# Properties

 Label 5054.2.a.d Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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
 1.61803 −0.618034
−1.00000 −1.61803 1.00000 −3.85410 1.61803 −1.00000 −1.00000 −0.381966 3.85410
1.2 −1.00000 0.618034 1.00000 2.85410 −0.618034 −1.00000 −1.00000 −2.61803 −2.85410
 $$n$$: 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.d 2
19.b odd 2 1 5054.2.a.o yes 2

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-1 + T + T^{2}$$
$5$ $$-11 + T + T^{2}$$
$7$ $$( 1 + T )^{2}$$
$11$ $$-1 - T + T^{2}$$
$13$ $$( 2 + T )^{2}$$
$17$ $$-4 + 2 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$T^{2}$$
$29$ $$-11 + 9 T + T^{2}$$
$31$ $$-4 + 8 T + T^{2}$$
$37$ $$-19 + 7 T + T^{2}$$
$41$ $$19 - 11 T + T^{2}$$
$43$ $$5 - 5 T + T^{2}$$
$47$ $$89 - 19 T + T^{2}$$
$53$ $$-31 - 11 T + T^{2}$$
$59$ $$109 + 21 T + T^{2}$$
$61$ $$-95 - 5 T + T^{2}$$
$67$ $$20 + 10 T + T^{2}$$
$71$ $$-95 - 5 T + T^{2}$$
$73$ $$-64 - 8 T + T^{2}$$
$79$ $$-1 + 11 T + T^{2}$$
$83$ $$4 - 14 T + T^{2}$$
$89$ $$-49 + 7 T + T^{2}$$
$97$ $$19 + 9 T + T^{2}$$