# Properties

 Label 5610.2.a.br Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 0 Dimension 2 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$5610 = 2 \cdot 3 \cdot 5 \cdot 11 \cdot 17$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 5610.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$44.7960755339$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 2 \beta q^{7} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{5} + q^{6} + 2 \beta q^{7} - q^{8} + q^{9} - q^{10} + q^{11} - q^{12} + ( 2 + 2 \beta ) q^{13} -2 \beta q^{14} - q^{15} + q^{16} + q^{17} - q^{18} + ( -2 - 2 \beta ) q^{19} + q^{20} -2 \beta q^{21} - q^{22} + ( 6 - 2 \beta ) q^{23} + q^{24} + q^{25} + ( -2 - 2 \beta ) q^{26} - q^{27} + 2 \beta q^{28} + 2 q^{29} + q^{30} + ( 4 - 4 \beta ) q^{31} - q^{32} - q^{33} - q^{34} + 2 \beta q^{35} + q^{36} + ( 2 - 4 \beta ) q^{37} + ( 2 + 2 \beta ) q^{38} + ( -2 - 2 \beta ) q^{39} - q^{40} + ( -4 + 8 \beta ) q^{41} + 2 \beta q^{42} + ( 4 + 2 \beta ) q^{43} + q^{44} + q^{45} + ( -6 + 2 \beta ) q^{46} + ( -4 + 8 \beta ) q^{47} - q^{48} + ( -3 + 4 \beta ) q^{49} - q^{50} - q^{51} + ( 2 + 2 \beta ) q^{52} + ( -4 + 6 \beta ) q^{53} + q^{54} + q^{55} -2 \beta q^{56} + ( 2 + 2 \beta ) q^{57} -2 q^{58} + ( 12 - 2 \beta ) q^{59} - q^{60} + ( -2 - 8 \beta ) q^{61} + ( -4 + 4 \beta ) q^{62} + 2 \beta q^{63} + q^{64} + ( 2 + 2 \beta ) q^{65} + q^{66} + ( -6 + 4 \beta ) q^{67} + q^{68} + ( -6 + 2 \beta ) q^{69} -2 \beta q^{70} + ( 4 - 2 \beta ) q^{71} - q^{72} + ( -2 - 6 \beta ) q^{73} + ( -2 + 4 \beta ) q^{74} - q^{75} + ( -2 - 2 \beta ) q^{76} + 2 \beta q^{77} + ( 2 + 2 \beta ) q^{78} + ( 6 + 2 \beta ) q^{79} + q^{80} + q^{81} + ( 4 - 8 \beta ) q^{82} + 8 \beta q^{83} -2 \beta q^{84} + q^{85} + ( -4 - 2 \beta ) q^{86} -2 q^{87} - q^{88} + 14 q^{89} - q^{90} + ( 4 + 8 \beta ) q^{91} + ( 6 - 2 \beta ) q^{92} + ( -4 + 4 \beta ) q^{93} + ( 4 - 8 \beta ) q^{94} + ( -2 - 2 \beta ) q^{95} + q^{96} -8 \beta q^{97} + ( 3 - 4 \beta ) q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 2q^{11} - 2q^{12} + 6q^{13} - 2q^{14} - 2q^{15} + 2q^{16} + 2q^{17} - 2q^{18} - 6q^{19} + 2q^{20} - 2q^{21} - 2q^{22} + 10q^{23} + 2q^{24} + 2q^{25} - 6q^{26} - 2q^{27} + 2q^{28} + 4q^{29} + 2q^{30} + 4q^{31} - 2q^{32} - 2q^{33} - 2q^{34} + 2q^{35} + 2q^{36} + 6q^{38} - 6q^{39} - 2q^{40} + 2q^{42} + 10q^{43} + 2q^{44} + 2q^{45} - 10q^{46} - 2q^{48} - 2q^{49} - 2q^{50} - 2q^{51} + 6q^{52} - 2q^{53} + 2q^{54} + 2q^{55} - 2q^{56} + 6q^{57} - 4q^{58} + 22q^{59} - 2q^{60} - 12q^{61} - 4q^{62} + 2q^{63} + 2q^{64} + 6q^{65} + 2q^{66} - 8q^{67} + 2q^{68} - 10q^{69} - 2q^{70} + 6q^{71} - 2q^{72} - 10q^{73} - 2q^{75} - 6q^{76} + 2q^{77} + 6q^{78} + 14q^{79} + 2q^{80} + 2q^{81} + 8q^{83} - 2q^{84} + 2q^{85} - 10q^{86} - 4q^{87} - 2q^{88} + 28q^{89} - 2q^{90} + 16q^{91} + 10q^{92} - 4q^{93} - 6q^{95} + 2q^{96} - 8q^{97} + 2q^{98} + 2q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
−1.00000 −1.00000 1.00000 1.00000 1.00000 −1.23607 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 3.23607 −1.00000 1.00000 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$-1$$
$$17$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{2} - 2 T_{7} - 4$$ $$T_{13}^{2} - 6 T_{13} + 4$$ $$T_{19}^{2} + 6 T_{19} + 4$$ $$T_{23}^{2} - 10 T_{23} + 20$$