# Properties

 Label 6042.2.a.o Level 6042 Weight 2 Character orbit 6042.a Self dual yes Analytic conductor 48.246 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6042 = 2 \cdot 3 \cdot 19 \cdot 53$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6042.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2456129013$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{33})$$ Defining polynomial: $$x^{2} - x - 8$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{33})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 3.37228 −2.37228
1.00000 −1.00000 1.00000 −4.37228 −1.00000 1.00000 1.00000 1.00000 −4.37228
1.2 1.00000 −1.00000 1.00000 1.37228 −1.00000 1.00000 1.00000 1.00000 1.37228
 $$n$$: 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 6042.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6042.2.a.o 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$19$$ $$-1$$
$$53$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6042))$$:

 $$T_{5}^{2} + 3 T_{5} - 6$$ $$T_{7} - 1$$ $$T_{11} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$1 + 3 T + 4 T^{2} + 15 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T + 7 T^{2} )^{2}$$
$11$ $$( 1 + 4 T + 11 T^{2} )^{2}$$
$13$ $$1 + 3 T + 20 T^{2} + 39 T^{3} + 169 T^{4}$$
$17$ $$1 - 7 T + 38 T^{2} - 119 T^{3} + 289 T^{4}$$
$19$ $$( 1 - T )^{2}$$
$23$ $$1 - 3 T + 40 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$1 + 8 T + 41 T^{2} + 232 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 7 T + 31 T^{2} )^{2}$$
$37$ $$1 - 9 T + 86 T^{2} - 333 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 15 T + 134 T^{2} - 645 T^{3} + 1849 T^{4}$$
$47$ $$1 + 3 T + 22 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + T )^{2}$$
$59$ $$1 - 16 T + 149 T^{2} - 944 T^{3} + 3481 T^{4}$$
$61$ $$1 + 5 T + 120 T^{2} + 305 T^{3} + 3721 T^{4}$$
$67$ $$1 + T + 126 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$1 + 19 T + 224 T^{2} + 1349 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - 4 T + 73 T^{2} )^{2}$$
$79$ $$( 1 + 8 T + 79 T^{2} )^{2}$$
$83$ $$1 - 25 T + 314 T^{2} - 2075 T^{3} + 6889 T^{4}$$
$89$ $$1 + 21 T + 280 T^{2} + 1869 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 14 T + 97 T^{2} )^{2}$$