# Properties

 Label 1680.2.bg.l 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: $$\Q(\sqrt{-3})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 105) 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.

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

## 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 0.500000 + 2.59808i 0 −0.500000 + 0.866025i 0
1201.1 0 0.500000 0.866025i 0 −0.500000 0.866025i 0 0.500000 2.59808i 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.l 2
4.b odd 2 1 105.2.i.b 2
7.c even 3 1 inner 1680.2.bg.l 2
12.b even 2 1 315.2.j.a 2
20.d odd 2 1 525.2.i.a 2
20.e even 4 2 525.2.r.d 4
28.d even 2 1 735.2.i.f 2
28.f even 6 1 735.2.a.a 1
28.f even 6 1 735.2.i.f 2
28.g odd 6 1 105.2.i.b 2
28.g odd 6 1 735.2.a.b 1
84.j odd 6 1 2205.2.a.m 1
84.n even 6 1 315.2.j.a 2
84.n even 6 1 2205.2.a.k 1
140.p odd 6 1 525.2.i.a 2
140.p odd 6 1 3675.2.a.o 1
140.s even 6 1 3675.2.a.p 1
140.w even 12 2 525.2.r.d 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.i.b 2 4.b odd 2 1
105.2.i.b 2 28.g odd 6 1
315.2.j.a 2 12.b even 2 1
315.2.j.a 2 84.n even 6 1
525.2.i.a 2 20.d odd 2 1
525.2.i.a 2 140.p odd 6 1
525.2.r.d 4 20.e even 4 2
525.2.r.d 4 140.w even 12 2
735.2.a.a 1 28.f even 6 1
735.2.a.b 1 28.g odd 6 1
735.2.i.f 2 28.d even 2 1
735.2.i.f 2 28.f even 6 1
1680.2.bg.l 2 1.a even 1 1 trivial
1680.2.bg.l 2 7.c even 3 1 inner
2205.2.a.k 1 84.n even 6 1
2205.2.a.m 1 84.j odd 6 1
3675.2.a.o 1 140.p odd 6 1
3675.2.a.p 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 $$S_{2}^{\mathrm{new}}(1680, [\chi])$$:

 $$T_{11}^{2} + 6 T_{11} + 36$$ $$T_{13} + 3$$ $$T_{17}^{2} - 4 T_{17} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 + 6 T + 25 T^{2} + 66 T^{3} + 121 T^{4}$$
$13$ $$( 1 + 3 T + 13 T^{2} )^{2}$$
$17$ $$1 - 4 T - T^{2} - 68 T^{3} + 289 T^{4}$$
$19$ $$( 1 - 8 T + 19 T^{2} )( 1 + 7 T + 19 T^{2} )$$
$23$ $$1 + 4 T - 7 T^{2} + 92 T^{3} + 529 T^{4}$$
$29$ $$( 1 + 8 T + 29 T^{2} )^{2}$$
$31$ $$1 - T - 30 T^{2} - 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 7 T + 12 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$( 1 + T + 43 T^{2} )^{2}$$
$47$ $$1 - 2 T - 43 T^{2} - 94 T^{3} + 2209 T^{4}$$
$53$ $$1 + 4 T - 37 T^{2} + 212 T^{3} + 2809 T^{4}$$
$59$ $$1 + 8 T + 5 T^{2} + 472 T^{3} + 3481 T^{4}$$
$61$ $$( 1 - 13 T + 61 T^{2} )( 1 - T + 61 T^{2} )$$
$67$ $$1 - 7 T - 18 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$( 1 + 6 T + 71 T^{2} )^{2}$$
$73$ $$1 + T - 72 T^{2} + 73 T^{3} + 5329 T^{4}$$
$79$ $$1 + T - 78 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$( 1 + 2 T + 83 T^{2} )^{2}$$
$89$ $$1 - 12 T + 55 T^{2} - 1068 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 6 T + 97 T^{2} )^{2}$$