Properties

Label 405.4.a.i.1.3
Level $405$
Weight $4$
Character 405.1
Self dual yes
Analytic conductor $23.896$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 405 = 3^{4} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 405.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(23.8957735523\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.7032.1
Defining polynomial: \( x^{3} - x^{2} - 14x + 18 \) 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 \(3.52348\) of defining polynomial
Character \(\chi\) \(=\) 405.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.52348 q^{2} +4.41489 q^{4} -5.00000 q^{5} +25.4325 q^{7} -12.6321 q^{8} +O(q^{10})\) \(q+3.52348 q^{2} +4.41489 q^{4} -5.00000 q^{5} +25.4325 q^{7} -12.6321 q^{8} -17.6174 q^{10} +71.3561 q^{11} -51.3935 q^{13} +89.6107 q^{14} -79.8279 q^{16} +33.3191 q^{17} +113.372 q^{19} -22.0744 q^{20} +251.422 q^{22} -81.9767 q^{23} +25.0000 q^{25} -181.084 q^{26} +112.281 q^{28} +246.827 q^{29} +222.679 q^{31} -180.215 q^{32} +117.399 q^{34} -127.162 q^{35} +22.3910 q^{37} +399.463 q^{38} +63.1603 q^{40} +434.225 q^{41} -236.850 q^{43} +315.029 q^{44} -288.843 q^{46} +107.984 q^{47} +303.810 q^{49} +88.0869 q^{50} -226.896 q^{52} +123.961 q^{53} -356.781 q^{55} -321.265 q^{56} +869.690 q^{58} -171.091 q^{59} -79.4391 q^{61} +784.604 q^{62} +3.63945 q^{64} +256.968 q^{65} -611.506 q^{67} +147.100 q^{68} -448.053 q^{70} -511.102 q^{71} -410.012 q^{73} +78.8941 q^{74} +500.524 q^{76} +1814.76 q^{77} -793.597 q^{79} +399.139 q^{80} +1529.98 q^{82} -270.081 q^{83} -166.595 q^{85} -834.534 q^{86} -901.376 q^{88} -177.400 q^{89} -1307.06 q^{91} -361.918 q^{92} +380.478 q^{94} -566.859 q^{95} -881.860 q^{97} +1070.47 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} + 5 q^{4} - 15 q^{5} - 25 q^{7} - 27 q^{8} - 5 q^{10} + 58 q^{11} - 47 q^{13} + 159 q^{14} - 127 q^{16} + 34 q^{17} - 5 q^{19} - 25 q^{20} + 260 q^{22} - 51 q^{23} + 75 q^{25} + 253 q^{26} + 83 q^{28} + 350 q^{29} + 638 q^{31} + 245 q^{32} - 154 q^{34} + 125 q^{35} - 414 q^{37} + 397 q^{38} + 135 q^{40} + 179 q^{41} - 836 q^{43} + 332 q^{44} + 261 q^{46} + 235 q^{47} + 892 q^{49} + 25 q^{50} - 1335 q^{52} + 505 q^{53} - 290 q^{55} + 15 q^{56} + 1876 q^{58} + 535 q^{59} - 104 q^{61} + 348 q^{62} - 303 q^{64} + 235 q^{65} - 40 q^{67} + 830 q^{68} - 795 q^{70} + 452 q^{71} - 710 q^{73} + 1394 q^{74} + 849 q^{76} + 2148 q^{77} - 634 q^{79} + 635 q^{80} + 613 q^{82} + 1734 q^{83} - 170 q^{85} + 460 q^{86} - 768 q^{88} - 852 q^{89} - 1229 q^{91} - 1839 q^{92} + 1751 q^{94} + 25 q^{95} + 38 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.52348 1.24574 0.622868 0.782327i \(-0.285967\pi\)
0.622868 + 0.782327i \(0.285967\pi\)
\(3\) 0 0
\(4\) 4.41489 0.551861
\(5\) −5.00000 −0.447214
\(6\) 0 0
\(7\) 25.4325 1.37322 0.686612 0.727024i \(-0.259097\pi\)
0.686612 + 0.727024i \(0.259097\pi\)
\(8\) −12.6321 −0.558264
\(9\) 0 0
\(10\) −17.6174 −0.557111
\(11\) 71.3561 1.95588 0.977940 0.208885i \(-0.0669834\pi\)
0.977940 + 0.208885i \(0.0669834\pi\)
\(12\) 0 0
\(13\) −51.3935 −1.09646 −0.548231 0.836327i \(-0.684698\pi\)
−0.548231 + 0.836327i \(0.684698\pi\)
\(14\) 89.6107 1.71068
\(15\) 0 0
\(16\) −79.8279 −1.24731
\(17\) 33.3191 0.475357 0.237678 0.971344i \(-0.423614\pi\)
0.237678 + 0.971344i \(0.423614\pi\)
\(18\) 0 0
\(19\) 113.372 1.36891 0.684455 0.729055i \(-0.260040\pi\)
0.684455 + 0.729055i \(0.260040\pi\)
\(20\) −22.0744 −0.246800
\(21\) 0 0
\(22\) 251.422 2.43651
\(23\) −81.9767 −0.743188 −0.371594 0.928395i \(-0.621189\pi\)
−0.371594 + 0.928395i \(0.621189\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) −181.084 −1.36590
\(27\) 0 0
\(28\) 112.281 0.757828
\(29\) 246.827 1.58051 0.790253 0.612781i \(-0.209949\pi\)
0.790253 + 0.612781i \(0.209949\pi\)
\(30\) 0 0
\(31\) 222.679 1.29014 0.645070 0.764124i \(-0.276828\pi\)
0.645070 + 0.764124i \(0.276828\pi\)
\(32\) −180.215 −0.995557
\(33\) 0 0
\(34\) 117.399 0.592169
\(35\) −127.162 −0.614124
\(36\) 0 0
\(37\) 22.3910 0.0994880 0.0497440 0.998762i \(-0.484159\pi\)
0.0497440 + 0.998762i \(0.484159\pi\)
\(38\) 399.463 1.70530
\(39\) 0 0
\(40\) 63.1603 0.249663
\(41\) 434.225 1.65401 0.827007 0.562191i \(-0.190041\pi\)
0.827007 + 0.562191i \(0.190041\pi\)
\(42\) 0 0
\(43\) −236.850 −0.839982 −0.419991 0.907528i \(-0.637967\pi\)
−0.419991 + 0.907528i \(0.637967\pi\)
\(44\) 315.029 1.07937
\(45\) 0 0
\(46\) −288.843 −0.925817
\(47\) 107.984 0.335129 0.167564 0.985861i \(-0.446410\pi\)
0.167564 + 0.985861i \(0.446410\pi\)
\(48\) 0 0
\(49\) 303.810 0.885744
\(50\) 88.0869 0.249147
\(51\) 0 0
\(52\) −226.896 −0.605094
\(53\) 123.961 0.321272 0.160636 0.987014i \(-0.448646\pi\)
0.160636 + 0.987014i \(0.448646\pi\)
\(54\) 0 0
\(55\) −356.781 −0.874696
\(56\) −321.265 −0.766621
\(57\) 0 0
\(58\) 869.690 1.96889
\(59\) −171.091 −0.377528 −0.188764 0.982023i \(-0.560448\pi\)
−0.188764 + 0.982023i \(0.560448\pi\)
\(60\) 0 0
\(61\) −79.4391 −0.166740 −0.0833700 0.996519i \(-0.526568\pi\)
−0.0833700 + 0.996519i \(0.526568\pi\)
\(62\) 784.604 1.60717
\(63\) 0 0
\(64\) 3.63945 0.00710830
\(65\) 256.968 0.490352
\(66\) 0 0
\(67\) −611.506 −1.11503 −0.557517 0.830165i \(-0.688246\pi\)
−0.557517 + 0.830165i \(0.688246\pi\)
\(68\) 147.100 0.262331
\(69\) 0 0
\(70\) −448.053 −0.765038
\(71\) −511.102 −0.854318 −0.427159 0.904176i \(-0.640486\pi\)
−0.427159 + 0.904176i \(0.640486\pi\)
\(72\) 0 0
\(73\) −410.012 −0.657373 −0.328686 0.944439i \(-0.606606\pi\)
−0.328686 + 0.944439i \(0.606606\pi\)
\(74\) 78.8941 0.123936
\(75\) 0 0
\(76\) 500.524 0.755447
\(77\) 1814.76 2.68586
\(78\) 0 0
\(79\) −793.597 −1.13021 −0.565105 0.825019i \(-0.691164\pi\)
−0.565105 + 0.825019i \(0.691164\pi\)
\(80\) 399.139 0.557814
\(81\) 0 0
\(82\) 1529.98 2.06047
\(83\) −270.081 −0.357171 −0.178586 0.983924i \(-0.557152\pi\)
−0.178586 + 0.983924i \(0.557152\pi\)
\(84\) 0 0
\(85\) −166.595 −0.212586
\(86\) −834.534 −1.04640
\(87\) 0 0
\(88\) −901.376 −1.09190
\(89\) −177.400 −0.211284 −0.105642 0.994404i \(-0.533690\pi\)
−0.105642 + 0.994404i \(0.533690\pi\)
\(90\) 0 0
\(91\) −1307.06 −1.50569
\(92\) −361.918 −0.410136
\(93\) 0 0
\(94\) 380.478 0.417482
\(95\) −566.859 −0.612195
\(96\) 0 0
\(97\) −881.860 −0.923086 −0.461543 0.887118i \(-0.652704\pi\)
−0.461543 + 0.887118i \(0.652704\pi\)
\(98\) 1070.47 1.10340
\(99\) 0 0
\(100\) 110.372 0.110372
\(101\) −1026.03 −1.01083 −0.505417 0.862875i \(-0.668661\pi\)
−0.505417 + 0.862875i \(0.668661\pi\)
\(102\) 0 0
\(103\) −1790.28 −1.71263 −0.856317 0.516451i \(-0.827253\pi\)
−0.856317 + 0.516451i \(0.827253\pi\)
\(104\) 649.206 0.612115
\(105\) 0 0
\(106\) 436.775 0.400220
\(107\) 2007.30 1.81358 0.906788 0.421587i \(-0.138527\pi\)
0.906788 + 0.421587i \(0.138527\pi\)
\(108\) 0 0
\(109\) −211.654 −0.185989 −0.0929945 0.995667i \(-0.529644\pi\)
−0.0929945 + 0.995667i \(0.529644\pi\)
\(110\) −1257.11 −1.08964
\(111\) 0 0
\(112\) −2030.22 −1.71284
\(113\) −1.94491 −0.00161913 −0.000809567 1.00000i \(-0.500258\pi\)
−0.000809567 1.00000i \(0.500258\pi\)
\(114\) 0 0
\(115\) 409.884 0.332364
\(116\) 1089.71 0.872219
\(117\) 0 0
\(118\) −602.835 −0.470300
\(119\) 847.386 0.652771
\(120\) 0 0
\(121\) 3760.70 2.82547
\(122\) −279.902 −0.207714
\(123\) 0 0
\(124\) 983.102 0.711977
\(125\) −125.000 −0.0894427
\(126\) 0 0
\(127\) −1178.02 −0.823089 −0.411544 0.911390i \(-0.635011\pi\)
−0.411544 + 0.911390i \(0.635011\pi\)
\(128\) 1454.54 1.00441
\(129\) 0 0
\(130\) 905.419 0.610850
\(131\) −521.218 −0.347626 −0.173813 0.984779i \(-0.555609\pi\)
−0.173813 + 0.984779i \(0.555609\pi\)
\(132\) 0 0
\(133\) 2883.32 1.87982
\(134\) −2154.63 −1.38904
\(135\) 0 0
\(136\) −420.889 −0.265374
\(137\) 449.532 0.280336 0.140168 0.990128i \(-0.455236\pi\)
0.140168 + 0.990128i \(0.455236\pi\)
\(138\) 0 0
\(139\) −2345.45 −1.43121 −0.715605 0.698505i \(-0.753849\pi\)
−0.715605 + 0.698505i \(0.753849\pi\)
\(140\) −561.407 −0.338911
\(141\) 0 0
\(142\) −1800.85 −1.06426
\(143\) −3667.24 −2.14455
\(144\) 0 0
\(145\) −1234.14 −0.706824
\(146\) −1444.67 −0.818914
\(147\) 0 0
\(148\) 98.8537 0.0549035
\(149\) 1269.38 0.697932 0.348966 0.937135i \(-0.386533\pi\)
0.348966 + 0.937135i \(0.386533\pi\)
\(150\) 0 0
\(151\) 2158.32 1.16319 0.581594 0.813480i \(-0.302429\pi\)
0.581594 + 0.813480i \(0.302429\pi\)
\(152\) −1432.12 −0.764213
\(153\) 0 0
\(154\) 6394.27 3.34588
\(155\) −1113.40 −0.576968
\(156\) 0 0
\(157\) −2343.31 −1.19119 −0.595593 0.803286i \(-0.703083\pi\)
−0.595593 + 0.803286i \(0.703083\pi\)
\(158\) −2796.22 −1.40794
\(159\) 0 0
\(160\) 901.075 0.445227
\(161\) −2084.87 −1.02056
\(162\) 0 0
\(163\) 3399.94 1.63376 0.816882 0.576804i \(-0.195700\pi\)
0.816882 + 0.576804i \(0.195700\pi\)
\(164\) 1917.06 0.912786
\(165\) 0 0
\(166\) −951.623 −0.444942
\(167\) 1809.22 0.838332 0.419166 0.907910i \(-0.362323\pi\)
0.419166 + 0.907910i \(0.362323\pi\)
\(168\) 0 0
\(169\) 444.292 0.202227
\(170\) −586.995 −0.264826
\(171\) 0 0
\(172\) −1045.66 −0.463553
\(173\) 1327.41 0.583358 0.291679 0.956516i \(-0.405786\pi\)
0.291679 + 0.956516i \(0.405786\pi\)
\(174\) 0 0
\(175\) 635.812 0.274645
\(176\) −5696.21 −2.43959
\(177\) 0 0
\(178\) −625.063 −0.263205
\(179\) 448.734 0.187374 0.0936871 0.995602i \(-0.470135\pi\)
0.0936871 + 0.995602i \(0.470135\pi\)
\(180\) 0 0
\(181\) −3450.55 −1.41700 −0.708502 0.705709i \(-0.750629\pi\)
−0.708502 + 0.705709i \(0.750629\pi\)
\(182\) −4605.41 −1.87569
\(183\) 0 0
\(184\) 1035.54 0.414895
\(185\) −111.955 −0.0444924
\(186\) 0 0
\(187\) 2377.52 0.929741
\(188\) 476.736 0.184944
\(189\) 0 0
\(190\) −1997.31 −0.762634
\(191\) 1592.09 0.603141 0.301570 0.953444i \(-0.402489\pi\)
0.301570 + 0.953444i \(0.402489\pi\)
\(192\) 0 0
\(193\) 443.299 0.165334 0.0826668 0.996577i \(-0.473656\pi\)
0.0826668 + 0.996577i \(0.473656\pi\)
\(194\) −3107.21 −1.14992
\(195\) 0 0
\(196\) 1341.29 0.488807
\(197\) 208.992 0.0755839 0.0377919 0.999286i \(-0.487968\pi\)
0.0377919 + 0.999286i \(0.487968\pi\)
\(198\) 0 0
\(199\) −479.916 −0.170957 −0.0854783 0.996340i \(-0.527242\pi\)
−0.0854783 + 0.996340i \(0.527242\pi\)
\(200\) −315.802 −0.111653
\(201\) 0 0
\(202\) −3615.21 −1.25923
\(203\) 6277.43 2.17039
\(204\) 0 0
\(205\) −2171.13 −0.739698
\(206\) −6308.00 −2.13349
\(207\) 0 0
\(208\) 4102.63 1.36763
\(209\) 8089.78 2.67742
\(210\) 0 0
\(211\) −2632.14 −0.858786 −0.429393 0.903118i \(-0.641273\pi\)
−0.429393 + 0.903118i \(0.641273\pi\)
\(212\) 547.275 0.177297
\(213\) 0 0
\(214\) 7072.66 2.25924
\(215\) 1184.25 0.375651
\(216\) 0 0
\(217\) 5663.28 1.77165
\(218\) −745.759 −0.231693
\(219\) 0 0
\(220\) −1575.15 −0.482710
\(221\) −1712.38 −0.521210
\(222\) 0 0
\(223\) 3353.22 1.00694 0.503471 0.864012i \(-0.332056\pi\)
0.503471 + 0.864012i \(0.332056\pi\)
\(224\) −4583.31 −1.36712
\(225\) 0 0
\(226\) −6.85286 −0.00201701
\(227\) −690.764 −0.201972 −0.100986 0.994888i \(-0.532200\pi\)
−0.100986 + 0.994888i \(0.532200\pi\)
\(228\) 0 0
\(229\) −668.131 −0.192801 −0.0964003 0.995343i \(-0.530733\pi\)
−0.0964003 + 0.995343i \(0.530733\pi\)
\(230\) 1444.22 0.414038
\(231\) 0 0
\(232\) −3117.94 −0.882339
\(233\) 940.537 0.264449 0.132225 0.991220i \(-0.457788\pi\)
0.132225 + 0.991220i \(0.457788\pi\)
\(234\) 0 0
\(235\) −539.919 −0.149874
\(236\) −755.347 −0.208343
\(237\) 0 0
\(238\) 2985.75 0.813181
\(239\) −1107.40 −0.299715 −0.149858 0.988708i \(-0.547882\pi\)
−0.149858 + 0.988708i \(0.547882\pi\)
\(240\) 0 0
\(241\) 1228.86 0.328456 0.164228 0.986422i \(-0.447487\pi\)
0.164228 + 0.986422i \(0.447487\pi\)
\(242\) 13250.7 3.51979
\(243\) 0 0
\(244\) −350.715 −0.0920172
\(245\) −1519.05 −0.396117
\(246\) 0 0
\(247\) −5826.58 −1.50096
\(248\) −2812.90 −0.720238
\(249\) 0 0
\(250\) −440.435 −0.111422
\(251\) −738.480 −0.185707 −0.0928534 0.995680i \(-0.529599\pi\)
−0.0928534 + 0.995680i \(0.529599\pi\)
\(252\) 0 0
\(253\) −5849.54 −1.45359
\(254\) −4150.72 −1.02535
\(255\) 0 0
\(256\) 5095.94 1.24412
\(257\) −6312.57 −1.53217 −0.766084 0.642740i \(-0.777797\pi\)
−0.766084 + 0.642740i \(0.777797\pi\)
\(258\) 0 0
\(259\) 569.458 0.136619
\(260\) 1134.48 0.270606
\(261\) 0 0
\(262\) −1836.50 −0.433051
\(263\) −5993.62 −1.40526 −0.702628 0.711558i \(-0.747990\pi\)
−0.702628 + 0.711558i \(0.747990\pi\)
\(264\) 0 0
\(265\) −619.807 −0.143677
\(266\) 10159.3 2.34176
\(267\) 0 0
\(268\) −2699.73 −0.615344
\(269\) 1749.70 0.396584 0.198292 0.980143i \(-0.436461\pi\)
0.198292 + 0.980143i \(0.436461\pi\)
\(270\) 0 0
\(271\) −1931.68 −0.432994 −0.216497 0.976283i \(-0.569463\pi\)
−0.216497 + 0.976283i \(0.569463\pi\)
\(272\) −2659.79 −0.592917
\(273\) 0 0
\(274\) 1583.91 0.349225
\(275\) 1783.90 0.391176
\(276\) 0 0
\(277\) −3791.55 −0.822427 −0.411213 0.911539i \(-0.634895\pi\)
−0.411213 + 0.911539i \(0.634895\pi\)
\(278\) −8264.13 −1.78291
\(279\) 0 0
\(280\) 1606.32 0.342843
\(281\) −4489.36 −0.953071 −0.476536 0.879155i \(-0.658108\pi\)
−0.476536 + 0.879155i \(0.658108\pi\)
\(282\) 0 0
\(283\) 3133.86 0.658264 0.329132 0.944284i \(-0.393244\pi\)
0.329132 + 0.944284i \(0.393244\pi\)
\(284\) −2256.45 −0.471465
\(285\) 0 0
\(286\) −12921.4 −2.67154
\(287\) 11043.4 2.27133
\(288\) 0 0
\(289\) −3802.84 −0.774036
\(290\) −4348.45 −0.880516
\(291\) 0 0
\(292\) −1810.15 −0.362778
\(293\) −2797.50 −0.557787 −0.278894 0.960322i \(-0.589968\pi\)
−0.278894 + 0.960322i \(0.589968\pi\)
\(294\) 0 0
\(295\) 855.454 0.168836
\(296\) −282.845 −0.0555406
\(297\) 0 0
\(298\) 4472.64 0.869439
\(299\) 4213.07 0.814877
\(300\) 0 0
\(301\) −6023.67 −1.15348
\(302\) 7604.77 1.44903
\(303\) 0 0
\(304\) −9050.23 −1.70746
\(305\) 397.196 0.0745684
\(306\) 0 0
\(307\) −6839.91 −1.27158 −0.635789 0.771863i \(-0.719325\pi\)
−0.635789 + 0.771863i \(0.719325\pi\)
\(308\) 8011.97 1.48222
\(309\) 0 0
\(310\) −3923.02 −0.718750
\(311\) −2419.41 −0.441132 −0.220566 0.975372i \(-0.570790\pi\)
−0.220566 + 0.975372i \(0.570790\pi\)
\(312\) 0 0
\(313\) 3903.18 0.704858 0.352429 0.935838i \(-0.385356\pi\)
0.352429 + 0.935838i \(0.385356\pi\)
\(314\) −8256.59 −1.48391
\(315\) 0 0
\(316\) −3503.64 −0.623718
\(317\) −9894.73 −1.75313 −0.876567 0.481280i \(-0.840172\pi\)
−0.876567 + 0.481280i \(0.840172\pi\)
\(318\) 0 0
\(319\) 17612.6 3.09128
\(320\) −18.1973 −0.00317893
\(321\) 0 0
\(322\) −7345.99 −1.27135
\(323\) 3777.44 0.650720
\(324\) 0 0
\(325\) −1284.84 −0.219292
\(326\) 11979.6 2.03524
\(327\) 0 0
\(328\) −5485.16 −0.923377
\(329\) 2746.29 0.460207
\(330\) 0 0
\(331\) 4163.04 0.691303 0.345651 0.938363i \(-0.387658\pi\)
0.345651 + 0.938363i \(0.387658\pi\)
\(332\) −1192.38 −0.197109
\(333\) 0 0
\(334\) 6374.73 1.04434
\(335\) 3057.53 0.498659
\(336\) 0 0
\(337\) 9704.93 1.56873 0.784364 0.620301i \(-0.212989\pi\)
0.784364 + 0.620301i \(0.212989\pi\)
\(338\) 1565.45 0.251921
\(339\) 0 0
\(340\) −735.500 −0.117318
\(341\) 15889.5 2.52336
\(342\) 0 0
\(343\) −996.691 −0.156899
\(344\) 2991.90 0.468931
\(345\) 0 0
\(346\) 4677.09 0.726711
\(347\) 5432.38 0.840420 0.420210 0.907427i \(-0.361956\pi\)
0.420210 + 0.907427i \(0.361956\pi\)
\(348\) 0 0
\(349\) 10571.9 1.62150 0.810749 0.585394i \(-0.199060\pi\)
0.810749 + 0.585394i \(0.199060\pi\)
\(350\) 2240.27 0.342135
\(351\) 0 0
\(352\) −12859.5 −1.94719
\(353\) 9642.98 1.45395 0.726975 0.686664i \(-0.240926\pi\)
0.726975 + 0.686664i \(0.240926\pi\)
\(354\) 0 0
\(355\) 2555.51 0.382063
\(356\) −783.198 −0.116600
\(357\) 0 0
\(358\) 1581.10 0.233419
\(359\) 3002.18 0.441362 0.220681 0.975346i \(-0.429172\pi\)
0.220681 + 0.975346i \(0.429172\pi\)
\(360\) 0 0
\(361\) 5994.17 0.873913
\(362\) −12157.9 −1.76521
\(363\) 0 0
\(364\) −5770.54 −0.830929
\(365\) 2050.06 0.293986
\(366\) 0 0
\(367\) −1149.28 −0.163465 −0.0817326 0.996654i \(-0.526045\pi\)
−0.0817326 + 0.996654i \(0.526045\pi\)
\(368\) 6544.03 0.926986
\(369\) 0 0
\(370\) −394.471 −0.0554258
\(371\) 3152.64 0.441178
\(372\) 0 0
\(373\) −513.214 −0.0712418 −0.0356209 0.999365i \(-0.511341\pi\)
−0.0356209 + 0.999365i \(0.511341\pi\)
\(374\) 8377.14 1.15821
\(375\) 0 0
\(376\) −1364.06 −0.187090
\(377\) −12685.3 −1.73296
\(378\) 0 0
\(379\) 12560.0 1.70228 0.851138 0.524942i \(-0.175913\pi\)
0.851138 + 0.524942i \(0.175913\pi\)
\(380\) −2502.62 −0.337846
\(381\) 0 0
\(382\) 5609.70 0.751355
\(383\) 273.990 0.0365542 0.0182771 0.999833i \(-0.494182\pi\)
0.0182771 + 0.999833i \(0.494182\pi\)
\(384\) 0 0
\(385\) −9073.81 −1.20115
\(386\) 1561.95 0.205962
\(387\) 0 0
\(388\) −3893.31 −0.509415
\(389\) 6846.43 0.892359 0.446179 0.894944i \(-0.352784\pi\)
0.446179 + 0.894944i \(0.352784\pi\)
\(390\) 0 0
\(391\) −2731.39 −0.353279
\(392\) −3837.75 −0.494479
\(393\) 0 0
\(394\) 736.377 0.0941577
\(395\) 3967.98 0.505445
\(396\) 0 0
\(397\) −12117.5 −1.53189 −0.765943 0.642909i \(-0.777727\pi\)
−0.765943 + 0.642909i \(0.777727\pi\)
\(398\) −1690.97 −0.212967
\(399\) 0 0
\(400\) −1995.70 −0.249462
\(401\) 3016.80 0.375691 0.187845 0.982199i \(-0.439850\pi\)
0.187845 + 0.982199i \(0.439850\pi\)
\(402\) 0 0
\(403\) −11444.3 −1.41459
\(404\) −4529.82 −0.557839
\(405\) 0 0
\(406\) 22118.4 2.70373
\(407\) 1597.74 0.194587
\(408\) 0 0
\(409\) 6535.13 0.790076 0.395038 0.918665i \(-0.370731\pi\)
0.395038 + 0.918665i \(0.370731\pi\)
\(410\) −7649.91 −0.921469
\(411\) 0 0
\(412\) −7903.87 −0.945135
\(413\) −4351.26 −0.518430
\(414\) 0 0
\(415\) 1350.40 0.159732
\(416\) 9261.88 1.09159
\(417\) 0 0
\(418\) 28504.1 3.33537
\(419\) −7148.45 −0.833471 −0.416736 0.909028i \(-0.636826\pi\)
−0.416736 + 0.909028i \(0.636826\pi\)
\(420\) 0 0
\(421\) 14801.5 1.71350 0.856749 0.515733i \(-0.172480\pi\)
0.856749 + 0.515733i \(0.172480\pi\)
\(422\) −9274.28 −1.06982
\(423\) 0 0
\(424\) −1565.89 −0.179354
\(425\) 832.977 0.0950713
\(426\) 0 0
\(427\) −2020.33 −0.228971
\(428\) 8861.98 1.00084
\(429\) 0 0
\(430\) 4172.67 0.467963
\(431\) −2284.19 −0.255280 −0.127640 0.991821i \(-0.540740\pi\)
−0.127640 + 0.991821i \(0.540740\pi\)
\(432\) 0 0
\(433\) 5529.26 0.613670 0.306835 0.951763i \(-0.400730\pi\)
0.306835 + 0.951763i \(0.400730\pi\)
\(434\) 19954.4 2.20701
\(435\) 0 0
\(436\) −934.429 −0.102640
\(437\) −9293.85 −1.01736
\(438\) 0 0
\(439\) −11861.6 −1.28958 −0.644788 0.764361i \(-0.723054\pi\)
−0.644788 + 0.764361i \(0.723054\pi\)
\(440\) 4506.88 0.488311
\(441\) 0 0
\(442\) −6033.55 −0.649291
\(443\) 15293.6 1.64023 0.820115 0.572199i \(-0.193910\pi\)
0.820115 + 0.572199i \(0.193910\pi\)
\(444\) 0 0
\(445\) 886.998 0.0944893
\(446\) 11815.0 1.25439
\(447\) 0 0
\(448\) 92.5602 0.00976129
\(449\) −12998.8 −1.36626 −0.683129 0.730297i \(-0.739381\pi\)
−0.683129 + 0.730297i \(0.739381\pi\)
\(450\) 0 0
\(451\) 30984.6 3.23506
\(452\) −8.58657 −0.000893536 0
\(453\) 0 0
\(454\) −2433.89 −0.251604
\(455\) 6535.32 0.673364
\(456\) 0 0
\(457\) 14311.9 1.46495 0.732477 0.680792i \(-0.238364\pi\)
0.732477 + 0.680792i \(0.238364\pi\)
\(458\) −2354.14 −0.240179
\(459\) 0 0
\(460\) 1809.59 0.183419
\(461\) −4290.34 −0.433451 −0.216725 0.976233i \(-0.569538\pi\)
−0.216725 + 0.976233i \(0.569538\pi\)
\(462\) 0 0
\(463\) −14510.3 −1.45648 −0.728240 0.685322i \(-0.759661\pi\)
−0.728240 + 0.685322i \(0.759661\pi\)
\(464\) −19703.7 −1.97138
\(465\) 0 0
\(466\) 3313.96 0.329434
\(467\) 7393.05 0.732569 0.366285 0.930503i \(-0.380630\pi\)
0.366285 + 0.930503i \(0.380630\pi\)
\(468\) 0 0
\(469\) −15552.1 −1.53119
\(470\) −1902.39 −0.186704
\(471\) 0 0
\(472\) 2161.23 0.210760
\(473\) −16900.7 −1.64290
\(474\) 0 0
\(475\) 2834.30 0.273782
\(476\) 3741.11 0.360239
\(477\) 0 0
\(478\) −3901.91 −0.373366
\(479\) −13509.9 −1.28869 −0.644344 0.764736i \(-0.722869\pi\)
−0.644344 + 0.764736i \(0.722869\pi\)
\(480\) 0 0
\(481\) −1150.75 −0.109085
\(482\) 4329.87 0.409170
\(483\) 0 0
\(484\) 16603.1 1.55926
\(485\) 4409.30 0.412817
\(486\) 0 0
\(487\) −11140.0 −1.03655 −0.518277 0.855213i \(-0.673427\pi\)
−0.518277 + 0.855213i \(0.673427\pi\)
\(488\) 1003.48 0.0930849
\(489\) 0 0
\(490\) −5352.34 −0.493457
\(491\) −2012.74 −0.184998 −0.0924988 0.995713i \(-0.529485\pi\)
−0.0924988 + 0.995713i \(0.529485\pi\)
\(492\) 0 0
\(493\) 8224.06 0.751304
\(494\) −20529.8 −1.86980
\(495\) 0 0
\(496\) −17776.0 −1.60920
\(497\) −12998.6 −1.17317
\(498\) 0 0
\(499\) 7352.98 0.659648 0.329824 0.944042i \(-0.393011\pi\)
0.329824 + 0.944042i \(0.393011\pi\)
\(500\) −551.861 −0.0493599
\(501\) 0 0
\(502\) −2602.02 −0.231342
\(503\) −16898.3 −1.49793 −0.748965 0.662609i \(-0.769449\pi\)
−0.748965 + 0.662609i \(0.769449\pi\)
\(504\) 0 0
\(505\) 5130.17 0.452059
\(506\) −20610.7 −1.81079
\(507\) 0 0
\(508\) −5200.82 −0.454230
\(509\) 19358.6 1.68577 0.842884 0.538095i \(-0.180856\pi\)
0.842884 + 0.538095i \(0.180856\pi\)
\(510\) 0 0
\(511\) −10427.6 −0.902720
\(512\) 6319.06 0.545440
\(513\) 0 0
\(514\) −22242.2 −1.90868
\(515\) 8951.39 0.765913
\(516\) 0 0
\(517\) 7705.30 0.655471
\(518\) 2006.47 0.170192
\(519\) 0 0
\(520\) −3246.03 −0.273746
\(521\) 146.772 0.0123420 0.00617100 0.999981i \(-0.498036\pi\)
0.00617100 + 0.999981i \(0.498036\pi\)
\(522\) 0 0
\(523\) −2872.03 −0.240125 −0.120062 0.992766i \(-0.538309\pi\)
−0.120062 + 0.992766i \(0.538309\pi\)
\(524\) −2301.12 −0.191841
\(525\) 0 0
\(526\) −21118.4 −1.75058
\(527\) 7419.46 0.613277
\(528\) 0 0
\(529\) −5446.82 −0.447671
\(530\) −2183.87 −0.178984
\(531\) 0 0
\(532\) 12729.5 1.03740
\(533\) −22316.4 −1.81356
\(534\) 0 0
\(535\) −10036.5 −0.811056
\(536\) 7724.58 0.622484
\(537\) 0 0
\(538\) 6165.03 0.494039
\(539\) 21678.7 1.73241
\(540\) 0 0
\(541\) −3810.28 −0.302804 −0.151402 0.988472i \(-0.548379\pi\)
−0.151402 + 0.988472i \(0.548379\pi\)
\(542\) −6806.24 −0.539397
\(543\) 0 0
\(544\) −6004.60 −0.473245
\(545\) 1058.27 0.0831768
\(546\) 0 0
\(547\) 5945.59 0.464744 0.232372 0.972627i \(-0.425351\pi\)
0.232372 + 0.972627i \(0.425351\pi\)
\(548\) 1984.63 0.154707
\(549\) 0 0
\(550\) 6285.54 0.487303
\(551\) 27983.3 2.16357
\(552\) 0 0
\(553\) −20183.1 −1.55203
\(554\) −13359.4 −1.02453
\(555\) 0 0
\(556\) −10354.9 −0.789828
\(557\) 10872.2 0.827059 0.413530 0.910491i \(-0.364296\pi\)
0.413530 + 0.910491i \(0.364296\pi\)
\(558\) 0 0
\(559\) 12172.5 0.921007
\(560\) 10151.1 0.766004
\(561\) 0 0
\(562\) −15818.2 −1.18728
\(563\) −13575.1 −1.01620 −0.508100 0.861298i \(-0.669652\pi\)
−0.508100 + 0.861298i \(0.669652\pi\)
\(564\) 0 0
\(565\) 9.72457 0.000724099 0
\(566\) 11042.1 0.820024
\(567\) 0 0
\(568\) 6456.27 0.476935
\(569\) −20642.6 −1.52088 −0.760442 0.649406i \(-0.775018\pi\)
−0.760442 + 0.649406i \(0.775018\pi\)
\(570\) 0 0
\(571\) −2730.77 −0.200139 −0.100069 0.994980i \(-0.531906\pi\)
−0.100069 + 0.994980i \(0.531906\pi\)
\(572\) −16190.5 −1.18349
\(573\) 0 0
\(574\) 38911.2 2.82948
\(575\) −2049.42 −0.148638
\(576\) 0 0
\(577\) 21953.4 1.58394 0.791970 0.610560i \(-0.209055\pi\)
0.791970 + 0.610560i \(0.209055\pi\)
\(578\) −13399.2 −0.964245
\(579\) 0 0
\(580\) −5448.57 −0.390068
\(581\) −6868.82 −0.490476
\(582\) 0 0
\(583\) 8845.40 0.628369
\(584\) 5179.29 0.366988
\(585\) 0 0
\(586\) −9856.92 −0.694856
\(587\) −11055.3 −0.777348 −0.388674 0.921375i \(-0.627067\pi\)
−0.388674 + 0.921375i \(0.627067\pi\)
\(588\) 0 0
\(589\) 25245.5 1.76608
\(590\) 3014.17 0.210325
\(591\) 0 0
\(592\) −1787.43 −0.124092
\(593\) −16538.0 −1.14525 −0.572625 0.819817i \(-0.694075\pi\)
−0.572625 + 0.819817i \(0.694075\pi\)
\(594\) 0 0
\(595\) −4236.93 −0.291928
\(596\) 5604.18 0.385161
\(597\) 0 0
\(598\) 14844.7 1.01512
\(599\) 6403.59 0.436800 0.218400 0.975859i \(-0.429916\pi\)
0.218400 + 0.975859i \(0.429916\pi\)
\(600\) 0 0
\(601\) −14210.4 −0.964480 −0.482240 0.876039i \(-0.660177\pi\)
−0.482240 + 0.876039i \(0.660177\pi\)
\(602\) −21224.3 −1.43694
\(603\) 0 0
\(604\) 9528.72 0.641917
\(605\) −18803.5 −1.26359
\(606\) 0 0
\(607\) −8915.72 −0.596174 −0.298087 0.954539i \(-0.596349\pi\)
−0.298087 + 0.954539i \(0.596349\pi\)
\(608\) −20431.3 −1.36283
\(609\) 0 0
\(610\) 1399.51 0.0928926
\(611\) −5549.66 −0.367455
\(612\) 0 0
\(613\) −15372.5 −1.01287 −0.506436 0.862277i \(-0.669037\pi\)
−0.506436 + 0.862277i \(0.669037\pi\)
\(614\) −24100.3 −1.58405
\(615\) 0 0
\(616\) −22924.2 −1.49942
\(617\) −17327.5 −1.13060 −0.565298 0.824887i \(-0.691239\pi\)
−0.565298 + 0.824887i \(0.691239\pi\)
\(618\) 0 0
\(619\) −28787.3 −1.86924 −0.934621 0.355645i \(-0.884261\pi\)
−0.934621 + 0.355645i \(0.884261\pi\)
\(620\) −4915.51 −0.318406
\(621\) 0 0
\(622\) −8524.73 −0.549535
\(623\) −4511.71 −0.290141
\(624\) 0 0
\(625\) 625.000 0.0400000
\(626\) 13752.8 0.878068
\(627\) 0 0
\(628\) −10345.4 −0.657369
\(629\) 746.047 0.0472923
\(630\) 0 0
\(631\) 18091.0 1.14135 0.570675 0.821176i \(-0.306682\pi\)
0.570675 + 0.821176i \(0.306682\pi\)
\(632\) 10024.8 0.630955
\(633\) 0 0
\(634\) −34863.8 −2.18394
\(635\) 5890.10 0.368097
\(636\) 0 0
\(637\) −15613.9 −0.971184
\(638\) 62057.7 3.85092
\(639\) 0 0
\(640\) −7272.72 −0.449187
\(641\) −19577.5 −1.20634 −0.603171 0.797612i \(-0.706096\pi\)
−0.603171 + 0.797612i \(0.706096\pi\)
\(642\) 0 0
\(643\) −29971.7 −1.83821 −0.919106 0.394011i \(-0.871087\pi\)
−0.919106 + 0.394011i \(0.871087\pi\)
\(644\) −9204.46 −0.563209
\(645\) 0 0
\(646\) 13309.7 0.810626
\(647\) 3635.64 0.220915 0.110457 0.993881i \(-0.464768\pi\)
0.110457 + 0.993881i \(0.464768\pi\)
\(648\) 0 0
\(649\) −12208.4 −0.738399
\(650\) −4527.09 −0.273180
\(651\) 0 0
\(652\) 15010.3 0.901610
\(653\) 19407.3 1.16304 0.581520 0.813532i \(-0.302458\pi\)
0.581520 + 0.813532i \(0.302458\pi\)
\(654\) 0 0
\(655\) 2606.09 0.155463
\(656\) −34663.3 −2.06307
\(657\) 0 0
\(658\) 9676.49 0.573296
\(659\) 5627.86 0.332671 0.166335 0.986069i \(-0.446807\pi\)
0.166335 + 0.986069i \(0.446807\pi\)
\(660\) 0 0
\(661\) −16663.0 −0.980507 −0.490253 0.871580i \(-0.663096\pi\)
−0.490253 + 0.871580i \(0.663096\pi\)
\(662\) 14668.4 0.861182
\(663\) 0 0
\(664\) 3411.68 0.199396
\(665\) −14416.6 −0.840681
\(666\) 0 0
\(667\) −20234.1 −1.17461
\(668\) 7987.48 0.462642
\(669\) 0 0
\(670\) 10773.1 0.621198
\(671\) −5668.47 −0.326124
\(672\) 0 0
\(673\) −12638.8 −0.723906 −0.361953 0.932196i \(-0.617890\pi\)
−0.361953 + 0.932196i \(0.617890\pi\)
\(674\) 34195.1 1.95422
\(675\) 0 0
\(676\) 1961.50 0.111601
\(677\) 25316.0 1.43718 0.718591 0.695433i \(-0.244787\pi\)
0.718591 + 0.695433i \(0.244787\pi\)
\(678\) 0 0
\(679\) −22427.9 −1.26760
\(680\) 2104.44 0.118679
\(681\) 0 0
\(682\) 55986.3 3.14344
\(683\) −24818.4 −1.39041 −0.695204 0.718812i \(-0.744686\pi\)
−0.695204 + 0.718812i \(0.744686\pi\)
\(684\) 0 0
\(685\) −2247.66 −0.125370
\(686\) −3511.82 −0.195455
\(687\) 0 0
\(688\) 18907.2 1.04772
\(689\) −6370.81 −0.352262
\(690\) 0 0
\(691\) 19958.0 1.09875 0.549375 0.835576i \(-0.314866\pi\)
0.549375 + 0.835576i \(0.314866\pi\)
\(692\) 5860.35 0.321932
\(693\) 0 0
\(694\) 19140.9 1.04694
\(695\) 11727.2 0.640057
\(696\) 0 0
\(697\) 14468.0 0.786247
\(698\) 37250.0 2.01996
\(699\) 0 0
\(700\) 2807.04 0.151566
\(701\) 5624.21 0.303029 0.151515 0.988455i \(-0.451585\pi\)
0.151515 + 0.988455i \(0.451585\pi\)
\(702\) 0 0
\(703\) 2538.51 0.136190
\(704\) 259.697 0.0139030
\(705\) 0 0
\(706\) 33976.8 1.81124
\(707\) −26094.6 −1.38810
\(708\) 0 0
\(709\) −7715.43 −0.408687 −0.204344 0.978899i \(-0.565506\pi\)
−0.204344 + 0.978899i \(0.565506\pi\)
\(710\) 9004.27 0.475950
\(711\) 0 0
\(712\) 2240.92 0.117952
\(713\) −18254.5 −0.958817
\(714\) 0 0
\(715\) 18336.2 0.959071
\(716\) 1981.11 0.103404
\(717\) 0 0
\(718\) 10578.1 0.549821
\(719\) −21647.8 −1.12285 −0.561424 0.827528i \(-0.689746\pi\)
−0.561424 + 0.827528i \(0.689746\pi\)
\(720\) 0 0
\(721\) −45531.2 −2.35183
\(722\) 21120.3 1.08867
\(723\) 0 0
\(724\) −15233.8 −0.781989
\(725\) 6170.68 0.316101
\(726\) 0 0
\(727\) −3071.30 −0.156682 −0.0783412 0.996927i \(-0.524962\pi\)
−0.0783412 + 0.996927i \(0.524962\pi\)
\(728\) 16510.9 0.840570
\(729\) 0 0
\(730\) 7223.33 0.366229
\(731\) −7891.61 −0.399291
\(732\) 0 0
\(733\) 14652.0 0.738314 0.369157 0.929367i \(-0.379646\pi\)
0.369157 + 0.929367i \(0.379646\pi\)
\(734\) −4049.45 −0.203635
\(735\) 0 0
\(736\) 14773.4 0.739886
\(737\) −43634.7 −2.18087
\(738\) 0 0
\(739\) 17610.9 0.876629 0.438314 0.898822i \(-0.355576\pi\)
0.438314 + 0.898822i \(0.355576\pi\)
\(740\) −494.268 −0.0245536
\(741\) 0 0
\(742\) 11108.3 0.549592
\(743\) 16788.2 0.828935 0.414468 0.910064i \(-0.363968\pi\)
0.414468 + 0.910064i \(0.363968\pi\)
\(744\) 0 0
\(745\) −6346.91 −0.312125
\(746\) −1808.30 −0.0887485
\(747\) 0 0
\(748\) 10496.5 0.513087
\(749\) 51050.5 2.49045
\(750\) 0 0
\(751\) 23514.2 1.14254 0.571269 0.820763i \(-0.306451\pi\)
0.571269 + 0.820763i \(0.306451\pi\)
\(752\) −8620.11 −0.418009
\(753\) 0 0
\(754\) −44696.4 −2.15882
\(755\) −10791.6 −0.520193
\(756\) 0 0
\(757\) 24817.4 1.19155 0.595775 0.803151i \(-0.296845\pi\)
0.595775 + 0.803151i \(0.296845\pi\)
\(758\) 44254.8 2.12059
\(759\) 0 0
\(760\) 7160.60 0.341766
\(761\) −32517.7 −1.54897 −0.774485 0.632592i \(-0.781991\pi\)
−0.774485 + 0.632592i \(0.781991\pi\)
\(762\) 0 0
\(763\) −5382.89 −0.255404
\(764\) 7028.91 0.332850
\(765\) 0 0
\(766\) 965.398 0.0455369
\(767\) 8792.96 0.413944
\(768\) 0 0
\(769\) 18057.8 0.846791 0.423395 0.905945i \(-0.360838\pi\)
0.423395 + 0.905945i \(0.360838\pi\)
\(770\) −31971.4 −1.49632
\(771\) 0 0
\(772\) 1957.11 0.0912411
\(773\) −38147.2 −1.77498 −0.887491 0.460825i \(-0.847553\pi\)
−0.887491 + 0.460825i \(0.847553\pi\)
\(774\) 0 0
\(775\) 5566.98 0.258028
\(776\) 11139.7 0.515326
\(777\) 0 0
\(778\) 24123.2 1.11164
\(779\) 49228.9 2.26420
\(780\) 0 0
\(781\) −36470.2 −1.67094
\(782\) −9623.98 −0.440093
\(783\) 0 0
\(784\) −24252.5 −1.10480
\(785\) 11716.5 0.532715
\(786\) 0 0
\(787\) −4801.76 −0.217489 −0.108745 0.994070i \(-0.534683\pi\)
−0.108745 + 0.994070i \(0.534683\pi\)
\(788\) 922.673 0.0417118
\(789\) 0 0
\(790\) 13981.1 0.629652
\(791\) −49.4640 −0.00222343
\(792\) 0 0
\(793\) 4082.66 0.182824
\(794\) −42695.6 −1.90833
\(795\) 0 0
\(796\) −2118.77 −0.0943442
\(797\) 15471.0 0.687592 0.343796 0.939044i \(-0.388287\pi\)
0.343796 + 0.939044i \(0.388287\pi\)
\(798\) 0 0
\(799\) 3597.92 0.159306
\(800\) −4505.38 −0.199111
\(801\) 0 0
\(802\) 10629.6 0.468012
\(803\) −29256.8 −1.28574
\(804\) 0 0
\(805\) 10424.4 0.456410
\(806\) −40323.6 −1.76220
\(807\) 0 0
\(808\) 12960.9 0.564312
\(809\) 17007.6 0.739129 0.369564 0.929205i \(-0.379507\pi\)
0.369564 + 0.929205i \(0.379507\pi\)
\(810\) 0 0
\(811\) −7552.52 −0.327010 −0.163505 0.986543i \(-0.552280\pi\)
−0.163505 + 0.986543i \(0.552280\pi\)
\(812\) 27714.1 1.19775
\(813\) 0 0
\(814\) 5629.58 0.242404
\(815\) −16999.7 −0.730642
\(816\) 0 0
\(817\) −26852.1 −1.14986
\(818\) 23026.4 0.984227
\(819\) 0 0
\(820\) −9585.28 −0.408210
\(821\) −3745.20 −0.159206 −0.0796031 0.996827i \(-0.525365\pi\)
−0.0796031 + 0.996827i \(0.525365\pi\)
\(822\) 0 0
\(823\) −30021.0 −1.27153 −0.635764 0.771884i \(-0.719315\pi\)
−0.635764 + 0.771884i \(0.719315\pi\)
\(824\) 22614.9 0.956101
\(825\) 0 0
\(826\) −15331.6 −0.645828
\(827\) 7918.68 0.332962 0.166481 0.986045i \(-0.446760\pi\)
0.166481 + 0.986045i \(0.446760\pi\)
\(828\) 0 0
\(829\) −6976.17 −0.292271 −0.146135 0.989265i \(-0.546683\pi\)
−0.146135 + 0.989265i \(0.546683\pi\)
\(830\) 4758.12 0.198984
\(831\) 0 0
\(832\) −187.044 −0.00779398
\(833\) 10122.7 0.421044
\(834\) 0 0
\(835\) −9046.08 −0.374913
\(836\) 35715.4 1.47756
\(837\) 0 0
\(838\) −25187.4 −1.03829
\(839\) −8657.91 −0.356263 −0.178131 0.984007i \(-0.557005\pi\)
−0.178131 + 0.984007i \(0.557005\pi\)
\(840\) 0 0
\(841\) 36534.7 1.49800
\(842\) 52152.9 2.13457
\(843\) 0 0
\(844\) −11620.6 −0.473930
\(845\) −2221.46 −0.0904386
\(846\) 0 0
\(847\) 95643.8 3.88000
\(848\) −9895.57 −0.400726
\(849\) 0 0
\(850\) 2934.97 0.118434
\(851\) −1835.54 −0.0739383
\(852\) 0 0
\(853\) 18984.8 0.762047 0.381024 0.924565i \(-0.375572\pi\)
0.381024 + 0.924565i \(0.375572\pi\)
\(854\) −7118.60 −0.285238
\(855\) 0 0
\(856\) −25356.3 −1.01245
\(857\) 36434.4 1.45225 0.726124 0.687564i \(-0.241320\pi\)
0.726124 + 0.687564i \(0.241320\pi\)
\(858\) 0 0
\(859\) −17485.0 −0.694507 −0.347253 0.937771i \(-0.612886\pi\)
−0.347253 + 0.937771i \(0.612886\pi\)
\(860\) 5228.32 0.207307
\(861\) 0 0
\(862\) −8048.29 −0.318011
\(863\) 10423.6 0.411152 0.205576 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(864\) 0 0
\(865\) −6637.04 −0.260886
\(866\) 19482.2 0.764472
\(867\) 0 0
\(868\) 25002.7 0.977704
\(869\) −56628.0 −2.21056
\(870\) 0 0
\(871\) 31427.4 1.22259
\(872\) 2673.63 0.103831
\(873\) 0 0
\(874\) −32746.7 −1.26736
\(875\) −3179.06 −0.122825
\(876\) 0 0
\(877\) 7873.24 0.303148 0.151574 0.988446i \(-0.451566\pi\)
0.151574 + 0.988446i \(0.451566\pi\)
\(878\) −41794.1 −1.60647
\(879\) 0 0
\(880\) 28481.0 1.09102
\(881\) 35135.0 1.34362 0.671809 0.740725i \(-0.265518\pi\)
0.671809 + 0.740725i \(0.265518\pi\)
\(882\) 0 0
\(883\) 33069.4 1.26033 0.630167 0.776460i \(-0.282987\pi\)
0.630167 + 0.776460i \(0.282987\pi\)
\(884\) −7559.98 −0.287635
\(885\) 0 0
\(886\) 53886.7 2.04329
\(887\) 51479.1 1.94870 0.974351 0.225031i \(-0.0722485\pi\)
0.974351 + 0.225031i \(0.0722485\pi\)
\(888\) 0 0
\(889\) −29959.9 −1.13029
\(890\) 3125.31 0.117709
\(891\) 0 0
\(892\) 14804.1 0.555692
\(893\) 12242.3 0.458761
\(894\) 0 0
\(895\) −2243.67 −0.0837963
\(896\) 36992.6 1.37928
\(897\) 0 0
\(898\) −45800.9 −1.70200
\(899\) 54963.3 2.03907
\(900\) 0 0
\(901\) 4130.28 0.152719
\(902\) 109174. 4.03003
\(903\) 0 0
\(904\) 24.5683 0.000903904 0
\(905\) 17252.8 0.633703
\(906\) 0 0
\(907\) −10809.8 −0.395737 −0.197868 0.980229i \(-0.563402\pi\)
−0.197868 + 0.980229i \(0.563402\pi\)
\(908\) −3049.64 −0.111460
\(909\) 0 0
\(910\) 23027.0 0.838834
\(911\) −31539.8 −1.14705 −0.573523 0.819190i \(-0.694424\pi\)
−0.573523 + 0.819190i \(0.694424\pi\)
\(912\) 0 0
\(913\) −19271.9 −0.698585
\(914\) 50427.8 1.82495
\(915\) 0 0
\(916\) −2949.72 −0.106399
\(917\) −13255.9 −0.477368
\(918\) 0 0
\(919\) 3592.53 0.128952 0.0644759 0.997919i \(-0.479462\pi\)
0.0644759 + 0.997919i \(0.479462\pi\)
\(920\) −5177.68 −0.185547
\(921\) 0 0
\(922\) −15116.9 −0.539966
\(923\) 26267.3 0.936727
\(924\) 0 0
\(925\) 559.775 0.0198976
\(926\) −51126.7 −1.81439
\(927\) 0 0
\(928\) −44482.0 −1.57348
\(929\) −31614.7 −1.11652 −0.558259 0.829667i \(-0.688530\pi\)
−0.558259 + 0.829667i \(0.688530\pi\)
\(930\) 0 0
\(931\) 34443.5 1.21250
\(932\) 4152.36 0.145939
\(933\) 0 0
\(934\) 26049.2 0.912588
\(935\) −11887.6 −0.415793
\(936\) 0 0
\(937\) −20891.5 −0.728382 −0.364191 0.931324i \(-0.618654\pi\)
−0.364191 + 0.931324i \(0.618654\pi\)
\(938\) −54797.5 −1.90746
\(939\) 0 0
\(940\) −2383.68 −0.0827096
\(941\) −37769.2 −1.30844 −0.654219 0.756305i \(-0.727002\pi\)
−0.654219 + 0.756305i \(0.727002\pi\)
\(942\) 0 0
\(943\) −35596.4 −1.22924
\(944\) 13657.8 0.470894
\(945\) 0 0
\(946\) −59549.1 −2.04663
\(947\) 10430.8 0.357927 0.178963 0.983856i \(-0.442726\pi\)
0.178963 + 0.983856i \(0.442726\pi\)
\(948\) 0 0
\(949\) 21071.9 0.720784
\(950\) 9986.57 0.341060
\(951\) 0 0
\(952\) −10704.2 −0.364419
\(953\) 33865.3 1.15111 0.575553 0.817764i \(-0.304787\pi\)
0.575553 + 0.817764i \(0.304787\pi\)
\(954\) 0 0
\(955\) −7960.47 −0.269733
\(956\) −4889.06 −0.165401
\(957\) 0 0
\(958\) −47601.7 −1.60537
\(959\) 11432.7 0.384965
\(960\) 0 0
\(961\) 19794.9 0.664461
\(962\) −4054.65 −0.135891
\(963\) 0 0
\(964\) 5425.28 0.181262
\(965\) −2216.50 −0.0739394
\(966\) 0 0
\(967\) −46581.8 −1.54909 −0.774544 0.632520i \(-0.782021\pi\)
−0.774544 + 0.632520i \(0.782021\pi\)
\(968\) −47505.4 −1.57736
\(969\) 0 0
\(970\) 15536.1 0.514261
\(971\) −13829.5 −0.457065 −0.228532 0.973536i \(-0.573393\pi\)
−0.228532 + 0.973536i \(0.573393\pi\)
\(972\) 0 0
\(973\) −59650.5 −1.96537
\(974\) −39251.6 −1.29127
\(975\) 0 0
\(976\) 6341.46 0.207977
\(977\) −15943.6 −0.522089 −0.261045 0.965327i \(-0.584067\pi\)
−0.261045 + 0.965327i \(0.584067\pi\)
\(978\) 0 0
\(979\) −12658.5 −0.413247
\(980\) −6706.44 −0.218601
\(981\) 0 0
\(982\) −7091.85 −0.230458
\(983\) 29111.6 0.944572 0.472286 0.881445i \(-0.343429\pi\)
0.472286 + 0.881445i \(0.343429\pi\)
\(984\) 0 0
\(985\) −1044.96 −0.0338021
\(986\) 28977.3 0.935927
\(987\) 0 0
\(988\) −25723.7 −0.828318
\(989\) 19416.1 0.624265
\(990\) 0 0
\(991\) 8745.85 0.280344 0.140172 0.990127i \(-0.455234\pi\)
0.140172 + 0.990127i \(0.455234\pi\)
\(992\) −40130.1 −1.28441
\(993\) 0 0
\(994\) −45800.2 −1.46146
\(995\) 2399.58 0.0764541
\(996\) 0 0
\(997\) 45026.9 1.43031 0.715153 0.698968i \(-0.246357\pi\)
0.715153 + 0.698968i \(0.246357\pi\)
\(998\) 25908.0 0.821748
\(999\) 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 405.4.a.i.1.3 yes 3
3.2 odd 2 405.4.a.g.1.1 3
5.4 even 2 2025.4.a.p.1.1 3
9.2 odd 6 405.4.e.u.271.3 6
9.4 even 3 405.4.e.s.136.1 6
9.5 odd 6 405.4.e.u.136.3 6
9.7 even 3 405.4.e.s.271.1 6
15.14 odd 2 2025.4.a.r.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
405.4.a.g.1.1 3 3.2 odd 2
405.4.a.i.1.3 yes 3 1.1 even 1 trivial
405.4.e.s.136.1 6 9.4 even 3
405.4.e.s.271.1 6 9.7 even 3
405.4.e.u.136.3 6 9.5 odd 6
405.4.e.u.271.3 6 9.2 odd 6
2025.4.a.p.1.1 3 5.4 even 2
2025.4.a.r.1.3 3 15.14 odd 2