Properties

Label 4005.2.a.o.1.6
Level 4005
Weight 2
Character 4005.1
Self dual yes
Analytic conductor 31.980
Analytic rank 0
Dimension 7
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4005 = 3^{2} \cdot 5 \cdot 89 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4005.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9800860095\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 445)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.885013\)
Character \(\chi\) = 4005.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.21675 q^{2} +2.91399 q^{4} -1.00000 q^{5} -3.75132 q^{7} +2.02609 q^{8} +O(q^{10})\) \(q+2.21675 q^{2} +2.91399 q^{4} -1.00000 q^{5} -3.75132 q^{7} +2.02609 q^{8} -2.21675 q^{10} +1.54195 q^{11} +4.15218 q^{13} -8.31575 q^{14} -1.33664 q^{16} -5.74043 q^{17} +7.03169 q^{19} -2.91399 q^{20} +3.41813 q^{22} +6.77641 q^{23} +1.00000 q^{25} +9.20436 q^{26} -10.9313 q^{28} +10.1743 q^{29} +0.0578180 q^{31} -7.01518 q^{32} -12.7251 q^{34} +3.75132 q^{35} -2.18339 q^{37} +15.5875 q^{38} -2.02609 q^{40} +8.82042 q^{41} -6.08017 q^{43} +4.49324 q^{44} +15.0216 q^{46} -4.08434 q^{47} +7.07241 q^{49} +2.21675 q^{50} +12.0994 q^{52} +3.49748 q^{53} -1.54195 q^{55} -7.60052 q^{56} +22.5540 q^{58} +9.20908 q^{59} +8.32403 q^{61} +0.128168 q^{62} -12.8776 q^{64} -4.15218 q^{65} +3.93525 q^{67} -16.7276 q^{68} +8.31575 q^{70} +5.49996 q^{71} -13.0277 q^{73} -4.84003 q^{74} +20.4903 q^{76} -5.78437 q^{77} +8.83960 q^{79} +1.33664 q^{80} +19.5527 q^{82} +9.09443 q^{83} +5.74043 q^{85} -13.4782 q^{86} +3.12414 q^{88} -1.00000 q^{89} -15.5762 q^{91} +19.7464 q^{92} -9.05396 q^{94} -7.03169 q^{95} -2.70331 q^{97} +15.6778 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} + O(q^{10}) \) \( 7q + 4q^{2} + 8q^{4} - 7q^{5} - 16q^{7} + 12q^{8} - 4q^{10} + 10q^{11} - 7q^{13} - 3q^{14} + 10q^{16} + 13q^{17} - 7q^{19} - 8q^{20} + 2q^{22} + 13q^{23} + 7q^{25} - q^{26} - 21q^{28} + 4q^{29} + q^{31} + 13q^{32} + 10q^{34} + 16q^{35} - 5q^{37} + 40q^{38} - 12q^{40} - 5q^{41} - 31q^{43} + 21q^{44} + 16q^{46} + 14q^{47} + 19q^{49} + 4q^{50} + 13q^{53} - 10q^{55} + q^{56} + 17q^{58} + 14q^{59} + 3q^{61} - 26q^{62} + 14q^{64} + 7q^{65} + q^{67} + 35q^{68} + 3q^{70} + 8q^{71} + 9q^{73} + 35q^{74} + 40q^{76} - 42q^{77} + 9q^{79} - 10q^{80} + 29q^{82} + 42q^{83} - 13q^{85} - 35q^{86} + 30q^{88} - 7q^{89} + 31q^{91} - 19q^{92} + 37q^{94} + 7q^{95} - 7q^{97} - 9q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.21675 1.56748 0.783740 0.621089i \(-0.213309\pi\)
0.783740 + 0.621089i \(0.213309\pi\)
\(3\) 0 0
\(4\) 2.91399 1.45700
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −3.75132 −1.41787 −0.708933 0.705276i \(-0.750823\pi\)
−0.708933 + 0.705276i \(0.750823\pi\)
\(8\) 2.02609 0.716332
\(9\) 0 0
\(10\) −2.21675 −0.700999
\(11\) 1.54195 0.464917 0.232458 0.972606i \(-0.425323\pi\)
0.232458 + 0.972606i \(0.425323\pi\)
\(12\) 0 0
\(13\) 4.15218 1.15161 0.575804 0.817588i \(-0.304689\pi\)
0.575804 + 0.817588i \(0.304689\pi\)
\(14\) −8.31575 −2.22248
\(15\) 0 0
\(16\) −1.33664 −0.334160
\(17\) −5.74043 −1.39226 −0.696129 0.717916i \(-0.745096\pi\)
−0.696129 + 0.717916i \(0.745096\pi\)
\(18\) 0 0
\(19\) 7.03169 1.61318 0.806590 0.591111i \(-0.201310\pi\)
0.806590 + 0.591111i \(0.201310\pi\)
\(20\) −2.91399 −0.651588
\(21\) 0 0
\(22\) 3.41813 0.728748
\(23\) 6.77641 1.41298 0.706490 0.707723i \(-0.250278\pi\)
0.706490 + 0.707723i \(0.250278\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 9.20436 1.80512
\(27\) 0 0
\(28\) −10.9313 −2.06582
\(29\) 10.1743 1.88933 0.944664 0.328041i \(-0.106388\pi\)
0.944664 + 0.328041i \(0.106388\pi\)
\(30\) 0 0
\(31\) 0.0578180 0.0103844 0.00519221 0.999987i \(-0.498347\pi\)
0.00519221 + 0.999987i \(0.498347\pi\)
\(32\) −7.01518 −1.24012
\(33\) 0 0
\(34\) −12.7251 −2.18234
\(35\) 3.75132 0.634089
\(36\) 0 0
\(37\) −2.18339 −0.358947 −0.179473 0.983763i \(-0.557439\pi\)
−0.179473 + 0.983763i \(0.557439\pi\)
\(38\) 15.5875 2.52863
\(39\) 0 0
\(40\) −2.02609 −0.320353
\(41\) 8.82042 1.37752 0.688759 0.724990i \(-0.258156\pi\)
0.688759 + 0.724990i \(0.258156\pi\)
\(42\) 0 0
\(43\) −6.08017 −0.927218 −0.463609 0.886040i \(-0.653446\pi\)
−0.463609 + 0.886040i \(0.653446\pi\)
\(44\) 4.49324 0.677382
\(45\) 0 0
\(46\) 15.0216 2.21482
\(47\) −4.08434 −0.595762 −0.297881 0.954603i \(-0.596280\pi\)
−0.297881 + 0.954603i \(0.596280\pi\)
\(48\) 0 0
\(49\) 7.07241 1.01034
\(50\) 2.21675 0.313496
\(51\) 0 0
\(52\) 12.0994 1.67789
\(53\) 3.49748 0.480415 0.240208 0.970722i \(-0.422784\pi\)
0.240208 + 0.970722i \(0.422784\pi\)
\(54\) 0 0
\(55\) −1.54195 −0.207917
\(56\) −7.60052 −1.01566
\(57\) 0 0
\(58\) 22.5540 2.96148
\(59\) 9.20908 1.19892 0.599460 0.800405i \(-0.295382\pi\)
0.599460 + 0.800405i \(0.295382\pi\)
\(60\) 0 0
\(61\) 8.32403 1.06578 0.532891 0.846184i \(-0.321105\pi\)
0.532891 + 0.846184i \(0.321105\pi\)
\(62\) 0.128168 0.0162774
\(63\) 0 0
\(64\) −12.8776 −1.60970
\(65\) −4.15218 −0.515015
\(66\) 0 0
\(67\) 3.93525 0.480767 0.240384 0.970678i \(-0.422727\pi\)
0.240384 + 0.970678i \(0.422727\pi\)
\(68\) −16.7276 −2.02851
\(69\) 0 0
\(70\) 8.31575 0.993922
\(71\) 5.49996 0.652726 0.326363 0.945245i \(-0.394177\pi\)
0.326363 + 0.945245i \(0.394177\pi\)
\(72\) 0 0
\(73\) −13.0277 −1.52478 −0.762388 0.647120i \(-0.775973\pi\)
−0.762388 + 0.647120i \(0.775973\pi\)
\(74\) −4.84003 −0.562642
\(75\) 0 0
\(76\) 20.4903 2.35040
\(77\) −5.78437 −0.659190
\(78\) 0 0
\(79\) 8.83960 0.994533 0.497266 0.867598i \(-0.334337\pi\)
0.497266 + 0.867598i \(0.334337\pi\)
\(80\) 1.33664 0.149441
\(81\) 0 0
\(82\) 19.5527 2.15923
\(83\) 9.09443 0.998243 0.499122 0.866532i \(-0.333656\pi\)
0.499122 + 0.866532i \(0.333656\pi\)
\(84\) 0 0
\(85\) 5.74043 0.622637
\(86\) −13.4782 −1.45340
\(87\) 0 0
\(88\) 3.12414 0.333035
\(89\) −1.00000 −0.106000
\(90\) 0 0
\(91\) −15.5762 −1.63283
\(92\) 19.7464 2.05870
\(93\) 0 0
\(94\) −9.05396 −0.933845
\(95\) −7.03169 −0.721436
\(96\) 0 0
\(97\) −2.70331 −0.274479 −0.137240 0.990538i \(-0.543823\pi\)
−0.137240 + 0.990538i \(0.543823\pi\)
\(98\) 15.6778 1.58369
\(99\) 0 0
\(100\) 2.91399 0.291399
\(101\) 10.6908 1.06378 0.531888 0.846815i \(-0.321483\pi\)
0.531888 + 0.846815i \(0.321483\pi\)
\(102\) 0 0
\(103\) 13.1321 1.29395 0.646973 0.762513i \(-0.276034\pi\)
0.646973 + 0.762513i \(0.276034\pi\)
\(104\) 8.41270 0.824933
\(105\) 0 0
\(106\) 7.75304 0.753042
\(107\) −11.1134 −1.07437 −0.537187 0.843463i \(-0.680513\pi\)
−0.537187 + 0.843463i \(0.680513\pi\)
\(108\) 0 0
\(109\) −1.18261 −0.113274 −0.0566368 0.998395i \(-0.518038\pi\)
−0.0566368 + 0.998395i \(0.518038\pi\)
\(110\) −3.41813 −0.325906
\(111\) 0 0
\(112\) 5.01416 0.473794
\(113\) 7.93908 0.746846 0.373423 0.927661i \(-0.378184\pi\)
0.373423 + 0.927661i \(0.378184\pi\)
\(114\) 0 0
\(115\) −6.77641 −0.631904
\(116\) 29.6479 2.75274
\(117\) 0 0
\(118\) 20.4142 1.87928
\(119\) 21.5342 1.97404
\(120\) 0 0
\(121\) −8.62238 −0.783852
\(122\) 18.4523 1.67059
\(123\) 0 0
\(124\) 0.168481 0.0151301
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −5.22603 −0.463735 −0.231868 0.972747i \(-0.574484\pi\)
−0.231868 + 0.972747i \(0.574484\pi\)
\(128\) −14.5162 −1.28306
\(129\) 0 0
\(130\) −9.20436 −0.807276
\(131\) 3.70672 0.323857 0.161929 0.986802i \(-0.448229\pi\)
0.161929 + 0.986802i \(0.448229\pi\)
\(132\) 0 0
\(133\) −26.3781 −2.28727
\(134\) 8.72348 0.753594
\(135\) 0 0
\(136\) −11.6306 −0.997319
\(137\) 15.4250 1.31785 0.658925 0.752209i \(-0.271011\pi\)
0.658925 + 0.752209i \(0.271011\pi\)
\(138\) 0 0
\(139\) −7.00287 −0.593975 −0.296988 0.954881i \(-0.595982\pi\)
−0.296988 + 0.954881i \(0.595982\pi\)
\(140\) 10.9313 0.923865
\(141\) 0 0
\(142\) 12.1921 1.02313
\(143\) 6.40248 0.535402
\(144\) 0 0
\(145\) −10.1743 −0.844933
\(146\) −28.8792 −2.39006
\(147\) 0 0
\(148\) −6.36237 −0.522984
\(149\) −12.5398 −1.02730 −0.513650 0.858000i \(-0.671707\pi\)
−0.513650 + 0.858000i \(0.671707\pi\)
\(150\) 0 0
\(151\) 0.236140 0.0192168 0.00960840 0.999954i \(-0.496942\pi\)
0.00960840 + 0.999954i \(0.496942\pi\)
\(152\) 14.2469 1.15557
\(153\) 0 0
\(154\) −12.8225 −1.03327
\(155\) −0.0578180 −0.00464406
\(156\) 0 0
\(157\) −4.98636 −0.397955 −0.198977 0.980004i \(-0.563762\pi\)
−0.198977 + 0.980004i \(0.563762\pi\)
\(158\) 19.5952 1.55891
\(159\) 0 0
\(160\) 7.01518 0.554599
\(161\) −25.4205 −2.00342
\(162\) 0 0
\(163\) −10.9166 −0.855055 −0.427528 0.904002i \(-0.640615\pi\)
−0.427528 + 0.904002i \(0.640615\pi\)
\(164\) 25.7026 2.00704
\(165\) 0 0
\(166\) 20.1601 1.56473
\(167\) −19.6561 −1.52104 −0.760518 0.649317i \(-0.775055\pi\)
−0.760518 + 0.649317i \(0.775055\pi\)
\(168\) 0 0
\(169\) 4.24062 0.326202
\(170\) 12.7251 0.975971
\(171\) 0 0
\(172\) −17.7176 −1.35095
\(173\) 13.1762 1.00177 0.500883 0.865515i \(-0.333009\pi\)
0.500883 + 0.865515i \(0.333009\pi\)
\(174\) 0 0
\(175\) −3.75132 −0.283573
\(176\) −2.06104 −0.155356
\(177\) 0 0
\(178\) −2.21675 −0.166153
\(179\) −9.74668 −0.728501 −0.364250 0.931301i \(-0.618675\pi\)
−0.364250 + 0.931301i \(0.618675\pi\)
\(180\) 0 0
\(181\) 6.93270 0.515304 0.257652 0.966238i \(-0.417051\pi\)
0.257652 + 0.966238i \(0.417051\pi\)
\(182\) −34.5285 −2.55942
\(183\) 0 0
\(184\) 13.7296 1.01216
\(185\) 2.18339 0.160526
\(186\) 0 0
\(187\) −8.85148 −0.647285
\(188\) −11.9017 −0.868022
\(189\) 0 0
\(190\) −15.5875 −1.13084
\(191\) −4.37217 −0.316359 −0.158180 0.987410i \(-0.550563\pi\)
−0.158180 + 0.987410i \(0.550563\pi\)
\(192\) 0 0
\(193\) 2.05206 0.147711 0.0738554 0.997269i \(-0.476470\pi\)
0.0738554 + 0.997269i \(0.476470\pi\)
\(194\) −5.99256 −0.430241
\(195\) 0 0
\(196\) 20.6089 1.47207
\(197\) 6.48648 0.462143 0.231071 0.972937i \(-0.425777\pi\)
0.231071 + 0.972937i \(0.425777\pi\)
\(198\) 0 0
\(199\) 13.4260 0.951745 0.475873 0.879514i \(-0.342132\pi\)
0.475873 + 0.879514i \(0.342132\pi\)
\(200\) 2.02609 0.143266
\(201\) 0 0
\(202\) 23.6989 1.66745
\(203\) −38.1672 −2.67881
\(204\) 0 0
\(205\) −8.82042 −0.616045
\(206\) 29.1107 2.02824
\(207\) 0 0
\(208\) −5.54997 −0.384821
\(209\) 10.8426 0.749995
\(210\) 0 0
\(211\) −20.1294 −1.38576 −0.692882 0.721051i \(-0.743659\pi\)
−0.692882 + 0.721051i \(0.743659\pi\)
\(212\) 10.1916 0.699963
\(213\) 0 0
\(214\) −24.6357 −1.68406
\(215\) 6.08017 0.414664
\(216\) 0 0
\(217\) −0.216894 −0.0147237
\(218\) −2.62155 −0.177554
\(219\) 0 0
\(220\) −4.49324 −0.302934
\(221\) −23.8353 −1.60334
\(222\) 0 0
\(223\) −8.26176 −0.553248 −0.276624 0.960978i \(-0.589216\pi\)
−0.276624 + 0.960978i \(0.589216\pi\)
\(224\) 26.3162 1.75832
\(225\) 0 0
\(226\) 17.5990 1.17067
\(227\) 21.1024 1.40061 0.700306 0.713842i \(-0.253047\pi\)
0.700306 + 0.713842i \(0.253047\pi\)
\(228\) 0 0
\(229\) −13.3006 −0.878930 −0.439465 0.898260i \(-0.644832\pi\)
−0.439465 + 0.898260i \(0.644832\pi\)
\(230\) −15.0216 −0.990497
\(231\) 0 0
\(232\) 20.6141 1.35338
\(233\) −5.42111 −0.355148 −0.177574 0.984107i \(-0.556825\pi\)
−0.177574 + 0.984107i \(0.556825\pi\)
\(234\) 0 0
\(235\) 4.08434 0.266433
\(236\) 26.8352 1.74682
\(237\) 0 0
\(238\) 47.7360 3.09426
\(239\) −10.4763 −0.677657 −0.338828 0.940848i \(-0.610031\pi\)
−0.338828 + 0.940848i \(0.610031\pi\)
\(240\) 0 0
\(241\) −13.1333 −0.845991 −0.422995 0.906132i \(-0.639021\pi\)
−0.422995 + 0.906132i \(0.639021\pi\)
\(242\) −19.1137 −1.22867
\(243\) 0 0
\(244\) 24.2561 1.55284
\(245\) −7.07241 −0.451839
\(246\) 0 0
\(247\) 29.1969 1.85775
\(248\) 0.117145 0.00743869
\(249\) 0 0
\(250\) −2.21675 −0.140200
\(251\) 27.1469 1.71350 0.856748 0.515735i \(-0.172481\pi\)
0.856748 + 0.515735i \(0.172481\pi\)
\(252\) 0 0
\(253\) 10.4489 0.656918
\(254\) −11.5848 −0.726896
\(255\) 0 0
\(256\) −6.42349 −0.401468
\(257\) 18.1588 1.13272 0.566358 0.824159i \(-0.308352\pi\)
0.566358 + 0.824159i \(0.308352\pi\)
\(258\) 0 0
\(259\) 8.19059 0.508938
\(260\) −12.0994 −0.750374
\(261\) 0 0
\(262\) 8.21687 0.507640
\(263\) 23.7946 1.46724 0.733619 0.679561i \(-0.237830\pi\)
0.733619 + 0.679561i \(0.237830\pi\)
\(264\) 0 0
\(265\) −3.49748 −0.214848
\(266\) −58.4738 −3.58526
\(267\) 0 0
\(268\) 11.4673 0.700476
\(269\) −0.159138 −0.00970280 −0.00485140 0.999988i \(-0.501544\pi\)
−0.00485140 + 0.999988i \(0.501544\pi\)
\(270\) 0 0
\(271\) −1.15143 −0.0699447 −0.0349723 0.999388i \(-0.511134\pi\)
−0.0349723 + 0.999388i \(0.511134\pi\)
\(272\) 7.67288 0.465237
\(273\) 0 0
\(274\) 34.1935 2.06570
\(275\) 1.54195 0.0929834
\(276\) 0 0
\(277\) −1.65637 −0.0995216 −0.0497608 0.998761i \(-0.515846\pi\)
−0.0497608 + 0.998761i \(0.515846\pi\)
\(278\) −15.5236 −0.931045
\(279\) 0 0
\(280\) 7.60052 0.454218
\(281\) −4.08763 −0.243848 −0.121924 0.992539i \(-0.538906\pi\)
−0.121924 + 0.992539i \(0.538906\pi\)
\(282\) 0 0
\(283\) −14.7662 −0.877757 −0.438879 0.898546i \(-0.644624\pi\)
−0.438879 + 0.898546i \(0.644624\pi\)
\(284\) 16.0268 0.951018
\(285\) 0 0
\(286\) 14.1927 0.839232
\(287\) −33.0882 −1.95314
\(288\) 0 0
\(289\) 15.9525 0.938384
\(290\) −22.5540 −1.32442
\(291\) 0 0
\(292\) −37.9626 −2.22159
\(293\) −3.43348 −0.200586 −0.100293 0.994958i \(-0.531978\pi\)
−0.100293 + 0.994958i \(0.531978\pi\)
\(294\) 0 0
\(295\) −9.20908 −0.536173
\(296\) −4.42374 −0.257125
\(297\) 0 0
\(298\) −27.7976 −1.61027
\(299\) 28.1369 1.62720
\(300\) 0 0
\(301\) 22.8087 1.31467
\(302\) 0.523464 0.0301220
\(303\) 0 0
\(304\) −9.39883 −0.539060
\(305\) −8.32403 −0.476633
\(306\) 0 0
\(307\) −22.5178 −1.28516 −0.642581 0.766218i \(-0.722136\pi\)
−0.642581 + 0.766218i \(0.722136\pi\)
\(308\) −16.8556 −0.960436
\(309\) 0 0
\(310\) −0.128168 −0.00727947
\(311\) −11.4828 −0.651129 −0.325564 0.945520i \(-0.605554\pi\)
−0.325564 + 0.945520i \(0.605554\pi\)
\(312\) 0 0
\(313\) −2.22350 −0.125679 −0.0628397 0.998024i \(-0.520016\pi\)
−0.0628397 + 0.998024i \(0.520016\pi\)
\(314\) −11.0535 −0.623786
\(315\) 0 0
\(316\) 25.7585 1.44903
\(317\) −32.5253 −1.82680 −0.913402 0.407059i \(-0.866554\pi\)
−0.913402 + 0.407059i \(0.866554\pi\)
\(318\) 0 0
\(319\) 15.6884 0.878380
\(320\) 12.8776 0.719882
\(321\) 0 0
\(322\) −56.3509 −3.14031
\(323\) −40.3649 −2.24597
\(324\) 0 0
\(325\) 4.15218 0.230322
\(326\) −24.1994 −1.34028
\(327\) 0 0
\(328\) 17.8710 0.986760
\(329\) 15.3217 0.844710
\(330\) 0 0
\(331\) −27.2476 −1.49766 −0.748831 0.662761i \(-0.769385\pi\)
−0.748831 + 0.662761i \(0.769385\pi\)
\(332\) 26.5011 1.45444
\(333\) 0 0
\(334\) −43.5727 −2.38419
\(335\) −3.93525 −0.215006
\(336\) 0 0
\(337\) 16.5317 0.900536 0.450268 0.892893i \(-0.351328\pi\)
0.450268 + 0.892893i \(0.351328\pi\)
\(338\) 9.40042 0.511315
\(339\) 0 0
\(340\) 16.7276 0.907179
\(341\) 0.0891528 0.00482789
\(342\) 0 0
\(343\) −0.271619 −0.0146661
\(344\) −12.3190 −0.664195
\(345\) 0 0
\(346\) 29.2083 1.57025
\(347\) 33.2231 1.78351 0.891756 0.452517i \(-0.149474\pi\)
0.891756 + 0.452517i \(0.149474\pi\)
\(348\) 0 0
\(349\) −5.87046 −0.314239 −0.157119 0.987580i \(-0.550221\pi\)
−0.157119 + 0.987580i \(0.550221\pi\)
\(350\) −8.31575 −0.444495
\(351\) 0 0
\(352\) −10.8171 −0.576553
\(353\) −31.2340 −1.66242 −0.831210 0.555958i \(-0.812351\pi\)
−0.831210 + 0.555958i \(0.812351\pi\)
\(354\) 0 0
\(355\) −5.49996 −0.291908
\(356\) −2.91399 −0.154441
\(357\) 0 0
\(358\) −21.6060 −1.14191
\(359\) 4.01656 0.211986 0.105993 0.994367i \(-0.466198\pi\)
0.105993 + 0.994367i \(0.466198\pi\)
\(360\) 0 0
\(361\) 30.4447 1.60235
\(362\) 15.3681 0.807729
\(363\) 0 0
\(364\) −45.3888 −2.37902
\(365\) 13.0277 0.681901
\(366\) 0 0
\(367\) 13.7718 0.718882 0.359441 0.933168i \(-0.382967\pi\)
0.359441 + 0.933168i \(0.382967\pi\)
\(368\) −9.05761 −0.472161
\(369\) 0 0
\(370\) 4.84003 0.251621
\(371\) −13.1202 −0.681165
\(372\) 0 0
\(373\) −24.2574 −1.25600 −0.628000 0.778214i \(-0.716126\pi\)
−0.628000 + 0.778214i \(0.716126\pi\)
\(374\) −19.6215 −1.01461
\(375\) 0 0
\(376\) −8.27524 −0.426763
\(377\) 42.2457 2.17577
\(378\) 0 0
\(379\) 29.7093 1.52606 0.763032 0.646361i \(-0.223710\pi\)
0.763032 + 0.646361i \(0.223710\pi\)
\(380\) −20.4903 −1.05113
\(381\) 0 0
\(382\) −9.69203 −0.495887
\(383\) −31.1003 −1.58915 −0.794575 0.607166i \(-0.792306\pi\)
−0.794575 + 0.607166i \(0.792306\pi\)
\(384\) 0 0
\(385\) 5.78437 0.294799
\(386\) 4.54892 0.231534
\(387\) 0 0
\(388\) −7.87741 −0.399915
\(389\) 2.70571 0.137185 0.0685924 0.997645i \(-0.478149\pi\)
0.0685924 + 0.997645i \(0.478149\pi\)
\(390\) 0 0
\(391\) −38.8995 −1.96723
\(392\) 14.3293 0.723741
\(393\) 0 0
\(394\) 14.3789 0.724400
\(395\) −8.83960 −0.444768
\(396\) 0 0
\(397\) −22.9150 −1.15007 −0.575036 0.818128i \(-0.695012\pi\)
−0.575036 + 0.818128i \(0.695012\pi\)
\(398\) 29.7622 1.49184
\(399\) 0 0
\(400\) −1.33664 −0.0668319
\(401\) −37.1991 −1.85764 −0.928818 0.370535i \(-0.879174\pi\)
−0.928818 + 0.370535i \(0.879174\pi\)
\(402\) 0 0
\(403\) 0.240071 0.0119588
\(404\) 31.1530 1.54992
\(405\) 0 0
\(406\) −84.6072 −4.19899
\(407\) −3.36669 −0.166880
\(408\) 0 0
\(409\) 25.6275 1.26720 0.633598 0.773663i \(-0.281577\pi\)
0.633598 + 0.773663i \(0.281577\pi\)
\(410\) −19.5527 −0.965639
\(411\) 0 0
\(412\) 38.2669 1.88527
\(413\) −34.5462 −1.69991
\(414\) 0 0
\(415\) −9.09443 −0.446428
\(416\) −29.1283 −1.42813
\(417\) 0 0
\(418\) 24.0353 1.17560
\(419\) 28.1678 1.37609 0.688043 0.725670i \(-0.258470\pi\)
0.688043 + 0.725670i \(0.258470\pi\)
\(420\) 0 0
\(421\) −20.4045 −0.994453 −0.497227 0.867621i \(-0.665648\pi\)
−0.497227 + 0.867621i \(0.665648\pi\)
\(422\) −44.6219 −2.17216
\(423\) 0 0
\(424\) 7.08621 0.344137
\(425\) −5.74043 −0.278452
\(426\) 0 0
\(427\) −31.2261 −1.51114
\(428\) −32.3844 −1.56536
\(429\) 0 0
\(430\) 13.4782 0.649978
\(431\) −20.4146 −0.983338 −0.491669 0.870782i \(-0.663613\pi\)
−0.491669 + 0.870782i \(0.663613\pi\)
\(432\) 0 0
\(433\) 16.1740 0.777275 0.388637 0.921391i \(-0.372946\pi\)
0.388637 + 0.921391i \(0.372946\pi\)
\(434\) −0.480800 −0.0230791
\(435\) 0 0
\(436\) −3.44612 −0.165039
\(437\) 47.6496 2.27939
\(438\) 0 0
\(439\) 9.85761 0.470478 0.235239 0.971938i \(-0.424413\pi\)
0.235239 + 0.971938i \(0.424413\pi\)
\(440\) −3.12414 −0.148938
\(441\) 0 0
\(442\) −52.8370 −2.51320
\(443\) −7.14208 −0.339331 −0.169665 0.985502i \(-0.554269\pi\)
−0.169665 + 0.985502i \(0.554269\pi\)
\(444\) 0 0
\(445\) 1.00000 0.0474045
\(446\) −18.3143 −0.867206
\(447\) 0 0
\(448\) 48.3082 2.28235
\(449\) 34.4416 1.62540 0.812701 0.582681i \(-0.197996\pi\)
0.812701 + 0.582681i \(0.197996\pi\)
\(450\) 0 0
\(451\) 13.6007 0.640432
\(452\) 23.1344 1.08815
\(453\) 0 0
\(454\) 46.7787 2.19543
\(455\) 15.5762 0.730222
\(456\) 0 0
\(457\) 28.5729 1.33658 0.668292 0.743899i \(-0.267026\pi\)
0.668292 + 0.743899i \(0.267026\pi\)
\(458\) −29.4842 −1.37771
\(459\) 0 0
\(460\) −19.7464 −0.920681
\(461\) −24.3926 −1.13608 −0.568038 0.823002i \(-0.692297\pi\)
−0.568038 + 0.823002i \(0.692297\pi\)
\(462\) 0 0
\(463\) −9.51283 −0.442099 −0.221049 0.975263i \(-0.570948\pi\)
−0.221049 + 0.975263i \(0.570948\pi\)
\(464\) −13.5994 −0.631337
\(465\) 0 0
\(466\) −12.0172 −0.556688
\(467\) 11.5000 0.532157 0.266079 0.963951i \(-0.414272\pi\)
0.266079 + 0.963951i \(0.414272\pi\)
\(468\) 0 0
\(469\) −14.7624 −0.681664
\(470\) 9.05396 0.417628
\(471\) 0 0
\(472\) 18.6584 0.858824
\(473\) −9.37535 −0.431079
\(474\) 0 0
\(475\) 7.03169 0.322636
\(476\) 62.7504 2.87616
\(477\) 0 0
\(478\) −23.2234 −1.06221
\(479\) 37.9454 1.73377 0.866884 0.498510i \(-0.166119\pi\)
0.866884 + 0.498510i \(0.166119\pi\)
\(480\) 0 0
\(481\) −9.06583 −0.413366
\(482\) −29.1133 −1.32607
\(483\) 0 0
\(484\) −25.1255 −1.14207
\(485\) 2.70331 0.122751
\(486\) 0 0
\(487\) 24.8040 1.12398 0.561988 0.827145i \(-0.310037\pi\)
0.561988 + 0.827145i \(0.310037\pi\)
\(488\) 16.8652 0.763454
\(489\) 0 0
\(490\) −15.6778 −0.708250
\(491\) 27.1395 1.22479 0.612395 0.790552i \(-0.290206\pi\)
0.612395 + 0.790552i \(0.290206\pi\)
\(492\) 0 0
\(493\) −58.4051 −2.63043
\(494\) 64.7222 2.91199
\(495\) 0 0
\(496\) −0.0772818 −0.00347006
\(497\) −20.6321 −0.925478
\(498\) 0 0
\(499\) −33.3081 −1.49108 −0.745538 0.666463i \(-0.767808\pi\)
−0.745538 + 0.666463i \(0.767808\pi\)
\(500\) −2.91399 −0.130318
\(501\) 0 0
\(502\) 60.1779 2.68587
\(503\) 26.6651 1.18894 0.594469 0.804119i \(-0.297362\pi\)
0.594469 + 0.804119i \(0.297362\pi\)
\(504\) 0 0
\(505\) −10.6908 −0.475735
\(506\) 23.1627 1.02971
\(507\) 0 0
\(508\) −15.2286 −0.675661
\(509\) −28.5259 −1.26439 −0.632193 0.774811i \(-0.717845\pi\)
−0.632193 + 0.774811i \(0.717845\pi\)
\(510\) 0 0
\(511\) 48.8710 2.16193
\(512\) 14.7931 0.653767
\(513\) 0 0
\(514\) 40.2536 1.77551
\(515\) −13.1321 −0.578671
\(516\) 0 0
\(517\) −6.29786 −0.276980
\(518\) 18.1565 0.797751
\(519\) 0 0
\(520\) −8.41270 −0.368921
\(521\) 3.47227 0.152123 0.0760614 0.997103i \(-0.475765\pi\)
0.0760614 + 0.997103i \(0.475765\pi\)
\(522\) 0 0
\(523\) −16.5706 −0.724584 −0.362292 0.932065i \(-0.618006\pi\)
−0.362292 + 0.932065i \(0.618006\pi\)
\(524\) 10.8013 0.471859
\(525\) 0 0
\(526\) 52.7467 2.29987
\(527\) −0.331900 −0.0144578
\(528\) 0 0
\(529\) 22.9197 0.996511
\(530\) −7.75304 −0.336771
\(531\) 0 0
\(532\) −76.8656 −3.33255
\(533\) 36.6240 1.58636
\(534\) 0 0
\(535\) 11.1134 0.480475
\(536\) 7.97318 0.344389
\(537\) 0 0
\(538\) −0.352769 −0.0152090
\(539\) 10.9053 0.469726
\(540\) 0 0
\(541\) −29.0138 −1.24740 −0.623700 0.781664i \(-0.714371\pi\)
−0.623700 + 0.781664i \(0.714371\pi\)
\(542\) −2.55245 −0.109637
\(543\) 0 0
\(544\) 40.2701 1.72657
\(545\) 1.18261 0.0506575
\(546\) 0 0
\(547\) 32.6731 1.39700 0.698500 0.715610i \(-0.253851\pi\)
0.698500 + 0.715610i \(0.253851\pi\)
\(548\) 44.9484 1.92010
\(549\) 0 0
\(550\) 3.41813 0.145750
\(551\) 71.5428 3.04783
\(552\) 0 0
\(553\) −33.1602 −1.41011
\(554\) −3.67176 −0.155998
\(555\) 0 0
\(556\) −20.4063 −0.865419
\(557\) 10.9818 0.465315 0.232658 0.972559i \(-0.425258\pi\)
0.232658 + 0.972559i \(0.425258\pi\)
\(558\) 0 0
\(559\) −25.2460 −1.06779
\(560\) −5.01416 −0.211887
\(561\) 0 0
\(562\) −9.06126 −0.382226
\(563\) 33.8545 1.42680 0.713399 0.700758i \(-0.247155\pi\)
0.713399 + 0.700758i \(0.247155\pi\)
\(564\) 0 0
\(565\) −7.93908 −0.333999
\(566\) −32.7329 −1.37587
\(567\) 0 0
\(568\) 11.1434 0.467568
\(569\) −9.27784 −0.388947 −0.194474 0.980908i \(-0.562300\pi\)
−0.194474 + 0.980908i \(0.562300\pi\)
\(570\) 0 0
\(571\) −16.6644 −0.697383 −0.348692 0.937238i \(-0.613374\pi\)
−0.348692 + 0.937238i \(0.613374\pi\)
\(572\) 18.6568 0.780078
\(573\) 0 0
\(574\) −73.3484 −3.06150
\(575\) 6.77641 0.282596
\(576\) 0 0
\(577\) 9.60454 0.399842 0.199921 0.979812i \(-0.435931\pi\)
0.199921 + 0.979812i \(0.435931\pi\)
\(578\) 35.3628 1.47090
\(579\) 0 0
\(580\) −29.6479 −1.23106
\(581\) −34.1161 −1.41538
\(582\) 0 0
\(583\) 5.39295 0.223353
\(584\) −26.3953 −1.09225
\(585\) 0 0
\(586\) −7.61118 −0.314415
\(587\) −42.5202 −1.75500 −0.877498 0.479580i \(-0.840789\pi\)
−0.877498 + 0.479580i \(0.840789\pi\)
\(588\) 0 0
\(589\) 0.406559 0.0167520
\(590\) −20.4142 −0.840441
\(591\) 0 0
\(592\) 2.91840 0.119946
\(593\) 16.5448 0.679412 0.339706 0.940532i \(-0.389672\pi\)
0.339706 + 0.940532i \(0.389672\pi\)
\(594\) 0 0
\(595\) −21.5342 −0.882816
\(596\) −36.5408 −1.49677
\(597\) 0 0
\(598\) 62.3725 2.55060
\(599\) −3.61576 −0.147736 −0.0738680 0.997268i \(-0.523534\pi\)
−0.0738680 + 0.997268i \(0.523534\pi\)
\(600\) 0 0
\(601\) −6.04796 −0.246701 −0.123351 0.992363i \(-0.539364\pi\)
−0.123351 + 0.992363i \(0.539364\pi\)
\(602\) 50.5612 2.06072
\(603\) 0 0
\(604\) 0.688110 0.0279988
\(605\) 8.62238 0.350549
\(606\) 0 0
\(607\) −13.4200 −0.544701 −0.272350 0.962198i \(-0.587801\pi\)
−0.272350 + 0.962198i \(0.587801\pi\)
\(608\) −49.3286 −2.00054
\(609\) 0 0
\(610\) −18.4523 −0.747112
\(611\) −16.9589 −0.686084
\(612\) 0 0
\(613\) 11.6728 0.471461 0.235731 0.971818i \(-0.424252\pi\)
0.235731 + 0.971818i \(0.424252\pi\)
\(614\) −49.9165 −2.01447
\(615\) 0 0
\(616\) −11.7197 −0.472198
\(617\) 28.0621 1.12974 0.564869 0.825180i \(-0.308927\pi\)
0.564869 + 0.825180i \(0.308927\pi\)
\(618\) 0 0
\(619\) 20.9899 0.843655 0.421827 0.906676i \(-0.361389\pi\)
0.421827 + 0.906676i \(0.361389\pi\)
\(620\) −0.168481 −0.00676637
\(621\) 0 0
\(622\) −25.4545 −1.02063
\(623\) 3.75132 0.150293
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −4.92894 −0.197000
\(627\) 0 0
\(628\) −14.5302 −0.579818
\(629\) 12.5336 0.499747
\(630\) 0 0
\(631\) 29.7837 1.18567 0.592836 0.805323i \(-0.298008\pi\)
0.592836 + 0.805323i \(0.298008\pi\)
\(632\) 17.9098 0.712415
\(633\) 0 0
\(634\) −72.1006 −2.86348
\(635\) 5.22603 0.207389
\(636\) 0 0
\(637\) 29.3659 1.16352
\(638\) 34.7772 1.37684
\(639\) 0 0
\(640\) 14.5162 0.573802
\(641\) −18.9560 −0.748715 −0.374358 0.927284i \(-0.622137\pi\)
−0.374358 + 0.927284i \(0.622137\pi\)
\(642\) 0 0
\(643\) 48.3147 1.90535 0.952673 0.303996i \(-0.0983211\pi\)
0.952673 + 0.303996i \(0.0983211\pi\)
\(644\) −74.0751 −2.91897
\(645\) 0 0
\(646\) −89.4791 −3.52051
\(647\) 10.0752 0.396096 0.198048 0.980192i \(-0.436540\pi\)
0.198048 + 0.980192i \(0.436540\pi\)
\(648\) 0 0
\(649\) 14.2000 0.557398
\(650\) 9.20436 0.361025
\(651\) 0 0
\(652\) −31.8109 −1.24581
\(653\) −10.2900 −0.402678 −0.201339 0.979522i \(-0.564529\pi\)
−0.201339 + 0.979522i \(0.564529\pi\)
\(654\) 0 0
\(655\) −3.70672 −0.144833
\(656\) −11.7897 −0.460311
\(657\) 0 0
\(658\) 33.9643 1.32407
\(659\) −32.9627 −1.28405 −0.642023 0.766686i \(-0.721905\pi\)
−0.642023 + 0.766686i \(0.721905\pi\)
\(660\) 0 0
\(661\) 40.9469 1.59265 0.796325 0.604869i \(-0.206775\pi\)
0.796325 + 0.604869i \(0.206775\pi\)
\(662\) −60.4011 −2.34756
\(663\) 0 0
\(664\) 18.4261 0.715073
\(665\) 26.3781 1.02290
\(666\) 0 0
\(667\) 68.9455 2.66958
\(668\) −57.2777 −2.21614
\(669\) 0 0
\(670\) −8.72348 −0.337017
\(671\) 12.8353 0.495500
\(672\) 0 0
\(673\) −19.3257 −0.744952 −0.372476 0.928042i \(-0.621491\pi\)
−0.372476 + 0.928042i \(0.621491\pi\)
\(674\) 36.6466 1.41157
\(675\) 0 0
\(676\) 12.3571 0.475275
\(677\) −13.6476 −0.524518 −0.262259 0.964998i \(-0.584467\pi\)
−0.262259 + 0.964998i \(0.584467\pi\)
\(678\) 0 0
\(679\) 10.1410 0.389175
\(680\) 11.6306 0.446015
\(681\) 0 0
\(682\) 0.197630 0.00756763
\(683\) 34.1585 1.30704 0.653520 0.756909i \(-0.273291\pi\)
0.653520 + 0.756909i \(0.273291\pi\)
\(684\) 0 0
\(685\) −15.4250 −0.589360
\(686\) −0.602113 −0.0229888
\(687\) 0 0
\(688\) 8.12700 0.309839
\(689\) 14.5222 0.553250
\(690\) 0 0
\(691\) −4.07591 −0.155055 −0.0775275 0.996990i \(-0.524703\pi\)
−0.0775275 + 0.996990i \(0.524703\pi\)
\(692\) 38.3953 1.45957
\(693\) 0 0
\(694\) 73.6475 2.79562
\(695\) 7.00287 0.265634
\(696\) 0 0
\(697\) −50.6330 −1.91786
\(698\) −13.0134 −0.492563
\(699\) 0 0
\(700\) −10.9313 −0.413165
\(701\) 4.59497 0.173550 0.0867749 0.996228i \(-0.472344\pi\)
0.0867749 + 0.996228i \(0.472344\pi\)
\(702\) 0 0
\(703\) −15.3529 −0.579046
\(704\) −19.8567 −0.748379
\(705\) 0 0
\(706\) −69.2381 −2.60581
\(707\) −40.1047 −1.50829
\(708\) 0 0
\(709\) −1.21881 −0.0457735 −0.0228867 0.999738i \(-0.507286\pi\)
−0.0228867 + 0.999738i \(0.507286\pi\)
\(710\) −12.1921 −0.457560
\(711\) 0 0
\(712\) −2.02609 −0.0759310
\(713\) 0.391799 0.0146730
\(714\) 0 0
\(715\) −6.40248 −0.239439
\(716\) −28.4017 −1.06142
\(717\) 0 0
\(718\) 8.90373 0.332284
\(719\) −19.6350 −0.732263 −0.366131 0.930563i \(-0.619318\pi\)
−0.366131 + 0.930563i \(0.619318\pi\)
\(720\) 0 0
\(721\) −49.2628 −1.83464
\(722\) 67.4884 2.51166
\(723\) 0 0
\(724\) 20.2018 0.750795
\(725\) 10.1743 0.377865
\(726\) 0 0
\(727\) 21.2231 0.787121 0.393561 0.919299i \(-0.371243\pi\)
0.393561 + 0.919299i \(0.371243\pi\)
\(728\) −31.5587 −1.16964
\(729\) 0 0
\(730\) 28.8792 1.06887
\(731\) 34.9028 1.29093
\(732\) 0 0
\(733\) 18.4194 0.680338 0.340169 0.940364i \(-0.389516\pi\)
0.340169 + 0.940364i \(0.389516\pi\)
\(734\) 30.5287 1.12683
\(735\) 0 0
\(736\) −47.5377 −1.75226
\(737\) 6.06798 0.223517
\(738\) 0 0
\(739\) −9.08147 −0.334067 −0.167034 0.985951i \(-0.553419\pi\)
−0.167034 + 0.985951i \(0.553419\pi\)
\(740\) 6.36237 0.233885
\(741\) 0 0
\(742\) −29.0841 −1.06771
\(743\) 0.0809652 0.00297033 0.00148516 0.999999i \(-0.499527\pi\)
0.00148516 + 0.999999i \(0.499527\pi\)
\(744\) 0 0
\(745\) 12.5398 0.459422
\(746\) −53.7726 −1.96875
\(747\) 0 0
\(748\) −25.7931 −0.943091
\(749\) 41.6900 1.52332
\(750\) 0 0
\(751\) −6.49039 −0.236838 −0.118419 0.992964i \(-0.537783\pi\)
−0.118419 + 0.992964i \(0.537783\pi\)
\(752\) 5.45928 0.199080
\(753\) 0 0
\(754\) 93.6483 3.41047
\(755\) −0.236140 −0.00859402
\(756\) 0 0
\(757\) −28.1185 −1.02199 −0.510993 0.859585i \(-0.670722\pi\)
−0.510993 + 0.859585i \(0.670722\pi\)
\(758\) 65.8582 2.39208
\(759\) 0 0
\(760\) −14.2469 −0.516788
\(761\) −2.54060 −0.0920966 −0.0460483 0.998939i \(-0.514663\pi\)
−0.0460483 + 0.998939i \(0.514663\pi\)
\(762\) 0 0
\(763\) 4.43635 0.160607
\(764\) −12.7405 −0.460934
\(765\) 0 0
\(766\) −68.9416 −2.49096
\(767\) 38.2378 1.38069
\(768\) 0 0
\(769\) 0.0116365 0.000419624 0 0.000209812 1.00000i \(-0.499933\pi\)
0.000209812 1.00000i \(0.499933\pi\)
\(770\) 12.8225 0.462091
\(771\) 0 0
\(772\) 5.97970 0.215214
\(773\) −12.9008 −0.464010 −0.232005 0.972715i \(-0.574529\pi\)
−0.232005 + 0.972715i \(0.574529\pi\)
\(774\) 0 0
\(775\) 0.0578180 0.00207688
\(776\) −5.47714 −0.196618
\(777\) 0 0
\(778\) 5.99788 0.215035
\(779\) 62.0225 2.22219
\(780\) 0 0
\(781\) 8.48070 0.303463
\(782\) −86.2306 −3.08360
\(783\) 0 0
\(784\) −9.45325 −0.337616
\(785\) 4.98636 0.177971
\(786\) 0 0
\(787\) 55.1951 1.96749 0.983746 0.179566i \(-0.0574694\pi\)
0.983746 + 0.179566i \(0.0574694\pi\)
\(788\) 18.9016 0.673340
\(789\) 0 0
\(790\) −19.5952 −0.697166
\(791\) −29.7820 −1.05893
\(792\) 0 0
\(793\) 34.5629 1.22736
\(794\) −50.7970 −1.80272
\(795\) 0 0
\(796\) 39.1233 1.38669
\(797\) −31.5728 −1.11837 −0.559183 0.829044i \(-0.688885\pi\)
−0.559183 + 0.829044i \(0.688885\pi\)
\(798\) 0 0
\(799\) 23.4458 0.829455
\(800\) −7.01518 −0.248024
\(801\) 0 0
\(802\) −82.4613 −2.91181
\(803\) −20.0881 −0.708894
\(804\) 0 0
\(805\) 25.4205 0.895955
\(806\) 0.532178 0.0187452
\(807\) 0 0
\(808\) 21.6606 0.762017
\(809\) −37.2161 −1.30845 −0.654225 0.756300i \(-0.727005\pi\)
−0.654225 + 0.756300i \(0.727005\pi\)
\(810\) 0 0
\(811\) 45.2306 1.58826 0.794131 0.607746i \(-0.207926\pi\)
0.794131 + 0.607746i \(0.207926\pi\)
\(812\) −111.219 −3.90302
\(813\) 0 0
\(814\) −7.46311 −0.261582
\(815\) 10.9166 0.382392
\(816\) 0 0
\(817\) −42.7539 −1.49577
\(818\) 56.8097 1.98630
\(819\) 0 0
\(820\) −25.7026 −0.897575
\(821\) −19.6062 −0.684261 −0.342131 0.939652i \(-0.611149\pi\)
−0.342131 + 0.939652i \(0.611149\pi\)
\(822\) 0 0
\(823\) −19.4397 −0.677625 −0.338812 0.940854i \(-0.610025\pi\)
−0.338812 + 0.940854i \(0.610025\pi\)
\(824\) 26.6069 0.926895
\(825\) 0 0
\(826\) −76.5804 −2.66457
\(827\) 15.8859 0.552407 0.276203 0.961099i \(-0.410924\pi\)
0.276203 + 0.961099i \(0.410924\pi\)
\(828\) 0 0
\(829\) 9.64772 0.335079 0.167539 0.985865i \(-0.446418\pi\)
0.167539 + 0.985865i \(0.446418\pi\)
\(830\) −20.1601 −0.699767
\(831\) 0 0
\(832\) −53.4703 −1.85375
\(833\) −40.5987 −1.40666
\(834\) 0 0
\(835\) 19.6561 0.680228
\(836\) 31.5951 1.09274
\(837\) 0 0
\(838\) 62.4410 2.15699
\(839\) −38.4916 −1.32888 −0.664439 0.747342i \(-0.731330\pi\)
−0.664439 + 0.747342i \(0.731330\pi\)
\(840\) 0 0
\(841\) 74.5172 2.56956
\(842\) −45.2317 −1.55879
\(843\) 0 0
\(844\) −58.6569 −2.01905
\(845\) −4.24062 −0.145882
\(846\) 0 0
\(847\) 32.3453 1.11140
\(848\) −4.67486 −0.160535
\(849\) 0 0
\(850\) −12.7251 −0.436468
\(851\) −14.7955 −0.507184
\(852\) 0 0
\(853\) 7.63504 0.261419 0.130709 0.991421i \(-0.458275\pi\)
0.130709 + 0.991421i \(0.458275\pi\)
\(854\) −69.2205 −2.36868
\(855\) 0 0
\(856\) −22.5168 −0.769609
\(857\) 4.22874 0.144451 0.0722255 0.997388i \(-0.476990\pi\)
0.0722255 + 0.997388i \(0.476990\pi\)
\(858\) 0 0
\(859\) −44.5369 −1.51958 −0.759789 0.650170i \(-0.774698\pi\)
−0.759789 + 0.650170i \(0.774698\pi\)
\(860\) 17.7176 0.604164
\(861\) 0 0
\(862\) −45.2542 −1.54136
\(863\) −3.64429 −0.124053 −0.0620266 0.998074i \(-0.519756\pi\)
−0.0620266 + 0.998074i \(0.519756\pi\)
\(864\) 0 0
\(865\) −13.1762 −0.448004
\(866\) 35.8538 1.21836
\(867\) 0 0
\(868\) −0.632027 −0.0214524
\(869\) 13.6303 0.462375
\(870\) 0 0
\(871\) 16.3399 0.553656
\(872\) −2.39608 −0.0811414
\(873\) 0 0
\(874\) 105.627 3.57290
\(875\) 3.75132 0.126818
\(876\) 0 0
\(877\) −40.7502 −1.37604 −0.688018 0.725694i \(-0.741519\pi\)
−0.688018 + 0.725694i \(0.741519\pi\)
\(878\) 21.8519 0.737465
\(879\) 0 0
\(880\) 2.06104 0.0694775
\(881\) 11.4553 0.385940 0.192970 0.981205i \(-0.438188\pi\)
0.192970 + 0.981205i \(0.438188\pi\)
\(882\) 0 0
\(883\) −5.39868 −0.181680 −0.0908401 0.995865i \(-0.528955\pi\)
−0.0908401 + 0.995865i \(0.528955\pi\)
\(884\) −69.4559 −2.33605
\(885\) 0 0
\(886\) −15.8322 −0.531894
\(887\) 25.7317 0.863987 0.431994 0.901877i \(-0.357810\pi\)
0.431994 + 0.901877i \(0.357810\pi\)
\(888\) 0 0
\(889\) 19.6045 0.657515
\(890\) 2.21675 0.0743057
\(891\) 0 0
\(892\) −24.0747 −0.806080
\(893\) −28.7198 −0.961071
\(894\) 0 0
\(895\) 9.74668 0.325796
\(896\) 54.4548 1.81921
\(897\) 0 0
\(898\) 76.3486 2.54779
\(899\) 0.588260 0.0196196
\(900\) 0 0
\(901\) −20.0770 −0.668862
\(902\) 30.1494 1.00386
\(903\) 0 0
\(904\) 16.0853 0.534989
\(905\) −6.93270 −0.230451
\(906\) 0 0
\(907\) 6.59646 0.219032 0.109516 0.993985i \(-0.465070\pi\)
0.109516 + 0.993985i \(0.465070\pi\)
\(908\) 61.4921 2.04069
\(909\) 0 0
\(910\) 34.5285 1.14461
\(911\) −22.1874 −0.735102 −0.367551 0.930003i \(-0.619804\pi\)
−0.367551 + 0.930003i \(0.619804\pi\)
\(912\) 0 0
\(913\) 14.0232 0.464100
\(914\) 63.3390 2.09507
\(915\) 0 0
\(916\) −38.7579 −1.28060
\(917\) −13.9051 −0.459186
\(918\) 0 0
\(919\) 15.0284 0.495742 0.247871 0.968793i \(-0.420269\pi\)
0.247871 + 0.968793i \(0.420269\pi\)
\(920\) −13.7296 −0.452652
\(921\) 0 0
\(922\) −54.0723 −1.78078
\(923\) 22.8369 0.751684
\(924\) 0 0
\(925\) −2.18339 −0.0717894
\(926\) −21.0876 −0.692981
\(927\) 0 0
\(928\) −71.3748 −2.34299
\(929\) 21.3719 0.701189 0.350594 0.936527i \(-0.385980\pi\)
0.350594 + 0.936527i \(0.385980\pi\)
\(930\) 0 0
\(931\) 49.7310 1.62987
\(932\) −15.7971 −0.517450
\(933\) 0 0
\(934\) 25.4927 0.834146
\(935\) 8.85148 0.289474
\(936\) 0 0
\(937\) −9.16254 −0.299327 −0.149664 0.988737i \(-0.547819\pi\)
−0.149664 + 0.988737i \(0.547819\pi\)
\(938\) −32.7246 −1.06849
\(939\) 0 0
\(940\) 11.9017 0.388191
\(941\) 49.0960 1.60048 0.800241 0.599678i \(-0.204705\pi\)
0.800241 + 0.599678i \(0.204705\pi\)
\(942\) 0 0
\(943\) 59.7708 1.94641
\(944\) −12.3092 −0.400631
\(945\) 0 0
\(946\) −20.7828 −0.675708
\(947\) −15.2650 −0.496045 −0.248023 0.968754i \(-0.579781\pi\)
−0.248023 + 0.968754i \(0.579781\pi\)
\(948\) 0 0
\(949\) −54.0934 −1.75594
\(950\) 15.5875 0.505726
\(951\) 0 0
\(952\) 43.6302 1.41406
\(953\) 4.11784 0.133390 0.0666949 0.997773i \(-0.478755\pi\)
0.0666949 + 0.997773i \(0.478755\pi\)
\(954\) 0 0
\(955\) 4.37217 0.141480
\(956\) −30.5279 −0.987343
\(957\) 0 0
\(958\) 84.1155 2.71765
\(959\) −57.8643 −1.86853
\(960\) 0 0
\(961\) −30.9967 −0.999892
\(962\) −20.0967 −0.647943
\(963\) 0 0
\(964\) −38.2703 −1.23260
\(965\) −2.05206 −0.0660583
\(966\) 0 0
\(967\) −34.1373 −1.09778 −0.548891 0.835894i \(-0.684950\pi\)
−0.548891 + 0.835894i \(0.684950\pi\)
\(968\) −17.4697 −0.561498
\(969\) 0 0
\(970\) 5.99256 0.192409
\(971\) 16.7233 0.536675 0.268338 0.963325i \(-0.413526\pi\)
0.268338 + 0.963325i \(0.413526\pi\)
\(972\) 0 0
\(973\) 26.2700 0.842177
\(974\) 54.9843 1.76181
\(975\) 0 0
\(976\) −11.1262 −0.356142
\(977\) −39.4703 −1.26277 −0.631384 0.775470i \(-0.717513\pi\)
−0.631384 + 0.775470i \(0.717513\pi\)
\(978\) 0 0
\(979\) −1.54195 −0.0492811
\(980\) −20.6089 −0.658328
\(981\) 0 0
\(982\) 60.1616 1.91983
\(983\) 24.8433 0.792380 0.396190 0.918169i \(-0.370332\pi\)
0.396190 + 0.918169i \(0.370332\pi\)
\(984\) 0 0
\(985\) −6.48648 −0.206677
\(986\) −129.470 −4.12315
\(987\) 0 0
\(988\) 85.0794 2.70674
\(989\) −41.2018 −1.31014
\(990\) 0 0
\(991\) −0.276840 −0.00879411 −0.00439706 0.999990i \(-0.501400\pi\)
−0.00439706 + 0.999990i \(0.501400\pi\)
\(992\) −0.405604 −0.0128779
\(993\) 0 0
\(994\) −45.7363 −1.45067
\(995\) −13.4260 −0.425633
\(996\) 0 0
\(997\) −61.1434 −1.93643 −0.968216 0.250117i \(-0.919531\pi\)
−0.968216 + 0.250117i \(0.919531\pi\)
\(998\) −73.8359 −2.33723
\(999\) 0 0
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 4005.2.a.o.1.6 7
3.2 odd 2 445.2.a.f.1.2 7
12.11 even 2 7120.2.a.bj.1.5 7
15.14 odd 2 2225.2.a.k.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
445.2.a.f.1.2 7 3.2 odd 2
2225.2.a.k.1.6 7 15.14 odd 2
4005.2.a.o.1.6 7 1.1 even 1 trivial
7120.2.a.bj.1.5 7 12.11 even 2