# Properties

 Label 4730.2.a.p 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

# Related objects

## 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: $$\Q(\sqrt{2})$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
1.00000 −2.41421 1.00000 −1.00000 −2.41421 0.414214 1.00000 2.82843 −1.00000
1.2 1.00000 0.414214 1.00000 −1.00000 0.414214 −2.41421 1.00000 −2.82843 −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.p 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.p 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4730))$$:

 $$T_{3}^{2} + 2 T_{3} - 1$$ $$T_{7}^{2} + 2 T_{7} - 1$$ $$T_{13}^{2} - 4 T_{13} + 2$$