Properties

Label 177.4.a.a
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: \(x^{7} - x^{6} - 34 x^{5} + 25 x^{4} + 315 x^{3} - 146 x^{2} - 736 x + 512\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
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.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{4} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} + ( -3 - 3 \beta_{1} ) q^{6} + ( -8 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( -15 - 4 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{8} + 9 q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} ) q^{2} + 3 q^{3} + ( 3 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} ) q^{4} + ( -5 + 2 \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{5} + ( -3 - 3 \beta_{1} ) q^{6} + ( -8 + 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} + \beta_{5} + \beta_{6} ) q^{7} + ( -15 - 4 \beta_{1} + 4 \beta_{2} - 7 \beta_{3} - 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{8} + 9 q^{9} + ( -15 + 5 \beta_{1} + 9 \beta_{2} - \beta_{3} - 3 \beta_{4} - 5 \beta_{5} - 5 \beta_{6} ) q^{10} + ( -15 - 10 \beta_{1} - \beta_{2} - 5 \beta_{3} - 4 \beta_{4} - \beta_{5} ) q^{11} + ( 9 + 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{12} + ( -16 + 5 \beta_{1} - \beta_{2} - 2 \beta_{3} + 2 \beta_{4} + 8 \beta_{5} + 3 \beta_{6} ) q^{13} + ( -17 + 12 \beta_{1} - 7 \beta_{2} + 2 \beta_{3} + 9 \beta_{4} + 3 \beta_{6} ) q^{14} + ( -15 + 6 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - 3 \beta_{6} ) q^{15} + ( 23 + 25 \beta_{1} - 14 \beta_{2} + 14 \beta_{3} + 12 \beta_{4} + 6 \beta_{5} + 4 \beta_{6} ) q^{16} + ( -48 + 4 \beta_{1} + 10 \beta_{2} + 20 \beta_{3} - 5 \beta_{4} - 13 \beta_{5} - 9 \beta_{6} ) q^{17} + ( -9 - 9 \beta_{1} ) q^{18} + ( -5 - 5 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} - 12 \beta_{4} - 10 \beta_{5} + 9 \beta_{6} ) q^{19} + ( 11 + 20 \beta_{1} - 18 \beta_{2} + 2 \beta_{3} + 19 \beta_{4} + 18 \beta_{5} + 8 \beta_{6} ) q^{20} + ( -24 + 9 \beta_{1} + 9 \beta_{2} - 6 \beta_{3} + 3 \beta_{5} + 3 \beta_{6} ) q^{21} + ( 99 + 23 \beta_{1} - 19 \beta_{2} + 11 \beta_{3} - 2 \beta_{4} + 19 \beta_{5} + 14 \beta_{6} ) q^{22} + ( -41 - 15 \beta_{1} - 2 \beta_{2} - 9 \beta_{3} - 6 \beta_{4} + 14 \beta_{5} - 19 \beta_{6} ) q^{23} + ( -45 - 12 \beta_{1} + 12 \beta_{2} - 21 \beta_{3} - 9 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} ) q^{24} + ( 39 - 21 \beta_{1} - 11 \beta_{2} - 2 \beta_{3} - 9 \beta_{4} + 18 \beta_{5} + 14 \beta_{6} ) q^{25} + ( -26 + 11 \beta_{2} - 4 \beta_{3} - 8 \beta_{4} - 20 \beta_{5} - \beta_{6} ) q^{26} + 27 q^{27} + ( -33 - 13 \beta_{1} + 33 \beta_{2} + 5 \beta_{3} - 18 \beta_{4} - 39 \beta_{5} - 26 \beta_{6} ) q^{28} + ( -58 - 7 \beta_{1} - 5 \beta_{2} + 28 \beta_{3} + 21 \beta_{4} + 3 \beta_{5} + 24 \beta_{6} ) q^{29} + ( -45 + 15 \beta_{1} + 27 \beta_{2} - 3 \beta_{3} - 9 \beta_{4} - 15 \beta_{5} - 15 \beta_{6} ) q^{30} + ( -34 - 3 \beta_{1} - 9 \beta_{2} - 16 \beta_{3} + 19 \beta_{4} - \beta_{5} + 16 \beta_{6} ) q^{31} + ( -137 - 52 \beta_{1} + 67 \beta_{2} + 5 \beta_{3} - 20 \beta_{4} - 43 \beta_{5} - 25 \beta_{6} ) q^{32} + ( -45 - 30 \beta_{1} - 3 \beta_{2} - 15 \beta_{3} - 12 \beta_{4} - 3 \beta_{5} ) q^{33} + ( 25 + 44 \beta_{1} - 32 \beta_{2} - 26 \beta_{3} + 34 \beta_{4} + 24 \beta_{5} - 23 \beta_{6} ) q^{34} + ( -29 - 78 \beta_{1} - 23 \beta_{2} - 11 \beta_{3} - 19 \beta_{4} + 20 \beta_{5} + 29 \beta_{6} ) q^{35} + ( 27 + 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} ) q^{36} + ( -37 + 9 \beta_{1} + \beta_{3} + 17 \beta_{4} + 44 \beta_{5} - 30 \beta_{6} ) q^{37} + ( 10 - 17 \beta_{1} - 38 \beta_{2} - 35 \beta_{3} + 7 \beta_{4} + 63 \beta_{5} + 13 \beta_{6} ) q^{38} + ( -48 + 15 \beta_{1} - 3 \beta_{2} - 6 \beta_{3} + 6 \beta_{4} + 24 \beta_{5} + 9 \beta_{6} ) q^{39} + ( -69 - 98 \beta_{1} + 51 \beta_{2} - 12 \beta_{3} - 40 \beta_{4} - 56 \beta_{5} + 8 \beta_{6} ) q^{40} + ( -112 + 19 \beta_{1} - 33 \beta_{2} + 20 \beta_{3} + 42 \beta_{4} - 15 \beta_{5} + 29 \beta_{6} ) q^{41} + ( -51 + 36 \beta_{1} - 21 \beta_{2} + 6 \beta_{3} + 27 \beta_{4} + 9 \beta_{6} ) q^{42} + ( -117 - 26 \beta_{1} + 33 \beta_{2} + 7 \beta_{3} + 16 \beta_{4} - 9 \beta_{5} + 8 \beta_{6} ) q^{43} + ( -221 - 160 \beta_{1} + 68 \beta_{2} - 56 \beta_{3} - 30 \beta_{4} - 40 \beta_{5} - 39 \beta_{6} ) q^{44} + ( -45 + 18 \beta_{1} - 9 \beta_{2} + 9 \beta_{3} + 9 \beta_{4} - 9 \beta_{5} - 9 \beta_{6} ) q^{45} + ( 180 + 43 \beta_{1} - 8 \beta_{2} + 57 \beta_{3} - 39 \beta_{4} - 41 \beta_{5} + 3 \beta_{6} ) q^{46} + ( -167 - 14 \beta_{1} - \beta_{2} + 17 \beta_{3} - 7 \beta_{4} + 28 \beta_{5} + 21 \beta_{6} ) q^{47} + ( 69 + 75 \beta_{1} - 42 \beta_{2} + 42 \beta_{3} + 36 \beta_{4} + 18 \beta_{5} + 12 \beta_{6} ) q^{48} + ( 111 - 60 \beta_{1} - 16 \beta_{2} + 24 \beta_{3} - 25 \beta_{4} - 5 \beta_{5} - 47 \beta_{6} ) q^{49} + ( 147 - 113 \beta_{1} - 29 \beta_{2} - 32 \beta_{3} - 37 \beta_{4} + 20 \beta_{5} + 26 \beta_{6} ) q^{50} + ( -144 + 12 \beta_{1} + 30 \beta_{2} + 60 \beta_{3} - 15 \beta_{4} - 39 \beta_{5} - 27 \beta_{6} ) q^{51} + ( 136 + 33 \beta_{1} - 48 \beta_{2} + 21 \beta_{3} + 36 \beta_{4} + \beta_{5} - 10 \beta_{6} ) q^{52} + ( -167 + 58 \beta_{1} + \beta_{2} - 5 \beta_{3} - 5 \beta_{4} - 59 \beta_{5} - 75 \beta_{6} ) q^{53} + ( -27 - 27 \beta_{1} ) q^{54} + ( -174 + 53 \beta_{1} + 109 \beta_{2} - 20 \beta_{3} - \beta_{4} - 70 \beta_{5} - 24 \beta_{6} ) q^{55} + ( 295 + 56 \beta_{1} - 84 \beta_{2} + 36 \beta_{3} + 40 \beta_{4} + 108 \beta_{5} - 9 \beta_{6} ) q^{56} + ( -15 - 15 \beta_{1} - 12 \beta_{2} - 9 \beta_{3} - 36 \beta_{4} - 30 \beta_{5} + 27 \beta_{6} ) q^{57} + ( 191 + 32 \beta_{1} + 47 \beta_{2} - 60 \beta_{3} + 6 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} ) q^{58} -59 q^{59} + ( 33 + 60 \beta_{1} - 54 \beta_{2} + 6 \beta_{3} + 57 \beta_{4} + 54 \beta_{5} + 24 \beta_{6} ) q^{60} + ( 48 + 67 \beta_{1} - 19 \beta_{2} + 6 \beta_{3} + 23 \beta_{4} - 63 \beta_{5} - 6 \beta_{6} ) q^{61} + ( 67 + 94 \beta_{1} + 65 \beta_{2} + 32 \beta_{3} - 10 \beta_{4} - 10 \beta_{5} + 26 \beta_{6} ) q^{62} + ( -72 + 27 \beta_{1} + 27 \beta_{2} - 18 \beta_{3} + 9 \beta_{5} + 9 \beta_{6} ) q^{63} + ( 529 + 164 \beta_{1} - 176 \beta_{2} + 7 \beta_{3} + 123 \beta_{4} + 147 \beta_{5} + 57 \beta_{6} ) q^{64} + ( 44 - 3 \beta_{1} - 31 \beta_{2} + 36 \beta_{3} + \beta_{4} + 48 \beta_{5} + 64 \beta_{6} ) q^{65} + ( 297 + 69 \beta_{1} - 57 \beta_{2} + 33 \beta_{3} - 6 \beta_{4} + 57 \beta_{5} + 42 \beta_{6} ) q^{66} + ( 214 - 35 \beta_{1} + 33 \beta_{2} - 60 \beta_{3} - 69 \beta_{4} - 4 \beta_{5} ) q^{67} + ( -79 - 36 \beta_{1} + 185 \beta_{2} - 70 \beta_{3} - 103 \beta_{4} - 134 \beta_{5} - \beta_{6} ) q^{68} + ( -123 - 45 \beta_{1} - 6 \beta_{2} - 27 \beta_{3} - 18 \beta_{4} + 42 \beta_{5} - 57 \beta_{6} ) q^{69} + ( 734 - 43 \beta_{1} - 95 \beta_{2} - 19 \beta_{3} - 60 \beta_{4} + 111 \beta_{5} + 95 \beta_{6} ) q^{70} + ( 26 + 23 \beta_{1} - 27 \beta_{2} - 80 \beta_{3} + 38 \beta_{4} + 90 \beta_{5} + 37 \beta_{6} ) q^{71} + ( -135 - 36 \beta_{1} + 36 \beta_{2} - 63 \beta_{3} - 27 \beta_{4} - 27 \beta_{5} - 27 \beta_{6} ) q^{72} + ( -29 - 40 \beta_{1} + 59 \beta_{2} - 83 \beta_{3} + 39 \beta_{4} - 40 \beta_{5} - 5 \beta_{6} ) q^{73} + ( 26 + 19 \beta_{1} + 90 \beta_{2} + 83 \beta_{3} - 74 \beta_{4} - 191 \beta_{5} - 40 \beta_{6} ) q^{74} + ( 117 - 63 \beta_{1} - 33 \beta_{2} - 6 \beta_{3} - 27 \beta_{4} + 54 \beta_{5} + 42 \beta_{6} ) q^{75} + ( 166 - 77 \beta_{1} + 137 \beta_{2} + 61 \beta_{3} - 68 \beta_{4} - 55 \beta_{5} - 45 \beta_{6} ) q^{76} + ( -204 + 177 \beta_{1} - 45 \beta_{2} + 104 \beta_{3} + 98 \beta_{4} - 9 \beta_{5} - 77 \beta_{6} ) q^{77} + ( -78 + 33 \beta_{2} - 12 \beta_{3} - 24 \beta_{4} - 60 \beta_{5} - 3 \beta_{6} ) q^{78} + ( 381 - 124 \beta_{1} - \beta_{2} + 77 \beta_{3} - 10 \beta_{4} + 7 \beta_{5} + 46 \beta_{6} ) q^{79} + ( 915 + 117 \beta_{1} - 235 \beta_{2} + 61 \beta_{3} + 65 \beta_{4} + 213 \beta_{5} + 105 \beta_{6} ) q^{80} + 81 q^{81} + ( -55 + 114 \beta_{1} + 215 \beta_{2} - 66 \beta_{3} - 55 \beta_{4} - 48 \beta_{5} - 43 \beta_{6} ) q^{82} + ( 44 + 93 \beta_{1} + 35 \beta_{2} - 98 \beta_{3} - 34 \beta_{4} - 5 \beta_{5} - 105 \beta_{6} ) q^{83} + ( -99 - 39 \beta_{1} + 99 \beta_{2} + 15 \beta_{3} - 54 \beta_{4} - 117 \beta_{5} - 78 \beta_{6} ) q^{84} + ( 267 - 307 \beta_{1} + 20 \beta_{2} - 145 \beta_{3} - 22 \beta_{4} + 136 \beta_{5} - 41 \beta_{6} ) q^{85} + ( 473 + 253 \beta_{1} - 85 \beta_{2} + 61 \beta_{3} + 116 \beta_{4} + 61 \beta_{5} + 60 \beta_{6} ) q^{86} + ( -174 - 21 \beta_{1} - 15 \beta_{2} + 84 \beta_{3} + 63 \beta_{4} + 9 \beta_{5} + 72 \beta_{6} ) q^{87} + ( 1009 + 474 \beta_{1} - 263 \beta_{2} + 270 \beta_{3} + 221 \beta_{4} + 138 \beta_{5} + 133 \beta_{6} ) q^{88} + ( -33 + 18 \beta_{1} + 7 \beta_{2} + 147 \beta_{3} - 71 \beta_{4} - 250 \beta_{5} - 119 \beta_{6} ) q^{89} + ( -135 + 45 \beta_{1} + 81 \beta_{2} - 9 \beta_{3} - 27 \beta_{4} - 45 \beta_{5} - 45 \beta_{6} ) q^{90} + ( 422 - 135 \beta_{1} + 15 \beta_{2} - 26 \beta_{3} - 40 \beta_{4} - 103 \beta_{5} - 141 \beta_{6} ) q^{91} + ( -360 - 271 \beta_{1} - 37 \beta_{2} - 177 \beta_{3} + 68 \beta_{4} + 3 \beta_{5} + 47 \beta_{6} ) q^{92} + ( -102 - 9 \beta_{1} - 27 \beta_{2} - 48 \beta_{3} + 57 \beta_{4} - 3 \beta_{5} + 48 \beta_{6} ) q^{93} + ( 336 + 47 \beta_{1} - 53 \beta_{2} - 77 \beta_{3} - 10 \beta_{4} + 13 \beta_{5} + 17 \beta_{6} ) q^{94} + ( -269 + 93 \beta_{1} + 80 \beta_{2} - 195 \beta_{3} + 100 \beta_{4} - 4 \beta_{5} - 15 \beta_{6} ) q^{95} + ( -411 - 156 \beta_{1} + 201 \beta_{2} + 15 \beta_{3} - 60 \beta_{4} - 129 \beta_{5} - 75 \beta_{6} ) q^{96} + ( -194 - 191 \beta_{1} - 27 \beta_{2} - 74 \beta_{3} - 7 \beta_{4} - 134 \beta_{5} + 36 \beta_{6} ) q^{97} + ( 426 - 149 \beta_{1} - 40 \beta_{2} + 40 \beta_{3} - 90 \beta_{4} + 10 \beta_{5} - 27 \beta_{6} ) q^{98} + ( -135 - 90 \beta_{1} - 9 \beta_{2} - 45 \beta_{3} - 36 \beta_{4} - 9 \beta_{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q - 8q^{2} + 21q^{3} + 22q^{4} - 28q^{5} - 24q^{6} - 59q^{7} - 117q^{8} + 63q^{9} + O(q^{10}) \) \( 7q - 8q^{2} + 21q^{3} + 22q^{4} - 28q^{5} - 24q^{6} - 59q^{7} - 117q^{8} + 63q^{9} - 79q^{10} - 131q^{11} + 66q^{12} - 123q^{13} - 117q^{14} - 84q^{15} + 202q^{16} - 235q^{17} - 72q^{18} - 80q^{19} + 61q^{20} - 177q^{21} + 688q^{22} - 274q^{23} - 351q^{24} + 193q^{25} - 180q^{26} + 189q^{27} - 118q^{28} - 406q^{29} - 237q^{30} - 346q^{31} - 854q^{32} - 393q^{33} + 178q^{34} - 424q^{35} + 198q^{36} - 157q^{37} - 129q^{38} - 369q^{39} - 590q^{40} - 825q^{41} - 351q^{42} - 815q^{43} - 1690q^{44} - 252q^{45} + 1457q^{46} - 1196q^{47} + 606q^{48} + 914q^{49} + 713q^{50} - 705q^{51} + 1030q^{52} - 900q^{53} - 216q^{54} - 1044q^{55} + 2172q^{56} - 240q^{57} + 1242q^{58} - 413q^{59} + 183q^{60} + 420q^{61} + 646q^{62} - 531q^{63} + 3541q^{64} + 190q^{65} + 2064q^{66} + 1316q^{67} - 611q^{68} - 822q^{69} + 4658q^{70} - 173q^{71} - 1053q^{72} - 418q^{73} + 660q^{74} + 579q^{75} + 1540q^{76} - 753q^{77} - 540q^{78} + 2635q^{79} + 6155q^{80} + 567q^{81} - 125q^{82} + 457q^{83} - 354q^{84} + 1270q^{85} + 3482q^{86} - 1218q^{87} + 7685q^{88} + 592q^{89} - 711q^{90} + 3179q^{91} - 3500q^{92} - 1038q^{93} + 2064q^{94} - 2250q^{95} - 2562q^{96} - 1906q^{97} + 2994q^{98} - 1179q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{7} - x^{6} - 34 x^{5} + 25 x^{4} + 315 x^{3} - 146 x^{2} - 736 x + 512\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 27 \nu^{5} - 154 \nu^{4} - 783 \nu^{3} + 2919 \nu^{2} + 4790 \nu - 7944 \)\()/584\)
\(\beta_{3}\)\(=\)\((\)\( 7 \nu^{6} - 30 \nu^{5} - 202 \nu^{4} + 797 \nu^{3} + 1380 \nu^{2} - 3992 \nu - 1296 \)\()/584\)
\(\beta_{4}\)\(=\)\((\)\( -9 \nu^{6} + 49 \nu^{5} + 218 \nu^{4} - 1129 \nu^{3} - 867 \nu^{2} + 3610 \nu - 920 \)\()/584\)
\(\beta_{5}\)\(=\)\((\)\( 24 \nu^{6} - 9 \nu^{5} - 776 \nu^{4} + 42 \nu^{3} + 6473 \nu^{2} + 2102 \nu - 10784 \)\()/584\)
\(\beta_{6}\)\(=\)\((\)\( -15 \nu^{6} + 33 \nu^{5} + 412 \nu^{4} - 811 \nu^{3} - 2175 \nu^{2} + 3340 \nu - 852 \)\()/292\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} + \beta_{5} + \beta_{3} - \beta_{2} + 10\)
\(\nu^{3}\)\(=\)\(3 \beta_{4} + 4 \beta_{3} - \beta_{2} + 17 \beta_{1}\)
\(\nu^{4}\)\(=\)\(22 \beta_{6} + 24 \beta_{5} + 16 \beta_{3} - 28 \beta_{2} + \beta_{1} + 162\)
\(\nu^{5}\)\(=\)\(\beta_{6} + 9 \beta_{5} + 86 \beta_{4} + 89 \beta_{3} - 35 \beta_{2} + 320 \beta_{1} + 26\)
\(\nu^{6}\)\(=\)\(442 \beta_{6} + 534 \beta_{5} + 27 \beta_{4} + 274 \beta_{3} - 647 \beta_{2} + 35 \beta_{1} + 3000\)

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
4.60512
3.58179
1.46227
0.775001
−2.29817
−2.68175
−4.44426
−5.60512 3.00000 23.4174 12.9748 −16.8154 −18.2988 −86.4162 9.00000 −72.7251
1.2 −4.58179 3.00000 12.9928 −12.2965 −13.7454 15.3493 −22.8759 9.00000 56.3401
1.3 −2.46227 3.00000 −1.93724 −11.2702 −7.38680 18.6588 24.4681 9.00000 27.7502
1.4 −1.77500 3.00000 −4.84937 6.23028 −5.32500 −18.0779 22.8076 9.00000 −11.0588
1.5 1.29817 3.00000 −6.31476 7.05496 3.89451 −33.4497 −18.5830 9.00000 9.15853
1.6 1.68175 3.00000 −5.17172 −9.78761 5.04525 6.87552 −22.1515 9.00000 −16.4603
1.7 3.44426 3.00000 3.86292 −20.9057 10.3328 −30.0572 −14.2492 9.00000 −72.0046
\(n\): 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.a 7
3.b odd 2 1 531.4.a.d 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
177.4.a.a 7 1.a even 1 1 trivial
531.4.a.d 7 3.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{7} + 8 T_{2}^{6} - 7 T_{2}^{5} - 145 T_{2}^{4} - 70 T_{2}^{3} + 637 T_{2}^{2} + 244 T_{2} - 844 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(177))\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -844 + 244 T + 637 T^{2} - 70 T^{3} - 145 T^{4} - 7 T^{5} + 8 T^{6} + T^{7} \)
$3$ \( ( -3 + T )^{7} \)
$5$ \( -16171712 + 958432 T + 610440 T^{2} - 10195 T^{3} - 7764 T^{4} - 142 T^{5} + 28 T^{6} + T^{7} \)
$7$ \( -654921472 + 59635248 T + 9177475 T^{2} - 321309 T^{3} - 41503 T^{4} + 83 T^{5} + 59 T^{6} + T^{7} \)
$11$ \( 35342635856 + 2354403200 T - 141567805 T^{2} - 14373439 T^{3} - 293782 T^{4} + 2494 T^{5} + 131 T^{6} + T^{7} \)
$13$ \( 17075623568 + 2329203068 T + 32473719 T^{2} - 5662483 T^{3} - 174306 T^{4} + 2422 T^{5} + 123 T^{6} + T^{7} \)
$17$ \( 83907218653452 + 3294049526520 T + 29824194343 T^{2} - 314280439 T^{3} - 5906085 T^{4} - 8859 T^{5} + 235 T^{6} + T^{7} \)
$19$ \( -83214285665216 - 1427637821040 T + 24485828652 T^{2} + 387062779 T^{3} - 2417914 T^{4} - 34385 T^{5} + 80 T^{6} + T^{7} \)
$23$ \( -498329696312416 - 1259373688656 T + 147129916878 T^{2} + 337086933 T^{3} - 11579010 T^{4} - 33975 T^{5} + 274 T^{6} + T^{7} \)
$29$ \( 102487884628576 - 117096267464 T - 161861671722 T^{2} - 3596301415 T^{3} - 26062698 T^{4} - 20611 T^{5} + 406 T^{6} + T^{7} \)
$31$ \( 71281602727072 + 13383479373228 T + 235299161864 T^{2} - 324188383 T^{3} - 18619338 T^{4} - 33649 T^{5} + 346 T^{6} + T^{7} \)
$37$ \( -21927969574050832 - 361082892436340 T + 928399161095 T^{2} + 22687930517 T^{3} - 25530531 T^{4} - 281925 T^{5} + 157 T^{6} + T^{7} \)
$41$ \( -156295416469202284 + 445297299970688 T + 7419838781185 T^{2} - 10238764799 T^{3} - 130782015 T^{4} - 10215 T^{5} + 825 T^{6} + T^{7} \)
$43$ \( -6798950817082112 - 294960143818768 T - 4748756395485 T^{2} - 34190255675 T^{3} - 94017410 T^{4} + 76666 T^{5} + 815 T^{6} + T^{7} \)
$47$ \( -290351607809472 - 60857559559632 T - 813748162324 T^{2} + 1540531513 T^{3} + 79369620 T^{4} + 492637 T^{5} + 1196 T^{6} + T^{7} \)
$53$ \( 15838543304612864 + 1155182562235504 T + 19768603062500 T^{2} + 13029308089 T^{3} - 285704744 T^{4} - 242202 T^{5} + 900 T^{6} + T^{7} \)
$59$ \( ( 59 + T )^{7} \)
$61$ \( 479182635497312 - 2594291052616 T - 1666891850746 T^{2} + 35140206215 T^{3} + 102285166 T^{4} - 473335 T^{5} - 420 T^{6} + T^{7} \)
$67$ \( 945626488765454528 + 4294495128486896 T - 29276318744892 T^{2} - 80288727931 T^{3} + 357722888 T^{4} + 146978 T^{5} - 1316 T^{6} + T^{7} \)
$71$ \( -115748174098770148 - 3282740776406564 T - 37859321451 T^{2} + 129577660545 T^{3} - 68753230 T^{4} - 704622 T^{5} + 173 T^{6} + T^{7} \)
$73$ \( -10406926504120742000 - 39610605611121700 T + 142358344390120 T^{2} + 410747344383 T^{3} - 484469968 T^{4} - 1226731 T^{5} + 418 T^{6} + T^{7} \)
$79$ \( 1105102469977771968 + 23677328181601344 T + 37085375396549 T^{2} - 392336169175 T^{3} - 18533418 T^{4} + 1973382 T^{5} - 2635 T^{6} + T^{7} \)
$83$ \( 312127819328844656 - 5650502907778152 T - 68923560686241 T^{2} + 290966352821 T^{3} + 296091211 T^{4} - 1260195 T^{5} - 457 T^{6} + T^{7} \)
$89$ \( -\)\(35\!\cdots\!48\)\( - 1667265158063312852 T - 667440385743808 T^{2} + 4501366123475 T^{3} + 1635330782 T^{4} - 3861133 T^{5} - 592 T^{6} + T^{7} \)
$97$ \( -\)\(14\!\cdots\!96\)\( + 353755240309251516 T + 1304722444420528 T^{2} - 1466526753865 T^{3} - 4794101130 T^{4} - 1615068 T^{5} + 1906 T^{6} + T^{7} \)
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