# Properties

 Label 6762.2.a.cp 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: 4.4.42048.1 Defining polynomial: $$x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34$$ Coefficient ring: $$\Z[a_1, \ldots, a_{13}]$$ Coefficient ring index: $$2$$ 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.

 $$f(q)$$ $$=$$ $$q + q^{2} + q^{3} + q^{4} + ( -1 + \beta_{2} ) q^{5} + q^{6} + q^{8} + q^{9} +O(q^{10})$$ $$q + q^{2} + q^{3} + q^{4} + ( -1 + \beta_{2} ) q^{5} + q^{6} + q^{8} + q^{9} + ( -1 + \beta_{2} ) q^{10} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{11} + q^{12} + ( 1 - \beta_{1} ) q^{13} + ( -1 + \beta_{2} ) q^{15} + q^{16} + ( -3 - \beta_{1} + \beta_{2} ) q^{17} + q^{18} + ( -1 - \beta_{2} + \beta_{3} ) q^{19} + ( -1 + \beta_{2} ) q^{20} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{22} + q^{23} + q^{24} + ( 4 - 2 \beta_{2} ) q^{25} + ( 1 - \beta_{1} ) q^{26} + q^{27} + ( 2 + \beta_{1} ) q^{29} + ( -1 + \beta_{2} ) q^{30} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{31} + q^{32} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{33} + ( -3 - \beta_{1} + \beta_{2} ) q^{34} + q^{36} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{37} + ( -1 - \beta_{2} + \beta_{3} ) q^{38} + ( 1 - \beta_{1} ) q^{39} + ( -1 + \beta_{2} ) q^{40} + ( 1 + \beta_{2} - \beta_{3} ) q^{41} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{43} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{44} + ( -1 + \beta_{2} ) q^{45} + q^{46} + ( 4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{47} + q^{48} + ( 4 - 2 \beta_{2} ) q^{50} + ( -3 - \beta_{1} + \beta_{2} ) q^{51} + ( 1 - \beta_{1} ) q^{52} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{53} + q^{54} + ( 5 + 5 \beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{55} + ( -1 - \beta_{2} + \beta_{3} ) q^{57} + ( 2 + \beta_{1} ) q^{58} + ( 2 + \beta_{1} - 2 \beta_{2} ) q^{59} + ( -1 + \beta_{2} ) q^{60} + ( -2 - 4 \beta_{1} ) q^{61} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{62} + q^{64} + ( -3 + \beta_{1} + \beta_{2} + 2 \beta_{3} ) q^{65} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{66} + ( 6 + 3 \beta_{1} - 2 \beta_{2} - \beta_{3} ) q^{67} + ( -3 - \beta_{1} + \beta_{2} ) q^{68} + q^{69} + ( 11 + \beta_{1} ) q^{71} + q^{72} + ( 2 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{73} + ( -1 - 2 \beta_{1} + \beta_{2} - \beta_{3} ) q^{74} + ( 4 - 2 \beta_{2} ) q^{75} + ( -1 - \beta_{2} + \beta_{3} ) q^{76} + ( 1 - \beta_{1} ) q^{78} + ( -2 - \beta_{1} ) q^{79} + ( -1 + \beta_{2} ) q^{80} + q^{81} + ( 1 + \beta_{2} - \beta_{3} ) q^{82} + ( -3 - \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{83} + ( 9 + \beta_{1} - 4 \beta_{2} + 2 \beta_{3} ) q^{85} + ( 6 + 2 \beta_{1} + 2 \beta_{3} ) q^{86} + ( 2 + \beta_{1} ) q^{87} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{88} + ( 1 + 2 \beta_{1} + \beta_{2} + \beta_{3} ) q^{89} + ( -1 + \beta_{2} ) q^{90} + q^{92} + ( 1 + \beta_{1} + \beta_{2} - \beta_{3} ) q^{93} + ( 4 + 3 \beta_{1} - \beta_{2} + \beta_{3} ) q^{94} + ( -7 - 4 \beta_{1} + \beta_{2} - \beta_{3} ) q^{95} + q^{96} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{97} + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} + O(q^{10})$$ $$4q + 4q^{2} + 4q^{3} + 4q^{4} - 4q^{5} + 4q^{6} + 4q^{8} + 4q^{9} - 4q^{10} + 6q^{11} + 4q^{12} + 6q^{13} - 4q^{15} + 4q^{16} - 10q^{17} + 4q^{18} - 4q^{19} - 4q^{20} + 6q^{22} + 4q^{23} + 4q^{24} + 16q^{25} + 6q^{26} + 4q^{27} + 6q^{29} - 4q^{30} + 2q^{31} + 4q^{32} + 6q^{33} - 10q^{34} + 4q^{36} - 4q^{38} + 6q^{39} - 4q^{40} + 4q^{41} + 20q^{43} + 6q^{44} - 4q^{45} + 4q^{46} + 10q^{47} + 4q^{48} + 16q^{50} - 10q^{51} + 6q^{52} + 4q^{54} + 10q^{55} - 4q^{57} + 6q^{58} + 6q^{59} - 4q^{60} + 2q^{62} + 4q^{64} - 14q^{65} + 6q^{66} + 18q^{67} - 10q^{68} + 4q^{69} + 42q^{71} + 4q^{72} + 6q^{73} + 16q^{75} - 4q^{76} + 6q^{78} - 6q^{79} - 4q^{80} + 4q^{81} + 4q^{82} - 10q^{83} + 34q^{85} + 20q^{86} + 6q^{87} + 6q^{88} - 4q^{90} + 4q^{92} + 2q^{93} + 10q^{94} - 20q^{95} + 4q^{96} - 10q^{97} + 6q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 2 x^{3} - 13 x^{2} + 8 x + 34$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 10 \nu + 10$$$$)/4$$ $$\beta_{2}$$ $$=$$ $$($$$$\nu^{3} - 3 \nu^{2} - 6 \nu + 10$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$($$$$-\nu^{3} + 7 \nu^{2} + 2 \nu - 38$$$$)/4$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} - 2 \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$\beta_{3} + \beta_{2} - \beta_{1} + 7$$ $$\nu^{3}$$ $$=$$ $$3 \beta_{3} + 8 \beta_{2} - 9 \beta_{1} + 11$$

## 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
 −2.35854 1.94433 4.18558 −1.77137
1.00000 1.00000 1.00000 −3.82843 1.00000 0 1.00000 1.00000 −3.82843
1.2 1.00000 1.00000 1.00000 −3.82843 1.00000 0 1.00000 1.00000 −3.82843
1.3 1.00000 1.00000 1.00000 1.82843 1.00000 0 1.00000 1.00000 1.82843
1.4 1.00000 1.00000 1.00000 1.82843 1.00000 0 1.00000 1.00000 1.82843
 $$n$$: 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.cp 4
7.b odd 2 1 6762.2.a.cm 4
7.c even 3 2 966.2.i.l 8

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

## Hecke kernels

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

 $$T_{5}^{2} + 2 T_{5} - 7$$ $$T_{11}^{4} - 6 T_{11}^{3} - 19 T_{11}^{2} + 156 T_{11} - 188$$ $$T_{13}^{4} - 6 T_{13}^{3} - T_{13}^{2} + 36 T_{13} - 2$$ $$T_{17}^{4} + 10 T_{17}^{3} + 15 T_{17}^{2} - 68 T_{17} - 146$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( -1 + T )^{4}$$
$3$ $$( -1 + T )^{4}$$
$5$ $$( -7 + 2 T + T^{2} )^{2}$$
$7$ $$T^{4}$$
$11$ $$-188 + 156 T - 19 T^{2} - 6 T^{3} + T^{4}$$
$13$ $$-2 + 36 T - T^{2} - 6 T^{3} + T^{4}$$
$17$ $$-146 - 68 T + 15 T^{2} + 10 T^{3} + T^{4}$$
$19$ $$64 - 32 T - 30 T^{2} + 4 T^{3} + T^{4}$$
$23$ $$( -1 + T )^{4}$$
$29$ $$16 + 24 T - T^{2} - 6 T^{3} + T^{4}$$
$31$ $$-188 - 220 T - 67 T^{2} - 2 T^{3} + T^{4}$$
$37$ $$136 + 72 T - 58 T^{2} + T^{4}$$
$41$ $$64 + 32 T - 30 T^{2} - 4 T^{3} + T^{4}$$
$43$ $$-5696 + 1376 T + 20 T^{2} - 20 T^{3} + T^{4}$$
$47$ $$1348 + 500 T - 75 T^{2} - 10 T^{3} + T^{4}$$
$53$ $$553 + 48 T - 58 T^{2} + T^{4}$$
$59$ $$112 + 216 T - 49 T^{2} - 6 T^{3} + T^{4}$$
$61$ $$8848 + 384 T - 232 T^{2} + T^{4}$$
$67$ $$-11196 + 2988 T - 99 T^{2} - 18 T^{3} + T^{4}$$
$71$ $$10654 - 4332 T + 647 T^{2} - 42 T^{3} + T^{4}$$
$73$ $$-32 - 96 T - 55 T^{2} - 6 T^{3} + T^{4}$$
$79$ $$16 - 24 T - T^{2} + 6 T^{3} + T^{4}$$
$83$ $$1252 - 220 T - 91 T^{2} + 10 T^{3} + T^{4}$$
$89$ $$-632 + 504 T - 106 T^{2} + T^{4}$$
$97$ $$64 - 208 T - 25 T^{2} + 10 T^{3} + T^{4}$$