Properties

Label 600.3.c.d.449.16
Level $600$
Weight $3$
Character 600.449
Analytic conductor $16.349$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,3,Mod(449,600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.c (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: \(16.3488158616\)
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: \( x^{16} + 138x^{12} + 3393x^{8} + 15208x^{4} + 1296 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.16
Root \(2.27869 - 2.27869i\) of defining polynomial
Character \(\chi\) \(=\) 600.449
Dual form 600.3.c.d.449.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.98579 + 0.291610i) q^{3} -4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} +O(q^{10})\) \(q+(2.98579 + 0.291610i) q^{3} -4.46268i q^{7} +(8.82993 + 1.74137i) q^{9} -17.8696i q^{11} -11.0107i q^{13} +0.794055 q^{17} -26.5852 q^{19} +(1.30136 - 13.3246i) q^{21} -14.9276 q^{23} +(25.8565 + 7.77428i) q^{27} -5.58545i q^{29} +53.1074 q^{31} +(5.21095 - 53.3549i) q^{33} -51.7565i q^{37} +(3.21084 - 32.8758i) q^{39} +67.8236i q^{41} -40.8243i q^{43} +12.3483 q^{47} +29.0845 q^{49} +(2.37088 + 0.231554i) q^{51} -37.0351 q^{53} +(-79.3779 - 7.75251i) q^{57} +61.0932i q^{59} +97.8289 q^{61} +(7.77119 - 39.4051i) q^{63} -3.02541i q^{67} +(-44.5708 - 4.35305i) q^{69} -57.0787i q^{71} -31.4690i q^{73} -79.7461 q^{77} +2.16053 q^{79} +(74.9352 + 30.7524i) q^{81} +13.0710 q^{83} +(1.62877 - 16.6770i) q^{87} +173.692i q^{89} -49.1374 q^{91} +(158.568 + 15.4866i) q^{93} +91.6381i q^{97} +(31.1176 - 157.787i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 40 q^{9} + 16 q^{19} + 56 q^{21} + 240 q^{31} + 144 q^{39} - 128 q^{49} + 128 q^{51} + 16 q^{61} - 200 q^{69} - 176 q^{79} + 448 q^{81} + 1120 q^{91} - 64 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/600\mathbb{Z}\right)^\times\).

\(n\) \(151\) \(301\) \(401\) \(577\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(3\) 2.98579 + 0.291610i 0.995265 + 0.0972033i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.46268i 0.637525i −0.947835 0.318763i \(-0.896733\pi\)
0.947835 0.318763i \(-0.103267\pi\)
\(8\) 0 0
\(9\) 8.82993 + 1.74137i 0.981103 + 0.193486i
\(10\) 0 0
\(11\) 17.8696i 1.62451i −0.583305 0.812253i \(-0.698241\pi\)
0.583305 0.812253i \(-0.301759\pi\)
\(12\) 0 0
\(13\) 11.0107i 0.846980i −0.905901 0.423490i \(-0.860805\pi\)
0.905901 0.423490i \(-0.139195\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.794055 0.0467091 0.0233546 0.999727i \(-0.492565\pi\)
0.0233546 + 0.999727i \(0.492565\pi\)
\(18\) 0 0
\(19\) −26.5852 −1.39922 −0.699611 0.714524i \(-0.746643\pi\)
−0.699611 + 0.714524i \(0.746643\pi\)
\(20\) 0 0
\(21\) 1.30136 13.3246i 0.0619696 0.634506i
\(22\) 0 0
\(23\) −14.9276 −0.649027 −0.324514 0.945881i \(-0.605201\pi\)
−0.324514 + 0.945881i \(0.605201\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 25.8565 + 7.77428i 0.957650 + 0.287936i
\(28\) 0 0
\(29\) 5.58545i 0.192602i −0.995352 0.0963008i \(-0.969299\pi\)
0.995352 0.0963008i \(-0.0307011\pi\)
\(30\) 0 0
\(31\) 53.1074 1.71314 0.856571 0.516030i \(-0.172591\pi\)
0.856571 + 0.516030i \(0.172591\pi\)
\(32\) 0 0
\(33\) 5.21095 53.3549i 0.157907 1.61681i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 51.7565i 1.39882i −0.714719 0.699412i \(-0.753445\pi\)
0.714719 0.699412i \(-0.246555\pi\)
\(38\) 0 0
\(39\) 3.21084 32.8758i 0.0823293 0.842969i
\(40\) 0 0
\(41\) 67.8236i 1.65423i 0.562030 + 0.827117i \(0.310020\pi\)
−0.562030 + 0.827117i \(0.689980\pi\)
\(42\) 0 0
\(43\) 40.8243i 0.949403i −0.880147 0.474701i \(-0.842556\pi\)
0.880147 0.474701i \(-0.157444\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.3483 0.262729 0.131365 0.991334i \(-0.458064\pi\)
0.131365 + 0.991334i \(0.458064\pi\)
\(48\) 0 0
\(49\) 29.0845 0.593562
\(50\) 0 0
\(51\) 2.37088 + 0.231554i 0.0464879 + 0.00454028i
\(52\) 0 0
\(53\) −37.0351 −0.698775 −0.349387 0.936978i \(-0.613610\pi\)
−0.349387 + 0.936978i \(0.613610\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −79.3779 7.75251i −1.39260 0.136009i
\(58\) 0 0
\(59\) 61.0932i 1.03548i 0.855539 + 0.517739i \(0.173226\pi\)
−0.855539 + 0.517739i \(0.826774\pi\)
\(60\) 0 0
\(61\) 97.8289 1.60375 0.801876 0.597490i \(-0.203835\pi\)
0.801876 + 0.597490i \(0.203835\pi\)
\(62\) 0 0
\(63\) 7.77119 39.4051i 0.123352 0.625478i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 3.02541i 0.0451553i −0.999745 0.0225777i \(-0.992813\pi\)
0.999745 0.0225777i \(-0.00718731\pi\)
\(68\) 0 0
\(69\) −44.5708 4.35305i −0.645954 0.0630876i
\(70\) 0 0
\(71\) 57.0787i 0.803926i −0.915656 0.401963i \(-0.868328\pi\)
0.915656 0.401963i \(-0.131672\pi\)
\(72\) 0 0
\(73\) 31.4690i 0.431082i −0.976495 0.215541i \(-0.930848\pi\)
0.976495 0.215541i \(-0.0691515\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −79.7461 −1.03566
\(78\) 0 0
\(79\) 2.16053 0.0273485 0.0136743 0.999907i \(-0.495647\pi\)
0.0136743 + 0.999907i \(0.495647\pi\)
\(80\) 0 0
\(81\) 74.9352 + 30.7524i 0.925126 + 0.379660i
\(82\) 0 0
\(83\) 13.0710 0.157482 0.0787411 0.996895i \(-0.474910\pi\)
0.0787411 + 0.996895i \(0.474910\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.62877 16.6770i 0.0187215 0.191690i
\(88\) 0 0
\(89\) 173.692i 1.95160i 0.218667 + 0.975800i \(0.429829\pi\)
−0.218667 + 0.975800i \(0.570171\pi\)
\(90\) 0 0
\(91\) −49.1374 −0.539971
\(92\) 0 0
\(93\) 158.568 + 15.4866i 1.70503 + 0.166523i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 91.6381i 0.944722i 0.881405 + 0.472361i \(0.156598\pi\)
−0.881405 + 0.472361i \(0.843402\pi\)
\(98\) 0 0
\(99\) 31.1176 157.787i 0.314319 1.59381i
\(100\) 0 0
\(101\) 116.353i 1.15201i 0.817446 + 0.576005i \(0.195389\pi\)
−0.817446 + 0.576005i \(0.804611\pi\)
\(102\) 0 0
\(103\) 182.071i 1.76768i −0.467786 0.883842i \(-0.654948\pi\)
0.467786 0.883842i \(-0.345052\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 102.828 0.961013 0.480506 0.876991i \(-0.340453\pi\)
0.480506 + 0.876991i \(0.340453\pi\)
\(108\) 0 0
\(109\) −75.4257 −0.691979 −0.345990 0.938238i \(-0.612457\pi\)
−0.345990 + 0.938238i \(0.612457\pi\)
\(110\) 0 0
\(111\) 15.0927 154.534i 0.135970 1.39220i
\(112\) 0 0
\(113\) −141.923 −1.25596 −0.627980 0.778230i \(-0.716118\pi\)
−0.627980 + 0.778230i \(0.716118\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 19.1738 97.2241i 0.163879 0.830975i
\(118\) 0 0
\(119\) 3.54361i 0.0297782i
\(120\) 0 0
\(121\) −198.322 −1.63902
\(122\) 0 0
\(123\) −19.7780 + 202.507i −0.160797 + 1.64640i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 112.920i 0.889132i −0.895746 0.444566i \(-0.853358\pi\)
0.895746 0.444566i \(-0.146642\pi\)
\(128\) 0 0
\(129\) 11.9048 121.893i 0.0922851 0.944907i
\(130\) 0 0
\(131\) 107.233i 0.818570i −0.912407 0.409285i \(-0.865778\pi\)
0.912407 0.409285i \(-0.134222\pi\)
\(132\) 0 0
\(133\) 118.641i 0.892039i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 67.7434 0.494477 0.247239 0.968955i \(-0.420477\pi\)
0.247239 + 0.968955i \(0.420477\pi\)
\(138\) 0 0
\(139\) 141.204 1.01586 0.507928 0.861399i \(-0.330411\pi\)
0.507928 + 0.861399i \(0.330411\pi\)
\(140\) 0 0
\(141\) 36.8694 + 3.60088i 0.261485 + 0.0255382i
\(142\) 0 0
\(143\) −196.757 −1.37593
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 86.8404 + 8.48134i 0.590751 + 0.0576962i
\(148\) 0 0
\(149\) 55.0831i 0.369685i −0.982768 0.184843i \(-0.940822\pi\)
0.982768 0.184843i \(-0.0591775\pi\)
\(150\) 0 0
\(151\) 56.1302 0.371723 0.185862 0.982576i \(-0.440492\pi\)
0.185862 + 0.982576i \(0.440492\pi\)
\(152\) 0 0
\(153\) 7.01145 + 1.38275i 0.0458265 + 0.00903756i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 274.266i 1.74692i 0.486896 + 0.873460i \(0.338129\pi\)
−0.486896 + 0.873460i \(0.661871\pi\)
\(158\) 0 0
\(159\) −110.579 10.7998i −0.695466 0.0679232i
\(160\) 0 0
\(161\) 66.6172i 0.413771i
\(162\) 0 0
\(163\) 260.316i 1.59703i 0.601974 + 0.798516i \(0.294381\pi\)
−0.601974 + 0.798516i \(0.705619\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 179.949 1.07754 0.538769 0.842453i \(-0.318889\pi\)
0.538769 + 0.842453i \(0.318889\pi\)
\(168\) 0 0
\(169\) 47.7635 0.282624
\(170\) 0 0
\(171\) −234.745 46.2948i −1.37278 0.270730i
\(172\) 0 0
\(173\) 111.265 0.643148 0.321574 0.946884i \(-0.395788\pi\)
0.321574 + 0.946884i \(0.395788\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −17.8154 + 182.412i −0.100652 + 1.03057i
\(178\) 0 0
\(179\) 77.8475i 0.434902i 0.976071 + 0.217451i \(0.0697743\pi\)
−0.976071 + 0.217451i \(0.930226\pi\)
\(180\) 0 0
\(181\) −238.852 −1.31963 −0.659813 0.751430i \(-0.729364\pi\)
−0.659813 + 0.751430i \(0.729364\pi\)
\(182\) 0 0
\(183\) 292.097 + 28.5279i 1.59616 + 0.155890i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 14.1894i 0.0758793i
\(188\) 0 0
\(189\) 34.6941 115.389i 0.183567 0.610526i
\(190\) 0 0
\(191\) 177.248i 0.928002i 0.885835 + 0.464001i \(0.153587\pi\)
−0.885835 + 0.464001i \(0.846413\pi\)
\(192\) 0 0
\(193\) 284.254i 1.47282i 0.676536 + 0.736409i \(0.263480\pi\)
−0.676536 + 0.736409i \(0.736520\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −244.341 −1.24031 −0.620156 0.784479i \(-0.712931\pi\)
−0.620156 + 0.784479i \(0.712931\pi\)
\(198\) 0 0
\(199\) −74.0122 −0.371921 −0.185960 0.982557i \(-0.559540\pi\)
−0.185960 + 0.982557i \(0.559540\pi\)
\(200\) 0 0
\(201\) 0.882239 9.03324i 0.00438925 0.0449415i
\(202\) 0 0
\(203\) −24.9260 −0.122788
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −131.810 25.9946i −0.636763 0.125578i
\(208\) 0 0
\(209\) 475.066i 2.27304i
\(210\) 0 0
\(211\) −31.0682 −0.147243 −0.0736214 0.997286i \(-0.523456\pi\)
−0.0736214 + 0.997286i \(0.523456\pi\)
\(212\) 0 0
\(213\) 16.6447 170.425i 0.0781443 0.800119i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 237.001i 1.09217i
\(218\) 0 0
\(219\) 9.17668 93.9600i 0.0419026 0.429041i
\(220\) 0 0
\(221\) 8.74314i 0.0395617i
\(222\) 0 0
\(223\) 100.432i 0.450366i 0.974316 + 0.225183i \(0.0722980\pi\)
−0.974316 + 0.225183i \(0.927702\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −349.117 −1.53796 −0.768980 0.639273i \(-0.779236\pi\)
−0.768980 + 0.639273i \(0.779236\pi\)
\(228\) 0 0
\(229\) 165.007 0.720555 0.360278 0.932845i \(-0.382682\pi\)
0.360278 + 0.932845i \(0.382682\pi\)
\(230\) 0 0
\(231\) −238.105 23.2548i −1.03076 0.100670i
\(232\) 0 0
\(233\) −124.800 −0.535620 −0.267810 0.963472i \(-0.586300\pi\)
−0.267810 + 0.963472i \(0.586300\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 6.45091 + 0.630033i 0.0272190 + 0.00265837i
\(238\) 0 0
\(239\) 61.8321i 0.258712i 0.991598 + 0.129356i \(0.0412910\pi\)
−0.991598 + 0.129356i \(0.958709\pi\)
\(240\) 0 0
\(241\) 2.25555 0.00935913 0.00467956 0.999989i \(-0.498510\pi\)
0.00467956 + 0.999989i \(0.498510\pi\)
\(242\) 0 0
\(243\) 214.773 + 113.672i 0.883841 + 0.467787i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 292.723i 1.18511i
\(248\) 0 0
\(249\) 39.0274 + 3.81164i 0.156736 + 0.0153078i
\(250\) 0 0
\(251\) 108.993i 0.434234i 0.976146 + 0.217117i \(0.0696654\pi\)
−0.976146 + 0.217117i \(0.930335\pi\)
\(252\) 0 0
\(253\) 266.750i 1.05435i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 31.4664 0.122437 0.0612187 0.998124i \(-0.480501\pi\)
0.0612187 + 0.998124i \(0.480501\pi\)
\(258\) 0 0
\(259\) −230.972 −0.891785
\(260\) 0 0
\(261\) 9.72636 49.3191i 0.0372657 0.188962i
\(262\) 0 0
\(263\) −235.190 −0.894260 −0.447130 0.894469i \(-0.647554\pi\)
−0.447130 + 0.894469i \(0.647554\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −50.6504 + 518.609i −0.189702 + 1.94236i
\(268\) 0 0
\(269\) 221.066i 0.821808i −0.911679 0.410904i \(-0.865213\pi\)
0.911679 0.410904i \(-0.134787\pi\)
\(270\) 0 0
\(271\) 268.830 0.991991 0.495995 0.868325i \(-0.334803\pi\)
0.495995 + 0.868325i \(0.334803\pi\)
\(272\) 0 0
\(273\) −146.714 14.3290i −0.537414 0.0524870i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 144.697i 0.522370i −0.965289 0.261185i \(-0.915887\pi\)
0.965289 0.261185i \(-0.0841133\pi\)
\(278\) 0 0
\(279\) 468.934 + 92.4798i 1.68077 + 0.331469i
\(280\) 0 0
\(281\) 189.426i 0.674112i 0.941485 + 0.337056i \(0.109431\pi\)
−0.941485 + 0.337056i \(0.890569\pi\)
\(282\) 0 0
\(283\) 420.621i 1.48629i −0.669129 0.743146i \(-0.733333\pi\)
0.669129 0.743146i \(-0.266667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 302.675 1.05462
\(288\) 0 0
\(289\) −288.369 −0.997818
\(290\) 0 0
\(291\) −26.7226 + 273.612i −0.0918302 + 0.940249i
\(292\) 0 0
\(293\) 283.828 0.968696 0.484348 0.874875i \(-0.339057\pi\)
0.484348 + 0.874875i \(0.339057\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 138.923 462.045i 0.467754 1.55571i
\(298\) 0 0
\(299\) 164.364i 0.549713i
\(300\) 0 0
\(301\) −182.186 −0.605268
\(302\) 0 0
\(303\) −33.9297 + 347.406i −0.111979 + 1.14656i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 167.979i 0.547162i −0.961849 0.273581i \(-0.911792\pi\)
0.961849 0.273581i \(-0.0882081\pi\)
\(308\) 0 0
\(309\) 53.0938 543.628i 0.171825 1.75931i
\(310\) 0 0
\(311\) 192.970i 0.620482i 0.950658 + 0.310241i \(0.100410\pi\)
−0.950658 + 0.310241i \(0.899590\pi\)
\(312\) 0 0
\(313\) 161.056i 0.514555i 0.966338 + 0.257278i \(0.0828255\pi\)
−0.966338 + 0.257278i \(0.917175\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 514.761 1.62385 0.811926 0.583761i \(-0.198419\pi\)
0.811926 + 0.583761i \(0.198419\pi\)
\(318\) 0 0
\(319\) −99.8096 −0.312883
\(320\) 0 0
\(321\) 307.024 + 29.9858i 0.956462 + 0.0934136i
\(322\) 0 0
\(323\) −21.1101 −0.0653564
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −225.206 21.9949i −0.688702 0.0672627i
\(328\) 0 0
\(329\) 55.1064i 0.167497i
\(330\) 0 0
\(331\) 63.7466 0.192588 0.0962939 0.995353i \(-0.469301\pi\)
0.0962939 + 0.995353i \(0.469301\pi\)
\(332\) 0 0
\(333\) 90.1274 457.006i 0.270653 1.37239i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 64.2444i 0.190636i 0.995447 + 0.0953181i \(0.0303868\pi\)
−0.995447 + 0.0953181i \(0.969613\pi\)
\(338\) 0 0
\(339\) −423.754 41.3863i −1.25001 0.122083i
\(340\) 0 0
\(341\) 949.006i 2.78301i
\(342\) 0 0
\(343\) 348.466i 1.01594i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 66.1416 0.190610 0.0953049 0.995448i \(-0.469617\pi\)
0.0953049 + 0.995448i \(0.469617\pi\)
\(348\) 0 0
\(349\) 188.752 0.540837 0.270419 0.962743i \(-0.412838\pi\)
0.270419 + 0.962743i \(0.412838\pi\)
\(350\) 0 0
\(351\) 85.6006 284.700i 0.243876 0.811110i
\(352\) 0 0
\(353\) −435.927 −1.23492 −0.617461 0.786602i \(-0.711839\pi\)
−0.617461 + 0.786602i \(0.711839\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.03335 10.5805i 0.00289454 0.0296372i
\(358\) 0 0
\(359\) 188.255i 0.524386i −0.965015 0.262193i \(-0.915554\pi\)
0.965015 0.262193i \(-0.0844457\pi\)
\(360\) 0 0
\(361\) 345.773 0.957820
\(362\) 0 0
\(363\) −592.148 57.8326i −1.63126 0.159318i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 276.868i 0.754409i −0.926130 0.377205i \(-0.876885\pi\)
0.926130 0.377205i \(-0.123115\pi\)
\(368\) 0 0
\(369\) −118.106 + 598.877i −0.320071 + 1.62297i
\(370\) 0 0
\(371\) 165.275i 0.445486i
\(372\) 0 0
\(373\) 510.098i 1.36756i −0.729691 0.683778i \(-0.760336\pi\)
0.729691 0.683778i \(-0.239664\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −61.4999 −0.163130
\(378\) 0 0
\(379\) 298.461 0.787497 0.393748 0.919218i \(-0.371178\pi\)
0.393748 + 0.919218i \(0.371178\pi\)
\(380\) 0 0
\(381\) 32.9285 337.155i 0.0864266 0.884922i
\(382\) 0 0
\(383\) 500.548 1.30691 0.653457 0.756963i \(-0.273318\pi\)
0.653457 + 0.756963i \(0.273318\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 71.0904 360.476i 0.183696 0.931462i
\(388\) 0 0
\(389\) 465.726i 1.19724i −0.801033 0.598620i \(-0.795716\pi\)
0.801033 0.598620i \(-0.204284\pi\)
\(390\) 0 0
\(391\) −11.8534 −0.0303155
\(392\) 0 0
\(393\) 31.2701 320.175i 0.0795678 0.814694i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 344.998i 0.869013i 0.900669 + 0.434507i \(0.143077\pi\)
−0.900669 + 0.434507i \(0.856923\pi\)
\(398\) 0 0
\(399\) −34.5969 + 354.238i −0.0867091 + 0.887815i
\(400\) 0 0
\(401\) 428.755i 1.06922i −0.845100 0.534608i \(-0.820459\pi\)
0.845100 0.534608i \(-0.179541\pi\)
\(402\) 0 0
\(403\) 584.752i 1.45100i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −924.866 −2.27240
\(408\) 0 0
\(409\) 349.445 0.854389 0.427194 0.904160i \(-0.359502\pi\)
0.427194 + 0.904160i \(0.359502\pi\)
\(410\) 0 0
\(411\) 202.268 + 19.7547i 0.492136 + 0.0480648i
\(412\) 0 0
\(413\) 272.639 0.660143
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 421.606 + 41.1765i 1.01105 + 0.0987447i
\(418\) 0 0
\(419\) 225.745i 0.538770i 0.963033 + 0.269385i \(0.0868205\pi\)
−0.963033 + 0.269385i \(0.913180\pi\)
\(420\) 0 0
\(421\) −138.931 −0.330003 −0.165002 0.986293i \(-0.552763\pi\)
−0.165002 + 0.986293i \(0.552763\pi\)
\(422\) 0 0
\(423\) 109.034 + 21.5030i 0.257765 + 0.0508345i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 436.579i 1.02243i
\(428\) 0 0
\(429\) −587.477 57.3764i −1.36941 0.133745i
\(430\) 0 0
\(431\) 289.802i 0.672394i 0.941792 + 0.336197i \(0.109141\pi\)
−0.941792 + 0.336197i \(0.890859\pi\)
\(432\) 0 0
\(433\) 53.8425i 0.124348i 0.998065 + 0.0621738i \(0.0198033\pi\)
−0.998065 + 0.0621738i \(0.980197\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 396.854 0.908133
\(438\) 0 0
\(439\) −333.179 −0.758949 −0.379474 0.925202i \(-0.623895\pi\)
−0.379474 + 0.925202i \(0.623895\pi\)
\(440\) 0 0
\(441\) 256.814 + 50.6471i 0.582345 + 0.114846i
\(442\) 0 0
\(443\) 652.015 1.47182 0.735909 0.677081i \(-0.236755\pi\)
0.735909 + 0.677081i \(0.236755\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 16.0628 164.467i 0.0359346 0.367935i
\(448\) 0 0
\(449\) 115.054i 0.256245i 0.991758 + 0.128122i \(0.0408950\pi\)
−0.991758 + 0.128122i \(0.959105\pi\)
\(450\) 0 0
\(451\) 1211.98 2.68731
\(452\) 0 0
\(453\) 167.593 + 16.3681i 0.369963 + 0.0361327i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 138.318i 0.302666i 0.988483 + 0.151333i \(0.0483566\pi\)
−0.988483 + 0.151333i \(0.951643\pi\)
\(458\) 0 0
\(459\) 20.5315 + 6.17321i 0.0447310 + 0.0134493i
\(460\) 0 0
\(461\) 300.475i 0.651789i 0.945406 + 0.325894i \(0.105665\pi\)
−0.945406 + 0.325894i \(0.894335\pi\)
\(462\) 0 0
\(463\) 91.5721i 0.197780i −0.995098 0.0988899i \(-0.968471\pi\)
0.995098 0.0988899i \(-0.0315292\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −308.136 −0.659821 −0.329910 0.944012i \(-0.607019\pi\)
−0.329910 + 0.944012i \(0.607019\pi\)
\(468\) 0 0
\(469\) −13.5014 −0.0287877
\(470\) 0 0
\(471\) −79.9788 + 818.903i −0.169806 + 1.73865i
\(472\) 0 0
\(473\) −729.513 −1.54231
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −327.017 64.4919i −0.685570 0.135203i
\(478\) 0 0
\(479\) 218.332i 0.455808i −0.973684 0.227904i \(-0.926813\pi\)
0.973684 0.227904i \(-0.0731873\pi\)
\(480\) 0 0
\(481\) −569.877 −1.18478
\(482\) 0 0
\(483\) −19.4262 + 198.905i −0.0402200 + 0.411812i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 538.368i 1.10548i 0.833354 + 0.552739i \(0.186417\pi\)
−0.833354 + 0.552739i \(0.813583\pi\)
\(488\) 0 0
\(489\) −75.9108 + 777.250i −0.155237 + 1.58947i
\(490\) 0 0
\(491\) 418.223i 0.851777i −0.904776 0.425889i \(-0.859962\pi\)
0.904776 0.425889i \(-0.140038\pi\)
\(492\) 0 0
\(493\) 4.43515i 0.00899625i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −254.724 −0.512523
\(498\) 0 0
\(499\) 732.199 1.46733 0.733667 0.679510i \(-0.237807\pi\)
0.733667 + 0.679510i \(0.237807\pi\)
\(500\) 0 0
\(501\) 537.291 + 52.4749i 1.07244 + 0.104740i
\(502\) 0 0
\(503\) −338.339 −0.672641 −0.336321 0.941748i \(-0.609183\pi\)
−0.336321 + 0.941748i \(0.609183\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 142.612 + 13.9283i 0.281286 + 0.0274720i
\(508\) 0 0
\(509\) 864.760i 1.69894i −0.527637 0.849470i \(-0.676922\pi\)
0.527637 0.849470i \(-0.323078\pi\)
\(510\) 0 0
\(511\) −140.436 −0.274826
\(512\) 0 0
\(513\) −687.401 206.681i −1.33996 0.402887i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 220.659i 0.426806i
\(518\) 0 0
\(519\) 332.213 + 32.4459i 0.640103 + 0.0625162i
\(520\) 0 0
\(521\) 294.647i 0.565541i 0.959188 + 0.282770i \(0.0912534\pi\)
−0.959188 + 0.282770i \(0.908747\pi\)
\(522\) 0 0
\(523\) 613.850i 1.17371i −0.809692 0.586855i \(-0.800366\pi\)
0.809692 0.586855i \(-0.199634\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.1702 0.0800193
\(528\) 0 0
\(529\) −306.166 −0.578763
\(530\) 0 0
\(531\) −106.386 + 539.448i −0.200351 + 1.01591i
\(532\) 0 0
\(533\) 746.788 1.40110
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −22.7011 + 232.437i −0.0422740 + 0.432843i
\(538\) 0 0
\(539\) 519.728i 0.964245i
\(540\) 0 0
\(541\) 188.436 0.348311 0.174156 0.984718i \(-0.444280\pi\)
0.174156 + 0.984718i \(0.444280\pi\)
\(542\) 0 0
\(543\) −713.163 69.6517i −1.31338 0.128272i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 637.744i 1.16589i −0.812510 0.582947i \(-0.801899\pi\)
0.812510 0.582947i \(-0.198101\pi\)
\(548\) 0 0
\(549\) 863.822 + 170.357i 1.57345 + 0.310304i
\(550\) 0 0
\(551\) 148.490i 0.269492i
\(552\) 0 0
\(553\) 9.64176i 0.0174354i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 77.2307 0.138655 0.0693274 0.997594i \(-0.477915\pi\)
0.0693274 + 0.997594i \(0.477915\pi\)
\(558\) 0 0
\(559\) −449.506 −0.804125
\(560\) 0 0
\(561\) 4.13778 42.3667i 0.00737572 0.0755200i
\(562\) 0 0
\(563\) −257.584 −0.457521 −0.228760 0.973483i \(-0.573467\pi\)
−0.228760 + 0.973483i \(0.573467\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 137.238 334.412i 0.242042 0.589791i
\(568\) 0 0
\(569\) 307.483i 0.540392i −0.962805 0.270196i \(-0.912911\pi\)
0.962805 0.270196i \(-0.0870886\pi\)
\(570\) 0 0
\(571\) −384.100 −0.672679 −0.336340 0.941741i \(-0.609189\pi\)
−0.336340 + 0.941741i \(0.609189\pi\)
\(572\) 0 0
\(573\) −51.6874 + 529.227i −0.0902049 + 0.923608i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 235.337i 0.407863i 0.978985 + 0.203932i \(0.0653720\pi\)
−0.978985 + 0.203932i \(0.934628\pi\)
\(578\) 0 0
\(579\) −82.8913 + 848.724i −0.143163 + 1.46584i
\(580\) 0 0
\(581\) 58.3317i 0.100399i
\(582\) 0 0
\(583\) 661.801i 1.13516i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −230.275 −0.392292 −0.196146 0.980575i \(-0.562843\pi\)
−0.196146 + 0.980575i \(0.562843\pi\)
\(588\) 0 0
\(589\) −1411.87 −2.39706
\(590\) 0 0
\(591\) −729.553 71.2524i −1.23444 0.120562i
\(592\) 0 0
\(593\) 1042.09 1.75731 0.878656 0.477455i \(-0.158441\pi\)
0.878656 + 0.477455i \(0.158441\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −220.985 21.5827i −0.370160 0.0361519i
\(598\) 0 0
\(599\) 136.181i 0.227348i 0.993518 + 0.113674i \(0.0362619\pi\)
−0.993518 + 0.113674i \(0.963738\pi\)
\(600\) 0 0
\(601\) 136.625 0.227329 0.113665 0.993519i \(-0.463741\pi\)
0.113665 + 0.993519i \(0.463741\pi\)
\(602\) 0 0
\(603\) 5.26837 26.7141i 0.00873693 0.0443020i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 426.666i 0.702910i 0.936205 + 0.351455i \(0.114313\pi\)
−0.936205 + 0.351455i \(0.885687\pi\)
\(608\) 0 0
\(609\) −74.4240 7.26868i −0.122207 0.0119354i
\(610\) 0 0
\(611\) 135.964i 0.222527i
\(612\) 0 0
\(613\) 665.317i 1.08535i 0.839944 + 0.542673i \(0.182588\pi\)
−0.839944 + 0.542673i \(0.817412\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 826.628 1.33975 0.669877 0.742473i \(-0.266347\pi\)
0.669877 + 0.742473i \(0.266347\pi\)
\(618\) 0 0
\(619\) −19.8176 −0.0320155 −0.0160078 0.999872i \(-0.505096\pi\)
−0.0160078 + 0.999872i \(0.505096\pi\)
\(620\) 0 0
\(621\) −385.977 116.052i −0.621541 0.186879i
\(622\) 0 0
\(623\) 775.133 1.24419
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −138.534 + 1418.45i −0.220948 + 2.26228i
\(628\) 0 0
\(629\) 41.0975i 0.0653378i
\(630\) 0 0
\(631\) −728.156 −1.15397 −0.576986 0.816754i \(-0.695771\pi\)
−0.576986 + 0.816754i \(0.695771\pi\)
\(632\) 0 0
\(633\) −92.7633 9.05981i −0.146546 0.0143125i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 320.242i 0.502735i
\(638\) 0 0
\(639\) 99.3955 504.001i 0.155548 0.788734i
\(640\) 0 0
\(641\) 438.969i 0.684819i 0.939551 + 0.342410i \(0.111243\pi\)
−0.939551 + 0.342410i \(0.888757\pi\)
\(642\) 0 0
\(643\) 483.535i 0.751998i 0.926620 + 0.375999i \(0.122700\pi\)
−0.926620 + 0.375999i \(0.877300\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 575.687 0.889780 0.444890 0.895585i \(-0.353243\pi\)
0.444890 + 0.895585i \(0.353243\pi\)
\(648\) 0 0
\(649\) 1091.71 1.68214
\(650\) 0 0
\(651\) 69.1119 707.636i 0.106163 1.08700i
\(652\) 0 0
\(653\) 407.885 0.624633 0.312317 0.949978i \(-0.398895\pi\)
0.312317 + 0.949978i \(0.398895\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 54.7993 277.869i 0.0834084 0.422936i
\(658\) 0 0
\(659\) 187.060i 0.283854i 0.989877 + 0.141927i \(0.0453298\pi\)
−0.989877 + 0.141927i \(0.954670\pi\)
\(660\) 0 0
\(661\) 483.965 0.732171 0.366085 0.930581i \(-0.380698\pi\)
0.366085 + 0.930581i \(0.380698\pi\)
\(662\) 0 0
\(663\) 2.54959 26.1052i 0.00384553 0.0393744i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 83.3775i 0.125004i
\(668\) 0 0
\(669\) −29.2869 + 299.868i −0.0437771 + 0.448233i
\(670\) 0 0
\(671\) 1748.16i 2.60531i
\(672\) 0 0
\(673\) 480.449i 0.713891i 0.934125 + 0.356945i \(0.116182\pi\)
−0.934125 + 0.356945i \(0.883818\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1245.12 −1.83918 −0.919589 0.392881i \(-0.871478\pi\)
−0.919589 + 0.392881i \(0.871478\pi\)
\(678\) 0 0
\(679\) 408.951 0.602284
\(680\) 0 0
\(681\) −1042.39 101.806i −1.53068 0.149495i
\(682\) 0 0
\(683\) 118.409 0.173366 0.0866832 0.996236i \(-0.472373\pi\)
0.0866832 + 0.996236i \(0.472373\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 492.677 + 48.1177i 0.717143 + 0.0700404i
\(688\) 0 0
\(689\) 407.784i 0.591848i
\(690\) 0 0
\(691\) −481.257 −0.696465 −0.348232 0.937408i \(-0.613218\pi\)
−0.348232 + 0.937408i \(0.613218\pi\)
\(692\) 0 0
\(693\) −704.152 138.868i −1.01609 0.200387i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 53.8556i 0.0772678i
\(698\) 0 0
\(699\) −372.626 36.3928i −0.533084 0.0520641i
\(700\) 0 0
\(701\) 644.879i 0.919941i −0.887934 0.459971i \(-0.847860\pi\)
0.887934 0.459971i \(-0.152140\pi\)
\(702\) 0 0
\(703\) 1375.96i 1.95726i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 519.246 0.734436
\(708\) 0 0
\(709\) 534.801 0.754303 0.377151 0.926152i \(-0.376904\pi\)
0.377151 + 0.926152i \(0.376904\pi\)
\(710\) 0 0
\(711\) 19.0774 + 3.76230i 0.0268317 + 0.00529156i
\(712\) 0 0
\(713\) −792.767 −1.11188
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −18.0309 + 184.618i −0.0251476 + 0.257487i
\(718\) 0 0
\(719\) 891.507i 1.23993i −0.784631 0.619963i \(-0.787148\pi\)
0.784631 0.619963i \(-0.212852\pi\)
\(720\) 0 0
\(721\) −812.526 −1.12694
\(722\) 0 0
\(723\) 6.73461 + 0.657741i 0.00931481 + 0.000909739i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 720.024i 0.990405i 0.868778 + 0.495202i \(0.164906\pi\)
−0.868778 + 0.495202i \(0.835094\pi\)
\(728\) 0 0
\(729\) 608.121 + 402.032i 0.834185 + 0.551484i
\(730\) 0 0
\(731\) 32.4167i 0.0443458i
\(732\) 0 0
\(733\) 953.335i 1.30059i 0.759681 + 0.650296i \(0.225355\pi\)
−0.759681 + 0.650296i \(0.774645\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −54.0627 −0.0733552
\(738\) 0 0
\(739\) −615.723 −0.833185 −0.416592 0.909093i \(-0.636776\pi\)
−0.416592 + 0.909093i \(0.636776\pi\)
\(740\) 0 0
\(741\) −85.3609 + 874.010i −0.115197 + 1.17950i
\(742\) 0 0
\(743\) 377.743 0.508403 0.254201 0.967151i \(-0.418187\pi\)
0.254201 + 0.967151i \(0.418187\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 115.416 + 22.7615i 0.154506 + 0.0304706i
\(748\) 0 0
\(749\) 458.890i 0.612670i
\(750\) 0 0
\(751\) 735.578 0.979465 0.489733 0.871873i \(-0.337094\pi\)
0.489733 + 0.871873i \(0.337094\pi\)
\(752\) 0 0
\(753\) −31.7834 + 325.430i −0.0422090 + 0.432178i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 921.601i 1.21744i −0.793386 0.608719i \(-0.791684\pi\)
0.793386 0.608719i \(-0.208316\pi\)
\(758\) 0 0
\(759\) −77.7871 + 796.462i −0.102486 + 1.04936i
\(760\) 0 0
\(761\) 240.158i 0.315582i −0.987473 0.157791i \(-0.949563\pi\)
0.987473 0.157791i \(-0.0504372\pi\)
\(762\) 0 0
\(763\) 336.601i 0.441154i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 672.682 0.877029
\(768\) 0 0
\(769\) 4.79672 0.00623760 0.00311880 0.999995i \(-0.499007\pi\)
0.00311880 + 0.999995i \(0.499007\pi\)
\(770\) 0 0
\(771\) 93.9522 + 9.17592i 0.121858 + 0.0119013i
\(772\) 0 0
\(773\) 140.044 0.181170 0.0905850 0.995889i \(-0.471126\pi\)
0.0905850 + 0.995889i \(0.471126\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −689.636 67.3539i −0.887562 0.0866845i
\(778\) 0 0
\(779\) 1803.10i 2.31464i
\(780\) 0 0
\(781\) −1019.97 −1.30598
\(782\) 0 0
\(783\) 43.4228 144.420i 0.0554570 0.184445i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 649.685i 0.825520i −0.910840 0.412760i \(-0.864565\pi\)
0.910840 0.412760i \(-0.135435\pi\)
\(788\) 0 0
\(789\) −702.230 68.5838i −0.890025 0.0869250i
\(790\) 0 0
\(791\) 633.358i 0.800706i
\(792\) 0 0
\(793\) 1077.17i 1.35835i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −547.802 −0.687329 −0.343665 0.939092i \(-0.611668\pi\)
−0.343665 + 0.939092i \(0.611668\pi\)
\(798\) 0 0
\(799\) 9.80522 0.0122719
\(800\) 0 0
\(801\) −302.463 + 1533.69i −0.377607 + 1.91472i
\(802\) 0 0
\(803\) −562.338 −0.700296
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 64.4652 660.059i 0.0798825 0.817916i
\(808\) 0 0
\(809\) 719.685i 0.889598i 0.895630 + 0.444799i \(0.146725\pi\)
−0.895630 + 0.444799i \(0.853275\pi\)
\(810\) 0 0
\(811\) −775.519 −0.956251 −0.478125 0.878292i \(-0.658684\pi\)
−0.478125 + 0.878292i \(0.658684\pi\)
\(812\) 0 0
\(813\) 802.669 + 78.3934i 0.987293 + 0.0964248i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1085.32i 1.32842i
\(818\) 0 0
\(819\) −433.879 85.5666i −0.529767 0.104477i
\(820\) 0 0
\(821\) 1383.11i 1.68466i 0.538960 + 0.842331i \(0.318817\pi\)
−0.538960 + 0.842331i \(0.681183\pi\)
\(822\) 0 0
\(823\) 536.474i 0.651852i 0.945395 + 0.325926i \(0.105676\pi\)
−0.945395 + 0.325926i \(0.894324\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1652.28 1.99792 0.998961 0.0455686i \(-0.0145100\pi\)
0.998961 + 0.0455686i \(0.0145100\pi\)
\(828\) 0 0
\(829\) 999.416 1.20557 0.602784 0.797904i \(-0.294058\pi\)
0.602784 + 0.797904i \(0.294058\pi\)
\(830\) 0 0
\(831\) 42.1950 432.034i 0.0507761 0.519897i
\(832\) 0 0
\(833\) 23.0947 0.0277247
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1373.17 + 412.872i 1.64059 + 0.493276i
\(838\) 0 0
\(839\) 1394.04i 1.66155i 0.556611 + 0.830773i \(0.312101\pi\)
−0.556611 + 0.830773i \(0.687899\pi\)
\(840\) 0 0
\(841\) 809.803 0.962905
\(842\) 0 0
\(843\) −55.2384 + 565.586i −0.0655260 + 0.670920i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 885.045i 1.04492i
\(848\) 0 0
\(849\) 122.657 1255.89i 0.144473 1.47925i
\(850\) 0 0
\(851\) 772.602i 0.907875i
\(852\) 0 0
\(853\) 1293.04i 1.51587i −0.652330 0.757935i \(-0.726208\pi\)
0.652330 0.757935i \(-0.273792\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1513.48 −1.76602 −0.883008 0.469357i \(-0.844486\pi\)
−0.883008 + 0.469357i \(0.844486\pi\)
\(858\) 0 0
\(859\) −767.814 −0.893846 −0.446923 0.894572i \(-0.647480\pi\)
−0.446923 + 0.894572i \(0.647480\pi\)
\(860\) 0 0
\(861\) 903.724 + 88.2629i 1.04962 + 0.102512i
\(862\) 0 0
\(863\) 29.2279 0.0338677 0.0169339 0.999857i \(-0.494610\pi\)
0.0169339 + 0.999857i \(0.494610\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −861.012 84.0914i −0.993093 0.0969913i
\(868\) 0 0
\(869\) 38.6078i 0.0444279i
\(870\) 0 0
\(871\) −33.3120 −0.0382457
\(872\) 0 0
\(873\) −159.576 + 809.157i −0.182791 + 0.926870i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1070.98i 1.22119i 0.791943 + 0.610595i \(0.209070\pi\)
−0.791943 + 0.610595i \(0.790930\pi\)
\(878\) 0 0
\(879\) 847.452 + 82.7671i 0.964109 + 0.0941605i
\(880\) 0 0
\(881\) 489.535i 0.555659i 0.960630 + 0.277829i \(0.0896150\pi\)
−0.960630 + 0.277829i \(0.910385\pi\)
\(882\) 0 0
\(883\) 742.175i 0.840515i −0.907405 0.420257i \(-0.861940\pi\)
0.907405 0.420257i \(-0.138060\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1441.82 −1.62550 −0.812751 0.582612i \(-0.802031\pi\)
−0.812751 + 0.582612i \(0.802031\pi\)
\(888\) 0 0
\(889\) −503.924 −0.566844
\(890\) 0 0
\(891\) 549.533 1339.06i 0.616759 1.50287i
\(892\) 0 0
\(893\) −328.282 −0.367617
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −47.9303 + 490.758i −0.0534340 + 0.547110i
\(898\) 0 0
\(899\) 296.629i 0.329954i
\(900\) 0 0
\(901\) −29.4079 −0.0326392
\(902\) 0 0
\(903\) −543.969 53.1272i −0.602402 0.0588341i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 352.397i 0.388531i 0.980949 + 0.194265i \(0.0622323\pi\)
−0.980949 + 0.194265i \(0.937768\pi\)
\(908\) 0 0
\(909\) −202.614 + 1027.39i −0.222898 + 1.13024i
\(910\) 0 0
\(911\) 184.505i 0.202531i −0.994859 0.101265i \(-0.967711\pi\)
0.994859 0.101265i \(-0.0322891\pi\)
\(912\) 0 0
\(913\) 233.574i 0.255831i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −478.545 −0.521859
\(918\) 0 0
\(919\) −666.062 −0.724769 −0.362384 0.932029i \(-0.618037\pi\)
−0.362384 + 0.932029i \(0.618037\pi\)
\(920\) 0 0
\(921\) 48.9843 501.550i 0.0531860 0.544571i
\(922\) 0 0
\(923\) −628.479 −0.680909
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 317.055 1607.68i 0.342022 1.73428i
\(928\) 0 0
\(929\) 957.310i 1.03047i 0.857048 + 0.515237i \(0.172296\pi\)
−0.857048 + 0.515237i \(0.827704\pi\)
\(930\) 0 0
\(931\) −773.218 −0.830524
\(932\) 0 0
\(933\) −56.2720 + 576.169i −0.0603130 + 0.617544i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1108.63i 1.18317i −0.806243 0.591585i \(-0.798502\pi\)
0.806243 0.591585i \(-0.201498\pi\)
\(938\) 0 0
\(939\) −46.9655 + 480.879i −0.0500165 + 0.512119i
\(940\) 0 0
\(941\) 50.4523i 0.0536157i −0.999641 0.0268078i \(-0.991466\pi\)
0.999641 0.0268078i \(-0.00853422\pi\)
\(942\) 0 0
\(943\) 1012.45i 1.07364i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −883.676 −0.933132 −0.466566 0.884486i \(-0.654509\pi\)
−0.466566 + 0.884486i \(0.654509\pi\)
\(948\) 0 0
\(949\) −346.497 −0.365118
\(950\) 0 0
\(951\) 1536.97 + 150.109i 1.61616 + 0.157844i
\(952\) 0 0
\(953\) 1238.78 1.29988 0.649938 0.759987i \(-0.274795\pi\)
0.649938 + 0.759987i \(0.274795\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −298.011 29.1055i −0.311401 0.0304132i
\(958\) 0 0
\(959\) 302.317i 0.315242i
\(960\) 0 0
\(961\) 1859.39 1.93485
\(962\) 0 0
\(963\) 907.967 + 179.063i 0.942852 + 0.185943i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1221.47i 1.26315i −0.775314 0.631575i \(-0.782409\pi\)
0.775314 0.631575i \(-0.217591\pi\)
\(968\) 0 0
\(969\) −63.0304 6.15592i −0.0650469 0.00635286i
\(970\) 0 0
\(971\) 1142.08i 1.17619i −0.808792 0.588095i \(-0.799878\pi\)
0.808792 0.588095i \(-0.200122\pi\)
\(972\) 0 0
\(973\) 630.148i 0.647634i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 364.738 0.373324 0.186662 0.982424i \(-0.440233\pi\)
0.186662 + 0.982424i \(0.440233\pi\)
\(978\) 0 0
\(979\) 3103.81 3.17039
\(980\) 0 0
\(981\) −666.004 131.344i −0.678903 0.133888i
\(982\) 0 0
\(983\) −40.0459 −0.0407384 −0.0203692 0.999793i \(-0.506484\pi\)
−0.0203692 + 0.999793i \(0.506484\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 16.0696 164.536i 0.0162812 0.166703i
\(988\) 0 0
\(989\) 609.410i 0.616188i
\(990\) 0 0
\(991\) −910.307 −0.918574 −0.459287 0.888288i \(-0.651895\pi\)
−0.459287 + 0.888288i \(0.651895\pi\)
\(992\) 0 0
\(993\) 190.334 + 18.5891i 0.191676 + 0.0187202i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 358.437i 0.359516i −0.983711 0.179758i \(-0.942469\pi\)
0.983711 0.179758i \(-0.0575314\pi\)
\(998\) 0 0
\(999\) 402.369 1338.24i 0.402772 1.33958i
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 600.3.c.d.449.16 16
3.2 odd 2 inner 600.3.c.d.449.2 16
4.3 odd 2 1200.3.c.m.449.1 16
5.2 odd 4 120.3.l.a.41.5 8
5.3 odd 4 600.3.l.f.401.4 8
5.4 even 2 inner 600.3.c.d.449.1 16
12.11 even 2 1200.3.c.m.449.15 16
15.2 even 4 120.3.l.a.41.6 yes 8
15.8 even 4 600.3.l.f.401.3 8
15.14 odd 2 inner 600.3.c.d.449.15 16
20.3 even 4 1200.3.l.x.401.5 8
20.7 even 4 240.3.l.d.161.4 8
20.19 odd 2 1200.3.c.m.449.16 16
40.27 even 4 960.3.l.g.641.5 8
40.37 odd 4 960.3.l.h.641.4 8
60.23 odd 4 1200.3.l.x.401.6 8
60.47 odd 4 240.3.l.d.161.3 8
60.59 even 2 1200.3.c.m.449.2 16
120.77 even 4 960.3.l.h.641.3 8
120.107 odd 4 960.3.l.g.641.6 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.5 8 5.2 odd 4
120.3.l.a.41.6 yes 8 15.2 even 4
240.3.l.d.161.3 8 60.47 odd 4
240.3.l.d.161.4 8 20.7 even 4
600.3.c.d.449.1 16 5.4 even 2 inner
600.3.c.d.449.2 16 3.2 odd 2 inner
600.3.c.d.449.15 16 15.14 odd 2 inner
600.3.c.d.449.16 16 1.1 even 1 trivial
600.3.l.f.401.3 8 15.8 even 4
600.3.l.f.401.4 8 5.3 odd 4
960.3.l.g.641.5 8 40.27 even 4
960.3.l.g.641.6 8 120.107 odd 4
960.3.l.h.641.3 8 120.77 even 4
960.3.l.h.641.4 8 40.37 odd 4
1200.3.c.m.449.1 16 4.3 odd 2
1200.3.c.m.449.2 16 60.59 even 2
1200.3.c.m.449.15 16 12.11 even 2
1200.3.c.m.449.16 16 20.19 odd 2
1200.3.l.x.401.5 8 20.3 even 4
1200.3.l.x.401.6 8 60.23 odd 4