# Properties

 Label 6762.2.a.y Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$6762 = 2 \cdot 3 \cdot 7^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 6762.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$53.9948418468$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

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

## 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
 0
1.00000 −1.00000 1.00000 −3.00000 −1.00000 0 1.00000 1.00000 −3.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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 6762.2.a.y 1
7.b odd 2 1 966.2.a.k 1
21.c even 2 1 2898.2.a.a 1
28.d even 2 1 7728.2.a.j 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.k 1 7.b odd 2 1
2898.2.a.a 1 21.c even 2 1
6762.2.a.y 1 1.a even 1 1 trivial
7728.2.a.j 1 28.d even 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5} + 3$$ $$T_{11} - 4$$ $$T_{13} - 3$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$1 + T$$
$5$ $$3 + T$$
$7$ $$T$$
$11$ $$-4 + T$$
$13$ $$-3 + T$$
$17$ $$-4 + T$$
$19$ $$T$$
$23$ $$-1 + T$$
$29$ $$-3 + T$$
$31$ $$-6 + T$$
$37$ $$9 + T$$
$41$ $$9 + T$$
$43$ $$3 + T$$
$47$ $$-7 + T$$
$53$ $$4 + T$$
$59$ $$6 + T$$
$61$ $$10 + T$$
$67$ $$-4 + T$$
$71$ $$6 + T$$
$73$ $$-8 + T$$
$79$ $$-8 + T$$
$83$ $$4 + T$$
$89$ $$-14 + T$$
$97$ $$-7 + T$$