Properties

Label 2738.2.a.l.1.1
Level $2738$
Weight $2$
Character 2738.1
Self dual yes
Analytic conductor $21.863$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2738 = 2 \cdot 37^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2738.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.8630400734\)
Analytic rank: \(0\)
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 1.1
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 2738.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} +4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.302776 q^{3} +1.00000 q^{4} -1.30278 q^{5} -0.302776 q^{6} +4.60555 q^{7} +1.00000 q^{8} -2.90833 q^{9} -1.30278 q^{10} +1.30278 q^{11} -0.302776 q^{12} +2.30278 q^{13} +4.60555 q^{14} +0.394449 q^{15} +1.00000 q^{16} +6.00000 q^{17} -2.90833 q^{18} -2.00000 q^{19} -1.30278 q^{20} -1.39445 q^{21} +1.30278 q^{22} +6.90833 q^{23} -0.302776 q^{24} -3.30278 q^{25} +2.30278 q^{26} +1.78890 q^{27} +4.60555 q^{28} -6.90833 q^{29} +0.394449 q^{30} -3.30278 q^{31} +1.00000 q^{32} -0.394449 q^{33} +6.00000 q^{34} -6.00000 q^{35} -2.90833 q^{36} -2.00000 q^{38} -0.697224 q^{39} -1.30278 q^{40} -0.908327 q^{41} -1.39445 q^{42} +6.60555 q^{43} +1.30278 q^{44} +3.78890 q^{45} +6.90833 q^{46} -2.60555 q^{47} -0.302776 q^{48} +14.2111 q^{49} -3.30278 q^{50} -1.81665 q^{51} +2.30278 q^{52} -6.00000 q^{53} +1.78890 q^{54} -1.69722 q^{55} +4.60555 q^{56} +0.605551 q^{57} -6.90833 q^{58} -3.39445 q^{59} +0.394449 q^{60} +10.5139 q^{61} -3.30278 q^{62} -13.3944 q^{63} +1.00000 q^{64} -3.00000 q^{65} -0.394449 q^{66} +14.5139 q^{67} +6.00000 q^{68} -2.09167 q^{69} -6.00000 q^{70} +6.00000 q^{71} -2.90833 q^{72} -8.69722 q^{73} +1.00000 q^{75} -2.00000 q^{76} +6.00000 q^{77} -0.697224 q^{78} +16.1194 q^{79} -1.30278 q^{80} +8.18335 q^{81} -0.908327 q^{82} +17.2111 q^{83} -1.39445 q^{84} -7.81665 q^{85} +6.60555 q^{86} +2.09167 q^{87} +1.30278 q^{88} -5.21110 q^{89} +3.78890 q^{90} +10.6056 q^{91} +6.90833 q^{92} +1.00000 q^{93} -2.60555 q^{94} +2.60555 q^{95} -0.302776 q^{96} -12.4222 q^{97} +14.2111 q^{98} -3.78890 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 3 q^{3} + 2 q^{4} + q^{5} + 3 q^{6} + 2 q^{7} + 2 q^{8} + 5 q^{9} + q^{10} - q^{11} + 3 q^{12} + q^{13} + 2 q^{14} + 8 q^{15} + 2 q^{16} + 12 q^{17} + 5 q^{18} - 4 q^{19} + q^{20} - 10 q^{21} - q^{22} + 3 q^{23} + 3 q^{24} - 3 q^{25} + q^{26} + 18 q^{27} + 2 q^{28} - 3 q^{29} + 8 q^{30} - 3 q^{31} + 2 q^{32} - 8 q^{33} + 12 q^{34} - 12 q^{35} + 5 q^{36} - 4 q^{38} - 5 q^{39} + q^{40} + 9 q^{41} - 10 q^{42} + 6 q^{43} - q^{44} + 22 q^{45} + 3 q^{46} + 2 q^{47} + 3 q^{48} + 14 q^{49} - 3 q^{50} + 18 q^{51} + q^{52} - 12 q^{53} + 18 q^{54} - 7 q^{55} + 2 q^{56} - 6 q^{57} - 3 q^{58} - 14 q^{59} + 8 q^{60} + 3 q^{61} - 3 q^{62} - 34 q^{63} + 2 q^{64} - 6 q^{65} - 8 q^{66} + 11 q^{67} + 12 q^{68} - 15 q^{69} - 12 q^{70} + 12 q^{71} + 5 q^{72} - 21 q^{73} + 2 q^{75} - 4 q^{76} + 12 q^{77} - 5 q^{78} + 7 q^{79} + q^{80} + 38 q^{81} + 9 q^{82} + 20 q^{83} - 10 q^{84} + 6 q^{85} + 6 q^{86} + 15 q^{87} - q^{88} + 4 q^{89} + 22 q^{90} + 14 q^{91} + 3 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{95} + 3 q^{96} + 4 q^{97} + 14 q^{98} - 22 q^{99}+O(q^{100}) \) 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 \(-0.527857\pi\)
−0.0874038 + 0.996173i \(0.527857\pi\)
\(4\) 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
\(7\) 4.60555 1.74073 0.870367 0.492403i \(-0.163881\pi\)
0.870367 + 0.492403i \(0.163881\pi\)
\(8\) 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 \(0.396540\pi\)
\(14\) 4.60555 1.23089
\(15\) 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
\(21\) −1.39445 −0.304294
\(22\) 1.30278 0.277753
\(23\) 6.90833 1.44049 0.720243 0.693722i \(-0.244030\pi\)
0.720243 + 0.693722i \(0.244030\pi\)
\(24\) −0.302776 −0.0618038
\(25\) −3.30278 −0.660555
\(26\) 2.30278 0.451611
\(27\) 1.78890 0.344273
\(28\) 4.60555 0.870367
\(29\) −6.90833 −1.28284 −0.641422 0.767188i \(-0.721655\pi\)
−0.641422 + 0.767188i \(0.721655\pi\)
\(30\) 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
\(35\) −6.00000 −1.01419
\(36\) −2.90833 −0.484721
\(37\) 0 0
\(38\) −2.00000 −0.324443
\(39\) −0.697224 −0.111645
\(40\) −1.30278 −0.205987
\(41\) −0.908327 −0.141857 −0.0709284 0.997481i \(-0.522596\pi\)
−0.0709284 + 0.997481i \(0.522596\pi\)
\(42\) −1.39445 −0.215168
\(43\) 6.60555 1.00734 0.503669 0.863897i \(-0.331983\pi\)
0.503669 + 0.863897i \(0.331983\pi\)
\(44\) 1.30278 0.196401
\(45\) 3.78890 0.564815
\(46\) 6.90833 1.01858
\(47\) −2.60555 −0.380059 −0.190029 0.981778i \(-0.560858\pi\)
−0.190029 + 0.981778i \(0.560858\pi\)
\(48\) −0.302776 −0.0437019
\(49\) 14.2111 2.03016
\(50\) −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
\(56\) 4.60555 0.615443
\(57\) 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
\(63\) −13.3944 −1.68754
\(64\) 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
\(70\) −6.00000 −0.717137
\(71\) 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 \(-0.669972\pi\)
−0.508967 + 0.860786i \(0.669972\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) −2.00000 −0.229416
\(77\) 6.00000 0.683763
\(78\) −0.697224 −0.0789451
\(79\) 16.1194 1.81358 0.906789 0.421585i \(-0.138526\pi\)
0.906789 + 0.421585i \(0.138526\pi\)
\(80\) −1.30278 −0.145655
\(81\) 8.18335 0.909261
\(82\) −0.908327 −0.100308
\(83\) 17.2111 1.88916 0.944582 0.328276i \(-0.106467\pi\)
0.944582 + 0.328276i \(0.106467\pi\)
\(84\) −1.39445 −0.152147
\(85\) −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
\(91\) 10.6056 1.11176
\(92\) 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 \(0.717208\pi\)
\(98\) 14.2111 1.43554
\(99\) −3.78890 −0.380799
\(100\) −3.30278 −0.330278
\(101\) 16.4222 1.63407 0.817035 0.576588i \(-0.195616\pi\)
0.817035 + 0.576588i \(0.195616\pi\)
\(102\) −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
\(105\) 1.81665 0.177287
\(106\) −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 \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) −1.69722 −0.161824
\(111\) 0 0
\(112\) 4.60555 0.435184
\(113\) −11.2111 −1.05465 −0.527326 0.849663i \(-0.676805\pi\)
−0.527326 + 0.849663i \(0.676805\pi\)
\(114\) 0.605551 0.0567151
\(115\) −9.00000 −0.839254
\(116\) −6.90833 −0.641422
\(117\) −6.69722 −0.619159
\(118\) −3.39445 −0.312484
\(119\) 27.6333 2.53314
\(120\) 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
\(126\) −13.3944 −1.19327
\(127\) −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
\(133\) −9.21110 −0.798704
\(134\) 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 \(-0.376682\pi\)
0.377797 + 0.925888i \(0.376682\pi\)
\(140\) −6.00000 −0.507093
\(141\) 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
\(147\) −4.30278 −0.354887
\(148\) 0 0
\(149\) −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
\(154\) 6.00000 0.483494
\(155\) 4.30278 0.345607
\(156\) −0.697224 −0.0558226
\(157\) 7.21110 0.575509 0.287754 0.957704i \(-0.407091\pi\)
0.287754 + 0.957704i \(0.407091\pi\)
\(158\) 16.1194 1.28239
\(159\) 1.81665 0.144070
\(160\) −1.30278 −0.102993
\(161\) 31.8167 2.50750
\(162\) 8.18335 0.642944
\(163\) 20.4222 1.59959 0.799795 0.600273i \(-0.204941\pi\)
0.799795 + 0.600273i \(0.204941\pi\)
\(164\) −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 \(0.660881\pi\)
\(168\) −1.39445 −0.107584
\(169\) −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 \(-0.844042\pi\)
−0.882354 + 0.470587i \(0.844042\pi\)
\(174\) 2.09167 0.158569
\(175\) −15.2111 −1.14985
\(176\) 1.30278 0.0982004
\(177\) 1.02776 0.0772509
\(178\) −5.21110 −0.390589
\(179\) −7.81665 −0.584244 −0.292122 0.956381i \(-0.594361\pi\)
−0.292122 + 0.956381i \(0.594361\pi\)
\(180\) 3.78890 0.282408
\(181\) 20.0000 1.48659 0.743294 0.668965i \(-0.233262\pi\)
0.743294 + 0.668965i \(0.233262\pi\)
\(182\) 10.6056 0.786136
\(183\) −3.18335 −0.235320
\(184\) 6.90833 0.509289
\(185\) 0 0
\(186\) 1.00000 0.0733236
\(187\) 7.81665 0.571610
\(188\) −2.60555 −0.190029
\(189\) 8.23886 0.599289
\(190\) 2.60555 0.189027
\(191\) −12.5139 −0.905472 −0.452736 0.891644i \(-0.649552\pi\)
−0.452736 + 0.891644i \(0.649552\pi\)
\(192\) −0.302776 −0.0218509
\(193\) 4.00000 0.287926 0.143963 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −12.4222 −0.891862
\(195\) 0.908327 0.0650466
\(196\) 14.2111 1.01508
\(197\) −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
\(203\) −31.8167 −2.23309
\(204\) −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
\(210\) 1.81665 0.125361
\(211\) 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
\(217\) −15.2111 −1.03260
\(218\) −2.00000 −0.135457
\(219\) 2.63331 0.177942
\(220\) −1.69722 −0.114427
\(221\) 13.8167 0.929409
\(222\) 0 0
\(223\) 15.8167 1.05916 0.529581 0.848260i \(-0.322349\pi\)
0.529581 + 0.848260i \(0.322349\pi\)
\(224\) 4.60555 0.307721
\(225\) 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 \(-0.305108\pi\)
0.574729 + 0.818344i \(0.305108\pi\)
\(230\) −9.00000 −0.593442
\(231\) −1.81665 −0.119527
\(232\) −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
\(238\) 27.6333 1.79120
\(239\) −0.513878 −0.0332400 −0.0166200 0.999862i \(-0.505291\pi\)
−0.0166200 + 0.999862i \(0.505291\pi\)
\(240\) 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
\(245\) −18.5139 −1.18281
\(246\) 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 \(0.431267\pi\)
\(252\) −13.3944 −0.843771
\(253\) 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
\(266\) −9.21110 −0.564769
\(267\) 1.57779 0.0965595
\(268\) 14.5139 0.886576
\(269\) −6.78890 −0.413926 −0.206963 0.978349i \(-0.566358\pi\)
−0.206963 + 0.978349i \(0.566358\pi\)
\(270\) −2.33053 −0.141832
\(271\) 6.42221 0.390121 0.195061 0.980791i \(-0.437510\pi\)
0.195061 + 0.980791i \(0.437510\pi\)
\(272\) 6.00000 0.363803
\(273\) −3.21110 −0.194345
\(274\) −9.90833 −0.598584
\(275\) −4.30278 −0.259467
\(276\) −2.09167 −0.125904
\(277\) 25.1194 1.50928 0.754640 0.656139i \(-0.227811\pi\)
0.754640 + 0.656139i \(0.227811\pi\)
\(278\) 8.90833 0.534286
\(279\) 9.60555 0.575069
\(280\) −6.00000 −0.358569
\(281\) 12.0000 0.715860 0.357930 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 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
\(287\) −4.18335 −0.246935
\(288\) −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 \(-0.760978\pi\)
−0.731069 + 0.682304i \(0.760978\pi\)
\(294\) −4.30278 −0.250943
\(295\) 4.42221 0.257471
\(296\) 0 0
\(297\) 2.33053 0.135231
\(298\) −1.81665 −0.105236
\(299\) 15.9083 0.920002
\(300\) 1.00000 0.0577350
\(301\) 30.4222 1.75351
\(302\) −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 \(-0.435135\pi\)
0.202372 + 0.979309i \(0.435135\pi\)
\(308\) 6.00000 0.341882
\(309\) 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
\(315\) 17.4500 0.983194
\(316\) 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
\(322\) 31.8167 1.77307
\(323\) −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
\(329\) −12.0000 −0.661581
\(330\) 0.513878 0.0282881
\(331\) −1.21110 −0.0665682 −0.0332841 0.999446i \(-0.510597\pi\)
−0.0332841 + 0.999446i \(0.510597\pi\)
\(332\) 17.2111 0.944582
\(333\) 0 0
\(334\) −12.5139 −0.684729
\(335\) −18.9083 −1.03307
\(336\) −1.39445 −0.0760734
\(337\) −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
\(343\) 33.2111 1.79323
\(344\) 6.60555 0.356147
\(345\) 2.72498 0.146708
\(346\) −23.2111 −1.24784
\(347\) −31.8167 −1.70801 −0.854004 0.520267i \(-0.825832\pi\)
−0.854004 + 0.520267i \(0.825832\pi\)
\(348\) 2.09167 0.112125
\(349\) −22.2389 −1.19042 −0.595209 0.803571i \(-0.702931\pi\)
−0.595209 + 0.803571i \(0.702931\pi\)
\(350\) −15.2111 −0.813068
\(351\) 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
\(357\) −8.36669 −0.442812
\(358\) −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
\(364\) 10.6056 0.555882
\(365\) 11.3305 0.593067
\(366\) −3.18335 −0.166396
\(367\) −17.8167 −0.930022 −0.465011 0.885305i \(-0.653950\pi\)
−0.465011 + 0.885305i \(0.653950\pi\)
\(368\) 6.90833 0.360121
\(369\) 2.64171 0.137522
\(370\) 0 0
\(371\) −27.6333 −1.43465
\(372\) 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
\(378\) 8.23886 0.423761
\(379\) −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
\(385\) −7.81665 −0.398374
\(386\) 4.00000 0.203595
\(387\) −19.2111 −0.976555
\(388\) −12.4222 −0.630642
\(389\) 11.8806 0.602369 0.301184 0.953566i \(-0.402618\pi\)
0.301184 + 0.953566i \(0.402618\pi\)
\(390\) 0.908327 0.0459949
\(391\) 41.4500 2.09621
\(392\) 14.2111 0.717769
\(393\) 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 \(-0.254056\pi\)
0.698039 + 0.716060i \(0.254056\pi\)
\(398\) 2.42221 0.121414
\(399\) 2.78890 0.139620
\(400\) −3.30278 −0.165139
\(401\) −13.8167 −0.689971 −0.344985 0.938608i \(-0.612116\pi\)
−0.344985 + 0.938608i \(0.612116\pi\)
\(402\) −4.39445 −0.219175
\(403\) −7.60555 −0.378859
\(404\) 16.4222 0.817035
\(405\) −10.6611 −0.529753
\(406\) −31.8167 −1.57903
\(407\) 0 0
\(408\) −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
\(413\) −15.6333 −0.769265
\(414\) −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 \(-0.710545\pi\)
−0.614260 + 0.789104i \(0.710545\pi\)
\(420\) 1.81665 0.0886436
\(421\) −28.7250 −1.39997 −0.699985 0.714158i \(-0.746810\pi\)
−0.699985 + 0.714158i \(0.746810\pi\)
\(422\) 6.69722 0.326016
\(423\) 7.57779 0.368445
\(424\) −6.00000 −0.291386
\(425\) −19.8167 −0.961249
\(426\) −1.81665 −0.0880172
\(427\) 48.4222 2.34331
\(428\) 4.30278 0.207983
\(429\) −0.908327 −0.0438544
\(430\) −8.60555 −0.414997
\(431\) 5.21110 0.251010 0.125505 0.992093i \(-0.459945\pi\)
0.125505 + 0.992093i \(0.459945\pi\)
\(432\) 1.78890 0.0860684
\(433\) −11.9361 −0.573612 −0.286806 0.957989i \(-0.592593\pi\)
−0.286806 + 0.957989i \(0.592593\pi\)
\(434\) −15.2111 −0.730156
\(435\) −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
\(441\) −41.3305 −1.96812
\(442\) 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 0
\(445\) 6.78890 0.321825
\(446\) 15.8167 0.748940
\(447\) 0.550039 0.0260159
\(448\) 4.60555 0.217592
\(449\) 0.788897 0.0372304 0.0186152 0.999827i \(-0.494074\pi\)
0.0186152 + 0.999827i \(0.494074\pi\)
\(450\) 9.60555 0.452810
\(451\) −1.18335 −0.0557216
\(452\) −11.2111 −0.527326
\(453\) 4.05551 0.190545
\(454\) 7.81665 0.366854
\(455\) −13.8167 −0.647735
\(456\) 0.605551 0.0283575
\(457\) −4.60555 −0.215439 −0.107719 0.994181i \(-0.534355\pi\)
−0.107719 + 0.994181i \(0.534355\pi\)
\(458\) 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 \(0.375088\pi\)
\(462\) −1.81665 −0.0845184
\(463\) −30.3028 −1.40829 −0.704145 0.710056i \(-0.748669\pi\)
−0.704145 + 0.710056i \(0.748669\pi\)
\(464\) −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
\(469\) 66.8444 3.08659
\(470\) 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
\(476\) 27.6333 1.26657
\(477\) 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
\(481\) 0 0
\(482\) −8.00000 −0.364390
\(483\) −9.63331 −0.438331
\(484\) −9.30278 −0.422853
\(485\) 16.1833 0.734848
\(486\) −7.84441 −0.355830
\(487\) 22.7889 1.03266 0.516332 0.856389i \(-0.327297\pi\)
0.516332 + 0.856389i \(0.327297\pi\)
\(488\) 10.5139 0.475941
\(489\) −6.18335 −0.279621
\(490\) −18.5139 −0.836372
\(491\) −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
\(497\) 27.6333 1.23952
\(498\) −5.21110 −0.233515
\(499\) −8.23886 −0.368822 −0.184411 0.982849i \(-0.559038\pi\)
−0.184411 + 0.982849i \(0.559038\pi\)
\(500\) 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 \(0.315956\pi\)
\(504\) −13.3944 −0.596636
\(505\) −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 \(-0.693891\pi\)
−0.572152 + 0.820148i \(0.693891\pi\)
\(510\) 2.36669 0.104799
\(511\) −40.0555 −1.77195
\(512\) 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 \(-0.567679\pi\)
−0.211021 + 0.977481i \(0.567679\pi\)
\(522\) 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
\(525\) 4.60555 0.201003
\(526\) −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
\(532\) −9.21110 −0.399352
\(533\) −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
\(539\) 18.5139 0.797449
\(540\) −2.33053 −0.100290
\(541\) 20.9361 0.900113 0.450056 0.893000i \(-0.351404\pi\)
0.450056 + 0.893000i \(0.351404\pi\)
\(542\) 6.42221 0.275857
\(543\) −6.05551 −0.259867
\(544\) 6.00000 0.257248
\(545\) 2.60555 0.111610
\(546\) −3.21110 −0.137423
\(547\) 13.3944 0.572705 0.286353 0.958124i \(-0.407557\pi\)
0.286353 + 0.958124i \(0.407557\pi\)
\(548\) −9.90833 −0.423263
\(549\) −30.5778 −1.30503
\(550\) −4.30278 −0.183471
\(551\) 13.8167 0.588609
\(552\) −2.09167 −0.0890275
\(553\) 74.2389 3.15696
\(554\) 25.1194 1.06722
\(555\) 0 0
\(556\) 8.90833 0.377797
\(557\) 6.51388 0.276002 0.138001 0.990432i \(-0.455932\pi\)
0.138001 + 0.990432i \(0.455932\pi\)
\(558\) 9.60555 0.406635
\(559\) 15.2111 0.643361
\(560\) −6.00000 −0.253546
\(561\) −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
\(567\) 37.6888 1.58278
\(568\) 6.00000 0.251754
\(569\) 10.4222 0.436922 0.218461 0.975846i \(-0.429896\pi\)
0.218461 + 0.975846i \(0.429896\pi\)
\(570\) −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
\(574\) −4.18335 −0.174609
\(575\) −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
\(581\) 79.2666 3.28853
\(582\) 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 \(0.484447\pi\)
\(588\) −4.30278 −0.177443
\(589\) 6.60555 0.272177
\(590\) 4.42221 0.182059
\(591\) 1.81665 0.0747272
\(592\) 0 0
\(593\) 36.5139 1.49945 0.749723 0.661752i \(-0.230187\pi\)
0.749723 + 0.661752i \(0.230187\pi\)
\(594\) 2.33053 0.0956229
\(595\) −36.0000 −1.47586
\(596\) −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 \(-0.638717\pi\)
−0.422129 + 0.906536i \(0.638717\pi\)
\(602\) 30.4222 1.23992
\(603\) −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
\(609\) 9.63331 0.390361
\(610\) −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
\(616\) 6.00000 0.241747
\(617\) 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 \(-0.517263\pi\)
−0.0542053 + 0.998530i \(0.517263\pi\)
\(620\) 4.30278 0.172804
\(621\) 12.3583 0.495921
\(622\) −5.09167 −0.204157
\(623\) −24.0000 −0.961540
\(624\) −0.697224 −0.0279113
\(625\) 2.42221 0.0968882
\(626\) −27.0278 −1.08025
\(627\) 0.788897 0.0315055
\(628\) 7.21110 0.287754
\(629\) 0 0
\(630\) 17.4500 0.695223
\(631\) −18.3028 −0.728622 −0.364311 0.931277i \(-0.618695\pi\)
−0.364311 + 0.931277i \(0.618695\pi\)
\(632\) 16.1194 0.641196
\(633\) −2.02776 −0.0805961
\(634\) −5.21110 −0.206959
\(635\) 6.23886 0.247582
\(636\) 1.81665 0.0720350
\(637\) 32.7250 1.29661
\(638\) −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 \(0.299944\pi\)
\(644\) 31.8167 1.25375
\(645\) 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
\(651\) 4.60555 0.180506
\(652\) 20.4222 0.799795
\(653\) −6.90833 −0.270344 −0.135172 0.990822i \(-0.543159\pi\)
−0.135172 + 0.990822i \(0.543159\pi\)
\(654\) 0.605551 0.0236789
\(655\) 4.42221 0.172790
\(656\) −0.908327 −0.0354642
\(657\) 25.2944 0.986827
\(658\) −12.0000 −0.467809
\(659\) −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
\(665\) 12.0000 0.465340
\(666\) 0 0
\(667\) −47.7250 −1.84792
\(668\) −12.5139 −0.484176
\(669\) −4.78890 −0.185149
\(670\) −18.9083 −0.730492
\(671\) 13.6972 0.528775
\(672\) −1.39445 −0.0537920
\(673\) 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 \(-0.745210\pi\)
−0.696386 + 0.717667i \(0.745210\pi\)
\(678\) 3.39445 0.130363
\(679\) −57.2111 −2.19556
\(680\) −7.81665 −0.299755
\(681\) −2.36669 −0.0906918
\(682\) −4.30278 −0.164762
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 5.81665 0.222405
\(685\) 12.9083 0.493202
\(686\) 33.2111 1.26801
\(687\) −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 \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) −23.2111 −0.882354
\(693\) −17.4500 −0.662869
\(694\) −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
\(700\) −15.2111 −0.574926
\(701\) −14.8806 −0.562031 −0.281016 0.959703i \(-0.590671\pi\)
−0.281016 + 0.959703i \(0.590671\pi\)
\(702\) 4.11943 0.155478
\(703\) 0 0
\(704\) 1.30278 0.0491002
\(705\) −1.02776 −0.0387075
\(706\) −31.8167 −1.19744
\(707\) 75.6333 2.84448
\(708\) 1.02776 0.0386254
\(709\) 1.66947 0.0626982 0.0313491 0.999508i \(-0.490020\pi\)
0.0313491 + 0.999508i \(0.490020\pi\)
\(710\) −7.81665 −0.293354
\(711\) −46.8806 −1.75816
\(712\) −5.21110 −0.195294
\(713\) −22.8167 −0.854490
\(714\) −8.36669 −0.313116
\(715\) −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 \(-0.549864\pi\)
−0.156012 + 0.987755i \(0.549864\pi\)
\(720\) 3.78890 0.141204
\(721\) −15.2111 −0.566491
\(722\) −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 \(0.687136\pi\)
\(728\) 10.6056 0.393068
\(729\) −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 \(-0.315669\pi\)
0.547266 + 0.836959i \(0.315669\pi\)
\(734\) −17.8167 −0.657625
\(735\) 5.60555 0.206764
\(736\) 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 \(0.784058\pi\)
\(740\) 0 0
\(741\) 1.39445 0.0512264
\(742\) −27.6333 −1.01445
\(743\) 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
\(749\) 19.8167 0.724085
\(750\) −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
\(756\) 8.23886 0.299644
\(757\) −9.30278 −0.338115 −0.169058 0.985606i \(-0.554072\pi\)
−0.169058 + 0.985606i \(0.554072\pi\)
\(758\) −15.3305 −0.556830
\(759\) −2.72498 −0.0989105
\(760\) 2.60555 0.0945133
\(761\) 42.1194 1.52683 0.763414 0.645909i \(-0.223522\pi\)
0.763414 + 0.645909i \(0.223522\pi\)
\(762\) 1.44996 0.0525266
\(763\) −9.21110 −0.333464
\(764\) −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 \(0.370166\pi\)
\(770\) −7.81665 −0.281693
\(771\) −3.39445 −0.122248
\(772\) 4.00000 0.143963
\(773\) −50.0555 −1.80037 −0.900186 0.435506i \(-0.856569\pi\)
−0.900186 + 0.435506i \(0.856569\pi\)
\(774\) −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
\(784\) 14.2111 0.507539
\(785\) −9.39445 −0.335302
\(786\) 1.02776 0.0366589
\(787\) 25.2111 0.898679 0.449339 0.893361i \(-0.351659\pi\)
0.449339 + 0.893361i \(0.351659\pi\)
\(788\) −6.00000 −0.213741
\(789\) 2.36669 0.0842565
\(790\) −21.0000 −0.747146
\(791\) −51.6333 −1.83587
\(792\) −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 \(0.400695\pi\)
\(798\) 2.78890 0.0987259
\(799\) −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
\(805\) −41.4500 −1.46092
\(806\) −7.60555 −0.267894
\(807\) 2.05551 0.0723575
\(808\) 16.4222 0.577731
\(809\) −29.4500 −1.03541 −0.517703 0.855561i \(-0.673213\pi\)
−0.517703 + 0.855561i \(0.673213\pi\)
\(810\) −10.6611 −0.374592
\(811\) 54.1472 1.90136 0.950682 0.310166i \(-0.100385\pi\)
0.950682 + 0.310166i \(0.100385\pi\)
\(812\) −31.8167 −1.11655
\(813\) −1.94449 −0.0681961
\(814\) 0 0
\(815\) −26.6056 −0.931952
\(816\) −1.81665 −0.0635956
\(817\) −13.2111 −0.462198
\(818\) 5.02776 0.175791
\(819\) −30.8444 −1.07779
\(820\) 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
\(826\) −15.6333 −0.543952
\(827\) −27.3944 −0.952598 −0.476299 0.879283i \(-0.658022\pi\)
−0.476299 + 0.879283i \(0.658022\pi\)
\(828\) −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
\(833\) 85.2666 2.95431
\(834\) −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 \(-0.821183\pi\)
−0.846313 + 0.532686i \(0.821183\pi\)
\(840\) 1.81665 0.0626805
\(841\) 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
\(847\) −42.8444 −1.47215
\(848\) −6.00000 −0.206041
\(849\) 5.26662 0.180750
\(850\) −19.8167 −0.679706
\(851\) 0 0
\(852\) −1.81665 −0.0622375
\(853\) 11.5416 0.395178 0.197589 0.980285i \(-0.436689\pi\)
0.197589 + 0.980285i \(0.436689\pi\)
\(854\) 48.4222 1.65697
\(855\) −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
\(861\) 1.26662 0.0431661
\(862\) 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
\(868\) −15.2111 −0.516298
\(869\) 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
\(875\) 49.8167 1.68411
\(876\) 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 \(-0.358422\pi\)
0.430260 + 0.902705i \(0.358422\pi\)
\(882\) −41.3305 −1.39167
\(883\) 2.42221 0.0815137 0.0407568 0.999169i \(-0.487023\pi\)
0.0407568 + 0.999169i \(0.487023\pi\)
\(884\) 13.8167 0.464704
\(885\) −1.33894 −0.0450078
\(886\) 0.275019 0.00923945
\(887\) 28.4222 0.954324 0.477162 0.878815i \(-0.341665\pi\)
0.477162 + 0.878815i \(0.341665\pi\)
\(888\) 0 0
\(889\) −22.0555 −0.739718
\(890\) 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
\(896\) 4.60555 0.153861
\(897\) −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
\(903\) −9.21110 −0.306526
\(904\) −11.2111 −0.372876
\(905\) −26.0555 −0.866115
\(906\) 4.05551 0.134735
\(907\) −26.0000 −0.863316 −0.431658 0.902037i \(-0.642071\pi\)
−0.431658 + 0.902037i \(0.642071\pi\)
\(908\) 7.81665 0.259405
\(909\) −47.7611 −1.58414
\(910\) −13.8167 −0.458018
\(911\) 46.4222 1.53804 0.769018 0.639227i \(-0.220746\pi\)
0.769018 + 0.639227i \(0.220746\pi\)
\(912\) 0.605551 0.0200518
\(913\) 22.4222 0.742067
\(914\) −4.60555 −0.152338
\(915\) 4.14719 0.137102
\(916\) 17.3944 0.574729
\(917\) −15.6333 −0.516257
\(918\) 10.7334 0.354254
\(919\) 38.4222 1.26743 0.633716 0.773566i \(-0.281529\pi\)
0.633716 + 0.773566i \(0.281529\pi\)
\(920\) −9.00000 −0.296721
\(921\) −2.14719 −0.0707522
\(922\) 16.4222 0.540837
\(923\) 13.8167 0.454781
\(924\) −1.81665 −0.0597635
\(925\) 0 0
\(926\) −30.3028 −0.995811
\(927\) 9.60555 0.315488
\(928\) −6.90833 −0.226777
\(929\) −36.5139 −1.19798 −0.598991 0.800756i \(-0.704431\pi\)
−0.598991 + 0.800756i \(0.704431\pi\)
\(930\) −1.30278 −0.0427197
\(931\) −28.4222 −0.931500
\(932\) −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 \(-0.656539\pi\)
−0.472197 + 0.881493i \(0.656539\pi\)
\(938\) 66.8444 2.18255
\(939\) 8.18335 0.267053
\(940\) 3.39445 0.110715
\(941\) −7.81665 −0.254816 −0.127408 0.991850i \(-0.540666\pi\)
−0.127408 + 0.991850i \(0.540666\pi\)
\(942\) −2.18335 −0.0711373
\(943\) −6.27502 −0.204343
\(944\) −3.39445 −0.110480
\(945\) −10.7334 −0.349157
\(946\) 8.60555 0.279791
\(947\) −39.6333 −1.28791 −0.643955 0.765064i \(-0.722707\pi\)
−0.643955 + 0.765064i \(0.722707\pi\)
\(948\) −4.88057 −0.158514
\(949\) −20.0278 −0.650128
\(950\) 6.60555 0.214312
\(951\) 1.57779 0.0511635
\(952\) 27.6333 0.895601
\(953\) −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
\(959\) −45.6333 −1.47358
\(960\) 0.394449 0.0127308
\(961\) −20.0917 −0.648118
\(962\) 0 0
\(963\) −12.5139 −0.403254
\(964\) −8.00000 −0.257663
\(965\) −5.21110 −0.167751
\(966\) −9.63331 −0.309947
\(967\) −25.7250 −0.827260 −0.413630 0.910445i \(-0.635739\pi\)
−0.413630 + 0.910445i \(0.635739\pi\)
\(968\) −9.30278 −0.299003
\(969\) 3.63331 0.116719
\(970\) 16.1833 0.519616
\(971\) 31.5416 1.01222 0.506110 0.862469i \(-0.331083\pi\)
0.506110 + 0.862469i \(0.331083\pi\)
\(972\) −7.84441 −0.251610
\(973\) 41.0278 1.31529
\(974\) 22.7889 0.730203
\(975\) 2.30278 0.0737478
\(976\) 10.5139 0.336541
\(977\) −18.0000 −0.575871 −0.287936 0.957650i \(-0.592969\pi\)
−0.287936 + 0.957650i \(0.592969\pi\)
\(978\) −6.18335 −0.197722
\(979\) −6.78890 −0.216974
\(980\) −18.5139 −0.591404
\(981\) 5.81665 0.185711
\(982\) −14.7250 −0.469893
\(983\) −12.0000 −0.382741 −0.191370 0.981518i \(-0.561293\pi\)
−0.191370 + 0.981518i \(0.561293\pi\)
\(984\) 0.275019 0.00876729
\(985\) 7.81665 0.249059
\(986\) −41.4500 −1.32004
\(987\) 3.63331 0.115649
\(988\) −4.60555 −0.146522
\(989\) 45.6333 1.45105
\(990\) 4.93608 0.156879
\(991\) −54.3028 −1.72498 −0.862492 0.506070i \(-0.831098\pi\)
−0.862492 + 0.506070i \(0.831098\pi\)
\(992\) −3.30278 −0.104863
\(993\) 0.366692 0.0116366
\(994\) 27.6333 0.876475
\(995\) −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
\(999\) 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 2738.2.a.l.1.1 2
37.36 even 2 74.2.a.a.1.1 2
111.110 odd 2 666.2.a.j.1.1 2
148.147 odd 2 592.2.a.f.1.2 2
185.73 odd 4 1850.2.b.i.149.3 4
185.147 odd 4 1850.2.b.i.149.2 4
185.184 even 2 1850.2.a.u.1.2 2
259.258 odd 2 3626.2.a.a.1.2 2
296.147 odd 2 2368.2.a.ba.1.1 2
296.221 even 2 2368.2.a.s.1.2 2
407.406 odd 2 8954.2.a.p.1.1 2
444.443 even 2 5328.2.a.bf.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
74.2.a.a.1.1 2 37.36 even 2
592.2.a.f.1.2 2 148.147 odd 2
666.2.a.j.1.1 2 111.110 odd 2
1850.2.a.u.1.2 2 185.184 even 2
1850.2.b.i.149.2 4 185.147 odd 4
1850.2.b.i.149.3 4 185.73 odd 4
2368.2.a.s.1.2 2 296.221 even 2
2368.2.a.ba.1.1 2 296.147 odd 2
2738.2.a.l.1.1 2 1.1 even 1 trivial
3626.2.a.a.1.2 2 259.258 odd 2
5328.2.a.bf.1.1 2 444.443 even 2
8954.2.a.p.1.1 2 407.406 odd 2