# Properties

 Label 8030.2.a.o Level 8030 Weight 2 Character orbit 8030.a Self dual yes Analytic conductor 64.120 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8030 = 2 \cdot 5 \cdot 11 \cdot 73$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8030.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1198728231$$ 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.

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

## 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
 1.61803 −0.618034
1.00000 −1.61803 1.00000 −1.00000 −1.61803 −1.23607 1.00000 −0.381966 −1.00000
1.2 1.00000 0.618034 1.00000 −1.00000 0.618034 3.23607 1.00000 −2.61803 −1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## 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 8030.2.a.o 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.o 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$5$$ $$1$$
$$11$$ $$-1$$
$$73$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8030))$$:

 $$T_{3}^{2} + T_{3} - 1$$ $$T_{7}^{2} - 2 T_{7} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + T + 5 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 - 2 T + 10 T^{2} - 14 T^{3} + 49 T^{4}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$1 - T + 25 T^{2} - 13 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 4 T + 17 T^{2} )^{2}$$
$19$ $$1 + 9 T + 57 T^{2} + 171 T^{3} + 361 T^{4}$$
$23$ $$( 1 - 2 T + 23 T^{2} )^{2}$$
$29$ $$1 + 2 T + 54 T^{2} + 58 T^{3} + 841 T^{4}$$
$31$ $$1 - 2 T + 18 T^{2} - 62 T^{3} + 961 T^{4}$$
$37$ $$1 - 8 T + 70 T^{2} - 296 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 2 T + 41 T^{2} )^{2}$$
$43$ $$1 + 9 T + 105 T^{2} + 387 T^{3} + 1849 T^{4}$$
$47$ $$1 + 9 T + 53 T^{2} + 423 T^{3} + 2209 T^{4}$$
$53$ $$1 - 20 T + 186 T^{2} - 1060 T^{3} + 2809 T^{4}$$
$59$ $$1 - T + 17 T^{2} - 59 T^{3} + 3481 T^{4}$$
$61$ $$1 + 5 T + 97 T^{2} + 305 T^{3} + 3721 T^{4}$$
$67$ $$1 + 7 T + 45 T^{2} + 469 T^{3} + 4489 T^{4}$$
$71$ $$1 + 3 T + 133 T^{2} + 213 T^{3} + 5041 T^{4}$$
$73$ $$( 1 + T )^{2}$$
$79$ $$1 + 2 T + 114 T^{2} + 158 T^{3} + 6241 T^{4}$$
$83$ $$1 + 27 T + 347 T^{2} + 2241 T^{3} + 6889 T^{4}$$
$89$ $$1 + 3 T + 29 T^{2} + 267 T^{3} + 7921 T^{4}$$
$97$ $$1 + 12 T + 210 T^{2} + 1164 T^{3} + 9409 T^{4}$$