Newspace parameters
Level: | \( N \) | \(=\) | \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \) |
Weight: | \( k \) | \(=\) | \( 4 \) |
Character orbit: | \([\chi]\) | \(=\) | 588.a (trivial) |
Newform invariants
Self dual: | yes |
Analytic conductor: | \(34.6931230834\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{193}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x - 48 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 84) |
Fricke sign: | \(1\) |
Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \frac{1}{2}(1 + \sqrt{193})\). 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 |
|
0 | 3.00000 | 0 | −1.44622 | 0 | 0 | 0 | 9.00000 | 0 | ||||||||||||||||||||||||
1.2 | 0 | 3.00000 | 0 | 12.4462 | 0 | 0 | 0 | 9.00000 | 0 | |||||||||||||||||||||||||
Atkin-Lehner signs
\( p \) | Sign |
---|---|
\(2\) | \(-1\) |
\(3\) | \(-1\) |
\(7\) | \(1\) |
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 | 588.4.a.i | 2 | |
3.b | odd | 2 | 1 | 1764.4.a.o | 2 | ||
4.b | odd | 2 | 1 | 2352.4.a.bt | 2 | ||
7.b | odd | 2 | 1 | 588.4.a.f | 2 | ||
7.c | even | 3 | 2 | 84.4.i.a | ✓ | 4 | |
7.d | odd | 6 | 2 | 588.4.i.j | 4 | ||
21.c | even | 2 | 1 | 1764.4.a.y | 2 | ||
21.g | even | 6 | 2 | 1764.4.k.q | 4 | ||
21.h | odd | 6 | 2 | 252.4.k.f | 4 | ||
28.d | even | 2 | 1 | 2352.4.a.bx | 2 | ||
28.g | odd | 6 | 2 | 336.4.q.i | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
84.4.i.a | ✓ | 4 | 7.c | even | 3 | 2 | |
252.4.k.f | 4 | 21.h | odd | 6 | 2 | ||
336.4.q.i | 4 | 28.g | odd | 6 | 2 | ||
588.4.a.f | 2 | 7.b | odd | 2 | 1 | ||
588.4.a.i | 2 | 1.a | even | 1 | 1 | trivial | |
588.4.i.j | 4 | 7.d | odd | 6 | 2 | ||
1764.4.a.o | 2 | 3.b | odd | 2 | 1 | ||
1764.4.a.y | 2 | 21.c | even | 2 | 1 | ||
1764.4.k.q | 4 | 21.g | even | 6 | 2 | ||
2352.4.a.bt | 2 | 4.b | odd | 2 | 1 | ||
2352.4.a.bx | 2 | 28.d | even | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{2} - 11T_{5} - 18 \)
acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(588))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( (T - 3)^{2} \)
$5$
\( T^{2} - 11T - 18 \)
$7$
\( T^{2} \)
$11$
\( T^{2} + 5T - 2358 \)
$13$
\( T^{2} - 5T - 1200 \)
$17$
\( T^{2} - 100T + 1728 \)
$19$
\( T^{2} - 67T + 688 \)
$23$
\( T^{2} - 76T - 17856 \)
$29$
\( T^{2} - 275T + 13068 \)
$31$
\( T^{2} - 362T + 13461 \)
$37$
\( T^{2} + 5T - 97700 \)
$41$
\( T^{2} + 162T - 9072 \)
$43$
\( T^{2} - 721T + 129526 \)
$47$
\( T^{2} + 216T - 50868 \)
$53$
\( T^{2} - 495T + 57348 \)
$59$
\( T^{2} - 173T - 128052 \)
$61$
\( T^{2} + 532T - 6444 \)
$67$
\( T^{2} - 111T - 282994 \)
$71$
\( T^{2} - 1600 T + 546588 \)
$73$
\( T^{2} - 1215 T + 233522 \)
$79$
\( T^{2} - 1460 T + 347427 \)
$83$
\( T^{2} + 1409T + 4122 \)
$89$
\( T^{2} + 1974 T + 889056 \)
$97$
\( T^{2} - 561T + 38102 \)
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