# Properties

 Label 6160.2.a.bs Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.111028.1 Defining polynomial: $$x^{4} - 10 x^{2} - 2 x + 4$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{3} + q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{3} + q^{5} + q^{7} + ( 2 + \beta_{2} ) q^{9} - q^{11} + ( 1 - \beta_{3} ) q^{13} + \beta_{1} q^{15} + ( 2 - \beta_{1} ) q^{17} + ( -1 + \beta_{1} - \beta_{3} ) q^{19} + \beta_{1} q^{21} + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + q^{25} + ( 1 + 3 \beta_{1} + \beta_{3} ) q^{27} + 2 q^{29} + ( -1 - \beta_{1} + \beta_{2} ) q^{31} -\beta_{1} q^{33} + q^{35} + ( 2 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{37} + ( -1 - \beta_{2} ) q^{39} + ( 3 - \beta_{1} - \beta_{2} ) q^{41} + ( \beta_{1} + \beta_{2} - \beta_{3} ) q^{43} + ( 2 + \beta_{2} ) q^{45} + ( 1 + 2 \beta_{1} + \beta_{3} ) q^{47} + q^{49} + ( -5 + 2 \beta_{1} - \beta_{2} ) q^{51} + ( 2 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{53} - q^{55} + ( 4 - 2 \beta_{1} ) q^{57} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{59} + ( 6 - 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{61} + ( 2 + \beta_{2} ) q^{63} + ( 1 - \beta_{3} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} ) q^{67} + ( -5 + 5 \beta_{1} - 2 \beta_{2} + \beta_{3} ) q^{69} + ( -2 + 2 \beta_{1} + 2 \beta_{3} ) q^{71} + ( 6 + \beta_{1} ) q^{73} + \beta_{1} q^{75} - q^{77} + ( -3 - 4 \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{79} + ( 10 + 2 \beta_{1} + \beta_{2} ) q^{81} + ( 1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{83} + ( 2 - \beta_{1} ) q^{85} + 2 \beta_{1} q^{87} + ( -1 - \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( 1 - \beta_{3} ) q^{91} + ( -4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{93} + ( -1 + \beta_{1} - \beta_{3} ) q^{95} + ( -1 + \beta_{1} + \beta_{3} ) q^{97} + ( -2 - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} + O(q^{10})$$ $$4 q + 4 q^{5} + 4 q^{7} + 8 q^{9} - 4 q^{11} + 2 q^{13} + 8 q^{17} - 6 q^{19} + 6 q^{23} + 4 q^{25} + 6 q^{27} + 8 q^{29} - 4 q^{31} + 4 q^{35} + 10 q^{37} - 4 q^{39} + 12 q^{41} - 2 q^{43} + 8 q^{45} + 6 q^{47} + 4 q^{49} - 20 q^{51} + 6 q^{53} - 4 q^{55} + 16 q^{57} - 4 q^{59} + 26 q^{61} + 8 q^{63} + 2 q^{65} + 10 q^{67} - 18 q^{69} - 4 q^{71} + 24 q^{73} - 4 q^{77} - 8 q^{79} + 40 q^{81} + 2 q^{83} + 8 q^{85} - 6 q^{89} + 2 q^{91} - 14 q^{93} - 6 q^{95} - 2 q^{97} - 8 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 10 x^{2} - 2 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 5$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 9 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 5$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 9 \beta_{1} + 1$$

## 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
 −2.97964 −0.766757 0.547280 3.19911
0 −2.97964 0 1.00000 0 1.00000 0 5.87824 0
1.2 0 −0.766757 0 1.00000 0 1.00000 0 −2.41208 0
1.3 0 0.547280 0 1.00000 0 1.00000 0 −2.70049 0
1.4 0 3.19911 0 1.00000 0 1.00000 0 7.23433 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.bs 4
4.b odd 2 1 1540.2.a.i 4
20.d odd 2 1 7700.2.a.bb 4
20.e even 4 2 7700.2.e.t 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.i 4 4.b odd 2 1
6160.2.a.bs 4 1.a even 1 1 trivial
7700.2.a.bb 4 20.d odd 2 1
7700.2.e.t 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6160))$$:

 $$T_{3}^{4} - 10 T_{3}^{2} - 2 T_{3} + 4$$ $$T_{13}^{4} - 2 T_{13}^{3} - 34 T_{13}^{2} - 2 T_{13} + 96$$ $$T_{17}^{4} - 8 T_{17}^{3} + 14 T_{17}^{2} + 10 T_{17} - 24$$ $$T_{19}^{4} + 6 T_{19}^{3} - 28 T_{19}^{2} - 152 T_{19} - 96$$ $$T_{23}^{4} - 6 T_{23}^{3} - 56 T_{23}^{2} + 236 T_{23} - 184$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$4 - 2 T - 10 T^{2} + T^{4}$$
$5$ $$( -1 + T )^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$( 1 + T )^{4}$$
$13$ $$96 - 2 T - 34 T^{2} - 2 T^{3} + T^{4}$$
$17$ $$-24 + 10 T + 14 T^{2} - 8 T^{3} + T^{4}$$
$19$ $$-96 - 152 T - 28 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$-184 + 236 T - 56 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$( -2 + T )^{4}$$
$31$ $$176 - 134 T - 40 T^{2} + 4 T^{3} + T^{4}$$
$37$ $$-264 + 244 T - 32 T^{2} - 10 T^{3} + T^{4}$$
$41$ $$-668 + 370 T - 4 T^{2} - 12 T^{3} + T^{4}$$
$43$ $$-144 + 196 T - 72 T^{2} + 2 T^{3} + T^{4}$$
$47$ $$1044 + 158 T - 70 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$1608 + 364 T - 144 T^{2} - 6 T^{3} + T^{4}$$
$59$ $$-1448 - 810 T - 108 T^{2} + 4 T^{3} + T^{4}$$
$61$ $$-5636 + 738 T + 140 T^{2} - 26 T^{3} + T^{4}$$
$67$ $$-8 + 36 T - 36 T^{2} - 10 T^{3} + T^{4}$$
$71$ $$8704 - 384 T - 192 T^{2} + 4 T^{3} + T^{4}$$
$73$ $$952 - 746 T + 206 T^{2} - 24 T^{3} + T^{4}$$
$79$ $$10528 - 1908 T - 280 T^{2} + 8 T^{3} + T^{4}$$
$83$ $$3936 + 488 T - 260 T^{2} - 2 T^{3} + T^{4}$$
$89$ $$2592 - 352 T - 160 T^{2} + 6 T^{3} + T^{4}$$
$97$ $$544 - 48 T - 48 T^{2} + 2 T^{3} + T^{4}$$