Newspace parameters
Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 3 \) |
Character orbit: | \([\chi]\) | \(=\) | 57.c (of order \(2\), degree \(1\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.55313750685\) |
Analytic rank: | \(0\) |
Dimension: | \(4\) |
Coefficient field: | \(\Q(\sqrt{-3}, \sqrt{-19})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
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Defining polynomial: | \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) |
Coefficient ring: | \(\Z[a_1, a_2, a_3]\) |
Coefficient ring index: | \( 2^{2} \) |
Twist minimal: | yes |
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 \( x^{4} - x^{3} - 4x^{2} - 5x + 25 \) :
\(\beta_{1}\) | \(=\) | \( ( \nu^{3} + 4\nu^{2} - 4\nu - 15 ) / 10 \) |
\(\beta_{2}\) | \(=\) | \( ( -\nu^{3} + \nu^{2} + 9\nu + 5 ) / 5 \) |
\(\beta_{3}\) | \(=\) | \( ( -3\nu^{3} - 2\nu^{2} + 2\nu + 25 ) / 10 \) |
\(\nu\) | \(=\) | \( ( -\beta_{3} + \beta_{2} - \beta_1 ) / 2 \) |
\(\nu^{2}\) | \(=\) | \( ( \beta_{3} + \beta_{2} + 5\beta _1 + 4 ) / 2 \) |
\(\nu^{3}\) | \(=\) | \( -4\beta_{3} - 2\beta _1 + 7 \) |
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
\(n\) | \(20\) | \(40\) |
\(\chi(n)\) | \(1\) | \(-1\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
37.1 |
|
− | 3.04547i | − | 1.73205i | −5.27492 | −6.27492 | −5.27492 | 12.2749 | 3.88273i | −3.00000 | 19.1101i | ||||||||||||||||||||||||||||
37.2 | − | 1.31342i | 1.73205i | 2.27492 | 1.27492 | 2.27492 | 4.72508 | − | 8.24163i | −3.00000 | − | 1.67451i | ||||||||||||||||||||||||||||
37.3 | 1.31342i | − | 1.73205i | 2.27492 | 1.27492 | 2.27492 | 4.72508 | 8.24163i | −3.00000 | 1.67451i | ||||||||||||||||||||||||||||||
37.4 | 3.04547i | 1.73205i | −5.27492 | −6.27492 | −5.27492 | 12.2749 | − | 3.88273i | −3.00000 | − | 19.1101i | |||||||||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.b | odd | 2 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 57.3.c.b | ✓ | 4 |
3.b | odd | 2 | 1 | 171.3.c.f | 4 | ||
4.b | odd | 2 | 1 | 912.3.o.b | 4 | ||
12.b | even | 2 | 1 | 2736.3.o.l | 4 | ||
19.b | odd | 2 | 1 | inner | 57.3.c.b | ✓ | 4 |
57.d | even | 2 | 1 | 171.3.c.f | 4 | ||
76.d | even | 2 | 1 | 912.3.o.b | 4 | ||
228.b | odd | 2 | 1 | 2736.3.o.l | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.3.c.b | ✓ | 4 | 1.a | even | 1 | 1 | trivial |
57.3.c.b | ✓ | 4 | 19.b | odd | 2 | 1 | inner |
171.3.c.f | 4 | 3.b | odd | 2 | 1 | ||
171.3.c.f | 4 | 57.d | even | 2 | 1 | ||
912.3.o.b | 4 | 4.b | odd | 2 | 1 | ||
912.3.o.b | 4 | 76.d | even | 2 | 1 | ||
2736.3.o.l | 4 | 12.b | even | 2 | 1 | ||
2736.3.o.l | 4 | 228.b | odd | 2 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{4} + 11T_{2}^{2} + 16 \)
acting on \(S_{3}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{4} + 11T^{2} + 16 \)
$3$
\( (T^{2} + 3)^{2} \)
$5$
\( (T^{2} + 5 T - 8)^{2} \)
$7$
\( (T^{2} - 17 T + 58)^{2} \)
$11$
\( (T^{2} + 7 T - 2)^{2} \)
$13$
\( T^{4} + 188T^{2} + 3136 \)
$17$
\( (T^{2} + 3 T - 354)^{2} \)
$19$
\( T^{4} + 494 T^{2} + 130321 \)
$23$
\( (T^{2} - 26 T + 112)^{2} \)
$29$
\( T^{4} + 752 T^{2} + 50176 \)
$31$
\( T^{4} + 956 T^{2} + 222784 \)
$37$
\( T^{4} + 6108 T^{2} + \cdots + 7354944 \)
$41$
\( T^{4} + 992 T^{2} + 12544 \)
$43$
\( (T^{2} - 17 T - 3134)^{2} \)
$47$
\( (T^{2} - 43 T - 692)^{2} \)
$53$
\( T^{4} + 6768 T^{2} + \cdots + 4064256 \)
$59$
\( T^{4} + 4832T^{2} + 256 \)
$61$
\( (T^{2} + 35 T + 178)^{2} \)
$67$
\( T^{4} + 25904 T^{2} + \cdots + 162205696 \)
$71$
\( T^{4} + 11616 T^{2} + \cdots + 112896 \)
$73$
\( (T^{2} - 45 T + 378)^{2} \)
$79$
\( T^{4} + 8396 T^{2} + \cdots + 5456896 \)
$83$
\( (T^{2} - 32 T - 10916)^{2} \)
$89$
\( T^{4} + 24288 T^{2} + \cdots + 94945536 \)
$97$
\( T^{4} + 14048 T^{2} + \cdots + 21086464 \)
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