# Properties

 Label 6160.2.a.b Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6160 = 2^{4} \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6160.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$49.1878476451$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1540) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - 2 q^{3} + q^{5} - q^{7} + q^{9} + O(q^{10})$$ $$q - 2 q^{3} + q^{5} - q^{7} + q^{9} + q^{11} + 4 q^{13} - 2 q^{15} + 8 q^{17} + 4 q^{19} + 2 q^{21} - 2 q^{23} + q^{25} + 4 q^{27} - 6 q^{29} + 4 q^{31} - 2 q^{33} - q^{35} + 10 q^{37} - 8 q^{39} + 2 q^{41} - 4 q^{43} + q^{45} - 2 q^{47} + q^{49} - 16 q^{51} + 2 q^{53} + q^{55} - 8 q^{57} - 2 q^{61} - q^{63} + 4 q^{65} - 14 q^{67} + 4 q^{69} + 4 q^{71} - 16 q^{73} - 2 q^{75} - q^{77} + 8 q^{79} - 11 q^{81} + 4 q^{83} + 8 q^{85} + 12 q^{87} - 2 q^{89} - 4 q^{91} - 8 q^{93} + 4 q^{95} - 2 q^{97} + q^{99} + O(q^{100})$$

## 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 −2.00000 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

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$-1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 6160.2.a.b 1
4.b odd 2 1 1540.2.a.c 1
20.d odd 2 1 7700.2.a.b 1
20.e even 4 2 7700.2.e.b 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.c 1 4.b odd 2 1
6160.2.a.b 1 1.a even 1 1 trivial
7700.2.a.b 1 20.d odd 2 1
7700.2.e.b 2 20.e even 4 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6160))$$:

 $$T_{3} + 2$$ $$T_{13} - 4$$ $$T_{17} - 8$$ $$T_{19} - 4$$ $$T_{23} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$2 + T$$
$5$ $$-1 + T$$
$7$ $$1 + T$$
$11$ $$-1 + T$$
$13$ $$-4 + T$$
$17$ $$-8 + T$$
$19$ $$-4 + T$$
$23$ $$2 + T$$
$29$ $$6 + T$$
$31$ $$-4 + T$$
$37$ $$-10 + T$$
$41$ $$-2 + T$$
$43$ $$4 + T$$
$47$ $$2 + T$$
$53$ $$-2 + T$$
$59$ $$T$$
$61$ $$2 + T$$
$67$ $$14 + T$$
$71$ $$-4 + T$$
$73$ $$16 + T$$
$79$ $$-8 + T$$
$83$ $$-4 + T$$
$89$ $$2 + T$$
$97$ $$2 + T$$