Properties

Label 1805.2.a.l.1.3
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4129975648\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{20})^+\)
Defining polynomial: \( x^{4} - 5x^{2} + 5 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(1.17557\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.17557 q^{2} +1.17557 q^{3} -0.618034 q^{4} -1.00000 q^{5} +1.38197 q^{6} +0.236068 q^{7} -3.07768 q^{8} -1.61803 q^{9} +O(q^{10})\) \(q+1.17557 q^{2} +1.17557 q^{3} -0.618034 q^{4} -1.00000 q^{5} +1.38197 q^{6} +0.236068 q^{7} -3.07768 q^{8} -1.61803 q^{9} -1.17557 q^{10} +0.854102 q^{11} -0.726543 q^{12} +0.726543 q^{13} +0.277515 q^{14} -1.17557 q^{15} -2.38197 q^{16} +0.763932 q^{17} -1.90211 q^{18} +0.618034 q^{20} +0.277515 q^{21} +1.00406 q^{22} -7.09017 q^{23} -3.61803 q^{24} +1.00000 q^{25} +0.854102 q^{26} -5.42882 q^{27} -0.145898 q^{28} -8.78402 q^{29} -1.38197 q^{30} -1.17557 q^{31} +3.35520 q^{32} +1.00406 q^{33} +0.898056 q^{34} -0.236068 q^{35} +1.00000 q^{36} -8.78402 q^{37} +0.854102 q^{39} +3.07768 q^{40} +1.62460 q^{41} +0.326238 q^{42} +2.61803 q^{43} -0.527864 q^{44} +1.61803 q^{45} -8.33499 q^{46} +7.47214 q^{47} -2.80017 q^{48} -6.94427 q^{49} +1.17557 q^{50} +0.898056 q^{51} -0.449028 q^{52} +1.00406 q^{53} -6.38197 q^{54} -0.854102 q^{55} -0.726543 q^{56} -10.3262 q^{58} -11.3067 q^{59} +0.726543 q^{60} -13.9443 q^{61} -1.38197 q^{62} -0.381966 q^{63} +8.70820 q^{64} -0.726543 q^{65} +1.18034 q^{66} -11.5842 q^{67} -0.472136 q^{68} -8.33499 q^{69} -0.277515 q^{70} +13.0373 q^{71} +4.97980 q^{72} -1.00000 q^{73} -10.3262 q^{74} +1.17557 q^{75} +0.201626 q^{77} +1.00406 q^{78} +8.50651 q^{79} +2.38197 q^{80} -1.52786 q^{81} +1.90983 q^{82} -13.2361 q^{83} -0.171513 q^{84} -0.763932 q^{85} +3.07768 q^{86} -10.3262 q^{87} -2.62866 q^{88} +8.33499 q^{89} +1.90211 q^{90} +0.171513 q^{91} +4.38197 q^{92} -1.38197 q^{93} +8.78402 q^{94} +3.94427 q^{96} +4.25325 q^{97} -8.16348 q^{98} -1.38197 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 4 q^{5} + 10 q^{6} - 8 q^{7} - 2 q^{9} - 10 q^{11} - 14 q^{16} + 12 q^{17} - 2 q^{20} - 6 q^{23} - 10 q^{24} + 4 q^{25} - 10 q^{26} - 14 q^{28} - 10 q^{30} + 8 q^{35} + 4 q^{36} - 10 q^{39} - 30 q^{42} + 6 q^{43} - 20 q^{44} + 2 q^{45} + 12 q^{47} + 8 q^{49} - 30 q^{54} + 10 q^{55} - 10 q^{58} - 20 q^{61} - 10 q^{62} - 6 q^{63} + 8 q^{64} - 40 q^{66} + 16 q^{68} - 4 q^{73} - 10 q^{74} + 50 q^{77} + 14 q^{80} - 24 q^{81} + 30 q^{82} - 44 q^{83} - 12 q^{85} - 10 q^{87} + 22 q^{92} - 10 q^{93} - 20 q^{96} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.17557 0.831254 0.415627 0.909535i \(-0.363562\pi\)
0.415627 + 0.909535i \(0.363562\pi\)
\(3\) 1.17557 0.678716 0.339358 0.940657i \(-0.389790\pi\)
0.339358 + 0.940657i \(0.389790\pi\)
\(4\) −0.618034 −0.309017
\(5\) −1.00000 −0.447214
\(6\) 1.38197 0.564185
\(7\) 0.236068 0.0892253 0.0446127 0.999004i \(-0.485795\pi\)
0.0446127 + 0.999004i \(0.485795\pi\)
\(8\) −3.07768 −1.08813
\(9\) −1.61803 −0.539345
\(10\) −1.17557 −0.371748
\(11\) 0.854102 0.257521 0.128761 0.991676i \(-0.458900\pi\)
0.128761 + 0.991676i \(0.458900\pi\)
\(12\) −0.726543 −0.209735
\(13\) 0.726543 0.201507 0.100753 0.994911i \(-0.467875\pi\)
0.100753 + 0.994911i \(0.467875\pi\)
\(14\) 0.277515 0.0741689
\(15\) −1.17557 −0.303531
\(16\) −2.38197 −0.595492
\(17\) 0.763932 0.185281 0.0926404 0.995700i \(-0.470469\pi\)
0.0926404 + 0.995700i \(0.470469\pi\)
\(18\) −1.90211 −0.448332
\(19\) 0 0
\(20\) 0.618034 0.138197
\(21\) 0.277515 0.0605586
\(22\) 1.00406 0.214066
\(23\) −7.09017 −1.47840 −0.739201 0.673485i \(-0.764797\pi\)
−0.739201 + 0.673485i \(0.764797\pi\)
\(24\) −3.61803 −0.738528
\(25\) 1.00000 0.200000
\(26\) 0.854102 0.167503
\(27\) −5.42882 −1.04478
\(28\) −0.145898 −0.0275721
\(29\) −8.78402 −1.63115 −0.815576 0.578650i \(-0.803580\pi\)
−0.815576 + 0.578650i \(0.803580\pi\)
\(30\) −1.38197 −0.252311
\(31\) −1.17557 −0.211139 −0.105569 0.994412i \(-0.533667\pi\)
−0.105569 + 0.994412i \(0.533667\pi\)
\(32\) 3.35520 0.593121
\(33\) 1.00406 0.174784
\(34\) 0.898056 0.154015
\(35\) −0.236068 −0.0399028
\(36\) 1.00000 0.166667
\(37\) −8.78402 −1.44408 −0.722042 0.691849i \(-0.756796\pi\)
−0.722042 + 0.691849i \(0.756796\pi\)
\(38\) 0 0
\(39\) 0.854102 0.136766
\(40\) 3.07768 0.486624
\(41\) 1.62460 0.253720 0.126860 0.991921i \(-0.459510\pi\)
0.126860 + 0.991921i \(0.459510\pi\)
\(42\) 0.326238 0.0503396
\(43\) 2.61803 0.399246 0.199623 0.979873i \(-0.436028\pi\)
0.199623 + 0.979873i \(0.436028\pi\)
\(44\) −0.527864 −0.0795785
\(45\) 1.61803 0.241202
\(46\) −8.33499 −1.22893
\(47\) 7.47214 1.08992 0.544962 0.838461i \(-0.316544\pi\)
0.544962 + 0.838461i \(0.316544\pi\)
\(48\) −2.80017 −0.404170
\(49\) −6.94427 −0.992039
\(50\) 1.17557 0.166251
\(51\) 0.898056 0.125753
\(52\) −0.449028 −0.0622690
\(53\) 1.00406 0.137918 0.0689589 0.997620i \(-0.478032\pi\)
0.0689589 + 0.997620i \(0.478032\pi\)
\(54\) −6.38197 −0.868476
\(55\) −0.854102 −0.115167
\(56\) −0.726543 −0.0970883
\(57\) 0 0
\(58\) −10.3262 −1.35590
\(59\) −11.3067 −1.47200 −0.736002 0.676979i \(-0.763289\pi\)
−0.736002 + 0.676979i \(0.763289\pi\)
\(60\) 0.726543 0.0937962
\(61\) −13.9443 −1.78538 −0.892691 0.450670i \(-0.851185\pi\)
−0.892691 + 0.450670i \(0.851185\pi\)
\(62\) −1.38197 −0.175510
\(63\) −0.381966 −0.0481232
\(64\) 8.70820 1.08853
\(65\) −0.726543 −0.0901165
\(66\) 1.18034 0.145290
\(67\) −11.5842 −1.41523 −0.707617 0.706596i \(-0.750230\pi\)
−0.707617 + 0.706596i \(0.750230\pi\)
\(68\) −0.472136 −0.0572549
\(69\) −8.33499 −1.00342
\(70\) −0.277515 −0.0331693
\(71\) 13.0373 1.54724 0.773620 0.633650i \(-0.218444\pi\)
0.773620 + 0.633650i \(0.218444\pi\)
\(72\) 4.97980 0.586875
\(73\) −1.00000 −0.117041 −0.0585206 0.998286i \(-0.518638\pi\)
−0.0585206 + 0.998286i \(0.518638\pi\)
\(74\) −10.3262 −1.20040
\(75\) 1.17557 0.135743
\(76\) 0 0
\(77\) 0.201626 0.0229774
\(78\) 1.00406 0.113687
\(79\) 8.50651 0.957057 0.478528 0.878072i \(-0.341170\pi\)
0.478528 + 0.878072i \(0.341170\pi\)
\(80\) 2.38197 0.266312
\(81\) −1.52786 −0.169763
\(82\) 1.90983 0.210905
\(83\) −13.2361 −1.45285 −0.726424 0.687247i \(-0.758819\pi\)
−0.726424 + 0.687247i \(0.758819\pi\)
\(84\) −0.171513 −0.0187136
\(85\) −0.763932 −0.0828601
\(86\) 3.07768 0.331875
\(87\) −10.3262 −1.10709
\(88\) −2.62866 −0.280216
\(89\) 8.33499 0.883508 0.441754 0.897136i \(-0.354356\pi\)
0.441754 + 0.897136i \(0.354356\pi\)
\(90\) 1.90211 0.200500
\(91\) 0.171513 0.0179795
\(92\) 4.38197 0.456852
\(93\) −1.38197 −0.143303
\(94\) 8.78402 0.906003
\(95\) 0 0
\(96\) 3.94427 0.402561
\(97\) 4.25325 0.431853 0.215926 0.976410i \(-0.430723\pi\)
0.215926 + 0.976410i \(0.430723\pi\)
\(98\) −8.16348 −0.824636
\(99\) −1.38197 −0.138893
\(100\) −0.618034 −0.0618034
\(101\) 16.7082 1.66253 0.831264 0.555878i \(-0.187618\pi\)
0.831264 + 0.555878i \(0.187618\pi\)
\(102\) 1.05573 0.104533
\(103\) 1.34708 0.132732 0.0663661 0.997795i \(-0.478859\pi\)
0.0663661 + 0.997795i \(0.478859\pi\)
\(104\) −2.23607 −0.219265
\(105\) −0.277515 −0.0270826
\(106\) 1.18034 0.114645
\(107\) 7.77997 0.752118 0.376059 0.926596i \(-0.377279\pi\)
0.376059 + 0.926596i \(0.377279\pi\)
\(108\) 3.35520 0.322854
\(109\) 9.23305 0.884366 0.442183 0.896925i \(-0.354204\pi\)
0.442183 + 0.896925i \(0.354204\pi\)
\(110\) −1.00406 −0.0957331
\(111\) −10.3262 −0.980123
\(112\) −0.562306 −0.0531329
\(113\) 4.35926 0.410084 0.205042 0.978753i \(-0.434267\pi\)
0.205042 + 0.978753i \(0.434267\pi\)
\(114\) 0 0
\(115\) 7.09017 0.661162
\(116\) 5.42882 0.504054
\(117\) −1.17557 −0.108682
\(118\) −13.2918 −1.22361
\(119\) 0.180340 0.0165317
\(120\) 3.61803 0.330280
\(121\) −10.2705 −0.933683
\(122\) −16.3925 −1.48410
\(123\) 1.90983 0.172204
\(124\) 0.726543 0.0652454
\(125\) −1.00000 −0.0894427
\(126\) −0.449028 −0.0400026
\(127\) 2.17963 0.193411 0.0967053 0.995313i \(-0.469170\pi\)
0.0967053 + 0.995313i \(0.469170\pi\)
\(128\) 3.52671 0.311720
\(129\) 3.07768 0.270975
\(130\) −0.854102 −0.0749097
\(131\) −13.3820 −1.16919 −0.584594 0.811326i \(-0.698746\pi\)
−0.584594 + 0.811326i \(0.698746\pi\)
\(132\) −0.620541 −0.0540112
\(133\) 0 0
\(134\) −13.6180 −1.17642
\(135\) 5.42882 0.467239
\(136\) −2.35114 −0.201609
\(137\) 14.2361 1.21627 0.608135 0.793834i \(-0.291918\pi\)
0.608135 + 0.793834i \(0.291918\pi\)
\(138\) −9.79837 −0.834093
\(139\) −10.9098 −0.925360 −0.462680 0.886525i \(-0.653112\pi\)
−0.462680 + 0.886525i \(0.653112\pi\)
\(140\) 0.145898 0.0123306
\(141\) 8.78402 0.739748
\(142\) 15.3262 1.28615
\(143\) 0.620541 0.0518923
\(144\) 3.85410 0.321175
\(145\) 8.78402 0.729473
\(146\) −1.17557 −0.0972909
\(147\) −8.16348 −0.673313
\(148\) 5.42882 0.446247
\(149\) −4.09017 −0.335080 −0.167540 0.985865i \(-0.553582\pi\)
−0.167540 + 0.985865i \(0.553582\pi\)
\(150\) 1.38197 0.112837
\(151\) 18.0171 1.46621 0.733104 0.680116i \(-0.238071\pi\)
0.733104 + 0.680116i \(0.238071\pi\)
\(152\) 0 0
\(153\) −1.23607 −0.0999302
\(154\) 0.237026 0.0191001
\(155\) 1.17557 0.0944241
\(156\) −0.527864 −0.0422629
\(157\) 17.2705 1.37834 0.689168 0.724601i \(-0.257976\pi\)
0.689168 + 0.724601i \(0.257976\pi\)
\(158\) 10.0000 0.795557
\(159\) 1.18034 0.0936070
\(160\) −3.35520 −0.265252
\(161\) −1.67376 −0.131911
\(162\) −1.79611 −0.141116
\(163\) −2.81966 −0.220853 −0.110426 0.993884i \(-0.535222\pi\)
−0.110426 + 0.993884i \(0.535222\pi\)
\(164\) −1.00406 −0.0784037
\(165\) −1.00406 −0.0781657
\(166\) −15.5599 −1.20768
\(167\) 13.0373 1.00885 0.504427 0.863454i \(-0.331704\pi\)
0.504427 + 0.863454i \(0.331704\pi\)
\(168\) −0.854102 −0.0658954
\(169\) −12.4721 −0.959395
\(170\) −0.898056 −0.0688777
\(171\) 0 0
\(172\) −1.61803 −0.123374
\(173\) −12.8658 −0.978166 −0.489083 0.872237i \(-0.662668\pi\)
−0.489083 + 0.872237i \(0.662668\pi\)
\(174\) −12.1392 −0.920272
\(175\) 0.236068 0.0178451
\(176\) −2.03444 −0.153352
\(177\) −13.2918 −0.999073
\(178\) 9.79837 0.734419
\(179\) 12.1392 0.907328 0.453664 0.891173i \(-0.350117\pi\)
0.453664 + 0.891173i \(0.350117\pi\)
\(180\) −1.00000 −0.0745356
\(181\) 14.6619 1.08981 0.544904 0.838498i \(-0.316566\pi\)
0.544904 + 0.838498i \(0.316566\pi\)
\(182\) 0.201626 0.0149455
\(183\) −16.3925 −1.21177
\(184\) 21.8213 1.60869
\(185\) 8.78402 0.645814
\(186\) −1.62460 −0.119121
\(187\) 0.652476 0.0477138
\(188\) −4.61803 −0.336805
\(189\) −1.28157 −0.0932206
\(190\) 0 0
\(191\) 13.0000 0.940647 0.470323 0.882494i \(-0.344137\pi\)
0.470323 + 0.882494i \(0.344137\pi\)
\(192\) 10.2371 0.738800
\(193\) −6.15537 −0.443073 −0.221537 0.975152i \(-0.571107\pi\)
−0.221537 + 0.975152i \(0.571107\pi\)
\(194\) 5.00000 0.358979
\(195\) −0.854102 −0.0611635
\(196\) 4.29180 0.306557
\(197\) −20.8885 −1.48825 −0.744124 0.668042i \(-0.767133\pi\)
−0.744124 + 0.668042i \(0.767133\pi\)
\(198\) −1.62460 −0.115455
\(199\) −13.4164 −0.951064 −0.475532 0.879698i \(-0.657744\pi\)
−0.475532 + 0.879698i \(0.657744\pi\)
\(200\) −3.07768 −0.217625
\(201\) −13.6180 −0.960542
\(202\) 19.6417 1.38198
\(203\) −2.07363 −0.145540
\(204\) −0.555029 −0.0388598
\(205\) −1.62460 −0.113467
\(206\) 1.58359 0.110334
\(207\) 11.4721 0.797369
\(208\) −1.73060 −0.119995
\(209\) 0 0
\(210\) −0.326238 −0.0225126
\(211\) 14.2128 0.978453 0.489226 0.872157i \(-0.337279\pi\)
0.489226 + 0.872157i \(0.337279\pi\)
\(212\) −0.620541 −0.0426190
\(213\) 15.3262 1.05014
\(214\) 9.14590 0.625201
\(215\) −2.61803 −0.178548
\(216\) 16.7082 1.13685
\(217\) −0.277515 −0.0188389
\(218\) 10.8541 0.735133
\(219\) −1.17557 −0.0794377
\(220\) 0.527864 0.0355886
\(221\) 0.555029 0.0373353
\(222\) −12.1392 −0.814731
\(223\) −19.7477 −1.32240 −0.661201 0.750209i \(-0.729953\pi\)
−0.661201 + 0.750209i \(0.729953\pi\)
\(224\) 0.792055 0.0529214
\(225\) −1.61803 −0.107869
\(226\) 5.12461 0.340884
\(227\) −18.6376 −1.23702 −0.618511 0.785776i \(-0.712264\pi\)
−0.618511 + 0.785776i \(0.712264\pi\)
\(228\) 0 0
\(229\) −24.8541 −1.64241 −0.821203 0.570637i \(-0.806696\pi\)
−0.821203 + 0.570637i \(0.806696\pi\)
\(230\) 8.33499 0.549593
\(231\) 0.237026 0.0155951
\(232\) 27.0344 1.77490
\(233\) 6.23607 0.408538 0.204269 0.978915i \(-0.434518\pi\)
0.204269 + 0.978915i \(0.434518\pi\)
\(234\) −1.38197 −0.0903419
\(235\) −7.47214 −0.487428
\(236\) 6.98791 0.454874
\(237\) 10.0000 0.649570
\(238\) 0.212002 0.0137421
\(239\) 28.2148 1.82506 0.912531 0.409007i \(-0.134125\pi\)
0.912531 + 0.409007i \(0.134125\pi\)
\(240\) 2.80017 0.180750
\(241\) −14.4904 −0.933406 −0.466703 0.884414i \(-0.654558\pi\)
−0.466703 + 0.884414i \(0.654558\pi\)
\(242\) −12.0737 −0.776127
\(243\) 14.4904 0.929557
\(244\) 8.61803 0.551713
\(245\) 6.94427 0.443653
\(246\) 2.24514 0.143145
\(247\) 0 0
\(248\) 3.61803 0.229745
\(249\) −15.5599 −0.986071
\(250\) −1.17557 −0.0743496
\(251\) −3.00000 −0.189358 −0.0946792 0.995508i \(-0.530183\pi\)
−0.0946792 + 0.995508i \(0.530183\pi\)
\(252\) 0.236068 0.0148709
\(253\) −6.05573 −0.380720
\(254\) 2.56231 0.160773
\(255\) −0.898056 −0.0562384
\(256\) −13.2705 −0.829407
\(257\) −13.7638 −0.858563 −0.429282 0.903171i \(-0.641233\pi\)
−0.429282 + 0.903171i \(0.641233\pi\)
\(258\) 3.61803 0.225249
\(259\) −2.07363 −0.128849
\(260\) 0.449028 0.0278475
\(261\) 14.2128 0.879753
\(262\) −15.7314 −0.971892
\(263\) −6.00000 −0.369976 −0.184988 0.982741i \(-0.559225\pi\)
−0.184988 + 0.982741i \(0.559225\pi\)
\(264\) −3.09017 −0.190187
\(265\) −1.00406 −0.0616787
\(266\) 0 0
\(267\) 9.79837 0.599651
\(268\) 7.15942 0.437331
\(269\) 6.26137 0.381762 0.190881 0.981613i \(-0.438865\pi\)
0.190881 + 0.981613i \(0.438865\pi\)
\(270\) 6.38197 0.388394
\(271\) 6.38197 0.387677 0.193838 0.981033i \(-0.437906\pi\)
0.193838 + 0.981033i \(0.437906\pi\)
\(272\) −1.81966 −0.110333
\(273\) 0.201626 0.0122030
\(274\) 16.7355 1.01103
\(275\) 0.854102 0.0515043
\(276\) 5.15131 0.310072
\(277\) −28.8328 −1.73240 −0.866198 0.499701i \(-0.833443\pi\)
−0.866198 + 0.499701i \(0.833443\pi\)
\(278\) −12.8253 −0.769209
\(279\) 1.90211 0.113877
\(280\) 0.726543 0.0434192
\(281\) −10.7921 −0.643805 −0.321902 0.946773i \(-0.604322\pi\)
−0.321902 + 0.946773i \(0.604322\pi\)
\(282\) 10.3262 0.614919
\(283\) 5.27051 0.313299 0.156650 0.987654i \(-0.449931\pi\)
0.156650 + 0.987654i \(0.449931\pi\)
\(284\) −8.05748 −0.478123
\(285\) 0 0
\(286\) 0.729490 0.0431357
\(287\) 0.383516 0.0226382
\(288\) −5.42882 −0.319897
\(289\) −16.4164 −0.965671
\(290\) 10.3262 0.606378
\(291\) 5.00000 0.293105
\(292\) 0.618034 0.0361677
\(293\) 30.8828 1.80419 0.902097 0.431533i \(-0.142027\pi\)
0.902097 + 0.431533i \(0.142027\pi\)
\(294\) −9.59675 −0.559694
\(295\) 11.3067 0.658300
\(296\) 27.0344 1.57135
\(297\) −4.63677 −0.269053
\(298\) −4.80828 −0.278536
\(299\) −5.15131 −0.297908
\(300\) −0.726543 −0.0419470
\(301\) 0.618034 0.0356229
\(302\) 21.1803 1.21879
\(303\) 19.6417 1.12838
\(304\) 0 0
\(305\) 13.9443 0.798447
\(306\) −1.45309 −0.0830673
\(307\) 25.4540 1.45274 0.726369 0.687305i \(-0.241207\pi\)
0.726369 + 0.687305i \(0.241207\pi\)
\(308\) −0.124612 −0.00710042
\(309\) 1.58359 0.0900874
\(310\) 1.38197 0.0784904
\(311\) −7.23607 −0.410320 −0.205160 0.978728i \(-0.565771\pi\)
−0.205160 + 0.978728i \(0.565771\pi\)
\(312\) −2.62866 −0.148818
\(313\) −30.2705 −1.71099 −0.855495 0.517811i \(-0.826747\pi\)
−0.855495 + 0.517811i \(0.826747\pi\)
\(314\) 20.3027 1.14575
\(315\) 0.381966 0.0215213
\(316\) −5.25731 −0.295747
\(317\) −26.6296 −1.49567 −0.747833 0.663887i \(-0.768906\pi\)
−0.747833 + 0.663887i \(0.768906\pi\)
\(318\) 1.38757 0.0778112
\(319\) −7.50245 −0.420057
\(320\) −8.70820 −0.486803
\(321\) 9.14590 0.510474
\(322\) −1.96763 −0.109651
\(323\) 0 0
\(324\) 0.944272 0.0524596
\(325\) 0.726543 0.0403013
\(326\) −3.31471 −0.183585
\(327\) 10.8541 0.600233
\(328\) −5.00000 −0.276079
\(329\) 1.76393 0.0972487
\(330\) −1.18034 −0.0649756
\(331\) 1.79611 0.0987232 0.0493616 0.998781i \(-0.484281\pi\)
0.0493616 + 0.998781i \(0.484281\pi\)
\(332\) 8.18034 0.448954
\(333\) 14.2128 0.778859
\(334\) 15.3262 0.838614
\(335\) 11.5842 0.632912
\(336\) −0.661030 −0.0360622
\(337\) 3.46120 0.188544 0.0942718 0.995547i \(-0.469948\pi\)
0.0942718 + 0.995547i \(0.469948\pi\)
\(338\) −14.6619 −0.797501
\(339\) 5.12461 0.278331
\(340\) 0.472136 0.0256052
\(341\) −1.00406 −0.0543727
\(342\) 0 0
\(343\) −3.29180 −0.177740
\(344\) −8.05748 −0.434430
\(345\) 8.33499 0.448741
\(346\) −15.1246 −0.813104
\(347\) 7.52786 0.404117 0.202058 0.979373i \(-0.435237\pi\)
0.202058 + 0.979373i \(0.435237\pi\)
\(348\) 6.38197 0.342109
\(349\) 17.5623 0.940089 0.470044 0.882643i \(-0.344238\pi\)
0.470044 + 0.882643i \(0.344238\pi\)
\(350\) 0.277515 0.0148338
\(351\) −3.94427 −0.210530
\(352\) 2.86568 0.152741
\(353\) 19.3262 1.02863 0.514316 0.857601i \(-0.328046\pi\)
0.514316 + 0.857601i \(0.328046\pi\)
\(354\) −15.6254 −0.830483
\(355\) −13.0373 −0.691947
\(356\) −5.15131 −0.273019
\(357\) 0.212002 0.0112203
\(358\) 14.2705 0.754220
\(359\) 9.61803 0.507620 0.253810 0.967254i \(-0.418316\pi\)
0.253810 + 0.967254i \(0.418316\pi\)
\(360\) −4.97980 −0.262458
\(361\) 0 0
\(362\) 17.2361 0.905908
\(363\) −12.0737 −0.633705
\(364\) −0.106001 −0.00555597
\(365\) 1.00000 0.0523424
\(366\) −19.2705 −1.00729
\(367\) −27.0689 −1.41298 −0.706492 0.707721i \(-0.749723\pi\)
−0.706492 + 0.707721i \(0.749723\pi\)
\(368\) 16.8885 0.880376
\(369\) −2.62866 −0.136842
\(370\) 10.3262 0.536836
\(371\) 0.237026 0.0123058
\(372\) 0.854102 0.0442831
\(373\) 4.53077 0.234594 0.117297 0.993097i \(-0.462577\pi\)
0.117297 + 0.993097i \(0.462577\pi\)
\(374\) 0.767031 0.0396622
\(375\) −1.17557 −0.0607062
\(376\) −22.9969 −1.18597
\(377\) −6.38197 −0.328688
\(378\) −1.50658 −0.0774900
\(379\) 13.2088 0.678490 0.339245 0.940698i \(-0.389828\pi\)
0.339245 + 0.940698i \(0.389828\pi\)
\(380\) 0 0
\(381\) 2.56231 0.131271
\(382\) 15.2824 0.781916
\(383\) −2.62866 −0.134318 −0.0671590 0.997742i \(-0.521393\pi\)
−0.0671590 + 0.997742i \(0.521393\pi\)
\(384\) 4.14590 0.211569
\(385\) −0.201626 −0.0102758
\(386\) −7.23607 −0.368306
\(387\) −4.23607 −0.215331
\(388\) −2.62866 −0.133450
\(389\) −14.1459 −0.717226 −0.358613 0.933486i \(-0.616750\pi\)
−0.358613 + 0.933486i \(0.616750\pi\)
\(390\) −1.00406 −0.0508424
\(391\) −5.41641 −0.273920
\(392\) 21.3723 1.07946
\(393\) −15.7314 −0.793546
\(394\) −24.5560 −1.23711
\(395\) −8.50651 −0.428009
\(396\) 0.854102 0.0429202
\(397\) 29.8328 1.49727 0.748633 0.662985i \(-0.230711\pi\)
0.748633 + 0.662985i \(0.230711\pi\)
\(398\) −15.7719 −0.790576
\(399\) 0 0
\(400\) −2.38197 −0.119098
\(401\) −10.8576 −0.542205 −0.271103 0.962550i \(-0.587388\pi\)
−0.271103 + 0.962550i \(0.587388\pi\)
\(402\) −16.0090 −0.798454
\(403\) −0.854102 −0.0425458
\(404\) −10.3262 −0.513750
\(405\) 1.52786 0.0759202
\(406\) −2.43769 −0.120981
\(407\) −7.50245 −0.371883
\(408\) −2.76393 −0.136835
\(409\) 18.1231 0.896128 0.448064 0.894001i \(-0.352114\pi\)
0.448064 + 0.894001i \(0.352114\pi\)
\(410\) −1.90983 −0.0943198
\(411\) 16.7355 0.825501
\(412\) −0.832544 −0.0410165
\(413\) −2.66914 −0.131340
\(414\) 13.4863 0.662816
\(415\) 13.2361 0.649733
\(416\) 2.43769 0.119518
\(417\) −12.8253 −0.628056
\(418\) 0 0
\(419\) 3.81966 0.186603 0.0933013 0.995638i \(-0.470258\pi\)
0.0933013 + 0.995638i \(0.470258\pi\)
\(420\) 0.171513 0.00836900
\(421\) −5.98385 −0.291635 −0.145818 0.989311i \(-0.546581\pi\)
−0.145818 + 0.989311i \(0.546581\pi\)
\(422\) 16.7082 0.813343
\(423\) −12.0902 −0.587844
\(424\) −3.09017 −0.150072
\(425\) 0.763932 0.0370561
\(426\) 18.0171 0.872930
\(427\) −3.29180 −0.159301
\(428\) −4.80828 −0.232417
\(429\) 0.729490 0.0352201
\(430\) −3.07768 −0.148419
\(431\) −1.62460 −0.0782542 −0.0391271 0.999234i \(-0.512458\pi\)
−0.0391271 + 0.999234i \(0.512458\pi\)
\(432\) 12.9313 0.622156
\(433\) −22.5478 −1.08358 −0.541790 0.840514i \(-0.682253\pi\)
−0.541790 + 0.840514i \(0.682253\pi\)
\(434\) −0.326238 −0.0156599
\(435\) 10.3262 0.495105
\(436\) −5.70634 −0.273284
\(437\) 0 0
\(438\) −1.38197 −0.0660329
\(439\) −37.6587 −1.79735 −0.898677 0.438611i \(-0.855471\pi\)
−0.898677 + 0.438611i \(0.855471\pi\)
\(440\) 2.62866 0.125316
\(441\) 11.2361 0.535051
\(442\) 0.652476 0.0310351
\(443\) −36.7771 −1.74733 −0.873666 0.486526i \(-0.838264\pi\)
−0.873666 + 0.486526i \(0.838264\pi\)
\(444\) 6.38197 0.302875
\(445\) −8.33499 −0.395117
\(446\) −23.2148 −1.09925
\(447\) −4.80828 −0.227424
\(448\) 2.05573 0.0971240
\(449\) −37.6587 −1.77723 −0.888613 0.458658i \(-0.848330\pi\)
−0.888613 + 0.458658i \(0.848330\pi\)
\(450\) −1.90211 −0.0896665
\(451\) 1.38757 0.0653382
\(452\) −2.69417 −0.126723
\(453\) 21.1803 0.995139
\(454\) −21.9098 −1.02828
\(455\) −0.171513 −0.00804067
\(456\) 0 0
\(457\) 0.763932 0.0357352 0.0178676 0.999840i \(-0.494312\pi\)
0.0178676 + 0.999840i \(0.494312\pi\)
\(458\) −29.2177 −1.36526
\(459\) −4.14725 −0.193577
\(460\) −4.38197 −0.204310
\(461\) −8.00000 −0.372597 −0.186299 0.982493i \(-0.559649\pi\)
−0.186299 + 0.982493i \(0.559649\pi\)
\(462\) 0.278640 0.0129635
\(463\) 6.85410 0.318537 0.159269 0.987235i \(-0.449086\pi\)
0.159269 + 0.987235i \(0.449086\pi\)
\(464\) 20.9232 0.971337
\(465\) 1.38197 0.0640871
\(466\) 7.33094 0.339599
\(467\) 13.1803 0.609913 0.304957 0.952366i \(-0.401358\pi\)
0.304957 + 0.952366i \(0.401358\pi\)
\(468\) 0.726543 0.0335844
\(469\) −2.73466 −0.126275
\(470\) −8.78402 −0.405177
\(471\) 20.3027 0.935499
\(472\) 34.7984 1.60172
\(473\) 2.23607 0.102815
\(474\) 11.7557 0.539957
\(475\) 0 0
\(476\) −0.111456 −0.00510859
\(477\) −1.62460 −0.0743853
\(478\) 33.1685 1.51709
\(479\) −31.5623 −1.44212 −0.721059 0.692873i \(-0.756344\pi\)
−0.721059 + 0.692873i \(0.756344\pi\)
\(480\) −3.94427 −0.180031
\(481\) −6.38197 −0.290993
\(482\) −17.0344 −0.775898
\(483\) −1.96763 −0.0895301
\(484\) 6.34752 0.288524
\(485\) −4.25325 −0.193130
\(486\) 17.0344 0.772698
\(487\) 34.0260 1.54187 0.770933 0.636916i \(-0.219790\pi\)
0.770933 + 0.636916i \(0.219790\pi\)
\(488\) 42.9161 1.94272
\(489\) −3.31471 −0.149896
\(490\) 8.16348 0.368788
\(491\) 3.72949 0.168310 0.0841548 0.996453i \(-0.473181\pi\)
0.0841548 + 0.996453i \(0.473181\pi\)
\(492\) −1.18034 −0.0532138
\(493\) −6.71040 −0.302221
\(494\) 0 0
\(495\) 1.38197 0.0621148
\(496\) 2.80017 0.125731
\(497\) 3.07768 0.138053
\(498\) −18.2918 −0.819675
\(499\) −38.4721 −1.72225 −0.861125 0.508394i \(-0.830239\pi\)
−0.861125 + 0.508394i \(0.830239\pi\)
\(500\) 0.618034 0.0276393
\(501\) 15.3262 0.684726
\(502\) −3.52671 −0.157405
\(503\) 0.347524 0.0154953 0.00774767 0.999970i \(-0.497534\pi\)
0.00774767 + 0.999970i \(0.497534\pi\)
\(504\) 1.17557 0.0523641
\(505\) −16.7082 −0.743505
\(506\) −7.11894 −0.316475
\(507\) −14.6619 −0.651157
\(508\) −1.34708 −0.0597672
\(509\) −10.7516 −0.476558 −0.238279 0.971197i \(-0.576583\pi\)
−0.238279 + 0.971197i \(0.576583\pi\)
\(510\) −1.05573 −0.0467484
\(511\) −0.236068 −0.0104430
\(512\) −22.6538 −1.00117
\(513\) 0 0
\(514\) −16.1803 −0.713684
\(515\) −1.34708 −0.0593596
\(516\) −1.90211 −0.0837359
\(517\) 6.38197 0.280679
\(518\) −2.43769 −0.107106
\(519\) −15.1246 −0.663897
\(520\) 2.23607 0.0980581
\(521\) −6.08985 −0.266801 −0.133401 0.991062i \(-0.542590\pi\)
−0.133401 + 0.991062i \(0.542590\pi\)
\(522\) 16.7082 0.731298
\(523\) 27.6992 1.21120 0.605600 0.795769i \(-0.292933\pi\)
0.605600 + 0.795769i \(0.292933\pi\)
\(524\) 8.27051 0.361299
\(525\) 0.277515 0.0121117
\(526\) −7.05342 −0.307544
\(527\) −0.898056 −0.0391199
\(528\) −2.39163 −0.104082
\(529\) 27.2705 1.18567
\(530\) −1.18034 −0.0512707
\(531\) 18.2946 0.793917
\(532\) 0 0
\(533\) 1.18034 0.0511262
\(534\) 11.5187 0.498462
\(535\) −7.77997 −0.336357
\(536\) 35.6525 1.53995
\(537\) 14.2705 0.615818
\(538\) 7.36068 0.317341
\(539\) −5.93112 −0.255471
\(540\) −3.35520 −0.144385
\(541\) 24.5967 1.05750 0.528748 0.848779i \(-0.322662\pi\)
0.528748 + 0.848779i \(0.322662\pi\)
\(542\) 7.50245 0.322258
\(543\) 17.2361 0.739670
\(544\) 2.56314 0.109894
\(545\) −9.23305 −0.395500
\(546\) 0.237026 0.0101438
\(547\) −13.1433 −0.561966 −0.280983 0.959713i \(-0.590660\pi\)
−0.280983 + 0.959713i \(0.590660\pi\)
\(548\) −8.79837 −0.375848
\(549\) 22.5623 0.962936
\(550\) 1.00406 0.0428131
\(551\) 0 0
\(552\) 25.6525 1.09184
\(553\) 2.00811 0.0853937
\(554\) −33.8950 −1.44006
\(555\) 10.3262 0.438324
\(556\) 6.74265 0.285952
\(557\) 27.0689 1.14695 0.573473 0.819225i \(-0.305596\pi\)
0.573473 + 0.819225i \(0.305596\pi\)
\(558\) 2.23607 0.0946603
\(559\) 1.90211 0.0804508
\(560\) 0.562306 0.0237618
\(561\) 0.767031 0.0323841
\(562\) −12.6869 −0.535165
\(563\) 18.4661 0.778253 0.389127 0.921184i \(-0.372777\pi\)
0.389127 + 0.921184i \(0.372777\pi\)
\(564\) −5.42882 −0.228595
\(565\) −4.35926 −0.183395
\(566\) 6.19586 0.260431
\(567\) −0.360680 −0.0151471
\(568\) −40.1246 −1.68359
\(569\) −7.71445 −0.323407 −0.161703 0.986839i \(-0.551699\pi\)
−0.161703 + 0.986839i \(0.551699\pi\)
\(570\) 0 0
\(571\) −19.1459 −0.801231 −0.400615 0.916246i \(-0.631204\pi\)
−0.400615 + 0.916246i \(0.631204\pi\)
\(572\) −0.383516 −0.0160356
\(573\) 15.2824 0.638432
\(574\) 0.450850 0.0188181
\(575\) −7.09017 −0.295681
\(576\) −14.0902 −0.587090
\(577\) 17.4721 0.727375 0.363687 0.931521i \(-0.381518\pi\)
0.363687 + 0.931521i \(0.381518\pi\)
\(578\) −19.2986 −0.802718
\(579\) −7.23607 −0.300721
\(580\) −5.42882 −0.225420
\(581\) −3.12461 −0.129631
\(582\) 5.87785 0.243645
\(583\) 0.857567 0.0355168
\(584\) 3.07768 0.127355
\(585\) 1.17557 0.0486039
\(586\) 36.3050 1.49974
\(587\) 9.58359 0.395557 0.197779 0.980247i \(-0.436627\pi\)
0.197779 + 0.980247i \(0.436627\pi\)
\(588\) 5.04531 0.208065
\(589\) 0 0
\(590\) 13.2918 0.547215
\(591\) −24.5560 −1.01010
\(592\) 20.9232 0.859940
\(593\) 7.29180 0.299438 0.149719 0.988729i \(-0.452163\pi\)
0.149719 + 0.988729i \(0.452163\pi\)
\(594\) −5.45085 −0.223651
\(595\) −0.180340 −0.00739321
\(596\) 2.52786 0.103545
\(597\) −15.7719 −0.645502
\(598\) −6.05573 −0.247637
\(599\) −10.3431 −0.422608 −0.211304 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(600\) −3.61803 −0.147706
\(601\) −17.0130 −0.693975 −0.346988 0.937870i \(-0.612795\pi\)
−0.346988 + 0.937870i \(0.612795\pi\)
\(602\) 0.726543 0.0296117
\(603\) 18.7436 0.763299
\(604\) −11.1352 −0.453083
\(605\) 10.2705 0.417556
\(606\) 23.0902 0.937974
\(607\) 20.5397 0.833682 0.416841 0.908979i \(-0.363137\pi\)
0.416841 + 0.908979i \(0.363137\pi\)
\(608\) 0 0
\(609\) −2.43769 −0.0987803
\(610\) 16.3925 0.663712
\(611\) 5.42882 0.219627
\(612\) 0.763932 0.0308801
\(613\) 35.4721 1.43271 0.716353 0.697738i \(-0.245810\pi\)
0.716353 + 0.697738i \(0.245810\pi\)
\(614\) 29.9230 1.20759
\(615\) −1.90983 −0.0770118
\(616\) −0.620541 −0.0250023
\(617\) 19.3820 0.780289 0.390144 0.920754i \(-0.372425\pi\)
0.390144 + 0.920754i \(0.372425\pi\)
\(618\) 1.86162 0.0748855
\(619\) −27.5410 −1.10697 −0.553484 0.832860i \(-0.686702\pi\)
−0.553484 + 0.832860i \(0.686702\pi\)
\(620\) −0.726543 −0.0291787
\(621\) 38.4913 1.54460
\(622\) −8.50651 −0.341080
\(623\) 1.96763 0.0788312
\(624\) −2.03444 −0.0814429
\(625\) 1.00000 0.0400000
\(626\) −35.5851 −1.42227
\(627\) 0 0
\(628\) −10.6738 −0.425929
\(629\) −6.71040 −0.267561
\(630\) 0.449028 0.0178897
\(631\) −18.8197 −0.749199 −0.374599 0.927187i \(-0.622220\pi\)
−0.374599 + 0.927187i \(0.622220\pi\)
\(632\) −26.1803 −1.04140
\(633\) 16.7082 0.664091
\(634\) −31.3050 −1.24328
\(635\) −2.17963 −0.0864959
\(636\) −0.729490 −0.0289262
\(637\) −5.04531 −0.199902
\(638\) −8.81966 −0.349174
\(639\) −21.0948 −0.834496
\(640\) −3.52671 −0.139406
\(641\) 38.4508 1.51872 0.759358 0.650673i \(-0.225513\pi\)
0.759358 + 0.650673i \(0.225513\pi\)
\(642\) 10.7516 0.424334
\(643\) −29.5279 −1.16447 −0.582233 0.813022i \(-0.697821\pi\)
−0.582233 + 0.813022i \(0.697821\pi\)
\(644\) 1.03444 0.0407627
\(645\) −3.07768 −0.121184
\(646\) 0 0
\(647\) 17.4721 0.686901 0.343450 0.939171i \(-0.388404\pi\)
0.343450 + 0.939171i \(0.388404\pi\)
\(648\) 4.70228 0.184723
\(649\) −9.65706 −0.379073
\(650\) 0.854102 0.0335006
\(651\) −0.326238 −0.0127863
\(652\) 1.74265 0.0682473
\(653\) 10.7984 0.422573 0.211287 0.977424i \(-0.432235\pi\)
0.211287 + 0.977424i \(0.432235\pi\)
\(654\) 12.7598 0.498946
\(655\) 13.3820 0.522877
\(656\) −3.86974 −0.151088
\(657\) 1.61803 0.0631255
\(658\) 2.07363 0.0808384
\(659\) −30.7768 −1.19890 −0.599448 0.800414i \(-0.704613\pi\)
−0.599448 + 0.800414i \(0.704613\pi\)
\(660\) 0.620541 0.0241545
\(661\) −13.4208 −0.522008 −0.261004 0.965338i \(-0.584054\pi\)
−0.261004 + 0.965338i \(0.584054\pi\)
\(662\) 2.11146 0.0820641
\(663\) 0.652476 0.0253401
\(664\) 40.7364 1.58088
\(665\) 0 0
\(666\) 16.7082 0.647430
\(667\) 62.2802 2.41150
\(668\) −8.05748 −0.311753
\(669\) −23.2148 −0.897535
\(670\) 13.6180 0.526111
\(671\) −11.9098 −0.459774
\(672\) 0.931116 0.0359186
\(673\) 34.9646 1.34779 0.673893 0.738829i \(-0.264621\pi\)
0.673893 + 0.738829i \(0.264621\pi\)
\(674\) 4.06888 0.156728
\(675\) −5.42882 −0.208956
\(676\) 7.70820 0.296469
\(677\) 16.0494 0.616830 0.308415 0.951252i \(-0.400201\pi\)
0.308415 + 0.951252i \(0.400201\pi\)
\(678\) 6.02434 0.231363
\(679\) 1.00406 0.0385322
\(680\) 2.35114 0.0901621
\(681\) −21.9098 −0.839587
\(682\) −1.18034 −0.0451976
\(683\) 32.6134 1.24792 0.623959 0.781457i \(-0.285523\pi\)
0.623959 + 0.781457i \(0.285523\pi\)
\(684\) 0 0
\(685\) −14.2361 −0.543932
\(686\) −3.86974 −0.147747
\(687\) −29.2177 −1.11473
\(688\) −6.23607 −0.237748
\(689\) 0.729490 0.0277914
\(690\) 9.79837 0.373018
\(691\) −7.76393 −0.295354 −0.147677 0.989036i \(-0.547180\pi\)
−0.147677 + 0.989036i \(0.547180\pi\)
\(692\) 7.95148 0.302270
\(693\) −0.326238 −0.0123928
\(694\) 8.84953 0.335924
\(695\) 10.9098 0.413833
\(696\) 31.7809 1.20465
\(697\) 1.24108 0.0470094
\(698\) 20.6457 0.781452
\(699\) 7.33094 0.277282
\(700\) −0.145898 −0.00551443
\(701\) −43.0902 −1.62749 −0.813747 0.581220i \(-0.802576\pi\)
−0.813747 + 0.581220i \(0.802576\pi\)
\(702\) −4.63677 −0.175004
\(703\) 0 0
\(704\) 7.43769 0.280319
\(705\) −8.78402 −0.330825
\(706\) 22.7194 0.855054
\(707\) 3.94427 0.148340
\(708\) 8.21478 0.308730
\(709\) 13.8197 0.519008 0.259504 0.965742i \(-0.416441\pi\)
0.259504 + 0.965742i \(0.416441\pi\)
\(710\) −15.3262 −0.575183
\(711\) −13.7638 −0.516184
\(712\) −25.6525 −0.961367
\(713\) 8.33499 0.312148
\(714\) 0.249224 0.00932696
\(715\) −0.620541 −0.0232069
\(716\) −7.50245 −0.280380
\(717\) 33.1685 1.23870
\(718\) 11.3067 0.421961
\(719\) −42.0902 −1.56970 −0.784849 0.619687i \(-0.787260\pi\)
−0.784849 + 0.619687i \(0.787260\pi\)
\(720\) −3.85410 −0.143634
\(721\) 0.318003 0.0118431
\(722\) 0 0
\(723\) −17.0344 −0.633518
\(724\) −9.06154 −0.336769
\(725\) −8.78402 −0.326230
\(726\) −14.1935 −0.526770
\(727\) −10.3607 −0.384256 −0.192128 0.981370i \(-0.561539\pi\)
−0.192128 + 0.981370i \(0.561539\pi\)
\(728\) −0.527864 −0.0195639
\(729\) 21.6180 0.800668
\(730\) 1.17557 0.0435098
\(731\) 2.00000 0.0739727
\(732\) 10.1311 0.374456
\(733\) −11.0000 −0.406294 −0.203147 0.979148i \(-0.565117\pi\)
−0.203147 + 0.979148i \(0.565117\pi\)
\(734\) −31.8214 −1.17455
\(735\) 8.16348 0.301115
\(736\) −23.7889 −0.876871
\(737\) −9.89408 −0.364453
\(738\) −3.09017 −0.113751
\(739\) −32.2361 −1.18582 −0.592911 0.805268i \(-0.702022\pi\)
−0.592911 + 0.805268i \(0.702022\pi\)
\(740\) −5.42882 −0.199568
\(741\) 0 0
\(742\) 0.278640 0.0102292
\(743\) 11.5842 0.424983 0.212491 0.977163i \(-0.431842\pi\)
0.212491 + 0.977163i \(0.431842\pi\)
\(744\) 4.25325 0.155932
\(745\) 4.09017 0.149852
\(746\) 5.32624 0.195007
\(747\) 21.4164 0.783585
\(748\) −0.403252 −0.0147444
\(749\) 1.83660 0.0671079
\(750\) −1.38197 −0.0504623
\(751\) −41.7000 −1.52165 −0.760827 0.648954i \(-0.775207\pi\)
−0.760827 + 0.648954i \(0.775207\pi\)
\(752\) −17.7984 −0.649040
\(753\) −3.52671 −0.128521
\(754\) −7.50245 −0.273223
\(755\) −18.0171 −0.655708
\(756\) 0.792055 0.0288068
\(757\) −43.9787 −1.59843 −0.799217 0.601043i \(-0.794752\pi\)
−0.799217 + 0.601043i \(0.794752\pi\)
\(758\) 15.5279 0.563997
\(759\) −7.11894 −0.258401
\(760\) 0 0
\(761\) −41.1803 −1.49279 −0.746393 0.665505i \(-0.768216\pi\)
−0.746393 + 0.665505i \(0.768216\pi\)
\(762\) 3.01217 0.109119
\(763\) 2.17963 0.0789078
\(764\) −8.03444 −0.290676
\(765\) 1.23607 0.0446901
\(766\) −3.09017 −0.111652
\(767\) −8.21478 −0.296619
\(768\) −15.6004 −0.562932
\(769\) 20.2705 0.730973 0.365487 0.930817i \(-0.380903\pi\)
0.365487 + 0.930817i \(0.380903\pi\)
\(770\) −0.237026 −0.00854181
\(771\) −16.1803 −0.582721
\(772\) 3.80423 0.136917
\(773\) 6.94742 0.249881 0.124941 0.992164i \(-0.460126\pi\)
0.124941 + 0.992164i \(0.460126\pi\)
\(774\) −4.97980 −0.178995
\(775\) −1.17557 −0.0422277
\(776\) −13.0902 −0.469910
\(777\) −2.43769 −0.0874518
\(778\) −16.6295 −0.596196
\(779\) 0 0
\(780\) 0.527864 0.0189006
\(781\) 11.1352 0.398447
\(782\) −6.36737 −0.227697
\(783\) 47.6869 1.70419
\(784\) 16.5410 0.590751
\(785\) −17.2705 −0.616411
\(786\) −18.4934 −0.659639
\(787\) −43.9856 −1.56792 −0.783959 0.620812i \(-0.786803\pi\)
−0.783959 + 0.620812i \(0.786803\pi\)
\(788\) 12.9098 0.459894
\(789\) −7.05342 −0.251109
\(790\) −10.0000 −0.355784
\(791\) 1.02908 0.0365899
\(792\) 4.25325 0.151133
\(793\) −10.1311 −0.359766
\(794\) 35.0706 1.24461
\(795\) −1.18034 −0.0418623
\(796\) 8.29180 0.293895
\(797\) 4.14725 0.146903 0.0734516 0.997299i \(-0.476599\pi\)
0.0734516 + 0.997299i \(0.476599\pi\)
\(798\) 0 0
\(799\) 5.70820 0.201942
\(800\) 3.35520 0.118624
\(801\) −13.4863 −0.476515
\(802\) −12.7639 −0.450710
\(803\) −0.854102 −0.0301406
\(804\) 8.41641 0.296824
\(805\) 1.67376 0.0589924
\(806\) −1.00406 −0.0353664
\(807\) 7.36068 0.259108
\(808\) −51.4226 −1.80904
\(809\) −1.78522 −0.0627649 −0.0313825 0.999507i \(-0.509991\pi\)
−0.0313825 + 0.999507i \(0.509991\pi\)
\(810\) 1.79611 0.0631089
\(811\) 21.2663 0.746760 0.373380 0.927679i \(-0.378199\pi\)
0.373380 + 0.927679i \(0.378199\pi\)
\(812\) 1.28157 0.0449743
\(813\) 7.50245 0.263122
\(814\) −8.81966 −0.309129
\(815\) 2.81966 0.0987684
\(816\) −2.13914 −0.0748848
\(817\) 0 0
\(818\) 21.3050 0.744910
\(819\) −0.277515 −0.00969714
\(820\) 1.00406 0.0350632
\(821\) 28.4508 0.992942 0.496471 0.868053i \(-0.334629\pi\)
0.496471 + 0.868053i \(0.334629\pi\)
\(822\) 19.6738 0.686201
\(823\) −21.2361 −0.740243 −0.370121 0.928983i \(-0.620684\pi\)
−0.370121 + 0.928983i \(0.620684\pi\)
\(824\) −4.14590 −0.144429
\(825\) 1.00406 0.0349568
\(826\) −3.13777 −0.109177
\(827\) −2.17963 −0.0757931 −0.0378965 0.999282i \(-0.512066\pi\)
−0.0378965 + 0.999282i \(0.512066\pi\)
\(828\) −7.09017 −0.246400
\(829\) 20.7517 0.720737 0.360369 0.932810i \(-0.382651\pi\)
0.360369 + 0.932810i \(0.382651\pi\)
\(830\) 15.5599 0.540093
\(831\) −33.8950 −1.17580
\(832\) 6.32688 0.219345
\(833\) −5.30495 −0.183806
\(834\) −15.0770 −0.522074
\(835\) −13.0373 −0.451174
\(836\) 0 0
\(837\) 6.38197 0.220593
\(838\) 4.49028 0.155114
\(839\) 10.4086 0.359346 0.179673 0.983726i \(-0.442496\pi\)
0.179673 + 0.983726i \(0.442496\pi\)
\(840\) 0.854102 0.0294693
\(841\) 48.1591 1.66066
\(842\) −7.03444 −0.242423
\(843\) −12.6869 −0.436961
\(844\) −8.78402 −0.302359
\(845\) 12.4721 0.429055
\(846\) −14.2128 −0.488648
\(847\) −2.42454 −0.0833081
\(848\) −2.39163 −0.0821289
\(849\) 6.19586 0.212641
\(850\) 0.898056 0.0308031
\(851\) 62.2802 2.13494
\(852\) −9.47214 −0.324510
\(853\) −6.52786 −0.223510 −0.111755 0.993736i \(-0.535647\pi\)
−0.111755 + 0.993736i \(0.535647\pi\)
\(854\) −3.86974 −0.132420
\(855\) 0 0
\(856\) −23.9443 −0.818398
\(857\) 35.5851 1.21556 0.607782 0.794104i \(-0.292059\pi\)
0.607782 + 0.794104i \(0.292059\pi\)
\(858\) 0.857567 0.0292769
\(859\) −27.2361 −0.929283 −0.464641 0.885499i \(-0.653817\pi\)
−0.464641 + 0.885499i \(0.653817\pi\)
\(860\) 1.61803 0.0551745
\(861\) 0.450850 0.0153649
\(862\) −1.90983 −0.0650491
\(863\) 24.8335 0.845341 0.422671 0.906283i \(-0.361093\pi\)
0.422671 + 0.906283i \(0.361093\pi\)
\(864\) −18.2148 −0.619679
\(865\) 12.8658 0.437449
\(866\) −26.5066 −0.900730
\(867\) −19.2986 −0.655416
\(868\) 0.171513 0.00582154
\(869\) 7.26543 0.246463
\(870\) 12.1392 0.411558
\(871\) −8.41641 −0.285179
\(872\) −28.4164 −0.962301
\(873\) −6.88191 −0.232917
\(874\) 0 0
\(875\) −0.236068 −0.00798055
\(876\) 0.726543 0.0245476
\(877\) 23.1684 0.782341 0.391170 0.920318i \(-0.372070\pi\)
0.391170 + 0.920318i \(0.372070\pi\)
\(878\) −44.2705 −1.49406
\(879\) 36.3050 1.22454
\(880\) 2.03444 0.0685810
\(881\) −5.32624 −0.179446 −0.0897228 0.995967i \(-0.528598\pi\)
−0.0897228 + 0.995967i \(0.528598\pi\)
\(882\) 13.2088 0.444763
\(883\) −57.2148 −1.92543 −0.962715 0.270516i \(-0.912806\pi\)
−0.962715 + 0.270516i \(0.912806\pi\)
\(884\) −0.343027 −0.0115372
\(885\) 13.2918 0.446799
\(886\) −43.2341 −1.45248
\(887\) 19.3642 0.650185 0.325092 0.945682i \(-0.394605\pi\)
0.325092 + 0.945682i \(0.394605\pi\)
\(888\) 31.7809 1.06650
\(889\) 0.514540 0.0172571
\(890\) −9.79837 −0.328442
\(891\) −1.30495 −0.0437175
\(892\) 12.2047 0.408645
\(893\) 0 0
\(894\) −5.65248 −0.189047
\(895\) −12.1392 −0.405769
\(896\) 0.832544 0.0278133
\(897\) −6.05573 −0.202195
\(898\) −44.2705 −1.47733
\(899\) 10.3262 0.344399
\(900\) 1.00000 0.0333333
\(901\) 0.767031 0.0255535
\(902\) 1.63119 0.0543127
\(903\) 0.726543 0.0241778
\(904\) −13.4164 −0.446223
\(905\) −14.6619 −0.487377
\(906\) 24.8990 0.827213
\(907\) −41.5035 −1.37810 −0.689050 0.724714i \(-0.741972\pi\)
−0.689050 + 0.724714i \(0.741972\pi\)
\(908\) 11.5187 0.382261
\(909\) −27.0344 −0.896676
\(910\) −0.201626 −0.00668384
\(911\) 12.9718 0.429774 0.214887 0.976639i \(-0.431062\pi\)
0.214887 + 0.976639i \(0.431062\pi\)
\(912\) 0 0
\(913\) −11.3050 −0.374139
\(914\) 0.898056 0.0297051
\(915\) 16.3925 0.541918
\(916\) 15.3607 0.507531
\(917\) −3.15905 −0.104321
\(918\) −4.87539 −0.160912
\(919\) −37.3607 −1.23242 −0.616208 0.787584i \(-0.711332\pi\)
−0.616208 + 0.787584i \(0.711332\pi\)
\(920\) −21.8213 −0.719427
\(921\) 29.9230 0.985996
\(922\) −9.40456 −0.309723
\(923\) 9.47214 0.311779
\(924\) −0.146490 −0.00481917
\(925\) −8.78402 −0.288817
\(926\) 8.05748 0.264785
\(927\) −2.17963 −0.0715884
\(928\) −29.4721 −0.967470
\(929\) 17.4377 0.572112 0.286056 0.958213i \(-0.407656\pi\)
0.286056 + 0.958213i \(0.407656\pi\)
\(930\) 1.62460 0.0532727
\(931\) 0 0
\(932\) −3.85410 −0.126245
\(933\) −8.50651 −0.278491
\(934\) 15.4944 0.506993
\(935\) −0.652476 −0.0213382
\(936\) 3.61803 0.118259
\(937\) 47.7984 1.56150 0.780752 0.624841i \(-0.214836\pi\)
0.780752 + 0.624841i \(0.214836\pi\)
\(938\) −3.21478 −0.104966
\(939\) −35.5851 −1.16128
\(940\) 4.61803 0.150624
\(941\) −39.6013 −1.29097 −0.645484 0.763774i \(-0.723344\pi\)
−0.645484 + 0.763774i \(0.723344\pi\)
\(942\) 23.8673 0.777637
\(943\) −11.5187 −0.375100
\(944\) 26.9321 0.876566
\(945\) 1.28157 0.0416895
\(946\) 2.62866 0.0854650
\(947\) −25.4164 −0.825922 −0.412961 0.910749i \(-0.635505\pi\)
−0.412961 + 0.910749i \(0.635505\pi\)
\(948\) −6.18034 −0.200728
\(949\) −0.726543 −0.0235846
\(950\) 0 0
\(951\) −31.3050 −1.01513
\(952\) −0.555029 −0.0179886
\(953\) −55.0148 −1.78210 −0.891052 0.453901i \(-0.850032\pi\)
−0.891052 + 0.453901i \(0.850032\pi\)
\(954\) −1.90983 −0.0618330
\(955\) −13.0000 −0.420670
\(956\) −17.4377 −0.563975
\(957\) −8.81966 −0.285099
\(958\) −37.1037 −1.19877
\(959\) 3.36068 0.108522
\(960\) −10.2371 −0.330401
\(961\) −29.6180 −0.955420
\(962\) −7.50245 −0.241889
\(963\) −12.5882 −0.405651
\(964\) 8.95554 0.288438
\(965\) 6.15537 0.198148
\(966\) −2.31308 −0.0744222
\(967\) −46.9443 −1.50963 −0.754813 0.655940i \(-0.772272\pi\)
−0.754813 + 0.655940i \(0.772272\pi\)
\(968\) 31.6094 1.01596
\(969\) 0 0
\(970\) −5.00000 −0.160540
\(971\) −14.5964 −0.468420 −0.234210 0.972186i \(-0.575250\pi\)
−0.234210 + 0.972186i \(0.575250\pi\)
\(972\) −8.95554 −0.287249
\(973\) −2.57546 −0.0825655
\(974\) 40.0000 1.28168
\(975\) 0.854102 0.0273532
\(976\) 33.2148 1.06318
\(977\) 4.31877 0.138170 0.0690848 0.997611i \(-0.477992\pi\)
0.0690848 + 0.997611i \(0.477992\pi\)
\(978\) −3.89667 −0.124602
\(979\) 7.11894 0.227522
\(980\) −4.29180 −0.137096
\(981\) −14.9394 −0.476978
\(982\) 4.38428 0.139908
\(983\) −6.26137 −0.199707 −0.0998533 0.995002i \(-0.531837\pi\)
−0.0998533 + 0.995002i \(0.531837\pi\)
\(984\) −5.87785 −0.187379
\(985\) 20.8885 0.665564
\(986\) −7.88854 −0.251222
\(987\) 2.07363 0.0660043
\(988\) 0 0
\(989\) −18.5623 −0.590247
\(990\) 1.62460 0.0516331
\(991\) −17.2500 −0.547966 −0.273983 0.961735i \(-0.588341\pi\)
−0.273983 + 0.961735i \(0.588341\pi\)
\(992\) −3.94427 −0.125231
\(993\) 2.11146 0.0670050
\(994\) 3.61803 0.114757
\(995\) 13.4164 0.425329
\(996\) 9.61657 0.304713
\(997\) −19.0557 −0.603501 −0.301750 0.953387i \(-0.597571\pi\)
−0.301750 + 0.953387i \(0.597571\pi\)
\(998\) −45.2267 −1.43163
\(999\) 47.6869 1.50875
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 1805.2.a.l.1.3 yes 4
5.4 even 2 9025.2.a.bl.1.2 4
19.18 odd 2 inner 1805.2.a.l.1.2 4
95.94 odd 2 9025.2.a.bl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1805.2.a.l.1.2 4 19.18 odd 2 inner
1805.2.a.l.1.3 yes 4 1.1 even 1 trivial
9025.2.a.bl.1.2 4 5.4 even 2
9025.2.a.bl.1.3 4 95.94 odd 2