# Properties

 Label 6160.2.a.bh Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $1$ Dimension $3$ CM no Inner twists $1$

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## 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: 3.3.148.1 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 1540) 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.

 $$f(q)$$ $$=$$ $$q -\beta_{2} q^{3} - q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q -\beta_{2} q^{3} - q^{5} + q^{7} + ( -\beta_{1} - \beta_{2} ) q^{9} - q^{11} -\beta_{2} q^{13} + \beta_{2} q^{15} + ( -2 + 4 \beta_{1} + \beta_{2} ) q^{17} + ( 2 \beta_{1} + 4 \beta_{2} ) q^{19} -\beta_{2} q^{21} + ( 3 - 5 \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( 2 + 2 \beta_{2} ) q^{27} + ( -4 - 2 \beta_{1} - 2 \beta_{2} ) q^{29} + ( -3 + 3 \beta_{1} ) q^{31} + \beta_{2} q^{33} - q^{35} + ( -1 - 5 \beta_{1} + 3 \beta_{2} ) q^{37} + ( 3 - \beta_{1} - \beta_{2} ) q^{39} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{41} + ( 7 + \beta_{1} + 3 \beta_{2} ) q^{43} + ( \beta_{1} + \beta_{2} ) q^{45} + ( -6 + 4 \beta_{1} + 3 \beta_{2} ) q^{47} + q^{49} + ( 1 - 3 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -7 + \beta_{1} + \beta_{2} ) q^{53} + q^{55} + ( -10 + 2 \beta_{1} + 4 \beta_{2} ) q^{57} + ( 5 - \beta_{1} - 2 \beta_{2} ) q^{59} + ( 3 + 5 \beta_{1} ) q^{61} + ( -\beta_{1} - \beta_{2} ) q^{63} + \beta_{2} q^{65} + ( 7 + 3 \beta_{1} - 3 \beta_{2} ) q^{67} + ( -2 + 4 \beta_{1} - 4 \beta_{2} ) q^{69} + ( 2 - 2 \beta_{1} + 2 \beta_{2} ) q^{71} + ( -2 - 4 \beta_{1} - 5 \beta_{2} ) q^{73} -\beta_{2} q^{75} - q^{77} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{79} + ( -6 + 5 \beta_{1} + 3 \beta_{2} ) q^{81} + ( -4 + 6 \beta_{1} - 2 \beta_{2} ) q^{83} + ( 2 - 4 \beta_{1} - \beta_{2} ) q^{85} + ( 4 + 2 \beta_{2} ) q^{87} + ( -8 + 4 \beta_{2} ) q^{89} -\beta_{2} q^{91} + ( 3 - 3 \beta_{1} + 3 \beta_{2} ) q^{93} + ( -2 \beta_{1} - 4 \beta_{2} ) q^{95} + ( -4 + 8 \beta_{1} + 4 \beta_{2} ) q^{97} + ( \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{5} + 3 q^{7} - q^{9} + O(q^{10})$$ $$3 q - 3 q^{5} + 3 q^{7} - q^{9} - 3 q^{11} - 2 q^{17} + 2 q^{19} + 4 q^{23} + 3 q^{25} + 6 q^{27} - 14 q^{29} - 6 q^{31} - 3 q^{35} - 8 q^{37} + 8 q^{39} - 12 q^{41} + 22 q^{43} + q^{45} - 14 q^{47} + 3 q^{49} - 20 q^{53} + 3 q^{55} - 28 q^{57} + 14 q^{59} + 14 q^{61} - q^{63} + 24 q^{67} - 2 q^{69} + 4 q^{71} - 10 q^{73} - 3 q^{77} - 12 q^{79} - 13 q^{81} - 6 q^{83} + 2 q^{85} + 12 q^{87} - 24 q^{89} + 6 q^{93} - 2 q^{95} - 4 q^{97} + q^{99} + O(q^{100})$$

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
 −1.48119 2.17009 0.311108
0 −1.67513 0 −1.00000 0 1.00000 0 −0.193937 0
1.2 0 −0.539189 0 −1.00000 0 1.00000 0 −2.70928 0
1.3 0 2.21432 0 −1.00000 0 1.00000 0 1.90321 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.bh 3
4.b odd 2 1 1540.2.a.g 3
20.d odd 2 1 7700.2.a.x 3
20.e even 4 2 7700.2.e.r 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.g 3 4.b odd 2 1
6160.2.a.bh 3 1.a even 1 1 trivial
7700.2.a.x 3 20.d odd 2 1
7700.2.e.r 6 20.e even 4 2

## Hecke kernels

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

 $$T_{3}^{3} - 4 T_{3} - 2$$ $$T_{13}^{3} - 4 T_{13} - 2$$ $$T_{17}^{3} + 2 T_{17}^{2} - 48 T_{17} - 134$$ $$T_{19}^{3} - 2 T_{19}^{2} - 60 T_{19} + 200$$ $$T_{23}^{3} - 4 T_{23}^{2} - 72 T_{23} + 268$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{3}$$
$3$ $$-2 - 4 T + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( -1 + T )^{3}$$
$11$ $$( 1 + T )^{3}$$
$13$ $$-2 - 4 T + T^{3}$$
$17$ $$-134 - 48 T + 2 T^{2} + T^{3}$$
$19$ $$200 - 60 T - 2 T^{2} + T^{3}$$
$23$ $$268 - 72 T - 4 T^{2} + T^{3}$$
$29$ $$8 + 44 T + 14 T^{2} + T^{3}$$
$31$ $$-54 - 18 T + 6 T^{2} + T^{3}$$
$37$ $$-1076 - 128 T + 8 T^{2} + T^{3}$$
$41$ $$-338 - 10 T + 12 T^{2} + T^{3}$$
$43$ $$-76 + 128 T - 22 T^{2} + T^{3}$$
$47$ $$-338 + 14 T^{2} + T^{3}$$
$53$ $$260 + 128 T + 20 T^{2} + T^{3}$$
$59$ $$-50 + 50 T - 14 T^{2} + T^{3}$$
$61$ $$278 - 18 T - 14 T^{2} + T^{3}$$
$67$ $$428 + 108 T - 24 T^{2} + T^{3}$$
$71$ $$-32 - 32 T - 4 T^{2} + T^{3}$$
$73$ $$-466 - 80 T + 10 T^{2} + T^{3}$$
$79$ $$4 + 20 T + 12 T^{2} + T^{3}$$
$83$ $$296 - 148 T + 6 T^{2} + T^{3}$$
$89$ $$128 + 128 T + 24 T^{2} + T^{3}$$
$97$ $$-1472 - 208 T + 4 T^{2} + T^{3}$$
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