# Properties

 Label 8030.2.a.l 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.

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

## 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
 −1.30278 2.30278
−1.00000 −1.30278 1.00000 −1.00000 1.30278 −3.30278 −1.00000 −1.30278 1.00000
1.2 −1.00000 2.30278 1.00000 −1.00000 −2.30278 0.302776 −1.00000 2.30278 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.l 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 - T + 3 T^{2} - 3 T^{3} + 9 T^{4}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 + 3 T + 13 T^{2} + 21 T^{3} + 49 T^{4}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$1 - T - 3 T^{2} - 13 T^{3} + 169 T^{4}$$
$17$ $$1 - 3 T + 7 T^{2} - 51 T^{3} + 289 T^{4}$$
$19$ $$( 1 + 4 T + 19 T^{2} )^{2}$$
$23$ $$1 + T + 43 T^{2} + 23 T^{3} + 529 T^{4}$$
$29$ $$1 + 7 T + 67 T^{2} + 203 T^{3} + 841 T^{4}$$
$31$ $$1 + 6 T + 58 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 - 7 T + 57 T^{2} - 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 6 T + 41 T^{2} )^{2}$$
$43$ $$1 + 4 T + 38 T^{2} + 172 T^{3} + 1849 T^{4}$$
$47$ $$1 + 8 T + 58 T^{2} + 376 T^{3} + 2209 T^{4}$$
$53$ $$( 1 + 53 T^{2} )^{2}$$
$59$ $$1 + 6 T + 10 T^{2} + 354 T^{3} + 3481 T^{4}$$
$61$ $$1 - 6 T + 118 T^{2} - 366 T^{3} + 3721 T^{4}$$
$67$ $$1 + T + 131 T^{2} + 67 T^{3} + 4489 T^{4}$$
$71$ $$1 - 11 T + 91 T^{2} - 781 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - T )^{2}$$
$79$ $$1 - 8 T + 122 T^{2} - 632 T^{3} + 6241 T^{4}$$
$83$ $$1 - 17 T + 235 T^{2} - 1411 T^{3} + 6889 T^{4}$$
$89$ $$1 - 3 T - 83 T^{2} - 267 T^{3} + 7921 T^{4}$$
$97$ $$1 + 21 T + 223 T^{2} + 2037 T^{3} + 9409 T^{4}$$