# Properties

 Label 4029.2.a.d Level 4029 Weight 2 Character orbit 4029.a Self dual yes Analytic conductor 32.172 Analytic rank 1 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4029 = 3 \cdot 17 \cdot 79$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4029.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1717269744$$ Analytic rank: $$1$$ Dimension: $$3$$ Coefficient field: 3.3.148.1 Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 0.311108 2.17009 −1.48119
−2.21432 −1.00000 2.90321 −1.68889 2.21432 1.00000 −2.00000 1.00000 3.73975
1.2 0.539189 −1.00000 −1.70928 0.170086 −0.539189 1.00000 −2.00000 1.00000 0.0917087
1.3 1.67513 −1.00000 0.806063 −3.48119 −1.67513 1.00000 −2.00000 1.00000 −5.83146
 $$n$$: 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 4029.2.a.d 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4029.2.a.d 3 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$17$$ $$-1$$
$$79$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4029))$$:

 $$T_{2}^{3} - 4 T_{2} + 2$$ $$T_{5}^{3} + 5 T_{5}^{2} + 5 T_{5} - 1$$