Properties

Label 177.8.a.b.1.11
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(4.11298\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.11298 q^{2} +27.0000 q^{3} -123.535 q^{4} -537.118 q^{5} +57.0506 q^{6} +1259.15 q^{7} -531.490 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+2.11298 q^{2} +27.0000 q^{3} -123.535 q^{4} -537.118 q^{5} +57.0506 q^{6} +1259.15 q^{7} -531.490 q^{8} +729.000 q^{9} -1134.92 q^{10} +7730.24 q^{11} -3335.45 q^{12} -8860.73 q^{13} +2660.56 q^{14} -14502.2 q^{15} +14689.5 q^{16} +22368.4 q^{17} +1540.37 q^{18} -24886.1 q^{19} +66353.0 q^{20} +33997.0 q^{21} +16333.9 q^{22} -26686.2 q^{23} -14350.2 q^{24} +210371. q^{25} -18722.6 q^{26} +19683.0 q^{27} -155549. q^{28} +10101.0 q^{29} -30642.9 q^{30} -265180. q^{31} +99069.4 q^{32} +208717. q^{33} +47264.0 q^{34} -676311. q^{35} -90057.2 q^{36} +122638. q^{37} -52583.9 q^{38} -239240. q^{39} +285473. q^{40} -741866. q^{41} +71835.1 q^{42} -166554. q^{43} -954958. q^{44} -391559. q^{45} -56387.4 q^{46} -545699. q^{47} +396616. q^{48} +761910. q^{49} +444510. q^{50} +603946. q^{51} +1.09461e6 q^{52} -1.54083e6 q^{53} +41589.9 q^{54} -4.15205e6 q^{55} -669225. q^{56} -671924. q^{57} +21343.2 q^{58} -205379. q^{59} +1.79153e6 q^{60} +2.74259e6 q^{61} -560320. q^{62} +917919. q^{63} -1.67092e6 q^{64} +4.75926e6 q^{65} +441015. q^{66} -41604.4 q^{67} -2.76328e6 q^{68} -720526. q^{69} -1.42903e6 q^{70} -1.57840e6 q^{71} -387456. q^{72} +5.59479e6 q^{73} +259132. q^{74} +5.68001e6 q^{75} +3.07431e6 q^{76} +9.73352e6 q^{77} -505510. q^{78} -3.74825e6 q^{79} -7.88999e6 q^{80} +531441. q^{81} -1.56755e6 q^{82} -3.37715e6 q^{83} -4.19983e6 q^{84} -1.20144e7 q^{85} -351926. q^{86} +272726. q^{87} -4.10855e6 q^{88} -5.43486e6 q^{89} -827358. q^{90} -1.11570e7 q^{91} +3.29668e6 q^{92} -7.15985e6 q^{93} -1.15305e6 q^{94} +1.33668e7 q^{95} +2.67487e6 q^{96} -1.14400e7 q^{97} +1.60990e6 q^{98} +5.63535e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.11298 0.186763 0.0933816 0.995630i \(-0.470232\pi\)
0.0933816 + 0.995630i \(0.470232\pi\)
\(3\) 27.0000 0.577350
\(4\) −123.535 −0.965120
\(5\) −537.118 −1.92165 −0.960826 0.277153i \(-0.910609\pi\)
−0.960826 + 0.277153i \(0.910609\pi\)
\(6\) 57.0506 0.107828
\(7\) 1259.15 1.38750 0.693751 0.720215i \(-0.255957\pi\)
0.693751 + 0.720215i \(0.255957\pi\)
\(8\) −531.490 −0.367012
\(9\) 729.000 0.333333
\(10\) −1134.92 −0.358894
\(11\) 7730.24 1.75113 0.875566 0.483099i \(-0.160489\pi\)
0.875566 + 0.483099i \(0.160489\pi\)
\(12\) −3335.45 −0.557212
\(13\) −8860.73 −1.11858 −0.559291 0.828971i \(-0.688927\pi\)
−0.559291 + 0.828971i \(0.688927\pi\)
\(14\) 2660.56 0.259134
\(15\) −14502.2 −1.10947
\(16\) 14689.5 0.896575
\(17\) 22368.4 1.10424 0.552119 0.833765i \(-0.313819\pi\)
0.552119 + 0.833765i \(0.313819\pi\)
\(18\) 1540.37 0.0622544
\(19\) −24886.1 −0.832375 −0.416187 0.909279i \(-0.636634\pi\)
−0.416187 + 0.909279i \(0.636634\pi\)
\(20\) 66353.0 1.85462
\(21\) 33997.0 0.801075
\(22\) 16333.9 0.327047
\(23\) −26686.2 −0.457339 −0.228670 0.973504i \(-0.573438\pi\)
−0.228670 + 0.973504i \(0.573438\pi\)
\(24\) −14350.2 −0.211894
\(25\) 210371. 2.69275
\(26\) −18722.6 −0.208910
\(27\) 19683.0 0.192450
\(28\) −155549. −1.33911
\(29\) 10101.0 0.0769077 0.0384539 0.999260i \(-0.487757\pi\)
0.0384539 + 0.999260i \(0.487757\pi\)
\(30\) −30642.9 −0.207207
\(31\) −265180. −1.59873 −0.799363 0.600848i \(-0.794830\pi\)
−0.799363 + 0.600848i \(0.794830\pi\)
\(32\) 99069.4 0.534459
\(33\) 208717. 1.01102
\(34\) 47264.0 0.206231
\(35\) −676311. −2.66630
\(36\) −90057.2 −0.321707
\(37\) 122638. 0.398032 0.199016 0.979996i \(-0.436225\pi\)
0.199016 + 0.979996i \(0.436225\pi\)
\(38\) −52583.9 −0.155457
\(39\) −239240. −0.645813
\(40\) 285473. 0.705269
\(41\) −741866. −1.68105 −0.840527 0.541769i \(-0.817755\pi\)
−0.840527 + 0.541769i \(0.817755\pi\)
\(42\) 71835.1 0.149611
\(43\) −166554. −0.319459 −0.159730 0.987161i \(-0.551062\pi\)
−0.159730 + 0.987161i \(0.551062\pi\)
\(44\) −954958. −1.69005
\(45\) −391559. −0.640551
\(46\) −56387.4 −0.0854142
\(47\) −545699. −0.766674 −0.383337 0.923608i \(-0.625225\pi\)
−0.383337 + 0.923608i \(0.625225\pi\)
\(48\) 396616. 0.517638
\(49\) 761910. 0.925161
\(50\) 444510. 0.502906
\(51\) 603946. 0.637532
\(52\) 1.09461e6 1.07957
\(53\) −1.54083e6 −1.42164 −0.710819 0.703375i \(-0.751676\pi\)
−0.710819 + 0.703375i \(0.751676\pi\)
\(54\) 41589.9 0.0359426
\(55\) −4.15205e6 −3.36506
\(56\) −669225. −0.509230
\(57\) −671924. −0.480572
\(58\) 21343.2 0.0143635
\(59\) −205379. −0.130189
\(60\) 1.79153e6 1.07077
\(61\) 2.74259e6 1.54706 0.773529 0.633761i \(-0.218490\pi\)
0.773529 + 0.633761i \(0.218490\pi\)
\(62\) −560320. −0.298583
\(63\) 917919. 0.462501
\(64\) −1.67092e6 −0.796758
\(65\) 4.75926e6 2.14952
\(66\) 441015. 0.188821
\(67\) −41604.4 −0.0168996 −0.00844981 0.999964i \(-0.502690\pi\)
−0.00844981 + 0.999964i \(0.502690\pi\)
\(68\) −2.76328e6 −1.06572
\(69\) −720526. −0.264045
\(70\) −1.42903e6 −0.497966
\(71\) −1.57840e6 −0.523376 −0.261688 0.965152i \(-0.584279\pi\)
−0.261688 + 0.965152i \(0.584279\pi\)
\(72\) −387456. −0.122337
\(73\) 5.59479e6 1.68327 0.841635 0.540047i \(-0.181594\pi\)
0.841635 + 0.540047i \(0.181594\pi\)
\(74\) 259132. 0.0743378
\(75\) 5.68001e6 1.55466
\(76\) 3.07431e6 0.803341
\(77\) 9.73352e6 2.42970
\(78\) −505510. −0.120614
\(79\) −3.74825e6 −0.855330 −0.427665 0.903937i \(-0.640664\pi\)
−0.427665 + 0.903937i \(0.640664\pi\)
\(80\) −7.88999e6 −1.72291
\(81\) 531441. 0.111111
\(82\) −1.56755e6 −0.313959
\(83\) −3.37715e6 −0.648302 −0.324151 0.946005i \(-0.605079\pi\)
−0.324151 + 0.946005i \(0.605079\pi\)
\(84\) −4.19983e6 −0.773133
\(85\) −1.20144e7 −2.12196
\(86\) −351926. −0.0596632
\(87\) 272726. 0.0444027
\(88\) −4.10855e6 −0.642686
\(89\) −5.43486e6 −0.817191 −0.408595 0.912716i \(-0.633981\pi\)
−0.408595 + 0.912716i \(0.633981\pi\)
\(90\) −827358. −0.119631
\(91\) −1.11570e7 −1.55203
\(92\) 3.29668e6 0.441387
\(93\) −7.15985e6 −0.923025
\(94\) −1.15305e6 −0.143187
\(95\) 1.33668e7 1.59953
\(96\) 2.67487e6 0.308570
\(97\) −1.14400e7 −1.27270 −0.636349 0.771401i \(-0.719556\pi\)
−0.636349 + 0.771401i \(0.719556\pi\)
\(98\) 1.60990e6 0.172786
\(99\) 5.63535e6 0.583710
\(100\) −2.59882e7 −2.59882
\(101\) −1.39505e7 −1.34730 −0.673652 0.739049i \(-0.735275\pi\)
−0.673652 + 0.739049i \(0.735275\pi\)
\(102\) 1.27613e6 0.119068
\(103\) −9.43458e6 −0.850731 −0.425366 0.905022i \(-0.639855\pi\)
−0.425366 + 0.905022i \(0.639855\pi\)
\(104\) 4.70939e6 0.410533
\(105\) −1.82604e7 −1.53939
\(106\) −3.25575e6 −0.265510
\(107\) 1.19603e7 0.943844 0.471922 0.881640i \(-0.343560\pi\)
0.471922 + 0.881640i \(0.343560\pi\)
\(108\) −2.43155e6 −0.185737
\(109\) 2.39265e6 0.176964 0.0884822 0.996078i \(-0.471798\pi\)
0.0884822 + 0.996078i \(0.471798\pi\)
\(110\) −8.77322e6 −0.628470
\(111\) 3.31122e6 0.229804
\(112\) 1.84962e7 1.24400
\(113\) −4.09793e6 −0.267171 −0.133586 0.991037i \(-0.542649\pi\)
−0.133586 + 0.991037i \(0.542649\pi\)
\(114\) −1.41976e6 −0.0897531
\(115\) 1.43336e7 0.878847
\(116\) −1.24783e6 −0.0742251
\(117\) −6.45947e6 −0.372861
\(118\) −433963. −0.0243145
\(119\) 2.81651e7 1.53213
\(120\) 7.70777e6 0.407187
\(121\) 4.02695e7 2.06646
\(122\) 5.79505e6 0.288933
\(123\) −2.00304e7 −0.970557
\(124\) 3.27590e7 1.54296
\(125\) −7.10316e7 −3.25287
\(126\) 1.93955e6 0.0863781
\(127\) 1.95190e7 0.845560 0.422780 0.906232i \(-0.361054\pi\)
0.422780 + 0.906232i \(0.361054\pi\)
\(128\) −1.62115e7 −0.683264
\(129\) −4.49696e6 −0.184440
\(130\) 1.00562e7 0.401452
\(131\) −3.90620e6 −0.151811 −0.0759057 0.997115i \(-0.524185\pi\)
−0.0759057 + 0.997115i \(0.524185\pi\)
\(132\) −2.57839e7 −0.975751
\(133\) −3.13352e7 −1.15492
\(134\) −87909.4 −0.00315623
\(135\) −1.05721e7 −0.369822
\(136\) −1.18886e7 −0.405269
\(137\) −2.53665e7 −0.842828 −0.421414 0.906868i \(-0.638466\pi\)
−0.421414 + 0.906868i \(0.638466\pi\)
\(138\) −1.52246e6 −0.0493139
\(139\) 4.42236e7 1.39670 0.698348 0.715758i \(-0.253919\pi\)
0.698348 + 0.715758i \(0.253919\pi\)
\(140\) 8.35483e7 2.57329
\(141\) −1.47339e7 −0.442640
\(142\) −3.33514e6 −0.0977474
\(143\) −6.84956e7 −1.95878
\(144\) 1.07086e7 0.298858
\(145\) −5.42541e6 −0.147790
\(146\) 1.18217e7 0.314373
\(147\) 2.05716e7 0.534142
\(148\) −1.51501e7 −0.384149
\(149\) 2.22130e6 0.0550117 0.0275059 0.999622i \(-0.491244\pi\)
0.0275059 + 0.999622i \(0.491244\pi\)
\(150\) 1.20018e7 0.290353
\(151\) −4.02533e7 −0.951443 −0.475721 0.879596i \(-0.657813\pi\)
−0.475721 + 0.879596i \(0.657813\pi\)
\(152\) 1.32267e7 0.305491
\(153\) 1.63065e7 0.368079
\(154\) 2.05668e7 0.453778
\(155\) 1.42433e8 3.07220
\(156\) 2.95545e7 0.623287
\(157\) 3.47322e7 0.716281 0.358140 0.933668i \(-0.383411\pi\)
0.358140 + 0.933668i \(0.383411\pi\)
\(158\) −7.91999e6 −0.159744
\(159\) −4.16024e7 −0.820783
\(160\) −5.32120e7 −1.02704
\(161\) −3.36018e7 −0.634559
\(162\) 1.12293e6 0.0207515
\(163\) −1.15628e7 −0.209126 −0.104563 0.994518i \(-0.533344\pi\)
−0.104563 + 0.994518i \(0.533344\pi\)
\(164\) 9.16466e7 1.62242
\(165\) −1.12105e8 −1.94282
\(166\) −7.13587e6 −0.121079
\(167\) −1.06887e8 −1.77590 −0.887949 0.459941i \(-0.847870\pi\)
−0.887949 + 0.459941i \(0.847870\pi\)
\(168\) −1.80691e7 −0.294004
\(169\) 1.57640e7 0.251225
\(170\) −2.53863e7 −0.396304
\(171\) −1.81419e7 −0.277458
\(172\) 2.05753e7 0.308316
\(173\) −7.33625e7 −1.07724 −0.538621 0.842548i \(-0.681054\pi\)
−0.538621 + 0.842548i \(0.681054\pi\)
\(174\) 576266. 0.00829279
\(175\) 2.64888e8 3.73619
\(176\) 1.13553e8 1.57002
\(177\) −5.54523e6 −0.0751646
\(178\) −1.14838e7 −0.152621
\(179\) −7.93198e7 −1.03370 −0.516852 0.856075i \(-0.672896\pi\)
−0.516852 + 0.856075i \(0.672896\pi\)
\(180\) 4.83714e7 0.618208
\(181\) 4.76795e7 0.597663 0.298831 0.954306i \(-0.403403\pi\)
0.298831 + 0.954306i \(0.403403\pi\)
\(182\) −2.35745e7 −0.289863
\(183\) 7.40500e7 0.893194
\(184\) 1.41834e7 0.167849
\(185\) −6.58710e7 −0.764880
\(186\) −1.51286e7 −0.172387
\(187\) 1.72913e8 1.93367
\(188\) 6.74131e7 0.739932
\(189\) 2.47838e7 0.267025
\(190\) 2.82437e7 0.298734
\(191\) −5.66760e7 −0.588549 −0.294274 0.955721i \(-0.595078\pi\)
−0.294274 + 0.955721i \(0.595078\pi\)
\(192\) −4.51149e7 −0.460008
\(193\) −1.76636e8 −1.76859 −0.884297 0.466924i \(-0.845362\pi\)
−0.884297 + 0.466924i \(0.845362\pi\)
\(194\) −2.41726e7 −0.237693
\(195\) 1.28500e8 1.24103
\(196\) −9.41228e7 −0.892891
\(197\) 1.27300e8 1.18630 0.593151 0.805091i \(-0.297884\pi\)
0.593151 + 0.805091i \(0.297884\pi\)
\(198\) 1.19074e7 0.109016
\(199\) 1.45827e8 1.31175 0.655877 0.754868i \(-0.272299\pi\)
0.655877 + 0.754868i \(0.272299\pi\)
\(200\) −1.11810e8 −0.988270
\(201\) −1.12332e6 −0.00975700
\(202\) −2.94772e7 −0.251627
\(203\) 1.27186e7 0.106710
\(204\) −7.46086e7 −0.615295
\(205\) 3.98469e8 3.23040
\(206\) −1.99351e7 −0.158885
\(207\) −1.94542e7 −0.152446
\(208\) −1.30160e8 −1.00289
\(209\) −1.92375e8 −1.45760
\(210\) −3.85839e7 −0.287501
\(211\) −1.57257e8 −1.15245 −0.576223 0.817292i \(-0.695474\pi\)
−0.576223 + 0.817292i \(0.695474\pi\)
\(212\) 1.90347e8 1.37205
\(213\) −4.26169e7 −0.302171
\(214\) 2.52720e7 0.176275
\(215\) 8.94591e7 0.613889
\(216\) −1.04613e7 −0.0706315
\(217\) −3.33900e8 −2.21824
\(218\) 5.05563e6 0.0330504
\(219\) 1.51059e8 0.971836
\(220\) 5.12925e8 3.24769
\(221\) −1.98200e8 −1.23518
\(222\) 6.99656e6 0.0429190
\(223\) −1.56191e8 −0.943169 −0.471584 0.881821i \(-0.656318\pi\)
−0.471584 + 0.881821i \(0.656318\pi\)
\(224\) 1.24743e8 0.741563
\(225\) 1.53360e8 0.897582
\(226\) −8.65886e6 −0.0498978
\(227\) −1.43058e8 −0.811748 −0.405874 0.913929i \(-0.633033\pi\)
−0.405874 + 0.913929i \(0.633033\pi\)
\(228\) 8.30063e7 0.463809
\(229\) −9.00963e7 −0.495773 −0.247886 0.968789i \(-0.579736\pi\)
−0.247886 + 0.968789i \(0.579736\pi\)
\(230\) 3.02867e7 0.164136
\(231\) 2.62805e8 1.40279
\(232\) −5.36856e6 −0.0282261
\(233\) 1.58220e8 0.819437 0.409718 0.912212i \(-0.365627\pi\)
0.409718 + 0.912212i \(0.365627\pi\)
\(234\) −1.36488e7 −0.0696366
\(235\) 2.93105e8 1.47328
\(236\) 2.53716e7 0.125648
\(237\) −1.01203e8 −0.493825
\(238\) 5.95123e7 0.286146
\(239\) −7.78834e7 −0.369022 −0.184511 0.982830i \(-0.559070\pi\)
−0.184511 + 0.982830i \(0.559070\pi\)
\(240\) −2.13030e8 −0.994720
\(241\) 7.51343e7 0.345763 0.172882 0.984943i \(-0.444692\pi\)
0.172882 + 0.984943i \(0.444692\pi\)
\(242\) 8.50887e7 0.385939
\(243\) 1.43489e7 0.0641500
\(244\) −3.38807e8 −1.49310
\(245\) −4.09236e8 −1.77784
\(246\) −4.23239e7 −0.181264
\(247\) 2.20509e8 0.931079
\(248\) 1.40940e8 0.586752
\(249\) −9.11832e7 −0.374297
\(250\) −1.50089e8 −0.607516
\(251\) 1.66455e8 0.664416 0.332208 0.943206i \(-0.392206\pi\)
0.332208 + 0.943206i \(0.392206\pi\)
\(252\) −1.13395e8 −0.446368
\(253\) −2.06290e8 −0.800861
\(254\) 4.12433e7 0.157919
\(255\) −3.24390e8 −1.22512
\(256\) 1.79623e8 0.669149
\(257\) −5.67936e7 −0.208705 −0.104353 0.994540i \(-0.533277\pi\)
−0.104353 + 0.994540i \(0.533277\pi\)
\(258\) −9.50200e6 −0.0344466
\(259\) 1.54419e8 0.552271
\(260\) −5.87936e8 −2.07455
\(261\) 7.36360e6 0.0256359
\(262\) −8.25373e6 −0.0283528
\(263\) −1.48859e8 −0.504581 −0.252291 0.967651i \(-0.581184\pi\)
−0.252291 + 0.967651i \(0.581184\pi\)
\(264\) −1.10931e8 −0.371055
\(265\) 8.27608e8 2.73189
\(266\) −6.62109e7 −0.215697
\(267\) −1.46741e8 −0.471805
\(268\) 5.13961e6 0.0163102
\(269\) −3.17458e7 −0.0994382 −0.0497191 0.998763i \(-0.515833\pi\)
−0.0497191 + 0.998763i \(0.515833\pi\)
\(270\) −2.23387e7 −0.0690691
\(271\) 470544. 0.00143618 0.000718089 1.00000i \(-0.499771\pi\)
0.000718089 1.00000i \(0.499771\pi\)
\(272\) 3.28580e8 0.990033
\(273\) −3.01238e8 −0.896067
\(274\) −5.35991e7 −0.157409
\(275\) 1.62622e9 4.71535
\(276\) 8.90104e7 0.254835
\(277\) 4.15036e8 1.17329 0.586646 0.809843i \(-0.300448\pi\)
0.586646 + 0.809843i \(0.300448\pi\)
\(278\) 9.34437e7 0.260851
\(279\) −1.93316e8 −0.532909
\(280\) 3.59453e8 0.978562
\(281\) 4.91128e8 1.32045 0.660226 0.751067i \(-0.270461\pi\)
0.660226 + 0.751067i \(0.270461\pi\)
\(282\) −3.11325e7 −0.0826688
\(283\) −3.10302e8 −0.813827 −0.406914 0.913467i \(-0.633395\pi\)
−0.406914 + 0.913467i \(0.633395\pi\)
\(284\) 1.94989e8 0.505121
\(285\) 3.60902e8 0.923491
\(286\) −1.44730e8 −0.365829
\(287\) −9.34119e8 −2.33247
\(288\) 7.22216e7 0.178153
\(289\) 9.00046e7 0.219342
\(290\) −1.14638e7 −0.0276017
\(291\) −3.08880e8 −0.734793
\(292\) −6.91154e8 −1.62456
\(293\) −2.94684e8 −0.684415 −0.342208 0.939624i \(-0.611175\pi\)
−0.342208 + 0.939624i \(0.611175\pi\)
\(294\) 4.34674e7 0.0997581
\(295\) 1.10313e8 0.250178
\(296\) −6.51808e7 −0.146083
\(297\) 1.52154e8 0.337005
\(298\) 4.69357e6 0.0102742
\(299\) 2.36459e8 0.511572
\(300\) −7.01682e8 −1.50043
\(301\) −2.09716e8 −0.443250
\(302\) −8.50547e7 −0.177694
\(303\) −3.76664e8 −0.777867
\(304\) −3.65564e8 −0.746286
\(305\) −1.47309e9 −2.97291
\(306\) 3.44554e7 0.0687437
\(307\) 2.26044e8 0.445871 0.222936 0.974833i \(-0.428436\pi\)
0.222936 + 0.974833i \(0.428436\pi\)
\(308\) −1.20243e9 −2.34495
\(309\) −2.54734e8 −0.491170
\(310\) 3.00958e8 0.573773
\(311\) −4.63267e8 −0.873313 −0.436656 0.899628i \(-0.643837\pi\)
−0.436656 + 0.899628i \(0.643837\pi\)
\(312\) 1.27154e8 0.237021
\(313\) 6.22167e8 1.14684 0.573418 0.819263i \(-0.305617\pi\)
0.573418 + 0.819263i \(0.305617\pi\)
\(314\) 7.33885e7 0.133775
\(315\) −4.93031e8 −0.888765
\(316\) 4.63041e8 0.825496
\(317\) 8.20133e8 1.44603 0.723014 0.690833i \(-0.242756\pi\)
0.723014 + 0.690833i \(0.242756\pi\)
\(318\) −8.79052e7 −0.153292
\(319\) 7.80829e7 0.134675
\(320\) 8.97483e8 1.53109
\(321\) 3.22929e8 0.544929
\(322\) −7.10001e7 −0.118512
\(323\) −5.56661e8 −0.919140
\(324\) −6.56517e7 −0.107236
\(325\) −1.86404e9 −3.01206
\(326\) −2.44321e7 −0.0390570
\(327\) 6.46015e7 0.102170
\(328\) 3.94294e8 0.616967
\(329\) −6.87116e8 −1.06376
\(330\) −2.36877e8 −0.362847
\(331\) −4.99025e8 −0.756352 −0.378176 0.925734i \(-0.623449\pi\)
−0.378176 + 0.925734i \(0.623449\pi\)
\(332\) 4.17198e8 0.625689
\(333\) 8.94030e7 0.132677
\(334\) −2.25851e8 −0.331672
\(335\) 2.23465e7 0.0324752
\(336\) 4.99398e8 0.718224
\(337\) −2.16438e8 −0.308056 −0.154028 0.988066i \(-0.549225\pi\)
−0.154028 + 0.988066i \(0.549225\pi\)
\(338\) 3.33091e7 0.0469196
\(339\) −1.10644e8 −0.154252
\(340\) 1.48421e9 2.04795
\(341\) −2.04990e9 −2.79958
\(342\) −3.83336e7 −0.0518190
\(343\) −7.76048e7 −0.103839
\(344\) 8.85218e7 0.117245
\(345\) 3.87008e8 0.507403
\(346\) −1.55014e8 −0.201189
\(347\) 2.49019e8 0.319948 0.159974 0.987121i \(-0.448859\pi\)
0.159974 + 0.987121i \(0.448859\pi\)
\(348\) −3.36913e7 −0.0428539
\(349\) −1.28064e9 −1.61264 −0.806319 0.591481i \(-0.798543\pi\)
−0.806319 + 0.591481i \(0.798543\pi\)
\(350\) 5.59704e8 0.697783
\(351\) −1.74406e8 −0.215271
\(352\) 7.65830e8 0.935908
\(353\) 1.01386e8 0.122678 0.0613390 0.998117i \(-0.480463\pi\)
0.0613390 + 0.998117i \(0.480463\pi\)
\(354\) −1.17170e7 −0.0140380
\(355\) 8.47790e8 1.00575
\(356\) 6.71397e8 0.788687
\(357\) 7.60457e8 0.884577
\(358\) −1.67601e8 −0.193058
\(359\) −9.38042e8 −1.07002 −0.535010 0.844846i \(-0.679692\pi\)
−0.535010 + 0.844846i \(0.679692\pi\)
\(360\) 2.08110e8 0.235090
\(361\) −2.74555e8 −0.307153
\(362\) 1.00746e8 0.111621
\(363\) 1.08728e9 1.19307
\(364\) 1.37828e9 1.49790
\(365\) −3.00506e9 −3.23466
\(366\) 1.56466e8 0.166816
\(367\) 7.40350e8 0.781818 0.390909 0.920429i \(-0.372161\pi\)
0.390909 + 0.920429i \(0.372161\pi\)
\(368\) −3.92006e8 −0.410039
\(369\) −5.40820e8 −0.560352
\(370\) −1.39184e8 −0.142851
\(371\) −1.94013e9 −1.97253
\(372\) 8.84494e8 0.890829
\(373\) 1.05390e9 1.05152 0.525761 0.850632i \(-0.323781\pi\)
0.525761 + 0.850632i \(0.323781\pi\)
\(374\) 3.65362e8 0.361138
\(375\) −1.91785e9 −1.87804
\(376\) 2.90034e8 0.281379
\(377\) −8.95019e7 −0.0860276
\(378\) 5.23678e7 0.0498704
\(379\) 1.16570e9 1.09989 0.549947 0.835199i \(-0.314648\pi\)
0.549947 + 0.835199i \(0.314648\pi\)
\(380\) −1.65127e9 −1.54374
\(381\) 5.27013e8 0.488184
\(382\) −1.19755e8 −0.109919
\(383\) −1.70280e7 −0.0154870 −0.00774349 0.999970i \(-0.502465\pi\)
−0.00774349 + 0.999970i \(0.502465\pi\)
\(384\) −4.37711e8 −0.394483
\(385\) −5.22805e9 −4.66903
\(386\) −3.73229e8 −0.330308
\(387\) −1.21418e8 −0.106486
\(388\) 1.41325e9 1.22831
\(389\) 3.59169e8 0.309368 0.154684 0.987964i \(-0.450564\pi\)
0.154684 + 0.987964i \(0.450564\pi\)
\(390\) 2.71518e8 0.231778
\(391\) −5.96925e8 −0.505012
\(392\) −4.04948e8 −0.339545
\(393\) −1.05467e8 −0.0876484
\(394\) 2.68982e8 0.221558
\(395\) 2.01325e9 1.64365
\(396\) −6.96164e8 −0.563350
\(397\) −1.33556e9 −1.07127 −0.535634 0.844450i \(-0.679927\pi\)
−0.535634 + 0.844450i \(0.679927\pi\)
\(398\) 3.08130e8 0.244987
\(399\) −8.46052e8 −0.666794
\(400\) 3.09024e9 2.41425
\(401\) 2.22573e9 1.72372 0.861862 0.507143i \(-0.169298\pi\)
0.861862 + 0.507143i \(0.169298\pi\)
\(402\) −2.37355e6 −0.00182225
\(403\) 2.34968e9 1.78831
\(404\) 1.72338e9 1.30031
\(405\) −2.85447e8 −0.213517
\(406\) 2.68742e7 0.0199294
\(407\) 9.48020e8 0.697007
\(408\) −3.20991e8 −0.233982
\(409\) 1.44669e9 1.04555 0.522774 0.852471i \(-0.324897\pi\)
0.522774 + 0.852471i \(0.324897\pi\)
\(410\) 8.41960e8 0.603320
\(411\) −6.84896e8 −0.486607
\(412\) 1.16550e9 0.821057
\(413\) −2.58603e8 −0.180637
\(414\) −4.11064e7 −0.0284714
\(415\) 1.81393e9 1.24581
\(416\) −8.77827e8 −0.597836
\(417\) 1.19404e9 0.806383
\(418\) −4.06486e8 −0.272225
\(419\) −1.91705e9 −1.27316 −0.636581 0.771210i \(-0.719652\pi\)
−0.636581 + 0.771210i \(0.719652\pi\)
\(420\) 2.25580e9 1.48569
\(421\) −1.78027e9 −1.16278 −0.581392 0.813623i \(-0.697492\pi\)
−0.581392 + 0.813623i \(0.697492\pi\)
\(422\) −3.32281e8 −0.215235
\(423\) −3.97815e8 −0.255558
\(424\) 8.18936e8 0.521758
\(425\) 4.70565e9 2.97343
\(426\) −9.00489e7 −0.0564345
\(427\) 3.45333e9 2.14655
\(428\) −1.47752e9 −0.910923
\(429\) −1.84938e9 −1.13090
\(430\) 1.89026e8 0.114652
\(431\) 3.29923e8 0.198492 0.0992458 0.995063i \(-0.468357\pi\)
0.0992458 + 0.995063i \(0.468357\pi\)
\(432\) 2.89133e8 0.172546
\(433\) −4.75888e7 −0.0281707 −0.0140853 0.999901i \(-0.504484\pi\)
−0.0140853 + 0.999901i \(0.504484\pi\)
\(434\) −7.05526e8 −0.414285
\(435\) −1.46486e8 −0.0853265
\(436\) −2.95576e8 −0.170792
\(437\) 6.64114e8 0.380678
\(438\) 3.19186e8 0.181503
\(439\) −3.04664e8 −0.171868 −0.0859342 0.996301i \(-0.527387\pi\)
−0.0859342 + 0.996301i \(0.527387\pi\)
\(440\) 2.20677e9 1.23502
\(441\) 5.55433e8 0.308387
\(442\) −4.18793e8 −0.230686
\(443\) 2.92263e9 1.59721 0.798603 0.601858i \(-0.205573\pi\)
0.798603 + 0.601858i \(0.205573\pi\)
\(444\) −4.09053e8 −0.221788
\(445\) 2.91916e9 1.57036
\(446\) −3.30029e8 −0.176149
\(447\) 5.99751e7 0.0317610
\(448\) −2.10394e9 −1.10550
\(449\) −3.10711e9 −1.61993 −0.809963 0.586481i \(-0.800513\pi\)
−0.809963 + 0.586481i \(0.800513\pi\)
\(450\) 3.24048e8 0.167635
\(451\) −5.73480e9 −2.94375
\(452\) 5.06239e8 0.257852
\(453\) −1.08684e9 −0.549316
\(454\) −3.02279e8 −0.151605
\(455\) 5.99261e9 2.98247
\(456\) 3.57121e8 0.176376
\(457\) −1.02063e8 −0.0500221 −0.0250110 0.999687i \(-0.507962\pi\)
−0.0250110 + 0.999687i \(0.507962\pi\)
\(458\) −1.90372e8 −0.0925921
\(459\) 4.40276e8 0.212511
\(460\) −1.77071e9 −0.848192
\(461\) −1.96777e9 −0.935452 −0.467726 0.883873i \(-0.654927\pi\)
−0.467726 + 0.883873i \(0.654927\pi\)
\(462\) 5.55303e8 0.261989
\(463\) 1.67779e9 0.785604 0.392802 0.919623i \(-0.371506\pi\)
0.392802 + 0.919623i \(0.371506\pi\)
\(464\) 1.48378e8 0.0689536
\(465\) 3.84568e9 1.77373
\(466\) 3.34316e8 0.153041
\(467\) −1.19888e9 −0.544709 −0.272355 0.962197i \(-0.587802\pi\)
−0.272355 + 0.962197i \(0.587802\pi\)
\(468\) 7.97973e8 0.359855
\(469\) −5.23860e7 −0.0234483
\(470\) 6.19326e8 0.275155
\(471\) 9.37769e8 0.413545
\(472\) 1.09157e8 0.0477809
\(473\) −1.28750e9 −0.559415
\(474\) −2.13840e8 −0.0922284
\(475\) −5.23530e9 −2.24137
\(476\) −3.47938e9 −1.47869
\(477\) −1.12327e9 −0.473880
\(478\) −1.64566e8 −0.0689197
\(479\) −3.41930e9 −1.42155 −0.710777 0.703418i \(-0.751656\pi\)
−0.710777 + 0.703418i \(0.751656\pi\)
\(480\) −1.43672e9 −0.592964
\(481\) −1.08666e9 −0.445232
\(482\) 1.58758e8 0.0645758
\(483\) −9.07249e8 −0.366363
\(484\) −4.97470e9 −1.99438
\(485\) 6.14464e9 2.44568
\(486\) 3.03190e7 0.0119809
\(487\) −1.02116e9 −0.400628 −0.200314 0.979732i \(-0.564196\pi\)
−0.200314 + 0.979732i \(0.564196\pi\)
\(488\) −1.45766e9 −0.567789
\(489\) −3.12196e8 −0.120739
\(490\) −8.64709e8 −0.332035
\(491\) 3.59695e9 1.37135 0.685677 0.727906i \(-0.259506\pi\)
0.685677 + 0.727906i \(0.259506\pi\)
\(492\) 2.47446e9 0.936704
\(493\) 2.25942e8 0.0849245
\(494\) 4.65932e8 0.173891
\(495\) −3.02685e9 −1.12169
\(496\) −3.89535e9 −1.43338
\(497\) −1.98744e9 −0.726186
\(498\) −1.92669e8 −0.0699050
\(499\) −1.17064e9 −0.421765 −0.210883 0.977511i \(-0.567634\pi\)
−0.210883 + 0.977511i \(0.567634\pi\)
\(500\) 8.77491e9 3.13941
\(501\) −2.88596e9 −1.02532
\(502\) 3.51718e8 0.124088
\(503\) −1.37007e9 −0.480014 −0.240007 0.970771i \(-0.577150\pi\)
−0.240007 + 0.970771i \(0.577150\pi\)
\(504\) −4.87865e8 −0.169743
\(505\) 7.49308e9 2.58905
\(506\) −4.35888e8 −0.149571
\(507\) 4.25628e8 0.145045
\(508\) −2.41129e9 −0.816067
\(509\) −6.94286e8 −0.233360 −0.116680 0.993170i \(-0.537225\pi\)
−0.116680 + 0.993170i \(0.537225\pi\)
\(510\) −6.85431e8 −0.228806
\(511\) 7.04466e9 2.33554
\(512\) 2.45462e9 0.808237
\(513\) −4.89833e8 −0.160191
\(514\) −1.20004e8 −0.0389785
\(515\) 5.06748e9 1.63481
\(516\) 5.55533e8 0.178006
\(517\) −4.21839e9 −1.34255
\(518\) 3.26285e8 0.103144
\(519\) −1.98079e9 −0.621946
\(520\) −2.52950e9 −0.788901
\(521\) −3.26480e9 −1.01140 −0.505702 0.862708i \(-0.668767\pi\)
−0.505702 + 0.862708i \(0.668767\pi\)
\(522\) 1.55592e7 0.00478784
\(523\) 1.44867e9 0.442806 0.221403 0.975182i \(-0.428936\pi\)
0.221403 + 0.975182i \(0.428936\pi\)
\(524\) 4.82553e8 0.146516
\(525\) 7.15197e9 2.15709
\(526\) −3.14538e8 −0.0942372
\(527\) −5.93163e9 −1.76537
\(528\) 3.06594e9 0.906452
\(529\) −2.69267e9 −0.790841
\(530\) 1.74872e9 0.510217
\(531\) −1.49721e8 −0.0433963
\(532\) 3.87101e9 1.11464
\(533\) 6.57347e9 1.88040
\(534\) −3.10062e8 −0.0881159
\(535\) −6.42412e9 −1.81374
\(536\) 2.21123e7 0.00620237
\(537\) −2.14163e9 −0.596809
\(538\) −6.70784e7 −0.0185714
\(539\) 5.88975e9 1.62008
\(540\) 1.30603e9 0.356922
\(541\) 5.30499e9 1.44044 0.720218 0.693748i \(-0.244042\pi\)
0.720218 + 0.693748i \(0.244042\pi\)
\(542\) 994253. 0.000268225 0
\(543\) 1.28735e9 0.345061
\(544\) 2.21602e9 0.590170
\(545\) −1.28513e9 −0.340064
\(546\) −6.36511e8 −0.167352
\(547\) 2.77315e9 0.724465 0.362232 0.932088i \(-0.382015\pi\)
0.362232 + 0.932088i \(0.382015\pi\)
\(548\) 3.13366e9 0.813430
\(549\) 1.99935e9 0.515686
\(550\) 3.43617e9 0.880654
\(551\) −2.51373e8 −0.0640160
\(552\) 3.82953e8 0.0969077
\(553\) −4.71960e9 −1.18677
\(554\) 8.76964e8 0.219128
\(555\) −1.77852e9 −0.441604
\(556\) −5.46317e9 −1.34798
\(557\) −2.33572e9 −0.572701 −0.286350 0.958125i \(-0.592442\pi\)
−0.286350 + 0.958125i \(0.592442\pi\)
\(558\) −4.08473e8 −0.0995277
\(559\) 1.47579e9 0.357341
\(560\) −9.93466e9 −2.39053
\(561\) 4.66864e9 1.11640
\(562\) 1.03775e9 0.246612
\(563\) 5.53231e9 1.30655 0.653276 0.757120i \(-0.273394\pi\)
0.653276 + 0.757120i \(0.273394\pi\)
\(564\) 1.82015e9 0.427200
\(565\) 2.20107e9 0.513411
\(566\) −6.55663e8 −0.151993
\(567\) 6.69163e8 0.154167
\(568\) 8.38906e8 0.192085
\(569\) 5.01743e8 0.114180 0.0570898 0.998369i \(-0.481818\pi\)
0.0570898 + 0.998369i \(0.481818\pi\)
\(570\) 7.62581e8 0.172474
\(571\) −1.45684e9 −0.327481 −0.163741 0.986503i \(-0.552356\pi\)
−0.163741 + 0.986503i \(0.552356\pi\)
\(572\) 8.46162e9 1.89046
\(573\) −1.53025e9 −0.339799
\(574\) −1.97378e9 −0.435619
\(575\) −5.61399e9 −1.23150
\(576\) −1.21810e9 −0.265586
\(577\) −5.26247e9 −1.14044 −0.570222 0.821490i \(-0.693143\pi\)
−0.570222 + 0.821490i \(0.693143\pi\)
\(578\) 1.90178e8 0.0409650
\(579\) −4.76917e9 −1.02110
\(580\) 6.70230e8 0.142635
\(581\) −4.25234e9 −0.899521
\(582\) −6.52659e8 −0.137232
\(583\) −1.19110e10 −2.48948
\(584\) −2.97357e9 −0.617780
\(585\) 3.46950e9 0.716508
\(586\) −6.22662e8 −0.127824
\(587\) −3.91337e9 −0.798579 −0.399290 0.916825i \(-0.630743\pi\)
−0.399290 + 0.916825i \(0.630743\pi\)
\(588\) −2.54132e9 −0.515511
\(589\) 6.59928e9 1.33074
\(590\) 2.33089e8 0.0467240
\(591\) 3.43709e9 0.684912
\(592\) 1.80149e9 0.356866
\(593\) 6.52849e9 1.28564 0.642822 0.766015i \(-0.277763\pi\)
0.642822 + 0.766015i \(0.277763\pi\)
\(594\) 3.21500e8 0.0629402
\(595\) −1.51280e10 −2.94423
\(596\) −2.74409e8 −0.0530929
\(597\) 3.93733e9 0.757342
\(598\) 4.99634e8 0.0955427
\(599\) 8.24579e9 1.56761 0.783805 0.621007i \(-0.213276\pi\)
0.783805 + 0.621007i \(0.213276\pi\)
\(600\) −3.01887e9 −0.570578
\(601\) −6.15461e9 −1.15648 −0.578242 0.815865i \(-0.696261\pi\)
−0.578242 + 0.815865i \(0.696261\pi\)
\(602\) −4.43127e8 −0.0827828
\(603\) −3.03296e7 −0.00563321
\(604\) 4.97271e9 0.918256
\(605\) −2.16295e10 −3.97102
\(606\) −7.95885e8 −0.145277
\(607\) 7.75348e6 0.00140714 0.000703569 1.00000i \(-0.499776\pi\)
0.000703569 1.00000i \(0.499776\pi\)
\(608\) −2.46545e9 −0.444870
\(609\) 3.43402e8 0.0616088
\(610\) −3.11263e9 −0.555230
\(611\) 4.83529e9 0.857588
\(612\) −2.01443e9 −0.355241
\(613\) 5.41748e9 0.949917 0.474958 0.880008i \(-0.342463\pi\)
0.474958 + 0.880008i \(0.342463\pi\)
\(614\) 4.77628e8 0.0832723
\(615\) 1.07587e10 1.86507
\(616\) −5.17327e9 −0.891728
\(617\) −1.53588e9 −0.263245 −0.131622 0.991300i \(-0.542019\pi\)
−0.131622 + 0.991300i \(0.542019\pi\)
\(618\) −5.38248e8 −0.0917325
\(619\) −1.24887e9 −0.211641 −0.105821 0.994385i \(-0.533747\pi\)
−0.105821 + 0.994385i \(0.533747\pi\)
\(620\) −1.75955e10 −2.96504
\(621\) −5.25264e8 −0.0880150
\(622\) −9.78875e8 −0.163103
\(623\) −6.84329e9 −1.13385
\(624\) −3.51431e9 −0.579020
\(625\) 2.17171e10 3.55813
\(626\) 1.31463e9 0.214187
\(627\) −5.19413e9 −0.841544
\(628\) −4.29065e9 −0.691296
\(629\) 2.74321e9 0.439523
\(630\) −1.04177e9 −0.165989
\(631\) −8.97235e9 −1.42169 −0.710843 0.703351i \(-0.751686\pi\)
−0.710843 + 0.703351i \(0.751686\pi\)
\(632\) 1.99216e9 0.313916
\(633\) −4.24593e9 −0.665365
\(634\) 1.73293e9 0.270065
\(635\) −1.04840e10 −1.62487
\(636\) 5.13937e9 0.792154
\(637\) −6.75108e9 −1.03487
\(638\) 1.64988e8 0.0251524
\(639\) −1.15066e9 −0.174459
\(640\) 8.70750e9 1.31300
\(641\) −5.94956e9 −0.892240 −0.446120 0.894973i \(-0.647195\pi\)
−0.446120 + 0.894973i \(0.647195\pi\)
\(642\) 6.82344e8 0.101773
\(643\) 8.03210e9 1.19149 0.595745 0.803173i \(-0.296857\pi\)
0.595745 + 0.803173i \(0.296857\pi\)
\(644\) 4.15101e9 0.612426
\(645\) 2.41540e9 0.354429
\(646\) −1.17621e9 −0.171661
\(647\) −8.88936e9 −1.29034 −0.645172 0.764038i \(-0.723214\pi\)
−0.645172 + 0.764038i \(0.723214\pi\)
\(648\) −2.82456e8 −0.0407791
\(649\) −1.58763e9 −0.227978
\(650\) −3.93868e9 −0.562541
\(651\) −9.01531e9 −1.28070
\(652\) 1.42842e9 0.201831
\(653\) −1.14448e10 −1.60846 −0.804231 0.594316i \(-0.797423\pi\)
−0.804231 + 0.594316i \(0.797423\pi\)
\(654\) 1.36502e8 0.0190817
\(655\) 2.09809e9 0.291729
\(656\) −1.08976e10 −1.50719
\(657\) 4.07860e9 0.561090
\(658\) −1.45187e9 −0.198672
\(659\) 9.40014e9 1.27949 0.639743 0.768589i \(-0.279041\pi\)
0.639743 + 0.768589i \(0.279041\pi\)
\(660\) 1.38490e10 1.87505
\(661\) 1.37618e9 0.185341 0.0926704 0.995697i \(-0.470460\pi\)
0.0926704 + 0.995697i \(0.470460\pi\)
\(662\) −1.05443e9 −0.141259
\(663\) −5.35140e9 −0.713132
\(664\) 1.79492e9 0.237935
\(665\) 1.68307e10 2.21936
\(666\) 1.88907e8 0.0247793
\(667\) −2.69556e8 −0.0351729
\(668\) 1.32043e10 1.71395
\(669\) −4.21716e9 −0.544539
\(670\) 4.72177e7 0.00606517
\(671\) 2.12009e10 2.70910
\(672\) 3.36806e9 0.428142
\(673\) 9.13119e9 1.15471 0.577357 0.816492i \(-0.304084\pi\)
0.577357 + 0.816492i \(0.304084\pi\)
\(674\) −4.57331e8 −0.0575335
\(675\) 4.14073e9 0.518219
\(676\) −1.94741e9 −0.242462
\(677\) 6.14968e9 0.761715 0.380857 0.924634i \(-0.375629\pi\)
0.380857 + 0.924634i \(0.375629\pi\)
\(678\) −2.33789e8 −0.0288085
\(679\) −1.44047e10 −1.76587
\(680\) 6.38556e9 0.778785
\(681\) −3.86256e9 −0.468663
\(682\) −4.33141e9 −0.522858
\(683\) −1.15096e10 −1.38225 −0.691127 0.722734i \(-0.742885\pi\)
−0.691127 + 0.722734i \(0.742885\pi\)
\(684\) 2.24117e9 0.267780
\(685\) 1.36248e10 1.61962
\(686\) −1.63978e8 −0.0193932
\(687\) −2.43260e9 −0.286235
\(688\) −2.44659e9 −0.286419
\(689\) 1.36529e10 1.59022
\(690\) 8.17741e8 0.0947641
\(691\) −6.76616e9 −0.780134 −0.390067 0.920786i \(-0.627548\pi\)
−0.390067 + 0.920786i \(0.627548\pi\)
\(692\) 9.06286e9 1.03967
\(693\) 7.09573e9 0.809899
\(694\) 5.26173e8 0.0597545
\(695\) −2.37533e10 −2.68396
\(696\) −1.44951e8 −0.0162963
\(697\) −1.65943e10 −1.85628
\(698\) −2.70596e9 −0.301181
\(699\) 4.27194e9 0.473102
\(700\) −3.27230e10 −3.60587
\(701\) 2.85225e9 0.312734 0.156367 0.987699i \(-0.450022\pi\)
0.156367 + 0.987699i \(0.450022\pi\)
\(702\) −3.68517e8 −0.0402047
\(703\) −3.05197e9 −0.331312
\(704\) −1.29166e10 −1.39523
\(705\) 7.91383e9 0.850599
\(706\) 2.14227e8 0.0229117
\(707\) −1.75658e10 −1.86939
\(708\) 6.85032e8 0.0725428
\(709\) 1.72710e10 1.81994 0.909969 0.414676i \(-0.136105\pi\)
0.909969 + 0.414676i \(0.136105\pi\)
\(710\) 1.79137e9 0.187837
\(711\) −2.73247e9 −0.285110
\(712\) 2.88858e9 0.299919
\(713\) 7.07662e9 0.731161
\(714\) 1.60683e9 0.165206
\(715\) 3.67902e10 3.76410
\(716\) 9.79879e9 0.997647
\(717\) −2.10285e9 −0.213055
\(718\) −1.98207e9 −0.199840
\(719\) 6.37828e9 0.639959 0.319980 0.947424i \(-0.396324\pi\)
0.319980 + 0.947424i \(0.396324\pi\)
\(720\) −5.75180e9 −0.574302
\(721\) −1.18795e10 −1.18039
\(722\) −5.80130e8 −0.0573648
\(723\) 2.02863e9 0.199626
\(724\) −5.89010e9 −0.576816
\(725\) 2.12495e9 0.207093
\(726\) 2.29740e9 0.222822
\(727\) 1.75274e10 1.69179 0.845895 0.533350i \(-0.179067\pi\)
0.845895 + 0.533350i \(0.179067\pi\)
\(728\) 5.92982e9 0.569615
\(729\) 3.87420e8 0.0370370
\(730\) −6.34965e9 −0.604115
\(731\) −3.72554e9 −0.352759
\(732\) −9.14778e9 −0.862039
\(733\) 6.90854e9 0.647922 0.323961 0.946070i \(-0.394985\pi\)
0.323961 + 0.946070i \(0.394985\pi\)
\(734\) 1.56435e9 0.146015
\(735\) −1.10494e10 −1.02644
\(736\) −2.64378e9 −0.244429
\(737\) −3.21612e8 −0.0295935
\(738\) −1.14274e9 −0.104653
\(739\) 1.05720e10 0.963610 0.481805 0.876278i \(-0.339981\pi\)
0.481805 + 0.876278i \(0.339981\pi\)
\(740\) 8.13739e9 0.738200
\(741\) 5.95374e9 0.537559
\(742\) −4.09947e9 −0.368395
\(743\) 1.14806e10 1.02685 0.513423 0.858136i \(-0.328377\pi\)
0.513423 + 0.858136i \(0.328377\pi\)
\(744\) 3.80539e9 0.338761
\(745\) −1.19310e9 −0.105713
\(746\) 2.22687e9 0.196386
\(747\) −2.46195e9 −0.216101
\(748\) −2.13608e10 −1.86622
\(749\) 1.50598e10 1.30959
\(750\) −4.05239e9 −0.350749
\(751\) −6.77468e9 −0.583645 −0.291823 0.956472i \(-0.594262\pi\)
−0.291823 + 0.956472i \(0.594262\pi\)
\(752\) −8.01604e9 −0.687381
\(753\) 4.49430e9 0.383601
\(754\) −1.89116e8 −0.0160668
\(755\) 2.16208e10 1.82834
\(756\) −3.06167e9 −0.257711
\(757\) −1.20655e10 −1.01090 −0.505451 0.862855i \(-0.668674\pi\)
−0.505451 + 0.862855i \(0.668674\pi\)
\(758\) 2.46311e9 0.205420
\(759\) −5.56984e9 −0.462377
\(760\) −7.10430e9 −0.587048
\(761\) −6.97304e9 −0.573556 −0.286778 0.957997i \(-0.592584\pi\)
−0.286778 + 0.957997i \(0.592584\pi\)
\(762\) 1.11357e9 0.0911749
\(763\) 3.01270e9 0.245539
\(764\) 7.00149e9 0.568020
\(765\) −8.75853e9 −0.707320
\(766\) −3.59798e7 −0.00289240
\(767\) 1.81981e9 0.145627
\(768\) 4.84983e9 0.386334
\(769\) −6.18686e9 −0.490600 −0.245300 0.969447i \(-0.578887\pi\)
−0.245300 + 0.969447i \(0.578887\pi\)
\(770\) −1.10468e10 −0.872003
\(771\) −1.53343e9 −0.120496
\(772\) 2.18208e10 1.70691
\(773\) 9.77041e9 0.760825 0.380412 0.924817i \(-0.375782\pi\)
0.380412 + 0.924817i \(0.375782\pi\)
\(774\) −2.56554e8 −0.0198877
\(775\) −5.57860e10 −4.30496
\(776\) 6.08025e9 0.467096
\(777\) 4.16932e9 0.318854
\(778\) 7.58918e8 0.0577785
\(779\) 1.84621e10 1.39927
\(780\) −1.58743e10 −1.19774
\(781\) −1.22014e10 −0.916501
\(782\) −1.26129e9 −0.0943176
\(783\) 1.98817e8 0.0148009
\(784\) 1.11921e10 0.829477
\(785\) −1.86553e10 −1.37644
\(786\) −2.22851e8 −0.0163695
\(787\) −3.17612e9 −0.232266 −0.116133 0.993234i \(-0.537050\pi\)
−0.116133 + 0.993234i \(0.537050\pi\)
\(788\) −1.57260e10 −1.14492
\(789\) −4.01921e9 −0.291320
\(790\) 4.25397e9 0.306973
\(791\) −5.15990e9 −0.370701
\(792\) −2.99513e9 −0.214229
\(793\) −2.43014e10 −1.73051
\(794\) −2.82203e9 −0.200073
\(795\) 2.23454e10 1.57726
\(796\) −1.80148e10 −1.26600
\(797\) −1.64367e10 −1.15003 −0.575016 0.818142i \(-0.695004\pi\)
−0.575016 + 0.818142i \(0.695004\pi\)
\(798\) −1.78769e9 −0.124533
\(799\) −1.22064e10 −0.846591
\(800\) 2.08413e10 1.43916
\(801\) −3.96201e9 −0.272397
\(802\) 4.70294e9 0.321928
\(803\) 4.32491e10 2.94762
\(804\) 1.38769e8 0.00941668
\(805\) 1.80481e10 1.21940
\(806\) 4.96485e9 0.333990
\(807\) −8.57137e8 −0.0574107
\(808\) 7.41457e9 0.494477
\(809\) 6.28475e9 0.417319 0.208659 0.977988i \(-0.433090\pi\)
0.208659 + 0.977988i \(0.433090\pi\)
\(810\) −6.03144e8 −0.0398771
\(811\) 2.31027e10 1.52086 0.760432 0.649418i \(-0.224987\pi\)
0.760432 + 0.649418i \(0.224987\pi\)
\(812\) −1.57120e9 −0.102988
\(813\) 1.27047e7 0.000829177 0
\(814\) 2.00315e9 0.130175
\(815\) 6.21060e9 0.401866
\(816\) 8.87165e9 0.571596
\(817\) 4.14487e9 0.265910
\(818\) 3.05683e9 0.195270
\(819\) −8.13343e9 −0.517345
\(820\) −4.92250e10 −3.11772
\(821\) −2.91799e10 −1.84027 −0.920137 0.391597i \(-0.871923\pi\)
−0.920137 + 0.391597i \(0.871923\pi\)
\(822\) −1.44717e9 −0.0908803
\(823\) 2.76539e10 1.72925 0.864624 0.502419i \(-0.167557\pi\)
0.864624 + 0.502419i \(0.167557\pi\)
\(824\) 5.01439e9 0.312229
\(825\) 4.39078e10 2.72241
\(826\) −5.46423e8 −0.0337364
\(827\) −5.03285e9 −0.309417 −0.154709 0.987960i \(-0.549444\pi\)
−0.154709 + 0.987960i \(0.549444\pi\)
\(828\) 2.40328e9 0.147129
\(829\) 1.94317e10 1.18459 0.592296 0.805720i \(-0.298222\pi\)
0.592296 + 0.805720i \(0.298222\pi\)
\(830\) 3.83281e9 0.232672
\(831\) 1.12060e10 0.677401
\(832\) 1.48056e10 0.891239
\(833\) 1.70427e10 1.02160
\(834\) 2.52298e9 0.150603
\(835\) 5.74111e10 3.41266
\(836\) 2.37651e10 1.40676
\(837\) −5.21953e9 −0.307675
\(838\) −4.05069e9 −0.237780
\(839\) 1.03412e10 0.604509 0.302255 0.953227i \(-0.402261\pi\)
0.302255 + 0.953227i \(0.402261\pi\)
\(840\) 9.70522e9 0.564973
\(841\) −1.71478e10 −0.994085
\(842\) −3.76169e9 −0.217165
\(843\) 1.32604e10 0.762363
\(844\) 1.94268e10 1.11225
\(845\) −8.46713e9 −0.482767
\(846\) −8.40576e8 −0.0477288
\(847\) 5.07052e10 2.86722
\(848\) −2.26340e10 −1.27461
\(849\) −8.37816e9 −0.469863
\(850\) 9.94296e9 0.555328
\(851\) −3.27273e9 −0.182036
\(852\) 5.26469e9 0.291632
\(853\) 2.35292e10 1.29803 0.649016 0.760775i \(-0.275181\pi\)
0.649016 + 0.760775i \(0.275181\pi\)
\(854\) 7.29683e9 0.400896
\(855\) 9.74437e9 0.533178
\(856\) −6.35680e9 −0.346402
\(857\) −2.62808e10 −1.42628 −0.713142 0.701020i \(-0.752728\pi\)
−0.713142 + 0.701020i \(0.752728\pi\)
\(858\) −3.90771e9 −0.211211
\(859\) 1.11154e9 0.0598339 0.0299170 0.999552i \(-0.490476\pi\)
0.0299170 + 0.999552i \(0.490476\pi\)
\(860\) −1.10514e10 −0.592476
\(861\) −2.52212e10 −1.34665
\(862\) 6.97122e8 0.0370709
\(863\) −1.39838e10 −0.740604 −0.370302 0.928911i \(-0.620746\pi\)
−0.370302 + 0.928911i \(0.620746\pi\)
\(864\) 1.94998e9 0.102857
\(865\) 3.94043e10 2.07008
\(866\) −1.00554e8 −0.00526125
\(867\) 2.43012e9 0.126637
\(868\) 4.12485e10 2.14086
\(869\) −2.89749e10 −1.49780
\(870\) −3.09523e8 −0.0159359
\(871\) 3.68645e8 0.0189036
\(872\) −1.27167e9 −0.0649481
\(873\) −8.33977e9 −0.424233
\(874\) 1.40326e9 0.0710966
\(875\) −8.94392e10 −4.51336
\(876\) −1.86611e10 −0.937938
\(877\) 3.43482e10 1.71951 0.859756 0.510705i \(-0.170615\pi\)
0.859756 + 0.510705i \(0.170615\pi\)
\(878\) −6.43751e8 −0.0320987
\(879\) −7.95646e9 −0.395147
\(880\) −6.09915e10 −3.01703
\(881\) 3.32281e10 1.63715 0.818576 0.574398i \(-0.194764\pi\)
0.818576 + 0.574398i \(0.194764\pi\)
\(882\) 1.17362e9 0.0575954
\(883\) −2.10543e10 −1.02915 −0.514576 0.857445i \(-0.672051\pi\)
−0.514576 + 0.857445i \(0.672051\pi\)
\(884\) 2.44847e10 1.19210
\(885\) 2.97844e9 0.144440
\(886\) 6.17547e9 0.298299
\(887\) 1.23601e10 0.594688 0.297344 0.954770i \(-0.403899\pi\)
0.297344 + 0.954770i \(0.403899\pi\)
\(888\) −1.75988e9 −0.0843409
\(889\) 2.45773e10 1.17322
\(890\) 6.16814e9 0.293285
\(891\) 4.10817e9 0.194570
\(892\) 1.92951e10 0.910271
\(893\) 1.35803e10 0.638160
\(894\) 1.26726e8 0.00593179
\(895\) 4.26041e10 1.98642
\(896\) −2.04127e10 −0.948030
\(897\) 6.38439e9 0.295356
\(898\) −6.56528e9 −0.302542
\(899\) −2.67857e9 −0.122954
\(900\) −1.89454e10 −0.866274
\(901\) −3.44658e10 −1.56983
\(902\) −1.21175e10 −0.549784
\(903\) −5.66233e9 −0.255911
\(904\) 2.17801e9 0.0980551
\(905\) −2.56095e10 −1.14850
\(906\) −2.29648e9 −0.102592
\(907\) 1.40907e10 0.627058 0.313529 0.949579i \(-0.398489\pi\)
0.313529 + 0.949579i \(0.398489\pi\)
\(908\) 1.76727e10 0.783433
\(909\) −1.01699e10 −0.449101
\(910\) 1.26623e10 0.557015
\(911\) 2.85261e10 1.25005 0.625027 0.780603i \(-0.285088\pi\)
0.625027 + 0.780603i \(0.285088\pi\)
\(912\) −9.87022e9 −0.430869
\(913\) −2.61062e10 −1.13526
\(914\) −2.15657e8 −0.00934228
\(915\) −3.97736e10 −1.71641
\(916\) 1.11301e10 0.478480
\(917\) −4.91848e9 −0.210639
\(918\) 9.30297e8 0.0396892
\(919\) −1.21759e10 −0.517485 −0.258742 0.965946i \(-0.583308\pi\)
−0.258742 + 0.965946i \(0.583308\pi\)
\(920\) −7.61817e9 −0.322547
\(921\) 6.10320e9 0.257424
\(922\) −4.15787e9 −0.174708
\(923\) 1.39858e10 0.585439
\(924\) −3.24657e10 −1.35386
\(925\) 2.57994e10 1.07180
\(926\) 3.54514e9 0.146722
\(927\) −6.87781e9 −0.283577
\(928\) 1.00070e9 0.0411040
\(929\) −4.26915e9 −0.174697 −0.0873487 0.996178i \(-0.527839\pi\)
−0.0873487 + 0.996178i \(0.527839\pi\)
\(930\) 8.12587e9 0.331268
\(931\) −1.89610e10 −0.770081
\(932\) −1.95457e10 −0.790854
\(933\) −1.25082e10 −0.504207
\(934\) −2.53320e9 −0.101732
\(935\) −9.28746e10 −3.71583
\(936\) 3.43315e9 0.136844
\(937\) 1.52673e10 0.606282 0.303141 0.952946i \(-0.401965\pi\)
0.303141 + 0.952946i \(0.401965\pi\)
\(938\) −1.10691e8 −0.00437927
\(939\) 1.67985e10 0.662126
\(940\) −3.62088e10 −1.42189
\(941\) 3.60621e10 1.41087 0.705435 0.708775i \(-0.250752\pi\)
0.705435 + 0.708775i \(0.250752\pi\)
\(942\) 1.98149e9 0.0772349
\(943\) 1.97975e10 0.768813
\(944\) −3.01691e9 −0.116724
\(945\) −1.33118e10 −0.513129
\(946\) −2.72047e9 −0.104478
\(947\) 1.21747e10 0.465836 0.232918 0.972496i \(-0.425173\pi\)
0.232918 + 0.972496i \(0.425173\pi\)
\(948\) 1.25021e10 0.476600
\(949\) −4.95739e10 −1.88287
\(950\) −1.10621e10 −0.418606
\(951\) 2.21436e10 0.834865
\(952\) −1.49695e10 −0.562311
\(953\) −2.94721e10 −1.10303 −0.551514 0.834166i \(-0.685950\pi\)
−0.551514 + 0.834166i \(0.685950\pi\)
\(954\) −2.37344e9 −0.0885032
\(955\) 3.04417e10 1.13099
\(956\) 9.62135e9 0.356150
\(957\) 2.10824e9 0.0777549
\(958\) −7.22493e9 −0.265494
\(959\) −3.19402e10 −1.16943
\(960\) 2.42320e10 0.883976
\(961\) 4.28076e10 1.55593
\(962\) −2.29610e9 −0.0831529
\(963\) 8.71909e9 0.314615
\(964\) −9.28174e9 −0.333703
\(965\) 9.48743e10 3.39862
\(966\) −1.91700e9 −0.0684231
\(967\) −3.39812e10 −1.20850 −0.604250 0.796795i \(-0.706527\pi\)
−0.604250 + 0.796795i \(0.706527\pi\)
\(968\) −2.14028e10 −0.758415
\(969\) −1.50298e10 −0.530666
\(970\) 1.29835e10 0.456764
\(971\) −1.42757e9 −0.0500415 −0.0250208 0.999687i \(-0.507965\pi\)
−0.0250208 + 0.999687i \(0.507965\pi\)
\(972\) −1.77260e9 −0.0619124
\(973\) 5.56840e10 1.93792
\(974\) −2.15769e9 −0.0748226
\(975\) −5.03290e10 −1.73901
\(976\) 4.02873e10 1.38705
\(977\) 2.69567e10 0.924775 0.462388 0.886678i \(-0.346993\pi\)
0.462388 + 0.886678i \(0.346993\pi\)
\(978\) −6.59665e8 −0.0225495
\(979\) −4.20128e10 −1.43101
\(980\) 5.05551e10 1.71583
\(981\) 1.74424e9 0.0589882
\(982\) 7.60030e9 0.256118
\(983\) −1.23730e10 −0.415469 −0.207735 0.978185i \(-0.566609\pi\)
−0.207735 + 0.978185i \(0.566609\pi\)
\(984\) 1.06459e10 0.356206
\(985\) −6.83749e10 −2.27966
\(986\) 4.77412e8 0.0158608
\(987\) −1.85521e10 −0.614163
\(988\) −2.72406e10 −0.898602
\(989\) 4.44468e9 0.146101
\(990\) −6.39568e9 −0.209490
\(991\) −6.08067e10 −1.98470 −0.992348 0.123474i \(-0.960597\pi\)
−0.992348 + 0.123474i \(0.960597\pi\)
\(992\) −2.62712e10 −0.854454
\(993\) −1.34737e10 −0.436680
\(994\) −4.19944e9 −0.135625
\(995\) −7.83264e10 −2.52074
\(996\) 1.12643e10 0.361242
\(997\) 4.42391e10 1.41375 0.706876 0.707338i \(-0.250104\pi\)
0.706876 + 0.707338i \(0.250104\pi\)
\(998\) −2.47354e9 −0.0787702
\(999\) 2.41388e9 0.0766014
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.b.1.11 17
3.2 odd 2 531.8.a.d.1.7 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.11 17 1.1 even 1 trivial
531.8.a.d.1.7 17 3.2 odd 2