# Properties

 Label 6762.2.a.f Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $1$ 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: $$1$$ 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

 $$f(q)$$ $$=$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} - q^{3} + q^{4} + q^{6} - q^{8} + q^{9} - 2q^{11} - q^{12} + 6q^{13} + q^{16} - 2q^{17} - q^{18} + 6q^{19} + 2q^{22} + q^{23} + q^{24} - 5q^{25} - 6q^{26} - q^{27} - 6q^{29} - q^{32} + 2q^{33} + 2q^{34} + q^{36} - 6q^{38} - 6q^{39} - 6q^{41} - 6q^{43} - 2q^{44} - q^{46} - 8q^{47} - q^{48} + 5q^{50} + 2q^{51} + 6q^{52} - 4q^{53} + q^{54} - 6q^{57} + 6q^{58} + 8q^{61} + q^{64} - 2q^{66} - 2q^{67} - 2q^{68} - q^{69} - q^{72} + 2q^{73} + 5q^{75} + 6q^{76} + 6q^{78} + 8q^{79} + q^{81} + 6q^{82} + 2q^{83} + 6q^{86} + 6q^{87} + 2q^{88} + 2q^{89} + q^{92} + 8q^{94} + q^{96} - 14q^{97} - 2q^{99} + O(q^{100})$$

## 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 0 1.00000 0 −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
$$2$$ $$1$$
$$3$$ $$1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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.f 1
7.b odd 2 1 966.2.a.e 1
21.c even 2 1 2898.2.a.n 1
28.d even 2 1 7728.2.a.g 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.a.e 1 7.b odd 2 1
2898.2.a.n 1 21.c even 2 1
6762.2.a.f 1 1.a even 1 1 trivial
7728.2.a.g 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5}$$ $$T_{11} + 2$$ $$T_{13} - 6$$ $$T_{17} + 2$$

## Hecke characteristic polynomials

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