# Properties

 Label 1680.2.di.a Level 1680 Weight 2 Character orbit 1680.di Analytic conductor 13.415 Analytic rank 0 Dimension 4 CM no Inner twists 4

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.di (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Relative dimension: $$2$$ over $$\Q(\zeta_{6})$$ Coefficient field: $$\Q(\zeta_{12})$$ Defining polynomial: $$x^{4} - x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})$$ $$q + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{3} + ( -2 + \zeta_{12} + 2 \zeta_{12}^{2} ) q^{5} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{7} + ( 1 - \zeta_{12}^{2} ) q^{9} -5 \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{13} + ( -1 - 2 \zeta_{12}^{3} ) q^{15} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{17} + ( -7 + 7 \zeta_{12}^{2} ) q^{19} + ( 2 - 3 \zeta_{12}^{2} ) q^{21} + 3 \zeta_{12} q^{23} + ( -4 \zeta_{12} - 3 \zeta_{12}^{2} + 4 \zeta_{12}^{3} ) q^{25} + \zeta_{12}^{3} q^{27} -6 \zeta_{12}^{2} q^{31} + 5 \zeta_{12} q^{33} + ( -3 - 2 \zeta_{12} + \zeta_{12}^{2} - 4 \zeta_{12}^{3} ) q^{35} -5 \zeta_{12} q^{37} -\zeta_{12}^{2} q^{39} -9 q^{41} -10 \zeta_{12}^{3} q^{43} + ( \zeta_{12} + 2 \zeta_{12}^{2} - \zeta_{12}^{3} ) q^{45} + 13 \zeta_{12} q^{47} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -2 + 2 \zeta_{12}^{2} ) q^{51} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{53} + ( 10 - 5 \zeta_{12}^{3} ) q^{55} -7 \zeta_{12}^{3} q^{57} -4 \zeta_{12}^{2} q^{59} + ( 2 - 2 \zeta_{12}^{2} ) q^{61} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{63} + ( -1 - 2 \zeta_{12} + \zeta_{12}^{2} ) q^{65} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{67} -3 q^{69} + 2 q^{71} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + ( 4 + 3 \zeta_{12} - 4 \zeta_{12}^{2} ) q^{75} + ( 15 \zeta_{12} - 5 \zeta_{12}^{3} ) q^{77} + ( 14 - 14 \zeta_{12}^{2} ) q^{79} -\zeta_{12}^{2} q^{81} + 10 \zeta_{12}^{3} q^{83} + ( 2 + 4 \zeta_{12}^{3} ) q^{85} + ( 10 - 10 \zeta_{12}^{2} ) q^{89} + ( -1 - 2 \zeta_{12}^{2} ) q^{91} + 6 \zeta_{12} q^{93} + ( -7 \zeta_{12} - 14 \zeta_{12}^{2} + 7 \zeta_{12}^{3} ) q^{95} -8 \zeta_{12}^{3} q^{97} -5 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{5} + 2q^{9} + O(q^{10})$$ $$4q - 4q^{5} + 2q^{9} - 10q^{11} - 4q^{15} - 14q^{19} + 2q^{21} - 6q^{25} - 12q^{31} - 10q^{35} - 2q^{39} - 36q^{41} + 4q^{45} - 4q^{49} - 4q^{51} + 40q^{55} - 8q^{59} + 4q^{61} - 2q^{65} - 12q^{69} + 8q^{71} + 8q^{75} + 28q^{79} - 2q^{81} + 8q^{85} + 20q^{89} - 8q^{91} - 28q^{95} - 20q^{99} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
289.1
 0.866025 + 0.500000i −0.866025 − 0.500000i 0.866025 − 0.500000i −0.866025 + 0.500000i
0 −0.866025 + 0.500000i 0 −0.133975 + 2.23205i 0 −1.73205 + 2.00000i 0 0.500000 0.866025i 0
289.2 0 0.866025 0.500000i 0 −1.86603 + 1.23205i 0 1.73205 2.00000i 0 0.500000 0.866025i 0
529.1 0 −0.866025 0.500000i 0 −0.133975 2.23205i 0 −1.73205 2.00000i 0 0.500000 + 0.866025i 0
529.2 0 0.866025 + 0.500000i 0 −1.86603 1.23205i 0 1.73205 + 2.00000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.di.a 4
4.b odd 2 1 210.2.n.a 4
5.b even 2 1 inner 1680.2.di.a 4
7.c even 3 1 inner 1680.2.di.a 4
12.b even 2 1 630.2.u.c 4
20.d odd 2 1 210.2.n.a 4
20.e even 4 1 1050.2.i.f 2
20.e even 4 1 1050.2.i.o 2
28.d even 2 1 1470.2.n.i 4
28.f even 6 1 1470.2.g.a 2
28.f even 6 1 1470.2.n.i 4
28.g odd 6 1 210.2.n.a 4
28.g odd 6 1 1470.2.g.f 2
35.j even 6 1 inner 1680.2.di.a 4
60.h even 2 1 630.2.u.c 4
84.n even 6 1 630.2.u.c 4
140.c even 2 1 1470.2.n.i 4
140.p odd 6 1 210.2.n.a 4
140.p odd 6 1 1470.2.g.f 2
140.s even 6 1 1470.2.g.a 2
140.s even 6 1 1470.2.n.i 4
140.w even 12 1 1050.2.i.f 2
140.w even 12 1 1050.2.i.o 2
140.w even 12 1 7350.2.a.t 1
140.w even 12 1 7350.2.a.bn 1
140.x odd 12 1 7350.2.a.b 1
140.x odd 12 1 7350.2.a.ch 1
420.ba even 6 1 630.2.u.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.n.a 4 4.b odd 2 1
210.2.n.a 4 20.d odd 2 1
210.2.n.a 4 28.g odd 6 1
210.2.n.a 4 140.p odd 6 1
630.2.u.c 4 12.b even 2 1
630.2.u.c 4 60.h even 2 1
630.2.u.c 4 84.n even 6 1
630.2.u.c 4 420.ba even 6 1
1050.2.i.f 2 20.e even 4 1
1050.2.i.f 2 140.w even 12 1
1050.2.i.o 2 20.e even 4 1
1050.2.i.o 2 140.w even 12 1
1470.2.g.a 2 28.f even 6 1
1470.2.g.a 2 140.s even 6 1
1470.2.g.f 2 28.g odd 6 1
1470.2.g.f 2 140.p odd 6 1
1470.2.n.i 4 28.d even 2 1
1470.2.n.i 4 28.f even 6 1
1470.2.n.i 4 140.c even 2 1
1470.2.n.i 4 140.s even 6 1
1680.2.di.a 4 1.a even 1 1 trivial
1680.2.di.a 4 5.b even 2 1 inner
1680.2.di.a 4 7.c even 3 1 inner
1680.2.di.a 4 35.j even 6 1 inner
7350.2.a.b 1 140.x odd 12 1
7350.2.a.t 1 140.w even 12 1
7350.2.a.bn 1 140.w even 12 1
7350.2.a.ch 1 140.x odd 12 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{11}^{2} + 5 T_{11} + 25$$ acting on $$S_{2}^{\mathrm{new}}(1680, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T^{2} + T^{4}$$
$5$ $$1 + 4 T + 11 T^{2} + 20 T^{3} + 25 T^{4}$$
$7$ $$1 + 2 T^{2} + 49 T^{4}$$
$11$ $$( 1 + 5 T + 14 T^{2} + 55 T^{3} + 121 T^{4} )^{2}$$
$13$ $$( 1 - 25 T^{2} + 169 T^{4} )^{2}$$
$17$ $$( 1 - 8 T + 47 T^{2} - 136 T^{3} + 289 T^{4} )( 1 + 8 T + 47 T^{2} + 136 T^{3} + 289 T^{4} )$$
$19$ $$( 1 - T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2}$$
$23$ $$1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8}$$
$29$ $$( 1 + 29 T^{2} )^{4}$$
$31$ $$( 1 + 6 T + 5 T^{2} + 186 T^{3} + 961 T^{4} )^{2}$$
$37$ $$1 + 49 T^{2} + 1032 T^{4} + 67081 T^{6} + 1874161 T^{8}$$
$41$ $$( 1 + 9 T + 41 T^{2} )^{4}$$
$43$ $$( 1 + 14 T^{2} + 1849 T^{4} )^{2}$$
$47$ $$1 - 75 T^{2} + 3416 T^{4} - 165675 T^{6} + 4879681 T^{8}$$
$53$ $$1 + 105 T^{2} + 8216 T^{4} + 294945 T^{6} + 7890481 T^{8}$$
$59$ $$( 1 + 4 T - 43 T^{2} + 236 T^{3} + 3481 T^{4} )^{2}$$
$61$ $$( 1 - 2 T - 57 T^{2} - 122 T^{3} + 3721 T^{4} )^{2}$$
$67$ $$1 + 98 T^{2} + 5115 T^{4} + 439922 T^{6} + 20151121 T^{8}$$
$71$ $$( 1 - 2 T + 71 T^{2} )^{4}$$
$73$ $$1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8}$$
$79$ $$( 1 - 14 T + 117 T^{2} - 1106 T^{3} + 6241 T^{4} )^{2}$$
$83$ $$( 1 - 66 T^{2} + 6889 T^{4} )^{2}$$
$89$ $$( 1 - 10 T + 11 T^{2} - 890 T^{3} + 7921 T^{4} )^{2}$$
$97$ $$( 1 - 18 T + 97 T^{2} )^{2}( 1 + 18 T + 97 T^{2} )^{2}$$