Properties

Label 1200.4.a.bl.1.1
Level $1200$
Weight $4$
Character 1200.1
Self dual yes
Analytic conductor $70.802$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1200,4,Mod(1,1200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1200.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1200 = 2^{4} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1200.a (trivial)

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: yes
Analytic conductor: \(70.8022920069\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 75)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.35890\) of defining polynomial
Character \(\chi\) \(=\) 1200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000 q^{3} -30.4356 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -30.4356 q^{7} +9.00000 q^{9} -31.4356 q^{11} -60.7424 q^{13} -121.178 q^{17} +14.4356 q^{19} +91.3068 q^{21} -13.6932 q^{23} -27.0000 q^{27} -76.0492 q^{29} -183.049 q^{31} +94.3068 q^{33} -37.3864 q^{37} +182.227 q^{39} -30.6627 q^{41} -327.564 q^{43} +449.485 q^{47} +583.325 q^{49} +363.534 q^{51} -301.951 q^{53} -43.3068 q^{57} -340.970 q^{59} +619.098 q^{61} -273.920 q^{63} -256.890 q^{67} +41.0796 q^{69} -499.178 q^{71} -19.1288 q^{73} +956.761 q^{77} -257.424 q^{79} +81.0000 q^{81} -914.909 q^{83} +228.148 q^{87} -1059.68 q^{89} +1848.73 q^{91} +549.148 q^{93} +521.000 q^{97} -282.920 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 6 q^{3} - 26 q^{7} + 18 q^{9} - 28 q^{11} + 18 q^{13} - 68 q^{17} - 6 q^{19} + 78 q^{21} - 132 q^{23} - 54 q^{27} + 92 q^{29} - 122 q^{31} + 84 q^{33} - 284 q^{37} - 54 q^{39} + 392 q^{41} - 690 q^{43} + 620 q^{47} + 260 q^{49} + 204 q^{51} - 848 q^{53} + 18 q^{57} - 124 q^{59} + 750 q^{61} - 234 q^{63} + 358 q^{67} + 396 q^{69} - 824 q^{71} - 108 q^{73} + 972 q^{77} + 880 q^{79} + 162 q^{81} - 156 q^{83} - 276 q^{87} - 864 q^{89} + 2198 q^{91} + 366 q^{93} + 1042 q^{97} - 252 q^{99}+O(q^{100}) \) 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 0
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −30.4356 −1.64337 −0.821684 0.569944i \(-0.806965\pi\)
−0.821684 + 0.569944i \(0.806965\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −31.4356 −0.861654 −0.430827 0.902435i \(-0.641778\pi\)
−0.430827 + 0.902435i \(0.641778\pi\)
\(12\) 0 0
\(13\) −60.7424 −1.29592 −0.647958 0.761676i \(-0.724377\pi\)
−0.647958 + 0.761676i \(0.724377\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −121.178 −1.72882 −0.864411 0.502786i \(-0.832308\pi\)
−0.864411 + 0.502786i \(0.832308\pi\)
\(18\) 0 0
\(19\) 14.4356 0.174303 0.0871514 0.996195i \(-0.472224\pi\)
0.0871514 + 0.996195i \(0.472224\pi\)
\(20\) 0 0
\(21\) 91.3068 0.948799
\(22\) 0 0
\(23\) −13.6932 −0.124141 −0.0620703 0.998072i \(-0.519770\pi\)
−0.0620703 + 0.998072i \(0.519770\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) −76.0492 −0.486965 −0.243482 0.969905i \(-0.578290\pi\)
−0.243482 + 0.969905i \(0.578290\pi\)
\(30\) 0 0
\(31\) −183.049 −1.06054 −0.530268 0.847830i \(-0.677909\pi\)
−0.530268 + 0.847830i \(0.677909\pi\)
\(32\) 0 0
\(33\) 94.3068 0.497476
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −37.3864 −0.166116 −0.0830580 0.996545i \(-0.526469\pi\)
−0.0830580 + 0.996545i \(0.526469\pi\)
\(38\) 0 0
\(39\) 182.227 0.748197
\(40\) 0 0
\(41\) −30.6627 −0.116798 −0.0583990 0.998293i \(-0.518600\pi\)
−0.0583990 + 0.998293i \(0.518600\pi\)
\(42\) 0 0
\(43\) −327.564 −1.16170 −0.580850 0.814011i \(-0.697280\pi\)
−0.580850 + 0.814011i \(0.697280\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 449.485 1.39498 0.697490 0.716594i \(-0.254300\pi\)
0.697490 + 0.716594i \(0.254300\pi\)
\(48\) 0 0
\(49\) 583.325 1.70066
\(50\) 0 0
\(51\) 363.534 0.998136
\(52\) 0 0
\(53\) −301.951 −0.782569 −0.391284 0.920270i \(-0.627969\pi\)
−0.391284 + 0.920270i \(0.627969\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −43.3068 −0.100634
\(58\) 0 0
\(59\) −340.970 −0.752381 −0.376190 0.926542i \(-0.622766\pi\)
−0.376190 + 0.926542i \(0.622766\pi\)
\(60\) 0 0
\(61\) 619.098 1.29947 0.649733 0.760163i \(-0.274881\pi\)
0.649733 + 0.760163i \(0.274881\pi\)
\(62\) 0 0
\(63\) −273.920 −0.547789
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −256.890 −0.468419 −0.234210 0.972186i \(-0.575250\pi\)
−0.234210 + 0.972186i \(0.575250\pi\)
\(68\) 0 0
\(69\) 41.0796 0.0716726
\(70\) 0 0
\(71\) −499.178 −0.834388 −0.417194 0.908818i \(-0.636986\pi\)
−0.417194 + 0.908818i \(0.636986\pi\)
\(72\) 0 0
\(73\) −19.1288 −0.0306693 −0.0153346 0.999882i \(-0.504881\pi\)
−0.0153346 + 0.999882i \(0.504881\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 956.761 1.41601
\(78\) 0 0
\(79\) −257.424 −0.366613 −0.183307 0.983056i \(-0.558680\pi\)
−0.183307 + 0.983056i \(0.558680\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) −914.909 −1.20993 −0.604965 0.796252i \(-0.706813\pi\)
−0.604965 + 0.796252i \(0.706813\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 228.148 0.281149
\(88\) 0 0
\(89\) −1059.68 −1.26209 −0.631045 0.775746i \(-0.717374\pi\)
−0.631045 + 0.775746i \(0.717374\pi\)
\(90\) 0 0
\(91\) 1848.73 2.12967
\(92\) 0 0
\(93\) 549.148 0.612300
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 521.000 0.545356 0.272678 0.962105i \(-0.412091\pi\)
0.272678 + 0.962105i \(0.412091\pi\)
\(98\) 0 0
\(99\) −282.920 −0.287218
\(100\) 0 0
\(101\) 347.080 0.341938 0.170969 0.985276i \(-0.445310\pi\)
0.170969 + 0.985276i \(0.445310\pi\)
\(102\) 0 0
\(103\) 770.749 0.737322 0.368661 0.929564i \(-0.379816\pi\)
0.368661 + 0.929564i \(0.379816\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1415.37 1.27878 0.639390 0.768883i \(-0.279187\pi\)
0.639390 + 0.768883i \(0.279187\pi\)
\(108\) 0 0
\(109\) 908.386 0.798235 0.399118 0.916900i \(-0.369317\pi\)
0.399118 + 0.916900i \(0.369317\pi\)
\(110\) 0 0
\(111\) 112.159 0.0959071
\(112\) 0 0
\(113\) −2049.94 −1.70657 −0.853283 0.521447i \(-0.825392\pi\)
−0.853283 + 0.521447i \(0.825392\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −546.681 −0.431972
\(118\) 0 0
\(119\) 3688.12 2.84109
\(120\) 0 0
\(121\) −342.803 −0.257553
\(122\) 0 0
\(123\) 91.9882 0.0674333
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −281.644 −0.196786 −0.0983932 0.995148i \(-0.531370\pi\)
−0.0983932 + 0.995148i \(0.531370\pi\)
\(128\) 0 0
\(129\) 982.693 0.670708
\(130\) 0 0
\(131\) −243.056 −0.162106 −0.0810531 0.996710i \(-0.525828\pi\)
−0.0810531 + 0.996710i \(0.525828\pi\)
\(132\) 0 0
\(133\) −439.356 −0.286444
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 909.386 0.567110 0.283555 0.958956i \(-0.408486\pi\)
0.283555 + 0.958956i \(0.408486\pi\)
\(138\) 0 0
\(139\) 2049.52 1.25063 0.625317 0.780371i \(-0.284970\pi\)
0.625317 + 0.780371i \(0.284970\pi\)
\(140\) 0 0
\(141\) −1348.45 −0.805392
\(142\) 0 0
\(143\) 1909.47 1.11663
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −1749.98 −0.981875
\(148\) 0 0
\(149\) 3601.14 1.97998 0.989990 0.141136i \(-0.0450753\pi\)
0.989990 + 0.141136i \(0.0450753\pi\)
\(150\) 0 0
\(151\) −1383.38 −0.745550 −0.372775 0.927922i \(-0.621594\pi\)
−0.372775 + 0.927922i \(0.621594\pi\)
\(152\) 0 0
\(153\) −1090.60 −0.576274
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −131.749 −0.0669729 −0.0334864 0.999439i \(-0.510661\pi\)
−0.0334864 + 0.999439i \(0.510661\pi\)
\(158\) 0 0
\(159\) 905.852 0.451816
\(160\) 0 0
\(161\) 416.761 0.204009
\(162\) 0 0
\(163\) −2897.74 −1.39244 −0.696222 0.717827i \(-0.745137\pi\)
−0.696222 + 0.717827i \(0.745137\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −260.283 −0.120607 −0.0603034 0.998180i \(-0.519207\pi\)
−0.0603034 + 0.998180i \(0.519207\pi\)
\(168\) 0 0
\(169\) 1492.64 0.679398
\(170\) 0 0
\(171\) 129.920 0.0581009
\(172\) 0 0
\(173\) −1935.83 −0.850742 −0.425371 0.905019i \(-0.639856\pi\)
−0.425371 + 0.905019i \(0.639856\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1022.91 0.434387
\(178\) 0 0
\(179\) −576.627 −0.240777 −0.120389 0.992727i \(-0.538414\pi\)
−0.120389 + 0.992727i \(0.538414\pi\)
\(180\) 0 0
\(181\) −1962.04 −0.805733 −0.402866 0.915259i \(-0.631986\pi\)
−0.402866 + 0.915259i \(0.631986\pi\)
\(182\) 0 0
\(183\) −1857.30 −0.750247
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 3809.30 1.48965
\(188\) 0 0
\(189\) 821.761 0.316266
\(190\) 0 0
\(191\) 4318.75 1.63609 0.818047 0.575152i \(-0.195057\pi\)
0.818047 + 0.575152i \(0.195057\pi\)
\(192\) 0 0
\(193\) 2.97647 0.00111011 0.000555054 1.00000i \(-0.499823\pi\)
0.000555054 1.00000i \(0.499823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 569.705 0.206040 0.103020 0.994679i \(-0.467149\pi\)
0.103020 + 0.994679i \(0.467149\pi\)
\(198\) 0 0
\(199\) 3050.73 1.08674 0.543368 0.839494i \(-0.317149\pi\)
0.543368 + 0.839494i \(0.317149\pi\)
\(200\) 0 0
\(201\) 770.670 0.270442
\(202\) 0 0
\(203\) 2314.60 0.800262
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −123.239 −0.0413802
\(208\) 0 0
\(209\) −453.792 −0.150189
\(210\) 0 0
\(211\) 50.5104 0.0164800 0.00824000 0.999966i \(-0.497377\pi\)
0.00824000 + 0.999966i \(0.497377\pi\)
\(212\) 0 0
\(213\) 1497.53 0.481734
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5571.21 1.74285
\(218\) 0 0
\(219\) 57.3864 0.0177069
\(220\) 0 0
\(221\) 7360.64 2.24041
\(222\) 0 0
\(223\) −5453.55 −1.63765 −0.818827 0.574040i \(-0.805375\pi\)
−0.818827 + 0.574040i \(0.805375\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4777.14 1.39678 0.698392 0.715715i \(-0.253899\pi\)
0.698392 + 0.715715i \(0.253899\pi\)
\(228\) 0 0
\(229\) −2085.51 −0.601808 −0.300904 0.953654i \(-0.597288\pi\)
−0.300904 + 0.953654i \(0.597288\pi\)
\(230\) 0 0
\(231\) −2870.28 −0.817536
\(232\) 0 0
\(233\) 6484.53 1.82324 0.911622 0.411030i \(-0.134831\pi\)
0.911622 + 0.411030i \(0.134831\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 772.271 0.211664
\(238\) 0 0
\(239\) −2234.62 −0.604792 −0.302396 0.953182i \(-0.597786\pi\)
−0.302396 + 0.953182i \(0.597786\pi\)
\(240\) 0 0
\(241\) −2393.01 −0.639616 −0.319808 0.947482i \(-0.603618\pi\)
−0.319808 + 0.947482i \(0.603618\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −876.852 −0.225882
\(248\) 0 0
\(249\) 2744.73 0.698554
\(250\) 0 0
\(251\) 612.661 0.154067 0.0770335 0.997029i \(-0.475455\pi\)
0.0770335 + 0.997029i \(0.475455\pi\)
\(252\) 0 0
\(253\) 430.454 0.106966
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −306.112 −0.0742987 −0.0371493 0.999310i \(-0.511828\pi\)
−0.0371493 + 0.999310i \(0.511828\pi\)
\(258\) 0 0
\(259\) 1137.88 0.272990
\(260\) 0 0
\(261\) −684.443 −0.162322
\(262\) 0 0
\(263\) −283.839 −0.0665484 −0.0332742 0.999446i \(-0.510593\pi\)
−0.0332742 + 0.999446i \(0.510593\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 3179.04 0.728668
\(268\) 0 0
\(269\) 2426.21 0.549920 0.274960 0.961456i \(-0.411335\pi\)
0.274960 + 0.961456i \(0.411335\pi\)
\(270\) 0 0
\(271\) 174.946 0.0392148 0.0196074 0.999808i \(-0.493758\pi\)
0.0196074 + 0.999808i \(0.493758\pi\)
\(272\) 0 0
\(273\) −5546.19 −1.22956
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −7807.07 −1.69344 −0.846718 0.532042i \(-0.821425\pi\)
−0.846718 + 0.532042i \(0.821425\pi\)
\(278\) 0 0
\(279\) −1647.44 −0.353512
\(280\) 0 0
\(281\) 584.171 0.124017 0.0620084 0.998076i \(-0.480249\pi\)
0.0620084 + 0.998076i \(0.480249\pi\)
\(282\) 0 0
\(283\) 5897.31 1.23872 0.619362 0.785106i \(-0.287391\pi\)
0.619362 + 0.785106i \(0.287391\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 933.239 0.191942
\(288\) 0 0
\(289\) 9771.10 1.98883
\(290\) 0 0
\(291\) −1563.00 −0.314861
\(292\) 0 0
\(293\) 1609.73 0.320960 0.160480 0.987039i \(-0.448696\pi\)
0.160480 + 0.987039i \(0.448696\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 848.761 0.165825
\(298\) 0 0
\(299\) 831.758 0.160876
\(300\) 0 0
\(301\) 9969.62 1.90910
\(302\) 0 0
\(303\) −1041.24 −0.197418
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −234.473 −0.0435898 −0.0217949 0.999762i \(-0.506938\pi\)
−0.0217949 + 0.999762i \(0.506938\pi\)
\(308\) 0 0
\(309\) −2312.25 −0.425693
\(310\) 0 0
\(311\) 1795.25 0.327329 0.163665 0.986516i \(-0.447669\pi\)
0.163665 + 0.986516i \(0.447669\pi\)
\(312\) 0 0
\(313\) −8440.61 −1.52425 −0.762127 0.647427i \(-0.775845\pi\)
−0.762127 + 0.647427i \(0.775845\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10551.7 1.86953 0.934766 0.355264i \(-0.115609\pi\)
0.934766 + 0.355264i \(0.115609\pi\)
\(318\) 0 0
\(319\) 2390.65 0.419595
\(320\) 0 0
\(321\) −4246.12 −0.738304
\(322\) 0 0
\(323\) −1749.28 −0.301339
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2725.16 −0.460861
\(328\) 0 0
\(329\) −13680.3 −2.29247
\(330\) 0 0
\(331\) −6743.17 −1.11975 −0.559876 0.828576i \(-0.689151\pi\)
−0.559876 + 0.828576i \(0.689151\pi\)
\(332\) 0 0
\(333\) −336.478 −0.0553720
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −8437.26 −1.36382 −0.681909 0.731437i \(-0.738850\pi\)
−0.681909 + 0.731437i \(0.738850\pi\)
\(338\) 0 0
\(339\) 6149.82 0.985287
\(340\) 0 0
\(341\) 5754.26 0.913814
\(342\) 0 0
\(343\) −7314.45 −1.15144
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1848.85 0.286027 0.143013 0.989721i \(-0.454321\pi\)
0.143013 + 0.989721i \(0.454321\pi\)
\(348\) 0 0
\(349\) −1148.38 −0.176136 −0.0880678 0.996114i \(-0.528069\pi\)
−0.0880678 + 0.996114i \(0.528069\pi\)
\(350\) 0 0
\(351\) 1640.04 0.249399
\(352\) 0 0
\(353\) −5753.60 −0.867516 −0.433758 0.901029i \(-0.642813\pi\)
−0.433758 + 0.901029i \(0.642813\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −11064.4 −1.64030
\(358\) 0 0
\(359\) −5452.01 −0.801521 −0.400761 0.916183i \(-0.631254\pi\)
−0.400761 + 0.916183i \(0.631254\pi\)
\(360\) 0 0
\(361\) −6650.61 −0.969619
\(362\) 0 0
\(363\) 1028.41 0.148698
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −8385.93 −1.19276 −0.596379 0.802703i \(-0.703394\pi\)
−0.596379 + 0.802703i \(0.703394\pi\)
\(368\) 0 0
\(369\) −275.965 −0.0389327
\(370\) 0 0
\(371\) 9190.05 1.28605
\(372\) 0 0
\(373\) 2728.30 0.378730 0.189365 0.981907i \(-0.439357\pi\)
0.189365 + 0.981907i \(0.439357\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4619.41 0.631065
\(378\) 0 0
\(379\) −3348.99 −0.453895 −0.226947 0.973907i \(-0.572875\pi\)
−0.226947 + 0.973907i \(0.572875\pi\)
\(380\) 0 0
\(381\) 844.932 0.113615
\(382\) 0 0
\(383\) −10430.3 −1.39155 −0.695774 0.718261i \(-0.744938\pi\)
−0.695774 + 0.718261i \(0.744938\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2948.08 −0.387233
\(388\) 0 0
\(389\) 9827.23 1.28088 0.640438 0.768010i \(-0.278753\pi\)
0.640438 + 0.768010i \(0.278753\pi\)
\(390\) 0 0
\(391\) 1659.32 0.214617
\(392\) 0 0
\(393\) 729.168 0.0935921
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −436.382 −0.0551672 −0.0275836 0.999619i \(-0.508781\pi\)
−0.0275836 + 0.999619i \(0.508781\pi\)
\(398\) 0 0
\(399\) 1318.07 0.165378
\(400\) 0 0
\(401\) −14501.5 −1.80591 −0.902955 0.429736i \(-0.858607\pi\)
−0.902955 + 0.429736i \(0.858607\pi\)
\(402\) 0 0
\(403\) 11118.8 1.37436
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1175.26 0.143134
\(408\) 0 0
\(409\) −12058.4 −1.45782 −0.728911 0.684609i \(-0.759973\pi\)
−0.728911 + 0.684609i \(0.759973\pi\)
\(410\) 0 0
\(411\) −2728.16 −0.327421
\(412\) 0 0
\(413\) 10377.6 1.23644
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −6148.57 −0.722054
\(418\) 0 0
\(419\) −6042.95 −0.704577 −0.352288 0.935892i \(-0.614596\pi\)
−0.352288 + 0.935892i \(0.614596\pi\)
\(420\) 0 0
\(421\) −9994.67 −1.15703 −0.578516 0.815671i \(-0.696368\pi\)
−0.578516 + 0.815671i \(0.696368\pi\)
\(422\) 0 0
\(423\) 4045.36 0.464994
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −18842.6 −2.13550
\(428\) 0 0
\(429\) −5728.42 −0.644687
\(430\) 0 0
\(431\) −9327.32 −1.04242 −0.521208 0.853430i \(-0.674518\pi\)
−0.521208 + 0.853430i \(0.674518\pi\)
\(432\) 0 0
\(433\) −7861.22 −0.872485 −0.436243 0.899829i \(-0.643691\pi\)
−0.436243 + 0.899829i \(0.643691\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −197.670 −0.0216380
\(438\) 0 0
\(439\) −7412.06 −0.805828 −0.402914 0.915238i \(-0.632003\pi\)
−0.402914 + 0.915238i \(0.632003\pi\)
\(440\) 0 0
\(441\) 5249.93 0.566886
\(442\) 0 0
\(443\) 3043.66 0.326430 0.163215 0.986591i \(-0.447814\pi\)
0.163215 + 0.986591i \(0.447814\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −10803.4 −1.14314
\(448\) 0 0
\(449\) −9547.87 −1.00355 −0.501773 0.864999i \(-0.667319\pi\)
−0.501773 + 0.864999i \(0.667319\pi\)
\(450\) 0 0
\(451\) 963.902 0.100639
\(452\) 0 0
\(453\) 4150.14 0.430443
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 13401.9 1.37180 0.685901 0.727695i \(-0.259408\pi\)
0.685901 + 0.727695i \(0.259408\pi\)
\(458\) 0 0
\(459\) 3271.81 0.332712
\(460\) 0 0
\(461\) −4137.03 −0.417962 −0.208981 0.977920i \(-0.567015\pi\)
−0.208981 + 0.977920i \(0.567015\pi\)
\(462\) 0 0
\(463\) 13976.3 1.40288 0.701439 0.712729i \(-0.252541\pi\)
0.701439 + 0.712729i \(0.252541\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 10796.5 1.06982 0.534908 0.844910i \(-0.320346\pi\)
0.534908 + 0.844910i \(0.320346\pi\)
\(468\) 0 0
\(469\) 7818.60 0.769785
\(470\) 0 0
\(471\) 395.248 0.0386668
\(472\) 0 0
\(473\) 10297.2 1.00098
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −2717.56 −0.260856
\(478\) 0 0
\(479\) −14568.4 −1.38966 −0.694830 0.719174i \(-0.744521\pi\)
−0.694830 + 0.719174i \(0.744521\pi\)
\(480\) 0 0
\(481\) 2270.94 0.215272
\(482\) 0 0
\(483\) −1250.28 −0.117784
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 11456.6 1.06601 0.533004 0.846113i \(-0.321063\pi\)
0.533004 + 0.846113i \(0.321063\pi\)
\(488\) 0 0
\(489\) 8693.21 0.803928
\(490\) 0 0
\(491\) 19666.5 1.80761 0.903804 0.427948i \(-0.140763\pi\)
0.903804 + 0.427948i \(0.140763\pi\)
\(492\) 0 0
\(493\) 9215.48 0.841875
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 15192.8 1.37121
\(498\) 0 0
\(499\) −8379.31 −0.751722 −0.375861 0.926676i \(-0.622653\pi\)
−0.375861 + 0.926676i \(0.622653\pi\)
\(500\) 0 0
\(501\) 780.850 0.0696323
\(502\) 0 0
\(503\) 15678.1 1.38976 0.694881 0.719124i \(-0.255457\pi\)
0.694881 + 0.719124i \(0.255457\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4477.91 −0.392251
\(508\) 0 0
\(509\) −17037.3 −1.48363 −0.741813 0.670606i \(-0.766034\pi\)
−0.741813 + 0.670606i \(0.766034\pi\)
\(510\) 0 0
\(511\) 582.197 0.0504009
\(512\) 0 0
\(513\) −389.761 −0.0335446
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −14129.8 −1.20199
\(518\) 0 0
\(519\) 5807.49 0.491176
\(520\) 0 0
\(521\) −8776.12 −0.737982 −0.368991 0.929433i \(-0.620297\pi\)
−0.368991 + 0.929433i \(0.620297\pi\)
\(522\) 0 0
\(523\) −11120.4 −0.929753 −0.464877 0.885375i \(-0.653901\pi\)
−0.464877 + 0.885375i \(0.653901\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 22181.5 1.83348
\(528\) 0 0
\(529\) −11979.5 −0.984589
\(530\) 0 0
\(531\) −3068.73 −0.250794
\(532\) 0 0
\(533\) 1862.53 0.151360
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1729.88 0.139013
\(538\) 0 0
\(539\) −18337.2 −1.46538
\(540\) 0 0
\(541\) 21730.6 1.72693 0.863467 0.504405i \(-0.168288\pi\)
0.863467 + 0.504405i \(0.168288\pi\)
\(542\) 0 0
\(543\) 5886.13 0.465190
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6926.17 0.541392 0.270696 0.962665i \(-0.412746\pi\)
0.270696 + 0.962665i \(0.412746\pi\)
\(548\) 0 0
\(549\) 5571.89 0.433155
\(550\) 0 0
\(551\) −1097.82 −0.0848793
\(552\) 0 0
\(553\) 7834.85 0.602480
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6589.22 −0.501246 −0.250623 0.968085i \(-0.580636\pi\)
−0.250623 + 0.968085i \(0.580636\pi\)
\(558\) 0 0
\(559\) 19897.0 1.50547
\(560\) 0 0
\(561\) −11427.9 −0.860047
\(562\) 0 0
\(563\) −3839.63 −0.287426 −0.143713 0.989619i \(-0.545904\pi\)
−0.143713 + 0.989619i \(0.545904\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −2465.28 −0.182596
\(568\) 0 0
\(569\) −20874.0 −1.53794 −0.768968 0.639287i \(-0.779229\pi\)
−0.768968 + 0.639287i \(0.779229\pi\)
\(570\) 0 0
\(571\) −21175.9 −1.55199 −0.775994 0.630740i \(-0.782752\pi\)
−0.775994 + 0.630740i \(0.782752\pi\)
\(572\) 0 0
\(573\) −12956.3 −0.944599
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −14924.2 −1.07678 −0.538391 0.842695i \(-0.680968\pi\)
−0.538391 + 0.842695i \(0.680968\pi\)
\(578\) 0 0
\(579\) −8.92941 −0.000640922 0
\(580\) 0 0
\(581\) 27845.8 1.98836
\(582\) 0 0
\(583\) 9492.00 0.674303
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −25218.0 −1.77318 −0.886592 0.462552i \(-0.846934\pi\)
−0.886592 + 0.462552i \(0.846934\pi\)
\(588\) 0 0
\(589\) −2642.42 −0.184854
\(590\) 0 0
\(591\) −1709.11 −0.118957
\(592\) 0 0
\(593\) 5011.77 0.347063 0.173532 0.984828i \(-0.444482\pi\)
0.173532 + 0.984828i \(0.444482\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −9152.19 −0.627428
\(598\) 0 0
\(599\) −4943.41 −0.337199 −0.168600 0.985685i \(-0.553925\pi\)
−0.168600 + 0.985685i \(0.553925\pi\)
\(600\) 0 0
\(601\) −24334.8 −1.65164 −0.825821 0.563932i \(-0.809288\pi\)
−0.825821 + 0.563932i \(0.809288\pi\)
\(602\) 0 0
\(603\) −2312.01 −0.156140
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −28973.8 −1.93741 −0.968707 0.248207i \(-0.920159\pi\)
−0.968707 + 0.248207i \(0.920159\pi\)
\(608\) 0 0
\(609\) −6943.81 −0.462032
\(610\) 0 0
\(611\) −27302.8 −1.80778
\(612\) 0 0
\(613\) 15139.1 0.997490 0.498745 0.866749i \(-0.333794\pi\)
0.498745 + 0.866749i \(0.333794\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 13894.8 0.906617 0.453309 0.891354i \(-0.350244\pi\)
0.453309 + 0.891354i \(0.350244\pi\)
\(618\) 0 0
\(619\) 4589.69 0.298021 0.149011 0.988836i \(-0.452391\pi\)
0.149011 + 0.988836i \(0.452391\pi\)
\(620\) 0 0
\(621\) 369.717 0.0238909
\(622\) 0 0
\(623\) 32252.0 2.07408
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1361.37 0.0867114
\(628\) 0 0
\(629\) 4530.41 0.287185
\(630\) 0 0
\(631\) −3005.77 −0.189632 −0.0948160 0.995495i \(-0.530226\pi\)
−0.0948160 + 0.995495i \(0.530226\pi\)
\(632\) 0 0
\(633\) −151.531 −0.00951473
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −35432.6 −2.20391
\(638\) 0 0
\(639\) −4492.60 −0.278129
\(640\) 0 0
\(641\) −5631.47 −0.347004 −0.173502 0.984834i \(-0.555508\pi\)
−0.173502 + 0.984834i \(0.555508\pi\)
\(642\) 0 0
\(643\) −11305.1 −0.693358 −0.346679 0.937984i \(-0.612691\pi\)
−0.346679 + 0.937984i \(0.612691\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8614.30 0.523436 0.261718 0.965144i \(-0.415711\pi\)
0.261718 + 0.965144i \(0.415711\pi\)
\(648\) 0 0
\(649\) 10718.6 0.648291
\(650\) 0 0
\(651\) −16713.6 −1.00623
\(652\) 0 0
\(653\) 12639.8 0.757479 0.378739 0.925503i \(-0.376358\pi\)
0.378739 + 0.925503i \(0.376358\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −172.159 −0.0102231
\(658\) 0 0
\(659\) −13640.4 −0.806306 −0.403153 0.915133i \(-0.632086\pi\)
−0.403153 + 0.915133i \(0.632086\pi\)
\(660\) 0 0
\(661\) −17052.0 −1.00340 −0.501699 0.865042i \(-0.667291\pi\)
−0.501699 + 0.865042i \(0.667291\pi\)
\(662\) 0 0
\(663\) −22081.9 −1.29350
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1041.36 0.0604521
\(668\) 0 0
\(669\) 16360.7 0.945500
\(670\) 0 0
\(671\) −19461.7 −1.11969
\(672\) 0 0
\(673\) −16419.7 −0.940467 −0.470234 0.882542i \(-0.655830\pi\)
−0.470234 + 0.882542i \(0.655830\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8670.47 −0.492221 −0.246110 0.969242i \(-0.579153\pi\)
−0.246110 + 0.969242i \(0.579153\pi\)
\(678\) 0 0
\(679\) −15856.9 −0.896220
\(680\) 0 0
\(681\) −14331.4 −0.806434
\(682\) 0 0
\(683\) −5973.36 −0.334647 −0.167324 0.985902i \(-0.553512\pi\)
−0.167324 + 0.985902i \(0.553512\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 6256.52 0.347454
\(688\) 0 0
\(689\) 18341.2 1.01414
\(690\) 0 0
\(691\) 15316.3 0.843212 0.421606 0.906779i \(-0.361467\pi\)
0.421606 + 0.906779i \(0.361467\pi\)
\(692\) 0 0
\(693\) 8610.85 0.472005
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3715.65 0.201923
\(698\) 0 0
\(699\) −19453.6 −1.05265
\(700\) 0 0
\(701\) 34583.1 1.86332 0.931660 0.363333i \(-0.118361\pi\)
0.931660 + 0.363333i \(0.118361\pi\)
\(702\) 0 0
\(703\) −539.695 −0.0289545
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10563.6 −0.561929
\(708\) 0 0
\(709\) 11194.1 0.592955 0.296477 0.955040i \(-0.404188\pi\)
0.296477 + 0.955040i \(0.404188\pi\)
\(710\) 0 0
\(711\) −2316.81 −0.122204
\(712\) 0 0
\(713\) 2506.53 0.131655
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 6703.85 0.349177
\(718\) 0 0
\(719\) 15491.9 0.803549 0.401774 0.915739i \(-0.368394\pi\)
0.401774 + 0.915739i \(0.368394\pi\)
\(720\) 0 0
\(721\) −23458.2 −1.21169
\(722\) 0 0
\(723\) 7179.04 0.369283
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 6272.72 0.320003 0.160002 0.987117i \(-0.448850\pi\)
0.160002 + 0.987117i \(0.448850\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 39693.6 2.00837
\(732\) 0 0
\(733\) 24980.5 1.25877 0.629383 0.777095i \(-0.283308\pi\)
0.629383 + 0.777095i \(0.283308\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 8075.49 0.403615
\(738\) 0 0
\(739\) 30660.7 1.52621 0.763107 0.646272i \(-0.223673\pi\)
0.763107 + 0.646272i \(0.223673\pi\)
\(740\) 0 0
\(741\) 2630.56 0.130413
\(742\) 0 0
\(743\) 17205.7 0.849551 0.424776 0.905299i \(-0.360353\pi\)
0.424776 + 0.905299i \(0.360353\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −8234.18 −0.403310
\(748\) 0 0
\(749\) −43077.8 −2.10151
\(750\) 0 0
\(751\) −18397.1 −0.893901 −0.446950 0.894559i \(-0.647490\pi\)
−0.446950 + 0.894559i \(0.647490\pi\)
\(752\) 0 0
\(753\) −1837.98 −0.0889506
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22305.1 1.07093 0.535465 0.844557i \(-0.320136\pi\)
0.535465 + 0.844557i \(0.320136\pi\)
\(758\) 0 0
\(759\) −1291.36 −0.0617569
\(760\) 0 0
\(761\) 14458.5 0.688727 0.344364 0.938836i \(-0.388095\pi\)
0.344364 + 0.938836i \(0.388095\pi\)
\(762\) 0 0
\(763\) −27647.3 −1.31179
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 20711.3 0.975022
\(768\) 0 0
\(769\) −39897.6 −1.87093 −0.935463 0.353424i \(-0.885017\pi\)
−0.935463 + 0.353424i \(0.885017\pi\)
\(770\) 0 0
\(771\) 918.337 0.0428964
\(772\) 0 0
\(773\) 20070.2 0.933863 0.466931 0.884294i \(-0.345359\pi\)
0.466931 + 0.884294i \(0.345359\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −3413.63 −0.157611
\(778\) 0 0
\(779\) −442.635 −0.0203582
\(780\) 0 0
\(781\) 15692.0 0.718953
\(782\) 0 0
\(783\) 2053.33 0.0937164
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −10733.0 −0.486137 −0.243069 0.970009i \(-0.578154\pi\)
−0.243069 + 0.970009i \(0.578154\pi\)
\(788\) 0 0
\(789\) 851.516 0.0384218
\(790\) 0 0
\(791\) 62391.1 2.80452
\(792\) 0 0
\(793\) −37605.5 −1.68400
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −14335.8 −0.637140 −0.318570 0.947899i \(-0.603203\pi\)
−0.318570 + 0.947899i \(0.603203\pi\)
\(798\) 0 0
\(799\) −54467.7 −2.41167
\(800\) 0 0
\(801\) −9537.13 −0.420697
\(802\) 0 0
\(803\) 601.325 0.0264263
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −7278.63 −0.317497
\(808\) 0 0
\(809\) 20920.7 0.909187 0.454593 0.890699i \(-0.349785\pi\)
0.454593 + 0.890699i \(0.349785\pi\)
\(810\) 0 0
\(811\) −12816.5 −0.554931 −0.277465 0.960736i \(-0.589494\pi\)
−0.277465 + 0.960736i \(0.589494\pi\)
\(812\) 0 0
\(813\) −524.838 −0.0226407
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −4728.59 −0.202488
\(818\) 0 0
\(819\) 16638.6 0.709889
\(820\) 0 0
\(821\) 7253.55 0.308344 0.154172 0.988044i \(-0.450729\pi\)
0.154172 + 0.988044i \(0.450729\pi\)
\(822\) 0 0
\(823\) −35288.1 −1.49461 −0.747307 0.664479i \(-0.768654\pi\)
−0.747307 + 0.664479i \(0.768654\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 32205.9 1.35418 0.677092 0.735899i \(-0.263240\pi\)
0.677092 + 0.735899i \(0.263240\pi\)
\(828\) 0 0
\(829\) 29993.3 1.25659 0.628294 0.777976i \(-0.283754\pi\)
0.628294 + 0.777976i \(0.283754\pi\)
\(830\) 0 0
\(831\) 23421.2 0.977706
\(832\) 0 0
\(833\) −70686.2 −2.94013
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 4942.33 0.204100
\(838\) 0 0
\(839\) 35608.7 1.46526 0.732628 0.680629i \(-0.238294\pi\)
0.732628 + 0.680629i \(0.238294\pi\)
\(840\) 0 0
\(841\) −18605.5 −0.762865
\(842\) 0 0
\(843\) −1752.51 −0.0716011
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 10433.4 0.423255
\(848\) 0 0
\(849\) −17691.9 −0.715177
\(850\) 0 0
\(851\) 511.940 0.0206217
\(852\) 0 0
\(853\) −11229.3 −0.450744 −0.225372 0.974273i \(-0.572360\pi\)
−0.225372 + 0.974273i \(0.572360\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 22136.1 0.882327 0.441164 0.897427i \(-0.354566\pi\)
0.441164 + 0.897427i \(0.354566\pi\)
\(858\) 0 0
\(859\) 820.727 0.0325994 0.0162997 0.999867i \(-0.494811\pi\)
0.0162997 + 0.999867i \(0.494811\pi\)
\(860\) 0 0
\(861\) −2799.72 −0.110818
\(862\) 0 0
\(863\) 245.223 0.00967264 0.00483632 0.999988i \(-0.498461\pi\)
0.00483632 + 0.999988i \(0.498461\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −29313.3 −1.14825
\(868\) 0 0
\(869\) 8092.27 0.315894
\(870\) 0 0
\(871\) 15604.1 0.607032
\(872\) 0 0
\(873\) 4689.00 0.181785
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 37727.8 1.45265 0.726326 0.687350i \(-0.241226\pi\)
0.726326 + 0.687350i \(0.241226\pi\)
\(878\) 0 0
\(879\) −4829.19 −0.185306
\(880\) 0 0
\(881\) 21738.8 0.831326 0.415663 0.909519i \(-0.363550\pi\)
0.415663 + 0.909519i \(0.363550\pi\)
\(882\) 0 0
\(883\) 44340.4 1.68989 0.844946 0.534852i \(-0.179632\pi\)
0.844946 + 0.534852i \(0.179632\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 681.008 0.0257790 0.0128895 0.999917i \(-0.495897\pi\)
0.0128895 + 0.999917i \(0.495897\pi\)
\(888\) 0 0
\(889\) 8572.00 0.323392
\(890\) 0 0
\(891\) −2546.28 −0.0957393
\(892\) 0 0
\(893\) 6488.58 0.243149
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2495.28 −0.0928816
\(898\) 0 0
\(899\) 13920.7 0.516443
\(900\) 0 0
\(901\) 36589.8 1.35292
\(902\) 0 0
\(903\) −29908.9 −1.10222
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5348.01 −0.195786 −0.0978929 0.995197i \(-0.531210\pi\)
−0.0978929 + 0.995197i \(0.531210\pi\)
\(908\) 0 0
\(909\) 3123.72 0.113979
\(910\) 0 0
\(911\) −14488.7 −0.526930 −0.263465 0.964669i \(-0.584865\pi\)
−0.263465 + 0.964669i \(0.584865\pi\)
\(912\) 0 0
\(913\) 28760.7 1.04254
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 7397.56 0.266400
\(918\) 0 0
\(919\) 22546.9 0.809308 0.404654 0.914470i \(-0.367392\pi\)
0.404654 + 0.914470i \(0.367392\pi\)
\(920\) 0 0
\(921\) 703.419 0.0251666
\(922\) 0 0
\(923\) 30321.3 1.08130
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 6936.74 0.245774
\(928\) 0 0
\(929\) −13980.7 −0.493747 −0.246874 0.969048i \(-0.579403\pi\)
−0.246874 + 0.969048i \(0.579403\pi\)
\(930\) 0 0
\(931\) 8420.65 0.296429
\(932\) 0 0
\(933\) −5385.75 −0.188984
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 26362.6 0.919133 0.459567 0.888143i \(-0.348005\pi\)
0.459567 + 0.888143i \(0.348005\pi\)
\(938\) 0 0
\(939\) 25321.8 0.880029
\(940\) 0 0
\(941\) −14715.6 −0.509792 −0.254896 0.966968i \(-0.582041\pi\)
−0.254896 + 0.966968i \(0.582041\pi\)
\(942\) 0 0
\(943\) 419.871 0.0144994
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 11818.2 0.405532 0.202766 0.979227i \(-0.435007\pi\)
0.202766 + 0.979227i \(0.435007\pi\)
\(948\) 0 0
\(949\) 1161.93 0.0397448
\(950\) 0 0
\(951\) −31655.1 −1.07937
\(952\) 0 0
\(953\) −43832.3 −1.48989 −0.744945 0.667125i \(-0.767524\pi\)
−0.744945 + 0.667125i \(0.767524\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −7171.95 −0.242253
\(958\) 0 0
\(959\) −27677.7 −0.931971
\(960\) 0 0
\(961\) 3716.00 0.124736
\(962\) 0 0
\(963\) 12738.4 0.426260
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −10696.2 −0.355706 −0.177853 0.984057i \(-0.556915\pi\)
−0.177853 + 0.984057i \(0.556915\pi\)
\(968\) 0 0
\(969\) 5247.83 0.173978
\(970\) 0 0
\(971\) −27933.0 −0.923187 −0.461593 0.887092i \(-0.652722\pi\)
−0.461593 + 0.887092i \(0.652722\pi\)
\(972\) 0 0
\(973\) −62378.4 −2.05525
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 24341.7 0.797094 0.398547 0.917148i \(-0.369515\pi\)
0.398547 + 0.917148i \(0.369515\pi\)
\(978\) 0 0
\(979\) 33311.7 1.08748
\(980\) 0 0
\(981\) 8175.48 0.266078
\(982\) 0 0
\(983\) −12553.0 −0.407301 −0.203651 0.979044i \(-0.565281\pi\)
−0.203651 + 0.979044i \(0.565281\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 41041.0 1.32356
\(988\) 0 0
\(989\) 4485.41 0.144214
\(990\) 0 0
\(991\) 45631.8 1.46271 0.731353 0.681999i \(-0.238889\pi\)
0.731353 + 0.681999i \(0.238889\pi\)
\(992\) 0 0
\(993\) 20229.5 0.646490
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −34499.0 −1.09588 −0.547941 0.836517i \(-0.684588\pi\)
−0.547941 + 0.836517i \(0.684588\pi\)
\(998\) 0 0
\(999\) 1009.43 0.0319690
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 1200.4.a.bl.1.1 2
4.3 odd 2 75.4.a.d.1.2 2
5.2 odd 4 1200.4.f.v.49.3 4
5.3 odd 4 1200.4.f.v.49.2 4
5.4 even 2 1200.4.a.bu.1.2 2
12.11 even 2 225.4.a.n.1.1 2
20.3 even 4 75.4.b.c.49.2 4
20.7 even 4 75.4.b.c.49.3 4
20.19 odd 2 75.4.a.e.1.1 yes 2
60.23 odd 4 225.4.b.h.199.3 4
60.47 odd 4 225.4.b.h.199.2 4
60.59 even 2 225.4.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.4.a.d.1.2 2 4.3 odd 2
75.4.a.e.1.1 yes 2 20.19 odd 2
75.4.b.c.49.2 4 20.3 even 4
75.4.b.c.49.3 4 20.7 even 4
225.4.a.j.1.2 2 60.59 even 2
225.4.a.n.1.1 2 12.11 even 2
225.4.b.h.199.2 4 60.47 odd 4
225.4.b.h.199.3 4 60.23 odd 4
1200.4.a.bl.1.1 2 1.1 even 1 trivial
1200.4.a.bu.1.2 2 5.4 even 2
1200.4.f.v.49.2 4 5.3 odd 4
1200.4.f.v.49.3 4 5.2 odd 4