# Properties

 Label 1680.2.bg.o Level $1680$ Weight $2$ Character orbit 1680.bg Analytic conductor $13.415$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{3})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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 $$\zeta_{12}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q -\zeta_{12}^{2} q^{3} + ( -1 + \zeta_{12}^{2} ) q^{5} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{7} + ( -1 + \zeta_{12}^{2} ) q^{9} + ( \zeta_{12} - \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{11} + ( 4 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{13} + q^{15} + ( \zeta_{12} - 5 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{17} + ( 1 + 2 \zeta_{12} - \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{19} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{21} + ( -3 - \zeta_{12} + 3 \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{23} -\zeta_{12}^{2} q^{25} + q^{27} + ( 1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{29} + ( -2 \zeta_{12} + 3 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{31} + ( -1 + \zeta_{12} + \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{33} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{35} + ( -2 + 3 \zeta_{12} + 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{37} + ( \zeta_{12} - 4 \zeta_{12}^{2} + \zeta_{12}^{3} ) q^{39} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{41} + ( 2 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{43} -\zeta_{12}^{2} q^{45} + ( 2 - 2 \zeta_{12}^{2} ) q^{47} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + ( -5 + \zeta_{12} + 5 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{51} + ( -6 \zeta_{12} - 2 \zeta_{12}^{2} - 6 \zeta_{12}^{3} ) q^{53} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{55} + ( -1 - 4 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{57} + ( -3 \zeta_{12} + 5 \zeta_{12}^{2} - 3 \zeta_{12}^{3} ) q^{59} + ( -4 + 4 \zeta_{12}^{2} ) q^{61} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{63} + ( -4 + \zeta_{12} + 4 \zeta_{12}^{2} - 2 \zeta_{12}^{3} ) q^{65} + ( -5 \zeta_{12} - 6 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{67} + ( 3 + 2 \zeta_{12} - \zeta_{12}^{3} ) q^{69} + ( -1 - 6 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{71} + ( -5 \zeta_{12} - 4 \zeta_{12}^{2} - 5 \zeta_{12}^{3} ) q^{73} + ( -1 + \zeta_{12}^{2} ) q^{75} + ( 5 - 2 \zeta_{12} - 4 \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{77} + ( 3 - 6 \zeta_{12} - 3 \zeta_{12}^{2} + 12 \zeta_{12}^{3} ) q^{79} -\zeta_{12}^{2} q^{81} + ( -3 - 14 \zeta_{12} + 7 \zeta_{12}^{3} ) q^{83} + ( 5 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{85} + ( 3 \zeta_{12} - \zeta_{12}^{2} + 3 \zeta_{12}^{3} ) q^{87} + ( -3 - 7 \zeta_{12} + 3 \zeta_{12}^{2} + 14 \zeta_{12}^{3} ) q^{89} + ( -1 - 4 \zeta_{12} + 5 \zeta_{12}^{2} - 8 \zeta_{12}^{3} ) q^{91} + ( 3 - 2 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{93} + ( 2 \zeta_{12} + \zeta_{12}^{2} + 2 \zeta_{12}^{3} ) q^{95} + ( 8 + 8 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{97} + ( 1 - 2 \zeta_{12} + \zeta_{12}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 2q^{3} - 2q^{5} - 2q^{9} + O(q^{10})$$ $$4q - 2q^{3} - 2q^{5} - 2q^{9} - 2q^{11} + 16q^{13} + 4q^{15} - 10q^{17} + 2q^{19} - 6q^{23} - 2q^{25} + 4q^{27} + 4q^{29} + 6q^{31} - 2q^{33} - 4q^{37} - 8q^{39} + 4q^{41} + 8q^{43} - 2q^{45} + 4q^{47} - 22q^{49} - 10q^{51} - 4q^{53} + 4q^{55} - 4q^{57} + 10q^{59} - 8q^{61} - 8q^{65} - 12q^{67} + 12q^{69} - 4q^{71} - 8q^{73} - 2q^{75} + 12q^{77} + 6q^{79} - 2q^{81} - 12q^{83} + 20q^{85} - 2q^{87} - 6q^{89} + 6q^{91} + 6q^{93} + 2q^{95} + 32q^{97} + 4q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1 + \zeta_{12}^{2}$$ $$1$$ $$1$$ $$1$$ $$1$$

## 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
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 −0.866025 2.50000i 0 −0.500000 + 0.866025i 0
961.2 0 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0.866025 + 2.50000i 0 −0.500000 + 0.866025i 0
1201.1 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 −0.866025 + 2.50000i 0 −0.500000 0.866025i 0
1201.2 0 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0.866025 2.50000i 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.o 4
4.b odd 2 1 105.2.i.d 4
7.c even 3 1 inner 1680.2.bg.o 4
12.b even 2 1 315.2.j.c 4
20.d odd 2 1 525.2.i.f 4
20.e even 4 1 525.2.r.a 4
20.e even 4 1 525.2.r.f 4
28.d even 2 1 735.2.i.l 4
28.f even 6 1 735.2.a.h 2
28.f even 6 1 735.2.i.l 4
28.g odd 6 1 105.2.i.d 4
28.g odd 6 1 735.2.a.g 2
84.j odd 6 1 2205.2.a.ba 2
84.n even 6 1 315.2.j.c 4
84.n even 6 1 2205.2.a.z 2
140.p odd 6 1 525.2.i.f 4
140.p odd 6 1 3675.2.a.bg 2
140.s even 6 1 3675.2.a.be 2
140.w even 12 1 525.2.r.a 4
140.w even 12 1 525.2.r.f 4

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

 $$T_{11}^{4} + 2 T_{11}^{3} + 6 T_{11}^{2} - 4 T_{11} + 4$$ $$T_{13}^{2} - 8 T_{13} + 13$$ $$T_{17}^{4} + 10 T_{17}^{3} + 78 T_{17}^{2} + 220 T_{17} + 484$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$( 1 + T + T^{2} )^{2}$$
$5$ $$( 1 + T + T^{2} )^{2}$$
$7$ $$49 + 11 T^{2} + T^{4}$$
$11$ $$4 - 4 T + 6 T^{2} + 2 T^{3} + T^{4}$$
$13$ $$( 13 - 8 T + T^{2} )^{2}$$
$17$ $$484 + 220 T + 78 T^{2} + 10 T^{3} + T^{4}$$
$19$ $$121 + 22 T + 15 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$36 + 36 T + 30 T^{2} + 6 T^{3} + T^{4}$$
$29$ $$( -26 - 2 T + T^{2} )^{2}$$
$31$ $$9 + 18 T + 39 T^{2} - 6 T^{3} + T^{4}$$
$37$ $$529 - 92 T + 39 T^{2} + 4 T^{3} + T^{4}$$
$41$ $$( -2 - 2 T + T^{2} )^{2}$$
$43$ $$( -23 - 4 T + T^{2} )^{2}$$
$47$ $$( 4 - 2 T + T^{2} )^{2}$$
$53$ $$10816 - 416 T + 120 T^{2} + 4 T^{3} + T^{4}$$
$59$ $$4 + 20 T + 102 T^{2} - 10 T^{3} + T^{4}$$
$61$ $$( 16 + 4 T + T^{2} )^{2}$$
$67$ $$1521 - 468 T + 183 T^{2} + 12 T^{3} + T^{4}$$
$71$ $$( -26 + 2 T + T^{2} )^{2}$$
$73$ $$3481 - 472 T + 123 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$9801 + 594 T + 135 T^{2} - 6 T^{3} + T^{4}$$
$83$ $$( -138 + 6 T + T^{2} )^{2}$$
$89$ $$19044 - 828 T + 174 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$( 16 - 16 T + T^{2} )^{2}$$