Properties

Label 177.2.d.c
Level $177$
Weight $2$
Character orbit 177.d
Analytic conductor $1.413$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 177.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.41335211578\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.19298288.1
Defining polynomial: \(x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 9 x^{2} - 9 x + 27\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 9 x^{2} - 9 x + 27\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -\nu^{4} + \nu^{3} - \nu - 3 \)\()/6\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{5} - \nu^{4} - 6 \nu^{3} + 7 \nu^{2} - 9 \nu \)\()/18\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{5} + 4 \nu^{4} + 3 \nu^{3} + 11 \nu^{2} - 6 \nu + 27 \)\()/18\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{5} + \nu^{4} - 3 \nu^{3} + 2 \nu^{2} - 9 \nu + 9 \)\()/9\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{4} + \beta_{3} + \beta_{2} + \beta_{1} - 1\)
\(\nu^{3}\)\(=\)\(-\beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_{1}\)
\(\nu^{4}\)\(=\)\(-\beta_{5} + \beta_{4} - \beta_{3} - 5 \beta_{2} - 2 \beta_{1} - 3\)
\(\nu^{5}\)\(=\)\(-7 \beta_{5} + 4 \beta_{3} - 6 \beta_{2} - 6 \beta_{1} + 4\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/177\mathbb{Z}\right)^\times\).

\(n\) \(61\) \(119\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
176.1
1.34067 + 1.09664i
1.34067 1.09664i
−1.16170 + 1.28470i
−1.16170 1.28470i
0.321037 + 1.70204i
0.321037 1.70204i
−1.93543 −1.34067 1.09664i 1.74590 3.21911i 2.59477 + 2.12247i 0.254102 0.491797 0.594767 + 2.94045i 6.23037i
176.2 −1.93543 −1.34067 + 1.09664i 1.74590 3.21911i 2.59477 2.12247i 0.254102 0.491797 0.594767 2.94045i 6.23037i
176.3 1.46260 1.16170 1.28470i 0.139194 0.594299i 1.69910 1.87900i 1.86081 −2.72161 −0.300896 2.98487i 0.869221i
176.4 1.46260 1.16170 + 1.28470i 0.139194 0.594299i 1.69910 + 1.87900i 1.86081 −2.72161 −0.300896 + 2.98487i 0.869221i
176.5 2.47283 −0.321037 1.70204i 4.11491 2.50682i −0.793871 4.20886i −2.11491 5.22982 −2.79387 + 1.09283i 6.19895i
176.6 2.47283 −0.321037 + 1.70204i 4.11491 2.50682i −0.793871 + 4.20886i −2.11491 5.22982 −2.79387 1.09283i 6.19895i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 176.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
177.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 177.2.d.c yes 6
3.b odd 2 1 177.2.d.a 6
59.b odd 2 1 177.2.d.a 6
177.d even 2 1 inner 177.2.d.c yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.2.d.a 6 3.b odd 2 1
177.2.d.a 6 59.b odd 2 1
177.2.d.c yes 6 1.a even 1 1 trivial
177.2.d.c yes 6 177.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{3} - 2 T_{2}^{2} - 4 T_{2} + 7 \) acting on \(S_{2}^{\mathrm{new}}(177, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 7 - 4 T - 2 T^{2} + T^{3} )^{2} \)
$3$ \( 27 + 9 T + 9 T^{2} + 2 T^{3} + 3 T^{4} + T^{5} + T^{6} \)
$5$ \( 23 + 71 T^{2} + 17 T^{4} + T^{6} \)
$7$ \( ( 1 - 4 T + T^{3} )^{2} \)
$11$ \( ( 14 + 27 T + 10 T^{2} + T^{3} )^{2} \)
$13$ \( 4508 + 997 T^{2} + 58 T^{4} + T^{6} \)
$17$ \( 368 + 197 T^{2} + 27 T^{4} + T^{6} \)
$19$ \( ( -37 - 26 T - 2 T^{2} + T^{3} )^{2} \)
$23$ \( ( 14 + 23 T + 9 T^{2} + T^{3} )^{2} \)
$29$ \( 1127 + 614 T^{2} + 54 T^{4} + T^{6} \)
$31$ \( 18032 + 3589 T^{2} + 115 T^{4} + T^{6} \)
$37$ \( 220892 + 13537 T^{2} + 223 T^{4} + T^{6} \)
$41$ \( 23 + 282 T^{2} + 58 T^{4} + T^{6} \)
$43$ \( 288512 + 13565 T^{2} + 206 T^{4} + T^{6} \)
$47$ \( ( -406 + 171 T - 23 T^{2} + T^{3} )^{2} \)
$53$ \( 3887 + 3015 T^{2} + 121 T^{4} + T^{6} \)
$59$ \( 205379 + 34810 T + 5015 T^{2} + 732 T^{3} + 85 T^{4} + 10 T^{5} + T^{6} \)
$61$ \( 4508 + 9945 T^{2} + 199 T^{4} + T^{6} \)
$67$ \( 4508 + 997 T^{2} + 58 T^{4} + T^{6} \)
$71$ \( 67068 + 11421 T^{2} + 306 T^{4} + T^{6} \)
$73$ \( 72128 + 13261 T^{2} + 247 T^{4} + T^{6} \)
$79$ \( ( -347 - 85 T + 7 T^{2} + T^{3} )^{2} \)
$83$ \( ( 14 + 151 T + 25 T^{2} + T^{3} )^{2} \)
$89$ \( ( 518 - 25 T - 13 T^{2} + T^{3} )^{2} \)
$97$ \( 18032 + 7349 T^{2} + 342 T^{4} + T^{6} \)
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