# Properties

 Label 4730.2.a.w Level 4730 Weight 2 Character orbit 4730.a Self dual yes Analytic conductor 37.769 Analytic rank 1 Dimension 8 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: $$8$$ Coefficient field: $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{8} - x^{7} - 12 x^{6} + 7 x^{5} + 41 x^{4} - 6 x^{3} - 28 x^{2} + 2 x + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{6} - 10 \nu^{4} - \nu^{3} + 24 \nu^{2} + 2 \nu - 8$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 39 \nu^{3} - 4 \nu^{2} - 16 \nu - 2$$$$)/2$$ $$\beta_{4}$$ $$=$$ $$($$$$-\nu^{7} + \nu^{6} + 12 \nu^{5} - 5 \nu^{4} - 39 \nu^{3} - 8 \nu^{2} + 12 \nu + 4$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 39 \nu^{3} - 6 \nu^{2} - 16 \nu + 4$$$$)/2$$ $$\beta_{6}$$ $$=$$ $$($$$$\nu^{7} - \nu^{6} - 12 \nu^{5} + 7 \nu^{4} + 41 \nu^{3} - 6 \nu^{2} - 26 \nu + 2$$$$)/2$$ $$\beta_{7}$$ $$=$$ $$($$$$\nu^{7} - 2 \nu^{6} - 10 \nu^{5} + 17 \nu^{4} + 26 \nu^{3} - 32 \nu^{2} - 8 \nu + 8$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{3} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{6} - \beta_{5} + 5 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$-6 \beta_{5} + \beta_{4} + 7 \beta_{3} + 2 \beta_{1} + 17$$ $$\nu^{5}$$ $$=$$ $$\beta_{7} + 7 \beta_{6} - 9 \beta_{5} + \beta_{3} + \beta_{2} + 30 \beta_{1} + 12$$ $$\nu^{6}$$ $$=$$ $$\beta_{6} - 37 \beta_{5} + 10 \beta_{4} + 46 \beta_{3} + 2 \beta_{2} + 23 \beta_{1} + 107$$ $$\nu^{7}$$ $$=$$ $$12 \beta_{7} + 46 \beta_{6} - 68 \beta_{5} + 3 \beta_{4} + 15 \beta_{3} + 14 \beta_{2} + 190 \beta_{1} + 107$$

## 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
 −2.42597 −1.86277 −0.843708 −0.402785 0.484643 0.771947 2.57663 2.70201
−1.00000 −3.42597 1.00000 1.00000 3.42597 −2.60155 −1.00000 8.73725 −1.00000
1.2 −1.00000 −2.86277 1.00000 1.00000 2.86277 −1.78911 −1.00000 5.19548 −1.00000
1.3 −1.00000 −1.84371 1.00000 1.00000 1.84371 0.526782 −1.00000 0.399257 −1.00000
1.4 −1.00000 −1.40279 1.00000 1.00000 1.40279 3.56264 −1.00000 −1.03219 −1.00000
1.5 −1.00000 −0.515357 1.00000 1.00000 0.515357 −4.64210 −1.00000 −2.73441 −1.00000
1.6 −1.00000 −0.228053 1.00000 1.00000 0.228053 −2.81890 −1.00000 −2.94799 −1.00000
1.7 −1.00000 1.57663 1.00000 1.00000 −1.57663 0.800428 −1.00000 −0.514223 −1.00000
1.8 −1.00000 1.70201 1.00000 1.00000 −1.70201 0.961819 −1.00000 −0.103166 −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.w 8

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

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