Properties

Label 177.4.a.b
Level $177$
Weight $4$
Character orbit 177.a
Self dual yes
Analytic conductor $10.443$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44\)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -8 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( -2 - \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q -\beta_{1} q^{2} -3 q^{3} + ( 4 + \beta_{2} ) q^{4} + ( -1 + \beta_{1} - \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} ) q^{5} + 3 \beta_{1} q^{6} + ( -8 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} + 2 \beta_{5} ) q^{7} + ( -2 - \beta_{1} + 3 \beta_{3} - 3 \beta_{4} + \beta_{5} + \beta_{6} ) q^{8} + 9 q^{9} + ( -12 + 4 \beta_{1} - 3 \beta_{2} - \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{10} + ( 9 \beta_{1} + 2 \beta_{2} - \beta_{4} + 3 \beta_{5} - 3 \beta_{6} ) q^{11} + ( -12 - 3 \beta_{2} ) q^{12} + ( -10 - 2 \beta_{1} + 2 \beta_{3} + \beta_{4} - 5 \beta_{5} + 4 \beta_{6} ) q^{13} + ( -9 + 14 \beta_{1} + 2 \beta_{2} - 3 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} ) q^{14} + ( 3 - 3 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} ) q^{15} + ( -14 + 9 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} - 4 \beta_{5} + 4 \beta_{6} ) q^{16} + ( -1 + 7 \beta_{1} + 4 \beta_{2} - 4 \beta_{4} - 5 \beta_{5} + 7 \beta_{6} ) q^{17} -9 \beta_{1} q^{18} + ( -28 + 8 \beta_{1} + \beta_{2} - 5 \beta_{3} + 2 \beta_{4} - 3 \beta_{5} - 7 \beta_{6} ) q^{19} + ( -32 + 19 \beta_{1} - 3 \beta_{3} + 7 \beta_{4} + \beta_{5} - 6 \beta_{6} ) q^{20} + ( 24 - 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 3 \beta_{4} - 6 \beta_{5} ) q^{21} + ( -103 - 10 \beta_{1} - 8 \beta_{2} + 6 \beta_{3} + 9 \beta_{4} + 3 \beta_{5} + \beta_{6} ) q^{22} + ( -26 - 12 \beta_{1} + 11 \beta_{2} + 15 \beta_{3} + 2 \beta_{4} + 13 \beta_{5} - 5 \beta_{6} ) q^{23} + ( 6 + 3 \beta_{1} - 9 \beta_{3} + 9 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{24} + ( -26 - 16 \beta_{1} + \beta_{2} - 3 \beta_{3} - 7 \beta_{4} - 7 \beta_{5} + 3 \beta_{6} ) q^{25} + ( 9 + 17 \beta_{1} - 10 \beta_{3} - 19 \beta_{4} - 7 \beta_{5} ) q^{26} -27 q^{27} + ( -137 - 20 \beta_{1} - 10 \beta_{2} + 14 \beta_{3} + \beta_{4} + 7 \beta_{5} - 7 \beta_{6} ) q^{28} + ( 31 - 15 \beta_{1} + 8 \beta_{2} + 8 \beta_{3} - 19 \beta_{4} + 6 \beta_{5} + \beta_{6} ) q^{29} + ( 36 - 12 \beta_{1} + 9 \beta_{2} + 3 \beta_{3} - 3 \beta_{4} + 3 \beta_{5} + 6 \beta_{6} ) q^{30} + ( -85 - 9 \beta_{1} + 2 \beta_{2} - 8 \beta_{3} + 13 \beta_{4} + 18 \beta_{5} - 9 \beta_{6} ) q^{31} + ( -68 + 8 \beta_{1} - 9 \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - 6 \beta_{5} + 2 \beta_{6} ) q^{32} + ( -27 \beta_{1} - 6 \beta_{2} + 3 \beta_{4} - 9 \beta_{5} + 9 \beta_{6} ) q^{33} + ( -77 - 21 \beta_{1} - \beta_{2} + 16 \beta_{3} - 45 \beta_{4} - 5 \beta_{5} + 19 \beta_{6} ) q^{34} + ( -35 - 24 \beta_{1} + 6 \beta_{2} + 14 \beta_{3} - 15 \beta_{4} + 12 \beta_{5} + 20 \beta_{6} ) q^{35} + ( 36 + 9 \beta_{2} ) q^{36} + ( -71 - 14 \beta_{1} + 2 \beta_{2} - 14 \beta_{3} + 18 \beta_{4} + 17 \beta_{5} - 6 \beta_{6} ) q^{37} + ( -78 + 18 \beta_{1} - 20 \beta_{2} + 3 \beta_{3} + 17 \beta_{4} - 13 \beta_{5} - 5 \beta_{6} ) q^{38} + ( 30 + 6 \beta_{1} - 6 \beta_{3} - 3 \beta_{4} + 15 \beta_{5} - 12 \beta_{6} ) q^{39} + ( -154 - 4 \beta_{1} - 7 \beta_{2} + 10 \beta_{3} + 14 \beta_{4} + 16 \beta_{5} - \beta_{6} ) q^{40} + ( -50 - 29 \beta_{1} - 11 \beta_{2} + 15 \beta_{3} + 39 \beta_{4} - 14 \beta_{5} + 2 \beta_{6} ) q^{41} + ( 27 - 42 \beta_{1} - 6 \beta_{2} + 9 \beta_{3} - 18 \beta_{4} - 6 \beta_{5} ) q^{42} + ( -68 - 43 \beta_{1} + 8 \beta_{2} + 12 \beta_{3} - 25 \beta_{4} - 7 \beta_{5} - 5 \beta_{6} ) q^{43} + ( 59 + 85 \beta_{1} - 11 \beta_{2} - 40 \beta_{3} + 37 \beta_{4} - 9 \beta_{5} - 7 \beta_{6} ) q^{44} + ( -9 + 9 \beta_{1} - 9 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} ) q^{45} + ( 62 + 8 \beta_{1} + 18 \beta_{2} - 11 \beta_{3} + 15 \beta_{4} + 29 \beta_{5} - 13 \beta_{6} ) q^{46} + ( -53 + 24 \beta_{1} - 40 \beta_{2} - 34 \beta_{3} - 21 \beta_{4} - 20 \beta_{5} - 14 \beta_{6} ) q^{47} + ( 42 - 27 \beta_{1} + 6 \beta_{2} + 24 \beta_{3} + 12 \beta_{5} - 12 \beta_{6} ) q^{48} + ( 90 + 21 \beta_{1} - 2 \beta_{2} - 50 \beta_{3} + 8 \beta_{4} - 53 \beta_{5} - 11 \beta_{6} ) q^{49} + ( 242 + 16 \beta_{1} + 19 \beta_{2} + 7 \beta_{3} - 25 \beta_{4} - 25 \beta_{5} + 21 \beta_{6} ) q^{50} + ( 3 - 21 \beta_{1} - 12 \beta_{2} + 12 \beta_{4} + 15 \beta_{5} - 21 \beta_{6} ) q^{51} + ( 27 - 16 \beta_{1} - 5 \beta_{2} + 10 \beta_{3} - 25 \beta_{4} - 9 \beta_{5} + 16 \beta_{6} ) q^{52} + ( 101 - 43 \beta_{1} - 11 \beta_{2} - 19 \beta_{3} + 19 \beta_{4} + 13 \beta_{5} + 18 \beta_{6} ) q^{53} + 27 \beta_{1} q^{54} + ( 3 - 46 \beta_{1} + 41 \beta_{2} + 13 \beta_{3} - 7 \beta_{4} - \beta_{5} + 25 \beta_{6} ) q^{55} + ( 291 + 117 \beta_{1} + 3 \beta_{2} - 62 \beta_{3} + 31 \beta_{4} - 31 \beta_{5} - 33 \beta_{6} ) q^{56} + ( 84 - 24 \beta_{1} - 3 \beta_{2} + 15 \beta_{3} - 6 \beta_{4} + 9 \beta_{5} + 21 \beta_{6} ) q^{57} + ( 246 - 54 \beta_{1} + 41 \beta_{2} + 6 \beta_{3} - 14 \beta_{4} - 18 \beta_{5} + 39 \beta_{6} ) q^{58} + 59 q^{59} + ( 96 - 57 \beta_{1} + 9 \beta_{3} - 21 \beta_{4} - 3 \beta_{5} + 18 \beta_{6} ) q^{60} + ( -135 - 85 \beta_{1} - 4 \beta_{2} + 52 \beta_{3} + 73 \beta_{4} + 6 \beta_{5} - 19 \beta_{6} ) q^{61} + ( 50 + 42 \beta_{1} + 5 \beta_{2} + 56 \beta_{3} + 40 \beta_{4} + 72 \beta_{5} - 25 \beta_{6} ) q^{62} + ( -72 + 9 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} - 9 \beta_{4} + 18 \beta_{5} ) q^{63} + ( 18 + 51 \beta_{1} + 2 \beta_{2} + 21 \beta_{3} + 15 \beta_{4} + 11 \beta_{5} - 45 \beta_{6} ) q^{64} + ( 67 + 24 \beta_{1} - \beta_{2} + 9 \beta_{3} + 3 \beta_{4} + 29 \beta_{5} - 37 \beta_{6} ) q^{65} + ( 309 + 30 \beta_{1} + 24 \beta_{2} - 18 \beta_{3} - 27 \beta_{4} - 9 \beta_{5} - 3 \beta_{6} ) q^{66} + ( -205 - 42 \beta_{1} - 9 \beta_{2} + 15 \beta_{3} - 47 \beta_{4} - 23 \beta_{5} + 43 \beta_{6} ) q^{67} + ( 451 + 60 \beta_{1} + 48 \beta_{2} - 39 \beta_{3} - 30 \beta_{4} - 44 \beta_{5} + 36 \beta_{6} ) q^{68} + ( 78 + 36 \beta_{1} - 33 \beta_{2} - 45 \beta_{3} - 6 \beta_{4} - 39 \beta_{5} + 15 \beta_{6} ) q^{69} + ( 272 + 15 \beta_{1} + 71 \beta_{2} + 26 \beta_{3} - 72 \beta_{4} + 38 \beta_{5} + 42 \beta_{6} ) q^{70} + ( -170 - 12 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} - 47 \beta_{4} + 13 \beta_{5} + 16 \beta_{6} ) q^{71} + ( -18 - 9 \beta_{1} + 27 \beta_{3} - 27 \beta_{4} + 9 \beta_{5} + 9 \beta_{6} ) q^{72} + ( -115 + 74 \beta_{1} - 4 \beta_{2} - 58 \beta_{3} - 29 \beta_{4} - 74 \beta_{5} - 6 \beta_{6} ) q^{73} + ( 93 + 8 \beta_{1} + 7 \beta_{2} + 84 \beta_{3} + 21 \beta_{4} + 91 \beta_{5} - 26 \beta_{6} ) q^{74} + ( 78 + 48 \beta_{1} - 3 \beta_{2} + 9 \beta_{3} + 21 \beta_{4} + 21 \beta_{5} - 9 \beta_{6} ) q^{75} + ( -40 + 141 \beta_{1} - 61 \beta_{2} - 68 \beta_{3} + 54 \beta_{4} - 14 \beta_{5} - 6 \beta_{6} ) q^{76} + ( 164 - 121 \beta_{1} + 3 \beta_{2} + 55 \beta_{3} - 85 \beta_{4} - 2 \beta_{5} - 24 \beta_{6} ) q^{77} + ( -27 - 51 \beta_{1} + 30 \beta_{3} + 57 \beta_{4} + 21 \beta_{5} ) q^{78} + ( -376 + 33 \beta_{1} + 32 \beta_{2} + 80 \beta_{3} + 27 \beta_{4} + 35 \beta_{5} + \beta_{6} ) q^{79} + ( 190 + 52 \beta_{1} + 5 \beta_{2} - 7 \beta_{3} - 5 \beta_{4} + 49 \beta_{5} + 2 \beta_{6} ) q^{80} + 81 q^{81} + ( 101 + 162 \beta_{1} - 22 \beta_{2} - 117 \beta_{3} + 26 \beta_{4} + 14 \beta_{5} - 102 \beta_{6} ) q^{82} + ( -280 + 51 \beta_{1} - 33 \beta_{2} - 7 \beta_{3} - 15 \beta_{4} - 14 \beta_{5} + 72 \beta_{6} ) q^{83} + ( 411 + 60 \beta_{1} + 30 \beta_{2} - 42 \beta_{3} - 3 \beta_{4} - 21 \beta_{5} + 21 \beta_{6} ) q^{84} + ( -178 + 132 \beta_{1} + 5 \beta_{2} + 5 \beta_{3} + 54 \beta_{4} + 89 \beta_{5} - 53 \beta_{6} ) q^{85} + ( 631 + 76 \beta_{1} + 56 \beta_{2} - 48 \beta_{3} + \beta_{4} - 85 \beta_{5} + 41 \beta_{6} ) q^{86} + ( -93 + 45 \beta_{1} - 24 \beta_{2} - 24 \beta_{3} + 57 \beta_{4} - 18 \beta_{5} - 3 \beta_{6} ) q^{87} + ( -253 - 28 \beta_{1} - 74 \beta_{2} + 47 \beta_{3} - 60 \beta_{4} + 38 \beta_{5} - 60 \beta_{6} ) q^{88} + ( 83 + 82 \beta_{1} + 14 \beta_{2} + 14 \beta_{3} - 13 \beta_{4} + 16 \beta_{5} - 54 \beta_{6} ) q^{89} + ( -108 + 36 \beta_{1} - 27 \beta_{2} - 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - 18 \beta_{6} ) q^{90} + ( -352 + 7 \beta_{1} + 7 \beta_{2} - 25 \beta_{3} + 67 \beta_{4} - 12 \beta_{5} + 14 \beta_{6} ) q^{91} + ( 16 - 105 \beta_{1} - 95 \beta_{2} + 10 \beta_{3} + 16 \beta_{5} + 26 \beta_{6} ) q^{92} + ( 255 + 27 \beta_{1} - 6 \beta_{2} + 24 \beta_{3} - 39 \beta_{4} - 54 \beta_{5} + 27 \beta_{6} ) q^{93} + ( 68 + 185 \beta_{1} - 37 \beta_{2} - 52 \beta_{3} + 122 \beta_{4} - 136 \beta_{5} + 22 \beta_{6} ) q^{94} + ( 374 - 154 \beta_{1} + 73 \beta_{2} + 11 \beta_{3} - 22 \beta_{4} - 45 \beta_{5} - 7 \beta_{6} ) q^{95} + ( 204 - 24 \beta_{1} + 27 \beta_{2} - 6 \beta_{3} - 6 \beta_{4} + 18 \beta_{5} - 6 \beta_{6} ) q^{96} + ( -195 + 82 \beta_{1} - 51 \beta_{2} - 111 \beta_{3} - 3 \beta_{4} - 19 \beta_{5} + 35 \beta_{6} ) q^{97} + ( -71 - 166 \beta_{1} - 93 \beta_{2} + 66 \beta_{3} - 53 \beta_{4} - 117 \beta_{5} + 21 \beta_{6} ) q^{98} + ( 81 \beta_{1} + 18 \beta_{2} - 9 \beta_{4} + 27 \beta_{5} - 27 \beta_{6} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 21q^{3} + 26q^{4} - 2q^{5} - 59q^{7} - 21q^{8} + 63q^{9} - 71q^{10} - 5q^{11} - 78q^{12} - 67q^{13} - 65q^{14} + 6q^{15} - 94q^{16} - 23q^{17} - 176q^{19} - 207q^{20} + 177q^{21} - 704q^{22} - 218q^{23} + 63q^{24} - 183q^{25} + 58q^{26} - 189q^{27} - 938q^{28} + 168q^{29} + 213q^{30} - 604q^{31} - 448q^{32} + 15q^{33} - 610q^{34} - 336q^{35} + 234q^{36} - 505q^{37} - 453q^{38} + 201q^{39} - 1080q^{40} - 265q^{41} + 195q^{42} - 493q^{43} + 504q^{44} - 18q^{45} + 381q^{46} - 244q^{47} + 282q^{48} + 770q^{49} + 1639q^{50} + 69q^{51} + 160q^{52} + 686q^{53} - 116q^{55} + 2190q^{56} + 528q^{57} + 1584q^{58} + 413q^{59} + 621q^{60} - 838q^{61} + 286q^{62} - 531q^{63} + 205q^{64} + 490q^{65} + 2112q^{66} - 1504q^{67} + 3047q^{68} + 654q^{69} + 1530q^{70} - 1267q^{71} - 189q^{72} - 666q^{73} + 528q^{74} + 549q^{75} - 64q^{76} + 1109q^{77} - 174q^{78} - 2741q^{79} + 1213q^{80} + 567q^{81} + 953q^{82} - 2025q^{83} + 2814q^{84} - 1274q^{85} + 4394q^{86} - 504q^{87} - 1639q^{88} + 616q^{89} - 639q^{90} - 2415q^{91} + 218q^{92} + 1812q^{93} + 900q^{94} + 2554q^{95} + 1344q^{96} - 1298q^{97} - 172q^{98} - 45q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - 41 x^{5} - 7 x^{4} + 484 x^{3} + 63 x^{2} - 1736 x - 44\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 12 \)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{6} - 9 \nu^{5} + 24 \nu^{4} + 287 \nu^{3} + 51 \nu^{2} - 1588 \nu - 716 \)\()/128\)
\(\beta_{4}\)\(=\)\((\)\( -\nu^{6} - \nu^{5} + 32 \nu^{4} + 47 \nu^{3} - 197 \nu^{2} - 332 \nu + 4 \)\()/32\)
\(\beta_{5}\)\(=\)\((\)\( -7 \nu^{6} + 33 \nu^{5} + 232 \nu^{4} - 999 \nu^{3} - 1915 \nu^{2} + 6420 \nu + 2156 \)\()/256\)
\(\beta_{6}\)\(=\)\((\)\( -11 \nu^{6} - 3 \nu^{5} + 392 \nu^{4} + 149 \nu^{3} - 3119 \nu^{2} - 508 \nu + 2748 \)\()/256\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2} + 12\)
\(\nu^{3}\)\(=\)\(-\beta_{6} - \beta_{5} + 3 \beta_{4} - 3 \beta_{3} + 17 \beta_{1} + 2\)
\(\nu^{4}\)\(=\)\(4 \beta_{6} - 4 \beta_{5} - 8 \beta_{3} + 22 \beta_{2} + 9 \beta_{1} + 210\)
\(\nu^{5}\)\(=\)\(-34 \beta_{6} - 26 \beta_{5} + 94 \beta_{4} - 98 \beta_{3} + 9 \beta_{2} + 344 \beta_{1} + 132\)
\(\nu^{6}\)\(=\)\(115 \beta_{6} - 149 \beta_{5} + 15 \beta_{4} - 299 \beta_{3} + 498 \beta_{2} + 411 \beta_{1} + 4322\)

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
5.07078
3.06012
2.61892
−0.0253269
−2.74916
−3.39497
−4.58037
−5.07078 −3.00000 17.7128 5.77165 15.2123 −31.1296 −49.2517 9.00000 −29.2668
1.2 −3.06012 −3.00000 1.36432 0.675250 9.18035 −3.38755 20.3060 9.00000 −2.06635
1.3 −2.61892 −3.00000 −1.14126 −7.96684 7.85676 16.8751 23.9402 9.00000 20.8645
1.4 0.0253269 −3.00000 −7.99936 8.85515 −0.0759806 13.8749 −0.405214 9.00000 0.224273
1.5 2.74916 −3.00000 −0.442134 13.7745 −8.24747 −35.6462 −23.2088 9.00000 37.8683
1.6 3.39497 −3.00000 3.52580 −6.09684 −10.1849 1.40309 −15.1898 9.00000 −20.6986
1.7 4.58037 −3.00000 12.9798 −17.0129 −13.7411 −20.9898 22.8093 9.00000 −77.9254
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
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.b 7
3.b odd 2 1 531.4.a.c 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.b 7 1.a even 1 1 trivial
531.4.a.c 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} - 41 T_{2}^{5} + 7 T_{2}^{4} + 484 T_{2}^{3} - 63 T_{2}^{2} - 1736 T_{2} + 44 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 44 - 1736 T - 63 T^{2} + 484 T^{3} + 7 T^{4} - 41 T^{5} + T^{7} \)
$3$ \( ( 3 + T )^{7} \)
$5$ \( 392832 - 585870 T - 12382 T^{2} + 27493 T^{3} + 28 T^{4} - 344 T^{5} + 2 T^{6} + T^{7} \)
$7$ \( -25920400 + 11431608 T + 5357659 T^{2} - 194685 T^{3} - 33559 T^{4} + 155 T^{5} + 59 T^{6} + T^{7} \)
$11$ \( -1989231628 + 1117706696 T + 222471693 T^{2} + 7482209 T^{3} - 97942 T^{4} - 5562 T^{5} + 5 T^{6} + T^{7} \)
$13$ \( -31480216830 - 1084049856 T + 205800691 T^{2} + 6203733 T^{3} - 295736 T^{4} - 5142 T^{5} + 67 T^{6} + T^{7} \)
$17$ \( -2416315770100 - 144632302612 T + 3862461779 T^{2} + 141472241 T^{3} - 533425 T^{4} - 23183 T^{5} + 23 T^{6} + T^{7} \)
$19$ \( -4741294131040 + 93764326076 T + 6441225308 T^{2} - 34162417 T^{3} - 2327842 T^{4} - 7173 T^{5} + 176 T^{6} + T^{7} \)
$23$ \( 3139469662768 + 546080146560 T + 23064610146 T^{2} - 52827111 T^{3} - 5323854 T^{4} - 18195 T^{5} + 218 T^{6} + T^{7} \)
$29$ \( -6671473883660 + 715774991038 T + 48732024116 T^{2} + 823172155 T^{3} + 1610880 T^{4} - 53677 T^{5} - 168 T^{6} + T^{7} \)
$31$ \( 161706929046080 + 3422990881634 T - 409110684710 T^{2} - 7073947977 T^{3} - 29985840 T^{4} + 54777 T^{5} + 604 T^{6} + T^{7} \)
$37$ \( -6072074830482974 + 79320227242320 T + 1023966709053 T^{2} - 4364011463 T^{3} - 56170389 T^{4} - 52013 T^{5} + 505 T^{6} + T^{7} \)
$41$ \( -50374936284264380 - 448270174458332 T + 6373895568029 T^{2} + 28780360621 T^{3} - 83812483 T^{4} - 336115 T^{5} + 265 T^{6} + T^{7} \)
$43$ \( -33913476387070128 + 33186504122760 T + 3924236332393 T^{2} + 6777926869 T^{3} - 86762774 T^{4} - 171214 T^{5} + 493 T^{6} + T^{7} \)
$47$ \( -1160151063849624384 - 3943903165571160 T + 23590276456540 T^{2} + 83605511557 T^{3} - 137872464 T^{4} - 521387 T^{5} + 244 T^{6} + T^{7} \)
$53$ \( 262371725884790104 - 614257571452974 T - 13409985392458 T^{2} + 23767807785 T^{3} + 181919104 T^{4} - 259804 T^{5} - 686 T^{6} + T^{7} \)
$59$ \( ( -59 + T )^{7} \)
$61$ \( -56018457114922274852 - 68219570280244910 T + 441270620215484 T^{2} + 526741153813 T^{3} - 1091233996 T^{4} - 1299901 T^{5} + 838 T^{6} + T^{7} \)
$67$ \( 458100199887257248 + 1885672833423416 T - 43078004484032 T^{2} - 308643298035 T^{3} - 569860168 T^{4} + 248370 T^{5} + 1504 T^{6} + T^{7} \)
$71$ \( -302257493806042742 + 1515434826117112 T + 11545600692447 T^{2} - 41378621991 T^{3} - 187475564 T^{4} + 234558 T^{5} + 1267 T^{6} + T^{7} \)
$73$ \( 712065321253374176 - 20212977040789028 T + 84789173687860 T^{2} + 344920694751 T^{3} - 511602404 T^{4} - 1203467 T^{5} + 666 T^{6} + T^{7} \)
$79$ \( -15688795466953353600 - 217660653935880624 T - 1011835955959003 T^{2} - 1922828974135 T^{3} - 838622730 T^{4} + 2016150 T^{5} + 2741 T^{6} + T^{7} \)
$83$ \( -45339559236848304 + 2541516204749880 T + 2876508233789 T^{2} - 614118278731 T^{3} - 1345684463 T^{4} + 267037 T^{5} + 2025 T^{6} + T^{7} \)
$89$ \( -42024068695984000 + 6583090481312280 T - 70534738277044 T^{2} + 156068554731 T^{3} + 392520102 T^{4} - 879761 T^{5} - 616 T^{6} + T^{7} \)
$97$ \( 61895303469400133648 + 528245527399525820 T + 1273155523159212 T^{2} + 75391709063 T^{3} - 2801122726 T^{4} - 1816092 T^{5} + 1298 T^{6} + T^{7} \)
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