# Properties

 Label 2997.1.o.a Level $2997$ Weight $1$ Character orbit 2997.o Analytic conductor $1.496$ Analytic rank $0$ Dimension $2$ Projective image $D_{6}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2997 = 3^{4} \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 2997.o (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: $$D_{6}$$ Projective field: Galois closure of 6.2.624095613.1

## $q$-expansion

The $$q$$-expansion and trace form are shown below.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{4} + q^{7} +O(q^{10})$$ $$q + \zeta_{6} q^{4} + q^{7} + ( 1 - \zeta_{6}^{2} ) q^{13} + \zeta_{6}^{2} q^{16} -\zeta_{6}^{2} q^{25} + \zeta_{6} q^{28} + ( -1 - \zeta_{6} ) q^{31} + q^{37} + ( -1 + \zeta_{6}^{2} ) q^{43} + ( 1 + \zeta_{6} ) q^{52} - q^{64} -\zeta_{6} q^{67} + q^{73} + ( \zeta_{6} + \zeta_{6}^{2} ) q^{79} + ( 1 - \zeta_{6}^{2} ) q^{91} + ( -1 + \zeta_{6}^{2} ) q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{4} + 2 q^{7} + O(q^{10})$$ $$2 q + q^{4} + 2 q^{7} + 3 q^{13} - q^{16} + q^{25} + q^{28} - 3 q^{31} + 2 q^{37} - 3 q^{43} + 3 q^{52} - 2 q^{64} - q^{67} + 2 q^{73} + 3 q^{91} - 3 q^{97} + O(q^{100})$$

## Character values

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1322.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0 0.500000 0.866025i 0 0 1.00000 0 0 0
2564.1 0 0 0.500000 + 0.866025i 0 0 1.00000 0 0 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 CM by $$\Q(\sqrt{-3})$$
333.o odd 6 1 inner
333.t 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.o.a 2
3.b odd 2 1 CM 2997.1.o.a 2
9.c even 3 1 111.1.h.a 2
9.c even 3 1 2997.1.v.a 2
9.d odd 6 1 111.1.h.a 2
9.d odd 6 1 2997.1.v.a 2
36.f odd 6 1 1776.1.bq.a 2
36.h even 6 1 1776.1.bq.a 2
37.e even 6 1 2997.1.v.a 2
45.h odd 6 1 2775.1.w.a 2
45.j even 6 1 2775.1.w.a 2
45.k odd 12 2 2775.1.bb.a 4
45.l even 12 2 2775.1.bb.a 4
111.h odd 6 1 2997.1.v.a 2
333.k even 6 1 111.1.h.a 2
333.o odd 6 1 inner 2997.1.o.a 2
333.t even 6 1 inner 2997.1.o.a 2
333.v odd 6 1 111.1.h.a 2
1332.t odd 6 1 1776.1.bq.a 2
1332.bq even 6 1 1776.1.bq.a 2
1665.bb odd 6 1 2775.1.w.a 2
1665.by even 6 1 2775.1.w.a 2
1665.cn odd 12 2 2775.1.bb.a 4
1665.cq even 12 2 2775.1.bb.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.h.a 2 9.c even 3 1
111.1.h.a 2 9.d odd 6 1
111.1.h.a 2 333.k even 6 1
111.1.h.a 2 333.v odd 6 1
1776.1.bq.a 2 36.f odd 6 1
1776.1.bq.a 2 36.h even 6 1
1776.1.bq.a 2 1332.t odd 6 1
1776.1.bq.a 2 1332.bq even 6 1
2775.1.w.a 2 45.h odd 6 1
2775.1.w.a 2 45.j even 6 1
2775.1.w.a 2 1665.bb odd 6 1
2775.1.w.a 2 1665.by even 6 1
2775.1.bb.a 4 45.k odd 12 2
2775.1.bb.a 4 45.l even 12 2
2775.1.bb.a 4 1665.cn odd 12 2
2775.1.bb.a 4 1665.cq even 12 2
2997.1.o.a 2 1.a even 1 1 trivial
2997.1.o.a 2 3.b odd 2 1 CM
2997.1.o.a 2 333.o odd 6 1 inner
2997.1.o.a 2 333.t even 6 1 inner
2997.1.v.a 2 9.c even 3 1
2997.1.v.a 2 9.d odd 6 1
2997.1.v.a 2 37.e even 6 1
2997.1.v.a 2 111.h odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(2997, [\chi])$$.

## Hecke characteristic polynomials

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