Properties

Label 2736.3.o.n.721.2
Level $2736$
Weight $3$
Character 2736.721
Analytic conductor $74.551$
Analytic rank $0$
Dimension $8$
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] = [2736,3,Mod(721,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.721");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2736.o (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: \(74.5506003290\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.184143974400.3
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 22x^{6} + 80x^{5} + 215x^{4} - 568x^{3} - 1022x^{2} + 1320x + 2628 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{2}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 114)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 721.2
Root \(2.51885 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.721
Dual form 2736.3.o.n.721.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.91350 q^{5} -3.01452 q^{7} +O(q^{10})\) \(q-9.91350 q^{5} -3.01452 q^{7} -10.8190 q^{11} +15.4135i q^{13} -0.115542 q^{17} +(-1.92003 + 18.9027i) q^{19} -26.9636 q^{23} +73.2775 q^{25} +34.0027i q^{29} +27.7631i q^{31} +29.8845 q^{35} +51.6116i q^{37} +15.3745i q^{41} -50.2194 q^{43} -25.0210 q^{47} -39.9127 q^{49} +84.8577i q^{53} +107.254 q^{55} -60.0742i q^{59} +16.6264 q^{61} -152.802i q^{65} +4.59832i q^{67} +73.9759i q^{71} -9.08734 q^{73} +32.6141 q^{77} -96.6761i q^{79} -61.7513 q^{83} +1.14542 q^{85} -39.9731i q^{89} -46.4643i q^{91} +(19.0342 - 187.392i) q^{95} -35.6440i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{5} + 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{5} + 12 q^{7} + 4 q^{11} - 4 q^{17} + 36 q^{19} - 56 q^{23} + 140 q^{25} + 236 q^{35} - 100 q^{43} - 188 q^{47} - 36 q^{49} - 28 q^{55} - 180 q^{61} - 356 q^{73} - 68 q^{77} + 136 q^{83} + 148 q^{85} - 140 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −9.91350 −1.98270 −0.991350 0.131244i \(-0.958103\pi\)
−0.991350 + 0.131244i \(0.958103\pi\)
\(6\) 0 0
\(7\) −3.01452 −0.430646 −0.215323 0.976543i \(-0.569080\pi\)
−0.215323 + 0.976543i \(0.569080\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −10.8190 −0.983546 −0.491773 0.870723i \(-0.663651\pi\)
−0.491773 + 0.870723i \(0.663651\pi\)
\(12\) 0 0
\(13\) 15.4135i 1.18565i 0.805330 + 0.592826i \(0.201988\pi\)
−0.805330 + 0.592826i \(0.798012\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −0.115542 −0.00679657 −0.00339829 0.999994i \(-0.501082\pi\)
−0.00339829 + 0.999994i \(0.501082\pi\)
\(18\) 0 0
\(19\) −1.92003 + 18.9027i −0.101054 + 0.994881i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −26.9636 −1.17233 −0.586165 0.810192i \(-0.699363\pi\)
−0.586165 + 0.810192i \(0.699363\pi\)
\(24\) 0 0
\(25\) 73.2775 2.93110
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 34.0027i 1.17251i 0.810127 + 0.586254i \(0.199398\pi\)
−0.810127 + 0.586254i \(0.800602\pi\)
\(30\) 0 0
\(31\) 27.7631i 0.895584i 0.894138 + 0.447792i \(0.147790\pi\)
−0.894138 + 0.447792i \(0.852210\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 29.8845 0.853842
\(36\) 0 0
\(37\) 51.6116i 1.39491i 0.716630 + 0.697454i \(0.245684\pi\)
−0.716630 + 0.697454i \(0.754316\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 15.3745i 0.374987i 0.982266 + 0.187494i \(0.0600364\pi\)
−0.982266 + 0.187494i \(0.939964\pi\)
\(42\) 0 0
\(43\) −50.2194 −1.16789 −0.583947 0.811792i \(-0.698492\pi\)
−0.583947 + 0.811792i \(0.698492\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −25.0210 −0.532363 −0.266181 0.963923i \(-0.585762\pi\)
−0.266181 + 0.963923i \(0.585762\pi\)
\(48\) 0 0
\(49\) −39.9127 −0.814544
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 84.8577i 1.60109i 0.599273 + 0.800545i \(0.295456\pi\)
−0.599273 + 0.800545i \(0.704544\pi\)
\(54\) 0 0
\(55\) 107.254 1.95008
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 60.0742i 1.01821i −0.860705 0.509104i \(-0.829977\pi\)
0.860705 0.509104i \(-0.170023\pi\)
\(60\) 0 0
\(61\) 16.6264 0.272564 0.136282 0.990670i \(-0.456485\pi\)
0.136282 + 0.990670i \(0.456485\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 152.802i 2.35079i
\(66\) 0 0
\(67\) 4.59832i 0.0686317i 0.999411 + 0.0343158i \(0.0109252\pi\)
−0.999411 + 0.0343158i \(0.989075\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 73.9759i 1.04191i 0.853583 + 0.520957i \(0.174425\pi\)
−0.853583 + 0.520957i \(0.825575\pi\)
\(72\) 0 0
\(73\) −9.08734 −0.124484 −0.0622420 0.998061i \(-0.519825\pi\)
−0.0622420 + 0.998061i \(0.519825\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 32.6141 0.423560
\(78\) 0 0
\(79\) 96.6761i 1.22375i −0.790955 0.611874i \(-0.790416\pi\)
0.790955 0.611874i \(-0.209584\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −61.7513 −0.743992 −0.371996 0.928234i \(-0.621327\pi\)
−0.371996 + 0.928234i \(0.621327\pi\)
\(84\) 0 0
\(85\) 1.14542 0.0134756
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 39.9731i 0.449136i −0.974458 0.224568i \(-0.927903\pi\)
0.974458 0.224568i \(-0.0720972\pi\)
\(90\) 0 0
\(91\) 46.4643i 0.510596i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 19.0342 187.392i 0.200360 1.97255i
\(96\) 0 0
\(97\) 35.6440i 0.367464i −0.982976 0.183732i \(-0.941182\pi\)
0.982976 0.183732i \(-0.0588179\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.89236 0.0682412 0.0341206 0.999418i \(-0.489137\pi\)
0.0341206 + 0.999418i \(0.489137\pi\)
\(102\) 0 0
\(103\) 101.969i 0.989992i 0.868895 + 0.494996i \(0.164831\pi\)
−0.868895 + 0.494996i \(0.835169\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 88.1128i 0.823484i 0.911300 + 0.411742i \(0.135080\pi\)
−0.911300 + 0.411742i \(0.864920\pi\)
\(108\) 0 0
\(109\) 11.4988i 0.105494i −0.998608 0.0527469i \(-0.983202\pi\)
0.998608 0.0527469i \(-0.0167977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 126.730i 1.12151i 0.827983 + 0.560753i \(0.189488\pi\)
−0.827983 + 0.560753i \(0.810512\pi\)
\(114\) 0 0
\(115\) 267.304 2.32438
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.348303 0.00292692
\(120\) 0 0
\(121\) −3.94908 −0.0326370
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −478.599 −3.82879
\(126\) 0 0
\(127\) 158.733i 1.24987i −0.780678 0.624934i \(-0.785126\pi\)
0.780678 0.624934i \(-0.214874\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −142.597 −1.08852 −0.544262 0.838916i \(-0.683190\pi\)
−0.544262 + 0.838916i \(0.683190\pi\)
\(132\) 0 0
\(133\) 5.78796 56.9827i 0.0435185 0.428441i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −166.791 −1.21745 −0.608727 0.793380i \(-0.708319\pi\)
−0.608727 + 0.793380i \(0.708319\pi\)
\(138\) 0 0
\(139\) 102.110 0.734606 0.367303 0.930101i \(-0.380281\pi\)
0.367303 + 0.930101i \(0.380281\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 166.759i 1.16614i
\(144\) 0 0
\(145\) 337.086i 2.32473i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 244.160 1.63866 0.819329 0.573323i \(-0.194346\pi\)
0.819329 + 0.573323i \(0.194346\pi\)
\(150\) 0 0
\(151\) 195.871i 1.29716i −0.761146 0.648580i \(-0.775363\pi\)
0.761146 0.648580i \(-0.224637\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 275.230i 1.77568i
\(156\) 0 0
\(157\) 117.369 0.747573 0.373787 0.927515i \(-0.378059\pi\)
0.373787 + 0.927515i \(0.378059\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 81.2823 0.504859
\(162\) 0 0
\(163\) −180.203 −1.10554 −0.552771 0.833333i \(-0.686430\pi\)
−0.552771 + 0.833333i \(0.686430\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 113.141i 0.677491i 0.940878 + 0.338745i \(0.110003\pi\)
−0.940878 + 0.338745i \(0.889997\pi\)
\(168\) 0 0
\(169\) −68.5755 −0.405772
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 298.411i 1.72492i −0.506127 0.862459i \(-0.668923\pi\)
0.506127 0.862459i \(-0.331077\pi\)
\(174\) 0 0
\(175\) −220.897 −1.26227
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 17.6491i 0.0985986i 0.998784 + 0.0492993i \(0.0156988\pi\)
−0.998784 + 0.0492993i \(0.984301\pi\)
\(180\) 0 0
\(181\) 297.751i 1.64503i 0.568740 + 0.822517i \(0.307431\pi\)
−0.568740 + 0.822517i \(0.692569\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 511.651i 2.76568i
\(186\) 0 0
\(187\) 1.25005 0.00668474
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −17.9862 −0.0941688 −0.0470844 0.998891i \(-0.514993\pi\)
−0.0470844 + 0.998891i \(0.514993\pi\)
\(192\) 0 0
\(193\) 104.090i 0.539327i −0.962955 0.269664i \(-0.913087\pi\)
0.962955 0.269664i \(-0.0869125\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 196.215 0.996017 0.498008 0.867172i \(-0.334065\pi\)
0.498008 + 0.867172i \(0.334065\pi\)
\(198\) 0 0
\(199\) −301.530 −1.51523 −0.757613 0.652704i \(-0.773634\pi\)
−0.757613 + 0.652704i \(0.773634\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 102.502i 0.504936i
\(204\) 0 0
\(205\) 152.415i 0.743488i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 20.7728 204.509i 0.0993914 0.978511i
\(210\) 0 0
\(211\) 153.409i 0.727055i −0.931584 0.363527i \(-0.881572\pi\)
0.931584 0.363527i \(-0.118428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 497.850 2.31558
\(216\) 0 0
\(217\) 83.6925i 0.385680i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.78090i 0.00805837i
\(222\) 0 0
\(223\) 272.878i 1.22367i 0.790987 + 0.611834i \(0.209568\pi\)
−0.790987 + 0.611834i \(0.790432\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 125.189i 0.551495i 0.961230 + 0.275748i \(0.0889254\pi\)
−0.961230 + 0.275748i \(0.911075\pi\)
\(228\) 0 0
\(229\) −274.350 −1.19803 −0.599017 0.800736i \(-0.704442\pi\)
−0.599017 + 0.800736i \(0.704442\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 115.719 0.496646 0.248323 0.968677i \(-0.420121\pi\)
0.248323 + 0.968677i \(0.420121\pi\)
\(234\) 0 0
\(235\) 248.046 1.05552
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 67.2090 0.281209 0.140605 0.990066i \(-0.455095\pi\)
0.140605 + 0.990066i \(0.455095\pi\)
\(240\) 0 0
\(241\) 124.189i 0.515307i 0.966237 + 0.257653i \(0.0829493\pi\)
−0.966237 + 0.257653i \(0.917051\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 395.674 1.61500
\(246\) 0 0
\(247\) −291.357 29.5943i −1.17958 0.119815i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.1645 −0.0484641 −0.0242321 0.999706i \(-0.507714\pi\)
−0.0242321 + 0.999706i \(0.507714\pi\)
\(252\) 0 0
\(253\) 291.719 1.15304
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 413.015i 1.60706i −0.595262 0.803532i \(-0.702952\pi\)
0.595262 0.803532i \(-0.297048\pi\)
\(258\) 0 0
\(259\) 155.584i 0.600711i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 244.309 0.928931 0.464466 0.885591i \(-0.346246\pi\)
0.464466 + 0.885591i \(0.346246\pi\)
\(264\) 0 0
\(265\) 841.237i 3.17448i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 221.977i 0.825193i 0.910914 + 0.412596i \(0.135378\pi\)
−0.910914 + 0.412596i \(0.864622\pi\)
\(270\) 0 0
\(271\) −464.627 −1.71449 −0.857245 0.514908i \(-0.827826\pi\)
−0.857245 + 0.514908i \(0.827826\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −792.790 −2.88287
\(276\) 0 0
\(277\) −338.509 −1.22206 −0.611028 0.791609i \(-0.709244\pi\)
−0.611028 + 0.791609i \(0.709244\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 171.878i 0.611666i −0.952085 0.305833i \(-0.901065\pi\)
0.952085 0.305833i \(-0.0989349\pi\)
\(282\) 0 0
\(283\) 394.594 1.39432 0.697162 0.716913i \(-0.254446\pi\)
0.697162 + 0.716913i \(0.254446\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 46.3467i 0.161487i
\(288\) 0 0
\(289\) −288.987 −0.999954
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 541.555i 1.84831i 0.382016 + 0.924156i \(0.375230\pi\)
−0.382016 + 0.924156i \(0.624770\pi\)
\(294\) 0 0
\(295\) 595.546i 2.01880i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 415.603i 1.38998i
\(300\) 0 0
\(301\) 151.387 0.502948
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −164.826 −0.540413
\(306\) 0 0
\(307\) 154.522i 0.503328i −0.967815 0.251664i \(-0.919022\pi\)
0.967815 0.251664i \(-0.0809778\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −200.544 −0.644836 −0.322418 0.946597i \(-0.604496\pi\)
−0.322418 + 0.946597i \(0.604496\pi\)
\(312\) 0 0
\(313\) 354.596 1.13289 0.566447 0.824098i \(-0.308317\pi\)
0.566447 + 0.824098i \(0.308317\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 172.282i 0.543477i 0.962371 + 0.271738i \(0.0875985\pi\)
−0.962371 + 0.271738i \(0.912401\pi\)
\(318\) 0 0
\(319\) 367.876i 1.15322i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.221843 2.18406i 0.000686822 0.00676178i
\(324\) 0 0
\(325\) 1129.46i 3.47527i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 75.4265 0.229260
\(330\) 0 0
\(331\) 142.309i 0.429936i 0.976621 + 0.214968i \(0.0689648\pi\)
−0.976621 + 0.214968i \(0.931035\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 45.5855i 0.136076i
\(336\) 0 0
\(337\) 38.4862i 0.114202i 0.998368 + 0.0571012i \(0.0181858\pi\)
−0.998368 + 0.0571012i \(0.981814\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 300.369i 0.880849i
\(342\) 0 0
\(343\) 268.029 0.781426
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 387.777 1.11751 0.558757 0.829331i \(-0.311278\pi\)
0.558757 + 0.829331i \(0.311278\pi\)
\(348\) 0 0
\(349\) 64.3080 0.184264 0.0921318 0.995747i \(-0.470632\pi\)
0.0921318 + 0.995747i \(0.470632\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 62.7329 0.177713 0.0888567 0.996044i \(-0.471679\pi\)
0.0888567 + 0.996044i \(0.471679\pi\)
\(354\) 0 0
\(355\) 733.360i 2.06580i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 609.306 1.69723 0.848616 0.529009i \(-0.177436\pi\)
0.848616 + 0.529009i \(0.177436\pi\)
\(360\) 0 0
\(361\) −353.627 72.5876i −0.979576 0.201074i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 90.0873 0.246815
\(366\) 0 0
\(367\) −111.317 −0.303316 −0.151658 0.988433i \(-0.548461\pi\)
−0.151658 + 0.988433i \(0.548461\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 255.805i 0.689503i
\(372\) 0 0
\(373\) 202.144i 0.541940i −0.962588 0.270970i \(-0.912656\pi\)
0.962588 0.270970i \(-0.0873445\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −524.101 −1.39019
\(378\) 0 0
\(379\) 490.422i 1.29399i 0.762495 + 0.646994i \(0.223974\pi\)
−0.762495 + 0.646994i \(0.776026\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 36.2292i 0.0945933i 0.998881 + 0.0472966i \(0.0150606\pi\)
−0.998881 + 0.0472966i \(0.984939\pi\)
\(384\) 0 0
\(385\) −323.320 −0.839793
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −185.177 −0.476033 −0.238016 0.971261i \(-0.576497\pi\)
−0.238016 + 0.971261i \(0.576497\pi\)
\(390\) 0 0
\(391\) 3.11542 0.00796783
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 958.398i 2.42632i
\(396\) 0 0
\(397\) −228.224 −0.574872 −0.287436 0.957800i \(-0.592803\pi\)
−0.287436 + 0.957800i \(0.592803\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 72.1432i 0.179908i −0.995946 0.0899541i \(-0.971328\pi\)
0.995946 0.0899541i \(-0.0286720\pi\)
\(402\) 0 0
\(403\) −427.926 −1.06185
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 558.386i 1.37196i
\(408\) 0 0
\(409\) 400.431i 0.979048i 0.871990 + 0.489524i \(0.162829\pi\)
−0.871990 + 0.489524i \(0.837171\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 181.095i 0.438487i
\(414\) 0 0
\(415\) 612.172 1.47511
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 456.141 1.08864 0.544321 0.838877i \(-0.316787\pi\)
0.544321 + 0.838877i \(0.316787\pi\)
\(420\) 0 0
\(421\) 163.425i 0.388182i −0.980983 0.194091i \(-0.937824\pi\)
0.980983 0.194091i \(-0.0621758\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −8.46661 −0.0199214
\(426\) 0 0
\(427\) −50.1207 −0.117379
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 223.091i 0.517613i −0.965929 0.258806i \(-0.916671\pi\)
0.965929 0.258806i \(-0.0833292\pi\)
\(432\) 0 0
\(433\) 60.0131i 0.138598i −0.997596 0.0692991i \(-0.977924\pi\)
0.997596 0.0692991i \(-0.0220763\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 51.7708 509.686i 0.118469 1.16633i
\(438\) 0 0
\(439\) 183.424i 0.417823i 0.977935 + 0.208911i \(0.0669920\pi\)
−0.977935 + 0.208911i \(0.933008\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −825.921 −1.86438 −0.932191 0.361967i \(-0.882105\pi\)
−0.932191 + 0.361967i \(0.882105\pi\)
\(444\) 0 0
\(445\) 396.274i 0.890503i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 559.499i 1.24610i 0.782182 + 0.623050i \(0.214107\pi\)
−0.782182 + 0.623050i \(0.785893\pi\)
\(450\) 0 0
\(451\) 166.337i 0.368817i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 460.624i 1.01236i
\(456\) 0 0
\(457\) 511.991 1.12033 0.560166 0.828381i \(-0.310737\pi\)
0.560166 + 0.828381i \(0.310737\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 26.8317 0.0582033 0.0291016 0.999576i \(-0.490735\pi\)
0.0291016 + 0.999576i \(0.490735\pi\)
\(462\) 0 0
\(463\) −431.887 −0.932802 −0.466401 0.884574i \(-0.654450\pi\)
−0.466401 + 0.884574i \(0.654450\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 64.2540 0.137589 0.0687944 0.997631i \(-0.478085\pi\)
0.0687944 + 0.997631i \(0.478085\pi\)
\(468\) 0 0
\(469\) 13.8617i 0.0295559i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 543.324 1.14868
\(474\) 0 0
\(475\) −140.695 + 1385.15i −0.296200 + 2.91610i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 640.901 1.33800 0.668999 0.743263i \(-0.266723\pi\)
0.668999 + 0.743263i \(0.266723\pi\)
\(480\) 0 0
\(481\) −795.514 −1.65388
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 353.357i 0.728571i
\(486\) 0 0
\(487\) 625.075i 1.28352i 0.766905 + 0.641760i \(0.221796\pi\)
−0.766905 + 0.641760i \(0.778204\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −257.709 −0.524867 −0.262433 0.964950i \(-0.584525\pi\)
−0.262433 + 0.964950i \(0.584525\pi\)
\(492\) 0 0
\(493\) 3.92873i 0.00796904i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 223.002i 0.448696i
\(498\) 0 0
\(499\) 658.482 1.31960 0.659801 0.751440i \(-0.270640\pi\)
0.659801 + 0.751440i \(0.270640\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 236.146 0.469474 0.234737 0.972059i \(-0.424577\pi\)
0.234737 + 0.972059i \(0.424577\pi\)
\(504\) 0 0
\(505\) −68.3274 −0.135302
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 68.2300i 0.134047i 0.997751 + 0.0670236i \(0.0213503\pi\)
−0.997751 + 0.0670236i \(0.978650\pi\)
\(510\) 0 0
\(511\) 27.3940 0.0536086
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1010.87i 1.96286i
\(516\) 0 0
\(517\) 270.703 0.523603
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 635.263i 1.21931i 0.792665 + 0.609657i \(0.208693\pi\)
−0.792665 + 0.609657i \(0.791307\pi\)
\(522\) 0 0
\(523\) 472.488i 0.903419i −0.892165 0.451710i \(-0.850814\pi\)
0.892165 0.451710i \(-0.149186\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 3.20780i 0.00608690i
\(528\) 0 0
\(529\) 198.035 0.374358
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −236.974 −0.444605
\(534\) 0 0
\(535\) 873.506i 1.63272i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 431.815 0.801142
\(540\) 0 0
\(541\) 548.295 1.01348 0.506742 0.862098i \(-0.330850\pi\)
0.506742 + 0.862098i \(0.330850\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 113.994i 0.209163i
\(546\) 0 0
\(547\) 369.232i 0.675012i −0.941323 0.337506i \(-0.890417\pi\)
0.941323 0.337506i \(-0.109583\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −642.745 65.2862i −1.16651 0.118487i
\(552\) 0 0
\(553\) 291.432i 0.527002i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −642.882 −1.15419 −0.577093 0.816678i \(-0.695813\pi\)
−0.577093 + 0.816678i \(0.695813\pi\)
\(558\) 0 0
\(559\) 774.056i 1.38472i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 267.879i 0.475807i −0.971289 0.237903i \(-0.923540\pi\)
0.971289 0.237903i \(-0.0764602\pi\)
\(564\) 0 0
\(565\) 1256.34i 2.22361i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 539.002i 0.947279i −0.880719 0.473640i \(-0.842940\pi\)
0.880719 0.473640i \(-0.157060\pi\)
\(570\) 0 0
\(571\) 136.244 0.238606 0.119303 0.992858i \(-0.461934\pi\)
0.119303 + 0.992858i \(0.461934\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1975.82 −3.43622
\(576\) 0 0
\(577\) 331.980 0.575356 0.287678 0.957727i \(-0.407117\pi\)
0.287678 + 0.957727i \(0.407117\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 186.151 0.320397
\(582\) 0 0
\(583\) 918.076i 1.57475i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −42.6626 −0.0726791 −0.0363395 0.999340i \(-0.511570\pi\)
−0.0363395 + 0.999340i \(0.511570\pi\)
\(588\) 0 0
\(589\) −524.799 53.3060i −0.891000 0.0905025i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 298.058 0.502628 0.251314 0.967906i \(-0.419137\pi\)
0.251314 + 0.967906i \(0.419137\pi\)
\(594\) 0 0
\(595\) −3.45290 −0.00580320
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 690.206i 1.15226i −0.817357 0.576132i \(-0.804561\pi\)
0.817357 0.576132i \(-0.195439\pi\)
\(600\) 0 0
\(601\) 596.790i 0.992995i 0.868038 + 0.496497i \(0.165381\pi\)
−0.868038 + 0.496497i \(0.834619\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 39.1492 0.0647094
\(606\) 0 0
\(607\) 334.973i 0.551851i 0.961179 + 0.275925i \(0.0889842\pi\)
−0.961179 + 0.275925i \(0.911016\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 385.662i 0.631197i
\(612\) 0 0
\(613\) −344.115 −0.561362 −0.280681 0.959801i \(-0.590560\pi\)
−0.280681 + 0.959801i \(0.590560\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 574.986 0.931907 0.465953 0.884809i \(-0.345711\pi\)
0.465953 + 0.884809i \(0.345711\pi\)
\(618\) 0 0
\(619\) −127.502 −0.205981 −0.102990 0.994682i \(-0.532841\pi\)
−0.102990 + 0.994682i \(0.532841\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 120.500i 0.193419i
\(624\) 0 0
\(625\) 2912.65 4.66025
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 5.96329i 0.00948059i
\(630\) 0 0
\(631\) 352.673 0.558912 0.279456 0.960159i \(-0.409846\pi\)
0.279456 + 0.960159i \(0.409846\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1573.60i 2.47811i
\(636\) 0 0
\(637\) 615.193i 0.965766i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1133.50i 1.76833i −0.467171 0.884167i \(-0.654727\pi\)
0.467171 0.884167i \(-0.345273\pi\)
\(642\) 0 0
\(643\) −469.085 −0.729526 −0.364763 0.931100i \(-0.618850\pi\)
−0.364763 + 0.931100i \(0.618850\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −203.611 −0.314701 −0.157350 0.987543i \(-0.550295\pi\)
−0.157350 + 0.987543i \(0.550295\pi\)
\(648\) 0 0
\(649\) 649.944i 1.00145i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −961.517 −1.47246 −0.736231 0.676731i \(-0.763396\pi\)
−0.736231 + 0.676731i \(0.763396\pi\)
\(654\) 0 0
\(655\) 1413.63 2.15822
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 230.320i 0.349499i −0.984613 0.174749i \(-0.944088\pi\)
0.984613 0.174749i \(-0.0559115\pi\)
\(660\) 0 0
\(661\) 359.931i 0.544525i 0.962223 + 0.272262i \(0.0877719\pi\)
−0.962223 + 0.272262i \(0.912228\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −57.3790 + 564.898i −0.0862842 + 0.849471i
\(666\) 0 0
\(667\) 916.836i 1.37457i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −179.881 −0.268080
\(672\) 0 0
\(673\) 1124.17i 1.67039i −0.549954 0.835195i \(-0.685355\pi\)
0.549954 0.835195i \(-0.314645\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 29.0597i 0.0429242i 0.999770 + 0.0214621i \(0.00683212\pi\)
−0.999770 + 0.0214621i \(0.993168\pi\)
\(678\) 0 0
\(679\) 107.450i 0.158247i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 724.655i 1.06099i 0.847689 + 0.530494i \(0.177994\pi\)
−0.847689 + 0.530494i \(0.822006\pi\)
\(684\) 0 0
\(685\) 1653.48 2.41385
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1307.95 −1.89834
\(690\) 0 0
\(691\) 705.183 1.02053 0.510263 0.860018i \(-0.329548\pi\)
0.510263 + 0.860018i \(0.329548\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1012.27 −1.45650
\(696\) 0 0
\(697\) 1.77639i 0.00254863i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 608.521 0.868076 0.434038 0.900895i \(-0.357088\pi\)
0.434038 + 0.900895i \(0.357088\pi\)
\(702\) 0 0
\(703\) −975.600 99.0957i −1.38777 0.140961i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −20.7772 −0.0293878
\(708\) 0 0
\(709\) −790.932 −1.11556 −0.557780 0.829989i \(-0.688347\pi\)
−0.557780 + 0.829989i \(0.688347\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 748.593i 1.04992i
\(714\) 0 0
\(715\) 1653.16i 2.31211i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 858.574 1.19412 0.597061 0.802196i \(-0.296335\pi\)
0.597061 + 0.802196i \(0.296335\pi\)
\(720\) 0 0
\(721\) 307.388i 0.426336i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2491.64i 3.43674i
\(726\) 0 0
\(727\) −1204.58 −1.65691 −0.828457 0.560053i \(-0.810781\pi\)
−0.828457 + 0.560053i \(0.810781\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.80244 0.00793767
\(732\) 0 0
\(733\) −676.835 −0.923377 −0.461689 0.887042i \(-0.652756\pi\)
−0.461689 + 0.887042i \(0.652756\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 49.7493i 0.0675024i
\(738\) 0 0
\(739\) 86.5075 0.117060 0.0585301 0.998286i \(-0.481359\pi\)
0.0585301 + 0.998286i \(0.481359\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1196.50i 1.61036i 0.593030 + 0.805180i \(0.297932\pi\)
−0.593030 + 0.805180i \(0.702068\pi\)
\(744\) 0 0
\(745\) −2420.48 −3.24897
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 265.618i 0.354630i
\(750\) 0 0
\(751\) 279.663i 0.372387i −0.982513 0.186194i \(-0.940385\pi\)
0.982513 0.186194i \(-0.0596152\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1941.77i 2.57188i
\(756\) 0 0
\(757\) 234.363 0.309594 0.154797 0.987946i \(-0.450528\pi\)
0.154797 + 0.987946i \(0.450528\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −201.790 −0.265164 −0.132582 0.991172i \(-0.542327\pi\)
−0.132582 + 0.991172i \(0.542327\pi\)
\(762\) 0 0
\(763\) 34.6635i 0.0454305i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 925.954 1.20724
\(768\) 0 0
\(769\) −130.331 −0.169481 −0.0847406 0.996403i \(-0.527006\pi\)
−0.0847406 + 0.996403i \(0.527006\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 590.469i 0.763867i −0.924190 0.381933i \(-0.875258\pi\)
0.924190 0.381933i \(-0.124742\pi\)
\(774\) 0 0
\(775\) 2034.41i 2.62505i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −290.620 29.5194i −0.373068 0.0378940i
\(780\) 0 0
\(781\) 800.346i 1.02477i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1163.54 −1.48221
\(786\) 0 0
\(787\) 571.324i 0.725952i −0.931799 0.362976i \(-0.881761\pi\)
0.931799 0.362976i \(-0.118239\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 382.030i 0.482972i
\(792\) 0 0
\(793\) 256.271i 0.323167i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 935.941i 1.17433i 0.809467 + 0.587165i \(0.199756\pi\)
−0.809467 + 0.587165i \(0.800244\pi\)
\(798\) 0 0
\(799\) 2.89098 0.00361824
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 98.3160 0.122436
\(804\) 0 0
\(805\) −805.792 −1.00098
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1456.28 1.80009 0.900047 0.435792i \(-0.143532\pi\)
0.900047 + 0.435792i \(0.143532\pi\)
\(810\) 0 0
\(811\) 1222.20i 1.50703i 0.657432 + 0.753514i \(0.271643\pi\)
−0.657432 + 0.753514i \(0.728357\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1786.45 2.19196
\(816\) 0 0
\(817\) 96.4227 949.284i 0.118020 1.16191i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −306.106 −0.372846 −0.186423 0.982470i \(-0.559689\pi\)
−0.186423 + 0.982470i \(0.559689\pi\)
\(822\) 0 0
\(823\) −217.902 −0.264766 −0.132383 0.991199i \(-0.542263\pi\)
−0.132383 + 0.991199i \(0.542263\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.2664i 0.0245059i 0.999925 + 0.0122529i \(0.00390033\pi\)
−0.999925 + 0.0122529i \(0.996100\pi\)
\(828\) 0 0
\(829\) 533.462i 0.643501i −0.946824 0.321751i \(-0.895729\pi\)
0.946824 0.321751i \(-0.104271\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.61158 0.00553611
\(834\) 0 0
\(835\) 1121.62i 1.34326i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1248.51i 1.48810i 0.668126 + 0.744048i \(0.267097\pi\)
−0.668126 + 0.744048i \(0.732903\pi\)
\(840\) 0 0
\(841\) −315.186 −0.374775
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 679.823 0.804525
\(846\) 0 0
\(847\) 11.9046 0.0140550
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1391.63i 1.63529i
\(852\) 0 0
\(853\) 1250.08 1.46551 0.732753 0.680495i \(-0.238235\pi\)
0.732753 + 0.680495i \(0.238235\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 845.935i 0.987088i 0.869721 + 0.493544i \(0.164299\pi\)
−0.869721 + 0.493544i \(0.835701\pi\)
\(858\) 0 0
\(859\) 829.883 0.966103 0.483052 0.875592i \(-0.339528\pi\)
0.483052 + 0.875592i \(0.339528\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 963.692i 1.11668i 0.829613 + 0.558338i \(0.188561\pi\)
−0.829613 + 0.558338i \(0.811439\pi\)
\(864\) 0 0
\(865\) 2958.30i 3.42000i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1045.94i 1.20361i
\(870\) 0 0
\(871\) −70.8761 −0.0813733
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1442.75 1.64885
\(876\) 0 0
\(877\) 145.023i 0.165363i −0.996576 0.0826815i \(-0.973652\pi\)
0.996576 0.0826815i \(-0.0263484\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −616.931 −0.700262 −0.350131 0.936701i \(-0.613863\pi\)
−0.350131 + 0.936701i \(0.613863\pi\)
\(882\) 0 0
\(883\) −462.198 −0.523440 −0.261720 0.965144i \(-0.584290\pi\)
−0.261720 + 0.965144i \(0.584290\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 359.598i 0.405409i 0.979240 + 0.202704i \(0.0649730\pi\)
−0.979240 + 0.202704i \(0.935027\pi\)
\(888\) 0 0
\(889\) 478.504i 0.538250i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 48.0411 472.966i 0.0537974 0.529638i
\(894\) 0 0
\(895\) 174.965i 0.195491i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −944.022 −1.05008
\(900\) 0 0
\(901\) 9.80461i 0.0108819i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2951.76i 3.26161i
\(906\) 0 0
\(907\) 1629.31i 1.79637i 0.439614 + 0.898187i \(0.355115\pi\)
−0.439614 + 0.898187i \(0.644885\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 979.434i 1.07512i −0.843225 0.537560i \(-0.819346\pi\)
0.843225 0.537560i \(-0.180654\pi\)
\(912\) 0 0
\(913\) 668.088 0.731751
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 429.860 0.468768
\(918\) 0 0
\(919\) 1229.87 1.33827 0.669133 0.743143i \(-0.266666\pi\)
0.669133 + 0.743143i \(0.266666\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1140.23 −1.23535
\(924\) 0 0
\(925\) 3781.97i 4.08861i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 904.746 0.973892 0.486946 0.873432i \(-0.338111\pi\)
0.486946 + 0.873432i \(0.338111\pi\)
\(930\) 0 0
\(931\) 76.6334 754.459i 0.0823130 0.810374i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −12.3923 −0.0132538
\(936\) 0 0
\(937\) −754.845 −0.805598 −0.402799 0.915289i \(-0.631963\pi\)
−0.402799 + 0.915289i \(0.631963\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1256.04i 1.33479i 0.744705 + 0.667394i \(0.232590\pi\)
−0.744705 + 0.667394i \(0.767410\pi\)
\(942\) 0 0
\(943\) 414.551i 0.439609i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −391.891 −0.413824 −0.206912 0.978360i \(-0.566341\pi\)
−0.206912 + 0.978360i \(0.566341\pi\)
\(948\) 0 0
\(949\) 140.068i 0.147595i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 255.958i 0.268581i −0.990942 0.134291i \(-0.957124\pi\)
0.990942 0.134291i \(-0.0428755\pi\)
\(954\) 0 0
\(955\) 178.307 0.186709
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 502.795 0.524291
\(960\) 0 0
\(961\) 190.209 0.197929
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1031.90i 1.06932i
\(966\) 0 0
\(967\) 1701.29 1.75935 0.879675 0.475575i \(-0.157760\pi\)
0.879675 + 0.475575i \(0.157760\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 416.242i 0.428674i 0.976760 + 0.214337i \(0.0687590\pi\)
−0.976760 + 0.214337i \(0.931241\pi\)
\(972\) 0 0
\(973\) −307.813 −0.316355
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 950.474i 0.972850i 0.873722 + 0.486425i \(0.161699\pi\)
−0.873722 + 0.486425i \(0.838301\pi\)
\(978\) 0 0
\(979\) 432.470i 0.441746i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1052.29i 1.07049i 0.844696 + 0.535246i \(0.179781\pi\)
−0.844696 + 0.535246i \(0.820219\pi\)
\(984\) 0 0
\(985\) −1945.18 −1.97480
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1354.10 1.36916
\(990\) 0 0
\(991\) 238.960i 0.241130i −0.992705 0.120565i \(-0.961529\pi\)
0.992705 0.120565i \(-0.0384706\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 2989.22 3.00424
\(996\) 0 0
\(997\) 35.0288 0.0351342 0.0175671 0.999846i \(-0.494408\pi\)
0.0175671 + 0.999846i \(0.494408\pi\)
\(998\) 0 0
\(999\) 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 2736.3.o.n.721.2 8
3.2 odd 2 912.3.o.d.721.4 8
4.3 odd 2 342.3.d.b.37.1 8
12.11 even 2 114.3.d.a.37.8 yes 8
19.18 odd 2 inner 2736.3.o.n.721.1 8
57.56 even 2 912.3.o.d.721.8 8
76.75 even 2 342.3.d.b.37.5 8
228.227 odd 2 114.3.d.a.37.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
114.3.d.a.37.2 8 228.227 odd 2
114.3.d.a.37.8 yes 8 12.11 even 2
342.3.d.b.37.1 8 4.3 odd 2
342.3.d.b.37.5 8 76.75 even 2
912.3.o.d.721.4 8 3.2 odd 2
912.3.o.d.721.8 8 57.56 even 2
2736.3.o.n.721.1 8 19.18 odd 2 inner
2736.3.o.n.721.2 8 1.1 even 1 trivial