# Properties

 Label 4026.2.a.q Level 4026 Weight 2 Character orbit 4026.a Self dual yes Analytic conductor 32.148 Analytic rank 0 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$4026 = 2 \cdot 3 \cdot 11 \cdot 61$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 4026.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$32.1477718538$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: 4.4.6809.1 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 5 x^{2} - x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 5 \nu - 1$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + 5 \beta_{1} + 1$$

## 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
 −2.06963 −0.582772 0.361989 2.29041
−1.00000 −1.00000 1.00000 −2.06963 1.00000 −4.35299 −1.00000 1.00000 2.06963
1.2 −1.00000 −1.00000 1.00000 −0.582772 1.00000 1.07760 −1.00000 1.00000 0.582772
1.3 −1.00000 −1.00000 1.00000 0.361989 1.00000 2.23095 −1.00000 1.00000 −0.361989
1.4 −1.00000 −1.00000 1.00000 2.29041 1.00000 −0.955570 −1.00000 1.00000 −2.29041
 $$n$$: 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 4026.2.a.q 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.q 4 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$11$$ $$1$$
$$61$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(4026))$$:

 $$T_{5}^{4} - 5 T_{5}^{2} - T_{5} + 1$$ $$T_{7}^{4} + 2 T_{7}^{3} - 11 T_{7}^{2} - T_{7} + 10$$ $$T_{13}^{4} + 4 T_{13}^{3} - 21 T_{13}^{2} + 9 T_{13} + 17$$