# Properties

 Label 1006.2.a.f Level 1006 Weight 2 Character orbit 1006.a Self dual yes Analytic conductor 8.033 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1006 = 2 \cdot 503$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1006.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.03295044334$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ 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 $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
−1.00000 −2.23607 1.00000 −3.23607 2.23607 0.236068 −1.00000 2.00000 3.23607
1.2 −1.00000 2.23607 1.00000 1.23607 −2.23607 −4.23607 −1.00000 2.00000 −1.23607
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$503$$ $$1$$

## 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 1006.2.a.f 2
3.b odd 2 1 9054.2.a.y 2
4.b odd 2 1 8048.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1006.2.a.f 2 1.a even 1 1 trivial
8048.2.a.l 2 4.b odd 2 1
9054.2.a.y 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(1006))$$:

 $$T_{3}^{2} - 5$$ $$T_{5}^{2} + 2 T_{5} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + T^{2} + 9 T^{4}$$
$5$ $$1 + 2 T + 6 T^{2} + 10 T^{3} + 25 T^{4}$$
$7$ $$1 + 4 T + 13 T^{2} + 28 T^{3} + 49 T^{4}$$
$11$ $$1 + 4 T + 21 T^{2} + 44 T^{3} + 121 T^{4}$$
$13$ $$1 + 2 T + 7 T^{2} + 26 T^{3} + 169 T^{4}$$
$17$ $$( 1 - 2 T + 17 T^{2} )^{2}$$
$19$ $$1 + 2 T - 6 T^{2} + 38 T^{3} + 361 T^{4}$$
$23$ $$1 + 41 T^{2} + 529 T^{4}$$
$29$ $$1 + 10 T + 78 T^{2} + 290 T^{3} + 841 T^{4}$$
$31$ $$1 - 2 T + 18 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 + 54 T^{2} + 1369 T^{4}$$
$41$ $$1 + 6 T + 86 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 - 16 T + 145 T^{2} - 688 T^{3} + 1849 T^{4}$$
$47$ $$1 - 4 T + 93 T^{2} - 188 T^{3} + 2209 T^{4}$$
$53$ $$1 - 8 T + 102 T^{2} - 424 T^{3} + 2809 T^{4}$$
$59$ $$1 + 38 T^{2} + 3481 T^{4}$$
$61$ $$1 - 6 T + 111 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T + 93 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$1 - 6 T + 146 T^{2} - 426 T^{3} + 5041 T^{4}$$
$73$ $$1 + 24 T + 270 T^{2} + 1752 T^{3} + 5329 T^{4}$$
$79$ $$1 - 8 T + 94 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 20 T + 221 T^{2} - 1660 T^{3} + 6889 T^{4}$$
$89$ $$1 + 20 T + 258 T^{2} + 1780 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 10 T + 97 T^{2} )^{2}$$