# Properties

 Label 5610.2.a.ch Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 0 Dimension 4 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: $$4$$ Coefficient field: 4.4.65905.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{3}$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + \beta_{2} q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + \beta_{2} q^{7} + q^{8} + q^{9} - q^{10} + q^{11} + q^{12} + ( 2 + \beta_{2} ) q^{13} + \beta_{2} q^{14} - q^{15} + q^{16} - q^{17} + q^{18} + ( 2 + \beta_{2} ) q^{19} - q^{20} + \beta_{2} q^{21} + q^{22} + ( 2 - \beta_{2} ) q^{23} + q^{24} + q^{25} + ( 2 + \beta_{2} ) q^{26} + q^{27} + \beta_{2} q^{28} -2 q^{29} - q^{30} + ( 1 + \beta_{3} ) q^{31} + q^{32} + q^{33} - q^{34} -\beta_{2} q^{35} + q^{36} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{37} + ( 2 + \beta_{2} ) q^{38} + ( 2 + \beta_{2} ) q^{39} - q^{40} + ( 2 - \beta_{1} - \beta_{2} ) q^{41} + \beta_{2} q^{42} + ( 1 - \beta_{2} + \beta_{3} ) q^{43} + q^{44} - q^{45} + ( 2 - \beta_{2} ) q^{46} + ( 2 - \beta_{1} - \beta_{2} ) q^{47} + q^{48} -\beta_{3} q^{49} + q^{50} - q^{51} + ( 2 + \beta_{2} ) q^{52} + ( -1 - \beta_{2} - \beta_{3} ) q^{53} + q^{54} - q^{55} + \beta_{2} q^{56} + ( 2 + \beta_{2} ) q^{57} -2 q^{58} + ( 1 + \beta_{1} - \beta_{3} ) q^{59} - q^{60} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{61} + ( 1 + \beta_{3} ) q^{62} + \beta_{2} q^{63} + q^{64} + ( -2 - \beta_{2} ) q^{65} + q^{66} + ( 3 + \beta_{3} ) q^{67} - q^{68} + ( 2 - \beta_{2} ) q^{69} -\beta_{2} q^{70} + ( 5 - 3 \beta_{2} + \beta_{3} ) q^{71} + q^{72} + ( 3 - \beta_{1} - \beta_{3} ) q^{73} + ( 3 + \beta_{1} - \beta_{2} - \beta_{3} ) q^{74} + q^{75} + ( 2 + \beta_{2} ) q^{76} + \beta_{2} q^{77} + ( 2 + \beta_{2} ) q^{78} + ( 1 - \beta_{1} + \beta_{3} ) q^{79} - q^{80} + q^{81} + ( 2 - \beta_{1} - \beta_{2} ) q^{82} + ( -5 + \beta_{1} + \beta_{2} + \beta_{3} ) q^{83} + \beta_{2} q^{84} + q^{85} + ( 1 - \beta_{2} + \beta_{3} ) q^{86} -2 q^{87} + q^{88} + ( -2 - 4 \beta_{2} ) q^{89} - q^{90} + ( 7 + 2 \beta_{2} - \beta_{3} ) q^{91} + ( 2 - \beta_{2} ) q^{92} + ( 1 + \beta_{3} ) q^{93} + ( 2 - \beta_{1} - \beta_{2} ) q^{94} + ( -2 - \beta_{2} ) q^{95} + q^{96} + ( 3 - \beta_{1} + \beta_{2} + \beta_{3} ) q^{97} -\beta_{3} q^{98} + q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + q^{7} + 4q^{8} + 4q^{9} - 4q^{10} + 4q^{11} + 4q^{12} + 9q^{13} + q^{14} - 4q^{15} + 4q^{16} - 4q^{17} + 4q^{18} + 9q^{19} - 4q^{20} + q^{21} + 4q^{22} + 7q^{23} + 4q^{24} + 4q^{25} + 9q^{26} + 4q^{27} + q^{28} - 8q^{29} - 4q^{30} + 3q^{31} + 4q^{32} + 4q^{33} - 4q^{34} - q^{35} + 4q^{36} + 13q^{37} + 9q^{38} + 9q^{39} - 4q^{40} + 6q^{41} + q^{42} + 2q^{43} + 4q^{44} - 4q^{45} + 7q^{46} + 6q^{47} + 4q^{48} + q^{49} + 4q^{50} - 4q^{51} + 9q^{52} - 4q^{53} + 4q^{54} - 4q^{55} + q^{56} + 9q^{57} - 8q^{58} + 6q^{59} - 4q^{60} + 13q^{61} + 3q^{62} + q^{63} + 4q^{64} - 9q^{65} + 4q^{66} + 11q^{67} - 4q^{68} + 7q^{69} - q^{70} + 16q^{71} + 4q^{72} + 12q^{73} + 13q^{74} + 4q^{75} + 9q^{76} + q^{77} + 9q^{78} + 2q^{79} - 4q^{80} + 4q^{81} + 6q^{82} - 19q^{83} + q^{84} + 4q^{85} + 2q^{86} - 8q^{87} + 4q^{88} - 12q^{89} - 4q^{90} + 31q^{91} + 7q^{92} + 3q^{93} + 6q^{94} - 9q^{95} + 4q^{96} + 11q^{97} + q^{98} + 4q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 7 x^{2} + x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu - 1$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 7 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 4 \nu^{2} + 3 \nu - 8$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 2$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 9$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$4 \beta_{3} - 3 \beta_{2} + 11 \beta_{1} + 46$$$$)/4$$

## 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
 0.573756 3.75534 −1.81872 −0.510373
1.00000 1.00000 1.00000 −1.00000 1.00000 −3.48580 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 −0.532575 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 1.09967 1.00000 1.00000 −1.00000
1.4 1.00000 1.00000 1.00000 −1.00000 1.00000 3.91871 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{4} - T_{7}^{3} - 14 T_{7}^{2} + 8 T_{7} + 8$$ $$T_{13}^{4} - 9 T_{13}^{3} + 16 T_{13}^{2} + 20 T_{13} - 40$$ $$T_{19}^{4} - 9 T_{19}^{3} + 16 T_{19}^{2} + 20 T_{19} - 40$$ $$T_{23}^{4} - 7 T_{23}^{3} + 4 T_{23}^{2} + 28 T_{23} - 24$$