# Properties

 Label 6762.2.a.ci Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $1$ 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: $$1$$ Dimension: $$4$$ Coefficient field: 4.4.2624.1 Defining polynomial: $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{3} - 2 \nu^{2} - 2 \nu + 1$$ $$\beta_{2}$$ $$=$$ $$2 \nu^{2} - 4 \nu - 3$$ $$\beta_{3}$$ $$=$$ $$-\nu^{3} + 2 \nu^{2} + 4 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{3} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$2 \beta_{3} + \beta_{2} + 2 \beta_{1} + 5$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + \beta_{2} + 4 \beta_{1} + 5$$

## 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
 −0.360409 2.77462 0.814115 −1.22833
1.00000 −1.00000 1.00000 −3.41421 −1.00000 0 1.00000 1.00000 −3.41421
1.2 1.00000 −1.00000 1.00000 −3.41421 −1.00000 0 1.00000 1.00000 −3.41421
1.3 1.00000 −1.00000 1.00000 −0.585786 −1.00000 0 1.00000 1.00000 −0.585786
1.4 1.00000 −1.00000 1.00000 −0.585786 −1.00000 0 1.00000 1.00000 −0.585786
 $$n$$: 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.ci 4
7.b odd 2 1 6762.2.a.ct yes 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
6762.2.a.ci 4 1.a even 1 1 trivial
6762.2.a.ct yes 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5}^{2} + 4 T_{5} + 2$$ $$T_{11}^{4} - 4 T_{11}^{3} - 20 T_{11}^{2} + 48 T_{11} + 16$$ $$T_{13}^{4} + 12 T_{13}^{3} + 36 T_{13}^{2} - 16 T_{13} - 112$$ $$T_{17}^{4} - 4 T_{17}^{3} - 12 T_{17}^{2} + 48 T_{17} - 16$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$( 2 + 4 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$16 + 48 T - 20 T^{2} - 4 T^{3} + T^{4}$$
$13$ $$-112 - 16 T + 36 T^{2} + 12 T^{3} + T^{4}$$
$17$ $$-16 + 48 T - 12 T^{2} - 4 T^{3} + T^{4}$$
$19$ $$-292 - 280 T - 56 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -1 + T )^{4}$$
$29$ $$784 + 736 T - 104 T^{2} - 8 T^{3} + T^{4}$$
$31$ $$28 - 40 T - 56 T^{2} - 4 T^{3} + T^{4}$$
$37$ $$-16 + 48 T - 12 T^{2} - 4 T^{3} + T^{4}$$
$41$ $$-476 + 656 T - 84 T^{2} - 8 T^{3} + T^{4}$$
$43$ $$3088 + 112 T - 124 T^{2} - 4 T^{3} + T^{4}$$
$47$ $$412 - 712 T - 56 T^{2} + 12 T^{3} + T^{4}$$
$53$ $$1648 + 272 T - 92 T^{2} - 4 T^{3} + T^{4}$$
$59$ $$-1024 - 1024 T - 32 T^{2} + 16 T^{3} + T^{4}$$
$61$ $$4 - 352 T - 12 T^{2} + 16 T^{3} + T^{4}$$
$67$ $$16 + 560 T + 36 T^{2} - 20 T^{3} + T^{4}$$
$71$ $$-1424 + 64 T + 160 T^{2} + 24 T^{3} + T^{4}$$
$73$ $$-316 - 464 T - 68 T^{2} + 8 T^{3} + T^{4}$$
$79$ $$-5056 + 256 T + 296 T^{2} + 32 T^{3} + T^{4}$$
$83$ $$28 - 8 T - 24 T^{2} - 4 T^{3} + T^{4}$$
$89$ $$-9968 - 4656 T - 188 T^{2} + 20 T^{3} + T^{4}$$
$97$ $$7952 - 528 T - 204 T^{2} + 4 T^{3} + T^{4}$$