# Properties

 Label 5054.2.a.a Level $5054$ Weight $2$ Character orbit 5054.a Self dual yes Analytic conductor $40.356$ Analytic rank $1$ Dimension $1$ 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: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 266) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - 3 q^{9}+O(q^{10})$$ q - q^2 + q^4 + q^5 + q^7 - q^8 - 3 * q^9 $$q - q^{2} + q^{4} + q^{5} + q^{7} - q^{8} - 3 q^{9} - q^{10} - 2 q^{11} + 5 q^{13} - q^{14} + q^{16} + 3 q^{18} + q^{20} + 2 q^{22} + q^{23} - 4 q^{25} - 5 q^{26} + q^{28} - 6 q^{29} - 4 q^{31} - q^{32} + q^{35} - 3 q^{36} + 4 q^{37} - q^{40} + 2 q^{41} - 8 q^{43} - 2 q^{44} - 3 q^{45} - q^{46} + q^{49} + 4 q^{50} + 5 q^{52} + 2 q^{53} - 2 q^{55} - q^{56} + 6 q^{58} + 7 q^{59} - 7 q^{61} + 4 q^{62} - 3 q^{63} + q^{64} + 5 q^{65} - 12 q^{67} - q^{70} + 15 q^{71} + 3 q^{72} - 14 q^{73} - 4 q^{74} - 2 q^{77} + 4 q^{79} + q^{80} + 9 q^{81} - 2 q^{82} - 7 q^{83} + 8 q^{86} + 2 q^{88} + 3 q^{90} + 5 q^{91} + q^{92} - 12 q^{97} - q^{98} + 6 q^{99}+O(q^{100})$$ q - q^2 + q^4 + q^5 + q^7 - q^8 - 3 * q^9 - q^10 - 2 * q^11 + 5 * q^13 - q^14 + q^16 + 3 * q^18 + q^20 + 2 * q^22 + q^23 - 4 * q^25 - 5 * q^26 + q^28 - 6 * q^29 - 4 * q^31 - q^32 + q^35 - 3 * q^36 + 4 * q^37 - q^40 + 2 * q^41 - 8 * q^43 - 2 * q^44 - 3 * q^45 - q^46 + q^49 + 4 * q^50 + 5 * q^52 + 2 * q^53 - 2 * q^55 - q^56 + 6 * q^58 + 7 * q^59 - 7 * q^61 + 4 * q^62 - 3 * q^63 + q^64 + 5 * q^65 - 12 * q^67 - q^70 + 15 * q^71 + 3 * q^72 - 14 * q^73 - 4 * q^74 - 2 * q^77 + 4 * q^79 + q^80 + 9 * q^81 - 2 * q^82 - 7 * q^83 + 8 * q^86 + 2 * q^88 + 3 * q^90 + 5 * q^91 + q^92 - 12 * q^97 - q^98 + 6 * q^99

## 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
−1.00000 0 1.00000 1.00000 0 1.00000 −1.00000 −3.00000 −1.00000
 $$n$$: 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.a 1
19.b odd 2 1 5054.2.a.b 1
19.d odd 6 2 266.2.f.a 2
57.f even 6 2 2394.2.o.i 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
266.2.f.a 2 19.d odd 6 2
2394.2.o.i 2 57.f even 6 2
5054.2.a.a 1 1.a even 1 1 trivial
5054.2.a.b 1 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5054))$$:

 $$T_{3}$$ T3 $$T_{5} - 1$$ T5 - 1 $$T_{13} - 5$$ T13 - 5

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T + 1$$
$3$ $$T$$
$5$ $$T - 1$$
$7$ $$T - 1$$
$11$ $$T + 2$$
$13$ $$T - 5$$
$17$ $$T$$
$19$ $$T$$
$23$ $$T - 1$$
$29$ $$T + 6$$
$31$ $$T + 4$$
$37$ $$T - 4$$
$41$ $$T - 2$$
$43$ $$T + 8$$
$47$ $$T$$
$53$ $$T - 2$$
$59$ $$T - 7$$
$61$ $$T + 7$$
$67$ $$T + 12$$
$71$ $$T - 15$$
$73$ $$T + 14$$
$79$ $$T - 4$$
$83$ $$T + 7$$
$89$ $$T$$
$97$ $$T + 12$$