# Properties

 Label 966.2.g.b 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: $$\Q(i, \sqrt{14})$$ Defining polynomial: $$x^{4} + 49$$ Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{4} + 49$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2}$$$$/7$$ $$\beta_{3}$$ $$=$$ $$\nu^{3}$$$$/7$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$7 \beta_{2}$$ $$\nu^{3}$$ $$=$$ $$7 \beta_{3}$$

## 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
 1.87083 − 1.87083i −1.87083 + 1.87083i 1.87083 + 1.87083i −1.87083 − 1.87083i
−1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 1.87083i −1.00000 −1.00000 3.74166
643.2 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 + 1.87083i −1.00000 −1.00000 −3.74166
643.3 −1.00000 1.00000i 1.00000 −3.74166 1.00000i 1.87083 + 1.87083i −1.00000 −1.00000 3.74166
643.4 −1.00000 1.00000i 1.00000 3.74166 1.00000i −1.87083 1.87083i −1.00000 −1.00000 −3.74166
 $$n$$: 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.b 4
3.b odd 2 1 2898.2.g.h 4
7.b odd 2 1 inner 966.2.g.b 4
21.c even 2 1 2898.2.g.h 4
23.b odd 2 1 inner 966.2.g.b 4
69.c even 2 1 2898.2.g.h 4
161.c even 2 1 inner 966.2.g.b 4
483.c odd 2 1 2898.2.g.h 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
966.2.g.b 4 1.a even 1 1 trivial
966.2.g.b 4 7.b odd 2 1 inner
966.2.g.b 4 23.b odd 2 1 inner
966.2.g.b 4 161.c even 2 1 inner
2898.2.g.h 4 3.b odd 2 1
2898.2.g.h 4 21.c even 2 1
2898.2.g.h 4 69.c even 2 1
2898.2.g.h 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 $$S_{2}^{\mathrm{new}}(966, [\chi])$$:

 $$T_{5}^{2} - 14$$ $$T_{11}$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$( -14 + T^{2} )^{2}$$
$7$ $$49 + T^{4}$$
$11$ $$T^{4}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -14 + T^{2} )^{2}$$
$19$ $$( -14 + T^{2} )^{2}$$
$23$ $$( 23 + 6 T + T^{2} )^{2}$$
$29$ $$( -6 + T )^{4}$$
$31$ $$( 16 + T^{2} )^{2}$$
$37$ $$T^{4}$$
$41$ $$( 4 + T^{2} )^{2}$$
$43$ $$( 126 + T^{2} )^{2}$$
$47$ $$( 36 + T^{2} )^{2}$$
$53$ $$( 14 + T^{2} )^{2}$$
$59$ $$( 196 + T^{2} )^{2}$$
$61$ $$T^{4}$$
$67$ $$( 14 + T^{2} )^{2}$$
$71$ $$( -6 + T )^{4}$$
$73$ $$( 16 + T^{2} )^{2}$$
$79$ $$( 14 + T^{2} )^{2}$$
$83$ $$( -56 + T^{2} )^{2}$$
$89$ $$( -14 + T^{2} )^{2}$$
$97$ $$( -56 + T^{2} )^{2}$$