# Properties

 Label 8030.2.a.k 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: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{13})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 2.30278 −1.30278
−1.00000 −2.30278 1.00000 −1.00000 2.30278 2.60555 −1.00000 2.30278 1.00000
1.2 −1.00000 1.30278 1.00000 −1.00000 −1.30278 −4.60555 −1.00000 −1.30278 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.k 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8030.2.a.k 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(8030))$$:

 $$T_{3}^{2} + T_{3} - 3$$ $$T_{7}^{2} + 2 T_{7} - 12$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 + T + 3 T^{2} + 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 + 2 T + 2 T^{2} + 14 T^{3} + 49 T^{4}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$1 - 3 T + 25 T^{2} - 39 T^{3} + 169 T^{4}$$
$17$ $$( 1 + 17 T^{2} )^{2}$$
$19$ $$1 + T + 35 T^{2} + 19 T^{3} + 361 T^{4}$$
$23$ $$( 1 - 2 T + 23 T^{2} )^{2}$$
$29$ $$1 - 2 T + 46 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$1 + 6 T + 58 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$( 1 - 2 T + 37 T^{2} )^{2}$$
$41$ $$( 1 - 6 T + 41 T^{2} )^{2}$$
$43$ $$1 - 5 T + 89 T^{2} - 215 T^{3} + 1849 T^{4}$$
$47$ $$1 + 3 T + 15 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 4 T + 58 T^{2} - 212 T^{3} + 2809 T^{4}$$
$59$ $$1 + 11 T + 145 T^{2} + 649 T^{3} + 3481 T^{4}$$
$61$ $$1 + 19 T + 183 T^{2} + 1159 T^{3} + 3721 T^{4}$$
$67$ $$1 + 3 T + 107 T^{2} + 201 T^{3} + 4489 T^{4}$$
$71$ $$1 - 5 T - 11 T^{2} - 355 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - T )^{2}$$
$79$ $$1 + 10 T + 170 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 11 T + 115 T^{2} - 913 T^{3} + 6889 T^{4}$$
$89$ $$1 + 15 T + 205 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 + 12 T + 178 T^{2} + 1164 T^{3} + 9409 T^{4}$$