# Properties

 Label 5082.2.a.q Level $5082$ Weight $2$ Character orbit 5082.a Self dual yes Analytic conductor $40.580$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} - q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{7} + q^{8} + q^{9} - 2 q^{10} - q^{12} - 2 q^{13} - q^{14} + 2 q^{15} + q^{16} + 6 q^{17} + q^{18} + 4 q^{19} - 2 q^{20} + q^{21} - 4 q^{23} - q^{24} - q^{25} - 2 q^{26} - q^{27} - q^{28} - 2 q^{29} + 2 q^{30} - 4 q^{31} + q^{32} + 6 q^{34} + 2 q^{35} + q^{36} - 2 q^{37} + 4 q^{38} + 2 q^{39} - 2 q^{40} + 6 q^{41} + q^{42} - 2 q^{45} - 4 q^{46} - 8 q^{47} - q^{48} + q^{49} - q^{50} - 6 q^{51} - 2 q^{52} - 14 q^{53} - q^{54} - q^{56} - 4 q^{57} - 2 q^{58} + 12 q^{59} + 2 q^{60} + 14 q^{61} - 4 q^{62} - q^{63} + q^{64} + 4 q^{65} + 4 q^{67} + 6 q^{68} + 4 q^{69} + 2 q^{70} + 12 q^{71} + q^{72} - 6 q^{73} - 2 q^{74} + q^{75} + 4 q^{76} + 2 q^{78} - 2 q^{80} + q^{81} + 6 q^{82} + q^{84} - 12 q^{85} + 2 q^{87} - 6 q^{89} - 2 q^{90} + 2 q^{91} - 4 q^{92} + 4 q^{93} - 8 q^{94} - 8 q^{95} - q^{96} - 14 q^{97} + q^{98} + O(q^{100})$$

## 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
 0
1.00000 −1.00000 1.00000 −2.00000 −1.00000 −1.00000 1.00000 1.00000 −2.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

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

## 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.q 1
11.b odd 2 1 462.2.a.a 1
33.d even 2 1 1386.2.a.k 1
44.c even 2 1 3696.2.a.s 1
77.b even 2 1 3234.2.a.n 1
231.h odd 2 1 9702.2.a.bf 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
462.2.a.a 1 11.b odd 2 1
1386.2.a.k 1 33.d even 2 1
3234.2.a.n 1 77.b even 2 1
3696.2.a.s 1 44.c even 2 1
5082.2.a.q 1 1.a even 1 1 trivial
9702.2.a.bf 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(5082))$$:

 $$T_{5} + 2$$ $$T_{13} + 2$$ $$T_{17} - 6$$ $$T_{19} - 4$$

## Hecke characteristic polynomials

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