# Properties

 Label 4730.2.a.l Level 4730 Weight 2 Character orbit 4730.a Self dual yes Analytic conductor 37.769 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$4730 = 2 \cdot 5 \cdot 11 \cdot 43$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4730.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$37.7692401561$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.56155 −1.56155
−1.00000 −2.56155 1.00000 1.00000 2.56155 −3.00000 −1.00000 3.56155 −1.00000
1.2 −1.00000 1.56155 1.00000 1.00000 −1.56155 −3.00000 −1.00000 −0.561553 −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.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4730.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.l 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$-1$$
$$11$$ $$1$$
$$43$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4730))$$:

 $$T_{3}^{2} + T_{3} - 4$$ $$T_{7} + 3$$ $$T_{13}^{2} - 3 T_{13} - 2$$