# Properties

 Label 5610.2.a.cc Level 5610 Weight 2 Character orbit 5610.a Self dual Yes Analytic conductor 44.796 Analytic rank 1 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: $$1$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ 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 root $$\beta$$ of the polynomial $$x^{3} - x^{2} - 2 x + 1$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.80194 0.445042 −1.24698
1.00000 −1.00000 1.00000 1.00000 −1.00000 −3.60388 1.00000 1.00000 1.00000
1.2 1.00000 −1.00000 1.00000 1.00000 −1.00000 −0.890084 1.00000 1.00000 1.00000
1.3 1.00000 −1.00000 1.00000 1.00000 −1.00000 2.49396 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5610))$$:

 $$T_{7}^{3} + 2 T_{7}^{2} - 8 T_{7} - 8$$ $$T_{13}^{3} + 4 T_{13}^{2} - 4 T_{13} - 8$$ $$T_{19}^{3} - 28 T_{19} - 56$$ $$T_{23}^{3} + 8 T_{23}^{2} - 44 T_{23} - 344$$