# Properties

 Label 1680.2.f.d Level $1680$ Weight $2$ Character orbit 1680.f 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.f (of order $$2$$, degree $$1$$, 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 420) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( 2 - \zeta_{6} ) q^{3} - q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} +O(q^{10})$$ $$q + ( 2 - \zeta_{6} ) q^{3} - q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} + ( 3 - 3 \zeta_{6} ) q^{9} + ( 3 - 6 \zeta_{6} ) q^{11} + ( -1 + 2 \zeta_{6} ) q^{13} + ( -2 + \zeta_{6} ) q^{15} + 3 q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( -4 + 5 \zeta_{6} ) q^{21} + q^{25} + ( 3 - 6 \zeta_{6} ) q^{27} + ( -3 + 6 \zeta_{6} ) q^{29} + ( 6 - 12 \zeta_{6} ) q^{31} -9 \zeta_{6} q^{33} + ( 3 - 2 \zeta_{6} ) q^{35} -8 q^{37} + 3 \zeta_{6} q^{39} + 6 q^{41} -10 q^{43} + ( -3 + 3 \zeta_{6} ) q^{45} -3 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 6 - 3 \zeta_{6} ) q^{51} + ( 6 - 12 \zeta_{6} ) q^{53} + ( -3 + 6 \zeta_{6} ) q^{55} -6 \zeta_{6} q^{57} -6 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} + ( -3 + 9 \zeta_{6} ) q^{63} + ( 1 - 2 \zeta_{6} ) q^{65} -2 q^{67} + ( 6 - 12 \zeta_{6} ) q^{71} + ( 4 - 8 \zeta_{6} ) q^{73} + ( 2 - \zeta_{6} ) q^{75} + ( 3 + 12 \zeta_{6} ) q^{77} + 13 q^{79} -9 \zeta_{6} q^{81} + 12 q^{83} -3 q^{85} + 9 \zeta_{6} q^{87} + ( -1 - 4 \zeta_{6} ) q^{91} -18 \zeta_{6} q^{93} + ( -2 + 4 \zeta_{6} ) q^{95} + ( -1 + 2 \zeta_{6} ) q^{97} + ( -9 - 9 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 3q^{3} - 2q^{5} - 4q^{7} + 3q^{9} + O(q^{10})$$ $$2q + 3q^{3} - 2q^{5} - 4q^{7} + 3q^{9} - 3q^{15} + 6q^{17} - 3q^{21} + 2q^{25} - 9q^{33} + 4q^{35} - 16q^{37} + 3q^{39} + 12q^{41} - 20q^{43} - 3q^{45} - 6q^{47} + 2q^{49} + 9q^{51} - 6q^{57} - 12q^{59} + 3q^{63} - 4q^{67} + 3q^{75} + 18q^{77} + 26q^{79} - 9q^{81} + 24q^{83} - 6q^{85} + 9q^{87} - 6q^{91} - 18q^{93} - 27q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$241$$ $$337$$ $$421$$ $$1121$$ $$1471$$ $$\chi(n)$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
881.1
 0.5 + 0.866025i 0.5 − 0.866025i
0 1.50000 0.866025i 0 −1.00000 0 −2.00000 + 1.73205i 0 1.50000 2.59808i 0
881.2 0 1.50000 + 0.866025i 0 −1.00000 0 −2.00000 1.73205i 0 1.50000 + 2.59808i 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
21.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.f.d 2
3.b odd 2 1 1680.2.f.a 2
4.b odd 2 1 420.2.d.a 2
7.b odd 2 1 1680.2.f.a 2
12.b even 2 1 420.2.d.b yes 2
20.d odd 2 1 2100.2.d.e 2
20.e even 4 2 2100.2.f.c 4
21.c even 2 1 inner 1680.2.f.d 2
28.d even 2 1 420.2.d.b yes 2
60.h even 2 1 2100.2.d.a 2
60.l odd 4 2 2100.2.f.d 4
84.h odd 2 1 420.2.d.a 2
140.c even 2 1 2100.2.d.a 2
140.j odd 4 2 2100.2.f.d 4
420.o odd 2 1 2100.2.d.e 2
420.w even 4 2 2100.2.f.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
420.2.d.a 2 4.b odd 2 1
420.2.d.a 2 84.h odd 2 1
420.2.d.b yes 2 12.b even 2 1
420.2.d.b yes 2 28.d even 2 1
1680.2.f.a 2 3.b odd 2 1
1680.2.f.a 2 7.b odd 2 1
1680.2.f.d 2 1.a even 1 1 trivial
1680.2.f.d 2 21.c even 2 1 inner
2100.2.d.a 2 60.h even 2 1
2100.2.d.a 2 140.c even 2 1
2100.2.d.e 2 20.d odd 2 1
2100.2.d.e 2 420.o odd 2 1
2100.2.f.c 4 20.e even 4 2
2100.2.f.c 4 420.w even 4 2
2100.2.f.d 4 60.l odd 4 2
2100.2.f.d 4 140.j odd 4 2

## Hecke kernels

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

 $$T_{11}^{2} + 27$$ $$T_{17} - 3$$ $$T_{41} - 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$3 - 3 T + T^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$7 + 4 T + T^{2}$$
$11$ $$27 + T^{2}$$
$13$ $$3 + T^{2}$$
$17$ $$( -3 + T )^{2}$$
$19$ $$12 + T^{2}$$
$23$ $$T^{2}$$
$29$ $$27 + T^{2}$$
$31$ $$108 + T^{2}$$
$37$ $$( 8 + T )^{2}$$
$41$ $$( -6 + T )^{2}$$
$43$ $$( 10 + T )^{2}$$
$47$ $$( 3 + T )^{2}$$
$53$ $$108 + T^{2}$$
$59$ $$( 6 + T )^{2}$$
$61$ $$48 + T^{2}$$
$67$ $$( 2 + T )^{2}$$
$71$ $$108 + T^{2}$$
$73$ $$48 + T^{2}$$
$79$ $$( -13 + T )^{2}$$
$83$ $$( -12 + T )^{2}$$
$89$ $$T^{2}$$
$97$ $$3 + T^{2}$$