# Properties

 Label 6762.2.a.cl 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: 4.4.473376.1 Defining polynomial: $$x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 30$$ Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 30$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2 \nu - 7$$ $$\beta_{3}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 8 \nu + 10$$$$)/2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2 \beta_{1} + 7$$ $$\nu^{3}$$ $$=$$ $$2 \beta_{3} + 3 \beta_{2} + 14 \beta_{1} + 11$$

## 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
 4.22985 1.83178 −1.52852 −2.53310
1.00000 −1.00000 1.00000 −3.22985 −1.00000 0 1.00000 1.00000 −3.22985
1.2 1.00000 −1.00000 1.00000 −0.831776 −1.00000 0 1.00000 1.00000 −0.831776
1.3 1.00000 −1.00000 1.00000 2.52852 −1.00000 0 1.00000 1.00000 2.52852
1.4 1.00000 −1.00000 1.00000 3.53310 −1.00000 0 1.00000 1.00000 3.53310
 $$n$$: 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.cl 4
7.b odd 2 1 6762.2.a.cr 4
7.c even 3 2 966.2.i.m 8

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.m 8 7.c even 3 2
6762.2.a.cl 4 1.a even 1 1 trivial
6762.2.a.cr 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5}^{4} - 2 T_{5}^{3} - 13 T_{5}^{2} + 20 T_{5} + 24$$ $$T_{11}^{4} - 31 T_{11}^{2} + 12 T_{11} + 180$$ $$T_{13}^{4} - 6 T_{13}^{3} - 27 T_{13}^{2} + 164 T_{13} - 60$$ $$T_{17}^{4} - 2 T_{17}^{3} - 28 T_{17}^{2} - 34 T_{17} + 3$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( 1 + T )^{4}$$
$5$ $$24 + 20 T - 13 T^{2} - 2 T^{3} + T^{4}$$
$7$ $$T^{4}$$
$11$ $$180 + 12 T - 31 T^{2} + T^{4}$$
$13$ $$-60 + 164 T - 27 T^{2} - 6 T^{3} + T^{4}$$
$17$ $$3 - 34 T - 28 T^{2} - 2 T^{3} + T^{4}$$
$19$ $$( 2 + T )^{4}$$
$23$ $$( 1 + T )^{4}$$
$29$ $$-40 + 56 T + 33 T^{2} - 14 T^{3} + T^{4}$$
$31$ $$( 2 + T )^{4}$$
$37$ $$-576 + 128 T + 92 T^{2} - 20 T^{3} + T^{4}$$
$41$ $$480 + 64 T - 52 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$( -2 + T )^{4}$$
$47$ $$2661 - 340 T - 118 T^{2} + 4 T^{3} + T^{4}$$
$53$ $$-324 + 1264 T - 33 T^{2} - 18 T^{3} + T^{4}$$
$59$ $$6144 + 1280 T - 208 T^{2} - 8 T^{3} + T^{4}$$
$61$ $$2048 - 128 T - 156 T^{2} - 4 T^{3} + T^{4}$$
$67$ $$3308 - 392 T - 153 T^{2} + 2 T^{3} + T^{4}$$
$71$ $$8181 + 612 T - 246 T^{2} - 4 T^{3} + T^{4}$$
$73$ $$2025 + 168 T - 138 T^{2} - 8 T^{3} + T^{4}$$
$79$ $$2388 - 1564 T + 353 T^{2} - 32 T^{3} + T^{4}$$
$83$ $$48 + 272 T - 112 T^{2} + 4 T^{3} + T^{4}$$
$89$ $$2304 - 960 T - 152 T^{2} + 8 T^{3} + T^{4}$$
$97$ $$( 8 + T )^{4}$$