# Properties

 Label 1183.2.c.d Level $1183$ Weight $2$ Character orbit 1183.c Analytic conductor $9.446$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: no Analytic conductor: $$9.44630255912$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{8})$$ Defining polynomial: $$x^{4} + 1$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 91) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a primitive root of unity $$\zeta_{8}$$. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} -2 \zeta_{8}^{2} q^{6} -\zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} - q^{9} +O(q^{10})$$ $$q + ( \zeta_{8} + \zeta_{8}^{3} ) q^{2} + ( -\zeta_{8} + \zeta_{8}^{3} ) q^{3} + ( \zeta_{8} + 3 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{5} -2 \zeta_{8}^{2} q^{6} -\zeta_{8}^{2} q^{7} + ( 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{8} - q^{9} + ( -2 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{10} + ( 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{11} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{14} + ( -3 \zeta_{8} - 2 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{15} -4 q^{16} + ( \zeta_{8} - \zeta_{8}^{3} ) q^{17} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{18} + ( 3 \zeta_{8} - 3 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{19} + ( \zeta_{8} + \zeta_{8}^{3} ) q^{21} -6 q^{22} + ( 3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{23} -4 \zeta_{8}^{2} q^{24} + ( -6 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{25} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{27} + ( 3 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{29} + ( 6 + 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{30} + ( -3 \zeta_{8} - \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{31} -6 \zeta_{8}^{2} q^{33} + 2 \zeta_{8}^{2} q^{34} + ( 3 + \zeta_{8} - \zeta_{8}^{3} ) q^{35} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{37} + ( -6 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{38} + ( -4 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{40} + ( -2 \zeta_{8} + 6 \zeta_{8}^{2} - 2 \zeta_{8}^{3} ) q^{41} -2 q^{42} + 5 q^{43} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{45} + ( 3 \zeta_{8} - 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{46} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{47} + ( 4 \zeta_{8} - 4 \zeta_{8}^{3} ) q^{48} - q^{49} + ( -6 \zeta_{8} - 12 \zeta_{8}^{2} - 6 \zeta_{8}^{3} ) q^{50} -2 q^{51} + ( -3 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{53} + 8 \zeta_{8}^{2} q^{54} + ( -6 - 9 \zeta_{8} + 9 \zeta_{8}^{3} ) q^{55} + ( 2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{56} + ( 3 \zeta_{8} - 6 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{57} + ( 3 \zeta_{8} + 4 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{58} + ( -4 \zeta_{8} - 6 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{59} + 6 q^{61} + ( 6 + \zeta_{8} - \zeta_{8}^{3} ) q^{62} + \zeta_{8}^{2} q^{63} -8 q^{64} + ( 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{66} + ( 6 \zeta_{8} - 6 \zeta_{8}^{2} + 6 \zeta_{8}^{3} ) q^{67} + ( 4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{69} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{70} + ( 5 \zeta_{8} - 6 \zeta_{8}^{2} + 5 \zeta_{8}^{3} ) q^{71} + ( -2 \zeta_{8} - 2 \zeta_{8}^{3} ) q^{72} + ( -3 \zeta_{8} + 5 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{73} + ( -6 - 2 \zeta_{8} + 2 \zeta_{8}^{3} ) q^{74} + ( 12 + 6 \zeta_{8} - 6 \zeta_{8}^{3} ) q^{75} + ( 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{77} + ( 7 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{79} + ( -4 \zeta_{8} - 12 \zeta_{8}^{2} - 4 \zeta_{8}^{3} ) q^{80} -5 q^{81} + ( 4 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{82} + ( -3 \zeta_{8} + 9 \zeta_{8}^{2} - 3 \zeta_{8}^{3} ) q^{83} + ( 3 \zeta_{8} + 2 \zeta_{8}^{2} + 3 \zeta_{8}^{3} ) q^{85} + ( 5 \zeta_{8} + 5 \zeta_{8}^{3} ) q^{86} + ( -4 - 3 \zeta_{8} + 3 \zeta_{8}^{3} ) q^{87} -12 q^{88} + ( -\zeta_{8} - 3 \zeta_{8}^{2} - \zeta_{8}^{3} ) q^{89} + ( 2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{90} + ( \zeta_{8} + 6 \zeta_{8}^{2} + \zeta_{8}^{3} ) q^{93} + ( 2 + 3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{94} + ( 3 - 6 \zeta_{8} + 6 \zeta_{8}^{3} ) q^{95} + ( -9 \zeta_{8} - \zeta_{8}^{2} - 9 \zeta_{8}^{3} ) q^{97} + ( -\zeta_{8} - \zeta_{8}^{3} ) q^{98} + ( -3 \zeta_{8} - 3 \zeta_{8}^{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 4q^{9} + O(q^{10})$$ $$4q - 4q^{9} - 8q^{10} - 16q^{16} - 24q^{22} + 12q^{23} - 24q^{25} + 12q^{29} + 24q^{30} + 12q^{35} - 24q^{38} - 16q^{40} - 8q^{42} + 20q^{43} - 4q^{49} - 8q^{51} - 12q^{53} - 24q^{55} + 24q^{61} + 24q^{62} - 32q^{64} + 16q^{69} - 24q^{74} + 48q^{75} + 28q^{79} - 20q^{81} + 16q^{82} - 16q^{87} - 48q^{88} + 8q^{90} + 8q^{94} + 12q^{95} + O(q^{100})$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 0.707107 − 0.707107i −0.707107 − 0.707107i 0.707107 + 0.707107i −0.707107 + 0.707107i
1.41421i −1.41421 0 4.41421i 2.00000i 1.00000i 2.82843i −1.00000 −6.24264
337.2 1.41421i 1.41421 0 1.58579i 2.00000i 1.00000i 2.82843i −1.00000 2.24264
337.3 1.41421i −1.41421 0 4.41421i 2.00000i 1.00000i 2.82843i −1.00000 −6.24264
337.4 1.41421i 1.41421 0 1.58579i 2.00000i 1.00000i 2.82843i −1.00000 2.24264
 $$n$$: 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
13.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.c.d 4
13.b even 2 1 inner 1183.2.c.d 4
13.d odd 4 1 91.2.a.c 2
13.d odd 4 1 1183.2.a.d 2
39.f even 4 1 819.2.a.h 2
52.f even 4 1 1456.2.a.q 2
65.g odd 4 1 2275.2.a.j 2
91.i even 4 1 637.2.a.g 2
91.i even 4 1 8281.2.a.v 2
91.z odd 12 2 637.2.e.f 4
91.bb even 12 2 637.2.e.g 4
104.j odd 4 1 5824.2.a.bl 2
104.m even 4 1 5824.2.a.bk 2
273.o odd 4 1 5733.2.a.s 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.a.c 2 13.d odd 4 1
637.2.a.g 2 91.i even 4 1
637.2.e.f 4 91.z odd 12 2
637.2.e.g 4 91.bb even 12 2
819.2.a.h 2 39.f even 4 1
1183.2.a.d 2 13.d odd 4 1
1183.2.c.d 4 1.a even 1 1 trivial
1183.2.c.d 4 13.b even 2 1 inner
1456.2.a.q 2 52.f even 4 1
2275.2.a.j 2 65.g odd 4 1
5733.2.a.s 2 273.o odd 4 1
5824.2.a.bk 2 104.m even 4 1
5824.2.a.bl 2 104.j odd 4 1
8281.2.a.v 2 91.i even 4 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$( 2 + T^{2} )^{2}$$
$3$ $$( -2 + T^{2} )^{2}$$
$5$ $$49 + 22 T^{2} + T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$( 18 + T^{2} )^{2}$$
$13$ $$T^{4}$$
$17$ $$( -2 + T^{2} )^{2}$$
$19$ $$81 + 54 T^{2} + T^{4}$$
$23$ $$( 1 - 6 T + T^{2} )^{2}$$
$29$ $$( 1 - 6 T + T^{2} )^{2}$$
$31$ $$289 + 38 T^{2} + T^{4}$$
$37$ $$196 + 44 T^{2} + T^{4}$$
$41$ $$784 + 88 T^{2} + T^{4}$$
$43$ $$( -5 + T )^{4}$$
$47$ $$49 + 22 T^{2} + T^{4}$$
$53$ $$( 1 + 6 T + T^{2} )^{2}$$
$59$ $$16 + 136 T^{2} + T^{4}$$
$61$ $$( -6 + T )^{4}$$
$67$ $$1296 + 216 T^{2} + T^{4}$$
$71$ $$196 + 172 T^{2} + T^{4}$$
$73$ $$49 + 86 T^{2} + T^{4}$$
$79$ $$( -23 - 14 T + T^{2} )^{2}$$
$83$ $$3969 + 198 T^{2} + T^{4}$$
$89$ $$49 + 22 T^{2} + T^{4}$$
$97$ $$25921 + 326 T^{2} + T^{4}$$