Properties

Label 1875.2.b.h.1249.4
Level $1875$
Weight $2$
Character 1875.1249
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 20x^{14} + 156x^{12} + 610x^{10} + 1286x^{8} + 1440x^{6} + 761x^{4} + 130x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: no (minimal twist has level 75)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1249.4
Root \(0.536547i\) of defining polynomial
Character \(\chi\) \(=\) 1875.1249
Dual form 1875.2.b.h.1249.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.53655i q^{2} -1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} -1.49550i q^{7} -2.51844i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.53655i q^{2} -1.00000i q^{3} -0.360976 q^{4} -1.53655 q^{6} -1.49550i q^{7} -2.51844i q^{8} -1.00000 q^{9} -2.35626 q^{11} +0.360976i q^{12} +1.34951i q^{13} -2.29790 q^{14} -4.59165 q^{16} +2.19405i q^{17} +1.53655i q^{18} -5.71069 q^{19} -1.49550 q^{21} +3.62050i q^{22} +8.79501i q^{23} -2.51844 q^{24} +2.07358 q^{26} +1.00000i q^{27} +0.539839i q^{28} -7.90017 q^{29} -3.69717 q^{31} +2.01841i q^{32} +2.35626i q^{33} +3.37126 q^{34} +0.360976 q^{36} -9.75097i q^{37} +8.77474i q^{38} +1.34951 q^{39} +1.85550 q^{41} +2.29790i q^{42} -8.01874i q^{43} +0.850553 q^{44} +13.5139 q^{46} -6.66298i q^{47} +4.59165i q^{48} +4.76349 q^{49} +2.19405 q^{51} -0.487140i q^{52} +4.17153i q^{53} +1.53655 q^{54} -3.76631 q^{56} +5.71069i q^{57} +12.1390i q^{58} -11.0647 q^{59} -12.2372 q^{61} +5.68088i q^{62} +1.49550i q^{63} -6.08192 q^{64} +3.62050 q^{66} -4.31358i q^{67} -0.792000i q^{68} +8.79501 q^{69} +5.77750 q^{71} +2.51844i q^{72} +6.92684i q^{73} -14.9828 q^{74} +2.06142 q^{76} +3.52377i q^{77} -2.07358i q^{78} +10.6687 q^{79} +1.00000 q^{81} -2.85106i q^{82} +0.224003i q^{83} +0.539839 q^{84} -12.3212 q^{86} +7.90017i q^{87} +5.93408i q^{88} -0.429167 q^{89} +2.01818 q^{91} -3.17479i q^{92} +3.69717i q^{93} -10.2380 q^{94} +2.01841 q^{96} -12.4945i q^{97} -7.31933i q^{98} +2.35626 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{4} - 8 q^{6} - 16 q^{9} + 4 q^{11} - 12 q^{14} + 28 q^{19} - 16 q^{21} + 24 q^{24} + 12 q^{26} - 4 q^{29} - 44 q^{31} + 24 q^{34} + 8 q^{36} + 32 q^{39} + 16 q^{41} - 44 q^{44} - 4 q^{46} - 32 q^{51} + 8 q^{54} + 60 q^{56} - 28 q^{59} - 40 q^{61} - 12 q^{64} - 24 q^{66} + 28 q^{69} + 32 q^{71} - 52 q^{74} - 32 q^{76} + 60 q^{79} + 16 q^{81} + 32 q^{84} + 64 q^{86} - 32 q^{89} - 24 q^{91} - 28 q^{94} + 4 q^{96} - 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.53655i − 1.08650i −0.839570 0.543251i \(-0.817193\pi\)
0.839570 0.543251i \(-0.182807\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) −0.360976 −0.180488
\(5\) 0 0
\(6\) −1.53655 −0.627293
\(7\) − 1.49550i − 0.565244i −0.959231 0.282622i \(-0.908796\pi\)
0.959231 0.282622i \(-0.0912043\pi\)
\(8\) − 2.51844i − 0.890402i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −2.35626 −0.710438 −0.355219 0.934783i \(-0.615594\pi\)
−0.355219 + 0.934783i \(0.615594\pi\)
\(12\) 0.360976i 0.104205i
\(13\) 1.34951i 0.374286i 0.982333 + 0.187143i \(0.0599227\pi\)
−0.982333 + 0.187143i \(0.940077\pi\)
\(14\) −2.29790 −0.614140
\(15\) 0 0
\(16\) −4.59165 −1.14791
\(17\) 2.19405i 0.532135i 0.963954 + 0.266068i \(0.0857245\pi\)
−0.963954 + 0.266068i \(0.914276\pi\)
\(18\) 1.53655i 0.362168i
\(19\) −5.71069 −1.31012 −0.655061 0.755576i \(-0.727357\pi\)
−0.655061 + 0.755576i \(0.727357\pi\)
\(20\) 0 0
\(21\) −1.49550 −0.326344
\(22\) 3.62050i 0.771892i
\(23\) 8.79501i 1.83389i 0.399018 + 0.916943i \(0.369351\pi\)
−0.399018 + 0.916943i \(0.630649\pi\)
\(24\) −2.51844 −0.514074
\(25\) 0 0
\(26\) 2.07358 0.406662
\(27\) 1.00000i 0.192450i
\(28\) 0.539839i 0.102020i
\(29\) −7.90017 −1.46702 −0.733512 0.679676i \(-0.762120\pi\)
−0.733512 + 0.679676i \(0.762120\pi\)
\(30\) 0 0
\(31\) −3.69717 −0.664031 −0.332016 0.943274i \(-0.607729\pi\)
−0.332016 + 0.943274i \(0.607729\pi\)
\(32\) 2.01841i 0.356808i
\(33\) 2.35626i 0.410171i
\(34\) 3.37126 0.578167
\(35\) 0 0
\(36\) 0.360976 0.0601627
\(37\) − 9.75097i − 1.60305i −0.597962 0.801525i \(-0.704023\pi\)
0.597962 0.801525i \(-0.295977\pi\)
\(38\) 8.77474i 1.42345i
\(39\) 1.34951 0.216094
\(40\) 0 0
\(41\) 1.85550 0.289780 0.144890 0.989448i \(-0.453717\pi\)
0.144890 + 0.989448i \(0.453717\pi\)
\(42\) 2.29790i 0.354574i
\(43\) − 8.01874i − 1.22285i −0.791304 0.611423i \(-0.790597\pi\)
0.791304 0.611423i \(-0.209403\pi\)
\(44\) 0.850553 0.128226
\(45\) 0 0
\(46\) 13.5139 1.99252
\(47\) − 6.66298i − 0.971895i −0.873988 0.485948i \(-0.838475\pi\)
0.873988 0.485948i \(-0.161525\pi\)
\(48\) 4.59165i 0.662747i
\(49\) 4.76349 0.680499
\(50\) 0 0
\(51\) 2.19405 0.307228
\(52\) − 0.487140i − 0.0675542i
\(53\) 4.17153i 0.573003i 0.958080 + 0.286502i \(0.0924924\pi\)
−0.958080 + 0.286502i \(0.907508\pi\)
\(54\) 1.53655 0.209098
\(55\) 0 0
\(56\) −3.76631 −0.503295
\(57\) 5.71069i 0.756399i
\(58\) 12.1390i 1.59393i
\(59\) −11.0647 −1.44050 −0.720248 0.693716i \(-0.755972\pi\)
−0.720248 + 0.693716i \(0.755972\pi\)
\(60\) 0 0
\(61\) −12.2372 −1.56682 −0.783408 0.621508i \(-0.786520\pi\)
−0.783408 + 0.621508i \(0.786520\pi\)
\(62\) 5.68088i 0.721472i
\(63\) 1.49550i 0.188415i
\(64\) −6.08192 −0.760239
\(65\) 0 0
\(66\) 3.62050 0.445652
\(67\) − 4.31358i − 0.526988i −0.964661 0.263494i \(-0.915125\pi\)
0.964661 0.263494i \(-0.0848749\pi\)
\(68\) − 0.792000i − 0.0960442i
\(69\) 8.79501 1.05879
\(70\) 0 0
\(71\) 5.77750 0.685663 0.342832 0.939397i \(-0.388614\pi\)
0.342832 + 0.939397i \(0.388614\pi\)
\(72\) 2.51844i 0.296801i
\(73\) 6.92684i 0.810725i 0.914156 + 0.405362i \(0.132855\pi\)
−0.914156 + 0.405362i \(0.867145\pi\)
\(74\) −14.9828 −1.74172
\(75\) 0 0
\(76\) 2.06142 0.236461
\(77\) 3.52377i 0.401571i
\(78\) − 2.07358i − 0.234787i
\(79\) 10.6687 1.20033 0.600163 0.799878i \(-0.295102\pi\)
0.600163 + 0.799878i \(0.295102\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 2.85106i − 0.314846i
\(83\) 0.224003i 0.0245875i 0.999924 + 0.0122938i \(0.00391332\pi\)
−0.999924 + 0.0122938i \(0.996087\pi\)
\(84\) 0.539839 0.0589012
\(85\) 0 0
\(86\) −12.3212 −1.32863
\(87\) 7.90017i 0.846987i
\(88\) 5.93408i 0.632575i
\(89\) −0.429167 −0.0454916 −0.0227458 0.999741i \(-0.507241\pi\)
−0.0227458 + 0.999741i \(0.507241\pi\)
\(90\) 0 0
\(91\) 2.01818 0.211563
\(92\) − 3.17479i − 0.330995i
\(93\) 3.69717i 0.383379i
\(94\) −10.2380 −1.05597
\(95\) 0 0
\(96\) 2.01841 0.206003
\(97\) − 12.4945i − 1.26863i −0.773075 0.634315i \(-0.781282\pi\)
0.773075 0.634315i \(-0.218718\pi\)
\(98\) − 7.31933i − 0.739364i
\(99\) 2.35626 0.236813
\(100\) 0 0
\(101\) 8.19767 0.815698 0.407849 0.913049i \(-0.366279\pi\)
0.407849 + 0.913049i \(0.366279\pi\)
\(102\) − 3.37126i − 0.333805i
\(103\) 2.50005i 0.246337i 0.992386 + 0.123169i \(0.0393056\pi\)
−0.992386 + 0.123169i \(0.960694\pi\)
\(104\) 3.39865 0.333265
\(105\) 0 0
\(106\) 6.40975 0.622570
\(107\) 1.81004i 0.174983i 0.996165 + 0.0874914i \(0.0278850\pi\)
−0.996165 + 0.0874914i \(0.972115\pi\)
\(108\) − 0.360976i − 0.0347350i
\(109\) 2.94778 0.282346 0.141173 0.989985i \(-0.454913\pi\)
0.141173 + 0.989985i \(0.454913\pi\)
\(110\) 0 0
\(111\) −9.75097 −0.925521
\(112\) 6.86679i 0.648851i
\(113\) 13.8365i 1.30163i 0.759238 + 0.650813i \(0.225572\pi\)
−0.759238 + 0.650813i \(0.774428\pi\)
\(114\) 8.77474 0.821829
\(115\) 0 0
\(116\) 2.85178 0.264781
\(117\) − 1.34951i − 0.124762i
\(118\) 17.0014i 1.56510i
\(119\) 3.28119 0.300787
\(120\) 0 0
\(121\) −5.44806 −0.495278
\(122\) 18.8031i 1.70235i
\(123\) − 1.85550i − 0.167304i
\(124\) 1.33459 0.119850
\(125\) 0 0
\(126\) 2.29790 0.204713
\(127\) − 5.73995i − 0.509338i −0.967028 0.254669i \(-0.918033\pi\)
0.967028 0.254669i \(-0.0819666\pi\)
\(128\) 13.3820i 1.18281i
\(129\) −8.01874 −0.706011
\(130\) 0 0
\(131\) −4.35756 −0.380722 −0.190361 0.981714i \(-0.560966\pi\)
−0.190361 + 0.981714i \(0.560966\pi\)
\(132\) − 0.850553i − 0.0740311i
\(133\) 8.54031i 0.740539i
\(134\) −6.62802 −0.572574
\(135\) 0 0
\(136\) 5.52558 0.473814
\(137\) − 1.97461i − 0.168702i −0.996436 0.0843510i \(-0.973118\pi\)
0.996436 0.0843510i \(-0.0268817\pi\)
\(138\) − 13.5139i − 1.15038i
\(139\) −1.67910 −0.142419 −0.0712095 0.997461i \(-0.522686\pi\)
−0.0712095 + 0.997461i \(0.522686\pi\)
\(140\) 0 0
\(141\) −6.66298 −0.561124
\(142\) − 8.87740i − 0.744975i
\(143\) − 3.17978i − 0.265907i
\(144\) 4.59165 0.382637
\(145\) 0 0
\(146\) 10.6434 0.880855
\(147\) − 4.76349i − 0.392886i
\(148\) 3.51987i 0.289331i
\(149\) 7.38524 0.605023 0.302511 0.953146i \(-0.402175\pi\)
0.302511 + 0.953146i \(0.402175\pi\)
\(150\) 0 0
\(151\) −4.26137 −0.346785 −0.173393 0.984853i \(-0.555473\pi\)
−0.173393 + 0.984853i \(0.555473\pi\)
\(152\) 14.3820i 1.16653i
\(153\) − 2.19405i − 0.177378i
\(154\) 5.41444 0.436308
\(155\) 0 0
\(156\) −0.487140 −0.0390024
\(157\) − 16.0573i − 1.28152i −0.767743 0.640758i \(-0.778620\pi\)
0.767743 0.640758i \(-0.221380\pi\)
\(158\) − 16.3930i − 1.30416i
\(159\) 4.17153 0.330824
\(160\) 0 0
\(161\) 13.1529 1.03659
\(162\) − 1.53655i − 0.120723i
\(163\) − 22.2938i − 1.74618i −0.487556 0.873092i \(-0.662111\pi\)
0.487556 0.873092i \(-0.337889\pi\)
\(164\) −0.669790 −0.0523018
\(165\) 0 0
\(166\) 0.344191 0.0267144
\(167\) 6.46601i 0.500355i 0.968200 + 0.250177i \(0.0804889\pi\)
−0.968200 + 0.250177i \(0.919511\pi\)
\(168\) 3.76631i 0.290577i
\(169\) 11.1788 0.859910
\(170\) 0 0
\(171\) 5.71069 0.436707
\(172\) 2.89458i 0.220709i
\(173\) − 11.8180i − 0.898509i −0.893404 0.449255i \(-0.851690\pi\)
0.893404 0.449255i \(-0.148310\pi\)
\(174\) 12.1390 0.920254
\(175\) 0 0
\(176\) 10.8191 0.815520
\(177\) 11.0647i 0.831671i
\(178\) 0.659435i 0.0494268i
\(179\) −15.5746 −1.16410 −0.582049 0.813154i \(-0.697749\pi\)
−0.582049 + 0.813154i \(0.697749\pi\)
\(180\) 0 0
\(181\) −14.5797 −1.08370 −0.541851 0.840475i \(-0.682276\pi\)
−0.541851 + 0.840475i \(0.682276\pi\)
\(182\) − 3.10103i − 0.229864i
\(183\) 12.2372i 0.904602i
\(184\) 22.1497 1.63290
\(185\) 0 0
\(186\) 5.68088 0.416542
\(187\) − 5.16974i − 0.378049i
\(188\) 2.40518i 0.175416i
\(189\) 1.49550 0.108781
\(190\) 0 0
\(191\) −20.9884 −1.51867 −0.759333 0.650702i \(-0.774475\pi\)
−0.759333 + 0.650702i \(0.774475\pi\)
\(192\) 6.08192i 0.438924i
\(193\) 22.7094i 1.63466i 0.576173 + 0.817328i \(0.304546\pi\)
−0.576173 + 0.817328i \(0.695454\pi\)
\(194\) −19.1985 −1.37837
\(195\) 0 0
\(196\) −1.71951 −0.122822
\(197\) − 1.35341i − 0.0964268i −0.998837 0.0482134i \(-0.984647\pi\)
0.998837 0.0482134i \(-0.0153528\pi\)
\(198\) − 3.62050i − 0.257297i
\(199\) 8.96061 0.635201 0.317600 0.948225i \(-0.397123\pi\)
0.317600 + 0.948225i \(0.397123\pi\)
\(200\) 0 0
\(201\) −4.31358 −0.304257
\(202\) − 12.5961i − 0.886259i
\(203\) 11.8147i 0.829228i
\(204\) −0.792000 −0.0554511
\(205\) 0 0
\(206\) 3.84145 0.267646
\(207\) − 8.79501i − 0.611295i
\(208\) − 6.19646i − 0.429647i
\(209\) 13.4558 0.930760
\(210\) 0 0
\(211\) 5.83983 0.402031 0.201015 0.979588i \(-0.435576\pi\)
0.201015 + 0.979588i \(0.435576\pi\)
\(212\) − 1.50582i − 0.103420i
\(213\) − 5.77750i − 0.395868i
\(214\) 2.78121 0.190119
\(215\) 0 0
\(216\) 2.51844 0.171358
\(217\) 5.52910i 0.375340i
\(218\) − 4.52939i − 0.306769i
\(219\) 6.92684 0.468072
\(220\) 0 0
\(221\) −2.96088 −0.199171
\(222\) 14.9828i 1.00558i
\(223\) − 7.82097i − 0.523731i −0.965104 0.261865i \(-0.915662\pi\)
0.965104 0.261865i \(-0.0843377\pi\)
\(224\) 3.01853 0.201684
\(225\) 0 0
\(226\) 21.2604 1.41422
\(227\) − 16.3090i − 1.08246i −0.840873 0.541232i \(-0.817958\pi\)
0.840873 0.541232i \(-0.182042\pi\)
\(228\) − 2.06142i − 0.136521i
\(229\) 21.7088 1.43456 0.717278 0.696787i \(-0.245388\pi\)
0.717278 + 0.696787i \(0.245388\pi\)
\(230\) 0 0
\(231\) 3.52377 0.231847
\(232\) 19.8961i 1.30624i
\(233\) − 13.5341i − 0.886646i −0.896362 0.443323i \(-0.853799\pi\)
0.896362 0.443323i \(-0.146201\pi\)
\(234\) −2.07358 −0.135554
\(235\) 0 0
\(236\) 3.99408 0.259993
\(237\) − 10.6687i − 0.693009i
\(238\) − 5.04171i − 0.326805i
\(239\) 10.5338 0.681377 0.340689 0.940176i \(-0.389340\pi\)
0.340689 + 0.940176i \(0.389340\pi\)
\(240\) 0 0
\(241\) 19.4838 1.25506 0.627530 0.778592i \(-0.284066\pi\)
0.627530 + 0.778592i \(0.284066\pi\)
\(242\) 8.37120i 0.538121i
\(243\) − 1.00000i − 0.0641500i
\(244\) 4.41735 0.282792
\(245\) 0 0
\(246\) −2.85106 −0.181777
\(247\) − 7.70661i − 0.490360i
\(248\) 9.31109i 0.591255i
\(249\) 0.224003 0.0141956
\(250\) 0 0
\(251\) −20.9446 −1.32201 −0.661007 0.750380i \(-0.729871\pi\)
−0.661007 + 0.750380i \(0.729871\pi\)
\(252\) − 0.539839i − 0.0340066i
\(253\) − 20.7233i − 1.30286i
\(254\) −8.81971 −0.553398
\(255\) 0 0
\(256\) 8.39819 0.524887
\(257\) 1.67121i 0.104247i 0.998641 + 0.0521237i \(0.0165990\pi\)
−0.998641 + 0.0521237i \(0.983401\pi\)
\(258\) 12.3212i 0.767083i
\(259\) −14.5825 −0.906115
\(260\) 0 0
\(261\) 7.90017 0.489008
\(262\) 6.69559i 0.413655i
\(263\) 7.55667i 0.465964i 0.972481 + 0.232982i \(0.0748483\pi\)
−0.972481 + 0.232982i \(0.925152\pi\)
\(264\) 5.93408 0.365217
\(265\) 0 0
\(266\) 13.1226 0.804597
\(267\) 0.429167i 0.0262646i
\(268\) 1.55710i 0.0951151i
\(269\) 11.2841 0.688005 0.344002 0.938969i \(-0.388217\pi\)
0.344002 + 0.938969i \(0.388217\pi\)
\(270\) 0 0
\(271\) −10.9752 −0.666696 −0.333348 0.942804i \(-0.608178\pi\)
−0.333348 + 0.942804i \(0.608178\pi\)
\(272\) − 10.0743i − 0.610845i
\(273\) − 2.01818i − 0.122146i
\(274\) −3.03407 −0.183295
\(275\) 0 0
\(276\) −3.17479 −0.191100
\(277\) 5.75112i 0.345551i 0.984961 + 0.172776i \(0.0552735\pi\)
−0.984961 + 0.172776i \(0.944726\pi\)
\(278\) 2.58001i 0.154739i
\(279\) 3.69717 0.221344
\(280\) 0 0
\(281\) 8.49962 0.507045 0.253522 0.967330i \(-0.418411\pi\)
0.253522 + 0.967330i \(0.418411\pi\)
\(282\) 10.2380i 0.609663i
\(283\) − 17.2248i − 1.02391i −0.859013 0.511955i \(-0.828922\pi\)
0.859013 0.511955i \(-0.171078\pi\)
\(284\) −2.08554 −0.123754
\(285\) 0 0
\(286\) −4.88588 −0.288908
\(287\) − 2.77489i − 0.163796i
\(288\) − 2.01841i − 0.118936i
\(289\) 12.1861 0.716832
\(290\) 0 0
\(291\) −12.4945 −0.732443
\(292\) − 2.50042i − 0.146326i
\(293\) 9.38764i 0.548432i 0.961668 + 0.274216i \(0.0884183\pi\)
−0.961668 + 0.274216i \(0.911582\pi\)
\(294\) −7.31933 −0.426872
\(295\) 0 0
\(296\) −24.5572 −1.42736
\(297\) − 2.35626i − 0.136724i
\(298\) − 11.3478i − 0.657359i
\(299\) −11.8689 −0.686397
\(300\) 0 0
\(301\) −11.9920 −0.691207
\(302\) 6.54779i 0.376783i
\(303\) − 8.19767i − 0.470944i
\(304\) 26.2215 1.50390
\(305\) 0 0
\(306\) −3.37126 −0.192722
\(307\) − 10.6465i − 0.607627i −0.952731 0.303814i \(-0.901740\pi\)
0.952731 0.303814i \(-0.0982600\pi\)
\(308\) − 1.27200i − 0.0724788i
\(309\) 2.50005 0.142223
\(310\) 0 0
\(311\) −27.3572 −1.55128 −0.775641 0.631174i \(-0.782573\pi\)
−0.775641 + 0.631174i \(0.782573\pi\)
\(312\) − 3.39865i − 0.192410i
\(313\) − 1.99509i − 0.112769i −0.998409 0.0563846i \(-0.982043\pi\)
0.998409 0.0563846i \(-0.0179573\pi\)
\(314\) −24.6729 −1.39237
\(315\) 0 0
\(316\) −3.85116 −0.216645
\(317\) − 8.64876i − 0.485763i −0.970056 0.242881i \(-0.921907\pi\)
0.970056 0.242881i \(-0.0780925\pi\)
\(318\) − 6.40975i − 0.359441i
\(319\) 18.6148 1.04223
\(320\) 0 0
\(321\) 1.81004 0.101026
\(322\) − 20.2101i − 1.12626i
\(323\) − 12.5295i − 0.697162i
\(324\) −0.360976 −0.0200542
\(325\) 0 0
\(326\) −34.2554 −1.89723
\(327\) − 2.94778i − 0.163012i
\(328\) − 4.67295i − 0.258020i
\(329\) −9.96446 −0.549358
\(330\) 0 0
\(331\) −3.94331 −0.216744 −0.108372 0.994110i \(-0.534564\pi\)
−0.108372 + 0.994110i \(0.534564\pi\)
\(332\) − 0.0808598i − 0.00443776i
\(333\) 9.75097i 0.534350i
\(334\) 9.93532 0.543637
\(335\) 0 0
\(336\) 6.86679 0.374614
\(337\) − 3.95471i − 0.215427i −0.994182 0.107713i \(-0.965647\pi\)
0.994182 0.107713i \(-0.0343529\pi\)
\(338\) − 17.1768i − 0.934295i
\(339\) 13.8365 0.751494
\(340\) 0 0
\(341\) 8.71148 0.471753
\(342\) − 8.77474i − 0.474483i
\(343\) − 17.5923i − 0.949893i
\(344\) −20.1947 −1.08882
\(345\) 0 0
\(346\) −18.1590 −0.976233
\(347\) 9.99596i 0.536611i 0.963334 + 0.268306i \(0.0864637\pi\)
−0.963334 + 0.268306i \(0.913536\pi\)
\(348\) − 2.85178i − 0.152871i
\(349\) −18.4534 −0.987789 −0.493895 0.869522i \(-0.664427\pi\)
−0.493895 + 0.869522i \(0.664427\pi\)
\(350\) 0 0
\(351\) −1.34951 −0.0720313
\(352\) − 4.75589i − 0.253490i
\(353\) 9.32398i 0.496265i 0.968726 + 0.248133i \(0.0798169\pi\)
−0.968726 + 0.248133i \(0.920183\pi\)
\(354\) 17.0014 0.903613
\(355\) 0 0
\(356\) 0.154919 0.00821070
\(357\) − 3.28119i − 0.173659i
\(358\) 23.9311i 1.26480i
\(359\) −28.9438 −1.52759 −0.763797 0.645457i \(-0.776667\pi\)
−0.763797 + 0.645457i \(0.776667\pi\)
\(360\) 0 0
\(361\) 13.6119 0.716418
\(362\) 22.4024i 1.17744i
\(363\) 5.44806i 0.285949i
\(364\) −0.728516 −0.0381846
\(365\) 0 0
\(366\) 18.8031 0.982852
\(367\) 3.99869i 0.208730i 0.994539 + 0.104365i \(0.0332810\pi\)
−0.994539 + 0.104365i \(0.966719\pi\)
\(368\) − 40.3836i − 2.10514i
\(369\) −1.85550 −0.0965932
\(370\) 0 0
\(371\) 6.23850 0.323887
\(372\) − 1.33459i − 0.0691953i
\(373\) − 3.17260i − 0.164271i −0.996621 0.0821354i \(-0.973826\pi\)
0.996621 0.0821354i \(-0.0261740\pi\)
\(374\) −7.94355 −0.410751
\(375\) 0 0
\(376\) −16.7803 −0.865377
\(377\) − 10.6613i − 0.549086i
\(378\) − 2.29790i − 0.118191i
\(379\) −28.5206 −1.46500 −0.732501 0.680766i \(-0.761647\pi\)
−0.732501 + 0.680766i \(0.761647\pi\)
\(380\) 0 0
\(381\) −5.73995 −0.294067
\(382\) 32.2497i 1.65004i
\(383\) − 11.4611i − 0.585636i −0.956168 0.292818i \(-0.905407\pi\)
0.956168 0.292818i \(-0.0945929\pi\)
\(384\) 13.3820 0.682896
\(385\) 0 0
\(386\) 34.8940 1.77606
\(387\) 8.01874i 0.407616i
\(388\) 4.51024i 0.228973i
\(389\) −34.2463 −1.73636 −0.868179 0.496252i \(-0.834709\pi\)
−0.868179 + 0.496252i \(0.834709\pi\)
\(390\) 0 0
\(391\) −19.2967 −0.975876
\(392\) − 11.9966i − 0.605917i
\(393\) 4.35756i 0.219810i
\(394\) −2.07958 −0.104768
\(395\) 0 0
\(396\) −0.850553 −0.0427419
\(397\) 20.0333i 1.00545i 0.864448 + 0.502723i \(0.167668\pi\)
−0.864448 + 0.502723i \(0.832332\pi\)
\(398\) − 13.7684i − 0.690148i
\(399\) 8.54031 0.427550
\(400\) 0 0
\(401\) 4.98200 0.248789 0.124395 0.992233i \(-0.460301\pi\)
0.124395 + 0.992233i \(0.460301\pi\)
\(402\) 6.62802i 0.330576i
\(403\) − 4.98935i − 0.248537i
\(404\) −2.95916 −0.147224
\(405\) 0 0
\(406\) 18.1538 0.900958
\(407\) 22.9758i 1.13887i
\(408\) − 5.52558i − 0.273557i
\(409\) −23.8591 −1.17976 −0.589878 0.807493i \(-0.700824\pi\)
−0.589878 + 0.807493i \(0.700824\pi\)
\(410\) 0 0
\(411\) −1.97461 −0.0974001
\(412\) − 0.902460i − 0.0444610i
\(413\) 16.5472i 0.814233i
\(414\) −13.5139 −0.664174
\(415\) 0 0
\(416\) −2.72386 −0.133548
\(417\) 1.67910i 0.0822257i
\(418\) − 20.6755i − 1.01127i
\(419\) −0.482550 −0.0235741 −0.0117871 0.999931i \(-0.503752\pi\)
−0.0117871 + 0.999931i \(0.503752\pi\)
\(420\) 0 0
\(421\) −17.7183 −0.863537 −0.431769 0.901984i \(-0.642110\pi\)
−0.431769 + 0.901984i \(0.642110\pi\)
\(422\) − 8.97317i − 0.436807i
\(423\) 6.66298i 0.323965i
\(424\) 10.5057 0.510203
\(425\) 0 0
\(426\) −8.87740 −0.430112
\(427\) 18.3007i 0.885634i
\(428\) − 0.653380i − 0.0315823i
\(429\) −3.17978 −0.153521
\(430\) 0 0
\(431\) 26.8061 1.29121 0.645603 0.763673i \(-0.276606\pi\)
0.645603 + 0.763673i \(0.276606\pi\)
\(432\) − 4.59165i − 0.220916i
\(433\) − 9.23115i − 0.443621i −0.975090 0.221810i \(-0.928803\pi\)
0.975090 0.221810i \(-0.0711966\pi\)
\(434\) 8.49573 0.407808
\(435\) 0 0
\(436\) −1.06408 −0.0509601
\(437\) − 50.2255i − 2.40261i
\(438\) − 10.6434i − 0.508562i
\(439\) −1.00240 −0.0478421 −0.0239211 0.999714i \(-0.507615\pi\)
−0.0239211 + 0.999714i \(0.507615\pi\)
\(440\) 0 0
\(441\) −4.76349 −0.226833
\(442\) 4.54954i 0.216399i
\(443\) − 26.2872i − 1.24894i −0.781048 0.624471i \(-0.785315\pi\)
0.781048 0.624471i \(-0.214685\pi\)
\(444\) 3.51987 0.167046
\(445\) 0 0
\(446\) −12.0173 −0.569035
\(447\) − 7.38524i − 0.349310i
\(448\) 9.09548i 0.429721i
\(449\) −4.75449 −0.224378 −0.112189 0.993687i \(-0.535786\pi\)
−0.112189 + 0.993687i \(0.535786\pi\)
\(450\) 0 0
\(451\) −4.37202 −0.205870
\(452\) − 4.99464i − 0.234928i
\(453\) 4.26137i 0.200216i
\(454\) −25.0595 −1.17610
\(455\) 0 0
\(456\) 14.3820 0.673499
\(457\) − 15.9703i − 0.747059i −0.927618 0.373529i \(-0.878148\pi\)
0.927618 0.373529i \(-0.121852\pi\)
\(458\) − 33.3566i − 1.55865i
\(459\) −2.19405 −0.102409
\(460\) 0 0
\(461\) 39.7558 1.85161 0.925806 0.377998i \(-0.123387\pi\)
0.925806 + 0.377998i \(0.123387\pi\)
\(462\) − 5.41444i − 0.251902i
\(463\) − 26.1209i − 1.21394i −0.794724 0.606971i \(-0.792384\pi\)
0.794724 0.606971i \(-0.207616\pi\)
\(464\) 36.2748 1.68402
\(465\) 0 0
\(466\) −20.7957 −0.963343
\(467\) − 3.85204i − 0.178251i −0.996020 0.0891256i \(-0.971593\pi\)
0.996020 0.0891256i \(-0.0284073\pi\)
\(468\) 0.487140i 0.0225181i
\(469\) −6.45095 −0.297877
\(470\) 0 0
\(471\) −16.0573 −0.739883
\(472\) 27.8657i 1.28262i
\(473\) 18.8942i 0.868756i
\(474\) −16.3930 −0.752956
\(475\) 0 0
\(476\) −1.18443 −0.0542884
\(477\) − 4.17153i − 0.191001i
\(478\) − 16.1857i − 0.740318i
\(479\) 14.2698 0.652004 0.326002 0.945369i \(-0.394298\pi\)
0.326002 + 0.945369i \(0.394298\pi\)
\(480\) 0 0
\(481\) 13.1590 0.599998
\(482\) − 29.9377i − 1.36363i
\(483\) − 13.1529i − 0.598478i
\(484\) 1.96662 0.0893919
\(485\) 0 0
\(486\) −1.53655 −0.0696992
\(487\) 24.8222i 1.12480i 0.826865 + 0.562401i \(0.190122\pi\)
−0.826865 + 0.562401i \(0.809878\pi\)
\(488\) 30.8187i 1.39510i
\(489\) −22.2938 −1.00816
\(490\) 0 0
\(491\) 11.4893 0.518507 0.259253 0.965809i \(-0.416524\pi\)
0.259253 + 0.965809i \(0.416524\pi\)
\(492\) 0.669790i 0.0301965i
\(493\) − 17.3334i − 0.780656i
\(494\) −11.8416 −0.532777
\(495\) 0 0
\(496\) 16.9761 0.762250
\(497\) − 8.64023i − 0.387567i
\(498\) − 0.344191i − 0.0154236i
\(499\) 4.17487 0.186893 0.0934465 0.995624i \(-0.470212\pi\)
0.0934465 + 0.995624i \(0.470212\pi\)
\(500\) 0 0
\(501\) 6.46601 0.288880
\(502\) 32.1824i 1.43637i
\(503\) − 38.7163i − 1.72627i −0.504970 0.863137i \(-0.668497\pi\)
0.504970 0.863137i \(-0.331503\pi\)
\(504\) 3.76631 0.167765
\(505\) 0 0
\(506\) −31.8423 −1.41556
\(507\) − 11.1788i − 0.496469i
\(508\) 2.07199i 0.0919296i
\(509\) −30.1797 −1.33769 −0.668845 0.743402i \(-0.733211\pi\)
−0.668845 + 0.743402i \(0.733211\pi\)
\(510\) 0 0
\(511\) 10.3591 0.458258
\(512\) 13.8597i 0.612519i
\(513\) − 5.71069i − 0.252133i
\(514\) 2.56790 0.113265
\(515\) 0 0
\(516\) 2.89458 0.127427
\(517\) 15.6997i 0.690471i
\(518\) 22.4067i 0.984496i
\(519\) −11.8180 −0.518755
\(520\) 0 0
\(521\) −25.4856 −1.11654 −0.558272 0.829658i \(-0.688536\pi\)
−0.558272 + 0.829658i \(0.688536\pi\)
\(522\) − 12.1390i − 0.531309i
\(523\) 3.89180i 0.170176i 0.996373 + 0.0850882i \(0.0271172\pi\)
−0.996373 + 0.0850882i \(0.972883\pi\)
\(524\) 1.57298 0.0687158
\(525\) 0 0
\(526\) 11.6112 0.506271
\(527\) − 8.11178i − 0.353355i
\(528\) − 10.8191i − 0.470841i
\(529\) −54.3522 −2.36314
\(530\) 0 0
\(531\) 11.0647 0.480166
\(532\) − 3.08285i − 0.133659i
\(533\) 2.50400i 0.108460i
\(534\) 0.659435 0.0285366
\(535\) 0 0
\(536\) −10.8635 −0.469231
\(537\) 15.5746i 0.672092i
\(538\) − 17.3386i − 0.747519i
\(539\) −11.2240 −0.483452
\(540\) 0 0
\(541\) 40.2148 1.72897 0.864484 0.502661i \(-0.167646\pi\)
0.864484 + 0.502661i \(0.167646\pi\)
\(542\) 16.8639i 0.724367i
\(543\) 14.5797i 0.625675i
\(544\) −4.42849 −0.189870
\(545\) 0 0
\(546\) −3.10103 −0.132712
\(547\) − 7.37923i − 0.315513i −0.987478 0.157757i \(-0.949574\pi\)
0.987478 0.157757i \(-0.0504261\pi\)
\(548\) 0.712786i 0.0304487i
\(549\) 12.2372 0.522272
\(550\) 0 0
\(551\) 45.1154 1.92198
\(552\) − 22.1497i − 0.942753i
\(553\) − 15.9550i − 0.678478i
\(554\) 8.83686 0.375442
\(555\) 0 0
\(556\) 0.606114 0.0257050
\(557\) − 1.52499i − 0.0646160i −0.999478 0.0323080i \(-0.989714\pi\)
0.999478 0.0323080i \(-0.0102857\pi\)
\(558\) − 5.68088i − 0.240491i
\(559\) 10.8213 0.457694
\(560\) 0 0
\(561\) −5.16974 −0.218267
\(562\) − 13.0601i − 0.550905i
\(563\) 13.7955i 0.581411i 0.956813 + 0.290706i \(0.0938901\pi\)
−0.956813 + 0.290706i \(0.906110\pi\)
\(564\) 2.40518 0.101276
\(565\) 0 0
\(566\) −26.4667 −1.11248
\(567\) − 1.49550i − 0.0628049i
\(568\) − 14.5503i − 0.610516i
\(569\) 8.49904 0.356298 0.178149 0.984004i \(-0.442989\pi\)
0.178149 + 0.984004i \(0.442989\pi\)
\(570\) 0 0
\(571\) 39.3230 1.64562 0.822809 0.568319i \(-0.192406\pi\)
0.822809 + 0.568319i \(0.192406\pi\)
\(572\) 1.14783i 0.0479930i
\(573\) 20.9884i 0.876803i
\(574\) −4.26374 −0.177965
\(575\) 0 0
\(576\) 6.08192 0.253413
\(577\) − 24.5832i − 1.02341i −0.859160 0.511707i \(-0.829013\pi\)
0.859160 0.511707i \(-0.170987\pi\)
\(578\) − 18.7246i − 0.778840i
\(579\) 22.7094 0.943769
\(580\) 0 0
\(581\) 0.334995 0.0138980
\(582\) 19.1985i 0.795802i
\(583\) − 9.82918i − 0.407083i
\(584\) 17.4448 0.721871
\(585\) 0 0
\(586\) 14.4245 0.595872
\(587\) 9.24270i 0.381487i 0.981640 + 0.190744i \(0.0610899\pi\)
−0.981640 + 0.190744i \(0.938910\pi\)
\(588\) 1.71951i 0.0709113i
\(589\) 21.1134 0.869962
\(590\) 0 0
\(591\) −1.35341 −0.0556720
\(592\) 44.7730i 1.84016i
\(593\) − 6.07888i − 0.249630i −0.992180 0.124815i \(-0.960166\pi\)
0.992180 0.124815i \(-0.0398337\pi\)
\(594\) −3.62050 −0.148551
\(595\) 0 0
\(596\) −2.66590 −0.109199
\(597\) − 8.96061i − 0.366733i
\(598\) 18.2372i 0.745773i
\(599\) −6.40129 −0.261550 −0.130775 0.991412i \(-0.541746\pi\)
−0.130775 + 0.991412i \(0.541746\pi\)
\(600\) 0 0
\(601\) −38.4675 −1.56912 −0.784560 0.620052i \(-0.787111\pi\)
−0.784560 + 0.620052i \(0.787111\pi\)
\(602\) 18.4263i 0.750999i
\(603\) 4.31358i 0.175663i
\(604\) 1.53825 0.0625906
\(605\) 0 0
\(606\) −12.5961 −0.511682
\(607\) − 5.22464i − 0.212062i −0.994363 0.106031i \(-0.966186\pi\)
0.994363 0.106031i \(-0.0338142\pi\)
\(608\) − 11.5265i − 0.467462i
\(609\) 11.8147 0.478755
\(610\) 0 0
\(611\) 8.99173 0.363766
\(612\) 0.792000i 0.0320147i
\(613\) 29.8153i 1.20423i 0.798410 + 0.602114i \(0.205675\pi\)
−0.798410 + 0.602114i \(0.794325\pi\)
\(614\) −16.3588 −0.660189
\(615\) 0 0
\(616\) 8.87439 0.357559
\(617\) − 23.3025i − 0.938124i −0.883165 0.469062i \(-0.844592\pi\)
0.883165 0.469062i \(-0.155408\pi\)
\(618\) − 3.84145i − 0.154526i
\(619\) 0.236492 0.00950543 0.00475272 0.999989i \(-0.498487\pi\)
0.00475272 + 0.999989i \(0.498487\pi\)
\(620\) 0 0
\(621\) −8.79501 −0.352932
\(622\) 42.0356i 1.68547i
\(623\) 0.641818i 0.0257139i
\(624\) −6.19646 −0.248057
\(625\) 0 0
\(626\) −3.06555 −0.122524
\(627\) − 13.4558i − 0.537374i
\(628\) 5.79632i 0.231298i
\(629\) 21.3941 0.853039
\(630\) 0 0
\(631\) −17.8789 −0.711748 −0.355874 0.934534i \(-0.615817\pi\)
−0.355874 + 0.934534i \(0.615817\pi\)
\(632\) − 26.8685i − 1.06877i
\(633\) − 5.83983i − 0.232112i
\(634\) −13.2892 −0.527782
\(635\) 0 0
\(636\) −1.50582 −0.0597098
\(637\) 6.42836i 0.254701i
\(638\) − 28.6025i − 1.13239i
\(639\) −5.77750 −0.228554
\(640\) 0 0
\(641\) 10.8680 0.429259 0.214630 0.976695i \(-0.431146\pi\)
0.214630 + 0.976695i \(0.431146\pi\)
\(642\) − 2.78121i − 0.109765i
\(643\) 3.09039i 0.121873i 0.998142 + 0.0609366i \(0.0194088\pi\)
−0.998142 + 0.0609366i \(0.980591\pi\)
\(644\) −4.74789 −0.187093
\(645\) 0 0
\(646\) −19.2522 −0.757468
\(647\) 5.53705i 0.217684i 0.994059 + 0.108842i \(0.0347142\pi\)
−0.994059 + 0.108842i \(0.965286\pi\)
\(648\) − 2.51844i − 0.0989335i
\(649\) 26.0712 1.02338
\(650\) 0 0
\(651\) 5.52910 0.216703
\(652\) 8.04753i 0.315166i
\(653\) 38.8754i 1.52131i 0.649154 + 0.760657i \(0.275123\pi\)
−0.649154 + 0.760657i \(0.724877\pi\)
\(654\) −4.52939 −0.177113
\(655\) 0 0
\(656\) −8.51978 −0.332642
\(657\) − 6.92684i − 0.270242i
\(658\) 15.3109i 0.596879i
\(659\) 20.5389 0.800082 0.400041 0.916497i \(-0.368996\pi\)
0.400041 + 0.916497i \(0.368996\pi\)
\(660\) 0 0
\(661\) 38.4254 1.49457 0.747287 0.664502i \(-0.231356\pi\)
0.747287 + 0.664502i \(0.231356\pi\)
\(662\) 6.05907i 0.235493i
\(663\) 2.96088i 0.114991i
\(664\) 0.564137 0.0218928
\(665\) 0 0
\(666\) 14.9828 0.580572
\(667\) − 69.4821i − 2.69036i
\(668\) − 2.33408i − 0.0903081i
\(669\) −7.82097 −0.302376
\(670\) 0 0
\(671\) 28.8340 1.11313
\(672\) − 3.01853i − 0.116442i
\(673\) − 12.9819i − 0.500417i −0.968192 0.250208i \(-0.919501\pi\)
0.968192 0.250208i \(-0.0804991\pi\)
\(674\) −6.07660 −0.234062
\(675\) 0 0
\(676\) −4.03529 −0.155204
\(677\) − 7.23957i − 0.278239i −0.990276 0.139120i \(-0.955573\pi\)
0.990276 0.139120i \(-0.0444273\pi\)
\(678\) − 21.2604i − 0.816500i
\(679\) −18.6855 −0.717086
\(680\) 0 0
\(681\) −16.3090 −0.624961
\(682\) − 13.3856i − 0.512561i
\(683\) 24.8800i 0.952005i 0.879444 + 0.476003i \(0.157915\pi\)
−0.879444 + 0.476003i \(0.842085\pi\)
\(684\) −2.06142 −0.0788205
\(685\) 0 0
\(686\) −27.0313 −1.03206
\(687\) − 21.7088i − 0.828241i
\(688\) 36.8192i 1.40372i
\(689\) −5.62950 −0.214467
\(690\) 0 0
\(691\) −33.2706 −1.26567 −0.632836 0.774286i \(-0.718109\pi\)
−0.632836 + 0.774286i \(0.718109\pi\)
\(692\) 4.26604i 0.162170i
\(693\) − 3.52377i − 0.133857i
\(694\) 15.3593 0.583029
\(695\) 0 0
\(696\) 19.8961 0.754159
\(697\) 4.07105i 0.154202i
\(698\) 28.3546i 1.07324i
\(699\) −13.5341 −0.511905
\(700\) 0 0
\(701\) −13.2163 −0.499173 −0.249586 0.968353i \(-0.580295\pi\)
−0.249586 + 0.968353i \(0.580295\pi\)
\(702\) 2.07358i 0.0782622i
\(703\) 55.6847i 2.10019i
\(704\) 14.3305 0.540103
\(705\) 0 0
\(706\) 14.3267 0.539194
\(707\) − 12.2596i − 0.461069i
\(708\) − 3.99408i − 0.150107i
\(709\) −6.25068 −0.234749 −0.117375 0.993088i \(-0.537448\pi\)
−0.117375 + 0.993088i \(0.537448\pi\)
\(710\) 0 0
\(711\) −10.6687 −0.400109
\(712\) 1.08083i 0.0405058i
\(713\) − 32.5167i − 1.21776i
\(714\) −5.04171 −0.188681
\(715\) 0 0
\(716\) 5.62205 0.210106
\(717\) − 10.5338i − 0.393393i
\(718\) 44.4735i 1.65973i
\(719\) −27.5016 −1.02564 −0.512818 0.858497i \(-0.671398\pi\)
−0.512818 + 0.858497i \(0.671398\pi\)
\(720\) 0 0
\(721\) 3.73882 0.139241
\(722\) − 20.9154i − 0.778390i
\(723\) − 19.4838i − 0.724610i
\(724\) 5.26293 0.195595
\(725\) 0 0
\(726\) 8.37120 0.310684
\(727\) 22.0397i 0.817406i 0.912668 + 0.408703i \(0.134019\pi\)
−0.912668 + 0.408703i \(0.865981\pi\)
\(728\) − 5.08266i − 0.188376i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 17.5935 0.650720
\(732\) − 4.41735i − 0.163270i
\(733\) 34.7134i 1.28217i 0.767470 + 0.641085i \(0.221515\pi\)
−0.767470 + 0.641085i \(0.778485\pi\)
\(734\) 6.14417 0.226785
\(735\) 0 0
\(736\) −17.7519 −0.654345
\(737\) 10.1639i 0.374392i
\(738\) 2.85106i 0.104949i
\(739\) 7.58593 0.279053 0.139526 0.990218i \(-0.455442\pi\)
0.139526 + 0.990218i \(0.455442\pi\)
\(740\) 0 0
\(741\) −7.70661 −0.283109
\(742\) − 9.58575i − 0.351904i
\(743\) 27.5328i 1.01008i 0.863096 + 0.505040i \(0.168522\pi\)
−0.863096 + 0.505040i \(0.831478\pi\)
\(744\) 9.31109 0.341361
\(745\) 0 0
\(746\) −4.87484 −0.178481
\(747\) − 0.224003i − 0.00819584i
\(748\) 1.86615i 0.0682334i
\(749\) 2.70690 0.0989081
\(750\) 0 0
\(751\) 4.24930 0.155059 0.0775296 0.996990i \(-0.475297\pi\)
0.0775296 + 0.996990i \(0.475297\pi\)
\(752\) 30.5940i 1.11565i
\(753\) 20.9446i 0.763265i
\(754\) −16.3816 −0.596584
\(755\) 0 0
\(756\) −0.539839 −0.0196337
\(757\) 45.6609i 1.65957i 0.558081 + 0.829787i \(0.311538\pi\)
−0.558081 + 0.829787i \(0.688462\pi\)
\(758\) 43.8232i 1.59173i
\(759\) −20.7233 −0.752208
\(760\) 0 0
\(761\) −40.1268 −1.45460 −0.727298 0.686322i \(-0.759224\pi\)
−0.727298 + 0.686322i \(0.759224\pi\)
\(762\) 8.81971i 0.319504i
\(763\) − 4.40839i − 0.159594i
\(764\) 7.57631 0.274101
\(765\) 0 0
\(766\) −17.6105 −0.636295
\(767\) − 14.9318i − 0.539157i
\(768\) − 8.39819i − 0.303044i
\(769\) 37.8350 1.36437 0.682183 0.731181i \(-0.261031\pi\)
0.682183 + 0.731181i \(0.261031\pi\)
\(770\) 0 0
\(771\) 1.67121 0.0601872
\(772\) − 8.19755i − 0.295036i
\(773\) − 26.2148i − 0.942881i −0.881898 0.471441i \(-0.843734\pi\)
0.881898 0.471441i \(-0.156266\pi\)
\(774\) 12.3212 0.442875
\(775\) 0 0
\(776\) −31.4667 −1.12959
\(777\) 14.5825i 0.523146i
\(778\) 52.6211i 1.88656i
\(779\) −10.5962 −0.379646
\(780\) 0 0
\(781\) −13.6133 −0.487121
\(782\) 29.6503i 1.06029i
\(783\) − 7.90017i − 0.282329i
\(784\) −21.8723 −0.781153
\(785\) 0 0
\(786\) 6.69559 0.238824
\(787\) 2.41254i 0.0859977i 0.999075 + 0.0429988i \(0.0136912\pi\)
−0.999075 + 0.0429988i \(0.986309\pi\)
\(788\) 0.488551i 0.0174039i
\(789\) 7.55667 0.269025
\(790\) 0 0
\(791\) 20.6924 0.735737
\(792\) − 5.93408i − 0.210858i
\(793\) − 16.5142i − 0.586437i
\(794\) 30.7822 1.09242
\(795\) 0 0
\(796\) −3.23457 −0.114646
\(797\) − 8.11230i − 0.287353i −0.989625 0.143676i \(-0.954108\pi\)
0.989625 0.143676i \(-0.0458924\pi\)
\(798\) − 13.1226i − 0.464534i
\(799\) 14.6189 0.517180
\(800\) 0 0
\(801\) 0.429167 0.0151639
\(802\) − 7.65507i − 0.270310i
\(803\) − 16.3214i − 0.575970i
\(804\) 1.55710 0.0549148
\(805\) 0 0
\(806\) −7.66638 −0.270037
\(807\) − 11.2841i − 0.397220i
\(808\) − 20.6453i − 0.726299i
\(809\) 40.7091 1.43125 0.715627 0.698482i \(-0.246141\pi\)
0.715627 + 0.698482i \(0.246141\pi\)
\(810\) 0 0
\(811\) −49.4990 −1.73814 −0.869072 0.494685i \(-0.835283\pi\)
−0.869072 + 0.494685i \(0.835283\pi\)
\(812\) − 4.26482i − 0.149666i
\(813\) 10.9752i 0.384917i
\(814\) 35.3033 1.23738
\(815\) 0 0
\(816\) −10.0743 −0.352671
\(817\) 45.7925i 1.60208i
\(818\) 36.6606i 1.28181i
\(819\) −2.01818 −0.0705210
\(820\) 0 0
\(821\) 42.2114 1.47319 0.736594 0.676335i \(-0.236433\pi\)
0.736594 + 0.676335i \(0.236433\pi\)
\(822\) 3.03407i 0.105825i
\(823\) − 49.2349i − 1.71622i −0.513464 0.858111i \(-0.671638\pi\)
0.513464 0.858111i \(-0.328362\pi\)
\(824\) 6.29622 0.219339
\(825\) 0 0
\(826\) 25.4255 0.884666
\(827\) − 51.0011i − 1.77348i −0.462266 0.886742i \(-0.652963\pi\)
0.462266 0.886742i \(-0.347037\pi\)
\(828\) 3.17479i 0.110332i
\(829\) 38.2342 1.32793 0.663964 0.747764i \(-0.268873\pi\)
0.663964 + 0.747764i \(0.268873\pi\)
\(830\) 0 0
\(831\) 5.75112 0.199504
\(832\) − 8.20758i − 0.284547i
\(833\) 10.4513i 0.362117i
\(834\) 2.58001 0.0893384
\(835\) 0 0
\(836\) −4.85724 −0.167991
\(837\) − 3.69717i − 0.127793i
\(838\) 0.741461i 0.0256133i
\(839\) 16.7086 0.576845 0.288422 0.957503i \(-0.406869\pi\)
0.288422 + 0.957503i \(0.406869\pi\)
\(840\) 0 0
\(841\) 33.4127 1.15216
\(842\) 27.2250i 0.938236i
\(843\) − 8.49962i − 0.292742i
\(844\) −2.10804 −0.0725618
\(845\) 0 0
\(846\) 10.2380 0.351989
\(847\) 8.14755i 0.279953i
\(848\) − 19.1542i − 0.657757i
\(849\) −17.2248 −0.591154
\(850\) 0 0
\(851\) 85.7599 2.93981
\(852\) 2.08554i 0.0714495i
\(853\) − 18.2644i − 0.625361i −0.949858 0.312681i \(-0.898773\pi\)
0.949858 0.312681i \(-0.101227\pi\)
\(854\) 28.1199 0.962244
\(855\) 0 0
\(856\) 4.55846 0.155805
\(857\) 53.4773i 1.82675i 0.407119 + 0.913375i \(0.366534\pi\)
−0.407119 + 0.913375i \(0.633466\pi\)
\(858\) 4.88588i 0.166801i
\(859\) 18.7575 0.639998 0.319999 0.947418i \(-0.396317\pi\)
0.319999 + 0.947418i \(0.396317\pi\)
\(860\) 0 0
\(861\) −2.77489 −0.0945678
\(862\) − 41.1889i − 1.40290i
\(863\) − 51.2363i − 1.74410i −0.489414 0.872051i \(-0.662789\pi\)
0.489414 0.872051i \(-0.337211\pi\)
\(864\) −2.01841 −0.0686677
\(865\) 0 0
\(866\) −14.1841 −0.481995
\(867\) − 12.1861i − 0.413863i
\(868\) − 1.99588i − 0.0677444i
\(869\) −25.1383 −0.852757
\(870\) 0 0
\(871\) 5.82121 0.197244
\(872\) − 7.42378i − 0.251401i
\(873\) 12.4945i 0.422876i
\(874\) −77.1739 −2.61045
\(875\) 0 0
\(876\) −2.50042 −0.0844815
\(877\) 6.06306i 0.204735i 0.994747 + 0.102368i \(0.0326418\pi\)
−0.994747 + 0.102368i \(0.967358\pi\)
\(878\) 1.54024i 0.0519806i
\(879\) 9.38764 0.316637
\(880\) 0 0
\(881\) 22.6698 0.763765 0.381883 0.924211i \(-0.375276\pi\)
0.381883 + 0.924211i \(0.375276\pi\)
\(882\) 7.31933i 0.246455i
\(883\) − 5.53899i − 0.186402i −0.995647 0.0932009i \(-0.970290\pi\)
0.995647 0.0932009i \(-0.0297099\pi\)
\(884\) 1.06881 0.0359480
\(885\) 0 0
\(886\) −40.3915 −1.35698
\(887\) 10.9716i 0.368390i 0.982890 + 0.184195i \(0.0589678\pi\)
−0.982890 + 0.184195i \(0.941032\pi\)
\(888\) 24.5572i 0.824086i
\(889\) −8.58408 −0.287901
\(890\) 0 0
\(891\) −2.35626 −0.0789375
\(892\) 2.82319i 0.0945272i
\(893\) 38.0502i 1.27330i
\(894\) −11.3478 −0.379526
\(895\) 0 0
\(896\) 20.0127 0.668577
\(897\) 11.8689i 0.396292i
\(898\) 7.30550i 0.243788i
\(899\) 29.2083 0.974151
\(900\) 0 0
\(901\) −9.15254 −0.304915
\(902\) 6.71781i 0.223679i
\(903\) 11.9920i 0.399069i
\(904\) 34.8463 1.15897
\(905\) 0 0
\(906\) 6.54779 0.217536
\(907\) 19.3907i 0.643856i 0.946764 + 0.321928i \(0.104331\pi\)
−0.946764 + 0.321928i \(0.895669\pi\)
\(908\) 5.88715i 0.195372i
\(909\) −8.19767 −0.271899
\(910\) 0 0
\(911\) −13.7190 −0.454530 −0.227265 0.973833i \(-0.572978\pi\)
−0.227265 + 0.973833i \(0.572978\pi\)
\(912\) − 26.2215i − 0.868280i
\(913\) − 0.527808i − 0.0174679i
\(914\) −24.5391 −0.811681
\(915\) 0 0
\(916\) −7.83636 −0.258920
\(917\) 6.51671i 0.215201i
\(918\) 3.37126i 0.111268i
\(919\) 32.4108 1.06913 0.534567 0.845126i \(-0.320475\pi\)
0.534567 + 0.845126i \(0.320475\pi\)
\(920\) 0 0
\(921\) −10.6465 −0.350814
\(922\) − 61.0867i − 2.01178i
\(923\) 7.79678i 0.256634i
\(924\) −1.27200 −0.0418457
\(925\) 0 0
\(926\) −40.1360 −1.31895
\(927\) − 2.50005i − 0.0821125i
\(928\) − 15.9458i − 0.523446i
\(929\) 4.33444 0.142208 0.0711042 0.997469i \(-0.477348\pi\)
0.0711042 + 0.997469i \(0.477348\pi\)
\(930\) 0 0
\(931\) −27.2028 −0.891536
\(932\) 4.88548i 0.160029i
\(933\) 27.3572i 0.895633i
\(934\) −5.91884 −0.193670
\(935\) 0 0
\(936\) −3.39865 −0.111088
\(937\) 54.5925i 1.78346i 0.452567 + 0.891730i \(0.350508\pi\)
−0.452567 + 0.891730i \(0.649492\pi\)
\(938\) 9.91218i 0.323644i
\(939\) −1.99509 −0.0651073
\(940\) 0 0
\(941\) 4.23853 0.138172 0.0690861 0.997611i \(-0.477992\pi\)
0.0690861 + 0.997611i \(0.477992\pi\)
\(942\) 24.6729i 0.803885i
\(943\) 16.3191i 0.531423i
\(944\) 50.8051 1.65356
\(945\) 0 0
\(946\) 29.0318 0.943906
\(947\) 12.6069i 0.409671i 0.978796 + 0.204835i \(0.0656659\pi\)
−0.978796 + 0.204835i \(0.934334\pi\)
\(948\) 3.85116i 0.125080i
\(949\) −9.34781 −0.303443
\(950\) 0 0
\(951\) −8.64876 −0.280455
\(952\) − 8.26348i − 0.267821i
\(953\) 31.1635i 1.00948i 0.863270 + 0.504742i \(0.168412\pi\)
−0.863270 + 0.504742i \(0.831588\pi\)
\(954\) −6.40975 −0.207523
\(955\) 0 0
\(956\) −3.80247 −0.122981
\(957\) − 18.6148i − 0.601732i
\(958\) − 21.9262i − 0.708405i
\(959\) −2.95301 −0.0953578
\(960\) 0 0
\(961\) −17.3309 −0.559062
\(962\) − 20.2194i − 0.651900i
\(963\) − 1.81004i − 0.0583276i
\(964\) −7.03319 −0.226524
\(965\) 0 0
\(966\) −20.2101 −0.650248
\(967\) 1.55308i 0.0499438i 0.999688 + 0.0249719i \(0.00794963\pi\)
−0.999688 + 0.0249719i \(0.992050\pi\)
\(968\) 13.7206i 0.440997i
\(969\) −12.5295 −0.402507
\(970\) 0 0
\(971\) −38.2623 −1.22790 −0.613948 0.789347i \(-0.710419\pi\)
−0.613948 + 0.789347i \(0.710419\pi\)
\(972\) 0.360976i 0.0115783i
\(973\) 2.51108i 0.0805016i
\(974\) 38.1405 1.22210
\(975\) 0 0
\(976\) 56.1890 1.79857
\(977\) − 20.3661i − 0.651570i −0.945444 0.325785i \(-0.894371\pi\)
0.945444 0.325785i \(-0.105629\pi\)
\(978\) 34.2554i 1.09537i
\(979\) 1.01123 0.0323190
\(980\) 0 0
\(981\) −2.94778 −0.0941152
\(982\) − 17.6539i − 0.563359i
\(983\) 13.2557i 0.422791i 0.977401 + 0.211395i \(0.0678008\pi\)
−0.977401 + 0.211395i \(0.932199\pi\)
\(984\) −4.67295 −0.148968
\(985\) 0 0
\(986\) −26.6335 −0.848185
\(987\) 9.96446i 0.317172i
\(988\) 2.78190i 0.0885041i
\(989\) 70.5249 2.24256
\(990\) 0 0
\(991\) −2.24081 −0.0711816 −0.0355908 0.999366i \(-0.511331\pi\)
−0.0355908 + 0.999366i \(0.511331\pi\)
\(992\) − 7.46241i − 0.236932i
\(993\) 3.94331i 0.125137i
\(994\) −13.2761 −0.421093
\(995\) 0 0
\(996\) −0.0808598 −0.00256214
\(997\) − 14.0404i − 0.444663i −0.974971 0.222331i \(-0.928633\pi\)
0.974971 0.222331i \(-0.0713667\pi\)
\(998\) − 6.41489i − 0.203060i
\(999\) 9.75097 0.308507
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 1875.2.b.h.1249.4 16
5.2 odd 4 1875.2.a.p.1.6 8
5.3 odd 4 1875.2.a.m.1.3 8
5.4 even 2 inner 1875.2.b.h.1249.13 16
15.2 even 4 5625.2.a.t.1.3 8
15.8 even 4 5625.2.a.bd.1.6 8
25.2 odd 20 375.2.g.d.226.3 16
25.9 even 10 375.2.i.c.349.1 16
25.11 even 5 375.2.i.c.274.1 16
25.12 odd 20 375.2.g.d.151.3 16
25.13 odd 20 375.2.g.e.151.2 16
25.14 even 10 75.2.i.a.4.4 16
25.16 even 5 75.2.i.a.19.4 yes 16
25.23 odd 20 375.2.g.e.226.2 16
75.14 odd 10 225.2.m.b.154.1 16
75.41 odd 10 225.2.m.b.19.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.2.i.a.4.4 16 25.14 even 10
75.2.i.a.19.4 yes 16 25.16 even 5
225.2.m.b.19.1 16 75.41 odd 10
225.2.m.b.154.1 16 75.14 odd 10
375.2.g.d.151.3 16 25.12 odd 20
375.2.g.d.226.3 16 25.2 odd 20
375.2.g.e.151.2 16 25.13 odd 20
375.2.g.e.226.2 16 25.23 odd 20
375.2.i.c.274.1 16 25.11 even 5
375.2.i.c.349.1 16 25.9 even 10
1875.2.a.m.1.3 8 5.3 odd 4
1875.2.a.p.1.6 8 5.2 odd 4
1875.2.b.h.1249.4 16 1.1 even 1 trivial
1875.2.b.h.1249.13 16 5.4 even 2 inner
5625.2.a.t.1.3 8 15.2 even 4
5625.2.a.bd.1.6 8 15.8 even 4