Properties

Label 3381.2.a.bi.1.6
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.864859\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.864859 q^{2} -1.00000 q^{3} -1.25202 q^{4} -2.49557 q^{5} -0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.864859 q^{2} -1.00000 q^{3} -1.25202 q^{4} -2.49557 q^{5} -0.864859 q^{6} -2.81254 q^{8} +1.00000 q^{9} -2.15832 q^{10} -3.98310 q^{11} +1.25202 q^{12} +2.80699 q^{13} +2.49557 q^{15} +0.0715855 q^{16} -6.98770 q^{17} +0.864859 q^{18} -5.98058 q^{19} +3.12450 q^{20} -3.44482 q^{22} +1.00000 q^{23} +2.81254 q^{24} +1.22789 q^{25} +2.42765 q^{26} -1.00000 q^{27} -4.06198 q^{29} +2.15832 q^{30} -3.44346 q^{31} +5.68699 q^{32} +3.98310 q^{33} -6.04338 q^{34} -1.25202 q^{36} -1.47693 q^{37} -5.17236 q^{38} -2.80699 q^{39} +7.01890 q^{40} -7.65339 q^{41} +7.07993 q^{43} +4.98691 q^{44} -2.49557 q^{45} +0.864859 q^{46} -3.04531 q^{47} -0.0715855 q^{48} +1.06195 q^{50} +6.98770 q^{51} -3.51440 q^{52} +9.57104 q^{53} -0.864859 q^{54} +9.94011 q^{55} +5.98058 q^{57} -3.51304 q^{58} -8.30618 q^{59} -3.12450 q^{60} -3.71282 q^{61} -2.97811 q^{62} +4.77528 q^{64} -7.00504 q^{65} +3.44482 q^{66} +4.46957 q^{67} +8.74872 q^{68} -1.00000 q^{69} -9.49028 q^{71} -2.81254 q^{72} +12.9021 q^{73} -1.27733 q^{74} -1.22789 q^{75} +7.48779 q^{76} -2.42765 q^{78} +7.12304 q^{79} -0.178647 q^{80} +1.00000 q^{81} -6.61911 q^{82} +1.05716 q^{83} +17.4383 q^{85} +6.12315 q^{86} +4.06198 q^{87} +11.2026 q^{88} -3.30053 q^{89} -2.15832 q^{90} -1.25202 q^{92} +3.44346 q^{93} -2.63377 q^{94} +14.9250 q^{95} -5.68699 q^{96} -15.9312 q^{97} -3.98310 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.864859 0.611548 0.305774 0.952104i \(-0.401085\pi\)
0.305774 + 0.952104i \(0.401085\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.25202 −0.626009
\(5\) −2.49557 −1.11605 −0.558027 0.829823i \(-0.688441\pi\)
−0.558027 + 0.829823i \(0.688441\pi\)
\(6\) −0.864859 −0.353077
\(7\) 0 0
\(8\) −2.81254 −0.994383
\(9\) 1.00000 0.333333
\(10\) −2.15832 −0.682521
\(11\) −3.98310 −1.20095 −0.600475 0.799644i \(-0.705022\pi\)
−0.600475 + 0.799644i \(0.705022\pi\)
\(12\) 1.25202 0.361426
\(13\) 2.80699 0.778518 0.389259 0.921128i \(-0.372731\pi\)
0.389259 + 0.921128i \(0.372731\pi\)
\(14\) 0 0
\(15\) 2.49557 0.644354
\(16\) 0.0715855 0.0178964
\(17\) −6.98770 −1.69477 −0.847383 0.530983i \(-0.821823\pi\)
−0.847383 + 0.530983i \(0.821823\pi\)
\(18\) 0.864859 0.203849
\(19\) −5.98058 −1.37204 −0.686019 0.727583i \(-0.740643\pi\)
−0.686019 + 0.727583i \(0.740643\pi\)
\(20\) 3.12450 0.698660
\(21\) 0 0
\(22\) −3.44482 −0.734438
\(23\) 1.00000 0.208514
\(24\) 2.81254 0.574107
\(25\) 1.22789 0.245577
\(26\) 2.42765 0.476101
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.06198 −0.754291 −0.377145 0.926154i \(-0.623094\pi\)
−0.377145 + 0.926154i \(0.623094\pi\)
\(30\) 2.15832 0.394054
\(31\) −3.44346 −0.618464 −0.309232 0.950987i \(-0.600072\pi\)
−0.309232 + 0.950987i \(0.600072\pi\)
\(32\) 5.68699 1.00533
\(33\) 3.98310 0.693368
\(34\) −6.04338 −1.03643
\(35\) 0 0
\(36\) −1.25202 −0.208670
\(37\) −1.47693 −0.242805 −0.121402 0.992603i \(-0.538739\pi\)
−0.121402 + 0.992603i \(0.538739\pi\)
\(38\) −5.17236 −0.839067
\(39\) −2.80699 −0.449477
\(40\) 7.01890 1.10978
\(41\) −7.65339 −1.19526 −0.597630 0.801772i \(-0.703891\pi\)
−0.597630 + 0.801772i \(0.703891\pi\)
\(42\) 0 0
\(43\) 7.07993 1.07968 0.539840 0.841768i \(-0.318485\pi\)
0.539840 + 0.841768i \(0.318485\pi\)
\(44\) 4.98691 0.751805
\(45\) −2.49557 −0.372018
\(46\) 0.864859 0.127517
\(47\) −3.04531 −0.444204 −0.222102 0.975023i \(-0.571292\pi\)
−0.222102 + 0.975023i \(0.571292\pi\)
\(48\) −0.0715855 −0.0103325
\(49\) 0 0
\(50\) 1.06195 0.150182
\(51\) 6.98770 0.978473
\(52\) −3.51440 −0.487359
\(53\) 9.57104 1.31468 0.657342 0.753593i \(-0.271681\pi\)
0.657342 + 0.753593i \(0.271681\pi\)
\(54\) −0.864859 −0.117692
\(55\) 9.94011 1.34032
\(56\) 0 0
\(57\) 5.98058 0.792147
\(58\) −3.51304 −0.461285
\(59\) −8.30618 −1.08137 −0.540686 0.841224i \(-0.681835\pi\)
−0.540686 + 0.841224i \(0.681835\pi\)
\(60\) −3.12450 −0.403372
\(61\) −3.71282 −0.475377 −0.237689 0.971341i \(-0.576390\pi\)
−0.237689 + 0.971341i \(0.576390\pi\)
\(62\) −2.97811 −0.378220
\(63\) 0 0
\(64\) 4.77528 0.596909
\(65\) −7.00504 −0.868868
\(66\) 3.44482 0.424028
\(67\) 4.46957 0.546045 0.273023 0.962008i \(-0.411977\pi\)
0.273023 + 0.962008i \(0.411977\pi\)
\(68\) 8.74872 1.06094
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −9.49028 −1.12629 −0.563144 0.826359i \(-0.690408\pi\)
−0.563144 + 0.826359i \(0.690408\pi\)
\(72\) −2.81254 −0.331461
\(73\) 12.9021 1.51008 0.755039 0.655680i \(-0.227618\pi\)
0.755039 + 0.655680i \(0.227618\pi\)
\(74\) −1.27733 −0.148487
\(75\) −1.22789 −0.141784
\(76\) 7.48779 0.858908
\(77\) 0 0
\(78\) −2.42765 −0.274877
\(79\) 7.12304 0.801405 0.400702 0.916208i \(-0.368766\pi\)
0.400702 + 0.916208i \(0.368766\pi\)
\(80\) −0.178647 −0.0199733
\(81\) 1.00000 0.111111
\(82\) −6.61911 −0.730958
\(83\) 1.05716 0.116038 0.0580192 0.998315i \(-0.481522\pi\)
0.0580192 + 0.998315i \(0.481522\pi\)
\(84\) 0 0
\(85\) 17.4383 1.89145
\(86\) 6.12315 0.660276
\(87\) 4.06198 0.435490
\(88\) 11.2026 1.19420
\(89\) −3.30053 −0.349856 −0.174928 0.984581i \(-0.555969\pi\)
−0.174928 + 0.984581i \(0.555969\pi\)
\(90\) −2.15832 −0.227507
\(91\) 0 0
\(92\) −1.25202 −0.130532
\(93\) 3.44346 0.357070
\(94\) −2.63377 −0.271652
\(95\) 14.9250 1.53127
\(96\) −5.68699 −0.580426
\(97\) −15.9312 −1.61757 −0.808786 0.588103i \(-0.799875\pi\)
−0.808786 + 0.588103i \(0.799875\pi\)
\(98\) 0 0
\(99\) −3.98310 −0.400316
\(100\) −1.53733 −0.153733
\(101\) 6.96429 0.692972 0.346486 0.938055i \(-0.387375\pi\)
0.346486 + 0.938055i \(0.387375\pi\)
\(102\) 6.04338 0.598383
\(103\) 6.00488 0.591678 0.295839 0.955238i \(-0.404401\pi\)
0.295839 + 0.955238i \(0.404401\pi\)
\(104\) −7.89475 −0.774144
\(105\) 0 0
\(106\) 8.27760 0.803992
\(107\) 5.71797 0.552777 0.276388 0.961046i \(-0.410862\pi\)
0.276388 + 0.961046i \(0.410862\pi\)
\(108\) 1.25202 0.120475
\(109\) 6.25586 0.599202 0.299601 0.954065i \(-0.403146\pi\)
0.299601 + 0.954065i \(0.403146\pi\)
\(110\) 8.59680 0.819673
\(111\) 1.47693 0.140184
\(112\) 0 0
\(113\) −3.79544 −0.357045 −0.178523 0.983936i \(-0.557132\pi\)
−0.178523 + 0.983936i \(0.557132\pi\)
\(114\) 5.17236 0.484436
\(115\) −2.49557 −0.232713
\(116\) 5.08567 0.472193
\(117\) 2.80699 0.259506
\(118\) −7.18368 −0.661311
\(119\) 0 0
\(120\) −7.01890 −0.640735
\(121\) 4.86506 0.442279
\(122\) −3.21106 −0.290716
\(123\) 7.65339 0.690083
\(124\) 4.31127 0.387164
\(125\) 9.41359 0.841977
\(126\) 0 0
\(127\) 20.8819 1.85297 0.926484 0.376333i \(-0.122815\pi\)
0.926484 + 0.376333i \(0.122815\pi\)
\(128\) −7.24404 −0.640288
\(129\) −7.07993 −0.623353
\(130\) −6.05837 −0.531354
\(131\) 12.1488 1.06144 0.530722 0.847546i \(-0.321921\pi\)
0.530722 + 0.847546i \(0.321921\pi\)
\(132\) −4.98691 −0.434055
\(133\) 0 0
\(134\) 3.86555 0.333933
\(135\) 2.49557 0.214785
\(136\) 19.6532 1.68525
\(137\) −0.828633 −0.0707949 −0.0353974 0.999373i \(-0.511270\pi\)
−0.0353974 + 0.999373i \(0.511270\pi\)
\(138\) −0.864859 −0.0736217
\(139\) 0.546099 0.0463195 0.0231598 0.999732i \(-0.492627\pi\)
0.0231598 + 0.999732i \(0.492627\pi\)
\(140\) 0 0
\(141\) 3.04531 0.256461
\(142\) −8.20775 −0.688779
\(143\) −11.1805 −0.934960
\(144\) 0.0715855 0.00596546
\(145\) 10.1370 0.841830
\(146\) 11.1585 0.923485
\(147\) 0 0
\(148\) 1.84914 0.151998
\(149\) −18.5487 −1.51957 −0.759783 0.650176i \(-0.774695\pi\)
−0.759783 + 0.650176i \(0.774695\pi\)
\(150\) −1.06195 −0.0867077
\(151\) −5.77251 −0.469760 −0.234880 0.972024i \(-0.575470\pi\)
−0.234880 + 0.972024i \(0.575470\pi\)
\(152\) 16.8206 1.36433
\(153\) −6.98770 −0.564922
\(154\) 0 0
\(155\) 8.59340 0.690239
\(156\) 3.51440 0.281377
\(157\) 14.1290 1.12761 0.563807 0.825906i \(-0.309336\pi\)
0.563807 + 0.825906i \(0.309336\pi\)
\(158\) 6.16043 0.490097
\(159\) −9.57104 −0.759033
\(160\) −14.1923 −1.12200
\(161\) 0 0
\(162\) 0.864859 0.0679498
\(163\) −16.7496 −1.31193 −0.655966 0.754790i \(-0.727739\pi\)
−0.655966 + 0.754790i \(0.727739\pi\)
\(164\) 9.58219 0.748243
\(165\) −9.94011 −0.773837
\(166\) 0.914295 0.0709631
\(167\) 17.8518 1.38141 0.690707 0.723135i \(-0.257299\pi\)
0.690707 + 0.723135i \(0.257299\pi\)
\(168\) 0 0
\(169\) −5.12084 −0.393910
\(170\) 15.0817 1.15671
\(171\) −5.98058 −0.457346
\(172\) −8.86420 −0.675889
\(173\) 2.65359 0.201749 0.100874 0.994899i \(-0.467836\pi\)
0.100874 + 0.994899i \(0.467836\pi\)
\(174\) 3.51304 0.266323
\(175\) 0 0
\(176\) −0.285132 −0.0214926
\(177\) 8.30618 0.624331
\(178\) −2.85450 −0.213954
\(179\) 11.4696 0.857281 0.428640 0.903475i \(-0.358993\pi\)
0.428640 + 0.903475i \(0.358993\pi\)
\(180\) 3.12450 0.232887
\(181\) −15.8393 −1.17732 −0.588662 0.808380i \(-0.700345\pi\)
−0.588662 + 0.808380i \(0.700345\pi\)
\(182\) 0 0
\(183\) 3.71282 0.274459
\(184\) −2.81254 −0.207343
\(185\) 3.68577 0.270984
\(186\) 2.97811 0.218365
\(187\) 27.8327 2.03533
\(188\) 3.81278 0.278076
\(189\) 0 0
\(190\) 12.9080 0.936445
\(191\) 21.7951 1.57704 0.788518 0.615012i \(-0.210849\pi\)
0.788518 + 0.615012i \(0.210849\pi\)
\(192\) −4.77528 −0.344626
\(193\) 16.0333 1.15411 0.577053 0.816707i \(-0.304203\pi\)
0.577053 + 0.816707i \(0.304203\pi\)
\(194\) −13.7783 −0.989223
\(195\) 7.00504 0.501641
\(196\) 0 0
\(197\) 10.6653 0.759870 0.379935 0.925013i \(-0.375946\pi\)
0.379935 + 0.925013i \(0.375946\pi\)
\(198\) −3.44482 −0.244813
\(199\) −3.65905 −0.259383 −0.129691 0.991554i \(-0.541399\pi\)
−0.129691 + 0.991554i \(0.541399\pi\)
\(200\) −3.45347 −0.244198
\(201\) −4.46957 −0.315259
\(202\) 6.02313 0.423786
\(203\) 0 0
\(204\) −8.74872 −0.612533
\(205\) 19.0996 1.33397
\(206\) 5.19338 0.361840
\(207\) 1.00000 0.0695048
\(208\) 0.200939 0.0139326
\(209\) 23.8212 1.64775
\(210\) 0 0
\(211\) −28.1857 −1.94039 −0.970193 0.242332i \(-0.922088\pi\)
−0.970193 + 0.242332i \(0.922088\pi\)
\(212\) −11.9831 −0.823004
\(213\) 9.49028 0.650263
\(214\) 4.94524 0.338050
\(215\) −17.6685 −1.20498
\(216\) 2.81254 0.191369
\(217\) 0 0
\(218\) 5.41044 0.366441
\(219\) −12.9021 −0.871844
\(220\) −12.4452 −0.839055
\(221\) −19.6144 −1.31940
\(222\) 1.27733 0.0857290
\(223\) 24.2280 1.62243 0.811215 0.584748i \(-0.198807\pi\)
0.811215 + 0.584748i \(0.198807\pi\)
\(224\) 0 0
\(225\) 1.22789 0.0818590
\(226\) −3.28252 −0.218350
\(227\) −8.03654 −0.533404 −0.266702 0.963779i \(-0.585934\pi\)
−0.266702 + 0.963779i \(0.585934\pi\)
\(228\) −7.48779 −0.495891
\(229\) 13.7790 0.910542 0.455271 0.890353i \(-0.349542\pi\)
0.455271 + 0.890353i \(0.349542\pi\)
\(230\) −2.15832 −0.142315
\(231\) 0 0
\(232\) 11.4245 0.750054
\(233\) −1.83601 −0.120281 −0.0601405 0.998190i \(-0.519155\pi\)
−0.0601405 + 0.998190i \(0.519155\pi\)
\(234\) 2.42765 0.158700
\(235\) 7.59979 0.495756
\(236\) 10.3995 0.676949
\(237\) −7.12304 −0.462691
\(238\) 0 0
\(239\) −4.42206 −0.286039 −0.143020 0.989720i \(-0.545681\pi\)
−0.143020 + 0.989720i \(0.545681\pi\)
\(240\) 0.178647 0.0115316
\(241\) −28.4589 −1.83320 −0.916598 0.399810i \(-0.869076\pi\)
−0.916598 + 0.399810i \(0.869076\pi\)
\(242\) 4.20760 0.270475
\(243\) −1.00000 −0.0641500
\(244\) 4.64851 0.297591
\(245\) 0 0
\(246\) 6.61911 0.422019
\(247\) −16.7874 −1.06816
\(248\) 9.68486 0.614989
\(249\) −1.05716 −0.0669948
\(250\) 8.14143 0.514909
\(251\) −26.7064 −1.68569 −0.842846 0.538156i \(-0.819121\pi\)
−0.842846 + 0.538156i \(0.819121\pi\)
\(252\) 0 0
\(253\) −3.98310 −0.250415
\(254\) 18.0599 1.13318
\(255\) −17.4383 −1.09203
\(256\) −15.8156 −0.988476
\(257\) 4.36432 0.272239 0.136119 0.990692i \(-0.456537\pi\)
0.136119 + 0.990692i \(0.456537\pi\)
\(258\) −6.12315 −0.381210
\(259\) 0 0
\(260\) 8.77043 0.543919
\(261\) −4.06198 −0.251430
\(262\) 10.5070 0.649124
\(263\) 1.54914 0.0955242 0.0477621 0.998859i \(-0.484791\pi\)
0.0477621 + 0.998859i \(0.484791\pi\)
\(264\) −11.2026 −0.689473
\(265\) −23.8852 −1.46726
\(266\) 0 0
\(267\) 3.30053 0.201989
\(268\) −5.59599 −0.341829
\(269\) −3.71532 −0.226527 −0.113264 0.993565i \(-0.536130\pi\)
−0.113264 + 0.993565i \(0.536130\pi\)
\(270\) 2.15832 0.131351
\(271\) 4.02009 0.244203 0.122101 0.992518i \(-0.461037\pi\)
0.122101 + 0.992518i \(0.461037\pi\)
\(272\) −0.500218 −0.0303302
\(273\) 0 0
\(274\) −0.716651 −0.0432945
\(275\) −4.89079 −0.294925
\(276\) 1.25202 0.0753626
\(277\) −11.3927 −0.684523 −0.342261 0.939605i \(-0.611193\pi\)
−0.342261 + 0.939605i \(0.611193\pi\)
\(278\) 0.472299 0.0283266
\(279\) −3.44346 −0.206155
\(280\) 0 0
\(281\) −27.9503 −1.66738 −0.833689 0.552235i \(-0.813775\pi\)
−0.833689 + 0.552235i \(0.813775\pi\)
\(282\) 2.63377 0.156838
\(283\) −29.9754 −1.78185 −0.890925 0.454150i \(-0.849943\pi\)
−0.890925 + 0.454150i \(0.849943\pi\)
\(284\) 11.8820 0.705067
\(285\) −14.9250 −0.884079
\(286\) −9.66956 −0.571773
\(287\) 0 0
\(288\) 5.68699 0.335109
\(289\) 31.8279 1.87223
\(290\) 8.76706 0.514819
\(291\) 15.9312 0.933906
\(292\) −16.1537 −0.945322
\(293\) −4.87346 −0.284711 −0.142355 0.989816i \(-0.545468\pi\)
−0.142355 + 0.989816i \(0.545468\pi\)
\(294\) 0 0
\(295\) 20.7287 1.20687
\(296\) 4.15391 0.241441
\(297\) 3.98310 0.231123
\(298\) −16.0420 −0.929288
\(299\) 2.80699 0.162332
\(300\) 1.53733 0.0887580
\(301\) 0 0
\(302\) −4.99241 −0.287281
\(303\) −6.96429 −0.400088
\(304\) −0.428122 −0.0245545
\(305\) 9.26560 0.530547
\(306\) −6.04338 −0.345477
\(307\) −27.9064 −1.59270 −0.796351 0.604835i \(-0.793239\pi\)
−0.796351 + 0.604835i \(0.793239\pi\)
\(308\) 0 0
\(309\) −6.00488 −0.341606
\(310\) 7.43209 0.422114
\(311\) 27.0086 1.53152 0.765758 0.643129i \(-0.222364\pi\)
0.765758 + 0.643129i \(0.222364\pi\)
\(312\) 7.89475 0.446952
\(313\) 4.21402 0.238190 0.119095 0.992883i \(-0.462001\pi\)
0.119095 + 0.992883i \(0.462001\pi\)
\(314\) 12.2196 0.689590
\(315\) 0 0
\(316\) −8.91818 −0.501687
\(317\) −7.99795 −0.449210 −0.224605 0.974450i \(-0.572109\pi\)
−0.224605 + 0.974450i \(0.572109\pi\)
\(318\) −8.27760 −0.464185
\(319\) 16.1793 0.905865
\(320\) −11.9170 −0.666183
\(321\) −5.71797 −0.319146
\(322\) 0 0
\(323\) 41.7905 2.32528
\(324\) −1.25202 −0.0695566
\(325\) 3.44666 0.191186
\(326\) −14.4861 −0.802309
\(327\) −6.25586 −0.345950
\(328\) 21.5255 1.18854
\(329\) 0 0
\(330\) −8.59680 −0.473238
\(331\) 24.0871 1.32395 0.661974 0.749527i \(-0.269719\pi\)
0.661974 + 0.749527i \(0.269719\pi\)
\(332\) −1.32358 −0.0726411
\(333\) −1.47693 −0.0809350
\(334\) 15.4393 0.844801
\(335\) −11.1541 −0.609416
\(336\) 0 0
\(337\) −6.91298 −0.376574 −0.188287 0.982114i \(-0.560294\pi\)
−0.188287 + 0.982114i \(0.560294\pi\)
\(338\) −4.42880 −0.240895
\(339\) 3.79544 0.206140
\(340\) −21.8331 −1.18406
\(341\) 13.7156 0.742743
\(342\) −5.17236 −0.279689
\(343\) 0 0
\(344\) −19.9126 −1.07361
\(345\) 2.49557 0.134357
\(346\) 2.29499 0.123379
\(347\) 12.1876 0.654266 0.327133 0.944978i \(-0.393917\pi\)
0.327133 + 0.944978i \(0.393917\pi\)
\(348\) −5.08567 −0.272621
\(349\) 27.0701 1.44903 0.724515 0.689259i \(-0.242064\pi\)
0.724515 + 0.689259i \(0.242064\pi\)
\(350\) 0 0
\(351\) −2.80699 −0.149826
\(352\) −22.6518 −1.20735
\(353\) −28.5772 −1.52101 −0.760505 0.649332i \(-0.775049\pi\)
−0.760505 + 0.649332i \(0.775049\pi\)
\(354\) 7.18368 0.381808
\(355\) 23.6837 1.25700
\(356\) 4.13233 0.219013
\(357\) 0 0
\(358\) 9.91962 0.524268
\(359\) −19.4627 −1.02720 −0.513601 0.858029i \(-0.671689\pi\)
−0.513601 + 0.858029i \(0.671689\pi\)
\(360\) 7.01890 0.369928
\(361\) 16.7673 0.882489
\(362\) −13.6987 −0.719990
\(363\) −4.86506 −0.255350
\(364\) 0 0
\(365\) −32.1982 −1.68533
\(366\) 3.21106 0.167845
\(367\) 4.81840 0.251518 0.125759 0.992061i \(-0.459863\pi\)
0.125759 + 0.992061i \(0.459863\pi\)
\(368\) 0.0715855 0.00373165
\(369\) −7.65339 −0.398420
\(370\) 3.18768 0.165719
\(371\) 0 0
\(372\) −4.31127 −0.223529
\(373\) −24.6352 −1.27556 −0.637782 0.770217i \(-0.720148\pi\)
−0.637782 + 0.770217i \(0.720148\pi\)
\(374\) 24.0714 1.24470
\(375\) −9.41359 −0.486116
\(376\) 8.56505 0.441709
\(377\) −11.4019 −0.587229
\(378\) 0 0
\(379\) 31.7983 1.63337 0.816684 0.577085i \(-0.195810\pi\)
0.816684 + 0.577085i \(0.195810\pi\)
\(380\) −18.6863 −0.958588
\(381\) −20.8819 −1.06981
\(382\) 18.8497 0.964433
\(383\) −22.3654 −1.14282 −0.571409 0.820666i \(-0.693603\pi\)
−0.571409 + 0.820666i \(0.693603\pi\)
\(384\) 7.24404 0.369671
\(385\) 0 0
\(386\) 13.8666 0.705791
\(387\) 7.07993 0.359893
\(388\) 19.9462 1.01262
\(389\) 19.9097 1.00946 0.504730 0.863277i \(-0.331592\pi\)
0.504730 + 0.863277i \(0.331592\pi\)
\(390\) 6.05837 0.306778
\(391\) −6.98770 −0.353383
\(392\) 0 0
\(393\) −12.1488 −0.612825
\(394\) 9.22397 0.464697
\(395\) −17.7761 −0.894411
\(396\) 4.98691 0.250602
\(397\) 27.1066 1.36044 0.680222 0.733006i \(-0.261883\pi\)
0.680222 + 0.733006i \(0.261883\pi\)
\(398\) −3.16456 −0.158625
\(399\) 0 0
\(400\) 0.0878987 0.00439494
\(401\) −34.1782 −1.70678 −0.853390 0.521274i \(-0.825457\pi\)
−0.853390 + 0.521274i \(0.825457\pi\)
\(402\) −3.86555 −0.192796
\(403\) −9.66574 −0.481485
\(404\) −8.71941 −0.433807
\(405\) −2.49557 −0.124006
\(406\) 0 0
\(407\) 5.88274 0.291596
\(408\) −19.6532 −0.972977
\(409\) 15.1477 0.749006 0.374503 0.927226i \(-0.377813\pi\)
0.374503 + 0.927226i \(0.377813\pi\)
\(410\) 16.5185 0.815789
\(411\) 0.828633 0.0408734
\(412\) −7.51822 −0.370396
\(413\) 0 0
\(414\) 0.864859 0.0425055
\(415\) −2.63822 −0.129505
\(416\) 15.9633 0.782665
\(417\) −0.546099 −0.0267426
\(418\) 20.6020 1.00768
\(419\) 14.0504 0.686405 0.343202 0.939262i \(-0.388488\pi\)
0.343202 + 0.939262i \(0.388488\pi\)
\(420\) 0 0
\(421\) −26.0914 −1.27161 −0.635807 0.771848i \(-0.719333\pi\)
−0.635807 + 0.771848i \(0.719333\pi\)
\(422\) −24.3767 −1.18664
\(423\) −3.04531 −0.148068
\(424\) −26.9189 −1.30730
\(425\) −8.58009 −0.416195
\(426\) 8.20775 0.397667
\(427\) 0 0
\(428\) −7.15900 −0.346043
\(429\) 11.1805 0.539799
\(430\) −15.2808 −0.736904
\(431\) 3.09446 0.149055 0.0745274 0.997219i \(-0.476255\pi\)
0.0745274 + 0.997219i \(0.476255\pi\)
\(432\) −0.0715855 −0.00344416
\(433\) 18.9836 0.912293 0.456146 0.889905i \(-0.349229\pi\)
0.456146 + 0.889905i \(0.349229\pi\)
\(434\) 0 0
\(435\) −10.1370 −0.486031
\(436\) −7.83244 −0.375106
\(437\) −5.98058 −0.286090
\(438\) −11.1585 −0.533174
\(439\) −11.9712 −0.571355 −0.285677 0.958326i \(-0.592219\pi\)
−0.285677 + 0.958326i \(0.592219\pi\)
\(440\) −27.9569 −1.33280
\(441\) 0 0
\(442\) −16.9637 −0.806879
\(443\) 19.4820 0.925616 0.462808 0.886458i \(-0.346842\pi\)
0.462808 + 0.886458i \(0.346842\pi\)
\(444\) −1.84914 −0.0877562
\(445\) 8.23672 0.390458
\(446\) 20.9539 0.992194
\(447\) 18.5487 0.877322
\(448\) 0 0
\(449\) −25.9358 −1.22399 −0.611993 0.790864i \(-0.709632\pi\)
−0.611993 + 0.790864i \(0.709632\pi\)
\(450\) 1.06195 0.0500607
\(451\) 30.4842 1.43545
\(452\) 4.75196 0.223513
\(453\) 5.77251 0.271216
\(454\) −6.95048 −0.326202
\(455\) 0 0
\(456\) −16.8206 −0.787697
\(457\) 4.61332 0.215802 0.107901 0.994162i \(-0.465587\pi\)
0.107901 + 0.994162i \(0.465587\pi\)
\(458\) 11.9169 0.556840
\(459\) 6.98770 0.326158
\(460\) 3.12450 0.145681
\(461\) 13.8181 0.643571 0.321786 0.946813i \(-0.395717\pi\)
0.321786 + 0.946813i \(0.395717\pi\)
\(462\) 0 0
\(463\) −34.1291 −1.58611 −0.793057 0.609147i \(-0.791512\pi\)
−0.793057 + 0.609147i \(0.791512\pi\)
\(464\) −0.290779 −0.0134991
\(465\) −8.59340 −0.398510
\(466\) −1.58789 −0.0735576
\(467\) −27.9716 −1.29437 −0.647185 0.762333i \(-0.724054\pi\)
−0.647185 + 0.762333i \(0.724054\pi\)
\(468\) −3.51440 −0.162453
\(469\) 0 0
\(470\) 6.57275 0.303179
\(471\) −14.1290 −0.651029
\(472\) 23.3614 1.07530
\(473\) −28.2001 −1.29664
\(474\) −6.16043 −0.282958
\(475\) −7.34346 −0.336941
\(476\) 0 0
\(477\) 9.57104 0.438228
\(478\) −3.82446 −0.174927
\(479\) −32.6269 −1.49076 −0.745380 0.666639i \(-0.767732\pi\)
−0.745380 + 0.666639i \(0.767732\pi\)
\(480\) 14.1923 0.647787
\(481\) −4.14571 −0.189028
\(482\) −24.6129 −1.12109
\(483\) 0 0
\(484\) −6.09115 −0.276870
\(485\) 39.7576 1.80530
\(486\) −0.864859 −0.0392308
\(487\) −25.9421 −1.17555 −0.587775 0.809025i \(-0.699996\pi\)
−0.587775 + 0.809025i \(0.699996\pi\)
\(488\) 10.4424 0.472707
\(489\) 16.7496 0.757444
\(490\) 0 0
\(491\) 26.0005 1.17339 0.586693 0.809810i \(-0.300430\pi\)
0.586693 + 0.809810i \(0.300430\pi\)
\(492\) −9.58219 −0.431998
\(493\) 28.3839 1.27835
\(494\) −14.5187 −0.653229
\(495\) 9.94011 0.446775
\(496\) −0.246502 −0.0110683
\(497\) 0 0
\(498\) −0.914295 −0.0409705
\(499\) −23.2072 −1.03890 −0.519448 0.854502i \(-0.673862\pi\)
−0.519448 + 0.854502i \(0.673862\pi\)
\(500\) −11.7860 −0.527085
\(501\) −17.8518 −0.797560
\(502\) −23.0973 −1.03088
\(503\) 12.5204 0.558256 0.279128 0.960254i \(-0.409955\pi\)
0.279128 + 0.960254i \(0.409955\pi\)
\(504\) 0 0
\(505\) −17.3799 −0.773395
\(506\) −3.44482 −0.153141
\(507\) 5.12084 0.227424
\(508\) −26.1445 −1.15998
\(509\) −24.2118 −1.07317 −0.536585 0.843847i \(-0.680286\pi\)
−0.536585 + 0.843847i \(0.680286\pi\)
\(510\) −15.0817 −0.667828
\(511\) 0 0
\(512\) 0.809780 0.0357875
\(513\) 5.98058 0.264049
\(514\) 3.77452 0.166487
\(515\) −14.9856 −0.660345
\(516\) 8.86420 0.390225
\(517\) 12.1298 0.533466
\(518\) 0 0
\(519\) −2.65359 −0.116480
\(520\) 19.7019 0.863987
\(521\) −6.82821 −0.299149 −0.149575 0.988750i \(-0.547790\pi\)
−0.149575 + 0.988750i \(0.547790\pi\)
\(522\) −3.51304 −0.153762
\(523\) −9.54004 −0.417157 −0.208579 0.978006i \(-0.566884\pi\)
−0.208579 + 0.978006i \(0.566884\pi\)
\(524\) −15.2105 −0.664474
\(525\) 0 0
\(526\) 1.33979 0.0584177
\(527\) 24.0619 1.04815
\(528\) 0.285132 0.0124088
\(529\) 1.00000 0.0434783
\(530\) −20.6574 −0.897298
\(531\) −8.30618 −0.360457
\(532\) 0 0
\(533\) −21.4830 −0.930530
\(534\) 2.85450 0.123526
\(535\) −14.2696 −0.616929
\(536\) −12.5708 −0.542978
\(537\) −11.4696 −0.494951
\(538\) −3.21323 −0.138532
\(539\) 0 0
\(540\) −3.12450 −0.134457
\(541\) 33.1748 1.42629 0.713147 0.701014i \(-0.247269\pi\)
0.713147 + 0.701014i \(0.247269\pi\)
\(542\) 3.47681 0.149342
\(543\) 15.8393 0.679728
\(544\) −39.7390 −1.70379
\(545\) −15.6119 −0.668742
\(546\) 0 0
\(547\) 17.2100 0.735848 0.367924 0.929856i \(-0.380069\pi\)
0.367924 + 0.929856i \(0.380069\pi\)
\(548\) 1.03746 0.0443182
\(549\) −3.71282 −0.158459
\(550\) −4.22984 −0.180361
\(551\) 24.2930 1.03492
\(552\) 2.81254 0.119710
\(553\) 0 0
\(554\) −9.85311 −0.418619
\(555\) −3.68577 −0.156452
\(556\) −0.683726 −0.0289964
\(557\) 17.8924 0.758125 0.379062 0.925371i \(-0.376247\pi\)
0.379062 + 0.925371i \(0.376247\pi\)
\(558\) −2.97811 −0.126073
\(559\) 19.8733 0.840550
\(560\) 0 0
\(561\) −27.8327 −1.17510
\(562\) −24.1731 −1.01968
\(563\) 16.4160 0.691852 0.345926 0.938262i \(-0.387565\pi\)
0.345926 + 0.938262i \(0.387565\pi\)
\(564\) −3.81278 −0.160547
\(565\) 9.47180 0.398482
\(566\) −25.9245 −1.08969
\(567\) 0 0
\(568\) 26.6918 1.11996
\(569\) 4.94134 0.207152 0.103576 0.994622i \(-0.466972\pi\)
0.103576 + 0.994622i \(0.466972\pi\)
\(570\) −12.9080 −0.540657
\(571\) −42.8618 −1.79371 −0.896855 0.442325i \(-0.854154\pi\)
−0.896855 + 0.442325i \(0.854154\pi\)
\(572\) 13.9982 0.585293
\(573\) −21.7951 −0.910502
\(574\) 0 0
\(575\) 1.22789 0.0512063
\(576\) 4.77528 0.198970
\(577\) 13.8130 0.575043 0.287522 0.957774i \(-0.407169\pi\)
0.287522 + 0.957774i \(0.407169\pi\)
\(578\) 27.5267 1.14496
\(579\) −16.0333 −0.666323
\(580\) −12.6917 −0.526993
\(581\) 0 0
\(582\) 13.7783 0.571128
\(583\) −38.1224 −1.57887
\(584\) −36.2877 −1.50160
\(585\) −7.00504 −0.289623
\(586\) −4.21486 −0.174114
\(587\) 19.9885 0.825012 0.412506 0.910955i \(-0.364654\pi\)
0.412506 + 0.910955i \(0.364654\pi\)
\(588\) 0 0
\(589\) 20.5939 0.848556
\(590\) 17.9274 0.738059
\(591\) −10.6653 −0.438711
\(592\) −0.105726 −0.00434533
\(593\) −2.72068 −0.111725 −0.0558625 0.998438i \(-0.517791\pi\)
−0.0558625 + 0.998438i \(0.517791\pi\)
\(594\) 3.44482 0.141343
\(595\) 0 0
\(596\) 23.2233 0.951262
\(597\) 3.65905 0.149755
\(598\) 2.42765 0.0992739
\(599\) 17.4698 0.713799 0.356899 0.934143i \(-0.383834\pi\)
0.356899 + 0.934143i \(0.383834\pi\)
\(600\) 3.45347 0.140987
\(601\) 27.6015 1.12589 0.562944 0.826495i \(-0.309669\pi\)
0.562944 + 0.826495i \(0.309669\pi\)
\(602\) 0 0
\(603\) 4.46957 0.182015
\(604\) 7.22728 0.294074
\(605\) −12.1411 −0.493607
\(606\) −6.02313 −0.244673
\(607\) −12.9173 −0.524295 −0.262148 0.965028i \(-0.584431\pi\)
−0.262148 + 0.965028i \(0.584431\pi\)
\(608\) −34.0115 −1.37935
\(609\) 0 0
\(610\) 8.01345 0.324455
\(611\) −8.54814 −0.345821
\(612\) 8.74872 0.353646
\(613\) −38.8541 −1.56930 −0.784651 0.619937i \(-0.787158\pi\)
−0.784651 + 0.619937i \(0.787158\pi\)
\(614\) −24.1351 −0.974013
\(615\) −19.0996 −0.770170
\(616\) 0 0
\(617\) −25.3417 −1.02022 −0.510108 0.860110i \(-0.670395\pi\)
−0.510108 + 0.860110i \(0.670395\pi\)
\(618\) −5.19338 −0.208908
\(619\) 35.5852 1.43029 0.715145 0.698976i \(-0.246361\pi\)
0.715145 + 0.698976i \(0.246361\pi\)
\(620\) −10.7591 −0.432096
\(621\) −1.00000 −0.0401286
\(622\) 23.3586 0.936596
\(623\) 0 0
\(624\) −0.200939 −0.00804401
\(625\) −29.6317 −1.18527
\(626\) 3.64453 0.145665
\(627\) −23.8212 −0.951328
\(628\) −17.6897 −0.705897
\(629\) 10.3203 0.411498
\(630\) 0 0
\(631\) −12.5559 −0.499842 −0.249921 0.968266i \(-0.580405\pi\)
−0.249921 + 0.968266i \(0.580405\pi\)
\(632\) −20.0338 −0.796903
\(633\) 28.1857 1.12028
\(634\) −6.91710 −0.274713
\(635\) −52.1123 −2.06801
\(636\) 11.9831 0.475161
\(637\) 0 0
\(638\) 13.9928 0.553980
\(639\) −9.49028 −0.375429
\(640\) 18.0780 0.714596
\(641\) −44.0488 −1.73982 −0.869912 0.493207i \(-0.835825\pi\)
−0.869912 + 0.493207i \(0.835825\pi\)
\(642\) −4.94524 −0.195173
\(643\) 27.1797 1.07186 0.535931 0.844262i \(-0.319961\pi\)
0.535931 + 0.844262i \(0.319961\pi\)
\(644\) 0 0
\(645\) 17.6685 0.695696
\(646\) 36.1429 1.42202
\(647\) 33.7648 1.32743 0.663716 0.747985i \(-0.268978\pi\)
0.663716 + 0.747985i \(0.268978\pi\)
\(648\) −2.81254 −0.110487
\(649\) 33.0843 1.29867
\(650\) 2.98087 0.116919
\(651\) 0 0
\(652\) 20.9708 0.821281
\(653\) 31.9330 1.24963 0.624817 0.780771i \(-0.285174\pi\)
0.624817 + 0.780771i \(0.285174\pi\)
\(654\) −5.41044 −0.211565
\(655\) −30.3182 −1.18463
\(656\) −0.547872 −0.0213908
\(657\) 12.9021 0.503359
\(658\) 0 0
\(659\) 47.4986 1.85028 0.925142 0.379622i \(-0.123946\pi\)
0.925142 + 0.379622i \(0.123946\pi\)
\(660\) 12.4452 0.484429
\(661\) −29.4292 −1.14466 −0.572332 0.820022i \(-0.693961\pi\)
−0.572332 + 0.820022i \(0.693961\pi\)
\(662\) 20.8320 0.809658
\(663\) 19.6144 0.761759
\(664\) −2.97330 −0.115387
\(665\) 0 0
\(666\) −1.27733 −0.0494956
\(667\) −4.06198 −0.157281
\(668\) −22.3508 −0.864778
\(669\) −24.2280 −0.936710
\(670\) −9.64677 −0.372687
\(671\) 14.7885 0.570904
\(672\) 0 0
\(673\) −21.8311 −0.841528 −0.420764 0.907170i \(-0.638238\pi\)
−0.420764 + 0.907170i \(0.638238\pi\)
\(674\) −5.97876 −0.230293
\(675\) −1.22789 −0.0472613
\(676\) 6.41138 0.246591
\(677\) 31.9593 1.22830 0.614148 0.789191i \(-0.289500\pi\)
0.614148 + 0.789191i \(0.289500\pi\)
\(678\) 3.28252 0.126065
\(679\) 0 0
\(680\) −49.0459 −1.88083
\(681\) 8.03654 0.307961
\(682\) 11.8621 0.454223
\(683\) 5.80239 0.222022 0.111011 0.993819i \(-0.464591\pi\)
0.111011 + 0.993819i \(0.464591\pi\)
\(684\) 7.48779 0.286303
\(685\) 2.06791 0.0790109
\(686\) 0 0
\(687\) −13.7790 −0.525702
\(688\) 0.506820 0.0193223
\(689\) 26.8658 1.02350
\(690\) 2.15832 0.0821658
\(691\) 50.4258 1.91829 0.959145 0.282916i \(-0.0913019\pi\)
0.959145 + 0.282916i \(0.0913019\pi\)
\(692\) −3.32235 −0.126297
\(693\) 0 0
\(694\) 10.5406 0.400115
\(695\) −1.36283 −0.0516951
\(696\) −11.4245 −0.433044
\(697\) 53.4796 2.02568
\(698\) 23.4118 0.886151
\(699\) 1.83601 0.0694443
\(700\) 0 0
\(701\) 28.8283 1.08883 0.544414 0.838817i \(-0.316752\pi\)
0.544414 + 0.838817i \(0.316752\pi\)
\(702\) −2.42765 −0.0916257
\(703\) 8.83286 0.333138
\(704\) −19.0204 −0.716858
\(705\) −7.59979 −0.286225
\(706\) −24.7153 −0.930171
\(707\) 0 0
\(708\) −10.3995 −0.390837
\(709\) 32.4556 1.21889 0.609447 0.792826i \(-0.291391\pi\)
0.609447 + 0.792826i \(0.291391\pi\)
\(710\) 20.4831 0.768715
\(711\) 7.12304 0.267135
\(712\) 9.28288 0.347891
\(713\) −3.44346 −0.128959
\(714\) 0 0
\(715\) 27.9017 1.04347
\(716\) −14.3602 −0.536665
\(717\) 4.42206 0.165145
\(718\) −16.8325 −0.628184
\(719\) −16.0351 −0.598009 −0.299004 0.954252i \(-0.596655\pi\)
−0.299004 + 0.954252i \(0.596655\pi\)
\(720\) −0.178647 −0.00665777
\(721\) 0 0
\(722\) 14.5014 0.539685
\(723\) 28.4589 1.05840
\(724\) 19.8310 0.737015
\(725\) −4.98765 −0.185237
\(726\) −4.20760 −0.156159
\(727\) 40.6150 1.50633 0.753164 0.657833i \(-0.228527\pi\)
0.753164 + 0.657833i \(0.228527\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −27.8469 −1.03066
\(731\) −49.4724 −1.82980
\(732\) −4.64851 −0.171814
\(733\) −2.41457 −0.0891843 −0.0445922 0.999005i \(-0.514199\pi\)
−0.0445922 + 0.999005i \(0.514199\pi\)
\(734\) 4.16724 0.153816
\(735\) 0 0
\(736\) 5.68699 0.209625
\(737\) −17.8027 −0.655772
\(738\) −6.61911 −0.243653
\(739\) 2.79635 0.102865 0.0514327 0.998676i \(-0.483621\pi\)
0.0514327 + 0.998676i \(0.483621\pi\)
\(740\) −4.61466 −0.169638
\(741\) 16.7874 0.616700
\(742\) 0 0
\(743\) 3.56515 0.130793 0.0653964 0.997859i \(-0.479169\pi\)
0.0653964 + 0.997859i \(0.479169\pi\)
\(744\) −9.68486 −0.355064
\(745\) 46.2896 1.69592
\(746\) −21.3060 −0.780069
\(747\) 1.05716 0.0386795
\(748\) −34.8470 −1.27413
\(749\) 0 0
\(750\) −8.14143 −0.297283
\(751\) −3.79314 −0.138413 −0.0692067 0.997602i \(-0.522047\pi\)
−0.0692067 + 0.997602i \(0.522047\pi\)
\(752\) −0.218000 −0.00794964
\(753\) 26.7064 0.973234
\(754\) −9.86106 −0.359119
\(755\) 14.4057 0.524278
\(756\) 0 0
\(757\) −6.45659 −0.234669 −0.117334 0.993092i \(-0.537435\pi\)
−0.117334 + 0.993092i \(0.537435\pi\)
\(758\) 27.5010 0.998883
\(759\) 3.98310 0.144577
\(760\) −41.9770 −1.52267
\(761\) −22.4004 −0.812014 −0.406007 0.913870i \(-0.633079\pi\)
−0.406007 + 0.913870i \(0.633079\pi\)
\(762\) −18.0599 −0.654241
\(763\) 0 0
\(764\) −27.2878 −0.987239
\(765\) 17.4383 0.630483
\(766\) −19.3429 −0.698888
\(767\) −23.3153 −0.841867
\(768\) 15.8156 0.570697
\(769\) 53.5732 1.93190 0.965950 0.258731i \(-0.0833042\pi\)
0.965950 + 0.258731i \(0.0833042\pi\)
\(770\) 0 0
\(771\) −4.36432 −0.157177
\(772\) −20.0740 −0.722481
\(773\) −44.5244 −1.60143 −0.800715 0.599045i \(-0.795547\pi\)
−0.800715 + 0.599045i \(0.795547\pi\)
\(774\) 6.12315 0.220092
\(775\) −4.22817 −0.151880
\(776\) 44.8072 1.60849
\(777\) 0 0
\(778\) 17.2191 0.617333
\(779\) 45.7717 1.63994
\(780\) −8.77043 −0.314032
\(781\) 37.8007 1.35262
\(782\) −6.04338 −0.216111
\(783\) 4.06198 0.145163
\(784\) 0 0
\(785\) −35.2599 −1.25848
\(786\) −10.5070 −0.374772
\(787\) −15.2713 −0.544364 −0.272182 0.962246i \(-0.587745\pi\)
−0.272182 + 0.962246i \(0.587745\pi\)
\(788\) −13.3531 −0.475685
\(789\) −1.54914 −0.0551509
\(790\) −15.3738 −0.546975
\(791\) 0 0
\(792\) 11.2026 0.398068
\(793\) −10.4218 −0.370090
\(794\) 23.4434 0.831977
\(795\) 23.8852 0.847122
\(796\) 4.58119 0.162376
\(797\) 38.5369 1.36505 0.682524 0.730863i \(-0.260882\pi\)
0.682524 + 0.730863i \(0.260882\pi\)
\(798\) 0 0
\(799\) 21.2797 0.752822
\(800\) 6.98297 0.246885
\(801\) −3.30053 −0.116619
\(802\) −29.5594 −1.04378
\(803\) −51.3904 −1.81353
\(804\) 5.59599 0.197355
\(805\) 0 0
\(806\) −8.35951 −0.294451
\(807\) 3.71532 0.130786
\(808\) −19.5873 −0.689080
\(809\) −36.3518 −1.27806 −0.639031 0.769181i \(-0.720664\pi\)
−0.639031 + 0.769181i \(0.720664\pi\)
\(810\) −2.15832 −0.0758356
\(811\) −21.9469 −0.770661 −0.385330 0.922779i \(-0.625912\pi\)
−0.385330 + 0.922779i \(0.625912\pi\)
\(812\) 0 0
\(813\) −4.02009 −0.140991
\(814\) 5.08774 0.178325
\(815\) 41.7999 1.46419
\(816\) 0.500218 0.0175111
\(817\) −42.3421 −1.48136
\(818\) 13.1006 0.458053
\(819\) 0 0
\(820\) −23.9130 −0.835080
\(821\) −13.0796 −0.456481 −0.228241 0.973605i \(-0.573297\pi\)
−0.228241 + 0.973605i \(0.573297\pi\)
\(822\) 0.716651 0.0249961
\(823\) 31.6053 1.10169 0.550845 0.834608i \(-0.314306\pi\)
0.550845 + 0.834608i \(0.314306\pi\)
\(824\) −16.8890 −0.588355
\(825\) 4.89079 0.170275
\(826\) 0 0
\(827\) 21.5412 0.749060 0.374530 0.927215i \(-0.377804\pi\)
0.374530 + 0.927215i \(0.377804\pi\)
\(828\) −1.25202 −0.0435106
\(829\) 33.3139 1.15704 0.578519 0.815669i \(-0.303631\pi\)
0.578519 + 0.815669i \(0.303631\pi\)
\(830\) −2.28169 −0.0791986
\(831\) 11.3927 0.395209
\(832\) 13.4041 0.464704
\(833\) 0 0
\(834\) −0.472299 −0.0163544
\(835\) −44.5505 −1.54173
\(836\) −29.8246 −1.03151
\(837\) 3.44346 0.119023
\(838\) 12.1516 0.419769
\(839\) 4.75994 0.164331 0.0821656 0.996619i \(-0.473816\pi\)
0.0821656 + 0.996619i \(0.473816\pi\)
\(840\) 0 0
\(841\) −12.5003 −0.431045
\(842\) −22.5654 −0.777653
\(843\) 27.9503 0.962661
\(844\) 35.2891 1.21470
\(845\) 12.7794 0.439625
\(846\) −2.63377 −0.0905507
\(847\) 0 0
\(848\) 0.685147 0.0235281
\(849\) 29.9754 1.02875
\(850\) −7.42057 −0.254524
\(851\) −1.47693 −0.0506283
\(852\) −11.8820 −0.407070
\(853\) 34.8330 1.19266 0.596330 0.802740i \(-0.296625\pi\)
0.596330 + 0.802740i \(0.296625\pi\)
\(854\) 0 0
\(855\) 14.9250 0.510423
\(856\) −16.0820 −0.549672
\(857\) 19.8576 0.678323 0.339161 0.940728i \(-0.389857\pi\)
0.339161 + 0.940728i \(0.389857\pi\)
\(858\) 9.66956 0.330113
\(859\) 33.7997 1.15323 0.576615 0.817016i \(-0.304373\pi\)
0.576615 + 0.817016i \(0.304373\pi\)
\(860\) 22.1213 0.754329
\(861\) 0 0
\(862\) 2.67627 0.0911541
\(863\) 35.6001 1.21184 0.605921 0.795525i \(-0.292805\pi\)
0.605921 + 0.795525i \(0.292805\pi\)
\(864\) −5.68699 −0.193475
\(865\) −6.62224 −0.225163
\(866\) 16.4181 0.557911
\(867\) −31.8279 −1.08093
\(868\) 0 0
\(869\) −28.3718 −0.962446
\(870\) −8.76706 −0.297231
\(871\) 12.5460 0.425106
\(872\) −17.5948 −0.595836
\(873\) −15.9312 −0.539191
\(874\) −5.17236 −0.174958
\(875\) 0 0
\(876\) 16.1537 0.545782
\(877\) 14.8802 0.502470 0.251235 0.967926i \(-0.419163\pi\)
0.251235 + 0.967926i \(0.419163\pi\)
\(878\) −10.3534 −0.349411
\(879\) 4.87346 0.164378
\(880\) 0.711568 0.0239869
\(881\) −25.2288 −0.849980 −0.424990 0.905198i \(-0.639722\pi\)
−0.424990 + 0.905198i \(0.639722\pi\)
\(882\) 0 0
\(883\) −5.13080 −0.172665 −0.0863325 0.996266i \(-0.527515\pi\)
−0.0863325 + 0.996266i \(0.527515\pi\)
\(884\) 24.5575 0.825959
\(885\) −20.7287 −0.696787
\(886\) 16.8492 0.566059
\(887\) −4.27219 −0.143446 −0.0717231 0.997425i \(-0.522850\pi\)
−0.0717231 + 0.997425i \(0.522850\pi\)
\(888\) −4.15391 −0.139396
\(889\) 0 0
\(890\) 7.12361 0.238784
\(891\) −3.98310 −0.133439
\(892\) −30.3340 −1.01566
\(893\) 18.2127 0.609465
\(894\) 16.0420 0.536525
\(895\) −28.6233 −0.956772
\(896\) 0 0
\(897\) −2.80699 −0.0937225
\(898\) −22.4308 −0.748526
\(899\) 13.9873 0.466501
\(900\) −1.53733 −0.0512445
\(901\) −66.8795 −2.22808
\(902\) 26.3646 0.877844
\(903\) 0 0
\(904\) 10.6748 0.355039
\(905\) 39.5280 1.31396
\(906\) 4.99241 0.165862
\(907\) −29.2234 −0.970346 −0.485173 0.874418i \(-0.661243\pi\)
−0.485173 + 0.874418i \(0.661243\pi\)
\(908\) 10.0619 0.333916
\(909\) 6.96429 0.230991
\(910\) 0 0
\(911\) −48.3422 −1.60165 −0.800824 0.598900i \(-0.795605\pi\)
−0.800824 + 0.598900i \(0.795605\pi\)
\(912\) 0.428122 0.0141766
\(913\) −4.21077 −0.139356
\(914\) 3.98987 0.131973
\(915\) −9.26560 −0.306311
\(916\) −17.2516 −0.570008
\(917\) 0 0
\(918\) 6.04338 0.199461
\(919\) 30.7928 1.01576 0.507881 0.861427i \(-0.330429\pi\)
0.507881 + 0.861427i \(0.330429\pi\)
\(920\) 7.01890 0.231406
\(921\) 27.9064 0.919547
\(922\) 11.9507 0.393575
\(923\) −26.6391 −0.876835
\(924\) 0 0
\(925\) −1.81349 −0.0596273
\(926\) −29.5169 −0.969985
\(927\) 6.00488 0.197226
\(928\) −23.1004 −0.758309
\(929\) −24.6571 −0.808975 −0.404487 0.914544i \(-0.632550\pi\)
−0.404487 + 0.914544i \(0.632550\pi\)
\(930\) −7.43209 −0.243708
\(931\) 0 0
\(932\) 2.29872 0.0752970
\(933\) −27.0086 −0.884221
\(934\) −24.1915 −0.791569
\(935\) −69.4585 −2.27154
\(936\) −7.89475 −0.258048
\(937\) 35.2046 1.15008 0.575042 0.818124i \(-0.304986\pi\)
0.575042 + 0.818124i \(0.304986\pi\)
\(938\) 0 0
\(939\) −4.21402 −0.137519
\(940\) −9.51508 −0.310348
\(941\) −22.0830 −0.719885 −0.359942 0.932975i \(-0.617204\pi\)
−0.359942 + 0.932975i \(0.617204\pi\)
\(942\) −12.2196 −0.398135
\(943\) −7.65339 −0.249229
\(944\) −0.594602 −0.0193526
\(945\) 0 0
\(946\) −24.3891 −0.792958
\(947\) 5.85753 0.190344 0.0951720 0.995461i \(-0.469660\pi\)
0.0951720 + 0.995461i \(0.469660\pi\)
\(948\) 8.91818 0.289649
\(949\) 36.2160 1.17562
\(950\) −6.35106 −0.206056
\(951\) 7.99795 0.259351
\(952\) 0 0
\(953\) −48.8701 −1.58306 −0.791530 0.611131i \(-0.790715\pi\)
−0.791530 + 0.611131i \(0.790715\pi\)
\(954\) 8.27760 0.267997
\(955\) −54.3912 −1.76006
\(956\) 5.53650 0.179063
\(957\) −16.1793 −0.523001
\(958\) −28.2177 −0.911672
\(959\) 0 0
\(960\) 11.9170 0.384621
\(961\) −19.1426 −0.617503
\(962\) −3.58545 −0.115600
\(963\) 5.71797 0.184259
\(964\) 35.6310 1.14760
\(965\) −40.0124 −1.28804
\(966\) 0 0
\(967\) 11.7300 0.377213 0.188606 0.982053i \(-0.439603\pi\)
0.188606 + 0.982053i \(0.439603\pi\)
\(968\) −13.6832 −0.439794
\(969\) −41.7905 −1.34250
\(970\) 34.3847 1.10403
\(971\) −24.3527 −0.781516 −0.390758 0.920493i \(-0.627787\pi\)
−0.390758 + 0.920493i \(0.627787\pi\)
\(972\) 1.25202 0.0401585
\(973\) 0 0
\(974\) −22.4363 −0.718905
\(975\) −3.44666 −0.110381
\(976\) −0.265784 −0.00850753
\(977\) 50.9094 1.62874 0.814368 0.580348i \(-0.197084\pi\)
0.814368 + 0.580348i \(0.197084\pi\)
\(978\) 14.4861 0.463214
\(979\) 13.1463 0.420159
\(980\) 0 0
\(981\) 6.25586 0.199734
\(982\) 22.4868 0.717581
\(983\) −33.5958 −1.07154 −0.535770 0.844364i \(-0.679979\pi\)
−0.535770 + 0.844364i \(0.679979\pi\)
\(984\) −21.5255 −0.686207
\(985\) −26.6160 −0.848056
\(986\) 24.5481 0.781770
\(987\) 0 0
\(988\) 21.0181 0.668675
\(989\) 7.07993 0.225129
\(990\) 8.59680 0.273224
\(991\) 10.4350 0.331480 0.165740 0.986169i \(-0.446999\pi\)
0.165740 + 0.986169i \(0.446999\pi\)
\(992\) −19.5829 −0.621758
\(993\) −24.0871 −0.764382
\(994\) 0 0
\(995\) 9.13142 0.289485
\(996\) 1.32358 0.0419394
\(997\) 15.1736 0.480553 0.240277 0.970704i \(-0.422762\pi\)
0.240277 + 0.970704i \(0.422762\pi\)
\(998\) −20.0710 −0.635335
\(999\) 1.47693 0.0467278
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 3381.2.a.bi.1.6 10
7.2 even 3 483.2.i.h.277.5 20
7.4 even 3 483.2.i.h.415.5 yes 20
7.6 odd 2 3381.2.a.bj.1.6 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.5 20 7.2 even 3
483.2.i.h.415.5 yes 20 7.4 even 3
3381.2.a.bi.1.6 10 1.1 even 1 trivial
3381.2.a.bj.1.6 10 7.6 odd 2