Properties

Label 425.4.b.f.324.6
Level $425$
Weight $4$
Character 425.324
Analytic conductor $25.076$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.0758117524\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.27793984.1
Defining polynomial: \( x^{6} - 2x^{3} + 49x^{2} - 14x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 17)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 324.6
Root \(1.79483 - 1.79483i\) of defining polynomial
Character \(\chi\) \(=\) 425.324
Dual form 425.4.b.f.324.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} +O(q^{10})\) \(q+5.03251i q^{2} +8.47535i q^{3} -17.3261 q^{4} -42.6523 q^{6} -3.81828i q^{7} -46.9339i q^{8} -44.8316 q^{9} -52.3720 q^{11} -146.845i q^{12} -8.06025i q^{13} +19.2156 q^{14} +97.5862 q^{16} +17.0000i q^{17} -225.616i q^{18} +66.5154 q^{19} +32.3613 q^{21} -263.563i q^{22} +180.226i q^{23} +397.782 q^{24} +40.5633 q^{26} -151.129i q^{27} +66.1562i q^{28} +41.2800 q^{29} -34.9114 q^{31} +115.632i q^{32} -443.871i q^{33} -85.5527 q^{34} +776.759 q^{36} -130.368i q^{37} +334.739i q^{38} +68.3134 q^{39} -17.9081 q^{41} +162.859i q^{42} +277.620i q^{43} +907.405 q^{44} -906.987 q^{46} -463.789i q^{47} +827.078i q^{48} +328.421 q^{49} -144.081 q^{51} +139.653i q^{52} -329.944i q^{53} +760.560 q^{54} -179.207 q^{56} +563.741i q^{57} +207.742i q^{58} -678.656 q^{59} +340.280 q^{61} -175.692i q^{62} +171.180i q^{63} +198.770 q^{64} +2233.79 q^{66} -15.3925i q^{67} -294.545i q^{68} -1527.48 q^{69} -670.203 q^{71} +2104.12i q^{72} +193.480i q^{73} +656.080 q^{74} -1152.46 q^{76} +199.971i q^{77} +343.788i q^{78} -1080.15 q^{79} +70.4207 q^{81} -90.1229i q^{82} -865.668i q^{83} -560.697 q^{84} -1397.13 q^{86} +349.863i q^{87} +2458.02i q^{88} -1129.46 q^{89} -30.7763 q^{91} -3122.61i q^{92} -295.886i q^{93} +2334.02 q^{94} -980.023 q^{96} +379.412i q^{97} +1652.78i q^{98} +2347.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 50 q^{4} - 148 q^{6} - 118 q^{9} - 56 q^{11} - 184 q^{14} + 274 q^{16} - 160 q^{19} - 384 q^{21} + 1332 q^{24} + 52 q^{26} + 912 q^{29} + 460 q^{31} + 34 q^{34} + 2626 q^{36} - 536 q^{39} - 588 q^{41} + 2244 q^{44} - 1408 q^{46} + 538 q^{49} - 136 q^{51} + 2200 q^{54} + 1368 q^{56} - 1272 q^{59} - 168 q^{61} + 1838 q^{64} + 4936 q^{66} - 1152 q^{69} - 804 q^{71} - 1672 q^{74} - 1816 q^{76} + 1188 q^{79} - 1010 q^{81} + 4080 q^{84} - 2528 q^{86} + 340 q^{89} - 2032 q^{91} + 4032 q^{94} + 1356 q^{96} + 5840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/425\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(326\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.03251i 1.77926i 0.456681 + 0.889630i \(0.349038\pi\)
−0.456681 + 0.889630i \(0.650962\pi\)
\(3\) 8.47535i 1.63108i 0.578699 + 0.815541i \(0.303561\pi\)
−0.578699 + 0.815541i \(0.696439\pi\)
\(4\) −17.3261 −2.16577
\(5\) 0 0
\(6\) −42.6523 −2.90212
\(7\) − 3.81828i − 0.206168i −0.994673 0.103084i \(-0.967129\pi\)
0.994673 0.103084i \(-0.0328711\pi\)
\(8\) − 46.9339i − 2.07421i
\(9\) −44.8316 −1.66043
\(10\) 0 0
\(11\) −52.3720 −1.43552 −0.717761 0.696289i \(-0.754833\pi\)
−0.717761 + 0.696289i \(0.754833\pi\)
\(12\) − 146.845i − 3.53255i
\(13\) − 8.06025i − 0.171962i −0.996297 0.0859811i \(-0.972598\pi\)
0.996297 0.0859811i \(-0.0274025\pi\)
\(14\) 19.2156 0.366827
\(15\) 0 0
\(16\) 97.5862 1.52478
\(17\) 17.0000i 0.242536i
\(18\) − 225.616i − 2.95434i
\(19\) 66.5154 0.803141 0.401570 0.915828i \(-0.368465\pi\)
0.401570 + 0.915828i \(0.368465\pi\)
\(20\) 0 0
\(21\) 32.3613 0.336277
\(22\) − 263.563i − 2.55417i
\(23\) 180.226i 1.63390i 0.576711 + 0.816948i \(0.304336\pi\)
−0.576711 + 0.816948i \(0.695664\pi\)
\(24\) 397.782 3.38320
\(25\) 0 0
\(26\) 40.5633 0.305966
\(27\) − 151.129i − 1.07722i
\(28\) 66.1562i 0.446512i
\(29\) 41.2800 0.264328 0.132164 0.991228i \(-0.457807\pi\)
0.132164 + 0.991228i \(0.457807\pi\)
\(30\) 0 0
\(31\) −34.9114 −0.202267 −0.101133 0.994873i \(-0.532247\pi\)
−0.101133 + 0.994873i \(0.532247\pi\)
\(32\) 115.632i 0.638783i
\(33\) − 443.871i − 2.34146i
\(34\) −85.5527 −0.431534
\(35\) 0 0
\(36\) 776.759 3.59611
\(37\) − 130.368i − 0.579255i −0.957139 0.289627i \(-0.906469\pi\)
0.957139 0.289627i \(-0.0935314\pi\)
\(38\) 334.739i 1.42900i
\(39\) 68.3134 0.280485
\(40\) 0 0
\(41\) −17.9081 −0.0682142 −0.0341071 0.999418i \(-0.510859\pi\)
−0.0341071 + 0.999418i \(0.510859\pi\)
\(42\) 162.859i 0.598325i
\(43\) 277.620i 0.984573i 0.870433 + 0.492287i \(0.163839\pi\)
−0.870433 + 0.492287i \(0.836161\pi\)
\(44\) 907.405 3.10901
\(45\) 0 0
\(46\) −906.987 −2.90713
\(47\) − 463.789i − 1.43937i −0.694299 0.719687i \(-0.744285\pi\)
0.694299 0.719687i \(-0.255715\pi\)
\(48\) 827.078i 2.48705i
\(49\) 328.421 0.957495
\(50\) 0 0
\(51\) −144.081 −0.395596
\(52\) 139.653i 0.372431i
\(53\) − 329.944i − 0.855118i −0.903987 0.427559i \(-0.859374\pi\)
0.903987 0.427559i \(-0.140626\pi\)
\(54\) 760.560 1.91665
\(55\) 0 0
\(56\) −179.207 −0.427635
\(57\) 563.741i 1.30999i
\(58\) 207.742i 0.470308i
\(59\) −678.656 −1.49752 −0.748759 0.662843i \(-0.769350\pi\)
−0.748759 + 0.662843i \(0.769350\pi\)
\(60\) 0 0
\(61\) 340.280 0.714237 0.357118 0.934059i \(-0.383759\pi\)
0.357118 + 0.934059i \(0.383759\pi\)
\(62\) − 175.692i − 0.359885i
\(63\) 171.180i 0.342328i
\(64\) 198.770 0.388223
\(65\) 0 0
\(66\) 2233.79 4.16606
\(67\) − 15.3925i − 0.0280671i −0.999902 0.0140336i \(-0.995533\pi\)
0.999902 0.0140336i \(-0.00446717\pi\)
\(68\) − 294.545i − 0.525276i
\(69\) −1527.48 −2.66502
\(70\) 0 0
\(71\) −670.203 −1.12026 −0.560130 0.828405i \(-0.689249\pi\)
−0.560130 + 0.828405i \(0.689249\pi\)
\(72\) 2104.12i 3.44408i
\(73\) 193.480i 0.310207i 0.987898 + 0.155103i \(0.0495711\pi\)
−0.987898 + 0.155103i \(0.950429\pi\)
\(74\) 656.080 1.03065
\(75\) 0 0
\(76\) −1152.46 −1.73942
\(77\) 199.971i 0.295959i
\(78\) 343.788i 0.499055i
\(79\) −1080.15 −1.53831 −0.769156 0.639061i \(-0.779323\pi\)
−0.769156 + 0.639061i \(0.779323\pi\)
\(80\) 0 0
\(81\) 70.4207 0.0965990
\(82\) − 90.1229i − 0.121371i
\(83\) − 865.668i − 1.14481i −0.819970 0.572406i \(-0.806010\pi\)
0.819970 0.572406i \(-0.193990\pi\)
\(84\) −560.697 −0.728298
\(85\) 0 0
\(86\) −1397.13 −1.75181
\(87\) 349.863i 0.431141i
\(88\) 2458.02i 2.97757i
\(89\) −1129.46 −1.34520 −0.672599 0.740008i \(-0.734822\pi\)
−0.672599 + 0.740008i \(0.734822\pi\)
\(90\) 0 0
\(91\) −30.7763 −0.0354531
\(92\) − 3122.61i − 3.53864i
\(93\) − 295.886i − 0.329914i
\(94\) 2334.02 2.56102
\(95\) 0 0
\(96\) −980.023 −1.04191
\(97\) 379.412i 0.397149i 0.980086 + 0.198574i \(0.0636311\pi\)
−0.980086 + 0.198574i \(0.936369\pi\)
\(98\) 1652.78i 1.70363i
\(99\) 2347.92 2.38359
\(100\) 0 0
\(101\) 131.732 0.129780 0.0648902 0.997892i \(-0.479330\pi\)
0.0648902 + 0.997892i \(0.479330\pi\)
\(102\) − 725.089i − 0.703868i
\(103\) 195.988i 0.187488i 0.995596 + 0.0937442i \(0.0298836\pi\)
−0.995596 + 0.0937442i \(0.970116\pi\)
\(104\) −378.299 −0.356685
\(105\) 0 0
\(106\) 1660.45 1.52148
\(107\) 485.147i 0.438326i 0.975688 + 0.219163i \(0.0703327\pi\)
−0.975688 + 0.219163i \(0.929667\pi\)
\(108\) 2618.49i 2.33300i
\(109\) 1255.12 1.10292 0.551460 0.834201i \(-0.314071\pi\)
0.551460 + 0.834201i \(0.314071\pi\)
\(110\) 0 0
\(111\) 1104.92 0.944812
\(112\) − 372.612i − 0.314362i
\(113\) − 1013.35i − 0.843612i −0.906686 0.421806i \(-0.861396\pi\)
0.906686 0.421806i \(-0.138604\pi\)
\(114\) −2837.03 −2.33081
\(115\) 0 0
\(116\) −715.224 −0.572473
\(117\) 361.354i 0.285531i
\(118\) − 3415.34i − 2.66447i
\(119\) 64.9108 0.0500031
\(120\) 0 0
\(121\) 1411.83 1.06073
\(122\) 1712.46i 1.27081i
\(123\) − 151.778i − 0.111263i
\(124\) 604.880 0.438063
\(125\) 0 0
\(126\) −861.464 −0.609090
\(127\) − 1927.72i − 1.34691i −0.739227 0.673456i \(-0.764809\pi\)
0.739227 0.673456i \(-0.235191\pi\)
\(128\) 1925.37i 1.32953i
\(129\) −2352.93 −1.60592
\(130\) 0 0
\(131\) −406.738 −0.271274 −0.135637 0.990759i \(-0.543308\pi\)
−0.135637 + 0.990759i \(0.543308\pi\)
\(132\) 7690.58i 5.07105i
\(133\) − 253.975i − 0.165582i
\(134\) 77.4631 0.0499387
\(135\) 0 0
\(136\) 797.877 0.503069
\(137\) 130.552i 0.0814149i 0.999171 + 0.0407074i \(0.0129612\pi\)
−0.999171 + 0.0407074i \(0.987039\pi\)
\(138\) − 7687.03i − 4.74177i
\(139\) −2073.54 −1.26529 −0.632644 0.774443i \(-0.718030\pi\)
−0.632644 + 0.774443i \(0.718030\pi\)
\(140\) 0 0
\(141\) 3930.78 2.34774
\(142\) − 3372.80i − 1.99323i
\(143\) 422.131i 0.246856i
\(144\) −4374.95 −2.53180
\(145\) 0 0
\(146\) −973.689 −0.551939
\(147\) 2783.48i 1.56175i
\(148\) 2258.78i 1.25453i
\(149\) 1852.73 1.01867 0.509334 0.860569i \(-0.329892\pi\)
0.509334 + 0.860569i \(0.329892\pi\)
\(150\) 0 0
\(151\) 2050.86 1.10527 0.552637 0.833422i \(-0.313622\pi\)
0.552637 + 0.833422i \(0.313622\pi\)
\(152\) − 3121.83i − 1.66588i
\(153\) − 762.138i − 0.402714i
\(154\) −1006.36 −0.526588
\(155\) 0 0
\(156\) −1183.61 −0.607465
\(157\) 262.991i 0.133688i 0.997763 + 0.0668438i \(0.0212929\pi\)
−0.997763 + 0.0668438i \(0.978707\pi\)
\(158\) − 5435.88i − 2.73706i
\(159\) 2796.39 1.39477
\(160\) 0 0
\(161\) 688.152 0.336857
\(162\) 354.393i 0.171875i
\(163\) − 1444.98i − 0.694354i −0.937800 0.347177i \(-0.887140\pi\)
0.937800 0.347177i \(-0.112860\pi\)
\(164\) 310.279 0.147736
\(165\) 0 0
\(166\) 4356.48 2.03692
\(167\) 501.565i 0.232409i 0.993225 + 0.116204i \(0.0370728\pi\)
−0.993225 + 0.116204i \(0.962927\pi\)
\(168\) − 1518.84i − 0.697508i
\(169\) 2132.03 0.970429
\(170\) 0 0
\(171\) −2981.99 −1.33356
\(172\) − 4810.08i − 2.13236i
\(173\) − 2590.14i − 1.13829i −0.822237 0.569146i \(-0.807274\pi\)
0.822237 0.569146i \(-0.192726\pi\)
\(174\) −1760.69 −0.767112
\(175\) 0 0
\(176\) −5110.79 −2.18886
\(177\) − 5751.85i − 2.44257i
\(178\) − 5684.02i − 2.39346i
\(179\) −2165.65 −0.904294 −0.452147 0.891943i \(-0.649342\pi\)
−0.452147 + 0.891943i \(0.649342\pi\)
\(180\) 0 0
\(181\) −1925.56 −0.790750 −0.395375 0.918520i \(-0.629385\pi\)
−0.395375 + 0.918520i \(0.629385\pi\)
\(182\) − 154.882i − 0.0630803i
\(183\) 2884.00i 1.16498i
\(184\) 8458.69 3.38904
\(185\) 0 0
\(186\) 1489.05 0.587003
\(187\) − 890.324i − 0.348165i
\(188\) 8035.68i 3.11735i
\(189\) −577.055 −0.222088
\(190\) 0 0
\(191\) −2783.52 −1.05449 −0.527247 0.849712i \(-0.676776\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(192\) 1684.65i 0.633223i
\(193\) 2258.27i 0.842246i 0.907004 + 0.421123i \(0.138364\pi\)
−0.907004 + 0.421123i \(0.861636\pi\)
\(194\) −1909.39 −0.706631
\(195\) 0 0
\(196\) −5690.27 −2.07371
\(197\) 1270.70i 0.459560i 0.973243 + 0.229780i \(0.0738007\pi\)
−0.973243 + 0.229780i \(0.926199\pi\)
\(198\) 11815.9i 4.24102i
\(199\) 4794.36 1.70786 0.853928 0.520392i \(-0.174214\pi\)
0.853928 + 0.520392i \(0.174214\pi\)
\(200\) 0 0
\(201\) 130.457 0.0457798
\(202\) 662.942i 0.230913i
\(203\) − 157.619i − 0.0544960i
\(204\) 2496.37 0.856769
\(205\) 0 0
\(206\) −986.313 −0.333591
\(207\) − 8079.80i − 2.71297i
\(208\) − 786.569i − 0.262205i
\(209\) −3483.54 −1.15293
\(210\) 0 0
\(211\) −2807.00 −0.915837 −0.457918 0.888994i \(-0.651405\pi\)
−0.457918 + 0.888994i \(0.651405\pi\)
\(212\) 5716.66i 1.85199i
\(213\) − 5680.21i − 1.82724i
\(214\) −2441.50 −0.779896
\(215\) 0 0
\(216\) −7093.09 −2.23437
\(217\) 133.302i 0.0417009i
\(218\) 6316.38i 1.96238i
\(219\) −1639.81 −0.505973
\(220\) 0 0
\(221\) 137.024 0.0417070
\(222\) 5560.51i 1.68107i
\(223\) 4684.30i 1.40665i 0.710866 + 0.703327i \(0.248303\pi\)
−0.710866 + 0.703327i \(0.751697\pi\)
\(224\) 441.516 0.131697
\(225\) 0 0
\(226\) 5099.70 1.50101
\(227\) 1395.72i 0.408095i 0.978961 + 0.204047i \(0.0654096\pi\)
−0.978961 + 0.204047i \(0.934590\pi\)
\(228\) − 9767.47i − 2.83713i
\(229\) −894.638 −0.258163 −0.129082 0.991634i \(-0.541203\pi\)
−0.129082 + 0.991634i \(0.541203\pi\)
\(230\) 0 0
\(231\) −1694.83 −0.482733
\(232\) − 1937.43i − 0.548270i
\(233\) 1196.13i 0.336313i 0.985760 + 0.168156i \(0.0537814\pi\)
−0.985760 + 0.168156i \(0.946219\pi\)
\(234\) −1818.52 −0.508035
\(235\) 0 0
\(236\) 11758.5 3.24328
\(237\) − 9154.67i − 2.50911i
\(238\) 326.664i 0.0889685i
\(239\) −4948.82 −1.33938 −0.669691 0.742639i \(-0.733574\pi\)
−0.669691 + 0.742639i \(0.733574\pi\)
\(240\) 0 0
\(241\) −6702.73 −1.79154 −0.895770 0.444518i \(-0.853375\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(242\) 7105.03i 1.88731i
\(243\) − 3483.65i − 0.919656i
\(244\) −5895.75 −1.54687
\(245\) 0 0
\(246\) 763.824 0.197966
\(247\) − 536.130i − 0.138110i
\(248\) 1638.53i 0.419543i
\(249\) 7336.85 1.86728
\(250\) 0 0
\(251\) −4756.08 −1.19602 −0.598010 0.801489i \(-0.704042\pi\)
−0.598010 + 0.801489i \(0.704042\pi\)
\(252\) − 2965.89i − 0.741402i
\(253\) − 9438.77i − 2.34550i
\(254\) 9701.29 2.39651
\(255\) 0 0
\(256\) −8099.28 −1.97736
\(257\) − 2892.84i − 0.702143i −0.936349 0.351071i \(-0.885817\pi\)
0.936349 0.351071i \(-0.114183\pi\)
\(258\) − 11841.1i − 2.85735i
\(259\) −497.784 −0.119424
\(260\) 0 0
\(261\) −1850.65 −0.438898
\(262\) − 2046.92i − 0.482667i
\(263\) 5415.48i 1.26971i 0.772633 + 0.634853i \(0.218939\pi\)
−0.772633 + 0.634853i \(0.781061\pi\)
\(264\) −20832.6 −4.85666
\(265\) 0 0
\(266\) 1278.13 0.294613
\(267\) − 9572.58i − 2.19413i
\(268\) 266.693i 0.0607869i
\(269\) −5787.00 −1.31167 −0.655835 0.754904i \(-0.727683\pi\)
−0.655835 + 0.754904i \(0.727683\pi\)
\(270\) 0 0
\(271\) 5465.13 1.22503 0.612515 0.790459i \(-0.290158\pi\)
0.612515 + 0.790459i \(0.290158\pi\)
\(272\) 1658.97i 0.369815i
\(273\) − 260.840i − 0.0578270i
\(274\) −657.006 −0.144858
\(275\) 0 0
\(276\) 26465.3 5.77182
\(277\) 1207.65i 0.261952i 0.991386 + 0.130976i \(0.0418111\pi\)
−0.991386 + 0.130976i \(0.958189\pi\)
\(278\) − 10435.1i − 2.25128i
\(279\) 1565.13 0.335850
\(280\) 0 0
\(281\) −1197.18 −0.254155 −0.127077 0.991893i \(-0.540560\pi\)
−0.127077 + 0.991893i \(0.540560\pi\)
\(282\) 19781.7i 4.17724i
\(283\) 3164.73i 0.664748i 0.943148 + 0.332374i \(0.107850\pi\)
−0.943148 + 0.332374i \(0.892150\pi\)
\(284\) 11612.0 2.42622
\(285\) 0 0
\(286\) −2124.38 −0.439221
\(287\) 68.3784i 0.0140636i
\(288\) − 5183.98i − 1.06066i
\(289\) −289.000 −0.0588235
\(290\) 0 0
\(291\) −3215.65 −0.647782
\(292\) − 3352.26i − 0.671836i
\(293\) 7456.21i 1.48668i 0.668915 + 0.743339i \(0.266759\pi\)
−0.668915 + 0.743339i \(0.733241\pi\)
\(294\) −14007.9 −2.77877
\(295\) 0 0
\(296\) −6118.70 −1.20149
\(297\) 7914.94i 1.54637i
\(298\) 9323.89i 1.81248i
\(299\) 1452.66 0.280969
\(300\) 0 0
\(301\) 1060.03 0.202988
\(302\) 10321.0i 1.96657i
\(303\) 1116.47i 0.211683i
\(304\) 6490.98 1.22462
\(305\) 0 0
\(306\) 3835.46 0.716532
\(307\) 6535.48i 1.21498i 0.794327 + 0.607491i \(0.207824\pi\)
−0.794327 + 0.607491i \(0.792176\pi\)
\(308\) − 3464.73i − 0.640978i
\(309\) −1661.07 −0.305809
\(310\) 0 0
\(311\) −8935.89 −1.62928 −0.814642 0.579963i \(-0.803067\pi\)
−0.814642 + 0.579963i \(0.803067\pi\)
\(312\) − 3206.22i − 0.581783i
\(313\) − 2628.71i − 0.474707i −0.971423 0.237353i \(-0.923720\pi\)
0.971423 0.237353i \(-0.0762799\pi\)
\(314\) −1323.50 −0.237865
\(315\) 0 0
\(316\) 18714.9 3.33163
\(317\) − 4268.54i − 0.756293i −0.925746 0.378147i \(-0.876562\pi\)
0.925746 0.378147i \(-0.123438\pi\)
\(318\) 14072.9i 2.48166i
\(319\) −2161.92 −0.379449
\(320\) 0 0
\(321\) −4111.79 −0.714946
\(322\) 3463.13i 0.599357i
\(323\) 1130.76i 0.194790i
\(324\) −1220.12 −0.209211
\(325\) 0 0
\(326\) 7271.89 1.23544
\(327\) 10637.5i 1.79895i
\(328\) 840.500i 0.141490i
\(329\) −1770.88 −0.296753
\(330\) 0 0
\(331\) 992.298 0.164778 0.0823892 0.996600i \(-0.473745\pi\)
0.0823892 + 0.996600i \(0.473745\pi\)
\(332\) 14998.7i 2.47940i
\(333\) 5844.63i 0.961812i
\(334\) −2524.13 −0.413516
\(335\) 0 0
\(336\) 3158.02 0.512750
\(337\) − 8042.26i − 1.29997i −0.759947 0.649985i \(-0.774775\pi\)
0.759947 0.649985i \(-0.225225\pi\)
\(338\) 10729.5i 1.72665i
\(339\) 8588.52 1.37600
\(340\) 0 0
\(341\) 1828.38 0.290359
\(342\) − 15006.9i − 2.37275i
\(343\) − 2563.68i − 0.403573i
\(344\) 13029.8 2.04221
\(345\) 0 0
\(346\) 13034.9 2.02532
\(347\) 7414.16i 1.14701i 0.819202 + 0.573506i \(0.194417\pi\)
−0.819202 + 0.573506i \(0.805583\pi\)
\(348\) − 6061.78i − 0.933751i
\(349\) 859.194 0.131781 0.0658905 0.997827i \(-0.479011\pi\)
0.0658905 + 0.997827i \(0.479011\pi\)
\(350\) 0 0
\(351\) −1218.14 −0.185241
\(352\) − 6055.89i − 0.916988i
\(353\) 569.084i 0.0858053i 0.999079 + 0.0429027i \(0.0136605\pi\)
−0.999079 + 0.0429027i \(0.986339\pi\)
\(354\) 28946.3 4.34598
\(355\) 0 0
\(356\) 19569.2 2.91339
\(357\) 550.142i 0.0815592i
\(358\) − 10898.7i − 1.60897i
\(359\) 5005.21 0.735835 0.367918 0.929858i \(-0.380071\pi\)
0.367918 + 0.929858i \(0.380071\pi\)
\(360\) 0 0
\(361\) −2434.71 −0.354965
\(362\) − 9690.40i − 1.40695i
\(363\) 11965.7i 1.73013i
\(364\) 533.235 0.0767833
\(365\) 0 0
\(366\) −14513.7 −2.07280
\(367\) 10975.3i 1.56105i 0.625127 + 0.780523i \(0.285047\pi\)
−0.625127 + 0.780523i \(0.714953\pi\)
\(368\) 17587.5i 2.49134i
\(369\) 802.851 0.113265
\(370\) 0 0
\(371\) −1259.82 −0.176298
\(372\) 5126.57i 0.714517i
\(373\) − 3211.72i − 0.445835i −0.974837 0.222918i \(-0.928442\pi\)
0.974837 0.222918i \(-0.0715581\pi\)
\(374\) 4480.56 0.619477
\(375\) 0 0
\(376\) −21767.4 −2.98556
\(377\) − 332.727i − 0.0454544i
\(378\) − 2904.03i − 0.395152i
\(379\) −8051.48 −1.09123 −0.545616 0.838035i \(-0.683704\pi\)
−0.545616 + 0.838035i \(0.683704\pi\)
\(380\) 0 0
\(381\) 16338.1 2.19692
\(382\) − 14008.1i − 1.87622i
\(383\) − 2584.16i − 0.344763i −0.985030 0.172382i \(-0.944854\pi\)
0.985030 0.172382i \(-0.0551462\pi\)
\(384\) −16318.2 −2.16858
\(385\) 0 0
\(386\) −11364.7 −1.49858
\(387\) − 12446.2i − 1.63482i
\(388\) − 6573.74i − 0.860132i
\(389\) 5174.31 0.674417 0.337208 0.941430i \(-0.390517\pi\)
0.337208 + 0.941430i \(0.390517\pi\)
\(390\) 0 0
\(391\) −3063.83 −0.396278
\(392\) − 15414.1i − 1.98604i
\(393\) − 3447.25i − 0.442470i
\(394\) −6394.79 −0.817677
\(395\) 0 0
\(396\) −40680.4 −5.16230
\(397\) 5149.36i 0.650980i 0.945545 + 0.325490i \(0.105529\pi\)
−0.945545 + 0.325490i \(0.894471\pi\)
\(398\) 24127.7i 3.03872i
\(399\) 2152.53 0.270078
\(400\) 0 0
\(401\) 8700.49 1.08350 0.541748 0.840541i \(-0.317763\pi\)
0.541748 + 0.840541i \(0.317763\pi\)
\(402\) 656.527i 0.0814542i
\(403\) 281.394i 0.0347823i
\(404\) −2282.41 −0.281074
\(405\) 0 0
\(406\) 793.219 0.0969625
\(407\) 6827.65i 0.831533i
\(408\) 6762.29i 0.820547i
\(409\) −12346.0 −1.49260 −0.746299 0.665611i \(-0.768171\pi\)
−0.746299 + 0.665611i \(0.768171\pi\)
\(410\) 0 0
\(411\) −1106.48 −0.132794
\(412\) − 3395.72i − 0.406056i
\(413\) 2591.30i 0.308740i
\(414\) 40661.7 4.82708
\(415\) 0 0
\(416\) 932.023 0.109847
\(417\) − 17574.0i − 2.06379i
\(418\) − 17531.0i − 2.05136i
\(419\) 5763.33 0.671974 0.335987 0.941867i \(-0.390930\pi\)
0.335987 + 0.941867i \(0.390930\pi\)
\(420\) 0 0
\(421\) −1876.12 −0.217188 −0.108594 0.994086i \(-0.534635\pi\)
−0.108594 + 0.994086i \(0.534635\pi\)
\(422\) − 14126.2i − 1.62951i
\(423\) 20792.4i 2.38998i
\(424\) −15485.6 −1.77369
\(425\) 0 0
\(426\) 28585.7 3.25113
\(427\) − 1299.29i − 0.147253i
\(428\) − 8405.72i − 0.949313i
\(429\) −3577.71 −0.402642
\(430\) 0 0
\(431\) 83.9299 0.00937996 0.00468998 0.999989i \(-0.498507\pi\)
0.00468998 + 0.999989i \(0.498507\pi\)
\(432\) − 14748.1i − 1.64252i
\(433\) − 15345.0i − 1.70308i −0.524291 0.851539i \(-0.675669\pi\)
0.524291 0.851539i \(-0.324331\pi\)
\(434\) −670.842 −0.0741968
\(435\) 0 0
\(436\) −21746.3 −2.38867
\(437\) 11987.8i 1.31225i
\(438\) − 8252.36i − 0.900258i
\(439\) −3064.74 −0.333194 −0.166597 0.986025i \(-0.553278\pi\)
−0.166597 + 0.986025i \(0.553278\pi\)
\(440\) 0 0
\(441\) −14723.6 −1.58985
\(442\) 689.575i 0.0742076i
\(443\) − 1792.97i − 0.192295i −0.995367 0.0961474i \(-0.969348\pi\)
0.995367 0.0961474i \(-0.0306520\pi\)
\(444\) −19144.0 −2.04624
\(445\) 0 0
\(446\) −23573.8 −2.50281
\(447\) 15702.6i 1.66153i
\(448\) − 758.960i − 0.0800391i
\(449\) −2499.19 −0.262681 −0.131341 0.991337i \(-0.541928\pi\)
−0.131341 + 0.991337i \(0.541928\pi\)
\(450\) 0 0
\(451\) 937.885 0.0979231
\(452\) 17557.5i 1.82707i
\(453\) 17381.7i 1.80279i
\(454\) −7024.00 −0.726107
\(455\) 0 0
\(456\) 26458.6 2.71719
\(457\) − 14784.4i − 1.51331i −0.653813 0.756656i \(-0.726832\pi\)
0.653813 0.756656i \(-0.273168\pi\)
\(458\) − 4502.28i − 0.459340i
\(459\) 2569.20 0.261263
\(460\) 0 0
\(461\) −17746.9 −1.79297 −0.896483 0.443078i \(-0.853887\pi\)
−0.896483 + 0.443078i \(0.853887\pi\)
\(462\) − 8529.23i − 0.858909i
\(463\) 18486.4i 1.85559i 0.373096 + 0.927793i \(0.378296\pi\)
−0.373096 + 0.927793i \(0.621704\pi\)
\(464\) 4028.36 0.403043
\(465\) 0 0
\(466\) −6019.52 −0.598388
\(467\) − 7406.57i − 0.733908i −0.930239 0.366954i \(-0.880401\pi\)
0.930239 0.366954i \(-0.119599\pi\)
\(468\) − 6260.87i − 0.618395i
\(469\) −58.7731 −0.00578655
\(470\) 0 0
\(471\) −2228.94 −0.218055
\(472\) 31852.0i 3.10616i
\(473\) − 14539.5i − 1.41338i
\(474\) 46071.0 4.46437
\(475\) 0 0
\(476\) −1124.65 −0.108295
\(477\) 14791.9i 1.41986i
\(478\) − 24905.0i − 2.38311i
\(479\) 18550.9 1.76955 0.884775 0.466019i \(-0.154312\pi\)
0.884775 + 0.466019i \(0.154312\pi\)
\(480\) 0 0
\(481\) −1050.80 −0.0996100
\(482\) − 33731.6i − 3.18762i
\(483\) 5832.34i 0.549442i
\(484\) −24461.5 −2.29729
\(485\) 0 0
\(486\) 17531.5 1.63631
\(487\) − 10203.4i − 0.949406i −0.880146 0.474703i \(-0.842556\pi\)
0.880146 0.474703i \(-0.157444\pi\)
\(488\) − 15970.7i − 1.48147i
\(489\) 12246.7 1.13255
\(490\) 0 0
\(491\) −1247.46 −0.114658 −0.0573290 0.998355i \(-0.518258\pi\)
−0.0573290 + 0.998355i \(0.518258\pi\)
\(492\) 2629.73i 0.240970i
\(493\) 701.760i 0.0641089i
\(494\) 2698.08 0.245734
\(495\) 0 0
\(496\) −3406.87 −0.308413
\(497\) 2559.03i 0.230962i
\(498\) 36922.7i 3.32238i
\(499\) −70.0303 −0.00628254 −0.00314127 0.999995i \(-0.501000\pi\)
−0.00314127 + 0.999995i \(0.501000\pi\)
\(500\) 0 0
\(501\) −4250.94 −0.379078
\(502\) − 23935.0i − 2.12803i
\(503\) 1444.29i 0.128028i 0.997949 + 0.0640138i \(0.0203902\pi\)
−0.997949 + 0.0640138i \(0.979610\pi\)
\(504\) 8034.15 0.710058
\(505\) 0 0
\(506\) 47500.7 4.17325
\(507\) 18069.7i 1.58285i
\(508\) 33400.0i 2.91710i
\(509\) −14272.8 −1.24289 −0.621445 0.783458i \(-0.713454\pi\)
−0.621445 + 0.783458i \(0.713454\pi\)
\(510\) 0 0
\(511\) 738.761 0.0639547
\(512\) − 25356.7i − 2.18871i
\(513\) − 10052.4i − 0.865157i
\(514\) 14558.3 1.24929
\(515\) 0 0
\(516\) 40767.2 3.47805
\(517\) 24289.6i 2.06625i
\(518\) − 2505.10i − 0.212486i
\(519\) 21952.3 1.85665
\(520\) 0 0
\(521\) 14874.0 1.25075 0.625376 0.780324i \(-0.284946\pi\)
0.625376 + 0.780324i \(0.284946\pi\)
\(522\) − 9313.42i − 0.780914i
\(523\) − 8142.90i − 0.680811i −0.940279 0.340406i \(-0.889436\pi\)
0.940279 0.340406i \(-0.110564\pi\)
\(524\) 7047.21 0.587517
\(525\) 0 0
\(526\) −27253.4 −2.25914
\(527\) − 593.494i − 0.0490569i
\(528\) − 43315.7i − 3.57022i
\(529\) −20314.2 −1.66962
\(530\) 0 0
\(531\) 30425.3 2.48652
\(532\) 4400.40i 0.358612i
\(533\) 144.344i 0.0117303i
\(534\) 48174.1 3.90393
\(535\) 0 0
\(536\) −722.432 −0.0582170
\(537\) − 18354.7i − 1.47498i
\(538\) − 29123.1i − 2.33380i
\(539\) −17200.0 −1.37451
\(540\) 0 0
\(541\) 3179.67 0.252689 0.126344 0.991986i \(-0.459676\pi\)
0.126344 + 0.991986i \(0.459676\pi\)
\(542\) 27503.3i 2.17965i
\(543\) − 16319.8i − 1.28978i
\(544\) −1965.75 −0.154928
\(545\) 0 0
\(546\) 1312.68 0.102889
\(547\) − 2107.07i − 0.164702i −0.996603 0.0823509i \(-0.973757\pi\)
0.996603 0.0823509i \(-0.0262428\pi\)
\(548\) − 2261.97i − 0.176326i
\(549\) −15255.3 −1.18594
\(550\) 0 0
\(551\) 2745.76 0.212292
\(552\) 71690.4i 5.52780i
\(553\) 4124.33i 0.317151i
\(554\) −6077.51 −0.466081
\(555\) 0 0
\(556\) 35926.4 2.74032
\(557\) − 467.382i − 0.0355540i −0.999842 0.0177770i \(-0.994341\pi\)
0.999842 0.0177770i \(-0.00565890\pi\)
\(558\) 7876.55i 0.597565i
\(559\) 2237.69 0.169309
\(560\) 0 0
\(561\) 7545.81 0.567887
\(562\) − 6024.80i − 0.452208i
\(563\) 14612.6i 1.09387i 0.837175 + 0.546935i \(0.184206\pi\)
−0.837175 + 0.546935i \(0.815794\pi\)
\(564\) −68105.2 −5.08466
\(565\) 0 0
\(566\) −15926.5 −1.18276
\(567\) − 268.886i − 0.0199156i
\(568\) 31455.3i 2.32365i
\(569\) −11602.3 −0.854821 −0.427410 0.904058i \(-0.640574\pi\)
−0.427410 + 0.904058i \(0.640574\pi\)
\(570\) 0 0
\(571\) −10534.9 −0.772104 −0.386052 0.922477i \(-0.626161\pi\)
−0.386052 + 0.922477i \(0.626161\pi\)
\(572\) − 7313.91i − 0.534633i
\(573\) − 23591.3i − 1.71997i
\(574\) −344.115 −0.0250228
\(575\) 0 0
\(576\) −8911.18 −0.644617
\(577\) − 14404.7i − 1.03930i −0.854379 0.519650i \(-0.826062\pi\)
0.854379 0.519650i \(-0.173938\pi\)
\(578\) − 1454.40i − 0.104662i
\(579\) −19139.6 −1.37377
\(580\) 0 0
\(581\) −3305.37 −0.236024
\(582\) − 16182.8i − 1.15257i
\(583\) 17279.8i 1.22754i
\(584\) 9080.77 0.643433
\(585\) 0 0
\(586\) −37523.5 −2.64519
\(587\) 11004.9i 0.773799i 0.922122 + 0.386900i \(0.126454\pi\)
−0.922122 + 0.386900i \(0.873546\pi\)
\(588\) − 48227.0i − 3.38240i
\(589\) −2322.14 −0.162449
\(590\) 0 0
\(591\) −10769.6 −0.749581
\(592\) − 12722.2i − 0.883239i
\(593\) 1853.59i 0.128361i 0.997938 + 0.0641804i \(0.0204433\pi\)
−0.997938 + 0.0641804i \(0.979557\pi\)
\(594\) −39832.0 −2.75139
\(595\) 0 0
\(596\) −32100.7 −2.20620
\(597\) 40633.9i 2.78565i
\(598\) 7310.53i 0.499916i
\(599\) −19074.7 −1.30112 −0.650559 0.759456i \(-0.725465\pi\)
−0.650559 + 0.759456i \(0.725465\pi\)
\(600\) 0 0
\(601\) −27776.0 −1.88520 −0.942600 0.333923i \(-0.891627\pi\)
−0.942600 + 0.333923i \(0.891627\pi\)
\(602\) 5334.62i 0.361168i
\(603\) 690.073i 0.0466035i
\(604\) −35533.4 −2.39377
\(605\) 0 0
\(606\) −5618.67 −0.376638
\(607\) − 18728.3i − 1.25232i −0.779695 0.626159i \(-0.784626\pi\)
0.779695 0.626159i \(-0.215374\pi\)
\(608\) 7691.32i 0.513033i
\(609\) 1335.88 0.0888874
\(610\) 0 0
\(611\) −3738.25 −0.247518
\(612\) 13204.9i 0.872184i
\(613\) − 24405.3i − 1.60802i −0.594613 0.804012i \(-0.702695\pi\)
0.594613 0.804012i \(-0.297305\pi\)
\(614\) −32889.8 −2.16177
\(615\) 0 0
\(616\) 9385.43 0.613880
\(617\) 22516.4i 1.46917i 0.678518 + 0.734584i \(0.262623\pi\)
−0.678518 + 0.734584i \(0.737377\pi\)
\(618\) − 8359.35i − 0.544114i
\(619\) 5146.53 0.334179 0.167089 0.985942i \(-0.446563\pi\)
0.167089 + 0.985942i \(0.446563\pi\)
\(620\) 0 0
\(621\) 27237.4 1.76006
\(622\) − 44969.9i − 2.89892i
\(623\) 4312.60i 0.277337i
\(624\) 6666.45 0.427679
\(625\) 0 0
\(626\) 13229.0 0.844627
\(627\) − 29524.3i − 1.88052i
\(628\) − 4556.62i − 0.289536i
\(629\) 2216.26 0.140490
\(630\) 0 0
\(631\) −3858.77 −0.243447 −0.121724 0.992564i \(-0.538842\pi\)
−0.121724 + 0.992564i \(0.538842\pi\)
\(632\) 50695.8i 3.19078i
\(633\) − 23790.3i − 1.49381i
\(634\) 21481.5 1.34564
\(635\) 0 0
\(636\) −48450.7 −3.02075
\(637\) − 2647.15i − 0.164653i
\(638\) − 10879.9i − 0.675138i
\(639\) 30046.3 1.86011
\(640\) 0 0
\(641\) 18689.3 1.15161 0.575805 0.817587i \(-0.304689\pi\)
0.575805 + 0.817587i \(0.304689\pi\)
\(642\) − 20692.6i − 1.27208i
\(643\) 26473.5i 1.62366i 0.583893 + 0.811831i \(0.301529\pi\)
−0.583893 + 0.811831i \(0.698471\pi\)
\(644\) −11923.0 −0.729555
\(645\) 0 0
\(646\) −5690.57 −0.346583
\(647\) − 14397.7i − 0.874855i −0.899254 0.437427i \(-0.855890\pi\)
0.899254 0.437427i \(-0.144110\pi\)
\(648\) − 3305.12i − 0.200366i
\(649\) 35542.6 2.14972
\(650\) 0 0
\(651\) −1129.78 −0.0680177
\(652\) 25036.0i 1.50381i
\(653\) 20939.5i 1.25486i 0.778672 + 0.627431i \(0.215893\pi\)
−0.778672 + 0.627431i \(0.784107\pi\)
\(654\) −53533.6 −3.20081
\(655\) 0 0
\(656\) −1747.59 −0.104012
\(657\) − 8674.02i − 0.515077i
\(658\) − 8911.96i − 0.528001i
\(659\) −4031.76 −0.238323 −0.119162 0.992875i \(-0.538021\pi\)
−0.119162 + 0.992875i \(0.538021\pi\)
\(660\) 0 0
\(661\) 6691.52 0.393752 0.196876 0.980428i \(-0.436920\pi\)
0.196876 + 0.980428i \(0.436920\pi\)
\(662\) 4993.75i 0.293184i
\(663\) 1161.33i 0.0680275i
\(664\) −40629.2 −2.37458
\(665\) 0 0
\(666\) −29413.1 −1.71131
\(667\) 7439.71i 0.431884i
\(668\) − 8690.19i − 0.503344i
\(669\) −39701.1 −2.29437
\(670\) 0 0
\(671\) −17821.2 −1.02530
\(672\) 3742.01i 0.214808i
\(673\) 10319.2i 0.591048i 0.955335 + 0.295524i \(0.0954942\pi\)
−0.955335 + 0.295524i \(0.904506\pi\)
\(674\) 40472.7 2.31298
\(675\) 0 0
\(676\) −36939.9 −2.10172
\(677\) 19813.3i 1.12480i 0.826866 + 0.562398i \(0.190121\pi\)
−0.826866 + 0.562398i \(0.809879\pi\)
\(678\) 43221.8i 2.44826i
\(679\) 1448.70 0.0818793
\(680\) 0 0
\(681\) −11829.3 −0.665636
\(682\) 9201.33i 0.516624i
\(683\) 5924.61i 0.331916i 0.986133 + 0.165958i \(0.0530717\pi\)
−0.986133 + 0.165958i \(0.946928\pi\)
\(684\) 51666.4 2.88818
\(685\) 0 0
\(686\) 12901.7 0.718061
\(687\) − 7582.38i − 0.421085i
\(688\) 27091.9i 1.50126i
\(689\) −2659.43 −0.147048
\(690\) 0 0
\(691\) 1973.16 0.108629 0.0543143 0.998524i \(-0.482703\pi\)
0.0543143 + 0.998524i \(0.482703\pi\)
\(692\) 44877.1i 2.46528i
\(693\) − 8965.03i − 0.491419i
\(694\) −37311.8 −2.04083
\(695\) 0 0
\(696\) 16420.4 0.894274
\(697\) − 304.439i − 0.0165444i
\(698\) 4323.90i 0.234473i
\(699\) −10137.6 −0.548554
\(700\) 0 0
\(701\) −12840.1 −0.691815 −0.345907 0.938269i \(-0.612429\pi\)
−0.345907 + 0.938269i \(0.612429\pi\)
\(702\) − 6130.30i − 0.329591i
\(703\) − 8671.50i − 0.465223i
\(704\) −10410.0 −0.557302
\(705\) 0 0
\(706\) −2863.92 −0.152670
\(707\) − 502.990i − 0.0267566i
\(708\) 99657.5i 5.29005i
\(709\) 27749.7 1.46990 0.734952 0.678119i \(-0.237204\pi\)
0.734952 + 0.678119i \(0.237204\pi\)
\(710\) 0 0
\(711\) 48425.0 2.55426
\(712\) 53010.0i 2.79022i
\(713\) − 6291.92i − 0.330483i
\(714\) −2768.60 −0.145115
\(715\) 0 0
\(716\) 37522.4 1.95849
\(717\) − 41943.0i − 2.18464i
\(718\) 25188.8i 1.30924i
\(719\) −16888.3 −0.875979 −0.437989 0.898980i \(-0.644309\pi\)
−0.437989 + 0.898980i \(0.644309\pi\)
\(720\) 0 0
\(721\) 748.339 0.0386541
\(722\) − 12252.7i − 0.631575i
\(723\) − 56808.0i − 2.92215i
\(724\) 33362.6 1.71258
\(725\) 0 0
\(726\) −60217.6 −3.07835
\(727\) − 2135.25i − 0.108930i −0.998516 0.0544649i \(-0.982655\pi\)
0.998516 0.0544649i \(-0.0173453\pi\)
\(728\) 1444.45i 0.0735371i
\(729\) 31426.5 1.59663
\(730\) 0 0
\(731\) −4719.54 −0.238794
\(732\) − 49968.6i − 2.52308i
\(733\) 4795.27i 0.241633i 0.992675 + 0.120817i \(0.0385513\pi\)
−0.992675 + 0.120817i \(0.961449\pi\)
\(734\) −55233.1 −2.77751
\(735\) 0 0
\(736\) −20839.9 −1.04371
\(737\) 806.138i 0.0402910i
\(738\) 4040.36i 0.201528i
\(739\) 32747.6 1.63010 0.815048 0.579393i \(-0.196710\pi\)
0.815048 + 0.579393i \(0.196710\pi\)
\(740\) 0 0
\(741\) 4543.89 0.225269
\(742\) − 6340.05i − 0.313680i
\(743\) 12299.4i 0.607298i 0.952784 + 0.303649i \(0.0982050\pi\)
−0.952784 + 0.303649i \(0.901795\pi\)
\(744\) −13887.1 −0.684309
\(745\) 0 0
\(746\) 16163.0 0.793257
\(747\) 38809.3i 1.90088i
\(748\) 15425.9i 0.754046i
\(749\) 1852.43 0.0903688
\(750\) 0 0
\(751\) 30102.6 1.46266 0.731332 0.682021i \(-0.238899\pi\)
0.731332 + 0.682021i \(0.238899\pi\)
\(752\) − 45259.4i − 2.19474i
\(753\) − 40309.4i − 1.95081i
\(754\) 1674.45 0.0808753
\(755\) 0 0
\(756\) 9998.14 0.480990
\(757\) − 38826.3i − 1.86416i −0.362257 0.932078i \(-0.617994\pi\)
0.362257 0.932078i \(-0.382006\pi\)
\(758\) − 40519.2i − 1.94159i
\(759\) 79996.9 3.82570
\(760\) 0 0
\(761\) 19981.6 0.951815 0.475907 0.879495i \(-0.342120\pi\)
0.475907 + 0.879495i \(0.342120\pi\)
\(762\) 82221.8i 3.90890i
\(763\) − 4792.39i − 0.227387i
\(764\) 48227.7 2.28379
\(765\) 0 0
\(766\) 13004.8 0.613424
\(767\) 5470.14i 0.257517i
\(768\) − 68644.2i − 3.22524i
\(769\) 22407.7 1.05077 0.525384 0.850865i \(-0.323922\pi\)
0.525384 + 0.850865i \(0.323922\pi\)
\(770\) 0 0
\(771\) 24517.9 1.14525
\(772\) − 39127.0i − 1.82411i
\(773\) − 6902.77i − 0.321184i −0.987021 0.160592i \(-0.948660\pi\)
0.987021 0.160592i \(-0.0513403\pi\)
\(774\) 62635.4 2.90876
\(775\) 0 0
\(776\) 17807.3 0.823768
\(777\) − 4218.89i − 0.194790i
\(778\) 26039.8i 1.19996i
\(779\) −1191.17 −0.0547856
\(780\) 0 0
\(781\) 35099.9 1.60816
\(782\) − 15418.8i − 0.705082i
\(783\) − 6238.62i − 0.284738i
\(784\) 32049.3 1.45997
\(785\) 0 0
\(786\) 17348.3 0.787270
\(787\) 22185.9i 1.00488i 0.864611 + 0.502442i \(0.167565\pi\)
−0.864611 + 0.502442i \(0.832435\pi\)
\(788\) − 22016.3i − 0.995301i
\(789\) −45898.1 −2.07099
\(790\) 0 0
\(791\) −3869.27 −0.173926
\(792\) − 110197.i − 4.94405i
\(793\) − 2742.74i − 0.122822i
\(794\) −25914.2 −1.15826
\(795\) 0 0
\(796\) −83067.8 −3.69882
\(797\) 16291.1i 0.724040i 0.932170 + 0.362020i \(0.117913\pi\)
−0.932170 + 0.362020i \(0.882087\pi\)
\(798\) 10832.6i 0.480539i
\(799\) 7884.41 0.349100
\(800\) 0 0
\(801\) 50635.5 2.23361
\(802\) 43785.3i 1.92782i
\(803\) − 10132.9i − 0.445309i
\(804\) −2260.32 −0.0991485
\(805\) 0 0
\(806\) −1416.12 −0.0618867
\(807\) − 49046.8i − 2.13944i
\(808\) − 6182.70i − 0.269191i
\(809\) −17696.8 −0.769082 −0.384541 0.923108i \(-0.625640\pi\)
−0.384541 + 0.923108i \(0.625640\pi\)
\(810\) 0 0
\(811\) −3095.34 −0.134022 −0.0670111 0.997752i \(-0.521346\pi\)
−0.0670111 + 0.997752i \(0.521346\pi\)
\(812\) 2730.93i 0.118026i
\(813\) 46318.9i 1.99812i
\(814\) −34360.2 −1.47951
\(815\) 0 0
\(816\) −14060.3 −0.603198
\(817\) 18466.0i 0.790751i
\(818\) − 62131.5i − 2.65572i
\(819\) 1379.75 0.0588674
\(820\) 0 0
\(821\) 12323.5 0.523864 0.261932 0.965086i \(-0.415640\pi\)
0.261932 + 0.965086i \(0.415640\pi\)
\(822\) − 5568.36i − 0.236276i
\(823\) − 34436.5i − 1.45854i −0.684225 0.729271i \(-0.739860\pi\)
0.684225 0.729271i \(-0.260140\pi\)
\(824\) 9198.50 0.388889
\(825\) 0 0
\(826\) −13040.8 −0.549329
\(827\) − 18761.6i − 0.788880i −0.918922 0.394440i \(-0.870939\pi\)
0.918922 0.394440i \(-0.129061\pi\)
\(828\) 139992.i 5.87567i
\(829\) −22423.8 −0.939457 −0.469728 0.882811i \(-0.655648\pi\)
−0.469728 + 0.882811i \(0.655648\pi\)
\(830\) 0 0
\(831\) −10235.3 −0.427265
\(832\) − 1602.13i − 0.0667596i
\(833\) 5583.15i 0.232227i
\(834\) 88441.1 3.67202
\(835\) 0 0
\(836\) 60356.4 2.49697
\(837\) 5276.13i 0.217885i
\(838\) 29004.0i 1.19562i
\(839\) −9128.63 −0.375632 −0.187816 0.982204i \(-0.560141\pi\)
−0.187816 + 0.982204i \(0.560141\pi\)
\(840\) 0 0
\(841\) −22685.0 −0.930131
\(842\) − 9441.58i − 0.386435i
\(843\) − 10146.5i − 0.414547i
\(844\) 48634.4 1.98349
\(845\) 0 0
\(846\) −104638. −4.25240
\(847\) − 5390.75i − 0.218688i
\(848\) − 32198.0i − 1.30387i
\(849\) −26822.2 −1.08426
\(850\) 0 0
\(851\) 23495.7 0.946442
\(852\) 98416.1i 3.95737i
\(853\) 27204.8i 1.09200i 0.837786 + 0.545999i \(0.183850\pi\)
−0.837786 + 0.545999i \(0.816150\pi\)
\(854\) 6538.68 0.262001
\(855\) 0 0
\(856\) 22769.8 0.909179
\(857\) 38060.0i 1.51704i 0.651649 + 0.758520i \(0.274077\pi\)
−0.651649 + 0.758520i \(0.725923\pi\)
\(858\) − 18004.9i − 0.716405i
\(859\) 33326.2 1.32372 0.661860 0.749627i \(-0.269767\pi\)
0.661860 + 0.749627i \(0.269767\pi\)
\(860\) 0 0
\(861\) −579.531 −0.0229389
\(862\) 422.378i 0.0166894i
\(863\) − 41724.2i − 1.64578i −0.568201 0.822890i \(-0.692360\pi\)
0.568201 0.822890i \(-0.307640\pi\)
\(864\) 17475.4 0.688108
\(865\) 0 0
\(866\) 77223.8 3.03022
\(867\) − 2449.38i − 0.0959460i
\(868\) − 2309.60i − 0.0903146i
\(869\) 56569.7 2.20828
\(870\) 0 0
\(871\) −124.068 −0.00482649
\(872\) − 58907.5i − 2.28768i
\(873\) − 17009.6i − 0.659438i
\(874\) −60328.6 −2.33483
\(875\) 0 0
\(876\) 28411.6 1.09582
\(877\) 49337.3i 1.89966i 0.312767 + 0.949830i \(0.398744\pi\)
−0.312767 + 0.949830i \(0.601256\pi\)
\(878\) − 15423.3i − 0.592838i
\(879\) −63194.0 −2.42489
\(880\) 0 0
\(881\) 8845.46 0.338265 0.169132 0.985593i \(-0.445903\pi\)
0.169132 + 0.985593i \(0.445903\pi\)
\(882\) − 74096.8i − 2.82876i
\(883\) 14724.2i 0.561165i 0.959830 + 0.280582i \(0.0905276\pi\)
−0.959830 + 0.280582i \(0.909472\pi\)
\(884\) −2374.10 −0.0903277
\(885\) 0 0
\(886\) 9023.14 0.342143
\(887\) − 3864.38i − 0.146283i −0.997322 0.0731415i \(-0.976698\pi\)
0.997322 0.0731415i \(-0.0233025\pi\)
\(888\) − 51858.1i − 1.95974i
\(889\) −7360.60 −0.277690
\(890\) 0 0
\(891\) −3688.07 −0.138670
\(892\) − 81160.9i − 3.04649i
\(893\) − 30849.1i − 1.15602i
\(894\) −79023.2 −2.95630
\(895\) 0 0
\(896\) 7351.61 0.274107
\(897\) 12311.8i 0.458283i
\(898\) − 12577.2i − 0.467379i
\(899\) −1441.14 −0.0534648
\(900\) 0 0
\(901\) 5609.04 0.207397
\(902\) 4719.92i 0.174231i
\(903\) 8984.15i 0.331089i
\(904\) −47560.6 −1.74982
\(905\) 0 0
\(906\) −87473.7 −3.20764
\(907\) 743.409i 0.0272155i 0.999907 + 0.0136078i \(0.00433162\pi\)
−0.999907 + 0.0136078i \(0.995668\pi\)
\(908\) − 24182.5i − 0.883839i
\(909\) −5905.76 −0.215491
\(910\) 0 0
\(911\) 16291.0 0.592475 0.296238 0.955114i \(-0.404268\pi\)
0.296238 + 0.955114i \(0.404268\pi\)
\(912\) 55013.4i 1.99745i
\(913\) 45336.8i 1.64340i
\(914\) 74402.5 2.69258
\(915\) 0 0
\(916\) 15500.6 0.559122
\(917\) 1553.04i 0.0559280i
\(918\) 12929.5i 0.464856i
\(919\) 6188.99 0.222150 0.111075 0.993812i \(-0.464571\pi\)
0.111075 + 0.993812i \(0.464571\pi\)
\(920\) 0 0
\(921\) −55390.5 −1.98174
\(922\) − 89311.6i − 3.19015i
\(923\) 5402.00i 0.192643i
\(924\) 29364.8 1.04549
\(925\) 0 0
\(926\) −93033.0 −3.30157
\(927\) − 8786.47i − 0.311311i
\(928\) 4773.30i 0.168848i
\(929\) 31661.7 1.11818 0.559089 0.829108i \(-0.311151\pi\)
0.559089 + 0.829108i \(0.311151\pi\)
\(930\) 0 0
\(931\) 21845.0 0.769003
\(932\) − 20724.3i − 0.728376i
\(933\) − 75734.8i − 2.65750i
\(934\) 37273.6 1.30581
\(935\) 0 0
\(936\) 16959.8 0.592251
\(937\) − 35010.5i − 1.22064i −0.792153 0.610322i \(-0.791040\pi\)
0.792153 0.610322i \(-0.208960\pi\)
\(938\) − 295.776i − 0.0102958i
\(939\) 22279.2 0.774286
\(940\) 0 0
\(941\) −45625.8 −1.58061 −0.790307 0.612711i \(-0.790079\pi\)
−0.790307 + 0.612711i \(0.790079\pi\)
\(942\) − 11217.2i − 0.387977i
\(943\) − 3227.51i − 0.111455i
\(944\) −66227.5 −2.28339
\(945\) 0 0
\(946\) 73170.2 2.51477
\(947\) 21508.4i 0.738044i 0.929421 + 0.369022i \(0.120307\pi\)
−0.929421 + 0.369022i \(0.879693\pi\)
\(948\) 158615.i 5.43416i
\(949\) 1559.50 0.0533439
\(950\) 0 0
\(951\) 36177.4 1.23358
\(952\) − 3046.52i − 0.103717i
\(953\) 35686.7i 1.21302i 0.795076 + 0.606509i \(0.207431\pi\)
−0.795076 + 0.606509i \(0.792569\pi\)
\(954\) −74440.5 −2.52631
\(955\) 0 0
\(956\) 85744.0 2.90079
\(957\) − 18323.0i − 0.618912i
\(958\) 93357.8i 3.14849i
\(959\) 498.486 0.0167851
\(960\) 0 0
\(961\) −28572.2 −0.959088
\(962\) − 5288.17i − 0.177232i
\(963\) − 21749.9i − 0.727810i
\(964\) 116133. 3.88006
\(965\) 0 0
\(966\) −29351.3 −0.977600
\(967\) 3731.33i 0.124086i 0.998073 + 0.0620432i \(0.0197617\pi\)
−0.998073 + 0.0620432i \(0.980238\pi\)
\(968\) − 66262.5i − 2.20016i
\(969\) −9583.60 −0.317719
\(970\) 0 0
\(971\) 17645.1 0.583171 0.291585 0.956545i \(-0.405817\pi\)
0.291585 + 0.956545i \(0.405817\pi\)
\(972\) 60358.3i 1.99176i
\(973\) 7917.35i 0.260862i
\(974\) 51348.7 1.68924
\(975\) 0 0
\(976\) 33206.7 1.08906
\(977\) − 24941.2i − 0.816723i −0.912820 0.408362i \(-0.866100\pi\)
0.912820 0.408362i \(-0.133900\pi\)
\(978\) 61631.8i 2.01510i
\(979\) 59152.1 1.93106
\(980\) 0 0
\(981\) −56268.9 −1.83132
\(982\) − 6277.85i − 0.204006i
\(983\) − 22506.2i − 0.730252i −0.930958 0.365126i \(-0.881026\pi\)
0.930958 0.365126i \(-0.118974\pi\)
\(984\) −7123.53 −0.230782
\(985\) 0 0
\(986\) −3531.62 −0.114066
\(987\) − 15008.8i − 0.484029i
\(988\) 9289.07i 0.299114i
\(989\) −50034.2 −1.60869
\(990\) 0 0
\(991\) 32694.1 1.04799 0.523997 0.851720i \(-0.324440\pi\)
0.523997 + 0.851720i \(0.324440\pi\)
\(992\) − 4036.88i − 0.129205i
\(993\) 8410.08i 0.268767i
\(994\) −12878.3 −0.410941
\(995\) 0 0
\(996\) −127119. −4.04410
\(997\) − 18248.8i − 0.579686i −0.957074 0.289843i \(-0.906397\pi\)
0.957074 0.289843i \(-0.0936030\pi\)
\(998\) − 352.428i − 0.0111783i
\(999\) −19702.5 −0.623983
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 425.4.b.f.324.6 6
5.2 odd 4 17.4.a.b.1.1 3
5.3 odd 4 425.4.a.g.1.3 3
5.4 even 2 inner 425.4.b.f.324.1 6
15.2 even 4 153.4.a.g.1.3 3
20.7 even 4 272.4.a.h.1.1 3
35.27 even 4 833.4.a.d.1.1 3
40.27 even 4 1088.4.a.x.1.3 3
40.37 odd 4 1088.4.a.v.1.1 3
55.32 even 4 2057.4.a.e.1.3 3
60.47 odd 4 2448.4.a.bi.1.2 3
85.47 odd 4 289.4.b.b.288.6 6
85.67 odd 4 289.4.a.b.1.1 3
85.72 odd 4 289.4.b.b.288.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 5.2 odd 4
153.4.a.g.1.3 3 15.2 even 4
272.4.a.h.1.1 3 20.7 even 4
289.4.a.b.1.1 3 85.67 odd 4
289.4.b.b.288.5 6 85.72 odd 4
289.4.b.b.288.6 6 85.47 odd 4
425.4.a.g.1.3 3 5.3 odd 4
425.4.b.f.324.1 6 5.4 even 2 inner
425.4.b.f.324.6 6 1.1 even 1 trivial
833.4.a.d.1.1 3 35.27 even 4
1088.4.a.v.1.1 3 40.37 odd 4
1088.4.a.x.1.3 3 40.27 even 4
2057.4.a.e.1.3 3 55.32 even 4
2448.4.a.bi.1.2 3 60.47 odd 4