Properties

Label 3525.2.a.bd.1.1
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3525.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(28.1472667125\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 3 x^{7} - 7 x^{6} + 24 x^{5} + 8 x^{4} - 47 x^{3} + 8 x^{2} + 13 x + 2\)
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 1.1
Root \(2.62510\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.62510 q^{2} +1.00000 q^{3} +4.89115 q^{4} -2.62510 q^{6} -5.24764 q^{7} -7.58955 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.62510 q^{2} +1.00000 q^{3} +4.89115 q^{4} -2.62510 q^{6} -5.24764 q^{7} -7.58955 q^{8} +1.00000 q^{9} +2.45327 q^{11} +4.89115 q^{12} +1.47869 q^{13} +13.7756 q^{14} +10.1410 q^{16} -4.14255 q^{17} -2.62510 q^{18} +4.25408 q^{19} -5.24764 q^{21} -6.44007 q^{22} +6.50248 q^{23} -7.58955 q^{24} -3.88171 q^{26} +1.00000 q^{27} -25.6670 q^{28} -8.17052 q^{29} -5.69350 q^{31} -11.4421 q^{32} +2.45327 q^{33} +10.8746 q^{34} +4.89115 q^{36} +1.84684 q^{37} -11.1674 q^{38} +1.47869 q^{39} -9.39994 q^{41} +13.7756 q^{42} +4.34441 q^{43} +11.9993 q^{44} -17.0697 q^{46} -1.00000 q^{47} +10.1410 q^{48} +20.5377 q^{49} -4.14255 q^{51} +7.23250 q^{52} +6.95955 q^{53} -2.62510 q^{54} +39.8272 q^{56} +4.25408 q^{57} +21.4484 q^{58} -7.70051 q^{59} +12.0670 q^{61} +14.9460 q^{62} -5.24764 q^{63} +9.75464 q^{64} -6.44007 q^{66} +5.84395 q^{67} -20.2618 q^{68} +6.50248 q^{69} -2.81951 q^{71} -7.58955 q^{72} -5.10213 q^{73} -4.84815 q^{74} +20.8074 q^{76} -12.8739 q^{77} -3.88171 q^{78} +5.56405 q^{79} +1.00000 q^{81} +24.6758 q^{82} -15.4973 q^{83} -25.6670 q^{84} -11.4045 q^{86} -8.17052 q^{87} -18.6192 q^{88} -3.59698 q^{89} -7.75963 q^{91} +31.8046 q^{92} -5.69350 q^{93} +2.62510 q^{94} -11.4421 q^{96} +3.34748 q^{97} -53.9135 q^{98} +2.45327 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.62510 −1.85623 −0.928113 0.372299i \(-0.878570\pi\)
−0.928113 + 0.372299i \(0.878570\pi\)
\(3\) 1.00000 0.577350
\(4\) 4.89115 2.44557
\(5\) 0 0
\(6\) −2.62510 −1.07169
\(7\) −5.24764 −1.98342 −0.991710 0.128494i \(-0.958986\pi\)
−0.991710 + 0.128494i \(0.958986\pi\)
\(8\) −7.58955 −2.68331
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 2.45327 0.739688 0.369844 0.929094i \(-0.379411\pi\)
0.369844 + 0.929094i \(0.379411\pi\)
\(12\) 4.89115 1.41195
\(13\) 1.47869 0.410115 0.205058 0.978750i \(-0.434262\pi\)
0.205058 + 0.978750i \(0.434262\pi\)
\(14\) 13.7756 3.68168
\(15\) 0 0
\(16\) 10.1410 2.53526
\(17\) −4.14255 −1.00471 −0.502357 0.864660i \(-0.667534\pi\)
−0.502357 + 0.864660i \(0.667534\pi\)
\(18\) −2.62510 −0.618742
\(19\) 4.25408 0.975954 0.487977 0.872857i \(-0.337735\pi\)
0.487977 + 0.872857i \(0.337735\pi\)
\(20\) 0 0
\(21\) −5.24764 −1.14513
\(22\) −6.44007 −1.37303
\(23\) 6.50248 1.35586 0.677931 0.735126i \(-0.262877\pi\)
0.677931 + 0.735126i \(0.262877\pi\)
\(24\) −7.58955 −1.54921
\(25\) 0 0
\(26\) −3.88171 −0.761266
\(27\) 1.00000 0.192450
\(28\) −25.6670 −4.85060
\(29\) −8.17052 −1.51723 −0.758614 0.651540i \(-0.774123\pi\)
−0.758614 + 0.651540i \(0.774123\pi\)
\(30\) 0 0
\(31\) −5.69350 −1.02258 −0.511291 0.859407i \(-0.670833\pi\)
−0.511291 + 0.859407i \(0.670833\pi\)
\(32\) −11.4421 −2.02270
\(33\) 2.45327 0.427059
\(34\) 10.8746 1.86498
\(35\) 0 0
\(36\) 4.89115 0.815191
\(37\) 1.84684 0.303619 0.151810 0.988410i \(-0.451490\pi\)
0.151810 + 0.988410i \(0.451490\pi\)
\(38\) −11.1674 −1.81159
\(39\) 1.47869 0.236780
\(40\) 0 0
\(41\) −9.39994 −1.46802 −0.734012 0.679137i \(-0.762354\pi\)
−0.734012 + 0.679137i \(0.762354\pi\)
\(42\) 13.7756 2.12562
\(43\) 4.34441 0.662517 0.331259 0.943540i \(-0.392527\pi\)
0.331259 + 0.943540i \(0.392527\pi\)
\(44\) 11.9993 1.80896
\(45\) 0 0
\(46\) −17.0697 −2.51679
\(47\) −1.00000 −0.145865
\(48\) 10.1410 1.46373
\(49\) 20.5377 2.93396
\(50\) 0 0
\(51\) −4.14255 −0.580072
\(52\) 7.23250 1.00297
\(53\) 6.95955 0.955967 0.477984 0.878369i \(-0.341368\pi\)
0.477984 + 0.878369i \(0.341368\pi\)
\(54\) −2.62510 −0.357231
\(55\) 0 0
\(56\) 39.8272 5.32214
\(57\) 4.25408 0.563467
\(58\) 21.4484 2.81632
\(59\) −7.70051 −1.00252 −0.501260 0.865297i \(-0.667130\pi\)
−0.501260 + 0.865297i \(0.667130\pi\)
\(60\) 0 0
\(61\) 12.0670 1.54501 0.772507 0.635006i \(-0.219002\pi\)
0.772507 + 0.635006i \(0.219002\pi\)
\(62\) 14.9460 1.89814
\(63\) −5.24764 −0.661140
\(64\) 9.75464 1.21933
\(65\) 0 0
\(66\) −6.44007 −0.792718
\(67\) 5.84395 0.713952 0.356976 0.934113i \(-0.383808\pi\)
0.356976 + 0.934113i \(0.383808\pi\)
\(68\) −20.2618 −2.45710
\(69\) 6.50248 0.782807
\(70\) 0 0
\(71\) −2.81951 −0.334614 −0.167307 0.985905i \(-0.553507\pi\)
−0.167307 + 0.985905i \(0.553507\pi\)
\(72\) −7.58955 −0.894437
\(73\) −5.10213 −0.597159 −0.298579 0.954385i \(-0.596513\pi\)
−0.298579 + 0.954385i \(0.596513\pi\)
\(74\) −4.84815 −0.563586
\(75\) 0 0
\(76\) 20.8074 2.38677
\(77\) −12.8739 −1.46711
\(78\) −3.88171 −0.439517
\(79\) 5.56405 0.626005 0.313002 0.949752i \(-0.398665\pi\)
0.313002 + 0.949752i \(0.398665\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 24.6758 2.72498
\(83\) −15.4973 −1.70105 −0.850523 0.525937i \(-0.823715\pi\)
−0.850523 + 0.525937i \(0.823715\pi\)
\(84\) −25.6670 −2.80050
\(85\) 0 0
\(86\) −11.4045 −1.22978
\(87\) −8.17052 −0.875972
\(88\) −18.6192 −1.98481
\(89\) −3.59698 −0.381280 −0.190640 0.981660i \(-0.561056\pi\)
−0.190640 + 0.981660i \(0.561056\pi\)
\(90\) 0 0
\(91\) −7.75963 −0.813431
\(92\) 31.8046 3.31586
\(93\) −5.69350 −0.590388
\(94\) 2.62510 0.270758
\(95\) 0 0
\(96\) −11.4421 −1.16781
\(97\) 3.34748 0.339886 0.169943 0.985454i \(-0.445642\pi\)
0.169943 + 0.985454i \(0.445642\pi\)
\(98\) −53.9135 −5.44609
\(99\) 2.45327 0.246563
\(100\) 0 0
\(101\) −1.29938 −0.129294 −0.0646468 0.997908i \(-0.520592\pi\)
−0.0646468 + 0.997908i \(0.520592\pi\)
\(102\) 10.8746 1.07675
\(103\) 16.5846 1.63413 0.817065 0.576546i \(-0.195600\pi\)
0.817065 + 0.576546i \(0.195600\pi\)
\(104\) −11.2226 −1.10047
\(105\) 0 0
\(106\) −18.2695 −1.77449
\(107\) −10.0354 −0.970163 −0.485081 0.874469i \(-0.661210\pi\)
−0.485081 + 0.874469i \(0.661210\pi\)
\(108\) 4.89115 0.470651
\(109\) −9.16258 −0.877616 −0.438808 0.898581i \(-0.644599\pi\)
−0.438808 + 0.898581i \(0.644599\pi\)
\(110\) 0 0
\(111\) 1.84684 0.175295
\(112\) −53.2165 −5.02848
\(113\) −11.4553 −1.07762 −0.538810 0.842427i \(-0.681126\pi\)
−0.538810 + 0.842427i \(0.681126\pi\)
\(114\) −11.1674 −1.04592
\(115\) 0 0
\(116\) −39.9632 −3.71049
\(117\) 1.47869 0.136705
\(118\) 20.2146 1.86090
\(119\) 21.7386 1.99277
\(120\) 0 0
\(121\) −4.98148 −0.452862
\(122\) −31.6770 −2.86790
\(123\) −9.39994 −0.847564
\(124\) −27.8477 −2.50080
\(125\) 0 0
\(126\) 13.7756 1.22723
\(127\) −7.50757 −0.666189 −0.333095 0.942893i \(-0.608093\pi\)
−0.333095 + 0.942893i \(0.608093\pi\)
\(128\) −2.72267 −0.240652
\(129\) 4.34441 0.382504
\(130\) 0 0
\(131\) 12.4292 1.08595 0.542974 0.839749i \(-0.317298\pi\)
0.542974 + 0.839749i \(0.317298\pi\)
\(132\) 11.9993 1.04440
\(133\) −22.3239 −1.93573
\(134\) −15.3410 −1.32526
\(135\) 0 0
\(136\) 31.4401 2.69596
\(137\) 12.7385 1.08832 0.544162 0.838980i \(-0.316848\pi\)
0.544162 + 0.838980i \(0.316848\pi\)
\(138\) −17.0697 −1.45307
\(139\) −13.7235 −1.16401 −0.582005 0.813185i \(-0.697732\pi\)
−0.582005 + 0.813185i \(0.697732\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) 7.40148 0.621119
\(143\) 3.62762 0.303357
\(144\) 10.1410 0.845086
\(145\) 0 0
\(146\) 13.3936 1.10846
\(147\) 20.5377 1.69392
\(148\) 9.03319 0.742524
\(149\) 6.30968 0.516909 0.258455 0.966023i \(-0.416787\pi\)
0.258455 + 0.966023i \(0.416787\pi\)
\(150\) 0 0
\(151\) 7.80524 0.635182 0.317591 0.948228i \(-0.397126\pi\)
0.317591 + 0.948228i \(0.397126\pi\)
\(152\) −32.2866 −2.61879
\(153\) −4.14255 −0.334905
\(154\) 33.7952 2.72329
\(155\) 0 0
\(156\) 7.23250 0.579063
\(157\) −21.1234 −1.68583 −0.842914 0.538048i \(-0.819162\pi\)
−0.842914 + 0.538048i \(0.819162\pi\)
\(158\) −14.6062 −1.16201
\(159\) 6.95955 0.551928
\(160\) 0 0
\(161\) −34.1227 −2.68924
\(162\) −2.62510 −0.206247
\(163\) −19.5801 −1.53363 −0.766816 0.641867i \(-0.778160\pi\)
−0.766816 + 0.641867i \(0.778160\pi\)
\(164\) −45.9765 −3.59016
\(165\) 0 0
\(166\) 40.6819 3.15753
\(167\) 2.84701 0.220308 0.110154 0.993915i \(-0.464866\pi\)
0.110154 + 0.993915i \(0.464866\pi\)
\(168\) 39.8272 3.07274
\(169\) −10.8135 −0.831806
\(170\) 0 0
\(171\) 4.25408 0.325318
\(172\) 21.2492 1.62023
\(173\) 7.99152 0.607584 0.303792 0.952738i \(-0.401747\pi\)
0.303792 + 0.952738i \(0.401747\pi\)
\(174\) 21.4484 1.62600
\(175\) 0 0
\(176\) 24.8787 1.87530
\(177\) −7.70051 −0.578806
\(178\) 9.44244 0.707741
\(179\) −11.6594 −0.871461 −0.435731 0.900077i \(-0.643510\pi\)
−0.435731 + 0.900077i \(0.643510\pi\)
\(180\) 0 0
\(181\) 19.5331 1.45188 0.725942 0.687756i \(-0.241404\pi\)
0.725942 + 0.687756i \(0.241404\pi\)
\(182\) 20.3698 1.50991
\(183\) 12.0670 0.892015
\(184\) −49.3509 −3.63820
\(185\) 0 0
\(186\) 14.9460 1.09589
\(187\) −10.1628 −0.743175
\(188\) −4.89115 −0.356724
\(189\) −5.24764 −0.381709
\(190\) 0 0
\(191\) −23.1126 −1.67237 −0.836185 0.548447i \(-0.815219\pi\)
−0.836185 + 0.548447i \(0.815219\pi\)
\(192\) 9.75464 0.703981
\(193\) −17.2364 −1.24070 −0.620351 0.784324i \(-0.713010\pi\)
−0.620351 + 0.784324i \(0.713010\pi\)
\(194\) −8.78748 −0.630904
\(195\) 0 0
\(196\) 100.453 7.17521
\(197\) −11.1056 −0.791240 −0.395620 0.918414i \(-0.629470\pi\)
−0.395620 + 0.918414i \(0.629470\pi\)
\(198\) −6.44007 −0.457676
\(199\) −23.0611 −1.63476 −0.817380 0.576098i \(-0.804575\pi\)
−0.817380 + 0.576098i \(0.804575\pi\)
\(200\) 0 0
\(201\) 5.84395 0.412201
\(202\) 3.41101 0.239998
\(203\) 42.8759 3.00930
\(204\) −20.2618 −1.41861
\(205\) 0 0
\(206\) −43.5362 −3.03331
\(207\) 6.50248 0.451954
\(208\) 14.9955 1.03975
\(209\) 10.4364 0.721901
\(210\) 0 0
\(211\) 4.09406 0.281847 0.140923 0.990021i \(-0.454993\pi\)
0.140923 + 0.990021i \(0.454993\pi\)
\(212\) 34.0402 2.33789
\(213\) −2.81951 −0.193189
\(214\) 26.3440 1.80084
\(215\) 0 0
\(216\) −7.58955 −0.516404
\(217\) 29.8774 2.02821
\(218\) 24.0527 1.62905
\(219\) −5.10213 −0.344770
\(220\) 0 0
\(221\) −6.12555 −0.412049
\(222\) −4.84815 −0.325387
\(223\) 21.0775 1.41146 0.705728 0.708483i \(-0.250620\pi\)
0.705728 + 0.708483i \(0.250620\pi\)
\(224\) 60.0441 4.01187
\(225\) 0 0
\(226\) 30.0712 2.00031
\(227\) 1.70842 0.113392 0.0566958 0.998391i \(-0.481943\pi\)
0.0566958 + 0.998391i \(0.481943\pi\)
\(228\) 20.8074 1.37800
\(229\) 5.07837 0.335588 0.167794 0.985822i \(-0.446336\pi\)
0.167794 + 0.985822i \(0.446336\pi\)
\(230\) 0 0
\(231\) −12.8739 −0.847037
\(232\) 62.0106 4.07120
\(233\) −0.738739 −0.0483964 −0.0241982 0.999707i \(-0.507703\pi\)
−0.0241982 + 0.999707i \(0.507703\pi\)
\(234\) −3.88171 −0.253755
\(235\) 0 0
\(236\) −37.6643 −2.45174
\(237\) 5.56405 0.361424
\(238\) −57.0659 −3.69904
\(239\) −3.36660 −0.217767 −0.108884 0.994055i \(-0.534728\pi\)
−0.108884 + 0.994055i \(0.534728\pi\)
\(240\) 0 0
\(241\) 6.51135 0.419433 0.209716 0.977762i \(-0.432746\pi\)
0.209716 + 0.977762i \(0.432746\pi\)
\(242\) 13.0769 0.840614
\(243\) 1.00000 0.0641500
\(244\) 59.0212 3.77845
\(245\) 0 0
\(246\) 24.6758 1.57327
\(247\) 6.29048 0.400253
\(248\) 43.2111 2.74391
\(249\) −15.4973 −0.982100
\(250\) 0 0
\(251\) 11.7004 0.738524 0.369262 0.929325i \(-0.379611\pi\)
0.369262 + 0.929325i \(0.379611\pi\)
\(252\) −25.6670 −1.61687
\(253\) 15.9523 1.00291
\(254\) 19.7081 1.23660
\(255\) 0 0
\(256\) −12.3620 −0.772626
\(257\) −28.1645 −1.75685 −0.878427 0.477876i \(-0.841407\pi\)
−0.878427 + 0.477876i \(0.841407\pi\)
\(258\) −11.4045 −0.710015
\(259\) −9.69157 −0.602205
\(260\) 0 0
\(261\) −8.17052 −0.505743
\(262\) −32.6280 −2.01577
\(263\) 15.2487 0.940276 0.470138 0.882593i \(-0.344204\pi\)
0.470138 + 0.882593i \(0.344204\pi\)
\(264\) −18.6192 −1.14593
\(265\) 0 0
\(266\) 58.6024 3.59315
\(267\) −3.59698 −0.220132
\(268\) 28.5836 1.74602
\(269\) −7.75275 −0.472693 −0.236347 0.971669i \(-0.575950\pi\)
−0.236347 + 0.971669i \(0.575950\pi\)
\(270\) 0 0
\(271\) −14.9314 −0.907020 −0.453510 0.891251i \(-0.649828\pi\)
−0.453510 + 0.891251i \(0.649828\pi\)
\(272\) −42.0097 −2.54721
\(273\) −7.75963 −0.469634
\(274\) −33.4398 −2.02018
\(275\) 0 0
\(276\) 31.8046 1.91441
\(277\) 19.0645 1.14548 0.572739 0.819738i \(-0.305881\pi\)
0.572739 + 0.819738i \(0.305881\pi\)
\(278\) 36.0255 2.16067
\(279\) −5.69350 −0.340861
\(280\) 0 0
\(281\) −11.2828 −0.673076 −0.336538 0.941670i \(-0.609256\pi\)
−0.336538 + 0.941670i \(0.609256\pi\)
\(282\) 2.62510 0.156322
\(283\) −16.9950 −1.01025 −0.505124 0.863047i \(-0.668553\pi\)
−0.505124 + 0.863047i \(0.668553\pi\)
\(284\) −13.7906 −0.818323
\(285\) 0 0
\(286\) −9.52287 −0.563099
\(287\) 49.3275 2.91171
\(288\) −11.4421 −0.674234
\(289\) 0.160687 0.00945218
\(290\) 0 0
\(291\) 3.34748 0.196233
\(292\) −24.9553 −1.46040
\(293\) −29.7556 −1.73834 −0.869170 0.494513i \(-0.835346\pi\)
−0.869170 + 0.494513i \(0.835346\pi\)
\(294\) −53.9135 −3.14430
\(295\) 0 0
\(296\) −14.0167 −0.814705
\(297\) 2.45327 0.142353
\(298\) −16.5635 −0.959500
\(299\) 9.61516 0.556059
\(300\) 0 0
\(301\) −22.7979 −1.31405
\(302\) −20.4895 −1.17904
\(303\) −1.29938 −0.0746477
\(304\) 43.1408 2.47430
\(305\) 0 0
\(306\) 10.8746 0.621659
\(307\) −24.4730 −1.39675 −0.698373 0.715734i \(-0.746092\pi\)
−0.698373 + 0.715734i \(0.746092\pi\)
\(308\) −62.9679 −3.58793
\(309\) 16.5846 0.943465
\(310\) 0 0
\(311\) −8.26842 −0.468859 −0.234430 0.972133i \(-0.575322\pi\)
−0.234430 + 0.972133i \(0.575322\pi\)
\(312\) −11.2226 −0.635355
\(313\) −26.7049 −1.50945 −0.754724 0.656043i \(-0.772229\pi\)
−0.754724 + 0.656043i \(0.772229\pi\)
\(314\) 55.4509 3.12928
\(315\) 0 0
\(316\) 27.2146 1.53094
\(317\) 12.1898 0.684649 0.342325 0.939582i \(-0.388786\pi\)
0.342325 + 0.939582i \(0.388786\pi\)
\(318\) −18.2695 −1.02450
\(319\) −20.0445 −1.12227
\(320\) 0 0
\(321\) −10.0354 −0.560124
\(322\) 89.5754 4.99184
\(323\) −17.6227 −0.980556
\(324\) 4.89115 0.271730
\(325\) 0 0
\(326\) 51.3997 2.84677
\(327\) −9.16258 −0.506692
\(328\) 71.3413 3.93916
\(329\) 5.24764 0.289312
\(330\) 0 0
\(331\) 8.91572 0.490052 0.245026 0.969516i \(-0.421203\pi\)
0.245026 + 0.969516i \(0.421203\pi\)
\(332\) −75.7995 −4.16004
\(333\) 1.84684 0.101206
\(334\) −7.47368 −0.408942
\(335\) 0 0
\(336\) −53.2165 −2.90320
\(337\) −0.427820 −0.0233049 −0.0116524 0.999932i \(-0.503709\pi\)
−0.0116524 + 0.999932i \(0.503709\pi\)
\(338\) 28.3864 1.54402
\(339\) −11.4553 −0.622165
\(340\) 0 0
\(341\) −13.9677 −0.756392
\(342\) −11.1674 −0.603864
\(343\) −71.0410 −3.83585
\(344\) −32.9722 −1.77774
\(345\) 0 0
\(346\) −20.9785 −1.12781
\(347\) −13.8379 −0.742859 −0.371430 0.928461i \(-0.621132\pi\)
−0.371430 + 0.928461i \(0.621132\pi\)
\(348\) −39.9632 −2.14225
\(349\) 22.3237 1.19496 0.597479 0.801884i \(-0.296169\pi\)
0.597479 + 0.801884i \(0.296169\pi\)
\(350\) 0 0
\(351\) 1.47869 0.0789267
\(352\) −28.0706 −1.49617
\(353\) −20.0786 −1.06868 −0.534339 0.845270i \(-0.679439\pi\)
−0.534339 + 0.845270i \(0.679439\pi\)
\(354\) 20.2146 1.07439
\(355\) 0 0
\(356\) −17.5934 −0.932448
\(357\) 21.7386 1.15053
\(358\) 30.6070 1.61763
\(359\) −18.8565 −0.995208 −0.497604 0.867404i \(-0.665787\pi\)
−0.497604 + 0.867404i \(0.665787\pi\)
\(360\) 0 0
\(361\) −0.902764 −0.0475139
\(362\) −51.2763 −2.69502
\(363\) −4.98148 −0.261460
\(364\) −37.9535 −1.98931
\(365\) 0 0
\(366\) −31.6770 −1.65578
\(367\) −21.5685 −1.12587 −0.562934 0.826502i \(-0.690328\pi\)
−0.562934 + 0.826502i \(0.690328\pi\)
\(368\) 65.9419 3.43746
\(369\) −9.39994 −0.489341
\(370\) 0 0
\(371\) −36.5212 −1.89609
\(372\) −27.8477 −1.44384
\(373\) 8.49759 0.439988 0.219994 0.975501i \(-0.429396\pi\)
0.219994 + 0.975501i \(0.429396\pi\)
\(374\) 26.6783 1.37950
\(375\) 0 0
\(376\) 7.58955 0.391401
\(377\) −12.0817 −0.622238
\(378\) 13.7756 0.708539
\(379\) 12.9777 0.666618 0.333309 0.942818i \(-0.391835\pi\)
0.333309 + 0.942818i \(0.391835\pi\)
\(380\) 0 0
\(381\) −7.50757 −0.384624
\(382\) 60.6729 3.10430
\(383\) 28.8351 1.47340 0.736701 0.676218i \(-0.236382\pi\)
0.736701 + 0.676218i \(0.236382\pi\)
\(384\) −2.72267 −0.138940
\(385\) 0 0
\(386\) 45.2472 2.30302
\(387\) 4.34441 0.220839
\(388\) 16.3730 0.831215
\(389\) 21.5803 1.09417 0.547084 0.837078i \(-0.315738\pi\)
0.547084 + 0.837078i \(0.315738\pi\)
\(390\) 0 0
\(391\) −26.9368 −1.36225
\(392\) −155.872 −7.87272
\(393\) 12.4292 0.626972
\(394\) 29.1533 1.46872
\(395\) 0 0
\(396\) 11.9993 0.602987
\(397\) 16.6366 0.834969 0.417484 0.908684i \(-0.362912\pi\)
0.417484 + 0.908684i \(0.362912\pi\)
\(398\) 60.5378 3.03449
\(399\) −22.3239 −1.11759
\(400\) 0 0
\(401\) −33.5127 −1.67354 −0.836772 0.547552i \(-0.815560\pi\)
−0.836772 + 0.547552i \(0.815560\pi\)
\(402\) −15.3410 −0.765137
\(403\) −8.41892 −0.419376
\(404\) −6.35548 −0.316197
\(405\) 0 0
\(406\) −112.554 −5.58594
\(407\) 4.53080 0.224583
\(408\) 31.4401 1.55652
\(409\) −11.8817 −0.587512 −0.293756 0.955880i \(-0.594905\pi\)
−0.293756 + 0.955880i \(0.594905\pi\)
\(410\) 0 0
\(411\) 12.7385 0.628344
\(412\) 81.1178 3.99639
\(413\) 40.4095 1.98842
\(414\) −17.0697 −0.838928
\(415\) 0 0
\(416\) −16.9194 −0.829540
\(417\) −13.7235 −0.672042
\(418\) −27.3966 −1.34001
\(419\) −2.52390 −0.123301 −0.0616503 0.998098i \(-0.519636\pi\)
−0.0616503 + 0.998098i \(0.519636\pi\)
\(420\) 0 0
\(421\) 14.7777 0.720220 0.360110 0.932910i \(-0.382739\pi\)
0.360110 + 0.932910i \(0.382739\pi\)
\(422\) −10.7473 −0.523171
\(423\) −1.00000 −0.0486217
\(424\) −52.8198 −2.56516
\(425\) 0 0
\(426\) 7.40148 0.358603
\(427\) −63.3230 −3.06441
\(428\) −49.0848 −2.37260
\(429\) 3.62762 0.175143
\(430\) 0 0
\(431\) −28.6446 −1.37976 −0.689880 0.723923i \(-0.742337\pi\)
−0.689880 + 0.723923i \(0.742337\pi\)
\(432\) 10.1410 0.487911
\(433\) 14.0010 0.672844 0.336422 0.941711i \(-0.390783\pi\)
0.336422 + 0.941711i \(0.390783\pi\)
\(434\) −78.4312 −3.76482
\(435\) 0 0
\(436\) −44.8155 −2.14627
\(437\) 27.6621 1.32326
\(438\) 13.3936 0.639971
\(439\) −23.0784 −1.10147 −0.550735 0.834680i \(-0.685653\pi\)
−0.550735 + 0.834680i \(0.685653\pi\)
\(440\) 0 0
\(441\) 20.5377 0.977986
\(442\) 16.0802 0.764856
\(443\) −17.2342 −0.818822 −0.409411 0.912350i \(-0.634266\pi\)
−0.409411 + 0.912350i \(0.634266\pi\)
\(444\) 9.03319 0.428696
\(445\) 0 0
\(446\) −55.3306 −2.61998
\(447\) 6.30968 0.298438
\(448\) −51.1888 −2.41845
\(449\) −14.4444 −0.681673 −0.340837 0.940123i \(-0.610710\pi\)
−0.340837 + 0.940123i \(0.610710\pi\)
\(450\) 0 0
\(451\) −23.0606 −1.08588
\(452\) −56.0294 −2.63540
\(453\) 7.80524 0.366722
\(454\) −4.48476 −0.210480
\(455\) 0 0
\(456\) −32.2866 −1.51196
\(457\) 29.3230 1.37167 0.685835 0.727757i \(-0.259437\pi\)
0.685835 + 0.727757i \(0.259437\pi\)
\(458\) −13.3312 −0.622928
\(459\) −4.14255 −0.193357
\(460\) 0 0
\(461\) −10.0074 −0.466091 −0.233045 0.972466i \(-0.574869\pi\)
−0.233045 + 0.972466i \(0.574869\pi\)
\(462\) 33.7952 1.57229
\(463\) −6.74306 −0.313377 −0.156688 0.987648i \(-0.550082\pi\)
−0.156688 + 0.987648i \(0.550082\pi\)
\(464\) −82.8576 −3.84657
\(465\) 0 0
\(466\) 1.93926 0.0898347
\(467\) −22.6006 −1.04583 −0.522915 0.852385i \(-0.675155\pi\)
−0.522915 + 0.852385i \(0.675155\pi\)
\(468\) 7.23250 0.334322
\(469\) −30.6669 −1.41607
\(470\) 0 0
\(471\) −21.1234 −0.973313
\(472\) 58.4434 2.69008
\(473\) 10.6580 0.490056
\(474\) −14.6062 −0.670885
\(475\) 0 0
\(476\) 106.327 4.87347
\(477\) 6.95955 0.318656
\(478\) 8.83765 0.404225
\(479\) 28.0031 1.27949 0.639747 0.768585i \(-0.279039\pi\)
0.639747 + 0.768585i \(0.279039\pi\)
\(480\) 0 0
\(481\) 2.73091 0.124519
\(482\) −17.0929 −0.778562
\(483\) −34.1227 −1.55264
\(484\) −24.3652 −1.10751
\(485\) 0 0
\(486\) −2.62510 −0.119077
\(487\) −29.5359 −1.33840 −0.669200 0.743082i \(-0.733363\pi\)
−0.669200 + 0.743082i \(0.733363\pi\)
\(488\) −91.5828 −4.14576
\(489\) −19.5801 −0.885443
\(490\) 0 0
\(491\) −10.1287 −0.457102 −0.228551 0.973532i \(-0.573399\pi\)
−0.228551 + 0.973532i \(0.573399\pi\)
\(492\) −45.9765 −2.07278
\(493\) 33.8468 1.52438
\(494\) −16.5131 −0.742961
\(495\) 0 0
\(496\) −57.7380 −2.59251
\(497\) 14.7957 0.663680
\(498\) 40.6819 1.82300
\(499\) −6.22375 −0.278613 −0.139307 0.990249i \(-0.544487\pi\)
−0.139307 + 0.990249i \(0.544487\pi\)
\(500\) 0 0
\(501\) 2.84701 0.127195
\(502\) −30.7148 −1.37087
\(503\) −31.8043 −1.41809 −0.709043 0.705166i \(-0.750873\pi\)
−0.709043 + 0.705166i \(0.750873\pi\)
\(504\) 39.8272 1.77405
\(505\) 0 0
\(506\) −41.8765 −1.86164
\(507\) −10.8135 −0.480243
\(508\) −36.7206 −1.62921
\(509\) 13.5256 0.599513 0.299756 0.954016i \(-0.403095\pi\)
0.299756 + 0.954016i \(0.403095\pi\)
\(510\) 0 0
\(511\) 26.7741 1.18442
\(512\) 37.8969 1.67482
\(513\) 4.25408 0.187822
\(514\) 73.9347 3.26112
\(515\) 0 0
\(516\) 21.2492 0.935443
\(517\) −2.45327 −0.107895
\(518\) 25.4413 1.11783
\(519\) 7.99152 0.350789
\(520\) 0 0
\(521\) −6.89142 −0.301918 −0.150959 0.988540i \(-0.548236\pi\)
−0.150959 + 0.988540i \(0.548236\pi\)
\(522\) 21.4484 0.938773
\(523\) 10.4160 0.455461 0.227730 0.973724i \(-0.426869\pi\)
0.227730 + 0.973724i \(0.426869\pi\)
\(524\) 60.7933 2.65577
\(525\) 0 0
\(526\) −40.0294 −1.74537
\(527\) 23.5856 1.02740
\(528\) 24.8787 1.08270
\(529\) 19.2823 0.838361
\(530\) 0 0
\(531\) −7.70051 −0.334174
\(532\) −109.189 −4.73396
\(533\) −13.8996 −0.602059
\(534\) 9.44244 0.408615
\(535\) 0 0
\(536\) −44.3530 −1.91576
\(537\) −11.6594 −0.503138
\(538\) 20.3517 0.877426
\(539\) 50.3845 2.17021
\(540\) 0 0
\(541\) −4.77543 −0.205312 −0.102656 0.994717i \(-0.532734\pi\)
−0.102656 + 0.994717i \(0.532734\pi\)
\(542\) 39.1965 1.68363
\(543\) 19.5331 0.838245
\(544\) 47.3995 2.03224
\(545\) 0 0
\(546\) 20.3698 0.871748
\(547\) 35.1733 1.50390 0.751950 0.659220i \(-0.229113\pi\)
0.751950 + 0.659220i \(0.229113\pi\)
\(548\) 62.3059 2.66158
\(549\) 12.0670 0.515005
\(550\) 0 0
\(551\) −34.7581 −1.48074
\(552\) −49.3509 −2.10052
\(553\) −29.1981 −1.24163
\(554\) −50.0463 −2.12626
\(555\) 0 0
\(556\) −67.1236 −2.84667
\(557\) 41.5731 1.76151 0.880755 0.473573i \(-0.157036\pi\)
0.880755 + 0.473573i \(0.157036\pi\)
\(558\) 14.9460 0.632715
\(559\) 6.42405 0.271708
\(560\) 0 0
\(561\) −10.1628 −0.429072
\(562\) 29.6185 1.24938
\(563\) 20.6694 0.871112 0.435556 0.900162i \(-0.356552\pi\)
0.435556 + 0.900162i \(0.356552\pi\)
\(564\) −4.89115 −0.205954
\(565\) 0 0
\(566\) 44.6135 1.87525
\(567\) −5.24764 −0.220380
\(568\) 21.3988 0.897873
\(569\) 33.7275 1.41393 0.706966 0.707248i \(-0.250063\pi\)
0.706966 + 0.707248i \(0.250063\pi\)
\(570\) 0 0
\(571\) 1.88647 0.0789464 0.0394732 0.999221i \(-0.487432\pi\)
0.0394732 + 0.999221i \(0.487432\pi\)
\(572\) 17.7432 0.741882
\(573\) −23.1126 −0.965544
\(574\) −129.490 −5.40479
\(575\) 0 0
\(576\) 9.75464 0.406443
\(577\) −15.4800 −0.644440 −0.322220 0.946665i \(-0.604429\pi\)
−0.322220 + 0.946665i \(0.604429\pi\)
\(578\) −0.421820 −0.0175454
\(579\) −17.2364 −0.716320
\(580\) 0 0
\(581\) 81.3241 3.37389
\(582\) −8.78748 −0.364253
\(583\) 17.0736 0.707117
\(584\) 38.7229 1.60236
\(585\) 0 0
\(586\) 78.1114 3.22675
\(587\) −24.5995 −1.01533 −0.507665 0.861554i \(-0.669491\pi\)
−0.507665 + 0.861554i \(0.669491\pi\)
\(588\) 100.453 4.14261
\(589\) −24.2206 −0.997993
\(590\) 0 0
\(591\) −11.1056 −0.456823
\(592\) 18.7289 0.769754
\(593\) 5.36195 0.220189 0.110094 0.993921i \(-0.464885\pi\)
0.110094 + 0.993921i \(0.464885\pi\)
\(594\) −6.44007 −0.264239
\(595\) 0 0
\(596\) 30.8616 1.26414
\(597\) −23.0611 −0.943830
\(598\) −25.2408 −1.03217
\(599\) 32.0888 1.31111 0.655556 0.755147i \(-0.272434\pi\)
0.655556 + 0.755147i \(0.272434\pi\)
\(600\) 0 0
\(601\) 10.4599 0.426668 0.213334 0.976979i \(-0.431568\pi\)
0.213334 + 0.976979i \(0.431568\pi\)
\(602\) 59.8468 2.43917
\(603\) 5.84395 0.237984
\(604\) 38.1766 1.55338
\(605\) 0 0
\(606\) 3.41101 0.138563
\(607\) −37.0898 −1.50543 −0.752714 0.658347i \(-0.771256\pi\)
−0.752714 + 0.658347i \(0.771256\pi\)
\(608\) −48.6758 −1.97406
\(609\) 42.8759 1.73742
\(610\) 0 0
\(611\) −1.47869 −0.0598214
\(612\) −20.2618 −0.819035
\(613\) 40.3328 1.62903 0.814513 0.580145i \(-0.197004\pi\)
0.814513 + 0.580145i \(0.197004\pi\)
\(614\) 64.2440 2.59268
\(615\) 0 0
\(616\) 97.7068 3.93672
\(617\) 5.61640 0.226108 0.113054 0.993589i \(-0.463937\pi\)
0.113054 + 0.993589i \(0.463937\pi\)
\(618\) −43.5362 −1.75128
\(619\) −26.8598 −1.07959 −0.539794 0.841797i \(-0.681498\pi\)
−0.539794 + 0.841797i \(0.681498\pi\)
\(620\) 0 0
\(621\) 6.50248 0.260936
\(622\) 21.7054 0.870308
\(623\) 18.8757 0.756238
\(624\) 14.9955 0.600299
\(625\) 0 0
\(626\) 70.1029 2.80188
\(627\) 10.4364 0.416790
\(628\) −103.318 −4.12282
\(629\) −7.65064 −0.305051
\(630\) 0 0
\(631\) −46.1673 −1.83789 −0.918946 0.394384i \(-0.870958\pi\)
−0.918946 + 0.394384i \(0.870958\pi\)
\(632\) −42.2287 −1.67977
\(633\) 4.09406 0.162724
\(634\) −31.9995 −1.27086
\(635\) 0 0
\(636\) 34.0402 1.34978
\(637\) 30.3689 1.20326
\(638\) 52.6187 2.08320
\(639\) −2.81951 −0.111538
\(640\) 0 0
\(641\) −18.1813 −0.718119 −0.359059 0.933315i \(-0.616902\pi\)
−0.359059 + 0.933315i \(0.616902\pi\)
\(642\) 26.3440 1.03972
\(643\) −23.3628 −0.921338 −0.460669 0.887572i \(-0.652391\pi\)
−0.460669 + 0.887572i \(0.652391\pi\)
\(644\) −166.899 −6.57675
\(645\) 0 0
\(646\) 46.2615 1.82013
\(647\) 36.0372 1.41677 0.708383 0.705828i \(-0.249425\pi\)
0.708383 + 0.705828i \(0.249425\pi\)
\(648\) −7.58955 −0.298146
\(649\) −18.8914 −0.741552
\(650\) 0 0
\(651\) 29.8774 1.17099
\(652\) −95.7692 −3.75061
\(653\) 39.5118 1.54622 0.773109 0.634274i \(-0.218701\pi\)
0.773109 + 0.634274i \(0.218701\pi\)
\(654\) 24.0527 0.940534
\(655\) 0 0
\(656\) −95.3251 −3.72182
\(657\) −5.10213 −0.199053
\(658\) −13.7756 −0.537028
\(659\) 2.13562 0.0831919 0.0415960 0.999135i \(-0.486756\pi\)
0.0415960 + 0.999135i \(0.486756\pi\)
\(660\) 0 0
\(661\) 0.344591 0.0134030 0.00670151 0.999978i \(-0.497867\pi\)
0.00670151 + 0.999978i \(0.497867\pi\)
\(662\) −23.4047 −0.909648
\(663\) −6.12555 −0.237896
\(664\) 117.617 4.56444
\(665\) 0 0
\(666\) −4.84815 −0.187862
\(667\) −53.1287 −2.05715
\(668\) 13.9251 0.538780
\(669\) 21.0775 0.814904
\(670\) 0 0
\(671\) 29.6034 1.14283
\(672\) 60.0441 2.31625
\(673\) −27.1613 −1.04699 −0.523495 0.852029i \(-0.675372\pi\)
−0.523495 + 0.852029i \(0.675372\pi\)
\(674\) 1.12307 0.0432591
\(675\) 0 0
\(676\) −52.8903 −2.03424
\(677\) −13.1729 −0.506275 −0.253137 0.967430i \(-0.581462\pi\)
−0.253137 + 0.967430i \(0.581462\pi\)
\(678\) 30.0712 1.15488
\(679\) −17.5664 −0.674136
\(680\) 0 0
\(681\) 1.70842 0.0654667
\(682\) 36.6665 1.40403
\(683\) −15.2104 −0.582011 −0.291006 0.956721i \(-0.593990\pi\)
−0.291006 + 0.956721i \(0.593990\pi\)
\(684\) 20.8074 0.795589
\(685\) 0 0
\(686\) 186.490 7.12021
\(687\) 5.07837 0.193752
\(688\) 44.0569 1.67965
\(689\) 10.2910 0.392057
\(690\) 0 0
\(691\) 13.3184 0.506656 0.253328 0.967380i \(-0.418475\pi\)
0.253328 + 0.967380i \(0.418475\pi\)
\(692\) 39.0877 1.48589
\(693\) −12.8739 −0.489037
\(694\) 36.3260 1.37891
\(695\) 0 0
\(696\) 62.0106 2.35051
\(697\) 38.9397 1.47495
\(698\) −58.6018 −2.21811
\(699\) −0.738739 −0.0279417
\(700\) 0 0
\(701\) −24.9303 −0.941604 −0.470802 0.882239i \(-0.656035\pi\)
−0.470802 + 0.882239i \(0.656035\pi\)
\(702\) −3.88171 −0.146506
\(703\) 7.85663 0.296318
\(704\) 23.9307 0.901924
\(705\) 0 0
\(706\) 52.7084 1.98371
\(707\) 6.81870 0.256444
\(708\) −37.6643 −1.41551
\(709\) 17.1322 0.643414 0.321707 0.946839i \(-0.395743\pi\)
0.321707 + 0.946839i \(0.395743\pi\)
\(710\) 0 0
\(711\) 5.56405 0.208668
\(712\) 27.2995 1.02309
\(713\) −37.0219 −1.38648
\(714\) −57.0659 −2.13564
\(715\) 0 0
\(716\) −57.0277 −2.13122
\(717\) −3.36660 −0.125728
\(718\) 49.5002 1.84733
\(719\) −36.7634 −1.37104 −0.685521 0.728053i \(-0.740426\pi\)
−0.685521 + 0.728053i \(0.740426\pi\)
\(720\) 0 0
\(721\) −87.0300 −3.24117
\(722\) 2.36985 0.0881965
\(723\) 6.51135 0.242160
\(724\) 95.5392 3.55069
\(725\) 0 0
\(726\) 13.0769 0.485329
\(727\) −46.9974 −1.74304 −0.871519 0.490362i \(-0.836865\pi\)
−0.871519 + 0.490362i \(0.836865\pi\)
\(728\) 58.8922 2.18269
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −17.9969 −0.665641
\(732\) 59.0212 2.18149
\(733\) −0.790148 −0.0291848 −0.0145924 0.999894i \(-0.504645\pi\)
−0.0145924 + 0.999894i \(0.504645\pi\)
\(734\) 56.6196 2.08987
\(735\) 0 0
\(736\) −74.4022 −2.74250
\(737\) 14.3368 0.528102
\(738\) 24.6758 0.908328
\(739\) 34.4106 1.26581 0.632907 0.774228i \(-0.281861\pi\)
0.632907 + 0.774228i \(0.281861\pi\)
\(740\) 0 0
\(741\) 6.29048 0.231086
\(742\) 95.8717 3.51956
\(743\) 26.4476 0.970267 0.485133 0.874440i \(-0.338771\pi\)
0.485133 + 0.874440i \(0.338771\pi\)
\(744\) 43.2111 1.58420
\(745\) 0 0
\(746\) −22.3070 −0.816718
\(747\) −15.4973 −0.567016
\(748\) −49.7076 −1.81749
\(749\) 52.6624 1.92424
\(750\) 0 0
\(751\) 47.6505 1.73879 0.869396 0.494117i \(-0.164508\pi\)
0.869396 + 0.494117i \(0.164508\pi\)
\(752\) −10.1410 −0.369806
\(753\) 11.7004 0.426387
\(754\) 31.7156 1.15501
\(755\) 0 0
\(756\) −25.6670 −0.933499
\(757\) 32.3523 1.17587 0.587933 0.808910i \(-0.299942\pi\)
0.587933 + 0.808910i \(0.299942\pi\)
\(758\) −34.0677 −1.23739
\(759\) 15.9523 0.579033
\(760\) 0 0
\(761\) −2.25280 −0.0816639 −0.0408319 0.999166i \(-0.513001\pi\)
−0.0408319 + 0.999166i \(0.513001\pi\)
\(762\) 19.7081 0.713950
\(763\) 48.0819 1.74068
\(764\) −113.047 −4.08991
\(765\) 0 0
\(766\) −75.6949 −2.73497
\(767\) −11.3867 −0.411149
\(768\) −12.3620 −0.446076
\(769\) 2.40249 0.0866361 0.0433181 0.999061i \(-0.486207\pi\)
0.0433181 + 0.999061i \(0.486207\pi\)
\(770\) 0 0
\(771\) −28.1645 −1.01432
\(772\) −84.3057 −3.03423
\(773\) −16.0770 −0.578248 −0.289124 0.957292i \(-0.593364\pi\)
−0.289124 + 0.957292i \(0.593364\pi\)
\(774\) −11.4045 −0.409927
\(775\) 0 0
\(776\) −25.4059 −0.912019
\(777\) −9.69157 −0.347683
\(778\) −56.6506 −2.03102
\(779\) −39.9881 −1.43272
\(780\) 0 0
\(781\) −6.91700 −0.247510
\(782\) 70.7119 2.52865
\(783\) −8.17052 −0.291991
\(784\) 208.274 7.43834
\(785\) 0 0
\(786\) −32.6280 −1.16380
\(787\) −18.5923 −0.662745 −0.331372 0.943500i \(-0.607512\pi\)
−0.331372 + 0.943500i \(0.607512\pi\)
\(788\) −54.3190 −1.93504
\(789\) 15.2487 0.542869
\(790\) 0 0
\(791\) 60.1131 2.13738
\(792\) −18.6192 −0.661604
\(793\) 17.8433 0.633634
\(794\) −43.6728 −1.54989
\(795\) 0 0
\(796\) −112.795 −3.99793
\(797\) 19.2123 0.680535 0.340267 0.940329i \(-0.389482\pi\)
0.340267 + 0.940329i \(0.389482\pi\)
\(798\) 58.6024 2.07450
\(799\) 4.14255 0.146553
\(800\) 0 0
\(801\) −3.59698 −0.127093
\(802\) 87.9741 3.10647
\(803\) −12.5169 −0.441711
\(804\) 28.5836 1.00807
\(805\) 0 0
\(806\) 22.1005 0.778457
\(807\) −7.75275 −0.272910
\(808\) 9.86175 0.346935
\(809\) 15.4036 0.541561 0.270781 0.962641i \(-0.412718\pi\)
0.270781 + 0.962641i \(0.412718\pi\)
\(810\) 0 0
\(811\) −16.0580 −0.563872 −0.281936 0.959433i \(-0.590977\pi\)
−0.281936 + 0.959433i \(0.590977\pi\)
\(812\) 209.713 7.35947
\(813\) −14.9314 −0.523668
\(814\) −11.8938 −0.416878
\(815\) 0 0
\(816\) −42.0097 −1.47063
\(817\) 18.4815 0.646586
\(818\) 31.1906 1.09055
\(819\) −7.75963 −0.271144
\(820\) 0 0
\(821\) −24.4199 −0.852259 −0.426130 0.904662i \(-0.640123\pi\)
−0.426130 + 0.904662i \(0.640123\pi\)
\(822\) −33.4398 −1.16635
\(823\) −36.9868 −1.28928 −0.644640 0.764486i \(-0.722993\pi\)
−0.644640 + 0.764486i \(0.722993\pi\)
\(824\) −125.870 −4.38488
\(825\) 0 0
\(826\) −106.079 −3.69096
\(827\) −11.7608 −0.408963 −0.204481 0.978870i \(-0.565551\pi\)
−0.204481 + 0.978870i \(0.565551\pi\)
\(828\) 31.8046 1.10529
\(829\) 1.09175 0.0379182 0.0189591 0.999820i \(-0.493965\pi\)
0.0189591 + 0.999820i \(0.493965\pi\)
\(830\) 0 0
\(831\) 19.0645 0.661342
\(832\) 14.4241 0.500066
\(833\) −85.0784 −2.94779
\(834\) 36.0255 1.24746
\(835\) 0 0
\(836\) 51.0460 1.76546
\(837\) −5.69350 −0.196796
\(838\) 6.62549 0.228874
\(839\) −9.05720 −0.312689 −0.156345 0.987703i \(-0.549971\pi\)
−0.156345 + 0.987703i \(0.549971\pi\)
\(840\) 0 0
\(841\) 37.7574 1.30198
\(842\) −38.7929 −1.33689
\(843\) −11.2828 −0.388601
\(844\) 20.0247 0.689277
\(845\) 0 0
\(846\) 2.62510 0.0902528
\(847\) 26.1410 0.898216
\(848\) 70.5770 2.42362
\(849\) −16.9950 −0.583266
\(850\) 0 0
\(851\) 12.0091 0.411666
\(852\) −13.7906 −0.472459
\(853\) −23.2497 −0.796053 −0.398027 0.917374i \(-0.630305\pi\)
−0.398027 + 0.917374i \(0.630305\pi\)
\(854\) 166.229 5.68824
\(855\) 0 0
\(856\) 76.1645 2.60325
\(857\) 22.9394 0.783596 0.391798 0.920051i \(-0.371853\pi\)
0.391798 + 0.920051i \(0.371853\pi\)
\(858\) −9.52287 −0.325105
\(859\) −1.31714 −0.0449402 −0.0224701 0.999748i \(-0.507153\pi\)
−0.0224701 + 0.999748i \(0.507153\pi\)
\(860\) 0 0
\(861\) 49.3275 1.68108
\(862\) 75.1949 2.56115
\(863\) −42.4780 −1.44597 −0.722983 0.690865i \(-0.757230\pi\)
−0.722983 + 0.690865i \(0.757230\pi\)
\(864\) −11.4421 −0.389269
\(865\) 0 0
\(866\) −36.7539 −1.24895
\(867\) 0.160687 0.00545722
\(868\) 146.135 4.96014
\(869\) 13.6501 0.463048
\(870\) 0 0
\(871\) 8.64140 0.292803
\(872\) 69.5399 2.35492
\(873\) 3.34748 0.113295
\(874\) −72.6158 −2.45627
\(875\) 0 0
\(876\) −24.9553 −0.843160
\(877\) −50.9867 −1.72170 −0.860850 0.508859i \(-0.830068\pi\)
−0.860850 + 0.508859i \(0.830068\pi\)
\(878\) 60.5830 2.04458
\(879\) −29.7556 −1.00363
\(880\) 0 0
\(881\) 0.851944 0.0287027 0.0143514 0.999897i \(-0.495432\pi\)
0.0143514 + 0.999897i \(0.495432\pi\)
\(882\) −53.9135 −1.81536
\(883\) 21.6486 0.728532 0.364266 0.931295i \(-0.381320\pi\)
0.364266 + 0.931295i \(0.381320\pi\)
\(884\) −29.9609 −1.00770
\(885\) 0 0
\(886\) 45.2415 1.51992
\(887\) 20.7832 0.697830 0.348915 0.937154i \(-0.386550\pi\)
0.348915 + 0.937154i \(0.386550\pi\)
\(888\) −14.0167 −0.470370
\(889\) 39.3970 1.32133
\(890\) 0 0
\(891\) 2.45327 0.0821875
\(892\) 103.093 3.45182
\(893\) −4.25408 −0.142358
\(894\) −16.5635 −0.553968
\(895\) 0 0
\(896\) 14.2876 0.477314
\(897\) 9.61516 0.321041
\(898\) 37.9180 1.26534
\(899\) 46.5189 1.55149
\(900\) 0 0
\(901\) −28.8302 −0.960475
\(902\) 60.5362 2.01564
\(903\) −22.7979 −0.758667
\(904\) 86.9404 2.89159
\(905\) 0 0
\(906\) −20.4895 −0.680719
\(907\) 31.4193 1.04326 0.521631 0.853171i \(-0.325324\pi\)
0.521631 + 0.853171i \(0.325324\pi\)
\(908\) 8.35612 0.277308
\(909\) −1.29938 −0.0430979
\(910\) 0 0
\(911\) 53.0020 1.75603 0.878017 0.478630i \(-0.158867\pi\)
0.878017 + 0.478630i \(0.158867\pi\)
\(912\) 43.1408 1.42854
\(913\) −38.0189 −1.25824
\(914\) −76.9757 −2.54613
\(915\) 0 0
\(916\) 24.8391 0.820706
\(917\) −65.2242 −2.15389
\(918\) 10.8746 0.358915
\(919\) −29.1849 −0.962720 −0.481360 0.876523i \(-0.659857\pi\)
−0.481360 + 0.876523i \(0.659857\pi\)
\(920\) 0 0
\(921\) −24.4730 −0.806412
\(922\) 26.2704 0.865170
\(923\) −4.16918 −0.137230
\(924\) −62.9679 −2.07149
\(925\) 0 0
\(926\) 17.7012 0.581698
\(927\) 16.5846 0.544710
\(928\) 93.4881 3.06890
\(929\) 3.60942 0.118421 0.0592106 0.998246i \(-0.481142\pi\)
0.0592106 + 0.998246i \(0.481142\pi\)
\(930\) 0 0
\(931\) 87.3691 2.86341
\(932\) −3.61328 −0.118357
\(933\) −8.26842 −0.270696
\(934\) 59.3287 1.94129
\(935\) 0 0
\(936\) −11.2226 −0.366822
\(937\) −2.39765 −0.0783279 −0.0391640 0.999233i \(-0.512469\pi\)
−0.0391640 + 0.999233i \(0.512469\pi\)
\(938\) 80.5038 2.62854
\(939\) −26.7049 −0.871480
\(940\) 0 0
\(941\) −2.61594 −0.0852774 −0.0426387 0.999091i \(-0.513576\pi\)
−0.0426387 + 0.999091i \(0.513576\pi\)
\(942\) 55.4509 1.80669
\(943\) −61.1229 −1.99044
\(944\) −78.0911 −2.54165
\(945\) 0 0
\(946\) −27.9783 −0.909654
\(947\) −49.1337 −1.59663 −0.798316 0.602239i \(-0.794276\pi\)
−0.798316 + 0.602239i \(0.794276\pi\)
\(948\) 27.2146 0.883889
\(949\) −7.54447 −0.244904
\(950\) 0 0
\(951\) 12.1898 0.395283
\(952\) −164.986 −5.34723
\(953\) 52.3473 1.69569 0.847847 0.530240i \(-0.177898\pi\)
0.847847 + 0.530240i \(0.177898\pi\)
\(954\) −18.2695 −0.591497
\(955\) 0 0
\(956\) −16.4665 −0.532565
\(957\) −20.0445 −0.647946
\(958\) −73.5109 −2.37503
\(959\) −66.8471 −2.15860
\(960\) 0 0
\(961\) 1.41592 0.0456750
\(962\) −7.16892 −0.231135
\(963\) −10.0354 −0.323388
\(964\) 31.8480 1.02575
\(965\) 0 0
\(966\) 89.5754 2.88204
\(967\) −34.5982 −1.11260 −0.556301 0.830981i \(-0.687780\pi\)
−0.556301 + 0.830981i \(0.687780\pi\)
\(968\) 37.8072 1.21517
\(969\) −17.6227 −0.566124
\(970\) 0 0
\(971\) 18.0689 0.579857 0.289929 0.957048i \(-0.406368\pi\)
0.289929 + 0.957048i \(0.406368\pi\)
\(972\) 4.89115 0.156884
\(973\) 72.0158 2.30872
\(974\) 77.5348 2.48437
\(975\) 0 0
\(976\) 122.371 3.91701
\(977\) 4.72708 0.151233 0.0756163 0.997137i \(-0.475908\pi\)
0.0756163 + 0.997137i \(0.475908\pi\)
\(978\) 51.3997 1.64358
\(979\) −8.82436 −0.282028
\(980\) 0 0
\(981\) −9.16258 −0.292539
\(982\) 26.5889 0.848485
\(983\) 53.3395 1.70126 0.850632 0.525761i \(-0.176219\pi\)
0.850632 + 0.525761i \(0.176219\pi\)
\(984\) 71.3413 2.27428
\(985\) 0 0
\(986\) −88.8511 −2.82960
\(987\) 5.24764 0.167034
\(988\) 30.7677 0.978849
\(989\) 28.2495 0.898282
\(990\) 0 0
\(991\) −17.8741 −0.567791 −0.283895 0.958855i \(-0.591627\pi\)
−0.283895 + 0.958855i \(0.591627\pi\)
\(992\) 65.1457 2.06838
\(993\) 8.91572 0.282932
\(994\) −38.8403 −1.23194
\(995\) 0 0
\(996\) −75.7995 −2.40180
\(997\) −26.5743 −0.841616 −0.420808 0.907150i \(-0.638253\pi\)
−0.420808 + 0.907150i \(0.638253\pi\)
\(998\) 16.3380 0.517169
\(999\) 1.84684 0.0584316
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 3525.2.a.bd.1.1 8
5.4 even 2 3525.2.a.be.1.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.1 8 1.1 even 1 trivial
3525.2.a.be.1.8 yes 8 5.4 even 2