# Properties

 Label 6762.2.a.bw Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{17})$$ Defining polynomial: $$x^{2} - x - 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 966) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \frac{1}{2}(1 + \sqrt{17})$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} -\beta q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} -\beta q^{5} - q^{6} - q^{8} + q^{9} + \beta q^{10} + ( 2 - 2 \beta ) q^{11} + q^{12} + ( 2 - \beta ) q^{13} -\beta q^{15} + q^{16} + ( -2 - 2 \beta ) q^{17} - q^{18} + ( 2 - 2 \beta ) q^{19} -\beta q^{20} + ( -2 + 2 \beta ) q^{22} + q^{23} - q^{24} + ( -1 + \beta ) q^{25} + ( -2 + \beta ) q^{26} + q^{27} + ( 6 - \beta ) q^{29} + \beta q^{30} + 2 \beta q^{31} - q^{32} + ( 2 - 2 \beta ) q^{33} + ( 2 + 2 \beta ) q^{34} + q^{36} + \beta q^{37} + ( -2 + 2 \beta ) q^{38} + ( 2 - \beta ) q^{39} + \beta q^{40} + ( -2 + \beta ) q^{41} + ( 2 - 3 \beta ) q^{43} + ( 2 - 2 \beta ) q^{44} -\beta q^{45} - q^{46} + ( -4 - \beta ) q^{47} + q^{48} + ( 1 - \beta ) q^{50} + ( -2 - 2 \beta ) q^{51} + ( 2 - \beta ) q^{52} + ( 8 - 4 \beta ) q^{53} - q^{54} + 8 q^{55} + ( 2 - 2 \beta ) q^{57} + ( -6 + \beta ) q^{58} + ( -8 - 2 \beta ) q^{59} -\beta q^{60} + ( -4 - 2 \beta ) q^{61} -2 \beta q^{62} + q^{64} + ( 4 - \beta ) q^{65} + ( -2 + 2 \beta ) q^{66} + ( 2 + 2 \beta ) q^{67} + ( -2 - 2 \beta ) q^{68} + q^{69} + 6 \beta q^{71} - q^{72} + ( -6 + 2 \beta ) q^{73} -\beta q^{74} + ( -1 + \beta ) q^{75} + ( 2 - 2 \beta ) q^{76} + ( -2 + \beta ) q^{78} + ( 8 - 4 \beta ) q^{79} -\beta q^{80} + q^{81} + ( 2 - \beta ) q^{82} + ( 6 - 2 \beta ) q^{83} + ( 8 + 4 \beta ) q^{85} + ( -2 + 3 \beta ) q^{86} + ( 6 - \beta ) q^{87} + ( -2 + 2 \beta ) q^{88} -14 q^{89} + \beta q^{90} + q^{92} + 2 \beta q^{93} + ( 4 + \beta ) q^{94} + 8 q^{95} - q^{96} + ( 14 + \beta ) q^{97} + ( 2 - 2 \beta ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + O(q^{10})$$ $$2q - 2q^{2} + 2q^{3} + 2q^{4} - q^{5} - 2q^{6} - 2q^{8} + 2q^{9} + q^{10} + 2q^{11} + 2q^{12} + 3q^{13} - q^{15} + 2q^{16} - 6q^{17} - 2q^{18} + 2q^{19} - q^{20} - 2q^{22} + 2q^{23} - 2q^{24} - q^{25} - 3q^{26} + 2q^{27} + 11q^{29} + q^{30} + 2q^{31} - 2q^{32} + 2q^{33} + 6q^{34} + 2q^{36} + q^{37} - 2q^{38} + 3q^{39} + q^{40} - 3q^{41} + q^{43} + 2q^{44} - q^{45} - 2q^{46} - 9q^{47} + 2q^{48} + q^{50} - 6q^{51} + 3q^{52} + 12q^{53} - 2q^{54} + 16q^{55} + 2q^{57} - 11q^{58} - 18q^{59} - q^{60} - 10q^{61} - 2q^{62} + 2q^{64} + 7q^{65} - 2q^{66} + 6q^{67} - 6q^{68} + 2q^{69} + 6q^{71} - 2q^{72} - 10q^{73} - q^{74} - q^{75} + 2q^{76} - 3q^{78} + 12q^{79} - q^{80} + 2q^{81} + 3q^{82} + 10q^{83} + 20q^{85} - q^{86} + 11q^{87} - 2q^{88} - 28q^{89} + q^{90} + 2q^{92} + 2q^{93} + 9q^{94} + 16q^{95} - 2q^{96} + 29q^{97} + 2q^{99} + O(q^{100})$$

## 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
 2.56155 −1.56155
−1.00000 1.00000 1.00000 −2.56155 −1.00000 0 −1.00000 1.00000 2.56155
1.2 −1.00000 1.00000 1.00000 1.56155 −1.00000 0 −1.00000 1.00000 −1.56155
 $$n$$: 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.bw 2
7.b odd 2 1 966.2.a.l 2
21.c even 2 1 2898.2.a.ba 2
28.d even 2 1 7728.2.a.bm 2

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

 $$T_{5}^{2} + T_{5} - 4$$ $$T_{11}^{2} - 2 T_{11} - 16$$ $$T_{13}^{2} - 3 T_{13} - 2$$ $$T_{17}^{2} + 6 T_{17} - 8$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( -1 + T )^{2}$$
$5$ $$-4 + T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$-16 - 2 T + T^{2}$$
$13$ $$-2 - 3 T + T^{2}$$
$17$ $$-8 + 6 T + T^{2}$$
$19$ $$-16 - 2 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$26 - 11 T + T^{2}$$
$31$ $$-16 - 2 T + T^{2}$$
$37$ $$-4 - T + T^{2}$$
$41$ $$-2 + 3 T + T^{2}$$
$43$ $$-38 - T + T^{2}$$
$47$ $$16 + 9 T + T^{2}$$
$53$ $$-32 - 12 T + T^{2}$$
$59$ $$64 + 18 T + T^{2}$$
$61$ $$8 + 10 T + T^{2}$$
$67$ $$-8 - 6 T + T^{2}$$
$71$ $$-144 - 6 T + T^{2}$$
$73$ $$8 + 10 T + T^{2}$$
$79$ $$-32 - 12 T + T^{2}$$
$83$ $$8 - 10 T + T^{2}$$
$89$ $$( 14 + T )^{2}$$
$97$ $$206 - 29 T + T^{2}$$