# Properties

 Label 6762.2.a.bt Level $6762$ Weight $2$ Character orbit 6762.a Self dual yes Analytic conductor $53.995$ Analytic rank $1$ 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{2})$$ Defining polynomial: $$x^{2} - 2$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ 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 $$\beta = \sqrt{2}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.41421 1.41421
−1.00000 −1.00000 1.00000 −0.414214 1.00000 0 −1.00000 1.00000 0.414214
1.2 −1.00000 −1.00000 1.00000 2.41421 1.00000 0 −1.00000 1.00000 −2.41421
 $$n$$: 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.bt 2
7.b odd 2 1 6762.2.a.bv 2
7.d odd 6 2 966.2.i.j 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.j 4 7.d odd 6 2
6762.2.a.bt 2 1.a even 1 1 trivial
6762.2.a.bv 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(6762))$$:

 $$T_{5}^{2} - 2 T_{5} - 1$$ $$T_{11} - 2$$ $$T_{13}^{2} + 2 T_{13} - 7$$ $$T_{17}^{2} + 2 T_{17} - 17$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$( 1 + T )^{2}$$
$5$ $$-1 - 2 T + T^{2}$$
$7$ $$T^{2}$$
$11$ $$( -2 + T )^{2}$$
$13$ $$-7 + 2 T + T^{2}$$
$17$ $$-17 + 2 T + T^{2}$$
$19$ $$-28 - 4 T + T^{2}$$
$23$ $$( -1 + T )^{2}$$
$29$ $$8 + 8 T + T^{2}$$
$31$ $$( -2 + T )^{2}$$
$37$ $$28 + 12 T + T^{2}$$
$41$ $$56 - 16 T + T^{2}$$
$43$ $$4 + 12 T + T^{2}$$
$47$ $$( 9 + T )^{2}$$
$53$ $$-41 + 6 T + T^{2}$$
$59$ $$-16 - 8 T + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-73 - 10 T + T^{2}$$
$71$ $$41 + 14 T + T^{2}$$
$73$ $$-7 - 10 T + T^{2}$$
$79$ $$( 10 + T )^{2}$$
$83$ $$92 + 20 T + T^{2}$$
$89$ $$8 + 8 T + T^{2}$$
$97$ $$-16 - 8 T + T^{2}$$