Properties

Label 825.4.c.l.199.4
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.2230106176.1
Defining polynomial: \(x^{6} + 41 x^{4} + 452 x^{2} + 676\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(1.32906i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.l.199.3

$q$-expansion

\(f(q)\) \(=\) \(q+2.32906i q^{2} -3.00000i q^{3} +2.57547 q^{4} +6.98719 q^{6} -22.4672i q^{7} +24.6309i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+2.32906i q^{2} -3.00000i q^{3} +2.57547 q^{4} +6.98719 q^{6} -22.4672i q^{7} +24.6309i q^{8} -9.00000 q^{9} +11.0000 q^{11} -7.72640i q^{12} -9.86030i q^{13} +52.3275 q^{14} -36.7633 q^{16} +128.137i q^{17} -20.9616i q^{18} -7.04001 q^{19} -67.4015 q^{21} +25.6197i q^{22} +0.654969i q^{23} +73.8928 q^{24} +22.9653 q^{26} +27.0000i q^{27} -57.8635i q^{28} +229.279 q^{29} +155.789 q^{31} +111.423i q^{32} -33.0000i q^{33} -298.438 q^{34} -23.1792 q^{36} +110.279i q^{37} -16.3966i q^{38} -29.5809 q^{39} +154.749 q^{41} -156.982i q^{42} -401.014i q^{43} +28.3301 q^{44} -1.52546 q^{46} +277.532i q^{47} +110.290i q^{48} -161.774 q^{49} +384.410 q^{51} -25.3949i q^{52} -651.566i q^{53} -62.8847 q^{54} +553.388 q^{56} +21.1200i q^{57} +534.005i q^{58} +423.869 q^{59} +681.851 q^{61} +362.842i q^{62} +202.205i q^{63} -553.618 q^{64} +76.8591 q^{66} -374.028i q^{67} +330.011i q^{68} +1.96491 q^{69} +96.6950 q^{71} -221.678i q^{72} -19.9460i q^{73} -256.848 q^{74} -18.1313 q^{76} -247.139i q^{77} -68.8958i q^{78} -24.4286 q^{79} +81.0000 q^{81} +360.419i q^{82} -1127.35i q^{83} -173.590 q^{84} +933.987 q^{86} -687.836i q^{87} +270.940i q^{88} +639.624 q^{89} -221.533 q^{91} +1.68685i q^{92} -467.366i q^{93} -646.389 q^{94} +334.270 q^{96} +730.865i q^{97} -376.783i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + O(q^{10}) \) \( 6q - 44q^{4} + 24q^{6} - 54q^{9} + 66q^{11} + 112q^{14} + 100q^{16} - 292q^{19} + 24q^{21} - 288q^{24} - 1016q^{26} - 136q^{29} - 136q^{31} + 352q^{34} + 396q^{36} - 392q^{41} - 484q^{44} + 2320q^{46} + 314q^{49} + 1308q^{51} - 216q^{54} + 2736q^{56} + 2088q^{59} + 1284q^{61} - 2332q^{64} + 264q^{66} - 1200q^{69} - 1088q^{71} - 3072q^{74} + 3992q^{76} + 3172q^{79} + 486q^{81} - 2040q^{84} + 7136q^{86} + 4244q^{89} - 16q^{91} + 4304q^{94} + 4128q^{96} - 594q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.32906i 0.823448i 0.911309 + 0.411724i \(0.135073\pi\)
−0.911309 + 0.411724i \(0.864927\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 2.57547 0.321933
\(5\) 0 0
\(6\) 6.98719 0.475418
\(7\) − 22.4672i − 1.21311i −0.795040 0.606557i \(-0.792550\pi\)
0.795040 0.606557i \(-0.207450\pi\)
\(8\) 24.6309i 1.08854i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 7.72640i − 0.185868i
\(13\) − 9.86030i − 0.210366i −0.994453 0.105183i \(-0.966457\pi\)
0.994453 0.105183i \(-0.0335428\pi\)
\(14\) 52.3275 0.998936
\(15\) 0 0
\(16\) −36.7633 −0.574426
\(17\) 128.137i 1.82810i 0.405603 + 0.914049i \(0.367062\pi\)
−0.405603 + 0.914049i \(0.632938\pi\)
\(18\) − 20.9616i − 0.274483i
\(19\) −7.04001 −0.0850047 −0.0425024 0.999096i \(-0.513533\pi\)
−0.0425024 + 0.999096i \(0.513533\pi\)
\(20\) 0 0
\(21\) −67.4015 −0.700392
\(22\) 25.6197i 0.248279i
\(23\) 0.654969i 0.00593785i 0.999996 + 0.00296892i \(0.000945039\pi\)
−0.999996 + 0.00296892i \(0.999055\pi\)
\(24\) 73.8928 0.628471
\(25\) 0 0
\(26\) 22.9653 0.173225
\(27\) 27.0000i 0.192450i
\(28\) − 57.8635i − 0.390542i
\(29\) 229.279 1.46814 0.734069 0.679075i \(-0.237619\pi\)
0.734069 + 0.679075i \(0.237619\pi\)
\(30\) 0 0
\(31\) 155.789 0.902596 0.451298 0.892373i \(-0.350961\pi\)
0.451298 + 0.892373i \(0.350961\pi\)
\(32\) 111.423i 0.615534i
\(33\) − 33.0000i − 0.174078i
\(34\) −298.438 −1.50534
\(35\) 0 0
\(36\) −23.1792 −0.107311
\(37\) 110.279i 0.489995i 0.969524 + 0.244998i \(0.0787872\pi\)
−0.969524 + 0.244998i \(0.921213\pi\)
\(38\) − 16.3966i − 0.0699970i
\(39\) −29.5809 −0.121455
\(40\) 0 0
\(41\) 154.749 0.589456 0.294728 0.955581i \(-0.404771\pi\)
0.294728 + 0.955581i \(0.404771\pi\)
\(42\) − 156.982i − 0.576736i
\(43\) − 401.014i − 1.42219i −0.703097 0.711094i \(-0.748200\pi\)
0.703097 0.711094i \(-0.251800\pi\)
\(44\) 28.3301 0.0970665
\(45\) 0 0
\(46\) −1.52546 −0.00488951
\(47\) 277.532i 0.861323i 0.902514 + 0.430661i \(0.141720\pi\)
−0.902514 + 0.430661i \(0.858280\pi\)
\(48\) 110.290i 0.331645i
\(49\) −161.774 −0.471645
\(50\) 0 0
\(51\) 384.410 1.05545
\(52\) − 25.3949i − 0.0677237i
\(53\) − 651.566i − 1.68867i −0.535817 0.844334i \(-0.679996\pi\)
0.535817 0.844334i \(-0.320004\pi\)
\(54\) −62.8847 −0.158473
\(55\) 0 0
\(56\) 553.388 1.32053
\(57\) 21.1200i 0.0490775i
\(58\) 534.005i 1.20894i
\(59\) 423.869 0.935307 0.467653 0.883912i \(-0.345100\pi\)
0.467653 + 0.883912i \(0.345100\pi\)
\(60\) 0 0
\(61\) 681.851 1.43118 0.715590 0.698520i \(-0.246158\pi\)
0.715590 + 0.698520i \(0.246158\pi\)
\(62\) 362.842i 0.743241i
\(63\) 202.205i 0.404371i
\(64\) −553.618 −1.08129
\(65\) 0 0
\(66\) 76.8591 0.143344
\(67\) − 374.028i − 0.682012i −0.940061 0.341006i \(-0.889232\pi\)
0.940061 0.341006i \(-0.110768\pi\)
\(68\) 330.011i 0.588526i
\(69\) 1.96491 0.00342822
\(70\) 0 0
\(71\) 96.6950 0.161628 0.0808140 0.996729i \(-0.474248\pi\)
0.0808140 + 0.996729i \(0.474248\pi\)
\(72\) − 221.678i − 0.362848i
\(73\) − 19.9460i − 0.0319795i −0.999872 0.0159897i \(-0.994910\pi\)
0.999872 0.0159897i \(-0.00508991\pi\)
\(74\) −256.848 −0.403486
\(75\) 0 0
\(76\) −18.1313 −0.0273658
\(77\) − 247.139i − 0.365768i
\(78\) − 68.8958i − 0.100012i
\(79\) −24.4286 −0.0347903 −0.0173951 0.999849i \(-0.505537\pi\)
−0.0173951 + 0.999849i \(0.505537\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 360.419i 0.485386i
\(83\) − 1127.35i − 1.49088i −0.666574 0.745439i \(-0.732240\pi\)
0.666574 0.745439i \(-0.267760\pi\)
\(84\) −173.590 −0.225479
\(85\) 0 0
\(86\) 933.987 1.17110
\(87\) − 687.836i − 0.847630i
\(88\) 270.940i 0.328208i
\(89\) 639.624 0.761798 0.380899 0.924617i \(-0.375615\pi\)
0.380899 + 0.924617i \(0.375615\pi\)
\(90\) 0 0
\(91\) −221.533 −0.255198
\(92\) 1.68685i 0.00191159i
\(93\) − 467.366i − 0.521114i
\(94\) −646.389 −0.709255
\(95\) 0 0
\(96\) 334.270 0.355378
\(97\) 730.865i 0.765032i 0.923949 + 0.382516i \(0.124942\pi\)
−0.923949 + 0.382516i \(0.875058\pi\)
\(98\) − 376.783i − 0.388375i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −810.342 −0.798337 −0.399168 0.916878i \(-0.630701\pi\)
−0.399168 + 0.916878i \(0.630701\pi\)
\(102\) 895.314i 0.869111i
\(103\) − 1461.89i − 1.39849i −0.714882 0.699245i \(-0.753520\pi\)
0.714882 0.699245i \(-0.246480\pi\)
\(104\) 242.868 0.228992
\(105\) 0 0
\(106\) 1517.54 1.39053
\(107\) − 1690.40i − 1.52726i −0.645654 0.763630i \(-0.723415\pi\)
0.645654 0.763630i \(-0.276585\pi\)
\(108\) 69.5376i 0.0619561i
\(109\) 1409.41 1.23851 0.619254 0.785190i \(-0.287435\pi\)
0.619254 + 0.785190i \(0.287435\pi\)
\(110\) 0 0
\(111\) 330.838 0.282899
\(112\) 825.967i 0.696844i
\(113\) 2185.67i 1.81956i 0.415090 + 0.909780i \(0.363750\pi\)
−0.415090 + 0.909780i \(0.636250\pi\)
\(114\) −49.1899 −0.0404128
\(115\) 0 0
\(116\) 590.499 0.472642
\(117\) 88.7427i 0.0701219i
\(118\) 987.219i 0.770177i
\(119\) 2878.87 2.21769
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1588.07i 1.17850i
\(123\) − 464.246i − 0.340322i
\(124\) 401.228 0.290576
\(125\) 0 0
\(126\) −470.947 −0.332979
\(127\) 1918.85i 1.34071i 0.742038 + 0.670357i \(0.233859\pi\)
−0.742038 + 0.670357i \(0.766141\pi\)
\(128\) − 398.024i − 0.274849i
\(129\) −1203.04 −0.821100
\(130\) 0 0
\(131\) 1339.41 0.893320 0.446660 0.894704i \(-0.352613\pi\)
0.446660 + 0.894704i \(0.352613\pi\)
\(132\) − 84.9904i − 0.0560414i
\(133\) 158.169i 0.103120i
\(134\) 871.135 0.561602
\(135\) 0 0
\(136\) −3156.12 −1.98996
\(137\) 1100.56i 0.686330i 0.939275 + 0.343165i \(0.111499\pi\)
−0.939275 + 0.343165i \(0.888501\pi\)
\(138\) 4.57639i 0.00282296i
\(139\) 1284.51 0.783819 0.391910 0.920004i \(-0.371815\pi\)
0.391910 + 0.920004i \(0.371815\pi\)
\(140\) 0 0
\(141\) 832.595 0.497285
\(142\) 225.209i 0.133092i
\(143\) − 108.463i − 0.0634277i
\(144\) 330.869 0.191475
\(145\) 0 0
\(146\) 46.4554 0.0263334
\(147\) 485.323i 0.272305i
\(148\) 284.021i 0.157746i
\(149\) −1277.21 −0.702236 −0.351118 0.936331i \(-0.614198\pi\)
−0.351118 + 0.936331i \(0.614198\pi\)
\(150\) 0 0
\(151\) 886.317 0.477665 0.238833 0.971061i \(-0.423235\pi\)
0.238833 + 0.971061i \(0.423235\pi\)
\(152\) − 173.402i − 0.0925313i
\(153\) − 1153.23i − 0.609366i
\(154\) 575.602 0.301191
\(155\) 0 0
\(156\) −76.1846 −0.0391003
\(157\) 1681.12i 0.854575i 0.904116 + 0.427288i \(0.140531\pi\)
−0.904116 + 0.427288i \(0.859469\pi\)
\(158\) − 56.8958i − 0.0286480i
\(159\) −1954.70 −0.974953
\(160\) 0 0
\(161\) 14.7153 0.00720329
\(162\) 188.654i 0.0914942i
\(163\) − 622.100i − 0.298937i −0.988767 0.149468i \(-0.952244\pi\)
0.988767 0.149468i \(-0.0477562\pi\)
\(164\) 398.550 0.189765
\(165\) 0 0
\(166\) 2625.67 1.22766
\(167\) 2611.82i 1.21023i 0.796138 + 0.605115i \(0.206873\pi\)
−0.796138 + 0.605115i \(0.793127\pi\)
\(168\) − 1660.16i − 0.762407i
\(169\) 2099.77 0.955746
\(170\) 0 0
\(171\) 63.3601 0.0283349
\(172\) − 1032.80i − 0.457849i
\(173\) 2342.97i 1.02967i 0.857290 + 0.514835i \(0.172147\pi\)
−0.857290 + 0.514835i \(0.827853\pi\)
\(174\) 1602.01 0.697979
\(175\) 0 0
\(176\) −404.396 −0.173196
\(177\) − 1271.61i − 0.540000i
\(178\) 1489.72i 0.627301i
\(179\) −1314.75 −0.548991 −0.274495 0.961588i \(-0.588511\pi\)
−0.274495 + 0.961588i \(0.588511\pi\)
\(180\) 0 0
\(181\) 8.69006 0.00356866 0.00178433 0.999998i \(-0.499432\pi\)
0.00178433 + 0.999998i \(0.499432\pi\)
\(182\) − 515.965i − 0.210142i
\(183\) − 2045.55i − 0.826292i
\(184\) −16.1325 −0.00646361
\(185\) 0 0
\(186\) 1088.53 0.429110
\(187\) 1409.50i 0.551192i
\(188\) 714.774i 0.277288i
\(189\) 606.614 0.233464
\(190\) 0 0
\(191\) 644.102 0.244008 0.122004 0.992530i \(-0.461068\pi\)
0.122004 + 0.992530i \(0.461068\pi\)
\(192\) 1660.85i 0.624281i
\(193\) 3970.76i 1.48094i 0.672089 + 0.740470i \(0.265397\pi\)
−0.672089 + 0.740470i \(0.734603\pi\)
\(194\) −1702.23 −0.629964
\(195\) 0 0
\(196\) −416.644 −0.151838
\(197\) − 3756.34i − 1.35852i −0.733898 0.679260i \(-0.762301\pi\)
0.733898 0.679260i \(-0.237699\pi\)
\(198\) − 230.577i − 0.0827596i
\(199\) −4825.48 −1.71894 −0.859470 0.511186i \(-0.829206\pi\)
−0.859470 + 0.511186i \(0.829206\pi\)
\(200\) 0 0
\(201\) −1122.08 −0.393760
\(202\) − 1887.34i − 0.657389i
\(203\) − 5151.25i − 1.78102i
\(204\) 990.034 0.339785
\(205\) 0 0
\(206\) 3404.84 1.15158
\(207\) − 5.89472i − 0.00197928i
\(208\) 362.497i 0.120840i
\(209\) −77.4401 −0.0256299
\(210\) 0 0
\(211\) −4394.02 −1.43363 −0.716817 0.697261i \(-0.754402\pi\)
−0.716817 + 0.697261i \(0.754402\pi\)
\(212\) − 1678.08i − 0.543638i
\(213\) − 290.085i − 0.0933160i
\(214\) 3937.04 1.25762
\(215\) 0 0
\(216\) −665.035 −0.209490
\(217\) − 3500.13i − 1.09495i
\(218\) 3282.62i 1.01985i
\(219\) −59.8379 −0.0184633
\(220\) 0 0
\(221\) 1263.46 0.384569
\(222\) 770.543i 0.232953i
\(223\) 2189.67i 0.657538i 0.944410 + 0.328769i \(0.106634\pi\)
−0.944410 + 0.328769i \(0.893366\pi\)
\(224\) 2503.37 0.746712
\(225\) 0 0
\(226\) −5090.56 −1.49831
\(227\) 1139.27i 0.333110i 0.986032 + 0.166555i \(0.0532643\pi\)
−0.986032 + 0.166555i \(0.946736\pi\)
\(228\) 54.3939i 0.0157997i
\(229\) −3416.10 −0.985773 −0.492886 0.870094i \(-0.664058\pi\)
−0.492886 + 0.870094i \(0.664058\pi\)
\(230\) 0 0
\(231\) −741.417 −0.211176
\(232\) 5647.35i 1.59813i
\(233\) 6147.08i 1.72836i 0.503181 + 0.864181i \(0.332163\pi\)
−0.503181 + 0.864181i \(0.667837\pi\)
\(234\) −206.687 −0.0577418
\(235\) 0 0
\(236\) 1091.66 0.301106
\(237\) 73.2858i 0.0200862i
\(238\) 6705.06i 1.82615i
\(239\) 2080.03 0.562954 0.281477 0.959568i \(-0.409176\pi\)
0.281477 + 0.959568i \(0.409176\pi\)
\(240\) 0 0
\(241\) 1846.28 0.493484 0.246742 0.969081i \(-0.420640\pi\)
0.246742 + 0.969081i \(0.420640\pi\)
\(242\) 281.817i 0.0748589i
\(243\) − 243.000i − 0.0641500i
\(244\) 1756.08 0.460745
\(245\) 0 0
\(246\) 1081.26 0.280238
\(247\) 69.4166i 0.0178821i
\(248\) 3837.22i 0.982515i
\(249\) −3382.05 −0.860758
\(250\) 0 0
\(251\) −2555.32 −0.642592 −0.321296 0.946979i \(-0.604118\pi\)
−0.321296 + 0.946979i \(0.604118\pi\)
\(252\) 520.771i 0.130181i
\(253\) 7.20466i 0.00179033i
\(254\) −4469.13 −1.10401
\(255\) 0 0
\(256\) −3501.92 −0.854962
\(257\) 1819.39i 0.441598i 0.975319 + 0.220799i \(0.0708664\pi\)
−0.975319 + 0.220799i \(0.929134\pi\)
\(258\) − 2801.96i − 0.676134i
\(259\) 2477.67 0.594420
\(260\) 0 0
\(261\) −2063.51 −0.489379
\(262\) 3119.57i 0.735603i
\(263\) − 6023.03i − 1.41215i −0.708136 0.706076i \(-0.750464\pi\)
0.708136 0.706076i \(-0.249536\pi\)
\(264\) 812.821 0.189491
\(265\) 0 0
\(266\) −368.386 −0.0849143
\(267\) − 1918.87i − 0.439824i
\(268\) − 963.297i − 0.219562i
\(269\) 2978.38 0.675075 0.337537 0.941312i \(-0.390406\pi\)
0.337537 + 0.941312i \(0.390406\pi\)
\(270\) 0 0
\(271\) −524.969 −0.117674 −0.0588369 0.998268i \(-0.518739\pi\)
−0.0588369 + 0.998268i \(0.518739\pi\)
\(272\) − 4710.72i − 1.05011i
\(273\) 664.600i 0.147338i
\(274\) −2563.28 −0.565157
\(275\) 0 0
\(276\) 5.06055 0.00110366
\(277\) 1693.07i 0.367245i 0.982997 + 0.183623i \(0.0587824\pi\)
−0.982997 + 0.183623i \(0.941218\pi\)
\(278\) 2991.71i 0.645435i
\(279\) −1402.10 −0.300865
\(280\) 0 0
\(281\) 7346.60 1.55965 0.779824 0.625998i \(-0.215308\pi\)
0.779824 + 0.625998i \(0.215308\pi\)
\(282\) 1939.17i 0.409488i
\(283\) − 1501.69i − 0.315429i −0.987485 0.157714i \(-0.949587\pi\)
0.987485 0.157714i \(-0.0504125\pi\)
\(284\) 249.035 0.0520334
\(285\) 0 0
\(286\) 252.618 0.0522294
\(287\) − 3476.77i − 0.715077i
\(288\) − 1002.81i − 0.205178i
\(289\) −11506.0 −2.34194
\(290\) 0 0
\(291\) 2192.59 0.441691
\(292\) − 51.3702i − 0.0102952i
\(293\) − 4481.03i − 0.893462i −0.894668 0.446731i \(-0.852588\pi\)
0.894668 0.446731i \(-0.147412\pi\)
\(294\) −1130.35 −0.224229
\(295\) 0 0
\(296\) −2716.28 −0.533381
\(297\) 297.000i 0.0580259i
\(298\) − 2974.71i − 0.578255i
\(299\) 6.45819 0.00124912
\(300\) 0 0
\(301\) −9009.66 −1.72528
\(302\) 2064.29i 0.393333i
\(303\) 2431.02i 0.460920i
\(304\) 258.814 0.0488289
\(305\) 0 0
\(306\) 2685.94 0.501781
\(307\) − 3052.17i − 0.567416i −0.958911 0.283708i \(-0.908435\pi\)
0.958911 0.283708i \(-0.0915646\pi\)
\(308\) − 636.498i − 0.117753i
\(309\) −4385.67 −0.807418
\(310\) 0 0
\(311\) 10255.1 1.86983 0.934913 0.354878i \(-0.115477\pi\)
0.934913 + 0.354878i \(0.115477\pi\)
\(312\) − 728.605i − 0.132209i
\(313\) − 6190.18i − 1.11786i −0.829215 0.558929i \(-0.811212\pi\)
0.829215 0.558929i \(-0.188788\pi\)
\(314\) −3915.44 −0.703698
\(315\) 0 0
\(316\) −62.9150 −0.0112001
\(317\) 6735.38i 1.19337i 0.802477 + 0.596683i \(0.203515\pi\)
−0.802477 + 0.596683i \(0.796485\pi\)
\(318\) − 4552.61i − 0.802823i
\(319\) 2522.07 0.442660
\(320\) 0 0
\(321\) −5071.19 −0.881764
\(322\) 34.2729i 0.00593153i
\(323\) − 902.083i − 0.155397i
\(324\) 208.613 0.0357704
\(325\) 0 0
\(326\) 1448.91 0.246159
\(327\) − 4228.24i − 0.715053i
\(328\) 3811.60i 0.641648i
\(329\) 6235.36 1.04488
\(330\) 0 0
\(331\) −4780.83 −0.793891 −0.396946 0.917842i \(-0.629930\pi\)
−0.396946 + 0.917842i \(0.629930\pi\)
\(332\) − 2903.45i − 0.479963i
\(333\) − 992.515i − 0.163332i
\(334\) −6083.09 −0.996562
\(335\) 0 0
\(336\) 2477.90 0.402323
\(337\) − 11890.3i − 1.92197i −0.276604 0.960984i \(-0.589209\pi\)
0.276604 0.960984i \(-0.410791\pi\)
\(338\) 4890.51i 0.787007i
\(339\) 6557.00 1.05052
\(340\) 0 0
\(341\) 1713.68 0.272143
\(342\) 147.570i 0.0233323i
\(343\) − 4071.63i − 0.640954i
\(344\) 9877.35 1.54811
\(345\) 0 0
\(346\) −5456.93 −0.847879
\(347\) − 8462.47i − 1.30919i −0.755979 0.654595i \(-0.772839\pi\)
0.755979 0.654595i \(-0.227161\pi\)
\(348\) − 1771.50i − 0.272880i
\(349\) 3291.90 0.504903 0.252452 0.967610i \(-0.418763\pi\)
0.252452 + 0.967610i \(0.418763\pi\)
\(350\) 0 0
\(351\) 266.228 0.0404849
\(352\) 1225.66i 0.185590i
\(353\) − 8193.52i − 1.23540i −0.786413 0.617701i \(-0.788064\pi\)
0.786413 0.617701i \(-0.211936\pi\)
\(354\) 2961.66 0.444662
\(355\) 0 0
\(356\) 1647.33 0.245248
\(357\) − 8636.60i − 1.28038i
\(358\) − 3062.15i − 0.452066i
\(359\) −12817.6 −1.88437 −0.942185 0.335093i \(-0.891232\pi\)
−0.942185 + 0.335093i \(0.891232\pi\)
\(360\) 0 0
\(361\) −6809.44 −0.992774
\(362\) 20.2397i 0.00293861i
\(363\) − 363.000i − 0.0524864i
\(364\) −570.551 −0.0821566
\(365\) 0 0
\(366\) 4764.22 0.680409
\(367\) 2801.22i 0.398427i 0.979956 + 0.199213i \(0.0638387\pi\)
−0.979956 + 0.199213i \(0.936161\pi\)
\(368\) − 24.0788i − 0.00341085i
\(369\) −1392.74 −0.196485
\(370\) 0 0
\(371\) −14638.8 −2.04855
\(372\) − 1203.69i − 0.167764i
\(373\) 6838.03i 0.949222i 0.880196 + 0.474611i \(0.157411\pi\)
−0.880196 + 0.474611i \(0.842589\pi\)
\(374\) −3282.82 −0.453878
\(375\) 0 0
\(376\) −6835.86 −0.937587
\(377\) − 2260.76i − 0.308846i
\(378\) 1412.84i 0.192245i
\(379\) 7465.79 1.01185 0.505926 0.862577i \(-0.331151\pi\)
0.505926 + 0.862577i \(0.331151\pi\)
\(380\) 0 0
\(381\) 5756.56 0.774062
\(382\) 1500.16i 0.200928i
\(383\) − 8646.55i − 1.15357i −0.816895 0.576786i \(-0.804307\pi\)
0.816895 0.576786i \(-0.195693\pi\)
\(384\) −1194.07 −0.158684
\(385\) 0 0
\(386\) −9248.15 −1.21948
\(387\) 3609.13i 0.474063i
\(388\) 1882.32i 0.246289i
\(389\) −4382.78 −0.571248 −0.285624 0.958342i \(-0.592201\pi\)
−0.285624 + 0.958342i \(0.592201\pi\)
\(390\) 0 0
\(391\) −83.9255 −0.0108550
\(392\) − 3984.65i − 0.513406i
\(393\) − 4018.24i − 0.515759i
\(394\) 8748.76 1.11867
\(395\) 0 0
\(396\) −254.971 −0.0323555
\(397\) 4432.58i 0.560365i 0.959947 + 0.280182i \(0.0903950\pi\)
−0.959947 + 0.280182i \(0.909605\pi\)
\(398\) − 11238.8i − 1.41546i
\(399\) 474.508 0.0595366
\(400\) 0 0
\(401\) −5034.93 −0.627013 −0.313507 0.949586i \(-0.601504\pi\)
−0.313507 + 0.949586i \(0.601504\pi\)
\(402\) − 2613.41i − 0.324241i
\(403\) − 1536.12i − 0.189875i
\(404\) −2087.01 −0.257011
\(405\) 0 0
\(406\) 11997.6 1.46658
\(407\) 1213.07i 0.147739i
\(408\) 9468.36i 1.14891i
\(409\) −6474.64 −0.782764 −0.391382 0.920228i \(-0.628003\pi\)
−0.391382 + 0.920228i \(0.628003\pi\)
\(410\) 0 0
\(411\) 3301.68 0.396253
\(412\) − 3765.05i − 0.450220i
\(413\) − 9523.15i − 1.13463i
\(414\) 13.7292 0.00162984
\(415\) 0 0
\(416\) 1098.67 0.129487
\(417\) − 3853.54i − 0.452538i
\(418\) − 180.363i − 0.0211049i
\(419\) 8257.80 0.962816 0.481408 0.876497i \(-0.340126\pi\)
0.481408 + 0.876497i \(0.340126\pi\)
\(420\) 0 0
\(421\) −3429.36 −0.397000 −0.198500 0.980101i \(-0.563607\pi\)
−0.198500 + 0.980101i \(0.563607\pi\)
\(422\) − 10233.9i − 1.18052i
\(423\) − 2497.79i − 0.287108i
\(424\) 16048.7 1.83819
\(425\) 0 0
\(426\) 675.626 0.0768408
\(427\) − 15319.3i − 1.73618i
\(428\) − 4353.56i − 0.491676i
\(429\) −325.390 −0.0366200
\(430\) 0 0
\(431\) −11260.4 −1.25846 −0.629230 0.777219i \(-0.716630\pi\)
−0.629230 + 0.777219i \(0.716630\pi\)
\(432\) − 992.608i − 0.110548i
\(433\) 12598.5i 1.39826i 0.714996 + 0.699128i \(0.246428\pi\)
−0.714996 + 0.699128i \(0.753572\pi\)
\(434\) 8152.03 0.901636
\(435\) 0 0
\(436\) 3629.90 0.398717
\(437\) − 4.61099i 0 0.000504745i
\(438\) − 139.366i − 0.0152036i
\(439\) −4176.90 −0.454106 −0.227053 0.973882i \(-0.572909\pi\)
−0.227053 + 0.973882i \(0.572909\pi\)
\(440\) 0 0
\(441\) 1455.97 0.157215
\(442\) 2942.69i 0.316673i
\(443\) 2354.16i 0.252482i 0.992000 + 0.126241i \(0.0402913\pi\)
−0.992000 + 0.126241i \(0.959709\pi\)
\(444\) 852.062 0.0910745
\(445\) 0 0
\(446\) −5099.88 −0.541449
\(447\) 3831.63i 0.405436i
\(448\) 12438.2i 1.31172i
\(449\) −9286.18 −0.976040 −0.488020 0.872832i \(-0.662281\pi\)
−0.488020 + 0.872832i \(0.662281\pi\)
\(450\) 0 0
\(451\) 1702.24 0.177728
\(452\) 5629.11i 0.585777i
\(453\) − 2658.95i − 0.275780i
\(454\) −2653.43 −0.274299
\(455\) 0 0
\(456\) −520.206 −0.0534230
\(457\) − 14378.4i − 1.47176i −0.677115 0.735878i \(-0.736770\pi\)
0.677115 0.735878i \(-0.263230\pi\)
\(458\) − 7956.30i − 0.811733i
\(459\) −3459.69 −0.351818
\(460\) 0 0
\(461\) −5383.19 −0.543861 −0.271931 0.962317i \(-0.587662\pi\)
−0.271931 + 0.962317i \(0.587662\pi\)
\(462\) − 1726.81i − 0.173892i
\(463\) 18360.5i 1.84294i 0.388446 + 0.921471i \(0.373012\pi\)
−0.388446 + 0.921471i \(0.626988\pi\)
\(464\) −8429.03 −0.843336
\(465\) 0 0
\(466\) −14316.9 −1.42322
\(467\) − 9063.65i − 0.898106i −0.893505 0.449053i \(-0.851761\pi\)
0.893505 0.449053i \(-0.148239\pi\)
\(468\) 228.554i 0.0225746i
\(469\) −8403.36 −0.827358
\(470\) 0 0
\(471\) 5043.37 0.493389
\(472\) 10440.3i 1.01812i
\(473\) − 4411.15i − 0.428806i
\(474\) −170.687 −0.0165399
\(475\) 0 0
\(476\) 7414.42 0.713949
\(477\) 5864.09i 0.562889i
\(478\) 4844.53i 0.463564i
\(479\) −12608.2 −1.20268 −0.601341 0.798992i \(-0.705367\pi\)
−0.601341 + 0.798992i \(0.705367\pi\)
\(480\) 0 0
\(481\) 1087.39 0.103078
\(482\) 4300.11i 0.406358i
\(483\) − 44.1459i − 0.00415882i
\(484\) 311.631 0.0292667
\(485\) 0 0
\(486\) 565.962 0.0528242
\(487\) 13214.2i 1.22955i 0.788703 + 0.614775i \(0.210753\pi\)
−0.788703 + 0.614775i \(0.789247\pi\)
\(488\) 16794.6i 1.55790i
\(489\) −1866.30 −0.172591
\(490\) 0 0
\(491\) 6553.27 0.602332 0.301166 0.953572i \(-0.402624\pi\)
0.301166 + 0.953572i \(0.402624\pi\)
\(492\) − 1195.65i − 0.109561i
\(493\) 29379.0i 2.68390i
\(494\) −161.676 −0.0147250
\(495\) 0 0
\(496\) −5727.30 −0.518474
\(497\) − 2172.46i − 0.196073i
\(498\) − 7877.01i − 0.708790i
\(499\) 2596.63 0.232948 0.116474 0.993194i \(-0.462841\pi\)
0.116474 + 0.993194i \(0.462841\pi\)
\(500\) 0 0
\(501\) 7835.45 0.698727
\(502\) − 5951.51i − 0.529141i
\(503\) − 659.714i − 0.0584795i −0.999572 0.0292398i \(-0.990691\pi\)
0.999572 0.0292398i \(-0.00930863\pi\)
\(504\) −4980.49 −0.440176
\(505\) 0 0
\(506\) −16.7801 −0.00147424
\(507\) − 6299.32i − 0.551800i
\(508\) 4941.94i 0.431621i
\(509\) −4825.41 −0.420201 −0.210101 0.977680i \(-0.567379\pi\)
−0.210101 + 0.977680i \(0.567379\pi\)
\(510\) 0 0
\(511\) −448.130 −0.0387947
\(512\) − 11340.4i − 0.978866i
\(513\) − 190.080i − 0.0163592i
\(514\) −4237.48 −0.363633
\(515\) 0 0
\(516\) −3098.39 −0.264339
\(517\) 3052.85i 0.259699i
\(518\) 5770.64i 0.489474i
\(519\) 7028.91 0.594480
\(520\) 0 0
\(521\) −2329.24 −0.195866 −0.0979328 0.995193i \(-0.531223\pi\)
−0.0979328 + 0.995193i \(0.531223\pi\)
\(522\) − 4806.04i − 0.402978i
\(523\) − 15104.6i − 1.26287i −0.775429 0.631434i \(-0.782467\pi\)
0.775429 0.631434i \(-0.217533\pi\)
\(524\) 3449.61 0.287589
\(525\) 0 0
\(526\) 14028.0 1.16283
\(527\) 19962.2i 1.65003i
\(528\) 1213.19i 0.0999947i
\(529\) 12166.6 0.999965
\(530\) 0 0
\(531\) −3814.82 −0.311769
\(532\) 407.359i 0.0331979i
\(533\) − 1525.87i − 0.124001i
\(534\) 4469.17 0.362172
\(535\) 0 0
\(536\) 9212.66 0.742400
\(537\) 3944.26i 0.316960i
\(538\) 6936.84i 0.555889i
\(539\) −1779.52 −0.142206
\(540\) 0 0
\(541\) 10712.1 0.851293 0.425647 0.904889i \(-0.360047\pi\)
0.425647 + 0.904889i \(0.360047\pi\)
\(542\) − 1222.69i − 0.0968983i
\(543\) − 26.0702i − 0.00206037i
\(544\) −14277.4 −1.12526
\(545\) 0 0
\(546\) −1547.89 −0.121326
\(547\) 17251.6i 1.34849i 0.738508 + 0.674245i \(0.235531\pi\)
−0.738508 + 0.674245i \(0.764469\pi\)
\(548\) 2834.46i 0.220953i
\(549\) −6136.65 −0.477060
\(550\) 0 0
\(551\) −1614.12 −0.124799
\(552\) 48.3975i 0.00373177i
\(553\) 548.842i 0.0422046i
\(554\) −3943.27 −0.302407
\(555\) 0 0
\(556\) 3308.22 0.252337
\(557\) 8179.34i 0.622208i 0.950376 + 0.311104i \(0.100699\pi\)
−0.950376 + 0.311104i \(0.899301\pi\)
\(558\) − 3265.58i − 0.247747i
\(559\) −3954.12 −0.299180
\(560\) 0 0
\(561\) 4228.51 0.318231
\(562\) 17110.7i 1.28429i
\(563\) 4939.38i 0.369752i 0.982762 + 0.184876i \(0.0591883\pi\)
−0.982762 + 0.184876i \(0.940812\pi\)
\(564\) 2144.32 0.160093
\(565\) 0 0
\(566\) 3497.54 0.259739
\(567\) − 1819.84i − 0.134790i
\(568\) 2381.69i 0.175939i
\(569\) 7658.76 0.564274 0.282137 0.959374i \(-0.408957\pi\)
0.282137 + 0.959374i \(0.408957\pi\)
\(570\) 0 0
\(571\) −1744.16 −0.127830 −0.0639149 0.997955i \(-0.520359\pi\)
−0.0639149 + 0.997955i \(0.520359\pi\)
\(572\) − 279.344i − 0.0204195i
\(573\) − 1932.31i − 0.140878i
\(574\) 8097.61 0.588829
\(575\) 0 0
\(576\) 4982.56 0.360429
\(577\) − 1264.61i − 0.0912414i −0.998959 0.0456207i \(-0.985473\pi\)
0.998959 0.0456207i \(-0.0145266\pi\)
\(578\) − 26798.1i − 1.92847i
\(579\) 11912.3 0.855022
\(580\) 0 0
\(581\) −25328.4 −1.80860
\(582\) 5106.69i 0.363710i
\(583\) − 7167.22i − 0.509153i
\(584\) 491.288 0.0348110
\(585\) 0 0
\(586\) 10436.6 0.735720
\(587\) 17167.4i 1.20711i 0.797320 + 0.603557i \(0.206250\pi\)
−0.797320 + 0.603557i \(0.793750\pi\)
\(588\) 1249.93i 0.0876639i
\(589\) −1096.75 −0.0767249
\(590\) 0 0
\(591\) −11269.0 −0.784342
\(592\) − 4054.23i − 0.281466i
\(593\) − 21429.5i − 1.48399i −0.670406 0.741995i \(-0.733880\pi\)
0.670406 0.741995i \(-0.266120\pi\)
\(594\) −691.732 −0.0477813
\(595\) 0 0
\(596\) −3289.41 −0.226073
\(597\) 14476.4i 0.992431i
\(598\) 15.0415i 0.00102859i
\(599\) 7994.61 0.545327 0.272664 0.962109i \(-0.412095\pi\)
0.272664 + 0.962109i \(0.412095\pi\)
\(600\) 0 0
\(601\) −24313.4 −1.65019 −0.825094 0.564996i \(-0.808878\pi\)
−0.825094 + 0.564996i \(0.808878\pi\)
\(602\) − 20984.1i − 1.42067i
\(603\) 3366.25i 0.227337i
\(604\) 2282.68 0.153776
\(605\) 0 0
\(606\) −5662.01 −0.379544
\(607\) − 24569.7i − 1.64292i −0.570266 0.821460i \(-0.693160\pi\)
0.570266 0.821460i \(-0.306840\pi\)
\(608\) − 784.423i − 0.0523232i
\(609\) −15453.7 −1.02827
\(610\) 0 0
\(611\) 2736.55 0.181193
\(612\) − 2970.10i − 0.196175i
\(613\) − 12746.7i − 0.839859i −0.907557 0.419929i \(-0.862055\pi\)
0.907557 0.419929i \(-0.137945\pi\)
\(614\) 7108.70 0.467237
\(615\) 0 0
\(616\) 6087.26 0.398154
\(617\) 15607.4i 1.01837i 0.860658 + 0.509183i \(0.170052\pi\)
−0.860658 + 0.509183i \(0.829948\pi\)
\(618\) − 10214.5i − 0.664867i
\(619\) 11909.7 0.773329 0.386665 0.922220i \(-0.373627\pi\)
0.386665 + 0.922220i \(0.373627\pi\)
\(620\) 0 0
\(621\) −17.6842 −0.00114274
\(622\) 23884.9i 1.53970i
\(623\) − 14370.5i − 0.924147i
\(624\) 1087.49 0.0697667
\(625\) 0 0
\(626\) 14417.3 0.920499
\(627\) 232.320i 0.0147974i
\(628\) 4329.68i 0.275116i
\(629\) −14130.8 −0.895759
\(630\) 0 0
\(631\) −19304.2 −1.21789 −0.608946 0.793212i \(-0.708407\pi\)
−0.608946 + 0.793212i \(0.708407\pi\)
\(632\) − 601.699i − 0.0378707i
\(633\) 13182.1i 0.827709i
\(634\) −15687.1 −0.982674
\(635\) 0 0
\(636\) −5034.25 −0.313870
\(637\) 1595.14i 0.0992180i
\(638\) 5874.05i 0.364508i
\(639\) −870.255 −0.0538760
\(640\) 0 0
\(641\) 26678.6 1.64390 0.821950 0.569560i \(-0.192886\pi\)
0.821950 + 0.569560i \(0.192886\pi\)
\(642\) − 11811.1i − 0.726087i
\(643\) − 26456.2i − 1.62260i −0.584631 0.811299i \(-0.698761\pi\)
0.584631 0.811299i \(-0.301239\pi\)
\(644\) 37.8988 0.00231898
\(645\) 0 0
\(646\) 2101.01 0.127961
\(647\) 23523.7i 1.42939i 0.699438 + 0.714694i \(0.253434\pi\)
−0.699438 + 0.714694i \(0.746566\pi\)
\(648\) 1995.10i 0.120949i
\(649\) 4662.56 0.282006
\(650\) 0 0
\(651\) −10500.4 −0.632171
\(652\) − 1602.20i − 0.0962376i
\(653\) 18071.1i 1.08296i 0.840712 + 0.541482i \(0.182137\pi\)
−0.840712 + 0.541482i \(0.817863\pi\)
\(654\) 9847.85 0.588809
\(655\) 0 0
\(656\) −5689.07 −0.338599
\(657\) 179.514i 0.0106598i
\(658\) 14522.5i 0.860407i
\(659\) −17023.1 −1.00626 −0.503130 0.864210i \(-0.667818\pi\)
−0.503130 + 0.864210i \(0.667818\pi\)
\(660\) 0 0
\(661\) 2137.74 0.125792 0.0628959 0.998020i \(-0.479966\pi\)
0.0628959 + 0.998020i \(0.479966\pi\)
\(662\) − 11134.8i − 0.653728i
\(663\) − 3790.39i − 0.222031i
\(664\) 27767.7 1.62288
\(665\) 0 0
\(666\) 2311.63 0.134495
\(667\) 150.171i 0.00871758i
\(668\) 6726.65i 0.389614i
\(669\) 6569.01 0.379630
\(670\) 0 0
\(671\) 7500.36 0.431517
\(672\) − 7510.11i − 0.431115i
\(673\) 31790.1i 1.82083i 0.413698 + 0.910414i \(0.364237\pi\)
−0.413698 + 0.910414i \(0.635763\pi\)
\(674\) 27693.1 1.58264
\(675\) 0 0
\(676\) 5407.90 0.307686
\(677\) 10225.1i 0.580476i 0.956955 + 0.290238i \(0.0937344\pi\)
−0.956955 + 0.290238i \(0.906266\pi\)
\(678\) 15271.7i 0.865052i
\(679\) 16420.5 0.928071
\(680\) 0 0
\(681\) 3417.81 0.192321
\(682\) 3991.26i 0.224096i
\(683\) 21274.0i 1.19184i 0.803044 + 0.595919i \(0.203212\pi\)
−0.803044 + 0.595919i \(0.796788\pi\)
\(684\) 163.182 0.00912195
\(685\) 0 0
\(686\) 9483.08 0.527793
\(687\) 10248.3i 0.569136i
\(688\) 14742.6i 0.816941i
\(689\) −6424.63 −0.355238
\(690\) 0 0
\(691\) −22568.1 −1.24245 −0.621224 0.783633i \(-0.713364\pi\)
−0.621224 + 0.783633i \(0.713364\pi\)
\(692\) 6034.24i 0.331485i
\(693\) 2224.25i 0.121923i
\(694\) 19709.6 1.07805
\(695\) 0 0
\(696\) 16942.0 0.922682
\(697\) 19829.0i 1.07758i
\(698\) 7667.03i 0.415761i
\(699\) 18441.2 0.997870
\(700\) 0 0
\(701\) −7735.03 −0.416759 −0.208380 0.978048i \(-0.566819\pi\)
−0.208380 + 0.978048i \(0.566819\pi\)
\(702\) 620.062i 0.0333372i
\(703\) − 776.368i − 0.0416519i
\(704\) −6089.80 −0.326020
\(705\) 0 0
\(706\) 19083.2 1.01729
\(707\) 18206.1i 0.968473i
\(708\) − 3274.98i − 0.173844i
\(709\) −35115.1 −1.86005 −0.930025 0.367497i \(-0.880215\pi\)
−0.930025 + 0.367497i \(0.880215\pi\)
\(710\) 0 0
\(711\) 219.857 0.0115968
\(712\) 15754.5i 0.829250i
\(713\) 102.037i 0.00535948i
\(714\) 20115.2 1.05433
\(715\) 0 0
\(716\) −3386.11 −0.176738
\(717\) − 6240.10i − 0.325022i
\(718\) − 29853.1i − 1.55168i
\(719\) −15334.3 −0.795370 −0.397685 0.917522i \(-0.630186\pi\)
−0.397685 + 0.917522i \(0.630186\pi\)
\(720\) 0 0
\(721\) −32844.6 −1.69653
\(722\) − 15859.6i − 0.817498i
\(723\) − 5538.85i − 0.284913i
\(724\) 22.3810 0.00114887
\(725\) 0 0
\(726\) 845.450 0.0432198
\(727\) 12360.4i 0.630567i 0.948997 + 0.315284i \(0.102100\pi\)
−0.948997 + 0.315284i \(0.897900\pi\)
\(728\) − 5456.57i − 0.277794i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 51384.6 2.59990
\(732\) − 5268.25i − 0.266011i
\(733\) − 15097.6i − 0.760769i −0.924828 0.380384i \(-0.875792\pi\)
0.924828 0.380384i \(-0.124208\pi\)
\(734\) −6524.23 −0.328084
\(735\) 0 0
\(736\) −72.9790 −0.00365495
\(737\) − 4114.31i − 0.205634i
\(738\) − 3243.78i − 0.161795i
\(739\) 3667.49 0.182559 0.0912793 0.995825i \(-0.470904\pi\)
0.0912793 + 0.995825i \(0.470904\pi\)
\(740\) 0 0
\(741\) 208.250 0.0103242
\(742\) − 34094.8i − 1.68687i
\(743\) 10172.1i 0.502257i 0.967954 + 0.251128i \(0.0808016\pi\)
−0.967954 + 0.251128i \(0.919198\pi\)
\(744\) 11511.7 0.567255
\(745\) 0 0
\(746\) −15926.2 −0.781635
\(747\) 10146.2i 0.496959i
\(748\) 3630.12i 0.177447i
\(749\) −37978.5 −1.85274
\(750\) 0 0
\(751\) 31430.3 1.52718 0.763588 0.645704i \(-0.223436\pi\)
0.763588 + 0.645704i \(0.223436\pi\)
\(752\) − 10203.0i − 0.494766i
\(753\) 7665.97i 0.371000i
\(754\) 5265.45 0.254319
\(755\) 0 0
\(756\) 1562.31 0.0751598
\(757\) 28362.0i 1.36174i 0.732406 + 0.680868i \(0.238397\pi\)
−0.732406 + 0.680868i \(0.761603\pi\)
\(758\) 17388.3i 0.833207i
\(759\) 21.6140 0.00103365
\(760\) 0 0
\(761\) 30722.6 1.46346 0.731731 0.681594i \(-0.238713\pi\)
0.731731 + 0.681594i \(0.238713\pi\)
\(762\) 13407.4i 0.637400i
\(763\) − 31665.6i − 1.50245i
\(764\) 1658.86 0.0785544
\(765\) 0 0
\(766\) 20138.4 0.949906
\(767\) − 4179.48i − 0.196757i
\(768\) 10505.8i 0.493612i
\(769\) 18443.1 0.864855 0.432427 0.901669i \(-0.357657\pi\)
0.432427 + 0.901669i \(0.357657\pi\)
\(770\) 0 0
\(771\) 5458.18 0.254956
\(772\) 10226.6i 0.476764i
\(773\) − 545.742i − 0.0253932i −0.999919 0.0126966i \(-0.995958\pi\)
0.999919 0.0126966i \(-0.00404156\pi\)
\(774\) −8405.88 −0.390366
\(775\) 0 0
\(776\) −18001.9 −0.832770
\(777\) − 7433.00i − 0.343189i
\(778\) − 10207.8i − 0.470393i
\(779\) −1089.43 −0.0501065
\(780\) 0 0
\(781\) 1063.65 0.0487327
\(782\) − 195.468i − 0.00893851i
\(783\) 6190.53i 0.282543i
\(784\) 5947.35 0.270925
\(785\) 0 0
\(786\) 9358.72 0.424701
\(787\) − 17365.2i − 0.786536i −0.919424 0.393268i \(-0.871345\pi\)
0.919424 0.393268i \(-0.128655\pi\)
\(788\) − 9674.33i − 0.437352i
\(789\) −18069.1 −0.815306
\(790\) 0 0
\(791\) 49105.8 2.20733
\(792\) − 2438.46i − 0.109403i
\(793\) − 6723.25i − 0.301071i
\(794\) −10323.8 −0.461431
\(795\) 0 0
\(796\) −12427.9 −0.553384
\(797\) − 7055.12i − 0.313557i −0.987634 0.156779i \(-0.949889\pi\)
0.987634 0.156779i \(-0.0501109\pi\)
\(798\) 1105.16i 0.0490253i
\(799\) −35562.0 −1.57458
\(800\) 0 0
\(801\) −5756.61 −0.253933
\(802\) − 11726.7i − 0.516313i
\(803\) − 219.406i − 0.00964217i
\(804\) −2889.89 −0.126764
\(805\) 0 0
\(806\) 3577.73 0.156353
\(807\) − 8935.14i − 0.389755i
\(808\) − 19959.5i − 0.869024i
\(809\) 6937.17 0.301481 0.150740 0.988573i \(-0.451834\pi\)
0.150740 + 0.988573i \(0.451834\pi\)
\(810\) 0 0
\(811\) 5610.44 0.242921 0.121461 0.992596i \(-0.461242\pi\)
0.121461 + 0.992596i \(0.461242\pi\)
\(812\) − 13266.9i − 0.573369i
\(813\) 1574.91i 0.0679390i
\(814\) −2825.32 −0.121655
\(815\) 0 0
\(816\) −14132.1 −0.606280
\(817\) 2823.14i 0.120893i
\(818\) − 15079.8i − 0.644565i
\(819\) 1993.80 0.0850659
\(820\) 0 0
\(821\) −17001.8 −0.722735 −0.361368 0.932423i \(-0.617690\pi\)
−0.361368 + 0.932423i \(0.617690\pi\)
\(822\) 7689.83i 0.326294i
\(823\) 14567.3i 0.616991i 0.951226 + 0.308496i \(0.0998255\pi\)
−0.951226 + 0.308496i \(0.900175\pi\)
\(824\) 36007.7 1.52232
\(825\) 0 0
\(826\) 22180.0 0.934312
\(827\) 7345.87i 0.308877i 0.988002 + 0.154438i \(0.0493568\pi\)
−0.988002 + 0.154438i \(0.950643\pi\)
\(828\) − 15.1817i 0 0.000637197i
\(829\) 13903.2 0.582482 0.291241 0.956650i \(-0.405932\pi\)
0.291241 + 0.956650i \(0.405932\pi\)
\(830\) 0 0
\(831\) 5079.22 0.212029
\(832\) 5458.84i 0.227466i
\(833\) − 20729.2i − 0.862214i
\(834\) 8975.13 0.372642
\(835\) 0 0
\(836\) −199.444 −0.00825111
\(837\) 4206.30i 0.173705i
\(838\) 19232.9i 0.792829i
\(839\) 25111.9 1.03332 0.516662 0.856190i \(-0.327175\pi\)
0.516662 + 0.856190i \(0.327175\pi\)
\(840\) 0 0
\(841\) 28179.7 1.15543
\(842\) − 7987.20i − 0.326909i
\(843\) − 22039.8i − 0.900464i
\(844\) −11316.6 −0.461534
\(845\) 0 0
\(846\) 5817.50 0.236418
\(847\) − 2718.53i − 0.110283i
\(848\) 23953.7i 0.970014i
\(849\) −4505.08 −0.182113
\(850\) 0 0
\(851\) −72.2296 −0.00290952
\(852\) − 747.104i − 0.0300415i
\(853\) − 27545.3i − 1.10567i −0.833292 0.552833i \(-0.813547\pi\)
0.833292 0.552833i \(-0.186453\pi\)
\(854\) 35679.5 1.42966
\(855\) 0 0
\(856\) 41636.1 1.66249
\(857\) − 1808.04i − 0.0720669i −0.999351 0.0360334i \(-0.988528\pi\)
0.999351 0.0360334i \(-0.0114723\pi\)
\(858\) − 757.854i − 0.0301547i
\(859\) −32160.8 −1.27743 −0.638716 0.769443i \(-0.720534\pi\)
−0.638716 + 0.769443i \(0.720534\pi\)
\(860\) 0 0
\(861\) −10430.3 −0.412850
\(862\) − 26226.3i − 1.03628i
\(863\) 33734.8i 1.33064i 0.746557 + 0.665321i \(0.231706\pi\)
−0.746557 + 0.665321i \(0.768294\pi\)
\(864\) −3008.43 −0.118459
\(865\) 0 0
\(866\) −29342.7 −1.15139
\(867\) 34517.9i 1.35212i
\(868\) − 9014.47i − 0.352501i
\(869\) −268.715 −0.0104897
\(870\) 0 0
\(871\) −3688.03 −0.143472
\(872\) 34715.2i 1.34817i
\(873\) − 6577.78i − 0.255011i
\(874\) 10.7393 0.000415631 0
\(875\) 0 0
\(876\) −154.111 −0.00594397
\(877\) − 46573.5i − 1.79325i −0.442795 0.896623i \(-0.646013\pi\)
0.442795 0.896623i \(-0.353987\pi\)
\(878\) − 9728.26i − 0.373933i
\(879\) −13443.1 −0.515841
\(880\) 0 0
\(881\) −9949.72 −0.380493 −0.190247 0.981736i \(-0.560929\pi\)
−0.190247 + 0.981736i \(0.560929\pi\)
\(882\) 3391.04i 0.129458i
\(883\) − 49269.1i − 1.87773i −0.344282 0.938866i \(-0.611878\pi\)
0.344282 0.938866i \(-0.388122\pi\)
\(884\) 3254.01 0.123806
\(885\) 0 0
\(886\) −5483.00 −0.207906
\(887\) − 27347.5i − 1.03522i −0.855617 0.517609i \(-0.826822\pi\)
0.855617 0.517609i \(-0.173178\pi\)
\(888\) 8148.85i 0.307948i
\(889\) 43111.2 1.62644
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) 5639.42i 0.211683i
\(893\) − 1953.83i − 0.0732165i
\(894\) −8924.12 −0.333856
\(895\) 0 0
\(896\) −8942.48 −0.333423
\(897\) − 19.3746i 0 0.000721180i
\(898\) − 21628.1i − 0.803718i
\(899\) 35719.0 1.32514
\(900\) 0 0
\(901\) 83489.4 3.08705
\(902\) 3964.61i 0.146349i
\(903\) 27029.0i 0.996088i
\(904\) −53835.0 −1.98067
\(905\) 0 0
\(906\) 6192.86 0.227091
\(907\) 25516.6i 0.934141i 0.884220 + 0.467071i \(0.154691\pi\)
−0.884220 + 0.467071i \(0.845309\pi\)
\(908\) 2934.15i 0.107239i
\(909\) 7293.07 0.266112
\(910\) 0 0
\(911\) 22379.0 0.813885 0.406942 0.913454i \(-0.366595\pi\)
0.406942 + 0.913454i \(0.366595\pi\)
\(912\) − 776.441i − 0.0281914i
\(913\) − 12400.9i − 0.449516i
\(914\) 33488.1 1.21191
\(915\) 0 0
\(916\) −8798.04 −0.317353
\(917\) − 30092.8i − 1.08370i
\(918\) − 8057.83i − 0.289704i
\(919\) 6244.97 0.224159 0.112080 0.993699i \(-0.464249\pi\)
0.112080 + 0.993699i \(0.464249\pi\)
\(920\) 0 0
\(921\) −9156.52 −0.327598
\(922\) − 12537.8i − 0.447841i
\(923\) − 953.442i − 0.0340010i
\(924\) −1909.49 −0.0679846
\(925\) 0 0
\(926\) −42762.6 −1.51757
\(927\) 13157.0i 0.466163i
\(928\) 25547.0i 0.903688i
\(929\) −16122.2 −0.569378 −0.284689 0.958620i \(-0.591890\pi\)
−0.284689 + 0.958620i \(0.591890\pi\)
\(930\) 0 0
\(931\) 1138.89 0.0400921
\(932\) 15831.6i 0.556417i
\(933\) − 30765.4i − 1.07954i
\(934\) 21109.8 0.739544
\(935\) 0 0
\(936\) −2185.82 −0.0763308
\(937\) − 56379.6i − 1.96568i −0.184466 0.982839i \(-0.559055\pi\)
0.184466 0.982839i \(-0.440945\pi\)
\(938\) − 19572.0i − 0.681287i
\(939\) −18570.5 −0.645396
\(940\) 0 0
\(941\) 25527.0 0.884332 0.442166 0.896933i \(-0.354210\pi\)
0.442166 + 0.896933i \(0.354210\pi\)
\(942\) 11746.3i 0.406280i
\(943\) 101.356i 0.00350010i
\(944\) −15582.8 −0.537264
\(945\) 0 0
\(946\) 10273.9 0.353099
\(947\) − 46411.3i − 1.59257i −0.604921 0.796285i \(-0.706795\pi\)
0.604921 0.796285i \(-0.293205\pi\)
\(948\) 188.745i 0.00646641i
\(949\) −196.673 −0.00672738
\(950\) 0 0
\(951\) 20206.1 0.688990
\(952\) 70909.2i 2.41405i
\(953\) − 21266.1i − 0.722850i −0.932401 0.361425i \(-0.882290\pi\)
0.932401 0.361425i \(-0.117710\pi\)
\(954\) −13657.8 −0.463510
\(955\) 0 0
\(956\) 5357.05 0.181234
\(957\) − 7566.20i − 0.255570i
\(958\) − 29365.4i − 0.990347i
\(959\) 24726.5 0.832597
\(960\) 0 0
\(961\) −5520.88 −0.185320
\(962\) 2532.60i 0.0848796i
\(963\) 15213.6i 0.509087i
\(964\) 4755.04 0.158869
\(965\) 0 0
\(966\) 102.819 0.00342457
\(967\) − 20035.9i − 0.666300i −0.942874 0.333150i \(-0.891888\pi\)
0.942874 0.333150i \(-0.108112\pi\)
\(968\) 2980.34i 0.0989585i
\(969\) −2706.25 −0.0897185
\(970\) 0 0
\(971\) 21354.1 0.705751 0.352876 0.935670i \(-0.385204\pi\)
0.352876 + 0.935670i \(0.385204\pi\)
\(972\) − 625.838i − 0.0206520i
\(973\) − 28859.4i − 0.950862i
\(974\) −30776.6 −1.01247
\(975\) 0 0
\(976\) −25067.0 −0.822107
\(977\) 34057.7i 1.11525i 0.830092 + 0.557626i \(0.188288\pi\)
−0.830092 + 0.557626i \(0.811712\pi\)
\(978\) − 4346.73i − 0.142120i
\(979\) 7035.86 0.229691
\(980\) 0 0
\(981\) −12684.7 −0.412836
\(982\) 15263.0i 0.495989i
\(983\) 31846.5i 1.03331i 0.856193 + 0.516657i \(0.172824\pi\)
−0.856193 + 0.516657i \(0.827176\pi\)
\(984\) 11434.8 0.370456
\(985\) 0 0
\(986\) −68425.5 −2.21005
\(987\) − 18706.1i − 0.603263i
\(988\) 178.780i 0.00575684i
\(989\) 262.652 0.00844474
\(990\) 0 0
\(991\) −20462.5 −0.655915 −0.327957 0.944693i \(-0.606360\pi\)
−0.327957 + 0.944693i \(0.606360\pi\)
\(992\) 17358.5i 0.555578i
\(993\) 14342.5i 0.458353i
\(994\) 5059.81 0.161456
\(995\) 0 0
\(996\) −8710.36 −0.277107
\(997\) − 35227.4i − 1.11902i −0.828824 0.559510i \(-0.810989\pi\)
0.828824 0.559510i \(-0.189011\pi\)
\(998\) 6047.71i 0.191820i
\(999\) −2977.54 −0.0942996
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 825.4.c.l.199.4 6
5.2 odd 4 165.4.a.d.1.2 3
5.3 odd 4 825.4.a.s.1.2 3
5.4 even 2 inner 825.4.c.l.199.3 6
15.2 even 4 495.4.a.l.1.2 3
15.8 even 4 2475.4.a.s.1.2 3
55.32 even 4 1815.4.a.s.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.d.1.2 3 5.2 odd 4
495.4.a.l.1.2 3 15.2 even 4
825.4.a.s.1.2 3 5.3 odd 4
825.4.c.l.199.3 6 5.4 even 2 inner
825.4.c.l.199.4 6 1.1 even 1 trivial
1815.4.a.s.1.2 3 55.32 even 4
2475.4.a.s.1.2 3 15.8 even 4