# Properties

 Label 966.2.g.c Level $966$ Weight $2$ Character orbit 966.g Analytic conductor $7.714$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$966 = 2 \cdot 3 \cdot 7 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 966.g (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$7.71354883526$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{7})$$ Defining polynomial: $$x^{4} - 3 x^{2} + 4$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$2^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} - 3 x^{2} + 4$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$($$$$\nu^{3} - \nu$$$$)/2$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{3} + 5 \nu$$$$)/2$$ $$\beta_{3}$$ $$=$$ $$2 \nu^{2} - 3$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$($$$$\beta_{2} + \beta_{1}$$$$)/2$$ $$\nu^{2}$$ $$=$$ $$($$$$\beta_{3} + 3$$$$)/2$$ $$\nu^{3}$$ $$=$$ $$($$$$\beta_{2} + 5 \beta_{1}$$$$)/2$$

## Character values

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

 $$n$$ $$323$$ $$829$$ $$925$$ $$\chi(n)$$ $$1$$ $$-1$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
643.1
 −1.32288 + 0.500000i 1.32288 + 0.500000i 1.32288 − 0.500000i −1.32288 − 0.500000i
−1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.2 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.3 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
643.4 −1.00000 1.00000i 1.00000 0 1.00000i 2.64575i −1.00000 −1.00000 0
 $$n$$: 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.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 966.2.g.c 4
3.b odd 2 1 2898.2.g.c 4
7.b odd 2 1 inner 966.2.g.c 4
21.c even 2 1 2898.2.g.c 4
23.b odd 2 1 inner 966.2.g.c 4
69.c even 2 1 2898.2.g.c 4
161.c even 2 1 inner 966.2.g.c 4
483.c odd 2 1 2898.2.g.c 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.c 4 1.a even 1 1 trivial
966.2.g.c 4 7.b odd 2 1 inner
966.2.g.c 4 23.b odd 2 1 inner
966.2.g.c 4 161.c even 2 1 inner
2898.2.g.c 4 3.b odd 2 1
2898.2.g.c 4 21.c even 2 1
2898.2.g.c 4 69.c even 2 1
2898.2.g.c 4 483.c odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}$$ $$T_{11}^{2} + 28$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$( 7 + T^{2} )^{2}$$
$11$ $$( 28 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -28 + T^{2} )^{2}$$
$19$ $$( -28 + T^{2} )^{2}$$
$23$ $$( 23 - 8 T + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 100 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 144 + T^{2} )^{2}$$
$43$ $$( 28 + T^{2} )^{2}$$
$47$ $$( 36 + T^{2} )^{2}$$
$53$ $$T^{4}$$
$59$ $$T^{4}$$
$61$ $$( -112 + T^{2} )^{2}$$
$67$ $$( 28 + T^{2} )^{2}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$( 16 + T^{2} )^{2}$$
$79$ $$( 28 + T^{2} )^{2}$$
$83$ $$( -252 + T^{2} )^{2}$$
$89$ $$( -28 + T^{2} )^{2}$$
$97$ $$( -28 + T^{2} )^{2}$$