# Properties

 Label 6762.2.a.bh 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

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} + O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} - q^{5} + q^{6} + q^{8} + q^{9} - q^{10} - 3 q^{11} + q^{12} - 2 q^{13} - q^{15} + q^{16} + 2 q^{17} + q^{18} + 2 q^{19} - q^{20} - 3 q^{22} + q^{23} + q^{24} - 4 q^{25} - 2 q^{26} + q^{27} - 7 q^{29} - q^{30} - 7 q^{31} + q^{32} - 3 q^{33} + 2 q^{34} + q^{36} - 2 q^{37} + 2 q^{38} - 2 q^{39} - q^{40} + 2 q^{41} - 6 q^{43} - 3 q^{44} - q^{45} + q^{46} - 6 q^{47} + q^{48} - 4 q^{50} + 2 q^{51} - 2 q^{52} - 9 q^{53} + q^{54} + 3 q^{55} + 2 q^{57} - 7 q^{58} + 9 q^{59} - q^{60} - 6 q^{61} - 7 q^{62} + q^{64} + 2 q^{65} - 3 q^{66} + 10 q^{67} + 2 q^{68} + q^{69} - 8 q^{71} + q^{72} + 10 q^{73} - 2 q^{74} - 4 q^{75} + 2 q^{76} - 2 q^{78} - 15 q^{79} - q^{80} + q^{81} + 2 q^{82} - q^{83} - 2 q^{85} - 6 q^{86} - 7 q^{87} - 3 q^{88} - q^{90} + q^{92} - 7 q^{93} - 6 q^{94} - 2 q^{95} + q^{96} - 5 q^{97} - 3 q^{99} + O(q^{100})$$

## 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 −1.00000 1.00000 0 1.00000 1.00000 −1.00000
 $$n$$: 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.bh 1
7.b odd 2 1 6762.2.a.bb 1
7.d odd 6 2 966.2.i.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.c 2 7.d odd 6 2
6762.2.a.bb 1 7.b odd 2 1
6762.2.a.bh 1 1.a even 1 1 trivial

## Hecke kernels

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

 $$T_{5} + 1$$ $$T_{11} + 3$$ $$T_{13} + 2$$ $$T_{17} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$-1 + T$$
$3$ $$-1 + T$$
$5$ $$1 + T$$
$7$ $$T$$
$11$ $$3 + T$$
$13$ $$2 + T$$
$17$ $$-2 + T$$
$19$ $$-2 + T$$
$23$ $$-1 + T$$
$29$ $$7 + T$$
$31$ $$7 + T$$
$37$ $$2 + T$$
$41$ $$-2 + T$$
$43$ $$6 + T$$
$47$ $$6 + T$$
$53$ $$9 + T$$
$59$ $$-9 + T$$
$61$ $$6 + T$$
$67$ $$-10 + T$$
$71$ $$8 + T$$
$73$ $$-10 + T$$
$79$ $$15 + T$$
$83$ $$1 + T$$
$89$ $$T$$
$97$ $$5 + T$$