# Properties

 Label 8034.2.a.l Level 8034 Weight 2 Character orbit 8034.a Self dual yes Analytic conductor 64.152 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$8034 = 2 \cdot 3 \cdot 13 \cdot 103$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 8034.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$64.1518129839$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.56155 2.56155
−1.00000 −1.00000 1.00000 −2.00000 1.00000 −0.561553 −1.00000 1.00000 2.00000
1.2 −1.00000 −1.00000 1.00000 −2.00000 1.00000 3.56155 −1.00000 1.00000 2.00000
 $$n$$: 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 8034.2.a.l 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8034.2.a.l 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$3$$ $$1$$
$$13$$ $$1$$
$$103$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(8034))$$:

 $$T_{5} + 2$$ $$T_{7}^{2} - 3 T_{7} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + 2 T + 5 T^{2} )^{2}$$
$7$ $$1 - 3 T + 12 T^{2} - 21 T^{3} + 49 T^{4}$$
$11$ $$1 + T + 18 T^{2} + 11 T^{3} + 121 T^{4}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$1 - 11 T + 60 T^{2} - 187 T^{3} + 289 T^{4}$$
$19$ $$( 1 + 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 2 T + 30 T^{2} + 46 T^{3} + 529 T^{4}$$
$29$ $$( 1 - 6 T + 29 T^{2} )^{2}$$
$31$ $$1 + 6 T + 54 T^{2} + 186 T^{3} + 961 T^{4}$$
$37$ $$1 + 10 T + 82 T^{2} + 370 T^{3} + 1369 T^{4}$$
$41$ $$( 1 - 2 T + 41 T^{2} )^{2}$$
$43$ $$1 + 6 T + 78 T^{2} + 258 T^{3} + 1849 T^{4}$$
$47$ $$1 + 4 T + 30 T^{2} + 188 T^{3} + 2209 T^{4}$$
$53$ $$1 - 9 T + 122 T^{2} - 477 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + 8 T + 59 T^{2} )^{2}$$
$61$ $$1 - 14 T + 154 T^{2} - 854 T^{3} + 3721 T^{4}$$
$67$ $$1 - 11 T + 126 T^{2} - 737 T^{3} + 4489 T^{4}$$
$71$ $$1 + 12 T + 110 T^{2} + 852 T^{3} + 5041 T^{4}$$
$73$ $$1 + 5 T + 114 T^{2} + 365 T^{3} + 5329 T^{4}$$
$79$ $$1 + 10 T + 30 T^{2} + 790 T^{3} + 6241 T^{4}$$
$83$ $$1 + 2 T + 14 T^{2} + 166 T^{3} + 6889 T^{4}$$
$89$ $$( 1 - 2 T + 89 T^{2} )^{2}$$
$97$ $$1 + 8 T + 142 T^{2} + 776 T^{3} + 9409 T^{4}$$