# Properties

 Label 1183.1.b.a Level $1183$ Weight $1$ Character orbit 1183.b Analytic conductor $0.590$ Analytic rank $0$ Dimension $6$ Projective image $D_{7}$ CM discriminant -7 Inner twists $4$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1183.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.590393909945$$ Analytic rank: $$0$$ Dimension: $$6$$ Coefficient field: 6.0.153664.1 Defining polynomial: $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$ 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 $$1,\beta_1,\ldots,\beta_{5}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{5} q^{7} + ( \beta_{1} - \beta_{3} ) q^{8} + q^{9} +O(q^{10})$$ $$q -\beta_{1} q^{2} + ( -1 + \beta_{2} ) q^{4} -\beta_{5} q^{7} + ( \beta_{1} - \beta_{3} ) q^{8} + q^{9} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{11} -\beta_{4} q^{14} + \beta_{4} q^{16} -\beta_{1} q^{18} + ( -1 + \beta_{2} ) q^{22} + ( 1 - \beta_{2} - \beta_{4} ) q^{23} - q^{25} + ( \beta_{1} - \beta_{3} ) q^{28} -\beta_{4} q^{29} -\beta_{5} q^{32} + ( -1 + \beta_{2} ) q^{36} -\beta_{3} q^{37} + \beta_{4} q^{43} + ( \beta_{1} + \beta_{5} ) q^{44} + ( -\beta_{1} + \beta_{5} ) q^{46} - q^{49} + \beta_{1} q^{50} + ( -1 + \beta_{2} + \beta_{4} ) q^{53} + ( 1 - \beta_{2} ) q^{56} + ( \beta_{1} - \beta_{3} + \beta_{5} ) q^{58} -\beta_{5} q^{63} + \beta_{3} q^{67} + \beta_{3} q^{71} + ( \beta_{1} - \beta_{3} ) q^{72} + ( -1 + \beta_{4} ) q^{74} + \beta_{2} q^{77} -\beta_{2} q^{79} + q^{81} + ( -\beta_{1} + \beta_{3} - \beta_{5} ) q^{86} + ( 1 + \beta_{4} ) q^{88} - q^{92} + \beta_{1} q^{98} + ( -\beta_{1} + \beta_{3} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 4q^{4} + 6q^{9} + O(q^{10})$$ $$6q - 4q^{4} + 6q^{9} - 2q^{14} + 2q^{16} - 4q^{22} + 2q^{23} - 6q^{25} - 2q^{29} - 4q^{36} + 2q^{43} - 6q^{49} - 2q^{53} + 4q^{56} - 4q^{74} + 2q^{77} - 2q^{79} + 6q^{81} + 8q^{88} - 6q^{92} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} + 5 x^{4} + 6 x^{2} + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} + 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} + 3 \nu$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} + 3 \nu^{2} + 1$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} + 4 \nu^{3} + 3 \nu$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} - 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{1}$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} - 3 \beta_{2} + 5$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{3} + 9 \beta_{1}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1182.1
 1.80194i 1.24698i 0.445042i − 0.445042i − 1.24698i − 1.80194i
1.80194i 0 −2.24698 0 0 1.00000i 2.24698i 1.00000 0
1182.2 1.24698i 0 −0.554958 0 0 1.00000i 0.554958i 1.00000 0
1182.3 0.445042i 0 0.801938 0 0 1.00000i 0.801938i 1.00000 0
1182.4 0.445042i 0 0.801938 0 0 1.00000i 0.801938i 1.00000 0
1182.5 1.24698i 0 −0.554958 0 0 1.00000i 0.554958i 1.00000 0
1182.6 1.80194i 0 −2.24698 0 0 1.00000i 2.24698i 1.00000 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1182.6 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 $$\Q(\sqrt{-7})$$
13.b even 2 1 inner
91.b odd 2 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace is the entire newspace $$S_{1}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$3$ $$T^{6}$$
$5$ $$T^{6}$$
$7$ $$( 1 + T^{2} )^{3}$$
$11$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$T^{6}$$
$19$ $$T^{6}$$
$23$ $$( 1 - 2 T - T^{2} + T^{3} )^{2}$$
$29$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$31$ $$T^{6}$$
$37$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$41$ $$T^{6}$$
$43$ $$( 1 - 2 T - T^{2} + T^{3} )^{2}$$
$47$ $$T^{6}$$
$53$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$59$ $$T^{6}$$
$61$ $$T^{6}$$
$67$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$71$ $$1 + 6 T^{2} + 5 T^{4} + T^{6}$$
$73$ $$T^{6}$$
$79$ $$( -1 - 2 T + T^{2} + T^{3} )^{2}$$
$83$ $$T^{6}$$
$89$ $$T^{6}$$
$97$ $$T^{6}$$
show more
show less