Properties

Label 245.4.a.k.1.1
Level $245$
Weight $4$
Character 245.1
Self dual yes
Analytic conductor $14.455$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 35)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +5.00000 q^{5} -17.2132 q^{6} -24.0833 q^{8} +17.3137 q^{9} +O(q^{10})\) \(q+2.58579 q^{2} -6.65685 q^{3} -1.31371 q^{4} +5.00000 q^{5} -17.2132 q^{6} -24.0833 q^{8} +17.3137 q^{9} +12.9289 q^{10} +38.2548 q^{11} +8.74517 q^{12} -19.3431 q^{13} -33.2843 q^{15} -51.7645 q^{16} +87.2254 q^{17} +44.7696 q^{18} +44.2254 q^{19} -6.56854 q^{20} +98.9188 q^{22} +218.167 q^{23} +160.319 q^{24} +25.0000 q^{25} -50.0172 q^{26} +64.4802 q^{27} -46.9411 q^{29} -86.0660 q^{30} -194.558 q^{31} +58.8141 q^{32} -254.657 q^{33} +225.546 q^{34} -22.7452 q^{36} +366.853 q^{37} +114.357 q^{38} +128.765 q^{39} -120.416 q^{40} +339.362 q^{41} -226.167 q^{43} -50.2557 q^{44} +86.5685 q^{45} +564.132 q^{46} -11.6762 q^{47} +344.589 q^{48} +64.6447 q^{50} -580.647 q^{51} +25.4113 q^{52} -209.019 q^{53} +166.732 q^{54} +191.274 q^{55} -294.402 q^{57} -121.380 q^{58} +616.000 q^{59} +43.7258 q^{60} -320.735 q^{61} -503.087 q^{62} +566.197 q^{64} -96.7157 q^{65} -658.488 q^{66} +14.5097 q^{67} -114.589 q^{68} -1452.30 q^{69} -952.000 q^{71} -416.971 q^{72} -824.489 q^{73} +948.603 q^{74} -166.421 q^{75} -58.0993 q^{76} +332.958 q^{78} +156.275 q^{79} -258.823 q^{80} -896.706 q^{81} +877.519 q^{82} +1036.53 q^{83} +436.127 q^{85} -584.818 q^{86} +312.480 q^{87} -921.301 q^{88} +170.225 q^{89} +223.848 q^{90} -286.607 q^{92} +1295.15 q^{93} -30.1921 q^{94} +221.127 q^{95} -391.517 q^{96} -1059.87 q^{97} +662.333 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9} + 40 q^{10} - 14 q^{11} + 108 q^{12} - 50 q^{13} - 10 q^{15} + 168 q^{16} + 50 q^{17} + 16 q^{18} - 36 q^{19} + 100 q^{20} - 184 q^{22} + 244 q^{23} + 496 q^{24} + 50 q^{25} - 216 q^{26} - 86 q^{27} - 26 q^{29} + 40 q^{30} + 120 q^{31} + 672 q^{32} - 498 q^{33} + 24 q^{34} - 136 q^{36} + 564 q^{37} - 320 q^{38} - 14 q^{39} + 240 q^{40} + 328 q^{41} - 260 q^{43} - 1164 q^{44} + 60 q^{45} + 704 q^{46} + 350 q^{47} + 1368 q^{48} + 200 q^{50} - 754 q^{51} - 628 q^{52} - 56 q^{53} - 648 q^{54} - 70 q^{55} - 668 q^{57} - 8 q^{58} + 1232 q^{59} + 540 q^{60} - 336 q^{61} + 1200 q^{62} + 2128 q^{64} - 250 q^{65} - 1976 q^{66} - 152 q^{67} - 908 q^{68} - 1332 q^{69} - 1904 q^{71} - 800 q^{72} - 676 q^{73} + 2016 q^{74} - 50 q^{75} - 1768 q^{76} - 440 q^{78} + 1014 q^{79} + 840 q^{80} - 1454 q^{81} + 816 q^{82} + 376 q^{83} + 250 q^{85} - 768 q^{86} + 410 q^{87} - 4688 q^{88} + 216 q^{89} + 80 q^{90} + 264 q^{92} + 2760 q^{93} + 1928 q^{94} - 180 q^{95} + 2464 q^{96} - 2742 q^{97} + 940 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.58579 0.914214 0.457107 0.889412i \(-0.348886\pi\)
0.457107 + 0.889412i \(0.348886\pi\)
\(3\) −6.65685 −1.28111 −0.640556 0.767911i \(-0.721296\pi\)
−0.640556 + 0.767911i \(0.721296\pi\)
\(4\) −1.31371 −0.164214
\(5\) 5.00000 0.447214
\(6\) −17.2132 −1.17121
\(7\) 0 0
\(8\) −24.0833 −1.06434
\(9\) 17.3137 0.641248
\(10\) 12.9289 0.408849
\(11\) 38.2548 1.04857 0.524285 0.851543i \(-0.324333\pi\)
0.524285 + 0.851543i \(0.324333\pi\)
\(12\) 8.74517 0.210376
\(13\) −19.3431 −0.412679 −0.206339 0.978480i \(-0.566155\pi\)
−0.206339 + 0.978480i \(0.566155\pi\)
\(14\) 0 0
\(15\) −33.2843 −0.572931
\(16\) −51.7645 −0.808820
\(17\) 87.2254 1.24443 0.622214 0.782847i \(-0.286233\pi\)
0.622214 + 0.782847i \(0.286233\pi\)
\(18\) 44.7696 0.586238
\(19\) 44.2254 0.534000 0.267000 0.963697i \(-0.413968\pi\)
0.267000 + 0.963697i \(0.413968\pi\)
\(20\) −6.56854 −0.0734385
\(21\) 0 0
\(22\) 98.9188 0.958617
\(23\) 218.167 1.97786 0.988932 0.148371i \(-0.0474028\pi\)
0.988932 + 0.148371i \(0.0474028\pi\)
\(24\) 160.319 1.36354
\(25\) 25.0000 0.200000
\(26\) −50.0172 −0.377276
\(27\) 64.4802 0.459601
\(28\) 0 0
\(29\) −46.9411 −0.300578 −0.150289 0.988642i \(-0.548020\pi\)
−0.150289 + 0.988642i \(0.548020\pi\)
\(30\) −86.0660 −0.523781
\(31\) −194.558 −1.12722 −0.563609 0.826042i \(-0.690587\pi\)
−0.563609 + 0.826042i \(0.690587\pi\)
\(32\) 58.8141 0.324905
\(33\) −254.657 −1.34334
\(34\) 225.546 1.13767
\(35\) 0 0
\(36\) −22.7452 −0.105302
\(37\) 366.853 1.63001 0.815003 0.579457i \(-0.196735\pi\)
0.815003 + 0.579457i \(0.196735\pi\)
\(38\) 114.357 0.488190
\(39\) 128.765 0.528688
\(40\) −120.416 −0.475987
\(41\) 339.362 1.29267 0.646336 0.763053i \(-0.276301\pi\)
0.646336 + 0.763053i \(0.276301\pi\)
\(42\) 0 0
\(43\) −226.167 −0.802095 −0.401047 0.916057i \(-0.631354\pi\)
−0.401047 + 0.916057i \(0.631354\pi\)
\(44\) −50.2557 −0.172189
\(45\) 86.5685 0.286775
\(46\) 564.132 1.80819
\(47\) −11.6762 −0.0362372 −0.0181186 0.999836i \(-0.505768\pi\)
−0.0181186 + 0.999836i \(0.505768\pi\)
\(48\) 344.589 1.03619
\(49\) 0 0
\(50\) 64.6447 0.182843
\(51\) −580.647 −1.59425
\(52\) 25.4113 0.0677674
\(53\) −209.019 −0.541717 −0.270859 0.962619i \(-0.587308\pi\)
−0.270859 + 0.962619i \(0.587308\pi\)
\(54\) 166.732 0.420173
\(55\) 191.274 0.468935
\(56\) 0 0
\(57\) −294.402 −0.684114
\(58\) −121.380 −0.274792
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 43.7258 0.0940830
\(61\) −320.735 −0.673212 −0.336606 0.941646i \(-0.609279\pi\)
−0.336606 + 0.941646i \(0.609279\pi\)
\(62\) −503.087 −1.03052
\(63\) 0 0
\(64\) 566.197 1.10585
\(65\) −96.7157 −0.184556
\(66\) −658.488 −1.22810
\(67\) 14.5097 0.0264573 0.0132286 0.999912i \(-0.495789\pi\)
0.0132286 + 0.999912i \(0.495789\pi\)
\(68\) −114.589 −0.204352
\(69\) −1452.30 −2.53387
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) −416.971 −0.682506
\(73\) −824.489 −1.32191 −0.660953 0.750427i \(-0.729848\pi\)
−0.660953 + 0.750427i \(0.729848\pi\)
\(74\) 948.603 1.49017
\(75\) −166.421 −0.256222
\(76\) −58.0993 −0.0876901
\(77\) 0 0
\(78\) 332.958 0.483334
\(79\) 156.275 0.222561 0.111280 0.993789i \(-0.464505\pi\)
0.111280 + 0.993789i \(0.464505\pi\)
\(80\) −258.823 −0.361715
\(81\) −896.706 −1.23005
\(82\) 877.519 1.18178
\(83\) 1036.53 1.37077 0.685384 0.728182i \(-0.259634\pi\)
0.685384 + 0.728182i \(0.259634\pi\)
\(84\) 0 0
\(85\) 436.127 0.556525
\(86\) −584.818 −0.733286
\(87\) 312.480 0.385074
\(88\) −921.301 −1.11603
\(89\) 170.225 0.202740 0.101370 0.994849i \(-0.467677\pi\)
0.101370 + 0.994849i \(0.467677\pi\)
\(90\) 223.848 0.262174
\(91\) 0 0
\(92\) −286.607 −0.324792
\(93\) 1295.15 1.44409
\(94\) −30.1921 −0.0331285
\(95\) 221.127 0.238812
\(96\) −391.517 −0.416240
\(97\) −1059.87 −1.10942 −0.554710 0.832044i \(-0.687171\pi\)
−0.554710 + 0.832044i \(0.687171\pi\)
\(98\) 0 0
\(99\) 662.333 0.672394
\(100\) −32.8427 −0.0328427
\(101\) 241.833 0.238251 0.119125 0.992879i \(-0.461991\pi\)
0.119125 + 0.992879i \(0.461991\pi\)
\(102\) −1501.43 −1.45749
\(103\) 1679.58 1.60673 0.803367 0.595484i \(-0.203040\pi\)
0.803367 + 0.595484i \(0.203040\pi\)
\(104\) 465.846 0.439230
\(105\) 0 0
\(106\) −540.479 −0.495245
\(107\) 1506.88 1.36146 0.680728 0.732537i \(-0.261664\pi\)
0.680728 + 0.732537i \(0.261664\pi\)
\(108\) −84.7082 −0.0754727
\(109\) −1252.41 −1.10054 −0.550271 0.834986i \(-0.685476\pi\)
−0.550271 + 0.834986i \(0.685476\pi\)
\(110\) 494.594 0.428706
\(111\) −2442.09 −2.08822
\(112\) 0 0
\(113\) 1370.20 1.14069 0.570345 0.821405i \(-0.306810\pi\)
0.570345 + 0.821405i \(0.306810\pi\)
\(114\) −761.261 −0.625426
\(115\) 1090.83 0.884528
\(116\) 61.6670 0.0493589
\(117\) −334.902 −0.264630
\(118\) 1592.84 1.24265
\(119\) 0 0
\(120\) 801.594 0.609793
\(121\) 132.432 0.0994984
\(122\) −829.352 −0.615459
\(123\) −2259.09 −1.65606
\(124\) 255.593 0.185104
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1213.49 0.847873 0.423936 0.905692i \(-0.360648\pi\)
0.423936 + 0.905692i \(0.360648\pi\)
\(128\) 993.551 0.686081
\(129\) 1505.56 1.02757
\(130\) −250.086 −0.168723
\(131\) 1982.42 1.32217 0.661087 0.750309i \(-0.270096\pi\)
0.661087 + 0.750309i \(0.270096\pi\)
\(132\) 334.545 0.220594
\(133\) 0 0
\(134\) 37.5189 0.0241876
\(135\) 322.401 0.205540
\(136\) −2100.67 −1.32449
\(137\) 2210.95 1.37879 0.689394 0.724386i \(-0.257877\pi\)
0.689394 + 0.724386i \(0.257877\pi\)
\(138\) −3755.34 −2.31649
\(139\) −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
\(142\) −2461.67 −1.45478
\(143\) −739.969 −0.432722
\(144\) −896.235 −0.518655
\(145\) −234.706 −0.134422
\(146\) −2131.95 −1.20851
\(147\) 0 0
\(148\) −481.938 −0.267669
\(149\) −328.372 −0.180545 −0.0902727 0.995917i \(-0.528774\pi\)
−0.0902727 + 0.995917i \(0.528774\pi\)
\(150\) −430.330 −0.234242
\(151\) 1029.43 0.554793 0.277396 0.960756i \(-0.410528\pi\)
0.277396 + 0.960756i \(0.410528\pi\)
\(152\) −1065.09 −0.568358
\(153\) 1510.20 0.797987
\(154\) 0 0
\(155\) −972.792 −0.504107
\(156\) −169.159 −0.0868177
\(157\) −525.098 −0.266926 −0.133463 0.991054i \(-0.542610\pi\)
−0.133463 + 0.991054i \(0.542610\pi\)
\(158\) 404.094 0.203468
\(159\) 1391.41 0.694001
\(160\) 294.071 0.145302
\(161\) 0 0
\(162\) −2318.69 −1.12453
\(163\) 1002.63 0.481790 0.240895 0.970551i \(-0.422559\pi\)
0.240895 + 0.970551i \(0.422559\pi\)
\(164\) −445.823 −0.212274
\(165\) −1273.28 −0.600758
\(166\) 2680.24 1.25317
\(167\) 359.422 0.166544 0.0832722 0.996527i \(-0.473463\pi\)
0.0832722 + 0.996527i \(0.473463\pi\)
\(168\) 0 0
\(169\) −1822.84 −0.829696
\(170\) 1127.73 0.508783
\(171\) 765.706 0.342427
\(172\) 297.117 0.131715
\(173\) −3293.65 −1.44747 −0.723733 0.690080i \(-0.757575\pi\)
−0.723733 + 0.690080i \(0.757575\pi\)
\(174\) 808.007 0.352039
\(175\) 0 0
\(176\) −1980.24 −0.848104
\(177\) −4100.62 −1.74137
\(178\) 440.167 0.185348
\(179\) 2978.82 1.24384 0.621921 0.783080i \(-0.286353\pi\)
0.621921 + 0.783080i \(0.286353\pi\)
\(180\) −113.726 −0.0470923
\(181\) −1462.31 −0.600514 −0.300257 0.953858i \(-0.597072\pi\)
−0.300257 + 0.953858i \(0.597072\pi\)
\(182\) 0 0
\(183\) 2135.09 0.862460
\(184\) −5254.16 −2.10512
\(185\) 1834.26 0.728961
\(186\) 3348.97 1.32021
\(187\) 3336.79 1.30487
\(188\) 15.3391 0.00595064
\(189\) 0 0
\(190\) 571.787 0.218325
\(191\) −374.923 −0.142034 −0.0710169 0.997475i \(-0.522624\pi\)
−0.0710169 + 0.997475i \(0.522624\pi\)
\(192\) −3769.09 −1.41672
\(193\) 733.028 0.273391 0.136696 0.990613i \(-0.456352\pi\)
0.136696 + 0.990613i \(0.456352\pi\)
\(194\) −2740.61 −1.01425
\(195\) 643.823 0.236436
\(196\) 0 0
\(197\) −2093.24 −0.757043 −0.378521 0.925593i \(-0.623567\pi\)
−0.378521 + 0.925593i \(0.623567\pi\)
\(198\) 1712.65 0.614711
\(199\) −2865.04 −1.02059 −0.510295 0.860000i \(-0.670464\pi\)
−0.510295 + 0.860000i \(0.670464\pi\)
\(200\) −602.082 −0.212868
\(201\) −96.5887 −0.0338948
\(202\) 625.330 0.217812
\(203\) 0 0
\(204\) 762.801 0.261798
\(205\) 1696.81 0.578100
\(206\) 4343.03 1.46890
\(207\) 3777.27 1.26830
\(208\) 1001.29 0.333783
\(209\) 1691.84 0.559936
\(210\) 0 0
\(211\) 5643.65 1.84135 0.920674 0.390331i \(-0.127640\pi\)
0.920674 + 0.390331i \(0.127640\pi\)
\(212\) 274.590 0.0889573
\(213\) 6337.33 2.03862
\(214\) 3896.47 1.24466
\(215\) −1130.83 −0.358708
\(216\) −1552.89 −0.489172
\(217\) 0 0
\(218\) −3238.46 −1.00613
\(219\) 5488.51 1.69351
\(220\) −251.279 −0.0770054
\(221\) −1687.21 −0.513549
\(222\) −6314.71 −1.90908
\(223\) 6369.16 1.91260 0.956302 0.292381i \(-0.0944477\pi\)
0.956302 + 0.292381i \(0.0944477\pi\)
\(224\) 0 0
\(225\) 432.843 0.128250
\(226\) 3543.05 1.04283
\(227\) 1015.67 0.296972 0.148486 0.988914i \(-0.452560\pi\)
0.148486 + 0.988914i \(0.452560\pi\)
\(228\) 386.758 0.112341
\(229\) −4108.35 −1.18554 −0.592768 0.805373i \(-0.701965\pi\)
−0.592768 + 0.805373i \(0.701965\pi\)
\(230\) 2820.66 0.808647
\(231\) 0 0
\(232\) 1130.50 0.319917
\(233\) 608.431 0.171071 0.0855357 0.996335i \(-0.472740\pi\)
0.0855357 + 0.996335i \(0.472740\pi\)
\(234\) −865.984 −0.241928
\(235\) −58.3810 −0.0162058
\(236\) −809.244 −0.223209
\(237\) −1040.30 −0.285126
\(238\) 0 0
\(239\) −5054.44 −1.36797 −0.683985 0.729496i \(-0.739755\pi\)
−0.683985 + 0.729496i \(0.739755\pi\)
\(240\) 1722.94 0.463398
\(241\) −4.86782 −0.00130109 −0.000650547 1.00000i \(-0.500207\pi\)
−0.000650547 1.00000i \(0.500207\pi\)
\(242\) 342.442 0.0909628
\(243\) 4228.27 1.11623
\(244\) 421.352 0.110551
\(245\) 0 0
\(246\) −5841.52 −1.51399
\(247\) −855.458 −0.220370
\(248\) 4685.60 1.19974
\(249\) −6900.02 −1.75611
\(250\) 323.223 0.0817697
\(251\) 547.921 0.137787 0.0688934 0.997624i \(-0.478053\pi\)
0.0688934 + 0.997624i \(0.478053\pi\)
\(252\) 0 0
\(253\) 8345.92 2.07393
\(254\) 3137.83 0.775137
\(255\) −2903.23 −0.712971
\(256\) −1960.46 −0.478629
\(257\) 1774.61 0.430729 0.215364 0.976534i \(-0.430906\pi\)
0.215364 + 0.976534i \(0.430906\pi\)
\(258\) 3893.05 0.939421
\(259\) 0 0
\(260\) 127.056 0.0303065
\(261\) −812.725 −0.192745
\(262\) 5126.11 1.20875
\(263\) −1199.09 −0.281138 −0.140569 0.990071i \(-0.544893\pi\)
−0.140569 + 0.990071i \(0.544893\pi\)
\(264\) 6132.97 1.42977
\(265\) −1045.10 −0.242263
\(266\) 0 0
\(267\) −1133.17 −0.259733
\(268\) −19.0615 −0.00434464
\(269\) −3250.29 −0.736706 −0.368353 0.929686i \(-0.620078\pi\)
−0.368353 + 0.929686i \(0.620078\pi\)
\(270\) 833.661 0.187907
\(271\) 896.143 0.200874 0.100437 0.994943i \(-0.467976\pi\)
0.100437 + 0.994943i \(0.467976\pi\)
\(272\) −4515.18 −1.00652
\(273\) 0 0
\(274\) 5717.04 1.26051
\(275\) 956.371 0.209714
\(276\) 1907.90 0.416095
\(277\) −386.562 −0.0838492 −0.0419246 0.999121i \(-0.513349\pi\)
−0.0419246 + 0.999121i \(0.513349\pi\)
\(278\) −1365.40 −0.294572
\(279\) −3368.53 −0.722826
\(280\) 0 0
\(281\) −3335.10 −0.708025 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(282\) 200.985 0.0424414
\(283\) −5412.26 −1.13684 −0.568419 0.822739i \(-0.692445\pi\)
−0.568419 + 0.822739i \(0.692445\pi\)
\(284\) 1250.65 0.261311
\(285\) −1472.01 −0.305945
\(286\) −1913.40 −0.395601
\(287\) 0 0
\(288\) 1018.29 0.208345
\(289\) 2695.27 0.548600
\(290\) −606.899 −0.122891
\(291\) 7055.42 1.42129
\(292\) 1083.14 0.217075
\(293\) −282.211 −0.0562695 −0.0281347 0.999604i \(-0.508957\pi\)
−0.0281347 + 0.999604i \(0.508957\pi\)
\(294\) 0 0
\(295\) 3080.00 0.607880
\(296\) −8835.01 −1.73488
\(297\) 2466.68 0.481924
\(298\) −849.099 −0.165057
\(299\) −4220.03 −0.816222
\(300\) 218.629 0.0420752
\(301\) 0 0
\(302\) 2661.88 0.507199
\(303\) −1609.85 −0.305226
\(304\) −2289.31 −0.431910
\(305\) −1603.68 −0.301069
\(306\) 3905.04 0.729531
\(307\) −1919.67 −0.356878 −0.178439 0.983951i \(-0.557105\pi\)
−0.178439 + 0.983951i \(0.557105\pi\)
\(308\) 0 0
\(309\) −11180.7 −2.05841
\(310\) −2515.43 −0.460861
\(311\) −1213.31 −0.221223 −0.110612 0.993864i \(-0.535281\pi\)
−0.110612 + 0.993864i \(0.535281\pi\)
\(312\) −3101.07 −0.562703
\(313\) 1434.00 0.258960 0.129480 0.991582i \(-0.458669\pi\)
0.129480 + 0.991582i \(0.458669\pi\)
\(314\) −1357.79 −0.244028
\(315\) 0 0
\(316\) −205.300 −0.0365475
\(317\) 6496.95 1.15112 0.575560 0.817760i \(-0.304784\pi\)
0.575560 + 0.817760i \(0.304784\pi\)
\(318\) 3597.89 0.634465
\(319\) −1795.72 −0.315176
\(320\) 2830.98 0.494553
\(321\) −10031.1 −1.74418
\(322\) 0 0
\(323\) 3857.58 0.664524
\(324\) 1178.01 0.201991
\(325\) −483.579 −0.0825357
\(326\) 2592.58 0.440459
\(327\) 8337.11 1.40992
\(328\) −8172.96 −1.37584
\(329\) 0 0
\(330\) −3292.44 −0.549221
\(331\) −9683.88 −1.60808 −0.804039 0.594576i \(-0.797320\pi\)
−0.804039 + 0.594576i \(0.797320\pi\)
\(332\) −1361.70 −0.225099
\(333\) 6351.58 1.04524
\(334\) 929.389 0.152257
\(335\) 72.5483 0.0118321
\(336\) 0 0
\(337\) 29.1319 0.00470895 0.00235447 0.999997i \(-0.499251\pi\)
0.00235447 + 0.999997i \(0.499251\pi\)
\(338\) −4713.48 −0.758520
\(339\) −9121.24 −1.46135
\(340\) −572.944 −0.0913889
\(341\) −7442.80 −1.18197
\(342\) 1979.95 0.313051
\(343\) 0 0
\(344\) 5446.83 0.853701
\(345\) −7261.51 −1.13318
\(346\) −8516.68 −1.32329
\(347\) −7848.58 −1.21422 −0.607110 0.794618i \(-0.707671\pi\)
−0.607110 + 0.794618i \(0.707671\pi\)
\(348\) −410.508 −0.0632343
\(349\) 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
\(352\) 2249.93 0.340686
\(353\) −2799.93 −0.422168 −0.211084 0.977468i \(-0.567699\pi\)
−0.211084 + 0.977468i \(0.567699\pi\)
\(354\) −10603.3 −1.59198
\(355\) −4760.00 −0.711647
\(356\) −223.627 −0.0332927
\(357\) 0 0
\(358\) 7702.60 1.13714
\(359\) −3163.29 −0.465048 −0.232524 0.972591i \(-0.574698\pi\)
−0.232524 + 0.972591i \(0.574698\pi\)
\(360\) −2084.85 −0.305226
\(361\) −4903.11 −0.714844
\(362\) −3781.23 −0.548998
\(363\) −881.583 −0.127469
\(364\) 0 0
\(365\) −4122.45 −0.591175
\(366\) 5520.88 0.788472
\(367\) −3182.85 −0.452706 −0.226353 0.974045i \(-0.572680\pi\)
−0.226353 + 0.974045i \(0.572680\pi\)
\(368\) −11293.3 −1.59974
\(369\) 5875.62 0.828923
\(370\) 4743.02 0.666426
\(371\) 0 0
\(372\) −1701.45 −0.237139
\(373\) −2615.14 −0.363021 −0.181510 0.983389i \(-0.558099\pi\)
−0.181510 + 0.983389i \(0.558099\pi\)
\(374\) 8628.23 1.19293
\(375\) −832.107 −0.114586
\(376\) 281.201 0.0385687
\(377\) 907.989 0.124042
\(378\) 0 0
\(379\) −672.434 −0.0911362 −0.0455681 0.998961i \(-0.514510\pi\)
−0.0455681 + 0.998961i \(0.514510\pi\)
\(380\) −290.496 −0.0392162
\(381\) −8078.03 −1.08622
\(382\) −969.470 −0.129849
\(383\) −1169.86 −0.156075 −0.0780377 0.996950i \(-0.524865\pi\)
−0.0780377 + 0.996950i \(0.524865\pi\)
\(384\) −6613.92 −0.878946
\(385\) 0 0
\(386\) 1895.45 0.249938
\(387\) −3915.78 −0.514342
\(388\) 1392.36 0.182182
\(389\) −1122.22 −0.146269 −0.0731347 0.997322i \(-0.523300\pi\)
−0.0731347 + 0.997322i \(0.523300\pi\)
\(390\) 1664.79 0.216153
\(391\) 19029.7 2.46131
\(392\) 0 0
\(393\) −13196.7 −1.69385
\(394\) −5412.68 −0.692099
\(395\) 781.375 0.0995323
\(396\) −870.113 −0.110416
\(397\) 1985.93 0.251060 0.125530 0.992090i \(-0.459937\pi\)
0.125530 + 0.992090i \(0.459937\pi\)
\(398\) −7408.38 −0.933037
\(399\) 0 0
\(400\) −1294.11 −0.161764
\(401\) −4172.38 −0.519597 −0.259799 0.965663i \(-0.583656\pi\)
−0.259799 + 0.965663i \(0.583656\pi\)
\(402\) −249.758 −0.0309870
\(403\) 3763.37 0.465178
\(404\) −317.699 −0.0391240
\(405\) −4483.53 −0.550095
\(406\) 0 0
\(407\) 14033.9 1.70918
\(408\) 13983.9 1.69682
\(409\) 11700.8 1.41459 0.707295 0.706919i \(-0.249915\pi\)
0.707295 + 0.706919i \(0.249915\pi\)
\(410\) 4387.59 0.528507
\(411\) −14718.0 −1.76638
\(412\) −2206.47 −0.263848
\(413\) 0 0
\(414\) 9767.22 1.15950
\(415\) 5182.64 0.613026
\(416\) −1137.65 −0.134082
\(417\) 3515.08 0.412791
\(418\) 4374.72 0.511901
\(419\) −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 \(0.213037\pi\)
\(422\) 14593.3 1.68339
\(423\) −202.158 −0.0232370
\(424\) 5033.87 0.576571
\(425\) 2180.63 0.248885
\(426\) 16387.0 1.86374
\(427\) 0 0
\(428\) −1979.60 −0.223569
\(429\) 4925.86 0.554366
\(430\) −2924.09 −0.327935
\(431\) −6429.25 −0.718530 −0.359265 0.933236i \(-0.616973\pi\)
−0.359265 + 0.933236i \(0.616973\pi\)
\(432\) −3337.79 −0.371735
\(433\) −8022.03 −0.890333 −0.445166 0.895448i \(-0.646855\pi\)
−0.445166 + 0.895448i \(0.646855\pi\)
\(434\) 0 0
\(435\) 1562.40 0.172210
\(436\) 1645.30 0.180724
\(437\) 9648.50 1.05618
\(438\) 14192.1 1.54823
\(439\) 5569.88 0.605549 0.302774 0.953062i \(-0.402087\pi\)
0.302774 + 0.953062i \(0.402087\pi\)
\(440\) −4606.51 −0.499106
\(441\) 0 0
\(442\) −4362.77 −0.469493
\(443\) −5486.21 −0.588392 −0.294196 0.955745i \(-0.595052\pi\)
−0.294196 + 0.955745i \(0.595052\pi\)
\(444\) 3208.19 0.342914
\(445\) 851.127 0.0906681
\(446\) 16469.3 1.74853
\(447\) 2185.92 0.231299
\(448\) 0 0
\(449\) −7232.67 −0.760203 −0.380101 0.924945i \(-0.624111\pi\)
−0.380101 + 0.924945i \(0.624111\pi\)
\(450\) 1119.24 0.117248
\(451\) 12982.3 1.35546
\(452\) −1800.05 −0.187317
\(453\) −6852.76 −0.710752
\(454\) 2626.32 0.271496
\(455\) 0 0
\(456\) 7090.16 0.728130
\(457\) −2900.51 −0.296893 −0.148446 0.988920i \(-0.547427\pi\)
−0.148446 + 0.988920i \(0.547427\pi\)
\(458\) −10623.3 −1.08383
\(459\) 5624.31 0.571940
\(460\) −1433.04 −0.145251
\(461\) −6073.57 −0.613611 −0.306805 0.951772i \(-0.599260\pi\)
−0.306805 + 0.951772i \(0.599260\pi\)
\(462\) 0 0
\(463\) −18922.8 −1.89939 −0.949693 0.313183i \(-0.898605\pi\)
−0.949693 + 0.313183i \(0.898605\pi\)
\(464\) 2429.88 0.243113
\(465\) 6475.74 0.645817
\(466\) 1573.27 0.156396
\(467\) 6776.71 0.671496 0.335748 0.941952i \(-0.391011\pi\)
0.335748 + 0.941952i \(0.391011\pi\)
\(468\) 439.963 0.0434558
\(469\) 0 0
\(470\) −150.961 −0.0148155
\(471\) 3495.50 0.341962
\(472\) −14835.3 −1.44672
\(473\) −8651.96 −0.841052
\(474\) −2689.99 −0.260666
\(475\) 1105.63 0.106800
\(476\) 0 0
\(477\) −3618.90 −0.347375
\(478\) −13069.7 −1.25062
\(479\) −2397.32 −0.228677 −0.114338 0.993442i \(-0.536475\pi\)
−0.114338 + 0.993442i \(0.536475\pi\)
\(480\) −1957.59 −0.186148
\(481\) −7096.09 −0.672669
\(482\) −12.5871 −0.00118948
\(483\) 0 0
\(484\) −173.977 −0.0163390
\(485\) −5299.37 −0.496148
\(486\) 10933.4 1.02047
\(487\) 5586.17 0.519781 0.259890 0.965638i \(-0.416314\pi\)
0.259890 + 0.965638i \(0.416314\pi\)
\(488\) 7724.35 0.716526
\(489\) −6674.33 −0.617227
\(490\) 0 0
\(491\) 537.392 0.0493934 0.0246967 0.999695i \(-0.492138\pi\)
0.0246967 + 0.999695i \(0.492138\pi\)
\(492\) 2967.78 0.271947
\(493\) −4094.46 −0.374047
\(494\) −2212.03 −0.201466
\(495\) 3311.67 0.300704
\(496\) 10071.2 0.911716
\(497\) 0 0
\(498\) −17842.0 −1.60546
\(499\) 598.965 0.0537342 0.0268671 0.999639i \(-0.491447\pi\)
0.0268671 + 0.999639i \(0.491447\pi\)
\(500\) −164.214 −0.0146877
\(501\) −2392.62 −0.213362
\(502\) 1416.81 0.125966
\(503\) −4426.76 −0.392405 −0.196202 0.980563i \(-0.562861\pi\)
−0.196202 + 0.980563i \(0.562861\pi\)
\(504\) 0 0
\(505\) 1209.17 0.106549
\(506\) 21580.8 1.89601
\(507\) 12134.4 1.06293
\(508\) −1594.17 −0.139232
\(509\) −17727.7 −1.54374 −0.771872 0.635779i \(-0.780679\pi\)
−0.771872 + 0.635779i \(0.780679\pi\)
\(510\) −7507.14 −0.651808
\(511\) 0 0
\(512\) −13017.7 −1.12365
\(513\) 2851.66 0.245427
\(514\) 4588.77 0.393778
\(515\) 8397.88 0.718553
\(516\) −1977.86 −0.168741
\(517\) −446.671 −0.0379972
\(518\) 0 0
\(519\) 21925.4 1.85437
\(520\) 2329.23 0.196430
\(521\) −8662.79 −0.728453 −0.364226 0.931310i \(-0.618667\pi\)
−0.364226 + 0.931310i \(0.618667\pi\)
\(522\) −2101.53 −0.176210
\(523\) 7770.40 0.649667 0.324833 0.945771i \(-0.394692\pi\)
0.324833 + 0.945771i \(0.394692\pi\)
\(524\) −2604.32 −0.217119
\(525\) 0 0
\(526\) −3100.60 −0.257020
\(527\) −16970.4 −1.40274
\(528\) 13182.2 1.08652
\(529\) 35429.6 2.91194
\(530\) −2702.40 −0.221480
\(531\) 10665.2 0.871624
\(532\) 0 0
\(533\) −6564.34 −0.533458
\(534\) −2930.12 −0.237451
\(535\) 7534.41 0.608861
\(536\) −349.440 −0.0281595
\(537\) −19829.6 −1.59350
\(538\) −8404.56 −0.673506
\(539\) 0 0
\(540\) −423.541 −0.0337524
\(541\) 21641.0 1.71981 0.859906 0.510453i \(-0.170522\pi\)
0.859906 + 0.510453i \(0.170522\pi\)
\(542\) 2317.23 0.183642
\(543\) 9734.41 0.769325
\(544\) 5130.09 0.404321
\(545\) −6262.05 −0.492177
\(546\) 0 0
\(547\) 7489.29 0.585409 0.292705 0.956203i \(-0.405445\pi\)
0.292705 + 0.956203i \(0.405445\pi\)
\(548\) −2904.54 −0.226416
\(549\) −5553.11 −0.431696
\(550\) 2472.97 0.191723
\(551\) −2075.99 −0.160508
\(552\) 34976.2 2.69689
\(553\) 0 0
\(554\) −999.566 −0.0766561
\(555\) −12210.4 −0.933881
\(556\) 693.689 0.0529118
\(557\) 25297.9 1.92443 0.962214 0.272295i \(-0.0877826\pi\)
0.962214 + 0.272295i \(0.0877826\pi\)
\(558\) −8710.29 −0.660817
\(559\) 4374.77 0.331007
\(560\) 0 0
\(561\) −22212.5 −1.67168
\(562\) −8623.85 −0.647286
\(563\) −15661.3 −1.17237 −0.586186 0.810177i \(-0.699371\pi\)
−0.586186 + 0.810177i \(0.699371\pi\)
\(564\) −102.110 −0.00762343
\(565\) 6851.02 0.510132
\(566\) −13994.9 −1.03931
\(567\) 0 0
\(568\) 22927.3 1.69367
\(569\) −9982.75 −0.735498 −0.367749 0.929925i \(-0.619871\pi\)
−0.367749 + 0.929925i \(0.619871\pi\)
\(570\) −3806.30 −0.279699
\(571\) −11583.6 −0.848966 −0.424483 0.905436i \(-0.639544\pi\)
−0.424483 + 0.905436i \(0.639544\pi\)
\(572\) 972.103 0.0710589
\(573\) 2495.81 0.181961
\(574\) 0 0
\(575\) 5454.16 0.395573
\(576\) 9802.97 0.709127
\(577\) 595.378 0.0429565 0.0214783 0.999769i \(-0.493163\pi\)
0.0214783 + 0.999769i \(0.493163\pi\)
\(578\) 6969.39 0.501537
\(579\) −4879.66 −0.350245
\(580\) 308.335 0.0220740
\(581\) 0 0
\(582\) 18243.8 1.29936
\(583\) −7996.00 −0.568028
\(584\) 19856.4 1.40696
\(585\) −1674.51 −0.118346
\(586\) −729.738 −0.0514423
\(587\) 15750.3 1.10747 0.553736 0.832693i \(-0.313202\pi\)
0.553736 + 0.832693i \(0.313202\pi\)
\(588\) 0 0
\(589\) −8604.42 −0.601934
\(590\) 7964.22 0.555732
\(591\) 13934.4 0.969856
\(592\) −18990.0 −1.31838
\(593\) 417.878 0.0289379 0.0144690 0.999895i \(-0.495394\pi\)
0.0144690 + 0.999895i \(0.495394\pi\)
\(594\) 6378.31 0.440581
\(595\) 0 0
\(596\) 431.385 0.0296480
\(597\) 19072.2 1.30749
\(598\) −10912.1 −0.746201
\(599\) −19997.3 −1.36406 −0.682028 0.731326i \(-0.738902\pi\)
−0.682028 + 0.731326i \(0.738902\pi\)
\(600\) 4007.97 0.272708
\(601\) 15992.6 1.08545 0.542723 0.839912i \(-0.317393\pi\)
0.542723 + 0.839912i \(0.317393\pi\)
\(602\) 0 0
\(603\) 251.216 0.0169657
\(604\) −1352.37 −0.0911045
\(605\) 662.162 0.0444970
\(606\) −4162.73 −0.279042
\(607\) −14159.2 −0.946793 −0.473396 0.880850i \(-0.656972\pi\)
−0.473396 + 0.880850i \(0.656972\pi\)
\(608\) 2601.08 0.173499
\(609\) 0 0
\(610\) −4146.76 −0.275242
\(611\) 225.854 0.0149543
\(612\) −1983.96 −0.131040
\(613\) −4629.41 −0.305025 −0.152512 0.988302i \(-0.548736\pi\)
−0.152512 + 0.988302i \(0.548736\pi\)
\(614\) −4963.87 −0.326263
\(615\) −11295.4 −0.740611
\(616\) 0 0
\(617\) −23165.3 −1.51151 −0.755753 0.654857i \(-0.772729\pi\)
−0.755753 + 0.654857i \(0.772729\pi\)
\(618\) −28910.9 −1.88182
\(619\) −12370.6 −0.803258 −0.401629 0.915803i \(-0.631556\pi\)
−0.401629 + 0.915803i \(0.631556\pi\)
\(620\) 1277.97 0.0827812
\(621\) 14067.4 0.909028
\(622\) −3137.36 −0.202245
\(623\) 0 0
\(624\) −6665.43 −0.427613
\(625\) 625.000 0.0400000
\(626\) 3708.02 0.236745
\(627\) −11262.3 −0.717341
\(628\) 689.826 0.0438329
\(629\) 31998.9 2.02842
\(630\) 0 0
\(631\) 13980.2 0.882002 0.441001 0.897507i \(-0.354623\pi\)
0.441001 + 0.897507i \(0.354623\pi\)
\(632\) −3763.61 −0.236880
\(633\) −37568.9 −2.35897
\(634\) 16799.7 1.05237
\(635\) 6067.45 0.379180
\(636\) −1827.91 −0.113964
\(637\) 0 0
\(638\) −4643.36 −0.288139
\(639\) −16482.7 −1.02041
\(640\) 4967.75 0.306825
\(641\) −16060.9 −0.989655 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(642\) −25938.3 −1.59455
\(643\) 4502.17 0.276125 0.138063 0.990424i \(-0.455913\pi\)
0.138063 + 0.990424i \(0.455913\pi\)
\(644\) 0 0
\(645\) 7527.79 0.459545
\(646\) 9974.87 0.607517
\(647\) −29414.8 −1.78735 −0.893675 0.448715i \(-0.851882\pi\)
−0.893675 + 0.448715i \(0.851882\pi\)
\(648\) 21595.6 1.30919
\(649\) 23565.0 1.42528
\(650\) −1250.43 −0.0754553
\(651\) 0 0
\(652\) −1317.16 −0.0791164
\(653\) 13013.6 0.779882 0.389941 0.920840i \(-0.372495\pi\)
0.389941 + 0.920840i \(0.372495\pi\)
\(654\) 21558.0 1.28897
\(655\) 9912.09 0.591294
\(656\) −17566.9 −1.04554
\(657\) −14275.0 −0.847671
\(658\) 0 0
\(659\) 23474.2 1.38759 0.693797 0.720171i \(-0.255937\pi\)
0.693797 + 0.720171i \(0.255937\pi\)
\(660\) 1672.72 0.0986526
\(661\) 9266.36 0.545264 0.272632 0.962118i \(-0.412106\pi\)
0.272632 + 0.962118i \(0.412106\pi\)
\(662\) −25040.4 −1.47013
\(663\) 11231.5 0.657914
\(664\) −24963.0 −1.45896
\(665\) 0 0
\(666\) 16423.8 0.955572
\(667\) −10241.0 −0.594501
\(668\) −472.176 −0.0273489
\(669\) −42398.6 −2.45026
\(670\) 187.595 0.0108170
\(671\) −12269.7 −0.705909
\(672\) 0 0
\(673\) −25067.2 −1.43576 −0.717882 0.696164i \(-0.754888\pi\)
−0.717882 + 0.696164i \(0.754888\pi\)
\(674\) 75.3288 0.00430498
\(675\) 1612.01 0.0919202
\(676\) 2394.68 0.136247
\(677\) 22409.6 1.27219 0.636093 0.771613i \(-0.280550\pi\)
0.636093 + 0.771613i \(0.280550\pi\)
\(678\) −23585.6 −1.33599
\(679\) 0 0
\(680\) −10503.4 −0.592332
\(681\) −6761.20 −0.380455
\(682\) −19245.5 −1.08057
\(683\) −8757.53 −0.490626 −0.245313 0.969444i \(-0.578891\pi\)
−0.245313 + 0.969444i \(0.578891\pi\)
\(684\) −1005.91 −0.0562311
\(685\) 11054.7 0.616613
\(686\) 0 0
\(687\) 27348.7 1.51880
\(688\) 11707.4 0.648750
\(689\) 4043.09 0.223555
\(690\) −18776.7 −1.03597
\(691\) −8468.42 −0.466214 −0.233107 0.972451i \(-0.574889\pi\)
−0.233107 + 0.972451i \(0.574889\pi\)
\(692\) 4326.90 0.237694
\(693\) 0 0
\(694\) −20294.8 −1.11006
\(695\) −2640.19 −0.144098
\(696\) −7525.54 −0.409849
\(697\) 29601.0 1.60864
\(698\) 26555.1 1.44001
\(699\) −4050.23 −0.219162
\(700\) 0 0
\(701\) 15996.9 0.861906 0.430953 0.902374i \(-0.358177\pi\)
0.430953 + 0.902374i \(0.358177\pi\)
\(702\) −3225.12 −0.173397
\(703\) 16224.2 0.870423
\(704\) 21659.8 1.15956
\(705\) 388.633 0.0207614
\(706\) −7240.03 −0.385952
\(707\) 0 0
\(708\) 5387.02 0.285956
\(709\) 19903.0 1.05426 0.527131 0.849784i \(-0.323268\pi\)
0.527131 + 0.849784i \(0.323268\pi\)
\(710\) −12308.3 −0.650597
\(711\) 2705.70 0.142717
\(712\) −4099.58 −0.215784
\(713\) −42446.1 −2.22948
\(714\) 0 0
\(715\) −3699.84 −0.193519
\(716\) −3913.30 −0.204256
\(717\) 33646.7 1.75252
\(718\) −8179.59 −0.425153
\(719\) 11073.1 0.574347 0.287174 0.957879i \(-0.407284\pi\)
0.287174 + 0.957879i \(0.407284\pi\)
\(720\) −4481.18 −0.231949
\(721\) 0 0
\(722\) −12678.4 −0.653520
\(723\) 32.4043 0.00166685
\(724\) 1921.06 0.0986125
\(725\) −1173.53 −0.0601155
\(726\) −2279.58 −0.116533
\(727\) −31652.7 −1.61476 −0.807382 0.590029i \(-0.799116\pi\)
−0.807382 + 0.590029i \(0.799116\pi\)
\(728\) 0 0
\(729\) −3935.94 −0.199967
\(730\) −10659.8 −0.540460
\(731\) −19727.5 −0.998149
\(732\) −2804.88 −0.141628
\(733\) −16958.3 −0.854528 −0.427264 0.904127i \(-0.640523\pi\)
−0.427264 + 0.904127i \(0.640523\pi\)
\(734\) −8230.16 −0.413870
\(735\) 0 0
\(736\) 12831.3 0.642618
\(737\) 555.065 0.0277423
\(738\) 15193.1 0.757813
\(739\) −11616.6 −0.578245 −0.289123 0.957292i \(-0.593364\pi\)
−0.289123 + 0.957292i \(0.593364\pi\)
\(740\) −2409.69 −0.119705
\(741\) 5694.66 0.282319
\(742\) 0 0
\(743\) 15928.0 0.786464 0.393232 0.919439i \(-0.371357\pi\)
0.393232 + 0.919439i \(0.371357\pi\)
\(744\) −31191.4 −1.53700
\(745\) −1641.86 −0.0807423
\(746\) −6762.19 −0.331879
\(747\) 17946.1 0.879003
\(748\) −4383.57 −0.214277
\(749\) 0 0
\(750\) −2151.65 −0.104756
\(751\) 25571.9 1.24252 0.621260 0.783604i \(-0.286621\pi\)
0.621260 + 0.783604i \(0.286621\pi\)
\(752\) 604.412 0.0293094
\(753\) −3647.43 −0.176520
\(754\) 2347.87 0.113401
\(755\) 5147.14 0.248111
\(756\) 0 0
\(757\) 6202.41 0.297794 0.148897 0.988853i \(-0.452428\pi\)
0.148897 + 0.988853i \(0.452428\pi\)
\(758\) −1738.77 −0.0833179
\(759\) −55557.6 −2.65693
\(760\) −5325.46 −0.254177
\(761\) 29199.1 1.39089 0.695444 0.718580i \(-0.255208\pi\)
0.695444 + 0.718580i \(0.255208\pi\)
\(762\) −20888.1 −0.993037
\(763\) 0 0
\(764\) 492.539 0.0233239
\(765\) 7550.98 0.356871
\(766\) −3025.00 −0.142686
\(767\) −11915.4 −0.560938
\(768\) 13050.5 0.613177
\(769\) −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
\(772\) −962.985 −0.0448945
\(773\) 25544.8 1.18859 0.594296 0.804246i \(-0.297431\pi\)
0.594296 + 0.804246i \(0.297431\pi\)
\(774\) −10125.4 −0.470218
\(775\) −4863.96 −0.225443
\(776\) 25525.2 1.18080
\(777\) 0 0
\(778\) −2901.82 −0.133721
\(779\) 15008.4 0.690286
\(780\) −845.795 −0.0388261
\(781\) −36418.6 −1.66858
\(782\) 49206.6 2.25016
\(783\) −3026.77 −0.138146
\(784\) 0 0
\(785\) −2625.49 −0.119373
\(786\) −34123.8 −1.54854
\(787\) 37223.2 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(788\) 2749.91 0.124317
\(789\) 7982.19 0.360169
\(790\) 2020.47 0.0909938
\(791\) 0 0
\(792\) −15951.1 −0.715655
\(793\) 6204.03 0.277820
\(794\) 5135.18 0.229522
\(795\) 6957.06 0.310366
\(796\) 3763.83 0.167595
\(797\) 40384.6 1.79485 0.897425 0.441168i \(-0.145436\pi\)
0.897425 + 0.441168i \(0.145436\pi\)
\(798\) 0 0
\(799\) −1018.46 −0.0450945
\(800\) 1470.35 0.0649811
\(801\) 2947.23 0.130007
\(802\) −10788.9 −0.475023
\(803\) −31540.7 −1.38611
\(804\) 126.889 0.00556598
\(805\) 0 0
\(806\) 9731.28 0.425272
\(807\) 21636.7 0.943803
\(808\) −5824.14 −0.253580
\(809\) −1955.76 −0.0849948 −0.0424974 0.999097i \(-0.513531\pi\)
−0.0424974 + 0.999097i \(0.513531\pi\)
\(810\) −11593.4 −0.502904
\(811\) 34301.8 1.48520 0.742600 0.669735i \(-0.233592\pi\)
0.742600 + 0.669735i \(0.233592\pi\)
\(812\) 0 0
\(813\) −5965.49 −0.257342
\(814\) 36288.7 1.56255
\(815\) 5013.13 0.215463
\(816\) 30056.9 1.28946
\(817\) −10002.3 −0.428319
\(818\) 30255.8 1.29324
\(819\) 0 0
\(820\) −2229.12 −0.0949319
\(821\) −13665.6 −0.580918 −0.290459 0.956887i \(-0.593808\pi\)
−0.290459 + 0.956887i \(0.593808\pi\)
\(822\) −38057.5 −1.61485
\(823\) −21519.5 −0.911449 −0.455724 0.890121i \(-0.650620\pi\)
−0.455724 + 0.890121i \(0.650620\pi\)
\(824\) −40449.7 −1.71011
\(825\) −6366.42 −0.268667
\(826\) 0 0
\(827\) 35220.6 1.48094 0.740471 0.672088i \(-0.234602\pi\)
0.740471 + 0.672088i \(0.234602\pi\)
\(828\) −4962.23 −0.208272
\(829\) −31365.5 −1.31408 −0.657039 0.753857i \(-0.728191\pi\)
−0.657039 + 0.753857i \(0.728191\pi\)
\(830\) 13401.2 0.560437
\(831\) 2573.28 0.107420
\(832\) −10952.0 −0.456362
\(833\) 0 0
\(834\) 9089.24 0.377380
\(835\) 1797.11 0.0744810
\(836\) −2222.58 −0.0919491
\(837\) −12545.2 −0.518070
\(838\) −7067.48 −0.291339
\(839\) 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
\(842\) 35035.8 1.43398
\(843\) 22201.2 0.907060
\(844\) −7414.11 −0.302374
\(845\) −9114.21 −0.371051
\(846\) −522.738 −0.0212436
\(847\) 0 0
\(848\) 10819.8 0.438152
\(849\) 36028.6 1.45642
\(850\) 5638.66 0.227534
\(851\) 80035.0 3.22393
\(852\) −8325.40 −0.334769
\(853\) −9405.41 −0.377533 −0.188766 0.982022i \(-0.560449\pi\)
−0.188766 + 0.982022i \(0.560449\pi\)
\(854\) 0 0
\(855\) 3828.53 0.153138
\(856\) −36290.6 −1.44905
\(857\) 27966.9 1.11474 0.557369 0.830265i \(-0.311811\pi\)
0.557369 + 0.830265i \(0.311811\pi\)
\(858\) 12737.2 0.506809
\(859\) −6281.11 −0.249486 −0.124743 0.992189i \(-0.539811\pi\)
−0.124743 + 0.992189i \(0.539811\pi\)
\(860\) 1485.58 0.0589047
\(861\) 0 0
\(862\) −16624.7 −0.656890
\(863\) −4757.13 −0.187642 −0.0938208 0.995589i \(-0.529908\pi\)
−0.0938208 + 0.995589i \(0.529908\pi\)
\(864\) 3792.35 0.149327
\(865\) −16468.3 −0.647327
\(866\) −20743.3 −0.813954
\(867\) −17942.0 −0.702818
\(868\) 0 0
\(869\) 5978.28 0.233371
\(870\) 4040.04 0.157437
\(871\) −280.663 −0.0109184
\(872\) 30162.1 1.17135
\(873\) −18350.3 −0.711414
\(874\) 24949.0 0.965574
\(875\) 0 0
\(876\) −7210.30 −0.278097
\(877\) −30240.5 −1.16437 −0.582184 0.813057i \(-0.697802\pi\)
−0.582184 + 0.813057i \(0.697802\pi\)
\(878\) 14402.5 0.553601
\(879\) 1878.64 0.0720875
\(880\) −9901.21 −0.379284
\(881\) −44875.5 −1.71611 −0.858056 0.513556i \(-0.828328\pi\)
−0.858056 + 0.513556i \(0.828328\pi\)
\(882\) 0 0
\(883\) 4892.13 0.186448 0.0932238 0.995645i \(-0.470283\pi\)
0.0932238 + 0.995645i \(0.470283\pi\)
\(884\) 2216.51 0.0843317
\(885\) −20503.1 −0.778762
\(886\) −14186.2 −0.537916
\(887\) −1761.40 −0.0666765 −0.0333382 0.999444i \(-0.510614\pi\)
−0.0333382 + 0.999444i \(0.510614\pi\)
\(888\) 58813.4 2.22258
\(889\) 0 0
\(890\) 2200.83 0.0828900
\(891\) −34303.3 −1.28979
\(892\) −8367.22 −0.314075
\(893\) −516.384 −0.0193507
\(894\) 5652.33 0.211457
\(895\) 14894.1 0.556263
\(896\) 0 0
\(897\) 28092.1 1.04567
\(898\) −18702.2 −0.694988
\(899\) 9132.79 0.338816
\(900\) −568.629 −0.0210603
\(901\) −18231.8 −0.674128
\(902\) 33569.3 1.23918
\(903\) 0 0
\(904\) −32999.0 −1.21408
\(905\) −7311.57 −0.268558
\(906\) −17719.8 −0.649779
\(907\) 23689.1 0.867238 0.433619 0.901096i \(-0.357236\pi\)
0.433619 + 0.901096i \(0.357236\pi\)
\(908\) −1334.30 −0.0487669
\(909\) 4187.03 0.152778
\(910\) 0 0
\(911\) −13877.3 −0.504692 −0.252346 0.967637i \(-0.581202\pi\)
−0.252346 + 0.967637i \(0.581202\pi\)
\(912\) 15239.6 0.553325
\(913\) 39652.2 1.43735
\(914\) −7500.09 −0.271423
\(915\) 10675.4 0.385704
\(916\) 5397.18 0.194681
\(917\) 0 0
\(918\) 14543.3 0.522875
\(919\) 14331.6 0.514426 0.257213 0.966355i \(-0.417196\pi\)
0.257213 + 0.966355i \(0.417196\pi\)
\(920\) −26270.8 −0.941438
\(921\) 12779.0 0.457201
\(922\) −15705.0 −0.560971
\(923\) 18414.7 0.656692
\(924\) 0 0
\(925\) 9171.32 0.326001
\(926\) −48930.2 −1.73644
\(927\) 29079.7 1.03032
\(928\) −2760.80 −0.0976592
\(929\) −16668.4 −0.588668 −0.294334 0.955703i \(-0.595098\pi\)
−0.294334 + 0.955703i \(0.595098\pi\)
\(930\) 16744.9 0.590415
\(931\) 0 0
\(932\) −799.300 −0.0280922
\(933\) 8076.82 0.283412
\(934\) 17523.1 0.613891
\(935\) 16684.0 0.583555
\(936\) 8065.52 0.281656
\(937\) −30384.9 −1.05937 −0.529685 0.848194i \(-0.677690\pi\)
−0.529685 + 0.848194i \(0.677690\pi\)
\(938\) 0 0
\(939\) −9545.94 −0.331757
\(940\) 76.6956 0.00266121
\(941\) 1196.35 0.0414452 0.0207226 0.999785i \(-0.493403\pi\)
0.0207226 + 0.999785i \(0.493403\pi\)
\(942\) 9038.63 0.312627
\(943\) 74037.5 2.55673
\(944\) −31886.9 −1.09940
\(945\) 0 0
\(946\) −22372.1 −0.768901
\(947\) 1788.41 0.0613681 0.0306840 0.999529i \(-0.490231\pi\)
0.0306840 + 0.999529i \(0.490231\pi\)
\(948\) 1366.65 0.0468215
\(949\) 15948.2 0.545523
\(950\) 2858.94 0.0976380
\(951\) −43249.2 −1.47471
\(952\) 0 0
\(953\) 8578.60 0.291593 0.145796 0.989315i \(-0.453426\pi\)
0.145796 + 0.989315i \(0.453426\pi\)
\(954\) −9357.70 −0.317575
\(955\) −1874.61 −0.0635194
\(956\) 6640.06 0.224639
\(957\) 11953.9 0.403776
\(958\) −6198.95 −0.209059
\(959\) 0 0
\(960\) −18845.4 −0.633577
\(961\) 8061.99 0.270618
\(962\) −18349.0 −0.614963
\(963\) 26089.7 0.873031
\(964\) 6.39489 0.000213657 0
\(965\) 3665.14 0.122264
\(966\) 0 0
\(967\) −55459.3 −1.84431 −0.922156 0.386818i \(-0.873574\pi\)
−0.922156 + 0.386818i \(0.873574\pi\)
\(968\) −3189.40 −0.105900
\(969\) −25679.3 −0.851330
\(970\) −13703.0 −0.453585
\(971\) −22047.3 −0.728662 −0.364331 0.931270i \(-0.618702\pi\)
−0.364331 + 0.931270i \(0.618702\pi\)
\(972\) −5554.72 −0.183300
\(973\) 0 0
\(974\) 14444.6 0.475191
\(975\) 3219.11 0.105738
\(976\) 16602.7 0.544507
\(977\) 14402.3 0.471617 0.235809 0.971800i \(-0.424226\pi\)
0.235809 + 0.971800i \(0.424226\pi\)
\(978\) −17258.4 −0.564277
\(979\) 6511.94 0.212587
\(980\) 0 0
\(981\) −21683.9 −0.705721
\(982\) 1389.58 0.0451561
\(983\) −7817.11 −0.253639 −0.126819 0.991926i \(-0.540477\pi\)
−0.126819 + 0.991926i \(0.540477\pi\)
\(984\) 54406.2 1.76261
\(985\) −10466.2 −0.338560
\(986\) −10587.4 −0.341959
\(987\) 0 0
\(988\) 1123.82 0.0361878
\(989\) −49342.0 −1.58643
\(990\) 8563.26 0.274907
\(991\) 24501.6 0.785386 0.392693 0.919670i \(-0.371543\pi\)
0.392693 + 0.919670i \(0.371543\pi\)
\(992\) −11442.8 −0.366239
\(993\) 64464.2 2.06013
\(994\) 0 0
\(995\) −14325.2 −0.456421
\(996\) 9064.61 0.288377
\(997\) 50696.0 1.61039 0.805195 0.593010i \(-0.202060\pi\)
0.805195 + 0.593010i \(0.202060\pi\)
\(998\) 1548.80 0.0491245
\(999\) 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 245.4.a.k.1.1 2
3.2 odd 2 2205.4.a.u.1.2 2
5.4 even 2 1225.4.a.m.1.2 2
7.2 even 3 245.4.e.i.116.2 4
7.3 odd 6 245.4.e.h.226.2 4
7.4 even 3 245.4.e.i.226.2 4
7.5 odd 6 245.4.e.h.116.2 4
7.6 odd 2 35.4.a.b.1.1 2
21.20 even 2 315.4.a.f.1.2 2
28.27 even 2 560.4.a.r.1.1 2
35.13 even 4 175.4.b.c.99.2 4
35.27 even 4 175.4.b.c.99.3 4
35.34 odd 2 175.4.a.c.1.2 2
56.13 odd 2 2240.4.a.bn.1.1 2
56.27 even 2 2240.4.a.bo.1.2 2
105.104 even 2 1575.4.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.1 2 7.6 odd 2
175.4.a.c.1.2 2 35.34 odd 2
175.4.b.c.99.2 4 35.13 even 4
175.4.b.c.99.3 4 35.27 even 4
245.4.a.k.1.1 2 1.1 even 1 trivial
245.4.e.h.116.2 4 7.5 odd 6
245.4.e.h.226.2 4 7.3 odd 6
245.4.e.i.116.2 4 7.2 even 3
245.4.e.i.226.2 4 7.4 even 3
315.4.a.f.1.2 2 21.20 even 2
560.4.a.r.1.1 2 28.27 even 2
1225.4.a.m.1.2 2 5.4 even 2
1575.4.a.z.1.1 2 105.104 even 2
2205.4.a.u.1.2 2 3.2 odd 2
2240.4.a.bn.1.1 2 56.13 odd 2
2240.4.a.bo.1.2 2 56.27 even 2