# Properties

 Label 1776.1.bw.a Level $1776$ Weight $1$ Character orbit 1776.bw Analytic conductor $0.886$ Analytic rank $0$ Dimension $2$ Projective image $D_{3}$ CM discriminant -3 Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1776 = 2^{4} \cdot 3 \cdot 37$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1776.bw (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.886339462436$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{6})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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: $C_6\times S_3$ Artin field: Galois closure of $$\mathbb{Q}[x]/(x^{12} - \cdots)$$

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{3} -\zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{3} -\zeta_{6} q^{7} + \zeta_{6}^{2} q^{9} + \zeta_{6} q^{13} + 2 \zeta_{6} q^{19} -\zeta_{6}^{2} q^{21} + \zeta_{6}^{2} q^{25} - q^{27} + q^{31} + q^{37} + \zeta_{6}^{2} q^{39} + q^{43} + 2 \zeta_{6}^{2} q^{57} -2 \zeta_{6} q^{61} + q^{63} -\zeta_{6} q^{67} - q^{73} - q^{75} -\zeta_{6} q^{79} -\zeta_{6} q^{81} -\zeta_{6}^{2} q^{91} + \zeta_{6} q^{93} - q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + q^{3} - q^{7} - q^{9} + O(q^{10})$$ $$2 q + q^{3} - q^{7} - q^{9} + q^{13} + 2 q^{19} + q^{21} - q^{25} - 2 q^{27} + 2 q^{31} + 2 q^{37} - q^{39} + 2 q^{43} - 2 q^{57} - 2 q^{61} + 2 q^{63} - q^{67} - 2 q^{73} - 2 q^{75} - q^{79} - q^{81} + q^{91} + q^{93} - 2 q^{97} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times$$.

 $$n$$ $$223$$ $$593$$ $$1297$$ $$1333$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1025.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 0.866025i 0 0 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0
1601.1 0 0.500000 + 0.866025i 0 0 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 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 $$\Q(\sqrt{-3})$$
37.c even 3 1 inner
111.i odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1776.1.bw.a 2
3.b odd 2 1 CM 1776.1.bw.a 2
4.b odd 2 1 111.1.i.a 2
12.b even 2 1 111.1.i.a 2
20.d odd 2 1 2775.1.z.a 2
20.e even 4 2 2775.1.x.a 4
36.f odd 6 1 2997.1.l.a 2
36.f odd 6 1 2997.1.u.a 2
36.h even 6 1 2997.1.l.a 2
36.h even 6 1 2997.1.u.a 2
37.c even 3 1 inner 1776.1.bw.a 2
60.h even 2 1 2775.1.z.a 2
60.l odd 4 2 2775.1.x.a 4
111.i odd 6 1 inner 1776.1.bw.a 2
148.i odd 6 1 111.1.i.a 2
444.t even 6 1 111.1.i.a 2
740.w odd 6 1 2775.1.z.a 2
740.bg even 12 2 2775.1.x.a 4
1332.s even 6 1 2997.1.u.a 2
1332.v odd 6 1 2997.1.l.a 2
1332.bd even 6 1 2997.1.l.a 2
1332.br odd 6 1 2997.1.u.a 2
2220.bu even 6 1 2775.1.z.a 2
2220.cu odd 12 2 2775.1.x.a 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
111.1.i.a 2 4.b odd 2 1
111.1.i.a 2 12.b even 2 1
111.1.i.a 2 148.i odd 6 1
111.1.i.a 2 444.t even 6 1
1776.1.bw.a 2 1.a even 1 1 trivial
1776.1.bw.a 2 3.b odd 2 1 CM
1776.1.bw.a 2 37.c even 3 1 inner
1776.1.bw.a 2 111.i odd 6 1 inner
2775.1.x.a 4 20.e even 4 2
2775.1.x.a 4 60.l odd 4 2
2775.1.x.a 4 740.bg even 12 2
2775.1.x.a 4 2220.cu odd 12 2
2775.1.z.a 2 20.d odd 2 1
2775.1.z.a 2 60.h even 2 1
2775.1.z.a 2 740.w odd 6 1
2775.1.z.a 2 2220.bu even 6 1
2997.1.l.a 2 36.f odd 6 1
2997.1.l.a 2 36.h even 6 1
2997.1.l.a 2 1332.v odd 6 1
2997.1.l.a 2 1332.bd even 6 1
2997.1.u.a 2 36.f odd 6 1
2997.1.u.a 2 36.h even 6 1
2997.1.u.a 2 1332.s even 6 1
2997.1.u.a 2 1332.br odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1776, [\chi])$$.

## Hecke characteristic polynomials

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