# Properties

 Label 4026.2.a.x Level 4026 Weight 2 Character orbit 4026.a Self dual yes Analytic conductor 32.148 Analytic rank 1 Dimension 6 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: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.46101901.1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

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

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$-\nu^{5} + 3 \nu^{4} + 5 \nu^{3} - 12 \nu^{2} - 5 \nu + 4$$ $$\beta_{3}$$ $$=$$ $$-\nu^{5} + 4 \nu^{4} + 2 \nu^{3} - 16 \nu^{2} + 4 \nu + 6$$ $$\beta_{4}$$ $$=$$ $$($$$$-3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 37 \nu^{2} - 3 \nu + 8$$$$)/2$$ $$\beta_{5}$$ $$=$$ $$($$$$-3 \nu^{5} + 10 \nu^{4} + 11 \nu^{3} - 39 \nu^{2} - \nu + 14$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$-3 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 6 \beta_{1} + 3$$ $$\nu^{4}$$ $$=$$ $$-13 \beta_{5} + 7 \beta_{4} + 4 \beta_{3} + 5 \beta_{2} + 13 \beta_{1} + 19$$ $$\nu^{5}$$ $$=$$ $$-42 \beta_{5} + 14 \beta_{4} + 17 \beta_{3} + 24 \beta_{2} + 52 \beta_{1} + 40$$

## 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
 −0.654438 1.90193 3.29072 −1.86911 −0.371781 0.702675
1.00000 −1.00000 1.00000 −4.17718 −1.00000 2.87039 1.00000 1.00000 −4.17718
1.2 1.00000 −1.00000 1.00000 −4.05628 −1.00000 −1.59539 1.00000 1.00000 −4.05628
1.3 1.00000 −1.00000 1.00000 −0.656263 −1.00000 4.00423 1.00000 1.00000 −0.656263
1.4 1.00000 −1.00000 1.00000 0.199215 −1.00000 −0.938542 1.00000 1.00000 0.199215
1.5 1.00000 −1.00000 1.00000 1.28209 −1.00000 −0.306395 1.00000 1.00000 1.28209
1.6 1.00000 −1.00000 1.00000 1.40842 −1.00000 −3.03430 1.00000 1.00000 1.40842
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 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.x 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4026.2.a.x 6 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(4026))$$:

 $$T_{5}^{6} + 6 T_{5}^{5} - T_{5}^{4} - 33 T_{5}^{3} + 17 T_{5}^{2} + 18 T_{5} - 4$$ $$T_{7}^{6} - T_{7}^{5} - 18 T_{7}^{4} + 76 T_{7}^{2} + 75 T_{7} + 16$$ $$T_{13}^{6} - 2 T_{13}^{5} - 18 T_{13}^{4} + 22 T_{13}^{3} + 69 T_{13}^{2} + 35 T_{13} + 2$$