# Properties

 Label 1183.2.a.n Level $1183$ Weight $2$ Character orbit 1183.a Self dual yes Analytic conductor $9.446$ Analytic rank $1$ Dimension $6$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1183 = 7 \cdot 13^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1183.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$9.44630255912$$ Analytic rank: $$1$$ Dimension: $$6$$ Coefficient field: 6.6.1279733.1 Defining polynomial: $$x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1$$ Coefficient ring: $$\Z[a_1, a_2, a_3]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-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.

 $$f(q)$$ $$=$$ $$q + ( -1 + \beta_{2} + \beta_{4} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{6} + q^{7} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{8} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{9} +O(q^{10})$$ $$q + ( -1 + \beta_{2} + \beta_{4} ) q^{2} + ( -\beta_{1} + \beta_{3} ) q^{3} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{4} -\beta_{2} q^{5} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{6} + q^{7} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{8} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{9} + ( -1 + 2 \beta_{3} - 2 \beta_{4} ) q^{10} + ( -\beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{11} + ( -4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{12} + ( -1 + \beta_{2} + \beta_{4} ) q^{14} + ( -1 + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} ) q^{15} + ( 2 - \beta_{1} - 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} ) q^{16} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{5} ) q^{17} + ( -2 - 2 \beta_{1} + 4 \beta_{3} - \beta_{4} ) q^{18} + ( 3 - \beta_{1} + \beta_{3} - \beta_{4} - \beta_{5} ) q^{19} + ( 3 - 3 \beta_{2} - 3 \beta_{4} + 2 \beta_{5} ) q^{20} + ( -\beta_{1} + \beta_{3} ) q^{21} + ( 1 + 4 \beta_{3} - 2 \beta_{4} - 2 \beta_{5} ) q^{22} + ( -2 + \beta_{1} + \beta_{3} - 3 \beta_{4} ) q^{23} + ( 5 - 4 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{24} + ( -2 - \beta_{1} - \beta_{3} + \beta_{4} ) q^{25} + ( -4 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} + 2 \beta_{4} ) q^{27} + ( \beta_{1} - 2 \beta_{3} + \beta_{4} ) q^{28} + ( -5 + 2 \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{29} + ( 2 + \beta_{1} - 2 \beta_{2} - 3 \beta_{3} + 2 \beta_{5} ) q^{30} + ( -5 + 2 \beta_{1} + 2 \beta_{2} - 6 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{31} + ( -4 + \beta_{1} + \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{32} + ( -1 + 2 \beta_{1} + 3 \beta_{2} + \beta_{3} - 3 \beta_{4} - \beta_{5} ) q^{33} + ( 7 + 2 \beta_{1} - 4 \beta_{2} + \beta_{3} - 3 \beta_{4} + \beta_{5} ) q^{34} -\beta_{2} q^{35} + ( 6 + \beta_{1} - 5 \beta_{2} - 2 \beta_{3} - 4 \beta_{4} + 2 \beta_{5} ) q^{36} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{5} ) q^{37} + ( -3 - \beta_{1} - 3 \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{38} + ( -2 - \beta_{1} + 2 \beta_{2} + 4 \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{40} + ( -1 - 2 \beta_{1} - 4 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{41} + ( 1 + \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} ) q^{42} + ( -5 - \beta_{2} - 3 \beta_{3} + 4 \beta_{4} ) q^{43} + ( 1 - 5 \beta_{2} - 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{44} + ( 2 - 3 \beta_{3} + \beta_{4} + \beta_{5} ) q^{45} + ( 3 - 2 \beta_{1} - 6 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} ) q^{46} + ( 5 - 2 \beta_{1} - 3 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{47} + ( -2 + \beta_{1} + 6 \beta_{2} + 6 \beta_{3} - 2 \beta_{4} - 3 \beta_{5} ) q^{48} + q^{49} + ( 1 - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{50} + ( 4 \beta_{1} - \beta_{2} - 6 \beta_{3} - 2 \beta_{4} ) q^{51} + ( -5 + 2 \beta_{1} + \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{53} + ( 5 + \beta_{1} - \beta_{2} + \beta_{4} + \beta_{5} ) q^{54} + ( -5 + 3 \beta_{1} + 2 \beta_{2} - 7 \beta_{3} + 3 \beta_{4} ) q^{55} + ( -\beta_{1} + \beta_{2} + 2 \beta_{3} - \beta_{5} ) q^{56} + ( 3 - 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + \beta_{4} + 2 \beta_{5} ) q^{57} + ( 2 - 3 \beta_{1} - 4 \beta_{2} - 7 \beta_{3} + \beta_{4} + \beta_{5} ) q^{58} + ( -5 + 5 \beta_{1} + \beta_{2} + \beta_{3} - 2 \beta_{5} ) q^{59} + ( -3 - \beta_{1} + 5 \beta_{2} + 6 \beta_{3} - \beta_{4} - 2 \beta_{5} ) q^{60} + ( 6 - \beta_{1} - 2 \beta_{2} + \beta_{3} - 4 \beta_{4} + 4 \beta_{5} ) q^{61} + ( -2 - 7 \beta_{1} - 3 \beta_{3} + 5 \beta_{4} - \beta_{5} ) q^{62} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} ) q^{63} + ( 1 - \beta_{1} - 5 \beta_{2} + \beta_{5} ) q^{64} + ( 4 - 3 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 4 \beta_{4} + 4 \beta_{5} ) q^{66} + ( -1 - 5 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - 2 \beta_{5} ) q^{67} + ( -1 - 3 \beta_{1} + 2 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} ) q^{68} + ( -1 + \beta_{1} - \beta_{2} - 3 \beta_{3} + \beta_{4} + 3 \beta_{5} ) q^{69} + ( -1 + 2 \beta_{3} - 2 \beta_{4} ) q^{70} + ( \beta_{2} + 7 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{71} + ( -7 + 6 \beta_{2} + 6 \beta_{3} - 3 \beta_{5} ) q^{72} + ( -2 + 5 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{5} ) q^{73} + ( -1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + 3 \beta_{4} + 4 \beta_{5} ) q^{74} + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} ) q^{75} + ( -5 + 2 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} ) q^{76} + ( -\beta_{2} - 2 \beta_{4} + 2 \beta_{5} ) q^{77} + ( -4 + \beta_{1} + 2 \beta_{2} - 5 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{79} + ( 3 \beta_{1} - \beta_{2} - 8 \beta_{3} + 4 \beta_{4} + \beta_{5} ) q^{80} + ( -3 - \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + 3 \beta_{4} ) q^{81} + ( -5 - 4 \beta_{1} + 3 \beta_{2} - 6 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} ) q^{82} + ( -6 + 4 \beta_{1} + 6 \beta_{2} - 4 \beta_{3} + 7 \beta_{4} - \beta_{5} ) q^{83} + ( -4 - \beta_{1} + 2 \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} ) q^{84} + ( -4 + 5 \beta_{2} - 2 \beta_{3} - \beta_{5} ) q^{85} + ( 1 + \beta_{1} + 2 \beta_{2} + 2 \beta_{3} - 4 \beta_{4} - 3 \beta_{5} ) q^{86} + ( 2 \beta_{1} - 3 \beta_{2} - 5 \beta_{3} + 4 \beta_{4} + 3 \beta_{5} ) q^{87} + ( -8 + \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} ) q^{88} + ( -2 \beta_{1} + \beta_{2} - 5 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} ) q^{89} + ( -4 - \beta_{1} + 7 \beta_{2} + \beta_{3} + 5 \beta_{4} - 4 \beta_{5} ) q^{90} + ( -1 - \beta_{1} + 8 \beta_{3} - 5 \beta_{4} - 2 \beta_{5} ) q^{92} + ( -8 + \beta_{2} - \beta_{3} + 6 \beta_{4} + 3 \beta_{5} ) q^{93} + ( -4 + 5 \beta_{1} + 6 \beta_{2} + 4 \beta_{3} - 3 \beta_{4} - 2 \beta_{5} ) q^{94} + ( -1 - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} ) q^{95} + ( 1 + \beta_{1} - 5 \beta_{2} - 7 \beta_{3} + 4 \beta_{4} + 6 \beta_{5} ) q^{96} + ( 5 + 2 \beta_{1} + 8 \beta_{3} - 5 \beta_{4} + 3 \beta_{5} ) q^{97} + ( -1 + \beta_{2} + \beta_{4} ) q^{98} + ( -2 \beta_{1} - 2 \beta_{2} - 5 \beta_{3} + 7 \beta_{4} + \beta_{5} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$6q - 2q^{2} - 4q^{3} + 8q^{4} - 2q^{5} + 8q^{6} + 6q^{7} - 3q^{8} + O(q^{10})$$ $$6q - 2q^{2} - 4q^{3} + 8q^{4} - 2q^{5} + 8q^{6} + 6q^{7} - 3q^{8} - 14q^{10} - 8q^{11} - 23q^{12} - 2q^{14} - 3q^{15} - 23q^{17} - 26q^{18} + 13q^{19} + 4q^{20} - 4q^{21} - 4q^{22} - 18q^{23} + 26q^{24} - 10q^{25} - 10q^{27} + 8q^{28} - 15q^{29} + 14q^{30} - 3q^{31} - 28q^{32} - 3q^{33} + 29q^{34} - 2q^{35} + 22q^{36} + 13q^{37} - 11q^{38} - 14q^{40} + 4q^{41} + 8q^{42} - 18q^{43} + 19q^{45} - 10q^{46} + 16q^{47} - 11q^{48} + 6q^{49} + 10q^{50} + 14q^{51} - 25q^{53} + 31q^{54} - 3q^{56} + 4q^{57} + 13q^{58} - 18q^{59} - 22q^{60} + 16q^{61} - 9q^{62} - 7q^{64} + 16q^{66} - 16q^{67} - 34q^{68} - q^{69} - 14q^{70} - 25q^{71} - 39q^{72} + 5q^{73} - 14q^{74} + 15q^{75} - 7q^{76} - 8q^{77} + 2q^{79} + 27q^{80} - 6q^{81} - 10q^{82} + 7q^{83} - 23q^{84} - 9q^{85} + 3q^{86} + 13q^{87} - 48q^{88} + 10q^{89} - 32q^{92} - 35q^{93} - 14q^{94} - 7q^{95} + 14q^{96} + 5q^{97} - 2q^{98} + 15q^{99} + O(q^{100})$$

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{6} - 2 x^{5} - 6 x^{4} + 10 x^{3} + 10 x^{2} - 11 x - 1$$:

 $$\beta_{0}$$ $$=$$ $$1$$ $$\beta_{1}$$ $$=$$ $$\nu$$ $$\beta_{2}$$ $$=$$ $$\nu^{2} - \nu - 2$$ $$\beta_{3}$$ $$=$$ $$\nu^{3} - \nu^{2} - 3 \nu + 1$$ $$\beta_{4}$$ $$=$$ $$\nu^{4} - \nu^{3} - 4 \nu^{2} + 2 \nu + 2$$ $$\beta_{5}$$ $$=$$ $$\nu^{5} - \nu^{4} - 5 \nu^{3} + 3 \nu^{2} + 5 \nu - 2$$
 $$1$$ $$=$$ $$\beta_0$$ $$\nu$$ $$=$$ $$\beta_{1}$$ $$\nu^{2}$$ $$=$$ $$\beta_{2} + \beta_{1} + 2$$ $$\nu^{3}$$ $$=$$ $$\beta_{3} + \beta_{2} + 4 \beta_{1} + 1$$ $$\nu^{4}$$ $$=$$ $$\beta_{4} + \beta_{3} + 5 \beta_{2} + 6 \beta_{1} + 7$$ $$\nu^{5}$$ $$=$$ $$\beta_{5} + \beta_{4} + 6 \beta_{3} + 7 \beta_{2} + 18 \beta_{1} + 8$$

## 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 $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 0.908891 2.10066 −0.0849355 −1.54570 2.33192 −1.71083
−2.63777 −2.71083 4.95781 2.08281 7.15053 1.00000 −7.80201 4.34860 −5.49396
1.2 −1.93488 −2.54570 1.74376 −0.312100 4.92562 1.00000 0.495793 3.48058 0.603875
1.3 −1.10591 1.33192 −0.776957 1.90785 −1.47298 1.00000 3.07107 −1.22600 −2.10992
1.4 −0.312100 1.10066 −1.90259 −1.93488 −0.343514 1.00000 1.21800 −1.78856 0.603875
1.5 1.90785 −1.08494 1.63989 −1.10591 −2.06989 1.00000 −0.687029 −1.82292 −2.10992
1.6 2.08281 −0.0911085 2.33809 −2.63777 −0.189762 1.00000 0.704173 −2.99170 −5.49396
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 1.6 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$7$$ $$-1$$
$$13$$ $$-1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1183.2.a.n 6
7.b odd 2 1 8281.2.a.cb 6
13.b even 2 1 1183.2.a.o yes 6
13.d odd 4 2 1183.2.c.h 12
91.b odd 2 1 8281.2.a.cg 6

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1183.2.a.n 6 1.a even 1 1 trivial
1183.2.a.o yes 6 13.b even 2 1
1183.2.c.h 12 13.d odd 4 2
8281.2.a.cb 6 7.b odd 2 1
8281.2.a.cg 6 91.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(1183))$$:

 $$T_{2}^{6} + 2 T_{2}^{5} - 8 T_{2}^{4} - 15 T_{2}^{3} + 14 T_{2}^{2} + 28 T_{2} + 7$$ $$T_{11}^{6} + 8 T_{11}^{5} - 16 T_{11}^{4} - 197 T_{11}^{3} - 28 T_{11}^{2} + 1204 T_{11} + 889$$

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$7 + 28 T + 14 T^{2} - 15 T^{3} - 8 T^{4} + 2 T^{5} + T^{6}$$
$3$ $$1 + 11 T - T^{2} - 14 T^{3} - T^{4} + 4 T^{5} + T^{6}$$
$5$ $$7 + 28 T + 14 T^{2} - 15 T^{3} - 8 T^{4} + 2 T^{5} + T^{6}$$
$7$ $$( -1 + T )^{6}$$
$11$ $$889 + 1204 T - 28 T^{2} - 197 T^{3} - 16 T^{4} + 8 T^{5} + T^{6}$$
$13$ $$T^{6}$$
$17$ $$-7351 - 5190 T - 319 T^{2} + 585 T^{3} + 190 T^{4} + 23 T^{5} + T^{6}$$
$19$ $$581 + 119 T - 322 T^{2} + 20 T^{3} + 47 T^{4} - 13 T^{5} + T^{6}$$
$23$ $$-587 - 2649 T - 1198 T^{2} + 21 T^{3} + 97 T^{4} + 18 T^{5} + T^{6}$$
$29$ $$3569 + 6707 T - 996 T^{2} - 598 T^{3} + 9 T^{4} + 15 T^{5} + T^{6}$$
$31$ $$-21463 + 24869 T + 6418 T^{2} - 564 T^{3} - 157 T^{4} + 3 T^{5} + T^{6}$$
$37$ $$-7 - 203 T - 14 T^{2} + 118 T^{3} + 5 T^{4} - 13 T^{5} + T^{6}$$
$41$ $$503 - 1164 T - 197 T^{2} + 528 T^{3} - 115 T^{4} - 4 T^{5} + T^{6}$$
$43$ $$181 - 19 T - 1107 T^{2} - 374 T^{3} + 53 T^{4} + 18 T^{5} + T^{6}$$
$47$ $$30233 - 7350 T - 4298 T^{2} + 1289 T^{3} - 22 T^{4} - 16 T^{5} + T^{6}$$
$53$ $$24193 + 14872 T - 7043 T^{2} - 1218 T^{3} + 111 T^{4} + 25 T^{5} + T^{6}$$
$59$ $$92911 + 73801 T - 2464 T^{2} - 3025 T^{3} - 123 T^{4} + 18 T^{5} + T^{6}$$
$61$ $$-12979 - 12037 T + 2271 T^{2} + 1088 T^{3} - 55 T^{4} - 16 T^{5} + T^{6}$$
$67$ $$26747 - 2737 T - 10556 T^{2} - 2612 T^{3} - 106 T^{4} + 16 T^{5} + T^{6}$$
$71$ $$563899 + 95935 T - 15813 T^{2} - 3242 T^{3} + 24 T^{4} + 25 T^{5} + T^{6}$$
$73$ $$-45367 + 1568 T + 5369 T^{2} + 13 T^{3} - 176 T^{4} - 5 T^{5} + T^{6}$$
$79$ $$-10277 + 5533 T + 7248 T^{2} + 101 T^{3} - 167 T^{4} - 2 T^{5} + T^{6}$$
$83$ $$-41203 + 70875 T + 22559 T^{2} + 420 T^{3} - 272 T^{4} - 7 T^{5} + T^{6}$$
$89$ $$-222257 - 73017 T + 15708 T^{2} + 1686 T^{3} - 242 T^{4} - 10 T^{5} + T^{6}$$
$97$ $$-43931 - 14930 T + 13618 T^{2} + 994 T^{3} - 296 T^{4} - 5 T^{5} + T^{6}$$