# Properties

 Label 574.2.a.h Level $574$ Weight $2$ Character orbit 574.a Self dual yes Analytic conductor $4.583$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$574 = 2 \cdot 7 \cdot 41$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 574.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$4.58341307602$$ 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

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

## 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 −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
$$2$$ $$-1$$
$$7$$ $$1$$
$$41$$ $$-1$$

## 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 574.2.a.h 1
3.b odd 2 1 5166.2.a.m 1
4.b odd 2 1 4592.2.a.h 1
7.b odd 2 1 4018.2.a.p 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
574.2.a.h 1 1.a even 1 1 trivial
4018.2.a.p 1 7.b odd 2 1
4592.2.a.h 1 4.b odd 2 1
5166.2.a.m 1 3.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(574))$$:

 $$T_{3} + 1$$ $$T_{5} + 1$$ $$T_{11} + 6$$

## Hecke characteristic polynomials

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