# Properties

 Label 983.1.b.b Level $983$ Weight $1$ Character orbit 983.b Self dual yes Analytic conductor $0.491$ Analytic rank $0$ Dimension $3$ Projective image $D_{9}$ CM discriminant -983 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$983$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 983.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.490580907418$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{18})^+$$ Defining polynomial: $$x^{3} - 3x - 1$$ x^3 - 3*x - 1 Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{9}$$ Projective field: Galois closure of 9.1.933714431521.1 Artin image: $D_9$ Artin field: Galois closure of 9.1.933714431521.1

## $q$-expansion

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

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 - 1) q^{6} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{8} + ( - \beta_{2} + \beta_1 + 1) q^{9}+O(q^{10})$$ q - b1 * q^2 + b2 * q^3 + (b2 + 1) * q^4 + (-b1 - 1) * q^6 + b2 * q^7 + (-b1 - 1) * q^8 + (-b2 + b1 + 1) * q^9 $$q - \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{2} + 1) q^{4} + ( - \beta_1 - 1) q^{6} + \beta_{2} q^{7} + ( - \beta_1 - 1) q^{8} + ( - \beta_{2} + \beta_1 + 1) q^{9} + (\beta_1 + 2) q^{12} + ( - \beta_1 - 1) q^{14} + (\beta_1 + 1) q^{16} + ( - \beta_{2} - 1) q^{18} - \beta_1 q^{19} + ( - \beta_{2} + \beta_1 + 2) q^{21} + ( - \beta_{2} + \beta_1) q^{23} + ( - \beta_{2} - \beta_1 - 1) q^{24} + q^{25} + (\beta_{2} - 1) q^{27} + (\beta_1 + 2) q^{28} - q^{31} + ( - \beta_{2} - 1) q^{32} + (\beta_{2} + \beta_1) q^{36} - q^{37} + (\beta_{2} + 2) q^{38} - q^{41} + ( - \beta_{2} - \beta_1 - 1) q^{42} - \beta_1 q^{43} + ( - \beta_{2} + \beta_1 - 1) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + (\beta_{2} + \beta_1 + 1) q^{48} + ( - \beta_{2} + \beta_1 + 1) q^{49} - \beta_1 q^{50} - \beta_1 q^{53} - q^{54} + ( - \beta_{2} - \beta_1 - 1) q^{56} + ( - \beta_1 - 1) q^{57} + 2 q^{59} + \beta_1 q^{62} + (2 \beta_{2} - 1) q^{63} + \beta_1 q^{64} - q^{67} + (\beta_{2} - 1) q^{69} + ( - \beta_1 - 2) q^{72} + \beta_1 q^{74} + \beta_{2} q^{75} + ( - 2 \beta_1 - 1) q^{76} + 2 q^{79} + ( - \beta_{2} + 1) q^{81} + \beta_1 q^{82} + (2 \beta_{2} + \beta_1 + 1) q^{84} + (\beta_{2} + 2) q^{86} + \beta_{2} q^{89} + (\beta_1 - 1) q^{92} - \beta_{2} q^{93} + ( - \beta_{2} + \beta_1 - 1) q^{94} + ( - \beta_1 - 2) q^{96} + ( - \beta_{2} - 1) q^{98}+O(q^{100})$$ q - b1 * q^2 + b2 * q^3 + (b2 + 1) * q^4 + (-b1 - 1) * q^6 + b2 * q^7 + (-b1 - 1) * q^8 + (-b2 + b1 + 1) * q^9 + (b1 + 2) * q^12 + (-b1 - 1) * q^14 + (b1 + 1) * q^16 + (-b2 - 1) * q^18 - b1 * q^19 + (-b2 + b1 + 2) * q^21 + (-b2 + b1) * q^23 + (-b2 - b1 - 1) * q^24 + q^25 + (b2 - 1) * q^27 + (b1 + 2) * q^28 - q^31 + (-b2 - 1) * q^32 + (b2 + b1) * q^36 - q^37 + (b2 + 2) * q^38 - q^41 + (-b2 - b1 - 1) * q^42 - b1 * q^43 + (-b2 + b1 - 1) * q^46 + (-b2 + b1) * q^47 + (b2 + b1 + 1) * q^48 + (-b2 + b1 + 1) * q^49 - b1 * q^50 - b1 * q^53 - q^54 + (-b2 - b1 - 1) * q^56 + (-b1 - 1) * q^57 + 2 * q^59 + b1 * q^62 + (2*b2 - 1) * q^63 + b1 * q^64 - q^67 + (b2 - 1) * q^69 + (-b1 - 2) * q^72 + b1 * q^74 + b2 * q^75 + (-2*b1 - 1) * q^76 + 2 * q^79 + (-b2 + 1) * q^81 + b1 * q^82 + (2*b2 + b1 + 1) * q^84 + (b2 + 2) * q^86 + b2 * q^89 + (b1 - 1) * q^92 - b2 * q^93 + (-b2 + b1 - 1) * q^94 + (-b1 - 2) * q^96 + (-b2 - 1) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9}+O(q^{10})$$ 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9 $$3 q + 3 q^{4} - 3 q^{6} - 3 q^{8} + 3 q^{9} + 6 q^{12} - 3 q^{14} + 3 q^{16} - 3 q^{18} + 6 q^{21} - 3 q^{24} + 3 q^{25} - 3 q^{27} + 6 q^{28} - 3 q^{31} - 3 q^{32} - 3 q^{37} + 6 q^{38} - 3 q^{41} - 3 q^{42} - 3 q^{46} + 3 q^{48} + 3 q^{49} - 3 q^{54} - 3 q^{56} - 3 q^{57} + 6 q^{59} - 3 q^{63} - 3 q^{67} - 3 q^{69} - 6 q^{72} - 3 q^{76} + 6 q^{79} + 3 q^{81} + 3 q^{84} + 6 q^{86} - 3 q^{92} - 3 q^{94} - 6 q^{96} - 3 q^{98}+O(q^{100})$$ 3 * q + 3 * q^4 - 3 * q^6 - 3 * q^8 + 3 * q^9 + 6 * q^12 - 3 * q^14 + 3 * q^16 - 3 * q^18 + 6 * q^21 - 3 * q^24 + 3 * q^25 - 3 * q^27 + 6 * q^28 - 3 * q^31 - 3 * q^32 - 3 * q^37 + 6 * q^38 - 3 * q^41 - 3 * q^42 - 3 * q^46 + 3 * q^48 + 3 * q^49 - 3 * q^54 - 3 * q^56 - 3 * q^57 + 6 * q^59 - 3 * q^63 - 3 * q^67 - 3 * q^69 - 6 * q^72 - 3 * q^76 + 6 * q^79 + 3 * q^81 + 3 * q^84 + 6 * q^86 - 3 * q^92 - 3 * q^94 - 6 * q^96 - 3 * q^98

Basis of coefficient ring in terms of $$\nu = \zeta_{18} + \zeta_{18}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 2$$ v^2 - 2
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 2$$ b2 + 2

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/983\mathbb{Z}\right)^\times$$.

 $$n$$ $$5$$ $$\chi(n)$$ $$-1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
982.1
 1.87939 −0.347296 −1.53209
−1.87939 1.53209 2.53209 0 −2.87939 1.53209 −2.87939 1.34730 0
982.2 0.347296 −1.87939 −0.879385 0 −0.652704 −1.87939 −0.652704 2.53209 0
982.3 1.53209 0.347296 1.34730 0 0.532089 0.347296 0.532089 −0.879385 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
983.b odd 2 1 CM by $$\Q(\sqrt{-983})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 983.1.b.b 3
983.b odd 2 1 CM 983.1.b.b 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.1.b.b 3 1.a even 1 1 trivial
983.1.b.b 3 983.b odd 2 1 CM

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 3T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(983, [\chi])$$.

## Hecke characteristic polynomials

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