# Properties

 Label 1680.2.f.g Level $1680$ Weight $2$ Character orbit 1680.f 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.f (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.4148675396$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\sqrt{-3}, \sqrt{-11})$$ Defining polynomial: $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{3} + q^{5} + ( 2 + \beta_{2} ) q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{3} + q^{5} + ( 2 + \beta_{2} ) q^{7} + ( 1 + \beta_{2} - \beta_{3} ) q^{9} + ( -\beta_{1} - \beta_{3} ) q^{11} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{13} -\beta_{1} q^{15} + ( 2 - \beta_{1} + \beta_{3} ) q^{17} -2 \beta_{2} q^{19} + ( -1 - 3 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{21} + ( 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} ) q^{23} + q^{25} + ( -4 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{27} + ( \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{29} + 2 \beta_{2} q^{31} + ( -2 + \beta_{2} - \beta_{3} ) q^{33} + ( 2 + \beta_{2} ) q^{35} + ( -2 + 2 \beta_{1} - 2 \beta_{3} ) q^{37} + ( -4 + 2 \beta_{1} + 5 \beta_{2} + \beta_{3} ) q^{39} -6 q^{41} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{43} + ( 1 + \beta_{2} - \beta_{3} ) q^{45} + ( 4 + \beta_{1} - \beta_{3} ) q^{47} + ( 1 + 4 \beta_{2} ) q^{49} + ( 4 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{51} + ( -2 \beta_{1} - 4 \beta_{2} - 2 \beta_{3} ) q^{53} + ( -\beta_{1} - \beta_{3} ) q^{55} + ( 2 + 2 \beta_{1} + 2 \beta_{2} + 4 \beta_{3} ) q^{57} + ( 4 - 2 \beta_{1} + 2 \beta_{3} ) q^{59} -4 \beta_{2} q^{61} + ( -2 + 2 \beta_{1} + 4 \beta_{2} - \beta_{3} ) q^{63} + ( -3 \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{65} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{67} + ( 6 + 2 \beta_{1} + 6 \beta_{3} ) q^{69} + ( -4 \beta_{1} - 6 \beta_{2} - 4 \beta_{3} ) q^{71} -4 \beta_{2} q^{73} -\beta_{1} q^{75} + ( -2 - \beta_{1} - 3 \beta_{3} ) q^{77} + ( -\beta_{1} + \beta_{3} ) q^{79} + ( -3 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( 8 - 4 \beta_{1} + 4 \beta_{3} ) q^{83} + ( 2 - \beta_{1} + \beta_{3} ) q^{85} + ( -2 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} ) q^{87} + ( 10 - 2 \beta_{1} + 2 \beta_{3} ) q^{89} + ( -3 \beta_{1} - 4 \beta_{2} - 9 \beta_{3} ) q^{91} + ( -2 - 2 \beta_{1} - 2 \beta_{2} - 4 \beta_{3} ) q^{93} -2 \beta_{2} q^{95} + ( -3 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} ) q^{97} + ( -4 + \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - q^{3} + 4q^{5} + 8q^{7} + 5q^{9} + O(q^{10})$$ $$4q - q^{3} + 4q^{5} + 8q^{7} + 5q^{9} - q^{15} + 6q^{17} - 5q^{21} + 4q^{25} - 16q^{27} - 7q^{33} + 8q^{35} - 4q^{37} - 15q^{39} - 24q^{41} + 4q^{43} + 5q^{45} + 18q^{47} + 4q^{49} + 15q^{51} + 6q^{57} + 12q^{59} - 5q^{63} + 4q^{67} + 20q^{69} - q^{75} - 6q^{77} - 2q^{79} - 7q^{81} + 24q^{83} + 6q^{85} + q^{87} + 36q^{89} + 6q^{91} - 6q^{93} - 13q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 2 x^{2} - 3 x + 9$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} + 2 \nu^{2} - 2 \nu - 6$$$$)/3$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 2 \nu - 3$$$$)/3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 1$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 4$$

## 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
 1.68614 + 0.396143i 1.68614 − 0.396143i −1.18614 + 1.26217i −1.18614 − 1.26217i
0 −1.68614 0.396143i 0 1.00000 0 2.00000 + 1.73205i 0 2.68614 + 1.33591i 0
881.2 0 −1.68614 + 0.396143i 0 1.00000 0 2.00000 1.73205i 0 2.68614 1.33591i 0
881.3 0 1.18614 1.26217i 0 1.00000 0 2.00000 1.73205i 0 −0.186141 2.99422i 0
881.4 0 1.18614 + 1.26217i 0 1.00000 0 2.00000 + 1.73205i 0 −0.186141 + 2.99422i 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.g 4
3.b odd 2 1 1680.2.f.h 4
4.b odd 2 1 105.2.b.d yes 4
7.b odd 2 1 1680.2.f.h 4
12.b even 2 1 105.2.b.c 4
20.d odd 2 1 525.2.b.e 4
20.e even 4 2 525.2.g.d 8
21.c even 2 1 inner 1680.2.f.g 4
28.d even 2 1 105.2.b.c 4
28.f even 6 1 735.2.s.h 4
28.f even 6 1 735.2.s.i 4
28.g odd 6 1 735.2.s.g 4
28.g odd 6 1 735.2.s.j 4
60.h even 2 1 525.2.b.g 4
60.l odd 4 2 525.2.g.e 8
84.h odd 2 1 105.2.b.d yes 4
84.j odd 6 1 735.2.s.g 4
84.j odd 6 1 735.2.s.j 4
84.n even 6 1 735.2.s.h 4
84.n even 6 1 735.2.s.i 4
140.c even 2 1 525.2.b.g 4
140.j odd 4 2 525.2.g.e 8
420.o odd 2 1 525.2.b.e 4
420.w even 4 2 525.2.g.d 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
105.2.b.c 4 12.b even 2 1
105.2.b.c 4 28.d even 2 1
105.2.b.d yes 4 4.b odd 2 1
105.2.b.d yes 4 84.h odd 2 1
525.2.b.e 4 20.d odd 2 1
525.2.b.e 4 420.o odd 2 1
525.2.b.g 4 60.h even 2 1
525.2.b.g 4 140.c even 2 1
525.2.g.d 8 20.e even 4 2
525.2.g.d 8 420.w even 4 2
525.2.g.e 8 60.l odd 4 2
525.2.g.e 8 140.j odd 4 2
735.2.s.g 4 28.g odd 6 1
735.2.s.g 4 84.j odd 6 1
735.2.s.h 4 28.f even 6 1
735.2.s.h 4 84.n even 6 1
735.2.s.i 4 28.f even 6 1
735.2.s.i 4 84.n even 6 1
735.2.s.j 4 28.g odd 6 1
735.2.s.j 4 84.j odd 6 1
1680.2.f.g 4 1.a even 1 1 trivial
1680.2.f.g 4 21.c even 2 1 inner
1680.2.f.h 4 3.b odd 2 1
1680.2.f.h 4 7.b odd 2 1

## Hecke kernels

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

 $$T_{11}^{4} + 7 T_{11}^{2} + 4$$ $$T_{17}^{2} - 3 T_{17} - 6$$ $$T_{41} + 6$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$9 + 3 T - 2 T^{2} + T^{3} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$( 7 - 4 T + T^{2} )^{2}$$
$11$ $$4 + 7 T^{2} + T^{4}$$
$13$ $$576 + 51 T^{2} + T^{4}$$
$17$ $$( -6 - 3 T + T^{2} )^{2}$$
$19$ $$( 12 + T^{2} )^{2}$$
$23$ $$256 + 76 T^{2} + T^{4}$$
$29$ $$16 + 19 T^{2} + T^{4}$$
$31$ $$( 12 + T^{2} )^{2}$$
$37$ $$( -32 + 2 T + T^{2} )^{2}$$
$41$ $$( 6 + T )^{4}$$
$43$ $$( -32 - 2 T + T^{2} )^{2}$$
$47$ $$( 12 - 9 T + T^{2} )^{2}$$
$53$ $$256 + 76 T^{2} + T^{4}$$
$59$ $$( -24 - 6 T + T^{2} )^{2}$$
$61$ $$( 48 + T^{2} )^{2}$$
$67$ $$( -32 - 2 T + T^{2} )^{2}$$
$71$ $$16 + 184 T^{2} + T^{4}$$
$73$ $$( 48 + T^{2} )^{2}$$
$79$ $$( -8 + T + T^{2} )^{2}$$
$83$ $$( -96 - 12 T + T^{2} )^{2}$$
$89$ $$( 48 - 18 T + T^{2} )^{2}$$
$97$ $$144 + 123 T^{2} + T^{4}$$