Properties

Label 5625.2.a.z.1.1
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 1.1
Root \(-2.66202\) of defining polynomial
Character \(\chi\) \(=\) 5625.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} -8.21589 q^{8} +O(q^{10})\) \(q-2.66202 q^{2} +5.08634 q^{4} -3.46831 q^{7} -8.21589 q^{8} +3.43248 q^{11} -5.70437 q^{13} +9.23269 q^{14} +11.6982 q^{16} +1.89155 q^{17} +2.46831 q^{19} -9.13733 q^{22} +8.84431 q^{23} +15.1851 q^{26} -17.6410 q^{28} -8.98636 q^{29} -1.90746 q^{31} -14.7090 q^{32} -5.03535 q^{34} -5.09254 q^{37} -6.57068 q^{38} -6.35103 q^{41} +3.43296 q^{43} +17.4588 q^{44} -23.5437 q^{46} +8.61447 q^{47} +5.02915 q^{49} -29.0144 q^{52} +2.03360 q^{53} +28.4952 q^{56} +23.9219 q^{58} +8.47242 q^{59} -7.90746 q^{61} +5.07770 q^{62} +15.7592 q^{64} -6.95846 q^{67} +9.62108 q^{68} -2.91855 q^{71} +11.9280 q^{73} +13.5564 q^{74} +12.5546 q^{76} -11.9049 q^{77} -6.60564 q^{79} +16.9066 q^{82} +12.9993 q^{83} -9.13860 q^{86} -28.2009 q^{88} +11.5873 q^{89} +19.7845 q^{91} +44.9852 q^{92} -22.9319 q^{94} +2.17888 q^{97} -13.3877 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 14 q^{4} - 10 q^{7}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.66202 −1.88233 −0.941166 0.337946i \(-0.890268\pi\)
−0.941166 + 0.337946i \(0.890268\pi\)
\(3\) 0 0
\(4\) 5.08634 2.54317
\(5\) 0 0
\(6\) 0 0
\(7\) −3.46831 −1.31090 −0.655448 0.755240i \(-0.727520\pi\)
−0.655448 + 0.755240i \(0.727520\pi\)
\(8\) −8.21589 −2.90476
\(9\) 0 0
\(10\) 0 0
\(11\) 3.43248 1.03493 0.517466 0.855703i \(-0.326875\pi\)
0.517466 + 0.855703i \(0.326875\pi\)
\(12\) 0 0
\(13\) −5.70437 −1.58211 −0.791054 0.611746i \(-0.790468\pi\)
−0.791054 + 0.611746i \(0.790468\pi\)
\(14\) 9.23269 2.46754
\(15\) 0 0
\(16\) 11.6982 2.92454
\(17\) 1.89155 0.458769 0.229384 0.973336i \(-0.426329\pi\)
0.229384 + 0.973336i \(0.426329\pi\)
\(18\) 0 0
\(19\) 2.46831 0.566268 0.283134 0.959080i \(-0.408626\pi\)
0.283134 + 0.959080i \(0.408626\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −9.13733 −1.94809
\(23\) 8.84431 1.84417 0.922083 0.386992i \(-0.126486\pi\)
0.922083 + 0.386992i \(0.126486\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 15.1851 2.97805
\(27\) 0 0
\(28\) −17.6410 −3.33383
\(29\) −8.98636 −1.66873 −0.834363 0.551216i \(-0.814164\pi\)
−0.834363 + 0.551216i \(0.814164\pi\)
\(30\) 0 0
\(31\) −1.90746 −0.342591 −0.171295 0.985220i \(-0.554795\pi\)
−0.171295 + 0.985220i \(0.554795\pi\)
\(32\) −14.7090 −2.60020
\(33\) 0 0
\(34\) −5.03535 −0.863555
\(35\) 0 0
\(36\) 0 0
\(37\) −5.09254 −0.837208 −0.418604 0.908169i \(-0.637480\pi\)
−0.418604 + 0.908169i \(0.637480\pi\)
\(38\) −6.57068 −1.06590
\(39\) 0 0
\(40\) 0 0
\(41\) −6.35103 −0.991864 −0.495932 0.868361i \(-0.665174\pi\)
−0.495932 + 0.868361i \(0.665174\pi\)
\(42\) 0 0
\(43\) 3.43296 0.523522 0.261761 0.965133i \(-0.415697\pi\)
0.261761 + 0.965133i \(0.415697\pi\)
\(44\) 17.4588 2.63201
\(45\) 0 0
\(46\) −23.5437 −3.47133
\(47\) 8.61447 1.25655 0.628275 0.777991i \(-0.283761\pi\)
0.628275 + 0.777991i \(0.283761\pi\)
\(48\) 0 0
\(49\) 5.02915 0.718450
\(50\) 0 0
\(51\) 0 0
\(52\) −29.0144 −4.02357
\(53\) 2.03360 0.279337 0.139668 0.990198i \(-0.455396\pi\)
0.139668 + 0.990198i \(0.455396\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 28.4952 3.80784
\(57\) 0 0
\(58\) 23.9219 3.14109
\(59\) 8.47242 1.10302 0.551508 0.834170i \(-0.314053\pi\)
0.551508 + 0.834170i \(0.314053\pi\)
\(60\) 0 0
\(61\) −7.90746 −1.01245 −0.506223 0.862402i \(-0.668959\pi\)
−0.506223 + 0.862402i \(0.668959\pi\)
\(62\) 5.07770 0.644869
\(63\) 0 0
\(64\) 15.7592 1.96990
\(65\) 0 0
\(66\) 0 0
\(67\) −6.95846 −0.850111 −0.425055 0.905167i \(-0.639745\pi\)
−0.425055 + 0.905167i \(0.639745\pi\)
\(68\) 9.62108 1.16673
\(69\) 0 0
\(70\) 0 0
\(71\) −2.91855 −0.346368 −0.173184 0.984890i \(-0.555405\pi\)
−0.173184 + 0.984890i \(0.555405\pi\)
\(72\) 0 0
\(73\) 11.9280 1.39607 0.698036 0.716062i \(-0.254057\pi\)
0.698036 + 0.716062i \(0.254057\pi\)
\(74\) 13.5564 1.57590
\(75\) 0 0
\(76\) 12.5546 1.44012
\(77\) −11.9049 −1.35669
\(78\) 0 0
\(79\) −6.60564 −0.743193 −0.371596 0.928394i \(-0.621189\pi\)
−0.371596 + 0.928394i \(0.621189\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 16.9066 1.86702
\(83\) 12.9993 1.42686 0.713429 0.700727i \(-0.247141\pi\)
0.713429 + 0.700727i \(0.247141\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −9.13860 −0.985441
\(87\) 0 0
\(88\) −28.2009 −3.00623
\(89\) 11.5873 1.22825 0.614124 0.789209i \(-0.289509\pi\)
0.614124 + 0.789209i \(0.289509\pi\)
\(90\) 0 0
\(91\) 19.7845 2.07398
\(92\) 44.9852 4.69003
\(93\) 0 0
\(94\) −22.9319 −2.36524
\(95\) 0 0
\(96\) 0 0
\(97\) 2.17888 0.221231 0.110616 0.993863i \(-0.464718\pi\)
0.110616 + 0.993863i \(0.464718\pi\)
\(98\) −13.3877 −1.35236
\(99\) 0 0
\(100\) 0 0
\(101\) −9.30399 −0.925782 −0.462891 0.886415i \(-0.653188\pi\)
−0.462891 + 0.886415i \(0.653188\pi\)
\(102\) 0 0
\(103\) 0.176510 0.0173921 0.00869604 0.999962i \(-0.497232\pi\)
0.00869604 + 0.999962i \(0.497232\pi\)
\(104\) 46.8665 4.59564
\(105\) 0 0
\(106\) −5.41348 −0.525804
\(107\) −1.37761 −0.133179 −0.0665895 0.997780i \(-0.521212\pi\)
−0.0665895 + 0.997780i \(0.521212\pi\)
\(108\) 0 0
\(109\) 1.89744 0.181742 0.0908708 0.995863i \(-0.471035\pi\)
0.0908708 + 0.995863i \(0.471035\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −40.5729 −3.83378
\(113\) 2.35123 0.221185 0.110593 0.993866i \(-0.464725\pi\)
0.110593 + 0.993866i \(0.464725\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −45.7077 −4.24385
\(117\) 0 0
\(118\) −22.5537 −2.07624
\(119\) −6.56048 −0.601398
\(120\) 0 0
\(121\) 0.781947 0.0710861
\(122\) 21.0498 1.90576
\(123\) 0 0
\(124\) −9.70201 −0.871266
\(125\) 0 0
\(126\) 0 0
\(127\) 3.78069 0.335482 0.167741 0.985831i \(-0.446353\pi\)
0.167741 + 0.985831i \(0.446353\pi\)
\(128\) −12.5333 −1.10780
\(129\) 0 0
\(130\) 0 0
\(131\) 0.797155 0.0696477 0.0348239 0.999393i \(-0.488913\pi\)
0.0348239 + 0.999393i \(0.488913\pi\)
\(132\) 0 0
\(133\) −8.56084 −0.742319
\(134\) 18.5235 1.60019
\(135\) 0 0
\(136\) −15.5408 −1.33261
\(137\) −12.4188 −1.06101 −0.530507 0.847681i \(-0.677998\pi\)
−0.530507 + 0.847681i \(0.677998\pi\)
\(138\) 0 0
\(139\) 12.2399 1.03817 0.519087 0.854721i \(-0.326272\pi\)
0.519087 + 0.854721i \(0.326272\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 7.76922 0.651979
\(143\) −19.5802 −1.63738
\(144\) 0 0
\(145\) 0 0
\(146\) −31.7527 −2.62787
\(147\) 0 0
\(148\) −25.9024 −2.12916
\(149\) 1.31109 0.107409 0.0537044 0.998557i \(-0.482897\pi\)
0.0537044 + 0.998557i \(0.482897\pi\)
\(150\) 0 0
\(151\) 2.57266 0.209360 0.104680 0.994506i \(-0.466618\pi\)
0.104680 + 0.994506i \(0.466618\pi\)
\(152\) −20.2793 −1.64487
\(153\) 0 0
\(154\) 31.6911 2.55374
\(155\) 0 0
\(156\) 0 0
\(157\) −15.7038 −1.25330 −0.626650 0.779301i \(-0.715574\pi\)
−0.626650 + 0.779301i \(0.715574\pi\)
\(158\) 17.5843 1.39893
\(159\) 0 0
\(160\) 0 0
\(161\) −30.6748 −2.41751
\(162\) 0 0
\(163\) 7.42913 0.581894 0.290947 0.956739i \(-0.406030\pi\)
0.290947 + 0.956739i \(0.406030\pi\)
\(164\) −32.3035 −2.52248
\(165\) 0 0
\(166\) −34.6044 −2.68582
\(167\) −4.12200 −0.318970 −0.159485 0.987200i \(-0.550983\pi\)
−0.159485 + 0.987200i \(0.550983\pi\)
\(168\) 0 0
\(169\) 19.5399 1.50307
\(170\) 0 0
\(171\) 0 0
\(172\) 17.4612 1.33140
\(173\) −19.7765 −1.50358 −0.751789 0.659404i \(-0.770809\pi\)
−0.751789 + 0.659404i \(0.770809\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 40.1538 3.02671
\(177\) 0 0
\(178\) −30.8455 −2.31197
\(179\) 7.18260 0.536853 0.268426 0.963300i \(-0.413496\pi\)
0.268426 + 0.963300i \(0.413496\pi\)
\(180\) 0 0
\(181\) 15.9100 1.18258 0.591292 0.806458i \(-0.298618\pi\)
0.591292 + 0.806458i \(0.298618\pi\)
\(182\) −52.6667 −3.90392
\(183\) 0 0
\(184\) −72.6639 −5.35686
\(185\) 0 0
\(186\) 0 0
\(187\) 6.49272 0.474795
\(188\) 43.8161 3.19562
\(189\) 0 0
\(190\) 0 0
\(191\) −14.5190 −1.05056 −0.525278 0.850931i \(-0.676039\pi\)
−0.525278 + 0.850931i \(0.676039\pi\)
\(192\) 0 0
\(193\) 13.0763 0.941254 0.470627 0.882332i \(-0.344028\pi\)
0.470627 + 0.882332i \(0.344028\pi\)
\(194\) −5.80021 −0.416431
\(195\) 0 0
\(196\) 25.5800 1.82714
\(197\) −17.8849 −1.27425 −0.637124 0.770761i \(-0.719876\pi\)
−0.637124 + 0.770761i \(0.719876\pi\)
\(198\) 0 0
\(199\) −21.1565 −1.49974 −0.749871 0.661584i \(-0.769884\pi\)
−0.749871 + 0.661584i \(0.769884\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 24.7674 1.74263
\(203\) 31.1675 2.18753
\(204\) 0 0
\(205\) 0 0
\(206\) −0.469874 −0.0327377
\(207\) 0 0
\(208\) −66.7308 −4.62695
\(209\) 8.47242 0.586050
\(210\) 0 0
\(211\) 13.4002 0.922507 0.461253 0.887268i \(-0.347400\pi\)
0.461253 + 0.887268i \(0.347400\pi\)
\(212\) 10.3436 0.710400
\(213\) 0 0
\(214\) 3.66724 0.250687
\(215\) 0 0
\(216\) 0 0
\(217\) 6.61567 0.449101
\(218\) −5.05101 −0.342098
\(219\) 0 0
\(220\) 0 0
\(221\) −10.7901 −0.725822
\(222\) 0 0
\(223\) −25.2555 −1.69124 −0.845618 0.533788i \(-0.820768\pi\)
−0.845618 + 0.533788i \(0.820768\pi\)
\(224\) 51.0152 3.40860
\(225\) 0 0
\(226\) −6.25902 −0.416344
\(227\) −1.89155 −0.125547 −0.0627734 0.998028i \(-0.519995\pi\)
−0.0627734 + 0.998028i \(0.519995\pi\)
\(228\) 0 0
\(229\) −4.78069 −0.315917 −0.157958 0.987446i \(-0.550491\pi\)
−0.157958 + 0.987446i \(0.550491\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 73.8310 4.84724
\(233\) −3.44563 −0.225731 −0.112865 0.993610i \(-0.536003\pi\)
−0.112865 + 0.993610i \(0.536003\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 43.0936 2.80516
\(237\) 0 0
\(238\) 17.4641 1.13203
\(239\) 24.8164 1.60524 0.802620 0.596490i \(-0.203439\pi\)
0.802620 + 0.596490i \(0.203439\pi\)
\(240\) 0 0
\(241\) −20.5045 −1.32081 −0.660407 0.750908i \(-0.729616\pi\)
−0.660407 + 0.750908i \(0.729616\pi\)
\(242\) −2.08156 −0.133808
\(243\) 0 0
\(244\) −40.2201 −2.57482
\(245\) 0 0
\(246\) 0 0
\(247\) −14.0801 −0.895898
\(248\) 15.6715 0.995142
\(249\) 0 0
\(250\) 0 0
\(251\) −2.10825 −0.133071 −0.0665357 0.997784i \(-0.521195\pi\)
−0.0665357 + 0.997784i \(0.521195\pi\)
\(252\) 0 0
\(253\) 30.3580 1.90859
\(254\) −10.0643 −0.631488
\(255\) 0 0
\(256\) 1.84554 0.115346
\(257\) −1.45314 −0.0906445 −0.0453222 0.998972i \(-0.514431\pi\)
−0.0453222 + 0.998972i \(0.514431\pi\)
\(258\) 0 0
\(259\) 17.6625 1.09749
\(260\) 0 0
\(261\) 0 0
\(262\) −2.12204 −0.131100
\(263\) −25.7688 −1.58897 −0.794485 0.607284i \(-0.792259\pi\)
−0.794485 + 0.607284i \(0.792259\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 22.7891 1.39729
\(267\) 0 0
\(268\) −35.3931 −2.16198
\(269\) 31.9778 1.94972 0.974859 0.222823i \(-0.0715272\pi\)
0.974859 + 0.222823i \(0.0715272\pi\)
\(270\) 0 0
\(271\) 25.1821 1.52971 0.764853 0.644205i \(-0.222812\pi\)
0.764853 + 0.644205i \(0.222812\pi\)
\(272\) 22.1277 1.34169
\(273\) 0 0
\(274\) 33.0592 1.99718
\(275\) 0 0
\(276\) 0 0
\(277\) −16.8847 −1.01450 −0.507252 0.861798i \(-0.669339\pi\)
−0.507252 + 0.861798i \(0.669339\pi\)
\(278\) −32.5828 −1.95419
\(279\) 0 0
\(280\) 0 0
\(281\) −7.35764 −0.438920 −0.219460 0.975622i \(-0.570430\pi\)
−0.219460 + 0.975622i \(0.570430\pi\)
\(282\) 0 0
\(283\) −12.1220 −0.720581 −0.360290 0.932840i \(-0.617322\pi\)
−0.360290 + 0.932840i \(0.617322\pi\)
\(284\) −14.8447 −0.880872
\(285\) 0 0
\(286\) 52.1228 3.08208
\(287\) 22.0273 1.30023
\(288\) 0 0
\(289\) −13.4220 −0.789531
\(290\) 0 0
\(291\) 0 0
\(292\) 60.6701 3.55045
\(293\) −24.6612 −1.44072 −0.720362 0.693598i \(-0.756024\pi\)
−0.720362 + 0.693598i \(0.756024\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 41.8397 2.43189
\(297\) 0 0
\(298\) −3.49015 −0.202179
\(299\) −50.4513 −2.91767
\(300\) 0 0
\(301\) −11.9066 −0.686283
\(302\) −6.84847 −0.394085
\(303\) 0 0
\(304\) 28.8747 1.65608
\(305\) 0 0
\(306\) 0 0
\(307\) 19.9964 1.14125 0.570627 0.821210i \(-0.306700\pi\)
0.570627 + 0.821210i \(0.306700\pi\)
\(308\) −60.5524 −3.45029
\(309\) 0 0
\(310\) 0 0
\(311\) 3.11485 0.176627 0.0883136 0.996093i \(-0.471852\pi\)
0.0883136 + 0.996093i \(0.471852\pi\)
\(312\) 0 0
\(313\) 10.1492 0.573664 0.286832 0.957981i \(-0.407398\pi\)
0.286832 + 0.957981i \(0.407398\pi\)
\(314\) 41.8038 2.35913
\(315\) 0 0
\(316\) −33.5985 −1.89007
\(317\) 8.18921 0.459952 0.229976 0.973196i \(-0.426135\pi\)
0.229976 + 0.973196i \(0.426135\pi\)
\(318\) 0 0
\(319\) −30.8455 −1.72702
\(320\) 0 0
\(321\) 0 0
\(322\) 81.6568 4.55056
\(323\) 4.66893 0.259786
\(324\) 0 0
\(325\) 0 0
\(326\) −19.7765 −1.09532
\(327\) 0 0
\(328\) 52.1794 2.88113
\(329\) −29.8776 −1.64721
\(330\) 0 0
\(331\) 19.3339 1.06269 0.531344 0.847156i \(-0.321687\pi\)
0.531344 + 0.847156i \(0.321687\pi\)
\(332\) 66.1189 3.62875
\(333\) 0 0
\(334\) 10.9728 0.600408
\(335\) 0 0
\(336\) 0 0
\(337\) 0.902375 0.0491555 0.0245778 0.999698i \(-0.492176\pi\)
0.0245778 + 0.999698i \(0.492176\pi\)
\(338\) −52.0155 −2.82927
\(339\) 0 0
\(340\) 0 0
\(341\) −6.54734 −0.354558
\(342\) 0 0
\(343\) 6.83551 0.369083
\(344\) −28.2048 −1.52070
\(345\) 0 0
\(346\) 52.6454 2.83023
\(347\) 0.676375 0.0363097 0.0181548 0.999835i \(-0.494221\pi\)
0.0181548 + 0.999835i \(0.494221\pi\)
\(348\) 0 0
\(349\) 10.0604 0.538523 0.269262 0.963067i \(-0.413220\pi\)
0.269262 + 0.963067i \(0.413220\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −50.4883 −2.69104
\(353\) 7.29023 0.388020 0.194010 0.981000i \(-0.437851\pi\)
0.194010 + 0.981000i \(0.437851\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 58.9368 3.12365
\(357\) 0 0
\(358\) −19.1202 −1.01053
\(359\) −15.8301 −0.835479 −0.417739 0.908567i \(-0.637177\pi\)
−0.417739 + 0.908567i \(0.637177\pi\)
\(360\) 0 0
\(361\) −12.9075 −0.679340
\(362\) −42.3528 −2.22601
\(363\) 0 0
\(364\) 100.631 5.27449
\(365\) 0 0
\(366\) 0 0
\(367\) −23.5116 −1.22730 −0.613649 0.789579i \(-0.710299\pi\)
−0.613649 + 0.789579i \(0.710299\pi\)
\(368\) 103.462 5.39335
\(369\) 0 0
\(370\) 0 0
\(371\) −7.05315 −0.366181
\(372\) 0 0
\(373\) 8.27467 0.428446 0.214223 0.976785i \(-0.431278\pi\)
0.214223 + 0.976785i \(0.431278\pi\)
\(374\) −17.2837 −0.893721
\(375\) 0 0
\(376\) −70.7756 −3.64997
\(377\) 51.2616 2.64010
\(378\) 0 0
\(379\) −27.3648 −1.40564 −0.702819 0.711369i \(-0.748075\pi\)
−0.702819 + 0.711369i \(0.748075\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 38.6498 1.97749
\(383\) 15.6880 0.801620 0.400810 0.916161i \(-0.368729\pi\)
0.400810 + 0.916161i \(0.368729\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −34.8094 −1.77175
\(387\) 0 0
\(388\) 11.0825 0.562629
\(389\) 20.6950 1.04928 0.524638 0.851325i \(-0.324201\pi\)
0.524638 + 0.851325i \(0.324201\pi\)
\(390\) 0 0
\(391\) 16.7295 0.846046
\(392\) −41.3190 −2.08692
\(393\) 0 0
\(394\) 47.6100 2.39856
\(395\) 0 0
\(396\) 0 0
\(397\) −10.3616 −0.520033 −0.260016 0.965604i \(-0.583728\pi\)
−0.260016 + 0.965604i \(0.583728\pi\)
\(398\) 56.3189 2.82301
\(399\) 0 0
\(400\) 0 0
\(401\) −0.600848 −0.0300049 −0.0150025 0.999887i \(-0.504776\pi\)
−0.0150025 + 0.999887i \(0.504776\pi\)
\(402\) 0 0
\(403\) 10.8809 0.542015
\(404\) −47.3233 −2.35442
\(405\) 0 0
\(406\) −82.9683 −4.11765
\(407\) −17.4801 −0.866454
\(408\) 0 0
\(409\) −20.0064 −0.989253 −0.494626 0.869106i \(-0.664695\pi\)
−0.494626 + 0.869106i \(0.664695\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.897792 0.0442310
\(413\) −29.3850 −1.44594
\(414\) 0 0
\(415\) 0 0
\(416\) 83.9055 4.11381
\(417\) 0 0
\(418\) −22.5537 −1.10314
\(419\) 10.8245 0.528813 0.264407 0.964411i \(-0.414824\pi\)
0.264407 + 0.964411i \(0.414824\pi\)
\(420\) 0 0
\(421\) −31.3321 −1.52703 −0.763516 0.645789i \(-0.776528\pi\)
−0.763516 + 0.645789i \(0.776528\pi\)
\(422\) −35.6715 −1.73646
\(423\) 0 0
\(424\) −16.7078 −0.811405
\(425\) 0 0
\(426\) 0 0
\(427\) 27.4255 1.32721
\(428\) −7.00702 −0.338697
\(429\) 0 0
\(430\) 0 0
\(431\) −31.6470 −1.52438 −0.762191 0.647353i \(-0.775876\pi\)
−0.762191 + 0.647353i \(0.775876\pi\)
\(432\) 0 0
\(433\) −19.9075 −0.956692 −0.478346 0.878172i \(-0.658763\pi\)
−0.478346 + 0.878172i \(0.658763\pi\)
\(434\) −17.6110 −0.845356
\(435\) 0 0
\(436\) 9.65101 0.462200
\(437\) 21.8305 1.04429
\(438\) 0 0
\(439\) −23.7707 −1.13451 −0.567256 0.823542i \(-0.691995\pi\)
−0.567256 + 0.823542i \(0.691995\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 28.7235 1.36624
\(443\) −8.36336 −0.397355 −0.198678 0.980065i \(-0.563665\pi\)
−0.198678 + 0.980065i \(0.563665\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 67.2307 3.18347
\(447\) 0 0
\(448\) −54.6577 −2.58234
\(449\) −35.2889 −1.66539 −0.832693 0.553734i \(-0.813202\pi\)
−0.832693 + 0.553734i \(0.813202\pi\)
\(450\) 0 0
\(451\) −21.7998 −1.02651
\(452\) 11.9592 0.562512
\(453\) 0 0
\(454\) 5.03535 0.236320
\(455\) 0 0
\(456\) 0 0
\(457\) −12.9534 −0.605933 −0.302967 0.953001i \(-0.597977\pi\)
−0.302967 + 0.953001i \(0.597977\pi\)
\(458\) 12.7263 0.594660
\(459\) 0 0
\(460\) 0 0
\(461\) 29.8908 1.39215 0.696076 0.717968i \(-0.254928\pi\)
0.696076 + 0.717968i \(0.254928\pi\)
\(462\) 0 0
\(463\) 14.1155 0.656002 0.328001 0.944677i \(-0.393625\pi\)
0.328001 + 0.944677i \(0.393625\pi\)
\(464\) −105.124 −4.88026
\(465\) 0 0
\(466\) 9.17233 0.424900
\(467\) 4.00068 0.185129 0.0925647 0.995707i \(-0.470494\pi\)
0.0925647 + 0.995707i \(0.470494\pi\)
\(468\) 0 0
\(469\) 24.1341 1.11441
\(470\) 0 0
\(471\) 0 0
\(472\) −69.6085 −3.20399
\(473\) 11.7836 0.541810
\(474\) 0 0
\(475\) 0 0
\(476\) −33.3688 −1.52946
\(477\) 0 0
\(478\) −66.0618 −3.02159
\(479\) −19.2413 −0.879156 −0.439578 0.898204i \(-0.644872\pi\)
−0.439578 + 0.898204i \(0.644872\pi\)
\(480\) 0 0
\(481\) 29.0497 1.32455
\(482\) 54.5835 2.48621
\(483\) 0 0
\(484\) 3.97725 0.180784
\(485\) 0 0
\(486\) 0 0
\(487\) 13.9219 0.630859 0.315430 0.948949i \(-0.397851\pi\)
0.315430 + 0.948949i \(0.397851\pi\)
\(488\) 64.9669 2.94091
\(489\) 0 0
\(490\) 0 0
\(491\) 14.5058 0.654639 0.327319 0.944914i \(-0.393855\pi\)
0.327319 + 0.944914i \(0.393855\pi\)
\(492\) 0 0
\(493\) −16.9982 −0.765559
\(494\) 37.4816 1.68638
\(495\) 0 0
\(496\) −22.3138 −1.00192
\(497\) 10.1224 0.454052
\(498\) 0 0
\(499\) 1.26905 0.0568103 0.0284052 0.999596i \(-0.490957\pi\)
0.0284052 + 0.999596i \(0.490957\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.61219 0.250484
\(503\) 8.50684 0.379301 0.189651 0.981852i \(-0.439264\pi\)
0.189651 + 0.981852i \(0.439264\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −80.8134 −3.59260
\(507\) 0 0
\(508\) 19.2299 0.853188
\(509\) 36.4250 1.61451 0.807254 0.590204i \(-0.200953\pi\)
0.807254 + 0.590204i \(0.200953\pi\)
\(510\) 0 0
\(511\) −41.3701 −1.83011
\(512\) 20.1538 0.890680
\(513\) 0 0
\(514\) 3.86829 0.170623
\(515\) 0 0
\(516\) 0 0
\(517\) 29.5690 1.30044
\(518\) −47.0178 −2.06585
\(519\) 0 0
\(520\) 0 0
\(521\) −31.9990 −1.40190 −0.700951 0.713209i \(-0.747241\pi\)
−0.700951 + 0.713209i \(0.747241\pi\)
\(522\) 0 0
\(523\) −29.0448 −1.27004 −0.635020 0.772496i \(-0.719008\pi\)
−0.635020 + 0.772496i \(0.719008\pi\)
\(524\) 4.05460 0.177126
\(525\) 0 0
\(526\) 68.5969 2.99097
\(527\) −3.60807 −0.157170
\(528\) 0 0
\(529\) 55.2218 2.40095
\(530\) 0 0
\(531\) 0 0
\(532\) −43.5434 −1.88784
\(533\) 36.2287 1.56924
\(534\) 0 0
\(535\) 0 0
\(536\) 57.1700 2.46937
\(537\) 0 0
\(538\) −85.1254 −3.67001
\(539\) 17.2625 0.743547
\(540\) 0 0
\(541\) −18.2546 −0.784827 −0.392414 0.919789i \(-0.628360\pi\)
−0.392414 + 0.919789i \(0.628360\pi\)
\(542\) −67.0353 −2.87941
\(543\) 0 0
\(544\) −27.8228 −1.19289
\(545\) 0 0
\(546\) 0 0
\(547\) −6.92805 −0.296222 −0.148111 0.988971i \(-0.547319\pi\)
−0.148111 + 0.988971i \(0.547319\pi\)
\(548\) −63.1665 −2.69834
\(549\) 0 0
\(550\) 0 0
\(551\) −22.1811 −0.944946
\(552\) 0 0
\(553\) 22.9104 0.974249
\(554\) 44.9474 1.90963
\(555\) 0 0
\(556\) 62.2563 2.64025
\(557\) 11.5873 0.490969 0.245484 0.969401i \(-0.421053\pi\)
0.245484 + 0.969401i \(0.421053\pi\)
\(558\) 0 0
\(559\) −19.5829 −0.828268
\(560\) 0 0
\(561\) 0 0
\(562\) 19.5862 0.826192
\(563\) 4.92004 0.207355 0.103677 0.994611i \(-0.466939\pi\)
0.103677 + 0.994611i \(0.466939\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.2691 1.35637
\(567\) 0 0
\(568\) 23.9785 1.00611
\(569\) −37.5316 −1.57341 −0.786704 0.617331i \(-0.788214\pi\)
−0.786704 + 0.617331i \(0.788214\pi\)
\(570\) 0 0
\(571\) −31.5956 −1.32224 −0.661118 0.750282i \(-0.729918\pi\)
−0.661118 + 0.750282i \(0.729918\pi\)
\(572\) −99.5914 −4.16413
\(573\) 0 0
\(574\) −58.6371 −2.44747
\(575\) 0 0
\(576\) 0 0
\(577\) −36.4055 −1.51558 −0.757790 0.652499i \(-0.773721\pi\)
−0.757790 + 0.652499i \(0.773721\pi\)
\(578\) 35.7297 1.48616
\(579\) 0 0
\(580\) 0 0
\(581\) −45.0856 −1.87046
\(582\) 0 0
\(583\) 6.98030 0.289095
\(584\) −97.9996 −4.05525
\(585\) 0 0
\(586\) 65.6486 2.71192
\(587\) −35.5061 −1.46550 −0.732748 0.680500i \(-0.761763\pi\)
−0.732748 + 0.680500i \(0.761763\pi\)
\(588\) 0 0
\(589\) −4.70820 −0.193998
\(590\) 0 0
\(591\) 0 0
\(592\) −59.5734 −2.44845
\(593\) −45.7086 −1.87703 −0.938513 0.345244i \(-0.887796\pi\)
−0.938513 + 0.345244i \(0.887796\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 6.66866 0.273159
\(597\) 0 0
\(598\) 134.302 5.49202
\(599\) −3.58625 −0.146530 −0.0732652 0.997312i \(-0.523342\pi\)
−0.0732652 + 0.997312i \(0.523342\pi\)
\(600\) 0 0
\(601\) 10.0366 0.409400 0.204700 0.978825i \(-0.434378\pi\)
0.204700 + 0.978825i \(0.434378\pi\)
\(602\) 31.6955 1.29181
\(603\) 0 0
\(604\) 13.0854 0.532439
\(605\) 0 0
\(606\) 0 0
\(607\) −34.8691 −1.41529 −0.707646 0.706567i \(-0.750243\pi\)
−0.707646 + 0.706567i \(0.750243\pi\)
\(608\) −36.3063 −1.47241
\(609\) 0 0
\(610\) 0 0
\(611\) −49.1402 −1.98800
\(612\) 0 0
\(613\) −23.8674 −0.963994 −0.481997 0.876173i \(-0.660088\pi\)
−0.481997 + 0.876173i \(0.660088\pi\)
\(614\) −53.2307 −2.14822
\(615\) 0 0
\(616\) 97.8095 3.94086
\(617\) 6.81159 0.274224 0.137112 0.990556i \(-0.456218\pi\)
0.137112 + 0.990556i \(0.456218\pi\)
\(618\) 0 0
\(619\) −24.7256 −0.993808 −0.496904 0.867806i \(-0.665530\pi\)
−0.496904 + 0.867806i \(0.665530\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.29180 −0.332471
\(623\) −40.1882 −1.61011
\(624\) 0 0
\(625\) 0 0
\(626\) −27.0172 −1.07983
\(627\) 0 0
\(628\) −79.8749 −3.18735
\(629\) −9.63280 −0.384085
\(630\) 0 0
\(631\) −23.4178 −0.932250 −0.466125 0.884719i \(-0.654350\pi\)
−0.466125 + 0.884719i \(0.654350\pi\)
\(632\) 54.2713 2.15879
\(633\) 0 0
\(634\) −21.7998 −0.865781
\(635\) 0 0
\(636\) 0 0
\(637\) −28.6882 −1.13667
\(638\) 82.1114 3.25082
\(639\) 0 0
\(640\) 0 0
\(641\) −26.9591 −1.06482 −0.532410 0.846487i \(-0.678713\pi\)
−0.532410 + 0.846487i \(0.678713\pi\)
\(642\) 0 0
\(643\) 9.56996 0.377403 0.188701 0.982035i \(-0.439572\pi\)
0.188701 + 0.982035i \(0.439572\pi\)
\(644\) −156.022 −6.14814
\(645\) 0 0
\(646\) −12.4288 −0.489004
\(647\) 20.0941 0.789981 0.394991 0.918685i \(-0.370748\pi\)
0.394991 + 0.918685i \(0.370748\pi\)
\(648\) 0 0
\(649\) 29.0815 1.14155
\(650\) 0 0
\(651\) 0 0
\(652\) 37.7871 1.47986
\(653\) 40.7211 1.59354 0.796770 0.604282i \(-0.206540\pi\)
0.796770 + 0.604282i \(0.206540\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −74.2955 −2.90075
\(657\) 0 0
\(658\) 79.5348 3.10059
\(659\) −11.4254 −0.445070 −0.222535 0.974925i \(-0.571433\pi\)
−0.222535 + 0.974925i \(0.571433\pi\)
\(660\) 0 0
\(661\) −33.5985 −1.30683 −0.653416 0.756999i \(-0.726665\pi\)
−0.653416 + 0.756999i \(0.726665\pi\)
\(662\) −51.4672 −2.00033
\(663\) 0 0
\(664\) −106.801 −4.14468
\(665\) 0 0
\(666\) 0 0
\(667\) −79.4782 −3.07741
\(668\) −20.9659 −0.811196
\(669\) 0 0
\(670\) 0 0
\(671\) −27.1422 −1.04781
\(672\) 0 0
\(673\) −27.7871 −1.07111 −0.535557 0.844499i \(-0.679898\pi\)
−0.535557 + 0.844499i \(0.679898\pi\)
\(674\) −2.40214 −0.0925270
\(675\) 0 0
\(676\) 99.3865 3.82256
\(677\) −31.3520 −1.20496 −0.602478 0.798135i \(-0.705820\pi\)
−0.602478 + 0.798135i \(0.705820\pi\)
\(678\) 0 0
\(679\) −7.55701 −0.290012
\(680\) 0 0
\(681\) 0 0
\(682\) 17.4291 0.667396
\(683\) −0.655106 −0.0250669 −0.0125335 0.999921i \(-0.503990\pi\)
−0.0125335 + 0.999921i \(0.503990\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −18.1963 −0.694736
\(687\) 0 0
\(688\) 40.1594 1.53106
\(689\) −11.6004 −0.441941
\(690\) 0 0
\(691\) 28.9381 1.10086 0.550428 0.834883i \(-0.314464\pi\)
0.550428 + 0.834883i \(0.314464\pi\)
\(692\) −100.590 −3.82385
\(693\) 0 0
\(694\) −1.80052 −0.0683469
\(695\) 0 0
\(696\) 0 0
\(697\) −12.0133 −0.455036
\(698\) −26.7811 −1.01368
\(699\) 0 0
\(700\) 0 0
\(701\) −31.0249 −1.17179 −0.585896 0.810386i \(-0.699257\pi\)
−0.585896 + 0.810386i \(0.699257\pi\)
\(702\) 0 0
\(703\) −12.5699 −0.474084
\(704\) 54.0932 2.03871
\(705\) 0 0
\(706\) −19.4067 −0.730382
\(707\) 32.2691 1.21360
\(708\) 0 0
\(709\) 41.2676 1.54984 0.774918 0.632062i \(-0.217791\pi\)
0.774918 + 0.632062i \(0.217791\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −95.1998 −3.56776
\(713\) −16.8702 −0.631794
\(714\) 0 0
\(715\) 0 0
\(716\) 36.5331 1.36531
\(717\) 0 0
\(718\) 42.1399 1.57265
\(719\) −2.15581 −0.0803980 −0.0401990 0.999192i \(-0.512799\pi\)
−0.0401990 + 0.999192i \(0.512799\pi\)
\(720\) 0 0
\(721\) −0.612192 −0.0227992
\(722\) 34.3599 1.27874
\(723\) 0 0
\(724\) 80.9239 3.00751
\(725\) 0 0
\(726\) 0 0
\(727\) −43.8091 −1.62479 −0.812396 0.583107i \(-0.801837\pi\)
−0.812396 + 0.583107i \(0.801837\pi\)
\(728\) −162.548 −6.02441
\(729\) 0 0
\(730\) 0 0
\(731\) 6.49362 0.240175
\(732\) 0 0
\(733\) 38.6326 1.42693 0.713464 0.700692i \(-0.247125\pi\)
0.713464 + 0.700692i \(0.247125\pi\)
\(734\) 62.5884 2.31018
\(735\) 0 0
\(736\) −130.091 −4.79521
\(737\) −23.8848 −0.879808
\(738\) 0 0
\(739\) 20.4743 0.753159 0.376579 0.926384i \(-0.377100\pi\)
0.376579 + 0.926384i \(0.377100\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 18.7756 0.689275
\(743\) 4.86490 0.178476 0.0892379 0.996010i \(-0.471557\pi\)
0.0892379 + 0.996010i \(0.471557\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −22.0273 −0.806478
\(747\) 0 0
\(748\) 33.0242 1.20748
\(749\) 4.77799 0.174584
\(750\) 0 0
\(751\) −36.7353 −1.34049 −0.670245 0.742140i \(-0.733811\pi\)
−0.670245 + 0.742140i \(0.733811\pi\)
\(752\) 100.774 3.67484
\(753\) 0 0
\(754\) −136.459 −4.96955
\(755\) 0 0
\(756\) 0 0
\(757\) −13.1609 −0.478340 −0.239170 0.970978i \(-0.576875\pi\)
−0.239170 + 0.970978i \(0.576875\pi\)
\(758\) 72.8457 2.64587
\(759\) 0 0
\(760\) 0 0
\(761\) −17.9859 −0.651987 −0.325994 0.945372i \(-0.605699\pi\)
−0.325994 + 0.945372i \(0.605699\pi\)
\(762\) 0 0
\(763\) −6.58089 −0.238244
\(764\) −73.8484 −2.67174
\(765\) 0 0
\(766\) −41.7618 −1.50891
\(767\) −48.3299 −1.74509
\(768\) 0 0
\(769\) 3.28562 0.118483 0.0592413 0.998244i \(-0.481132\pi\)
0.0592413 + 0.998244i \(0.481132\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 66.5106 2.39377
\(773\) 10.0265 0.360628 0.180314 0.983609i \(-0.442289\pi\)
0.180314 + 0.983609i \(0.442289\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −17.9014 −0.642624
\(777\) 0 0
\(778\) −55.0904 −1.97509
\(779\) −15.6763 −0.561661
\(780\) 0 0
\(781\) −10.0179 −0.358467
\(782\) −44.5342 −1.59254
\(783\) 0 0
\(784\) 58.8319 2.10114
\(785\) 0 0
\(786\) 0 0
\(787\) −43.1508 −1.53816 −0.769080 0.639153i \(-0.779285\pi\)
−0.769080 + 0.639153i \(0.779285\pi\)
\(788\) −90.9688 −3.24063
\(789\) 0 0
\(790\) 0 0
\(791\) −8.15479 −0.289951
\(792\) 0 0
\(793\) 45.1071 1.60180
\(794\) 27.5827 0.978874
\(795\) 0 0
\(796\) −107.609 −3.81410
\(797\) −23.2836 −0.824748 −0.412374 0.911015i \(-0.635300\pi\)
−0.412374 + 0.911015i \(0.635300\pi\)
\(798\) 0 0
\(799\) 16.2947 0.576466
\(800\) 0 0
\(801\) 0 0
\(802\) 1.59947 0.0564792
\(803\) 40.9428 1.44484
\(804\) 0 0
\(805\) 0 0
\(806\) −28.9651 −1.02025
\(807\) 0 0
\(808\) 76.4406 2.68917
\(809\) 6.18914 0.217598 0.108799 0.994064i \(-0.465299\pi\)
0.108799 + 0.994064i \(0.465299\pi\)
\(810\) 0 0
\(811\) −2.53809 −0.0891245 −0.0445623 0.999007i \(-0.514189\pi\)
−0.0445623 + 0.999007i \(0.514189\pi\)
\(812\) 158.528 5.56325
\(813\) 0 0
\(814\) 46.5322 1.63095
\(815\) 0 0
\(816\) 0 0
\(817\) 8.47360 0.296454
\(818\) 53.2574 1.86210
\(819\) 0 0
\(820\) 0 0
\(821\) 44.4179 1.55019 0.775097 0.631842i \(-0.217701\pi\)
0.775097 + 0.631842i \(0.217701\pi\)
\(822\) 0 0
\(823\) −19.6434 −0.684724 −0.342362 0.939568i \(-0.611227\pi\)
−0.342362 + 0.939568i \(0.611227\pi\)
\(824\) −1.45019 −0.0505198
\(825\) 0 0
\(826\) 78.2233 2.72174
\(827\) −31.1557 −1.08339 −0.541695 0.840575i \(-0.682217\pi\)
−0.541695 + 0.840575i \(0.682217\pi\)
\(828\) 0 0
\(829\) 2.46831 0.0857278 0.0428639 0.999081i \(-0.486352\pi\)
0.0428639 + 0.999081i \(0.486352\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −89.8964 −3.11660
\(833\) 9.51290 0.329602
\(834\) 0 0
\(835\) 0 0
\(836\) 43.0936 1.49042
\(837\) 0 0
\(838\) −28.8151 −0.995401
\(839\) 46.0334 1.58925 0.794625 0.607100i \(-0.207667\pi\)
0.794625 + 0.607100i \(0.207667\pi\)
\(840\) 0 0
\(841\) 51.7547 1.78464
\(842\) 83.4065 2.87438
\(843\) 0 0
\(844\) 68.1579 2.34609
\(845\) 0 0
\(846\) 0 0
\(847\) −2.71203 −0.0931866
\(848\) 23.7894 0.816932
\(849\) 0 0
\(850\) 0 0
\(851\) −45.0400 −1.54395
\(852\) 0 0
\(853\) 9.24700 0.316611 0.158306 0.987390i \(-0.449397\pi\)
0.158306 + 0.987390i \(0.449397\pi\)
\(854\) −73.0072 −2.49825
\(855\) 0 0
\(856\) 11.3183 0.386853
\(857\) 32.3505 1.10507 0.552536 0.833489i \(-0.313660\pi\)
0.552536 + 0.833489i \(0.313660\pi\)
\(858\) 0 0
\(859\) −14.0883 −0.480688 −0.240344 0.970688i \(-0.577260\pi\)
−0.240344 + 0.970688i \(0.577260\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 84.2448 2.86939
\(863\) 52.5301 1.78815 0.894073 0.447921i \(-0.147836\pi\)
0.894073 + 0.447921i \(0.147836\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 52.9940 1.80081
\(867\) 0 0
\(868\) 33.6495 1.14214
\(869\) −22.6738 −0.769155
\(870\) 0 0
\(871\) 39.6936 1.34497
\(872\) −15.5891 −0.527915
\(873\) 0 0
\(874\) −58.1131 −1.96571
\(875\) 0 0
\(876\) 0 0
\(877\) 6.58359 0.222312 0.111156 0.993803i \(-0.464545\pi\)
0.111156 + 0.993803i \(0.464545\pi\)
\(878\) 63.2779 2.13553
\(879\) 0 0
\(880\) 0 0
\(881\) 36.4594 1.22835 0.614174 0.789171i \(-0.289489\pi\)
0.614174 + 0.789171i \(0.289489\pi\)
\(882\) 0 0
\(883\) −17.5190 −0.589560 −0.294780 0.955565i \(-0.595246\pi\)
−0.294780 + 0.955565i \(0.595246\pi\)
\(884\) −54.8822 −1.84589
\(885\) 0 0
\(886\) 22.2634 0.747954
\(887\) 24.6277 0.826917 0.413459 0.910523i \(-0.364321\pi\)
0.413459 + 0.910523i \(0.364321\pi\)
\(888\) 0 0
\(889\) −13.1126 −0.439782
\(890\) 0 0
\(891\) 0 0
\(892\) −128.458 −4.30110
\(893\) 21.2632 0.711544
\(894\) 0 0
\(895\) 0 0
\(896\) 43.4694 1.45221
\(897\) 0 0
\(898\) 93.9397 3.13481
\(899\) 17.1412 0.571689
\(900\) 0 0
\(901\) 3.84666 0.128151
\(902\) 58.0315 1.93224
\(903\) 0 0
\(904\) −19.3175 −0.642490
\(905\) 0 0
\(906\) 0 0
\(907\) 4.47955 0.148741 0.0743705 0.997231i \(-0.476305\pi\)
0.0743705 + 0.997231i \(0.476305\pi\)
\(908\) −9.62108 −0.319287
\(909\) 0 0
\(910\) 0 0
\(911\) −41.9576 −1.39012 −0.695058 0.718953i \(-0.744621\pi\)
−0.695058 + 0.718953i \(0.744621\pi\)
\(912\) 0 0
\(913\) 44.6199 1.47670
\(914\) 34.4821 1.14057
\(915\) 0 0
\(916\) −24.3162 −0.803430
\(917\) −2.76478 −0.0913010
\(918\) 0 0
\(919\) −37.6692 −1.24259 −0.621297 0.783575i \(-0.713394\pi\)
−0.621297 + 0.783575i \(0.713394\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −79.5698 −2.62049
\(923\) 16.6485 0.547992
\(924\) 0 0
\(925\) 0 0
\(926\) −37.5757 −1.23481
\(927\) 0 0
\(928\) 132.180 4.33903
\(929\) −17.1493 −0.562649 −0.281325 0.959613i \(-0.590774\pi\)
−0.281325 + 0.959613i \(0.590774\pi\)
\(930\) 0 0
\(931\) 12.4135 0.406835
\(932\) −17.5256 −0.574071
\(933\) 0 0
\(934\) −10.6499 −0.348475
\(935\) 0 0
\(936\) 0 0
\(937\) −37.1944 −1.21509 −0.607544 0.794286i \(-0.707845\pi\)
−0.607544 + 0.794286i \(0.707845\pi\)
\(938\) −64.2453 −2.09768
\(939\) 0 0
\(940\) 0 0
\(941\) 29.5387 0.962935 0.481468 0.876464i \(-0.340104\pi\)
0.481468 + 0.876464i \(0.340104\pi\)
\(942\) 0 0
\(943\) −56.1705 −1.82916
\(944\) 99.1119 3.22582
\(945\) 0 0
\(946\) −31.3681 −1.01987
\(947\) −34.0869 −1.10767 −0.553837 0.832625i \(-0.686837\pi\)
−0.553837 + 0.832625i \(0.686837\pi\)
\(948\) 0 0
\(949\) −68.0421 −2.20874
\(950\) 0 0
\(951\) 0 0
\(952\) 53.9002 1.74692
\(953\) 9.67588 0.313432 0.156716 0.987644i \(-0.449909\pi\)
0.156716 + 0.987644i \(0.449909\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 126.225 4.08240
\(957\) 0 0
\(958\) 51.2206 1.65486
\(959\) 43.0724 1.39088
\(960\) 0 0
\(961\) −27.3616 −0.882632
\(962\) −77.3309 −2.49325
\(963\) 0 0
\(964\) −104.293 −3.35905
\(965\) 0 0
\(966\) 0 0
\(967\) 17.2537 0.554842 0.277421 0.960748i \(-0.410520\pi\)
0.277421 + 0.960748i \(0.410520\pi\)
\(968\) −6.42440 −0.206488
\(969\) 0 0
\(970\) 0 0
\(971\) −1.92509 −0.0617789 −0.0308895 0.999523i \(-0.509834\pi\)
−0.0308895 + 0.999523i \(0.509834\pi\)
\(972\) 0 0
\(973\) −42.4517 −1.36094
\(974\) −37.0602 −1.18749
\(975\) 0 0
\(976\) −92.5029 −2.96095
\(977\) −26.1185 −0.835605 −0.417803 0.908538i \(-0.637200\pi\)
−0.417803 + 0.908538i \(0.637200\pi\)
\(978\) 0 0
\(979\) 39.7731 1.27116
\(980\) 0 0
\(981\) 0 0
\(982\) −38.6148 −1.23225
\(983\) −9.28360 −0.296101 −0.148050 0.988980i \(-0.547300\pi\)
−0.148050 + 0.988980i \(0.547300\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 45.2494 1.44104
\(987\) 0 0
\(988\) −71.6164 −2.27842
\(989\) 30.3622 0.965461
\(990\) 0 0
\(991\) 41.1337 1.30666 0.653328 0.757075i \(-0.273372\pi\)
0.653328 + 0.757075i \(0.273372\pi\)
\(992\) 28.0568 0.890805
\(993\) 0 0
\(994\) −26.9461 −0.854677
\(995\) 0 0
\(996\) 0 0
\(997\) −4.60496 −0.145840 −0.0729202 0.997338i \(-0.523232\pi\)
−0.0729202 + 0.997338i \(0.523232\pi\)
\(998\) −3.37823 −0.106936
\(999\) 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.1.1 8
3.2 odd 2 inner 5625.2.a.z.1.8 yes 8
5.4 even 2 5625.2.a.bb.1.8 yes 8
15.14 odd 2 5625.2.a.bb.1.1 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5625.2.a.z.1.1 8 1.1 even 1 trivial
5625.2.a.z.1.8 yes 8 3.2 odd 2 inner
5625.2.a.bb.1.1 yes 8 15.14 odd 2
5625.2.a.bb.1.8 yes 8 5.4 even 2