Properties

 Label 858.2.a.k Level 858 Weight 2 Character orbit 858.a Self dual yes Analytic conductor 6.851 Analytic rank 0 Dimension 1 CM no Inner twists 1

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$858 = 2 \cdot 3 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 858.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$6.85116449343$$ 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

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

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 1.00000 1.00000 1.00000 −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
$$2$$ $$-1$$
$$3$$ $$-1$$
$$11$$ $$-1$$
$$13$$ $$1$$

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 858.2.a.k 1
3.b odd 2 1 2574.2.a.j 1
4.b odd 2 1 6864.2.a.g 1
11.b odd 2 1 9438.2.a.m 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
858.2.a.k 1 1.a even 1 1 trivial
2574.2.a.j 1 3.b odd 2 1
6864.2.a.g 1 4.b odd 2 1
9438.2.a.m 1 11.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(858))$$:

 $$T_{5} + 1$$ $$T_{7} - 1$$ $$T_{17} - 4$$

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ $$1 - T$$
$5$ $$1 + T + 5 T^{2}$$
$7$ $$1 - T + 7 T^{2}$$
$11$ $$1 - T$$
$13$ $$1 + T$$
$17$ $$1 - 4 T + 17 T^{2}$$
$19$ $$1 - 6 T + 19 T^{2}$$
$23$ $$1 - 3 T + 23 T^{2}$$
$29$ $$1 + 5 T + 29 T^{2}$$
$31$ $$1 - 4 T + 31 T^{2}$$
$37$ $$1 - 10 T + 37 T^{2}$$
$41$ $$1 + 7 T + 41 T^{2}$$
$43$ $$1 + 5 T + 43 T^{2}$$
$47$ $$1 + 8 T + 47 T^{2}$$
$53$ $$1 + 2 T + 53 T^{2}$$
$59$ $$1 + 3 T + 59 T^{2}$$
$61$ $$1 - 13 T + 61 T^{2}$$
$67$ $$1 + 9 T + 67 T^{2}$$
$71$ $$1 - 2 T + 71 T^{2}$$
$73$ $$1 + 3 T + 73 T^{2}$$
$79$ $$1 - 10 T + 79 T^{2}$$
$83$ $$1 + 14 T + 83 T^{2}$$
$89$ $$1 + 8 T + 89 T^{2}$$
$97$ $$1 + 14 T + 97 T^{2}$$