# Properties

 Label 1680.2.bg.k Level $1680$ Weight $2$ Character orbit 1680.bg Analytic conductor $13.415$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.bg (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ 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 210) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( 1 - \zeta_{6} ) q^{3} -\zeta_{6} q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + 7 q^{13} - q^{15} + ( 4 - 4 \zeta_{6} ) q^{17} + \zeta_{6} q^{19} + ( -1 + 3 \zeta_{6} ) q^{21} + \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} - q^{27} -8 q^{29} + ( 6 - 6 \zeta_{6} ) q^{31} + \zeta_{6} q^{33} + ( 2 + \zeta_{6} ) q^{35} + 3 \zeta_{6} q^{37} + ( 7 - 7 \zeta_{6} ) q^{39} + 9 q^{41} + 4 q^{43} + ( -1 + \zeta_{6} ) q^{45} -3 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} -4 \zeta_{6} q^{51} + ( 1 - \zeta_{6} ) q^{53} + q^{55} + q^{57} + ( 12 - 12 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} + ( 2 + \zeta_{6} ) q^{63} -7 \zeta_{6} q^{65} + ( 12 - 12 \zeta_{6} ) q^{67} + q^{69} + 14 q^{71} + ( 14 - 14 \zeta_{6} ) q^{73} + \zeta_{6} q^{75} + ( 1 - 3 \zeta_{6} ) q^{77} + 4 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -12 q^{83} -4 q^{85} + ( -8 + 8 \zeta_{6} ) q^{87} + 2 \zeta_{6} q^{89} + ( -21 + 14 \zeta_{6} ) q^{91} -6 \zeta_{6} q^{93} + ( 1 - \zeta_{6} ) q^{95} -16 q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} - q^{5} - 4q^{7} - q^{9} + O(q^{10})$$ $$2q + q^{3} - q^{5} - 4q^{7} - q^{9} - q^{11} + 14q^{13} - 2q^{15} + 4q^{17} + q^{19} + q^{21} + q^{23} - q^{25} - 2q^{27} - 16q^{29} + 6q^{31} + q^{33} + 5q^{35} + 3q^{37} + 7q^{39} + 18q^{41} + 8q^{43} - q^{45} - 3q^{47} + 2q^{49} - 4q^{51} + q^{53} + 2q^{55} + 2q^{57} + 12q^{59} + 4q^{61} + 5q^{63} - 7q^{65} + 12q^{67} + 2q^{69} + 28q^{71} + 14q^{73} + q^{75} - q^{77} + 4q^{79} - q^{81} - 24q^{83} - 8q^{85} - 8q^{87} + 2q^{89} - 28q^{91} - 6q^{93} + q^{95} - 32q^{97} + 2q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$$.

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
961.1
 0.5 − 0.866025i 0.5 + 0.866025i
0 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0 −2.00000 1.73205i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 −2.00000 + 1.73205i 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
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.bg.k 2
4.b odd 2 1 210.2.i.a 2
7.c even 3 1 inner 1680.2.bg.k 2
12.b even 2 1 630.2.k.h 2
20.d odd 2 1 1050.2.i.s 2
20.e even 4 2 1050.2.o.j 4
28.d even 2 1 1470.2.i.i 2
28.f even 6 1 1470.2.a.k 1
28.f even 6 1 1470.2.i.i 2
28.g odd 6 1 210.2.i.a 2
28.g odd 6 1 1470.2.a.r 1
84.j odd 6 1 4410.2.a.q 1
84.n even 6 1 630.2.k.h 2
84.n even 6 1 4410.2.a.g 1
140.p odd 6 1 1050.2.i.s 2
140.p odd 6 1 7350.2.a.j 1
140.s even 6 1 7350.2.a.ba 1
140.w even 12 2 1050.2.o.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.i.a 2 4.b odd 2 1
210.2.i.a 2 28.g odd 6 1
630.2.k.h 2 12.b even 2 1
630.2.k.h 2 84.n even 6 1
1050.2.i.s 2 20.d odd 2 1
1050.2.i.s 2 140.p odd 6 1
1050.2.o.j 4 20.e even 4 2
1050.2.o.j 4 140.w even 12 2
1470.2.a.k 1 28.f even 6 1
1470.2.a.r 1 28.g odd 6 1
1470.2.i.i 2 28.d even 2 1
1470.2.i.i 2 28.f even 6 1
1680.2.bg.k 2 1.a even 1 1 trivial
1680.2.bg.k 2 7.c even 3 1 inner
4410.2.a.g 1 84.n even 6 1
4410.2.a.q 1 84.j odd 6 1
7350.2.a.j 1 140.p odd 6 1
7350.2.a.ba 1 140.s even 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{2} + T_{11} + 1$$ $$T_{13} - 7$$ $$T_{17}^{2} - 4 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( -7 + T )^{2}$$
$17$ $$16 - 4 T + T^{2}$$
$19$ $$1 - T + T^{2}$$
$23$ $$1 - T + T^{2}$$
$29$ $$( 8 + T )^{2}$$
$31$ $$36 - 6 T + T^{2}$$
$37$ $$9 - 3 T + T^{2}$$
$41$ $$( -9 + T )^{2}$$
$43$ $$( -4 + T )^{2}$$
$47$ $$9 + 3 T + T^{2}$$
$53$ $$1 - T + T^{2}$$
$59$ $$144 - 12 T + T^{2}$$
$61$ $$16 - 4 T + T^{2}$$
$67$ $$144 - 12 T + T^{2}$$
$71$ $$( -14 + T )^{2}$$
$73$ $$196 - 14 T + T^{2}$$
$79$ $$16 - 4 T + T^{2}$$
$83$ $$( 12 + T )^{2}$$
$89$ $$4 - 2 T + T^{2}$$
$97$ $$( 16 + T )^{2}$$