# Properties

 Label 6762.2.a.bl 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + 2q^{5} + q^{6} + q^{8} + q^{9} + 2q^{10} - 4q^{11} + q^{12} + 2q^{13} + 2q^{15} + q^{16} + 6q^{17} + q^{18} - 4q^{19} + 2q^{20} - 4q^{22} - q^{23} + q^{24} - q^{25} + 2q^{26} + q^{27} - 2q^{29} + 2q^{30} + 8q^{31} + q^{32} - 4q^{33} + 6q^{34} + q^{36} + 6q^{37} - 4q^{38} + 2q^{39} + 2q^{40} + 6q^{41} - 4q^{43} - 4q^{44} + 2q^{45} - q^{46} + 8q^{47} + q^{48} - q^{50} + 6q^{51} + 2q^{52} + 6q^{53} + q^{54} - 8q^{55} - 4q^{57} - 2q^{58} - 4q^{59} + 2q^{60} + 10q^{61} + 8q^{62} + q^{64} + 4q^{65} - 4q^{66} + 4q^{67} + 6q^{68} - q^{69} - 8q^{71} + q^{72} + 6q^{73} + 6q^{74} - q^{75} - 4q^{76} + 2q^{78} + 2q^{80} + q^{81} + 6q^{82} + 12q^{83} + 12q^{85} - 4q^{86} - 2q^{87} - 4q^{88} - 2q^{89} + 2q^{90} - q^{92} + 8q^{93} + 8q^{94} - 8q^{95} + q^{96} - 10q^{97} - 4q^{99} + O(q^{100})$$

## 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 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.bl 1
7.b odd 2 1 966.2.a.g 1
21.c even 2 1 2898.2.a.i 1
28.d even 2 1 7728.2.a.o 1

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

 $$T_{5} - 2$$ $$T_{11} + 4$$ $$T_{13} - 2$$ $$T_{17} - 6$$

## Hecke characteristic polynomials

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