# Properties

 Label 6160.2.a.j Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $1$ 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: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 770) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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.j 1
4.b odd 2 1 770.2.a.f 1
12.b even 2 1 6930.2.a.o 1
20.d odd 2 1 3850.2.a.k 1
20.e even 4 2 3850.2.c.b 2
28.d even 2 1 5390.2.a.bj 1
44.c even 2 1 8470.2.a.c 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.f 1 4.b odd 2 1
3850.2.a.k 1 20.d odd 2 1
3850.2.c.b 2 20.e even 4 2
5390.2.a.bj 1 28.d even 2 1
6160.2.a.j 1 1.a even 1 1 trivial
6930.2.a.o 1 12.b even 2 1
8470.2.a.c 1 44.c even 2 1

## Hecke kernels

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T$$
$3$ $$-2 + T$$
$5$ $$1 + T$$
$7$ $$1 + T$$
$11$ $$-1 + T$$
$13$ $$4 + T$$
$17$ $$T$$
$19$ $$-4 + T$$
$23$ $$T$$
$29$ $$6 + T$$
$31$ $$-10 + T$$
$37$ $$-2 + T$$
$41$ $$12 + T$$
$43$ $$-4 + T$$
$47$ $$6 + T$$
$53$ $$6 + T$$
$59$ $$-6 + T$$
$61$ $$4 + T$$
$67$ $$-4 + T$$
$71$ $$12 + T$$
$73$ $$4 + T$$
$79$ $$8 + T$$
$83$ $$12 + T$$
$89$ $$-18 + T$$
$97$ $$10 + T$$