Properties

Label 45.4.a.c.1.1
Level $45$
Weight $4$
Character 45.1
Self dual yes
Analytic conductor $2.655$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 45 = 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 45.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.65508595026\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 45.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -24.0000 q^{7} +15.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -7.00000 q^{4} -5.00000 q^{5} -24.0000 q^{7} +15.0000 q^{8} +5.00000 q^{10} -52.0000 q^{11} +22.0000 q^{13} +24.0000 q^{14} +41.0000 q^{16} +14.0000 q^{17} -20.0000 q^{19} +35.0000 q^{20} +52.0000 q^{22} +168.000 q^{23} +25.0000 q^{25} -22.0000 q^{26} +168.000 q^{28} -230.000 q^{29} -288.000 q^{31} -161.000 q^{32} -14.0000 q^{34} +120.000 q^{35} -34.0000 q^{37} +20.0000 q^{38} -75.0000 q^{40} -122.000 q^{41} -188.000 q^{43} +364.000 q^{44} -168.000 q^{46} -256.000 q^{47} +233.000 q^{49} -25.0000 q^{50} -154.000 q^{52} +338.000 q^{53} +260.000 q^{55} -360.000 q^{56} +230.000 q^{58} -100.000 q^{59} +742.000 q^{61} +288.000 q^{62} -167.000 q^{64} -110.000 q^{65} -84.0000 q^{67} -98.0000 q^{68} -120.000 q^{70} +328.000 q^{71} -38.0000 q^{73} +34.0000 q^{74} +140.000 q^{76} +1248.00 q^{77} -240.000 q^{79} -205.000 q^{80} +122.000 q^{82} -1212.00 q^{83} -70.0000 q^{85} +188.000 q^{86} -780.000 q^{88} -330.000 q^{89} -528.000 q^{91} -1176.00 q^{92} +256.000 q^{94} +100.000 q^{95} +866.000 q^{97} -233.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) −24.0000 −1.29588 −0.647939 0.761692i \(-0.724369\pi\)
−0.647939 + 0.761692i \(0.724369\pi\)
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) 5.00000 0.158114
\(11\) −52.0000 −1.42533 −0.712663 0.701506i \(-0.752511\pi\)
−0.712663 + 0.701506i \(0.752511\pi\)
\(12\) 0 0
\(13\) 22.0000 0.469362 0.234681 0.972072i \(-0.424595\pi\)
0.234681 + 0.972072i \(0.424595\pi\)
\(14\) 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 \(0.468158\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 35.0000 0.391312
\(21\) 0 0
\(22\) 52.0000 0.503929
\(23\) 168.000 1.52306 0.761531 0.648129i \(-0.224448\pi\)
0.761531 + 0.648129i \(0.224448\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −22.0000 −0.165944
\(27\) 0 0
\(28\) 168.000 1.13389
\(29\) −230.000 −1.47276 −0.736378 0.676570i \(-0.763465\pi\)
−0.736378 + 0.676570i \(0.763465\pi\)
\(30\) 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
\(33\) 0 0
\(34\) −14.0000 −0.0706171
\(35\) 120.000 0.579534
\(36\) 0 0
\(37\) −34.0000 −0.151069 −0.0755347 0.997143i \(-0.524066\pi\)
−0.0755347 + 0.997143i \(0.524066\pi\)
\(38\) 20.0000 0.0853797
\(39\) 0 0
\(40\) −75.0000 −0.296464
\(41\) −122.000 −0.464712 −0.232356 0.972631i \(-0.574643\pi\)
−0.232356 + 0.972631i \(0.574643\pi\)
\(42\) 0 0
\(43\) −188.000 −0.666738 −0.333369 0.942796i \(-0.608185\pi\)
−0.333369 + 0.942796i \(0.608185\pi\)
\(44\) 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 \(0.630035\pi\)
\(48\) 0 0
\(49\) 233.000 0.679300
\(50\) −25.0000 −0.0707107
\(51\) 0 0
\(52\) −154.000 −0.410691
\(53\) 338.000 0.875998 0.437999 0.898976i \(-0.355687\pi\)
0.437999 + 0.898976i \(0.355687\pi\)
\(54\) 0 0
\(55\) 260.000 0.637425
\(56\) −360.000 −0.859054
\(57\) 0 0
\(58\) 230.000 0.520698
\(59\) −100.000 −0.220659 −0.110330 0.993895i \(-0.535191\pi\)
−0.110330 + 0.993895i \(0.535191\pi\)
\(60\) 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
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) −110.000 −0.209905
\(66\) 0 0
\(67\) −84.0000 −0.153168 −0.0765838 0.997063i \(-0.524401\pi\)
−0.0765838 + 0.997063i \(0.524401\pi\)
\(68\) −98.0000 −0.174768
\(69\) 0 0
\(70\) −120.000 −0.204896
\(71\) 328.000 0.548260 0.274130 0.961693i \(-0.411610\pi\)
0.274130 + 0.961693i \(0.411610\pi\)
\(72\) 0 0
\(73\) −38.0000 −0.0609255 −0.0304628 0.999536i \(-0.509698\pi\)
−0.0304628 + 0.999536i \(0.509698\pi\)
\(74\) 34.0000 0.0534111
\(75\) 0 0
\(76\) 140.000 0.211304
\(77\) 1248.00 1.84705
\(78\) 0 0
\(79\) −240.000 −0.341799 −0.170899 0.985288i \(-0.554667\pi\)
−0.170899 + 0.985288i \(0.554667\pi\)
\(80\) −205.000 −0.286496
\(81\) 0 0
\(82\) 122.000 0.164301
\(83\) −1212.00 −1.60282 −0.801411 0.598114i \(-0.795917\pi\)
−0.801411 + 0.598114i \(0.795917\pi\)
\(84\) 0 0
\(85\) −70.0000 −0.0893243
\(86\) 188.000 0.235727
\(87\) 0 0
\(88\) −780.000 −0.944867
\(89\) −330.000 −0.393033 −0.196516 0.980501i \(-0.562963\pi\)
−0.196516 + 0.980501i \(0.562963\pi\)
\(90\) 0 0
\(91\) −528.000 −0.608236
\(92\) −1176.00 −1.33268
\(93\) 0 0
\(94\) 256.000 0.280898
\(95\) 100.000 0.107998
\(96\) 0 0
\(97\) 866.000 0.906484 0.453242 0.891387i \(-0.350267\pi\)
0.453242 + 0.891387i \(0.350267\pi\)
\(98\) −233.000 −0.240169
\(99\) 0 0
\(100\) −175.000 −0.175000
\(101\) 1218.00 1.19996 0.599978 0.800017i \(-0.295176\pi\)
0.599978 + 0.800017i \(0.295176\pi\)
\(102\) 0 0
\(103\) −88.0000 −0.0841835 −0.0420917 0.999114i \(-0.513402\pi\)
−0.0420917 + 0.999114i \(0.513402\pi\)
\(104\) 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 \(0.505177\pi\)
\(108\) 0 0
\(109\) −970.000 −0.852378 −0.426189 0.904634i \(-0.640144\pi\)
−0.426189 + 0.904634i \(0.640144\pi\)
\(110\) −260.000 −0.225364
\(111\) 0 0
\(112\) −984.000 −0.830172
\(113\) −1042.00 −0.867461 −0.433731 0.901043i \(-0.642803\pi\)
−0.433731 + 0.901043i \(0.642803\pi\)
\(114\) 0 0
\(115\) −840.000 −0.681134
\(116\) 1610.00 1.28866
\(117\) 0 0
\(118\) 100.000 0.0780148
\(119\) −336.000 −0.258833
\(120\) 0 0
\(121\) 1373.00 1.03156
\(122\) −742.000 −0.550635
\(123\) 0 0
\(124\) 2016.00 1.46002
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) 1936.00 1.35269 0.676347 0.736583i \(-0.263562\pi\)
0.676347 + 0.736583i \(0.263562\pi\)
\(128\) 1455.00 1.00473
\(129\) 0 0
\(130\) 110.000 0.0742126
\(131\) −732.000 −0.488207 −0.244104 0.969749i \(-0.578494\pi\)
−0.244104 + 0.969749i \(0.578494\pi\)
\(132\) 0 0
\(133\) 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 \(0.257458\pi\)
\(138\) 0 0
\(139\) 20.0000 0.0122042 0.00610208 0.999981i \(-0.498058\pi\)
0.00610208 + 0.999981i \(0.498058\pi\)
\(140\) −840.000 −0.507093
\(141\) 0 0
\(142\) −328.000 −0.193839
\(143\) −1144.00 −0.668994
\(144\) 0 0
\(145\) 1150.00 0.658637
\(146\) 38.0000 0.0215404
\(147\) 0 0
\(148\) 238.000 0.132186
\(149\) 1330.00 0.731261 0.365630 0.930760i \(-0.380853\pi\)
0.365630 + 0.930760i \(0.380853\pi\)
\(150\) 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
\(153\) 0 0
\(154\) −1248.00 −0.653031
\(155\) 1440.00 0.746217
\(156\) 0 0
\(157\) −3514.00 −1.78629 −0.893146 0.449768i \(-0.851507\pi\)
−0.893146 + 0.449768i \(0.851507\pi\)
\(158\) 240.000 0.120844
\(159\) 0 0
\(160\) 805.000 0.397755
\(161\) −4032.00 −1.97370
\(162\) 0 0
\(163\) −2068.00 −0.993732 −0.496866 0.867827i \(-0.665516\pi\)
−0.496866 + 0.867827i \(0.665516\pi\)
\(164\) 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 \(0.498230\pi\)
\(168\) 0 0
\(169\) −1713.00 −0.779700
\(170\) 70.0000 0.0315809
\(171\) 0 0
\(172\) 1316.00 0.583396
\(173\) 618.000 0.271593 0.135797 0.990737i \(-0.456641\pi\)
0.135797 + 0.990737i \(0.456641\pi\)
\(174\) 0 0
\(175\) −600.000 −0.259176
\(176\) −2132.00 −0.913100
\(177\) 0 0
\(178\) 330.000 0.138958
\(179\) −3340.00 −1.39466 −0.697328 0.716752i \(-0.745628\pi\)
−0.697328 + 0.716752i \(0.745628\pi\)
\(180\) 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
\(183\) 0 0
\(184\) 2520.00 1.00966
\(185\) 170.000 0.0675603
\(186\) 0 0
\(187\) −728.000 −0.284688
\(188\) 1792.00 0.695186
\(189\) 0 0
\(190\) −100.000 −0.0381830
\(191\) 1888.00 0.715240 0.357620 0.933867i \(-0.383588\pi\)
0.357620 + 0.933867i \(0.383588\pi\)
\(192\) 0 0
\(193\) 1922.00 0.716832 0.358416 0.933562i \(-0.383317\pi\)
0.358416 + 0.933562i \(0.383317\pi\)
\(194\) −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 \(0.650996\pi\)
\(198\) 0 0
\(199\) −1160.00 −0.413217 −0.206609 0.978424i \(-0.566243\pi\)
−0.206609 + 0.978424i \(0.566243\pi\)
\(200\) 375.000 0.132583
\(201\) 0 0
\(202\) −1218.00 −0.424248
\(203\) 5520.00 1.90851
\(204\) 0 0
\(205\) 610.000 0.207826
\(206\) 88.0000 0.0297634
\(207\) 0 0
\(208\) 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
\(213\) 0 0
\(214\) 36.0000 0.0114996
\(215\) 940.000 0.298174
\(216\) 0 0
\(217\) 6912.00 2.16229
\(218\) 970.000 0.301361
\(219\) 0 0
\(220\) −1820.00 −0.557747
\(221\) 308.000 0.0937481
\(222\) 0 0
\(223\) 6032.00 1.81136 0.905678 0.423965i \(-0.139362\pi\)
0.905678 + 0.423965i \(0.139362\pi\)
\(224\) 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 \(0.625926\pi\)
\(228\) 0 0
\(229\) 4830.00 1.39378 0.696889 0.717179i \(-0.254567\pi\)
0.696889 + 0.717179i \(0.254567\pi\)
\(230\) 840.000 0.240817
\(231\) 0 0
\(232\) −3450.00 −0.976309
\(233\) −2682.00 −0.754093 −0.377046 0.926194i \(-0.623060\pi\)
−0.377046 + 0.926194i \(0.623060\pi\)
\(234\) 0 0
\(235\) 1280.00 0.355311
\(236\) 700.000 0.193077
\(237\) 0 0
\(238\) 336.000 0.0915111
\(239\) −2320.00 −0.627901 −0.313950 0.949439i \(-0.601653\pi\)
−0.313950 + 0.949439i \(0.601653\pi\)
\(240\) 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
\(243\) 0 0
\(244\) −5194.00 −1.36275
\(245\) −1165.00 −0.303792
\(246\) 0 0
\(247\) −440.000 −0.113346
\(248\) −4320.00 −1.10613
\(249\) 0 0
\(250\) 125.000 0.0316228
\(251\) −132.000 −0.0331943 −0.0165971 0.999862i \(-0.505283\pi\)
−0.0165971 + 0.999862i \(0.505283\pi\)
\(252\) 0 0
\(253\) −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 \(0.124880\pi\)
\(258\) 0 0
\(259\) 816.000 0.195767
\(260\) 770.000 0.183667
\(261\) 0 0
\(262\) 732.000 0.172607
\(263\) 4888.00 1.14603 0.573017 0.819543i \(-0.305773\pi\)
0.573017 + 0.819543i \(0.305773\pi\)
\(264\) 0 0
\(265\) −1690.00 −0.391758
\(266\) −480.000 −0.110642
\(267\) 0 0
\(268\) 588.000 0.134022
\(269\) −1270.00 −0.287856 −0.143928 0.989588i \(-0.545973\pi\)
−0.143928 + 0.989588i \(0.545973\pi\)
\(270\) 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
\(273\) 0 0
\(274\) −2214.00 −0.488148
\(275\) −1300.00 −0.285065
\(276\) 0 0
\(277\) −5394.00 −1.17001 −0.585007 0.811028i \(-0.698908\pi\)
−0.585007 + 0.811028i \(0.698908\pi\)
\(278\) −20.0000 −0.00431482
\(279\) 0 0
\(280\) 1800.00 0.384181
\(281\) −2442.00 −0.518425 −0.259213 0.965820i \(-0.583463\pi\)
−0.259213 + 0.965820i \(0.583463\pi\)
\(282\) 0 0
\(283\) 2772.00 0.582255 0.291128 0.956684i \(-0.405970\pi\)
0.291128 + 0.956684i \(0.405970\pi\)
\(284\) −2296.00 −0.479727
\(285\) 0 0
\(286\) 1144.00 0.236525
\(287\) 2928.00 0.602210
\(288\) 0 0
\(289\) −4717.00 −0.960106
\(290\) −1150.00 −0.232863
\(291\) 0 0
\(292\) 266.000 0.0533098
\(293\) −4542.00 −0.905619 −0.452810 0.891607i \(-0.649578\pi\)
−0.452810 + 0.891607i \(0.649578\pi\)
\(294\) 0 0
\(295\) 500.000 0.0986818
\(296\) −510.000 −0.100146
\(297\) 0 0
\(298\) −1330.00 −0.258540
\(299\) 3696.00 0.714867
\(300\) 0 0
\(301\) 4512.00 0.864011
\(302\) 1208.00 0.230174
\(303\) 0 0
\(304\) −820.000 −0.154705
\(305\) −3710.00 −0.696505
\(306\) 0 0
\(307\) 5116.00 0.951093 0.475546 0.879691i \(-0.342250\pi\)
0.475546 + 0.879691i \(0.342250\pi\)
\(308\) −8736.00 −1.61617
\(309\) 0 0
\(310\) −1440.00 −0.263827
\(311\) 2808.00 0.511984 0.255992 0.966679i \(-0.417598\pi\)
0.255992 + 0.966679i \(0.417598\pi\)
\(312\) 0 0
\(313\) −7318.00 −1.32153 −0.660763 0.750594i \(-0.729767\pi\)
−0.660763 + 0.750594i \(0.729767\pi\)
\(314\) 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 \(0.563760\pi\)
\(318\) 0 0
\(319\) 11960.0 2.09916
\(320\) 835.000 0.145868
\(321\) 0 0
\(322\) 4032.00 0.697809
\(323\) −280.000 −0.0482341
\(324\) 0 0
\(325\) 550.000 0.0938723
\(326\) 2068.00 0.351337
\(327\) 0 0
\(328\) −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
\(333\) 0 0
\(334\) −24.0000 −0.00393180
\(335\) 420.000 0.0684987
\(336\) 0 0
\(337\) −11534.0 −1.86438 −0.932191 0.361966i \(-0.882106\pi\)
−0.932191 + 0.361966i \(0.882106\pi\)
\(338\) 1713.00 0.275665
\(339\) 0 0
\(340\) 490.000 0.0781588
\(341\) 14976.0 2.37829
\(342\) 0 0
\(343\) 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 \(0.875793\pi\)
\(348\) 0 0
\(349\) 4870.00 0.746949 0.373474 0.927640i \(-0.378166\pi\)
0.373474 + 0.927640i \(0.378166\pi\)
\(350\) 600.000 0.0916324
\(351\) 0 0
\(352\) 8372.00 1.26770
\(353\) −10722.0 −1.61664 −0.808321 0.588742i \(-0.799623\pi\)
−0.808321 + 0.588742i \(0.799623\pi\)
\(354\) 0 0
\(355\) −1640.00 −0.245189
\(356\) 2310.00 0.343904
\(357\) 0 0
\(358\) 3340.00 0.493085
\(359\) −120.000 −0.0176417 −0.00882083 0.999961i \(-0.502808\pi\)
−0.00882083 + 0.999961i \(0.502808\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) 178.000 0.0258438
\(363\) 0 0
\(364\) 3696.00 0.532206
\(365\) 190.000 0.0272467
\(366\) 0 0
\(367\) 3936.00 0.559830 0.279915 0.960025i \(-0.409694\pi\)
0.279915 + 0.960025i \(0.409694\pi\)
\(368\) 6888.00 0.975711
\(369\) 0 0
\(370\) −170.000 −0.0238862
\(371\) −8112.00 −1.13519
\(372\) 0 0
\(373\) 3022.00 0.419499 0.209750 0.977755i \(-0.432735\pi\)
0.209750 + 0.977755i \(0.432735\pi\)
\(374\) 728.000 0.100652
\(375\) 0 0
\(376\) −3840.00 −0.526683
\(377\) −5060.00 −0.691255
\(378\) 0 0
\(379\) −13340.0 −1.80799 −0.903997 0.427539i \(-0.859381\pi\)
−0.903997 + 0.427539i \(0.859381\pi\)
\(380\) −700.000 −0.0944980
\(381\) 0 0
\(382\) −1888.00 −0.252876
\(383\) 1008.00 0.134481 0.0672407 0.997737i \(-0.478580\pi\)
0.0672407 + 0.997737i \(0.478580\pi\)
\(384\) 0 0
\(385\) −6240.00 −0.826026
\(386\) −1922.00 −0.253438
\(387\) 0 0
\(388\) −6062.00 −0.793174
\(389\) −9630.00 −1.25517 −0.627584 0.778549i \(-0.715956\pi\)
−0.627584 + 0.778549i \(0.715956\pi\)
\(390\) 0 0
\(391\) 2352.00 0.304209
\(392\) 3495.00 0.450317
\(393\) 0 0
\(394\) 2526.00 0.322990
\(395\) 1200.00 0.152857
\(396\) 0 0
\(397\) 7126.00 0.900866 0.450433 0.892810i \(-0.351270\pi\)
0.450433 + 0.892810i \(0.351270\pi\)
\(398\) 1160.00 0.146094
\(399\) 0 0
\(400\) 1025.00 0.128125
\(401\) 8718.00 1.08568 0.542838 0.839837i \(-0.317350\pi\)
0.542838 + 0.839837i \(0.317350\pi\)
\(402\) 0 0
\(403\) −6336.00 −0.783173
\(404\) −8526.00 −1.04996
\(405\) 0 0
\(406\) −5520.00 −0.674761
\(407\) 1768.00 0.215323
\(408\) 0 0
\(409\) −10870.0 −1.31415 −0.657074 0.753826i \(-0.728206\pi\)
−0.657074 + 0.753826i \(0.728206\pi\)
\(410\) −610.000 −0.0734774
\(411\) 0 0
\(412\) 616.000 0.0736605
\(413\) 2400.00 0.285947
\(414\) 0 0
\(415\) 6060.00 0.716804
\(416\) −3542.00 −0.417454
\(417\) 0 0
\(418\) −1040.00 −0.121694
\(419\) 9700.00 1.13097 0.565484 0.824759i \(-0.308689\pi\)
0.565484 + 0.824759i \(0.308689\pi\)
\(420\) 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
\(423\) 0 0
\(424\) 5070.00 0.580710
\(425\) 350.000 0.0399470
\(426\) 0 0
\(427\) −17808.0 −2.01824
\(428\) 252.000 0.0284600
\(429\) 0 0
\(430\) −940.000 −0.105421
\(431\) −15792.0 −1.76490 −0.882452 0.470402i \(-0.844109\pi\)
−0.882452 + 0.470402i \(0.844109\pi\)
\(432\) 0 0
\(433\) 11602.0 1.28766 0.643830 0.765169i \(-0.277345\pi\)
0.643830 + 0.765169i \(0.277345\pi\)
\(434\) −6912.00 −0.764485
\(435\) 0 0
\(436\) 6790.00 0.745830
\(437\) −3360.00 −0.367805
\(438\) 0 0
\(439\) −440.000 −0.0478361 −0.0239181 0.999714i \(-0.507614\pi\)
−0.0239181 + 0.999714i \(0.507614\pi\)
\(440\) 3900.00 0.422557
\(441\) 0 0
\(442\) −308.000 −0.0331449
\(443\) 10188.0 1.09266 0.546328 0.837571i \(-0.316025\pi\)
0.546328 + 0.837571i \(0.316025\pi\)
\(444\) 0 0
\(445\) 1650.00 0.175770
\(446\) −6032.00 −0.640411
\(447\) 0 0
\(448\) 4008.00 0.422679
\(449\) 13310.0 1.39897 0.699485 0.714647i \(-0.253413\pi\)
0.699485 + 0.714647i \(0.253413\pi\)
\(450\) 0 0
\(451\) 6344.00 0.662367
\(452\) 7294.00 0.759029
\(453\) 0 0
\(454\) 2636.00 0.272497
\(455\) 2640.00 0.272011
\(456\) 0 0
\(457\) 3226.00 0.330210 0.165105 0.986276i \(-0.447204\pi\)
0.165105 + 0.986276i \(0.447204\pi\)
\(458\) −4830.00 −0.492775
\(459\) 0 0
\(460\) 5880.00 0.595992
\(461\) −6582.00 −0.664977 −0.332488 0.943107i \(-0.607888\pi\)
−0.332488 + 0.943107i \(0.607888\pi\)
\(462\) 0 0
\(463\) 15072.0 1.51286 0.756431 0.654073i \(-0.226941\pi\)
0.756431 + 0.654073i \(0.226941\pi\)
\(464\) −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 \(0.507507\pi\)
\(468\) 0 0
\(469\) 2016.00 0.198487
\(470\) −1280.00 −0.125621
\(471\) 0 0
\(472\) −1500.00 −0.146278
\(473\) 9776.00 0.950319
\(474\) 0 0
\(475\) −500.000 −0.0482980
\(476\) 2352.00 0.226478
\(477\) 0 0
\(478\) 2320.00 0.221997
\(479\) 19680.0 1.87725 0.938624 0.344941i \(-0.112101\pi\)
0.938624 + 0.344941i \(0.112101\pi\)
\(480\) 0 0
\(481\) −748.000 −0.0709062
\(482\) −2002.00 −0.189188
\(483\) 0 0
\(484\) −9611.00 −0.902611
\(485\) −4330.00 −0.405392
\(486\) 0 0
\(487\) −5944.00 −0.553077 −0.276538 0.961003i \(-0.589187\pi\)
−0.276538 + 0.961003i \(0.589187\pi\)
\(488\) 11130.0 1.03244
\(489\) 0 0
\(490\) 1165.00 0.107407
\(491\) −10772.0 −0.990089 −0.495044 0.868868i \(-0.664848\pi\)
−0.495044 + 0.868868i \(0.664848\pi\)
\(492\) 0 0
\(493\) −3220.00 −0.294161
\(494\) 440.000 0.0400740
\(495\) 0 0
\(496\) −11808.0 −1.06894
\(497\) −7872.00 −0.710478
\(498\) 0 0
\(499\) 8140.00 0.730253 0.365127 0.930958i \(-0.381026\pi\)
0.365127 + 0.930958i \(0.381026\pi\)
\(500\) 875.000 0.0782624
\(501\) 0 0
\(502\) 132.000 0.0117360
\(503\) 13768.0 1.22045 0.610223 0.792229i \(-0.291080\pi\)
0.610223 + 0.792229i \(0.291080\pi\)
\(504\) 0 0
\(505\) −6090.00 −0.536637
\(506\) 8736.00 0.767515
\(507\) 0 0
\(508\) −13552.0 −1.18361
\(509\) −22150.0 −1.92884 −0.964422 0.264368i \(-0.914837\pi\)
−0.964422 + 0.264368i \(0.914837\pi\)
\(510\) 0 0
\(511\) 912.000 0.0789521
\(512\) −11521.0 −0.994455
\(513\) 0 0
\(514\) −7614.00 −0.653384
\(515\) 440.000 0.0376480
\(516\) 0 0
\(517\) 13312.0 1.13242
\(518\) −816.000 −0.0692143
\(519\) 0 0
\(520\) −1650.00 −0.139149
\(521\) −1562.00 −0.131348 −0.0656741 0.997841i \(-0.520920\pi\)
−0.0656741 + 0.997841i \(0.520920\pi\)
\(522\) 0 0
\(523\) −668.000 −0.0558501 −0.0279250 0.999610i \(-0.508890\pi\)
−0.0279250 + 0.999610i \(0.508890\pi\)
\(524\) 5124.00 0.427181
\(525\) 0 0
\(526\) −4888.00 −0.405184
\(527\) −4032.00 −0.333276
\(528\) 0 0
\(529\) 16057.0 1.31972
\(530\) 1690.00 0.138507
\(531\) 0 0
\(532\) −3360.00 −0.273824
\(533\) −2684.00 −0.218118
\(534\) 0 0
\(535\) 180.000 0.0145459
\(536\) −1260.00 −0.101537
\(537\) 0 0
\(538\) 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
\(543\) 0 0
\(544\) −2254.00 −0.177646
\(545\) 4850.00 0.381195
\(546\) 0 0
\(547\) −10484.0 −0.819494 −0.409747 0.912199i \(-0.634383\pi\)
−0.409747 + 0.912199i \(0.634383\pi\)
\(548\) −15498.0 −1.20811
\(549\) 0 0
\(550\) 1300.00 0.100786
\(551\) 4600.00 0.355656
\(552\) 0 0
\(553\) 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 \(0.543796\pi\)
\(558\) 0 0
\(559\) −4136.00 −0.312941
\(560\) 4920.00 0.371264
\(561\) 0 0
\(562\) 2442.00 0.183291
\(563\) −12252.0 −0.917159 −0.458579 0.888654i \(-0.651641\pi\)
−0.458579 + 0.888654i \(0.651641\pi\)
\(564\) 0 0
\(565\) 5210.00 0.387940
\(566\) −2772.00 −0.205858
\(567\) 0 0
\(568\) 4920.00 0.363448
\(569\) 14550.0 1.07200 0.536000 0.844218i \(-0.319935\pi\)
0.536000 + 0.844218i \(0.319935\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −2928.00 −0.212914
\(575\) 4200.00 0.304612
\(576\) 0 0
\(577\) 12866.0 0.928282 0.464141 0.885761i \(-0.346363\pi\)
0.464141 + 0.885761i \(0.346363\pi\)
\(578\) 4717.00 0.339449
\(579\) 0 0
\(580\) −8050.00 −0.576307
\(581\) 29088.0 2.07706
\(582\) 0 0
\(583\) −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 \(0.325234\pi\)
\(588\) 0 0
\(589\) 5760.00 0.402948
\(590\) −500.000 −0.0348893
\(591\) 0 0
\(592\) −1394.00 −0.0967788
\(593\) −20402.0 −1.41283 −0.706416 0.707797i \(-0.749689\pi\)
−0.706416 + 0.707797i \(0.749689\pi\)
\(594\) 0 0
\(595\) 1680.00 0.115753
\(596\) −9310.00 −0.639853
\(597\) 0 0
\(598\) −3696.00 −0.252744
\(599\) −10760.0 −0.733959 −0.366980 0.930229i \(-0.619608\pi\)
−0.366980 + 0.930229i \(0.619608\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 8456.00 0.569652
\(605\) −6865.00 −0.461326
\(606\) 0 0
\(607\) 11056.0 0.739290 0.369645 0.929173i \(-0.379479\pi\)
0.369645 + 0.929173i \(0.379479\pi\)
\(608\) 3220.00 0.214783
\(609\) 0 0
\(610\) 3710.00 0.246252
\(611\) −5632.00 −0.372907
\(612\) 0 0
\(613\) −16418.0 −1.08176 −0.540878 0.841101i \(-0.681908\pi\)
−0.540878 + 0.841101i \(0.681908\pi\)
\(614\) −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 \(0.390099\pi\)
\(618\) 0 0
\(619\) −5260.00 −0.341546 −0.170773 0.985310i \(-0.554627\pi\)
−0.170773 + 0.985310i \(0.554627\pi\)
\(620\) −10080.0 −0.652940
\(621\) 0 0
\(622\) −2808.00 −0.181014
\(623\) 7920.00 0.509323
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 7318.00 0.467230
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) 2246.00 0.140694
\(635\) −9680.00 −0.604943
\(636\) 0 0
\(637\) 5126.00 0.318838
\(638\) −11960.0 −0.742164
\(639\) 0 0
\(640\) −7275.00 −0.449328
\(641\) 29118.0 1.79422 0.897108 0.441812i \(-0.145664\pi\)
0.897108 + 0.441812i \(0.145664\pi\)
\(642\) 0 0
\(643\) 5772.00 0.354005 0.177003 0.984210i \(-0.443360\pi\)
0.177003 + 0.984210i \(0.443360\pi\)
\(644\) 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 \(0.357326\pi\)
\(648\) 0 0
\(649\) 5200.00 0.314511
\(650\) −550.000 −0.0331889
\(651\) 0 0
\(652\) 14476.0 0.869515
\(653\) −6902.00 −0.413623 −0.206812 0.978381i \(-0.566309\pi\)
−0.206812 + 0.978381i \(0.566309\pi\)
\(654\) 0 0
\(655\) 3660.00 0.218333
\(656\) −5002.00 −0.297706
\(657\) 0 0
\(658\) −6144.00 −0.364009
\(659\) −20140.0 −1.19051 −0.595253 0.803539i \(-0.702948\pi\)
−0.595253 + 0.803539i \(0.702948\pi\)
\(660\) 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
\(663\) 0 0
\(664\) −18180.0 −1.06253
\(665\) −2400.00 −0.139952
\(666\) 0 0
\(667\) −38640.0 −2.24310
\(668\) −168.000 −0.00973071
\(669\) 0 0
\(670\) −420.000 −0.0242179
\(671\) −38584.0 −2.21985
\(672\) 0 0
\(673\) −7518.00 −0.430606 −0.215303 0.976547i \(-0.569074\pi\)
−0.215303 + 0.976547i \(0.569074\pi\)
\(674\) 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 \(0.328102\pi\)
\(678\) 0 0
\(679\) −20784.0 −1.17469
\(680\) −1050.00 −0.0592142
\(681\) 0 0
\(682\) −14976.0 −0.840851
\(683\) 23868.0 1.33716 0.668582 0.743638i \(-0.266901\pi\)
0.668582 + 0.743638i \(0.266901\pi\)
\(684\) 0 0
\(685\) −11070.0 −0.617464
\(686\) −2640.00 −0.146932
\(687\) 0 0
\(688\) −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
\(693\) 0 0
\(694\) 11956.0 0.653953
\(695\) −100.000 −0.00545787
\(696\) 0 0
\(697\) −1708.00 −0.0928194
\(698\) −4870.00 −0.264086
\(699\) 0 0
\(700\) 4200.00 0.226779
\(701\) 22138.0 1.19278 0.596391 0.802694i \(-0.296601\pi\)
0.596391 + 0.802694i \(0.296601\pi\)
\(702\) 0 0
\(703\) 680.000 0.0364818
\(704\) 8684.00 0.464901
\(705\) 0 0
\(706\) 10722.0 0.571569
\(707\) −29232.0 −1.55500
\(708\) 0 0
\(709\) 3070.00 0.162618 0.0813091 0.996689i \(-0.474090\pi\)
0.0813091 + 0.996689i \(0.474090\pi\)
\(710\) 1640.00 0.0866875
\(711\) 0 0
\(712\) −4950.00 −0.260546
\(713\) −48384.0 −2.54137
\(714\) 0 0
\(715\) 5720.00 0.299183
\(716\) 23380.0 1.22032
\(717\) 0 0
\(718\) 120.000 0.00623727
\(719\) −15600.0 −0.809154 −0.404577 0.914504i \(-0.632581\pi\)
−0.404577 + 0.914504i \(0.632581\pi\)
\(720\) 0 0
\(721\) 2112.00 0.109092
\(722\) 6459.00 0.332935
\(723\) 0 0
\(724\) 1246.00 0.0639603
\(725\) −5750.00 −0.294551
\(726\) 0 0
\(727\) 20696.0 1.05581 0.527904 0.849304i \(-0.322978\pi\)
0.527904 + 0.849304i \(0.322978\pi\)
\(728\) −7920.00 −0.403207
\(729\) 0 0
\(730\) −190.000 −0.00963317
\(731\) −2632.00 −0.133171
\(732\) 0 0
\(733\) −30778.0 −1.55090 −0.775451 0.631408i \(-0.782478\pi\)
−0.775451 + 0.631408i \(0.782478\pi\)
\(734\) −3936.00 −0.197930
\(735\) 0 0
\(736\) −27048.0 −1.35462
\(737\) 4368.00 0.218314
\(738\) 0 0
\(739\) 11740.0 0.584388 0.292194 0.956359i \(-0.405615\pi\)
0.292194 + 0.956359i \(0.405615\pi\)
\(740\) −1190.00 −0.0591152
\(741\) 0 0
\(742\) 8112.00 0.401349
\(743\) −2632.00 −0.129958 −0.0649789 0.997887i \(-0.520698\pi\)
−0.0649789 + 0.997887i \(0.520698\pi\)
\(744\) 0 0
\(745\) −6650.00 −0.327030
\(746\) −3022.00 −0.148315
\(747\) 0 0
\(748\) 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
\(753\) 0 0
\(754\) 5060.00 0.244396
\(755\) 6040.00 0.291150
\(756\) 0 0
\(757\) 21646.0 1.03928 0.519642 0.854384i \(-0.326066\pi\)
0.519642 + 0.854384i \(0.326066\pi\)
\(758\) 13340.0 0.639222
\(759\) 0 0
\(760\) 1500.00 0.0715931
\(761\) −18282.0 −0.870857 −0.435428 0.900223i \(-0.643403\pi\)
−0.435428 + 0.900223i \(0.643403\pi\)
\(762\) 0 0
\(763\) 23280.0 1.10458
\(764\) −13216.0 −0.625835
\(765\) 0 0
\(766\) −1008.00 −0.0475464
\(767\) −2200.00 −0.103569
\(768\) 0 0
\(769\) −24190.0 −1.13435 −0.567174 0.823598i \(-0.691963\pi\)
−0.567174 + 0.823598i \(0.691963\pi\)
\(770\) 6240.00 0.292044
\(771\) 0 0
\(772\) −13454.0 −0.627228
\(773\) 25698.0 1.19572 0.597861 0.801600i \(-0.296018\pi\)
0.597861 + 0.801600i \(0.296018\pi\)
\(774\) 0 0
\(775\) −7200.00 −0.333718
\(776\) 12990.0 0.600920
\(777\) 0 0
\(778\) 9630.00 0.443769
\(779\) 2440.00 0.112223
\(780\) 0 0
\(781\) −17056.0 −0.781449
\(782\) −2352.00 −0.107554
\(783\) 0 0
\(784\) 9553.00 0.435177
\(785\) 17570.0 0.798854
\(786\) 0 0
\(787\) 33436.0 1.51444 0.757220 0.653160i \(-0.226557\pi\)
0.757220 + 0.653160i \(0.226557\pi\)
\(788\) 17682.0 0.799359
\(789\) 0 0
\(790\) −1200.00 −0.0540431
\(791\) 25008.0 1.12412
\(792\) 0 0
\(793\) 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 \(0.185229\pi\)
\(798\) 0 0
\(799\) −3584.00 −0.158689
\(800\) −4025.00 −0.177882
\(801\) 0 0
\(802\) −8718.00 −0.383844
\(803\) 1976.00 0.0868388
\(804\) 0 0
\(805\) 20160.0 0.882667
\(806\) 6336.00 0.276893
\(807\) 0 0
\(808\) 18270.0 0.795466
\(809\) −4730.00 −0.205560 −0.102780 0.994704i \(-0.532774\pi\)
−0.102780 + 0.994704i \(0.532774\pi\)
\(810\) 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
\(813\) 0 0
\(814\) −1768.00 −0.0761282
\(815\) 10340.0 0.444410
\(816\) 0 0
\(817\) 3760.00 0.161011
\(818\) 10870.0 0.464622
\(819\) 0 0
\(820\) −4270.00 −0.181847
\(821\) −44142.0 −1.87645 −0.938226 0.346024i \(-0.887532\pi\)
−0.938226 + 0.346024i \(0.887532\pi\)
\(822\) 0 0
\(823\) 3992.00 0.169079 0.0845397 0.996420i \(-0.473058\pi\)
0.0845397 + 0.996420i \(0.473058\pi\)
\(824\) −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 \(0.401789\pi\)
\(828\) 0 0
\(829\) 42150.0 1.76590 0.882949 0.469468i \(-0.155554\pi\)
0.882949 + 0.469468i \(0.155554\pi\)
\(830\) −6060.00 −0.253429
\(831\) 0 0
\(832\) −3674.00 −0.153093
\(833\) 3262.00 0.135680
\(834\) 0 0
\(835\) −120.000 −0.00497338
\(836\) −7280.00 −0.301177
\(837\) 0 0
\(838\) −9700.00 −0.399858
\(839\) −13400.0 −0.551394 −0.275697 0.961245i \(-0.588909\pi\)
−0.275697 + 0.961245i \(0.588909\pi\)
\(840\) 0 0
\(841\) 28511.0 1.16901
\(842\) −862.000 −0.0352809
\(843\) 0 0
\(844\) 31276.0 1.27555
\(845\) 8565.00 0.348692
\(846\) 0 0
\(847\) −32952.0 −1.33677
\(848\) 13858.0 0.561186
\(849\) 0 0
\(850\) −350.000 −0.0141234
\(851\) −5712.00 −0.230088
\(852\) 0 0
\(853\) −8658.00 −0.347531 −0.173766 0.984787i \(-0.555594\pi\)
−0.173766 + 0.984787i \(0.555594\pi\)
\(854\) 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 \(0.825528\pi\)
\(858\) 0 0
\(859\) −35900.0 −1.42595 −0.712976 0.701189i \(-0.752653\pi\)
−0.712976 + 0.701189i \(0.752653\pi\)
\(860\) −6580.00 −0.260902
\(861\) 0 0
\(862\) 15792.0 0.623988
\(863\) 3088.00 0.121804 0.0609019 0.998144i \(-0.480602\pi\)
0.0609019 + 0.998144i \(0.480602\pi\)
\(864\) 0 0
\(865\) −3090.00 −0.121460
\(866\) −11602.0 −0.455256
\(867\) 0 0
\(868\) −48384.0 −1.89200
\(869\) 12480.0 0.487175
\(870\) 0 0
\(871\) −1848.00 −0.0718910
\(872\) −14550.0 −0.565052
\(873\) 0 0
\(874\) 3360.00 0.130039
\(875\) 3000.00 0.115907
\(876\) 0 0
\(877\) −35274.0 −1.35817 −0.679087 0.734058i \(-0.737624\pi\)
−0.679087 + 0.734058i \(0.737624\pi\)
\(878\) 440.000 0.0169126
\(879\) 0 0
\(880\) 10660.0 0.408351
\(881\) −25042.0 −0.957646 −0.478823 0.877911i \(-0.658936\pi\)
−0.478823 + 0.877911i \(0.658936\pi\)
\(882\) 0 0
\(883\) 12572.0 0.479141 0.239570 0.970879i \(-0.422993\pi\)
0.239570 + 0.970879i \(0.422993\pi\)
\(884\) −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 \(0.364193\pi\)
\(888\) 0 0
\(889\) −46464.0 −1.75293
\(890\) −1650.00 −0.0621440
\(891\) 0 0
\(892\) −42224.0 −1.58494
\(893\) 5120.00 0.191864
\(894\) 0 0
\(895\) 16700.0 0.623709
\(896\) −34920.0 −1.30200
\(897\) 0 0
\(898\) −13310.0 −0.494611
\(899\) 66240.0 2.45743
\(900\) 0 0
\(901\) 4732.00 0.174968
\(902\) −6344.00 −0.234182
\(903\) 0 0
\(904\) −15630.0 −0.575051
\(905\) 890.000 0.0326902
\(906\) 0 0
\(907\) 31236.0 1.14352 0.571761 0.820420i \(-0.306260\pi\)
0.571761 + 0.820420i \(0.306260\pi\)
\(908\) 18452.0 0.674396
\(909\) 0 0
\(910\) −2640.00 −0.0961705
\(911\) −8272.00 −0.300838 −0.150419 0.988622i \(-0.548062\pi\)
−0.150419 + 0.988622i \(0.548062\pi\)
\(912\) 0 0
\(913\) 63024.0 2.28455
\(914\) −3226.00 −0.116747
\(915\) 0 0
\(916\) −33810.0 −1.21956
\(917\) 17568.0 0.632657
\(918\) 0 0
\(919\) 20200.0 0.725067 0.362533 0.931971i \(-0.381912\pi\)
0.362533 + 0.931971i \(0.381912\pi\)
\(920\) −12600.0 −0.451532
\(921\) 0 0
\(922\) 6582.00 0.235105
\(923\) 7216.00 0.257332
\(924\) 0 0
\(925\) −850.000 −0.0302139
\(926\) −15072.0 −0.534878
\(927\) 0 0
\(928\) 37030.0 1.30988
\(929\) −31010.0 −1.09516 −0.547581 0.836753i \(-0.684451\pi\)
−0.547581 + 0.836753i \(0.684451\pi\)
\(930\) 0 0
\(931\) −4660.00 −0.164044
\(932\) 18774.0 0.659831
\(933\) 0 0
\(934\) 476.000 0.0166758
\(935\) 3640.00 0.127316
\(936\) 0 0
\(937\) −39174.0 −1.36580 −0.682902 0.730510i \(-0.739283\pi\)
−0.682902 + 0.730510i \(0.739283\pi\)
\(938\) −2016.00 −0.0701756
\(939\) 0 0
\(940\) −8960.00 −0.310897
\(941\) 4138.00 0.143353 0.0716764 0.997428i \(-0.477165\pi\)
0.0716764 + 0.997428i \(0.477165\pi\)
\(942\) 0 0
\(943\) −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 \(0.633151\pi\)
\(948\) 0 0
\(949\) −836.000 −0.0285961
\(950\) 500.000 0.0170759
\(951\) 0 0
\(952\) −5040.00 −0.171583
\(953\) −18922.0 −0.643173 −0.321586 0.946880i \(-0.604216\pi\)
−0.321586 + 0.946880i \(0.604216\pi\)
\(954\) 0 0
\(955\) −9440.00 −0.319865
\(956\) 16240.0 0.549413
\(957\) 0 0
\(958\) −19680.0 −0.663708
\(959\) −53136.0 −1.78921
\(960\) 0 0
\(961\) 53153.0 1.78420
\(962\) 748.000 0.0250691
\(963\) 0 0
\(964\) −14014.0 −0.468216
\(965\) −9610.00 −0.320577
\(966\) 0 0
\(967\) 39656.0 1.31877 0.659385 0.751805i \(-0.270817\pi\)
0.659385 + 0.751805i \(0.270817\pi\)
\(968\) 20595.0 0.683831
\(969\) 0 0
\(970\) 4330.00 0.143328
\(971\) 33228.0 1.09818 0.549092 0.835762i \(-0.314974\pi\)
0.549092 + 0.835762i \(0.314974\pi\)
\(972\) 0 0
\(973\) −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 \(0.494924\pi\)
\(978\) 0 0
\(979\) 17160.0 0.560200
\(980\) 8155.00 0.265818
\(981\) 0 0
\(982\) 10772.0 0.350049
\(983\) 13608.0 0.441534 0.220767 0.975327i \(-0.429144\pi\)
0.220767 + 0.975327i \(0.429144\pi\)
\(984\) 0 0
\(985\) 12630.0 0.408554
\(986\) 3220.00 0.104002
\(987\) 0 0
\(988\) 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
\(993\) 0 0
\(994\) 7872.00 0.251192
\(995\) 5800.00 0.184796
\(996\) 0 0
\(997\) −3234.00 −0.102730 −0.0513650 0.998680i \(-0.516357\pi\)
−0.0513650 + 0.998680i \(0.516357\pi\)
\(998\) −8140.00 −0.258184
\(999\) 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 45.4.a.c.1.1 1
3.2 odd 2 15.4.a.a.1.1 1
4.3 odd 2 720.4.a.n.1.1 1
5.2 odd 4 225.4.b.e.199.1 2
5.3 odd 4 225.4.b.e.199.2 2
5.4 even 2 225.4.a.f.1.1 1
7.6 odd 2 2205.4.a.l.1.1 1
9.2 odd 6 405.4.e.g.271.1 2
9.4 even 3 405.4.e.i.136.1 2
9.5 odd 6 405.4.e.g.136.1 2
9.7 even 3 405.4.e.i.271.1 2
12.11 even 2 240.4.a.e.1.1 1
15.2 even 4 75.4.b.b.49.2 2
15.8 even 4 75.4.b.b.49.1 2
15.14 odd 2 75.4.a.b.1.1 1
21.20 even 2 735.4.a.e.1.1 1
24.5 odd 2 960.4.a.b.1.1 1
24.11 even 2 960.4.a.ba.1.1 1
33.32 even 2 1815.4.a.e.1.1 1
60.23 odd 4 1200.4.f.b.49.1 2
60.47 odd 4 1200.4.f.b.49.2 2
60.59 even 2 1200.4.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.4.a.a.1.1 1 3.2 odd 2
45.4.a.c.1.1 1 1.1 even 1 trivial
75.4.a.b.1.1 1 15.14 odd 2
75.4.b.b.49.1 2 15.8 even 4
75.4.b.b.49.2 2 15.2 even 4
225.4.a.f.1.1 1 5.4 even 2
225.4.b.e.199.1 2 5.2 odd 4
225.4.b.e.199.2 2 5.3 odd 4
240.4.a.e.1.1 1 12.11 even 2
405.4.e.g.136.1 2 9.5 odd 6
405.4.e.g.271.1 2 9.2 odd 6
405.4.e.i.136.1 2 9.4 even 3
405.4.e.i.271.1 2 9.7 even 3
720.4.a.n.1.1 1 4.3 odd 2
735.4.a.e.1.1 1 21.20 even 2
960.4.a.b.1.1 1 24.5 odd 2
960.4.a.ba.1.1 1 24.11 even 2
1200.4.a.t.1.1 1 60.59 even 2
1200.4.f.b.49.1 2 60.23 odd 4
1200.4.f.b.49.2 2 60.47 odd 4
1815.4.a.e.1.1 1 33.32 even 2
2205.4.a.l.1.1 1 7.6 odd 2