# Properties

 Label 6045.2.a.n Level 6045 Weight 2 Character orbit 6045.a Self dual yes Analytic conductor 48.270 Analytic rank 1 Dimension 2 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6045 = 3 \cdot 5 \cdot 13 \cdot 31$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6045.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$48.2695680219$$ Analytic rank: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

 $$f(q)$$ $$=$$ $$q - q^{3} -2 q^{4} - q^{5} + ( -1 + 2 \beta ) q^{7} + q^{9} +O(q^{10})$$ $$q - q^{3} -2 q^{4} - q^{5} + ( -1 + 2 \beta ) q^{7} + q^{9} + ( -2 - \beta ) q^{11} + 2 q^{12} - q^{13} + q^{15} + 4 q^{16} + ( 2 + \beta ) q^{17} -2 q^{19} + 2 q^{20} + ( 1 - 2 \beta ) q^{21} + ( -2 - \beta ) q^{23} + q^{25} - q^{27} + ( 2 - 4 \beta ) q^{28} + ( 3 + \beta ) q^{29} - q^{31} + ( 2 + \beta ) q^{33} + ( 1 - 2 \beta ) q^{35} -2 q^{36} + ( -2 - 3 \beta ) q^{37} + q^{39} -5 q^{41} + ( 1 + \beta ) q^{43} + ( 4 + 2 \beta ) q^{44} - q^{45} + ( 6 + 2 \beta ) q^{47} -4 q^{48} + 10 q^{49} + ( -2 - \beta ) q^{51} + 2 q^{52} + ( 4 - \beta ) q^{53} + ( 2 + \beta ) q^{55} + 2 q^{57} + ( 3 - 3 \beta ) q^{59} -2 q^{60} + ( -4 - \beta ) q^{61} + ( -1 + 2 \beta ) q^{63} -8 q^{64} + q^{65} + ( 9 - \beta ) q^{67} + ( -4 - 2 \beta ) q^{68} + ( 2 + \beta ) q^{69} + ( 4 + 3 \beta ) q^{71} + ( 4 - 6 \beta ) q^{73} - q^{75} + 4 q^{76} + ( -6 - 5 \beta ) q^{77} + ( 8 - 5 \beta ) q^{79} -4 q^{80} + q^{81} + ( 3 - 5 \beta ) q^{83} + ( -2 + 4 \beta ) q^{84} + ( -2 - \beta ) q^{85} + ( -3 - \beta ) q^{87} + ( 2 + 5 \beta ) q^{89} + ( 1 - 2 \beta ) q^{91} + ( 4 + 2 \beta ) q^{92} + q^{93} + 2 q^{95} + ( 1 - 2 \beta ) q^{97} + ( -2 - \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} - 5q^{11} + 4q^{12} - 2q^{13} + 2q^{15} + 8q^{16} + 5q^{17} - 4q^{19} + 4q^{20} - 5q^{23} + 2q^{25} - 2q^{27} + 7q^{29} - 2q^{31} + 5q^{33} - 4q^{36} - 7q^{37} + 2q^{39} - 10q^{41} + 3q^{43} + 10q^{44} - 2q^{45} + 14q^{47} - 8q^{48} + 20q^{49} - 5q^{51} + 4q^{52} + 7q^{53} + 5q^{55} + 4q^{57} + 3q^{59} - 4q^{60} - 9q^{61} - 16q^{64} + 2q^{65} + 17q^{67} - 10q^{68} + 5q^{69} + 11q^{71} + 2q^{73} - 2q^{75} + 8q^{76} - 17q^{77} + 11q^{79} - 8q^{80} + 2q^{81} + q^{83} - 5q^{85} - 7q^{87} + 9q^{89} + 10q^{92} + 2q^{93} + 4q^{95} - 5q^{99} + O(q^{100})$$

## 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
 −1.56155 2.56155
0 −1.00000 −2.00000 −1.00000 0 −4.12311 0 1.00000 0
1.2 0 −1.00000 −2.00000 −1.00000 0 4.12311 0 1.00000 0
 $$n$$: 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 6045.2.a.n 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6045.2.a.n 2 1.a even 1 1 trivial

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$5$$ $$1$$
$$13$$ $$1$$
$$31$$ $$1$$

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6045))$$:

 $$T_{2}$$ $$T_{7}^{2} - 17$$ $$T_{11}^{2} + 5 T_{11} + 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + 2 T^{2} )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$( 1 + T )^{2}$$
$7$ $$1 - 3 T^{2} + 49 T^{4}$$
$11$ $$1 + 5 T + 24 T^{2} + 55 T^{3} + 121 T^{4}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$1 - 5 T + 36 T^{2} - 85 T^{3} + 289 T^{4}$$
$19$ $$( 1 + 2 T + 19 T^{2} )^{2}$$
$23$ $$1 + 5 T + 48 T^{2} + 115 T^{3} + 529 T^{4}$$
$29$ $$1 - 7 T + 66 T^{2} - 203 T^{3} + 841 T^{4}$$
$31$ $$( 1 + T )^{2}$$
$37$ $$1 + 7 T + 48 T^{2} + 259 T^{3} + 1369 T^{4}$$
$41$ $$( 1 + 5 T + 41 T^{2} )^{2}$$
$43$ $$1 - 3 T + 84 T^{2} - 129 T^{3} + 1849 T^{4}$$
$47$ $$1 - 14 T + 126 T^{2} - 658 T^{3} + 2209 T^{4}$$
$53$ $$1 - 7 T + 114 T^{2} - 371 T^{3} + 2809 T^{4}$$
$59$ $$1 - 3 T + 82 T^{2} - 177 T^{3} + 3481 T^{4}$$
$61$ $$1 + 9 T + 138 T^{2} + 549 T^{3} + 3721 T^{4}$$
$67$ $$1 - 17 T + 202 T^{2} - 1139 T^{3} + 4489 T^{4}$$
$71$ $$1 - 11 T + 134 T^{2} - 781 T^{3} + 5041 T^{4}$$
$73$ $$1 - 2 T - 6 T^{2} - 146 T^{3} + 5329 T^{4}$$
$79$ $$1 - 11 T + 82 T^{2} - 869 T^{3} + 6241 T^{4}$$
$83$ $$1 - T + 60 T^{2} - 83 T^{3} + 6889 T^{4}$$
$89$ $$1 - 9 T + 92 T^{2} - 801 T^{3} + 7921 T^{4}$$
$97$ $$1 + 177 T^{2} + 9409 T^{4}$$