Properties

Label 2997.1.n.b
Level $2997$
Weight $1$
Character orbit 2997.n
Analytic conductor $1.496$
Analytic rank $0$
Dimension $2$
Projective image $D_{2}$
CM/RM discs -3, -111, 37
Inner twists $8$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 2997 = 3^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2997.n (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: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 111)
Projective image: \(D_{2}\)
Projective field: Galois closure of \(\Q(\sqrt{-3}, \sqrt{37})\)
Artin image: $C_3\times D_4$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{12} - \cdots)\)

$q$-expansion

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

\(f(q)\) \(=\) \( q + \zeta_{6} q^{4} -2 \zeta_{6}^{2} q^{7} +O(q^{10})\) \( q + \zeta_{6} q^{4} -2 \zeta_{6}^{2} q^{7} + \zeta_{6}^{2} q^{16} -\zeta_{6}^{2} q^{25} + 2 q^{28} + q^{37} -3 \zeta_{6} q^{49} - q^{64} + 2 \zeta_{6} q^{67} -2 q^{73} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{4} + 2q^{7} + O(q^{10}) \) \( 2q + q^{4} + 2q^{7} - q^{16} + q^{25} + 4q^{28} + 2q^{37} - 3q^{49} - 2q^{64} + 2q^{67} - 4q^{73} + O(q^{100}) \)

Character values

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

\(n\) \(1297\) \(1703\)
\(\chi(n)\) \(-1\) \(\zeta_{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
998.1
0.500000 + 0.866025i
0.500000 0.866025i
0 0 0.500000 + 0.866025i 0 0 1.00000 1.73205i 0 0 0
1997.1 0 0 0.500000 0.866025i 0 0 1.00000 + 1.73205i 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 \(\Q(\sqrt{-3}) \)
37.b even 2 1 RM by \(\Q(\sqrt{37}) \)
111.d odd 2 1 CM by \(\Q(\sqrt{-111}) \)
9.c even 3 1 inner
9.d odd 6 1 inner
333.n odd 6 1 inner
333.q even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2997.1.n.b 2
3.b odd 2 1 CM 2997.1.n.b 2
9.c even 3 1 111.1.d.a 1
9.c even 3 1 inner 2997.1.n.b 2
9.d odd 6 1 111.1.d.a 1
9.d odd 6 1 inner 2997.1.n.b 2
36.f odd 6 1 1776.1.n.a 1
36.h even 6 1 1776.1.n.a 1
37.b even 2 1 RM 2997.1.n.b 2
45.h odd 6 1 2775.1.h.a 1
45.j even 6 1 2775.1.h.a 1
45.k odd 12 2 2775.1.b.a 2
45.l even 12 2 2775.1.b.a 2
111.d odd 2 1 CM 2997.1.n.b 2
333.n odd 6 1 111.1.d.a 1
333.n odd 6 1 inner 2997.1.n.b 2
333.q even 6 1 111.1.d.a 1
333.q even 6 1 inner 2997.1.n.b 2
1332.z even 6 1 1776.1.n.a 1
1332.bl odd 6 1 1776.1.n.a 1
1665.bh even 6 1 2775.1.h.a 1
1665.bt odd 6 1 2775.1.h.a 1
1665.dc odd 12 2 2775.1.b.a 2
1665.di even 12 2 2775.1.b.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.d.a 1 9.c even 3 1
111.1.d.a 1 9.d odd 6 1
111.1.d.a 1 333.n odd 6 1
111.1.d.a 1 333.q even 6 1
1776.1.n.a 1 36.f odd 6 1
1776.1.n.a 1 36.h even 6 1
1776.1.n.a 1 1332.z even 6 1
1776.1.n.a 1 1332.bl odd 6 1
2775.1.b.a 2 45.k odd 12 2
2775.1.b.a 2 45.l even 12 2
2775.1.b.a 2 1665.dc odd 12 2
2775.1.b.a 2 1665.di even 12 2
2775.1.h.a 1 45.h odd 6 1
2775.1.h.a 1 45.j even 6 1
2775.1.h.a 1 1665.bh even 6 1
2775.1.h.a 1 1665.bt odd 6 1
2997.1.n.b 2 1.a even 1 1 trivial
2997.1.n.b 2 3.b odd 2 1 CM
2997.1.n.b 2 9.c even 3 1 inner
2997.1.n.b 2 9.d odd 6 1 inner
2997.1.n.b 2 37.b even 2 1 RM
2997.1.n.b 2 111.d odd 2 1 CM
2997.1.n.b 2 333.n odd 6 1 inner
2997.1.n.b 2 333.q even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(2997, [\chi])\):

\( T_{2} \)
\( T_{5} \)

Hecke characteristic polynomials

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