# Properties

 Label 6160.2.a.bd Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $1$ 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: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.404.1 Defining polynomial: $$x^{3} - x^{2} - 5 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{1} ) q^{3} - q^{5} + q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{1} ) q^{3} - q^{5} + q^{7} + ( 1 - \beta_{1} + \beta_{2} ) q^{9} + q^{11} + ( -1 + \beta_{1} ) q^{13} + ( 1 - \beta_{1} ) q^{15} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{17} + ( -2 - 2 \beta_{1} + 2 \beta_{2} ) q^{19} + ( -1 + \beta_{1} ) q^{21} + ( -\beta_{1} - \beta_{2} ) q^{23} + q^{25} -2 \beta_{2} q^{27} -2 q^{29} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{31} + ( -1 + \beta_{1} ) q^{33} - q^{35} + ( -2 + \beta_{1} + \beta_{2} ) q^{37} + ( 4 - \beta_{1} + \beta_{2} ) q^{39} + ( -3 - 2 \beta_{1} - \beta_{2} ) q^{41} + ( 2 + \beta_{1} + 3 \beta_{2} ) q^{43} + ( -1 + \beta_{1} - \beta_{2} ) q^{45} + ( -3 - \beta_{1} - 2 \beta_{2} ) q^{47} + q^{49} + ( 2 - 5 \beta_{1} + 3 \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} - \beta_{2} ) q^{53} - q^{55} + ( -2 + 2 \beta_{1} - 4 \beta_{2} ) q^{57} + ( 1 - 5 \beta_{2} ) q^{59} + ( -7 - 4 \beta_{1} + 3 \beta_{2} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} ) q^{63} + ( 1 - \beta_{1} ) q^{65} + ( 2 + 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -4 - 2 \beta_{1} ) q^{69} + ( 4 - 4 \beta_{1} + 4 \beta_{2} ) q^{71} + ( -7 - \beta_{1} + 2 \beta_{2} ) q^{73} + ( -1 + \beta_{1} ) q^{75} + q^{77} + ( 10 + 3 \beta_{1} - \beta_{2} ) q^{79} + ( -5 - \beta_{1} - \beta_{2} ) q^{81} + ( -4 - 2 \beta_{2} ) q^{83} + ( 1 - \beta_{1} + 2 \beta_{2} ) q^{85} + ( 2 - 2 \beta_{1} ) q^{87} + ( 2 - 2 \beta_{1} ) q^{89} + ( -1 + \beta_{1} ) q^{91} + ( -10 + \beta_{1} - \beta_{2} ) q^{93} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} + ( 2 - 6 \beta_{1} + 4 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10})$$ $$3 q - 2 q^{3} - 3 q^{5} + 3 q^{7} + 3 q^{9} + 3 q^{11} - 2 q^{13} + 2 q^{15} - 4 q^{17} - 6 q^{19} - 2 q^{21} - 2 q^{23} + 3 q^{25} - 2 q^{27} - 6 q^{29} + 6 q^{31} - 2 q^{33} - 3 q^{35} - 4 q^{37} + 12 q^{39} - 12 q^{41} + 10 q^{43} - 3 q^{45} - 12 q^{47} + 3 q^{49} + 4 q^{51} + 8 q^{53} - 3 q^{55} - 8 q^{57} - 2 q^{59} - 22 q^{61} + 3 q^{63} + 2 q^{65} + 6 q^{67} - 14 q^{69} + 12 q^{71} - 20 q^{73} - 2 q^{75} + 3 q^{77} + 32 q^{79} - 17 q^{81} - 14 q^{83} + 4 q^{85} + 4 q^{87} + 4 q^{89} - 2 q^{91} - 30 q^{93} + 6 q^{95} + 4 q^{97} + 3 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 5 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 3$$

## 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
 −1.65544 −0.210756 2.86620
0 −2.65544 0 −1.00000 0 1.00000 0 4.05137 0
1.2 0 −1.21076 0 −1.00000 0 1.00000 0 −1.53407 0
1.3 0 1.86620 0 −1.00000 0 1.00000 0 0.482696 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.bd 3
4.b odd 2 1 3080.2.a.m 3

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

 $$T_{3}^{3} + 2 T_{3}^{2} - 4 T_{3} - 6$$ $$T_{13}^{3} + 2 T_{13}^{2} - 4 T_{13} - 6$$ $$T_{17}^{3} + 4 T_{17}^{2} - 20 T_{17} - 66$$ $$T_{19}^{3} + 6 T_{19}^{2} - 20 T_{19} - 88$$ $$T_{23}^{3} + 2 T_{23}^{2} - 16 T_{23} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-6 - 4 T + 2 T^{2} + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-6 - 4 T + 2 T^{2} + T^{3}$$
$17$ $$-66 - 20 T + 4 T^{2} + T^{3}$$
$19$ $$-88 - 20 T + 6 T^{2} + T^{3}$$
$23$ $$4 - 16 T + 2 T^{2} + T^{3}$$
$29$ $$( 2 + T )^{3}$$
$31$ $$154 - 26 T - 6 T^{2} + T^{3}$$
$37$ $$-36 - 12 T + 4 T^{2} + T^{3}$$
$41$ $$-2 + 10 T + 12 T^{2} + T^{3}$$
$43$ $$348 - 52 T - 10 T^{2} + T^{3}$$
$47$ $$-118 + 4 T + 12 T^{2} + T^{3}$$
$53$ $$148 - 20 T - 8 T^{2} + T^{3}$$
$59$ $$-946 - 182 T + 2 T^{2} + T^{3}$$
$61$ $$-626 + 66 T + 22 T^{2} + T^{3}$$
$67$ $$244 - 60 T - 6 T^{2} + T^{3}$$
$71$ $$192 - 80 T - 12 T^{2} + T^{3}$$
$73$ $$162 + 108 T + 20 T^{2} + T^{3}$$
$79$ $$-716 + 300 T - 32 T^{2} + T^{3}$$
$83$ $$-88 + 36 T + 14 T^{2} + T^{3}$$
$89$ $$48 - 16 T - 4 T^{2} + T^{3}$$
$97$ $$-784 - 192 T - 4 T^{2} + T^{3}$$