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 $1$

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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:

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

$p$ $F_p(T)$
$2$ \( -1 - T + T^{2} \)
$3$ \( ( -1 + T )^{2} \)
$5$ \( ( -1 + T )^{2} \)
$7$ \( -1 - T + T^{2} \)
$11$ \( -1 - 4 T + T^{2} \)
$13$ \( ( 1 + T )^{2} \)
$17$ \( -11 - T + T^{2} \)
$19$ \( -5 + 5 T + T^{2} \)
$23$ \( -9 - 3 T + T^{2} \)
$29$ \( 5 - 5 T + T^{2} \)
$31$ \( -11 + T + T^{2} \)
$37$ \( 19 + 9 T + T^{2} \)
$41$ \( -31 + T + T^{2} \)
$43$ \( -79 + 2 T + T^{2} \)
$47$ \( 29 - 11 T + T^{2} \)
$53$ \( -79 + 2 T + T^{2} \)
$59$ \( ( 1 + T )^{2} \)
$61$ \( -11 + T + T^{2} \)
$67$ \( -71 - 6 T + T^{2} \)
$71$ \( -41 - 4 T + T^{2} \)
$73$ \( 1 - 3 T + T^{2} \)
$79$ \( -55 - 10 T + T^{2} \)
$83$ \( 31 - 13 T + T^{2} \)
$89$ \( -25 + 5 T + T^{2} \)
$97$ \( -41 + 4 T + T^{2} \)
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