Properties

 Label 5610.2.a.cd Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 0 Dimension 3 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: $$3$$ Coefficient field: 3.3.2089.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2$$ Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 13 x - 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$2 \nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$$$/2$$ $$\nu^{2}$$ $$=$$ $$($$$$4 \beta_{2} + \beta_{1} + 16$$$$)/2$$

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
 −0.309984 3.75054 −3.44055
1.00000 1.00000 1.00000 −1.00000 1.00000 −3.79696 1.00000 1.00000 −1.00000
1.2 1.00000 1.00000 1.00000 −1.00000 1.00000 1.15799 1.00000 1.00000 −1.00000
1.3 1.00000 1.00000 1.00000 −1.00000 1.00000 3.63897 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{3} - T_{7}^{2} - 14 T_{7} + 16$$ $$T_{13}^{3} - 5 T_{13}^{2} - 6 T_{13} + 8$$ $$T_{19}^{3} - 13 T_{19}^{2} + 42 T_{19} - 8$$ $$T_{23}^{3} + 11 T_{23}^{2} + 26 T_{23} + 8$$