# Properties

 Label 6160.2.a.bn Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 385) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( 1 - \beta_{1} ) q^{3} - q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( 1 - \beta_{1} ) q^{3} - q^{5} + q^{7} + ( -\beta_{1} + \beta_{2} ) q^{9} + q^{11} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{13} + ( -1 + \beta_{1} ) q^{15} + ( 1 - 3 \beta_{1} ) q^{17} + ( -2 + 2 \beta_{2} ) q^{19} + ( 1 - \beta_{1} ) q^{21} + ( 3 + \beta_{1} + \beta_{2} ) q^{23} + q^{25} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{27} + ( -4 + 2 \beta_{1} + 2 \beta_{2} ) q^{29} + ( 4 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( 1 - \beta_{1} ) q^{33} - q^{35} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{37} + ( 5 - 3 \beta_{1} + 3 \beta_{2} ) q^{39} -3 \beta_{2} q^{41} + ( 1 - \beta_{1} - 5 \beta_{2} ) q^{43} + ( \beta_{1} - \beta_{2} ) q^{45} + ( 7 - \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 7 - \beta_{1} + 3 \beta_{2} ) q^{51} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{53} - q^{55} + 2 \beta_{2} q^{57} + ( -6 + 4 \beta_{1} - \beta_{2} ) q^{59} + ( 4 - 2 \beta_{1} - 5 \beta_{2} ) q^{61} + ( -\beta_{1} + \beta_{2} ) q^{63} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{65} + ( 1 - \beta_{1} + 5 \beta_{2} ) q^{67} + ( 2 - 4 \beta_{1} ) q^{69} + ( 10 - 6 \beta_{1} - 2 \beta_{2} ) q^{71} + ( -1 + 7 \beta_{1} ) q^{73} + ( 1 - \beta_{1} ) q^{75} + q^{77} + ( -1 - 5 \beta_{1} - 5 \beta_{2} ) q^{79} + ( -2 + \beta_{1} - 3 \beta_{2} ) q^{81} + ( 2 + 4 \beta_{1} - 4 \beta_{2} ) q^{83} + ( -1 + 3 \beta_{1} ) q^{85} + ( -6 + 2 \beta_{1} ) q^{87} + ( 8 - 4 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{91} + ( 7 - 3 \beta_{1} + \beta_{2} ) q^{93} + ( 2 - 2 \beta_{2} ) q^{95} + ( -2 + 6 \beta_{1} + 6 \beta_{2} ) q^{97} + ( -\beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9} + O(q^{10})$$ $$3 q + 2 q^{3} - 3 q^{5} + 3 q^{7} - q^{9} + 3 q^{11} + 2 q^{13} - 2 q^{15} - 6 q^{19} + 2 q^{21} + 10 q^{23} + 3 q^{25} + 2 q^{27} - 10 q^{29} + 10 q^{31} + 2 q^{33} - 3 q^{35} - 16 q^{37} + 12 q^{39} + 2 q^{43} + q^{45} + 20 q^{47} + 3 q^{49} + 20 q^{51} - 12 q^{53} - 3 q^{55} - 14 q^{59} + 10 q^{61} - q^{63} - 2 q^{65} + 2 q^{67} + 2 q^{69} + 24 q^{71} + 4 q^{73} + 2 q^{75} + 3 q^{77} - 8 q^{79} - 5 q^{81} + 10 q^{83} - 16 q^{87} + 20 q^{89} + 2 q^{91} + 18 q^{93} + 6 q^{95} - q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 2.17009 0.311108 −1.48119
0 −1.17009 0 −1.00000 0 1.00000 0 −1.63090 0
1.2 0 0.688892 0 −1.00000 0 1.00000 0 −2.52543 0
1.3 0 2.48119 0 −1.00000 0 1.00000 0 3.15633 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.bn 3
4.b odd 2 1 385.2.a.f 3
12.b even 2 1 3465.2.a.bh 3
20.d odd 2 1 1925.2.a.v 3
20.e even 4 2 1925.2.b.n 6
28.d even 2 1 2695.2.a.g 3
44.c even 2 1 4235.2.a.q 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.f 3 4.b odd 2 1
1925.2.a.v 3 20.d odd 2 1
1925.2.b.n 6 20.e even 4 2
2695.2.a.g 3 28.d even 2 1
3465.2.a.bh 3 12.b even 2 1
4235.2.a.q 3 44.c even 2 1
6160.2.a.bn 3 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{3} - 2 T_{3}^{2} - 2 T_{3} + 2$$ $$T_{13}^{3} - 2 T_{13}^{2} - 22 T_{13} - 2$$ $$T_{17}^{3} - 30 T_{17} + 2$$ $$T_{19}^{3} + 6 T_{19}^{2} - 4 T_{19} - 8$$ $$T_{23}^{3} - 10 T_{23}^{2} + 28 T_{23} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$2 - 2 T - 2 T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-2 - 22 T - 2 T^{2} + T^{3}$$
$17$ $$2 - 30 T + T^{3}$$
$19$ $$-8 - 4 T + 6 T^{2} + T^{3}$$
$23$ $$-20 + 28 T - 10 T^{2} + T^{3}$$
$29$ $$-40 + 12 T + 10 T^{2} + T^{3}$$
$31$ $$26 + 20 T - 10 T^{2} + T^{3}$$
$37$ $$-100 + 52 T + 16 T^{2} + T^{3}$$
$41$ $$-54 - 36 T + T^{3}$$
$43$ $$-268 - 92 T - 2 T^{2} + T^{3}$$
$47$ $$-158 + 110 T - 20 T^{2} + T^{3}$$
$53$ $$4 + 20 T + 12 T^{2} + T^{3}$$
$59$ $$-74 + 14 T^{2} + T^{3}$$
$61$ $$-62 - 60 T - 10 T^{2} + T^{3}$$
$67$ $$172 - 112 T - 2 T^{2} + T^{3}$$
$71$ $$800 + 80 T - 24 T^{2} + T^{3}$$
$73$ $$190 - 158 T - 4 T^{2} + T^{3}$$
$79$ $$-244 - 112 T + 8 T^{2} + T^{3}$$
$83$ $$1096 - 116 T - 10 T^{2} + T^{3}$$
$89$ $$320 + 48 T - 20 T^{2} + T^{3}$$
$97$ $$-160 - 192 T + T^{3}$$