# Properties

 Label 5054.2.a.g 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.

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

## 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 −0.618034 1.00000 −0.381966 0.618034 1.00000 −1.00000 −2.61803 0.381966
1.2 −1.00000 1.61803 1.00000 −2.61803 −1.61803 1.00000 −1.00000 −0.381966 2.61803
 $$n$$: 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.g 2
19.b odd 2 1 5054.2.a.l yes 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.g 2 1.a even 1 1 trivial
5054.2.a.l 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3}^{2} - T_{3} - 1$$ $$T_{5}^{2} + 3 T_{5} + 1$$ $$T_{13}^{2} - 20$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-1 - T + T^{2}$$
$5$ $$1 + 3 T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$-9 + 3 T + T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$4 - 6 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-80 + T^{2}$$
$29$ $$1 - 7 T + T^{2}$$
$31$ $$( -6 + T )^{2}$$
$37$ $$9 - 9 T + T^{2}$$
$41$ $$-101 - T + T^{2}$$
$43$ $$-11 - T + T^{2}$$
$47$ $$45 + 15 T + T^{2}$$
$53$ $$61 + 17 T + T^{2}$$
$59$ $$-99 + 3 T + T^{2}$$
$61$ $$-99 - 3 T + T^{2}$$
$67$ $$4 - 6 T + T^{2}$$
$71$ $$-99 + 3 T + T^{2}$$
$73$ $$T^{2}$$
$79$ $$-5 + 15 T + T^{2}$$
$83$ $$76 - 18 T + T^{2}$$
$89$ $$31 + 13 T + T^{2}$$
$97$ $$99 - 21 T + T^{2}$$