# Properties

 Label 471.2.a.b Level $471$ Weight $2$ Character orbit 471.a Self dual yes Analytic conductor $3.761$ Analytic rank $1$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$471 = 3 \cdot 157$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 471.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$3.76095393520$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−1.61803 1.00000 0.618034 −1.00000 −1.61803 −3.00000 2.23607 1.00000 1.61803
1.2 0.618034 1.00000 −1.61803 −1.00000 0.618034 −3.00000 −2.23607 1.00000 −0.618034
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$-1$$
$$157$$ $$-1$$

## 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 471.2.a.b 2
3.b odd 2 1 1413.2.a.b 2
4.b odd 2 1 7536.2.a.m 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
471.2.a.b 2 1.a even 1 1 trivial
1413.2.a.b 2 3.b odd 2 1
7536.2.a.m 2 4.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{2} + T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(471))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T + T^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$( 3 + T )^{2}$$
$11$ $$-11 + T + T^{2}$$
$13$ $$1 + 3 T + T^{2}$$
$17$ $$-1 + 4 T + T^{2}$$
$19$ $$4 + 6 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-1 + T + T^{2}$$
$31$ $$-59 + 3 T + T^{2}$$
$37$ $$4 - 6 T + T^{2}$$
$41$ $$59 + 16 T + T^{2}$$
$43$ $$76 + 18 T + T^{2}$$
$47$ $$61 - 17 T + T^{2}$$
$53$ $$-20 + T^{2}$$
$59$ $$-124 - 2 T + T^{2}$$
$61$ $$59 + 16 T + T^{2}$$
$67$ $$-29 - 3 T + T^{2}$$
$71$ $$-11 + T + T^{2}$$
$73$ $$-71 + 11 T + T^{2}$$
$79$ $$-125 + T^{2}$$
$83$ $$-19 - 13 T + T^{2}$$
$89$ $$-211 - T + T^{2}$$
$97$ $$11 + 8 T + T^{2}$$