# Properties

 Label 5054.2.a.u Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.148.1 Defining polynomial: $$x^{3} - x^{2} - 3 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + q^{2} + ( 1 - \beta_{1} - \beta_{2} ) q^{3} + q^{4} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} - q^{7} + q^{8} + ( 1 - 2 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + q^{2} + ( 1 - \beta_{1} - \beta_{2} ) q^{3} + q^{4} + ( -\beta_{1} - 2 \beta_{2} ) q^{5} + ( 1 - \beta_{1} - \beta_{2} ) q^{6} - q^{7} + q^{8} + ( 1 - 2 \beta_{2} ) q^{9} + ( -\beta_{1} - 2 \beta_{2} ) q^{10} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{11} + ( 1 - \beta_{1} - \beta_{2} ) q^{12} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{13} - q^{14} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{15} + q^{16} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{17} + ( 1 - 2 \beta_{2} ) q^{18} + ( -\beta_{1} - 2 \beta_{2} ) q^{20} + ( -1 + \beta_{1} + \beta_{2} ) q^{21} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{22} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{23} + ( 1 - \beta_{1} - \beta_{2} ) q^{24} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{25} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{26} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{27} - q^{28} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{29} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{30} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{31} + q^{32} + ( 8 - 2 \beta_{2} ) q^{33} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{34} + ( \beta_{1} + 2 \beta_{2} ) q^{35} + ( 1 - 2 \beta_{2} ) q^{36} + ( -2 - 2 \beta_{1} ) q^{37} + ( -5 - \beta_{1} + \beta_{2} ) q^{39} + ( -\beta_{1} - 2 \beta_{2} ) q^{40} + ( -1 - \beta_{1} + \beta_{2} ) q^{41} + ( -1 + \beta_{1} + \beta_{2} ) q^{42} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{43} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{44} + ( 10 - 3 \beta_{1} - 6 \beta_{2} ) q^{45} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{46} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{47} + ( 1 - \beta_{1} - \beta_{2} ) q^{48} + q^{49} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{50} + ( 4 \beta_{1} + 6 \beta_{2} ) q^{51} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{52} + ( -1 - 3 \beta_{1} + 3 \beta_{2} ) q^{53} + ( 2 + 2 \beta_{1} - 2 \beta_{2} ) q^{54} + ( 5 + 5 \beta_{1} - 5 \beta_{2} ) q^{55} - q^{56} + ( -3 - \beta_{1} - 3 \beta_{2} ) q^{58} + ( 6 - 3 \beta_{1} ) q^{59} + ( 5 + \beta_{1} - 3 \beta_{2} ) q^{60} + ( 10 + \beta_{1} ) q^{61} + ( -1 + \beta_{1} + 3 \beta_{2} ) q^{62} + ( -1 + 2 \beta_{2} ) q^{63} + q^{64} + ( -5 - 4 \beta_{1} + 2 \beta_{2} ) q^{65} + ( 8 - 2 \beta_{2} ) q^{66} + ( -8 + 2 \beta_{2} ) q^{67} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{68} + ( -3 + 3 \beta_{1} + 7 \beta_{2} ) q^{69} + ( \beta_{1} + 2 \beta_{2} ) q^{70} + ( 10 - 3 \beta_{1} + \beta_{2} ) q^{71} + ( 1 - 2 \beta_{2} ) q^{72} + ( -4 + 6 \beta_{1} + 2 \beta_{2} ) q^{73} + ( -2 - 2 \beta_{1} ) q^{74} + ( 10 - 6 \beta_{1} - 12 \beta_{2} ) q^{75} + ( -3 + 3 \beta_{1} + \beta_{2} ) q^{77} + ( -5 - \beta_{1} + \beta_{2} ) q^{78} + ( -2 + 6 \beta_{1} - 2 \beta_{2} ) q^{79} + ( -\beta_{1} - 2 \beta_{2} ) q^{80} + ( 1 - 4 \beta_{1} - 2 \beta_{2} ) q^{81} + ( -1 - \beta_{1} + \beta_{2} ) q^{82} + ( -\beta_{1} - 2 \beta_{2} ) q^{83} + ( -1 + \beta_{1} + \beta_{2} ) q^{84} + ( -5 + 11 \beta_{1} + 7 \beta_{2} ) q^{85} + ( -5 - \beta_{1} - 3 \beta_{2} ) q^{86} + ( 4 + 4 \beta_{1} - 2 \beta_{2} ) q^{87} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{88} + ( -2 - 6 \beta_{1} + 2 \beta_{2} ) q^{89} + ( 10 - 3 \beta_{1} - 6 \beta_{2} ) q^{90} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{91} + ( -1 - 2 \beta_{1} + 2 \beta_{2} ) q^{92} + ( -8 + 6 \beta_{2} ) q^{93} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{94} + ( 1 - \beta_{1} - \beta_{2} ) q^{96} + ( 3 + \beta_{1} + 3 \beta_{2} ) q^{97} + q^{98} + ( 3 + \beta_{1} - 9 \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} - q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + O(q^{10})$$ $$3 q + 3 q^{2} + 2 q^{3} + 3 q^{4} - q^{5} + 2 q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} - q^{10} + 6 q^{11} + 2 q^{12} - q^{13} - 3 q^{14} + 16 q^{15} + 3 q^{16} - 6 q^{17} + 3 q^{18} - q^{20} - 2 q^{21} + 6 q^{22} - 5 q^{23} + 2 q^{24} + 16 q^{25} - q^{26} + 8 q^{27} - 3 q^{28} - 10 q^{29} + 16 q^{30} - 2 q^{31} + 3 q^{32} + 24 q^{33} - 6 q^{34} + q^{35} + 3 q^{36} - 8 q^{37} - 16 q^{39} - q^{40} - 4 q^{41} - 2 q^{42} - 16 q^{43} + 6 q^{44} + 27 q^{45} - 5 q^{46} - 2 q^{47} + 2 q^{48} + 3 q^{49} + 16 q^{50} + 4 q^{51} - q^{52} - 6 q^{53} + 8 q^{54} + 20 q^{55} - 3 q^{56} - 10 q^{58} + 15 q^{59} + 16 q^{60} + 31 q^{61} - 2 q^{62} - 3 q^{63} + 3 q^{64} - 19 q^{65} + 24 q^{66} - 24 q^{67} - 6 q^{68} - 6 q^{69} + q^{70} + 27 q^{71} + 3 q^{72} - 6 q^{73} - 8 q^{74} + 24 q^{75} - 6 q^{77} - 16 q^{78} - q^{80} - q^{81} - 4 q^{82} - q^{83} - 2 q^{84} - 4 q^{85} - 16 q^{86} + 16 q^{87} + 6 q^{88} - 12 q^{89} + 27 q^{90} + q^{91} - 5 q^{92} - 24 q^{93} - 2 q^{94} + 2 q^{96} + 10 q^{97} + 3 q^{98} + 10 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 3 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$

## 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
 2.17009 −1.48119 0.311108
1.00000 −1.70928 1.00000 −3.24846 −1.70928 −1.00000 1.00000 −0.0783777 −3.24846
1.2 1.00000 0.806063 1.00000 −1.86907 0.806063 −1.00000 1.00000 −2.35026 −1.86907
1.3 1.00000 2.90321 1.00000 4.11753 2.90321 −1.00000 1.00000 5.42864 4.11753
 $$n$$: 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.u 3
19.b odd 2 1 5054.2.a.q 3
19.c even 3 2 266.2.f.c 6
57.h odd 6 2 2394.2.o.s 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.c 6 19.c even 3 2
2394.2.o.s 6 57.h odd 6 2
5054.2.a.q 3 19.b odd 2 1
5054.2.a.u 3 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{3}^{3} - 2 T_{3}^{2} - 4 T_{3} + 4$$ $$T_{5}^{3} + T_{5}^{2} - 15 T_{5} - 25$$ $$T_{13}^{3} + T_{13}^{2} - 13 T_{13} - 23$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{3}$$
$3$ $$4 - 4 T - 2 T^{2} + T^{3}$$
$5$ $$-25 - 15 T + T^{2} + T^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$100 - 16 T - 6 T^{2} + T^{3}$$
$13$ $$-23 - 13 T + T^{2} + T^{3}$$
$17$ $$-148 - 28 T + 6 T^{2} + T^{3}$$
$19$ $$T^{3}$$
$23$ $$-137 - 29 T + 5 T^{2} + T^{3}$$
$29$ $$-148 + 10 T^{2} + T^{3}$$
$31$ $$52 - 32 T + 2 T^{2} + T^{3}$$
$37$ $$-16 + 8 T + 8 T^{2} + T^{3}$$
$41$ $$-20 - 4 T + 4 T^{2} + T^{3}$$
$43$ $$-100 + 52 T + 16 T^{2} + T^{3}$$
$47$ $$-260 - 96 T + 2 T^{2} + T^{3}$$
$53$ $$-428 - 72 T + 6 T^{2} + T^{3}$$
$59$ $$27 + 45 T - 15 T^{2} + T^{3}$$
$61$ $$-1069 + 317 T - 31 T^{2} + T^{3}$$
$67$ $$400 + 176 T + 24 T^{2} + T^{3}$$
$71$ $$-445 + 203 T - 27 T^{2} + T^{3}$$
$73$ $$-632 - 100 T + 6 T^{2} + T^{3}$$
$79$ $$608 - 160 T + T^{3}$$
$83$ $$-25 - 15 T + T^{2} + T^{3}$$
$89$ $$-1184 - 112 T + 12 T^{2} + T^{3}$$
$97$ $$148 - 10 T^{2} + T^{3}$$