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

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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: \(1\)
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:

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

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