# Properties

 Label 6160.2.a.bm 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: $$2$$ Twist minimal: no (minimal twist has level 3080) 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.

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

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

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

## 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 −1.48119 0.311108
0 −1.70928 0 −1.00000 0 −1.00000 0 −0.0783777 0
1.2 0 0.806063 0 −1.00000 0 −1.00000 0 −2.35026 0
1.3 0 2.90321 0 −1.00000 0 −1.00000 0 5.42864 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.bm 3
4.b odd 2 1 3080.2.a.k 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3080.2.a.k 3 4.b odd 2 1
6160.2.a.bm 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6160))$$:

 $$T_{3}^{3} - 2 T_{3}^{2} - 4 T_{3} + 4$$ $$T_{13}^{3} + 4 T_{13}^{2} - 4$$ $$T_{17}^{3} - 4 T_{17}^{2} + 4$$ $$T_{19}^{3} - 8 T_{19}^{2} - 16 T_{19} + 160$$ $$T_{23}^{3} + 4 T_{23}^{2} - 8 T_{23} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$4 - 4 T - 2 T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-4 + 4 T^{2} + T^{3}$$
$17$ $$4 - 4 T^{2} + T^{3}$$
$19$ $$160 - 16 T - 8 T^{2} + T^{3}$$
$23$ $$-16 - 8 T + 4 T^{2} + T^{3}$$
$29$ $$-8 - 20 T + 2 T^{2} + T^{3}$$
$31$ $$4 - 16 T + 6 T^{2} + T^{3}$$
$37$ $$-40 - 4 T + 6 T^{2} + T^{3}$$
$41$ $$52 - 28 T + T^{3}$$
$43$ $$304 - 40 T - 8 T^{2} + T^{3}$$
$47$ $$116 - 76 T + 2 T^{2} + T^{3}$$
$53$ $$184 - 52 T - 2 T^{2} + T^{3}$$
$59$ $$-68 + 56 T - 14 T^{2} + T^{3}$$
$61$ $$-540 - 36 T + 12 T^{2} + T^{3}$$
$67$ $$208 - 8 T - 16 T^{2} + T^{3}$$
$71$ $$-32 - 16 T + 4 T^{2} + T^{3}$$
$73$ $$-100 - 72 T - 8 T^{2} + T^{3}$$
$79$ $$496 - 136 T + T^{3}$$
$83$ $$-32 + 48 T - 16 T^{2} + T^{3}$$
$89$ $$40 - 84 T + 14 T^{2} + T^{3}$$
$97$ $$632 - 100 T - 6 T^{2} + T^{3}$$