Properties

Label 3525.2.a.bd.1.2
Level $3525$
Weight $2$
Character 3525.1
Self dual yes
Analytic conductor $28.147$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.2
Root \(2.19791\) of defining polynomial
Character \(\chi\) \(=\) 3525.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.19791 q^{2} +1.00000 q^{3} +2.83083 q^{4} -2.19791 q^{6} +1.05086 q^{7} -1.82609 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.19791 q^{2} +1.00000 q^{3} +2.83083 q^{4} -2.19791 q^{6} +1.05086 q^{7} -1.82609 q^{8} +1.00000 q^{9} -3.22580 q^{11} +2.83083 q^{12} +2.58899 q^{13} -2.30971 q^{14} -1.64807 q^{16} -7.50374 q^{17} -2.19791 q^{18} +1.70135 q^{19} +1.05086 q^{21} +7.09002 q^{22} -2.21428 q^{23} -1.82609 q^{24} -5.69038 q^{26} +1.00000 q^{27} +2.97481 q^{28} +6.74234 q^{29} +10.8019 q^{31} +7.27449 q^{32} -3.22580 q^{33} +16.4926 q^{34} +2.83083 q^{36} -11.7806 q^{37} -3.73942 q^{38} +2.58899 q^{39} -5.81601 q^{41} -2.30971 q^{42} -3.39497 q^{43} -9.13167 q^{44} +4.86680 q^{46} -1.00000 q^{47} -1.64807 q^{48} -5.89569 q^{49} -7.50374 q^{51} +7.32898 q^{52} -11.1690 q^{53} -2.19791 q^{54} -1.91897 q^{56} +1.70135 q^{57} -14.8191 q^{58} +12.3261 q^{59} -6.55166 q^{61} -23.7417 q^{62} +1.05086 q^{63} -12.6926 q^{64} +7.09002 q^{66} +0.750116 q^{67} -21.2418 q^{68} -2.21428 q^{69} +2.09231 q^{71} -1.82609 q^{72} +4.80219 q^{73} +25.8928 q^{74} +4.81623 q^{76} -3.38987 q^{77} -5.69038 q^{78} +6.59991 q^{79} +1.00000 q^{81} +12.7831 q^{82} -15.6214 q^{83} +2.97481 q^{84} +7.46185 q^{86} +6.74234 q^{87} +5.89058 q^{88} -15.4144 q^{89} +2.72067 q^{91} -6.26825 q^{92} +10.8019 q^{93} +2.19791 q^{94} +7.27449 q^{96} +11.6829 q^{97} +12.9582 q^{98} -3.22580 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} + O(q^{10}) \) \( 8q - 3q^{2} + 8q^{3} + 7q^{4} - 3q^{6} - 8q^{7} - 6q^{8} + 8q^{9} - 8q^{11} + 7q^{12} - 10q^{13} + q^{14} + 5q^{16} - 6q^{17} - 3q^{18} - 2q^{19} - 8q^{21} - 10q^{23} - 6q^{24} - 14q^{26} + 8q^{27} - 44q^{28} - 13q^{29} - 10q^{32} - 8q^{33} + 28q^{34} + 7q^{36} - 3q^{37} - 36q^{38} - 10q^{39} - 16q^{41} + q^{42} - 25q^{43} - 17q^{44} - 5q^{46} - 8q^{47} + 5q^{48} + 16q^{49} - 6q^{51} + 17q^{52} - 4q^{53} - 3q^{54} + 37q^{56} - 2q^{57} - 15q^{58} - 8q^{59} + 15q^{61} - 6q^{62} - 8q^{63} - 14q^{64} - 27q^{67} - 14q^{68} - 10q^{69} + 14q^{71} - 6q^{72} - 28q^{73} - 21q^{74} + 6q^{76} - 4q^{77} - 14q^{78} + 7q^{79} + 8q^{81} + 53q^{82} - 60q^{83} - 44q^{84} - 3q^{86} - 13q^{87} - 54q^{88} - 34q^{89} + 23q^{91} + 43q^{92} + 3q^{94} - 10q^{96} - 7q^{97} - 40q^{98} - 8q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.19791 −1.55416 −0.777080 0.629402i \(-0.783300\pi\)
−0.777080 + 0.629402i \(0.783300\pi\)
\(3\) 1.00000 0.577350
\(4\) 2.83083 1.41541
\(5\) 0 0
\(6\) −2.19791 −0.897295
\(7\) 1.05086 0.397189 0.198595 0.980082i \(-0.436362\pi\)
0.198595 + 0.980082i \(0.436362\pi\)
\(8\) −1.82609 −0.645619
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.22580 −0.972614 −0.486307 0.873788i \(-0.661656\pi\)
−0.486307 + 0.873788i \(0.661656\pi\)
\(12\) 2.83083 0.817189
\(13\) 2.58899 0.718056 0.359028 0.933327i \(-0.383108\pi\)
0.359028 + 0.933327i \(0.383108\pi\)
\(14\) −2.30971 −0.617295
\(15\) 0 0
\(16\) −1.64807 −0.412018
\(17\) −7.50374 −1.81992 −0.909962 0.414691i \(-0.863890\pi\)
−0.909962 + 0.414691i \(0.863890\pi\)
\(18\) −2.19791 −0.518053
\(19\) 1.70135 0.390316 0.195158 0.980772i \(-0.437478\pi\)
0.195158 + 0.980772i \(0.437478\pi\)
\(20\) 0 0
\(21\) 1.05086 0.229317
\(22\) 7.09002 1.51160
\(23\) −2.21428 −0.461710 −0.230855 0.972988i \(-0.574152\pi\)
−0.230855 + 0.972988i \(0.574152\pi\)
\(24\) −1.82609 −0.372748
\(25\) 0 0
\(26\) −5.69038 −1.11597
\(27\) 1.00000 0.192450
\(28\) 2.97481 0.562187
\(29\) 6.74234 1.25202 0.626010 0.779815i \(-0.284687\pi\)
0.626010 + 0.779815i \(0.284687\pi\)
\(30\) 0 0
\(31\) 10.8019 1.94008 0.970041 0.242942i \(-0.0781123\pi\)
0.970041 + 0.242942i \(0.0781123\pi\)
\(32\) 7.27449 1.28596
\(33\) −3.22580 −0.561539
\(34\) 16.4926 2.82845
\(35\) 0 0
\(36\) 2.83083 0.471804
\(37\) −11.7806 −1.93672 −0.968361 0.249552i \(-0.919717\pi\)
−0.968361 + 0.249552i \(0.919717\pi\)
\(38\) −3.73942 −0.606614
\(39\) 2.58899 0.414570
\(40\) 0 0
\(41\) −5.81601 −0.908308 −0.454154 0.890923i \(-0.650058\pi\)
−0.454154 + 0.890923i \(0.650058\pi\)
\(42\) −2.30971 −0.356396
\(43\) −3.39497 −0.517728 −0.258864 0.965914i \(-0.583348\pi\)
−0.258864 + 0.965914i \(0.583348\pi\)
\(44\) −9.13167 −1.37665
\(45\) 0 0
\(46\) 4.86680 0.717571
\(47\) −1.00000 −0.145865
\(48\) −1.64807 −0.237879
\(49\) −5.89569 −0.842241
\(50\) 0 0
\(51\) −7.50374 −1.05073
\(52\) 7.32898 1.01635
\(53\) −11.1690 −1.53418 −0.767090 0.641539i \(-0.778296\pi\)
−0.767090 + 0.641539i \(0.778296\pi\)
\(54\) −2.19791 −0.299098
\(55\) 0 0
\(56\) −1.91897 −0.256433
\(57\) 1.70135 0.225349
\(58\) −14.8191 −1.94584
\(59\) 12.3261 1.60472 0.802361 0.596839i \(-0.203577\pi\)
0.802361 + 0.596839i \(0.203577\pi\)
\(60\) 0 0
\(61\) −6.55166 −0.838854 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(62\) −23.7417 −3.01520
\(63\) 1.05086 0.132396
\(64\) −12.6926 −1.58657
\(65\) 0 0
\(66\) 7.09002 0.872722
\(67\) 0.750116 0.0916412 0.0458206 0.998950i \(-0.485410\pi\)
0.0458206 + 0.998950i \(0.485410\pi\)
\(68\) −21.2418 −2.57595
\(69\) −2.21428 −0.266568
\(70\) 0 0
\(71\) 2.09231 0.248311 0.124156 0.992263i \(-0.460378\pi\)
0.124156 + 0.992263i \(0.460378\pi\)
\(72\) −1.82609 −0.215206
\(73\) 4.80219 0.562054 0.281027 0.959700i \(-0.409325\pi\)
0.281027 + 0.959700i \(0.409325\pi\)
\(74\) 25.8928 3.00998
\(75\) 0 0
\(76\) 4.81623 0.552459
\(77\) −3.38987 −0.386312
\(78\) −5.69038 −0.644308
\(79\) 6.59991 0.742548 0.371274 0.928523i \(-0.378921\pi\)
0.371274 + 0.928523i \(0.378921\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 12.7831 1.41166
\(83\) −15.6214 −1.71467 −0.857334 0.514761i \(-0.827881\pi\)
−0.857334 + 0.514761i \(0.827881\pi\)
\(84\) 2.97481 0.324579
\(85\) 0 0
\(86\) 7.46185 0.804632
\(87\) 6.74234 0.722854
\(88\) 5.89058 0.627938
\(89\) −15.4144 −1.63392 −0.816960 0.576694i \(-0.804343\pi\)
−0.816960 + 0.576694i \(0.804343\pi\)
\(90\) 0 0
\(91\) 2.72067 0.285204
\(92\) −6.26825 −0.653510
\(93\) 10.8019 1.12011
\(94\) 2.19791 0.226698
\(95\) 0 0
\(96\) 7.27449 0.742450
\(97\) 11.6829 1.18622 0.593109 0.805122i \(-0.297900\pi\)
0.593109 + 0.805122i \(0.297900\pi\)
\(98\) 12.9582 1.30898
\(99\) −3.22580 −0.324205
\(100\) 0 0
\(101\) −3.43631 −0.341925 −0.170963 0.985278i \(-0.554688\pi\)
−0.170963 + 0.985278i \(0.554688\pi\)
\(102\) 16.4926 1.63301
\(103\) −12.4894 −1.23061 −0.615307 0.788288i \(-0.710968\pi\)
−0.615307 + 0.788288i \(0.710968\pi\)
\(104\) −4.72772 −0.463591
\(105\) 0 0
\(106\) 24.5485 2.38436
\(107\) 20.2622 1.95882 0.979410 0.201879i \(-0.0647048\pi\)
0.979410 + 0.201879i \(0.0647048\pi\)
\(108\) 2.83083 0.272396
\(109\) −12.9693 −1.24224 −0.621118 0.783717i \(-0.713321\pi\)
−0.621118 + 0.783717i \(0.713321\pi\)
\(110\) 0 0
\(111\) −11.7806 −1.11817
\(112\) −1.73190 −0.163649
\(113\) −14.6725 −1.38027 −0.690136 0.723680i \(-0.742449\pi\)
−0.690136 + 0.723680i \(0.742449\pi\)
\(114\) −3.73942 −0.350229
\(115\) 0 0
\(116\) 19.0864 1.77213
\(117\) 2.58899 0.239352
\(118\) −27.0917 −2.49400
\(119\) −7.88541 −0.722854
\(120\) 0 0
\(121\) −0.594237 −0.0540216
\(122\) 14.4000 1.30371
\(123\) −5.81601 −0.524412
\(124\) 30.5784 2.74602
\(125\) 0 0
\(126\) −2.30971 −0.205765
\(127\) −14.6756 −1.30225 −0.651123 0.758972i \(-0.725702\pi\)
−0.651123 + 0.758972i \(0.725702\pi\)
\(128\) 13.3482 1.17982
\(129\) −3.39497 −0.298910
\(130\) 0 0
\(131\) −15.1152 −1.32062 −0.660309 0.750994i \(-0.729575\pi\)
−0.660309 + 0.750994i \(0.729575\pi\)
\(132\) −9.13167 −0.794810
\(133\) 1.78789 0.155029
\(134\) −1.64869 −0.142425
\(135\) 0 0
\(136\) 13.7025 1.17498
\(137\) 17.6256 1.50585 0.752927 0.658104i \(-0.228641\pi\)
0.752927 + 0.658104i \(0.228641\pi\)
\(138\) 4.86680 0.414290
\(139\) 9.44352 0.800989 0.400495 0.916299i \(-0.368838\pi\)
0.400495 + 0.916299i \(0.368838\pi\)
\(140\) 0 0
\(141\) −1.00000 −0.0842152
\(142\) −4.59871 −0.385916
\(143\) −8.35155 −0.698392
\(144\) −1.64807 −0.137339
\(145\) 0 0
\(146\) −10.5548 −0.873521
\(147\) −5.89569 −0.486268
\(148\) −33.3489 −2.74126
\(149\) 5.63097 0.461307 0.230653 0.973036i \(-0.425914\pi\)
0.230653 + 0.973036i \(0.425914\pi\)
\(150\) 0 0
\(151\) 13.8349 1.12587 0.562933 0.826503i \(-0.309673\pi\)
0.562933 + 0.826503i \(0.309673\pi\)
\(152\) −3.10681 −0.251996
\(153\) −7.50374 −0.606641
\(154\) 7.45065 0.600390
\(155\) 0 0
\(156\) 7.32898 0.586788
\(157\) −4.25611 −0.339675 −0.169837 0.985472i \(-0.554324\pi\)
−0.169837 + 0.985472i \(0.554324\pi\)
\(158\) −14.5060 −1.15404
\(159\) −11.1690 −0.885760
\(160\) 0 0
\(161\) −2.32691 −0.183386
\(162\) −2.19791 −0.172684
\(163\) 7.80011 0.610952 0.305476 0.952200i \(-0.401184\pi\)
0.305476 + 0.952200i \(0.401184\pi\)
\(164\) −16.4641 −1.28563
\(165\) 0 0
\(166\) 34.3344 2.66487
\(167\) −11.0275 −0.853336 −0.426668 0.904408i \(-0.640313\pi\)
−0.426668 + 0.904408i \(0.640313\pi\)
\(168\) −1.91897 −0.148052
\(169\) −6.29713 −0.484395
\(170\) 0 0
\(171\) 1.70135 0.130105
\(172\) −9.61057 −0.732799
\(173\) 3.88852 0.295639 0.147819 0.989014i \(-0.452775\pi\)
0.147819 + 0.989014i \(0.452775\pi\)
\(174\) −14.8191 −1.12343
\(175\) 0 0
\(176\) 5.31635 0.400735
\(177\) 12.3261 0.926487
\(178\) 33.8795 2.53937
\(179\) 0.395968 0.0295961 0.0147980 0.999891i \(-0.495289\pi\)
0.0147980 + 0.999891i \(0.495289\pi\)
\(180\) 0 0
\(181\) −13.9286 −1.03530 −0.517651 0.855592i \(-0.673194\pi\)
−0.517651 + 0.855592i \(0.673194\pi\)
\(182\) −5.97981 −0.443253
\(183\) −6.55166 −0.484312
\(184\) 4.04347 0.298089
\(185\) 0 0
\(186\) −23.7417 −1.74083
\(187\) 24.2055 1.77008
\(188\) −2.83083 −0.206459
\(189\) 1.05086 0.0764391
\(190\) 0 0
\(191\) −7.94141 −0.574620 −0.287310 0.957838i \(-0.592761\pi\)
−0.287310 + 0.957838i \(0.592761\pi\)
\(192\) −12.6926 −0.916007
\(193\) 12.8812 0.927211 0.463606 0.886042i \(-0.346555\pi\)
0.463606 + 0.886042i \(0.346555\pi\)
\(194\) −25.6780 −1.84357
\(195\) 0 0
\(196\) −16.6897 −1.19212
\(197\) −8.58344 −0.611545 −0.305772 0.952105i \(-0.598915\pi\)
−0.305772 + 0.952105i \(0.598915\pi\)
\(198\) 7.09002 0.503866
\(199\) 11.9505 0.847149 0.423574 0.905861i \(-0.360775\pi\)
0.423574 + 0.905861i \(0.360775\pi\)
\(200\) 0 0
\(201\) 0.750116 0.0529091
\(202\) 7.55271 0.531407
\(203\) 7.08528 0.497289
\(204\) −21.2418 −1.48722
\(205\) 0 0
\(206\) 27.4506 1.91257
\(207\) −2.21428 −0.153903
\(208\) −4.26684 −0.295852
\(209\) −5.48821 −0.379627
\(210\) 0 0
\(211\) 5.97104 0.411063 0.205532 0.978650i \(-0.434108\pi\)
0.205532 + 0.978650i \(0.434108\pi\)
\(212\) −31.6175 −2.17150
\(213\) 2.09231 0.143363
\(214\) −44.5346 −3.04432
\(215\) 0 0
\(216\) −1.82609 −0.124249
\(217\) 11.3513 0.770579
\(218\) 28.5055 1.93063
\(219\) 4.80219 0.324502
\(220\) 0 0
\(221\) −19.4271 −1.30681
\(222\) 25.8928 1.73781
\(223\) −12.1892 −0.816247 −0.408124 0.912927i \(-0.633817\pi\)
−0.408124 + 0.912927i \(0.633817\pi\)
\(224\) 7.64450 0.510770
\(225\) 0 0
\(226\) 32.2489 2.14516
\(227\) −25.3158 −1.68027 −0.840136 0.542376i \(-0.817525\pi\)
−0.840136 + 0.542376i \(0.817525\pi\)
\(228\) 4.81623 0.318962
\(229\) 13.1238 0.867247 0.433623 0.901094i \(-0.357235\pi\)
0.433623 + 0.901094i \(0.357235\pi\)
\(230\) 0 0
\(231\) −3.38987 −0.223037
\(232\) −12.3121 −0.808328
\(233\) 20.8072 1.36313 0.681563 0.731759i \(-0.261300\pi\)
0.681563 + 0.731759i \(0.261300\pi\)
\(234\) −5.69038 −0.371992
\(235\) 0 0
\(236\) 34.8931 2.27135
\(237\) 6.59991 0.428710
\(238\) 17.3314 1.12343
\(239\) 0.988409 0.0639349 0.0319674 0.999489i \(-0.489823\pi\)
0.0319674 + 0.999489i \(0.489823\pi\)
\(240\) 0 0
\(241\) 16.0839 1.03606 0.518028 0.855364i \(-0.326666\pi\)
0.518028 + 0.855364i \(0.326666\pi\)
\(242\) 1.30608 0.0839582
\(243\) 1.00000 0.0641500
\(244\) −18.5466 −1.18732
\(245\) 0 0
\(246\) 12.7831 0.815020
\(247\) 4.40478 0.280269
\(248\) −19.7252 −1.25255
\(249\) −15.6214 −0.989964
\(250\) 0 0
\(251\) −10.5672 −0.666997 −0.333498 0.942751i \(-0.608229\pi\)
−0.333498 + 0.942751i \(0.608229\pi\)
\(252\) 2.97481 0.187396
\(253\) 7.14282 0.449066
\(254\) 32.2556 2.02390
\(255\) 0 0
\(256\) −3.95304 −0.247065
\(257\) −23.9740 −1.49546 −0.747728 0.664005i \(-0.768855\pi\)
−0.747728 + 0.664005i \(0.768855\pi\)
\(258\) 7.46185 0.464555
\(259\) −12.3798 −0.769245
\(260\) 0 0
\(261\) 6.74234 0.417340
\(262\) 33.2218 2.05245
\(263\) −2.96013 −0.182530 −0.0912649 0.995827i \(-0.529091\pi\)
−0.0912649 + 0.995827i \(0.529091\pi\)
\(264\) 5.89058 0.362540
\(265\) 0 0
\(266\) −3.92962 −0.240941
\(267\) −15.4144 −0.943345
\(268\) 2.12345 0.129710
\(269\) 30.7826 1.87685 0.938423 0.345489i \(-0.112287\pi\)
0.938423 + 0.345489i \(0.112287\pi\)
\(270\) 0 0
\(271\) −11.6106 −0.705296 −0.352648 0.935756i \(-0.614719\pi\)
−0.352648 + 0.935756i \(0.614719\pi\)
\(272\) 12.3667 0.749842
\(273\) 2.72067 0.164663
\(274\) −38.7395 −2.34034
\(275\) 0 0
\(276\) −6.26825 −0.377304
\(277\) −0.511633 −0.0307410 −0.0153705 0.999882i \(-0.504893\pi\)
−0.0153705 + 0.999882i \(0.504893\pi\)
\(278\) −20.7561 −1.24487
\(279\) 10.8019 0.646694
\(280\) 0 0
\(281\) −25.0058 −1.49172 −0.745862 0.666101i \(-0.767962\pi\)
−0.745862 + 0.666101i \(0.767962\pi\)
\(282\) 2.19791 0.130884
\(283\) −31.8811 −1.89513 −0.947567 0.319556i \(-0.896466\pi\)
−0.947567 + 0.319556i \(0.896466\pi\)
\(284\) 5.92296 0.351463
\(285\) 0 0
\(286\) 18.3560 1.08541
\(287\) −6.11183 −0.360770
\(288\) 7.27449 0.428654
\(289\) 39.3061 2.31212
\(290\) 0 0
\(291\) 11.6829 0.684863
\(292\) 13.5942 0.795538
\(293\) 11.7501 0.686447 0.343223 0.939254i \(-0.388481\pi\)
0.343223 + 0.939254i \(0.388481\pi\)
\(294\) 12.9582 0.755738
\(295\) 0 0
\(296\) 21.5124 1.25039
\(297\) −3.22580 −0.187180
\(298\) −12.3764 −0.716944
\(299\) −5.73275 −0.331534
\(300\) 0 0
\(301\) −3.56765 −0.205636
\(302\) −30.4078 −1.74978
\(303\) −3.43631 −0.197411
\(304\) −2.80395 −0.160817
\(305\) 0 0
\(306\) 16.4926 0.942818
\(307\) −14.3166 −0.817092 −0.408546 0.912738i \(-0.633964\pi\)
−0.408546 + 0.912738i \(0.633964\pi\)
\(308\) −9.59614 −0.546791
\(309\) −12.4894 −0.710495
\(310\) 0 0
\(311\) −15.9784 −0.906050 −0.453025 0.891498i \(-0.649655\pi\)
−0.453025 + 0.891498i \(0.649655\pi\)
\(312\) −4.72772 −0.267654
\(313\) 17.1969 0.972025 0.486013 0.873952i \(-0.338451\pi\)
0.486013 + 0.873952i \(0.338451\pi\)
\(314\) 9.35458 0.527909
\(315\) 0 0
\(316\) 18.6832 1.05101
\(317\) −24.6354 −1.38366 −0.691830 0.722061i \(-0.743195\pi\)
−0.691830 + 0.722061i \(0.743195\pi\)
\(318\) 24.5485 1.37661
\(319\) −21.7494 −1.21773
\(320\) 0 0
\(321\) 20.2622 1.13093
\(322\) 5.11435 0.285011
\(323\) −12.7665 −0.710346
\(324\) 2.83083 0.157268
\(325\) 0 0
\(326\) −17.1440 −0.949517
\(327\) −12.9693 −0.717205
\(328\) 10.6205 0.586421
\(329\) −1.05086 −0.0579360
\(330\) 0 0
\(331\) −7.11931 −0.391313 −0.195656 0.980673i \(-0.562684\pi\)
−0.195656 + 0.980673i \(0.562684\pi\)
\(332\) −44.2214 −2.42696
\(333\) −11.7806 −0.645574
\(334\) 24.2376 1.32622
\(335\) 0 0
\(336\) −1.73190 −0.0944828
\(337\) −7.95185 −0.433165 −0.216582 0.976264i \(-0.569491\pi\)
−0.216582 + 0.976264i \(0.569491\pi\)
\(338\) 13.8406 0.752827
\(339\) −14.6725 −0.796900
\(340\) 0 0
\(341\) −34.8448 −1.88695
\(342\) −3.73942 −0.202205
\(343\) −13.5516 −0.731718
\(344\) 6.19951 0.334255
\(345\) 0 0
\(346\) −8.54664 −0.459470
\(347\) −1.56985 −0.0842738 −0.0421369 0.999112i \(-0.513417\pi\)
−0.0421369 + 0.999112i \(0.513417\pi\)
\(348\) 19.0864 1.02314
\(349\) −20.9148 −1.11954 −0.559771 0.828647i \(-0.689111\pi\)
−0.559771 + 0.828647i \(0.689111\pi\)
\(350\) 0 0
\(351\) 2.58899 0.138190
\(352\) −23.4660 −1.25074
\(353\) 11.0677 0.589075 0.294537 0.955640i \(-0.404834\pi\)
0.294537 + 0.955640i \(0.404834\pi\)
\(354\) −27.0917 −1.43991
\(355\) 0 0
\(356\) −43.6354 −2.31267
\(357\) −7.88541 −0.417340
\(358\) −0.870304 −0.0459970
\(359\) −21.3955 −1.12921 −0.564607 0.825360i \(-0.690972\pi\)
−0.564607 + 0.825360i \(0.690972\pi\)
\(360\) 0 0
\(361\) −16.1054 −0.847653
\(362\) 30.6138 1.60903
\(363\) −0.594237 −0.0311894
\(364\) 7.70176 0.403682
\(365\) 0 0
\(366\) 14.4000 0.752699
\(367\) 22.9905 1.20009 0.600046 0.799965i \(-0.295149\pi\)
0.600046 + 0.799965i \(0.295149\pi\)
\(368\) 3.64930 0.190233
\(369\) −5.81601 −0.302769
\(370\) 0 0
\(371\) −11.7371 −0.609360
\(372\) 30.5784 1.58541
\(373\) −30.8418 −1.59693 −0.798464 0.602042i \(-0.794354\pi\)
−0.798464 + 0.602042i \(0.794354\pi\)
\(374\) −53.2017 −2.75099
\(375\) 0 0
\(376\) 1.82609 0.0941732
\(377\) 17.4558 0.899021
\(378\) −2.30971 −0.118799
\(379\) −18.6754 −0.959290 −0.479645 0.877463i \(-0.659235\pi\)
−0.479645 + 0.877463i \(0.659235\pi\)
\(380\) 0 0
\(381\) −14.6756 −0.751852
\(382\) 17.4545 0.893052
\(383\) −11.9269 −0.609438 −0.304719 0.952442i \(-0.598562\pi\)
−0.304719 + 0.952442i \(0.598562\pi\)
\(384\) 13.3482 0.681172
\(385\) 0 0
\(386\) −28.3118 −1.44104
\(387\) −3.39497 −0.172576
\(388\) 33.0722 1.67899
\(389\) 3.23480 0.164011 0.0820054 0.996632i \(-0.473868\pi\)
0.0820054 + 0.996632i \(0.473868\pi\)
\(390\) 0 0
\(391\) 16.6154 0.840277
\(392\) 10.7660 0.543767
\(393\) −15.1152 −0.762459
\(394\) 18.8657 0.950438
\(395\) 0 0
\(396\) −9.13167 −0.458884
\(397\) −0.211311 −0.0106054 −0.00530270 0.999986i \(-0.501688\pi\)
−0.00530270 + 0.999986i \(0.501688\pi\)
\(398\) −26.2662 −1.31661
\(399\) 1.78789 0.0895063
\(400\) 0 0
\(401\) 28.7778 1.43710 0.718548 0.695478i \(-0.244807\pi\)
0.718548 + 0.695478i \(0.244807\pi\)
\(402\) −1.64869 −0.0822292
\(403\) 27.9661 1.39309
\(404\) −9.72759 −0.483966
\(405\) 0 0
\(406\) −15.5728 −0.772866
\(407\) 38.0019 1.88368
\(408\) 13.7025 0.678374
\(409\) 3.37349 0.166808 0.0834042 0.996516i \(-0.473421\pi\)
0.0834042 + 0.996516i \(0.473421\pi\)
\(410\) 0 0
\(411\) 17.6256 0.869406
\(412\) −35.3552 −1.74183
\(413\) 12.9531 0.637378
\(414\) 4.86680 0.239190
\(415\) 0 0
\(416\) 18.8336 0.923393
\(417\) 9.44352 0.462451
\(418\) 12.0626 0.590002
\(419\) 8.59815 0.420047 0.210024 0.977696i \(-0.432646\pi\)
0.210024 + 0.977696i \(0.432646\pi\)
\(420\) 0 0
\(421\) −33.4340 −1.62947 −0.814737 0.579831i \(-0.803118\pi\)
−0.814737 + 0.579831i \(0.803118\pi\)
\(422\) −13.1238 −0.638858
\(423\) −1.00000 −0.0486217
\(424\) 20.3956 0.990496
\(425\) 0 0
\(426\) −4.59871 −0.222808
\(427\) −6.88490 −0.333183
\(428\) 57.3588 2.77254
\(429\) −8.35155 −0.403217
\(430\) 0 0
\(431\) −20.7133 −0.997726 −0.498863 0.866681i \(-0.666249\pi\)
−0.498863 + 0.866681i \(0.666249\pi\)
\(432\) −1.64807 −0.0792929
\(433\) 25.6067 1.23058 0.615291 0.788300i \(-0.289039\pi\)
0.615291 + 0.788300i \(0.289039\pi\)
\(434\) −24.9493 −1.19760
\(435\) 0 0
\(436\) −36.7139 −1.75828
\(437\) −3.76727 −0.180213
\(438\) −10.5548 −0.504328
\(439\) −2.07924 −0.0992367 −0.0496183 0.998768i \(-0.515800\pi\)
−0.0496183 + 0.998768i \(0.515800\pi\)
\(440\) 0 0
\(441\) −5.89569 −0.280747
\(442\) 42.6991 2.03099
\(443\) 21.9500 1.04287 0.521437 0.853290i \(-0.325396\pi\)
0.521437 + 0.853290i \(0.325396\pi\)
\(444\) −33.3489 −1.58267
\(445\) 0 0
\(446\) 26.7907 1.26858
\(447\) 5.63097 0.266336
\(448\) −13.3382 −0.630169
\(449\) 2.61869 0.123583 0.0617917 0.998089i \(-0.480319\pi\)
0.0617917 + 0.998089i \(0.480319\pi\)
\(450\) 0 0
\(451\) 18.7613 0.883433
\(452\) −41.5353 −1.95365
\(453\) 13.8349 0.650019
\(454\) 55.6421 2.61141
\(455\) 0 0
\(456\) −3.10681 −0.145490
\(457\) −31.5801 −1.47726 −0.738628 0.674113i \(-0.764526\pi\)
−0.738628 + 0.674113i \(0.764526\pi\)
\(458\) −28.8450 −1.34784
\(459\) −7.50374 −0.350245
\(460\) 0 0
\(461\) 0.891136 0.0415043 0.0207522 0.999785i \(-0.493394\pi\)
0.0207522 + 0.999785i \(0.493394\pi\)
\(462\) 7.45065 0.346635
\(463\) −15.1369 −0.703470 −0.351735 0.936100i \(-0.614408\pi\)
−0.351735 + 0.936100i \(0.614408\pi\)
\(464\) −11.1119 −0.515855
\(465\) 0 0
\(466\) −45.7325 −2.11852
\(467\) −2.92279 −0.135251 −0.0676253 0.997711i \(-0.521542\pi\)
−0.0676253 + 0.997711i \(0.521542\pi\)
\(468\) 7.32898 0.338782
\(469\) 0.788269 0.0363989
\(470\) 0 0
\(471\) −4.25611 −0.196111
\(472\) −22.5085 −1.03604
\(473\) 10.9515 0.503550
\(474\) −14.5060 −0.666284
\(475\) 0 0
\(476\) −22.3222 −1.02314
\(477\) −11.1690 −0.511394
\(478\) −2.17244 −0.0993650
\(479\) −13.3779 −0.611252 −0.305626 0.952152i \(-0.598866\pi\)
−0.305626 + 0.952152i \(0.598866\pi\)
\(480\) 0 0
\(481\) −30.4999 −1.39068
\(482\) −35.3511 −1.61020
\(483\) −2.32691 −0.105878
\(484\) −1.68218 −0.0764629
\(485\) 0 0
\(486\) −2.19791 −0.0996994
\(487\) −36.9084 −1.67248 −0.836239 0.548365i \(-0.815251\pi\)
−0.836239 + 0.548365i \(0.815251\pi\)
\(488\) 11.9639 0.541580
\(489\) 7.80011 0.352733
\(490\) 0 0
\(491\) 6.79535 0.306670 0.153335 0.988174i \(-0.450999\pi\)
0.153335 + 0.988174i \(0.450999\pi\)
\(492\) −16.4641 −0.742259
\(493\) −50.5927 −2.27858
\(494\) −9.68132 −0.435583
\(495\) 0 0
\(496\) −17.8023 −0.799349
\(497\) 2.19873 0.0986265
\(498\) 34.3344 1.53856
\(499\) −5.04952 −0.226047 −0.113024 0.993592i \(-0.536054\pi\)
−0.113024 + 0.993592i \(0.536054\pi\)
\(500\) 0 0
\(501\) −11.0275 −0.492674
\(502\) 23.2258 1.03662
\(503\) 18.6865 0.833191 0.416595 0.909092i \(-0.363223\pi\)
0.416595 + 0.909092i \(0.363223\pi\)
\(504\) −1.91897 −0.0854776
\(505\) 0 0
\(506\) −15.6993 −0.697920
\(507\) −6.29713 −0.279666
\(508\) −41.5440 −1.84322
\(509\) 26.5029 1.17472 0.587360 0.809326i \(-0.300167\pi\)
0.587360 + 0.809326i \(0.300167\pi\)
\(510\) 0 0
\(511\) 5.04644 0.223242
\(512\) −18.0079 −0.795846
\(513\) 1.70135 0.0751164
\(514\) 52.6928 2.32418
\(515\) 0 0
\(516\) −9.61057 −0.423082
\(517\) 3.22580 0.141870
\(518\) 27.2098 1.19553
\(519\) 3.88852 0.170687
\(520\) 0 0
\(521\) −14.1654 −0.620600 −0.310300 0.950639i \(-0.600429\pi\)
−0.310300 + 0.950639i \(0.600429\pi\)
\(522\) −14.8191 −0.648613
\(523\) 14.9461 0.653549 0.326775 0.945102i \(-0.394038\pi\)
0.326775 + 0.945102i \(0.394038\pi\)
\(524\) −42.7884 −1.86922
\(525\) 0 0
\(526\) 6.50612 0.283680
\(527\) −81.0548 −3.53080
\(528\) 5.31635 0.231364
\(529\) −18.0970 −0.786824
\(530\) 0 0
\(531\) 12.3261 0.534907
\(532\) 5.06120 0.219431
\(533\) −15.0576 −0.652216
\(534\) 33.8795 1.46611
\(535\) 0 0
\(536\) −1.36978 −0.0591653
\(537\) 0.395968 0.0170873
\(538\) −67.6574 −2.91692
\(539\) 19.0183 0.819175
\(540\) 0 0
\(541\) 8.91744 0.383391 0.191695 0.981454i \(-0.438601\pi\)
0.191695 + 0.981454i \(0.438601\pi\)
\(542\) 25.5192 1.09614
\(543\) −13.9286 −0.597732
\(544\) −54.5859 −2.34035
\(545\) 0 0
\(546\) −5.97981 −0.255912
\(547\) −8.89021 −0.380118 −0.190059 0.981773i \(-0.560868\pi\)
−0.190059 + 0.981773i \(0.560868\pi\)
\(548\) 49.8950 2.13141
\(549\) −6.55166 −0.279618
\(550\) 0 0
\(551\) 11.4711 0.488684
\(552\) 4.04347 0.172102
\(553\) 6.93560 0.294932
\(554\) 1.12452 0.0477765
\(555\) 0 0
\(556\) 26.7330 1.13373
\(557\) 35.1323 1.48860 0.744301 0.667844i \(-0.232783\pi\)
0.744301 + 0.667844i \(0.232783\pi\)
\(558\) −23.7417 −1.00507
\(559\) −8.78954 −0.371758
\(560\) 0 0
\(561\) 24.2055 1.02196
\(562\) 54.9607 2.31838
\(563\) 29.2562 1.23300 0.616502 0.787354i \(-0.288549\pi\)
0.616502 + 0.787354i \(0.288549\pi\)
\(564\) −2.83083 −0.119199
\(565\) 0 0
\(566\) 70.0719 2.94534
\(567\) 1.05086 0.0441321
\(568\) −3.82074 −0.160315
\(569\) −9.51183 −0.398756 −0.199378 0.979923i \(-0.563892\pi\)
−0.199378 + 0.979923i \(0.563892\pi\)
\(570\) 0 0
\(571\) 1.71387 0.0717231 0.0358615 0.999357i \(-0.488582\pi\)
0.0358615 + 0.999357i \(0.488582\pi\)
\(572\) −23.6418 −0.988513
\(573\) −7.94141 −0.331757
\(574\) 13.4333 0.560694
\(575\) 0 0
\(576\) −12.6926 −0.528857
\(577\) 22.3271 0.929491 0.464746 0.885444i \(-0.346146\pi\)
0.464746 + 0.885444i \(0.346146\pi\)
\(578\) −86.3915 −3.59341
\(579\) 12.8812 0.535326
\(580\) 0 0
\(581\) −16.4159 −0.681047
\(582\) −25.6780 −1.06439
\(583\) 36.0289 1.49217
\(584\) −8.76921 −0.362873
\(585\) 0 0
\(586\) −25.8257 −1.06685
\(587\) 15.3439 0.633311 0.316655 0.948541i \(-0.397440\pi\)
0.316655 + 0.948541i \(0.397440\pi\)
\(588\) −16.6897 −0.688270
\(589\) 18.3778 0.757246
\(590\) 0 0
\(591\) −8.58344 −0.353075
\(592\) 19.4153 0.797965
\(593\) −25.0738 −1.02966 −0.514830 0.857293i \(-0.672145\pi\)
−0.514830 + 0.857293i \(0.672145\pi\)
\(594\) 7.09002 0.290907
\(595\) 0 0
\(596\) 15.9403 0.652940
\(597\) 11.9505 0.489102
\(598\) 12.6001 0.515256
\(599\) −45.9704 −1.87830 −0.939150 0.343506i \(-0.888385\pi\)
−0.939150 + 0.343506i \(0.888385\pi\)
\(600\) 0 0
\(601\) −20.2362 −0.825453 −0.412727 0.910855i \(-0.635423\pi\)
−0.412727 + 0.910855i \(0.635423\pi\)
\(602\) 7.84139 0.319591
\(603\) 0.750116 0.0305471
\(604\) 39.1641 1.59357
\(605\) 0 0
\(606\) 7.55271 0.306808
\(607\) 2.94855 0.119678 0.0598389 0.998208i \(-0.480941\pi\)
0.0598389 + 0.998208i \(0.480941\pi\)
\(608\) 12.3765 0.501932
\(609\) 7.08528 0.287110
\(610\) 0 0
\(611\) −2.58899 −0.104739
\(612\) −21.2418 −0.858649
\(613\) 21.0447 0.849987 0.424993 0.905196i \(-0.360276\pi\)
0.424993 + 0.905196i \(0.360276\pi\)
\(614\) 31.4667 1.26989
\(615\) 0 0
\(616\) 6.19020 0.249410
\(617\) 16.6437 0.670051 0.335026 0.942209i \(-0.391255\pi\)
0.335026 + 0.942209i \(0.391255\pi\)
\(618\) 27.4506 1.10422
\(619\) 38.9234 1.56446 0.782232 0.622987i \(-0.214081\pi\)
0.782232 + 0.622987i \(0.214081\pi\)
\(620\) 0 0
\(621\) −2.21428 −0.0888561
\(622\) 35.1191 1.40815
\(623\) −16.1984 −0.648975
\(624\) −4.26684 −0.170810
\(625\) 0 0
\(626\) −37.7973 −1.51068
\(627\) −5.48821 −0.219178
\(628\) −12.0483 −0.480780
\(629\) 88.3987 3.52469
\(630\) 0 0
\(631\) −19.6963 −0.784097 −0.392049 0.919945i \(-0.628233\pi\)
−0.392049 + 0.919945i \(0.628233\pi\)
\(632\) −12.0520 −0.479403
\(633\) 5.97104 0.237328
\(634\) 54.1464 2.15043
\(635\) 0 0
\(636\) −31.6175 −1.25372
\(637\) −15.2639 −0.604776
\(638\) 47.8033 1.89255
\(639\) 2.09231 0.0827704
\(640\) 0 0
\(641\) −6.39022 −0.252399 −0.126199 0.992005i \(-0.540278\pi\)
−0.126199 + 0.992005i \(0.540278\pi\)
\(642\) −44.5346 −1.75764
\(643\) −39.9323 −1.57477 −0.787387 0.616459i \(-0.788567\pi\)
−0.787387 + 0.616459i \(0.788567\pi\)
\(644\) −6.58708 −0.259567
\(645\) 0 0
\(646\) 28.0596 1.10399
\(647\) 28.4879 1.11998 0.559988 0.828501i \(-0.310806\pi\)
0.559988 + 0.828501i \(0.310806\pi\)
\(648\) −1.82609 −0.0717355
\(649\) −39.7615 −1.56078
\(650\) 0 0
\(651\) 11.3513 0.444894
\(652\) 22.0808 0.864750
\(653\) 38.6045 1.51071 0.755355 0.655316i \(-0.227465\pi\)
0.755355 + 0.655316i \(0.227465\pi\)
\(654\) 28.5055 1.11465
\(655\) 0 0
\(656\) 9.58520 0.374239
\(657\) 4.80219 0.187351
\(658\) 2.30971 0.0900418
\(659\) −18.9279 −0.737324 −0.368662 0.929563i \(-0.620184\pi\)
−0.368662 + 0.929563i \(0.620184\pi\)
\(660\) 0 0
\(661\) −0.518381 −0.0201627 −0.0100813 0.999949i \(-0.503209\pi\)
−0.0100813 + 0.999949i \(0.503209\pi\)
\(662\) 15.6476 0.608162
\(663\) −19.4271 −0.754486
\(664\) 28.5260 1.10702
\(665\) 0 0
\(666\) 25.8928 1.00333
\(667\) −14.9294 −0.578070
\(668\) −31.2170 −1.20782
\(669\) −12.1892 −0.471260
\(670\) 0 0
\(671\) 21.1343 0.815881
\(672\) 7.64450 0.294893
\(673\) −17.7260 −0.683288 −0.341644 0.939829i \(-0.610984\pi\)
−0.341644 + 0.939829i \(0.610984\pi\)
\(674\) 17.4775 0.673207
\(675\) 0 0
\(676\) −17.8261 −0.685619
\(677\) 3.62115 0.139172 0.0695861 0.997576i \(-0.477832\pi\)
0.0695861 + 0.997576i \(0.477832\pi\)
\(678\) 32.2489 1.23851
\(679\) 12.2771 0.471153
\(680\) 0 0
\(681\) −25.3158 −0.970105
\(682\) 76.5859 2.93262
\(683\) −1.75071 −0.0669889 −0.0334944 0.999439i \(-0.510664\pi\)
−0.0334944 + 0.999439i \(0.510664\pi\)
\(684\) 4.81623 0.184153
\(685\) 0 0
\(686\) 29.7853 1.13721
\(687\) 13.1238 0.500705
\(688\) 5.59516 0.213313
\(689\) −28.9164 −1.10163
\(690\) 0 0
\(691\) 11.3812 0.432963 0.216481 0.976287i \(-0.430542\pi\)
0.216481 + 0.976287i \(0.430542\pi\)
\(692\) 11.0077 0.418451
\(693\) −3.38987 −0.128771
\(694\) 3.45039 0.130975
\(695\) 0 0
\(696\) −12.3121 −0.466689
\(697\) 43.6418 1.65305
\(698\) 45.9689 1.73995
\(699\) 20.8072 0.787001
\(700\) 0 0
\(701\) 14.5746 0.550476 0.275238 0.961376i \(-0.411243\pi\)
0.275238 + 0.961376i \(0.411243\pi\)
\(702\) −5.69038 −0.214769
\(703\) −20.0430 −0.755935
\(704\) 40.9437 1.54312
\(705\) 0 0
\(706\) −24.3259 −0.915516
\(707\) −3.61109 −0.135809
\(708\) 34.8931 1.31136
\(709\) −18.7870 −0.705561 −0.352780 0.935706i \(-0.614764\pi\)
−0.352780 + 0.935706i \(0.614764\pi\)
\(710\) 0 0
\(711\) 6.59991 0.247516
\(712\) 28.1480 1.05489
\(713\) −23.9185 −0.895755
\(714\) 17.3314 0.648613
\(715\) 0 0
\(716\) 1.12092 0.0418907
\(717\) 0.988409 0.0369128
\(718\) 47.0256 1.75498
\(719\) −9.79963 −0.365465 −0.182732 0.983163i \(-0.558494\pi\)
−0.182732 + 0.983163i \(0.558494\pi\)
\(720\) 0 0
\(721\) −13.1246 −0.488786
\(722\) 35.3983 1.31739
\(723\) 16.0839 0.598167
\(724\) −39.4294 −1.46538
\(725\) 0 0
\(726\) 1.30608 0.0484733
\(727\) −25.5912 −0.949126 −0.474563 0.880221i \(-0.657394\pi\)
−0.474563 + 0.880221i \(0.657394\pi\)
\(728\) −4.96819 −0.184133
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 25.4750 0.942226
\(732\) −18.5466 −0.685502
\(733\) −42.3314 −1.56354 −0.781772 0.623564i \(-0.785684\pi\)
−0.781772 + 0.623564i \(0.785684\pi\)
\(734\) −50.5311 −1.86514
\(735\) 0 0
\(736\) −16.1078 −0.593741
\(737\) −2.41972 −0.0891316
\(738\) 12.7831 0.470552
\(739\) −1.65177 −0.0607612 −0.0303806 0.999538i \(-0.509672\pi\)
−0.0303806 + 0.999538i \(0.509672\pi\)
\(740\) 0 0
\(741\) 4.40478 0.161814
\(742\) 25.7971 0.947043
\(743\) 30.1467 1.10597 0.552987 0.833190i \(-0.313488\pi\)
0.552987 + 0.833190i \(0.313488\pi\)
\(744\) −19.7252 −0.723162
\(745\) 0 0
\(746\) 67.7877 2.48188
\(747\) −15.6214 −0.571556
\(748\) 68.5217 2.50540
\(749\) 21.2928 0.778022
\(750\) 0 0
\(751\) −4.44762 −0.162296 −0.0811479 0.996702i \(-0.525859\pi\)
−0.0811479 + 0.996702i \(0.525859\pi\)
\(752\) 1.64807 0.0600990
\(753\) −10.5672 −0.385091
\(754\) −38.3664 −1.39722
\(755\) 0 0
\(756\) 2.97481 0.108193
\(757\) 41.9419 1.52441 0.762203 0.647338i \(-0.224118\pi\)
0.762203 + 0.647338i \(0.224118\pi\)
\(758\) 41.0469 1.49089
\(759\) 7.14282 0.259268
\(760\) 0 0
\(761\) −27.1036 −0.982504 −0.491252 0.871018i \(-0.663461\pi\)
−0.491252 + 0.871018i \(0.663461\pi\)
\(762\) 32.2556 1.16850
\(763\) −13.6290 −0.493402
\(764\) −22.4808 −0.813325
\(765\) 0 0
\(766\) 26.2144 0.947163
\(767\) 31.9122 1.15228
\(768\) −3.95304 −0.142643
\(769\) 32.8683 1.18526 0.592631 0.805474i \(-0.298089\pi\)
0.592631 + 0.805474i \(0.298089\pi\)
\(770\) 0 0
\(771\) −23.9740 −0.863402
\(772\) 36.4645 1.31239
\(773\) −21.9310 −0.788804 −0.394402 0.918938i \(-0.629048\pi\)
−0.394402 + 0.918938i \(0.629048\pi\)
\(774\) 7.46185 0.268211
\(775\) 0 0
\(776\) −21.3340 −0.765845
\(777\) −12.3798 −0.444124
\(778\) −7.10981 −0.254899
\(779\) −9.89506 −0.354527
\(780\) 0 0
\(781\) −6.74936 −0.241511
\(782\) −36.5192 −1.30592
\(783\) 6.74234 0.240951
\(784\) 9.71652 0.347018
\(785\) 0 0
\(786\) 33.2218 1.18498
\(787\) −53.6288 −1.91166 −0.955830 0.293920i \(-0.905040\pi\)
−0.955830 + 0.293920i \(0.905040\pi\)
\(788\) −24.2982 −0.865588
\(789\) −2.96013 −0.105384
\(790\) 0 0
\(791\) −15.4188 −0.548229
\(792\) 5.89058 0.209313
\(793\) −16.9622 −0.602344
\(794\) 0.464444 0.0164825
\(795\) 0 0
\(796\) 33.8298 1.19907
\(797\) 4.21744 0.149389 0.0746946 0.997206i \(-0.476202\pi\)
0.0746946 + 0.997206i \(0.476202\pi\)
\(798\) −3.92962 −0.139107
\(799\) 7.50374 0.265463
\(800\) 0 0
\(801\) −15.4144 −0.544640
\(802\) −63.2512 −2.23348
\(803\) −15.4909 −0.546661
\(804\) 2.12345 0.0748883
\(805\) 0 0
\(806\) −61.4670 −2.16508
\(807\) 30.7826 1.08360
\(808\) 6.27500 0.220754
\(809\) −9.77548 −0.343688 −0.171844 0.985124i \(-0.554972\pi\)
−0.171844 + 0.985124i \(0.554972\pi\)
\(810\) 0 0
\(811\) 13.9279 0.489075 0.244538 0.969640i \(-0.421364\pi\)
0.244538 + 0.969640i \(0.421364\pi\)
\(812\) 20.0572 0.703869
\(813\) −11.6106 −0.407203
\(814\) −83.5249 −2.92755
\(815\) 0 0
\(816\) 12.3667 0.432921
\(817\) −5.77603 −0.202078
\(818\) −7.41465 −0.259247
\(819\) 2.72067 0.0950681
\(820\) 0 0
\(821\) 0.631310 0.0220329 0.0110164 0.999939i \(-0.496493\pi\)
0.0110164 + 0.999939i \(0.496493\pi\)
\(822\) −38.7395 −1.35120
\(823\) 26.9584 0.939711 0.469855 0.882743i \(-0.344306\pi\)
0.469855 + 0.882743i \(0.344306\pi\)
\(824\) 22.8067 0.794508
\(825\) 0 0
\(826\) −28.4697 −0.990588
\(827\) 19.3808 0.673937 0.336968 0.941516i \(-0.390598\pi\)
0.336968 + 0.941516i \(0.390598\pi\)
\(828\) −6.26825 −0.217837
\(829\) 44.6224 1.54980 0.774900 0.632083i \(-0.217800\pi\)
0.774900 + 0.632083i \(0.217800\pi\)
\(830\) 0 0
\(831\) −0.511633 −0.0177483
\(832\) −32.8609 −1.13925
\(833\) 44.2397 1.53281
\(834\) −20.7561 −0.718723
\(835\) 0 0
\(836\) −15.5362 −0.537330
\(837\) 10.8019 0.373369
\(838\) −18.8980 −0.652820
\(839\) 25.8332 0.891860 0.445930 0.895068i \(-0.352873\pi\)
0.445930 + 0.895068i \(0.352873\pi\)
\(840\) 0 0
\(841\) 16.4591 0.567555
\(842\) 73.4851 2.53246
\(843\) −25.0058 −0.861247
\(844\) 16.9030 0.581825
\(845\) 0 0
\(846\) 2.19791 0.0755658
\(847\) −0.624462 −0.0214568
\(848\) 18.4073 0.632110
\(849\) −31.8811 −1.09416
\(850\) 0 0
\(851\) 26.0856 0.894204
\(852\) 5.92296 0.202917
\(853\) −1.16342 −0.0398346 −0.0199173 0.999802i \(-0.506340\pi\)
−0.0199173 + 0.999802i \(0.506340\pi\)
\(854\) 15.1324 0.517820
\(855\) 0 0
\(856\) −37.0005 −1.26465
\(857\) −26.9612 −0.920976 −0.460488 0.887666i \(-0.652326\pi\)
−0.460488 + 0.887666i \(0.652326\pi\)
\(858\) 18.3560 0.626663
\(859\) −16.1012 −0.549367 −0.274683 0.961535i \(-0.588573\pi\)
−0.274683 + 0.961535i \(0.588573\pi\)
\(860\) 0 0
\(861\) −6.11183 −0.208291
\(862\) 45.5261 1.55063
\(863\) 9.69771 0.330114 0.165057 0.986284i \(-0.447219\pi\)
0.165057 + 0.986284i \(0.447219\pi\)
\(864\) 7.27449 0.247483
\(865\) 0 0
\(866\) −56.2814 −1.91252
\(867\) 39.3061 1.33491
\(868\) 32.1337 1.09069
\(869\) −21.2900 −0.722212
\(870\) 0 0
\(871\) 1.94204 0.0658036
\(872\) 23.6831 0.802011
\(873\) 11.6829 0.395406
\(874\) 8.28013 0.280080
\(875\) 0 0
\(876\) 13.5942 0.459304
\(877\) −2.85277 −0.0963312 −0.0481656 0.998839i \(-0.515338\pi\)
−0.0481656 + 0.998839i \(0.515338\pi\)
\(878\) 4.56999 0.154230
\(879\) 11.7501 0.396320
\(880\) 0 0
\(881\) 31.9431 1.07619 0.538095 0.842884i \(-0.319144\pi\)
0.538095 + 0.842884i \(0.319144\pi\)
\(882\) 12.9582 0.436326
\(883\) −21.5997 −0.726886 −0.363443 0.931616i \(-0.618399\pi\)
−0.363443 + 0.931616i \(0.618399\pi\)
\(884\) −54.9948 −1.84967
\(885\) 0 0
\(886\) −48.2442 −1.62079
\(887\) 32.1151 1.07832 0.539160 0.842203i \(-0.318742\pi\)
0.539160 + 0.842203i \(0.318742\pi\)
\(888\) 21.5124 0.721910
\(889\) −15.4220 −0.517238
\(890\) 0 0
\(891\) −3.22580 −0.108068
\(892\) −34.5054 −1.15533
\(893\) −1.70135 −0.0569335
\(894\) −12.3764 −0.413928
\(895\) 0 0
\(896\) 14.0271 0.468613
\(897\) −5.73275 −0.191411
\(898\) −5.75565 −0.192068
\(899\) 72.8302 2.42902
\(900\) 0 0
\(901\) 83.8093 2.79209
\(902\) −41.2356 −1.37300
\(903\) −3.56765 −0.118724
\(904\) 26.7932 0.891129
\(905\) 0 0
\(906\) −30.4078 −1.01023
\(907\) 23.2655 0.772519 0.386259 0.922390i \(-0.373767\pi\)
0.386259 + 0.922390i \(0.373767\pi\)
\(908\) −71.6648 −2.37828
\(909\) −3.43631 −0.113975
\(910\) 0 0
\(911\) −27.7030 −0.917841 −0.458920 0.888477i \(-0.651764\pi\)
−0.458920 + 0.888477i \(0.651764\pi\)
\(912\) −2.80395 −0.0928480
\(913\) 50.3913 1.66771
\(914\) 69.4104 2.29589
\(915\) 0 0
\(916\) 37.1513 1.22751
\(917\) −15.8840 −0.524535
\(918\) 16.4926 0.544336
\(919\) −3.21697 −0.106118 −0.0530590 0.998591i \(-0.516897\pi\)
−0.0530590 + 0.998591i \(0.516897\pi\)
\(920\) 0 0
\(921\) −14.3166 −0.471748
\(922\) −1.95864 −0.0645044
\(923\) 5.41696 0.178302
\(924\) −9.59614 −0.315690
\(925\) 0 0
\(926\) 33.2695 1.09330
\(927\) −12.4894 −0.410205
\(928\) 49.0471 1.61005
\(929\) −5.48067 −0.179815 −0.0899075 0.995950i \(-0.528657\pi\)
−0.0899075 + 0.995950i \(0.528657\pi\)
\(930\) 0 0
\(931\) −10.0306 −0.328740
\(932\) 58.9016 1.92939
\(933\) −15.9784 −0.523108
\(934\) 6.42404 0.210201
\(935\) 0 0
\(936\) −4.72772 −0.154530
\(937\) 3.03424 0.0991242 0.0495621 0.998771i \(-0.484217\pi\)
0.0495621 + 0.998771i \(0.484217\pi\)
\(938\) −1.73255 −0.0565697
\(939\) 17.1969 0.561199
\(940\) 0 0
\(941\) −47.1897 −1.53834 −0.769170 0.639045i \(-0.779330\pi\)
−0.769170 + 0.639045i \(0.779330\pi\)
\(942\) 9.35458 0.304789
\(943\) 12.8783 0.419375
\(944\) −20.3143 −0.661175
\(945\) 0 0
\(946\) −24.0704 −0.782597
\(947\) 38.9579 1.26596 0.632982 0.774167i \(-0.281831\pi\)
0.632982 + 0.774167i \(0.281831\pi\)
\(948\) 18.6832 0.606802
\(949\) 12.4328 0.403586
\(950\) 0 0
\(951\) −24.6354 −0.798856
\(952\) 14.3994 0.466688
\(953\) 21.3170 0.690524 0.345262 0.938506i \(-0.387790\pi\)
0.345262 + 0.938506i \(0.387790\pi\)
\(954\) 24.5485 0.794788
\(955\) 0 0
\(956\) 2.79801 0.0904943
\(957\) −21.7494 −0.703058
\(958\) 29.4035 0.949983
\(959\) 18.5221 0.598109
\(960\) 0 0
\(961\) 85.6814 2.76392
\(962\) 67.0362 2.16133
\(963\) 20.2622 0.652940
\(964\) 45.5308 1.46645
\(965\) 0 0
\(966\) 5.11435 0.164551
\(967\) −48.1797 −1.54935 −0.774677 0.632358i \(-0.782087\pi\)
−0.774677 + 0.632358i \(0.782087\pi\)
\(968\) 1.08513 0.0348774
\(969\) −12.7665 −0.410119
\(970\) 0 0
\(971\) 29.4387 0.944732 0.472366 0.881402i \(-0.343400\pi\)
0.472366 + 0.881402i \(0.343400\pi\)
\(972\) 2.83083 0.0907988
\(973\) 9.92385 0.318144
\(974\) 81.1215 2.59930
\(975\) 0 0
\(976\) 10.7976 0.345623
\(977\) 44.7657 1.43218 0.716091 0.698007i \(-0.245930\pi\)
0.716091 + 0.698007i \(0.245930\pi\)
\(978\) −17.1440 −0.548204
\(979\) 49.7236 1.58917
\(980\) 0 0
\(981\) −12.9693 −0.414079
\(982\) −14.9356 −0.476614
\(983\) 41.8646 1.33527 0.667637 0.744487i \(-0.267306\pi\)
0.667637 + 0.744487i \(0.267306\pi\)
\(984\) 10.6205 0.338570
\(985\) 0 0
\(986\) 111.199 3.54128
\(987\) −1.05086 −0.0334494
\(988\) 12.4692 0.396697
\(989\) 7.51742 0.239040
\(990\) 0 0
\(991\) −28.5991 −0.908480 −0.454240 0.890879i \(-0.650089\pi\)
−0.454240 + 0.890879i \(0.650089\pi\)
\(992\) 78.5785 2.49487
\(993\) −7.11931 −0.225924
\(994\) −4.83262 −0.153281
\(995\) 0 0
\(996\) −44.2214 −1.40121
\(997\) 24.6378 0.780287 0.390144 0.920754i \(-0.372425\pi\)
0.390144 + 0.920754i \(0.372425\pi\)
\(998\) 11.0984 0.351314
\(999\) −11.7806 −0.372722
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.1.2 8
5.4 even 2 3525.2.a.be.1.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3525.2.a.bd.1.2 8 1.1 even 1 trivial
3525.2.a.be.1.7 yes 8 5.4 even 2