# Properties

 Label 6160.2.a.bf 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: 3.3.316.1 Defining polynomial: $$x^{3} - x^{2} - 4 x + 2$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 770) 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.

 $$f(q)$$ $$=$$ $$q + ( -1 - \beta_{2} ) q^{3} + q^{5} + q^{7} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 - \beta_{2} ) q^{3} + q^{5} + q^{7} + ( 3 - \beta_{1} + \beta_{2} ) q^{9} - q^{11} + ( \beta_{1} - \beta_{2} ) q^{13} + ( -1 - \beta_{2} ) q^{15} -2 \beta_{2} q^{17} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{19} + ( -1 - \beta_{2} ) q^{21} + ( -3 + \beta_{2} ) q^{23} + q^{25} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{27} + ( -1 - 3 \beta_{2} ) q^{29} + ( 4 - \beta_{1} - \beta_{2} ) q^{31} + ( 1 + \beta_{2} ) q^{33} + q^{35} + ( 1 - \beta_{2} ) q^{37} + ( 4 - 2 \beta_{1} + 2 \beta_{2} ) q^{39} + ( 3 - 2 \beta_{1} + 3 \beta_{2} ) q^{41} + ( -4 + 3 \beta_{1} - \beta_{2} ) q^{43} + ( 3 - \beta_{1} + \beta_{2} ) q^{45} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{47} + q^{49} + ( 10 - 2 \beta_{1} ) q^{51} + ( 5 + 3 \beta_{2} ) q^{53} - q^{55} + ( -6 + 3 \beta_{1} - 7 \beta_{2} ) q^{57} + ( 2 \beta_{1} - 2 \beta_{2} ) q^{59} + ( -2 - 2 \beta_{1} - 2 \beta_{2} ) q^{61} + ( 3 - \beta_{1} + \beta_{2} ) q^{63} + ( \beta_{1} - \beta_{2} ) q^{65} + 4 \beta_{2} q^{67} + ( -2 + \beta_{1} + 3 \beta_{2} ) q^{69} + ( 10 - 2 \beta_{1} ) q^{71} + ( -4 + 2 \beta_{2} ) q^{73} + ( -1 - \beta_{2} ) q^{75} - q^{77} + ( -7 + 2 \beta_{1} - \beta_{2} ) q^{79} + ( 3 - \beta_{1} + 5 \beta_{2} ) q^{81} + ( 2 - 2 \beta_{2} ) q^{83} -2 \beta_{2} q^{85} + ( 16 - 3 \beta_{1} + \beta_{2} ) q^{87} + ( 4 + 2 \beta_{1} - 4 \beta_{2} ) q^{89} + ( \beta_{1} - \beta_{2} ) q^{91} + ( 2 - 6 \beta_{2} ) q^{93} + ( 3 - 2 \beta_{1} + \beta_{2} ) q^{95} + ( -1 + 2 \beta_{1} + 3 \beta_{2} ) q^{97} + ( -3 + \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} + O(q^{10})$$ $$3 q - 2 q^{3} + 3 q^{5} + 3 q^{7} + 7 q^{9} - 3 q^{11} + 2 q^{13} - 2 q^{15} + 2 q^{17} + 6 q^{19} - 2 q^{21} - 10 q^{23} + 3 q^{25} - 8 q^{27} + 12 q^{31} + 2 q^{33} + 3 q^{35} + 4 q^{37} + 8 q^{39} + 4 q^{41} - 8 q^{43} + 7 q^{45} + 3 q^{49} + 28 q^{51} + 12 q^{53} - 3 q^{55} - 8 q^{57} + 4 q^{59} - 6 q^{61} + 7 q^{63} + 2 q^{65} - 4 q^{67} - 8 q^{69} + 28 q^{71} - 14 q^{73} - 2 q^{75} - 3 q^{77} - 18 q^{79} + 3 q^{81} + 8 q^{83} + 2 q^{85} + 44 q^{87} + 18 q^{89} + 2 q^{91} + 12 q^{93} + 6 q^{95} - 4 q^{97} - 7 q^{99} + O(q^{100})$$

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

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

## 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
 −1.81361 2.34292 0.470683
0 −3.10278 0 1.00000 0 1.00000 0 6.62721 0
1.2 0 −1.14637 0 1.00000 0 1.00000 0 −1.68585 0
1.3 0 2.24914 0 1.00000 0 1.00000 0 2.05863 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.bf 3
4.b odd 2 1 770.2.a.m 3
12.b even 2 1 6930.2.a.ce 3
20.d odd 2 1 3850.2.a.bt 3
20.e even 4 2 3850.2.c.ba 6
28.d even 2 1 5390.2.a.ca 3
44.c even 2 1 8470.2.a.ci 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.m 3 4.b odd 2 1
3850.2.a.bt 3 20.d odd 2 1
3850.2.c.ba 6 20.e even 4 2
5390.2.a.ca 3 28.d even 2 1
6160.2.a.bf 3 1.a even 1 1 trivial
6930.2.a.ce 3 12.b even 2 1
8470.2.a.ci 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6160))$$:

 $$T_{3}^{3} + 2 T_{3}^{2} - 6 T_{3} - 8$$ $$T_{13}^{3} - 2 T_{13}^{2} - 16 T_{13} + 16$$ $$T_{17}^{3} - 2 T_{17}^{2} - 28 T_{17} - 8$$ $$T_{19}^{3} - 6 T_{19}^{2} - 46 T_{19} + 232$$ $$T_{23}^{3} + 10 T_{23}^{2} + 26 T_{23} + 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-8 - 6 T + 2 T^{2} + T^{3}$$
$5$ $$( -1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$16 - 16 T - 2 T^{2} + T^{3}$$
$17$ $$-8 - 28 T - 2 T^{2} + T^{3}$$
$19$ $$232 - 46 T - 6 T^{2} + T^{3}$$
$23$ $$16 + 26 T + 10 T^{2} + T^{3}$$
$29$ $$-92 - 66 T + T^{3}$$
$31$ $$32 + 20 T - 12 T^{2} + T^{3}$$
$37$ $$4 - 2 T - 4 T^{2} + T^{3}$$
$41$ $$-164 - 90 T - 4 T^{2} + T^{3}$$
$43$ $$-848 - 108 T + 8 T^{2} + T^{3}$$
$47$ $$128 - 112 T + T^{3}$$
$53$ $$292 - 18 T - 12 T^{2} + T^{3}$$
$59$ $$128 - 64 T - 4 T^{2} + T^{3}$$
$61$ $$-344 - 100 T + 6 T^{2} + T^{3}$$
$67$ $$64 - 112 T + 4 T^{2} + T^{3}$$
$71$ $$-64 + 200 T - 28 T^{2} + T^{3}$$
$73$ $$-8 + 36 T + 14 T^{2} + T^{3}$$
$79$ $$-256 + 50 T + 18 T^{2} + T^{3}$$
$83$ $$32 - 8 T - 8 T^{2} + T^{3}$$
$89$ $$1208 - 28 T - 18 T^{2} + T^{3}$$
$97$ $$316 - 154 T + 4 T^{2} + T^{3}$$