# Properties

 Label 6018.2.a.o Level 6018 Weight 2 Character orbit 6018.a Self dual yes Analytic conductor 48.054 Analytic rank 1 Dimension 4 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ = $$6018 = 2 \cdot 3 \cdot 17 \cdot 59$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 6018.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.0539719364$$ Analytic rank: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.725.1 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 3 x^{2} + x + 1$$:

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

## 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
 −1.35567 0.737640 −0.477260 2.09529
1.00000 1.00000 1.00000 −2.97371 1.00000 4.16724 1.00000 1.00000 −2.97371
1.2 1.00000 1.00000 1.00000 −0.880394 1.00000 −1.31313 1.00000 1.00000 −0.880394
1.3 1.00000 1.00000 1.00000 0.140774 1.00000 −1.43574 1.00000 1.00000 0.140774
1.4 1.00000 1.00000 1.00000 2.71333 1.00000 −2.41837 1.00000 1.00000 2.71333
 $$n$$: 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 6018.2.a.o 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6018.2.a.o 4 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$17$$ $$-1$$
$$59$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6018))$$:

 $$T_{5}^{4} + T_{5}^{3} - 8 T_{5}^{2} - 6 T_{5} + 1$$ $$T_{7}^{4} + T_{7}^{3} - 13 T_{7}^{2} - 31 T_{7} - 19$$