# Properties

 Label 6040.2.a.j Level 6040 Weight 2 Character orbit 6040.a Self dual yes Analytic conductor 48.230 Analytic rank 1 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$6040 = 2^{3} \cdot 5 \cdot 151$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6040.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2296428209$$ Analytic rank: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + 2q^{3} - q^{5} + 2q^{7} + q^{9} + O(q^{10})$$ $$q + 2q^{3} - q^{5} + 2q^{7} + q^{9} + 4q^{11} - 4q^{13} - 2q^{15} - 2q^{17} - 4q^{19} + 4q^{21} - 6q^{23} + q^{25} - 4q^{27} - 6q^{29} + 8q^{33} - 2q^{35} - 2q^{37} - 8q^{39} - 10q^{41} - q^{45} - 3q^{49} - 4q^{51} + 4q^{53} - 4q^{55} - 8q^{57} - 12q^{59} + 14q^{61} + 2q^{63} + 4q^{65} - 6q^{67} - 12q^{69} + 8q^{71} - 4q^{73} + 2q^{75} + 8q^{77} - 4q^{79} - 11q^{81} - 10q^{83} + 2q^{85} - 12q^{87} + 14q^{89} - 8q^{91} + 4q^{95} - 6q^{97} + 4q^{99} + O(q^{100})$$

## 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
 0
0 2.00000 0 −1.00000 0 2.00000 0 1.00000 0
 $$n$$: 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 6040.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6040.2.a.j 1 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$151$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6040))$$:

 $$T_{3} - 2$$ $$T_{7} - 2$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

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