Properties

Label 147.4.a.j.1.2
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+2.41421 q^{2} -3.00000 q^{3} -2.17157 q^{4} +19.8995 q^{5} -7.24264 q^{6} -24.5563 q^{8} +9.00000 q^{9} +48.0416 q^{10} +23.9411 q^{11} +6.51472 q^{12} +87.3553 q^{13} -59.6985 q^{15} -41.9117 q^{16} -5.63961 q^{17} +21.7279 q^{18} +64.8873 q^{19} -43.2132 q^{20} +57.7990 q^{22} -25.5980 q^{23} +73.6690 q^{24} +270.990 q^{25} +210.894 q^{26} -27.0000 q^{27} +60.3188 q^{29} -144.125 q^{30} -122.711 q^{31} +95.2670 q^{32} -71.8234 q^{33} -13.6152 q^{34} -19.5442 q^{36} -56.1177 q^{37} +156.652 q^{38} -262.066 q^{39} -488.659 q^{40} -299.713 q^{41} -501.421 q^{43} -51.9899 q^{44} +179.095 q^{45} -61.7990 q^{46} +305.553 q^{47} +125.735 q^{48} +654.227 q^{50} +16.9188 q^{51} -189.698 q^{52} -375.117 q^{53} -65.1838 q^{54} +476.416 q^{55} -194.662 q^{57} +145.622 q^{58} -627.612 q^{59} +129.640 q^{60} +3.75736 q^{61} -296.250 q^{62} +565.288 q^{64} +1738.33 q^{65} -173.397 q^{66} -813.048 q^{67} +12.2468 q^{68} +76.7939 q^{69} +165.902 q^{71} -221.007 q^{72} +619.100 q^{73} -135.480 q^{74} -812.970 q^{75} -140.908 q^{76} -632.683 q^{78} -138.246 q^{79} -834.021 q^{80} +81.0000 q^{81} -723.571 q^{82} +621.137 q^{83} -112.225 q^{85} -1210.54 q^{86} -180.956 q^{87} -587.907 q^{88} +285.418 q^{89} +432.375 q^{90} +55.5879 q^{92} +368.132 q^{93} +737.671 q^{94} +1291.22 q^{95} -285.801 q^{96} -603.114 q^{97} +215.470 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + O(q^{10}) \) \( 2q + 2q^{2} - 6q^{3} - 10q^{4} + 20q^{5} - 6q^{6} - 18q^{8} + 18q^{9} + 48q^{10} - 20q^{11} + 30q^{12} + 104q^{13} - 60q^{15} + 18q^{16} + 116q^{17} + 18q^{18} + 192q^{19} - 44q^{20} + 76q^{22} + 28q^{23} + 54q^{24} + 146q^{25} + 204q^{26} - 54q^{27} + 296q^{29} - 144q^{30} - 104q^{31} + 18q^{32} + 60q^{33} - 64q^{34} - 90q^{36} - 248q^{37} + 104q^{38} - 312q^{39} - 488q^{40} + 20q^{41} - 720q^{43} + 292q^{44} + 180q^{45} - 84q^{46} - 96q^{47} - 54q^{48} + 706q^{50} - 348q^{51} - 320q^{52} + 268q^{53} - 54q^{54} + 472q^{55} - 576q^{57} + 48q^{58} - 616q^{59} + 132q^{60} + 16q^{61} - 304q^{62} + 118q^{64} + 1740q^{65} - 228q^{66} - 144q^{67} - 940q^{68} - 84q^{69} + 988q^{71} - 162q^{72} + 104q^{73} - 56q^{74} - 438q^{75} - 1136q^{76} - 612q^{78} - 944q^{79} - 828q^{80} + 162q^{81} - 856q^{82} + 1016q^{83} - 100q^{85} - 1120q^{86} - 888q^{87} - 876q^{88} - 388q^{89} + 432q^{90} - 364q^{92} + 312q^{93} + 904q^{94} + 1304q^{95} - 54q^{96} + 488q^{97} - 180q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421 0.853553 0.426777 0.904357i \(-0.359649\pi\)
0.426777 + 0.904357i \(0.359649\pi\)
\(3\) −3.00000 −0.577350
\(4\) −2.17157 −0.271447
\(5\) 19.8995 1.77986 0.889932 0.456092i \(-0.150751\pi\)
0.889932 + 0.456092i \(0.150751\pi\)
\(6\) −7.24264 −0.492799
\(7\) 0 0
\(8\) −24.5563 −1.08525
\(9\) 9.00000 0.333333
\(10\) 48.0416 1.51921
\(11\) 23.9411 0.656229 0.328115 0.944638i \(-0.393587\pi\)
0.328115 + 0.944638i \(0.393587\pi\)
\(12\) 6.51472 0.156720
\(13\) 87.3553 1.86369 0.931847 0.362852i \(-0.118197\pi\)
0.931847 + 0.362852i \(0.118197\pi\)
\(14\) 0 0
\(15\) −59.6985 −1.02761
\(16\) −41.9117 −0.654870
\(17\) −5.63961 −0.0804592 −0.0402296 0.999190i \(-0.512809\pi\)
−0.0402296 + 0.999190i \(0.512809\pi\)
\(18\) 21.7279 0.284518
\(19\) 64.8873 0.783483 0.391741 0.920075i \(-0.371873\pi\)
0.391741 + 0.920075i \(0.371873\pi\)
\(20\) −43.2132 −0.483138
\(21\) 0 0
\(22\) 57.7990 0.560127
\(23\) −25.5980 −0.232067 −0.116034 0.993245i \(-0.537018\pi\)
−0.116034 + 0.993245i \(0.537018\pi\)
\(24\) 73.6690 0.626568
\(25\) 270.990 2.16792
\(26\) 210.894 1.59076
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 60.3188 0.386238 0.193119 0.981175i \(-0.438140\pi\)
0.193119 + 0.981175i \(0.438140\pi\)
\(30\) −144.125 −0.877116
\(31\) −122.711 −0.710951 −0.355476 0.934686i \(-0.615681\pi\)
−0.355476 + 0.934686i \(0.615681\pi\)
\(32\) 95.2670 0.526281
\(33\) −71.8234 −0.378874
\(34\) −13.6152 −0.0686762
\(35\) 0 0
\(36\) −19.5442 −0.0904822
\(37\) −56.1177 −0.249343 −0.124672 0.992198i \(-0.539788\pi\)
−0.124672 + 0.992198i \(0.539788\pi\)
\(38\) 156.652 0.668744
\(39\) −262.066 −1.07600
\(40\) −488.659 −1.93159
\(41\) −299.713 −1.14164 −0.570820 0.821075i \(-0.693375\pi\)
−0.570820 + 0.821075i \(0.693375\pi\)
\(42\) 0 0
\(43\) −501.421 −1.77828 −0.889140 0.457635i \(-0.848697\pi\)
−0.889140 + 0.457635i \(0.848697\pi\)
\(44\) −51.9899 −0.178131
\(45\) 179.095 0.593288
\(46\) −61.7990 −0.198082
\(47\) 305.553 0.948288 0.474144 0.880447i \(-0.342758\pi\)
0.474144 + 0.880447i \(0.342758\pi\)
\(48\) 125.735 0.378089
\(49\) 0 0
\(50\) 654.227 1.85043
\(51\) 16.9188 0.0464531
\(52\) −189.698 −0.505893
\(53\) −375.117 −0.972194 −0.486097 0.873905i \(-0.661580\pi\)
−0.486097 + 0.873905i \(0.661580\pi\)
\(54\) −65.1838 −0.164266
\(55\) 476.416 1.16800
\(56\) 0 0
\(57\) −194.662 −0.452344
\(58\) 145.622 0.329675
\(59\) −627.612 −1.38488 −0.692442 0.721474i \(-0.743465\pi\)
−0.692442 + 0.721474i \(0.743465\pi\)
\(60\) 129.640 0.278940
\(61\) 3.75736 0.00788657 0.00394328 0.999992i \(-0.498745\pi\)
0.00394328 + 0.999992i \(0.498745\pi\)
\(62\) −296.250 −0.606835
\(63\) 0 0
\(64\) 565.288 1.10408
\(65\) 1738.33 3.31712
\(66\) −173.397 −0.323389
\(67\) −813.048 −1.48253 −0.741266 0.671212i \(-0.765774\pi\)
−0.741266 + 0.671212i \(0.765774\pi\)
\(68\) 12.2468 0.0218404
\(69\) 76.7939 0.133984
\(70\) 0 0
\(71\) 165.902 0.277310 0.138655 0.990341i \(-0.455722\pi\)
0.138655 + 0.990341i \(0.455722\pi\)
\(72\) −221.007 −0.361749
\(73\) 619.100 0.992605 0.496302 0.868150i \(-0.334691\pi\)
0.496302 + 0.868150i \(0.334691\pi\)
\(74\) −135.480 −0.212828
\(75\) −812.970 −1.25165
\(76\) −140.908 −0.212674
\(77\) 0 0
\(78\) −632.683 −0.918427
\(79\) −138.246 −0.196884 −0.0984421 0.995143i \(-0.531386\pi\)
−0.0984421 + 0.995143i \(0.531386\pi\)
\(80\) −834.021 −1.16558
\(81\) 81.0000 0.111111
\(82\) −723.571 −0.974451
\(83\) 621.137 0.821430 0.410715 0.911764i \(-0.365279\pi\)
0.410715 + 0.911764i \(0.365279\pi\)
\(84\) 0 0
\(85\) −112.225 −0.143207
\(86\) −1210.54 −1.51786
\(87\) −180.956 −0.222995
\(88\) −587.907 −0.712171
\(89\) 285.418 0.339936 0.169968 0.985450i \(-0.445634\pi\)
0.169968 + 0.985450i \(0.445634\pi\)
\(90\) 432.375 0.506403
\(91\) 0 0
\(92\) 55.5879 0.0629939
\(93\) 368.132 0.410468
\(94\) 737.671 0.809415
\(95\) 1291.22 1.39449
\(96\) −285.801 −0.303848
\(97\) −603.114 −0.631309 −0.315654 0.948874i \(-0.602224\pi\)
−0.315654 + 0.948874i \(0.602224\pi\)
\(98\) 0 0
\(99\) 215.470 0.218743
\(100\) −588.474 −0.588474
\(101\) 457.209 0.450436 0.225218 0.974308i \(-0.427691\pi\)
0.225218 + 0.974308i \(0.427691\pi\)
\(102\) 40.8457 0.0396502
\(103\) 786.045 0.751954 0.375977 0.926629i \(-0.377307\pi\)
0.375977 + 0.926629i \(0.377307\pi\)
\(104\) −2145.13 −2.02257
\(105\) 0 0
\(106\) −905.612 −0.829819
\(107\) −196.461 −0.177501 −0.0887504 0.996054i \(-0.528287\pi\)
−0.0887504 + 0.996054i \(0.528287\pi\)
\(108\) 58.6325 0.0522399
\(109\) −306.343 −0.269196 −0.134598 0.990900i \(-0.542974\pi\)
−0.134598 + 0.990900i \(0.542974\pi\)
\(110\) 1150.17 0.996950
\(111\) 168.353 0.143958
\(112\) 0 0
\(113\) 1997.63 1.66302 0.831508 0.555512i \(-0.187478\pi\)
0.831508 + 0.555512i \(0.187478\pi\)
\(114\) −469.955 −0.386100
\(115\) −509.387 −0.413048
\(116\) −130.987 −0.104843
\(117\) 786.198 0.621231
\(118\) −1515.19 −1.18207
\(119\) 0 0
\(120\) 1465.98 1.11521
\(121\) −757.823 −0.569363
\(122\) 9.07107 0.00673161
\(123\) 899.138 0.659127
\(124\) 266.475 0.192985
\(125\) 2905.13 2.07874
\(126\) 0 0
\(127\) −2311.40 −1.61499 −0.807494 0.589875i \(-0.799177\pi\)
−0.807494 + 0.589875i \(0.799177\pi\)
\(128\) 602.591 0.416109
\(129\) 1504.26 1.02669
\(130\) 4196.69 2.83134
\(131\) −155.018 −0.103389 −0.0516945 0.998663i \(-0.516462\pi\)
−0.0516945 + 0.998663i \(0.516462\pi\)
\(132\) 155.970 0.102844
\(133\) 0 0
\(134\) −1962.87 −1.26542
\(135\) −537.286 −0.342535
\(136\) 138.488 0.0873182
\(137\) 516.936 0.322371 0.161186 0.986924i \(-0.448468\pi\)
0.161186 + 0.986924i \(0.448468\pi\)
\(138\) 185.397 0.114363
\(139\) 958.067 0.584620 0.292310 0.956324i \(-0.405576\pi\)
0.292310 + 0.956324i \(0.405576\pi\)
\(140\) 0 0
\(141\) −916.660 −0.547494
\(142\) 400.524 0.236699
\(143\) 2091.39 1.22301
\(144\) −377.205 −0.218290
\(145\) 1200.31 0.687452
\(146\) 1494.64 0.847241
\(147\) 0 0
\(148\) 121.864 0.0676834
\(149\) −1770.63 −0.973526 −0.486763 0.873534i \(-0.661822\pi\)
−0.486763 + 0.873534i \(0.661822\pi\)
\(150\) −1962.68 −1.06835
\(151\) −2540.24 −1.36902 −0.684508 0.729005i \(-0.739983\pi\)
−0.684508 + 0.729005i \(0.739983\pi\)
\(152\) −1593.40 −0.850272
\(153\) −50.7565 −0.0268197
\(154\) 0 0
\(155\) −2441.88 −1.26540
\(156\) 569.095 0.292078
\(157\) −1083.34 −0.550702 −0.275351 0.961344i \(-0.588794\pi\)
−0.275351 + 0.961344i \(0.588794\pi\)
\(158\) −333.754 −0.168051
\(159\) 1125.35 0.561296
\(160\) 1895.77 0.936709
\(161\) 0 0
\(162\) 195.551 0.0948393
\(163\) 2968.72 1.42655 0.713277 0.700882i \(-0.247210\pi\)
0.713277 + 0.700882i \(0.247210\pi\)
\(164\) 650.848 0.309895
\(165\) −1429.25 −0.674345
\(166\) 1499.56 0.701134
\(167\) −2091.53 −0.969149 −0.484574 0.874750i \(-0.661025\pi\)
−0.484574 + 0.874750i \(0.661025\pi\)
\(168\) 0 0
\(169\) 5433.96 2.47335
\(170\) −270.936 −0.122234
\(171\) 583.986 0.261161
\(172\) 1088.87 0.482708
\(173\) −470.148 −0.206617 −0.103308 0.994649i \(-0.532943\pi\)
−0.103308 + 0.994649i \(0.532943\pi\)
\(174\) −436.867 −0.190338
\(175\) 0 0
\(176\) −1003.41 −0.429745
\(177\) 1882.84 0.799563
\(178\) 689.061 0.290153
\(179\) 1056.46 0.441137 0.220569 0.975371i \(-0.429209\pi\)
0.220569 + 0.975371i \(0.429209\pi\)
\(180\) −388.919 −0.161046
\(181\) −406.470 −0.166921 −0.0834605 0.996511i \(-0.526597\pi\)
−0.0834605 + 0.996511i \(0.526597\pi\)
\(182\) 0 0
\(183\) −11.2721 −0.00455331
\(184\) 628.593 0.251850
\(185\) −1116.71 −0.443797
\(186\) 888.749 0.350356
\(187\) −135.019 −0.0527997
\(188\) −663.531 −0.257410
\(189\) 0 0
\(190\) 3117.29 1.19027
\(191\) 179.410 0.0679666 0.0339833 0.999422i \(-0.489181\pi\)
0.0339833 + 0.999422i \(0.489181\pi\)
\(192\) −1695.87 −0.637440
\(193\) −2388.37 −0.890769 −0.445385 0.895339i \(-0.646933\pi\)
−0.445385 + 0.895339i \(0.646933\pi\)
\(194\) −1456.05 −0.538856
\(195\) −5214.98 −1.91514
\(196\) 0 0
\(197\) −2665.35 −0.963952 −0.481976 0.876184i \(-0.660081\pi\)
−0.481976 + 0.876184i \(0.660081\pi\)
\(198\) 520.191 0.186709
\(199\) −1342.31 −0.478159 −0.239079 0.971000i \(-0.576846\pi\)
−0.239079 + 0.971000i \(0.576846\pi\)
\(200\) −6654.52 −2.35273
\(201\) 2439.14 0.855940
\(202\) 1103.80 0.384471
\(203\) 0 0
\(204\) −36.7405 −0.0126095
\(205\) −5964.13 −2.03197
\(206\) 1897.68 0.641833
\(207\) −230.382 −0.0773558
\(208\) −3661.21 −1.22048
\(209\) 1553.48 0.514144
\(210\) 0 0
\(211\) 628.442 0.205042 0.102521 0.994731i \(-0.467309\pi\)
0.102521 + 0.994731i \(0.467309\pi\)
\(212\) 814.594 0.263899
\(213\) −497.707 −0.160105
\(214\) −474.299 −0.151506
\(215\) −9978.03 −3.16510
\(216\) 663.021 0.208856
\(217\) 0 0
\(218\) −739.578 −0.229773
\(219\) −1857.30 −0.573081
\(220\) −1034.57 −0.317049
\(221\) −492.650 −0.149951
\(222\) 406.441 0.122876
\(223\) −969.970 −0.291273 −0.145637 0.989338i \(-0.546523\pi\)
−0.145637 + 0.989338i \(0.546523\pi\)
\(224\) 0 0
\(225\) 2438.91 0.722640
\(226\) 4822.70 1.41947
\(227\) −4748.64 −1.38845 −0.694225 0.719758i \(-0.744253\pi\)
−0.694225 + 0.719758i \(0.744253\pi\)
\(228\) 422.723 0.122787
\(229\) −4801.99 −1.38570 −0.692848 0.721083i \(-0.743644\pi\)
−0.692848 + 0.721083i \(0.743644\pi\)
\(230\) −1229.77 −0.352559
\(231\) 0 0
\(232\) −1481.21 −0.419164
\(233\) −3155.29 −0.887166 −0.443583 0.896233i \(-0.646293\pi\)
−0.443583 + 0.896233i \(0.646293\pi\)
\(234\) 1898.05 0.530254
\(235\) 6080.36 1.68782
\(236\) 1362.91 0.375922
\(237\) 414.737 0.113671
\(238\) 0 0
\(239\) 4241.93 1.14806 0.574032 0.818833i \(-0.305378\pi\)
0.574032 + 0.818833i \(0.305378\pi\)
\(240\) 2502.06 0.672948
\(241\) −4342.99 −1.16081 −0.580407 0.814326i \(-0.697107\pi\)
−0.580407 + 0.814326i \(0.697107\pi\)
\(242\) −1829.55 −0.485982
\(243\) −243.000 −0.0641500
\(244\) −8.15938 −0.00214078
\(245\) 0 0
\(246\) 2170.71 0.562600
\(247\) 5668.25 1.46017
\(248\) 3013.33 0.771558
\(249\) −1863.41 −0.474253
\(250\) 7013.59 1.77431
\(251\) 3003.01 0.755172 0.377586 0.925974i \(-0.376754\pi\)
0.377586 + 0.925974i \(0.376754\pi\)
\(252\) 0 0
\(253\) −612.844 −0.152289
\(254\) −5580.21 −1.37848
\(255\) 336.676 0.0826803
\(256\) −3067.52 −0.748907
\(257\) 4468.84 1.08466 0.542332 0.840164i \(-0.317541\pi\)
0.542332 + 0.840164i \(0.317541\pi\)
\(258\) 3631.61 0.876335
\(259\) 0 0
\(260\) −3774.90 −0.900422
\(261\) 542.869 0.128746
\(262\) −374.246 −0.0882480
\(263\) 6160.13 1.44430 0.722148 0.691738i \(-0.243155\pi\)
0.722148 + 0.691738i \(0.243155\pi\)
\(264\) 1763.72 0.411172
\(265\) −7464.64 −1.73037
\(266\) 0 0
\(267\) −856.255 −0.196262
\(268\) 1765.59 0.402428
\(269\) 4988.90 1.13078 0.565388 0.824825i \(-0.308726\pi\)
0.565388 + 0.824825i \(0.308726\pi\)
\(270\) −1297.12 −0.292372
\(271\) 4433.73 0.993837 0.496918 0.867797i \(-0.334465\pi\)
0.496918 + 0.867797i \(0.334465\pi\)
\(272\) 236.366 0.0526903
\(273\) 0 0
\(274\) 1247.99 0.275161
\(275\) 6487.80 1.42265
\(276\) −166.764 −0.0363695
\(277\) 1112.37 0.241284 0.120642 0.992696i \(-0.461505\pi\)
0.120642 + 0.992696i \(0.461505\pi\)
\(278\) 2312.98 0.499005
\(279\) −1104.40 −0.236984
\(280\) 0 0
\(281\) 2813.22 0.597233 0.298616 0.954373i \(-0.403475\pi\)
0.298616 + 0.954373i \(0.403475\pi\)
\(282\) −2213.01 −0.467316
\(283\) 3147.54 0.661137 0.330569 0.943782i \(-0.392759\pi\)
0.330569 + 0.943782i \(0.392759\pi\)
\(284\) −360.269 −0.0752748
\(285\) −3873.67 −0.805111
\(286\) 5049.05 1.04390
\(287\) 0 0
\(288\) 857.403 0.175427
\(289\) −4881.19 −0.993526
\(290\) 2897.81 0.586777
\(291\) 1809.34 0.364486
\(292\) −1344.42 −0.269439
\(293\) 9143.04 1.82301 0.911505 0.411289i \(-0.134921\pi\)
0.911505 + 0.411289i \(0.134921\pi\)
\(294\) 0 0
\(295\) −12489.2 −2.46491
\(296\) 1378.05 0.270599
\(297\) −646.410 −0.126291
\(298\) −4274.67 −0.830956
\(299\) −2236.12 −0.432502
\(300\) 1765.42 0.339756
\(301\) 0 0
\(302\) −6132.67 −1.16853
\(303\) −1371.63 −0.260059
\(304\) −2719.54 −0.513079
\(305\) 74.7696 0.0140370
\(306\) −122.537 −0.0228921
\(307\) −4648.90 −0.864257 −0.432129 0.901812i \(-0.642237\pi\)
−0.432129 + 0.901812i \(0.642237\pi\)
\(308\) 0 0
\(309\) −2358.13 −0.434141
\(310\) −5895.22 −1.08008
\(311\) −6417.18 −1.17005 −0.585024 0.811016i \(-0.698915\pi\)
−0.585024 + 0.811016i \(0.698915\pi\)
\(312\) 6435.38 1.16773
\(313\) 5868.22 1.05972 0.529858 0.848086i \(-0.322245\pi\)
0.529858 + 0.848086i \(0.322245\pi\)
\(314\) −2615.42 −0.470054
\(315\) 0 0
\(316\) 300.210 0.0534435
\(317\) −3974.52 −0.704200 −0.352100 0.935962i \(-0.614532\pi\)
−0.352100 + 0.935962i \(0.614532\pi\)
\(318\) 2716.84 0.479096
\(319\) 1444.10 0.253461
\(320\) 11249.0 1.96511
\(321\) 589.383 0.102480
\(322\) 0 0
\(323\) −365.939 −0.0630384
\(324\) −175.897 −0.0301607
\(325\) 23672.4 4.04034
\(326\) 7167.13 1.21764
\(327\) 919.029 0.155420
\(328\) 7359.85 1.23896
\(329\) 0 0
\(330\) −3450.51 −0.575589
\(331\) −8912.82 −1.48004 −0.740020 0.672585i \(-0.765184\pi\)
−0.740020 + 0.672585i \(0.765184\pi\)
\(332\) −1348.84 −0.222974
\(333\) −505.060 −0.0831144
\(334\) −5049.41 −0.827220
\(335\) −16179.2 −2.63871
\(336\) 0 0
\(337\) 3977.06 0.642862 0.321431 0.946933i \(-0.395836\pi\)
0.321431 + 0.946933i \(0.395836\pi\)
\(338\) 13118.7 2.11114
\(339\) −5992.88 −0.960143
\(340\) 243.706 0.0388729
\(341\) −2937.83 −0.466547
\(342\) 1409.87 0.222915
\(343\) 0 0
\(344\) 12313.1 1.92987
\(345\) 1528.16 0.238474
\(346\) −1135.04 −0.176358
\(347\) 6826.43 1.05609 0.528043 0.849218i \(-0.322926\pi\)
0.528043 + 0.849218i \(0.322926\pi\)
\(348\) 392.960 0.0605312
\(349\) −807.342 −0.123828 −0.0619141 0.998081i \(-0.519720\pi\)
−0.0619141 + 0.998081i \(0.519720\pi\)
\(350\) 0 0
\(351\) −2358.59 −0.358668
\(352\) 2280.80 0.345361
\(353\) −7919.20 −1.19404 −0.597020 0.802226i \(-0.703649\pi\)
−0.597020 + 0.802226i \(0.703649\pi\)
\(354\) 4545.57 0.682470
\(355\) 3301.38 0.493574
\(356\) −619.807 −0.0922744
\(357\) 0 0
\(358\) 2550.52 0.376534
\(359\) −8819.21 −1.29655 −0.648273 0.761408i \(-0.724509\pi\)
−0.648273 + 0.761408i \(0.724509\pi\)
\(360\) −4397.93 −0.643865
\(361\) −2648.64 −0.386155
\(362\) −981.307 −0.142476
\(363\) 2273.47 0.328722
\(364\) 0 0
\(365\) 12319.8 1.76670
\(366\) −27.2132 −0.00388649
\(367\) −11161.8 −1.58758 −0.793788 0.608194i \(-0.791894\pi\)
−0.793788 + 0.608194i \(0.791894\pi\)
\(368\) 1072.85 0.151974
\(369\) −2697.41 −0.380547
\(370\) −2695.99 −0.378805
\(371\) 0 0
\(372\) −799.426 −0.111420
\(373\) 2727.86 0.378668 0.189334 0.981913i \(-0.439367\pi\)
0.189334 + 0.981913i \(0.439367\pi\)
\(374\) −325.964 −0.0450673
\(375\) −8715.38 −1.20016
\(376\) −7503.28 −1.02913
\(377\) 5269.17 0.719830
\(378\) 0 0
\(379\) 4086.49 0.553849 0.276924 0.960892i \(-0.410685\pi\)
0.276924 + 0.960892i \(0.410685\pi\)
\(380\) −2803.99 −0.378530
\(381\) 6934.20 0.932414
\(382\) 433.133 0.0580131
\(383\) 13032.9 1.73878 0.869389 0.494129i \(-0.164513\pi\)
0.869389 + 0.494129i \(0.164513\pi\)
\(384\) −1807.77 −0.240241
\(385\) 0 0
\(386\) −5766.03 −0.760319
\(387\) −4512.79 −0.592760
\(388\) 1309.71 0.171367
\(389\) −194.991 −0.0254151 −0.0127075 0.999919i \(-0.504045\pi\)
−0.0127075 + 0.999919i \(0.504045\pi\)
\(390\) −12590.1 −1.63468
\(391\) 144.363 0.0186719
\(392\) 0 0
\(393\) 465.053 0.0596916
\(394\) −6434.73 −0.822785
\(395\) −2751.02 −0.350427
\(396\) −467.909 −0.0593771
\(397\) −14183.0 −1.79300 −0.896501 0.443042i \(-0.853899\pi\)
−0.896501 + 0.443042i \(0.853899\pi\)
\(398\) −3240.61 −0.408134
\(399\) 0 0
\(400\) −11357.6 −1.41971
\(401\) 10005.0 1.24596 0.622978 0.782239i \(-0.285923\pi\)
0.622978 + 0.782239i \(0.285923\pi\)
\(402\) 5888.61 0.730590
\(403\) −10719.4 −1.32500
\(404\) −992.863 −0.122269
\(405\) 1611.86 0.197763
\(406\) 0 0
\(407\) −1343.52 −0.163626
\(408\) −415.465 −0.0504132
\(409\) 4634.93 0.560349 0.280174 0.959949i \(-0.409608\pi\)
0.280174 + 0.959949i \(0.409608\pi\)
\(410\) −14398.7 −1.73439
\(411\) −1550.81 −0.186121
\(412\) −1706.95 −0.204115
\(413\) 0 0
\(414\) −556.191 −0.0660273
\(415\) 12360.3 1.46203
\(416\) 8322.08 0.980826
\(417\) −2874.20 −0.337531
\(418\) 3750.42 0.438849
\(419\) 4998.31 0.582777 0.291388 0.956605i \(-0.405883\pi\)
0.291388 + 0.956605i \(0.405883\pi\)
\(420\) 0 0
\(421\) −704.160 −0.0815170 −0.0407585 0.999169i \(-0.512977\pi\)
−0.0407585 + 0.999169i \(0.512977\pi\)
\(422\) 1517.19 0.175014
\(423\) 2749.98 0.316096
\(424\) 9211.50 1.05507
\(425\) −1528.28 −0.174429
\(426\) −1201.57 −0.136658
\(427\) 0 0
\(428\) 426.629 0.0481820
\(429\) −6274.16 −0.706105
\(430\) −24089.1 −2.70158
\(431\) −10332.8 −1.15479 −0.577393 0.816466i \(-0.695930\pi\)
−0.577393 + 0.816466i \(0.695930\pi\)
\(432\) 1131.62 0.126030
\(433\) 11106.8 1.23270 0.616348 0.787474i \(-0.288611\pi\)
0.616348 + 0.787474i \(0.288611\pi\)
\(434\) 0 0
\(435\) −3600.94 −0.396901
\(436\) 665.246 0.0730723
\(437\) −1660.98 −0.181821
\(438\) −4483.92 −0.489155
\(439\) −7299.28 −0.793566 −0.396783 0.917912i \(-0.629874\pi\)
−0.396783 + 0.917912i \(0.629874\pi\)
\(440\) −11699.0 −1.26757
\(441\) 0 0
\(442\) −1189.36 −0.127991
\(443\) 16089.7 1.72560 0.862802 0.505542i \(-0.168707\pi\)
0.862802 + 0.505542i \(0.168707\pi\)
\(444\) −365.591 −0.0390770
\(445\) 5679.68 0.605040
\(446\) −2341.71 −0.248617
\(447\) 5311.88 0.562065
\(448\) 0 0
\(449\) 13561.7 1.42543 0.712715 0.701454i \(-0.247466\pi\)
0.712715 + 0.701454i \(0.247466\pi\)
\(450\) 5888.05 0.616812
\(451\) −7175.46 −0.749178
\(452\) −4337.99 −0.451420
\(453\) 7620.71 0.790402
\(454\) −11464.2 −1.18512
\(455\) 0 0
\(456\) 4780.19 0.490905
\(457\) 10848.6 1.11045 0.555224 0.831701i \(-0.312633\pi\)
0.555224 + 0.831701i \(0.312633\pi\)
\(458\) −11593.0 −1.18277
\(459\) 152.269 0.0154844
\(460\) 1106.17 0.112121
\(461\) −1758.69 −0.177679 −0.0888397 0.996046i \(-0.528316\pi\)
−0.0888397 + 0.996046i \(0.528316\pi\)
\(462\) 0 0
\(463\) −5411.95 −0.543228 −0.271614 0.962406i \(-0.587557\pi\)
−0.271614 + 0.962406i \(0.587557\pi\)
\(464\) −2528.06 −0.252936
\(465\) 7325.64 0.730577
\(466\) −7617.54 −0.757244
\(467\) 8111.34 0.803744 0.401872 0.915696i \(-0.368360\pi\)
0.401872 + 0.915696i \(0.368360\pi\)
\(468\) −1707.29 −0.168631
\(469\) 0 0
\(470\) 14679.3 1.44065
\(471\) 3250.03 0.317948
\(472\) 15411.9 1.50294
\(473\) −12004.6 −1.16696
\(474\) 1001.26 0.0970244
\(475\) 17583.8 1.69853
\(476\) 0 0
\(477\) −3376.05 −0.324065
\(478\) 10240.9 0.979935
\(479\) 2095.76 0.199912 0.0999559 0.994992i \(-0.468130\pi\)
0.0999559 + 0.994992i \(0.468130\pi\)
\(480\) −5687.30 −0.540809
\(481\) −4902.18 −0.464699
\(482\) −10484.9 −0.990817
\(483\) 0 0
\(484\) 1645.67 0.154552
\(485\) −12001.7 −1.12364
\(486\) −586.654 −0.0547555
\(487\) 9610.08 0.894197 0.447099 0.894485i \(-0.352457\pi\)
0.447099 + 0.894485i \(0.352457\pi\)
\(488\) −92.2670 −0.00855888
\(489\) −8906.17 −0.823622
\(490\) 0 0
\(491\) −11717.3 −1.07698 −0.538488 0.842633i \(-0.681004\pi\)
−0.538488 + 0.842633i \(0.681004\pi\)
\(492\) −1952.54 −0.178918
\(493\) −340.174 −0.0310764
\(494\) 13684.4 1.24633
\(495\) 4287.75 0.389333
\(496\) 5143.01 0.465581
\(497\) 0 0
\(498\) −4498.67 −0.404800
\(499\) −9195.19 −0.824916 −0.412458 0.910977i \(-0.635330\pi\)
−0.412458 + 0.910977i \(0.635330\pi\)
\(500\) −6308.69 −0.564266
\(501\) 6274.60 0.559538
\(502\) 7249.90 0.644580
\(503\) 16118.8 1.42883 0.714414 0.699724i \(-0.246693\pi\)
0.714414 + 0.699724i \(0.246693\pi\)
\(504\) 0 0
\(505\) 9098.23 0.801715
\(506\) −1479.54 −0.129987
\(507\) −16301.9 −1.42799
\(508\) 5019.37 0.438383
\(509\) 4918.78 0.428333 0.214166 0.976797i \(-0.431297\pi\)
0.214166 + 0.976797i \(0.431297\pi\)
\(510\) 812.808 0.0705721
\(511\) 0 0
\(512\) −12226.4 −1.05534
\(513\) −1751.96 −0.150781
\(514\) 10788.7 0.925819
\(515\) 15641.9 1.33838
\(516\) −3266.62 −0.278692
\(517\) 7315.29 0.622294
\(518\) 0 0
\(519\) 1410.44 0.119290
\(520\) −42687.0 −3.59990
\(521\) 13963.4 1.17418 0.587089 0.809522i \(-0.300274\pi\)
0.587089 + 0.809522i \(0.300274\pi\)
\(522\) 1310.60 0.109892
\(523\) −13755.3 −1.15005 −0.575024 0.818136i \(-0.695007\pi\)
−0.575024 + 0.818136i \(0.695007\pi\)
\(524\) 336.632 0.0280646
\(525\) 0 0
\(526\) 14871.9 1.23278
\(527\) 692.040 0.0572026
\(528\) 3010.24 0.248113
\(529\) −11511.7 −0.946145
\(530\) −18021.2 −1.47697
\(531\) −5648.51 −0.461628
\(532\) 0 0
\(533\) −26181.5 −2.12767
\(534\) −2067.18 −0.167520
\(535\) −3909.47 −0.315928
\(536\) 19965.5 1.60891
\(537\) −3169.38 −0.254691
\(538\) 12044.3 0.965178
\(539\) 0 0
\(540\) 1166.76 0.0929800
\(541\) 14462.7 1.14935 0.574676 0.818381i \(-0.305128\pi\)
0.574676 + 0.818381i \(0.305128\pi\)
\(542\) 10704.0 0.848293
\(543\) 1219.41 0.0963719
\(544\) −537.269 −0.0423441
\(545\) −6096.07 −0.479132
\(546\) 0 0
\(547\) 13682.5 1.06951 0.534755 0.845007i \(-0.320404\pi\)
0.534755 + 0.845007i \(0.320404\pi\)
\(548\) −1122.56 −0.0875065
\(549\) 33.8162 0.00262886
\(550\) 15662.9 1.21431
\(551\) 3913.92 0.302611
\(552\) −1885.78 −0.145406
\(553\) 0 0
\(554\) 2685.49 0.205949
\(555\) 3350.14 0.256227
\(556\) −2080.51 −0.158693
\(557\) −7663.13 −0.582939 −0.291470 0.956580i \(-0.594144\pi\)
−0.291470 + 0.956580i \(0.594144\pi\)
\(558\) −2666.25 −0.202278
\(559\) −43801.8 −3.31417
\(560\) 0 0
\(561\) 405.056 0.0304839
\(562\) 6791.71 0.509770
\(563\) 17470.5 1.30780 0.653902 0.756580i \(-0.273131\pi\)
0.653902 + 0.756580i \(0.273131\pi\)
\(564\) 1990.59 0.148616
\(565\) 39751.8 2.95995
\(566\) 7598.83 0.564316
\(567\) 0 0
\(568\) −4073.96 −0.300950
\(569\) −13873.9 −1.02219 −0.511094 0.859525i \(-0.670760\pi\)
−0.511094 + 0.859525i \(0.670760\pi\)
\(570\) −9351.88 −0.687205
\(571\) −3777.52 −0.276855 −0.138428 0.990373i \(-0.544205\pi\)
−0.138428 + 0.990373i \(0.544205\pi\)
\(572\) −4541.60 −0.331982
\(573\) −538.229 −0.0392405
\(574\) 0 0
\(575\) −6936.79 −0.503103
\(576\) 5087.60 0.368026
\(577\) 13880.5 1.00148 0.500738 0.865599i \(-0.333062\pi\)
0.500738 + 0.865599i \(0.333062\pi\)
\(578\) −11784.2 −0.848028
\(579\) 7165.10 0.514286
\(580\) −2606.57 −0.186607
\(581\) 0 0
\(582\) 4368.14 0.311108
\(583\) −8980.72 −0.637982
\(584\) −15202.8 −1.07722
\(585\) 15644.9 1.10571
\(586\) 22073.2 1.55604
\(587\) −2395.61 −0.168445 −0.0842227 0.996447i \(-0.526841\pi\)
−0.0842227 + 0.996447i \(0.526841\pi\)
\(588\) 0 0
\(589\) −7962.36 −0.557018
\(590\) −30151.5 −2.10393
\(591\) 7996.06 0.556538
\(592\) 2351.99 0.163287
\(593\) 6603.50 0.457290 0.228645 0.973510i \(-0.426570\pi\)
0.228645 + 0.973510i \(0.426570\pi\)
\(594\) −1560.57 −0.107796
\(595\) 0 0
\(596\) 3845.04 0.264260
\(597\) 4026.92 0.276065
\(598\) −5398.47 −0.369164
\(599\) 17252.1 1.17680 0.588399 0.808571i \(-0.299759\pi\)
0.588399 + 0.808571i \(0.299759\pi\)
\(600\) 19963.6 1.35835
\(601\) 12833.1 0.871005 0.435503 0.900187i \(-0.356571\pi\)
0.435503 + 0.900187i \(0.356571\pi\)
\(602\) 0 0
\(603\) −7317.43 −0.494177
\(604\) 5516.31 0.371615
\(605\) −15080.3 −1.01339
\(606\) −3311.40 −0.221974
\(607\) 8620.16 0.576411 0.288206 0.957569i \(-0.406941\pi\)
0.288206 + 0.957569i \(0.406941\pi\)
\(608\) 6181.62 0.412332
\(609\) 0 0
\(610\) 180.510 0.0119813
\(611\) 26691.7 1.76732
\(612\) 110.221 0.00728013
\(613\) 5136.73 0.338451 0.169226 0.985577i \(-0.445873\pi\)
0.169226 + 0.985577i \(0.445873\pi\)
\(614\) −11223.4 −0.737690
\(615\) 17892.4 1.17316
\(616\) 0 0
\(617\) −1759.82 −0.114826 −0.0574131 0.998351i \(-0.518285\pi\)
−0.0574131 + 0.998351i \(0.518285\pi\)
\(618\) −5693.04 −0.370562
\(619\) −3560.24 −0.231176 −0.115588 0.993297i \(-0.536875\pi\)
−0.115588 + 0.993297i \(0.536875\pi\)
\(620\) 5302.72 0.343488
\(621\) 691.145 0.0446614
\(622\) −15492.4 −0.998698
\(623\) 0 0
\(624\) 10983.6 0.704643
\(625\) 23936.8 1.53195
\(626\) 14167.1 0.904524
\(627\) −4660.43 −0.296841
\(628\) 2352.56 0.149486
\(629\) 316.482 0.0200620
\(630\) 0 0
\(631\) −27321.4 −1.72369 −0.861845 0.507172i \(-0.830691\pi\)
−0.861845 + 0.507172i \(0.830691\pi\)
\(632\) 3394.81 0.213668
\(633\) −1885.33 −0.118381
\(634\) −9595.34 −0.601072
\(635\) −45995.7 −2.87446
\(636\) −2443.78 −0.152362
\(637\) 0 0
\(638\) 3486.36 0.216342
\(639\) 1493.12 0.0924366
\(640\) 11991.3 0.740619
\(641\) −21927.0 −1.35112 −0.675558 0.737307i \(-0.736097\pi\)
−0.675558 + 0.737307i \(0.736097\pi\)
\(642\) 1422.90 0.0874723
\(643\) −5826.04 −0.357320 −0.178660 0.983911i \(-0.557176\pi\)
−0.178660 + 0.983911i \(0.557176\pi\)
\(644\) 0 0
\(645\) 29934.1 1.82737
\(646\) −883.455 −0.0538066
\(647\) 24210.7 1.47113 0.735565 0.677454i \(-0.236917\pi\)
0.735565 + 0.677454i \(0.236917\pi\)
\(648\) −1989.06 −0.120583
\(649\) −15025.7 −0.908801
\(650\) 57150.3 3.44864
\(651\) 0 0
\(652\) −6446.80 −0.387233
\(653\) −25623.3 −1.53556 −0.767778 0.640716i \(-0.778637\pi\)
−0.767778 + 0.640716i \(0.778637\pi\)
\(654\) 2218.73 0.132660
\(655\) −3084.77 −0.184018
\(656\) 12561.5 0.747626
\(657\) 5571.90 0.330868
\(658\) 0 0
\(659\) −23273.7 −1.37574 −0.687871 0.725833i \(-0.741455\pi\)
−0.687871 + 0.725833i \(0.741455\pi\)
\(660\) 3103.72 0.183049
\(661\) −20036.5 −1.17902 −0.589508 0.807763i \(-0.700678\pi\)
−0.589508 + 0.807763i \(0.700678\pi\)
\(662\) −21517.5 −1.26329
\(663\) 1477.95 0.0865744
\(664\) −15252.9 −0.891454
\(665\) 0 0
\(666\) −1219.32 −0.0709426
\(667\) −1544.04 −0.0896333
\(668\) 4541.92 0.263072
\(669\) 2909.91 0.168167
\(670\) −39060.1 −2.25228
\(671\) 89.9554 0.00517540
\(672\) 0 0
\(673\) −18127.8 −1.03830 −0.519149 0.854684i \(-0.673751\pi\)
−0.519149 + 0.854684i \(0.673751\pi\)
\(674\) 9601.48 0.548717
\(675\) −7316.73 −0.417216
\(676\) −11800.2 −0.671383
\(677\) 13815.5 0.784301 0.392150 0.919901i \(-0.371731\pi\)
0.392150 + 0.919901i \(0.371731\pi\)
\(678\) −14468.1 −0.819533
\(679\) 0 0
\(680\) 2755.85 0.155415
\(681\) 14245.9 0.801622
\(682\) −7092.55 −0.398223
\(683\) −4604.23 −0.257944 −0.128972 0.991648i \(-0.541168\pi\)
−0.128972 + 0.991648i \(0.541168\pi\)
\(684\) −1268.17 −0.0708912
\(685\) 10286.8 0.573777
\(686\) 0 0
\(687\) 14406.0 0.800032
\(688\) 21015.4 1.16454
\(689\) −32768.5 −1.81187
\(690\) 3689.31 0.203550
\(691\) 17913.7 0.986205 0.493103 0.869971i \(-0.335863\pi\)
0.493103 + 0.869971i \(0.335863\pi\)
\(692\) 1020.96 0.0560854
\(693\) 0 0
\(694\) 16480.5 0.901426
\(695\) 19065.1 1.04054
\(696\) 4443.63 0.242005
\(697\) 1690.26 0.0918555
\(698\) −1949.10 −0.105694
\(699\) 9465.86 0.512206
\(700\) 0 0
\(701\) 11303.7 0.609035 0.304518 0.952507i \(-0.401505\pi\)
0.304518 + 0.952507i \(0.401505\pi\)
\(702\) −5694.15 −0.306142
\(703\) −3641.33 −0.195356
\(704\) 13533.6 0.724529
\(705\) −18241.1 −0.974466
\(706\) −19118.6 −1.01918
\(707\) 0 0
\(708\) −4088.72 −0.217039
\(709\) −16046.3 −0.849973 −0.424987 0.905200i \(-0.639721\pi\)
−0.424987 + 0.905200i \(0.639721\pi\)
\(710\) 7970.22 0.421292
\(711\) −1244.21 −0.0656280
\(712\) −7008.83 −0.368915
\(713\) 3141.15 0.164989
\(714\) 0 0
\(715\) 41617.5 2.17679
\(716\) −2294.18 −0.119745
\(717\) −12725.8 −0.662836
\(718\) −21291.5 −1.10667
\(719\) −25190.5 −1.30660 −0.653300 0.757099i \(-0.726616\pi\)
−0.653300 + 0.757099i \(0.726616\pi\)
\(720\) −7506.19 −0.388527
\(721\) 0 0
\(722\) −6394.38 −0.329604
\(723\) 13029.0 0.670197
\(724\) 882.680 0.0453102
\(725\) 16345.8 0.837334
\(726\) 5488.64 0.280582
\(727\) 11277.2 0.575307 0.287653 0.957735i \(-0.407125\pi\)
0.287653 + 0.957735i \(0.407125\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 29742.6 1.50797
\(731\) 2827.82 0.143079
\(732\) 24.4781 0.00123598
\(733\) 23720.0 1.19525 0.597626 0.801775i \(-0.296111\pi\)
0.597626 + 0.801775i \(0.296111\pi\)
\(734\) −26946.9 −1.35508
\(735\) 0 0
\(736\) −2438.64 −0.122133
\(737\) −19465.3 −0.972880
\(738\) −6512.14 −0.324817
\(739\) 8124.72 0.404428 0.202214 0.979341i \(-0.435186\pi\)
0.202214 + 0.979341i \(0.435186\pi\)
\(740\) 2425.03 0.120467
\(741\) −17004.8 −0.843030
\(742\) 0 0
\(743\) 20955.3 1.03469 0.517346 0.855777i \(-0.326920\pi\)
0.517346 + 0.855777i \(0.326920\pi\)
\(744\) −9039.98 −0.445459
\(745\) −35234.6 −1.73274
\(746\) 6585.63 0.323213
\(747\) 5590.23 0.273810
\(748\) 293.203 0.0143323
\(749\) 0 0
\(750\) −21040.8 −1.02440
\(751\) −38208.1 −1.85650 −0.928251 0.371954i \(-0.878688\pi\)
−0.928251 + 0.371954i \(0.878688\pi\)
\(752\) −12806.3 −0.621006
\(753\) −9009.02 −0.435999
\(754\) 12720.9 0.614413
\(755\) −50549.4 −2.43666
\(756\) 0 0
\(757\) 30958.1 1.48638 0.743191 0.669079i \(-0.233311\pi\)
0.743191 + 0.669079i \(0.233311\pi\)
\(758\) 9865.65 0.472740
\(759\) 1838.53 0.0879243
\(760\) −31707.8 −1.51337
\(761\) −40049.4 −1.90774 −0.953871 0.300218i \(-0.902941\pi\)
−0.953871 + 0.300218i \(0.902941\pi\)
\(762\) 16740.6 0.795865
\(763\) 0 0
\(764\) −389.601 −0.0184493
\(765\) −1010.03 −0.0477355
\(766\) 31464.3 1.48414
\(767\) −54825.3 −2.58100
\(768\) 9202.57 0.432382
\(769\) −8002.01 −0.375240 −0.187620 0.982242i \(-0.560077\pi\)
−0.187620 + 0.982242i \(0.560077\pi\)
\(770\) 0 0
\(771\) −13406.5 −0.626231
\(772\) 5186.52 0.241796
\(773\) −13933.4 −0.648316 −0.324158 0.946003i \(-0.605081\pi\)
−0.324158 + 0.946003i \(0.605081\pi\)
\(774\) −10894.8 −0.505952
\(775\) −33253.4 −1.54128
\(776\) 14810.3 0.685126
\(777\) 0 0
\(778\) −470.751 −0.0216931
\(779\) −19447.6 −0.894456
\(780\) 11324.7 0.519859
\(781\) 3971.89 0.181979
\(782\) 348.522 0.0159375
\(783\) −1628.61 −0.0743316
\(784\) 0 0
\(785\) −21558.0 −0.980175
\(786\) 1122.74 0.0509500
\(787\) 37581.7 1.70222 0.851108 0.524991i \(-0.175931\pi\)
0.851108 + 0.524991i \(0.175931\pi\)
\(788\) 5788.01 0.261662
\(789\) −18480.4 −0.833865
\(790\) −6641.54 −0.299108
\(791\) 0 0
\(792\) −5291.16 −0.237390
\(793\) 328.225 0.0146981
\(794\) −34240.7 −1.53042
\(795\) 22393.9 0.999032
\(796\) 2914.92 0.129795
\(797\) −9458.78 −0.420385 −0.210193 0.977660i \(-0.567409\pi\)
−0.210193 + 0.977660i \(0.567409\pi\)
\(798\) 0 0
\(799\) −1723.20 −0.0762985
\(800\) 25816.4 1.14093
\(801\) 2568.77 0.113312
\(802\) 24154.3 1.06349
\(803\) 14821.9 0.651376
\(804\) −5296.78 −0.232342
\(805\) 0 0
\(806\) −25879.0 −1.13095
\(807\) −14966.7 −0.652854
\(808\) −11227.4 −0.488834
\(809\) −1909.94 −0.0830034 −0.0415017 0.999138i \(-0.513214\pi\)
−0.0415017 + 0.999138i \(0.513214\pi\)
\(810\) 3891.37 0.168801
\(811\) 43110.6 1.86661 0.933303 0.359091i \(-0.116913\pi\)
0.933303 + 0.359091i \(0.116913\pi\)
\(812\) 0 0
\(813\) −13301.2 −0.573792
\(814\) −3243.55 −0.139664
\(815\) 59076.1 2.53907
\(816\) −709.097 −0.0304208
\(817\) −32535.9 −1.39325
\(818\) 11189.7 0.478287
\(819\) 0 0
\(820\) 12951.5 0.551570
\(821\) 4026.56 0.171167 0.0855834 0.996331i \(-0.472725\pi\)
0.0855834 + 0.996331i \(0.472725\pi\)
\(822\) −3743.98 −0.158864
\(823\) 39668.1 1.68012 0.840062 0.542491i \(-0.182519\pi\)
0.840062 + 0.542491i \(0.182519\pi\)
\(824\) −19302.4 −0.816056
\(825\) −19463.4 −0.821368
\(826\) 0 0
\(827\) 30137.7 1.26722 0.633611 0.773652i \(-0.281572\pi\)
0.633611 + 0.773652i \(0.281572\pi\)
\(828\) 500.291 0.0209980
\(829\) 23278.0 0.975245 0.487622 0.873055i \(-0.337864\pi\)
0.487622 + 0.873055i \(0.337864\pi\)
\(830\) 29840.4 1.24792
\(831\) −3337.10 −0.139306
\(832\) 49381.0 2.05766
\(833\) 0 0
\(834\) −6938.94 −0.288100
\(835\) −41620.5 −1.72495
\(836\) −3373.48 −0.139563
\(837\) 3313.19 0.136823
\(838\) 12067.0 0.497431
\(839\) −9494.43 −0.390684 −0.195342 0.980735i \(-0.562582\pi\)
−0.195342 + 0.980735i \(0.562582\pi\)
\(840\) 0 0
\(841\) −20750.6 −0.850820
\(842\) −1699.99 −0.0695791
\(843\) −8439.65 −0.344813
\(844\) −1364.71 −0.0556578
\(845\) 108133. 4.40223
\(846\) 6639.04 0.269805
\(847\) 0 0
\(848\) 15721.8 0.636661
\(849\) −9442.62 −0.381708
\(850\) −3689.59 −0.148885
\(851\) 1436.50 0.0578644
\(852\) 1080.81 0.0434599
\(853\) −12692.4 −0.509471 −0.254736 0.967011i \(-0.581988\pi\)
−0.254736 + 0.967011i \(0.581988\pi\)
\(854\) 0 0
\(855\) 11621.0 0.464831
\(856\) 4824.36 0.192632
\(857\) 22206.0 0.885114 0.442557 0.896740i \(-0.354071\pi\)
0.442557 + 0.896740i \(0.354071\pi\)
\(858\) −15147.2 −0.602698
\(859\) 19820.5 0.787271 0.393636 0.919267i \(-0.371217\pi\)
0.393636 + 0.919267i \(0.371217\pi\)
\(860\) 21668.0 0.859155
\(861\) 0 0
\(862\) −24945.5 −0.985671
\(863\) 36413.8 1.43631 0.718157 0.695881i \(-0.244986\pi\)
0.718157 + 0.695881i \(0.244986\pi\)
\(864\) −2572.21 −0.101283
\(865\) −9355.70 −0.367749
\(866\) 26814.1 1.05217
\(867\) 14643.6 0.573613
\(868\) 0 0
\(869\) −3309.76 −0.129201
\(870\) −8693.43 −0.338776
\(871\) −71024.1 −2.76298
\(872\) 7522.67 0.292144
\(873\) −5428.03 −0.210436
\(874\) −4009.97 −0.155194
\(875\) 0 0
\(876\) 4033.26 0.155561
\(877\) 19442.6 0.748609 0.374305 0.927306i \(-0.377881\pi\)
0.374305 + 0.927306i \(0.377881\pi\)
\(878\) −17622.0 −0.677351
\(879\) −27429.1 −1.05252
\(880\) −19967.4 −0.764888
\(881\) −25184.2 −0.963082 −0.481541 0.876423i \(-0.659923\pi\)
−0.481541 + 0.876423i \(0.659923\pi\)
\(882\) 0 0
\(883\) −4050.03 −0.154354 −0.0771769 0.997017i \(-0.524591\pi\)
−0.0771769 + 0.997017i \(0.524591\pi\)
\(884\) 1069.83 0.0407038
\(885\) 37467.5 1.42311
\(886\) 38843.9 1.47290
\(887\) 41604.2 1.57489 0.787447 0.616383i \(-0.211402\pi\)
0.787447 + 0.616383i \(0.211402\pi\)
\(888\) −4134.14 −0.156231
\(889\) 0 0
\(890\) 13712.0 0.516434
\(891\) 1939.23 0.0729144
\(892\) 2106.36 0.0790652
\(893\) 19826.5 0.742967
\(894\) 12824.0 0.479753
\(895\) 21023.0 0.785165
\(896\) 0 0
\(897\) 6708.36 0.249705
\(898\) 32740.9 1.21668
\(899\) −7401.76 −0.274597
\(900\) −5296.27 −0.196158
\(901\) 2115.51 0.0782219
\(902\) −17323.1 −0.639463
\(903\) 0 0
\(904\) −49054.4 −1.80478
\(905\) −8088.56 −0.297097
\(906\) 18398.0 0.674650
\(907\) −3572.60 −0.130790 −0.0653949 0.997859i \(-0.520831\pi\)
−0.0653949 + 0.997859i \(0.520831\pi\)
\(908\) 10312.0 0.376890
\(909\) 4114.88 0.150145
\(910\) 0 0
\(911\) 29457.7 1.07133 0.535663 0.844432i \(-0.320062\pi\)
0.535663 + 0.844432i \(0.320062\pi\)
\(912\) 8158.61 0.296226
\(913\) 14870.7 0.539046
\(914\) 26190.8 0.947827
\(915\) −224.309 −0.00810428
\(916\) 10427.9 0.376143
\(917\) 0 0
\(918\) 367.611 0.0132167
\(919\) −3310.65 −0.118834 −0.0594168 0.998233i \(-0.518924\pi\)
−0.0594168 + 0.998233i \(0.518924\pi\)
\(920\) 12508.7 0.448260
\(921\) 13946.7 0.498979
\(922\) −4245.85 −0.151659
\(923\) 14492.5 0.516820
\(924\) 0 0
\(925\) −15207.3 −0.540556
\(926\) −13065.6 −0.463674
\(927\) 7074.40 0.250651
\(928\) 5746.39 0.203270
\(929\) −31467.5 −1.11132 −0.555660 0.831410i \(-0.687534\pi\)
−0.555660 + 0.831410i \(0.687534\pi\)
\(930\) 17685.7 0.623587
\(931\) 0 0
\(932\) 6851.94 0.240818
\(933\) 19251.5 0.675527
\(934\) 19582.5 0.686038
\(935\) −2686.80 −0.0939763
\(936\) −19306.2 −0.674190
\(937\) −17363.4 −0.605375 −0.302688 0.953090i \(-0.597884\pi\)
−0.302688 + 0.953090i \(0.597884\pi\)
\(938\) 0 0
\(939\) −17604.6 −0.611828
\(940\) −13203.9 −0.458154
\(941\) −5547.77 −0.192192 −0.0960958 0.995372i \(-0.530636\pi\)
−0.0960958 + 0.995372i \(0.530636\pi\)
\(942\) 7846.27 0.271386
\(943\) 7672.04 0.264937
\(944\) 26304.3 0.906919
\(945\) 0 0
\(946\) −28981.6 −0.996062
\(947\) −37960.3 −1.30258 −0.651290 0.758829i \(-0.725772\pi\)
−0.651290 + 0.758829i \(0.725772\pi\)
\(948\) −900.631 −0.0308556
\(949\) 54081.7 1.84991
\(950\) 42451.1 1.44978
\(951\) 11923.6 0.406570
\(952\) 0 0
\(953\) −10019.3 −0.340563 −0.170282 0.985395i \(-0.554468\pi\)
−0.170282 + 0.985395i \(0.554468\pi\)
\(954\) −8150.51 −0.276606
\(955\) 3570.16 0.120971
\(956\) −9211.65 −0.311638
\(957\) −4332.30 −0.146336
\(958\) 5059.61 0.170635
\(959\) 0 0
\(960\) −33746.9 −1.13456
\(961\) −14733.1 −0.494548
\(962\) −11834.9 −0.396646
\(963\) −1768.15 −0.0591670
\(964\) 9431.11 0.315099
\(965\) −47527.3 −1.58545
\(966\) 0 0
\(967\) 27834.4 0.925641 0.462820 0.886452i \(-0.346837\pi\)
0.462820 + 0.886452i \(0.346837\pi\)
\(968\) 18609.4 0.617900
\(969\) 1097.82 0.0363952
\(970\) −28974.6 −0.959090
\(971\) −18275.3 −0.604000 −0.302000 0.953308i \(-0.597654\pi\)
−0.302000 + 0.953308i \(0.597654\pi\)
\(972\) 527.692 0.0174133
\(973\) 0 0
\(974\) 23200.8 0.763245
\(975\) −71017.2 −2.33269
\(976\) −157.477 −0.00516468
\(977\) −43028.9 −1.40903 −0.704513 0.709691i \(-0.748834\pi\)
−0.704513 + 0.709691i \(0.748834\pi\)
\(978\) −21501.4 −0.703005
\(979\) 6833.24 0.223076
\(980\) 0 0
\(981\) −2757.09 −0.0897320
\(982\) −28288.1 −0.919256
\(983\) 30559.2 0.991544 0.495772 0.868453i \(-0.334885\pi\)
0.495772 + 0.868453i \(0.334885\pi\)
\(984\) −22079.6 −0.715316
\(985\) −53039.2 −1.71571
\(986\) −821.253 −0.0265254
\(987\) 0 0
\(988\) −12309.0 −0.396358
\(989\) 12835.4 0.412681
\(990\) 10351.5 0.332317
\(991\) −44945.9 −1.44072 −0.720360 0.693600i \(-0.756024\pi\)
−0.720360 + 0.693600i \(0.756024\pi\)
\(992\) −11690.3 −0.374160
\(993\) 26738.5 0.854501
\(994\) 0 0
\(995\) −26711.2 −0.851058
\(996\) 4046.53 0.128734
\(997\) 29006.1 0.921397 0.460698 0.887557i \(-0.347599\pi\)
0.460698 + 0.887557i \(0.347599\pi\)
\(998\) −22199.1 −0.704110
\(999\) 1515.18 0.0479861
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 147.4.a.j.1.2 2
3.2 odd 2 441.4.a.n.1.1 2
4.3 odd 2 2352.4.a.cf.1.2 2
7.2 even 3 147.4.e.k.67.1 4
7.3 odd 6 147.4.e.j.79.1 4
7.4 even 3 147.4.e.k.79.1 4
7.5 odd 6 147.4.e.j.67.1 4
7.6 odd 2 147.4.a.k.1.2 yes 2
21.2 odd 6 441.4.e.v.361.2 4
21.5 even 6 441.4.e.u.361.2 4
21.11 odd 6 441.4.e.v.226.2 4
21.17 even 6 441.4.e.u.226.2 4
21.20 even 2 441.4.a.o.1.1 2
28.27 even 2 2352.4.a.bl.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.4.a.j.1.2 2 1.1 even 1 trivial
147.4.a.k.1.2 yes 2 7.6 odd 2
147.4.e.j.67.1 4 7.5 odd 6
147.4.e.j.79.1 4 7.3 odd 6
147.4.e.k.67.1 4 7.2 even 3
147.4.e.k.79.1 4 7.4 even 3
441.4.a.n.1.1 2 3.2 odd 2
441.4.a.o.1.1 2 21.20 even 2
441.4.e.u.226.2 4 21.17 even 6
441.4.e.u.361.2 4 21.5 even 6
441.4.e.v.226.2 4 21.11 odd 6
441.4.e.v.361.2 4 21.2 odd 6
2352.4.a.bl.1.1 2 28.27 even 2
2352.4.a.cf.1.2 2 4.3 odd 2