# Properties

 Label 770.2.i.e Level $770$ Weight $2$ Character orbit 770.i Analytic conductor $6.148$ Analytic rank $1$ 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: $$1$$ 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.

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

## 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 −0.500000 + 0.866025i −0.500000 + 0.866025i −0.500000 0.866025i −1.00000 −2.50000 0.866025i −1.00000 1.00000 + 1.73205i 0.500000 0.866025i
331.1 0.500000 0.866025i −0.500000 0.866025i −0.500000 0.866025i −0.500000 + 0.866025i −1.00000 −2.50000 + 0.866025i −1.00000 1.00000 1.73205i 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.e 2
7.c even 3 1 inner 770.2.i.e 2
7.c even 3 1 5390.2.a.o 1
7.d odd 6 1 5390.2.a.f 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
770.2.i.e 2 1.a even 1 1 trivial
770.2.i.e 2 7.c even 3 1 inner
5390.2.a.f 1 7.d odd 6 1
5390.2.a.o 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 $$S_{2}^{\mathrm{new}}(770, [\chi])$$:

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$1 + T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$1 + T + T^{2}$$
$13$ $$( 3 + T )^{2}$$
$17$ $$T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$4 + 2 T + T^{2}$$
$29$ $$( 7 + T )^{2}$$
$31$ $$64 - 8 T + T^{2}$$
$37$ $$T^{2}$$
$41$ $$( 4 + T )^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$T^{2}$$
$53$ $$36 + 6 T + T^{2}$$
$59$ $$1 - T + T^{2}$$
$61$ $$1 - T + T^{2}$$
$67$ $$25 - 5 T + T^{2}$$
$71$ $$( -8 + T )^{2}$$
$73$ $$196 + 14 T + T^{2}$$
$79$ $$169 - 13 T + T^{2}$$
$83$ $$( 6 + T )^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( 7 + T )^{2}$$