# Properties

 Label 5054.2.a.i Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5054 = 2 \cdot 7 \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5054.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.3563931816$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
−1.00000 0.381966 1.00000 −4.23607 −0.381966 −1.00000 −1.00000 −2.85410 4.23607
1.2 −1.00000 2.61803 1.00000 0.236068 −2.61803 −1.00000 −1.00000 3.85410 −0.236068
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$7$$ $$1$$
$$19$$ $$-1$$

## 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 5054.2.a.i 2
19.b odd 2 1 5054.2.a.j yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.i 2 1.a even 1 1 trivial
5054.2.a.j yes 2 19.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3}^{2} - 3 T_{3} + 1$$ $$T_{5}^{2} + 4 T_{5} - 1$$ $$T_{13}^{2} + T_{13} - 11$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$1 - 3 T + T^{2}$$
$5$ $$-1 + 4 T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$( -3 + T )^{2}$$
$13$ $$-11 + T + T^{2}$$
$17$ $$-9 + 3 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-45 + T^{2}$$
$29$ $$31 + 12 T + T^{2}$$
$31$ $$( -5 + T )^{2}$$
$37$ $$( 5 + T )^{2}$$
$41$ $$-29 - 3 T + T^{2}$$
$43$ $$-99 + 3 T + T^{2}$$
$47$ $$11 + 8 T + T^{2}$$
$53$ $$-9 + 12 T + T^{2}$$
$59$ $$59 - 16 T + T^{2}$$
$61$ $$-45 + T^{2}$$
$67$ $$-71 - 11 T + T^{2}$$
$71$ $$44 - 16 T + T^{2}$$
$73$ $$-19 + 7 T + T^{2}$$
$79$ $$171 - 27 T + T^{2}$$
$83$ $$-16 + 4 T + T^{2}$$
$89$ $$5 + 5 T + T^{2}$$
$97$ $$99 + 24 T + T^{2}$$