Properties

Label 5625.2.a.z.1.4
Level $5625$
Weight $2$
Character 5625.1
Self dual yes
Analytic conductor $44.916$
Analytic rank $1$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 5625 = 3^{2} \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5625.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(44.9158511370\)
Analytic rank: \(1\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \( x^{8} - 15x^{6} + 70x^{4} - 105x^{2} + 45 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.856773\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.856773 q^{2} -1.26594 q^{4} +0.647907 q^{7} +2.79817 q^{8} +O(q^{10})\) \(q-0.856773 q^{2} -1.26594 q^{4} +0.647907 q^{7} +2.79817 q^{8} +5.91382 q^{11} +2.88397 q^{13} -0.555109 q^{14} +0.134488 q^{16} -4.20027 q^{17} -1.64791 q^{19} -5.06680 q^{22} -6.42752 q^{23} -2.47091 q^{26} -0.820212 q^{28} -2.25888 q^{29} -5.28440 q^{31} -5.71156 q^{32} +3.59868 q^{34} -1.71560 q^{37} +1.41188 q^{38} +0.176662 q^{41} +7.95077 q^{43} -7.48654 q^{44} +5.50692 q^{46} -1.05903 q^{47} -6.58022 q^{49} -3.65094 q^{52} +4.48612 q^{53} +1.81295 q^{56} +1.93534 q^{58} -9.74542 q^{59} -11.2844 q^{61} +4.52753 q^{62} +4.62453 q^{64} -12.6171 q^{67} +5.31730 q^{68} +6.09048 q^{71} -7.08312 q^{73} +1.46988 q^{74} +2.08615 q^{76} +3.83160 q^{77} +1.58111 q^{79} -0.151359 q^{82} -11.5102 q^{83} -6.81200 q^{86} +16.5479 q^{88} +15.5918 q^{89} +1.86855 q^{91} +8.13685 q^{92} +0.907347 q^{94} -7.55034 q^{97} +5.63775 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 14 q^{4} - 10 q^{7} - 10 q^{13} + 22 q^{16} + 2 q^{19} - 10 q^{22} - 70 q^{28} - 6 q^{31} - 50 q^{34} - 50 q^{37} + 14 q^{49} - 80 q^{52} + 30 q^{58} - 54 q^{61} + 36 q^{64} - 10 q^{67} - 30 q^{73} + 56 q^{76} + 28 q^{79} - 20 q^{88} + 60 q^{91} - 40 q^{94}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.856773 −0.605830 −0.302915 0.953018i \(-0.597960\pi\)
−0.302915 + 0.953018i \(0.597960\pi\)
\(3\) 0 0
\(4\) −1.26594 −0.632970
\(5\) 0 0
\(6\) 0 0
\(7\) 0.647907 0.244886 0.122443 0.992476i \(-0.460927\pi\)
0.122443 + 0.992476i \(0.460927\pi\)
\(8\) 2.79817 0.989302
\(9\) 0 0
\(10\) 0 0
\(11\) 5.91382 1.78308 0.891542 0.452939i \(-0.149624\pi\)
0.891542 + 0.452939i \(0.149624\pi\)
\(12\) 0 0
\(13\) 2.88397 0.799871 0.399935 0.916543i \(-0.369033\pi\)
0.399935 + 0.916543i \(0.369033\pi\)
\(14\) −0.555109 −0.148359
\(15\) 0 0
\(16\) 0.134488 0.0336219
\(17\) −4.20027 −1.01872 −0.509358 0.860555i \(-0.670117\pi\)
−0.509358 + 0.860555i \(0.670117\pi\)
\(18\) 0 0
\(19\) −1.64791 −0.378056 −0.189028 0.981972i \(-0.560534\pi\)
−0.189028 + 0.981972i \(0.560534\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −5.06680 −1.08024
\(23\) −6.42752 −1.34023 −0.670115 0.742257i \(-0.733755\pi\)
−0.670115 + 0.742257i \(0.733755\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.47091 −0.484585
\(27\) 0 0
\(28\) −0.820212 −0.155005
\(29\) −2.25888 −0.419463 −0.209732 0.977759i \(-0.567259\pi\)
−0.209732 + 0.977759i \(0.567259\pi\)
\(30\) 0 0
\(31\) −5.28440 −0.949107 −0.474553 0.880227i \(-0.657390\pi\)
−0.474553 + 0.880227i \(0.657390\pi\)
\(32\) −5.71156 −1.00967
\(33\) 0 0
\(34\) 3.59868 0.617168
\(35\) 0 0
\(36\) 0 0
\(37\) −1.71560 −0.282042 −0.141021 0.990007i \(-0.545039\pi\)
−0.141021 + 0.990007i \(0.545039\pi\)
\(38\) 1.41188 0.229037
\(39\) 0 0
\(40\) 0 0
\(41\) 0.176662 0.0275900 0.0137950 0.999905i \(-0.495609\pi\)
0.0137950 + 0.999905i \(0.495609\pi\)
\(42\) 0 0
\(43\) 7.95077 1.21248 0.606241 0.795281i \(-0.292677\pi\)
0.606241 + 0.795281i \(0.292677\pi\)
\(44\) −7.48654 −1.12864
\(45\) 0 0
\(46\) 5.50692 0.811951
\(47\) −1.05903 −0.154475 −0.0772376 0.997013i \(-0.524610\pi\)
−0.0772376 + 0.997013i \(0.524610\pi\)
\(48\) 0 0
\(49\) −6.58022 −0.940031
\(50\) 0 0
\(51\) 0 0
\(52\) −3.65094 −0.506294
\(53\) 4.48612 0.616216 0.308108 0.951351i \(-0.400304\pi\)
0.308108 + 0.951351i \(0.400304\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.81295 0.242266
\(57\) 0 0
\(58\) 1.93534 0.254123
\(59\) −9.74542 −1.26875 −0.634373 0.773027i \(-0.718742\pi\)
−0.634373 + 0.773027i \(0.718742\pi\)
\(60\) 0 0
\(61\) −11.2844 −1.44482 −0.722410 0.691465i \(-0.756966\pi\)
−0.722410 + 0.691465i \(0.756966\pi\)
\(62\) 4.52753 0.574997
\(63\) 0 0
\(64\) 4.62453 0.578067
\(65\) 0 0
\(66\) 0 0
\(67\) −12.6171 −1.54143 −0.770715 0.637181i \(-0.780101\pi\)
−0.770715 + 0.637181i \(0.780101\pi\)
\(68\) 5.31730 0.644817
\(69\) 0 0
\(70\) 0 0
\(71\) 6.09048 0.722807 0.361404 0.932409i \(-0.382298\pi\)
0.361404 + 0.932409i \(0.382298\pi\)
\(72\) 0 0
\(73\) −7.08312 −0.829016 −0.414508 0.910046i \(-0.636046\pi\)
−0.414508 + 0.910046i \(0.636046\pi\)
\(74\) 1.46988 0.170870
\(75\) 0 0
\(76\) 2.08615 0.239298
\(77\) 3.83160 0.436652
\(78\) 0 0
\(79\) 1.58111 0.177889 0.0889443 0.996037i \(-0.471651\pi\)
0.0889443 + 0.996037i \(0.471651\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −0.151359 −0.0167149
\(83\) −11.5102 −1.26340 −0.631702 0.775211i \(-0.717643\pi\)
−0.631702 + 0.775211i \(0.717643\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −6.81200 −0.734557
\(87\) 0 0
\(88\) 16.5479 1.76401
\(89\) 15.5918 1.65272 0.826362 0.563140i \(-0.190407\pi\)
0.826362 + 0.563140i \(0.190407\pi\)
\(90\) 0 0
\(91\) 1.86855 0.195877
\(92\) 8.13685 0.848326
\(93\) 0 0
\(94\) 0.907347 0.0935857
\(95\) 0 0
\(96\) 0 0
\(97\) −7.55034 −0.766621 −0.383311 0.923619i \(-0.625216\pi\)
−0.383311 + 0.923619i \(0.625216\pi\)
\(98\) 5.63775 0.569499
\(99\) 0 0
\(100\) 0 0
\(101\) 17.1645 1.70793 0.853965 0.520330i \(-0.174191\pi\)
0.853965 + 0.520330i \(0.174191\pi\)
\(102\) 0 0
\(103\) −17.3561 −1.71015 −0.855074 0.518506i \(-0.826489\pi\)
−0.855074 + 0.518506i \(0.826489\pi\)
\(104\) 8.06985 0.791314
\(105\) 0 0
\(106\) −3.84358 −0.373322
\(107\) 16.2046 1.56656 0.783278 0.621672i \(-0.213546\pi\)
0.783278 + 0.621672i \(0.213546\pi\)
\(108\) 0 0
\(109\) 7.12709 0.682652 0.341326 0.939945i \(-0.389124\pi\)
0.341326 + 0.939945i \(0.389124\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.0871355 0.00823353
\(113\) −14.9372 −1.40518 −0.702589 0.711596i \(-0.747973\pi\)
−0.702589 + 0.711596i \(0.747973\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2.85961 0.265508
\(117\) 0 0
\(118\) 8.34961 0.768644
\(119\) −2.72139 −0.249469
\(120\) 0 0
\(121\) 23.9733 2.17939
\(122\) 9.66817 0.875315
\(123\) 0 0
\(124\) 6.68974 0.600757
\(125\) 0 0
\(126\) 0 0
\(127\) −9.30722 −0.825883 −0.412941 0.910758i \(-0.635499\pi\)
−0.412941 + 0.910758i \(0.635499\pi\)
\(128\) 7.46095 0.659461
\(129\) 0 0
\(130\) 0 0
\(131\) 3.47828 0.303899 0.151949 0.988388i \(-0.451445\pi\)
0.151949 + 0.988388i \(0.451445\pi\)
\(132\) 0 0
\(133\) −1.06769 −0.0925805
\(134\) 10.8100 0.933844
\(135\) 0 0
\(136\) −11.7531 −1.00782
\(137\) −8.17270 −0.698241 −0.349120 0.937078i \(-0.613520\pi\)
−0.349120 + 0.937078i \(0.613520\pi\)
\(138\) 0 0
\(139\) 2.93970 0.249342 0.124671 0.992198i \(-0.460212\pi\)
0.124671 + 0.992198i \(0.460212\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.21816 −0.437898
\(143\) 17.0553 1.42624
\(144\) 0 0
\(145\) 0 0
\(146\) 6.06862 0.502243
\(147\) 0 0
\(148\) 2.17184 0.178524
\(149\) 15.4826 1.26838 0.634191 0.773176i \(-0.281333\pi\)
0.634191 + 0.773176i \(0.281333\pi\)
\(150\) 0 0
\(151\) 10.8233 0.880791 0.440395 0.897804i \(-0.354838\pi\)
0.440395 + 0.897804i \(0.354838\pi\)
\(152\) −4.61112 −0.374011
\(153\) 0 0
\(154\) −3.28281 −0.264537
\(155\) 0 0
\(156\) 0 0
\(157\) −16.9086 −1.34945 −0.674726 0.738068i \(-0.735738\pi\)
−0.674726 + 0.738068i \(0.735738\pi\)
\(158\) −1.35465 −0.107770
\(159\) 0 0
\(160\) 0 0
\(161\) −4.16443 −0.328203
\(162\) 0 0
\(163\) 16.7750 1.31392 0.656960 0.753926i \(-0.271842\pi\)
0.656960 + 0.753926i \(0.271842\pi\)
\(164\) −0.223644 −0.0174637
\(165\) 0 0
\(166\) 9.86159 0.765407
\(167\) 10.1916 0.788653 0.394326 0.918970i \(-0.370978\pi\)
0.394326 + 0.918970i \(0.370978\pi\)
\(168\) 0 0
\(169\) −4.68269 −0.360207
\(170\) 0 0
\(171\) 0 0
\(172\) −10.0652 −0.767465
\(173\) −14.3724 −1.09271 −0.546355 0.837554i \(-0.683985\pi\)
−0.546355 + 0.837554i \(0.683985\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0.795336 0.0599507
\(177\) 0 0
\(178\) −13.3586 −1.00127
\(179\) −7.59573 −0.567731 −0.283866 0.958864i \(-0.591617\pi\)
−0.283866 + 0.958864i \(0.591617\pi\)
\(180\) 0 0
\(181\) −21.8203 −1.62189 −0.810945 0.585122i \(-0.801047\pi\)
−0.810945 + 0.585122i \(0.801047\pi\)
\(182\) −1.60092 −0.118668
\(183\) 0 0
\(184\) −17.9853 −1.32589
\(185\) 0 0
\(186\) 0 0
\(187\) −24.8397 −1.81646
\(188\) 1.34067 0.0977783
\(189\) 0 0
\(190\) 0 0
\(191\) 19.0283 1.37684 0.688421 0.725311i \(-0.258304\pi\)
0.688421 + 0.725311i \(0.258304\pi\)
\(192\) 0 0
\(193\) 8.57675 0.617368 0.308684 0.951165i \(-0.400111\pi\)
0.308684 + 0.951165i \(0.400111\pi\)
\(194\) 6.46893 0.464442
\(195\) 0 0
\(196\) 8.33016 0.595012
\(197\) −18.5726 −1.32325 −0.661623 0.749837i \(-0.730132\pi\)
−0.661623 + 0.749837i \(0.730132\pi\)
\(198\) 0 0
\(199\) −7.32927 −0.519558 −0.259779 0.965668i \(-0.583650\pi\)
−0.259779 + 0.965668i \(0.583650\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −14.7061 −1.03471
\(203\) −1.46354 −0.102721
\(204\) 0 0
\(205\) 0 0
\(206\) 14.8702 1.03606
\(207\) 0 0
\(208\) 0.387859 0.0268932
\(209\) −9.74542 −0.674105
\(210\) 0 0
\(211\) −14.5553 −1.00203 −0.501013 0.865440i \(-0.667039\pi\)
−0.501013 + 0.865440i \(0.667039\pi\)
\(212\) −5.67916 −0.390046
\(213\) 0 0
\(214\) −13.8836 −0.949066
\(215\) 0 0
\(216\) 0 0
\(217\) −3.42380 −0.232423
\(218\) −6.10630 −0.413571
\(219\) 0 0
\(220\) 0 0
\(221\) −12.1135 −0.814841
\(222\) 0 0
\(223\) −26.8711 −1.79942 −0.899712 0.436485i \(-0.856223\pi\)
−0.899712 + 0.436485i \(0.856223\pi\)
\(224\) −3.70056 −0.247254
\(225\) 0 0
\(226\) 12.7978 0.851298
\(227\) 4.20027 0.278782 0.139391 0.990237i \(-0.455486\pi\)
0.139391 + 0.990237i \(0.455486\pi\)
\(228\) 0 0
\(229\) 8.30722 0.548957 0.274478 0.961593i \(-0.411495\pi\)
0.274478 + 0.961593i \(0.411495\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −6.32072 −0.414976
\(233\) 22.6158 1.48161 0.740805 0.671720i \(-0.234444\pi\)
0.740805 + 0.671720i \(0.234444\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 12.3371 0.803079
\(237\) 0 0
\(238\) 2.33161 0.151136
\(239\) −1.28688 −0.0832413 −0.0416207 0.999133i \(-0.513252\pi\)
−0.0416207 + 0.999133i \(0.513252\pi\)
\(240\) 0 0
\(241\) −4.91599 −0.316667 −0.158333 0.987386i \(-0.550612\pi\)
−0.158333 + 0.987386i \(0.550612\pi\)
\(242\) −20.5396 −1.32034
\(243\) 0 0
\(244\) 14.2854 0.914528
\(245\) 0 0
\(246\) 0 0
\(247\) −4.75252 −0.302396
\(248\) −14.7867 −0.938953
\(249\) 0 0
\(250\) 0 0
\(251\) −18.9609 −1.19680 −0.598399 0.801198i \(-0.704196\pi\)
−0.598399 + 0.801198i \(0.704196\pi\)
\(252\) 0 0
\(253\) −38.0112 −2.38974
\(254\) 7.97417 0.500344
\(255\) 0 0
\(256\) −15.6414 −0.977588
\(257\) −24.1690 −1.50762 −0.753810 0.657093i \(-0.771786\pi\)
−0.753810 + 0.657093i \(0.771786\pi\)
\(258\) 0 0
\(259\) −1.11155 −0.0690682
\(260\) 0 0
\(261\) 0 0
\(262\) −2.98009 −0.184111
\(263\) 17.6518 1.08846 0.544229 0.838937i \(-0.316822\pi\)
0.544229 + 0.838937i \(0.316822\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0.914768 0.0560880
\(267\) 0 0
\(268\) 15.9726 0.975679
\(269\) −26.5149 −1.61664 −0.808320 0.588743i \(-0.799623\pi\)
−0.808320 + 0.588743i \(0.799623\pi\)
\(270\) 0 0
\(271\) 20.4180 1.24031 0.620153 0.784481i \(-0.287071\pi\)
0.620153 + 0.784481i \(0.287071\pi\)
\(272\) −0.564885 −0.0342512
\(273\) 0 0
\(274\) 7.00214 0.423015
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0643 0.845043 0.422521 0.906353i \(-0.361145\pi\)
0.422521 + 0.906353i \(0.361145\pi\)
\(278\) −2.51866 −0.151059
\(279\) 0 0
\(280\) 0 0
\(281\) −6.19966 −0.369841 −0.184920 0.982753i \(-0.559203\pi\)
−0.184920 + 0.982753i \(0.559203\pi\)
\(282\) 0 0
\(283\) −12.9801 −0.771586 −0.385793 0.922585i \(-0.626072\pi\)
−0.385793 + 0.922585i \(0.626072\pi\)
\(284\) −7.71019 −0.457516
\(285\) 0 0
\(286\) −14.6125 −0.864056
\(287\) 0.114461 0.00675640
\(288\) 0 0
\(289\) 0.642299 0.0377823
\(290\) 0 0
\(291\) 0 0
\(292\) 8.96681 0.524743
\(293\) −18.5563 −1.08407 −0.542037 0.840355i \(-0.682347\pi\)
−0.542037 + 0.840355i \(0.682347\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4.80053 −0.279025
\(297\) 0 0
\(298\) −13.2650 −0.768424
\(299\) −18.5368 −1.07201
\(300\) 0 0
\(301\) 5.15136 0.296919
\(302\) −9.27314 −0.533609
\(303\) 0 0
\(304\) −0.221623 −0.0127110
\(305\) 0 0
\(306\) 0 0
\(307\) −24.0862 −1.37467 −0.687337 0.726338i \(-0.741221\pi\)
−0.687337 + 0.726338i \(0.741221\pi\)
\(308\) −4.85058 −0.276388
\(309\) 0 0
\(310\) 0 0
\(311\) 25.3372 1.43674 0.718370 0.695661i \(-0.244889\pi\)
0.718370 + 0.695661i \(0.244889\pi\)
\(312\) 0 0
\(313\) 21.8224 1.23348 0.616739 0.787168i \(-0.288454\pi\)
0.616739 + 0.787168i \(0.288454\pi\)
\(314\) 14.4868 0.817539
\(315\) 0 0
\(316\) −2.00159 −0.112598
\(317\) −1.21940 −0.0684884 −0.0342442 0.999413i \(-0.510902\pi\)
−0.0342442 + 0.999413i \(0.510902\pi\)
\(318\) 0 0
\(319\) −13.3586 −0.747938
\(320\) 0 0
\(321\) 0 0
\(322\) 3.56797 0.198835
\(323\) 6.92166 0.385131
\(324\) 0 0
\(325\) 0 0
\(326\) −14.3724 −0.796011
\(327\) 0 0
\(328\) 0.494331 0.0272949
\(329\) −0.686152 −0.0378288
\(330\) 0 0
\(331\) −5.97470 −0.328399 −0.164200 0.986427i \(-0.552504\pi\)
−0.164200 + 0.986427i \(0.552504\pi\)
\(332\) 14.5712 0.799697
\(333\) 0 0
\(334\) −8.73192 −0.477789
\(335\) 0 0
\(336\) 0 0
\(337\) −27.1718 −1.48014 −0.740072 0.672527i \(-0.765209\pi\)
−0.740072 + 0.672527i \(0.765209\pi\)
\(338\) 4.01200 0.218224
\(339\) 0 0
\(340\) 0 0
\(341\) −31.2510 −1.69234
\(342\) 0 0
\(343\) −8.79871 −0.475086
\(344\) 22.2476 1.19951
\(345\) 0 0
\(346\) 12.3138 0.661996
\(347\) 12.4242 0.666964 0.333482 0.942757i \(-0.391776\pi\)
0.333482 + 0.942757i \(0.391776\pi\)
\(348\) 0 0
\(349\) 20.2826 1.08570 0.542852 0.839828i \(-0.317345\pi\)
0.542852 + 0.839828i \(0.317345\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −33.7771 −1.80033
\(353\) 11.9880 0.638057 0.319029 0.947745i \(-0.396643\pi\)
0.319029 + 0.947745i \(0.396643\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −19.7382 −1.04613
\(357\) 0 0
\(358\) 6.50781 0.343949
\(359\) 3.54576 0.187138 0.0935690 0.995613i \(-0.470172\pi\)
0.0935690 + 0.995613i \(0.470172\pi\)
\(360\) 0 0
\(361\) −16.2844 −0.857074
\(362\) 18.6950 0.982589
\(363\) 0 0
\(364\) −2.36547 −0.123984
\(365\) 0 0
\(366\) 0 0
\(367\) −31.3333 −1.63558 −0.817792 0.575514i \(-0.804802\pi\)
−0.817792 + 0.575514i \(0.804802\pi\)
\(368\) −0.864421 −0.0450611
\(369\) 0 0
\(370\) 0 0
\(371\) 2.90659 0.150902
\(372\) 0 0
\(373\) 0.133595 0.00691730 0.00345865 0.999994i \(-0.498899\pi\)
0.00345865 + 0.999994i \(0.498899\pi\)
\(374\) 21.2819 1.10046
\(375\) 0 0
\(376\) −2.96334 −0.152823
\(377\) −6.51455 −0.335516
\(378\) 0 0
\(379\) −8.04343 −0.413163 −0.206582 0.978429i \(-0.566234\pi\)
−0.206582 + 0.978429i \(0.566234\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −16.3030 −0.834132
\(383\) −12.2322 −0.625034 −0.312517 0.949912i \(-0.601172\pi\)
−0.312517 + 0.949912i \(0.601172\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −7.34832 −0.374020
\(387\) 0 0
\(388\) 9.55829 0.485249
\(389\) −33.0004 −1.67319 −0.836593 0.547825i \(-0.815456\pi\)
−0.836593 + 0.547825i \(0.815456\pi\)
\(390\) 0 0
\(391\) 26.9973 1.36531
\(392\) −18.4126 −0.929974
\(393\) 0 0
\(394\) 15.9125 0.801661
\(395\) 0 0
\(396\) 0 0
\(397\) 13.9249 0.698872 0.349436 0.936960i \(-0.386373\pi\)
0.349436 + 0.936960i \(0.386373\pi\)
\(398\) 6.27952 0.314764
\(399\) 0 0
\(400\) 0 0
\(401\) 27.9494 1.39573 0.697863 0.716231i \(-0.254134\pi\)
0.697863 + 0.716231i \(0.254134\pi\)
\(402\) 0 0
\(403\) −15.2401 −0.759163
\(404\) −21.7292 −1.08107
\(405\) 0 0
\(406\) 1.25392 0.0622311
\(407\) −10.1457 −0.502905
\(408\) 0 0
\(409\) 25.9289 1.28210 0.641052 0.767498i \(-0.278498\pi\)
0.641052 + 0.767498i \(0.278498\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 21.9718 1.08247
\(413\) −6.31413 −0.310698
\(414\) 0 0
\(415\) 0 0
\(416\) −16.4720 −0.807606
\(417\) 0 0
\(418\) 8.34961 0.408393
\(419\) 1.21614 0.0594123 0.0297061 0.999559i \(-0.490543\pi\)
0.0297061 + 0.999559i \(0.490543\pi\)
\(420\) 0 0
\(421\) 20.4626 0.997286 0.498643 0.866807i \(-0.333832\pi\)
0.498643 + 0.866807i \(0.333832\pi\)
\(422\) 12.4705 0.607056
\(423\) 0 0
\(424\) 12.5529 0.609624
\(425\) 0 0
\(426\) 0 0
\(427\) −7.31124 −0.353816
\(428\) −20.5140 −0.991583
\(429\) 0 0
\(430\) 0 0
\(431\) −21.4381 −1.03264 −0.516319 0.856397i \(-0.672698\pi\)
−0.516319 + 0.856397i \(0.672698\pi\)
\(432\) 0 0
\(433\) −23.2844 −1.11898 −0.559489 0.828838i \(-0.689002\pi\)
−0.559489 + 0.828838i \(0.689002\pi\)
\(434\) 2.93342 0.140809
\(435\) 0 0
\(436\) −9.02248 −0.432098
\(437\) 10.5919 0.506681
\(438\) 0 0
\(439\) −12.5355 −0.598285 −0.299143 0.954208i \(-0.596701\pi\)
−0.299143 + 0.954208i \(0.596701\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 10.3785 0.493655
\(443\) 13.3228 0.632986 0.316493 0.948595i \(-0.397495\pi\)
0.316493 + 0.948595i \(0.397495\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 23.0224 1.09014
\(447\) 0 0
\(448\) 2.99627 0.141560
\(449\) −30.2500 −1.42758 −0.713792 0.700358i \(-0.753024\pi\)
−0.713792 + 0.700358i \(0.753024\pi\)
\(450\) 0 0
\(451\) 1.04475 0.0491953
\(452\) 18.9097 0.889436
\(453\) 0 0
\(454\) −3.59868 −0.168894
\(455\) 0 0
\(456\) 0 0
\(457\) 12.8391 0.600588 0.300294 0.953847i \(-0.402915\pi\)
0.300294 + 0.953847i \(0.402915\pi\)
\(458\) −7.11740 −0.332574
\(459\) 0 0
\(460\) 0 0
\(461\) −27.8435 −1.29680 −0.648400 0.761300i \(-0.724561\pi\)
−0.648400 + 0.761300i \(0.724561\pi\)
\(462\) 0 0
\(463\) −3.84616 −0.178746 −0.0893731 0.995998i \(-0.528486\pi\)
−0.0893731 + 0.995998i \(0.528486\pi\)
\(464\) −0.303791 −0.0141031
\(465\) 0 0
\(466\) −19.3766 −0.897603
\(467\) 40.6594 1.88149 0.940746 0.339112i \(-0.110126\pi\)
0.940746 + 0.339112i \(0.110126\pi\)
\(468\) 0 0
\(469\) −8.17473 −0.377474
\(470\) 0 0
\(471\) 0 0
\(472\) −27.2693 −1.25517
\(473\) 47.0194 2.16196
\(474\) 0 0
\(475\) 0 0
\(476\) 3.44511 0.157907
\(477\) 0 0
\(478\) 1.10256 0.0504301
\(479\) 15.2642 0.697440 0.348720 0.937227i \(-0.386616\pi\)
0.348720 + 0.937227i \(0.386616\pi\)
\(480\) 0 0
\(481\) −4.94774 −0.225597
\(482\) 4.21189 0.191846
\(483\) 0 0
\(484\) −30.3487 −1.37949
\(485\) 0 0
\(486\) 0 0
\(487\) −8.06466 −0.365444 −0.182722 0.983165i \(-0.558491\pi\)
−0.182722 + 0.983165i \(0.558491\pi\)
\(488\) −31.5757 −1.42936
\(489\) 0 0
\(490\) 0 0
\(491\) 9.50128 0.428787 0.214393 0.976747i \(-0.431223\pi\)
0.214393 + 0.976747i \(0.431223\pi\)
\(492\) 0 0
\(493\) 9.48790 0.427314
\(494\) 4.07183 0.183200
\(495\) 0 0
\(496\) −0.710687 −0.0319108
\(497\) 3.94606 0.177005
\(498\) 0 0
\(499\) −19.6405 −0.879230 −0.439615 0.898186i \(-0.644885\pi\)
−0.439615 + 0.898186i \(0.644885\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 16.2451 0.725056
\(503\) −20.6428 −0.920415 −0.460208 0.887811i \(-0.652225\pi\)
−0.460208 + 0.887811i \(0.652225\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 32.5669 1.44778
\(507\) 0 0
\(508\) 11.7824 0.522759
\(509\) 31.9372 1.41559 0.707795 0.706418i \(-0.249690\pi\)
0.707795 + 0.706418i \(0.249690\pi\)
\(510\) 0 0
\(511\) −4.58920 −0.203014
\(512\) −1.52077 −0.0672093
\(513\) 0 0
\(514\) 20.7073 0.913360
\(515\) 0 0
\(516\) 0 0
\(517\) −6.26291 −0.275442
\(518\) 0.952343 0.0418435
\(519\) 0 0
\(520\) 0 0
\(521\) 8.88261 0.389154 0.194577 0.980887i \(-0.437667\pi\)
0.194577 + 0.980887i \(0.437667\pi\)
\(522\) 0 0
\(523\) −28.3512 −1.23971 −0.619856 0.784716i \(-0.712809\pi\)
−0.619856 + 0.784716i \(0.712809\pi\)
\(524\) −4.40329 −0.192359
\(525\) 0 0
\(526\) −15.1236 −0.659420
\(527\) 22.1959 0.966870
\(528\) 0 0
\(529\) 18.3130 0.796215
\(530\) 0 0
\(531\) 0 0
\(532\) 1.35163 0.0586007
\(533\) 0.509490 0.0220685
\(534\) 0 0
\(535\) 0 0
\(536\) −35.3049 −1.52494
\(537\) 0 0
\(538\) 22.7172 0.979409
\(539\) −38.9142 −1.67615
\(540\) 0 0
\(541\) −13.8190 −0.594124 −0.297062 0.954858i \(-0.596007\pi\)
−0.297062 + 0.954858i \(0.596007\pi\)
\(542\) −17.4936 −0.751414
\(543\) 0 0
\(544\) 23.9901 1.02857
\(545\) 0 0
\(546\) 0 0
\(547\) 12.0831 0.516637 0.258318 0.966060i \(-0.416832\pi\)
0.258318 + 0.966060i \(0.416832\pi\)
\(548\) 10.3461 0.441966
\(549\) 0 0
\(550\) 0 0
\(551\) 3.72242 0.158580
\(552\) 0 0
\(553\) 1.02441 0.0435624
\(554\) −12.0499 −0.511952
\(555\) 0 0
\(556\) −3.72149 −0.157826
\(557\) 15.5918 0.660644 0.330322 0.943868i \(-0.392843\pi\)
0.330322 + 0.943868i \(0.392843\pi\)
\(558\) 0 0
\(559\) 22.9298 0.969828
\(560\) 0 0
\(561\) 0 0
\(562\) 5.31170 0.224061
\(563\) 19.1854 0.808570 0.404285 0.914633i \(-0.367520\pi\)
0.404285 + 0.914633i \(0.367520\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 11.1210 0.467450
\(567\) 0 0
\(568\) 17.0422 0.715074
\(569\) 30.1698 1.26478 0.632392 0.774648i \(-0.282073\pi\)
0.632392 + 0.774648i \(0.282073\pi\)
\(570\) 0 0
\(571\) −25.2616 −1.05716 −0.528582 0.848882i \(-0.677276\pi\)
−0.528582 + 0.848882i \(0.677276\pi\)
\(572\) −21.5910 −0.902765
\(573\) 0 0
\(574\) −0.0980668 −0.00409323
\(575\) 0 0
\(576\) 0 0
\(577\) 9.00948 0.375070 0.187535 0.982258i \(-0.439950\pi\)
0.187535 + 0.982258i \(0.439950\pi\)
\(578\) −0.550304 −0.0228896
\(579\) 0 0
\(580\) 0 0
\(581\) −7.45751 −0.309390
\(582\) 0 0
\(583\) 26.5301 1.09876
\(584\) −19.8198 −0.820147
\(585\) 0 0
\(586\) 15.8986 0.656764
\(587\) −11.5059 −0.474901 −0.237451 0.971400i \(-0.576312\pi\)
−0.237451 + 0.971400i \(0.576312\pi\)
\(588\) 0 0
\(589\) 8.70820 0.358815
\(590\) 0 0
\(591\) 0 0
\(592\) −0.230727 −0.00948280
\(593\) −23.0392 −0.946107 −0.473053 0.881034i \(-0.656848\pi\)
−0.473053 + 0.881034i \(0.656848\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −19.6000 −0.802848
\(597\) 0 0
\(598\) 15.8818 0.649456
\(599\) −2.07694 −0.0848614 −0.0424307 0.999099i \(-0.513510\pi\)
−0.0424307 + 0.999099i \(0.513510\pi\)
\(600\) 0 0
\(601\) 28.7922 1.17446 0.587230 0.809420i \(-0.300218\pi\)
0.587230 + 0.809420i \(0.300218\pi\)
\(602\) −4.41354 −0.179883
\(603\) 0 0
\(604\) −13.7017 −0.557514
\(605\) 0 0
\(606\) 0 0
\(607\) 6.99573 0.283948 0.141974 0.989870i \(-0.454655\pi\)
0.141974 + 0.989870i \(0.454655\pi\)
\(608\) 9.41212 0.381712
\(609\) 0 0
\(610\) 0 0
\(611\) −3.05421 −0.123560
\(612\) 0 0
\(613\) −34.6552 −1.39971 −0.699855 0.714285i \(-0.746752\pi\)
−0.699855 + 0.714285i \(0.746752\pi\)
\(614\) 20.6364 0.832819
\(615\) 0 0
\(616\) 10.7215 0.431980
\(617\) 14.9852 0.603280 0.301640 0.953422i \(-0.402466\pi\)
0.301640 + 0.953422i \(0.402466\pi\)
\(618\) 0 0
\(619\) −39.8216 −1.60056 −0.800282 0.599624i \(-0.795317\pi\)
−0.800282 + 0.599624i \(0.795317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −21.7082 −0.870420
\(623\) 10.1020 0.404728
\(624\) 0 0
\(625\) 0 0
\(626\) −18.6969 −0.747277
\(627\) 0 0
\(628\) 21.4053 0.854164
\(629\) 7.20598 0.287321
\(630\) 0 0
\(631\) 1.66278 0.0661944 0.0330972 0.999452i \(-0.489463\pi\)
0.0330972 + 0.999452i \(0.489463\pi\)
\(632\) 4.42421 0.175986
\(633\) 0 0
\(634\) 1.04475 0.0414923
\(635\) 0 0
\(636\) 0 0
\(637\) −18.9772 −0.751903
\(638\) 11.4453 0.453123
\(639\) 0 0
\(640\) 0 0
\(641\) −6.77663 −0.267661 −0.133830 0.991004i \(-0.542728\pi\)
−0.133830 + 0.991004i \(0.542728\pi\)
\(642\) 0 0
\(643\) −20.2108 −0.797035 −0.398517 0.917161i \(-0.630475\pi\)
−0.398517 + 0.917161i \(0.630475\pi\)
\(644\) 5.27192 0.207743
\(645\) 0 0
\(646\) −5.93029 −0.233324
\(647\) −5.05100 −0.198575 −0.0992877 0.995059i \(-0.531656\pi\)
−0.0992877 + 0.995059i \(0.531656\pi\)
\(648\) 0 0
\(649\) −57.6327 −2.26228
\(650\) 0 0
\(651\) 0 0
\(652\) −21.2362 −0.831672
\(653\) 9.64210 0.377325 0.188662 0.982042i \(-0.439585\pi\)
0.188662 + 0.982042i \(0.439585\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.0237589 0.000927629 0
\(657\) 0 0
\(658\) 0.587876 0.0229178
\(659\) 26.7332 1.04138 0.520690 0.853746i \(-0.325675\pi\)
0.520690 + 0.853746i \(0.325675\pi\)
\(660\) 0 0
\(661\) −2.00159 −0.0778529 −0.0389264 0.999242i \(-0.512394\pi\)
−0.0389264 + 0.999242i \(0.512394\pi\)
\(662\) 5.11896 0.198954
\(663\) 0 0
\(664\) −32.2074 −1.24989
\(665\) 0 0
\(666\) 0 0
\(667\) 14.5190 0.562177
\(668\) −12.9020 −0.499194
\(669\) 0 0
\(670\) 0 0
\(671\) −66.7339 −2.57623
\(672\) 0 0
\(673\) 31.2362 1.20407 0.602033 0.798471i \(-0.294358\pi\)
0.602033 + 0.798471i \(0.294358\pi\)
\(674\) 23.2801 0.896716
\(675\) 0 0
\(676\) 5.92801 0.228000
\(677\) −42.4874 −1.63292 −0.816462 0.577400i \(-0.804067\pi\)
−0.816462 + 0.577400i \(0.804067\pi\)
\(678\) 0 0
\(679\) −4.89192 −0.187735
\(680\) 0 0
\(681\) 0 0
\(682\) 26.7750 1.02527
\(683\) 5.20811 0.199283 0.0996415 0.995023i \(-0.468230\pi\)
0.0996415 + 0.995023i \(0.468230\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 7.53850 0.287821
\(687\) 0 0
\(688\) 1.06928 0.0407659
\(689\) 12.9379 0.492893
\(690\) 0 0
\(691\) 8.07419 0.307157 0.153578 0.988136i \(-0.450920\pi\)
0.153578 + 0.988136i \(0.450920\pi\)
\(692\) 18.1946 0.691653
\(693\) 0 0
\(694\) −10.6447 −0.404066
\(695\) 0 0
\(696\) 0 0
\(697\) −0.742030 −0.0281064
\(698\) −17.3776 −0.657752
\(699\) 0 0
\(700\) 0 0
\(701\) −31.7552 −1.19938 −0.599689 0.800233i \(-0.704709\pi\)
−0.599689 + 0.800233i \(0.704709\pi\)
\(702\) 0 0
\(703\) 2.82714 0.106628
\(704\) 27.3487 1.03074
\(705\) 0 0
\(706\) −10.2710 −0.386554
\(707\) 11.1210 0.418248
\(708\) 0 0
\(709\) 9.71630 0.364903 0.182452 0.983215i \(-0.441597\pi\)
0.182452 + 0.983215i \(0.441597\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 43.6284 1.63504
\(713\) 33.9656 1.27202
\(714\) 0 0
\(715\) 0 0
\(716\) 9.61574 0.359357
\(717\) 0 0
\(718\) −3.03791 −0.113374
\(719\) 20.4661 0.763257 0.381628 0.924316i \(-0.375363\pi\)
0.381628 + 0.924316i \(0.375363\pi\)
\(720\) 0 0
\(721\) −11.2451 −0.418791
\(722\) 13.9520 0.519241
\(723\) 0 0
\(724\) 27.6232 1.02661
\(725\) 0 0
\(726\) 0 0
\(727\) 50.2337 1.86306 0.931532 0.363659i \(-0.118473\pi\)
0.931532 + 0.363659i \(0.118473\pi\)
\(728\) 5.22851 0.193781
\(729\) 0 0
\(730\) 0 0
\(731\) −33.3954 −1.23517
\(732\) 0 0
\(733\) −37.8776 −1.39904 −0.699520 0.714613i \(-0.746603\pi\)
−0.699520 + 0.714613i \(0.746603\pi\)
\(734\) 26.8455 0.990886
\(735\) 0 0
\(736\) 36.7112 1.35319
\(737\) −74.6155 −2.74850
\(738\) 0 0
\(739\) 44.9709 1.65428 0.827141 0.561995i \(-0.189966\pi\)
0.827141 + 0.561995i \(0.189966\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −2.49028 −0.0914212
\(743\) −29.4546 −1.08059 −0.540293 0.841477i \(-0.681687\pi\)
−0.540293 + 0.841477i \(0.681687\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.114461 −0.00419071
\(747\) 0 0
\(748\) 31.4455 1.14976
\(749\) 10.4991 0.383627
\(750\) 0 0
\(751\) −34.1341 −1.24557 −0.622786 0.782392i \(-0.713999\pi\)
−0.622786 + 0.782392i \(0.713999\pi\)
\(752\) −0.142426 −0.00519375
\(753\) 0 0
\(754\) 5.58148 0.203266
\(755\) 0 0
\(756\) 0 0
\(757\) 15.2875 0.555635 0.277817 0.960634i \(-0.410389\pi\)
0.277817 + 0.960634i \(0.410389\pi\)
\(758\) 6.89139 0.250306
\(759\) 0 0
\(760\) 0 0
\(761\) 24.0119 0.870429 0.435215 0.900327i \(-0.356672\pi\)
0.435215 + 0.900327i \(0.356672\pi\)
\(762\) 0 0
\(763\) 4.61769 0.167172
\(764\) −24.0887 −0.871500
\(765\) 0 0
\(766\) 10.4802 0.378664
\(767\) −28.1056 −1.01483
\(768\) 0 0
\(769\) −0.656954 −0.0236904 −0.0118452 0.999930i \(-0.503771\pi\)
−0.0118452 + 0.999930i \(0.503771\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −10.8577 −0.390776
\(773\) −28.1609 −1.01288 −0.506439 0.862276i \(-0.669039\pi\)
−0.506439 + 0.862276i \(0.669039\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −21.1271 −0.758420
\(777\) 0 0
\(778\) 28.2738 1.01367
\(779\) −0.291123 −0.0104306
\(780\) 0 0
\(781\) 36.0180 1.28883
\(782\) −23.1306 −0.827147
\(783\) 0 0
\(784\) −0.884958 −0.0316056
\(785\) 0 0
\(786\) 0 0
\(787\) −7.66560 −0.273249 −0.136625 0.990623i \(-0.543625\pi\)
−0.136625 + 0.990623i \(0.543625\pi\)
\(788\) 23.5119 0.837575
\(789\) 0 0
\(790\) 0 0
\(791\) −9.67794 −0.344108
\(792\) 0 0
\(793\) −32.5439 −1.15567
\(794\) −11.9305 −0.423397
\(795\) 0 0
\(796\) 9.27842 0.328865
\(797\) −34.7609 −1.23129 −0.615647 0.788022i \(-0.711105\pi\)
−0.615647 + 0.788022i \(0.711105\pi\)
\(798\) 0 0
\(799\) 4.44821 0.157366
\(800\) 0 0
\(801\) 0 0
\(802\) −23.9463 −0.845572
\(803\) −41.8883 −1.47821
\(804\) 0 0
\(805\) 0 0
\(806\) 13.0573 0.459923
\(807\) 0 0
\(808\) 48.0291 1.68966
\(809\) −42.5017 −1.49428 −0.747140 0.664667i \(-0.768573\pi\)
−0.747140 + 0.664667i \(0.768573\pi\)
\(810\) 0 0
\(811\) 39.2810 1.37934 0.689672 0.724122i \(-0.257755\pi\)
0.689672 + 0.724122i \(0.257755\pi\)
\(812\) 1.85276 0.0650191
\(813\) 0 0
\(814\) 8.69258 0.304675
\(815\) 0 0
\(816\) 0 0
\(817\) −13.1021 −0.458386
\(818\) −22.2152 −0.776736
\(819\) 0 0
\(820\) 0 0
\(821\) −0.709911 −0.0247761 −0.0123880 0.999923i \(-0.503943\pi\)
−0.0123880 + 0.999923i \(0.503943\pi\)
\(822\) 0 0
\(823\) −10.6260 −0.370398 −0.185199 0.982701i \(-0.559293\pi\)
−0.185199 + 0.982701i \(0.559293\pi\)
\(824\) −48.5653 −1.69185
\(825\) 0 0
\(826\) 5.40977 0.188230
\(827\) −11.0597 −0.384585 −0.192292 0.981338i \(-0.561592\pi\)
−0.192292 + 0.981338i \(0.561592\pi\)
\(828\) 0 0
\(829\) −1.64791 −0.0572342 −0.0286171 0.999590i \(-0.509110\pi\)
−0.0286171 + 0.999590i \(0.509110\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 13.3370 0.462379
\(833\) 27.6387 0.957625
\(834\) 0 0
\(835\) 0 0
\(836\) 12.3371 0.426688
\(837\) 0 0
\(838\) −1.04195 −0.0359937
\(839\) 23.8789 0.824392 0.412196 0.911095i \(-0.364762\pi\)
0.412196 + 0.911095i \(0.364762\pi\)
\(840\) 0 0
\(841\) −23.8975 −0.824051
\(842\) −17.5318 −0.604186
\(843\) 0 0
\(844\) 18.4261 0.634252
\(845\) 0 0
\(846\) 0 0
\(847\) 15.5324 0.533701
\(848\) 0.603328 0.0207184
\(849\) 0 0
\(850\) 0 0
\(851\) 11.0270 0.378002
\(852\) 0 0
\(853\) 23.3570 0.799729 0.399864 0.916574i \(-0.369057\pi\)
0.399864 + 0.916574i \(0.369057\pi\)
\(854\) 6.26407 0.214352
\(855\) 0 0
\(856\) 45.3431 1.54980
\(857\) 2.70183 0.0922929 0.0461464 0.998935i \(-0.485306\pi\)
0.0461464 + 0.998935i \(0.485306\pi\)
\(858\) 0 0
\(859\) 23.5781 0.804474 0.402237 0.915536i \(-0.368233\pi\)
0.402237 + 0.915536i \(0.368233\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.3676 0.625602
\(863\) −26.2965 −0.895144 −0.447572 0.894248i \(-0.647711\pi\)
−0.447572 + 0.894248i \(0.647711\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 19.9494 0.677909
\(867\) 0 0
\(868\) 4.33433 0.147117
\(869\) 9.35039 0.317190
\(870\) 0 0
\(871\) −36.3875 −1.23294
\(872\) 19.9428 0.675349
\(873\) 0 0
\(874\) −9.07489 −0.306963
\(875\) 0 0
\(876\) 0 0
\(877\) 33.4164 1.12839 0.564196 0.825641i \(-0.309186\pi\)
0.564196 + 0.825641i \(0.309186\pi\)
\(878\) 10.7400 0.362459
\(879\) 0 0
\(880\) 0 0
\(881\) 21.0398 0.708849 0.354425 0.935085i \(-0.384677\pi\)
0.354425 + 0.935085i \(0.384677\pi\)
\(882\) 0 0
\(883\) 14.5435 0.489428 0.244714 0.969595i \(-0.421306\pi\)
0.244714 + 0.969595i \(0.421306\pi\)
\(884\) 15.3350 0.515770
\(885\) 0 0
\(886\) −11.4146 −0.383482
\(887\) 55.3525 1.85855 0.929277 0.369382i \(-0.120431\pi\)
0.929277 + 0.369382i \(0.120431\pi\)
\(888\) 0 0
\(889\) −6.03021 −0.202247
\(890\) 0 0
\(891\) 0 0
\(892\) 34.0172 1.13898
\(893\) 1.74518 0.0584003
\(894\) 0 0
\(895\) 0 0
\(896\) 4.83400 0.161493
\(897\) 0 0
\(898\) 25.9173 0.864873
\(899\) 11.9368 0.398115
\(900\) 0 0
\(901\) −18.8429 −0.627749
\(902\) −0.895112 −0.0298040
\(903\) 0 0
\(904\) −41.7969 −1.39015
\(905\) 0 0
\(906\) 0 0
\(907\) 25.9003 0.860006 0.430003 0.902828i \(-0.358513\pi\)
0.430003 + 0.902828i \(0.358513\pi\)
\(908\) −5.31730 −0.176461
\(909\) 0 0
\(910\) 0 0
\(911\) −10.6499 −0.352848 −0.176424 0.984314i \(-0.556453\pi\)
−0.176424 + 0.984314i \(0.556453\pi\)
\(912\) 0 0
\(913\) −68.0690 −2.25275
\(914\) −11.0002 −0.363854
\(915\) 0 0
\(916\) −10.5165 −0.347473
\(917\) 2.25360 0.0744204
\(918\) 0 0
\(919\) 11.1958 0.369314 0.184657 0.982803i \(-0.440883\pi\)
0.184657 + 0.982803i \(0.440883\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 23.8555 0.785640
\(923\) 17.5648 0.578152
\(924\) 0 0
\(925\) 0 0
\(926\) 3.29528 0.108290
\(927\) 0 0
\(928\) 12.9017 0.423520
\(929\) −58.0987 −1.90616 −0.953078 0.302723i \(-0.902104\pi\)
−0.953078 + 0.302723i \(0.902104\pi\)
\(930\) 0 0
\(931\) 10.8436 0.355384
\(932\) −28.6303 −0.937815
\(933\) 0 0
\(934\) −34.8359 −1.13986
\(935\) 0 0
\(936\) 0 0
\(937\) 40.7577 1.33150 0.665749 0.746176i \(-0.268112\pi\)
0.665749 + 0.746176i \(0.268112\pi\)
\(938\) 7.00389 0.228685
\(939\) 0 0
\(940\) 0 0
\(941\) 2.47724 0.0807559 0.0403779 0.999184i \(-0.487144\pi\)
0.0403779 + 0.999184i \(0.487144\pi\)
\(942\) 0 0
\(943\) −1.13550 −0.0369770
\(944\) −1.31064 −0.0426577
\(945\) 0 0
\(946\) −40.2850 −1.30978
\(947\) −18.3448 −0.596125 −0.298063 0.954546i \(-0.596340\pi\)
−0.298063 + 0.954546i \(0.596340\pi\)
\(948\) 0 0
\(949\) −20.4275 −0.663106
\(950\) 0 0
\(951\) 0 0
\(952\) −7.61490 −0.246800
\(953\) −13.8466 −0.448535 −0.224267 0.974528i \(-0.571999\pi\)
−0.224267 + 0.974528i \(0.571999\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 1.62911 0.0526893
\(957\) 0 0
\(958\) −13.0780 −0.422530
\(959\) −5.29515 −0.170989
\(960\) 0 0
\(961\) −3.07508 −0.0991962
\(962\) 4.23909 0.136674
\(963\) 0 0
\(964\) 6.22335 0.200441
\(965\) 0 0
\(966\) 0 0
\(967\) −7.61678 −0.244939 −0.122470 0.992472i \(-0.539081\pi\)
−0.122470 + 0.992472i \(0.539081\pi\)
\(968\) 67.0812 2.15607
\(969\) 0 0
\(970\) 0 0
\(971\) 40.9964 1.31564 0.657819 0.753176i \(-0.271479\pi\)
0.657819 + 0.753176i \(0.271479\pi\)
\(972\) 0 0
\(973\) 1.90465 0.0610604
\(974\) 6.90958 0.221397
\(975\) 0 0
\(976\) −1.51761 −0.0485776
\(977\) 57.8650 1.85127 0.925633 0.378423i \(-0.123534\pi\)
0.925633 + 0.378423i \(0.123534\pi\)
\(978\) 0 0
\(979\) 92.2069 2.94694
\(980\) 0 0
\(981\) 0 0
\(982\) −8.14044 −0.259772
\(983\) 8.89795 0.283801 0.141900 0.989881i \(-0.454679\pi\)
0.141900 + 0.989881i \(0.454679\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −8.12898 −0.258879
\(987\) 0 0
\(988\) 6.01641 0.191408
\(989\) −51.1037 −1.62500
\(990\) 0 0
\(991\) −7.01945 −0.222980 −0.111490 0.993766i \(-0.535562\pi\)
−0.111490 + 0.993766i \(0.535562\pi\)
\(992\) 30.1822 0.958286
\(993\) 0 0
\(994\) −3.38088 −0.107235
\(995\) 0 0
\(996\) 0 0
\(997\) 49.6542 1.57256 0.786281 0.617868i \(-0.212004\pi\)
0.786281 + 0.617868i \(0.212004\pi\)
\(998\) 16.8275 0.532664
\(999\) 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 5625.2.a.z.1.4 8
3.2 odd 2 inner 5625.2.a.z.1.5 yes 8
5.4 even 2 5625.2.a.bb.1.5 yes 8
15.14 odd 2 5625.2.a.bb.1.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.4 8 1.1 even 1 trivial
5625.2.a.z.1.5 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.4 yes 8 15.14 odd 2
5625.2.a.bb.1.5 yes 8 5.4 even 2