# Properties

 Label 177.2.a.c Level 177 Weight 2 Character orbit 177.a Self dual yes Analytic conductor 1.413 Analytic rank 0 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: $$0$$ 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.

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

## 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
 −0.618034 1.61803
−0.618034 1.00000 −1.61803 1.00000 −0.618034 1.61803 2.23607 1.00000 −0.618034
1.2 1.61803 1.00000 0.618034 1.00000 1.61803 −0.618034 −2.23607 1.00000 1.61803
 $$n$$: 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.c 2
3.b odd 2 1 531.2.a.a 2
4.b odd 2 1 2832.2.a.m 2
5.b even 2 1 4425.2.a.o 2
7.b odd 2 1 8673.2.a.n 2
12.b even 2 1 8496.2.a.ba 2

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

## Hecke kernels

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

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$1 - T + 3 T^{2} - 2 T^{3} + 4 T^{4}$$
$3$ $$( 1 - T )^{2}$$
$5$ $$( 1 - T + 5 T^{2} )^{2}$$
$7$ $$1 - T + 13 T^{2} - 7 T^{3} + 49 T^{4}$$
$11$ $$1 - 4 T + 21 T^{2} - 44 T^{3} + 121 T^{4}$$
$13$ $$( 1 + T + 13 T^{2} )^{2}$$
$17$ $$1 - T + 23 T^{2} - 17 T^{3} + 289 T^{4}$$
$19$ $$1 + 5 T + 33 T^{2} + 95 T^{3} + 361 T^{4}$$
$23$ $$1 - 3 T + 37 T^{2} - 69 T^{3} + 529 T^{4}$$
$29$ $$1 - 5 T + 63 T^{2} - 145 T^{3} + 841 T^{4}$$
$31$ $$1 + T + 51 T^{2} + 31 T^{3} + 961 T^{4}$$
$37$ $$1 + 9 T + 93 T^{2} + 333 T^{3} + 1369 T^{4}$$
$41$ $$1 + T + 51 T^{2} + 41 T^{3} + 1681 T^{4}$$
$43$ $$1 + 2 T + 7 T^{2} + 86 T^{3} + 1849 T^{4}$$
$47$ $$1 - 11 T + 123 T^{2} - 517 T^{3} + 2209 T^{4}$$
$53$ $$1 + 2 T + 27 T^{2} + 106 T^{3} + 2809 T^{4}$$
$59$ $$( 1 + T )^{2}$$
$61$ $$1 + T + 111 T^{2} + 61 T^{3} + 3721 T^{4}$$
$67$ $$1 - 6 T + 63 T^{2} - 402 T^{3} + 4489 T^{4}$$
$71$ $$1 - 4 T + 101 T^{2} - 284 T^{3} + 5041 T^{4}$$
$73$ $$1 - 3 T + 147 T^{2} - 219 T^{3} + 5329 T^{4}$$
$79$ $$1 - 10 T + 103 T^{2} - 790 T^{3} + 6241 T^{4}$$
$83$ $$1 - 13 T + 197 T^{2} - 1079 T^{3} + 6889 T^{4}$$
$89$ $$1 + 5 T + 153 T^{2} + 445 T^{3} + 7921 T^{4}$$
$97$ $$1 + 4 T + 153 T^{2} + 388 T^{3} + 9409 T^{4}$$