# Properties

 Label 6762.2.a.cf Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $1$ Dimension $3$ 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: $$3$$ Coefficient field: 3.3.3132.1 Defining polynomial: $$x^{3} - 15 x - 6$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ 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 a basis $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} +O(q^{10})$$ $$q - q^{2} + q^{3} + q^{4} - q^{5} - q^{6} - q^{8} + q^{9} + q^{10} + \beta_{1} q^{11} + q^{12} + ( 1 + \beta_{1} + \beta_{2} ) q^{13} - q^{15} + q^{16} + ( -1 - \beta_{1} + \beta_{2} ) q^{17} - q^{18} + ( -2 - \beta_{2} ) q^{19} - q^{20} -\beta_{1} q^{22} - q^{23} - q^{24} -4 q^{25} + ( -1 - \beta_{1} - \beta_{2} ) q^{26} + q^{27} + ( 4 - \beta_{1} - \beta_{2} ) q^{29} + q^{30} + ( -\beta_{1} - 2 \beta_{2} ) q^{31} - q^{32} + \beta_{1} q^{33} + ( 1 + \beta_{1} - \beta_{2} ) q^{34} + q^{36} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{37} + ( 2 + \beta_{2} ) q^{38} + ( 1 + \beta_{1} + \beta_{2} ) q^{39} + q^{40} + ( 6 - \beta_{2} ) q^{41} -2 \beta_{1} q^{43} + \beta_{1} q^{44} - q^{45} + q^{46} + ( 1 - \beta_{1} - 2 \beta_{2} ) q^{47} + q^{48} + 4 q^{50} + ( -1 - \beta_{1} + \beta_{2} ) q^{51} + ( 1 + \beta_{1} + \beta_{2} ) q^{52} + ( -1 - 2 \beta_{1} ) q^{53} - q^{54} -\beta_{1} q^{55} + ( -2 - \beta_{2} ) q^{57} + ( -4 + \beta_{1} + \beta_{2} ) q^{58} + ( 4 + \beta_{1} - \beta_{2} ) q^{59} - q^{60} -10 q^{61} + ( \beta_{1} + 2 \beta_{2} ) q^{62} + q^{64} + ( -1 - \beta_{1} - \beta_{2} ) q^{65} -\beta_{1} q^{66} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{67} + ( -1 - \beta_{1} + \beta_{2} ) q^{68} - q^{69} + ( -1 + 3 \beta_{1} + \beta_{2} ) q^{71} - q^{72} + ( -5 - \beta_{1} ) q^{73} + ( 4 + 2 \beta_{1} - \beta_{2} ) q^{74} -4 q^{75} + ( -2 - \beta_{2} ) q^{76} + ( -1 - \beta_{1} - \beta_{2} ) q^{78} + ( -4 + 3 \beta_{1} + \beta_{2} ) q^{79} - q^{80} + q^{81} + ( -6 + \beta_{2} ) q^{82} + ( 4 - \beta_{1} ) q^{83} + ( 1 + \beta_{1} - \beta_{2} ) q^{85} + 2 \beta_{1} q^{86} + ( 4 - \beta_{1} - \beta_{2} ) q^{87} -\beta_{1} q^{88} + ( -4 - 2 \beta_{1} + \beta_{2} ) q^{89} + q^{90} - q^{92} + ( -\beta_{1} - 2 \beta_{2} ) q^{93} + ( -1 + \beta_{1} + 2 \beta_{2} ) q^{94} + ( 2 + \beta_{2} ) q^{95} - q^{96} + ( 3 \beta_{1} + 3 \beta_{2} ) q^{97} + \beta_{1} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{2} + 3q^{3} + 3q^{4} - 3q^{5} - 3q^{6} - 3q^{8} + 3q^{9} + 3q^{10} + 3q^{12} + 3q^{13} - 3q^{15} + 3q^{16} - 3q^{17} - 3q^{18} - 6q^{19} - 3q^{20} - 3q^{23} - 3q^{24} - 12q^{25} - 3q^{26} + 3q^{27} + 12q^{29} + 3q^{30} - 3q^{32} + 3q^{34} + 3q^{36} - 12q^{37} + 6q^{38} + 3q^{39} + 3q^{40} + 18q^{41} - 3q^{45} + 3q^{46} + 3q^{47} + 3q^{48} + 12q^{50} - 3q^{51} + 3q^{52} - 3q^{53} - 3q^{54} - 6q^{57} - 12q^{58} + 12q^{59} - 3q^{60} - 30q^{61} + 3q^{64} - 3q^{65} - 21q^{67} - 3q^{68} - 3q^{69} - 3q^{71} - 3q^{72} - 15q^{73} + 12q^{74} - 12q^{75} - 6q^{76} - 3q^{78} - 12q^{79} - 3q^{80} + 3q^{81} - 18q^{82} + 12q^{83} + 3q^{85} + 12q^{87} - 12q^{89} + 3q^{90} - 3q^{92} - 3q^{94} + 6q^{95} - 3q^{96} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - 15 x - 6$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 10$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 10$$

## 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
 −3.65491 −0.404409 4.05932
−1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.2 −1.00000 1.00000 1.00000 −1.00000 −1.00000 0 −1.00000 1.00000 1.00000
1.3 −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.cf 3
7.b odd 2 1 6762.2.a.ce 3
7.d odd 6 2 966.2.i.k 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.k 6 7.d odd 6 2
6762.2.a.ce 3 7.b odd 2 1
6762.2.a.cf 3 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5} + 1$$ $$T_{11}^{3} - 15 T_{11} - 6$$ $$T_{13}^{3} - 3 T_{13}^{2} - 24 T_{13} + 22$$ $$T_{17}^{3} + 3 T_{17}^{2} - 36 T_{17} - 126$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$( -1 + T )^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$T^{3}$$
$11$ $$-6 - 15 T + T^{3}$$
$13$ $$22 - 24 T - 3 T^{2} + T^{3}$$
$17$ $$-126 - 36 T + 3 T^{2} + T^{3}$$
$19$ $$-48 - 6 T + 6 T^{2} + T^{3}$$
$23$ $$( 1 + T )^{3}$$
$29$ $$48 + 21 T - 12 T^{2} + T^{3}$$
$31$ $$-214 - 75 T + T^{3}$$
$37$ $$-588 - 42 T + 12 T^{2} + T^{3}$$
$41$ $$-128 + 90 T - 18 T^{2} + T^{3}$$
$43$ $$48 - 60 T + T^{3}$$
$47$ $$-140 - 72 T - 3 T^{2} + T^{3}$$
$53$ $$-11 - 57 T + 3 T^{2} + T^{3}$$
$59$ $$180 + 9 T - 12 T^{2} + T^{3}$$
$61$ $$( 10 + T )^{3}$$
$67$ $$-396 + 72 T + 21 T^{2} + T^{3}$$
$71$ $$-726 - 132 T + 3 T^{2} + T^{3}$$
$73$ $$56 + 60 T + 15 T^{2} + T^{3}$$
$79$ $$-1068 - 87 T + 12 T^{2} + T^{3}$$
$83$ $$2 + 33 T - 12 T^{2} + T^{3}$$
$89$ $$-588 - 42 T + 12 T^{2} + T^{3}$$
$97$ $$-108 - 243 T + T^{3}$$