# Properties

 Label 6042.2.a.m Level 6042 Weight 2 Character orbit 6042.a Self dual yes Analytic conductor 48.246 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## 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: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 1.00000 1.00000 2.00000 1.00000 4.00000 1.00000 1.00000 2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.m 1

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

## Hecke kernels

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

 $$T_{5} - 2$$ $$T_{7} - 4$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - T$$
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 11 T^{2}$$
$13$ $$1 - 4 T + 13 T^{2}$$
$17$ $$1 + 6 T + 17 T^{2}$$
$19$ $$1 + T$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 - 6 T + 29 T^{2}$$
$31$ $$1 - 4 T + 31 T^{2}$$
$37$ $$1 + 4 T + 37 T^{2}$$
$41$ $$1 - 10 T + 41 T^{2}$$
$43$ $$1 + 4 T + 43 T^{2}$$
$47$ $$1 + 8 T + 47 T^{2}$$
$53$ $$1 - T$$
$59$ $$1 + 14 T + 59 T^{2}$$
$61$ $$1 + 4 T + 61 T^{2}$$
$67$ $$1 - 4 T + 67 T^{2}$$
$71$ $$1 + 71 T^{2}$$
$73$ $$1 + 10 T + 73 T^{2}$$
$79$ $$1 - 12 T + 79 T^{2}$$
$83$ $$1 + 6 T + 83 T^{2}$$
$89$ $$1 + 89 T^{2}$$
$97$ $$1 - 18 T + 97 T^{2}$$
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