# Properties

 Label 770.2.a.h Level $770$ Weight $2$ Character orbit 770.a Self dual yes Analytic conductor $6.148$ Analytic rank $0$ Dimension $2$ 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: $$2$$ Coefficient field: $$\Q(\sqrt{3})$$ Defining polynomial: $$x^{2} - 3$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of $$\beta = \sqrt{3}$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 −1.73205 1.73205
−1.00000 −0.732051 1.00000 1.00000 0.732051 1.00000 −1.00000 −2.46410 −1.00000
1.2 −1.00000 2.73205 1.00000 1.00000 −2.73205 1.00000 −1.00000 4.46410 −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.h 2
3.b odd 2 1 6930.2.a.ca 2
4.b odd 2 1 6160.2.a.v 2
5.b even 2 1 3850.2.a.bm 2
5.c odd 4 2 3850.2.c.s 4
7.b odd 2 1 5390.2.a.bk 2
11.b odd 2 1 8470.2.a.ce 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.a.h 2 1.a even 1 1 trivial
3850.2.a.bm 2 5.b even 2 1
3850.2.c.s 4 5.c odd 4 2
5390.2.a.bk 2 7.b odd 2 1
6160.2.a.v 2 4.b odd 2 1
6930.2.a.ca 2 3.b odd 2 1
8470.2.a.ce 2 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 $$S_{2}^{\mathrm{new}}(\Gamma_0(770))$$:

 $$T_{3}^{2} - 2 T_{3} - 2$$ $$T_{13}^{2} - 4 T_{13} - 8$$ $$T_{17}^{2} - 12$$ $$T_{19}^{2} + 2 T_{19} - 2$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{2}$$
$3$ $$-2 - 2 T + T^{2}$$
$5$ $$( -1 + T )^{2}$$
$7$ $$( -1 + T )^{2}$$
$11$ $$( -1 + T )^{2}$$
$13$ $$-8 - 4 T + T^{2}$$
$17$ $$-12 + T^{2}$$
$19$ $$-2 + 2 T + T^{2}$$
$23$ $$6 - 6 T + T^{2}$$
$29$ $$6 + 6 T + T^{2}$$
$31$ $$-44 - 4 T + T^{2}$$
$37$ $$22 - 10 T + T^{2}$$
$41$ $$6 + 6 T + T^{2}$$
$43$ $$-44 - 4 T + T^{2}$$
$47$ $$T^{2}$$
$53$ $$6 + 6 T + T^{2}$$
$59$ $$-192 + T^{2}$$
$61$ $$( -2 + T )^{2}$$
$67$ $$-32 + 8 T + T^{2}$$
$71$ $$24 - 12 T + T^{2}$$
$73$ $$52 - 16 T + T^{2}$$
$79$ $$22 - 10 T + T^{2}$$
$83$ $$-72 - 12 T + T^{2}$$
$89$ $$132 + 24 T + T^{2}$$
$97$ $$94 - 22 T + T^{2}$$