Properties

Label 177.4.a.c
Level $177$
Weight $4$
Character orbit 177.a
Self dual yes
Analytic conductor $10.443$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.4433380710\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{2} ) q^{4} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( 7 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} + ( 7 \beta_{1} - 3 \beta_{3} + \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} -3 q^{3} + ( 5 + \beta_{2} ) q^{4} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} ) q^{5} -3 \beta_{1} q^{6} + ( 7 - \beta_{1} + \beta_{2} + 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{7} + ( 7 \beta_{1} - 3 \beta_{3} + \beta_{6} ) q^{8} + 9 q^{9} + ( 3 + 3 \beta_{1} + 2 \beta_{2} - 3 \beta_{4} - \beta_{5} + \beta_{6} ) q^{10} + ( -7 + 6 \beta_{1} - \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} - 3 \beta_{6} + 2 \beta_{7} ) q^{11} + ( -15 - 3 \beta_{2} ) q^{12} + ( 9 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} ) q^{13} + ( -8 + 12 \beta_{1} - 3 \beta_{2} + 2 \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{14} + ( 6 - 3 \beta_{1} - 3 \beta_{4} - 3 \beta_{7} ) q^{15} + ( 48 - \beta_{1} + 6 \beta_{2} - 2 \beta_{3} - 8 \beta_{4} + 2 \beta_{5} - 2 \beta_{7} ) q^{16} + ( 7 + 6 \beta_{1} + \beta_{3} - 2 \beta_{4} - 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{17} + 9 \beta_{1} q^{18} + ( 36 + 7 \beta_{1} - 4 \beta_{2} - \beta_{3} + 6 \beta_{4} + 5 \beta_{6} - \beta_{7} ) q^{19} + ( 60 + 10 \beta_{1} - \beta_{2} - 8 \beta_{3} - 3 \beta_{4} + 11 \beta_{5} + \beta_{6} ) q^{20} + ( -21 + 3 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} - 3 \beta_{5} + 3 \beta_{7} ) q^{21} + ( 74 - 2 \beta_{1} + \beta_{2} + \beta_{3} + 6 \beta_{4} - 8 \beta_{5} - \beta_{7} ) q^{22} + ( 28 - 7 \beta_{1} + 2 \beta_{2} + 3 \beta_{3} - 8 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} - \beta_{7} ) q^{23} + ( -21 \beta_{1} + 9 \beta_{3} - 3 \beta_{6} ) q^{24} + ( 41 - 11 \beta_{1} + \beta_{2} + 3 \beta_{3} - 6 \beta_{4} + 17 \beta_{5} - 7 \beta_{7} ) q^{25} + ( 30 - 2 \beta_{1} - 2 \beta_{2} + 8 \beta_{3} + 7 \beta_{4} - 15 \beta_{5} - 2 \beta_{6} - 7 \beta_{7} ) q^{26} -27 q^{27} + ( 90 - 26 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} + 5 \beta_{7} ) q^{28} + ( -18 - 10 \beta_{1} - 2 \beta_{2} + 15 \beta_{3} + 5 \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{29} + ( -9 - 9 \beta_{1} - 6 \beta_{2} + 9 \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{30} + ( 68 - 16 \beta_{1} - 4 \beta_{2} - 11 \beta_{3} + \beta_{4} - 2 \beta_{5} - 3 \beta_{6} + 14 \beta_{7} ) q^{31} + ( 35 + 26 \beta_{1} - \beta_{2} - 2 \beta_{3} + 20 \beta_{5} - 2 \beta_{6} + 8 \beta_{7} ) q^{32} + ( 21 - 18 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} - 6 \beta_{4} + 3 \beta_{5} + 9 \beta_{6} - 6 \beta_{7} ) q^{33} + ( 59 + 11 \beta_{1} - \beta_{3} + 4 \beta_{4} + 14 \beta_{5} + 13 \beta_{7} ) q^{34} + ( -40 + 8 \beta_{1} - 17 \beta_{2} - 14 \beta_{3} + 26 \beta_{4} - 19 \beta_{5} - 7 \beta_{6} + 30 \beta_{7} ) q^{35} + ( 45 + 9 \beta_{2} ) q^{36} + ( 57 - 24 \beta_{1} - \beta_{2} - 8 \beta_{3} + 9 \beta_{4} - 25 \beta_{5} + 4 \beta_{6} - 8 \beta_{7} ) q^{37} + ( 44 + 20 \beta_{2} + 23 \beta_{3} - 16 \beta_{4} - 10 \beta_{5} - 5 \beta_{6} - 10 \beta_{7} ) q^{38} + ( -27 - 6 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} - 9 \beta_{4} + 9 \beta_{6} ) q^{39} + ( 131 - 3 \beta_{1} + 30 \beta_{2} - 7 \beta_{3} - 35 \beta_{4} + 19 \beta_{5} + 2 \beta_{6} - 12 \beta_{7} ) q^{40} + ( 13 - 15 \beta_{1} + 23 \beta_{2} + 4 \beta_{3} - 26 \beta_{4} + 5 \beta_{5} + 4 \beta_{6} - 29 \beta_{7} ) q^{41} + ( 24 - 36 \beta_{1} + 9 \beta_{2} - 6 \beta_{4} + 6 \beta_{5} - 3 \beta_{6} + 3 \beta_{7} ) q^{42} + ( 63 - 56 \beta_{1} - 15 \beta_{2} + 3 \beta_{3} + 14 \beta_{4} - 25 \beta_{5} - 3 \beta_{6} + 18 \beta_{7} ) q^{43} + ( 3 + 61 \beta_{1} - 6 \beta_{2} - 3 \beta_{3} + 14 \beta_{4} - 12 \beta_{5} + 16 \beta_{6} - 21 \beta_{7} ) q^{44} + ( -18 + 9 \beta_{1} + 9 \beta_{4} + 9 \beta_{7} ) q^{45} + ( -68 + 14 \beta_{1} - 6 \beta_{2} - 7 \beta_{3} - 14 \beta_{4} + 28 \beta_{5} + 7 \beta_{6} + 14 \beta_{7} ) q^{46} + ( 126 - 8 \beta_{1} + 5 \beta_{2} - 26 \beta_{3} + 21 \beta_{5} - 11 \beta_{6} - 8 \beta_{7} ) q^{47} + ( -144 + 3 \beta_{1} - 18 \beta_{2} + 6 \beta_{3} + 24 \beta_{4} - 6 \beta_{5} + 6 \beta_{7} ) q^{48} + ( 58 - 36 \beta_{1} - 2 \beta_{2} + 7 \beta_{3} - 20 \beta_{4} + 14 \beta_{5} - 2 \beta_{6} - 22 \beta_{7} ) q^{49} + ( -63 - 16 \beta_{1} + 11 \beta_{2} + \beta_{3} - 42 \beta_{4} + 4 \beta_{5} + 11 \beta_{6} - 16 \beta_{7} ) q^{50} + ( -21 - 18 \beta_{1} - 3 \beta_{3} + 6 \beta_{4} + 6 \beta_{5} - 6 \beta_{6} - 6 \beta_{7} ) q^{51} + ( -95 + 19 \beta_{1} - 27 \beta_{2} + 10 \beta_{3} + 63 \beta_{4} - 39 \beta_{5} - 7 \beta_{7} ) q^{52} + ( -102 - 21 \beta_{1} - 6 \beta_{2} - 2 \beta_{3} + \beta_{4} + 18 \beta_{5} - 16 \beta_{6} + 15 \beta_{7} ) q^{53} -27 \beta_{1} q^{54} + ( 86 - 17 \beta_{1} + 11 \beta_{2} + 17 \beta_{3} - 26 \beta_{4} + 9 \beta_{5} + 28 \beta_{6} - 33 \beta_{7} ) q^{55} + ( -303 + 85 \beta_{1} - 24 \beta_{2} - 17 \beta_{3} + 30 \beta_{4} - 4 \beta_{5} - 14 \beta_{6} + 11 \beta_{7} ) q^{56} + ( -108 - 21 \beta_{1} + 12 \beta_{2} + 3 \beta_{3} - 18 \beta_{4} - 15 \beta_{6} + 3 \beta_{7} ) q^{57} + ( -161 - 23 \beta_{1} - 40 \beta_{2} + 25 \beta_{3} + 35 \beta_{4} - 17 \beta_{5} - 4 \beta_{6} + 6 \beta_{7} ) q^{58} -59 q^{59} + ( -180 - 30 \beta_{1} + 3 \beta_{2} + 24 \beta_{3} + 9 \beta_{4} - 33 \beta_{5} - 3 \beta_{6} ) q^{60} + ( -72 - 32 \beta_{1} - 32 \beta_{2} + 19 \beta_{3} + 11 \beta_{4} + 22 \beta_{5} - 33 \beta_{6} + 16 \beta_{7} ) q^{61} + ( -269 + 79 \beta_{1} - 15 \beta_{3} - 37 \beta_{4} + 19 \beta_{5} + 8 \beta_{6} + 14 \beta_{7} ) q^{62} + ( 63 - 9 \beta_{1} + 9 \beta_{2} + 18 \beta_{3} + 9 \beta_{5} - 9 \beta_{7} ) q^{63} + ( -60 + 19 \beta_{1} + 20 \beta_{2} + 7 \beta_{3} - 32 \beta_{4} - 4 \beta_{5} + 27 \beta_{6} + 8 \beta_{7} ) q^{64} + ( -118 - 65 \beta_{1} - 9 \beta_{2} + 7 \beta_{3} + 38 \beta_{4} - 25 \beta_{5} + 10 \beta_{6} + 19 \beta_{7} ) q^{65} + ( -222 + 6 \beta_{1} - 3 \beta_{2} - 3 \beta_{3} - 18 \beta_{4} + 24 \beta_{5} + 3 \beta_{7} ) q^{66} + ( 246 + 3 \beta_{1} + 7 \beta_{2} - 21 \beta_{3} - 18 \beta_{4} + 23 \beta_{5} - 16 \beta_{6} + 11 \beta_{7} ) q^{67} + ( 8 + 30 \beta_{1} + 45 \beta_{2} - 18 \beta_{3} - 72 \beta_{4} + 30 \beta_{5} + 11 \beta_{6} - 13 \beta_{7} ) q^{68} + ( -84 + 21 \beta_{1} - 6 \beta_{2} - 9 \beta_{3} + 24 \beta_{4} - 18 \beta_{5} - 9 \beta_{6} + 3 \beta_{7} ) q^{69} + ( -143 - 23 \beta_{1} + 17 \beta_{2} + 26 \beta_{3} - 24 \beta_{4} - 32 \beta_{5} - 6 \beta_{6} + 6 \beta_{7} ) q^{70} + ( -211 + 6 \beta_{1} + 2 \beta_{2} + 8 \beta_{3} - 21 \beta_{4} - 22 \beta_{5} + 33 \beta_{6} - 10 \beta_{7} ) q^{71} + ( 63 \beta_{1} - 27 \beta_{3} + 9 \beta_{6} ) q^{72} + ( 272 - 52 \beta_{1} + 23 \beta_{2} - 6 \beta_{3} - 62 \beta_{4} + 25 \beta_{5} - 13 \beta_{6} - 22 \beta_{7} ) q^{73} + ( -341 + 87 \beta_{1} - 45 \beta_{2} + 16 \beta_{3} + 83 \beta_{4} - 35 \beta_{5} - 34 \beta_{6} - 13 \beta_{7} ) q^{74} + ( -123 + 33 \beta_{1} - 3 \beta_{2} - 9 \beta_{3} + 18 \beta_{4} - 51 \beta_{5} + 21 \beta_{7} ) q^{75} + ( -167 + 151 \beta_{1} - 55 \beta_{2} - 40 \beta_{3} + 84 \beta_{4} + 18 \beta_{5} - 40 \beta_{6} + 48 \beta_{7} ) q^{76} + ( -125 + 99 \beta_{1} - 13 \beta_{2} - 20 \beta_{3} + 56 \beta_{4} - 73 \beta_{5} + 48 \beta_{6} + 7 \beta_{7} ) q^{77} + ( -90 + 6 \beta_{1} + 6 \beta_{2} - 24 \beta_{3} - 21 \beta_{4} + 45 \beta_{5} + 6 \beta_{6} + 21 \beta_{7} ) q^{78} + ( 413 + 76 \beta_{1} - 21 \beta_{2} - 11 \beta_{3} + 30 \beta_{4} + 13 \beta_{5} + 33 \beta_{6} - 8 \beta_{7} ) q^{79} + ( -293 + 205 \beta_{1} + 24 \beta_{2} - 54 \beta_{3} - 11 \beta_{4} - 3 \beta_{5} + 29 \beta_{6} + 22 \beta_{7} ) q^{80} + 81 q^{81} + ( 60 + 90 \beta_{1} - 35 \beta_{2} - 58 \beta_{3} + 64 \beta_{4} + 28 \beta_{5} - \beta_{6} - 3 \beta_{7} ) q^{82} + ( 279 - 117 \beta_{1} + 35 \beta_{2} + 60 \beta_{3} + 40 \beta_{4} + \beta_{5} - 24 \beta_{6} - 19 \beta_{7} ) q^{83} + ( -270 + 78 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 18 \beta_{4} + 42 \beta_{5} + 18 \beta_{6} - 15 \beta_{7} ) q^{84} + ( 28 + 137 \beta_{1} + 12 \beta_{2} + 37 \beta_{3} - 40 \beta_{4} + 6 \beta_{5} + 7 \beta_{6} - 11 \beta_{7} ) q^{85} + ( -902 + 48 \beta_{1} - 101 \beta_{2} + 41 \beta_{3} + 62 \beta_{4} - 12 \beta_{5} - 22 \beta_{6} + 33 \beta_{7} ) q^{86} + ( 54 + 30 \beta_{1} + 6 \beta_{2} - 45 \beta_{3} - 15 \beta_{4} + 6 \beta_{5} + 3 \beta_{6} ) q^{87} + ( 166 - 68 \beta_{1} + 65 \beta_{2} + 58 \beta_{3} - 10 \beta_{4} + 12 \beta_{5} - 39 \beta_{6} - 37 \beta_{7} ) q^{88} + ( 244 - 98 \beta_{1} + 67 \beta_{2} + 18 \beta_{3} - 88 \beta_{4} + 85 \beta_{5} - 31 \beta_{6} + 4 \beta_{7} ) q^{89} + ( 27 + 27 \beta_{1} + 18 \beta_{2} - 27 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} ) q^{90} + ( 315 + 79 \beta_{1} + 3 \beta_{2} + 10 \beta_{3} + 28 \beta_{4} + \beta_{5} + 46 \beta_{6} - 51 \beta_{7} ) q^{91} + ( -77 - 121 \beta_{1} + 61 \beta_{2} - 34 \beta_{3} - 90 \beta_{4} + 36 \beta_{5} + 12 \beta_{6} + 36 \beta_{7} ) q^{92} + ( -204 + 48 \beta_{1} + 12 \beta_{2} + 33 \beta_{3} - 3 \beta_{4} + 6 \beta_{5} + 9 \beta_{6} - 42 \beta_{7} ) q^{93} + ( 57 + 121 \beta_{1} + 75 \beta_{2} - 44 \beta_{3} - 98 \beta_{4} - 38 \beta_{5} + 18 \beta_{6} - 74 \beta_{7} ) q^{94} + ( 242 - 99 \beta_{1} - 6 \beta_{2} + 15 \beta_{3} + 76 \beta_{4} - 2 \beta_{5} - 63 \beta_{6} + 99 \beta_{7} ) q^{95} + ( -105 - 78 \beta_{1} + 3 \beta_{2} + 6 \beta_{3} - 60 \beta_{5} + 6 \beta_{6} - 24 \beta_{7} ) q^{96} + ( 172 + 27 \beta_{1} + 15 \beta_{2} + 67 \beta_{3} + 42 \beta_{4} - 101 \beta_{5} - 30 \beta_{6} + 45 \beta_{7} ) q^{97} + ( -237 - 94 \beta_{1} - 44 \beta_{2} + 13 \beta_{3} + 26 \beta_{4} + 12 \beta_{5} - 10 \beta_{6} - 13 \beta_{7} ) q^{98} + ( -63 + 54 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 18 \beta_{4} - 9 \beta_{5} - 27 \beta_{6} + 18 \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + O(q^{10}) \) \( 8q + 2q^{2} - 24q^{3} + 38q^{4} - 12q^{5} - 6q^{6} + 53q^{7} + 3q^{8} + 72q^{9} + 29q^{10} - 27q^{11} - 114q^{12} + 89q^{13} - 37q^{14} + 36q^{15} + 362q^{16} + 79q^{17} + 18q^{18} + 288q^{19} + 457q^{20} - 159q^{21} + 596q^{22} + 202q^{23} - 9q^{24} + 264q^{25} + 270q^{26} - 216q^{27} + 702q^{28} - 114q^{29} - 87q^{30} + 538q^{31} + 316q^{32} + 81q^{33} + 498q^{34} - 196q^{35} + 342q^{36} + 395q^{37} + 397q^{38} - 267q^{39} + 918q^{40} - 39q^{41} + 111q^{42} + 527q^{43} + 64q^{44} - 108q^{45} - 539q^{46} + 860q^{47} - 1086q^{48} + 347q^{49} - 591q^{50} - 237q^{51} - 644q^{52} - 812q^{53} - 54q^{54} + 536q^{55} - 2218q^{56} - 864q^{57} - 1154q^{58} - 472q^{59} - 1371q^{60} - 460q^{61} - 2014q^{62} + 477q^{63} - 451q^{64} - 986q^{65} - 1788q^{66} + 1934q^{67} - 69q^{68} - 606q^{69} - 1028q^{70} - 1687q^{71} + 27q^{72} + 1980q^{73} - 2400q^{74} - 792q^{75} - 940q^{76} - 821q^{77} - 810q^{78} + 3319q^{79} - 2119q^{80} + 648q^{81} + 429q^{82} + 2057q^{83} - 2106q^{84} + 566q^{85} - 6690q^{86} + 342q^{87} + 1189q^{88} + 1668q^{89} + 261q^{90} + 2427q^{91} - 980q^{92} - 1614q^{93} + 332q^{94} + 2146q^{95} - 948q^{96} + 1956q^{97} - 2026q^{98} - 243q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 2 x^{7} - 49 x^{6} + 89 x^{5} + 648 x^{4} - 1023 x^{3} - 1476 x^{2} + 1940 x - 384\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 13 \)
\(\beta_{3}\)\(=\)\((\)\( -13 \nu^{7} + 34 \nu^{6} + 647 \nu^{5} - 1483 \nu^{4} - 9408 \nu^{3} + 16089 \nu^{2} + 36118 \nu - 20832 \)\()/2144\)
\(\beta_{4}\)\(=\)\((\)\( -18 \nu^{7} + 11 \nu^{6} + 901 \nu^{5} - 734 \nu^{4} - 12207 \nu^{3} + 12861 \nu^{2} + 28786 \nu - 33792 \)\()/2144\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{7} + 8 \nu^{6} + 369 \nu^{5} - 345 \nu^{4} - 5406 \nu^{3} + 3695 \nu^{2} + 15758 \nu - 6352 \)\()/536\)
\(\beta_{6}\)\(=\)\((\)\( -39 \nu^{7} + 102 \nu^{6} + 1941 \nu^{5} - 4449 \nu^{4} - 26080 \nu^{3} + 48267 \nu^{2} + 59042 \nu - 62496 \)\()/2144\)
\(\beta_{7}\)\(=\)\((\)\( 57 \nu^{7} - 46 \nu^{6} - 2775 \nu^{5} + 1967 \nu^{4} + 36612 \nu^{3} - 20593 \nu^{2} - 89302 \nu + 29824 \)\()/2144\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 13\)
\(\nu^{3}\)\(=\)\(\beta_{6} - 3 \beta_{3} + 23 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-2 \beta_{7} + 2 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 30 \beta_{2} - \beta_{1} + 296\)
\(\nu^{5}\)\(=\)\(8 \beta_{7} + 30 \beta_{6} + 20 \beta_{5} - 98 \beta_{3} - \beta_{2} + 570 \beta_{1} + 35\)
\(\nu^{6}\)\(=\)\(-72 \beta_{7} + 27 \beta_{6} + 76 \beta_{5} - 352 \beta_{4} - 73 \beta_{3} + 836 \beta_{2} - 21 \beta_{1} + 7300\)
\(\nu^{7}\)\(=\)\(438 \beta_{7} + 840 \beta_{6} + 966 \beta_{5} - 8 \beta_{4} - 2834 \beta_{3} - 48 \beta_{2} + 14561 \beta_{1} + 1554\)

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
−5.19624
−4.21744
−1.67303
0.254436
0.780043
2.17127
4.61734
5.26363
−5.19624 −3.00000 19.0009 −8.30102 15.5887 21.5539 −57.1634 9.00000 43.1341
1.2 −4.21744 −3.00000 9.78684 17.5635 12.6523 14.8996 −7.53588 9.00000 −74.0731
1.3 −1.67303 −3.00000 −5.20096 −6.76323 5.01910 −19.0526 22.0856 9.00000 11.3151
1.4 0.254436 −3.00000 −7.93526 −10.8225 −0.763309 −23.2950 −4.05451 9.00000 −2.75362
1.5 0.780043 −3.00000 −7.39153 −21.9196 −2.34013 32.1153 −12.0061 9.00000 −17.0983
1.6 2.17127 −3.00000 −3.28558 9.58086 −6.51382 14.1591 −24.5041 9.00000 20.8027
1.7 4.61734 −3.00000 13.3198 −3.21787 −13.8520 15.9864 24.5634 9.00000 −14.8580
1.8 5.26363 −3.00000 19.7058 11.8799 −15.7909 −3.36662 61.6149 9.00000 62.5311
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
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 177.4.a.c 8
3.b odd 2 1 531.4.a.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.c 8 1.a even 1 1 trivial
531.4.a.f 8 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{2}^{8} - \cdots\) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -384 + 1940 T - 1476 T^{2} - 1023 T^{3} + 648 T^{4} + 89 T^{5} - 49 T^{6} - 2 T^{7} + T^{8} \)
$3$ \( ( 3 + T )^{8} \)
$5$ \( -85672464 - 40405196 T - 2469078 T^{2} + 861524 T^{3} + 80277 T^{4} - 5716 T^{5} - 560 T^{6} + 12 T^{7} + T^{8} \)
$7$ \( -3488279296 - 399824208 T + 167815056 T^{2} - 8549781 T^{3} - 516133 T^{4} + 45841 T^{5} - 141 T^{6} - 53 T^{7} + T^{8} \)
$11$ \( -223404635568 - 74917280804 T - 1963574268 T^{2} + 255710059 T^{3} + 9043745 T^{4} - 148898 T^{5} - 5942 T^{6} + 27 T^{7} + T^{8} \)
$13$ \( -5466828930044 + 441869764290 T + 5449126378 T^{2} - 909465641 T^{3} + 5431885 T^{4} + 519680 T^{5} - 5244 T^{6} - 89 T^{7} + T^{8} \)
$17$ \( 34686030744 + 821633910388 T - 18768055686 T^{2} - 1716992763 T^{3} + 27757319 T^{4} + 771505 T^{5} - 10801 T^{6} - 79 T^{7} + T^{8} \)
$19$ \( 612675702830144 - 91813882891136 T + 2714482097692 T^{2} - 12538959788 T^{3} - 466181345 T^{4} + 5561678 T^{5} + 3875 T^{6} - 288 T^{7} + T^{8} \)
$23$ \( -96081715816128 - 9355764769776 T + 383069668344 T^{2} + 12488762998 T^{3} - 608438203 T^{4} + 7421234 T^{5} - 18459 T^{6} - 202 T^{7} + T^{8} \)
$29$ \( -238656589193400 - 31895285211880 T - 729683982206 T^{2} + 29947346050 T^{3} + 450175903 T^{4} - 4361778 T^{5} - 52705 T^{6} + 114 T^{7} + T^{8} \)
$31$ \( 373989746307029344 - 14671727528641988 T + 177146746026986 T^{2} - 394299187352 T^{3} - 6572943625 T^{4} + 40216226 T^{5} + 9513 T^{6} - 538 T^{7} + T^{8} \)
$37$ \( -17609718565325908 - 6802879059905102 T - 545645679592918 T^{2} - 2581093421219 T^{3} + 17078307635 T^{4} + 66279559 T^{5} - 212175 T^{6} - 395 T^{7} + T^{8} \)
$41$ \( 3467857828629798168 - 54596984145333748 T - 1094937506543542 T^{2} + 1329506773555 T^{3} + 36119965003 T^{4} - 12235701 T^{5} - 343149 T^{6} + 39 T^{7} + T^{8} \)
$43$ \( -95383866842213663104 + 769097138033222688 T + 1891831228827066 T^{2} - 26244588674487 T^{3} + 21254212085 T^{4} + 219020278 T^{5} - 345408 T^{6} - 527 T^{7} + T^{8} \)
$47$ \( -61510955775947758848 + 14995869117559264 T + 4466702502723240 T^{2} - 10608638398880 T^{3} - 83816564919 T^{4} + 369668772 T^{5} - 213287 T^{6} - 860 T^{7} + T^{8} \)
$53$ \( 2415145722211600416 + 80808123212097596 T + 886827150577794 T^{2} + 2270325034168 T^{3} - 19636881903 T^{4} - 107112256 T^{5} + 29228 T^{6} + 812 T^{7} + T^{8} \)
$59$ \( ( 59 + T )^{8} \)
$61$ \( \)\(12\!\cdots\!88\)\( - 4148264243603022992 T - 27517213686627650 T^{2} + 72787451003566 T^{3} + 240158206629 T^{4} - 377447970 T^{5} - 909985 T^{6} + 460 T^{7} + T^{8} \)
$67$ \( -7430708718423391936 - 110334409032439696 T + 783050030355816 T^{2} + 9528145887250 T^{3} - 49162745095 T^{4} - 164546676 T^{5} + 1170254 T^{6} - 1934 T^{7} + T^{8} \)
$71$ \( -90426436968280392288 + 571114471027464574 T + 15561798577343672 T^{2} + 29934011829203 T^{3} - 245163194671 T^{4} - 617779352 T^{5} + 331910 T^{6} + 1687 T^{7} + T^{8} \)
$73$ \( \)\(70\!\cdots\!32\)\( + 9801783114216604168 T + 21707139357947164 T^{2} - 118754585627498 T^{3} - 273813192733 T^{4} + 691676814 T^{5} + 653565 T^{6} - 1980 T^{7} + T^{8} \)
$79$ \( \)\(47\!\cdots\!16\)\( - \)\(21\!\cdots\!52\)\( T - 30273778868140660 T^{2} + 1580910686454401 T^{3} - 2783070666087 T^{4} + 396292006 T^{5} + 3319402 T^{6} - 3319 T^{7} + T^{8} \)
$83$ \( \)\(11\!\cdots\!52\)\( + \)\(16\!\cdots\!52\)\( T - 316851294553911156 T^{2} - 1753324230030757 T^{3} + 626066347589 T^{4} + 3409775871 T^{5} - 1145959 T^{6} - 2057 T^{7} + T^{8} \)
$89$ \( -\)\(59\!\cdots\!44\)\( - 53671090805958449312 T + 2286036609406369528 T^{2} - 5387443498831848 T^{3} + 1080354579879 T^{4} + 6020684798 T^{5} - 3091237 T^{6} - 1668 T^{7} + T^{8} \)
$97$ \( \)\(11\!\cdots\!68\)\( + \)\(47\!\cdots\!72\)\( T - 1418207248608454620 T^{2} - 5565204479475238 T^{3} + 3092278816143 T^{4} + 7008888906 T^{5} - 3509564 T^{6} - 1956 T^{7} + T^{8} \)
show more
show less