# Properties

 Label 4730.2.a.y Level 4730 Weight 2 Character orbit 4730.a Self dual yes Analytic conductor 37.769 Analytic rank 0 Dimension 8 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: $$0$$ Dimension: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{7}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - 13 x^{6} - 6 x^{5} + 46 x^{4} + 26 x^{3} - 52 x^{2} - 20 x + 20$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$-2 \nu^{7} + 2 \nu^{6} + 23 \nu^{5} - 10 \nu^{4} - 71 \nu^{3} + 16 \nu^{2} + 56 \nu - 18$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$2 \nu^{7} - 3 \nu^{6} - 22 \nu^{5} + 21 \nu^{4} + 66 \nu^{3} - 44 \nu^{2} - 50 \nu + 30$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} + 23 \nu^{5} - 23 \nu^{4} - 75 \nu^{3} + 56 \nu^{2} + 68 \nu - 42$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-2 \nu^{7} + 3 \nu^{6} + 23 \nu^{5} - 23 \nu^{4} - 75 \nu^{3} + 58 \nu^{2} + 68 \nu - 50$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{7} + 3 \nu^{6} + 35 \nu^{5} - 17 \nu^{4} - 108 \nu^{3} + 34 \nu^{2} + 82 \nu - 34$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$3 \nu^{7} - 4 \nu^{6} - 34 \nu^{5} + 28 \nu^{4} + 105 \nu^{3} - 66 \nu^{2} - 86 \nu + 54$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$3 \nu^{7} - 4 \nu^{6} - 34 \nu^{5} + 28 \nu^{4} + 105 \nu^{3} - 68 \nu^{2} - 82 \nu + 62$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{7} - \beta_{6} + \beta_{4} - \beta_{3}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{4} - \beta_{3} + 4$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{7} - 3 \beta_{6} + 2 \beta_{5} + 9 \beta_{4} - 9 \beta_{3} - 2 \beta_{2} - 2 \beta_{1} + 8$$$$)/2$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{7} + \beta_{6} + \beta_{5} + 12 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 27$$ $$\nu^{5}$$ $$=$$ $$($$$$35 \beta_{7} - 5 \beta_{6} + 22 \beta_{5} + 87 \beta_{4} - 79 \beta_{3} - 22 \beta_{2} - 18 \beta_{1} + 108$$$$)/2$$ $$\nu^{6}$$ $$=$$ $$30 \beta_{7} + 13 \beta_{6} + 17 \beta_{5} + 128 \beta_{4} - 113 \beta_{3} - 30 \beta_{2} - 6 \beta_{1} + 231$$ $$\nu^{7}$$ $$=$$ $$($$$$293 \beta_{7} + 37 \beta_{6} + 206 \beta_{5} + 861 \beta_{4} - 749 \beta_{3} - 222 \beta_{2} - 150 \beta_{1} + 1196$$$$)/2$$

## 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
 −1.19586 −1.81167 0.939990 −1.27816 1.82453 3.17194 −2.24275 0.591975
−1.00000 −2.26092 1.00000 1.00000 2.26092 1.24841 −1.00000 2.11175 −1.00000
1.2 −1.00000 −1.26740 1.00000 1.00000 1.26740 −2.82729 −1.00000 −1.39369 −1.00000
1.3 −1.00000 −0.519165 1.00000 1.00000 0.519165 1.27228 −1.00000 −2.73047 −1.00000
1.4 −1.00000 −0.232604 1.00000 1.00000 0.232604 5.22723 −1.00000 −2.94590 −1.00000
1.5 −1.00000 −0.206773 1.00000 1.00000 0.206773 −1.71176 −1.00000 −2.95724 −1.00000
1.6 −1.00000 1.68714 1.00000 1.00000 −1.68714 1.43903 −1.00000 −0.153574 −1.00000
1.7 −1.00000 2.54582 1.00000 1.00000 −2.54582 1.88573 −1.00000 3.48118 −1.00000
1.8 −1.00000 3.25391 1.00000 1.00000 −3.25391 4.46635 −1.00000 7.58795 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.8 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.y 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4730.2.a.y 8 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4730))$$:

 $$T_{3}^{8} - \cdots$$ $$T_{7}^{8} - \cdots$$ $$T_{13}^{8} - \cdots$$