Properties

Label 75.4.a.b.1.1
Level $75$
Weight $4$
Character 75.1
Self dual yes
Analytic conductor $4.425$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.42514325043\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 75.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +24.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -3.00000 q^{3} -7.00000 q^{4} +3.00000 q^{6} +24.0000 q^{7} +15.0000 q^{8} +9.00000 q^{9} +52.0000 q^{11} +21.0000 q^{12} -22.0000 q^{13} -24.0000 q^{14} +41.0000 q^{16} +14.0000 q^{17} -9.00000 q^{18} -20.0000 q^{19} -72.0000 q^{21} -52.0000 q^{22} +168.000 q^{23} -45.0000 q^{24} +22.0000 q^{26} -27.0000 q^{27} -168.000 q^{28} +230.000 q^{29} -288.000 q^{31} -161.000 q^{32} -156.000 q^{33} -14.0000 q^{34} -63.0000 q^{36} +34.0000 q^{37} +20.0000 q^{38} +66.0000 q^{39} +122.000 q^{41} +72.0000 q^{42} +188.000 q^{43} -364.000 q^{44} -168.000 q^{46} -256.000 q^{47} -123.000 q^{48} +233.000 q^{49} -42.0000 q^{51} +154.000 q^{52} +338.000 q^{53} +27.0000 q^{54} +360.000 q^{56} +60.0000 q^{57} -230.000 q^{58} +100.000 q^{59} +742.000 q^{61} +288.000 q^{62} +216.000 q^{63} -167.000 q^{64} +156.000 q^{66} +84.0000 q^{67} -98.0000 q^{68} -504.000 q^{69} -328.000 q^{71} +135.000 q^{72} +38.0000 q^{73} -34.0000 q^{74} +140.000 q^{76} +1248.00 q^{77} -66.0000 q^{78} -240.000 q^{79} +81.0000 q^{81} -122.000 q^{82} -1212.00 q^{83} +504.000 q^{84} -188.000 q^{86} -690.000 q^{87} +780.000 q^{88} +330.000 q^{89} -528.000 q^{91} -1176.00 q^{92} +864.000 q^{93} +256.000 q^{94} +483.000 q^{96} -866.000 q^{97} -233.000 q^{98} +468.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) −3.00000 −0.577350
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) 3.00000 0.204124
\(7\) 24.0000 1.29588 0.647939 0.761692i \(-0.275631\pi\)
0.647939 + 0.761692i \(0.275631\pi\)
\(8\) 15.0000 0.662913
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 52.0000 1.42533 0.712663 0.701506i \(-0.247489\pi\)
0.712663 + 0.701506i \(0.247489\pi\)
\(12\) 21.0000 0.505181
\(13\) −22.0000 −0.469362 −0.234681 0.972072i \(-0.575405\pi\)
−0.234681 + 0.972072i \(0.575405\pi\)
\(14\) −24.0000 −0.458162
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) 14.0000 0.199735 0.0998676 0.995001i \(-0.468158\pi\)
0.0998676 + 0.995001i \(0.468158\pi\)
\(18\) −9.00000 −0.117851
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) −72.0000 −0.748176
\(22\) −52.0000 −0.503929
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) −45.0000 −0.382733
\(25\) 0 0
\(26\) 22.0000 0.165944
\(27\) −27.0000 −0.192450
\(28\) −168.000 −1.13389
\(29\) 230.000 1.47276 0.736378 0.676570i \(-0.236535\pi\)
0.736378 + 0.676570i \(0.236535\pi\)
\(30\) 0 0
\(31\) −288.000 −1.66859 −0.834296 0.551317i \(-0.814125\pi\)
−0.834296 + 0.551317i \(0.814125\pi\)
\(32\) −161.000 −0.889408
\(33\) −156.000 −0.822913
\(34\) −14.0000 −0.0706171
\(35\) 0 0
\(36\) −63.0000 −0.291667
\(37\) 34.0000 0.151069 0.0755347 0.997143i \(-0.475934\pi\)
0.0755347 + 0.997143i \(0.475934\pi\)
\(38\) 20.0000 0.0853797
\(39\) 66.0000 0.270986
\(40\) 0 0
\(41\) 122.000 0.464712 0.232356 0.972631i \(-0.425357\pi\)
0.232356 + 0.972631i \(0.425357\pi\)
\(42\) 72.0000 0.264520
\(43\) 188.000 0.666738 0.333369 0.942796i \(-0.391815\pi\)
0.333369 + 0.942796i \(0.391815\pi\)
\(44\) −364.000 −1.24716
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) −256.000 −0.794499 −0.397249 0.917711i \(-0.630035\pi\)
−0.397249 + 0.917711i \(0.630035\pi\)
\(48\) −123.000 −0.369865
\(49\) 233.000 0.679300
\(50\) 0 0
\(51\) −42.0000 −0.115317
\(52\) 154.000 0.410691
\(53\) 338.000 0.875998 0.437999 0.898976i \(-0.355687\pi\)
0.437999 + 0.898976i \(0.355687\pi\)
\(54\) 27.0000 0.0680414
\(55\) 0 0
\(56\) 360.000 0.859054
\(57\) 60.0000 0.139424
\(58\) −230.000 −0.520698
\(59\) 100.000 0.220659 0.110330 0.993895i \(-0.464809\pi\)
0.110330 + 0.993895i \(0.464809\pi\)
\(60\) 0 0
\(61\) 742.000 1.55743 0.778716 0.627376i \(-0.215871\pi\)
0.778716 + 0.627376i \(0.215871\pi\)
\(62\) 288.000 0.589936
\(63\) 216.000 0.431959
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 156.000 0.290944
\(67\) 84.0000 0.153168 0.0765838 0.997063i \(-0.475599\pi\)
0.0765838 + 0.997063i \(0.475599\pi\)
\(68\) −98.0000 −0.174768
\(69\) −504.000 −0.879340
\(70\) 0 0
\(71\) −328.000 −0.548260 −0.274130 0.961693i \(-0.588390\pi\)
−0.274130 + 0.961693i \(0.588390\pi\)
\(72\) 135.000 0.220971
\(73\) 38.0000 0.0609255 0.0304628 0.999536i \(-0.490302\pi\)
0.0304628 + 0.999536i \(0.490302\pi\)
\(74\) −34.0000 −0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) 1248.00 1.84705
\(78\) −66.0000 −0.0958081
\(79\) −240.000 −0.341799 −0.170899 0.985288i \(-0.554667\pi\)
−0.170899 + 0.985288i \(0.554667\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) −122.000 −0.164301
\(83\) −1212.00 −1.60282 −0.801411 0.598114i \(-0.795917\pi\)
−0.801411 + 0.598114i \(0.795917\pi\)
\(84\) 504.000 0.654654
\(85\) 0 0
\(86\) −188.000 −0.235727
\(87\) −690.000 −0.850296
\(88\) 780.000 0.944867
\(89\) 330.000 0.393033 0.196516 0.980501i \(-0.437037\pi\)
0.196516 + 0.980501i \(0.437037\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) −1176.00 −1.33268
\(93\) 864.000 0.963362
\(94\) 256.000 0.280898
\(95\) 0 0
\(96\) 483.000 0.513500
\(97\) −866.000 −0.906484 −0.453242 0.891387i \(-0.649733\pi\)
−0.453242 + 0.891387i \(0.649733\pi\)
\(98\) −233.000 −0.240169
\(99\) 468.000 0.475109
\(100\) 0 0
\(101\) −1218.00 −1.19996 −0.599978 0.800017i \(-0.704824\pi\)
−0.599978 + 0.800017i \(0.704824\pi\)
\(102\) 42.0000 0.0407708
\(103\) 88.0000 0.0841835 0.0420917 0.999114i \(-0.486598\pi\)
0.0420917 + 0.999114i \(0.486598\pi\)
\(104\) −330.000 −0.311146
\(105\) 0 0
\(106\) −338.000 −0.309712
\(107\) −36.0000 −0.0325257 −0.0162629 0.999868i \(-0.505177\pi\)
−0.0162629 + 0.999868i \(0.505177\pi\)
\(108\) 189.000 0.168394
\(109\) −970.000 −0.852378 −0.426189 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(110\) 0 0
\(111\) −102.000 −0.0872199
\(112\) 984.000 0.830172
\(113\) −1042.00 −0.867461 −0.433731 0.901043i \(-0.642803\pi\)
−0.433731 + 0.901043i \(0.642803\pi\)
\(114\) −60.0000 −0.0492940
\(115\) 0 0
\(116\) −1610.00 −1.28866
\(117\) −198.000 −0.156454
\(118\) −100.000 −0.0780148
\(119\) 336.000 0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) −742.000 −0.550635
\(123\) −366.000 −0.268302
\(124\) 2016.00 1.46002
\(125\) 0 0
\(126\) −216.000 −0.152721
\(127\) −1936.00 −1.35269 −0.676347 0.736583i \(-0.736438\pi\)
−0.676347 + 0.736583i \(0.736438\pi\)
\(128\) 1455.00 1.00473
\(129\) −564.000 −0.384941
\(130\) 0 0
\(131\) 732.000 0.488207 0.244104 0.969749i \(-0.421506\pi\)
0.244104 + 0.969749i \(0.421506\pi\)
\(132\) 1092.00 0.720048
\(133\) −480.000 −0.312942
\(134\) −84.0000 −0.0541529
\(135\) 0 0
\(136\) 210.000 0.132407
\(137\) 2214.00 1.38069 0.690346 0.723479i \(-0.257458\pi\)
0.690346 + 0.723479i \(0.257458\pi\)
\(138\) 504.000 0.310894
\(139\) 20.0000 0.0122042 0.00610208 0.999981i \(-0.498058\pi\)
0.00610208 + 0.999981i \(0.498058\pi\)
\(140\) 0 0
\(141\) 768.000 0.458704
\(142\) 328.000 0.193839
\(143\) −1144.00 −0.668994
\(144\) 369.000 0.213542
\(145\) 0 0
\(146\) −38.0000 −0.0215404
\(147\) −699.000 −0.392194
\(148\) −238.000 −0.132186
\(149\) −1330.00 −0.731261 −0.365630 0.930760i \(-0.619147\pi\)
−0.365630 + 0.930760i \(0.619147\pi\)
\(150\) 0 0
\(151\) −1208.00 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) −300.000 −0.160087
\(153\) 126.000 0.0665784
\(154\) −1248.00 −0.653031
\(155\) 0 0
\(156\) −462.000 −0.237113
\(157\) 3514.00 1.78629 0.893146 0.449768i \(-0.148493\pi\)
0.893146 + 0.449768i \(0.148493\pi\)
\(158\) 240.000 0.120844
\(159\) −1014.00 −0.505757
\(160\) 0 0
\(161\) 4032.00 1.97370
\(162\) −81.0000 −0.0392837
\(163\) 2068.00 0.993732 0.496866 0.867827i \(-0.334484\pi\)
0.496866 + 0.867827i \(0.334484\pi\)
\(164\) −854.000 −0.406623
\(165\) 0 0
\(166\) 1212.00 0.566683
\(167\) 24.0000 0.0111208 0.00556041 0.999985i \(-0.498230\pi\)
0.00556041 + 0.999985i \(0.498230\pi\)
\(168\) −1080.00 −0.495975
\(169\) −1713.00 −0.779700
\(170\) 0 0
\(171\) −180.000 −0.0804967
\(172\) −1316.00 −0.583396
\(173\) 618.000 0.271593 0.135797 0.990737i \(-0.456641\pi\)
0.135797 + 0.990737i \(0.456641\pi\)
\(174\) 690.000 0.300625
\(175\) 0 0
\(176\) 2132.00 0.913100
\(177\) −300.000 −0.127398
\(178\) −330.000 −0.138958
\(179\) 3340.00 1.39466 0.697328 0.716752i \(-0.254372\pi\)
0.697328 + 0.716752i \(0.254372\pi\)
\(180\) 0 0
\(181\) −178.000 −0.0730974 −0.0365487 0.999332i \(-0.511636\pi\)
−0.0365487 + 0.999332i \(0.511636\pi\)
\(182\) 528.000 0.215044
\(183\) −2226.00 −0.899184
\(184\) 2520.00 1.00966
\(185\) 0 0
\(186\) −864.000 −0.340600
\(187\) 728.000 0.284688
\(188\) 1792.00 0.695186
\(189\) −648.000 −0.249392
\(190\) 0 0
\(191\) −1888.00 −0.715240 −0.357620 0.933867i \(-0.616412\pi\)
−0.357620 + 0.933867i \(0.616412\pi\)
\(192\) 501.000 0.188315
\(193\) −1922.00 −0.716832 −0.358416 0.933562i \(-0.616683\pi\)
−0.358416 + 0.933562i \(0.616683\pi\)
\(194\) 866.000 0.320491
\(195\) 0 0
\(196\) −1631.00 −0.594388
\(197\) −2526.00 −0.913554 −0.456777 0.889581i \(-0.650996\pi\)
−0.456777 + 0.889581i \(0.650996\pi\)
\(198\) −468.000 −0.167976
\(199\) −1160.00 −0.413217 −0.206609 0.978424i \(-0.566243\pi\)
−0.206609 + 0.978424i \(0.566243\pi\)
\(200\) 0 0
\(201\) −252.000 −0.0884314
\(202\) 1218.00 0.424248
\(203\) 5520.00 1.90851
\(204\) 294.000 0.100903
\(205\) 0 0
\(206\) −88.0000 −0.0297634
\(207\) 1512.00 0.507687
\(208\) −902.000 −0.300685
\(209\) −1040.00 −0.344202
\(210\) 0 0
\(211\) −4468.00 −1.45777 −0.728886 0.684635i \(-0.759961\pi\)
−0.728886 + 0.684635i \(0.759961\pi\)
\(212\) −2366.00 −0.766498
\(213\) 984.000 0.316538
\(214\) 36.0000 0.0114996
\(215\) 0 0
\(216\) −405.000 −0.127578
\(217\) −6912.00 −2.16229
\(218\) 970.000 0.301361
\(219\) −114.000 −0.0351754
\(220\) 0 0
\(221\) −308.000 −0.0937481
\(222\) 102.000 0.0308369
\(223\) −6032.00 −1.81136 −0.905678 0.423965i \(-0.860638\pi\)
−0.905678 + 0.423965i \(0.860638\pi\)
\(224\) −3864.00 −1.15256
\(225\) 0 0
\(226\) 1042.00 0.306694
\(227\) −2636.00 −0.770738 −0.385369 0.922763i \(-0.625926\pi\)
−0.385369 + 0.922763i \(0.625926\pi\)
\(228\) −420.000 −0.121996
\(229\) 4830.00 1.39378 0.696889 0.717179i \(-0.254567\pi\)
0.696889 + 0.717179i \(0.254567\pi\)
\(230\) 0 0
\(231\) −3744.00 −1.06639
\(232\) 3450.00 0.976309
\(233\) −2682.00 −0.754093 −0.377046 0.926194i \(-0.623060\pi\)
−0.377046 + 0.926194i \(0.623060\pi\)
\(234\) 198.000 0.0553148
\(235\) 0 0
\(236\) −700.000 −0.193077
\(237\) 720.000 0.197338
\(238\) −336.000 −0.0915111
\(239\) 2320.00 0.627901 0.313950 0.949439i \(-0.398347\pi\)
0.313950 + 0.949439i \(0.398347\pi\)
\(240\) 0 0
\(241\) 2002.00 0.535104 0.267552 0.963543i \(-0.413785\pi\)
0.267552 + 0.963543i \(0.413785\pi\)
\(242\) −1373.00 −0.364710
\(243\) −243.000 −0.0641500
\(244\) −5194.00 −1.36275
\(245\) 0 0
\(246\) 366.000 0.0948590
\(247\) 440.000 0.113346
\(248\) −4320.00 −1.10613
\(249\) 3636.00 0.925390
\(250\) 0 0
\(251\) 132.000 0.0331943 0.0165971 0.999862i \(-0.494717\pi\)
0.0165971 + 0.999862i \(0.494717\pi\)
\(252\) −1512.00 −0.377964
\(253\) 8736.00 2.17086
\(254\) 1936.00 0.478250
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) 7614.00 1.84805 0.924024 0.382335i \(-0.124880\pi\)
0.924024 + 0.382335i \(0.124880\pi\)
\(258\) 564.000 0.136097
\(259\) 816.000 0.195767
\(260\) 0 0
\(261\) 2070.00 0.490919
\(262\) −732.000 −0.172607
\(263\) 4888.00 1.14603 0.573017 0.819543i \(-0.305773\pi\)
0.573017 + 0.819543i \(0.305773\pi\)
\(264\) −2340.00 −0.545519
\(265\) 0 0
\(266\) 480.000 0.110642
\(267\) −990.000 −0.226918
\(268\) −588.000 −0.134022
\(269\) 1270.00 0.287856 0.143928 0.989588i \(-0.454027\pi\)
0.143928 + 0.989588i \(0.454027\pi\)
\(270\) 0 0
\(271\) 1072.00 0.240293 0.120146 0.992756i \(-0.461664\pi\)
0.120146 + 0.992756i \(0.461664\pi\)
\(272\) 574.000 0.127955
\(273\) 1584.00 0.351165
\(274\) −2214.00 −0.488148
\(275\) 0 0
\(276\) 3528.00 0.769423
\(277\) 5394.00 1.17001 0.585007 0.811028i \(-0.301092\pi\)
0.585007 + 0.811028i \(0.301092\pi\)
\(278\) −20.0000 −0.00431482
\(279\) −2592.00 −0.556197
\(280\) 0 0
\(281\) 2442.00 0.518425 0.259213 0.965820i \(-0.416537\pi\)
0.259213 + 0.965820i \(0.416537\pi\)
\(282\) −768.000 −0.162176
\(283\) −2772.00 −0.582255 −0.291128 0.956684i \(-0.594030\pi\)
−0.291128 + 0.956684i \(0.594030\pi\)
\(284\) 2296.00 0.479727
\(285\) 0 0
\(286\) 1144.00 0.236525
\(287\) 2928.00 0.602210
\(288\) −1449.00 −0.296469
\(289\) −4717.00 −0.960106
\(290\) 0 0
\(291\) 2598.00 0.523359
\(292\) −266.000 −0.0533098
\(293\) −4542.00 −0.905619 −0.452810 0.891607i \(-0.649578\pi\)
−0.452810 + 0.891607i \(0.649578\pi\)
\(294\) 699.000 0.138662
\(295\) 0 0
\(296\) 510.000 0.100146
\(297\) −1404.00 −0.274304
\(298\) 1330.00 0.258540
\(299\) −3696.00 −0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 1208.00 0.230174
\(303\) 3654.00 0.692795
\(304\) −820.000 −0.154705
\(305\) 0 0
\(306\) −126.000 −0.0235390
\(307\) −5116.00 −0.951093 −0.475546 0.879691i \(-0.657750\pi\)
−0.475546 + 0.879691i \(0.657750\pi\)
\(308\) −8736.00 −1.61617
\(309\) −264.000 −0.0486034
\(310\) 0 0
\(311\) −2808.00 −0.511984 −0.255992 0.966679i \(-0.582402\pi\)
−0.255992 + 0.966679i \(0.582402\pi\)
\(312\) 990.000 0.179640
\(313\) 7318.00 1.32153 0.660763 0.750594i \(-0.270233\pi\)
0.660763 + 0.750594i \(0.270233\pi\)
\(314\) −3514.00 −0.631549
\(315\) 0 0
\(316\) 1680.00 0.299074
\(317\) −2246.00 −0.397943 −0.198971 0.980005i \(-0.563760\pi\)
−0.198971 + 0.980005i \(0.563760\pi\)
\(318\) 1014.00 0.178812
\(319\) 11960.0 2.09916
\(320\) 0 0
\(321\) 108.000 0.0187787
\(322\) −4032.00 −0.697809
\(323\) −280.000 −0.0482341
\(324\) −567.000 −0.0972222
\(325\) 0 0
\(326\) −2068.00 −0.351337
\(327\) 2910.00 0.492120
\(328\) 1830.00 0.308064
\(329\) −6144.00 −1.02957
\(330\) 0 0
\(331\) 1332.00 0.221188 0.110594 0.993866i \(-0.464725\pi\)
0.110594 + 0.993866i \(0.464725\pi\)
\(332\) 8484.00 1.40247
\(333\) 306.000 0.0503564
\(334\) −24.0000 −0.00393180
\(335\) 0 0
\(336\) −2952.00 −0.479300
\(337\) 11534.0 1.86438 0.932191 0.361966i \(-0.117894\pi\)
0.932191 + 0.361966i \(0.117894\pi\)
\(338\) 1713.00 0.275665
\(339\) 3126.00 0.500829
\(340\) 0 0
\(341\) −14976.0 −2.37829
\(342\) 180.000 0.0284599
\(343\) −2640.00 −0.415588
\(344\) 2820.00 0.441989
\(345\) 0 0
\(346\) −618.000 −0.0960228
\(347\) −11956.0 −1.84966 −0.924830 0.380382i \(-0.875793\pi\)
−0.924830 + 0.380382i \(0.875793\pi\)
\(348\) 4830.00 0.744009
\(349\) 4870.00 0.746949 0.373474 0.927640i \(-0.378166\pi\)
0.373474 + 0.927640i \(0.378166\pi\)
\(350\) 0 0
\(351\) 594.000 0.0903287
\(352\) −8372.00 −1.26770
\(353\) −10722.0 −1.61664 −0.808321 0.588742i \(-0.799623\pi\)
−0.808321 + 0.588742i \(0.799623\pi\)
\(354\) 300.000 0.0450419
\(355\) 0 0
\(356\) −2310.00 −0.343904
\(357\) −1008.00 −0.149437
\(358\) −3340.00 −0.493085
\(359\) 120.000 0.0176417 0.00882083 0.999961i \(-0.497192\pi\)
0.00882083 + 0.999961i \(0.497192\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 178.000 0.0258438
\(363\) −4119.00 −0.595569
\(364\) 3696.00 0.532206
\(365\) 0 0
\(366\) 2226.00 0.317910
\(367\) −3936.00 −0.559830 −0.279915 0.960025i \(-0.590306\pi\)
−0.279915 + 0.960025i \(0.590306\pi\)
\(368\) 6888.00 0.975711
\(369\) 1098.00 0.154904
\(370\) 0 0
\(371\) 8112.00 1.13519
\(372\) −6048.00 −0.842941
\(373\) −3022.00 −0.419499 −0.209750 0.977755i \(-0.567265\pi\)
−0.209750 + 0.977755i \(0.567265\pi\)
\(374\) −728.000 −0.100652
\(375\) 0 0
\(376\) −3840.00 −0.526683
\(377\) −5060.00 −0.691255
\(378\) 648.000 0.0881733
\(379\) −13340.0 −1.80799 −0.903997 0.427539i \(-0.859381\pi\)
−0.903997 + 0.427539i \(0.859381\pi\)
\(380\) 0 0
\(381\) 5808.00 0.780979
\(382\) 1888.00 0.252876
\(383\) 1008.00 0.134481 0.0672407 0.997737i \(-0.478580\pi\)
0.0672407 + 0.997737i \(0.478580\pi\)
\(384\) −4365.00 −0.580079
\(385\) 0 0
\(386\) 1922.00 0.253438
\(387\) 1692.00 0.222246
\(388\) 6062.00 0.793174
\(389\) 9630.00 1.25517 0.627584 0.778549i \(-0.284044\pi\)
0.627584 + 0.778549i \(0.284044\pi\)
\(390\) 0 0
\(391\) 2352.00 0.304209
\(392\) 3495.00 0.450317
\(393\) −2196.00 −0.281867
\(394\) 2526.00 0.322990
\(395\) 0 0
\(396\) −3276.00 −0.415720
\(397\) −7126.00 −0.900866 −0.450433 0.892810i \(-0.648730\pi\)
−0.450433 + 0.892810i \(0.648730\pi\)
\(398\) 1160.00 0.146094
\(399\) 1440.00 0.180677
\(400\) 0 0
\(401\) −8718.00 −1.08568 −0.542838 0.839837i \(-0.682650\pi\)
−0.542838 + 0.839837i \(0.682650\pi\)
\(402\) 252.000 0.0312652
\(403\) 6336.00 0.783173
\(404\) 8526.00 1.04996
\(405\) 0 0
\(406\) −5520.00 −0.674761
\(407\) 1768.00 0.215323
\(408\) −630.000 −0.0764452
\(409\) −10870.0 −1.31415 −0.657074 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) 0 0
\(411\) −6642.00 −0.797143
\(412\) −616.000 −0.0736605
\(413\) 2400.00 0.285947
\(414\) −1512.00 −0.179495
\(415\) 0 0
\(416\) 3542.00 0.417454
\(417\) −60.0000 −0.00704607
\(418\) 1040.00 0.121694
\(419\) −9700.00 −1.13097 −0.565484 0.824759i \(-0.691311\pi\)
−0.565484 + 0.824759i \(0.691311\pi\)
\(420\) 0 0
\(421\) 862.000 0.0997893 0.0498947 0.998754i \(-0.484111\pi\)
0.0498947 + 0.998754i \(0.484111\pi\)
\(422\) 4468.00 0.515400
\(423\) −2304.00 −0.264833
\(424\) 5070.00 0.580710
\(425\) 0 0
\(426\) −984.000 −0.111913
\(427\) 17808.0 2.01824
\(428\) 252.000 0.0284600
\(429\) 3432.00 0.386244
\(430\) 0 0
\(431\) 15792.0 1.76490 0.882452 0.470402i \(-0.155891\pi\)
0.882452 + 0.470402i \(0.155891\pi\)
\(432\) −1107.00 −0.123288
\(433\) −11602.0 −1.28766 −0.643830 0.765169i \(-0.722655\pi\)
−0.643830 + 0.765169i \(0.722655\pi\)
\(434\) 6912.00 0.764485
\(435\) 0 0
\(436\) 6790.00 0.745830
\(437\) −3360.00 −0.367805
\(438\) 114.000 0.0124364
\(439\) −440.000 −0.0478361 −0.0239181 0.999714i \(-0.507614\pi\)
−0.0239181 + 0.999714i \(0.507614\pi\)
\(440\) 0 0
\(441\) 2097.00 0.226433
\(442\) 308.000 0.0331449
\(443\) 10188.0 1.09266 0.546328 0.837571i \(-0.316025\pi\)
0.546328 + 0.837571i \(0.316025\pi\)
\(444\) 714.000 0.0763174
\(445\) 0 0
\(446\) 6032.00 0.640411
\(447\) 3990.00 0.422194
\(448\) −4008.00 −0.422679
\(449\) −13310.0 −1.39897 −0.699485 0.714647i \(-0.746587\pi\)
−0.699485 + 0.714647i \(0.746587\pi\)
\(450\) 0 0
\(451\) 6344.00 0.662367
\(452\) 7294.00 0.759029
\(453\) 3624.00 0.375873
\(454\) 2636.00 0.272497
\(455\) 0 0
\(456\) 900.000 0.0924262
\(457\) −3226.00 −0.330210 −0.165105 0.986276i \(-0.552796\pi\)
−0.165105 + 0.986276i \(0.552796\pi\)
\(458\) −4830.00 −0.492775
\(459\) −378.000 −0.0384391
\(460\) 0 0
\(461\) 6582.00 0.664977 0.332488 0.943107i \(-0.392112\pi\)
0.332488 + 0.943107i \(0.392112\pi\)
\(462\) 3744.00 0.377027
\(463\) −15072.0 −1.51286 −0.756431 0.654073i \(-0.773059\pi\)
−0.756431 + 0.654073i \(0.773059\pi\)
\(464\) 9430.00 0.943484
\(465\) 0 0
\(466\) 2682.00 0.266612
\(467\) −476.000 −0.0471663 −0.0235831 0.999722i \(-0.507507\pi\)
−0.0235831 + 0.999722i \(0.507507\pi\)
\(468\) 1386.00 0.136897
\(469\) 2016.00 0.198487
\(470\) 0 0
\(471\) −10542.0 −1.03132
\(472\) 1500.00 0.146278
\(473\) 9776.00 0.950319
\(474\) −720.000 −0.0697694
\(475\) 0 0
\(476\) −2352.00 −0.226478
\(477\) 3042.00 0.291999
\(478\) −2320.00 −0.221997
\(479\) −19680.0 −1.87725 −0.938624 0.344941i \(-0.887899\pi\)
−0.938624 + 0.344941i \(0.887899\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) −2002.00 −0.189188
\(483\) −12096.0 −1.13952
\(484\) −9611.00 −0.902611
\(485\) 0 0
\(486\) 243.000 0.0226805
\(487\) 5944.00 0.553077 0.276538 0.961003i \(-0.410813\pi\)
0.276538 + 0.961003i \(0.410813\pi\)
\(488\) 11130.0 1.03244
\(489\) −6204.00 −0.573731
\(490\) 0 0
\(491\) 10772.0 0.990089 0.495044 0.868868i \(-0.335152\pi\)
0.495044 + 0.868868i \(0.335152\pi\)
\(492\) 2562.00 0.234764
\(493\) 3220.00 0.294161
\(494\) −440.000 −0.0400740
\(495\) 0 0
\(496\) −11808.0 −1.06894
\(497\) −7872.00 −0.710478
\(498\) −3636.00 −0.327175
\(499\) 8140.00 0.730253 0.365127 0.930958i \(-0.381026\pi\)
0.365127 + 0.930958i \(0.381026\pi\)
\(500\) 0 0
\(501\) −72.0000 −0.00642060
\(502\) −132.000 −0.0117360
\(503\) 13768.0 1.22045 0.610223 0.792229i \(-0.291080\pi\)
0.610223 + 0.792229i \(0.291080\pi\)
\(504\) 3240.00 0.286351
\(505\) 0 0
\(506\) −8736.00 −0.767515
\(507\) 5139.00 0.450160
\(508\) 13552.0 1.18361
\(509\) 22150.0 1.92884 0.964422 0.264368i \(-0.0851633\pi\)
0.964422 + 0.264368i \(0.0851633\pi\)
\(510\) 0 0
\(511\) 912.000 0.0789521
\(512\) −11521.0 −0.994455
\(513\) 540.000 0.0464748
\(514\) −7614.00 −0.653384
\(515\) 0 0
\(516\) 3948.00 0.336824
\(517\) −13312.0 −1.13242
\(518\) −816.000 −0.0692143
\(519\) −1854.00 −0.156805
\(520\) 0 0
\(521\) 1562.00 0.131348 0.0656741 0.997841i \(-0.479080\pi\)
0.0656741 + 0.997841i \(0.479080\pi\)
\(522\) −2070.00 −0.173566
\(523\) 668.000 0.0558501 0.0279250 0.999610i \(-0.491110\pi\)
0.0279250 + 0.999610i \(0.491110\pi\)
\(524\) −5124.00 −0.427181
\(525\) 0 0
\(526\) −4888.00 −0.405184
\(527\) −4032.00 −0.333276
\(528\) −6396.00 −0.527178
\(529\) 16057.0 1.31972
\(530\) 0 0
\(531\) 900.000 0.0735531
\(532\) 3360.00 0.273824
\(533\) −2684.00 −0.218118
\(534\) 990.000 0.0802275
\(535\) 0 0
\(536\) 1260.00 0.101537
\(537\) −10020.0 −0.805205
\(538\) −1270.00 −0.101772
\(539\) 12116.0 0.968225
\(540\) 0 0
\(541\) −6138.00 −0.487788 −0.243894 0.969802i \(-0.578425\pi\)
−0.243894 + 0.969802i \(0.578425\pi\)
\(542\) −1072.00 −0.0849564
\(543\) 534.000 0.0422028
\(544\) −2254.00 −0.177646
\(545\) 0 0
\(546\) −1584.00 −0.124156
\(547\) 10484.0 0.819494 0.409747 0.912199i \(-0.365617\pi\)
0.409747 + 0.912199i \(0.365617\pi\)
\(548\) −15498.0 −1.20811
\(549\) 6678.00 0.519144
\(550\) 0 0
\(551\) −4600.00 −0.355656
\(552\) −7560.00 −0.582926
\(553\) −5760.00 −0.442930
\(554\) −5394.00 −0.413663
\(555\) 0 0
\(556\) −140.000 −0.0106786
\(557\) −3606.00 −0.274311 −0.137155 0.990550i \(-0.543796\pi\)
−0.137155 + 0.990550i \(0.543796\pi\)
\(558\) 2592.00 0.196645
\(559\) −4136.00 −0.312941
\(560\) 0 0
\(561\) −2184.00 −0.164365
\(562\) −2442.00 −0.183291
\(563\) −12252.0 −0.917159 −0.458579 0.888654i \(-0.651641\pi\)
−0.458579 + 0.888654i \(0.651641\pi\)
\(564\) −5376.00 −0.401366
\(565\) 0 0
\(566\) 2772.00 0.205858
\(567\) 1944.00 0.143986
\(568\) −4920.00 −0.363448
\(569\) −14550.0 −1.07200 −0.536000 0.844218i \(-0.680065\pi\)
−0.536000 + 0.844218i \(0.680065\pi\)
\(570\) 0 0
\(571\) −25468.0 −1.86655 −0.933277 0.359157i \(-0.883064\pi\)
−0.933277 + 0.359157i \(0.883064\pi\)
\(572\) 8008.00 0.585369
\(573\) 5664.00 0.412944
\(574\) −2928.00 −0.212914
\(575\) 0 0
\(576\) −1503.00 −0.108724
\(577\) −12866.0 −0.928282 −0.464141 0.885761i \(-0.653637\pi\)
−0.464141 + 0.885761i \(0.653637\pi\)
\(578\) 4717.00 0.339449
\(579\) 5766.00 0.413863
\(580\) 0 0
\(581\) −29088.0 −2.07706
\(582\) −2598.00 −0.185035
\(583\) 17576.0 1.24858
\(584\) 570.000 0.0403883
\(585\) 0 0
\(586\) 4542.00 0.320185
\(587\) 14844.0 1.04374 0.521872 0.853024i \(-0.325234\pi\)
0.521872 + 0.853024i \(0.325234\pi\)
\(588\) 4893.00 0.343170
\(589\) 5760.00 0.402948
\(590\) 0 0
\(591\) 7578.00 0.527440
\(592\) 1394.00 0.0967788
\(593\) −20402.0 −1.41283 −0.706416 0.707797i \(-0.749689\pi\)
−0.706416 + 0.707797i \(0.749689\pi\)
\(594\) 1404.00 0.0969812
\(595\) 0 0
\(596\) 9310.00 0.639853
\(597\) 3480.00 0.238571
\(598\) 3696.00 0.252744
\(599\) 10760.0 0.733959 0.366980 0.930229i \(-0.380392\pi\)
0.366980 + 0.930229i \(0.380392\pi\)
\(600\) 0 0
\(601\) 14282.0 0.969343 0.484671 0.874696i \(-0.338939\pi\)
0.484671 + 0.874696i \(0.338939\pi\)
\(602\) −4512.00 −0.305474
\(603\) 756.000 0.0510559
\(604\) 8456.00 0.569652
\(605\) 0 0
\(606\) −3654.00 −0.244940
\(607\) −11056.0 −0.739290 −0.369645 0.929173i \(-0.620521\pi\)
−0.369645 + 0.929173i \(0.620521\pi\)
\(608\) 3220.00 0.214783
\(609\) −16560.0 −1.10188
\(610\) 0 0
\(611\) 5632.00 0.372907
\(612\) −882.000 −0.0582561
\(613\) 16418.0 1.08176 0.540878 0.841101i \(-0.318092\pi\)
0.540878 + 0.841101i \(0.318092\pi\)
\(614\) 5116.00 0.336262
\(615\) 0 0
\(616\) 18720.0 1.22443
\(617\) 10374.0 0.676891 0.338445 0.940986i \(-0.390099\pi\)
0.338445 + 0.940986i \(0.390099\pi\)
\(618\) 264.000 0.0171839
\(619\) −5260.00 −0.341546 −0.170773 0.985310i \(-0.554627\pi\)
−0.170773 + 0.985310i \(0.554627\pi\)
\(620\) 0 0
\(621\) −4536.00 −0.293113
\(622\) 2808.00 0.181014
\(623\) 7920.00 0.509323
\(624\) 2706.00 0.173600
\(625\) 0 0
\(626\) −7318.00 −0.467230
\(627\) 3120.00 0.198725
\(628\) −24598.0 −1.56300
\(629\) 476.000 0.0301739
\(630\) 0 0
\(631\) 21352.0 1.34708 0.673542 0.739149i \(-0.264772\pi\)
0.673542 + 0.739149i \(0.264772\pi\)
\(632\) −3600.00 −0.226583
\(633\) 13404.0 0.841645
\(634\) 2246.00 0.140694
\(635\) 0 0
\(636\) 7098.00 0.442538
\(637\) −5126.00 −0.318838
\(638\) −11960.0 −0.742164
\(639\) −2952.00 −0.182753
\(640\) 0 0
\(641\) −29118.0 −1.79422 −0.897108 0.441812i \(-0.854336\pi\)
−0.897108 + 0.441812i \(0.854336\pi\)
\(642\) −108.000 −0.00663928
\(643\) −5772.00 −0.354005 −0.177003 0.984210i \(-0.556640\pi\)
−0.177003 + 0.984210i \(0.556640\pi\)
\(644\) −28224.0 −1.72699
\(645\) 0 0
\(646\) 280.000 0.0170533
\(647\) 14264.0 0.866732 0.433366 0.901218i \(-0.357326\pi\)
0.433366 + 0.901218i \(0.357326\pi\)
\(648\) 1215.00 0.0736570
\(649\) 5200.00 0.314511
\(650\) 0 0
\(651\) 20736.0 1.24840
\(652\) −14476.0 −0.869515
\(653\) −6902.00 −0.413623 −0.206812 0.978381i \(-0.566309\pi\)
−0.206812 + 0.978381i \(0.566309\pi\)
\(654\) −2910.00 −0.173991
\(655\) 0 0
\(656\) 5002.00 0.297706
\(657\) 342.000 0.0203085
\(658\) 6144.00 0.364009
\(659\) 20140.0 1.19051 0.595253 0.803539i \(-0.297052\pi\)
0.595253 + 0.803539i \(0.297052\pi\)
\(660\) 0 0
\(661\) −3218.00 −0.189358 −0.0946790 0.995508i \(-0.530182\pi\)
−0.0946790 + 0.995508i \(0.530182\pi\)
\(662\) −1332.00 −0.0782019
\(663\) 924.000 0.0541255
\(664\) −18180.0 −1.06253
\(665\) 0 0
\(666\) −306.000 −0.0178037
\(667\) 38640.0 2.24310
\(668\) −168.000 −0.00973071
\(669\) 18096.0 1.04579
\(670\) 0 0
\(671\) 38584.0 2.21985
\(672\) 11592.0 0.665433
\(673\) 7518.00 0.430606 0.215303 0.976547i \(-0.430926\pi\)
0.215303 + 0.976547i \(0.430926\pi\)
\(674\) −11534.0 −0.659159
\(675\) 0 0
\(676\) 11991.0 0.682237
\(677\) 18114.0 1.02833 0.514164 0.857692i \(-0.328102\pi\)
0.514164 + 0.857692i \(0.328102\pi\)
\(678\) −3126.00 −0.177070
\(679\) −20784.0 −1.17469
\(680\) 0 0
\(681\) 7908.00 0.444986
\(682\) 14976.0 0.840851
\(683\) 23868.0 1.33716 0.668582 0.743638i \(-0.266901\pi\)
0.668582 + 0.743638i \(0.266901\pi\)
\(684\) 1260.00 0.0704347
\(685\) 0 0
\(686\) 2640.00 0.146932
\(687\) −14490.0 −0.804699
\(688\) 7708.00 0.427129
\(689\) −7436.00 −0.411160
\(690\) 0 0
\(691\) 172.000 0.00946916 0.00473458 0.999989i \(-0.498493\pi\)
0.00473458 + 0.999989i \(0.498493\pi\)
\(692\) −4326.00 −0.237644
\(693\) 11232.0 0.615683
\(694\) 11956.0 0.653953
\(695\) 0 0
\(696\) −10350.0 −0.563672
\(697\) 1708.00 0.0928194
\(698\) −4870.00 −0.264086
\(699\) 8046.00 0.435376
\(700\) 0 0
\(701\) −22138.0 −1.19278 −0.596391 0.802694i \(-0.703399\pi\)
−0.596391 + 0.802694i \(0.703399\pi\)
\(702\) −594.000 −0.0319360
\(703\) −680.000 −0.0364818
\(704\) −8684.00 −0.464901
\(705\) 0 0
\(706\) 10722.0 0.571569
\(707\) −29232.0 −1.55500
\(708\) 2100.00 0.111473
\(709\) 3070.00 0.162618 0.0813091 0.996689i \(-0.474090\pi\)
0.0813091 + 0.996689i \(0.474090\pi\)
\(710\) 0 0
\(711\) −2160.00 −0.113933
\(712\) 4950.00 0.260546
\(713\) −48384.0 −2.54137
\(714\) 1008.00 0.0528340
\(715\) 0 0
\(716\) −23380.0 −1.22032
\(717\) −6960.00 −0.362519
\(718\) −120.000 −0.00623727
\(719\) 15600.0 0.809154 0.404577 0.914504i \(-0.367419\pi\)
0.404577 + 0.914504i \(0.367419\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 6459.00 0.332935
\(723\) −6006.00 −0.308943
\(724\) 1246.00 0.0639603
\(725\) 0 0
\(726\) 4119.00 0.210565
\(727\) −20696.0 −1.05581 −0.527904 0.849304i \(-0.677022\pi\)
−0.527904 + 0.849304i \(0.677022\pi\)
\(728\) −7920.00 −0.403207
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2632.00 0.133171
\(732\) 15582.0 0.786786
\(733\) 30778.0 1.55090 0.775451 0.631408i \(-0.217522\pi\)
0.775451 + 0.631408i \(0.217522\pi\)
\(734\) 3936.00 0.197930
\(735\) 0 0
\(736\) −27048.0 −1.35462
\(737\) 4368.00 0.218314
\(738\) −1098.00 −0.0547669
\(739\) 11740.0 0.584388 0.292194 0.956359i \(-0.405615\pi\)
0.292194 + 0.956359i \(0.405615\pi\)
\(740\) 0 0
\(741\) −1320.00 −0.0654405
\(742\) −8112.00 −0.401349
\(743\) −2632.00 −0.129958 −0.0649789 0.997887i \(-0.520698\pi\)
−0.0649789 + 0.997887i \(0.520698\pi\)
\(744\) 12960.0 0.638625
\(745\) 0 0
\(746\) 3022.00 0.148315
\(747\) −10908.0 −0.534274
\(748\) −5096.00 −0.249102
\(749\) −864.000 −0.0421494
\(750\) 0 0
\(751\) −20528.0 −0.997440 −0.498720 0.866763i \(-0.666196\pi\)
−0.498720 + 0.866763i \(0.666196\pi\)
\(752\) −10496.0 −0.508976
\(753\) −396.000 −0.0191647
\(754\) 5060.00 0.244396
\(755\) 0 0
\(756\) 4536.00 0.218218
\(757\) −21646.0 −1.03928 −0.519642 0.854384i \(-0.673934\pi\)
−0.519642 + 0.854384i \(0.673934\pi\)
\(758\) 13340.0 0.639222
\(759\) −26208.0 −1.25335
\(760\) 0 0
\(761\) 18282.0 0.870857 0.435428 0.900223i \(-0.356597\pi\)
0.435428 + 0.900223i \(0.356597\pi\)
\(762\) −5808.00 −0.276118
\(763\) −23280.0 −1.10458
\(764\) 13216.0 0.625835
\(765\) 0 0
\(766\) −1008.00 −0.0475464
\(767\) −2200.00 −0.103569
\(768\) 357.000 0.0167736
\(769\) −24190.0 −1.13435 −0.567174 0.823598i \(-0.691963\pi\)
−0.567174 + 0.823598i \(0.691963\pi\)
\(770\) 0 0
\(771\) −22842.0 −1.06697
\(772\) 13454.0 0.627228
\(773\) 25698.0 1.19572 0.597861 0.801600i \(-0.296018\pi\)
0.597861 + 0.801600i \(0.296018\pi\)
\(774\) −1692.00 −0.0785758
\(775\) 0 0
\(776\) −12990.0 −0.600920
\(777\) −2448.00 −0.113026
\(778\) −9630.00 −0.443769
\(779\) −2440.00 −0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) −2352.00 −0.107554
\(783\) −6210.00 −0.283432
\(784\) 9553.00 0.435177
\(785\) 0 0
\(786\) 2196.00 0.0996549
\(787\) −33436.0 −1.51444 −0.757220 0.653160i \(-0.773443\pi\)
−0.757220 + 0.653160i \(0.773443\pi\)
\(788\) 17682.0 0.799359
\(789\) −14664.0 −0.661663
\(790\) 0 0
\(791\) −25008.0 −1.12412
\(792\) 7020.00 0.314956
\(793\) −16324.0 −0.730999
\(794\) 7126.00 0.318504
\(795\) 0 0
\(796\) 8120.00 0.361565
\(797\) 37594.0 1.67083 0.835413 0.549623i \(-0.185229\pi\)
0.835413 + 0.549623i \(0.185229\pi\)
\(798\) −1440.00 −0.0638790
\(799\) −3584.00 −0.158689
\(800\) 0 0
\(801\) 2970.00 0.131011
\(802\) 8718.00 0.383844
\(803\) 1976.00 0.0868388
\(804\) 1764.00 0.0773775
\(805\) 0 0
\(806\) −6336.00 −0.276893
\(807\) −3810.00 −0.166194
\(808\) −18270.0 −0.795466
\(809\) 4730.00 0.205560 0.102780 0.994704i \(-0.467226\pi\)
0.102780 + 0.994704i \(0.467226\pi\)
\(810\) 0 0
\(811\) −8748.00 −0.378772 −0.189386 0.981903i \(-0.560650\pi\)
−0.189386 + 0.981903i \(0.560650\pi\)
\(812\) −38640.0 −1.66995
\(813\) −3216.00 −0.138733
\(814\) −1768.00 −0.0761282
\(815\) 0 0
\(816\) −1722.00 −0.0738751
\(817\) −3760.00 −0.161011
\(818\) 10870.0 0.464622
\(819\) −4752.00 −0.202745
\(820\) 0 0
\(821\) 44142.0 1.87645 0.938226 0.346024i \(-0.112468\pi\)
0.938226 + 0.346024i \(0.112468\pi\)
\(822\) 6642.00 0.281833
\(823\) −3992.00 −0.169079 −0.0845397 0.996420i \(-0.526942\pi\)
−0.0845397 + 0.996420i \(0.526942\pi\)
\(824\) 1320.00 0.0558063
\(825\) 0 0
\(826\) −2400.00 −0.101098
\(827\) 14444.0 0.607336 0.303668 0.952778i \(-0.401789\pi\)
0.303668 + 0.952778i \(0.401789\pi\)
\(828\) −10584.0 −0.444226
\(829\) 42150.0 1.76590 0.882949 0.469468i \(-0.155554\pi\)
0.882949 + 0.469468i \(0.155554\pi\)
\(830\) 0 0
\(831\) −16182.0 −0.675508
\(832\) 3674.00 0.153093
\(833\) 3262.00 0.135680
\(834\) 60.0000 0.00249116
\(835\) 0 0
\(836\) 7280.00 0.301177
\(837\) 7776.00 0.321121
\(838\) 9700.00 0.399858
\(839\) 13400.0 0.551394 0.275697 0.961245i \(-0.411091\pi\)
0.275697 + 0.961245i \(0.411091\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) −862.000 −0.0352809
\(843\) −7326.00 −0.299313
\(844\) 31276.0 1.27555
\(845\) 0 0
\(846\) 2304.00 0.0936326
\(847\) 32952.0 1.33677
\(848\) 13858.0 0.561186
\(849\) 8316.00 0.336165
\(850\) 0 0
\(851\) 5712.00 0.230088
\(852\) −6888.00 −0.276971
\(853\) 8658.00 0.347531 0.173766 0.984787i \(-0.444406\pi\)
0.173766 + 0.984787i \(0.444406\pi\)
\(854\) −17808.0 −0.713556
\(855\) 0 0
\(856\) −540.000 −0.0215617
\(857\) −42826.0 −1.70701 −0.853505 0.521084i \(-0.825528\pi\)
−0.853505 + 0.521084i \(0.825528\pi\)
\(858\) −3432.00 −0.136558
\(859\) −35900.0 −1.42595 −0.712976 0.701189i \(-0.752653\pi\)
−0.712976 + 0.701189i \(0.752653\pi\)
\(860\) 0 0
\(861\) −8784.00 −0.347686
\(862\) −15792.0 −0.623988
\(863\) 3088.00 0.121804 0.0609019 0.998144i \(-0.480602\pi\)
0.0609019 + 0.998144i \(0.480602\pi\)
\(864\) 4347.00 0.171167
\(865\) 0 0
\(866\) 11602.0 0.455256
\(867\) 14151.0 0.554317
\(868\) 48384.0 1.89200
\(869\) −12480.0 −0.487175
\(870\) 0 0
\(871\) −1848.00 −0.0718910
\(872\) −14550.0 −0.565052
\(873\) −7794.00 −0.302161
\(874\) 3360.00 0.130039
\(875\) 0 0
\(876\) 798.000 0.0307784
\(877\) 35274.0 1.35817 0.679087 0.734058i \(-0.262376\pi\)
0.679087 + 0.734058i \(0.262376\pi\)
\(878\) 440.000 0.0169126
\(879\) 13626.0 0.522860
\(880\) 0 0
\(881\) 25042.0 0.957646 0.478823 0.877911i \(-0.341064\pi\)
0.478823 + 0.877911i \(0.341064\pi\)
\(882\) −2097.00 −0.0800563
\(883\) −12572.0 −0.479141 −0.239570 0.970879i \(-0.577007\pi\)
−0.239570 + 0.970879i \(0.577007\pi\)
\(884\) 2156.00 0.0820296
\(885\) 0 0
\(886\) −10188.0 −0.386312
\(887\) 21864.0 0.827645 0.413823 0.910358i \(-0.364193\pi\)
0.413823 + 0.910358i \(0.364193\pi\)
\(888\) −1530.00 −0.0578192
\(889\) −46464.0 −1.75293
\(890\) 0 0
\(891\) 4212.00 0.158370
\(892\) 42224.0 1.58494
\(893\) 5120.00 0.191864
\(894\) −3990.00 −0.149268
\(895\) 0 0
\(896\) 34920.0 1.30200
\(897\) 11088.0 0.412729
\(898\) 13310.0 0.494611
\(899\) −66240.0 −2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) −6344.00 −0.234182
\(903\) −13536.0 −0.498837
\(904\) −15630.0 −0.575051
\(905\) 0 0
\(906\) −3624.00 −0.132891
\(907\) −31236.0 −1.14352 −0.571761 0.820420i \(-0.693740\pi\)
−0.571761 + 0.820420i \(0.693740\pi\)
\(908\) 18452.0 0.674396
\(909\) −10962.0 −0.399985
\(910\) 0 0
\(911\) 8272.00 0.300838 0.150419 0.988622i \(-0.451938\pi\)
0.150419 + 0.988622i \(0.451938\pi\)
\(912\) 2460.00 0.0893188
\(913\) −63024.0 −2.28455
\(914\) 3226.00 0.116747
\(915\) 0 0
\(916\) −33810.0 −1.21956
\(917\) 17568.0 0.632657
\(918\) 378.000 0.0135903
\(919\) 20200.0 0.725067 0.362533 0.931971i \(-0.381912\pi\)
0.362533 + 0.931971i \(0.381912\pi\)
\(920\) 0 0
\(921\) 15348.0 0.549114
\(922\) −6582.00 −0.235105
\(923\) 7216.00 0.257332
\(924\) 26208.0 0.933095
\(925\) 0 0
\(926\) 15072.0 0.534878
\(927\) 792.000 0.0280612
\(928\) −37030.0 −1.30988
\(929\) 31010.0 1.09516 0.547581 0.836753i \(-0.315549\pi\)
0.547581 + 0.836753i \(0.315549\pi\)
\(930\) 0 0
\(931\) −4660.00 −0.164044
\(932\) 18774.0 0.659831
\(933\) 8424.00 0.295594
\(934\) 476.000 0.0166758
\(935\) 0 0
\(936\) −2970.00 −0.103715
\(937\) 39174.0 1.36580 0.682902 0.730510i \(-0.260717\pi\)
0.682902 + 0.730510i \(0.260717\pi\)
\(938\) −2016.00 −0.0701756
\(939\) −21954.0 −0.762984
\(940\) 0 0
\(941\) −4138.00 −0.143353 −0.0716764 0.997428i \(-0.522835\pi\)
−0.0716764 + 0.997428i \(0.522835\pi\)
\(942\) 10542.0 0.364625
\(943\) 20496.0 0.707785
\(944\) 4100.00 0.141360
\(945\) 0 0
\(946\) −9776.00 −0.335989
\(947\) −23676.0 −0.812425 −0.406213 0.913779i \(-0.633151\pi\)
−0.406213 + 0.913779i \(0.633151\pi\)
\(948\) −5040.00 −0.172670
\(949\) −836.000 −0.0285961
\(950\) 0 0
\(951\) 6738.00 0.229752
\(952\) 5040.00 0.171583
\(953\) −18922.0 −0.643173 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(954\) −3042.00 −0.103237
\(955\) 0 0
\(956\) −16240.0 −0.549413
\(957\) −35880.0 −1.21195
\(958\) 19680.0 0.663708
\(959\) 53136.0 1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 748.000 0.0250691
\(963\) −324.000 −0.0108419
\(964\) −14014.0 −0.468216
\(965\) 0 0
\(966\) 12096.0 0.402880
\(967\) −39656.0 −1.31877 −0.659385 0.751805i \(-0.729183\pi\)
−0.659385 + 0.751805i \(0.729183\pi\)
\(968\) 20595.0 0.683831
\(969\) 840.000 0.0278480
\(970\) 0 0
\(971\) −33228.0 −1.09818 −0.549092 0.835762i \(-0.685026\pi\)
−0.549092 + 0.835762i \(0.685026\pi\)
\(972\) 1701.00 0.0561313
\(973\) 480.000 0.0158151
\(974\) −5944.00 −0.195542
\(975\) 0 0
\(976\) 30422.0 0.997730
\(977\) 974.000 0.0318946 0.0159473 0.999873i \(-0.494924\pi\)
0.0159473 + 0.999873i \(0.494924\pi\)
\(978\) 6204.00 0.202845
\(979\) 17160.0 0.560200
\(980\) 0 0
\(981\) −8730.00 −0.284126
\(982\) −10772.0 −0.350049
\(983\) 13608.0 0.441534 0.220767 0.975327i \(-0.429144\pi\)
0.220767 + 0.975327i \(0.429144\pi\)
\(984\) −5490.00 −0.177861
\(985\) 0 0
\(986\) −3220.00 −0.104002
\(987\) 18432.0 0.594425
\(988\) −3080.00 −0.0991780
\(989\) 31584.0 1.01548
\(990\) 0 0
\(991\) 13472.0 0.431839 0.215919 0.976411i \(-0.430725\pi\)
0.215919 + 0.976411i \(0.430725\pi\)
\(992\) 46368.0 1.48406
\(993\) −3996.00 −0.127703
\(994\) 7872.00 0.251192
\(995\) 0 0
\(996\) −25452.0 −0.809716
\(997\) 3234.00 0.102730 0.0513650 0.998680i \(-0.483643\pi\)
0.0513650 + 0.998680i \(0.483643\pi\)
\(998\) −8140.00 −0.258184
\(999\) −918.000 −0.0290733
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 75.4.a.b.1.1 1
3.2 odd 2 225.4.a.f.1.1 1
4.3 odd 2 1200.4.a.t.1.1 1
5.2 odd 4 75.4.b.b.49.1 2
5.3 odd 4 75.4.b.b.49.2 2
5.4 even 2 15.4.a.a.1.1 1
15.2 even 4 225.4.b.e.199.2 2
15.8 even 4 225.4.b.e.199.1 2
15.14 odd 2 45.4.a.c.1.1 1
20.3 even 4 1200.4.f.b.49.2 2
20.7 even 4 1200.4.f.b.49.1 2
20.19 odd 2 240.4.a.e.1.1 1
35.34 odd 2 735.4.a.e.1.1 1
40.19 odd 2 960.4.a.ba.1.1 1
40.29 even 2 960.4.a.b.1.1 1
45.4 even 6 405.4.e.g.136.1 2
45.14 odd 6 405.4.e.i.136.1 2
45.29 odd 6 405.4.e.i.271.1 2
45.34 even 6 405.4.e.g.271.1 2
55.54 odd 2 1815.4.a.e.1.1 1
60.59 even 2 720.4.a.n.1.1 1
105.104 even 2 2205.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 5.4 even 2
45.4.a.c.1.1 1 15.14 odd 2
75.4.a.b.1.1 1 1.1 even 1 trivial
75.4.b.b.49.1 2 5.2 odd 4
75.4.b.b.49.2 2 5.3 odd 4
225.4.a.f.1.1 1 3.2 odd 2
225.4.b.e.199.1 2 15.8 even 4
225.4.b.e.199.2 2 15.2 even 4
240.4.a.e.1.1 1 20.19 odd 2
405.4.e.g.136.1 2 45.4 even 6
405.4.e.g.271.1 2 45.34 even 6
405.4.e.i.136.1 2 45.14 odd 6
405.4.e.i.271.1 2 45.29 odd 6
720.4.a.n.1.1 1 60.59 even 2
735.4.a.e.1.1 1 35.34 odd 2
960.4.a.b.1.1 1 40.29 even 2
960.4.a.ba.1.1 1 40.19 odd 2
1200.4.a.t.1.1 1 4.3 odd 2
1200.4.f.b.49.1 2 20.7 even 4
1200.4.f.b.49.2 2 20.3 even 4
1815.4.a.e.1.1 1 55.54 odd 2
2205.4.a.l.1.1 1 105.104 even 2