Properties

Label 245.4.a.k.1.2
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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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [245,4,Mod(1,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(14.4554679514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 245.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.41421 q^{2} +4.65685 q^{3} +21.3137 q^{4} +5.00000 q^{5} +25.2132 q^{6} +72.0833 q^{8} -5.31371 q^{9} +O(q^{10})\) \(q+5.41421 q^{2} +4.65685 q^{3} +21.3137 q^{4} +5.00000 q^{5} +25.2132 q^{6} +72.0833 q^{8} -5.31371 q^{9} +27.0711 q^{10} -52.2548 q^{11} +99.2548 q^{12} -30.6569 q^{13} +23.2843 q^{15} +219.765 q^{16} -37.2254 q^{17} -28.7696 q^{18} -80.2254 q^{19} +106.569 q^{20} -282.919 q^{22} +25.8335 q^{23} +335.681 q^{24} +25.0000 q^{25} -165.983 q^{26} -150.480 q^{27} +20.9411 q^{29} +126.066 q^{30} +314.558 q^{31} +613.186 q^{32} -243.343 q^{33} -201.546 q^{34} -113.255 q^{36} +197.147 q^{37} -434.357 q^{38} -142.765 q^{39} +360.416 q^{40} -11.3625 q^{41} -33.8335 q^{43} -1113.74 q^{44} -26.5685 q^{45} +139.868 q^{46} +361.676 q^{47} +1023.41 q^{48} +135.355 q^{50} -173.353 q^{51} -653.411 q^{52} +153.019 q^{53} -814.732 q^{54} -261.274 q^{55} -373.598 q^{57} +113.380 q^{58} +616.000 q^{59} +496.274 q^{60} -15.2649 q^{61} +1703.09 q^{62} +1561.80 q^{64} -153.284 q^{65} -1317.51 q^{66} -166.510 q^{67} -793.411 q^{68} +120.303 q^{69} -952.000 q^{71} -383.029 q^{72} +148.489 q^{73} +1067.40 q^{74} +116.421 q^{75} -1709.90 q^{76} -772.958 q^{78} +857.725 q^{79} +1098.82 q^{80} -557.294 q^{81} -61.5189 q^{82} -660.528 q^{83} -186.127 q^{85} -183.182 q^{86} +97.5198 q^{87} -3766.70 q^{88} +45.7746 q^{89} -143.848 q^{90} +550.607 q^{92} +1464.85 q^{93} +1958.19 q^{94} -401.127 q^{95} +2855.52 q^{96} -1682.13 q^{97} +277.667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} - 2 q^{3} + 20 q^{4} + 10 q^{5} + 8 q^{6} + 48 q^{8} + 12 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.41421 1.91421 0.957107 0.289735i \(-0.0935673\pi\)
0.957107 + 0.289735i \(0.0935673\pi\)
\(3\) 4.65685 0.896212 0.448106 0.893980i \(-0.352099\pi\)
0.448106 + 0.893980i \(0.352099\pi\)
\(4\) 21.3137 2.66421
\(5\) 5.00000 0.447214
\(6\) 25.2132 1.71554
\(7\) 0 0
\(8\) 72.0833 3.18566
\(9\) −5.31371 −0.196804
\(10\) 27.0711 0.856062
\(11\) −52.2548 −1.43231 −0.716156 0.697941i \(-0.754100\pi\)
−0.716156 + 0.697941i \(0.754100\pi\)
\(12\) 99.2548 2.38770
\(13\) −30.6569 −0.654052 −0.327026 0.945015i \(-0.606047\pi\)
−0.327026 + 0.945015i \(0.606047\pi\)
\(14\) 0 0
\(15\) 23.2843 0.400798
\(16\) 219.765 3.43382
\(17\) −37.2254 −0.531087 −0.265544 0.964099i \(-0.585551\pi\)
−0.265544 + 0.964099i \(0.585551\pi\)
\(18\) −28.7696 −0.376725
\(19\) −80.2254 −0.968683 −0.484341 0.874879i \(-0.660941\pi\)
−0.484341 + 0.874879i \(0.660941\pi\)
\(20\) 106.569 1.19147
\(21\) 0 0
\(22\) −282.919 −2.74175
\(23\) 25.8335 0.234202 0.117101 0.993120i \(-0.462640\pi\)
0.117101 + 0.993120i \(0.462640\pi\)
\(24\) 335.681 2.85503
\(25\) 25.0000 0.200000
\(26\) −165.983 −1.25200
\(27\) −150.480 −1.07259
\(28\) 0 0
\(29\) 20.9411 0.134092 0.0670460 0.997750i \(-0.478643\pi\)
0.0670460 + 0.997750i \(0.478643\pi\)
\(30\) 126.066 0.767213
\(31\) 314.558 1.82246 0.911232 0.411894i \(-0.135133\pi\)
0.911232 + 0.411894i \(0.135133\pi\)
\(32\) 613.186 3.38741
\(33\) −243.343 −1.28365
\(34\) −201.546 −1.01661
\(35\) 0 0
\(36\) −113.255 −0.524328
\(37\) 197.147 0.875968 0.437984 0.898983i \(-0.355693\pi\)
0.437984 + 0.898983i \(0.355693\pi\)
\(38\) −434.357 −1.85427
\(39\) −142.765 −0.586170
\(40\) 360.416 1.42467
\(41\) −11.3625 −0.0432810 −0.0216405 0.999766i \(-0.506889\pi\)
−0.0216405 + 0.999766i \(0.506889\pi\)
\(42\) 0 0
\(43\) −33.8335 −0.119990 −0.0599948 0.998199i \(-0.519108\pi\)
−0.0599948 + 0.998199i \(0.519108\pi\)
\(44\) −1113.74 −3.81598
\(45\) −26.5685 −0.0880134
\(46\) 139.868 0.448313
\(47\) 361.676 1.12247 0.561233 0.827658i \(-0.310327\pi\)
0.561233 + 0.827658i \(0.310327\pi\)
\(48\) 1023.41 3.07743
\(49\) 0 0
\(50\) 135.355 0.382843
\(51\) −173.353 −0.475967
\(52\) −653.411 −1.74254
\(53\) 153.019 0.396582 0.198291 0.980143i \(-0.436461\pi\)
0.198291 + 0.980143i \(0.436461\pi\)
\(54\) −814.732 −2.05317
\(55\) −261.274 −0.640549
\(56\) 0 0
\(57\) −373.598 −0.868145
\(58\) 113.380 0.256681
\(59\) 616.000 1.35926 0.679630 0.733555i \(-0.262140\pi\)
0.679630 + 0.733555i \(0.262140\pi\)
\(60\) 496.274 1.06781
\(61\) −15.2649 −0.0320406 −0.0160203 0.999872i \(-0.505100\pi\)
−0.0160203 + 0.999872i \(0.505100\pi\)
\(62\) 1703.09 3.48858
\(63\) 0 0
\(64\) 1561.80 3.05040
\(65\) −153.284 −0.292501
\(66\) −1317.51 −2.45719
\(67\) −166.510 −0.303618 −0.151809 0.988410i \(-0.548510\pi\)
−0.151809 + 0.988410i \(0.548510\pi\)
\(68\) −793.411 −1.41493
\(69\) 120.303 0.209895
\(70\) 0 0
\(71\) −952.000 −1.59129 −0.795645 0.605763i \(-0.792868\pi\)
−0.795645 + 0.605763i \(0.792868\pi\)
\(72\) −383.029 −0.626951
\(73\) 148.489 0.238074 0.119037 0.992890i \(-0.462019\pi\)
0.119037 + 0.992890i \(0.462019\pi\)
\(74\) 1067.40 1.67679
\(75\) 116.421 0.179242
\(76\) −1709.90 −2.58078
\(77\) 0 0
\(78\) −772.958 −1.12205
\(79\) 857.725 1.22154 0.610770 0.791808i \(-0.290860\pi\)
0.610770 + 0.791808i \(0.290860\pi\)
\(80\) 1098.82 1.53565
\(81\) −557.294 −0.764464
\(82\) −61.5189 −0.0828491
\(83\) −660.528 −0.873523 −0.436761 0.899577i \(-0.643875\pi\)
−0.436761 + 0.899577i \(0.643875\pi\)
\(84\) 0 0
\(85\) −186.127 −0.237509
\(86\) −183.182 −0.229686
\(87\) 97.5198 0.120175
\(88\) −3766.70 −4.56286
\(89\) 45.7746 0.0545180 0.0272590 0.999628i \(-0.491322\pi\)
0.0272590 + 0.999628i \(0.491322\pi\)
\(90\) −143.848 −0.168477
\(91\) 0 0
\(92\) 550.607 0.623965
\(93\) 1464.85 1.63331
\(94\) 1958.19 2.14864
\(95\) −401.127 −0.433208
\(96\) 2855.52 3.03583
\(97\) −1682.13 −1.76076 −0.880382 0.474265i \(-0.842714\pi\)
−0.880382 + 0.474265i \(0.842714\pi\)
\(98\) 0 0
\(99\) 277.667 0.281885
\(100\) 532.843 0.532843
\(101\) 434.167 0.427734 0.213867 0.976863i \(-0.431394\pi\)
0.213867 + 0.976863i \(0.431394\pi\)
\(102\) −938.572 −0.911102
\(103\) −345.577 −0.330589 −0.165295 0.986244i \(-0.552858\pi\)
−0.165295 + 0.986244i \(0.552858\pi\)
\(104\) −2209.85 −2.08359
\(105\) 0 0
\(106\) 828.479 0.759142
\(107\) 217.119 0.196165 0.0980825 0.995178i \(-0.468729\pi\)
0.0980825 + 0.995178i \(0.468729\pi\)
\(108\) −3207.29 −2.85761
\(109\) 1734.41 1.52409 0.762047 0.647521i \(-0.224194\pi\)
0.762047 + 0.647521i \(0.224194\pi\)
\(110\) −1414.59 −1.22615
\(111\) 918.086 0.785053
\(112\) 0 0
\(113\) −1854.20 −1.54362 −0.771809 0.635855i \(-0.780648\pi\)
−0.771809 + 0.635855i \(0.780648\pi\)
\(114\) −2022.74 −1.66181
\(115\) 129.167 0.104738
\(116\) 446.333 0.357250
\(117\) 162.902 0.128720
\(118\) 3335.16 2.60191
\(119\) 0 0
\(120\) 1678.41 1.27681
\(121\) 1399.57 1.05152
\(122\) −82.6476 −0.0613325
\(123\) −52.9134 −0.0387890
\(124\) 6704.41 4.85543
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) 1394.51 0.974352 0.487176 0.873304i \(-0.338027\pi\)
0.487176 + 0.873304i \(0.338027\pi\)
\(128\) 3550.45 2.45171
\(129\) −157.558 −0.107536
\(130\) −829.914 −0.559910
\(131\) −1762.42 −1.17544 −0.587722 0.809063i \(-0.699975\pi\)
−0.587722 + 0.809063i \(0.699975\pi\)
\(132\) −5186.54 −3.41993
\(133\) 0 0
\(134\) −901.519 −0.581189
\(135\) −752.401 −0.479677
\(136\) −2683.33 −1.69186
\(137\) −922.949 −0.575568 −0.287784 0.957695i \(-0.592919\pi\)
−0.287784 + 0.957695i \(0.592919\pi\)
\(138\) 651.345 0.401784
\(139\) 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
\(142\) −5154.33 −3.04607
\(143\) 1601.97 0.936807
\(144\) −1167.76 −0.675790
\(145\) 104.706 0.0599678
\(146\) 803.954 0.455724
\(147\) 0 0
\(148\) 4201.94 2.33376
\(149\) 780.372 0.429064 0.214532 0.976717i \(-0.431177\pi\)
0.214532 + 0.976717i \(0.431177\pi\)
\(150\) 630.330 0.343108
\(151\) −2319.43 −1.25002 −0.625008 0.780618i \(-0.714904\pi\)
−0.625008 + 0.780618i \(0.714904\pi\)
\(152\) −5782.91 −3.08589
\(153\) 197.805 0.104520
\(154\) 0 0
\(155\) 1572.79 0.815030
\(156\) −3042.84 −1.56168
\(157\) −1022.90 −0.519977 −0.259989 0.965612i \(-0.583719\pi\)
−0.259989 + 0.965612i \(0.583719\pi\)
\(158\) 4643.91 2.33829
\(159\) 712.589 0.355421
\(160\) 3065.93 1.51489
\(161\) 0 0
\(162\) −3017.31 −1.46335
\(163\) −1350.63 −0.649013 −0.324507 0.945883i \(-0.605198\pi\)
−0.324507 + 0.945883i \(0.605198\pi\)
\(164\) −242.177 −0.115310
\(165\) −1216.72 −0.574068
\(166\) −3576.24 −1.67211
\(167\) 1230.58 0.570209 0.285105 0.958496i \(-0.407972\pi\)
0.285105 + 0.958496i \(0.407972\pi\)
\(168\) 0 0
\(169\) −1257.16 −0.572215
\(170\) −1007.73 −0.454644
\(171\) 426.294 0.190641
\(172\) −721.117 −0.319678
\(173\) 2487.65 1.09325 0.546626 0.837377i \(-0.315912\pi\)
0.546626 + 0.837377i \(0.315912\pi\)
\(174\) 527.993 0.230040
\(175\) 0 0
\(176\) −11483.8 −4.91830
\(177\) 2868.62 1.21819
\(178\) 247.833 0.104359
\(179\) 1621.18 0.676941 0.338471 0.940977i \(-0.390090\pi\)
0.338471 + 0.940977i \(0.390090\pi\)
\(180\) −566.274 −0.234487
\(181\) −2593.69 −1.06512 −0.532561 0.846392i \(-0.678770\pi\)
−0.532561 + 0.846392i \(0.678770\pi\)
\(182\) 0 0
\(183\) −71.0866 −0.0287151
\(184\) 1862.16 0.746089
\(185\) 985.736 0.391745
\(186\) 7931.03 3.12651
\(187\) 1945.21 0.760682
\(188\) 7708.66 2.99049
\(189\) 0 0
\(190\) −2171.79 −0.829253
\(191\) −1823.08 −0.690645 −0.345323 0.938484i \(-0.612231\pi\)
−0.345323 + 0.938484i \(0.612231\pi\)
\(192\) 7273.09 2.73380
\(193\) −1541.03 −0.574744 −0.287372 0.957819i \(-0.592782\pi\)
−0.287372 + 0.957819i \(0.592782\pi\)
\(194\) −9107.39 −3.37048
\(195\) −713.823 −0.262143
\(196\) 0 0
\(197\) 701.243 0.253612 0.126806 0.991928i \(-0.459527\pi\)
0.126806 + 0.991928i \(0.459527\pi\)
\(198\) 1503.35 0.539587
\(199\) −3294.96 −1.17374 −0.586868 0.809682i \(-0.699639\pi\)
−0.586868 + 0.809682i \(0.699639\pi\)
\(200\) 1802.08 0.637132
\(201\) −775.411 −0.272106
\(202\) 2350.67 0.818775
\(203\) 0 0
\(204\) −3694.80 −1.26808
\(205\) −56.8124 −0.0193559
\(206\) −1871.03 −0.632819
\(207\) −137.272 −0.0460920
\(208\) −6737.29 −2.24590
\(209\) 4192.16 1.38746
\(210\) 0 0
\(211\) 4082.35 1.33195 0.665974 0.745975i \(-0.268016\pi\)
0.665974 + 0.745975i \(0.268016\pi\)
\(212\) 3261.41 1.05658
\(213\) −4433.33 −1.42613
\(214\) 1175.53 0.375502
\(215\) −169.167 −0.0536610
\(216\) −10847.1 −3.41691
\(217\) 0 0
\(218\) 9390.46 2.91744
\(219\) 691.494 0.213364
\(220\) −5568.72 −1.70656
\(221\) 1141.21 0.347359
\(222\) 4970.71 1.50276
\(223\) −747.161 −0.224366 −0.112183 0.993688i \(-0.535784\pi\)
−0.112183 + 0.993688i \(0.535784\pi\)
\(224\) 0 0
\(225\) −132.843 −0.0393608
\(226\) −10039.1 −2.95481
\(227\) −1665.67 −0.487025 −0.243513 0.969898i \(-0.578300\pi\)
−0.243513 + 0.969898i \(0.578300\pi\)
\(228\) −7962.76 −2.31292
\(229\) 6628.35 1.91272 0.956362 0.292183i \(-0.0943816\pi\)
0.956362 + 0.292183i \(0.0943816\pi\)
\(230\) 699.340 0.200492
\(231\) 0 0
\(232\) 1509.50 0.427172
\(233\) −432.431 −0.121586 −0.0607929 0.998150i \(-0.519363\pi\)
−0.0607929 + 0.998150i \(0.519363\pi\)
\(234\) 881.984 0.246398
\(235\) 1808.38 0.501982
\(236\) 13129.2 3.62136
\(237\) 3994.30 1.09476
\(238\) 0 0
\(239\) 5580.44 1.51033 0.755165 0.655535i \(-0.227557\pi\)
0.755165 + 0.655535i \(0.227557\pi\)
\(240\) 5117.06 1.37627
\(241\) 6296.87 1.68306 0.841529 0.540212i \(-0.181656\pi\)
0.841529 + 0.540212i \(0.181656\pi\)
\(242\) 7577.56 2.01283
\(243\) 1467.73 0.387468
\(244\) −325.352 −0.0853629
\(245\) 0 0
\(246\) −286.485 −0.0742504
\(247\) 2459.46 0.633569
\(248\) 22674.4 5.80575
\(249\) −3075.98 −0.782862
\(250\) 676.777 0.171212
\(251\) −311.921 −0.0784393 −0.0392197 0.999231i \(-0.512487\pi\)
−0.0392197 + 0.999231i \(0.512487\pi\)
\(252\) 0 0
\(253\) −1349.92 −0.335451
\(254\) 7550.17 1.86512
\(255\) −866.766 −0.212859
\(256\) 6728.46 1.64269
\(257\) 7861.39 1.90809 0.954046 0.299659i \(-0.0968728\pi\)
0.954046 + 0.299659i \(0.0968728\pi\)
\(258\) −853.050 −0.205847
\(259\) 0 0
\(260\) −3267.06 −0.779285
\(261\) −111.275 −0.0263899
\(262\) −9542.11 −2.25005
\(263\) 5227.09 1.22554 0.612769 0.790262i \(-0.290056\pi\)
0.612769 + 0.790262i \(0.290056\pi\)
\(264\) −17541.0 −4.08929
\(265\) 765.097 0.177357
\(266\) 0 0
\(267\) 213.166 0.0488596
\(268\) −3548.94 −0.808903
\(269\) −1281.71 −0.290510 −0.145255 0.989394i \(-0.546400\pi\)
−0.145255 + 0.989394i \(0.546400\pi\)
\(270\) −4073.66 −0.918204
\(271\) −4704.14 −1.05445 −0.527226 0.849725i \(-0.676768\pi\)
−0.527226 + 0.849725i \(0.676768\pi\)
\(272\) −8180.82 −1.82366
\(273\) 0 0
\(274\) −4997.04 −1.10176
\(275\) −1306.37 −0.286462
\(276\) 2564.10 0.559205
\(277\) 8958.56 1.94321 0.971603 0.236619i \(-0.0760393\pi\)
0.971603 + 0.236619i \(0.0760393\pi\)
\(278\) 1061.40 0.228987
\(279\) −1671.47 −0.358668
\(280\) 0 0
\(281\) −370.904 −0.0787412 −0.0393706 0.999225i \(-0.512535\pi\)
−0.0393706 + 0.999225i \(0.512535\pi\)
\(282\) 9119.02 1.92564
\(283\) 5822.26 1.22296 0.611479 0.791261i \(-0.290575\pi\)
0.611479 + 0.791261i \(0.290575\pi\)
\(284\) −20290.7 −4.23954
\(285\) −1867.99 −0.388246
\(286\) 8673.40 1.79325
\(287\) 0 0
\(288\) −3258.29 −0.666655
\(289\) −3527.27 −0.717946
\(290\) 566.899 0.114791
\(291\) −7833.42 −1.57802
\(292\) 3164.86 0.634279
\(293\) −7443.79 −1.48420 −0.742100 0.670289i \(-0.766170\pi\)
−0.742100 + 0.670289i \(0.766170\pi\)
\(294\) 0 0
\(295\) 3080.00 0.607880
\(296\) 14211.0 2.79053
\(297\) 7863.32 1.53628
\(298\) 4225.10 0.821320
\(299\) −791.973 −0.153181
\(300\) 2481.37 0.477540
\(301\) 0 0
\(302\) −12557.9 −2.39280
\(303\) 2021.85 0.383341
\(304\) −17630.7 −3.32628
\(305\) −76.3247 −0.0143290
\(306\) 1070.96 0.200074
\(307\) 761.674 0.141600 0.0707998 0.997491i \(-0.477445\pi\)
0.0707998 + 0.997491i \(0.477445\pi\)
\(308\) 0 0
\(309\) −1609.30 −0.296278
\(310\) 8515.43 1.56014
\(311\) −7718.69 −1.40735 −0.703677 0.710520i \(-0.748460\pi\)
−0.703677 + 0.710520i \(0.748460\pi\)
\(312\) −10290.9 −1.86734
\(313\) −8556.00 −1.54509 −0.772546 0.634959i \(-0.781017\pi\)
−0.772546 + 0.634959i \(0.781017\pi\)
\(314\) −5538.21 −0.995348
\(315\) 0 0
\(316\) 18281.3 3.25444
\(317\) −7780.95 −1.37862 −0.689309 0.724468i \(-0.742086\pi\)
−0.689309 + 0.724468i \(0.742086\pi\)
\(318\) 3858.11 0.680352
\(319\) −1094.28 −0.192062
\(320\) 7809.02 1.36418
\(321\) 1011.09 0.175805
\(322\) 0 0
\(323\) 2986.42 0.514455
\(324\) −11878.0 −2.03670
\(325\) −766.421 −0.130810
\(326\) −7312.58 −1.24235
\(327\) 8076.89 1.36591
\(328\) −819.045 −0.137879
\(329\) 0 0
\(330\) −6587.56 −1.09889
\(331\) −4932.12 −0.819015 −0.409507 0.912307i \(-0.634299\pi\)
−0.409507 + 0.912307i \(0.634299\pi\)
\(332\) −14078.3 −2.32725
\(333\) −1047.58 −0.172394
\(334\) 6662.61 1.09150
\(335\) −832.548 −0.135782
\(336\) 0 0
\(337\) −7121.13 −1.15108 −0.575538 0.817775i \(-0.695207\pi\)
−0.575538 + 0.817775i \(0.695207\pi\)
\(338\) −6806.52 −1.09534
\(339\) −8634.76 −1.38341
\(340\) −3967.06 −0.632776
\(341\) −16437.2 −2.61034
\(342\) 2308.05 0.364927
\(343\) 0 0
\(344\) −2438.83 −0.382246
\(345\) 601.514 0.0938679
\(346\) 13468.7 2.09272
\(347\) 9540.58 1.47598 0.737991 0.674811i \(-0.235775\pi\)
0.737991 + 0.674811i \(0.235775\pi\)
\(348\) 2078.51 0.320172
\(349\) −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
\(352\) −32041.9 −4.85182
\(353\) −5798.07 −0.874221 −0.437110 0.899408i \(-0.643998\pi\)
−0.437110 + 0.899408i \(0.643998\pi\)
\(354\) 15531.3 2.33187
\(355\) −4760.00 −0.711647
\(356\) 975.627 0.145247
\(357\) 0 0
\(358\) 8777.40 1.29581
\(359\) 2267.29 0.333323 0.166662 0.986014i \(-0.446701\pi\)
0.166662 + 0.986014i \(0.446701\pi\)
\(360\) −1915.15 −0.280381
\(361\) −422.886 −0.0616541
\(362\) −14042.8 −2.03887
\(363\) 6517.58 0.942381
\(364\) 0 0
\(365\) 742.447 0.106470
\(366\) −384.878 −0.0549669
\(367\) 7372.85 1.04866 0.524332 0.851514i \(-0.324315\pi\)
0.524332 + 0.851514i \(0.324315\pi\)
\(368\) 5677.28 0.804209
\(369\) 60.3769 0.00851788
\(370\) 5336.98 0.749883
\(371\) 0 0
\(372\) 31221.4 4.35150
\(373\) 6447.14 0.894961 0.447480 0.894294i \(-0.352321\pi\)
0.447480 + 0.894294i \(0.352321\pi\)
\(374\) 10531.8 1.45611
\(375\) 582.107 0.0801596
\(376\) 26070.8 3.57579
\(377\) −641.989 −0.0877032
\(378\) 0 0
\(379\) −4247.57 −0.575680 −0.287840 0.957678i \(-0.592937\pi\)
−0.287840 + 0.957678i \(0.592937\pi\)
\(380\) −8549.50 −1.15416
\(381\) 6494.03 0.873226
\(382\) −9870.53 −1.32204
\(383\) 6681.86 0.891454 0.445727 0.895169i \(-0.352945\pi\)
0.445727 + 0.895169i \(0.352945\pi\)
\(384\) 16533.9 2.19725
\(385\) 0 0
\(386\) −8343.45 −1.10018
\(387\) 179.781 0.0236145
\(388\) −35852.4 −4.69105
\(389\) −6371.78 −0.830494 −0.415247 0.909709i \(-0.636305\pi\)
−0.415247 + 0.909709i \(0.636305\pi\)
\(390\) −3864.79 −0.501798
\(391\) −961.661 −0.124382
\(392\) 0 0
\(393\) −8207.33 −1.05345
\(394\) 3796.68 0.485467
\(395\) 4288.62 0.546289
\(396\) 5918.11 0.751001
\(397\) −4247.93 −0.537021 −0.268510 0.963277i \(-0.586531\pi\)
−0.268510 + 0.963277i \(0.586531\pi\)
\(398\) −17839.6 −2.24678
\(399\) 0 0
\(400\) 5494.11 0.686764
\(401\) −8833.62 −1.10008 −0.550038 0.835140i \(-0.685387\pi\)
−0.550038 + 0.835140i \(0.685387\pi\)
\(402\) −4198.24 −0.520869
\(403\) −9643.37 −1.19199
\(404\) 9253.70 1.13958
\(405\) −2786.47 −0.341879
\(406\) 0 0
\(407\) −10301.9 −1.25466
\(408\) −12495.9 −1.51627
\(409\) 319.205 0.0385908 0.0192954 0.999814i \(-0.493858\pi\)
0.0192954 + 0.999814i \(0.493858\pi\)
\(410\) −307.595 −0.0370512
\(411\) −4298.04 −0.515831
\(412\) −7365.53 −0.880761
\(413\) 0 0
\(414\) −743.218 −0.0882298
\(415\) −3302.64 −0.390651
\(416\) −18798.3 −2.21554
\(417\) 912.924 0.107209
\(418\) 22697.3 2.65589
\(419\) 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 \(0.627717\pi\)
\(422\) 22102.7 2.54963
\(423\) −1921.84 −0.220906
\(424\) 11030.1 1.26337
\(425\) −930.635 −0.106217
\(426\) −24003.0 −2.72992
\(427\) 0 0
\(428\) 4627.60 0.522625
\(429\) 7460.14 0.839577
\(430\) −915.908 −0.102719
\(431\) −5184.75 −0.579444 −0.289722 0.957111i \(-0.593563\pi\)
−0.289722 + 0.957111i \(0.593563\pi\)
\(432\) −33070.2 −3.68308
\(433\) 4242.03 0.470806 0.235403 0.971898i \(-0.424359\pi\)
0.235403 + 0.971898i \(0.424359\pi\)
\(434\) 0 0
\(435\) 487.599 0.0537439
\(436\) 36966.7 4.06051
\(437\) −2072.50 −0.226868
\(438\) 3743.90 0.408425
\(439\) 5434.12 0.590789 0.295394 0.955375i \(-0.404549\pi\)
0.295394 + 0.955375i \(0.404549\pi\)
\(440\) −18833.5 −2.04057
\(441\) 0 0
\(442\) 6178.77 0.664919
\(443\) −11493.8 −1.23270 −0.616350 0.787472i \(-0.711389\pi\)
−0.616350 + 0.787472i \(0.711389\pi\)
\(444\) 19567.8 2.09155
\(445\) 228.873 0.0243812
\(446\) −4045.29 −0.429484
\(447\) 3634.08 0.384532
\(448\) 0 0
\(449\) −16849.3 −1.77098 −0.885489 0.464661i \(-0.846176\pi\)
−0.885489 + 0.464661i \(0.846176\pi\)
\(450\) −719.239 −0.0753450
\(451\) 593.745 0.0619919
\(452\) −39520.0 −4.11253
\(453\) −10801.2 −1.12028
\(454\) −9018.32 −0.932270
\(455\) 0 0
\(456\) −26930.2 −2.76561
\(457\) 15348.5 1.57106 0.785528 0.618826i \(-0.212391\pi\)
0.785528 + 0.618826i \(0.212391\pi\)
\(458\) 35887.3 3.66136
\(459\) 5601.69 0.569639
\(460\) 2753.04 0.279046
\(461\) −14038.4 −1.41830 −0.709148 0.705059i \(-0.750920\pi\)
−0.709148 + 0.705059i \(0.750920\pi\)
\(462\) 0 0
\(463\) −8661.23 −0.869377 −0.434689 0.900581i \(-0.643142\pi\)
−0.434689 + 0.900581i \(0.643142\pi\)
\(464\) 4602.12 0.460448
\(465\) 7324.26 0.730440
\(466\) −2341.27 −0.232741
\(467\) −7014.71 −0.695079 −0.347539 0.937665i \(-0.612983\pi\)
−0.347539 + 0.937665i \(0.612983\pi\)
\(468\) 3472.04 0.342938
\(469\) 0 0
\(470\) 9790.96 0.960901
\(471\) −4763.50 −0.466010
\(472\) 44403.3 4.33014
\(473\) 1767.96 0.171863
\(474\) 21626.0 2.09560
\(475\) −2005.63 −0.193737
\(476\) 0 0
\(477\) −813.100 −0.0780488
\(478\) 30213.7 2.89109
\(479\) −18134.7 −1.72984 −0.864922 0.501907i \(-0.832632\pi\)
−0.864922 + 0.501907i \(0.832632\pi\)
\(480\) 14277.6 1.35767
\(481\) −6043.91 −0.572929
\(482\) 34092.6 3.22173
\(483\) 0 0
\(484\) 29830.0 2.80146
\(485\) −8410.63 −0.787438
\(486\) 7946.59 0.741697
\(487\) 16537.8 1.53881 0.769405 0.638761i \(-0.220553\pi\)
0.769405 + 0.638761i \(0.220553\pi\)
\(488\) −1100.35 −0.102070
\(489\) −6289.67 −0.581654
\(490\) 0 0
\(491\) 220.608 0.0202768 0.0101384 0.999949i \(-0.496773\pi\)
0.0101384 + 0.999949i \(0.496773\pi\)
\(492\) −1127.78 −0.103342
\(493\) −779.542 −0.0712146
\(494\) 13316.0 1.21279
\(495\) 1388.33 0.126063
\(496\) 69128.8 6.25801
\(497\) 0 0
\(498\) −16654.0 −1.49856
\(499\) 5939.04 0.532801 0.266401 0.963862i \(-0.414166\pi\)
0.266401 + 0.963862i \(0.414166\pi\)
\(500\) 2664.21 0.238295
\(501\) 5730.62 0.511029
\(502\) −1688.81 −0.150150
\(503\) 11604.8 1.02869 0.514345 0.857584i \(-0.328035\pi\)
0.514345 + 0.857584i \(0.328035\pi\)
\(504\) 0 0
\(505\) 2170.83 0.191289
\(506\) −7308.78 −0.642124
\(507\) −5854.40 −0.512826
\(508\) 29722.2 2.59588
\(509\) 1867.67 0.162639 0.0813193 0.996688i \(-0.474087\pi\)
0.0813193 + 0.996688i \(0.474087\pi\)
\(510\) −4692.86 −0.407457
\(511\) 0 0
\(512\) 8025.75 0.692757
\(513\) 12072.3 1.03900
\(514\) 42563.2 3.65250
\(515\) −1727.88 −0.147844
\(516\) −3358.14 −0.286499
\(517\) −18899.3 −1.60772
\(518\) 0 0
\(519\) 11584.6 0.979786
\(520\) −11049.2 −0.931809
\(521\) −6117.21 −0.514395 −0.257197 0.966359i \(-0.582799\pi\)
−0.257197 + 0.966359i \(0.582799\pi\)
\(522\) −602.467 −0.0505158
\(523\) 16685.6 1.39505 0.697524 0.716561i \(-0.254285\pi\)
0.697524 + 0.716561i \(0.254285\pi\)
\(524\) −37563.7 −3.13164
\(525\) 0 0
\(526\) 28300.6 2.34594
\(527\) −11709.6 −0.967887
\(528\) −53478.2 −4.40784
\(529\) −11499.6 −0.945149
\(530\) 4142.40 0.339499
\(531\) −3273.24 −0.267508
\(532\) 0 0
\(533\) 348.338 0.0283081
\(534\) 1154.12 0.0935278
\(535\) 1085.59 0.0877276
\(536\) −12002.6 −0.967223
\(537\) 7549.59 0.606683
\(538\) −6939.44 −0.556097
\(539\) 0 0
\(540\) −16036.5 −1.27796
\(541\) 9309.03 0.739790 0.369895 0.929074i \(-0.379394\pi\)
0.369895 + 0.929074i \(0.379394\pi\)
\(542\) −25469.2 −2.01845
\(543\) −12078.4 −0.954575
\(544\) −22826.1 −1.79901
\(545\) 8672.05 0.681596
\(546\) 0 0
\(547\) 10894.7 0.851598 0.425799 0.904818i \(-0.359993\pi\)
0.425799 + 0.904818i \(0.359993\pi\)
\(548\) −19671.5 −1.53344
\(549\) 81.1134 0.00630571
\(550\) −7072.97 −0.548350
\(551\) −1680.01 −0.129893
\(552\) 8671.81 0.668654
\(553\) 0 0
\(554\) 48503.6 3.71971
\(555\) 4590.43 0.351086
\(556\) 4178.31 0.318705
\(557\) −7873.90 −0.598973 −0.299486 0.954101i \(-0.596815\pi\)
−0.299486 + 0.954101i \(0.596815\pi\)
\(558\) −9049.71 −0.686567
\(559\) 1037.23 0.0784796
\(560\) 0 0
\(561\) 9058.55 0.681733
\(562\) −2008.15 −0.150728
\(563\) −21770.7 −1.62971 −0.814854 0.579666i \(-0.803183\pi\)
−0.814854 + 0.579666i \(0.803183\pi\)
\(564\) 35898.1 2.68011
\(565\) −9271.02 −0.690327
\(566\) 31522.9 2.34100
\(567\) 0 0
\(568\) −68623.3 −5.06931
\(569\) −12381.3 −0.912213 −0.456106 0.889925i \(-0.650756\pi\)
−0.456106 + 0.889925i \(0.650756\pi\)
\(570\) −10113.7 −0.743186
\(571\) −5768.38 −0.422765 −0.211383 0.977403i \(-0.567797\pi\)
−0.211383 + 0.977403i \(0.567797\pi\)
\(572\) 34143.9 2.49585
\(573\) −8489.81 −0.618965
\(574\) 0 0
\(575\) 645.837 0.0468405
\(576\) −8298.97 −0.600330
\(577\) −4733.38 −0.341513 −0.170757 0.985313i \(-0.554621\pi\)
−0.170757 + 0.985313i \(0.554621\pi\)
\(578\) −19097.4 −1.37430
\(579\) −7176.34 −0.515093
\(580\) 2231.67 0.159767
\(581\) 0 0
\(582\) −42411.8 −3.02066
\(583\) −7996.00 −0.568028
\(584\) 10703.6 0.758422
\(585\) 814.508 0.0575654
\(586\) −40302.3 −2.84108
\(587\) 8441.67 0.593569 0.296785 0.954944i \(-0.404086\pi\)
0.296785 + 0.954944i \(0.404086\pi\)
\(588\) 0 0
\(589\) −25235.6 −1.76539
\(590\) 16675.8 1.16361
\(591\) 3265.59 0.227290
\(592\) 43326.0 3.00792
\(593\) −18939.9 −1.31158 −0.655791 0.754943i \(-0.727665\pi\)
−0.655791 + 0.754943i \(0.727665\pi\)
\(594\) 42573.7 2.94077
\(595\) 0 0
\(596\) 16632.6 1.14312
\(597\) −15344.2 −1.05192
\(598\) −4287.91 −0.293220
\(599\) 22655.3 1.54536 0.772681 0.634794i \(-0.218915\pi\)
0.772681 + 0.634794i \(0.218915\pi\)
\(600\) 8392.03 0.571005
\(601\) 15947.4 1.08237 0.541187 0.840902i \(-0.317975\pi\)
0.541187 + 0.840902i \(0.317975\pi\)
\(602\) 0 0
\(603\) 884.784 0.0597532
\(604\) −49435.6 −3.33031
\(605\) 6997.84 0.470252
\(606\) 10946.7 0.733796
\(607\) 25993.2 1.73811 0.869053 0.494719i \(-0.164729\pi\)
0.869053 + 0.494719i \(0.164729\pi\)
\(608\) −49193.1 −3.28132
\(609\) 0 0
\(610\) −413.238 −0.0274287
\(611\) −11087.9 −0.734152
\(612\) 4215.96 0.278464
\(613\) 665.408 0.0438427 0.0219213 0.999760i \(-0.493022\pi\)
0.0219213 + 0.999760i \(0.493022\pi\)
\(614\) 4123.87 0.271052
\(615\) −264.567 −0.0173470
\(616\) 0 0
\(617\) 18401.3 1.20066 0.600330 0.799752i \(-0.295036\pi\)
0.600330 + 0.799752i \(0.295036\pi\)
\(618\) −8713.10 −0.567140
\(619\) 11150.6 0.724040 0.362020 0.932170i \(-0.382087\pi\)
0.362020 + 0.932170i \(0.382087\pi\)
\(620\) 33522.0 2.17142
\(621\) −3887.43 −0.251203
\(622\) −41790.6 −2.69397
\(623\) 0 0
\(624\) −31374.6 −2.01280
\(625\) 625.000 0.0400000
\(626\) −46324.0 −2.95764
\(627\) 19522.3 1.24345
\(628\) −21801.8 −1.38533
\(629\) −7338.88 −0.465215
\(630\) 0 0
\(631\) 5381.79 0.339534 0.169767 0.985484i \(-0.445699\pi\)
0.169767 + 0.985484i \(0.445699\pi\)
\(632\) 61827.6 3.89141
\(633\) 19010.9 1.19371
\(634\) −42127.7 −2.63897
\(635\) 6972.55 0.435744
\(636\) 15187.9 0.946918
\(637\) 0 0
\(638\) −5924.64 −0.367647
\(639\) 5058.65 0.313172
\(640\) 17752.2 1.09644
\(641\) −19455.1 −1.19880 −0.599398 0.800451i \(-0.704593\pi\)
−0.599398 + 0.800451i \(0.704593\pi\)
\(642\) 5474.26 0.336529
\(643\) 14695.8 0.901317 0.450658 0.892696i \(-0.351189\pi\)
0.450658 + 0.892696i \(0.351189\pi\)
\(644\) 0 0
\(645\) −787.788 −0.0480917
\(646\) 16169.1 0.984777
\(647\) 12694.8 0.771383 0.385691 0.922628i \(-0.373963\pi\)
0.385691 + 0.922628i \(0.373963\pi\)
\(648\) −40171.6 −2.43532
\(649\) −32189.0 −1.94688
\(650\) −4149.57 −0.250399
\(651\) 0 0
\(652\) −28786.8 −1.72911
\(653\) −12385.6 −0.742247 −0.371124 0.928583i \(-0.621027\pi\)
−0.371124 + 0.928583i \(0.621027\pi\)
\(654\) 43730.0 2.61465
\(655\) −8812.09 −0.525675
\(656\) −2497.07 −0.148619
\(657\) −789.030 −0.0468539
\(658\) 0 0
\(659\) −2072.18 −0.122489 −0.0612447 0.998123i \(-0.519507\pi\)
−0.0612447 + 0.998123i \(0.519507\pi\)
\(660\) −25932.7 −1.52944
\(661\) −1074.36 −0.0632193 −0.0316096 0.999500i \(-0.510063\pi\)
−0.0316096 + 0.999500i \(0.510063\pi\)
\(662\) −26703.6 −1.56777
\(663\) 5314.47 0.311307
\(664\) −47613.0 −2.78275
\(665\) 0 0
\(666\) −5671.84 −0.329999
\(667\) 540.982 0.0314047
\(668\) 26228.2 1.51916
\(669\) −3479.42 −0.201080
\(670\) −4507.59 −0.259916
\(671\) 797.667 0.0458921
\(672\) 0 0
\(673\) 26195.2 1.50037 0.750186 0.661226i \(-0.229964\pi\)
0.750186 + 0.661226i \(0.229964\pi\)
\(674\) −38555.3 −2.20341
\(675\) −3762.01 −0.214518
\(676\) −26794.7 −1.52450
\(677\) 4228.44 0.240047 0.120024 0.992771i \(-0.461703\pi\)
0.120024 + 0.992771i \(0.461703\pi\)
\(678\) −46750.4 −2.64814
\(679\) 0 0
\(680\) −13416.6 −0.756624
\(681\) −7756.80 −0.436478
\(682\) −88994.5 −4.99674
\(683\) 27525.5 1.54207 0.771036 0.636792i \(-0.219739\pi\)
0.771036 + 0.636792i \(0.219739\pi\)
\(684\) 9085.91 0.507907
\(685\) −4614.74 −0.257402
\(686\) 0 0
\(687\) 30867.3 1.71421
\(688\) −7435.40 −0.412023
\(689\) −4691.09 −0.259385
\(690\) 3256.72 0.179683
\(691\) 33324.4 1.83462 0.917309 0.398177i \(-0.130357\pi\)
0.917309 + 0.398177i \(0.130357\pi\)
\(692\) 53021.1 2.91266
\(693\) 0 0
\(694\) 51654.8 2.82534
\(695\) 980.193 0.0534976
\(696\) 7029.54 0.382836
\(697\) 422.973 0.0229860
\(698\) −6939.12 −0.376289
\(699\) −2013.77 −0.108967
\(700\) 0 0
\(701\) −33262.9 −1.79219 −0.896094 0.443864i \(-0.853607\pi\)
−0.896094 + 0.443864i \(0.853607\pi\)
\(702\) 24977.1 1.34288
\(703\) −15816.2 −0.848534
\(704\) −81611.8 −4.36912
\(705\) 8421.37 0.449882
\(706\) −31392.0 −1.67345
\(707\) 0 0
\(708\) 61141.0 3.24551
\(709\) 13703.0 0.725851 0.362926 0.931818i \(-0.381778\pi\)
0.362926 + 0.931818i \(0.381778\pi\)
\(710\) −25771.7 −1.36224
\(711\) −4557.70 −0.240404
\(712\) 3299.58 0.173676
\(713\) 8126.14 0.426825
\(714\) 0 0
\(715\) 8009.84 0.418953
\(716\) 34553.3 1.80352
\(717\) 25987.3 1.35358
\(718\) 12275.6 0.638052
\(719\) 8074.93 0.418838 0.209419 0.977826i \(-0.432843\pi\)
0.209419 + 0.977826i \(0.432843\pi\)
\(720\) −5838.82 −0.302222
\(721\) 0 0
\(722\) −2289.59 −0.118019
\(723\) 29323.6 1.50838
\(724\) −55281.1 −2.83771
\(725\) 523.528 0.0268184
\(726\) 35287.6 1.80392
\(727\) 3668.70 0.187159 0.0935794 0.995612i \(-0.470169\pi\)
0.0935794 + 0.995612i \(0.470169\pi\)
\(728\) 0 0
\(729\) 21881.9 1.11172
\(730\) 4019.77 0.203806
\(731\) 1259.46 0.0637250
\(732\) −1515.12 −0.0765033
\(733\) 14980.3 0.754857 0.377428 0.926039i \(-0.376808\pi\)
0.377428 + 0.926039i \(0.376808\pi\)
\(734\) 39918.2 2.00737
\(735\) 0 0
\(736\) 15840.7 0.793338
\(737\) 8700.94 0.434875
\(738\) 326.894 0.0163050
\(739\) 6530.59 0.325077 0.162538 0.986702i \(-0.448032\pi\)
0.162538 + 0.986702i \(0.448032\pi\)
\(740\) 21009.7 1.04369
\(741\) 11453.3 0.567812
\(742\) 0 0
\(743\) 25952.0 1.28141 0.640704 0.767788i \(-0.278643\pi\)
0.640704 + 0.767788i \(0.278643\pi\)
\(744\) 105591. 5.20318
\(745\) 3901.86 0.191883
\(746\) 34906.2 1.71315
\(747\) 3509.85 0.171913
\(748\) 41459.6 2.02662
\(749\) 0 0
\(750\) 3151.65 0.153443
\(751\) −14093.9 −0.684813 −0.342407 0.939552i \(-0.611242\pi\)
−0.342407 + 0.939552i \(0.611242\pi\)
\(752\) 79483.6 3.85435
\(753\) −1452.57 −0.0702983
\(754\) −3475.87 −0.167883
\(755\) −11597.1 −0.559024
\(756\) 0 0
\(757\) −2554.41 −0.122644 −0.0613220 0.998118i \(-0.519532\pi\)
−0.0613220 + 0.998118i \(0.519532\pi\)
\(758\) −22997.2 −1.10197
\(759\) −6286.40 −0.300635
\(760\) −28914.5 −1.38005
\(761\) −2219.08 −0.105705 −0.0528527 0.998602i \(-0.516831\pi\)
−0.0528527 + 0.998602i \(0.516831\pi\)
\(762\) 35160.1 1.67154
\(763\) 0 0
\(764\) −38856.5 −1.84003
\(765\) 989.025 0.0467428
\(766\) 36177.0 1.70643
\(767\) −18884.6 −0.889028
\(768\) 31333.5 1.47220
\(769\) 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
\(772\) −32845.0 −1.53124
\(773\) −9674.79 −0.450165 −0.225083 0.974340i \(-0.572265\pi\)
−0.225083 + 0.974340i \(0.572265\pi\)
\(774\) 973.374 0.0452031
\(775\) 7863.96 0.364493
\(776\) −121253. −5.60920
\(777\) 0 0
\(778\) −34498.2 −1.58974
\(779\) 911.560 0.0419256
\(780\) −15214.2 −0.698405
\(781\) 49746.6 2.27922
\(782\) −5206.64 −0.238093
\(783\) −3151.23 −0.143826
\(784\) 0 0
\(785\) −5114.51 −0.232541
\(786\) −44436.2 −2.01652
\(787\) 20942.8 0.948577 0.474288 0.880370i \(-0.342705\pi\)
0.474288 + 0.880370i \(0.342705\pi\)
\(788\) 14946.1 0.675676
\(789\) 24341.8 1.09834
\(790\) 23219.5 1.04571
\(791\) 0 0
\(792\) 20015.1 0.897989
\(793\) 467.975 0.0209562
\(794\) −22999.2 −1.02797
\(795\) 3562.94 0.158949
\(796\) −70227.8 −3.12708
\(797\) −23526.6 −1.04561 −0.522807 0.852451i \(-0.675115\pi\)
−0.522807 + 0.852451i \(0.675115\pi\)
\(798\) 0 0
\(799\) −13463.5 −0.596127
\(800\) 15329.6 0.677481
\(801\) −243.233 −0.0107294
\(802\) −47827.1 −2.10578
\(803\) −7759.29 −0.340996
\(804\) −16526.9 −0.724948
\(805\) 0 0
\(806\) −52211.3 −2.28172
\(807\) −5968.72 −0.260358
\(808\) 31296.1 1.36262
\(809\) −18202.2 −0.791047 −0.395523 0.918456i \(-0.629437\pi\)
−0.395523 + 0.918456i \(0.629437\pi\)
\(810\) −15086.6 −0.654429
\(811\) 2510.24 0.108689 0.0543443 0.998522i \(-0.482693\pi\)
0.0543443 + 0.998522i \(0.482693\pi\)
\(812\) 0 0
\(813\) −21906.5 −0.945012
\(814\) −55776.7 −2.40168
\(815\) −6753.13 −0.290248
\(816\) −38096.9 −1.63438
\(817\) 2714.30 0.116232
\(818\) 1728.24 0.0738711
\(819\) 0 0
\(820\) −1210.88 −0.0515681
\(821\) 17899.6 0.760903 0.380451 0.924801i \(-0.375769\pi\)
0.380451 + 0.924801i \(0.375769\pi\)
\(822\) −23270.5 −0.987411
\(823\) 14039.5 0.594637 0.297318 0.954778i \(-0.403908\pi\)
0.297318 + 0.954778i \(0.403908\pi\)
\(824\) −24910.3 −1.05315
\(825\) −6083.58 −0.256731
\(826\) 0 0
\(827\) 15127.4 0.636073 0.318036 0.948079i \(-0.396977\pi\)
0.318036 + 0.948079i \(0.396977\pi\)
\(828\) −2925.77 −0.122799
\(829\) −21986.5 −0.921136 −0.460568 0.887624i \(-0.652354\pi\)
−0.460568 + 0.887624i \(0.652354\pi\)
\(830\) −17881.2 −0.747790
\(831\) 41718.7 1.74152
\(832\) −47880.0 −1.99512
\(833\) 0 0
\(834\) 4942.76 0.205220
\(835\) 6152.89 0.255005
\(836\) 89350.6 3.69648
\(837\) −47334.8 −1.95476
\(838\) 69243.5 2.85439
\(839\) 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
\(842\) −36531.8 −1.49521
\(843\) −1727.25 −0.0705688
\(844\) 87010.1 3.54859
\(845\) −6285.79 −0.255903
\(846\) −10405.3 −0.422861
\(847\) 0 0
\(848\) 33628.2 1.36179
\(849\) 27113.4 1.09603
\(850\) −5038.66 −0.203323
\(851\) 5093.00 0.205154
\(852\) −94490.6 −3.79952
\(853\) −13342.6 −0.535570 −0.267785 0.963479i \(-0.586292\pi\)
−0.267785 + 0.963479i \(0.586292\pi\)
\(854\) 0 0
\(855\) 2131.47 0.0852571
\(856\) 15650.6 0.624915
\(857\) −18690.9 −0.745003 −0.372502 0.928032i \(-0.621500\pi\)
−0.372502 + 0.928032i \(0.621500\pi\)
\(858\) 40390.8 1.60713
\(859\) −18318.9 −0.727628 −0.363814 0.931472i \(-0.618526\pi\)
−0.363814 + 0.931472i \(0.618526\pi\)
\(860\) −3605.58 −0.142964
\(861\) 0 0
\(862\) −28071.3 −1.10918
\(863\) 38133.1 1.50413 0.752067 0.659087i \(-0.229057\pi\)
0.752067 + 0.659087i \(0.229057\pi\)
\(864\) −92272.3 −3.63330
\(865\) 12438.3 0.488917
\(866\) 22967.3 0.901223
\(867\) −16426.0 −0.643432
\(868\) 0 0
\(869\) −44820.3 −1.74962
\(870\) 2639.96 0.102877
\(871\) 5104.66 0.198582
\(872\) 125022. 4.85525
\(873\) 8938.33 0.346525
\(874\) −11221.0 −0.434273
\(875\) 0 0
\(876\) 14738.3 0.568449
\(877\) −19707.5 −0.758807 −0.379404 0.925231i \(-0.623871\pi\)
−0.379404 + 0.925231i \(0.623871\pi\)
\(878\) 29421.5 1.13090
\(879\) −34664.6 −1.33016
\(880\) −57418.8 −2.19953
\(881\) 14091.5 0.538883 0.269441 0.963017i \(-0.413161\pi\)
0.269441 + 0.963017i \(0.413161\pi\)
\(882\) 0 0
\(883\) 3115.87 0.118751 0.0593757 0.998236i \(-0.481089\pi\)
0.0593757 + 0.998236i \(0.481089\pi\)
\(884\) 24323.5 0.925438
\(885\) 14343.1 0.544789
\(886\) −62229.8 −2.35965
\(887\) −38734.6 −1.46627 −0.733134 0.680084i \(-0.761943\pi\)
−0.733134 + 0.680084i \(0.761943\pi\)
\(888\) 66178.6 2.50091
\(889\) 0 0
\(890\) 1239.17 0.0466708
\(891\) 29121.3 1.09495
\(892\) −15924.8 −0.597759
\(893\) −29015.6 −1.08731
\(894\) 19675.7 0.736077
\(895\) 8105.89 0.302737
\(896\) 0 0
\(897\) −3688.10 −0.137282
\(898\) −91225.8 −3.39003
\(899\) 6587.21 0.244378
\(900\) −2831.37 −0.104866
\(901\) −5696.21 −0.210619
\(902\) 3214.66 0.118666
\(903\) 0 0
\(904\) −133657. −4.91744
\(905\) −12968.4 −0.476337
\(906\) −58480.2 −2.14445
\(907\) 19242.9 0.704464 0.352232 0.935913i \(-0.385423\pi\)
0.352232 + 0.935913i \(0.385423\pi\)
\(908\) −35501.7 −1.29754
\(909\) −2307.03 −0.0841799
\(910\) 0 0
\(911\) 34613.3 1.25882 0.629412 0.777072i \(-0.283296\pi\)
0.629412 + 0.777072i \(0.283296\pi\)
\(912\) −82103.6 −2.98105
\(913\) 34515.8 1.25116
\(914\) 83100.1 3.00734
\(915\) −355.433 −0.0128418
\(916\) 141275. 5.09591
\(917\) 0 0
\(918\) 30328.7 1.09041
\(919\) 25826.4 0.927022 0.463511 0.886091i \(-0.346589\pi\)
0.463511 + 0.886091i \(0.346589\pi\)
\(920\) 9310.81 0.333661
\(921\) 3547.01 0.126903
\(922\) −76007.0 −2.71492
\(923\) 29185.3 1.04079
\(924\) 0 0
\(925\) 4928.68 0.175194
\(926\) −46893.8 −1.66417
\(927\) 1836.29 0.0650613
\(928\) 12840.8 0.454224
\(929\) −19451.6 −0.686960 −0.343480 0.939160i \(-0.611606\pi\)
−0.343480 + 0.939160i \(0.611606\pi\)
\(930\) 39655.1 1.39822
\(931\) 0 0
\(932\) −9216.70 −0.323930
\(933\) −35944.8 −1.26129
\(934\) −37979.1 −1.33053
\(935\) 9726.03 0.340188
\(936\) 11742.5 0.410059
\(937\) −34469.1 −1.20177 −0.600884 0.799336i \(-0.705185\pi\)
−0.600884 + 0.799336i \(0.705185\pi\)
\(938\) 0 0
\(939\) −39844.1 −1.38473
\(940\) 38543.3 1.33739
\(941\) −14156.4 −0.490419 −0.245209 0.969470i \(-0.578857\pi\)
−0.245209 + 0.969470i \(0.578857\pi\)
\(942\) −25790.6 −0.892042
\(943\) −293.532 −0.0101365
\(944\) 135375. 4.66746
\(945\) 0 0
\(946\) 9572.13 0.328982
\(947\) −38092.4 −1.30711 −0.653557 0.756877i \(-0.726724\pi\)
−0.653557 + 0.756877i \(0.726724\pi\)
\(948\) 85133.3 2.91667
\(949\) −4552.22 −0.155713
\(950\) −10858.9 −0.370853
\(951\) −36234.8 −1.23553
\(952\) 0 0
\(953\) 5037.40 0.171225 0.0856126 0.996329i \(-0.472715\pi\)
0.0856126 + 0.996329i \(0.472715\pi\)
\(954\) −4402.30 −0.149402
\(955\) −9115.39 −0.308866
\(956\) 118940. 4.02384
\(957\) −5095.88 −0.172128
\(958\) −98185.1 −3.31129
\(959\) 0 0
\(960\) 36365.4 1.22259
\(961\) 69156.0 2.32137
\(962\) −32723.0 −1.09671
\(963\) −1153.71 −0.0386060
\(964\) 134210. 4.48403
\(965\) −7705.14 −0.257033
\(966\) 0 0
\(967\) 11495.3 0.382278 0.191139 0.981563i \(-0.438782\pi\)
0.191139 + 0.981563i \(0.438782\pi\)
\(968\) 100885. 3.34977
\(969\) 13907.3 0.461061
\(970\) −45537.0 −1.50732
\(971\) −22352.7 −0.738757 −0.369379 0.929279i \(-0.620429\pi\)
−0.369379 + 0.929279i \(0.620429\pi\)
\(972\) 31282.7 1.03230
\(973\) 0 0
\(974\) 89539.4 2.94561
\(975\) −3569.11 −0.117234
\(976\) −3354.69 −0.110022
\(977\) 14345.7 0.469765 0.234882 0.972024i \(-0.424530\pi\)
0.234882 + 0.972024i \(0.424530\pi\)
\(978\) −34053.6 −1.11341
\(979\) −2391.94 −0.0780867
\(980\) 0 0
\(981\) −9216.15 −0.299948
\(982\) 1194.42 0.0388141
\(983\) −34460.9 −1.11814 −0.559070 0.829120i \(-0.688842\pi\)
−0.559070 + 0.829120i \(0.688842\pi\)
\(984\) −3814.17 −0.123568
\(985\) 3506.21 0.113419
\(986\) −4220.61 −0.136320
\(987\) 0 0
\(988\) 52420.2 1.68796
\(989\) −874.036 −0.0281019
\(990\) 7516.74 0.241311
\(991\) −35189.6 −1.12799 −0.563993 0.825780i \(-0.690735\pi\)
−0.563993 + 0.825780i \(0.690735\pi\)
\(992\) 192883. 6.17342
\(993\) −22968.2 −0.734011
\(994\) 0 0
\(995\) −16474.8 −0.524911
\(996\) −65560.6 −2.08571
\(997\) 50730.0 1.61147 0.805734 0.592277i \(-0.201771\pi\)
0.805734 + 0.592277i \(0.201771\pi\)
\(998\) 32155.2 1.01990
\(999\) −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 245.4.a.k.1.2 2
3.2 odd 2 2205.4.a.u.1.1 2
5.4 even 2 1225.4.a.m.1.1 2
7.2 even 3 245.4.e.i.116.1 4
7.3 odd 6 245.4.e.h.226.1 4
7.4 even 3 245.4.e.i.226.1 4
7.5 odd 6 245.4.e.h.116.1 4
7.6 odd 2 35.4.a.b.1.2 2
21.20 even 2 315.4.a.f.1.1 2
28.27 even 2 560.4.a.r.1.2 2
35.13 even 4 175.4.b.c.99.1 4
35.27 even 4 175.4.b.c.99.4 4
35.34 odd 2 175.4.a.c.1.1 2
56.13 odd 2 2240.4.a.bn.1.2 2
56.27 even 2 2240.4.a.bo.1.1 2
105.104 even 2 1575.4.a.z.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.4.a.b.1.2 2 7.6 odd 2
175.4.a.c.1.1 2 35.34 odd 2
175.4.b.c.99.1 4 35.13 even 4
175.4.b.c.99.4 4 35.27 even 4
245.4.a.k.1.2 2 1.1 even 1 trivial
245.4.e.h.116.1 4 7.5 odd 6
245.4.e.h.226.1 4 7.3 odd 6
245.4.e.i.116.1 4 7.2 even 3
245.4.e.i.226.1 4 7.4 even 3
315.4.a.f.1.1 2 21.20 even 2
560.4.a.r.1.2 2 28.27 even 2
1225.4.a.m.1.1 2 5.4 even 2
1575.4.a.z.1.2 2 105.104 even 2
2205.4.a.u.1.1 2 3.2 odd 2
2240.4.a.bn.1.2 2 56.13 odd 2
2240.4.a.bo.1.1 2 56.27 even 2