# Properties

 Label 6762.2.a.m 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

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - 2 q^{5} - q^{6} - q^{8} + q^{9} + 2 q^{10} - 4 q^{11} + q^{12} + 4 q^{13} - 2 q^{15} + q^{16} + 4 q^{17} - q^{18} + 2 q^{19} - 2 q^{20} + 4 q^{22} + q^{23} - q^{24} - q^{25} - 4 q^{26} + q^{27} - 6 q^{29} + 2 q^{30} - 6 q^{31} - q^{32} - 4 q^{33} - 4 q^{34} + q^{36} - 2 q^{37} - 2 q^{38} + 4 q^{39} + 2 q^{40} + 10 q^{41} + 8 q^{43} - 4 q^{44} - 2 q^{45} - q^{46} - 10 q^{47} + q^{48} + q^{50} + 4 q^{51} + 4 q^{52} - 6 q^{53} - q^{54} + 8 q^{55} + 2 q^{57} + 6 q^{58} + 12 q^{59} - 2 q^{60} + 10 q^{61} + 6 q^{62} + q^{64} - 8 q^{65} + 4 q^{66} + 8 q^{67} + 4 q^{68} + q^{69} - 4 q^{71} - q^{72} + 2 q^{73} + 2 q^{74} - q^{75} + 2 q^{76} - 4 q^{78} - 8 q^{79} - 2 q^{80} + q^{81} - 10 q^{82} + 6 q^{83} - 8 q^{85} - 8 q^{86} - 6 q^{87} + 4 q^{88} + 2 q^{90} + q^{92} - 6 q^{93} + 10 q^{94} - 4 q^{95} - q^{96} + 8 q^{97} - 4 q^{99} + O(q^{100})$$

## 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 −2.00000 −1.00000 0 −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
$$2$$ $$1$$
$$3$$ $$-1$$
$$7$$ $$-1$$
$$23$$ $$-1$$

## 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.m 1
7.b odd 2 1 966.2.a.c 1
21.c even 2 1 2898.2.a.l 1
28.d even 2 1 7728.2.a.t 1

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

 $$T_{5} + 2$$ $$T_{11} + 4$$ $$T_{13} - 4$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

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