# Properties

 Label 5610.2.a.cg 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: 4.4.54764.1 Coefficient ring: $$\Z[a_1, \ldots, a_{19}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 9 x^{2} + 3 x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 11 \nu - 2$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 2 \nu^{2} + 11 \nu - 8$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 7 \nu + 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{2} - \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 5$$ $$\nu^{3}$$ $$=$$ $$($$$$11 \beta_{3} + 11 \beta_{2} - 3 \beta_{1} + 19$$$$)/4$$

## 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
 0.655762 3.36007 −0.339102 −2.67673
1.00000 −1.00000 1.00000 −1.00000 −1.00000 −3.46569 1.00000 1.00000 −1.00000
1.2 1.00000 −1.00000 1.00000 −1.00000 −1.00000 0.487359 1.00000 1.00000 −1.00000
1.3 1.00000 −1.00000 1.00000 −1.00000 −1.00000 1.84556 1.00000 1.00000 −1.00000
1.4 1.00000 −1.00000 1.00000 −1.00000 −1.00000 5.13277 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{4} - 4 T_{7}^{3} - 13 T_{7}^{2} + 40 T_{7} - 16$$ $$T_{13}^{4} + T_{13}^{3} - 44 T_{13}^{2} - 60 T_{13} + 304$$ $$T_{19}^{4} - 5 T_{19}^{3} - 32 T_{19}^{2} + 112 T_{19} + 256$$ $$T_{23}^{4} - 14 T_{23}^{3} + 13 T_{23}^{2} + 496 T_{23} - 1712$$