# Properties

 Label 966.2.i.a Level $966$ Weight $2$ Character orbit 966.i Analytic conductor $7.714$ Analytic rank $1$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.i (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ 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.

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

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/966\mathbb{Z}\right)^\times$$.

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
277.1
 0.5 + 0.866025i 0.5 − 0.866025i
−0.500000 0.866025i −0.500000 + 0.866025i −0.500000 + 0.866025i −2.00000 3.46410i 1.00000 −2.50000 + 0.866025i 1.00000 −0.500000 0.866025i −2.00000 + 3.46410i
415.1 −0.500000 + 0.866025i −0.500000 0.866025i −0.500000 0.866025i −2.00000 + 3.46410i 1.00000 −2.50000 0.866025i 1.00000 −0.500000 + 0.866025i −2.00000 3.46410i
 $$n$$: 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 966.2.i.a 2
7.c even 3 1 inner 966.2.i.a 2
7.c even 3 1 6762.2.a.bn 1
7.d odd 6 1 6762.2.a.w 1

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.i.a 2 1.a even 1 1 trivial
966.2.i.a 2 7.c even 3 1 inner
6762.2.a.w 1 7.d odd 6 1
6762.2.a.bn 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 $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}^{2} + 4 T_{5} + 16$$ $$T_{11}^{2} - 5 T_{11} + 25$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + T + T^{2}$$
$3$ $$1 + T + T^{2}$$
$5$ $$16 + 4 T + T^{2}$$
$7$ $$7 + 5 T + T^{2}$$
$11$ $$25 - 5 T + T^{2}$$
$13$ $$( -2 + T )^{2}$$
$17$ $$9 + 3 T + T^{2}$$
$19$ $$36 + 6 T + T^{2}$$
$23$ $$1 + T + T^{2}$$
$29$ $$( -1 + T )^{2}$$
$31$ $$100 + 10 T + T^{2}$$
$37$ $$36 - 6 T + T^{2}$$
$41$ $$( 6 + T )^{2}$$
$43$ $$( 6 + T )^{2}$$
$47$ $$81 - 9 T + T^{2}$$
$53$ $$36 - 6 T + T^{2}$$
$59$ $$64 + 8 T + T^{2}$$
$61$ $$4 + 2 T + T^{2}$$
$67$ $$4 - 2 T + T^{2}$$
$71$ $$( 7 + T )^{2}$$
$73$ $$121 + 11 T + T^{2}$$
$79$ $$1 - T + T^{2}$$
$83$ $$( 16 + T )^{2}$$
$89$ $$324 - 18 T + T^{2}$$
$97$ $$( -4 + T )^{2}$$