# Properties

 Label 966.2.g.a 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.

 $$f(q)$$ $$=$$ $$q - q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{2} q^{6} + \beta_{1} q^{7} - q^{8} - q^{9} +O(q^{10})$$ $$q - q^{2} -\beta_{2} q^{3} + q^{4} + \beta_{2} q^{6} + \beta_{1} q^{7} - q^{8} - q^{9} + ( \beta_{1} + \beta_{3} ) q^{11} -\beta_{2} q^{12} + 2 \beta_{2} q^{13} -\beta_{1} q^{14} + q^{16} + ( -\beta_{1} + \beta_{3} ) q^{17} + q^{18} -\beta_{3} q^{21} + ( -\beta_{1} - \beta_{3} ) q^{22} + ( -3 + \beta_{1} + \beta_{3} ) q^{23} + \beta_{2} q^{24} -5 q^{25} -2 \beta_{2} q^{26} + \beta_{2} q^{27} + \beta_{1} q^{28} -8 q^{29} + 4 \beta_{2} q^{31} - q^{32} + ( \beta_{1} - \beta_{3} ) q^{33} + ( \beta_{1} - \beta_{3} ) q^{34} - q^{36} + ( -\beta_{1} - \beta_{3} ) q^{37} + 2 q^{39} + 2 \beta_{2} q^{41} + \beta_{3} q^{42} + ( \beta_{1} + \beta_{3} ) q^{44} + ( 3 - \beta_{1} - \beta_{3} ) q^{46} -8 \beta_{2} q^{47} -\beta_{2} q^{48} + 7 \beta_{2} q^{49} + 5 q^{50} + ( \beta_{1} + \beta_{3} ) q^{51} + 2 \beta_{2} q^{52} + ( 2 \beta_{1} + 2 \beta_{3} ) q^{53} -\beta_{2} q^{54} -\beta_{1} q^{56} + 8 q^{58} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{61} -4 \beta_{2} q^{62} -\beta_{1} q^{63} + q^{64} + ( -\beta_{1} + \beta_{3} ) q^{66} + ( -\beta_{1} + \beta_{3} ) q^{68} + ( \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{69} -8 q^{71} + q^{72} + 10 \beta_{2} q^{73} + ( \beta_{1} + \beta_{3} ) q^{74} + 5 \beta_{2} q^{75} + ( -7 + 7 \beta_{2} ) q^{77} -2 q^{78} + ( \beta_{1} + \beta_{3} ) q^{79} + q^{81} -2 \beta_{2} q^{82} + ( \beta_{1} - \beta_{3} ) q^{83} -\beta_{3} q^{84} + 8 \beta_{2} q^{87} + ( -\beta_{1} - \beta_{3} ) q^{88} + ( 3 \beta_{1} - 3 \beta_{3} ) q^{89} + 2 \beta_{3} q^{91} + ( -3 + \beta_{1} + \beta_{3} ) q^{92} + 4 q^{93} + 8 \beta_{2} q^{94} + \beta_{2} q^{96} -7 \beta_{2} q^{98} + ( -\beta_{1} - \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + O(q^{10})$$ $$4q - 4q^{2} + 4q^{4} - 4q^{8} - 4q^{9} + 4q^{16} + 4q^{18} - 12q^{23} - 20q^{25} - 32q^{29} - 4q^{32} - 4q^{36} + 8q^{39} + 12q^{46} + 20q^{50} + 32q^{58} + 4q^{64} - 32q^{71} + 4q^{72} - 28q^{77} - 8q^{78} + 4q^{81} - 12q^{92} + 16q^{93} + O(q^{100})$$

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 0 1.00000i −1.87083 1.87083i −1.00000 −1.00000 0
643.2 −1.00000 1.00000i 1.00000 0 1.00000i 1.87083 + 1.87083i −1.00000 −1.00000 0
643.3 −1.00000 1.00000i 1.00000 0 1.00000i −1.87083 + 1.87083i −1.00000 −1.00000 0
643.4 −1.00000 1.00000i 1.00000 0 1.00000i 1.87083 1.87083i −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.a 4
3.b odd 2 1 2898.2.g.g 4
7.b odd 2 1 inner 966.2.g.a 4
21.c even 2 1 2898.2.g.g 4
23.b odd 2 1 inner 966.2.g.a 4
69.c even 2 1 2898.2.g.g 4
161.c even 2 1 inner 966.2.g.a 4
483.c odd 2 1 2898.2.g.g 4

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

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 1 + T )^{4}$$
$3$ $$( 1 + T^{2} )^{2}$$
$5$ $$T^{4}$$
$7$ $$49 + T^{4}$$
$11$ $$( 14 + T^{2} )^{2}$$
$13$ $$( 4 + T^{2} )^{2}$$
$17$ $$( -14 + T^{2} )^{2}$$
$19$ $$T^{4}$$
$23$ $$( 23 + 6 T + T^{2} )^{2}$$
$29$ $$( 8 + T )^{4}$$
$31$ $$( 16 + T^{2} )^{2}$$
$37$ $$( 14 + T^{2} )^{2}$$
$41$ $$( 4 + T^{2} )^{2}$$
$43$ $$T^{4}$$
$47$ $$( 64 + T^{2} )^{2}$$
$53$ $$( 56 + T^{2} )^{2}$$
$59$ $$T^{4}$$
$61$ $$( -126 + T^{2} )^{2}$$
$67$ $$T^{4}$$
$71$ $$( 8 + T )^{4}$$
$73$ $$( 100 + T^{2} )^{2}$$
$79$ $$( 14 + T^{2} )^{2}$$
$83$ $$( -14 + T^{2} )^{2}$$
$89$ $$( -126 + T^{2} )^{2}$$
$97$ $$T^{4}$$