Newspace parameters
Level: | \( N \) | \(=\) | \( 2997 = 3^{4} \cdot 37 \) |
Weight: | \( k \) | \(=\) | \( 1 \) |
Character orbit: | \([\chi]\) | \(=\) | 2997.u (of order \(6\), degree \(2\), not minimal) |
Newform invariants
Self dual: | no |
Analytic conductor: | \(1.49569784286\) |
Analytic rank: | \(0\) |
Dimension: | \(2\) |
Coefficient field: | \(\Q(\zeta_{6})\) |
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
Defining polynomial: | \( x^{2} - x + 1 \) |
Coefficient ring: | \(\Z[a_1, \ldots, a_{4}]\) |
Coefficient ring index: | \( 1 \) |
Twist minimal: | no (minimal twist has level 111) |
Projective image: | \(D_{3}\) |
Projective field: | Galois closure of 3.1.4107.1 |
Artin image: | $C_3\times S_3$ |
Artin field: | Galois closure of 6.0.26946027.2 |
$q$-expansion
The \(q\)-expansion and trace form are shown below.
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).
\(n\) | \(1297\) | \(1703\) |
\(\chi(n)\) | \(-\zeta_{6}\) | \(\zeta_{6}\) |
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 | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
269.1 |
|
0 | 0 | −0.500000 | + | 0.866025i | 0 | 0 | −1.00000 | 0 | 0 | 0 | ||||||||||||||||||||||
1025.1 | 0 | 0 | −0.500000 | − | 0.866025i | 0 | 0 | −1.00000 | 0 | 0 | 0 | |||||||||||||||||||||||
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
3.b | odd | 2 | 1 | CM by \(\Q(\sqrt{-3}) \) |
333.g | even | 3 | 1 | inner |
333.u | odd | 6 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 2997.1.u.a | 2 | |
3.b | odd | 2 | 1 | CM | 2997.1.u.a | 2 | |
9.c | even | 3 | 1 | 111.1.i.a | ✓ | 2 | |
9.c | even | 3 | 1 | 2997.1.l.a | 2 | ||
9.d | odd | 6 | 1 | 111.1.i.a | ✓ | 2 | |
9.d | odd | 6 | 1 | 2997.1.l.a | 2 | ||
36.f | odd | 6 | 1 | 1776.1.bw.a | 2 | ||
36.h | even | 6 | 1 | 1776.1.bw.a | 2 | ||
37.c | even | 3 | 1 | 2997.1.l.a | 2 | ||
45.h | odd | 6 | 1 | 2775.1.z.a | 2 | ||
45.j | even | 6 | 1 | 2775.1.z.a | 2 | ||
45.k | odd | 12 | 2 | 2775.1.x.a | 4 | ||
45.l | even | 12 | 2 | 2775.1.x.a | 4 | ||
111.i | odd | 6 | 1 | 2997.1.l.a | 2 | ||
333.g | even | 3 | 1 | inner | 2997.1.u.a | 2 | |
333.h | even | 3 | 1 | 111.1.i.a | ✓ | 2 | |
333.l | odd | 6 | 1 | 111.1.i.a | ✓ | 2 | |
333.u | odd | 6 | 1 | inner | 2997.1.u.a | 2 | |
1332.s | even | 6 | 1 | 1776.1.bw.a | 2 | ||
1332.br | odd | 6 | 1 | 1776.1.bw.a | 2 | ||
1665.ba | even | 6 | 1 | 2775.1.z.a | 2 | ||
1665.bz | odd | 6 | 1 | 2775.1.z.a | 2 | ||
1665.co | odd | 12 | 2 | 2775.1.x.a | 4 | ||
1665.cp | even | 12 | 2 | 2775.1.x.a | 4 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
111.1.i.a | ✓ | 2 | 9.c | even | 3 | 1 | |
111.1.i.a | ✓ | 2 | 9.d | odd | 6 | 1 | |
111.1.i.a | ✓ | 2 | 333.h | even | 3 | 1 | |
111.1.i.a | ✓ | 2 | 333.l | odd | 6 | 1 | |
1776.1.bw.a | 2 | 36.f | odd | 6 | 1 | ||
1776.1.bw.a | 2 | 36.h | even | 6 | 1 | ||
1776.1.bw.a | 2 | 1332.s | even | 6 | 1 | ||
1776.1.bw.a | 2 | 1332.br | odd | 6 | 1 | ||
2775.1.x.a | 4 | 45.k | odd | 12 | 2 | ||
2775.1.x.a | 4 | 45.l | even | 12 | 2 | ||
2775.1.x.a | 4 | 1665.co | odd | 12 | 2 | ||
2775.1.x.a | 4 | 1665.cp | even | 12 | 2 | ||
2775.1.z.a | 2 | 45.h | odd | 6 | 1 | ||
2775.1.z.a | 2 | 45.j | even | 6 | 1 | ||
2775.1.z.a | 2 | 1665.ba | even | 6 | 1 | ||
2775.1.z.a | 2 | 1665.bz | odd | 6 | 1 | ||
2997.1.l.a | 2 | 9.c | even | 3 | 1 | ||
2997.1.l.a | 2 | 9.d | odd | 6 | 1 | ||
2997.1.l.a | 2 | 37.c | even | 3 | 1 | ||
2997.1.l.a | 2 | 111.i | odd | 6 | 1 | ||
2997.1.u.a | 2 | 1.a | even | 1 | 1 | trivial | |
2997.1.u.a | 2 | 3.b | odd | 2 | 1 | CM | |
2997.1.u.a | 2 | 333.g | even | 3 | 1 | inner | |
2997.1.u.a | 2 | 333.u | odd | 6 | 1 | inner |
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2} \)
acting on \(S_{1}^{\mathrm{new}}(2997, [\chi])\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{2} \)
$3$
\( T^{2} \)
$5$
\( T^{2} \)
$7$
\( (T + 1)^{2} \)
$11$
\( T^{2} \)
$13$
\( T^{2} - T + 1 \)
$17$
\( T^{2} \)
$19$
\( T^{2} + 2T + 4 \)
$23$
\( T^{2} \)
$29$
\( T^{2} \)
$31$
\( T^{2} - T + 1 \)
$37$
\( (T - 1)^{2} \)
$41$
\( T^{2} \)
$43$
\( T^{2} - T + 1 \)
$47$
\( T^{2} \)
$53$
\( T^{2} \)
$59$
\( T^{2} \)
$61$
\( (T - 2)^{2} \)
$67$
\( T^{2} - T + 1 \)
$71$
\( T^{2} \)
$73$
\( (T + 1)^{2} \)
$79$
\( (T + 1)^{2} \)
$83$
\( T^{2} \)
$89$
\( T^{2} \)
$97$
\( T^{2} - T + 1 \)
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