# Properties

 Label 5054.2.a.k 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: no (minimal twist has level 266) 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.

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

## 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 −2.61803 1.00000 −2.85410 −2.61803 1.00000 1.00000 3.85410 −2.85410
1.2 1.00000 −0.381966 1.00000 3.85410 −0.381966 1.00000 1.00000 −2.85410 3.85410
 $$n$$: 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.k 2
19.b odd 2 1 266.2.a.b 2
57.d even 2 1 2394.2.a.w 2
76.d even 2 1 2128.2.a.b 2
95.d odd 2 1 6650.2.a.bq 2
133.c even 2 1 1862.2.a.g 2
152.b even 2 1 8512.2.a.bc 2
152.g odd 2 1 8512.2.a.h 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.a.b 2 19.b odd 2 1
1862.2.a.g 2 133.c even 2 1
2128.2.a.b 2 76.d even 2 1
2394.2.a.w 2 57.d even 2 1
5054.2.a.k 2 1.a even 1 1 trivial
6650.2.a.bq 2 95.d odd 2 1
8512.2.a.h 2 152.g odd 2 1
8512.2.a.bc 2 152.b even 2 1

## Hecke kernels

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

 $$T_{3}^{2} + 3 T_{3} + 1$$ $$T_{5}^{2} - T_{5} - 11$$ $$T_{13}^{2} + 6 T_{13} + 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$1 + 3 T + T^{2}$$
$5$ $$-11 - T + T^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$11 - 7 T + T^{2}$$
$13$ $$4 + 6 T + T^{2}$$
$17$ $$-16 + 4 T + T^{2}$$
$19$ $$T^{2}$$
$23$ $$-44 - 2 T + T^{2}$$
$29$ $$19 - 11 T + T^{2}$$
$31$ $$-4 + 8 T + T^{2}$$
$37$ $$-9 + 3 T + T^{2}$$
$41$ $$-55 - 5 T + T^{2}$$
$43$ $$41 + 13 T + T^{2}$$
$47$ $$-11 + T + T^{2}$$
$53$ $$155 - 25 T + T^{2}$$
$59$ $$19 - 11 T + T^{2}$$
$61$ $$-31 - 11 T + T^{2}$$
$67$ $$-76 + 4 T + T^{2}$$
$71$ $$-209 + 3 T + T^{2}$$
$73$ $$20 + 10 T + T^{2}$$
$79$ $$-31 - 11 T + T^{2}$$
$83$ $$-80 + T^{2}$$
$89$ $$5 - 5 T + T^{2}$$
$97$ $$-11 + T + T^{2}$$