# Properties

 Label 6762.2.a.cg Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $0$ Dimension $4$ 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: $$4$$ Coefficient field: 4.4.138768.1 Defining polynomial: $$x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8$$ 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 $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 8$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{2} - \nu - 6$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - \nu^{2} - 10 \nu + 2$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$2 \beta_{2} + \beta_{1} + 6$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 2 \beta_{2} + 11 \beta_{1} + 4$$

## 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
 −2.89728 −0.479029 1.47903 3.89728
−1.00000 −1.00000 1.00000 −2.89728 1.00000 0 −1.00000 1.00000 2.89728
1.2 −1.00000 −1.00000 1.00000 −0.479029 1.00000 0 −1.00000 1.00000 0.479029
1.3 −1.00000 −1.00000 1.00000 1.47903 1.00000 0 −1.00000 1.00000 −1.47903
1.4 −1.00000 −1.00000 1.00000 3.89728 1.00000 0 −1.00000 1.00000 −3.89728
 $$n$$: 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.cg 4
7.b odd 2 1 6762.2.a.ch 4
7.c even 3 2 966.2.i.n 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.n 8 7.c even 3 2
6762.2.a.cg 4 1.a even 1 1 trivial
6762.2.a.ch 4 7.b odd 2 1

## Hecke kernels

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

 $$T_{5}^{4} - 2 T_{5}^{3} - 11 T_{5}^{2} + 12 T_{5} + 8$$ $$T_{11}^{4} - 6 T_{11}^{3} - 22 T_{11}^{2} + 170 T_{11} - 199$$ $$T_{13}^{4} + 2 T_{13}^{3} - 34 T_{13}^{2} - 112 T_{13} - 56$$ $$T_{17}^{4} + 2 T_{17}^{3} - 25 T_{17}^{2} - 82 T_{17} - 62$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$8 + 12 T - 11 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$-199 + 170 T - 22 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$-56 - 112 T - 34 T^{2} + 2 T^{3} + T^{4}$$
$17$ $$-62 - 82 T - 25 T^{2} + 2 T^{3} + T^{4}$$
$19$ $$296 - 80 T - 34 T^{2} + 6 T^{3} + T^{4}$$
$23$ $$( -1 + T )^{4}$$
$29$ $$189 - 126 T - 10 T^{2} + 10 T^{3} + T^{4}$$
$31$ $$-636 - 420 T + 5 T^{2} + 14 T^{3} + T^{4}$$
$37$ $$32 + 16 T - 22 T^{2} - 6 T^{3} + T^{4}$$
$41$ $$-56 + 224 T - 10 T^{2} - 10 T^{3} + T^{4}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$-2588 - 1052 T - 79 T^{2} + 10 T^{3} + T^{4}$$
$53$ $$-12 + 24 T + T^{2} - 6 T^{3} + T^{4}$$
$59$ $$-7104 + 1152 T + 97 T^{2} - 24 T^{3} + T^{4}$$
$61$ $$128 + 672 T + 244 T^{2} + 28 T^{3} + T^{4}$$
$67$ $$-9008 + 1232 T + 152 T^{2} - 28 T^{3} + T^{4}$$
$71$ $$18 - 66 T + 67 T^{2} - 16 T^{3} + T^{4}$$
$73$ $$-1624 + 868 T - 103 T^{2} - 6 T^{3} + T^{4}$$
$79$ $$1213 - 1244 T - 226 T^{2} + 4 T^{3} + T^{4}$$
$83$ $$-3708 + 1176 T - 43 T^{2} - 14 T^{3} + T^{4}$$
$89$ $$-3072 - 1344 T - 86 T^{2} + 14 T^{3} + T^{4}$$
$97$ $$23888 + 112 T - 319 T^{2} + T^{4}$$