# Properties

 Label 770.2.i.d Level $770$ Weight $2$ Character orbit 770.i Analytic conductor $6.148$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$770 = 2 \cdot 5 \cdot 7 \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 770.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$6.14848095564$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/770\mathbb{Z}\right)^\times$$.

 $$n$$ $$211$$ $$617$$ $$661$$ $$\chi(n)$$ $$1$$ $$1$$ $$-\zeta_{6}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
221.1
 0.5 + 0.866025i 0.5 − 0.866025i
0.500000 + 0.866025i −1.50000 + 2.59808i −0.500000 + 0.866025i −0.500000 0.866025i −3.00000 0.500000 2.59808i −1.00000 −3.00000 5.19615i 0.500000 0.866025i
331.1 0.500000 0.866025i −1.50000 2.59808i −0.500000 0.866025i −0.500000 + 0.866025i −3.00000 0.500000 + 2.59808i −1.00000 −3.00000 + 5.19615i 0.500000 + 0.866025i
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 770.2.i.d 2
7.c even 3 1 inner 770.2.i.d 2
7.c even 3 1 5390.2.a.t 1
7.d odd 6 1 5390.2.a.a 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.d 2 1.a even 1 1 trivial
770.2.i.d 2 7.c even 3 1 inner
5390.2.a.a 1 7.d odd 6 1
5390.2.a.t 1 7.c even 3 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(770, [\chi])$$:

 $$T_{3}^{2} + 3 T_{3} + 9$$ $$T_{13} + 1$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$9 + 3 T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 - T + T^{2}$$
$11$ $$1 - T + T^{2}$$
$13$ $$( 1 + T )^{2}$$
$17$ $$36 - 6 T + T^{2}$$
$19$ $$16 + 4 T + T^{2}$$
$23$ $$16 + 4 T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$64 + 8 T + T^{2}$$
$37$ $$64 + 8 T + T^{2}$$
$41$ $$( 12 + T )^{2}$$
$43$ $$( -12 + T )^{2}$$
$47$ $$36 - 6 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$225 + 15 T + T^{2}$$
$61$ $$9 + 3 T + T^{2}$$
$67$ $$49 - 7 T + T^{2}$$
$71$ $$( -10 + T )^{2}$$
$73$ $$16 - 4 T + T^{2}$$
$79$ $$225 + 15 T + T^{2}$$
$83$ $$( -10 + T )^{2}$$
$89$ $$36 + 6 T + T^{2}$$
$97$ $$( 5 + T )^{2}$$