# Properties

 Label 6762.2.a.cb 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: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 138) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{5}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −0.618034 1.61803
1.00000 −1.00000 1.00000 −1.23607 −1.00000 0 1.00000 1.00000 −1.23607
1.2 1.00000 −1.00000 1.00000 3.23607 −1.00000 0 1.00000 1.00000 3.23607
 $$n$$: 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.cb 2
7.b odd 2 1 138.2.a.d 2
21.c even 2 1 414.2.a.f 2
28.d even 2 1 1104.2.a.j 2
35.c odd 2 1 3450.2.a.be 2
35.f even 4 2 3450.2.d.x 4
56.e even 2 1 4416.2.a.bl 2
56.h odd 2 1 4416.2.a.bh 2
84.h odd 2 1 3312.2.a.bc 2
161.c even 2 1 3174.2.a.s 2
483.c odd 2 1 9522.2.a.q 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.2.a.d 2 7.b odd 2 1
414.2.a.f 2 21.c even 2 1
1104.2.a.j 2 28.d even 2 1
3174.2.a.s 2 161.c even 2 1
3312.2.a.bc 2 84.h odd 2 1
3450.2.a.be 2 35.c odd 2 1
3450.2.d.x 4 35.f even 4 2
4416.2.a.bh 2 56.h odd 2 1
4416.2.a.bl 2 56.e even 2 1
6762.2.a.cb 2 1.a even 1 1 trivial
9522.2.a.q 2 483.c odd 2 1

## Hecke kernels

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

 $$T_{5}^{2} - 2 T_{5} - 4$$ $$T_{11}^{2} + 6 T_{11} + 4$$ $$T_{13}^{2} - 20$$ $$T_{17} - 4$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-4 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$4 + 6 T + T^{2}$$
$13$ $$-20 + T^{2}$$
$17$ $$( -4 + T )^{2}$$
$19$ $$-44 - 2 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$-20 + T^{2}$$
$31$ $$-16 + 4 T + T^{2}$$
$37$ $$76 - 18 T + T^{2}$$
$41$ $$( -2 + T )^{2}$$
$43$ $$44 + 14 T + T^{2}$$
$47$ $$( 4 + T )^{2}$$
$53$ $$4 - 6 T + T^{2}$$
$59$ $$-80 + T^{2}$$
$61$ $$4 + 6 T + T^{2}$$
$67$ $$-36 - 6 T + T^{2}$$
$71$ $$-80 + T^{2}$$
$73$ $$-20 + T^{2}$$
$79$ $$-20 + T^{2}$$
$83$ $$116 + 22 T + T^{2}$$
$89$ $$16 - 12 T + T^{2}$$
$97$ $$-4 - 8 T + T^{2}$$