# Properties

 Label 177.2.a.d Level 177 Weight 2 Character orbit 177.a Self dual yes Analytic conductor 1.413 Analytic rank 0 Dimension 3 CM no Inner twists 1

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$177 = 3 \cdot 59$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 177.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$1.41335211578$$ Analytic rank: $$0$$ Dimension: $$3$$ Coefficient field: 3.3.229.1 Defining polynomial: $$x^{3} - 4 x - 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ 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 $$1,\beta_1,\beta_2$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{1} q^{6} + ( 3 + \beta_{1} ) q^{7} + q^{8} + q^{9} +O(q^{10})$$ $$q + \beta_{1} q^{2} - q^{3} + ( 1 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} ) q^{5} -\beta_{1} q^{6} + ( 3 + \beta_{1} ) q^{7} + q^{8} + q^{9} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{10} + ( -1 - \beta_{1} - \beta_{2} ) q^{11} + ( -1 - \beta_{2} ) q^{12} + ( 1 - \beta_{1} - \beta_{2} ) q^{13} + ( 3 + 3 \beta_{1} + \beta_{2} ) q^{14} + ( 1 - \beta_{1} + \beta_{2} ) q^{15} + ( -2 + \beta_{1} - 2 \beta_{2} ) q^{16} + ( 2 - 2 \beta_{1} + 3 \beta_{2} ) q^{17} + \beta_{1} q^{18} + ( 2 - \beta_{2} ) q^{19} + ( -3 + \beta_{1} ) q^{20} + ( -3 - \beta_{1} ) q^{21} + ( -4 - 2 \beta_{1} - \beta_{2} ) q^{22} + ( -2 \beta_{1} - \beta_{2} ) q^{23} - q^{24} + ( -3 \beta_{1} + \beta_{2} ) q^{25} + ( -4 - \beta_{2} ) q^{26} - q^{27} + ( 4 + 2 \beta_{1} + 3 \beta_{2} ) q^{28} + ( -3 + \beta_{1} + 2 \beta_{2} ) q^{29} + ( -2 + 2 \beta_{1} - \beta_{2} ) q^{30} + ( 5 - \beta_{1} + 2 \beta_{2} ) q^{31} + ( -1 - 4 \beta_{1} + \beta_{2} ) q^{32} + ( 1 + \beta_{1} + \beta_{2} ) q^{33} + ( -3 + 5 \beta_{1} - 2 \beta_{2} ) q^{34} + ( -1 + \beta_{1} - 2 \beta_{2} ) q^{35} + ( 1 + \beta_{2} ) q^{36} + ( -2 - 2 \beta_{1} - \beta_{2} ) q^{37} + ( -1 + \beta_{1} ) q^{38} + ( -1 + \beta_{1} + \beta_{2} ) q^{39} + ( -1 + \beta_{1} - \beta_{2} ) q^{40} + ( -1 + 3 \beta_{1} - 2 \beta_{2} ) q^{41} + ( -3 - 3 \beta_{1} - \beta_{2} ) q^{42} + ( 1 + 5 \beta_{1} - 3 \beta_{2} ) q^{43} + ( -5 - 3 \beta_{1} ) q^{44} + ( -1 + \beta_{1} - \beta_{2} ) q^{45} + ( -7 - \beta_{1} - 2 \beta_{2} ) q^{46} + ( 5 - \beta_{1} + 4 \beta_{2} ) q^{47} + ( 2 - \beta_{1} + 2 \beta_{2} ) q^{48} + ( 5 + 6 \beta_{1} + \beta_{2} ) q^{49} + ( -8 + \beta_{1} - 3 \beta_{2} ) q^{50} + ( -2 + 2 \beta_{1} - 3 \beta_{2} ) q^{51} + ( -3 - 3 \beta_{1} + 2 \beta_{2} ) q^{52} + ( 1 - 5 \beta_{1} + \beta_{2} ) q^{53} -\beta_{1} q^{54} + ( 1 + \beta_{1} - \beta_{2} ) q^{55} + ( 3 + \beta_{1} ) q^{56} + ( -2 + \beta_{2} ) q^{57} + ( 5 - \beta_{1} + \beta_{2} ) q^{58} + q^{59} + ( 3 - \beta_{1} ) q^{60} + ( 1 + \beta_{1} + 4 \beta_{2} ) q^{61} + ( -1 + 7 \beta_{1} - \beta_{2} ) q^{62} + ( 3 + \beta_{1} ) q^{63} + ( -7 - 2 \beta_{1} ) q^{64} + ( -1 + 3 \beta_{1} - 3 \beta_{2} ) q^{65} + ( 4 + 2 \beta_{1} + \beta_{2} ) q^{66} + ( 5 + \beta_{1} + 5 \beta_{2} ) q^{67} + ( 9 - \beta_{1} - \beta_{2} ) q^{68} + ( 2 \beta_{1} + \beta_{2} ) q^{69} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{70} + ( 9 - 3 \beta_{1} + \beta_{2} ) q^{71} + q^{72} + ( 1 - 3 \beta_{1} - 4 \beta_{2} ) q^{73} + ( -7 - 3 \beta_{1} - 2 \beta_{2} ) q^{74} + ( 3 \beta_{1} - \beta_{2} ) q^{75} + ( -1 - \beta_{1} + 3 \beta_{2} ) q^{76} + ( -7 - 5 \beta_{1} - 4 \beta_{2} ) q^{77} + ( 4 + \beta_{2} ) q^{78} + ( 1 - 3 \beta_{1} + \beta_{2} ) q^{79} + ( 8 - 4 \beta_{1} + \beta_{2} ) q^{80} + q^{81} + ( 7 - 3 \beta_{1} + 3 \beta_{2} ) q^{82} + ( 1 - 5 \beta_{1} + 6 \beta_{2} ) q^{83} + ( -4 - 2 \beta_{1} - 3 \beta_{2} ) q^{84} + ( -12 + 6 \beta_{1} - \beta_{2} ) q^{85} + ( 12 - 2 \beta_{1} + 5 \beta_{2} ) q^{86} + ( 3 - \beta_{1} - 2 \beta_{2} ) q^{87} + ( -1 - \beta_{1} - \beta_{2} ) q^{88} + ( -9 + 3 \beta_{1} - 4 \beta_{2} ) q^{89} + ( 2 - 2 \beta_{1} + \beta_{2} ) q^{90} + ( -1 - 3 \beta_{1} - 4 \beta_{2} ) q^{91} + ( -5 - 5 \beta_{1} + \beta_{2} ) q^{92} + ( -5 + \beta_{1} - 2 \beta_{2} ) q^{93} + ( 1 + 9 \beta_{1} - \beta_{2} ) q^{94} + ( 2 \beta_{1} - 3 \beta_{2} ) q^{95} + ( 1 + 4 \beta_{1} - \beta_{2} ) q^{96} + ( 5 - 5 \beta_{1} + \beta_{2} ) q^{97} + ( 19 + 6 \beta_{1} + 6 \beta_{2} ) q^{98} + ( -1 - \beta_{1} - \beta_{2} ) q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + O(q^{10})$$ $$3q - 3q^{3} + 2q^{4} - 2q^{5} + 9q^{7} + 3q^{8} + 3q^{9} + 5q^{10} - 2q^{11} - 2q^{12} + 4q^{13} + 8q^{14} + 2q^{15} - 4q^{16} + 3q^{17} + 7q^{19} - 9q^{20} - 9q^{21} - 11q^{22} + q^{23} - 3q^{24} - q^{25} - 11q^{26} - 3q^{27} + 9q^{28} - 11q^{29} - 5q^{30} + 13q^{31} - 4q^{32} + 2q^{33} - 7q^{34} - q^{35} + 2q^{36} - 5q^{37} - 3q^{38} - 4q^{39} - 2q^{40} - q^{41} - 8q^{42} + 6q^{43} - 15q^{44} - 2q^{45} - 19q^{46} + 11q^{47} + 4q^{48} + 14q^{49} - 21q^{50} - 3q^{51} - 11q^{52} + 2q^{53} + 4q^{55} + 9q^{56} - 7q^{57} + 14q^{58} + 3q^{59} + 9q^{60} - q^{61} - 2q^{62} + 9q^{63} - 21q^{64} + 11q^{66} + 10q^{67} + 28q^{68} - q^{69} + 2q^{70} + 26q^{71} + 3q^{72} + 7q^{73} - 19q^{74} + q^{75} - 6q^{76} - 17q^{77} + 11q^{78} + 2q^{79} + 23q^{80} + 3q^{81} + 18q^{82} - 3q^{83} - 9q^{84} - 35q^{85} + 31q^{86} + 11q^{87} - 2q^{88} - 23q^{89} + 5q^{90} + q^{91} - 16q^{92} - 13q^{93} + 4q^{94} + 3q^{95} + 4q^{96} + 14q^{97} + 51q^{98} - 2q^{99} + O(q^{100})$$

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

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

## 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
 −1.86081 −0.254102 2.11491
−1.86081 −1.00000 1.46260 −3.32340 1.86081 1.13919 1.00000 1.00000 6.18421
1.2 −0.254102 −1.00000 −1.93543 1.68133 0.254102 2.74590 1.00000 1.00000 −0.427229
1.3 2.11491 −1.00000 2.47283 −0.357926 −2.11491 5.11491 1.00000 1.00000 −0.756981
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$3$$ $$1$$
$$59$$ $$-1$$

## 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 177.2.a.d 3
3.b odd 2 1 531.2.a.d 3
4.b odd 2 1 2832.2.a.t 3
5.b even 2 1 4425.2.a.w 3
7.b odd 2 1 8673.2.a.s 3
12.b even 2 1 8496.2.a.bl 3

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.d 3 1.a even 1 1 trivial
531.2.a.d 3 3.b odd 2 1
2832.2.a.t 3 4.b odd 2 1
4425.2.a.w 3 5.b even 2 1
8496.2.a.bl 3 12.b even 2 1
8673.2.a.s 3 7.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{3} - 4 T_{2} - 1$$ acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(177))$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 2 T^{2} - T^{3} + 4 T^{4} + 8 T^{6}$$
$3$ $$( 1 + T )^{3}$$
$5$ $$1 + 2 T + 10 T^{2} + 18 T^{3} + 50 T^{4} + 50 T^{5} + 125 T^{6}$$
$7$ $$1 - 9 T + 44 T^{2} - 142 T^{3} + 308 T^{4} - 441 T^{5} + 343 T^{6}$$
$11$ $$1 + 2 T + 22 T^{2} + 48 T^{3} + 242 T^{4} + 242 T^{5} + 1331 T^{6}$$
$13$ $$1 - 4 T + 32 T^{2} - 78 T^{3} + 416 T^{4} - 676 T^{5} + 2197 T^{6}$$
$17$ $$1 - 3 T + 8 T^{2} - 4 T^{3} + 136 T^{4} - 867 T^{5} + 4913 T^{6}$$
$19$ $$1 - 7 T + 68 T^{2} - 270 T^{3} + 1292 T^{4} - 2527 T^{5} + 6859 T^{6}$$
$23$ $$1 - T + 42 T^{2} + 18 T^{3} + 966 T^{4} - 529 T^{5} + 12167 T^{6}$$
$29$ $$1 + 11 T + 96 T^{2} + 564 T^{3} + 2784 T^{4} + 9251 T^{5} + 24389 T^{6}$$
$31$ $$1 - 13 T + 130 T^{2} - 778 T^{3} + 4030 T^{4} - 12493 T^{5} + 29791 T^{6}$$
$37$ $$1 + 5 T + 92 T^{2} + 384 T^{3} + 3404 T^{4} + 6845 T^{5} + 50653 T^{6}$$
$41$ $$1 + T + 84 T^{2} + 156 T^{3} + 3444 T^{4} + 1681 T^{5} + 68921 T^{6}$$
$43$ $$1 - 6 T + 38 T^{2} + 76 T^{3} + 1634 T^{4} - 11094 T^{5} + 79507 T^{6}$$
$47$ $$1 - 11 T + 104 T^{2} - 538 T^{3} + 4888 T^{4} - 24299 T^{5} + 103823 T^{6}$$
$53$ $$1 - 2 T + 70 T^{2} - 270 T^{3} + 3710 T^{4} - 5618 T^{5} + 148877 T^{6}$$
$59$ $$( 1 - T )^{3}$$
$61$ $$1 + T + 82 T^{2} + 220 T^{3} + 5002 T^{4} + 3721 T^{5} + 226981 T^{6}$$
$67$ $$1 - 10 T + 82 T^{2} - 556 T^{3} + 5494 T^{4} - 44890 T^{5} + 300763 T^{6}$$
$71$ $$1 - 26 T + 406 T^{2} - 4116 T^{3} + 28826 T^{4} - 131066 T^{5} + 357911 T^{6}$$
$73$ $$1 - 7 T + 78 T^{2} - 304 T^{3} + 5694 T^{4} - 37303 T^{5} + 389017 T^{6}$$
$79$ $$1 - 2 T + 206 T^{2} - 348 T^{3} + 16274 T^{4} - 12482 T^{5} + 493039 T^{6}$$
$83$ $$1 + 3 T + 50 T^{2} + 350 T^{3} + 4150 T^{4} + 20667 T^{5} + 571787 T^{6}$$
$89$ $$1 + 23 T + 358 T^{2} + 3816 T^{3} + 31862 T^{4} + 182183 T^{5} + 704969 T^{6}$$
$97$ $$1 - 14 T + 266 T^{2} - 2514 T^{3} + 25802 T^{4} - 131726 T^{5} + 912673 T^{6}$$