Properties

Label 3549.2.a.bh.1.4
Level $3549$
Weight $2$
Character 3549.1
Self dual yes
Analytic conductor $28.339$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3549.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.3389076774\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Defining polynomial: \(x^{15} - 2 x^{14} - 27 x^{13} + 51 x^{12} + 290 x^{11} - 510 x^{10} - 1575 x^{9} + 2522 x^{8} + 4553 x^{7} - 6393 x^{6} - 6785 x^{5} + 7806 x^{4} + 4632 x^{3} - 3811 x^{2} - 1041 x + 281\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-1.89376\) of defining polynomial
Character \(\chi\) \(=\) 3549.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.89376 q^{2} +1.00000 q^{3} +1.58633 q^{4} +0.892462 q^{5} -1.89376 q^{6} -1.00000 q^{7} +0.783390 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.89376 q^{2} +1.00000 q^{3} +1.58633 q^{4} +0.892462 q^{5} -1.89376 q^{6} -1.00000 q^{7} +0.783390 q^{8} +1.00000 q^{9} -1.69011 q^{10} -5.27707 q^{11} +1.58633 q^{12} +1.89376 q^{14} +0.892462 q^{15} -4.65622 q^{16} +4.75966 q^{17} -1.89376 q^{18} +0.639234 q^{19} +1.41574 q^{20} -1.00000 q^{21} +9.99350 q^{22} +4.64933 q^{23} +0.783390 q^{24} -4.20351 q^{25} +1.00000 q^{27} -1.58633 q^{28} -3.84995 q^{29} -1.69011 q^{30} +2.03627 q^{31} +7.25098 q^{32} -5.27707 q^{33} -9.01366 q^{34} -0.892462 q^{35} +1.58633 q^{36} +0.645805 q^{37} -1.21056 q^{38} +0.699146 q^{40} -7.73993 q^{41} +1.89376 q^{42} +10.5241 q^{43} -8.37118 q^{44} +0.892462 q^{45} -8.80472 q^{46} -7.04313 q^{47} -4.65622 q^{48} +1.00000 q^{49} +7.96045 q^{50} +4.75966 q^{51} +8.06403 q^{53} -1.89376 q^{54} -4.70958 q^{55} -0.783390 q^{56} +0.639234 q^{57} +7.29089 q^{58} -11.1500 q^{59} +1.41574 q^{60} +0.832403 q^{61} -3.85621 q^{62} -1.00000 q^{63} -4.41919 q^{64} +9.99350 q^{66} +11.8274 q^{67} +7.55039 q^{68} +4.64933 q^{69} +1.69011 q^{70} +7.42431 q^{71} +0.783390 q^{72} +9.57578 q^{73} -1.22300 q^{74} -4.20351 q^{75} +1.01404 q^{76} +5.27707 q^{77} -0.136717 q^{79} -4.15549 q^{80} +1.00000 q^{81} +14.6576 q^{82} +14.7716 q^{83} -1.58633 q^{84} +4.24781 q^{85} -19.9301 q^{86} -3.84995 q^{87} -4.13400 q^{88} -5.50508 q^{89} -1.69011 q^{90} +7.37537 q^{92} +2.03627 q^{93} +13.3380 q^{94} +0.570492 q^{95} +7.25098 q^{96} +7.63761 q^{97} -1.89376 q^{98} -5.27707 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + O(q^{10}) \) \( 15q + 2q^{2} + 15q^{3} + 28q^{4} - 9q^{5} + 2q^{6} - 15q^{7} + 9q^{8} + 15q^{9} + 21q^{10} + 5q^{11} + 28q^{12} - 2q^{14} - 9q^{15} + 50q^{16} - q^{17} + 2q^{18} + 3q^{19} - 23q^{20} - 15q^{21} + 21q^{22} + 4q^{23} + 9q^{24} + 50q^{25} + 15q^{27} - 28q^{28} + 9q^{29} + 21q^{30} + 7q^{31} + 35q^{32} + 5q^{33} - 2q^{34} + 9q^{35} + 28q^{36} + 17q^{37} - 12q^{38} + 46q^{40} - 22q^{41} - 2q^{42} + 36q^{43} + 29q^{44} - 9q^{45} - q^{46} - 12q^{47} + 50q^{48} + 15q^{49} + 53q^{50} - q^{51} - 5q^{53} + 2q^{54} + 43q^{55} - 9q^{56} + 3q^{57} + 29q^{58} - 29q^{59} - 23q^{60} + 12q^{61} + 14q^{62} - 15q^{63} + 95q^{64} + 21q^{66} - 12q^{67} - 16q^{68} + 4q^{69} - 21q^{70} + 36q^{71} + 9q^{72} - 29q^{73} - 5q^{74} + 50q^{75} + 25q^{76} - 5q^{77} + 35q^{79} - 89q^{80} + 15q^{81} + 51q^{82} - 10q^{83} - 28q^{84} + 23q^{85} + 19q^{86} + 9q^{87} + 73q^{88} + 25q^{89} + 21q^{90} - 31q^{92} + 7q^{93} - 19q^{94} - 7q^{95} + 35q^{96} - 26q^{97} + 2q^{98} + 5q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.89376 −1.33909 −0.669546 0.742771i \(-0.733511\pi\)
−0.669546 + 0.742771i \(0.733511\pi\)
\(3\) 1.00000 0.577350
\(4\) 1.58633 0.793165
\(5\) 0.892462 0.399121 0.199561 0.979886i \(-0.436049\pi\)
0.199561 + 0.979886i \(0.436049\pi\)
\(6\) −1.89376 −0.773125
\(7\) −1.00000 −0.377964
\(8\) 0.783390 0.276970
\(9\) 1.00000 0.333333
\(10\) −1.69011 −0.534459
\(11\) −5.27707 −1.59110 −0.795548 0.605891i \(-0.792817\pi\)
−0.795548 + 0.605891i \(0.792817\pi\)
\(12\) 1.58633 0.457934
\(13\) 0 0
\(14\) 1.89376 0.506129
\(15\) 0.892462 0.230433
\(16\) −4.65622 −1.16405
\(17\) 4.75966 1.15439 0.577193 0.816608i \(-0.304148\pi\)
0.577193 + 0.816608i \(0.304148\pi\)
\(18\) −1.89376 −0.446364
\(19\) 0.639234 0.146650 0.0733251 0.997308i \(-0.476639\pi\)
0.0733251 + 0.997308i \(0.476639\pi\)
\(20\) 1.41574 0.316569
\(21\) −1.00000 −0.218218
\(22\) 9.99350 2.13062
\(23\) 4.64933 0.969452 0.484726 0.874666i \(-0.338919\pi\)
0.484726 + 0.874666i \(0.338919\pi\)
\(24\) 0.783390 0.159909
\(25\) −4.20351 −0.840702
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −1.58633 −0.299788
\(29\) −3.84995 −0.714918 −0.357459 0.933929i \(-0.616357\pi\)
−0.357459 + 0.933929i \(0.616357\pi\)
\(30\) −1.69011 −0.308570
\(31\) 2.03627 0.365725 0.182862 0.983139i \(-0.441464\pi\)
0.182862 + 0.983139i \(0.441464\pi\)
\(32\) 7.25098 1.28180
\(33\) −5.27707 −0.918619
\(34\) −9.01366 −1.54583
\(35\) −0.892462 −0.150854
\(36\) 1.58633 0.264388
\(37\) 0.645805 0.106170 0.0530849 0.998590i \(-0.483095\pi\)
0.0530849 + 0.998590i \(0.483095\pi\)
\(38\) −1.21056 −0.196378
\(39\) 0 0
\(40\) 0.699146 0.110545
\(41\) −7.73993 −1.20877 −0.604387 0.796691i \(-0.706582\pi\)
−0.604387 + 0.796691i \(0.706582\pi\)
\(42\) 1.89376 0.292214
\(43\) 10.5241 1.60491 0.802454 0.596713i \(-0.203527\pi\)
0.802454 + 0.596713i \(0.203527\pi\)
\(44\) −8.37118 −1.26200
\(45\) 0.892462 0.133040
\(46\) −8.80472 −1.29818
\(47\) −7.04313 −1.02735 −0.513673 0.857986i \(-0.671716\pi\)
−0.513673 + 0.857986i \(0.671716\pi\)
\(48\) −4.65622 −0.672067
\(49\) 1.00000 0.142857
\(50\) 7.96045 1.12578
\(51\) 4.75966 0.666485
\(52\) 0 0
\(53\) 8.06403 1.10768 0.553840 0.832623i \(-0.313162\pi\)
0.553840 + 0.832623i \(0.313162\pi\)
\(54\) −1.89376 −0.257708
\(55\) −4.70958 −0.635040
\(56\) −0.783390 −0.104685
\(57\) 0.639234 0.0846686
\(58\) 7.29089 0.957340
\(59\) −11.1500 −1.45161 −0.725805 0.687900i \(-0.758533\pi\)
−0.725805 + 0.687900i \(0.758533\pi\)
\(60\) 1.41574 0.182771
\(61\) 0.832403 0.106578 0.0532891 0.998579i \(-0.483030\pi\)
0.0532891 + 0.998579i \(0.483030\pi\)
\(62\) −3.85621 −0.489739
\(63\) −1.00000 −0.125988
\(64\) −4.41919 −0.552399
\(65\) 0 0
\(66\) 9.99350 1.23012
\(67\) 11.8274 1.44495 0.722476 0.691396i \(-0.243004\pi\)
0.722476 + 0.691396i \(0.243004\pi\)
\(68\) 7.55039 0.915620
\(69\) 4.64933 0.559713
\(70\) 1.69011 0.202007
\(71\) 7.42431 0.881103 0.440552 0.897727i \(-0.354783\pi\)
0.440552 + 0.897727i \(0.354783\pi\)
\(72\) 0.783390 0.0923234
\(73\) 9.57578 1.12076 0.560380 0.828236i \(-0.310655\pi\)
0.560380 + 0.828236i \(0.310655\pi\)
\(74\) −1.22300 −0.142171
\(75\) −4.20351 −0.485380
\(76\) 1.01404 0.116318
\(77\) 5.27707 0.601378
\(78\) 0 0
\(79\) −0.136717 −0.0153819 −0.00769094 0.999970i \(-0.502448\pi\)
−0.00769094 + 0.999970i \(0.502448\pi\)
\(80\) −4.15549 −0.464598
\(81\) 1.00000 0.111111
\(82\) 14.6576 1.61866
\(83\) 14.7716 1.62139 0.810696 0.585468i \(-0.199089\pi\)
0.810696 + 0.585468i \(0.199089\pi\)
\(84\) −1.58633 −0.173083
\(85\) 4.24781 0.460740
\(86\) −19.9301 −2.14912
\(87\) −3.84995 −0.412758
\(88\) −4.13400 −0.440686
\(89\) −5.50508 −0.583537 −0.291769 0.956489i \(-0.594244\pi\)
−0.291769 + 0.956489i \(0.594244\pi\)
\(90\) −1.69011 −0.178153
\(91\) 0 0
\(92\) 7.37537 0.768936
\(93\) 2.03627 0.211151
\(94\) 13.3380 1.37571
\(95\) 0.570492 0.0585312
\(96\) 7.25098 0.740050
\(97\) 7.63761 0.775482 0.387741 0.921768i \(-0.373256\pi\)
0.387741 + 0.921768i \(0.373256\pi\)
\(98\) −1.89376 −0.191299
\(99\) −5.27707 −0.530365
\(100\) −6.66816 −0.666816
\(101\) 13.9998 1.39303 0.696516 0.717541i \(-0.254732\pi\)
0.696516 + 0.717541i \(0.254732\pi\)
\(102\) −9.01366 −0.892485
\(103\) 13.6978 1.34968 0.674841 0.737963i \(-0.264212\pi\)
0.674841 + 0.737963i \(0.264212\pi\)
\(104\) 0 0
\(105\) −0.892462 −0.0870953
\(106\) −15.2713 −1.48328
\(107\) −14.0091 −1.35432 −0.677158 0.735838i \(-0.736788\pi\)
−0.677158 + 0.735838i \(0.736788\pi\)
\(108\) 1.58633 0.152645
\(109\) −14.7520 −1.41298 −0.706492 0.707721i \(-0.749723\pi\)
−0.706492 + 0.707721i \(0.749723\pi\)
\(110\) 8.91882 0.850376
\(111\) 0.645805 0.0612971
\(112\) 4.65622 0.439971
\(113\) −5.81164 −0.546713 −0.273356 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(114\) −1.21056 −0.113379
\(115\) 4.14935 0.386929
\(116\) −6.10730 −0.567048
\(117\) 0 0
\(118\) 21.1155 1.94384
\(119\) −4.75966 −0.436317
\(120\) 0.699146 0.0638230
\(121\) 16.8474 1.53159
\(122\) −1.57637 −0.142718
\(123\) −7.73993 −0.697886
\(124\) 3.23020 0.290080
\(125\) −8.21378 −0.734663
\(126\) 1.89376 0.168710
\(127\) −1.69886 −0.150750 −0.0753748 0.997155i \(-0.524015\pi\)
−0.0753748 + 0.997155i \(0.524015\pi\)
\(128\) −6.13307 −0.542092
\(129\) 10.5241 0.926594
\(130\) 0 0
\(131\) 1.19269 0.104206 0.0521030 0.998642i \(-0.483408\pi\)
0.0521030 + 0.998642i \(0.483408\pi\)
\(132\) −8.37118 −0.728617
\(133\) −0.639234 −0.0554286
\(134\) −22.3983 −1.93492
\(135\) 0.892462 0.0768109
\(136\) 3.72867 0.319731
\(137\) 4.03058 0.344356 0.172178 0.985066i \(-0.444920\pi\)
0.172178 + 0.985066i \(0.444920\pi\)
\(138\) −8.80472 −0.749507
\(139\) 7.79669 0.661307 0.330653 0.943752i \(-0.392731\pi\)
0.330653 + 0.943752i \(0.392731\pi\)
\(140\) −1.41574 −0.119652
\(141\) −7.04313 −0.593139
\(142\) −14.0599 −1.17988
\(143\) 0 0
\(144\) −4.65622 −0.388018
\(145\) −3.43593 −0.285339
\(146\) −18.1342 −1.50080
\(147\) 1.00000 0.0824786
\(148\) 1.02446 0.0842102
\(149\) 15.8997 1.30255 0.651277 0.758840i \(-0.274233\pi\)
0.651277 + 0.758840i \(0.274233\pi\)
\(150\) 7.96045 0.649968
\(151\) −5.74035 −0.467143 −0.233571 0.972340i \(-0.575041\pi\)
−0.233571 + 0.972340i \(0.575041\pi\)
\(152\) 0.500769 0.0406178
\(153\) 4.75966 0.384796
\(154\) −9.99350 −0.805300
\(155\) 1.81729 0.145968
\(156\) 0 0
\(157\) 18.2220 1.45427 0.727135 0.686494i \(-0.240851\pi\)
0.727135 + 0.686494i \(0.240851\pi\)
\(158\) 0.258909 0.0205977
\(159\) 8.06403 0.639519
\(160\) 6.47122 0.511595
\(161\) −4.64933 −0.366418
\(162\) −1.89376 −0.148788
\(163\) 17.2611 1.35200 0.675998 0.736904i \(-0.263713\pi\)
0.675998 + 0.736904i \(0.263713\pi\)
\(164\) −12.2781 −0.958758
\(165\) −4.70958 −0.366640
\(166\) −27.9738 −2.17119
\(167\) 12.7165 0.984032 0.492016 0.870586i \(-0.336260\pi\)
0.492016 + 0.870586i \(0.336260\pi\)
\(168\) −0.783390 −0.0604399
\(169\) 0 0
\(170\) −8.04434 −0.616973
\(171\) 0.639234 0.0488834
\(172\) 16.6947 1.27296
\(173\) −14.1113 −1.07286 −0.536432 0.843944i \(-0.680228\pi\)
−0.536432 + 0.843944i \(0.680228\pi\)
\(174\) 7.29089 0.552721
\(175\) 4.20351 0.317756
\(176\) 24.5712 1.85212
\(177\) −11.1500 −0.838088
\(178\) 10.4253 0.781410
\(179\) −22.2581 −1.66365 −0.831824 0.555039i \(-0.812703\pi\)
−0.831824 + 0.555039i \(0.812703\pi\)
\(180\) 1.41574 0.105523
\(181\) −7.25474 −0.539241 −0.269620 0.962967i \(-0.586898\pi\)
−0.269620 + 0.962967i \(0.586898\pi\)
\(182\) 0 0
\(183\) 0.832403 0.0615330
\(184\) 3.64224 0.268509
\(185\) 0.576356 0.0423746
\(186\) −3.85621 −0.282751
\(187\) −25.1170 −1.83674
\(188\) −11.1727 −0.814856
\(189\) −1.00000 −0.0727393
\(190\) −1.08037 −0.0783786
\(191\) −4.31005 −0.311864 −0.155932 0.987768i \(-0.549838\pi\)
−0.155932 + 0.987768i \(0.549838\pi\)
\(192\) −4.41919 −0.318928
\(193\) −12.3043 −0.885684 −0.442842 0.896600i \(-0.646030\pi\)
−0.442842 + 0.896600i \(0.646030\pi\)
\(194\) −14.4638 −1.03844
\(195\) 0 0
\(196\) 1.58633 0.113309
\(197\) 3.20784 0.228549 0.114275 0.993449i \(-0.463546\pi\)
0.114275 + 0.993449i \(0.463546\pi\)
\(198\) 9.99350 0.710207
\(199\) 2.13013 0.151001 0.0755005 0.997146i \(-0.475945\pi\)
0.0755005 + 0.997146i \(0.475945\pi\)
\(200\) −3.29299 −0.232850
\(201\) 11.8274 0.834243
\(202\) −26.5123 −1.86540
\(203\) 3.84995 0.270214
\(204\) 7.55039 0.528633
\(205\) −6.90759 −0.482447
\(206\) −25.9403 −1.80735
\(207\) 4.64933 0.323151
\(208\) 0 0
\(209\) −3.37328 −0.233335
\(210\) 1.69011 0.116629
\(211\) 12.9048 0.888404 0.444202 0.895927i \(-0.353487\pi\)
0.444202 + 0.895927i \(0.353487\pi\)
\(212\) 12.7922 0.878573
\(213\) 7.42431 0.508705
\(214\) 26.5300 1.81355
\(215\) 9.39235 0.640553
\(216\) 0.783390 0.0533030
\(217\) −2.03627 −0.138231
\(218\) 27.9367 1.89211
\(219\) 9.57578 0.647071
\(220\) −7.47095 −0.503692
\(221\) 0 0
\(222\) −1.22300 −0.0820824
\(223\) 25.4618 1.70505 0.852524 0.522688i \(-0.175071\pi\)
0.852524 + 0.522688i \(0.175071\pi\)
\(224\) −7.25098 −0.484476
\(225\) −4.20351 −0.280234
\(226\) 11.0059 0.732098
\(227\) −20.5447 −1.36360 −0.681799 0.731540i \(-0.738802\pi\)
−0.681799 + 0.731540i \(0.738802\pi\)
\(228\) 1.01404 0.0671562
\(229\) 13.1613 0.869722 0.434861 0.900498i \(-0.356798\pi\)
0.434861 + 0.900498i \(0.356798\pi\)
\(230\) −7.85787 −0.518133
\(231\) 5.27707 0.347206
\(232\) −3.01601 −0.198011
\(233\) 5.59262 0.366385 0.183192 0.983077i \(-0.441357\pi\)
0.183192 + 0.983077i \(0.441357\pi\)
\(234\) 0 0
\(235\) −6.28573 −0.410036
\(236\) −17.6876 −1.15137
\(237\) −0.136717 −0.00888073
\(238\) 9.01366 0.584268
\(239\) 25.9477 1.67842 0.839210 0.543808i \(-0.183018\pi\)
0.839210 + 0.543808i \(0.183018\pi\)
\(240\) −4.15549 −0.268236
\(241\) −0.242505 −0.0156211 −0.00781056 0.999969i \(-0.502486\pi\)
−0.00781056 + 0.999969i \(0.502486\pi\)
\(242\) −31.9050 −2.05093
\(243\) 1.00000 0.0641500
\(244\) 1.32047 0.0845342
\(245\) 0.892462 0.0570173
\(246\) 14.6576 0.934534
\(247\) 0 0
\(248\) 1.59519 0.101295
\(249\) 14.7716 0.936111
\(250\) 15.5549 0.983781
\(251\) 16.9718 1.07125 0.535626 0.844455i \(-0.320076\pi\)
0.535626 + 0.844455i \(0.320076\pi\)
\(252\) −1.58633 −0.0999295
\(253\) −24.5348 −1.54249
\(254\) 3.21724 0.201867
\(255\) 4.24781 0.266008
\(256\) 20.4529 1.27831
\(257\) −1.57975 −0.0985419 −0.0492710 0.998785i \(-0.515690\pi\)
−0.0492710 + 0.998785i \(0.515690\pi\)
\(258\) −19.9301 −1.24079
\(259\) −0.645805 −0.0401284
\(260\) 0 0
\(261\) −3.84995 −0.238306
\(262\) −2.25868 −0.139541
\(263\) 9.40336 0.579836 0.289918 0.957052i \(-0.406372\pi\)
0.289918 + 0.957052i \(0.406372\pi\)
\(264\) −4.13400 −0.254430
\(265\) 7.19684 0.442098
\(266\) 1.21056 0.0742239
\(267\) −5.50508 −0.336905
\(268\) 18.7622 1.14609
\(269\) −28.4573 −1.73507 −0.867535 0.497376i \(-0.834297\pi\)
−0.867535 + 0.497376i \(0.834297\pi\)
\(270\) −1.69011 −0.102857
\(271\) 7.15660 0.434733 0.217366 0.976090i \(-0.430253\pi\)
0.217366 + 0.976090i \(0.430253\pi\)
\(272\) −22.1620 −1.34377
\(273\) 0 0
\(274\) −7.63295 −0.461124
\(275\) 22.1822 1.33764
\(276\) 7.37537 0.443945
\(277\) −30.4598 −1.83015 −0.915075 0.403283i \(-0.867869\pi\)
−0.915075 + 0.403283i \(0.867869\pi\)
\(278\) −14.7651 −0.885550
\(279\) 2.03627 0.121908
\(280\) −0.699146 −0.0417820
\(281\) 8.06012 0.480827 0.240413 0.970671i \(-0.422717\pi\)
0.240413 + 0.970671i \(0.422717\pi\)
\(282\) 13.3380 0.794267
\(283\) 19.6060 1.16546 0.582729 0.812666i \(-0.301985\pi\)
0.582729 + 0.812666i \(0.301985\pi\)
\(284\) 11.7774 0.698861
\(285\) 0.570492 0.0337930
\(286\) 0 0
\(287\) 7.73993 0.456874
\(288\) 7.25098 0.427268
\(289\) 5.65435 0.332609
\(290\) 6.50684 0.382095
\(291\) 7.63761 0.447724
\(292\) 15.1904 0.888948
\(293\) 22.0402 1.28760 0.643801 0.765193i \(-0.277356\pi\)
0.643801 + 0.765193i \(0.277356\pi\)
\(294\) −1.89376 −0.110446
\(295\) −9.95098 −0.579368
\(296\) 0.505918 0.0294059
\(297\) −5.27707 −0.306206
\(298\) −30.1103 −1.74424
\(299\) 0 0
\(300\) −6.66816 −0.384986
\(301\) −10.5241 −0.606598
\(302\) 10.8708 0.625547
\(303\) 13.9998 0.804268
\(304\) −2.97641 −0.170709
\(305\) 0.742888 0.0425376
\(306\) −9.01366 −0.515276
\(307\) 31.2987 1.78631 0.893154 0.449750i \(-0.148487\pi\)
0.893154 + 0.449750i \(0.148487\pi\)
\(308\) 8.37118 0.476992
\(309\) 13.6978 0.779239
\(310\) −3.44152 −0.195465
\(311\) −14.2805 −0.809771 −0.404886 0.914367i \(-0.632689\pi\)
−0.404886 + 0.914367i \(0.632689\pi\)
\(312\) 0 0
\(313\) 30.3102 1.71323 0.856617 0.515953i \(-0.172562\pi\)
0.856617 + 0.515953i \(0.172562\pi\)
\(314\) −34.5080 −1.94740
\(315\) −0.892462 −0.0502845
\(316\) −0.216879 −0.0122004
\(317\) −6.17539 −0.346845 −0.173422 0.984848i \(-0.555483\pi\)
−0.173422 + 0.984848i \(0.555483\pi\)
\(318\) −15.2713 −0.856374
\(319\) 20.3164 1.13750
\(320\) −3.94396 −0.220474
\(321\) −14.0091 −0.781914
\(322\) 8.80472 0.490668
\(323\) 3.04253 0.169291
\(324\) 1.58633 0.0881295
\(325\) 0 0
\(326\) −32.6885 −1.81045
\(327\) −14.7520 −0.815787
\(328\) −6.06339 −0.334795
\(329\) 7.04313 0.388301
\(330\) 8.91882 0.490965
\(331\) 10.5567 0.580249 0.290125 0.956989i \(-0.406303\pi\)
0.290125 + 0.956989i \(0.406303\pi\)
\(332\) 23.4326 1.28603
\(333\) 0.645805 0.0353899
\(334\) −24.0820 −1.31771
\(335\) 10.5555 0.576711
\(336\) 4.65622 0.254017
\(337\) −14.9855 −0.816313 −0.408157 0.912912i \(-0.633828\pi\)
−0.408157 + 0.912912i \(0.633828\pi\)
\(338\) 0 0
\(339\) −5.81164 −0.315645
\(340\) 6.73844 0.365443
\(341\) −10.7455 −0.581903
\(342\) −1.21056 −0.0654594
\(343\) −1.00000 −0.0539949
\(344\) 8.24447 0.444512
\(345\) 4.14935 0.223393
\(346\) 26.7235 1.43666
\(347\) −29.4048 −1.57853 −0.789266 0.614052i \(-0.789538\pi\)
−0.789266 + 0.614052i \(0.789538\pi\)
\(348\) −6.10730 −0.327385
\(349\) 10.4048 0.556955 0.278477 0.960443i \(-0.410170\pi\)
0.278477 + 0.960443i \(0.410170\pi\)
\(350\) −7.96045 −0.425504
\(351\) 0 0
\(352\) −38.2639 −2.03947
\(353\) 15.0595 0.801537 0.400769 0.916179i \(-0.368743\pi\)
0.400769 + 0.916179i \(0.368743\pi\)
\(354\) 21.1155 1.12228
\(355\) 6.62591 0.351667
\(356\) −8.73288 −0.462842
\(357\) −4.75966 −0.251908
\(358\) 42.1515 2.22778
\(359\) −16.3983 −0.865471 −0.432736 0.901521i \(-0.642452\pi\)
−0.432736 + 0.901521i \(0.642452\pi\)
\(360\) 0.699146 0.0368482
\(361\) −18.5914 −0.978494
\(362\) 13.7387 0.722092
\(363\) 16.8474 0.884261
\(364\) 0 0
\(365\) 8.54602 0.447319
\(366\) −1.57637 −0.0823983
\(367\) −12.9004 −0.673393 −0.336697 0.941613i \(-0.609310\pi\)
−0.336697 + 0.941613i \(0.609310\pi\)
\(368\) −21.6483 −1.12849
\(369\) −7.73993 −0.402925
\(370\) −1.09148 −0.0567434
\(371\) −8.06403 −0.418664
\(372\) 3.23020 0.167478
\(373\) 28.8685 1.49475 0.747377 0.664400i \(-0.231313\pi\)
0.747377 + 0.664400i \(0.231313\pi\)
\(374\) 47.5657 2.45956
\(375\) −8.21378 −0.424158
\(376\) −5.51752 −0.284545
\(377\) 0 0
\(378\) 1.89376 0.0974046
\(379\) 12.3485 0.634300 0.317150 0.948375i \(-0.397274\pi\)
0.317150 + 0.948375i \(0.397274\pi\)
\(380\) 0.904989 0.0464249
\(381\) −1.69886 −0.0870353
\(382\) 8.16221 0.417615
\(383\) −3.75999 −0.192127 −0.0960633 0.995375i \(-0.530625\pi\)
−0.0960633 + 0.995375i \(0.530625\pi\)
\(384\) −6.13307 −0.312977
\(385\) 4.70958 0.240022
\(386\) 23.3014 1.18601
\(387\) 10.5241 0.534970
\(388\) 12.1158 0.615085
\(389\) −16.7332 −0.848407 −0.424203 0.905567i \(-0.639446\pi\)
−0.424203 + 0.905567i \(0.639446\pi\)
\(390\) 0 0
\(391\) 22.1292 1.11912
\(392\) 0.783390 0.0395672
\(393\) 1.19269 0.0601634
\(394\) −6.07488 −0.306048
\(395\) −0.122015 −0.00613923
\(396\) −8.37118 −0.420667
\(397\) 4.04515 0.203021 0.101510 0.994834i \(-0.467633\pi\)
0.101510 + 0.994834i \(0.467633\pi\)
\(398\) −4.03396 −0.202204
\(399\) −0.639234 −0.0320017
\(400\) 19.5725 0.978623
\(401\) −8.62134 −0.430529 −0.215265 0.976556i \(-0.569061\pi\)
−0.215265 + 0.976556i \(0.569061\pi\)
\(402\) −22.3983 −1.11713
\(403\) 0 0
\(404\) 22.2083 1.10491
\(405\) 0.892462 0.0443468
\(406\) −7.29089 −0.361841
\(407\) −3.40796 −0.168926
\(408\) 3.72867 0.184597
\(409\) −25.0049 −1.23641 −0.618206 0.786016i \(-0.712140\pi\)
−0.618206 + 0.786016i \(0.712140\pi\)
\(410\) 13.0813 0.646041
\(411\) 4.03058 0.198814
\(412\) 21.7292 1.07052
\(413\) 11.1500 0.548657
\(414\) −8.80472 −0.432728
\(415\) 13.1831 0.647131
\(416\) 0 0
\(417\) 7.79669 0.381806
\(418\) 6.38818 0.312456
\(419\) 5.95683 0.291010 0.145505 0.989358i \(-0.453519\pi\)
0.145505 + 0.989358i \(0.453519\pi\)
\(420\) −1.41574 −0.0690810
\(421\) 5.52313 0.269181 0.134591 0.990901i \(-0.457028\pi\)
0.134591 + 0.990901i \(0.457028\pi\)
\(422\) −24.4386 −1.18965
\(423\) −7.04313 −0.342449
\(424\) 6.31728 0.306794
\(425\) −20.0073 −0.970496
\(426\) −14.0599 −0.681203
\(427\) −0.832403 −0.0402828
\(428\) −22.2231 −1.07420
\(429\) 0 0
\(430\) −17.7869 −0.857759
\(431\) 24.0748 1.15964 0.579821 0.814744i \(-0.303122\pi\)
0.579821 + 0.814744i \(0.303122\pi\)
\(432\) −4.65622 −0.224022
\(433\) 36.8690 1.77181 0.885906 0.463865i \(-0.153538\pi\)
0.885906 + 0.463865i \(0.153538\pi\)
\(434\) 3.85621 0.185104
\(435\) −3.43593 −0.164740
\(436\) −23.4015 −1.12073
\(437\) 2.97201 0.142170
\(438\) −18.1342 −0.866487
\(439\) −16.6516 −0.794736 −0.397368 0.917659i \(-0.630076\pi\)
−0.397368 + 0.917659i \(0.630076\pi\)
\(440\) −3.68944 −0.175887
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 15.0276 0.713983 0.356991 0.934108i \(-0.383803\pi\)
0.356991 + 0.934108i \(0.383803\pi\)
\(444\) 1.02446 0.0486188
\(445\) −4.91307 −0.232902
\(446\) −48.2185 −2.28321
\(447\) 15.8997 0.752030
\(448\) 4.41919 0.208787
\(449\) 5.93074 0.279889 0.139944 0.990159i \(-0.455308\pi\)
0.139944 + 0.990159i \(0.455308\pi\)
\(450\) 7.96045 0.375259
\(451\) 40.8442 1.92328
\(452\) −9.21918 −0.433634
\(453\) −5.74035 −0.269705
\(454\) 38.9067 1.82598
\(455\) 0 0
\(456\) 0.500769 0.0234507
\(457\) −0.542892 −0.0253954 −0.0126977 0.999919i \(-0.504042\pi\)
−0.0126977 + 0.999919i \(0.504042\pi\)
\(458\) −24.9243 −1.16464
\(459\) 4.75966 0.222162
\(460\) 6.58224 0.306898
\(461\) −12.9687 −0.604015 −0.302007 0.953306i \(-0.597657\pi\)
−0.302007 + 0.953306i \(0.597657\pi\)
\(462\) −9.99350 −0.464940
\(463\) 41.0461 1.90758 0.953788 0.300481i \(-0.0971472\pi\)
0.953788 + 0.300481i \(0.0971472\pi\)
\(464\) 17.9262 0.832203
\(465\) 1.81729 0.0842749
\(466\) −10.5911 −0.490623
\(467\) 39.9232 1.84743 0.923714 0.383083i \(-0.125138\pi\)
0.923714 + 0.383083i \(0.125138\pi\)
\(468\) 0 0
\(469\) −11.8274 −0.546141
\(470\) 11.9037 0.549075
\(471\) 18.2220 0.839623
\(472\) −8.73483 −0.402053
\(473\) −55.5363 −2.55356
\(474\) 0.258909 0.0118921
\(475\) −2.68703 −0.123289
\(476\) −7.55039 −0.346072
\(477\) 8.06403 0.369227
\(478\) −49.1388 −2.24756
\(479\) 40.2182 1.83762 0.918808 0.394705i \(-0.129153\pi\)
0.918808 + 0.394705i \(0.129153\pi\)
\(480\) 6.47122 0.295370
\(481\) 0 0
\(482\) 0.459246 0.0209181
\(483\) −4.64933 −0.211552
\(484\) 26.7256 1.21480
\(485\) 6.81627 0.309511
\(486\) −1.89376 −0.0859027
\(487\) 17.2531 0.781813 0.390906 0.920430i \(-0.372162\pi\)
0.390906 + 0.920430i \(0.372162\pi\)
\(488\) 0.652096 0.0295190
\(489\) 17.2611 0.780575
\(490\) −1.69011 −0.0763514
\(491\) 31.2955 1.41235 0.706173 0.708039i \(-0.250420\pi\)
0.706173 + 0.708039i \(0.250420\pi\)
\(492\) −12.2781 −0.553539
\(493\) −18.3244 −0.825292
\(494\) 0 0
\(495\) −4.70958 −0.211680
\(496\) −9.48131 −0.425724
\(497\) −7.42431 −0.333026
\(498\) −27.9738 −1.25354
\(499\) 16.8461 0.754134 0.377067 0.926186i \(-0.376933\pi\)
0.377067 + 0.926186i \(0.376933\pi\)
\(500\) −13.0298 −0.582709
\(501\) 12.7165 0.568131
\(502\) −32.1406 −1.43450
\(503\) 20.8858 0.931252 0.465626 0.884982i \(-0.345829\pi\)
0.465626 + 0.884982i \(0.345829\pi\)
\(504\) −0.783390 −0.0348950
\(505\) 12.4943 0.555988
\(506\) 46.4631 2.06554
\(507\) 0 0
\(508\) −2.69496 −0.119569
\(509\) 36.3074 1.60930 0.804650 0.593750i \(-0.202353\pi\)
0.804650 + 0.593750i \(0.202353\pi\)
\(510\) −8.04434 −0.356209
\(511\) −9.57578 −0.423607
\(512\) −26.4669 −1.16968
\(513\) 0.639234 0.0282229
\(514\) 2.99166 0.131957
\(515\) 12.2247 0.538687
\(516\) 16.6947 0.734943
\(517\) 37.1671 1.63461
\(518\) 1.22300 0.0537356
\(519\) −14.1113 −0.619418
\(520\) 0 0
\(521\) 19.2511 0.843408 0.421704 0.906734i \(-0.361432\pi\)
0.421704 + 0.906734i \(0.361432\pi\)
\(522\) 7.29089 0.319113
\(523\) 14.6001 0.638419 0.319210 0.947684i \(-0.396583\pi\)
0.319210 + 0.947684i \(0.396583\pi\)
\(524\) 1.89201 0.0826527
\(525\) 4.20351 0.183456
\(526\) −17.8077 −0.776453
\(527\) 9.69195 0.422188
\(528\) 24.5712 1.06932
\(529\) −1.38376 −0.0601633
\(530\) −13.6291 −0.592010
\(531\) −11.1500 −0.483870
\(532\) −1.01404 −0.0439640
\(533\) 0 0
\(534\) 10.4253 0.451147
\(535\) −12.5026 −0.540536
\(536\) 9.26550 0.400209
\(537\) −22.2581 −0.960508
\(538\) 53.8913 2.32342
\(539\) −5.27707 −0.227299
\(540\) 1.41574 0.0609237
\(541\) −21.6960 −0.932783 −0.466392 0.884578i \(-0.654446\pi\)
−0.466392 + 0.884578i \(0.654446\pi\)
\(542\) −13.5529 −0.582147
\(543\) −7.25474 −0.311331
\(544\) 34.5122 1.47970
\(545\) −13.1656 −0.563952
\(546\) 0 0
\(547\) 18.1297 0.775170 0.387585 0.921834i \(-0.373309\pi\)
0.387585 + 0.921834i \(0.373309\pi\)
\(548\) 6.39383 0.273131
\(549\) 0.832403 0.0355261
\(550\) −42.0078 −1.79122
\(551\) −2.46102 −0.104843
\(552\) 3.64224 0.155024
\(553\) 0.136717 0.00581380
\(554\) 57.6835 2.45074
\(555\) 0.576356 0.0244650
\(556\) 12.3681 0.524526
\(557\) 30.1341 1.27682 0.638412 0.769695i \(-0.279592\pi\)
0.638412 + 0.769695i \(0.279592\pi\)
\(558\) −3.85621 −0.163246
\(559\) 0 0
\(560\) 4.15549 0.175602
\(561\) −25.1170 −1.06044
\(562\) −15.2639 −0.643871
\(563\) −34.5589 −1.45648 −0.728242 0.685320i \(-0.759662\pi\)
−0.728242 + 0.685320i \(0.759662\pi\)
\(564\) −11.1727 −0.470457
\(565\) −5.18666 −0.218205
\(566\) −37.1292 −1.56066
\(567\) −1.00000 −0.0419961
\(568\) 5.81613 0.244039
\(569\) 4.59779 0.192749 0.0963746 0.995345i \(-0.469275\pi\)
0.0963746 + 0.995345i \(0.469275\pi\)
\(570\) −1.08037 −0.0452519
\(571\) 19.8023 0.828699 0.414349 0.910118i \(-0.364009\pi\)
0.414349 + 0.910118i \(0.364009\pi\)
\(572\) 0 0
\(573\) −4.31005 −0.180055
\(574\) −14.6576 −0.611796
\(575\) −19.5435 −0.815020
\(576\) −4.41919 −0.184133
\(577\) −20.8218 −0.866822 −0.433411 0.901196i \(-0.642690\pi\)
−0.433411 + 0.901196i \(0.642690\pi\)
\(578\) −10.7080 −0.445393
\(579\) −12.3043 −0.511350
\(580\) −5.45053 −0.226321
\(581\) −14.7716 −0.612828
\(582\) −14.4638 −0.599544
\(583\) −42.5544 −1.76242
\(584\) 7.50157 0.310417
\(585\) 0 0
\(586\) −41.7389 −1.72422
\(587\) −40.6743 −1.67881 −0.839404 0.543508i \(-0.817096\pi\)
−0.839404 + 0.543508i \(0.817096\pi\)
\(588\) 1.58633 0.0654192
\(589\) 1.30165 0.0536337
\(590\) 18.8448 0.775827
\(591\) 3.20784 0.131953
\(592\) −3.00701 −0.123587
\(593\) −22.3890 −0.919405 −0.459702 0.888073i \(-0.652044\pi\)
−0.459702 + 0.888073i \(0.652044\pi\)
\(594\) 9.99350 0.410038
\(595\) −4.24781 −0.174143
\(596\) 25.2222 1.03314
\(597\) 2.13013 0.0871805
\(598\) 0 0
\(599\) −40.4388 −1.65229 −0.826143 0.563461i \(-0.809470\pi\)
−0.826143 + 0.563461i \(0.809470\pi\)
\(600\) −3.29299 −0.134436
\(601\) −20.2325 −0.825301 −0.412650 0.910890i \(-0.635397\pi\)
−0.412650 + 0.910890i \(0.635397\pi\)
\(602\) 19.9301 0.812291
\(603\) 11.8274 0.481651
\(604\) −9.10609 −0.370522
\(605\) 15.0357 0.611288
\(606\) −26.5123 −1.07699
\(607\) −31.5200 −1.27936 −0.639680 0.768642i \(-0.720933\pi\)
−0.639680 + 0.768642i \(0.720933\pi\)
\(608\) 4.63507 0.187977
\(609\) 3.84995 0.156008
\(610\) −1.40685 −0.0569618
\(611\) 0 0
\(612\) 7.55039 0.305207
\(613\) 30.1560 1.21799 0.608994 0.793175i \(-0.291573\pi\)
0.608994 + 0.793175i \(0.291573\pi\)
\(614\) −59.2722 −2.39203
\(615\) −6.90759 −0.278541
\(616\) 4.13400 0.166564
\(617\) −21.0232 −0.846361 −0.423180 0.906045i \(-0.639086\pi\)
−0.423180 + 0.906045i \(0.639086\pi\)
\(618\) −25.9403 −1.04347
\(619\) 26.2525 1.05518 0.527589 0.849500i \(-0.323096\pi\)
0.527589 + 0.849500i \(0.323096\pi\)
\(620\) 2.88283 0.115777
\(621\) 4.64933 0.186571
\(622\) 27.0438 1.08436
\(623\) 5.50508 0.220556
\(624\) 0 0
\(625\) 13.6871 0.547483
\(626\) −57.4003 −2.29418
\(627\) −3.37328 −0.134716
\(628\) 28.9061 1.15348
\(629\) 3.07381 0.122561
\(630\) 1.69011 0.0673356
\(631\) 3.88077 0.154491 0.0772455 0.997012i \(-0.475387\pi\)
0.0772455 + 0.997012i \(0.475387\pi\)
\(632\) −0.107103 −0.00426032
\(633\) 12.9048 0.512920
\(634\) 11.6947 0.464457
\(635\) −1.51617 −0.0601673
\(636\) 12.7922 0.507244
\(637\) 0 0
\(638\) −38.4745 −1.52322
\(639\) 7.42431 0.293701
\(640\) −5.47353 −0.216360
\(641\) 11.2641 0.444905 0.222452 0.974944i \(-0.428594\pi\)
0.222452 + 0.974944i \(0.428594\pi\)
\(642\) 26.5300 1.04705
\(643\) −30.1300 −1.18821 −0.594106 0.804386i \(-0.702494\pi\)
−0.594106 + 0.804386i \(0.702494\pi\)
\(644\) −7.37537 −0.290630
\(645\) 9.39235 0.369823
\(646\) −5.76183 −0.226696
\(647\) 1.03066 0.0405193 0.0202596 0.999795i \(-0.493551\pi\)
0.0202596 + 0.999795i \(0.493551\pi\)
\(648\) 0.783390 0.0307745
\(649\) 58.8395 2.30965
\(650\) 0 0
\(651\) −2.03627 −0.0798077
\(652\) 27.3819 1.07236
\(653\) 3.88842 0.152165 0.0760827 0.997102i \(-0.475759\pi\)
0.0760827 + 0.997102i \(0.475759\pi\)
\(654\) 27.9367 1.09241
\(655\) 1.06443 0.0415908
\(656\) 36.0388 1.40708
\(657\) 9.57578 0.373587
\(658\) −13.3380 −0.519970
\(659\) −11.5306 −0.449169 −0.224584 0.974455i \(-0.572102\pi\)
−0.224584 + 0.974455i \(0.572102\pi\)
\(660\) −7.47095 −0.290806
\(661\) −9.54278 −0.371171 −0.185586 0.982628i \(-0.559418\pi\)
−0.185586 + 0.982628i \(0.559418\pi\)
\(662\) −19.9919 −0.777007
\(663\) 0 0
\(664\) 11.5719 0.449077
\(665\) −0.570492 −0.0221227
\(666\) −1.22300 −0.0473903
\(667\) −17.8997 −0.693078
\(668\) 20.1726 0.780501
\(669\) 25.4618 0.984410
\(670\) −19.9897 −0.772268
\(671\) −4.39265 −0.169576
\(672\) −7.25098 −0.279713
\(673\) −35.1621 −1.35540 −0.677700 0.735339i \(-0.737023\pi\)
−0.677700 + 0.735339i \(0.737023\pi\)
\(674\) 28.3790 1.09312
\(675\) −4.20351 −0.161793
\(676\) 0 0
\(677\) −29.3436 −1.12777 −0.563884 0.825854i \(-0.690693\pi\)
−0.563884 + 0.825854i \(0.690693\pi\)
\(678\) 11.0059 0.422677
\(679\) −7.63761 −0.293104
\(680\) 3.32770 0.127611
\(681\) −20.5447 −0.787274
\(682\) 20.3495 0.779222
\(683\) −23.3475 −0.893369 −0.446685 0.894692i \(-0.647395\pi\)
−0.446685 + 0.894692i \(0.647395\pi\)
\(684\) 1.01404 0.0387726
\(685\) 3.59714 0.137440
\(686\) 1.89376 0.0723041
\(687\) 13.1613 0.502134
\(688\) −49.0024 −1.86820
\(689\) 0 0
\(690\) −7.85787 −0.299144
\(691\) 25.1221 0.955692 0.477846 0.878444i \(-0.341418\pi\)
0.477846 + 0.878444i \(0.341418\pi\)
\(692\) −22.3852 −0.850958
\(693\) 5.27707 0.200459
\(694\) 55.6856 2.11380
\(695\) 6.95825 0.263942
\(696\) −3.01601 −0.114322
\(697\) −36.8394 −1.39539
\(698\) −19.7042 −0.745813
\(699\) 5.59262 0.211532
\(700\) 6.66816 0.252033
\(701\) −21.1109 −0.797347 −0.398673 0.917093i \(-0.630529\pi\)
−0.398673 + 0.917093i \(0.630529\pi\)
\(702\) 0 0
\(703\) 0.412820 0.0155698
\(704\) 23.3204 0.878920
\(705\) −6.28573 −0.236734
\(706\) −28.5191 −1.07333
\(707\) −13.9998 −0.526517
\(708\) −17.6876 −0.664742
\(709\) −43.9044 −1.64886 −0.824431 0.565962i \(-0.808505\pi\)
−0.824431 + 0.565962i \(0.808505\pi\)
\(710\) −12.5479 −0.470914
\(711\) −0.136717 −0.00512729
\(712\) −4.31263 −0.161622
\(713\) 9.46729 0.354553
\(714\) 9.01366 0.337328
\(715\) 0 0
\(716\) −35.3087 −1.31955
\(717\) 25.9477 0.969036
\(718\) 31.0545 1.15895
\(719\) 20.6551 0.770305 0.385152 0.922853i \(-0.374149\pi\)
0.385152 + 0.922853i \(0.374149\pi\)
\(720\) −4.15549 −0.154866
\(721\) −13.6978 −0.510132
\(722\) 35.2076 1.31029
\(723\) −0.242505 −0.00901885
\(724\) −11.5084 −0.427707
\(725\) 16.1833 0.601033
\(726\) −31.9050 −1.18411
\(727\) 39.8996 1.47979 0.739897 0.672720i \(-0.234874\pi\)
0.739897 + 0.672720i \(0.234874\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −16.1841 −0.599001
\(731\) 50.0911 1.85269
\(732\) 1.32047 0.0488058
\(733\) −40.2926 −1.48824 −0.744120 0.668046i \(-0.767131\pi\)
−0.744120 + 0.668046i \(0.767131\pi\)
\(734\) 24.4302 0.901735
\(735\) 0.892462 0.0329189
\(736\) 33.7122 1.24265
\(737\) −62.4142 −2.29906
\(738\) 14.6576 0.539553
\(739\) −34.8101 −1.28051 −0.640255 0.768162i \(-0.721171\pi\)
−0.640255 + 0.768162i \(0.721171\pi\)
\(740\) 0.914292 0.0336100
\(741\) 0 0
\(742\) 15.2713 0.560629
\(743\) 36.6389 1.34415 0.672076 0.740483i \(-0.265403\pi\)
0.672076 + 0.740483i \(0.265403\pi\)
\(744\) 1.59519 0.0584827
\(745\) 14.1899 0.519877
\(746\) −54.6700 −2.00161
\(747\) 14.7716 0.540464
\(748\) −39.8439 −1.45684
\(749\) 14.0091 0.511883
\(750\) 15.5549 0.567986
\(751\) −23.6488 −0.862958 −0.431479 0.902123i \(-0.642008\pi\)
−0.431479 + 0.902123i \(0.642008\pi\)
\(752\) 32.7944 1.19589
\(753\) 16.9718 0.618487
\(754\) 0 0
\(755\) −5.12304 −0.186447
\(756\) −1.58633 −0.0576943
\(757\) 31.0041 1.12686 0.563432 0.826163i \(-0.309481\pi\)
0.563432 + 0.826163i \(0.309481\pi\)
\(758\) −23.3851 −0.849385
\(759\) −24.5348 −0.890557
\(760\) 0.446918 0.0162114
\(761\) −25.6436 −0.929579 −0.464790 0.885421i \(-0.653870\pi\)
−0.464790 + 0.885421i \(0.653870\pi\)
\(762\) 3.21724 0.116548
\(763\) 14.7520 0.534058
\(764\) −6.83717 −0.247360
\(765\) 4.24781 0.153580
\(766\) 7.12053 0.257275
\(767\) 0 0
\(768\) 20.4529 0.738032
\(769\) −28.8868 −1.04169 −0.520843 0.853653i \(-0.674382\pi\)
−0.520843 + 0.853653i \(0.674382\pi\)
\(770\) −8.91882 −0.321412
\(771\) −1.57975 −0.0568932
\(772\) −19.5187 −0.702494
\(773\) −28.1876 −1.01384 −0.506919 0.861993i \(-0.669216\pi\)
−0.506919 + 0.861993i \(0.669216\pi\)
\(774\) −19.9301 −0.716373
\(775\) −8.55949 −0.307466
\(776\) 5.98323 0.214785
\(777\) −0.645805 −0.0231681
\(778\) 31.6887 1.13609
\(779\) −4.94763 −0.177267
\(780\) 0 0
\(781\) −39.1786 −1.40192
\(782\) −41.9074 −1.49861
\(783\) −3.84995 −0.137586
\(784\) −4.65622 −0.166293
\(785\) 16.2624 0.580430
\(786\) −2.25868 −0.0805643
\(787\) −8.76865 −0.312569 −0.156284 0.987712i \(-0.549952\pi\)
−0.156284 + 0.987712i \(0.549952\pi\)
\(788\) 5.08869 0.181277
\(789\) 9.40336 0.334768
\(790\) 0.231067 0.00822099
\(791\) 5.81164 0.206638
\(792\) −4.13400 −0.146895
\(793\) 0 0
\(794\) −7.66056 −0.271863
\(795\) 7.19684 0.255246
\(796\) 3.37909 0.119769
\(797\) 41.2911 1.46261 0.731304 0.682052i \(-0.238912\pi\)
0.731304 + 0.682052i \(0.238912\pi\)
\(798\) 1.21056 0.0428532
\(799\) −33.5229 −1.18596
\(800\) −30.4796 −1.07762
\(801\) −5.50508 −0.194512
\(802\) 16.3268 0.576518
\(803\) −50.5320 −1.78324
\(804\) 18.7622 0.661693
\(805\) −4.14935 −0.146245
\(806\) 0 0
\(807\) −28.4573 −1.00174
\(808\) 10.9673 0.385829
\(809\) 14.9921 0.527094 0.263547 0.964647i \(-0.415108\pi\)
0.263547 + 0.964647i \(0.415108\pi\)
\(810\) −1.69011 −0.0593844
\(811\) −51.5182 −1.80905 −0.904525 0.426421i \(-0.859774\pi\)
−0.904525 + 0.426421i \(0.859774\pi\)
\(812\) 6.10730 0.214324
\(813\) 7.15660 0.250993
\(814\) 6.45386 0.226208
\(815\) 15.4049 0.539610
\(816\) −22.1620 −0.775825
\(817\) 6.72735 0.235360
\(818\) 47.3533 1.65567
\(819\) 0 0
\(820\) −10.9577 −0.382661
\(821\) 19.1154 0.667133 0.333567 0.942727i \(-0.391748\pi\)
0.333567 + 0.942727i \(0.391748\pi\)
\(822\) −7.63295 −0.266230
\(823\) −35.2300 −1.22804 −0.614021 0.789290i \(-0.710449\pi\)
−0.614021 + 0.789290i \(0.710449\pi\)
\(824\) 10.7307 0.373822
\(825\) 22.1822 0.772286
\(826\) −21.1155 −0.734702
\(827\) −0.921236 −0.0320345 −0.0160173 0.999872i \(-0.505099\pi\)
−0.0160173 + 0.999872i \(0.505099\pi\)
\(828\) 7.37537 0.256312
\(829\) 3.57666 0.124222 0.0621112 0.998069i \(-0.480217\pi\)
0.0621112 + 0.998069i \(0.480217\pi\)
\(830\) −24.9656 −0.866568
\(831\) −30.4598 −1.05664
\(832\) 0 0
\(833\) 4.75966 0.164912
\(834\) −14.7651 −0.511273
\(835\) 11.3490 0.392748
\(836\) −5.35114 −0.185073
\(837\) 2.03627 0.0703838
\(838\) −11.2808 −0.389689
\(839\) −46.3478 −1.60010 −0.800052 0.599930i \(-0.795195\pi\)
−0.800052 + 0.599930i \(0.795195\pi\)
\(840\) −0.699146 −0.0241228
\(841\) −14.1779 −0.488893
\(842\) −10.4595 −0.360458
\(843\) 8.06012 0.277605
\(844\) 20.4713 0.704651
\(845\) 0 0
\(846\) 13.3380 0.458570
\(847\) −16.8474 −0.578885
\(848\) −37.5479 −1.28940
\(849\) 19.6060 0.672878
\(850\) 37.8890 1.29958
\(851\) 3.00256 0.102926
\(852\) 11.7774 0.403487
\(853\) −57.5650 −1.97099 −0.985495 0.169706i \(-0.945718\pi\)
−0.985495 + 0.169706i \(0.945718\pi\)
\(854\) 1.57637 0.0539423
\(855\) 0.570492 0.0195104
\(856\) −10.9746 −0.375105
\(857\) 34.8463 1.19033 0.595164 0.803604i \(-0.297087\pi\)
0.595164 + 0.803604i \(0.297087\pi\)
\(858\) 0 0
\(859\) −32.4716 −1.10792 −0.553959 0.832544i \(-0.686883\pi\)
−0.553959 + 0.832544i \(0.686883\pi\)
\(860\) 14.8994 0.508064
\(861\) 7.73993 0.263776
\(862\) −45.5919 −1.55287
\(863\) −24.8269 −0.845117 −0.422559 0.906336i \(-0.638868\pi\)
−0.422559 + 0.906336i \(0.638868\pi\)
\(864\) 7.25098 0.246683
\(865\) −12.5938 −0.428202
\(866\) −69.8211 −2.37262
\(867\) 5.65435 0.192032
\(868\) −3.23020 −0.109640
\(869\) 0.721465 0.0244740
\(870\) 6.50684 0.220602
\(871\) 0 0
\(872\) −11.5566 −0.391355
\(873\) 7.63761 0.258494
\(874\) −5.62827 −0.190379
\(875\) 8.21378 0.277677
\(876\) 15.1904 0.513235
\(877\) 9.37789 0.316669 0.158334 0.987386i \(-0.449388\pi\)
0.158334 + 0.987386i \(0.449388\pi\)
\(878\) 31.5341 1.06422
\(879\) 22.0402 0.743398
\(880\) 21.9288 0.739220
\(881\) −36.4979 −1.22964 −0.614822 0.788666i \(-0.710772\pi\)
−0.614822 + 0.788666i \(0.710772\pi\)
\(882\) −1.89376 −0.0637663
\(883\) 43.9023 1.47743 0.738716 0.674017i \(-0.235433\pi\)
0.738716 + 0.674017i \(0.235433\pi\)
\(884\) 0 0
\(885\) −9.95098 −0.334498
\(886\) −28.4587 −0.956088
\(887\) −35.1542 −1.18036 −0.590181 0.807271i \(-0.700944\pi\)
−0.590181 + 0.807271i \(0.700944\pi\)
\(888\) 0.505918 0.0169775
\(889\) 1.69886 0.0569780
\(890\) 9.30419 0.311877
\(891\) −5.27707 −0.176788
\(892\) 40.3908 1.35238
\(893\) −4.50221 −0.150661
\(894\) −30.1103 −1.00704
\(895\) −19.8645 −0.663997
\(896\) 6.13307 0.204891
\(897\) 0 0
\(898\) −11.2314 −0.374797
\(899\) −7.83954 −0.261463
\(900\) −6.66816 −0.222272
\(901\) 38.3820 1.27869
\(902\) −77.3491 −2.57544
\(903\) −10.5241 −0.350220
\(904\) −4.55278 −0.151423
\(905\) −6.47458 −0.215222
\(906\) 10.8708 0.361160
\(907\) −50.8281 −1.68772 −0.843859 0.536564i \(-0.819722\pi\)
−0.843859 + 0.536564i \(0.819722\pi\)
\(908\) −32.5907 −1.08156
\(909\) 13.9998 0.464344
\(910\) 0 0
\(911\) 30.5332 1.01161 0.505806 0.862647i \(-0.331195\pi\)
0.505806 + 0.862647i \(0.331195\pi\)
\(912\) −2.97641 −0.0985588
\(913\) −77.9506 −2.57979
\(914\) 1.02811 0.0340068
\(915\) 0.742888 0.0245591
\(916\) 20.8781 0.689833
\(917\) −1.19269 −0.0393862
\(918\) −9.01366 −0.297495
\(919\) −10.1569 −0.335046 −0.167523 0.985868i \(-0.553577\pi\)
−0.167523 + 0.985868i \(0.553577\pi\)
\(920\) 3.25056 0.107168
\(921\) 31.2987 1.03133
\(922\) 24.5597 0.808831
\(923\) 0 0
\(924\) 8.37118 0.275391
\(925\) −2.71465 −0.0892571
\(926\) −77.7316 −2.55442
\(927\) 13.6978 0.449894
\(928\) −27.9159 −0.916385
\(929\) 30.9183 1.01440 0.507198 0.861829i \(-0.330681\pi\)
0.507198 + 0.861829i \(0.330681\pi\)
\(930\) −3.44152 −0.112852
\(931\) 0.639234 0.0209500
\(932\) 8.87175 0.290604
\(933\) −14.2805 −0.467522
\(934\) −75.6051 −2.47387
\(935\) −22.4160 −0.733081
\(936\) 0 0
\(937\) −25.1820 −0.822660 −0.411330 0.911487i \(-0.634936\pi\)
−0.411330 + 0.911487i \(0.634936\pi\)
\(938\) 22.3983 0.731332
\(939\) 30.3102 0.989136
\(940\) −9.97125 −0.325226
\(941\) 21.4880 0.700488 0.350244 0.936659i \(-0.386099\pi\)
0.350244 + 0.936659i \(0.386099\pi\)
\(942\) −34.5080 −1.12433
\(943\) −35.9855 −1.17185
\(944\) 51.9170 1.68975
\(945\) −0.892462 −0.0290318
\(946\) 105.173 3.41945
\(947\) 6.75721 0.219580 0.109790 0.993955i \(-0.464982\pi\)
0.109790 + 0.993955i \(0.464982\pi\)
\(948\) −0.216879 −0.00704389
\(949\) 0 0
\(950\) 5.08859 0.165096
\(951\) −6.17539 −0.200251
\(952\) −3.72867 −0.120847
\(953\) 12.7022 0.411465 0.205733 0.978608i \(-0.434042\pi\)
0.205733 + 0.978608i \(0.434042\pi\)
\(954\) −15.2713 −0.494428
\(955\) −3.84656 −0.124472
\(956\) 41.1617 1.33126
\(957\) 20.3164 0.656737
\(958\) −76.1636 −2.46074
\(959\) −4.03058 −0.130154
\(960\) −3.94396 −0.127291
\(961\) −26.8536 −0.866245
\(962\) 0 0
\(963\) −14.0091 −0.451438
\(964\) −0.384693 −0.0123901
\(965\) −10.9811 −0.353495
\(966\) 8.80472 0.283287
\(967\) −30.7402 −0.988537 −0.494268 0.869309i \(-0.664564\pi\)
−0.494268 + 0.869309i \(0.664564\pi\)
\(968\) 13.1981 0.424204
\(969\) 3.04253 0.0977403
\(970\) −12.9084 −0.414463
\(971\) 39.8444 1.27867 0.639333 0.768930i \(-0.279210\pi\)
0.639333 + 0.768930i \(0.279210\pi\)
\(972\) 1.58633 0.0508816
\(973\) −7.79669 −0.249951
\(974\) −32.6733 −1.04692
\(975\) 0 0
\(976\) −3.87585 −0.124063
\(977\) 41.5811 1.33030 0.665149 0.746711i \(-0.268368\pi\)
0.665149 + 0.746711i \(0.268368\pi\)
\(978\) −32.6885 −1.04526
\(979\) 29.0507 0.928464
\(980\) 1.41574 0.0452241
\(981\) −14.7520 −0.470995
\(982\) −59.2662 −1.89126
\(983\) −18.8221 −0.600332 −0.300166 0.953887i \(-0.597042\pi\)
−0.300166 + 0.953887i \(0.597042\pi\)
\(984\) −6.06339 −0.193294
\(985\) 2.86287 0.0912187
\(986\) 34.7021 1.10514
\(987\) 7.04313 0.224185
\(988\) 0 0
\(989\) 48.9299 1.55588
\(990\) 8.91882 0.283459
\(991\) −42.2885 −1.34334 −0.671669 0.740851i \(-0.734422\pi\)
−0.671669 + 0.740851i \(0.734422\pi\)
\(992\) 14.7650 0.468788
\(993\) 10.5567 0.335007
\(994\) 14.0599 0.445952
\(995\) 1.90106 0.0602677
\(996\) 23.4326 0.742491
\(997\) 3.60346 0.114123 0.0570614 0.998371i \(-0.481827\pi\)
0.0570614 + 0.998371i \(0.481827\pi\)
\(998\) −31.9024 −1.00985
\(999\) 0.645805 0.0204324
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 3549.2.a.bh.1.4 yes 15
13.12 even 2 3549.2.a.bg.1.12 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.2.a.bg.1.12 15 13.12 even 2
3549.2.a.bh.1.4 yes 15 1.1 even 1 trivial