# Properties

 Label 1183.2.c.c 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: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ 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 basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} + ( -1 - 3 \beta_{2} ) q^{9} +O(q^{10})$$ $$q + ( \beta_{1} - \beta_{3} ) q^{2} + ( -1 + \beta_{2} ) q^{3} + 3 \beta_{2} q^{4} + ( \beta_{1} - \beta_{3} ) q^{5} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{6} + \beta_{3} q^{7} + ( -4 \beta_{1} + \beta_{3} ) q^{8} + ( -1 - 3 \beta_{2} ) q^{9} + ( -2 + 3 \beta_{2} ) q^{10} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{11} + ( 3 - 6 \beta_{2} ) q^{12} + ( 1 - \beta_{2} ) q^{14} + ( -3 \beta_{1} + 2 \beta_{3} ) q^{15} + ( 5 - 3 \beta_{2} ) q^{16} + ( -5 - 4 \beta_{2} ) q^{17} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{18} + ( -3 \beta_{1} - 3 \beta_{3} ) q^{19} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{20} + ( \beta_{1} - \beta_{3} ) q^{21} -3 \beta_{2} q^{22} + ( -2 - 4 \beta_{2} ) q^{23} + ( 9 \beta_{1} - 5 \beta_{3} ) q^{24} + ( 3 + 3 \beta_{2} ) q^{25} + ( 1 + 2 \beta_{2} ) q^{27} + 3 \beta_{1} q^{28} + ( -1 - 5 \beta_{2} ) q^{29} + ( 5 - 8 \beta_{2} ) q^{30} + ( -6 \beta_{1} - 5 \beta_{3} ) q^{31} + ( 3 \beta_{1} - 6 \beta_{3} ) q^{32} + 3 \beta_{1} q^{33} + ( 3 \beta_{1} + \beta_{3} ) q^{34} + ( 1 - \beta_{2} ) q^{35} + ( -9 + 6 \beta_{2} ) q^{36} + 4 \beta_{3} q^{37} -3 \beta_{2} q^{38} + ( 5 - 9 \beta_{2} ) q^{40} + ( -2 \beta_{1} - 4 \beta_{3} ) q^{41} + ( -2 + 3 \beta_{2} ) q^{42} + ( 2 + 9 \beta_{2} ) q^{43} -9 \beta_{3} q^{44} + ( 5 \beta_{1} - 2 \beta_{3} ) q^{45} + ( 6 \beta_{1} - 2 \beta_{3} ) q^{46} + ( 2 \beta_{1} + \beta_{3} ) q^{47} + ( -8 + 11 \beta_{2} ) q^{48} - q^{49} -3 \beta_{1} q^{50} + ( 1 + 3 \beta_{2} ) q^{51} + ( 7 + 2 \beta_{2} ) q^{53} + ( -3 \beta_{1} + \beta_{3} ) q^{54} -3 \beta_{2} q^{55} + ( -1 + 4 \beta_{2} ) q^{56} + 3 \beta_{1} q^{57} + ( 9 \beta_{1} - 4 \beta_{3} ) q^{58} + ( 2 \beta_{1} + \beta_{3} ) q^{59} + ( 15 \beta_{1} - 9 \beta_{3} ) q^{60} -6 q^{61} + ( 1 - 7 \beta_{2} ) q^{62} + ( -3 \beta_{1} - \beta_{3} ) q^{63} + ( 1 + 6 \beta_{2} ) q^{64} + ( -3 + 6 \beta_{2} ) q^{66} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{67} + ( -12 - 3 \beta_{2} ) q^{68} + ( -2 + 6 \beta_{2} ) q^{69} + ( 3 \beta_{1} - 2 \beta_{3} ) q^{70} + ( -10 \beta_{1} - 2 \beta_{3} ) q^{71} + ( -11 \beta_{1} + 11 \beta_{3} ) q^{72} -2 \beta_{3} q^{73} + ( 4 - 4 \beta_{2} ) q^{74} -3 \beta_{2} q^{75} -9 \beta_{3} q^{76} + ( 3 + 3 \beta_{2} ) q^{77} + 4 q^{79} + ( 11 \beta_{1} - 8 \beta_{3} ) q^{80} + ( 4 + 6 \beta_{2} ) q^{81} -2 q^{82} + ( 6 \beta_{1} + 3 \beta_{3} ) q^{83} + ( -6 \beta_{1} + 3 \beta_{3} ) q^{84} + ( 3 \beta_{1} + \beta_{3} ) q^{85} + ( -16 \beta_{1} + 7 \beta_{3} ) q^{86} + ( -4 + 9 \beta_{2} ) q^{87} + ( -9 + 3 \beta_{2} ) q^{88} + ( 5 \beta_{1} + 13 \beta_{3} ) q^{89} + ( -7 + 12 \beta_{2} ) q^{90} + ( -12 + 6 \beta_{2} ) q^{92} + ( 7 \beta_{1} - \beta_{3} ) q^{93} + ( -1 + 3 \beta_{2} ) q^{94} -3 \beta_{2} q^{95} + ( -12 \beta_{1} + 9 \beta_{3} ) q^{96} + ( 3 \beta_{1} - 14 \beta_{3} ) q^{97} + ( -\beta_{1} + \beta_{3} ) q^{98} + ( 3 \beta_{1} + 12 \beta_{3} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4q - 6q^{3} - 6q^{4} + 2q^{9} + O(q^{10})$$ $$4q - 6q^{3} - 6q^{4} + 2q^{9} - 14q^{10} + 24q^{12} + 6q^{14} + 26q^{16} - 12q^{17} + 6q^{22} + 6q^{25} + 6q^{29} + 36q^{30} + 6q^{35} - 48q^{36} + 6q^{38} + 38q^{40} - 14q^{42} - 10q^{43} - 54q^{48} - 4q^{49} - 2q^{51} + 24q^{53} + 6q^{55} - 12q^{56} - 24q^{61} + 18q^{62} - 8q^{64} - 24q^{66} - 42q^{68} - 20q^{69} + 24q^{74} + 6q^{75} + 6q^{77} + 16q^{79} + 4q^{81} - 8q^{82} - 34q^{87} - 42q^{88} - 52q^{90} - 60q^{92} - 10q^{94} + 6q^{95} + O(q^{100})$$

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

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

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
337.1
 − 1.61803i 0.618034i − 0.618034i 1.61803i
2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
337.2 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.3 0.381966i −0.381966 1.85410 0.381966i 0.145898i 1.00000i 1.47214i −2.85410 −0.145898
337.4 2.61803i −2.61803 −4.85410 2.61803i 6.85410i 1.00000i 7.47214i 3.85410 −6.85410
 $$n$$: 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.c 4
13.b even 2 1 inner 1183.2.c.c 4
13.d odd 4 1 1183.2.a.c 2
13.d odd 4 1 1183.2.a.g 2
13.f odd 12 2 91.2.f.a 4
39.k even 12 2 819.2.o.c 4
52.l even 12 2 1456.2.s.h 4
91.i even 4 1 8281.2.a.n 2
91.i even 4 1 8281.2.a.bb 2
91.w even 12 2 637.2.g.c 4
91.x odd 12 2 637.2.h.g 4
91.ba even 12 2 637.2.h.f 4
91.bc even 12 2 637.2.f.c 4
91.bd odd 12 2 637.2.g.b 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.2.f.a 4 13.f odd 12 2
637.2.f.c 4 91.bc even 12 2
637.2.g.b 4 91.bd odd 12 2
637.2.g.c 4 91.w even 12 2
637.2.h.f 4 91.ba even 12 2
637.2.h.g 4 91.x odd 12 2
819.2.o.c 4 39.k even 12 2
1183.2.a.c 2 13.d odd 4 1
1183.2.a.g 2 13.d odd 4 1
1183.2.c.c 4 1.a even 1 1 trivial
1183.2.c.c 4 13.b even 2 1 inner
1456.2.s.h 4 52.l even 12 2
8281.2.a.n 2 91.i even 4 1
8281.2.a.bb 2 91.i even 4 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{4} + 7 T_{2}^{2} + 1$$ acting on $$S_{2}^{\mathrm{new}}(1183, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 7 T^{2} + T^{4}$$
$3$ $$( 1 + 3 T + T^{2} )^{2}$$
$5$ $$1 + 7 T^{2} + T^{4}$$
$7$ $$( 1 + T^{2} )^{2}$$
$11$ $$81 + 27 T^{2} + T^{4}$$
$13$ $$T^{4}$$
$17$ $$( -11 + 6 T + T^{2} )^{2}$$
$19$ $$81 + 27 T^{2} + T^{4}$$
$23$ $$( -20 + T^{2} )^{2}$$
$29$ $$( -29 - 3 T + T^{2} )^{2}$$
$31$ $$1681 + 98 T^{2} + T^{4}$$
$37$ $$( 16 + T^{2} )^{2}$$
$41$ $$16 + 28 T^{2} + T^{4}$$
$43$ $$( -95 + 5 T + T^{2} )^{2}$$
$47$ $$( 5 + T^{2} )^{2}$$
$53$ $$( 31 - 12 T + T^{2} )^{2}$$
$59$ $$( 5 + T^{2} )^{2}$$
$61$ $$( 6 + T )^{4}$$
$67$ $$81 + 162 T^{2} + T^{4}$$
$71$ $$13456 + 268 T^{2} + T^{4}$$
$73$ $$( 4 + T^{2} )^{2}$$
$79$ $$( -4 + T )^{4}$$
$83$ $$( 45 + T^{2} )^{2}$$
$89$ $$6241 + 283 T^{2} + T^{4}$$
$97$ $$52441 + 503 T^{2} + T^{4}$$