# Properties

 Label 8008.2.a.g Level 8008 Weight 2 Character orbit 8008.a Self dual yes Analytic conductor 63.944 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8008 = 2^{3} \cdot 7 \cdot 11 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8008.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$63.9442019386$$ 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: $$1$$ 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.

 $$f(q)$$ $$=$$ $$q + \beta q^{3} + ( 1 - \beta ) q^{5} + q^{7} + ( -2 + \beta ) q^{9} +O(q^{10})$$ $$q + \beta q^{3} + ( 1 - \beta ) q^{5} + q^{7} + ( -2 + \beta ) q^{9} + q^{11} + q^{13} - q^{15} + ( -6 + 3 \beta ) q^{17} + ( 1 + \beta ) q^{19} + \beta q^{21} + ( 2 - 6 \beta ) q^{23} + ( -3 - \beta ) q^{25} + ( 1 - 4 \beta ) q^{27} + ( 2 + 4 \beta ) q^{29} + ( -4 - 2 \beta ) q^{31} + \beta q^{33} + ( 1 - \beta ) q^{35} + ( 8 - 6 \beta ) q^{37} + \beta q^{39} + ( -4 + 2 \beta ) q^{41} + ( -2 + \beta ) q^{43} + ( -3 + 2 \beta ) q^{45} + ( -6 - 2 \beta ) q^{47} + q^{49} + ( 3 - 3 \beta ) q^{51} + ( 2 - \beta ) q^{53} + ( 1 - \beta ) q^{55} + ( 1 + 2 \beta ) q^{57} -2 \beta q^{59} + ( -8 - 3 \beta ) q^{61} + ( -2 + \beta ) q^{63} + ( 1 - \beta ) q^{65} + ( 10 - 5 \beta ) q^{67} + ( -6 - 4 \beta ) q^{69} + ( -7 + 9 \beta ) q^{71} + ( 2 - 4 \beta ) q^{73} + ( -1 - 4 \beta ) q^{75} + q^{77} + ( -7 + 13 \beta ) q^{79} + ( 2 - 6 \beta ) q^{81} + ( 1 + \beta ) q^{83} + ( -9 + 6 \beta ) q^{85} + ( 4 + 6 \beta ) q^{87} + ( -18 + \beta ) q^{89} + q^{91} + ( -2 - 6 \beta ) q^{93} -\beta q^{95} -2 q^{97} + ( -2 + \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{3} + q^{5} + 2q^{7} - 3q^{9} + O(q^{10})$$ $$2q + q^{3} + q^{5} + 2q^{7} - 3q^{9} + 2q^{11} + 2q^{13} - 2q^{15} - 9q^{17} + 3q^{19} + q^{21} - 2q^{23} - 7q^{25} - 2q^{27} + 8q^{29} - 10q^{31} + q^{33} + q^{35} + 10q^{37} + q^{39} - 6q^{41} - 3q^{43} - 4q^{45} - 14q^{47} + 2q^{49} + 3q^{51} + 3q^{53} + q^{55} + 4q^{57} - 2q^{59} - 19q^{61} - 3q^{63} + q^{65} + 15q^{67} - 16q^{69} - 5q^{71} - 6q^{75} + 2q^{77} - q^{79} - 2q^{81} + 3q^{83} - 12q^{85} + 14q^{87} - 35q^{89} + 2q^{91} - 10q^{93} - q^{95} - 4q^{97} - 3q^{99} + O(q^{100})$$

## 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
 −0.618034 1.61803
0 −0.618034 0 1.61803 0 1.00000 0 −2.61803 0
1.2 0 1.61803 0 −0.618034 0 1.00000 0 −0.381966 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$7$$ $$-1$$
$$11$$ $$-1$$
$$13$$ $$-1$$

## 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 8008.2.a.g 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8008.2.a.g 2 1.a even 1 1 trivial

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8008))$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{5}^{2} - T_{5} - 1$$ $$T_{17}^{2} + 9 T_{17} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ $$1 - T + 5 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$1 - T + 9 T^{2} - 5 T^{3} + 25 T^{4}$$
$7$ $$( 1 - T )^{2}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$( 1 - T )^{2}$$
$17$ $$1 + 9 T + 43 T^{2} + 153 T^{3} + 289 T^{4}$$
$19$ $$1 - 3 T + 39 T^{2} - 57 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 2 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 - 8 T + 54 T^{2} - 232 T^{3} + 841 T^{4}$$
$31$ $$1 + 10 T + 82 T^{2} + 310 T^{3} + 961 T^{4}$$
$37$ $$1 - 10 T + 54 T^{2} - 370 T^{3} + 1369 T^{4}$$
$41$ $$1 + 6 T + 86 T^{2} + 246 T^{3} + 1681 T^{4}$$
$43$ $$1 + 3 T + 87 T^{2} + 129 T^{3} + 1849 T^{4}$$
$47$ $$1 + 14 T + 138 T^{2} + 658 T^{3} + 2209 T^{4}$$
$53$ $$1 - 3 T + 107 T^{2} - 159 T^{3} + 2809 T^{4}$$
$59$ $$1 + 2 T + 114 T^{2} + 118 T^{3} + 3481 T^{4}$$
$61$ $$1 + 19 T + 201 T^{2} + 1159 T^{3} + 3721 T^{4}$$
$67$ $$1 - 15 T + 159 T^{2} - 1005 T^{3} + 4489 T^{4}$$
$71$ $$1 + 5 T + 47 T^{2} + 355 T^{3} + 5041 T^{4}$$
$73$ $$1 + 126 T^{2} + 5329 T^{4}$$
$79$ $$1 + T - 53 T^{2} + 79 T^{3} + 6241 T^{4}$$
$83$ $$1 - 3 T + 167 T^{2} - 249 T^{3} + 6889 T^{4}$$
$89$ $$1 + 35 T + 483 T^{2} + 3115 T^{3} + 7921 T^{4}$$
$97$ $$( 1 + 2 T + 97 T^{2} )^{2}$$