Properties

Label 175.4.b.c.99.1
Level $175$
Weight $4$
Character 175.99
Analytic conductor $10.325$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 175 = 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 175.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.3253342510\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.c.99.4

$q$-expansion

\(f(q)\) \(=\) \(q-5.41421i q^{2} -4.65685i q^{3} -21.3137 q^{4} -25.2132 q^{6} +7.00000i q^{7} +72.0833i q^{8} +5.31371 q^{9} +O(q^{10})\) \(q-5.41421i q^{2} -4.65685i q^{3} -21.3137 q^{4} -25.2132 q^{6} +7.00000i q^{7} +72.0833i q^{8} +5.31371 q^{9} -52.2548 q^{11} +99.2548i q^{12} +30.6569i q^{13} +37.8995 q^{14} +219.765 q^{16} -37.2254i q^{17} -28.7696i q^{18} -80.2254 q^{19} +32.5980 q^{21} +282.919i q^{22} +25.8335i q^{23} +335.681 q^{24} +165.983 q^{26} -150.480i q^{27} -149.196i q^{28} -20.9411 q^{29} -314.558 q^{31} -613.186i q^{32} +243.343i q^{33} -201.546 q^{34} -113.255 q^{36} -197.147i q^{37} +434.357i q^{38} +142.765 q^{39} +11.3625 q^{41} -176.492i q^{42} -33.8335i q^{43} +1113.74 q^{44} +139.868 q^{46} +361.676i q^{47} -1023.41i q^{48} -49.0000 q^{49} -173.353 q^{51} -653.411i q^{52} +153.019i q^{53} -814.732 q^{54} -504.583 q^{56} +373.598i q^{57} +113.380i q^{58} +616.000 q^{59} +15.2649 q^{61} +1703.09i q^{62} +37.1960i q^{63} -1561.80 q^{64} +1317.51 q^{66} +166.510i q^{67} +793.411i q^{68} +120.303 q^{69} -952.000 q^{71} +383.029i q^{72} -148.489i q^{73} -1067.40 q^{74} +1709.90 q^{76} -365.784i q^{77} -772.958i q^{78} -857.725 q^{79} -557.294 q^{81} -61.5189i q^{82} +660.528i q^{83} -694.784 q^{84} -183.182 q^{86} +97.5198i q^{87} -3766.70i q^{88} +45.7746 q^{89} -214.598 q^{91} -550.607i q^{92} +1464.85i q^{93} +1958.19 q^{94} -2855.52 q^{96} -1682.13i q^{97} +265.296i q^{98} -277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{4} - 16 q^{6} - 24 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 40 q^{4} - 16 q^{6} - 24 q^{9} - 28 q^{11} + 112 q^{14} + 336 q^{16} - 72 q^{19} - 28 q^{21} + 992 q^{24} + 432 q^{26} + 52 q^{29} - 240 q^{31} + 48 q^{34} - 272 q^{36} + 28 q^{39} - 656 q^{41} + 2328 q^{44} + 1408 q^{46} - 196 q^{49} - 1508 q^{51} - 1296 q^{54} - 672 q^{56} + 2464 q^{59} + 672 q^{61} - 4256 q^{64} + 3952 q^{66} - 2664 q^{69} - 3808 q^{71} - 4032 q^{74} + 3536 q^{76} - 2028 q^{79} - 2908 q^{81} - 1512 q^{84} - 1536 q^{86} + 432 q^{89} - 700 q^{91} + 3856 q^{94} - 4928 q^{96} - 1880 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/175\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.41421i − 1.91421i −0.289735 0.957107i \(-0.593567\pi\)
0.289735 0.957107i \(-0.406433\pi\)
\(3\) − 4.65685i − 0.896212i −0.893980 0.448106i \(-0.852099\pi\)
0.893980 0.448106i \(-0.147901\pi\)
\(4\) −21.3137 −2.66421
\(5\) 0 0
\(6\) −25.2132 −1.71554
\(7\) 7.00000i 0.377964i
\(8\) 72.0833i 3.18566i
\(9\) 5.31371 0.196804
\(10\) 0 0
\(11\) −52.2548 −1.43231 −0.716156 0.697941i \(-0.754100\pi\)
−0.716156 + 0.697941i \(0.754100\pi\)
\(12\) 99.2548i 2.38770i
\(13\) 30.6569i 0.654052i 0.945015 + 0.327026i \(0.106047\pi\)
−0.945015 + 0.327026i \(0.893953\pi\)
\(14\) 37.8995 0.723505
\(15\) 0 0
\(16\) 219.765 3.43382
\(17\) − 37.2254i − 0.531087i −0.964099 0.265544i \(-0.914449\pi\)
0.964099 0.265544i \(-0.0855514\pi\)
\(18\) − 28.7696i − 0.376725i
\(19\) −80.2254 −0.968683 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(20\) 0 0
\(21\) 32.5980 0.338736
\(22\) 282.919i 2.74175i
\(23\) 25.8335i 0.234202i 0.993120 + 0.117101i \(0.0373602\pi\)
−0.993120 + 0.117101i \(0.962640\pi\)
\(24\) 335.681 2.85503
\(25\) 0 0
\(26\) 165.983 1.25200
\(27\) − 150.480i − 1.07259i
\(28\) − 149.196i − 1.00698i
\(29\) −20.9411 −0.134092 −0.0670460 0.997750i \(-0.521357\pi\)
−0.0670460 + 0.997750i \(0.521357\pi\)
\(30\) 0 0
\(31\) −314.558 −1.82246 −0.911232 0.411894i \(-0.864867\pi\)
−0.911232 + 0.411894i \(0.864867\pi\)
\(32\) − 613.186i − 3.38741i
\(33\) 243.343i 1.28365i
\(34\) −201.546 −1.01661
\(35\) 0 0
\(36\) −113.255 −0.524328
\(37\) − 197.147i − 0.875968i −0.898983 0.437984i \(-0.855693\pi\)
0.898983 0.437984i \(-0.144307\pi\)
\(38\) 434.357i 1.85427i
\(39\) 142.765 0.586170
\(40\) 0 0
\(41\) 11.3625 0.0432810 0.0216405 0.999766i \(-0.493111\pi\)
0.0216405 + 0.999766i \(0.493111\pi\)
\(42\) − 176.492i − 0.648414i
\(43\) − 33.8335i − 0.119990i −0.998199 0.0599948i \(-0.980892\pi\)
0.998199 0.0599948i \(-0.0191084\pi\)
\(44\) 1113.74 3.81598
\(45\) 0 0
\(46\) 139.868 0.448313
\(47\) 361.676i 1.12247i 0.827658 + 0.561233i \(0.189673\pi\)
−0.827658 + 0.561233i \(0.810327\pi\)
\(48\) − 1023.41i − 3.07743i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −173.353 −0.475967
\(52\) − 653.411i − 1.74254i
\(53\) 153.019i 0.396582i 0.980143 + 0.198291i \(0.0635390\pi\)
−0.980143 + 0.198291i \(0.936461\pi\)
\(54\) −814.732 −2.05317
\(55\) 0 0
\(56\) −504.583 −1.20407
\(57\) 373.598i 0.868145i
\(58\) 113.380i 0.256681i
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 15.2649 0.0320406 0.0160203 0.999872i \(-0.494900\pi\)
0.0160203 + 0.999872i \(0.494900\pi\)
\(62\) 1703.09i 3.48858i
\(63\) 37.1960i 0.0743849i
\(64\) −1561.80 −3.05040
\(65\) 0 0
\(66\) 1317.51 2.45719
\(67\) 166.510i 0.303618i 0.988410 + 0.151809i \(0.0485098\pi\)
−0.988410 + 0.151809i \(0.951490\pi\)
\(68\) 793.411i 1.41493i
\(69\) 120.303 0.209895
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) 383.029i 0.626951i
\(73\) − 148.489i − 0.238074i −0.992890 0.119037i \(-0.962019\pi\)
0.992890 0.119037i \(-0.0379807\pi\)
\(74\) −1067.40 −1.67679
\(75\) 0 0
\(76\) 1709.90 2.58078
\(77\) − 365.784i − 0.541363i
\(78\) − 772.958i − 1.12205i
\(79\) −857.725 −1.22154 −0.610770 0.791808i \(-0.709140\pi\)
−0.610770 + 0.791808i \(0.709140\pi\)
\(80\) 0 0
\(81\) −557.294 −0.764464
\(82\) − 61.5189i − 0.0828491i
\(83\) 660.528i 0.873523i 0.899577 + 0.436761i \(0.143875\pi\)
−0.899577 + 0.436761i \(0.856125\pi\)
\(84\) −694.784 −0.902466
\(85\) 0 0
\(86\) −183.182 −0.229686
\(87\) 97.5198i 0.120175i
\(88\) − 3766.70i − 4.56286i
\(89\) 45.7746 0.0545180 0.0272590 0.999628i \(-0.491322\pi\)
0.0272590 + 0.999628i \(0.491322\pi\)
\(90\) 0 0
\(91\) −214.598 −0.247209
\(92\) − 550.607i − 0.623965i
\(93\) 1464.85i 1.63331i
\(94\) 1958.19 2.14864
\(95\) 0 0
\(96\) −2855.52 −3.03583
\(97\) − 1682.13i − 1.76076i −0.474265 0.880382i \(-0.657286\pi\)
0.474265 0.880382i \(-0.342714\pi\)
\(98\) 265.296i 0.273459i
\(99\) −277.667 −0.281885
\(100\) 0 0
\(101\) −434.167 −0.427734 −0.213867 0.976863i \(-0.568606\pi\)
−0.213867 + 0.976863i \(0.568606\pi\)
\(102\) 938.572i 0.911102i
\(103\) 345.577i 0.330589i 0.986244 + 0.165295i \(0.0528575\pi\)
−0.986244 + 0.165295i \(0.947142\pi\)
\(104\) −2209.85 −2.08359
\(105\) 0 0
\(106\) 828.479 0.759142
\(107\) − 217.119i − 0.196165i −0.995178 0.0980825i \(-0.968729\pi\)
0.995178 0.0980825i \(-0.0312709\pi\)
\(108\) 3207.29i 2.85761i
\(109\) −1734.41 −1.52409 −0.762047 0.647521i \(-0.775806\pi\)
−0.762047 + 0.647521i \(0.775806\pi\)
\(110\) 0 0
\(111\) −918.086 −0.785053
\(112\) 1538.35i 1.29786i
\(113\) − 1854.20i − 1.54362i −0.635855 0.771809i \(-0.719352\pi\)
0.635855 0.771809i \(-0.280648\pi\)
\(114\) 2022.74 1.66181
\(115\) 0 0
\(116\) 446.333 0.357250
\(117\) 162.902i 0.128720i
\(118\) − 3335.16i − 2.60191i
\(119\) 260.578 0.200732
\(120\) 0 0
\(121\) 1399.57 1.05152
\(122\) − 82.6476i − 0.0613325i
\(123\) − 52.9134i − 0.0387890i
\(124\) 6704.41 4.85543
\(125\) 0 0
\(126\) 201.387 0.142389
\(127\) − 1394.51i − 0.974352i −0.873304 0.487176i \(-0.838027\pi\)
0.873304 0.487176i \(-0.161973\pi\)
\(128\) 3550.45i 2.45171i
\(129\) −157.558 −0.107536
\(130\) 0 0
\(131\) 1762.42 1.17544 0.587722 0.809063i \(-0.300025\pi\)
0.587722 + 0.809063i \(0.300025\pi\)
\(132\) − 5186.54i − 3.41993i
\(133\) − 561.578i − 0.366128i
\(134\) 901.519 0.581189
\(135\) 0 0
\(136\) 2683.33 1.69186
\(137\) 922.949i 0.575568i 0.957695 + 0.287784i \(0.0929185\pi\)
−0.957695 + 0.287784i \(0.907081\pi\)
\(138\) − 651.345i − 0.401784i
\(139\) 196.039 0.119624 0.0598122 0.998210i \(-0.480950\pi\)
0.0598122 + 0.998210i \(0.480950\pi\)
\(140\) 0 0
\(141\) 1684.27 1.00597
\(142\) 5154.33i 3.04607i
\(143\) − 1601.97i − 0.936807i
\(144\) 1167.76 0.675790
\(145\) 0 0
\(146\) −803.954 −0.455724
\(147\) 228.186i 0.128030i
\(148\) 4201.94i 2.33376i
\(149\) −780.372 −0.429064 −0.214532 0.976717i \(-0.568823\pi\)
−0.214532 + 0.976717i \(0.568823\pi\)
\(150\) 0 0
\(151\) −2319.43 −1.25002 −0.625008 0.780618i \(-0.714904\pi\)
−0.625008 + 0.780618i \(0.714904\pi\)
\(152\) − 5782.91i − 3.08589i
\(153\) − 197.805i − 0.104520i
\(154\) −1980.43 −1.03628
\(155\) 0 0
\(156\) −3042.84 −1.56168
\(157\) − 1022.90i − 0.519977i −0.965612 0.259989i \(-0.916281\pi\)
0.965612 0.259989i \(-0.0837188\pi\)
\(158\) 4643.91i 2.33829i
\(159\) 712.589 0.355421
\(160\) 0 0
\(161\) −180.834 −0.0885201
\(162\) 3017.31i 1.46335i
\(163\) − 1350.63i − 0.649013i −0.945883 0.324507i \(-0.894802\pi\)
0.945883 0.324507i \(-0.105198\pi\)
\(164\) −242.177 −0.115310
\(165\) 0 0
\(166\) 3576.24 1.67211
\(167\) 1230.58i 0.570209i 0.958496 + 0.285105i \(0.0920284\pi\)
−0.958496 + 0.285105i \(0.907972\pi\)
\(168\) 2349.77i 1.07910i
\(169\) 1257.16 0.572215
\(170\) 0 0
\(171\) −426.294 −0.190641
\(172\) 721.117i 0.319678i
\(173\) − 2487.65i − 1.09325i −0.837377 0.546626i \(-0.815912\pi\)
0.837377 0.546626i \(-0.184088\pi\)
\(174\) 527.993 0.230040
\(175\) 0 0
\(176\) −11483.8 −4.91830
\(177\) − 2868.62i − 1.21819i
\(178\) − 247.833i − 0.104359i
\(179\) −1621.18 −0.676941 −0.338471 0.940977i \(-0.609910\pi\)
−0.338471 + 0.940977i \(0.609910\pi\)
\(180\) 0 0
\(181\) 2593.69 1.06512 0.532561 0.846392i \(-0.321230\pi\)
0.532561 + 0.846392i \(0.321230\pi\)
\(182\) 1161.88i 0.473210i
\(183\) − 71.0866i − 0.0287151i
\(184\) −1862.16 −0.746089
\(185\) 0 0
\(186\) 7931.03 3.12651
\(187\) 1945.21i 0.760682i
\(188\) − 7708.66i − 2.99049i
\(189\) 1053.36 0.405401
\(190\) 0 0
\(191\) −1823.08 −0.690645 −0.345323 0.938484i \(-0.612231\pi\)
−0.345323 + 0.938484i \(0.612231\pi\)
\(192\) 7273.09i 2.73380i
\(193\) − 1541.03i − 0.574744i −0.957819 0.287372i \(-0.907218\pi\)
0.957819 0.287372i \(-0.0927816\pi\)
\(194\) −9107.39 −3.37048
\(195\) 0 0
\(196\) 1044.37 0.380602
\(197\) − 701.243i − 0.253612i −0.991928 0.126806i \(-0.959527\pi\)
0.991928 0.126806i \(-0.0404725\pi\)
\(198\) 1503.35i 0.539587i
\(199\) −3294.96 −1.17374 −0.586868 0.809682i \(-0.699639\pi\)
−0.586868 + 0.809682i \(0.699639\pi\)
\(200\) 0 0
\(201\) 775.411 0.272106
\(202\) 2350.67i 0.818775i
\(203\) − 146.588i − 0.0506820i
\(204\) 3694.80 1.26808
\(205\) 0 0
\(206\) 1871.03 0.632819
\(207\) 137.272i 0.0460920i
\(208\) 6737.29i 2.24590i
\(209\) 4192.16 1.38746
\(210\) 0 0
\(211\) 4082.35 1.33195 0.665974 0.745975i \(-0.268016\pi\)
0.665974 + 0.745975i \(0.268016\pi\)
\(212\) − 3261.41i − 1.05658i
\(213\) 4433.33i 1.42613i
\(214\) −1175.53 −0.375502
\(215\) 0 0
\(216\) 10847.1 3.41691
\(217\) − 2201.91i − 0.688826i
\(218\) 9390.46i 2.91744i
\(219\) −691.494 −0.213364
\(220\) 0 0
\(221\) 1141.21 0.347359
\(222\) 4970.71i 1.50276i
\(223\) 747.161i 0.224366i 0.993688 + 0.112183i \(0.0357843\pi\)
−0.993688 + 0.112183i \(0.964216\pi\)
\(224\) 4292.30 1.28032
\(225\) 0 0
\(226\) −10039.1 −2.95481
\(227\) − 1665.67i − 0.487025i −0.969898 0.243513i \(-0.921700\pi\)
0.969898 0.243513i \(-0.0782997\pi\)
\(228\) − 7962.76i − 2.31292i
\(229\) 6628.35 1.91272 0.956362 0.292183i \(-0.0943816\pi\)
0.956362 + 0.292183i \(0.0943816\pi\)
\(230\) 0 0
\(231\) −1703.40 −0.485176
\(232\) − 1509.50i − 0.427172i
\(233\) − 432.431i − 0.121586i −0.998150 0.0607929i \(-0.980637\pi\)
0.998150 0.0607929i \(-0.0193629\pi\)
\(234\) 881.984 0.246398
\(235\) 0 0
\(236\) −13129.2 −3.62136
\(237\) 3994.30i 1.09476i
\(238\) − 1410.82i − 0.384244i
\(239\) −5580.44 −1.51033 −0.755165 0.655535i \(-0.772443\pi\)
−0.755165 + 0.655535i \(0.772443\pi\)
\(240\) 0 0
\(241\) −6296.87 −1.68306 −0.841529 0.540212i \(-0.818344\pi\)
−0.841529 + 0.540212i \(0.818344\pi\)
\(242\) − 7577.56i − 2.01283i
\(243\) − 1467.73i − 0.387468i
\(244\) −325.352 −0.0853629
\(245\) 0 0
\(246\) −286.485 −0.0742504
\(247\) − 2459.46i − 0.633569i
\(248\) − 22674.4i − 5.80575i
\(249\) 3075.98 0.782862
\(250\) 0 0
\(251\) 311.921 0.0784393 0.0392197 0.999231i \(-0.487513\pi\)
0.0392197 + 0.999231i \(0.487513\pi\)
\(252\) − 792.784i − 0.198177i
\(253\) − 1349.92i − 0.335451i
\(254\) −7550.17 −1.86512
\(255\) 0 0
\(256\) 6728.46 1.64269
\(257\) 7861.39i 1.90809i 0.299659 + 0.954046i \(0.403127\pi\)
−0.299659 + 0.954046i \(0.596873\pi\)
\(258\) 853.050i 0.205847i
\(259\) 1380.03 0.331085
\(260\) 0 0
\(261\) −111.275 −0.0263899
\(262\) − 9542.11i − 2.25005i
\(263\) 5227.09i 1.22554i 0.790262 + 0.612769i \(0.209944\pi\)
−0.790262 + 0.612769i \(0.790056\pi\)
\(264\) −17541.0 −4.08929
\(265\) 0 0
\(266\) −3040.50 −0.700846
\(267\) − 213.166i − 0.0488596i
\(268\) − 3548.94i − 0.808903i
\(269\) −1281.71 −0.290510 −0.145255 0.989394i \(-0.546400\pi\)
−0.145255 + 0.989394i \(0.546400\pi\)
\(270\) 0 0
\(271\) 4704.14 1.05445 0.527226 0.849725i \(-0.323232\pi\)
0.527226 + 0.849725i \(0.323232\pi\)
\(272\) − 8180.82i − 1.82366i
\(273\) 999.352i 0.221551i
\(274\) 4997.04 1.10176
\(275\) 0 0
\(276\) −2564.10 −0.559205
\(277\) − 8958.56i − 1.94321i −0.236619 0.971603i \(-0.576039\pi\)
0.236619 0.971603i \(-0.423961\pi\)
\(278\) − 1061.40i − 0.228987i
\(279\) −1671.47 −0.358668
\(280\) 0 0
\(281\) −370.904 −0.0787412 −0.0393706 0.999225i \(-0.512535\pi\)
−0.0393706 + 0.999225i \(0.512535\pi\)
\(282\) − 9119.02i − 1.92564i
\(283\) − 5822.26i − 1.22296i −0.791261 0.611479i \(-0.790575\pi\)
0.791261 0.611479i \(-0.209425\pi\)
\(284\) 20290.7 4.23954
\(285\) 0 0
\(286\) −8673.40 −1.79325
\(287\) 79.5374i 0.0163587i
\(288\) − 3258.29i − 0.666655i
\(289\) 3527.27 0.717946
\(290\) 0 0
\(291\) −7833.42 −1.57802
\(292\) 3164.86i 0.634279i
\(293\) 7443.79i 1.48420i 0.670289 + 0.742100i \(0.266170\pi\)
−0.670289 + 0.742100i \(0.733830\pi\)
\(294\) 1235.45 0.245077
\(295\) 0 0
\(296\) 14211.0 2.79053
\(297\) 7863.32i 1.53628i
\(298\) 4225.10i 0.821320i
\(299\) −791.973 −0.153181
\(300\) 0 0
\(301\) 236.834 0.0453518
\(302\) 12557.9i 2.39280i
\(303\) 2021.85i 0.383341i
\(304\) −17630.7 −3.32628
\(305\) 0 0
\(306\) −1070.96 −0.200074
\(307\) 761.674i 0.141600i 0.997491 + 0.0707998i \(0.0225551\pi\)
−0.997491 + 0.0707998i \(0.977445\pi\)
\(308\) 7796.21i 1.44231i
\(309\) 1609.30 0.296278
\(310\) 0 0
\(311\) 7718.69 1.40735 0.703677 0.710520i \(-0.251540\pi\)
0.703677 + 0.710520i \(0.251540\pi\)
\(312\) 10290.9i 1.86734i
\(313\) 8556.00i 1.54509i 0.634959 + 0.772546i \(0.281017\pi\)
−0.634959 + 0.772546i \(0.718983\pi\)
\(314\) −5538.21 −0.995348
\(315\) 0 0
\(316\) 18281.3 3.25444
\(317\) 7780.95i 1.37862i 0.724468 + 0.689309i \(0.242086\pi\)
−0.724468 + 0.689309i \(0.757914\pi\)
\(318\) − 3858.11i − 0.680352i
\(319\) 1094.28 0.192062
\(320\) 0 0
\(321\) −1011.09 −0.175805
\(322\) 979.076i 0.169446i
\(323\) 2986.42i 0.514455i
\(324\) 11878.0 2.03670
\(325\) 0 0
\(326\) −7312.58 −1.24235
\(327\) 8076.89i 1.36591i
\(328\) 819.045i 0.137879i
\(329\) −2531.73 −0.424252
\(330\) 0 0
\(331\) −4932.12 −0.819015 −0.409507 0.912307i \(-0.634299\pi\)
−0.409507 + 0.912307i \(0.634299\pi\)
\(332\) − 14078.3i − 2.32725i
\(333\) − 1047.58i − 0.172394i
\(334\) 6662.61 1.09150
\(335\) 0 0
\(336\) 7163.88 1.16316
\(337\) 7121.13i 1.15108i 0.817775 + 0.575538i \(0.195207\pi\)
−0.817775 + 0.575538i \(0.804793\pi\)
\(338\) − 6806.52i − 1.09534i
\(339\) −8634.76 −1.38341
\(340\) 0 0
\(341\) 16437.2 2.61034
\(342\) 2308.05i 0.364927i
\(343\) − 343.000i − 0.0539949i
\(344\) 2438.83 0.382246
\(345\) 0 0
\(346\) −13468.7 −2.09272
\(347\) − 9540.58i − 1.47598i −0.674811 0.737991i \(-0.735775\pi\)
0.674811 0.737991i \(-0.264225\pi\)
\(348\) − 2078.51i − 0.320172i
\(349\) −1281.65 −0.196576 −0.0982880 0.995158i \(-0.531337\pi\)
−0.0982880 + 0.995158i \(0.531337\pi\)
\(350\) 0 0
\(351\) 4613.25 0.701530
\(352\) 32041.9i 4.85182i
\(353\) 5798.07i 0.874221i 0.899408 + 0.437110i \(0.143998\pi\)
−0.899408 + 0.437110i \(0.856002\pi\)
\(354\) −15531.3 −2.33187
\(355\) 0 0
\(356\) −975.627 −0.145247
\(357\) − 1213.47i − 0.179899i
\(358\) 8777.40i 1.29581i
\(359\) −2267.29 −0.333323 −0.166662 0.986014i \(-0.553299\pi\)
−0.166662 + 0.986014i \(0.553299\pi\)
\(360\) 0 0
\(361\) −422.886 −0.0616541
\(362\) − 14042.8i − 2.03887i
\(363\) − 6517.58i − 0.942381i
\(364\) 4573.88 0.658616
\(365\) 0 0
\(366\) −384.878 −0.0549669
\(367\) 7372.85i 1.04866i 0.851514 + 0.524332i \(0.175685\pi\)
−0.851514 + 0.524332i \(0.824315\pi\)
\(368\) 5677.28i 0.804209i
\(369\) 60.3769 0.00851788
\(370\) 0 0
\(371\) −1071.14 −0.149894
\(372\) − 31221.4i − 4.35150i
\(373\) 6447.14i 0.894961i 0.894294 + 0.447480i \(0.147679\pi\)
−0.894294 + 0.447480i \(0.852321\pi\)
\(374\) 10531.8 1.45611
\(375\) 0 0
\(376\) −26070.8 −3.57579
\(377\) − 641.989i − 0.0877032i
\(378\) − 5703.12i − 0.776024i
\(379\) 4247.57 0.575680 0.287840 0.957678i \(-0.407063\pi\)
0.287840 + 0.957678i \(0.407063\pi\)
\(380\) 0 0
\(381\) −6494.03 −0.873226
\(382\) 9870.53i 1.32204i
\(383\) − 6681.86i − 0.891454i −0.895169 0.445727i \(-0.852945\pi\)
0.895169 0.445727i \(-0.147055\pi\)
\(384\) 16533.9 2.19725
\(385\) 0 0
\(386\) −8343.45 −1.10018
\(387\) − 179.781i − 0.0236145i
\(388\) 35852.4i 4.69105i
\(389\) 6371.78 0.830494 0.415247 0.909709i \(-0.363695\pi\)
0.415247 + 0.909709i \(0.363695\pi\)
\(390\) 0 0
\(391\) 961.661 0.124382
\(392\) − 3532.08i − 0.455094i
\(393\) − 8207.33i − 1.05345i
\(394\) −3796.68 −0.485467
\(395\) 0 0
\(396\) 5918.11 0.751001
\(397\) − 4247.93i − 0.537021i −0.963277 0.268510i \(-0.913469\pi\)
0.963277 0.268510i \(-0.0865314\pi\)
\(398\) 17839.6i 2.24678i
\(399\) −2615.19 −0.328128
\(400\) 0 0
\(401\) −8833.62 −1.10008 −0.550038 0.835140i \(-0.685387\pi\)
−0.550038 + 0.835140i \(0.685387\pi\)
\(402\) − 4198.24i − 0.520869i
\(403\) − 9643.37i − 1.19199i
\(404\) 9253.70 1.13958
\(405\) 0 0
\(406\) −793.658 −0.0970162
\(407\) 10301.9i 1.25466i
\(408\) − 12495.9i − 1.51627i
\(409\) 319.205 0.0385908 0.0192954 0.999814i \(-0.493858\pi\)
0.0192954 + 0.999814i \(0.493858\pi\)
\(410\) 0 0
\(411\) 4298.04 0.515831
\(412\) − 7365.53i − 0.880761i
\(413\) 4312.00i 0.513752i
\(414\) 743.218 0.0882298
\(415\) 0 0
\(416\) 18798.3 2.21554
\(417\) − 912.924i − 0.107209i
\(418\) − 22697.3i − 2.65589i
\(419\) 12789.2 1.49115 0.745577 0.666420i \(-0.232174\pi\)
0.745577 + 0.666420i \(0.232174\pi\)
\(420\) 0 0
\(421\) −6747.40 −0.781112 −0.390556 0.920579i \(-0.627717\pi\)
−0.390556 + 0.920579i \(0.627717\pi\)
\(422\) − 22102.7i − 2.54963i
\(423\) 1921.84i 0.220906i
\(424\) −11030.1 −1.26337
\(425\) 0 0
\(426\) 24003.0 2.72992
\(427\) 106.855i 0.0121102i
\(428\) 4627.60i 0.522625i
\(429\) −7460.14 −0.839577
\(430\) 0 0
\(431\) −5184.75 −0.579444 −0.289722 0.957111i \(-0.593563\pi\)
−0.289722 + 0.957111i \(0.593563\pi\)
\(432\) − 33070.2i − 3.68308i
\(433\) − 4242.03i − 0.470806i −0.971898 0.235403i \(-0.924359\pi\)
0.971898 0.235403i \(-0.0756410\pi\)
\(434\) −11921.6 −1.31856
\(435\) 0 0
\(436\) 36966.7 4.06051
\(437\) − 2072.50i − 0.226868i
\(438\) 3743.90i 0.408425i
\(439\) 5434.12 0.590789 0.295394 0.955375i \(-0.404549\pi\)
0.295394 + 0.955375i \(0.404549\pi\)
\(440\) 0 0
\(441\) −260.372 −0.0281149
\(442\) − 6178.77i − 0.664919i
\(443\) − 11493.8i − 1.23270i −0.787472 0.616350i \(-0.788611\pi\)
0.787472 0.616350i \(-0.211389\pi\)
\(444\) 19567.8 2.09155
\(445\) 0 0
\(446\) 4045.29 0.429484
\(447\) 3634.08i 0.384532i
\(448\) − 10932.6i − 1.15294i
\(449\) 16849.3 1.77098 0.885489 0.464661i \(-0.153824\pi\)
0.885489 + 0.464661i \(0.153824\pi\)
\(450\) 0 0
\(451\) −593.745 −0.0619919
\(452\) 39520.0i 4.11253i
\(453\) 10801.2i 1.12028i
\(454\) −9018.32 −0.932270
\(455\) 0 0
\(456\) −26930.2 −2.76561
\(457\) − 15348.5i − 1.57106i −0.618826 0.785528i \(-0.712391\pi\)
0.618826 0.785528i \(-0.287609\pi\)
\(458\) − 35887.3i − 3.66136i
\(459\) −5601.69 −0.569639
\(460\) 0 0
\(461\) 14038.4 1.41830 0.709148 0.705059i \(-0.249080\pi\)
0.709148 + 0.705059i \(0.249080\pi\)
\(462\) 9222.58i 0.928730i
\(463\) − 8661.23i − 0.869377i −0.900581 0.434689i \(-0.856858\pi\)
0.900581 0.434689i \(-0.143142\pi\)
\(464\) −4602.12 −0.460448
\(465\) 0 0
\(466\) −2341.27 −0.232741
\(467\) − 7014.71i − 0.695079i −0.937665 0.347539i \(-0.887017\pi\)
0.937665 0.347539i \(-0.112983\pi\)
\(468\) − 3472.04i − 0.342938i
\(469\) −1165.57 −0.114757
\(470\) 0 0
\(471\) −4763.50 −0.466010
\(472\) 44403.3i 4.33014i
\(473\) 1767.96i 0.171863i
\(474\) 21626.0 2.09560
\(475\) 0 0
\(476\) −5553.88 −0.534793
\(477\) 813.100i 0.0780488i
\(478\) 30213.7i 2.89109i
\(479\) −18134.7 −1.72984 −0.864922 0.501907i \(-0.832632\pi\)
−0.864922 + 0.501907i \(0.832632\pi\)
\(480\) 0 0
\(481\) 6043.91 0.572929
\(482\) 34092.6i 3.22173i
\(483\) 842.119i 0.0793328i
\(484\) −29830.0 −2.80146
\(485\) 0 0
\(486\) −7946.59 −0.741697
\(487\) − 16537.8i − 1.53881i −0.638761 0.769405i \(-0.720553\pi\)
0.638761 0.769405i \(-0.279447\pi\)
\(488\) 1100.35i 0.102070i
\(489\) −6289.67 −0.581654
\(490\) 0 0
\(491\) 220.608 0.0202768 0.0101384 0.999949i \(-0.496773\pi\)
0.0101384 + 0.999949i \(0.496773\pi\)
\(492\) 1127.78i 0.103342i
\(493\) 779.542i 0.0712146i
\(494\) −13316.0 −1.21279
\(495\) 0 0
\(496\) −69128.8 −6.25801
\(497\) − 6664.00i − 0.601451i
\(498\) − 16654.0i − 1.49856i
\(499\) −5939.04 −0.532801 −0.266401 0.963862i \(-0.585834\pi\)
−0.266401 + 0.963862i \(0.585834\pi\)
\(500\) 0 0
\(501\) 5730.62 0.511029
\(502\) − 1688.81i − 0.150150i
\(503\) − 11604.8i − 1.02869i −0.857584 0.514345i \(-0.828035\pi\)
0.857584 0.514345i \(-0.171965\pi\)
\(504\) −2681.21 −0.236965
\(505\) 0 0
\(506\) −7308.78 −0.642124
\(507\) − 5854.40i − 0.512826i
\(508\) 29722.2i 2.59588i
\(509\) 1867.67 0.162639 0.0813193 0.996688i \(-0.474087\pi\)
0.0813193 + 0.996688i \(0.474087\pi\)
\(510\) 0 0
\(511\) 1039.43 0.0899834
\(512\) − 8025.75i − 0.692757i
\(513\) 12072.3i 1.03900i
\(514\) 42563.2 3.65250
\(515\) 0 0
\(516\) 3358.14 0.286499
\(517\) − 18899.3i − 1.60772i
\(518\) − 7471.78i − 0.633767i
\(519\) −11584.6 −0.979786
\(520\) 0 0
\(521\) 6117.21 0.514395 0.257197 0.966359i \(-0.417201\pi\)
0.257197 + 0.966359i \(0.417201\pi\)
\(522\) 602.467i 0.0505158i
\(523\) − 16685.6i − 1.39505i −0.716561 0.697524i \(-0.754285\pi\)
0.716561 0.697524i \(-0.245715\pi\)
\(524\) −37563.7 −3.13164
\(525\) 0 0
\(526\) 28300.6 2.34594
\(527\) 11709.6i 0.967887i
\(528\) 53478.2i 4.40784i
\(529\) 11499.6 0.945149
\(530\) 0 0
\(531\) 3273.24 0.267508
\(532\) 11969.3i 0.975442i
\(533\) 348.338i 0.0283081i
\(534\) −1154.12 −0.0935278
\(535\) 0 0
\(536\) −12002.6 −0.967223
\(537\) 7549.59i 0.606683i
\(538\) 6939.44i 0.556097i
\(539\) 2560.49 0.204616
\(540\) 0 0
\(541\) 9309.03 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(542\) − 25469.2i − 2.01845i
\(543\) − 12078.4i − 0.954575i
\(544\) −22826.1 −1.79901
\(545\) 0 0
\(546\) 5410.70 0.424097
\(547\) − 10894.7i − 0.851598i −0.904818 0.425799i \(-0.859993\pi\)
0.904818 0.425799i \(-0.140007\pi\)
\(548\) − 19671.5i − 1.53344i
\(549\) 81.1134 0.00630571
\(550\) 0 0
\(551\) 1680.01 0.129893
\(552\) 8671.81i 0.668654i
\(553\) − 6004.07i − 0.461698i
\(554\) −48503.6 −3.71971
\(555\) 0 0
\(556\) −4178.31 −0.318705
\(557\) 7873.90i 0.598973i 0.954101 + 0.299486i \(0.0968153\pi\)
−0.954101 + 0.299486i \(0.903185\pi\)
\(558\) 9049.71i 0.686567i
\(559\) 1037.23 0.0784796
\(560\) 0 0
\(561\) 9058.55 0.681733
\(562\) 2008.15i 0.150728i
\(563\) 21770.7i 1.62971i 0.579666 + 0.814854i \(0.303183\pi\)
−0.579666 + 0.814854i \(0.696817\pi\)
\(564\) −35898.1 −2.68011
\(565\) 0 0
\(566\) −31522.9 −2.34100
\(567\) − 3901.06i − 0.288940i
\(568\) − 68623.3i − 5.06931i
\(569\) 12381.3 0.912213 0.456106 0.889925i \(-0.349244\pi\)
0.456106 + 0.889925i \(0.349244\pi\)
\(570\) 0 0
\(571\) −5768.38 −0.422765 −0.211383 0.977403i \(-0.567797\pi\)
−0.211383 + 0.977403i \(0.567797\pi\)
\(572\) 34143.9i 2.49585i
\(573\) 8489.81i 0.618965i
\(574\) 430.632 0.0313140
\(575\) 0 0
\(576\) −8298.97 −0.600330
\(577\) − 4733.38i − 0.341513i −0.985313 0.170757i \(-0.945379\pi\)
0.985313 0.170757i \(-0.0546212\pi\)
\(578\) − 19097.4i − 1.37430i
\(579\) −7176.34 −0.515093
\(580\) 0 0
\(581\) −4623.70 −0.330161
\(582\) 42411.8i 3.02066i
\(583\) − 7996.00i − 0.568028i
\(584\) 10703.6 0.758422
\(585\) 0 0
\(586\) 40302.3 2.84108
\(587\) 8441.67i 0.593569i 0.954944 + 0.296785i \(0.0959143\pi\)
−0.954944 + 0.296785i \(0.904086\pi\)
\(588\) − 4863.49i − 0.341100i
\(589\) 25235.6 1.76539
\(590\) 0 0
\(591\) −3265.59 −0.227290
\(592\) − 43326.0i − 3.00792i
\(593\) 18939.9i 1.31158i 0.754943 + 0.655791i \(0.227665\pi\)
−0.754943 + 0.655791i \(0.772335\pi\)
\(594\) 42573.7 2.94077
\(595\) 0 0
\(596\) 16632.6 1.14312
\(597\) 15344.2i 1.05192i
\(598\) 4287.91i 0.293220i
\(599\) −22655.3 −1.54536 −0.772681 0.634794i \(-0.781085\pi\)
−0.772681 + 0.634794i \(0.781085\pi\)
\(600\) 0 0
\(601\) −15947.4 −1.08237 −0.541187 0.840902i \(-0.682025\pi\)
−0.541187 + 0.840902i \(0.682025\pi\)
\(602\) − 1282.27i − 0.0868131i
\(603\) 884.784i 0.0597532i
\(604\) 49435.6 3.33031
\(605\) 0 0
\(606\) 10946.7 0.733796
\(607\) 25993.2i 1.73811i 0.494719 + 0.869053i \(0.335271\pi\)
−0.494719 + 0.869053i \(0.664729\pi\)
\(608\) 49193.1i 3.28132i
\(609\) −682.638 −0.0454218
\(610\) 0 0
\(611\) −11087.9 −0.734152
\(612\) 4215.96i 0.278464i
\(613\) 665.408i 0.0438427i 0.999760 + 0.0219213i \(0.00697834\pi\)
−0.999760 + 0.0219213i \(0.993022\pi\)
\(614\) 4123.87 0.271052
\(615\) 0 0
\(616\) 26366.9 1.72460
\(617\) − 18401.3i − 1.20066i −0.799752 0.600330i \(-0.795036\pi\)
0.799752 0.600330i \(-0.204964\pi\)
\(618\) − 8713.10i − 0.567140i
\(619\) 11150.6 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(620\) 0 0
\(621\) 3887.43 0.251203
\(622\) − 41790.6i − 2.69397i
\(623\) 320.422i 0.0206059i
\(624\) 31374.6 2.01280
\(625\) 0 0
\(626\) 46324.0 2.95764
\(627\) − 19522.3i − 1.24345i
\(628\) 21801.8i 1.38533i
\(629\) −7338.88 −0.465215
\(630\) 0 0
\(631\) 5381.79 0.339534 0.169767 0.985484i \(-0.445699\pi\)
0.169767 + 0.985484i \(0.445699\pi\)
\(632\) − 61827.6i − 3.89141i
\(633\) − 19010.9i − 1.19371i
\(634\) 42127.7 2.63897
\(635\) 0 0
\(636\) −15187.9 −0.946918
\(637\) − 1502.19i − 0.0934361i
\(638\) − 5924.64i − 0.367647i
\(639\) −5058.65 −0.313172
\(640\) 0 0
\(641\) −19455.1 −1.19880 −0.599398 0.800451i \(-0.704593\pi\)
−0.599398 + 0.800451i \(0.704593\pi\)
\(642\) 5474.26i 0.336529i
\(643\) − 14695.8i − 0.901317i −0.892696 0.450658i \(-0.851189\pi\)
0.892696 0.450658i \(-0.148811\pi\)
\(644\) 3854.25 0.235837
\(645\) 0 0
\(646\) 16169.1 0.984777
\(647\) 12694.8i 0.771383i 0.922628 + 0.385691i \(0.126037\pi\)
−0.922628 + 0.385691i \(0.873963\pi\)
\(648\) − 40171.6i − 2.43532i
\(649\) −32189.0 −1.94688
\(650\) 0 0
\(651\) −10254.0 −0.617334
\(652\) 28786.8i 1.72911i
\(653\) − 12385.6i − 0.742247i −0.928583 0.371124i \(-0.878973\pi\)
0.928583 0.371124i \(-0.121027\pi\)
\(654\) 43730.0 2.61465
\(655\) 0 0
\(656\) 2497.07 0.148619
\(657\) − 789.030i − 0.0468539i
\(658\) 13707.3i 0.812109i
\(659\) 2072.18 0.122489 0.0612447 0.998123i \(-0.480493\pi\)
0.0612447 + 0.998123i \(0.480493\pi\)
\(660\) 0 0
\(661\) 1074.36 0.0632193 0.0316096 0.999500i \(-0.489937\pi\)
0.0316096 + 0.999500i \(0.489937\pi\)
\(662\) 26703.6i 1.56777i
\(663\) − 5314.47i − 0.311307i
\(664\) −47613.0 −2.78275
\(665\) 0 0
\(666\) −5671.84 −0.329999
\(667\) − 540.982i − 0.0314047i
\(668\) − 26228.2i − 1.51916i
\(669\) 3479.42 0.201080
\(670\) 0 0
\(671\) −797.667 −0.0458921
\(672\) − 19988.6i − 1.14744i
\(673\) 26195.2i 1.50037i 0.661226 + 0.750186i \(0.270036\pi\)
−0.661226 + 0.750186i \(0.729964\pi\)
\(674\) 38555.3 2.20341
\(675\) 0 0
\(676\) −26794.7 −1.52450
\(677\) 4228.44i 0.240047i 0.992771 + 0.120024i \(0.0382970\pi\)
−0.992771 + 0.120024i \(0.961703\pi\)
\(678\) 46750.4i 2.64814i
\(679\) 11774.9 0.665506
\(680\) 0 0
\(681\) −7756.80 −0.436478
\(682\) − 88994.5i − 4.99674i
\(683\) 27525.5i 1.54207i 0.636792 + 0.771036i \(0.280261\pi\)
−0.636792 + 0.771036i \(0.719739\pi\)
\(684\) 9085.91 0.507907
\(685\) 0 0
\(686\) −1857.08 −0.103358
\(687\) − 30867.3i − 1.71421i
\(688\) − 7435.40i − 0.412023i
\(689\) −4691.09 −0.259385
\(690\) 0 0
\(691\) −33324.4 −1.83462 −0.917309 0.398177i \(-0.869643\pi\)
−0.917309 + 0.398177i \(0.869643\pi\)
\(692\) 53021.1i 2.91266i
\(693\) − 1943.67i − 0.106542i
\(694\) −51654.8 −2.82534
\(695\) 0 0
\(696\) −7029.54 −0.382836
\(697\) − 422.973i − 0.0229860i
\(698\) 6939.12i 0.376289i
\(699\) −2013.77 −0.108967
\(700\) 0 0
\(701\) −33262.9 −1.79219 −0.896094 0.443864i \(-0.853607\pi\)
−0.896094 + 0.443864i \(0.853607\pi\)
\(702\) − 24977.1i − 1.34288i
\(703\) 15816.2i 0.848534i
\(704\) 81611.8 4.36912
\(705\) 0 0
\(706\) 31392.0 1.67345
\(707\) − 3039.17i − 0.161668i
\(708\) 61141.0i 3.24551i
\(709\) −13703.0 −0.725851 −0.362926 0.931818i \(-0.618222\pi\)
−0.362926 + 0.931818i \(0.618222\pi\)
\(710\) 0 0
\(711\) −4557.70 −0.240404
\(712\) 3299.58i 0.173676i
\(713\) − 8126.14i − 0.426825i
\(714\) −6570.00 −0.344364
\(715\) 0 0
\(716\) 34553.3 1.80352
\(717\) 25987.3i 1.35358i
\(718\) 12275.6i 0.638052i
\(719\) 8074.93 0.418838 0.209419 0.977826i \(-0.432843\pi\)
0.209419 + 0.977826i \(0.432843\pi\)
\(720\) 0 0
\(721\) −2419.04 −0.124951
\(722\) 2289.59i 0.118019i
\(723\) 29323.6i 1.50838i
\(724\) −55281.1 −2.83771
\(725\) 0 0
\(726\) −35287.6 −1.80392
\(727\) 3668.70i 0.187159i 0.995612 + 0.0935794i \(0.0298309\pi\)
−0.995612 + 0.0935794i \(0.970169\pi\)
\(728\) − 15468.9i − 0.787523i
\(729\) −21881.9 −1.11172
\(730\) 0 0
\(731\) −1259.46 −0.0637250
\(732\) 1515.12i 0.0765033i
\(733\) − 14980.3i − 0.754857i −0.926039 0.377428i \(-0.876808\pi\)
0.926039 0.377428i \(-0.123192\pi\)
\(734\) 39918.2 2.00737
\(735\) 0 0
\(736\) 15840.7 0.793338
\(737\) − 8700.94i − 0.434875i
\(738\) − 326.894i − 0.0163050i
\(739\) −6530.59 −0.325077 −0.162538 0.986702i \(-0.551968\pi\)
−0.162538 + 0.986702i \(0.551968\pi\)
\(740\) 0 0
\(741\) −11453.3 −0.567812
\(742\) 5799.36i 0.286929i
\(743\) 25952.0i 1.28141i 0.767788 + 0.640704i \(0.221357\pi\)
−0.767788 + 0.640704i \(0.778643\pi\)
\(744\) −105591. −5.20318
\(745\) 0 0
\(746\) 34906.2 1.71315
\(747\) 3509.85i 0.171913i
\(748\) − 41459.6i − 2.02662i
\(749\) 1519.83 0.0741434
\(750\) 0 0
\(751\) −14093.9 −0.684813 −0.342407 0.939552i \(-0.611242\pi\)
−0.342407 + 0.939552i \(0.611242\pi\)
\(752\) 79483.6i 3.85435i
\(753\) − 1452.57i − 0.0702983i
\(754\) −3475.87 −0.167883
\(755\) 0 0
\(756\) −22451.0 −1.08007
\(757\) 2554.41i 0.122644i 0.998118 + 0.0613220i \(0.0195316\pi\)
−0.998118 + 0.0613220i \(0.980468\pi\)
\(758\) − 22997.2i − 1.10197i
\(759\) −6286.40 −0.300635
\(760\) 0 0
\(761\) 2219.08 0.105705 0.0528527 0.998602i \(-0.483169\pi\)
0.0528527 + 0.998602i \(0.483169\pi\)
\(762\) 35160.1i 1.67154i
\(763\) − 12140.9i − 0.576054i
\(764\) 38856.5 1.84003
\(765\) 0 0
\(766\) −36177.0 −1.70643
\(767\) 18884.6i 0.889028i
\(768\) − 31333.5i − 1.47220i
\(769\) 22466.2 1.05352 0.526758 0.850015i \(-0.323408\pi\)
0.526758 + 0.850015i \(0.323408\pi\)
\(770\) 0 0
\(771\) 36609.3 1.71006
\(772\) 32845.0i 1.53124i
\(773\) 9674.79i 0.450165i 0.974340 + 0.225083i \(0.0722652\pi\)
−0.974340 + 0.225083i \(0.927735\pi\)
\(774\) −973.374 −0.0452031
\(775\) 0 0
\(776\) 121253. 5.60920
\(777\) − 6426.60i − 0.296722i
\(778\) − 34498.2i − 1.58974i
\(779\) −911.560 −0.0419256
\(780\) 0 0
\(781\) 49746.6 2.27922
\(782\) − 5206.64i − 0.238093i
\(783\) 3151.23i 0.143826i
\(784\) −10768.5 −0.490546
\(785\) 0 0
\(786\) −44436.2 −2.01652
\(787\) 20942.8i 0.948577i 0.880370 + 0.474288i \(0.157295\pi\)
−0.880370 + 0.474288i \(0.842705\pi\)
\(788\) 14946.1i 0.675676i
\(789\) 24341.8 1.09834
\(790\) 0 0
\(791\) 12979.4 0.583433
\(792\) − 20015.1i − 0.897989i
\(793\) 467.975i 0.0209562i
\(794\) −22999.2 −1.02797
\(795\) 0 0
\(796\) 70227.8 3.12708
\(797\) − 23526.6i − 1.04561i −0.852451 0.522807i \(-0.824885\pi\)
0.852451 0.522807i \(-0.175115\pi\)
\(798\) 14159.2i 0.628107i
\(799\) 13463.5 0.596127
\(800\) 0 0
\(801\) 243.233 0.0107294
\(802\) 47827.1i 2.10578i
\(803\) 7759.29i 0.340996i
\(804\) −16526.9 −0.724948
\(805\) 0 0
\(806\) −52211.3 −2.28172
\(807\) 5968.72i 0.260358i
\(808\) − 31296.1i − 1.36262i
\(809\) 18202.2 0.791047 0.395523 0.918456i \(-0.370563\pi\)
0.395523 + 0.918456i \(0.370563\pi\)
\(810\) 0 0
\(811\) −2510.24 −0.108689 −0.0543443 0.998522i \(-0.517307\pi\)
−0.0543443 + 0.998522i \(0.517307\pi\)
\(812\) 3124.33i 0.135028i
\(813\) − 21906.5i − 0.945012i
\(814\) 55776.7 2.40168
\(815\) 0 0
\(816\) −38096.9 −1.63438
\(817\) 2714.30i 0.116232i
\(818\) − 1728.24i − 0.0738711i
\(819\) −1140.31 −0.0486516
\(820\) 0 0
\(821\) 17899.6 0.760903 0.380451 0.924801i \(-0.375769\pi\)
0.380451 + 0.924801i \(0.375769\pi\)
\(822\) − 23270.5i − 0.987411i
\(823\) 14039.5i 0.594637i 0.954778 + 0.297318i \(0.0960923\pi\)
−0.954778 + 0.297318i \(0.903908\pi\)
\(824\) −24910.3 −1.05315
\(825\) 0 0
\(826\) 23346.1 0.983431
\(827\) − 15127.4i − 0.636073i −0.948079 0.318036i \(-0.896977\pi\)
0.948079 0.318036i \(-0.103023\pi\)
\(828\) − 2925.77i − 0.122799i
\(829\) −21986.5 −0.921136 −0.460568 0.887624i \(-0.652354\pi\)
−0.460568 + 0.887624i \(0.652354\pi\)
\(830\) 0 0
\(831\) −41718.7 −1.74152
\(832\) − 47880.0i − 1.99512i
\(833\) 1824.04i 0.0758696i
\(834\) −4942.76 −0.205220
\(835\) 0 0
\(836\) −89350.6 −3.69648
\(837\) 47334.8i 1.95476i
\(838\) − 69243.5i − 2.85439i
\(839\) 2276.89 0.0936914 0.0468457 0.998902i \(-0.485083\pi\)
0.0468457 + 0.998902i \(0.485083\pi\)
\(840\) 0 0
\(841\) −23950.5 −0.982019
\(842\) 36531.8i 1.49521i
\(843\) 1727.25i 0.0705688i
\(844\) −87010.1 −3.54859
\(845\) 0 0
\(846\) 10405.3 0.422861
\(847\) 9796.97i 0.397436i
\(848\) 33628.2i 1.36179i
\(849\) −27113.4 −1.09603
\(850\) 0 0
\(851\) 5093.00 0.205154
\(852\) − 94490.6i − 3.79952i
\(853\) 13342.6i 0.535570i 0.963479 + 0.267785i \(0.0862917\pi\)
−0.963479 + 0.267785i \(0.913708\pi\)
\(854\) 578.533 0.0231815
\(855\) 0 0
\(856\) 15650.6 0.624915
\(857\) − 18690.9i − 0.745003i −0.928032 0.372502i \(-0.878500\pi\)
0.928032 0.372502i \(-0.121500\pi\)
\(858\) 40390.8i 1.60713i
\(859\) −18318.9 −0.727628 −0.363814 0.931472i \(-0.618526\pi\)
−0.363814 + 0.931472i \(0.618526\pi\)
\(860\) 0 0
\(861\) 370.394 0.0146609
\(862\) 28071.3i 1.10918i
\(863\) 38133.1i 1.50413i 0.659087 + 0.752067i \(0.270943\pi\)
−0.659087 + 0.752067i \(0.729057\pi\)
\(864\) −92272.3 −3.63330
\(865\) 0 0
\(866\) −22967.3 −0.901223
\(867\) − 16426.0i − 0.643432i
\(868\) 46930.8i 1.83518i
\(869\) 44820.3 1.74962
\(870\) 0 0
\(871\) −5104.66 −0.198582
\(872\) − 125022.i − 4.85525i
\(873\) − 8938.33i − 0.346525i
\(874\) −11221.0 −0.434273
\(875\) 0 0
\(876\) 14738.3 0.568449
\(877\) 19707.5i 0.758807i 0.925231 + 0.379404i \(0.123871\pi\)
−0.925231 + 0.379404i \(0.876129\pi\)
\(878\) − 29421.5i − 1.13090i
\(879\) 34664.6 1.33016
\(880\) 0 0
\(881\) −14091.5 −0.538883 −0.269441 0.963017i \(-0.586839\pi\)
−0.269441 + 0.963017i \(0.586839\pi\)
\(882\) 1409.71i 0.0538178i
\(883\) 3115.87i 0.118751i 0.998236 + 0.0593757i \(0.0189110\pi\)
−0.998236 + 0.0593757i \(0.981089\pi\)
\(884\) −24323.5 −0.925438
\(885\) 0 0
\(886\) −62229.8 −2.35965
\(887\) − 38734.6i − 1.46627i −0.680084 0.733134i \(-0.738057\pi\)
0.680084 0.733134i \(-0.261943\pi\)
\(888\) − 66178.6i − 2.50091i
\(889\) 9761.57 0.368270
\(890\) 0 0
\(891\) 29121.3 1.09495
\(892\) − 15924.8i − 0.597759i
\(893\) − 29015.6i − 1.08731i
\(894\) 19675.7 0.736077
\(895\) 0 0
\(896\) −24853.1 −0.926658
\(897\) 3688.10i 0.137282i
\(898\) − 91225.8i − 3.39003i
\(899\) 6587.21 0.244378
\(900\) 0 0
\(901\) 5696.21 0.210619
\(902\) 3214.66i 0.118666i
\(903\) − 1102.90i − 0.0406449i
\(904\) 133657. 4.91744
\(905\) 0 0
\(906\) 58480.2 2.14445
\(907\) − 19242.9i − 0.704464i −0.935913 0.352232i \(-0.885423\pi\)
0.935913 0.352232i \(-0.114577\pi\)
\(908\) 35501.7i 1.29754i
\(909\) −2307.03 −0.0841799
\(910\) 0 0
\(911\) 34613.3 1.25882 0.629412 0.777072i \(-0.283296\pi\)
0.629412 + 0.777072i \(0.283296\pi\)
\(912\) 82103.6i 2.98105i
\(913\) − 34515.8i − 1.25116i
\(914\) −83100.1 −3.00734
\(915\) 0 0
\(916\) −141275. −5.09591
\(917\) 12336.9i 0.444276i
\(918\) 30328.7i 1.09041i
\(919\) −25826.4 −0.927022 −0.463511 0.886091i \(-0.653411\pi\)
−0.463511 + 0.886091i \(0.653411\pi\)
\(920\) 0 0
\(921\) 3547.01 0.126903
\(922\) − 76007.0i − 2.71492i
\(923\) − 29185.3i − 1.04079i
\(924\) 36305.8 1.29261
\(925\) 0 0
\(926\) −46893.8 −1.66417
\(927\) 1836.29i 0.0650613i
\(928\) 12840.8i 0.454224i
\(929\) −19451.6 −0.686960 −0.343480 0.939160i \(-0.611606\pi\)
−0.343480 + 0.939160i \(0.611606\pi\)
\(930\) 0 0
\(931\) 3931.04 0.138383
\(932\) 9216.70i 0.323930i
\(933\) − 35944.8i − 1.26129i
\(934\) −37979.1 −1.33053
\(935\) 0 0
\(936\) −11742.5 −0.410059
\(937\) − 34469.1i − 1.20177i −0.799336 0.600884i \(-0.794815\pi\)
0.799336 0.600884i \(-0.205185\pi\)
\(938\) 6310.63i 0.219669i
\(939\) 39844.1 1.38473
\(940\) 0 0
\(941\) 14156.4 0.490419 0.245209 0.969470i \(-0.421143\pi\)
0.245209 + 0.969470i \(0.421143\pi\)
\(942\) 25790.6i 0.892042i
\(943\) 293.532i 0.0101365i
\(944\) 135375. 4.66746
\(945\) 0 0
\(946\) 9572.13 0.328982
\(947\) 38092.4i 1.30711i 0.756877 + 0.653557i \(0.226724\pi\)
−0.756877 + 0.653557i \(0.773276\pi\)
\(948\) − 85133.3i − 2.91667i
\(949\) 4552.22 0.155713
\(950\) 0 0
\(951\) 36234.8 1.23553
\(952\) 18783.3i 0.639464i
\(953\) 5037.40i 0.171225i 0.996329 + 0.0856126i \(0.0272847\pi\)
−0.996329 + 0.0856126i \(0.972715\pi\)
\(954\) 4402.30 0.149402
\(955\) 0 0
\(956\) 118940. 4.02384
\(957\) − 5095.88i − 0.172128i
\(958\) 98185.1i 3.31129i
\(959\) −6460.64 −0.217544
\(960\) 0 0
\(961\) 69156.0 2.32137
\(962\) − 32723.0i − 1.09671i
\(963\) − 1153.71i − 0.0386060i
\(964\) 134210. 4.48403
\(965\) 0 0
\(966\) 4559.41 0.151860
\(967\) − 11495.3i − 0.382278i −0.981563 0.191139i \(-0.938782\pi\)
0.981563 0.191139i \(-0.0612181\pi\)
\(968\) 100885.i 3.34977i
\(969\) 13907.3 0.461061
\(970\) 0 0
\(971\) 22352.7 0.738757 0.369379 0.929279i \(-0.379571\pi\)
0.369379 + 0.929279i \(0.379571\pi\)
\(972\) 31282.7i 1.03230i
\(973\) 1372.27i 0.0452138i
\(974\) −89539.4 −2.94561
\(975\) 0 0
\(976\) 3354.69 0.110022
\(977\) − 14345.7i − 0.469765i −0.972024 0.234882i \(-0.924530\pi\)
0.972024 0.234882i \(-0.0754705\pi\)
\(978\) 34053.6i 1.11341i
\(979\) −2391.94 −0.0780867
\(980\) 0 0
\(981\) −9216.15 −0.299948
\(982\) − 1194.42i − 0.0388141i
\(983\) 34460.9i 1.11814i 0.829120 + 0.559070i \(0.188842\pi\)
−0.829120 + 0.559070i \(0.811158\pi\)
\(984\) 3814.17 0.123568
\(985\) 0 0
\(986\) 4220.61 0.136320
\(987\) 11789.9i 0.380220i
\(988\) 52420.2i 1.68796i
\(989\) 874.036 0.0281019
\(990\) 0 0
\(991\) −35189.6 −1.12799 −0.563993 0.825780i \(-0.690735\pi\)
−0.563993 + 0.825780i \(0.690735\pi\)
\(992\) 192883.i 6.17342i
\(993\) 22968.2i 0.734011i
\(994\) −36080.3 −1.15131
\(995\) 0 0
\(996\) −65560.6 −2.08571
\(997\) 50730.0i 1.61147i 0.592277 + 0.805734i \(0.298229\pi\)
−0.592277 + 0.805734i \(0.701771\pi\)
\(998\) 32155.2i 1.01990i
\(999\) −29666.8 −0.939554
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 175.4.b.c.99.1 4
5.2 odd 4 35.4.a.b.1.2 2
5.3 odd 4 175.4.a.c.1.1 2
5.4 even 2 inner 175.4.b.c.99.4 4
15.2 even 4 315.4.a.f.1.1 2
15.8 even 4 1575.4.a.z.1.2 2
20.7 even 4 560.4.a.r.1.2 2
35.2 odd 12 245.4.e.h.116.1 4
35.12 even 12 245.4.e.i.116.1 4
35.13 even 4 1225.4.a.m.1.1 2
35.17 even 12 245.4.e.i.226.1 4
35.27 even 4 245.4.a.k.1.2 2
35.32 odd 12 245.4.e.h.226.1 4
40.27 even 4 2240.4.a.bo.1.1 2
40.37 odd 4 2240.4.a.bn.1.2 2
105.62 odd 4 2205.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 5.2 odd 4
175.4.a.c.1.1 2 5.3 odd 4
175.4.b.c.99.1 4 1.1 even 1 trivial
175.4.b.c.99.4 4 5.4 even 2 inner
245.4.a.k.1.2 2 35.27 even 4
245.4.e.h.116.1 4 35.2 odd 12
245.4.e.h.226.1 4 35.32 odd 12
245.4.e.i.116.1 4 35.12 even 12
245.4.e.i.226.1 4 35.17 even 12
315.4.a.f.1.1 2 15.2 even 4
560.4.a.r.1.2 2 20.7 even 4
1225.4.a.m.1.1 2 35.13 even 4
1575.4.a.z.1.2 2 15.8 even 4
2205.4.a.u.1.1 2 105.62 odd 4
2240.4.a.bn.1.2 2 40.37 odd 4
2240.4.a.bo.1.1 2 40.27 even 4