# Properties

 Label 574.2.a.l Level $574$ Weight $2$ Character orbit 574.a Self dual yes Analytic conductor $4.583$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.568.1 Defining polynomial: $$x^{3} - x^{2} - 6 x - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{1} q^{5} -\beta_{2} q^{6} - q^{7} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{1} q^{5} -\beta_{2} q^{6} - q^{7} + q^{8} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{9} + \beta_{1} q^{10} + ( 2 + \beta_{1} + \beta_{2} ) q^{11} -\beta_{2} q^{12} - q^{14} + ( 2 + \beta_{2} ) q^{15} + q^{16} + ( -2 + \beta_{2} ) q^{17} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{18} + ( 2 + 2 \beta_{2} ) q^{19} + \beta_{1} q^{20} + \beta_{2} q^{21} + ( 2 + \beta_{1} + \beta_{2} ) q^{22} + ( 2 - 2 \beta_{1} ) q^{23} -\beta_{2} q^{24} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{25} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{27} - q^{28} + ( 2 - \beta_{1} ) q^{29} + ( 2 + \beta_{2} ) q^{30} + ( 2 - \beta_{2} ) q^{31} + q^{32} + ( -4 + 2 \beta_{1} ) q^{33} + ( -2 + \beta_{2} ) q^{34} -\beta_{1} q^{35} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{36} + 2 q^{37} + ( 2 + 2 \beta_{2} ) q^{38} + \beta_{1} q^{40} - q^{41} + \beta_{2} q^{42} + ( -2 + 3 \beta_{2} ) q^{43} + ( 2 + \beta_{1} + \beta_{2} ) q^{44} + ( -6 - \beta_{1} - \beta_{2} ) q^{45} + ( 2 - 2 \beta_{1} ) q^{46} + 2 q^{47} -\beta_{2} q^{48} + q^{49} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{50} + ( -6 + 2 \beta_{1} + 3 \beta_{2} ) q^{51} + ( -2 - 3 \beta_{1} - 2 \beta_{2} ) q^{53} + ( 2 - 2 \beta_{1} - 3 \beta_{2} ) q^{54} + ( 2 + 4 \beta_{1} ) q^{55} - q^{56} + ( -12 + 4 \beta_{1} ) q^{57} + ( 2 - \beta_{1} ) q^{58} + ( -3 \beta_{1} - \beta_{2} ) q^{59} + ( 2 + \beta_{2} ) q^{60} + ( -4 + \beta_{1} + 2 \beta_{2} ) q^{61} + ( 2 - \beta_{2} ) q^{62} + ( -3 + 2 \beta_{1} + \beta_{2} ) q^{63} + q^{64} + ( -4 + 2 \beta_{1} ) q^{66} + ( 6 + \beta_{1} - 3 \beta_{2} ) q^{67} + ( -2 + \beta_{2} ) q^{68} + ( -4 - 4 \beta_{2} ) q^{69} -\beta_{1} q^{70} + ( 2 + 4 \beta_{1} + 3 \beta_{2} ) q^{71} + ( 3 - 2 \beta_{1} - \beta_{2} ) q^{72} + ( -10 - 2 \beta_{2} ) q^{73} + 2 q^{74} + ( -2 + 2 \beta_{1} + 4 \beta_{2} ) q^{75} + ( 2 + 2 \beta_{2} ) q^{76} + ( -2 - \beta_{1} - \beta_{2} ) q^{77} + ( 6 + 2 \beta_{1} + \beta_{2} ) q^{79} + \beta_{1} q^{80} + ( 5 - 4 \beta_{2} ) q^{81} - q^{82} + ( -4 - \beta_{1} + \beta_{2} ) q^{83} + \beta_{2} q^{84} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{85} + ( -2 + 3 \beta_{2} ) q^{86} + ( -2 - 3 \beta_{2} ) q^{87} + ( 2 + \beta_{1} + \beta_{2} ) q^{88} + ( 8 - 2 \beta_{1} - \beta_{2} ) q^{89} + ( -6 - \beta_{1} - \beta_{2} ) q^{90} + ( 2 - 2 \beta_{1} ) q^{92} + ( 6 - 2 \beta_{1} - 3 \beta_{2} ) q^{93} + 2 q^{94} + ( -4 + 2 \beta_{1} - 2 \beta_{2} ) q^{95} -\beta_{2} q^{96} + ( -8 - 2 \beta_{1} - \beta_{2} ) q^{97} + q^{98} + ( -2 - 3 \beta_{1} + 3 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + 3q^{2} + q^{3} + 3q^{4} + q^{5} + q^{6} - 3q^{7} + 3q^{8} + 8q^{9} + O(q^{10})$$ $$3q + 3q^{2} + q^{3} + 3q^{4} + q^{5} + q^{6} - 3q^{7} + 3q^{8} + 8q^{9} + q^{10} + 6q^{11} + q^{12} - 3q^{14} + 5q^{15} + 3q^{16} - 7q^{17} + 8q^{18} + 4q^{19} + q^{20} - q^{21} + 6q^{22} + 4q^{23} + q^{24} - 2q^{25} + 7q^{27} - 3q^{28} + 5q^{29} + 5q^{30} + 7q^{31} + 3q^{32} - 10q^{33} - 7q^{34} - q^{35} + 8q^{36} + 6q^{37} + 4q^{38} + q^{40} - 3q^{41} - q^{42} - 9q^{43} + 6q^{44} - 18q^{45} + 4q^{46} + 6q^{47} + q^{48} + 3q^{49} - 2q^{50} - 19q^{51} - 7q^{53} + 7q^{54} + 10q^{55} - 3q^{56} - 32q^{57} + 5q^{58} - 2q^{59} + 5q^{60} - 13q^{61} + 7q^{62} - 8q^{63} + 3q^{64} - 10q^{66} + 22q^{67} - 7q^{68} - 8q^{69} - q^{70} + 7q^{71} + 8q^{72} - 28q^{73} + 6q^{74} - 8q^{75} + 4q^{76} - 6q^{77} + 19q^{79} + q^{80} + 19q^{81} - 3q^{82} - 14q^{83} - q^{84} - 7q^{85} - 9q^{86} - 3q^{87} + 6q^{88} + 23q^{89} - 18q^{90} + 4q^{92} + 19q^{93} + 6q^{94} - 8q^{95} + q^{96} - 25q^{97} + 3q^{98} - 12q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 6 x - 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 4$$

## 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
 −1.76156 3.12489 −0.363328
1.00000 −2.62620 1.00000 −1.76156 −2.62620 −1.00000 1.00000 3.89692 −1.76156
1.2 1.00000 0.484862 1.00000 3.12489 0.484862 −1.00000 1.00000 −2.76491 3.12489
1.3 1.00000 3.14134 1.00000 −0.363328 3.14134 −1.00000 1.00000 6.86799 −0.363328
 $$n$$: 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.l 3
3.b odd 2 1 5166.2.a.bt 3
4.b odd 2 1 4592.2.a.s 3
7.b odd 2 1 4018.2.a.bg 3

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

 $$T_{3}^{3} - T_{3}^{2} - 8 T_{3} + 4$$ $$T_{5}^{3} - T_{5}^{2} - 6 T_{5} - 2$$ $$T_{11}^{3} - 6 T_{11}^{2} + 2 T_{11} + 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$4 - 8 T - T^{2} + T^{3}$$
$5$ $$-2 - 6 T - T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$20 + 2 T - 6 T^{2} + T^{3}$$
$13$ $$T^{3}$$
$17$ $$-8 + 8 T + 7 T^{2} + T^{3}$$
$19$ $$32 - 28 T - 4 T^{2} + T^{3}$$
$23$ $$64 - 20 T - 4 T^{2} + T^{3}$$
$29$ $$10 + 2 T - 5 T^{2} + T^{3}$$
$31$ $$8 + 8 T - 7 T^{2} + T^{3}$$
$37$ $$( -2 + T )^{3}$$
$41$ $$( 1 + T )^{3}$$
$43$ $$-232 - 48 T + 9 T^{2} + T^{3}$$
$47$ $$( -2 + T )^{3}$$
$53$ $$-110 - 46 T + 7 T^{2} + T^{3}$$
$59$ $$100 - 50 T + 2 T^{2} + T^{3}$$
$61$ $$10 + 26 T + 13 T^{2} + T^{3}$$
$67$ $$580 + 66 T - 22 T^{2} + T^{3}$$
$71$ $$328 - 104 T - 7 T^{2} + T^{3}$$
$73$ $$512 + 228 T + 28 T^{2} + T^{3}$$
$79$ $$-128 + 96 T - 19 T^{2} + T^{3}$$
$83$ $$-20 + 46 T + 14 T^{2} + T^{3}$$
$89$ $$-236 + 152 T - 23 T^{2} + T^{3}$$
$97$ $$404 + 184 T + 25 T^{2} + T^{3}$$