# Properties

 Label 5054.2.a.y Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5 x^{2} + 5$$ 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 a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.90211 −1.17557 1.17557 1.90211
1.00000 −0.902113 1.00000 0.557537 −0.902113 1.00000 1.00000 −2.18619 0.557537
1.2 1.00000 −0.175571 1.00000 −0.284079 −0.175571 1.00000 1.00000 −2.96917 −0.284079
1.3 1.00000 2.17557 1.00000 3.52015 2.17557 1.00000 1.00000 1.73311 3.52015
1.4 1.00000 2.90211 1.00000 −1.79360 2.90211 1.00000 1.00000 5.42226 −1.79360
 $$n$$: 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.y yes 4
19.b odd 2 1 5054.2.a.v 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
5054.2.a.v 4 19.b odd 2 1
5054.2.a.y yes 4 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3}^{4} - 4 T_{3}^{3} + T_{3}^{2} + 6 T_{3} + 1$$ $$T_{5}^{4} - 2 T_{5}^{3} - 6 T_{5}^{2} + 2 T_{5} + 1$$ $$T_{13}^{4} + 2 T_{13}^{3} - 11 T_{13}^{2} + 8 T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$1 + 6 T + T^{2} - 4 T^{3} + T^{4}$$
$5$ $$1 + 2 T - 6 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$( -1 + T )^{4}$$
$11$ $$61 + 36 T - 14 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$1 + 8 T - 11 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$-19 + 32 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$T^{4}$$
$23$ $$61 + 36 T - 14 T^{2} - 4 T^{3} + T^{4}$$
$29$ $$-19 + 12 T + 34 T^{2} - 12 T^{3} + T^{4}$$
$31$ $$-239 + 206 T + 26 T^{2} - 14 T^{3} + T^{4}$$
$37$ $$-1559 - 104 T + 166 T^{2} - 24 T^{3} + T^{4}$$
$41$ $$205 - 310 T - 95 T^{2} + T^{4}$$
$43$ $$-179 + 66 T + 31 T^{2} - 14 T^{3} + T^{4}$$
$47$ $$-719 - 602 T - 126 T^{2} + 2 T^{3} + T^{4}$$
$53$ $$905 + 160 T - 90 T^{2} + T^{4}$$
$59$ $$121 - 198 T - 86 T^{2} - 2 T^{3} + T^{4}$$
$61$ $$3401 + 162 T - 126 T^{2} - 2 T^{3} + T^{4}$$
$67$ $$-12119 + 2222 T + 19 T^{2} - 22 T^{3} + T^{4}$$
$71$ $$-944 + 816 T - 164 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$1805 + 760 T - 115 T^{2} - 10 T^{3} + T^{4}$$
$79$ $$2221 - 658 T - 241 T^{2} - 2 T^{3} + T^{4}$$
$83$ $$9680 - 200 T^{2} + T^{4}$$
$89$ $$1205 + 40 T - 115 T^{2} - 10 T^{3} + T^{4}$$
$97$ $$-16739 - 3542 T - 66 T^{2} + 22 T^{3} + T^{4}$$