Newspace parameters
Level: | \( N \) | \(=\) | \( 57 = 3 \cdot 19 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 57.e (of order \(3\), degree \(2\), minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(0.455147291521\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\sqrt{-3}) \) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: |
\( x^{2} - x + 1 \)
|
Coefficient ring: | \(\Z[a_1, a_2]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{3}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/57\mathbb{Z}\right)^\times\).
\(n\) | \(20\) | \(40\) |
\(\chi(n)\) | \(1\) | \(-\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 |
|
−0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0 | 0.500000 | − | 0.866025i | 1.00000 | −3.00000 | −0.500000 | + | 0.866025i | 0 | ||||||||||||||
49.1 | −0.500000 | + | 0.866025i | 0.500000 | − | 0.866025i | 0.500000 | + | 0.866025i | 0 | 0.500000 | + | 0.866025i | 1.00000 | −3.00000 | −0.500000 | − | 0.866025i | 0 | |||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
19.c | even | 3 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 57.2.e.a | ✓ | 2 |
3.b | odd | 2 | 1 | 171.2.f.a | 2 | ||
4.b | odd | 2 | 1 | 912.2.q.a | 2 | ||
12.b | even | 2 | 1 | 2736.2.s.j | 2 | ||
19.c | even | 3 | 1 | inner | 57.2.e.a | ✓ | 2 |
19.c | even | 3 | 1 | 1083.2.a.c | 1 | ||
19.d | odd | 6 | 1 | 1083.2.a.b | 1 | ||
57.f | even | 6 | 1 | 3249.2.a.f | 1 | ||
57.h | odd | 6 | 1 | 171.2.f.a | 2 | ||
57.h | odd | 6 | 1 | 3249.2.a.c | 1 | ||
76.g | odd | 6 | 1 | 912.2.q.a | 2 | ||
228.m | even | 6 | 1 | 2736.2.s.j | 2 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
57.2.e.a | ✓ | 2 | 1.a | even | 1 | 1 | trivial |
57.2.e.a | ✓ | 2 | 19.c | even | 3 | 1 | inner |
171.2.f.a | 2 | 3.b | odd | 2 | 1 | ||
171.2.f.a | 2 | 57.h | odd | 6 | 1 | ||
912.2.q.a | 2 | 4.b | odd | 2 | 1 | ||
912.2.q.a | 2 | 76.g | odd | 6 | 1 | ||
1083.2.a.b | 1 | 19.d | odd | 6 | 1 | ||
1083.2.a.c | 1 | 19.c | even | 3 | 1 | ||
2736.2.s.j | 2 | 12.b | even | 2 | 1 | ||
2736.2.s.j | 2 | 228.m | even | 6 | 1 | ||
3249.2.a.c | 1 | 57.h | odd | 6 | 1 | ||
3249.2.a.f | 1 | 57.f | even | 6 | 1 |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{2} + T_{2} + 1 \)
acting on \(S_{2}^{\mathrm{new}}(57, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} + T + 1 \)
$3$
\( T^{2} - T + 1 \)
$5$
\( T^{2} \)
$7$
\( (T - 1)^{2} \)
$11$
\( (T + 2)^{2} \)
$13$
\( T^{2} + 5T + 25 \)
$17$
\( T^{2} - 4T + 16 \)
$19$
\( T^{2} + 8T + 19 \)
$23$
\( T^{2} - 4T + 16 \)
$29$
\( T^{2} - 8T + 64 \)
$31$
\( (T + 3)^{2} \)
$37$
\( (T - 3)^{2} \)
$41$
\( T^{2} - 12T + 144 \)
$43$
\( T^{2} - T + 1 \)
$47$
\( T^{2} - 6T + 36 \)
$53$
\( T^{2} + 4T + 16 \)
$59$
\( T^{2} + 10T + 100 \)
$61$
\( T^{2} - 13T + 169 \)
$67$
\( T^{2} + 11T + 121 \)
$71$
\( T^{2} + 6T + 36 \)
$73$
\( T^{2} - 11T + 121 \)
$79$
\( T^{2} + T + 1 \)
$83$
\( T^{2} \)
$89$
\( T^{2} - 6T + 36 \)
$97$
\( T^{2} + 2T + 4 \)
show more
show less