# Properties

 Label 770.2.a.l Level $770$ Weight $2$ Character orbit 770.a Self dual yes Analytic conductor $6.148$ Analytic rank $0$ Dimension $3$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.892.1 Defining polynomial: $$x^{3} - x^{2} - 8 x + 10$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$2$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} - q^{7} - q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} +O(q^{10})$$ $$q - q^{2} + \beta_{1} q^{3} + q^{4} - q^{5} -\beta_{1} q^{6} - q^{7} - q^{8} + ( 4 + \beta_{1} + \beta_{2} ) q^{9} + q^{10} + q^{11} + \beta_{1} q^{12} + 2 \beta_{1} q^{13} + q^{14} -\beta_{1} q^{15} + q^{16} + ( -3 - \beta_{1} - \beta_{2} ) q^{17} + ( -4 - \beta_{1} - \beta_{2} ) q^{18} + ( -1 - \beta_{2} ) q^{19} - q^{20} -\beta_{1} q^{21} - q^{22} + ( 1 + \beta_{2} ) q^{23} -\beta_{1} q^{24} + q^{25} -2 \beta_{1} q^{26} + ( 4 + 4 \beta_{1} ) q^{27} - q^{28} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{29} + \beta_{1} q^{30} + 6 q^{31} - q^{32} + \beta_{1} q^{33} + ( 3 + \beta_{1} + \beta_{2} ) q^{34} + q^{35} + ( 4 + \beta_{1} + \beta_{2} ) q^{36} + ( 1 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( 1 + \beta_{2} ) q^{38} + ( 14 + 2 \beta_{1} + 2 \beta_{2} ) q^{39} + q^{40} + ( -6 - \beta_{1} ) q^{41} + \beta_{1} q^{42} + ( 3 - \beta_{1} + \beta_{2} ) q^{43} + q^{44} + ( -4 - \beta_{1} - \beta_{2} ) q^{45} + ( -1 - \beta_{2} ) q^{46} + ( 1 - \beta_{1} + \beta_{2} ) q^{47} + \beta_{1} q^{48} + q^{49} - q^{50} + ( -4 - 6 \beta_{1} ) q^{51} + 2 \beta_{1} q^{52} + ( 1 + 2 \beta_{1} - \beta_{2} ) q^{53} + ( -4 - 4 \beta_{1} ) q^{54} - q^{55} + q^{56} + ( 3 - 3 \beta_{1} + \beta_{2} ) q^{57} + ( -1 - 2 \beta_{1} + \beta_{2} ) q^{58} + ( 3 - 3 \beta_{1} - \beta_{2} ) q^{59} -\beta_{1} q^{60} + ( 7 - \beta_{1} + \beta_{2} ) q^{61} -6 q^{62} + ( -4 - \beta_{1} - \beta_{2} ) q^{63} + q^{64} -2 \beta_{1} q^{65} -\beta_{1} q^{66} + 8 q^{67} + ( -3 - \beta_{1} - \beta_{2} ) q^{68} + ( -3 + 3 \beta_{1} - \beta_{2} ) q^{69} - q^{70} + ( 2 - 2 \beta_{2} ) q^{71} + ( -4 - \beta_{1} - \beta_{2} ) q^{72} + ( -1 - 3 \beta_{1} + \beta_{2} ) q^{73} + ( -1 + 2 \beta_{1} + \beta_{2} ) q^{74} + \beta_{1} q^{75} + ( -1 - \beta_{2} ) q^{76} - q^{77} + ( -14 - 2 \beta_{1} - 2 \beta_{2} ) q^{78} + ( -3 - 2 \beta_{1} - 3 \beta_{2} ) q^{79} - q^{80} + ( 16 + 5 \beta_{1} + \beta_{2} ) q^{81} + ( 6 + \beta_{1} ) q^{82} + ( -8 + 2 \beta_{1} ) q^{83} -\beta_{1} q^{84} + ( 3 + \beta_{1} + \beta_{2} ) q^{85} + ( -3 + \beta_{1} - \beta_{2} ) q^{86} + ( 17 + \beta_{1} + 3 \beta_{2} ) q^{87} - q^{88} + ( -2 - 2 \beta_{1} ) q^{89} + ( 4 + \beta_{1} + \beta_{2} ) q^{90} -2 \beta_{1} q^{91} + ( 1 + \beta_{2} ) q^{92} + 6 \beta_{1} q^{93} + ( -1 + \beta_{1} - \beta_{2} ) q^{94} + ( 1 + \beta_{2} ) q^{95} -\beta_{1} q^{96} + ( 3 + \beta_{2} ) q^{97} - q^{98} + ( 4 + \beta_{1} + \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8} + 11 q^{9} + O(q^{10})$$ $$3 q - 3 q^{2} + 3 q^{4} - 3 q^{5} - 3 q^{7} - 3 q^{8} + 11 q^{9} + 3 q^{10} + 3 q^{11} + 3 q^{14} + 3 q^{16} - 8 q^{17} - 11 q^{18} - 2 q^{19} - 3 q^{20} - 3 q^{22} + 2 q^{23} + 3 q^{25} + 12 q^{27} - 3 q^{28} + 4 q^{29} + 18 q^{31} - 3 q^{32} + 8 q^{34} + 3 q^{35} + 11 q^{36} + 4 q^{37} + 2 q^{38} + 40 q^{39} + 3 q^{40} - 18 q^{41} + 8 q^{43} + 3 q^{44} - 11 q^{45} - 2 q^{46} + 2 q^{47} + 3 q^{49} - 3 q^{50} - 12 q^{51} + 4 q^{53} - 12 q^{54} - 3 q^{55} + 3 q^{56} + 8 q^{57} - 4 q^{58} + 10 q^{59} + 20 q^{61} - 18 q^{62} - 11 q^{63} + 3 q^{64} + 24 q^{67} - 8 q^{68} - 8 q^{69} - 3 q^{70} + 8 q^{71} - 11 q^{72} - 4 q^{73} - 4 q^{74} - 2 q^{76} - 3 q^{77} - 40 q^{78} - 6 q^{79} - 3 q^{80} + 47 q^{81} + 18 q^{82} - 24 q^{83} + 8 q^{85} - 8 q^{86} + 48 q^{87} - 3 q^{88} - 6 q^{89} + 11 q^{90} + 2 q^{92} - 2 q^{94} + 2 q^{95} + 8 q^{97} - 3 q^{98} + 11 q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{3} - x^{2} - 8 x + 10$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu^{2} + \nu - 6$$ $$\beta_{2}$$ $$=$$ $$-\nu^{2} + \nu + 5$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1} + 1$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$-\beta_{2} + \beta_{1} + 11$$$$)/2$$

## 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
 1.31955 −2.91729 2.59774
−1.00000 −2.93923 1.00000 −1.00000 2.93923 −1.00000 −1.00000 5.63910 1.00000
1.2 −1.00000 −0.406728 1.00000 −1.00000 0.406728 −1.00000 −1.00000 −2.83457 1.00000
1.3 −1.00000 3.34596 1.00000 −1.00000 −3.34596 −1.00000 −1.00000 8.19547 1.00000
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$1$$
$$5$$ $$1$$
$$7$$ $$1$$
$$11$$ $$-1$$

## 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 770.2.a.l 3
3.b odd 2 1 6930.2.a.cl 3
4.b odd 2 1 6160.2.a.bi 3
5.b even 2 1 3850.2.a.bu 3
5.c odd 4 2 3850.2.c.z 6
7.b odd 2 1 5390.2.a.bz 3
11.b odd 2 1 8470.2.a.cl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.l 3 1.a even 1 1 trivial
3850.2.a.bu 3 5.b even 2 1
3850.2.c.z 6 5.c odd 4 2
5390.2.a.bz 3 7.b odd 2 1
6160.2.a.bi 3 4.b odd 2 1
6930.2.a.cl 3 3.b odd 2 1
8470.2.a.cl 3 11.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 $$S_{2}^{\mathrm{new}}(\Gamma_0(770))$$:

 $$T_{3}^{3} - 10 T_{3} - 4$$ $$T_{13}^{3} - 40 T_{13} - 32$$ $$T_{17}^{3} + 8 T_{17}^{2} - 12 T_{17} - 128$$ $$T_{19}^{3} + 2 T_{19}^{2} - 30 T_{19} - 56$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{3}$$
$3$ $$-4 - 10 T + T^{3}$$
$5$ $$( 1 + T )^{3}$$
$7$ $$( 1 + T )^{3}$$
$11$ $$( -1 + T )^{3}$$
$13$ $$-32 - 40 T + T^{3}$$
$17$ $$-128 - 12 T + 8 T^{2} + T^{3}$$
$19$ $$-56 - 30 T + 2 T^{2} + T^{3}$$
$23$ $$56 - 30 T - 2 T^{2} + T^{3}$$
$29$ $$428 - 82 T - 4 T^{2} + T^{3}$$
$31$ $$( -6 + T )^{3}$$
$37$ $$124 - 50 T - 4 T^{2} + T^{3}$$
$41$ $$160 + 98 T + 18 T^{2} + T^{3}$$
$43$ $$16 - 28 T - 8 T^{2} + T^{3}$$
$47$ $$-64 - 48 T - 2 T^{2} + T^{3}$$
$53$ $$428 - 82 T - 4 T^{2} + T^{3}$$
$59$ $$608 - 64 T - 10 T^{2} + T^{3}$$
$61$ $$-64 + 84 T - 20 T^{2} + T^{3}$$
$67$ $$( -8 + T )^{3}$$
$71$ $$32 - 104 T - 8 T^{2} + T^{3}$$
$73$ $$-784 - 140 T + 4 T^{2} + T^{3}$$
$79$ $$-2272 - 262 T + 6 T^{2} + T^{3}$$
$83$ $$160 + 152 T + 24 T^{2} + T^{3}$$
$89$ $$-40 - 28 T + 6 T^{2} + T^{3}$$
$97$ $$100 - 10 T - 8 T^{2} + T^{3}$$