# Properties

 Label 1183.1.d.b Level $1183$ Weight $1$ Character orbit 1183.d Self dual yes Analytic conductor $0.590$ Analytic rank $0$ Dimension $3$ Projective image $D_{7}$ CM discriminant -7 Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1183.d (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: yes Analytic conductor: $$0.590393909945$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: $$\Q(\zeta_{14})^+$$ Defining polynomial: $$x^{3} - x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Projective image: $$D_{7}$$ Projective field: Galois closure of 7.1.1655595487.1

## $q$-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.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} + ( 1 + \beta_{2} ) q^{4} - q^{7} + ( 1 + \beta_{2} ) q^{8} + q^{9} -\beta_{2} q^{11} -\beta_{1} q^{14} + \beta_{1} q^{16} + \beta_{1} q^{18} + ( -1 - \beta_{2} ) q^{22} + ( -1 + \beta_{1} - \beta_{2} ) q^{23} + q^{25} + ( -1 - \beta_{2} ) q^{28} -\beta_{1} q^{29} + q^{32} + ( 1 + \beta_{2} ) q^{36} + ( 1 - \beta_{1} + \beta_{2} ) q^{37} -\beta_{1} q^{43} + ( -1 - \beta_{1} ) q^{44} + ( 1 - \beta_{1} ) q^{46} + q^{49} + \beta_{1} q^{50} + ( -1 + \beta_{1} - \beta_{2} ) q^{53} + ( -1 - \beta_{2} ) q^{56} + ( -2 - \beta_{2} ) q^{58} - q^{63} + ( 1 - \beta_{1} + \beta_{2} ) q^{67} + ( 1 - \beta_{1} + \beta_{2} ) q^{71} + ( 1 + \beta_{2} ) q^{72} + ( -1 + \beta_{1} ) q^{74} + \beta_{2} q^{77} + \beta_{2} q^{79} + q^{81} + ( -2 - \beta_{2} ) q^{86} + ( -1 - \beta_{1} ) q^{88} - q^{92} + \beta_{1} q^{98} -\beta_{2} q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q + q^{2} + 2q^{4} - 3q^{7} + 2q^{8} + 3q^{9} + O(q^{10})$$ $$3q + q^{2} + 2q^{4} - 3q^{7} + 2q^{8} + 3q^{9} + q^{11} - q^{14} + q^{16} + q^{18} - 2q^{22} - q^{23} + 3q^{25} - 2q^{28} - q^{29} + 3q^{32} + 2q^{36} + q^{37} - q^{43} - 4q^{44} + 2q^{46} + 3q^{49} + q^{50} - q^{53} - 2q^{56} - 5q^{58} - 3q^{63} + q^{67} + q^{71} + 2q^{72} - 2q^{74} - q^{77} - q^{79} + 3q^{81} - 5q^{86} - 4q^{88} - 3q^{92} + q^{98} + q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/1183\mathbb{Z}\right)^\times$$.

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-1$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
846.1
 −1.24698 0.445042 1.80194
−1.24698 0 0.554958 0 0 −1.00000 0.554958 1.00000 0
846.2 0.445042 0 −0.801938 0 0 −1.00000 −0.801938 1.00000 0
846.3 1.80194 0 2.24698 0 0 −1.00000 2.24698 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 CM by $$\Q(\sqrt{-7})$$

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.1.d.b yes 3
7.b odd 2 1 CM 1183.1.d.b yes 3
13.b even 2 1 1183.1.d.a 3
13.c even 3 2 1183.1.n.a 6
13.d odd 4 2 1183.1.b.a 6
13.e even 6 2 1183.1.n.b 6
13.f odd 12 4 1183.1.t.a 12
91.b odd 2 1 1183.1.d.a 3
91.i even 4 2 1183.1.b.a 6
91.n odd 6 2 1183.1.n.a 6
91.t odd 6 2 1183.1.n.b 6
91.bc even 12 4 1183.1.t.a 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.1.b.a 6 13.d odd 4 2
1183.1.b.a 6 91.i even 4 2
1183.1.d.a 3 13.b even 2 1
1183.1.d.a 3 91.b odd 2 1
1183.1.d.b yes 3 1.a even 1 1 trivial
1183.1.d.b yes 3 7.b odd 2 1 CM
1183.1.n.a 6 13.c even 3 2
1183.1.n.a 6 91.n odd 6 2
1183.1.n.b 6 13.e even 6 2
1183.1.n.b 6 91.t odd 6 2
1183.1.t.a 12 13.f odd 12 4
1183.1.t.a 12 91.bc even 12 4

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - T_{2}^{2} - 2 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

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