# Properties

 Label 5082.2.a.s Level $5082$ Weight $2$ Character orbit 5082.a Self dual yes Analytic conductor $40.580$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5082.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.5799743072$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 462) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9}+O(q^{10})$$ q + q^2 - q^3 + q^4 - q^6 + q^7 + q^8 + q^9 $$q + q^{2} - q^{3} + q^{4} - q^{6} + q^{7} + q^{8} + q^{9} - q^{12} + 2 q^{13} + q^{14} + q^{16} + 4 q^{17} + q^{18} - 6 q^{19} - q^{21} - 4 q^{23} - q^{24} - 5 q^{25} + 2 q^{26} - q^{27} + q^{28} + 10 q^{29} + 6 q^{31} + q^{32} + 4 q^{34} + q^{36} - 6 q^{37} - 6 q^{38} - 2 q^{39} + 12 q^{41} - q^{42} + 8 q^{43} - 4 q^{46} + 2 q^{47} - q^{48} + q^{49} - 5 q^{50} - 4 q^{51} + 2 q^{52} + 6 q^{53} - q^{54} + q^{56} + 6 q^{57} + 10 q^{58} - 8 q^{59} - 6 q^{61} + 6 q^{62} + q^{63} + q^{64} - 4 q^{67} + 4 q^{68} + 4 q^{69} + q^{72} + 12 q^{73} - 6 q^{74} + 5 q^{75} - 6 q^{76} - 2 q^{78} + q^{81} + 12 q^{82} - 14 q^{83} - q^{84} + 8 q^{86} - 10 q^{87} + 10 q^{89} + 2 q^{91} - 4 q^{92} - 6 q^{93} + 2 q^{94} - q^{96} + 10 q^{97} + q^{98}+O(q^{100})$$ q + q^2 - q^3 + q^4 - q^6 + q^7 + q^8 + q^9 - q^12 + 2 * q^13 + q^14 + q^16 + 4 * q^17 + q^18 - 6 * q^19 - q^21 - 4 * q^23 - q^24 - 5 * q^25 + 2 * q^26 - q^27 + q^28 + 10 * q^29 + 6 * q^31 + q^32 + 4 * q^34 + q^36 - 6 * q^37 - 6 * q^38 - 2 * q^39 + 12 * q^41 - q^42 + 8 * q^43 - 4 * q^46 + 2 * q^47 - q^48 + q^49 - 5 * q^50 - 4 * q^51 + 2 * q^52 + 6 * q^53 - q^54 + q^56 + 6 * q^57 + 10 * q^58 - 8 * q^59 - 6 * q^61 + 6 * q^62 + q^63 + q^64 - 4 * q^67 + 4 * q^68 + 4 * q^69 + q^72 + 12 * q^73 - 6 * q^74 + 5 * q^75 - 6 * q^76 - 2 * q^78 + q^81 + 12 * q^82 - 14 * q^83 - q^84 + 8 * q^86 - 10 * q^87 + 10 * q^89 + 2 * q^91 - 4 * q^92 - 6 * q^93 + 2 * q^94 - q^96 + 10 * q^97 + q^98

## 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 −1.00000 1.00000 0 −1.00000 1.00000 1.00000 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$11$$ $$-1$$

## 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 5082.2.a.s 1
11.b odd 2 1 462.2.a.b 1
33.d even 2 1 1386.2.a.i 1
44.c even 2 1 3696.2.a.y 1
77.b even 2 1 3234.2.a.k 1
231.h odd 2 1 9702.2.a.bt 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.b 1 11.b odd 2 1
1386.2.a.i 1 33.d even 2 1
3234.2.a.k 1 77.b even 2 1
3696.2.a.y 1 44.c even 2 1
5082.2.a.s 1 1.a even 1 1 trivial
9702.2.a.bt 1 231.h odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5082))$$:

 $$T_{5}$$ T5 $$T_{13} - 2$$ T13 - 2 $$T_{17} - 4$$ T17 - 4 $$T_{19} + 6$$ T19 + 6

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T - 1$$
$3$ $$T + 1$$
$5$ $$T$$
$7$ $$T - 1$$
$11$ $$T$$
$13$ $$T - 2$$
$17$ $$T - 4$$
$19$ $$T + 6$$
$23$ $$T + 4$$
$29$ $$T - 10$$
$31$ $$T - 6$$
$37$ $$T + 6$$
$41$ $$T - 12$$
$43$ $$T - 8$$
$47$ $$T - 2$$
$53$ $$T - 6$$
$59$ $$T + 8$$
$61$ $$T + 6$$
$67$ $$T + 4$$
$71$ $$T$$
$73$ $$T - 12$$
$79$ $$T$$
$83$ $$T + 14$$
$89$ $$T - 10$$
$97$ $$T - 10$$