# Properties

 Label 177.2.a.a Level 177 Weight 2 Character orbit 177.a Self dual yes Analytic conductor 1.413 Analytic rank 1 Dimension 2 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: $$1$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{5})$$ Defining polynomial: $$x^{2} - x - 1$$ 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 $$\beta = \frac{1}{2}(1 + \sqrt{5})$$. We also show the integral $$q$$-expansion of the trace form.

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

## 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
 1.61803 −0.618034
−2.61803 1.00000 4.85410 −3.00000 −2.61803 −2.38197 −7.47214 1.00000 7.85410
1.2 −0.381966 1.00000 −1.85410 −3.00000 −0.381966 −4.61803 1.47214 1.00000 1.14590
 $$n$$: 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.a 2
3.b odd 2 1 531.2.a.c 2
4.b odd 2 1 2832.2.a.h 2
5.b even 2 1 4425.2.a.u 2
7.b odd 2 1 8673.2.a.j 2
12.b even 2 1 8496.2.a.bg 2

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.a.a 2 1.a even 1 1 trivial
531.2.a.c 2 3.b odd 2 1
2832.2.a.h 2 4.b odd 2 1
4425.2.a.u 2 5.b even 2 1
8496.2.a.bg 2 12.b even 2 1
8673.2.a.j 2 7.b odd 2 1

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 + 3 T + 5 T^{2} + 6 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 + 3 T + 5 T^{2} )^{2}$$
$7$ $$1 + 7 T + 25 T^{2} + 49 T^{3} + 49 T^{4}$$
$11$ $$1 + 2 T + 3 T^{2} + 22 T^{3} + 121 T^{4}$$
$13$ $$1 - 19 T^{2} + 169 T^{4}$$
$17$ $$1 + 3 T + 35 T^{2} + 51 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 + 5 T + 51 T^{2} + 115 T^{3} + 529 T^{4}$$
$29$ $$1 + 11 T + 87 T^{2} + 319 T^{3} + 841 T^{4}$$
$31$ $$1 + 7 T + 73 T^{2} + 217 T^{3} + 961 T^{4}$$
$37$ $$1 + T + 43 T^{2} + 37 T^{3} + 1369 T^{4}$$
$41$ $$1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4}$$
$43$ $$1 - 12 T + 77 T^{2} - 516 T^{3} + 1849 T^{4}$$
$47$ $$1 + 3 T + 85 T^{2} + 141 T^{3} + 2209 T^{4}$$
$53$ $$1 - 2 T + 87 T^{2} - 106 T^{3} + 2809 T^{4}$$
$59$ $$( 1 - T )^{2}$$
$61$ $$1 - 13 T + 133 T^{2} - 793 T^{3} + 3721 T^{4}$$
$67$ $$1 + 4 T + 133 T^{2} + 268 T^{3} + 4489 T^{4}$$
$71$ $$1 - 4 T + 141 T^{2} - 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 5 T + 121 T^{2} - 365 T^{3} + 5329 T^{4}$$
$79$ $$( 1 + 3 T + 79 T^{2} )^{2}$$
$83$ $$1 + 9 T + 175 T^{2} + 747 T^{3} + 6889 T^{4}$$
$89$ $$1 - T + 147 T^{2} - 89 T^{3} + 7921 T^{4}$$
$97$ $$1 + 10 T + 199 T^{2} + 970 T^{3} + 9409 T^{4}$$