# Properties

 Label 1680.2.bg.g 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} +O(q^{10})$$ $$q + ( -1 + \zeta_{6} ) q^{3} + \zeta_{6} q^{5} + ( -1 - 2 \zeta_{6} ) q^{7} -\zeta_{6} q^{9} + ( -1 + \zeta_{6} ) q^{11} + q^{13} - q^{15} -3 \zeta_{6} q^{19} + ( 3 - \zeta_{6} ) q^{21} + 7 \zeta_{6} q^{23} + ( -1 + \zeta_{6} ) q^{25} + q^{27} -8 q^{29} + ( -2 + 2 \zeta_{6} ) q^{31} -\zeta_{6} q^{33} + ( 2 - 3 \zeta_{6} ) q^{35} -11 \zeta_{6} q^{37} + ( -1 + \zeta_{6} ) q^{39} -11 q^{41} -8 q^{43} + ( 1 - \zeta_{6} ) q^{45} -5 \zeta_{6} q^{47} + ( -3 + 8 \zeta_{6} ) q^{49} + ( 11 - 11 \zeta_{6} ) q^{53} - q^{55} + 3 q^{57} + ( 4 - 4 \zeta_{6} ) q^{59} + ( -2 + 3 \zeta_{6} ) q^{63} + \zeta_{6} q^{65} -7 q^{69} + 6 q^{71} + ( 6 - 6 \zeta_{6} ) q^{73} -\zeta_{6} q^{75} + ( 3 - \zeta_{6} ) q^{77} -8 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -8 q^{83} + ( 8 - 8 \zeta_{6} ) q^{87} + 10 \zeta_{6} q^{89} + ( -1 - 2 \zeta_{6} ) q^{91} -2 \zeta_{6} q^{93} + ( 3 - 3 \zeta_{6} ) q^{95} -16 q^{97} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - q^{3} + q^{5} - 4q^{7} - q^{9} + O(q^{10})$$ $$2q - q^{3} + q^{5} - 4q^{7} - q^{9} - q^{11} + 2q^{13} - 2q^{15} - 3q^{19} + 5q^{21} + 7q^{23} - q^{25} + 2q^{27} - 16q^{29} - 2q^{31} - q^{33} + q^{35} - 11q^{37} - q^{39} - 22q^{41} - 16q^{43} + q^{45} - 5q^{47} + 2q^{49} + 11q^{53} - 2q^{55} + 6q^{57} + 4q^{59} - q^{63} + q^{65} - 14q^{69} + 12q^{71} + 6q^{73} - q^{75} + 5q^{77} - 8q^{79} - q^{81} - 16q^{83} + 8q^{87} + 10q^{89} - 4q^{91} - 2q^{93} + 3q^{95} - 32q^{97} + 2q^{99} + O(q^{100})$$

## 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.g 2
4.b odd 2 1 210.2.i.d 2
7.c even 3 1 inner 1680.2.bg.g 2
12.b even 2 1 630.2.k.c 2
20.d odd 2 1 1050.2.i.b 2
20.e even 4 2 1050.2.o.i 4
28.d even 2 1 1470.2.i.m 2
28.f even 6 1 1470.2.a.h 1
28.f even 6 1 1470.2.i.m 2
28.g odd 6 1 210.2.i.d 2
28.g odd 6 1 1470.2.a.a 1
84.j odd 6 1 4410.2.a.ba 1
84.n even 6 1 630.2.k.c 2
84.n even 6 1 4410.2.a.bj 1
140.p odd 6 1 1050.2.i.b 2
140.p odd 6 1 7350.2.a.cp 1
140.s even 6 1 7350.2.a.bu 1
140.w even 12 2 1050.2.o.i 4

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

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 + T + T^{2}$$
$5$ $$1 - T + T^{2}$$
$7$ $$1 + 4 T + 7 T^{2}$$
$11$ $$1 + T - 10 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 - T + 13 T^{2} )^{2}$$
$17$ $$1 - 17 T^{2} + 289 T^{4}$$
$19$ $$1 + 3 T - 10 T^{2} + 57 T^{3} + 361 T^{4}$$
$23$ $$1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$1 + 2 T - 27 T^{2} + 62 T^{3} + 961 T^{4}$$
$37$ $$( 1 + T + 37 T^{2} )( 1 + 10 T + 37 T^{2} )$$
$41$ $$( 1 + 11 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + 8 T + 43 T^{2} )^{2}$$
$47$ $$1 + 5 T - 22 T^{2} + 235 T^{3} + 2209 T^{4}$$
$53$ $$1 - 11 T + 68 T^{2} - 583 T^{3} + 2809 T^{4}$$
$59$ $$1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4}$$
$61$ $$1 - 61 T^{2} + 3721 T^{4}$$
$67$ $$1 - 67 T^{2} + 4489 T^{4}$$
$71$ $$( 1 - 6 T + 71 T^{2} )^{2}$$
$73$ $$1 - 6 T - 37 T^{2} - 438 T^{3} + 5329 T^{4}$$
$79$ $$1 + 8 T - 15 T^{2} + 632 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 8 T + 83 T^{2} )^{2}$$
$89$ $$1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 16 T + 97 T^{2} )^{2}$$