# Properties

 Label 6160.2.a.bb Level $6160$ Weight $2$ Character orbit 6160.a Self dual yes Analytic conductor $49.188$ Analytic rank $1$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ x^2 - 2 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
0 −1.41421 0 1.00000 0 −1.00000 0 −1.00000 0
1.2 0 1.41421 0 1.00000 0 −1.00000 0 −1.00000 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.bb 2
4.b odd 2 1 3080.2.a.i 2

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

 $$T_{3}^{2} - 2$$ T3^2 - 2 $$T_{13}^{2} + 4T_{13} + 2$$ T13^2 + 4*T13 + 2 $$T_{17}^{2} + 4T_{17} - 14$$ T17^2 + 4*T17 - 14 $$T_{19}^{2} - 32$$ T19^2 - 32 $$T_{23} - 2$$ T23 - 2

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{2}$$
$3$ $$T^{2} - 2$$
$5$ $$(T - 1)^{2}$$
$7$ $$(T + 1)^{2}$$
$11$ $$(T - 1)^{2}$$
$13$ $$T^{2} + 4T + 2$$
$17$ $$T^{2} + 4T - 14$$
$19$ $$T^{2} - 32$$
$23$ $$(T - 2)^{2}$$
$29$ $$T^{2} + 12T + 28$$
$31$ $$T^{2} - 12T + 34$$
$37$ $$T^{2} - 8$$
$41$ $$T^{2} - 18$$
$43$ $$(T - 2)^{2}$$
$47$ $$T^{2} - 8T - 2$$
$53$ $$T^{2} - 8T - 56$$
$59$ $$T^{2} + 12T + 34$$
$61$ $$T^{2} + 16T + 46$$
$67$ $$T^{2} - 4T - 124$$
$71$ $$T^{2} + 8T - 56$$
$73$ $$T^{2} + 12T + 18$$
$79$ $$T^{2} + 20T + 92$$
$83$ $$T^{2} + 24T + 112$$
$89$ $$T^{2} + 12T + 28$$
$97$ $$T^{2} - 4T - 124$$