Properties

Label 175.4.b.c.99.2
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 99.2
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 175.99
Dual form 175.4.b.c.99.3

$q$-expansion

\(f(q)\) \(=\) \(q-2.58579i q^{2} +6.65685i q^{3} +1.31371 q^{4} +17.2132 q^{6} +7.00000i q^{7} -24.0833i q^{8} -17.3137 q^{9} +O(q^{10})\) \(q-2.58579i q^{2} +6.65685i q^{3} +1.31371 q^{4} +17.2132 q^{6} +7.00000i q^{7} -24.0833i q^{8} -17.3137 q^{9} +38.2548 q^{11} +8.74517i q^{12} +19.3431i q^{13} +18.1005 q^{14} -51.7645 q^{16} +87.2254i q^{17} +44.7696i q^{18} +44.2254 q^{19} -46.5980 q^{21} -98.9188i q^{22} +218.167i q^{23} +160.319 q^{24} +50.0172 q^{26} +64.4802i q^{27} +9.19596i q^{28} +46.9411 q^{29} +194.558 q^{31} -58.8141i q^{32} +254.657i q^{33} +225.546 q^{34} -22.7452 q^{36} -366.853i q^{37} -114.357i q^{38} -128.765 q^{39} -339.362 q^{41} +120.492i q^{42} -226.167i q^{43} +50.2557 q^{44} +564.132 q^{46} -11.6762i q^{47} -344.589i q^{48} -49.0000 q^{49} -580.647 q^{51} +25.4113i q^{52} -209.019i q^{53} +166.732 q^{54} +168.583 q^{56} +294.402i q^{57} -121.380i q^{58} +616.000 q^{59} +320.735 q^{61} -503.087i q^{62} -121.196i q^{63} -566.197 q^{64} +658.488 q^{66} -14.5097i q^{67} +114.589i q^{68} -1452.30 q^{69} -952.000 q^{71} +416.971i q^{72} +824.489i q^{73} -948.603 q^{74} +58.0993 q^{76} +267.784i q^{77} +332.958i q^{78} -156.275 q^{79} -896.706 q^{81} +877.519i q^{82} -1036.53i q^{83} -61.2162 q^{84} -584.818 q^{86} +312.480i q^{87} -921.301i q^{88} +170.225 q^{89} -135.402 q^{91} +286.607i q^{92} +1295.15i q^{93} -30.1921 q^{94} +391.517 q^{96} -1059.87i q^{97} +126.704i q^{98} -662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 40 q^{4} - 16 q^{6} - 24 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.58579i − 0.914214i −0.889412 0.457107i \(-0.848886\pi\)
0.889412 0.457107i \(-0.151114\pi\)
\(3\) 6.65685i 1.28111i 0.767911 + 0.640556i \(0.221296\pi\)
−0.767911 + 0.640556i \(0.778704\pi\)
\(4\) 1.31371 0.164214
\(5\) 0 0
\(6\) 17.2132 1.17121
\(7\) 7.00000i 0.377964i
\(8\) − 24.0833i − 1.06434i
\(9\) −17.3137 −0.641248
\(10\) 0 0
\(11\) 38.2548 1.04857 0.524285 0.851543i \(-0.324333\pi\)
0.524285 + 0.851543i \(0.324333\pi\)
\(12\) 8.74517i 0.210376i
\(13\) 19.3431i 0.412679i 0.978480 + 0.206339i \(0.0661551\pi\)
−0.978480 + 0.206339i \(0.933845\pi\)
\(14\) 18.1005 0.345540
\(15\) 0 0
\(16\) −51.7645 −0.808820
\(17\) 87.2254i 1.24443i 0.782847 + 0.622214i \(0.213767\pi\)
−0.782847 + 0.622214i \(0.786233\pi\)
\(18\) 44.7696i 0.586238i
\(19\) 44.2254 0.534000 0.267000 0.963697i \(-0.413968\pi\)
0.267000 + 0.963697i \(0.413968\pi\)
\(20\) 0 0
\(21\) −46.5980 −0.484215
\(22\) − 98.9188i − 0.958617i
\(23\) 218.167i 1.97786i 0.148371 + 0.988932i \(0.452597\pi\)
−0.148371 + 0.988932i \(0.547403\pi\)
\(24\) 160.319 1.36354
\(25\) 0 0
\(26\) 50.0172 0.377276
\(27\) 64.4802i 0.459601i
\(28\) 9.19596i 0.0620669i
\(29\) 46.9411 0.300578 0.150289 0.988642i \(-0.451980\pi\)
0.150289 + 0.988642i \(0.451980\pi\)
\(30\) 0 0
\(31\) 194.558 1.12722 0.563609 0.826042i \(-0.309413\pi\)
0.563609 + 0.826042i \(0.309413\pi\)
\(32\) − 58.8141i − 0.324905i
\(33\) 254.657i 1.34334i
\(34\) 225.546 1.13767
\(35\) 0 0
\(36\) −22.7452 −0.105302
\(37\) − 366.853i − 1.63001i −0.579457 0.815003i \(-0.696735\pi\)
0.579457 0.815003i \(-0.303265\pi\)
\(38\) − 114.357i − 0.488190i
\(39\) −128.765 −0.528688
\(40\) 0 0
\(41\) −339.362 −1.29267 −0.646336 0.763053i \(-0.723699\pi\)
−0.646336 + 0.763053i \(0.723699\pi\)
\(42\) 120.492i 0.442676i
\(43\) − 226.167i − 0.802095i −0.916057 0.401047i \(-0.868646\pi\)
0.916057 0.401047i \(-0.131354\pi\)
\(44\) 50.2557 0.172189
\(45\) 0 0
\(46\) 564.132 1.80819
\(47\) − 11.6762i − 0.0362372i −0.999836 0.0181186i \(-0.994232\pi\)
0.999836 0.0181186i \(-0.00576764\pi\)
\(48\) − 344.589i − 1.03619i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −580.647 −1.59425
\(52\) 25.4113i 0.0677674i
\(53\) − 209.019i − 0.541717i −0.962619 0.270859i \(-0.912692\pi\)
0.962619 0.270859i \(-0.0873076\pi\)
\(54\) 166.732 0.420173
\(55\) 0 0
\(56\) 168.583 0.402283
\(57\) 294.402i 0.684114i
\(58\) − 121.380i − 0.274792i
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 0 0
\(61\) 320.735 0.673212 0.336606 0.941646i \(-0.390721\pi\)
0.336606 + 0.941646i \(0.390721\pi\)
\(62\) − 503.087i − 1.03052i
\(63\) − 121.196i − 0.242369i
\(64\) −566.197 −1.10585
\(65\) 0 0
\(66\) 658.488 1.22810
\(67\) − 14.5097i − 0.0264573i −0.999912 0.0132286i \(-0.995789\pi\)
0.999912 0.0132286i \(-0.00421093\pi\)
\(68\) 114.589i 0.204352i
\(69\) −1452.30 −2.53387
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) 416.971i 0.682506i
\(73\) 824.489i 1.32191i 0.750427 + 0.660953i \(0.229848\pi\)
−0.750427 + 0.660953i \(0.770152\pi\)
\(74\) −948.603 −1.49017
\(75\) 0 0
\(76\) 58.0993 0.0876901
\(77\) 267.784i 0.396322i
\(78\) 332.958i 0.483334i
\(79\) −156.275 −0.222561 −0.111280 0.993789i \(-0.535495\pi\)
−0.111280 + 0.993789i \(0.535495\pi\)
\(80\) 0 0
\(81\) −896.706 −1.23005
\(82\) 877.519i 1.18178i
\(83\) − 1036.53i − 1.37077i −0.728182 0.685384i \(-0.759634\pi\)
0.728182 0.685384i \(-0.240366\pi\)
\(84\) −61.2162 −0.0795147
\(85\) 0 0
\(86\) −584.818 −0.733286
\(87\) 312.480i 0.385074i
\(88\) − 921.301i − 1.11603i
\(89\) 170.225 0.202740 0.101370 0.994849i \(-0.467677\pi\)
0.101370 + 0.994849i \(0.467677\pi\)
\(90\) 0 0
\(91\) −135.402 −0.155978
\(92\) 286.607i 0.324792i
\(93\) 1295.15i 1.44409i
\(94\) −30.1921 −0.0331285
\(95\) 0 0
\(96\) 391.517 0.416240
\(97\) − 1059.87i − 1.10942i −0.832044 0.554710i \(-0.812829\pi\)
0.832044 0.554710i \(-0.187171\pi\)
\(98\) 126.704i 0.130602i
\(99\) −662.333 −0.672394
\(100\) 0 0
\(101\) −241.833 −0.238251 −0.119125 0.992879i \(-0.538009\pi\)
−0.119125 + 0.992879i \(0.538009\pi\)
\(102\) 1501.43i 1.45749i
\(103\) − 1679.58i − 1.60673i −0.595484 0.803367i \(-0.703040\pi\)
0.595484 0.803367i \(-0.296960\pi\)
\(104\) 465.846 0.439230
\(105\) 0 0
\(106\) −540.479 −0.495245
\(107\) − 1506.88i − 1.36146i −0.732537 0.680728i \(-0.761664\pi\)
0.732537 0.680728i \(-0.238336\pi\)
\(108\) 84.7082i 0.0754727i
\(109\) 1252.41 1.10054 0.550271 0.834986i \(-0.314524\pi\)
0.550271 + 0.834986i \(0.314524\pi\)
\(110\) 0 0
\(111\) 2442.09 2.08822
\(112\) − 362.352i − 0.305705i
\(113\) 1370.20i 1.14069i 0.821405 + 0.570345i \(0.193190\pi\)
−0.821405 + 0.570345i \(0.806810\pi\)
\(114\) 761.261 0.625426
\(115\) 0 0
\(116\) 61.6670 0.0493589
\(117\) − 334.902i − 0.264630i
\(118\) − 1592.84i − 1.24265i
\(119\) −610.578 −0.470349
\(120\) 0 0
\(121\) 132.432 0.0994984
\(122\) − 829.352i − 0.615459i
\(123\) − 2259.09i − 1.65606i
\(124\) 255.593 0.185104
\(125\) 0 0
\(126\) −313.387 −0.221577
\(127\) − 1213.49i − 0.847873i −0.905692 0.423936i \(-0.860648\pi\)
0.905692 0.423936i \(-0.139352\pi\)
\(128\) 993.551i 0.686081i
\(129\) 1505.56 1.02757
\(130\) 0 0
\(131\) −1982.42 −1.32217 −0.661087 0.750309i \(-0.729904\pi\)
−0.661087 + 0.750309i \(0.729904\pi\)
\(132\) 334.545i 0.220594i
\(133\) 309.578i 0.201833i
\(134\) −37.5189 −0.0241876
\(135\) 0 0
\(136\) 2100.67 1.32449
\(137\) − 2210.95i − 1.37879i −0.724386 0.689394i \(-0.757877\pi\)
0.724386 0.689394i \(-0.242123\pi\)
\(138\) 3755.34i 2.31649i
\(139\) −528.039 −0.322213 −0.161107 0.986937i \(-0.551506\pi\)
−0.161107 + 0.986937i \(0.551506\pi\)
\(140\) 0 0
\(141\) 77.7267 0.0464239
\(142\) 2461.67i 1.45478i
\(143\) 739.969i 0.432722i
\(144\) 896.235 0.518655
\(145\) 0 0
\(146\) 2131.95 1.20851
\(147\) − 326.186i − 0.183016i
\(148\) − 481.938i − 0.267669i
\(149\) 328.372 0.180545 0.0902727 0.995917i \(-0.471226\pi\)
0.0902727 + 0.995917i \(0.471226\pi\)
\(150\) 0 0
\(151\) 1029.43 0.554793 0.277396 0.960756i \(-0.410528\pi\)
0.277396 + 0.960756i \(0.410528\pi\)
\(152\) − 1065.09i − 0.568358i
\(153\) − 1510.20i − 0.797987i
\(154\) 692.432 0.362323
\(155\) 0 0
\(156\) −169.159 −0.0868177
\(157\) − 525.098i − 0.266926i −0.991054 0.133463i \(-0.957390\pi\)
0.991054 0.133463i \(-0.0426098\pi\)
\(158\) 404.094i 0.203468i
\(159\) 1391.41 0.694001
\(160\) 0 0
\(161\) −1527.17 −0.747562
\(162\) 2318.69i 1.12453i
\(163\) 1002.63i 0.481790i 0.970551 + 0.240895i \(0.0774409\pi\)
−0.970551 + 0.240895i \(0.922559\pi\)
\(164\) −445.823 −0.212274
\(165\) 0 0
\(166\) −2680.24 −1.25317
\(167\) 359.422i 0.166544i 0.996527 + 0.0832722i \(0.0265371\pi\)
−0.996527 + 0.0832722i \(0.973463\pi\)
\(168\) 1122.23i 0.515369i
\(169\) 1822.84 0.829696
\(170\) 0 0
\(171\) −765.706 −0.342427
\(172\) − 297.117i − 0.131715i
\(173\) 3293.65i 1.44747i 0.690080 + 0.723733i \(0.257575\pi\)
−0.690080 + 0.723733i \(0.742425\pi\)
\(174\) 808.007 0.352039
\(175\) 0 0
\(176\) −1980.24 −0.848104
\(177\) 4100.62i 1.74137i
\(178\) − 440.167i − 0.185348i
\(179\) −2978.82 −1.24384 −0.621921 0.783080i \(-0.713647\pi\)
−0.621921 + 0.783080i \(0.713647\pi\)
\(180\) 0 0
\(181\) 1462.31 0.600514 0.300257 0.953858i \(-0.402928\pi\)
0.300257 + 0.953858i \(0.402928\pi\)
\(182\) 350.121i 0.142597i
\(183\) 2135.09i 0.862460i
\(184\) 5254.16 2.10512
\(185\) 0 0
\(186\) 3348.97 1.32021
\(187\) 3336.79i 1.30487i
\(188\) − 15.3391i − 0.00595064i
\(189\) −451.362 −0.173713
\(190\) 0 0
\(191\) −374.923 −0.142034 −0.0710169 0.997475i \(-0.522624\pi\)
−0.0710169 + 0.997475i \(0.522624\pi\)
\(192\) − 3769.09i − 1.41672i
\(193\) 733.028i 0.273391i 0.990613 + 0.136696i \(0.0436482\pi\)
−0.990613 + 0.136696i \(0.956352\pi\)
\(194\) −2740.61 −1.01425
\(195\) 0 0
\(196\) −64.3717 −0.0234591
\(197\) 2093.24i 0.757043i 0.925593 + 0.378521i \(0.123567\pi\)
−0.925593 + 0.378521i \(0.876433\pi\)
\(198\) 1712.65i 0.614711i
\(199\) −2865.04 −1.02059 −0.510295 0.860000i \(-0.670464\pi\)
−0.510295 + 0.860000i \(0.670464\pi\)
\(200\) 0 0
\(201\) 96.5887 0.0338948
\(202\) 625.330i 0.217812i
\(203\) 328.588i 0.113608i
\(204\) −762.801 −0.261798
\(205\) 0 0
\(206\) −4343.03 −1.46890
\(207\) − 3777.27i − 1.26830i
\(208\) − 1001.29i − 0.333783i
\(209\) 1691.84 0.559936
\(210\) 0 0
\(211\) 5643.65 1.84135 0.920674 0.390331i \(-0.127640\pi\)
0.920674 + 0.390331i \(0.127640\pi\)
\(212\) − 274.590i − 0.0889573i
\(213\) − 6337.33i − 2.03862i
\(214\) −3896.47 −1.24466
\(215\) 0 0
\(216\) 1552.89 0.489172
\(217\) 1361.91i 0.426048i
\(218\) − 3238.46i − 1.00613i
\(219\) −5488.51 −1.69351
\(220\) 0 0
\(221\) −1687.21 −0.513549
\(222\) − 6314.71i − 1.90908i
\(223\) − 6369.16i − 1.91260i −0.292381 0.956302i \(-0.594448\pi\)
0.292381 0.956302i \(-0.405552\pi\)
\(224\) 411.699 0.122803
\(225\) 0 0
\(226\) 3543.05 1.04283
\(227\) 1015.67i 0.296972i 0.988914 + 0.148486i \(0.0474400\pi\)
−0.988914 + 0.148486i \(0.952560\pi\)
\(228\) 386.758i 0.112341i
\(229\) −4108.35 −1.18554 −0.592768 0.805373i \(-0.701965\pi\)
−0.592768 + 0.805373i \(0.701965\pi\)
\(230\) 0 0
\(231\) −1782.60 −0.507733
\(232\) − 1130.50i − 0.319917i
\(233\) 608.431i 0.171071i 0.996335 + 0.0855357i \(0.0272602\pi\)
−0.996335 + 0.0855357i \(0.972740\pi\)
\(234\) −865.984 −0.241928
\(235\) 0 0
\(236\) 809.244 0.223209
\(237\) − 1040.30i − 0.285126i
\(238\) 1578.82i 0.430000i
\(239\) 5054.44 1.36797 0.683985 0.729496i \(-0.260245\pi\)
0.683985 + 0.729496i \(0.260245\pi\)
\(240\) 0 0
\(241\) 4.86782 0.00130109 0.000650547 1.00000i \(-0.499793\pi\)
0.000650547 1.00000i \(0.499793\pi\)
\(242\) − 342.442i − 0.0909628i
\(243\) − 4228.27i − 1.11623i
\(244\) 421.352 0.110551
\(245\) 0 0
\(246\) −5841.52 −1.51399
\(247\) 855.458i 0.220370i
\(248\) − 4685.60i − 1.19974i
\(249\) 6900.02 1.75611
\(250\) 0 0
\(251\) −547.921 −0.137787 −0.0688934 0.997624i \(-0.521947\pi\)
−0.0688934 + 0.997624i \(0.521947\pi\)
\(252\) − 159.216i − 0.0398003i
\(253\) 8345.92i 2.07393i
\(254\) −3137.83 −0.775137
\(255\) 0 0
\(256\) −1960.46 −0.478629
\(257\) 1774.61i 0.430729i 0.976534 + 0.215364i \(0.0690939\pi\)
−0.976534 + 0.215364i \(0.930906\pi\)
\(258\) − 3893.05i − 0.939421i
\(259\) 2567.97 0.616084
\(260\) 0 0
\(261\) −812.725 −0.192745
\(262\) 5126.11i 1.20875i
\(263\) − 1199.09i − 0.281138i −0.990071 0.140569i \(-0.955107\pi\)
0.990071 0.140569i \(-0.0448932\pi\)
\(264\) 6132.97 1.42977
\(265\) 0 0
\(266\) 800.502 0.184519
\(267\) 1133.17i 0.259733i
\(268\) − 19.0615i − 0.00434464i
\(269\) −3250.29 −0.736706 −0.368353 0.929686i \(-0.620078\pi\)
−0.368353 + 0.929686i \(0.620078\pi\)
\(270\) 0 0
\(271\) −896.143 −0.200874 −0.100437 0.994943i \(-0.532024\pi\)
−0.100437 + 0.994943i \(0.532024\pi\)
\(272\) − 4515.18i − 1.00652i
\(273\) − 901.352i − 0.199825i
\(274\) −5717.04 −1.26051
\(275\) 0 0
\(276\) −1907.90 −0.416095
\(277\) 386.562i 0.0838492i 0.999121 + 0.0419246i \(0.0133489\pi\)
−0.999121 + 0.0419246i \(0.986651\pi\)
\(278\) 1365.40i 0.294572i
\(279\) −3368.53 −0.722826
\(280\) 0 0
\(281\) −3335.10 −0.708025 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(282\) − 200.985i − 0.0424414i
\(283\) 5412.26i 1.13684i 0.822739 + 0.568419i \(0.192445\pi\)
−0.822739 + 0.568419i \(0.807555\pi\)
\(284\) −1250.65 −0.261311
\(285\) 0 0
\(286\) 1913.40 0.395601
\(287\) − 2375.54i − 0.488584i
\(288\) 1018.29i 0.208345i
\(289\) −2695.27 −0.548600
\(290\) 0 0
\(291\) 7055.42 1.42129
\(292\) 1083.14i 0.217075i
\(293\) 282.211i 0.0562695i 0.999604 + 0.0281347i \(0.00895675\pi\)
−0.999604 + 0.0281347i \(0.991043\pi\)
\(294\) −843.447 −0.167316
\(295\) 0 0
\(296\) −8835.01 −1.73488
\(297\) 2466.68i 0.481924i
\(298\) − 849.099i − 0.165057i
\(299\) −4220.03 −0.816222
\(300\) 0 0
\(301\) 1583.17 0.303163
\(302\) − 2661.88i − 0.507199i
\(303\) − 1609.85i − 0.305226i
\(304\) −2289.31 −0.431910
\(305\) 0 0
\(306\) −3905.04 −0.729531
\(307\) − 1919.67i − 0.356878i −0.983951 0.178439i \(-0.942895\pi\)
0.983951 0.178439i \(-0.0571048\pi\)
\(308\) 351.790i 0.0650815i
\(309\) 11180.7 2.05841
\(310\) 0 0
\(311\) 1213.31 0.221223 0.110612 0.993864i \(-0.464719\pi\)
0.110612 + 0.993864i \(0.464719\pi\)
\(312\) 3101.07i 0.562703i
\(313\) − 1434.00i − 0.258960i −0.991582 0.129480i \(-0.958669\pi\)
0.991582 0.129480i \(-0.0413309\pi\)
\(314\) −1357.79 −0.244028
\(315\) 0 0
\(316\) −205.300 −0.0365475
\(317\) − 6496.95i − 1.15112i −0.817760 0.575560i \(-0.804784\pi\)
0.817760 0.575560i \(-0.195216\pi\)
\(318\) − 3597.89i − 0.634465i
\(319\) 1795.72 0.315176
\(320\) 0 0
\(321\) 10031.1 1.74418
\(322\) 3948.92i 0.683432i
\(323\) 3857.58i 0.664524i
\(324\) −1178.01 −0.201991
\(325\) 0 0
\(326\) 2592.58 0.440459
\(327\) 8337.11i 1.40992i
\(328\) 8172.96i 1.37584i
\(329\) 81.7333 0.0136964
\(330\) 0 0
\(331\) −9683.88 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(332\) − 1361.70i − 0.225099i
\(333\) 6351.58i 1.04524i
\(334\) 929.389 0.152257
\(335\) 0 0
\(336\) 2412.12 0.391643
\(337\) − 29.1319i − 0.00470895i −0.999997 0.00235447i \(-0.999251\pi\)
0.999997 0.00235447i \(-0.000749453\pi\)
\(338\) − 4713.48i − 0.758520i
\(339\) −9121.24 −1.46135
\(340\) 0 0
\(341\) 7442.80 1.18197
\(342\) 1979.95i 0.313051i
\(343\) − 343.000i − 0.0539949i
\(344\) −5446.83 −0.853701
\(345\) 0 0
\(346\) 8516.68 1.32329
\(347\) 7848.58i 1.21422i 0.794618 + 0.607110i \(0.207671\pi\)
−0.794618 + 0.607110i \(0.792329\pi\)
\(348\) 410.508i 0.0632343i
\(349\) 10269.6 1.57513 0.787567 0.616229i \(-0.211341\pi\)
0.787567 + 0.616229i \(0.211341\pi\)
\(350\) 0 0
\(351\) −1247.25 −0.189668
\(352\) − 2249.93i − 0.340686i
\(353\) 2799.93i 0.422168i 0.977468 + 0.211084i \(0.0676994\pi\)
−0.977468 + 0.211084i \(0.932301\pi\)
\(354\) 10603.3 1.59198
\(355\) 0 0
\(356\) 223.627 0.0332927
\(357\) − 4064.53i − 0.602570i
\(358\) 7702.60i 1.13714i
\(359\) 3163.29 0.465048 0.232524 0.972591i \(-0.425302\pi\)
0.232524 + 0.972591i \(0.425302\pi\)
\(360\) 0 0
\(361\) −4903.11 −0.714844
\(362\) − 3781.23i − 0.548998i
\(363\) 881.583i 0.127469i
\(364\) −177.879 −0.0256137
\(365\) 0 0
\(366\) 5520.88 0.788472
\(367\) − 3182.85i − 0.452706i −0.974045 0.226353i \(-0.927320\pi\)
0.974045 0.226353i \(-0.0726803\pi\)
\(368\) − 11293.3i − 1.59974i
\(369\) 5875.62 0.828923
\(370\) 0 0
\(371\) 1463.14 0.204750
\(372\) 1701.45i 0.237139i
\(373\) − 2615.14i − 0.363021i −0.983389 0.181510i \(-0.941901\pi\)
0.983389 0.181510i \(-0.0580986\pi\)
\(374\) 8628.23 1.19293
\(375\) 0 0
\(376\) −281.201 −0.0385687
\(377\) 907.989i 0.124042i
\(378\) 1167.12i 0.158811i
\(379\) 672.434 0.0911362 0.0455681 0.998961i \(-0.485490\pi\)
0.0455681 + 0.998961i \(0.485490\pi\)
\(380\) 0 0
\(381\) 8078.03 1.08622
\(382\) 969.470i 0.129849i
\(383\) 1169.86i 0.156075i 0.996950 + 0.0780377i \(0.0248655\pi\)
−0.996950 + 0.0780377i \(0.975135\pi\)
\(384\) −6613.92 −0.878946
\(385\) 0 0
\(386\) 1895.45 0.249938
\(387\) 3915.78i 0.514342i
\(388\) − 1392.36i − 0.182182i
\(389\) 1122.22 0.146269 0.0731347 0.997322i \(-0.476700\pi\)
0.0731347 + 0.997322i \(0.476700\pi\)
\(390\) 0 0
\(391\) −19029.7 −2.46131
\(392\) 1180.08i 0.152049i
\(393\) − 13196.7i − 1.69385i
\(394\) 5412.68 0.692099
\(395\) 0 0
\(396\) −870.113 −0.110416
\(397\) 1985.93i 0.251060i 0.992090 + 0.125530i \(0.0400631\pi\)
−0.992090 + 0.125530i \(0.959937\pi\)
\(398\) 7408.38i 0.933037i
\(399\) −2060.81 −0.258571
\(400\) 0 0
\(401\) −4172.38 −0.519597 −0.259799 0.965663i \(-0.583656\pi\)
−0.259799 + 0.965663i \(0.583656\pi\)
\(402\) − 249.758i − 0.0309870i
\(403\) 3763.37i 0.465178i
\(404\) −317.699 −0.0391240
\(405\) 0 0
\(406\) 849.658 0.103862
\(407\) − 14033.9i − 1.70918i
\(408\) 13983.9i 1.69682i
\(409\) 11700.8 1.41459 0.707295 0.706919i \(-0.249915\pi\)
0.707295 + 0.706919i \(0.249915\pi\)
\(410\) 0 0
\(411\) 14718.0 1.76638
\(412\) − 2206.47i − 0.263848i
\(413\) 4312.00i 0.513752i
\(414\) −9767.22 −1.15950
\(415\) 0 0
\(416\) 1137.65 0.134082
\(417\) − 3515.08i − 0.412791i
\(418\) − 4374.72i − 0.511901i
\(419\) −2733.20 −0.318677 −0.159339 0.987224i \(-0.550936\pi\)
−0.159339 + 0.987224i \(0.550936\pi\)
\(420\) 0 0
\(421\) 13549.4 1.56854 0.784272 0.620417i \(-0.213037\pi\)
0.784272 + 0.620417i \(0.213037\pi\)
\(422\) − 14593.3i − 1.68339i
\(423\) 202.158i 0.0232370i
\(424\) −5033.87 −0.576571
\(425\) 0 0
\(426\) −16387.0 −1.86374
\(427\) 2245.15i 0.254450i
\(428\) − 1979.60i − 0.223569i
\(429\) −4925.86 −0.554366
\(430\) 0 0
\(431\) −6429.25 −0.718530 −0.359265 0.933236i \(-0.616973\pi\)
−0.359265 + 0.933236i \(0.616973\pi\)
\(432\) − 3337.79i − 0.371735i
\(433\) 8022.03i 0.890333i 0.895448 + 0.445166i \(0.146855\pi\)
−0.895448 + 0.445166i \(0.853145\pi\)
\(434\) 3521.61 0.389499
\(435\) 0 0
\(436\) 1645.30 0.180724
\(437\) 9648.50i 1.05618i
\(438\) 14192.1i 1.54823i
\(439\) 5569.88 0.605549 0.302774 0.953062i \(-0.402087\pi\)
0.302774 + 0.953062i \(0.402087\pi\)
\(440\) 0 0
\(441\) 848.372 0.0916069
\(442\) 4362.77i 0.469493i
\(443\) − 5486.21i − 0.588392i −0.955745 0.294196i \(-0.904948\pi\)
0.955745 0.294196i \(-0.0950519\pi\)
\(444\) 3208.19 0.342914
\(445\) 0 0
\(446\) −16469.3 −1.74853
\(447\) 2185.92i 0.231299i
\(448\) − 3963.38i − 0.417973i
\(449\) 7232.67 0.760203 0.380101 0.924945i \(-0.375889\pi\)
0.380101 + 0.924945i \(0.375889\pi\)
\(450\) 0 0
\(451\) −12982.3 −1.35546
\(452\) 1800.05i 0.187317i
\(453\) 6852.76i 0.710752i
\(454\) 2626.32 0.271496
\(455\) 0 0
\(456\) 7090.16 0.728130
\(457\) 2900.51i 0.296893i 0.988920 + 0.148446i \(0.0474272\pi\)
−0.988920 + 0.148446i \(0.952573\pi\)
\(458\) 10623.3i 1.08383i
\(459\) −5624.31 −0.571940
\(460\) 0 0
\(461\) 6073.57 0.613611 0.306805 0.951772i \(-0.400740\pi\)
0.306805 + 0.951772i \(0.400740\pi\)
\(462\) 4609.42i 0.464176i
\(463\) − 18922.8i − 1.89939i −0.313183 0.949693i \(-0.601395\pi\)
0.313183 0.949693i \(-0.398605\pi\)
\(464\) −2429.88 −0.243113
\(465\) 0 0
\(466\) 1573.27 0.156396
\(467\) 6776.71i 0.671496i 0.941952 + 0.335748i \(0.108989\pi\)
−0.941952 + 0.335748i \(0.891011\pi\)
\(468\) − 439.963i − 0.0434558i
\(469\) 101.568 0.00999991
\(470\) 0 0
\(471\) 3495.50 0.341962
\(472\) − 14835.3i − 1.44672i
\(473\) − 8651.96i − 0.841052i
\(474\) −2689.99 −0.260666
\(475\) 0 0
\(476\) −802.121 −0.0772377
\(477\) 3618.90i 0.347375i
\(478\) − 13069.7i − 1.25062i
\(479\) −2397.32 −0.228677 −0.114338 0.993442i \(-0.536475\pi\)
−0.114338 + 0.993442i \(0.536475\pi\)
\(480\) 0 0
\(481\) 7096.09 0.672669
\(482\) − 12.5871i − 0.00118948i
\(483\) − 10166.1i − 0.957711i
\(484\) 173.977 0.0163390
\(485\) 0 0
\(486\) −10933.4 −1.02047
\(487\) − 5586.17i − 0.519781i −0.965638 0.259890i \(-0.916314\pi\)
0.965638 0.259890i \(-0.0836865\pi\)
\(488\) − 7724.35i − 0.716526i
\(489\) −6674.33 −0.617227
\(490\) 0 0
\(491\) 537.392 0.0493934 0.0246967 0.999695i \(-0.492138\pi\)
0.0246967 + 0.999695i \(0.492138\pi\)
\(492\) − 2967.78i − 0.271947i
\(493\) 4094.46i 0.374047i
\(494\) 2212.03 0.201466
\(495\) 0 0
\(496\) −10071.2 −0.911716
\(497\) − 6664.00i − 0.601451i
\(498\) − 17842.0i − 1.60546i
\(499\) −598.965 −0.0537342 −0.0268671 0.999639i \(-0.508553\pi\)
−0.0268671 + 0.999639i \(0.508553\pi\)
\(500\) 0 0
\(501\) −2392.62 −0.213362
\(502\) 1416.81i 0.125966i
\(503\) 4426.76i 0.392405i 0.980563 + 0.196202i \(0.0628610\pi\)
−0.980563 + 0.196202i \(0.937139\pi\)
\(504\) −2918.79 −0.257963
\(505\) 0 0
\(506\) 21580.8 1.89601
\(507\) 12134.4i 1.06293i
\(508\) − 1594.17i − 0.139232i
\(509\) −17727.7 −1.54374 −0.771872 0.635779i \(-0.780679\pi\)
−0.771872 + 0.635779i \(0.780679\pi\)
\(510\) 0 0
\(511\) −5771.43 −0.499634
\(512\) 13017.7i 1.12365i
\(513\) 2851.66i 0.245427i
\(514\) 4588.77 0.393778
\(515\) 0 0
\(516\) 1977.86 0.168741
\(517\) − 446.671i − 0.0379972i
\(518\) − 6640.22i − 0.563233i
\(519\) −21925.4 −1.85437
\(520\) 0 0
\(521\) 8662.79 0.728453 0.364226 0.931310i \(-0.381333\pi\)
0.364226 + 0.931310i \(0.381333\pi\)
\(522\) 2101.53i 0.176210i
\(523\) − 7770.40i − 0.649667i −0.945771 0.324833i \(-0.894692\pi\)
0.945771 0.324833i \(-0.105308\pi\)
\(524\) −2604.32 −0.217119
\(525\) 0 0
\(526\) −3100.60 −0.257020
\(527\) 16970.4i 1.40274i
\(528\) − 13182.2i − 1.08652i
\(529\) −35429.6 −2.91194
\(530\) 0 0
\(531\) −10665.2 −0.871624
\(532\) 406.695i 0.0331437i
\(533\) − 6564.34i − 0.533458i
\(534\) 2930.12 0.237451
\(535\) 0 0
\(536\) −349.440 −0.0281595
\(537\) − 19829.6i − 1.59350i
\(538\) 8404.56i 0.673506i
\(539\) −1874.49 −0.149796
\(540\) 0 0
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) 2317.23i 0.183642i
\(543\) 9734.41i 0.769325i
\(544\) 5130.09 0.404321
\(545\) 0 0
\(546\) −2330.70 −0.182683
\(547\) − 7489.29i − 0.585409i −0.956203 0.292705i \(-0.905445\pi\)
0.956203 0.292705i \(-0.0945552\pi\)
\(548\) − 2904.54i − 0.226416i
\(549\) −5553.11 −0.431696
\(550\) 0 0
\(551\) 2075.99 0.160508
\(552\) 34976.2i 2.69689i
\(553\) − 1093.93i − 0.0841201i
\(554\) 999.566 0.0766561
\(555\) 0 0
\(556\) −693.689 −0.0529118
\(557\) − 25297.9i − 1.92443i −0.272295 0.962214i \(-0.587783\pi\)
0.272295 0.962214i \(-0.412217\pi\)
\(558\) 8710.29i 0.660817i
\(559\) 4374.77 0.331007
\(560\) 0 0
\(561\) −22212.5 −1.67168
\(562\) 8623.85i 0.647286i
\(563\) 15661.3i 1.17237i 0.810177 + 0.586186i \(0.199371\pi\)
−0.810177 + 0.586186i \(0.800629\pi\)
\(564\) 102.110 0.00762343
\(565\) 0 0
\(566\) 13994.9 1.03931
\(567\) − 6276.94i − 0.464915i
\(568\) 22927.3i 1.69367i
\(569\) 9982.75 0.735498 0.367749 0.929925i \(-0.380129\pi\)
0.367749 + 0.929925i \(0.380129\pi\)
\(570\) 0 0
\(571\) −11583.6 −0.848966 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(572\) 972.103i 0.0710589i
\(573\) − 2495.81i − 0.181961i
\(574\) −6142.63 −0.446670
\(575\) 0 0
\(576\) 9802.97 0.709127
\(577\) 595.378i 0.0429565i 0.999769 + 0.0214783i \(0.00683727\pi\)
−0.999769 + 0.0214783i \(0.993163\pi\)
\(578\) 6969.39i 0.501537i
\(579\) −4879.66 −0.350245
\(580\) 0 0
\(581\) 7255.70 0.518102
\(582\) − 18243.8i − 1.29936i
\(583\) − 7996.00i − 0.568028i
\(584\) 19856.4 1.40696
\(585\) 0 0
\(586\) 729.738 0.0514423
\(587\) 15750.3i 1.10747i 0.832693 + 0.553736i \(0.186798\pi\)
−0.832693 + 0.553736i \(0.813202\pi\)
\(588\) − 428.513i − 0.0300537i
\(589\) 8604.42 0.601934
\(590\) 0 0
\(591\) −13934.4 −0.969856
\(592\) 18990.0i 1.31838i
\(593\) − 417.878i − 0.0289379i −0.999895 0.0144690i \(-0.995394\pi\)
0.999895 0.0144690i \(-0.00460577\pi\)
\(594\) 6378.31 0.440581
\(595\) 0 0
\(596\) 431.385 0.0296480
\(597\) − 19072.2i − 1.30749i
\(598\) 10912.1i 0.746201i
\(599\) 19997.3 1.36406 0.682028 0.731326i \(-0.261098\pi\)
0.682028 + 0.731326i \(0.261098\pi\)
\(600\) 0 0
\(601\) −15992.6 −1.08545 −0.542723 0.839912i \(-0.682607\pi\)
−0.542723 + 0.839912i \(0.682607\pi\)
\(602\) − 4093.73i − 0.277156i
\(603\) 251.216i 0.0169657i
\(604\) 1352.37 0.0911045
\(605\) 0 0
\(606\) −4162.73 −0.279042
\(607\) − 14159.2i − 0.946793i −0.880850 0.473396i \(-0.843028\pi\)
0.880850 0.473396i \(-0.156972\pi\)
\(608\) − 2601.08i − 0.173499i
\(609\) −2187.36 −0.145544
\(610\) 0 0
\(611\) 225.854 0.0149543
\(612\) − 1983.96i − 0.131040i
\(613\) − 4629.41i − 0.305025i −0.988302 0.152512i \(-0.951264\pi\)
0.988302 0.152512i \(-0.0487364\pi\)
\(614\) −4963.87 −0.326263
\(615\) 0 0
\(616\) 6449.11 0.421821
\(617\) 23165.3i 1.51151i 0.654857 + 0.755753i \(0.272729\pi\)
−0.654857 + 0.755753i \(0.727271\pi\)
\(618\) − 28910.9i − 1.88182i
\(619\) −12370.6 −0.803258 −0.401629 0.915803i \(-0.631556\pi\)
−0.401629 + 0.915803i \(0.631556\pi\)
\(620\) 0 0
\(621\) −14067.4 −0.909028
\(622\) − 3137.36i − 0.202245i
\(623\) 1191.58i 0.0766285i
\(624\) 6665.43 0.427613
\(625\) 0 0
\(626\) −3708.02 −0.236745
\(627\) 11262.3i 0.717341i
\(628\) − 689.826i − 0.0438329i
\(629\) 31998.9 2.02842
\(630\) 0 0
\(631\) 13980.2 0.882002 0.441001 0.897507i \(-0.354623\pi\)
0.441001 + 0.897507i \(0.354623\pi\)
\(632\) 3763.61i 0.236880i
\(633\) 37568.9i 2.35897i
\(634\) −16799.7 −1.05237
\(635\) 0 0
\(636\) 1827.91 0.113964
\(637\) − 947.814i − 0.0589541i
\(638\) − 4643.36i − 0.288139i
\(639\) 16482.7 1.02041
\(640\) 0 0
\(641\) −16060.9 −0.989655 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(642\) − 25938.3i − 1.59455i
\(643\) − 4502.17i − 0.276125i −0.990424 0.138063i \(-0.955913\pi\)
0.990424 0.138063i \(-0.0440875\pi\)
\(644\) −2006.25 −0.122760
\(645\) 0 0
\(646\) 9974.87 0.607517
\(647\) − 29414.8i − 1.78735i −0.448715 0.893675i \(-0.648118\pi\)
0.448715 0.893675i \(-0.351882\pi\)
\(648\) 21595.6i 1.30919i
\(649\) 23565.0 1.42528
\(650\) 0 0
\(651\) −9066.03 −0.545815
\(652\) 1317.16i 0.0791164i
\(653\) 13013.6i 0.779882i 0.920840 + 0.389941i \(0.127505\pi\)
−0.920840 + 0.389941i \(0.872495\pi\)
\(654\) 21558.0 1.28897
\(655\) 0 0
\(656\) 17566.9 1.04554
\(657\) − 14275.0i − 0.847671i
\(658\) − 211.345i − 0.0125214i
\(659\) −23474.2 −1.38759 −0.693797 0.720171i \(-0.744063\pi\)
−0.693797 + 0.720171i \(0.744063\pi\)
\(660\) 0 0
\(661\) −9266.36 −0.545264 −0.272632 0.962118i \(-0.587894\pi\)
−0.272632 + 0.962118i \(0.587894\pi\)
\(662\) 25040.4i 1.47013i
\(663\) − 11231.5i − 0.657914i
\(664\) −24963.0 −1.45896
\(665\) 0 0
\(666\) 16423.8 0.955572
\(667\) 10241.0i 0.594501i
\(668\) 472.176i 0.0273489i
\(669\) 42398.6 2.45026
\(670\) 0 0
\(671\) 12269.7 0.705909
\(672\) 2740.62i 0.157324i
\(673\) − 25067.2i − 1.43576i −0.696164 0.717882i \(-0.745112\pi\)
0.696164 0.717882i \(-0.254888\pi\)
\(674\) −75.3288 −0.00430498
\(675\) 0 0
\(676\) 2394.68 0.136247
\(677\) 22409.6i 1.27219i 0.771613 + 0.636093i \(0.219450\pi\)
−0.771613 + 0.636093i \(0.780550\pi\)
\(678\) 23585.6i 1.33599i
\(679\) 7419.11 0.419322
\(680\) 0 0
\(681\) −6761.20 −0.380455
\(682\) − 19245.5i − 1.08057i
\(683\) − 8757.53i − 0.490626i −0.969444 0.245313i \(-0.921109\pi\)
0.969444 0.245313i \(-0.0788908\pi\)
\(684\) −1005.91 −0.0562311
\(685\) 0 0
\(686\) −886.925 −0.0493629
\(687\) − 27348.7i − 1.51880i
\(688\) 11707.4i 0.648750i
\(689\) 4043.09 0.223555
\(690\) 0 0
\(691\) 8468.42 0.466214 0.233107 0.972451i \(-0.425111\pi\)
0.233107 + 0.972451i \(0.425111\pi\)
\(692\) 4326.90i 0.237694i
\(693\) − 4636.33i − 0.254141i
\(694\) 20294.8 1.11006
\(695\) 0 0
\(696\) 7525.54 0.409849
\(697\) − 29601.0i − 1.60864i
\(698\) − 26555.1i − 1.44001i
\(699\) −4050.23 −0.219162
\(700\) 0 0
\(701\) 15996.9 0.861906 0.430953 0.902374i \(-0.358177\pi\)
0.430953 + 0.902374i \(0.358177\pi\)
\(702\) 3225.12i 0.173397i
\(703\) − 16224.2i − 0.870423i
\(704\) −21659.8 −1.15956
\(705\) 0 0
\(706\) 7240.03 0.385952
\(707\) − 1692.83i − 0.0900503i
\(708\) 5387.02i 0.285956i
\(709\) −19903.0 −1.05426 −0.527131 0.849784i \(-0.676732\pi\)
−0.527131 + 0.849784i \(0.676732\pi\)
\(710\) 0 0
\(711\) 2705.70 0.142717
\(712\) − 4099.58i − 0.215784i
\(713\) 42446.1i 2.22948i
\(714\) −10510.0 −0.550878
\(715\) 0 0
\(716\) −3913.30 −0.204256
\(717\) 33646.7i 1.75252i
\(718\) − 8179.59i − 0.425153i
\(719\) 11073.1 0.574347 0.287174 0.957879i \(-0.407284\pi\)
0.287174 + 0.957879i \(0.407284\pi\)
\(720\) 0 0
\(721\) 11757.0 0.607288
\(722\) 12678.4i 0.653520i
\(723\) 32.4043i 0.00166685i
\(724\) 1921.06 0.0986125
\(725\) 0 0
\(726\) 2279.58 0.116533
\(727\) − 31652.7i − 1.61476i −0.590029 0.807382i \(-0.700884\pi\)
0.590029 0.807382i \(-0.299116\pi\)
\(728\) 3260.92i 0.166013i
\(729\) 3935.94 0.199967
\(730\) 0 0
\(731\) 19727.5 0.998149
\(732\) 2804.88i 0.141628i
\(733\) 16958.3i 0.854528i 0.904127 + 0.427264i \(0.140523\pi\)
−0.904127 + 0.427264i \(0.859477\pi\)
\(734\) −8230.16 −0.413870
\(735\) 0 0
\(736\) 12831.3 0.642618
\(737\) − 555.065i − 0.0277423i
\(738\) − 15193.1i − 0.757813i
\(739\) 11616.6 0.578245 0.289123 0.957292i \(-0.406636\pi\)
0.289123 + 0.957292i \(0.406636\pi\)
\(740\) 0 0
\(741\) −5694.66 −0.282319
\(742\) − 3783.36i − 0.187185i
\(743\) 15928.0i 0.786464i 0.919439 + 0.393232i \(0.128643\pi\)
−0.919439 + 0.393232i \(0.871357\pi\)
\(744\) 31191.4 1.53700
\(745\) 0 0
\(746\) −6762.19 −0.331879
\(747\) 17946.1i 0.879003i
\(748\) 4383.57i 0.214277i
\(749\) 10548.2 0.514582
\(750\) 0 0
\(751\) 25571.9 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(752\) 604.412i 0.0293094i
\(753\) − 3647.43i − 0.176520i
\(754\) 2347.87 0.113401
\(755\) 0 0
\(756\) −592.958 −0.0285260
\(757\) − 6202.41i − 0.297794i −0.988853 0.148897i \(-0.952428\pi\)
0.988853 0.148897i \(-0.0475723\pi\)
\(758\) − 1738.77i − 0.0833179i
\(759\) −55557.6 −2.65693
\(760\) 0 0
\(761\) −29199.1 −1.39089 −0.695444 0.718580i \(-0.744792\pi\)
−0.695444 + 0.718580i \(0.744792\pi\)
\(762\) − 20888.1i − 0.993037i
\(763\) 8766.87i 0.415966i
\(764\) −492.539 −0.0233239
\(765\) 0 0
\(766\) 3025.00 0.142686
\(767\) 11915.4i 0.560938i
\(768\) − 13050.5i − 0.613177i
\(769\) −21838.2 −1.02407 −0.512033 0.858966i \(-0.671108\pi\)
−0.512033 + 0.858966i \(0.671108\pi\)
\(770\) 0 0
\(771\) −11813.3 −0.551812
\(772\) 962.985i 0.0448945i
\(773\) − 25544.8i − 1.18859i −0.804246 0.594296i \(-0.797431\pi\)
0.804246 0.594296i \(-0.202569\pi\)
\(774\) 10125.4 0.470218
\(775\) 0 0
\(776\) −25525.2 −1.18080
\(777\) 17094.6i 0.789273i
\(778\) − 2901.82i − 0.133721i
\(779\) −15008.4 −0.690286
\(780\) 0 0
\(781\) −36418.6 −1.66858
\(782\) 49206.6i 2.25016i
\(783\) 3026.77i 0.138146i
\(784\) 2536.46 0.115546
\(785\) 0 0
\(786\) −34123.8 −1.54854
\(787\) 37223.2i 1.68598i 0.537931 + 0.842989i \(0.319206\pi\)
−0.537931 + 0.842989i \(0.680794\pi\)
\(788\) 2749.91i 0.124317i
\(789\) 7982.19 0.360169
\(790\) 0 0
\(791\) −9591.42 −0.431140
\(792\) 15951.1i 0.715655i
\(793\) 6204.03i 0.277820i
\(794\) 5135.18 0.229522
\(795\) 0 0
\(796\) −3763.83 −0.167595
\(797\) 40384.6i 1.79485i 0.441168 + 0.897425i \(0.354564\pi\)
−0.441168 + 0.897425i \(0.645436\pi\)
\(798\) 5328.83i 0.236389i
\(799\) 1018.46 0.0450945
\(800\) 0 0
\(801\) −2947.23 −0.130007
\(802\) 10788.9i 0.475023i
\(803\) 31540.7i 1.38611i
\(804\) 126.889 0.00556598
\(805\) 0 0
\(806\) 9731.28 0.425272
\(807\) − 21636.7i − 0.943803i
\(808\) 5824.14i 0.253580i
\(809\) 1955.76 0.0849948 0.0424974 0.999097i \(-0.486469\pi\)
0.0424974 + 0.999097i \(0.486469\pi\)
\(810\) 0 0
\(811\) −34301.8 −1.48520 −0.742600 0.669735i \(-0.766408\pi\)
−0.742600 + 0.669735i \(0.766408\pi\)
\(812\) 431.669i 0.0186559i
\(813\) − 5965.49i − 0.257342i
\(814\) −36288.7 −1.56255
\(815\) 0 0
\(816\) 30056.9 1.28946
\(817\) − 10002.3i − 0.428319i
\(818\) − 30255.8i − 1.29324i
\(819\) 2344.31 0.100021
\(820\) 0 0
\(821\) −13665.6 −0.580918 −0.290459 0.956887i \(-0.593808\pi\)
−0.290459 + 0.956887i \(0.593808\pi\)
\(822\) − 38057.5i − 1.61485i
\(823\) − 21519.5i − 0.911449i −0.890121 0.455724i \(-0.849380\pi\)
0.890121 0.455724i \(-0.150620\pi\)
\(824\) −40449.7 −1.71011
\(825\) 0 0
\(826\) 11149.9 0.469679
\(827\) − 35220.6i − 1.48094i −0.672088 0.740471i \(-0.734602\pi\)
0.672088 0.740471i \(-0.265398\pi\)
\(828\) − 4962.23i − 0.208272i
\(829\) −31365.5 −1.31408 −0.657039 0.753857i \(-0.728191\pi\)
−0.657039 + 0.753857i \(0.728191\pi\)
\(830\) 0 0
\(831\) −2573.28 −0.107420
\(832\) − 10952.0i − 0.456362i
\(833\) − 4274.04i − 0.177775i
\(834\) −9089.24 −0.377380
\(835\) 0 0
\(836\) 2222.58 0.0919491
\(837\) 12545.2i 0.518070i
\(838\) 7067.48i 0.291339i
\(839\) 28287.1 1.16398 0.581990 0.813196i \(-0.302274\pi\)
0.581990 + 0.813196i \(0.302274\pi\)
\(840\) 0 0
\(841\) −22185.5 −0.909653
\(842\) − 35035.8i − 1.43398i
\(843\) − 22201.2i − 0.907060i
\(844\) 7414.11 0.302374
\(845\) 0 0
\(846\) 522.738 0.0212436
\(847\) 927.026i 0.0376068i
\(848\) 10819.8i 0.438152i
\(849\) −36028.6 −1.45642
\(850\) 0 0
\(851\) 80035.0 3.22393
\(852\) − 8325.40i − 0.334769i
\(853\) 9405.41i 0.377533i 0.982022 + 0.188766i \(0.0604489\pi\)
−0.982022 + 0.188766i \(0.939551\pi\)
\(854\) 5805.47 0.232622
\(855\) 0 0
\(856\) −36290.6 −1.44905
\(857\) 27966.9i 1.11474i 0.830265 + 0.557369i \(0.188189\pi\)
−0.830265 + 0.557369i \(0.811811\pi\)
\(858\) 12737.2i 0.506809i
\(859\) −6281.11 −0.249486 −0.124743 0.992189i \(-0.539811\pi\)
−0.124743 + 0.992189i \(0.539811\pi\)
\(860\) 0 0
\(861\) 15813.6 0.625931
\(862\) 16624.7i 0.656890i
\(863\) − 4757.13i − 0.187642i −0.995589 0.0938208i \(-0.970092\pi\)
0.995589 0.0938208i \(-0.0299081\pi\)
\(864\) 3792.35 0.149327
\(865\) 0 0
\(866\) 20743.3 0.813954
\(867\) − 17942.0i − 0.702818i
\(868\) 1789.15i 0.0699629i
\(869\) −5978.28 −0.233371
\(870\) 0 0
\(871\) 280.663 0.0109184
\(872\) − 30162.1i − 1.17135i
\(873\) 18350.3i 0.711414i
\(874\) 24949.0 0.965574
\(875\) 0 0
\(876\) −7210.30 −0.278097
\(877\) 30240.5i 1.16437i 0.813057 + 0.582184i \(0.197802\pi\)
−0.813057 + 0.582184i \(0.802198\pi\)
\(878\) − 14402.5i − 0.553601i
\(879\) −1878.64 −0.0720875
\(880\) 0 0
\(881\) 44875.5 1.71611 0.858056 0.513556i \(-0.171672\pi\)
0.858056 + 0.513556i \(0.171672\pi\)
\(882\) − 2193.71i − 0.0837483i
\(883\) 4892.13i 0.186448i 0.995645 + 0.0932238i \(0.0297172\pi\)
−0.995645 + 0.0932238i \(0.970283\pi\)
\(884\) −2216.51 −0.0843317
\(885\) 0 0
\(886\) −14186.2 −0.537916
\(887\) − 1761.40i − 0.0666765i −0.999444 0.0333382i \(-0.989386\pi\)
0.999444 0.0333382i \(-0.0106139\pi\)
\(888\) − 58813.4i − 2.22258i
\(889\) 8494.43 0.320466
\(890\) 0 0
\(891\) −34303.3 −1.28979
\(892\) − 8367.22i − 0.314075i
\(893\) − 516.384i − 0.0193507i
\(894\) 5652.33 0.211457
\(895\) 0 0
\(896\) −6954.86 −0.259314
\(897\) − 28092.1i − 1.04567i
\(898\) − 18702.2i − 0.694988i
\(899\) 9132.79 0.338816
\(900\) 0 0
\(901\) 18231.8 0.674128
\(902\) 33569.3i 1.23918i
\(903\) 10538.9i 0.388386i
\(904\) 32999.0 1.21408
\(905\) 0 0
\(906\) 17719.8 0.649779
\(907\) − 23689.1i − 0.867238i −0.901096 0.433619i \(-0.857236\pi\)
0.901096 0.433619i \(-0.142764\pi\)
\(908\) 1334.30i 0.0487669i
\(909\) 4187.03 0.152778
\(910\) 0 0
\(911\) −13877.3 −0.504692 −0.252346 0.967637i \(-0.581202\pi\)
−0.252346 + 0.967637i \(0.581202\pi\)
\(912\) − 15239.6i − 0.553325i
\(913\) − 39652.2i − 1.43735i
\(914\) 7500.09 0.271423
\(915\) 0 0
\(916\) −5397.18 −0.194681
\(917\) − 13876.9i − 0.499735i
\(918\) 14543.3i 0.522875i
\(919\) −14331.6 −0.514426 −0.257213 0.966355i \(-0.582804\pi\)
−0.257213 + 0.966355i \(0.582804\pi\)
\(920\) 0 0
\(921\) 12779.0 0.457201
\(922\) − 15705.0i − 0.560971i
\(923\) − 18414.7i − 0.656692i
\(924\) −2341.81 −0.0833767
\(925\) 0 0
\(926\) −48930.2 −1.73644
\(927\) 29079.7i 1.03032i
\(928\) − 2760.80i − 0.0976592i
\(929\) −16668.4 −0.588668 −0.294334 0.955703i \(-0.595098\pi\)
−0.294334 + 0.955703i \(0.595098\pi\)
\(930\) 0 0
\(931\) −2167.04 −0.0762857
\(932\) 799.300i 0.0280922i
\(933\) 8076.82i 0.283412i
\(934\) 17523.1 0.613891
\(935\) 0 0
\(936\) −8065.52 −0.281656
\(937\) − 30384.9i − 1.05937i −0.848194 0.529685i \(-0.822310\pi\)
0.848194 0.529685i \(-0.177690\pi\)
\(938\) − 262.632i − 0.00914206i
\(939\) 9545.94 0.331757
\(940\) 0 0
\(941\) −1196.35 −0.0414452 −0.0207226 0.999785i \(-0.506597\pi\)
−0.0207226 + 0.999785i \(0.506597\pi\)
\(942\) − 9038.63i − 0.312627i
\(943\) − 74037.5i − 2.55673i
\(944\) −31886.9 −1.09940
\(945\) 0 0
\(946\) −22372.1 −0.768901
\(947\) − 1788.41i − 0.0613681i −0.999529 0.0306840i \(-0.990231\pi\)
0.999529 0.0306840i \(-0.00976856\pi\)
\(948\) − 1366.65i − 0.0468215i
\(949\) −15948.2 −0.545523
\(950\) 0 0
\(951\) 43249.2 1.47471
\(952\) 14704.7i 0.500612i
\(953\) 8578.60i 0.291593i 0.989315 + 0.145796i \(0.0465744\pi\)
−0.989315 + 0.145796i \(0.953426\pi\)
\(954\) 9357.70 0.317575
\(955\) 0 0
\(956\) 6640.06 0.224639
\(957\) 11953.9i 0.403776i
\(958\) 6198.95i 0.209059i
\(959\) 15476.6 0.521133
\(960\) 0 0
\(961\) 8061.99 0.270618
\(962\) − 18349.0i − 0.614963i
\(963\) 26089.7i 0.873031i
\(964\) 6.39489 0.000213657 0
\(965\) 0 0
\(966\) −26287.4 −0.875552
\(967\) 55459.3i 1.84431i 0.386818 + 0.922156i \(0.373574\pi\)
−0.386818 + 0.922156i \(0.626426\pi\)
\(968\) − 3189.40i − 0.105900i
\(969\) −25679.3 −0.851330
\(970\) 0 0
\(971\) 22047.3 0.728662 0.364331 0.931270i \(-0.381298\pi\)
0.364331 + 0.931270i \(0.381298\pi\)
\(972\) − 5554.72i − 0.183300i
\(973\) − 3696.27i − 0.121785i
\(974\) −14444.6 −0.475191
\(975\) 0 0
\(976\) −16602.7 −0.544507
\(977\) − 14402.3i − 0.471617i −0.971800 0.235809i \(-0.924226\pi\)
0.971800 0.235809i \(-0.0757738\pi\)
\(978\) 17258.4i 0.564277i
\(979\) 6511.94 0.212587
\(980\) 0 0
\(981\) −21683.9 −0.705721
\(982\) − 1389.58i − 0.0451561i
\(983\) 7817.11i 0.253639i 0.991926 + 0.126819i \(0.0404769\pi\)
−0.991926 + 0.126819i \(0.959523\pi\)
\(984\) −54406.2 −1.76261
\(985\) 0 0
\(986\) 10587.4 0.341959
\(987\) 544.087i 0.0175466i
\(988\) 1123.82i 0.0361878i
\(989\) 49342.0 1.58643
\(990\) 0 0
\(991\) 24501.6 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(992\) − 11442.8i − 0.366239i
\(993\) − 64464.2i − 2.06013i
\(994\) −17231.7 −0.549855
\(995\) 0 0
\(996\) 9064.61 0.288377
\(997\) 50696.0i 1.61039i 0.593010 + 0.805195i \(0.297940\pi\)
−0.593010 + 0.805195i \(0.702060\pi\)
\(998\) 1548.80i 0.0491245i
\(999\) 23654.8 0.749152
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.99.2 4
5.2 odd 4 35.4.a.b.1.1 2
5.3 odd 4 175.4.a.c.1.2 2
5.4 even 2 inner 175.4.b.c.99.3 4
15.2 even 4 315.4.a.f.1.2 2
15.8 even 4 1575.4.a.z.1.1 2
20.7 even 4 560.4.a.r.1.1 2
35.2 odd 12 245.4.e.h.116.2 4
35.12 even 12 245.4.e.i.116.2 4
35.13 even 4 1225.4.a.m.1.2 2
35.17 even 12 245.4.e.i.226.2 4
35.27 even 4 245.4.a.k.1.1 2
35.32 odd 12 245.4.e.h.226.2 4
40.27 even 4 2240.4.a.bo.1.2 2
40.37 odd 4 2240.4.a.bn.1.1 2
105.62 odd 4 2205.4.a.u.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 5.2 odd 4
175.4.a.c.1.2 2 5.3 odd 4
175.4.b.c.99.2 4 1.1 even 1 trivial
175.4.b.c.99.3 4 5.4 even 2 inner
245.4.a.k.1.1 2 35.27 even 4
245.4.e.h.116.2 4 35.2 odd 12
245.4.e.h.226.2 4 35.32 odd 12
245.4.e.i.116.2 4 35.12 even 12
245.4.e.i.226.2 4 35.17 even 12
315.4.a.f.1.2 2 15.2 even 4
560.4.a.r.1.1 2 20.7 even 4
1225.4.a.m.1.2 2 35.13 even 4
1575.4.a.z.1.1 2 15.8 even 4
2205.4.a.u.1.2 2 105.62 odd 4
2240.4.a.bn.1.1 2 40.37 odd 4
2240.4.a.bo.1.2 2 40.27 even 4