# Properties

 Label 1680.2.a.g Level 1680 Weight 2 Character orbit 1680.a Self dual yes Analytic conductor 13.415 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.4148675396$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 210) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$q - q^{3} + q^{5} - q^{7} + q^{9} + 2q^{13} - q^{15} - 6q^{17} - 8q^{19} + q^{21} + q^{25} - q^{27} + 6q^{29} + 4q^{31} - q^{35} - 10q^{37} - 2q^{39} - 6q^{41} + 4q^{43} + q^{45} + q^{49} + 6q^{51} - 6q^{53} + 8q^{57} + 12q^{59} - 10q^{61} - q^{63} + 2q^{65} + 4q^{67} - 12q^{71} - 10q^{73} - q^{75} - 8q^{79} + q^{81} - 12q^{83} - 6q^{85} - 6q^{87} - 6q^{89} - 2q^{91} - 4q^{93} - 8q^{95} - 10q^{97} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0
0 −1.00000 0 1.00000 0 −1.00000 0 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1680.2.a.g 1
3.b odd 2 1 5040.2.a.g 1
4.b odd 2 1 210.2.a.b 1
5.b even 2 1 8400.2.a.cm 1
8.b even 2 1 6720.2.a.bi 1
8.d odd 2 1 6720.2.a.n 1
12.b even 2 1 630.2.a.h 1
20.d odd 2 1 1050.2.a.k 1
20.e even 4 2 1050.2.g.c 2
28.d even 2 1 1470.2.a.b 1
28.f even 6 2 1470.2.i.s 2
28.g odd 6 2 1470.2.i.l 2
60.h even 2 1 3150.2.a.f 1
60.l odd 4 2 3150.2.g.i 2
84.h odd 2 1 4410.2.a.bi 1
140.c even 2 1 7350.2.a.cs 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
210.2.a.b 1 4.b odd 2 1
630.2.a.h 1 12.b even 2 1
1050.2.a.k 1 20.d odd 2 1
1050.2.g.c 2 20.e even 4 2
1470.2.a.b 1 28.d even 2 1
1470.2.i.l 2 28.g odd 6 2
1470.2.i.s 2 28.f even 6 2
1680.2.a.g 1 1.a even 1 1 trivial
3150.2.a.f 1 60.h even 2 1
3150.2.g.i 2 60.l odd 4 2
4410.2.a.bi 1 84.h odd 2 1
5040.2.a.g 1 3.b odd 2 1
6720.2.a.n 1 8.d odd 2 1
6720.2.a.bi 1 8.b even 2 1
7350.2.a.cs 1 140.c even 2 1
8400.2.a.cm 1 5.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$7$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1680))$$:

 $$T_{11}$$ $$T_{13} - 2$$ $$T_{17} + 6$$ $$T_{19} + 8$$

## Hecke characteristic polynomials

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