# Properties

 Label 574.2.a.f Level $574$ Weight $2$ Character orbit 574.a Self dual yes Analytic conductor $4.583$ Analytic rank $0$ 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: $$0$$ 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

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

## 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 3.00000 1.00000 −1.00000 −3.00000 1.00000 −1.00000 6.00000 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.f 1
3.b odd 2 1 5166.2.a.bg 1
4.b odd 2 1 4592.2.a.a 1
7.b odd 2 1 4018.2.a.b 1

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

 $$T_{3} - 3$$ $$T_{5} + 1$$ $$T_{11} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T$$
$3$ $$-3 + T$$
$5$ $$1 + T$$
$7$ $$-1 + T$$
$11$ $$-4 + T$$
$13$ $$6 + T$$
$17$ $$-3 + T$$
$19$ $$-4 + T$$
$23$ $$-2 + T$$
$29$ $$-1 + T$$
$31$ $$-9 + T$$
$37$ $$8 + T$$
$41$ $$1 + T$$
$43$ $$5 + T$$
$47$ $$6 + T$$
$53$ $$3 + T$$
$59$ $$-14 + T$$
$61$ $$11 + T$$
$67$ $$8 + T$$
$71$ $$-3 + T$$
$73$ $$14 + T$$
$79$ $$-7 + T$$
$83$ $$-16 + T$$
$89$ $$-5 + T$$
$97$ $$-1 + T$$