# Properties

 Label 5610.2.a.ce Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 1 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2777.1 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.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - \nu^{2} - 4 \nu + 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 2 \nu - 4$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{3} - 2 \nu^{2} - 4 \nu + 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - 2 \beta_{1} + 1$$$$)/4$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 2 \beta_{2} - 2 \beta_{1} + 9$$$$)/4$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} + 2 \beta_{2} - 6 \beta_{1} + 9$$$$)/4$$

## 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
 −0.679643 −1.50848 2.36234 0.825785
−1.00000 −1.00000 1.00000 1.00000 1.00000 −2.94272 −1.00000 1.00000 −1.00000
1.2 −1.00000 −1.00000 1.00000 1.00000 1.00000 −1.32584 −1.00000 1.00000 −1.00000
1.3 −1.00000 −1.00000 1.00000 1.00000 1.00000 0.846618 −1.00000 1.00000 −1.00000
1.4 −1.00000 −1.00000 1.00000 1.00000 1.00000 2.42194 −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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{4} + T_{7}^{3} - 8 T_{7}^{2} - 4 T_{7} + 8$$ $$T_{13}^{4} - 3 T_{13}^{3} - 42 T_{13}^{2} + 128 T_{13} - 88$$ $$T_{19}^{4} + 9 T_{19}^{3} + 22 T_{19}^{2} + 8 T_{19} - 8$$ $$T_{23}^{4} + T_{23}^{3} - 46 T_{23}^{2} - 16 T_{23} + 488$$