Properties

Label 3381.2.a.bi.1.1
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 1.1
Root \(-2.73999\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.73999 q^{2} -1.00000 q^{3} +5.50755 q^{4} -3.38181 q^{5} +2.73999 q^{6} -9.61066 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.73999 q^{2} -1.00000 q^{3} +5.50755 q^{4} -3.38181 q^{5} +2.73999 q^{6} -9.61066 q^{8} +1.00000 q^{9} +9.26614 q^{10} +1.88341 q^{11} -5.50755 q^{12} +1.76938 q^{13} +3.38181 q^{15} +15.3180 q^{16} -5.89595 q^{17} -2.73999 q^{18} -6.62660 q^{19} -18.6255 q^{20} -5.16053 q^{22} +1.00000 q^{23} +9.61066 q^{24} +6.43667 q^{25} -4.84807 q^{26} -1.00000 q^{27} +8.24468 q^{29} -9.26614 q^{30} +2.94359 q^{31} -22.7499 q^{32} -1.88341 q^{33} +16.1549 q^{34} +5.50755 q^{36} -4.43562 q^{37} +18.1568 q^{38} -1.76938 q^{39} +32.5015 q^{40} -5.90495 q^{41} +0.669028 q^{43} +10.3730 q^{44} -3.38181 q^{45} -2.73999 q^{46} +1.79517 q^{47} -15.3180 q^{48} -17.6364 q^{50} +5.89595 q^{51} +9.74492 q^{52} -9.68500 q^{53} +2.73999 q^{54} -6.36935 q^{55} +6.62660 q^{57} -22.5904 q^{58} +1.36723 q^{59} +18.6255 q^{60} +9.36339 q^{61} -8.06540 q^{62} +31.6985 q^{64} -5.98370 q^{65} +5.16053 q^{66} -8.17894 q^{67} -32.4723 q^{68} -1.00000 q^{69} -0.295052 q^{71} -9.61066 q^{72} -4.45751 q^{73} +12.1536 q^{74} -6.43667 q^{75} -36.4963 q^{76} +4.84807 q^{78} -3.02768 q^{79} -51.8027 q^{80} +1.00000 q^{81} +16.1795 q^{82} -10.5300 q^{83} +19.9390 q^{85} -1.83313 q^{86} -8.24468 q^{87} -18.1008 q^{88} -10.7196 q^{89} +9.26614 q^{90} +5.50755 q^{92} -2.94359 q^{93} -4.91874 q^{94} +22.4099 q^{95} +22.7499 q^{96} +15.6290 q^{97} +1.88341 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.73999 −1.93747 −0.968733 0.248105i \(-0.920192\pi\)
−0.968733 + 0.248105i \(0.920192\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.50755 2.75378
\(5\) −3.38181 −1.51239 −0.756197 0.654345i \(-0.772945\pi\)
−0.756197 + 0.654345i \(0.772945\pi\)
\(6\) 2.73999 1.11860
\(7\) 0 0
\(8\) −9.61066 −3.39788
\(9\) 1.00000 0.333333
\(10\) 9.26614 2.93021
\(11\) 1.88341 0.567870 0.283935 0.958844i \(-0.408360\pi\)
0.283935 + 0.958844i \(0.408360\pi\)
\(12\) −5.50755 −1.58989
\(13\) 1.76938 0.490736 0.245368 0.969430i \(-0.421091\pi\)
0.245368 + 0.969430i \(0.421091\pi\)
\(14\) 0 0
\(15\) 3.38181 0.873181
\(16\) 15.3180 3.82950
\(17\) −5.89595 −1.42998 −0.714989 0.699135i \(-0.753569\pi\)
−0.714989 + 0.699135i \(0.753569\pi\)
\(18\) −2.73999 −0.645822
\(19\) −6.62660 −1.52025 −0.760123 0.649779i \(-0.774862\pi\)
−0.760123 + 0.649779i \(0.774862\pi\)
\(20\) −18.6255 −4.16479
\(21\) 0 0
\(22\) −5.16053 −1.10023
\(23\) 1.00000 0.208514
\(24\) 9.61066 1.96177
\(25\) 6.43667 1.28733
\(26\) −4.84807 −0.950785
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 8.24468 1.53100 0.765500 0.643436i \(-0.222492\pi\)
0.765500 + 0.643436i \(0.222492\pi\)
\(30\) −9.26614 −1.69176
\(31\) 2.94359 0.528684 0.264342 0.964429i \(-0.414845\pi\)
0.264342 + 0.964429i \(0.414845\pi\)
\(32\) −22.7499 −4.02165
\(33\) −1.88341 −0.327860
\(34\) 16.1549 2.77054
\(35\) 0 0
\(36\) 5.50755 0.917925
\(37\) −4.43562 −0.729212 −0.364606 0.931162i \(-0.618796\pi\)
−0.364606 + 0.931162i \(0.618796\pi\)
\(38\) 18.1568 2.94543
\(39\) −1.76938 −0.283327
\(40\) 32.5015 5.13893
\(41\) −5.90495 −0.922198 −0.461099 0.887349i \(-0.652545\pi\)
−0.461099 + 0.887349i \(0.652545\pi\)
\(42\) 0 0
\(43\) 0.669028 0.102026 0.0510129 0.998698i \(-0.483755\pi\)
0.0510129 + 0.998698i \(0.483755\pi\)
\(44\) 10.3730 1.56379
\(45\) −3.38181 −0.504131
\(46\) −2.73999 −0.403990
\(47\) 1.79517 0.261852 0.130926 0.991392i \(-0.458205\pi\)
0.130926 + 0.991392i \(0.458205\pi\)
\(48\) −15.3180 −2.21096
\(49\) 0 0
\(50\) −17.6364 −2.49416
\(51\) 5.89595 0.825599
\(52\) 9.74492 1.35138
\(53\) −9.68500 −1.33034 −0.665169 0.746693i \(-0.731640\pi\)
−0.665169 + 0.746693i \(0.731640\pi\)
\(54\) 2.73999 0.372866
\(55\) −6.36935 −0.858843
\(56\) 0 0
\(57\) 6.62660 0.877715
\(58\) −22.5904 −2.96626
\(59\) 1.36723 0.177998 0.0889988 0.996032i \(-0.471633\pi\)
0.0889988 + 0.996032i \(0.471633\pi\)
\(60\) 18.6255 2.40454
\(61\) 9.36339 1.19886 0.599430 0.800427i \(-0.295394\pi\)
0.599430 + 0.800427i \(0.295394\pi\)
\(62\) −8.06540 −1.02431
\(63\) 0 0
\(64\) 31.6985 3.96231
\(65\) −5.98370 −0.742186
\(66\) 5.16053 0.635218
\(67\) −8.17894 −0.999216 −0.499608 0.866252i \(-0.666523\pi\)
−0.499608 + 0.866252i \(0.666523\pi\)
\(68\) −32.4723 −3.93784
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −0.295052 −0.0350162 −0.0175081 0.999847i \(-0.505573\pi\)
−0.0175081 + 0.999847i \(0.505573\pi\)
\(72\) −9.61066 −1.13263
\(73\) −4.45751 −0.521712 −0.260856 0.965378i \(-0.584005\pi\)
−0.260856 + 0.965378i \(0.584005\pi\)
\(74\) 12.1536 1.41282
\(75\) −6.43667 −0.743242
\(76\) −36.4963 −4.18642
\(77\) 0 0
\(78\) 4.84807 0.548936
\(79\) −3.02768 −0.340641 −0.170320 0.985389i \(-0.554480\pi\)
−0.170320 + 0.985389i \(0.554480\pi\)
\(80\) −51.8027 −5.79171
\(81\) 1.00000 0.111111
\(82\) 16.1795 1.78673
\(83\) −10.5300 −1.15582 −0.577910 0.816100i \(-0.696132\pi\)
−0.577910 + 0.816100i \(0.696132\pi\)
\(84\) 0 0
\(85\) 19.9390 2.16269
\(86\) −1.83313 −0.197672
\(87\) −8.24468 −0.883923
\(88\) −18.1008 −1.92955
\(89\) −10.7196 −1.13628 −0.568138 0.822933i \(-0.692336\pi\)
−0.568138 + 0.822933i \(0.692336\pi\)
\(90\) 9.26614 0.976737
\(91\) 0 0
\(92\) 5.50755 0.574202
\(93\) −2.94359 −0.305236
\(94\) −4.91874 −0.507329
\(95\) 22.4099 2.29921
\(96\) 22.7499 2.32190
\(97\) 15.6290 1.58688 0.793441 0.608648i \(-0.208288\pi\)
0.793441 + 0.608648i \(0.208288\pi\)
\(98\) 0 0
\(99\) 1.88341 0.189290
\(100\) 35.4503 3.54503
\(101\) 7.56916 0.753159 0.376580 0.926384i \(-0.377100\pi\)
0.376580 + 0.926384i \(0.377100\pi\)
\(102\) −16.1549 −1.59957
\(103\) −2.62127 −0.258281 −0.129141 0.991626i \(-0.541222\pi\)
−0.129141 + 0.991626i \(0.541222\pi\)
\(104\) −17.0049 −1.66746
\(105\) 0 0
\(106\) 26.5368 2.57748
\(107\) 17.3322 1.67557 0.837783 0.546004i \(-0.183852\pi\)
0.837783 + 0.546004i \(0.183852\pi\)
\(108\) −5.50755 −0.529964
\(109\) −13.3681 −1.28043 −0.640214 0.768197i \(-0.721154\pi\)
−0.640214 + 0.768197i \(0.721154\pi\)
\(110\) 17.4520 1.66398
\(111\) 4.43562 0.421011
\(112\) 0 0
\(113\) −1.18709 −0.111672 −0.0558360 0.998440i \(-0.517782\pi\)
−0.0558360 + 0.998440i \(0.517782\pi\)
\(114\) −18.1568 −1.70054
\(115\) −3.38181 −0.315356
\(116\) 45.4080 4.21603
\(117\) 1.76938 0.163579
\(118\) −3.74618 −0.344864
\(119\) 0 0
\(120\) −32.5015 −2.96696
\(121\) −7.45276 −0.677523
\(122\) −25.6556 −2.32275
\(123\) 5.90495 0.532431
\(124\) 16.2120 1.45588
\(125\) −4.85853 −0.434560
\(126\) 0 0
\(127\) −12.2311 −1.08533 −0.542667 0.839948i \(-0.682585\pi\)
−0.542667 + 0.839948i \(0.682585\pi\)
\(128\) −41.3538 −3.65519
\(129\) −0.669028 −0.0589046
\(130\) 16.3953 1.43796
\(131\) −6.05670 −0.529176 −0.264588 0.964362i \(-0.585236\pi\)
−0.264588 + 0.964362i \(0.585236\pi\)
\(132\) −10.3730 −0.902853
\(133\) 0 0
\(134\) 22.4102 1.93595
\(135\) 3.38181 0.291060
\(136\) 56.6640 4.85890
\(137\) −14.2926 −1.22110 −0.610550 0.791978i \(-0.709052\pi\)
−0.610550 + 0.791978i \(0.709052\pi\)
\(138\) 2.73999 0.233244
\(139\) −16.4290 −1.39349 −0.696743 0.717321i \(-0.745368\pi\)
−0.696743 + 0.717321i \(0.745368\pi\)
\(140\) 0 0
\(141\) −1.79517 −0.151180
\(142\) 0.808440 0.0678427
\(143\) 3.33246 0.278675
\(144\) 15.3180 1.27650
\(145\) −27.8820 −2.31547
\(146\) 12.2135 1.01080
\(147\) 0 0
\(148\) −24.4294 −2.00809
\(149\) 13.8036 1.13084 0.565418 0.824805i \(-0.308715\pi\)
0.565418 + 0.824805i \(0.308715\pi\)
\(150\) 17.6364 1.44001
\(151\) −4.24714 −0.345628 −0.172814 0.984955i \(-0.555286\pi\)
−0.172814 + 0.984955i \(0.555286\pi\)
\(152\) 63.6860 5.16562
\(153\) −5.89595 −0.476660
\(154\) 0 0
\(155\) −9.95467 −0.799578
\(156\) −9.74492 −0.780218
\(157\) −0.667792 −0.0532956 −0.0266478 0.999645i \(-0.508483\pi\)
−0.0266478 + 0.999645i \(0.508483\pi\)
\(158\) 8.29583 0.659980
\(159\) 9.68500 0.768071
\(160\) 76.9359 6.08232
\(161\) 0 0
\(162\) −2.73999 −0.215274
\(163\) −0.727180 −0.0569572 −0.0284786 0.999594i \(-0.509066\pi\)
−0.0284786 + 0.999594i \(0.509066\pi\)
\(164\) −32.5218 −2.53952
\(165\) 6.36935 0.495853
\(166\) 28.8522 2.23936
\(167\) 9.42526 0.729349 0.364674 0.931135i \(-0.381180\pi\)
0.364674 + 0.931135i \(0.381180\pi\)
\(168\) 0 0
\(169\) −9.86931 −0.759178
\(170\) −54.6327 −4.19014
\(171\) −6.62660 −0.506749
\(172\) 3.68470 0.280956
\(173\) 10.8514 0.825014 0.412507 0.910954i \(-0.364653\pi\)
0.412507 + 0.910954i \(0.364653\pi\)
\(174\) 22.5904 1.71257
\(175\) 0 0
\(176\) 28.8501 2.17466
\(177\) −1.36723 −0.102767
\(178\) 29.3716 2.20150
\(179\) −9.94360 −0.743220 −0.371610 0.928389i \(-0.621194\pi\)
−0.371610 + 0.928389i \(0.621194\pi\)
\(180\) −18.6255 −1.38826
\(181\) −0.975614 −0.0725168 −0.0362584 0.999342i \(-0.511544\pi\)
−0.0362584 + 0.999342i \(0.511544\pi\)
\(182\) 0 0
\(183\) −9.36339 −0.692162
\(184\) −9.61066 −0.708507
\(185\) 15.0004 1.10285
\(186\) 8.06540 0.591384
\(187\) −11.1045 −0.812042
\(188\) 9.88696 0.721081
\(189\) 0 0
\(190\) −61.4030 −4.45464
\(191\) 7.00670 0.506987 0.253494 0.967337i \(-0.418420\pi\)
0.253494 + 0.967337i \(0.418420\pi\)
\(192\) −31.6985 −2.28764
\(193\) 12.5202 0.901225 0.450613 0.892720i \(-0.351206\pi\)
0.450613 + 0.892720i \(0.351206\pi\)
\(194\) −42.8232 −3.07453
\(195\) 5.98370 0.428502
\(196\) 0 0
\(197\) −25.3287 −1.80459 −0.902296 0.431116i \(-0.858120\pi\)
−0.902296 + 0.431116i \(0.858120\pi\)
\(198\) −5.16053 −0.366743
\(199\) 0.648559 0.0459751 0.0229876 0.999736i \(-0.492682\pi\)
0.0229876 + 0.999736i \(0.492682\pi\)
\(200\) −61.8606 −4.37420
\(201\) 8.17894 0.576898
\(202\) −20.7394 −1.45922
\(203\) 0 0
\(204\) 32.4723 2.27351
\(205\) 19.9694 1.39473
\(206\) 7.18225 0.500411
\(207\) 1.00000 0.0695048
\(208\) 27.1033 1.87928
\(209\) −12.4806 −0.863303
\(210\) 0 0
\(211\) 14.4647 0.995790 0.497895 0.867237i \(-0.334106\pi\)
0.497895 + 0.867237i \(0.334106\pi\)
\(212\) −53.3406 −3.66345
\(213\) 0.295052 0.0202166
\(214\) −47.4900 −3.24635
\(215\) −2.26253 −0.154303
\(216\) 9.61066 0.653922
\(217\) 0 0
\(218\) 36.6284 2.48079
\(219\) 4.45751 0.301210
\(220\) −35.0795 −2.36506
\(221\) −10.4322 −0.701743
\(222\) −12.1536 −0.815694
\(223\) −29.7327 −1.99105 −0.995525 0.0945015i \(-0.969874\pi\)
−0.995525 + 0.0945015i \(0.969874\pi\)
\(224\) 0 0
\(225\) 6.43667 0.429111
\(226\) 3.25261 0.216361
\(227\) −6.49019 −0.430769 −0.215385 0.976529i \(-0.569101\pi\)
−0.215385 + 0.976529i \(0.569101\pi\)
\(228\) 36.4963 2.41703
\(229\) −1.23608 −0.0816824 −0.0408412 0.999166i \(-0.513004\pi\)
−0.0408412 + 0.999166i \(0.513004\pi\)
\(230\) 9.26614 0.610991
\(231\) 0 0
\(232\) −79.2368 −5.20215
\(233\) 24.3562 1.59563 0.797814 0.602904i \(-0.205990\pi\)
0.797814 + 0.602904i \(0.205990\pi\)
\(234\) −4.84807 −0.316928
\(235\) −6.07091 −0.396023
\(236\) 7.53006 0.490165
\(237\) 3.02768 0.196669
\(238\) 0 0
\(239\) 15.2471 0.986251 0.493126 0.869958i \(-0.335854\pi\)
0.493126 + 0.869958i \(0.335854\pi\)
\(240\) 51.8027 3.34385
\(241\) 4.12703 0.265845 0.132923 0.991126i \(-0.457564\pi\)
0.132923 + 0.991126i \(0.457564\pi\)
\(242\) 20.4205 1.31268
\(243\) −1.00000 −0.0641500
\(244\) 51.5694 3.30139
\(245\) 0 0
\(246\) −16.1795 −1.03157
\(247\) −11.7249 −0.746040
\(248\) −28.2898 −1.79640
\(249\) 10.5300 0.667314
\(250\) 13.3123 0.841946
\(251\) 4.82682 0.304666 0.152333 0.988329i \(-0.451321\pi\)
0.152333 + 0.988329i \(0.451321\pi\)
\(252\) 0 0
\(253\) 1.88341 0.118409
\(254\) 33.5131 2.10280
\(255\) −19.9390 −1.24863
\(256\) 49.9120 3.11950
\(257\) −4.75140 −0.296384 −0.148192 0.988959i \(-0.547345\pi\)
−0.148192 + 0.988959i \(0.547345\pi\)
\(258\) 1.83313 0.114126
\(259\) 0 0
\(260\) −32.9555 −2.04381
\(261\) 8.24468 0.510333
\(262\) 16.5953 1.02526
\(263\) −16.1415 −0.995326 −0.497663 0.867370i \(-0.665808\pi\)
−0.497663 + 0.867370i \(0.665808\pi\)
\(264\) 18.1008 1.11403
\(265\) 32.7529 2.01199
\(266\) 0 0
\(267\) 10.7196 0.656030
\(268\) −45.0459 −2.75162
\(269\) 9.59563 0.585056 0.292528 0.956257i \(-0.405504\pi\)
0.292528 + 0.956257i \(0.405504\pi\)
\(270\) −9.26614 −0.563919
\(271\) 20.9621 1.27335 0.636677 0.771130i \(-0.280308\pi\)
0.636677 + 0.771130i \(0.280308\pi\)
\(272\) −90.3143 −5.47611
\(273\) 0 0
\(274\) 39.1616 2.36584
\(275\) 12.1229 0.731038
\(276\) −5.50755 −0.331516
\(277\) 14.5824 0.876169 0.438085 0.898934i \(-0.355657\pi\)
0.438085 + 0.898934i \(0.355657\pi\)
\(278\) 45.0152 2.69983
\(279\) 2.94359 0.176228
\(280\) 0 0
\(281\) −10.2706 −0.612694 −0.306347 0.951920i \(-0.599107\pi\)
−0.306347 + 0.951920i \(0.599107\pi\)
\(282\) 4.91874 0.292907
\(283\) 17.0930 1.01607 0.508037 0.861335i \(-0.330371\pi\)
0.508037 + 0.861335i \(0.330371\pi\)
\(284\) −1.62501 −0.0964268
\(285\) −22.4099 −1.32745
\(286\) −9.13092 −0.539923
\(287\) 0 0
\(288\) −22.7499 −1.34055
\(289\) 17.7623 1.04484
\(290\) 76.3964 4.48615
\(291\) −15.6290 −0.916186
\(292\) −24.5499 −1.43668
\(293\) 10.9029 0.636955 0.318478 0.947930i \(-0.396828\pi\)
0.318478 + 0.947930i \(0.396828\pi\)
\(294\) 0 0
\(295\) −4.62370 −0.269202
\(296\) 42.6292 2.47777
\(297\) −1.88341 −0.109287
\(298\) −37.8218 −2.19096
\(299\) 1.76938 0.102326
\(300\) −35.4503 −2.04672
\(301\) 0 0
\(302\) 11.6371 0.669642
\(303\) −7.56916 −0.434837
\(304\) −101.506 −5.82179
\(305\) −31.6653 −1.81315
\(306\) 16.1549 0.923512
\(307\) 26.7973 1.52940 0.764700 0.644386i \(-0.222887\pi\)
0.764700 + 0.644386i \(0.222887\pi\)
\(308\) 0 0
\(309\) 2.62127 0.149119
\(310\) 27.2757 1.54916
\(311\) 17.8615 1.01283 0.506417 0.862289i \(-0.330970\pi\)
0.506417 + 0.862289i \(0.330970\pi\)
\(312\) 17.0049 0.962711
\(313\) −29.1299 −1.64652 −0.823260 0.567665i \(-0.807847\pi\)
−0.823260 + 0.567665i \(0.807847\pi\)
\(314\) 1.82975 0.103258
\(315\) 0 0
\(316\) −16.6751 −0.938049
\(317\) 21.9373 1.23212 0.616062 0.787698i \(-0.288727\pi\)
0.616062 + 0.787698i \(0.288727\pi\)
\(318\) −26.5368 −1.48811
\(319\) 15.5281 0.869409
\(320\) −107.198 −5.99257
\(321\) −17.3322 −0.967388
\(322\) 0 0
\(323\) 39.0701 2.17392
\(324\) 5.50755 0.305975
\(325\) 11.3889 0.631741
\(326\) 1.99247 0.110353
\(327\) 13.3681 0.739255
\(328\) 56.7504 3.13352
\(329\) 0 0
\(330\) −17.4520 −0.960699
\(331\) −5.63045 −0.309477 −0.154739 0.987955i \(-0.549454\pi\)
−0.154739 + 0.987955i \(0.549454\pi\)
\(332\) −57.9947 −3.18287
\(333\) −4.43562 −0.243071
\(334\) −25.8251 −1.41309
\(335\) 27.6596 1.51121
\(336\) 0 0
\(337\) 21.7330 1.18387 0.591936 0.805985i \(-0.298364\pi\)
0.591936 + 0.805985i \(0.298364\pi\)
\(338\) 27.0418 1.47088
\(339\) 1.18709 0.0644738
\(340\) 109.815 5.95556
\(341\) 5.54399 0.300224
\(342\) 18.1568 0.981809
\(343\) 0 0
\(344\) −6.42980 −0.346671
\(345\) 3.38181 0.182071
\(346\) −29.7326 −1.59844
\(347\) 32.4805 1.74364 0.871822 0.489822i \(-0.162938\pi\)
0.871822 + 0.489822i \(0.162938\pi\)
\(348\) −45.4080 −2.43412
\(349\) 0.230635 0.0123456 0.00617281 0.999981i \(-0.498035\pi\)
0.00617281 + 0.999981i \(0.498035\pi\)
\(350\) 0 0
\(351\) −1.76938 −0.0944423
\(352\) −42.8474 −2.28378
\(353\) 27.8964 1.48478 0.742388 0.669970i \(-0.233693\pi\)
0.742388 + 0.669970i \(0.233693\pi\)
\(354\) 3.74618 0.199107
\(355\) 0.997811 0.0529583
\(356\) −59.0388 −3.12905
\(357\) 0 0
\(358\) 27.2454 1.43996
\(359\) 16.3625 0.863581 0.431790 0.901974i \(-0.357882\pi\)
0.431790 + 0.901974i \(0.357882\pi\)
\(360\) 32.5015 1.71298
\(361\) 24.9119 1.31115
\(362\) 2.67317 0.140499
\(363\) 7.45276 0.391168
\(364\) 0 0
\(365\) 15.0745 0.789033
\(366\) 25.6556 1.34104
\(367\) 15.4796 0.808031 0.404016 0.914752i \(-0.367614\pi\)
0.404016 + 0.914752i \(0.367614\pi\)
\(368\) 15.3180 0.798507
\(369\) −5.90495 −0.307399
\(370\) −41.1011 −2.13674
\(371\) 0 0
\(372\) −16.2120 −0.840551
\(373\) −29.4072 −1.52265 −0.761325 0.648371i \(-0.775451\pi\)
−0.761325 + 0.648371i \(0.775451\pi\)
\(374\) 30.4263 1.57330
\(375\) 4.85853 0.250894
\(376\) −17.2527 −0.889741
\(377\) 14.5879 0.751317
\(378\) 0 0
\(379\) 23.7449 1.21969 0.609846 0.792520i \(-0.291231\pi\)
0.609846 + 0.792520i \(0.291231\pi\)
\(380\) 123.424 6.33151
\(381\) 12.2311 0.626618
\(382\) −19.1983 −0.982270
\(383\) 4.79000 0.244758 0.122379 0.992483i \(-0.460948\pi\)
0.122379 + 0.992483i \(0.460948\pi\)
\(384\) 41.3538 2.11033
\(385\) 0 0
\(386\) −34.3053 −1.74609
\(387\) 0.669028 0.0340086
\(388\) 86.0773 4.36991
\(389\) 22.5553 1.14360 0.571798 0.820394i \(-0.306246\pi\)
0.571798 + 0.820394i \(0.306246\pi\)
\(390\) −16.3953 −0.830207
\(391\) −5.89595 −0.298171
\(392\) 0 0
\(393\) 6.05670 0.305520
\(394\) 69.4003 3.49634
\(395\) 10.2391 0.515183
\(396\) 10.3730 0.521262
\(397\) 3.58874 0.180114 0.0900570 0.995937i \(-0.471295\pi\)
0.0900570 + 0.995937i \(0.471295\pi\)
\(398\) −1.77705 −0.0890753
\(399\) 0 0
\(400\) 98.5969 4.92985
\(401\) 10.6336 0.531015 0.265508 0.964109i \(-0.414460\pi\)
0.265508 + 0.964109i \(0.414460\pi\)
\(402\) −22.4102 −1.11772
\(403\) 5.20831 0.259444
\(404\) 41.6875 2.07403
\(405\) −3.38181 −0.168044
\(406\) 0 0
\(407\) −8.35410 −0.414098
\(408\) −56.6640 −2.80529
\(409\) 14.5323 0.718575 0.359288 0.933227i \(-0.383020\pi\)
0.359288 + 0.933227i \(0.383020\pi\)
\(410\) −54.7161 −2.70223
\(411\) 14.2926 0.705002
\(412\) −14.4368 −0.711248
\(413\) 0 0
\(414\) −2.73999 −0.134663
\(415\) 35.6106 1.74806
\(416\) −40.2531 −1.97357
\(417\) 16.4290 0.804529
\(418\) 34.1968 1.67262
\(419\) −28.8287 −1.40837 −0.704187 0.710015i \(-0.748688\pi\)
−0.704187 + 0.710015i \(0.748688\pi\)
\(420\) 0 0
\(421\) 25.2897 1.23254 0.616271 0.787534i \(-0.288643\pi\)
0.616271 + 0.787534i \(0.288643\pi\)
\(422\) −39.6331 −1.92931
\(423\) 1.79517 0.0872839
\(424\) 93.0792 4.52033
\(425\) −37.9503 −1.84086
\(426\) −0.808440 −0.0391690
\(427\) 0 0
\(428\) 95.4579 4.61413
\(429\) −3.33246 −0.160893
\(430\) 6.19931 0.298957
\(431\) −1.33330 −0.0642226 −0.0321113 0.999484i \(-0.510223\pi\)
−0.0321113 + 0.999484i \(0.510223\pi\)
\(432\) −15.3180 −0.736988
\(433\) −12.4620 −0.598887 −0.299444 0.954114i \(-0.596801\pi\)
−0.299444 + 0.954114i \(0.596801\pi\)
\(434\) 0 0
\(435\) 27.8820 1.33684
\(436\) −73.6253 −3.52601
\(437\) −6.62660 −0.316993
\(438\) −12.2135 −0.583585
\(439\) 13.7656 0.656994 0.328497 0.944505i \(-0.393458\pi\)
0.328497 + 0.944505i \(0.393458\pi\)
\(440\) 61.2136 2.91825
\(441\) 0 0
\(442\) 28.5840 1.35960
\(443\) −7.06812 −0.335816 −0.167908 0.985803i \(-0.553701\pi\)
−0.167908 + 0.985803i \(0.553701\pi\)
\(444\) 24.4294 1.15937
\(445\) 36.2517 1.71850
\(446\) 81.4674 3.85759
\(447\) −13.8036 −0.652888
\(448\) 0 0
\(449\) −22.4140 −1.05778 −0.528892 0.848689i \(-0.677392\pi\)
−0.528892 + 0.848689i \(0.677392\pi\)
\(450\) −17.6364 −0.831388
\(451\) −11.1214 −0.523689
\(452\) −6.53795 −0.307519
\(453\) 4.24714 0.199548
\(454\) 17.7831 0.834601
\(455\) 0 0
\(456\) −63.6860 −2.98237
\(457\) −2.69034 −0.125849 −0.0629244 0.998018i \(-0.520043\pi\)
−0.0629244 + 0.998018i \(0.520043\pi\)
\(458\) 3.38684 0.158257
\(459\) 5.89595 0.275200
\(460\) −18.6255 −0.868419
\(461\) 30.1064 1.40220 0.701098 0.713065i \(-0.252693\pi\)
0.701098 + 0.713065i \(0.252693\pi\)
\(462\) 0 0
\(463\) −14.0463 −0.652787 −0.326394 0.945234i \(-0.605834\pi\)
−0.326394 + 0.945234i \(0.605834\pi\)
\(464\) 126.292 5.86296
\(465\) 9.95467 0.461637
\(466\) −66.7358 −3.09148
\(467\) −10.6041 −0.490699 −0.245350 0.969435i \(-0.578903\pi\)
−0.245350 + 0.969435i \(0.578903\pi\)
\(468\) 9.74492 0.450459
\(469\) 0 0
\(470\) 16.6343 0.767281
\(471\) 0.667792 0.0307703
\(472\) −13.1399 −0.604814
\(473\) 1.26006 0.0579374
\(474\) −8.29583 −0.381040
\(475\) −42.6532 −1.95706
\(476\) 0 0
\(477\) −9.68500 −0.443446
\(478\) −41.7768 −1.91083
\(479\) −11.2639 −0.514661 −0.257331 0.966323i \(-0.582843\pi\)
−0.257331 + 0.966323i \(0.582843\pi\)
\(480\) −76.9359 −3.51163
\(481\) −7.84828 −0.357851
\(482\) −11.3080 −0.515066
\(483\) 0 0
\(484\) −41.0464 −1.86575
\(485\) −52.8543 −2.39999
\(486\) 2.73999 0.124289
\(487\) 1.90335 0.0862488 0.0431244 0.999070i \(-0.486269\pi\)
0.0431244 + 0.999070i \(0.486269\pi\)
\(488\) −89.9884 −4.07358
\(489\) 0.727180 0.0328842
\(490\) 0 0
\(491\) 20.8742 0.942040 0.471020 0.882123i \(-0.343886\pi\)
0.471020 + 0.882123i \(0.343886\pi\)
\(492\) 32.5218 1.46620
\(493\) −48.6103 −2.18930
\(494\) 32.1262 1.44543
\(495\) −6.36935 −0.286281
\(496\) 45.0899 2.02460
\(497\) 0 0
\(498\) −28.8522 −1.29290
\(499\) −37.0935 −1.66053 −0.830266 0.557368i \(-0.811811\pi\)
−0.830266 + 0.557368i \(0.811811\pi\)
\(500\) −26.7586 −1.19668
\(501\) −9.42526 −0.421090
\(502\) −13.2254 −0.590280
\(503\) 0.702967 0.0313437 0.0156719 0.999877i \(-0.495011\pi\)
0.0156719 + 0.999877i \(0.495011\pi\)
\(504\) 0 0
\(505\) −25.5975 −1.13907
\(506\) −5.16053 −0.229414
\(507\) 9.86931 0.438311
\(508\) −67.3633 −2.98876
\(509\) 18.9239 0.838785 0.419393 0.907805i \(-0.362243\pi\)
0.419393 + 0.907805i \(0.362243\pi\)
\(510\) 54.6327 2.41918
\(511\) 0 0
\(512\) −54.0509 −2.38874
\(513\) 6.62660 0.292572
\(514\) 13.0188 0.574234
\(515\) 8.86464 0.390623
\(516\) −3.68470 −0.162210
\(517\) 3.38104 0.148698
\(518\) 0 0
\(519\) −10.8514 −0.476322
\(520\) 57.5073 2.52186
\(521\) 19.1445 0.838737 0.419369 0.907816i \(-0.362251\pi\)
0.419369 + 0.907816i \(0.362251\pi\)
\(522\) −22.5904 −0.988753
\(523\) 13.9178 0.608581 0.304291 0.952579i \(-0.401581\pi\)
0.304291 + 0.952579i \(0.401581\pi\)
\(524\) −33.3576 −1.45723
\(525\) 0 0
\(526\) 44.2275 1.92841
\(527\) −17.3553 −0.756007
\(528\) −28.8501 −1.25554
\(529\) 1.00000 0.0434783
\(530\) −89.7426 −3.89817
\(531\) 1.36723 0.0593325
\(532\) 0 0
\(533\) −10.4481 −0.452556
\(534\) −29.3716 −1.27104
\(535\) −58.6142 −2.53411
\(536\) 78.6049 3.39522
\(537\) 9.94360 0.429098
\(538\) −26.2919 −1.13353
\(539\) 0 0
\(540\) 18.6255 0.801514
\(541\) 28.7511 1.23611 0.618053 0.786136i \(-0.287922\pi\)
0.618053 + 0.786136i \(0.287922\pi\)
\(542\) −57.4359 −2.46708
\(543\) 0.975614 0.0418676
\(544\) 134.132 5.75088
\(545\) 45.2083 1.93651
\(546\) 0 0
\(547\) 25.9067 1.10769 0.553845 0.832620i \(-0.313160\pi\)
0.553845 + 0.832620i \(0.313160\pi\)
\(548\) −78.7172 −3.36263
\(549\) 9.36339 0.399620
\(550\) −33.2166 −1.41636
\(551\) −54.6342 −2.32750
\(552\) 9.61066 0.409057
\(553\) 0 0
\(554\) −39.9555 −1.69755
\(555\) −15.0004 −0.636733
\(556\) −90.4833 −3.83735
\(557\) −4.63598 −0.196433 −0.0982164 0.995165i \(-0.531314\pi\)
−0.0982164 + 0.995165i \(0.531314\pi\)
\(558\) −8.06540 −0.341436
\(559\) 1.18376 0.0500678
\(560\) 0 0
\(561\) 11.1045 0.468833
\(562\) 28.1414 1.18707
\(563\) 5.23116 0.220467 0.110233 0.993906i \(-0.464840\pi\)
0.110233 + 0.993906i \(0.464840\pi\)
\(564\) −9.88696 −0.416316
\(565\) 4.01451 0.168892
\(566\) −46.8347 −1.96861
\(567\) 0 0
\(568\) 2.83564 0.118981
\(569\) 7.71135 0.323277 0.161638 0.986850i \(-0.448322\pi\)
0.161638 + 0.986850i \(0.448322\pi\)
\(570\) 61.4030 2.57189
\(571\) 17.5515 0.734508 0.367254 0.930121i \(-0.380298\pi\)
0.367254 + 0.930121i \(0.380298\pi\)
\(572\) 18.3537 0.767407
\(573\) −7.00670 −0.292709
\(574\) 0 0
\(575\) 6.43667 0.268427
\(576\) 31.6985 1.32077
\(577\) 15.6526 0.651628 0.325814 0.945434i \(-0.394362\pi\)
0.325814 + 0.945434i \(0.394362\pi\)
\(578\) −48.6685 −2.02434
\(579\) −12.5202 −0.520323
\(580\) −153.561 −6.37629
\(581\) 0 0
\(582\) 42.8232 1.77508
\(583\) −18.2409 −0.755459
\(584\) 42.8396 1.77271
\(585\) −5.98370 −0.247395
\(586\) −29.8739 −1.23408
\(587\) 29.3900 1.21305 0.606527 0.795063i \(-0.292562\pi\)
0.606527 + 0.795063i \(0.292562\pi\)
\(588\) 0 0
\(589\) −19.5060 −0.803730
\(590\) 12.6689 0.521570
\(591\) 25.3287 1.04188
\(592\) −67.9449 −2.79252
\(593\) −18.0072 −0.739469 −0.369734 0.929137i \(-0.620551\pi\)
−0.369734 + 0.929137i \(0.620551\pi\)
\(594\) 5.16053 0.211739
\(595\) 0 0
\(596\) 76.0241 3.11407
\(597\) −0.648559 −0.0265438
\(598\) −4.84807 −0.198252
\(599\) −12.3391 −0.504162 −0.252081 0.967706i \(-0.581115\pi\)
−0.252081 + 0.967706i \(0.581115\pi\)
\(600\) 61.8606 2.52545
\(601\) 0.594459 0.0242485 0.0121243 0.999926i \(-0.496141\pi\)
0.0121243 + 0.999926i \(0.496141\pi\)
\(602\) 0 0
\(603\) −8.17894 −0.333072
\(604\) −23.3914 −0.951781
\(605\) 25.2038 1.02468
\(606\) 20.7394 0.842481
\(607\) −0.0677745 −0.00275088 −0.00137544 0.999999i \(-0.500438\pi\)
−0.00137544 + 0.999999i \(0.500438\pi\)
\(608\) 150.755 6.11390
\(609\) 0 0
\(610\) 86.7625 3.51291
\(611\) 3.17632 0.128500
\(612\) −32.4723 −1.31261
\(613\) 21.5033 0.868508 0.434254 0.900790i \(-0.357012\pi\)
0.434254 + 0.900790i \(0.357012\pi\)
\(614\) −73.4243 −2.96316
\(615\) −19.9694 −0.805245
\(616\) 0 0
\(617\) 10.0024 0.402683 0.201341 0.979521i \(-0.435470\pi\)
0.201341 + 0.979521i \(0.435470\pi\)
\(618\) −7.18225 −0.288912
\(619\) −13.1494 −0.528518 −0.264259 0.964452i \(-0.585127\pi\)
−0.264259 + 0.964452i \(0.585127\pi\)
\(620\) −54.8258 −2.20186
\(621\) −1.00000 −0.0401286
\(622\) −48.9404 −1.96233
\(623\) 0 0
\(624\) −27.1033 −1.08500
\(625\) −15.7527 −0.630107
\(626\) 79.8157 3.19008
\(627\) 12.4806 0.498428
\(628\) −3.67790 −0.146764
\(629\) 26.1522 1.04276
\(630\) 0 0
\(631\) 27.0831 1.07816 0.539081 0.842254i \(-0.318772\pi\)
0.539081 + 0.842254i \(0.318772\pi\)
\(632\) 29.0980 1.15746
\(633\) −14.4647 −0.574920
\(634\) −60.1081 −2.38720
\(635\) 41.3633 1.64145
\(636\) 53.3406 2.11509
\(637\) 0 0
\(638\) −42.5469 −1.68445
\(639\) −0.295052 −0.0116721
\(640\) 139.851 5.52809
\(641\) −37.6596 −1.48747 −0.743733 0.668477i \(-0.766947\pi\)
−0.743733 + 0.668477i \(0.766947\pi\)
\(642\) 47.4900 1.87428
\(643\) 0.584100 0.0230347 0.0115173 0.999934i \(-0.496334\pi\)
0.0115173 + 0.999934i \(0.496334\pi\)
\(644\) 0 0
\(645\) 2.26253 0.0890869
\(646\) −107.052 −4.21190
\(647\) 5.80803 0.228337 0.114169 0.993461i \(-0.463580\pi\)
0.114169 + 0.993461i \(0.463580\pi\)
\(648\) −9.61066 −0.377542
\(649\) 2.57505 0.101080
\(650\) −31.2054 −1.22398
\(651\) 0 0
\(652\) −4.00498 −0.156847
\(653\) 24.7043 0.966756 0.483378 0.875412i \(-0.339410\pi\)
0.483378 + 0.875412i \(0.339410\pi\)
\(654\) −36.6284 −1.43228
\(655\) 20.4826 0.800323
\(656\) −90.4520 −3.53156
\(657\) −4.45751 −0.173904
\(658\) 0 0
\(659\) 24.0716 0.937697 0.468848 0.883279i \(-0.344669\pi\)
0.468848 + 0.883279i \(0.344669\pi\)
\(660\) 35.0795 1.36547
\(661\) 10.1639 0.395330 0.197665 0.980270i \(-0.436664\pi\)
0.197665 + 0.980270i \(0.436664\pi\)
\(662\) 15.4274 0.599602
\(663\) 10.4322 0.405151
\(664\) 101.200 3.92734
\(665\) 0 0
\(666\) 12.1536 0.470941
\(667\) 8.24468 0.319235
\(668\) 51.9101 2.00846
\(669\) 29.7327 1.14953
\(670\) −75.7872 −2.92791
\(671\) 17.6351 0.680797
\(672\) 0 0
\(673\) 21.4374 0.826350 0.413175 0.910652i \(-0.364420\pi\)
0.413175 + 0.910652i \(0.364420\pi\)
\(674\) −59.5482 −2.29371
\(675\) −6.43667 −0.247747
\(676\) −54.3557 −2.09060
\(677\) 29.1868 1.12174 0.560869 0.827905i \(-0.310467\pi\)
0.560869 + 0.827905i \(0.310467\pi\)
\(678\) −3.25261 −0.124916
\(679\) 0 0
\(680\) −191.627 −7.34856
\(681\) 6.49019 0.248705
\(682\) −15.1905 −0.581674
\(683\) −49.6777 −1.90086 −0.950432 0.310934i \(-0.899358\pi\)
−0.950432 + 0.310934i \(0.899358\pi\)
\(684\) −36.4963 −1.39547
\(685\) 48.3349 1.84678
\(686\) 0 0
\(687\) 1.23608 0.0471593
\(688\) 10.2482 0.390708
\(689\) −17.1364 −0.652845
\(690\) −9.26614 −0.352756
\(691\) 7.22139 0.274715 0.137357 0.990522i \(-0.456139\pi\)
0.137357 + 0.990522i \(0.456139\pi\)
\(692\) 59.7644 2.27190
\(693\) 0 0
\(694\) −88.9963 −3.37825
\(695\) 55.5597 2.10750
\(696\) 79.2368 3.00346
\(697\) 34.8153 1.31872
\(698\) −0.631938 −0.0239192
\(699\) −24.3562 −0.921236
\(700\) 0 0
\(701\) −43.4759 −1.64206 −0.821031 0.570884i \(-0.806601\pi\)
−0.821031 + 0.570884i \(0.806601\pi\)
\(702\) 4.84807 0.182979
\(703\) 29.3931 1.10858
\(704\) 59.7013 2.25008
\(705\) 6.07091 0.228644
\(706\) −76.4359 −2.87670
\(707\) 0 0
\(708\) −7.53006 −0.282997
\(709\) 23.2264 0.872286 0.436143 0.899877i \(-0.356344\pi\)
0.436143 + 0.899877i \(0.356344\pi\)
\(710\) −2.73399 −0.102605
\(711\) −3.02768 −0.113547
\(712\) 103.022 3.86093
\(713\) 2.94359 0.110238
\(714\) 0 0
\(715\) −11.2698 −0.421466
\(716\) −54.7649 −2.04666
\(717\) −15.2471 −0.569412
\(718\) −44.8332 −1.67316
\(719\) 39.8039 1.48444 0.742218 0.670158i \(-0.233774\pi\)
0.742218 + 0.670158i \(0.233774\pi\)
\(720\) −51.8027 −1.93057
\(721\) 0 0
\(722\) −68.2582 −2.54031
\(723\) −4.12703 −0.153486
\(724\) −5.37324 −0.199695
\(725\) 53.0683 1.97091
\(726\) −20.4205 −0.757875
\(727\) −22.3786 −0.829978 −0.414989 0.909826i \(-0.636215\pi\)
−0.414989 + 0.909826i \(0.636215\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −41.3039 −1.52872
\(731\) −3.94456 −0.145895
\(732\) −51.5694 −1.90606
\(733\) −31.2261 −1.15336 −0.576681 0.816970i \(-0.695652\pi\)
−0.576681 + 0.816970i \(0.695652\pi\)
\(734\) −42.4141 −1.56553
\(735\) 0 0
\(736\) −22.7499 −0.838572
\(737\) −15.4043 −0.567425
\(738\) 16.1795 0.595576
\(739\) 34.6438 1.27439 0.637197 0.770701i \(-0.280094\pi\)
0.637197 + 0.770701i \(0.280094\pi\)
\(740\) 82.6157 3.03701
\(741\) 11.7249 0.430727
\(742\) 0 0
\(743\) −23.7961 −0.872996 −0.436498 0.899705i \(-0.643781\pi\)
−0.436498 + 0.899705i \(0.643781\pi\)
\(744\) 28.2898 1.03715
\(745\) −46.6812 −1.71027
\(746\) 80.5756 2.95008
\(747\) −10.5300 −0.385274
\(748\) −61.1587 −2.23618
\(749\) 0 0
\(750\) −13.3123 −0.486098
\(751\) −2.54860 −0.0929997 −0.0464999 0.998918i \(-0.514807\pi\)
−0.0464999 + 0.998918i \(0.514807\pi\)
\(752\) 27.4984 1.00276
\(753\) −4.82682 −0.175899
\(754\) −39.9708 −1.45565
\(755\) 14.3630 0.522725
\(756\) 0 0
\(757\) −35.5043 −1.29042 −0.645212 0.764004i \(-0.723231\pi\)
−0.645212 + 0.764004i \(0.723231\pi\)
\(758\) −65.0607 −2.36311
\(759\) −1.88341 −0.0683635
\(760\) −215.374 −7.81244
\(761\) −30.0000 −1.08750 −0.543749 0.839248i \(-0.682996\pi\)
−0.543749 + 0.839248i \(0.682996\pi\)
\(762\) −33.5131 −1.21405
\(763\) 0 0
\(764\) 38.5898 1.39613
\(765\) 19.9390 0.720897
\(766\) −13.1246 −0.474209
\(767\) 2.41913 0.0873499
\(768\) −49.9120 −1.80105
\(769\) 39.7222 1.43242 0.716210 0.697885i \(-0.245875\pi\)
0.716210 + 0.697885i \(0.245875\pi\)
\(770\) 0 0
\(771\) 4.75140 0.171117
\(772\) 68.9558 2.48177
\(773\) 11.9933 0.431369 0.215684 0.976463i \(-0.430802\pi\)
0.215684 + 0.976463i \(0.430802\pi\)
\(774\) −1.83313 −0.0658905
\(775\) 18.9469 0.680592
\(776\) −150.205 −5.39203
\(777\) 0 0
\(778\) −61.8012 −2.21568
\(779\) 39.1297 1.40197
\(780\) 32.9555 1.18000
\(781\) −0.555704 −0.0198847
\(782\) 16.1549 0.577697
\(783\) −8.24468 −0.294641
\(784\) 0 0
\(785\) 2.25835 0.0806040
\(786\) −16.5953 −0.591935
\(787\) −8.04112 −0.286635 −0.143318 0.989677i \(-0.545777\pi\)
−0.143318 + 0.989677i \(0.545777\pi\)
\(788\) −139.499 −4.96944
\(789\) 16.1415 0.574652
\(790\) −28.0549 −0.998150
\(791\) 0 0
\(792\) −18.1008 −0.643185
\(793\) 16.5674 0.588324
\(794\) −9.83313 −0.348965
\(795\) −32.7529 −1.16162
\(796\) 3.57197 0.126605
\(797\) −27.9672 −0.990651 −0.495325 0.868708i \(-0.664951\pi\)
−0.495325 + 0.868708i \(0.664951\pi\)
\(798\) 0 0
\(799\) −10.5842 −0.374443
\(800\) −146.433 −5.17721
\(801\) −10.7196 −0.378759
\(802\) −29.1359 −1.02882
\(803\) −8.39532 −0.296264
\(804\) 45.0459 1.58865
\(805\) 0 0
\(806\) −14.2707 −0.502665
\(807\) −9.59563 −0.337782
\(808\) −72.7446 −2.55914
\(809\) 9.31700 0.327568 0.163784 0.986496i \(-0.447630\pi\)
0.163784 + 0.986496i \(0.447630\pi\)
\(810\) 9.26614 0.325579
\(811\) −18.5550 −0.651553 −0.325776 0.945447i \(-0.605626\pi\)
−0.325776 + 0.945447i \(0.605626\pi\)
\(812\) 0 0
\(813\) −20.9621 −0.735172
\(814\) 22.8902 0.802300
\(815\) 2.45919 0.0861416
\(816\) 90.3143 3.16163
\(817\) −4.43338 −0.155104
\(818\) −39.8183 −1.39222
\(819\) 0 0
\(820\) 109.983 3.84076
\(821\) −21.6775 −0.756549 −0.378275 0.925693i \(-0.623483\pi\)
−0.378275 + 0.925693i \(0.623483\pi\)
\(822\) −39.1616 −1.36592
\(823\) 3.22962 0.112578 0.0562888 0.998415i \(-0.482073\pi\)
0.0562888 + 0.998415i \(0.482073\pi\)
\(824\) 25.1921 0.877609
\(825\) −12.1229 −0.422065
\(826\) 0 0
\(827\) 34.5890 1.20278 0.601388 0.798957i \(-0.294614\pi\)
0.601388 + 0.798957i \(0.294614\pi\)
\(828\) 5.50755 0.191401
\(829\) 17.1078 0.594179 0.297090 0.954850i \(-0.403984\pi\)
0.297090 + 0.954850i \(0.403984\pi\)
\(830\) −97.5727 −3.38680
\(831\) −14.5824 −0.505856
\(832\) 56.0865 1.94445
\(833\) 0 0
\(834\) −45.0152 −1.55875
\(835\) −31.8745 −1.10306
\(836\) −68.7377 −2.37734
\(837\) −2.94359 −0.101745
\(838\) 78.9904 2.72868
\(839\) −16.7696 −0.578951 −0.289476 0.957185i \(-0.593481\pi\)
−0.289476 + 0.957185i \(0.593481\pi\)
\(840\) 0 0
\(841\) 38.9748 1.34396
\(842\) −69.2934 −2.38801
\(843\) 10.2706 0.353739
\(844\) 79.6650 2.74218
\(845\) 33.3762 1.14818
\(846\) −4.91874 −0.169110
\(847\) 0 0
\(848\) −148.355 −5.09453
\(849\) −17.0930 −0.586631
\(850\) 103.983 3.56660
\(851\) −4.43562 −0.152051
\(852\) 1.62501 0.0556720
\(853\) 51.7879 1.77318 0.886591 0.462554i \(-0.153067\pi\)
0.886591 + 0.462554i \(0.153067\pi\)
\(854\) 0 0
\(855\) 22.4099 0.766404
\(856\) −166.574 −5.69337
\(857\) −43.1996 −1.47567 −0.737835 0.674982i \(-0.764152\pi\)
−0.737835 + 0.674982i \(0.764152\pi\)
\(858\) 9.13092 0.311724
\(859\) −2.30553 −0.0786636 −0.0393318 0.999226i \(-0.512523\pi\)
−0.0393318 + 0.999226i \(0.512523\pi\)
\(860\) −12.4610 −0.424916
\(861\) 0 0
\(862\) 3.65322 0.124429
\(863\) −25.4718 −0.867069 −0.433535 0.901137i \(-0.642734\pi\)
−0.433535 + 0.901137i \(0.642734\pi\)
\(864\) 22.7499 0.773967
\(865\) −36.6973 −1.24775
\(866\) 34.1459 1.16032
\(867\) −17.7623 −0.603238
\(868\) 0 0
\(869\) −5.70238 −0.193440
\(870\) −76.3964 −2.59008
\(871\) −14.4716 −0.490352
\(872\) 128.476 4.35074
\(873\) 15.6290 0.528960
\(874\) 18.1568 0.614164
\(875\) 0 0
\(876\) 24.5499 0.829465
\(877\) −7.10061 −0.239771 −0.119885 0.992788i \(-0.538253\pi\)
−0.119885 + 0.992788i \(0.538253\pi\)
\(878\) −37.7175 −1.27290
\(879\) −10.9029 −0.367746
\(880\) −97.5658 −3.28894
\(881\) −29.0434 −0.978497 −0.489249 0.872144i \(-0.662729\pi\)
−0.489249 + 0.872144i \(0.662729\pi\)
\(882\) 0 0
\(883\) −30.2662 −1.01854 −0.509269 0.860608i \(-0.670084\pi\)
−0.509269 + 0.860608i \(0.670084\pi\)
\(884\) −57.4556 −1.93244
\(885\) 4.62370 0.155424
\(886\) 19.3666 0.650633
\(887\) −31.3757 −1.05349 −0.526746 0.850023i \(-0.676588\pi\)
−0.526746 + 0.850023i \(0.676588\pi\)
\(888\) −42.6292 −1.43054
\(889\) 0 0
\(890\) −99.3294 −3.32953
\(891\) 1.88341 0.0630967
\(892\) −163.754 −5.48290
\(893\) −11.8958 −0.398079
\(894\) 37.8218 1.26495
\(895\) 33.6274 1.12404
\(896\) 0 0
\(897\) −1.76938 −0.0590777
\(898\) 61.4143 2.04942
\(899\) 24.2689 0.809415
\(900\) 35.4503 1.18168
\(901\) 57.1023 1.90235
\(902\) 30.4727 1.01463
\(903\) 0 0
\(904\) 11.4087 0.379448
\(905\) 3.29934 0.109674
\(906\) −11.6371 −0.386618
\(907\) 22.5635 0.749208 0.374604 0.927185i \(-0.377779\pi\)
0.374604 + 0.927185i \(0.377779\pi\)
\(908\) −35.7451 −1.18624
\(909\) 7.56916 0.251053
\(910\) 0 0
\(911\) 55.2986 1.83212 0.916062 0.401038i \(-0.131350\pi\)
0.916062 + 0.401038i \(0.131350\pi\)
\(912\) 101.506 3.36121
\(913\) −19.8324 −0.656356
\(914\) 7.37151 0.243828
\(915\) 31.6653 1.04682
\(916\) −6.80776 −0.224935
\(917\) 0 0
\(918\) −16.1549 −0.533190
\(919\) 25.4246 0.838682 0.419341 0.907829i \(-0.362261\pi\)
0.419341 + 0.907829i \(0.362261\pi\)
\(920\) 32.5015 1.07154
\(921\) −26.7973 −0.883000
\(922\) −82.4914 −2.71671
\(923\) −0.522058 −0.0171837
\(924\) 0 0
\(925\) −28.5506 −0.938738
\(926\) 38.4868 1.26475
\(927\) −2.62127 −0.0860937
\(928\) −187.566 −6.15715
\(929\) −34.2563 −1.12391 −0.561956 0.827167i \(-0.689951\pi\)
−0.561956 + 0.827167i \(0.689951\pi\)
\(930\) −27.2757 −0.894405
\(931\) 0 0
\(932\) 134.143 4.39400
\(933\) −17.8615 −0.584760
\(934\) 29.0552 0.950714
\(935\) 37.5534 1.22813
\(936\) −17.0049 −0.555821
\(937\) 54.0699 1.76639 0.883193 0.469010i \(-0.155389\pi\)
0.883193 + 0.469010i \(0.155389\pi\)
\(938\) 0 0
\(939\) 29.1299 0.950619
\(940\) −33.4359 −1.09056
\(941\) −17.0577 −0.556064 −0.278032 0.960572i \(-0.589682\pi\)
−0.278032 + 0.960572i \(0.589682\pi\)
\(942\) −1.82975 −0.0596163
\(943\) −5.90495 −0.192291
\(944\) 20.9432 0.681642
\(945\) 0 0
\(946\) −3.45254 −0.112252
\(947\) 33.9477 1.10315 0.551575 0.834125i \(-0.314027\pi\)
0.551575 + 0.834125i \(0.314027\pi\)
\(948\) 16.6751 0.541583
\(949\) −7.88700 −0.256023
\(950\) 116.869 3.79174
\(951\) −21.9373 −0.711367
\(952\) 0 0
\(953\) −22.1762 −0.718358 −0.359179 0.933269i \(-0.616943\pi\)
−0.359179 + 0.933269i \(0.616943\pi\)
\(954\) 26.5368 0.859161
\(955\) −23.6954 −0.766764
\(956\) 83.9740 2.71591
\(957\) −15.5281 −0.501953
\(958\) 30.8630 0.997139
\(959\) 0 0
\(960\) 107.198 3.45981
\(961\) −22.3353 −0.720493
\(962\) 21.5042 0.693324
\(963\) 17.3322 0.558522
\(964\) 22.7298 0.732078
\(965\) −42.3411 −1.36301
\(966\) 0 0
\(967\) 15.9540 0.513045 0.256523 0.966538i \(-0.417423\pi\)
0.256523 + 0.966538i \(0.417423\pi\)
\(968\) 71.6259 2.30214
\(969\) −39.0701 −1.25511
\(970\) 144.820 4.64990
\(971\) 37.3099 1.19733 0.598665 0.800999i \(-0.295698\pi\)
0.598665 + 0.800999i \(0.295698\pi\)
\(972\) −5.50755 −0.176655
\(973\) 0 0
\(974\) −5.21515 −0.167104
\(975\) −11.3889 −0.364736
\(976\) 143.429 4.59104
\(977\) −37.1289 −1.18786 −0.593930 0.804517i \(-0.702424\pi\)
−0.593930 + 0.804517i \(0.702424\pi\)
\(978\) −1.99247 −0.0637121
\(979\) −20.1894 −0.645258
\(980\) 0 0
\(981\) −13.3681 −0.426809
\(982\) −57.1951 −1.82517
\(983\) −20.0403 −0.639188 −0.319594 0.947555i \(-0.603546\pi\)
−0.319594 + 0.947555i \(0.603546\pi\)
\(984\) −56.7504 −1.80914
\(985\) 85.6568 2.72925
\(986\) 133.192 4.24169
\(987\) 0 0
\(988\) −64.5757 −2.05443
\(989\) 0.669028 0.0212738
\(990\) 17.4520 0.554660
\(991\) 40.3456 1.28162 0.640810 0.767699i \(-0.278598\pi\)
0.640810 + 0.767699i \(0.278598\pi\)
\(992\) −66.9663 −2.12618
\(993\) 5.63045 0.178677
\(994\) 0 0
\(995\) −2.19331 −0.0695325
\(996\) 57.9947 1.83763
\(997\) 4.57100 0.144765 0.0723826 0.997377i \(-0.476940\pi\)
0.0723826 + 0.997377i \(0.476940\pi\)
\(998\) 101.636 3.21722
\(999\) 4.43562 0.140337
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.1.1 10
7.2 even 3 483.2.i.h.277.10 20
7.4 even 3 483.2.i.h.415.10 yes 20
7.6 odd 2 3381.2.a.bj.1.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.10 20 7.2 even 3
483.2.i.h.415.10 yes 20 7.4 even 3
3381.2.a.bi.1.1 10 1.1 even 1 trivial
3381.2.a.bj.1.1 10 7.6 odd 2