# Properties

 Label 5610.2.a.ci Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 0 Dimension 5 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: $$5$$ Coefficient field: 5.5.18569692.1 Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2^{2}$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3,\beta_4$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{5} - 23 x^{3} - 32 x^{2} + 26 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-\nu^{4} + \nu^{3} + 20 \nu^{2} + 18 \nu - 12$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{4} - \nu^{3} + 26 \nu^{2} + 46 \nu - 30$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$-\nu^{4} + 22 \nu^{2} + 34 \nu - 12$$ $$\beta_{4}$$ $$=$$ $$-\nu^{4} + 22 \nu^{2} + 36 \nu - 12$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - 2 \beta_{3} + \beta_{2} + \beta_{1} + 9$$ $$\nu^{3}$$ $$=$$ $$10 \beta_{4} - 13 \beta_{3} + 2 \beta_{2} + 4 \beta_{1} + 18$$ $$\nu^{4}$$ $$=$$ $$39 \beta_{4} - 62 \beta_{3} + 22 \beta_{2} + 22 \beta_{1} + 186$$

## 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
 0.380617 −2.65676 −3.25711 0.228960 5.30430
1.00000 −1.00000 1.00000 1.00000 −1.00000 −4.10711 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.13372 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 1.89495 1.00000 1.00000 1.00000
1.4 1.00000 −1.00000 1.00000 1.00000 −1.00000 3.06482 1.00000 1.00000 1.00000
1.5 1.00000 −1.00000 1.00000 1.00000 −1.00000 4.28107 1.00000 1.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.5 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{5} - 2 T_{7}^{4} - 27 T_{7}^{3} + 52 T_{7}^{2} + 168 T_{7} - 320$$ $$T_{13}^{5} + T_{13}^{4} - 32 T_{13}^{3} + 36 T_{13}^{2} + 112 T_{13} - 128$$ $$T_{19}^{5} + T_{19}^{4} - 32 T_{19}^{3} + 36 T_{19}^{2} + 112 T_{19} - 128$$ $$T_{23}^{5} - 97 T_{23}^{3} + 76 T_{23}^{2} + 2236 T_{23} - 4720$$