# Properties

 Label 2358.2.a.w Level 2358 Weight 2 Character orbit 2358.a Self dual yes Analytic conductor 18.829 Analytic rank 0 Dimension 1 CM no Inner twists 1

# Learn more about

## Newspace parameters

 Level: $$N$$ = $$2358 = 2 \cdot 3^{2} \cdot 131$$ Weight: $$k$$ = $$2$$ Character orbit: $$[\chi]$$ = 2358.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$18.8287247966$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 786) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + q^{8} + O(q^{10})$$ $$q + q^{2} + q^{4} + 2q^{5} + 2q^{7} + q^{8} + 2q^{10} + 3q^{11} + 3q^{13} + 2q^{14} + q^{16} + 5q^{17} + q^{19} + 2q^{20} + 3q^{22} - 4q^{23} - q^{25} + 3q^{26} + 2q^{28} - 9q^{29} - 5q^{31} + q^{32} + 5q^{34} + 4q^{35} - 8q^{37} + q^{38} + 2q^{40} + 12q^{41} - 6q^{43} + 3q^{44} - 4q^{46} + 8q^{47} - 3q^{49} - q^{50} + 3q^{52} - 12q^{53} + 6q^{55} + 2q^{56} - 9q^{58} + 5q^{59} - 3q^{61} - 5q^{62} + q^{64} + 6q^{65} + 5q^{68} + 4q^{70} + 8q^{71} - 2q^{73} - 8q^{74} + q^{76} + 6q^{77} - 8q^{79} + 2q^{80} + 12q^{82} + 14q^{83} + 10q^{85} - 6q^{86} + 3q^{88} + 14q^{89} + 6q^{91} - 4q^{92} + 8q^{94} + 2q^{95} + 12q^{97} - 3q^{98} + O(q^{100})$$

## 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 0 1.00000 2.00000 0 2.00000 1.00000 0 2.00000
 $$n$$: 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 2358.2.a.w 1
3.b odd 2 1 786.2.a.a 1
12.b even 2 1 6288.2.a.i 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
786.2.a.a 1 3.b odd 2 1
2358.2.a.w 1 1.a even 1 1 trivial
6288.2.a.i 1 12.b even 2 1

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$-1$$
$$131$$ $$-1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(2358))$$:

 $$T_{5} - 2$$ $$T_{7} - 2$$ $$T_{11} - 3$$ $$T_{17} - 5$$

## Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ $$1 - T$$
$3$ 
$5$ $$1 - 2 T + 5 T^{2}$$
$7$ $$1 - 2 T + 7 T^{2}$$
$11$ $$1 - 3 T + 11 T^{2}$$
$13$ $$1 - 3 T + 13 T^{2}$$
$17$ $$1 - 5 T + 17 T^{2}$$
$19$ $$1 - T + 19 T^{2}$$
$23$ $$1 + 4 T + 23 T^{2}$$
$29$ $$1 + 9 T + 29 T^{2}$$
$31$ $$1 + 5 T + 31 T^{2}$$
$37$ $$1 + 8 T + 37 T^{2}$$
$41$ $$1 - 12 T + 41 T^{2}$$
$43$ $$1 + 6 T + 43 T^{2}$$
$47$ $$1 - 8 T + 47 T^{2}$$
$53$ $$1 + 12 T + 53 T^{2}$$
$59$ $$1 - 5 T + 59 T^{2}$$
$61$ $$1 + 3 T + 61 T^{2}$$
$67$ $$1 + 67 T^{2}$$
$71$ $$1 - 8 T + 71 T^{2}$$
$73$ $$1 + 2 T + 73 T^{2}$$
$79$ $$1 + 8 T + 79 T^{2}$$
$83$ $$1 - 14 T + 83 T^{2}$$
$89$ $$1 - 14 T + 89 T^{2}$$
$97$ $$1 - 12 T + 97 T^{2}$$
show more
show less