# Properties

 Label 1680.2.f.c Level $1680$ Weight $2$ Character orbit 1680.f Analytic conductor $13.415$ Analytic rank $1$ 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: $$1$$ 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: $$2$$ Twist minimal: no (minimal twist has level 105) 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.

 $$f(q)$$ $$=$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} +O(q^{10})$$ $$q + ( 1 - 2 \zeta_{6} ) q^{3} + q^{5} + ( -3 + 2 \zeta_{6} ) q^{7} -3 q^{9} + ( 2 - 4 \zeta_{6} ) q^{11} + ( 1 - 2 \zeta_{6} ) q^{15} -6 q^{17} + ( 2 - 4 \zeta_{6} ) q^{19} + ( 1 + 4 \zeta_{6} ) q^{21} + ( -2 + 4 \zeta_{6} ) q^{23} + q^{25} + ( -3 + 6 \zeta_{6} ) q^{27} + ( -4 + 8 \zeta_{6} ) q^{29} + ( -2 + 4 \zeta_{6} ) q^{31} -6 q^{33} + ( -3 + 2 \zeta_{6} ) q^{35} -2 q^{37} -6 q^{41} + 8 q^{43} -3 q^{45} -12 q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( -6 + 12 \zeta_{6} ) q^{51} + ( 2 - 4 \zeta_{6} ) q^{55} -6 q^{57} -12 q^{59} + ( 4 - 8 \zeta_{6} ) q^{61} + ( 9 - 6 \zeta_{6} ) q^{63} -8 q^{67} + 6 q^{69} + ( -2 + 4 \zeta_{6} ) q^{71} + ( -4 + 8 \zeta_{6} ) q^{73} + ( 1 - 2 \zeta_{6} ) q^{75} + ( 2 + 8 \zeta_{6} ) q^{77} -8 q^{79} + 9 q^{81} -6 q^{85} + 12 q^{87} -6 q^{89} + 6 q^{93} + ( 2 - 4 \zeta_{6} ) q^{95} + ( -4 + 8 \zeta_{6} ) q^{97} + ( -6 + 12 \zeta_{6} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + 2q^{5} - 4q^{7} - 6q^{9} + O(q^{10})$$ $$2q + 2q^{5} - 4q^{7} - 6q^{9} - 12q^{17} + 6q^{21} + 2q^{25} - 12q^{33} - 4q^{35} - 4q^{37} - 12q^{41} + 16q^{43} - 6q^{45} - 24q^{47} + 2q^{49} - 12q^{57} - 24q^{59} + 12q^{63} - 16q^{67} + 12q^{69} + 12q^{77} - 16q^{79} + 18q^{81} - 12q^{85} + 24q^{87} - 12q^{89} + 12q^{93} + O(q^{100})$$

## 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 1.73205i 0 1.00000 0 −2.00000 + 1.73205i 0 −3.00000 0
881.2 0 1.73205i 0 1.00000 0 −2.00000 1.73205i 0 −3.00000 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.c 2
3.b odd 2 1 1680.2.f.b 2
4.b odd 2 1 105.2.b.b yes 2
7.b odd 2 1 1680.2.f.b 2
12.b even 2 1 105.2.b.a 2
20.d odd 2 1 525.2.b.b 2
20.e even 4 2 525.2.g.c 4
21.c even 2 1 inner 1680.2.f.c 2
28.d even 2 1 105.2.b.a 2
28.f even 6 1 735.2.s.b 2
28.f even 6 1 735.2.s.d 2
28.g odd 6 1 735.2.s.a 2
28.g odd 6 1 735.2.s.f 2
60.h even 2 1 525.2.b.a 2
60.l odd 4 2 525.2.g.b 4
84.h odd 2 1 105.2.b.b yes 2
84.j odd 6 1 735.2.s.a 2
84.j odd 6 1 735.2.s.f 2
84.n even 6 1 735.2.s.b 2
84.n even 6 1 735.2.s.d 2
140.c even 2 1 525.2.b.a 2
140.j odd 4 2 525.2.g.b 4
420.o odd 2 1 525.2.b.b 2
420.w even 4 2 525.2.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.a 2 12.b even 2 1
105.2.b.a 2 28.d even 2 1
105.2.b.b yes 2 4.b odd 2 1
105.2.b.b yes 2 84.h odd 2 1
525.2.b.a 2 60.h even 2 1
525.2.b.a 2 140.c even 2 1
525.2.b.b 2 20.d odd 2 1
525.2.b.b 2 420.o odd 2 1
525.2.g.b 4 60.l odd 4 2
525.2.g.b 4 140.j odd 4 2
525.2.g.c 4 20.e even 4 2
525.2.g.c 4 420.w even 4 2
735.2.s.a 2 28.g odd 6 1
735.2.s.a 2 84.j odd 6 1
735.2.s.b 2 28.f even 6 1
735.2.s.b 2 84.n even 6 1
735.2.s.d 2 28.f even 6 1
735.2.s.d 2 84.n even 6 1
735.2.s.f 2 28.g odd 6 1
735.2.s.f 2 84.j odd 6 1
1680.2.f.b 2 3.b odd 2 1
1680.2.f.b 2 7.b odd 2 1
1680.2.f.c 2 1.a even 1 1 trivial
1680.2.f.c 2 21.c even 2 1 inner

## Hecke kernels

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

 $$T_{11}^{2} + 12$$ $$T_{17} + 6$$ $$T_{41} + 6$$

## Hecke characteristic polynomials

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