# Properties

 Label 6034.2.a.j Level 6034 Weight 2 Character orbit 6034.a Self dual Yes Analytic conductor 48.182 Analytic rank 1 Dimension 4 CM No Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6034 = 2 \cdot 7 \cdot 431$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6034.a (trivial)

## Newform invariants

 Self dual: Yes Analytic conductor: $$48.1817325796$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.10273.1 Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 5 x^{2} + x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 4 \nu$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3 \nu^{2} - 2 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{3} + \beta_{2} + 2 \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-2 \beta_{3} + 3 \beta_{2} + 8 \beta_{1} + 6$$

## 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
 −1.38266 −0.641043 0.673533 3.35017
−1.00000 −2.38266 1.00000 −2.61326 2.38266 1.00000 −1.00000 2.67708 2.61326
1.2 −1.00000 −1.64104 1.00000 2.78585 1.64104 1.00000 −1.00000 −0.306978 −2.78585
1.3 −1.00000 −0.326467 1.00000 0.597538 0.326467 1.00000 −1.00000 −2.89342 −0.597538
1.4 −1.00000 2.35017 1.00000 0.229876 −2.35017 1.00000 −1.00000 2.52331 −0.229876
 $$n$$: 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
$$2$$ $$1$$
$$7$$ $$-1$$
$$431$$ $$-1$$

## Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6034))$$:

 $$T_{3}^{4} + 2 T_{3}^{3} - 5 T_{3}^{2} - 11 T_{3} - 3$$ $$T_{5}^{4} - T_{5}^{3} - 7 T_{5}^{2} + 6 T_{5} - 1$$ $$T_{11}^{4} + 13 T_{11}^{3} + 55 T_{11}^{2} + 88 T_{11} + 47$$