# Properties

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

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$1$$ Character orbit: $$[\chi]$$ $$=$$ 1183.n (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$0.590393909945$$ Analytic rank: $$0$$ Dimension: $$6$$ Relative dimension: $$3$$ over $$\Q(\zeta_{6})$$ Coefficient field: 6.0.64827.1 Defining polynomial: $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ 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.

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

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - x^{5} + 3 x^{4} + 5 x^{2} - 2 x + 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$($$$$-\nu^{5} + 3 \nu^{4} - 9 \nu^{3} + 5 \nu^{2} - 2 \nu + 6$$$$)/13$$ $$\beta_{3}$$ $$=$$ $$($$$$-3 \nu^{5} + 9 \nu^{4} - 14 \nu^{3} + 15 \nu^{2} - 6 \nu + 18$$$$)/13$$ $$\beta_{4}$$ $$=$$ $$($$$$-4 \nu^{5} - \nu^{4} - 10 \nu^{3} - 6 \nu^{2} - 34 \nu - 2$$$$)/13$$ $$\beta_{5}$$ $$=$$ $$($$$$-6 \nu^{5} + 5 \nu^{4} - 15 \nu^{3} - 9 \nu^{2} - 25 \nu + 10$$$$)/13$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$-\beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_{1}$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} - 3 \beta_{2}$$ $$\nu^{4}$$ $$=$$ $$2 \beta_{5} - 3 \beta_{4} - 4 \beta_{1} - 2$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} - 4 \beta_{4} - 4 \beta_{3} + 9 \beta_{2} - 9 \beta_{1}$$

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-1$$ $$-\beta_{5}$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
146.1
 0.900969 + 1.56052i −0.623490 − 1.07992i 0.222521 + 0.385418i 0.900969 − 1.56052i −0.623490 + 1.07992i 0.222521 − 0.385418i
−0.623490 1.07992i 0 −0.277479 + 0.480608i 0 0 −0.500000 + 0.866025i −0.554958 −0.500000 + 0.866025i 0
146.2 0.222521 + 0.385418i 0 0.400969 0.694498i 0 0 −0.500000 + 0.866025i 0.801938 −0.500000 + 0.866025i 0
146.3 0.900969 + 1.56052i 0 −1.12349 + 1.94594i 0 0 −0.500000 + 0.866025i −2.24698 −0.500000 + 0.866025i 0
867.1 −0.623490 + 1.07992i 0 −0.277479 0.480608i 0 0 −0.500000 0.866025i −0.554958 −0.500000 0.866025i 0
867.2 0.222521 0.385418i 0 0.400969 + 0.694498i 0 0 −0.500000 0.866025i 0.801938 −0.500000 0.866025i 0
867.3 0.900969 1.56052i 0 −1.12349 1.94594i 0 0 −0.500000 0.866025i −2.24698 −0.500000 0.866025i 0
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 867.3 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.c even 3 1 inner
91.n odd 6 1 inner

## Twists

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

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

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{6} - T_{2}^{5} + 3 T_{2}^{4} + 5 T_{2}^{2} - 2 T_{2} + 1$$ acting on $$S_{1}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

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