Properties

Label 2997.1.l.a
Level $2997$
Weight $1$
Character orbit 2997.l
Analytic conductor $1.496$
Analytic rank $0$
Dimension $2$
Projective image $D_{3}$
CM discriminant -3
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 2997 = 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2997.l (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.49569784286\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 6.0.26946027.1

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q + q^{4} -\zeta_{6}^{2} q^{7} +O(q^{10})\) \( q + q^{4} -\zeta_{6}^{2} q^{7} - q^{13} + q^{16} -2 \zeta_{6} q^{19} + q^{25} -\zeta_{6}^{2} q^{28} -\zeta_{6}^{2} q^{31} + q^{37} + \zeta_{6} q^{43} - q^{52} + 2 \zeta_{6}^{2} q^{61} + q^{64} - q^{67} - q^{73} -2 \zeta_{6} q^{76} -\zeta_{6}^{2} q^{79} + \zeta_{6}^{2} q^{91} + \zeta_{6} q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{4} + q^{7} + O(q^{10}) \) \( 2q + 2q^{4} + q^{7} - 2q^{13} + 2q^{16} - 2q^{19} + 2q^{25} + q^{28} + q^{31} + 2q^{37} + q^{43} - 2q^{52} - 2q^{61} + 2q^{64} - 2q^{67} - 2q^{73} - 2q^{76} + q^{79} - q^{91} + q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2997\mathbb{Z}\right)^\times\).

\(n\) \(1297\) \(1703\)
\(\chi(n)\) \(-\zeta_{6}\) \(-\zeta_{6}^{2}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 0.866025i
0.500000 + 0.866025i
0 0 1.00000 0 0 0.500000 + 0.866025i 0 0 0
1268.1 0 0 1.00000 0 0 0.500000 0.866025i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by \(\Q(\sqrt{-3}) \)
333.h even 3 1 inner
333.l 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.l.a 2
3.b odd 2 1 CM 2997.1.l.a 2
9.c even 3 1 111.1.i.a 2
9.c even 3 1 2997.1.u.a 2
9.d odd 6 1 111.1.i.a 2
9.d odd 6 1 2997.1.u.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.u.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.u.a 2
333.g even 3 1 111.1.i.a 2
333.h even 3 1 inner 2997.1.l.a 2
333.l odd 6 1 inner 2997.1.l.a 2
333.u odd 6 1 111.1.i.a 2
1332.v odd 6 1 1776.1.bw.a 2
1332.bd even 6 1 1776.1.bw.a 2
1665.bd odd 6 1 2775.1.z.a 2
1665.bl even 6 1 2775.1.z.a 2
1665.de odd 12 2 2775.1.x.a 4
1665.dg 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.g even 3 1
111.1.i.a 2 333.u 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.v odd 6 1
1776.1.bw.a 2 1332.bd even 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.de odd 12 2
2775.1.x.a 4 1665.dg 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.bd odd 6 1
2775.1.z.a 2 1665.bl even 6 1
2997.1.l.a 2 1.a even 1 1 trivial
2997.1.l.a 2 3.b odd 2 1 CM
2997.1.l.a 2 333.h even 3 1 inner
2997.1.l.a 2 333.l odd 6 1 inner
2997.1.u.a 2 9.c even 3 1
2997.1.u.a 2 9.d odd 6 1
2997.1.u.a 2 37.c even 3 1
2997.1.u.a 2 111.i odd 6 1

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

$p$ $F_p(T)$
$2$ \( T^{2} \)
$3$ \( T^{2} \)
$5$ \( T^{2} \)
$7$ \( 1 - T + T^{2} \)
$11$ \( T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( T^{2} \)
$19$ \( 4 + 2 T + T^{2} \)
$23$ \( T^{2} \)
$29$ \( T^{2} \)
$31$ \( 1 - T + T^{2} \)
$37$ \( ( -1 + T )^{2} \)
$41$ \( T^{2} \)
$43$ \( 1 - T + T^{2} \)
$47$ \( T^{2} \)
$53$ \( T^{2} \)
$59$ \( T^{2} \)
$61$ \( 4 + 2 T + T^{2} \)
$67$ \( ( 1 + T )^{2} \)
$71$ \( T^{2} \)
$73$ \( ( 1 + T )^{2} \)
$79$ \( 1 - T + T^{2} \)
$83$ \( T^{2} \)
$89$ \( T^{2} \)
$97$ \( 1 - T + T^{2} \)
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