# Properties

 Label 1339.2.a.a Level $1339$ Weight $2$ Character orbit 1339.a Self dual yes Analytic conductor $10.692$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

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## Newspace parameters

 Level: $$N$$ $$=$$ $$1339 = 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1339.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$10.6919688306$$ 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

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

## 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
 0
1.00000 −1.00000 −1.00000 1.00000 −1.00000 4.00000 −3.00000 −2.00000 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$13$$ $$1$$
$$103$$ $$1$$

## 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 1339.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1339.2.a.a 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1339))$$:

 $$T_{2} - 1$$ $$T_{3} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 2 T^{2}$$
$3$ $$1 + T + 3 T^{2}$$
$5$ $$1 - T + 5 T^{2}$$
$7$ $$1 - 4 T + 7 T^{2}$$
$11$ $$1 + 4 T + 11 T^{2}$$
$13$ $$1 + T$$
$17$ $$1 - 3 T + 17 T^{2}$$
$19$ $$1 + 5 T + 19 T^{2}$$
$23$ $$1 + 23 T^{2}$$
$29$ $$1 + 3 T + 29 T^{2}$$
$31$ $$1 + 31 T^{2}$$
$37$ $$1 + 5 T + 37 T^{2}$$
$41$ $$1 + 41 T^{2}$$
$43$ $$1 + 12 T + 43 T^{2}$$
$47$ $$1 - 6 T + 47 T^{2}$$
$53$ $$1 + 12 T + 53 T^{2}$$
$59$ $$1 + 5 T + 59 T^{2}$$
$61$ $$1 + 5 T + 61 T^{2}$$
$67$ $$1 + 4 T + 67 T^{2}$$
$71$ $$1 + 6 T + 71 T^{2}$$
$73$ $$1 - 5 T + 73 T^{2}$$
$79$ $$1 - 6 T + 79 T^{2}$$
$83$ $$1 + T + 83 T^{2}$$
$89$ $$1 + 9 T + 89 T^{2}$$
$97$ $$1 - 10 T + 97 T^{2}$$
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