# Properties

 Label 8030.2.a.n 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.

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

## 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 −3.30278 1.00000 1.00000 −3.30278 0 1.00000 7.90833 1.00000
1.2 1.00000 0.302776 1.00000 1.00000 0.302776 0 1.00000 −2.90833 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.n 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 - T )^{2}$$
$3$ $$1 + 3 T + 5 T^{2} + 9 T^{3} + 9 T^{4}$$
$5$ $$( 1 - T )^{2}$$
$7$ $$( 1 + 7 T^{2} )^{2}$$
$11$ $$( 1 - T )^{2}$$
$13$ $$1 - 7 T + 35 T^{2} - 91 T^{3} + 169 T^{4}$$
$17$ $$1 + 6 T + 30 T^{2} + 102 T^{3} + 289 T^{4}$$
$19$ $$1 + 7 T + 47 T^{2} + 133 T^{3} + 361 T^{4}$$
$23$ $$1 + 2 T + 34 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$1 - 2 T + 46 T^{2} - 58 T^{3} + 841 T^{4}$$
$31$ $$( 1 + 6 T + 31 T^{2} )^{2}$$
$37$ $$( 1 + 6 T + 37 T^{2} )^{2}$$
$41$ $$1 + 10 T + 94 T^{2} + 410 T^{3} + 1681 T^{4}$$
$43$ $$1 + 5 T + 89 T^{2} + 215 T^{3} + 1849 T^{4}$$
$47$ $$1 + 7 T + 77 T^{2} + 329 T^{3} + 2209 T^{4}$$
$53$ $$1 - 18 T + 174 T^{2} - 954 T^{3} + 2809 T^{4}$$
$59$ $$1 + T + 115 T^{2} + 59 T^{3} + 3481 T^{4}$$
$61$ $$1 + 7 T + 105 T^{2} + 427 T^{3} + 3721 T^{4}$$
$67$ $$1 - 7 T + 117 T^{2} - 469 T^{3} + 4489 T^{4}$$
$71$ $$1 + 11 T + 91 T^{2} + 781 T^{3} + 5041 T^{4}$$
$73$ $$( 1 - T )^{2}$$
$79$ $$( 1 + 2 T + 79 T^{2} )^{2}$$
$83$ $$1 + 7 T + 175 T^{2} + 581 T^{3} + 6889 T^{4}$$
$89$ $$1 + 15 T + 205 T^{2} + 1335 T^{3} + 7921 T^{4}$$
$97$ $$1 + 22 T + 302 T^{2} + 2134 T^{3} + 9409 T^{4}$$