# Properties

 Label 574.2.a.m Level $574$ Weight $2$ Character orbit 574.a Self dual yes Analytic conductor $4.583$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.11348.1 Defining polynomial: $$x^{4} - x^{3} - 5 x^{2} + x + 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{2} q^{3} + q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{2} q^{3} + q^{4} + ( 1 - \beta_{2} + \beta_{3} ) q^{5} + \beta_{2} q^{6} - q^{7} - q^{8} + ( 2 - \beta_{1} + \beta_{3} ) q^{9} + ( -1 + \beta_{2} - \beta_{3} ) q^{10} + ( 1 - \beta_{3} ) q^{11} -\beta_{2} q^{12} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{13} + q^{14} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{15} + q^{16} + ( 2 \beta_{1} + \beta_{2} ) q^{17} + ( -2 + \beta_{1} - \beta_{3} ) q^{18} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{19} + ( 1 - \beta_{2} + \beta_{3} ) q^{20} + \beta_{2} q^{21} + ( -1 + \beta_{3} ) q^{22} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{23} + \beta_{2} q^{24} + ( 3 - 2 \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{25} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{26} + ( -3 - \beta_{1} - 2 \beta_{2} - 3 \beta_{3} ) q^{27} - q^{28} + ( 4 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{30} + ( 1 + \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{31} - q^{32} + ( 2 + 2 \beta_{3} ) q^{33} + ( -2 \beta_{1} - \beta_{2} ) q^{34} + ( -1 + \beta_{2} - \beta_{3} ) q^{35} + ( 2 - \beta_{1} + \beta_{3} ) q^{36} + ( -2 + 2 \beta_{2} ) q^{37} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + ( 4 + 2 \beta_{2} ) q^{39} + ( -1 + \beta_{2} - \beta_{3} ) q^{40} + q^{41} -\beta_{2} q^{42} + ( -3 + \beta_{1} + \beta_{3} ) q^{43} + ( 1 - \beta_{3} ) q^{44} + ( 8 - 3 \beta_{1} - \beta_{2} ) q^{45} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{46} + ( 1 + \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{47} -\beta_{2} q^{48} + q^{49} + ( -3 + 2 \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{50} + ( -3 + 3 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{51} + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 8 + \beta_{1} - 2 \beta_{2} + 4 \beta_{3} ) q^{53} + ( 3 + \beta_{1} + 2 \beta_{2} + 3 \beta_{3} ) q^{54} + ( -3 + \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{55} + q^{56} + 4 q^{57} + ( -4 - \beta_{1} - 2 \beta_{2} ) q^{58} + ( 4 - \beta_{1} - \beta_{2} - 2 \beta_{3} ) q^{59} + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{60} + ( 5 - 2 \beta_{1} - \beta_{2} - 3 \beta_{3} ) q^{61} + ( -1 - \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{62} + ( -2 + \beta_{1} - \beta_{3} ) q^{63} + q^{64} + ( 4 + 2 \beta_{2} - 4 \beta_{3} ) q^{65} + ( -2 - 2 \beta_{3} ) q^{66} + ( 1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{67} + ( 2 \beta_{1} + \beta_{2} ) q^{68} + ( -4 - 2 \beta_{2} ) q^{69} + ( 1 - \beta_{2} + \beta_{3} ) q^{70} + ( 3 - 5 \beta_{1} - \beta_{3} ) q^{71} + ( -2 + \beta_{1} - \beta_{3} ) q^{72} + ( 2 - 4 \beta_{1} ) q^{73} + ( 2 - 2 \beta_{2} ) q^{74} + ( 7 - 3 \beta_{1} - 5 \beta_{2} + 3 \beta_{3} ) q^{75} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} ) q^{76} + ( -1 + \beta_{3} ) q^{77} + ( -4 - 2 \beta_{2} ) q^{78} + ( -7 - \beta_{1} + \beta_{3} ) q^{79} + ( 1 - \beta_{2} + \beta_{3} ) q^{80} + ( 9 + 4 \beta_{2} + 4 \beta_{3} ) q^{81} - q^{82} + ( -4 + \beta_{1} - \beta_{2} + 2 \beta_{3} ) q^{83} + \beta_{2} q^{84} + ( -7 + 5 \beta_{1} + 4 \beta_{2} + \beta_{3} ) q^{85} + ( 3 - \beta_{1} - \beta_{3} ) q^{86} + ( -9 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{87} + ( -1 + \beta_{3} ) q^{88} + ( 9 + \beta_{1} + 2 \beta_{2} - \beta_{3} ) q^{89} + ( -8 + 3 \beta_{1} + \beta_{2} ) q^{90} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{91} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{92} + ( -11 + 3 \beta_{1} - 3 \beta_{3} ) q^{93} + ( -1 - \beta_{1} - 3 \beta_{2} + \beta_{3} ) q^{94} + ( 6 - 2 \beta_{3} ) q^{95} + \beta_{2} q^{96} + ( -1 - \beta_{1} - 3 \beta_{3} ) q^{97} - q^{98} + ( -7 - 4 \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} - q^{3} + 4q^{4} + 3q^{5} + q^{6} - 4q^{7} - 4q^{8} + 9q^{9} + O(q^{10})$$ $$4q - 4q^{2} - q^{3} + 4q^{4} + 3q^{5} + q^{6} - 4q^{7} - 4q^{8} + 9q^{9} - 3q^{10} + 4q^{11} - q^{12} - 6q^{13} + 4q^{14} + 11q^{15} + 4q^{16} - q^{17} - 9q^{18} + 2q^{19} + 3q^{20} + q^{21} - 4q^{22} + 6q^{23} + q^{24} + 13q^{25} + 6q^{26} - 13q^{27} - 4q^{28} + 17q^{29} - 11q^{30} + 5q^{31} - 4q^{32} + 8q^{33} + q^{34} - 3q^{35} + 9q^{36} - 6q^{37} - 2q^{38} + 18q^{39} - 3q^{40} + 4q^{41} - q^{42} - 13q^{43} + 4q^{44} + 34q^{45} - 6q^{46} + 6q^{47} - q^{48} + 4q^{49} - 13q^{50} - 11q^{51} - 6q^{52} + 29q^{53} + 13q^{54} - 16q^{55} + 4q^{56} + 16q^{57} - 17q^{58} + 16q^{59} + 11q^{60} + 21q^{61} - 5q^{62} - 9q^{63} + 4q^{64} + 18q^{65} - 8q^{66} + 4q^{67} - q^{68} - 18q^{69} + 3q^{70} + 17q^{71} - 9q^{72} + 12q^{73} + 6q^{74} + 26q^{75} + 2q^{76} - 4q^{77} - 18q^{78} - 27q^{79} + 3q^{80} + 40q^{81} - 4q^{82} - 18q^{83} + q^{84} - 29q^{85} + 13q^{86} - 41q^{87} - 4q^{88} + 37q^{89} - 34q^{90} + 6q^{91} + 6q^{92} - 47q^{93} - 6q^{94} + 24q^{95} + q^{96} - 3q^{97} - 4q^{98} - 32q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - x^{3} - 5 x^{2} + x + 2$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} - 3$$ $$\beta_{2}$$ $$=$$ $$\nu^{3} - \nu^{2} - 5 \nu + 1$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 3 \nu + 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} - \beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{1} + 3$$ $$\nu^{3}$$ $$=$$ $$($$$$5 \beta_{3} - 3 \beta_{2} + 7 \beta_{1} + 9$$$$)/2$$

## 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
 −0.589216 −1.77571 2.64119 0.723742
−1.00000 −3.39434 1.00000 1.47439 3.39434 −1.00000 −1.00000 8.52156 −1.47439
1.2 −1.00000 −1.12631 1.00000 −3.70458 1.12631 −1.00000 −1.00000 −1.73143 3.70458
1.3 −1.00000 0.757235 1.00000 1.30651 −0.757235 −1.00000 −1.00000 −2.42659 −1.30651
1.4 −1.00000 2.76342 1.00000 3.92368 −2.76342 −1.00000 −1.00000 4.63646 −3.92368
 $$n$$: 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.m 4
3.b odd 2 1 5166.2.a.bx 4
4.b odd 2 1 4592.2.a.ba 4
7.b odd 2 1 4018.2.a.bj 4

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

 $$T_{3}^{4} + T_{3}^{3} - 10 T_{3}^{2} - 4 T_{3} + 8$$ $$T_{5}^{4} - 3 T_{5}^{3} - 12 T_{5}^{2} + 40 T_{5} - 28$$ $$T_{11}^{4} - 4 T_{11}^{3} - 8 T_{11}^{2} + 28 T_{11} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$8 - 4 T - 10 T^{2} + T^{3} + T^{4}$$
$5$ $$-28 + 40 T - 12 T^{2} - 3 T^{3} + T^{4}$$
$7$ $$( 1 + T )^{4}$$
$11$ $$-16 + 28 T - 8 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$-32 - 64 T - 8 T^{2} + 6 T^{3} + T^{4}$$
$17$ $$152 - 28 T - 58 T^{2} + T^{3} + T^{4}$$
$19$ $$32 + 8 T - 20 T^{2} - 2 T^{3} + T^{4}$$
$23$ $$-32 + 64 T - 8 T^{2} - 6 T^{3} + T^{4}$$
$29$ $$-1348 + 248 T + 62 T^{2} - 17 T^{3} + T^{4}$$
$31$ $$32 + 176 T - 56 T^{2} - 5 T^{3} + T^{4}$$
$37$ $$32 - 120 T - 28 T^{2} + 6 T^{3} + T^{4}$$
$41$ $$( -1 + T )^{4}$$
$43$ $$-32 + 32 T + 48 T^{2} + 13 T^{3} + T^{4}$$
$47$ $$-1184 + 656 T - 80 T^{2} - 6 T^{3} + T^{4}$$
$53$ $$-16132 + 1992 T + 150 T^{2} - 29 T^{3} + T^{4}$$
$59$ $$-664 + 388 T + 28 T^{2} - 16 T^{3} + T^{4}$$
$61$ $$692 + 864 T + 32 T^{2} - 21 T^{3} + T^{4}$$
$67$ $$752 + 420 T - 96 T^{2} - 4 T^{3} + T^{4}$$
$71$ $$-18016 + 4640 T - 200 T^{2} - 17 T^{3} + T^{4}$$
$73$ $$-2896 + 2352 T - 176 T^{2} - 12 T^{3} + T^{4}$$
$79$ $$256 + 640 T + 232 T^{2} + 27 T^{3} + T^{4}$$
$83$ $$16 + 108 T + 80 T^{2} + 18 T^{3} + T^{4}$$
$89$ $$1384 - 2012 T + 458 T^{2} - 37 T^{3} + T^{4}$$
$97$ $$344 - 252 T - 98 T^{2} + 3 T^{3} + T^{4}$$