# Properties

 Label 1183.2.e.c Level $1183$ Weight $2$ Character orbit 1183.e Analytic conductor $9.446$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.e (of order $$3$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{-3})$$ Defining polynomial: $$x^{2} - x + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{6}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + 3 q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} + 3 q^{8} -6 \zeta_{6} q^{9} +O(q^{10})$$ $$q + \zeta_{6} q^{2} + ( 3 - 3 \zeta_{6} ) q^{3} + ( 1 - \zeta_{6} ) q^{4} + 3 \zeta_{6} q^{5} + 3 q^{6} + ( 1 - 3 \zeta_{6} ) q^{7} + 3 q^{8} -6 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{10} + ( -3 + 3 \zeta_{6} ) q^{11} -3 \zeta_{6} q^{12} + ( 3 - 2 \zeta_{6} ) q^{14} + 9 q^{15} + \zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 6 - 6 \zeta_{6} ) q^{18} -\zeta_{6} q^{19} + 3 q^{20} + ( -6 - 3 \zeta_{6} ) q^{21} -3 q^{22} + ( 9 - 9 \zeta_{6} ) q^{24} + ( -4 + 4 \zeta_{6} ) q^{25} -9 q^{27} + ( -2 - \zeta_{6} ) q^{28} + 7 q^{29} + 9 \zeta_{6} q^{30} + ( 3 - 3 \zeta_{6} ) q^{31} + ( 5 - 5 \zeta_{6} ) q^{32} + 9 \zeta_{6} q^{33} + 2 q^{34} + ( 9 - 6 \zeta_{6} ) q^{35} -6 q^{36} + 2 \zeta_{6} q^{37} + ( 1 - \zeta_{6} ) q^{38} + 9 \zeta_{6} q^{40} -3 q^{41} + ( 3 - 9 \zeta_{6} ) q^{42} -7 q^{43} + 3 \zeta_{6} q^{44} + ( 18 - 18 \zeta_{6} ) q^{45} + \zeta_{6} q^{47} + 3 q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} -4 q^{50} -6 \zeta_{6} q^{51} + ( -3 + 3 \zeta_{6} ) q^{53} -9 \zeta_{6} q^{54} -9 q^{55} + ( 3 - 9 \zeta_{6} ) q^{56} -3 q^{57} + 7 \zeta_{6} q^{58} + ( -4 + 4 \zeta_{6} ) q^{59} + ( 9 - 9 \zeta_{6} ) q^{60} + 13 \zeta_{6} q^{61} + 3 q^{62} + ( -18 + 12 \zeta_{6} ) q^{63} + 7 q^{64} + ( -9 + 9 \zeta_{6} ) q^{66} + ( -3 + 3 \zeta_{6} ) q^{67} -2 \zeta_{6} q^{68} + ( 6 + 3 \zeta_{6} ) q^{70} -13 q^{71} -18 \zeta_{6} q^{72} + ( -13 + 13 \zeta_{6} ) q^{73} + ( -2 + 2 \zeta_{6} ) q^{74} + 12 \zeta_{6} q^{75} - q^{76} + ( 6 + 3 \zeta_{6} ) q^{77} + 3 \zeta_{6} q^{79} + ( -3 + 3 \zeta_{6} ) q^{80} + ( -9 + 9 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} + ( -9 + 6 \zeta_{6} ) q^{84} + 6 q^{85} -7 \zeta_{6} q^{86} + ( 21 - 21 \zeta_{6} ) q^{87} + ( -9 + 9 \zeta_{6} ) q^{88} + 6 \zeta_{6} q^{89} + 18 q^{90} -9 \zeta_{6} q^{93} + ( -1 + \zeta_{6} ) q^{94} + ( 3 - 3 \zeta_{6} ) q^{95} -15 \zeta_{6} q^{96} + 5 q^{97} + ( -3 - 5 \zeta_{6} ) q^{98} + 18 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q + q^{2} + 3q^{3} + q^{4} + 3q^{5} + 6q^{6} - q^{7} + 6q^{8} - 6q^{9} + O(q^{10})$$ $$2q + q^{2} + 3q^{3} + q^{4} + 3q^{5} + 6q^{6} - q^{7} + 6q^{8} - 6q^{9} - 3q^{10} - 3q^{11} - 3q^{12} + 4q^{14} + 18q^{15} + q^{16} + 2q^{17} + 6q^{18} - q^{19} + 6q^{20} - 15q^{21} - 6q^{22} + 9q^{24} - 4q^{25} - 18q^{27} - 5q^{28} + 14q^{29} + 9q^{30} + 3q^{31} + 5q^{32} + 9q^{33} + 4q^{34} + 12q^{35} - 12q^{36} + 2q^{37} + q^{38} + 9q^{40} - 6q^{41} - 3q^{42} - 14q^{43} + 3q^{44} + 18q^{45} + q^{47} + 6q^{48} - 13q^{49} - 8q^{50} - 6q^{51} - 3q^{53} - 9q^{54} - 18q^{55} - 3q^{56} - 6q^{57} + 7q^{58} - 4q^{59} + 9q^{60} + 13q^{61} + 6q^{62} - 24q^{63} + 14q^{64} - 9q^{66} - 3q^{67} - 2q^{68} + 15q^{70} - 26q^{71} - 18q^{72} - 13q^{73} - 2q^{74} + 12q^{75} - 2q^{76} + 15q^{77} + 3q^{79} - 3q^{80} - 9q^{81} - 3q^{82} - 12q^{84} + 12q^{85} - 7q^{86} + 21q^{87} - 9q^{88} + 6q^{89} + 36q^{90} - 9q^{93} - q^{94} + 3q^{95} - 15q^{96} + 10q^{97} - 11q^{98} + 36q^{99} + O(q^{100})$$

## Character values

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

 $$n$$ $$339$$ $$1016$$ $$\chi(n)$$ $$-\zeta_{6}$$ $$1$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
170.1
 0.5 − 0.866025i 0.5 + 0.866025i
0.500000 0.866025i 1.50000 + 2.59808i 0.500000 + 0.866025i 1.50000 2.59808i 3.00000 −0.500000 + 2.59808i 3.00000 −3.00000 + 5.19615i −1.50000 2.59808i
508.1 0.500000 + 0.866025i 1.50000 2.59808i 0.500000 0.866025i 1.50000 + 2.59808i 3.00000 −0.500000 2.59808i 3.00000 −3.00000 5.19615i −1.50000 + 2.59808i
 $$n$$: 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.c even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.e.c 2
7.c even 3 1 inner 1183.2.e.c 2
7.c even 3 1 8281.2.a.c 1
7.d odd 6 1 8281.2.a.g 1
13.b even 2 1 1183.2.e.a 2
13.e even 6 1 91.2.g.a 2
13.e even 6 1 91.2.h.a yes 2
39.h odd 6 1 819.2.n.c 2
39.h odd 6 1 819.2.s.a 2
91.k even 6 1 91.2.g.a 2
91.k even 6 1 637.2.f.b 2
91.l odd 6 1 637.2.f.a 2
91.l odd 6 1 637.2.g.a 2
91.p odd 6 1 637.2.f.a 2
91.p odd 6 1 637.2.h.a 2
91.r even 6 1 1183.2.e.a 2
91.r even 6 1 8281.2.a.i 1
91.s odd 6 1 8281.2.a.j 1
91.t odd 6 1 637.2.g.a 2
91.t odd 6 1 637.2.h.a 2
91.u even 6 1 91.2.h.a yes 2
91.u even 6 1 637.2.f.b 2
273.x odd 6 1 819.2.s.a 2
273.bp odd 6 1 819.2.n.c 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.g.a 2 13.e even 6 1
91.2.g.a 2 91.k even 6 1
91.2.h.a yes 2 13.e even 6 1
91.2.h.a yes 2 91.u even 6 1
637.2.f.a 2 91.l odd 6 1
637.2.f.a 2 91.p odd 6 1
637.2.f.b 2 91.k even 6 1
637.2.f.b 2 91.u even 6 1
637.2.g.a 2 91.l odd 6 1
637.2.g.a 2 91.t odd 6 1
637.2.h.a 2 91.p odd 6 1
637.2.h.a 2 91.t odd 6 1
819.2.n.c 2 39.h odd 6 1
819.2.n.c 2 273.bp odd 6 1
819.2.s.a 2 39.h odd 6 1
819.2.s.a 2 273.x odd 6 1
1183.2.e.a 2 13.b even 2 1
1183.2.e.a 2 91.r even 6 1
1183.2.e.c 2 1.a even 1 1 trivial
1183.2.e.c 2 7.c even 3 1 inner
8281.2.a.c 1 7.c even 3 1
8281.2.a.g 1 7.d odd 6 1
8281.2.a.i 1 91.r even 6 1
8281.2.a.j 1 91.s odd 6 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$:

 $$T_{2}^{2} - T_{2} + 1$$ $$T_{3}^{2} - 3 T_{3} + 9$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + T^{2}$$
$3$ $$9 - 3 T + T^{2}$$
$5$ $$9 - 3 T + T^{2}$$
$7$ $$7 + T + T^{2}$$
$11$ $$9 + 3 T + T^{2}$$
$13$ $$T^{2}$$
$17$ $$4 - 2 T + T^{2}$$
$19$ $$1 + T + T^{2}$$
$23$ $$T^{2}$$
$29$ $$( -7 + T )^{2}$$
$31$ $$9 - 3 T + T^{2}$$
$37$ $$4 - 2 T + T^{2}$$
$41$ $$( 3 + T )^{2}$$
$43$ $$( 7 + T )^{2}$$
$47$ $$1 - T + T^{2}$$
$53$ $$9 + 3 T + T^{2}$$
$59$ $$16 + 4 T + T^{2}$$
$61$ $$169 - 13 T + T^{2}$$
$67$ $$9 + 3 T + T^{2}$$
$71$ $$( 13 + T )^{2}$$
$73$ $$169 + 13 T + T^{2}$$
$79$ $$9 - 3 T + T^{2}$$
$83$ $$T^{2}$$
$89$ $$36 - 6 T + T^{2}$$
$97$ $$( -5 + T )^{2}$$