Properties

Label 425.4.a.g.1.3
Level $425$
Weight $4$
Character 425.1
Self dual yes
Analytic conductor $25.076$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.0758117524\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.2636.1
Defining polynomial: \( x^{3} - 14x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-3.58966\) of defining polynomial
Character \(\chi\) \(=\) 425.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.03251 q^{2} -8.47535 q^{3} +17.3261 q^{4} -42.6523 q^{6} -3.81828 q^{7} +46.9339 q^{8} +44.8316 q^{9} +O(q^{10})\) \(q+5.03251 q^{2} -8.47535 q^{3} +17.3261 q^{4} -42.6523 q^{6} -3.81828 q^{7} +46.9339 q^{8} +44.8316 q^{9} -52.3720 q^{11} -146.845 q^{12} +8.06025 q^{13} -19.2156 q^{14} +97.5862 q^{16} +17.0000 q^{17} +225.616 q^{18} -66.5154 q^{19} +32.3613 q^{21} -263.563 q^{22} -180.226 q^{23} -397.782 q^{24} +40.5633 q^{26} -151.129 q^{27} -66.1562 q^{28} -41.2800 q^{29} -34.9114 q^{31} +115.632 q^{32} +443.871 q^{33} +85.5527 q^{34} +776.759 q^{36} -130.368 q^{37} -334.739 q^{38} -68.3134 q^{39} -17.9081 q^{41} +162.859 q^{42} -277.620 q^{43} -907.405 q^{44} -906.987 q^{46} -463.789 q^{47} -827.078 q^{48} -328.421 q^{49} -144.081 q^{51} +139.653 q^{52} +329.944 q^{53} -760.560 q^{54} -179.207 q^{56} +563.741 q^{57} -207.742 q^{58} +678.656 q^{59} +340.280 q^{61} -175.692 q^{62} -171.180 q^{63} -198.770 q^{64} +2233.79 q^{66} -15.3925 q^{67} +294.545 q^{68} +1527.48 q^{69} -670.203 q^{71} +2104.12 q^{72} -193.480 q^{73} -656.080 q^{74} -1152.46 q^{76} +199.971 q^{77} -343.788 q^{78} +1080.15 q^{79} +70.4207 q^{81} -90.1229 q^{82} +865.668 q^{83} +560.697 q^{84} -1397.13 q^{86} +349.863 q^{87} -2458.02 q^{88} +1129.46 q^{89} -30.7763 q^{91} -3122.61 q^{92} +295.886 q^{93} -2334.02 q^{94} -980.023 q^{96} +379.412 q^{97} -1652.78 q^{98} -2347.92 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} - 4 q^{3} + 25 q^{4} - 74 q^{6} - 22 q^{7} + 39 q^{8} + 59 q^{9} - 28 q^{11} - 22 q^{12} - 30 q^{13} + 92 q^{14} + 137 q^{16} + 51 q^{17} + 103 q^{18} + 80 q^{19} - 192 q^{21} - 286 q^{22} - 142 q^{23} - 666 q^{24} + 26 q^{26} + 20 q^{27} - 476 q^{28} - 456 q^{29} + 230 q^{31} + 71 q^{32} + 332 q^{33} - 17 q^{34} + 1313 q^{36} - 356 q^{37} - 724 q^{38} + 268 q^{39} - 294 q^{41} + 1128 q^{42} - 556 q^{43} - 1122 q^{44} - 704 q^{46} - 640 q^{47} - 774 q^{48} - 269 q^{49} - 68 q^{51} + 774 q^{52} - 302 q^{53} - 1100 q^{54} + 684 q^{56} + 720 q^{57} + 1304 q^{58} + 636 q^{59} - 84 q^{61} - 508 q^{62} - 1122 q^{63} - 919 q^{64} + 2468 q^{66} - 1008 q^{67} + 425 q^{68} + 576 q^{69} - 402 q^{71} + 927 q^{72} - 838 q^{73} + 836 q^{74} - 908 q^{76} + 504 q^{77} - 1308 q^{78} - 594 q^{79} - 505 q^{81} - 358 q^{82} + 2396 q^{83} - 2040 q^{84} - 1264 q^{86} - 1428 q^{87} - 1838 q^{88} - 170 q^{89} - 1016 q^{91} - 4896 q^{92} - 632 q^{93} - 2016 q^{94} + 678 q^{96} + 270 q^{97} - 2857 q^{98} - 2920 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.03251 1.77926 0.889630 0.456681i \(-0.150962\pi\)
0.889630 + 0.456681i \(0.150962\pi\)
\(3\) −8.47535 −1.63108 −0.815541 0.578699i \(-0.803561\pi\)
−0.815541 + 0.578699i \(0.803561\pi\)
\(4\) 17.3261 2.16577
\(5\) 0 0
\(6\) −42.6523 −2.90212
\(7\) −3.81828 −0.206168 −0.103084 0.994673i \(-0.532871\pi\)
−0.103084 + 0.994673i \(0.532871\pi\)
\(8\) 46.9339 2.07421
\(9\) 44.8316 1.66043
\(10\) 0 0
\(11\) −52.3720 −1.43552 −0.717761 0.696289i \(-0.754833\pi\)
−0.717761 + 0.696289i \(0.754833\pi\)
\(12\) −146.845 −3.53255
\(13\) 8.06025 0.171962 0.0859811 0.996297i \(-0.472598\pi\)
0.0859811 + 0.996297i \(0.472598\pi\)
\(14\) −19.2156 −0.366827
\(15\) 0 0
\(16\) 97.5862 1.52478
\(17\) 17.0000 0.242536
\(18\) 225.616 2.95434
\(19\) −66.5154 −0.803141 −0.401570 0.915828i \(-0.631535\pi\)
−0.401570 + 0.915828i \(0.631535\pi\)
\(20\) 0 0
\(21\) 32.3613 0.336277
\(22\) −263.563 −2.55417
\(23\) −180.226 −1.63390 −0.816948 0.576711i \(-0.804336\pi\)
−0.816948 + 0.576711i \(0.804336\pi\)
\(24\) −397.782 −3.38320
\(25\) 0 0
\(26\) 40.5633 0.305966
\(27\) −151.129 −1.07722
\(28\) −66.1562 −0.446512
\(29\) −41.2800 −0.264328 −0.132164 0.991228i \(-0.542193\pi\)
−0.132164 + 0.991228i \(0.542193\pi\)
\(30\) 0 0
\(31\) −34.9114 −0.202267 −0.101133 0.994873i \(-0.532247\pi\)
−0.101133 + 0.994873i \(0.532247\pi\)
\(32\) 115.632 0.638783
\(33\) 443.871 2.34146
\(34\) 85.5527 0.431534
\(35\) 0 0
\(36\) 776.759 3.59611
\(37\) −130.368 −0.579255 −0.289627 0.957139i \(-0.593531\pi\)
−0.289627 + 0.957139i \(0.593531\pi\)
\(38\) −334.739 −1.42900
\(39\) −68.3134 −0.280485
\(40\) 0 0
\(41\) −17.9081 −0.0682142 −0.0341071 0.999418i \(-0.510859\pi\)
−0.0341071 + 0.999418i \(0.510859\pi\)
\(42\) 162.859 0.598325
\(43\) −277.620 −0.984573 −0.492287 0.870433i \(-0.663839\pi\)
−0.492287 + 0.870433i \(0.663839\pi\)
\(44\) −907.405 −3.10901
\(45\) 0 0
\(46\) −906.987 −2.90713
\(47\) −463.789 −1.43937 −0.719687 0.694299i \(-0.755715\pi\)
−0.719687 + 0.694299i \(0.755715\pi\)
\(48\) −827.078 −2.48705
\(49\) −328.421 −0.957495
\(50\) 0 0
\(51\) −144.081 −0.395596
\(52\) 139.653 0.372431
\(53\) 329.944 0.855118 0.427559 0.903987i \(-0.359374\pi\)
0.427559 + 0.903987i \(0.359374\pi\)
\(54\) −760.560 −1.91665
\(55\) 0 0
\(56\) −179.207 −0.427635
\(57\) 563.741 1.30999
\(58\) −207.742 −0.470308
\(59\) 678.656 1.49752 0.748759 0.662843i \(-0.230650\pi\)
0.748759 + 0.662843i \(0.230650\pi\)
\(60\) 0 0
\(61\) 340.280 0.714237 0.357118 0.934059i \(-0.383759\pi\)
0.357118 + 0.934059i \(0.383759\pi\)
\(62\) −175.692 −0.359885
\(63\) −171.180 −0.342328
\(64\) −198.770 −0.388223
\(65\) 0 0
\(66\) 2233.79 4.16606
\(67\) −15.3925 −0.0280671 −0.0140336 0.999902i \(-0.504467\pi\)
−0.0140336 + 0.999902i \(0.504467\pi\)
\(68\) 294.545 0.525276
\(69\) 1527.48 2.66502
\(70\) 0 0
\(71\) −670.203 −1.12026 −0.560130 0.828405i \(-0.689249\pi\)
−0.560130 + 0.828405i \(0.689249\pi\)
\(72\) 2104.12 3.44408
\(73\) −193.480 −0.310207 −0.155103 0.987898i \(-0.549571\pi\)
−0.155103 + 0.987898i \(0.549571\pi\)
\(74\) −656.080 −1.03065
\(75\) 0 0
\(76\) −1152.46 −1.73942
\(77\) 199.971 0.295959
\(78\) −343.788 −0.499055
\(79\) 1080.15 1.53831 0.769156 0.639061i \(-0.220677\pi\)
0.769156 + 0.639061i \(0.220677\pi\)
\(80\) 0 0
\(81\) 70.4207 0.0965990
\(82\) −90.1229 −0.121371
\(83\) 865.668 1.14481 0.572406 0.819970i \(-0.306010\pi\)
0.572406 + 0.819970i \(0.306010\pi\)
\(84\) 560.697 0.728298
\(85\) 0 0
\(86\) −1397.13 −1.75181
\(87\) 349.863 0.431141
\(88\) −2458.02 −2.97757
\(89\) 1129.46 1.34520 0.672599 0.740008i \(-0.265178\pi\)
0.672599 + 0.740008i \(0.265178\pi\)
\(90\) 0 0
\(91\) −30.7763 −0.0354531
\(92\) −3122.61 −3.53864
\(93\) 295.886 0.329914
\(94\) −2334.02 −2.56102
\(95\) 0 0
\(96\) −980.023 −1.04191
\(97\) 379.412 0.397149 0.198574 0.980086i \(-0.436369\pi\)
0.198574 + 0.980086i \(0.436369\pi\)
\(98\) −1652.78 −1.70363
\(99\) −2347.92 −2.38359
\(100\) 0 0
\(101\) 131.732 0.129780 0.0648902 0.997892i \(-0.479330\pi\)
0.0648902 + 0.997892i \(0.479330\pi\)
\(102\) −725.089 −0.703868
\(103\) −195.988 −0.187488 −0.0937442 0.995596i \(-0.529884\pi\)
−0.0937442 + 0.995596i \(0.529884\pi\)
\(104\) 378.299 0.356685
\(105\) 0 0
\(106\) 1660.45 1.52148
\(107\) 485.147 0.438326 0.219163 0.975688i \(-0.429667\pi\)
0.219163 + 0.975688i \(0.429667\pi\)
\(108\) −2618.49 −2.33300
\(109\) −1255.12 −1.10292 −0.551460 0.834201i \(-0.685929\pi\)
−0.551460 + 0.834201i \(0.685929\pi\)
\(110\) 0 0
\(111\) 1104.92 0.944812
\(112\) −372.612 −0.314362
\(113\) 1013.35 0.843612 0.421806 0.906686i \(-0.361396\pi\)
0.421806 + 0.906686i \(0.361396\pi\)
\(114\) 2837.03 2.33081
\(115\) 0 0
\(116\) −715.224 −0.572473
\(117\) 361.354 0.285531
\(118\) 3415.34 2.66447
\(119\) −64.9108 −0.0500031
\(120\) 0 0
\(121\) 1411.83 1.06073
\(122\) 1712.46 1.27081
\(123\) 151.778 0.111263
\(124\) −604.880 −0.438063
\(125\) 0 0
\(126\) −861.464 −0.609090
\(127\) −1927.72 −1.34691 −0.673456 0.739227i \(-0.735191\pi\)
−0.673456 + 0.739227i \(0.735191\pi\)
\(128\) −1925.37 −1.32953
\(129\) 2352.93 1.60592
\(130\) 0 0
\(131\) −406.738 −0.271274 −0.135637 0.990759i \(-0.543308\pi\)
−0.135637 + 0.990759i \(0.543308\pi\)
\(132\) 7690.58 5.07105
\(133\) 253.975 0.165582
\(134\) −77.4631 −0.0499387
\(135\) 0 0
\(136\) 797.877 0.503069
\(137\) 130.552 0.0814149 0.0407074 0.999171i \(-0.487039\pi\)
0.0407074 + 0.999171i \(0.487039\pi\)
\(138\) 7687.03 4.74177
\(139\) 2073.54 1.26529 0.632644 0.774443i \(-0.281970\pi\)
0.632644 + 0.774443i \(0.281970\pi\)
\(140\) 0 0
\(141\) 3930.78 2.34774
\(142\) −3372.80 −1.99323
\(143\) −422.131 −0.246856
\(144\) 4374.95 2.53180
\(145\) 0 0
\(146\) −973.689 −0.551939
\(147\) 2783.48 1.56175
\(148\) −2258.78 −1.25453
\(149\) −1852.73 −1.01867 −0.509334 0.860569i \(-0.670108\pi\)
−0.509334 + 0.860569i \(0.670108\pi\)
\(150\) 0 0
\(151\) 2050.86 1.10527 0.552637 0.833422i \(-0.313622\pi\)
0.552637 + 0.833422i \(0.313622\pi\)
\(152\) −3121.83 −1.66588
\(153\) 762.138 0.402714
\(154\) 1006.36 0.526588
\(155\) 0 0
\(156\) −1183.61 −0.607465
\(157\) 262.991 0.133688 0.0668438 0.997763i \(-0.478707\pi\)
0.0668438 + 0.997763i \(0.478707\pi\)
\(158\) 5435.88 2.73706
\(159\) −2796.39 −1.39477
\(160\) 0 0
\(161\) 688.152 0.336857
\(162\) 354.393 0.171875
\(163\) 1444.98 0.694354 0.347177 0.937800i \(-0.387140\pi\)
0.347177 + 0.937800i \(0.387140\pi\)
\(164\) −310.279 −0.147736
\(165\) 0 0
\(166\) 4356.48 2.03692
\(167\) 501.565 0.232409 0.116204 0.993225i \(-0.462927\pi\)
0.116204 + 0.993225i \(0.462927\pi\)
\(168\) 1518.84 0.697508
\(169\) −2132.03 −0.970429
\(170\) 0 0
\(171\) −2981.99 −1.33356
\(172\) −4810.08 −2.13236
\(173\) 2590.14 1.13829 0.569146 0.822237i \(-0.307274\pi\)
0.569146 + 0.822237i \(0.307274\pi\)
\(174\) 1760.69 0.767112
\(175\) 0 0
\(176\) −5110.79 −2.18886
\(177\) −5751.85 −2.44257
\(178\) 5684.02 2.39346
\(179\) 2165.65 0.904294 0.452147 0.891943i \(-0.350658\pi\)
0.452147 + 0.891943i \(0.350658\pi\)
\(180\) 0 0
\(181\) −1925.56 −0.790750 −0.395375 0.918520i \(-0.629385\pi\)
−0.395375 + 0.918520i \(0.629385\pi\)
\(182\) −154.882 −0.0630803
\(183\) −2884.00 −1.16498
\(184\) −8458.69 −3.38904
\(185\) 0 0
\(186\) 1489.05 0.587003
\(187\) −890.324 −0.348165
\(188\) −8035.68 −3.11735
\(189\) 577.055 0.222088
\(190\) 0 0
\(191\) −2783.52 −1.05449 −0.527247 0.849712i \(-0.676776\pi\)
−0.527247 + 0.849712i \(0.676776\pi\)
\(192\) 1684.65 0.633223
\(193\) −2258.27 −0.842246 −0.421123 0.907004i \(-0.638364\pi\)
−0.421123 + 0.907004i \(0.638364\pi\)
\(194\) 1909.39 0.706631
\(195\) 0 0
\(196\) −5690.27 −2.07371
\(197\) 1270.70 0.459560 0.229780 0.973243i \(-0.426199\pi\)
0.229780 + 0.973243i \(0.426199\pi\)
\(198\) −11815.9 −4.24102
\(199\) −4794.36 −1.70786 −0.853928 0.520392i \(-0.825786\pi\)
−0.853928 + 0.520392i \(0.825786\pi\)
\(200\) 0 0
\(201\) 130.457 0.0457798
\(202\) 662.942 0.230913
\(203\) 157.619 0.0544960
\(204\) −2496.37 −0.856769
\(205\) 0 0
\(206\) −986.313 −0.333591
\(207\) −8079.80 −2.71297
\(208\) 786.569 0.262205
\(209\) 3483.54 1.15293
\(210\) 0 0
\(211\) −2807.00 −0.915837 −0.457918 0.888994i \(-0.651405\pi\)
−0.457918 + 0.888994i \(0.651405\pi\)
\(212\) 5716.66 1.85199
\(213\) 5680.21 1.82724
\(214\) 2441.50 0.779896
\(215\) 0 0
\(216\) −7093.09 −2.23437
\(217\) 133.302 0.0417009
\(218\) −6316.38 −1.96238
\(219\) 1639.81 0.505973
\(220\) 0 0
\(221\) 137.024 0.0417070
\(222\) 5560.51 1.68107
\(223\) −4684.30 −1.40665 −0.703327 0.710866i \(-0.748303\pi\)
−0.703327 + 0.710866i \(0.748303\pi\)
\(224\) −441.516 −0.131697
\(225\) 0 0
\(226\) 5099.70 1.50101
\(227\) 1395.72 0.408095 0.204047 0.978961i \(-0.434590\pi\)
0.204047 + 0.978961i \(0.434590\pi\)
\(228\) 9767.47 2.83713
\(229\) 894.638 0.258163 0.129082 0.991634i \(-0.458797\pi\)
0.129082 + 0.991634i \(0.458797\pi\)
\(230\) 0 0
\(231\) −1694.83 −0.482733
\(232\) −1937.43 −0.548270
\(233\) −1196.13 −0.336313 −0.168156 0.985760i \(-0.553781\pi\)
−0.168156 + 0.985760i \(0.553781\pi\)
\(234\) 1818.52 0.508035
\(235\) 0 0
\(236\) 11758.5 3.24328
\(237\) −9154.67 −2.50911
\(238\) −326.664 −0.0889685
\(239\) 4948.82 1.33938 0.669691 0.742639i \(-0.266426\pi\)
0.669691 + 0.742639i \(0.266426\pi\)
\(240\) 0 0
\(241\) −6702.73 −1.79154 −0.895770 0.444518i \(-0.853375\pi\)
−0.895770 + 0.444518i \(0.853375\pi\)
\(242\) 7105.03 1.88731
\(243\) 3483.65 0.919656
\(244\) 5895.75 1.54687
\(245\) 0 0
\(246\) 763.824 0.197966
\(247\) −536.130 −0.138110
\(248\) −1638.53 −0.419543
\(249\) −7336.85 −1.86728
\(250\) 0 0
\(251\) −4756.08 −1.19602 −0.598010 0.801489i \(-0.704042\pi\)
−0.598010 + 0.801489i \(0.704042\pi\)
\(252\) −2965.89 −0.741402
\(253\) 9438.77 2.34550
\(254\) −9701.29 −2.39651
\(255\) 0 0
\(256\) −8099.28 −1.97736
\(257\) −2892.84 −0.702143 −0.351071 0.936349i \(-0.614183\pi\)
−0.351071 + 0.936349i \(0.614183\pi\)
\(258\) 11841.1 2.85735
\(259\) 497.784 0.119424
\(260\) 0 0
\(261\) −1850.65 −0.438898
\(262\) −2046.92 −0.482667
\(263\) −5415.48 −1.26971 −0.634853 0.772633i \(-0.718939\pi\)
−0.634853 + 0.772633i \(0.718939\pi\)
\(264\) 20832.6 4.85666
\(265\) 0 0
\(266\) 1278.13 0.294613
\(267\) −9572.58 −2.19413
\(268\) −266.693 −0.0607869
\(269\) 5787.00 1.31167 0.655835 0.754904i \(-0.272317\pi\)
0.655835 + 0.754904i \(0.272317\pi\)
\(270\) 0 0
\(271\) 5465.13 1.22503 0.612515 0.790459i \(-0.290158\pi\)
0.612515 + 0.790459i \(0.290158\pi\)
\(272\) 1658.97 0.369815
\(273\) 260.840 0.0578270
\(274\) 657.006 0.144858
\(275\) 0 0
\(276\) 26465.3 5.77182
\(277\) 1207.65 0.261952 0.130976 0.991386i \(-0.458189\pi\)
0.130976 + 0.991386i \(0.458189\pi\)
\(278\) 10435.1 2.25128
\(279\) −1565.13 −0.335850
\(280\) 0 0
\(281\) −1197.18 −0.254155 −0.127077 0.991893i \(-0.540560\pi\)
−0.127077 + 0.991893i \(0.540560\pi\)
\(282\) 19781.7 4.17724
\(283\) −3164.73 −0.664748 −0.332374 0.943148i \(-0.607850\pi\)
−0.332374 + 0.943148i \(0.607850\pi\)
\(284\) −11612.0 −2.42622
\(285\) 0 0
\(286\) −2124.38 −0.439221
\(287\) 68.3784 0.0140636
\(288\) 5183.98 1.06066
\(289\) 289.000 0.0588235
\(290\) 0 0
\(291\) −3215.65 −0.647782
\(292\) −3352.26 −0.671836
\(293\) −7456.21 −1.48668 −0.743339 0.668915i \(-0.766759\pi\)
−0.743339 + 0.668915i \(0.766759\pi\)
\(294\) 14007.9 2.77877
\(295\) 0 0
\(296\) −6118.70 −1.20149
\(297\) 7914.94 1.54637
\(298\) −9323.89 −1.81248
\(299\) −1452.66 −0.280969
\(300\) 0 0
\(301\) 1060.03 0.202988
\(302\) 10321.0 1.96657
\(303\) −1116.47 −0.211683
\(304\) −6490.98 −1.22462
\(305\) 0 0
\(306\) 3835.46 0.716532
\(307\) 6535.48 1.21498 0.607491 0.794327i \(-0.292176\pi\)
0.607491 + 0.794327i \(0.292176\pi\)
\(308\) 3464.73 0.640978
\(309\) 1661.07 0.305809
\(310\) 0 0
\(311\) −8935.89 −1.62928 −0.814642 0.579963i \(-0.803067\pi\)
−0.814642 + 0.579963i \(0.803067\pi\)
\(312\) −3206.22 −0.581783
\(313\) 2628.71 0.474707 0.237353 0.971423i \(-0.423720\pi\)
0.237353 + 0.971423i \(0.423720\pi\)
\(314\) 1323.50 0.237865
\(315\) 0 0
\(316\) 18714.9 3.33163
\(317\) −4268.54 −0.756293 −0.378147 0.925746i \(-0.623438\pi\)
−0.378147 + 0.925746i \(0.623438\pi\)
\(318\) −14072.9 −2.48166
\(319\) 2161.92 0.379449
\(320\) 0 0
\(321\) −4111.79 −0.714946
\(322\) 3463.13 0.599357
\(323\) −1130.76 −0.194790
\(324\) 1220.12 0.209211
\(325\) 0 0
\(326\) 7271.89 1.23544
\(327\) 10637.5 1.79895
\(328\) −840.500 −0.141490
\(329\) 1770.88 0.296753
\(330\) 0 0
\(331\) 992.298 0.164778 0.0823892 0.996600i \(-0.473745\pi\)
0.0823892 + 0.996600i \(0.473745\pi\)
\(332\) 14998.7 2.47940
\(333\) −5844.63 −0.961812
\(334\) 2524.13 0.413516
\(335\) 0 0
\(336\) 3158.02 0.512750
\(337\) −8042.26 −1.29997 −0.649985 0.759947i \(-0.725225\pi\)
−0.649985 + 0.759947i \(0.725225\pi\)
\(338\) −10729.5 −1.72665
\(339\) −8588.52 −1.37600
\(340\) 0 0
\(341\) 1828.38 0.290359
\(342\) −15006.9 −2.37275
\(343\) 2563.68 0.403573
\(344\) −13029.8 −2.04221
\(345\) 0 0
\(346\) 13034.9 2.02532
\(347\) 7414.16 1.14701 0.573506 0.819202i \(-0.305583\pi\)
0.573506 + 0.819202i \(0.305583\pi\)
\(348\) 6061.78 0.933751
\(349\) −859.194 −0.131781 −0.0658905 0.997827i \(-0.520989\pi\)
−0.0658905 + 0.997827i \(0.520989\pi\)
\(350\) 0 0
\(351\) −1218.14 −0.185241
\(352\) −6055.89 −0.916988
\(353\) −569.084 −0.0858053 −0.0429027 0.999079i \(-0.513661\pi\)
−0.0429027 + 0.999079i \(0.513661\pi\)
\(354\) −28946.3 −4.34598
\(355\) 0 0
\(356\) 19569.2 2.91339
\(357\) 550.142 0.0815592
\(358\) 10898.7 1.60897
\(359\) −5005.21 −0.735835 −0.367918 0.929858i \(-0.619929\pi\)
−0.367918 + 0.929858i \(0.619929\pi\)
\(360\) 0 0
\(361\) −2434.71 −0.354965
\(362\) −9690.40 −1.40695
\(363\) −11965.7 −1.73013
\(364\) −533.235 −0.0767833
\(365\) 0 0
\(366\) −14513.7 −2.07280
\(367\) 10975.3 1.56105 0.780523 0.625127i \(-0.214953\pi\)
0.780523 + 0.625127i \(0.214953\pi\)
\(368\) −17587.5 −2.49134
\(369\) −802.851 −0.113265
\(370\) 0 0
\(371\) −1259.82 −0.176298
\(372\) 5126.57 0.714517
\(373\) 3211.72 0.445835 0.222918 0.974837i \(-0.428442\pi\)
0.222918 + 0.974837i \(0.428442\pi\)
\(374\) −4480.56 −0.619477
\(375\) 0 0
\(376\) −21767.4 −2.98556
\(377\) −332.727 −0.0454544
\(378\) 2904.03 0.395152
\(379\) 8051.48 1.09123 0.545616 0.838035i \(-0.316296\pi\)
0.545616 + 0.838035i \(0.316296\pi\)
\(380\) 0 0
\(381\) 16338.1 2.19692
\(382\) −14008.1 −1.87622
\(383\) 2584.16 0.344763 0.172382 0.985030i \(-0.444854\pi\)
0.172382 + 0.985030i \(0.444854\pi\)
\(384\) 16318.2 2.16858
\(385\) 0 0
\(386\) −11364.7 −1.49858
\(387\) −12446.2 −1.63482
\(388\) 6573.74 0.860132
\(389\) −5174.31 −0.674417 −0.337208 0.941430i \(-0.609483\pi\)
−0.337208 + 0.941430i \(0.609483\pi\)
\(390\) 0 0
\(391\) −3063.83 −0.396278
\(392\) −15414.1 −1.98604
\(393\) 3447.25 0.442470
\(394\) 6394.79 0.817677
\(395\) 0 0
\(396\) −40680.4 −5.16230
\(397\) 5149.36 0.650980 0.325490 0.945545i \(-0.394471\pi\)
0.325490 + 0.945545i \(0.394471\pi\)
\(398\) −24127.7 −3.03872
\(399\) −2152.53 −0.270078
\(400\) 0 0
\(401\) 8700.49 1.08350 0.541748 0.840541i \(-0.317763\pi\)
0.541748 + 0.840541i \(0.317763\pi\)
\(402\) 656.527 0.0814542
\(403\) −281.394 −0.0347823
\(404\) 2282.41 0.281074
\(405\) 0 0
\(406\) 793.219 0.0969625
\(407\) 6827.65 0.831533
\(408\) −6762.29 −0.820547
\(409\) 12346.0 1.49260 0.746299 0.665611i \(-0.231829\pi\)
0.746299 + 0.665611i \(0.231829\pi\)
\(410\) 0 0
\(411\) −1106.48 −0.132794
\(412\) −3395.72 −0.406056
\(413\) −2591.30 −0.308740
\(414\) −40661.7 −4.82708
\(415\) 0 0
\(416\) 932.023 0.109847
\(417\) −17574.0 −2.06379
\(418\) 17531.0 2.05136
\(419\) −5763.33 −0.671974 −0.335987 0.941867i \(-0.609070\pi\)
−0.335987 + 0.941867i \(0.609070\pi\)
\(420\) 0 0
\(421\) −1876.12 −0.217188 −0.108594 0.994086i \(-0.534635\pi\)
−0.108594 + 0.994086i \(0.534635\pi\)
\(422\) −14126.2 −1.62951
\(423\) −20792.4 −2.38998
\(424\) 15485.6 1.77369
\(425\) 0 0
\(426\) 28585.7 3.25113
\(427\) −1299.29 −0.147253
\(428\) 8405.72 0.949313
\(429\) 3577.71 0.402642
\(430\) 0 0
\(431\) 83.9299 0.00937996 0.00468998 0.999989i \(-0.498507\pi\)
0.00468998 + 0.999989i \(0.498507\pi\)
\(432\) −14748.1 −1.64252
\(433\) 15345.0 1.70308 0.851539 0.524291i \(-0.175669\pi\)
0.851539 + 0.524291i \(0.175669\pi\)
\(434\) 670.842 0.0741968
\(435\) 0 0
\(436\) −21746.3 −2.38867
\(437\) 11987.8 1.31225
\(438\) 8252.36 0.900258
\(439\) 3064.74 0.333194 0.166597 0.986025i \(-0.446722\pi\)
0.166597 + 0.986025i \(0.446722\pi\)
\(440\) 0 0
\(441\) −14723.6 −1.58985
\(442\) 689.575 0.0742076
\(443\) 1792.97 0.192295 0.0961474 0.995367i \(-0.469348\pi\)
0.0961474 + 0.995367i \(0.469348\pi\)
\(444\) 19144.0 2.04624
\(445\) 0 0
\(446\) −23573.8 −2.50281
\(447\) 15702.6 1.66153
\(448\) 758.960 0.0800391
\(449\) 2499.19 0.262681 0.131341 0.991337i \(-0.458072\pi\)
0.131341 + 0.991337i \(0.458072\pi\)
\(450\) 0 0
\(451\) 937.885 0.0979231
\(452\) 17557.5 1.82707
\(453\) −17381.7 −1.80279
\(454\) 7024.00 0.726107
\(455\) 0 0
\(456\) 26458.6 2.71719
\(457\) −14784.4 −1.51331 −0.756656 0.653813i \(-0.773168\pi\)
−0.756656 + 0.653813i \(0.773168\pi\)
\(458\) 4502.28 0.459340
\(459\) −2569.20 −0.261263
\(460\) 0 0
\(461\) −17746.9 −1.79297 −0.896483 0.443078i \(-0.853887\pi\)
−0.896483 + 0.443078i \(0.853887\pi\)
\(462\) −8529.23 −0.858909
\(463\) −18486.4 −1.85559 −0.927793 0.373096i \(-0.878296\pi\)
−0.927793 + 0.373096i \(0.878296\pi\)
\(464\) −4028.36 −0.403043
\(465\) 0 0
\(466\) −6019.52 −0.598388
\(467\) −7406.57 −0.733908 −0.366954 0.930239i \(-0.619599\pi\)
−0.366954 + 0.930239i \(0.619599\pi\)
\(468\) 6260.87 0.618395
\(469\) 58.7731 0.00578655
\(470\) 0 0
\(471\) −2228.94 −0.218055
\(472\) 31852.0 3.10616
\(473\) 14539.5 1.41338
\(474\) −46071.0 −4.46437
\(475\) 0 0
\(476\) −1124.65 −0.108295
\(477\) 14791.9 1.41986
\(478\) 24905.0 2.38311
\(479\) −18550.9 −1.76955 −0.884775 0.466019i \(-0.845688\pi\)
−0.884775 + 0.466019i \(0.845688\pi\)
\(480\) 0 0
\(481\) −1050.80 −0.0996100
\(482\) −33731.6 −3.18762
\(483\) −5832.34 −0.549442
\(484\) 24461.5 2.29729
\(485\) 0 0
\(486\) 17531.5 1.63631
\(487\) −10203.4 −0.949406 −0.474703 0.880146i \(-0.657444\pi\)
−0.474703 + 0.880146i \(0.657444\pi\)
\(488\) 15970.7 1.48147
\(489\) −12246.7 −1.13255
\(490\) 0 0
\(491\) −1247.46 −0.114658 −0.0573290 0.998355i \(-0.518258\pi\)
−0.0573290 + 0.998355i \(0.518258\pi\)
\(492\) 2629.73 0.240970
\(493\) −701.760 −0.0641089
\(494\) −2698.08 −0.245734
\(495\) 0 0
\(496\) −3406.87 −0.308413
\(497\) 2559.03 0.230962
\(498\) −36922.7 −3.32238
\(499\) 70.0303 0.00628254 0.00314127 0.999995i \(-0.499000\pi\)
0.00314127 + 0.999995i \(0.499000\pi\)
\(500\) 0 0
\(501\) −4250.94 −0.379078
\(502\) −23935.0 −2.12803
\(503\) −1444.29 −0.128028 −0.0640138 0.997949i \(-0.520390\pi\)
−0.0640138 + 0.997949i \(0.520390\pi\)
\(504\) −8034.15 −0.710058
\(505\) 0 0
\(506\) 47500.7 4.17325
\(507\) 18069.7 1.58285
\(508\) −33400.0 −2.91710
\(509\) 14272.8 1.24289 0.621445 0.783458i \(-0.286546\pi\)
0.621445 + 0.783458i \(0.286546\pi\)
\(510\) 0 0
\(511\) 738.761 0.0639547
\(512\) −25356.7 −2.18871
\(513\) 10052.4 0.865157
\(514\) −14558.3 −1.24929
\(515\) 0 0
\(516\) 40767.2 3.47805
\(517\) 24289.6 2.06625
\(518\) 2505.10 0.212486
\(519\) −21952.3 −1.85665
\(520\) 0 0
\(521\) 14874.0 1.25075 0.625376 0.780324i \(-0.284946\pi\)
0.625376 + 0.780324i \(0.284946\pi\)
\(522\) −9313.42 −0.780914
\(523\) 8142.90 0.680811 0.340406 0.940279i \(-0.389436\pi\)
0.340406 + 0.940279i \(0.389436\pi\)
\(524\) −7047.21 −0.587517
\(525\) 0 0
\(526\) −27253.4 −2.25914
\(527\) −593.494 −0.0490569
\(528\) 43315.7 3.57022
\(529\) 20314.2 1.66962
\(530\) 0 0
\(531\) 30425.3 2.48652
\(532\) 4400.40 0.358612
\(533\) −144.344 −0.0117303
\(534\) −48174.1 −3.90393
\(535\) 0 0
\(536\) −722.432 −0.0582170
\(537\) −18354.7 −1.47498
\(538\) 29123.1 2.33380
\(539\) 17200.0 1.37451
\(540\) 0 0
\(541\) 3179.67 0.252689 0.126344 0.991986i \(-0.459676\pi\)
0.126344 + 0.991986i \(0.459676\pi\)
\(542\) 27503.3 2.17965
\(543\) 16319.8 1.28978
\(544\) 1965.75 0.154928
\(545\) 0 0
\(546\) 1312.68 0.102889
\(547\) −2107.07 −0.164702 −0.0823509 0.996603i \(-0.526243\pi\)
−0.0823509 + 0.996603i \(0.526243\pi\)
\(548\) 2261.97 0.176326
\(549\) 15255.3 1.18594
\(550\) 0 0
\(551\) 2745.76 0.212292
\(552\) 71690.4 5.52780
\(553\) −4124.33 −0.317151
\(554\) 6077.51 0.466081
\(555\) 0 0
\(556\) 35926.4 2.74032
\(557\) −467.382 −0.0355540 −0.0177770 0.999842i \(-0.505659\pi\)
−0.0177770 + 0.999842i \(0.505659\pi\)
\(558\) −7876.55 −0.597565
\(559\) −2237.69 −0.169309
\(560\) 0 0
\(561\) 7545.81 0.567887
\(562\) −6024.80 −0.452208
\(563\) −14612.6 −1.09387 −0.546935 0.837175i \(-0.684206\pi\)
−0.546935 + 0.837175i \(0.684206\pi\)
\(564\) 68105.2 5.08466
\(565\) 0 0
\(566\) −15926.5 −1.18276
\(567\) −268.886 −0.0199156
\(568\) −31455.3 −2.32365
\(569\) 11602.3 0.854821 0.427410 0.904058i \(-0.359426\pi\)
0.427410 + 0.904058i \(0.359426\pi\)
\(570\) 0 0
\(571\) −10534.9 −0.772104 −0.386052 0.922477i \(-0.626161\pi\)
−0.386052 + 0.922477i \(0.626161\pi\)
\(572\) −7313.91 −0.534633
\(573\) 23591.3 1.71997
\(574\) 344.115 0.0250228
\(575\) 0 0
\(576\) −8911.18 −0.644617
\(577\) −14404.7 −1.03930 −0.519650 0.854379i \(-0.673938\pi\)
−0.519650 + 0.854379i \(0.673938\pi\)
\(578\) 1454.40 0.104662
\(579\) 19139.6 1.37377
\(580\) 0 0
\(581\) −3305.37 −0.236024
\(582\) −16182.8 −1.15257
\(583\) −17279.8 −1.22754
\(584\) −9080.77 −0.643433
\(585\) 0 0
\(586\) −37523.5 −2.64519
\(587\) 11004.9 0.773799 0.386900 0.922122i \(-0.373546\pi\)
0.386900 + 0.922122i \(0.373546\pi\)
\(588\) 48227.0 3.38240
\(589\) 2322.14 0.162449
\(590\) 0 0
\(591\) −10769.6 −0.749581
\(592\) −12722.2 −0.883239
\(593\) −1853.59 −0.128361 −0.0641804 0.997938i \(-0.520443\pi\)
−0.0641804 + 0.997938i \(0.520443\pi\)
\(594\) 39832.0 2.75139
\(595\) 0 0
\(596\) −32100.7 −2.20620
\(597\) 40633.9 2.78565
\(598\) −7310.53 −0.499916
\(599\) 19074.7 1.30112 0.650559 0.759456i \(-0.274535\pi\)
0.650559 + 0.759456i \(0.274535\pi\)
\(600\) 0 0
\(601\) −27776.0 −1.88520 −0.942600 0.333923i \(-0.891627\pi\)
−0.942600 + 0.333923i \(0.891627\pi\)
\(602\) 5334.62 0.361168
\(603\) −690.073 −0.0466035
\(604\) 35533.4 2.39377
\(605\) 0 0
\(606\) −5618.67 −0.376638
\(607\) −18728.3 −1.25232 −0.626159 0.779695i \(-0.715374\pi\)
−0.626159 + 0.779695i \(0.715374\pi\)
\(608\) −7691.32 −0.513033
\(609\) −1335.88 −0.0888874
\(610\) 0 0
\(611\) −3738.25 −0.247518
\(612\) 13204.9 0.872184
\(613\) 24405.3 1.60802 0.804012 0.594613i \(-0.202695\pi\)
0.804012 + 0.594613i \(0.202695\pi\)
\(614\) 32889.8 2.16177
\(615\) 0 0
\(616\) 9385.43 0.613880
\(617\) 22516.4 1.46917 0.734584 0.678518i \(-0.237377\pi\)
0.734584 + 0.678518i \(0.237377\pi\)
\(618\) 8359.35 0.544114
\(619\) −5146.53 −0.334179 −0.167089 0.985942i \(-0.553437\pi\)
−0.167089 + 0.985942i \(0.553437\pi\)
\(620\) 0 0
\(621\) 27237.4 1.76006
\(622\) −44969.9 −2.89892
\(623\) −4312.60 −0.277337
\(624\) −6666.45 −0.427679
\(625\) 0 0
\(626\) 13229.0 0.844627
\(627\) −29524.3 −1.88052
\(628\) 4556.62 0.289536
\(629\) −2216.26 −0.140490
\(630\) 0 0
\(631\) −3858.77 −0.243447 −0.121724 0.992564i \(-0.538842\pi\)
−0.121724 + 0.992564i \(0.538842\pi\)
\(632\) 50695.8 3.19078
\(633\) 23790.3 1.49381
\(634\) −21481.5 −1.34564
\(635\) 0 0
\(636\) −48450.7 −3.02075
\(637\) −2647.15 −0.164653
\(638\) 10879.9 0.675138
\(639\) −30046.3 −1.86011
\(640\) 0 0
\(641\) 18689.3 1.15161 0.575805 0.817587i \(-0.304689\pi\)
0.575805 + 0.817587i \(0.304689\pi\)
\(642\) −20692.6 −1.27208
\(643\) −26473.5 −1.62366 −0.811831 0.583893i \(-0.801529\pi\)
−0.811831 + 0.583893i \(0.801529\pi\)
\(644\) 11923.0 0.729555
\(645\) 0 0
\(646\) −5690.57 −0.346583
\(647\) −14397.7 −0.874855 −0.437427 0.899254i \(-0.644110\pi\)
−0.437427 + 0.899254i \(0.644110\pi\)
\(648\) 3305.12 0.200366
\(649\) −35542.6 −2.14972
\(650\) 0 0
\(651\) −1129.78 −0.0680177
\(652\) 25036.0 1.50381
\(653\) −20939.5 −1.25486 −0.627431 0.778672i \(-0.715893\pi\)
−0.627431 + 0.778672i \(0.715893\pi\)
\(654\) 53533.6 3.20081
\(655\) 0 0
\(656\) −1747.59 −0.104012
\(657\) −8674.02 −0.515077
\(658\) 8911.96 0.528001
\(659\) 4031.76 0.238323 0.119162 0.992875i \(-0.461979\pi\)
0.119162 + 0.992875i \(0.461979\pi\)
\(660\) 0 0
\(661\) 6691.52 0.393752 0.196876 0.980428i \(-0.436920\pi\)
0.196876 + 0.980428i \(0.436920\pi\)
\(662\) 4993.75 0.293184
\(663\) −1161.33 −0.0680275
\(664\) 40629.2 2.37458
\(665\) 0 0
\(666\) −29413.1 −1.71131
\(667\) 7439.71 0.431884
\(668\) 8690.19 0.503344
\(669\) 39701.1 2.29437
\(670\) 0 0
\(671\) −17821.2 −1.02530
\(672\) 3742.01 0.214808
\(673\) −10319.2 −0.591048 −0.295524 0.955335i \(-0.595494\pi\)
−0.295524 + 0.955335i \(0.595494\pi\)
\(674\) −40472.7 −2.31298
\(675\) 0 0
\(676\) −36939.9 −2.10172
\(677\) 19813.3 1.12480 0.562398 0.826866i \(-0.309879\pi\)
0.562398 + 0.826866i \(0.309879\pi\)
\(678\) −43221.8 −2.44826
\(679\) −1448.70 −0.0818793
\(680\) 0 0
\(681\) −11829.3 −0.665636
\(682\) 9201.33 0.516624
\(683\) −5924.61 −0.331916 −0.165958 0.986133i \(-0.553072\pi\)
−0.165958 + 0.986133i \(0.553072\pi\)
\(684\) −51666.4 −2.88818
\(685\) 0 0
\(686\) 12901.7 0.718061
\(687\) −7582.38 −0.421085
\(688\) −27091.9 −1.50126
\(689\) 2659.43 0.147048
\(690\) 0 0
\(691\) 1973.16 0.108629 0.0543143 0.998524i \(-0.482703\pi\)
0.0543143 + 0.998524i \(0.482703\pi\)
\(692\) 44877.1 2.46528
\(693\) 8965.03 0.491419
\(694\) 37311.8 2.04083
\(695\) 0 0
\(696\) 16420.4 0.894274
\(697\) −304.439 −0.0165444
\(698\) −4323.90 −0.234473
\(699\) 10137.6 0.548554
\(700\) 0 0
\(701\) −12840.1 −0.691815 −0.345907 0.938269i \(-0.612429\pi\)
−0.345907 + 0.938269i \(0.612429\pi\)
\(702\) −6130.30 −0.329591
\(703\) 8671.50 0.465223
\(704\) 10410.0 0.557302
\(705\) 0 0
\(706\) −2863.92 −0.152670
\(707\) −502.990 −0.0267566
\(708\) −99657.5 −5.29005
\(709\) −27749.7 −1.46990 −0.734952 0.678119i \(-0.762796\pi\)
−0.734952 + 0.678119i \(0.762796\pi\)
\(710\) 0 0
\(711\) 48425.0 2.55426
\(712\) 53010.0 2.79022
\(713\) 6291.92 0.330483
\(714\) 2768.60 0.145115
\(715\) 0 0
\(716\) 37522.4 1.95849
\(717\) −41943.0 −2.18464
\(718\) −25188.8 −1.30924
\(719\) 16888.3 0.875979 0.437989 0.898980i \(-0.355691\pi\)
0.437989 + 0.898980i \(0.355691\pi\)
\(720\) 0 0
\(721\) 748.339 0.0386541
\(722\) −12252.7 −0.631575
\(723\) 56808.0 2.92215
\(724\) −33362.6 −1.71258
\(725\) 0 0
\(726\) −60217.6 −3.07835
\(727\) −2135.25 −0.108930 −0.0544649 0.998516i \(-0.517345\pi\)
−0.0544649 + 0.998516i \(0.517345\pi\)
\(728\) −1444.45 −0.0735371
\(729\) −31426.5 −1.59663
\(730\) 0 0
\(731\) −4719.54 −0.238794
\(732\) −49968.6 −2.52308
\(733\) −4795.27 −0.241633 −0.120817 0.992675i \(-0.538551\pi\)
−0.120817 + 0.992675i \(0.538551\pi\)
\(734\) 55233.1 2.77751
\(735\) 0 0
\(736\) −20839.9 −1.04371
\(737\) 806.138 0.0402910
\(738\) −4040.36 −0.201528
\(739\) −32747.6 −1.63010 −0.815048 0.579393i \(-0.803290\pi\)
−0.815048 + 0.579393i \(0.803290\pi\)
\(740\) 0 0
\(741\) 4543.89 0.225269
\(742\) −6340.05 −0.313680
\(743\) −12299.4 −0.607298 −0.303649 0.952784i \(-0.598205\pi\)
−0.303649 + 0.952784i \(0.598205\pi\)
\(744\) 13887.1 0.684309
\(745\) 0 0
\(746\) 16163.0 0.793257
\(747\) 38809.3 1.90088
\(748\) −15425.9 −0.754046
\(749\) −1852.43 −0.0903688
\(750\) 0 0
\(751\) 30102.6 1.46266 0.731332 0.682021i \(-0.238899\pi\)
0.731332 + 0.682021i \(0.238899\pi\)
\(752\) −45259.4 −2.19474
\(753\) 40309.4 1.95081
\(754\) −1674.45 −0.0808753
\(755\) 0 0
\(756\) 9998.14 0.480990
\(757\) −38826.3 −1.86416 −0.932078 0.362257i \(-0.882006\pi\)
−0.932078 + 0.362257i \(0.882006\pi\)
\(758\) 40519.2 1.94159
\(759\) −79996.9 −3.82570
\(760\) 0 0
\(761\) 19981.6 0.951815 0.475907 0.879495i \(-0.342120\pi\)
0.475907 + 0.879495i \(0.342120\pi\)
\(762\) 82221.8 3.90890
\(763\) 4792.39 0.227387
\(764\) −48227.7 −2.28379
\(765\) 0 0
\(766\) 13004.8 0.613424
\(767\) 5470.14 0.257517
\(768\) 68644.2 3.22524
\(769\) −22407.7 −1.05077 −0.525384 0.850865i \(-0.676078\pi\)
−0.525384 + 0.850865i \(0.676078\pi\)
\(770\) 0 0
\(771\) 24517.9 1.14525
\(772\) −39127.0 −1.82411
\(773\) 6902.77 0.321184 0.160592 0.987021i \(-0.448660\pi\)
0.160592 + 0.987021i \(0.448660\pi\)
\(774\) −62635.4 −2.90876
\(775\) 0 0
\(776\) 17807.3 0.823768
\(777\) −4218.89 −0.194790
\(778\) −26039.8 −1.19996
\(779\) 1191.17 0.0547856
\(780\) 0 0
\(781\) 35099.9 1.60816
\(782\) −15418.8 −0.705082
\(783\) 6238.62 0.284738
\(784\) −32049.3 −1.45997
\(785\) 0 0
\(786\) 17348.3 0.787270
\(787\) 22185.9 1.00488 0.502442 0.864611i \(-0.332435\pi\)
0.502442 + 0.864611i \(0.332435\pi\)
\(788\) 22016.3 0.995301
\(789\) 45898.1 2.07099
\(790\) 0 0
\(791\) −3869.27 −0.173926
\(792\) −110197. −4.94405
\(793\) 2742.74 0.122822
\(794\) 25914.2 1.15826
\(795\) 0 0
\(796\) −83067.8 −3.69882
\(797\) 16291.1 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(798\) −10832.6 −0.480539
\(799\) −7884.41 −0.349100
\(800\) 0 0
\(801\) 50635.5 2.23361
\(802\) 43785.3 1.92782
\(803\) 10132.9 0.445309
\(804\) 2260.32 0.0991485
\(805\) 0 0
\(806\) −1416.12 −0.0618867
\(807\) −49046.8 −2.13944
\(808\) 6182.70 0.269191
\(809\) 17696.8 0.769082 0.384541 0.923108i \(-0.374360\pi\)
0.384541 + 0.923108i \(0.374360\pi\)
\(810\) 0 0
\(811\) −3095.34 −0.134022 −0.0670111 0.997752i \(-0.521346\pi\)
−0.0670111 + 0.997752i \(0.521346\pi\)
\(812\) 2730.93 0.118026
\(813\) −46318.9 −1.99812
\(814\) 34360.2 1.47951
\(815\) 0 0
\(816\) −14060.3 −0.603198
\(817\) 18466.0 0.790751
\(818\) 62131.5 2.65572
\(819\) −1379.75 −0.0588674
\(820\) 0 0
\(821\) 12323.5 0.523864 0.261932 0.965086i \(-0.415640\pi\)
0.261932 + 0.965086i \(0.415640\pi\)
\(822\) −5568.36 −0.236276
\(823\) 34436.5 1.45854 0.729271 0.684225i \(-0.239860\pi\)
0.729271 + 0.684225i \(0.239860\pi\)
\(824\) −9198.50 −0.388889
\(825\) 0 0
\(826\) −13040.8 −0.549329
\(827\) −18761.6 −0.788880 −0.394440 0.918922i \(-0.629061\pi\)
−0.394440 + 0.918922i \(0.629061\pi\)
\(828\) −139992. −5.87567
\(829\) 22423.8 0.939457 0.469728 0.882811i \(-0.344352\pi\)
0.469728 + 0.882811i \(0.344352\pi\)
\(830\) 0 0
\(831\) −10235.3 −0.427265
\(832\) −1602.13 −0.0667596
\(833\) −5583.15 −0.232227
\(834\) −88441.1 −3.67202
\(835\) 0 0
\(836\) 60356.4 2.49697
\(837\) 5276.13 0.217885
\(838\) −29004.0 −1.19562
\(839\) 9128.63 0.375632 0.187816 0.982204i \(-0.439859\pi\)
0.187816 + 0.982204i \(0.439859\pi\)
\(840\) 0 0
\(841\) −22685.0 −0.930131
\(842\) −9441.58 −0.386435
\(843\) 10146.5 0.414547
\(844\) −48634.4 −1.98349
\(845\) 0 0
\(846\) −104638. −4.25240
\(847\) −5390.75 −0.218688
\(848\) 32198.0 1.30387
\(849\) 26822.2 1.08426
\(850\) 0 0
\(851\) 23495.7 0.946442
\(852\) 98416.1 3.95737
\(853\) −27204.8 −1.09200 −0.545999 0.837786i \(-0.683850\pi\)
−0.545999 + 0.837786i \(0.683850\pi\)
\(854\) −6538.68 −0.262001
\(855\) 0 0
\(856\) 22769.8 0.909179
\(857\) 38060.0 1.51704 0.758520 0.651649i \(-0.225923\pi\)
0.758520 + 0.651649i \(0.225923\pi\)
\(858\) 18004.9 0.716405
\(859\) −33326.2 −1.32372 −0.661860 0.749627i \(-0.730233\pi\)
−0.661860 + 0.749627i \(0.730233\pi\)
\(860\) 0 0
\(861\) −579.531 −0.0229389
\(862\) 422.378 0.0166894
\(863\) 41724.2 1.64578 0.822890 0.568201i \(-0.192360\pi\)
0.822890 + 0.568201i \(0.192360\pi\)
\(864\) −17475.4 −0.688108
\(865\) 0 0
\(866\) 77223.8 3.03022
\(867\) −2449.38 −0.0959460
\(868\) 2309.60 0.0903146
\(869\) −56569.7 −2.20828
\(870\) 0 0
\(871\) −124.068 −0.00482649
\(872\) −58907.5 −2.28768
\(873\) 17009.6 0.659438
\(874\) 60328.6 2.33483
\(875\) 0 0
\(876\) 28411.6 1.09582
\(877\) 49337.3 1.89966 0.949830 0.312767i \(-0.101256\pi\)
0.949830 + 0.312767i \(0.101256\pi\)
\(878\) 15423.3 0.592838
\(879\) 63194.0 2.42489
\(880\) 0 0
\(881\) 8845.46 0.338265 0.169132 0.985593i \(-0.445903\pi\)
0.169132 + 0.985593i \(0.445903\pi\)
\(882\) −74096.8 −2.82876
\(883\) −14724.2 −0.561165 −0.280582 0.959830i \(-0.590528\pi\)
−0.280582 + 0.959830i \(0.590528\pi\)
\(884\) 2374.10 0.0903277
\(885\) 0 0
\(886\) 9023.14 0.342143
\(887\) −3864.38 −0.146283 −0.0731415 0.997322i \(-0.523302\pi\)
−0.0731415 + 0.997322i \(0.523302\pi\)
\(888\) 51858.1 1.95974
\(889\) 7360.60 0.277690
\(890\) 0 0
\(891\) −3688.07 −0.138670
\(892\) −81160.9 −3.04649
\(893\) 30849.1 1.15602
\(894\) 79023.2 2.95630
\(895\) 0 0
\(896\) 7351.61 0.274107
\(897\) 12311.8 0.458283
\(898\) 12577.2 0.467379
\(899\) 1441.14 0.0534648
\(900\) 0 0
\(901\) 5609.04 0.207397
\(902\) 4719.92 0.174231
\(903\) −8984.15 −0.331089
\(904\) 47560.6 1.74982
\(905\) 0 0
\(906\) −87473.7 −3.20764
\(907\) 743.409 0.0272155 0.0136078 0.999907i \(-0.495668\pi\)
0.0136078 + 0.999907i \(0.495668\pi\)
\(908\) 24182.5 0.883839
\(909\) 5905.76 0.215491
\(910\) 0 0
\(911\) 16291.0 0.592475 0.296238 0.955114i \(-0.404268\pi\)
0.296238 + 0.955114i \(0.404268\pi\)
\(912\) 55013.4 1.99745
\(913\) −45336.8 −1.64340
\(914\) −74402.5 −2.69258
\(915\) 0 0
\(916\) 15500.6 0.559122
\(917\) 1553.04 0.0559280
\(918\) −12929.5 −0.464856
\(919\) −6188.99 −0.222150 −0.111075 0.993812i \(-0.535429\pi\)
−0.111075 + 0.993812i \(0.535429\pi\)
\(920\) 0 0
\(921\) −55390.5 −1.98174
\(922\) −89311.6 −3.19015
\(923\) −5402.00 −0.192643
\(924\) −29364.8 −1.04549
\(925\) 0 0
\(926\) −93033.0 −3.30157
\(927\) −8786.47 −0.311311
\(928\) −4773.30 −0.168848
\(929\) −31661.7 −1.11818 −0.559089 0.829108i \(-0.688849\pi\)
−0.559089 + 0.829108i \(0.688849\pi\)
\(930\) 0 0
\(931\) 21845.0 0.769003
\(932\) −20724.3 −0.728376
\(933\) 75734.8 2.65750
\(934\) −37273.6 −1.30581
\(935\) 0 0
\(936\) 16959.8 0.592251
\(937\) −35010.5 −1.22064 −0.610322 0.792153i \(-0.708960\pi\)
−0.610322 + 0.792153i \(0.708960\pi\)
\(938\) 295.776 0.0102958
\(939\) −22279.2 −0.774286
\(940\) 0 0
\(941\) −45625.8 −1.58061 −0.790307 0.612711i \(-0.790079\pi\)
−0.790307 + 0.612711i \(0.790079\pi\)
\(942\) −11217.2 −0.387977
\(943\) 3227.51 0.111455
\(944\) 66227.5 2.28339
\(945\) 0 0
\(946\) 73170.2 2.51477
\(947\) 21508.4 0.738044 0.369022 0.929421i \(-0.379693\pi\)
0.369022 + 0.929421i \(0.379693\pi\)
\(948\) −158615. −5.43416
\(949\) −1559.50 −0.0533439
\(950\) 0 0
\(951\) 36177.4 1.23358
\(952\) −3046.52 −0.103717
\(953\) −35686.7 −1.21302 −0.606509 0.795076i \(-0.707431\pi\)
−0.606509 + 0.795076i \(0.707431\pi\)
\(954\) 74440.5 2.52631
\(955\) 0 0
\(956\) 85744.0 2.90079
\(957\) −18323.0 −0.618912
\(958\) −93357.8 −3.14849
\(959\) −498.486 −0.0167851
\(960\) 0 0
\(961\) −28572.2 −0.959088
\(962\) −5288.17 −0.177232
\(963\) 21749.9 0.727810
\(964\) −116133. −3.88006
\(965\) 0 0
\(966\) −29351.3 −0.977600
\(967\) 3731.33 0.124086 0.0620432 0.998073i \(-0.480238\pi\)
0.0620432 + 0.998073i \(0.480238\pi\)
\(968\) 66262.5 2.20016
\(969\) 9583.60 0.317719
\(970\) 0 0
\(971\) 17645.1 0.583171 0.291585 0.956545i \(-0.405817\pi\)
0.291585 + 0.956545i \(0.405817\pi\)
\(972\) 60358.3 1.99176
\(973\) −7917.35 −0.260862
\(974\) −51348.7 −1.68924
\(975\) 0 0
\(976\) 33206.7 1.08906
\(977\) −24941.2 −0.816723 −0.408362 0.912820i \(-0.633900\pi\)
−0.408362 + 0.912820i \(0.633900\pi\)
\(978\) −61631.8 −2.01510
\(979\) −59152.1 −1.93106
\(980\) 0 0
\(981\) −56268.9 −1.83132
\(982\) −6277.85 −0.204006
\(983\) 22506.2 0.730252 0.365126 0.930958i \(-0.381026\pi\)
0.365126 + 0.930958i \(0.381026\pi\)
\(984\) 7123.53 0.230782
\(985\) 0 0
\(986\) −3531.62 −0.114066
\(987\) −15008.8 −0.484029
\(988\) −9289.07 −0.299114
\(989\) 50034.2 1.60869
\(990\) 0 0
\(991\) 32694.1 1.04799 0.523997 0.851720i \(-0.324440\pi\)
0.523997 + 0.851720i \(0.324440\pi\)
\(992\) −4036.88 −0.129205
\(993\) −8410.08 −0.268767
\(994\) 12878.3 0.410941
\(995\) 0 0
\(996\) −127119. −4.04410
\(997\) −18248.8 −0.579686 −0.289843 0.957074i \(-0.593603\pi\)
−0.289843 + 0.957074i \(0.593603\pi\)
\(998\) 352.428 0.0111783
\(999\) 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.a.g.1.3 3
5.2 odd 4 425.4.b.f.324.6 6
5.3 odd 4 425.4.b.f.324.1 6
5.4 even 2 17.4.a.b.1.1 3
15.14 odd 2 153.4.a.g.1.3 3
20.19 odd 2 272.4.a.h.1.1 3
35.34 odd 2 833.4.a.d.1.1 3
40.19 odd 2 1088.4.a.x.1.3 3
40.29 even 2 1088.4.a.v.1.1 3
55.54 odd 2 2057.4.a.e.1.3 3
60.59 even 2 2448.4.a.bi.1.2 3
85.4 even 4 289.4.b.b.288.5 6
85.64 even 4 289.4.b.b.288.6 6
85.84 even 2 289.4.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.4.a.b.1.1 3 5.4 even 2
153.4.a.g.1.3 3 15.14 odd 2
272.4.a.h.1.1 3 20.19 odd 2
289.4.a.b.1.1 3 85.84 even 2
289.4.b.b.288.5 6 85.4 even 4
289.4.b.b.288.6 6 85.64 even 4
425.4.a.g.1.3 3 1.1 even 1 trivial
425.4.b.f.324.1 6 5.3 odd 4
425.4.b.f.324.6 6 5.2 odd 4
833.4.a.d.1.1 3 35.34 odd 2
1088.4.a.v.1.1 3 40.29 even 2
1088.4.a.x.1.3 3 40.19 odd 2
2057.4.a.e.1.3 3 55.54 odd 2
2448.4.a.bi.1.2 3 60.59 even 2