# Properties

 Label 5610.2.a.cf 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: 4.4.17428.1 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 6 x^{2} + 4 x + 6$$:

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

## 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
 2.36865 −0.787711 −2.10710 1.52616
−1.00000 1.00000 1.00000 −1.00000 −1.00000 −1.81471 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 −0.662077 −1.00000 1.00000 1.00000
1.3 −1.00000 1.00000 1.00000 −1.00000 −1.00000 2.92682 −1.00000 1.00000 1.00000
1.4 −1.00000 1.00000 1.00000 −1.00000 −1.00000 4.54997 −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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{4} - 5 T_{7}^{3} - 4 T_{7}^{2} + 24 T_{7} + 16$$ $$T_{13}^{4} - 6 T_{13}^{3} - 12 T_{13}^{2} + 56 T_{13} + 64$$ $$T_{19}^{4} - 8 T_{19}^{3} - 16 T_{19}^{2} + 208 T_{19} - 256$$ $$T_{23}^{4} - T_{23}^{3} - 36 T_{23}^{2} - 80 T_{23} - 48$$