^{2} \)
$83$
\( T^{4} + 5436 T^{2} + \cdots + 4981824 \)
$89$
\( T^{4} + 20787 T^{2} + \cdots + 4112784 \)
$97$
\( (T^{2} - 109 T - 6014)^{2} \)
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