Properties

Label 3626.2.a.a.1.2
Level $3626$
Weight $2$
Character 3626.1
Self dual yes
Analytic conductor $28.954$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3626 = 2 \cdot 7^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3626.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.9537557729\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 74)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 3626.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} -1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} -1.00000 q^{8} -2.90833 q^{9} +1.30278 q^{10} +1.30278 q^{11} +0.302776 q^{12} +2.30278 q^{13} -0.394449 q^{15} +1.00000 q^{16} +6.00000 q^{17} +2.90833 q^{18} -2.00000 q^{19} -1.30278 q^{20} -1.30278 q^{22} -6.90833 q^{23} -0.302776 q^{24} -3.30278 q^{25} -2.30278 q^{26} -1.78890 q^{27} +6.90833 q^{29} +0.394449 q^{30} -3.30278 q^{31} -1.00000 q^{32} +0.394449 q^{33} -6.00000 q^{34} -2.90833 q^{36} +1.00000 q^{37} +2.00000 q^{38} +0.697224 q^{39} +1.30278 q^{40} +0.908327 q^{41} -6.60555 q^{43} +1.30278 q^{44} +3.78890 q^{45} +6.90833 q^{46} +2.60555 q^{47} +0.302776 q^{48} +3.30278 q^{50} +1.81665 q^{51} +2.30278 q^{52} -6.00000 q^{53} +1.78890 q^{54} -1.69722 q^{55} -0.605551 q^{57} -6.90833 q^{58} -3.39445 q^{59} -0.394449 q^{60} +10.5139 q^{61} +3.30278 q^{62} +1.00000 q^{64} -3.00000 q^{65} -0.394449 q^{66} +14.5139 q^{67} +6.00000 q^{68} -2.09167 q^{69} +6.00000 q^{71} +2.90833 q^{72} +8.69722 q^{73} -1.00000 q^{74} -1.00000 q^{75} -2.00000 q^{76} -0.697224 q^{78} -16.1194 q^{79} -1.30278 q^{80} +8.18335 q^{81} -0.908327 q^{82} -17.2111 q^{83} -7.81665 q^{85} +6.60555 q^{86} +2.09167 q^{87} -1.30278 q^{88} -5.21110 q^{89} -3.78890 q^{90} -6.90833 q^{92} -1.00000 q^{93} -2.60555 q^{94} +2.60555 q^{95} -0.302776 q^{96} -12.4222 q^{97} -3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} - 2 q^{8} + 5 q^{9} - q^{10} - q^{11} - 3 q^{12} + q^{13} - 8 q^{15} + 2 q^{16} + 12 q^{17} - 5 q^{18} - 4 q^{19} + q^{20} + q^{22} - 3 q^{23} + 3 q^{24} - 3 q^{25} - q^{26} - 18 q^{27} + 3 q^{29} + 8 q^{30} - 3 q^{31} - 2 q^{32} + 8 q^{33} - 12 q^{34} + 5 q^{36} + 2 q^{37} + 4 q^{38} + 5 q^{39} - q^{40} - 9 q^{41} - 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} - 2 q^{47} - 3 q^{48} + 3 q^{50} - 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 6 q^{57} - 3 q^{58} - 14 q^{59} - 8 q^{60} + 3 q^{61} + 3 q^{62} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} + 12 q^{71} - 5 q^{72} + 21 q^{73} - 2 q^{74} - 2 q^{75} - 4 q^{76} - 5 q^{78} - 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} - 20 q^{83} + 6 q^{85} + 6 q^{86} + 15 q^{87} + q^{88} + 4 q^{89} - 22 q^{90} - 3 q^{92} - 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} - 22 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0.302776 0.174808 0.0874038 0.996173i \(-0.472143\pi\)
0.0874038 + 0.996173i \(0.472143\pi\)
\(4\) 1.00000 0.500000
\(5\) −1.30278 −0.582619 −0.291309 0.956629i \(-0.594091\pi\)
−0.291309 + 0.956629i \(0.594091\pi\)
\(6\) −0.302776 −0.123608
\(7\) 0 0
\(8\) −1.00000 −0.353553
\(9\) −2.90833 −0.969442
\(10\) 1.30278 0.411974
\(11\) 1.30278 0.392802 0.196401 0.980524i \(-0.437075\pi\)
0.196401 + 0.980524i \(0.437075\pi\)
\(12\) 0.302776 0.0874038
\(13\) 2.30278 0.638675 0.319338 0.947641i \(-0.396540\pi\)
0.319338 + 0.947641i \(0.396540\pi\)
\(14\) 0 0
\(15\) −0.394449 −0.101846
\(16\) 1.00000 0.250000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) 2.90833 0.685499
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) −1.30278 −0.291309
\(21\) 0 0
\(22\) −1.30278 −0.277753
\(23\) −6.90833 −1.44049 −0.720243 0.693722i \(-0.755970\pi\)
−0.720243 + 0.693722i \(0.755970\pi\)
\(24\) −0.302776 −0.0618038
\(25\) −3.30278 −0.660555
\(26\) −2.30278 −0.451611
\(27\) −1.78890 −0.344273
\(28\) 0 0
\(29\) 6.90833 1.28284 0.641422 0.767188i \(-0.278345\pi\)
0.641422 + 0.767188i \(0.278345\pi\)
\(30\) 0.394449 0.0720162
\(31\) −3.30278 −0.593196 −0.296598 0.955002i \(-0.595852\pi\)
−0.296598 + 0.955002i \(0.595852\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0.394449 0.0686647
\(34\) −6.00000 −1.02899
\(35\) 0 0
\(36\) −2.90833 −0.484721
\(37\) 1.00000 0.164399
\(38\) 2.00000 0.324443
\(39\) 0.697224 0.111645
\(40\) 1.30278 0.205987
\(41\) 0.908327 0.141857 0.0709284 0.997481i \(-0.477404\pi\)
0.0709284 + 0.997481i \(0.477404\pi\)
\(42\) 0 0
\(43\) −6.60555 −1.00734 −0.503669 0.863897i \(-0.668017\pi\)
−0.503669 + 0.863897i \(0.668017\pi\)
\(44\) 1.30278 0.196401
\(45\) 3.78890 0.564815
\(46\) 6.90833 1.01858
\(47\) 2.60555 0.380059 0.190029 0.981778i \(-0.439142\pi\)
0.190029 + 0.981778i \(0.439142\pi\)
\(48\) 0.302776 0.0437019
\(49\) 0 0
\(50\) 3.30278 0.467083
\(51\) 1.81665 0.254382
\(52\) 2.30278 0.319338
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) 1.78890 0.243438
\(55\) −1.69722 −0.228854
\(56\) 0 0
\(57\) −0.605551 −0.0802072
\(58\) −6.90833 −0.907108
\(59\) −3.39445 −0.441920 −0.220960 0.975283i \(-0.570919\pi\)
−0.220960 + 0.975283i \(0.570919\pi\)
\(60\) −0.394449 −0.0509231
\(61\) 10.5139 1.34616 0.673082 0.739568i \(-0.264970\pi\)
0.673082 + 0.739568i \(0.264970\pi\)
\(62\) 3.30278 0.419453
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −3.00000 −0.372104
\(66\) −0.394449 −0.0485533
\(67\) 14.5139 1.77315 0.886576 0.462583i \(-0.153077\pi\)
0.886576 + 0.462583i \(0.153077\pi\)
\(68\) 6.00000 0.727607
\(69\) −2.09167 −0.251808
\(70\) 0 0
\(71\) 6.00000 0.712069 0.356034 0.934473i \(-0.384129\pi\)
0.356034 + 0.934473i \(0.384129\pi\)
\(72\) 2.90833 0.342750
\(73\) 8.69722 1.01793 0.508967 0.860786i \(-0.330028\pi\)
0.508967 + 0.860786i \(0.330028\pi\)
\(74\) −1.00000 −0.116248
\(75\) −1.00000 −0.115470
\(76\) −2.00000 −0.229416
\(77\) 0 0
\(78\) −0.697224 −0.0789451
\(79\) −16.1194 −1.81358 −0.906789 0.421585i \(-0.861474\pi\)
−0.906789 + 0.421585i \(0.861474\pi\)
\(80\) −1.30278 −0.145655
\(81\) 8.18335 0.909261
\(82\) −0.908327 −0.100308
\(83\) −17.2111 −1.88916 −0.944582 0.328276i \(-0.893533\pi\)
−0.944582 + 0.328276i \(0.893533\pi\)
\(84\) 0 0
\(85\) −7.81665 −0.847835
\(86\) 6.60555 0.712295
\(87\) 2.09167 0.224251
\(88\) −1.30278 −0.138876
\(89\) −5.21110 −0.552376 −0.276188 0.961104i \(-0.589071\pi\)
−0.276188 + 0.961104i \(0.589071\pi\)
\(90\) −3.78890 −0.399385
\(91\) 0 0
\(92\) −6.90833 −0.720243
\(93\) −1.00000 −0.103695
\(94\) −2.60555 −0.268742
\(95\) 2.60555 0.267324
\(96\) −0.302776 −0.0309019
\(97\) −12.4222 −1.26128 −0.630642 0.776074i \(-0.717208\pi\)
−0.630642 + 0.776074i \(0.717208\pi\)
\(98\) 0 0
\(99\) −3.78890 −0.380799
\(100\) −3.30278 −0.330278
\(101\) −16.4222 −1.63407 −0.817035 0.576588i \(-0.804384\pi\)
−0.817035 + 0.576588i \(0.804384\pi\)
\(102\) −1.81665 −0.179876
\(103\) −3.30278 −0.325432 −0.162716 0.986673i \(-0.552025\pi\)
−0.162716 + 0.986673i \(0.552025\pi\)
\(104\) −2.30278 −0.225806
\(105\) 0 0
\(106\) 6.00000 0.582772
\(107\) 4.30278 0.415965 0.207983 0.978133i \(-0.433310\pi\)
0.207983 + 0.978133i \(0.433310\pi\)
\(108\) −1.78890 −0.172137
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 1.69722 0.161824
\(111\) 0.302776 0.0287382
\(112\) 0 0
\(113\) 11.2111 1.05465 0.527326 0.849663i \(-0.323195\pi\)
0.527326 + 0.849663i \(0.323195\pi\)
\(114\) 0.605551 0.0567151
\(115\) 9.00000 0.839254
\(116\) 6.90833 0.641422
\(117\) −6.69722 −0.619159
\(118\) 3.39445 0.312484
\(119\) 0 0
\(120\) 0.394449 0.0360081
\(121\) −9.30278 −0.845707
\(122\) −10.5139 −0.951882
\(123\) 0.275019 0.0247977
\(124\) −3.30278 −0.296598
\(125\) 10.8167 0.967471
\(126\) 0 0
\(127\) −4.78890 −0.424946 −0.212473 0.977167i \(-0.568152\pi\)
−0.212473 + 0.977167i \(0.568152\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −2.00000 −0.176090
\(130\) 3.00000 0.263117
\(131\) −3.39445 −0.296574 −0.148287 0.988944i \(-0.547376\pi\)
−0.148287 + 0.988944i \(0.547376\pi\)
\(132\) 0.394449 0.0343324
\(133\) 0 0
\(134\) −14.5139 −1.25381
\(135\) 2.33053 0.200580
\(136\) −6.00000 −0.514496
\(137\) −9.90833 −0.846525 −0.423263 0.906007i \(-0.639115\pi\)
−0.423263 + 0.906007i \(0.639115\pi\)
\(138\) 2.09167 0.178055
\(139\) −8.90833 −0.755594 −0.377797 0.925888i \(-0.623318\pi\)
−0.377797 + 0.925888i \(0.623318\pi\)
\(140\) 0 0
\(141\) 0.788897 0.0664372
\(142\) −6.00000 −0.503509
\(143\) 3.00000 0.250873
\(144\) −2.90833 −0.242361
\(145\) −9.00000 −0.747409
\(146\) −8.69722 −0.719787
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) −1.81665 −0.148826 −0.0744130 0.997228i \(-0.523708\pi\)
−0.0744130 + 0.997228i \(0.523708\pi\)
\(150\) 1.00000 0.0816497
\(151\) −13.3944 −1.09002 −0.545012 0.838428i \(-0.683475\pi\)
−0.545012 + 0.838428i \(0.683475\pi\)
\(152\) 2.00000 0.162221
\(153\) −17.4500 −1.41075
\(154\) 0 0
\(155\) 4.30278 0.345607
\(156\) 0.697224 0.0558226
\(157\) −7.21110 −0.575509 −0.287754 0.957704i \(-0.592909\pi\)
−0.287754 + 0.957704i \(0.592909\pi\)
\(158\) 16.1194 1.28239
\(159\) −1.81665 −0.144070
\(160\) 1.30278 0.102993
\(161\) 0 0
\(162\) −8.18335 −0.642944
\(163\) −20.4222 −1.59959 −0.799795 0.600273i \(-0.795059\pi\)
−0.799795 + 0.600273i \(0.795059\pi\)
\(164\) 0.908327 0.0709284
\(165\) −0.513878 −0.0400054
\(166\) 17.2111 1.33584
\(167\) −12.5139 −0.968353 −0.484176 0.874970i \(-0.660881\pi\)
−0.484176 + 0.874970i \(0.660881\pi\)
\(168\) 0 0
\(169\) −7.69722 −0.592094
\(170\) 7.81665 0.599510
\(171\) 5.81665 0.444811
\(172\) −6.60555 −0.503669
\(173\) 23.2111 1.76471 0.882354 0.470587i \(-0.155958\pi\)
0.882354 + 0.470587i \(0.155958\pi\)
\(174\) −2.09167 −0.158569
\(175\) 0 0
\(176\) 1.30278 0.0982004
\(177\) −1.02776 −0.0772509
\(178\) 5.21110 0.390589
\(179\) 7.81665 0.584244 0.292122 0.956381i \(-0.405639\pi\)
0.292122 + 0.956381i \(0.405639\pi\)
\(180\) 3.78890 0.282408
\(181\) −20.0000 −1.48659 −0.743294 0.668965i \(-0.766738\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 3.18335 0.235320
\(184\) 6.90833 0.509289
\(185\) −1.30278 −0.0957820
\(186\) 1.00000 0.0733236
\(187\) 7.81665 0.571610
\(188\) 2.60555 0.190029
\(189\) 0 0
\(190\) −2.60555 −0.189027
\(191\) 12.5139 0.905472 0.452736 0.891644i \(-0.350448\pi\)
0.452736 + 0.891644i \(0.350448\pi\)
\(192\) 0.302776 0.0218509
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 12.4222 0.891862
\(195\) −0.908327 −0.0650466
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 3.78890 0.269265
\(199\) 2.42221 0.171706 0.0858528 0.996308i \(-0.472639\pi\)
0.0858528 + 0.996308i \(0.472639\pi\)
\(200\) 3.30278 0.233542
\(201\) 4.39445 0.309961
\(202\) 16.4222 1.15546
\(203\) 0 0
\(204\) 1.81665 0.127191
\(205\) −1.18335 −0.0826485
\(206\) 3.30278 0.230115
\(207\) 20.0917 1.39647
\(208\) 2.30278 0.159669
\(209\) −2.60555 −0.180230
\(210\) 0 0
\(211\) 6.69722 0.461056 0.230528 0.973066i \(-0.425955\pi\)
0.230528 + 0.973066i \(0.425955\pi\)
\(212\) −6.00000 −0.412082
\(213\) 1.81665 0.124475
\(214\) −4.30278 −0.294132
\(215\) 8.60555 0.586894
\(216\) 1.78890 0.121719
\(217\) 0 0
\(218\) −2.00000 −0.135457
\(219\) 2.63331 0.177942
\(220\) −1.69722 −0.114427
\(221\) 13.8167 0.929409
\(222\) −0.302776 −0.0203210
\(223\) −15.8167 −1.05916 −0.529581 0.848260i \(-0.677651\pi\)
−0.529581 + 0.848260i \(0.677651\pi\)
\(224\) 0 0
\(225\) 9.60555 0.640370
\(226\) −11.2111 −0.745751
\(227\) 7.81665 0.518810 0.259405 0.965769i \(-0.416474\pi\)
0.259405 + 0.965769i \(0.416474\pi\)
\(228\) −0.605551 −0.0401036
\(229\) −17.3944 −1.14946 −0.574729 0.818344i \(-0.694892\pi\)
−0.574729 + 0.818344i \(0.694892\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) −6.90833 −0.453554
\(233\) −9.51388 −0.623275 −0.311637 0.950201i \(-0.600877\pi\)
−0.311637 + 0.950201i \(0.600877\pi\)
\(234\) 6.69722 0.437811
\(235\) −3.39445 −0.221429
\(236\) −3.39445 −0.220960
\(237\) −4.88057 −0.317027
\(238\) 0 0
\(239\) 0.513878 0.0332400 0.0166200 0.999862i \(-0.494709\pi\)
0.0166200 + 0.999862i \(0.494709\pi\)
\(240\) −0.394449 −0.0254616
\(241\) −8.00000 −0.515325 −0.257663 0.966235i \(-0.582952\pi\)
−0.257663 + 0.966235i \(0.582952\pi\)
\(242\) 9.30278 0.598005
\(243\) 7.84441 0.503219
\(244\) 10.5139 0.673082
\(245\) 0 0
\(246\) −0.275019 −0.0175346
\(247\) −4.60555 −0.293044
\(248\) 3.30278 0.209726
\(249\) −5.21110 −0.330240
\(250\) −10.8167 −0.684105
\(251\) 6.78890 0.428511 0.214256 0.976778i \(-0.431267\pi\)
0.214256 + 0.976778i \(0.431267\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) 4.78890 0.300482
\(255\) −2.36669 −0.148208
\(256\) 1.00000 0.0625000
\(257\) 11.2111 0.699329 0.349665 0.936875i \(-0.386296\pi\)
0.349665 + 0.936875i \(0.386296\pi\)
\(258\) 2.00000 0.124515
\(259\) 0 0
\(260\) −3.00000 −0.186052
\(261\) −20.0917 −1.24364
\(262\) 3.39445 0.209710
\(263\) −7.81665 −0.481996 −0.240998 0.970526i \(-0.577475\pi\)
−0.240998 + 0.970526i \(0.577475\pi\)
\(264\) −0.394449 −0.0242766
\(265\) 7.81665 0.480173
\(266\) 0 0
\(267\) −1.57779 −0.0965595
\(268\) 14.5139 0.886576
\(269\) 6.78890 0.413926 0.206963 0.978349i \(-0.433642\pi\)
0.206963 + 0.978349i \(0.433642\pi\)
\(270\) −2.33053 −0.141832
\(271\) −6.42221 −0.390121 −0.195061 0.980791i \(-0.562490\pi\)
−0.195061 + 0.980791i \(0.562490\pi\)
\(272\) 6.00000 0.363803
\(273\) 0 0
\(274\) 9.90833 0.598584
\(275\) −4.30278 −0.259467
\(276\) −2.09167 −0.125904
\(277\) −25.1194 −1.50928 −0.754640 0.656139i \(-0.772189\pi\)
−0.754640 + 0.656139i \(0.772189\pi\)
\(278\) 8.90833 0.534286
\(279\) 9.60555 0.575069
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) −0.788897 −0.0469782
\(283\) −17.3944 −1.03399 −0.516996 0.855988i \(-0.672950\pi\)
−0.516996 + 0.855988i \(0.672950\pi\)
\(284\) 6.00000 0.356034
\(285\) 0.788897 0.0467303
\(286\) −3.00000 −0.177394
\(287\) 0 0
\(288\) 2.90833 0.171375
\(289\) 19.0000 1.11765
\(290\) 9.00000 0.528498
\(291\) −3.76114 −0.220482
\(292\) 8.69722 0.508967
\(293\) 25.0278 1.46214 0.731069 0.682304i \(-0.239022\pi\)
0.731069 + 0.682304i \(0.239022\pi\)
\(294\) 0 0
\(295\) 4.42221 0.257471
\(296\) −1.00000 −0.0581238
\(297\) −2.33053 −0.135231
\(298\) 1.81665 0.105236
\(299\) −15.9083 −0.920002
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 13.3944 0.770764
\(303\) −4.97224 −0.285648
\(304\) −2.00000 −0.114708
\(305\) −13.6972 −0.784301
\(306\) 17.4500 0.997548
\(307\) −7.09167 −0.404743 −0.202372 0.979309i \(-0.564865\pi\)
−0.202372 + 0.979309i \(0.564865\pi\)
\(308\) 0 0
\(309\) −1.00000 −0.0568880
\(310\) −4.30278 −0.244381
\(311\) −5.09167 −0.288722 −0.144361 0.989525i \(-0.546113\pi\)
−0.144361 + 0.989525i \(0.546113\pi\)
\(312\) −0.697224 −0.0394726
\(313\) −27.0278 −1.52770 −0.763850 0.645394i \(-0.776693\pi\)
−0.763850 + 0.645394i \(0.776693\pi\)
\(314\) 7.21110 0.406946
\(315\) 0 0
\(316\) −16.1194 −0.906789
\(317\) −5.21110 −0.292685 −0.146342 0.989234i \(-0.546750\pi\)
−0.146342 + 0.989234i \(0.546750\pi\)
\(318\) 1.81665 0.101873
\(319\) 9.00000 0.503903
\(320\) −1.30278 −0.0728274
\(321\) 1.30278 0.0727138
\(322\) 0 0
\(323\) −12.0000 −0.667698
\(324\) 8.18335 0.454630
\(325\) −7.60555 −0.421880
\(326\) 20.4222 1.13108
\(327\) 0.605551 0.0334871
\(328\) −0.908327 −0.0501540
\(329\) 0 0
\(330\) 0.513878 0.0282881
\(331\) 1.21110 0.0665682 0.0332841 0.999446i \(-0.489403\pi\)
0.0332841 + 0.999446i \(0.489403\pi\)
\(332\) −17.2111 −0.944582
\(333\) −2.90833 −0.159375
\(334\) 12.5139 0.684729
\(335\) −18.9083 −1.03307
\(336\) 0 0
\(337\) −19.1194 −1.04150 −0.520751 0.853709i \(-0.674348\pi\)
−0.520751 + 0.853709i \(0.674348\pi\)
\(338\) 7.69722 0.418674
\(339\) 3.39445 0.184361
\(340\) −7.81665 −0.423918
\(341\) −4.30278 −0.233008
\(342\) −5.81665 −0.314529
\(343\) 0 0
\(344\) 6.60555 0.356147
\(345\) 2.72498 0.146708
\(346\) −23.2111 −1.24784
\(347\) 31.8167 1.70801 0.854004 0.520267i \(-0.174168\pi\)
0.854004 + 0.520267i \(0.174168\pi\)
\(348\) 2.09167 0.112125
\(349\) 22.2389 1.19042 0.595209 0.803571i \(-0.297069\pi\)
0.595209 + 0.803571i \(0.297069\pi\)
\(350\) 0 0
\(351\) −4.11943 −0.219879
\(352\) −1.30278 −0.0694382
\(353\) −31.8167 −1.69343 −0.846715 0.532047i \(-0.821423\pi\)
−0.846715 + 0.532047i \(0.821423\pi\)
\(354\) 1.02776 0.0546246
\(355\) −7.81665 −0.414865
\(356\) −5.21110 −0.276188
\(357\) 0 0
\(358\) −7.81665 −0.413123
\(359\) −11.2111 −0.591699 −0.295850 0.955235i \(-0.595603\pi\)
−0.295850 + 0.955235i \(0.595603\pi\)
\(360\) −3.78890 −0.199692
\(361\) −15.0000 −0.789474
\(362\) 20.0000 1.05118
\(363\) −2.81665 −0.147836
\(364\) 0 0
\(365\) −11.3305 −0.593067
\(366\) −3.18335 −0.166396
\(367\) 17.8167 0.930022 0.465011 0.885305i \(-0.346050\pi\)
0.465011 + 0.885305i \(0.346050\pi\)
\(368\) −6.90833 −0.360121
\(369\) −2.64171 −0.137522
\(370\) 1.30278 0.0677281
\(371\) 0 0
\(372\) −1.00000 −0.0518476
\(373\) 3.81665 0.197619 0.0988094 0.995106i \(-0.468497\pi\)
0.0988094 + 0.995106i \(0.468497\pi\)
\(374\) −7.81665 −0.404190
\(375\) 3.27502 0.169121
\(376\) −2.60555 −0.134371
\(377\) 15.9083 0.819321
\(378\) 0 0
\(379\) −15.3305 −0.787477 −0.393738 0.919223i \(-0.628818\pi\)
−0.393738 + 0.919223i \(0.628818\pi\)
\(380\) 2.60555 0.133662
\(381\) −1.44996 −0.0742838
\(382\) −12.5139 −0.640266
\(383\) −20.8444 −1.06510 −0.532550 0.846399i \(-0.678766\pi\)
−0.532550 + 0.846399i \(0.678766\pi\)
\(384\) −0.302776 −0.0154510
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 19.2111 0.976555
\(388\) −12.4222 −0.630642
\(389\) −11.8806 −0.602369 −0.301184 0.953566i \(-0.597382\pi\)
−0.301184 + 0.953566i \(0.597382\pi\)
\(390\) 0.908327 0.0459949
\(391\) −41.4500 −2.09621
\(392\) 0 0
\(393\) −1.02776 −0.0518435
\(394\) 6.00000 0.302276
\(395\) 21.0000 1.05662
\(396\) −3.78890 −0.190399
\(397\) −27.8167 −1.39608 −0.698039 0.716060i \(-0.745944\pi\)
−0.698039 + 0.716060i \(0.745944\pi\)
\(398\) −2.42221 −0.121414
\(399\) 0 0
\(400\) −3.30278 −0.165139
\(401\) 13.8167 0.689971 0.344985 0.938608i \(-0.387884\pi\)
0.344985 + 0.938608i \(0.387884\pi\)
\(402\) −4.39445 −0.219175
\(403\) −7.60555 −0.378859
\(404\) −16.4222 −0.817035
\(405\) −10.6611 −0.529753
\(406\) 0 0
\(407\) 1.30278 0.0645762
\(408\) −1.81665 −0.0899378
\(409\) 5.02776 0.248607 0.124303 0.992244i \(-0.460330\pi\)
0.124303 + 0.992244i \(0.460330\pi\)
\(410\) 1.18335 0.0584413
\(411\) −3.00000 −0.147979
\(412\) −3.30278 −0.162716
\(413\) 0 0
\(414\) −20.0917 −0.987452
\(415\) 22.4222 1.10066
\(416\) −2.30278 −0.112903
\(417\) −2.69722 −0.132084
\(418\) 2.60555 0.127442
\(419\) 25.1472 1.22852 0.614260 0.789104i \(-0.289455\pi\)
0.614260 + 0.789104i \(0.289455\pi\)
\(420\) 0 0
\(421\) 28.7250 1.39997 0.699985 0.714158i \(-0.253190\pi\)
0.699985 + 0.714158i \(0.253190\pi\)
\(422\) −6.69722 −0.326016
\(423\) −7.57779 −0.368445
\(424\) 6.00000 0.291386
\(425\) −19.8167 −0.961249
\(426\) −1.81665 −0.0880172
\(427\) 0 0
\(428\) 4.30278 0.207983
\(429\) 0.908327 0.0438544
\(430\) −8.60555 −0.414997
\(431\) −5.21110 −0.251010 −0.125505 0.992093i \(-0.540055\pi\)
−0.125505 + 0.992093i \(0.540055\pi\)
\(432\) −1.78890 −0.0860684
\(433\) 11.9361 0.573612 0.286806 0.957989i \(-0.407407\pi\)
0.286806 + 0.957989i \(0.407407\pi\)
\(434\) 0 0
\(435\) −2.72498 −0.130653
\(436\) 2.00000 0.0957826
\(437\) 13.8167 0.660940
\(438\) −2.63331 −0.125824
\(439\) 9.33053 0.445322 0.222661 0.974896i \(-0.428526\pi\)
0.222661 + 0.974896i \(0.428526\pi\)
\(440\) 1.69722 0.0809120
\(441\) 0 0
\(442\) −13.8167 −0.657191
\(443\) 0.275019 0.0130666 0.00653328 0.999979i \(-0.497920\pi\)
0.00653328 + 0.999979i \(0.497920\pi\)
\(444\) 0.302776 0.0143691
\(445\) 6.78890 0.321825
\(446\) 15.8167 0.748940
\(447\) −0.550039 −0.0260159
\(448\) 0 0
\(449\) −0.788897 −0.0372304 −0.0186152 0.999827i \(-0.505926\pi\)
−0.0186152 + 0.999827i \(0.505926\pi\)
\(450\) −9.60555 −0.452810
\(451\) 1.18335 0.0557216
\(452\) 11.2111 0.527326
\(453\) −4.05551 −0.190545
\(454\) −7.81665 −0.366854
\(455\) 0 0
\(456\) 0.605551 0.0283575
\(457\) 4.60555 0.215439 0.107719 0.994181i \(-0.465645\pi\)
0.107719 + 0.994181i \(0.465645\pi\)
\(458\) 17.3944 0.812789
\(459\) −10.7334 −0.500991
\(460\) 9.00000 0.419627
\(461\) 16.4222 0.764858 0.382429 0.923985i \(-0.375088\pi\)
0.382429 + 0.923985i \(0.375088\pi\)
\(462\) 0 0
\(463\) 30.3028 1.40829 0.704145 0.710056i \(-0.251331\pi\)
0.704145 + 0.710056i \(0.251331\pi\)
\(464\) 6.90833 0.320711
\(465\) 1.30278 0.0604148
\(466\) 9.51388 0.440722
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) −6.69722 −0.309579
\(469\) 0 0
\(470\) 3.39445 0.156574
\(471\) −2.18335 −0.100603
\(472\) 3.39445 0.156242
\(473\) −8.60555 −0.395684
\(474\) 4.88057 0.224172
\(475\) 6.60555 0.303083
\(476\) 0 0
\(477\) 17.4500 0.798979
\(478\) −0.513878 −0.0235042
\(479\) −12.1194 −0.553751 −0.276875 0.960906i \(-0.589299\pi\)
−0.276875 + 0.960906i \(0.589299\pi\)
\(480\) 0.394449 0.0180040
\(481\) 2.30278 0.104998
\(482\) 8.00000 0.364390
\(483\) 0 0
\(484\) −9.30278 −0.422853
\(485\) 16.1833 0.734848
\(486\) −7.84441 −0.355830
\(487\) −22.7889 −1.03266 −0.516332 0.856389i \(-0.672703\pi\)
−0.516332 + 0.856389i \(0.672703\pi\)
\(488\) −10.5139 −0.475941
\(489\) −6.18335 −0.279621
\(490\) 0 0
\(491\) −14.7250 −0.664529 −0.332265 0.943186i \(-0.607813\pi\)
−0.332265 + 0.943186i \(0.607813\pi\)
\(492\) 0.275019 0.0123988
\(493\) 41.4500 1.86681
\(494\) 4.60555 0.207214
\(495\) 4.93608 0.221860
\(496\) −3.30278 −0.148299
\(497\) 0 0
\(498\) 5.21110 0.233515
\(499\) 8.23886 0.368822 0.184411 0.982849i \(-0.440962\pi\)
0.184411 + 0.982849i \(0.440962\pi\)
\(500\) 10.8167 0.483735
\(501\) −3.78890 −0.169275
\(502\) −6.78890 −0.303003
\(503\) 24.5139 1.09302 0.546510 0.837453i \(-0.315956\pi\)
0.546510 + 0.837453i \(0.315956\pi\)
\(504\) 0 0
\(505\) 21.3944 0.952040
\(506\) 9.00000 0.400099
\(507\) −2.33053 −0.103503
\(508\) −4.78890 −0.212473
\(509\) 25.8167 1.14430 0.572152 0.820148i \(-0.306109\pi\)
0.572152 + 0.820148i \(0.306109\pi\)
\(510\) 2.36669 0.104799
\(511\) 0 0
\(512\) −1.00000 −0.0441942
\(513\) 3.57779 0.157964
\(514\) −11.2111 −0.494501
\(515\) 4.30278 0.189603
\(516\) −2.00000 −0.0880451
\(517\) 3.39445 0.149288
\(518\) 0 0
\(519\) 7.02776 0.308484
\(520\) 3.00000 0.131559
\(521\) 9.63331 0.422043 0.211021 0.977481i \(-0.432321\pi\)
0.211021 + 0.977481i \(0.432321\pi\)
\(522\) 20.0917 0.879389
\(523\) −32.2389 −1.40971 −0.704853 0.709353i \(-0.748987\pi\)
−0.704853 + 0.709353i \(0.748987\pi\)
\(524\) −3.39445 −0.148287
\(525\) 0 0
\(526\) 7.81665 0.340822
\(527\) −19.8167 −0.863227
\(528\) 0.394449 0.0171662
\(529\) 24.7250 1.07500
\(530\) −7.81665 −0.339534
\(531\) 9.87217 0.428416
\(532\) 0 0
\(533\) 2.09167 0.0906004
\(534\) 1.57779 0.0682779
\(535\) −5.60555 −0.242349
\(536\) −14.5139 −0.626904
\(537\) 2.36669 0.102130
\(538\) −6.78890 −0.292690
\(539\) 0 0
\(540\) 2.33053 0.100290
\(541\) −20.9361 −0.900113 −0.450056 0.893000i \(-0.648596\pi\)
−0.450056 + 0.893000i \(0.648596\pi\)
\(542\) 6.42221 0.275857
\(543\) −6.05551 −0.259867
\(544\) −6.00000 −0.257248
\(545\) −2.60555 −0.111610
\(546\) 0 0
\(547\) −13.3944 −0.572705 −0.286353 0.958124i \(-0.592443\pi\)
−0.286353 + 0.958124i \(0.592443\pi\)
\(548\) −9.90833 −0.423263
\(549\) −30.5778 −1.30503
\(550\) 4.30278 0.183471
\(551\) −13.8167 −0.588609
\(552\) 2.09167 0.0890275
\(553\) 0 0
\(554\) 25.1194 1.06722
\(555\) −0.394449 −0.0167434
\(556\) −8.90833 −0.377797
\(557\) −6.51388 −0.276002 −0.138001 0.990432i \(-0.544068\pi\)
−0.138001 + 0.990432i \(0.544068\pi\)
\(558\) −9.60555 −0.406635
\(559\) −15.2111 −0.643361
\(560\) 0 0
\(561\) 2.36669 0.0999218
\(562\) 12.0000 0.506189
\(563\) −44.0555 −1.85672 −0.928359 0.371684i \(-0.878780\pi\)
−0.928359 + 0.371684i \(0.878780\pi\)
\(564\) 0.788897 0.0332186
\(565\) −14.6056 −0.614460
\(566\) 17.3944 0.731143
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −10.4222 −0.436922 −0.218461 0.975846i \(-0.570104\pi\)
−0.218461 + 0.975846i \(0.570104\pi\)
\(570\) −0.788897 −0.0330433
\(571\) −20.3028 −0.849645 −0.424822 0.905277i \(-0.639663\pi\)
−0.424822 + 0.905277i \(0.639663\pi\)
\(572\) 3.00000 0.125436
\(573\) 3.78890 0.158283
\(574\) 0 0
\(575\) 22.8167 0.951520
\(576\) −2.90833 −0.121180
\(577\) 28.2389 1.17560 0.587800 0.809007i \(-0.299994\pi\)
0.587800 + 0.809007i \(0.299994\pi\)
\(578\) −19.0000 −0.790296
\(579\) −1.21110 −0.0503317
\(580\) −9.00000 −0.373705
\(581\) 0 0
\(582\) 3.76114 0.155904
\(583\) −7.81665 −0.323733
\(584\) −8.69722 −0.359894
\(585\) 8.72498 0.360734
\(586\) −25.0278 −1.03389
\(587\) 2.36669 0.0976838 0.0488419 0.998807i \(-0.484447\pi\)
0.0488419 + 0.998807i \(0.484447\pi\)
\(588\) 0 0
\(589\) 6.60555 0.272177
\(590\) −4.42221 −0.182059
\(591\) −1.81665 −0.0747272
\(592\) 1.00000 0.0410997
\(593\) −36.5139 −1.49945 −0.749723 0.661752i \(-0.769813\pi\)
−0.749723 + 0.661752i \(0.769813\pi\)
\(594\) 2.33053 0.0956229
\(595\) 0 0
\(596\) −1.81665 −0.0744130
\(597\) 0.733385 0.0300154
\(598\) 15.9083 0.650540
\(599\) −35.2111 −1.43869 −0.719343 0.694655i \(-0.755557\pi\)
−0.719343 + 0.694655i \(0.755557\pi\)
\(600\) 1.00000 0.0408248
\(601\) 20.6972 0.844257 0.422129 0.906536i \(-0.361283\pi\)
0.422129 + 0.906536i \(0.361283\pi\)
\(602\) 0 0
\(603\) −42.2111 −1.71897
\(604\) −13.3944 −0.545012
\(605\) 12.1194 0.492725
\(606\) 4.97224 0.201984
\(607\) 31.5139 1.27911 0.639554 0.768746i \(-0.279119\pi\)
0.639554 + 0.768746i \(0.279119\pi\)
\(608\) 2.00000 0.0811107
\(609\) 0 0
\(610\) 13.6972 0.554584
\(611\) 6.00000 0.242734
\(612\) −17.4500 −0.705373
\(613\) −8.18335 −0.330522 −0.165261 0.986250i \(-0.552847\pi\)
−0.165261 + 0.986250i \(0.552847\pi\)
\(614\) 7.09167 0.286197
\(615\) −0.358288 −0.0144476
\(616\) 0 0
\(617\) 47.5694 1.91507 0.957536 0.288314i \(-0.0930948\pi\)
0.957536 + 0.288314i \(0.0930948\pi\)
\(618\) 1.00000 0.0402259
\(619\) 2.69722 0.108411 0.0542053 0.998530i \(-0.482737\pi\)
0.0542053 + 0.998530i \(0.482737\pi\)
\(620\) 4.30278 0.172804
\(621\) 12.3583 0.495921
\(622\) 5.09167 0.204157
\(623\) 0 0
\(624\) 0.697224 0.0279113
\(625\) 2.42221 0.0968882
\(626\) 27.0278 1.08025
\(627\) −0.788897 −0.0315055
\(628\) −7.21110 −0.287754
\(629\) 6.00000 0.239236
\(630\) 0 0
\(631\) 18.3028 0.728622 0.364311 0.931277i \(-0.381305\pi\)
0.364311 + 0.931277i \(0.381305\pi\)
\(632\) 16.1194 0.641196
\(633\) 2.02776 0.0805961
\(634\) 5.21110 0.206959
\(635\) 6.23886 0.247582
\(636\) −1.81665 −0.0720350
\(637\) 0 0
\(638\) −9.00000 −0.356313
\(639\) −17.4500 −0.690310
\(640\) 1.30278 0.0514967
\(641\) −2.48612 −0.0981959 −0.0490980 0.998794i \(-0.515635\pi\)
−0.0490980 + 0.998794i \(0.515635\pi\)
\(642\) −1.30278 −0.0514165
\(643\) 29.8167 1.17585 0.587927 0.808914i \(-0.299944\pi\)
0.587927 + 0.808914i \(0.299944\pi\)
\(644\) 0 0
\(645\) 2.60555 0.102593
\(646\) 12.0000 0.472134
\(647\) 25.9361 1.01965 0.509826 0.860277i \(-0.329710\pi\)
0.509826 + 0.860277i \(0.329710\pi\)
\(648\) −8.18335 −0.321472
\(649\) −4.42221 −0.173587
\(650\) 7.60555 0.298314
\(651\) 0 0
\(652\) −20.4222 −0.799795
\(653\) 6.90833 0.270344 0.135172 0.990822i \(-0.456841\pi\)
0.135172 + 0.990822i \(0.456841\pi\)
\(654\) −0.605551 −0.0236789
\(655\) 4.42221 0.172790
\(656\) 0.908327 0.0354642
\(657\) −25.2944 −0.986827
\(658\) 0 0
\(659\) −42.1194 −1.64074 −0.820370 0.571833i \(-0.806233\pi\)
−0.820370 + 0.571833i \(0.806233\pi\)
\(660\) −0.513878 −0.0200027
\(661\) 12.4861 0.485654 0.242827 0.970070i \(-0.421925\pi\)
0.242827 + 0.970070i \(0.421925\pi\)
\(662\) −1.21110 −0.0470708
\(663\) 4.18335 0.162468
\(664\) 17.2111 0.667920
\(665\) 0 0
\(666\) 2.90833 0.112695
\(667\) −47.7250 −1.84792
\(668\) −12.5139 −0.484176
\(669\) −4.78890 −0.185149
\(670\) 18.9083 0.730492
\(671\) 13.6972 0.528775
\(672\) 0 0
\(673\) 24.3028 0.936803 0.468402 0.883516i \(-0.344830\pi\)
0.468402 + 0.883516i \(0.344830\pi\)
\(674\) 19.1194 0.736453
\(675\) 5.90833 0.227412
\(676\) −7.69722 −0.296047
\(677\) 36.2389 1.39277 0.696386 0.717667i \(-0.254790\pi\)
0.696386 + 0.717667i \(0.254790\pi\)
\(678\) −3.39445 −0.130363
\(679\) 0 0
\(680\) 7.81665 0.299755
\(681\) 2.36669 0.0906918
\(682\) 4.30278 0.164762
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) 5.81665 0.222405
\(685\) 12.9083 0.493202
\(686\) 0 0
\(687\) −5.26662 −0.200934
\(688\) −6.60555 −0.251834
\(689\) −13.8167 −0.526373
\(690\) −2.72498 −0.103738
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 23.2111 0.882354
\(693\) 0 0
\(694\) −31.8167 −1.20774
\(695\) 11.6056 0.440224
\(696\) −2.09167 −0.0792847
\(697\) 5.44996 0.206432
\(698\) −22.2389 −0.841753
\(699\) −2.88057 −0.108953
\(700\) 0 0
\(701\) 14.8806 0.562031 0.281016 0.959703i \(-0.409329\pi\)
0.281016 + 0.959703i \(0.409329\pi\)
\(702\) 4.11943 0.155478
\(703\) −2.00000 −0.0754314
\(704\) 1.30278 0.0491002
\(705\) −1.02776 −0.0387075
\(706\) 31.8167 1.19744
\(707\) 0 0
\(708\) −1.02776 −0.0386254
\(709\) −1.66947 −0.0626982 −0.0313491 0.999508i \(-0.509980\pi\)
−0.0313491 + 0.999508i \(0.509980\pi\)
\(710\) 7.81665 0.293354
\(711\) 46.8806 1.75816
\(712\) 5.21110 0.195294
\(713\) 22.8167 0.854490
\(714\) 0 0
\(715\) −3.90833 −0.146163
\(716\) 7.81665 0.292122
\(717\) 0.155590 0.00581061
\(718\) 11.2111 0.418395
\(719\) 8.36669 0.312025 0.156012 0.987755i \(-0.450136\pi\)
0.156012 + 0.987755i \(0.450136\pi\)
\(720\) 3.78890 0.141204
\(721\) 0 0
\(722\) 15.0000 0.558242
\(723\) −2.42221 −0.0900828
\(724\) −20.0000 −0.743294
\(725\) −22.8167 −0.847389
\(726\) 2.81665 0.104536
\(727\) −29.9083 −1.10924 −0.554619 0.832104i \(-0.687136\pi\)
−0.554619 + 0.832104i \(0.687136\pi\)
\(728\) 0 0
\(729\) −22.1749 −0.821294
\(730\) 11.3305 0.419362
\(731\) −39.6333 −1.46589
\(732\) 3.18335 0.117660
\(733\) −29.6333 −1.09453 −0.547266 0.836959i \(-0.684331\pi\)
−0.547266 + 0.836959i \(0.684331\pi\)
\(734\) −17.8167 −0.657625
\(735\) 0 0
\(736\) 6.90833 0.254644
\(737\) 18.9083 0.696497
\(738\) 2.64171 0.0972427
\(739\) −42.3305 −1.55715 −0.778577 0.627549i \(-0.784058\pi\)
−0.778577 + 0.627549i \(0.784058\pi\)
\(740\) −1.30278 −0.0478910
\(741\) −1.39445 −0.0512264
\(742\) 0 0
\(743\) 35.4500 1.30053 0.650266 0.759706i \(-0.274657\pi\)
0.650266 + 0.759706i \(0.274657\pi\)
\(744\) 1.00000 0.0366618
\(745\) 2.36669 0.0867089
\(746\) −3.81665 −0.139738
\(747\) 50.0555 1.83144
\(748\) 7.81665 0.285805
\(749\) 0 0
\(750\) −3.27502 −0.119587
\(751\) 14.0000 0.510867 0.255434 0.966827i \(-0.417782\pi\)
0.255434 + 0.966827i \(0.417782\pi\)
\(752\) 2.60555 0.0950147
\(753\) 2.05551 0.0749070
\(754\) −15.9083 −0.579347
\(755\) 17.4500 0.635069
\(756\) 0 0
\(757\) 9.30278 0.338115 0.169058 0.985606i \(-0.445928\pi\)
0.169058 + 0.985606i \(0.445928\pi\)
\(758\) 15.3305 0.556830
\(759\) −2.72498 −0.0989105
\(760\) −2.60555 −0.0945133
\(761\) −42.1194 −1.52683 −0.763414 0.645909i \(-0.776478\pi\)
−0.763414 + 0.645909i \(0.776478\pi\)
\(762\) 1.44996 0.0525266
\(763\) 0 0
\(764\) 12.5139 0.452736
\(765\) 22.7334 0.821927
\(766\) 20.8444 0.753139
\(767\) −7.81665 −0.282243
\(768\) 0.302776 0.0109255
\(769\) 22.0000 0.793340 0.396670 0.917961i \(-0.370166\pi\)
0.396670 + 0.917961i \(0.370166\pi\)
\(770\) 0 0
\(771\) 3.39445 0.122248
\(772\) −4.00000 −0.143963
\(773\) 50.0555 1.80037 0.900186 0.435506i \(-0.143431\pi\)
0.900186 + 0.435506i \(0.143431\pi\)
\(774\) −19.2111 −0.690529
\(775\) 10.9083 0.391839
\(776\) 12.4222 0.445931
\(777\) 0 0
\(778\) 11.8806 0.425939
\(779\) −1.81665 −0.0650884
\(780\) −0.908327 −0.0325233
\(781\) 7.81665 0.279702
\(782\) 41.4500 1.48225
\(783\) −12.3583 −0.441649
\(784\) 0 0
\(785\) 9.39445 0.335302
\(786\) 1.02776 0.0366589
\(787\) −25.2111 −0.898679 −0.449339 0.893361i \(-0.648341\pi\)
−0.449339 + 0.893361i \(0.648341\pi\)
\(788\) −6.00000 −0.213741
\(789\) −2.36669 −0.0842565
\(790\) −21.0000 −0.747146
\(791\) 0 0
\(792\) 3.78890 0.134633
\(793\) 24.2111 0.859761
\(794\) 27.8167 0.987176
\(795\) 2.36669 0.0839379
\(796\) 2.42221 0.0858528
\(797\) 17.3305 0.613879 0.306939 0.951729i \(-0.400695\pi\)
0.306939 + 0.951729i \(0.400695\pi\)
\(798\) 0 0
\(799\) 15.6333 0.553067
\(800\) 3.30278 0.116771
\(801\) 15.1556 0.535496
\(802\) −13.8167 −0.487883
\(803\) 11.3305 0.399846
\(804\) 4.39445 0.154980
\(805\) 0 0
\(806\) 7.60555 0.267894
\(807\) 2.05551 0.0723575
\(808\) 16.4222 0.577731
\(809\) 29.4500 1.03541 0.517703 0.855561i \(-0.326787\pi\)
0.517703 + 0.855561i \(0.326787\pi\)
\(810\) 10.6611 0.374592
\(811\) −54.1472 −1.90136 −0.950682 0.310166i \(-0.899615\pi\)
−0.950682 + 0.310166i \(0.899615\pi\)
\(812\) 0 0
\(813\) −1.94449 −0.0681961
\(814\) −1.30278 −0.0456623
\(815\) 26.6056 0.931952
\(816\) 1.81665 0.0635956
\(817\) 13.2111 0.462198
\(818\) −5.02776 −0.175791
\(819\) 0 0
\(820\) −1.18335 −0.0413242
\(821\) 11.2111 0.391270 0.195635 0.980677i \(-0.437323\pi\)
0.195635 + 0.980677i \(0.437323\pi\)
\(822\) 3.00000 0.104637
\(823\) −12.8444 −0.447728 −0.223864 0.974620i \(-0.571867\pi\)
−0.223864 + 0.974620i \(0.571867\pi\)
\(824\) 3.30278 0.115058
\(825\) −1.30278 −0.0453568
\(826\) 0 0
\(827\) 27.3944 0.952598 0.476299 0.879283i \(-0.341978\pi\)
0.476299 + 0.879283i \(0.341978\pi\)
\(828\) 20.0917 0.698234
\(829\) −4.72498 −0.164105 −0.0820527 0.996628i \(-0.526148\pi\)
−0.0820527 + 0.996628i \(0.526148\pi\)
\(830\) −22.4222 −0.778286
\(831\) −7.60555 −0.263834
\(832\) 2.30278 0.0798344
\(833\) 0 0
\(834\) 2.69722 0.0933972
\(835\) 16.3028 0.564181
\(836\) −2.60555 −0.0901149
\(837\) 5.90833 0.204222
\(838\) −25.1472 −0.868695
\(839\) 49.0278 1.69263 0.846313 0.532686i \(-0.178817\pi\)
0.846313 + 0.532686i \(0.178817\pi\)
\(840\) 0 0
\(841\) 18.7250 0.645689
\(842\) −28.7250 −0.989928
\(843\) −3.63331 −0.125138
\(844\) 6.69722 0.230528
\(845\) 10.0278 0.344965
\(846\) 7.57779 0.260530
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −5.26662 −0.180750
\(850\) 19.8167 0.679706
\(851\) −6.90833 −0.236814
\(852\) 1.81665 0.0622375
\(853\) 11.5416 0.395178 0.197589 0.980285i \(-0.436689\pi\)
0.197589 + 0.980285i \(0.436689\pi\)
\(854\) 0 0
\(855\) −7.57779 −0.259155
\(856\) −4.30278 −0.147066
\(857\) 14.8444 0.507075 0.253538 0.967326i \(-0.418406\pi\)
0.253538 + 0.967326i \(0.418406\pi\)
\(858\) −0.908327 −0.0310098
\(859\) 24.0555 0.820764 0.410382 0.911914i \(-0.365395\pi\)
0.410382 + 0.911914i \(0.365395\pi\)
\(860\) 8.60555 0.293447
\(861\) 0 0
\(862\) 5.21110 0.177491
\(863\) 12.0000 0.408485 0.204242 0.978920i \(-0.434527\pi\)
0.204242 + 0.978920i \(0.434527\pi\)
\(864\) 1.78890 0.0608595
\(865\) −30.2389 −1.02815
\(866\) −11.9361 −0.405605
\(867\) 5.75274 0.195373
\(868\) 0 0
\(869\) −21.0000 −0.712376
\(870\) 2.72498 0.0923855
\(871\) 33.4222 1.13247
\(872\) −2.00000 −0.0677285
\(873\) 36.1278 1.22274
\(874\) −13.8167 −0.467355
\(875\) 0 0
\(876\) 2.63331 0.0889712
\(877\) 7.21110 0.243502 0.121751 0.992561i \(-0.461149\pi\)
0.121751 + 0.992561i \(0.461149\pi\)
\(878\) −9.33053 −0.314890
\(879\) 7.57779 0.255593
\(880\) −1.69722 −0.0572134
\(881\) −25.5416 −0.860520 −0.430260 0.902705i \(-0.641578\pi\)
−0.430260 + 0.902705i \(0.641578\pi\)
\(882\) 0 0
\(883\) −2.42221 −0.0815137 −0.0407568 0.999169i \(-0.512977\pi\)
−0.0407568 + 0.999169i \(0.512977\pi\)
\(884\) 13.8167 0.464704
\(885\) 1.33894 0.0450078
\(886\) −0.275019 −0.00923945
\(887\) −28.4222 −0.954324 −0.477162 0.878815i \(-0.658335\pi\)
−0.477162 + 0.878815i \(0.658335\pi\)
\(888\) −0.302776 −0.0101605
\(889\) 0 0
\(890\) −6.78890 −0.227564
\(891\) 10.6611 0.357159
\(892\) −15.8167 −0.529581
\(893\) −5.21110 −0.174383
\(894\) 0.550039 0.0183960
\(895\) −10.1833 −0.340392
\(896\) 0 0
\(897\) −4.81665 −0.160823
\(898\) 0.788897 0.0263258
\(899\) −22.8167 −0.760978
\(900\) 9.60555 0.320185
\(901\) −36.0000 −1.19933
\(902\) −1.18335 −0.0394011
\(903\) 0 0
\(904\) −11.2111 −0.372876
\(905\) 26.0555 0.866115
\(906\) 4.05551 0.134735
\(907\) 26.0000 0.863316 0.431658 0.902037i \(-0.357929\pi\)
0.431658 + 0.902037i \(0.357929\pi\)
\(908\) 7.81665 0.259405
\(909\) 47.7611 1.58414
\(910\) 0 0
\(911\) −46.4222 −1.53804 −0.769018 0.639227i \(-0.779254\pi\)
−0.769018 + 0.639227i \(0.779254\pi\)
\(912\) −0.605551 −0.0200518
\(913\) −22.4222 −0.742067
\(914\) −4.60555 −0.152338
\(915\) −4.14719 −0.137102
\(916\) −17.3944 −0.574729
\(917\) 0 0
\(918\) 10.7334 0.354254
\(919\) −38.4222 −1.26743 −0.633716 0.773566i \(-0.718471\pi\)
−0.633716 + 0.773566i \(0.718471\pi\)
\(920\) −9.00000 −0.296721
\(921\) −2.14719 −0.0707522
\(922\) −16.4222 −0.540837
\(923\) 13.8167 0.454781
\(924\) 0 0
\(925\) −3.30278 −0.108595
\(926\) −30.3028 −0.995811
\(927\) 9.60555 0.315488
\(928\) −6.90833 −0.226777
\(929\) 36.5139 1.19798 0.598991 0.800756i \(-0.295569\pi\)
0.598991 + 0.800756i \(0.295569\pi\)
\(930\) −1.30278 −0.0427197
\(931\) 0 0
\(932\) −9.51388 −0.311637
\(933\) −1.54163 −0.0504708
\(934\) 0 0
\(935\) −10.1833 −0.333031
\(936\) 6.69722 0.218906
\(937\) 28.9083 0.944394 0.472197 0.881493i \(-0.343461\pi\)
0.472197 + 0.881493i \(0.343461\pi\)
\(938\) 0 0
\(939\) −8.18335 −0.267053
\(940\) −3.39445 −0.110715
\(941\) 7.81665 0.254816 0.127408 0.991850i \(-0.459334\pi\)
0.127408 + 0.991850i \(0.459334\pi\)
\(942\) 2.18335 0.0711373
\(943\) −6.27502 −0.204343
\(944\) −3.39445 −0.110480
\(945\) 0 0
\(946\) 8.60555 0.279791
\(947\) 39.6333 1.28791 0.643955 0.765064i \(-0.277293\pi\)
0.643955 + 0.765064i \(0.277293\pi\)
\(948\) −4.88057 −0.158514
\(949\) 20.0278 0.650128
\(950\) −6.60555 −0.214312
\(951\) −1.57779 −0.0511635
\(952\) 0 0
\(953\) −18.7527 −0.607461 −0.303730 0.952758i \(-0.598232\pi\)
−0.303730 + 0.952758i \(0.598232\pi\)
\(954\) −17.4500 −0.564963
\(955\) −16.3028 −0.527545
\(956\) 0.513878 0.0166200
\(957\) 2.72498 0.0880861
\(958\) 12.1194 0.391561
\(959\) 0 0
\(960\) −0.394449 −0.0127308
\(961\) −20.0917 −0.648118
\(962\) −2.30278 −0.0742445
\(963\) −12.5139 −0.403254
\(964\) −8.00000 −0.257663
\(965\) 5.21110 0.167751
\(966\) 0 0
\(967\) 25.7250 0.827260 0.413630 0.910445i \(-0.364261\pi\)
0.413630 + 0.910445i \(0.364261\pi\)
\(968\) 9.30278 0.299003
\(969\) −3.63331 −0.116719
\(970\) −16.1833 −0.519616
\(971\) −31.5416 −1.01222 −0.506110 0.862469i \(-0.668917\pi\)
−0.506110 + 0.862469i \(0.668917\pi\)
\(972\) 7.84441 0.251610
\(973\) 0 0
\(974\) 22.7889 0.730203
\(975\) −2.30278 −0.0737478
\(976\) 10.5139 0.336541
\(977\) 18.0000 0.575871 0.287936 0.957650i \(-0.407031\pi\)
0.287936 + 0.957650i \(0.407031\pi\)
\(978\) 6.18335 0.197722
\(979\) −6.78890 −0.216974
\(980\) 0 0
\(981\) −5.81665 −0.185711
\(982\) 14.7250 0.469893
\(983\) 12.0000 0.382741 0.191370 0.981518i \(-0.438707\pi\)
0.191370 + 0.981518i \(0.438707\pi\)
\(984\) −0.275019 −0.00876729
\(985\) 7.81665 0.249059
\(986\) −41.4500 −1.32004
\(987\) 0 0
\(988\) −4.60555 −0.146522
\(989\) 45.6333 1.45105
\(990\) −4.93608 −0.156879
\(991\) 54.3028 1.72498 0.862492 0.506070i \(-0.168902\pi\)
0.862492 + 0.506070i \(0.168902\pi\)
\(992\) 3.30278 0.104863
\(993\) 0.366692 0.0116366
\(994\) 0 0
\(995\) −3.15559 −0.100039
\(996\) −5.21110 −0.165120
\(997\) 23.5778 0.746716 0.373358 0.927687i \(-0.378206\pi\)
0.373358 + 0.927687i \(0.378206\pi\)
\(998\) −8.23886 −0.260797
\(999\) −1.78890 −0.0565982
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 3626.2.a.a.1.2 2
7.6 odd 2 74.2.a.a.1.1 2
21.20 even 2 666.2.a.j.1.1 2
28.27 even 2 592.2.a.f.1.2 2
35.13 even 4 1850.2.b.i.149.3 4
35.27 even 4 1850.2.b.i.149.2 4
35.34 odd 2 1850.2.a.u.1.2 2
56.13 odd 2 2368.2.a.s.1.2 2
56.27 even 2 2368.2.a.ba.1.1 2
77.76 even 2 8954.2.a.p.1.1 2
84.83 odd 2 5328.2.a.bf.1.1 2
259.258 odd 2 2738.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 7.6 odd 2
592.2.a.f.1.2 2 28.27 even 2
666.2.a.j.1.1 2 21.20 even 2
1850.2.a.u.1.2 2 35.34 odd 2
1850.2.b.i.149.2 4 35.27 even 4
1850.2.b.i.149.3 4 35.13 even 4
2368.2.a.s.1.2 2 56.13 odd 2
2368.2.a.ba.1.1 2 56.27 even 2
2738.2.a.l.1.1 2 259.258 odd 2
3626.2.a.a.1.2 2 1.1 even 1 trivial
5328.2.a.bf.1.1 2 84.83 odd 2
8954.2.a.p.1.1 2 77.76 even 2