Properties

Label 600.3.c.d.449.5
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 449.5
Root \(-1.57158 + 1.57158i\) of defining polynomial
Character \(\chi\) \(=\) 600.449
Dual form 600.3.c.d.449.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.864473 - 2.87275i) q^{3} -9.02416i q^{7} +(-7.50537 + 4.96683i) q^{9} +O(q^{10})\) \(q+(-0.864473 - 2.87275i) q^{3} -9.02416i q^{7} +(-7.50537 + 4.96683i) q^{9} +21.8827i q^{11} +21.6599i q^{13} -12.1078 q^{17} -3.03757 q^{19} +(-25.9241 + 7.80114i) q^{21} +28.5735 q^{23} +(20.7566 + 17.2674i) q^{27} -12.0364i q^{29} +2.19085 q^{31} +(62.8636 - 18.9170i) q^{33} -0.839959i q^{37} +(62.2233 - 18.7244i) q^{39} +35.5690i q^{41} -12.7152i q^{43} +22.5481 q^{47} -32.4354 q^{49} +(10.4668 + 34.7826i) q^{51} -9.13775 q^{53} +(2.62590 + 8.72618i) q^{57} +80.4459i q^{59} -57.8816 q^{61} +(44.8214 + 67.7297i) q^{63} +63.0560i q^{67} +(-24.7011 - 82.0846i) q^{69} +17.0218i q^{71} +52.1181i q^{73} +197.473 q^{77} +7.46224 q^{79} +(31.6612 - 74.5558i) q^{81} +82.3758 q^{83} +(-34.5774 + 10.4051i) q^{87} +27.5850i q^{89} +195.462 q^{91} +(-1.89393 - 6.29376i) q^{93} -114.989i q^{97} +(-108.688 - 164.238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{9}+O(q^{10}) \) 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
\(3\) −0.864473 2.87275i −0.288158 0.957583i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 9.02416i 1.28917i −0.764535 0.644583i \(-0.777031\pi\)
0.764535 0.644583i \(-0.222969\pi\)
\(8\) 0 0
\(9\) −7.50537 + 4.96683i −0.833930 + 0.551870i
\(10\) 0 0
\(11\) 21.8827i 1.98934i 0.103119 + 0.994669i \(0.467118\pi\)
−0.103119 + 0.994669i \(0.532882\pi\)
\(12\) 0 0
\(13\) 21.6599i 1.66614i 0.553166 + 0.833071i \(0.313420\pi\)
−0.553166 + 0.833071i \(0.686580\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −12.1078 −0.712221 −0.356111 0.934444i \(-0.615897\pi\)
−0.356111 + 0.934444i \(0.615897\pi\)
\(18\) 0 0
\(19\) −3.03757 −0.159872 −0.0799361 0.996800i \(-0.525472\pi\)
−0.0799361 + 0.996800i \(0.525472\pi\)
\(20\) 0 0
\(21\) −25.9241 + 7.80114i −1.23448 + 0.371483i
\(22\) 0 0
\(23\) 28.5735 1.24233 0.621164 0.783681i \(-0.286660\pi\)
0.621164 + 0.783681i \(0.286660\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 20.7566 + 17.2674i 0.768765 + 0.639532i
\(28\) 0 0
\(29\) 12.0364i 0.415047i −0.978230 0.207523i \(-0.933460\pi\)
0.978230 0.207523i \(-0.0665403\pi\)
\(30\) 0 0
\(31\) 2.19085 0.0706725 0.0353363 0.999375i \(-0.488750\pi\)
0.0353363 + 0.999375i \(0.488750\pi\)
\(32\) 0 0
\(33\) 62.8636 18.9170i 1.90496 0.573243i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.839959i 0.0227016i −0.999936 0.0113508i \(-0.996387\pi\)
0.999936 0.0113508i \(-0.00361315\pi\)
\(38\) 0 0
\(39\) 62.2233 18.7244i 1.59547 0.480112i
\(40\) 0 0
\(41\) 35.5690i 0.867537i 0.901024 + 0.433769i \(0.142816\pi\)
−0.901024 + 0.433769i \(0.857184\pi\)
\(42\) 0 0
\(43\) 12.7152i 0.295702i −0.989010 0.147851i \(-0.952764\pi\)
0.989010 0.147851i \(-0.0472356\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 22.5481 0.479746 0.239873 0.970804i \(-0.422894\pi\)
0.239873 + 0.970804i \(0.422894\pi\)
\(48\) 0 0
\(49\) −32.4354 −0.661947
\(50\) 0 0
\(51\) 10.4668 + 34.7826i 0.205232 + 0.682011i
\(52\) 0 0
\(53\) −9.13775 −0.172410 −0.0862052 0.996277i \(-0.527474\pi\)
−0.0862052 + 0.996277i \(0.527474\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.62590 + 8.72618i 0.0460684 + 0.153091i
\(58\) 0 0
\(59\) 80.4459i 1.36349i 0.731590 + 0.681745i \(0.238779\pi\)
−0.731590 + 0.681745i \(0.761221\pi\)
\(60\) 0 0
\(61\) −57.8816 −0.948878 −0.474439 0.880288i \(-0.657349\pi\)
−0.474439 + 0.880288i \(0.657349\pi\)
\(62\) 0 0
\(63\) 44.8214 + 67.7297i 0.711452 + 1.07507i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 63.0560i 0.941135i 0.882364 + 0.470567i \(0.155951\pi\)
−0.882364 + 0.470567i \(0.844049\pi\)
\(68\) 0 0
\(69\) −24.7011 82.0846i −0.357986 1.18963i
\(70\) 0 0
\(71\) 17.0218i 0.239743i 0.992789 + 0.119872i \(0.0382483\pi\)
−0.992789 + 0.119872i \(0.961752\pi\)
\(72\) 0 0
\(73\) 52.1181i 0.713947i 0.934115 + 0.356973i \(0.116191\pi\)
−0.934115 + 0.356973i \(0.883809\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 197.473 2.56459
\(78\) 0 0
\(79\) 7.46224 0.0944588 0.0472294 0.998884i \(-0.484961\pi\)
0.0472294 + 0.998884i \(0.484961\pi\)
\(80\) 0 0
\(81\) 31.6612 74.5558i 0.390879 0.920442i
\(82\) 0 0
\(83\) 82.3758 0.992480 0.496240 0.868185i \(-0.334714\pi\)
0.496240 + 0.868185i \(0.334714\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −34.5774 + 10.4051i −0.397442 + 0.119599i
\(88\) 0 0
\(89\) 27.5850i 0.309944i 0.987919 + 0.154972i \(0.0495287\pi\)
−0.987919 + 0.154972i \(0.950471\pi\)
\(90\) 0 0
\(91\) 195.462 2.14793
\(92\) 0 0
\(93\) −1.89393 6.29376i −0.0203648 0.0676748i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 114.989i 1.18545i −0.805404 0.592727i \(-0.798051\pi\)
0.805404 0.592727i \(-0.201949\pi\)
\(98\) 0 0
\(99\) −108.688 164.238i −1.09786 1.65897i
\(100\) 0 0
\(101\) 122.804i 1.21588i 0.793982 + 0.607941i \(0.208004\pi\)
−0.793982 + 0.607941i \(0.791996\pi\)
\(102\) 0 0
\(103\) 46.8275i 0.454636i −0.973821 0.227318i \(-0.927004\pi\)
0.973821 0.227318i \(-0.0729957\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −105.086 −0.982113 −0.491056 0.871128i \(-0.663389\pi\)
−0.491056 + 0.871128i \(0.663389\pi\)
\(108\) 0 0
\(109\) 116.777 1.07135 0.535673 0.844426i \(-0.320058\pi\)
0.535673 + 0.844426i \(0.320058\pi\)
\(110\) 0 0
\(111\) −2.41299 + 0.726122i −0.0217387 + 0.00654164i
\(112\) 0 0
\(113\) −10.8116 −0.0956779 −0.0478389 0.998855i \(-0.515233\pi\)
−0.0478389 + 0.998855i \(0.515233\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −107.581 162.565i −0.919494 1.38945i
\(118\) 0 0
\(119\) 109.262i 0.918171i
\(120\) 0 0
\(121\) −357.853 −2.95747
\(122\) 0 0
\(123\) 102.181 30.7485i 0.830739 0.249988i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 192.459i 1.51543i 0.652587 + 0.757714i \(0.273684\pi\)
−0.652587 + 0.757714i \(0.726316\pi\)
\(128\) 0 0
\(129\) −36.5276 + 10.9919i −0.283159 + 0.0852089i
\(130\) 0 0
\(131\) 48.6360i 0.371267i 0.982619 + 0.185633i \(0.0594337\pi\)
−0.982619 + 0.185633i \(0.940566\pi\)
\(132\) 0 0
\(133\) 27.4115i 0.206102i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −157.869 −1.15233 −0.576163 0.817335i \(-0.695451\pi\)
−0.576163 + 0.817335i \(0.695451\pi\)
\(138\) 0 0
\(139\) 164.752 1.18526 0.592632 0.805473i \(-0.298089\pi\)
0.592632 + 0.805473i \(0.298089\pi\)
\(140\) 0 0
\(141\) −19.4922 64.7749i −0.138243 0.459397i
\(142\) 0 0
\(143\) −473.977 −3.31452
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 28.0396 + 93.1788i 0.190745 + 0.633869i
\(148\) 0 0
\(149\) 262.935i 1.76467i 0.470626 + 0.882333i \(0.344028\pi\)
−0.470626 + 0.882333i \(0.655972\pi\)
\(150\) 0 0
\(151\) 15.8171 0.104749 0.0523745 0.998628i \(-0.483321\pi\)
0.0523745 + 0.998628i \(0.483321\pi\)
\(152\) 0 0
\(153\) 90.8733 60.1372i 0.593943 0.393054i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.11941i 0.0389771i 0.999810 + 0.0194886i \(0.00620380\pi\)
−0.999810 + 0.0194886i \(0.993796\pi\)
\(158\) 0 0
\(159\) 7.89934 + 26.2505i 0.0496814 + 0.165097i
\(160\) 0 0
\(161\) 257.852i 1.60157i
\(162\) 0 0
\(163\) 170.444i 1.04567i −0.852435 0.522833i \(-0.824875\pi\)
0.852435 0.522833i \(-0.175125\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −61.2668 −0.366867 −0.183434 0.983032i \(-0.558721\pi\)
−0.183434 + 0.983032i \(0.558721\pi\)
\(168\) 0 0
\(169\) −300.149 −1.77603
\(170\) 0 0
\(171\) 22.7981 15.0871i 0.133322 0.0882286i
\(172\) 0 0
\(173\) −262.548 −1.51762 −0.758810 0.651312i \(-0.774219\pi\)
−0.758810 + 0.651312i \(0.774219\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 231.101 69.5434i 1.30566 0.392900i
\(178\) 0 0
\(179\) 6.88752i 0.0384778i 0.999815 + 0.0192389i \(0.00612431\pi\)
−0.999815 + 0.0192389i \(0.993876\pi\)
\(180\) 0 0
\(181\) 218.536 1.20738 0.603691 0.797218i \(-0.293696\pi\)
0.603691 + 0.797218i \(0.293696\pi\)
\(182\) 0 0
\(183\) 50.0371 + 166.279i 0.273427 + 0.908630i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 264.951i 1.41685i
\(188\) 0 0
\(189\) 155.823 187.311i 0.824462 0.991065i
\(190\) 0 0
\(191\) 75.2506i 0.393982i 0.980405 + 0.196991i \(0.0631170\pi\)
−0.980405 + 0.196991i \(0.936883\pi\)
\(192\) 0 0
\(193\) 212.587i 1.10149i −0.834674 0.550744i \(-0.814344\pi\)
0.834674 0.550744i \(-0.185656\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −190.640 −0.967718 −0.483859 0.875146i \(-0.660765\pi\)
−0.483859 + 0.875146i \(0.660765\pi\)
\(198\) 0 0
\(199\) −209.996 −1.05526 −0.527629 0.849475i \(-0.676919\pi\)
−0.527629 + 0.849475i \(0.676919\pi\)
\(200\) 0 0
\(201\) 181.144 54.5102i 0.901214 0.271195i
\(202\) 0 0
\(203\) −108.618 −0.535064
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −214.455 + 141.920i −1.03601 + 0.685603i
\(208\) 0 0
\(209\) 66.4703i 0.318040i
\(210\) 0 0
\(211\) 176.419 0.836110 0.418055 0.908422i \(-0.362712\pi\)
0.418055 + 0.908422i \(0.362712\pi\)
\(212\) 0 0
\(213\) 48.8993 14.7149i 0.229574 0.0690839i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 19.7706i 0.0911086i
\(218\) 0 0
\(219\) 149.722 45.0547i 0.683663 0.205729i
\(220\) 0 0
\(221\) 262.252i 1.18666i
\(222\) 0 0
\(223\) 132.362i 0.593552i 0.954947 + 0.296776i \(0.0959114\pi\)
−0.954947 + 0.296776i \(0.904089\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 187.624 0.826537 0.413268 0.910609i \(-0.364387\pi\)
0.413268 + 0.910609i \(0.364387\pi\)
\(228\) 0 0
\(229\) 178.571 0.779788 0.389894 0.920860i \(-0.372512\pi\)
0.389894 + 0.920860i \(0.372512\pi\)
\(230\) 0 0
\(231\) −170.710 567.291i −0.739005 2.45580i
\(232\) 0 0
\(233\) −296.711 −1.27344 −0.636718 0.771097i \(-0.719708\pi\)
−0.636718 + 0.771097i \(0.719708\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −6.45091 21.4371i −0.0272190 0.0904521i
\(238\) 0 0
\(239\) 137.976i 0.577306i −0.957434 0.288653i \(-0.906793\pi\)
0.957434 0.288653i \(-0.0932074\pi\)
\(240\) 0 0
\(241\) 42.5687 0.176633 0.0883167 0.996092i \(-0.471851\pi\)
0.0883167 + 0.996092i \(0.471851\pi\)
\(242\) 0 0
\(243\) −241.550 26.5032i −0.994034 0.109067i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 65.7933i 0.266370i
\(248\) 0 0
\(249\) −71.2117 236.645i −0.285991 0.950382i
\(250\) 0 0
\(251\) 205.885i 0.820259i −0.912027 0.410130i \(-0.865483\pi\)
0.912027 0.410130i \(-0.134517\pi\)
\(252\) 0 0
\(253\) 625.267i 2.47141i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 188.382 0.733004 0.366502 0.930417i \(-0.380555\pi\)
0.366502 + 0.930417i \(0.380555\pi\)
\(258\) 0 0
\(259\) −7.57992 −0.0292661
\(260\) 0 0
\(261\) 59.7825 + 90.3373i 0.229052 + 0.346120i
\(262\) 0 0
\(263\) 188.745 0.717660 0.358830 0.933403i \(-0.383176\pi\)
0.358830 + 0.933403i \(0.383176\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 79.2447 23.8465i 0.296797 0.0893127i
\(268\) 0 0
\(269\) 333.372i 1.23930i 0.784878 + 0.619651i \(0.212726\pi\)
−0.784878 + 0.619651i \(0.787274\pi\)
\(270\) 0 0
\(271\) 262.047 0.966964 0.483482 0.875354i \(-0.339372\pi\)
0.483482 + 0.875354i \(0.339372\pi\)
\(272\) 0 0
\(273\) −168.972 561.513i −0.618944 2.05682i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 400.100i 1.44440i 0.691682 + 0.722202i \(0.256870\pi\)
−0.691682 + 0.722202i \(0.743130\pi\)
\(278\) 0 0
\(279\) −16.4431 + 10.8816i −0.0589360 + 0.0390020i
\(280\) 0 0
\(281\) 350.698i 1.24804i 0.781410 + 0.624018i \(0.214501\pi\)
−0.781410 + 0.624018i \(0.785499\pi\)
\(282\) 0 0
\(283\) 464.015i 1.63963i −0.572630 0.819814i \(-0.694077\pi\)
0.572630 0.819814i \(-0.305923\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 320.981 1.11840
\(288\) 0 0
\(289\) −142.402 −0.492741
\(290\) 0 0
\(291\) −330.334 + 99.4049i −1.13517 + 0.341598i
\(292\) 0 0
\(293\) 47.4080 0.161802 0.0809009 0.996722i \(-0.474220\pi\)
0.0809009 + 0.996722i \(0.474220\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −377.857 + 454.212i −1.27224 + 1.52933i
\(298\) 0 0
\(299\) 618.899i 2.06990i
\(300\) 0 0
\(301\) −114.744 −0.381209
\(302\) 0 0
\(303\) 352.785 106.161i 1.16431 0.350366i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 461.894i 1.50454i −0.658855 0.752270i \(-0.728959\pi\)
0.658855 0.752270i \(-0.271041\pi\)
\(308\) 0 0
\(309\) −134.524 + 40.4812i −0.435352 + 0.131007i
\(310\) 0 0
\(311\) 123.057i 0.395681i 0.980234 + 0.197841i \(0.0633929\pi\)
−0.980234 + 0.197841i \(0.936607\pi\)
\(312\) 0 0
\(313\) 97.4353i 0.311295i 0.987813 + 0.155647i \(0.0497464\pi\)
−0.987813 + 0.155647i \(0.950254\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −252.388 −0.796175 −0.398088 0.917347i \(-0.630326\pi\)
−0.398088 + 0.917347i \(0.630326\pi\)
\(318\) 0 0
\(319\) 263.388 0.825668
\(320\) 0 0
\(321\) 90.8441 + 301.886i 0.283003 + 0.940454i
\(322\) 0 0
\(323\) 36.7782 0.113864
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −100.950 335.470i −0.308716 1.02590i
\(328\) 0 0
\(329\) 203.477i 0.618472i
\(330\) 0 0
\(331\) −303.273 −0.916231 −0.458116 0.888893i \(-0.651475\pi\)
−0.458116 + 0.888893i \(0.651475\pi\)
\(332\) 0 0
\(333\) 4.17193 + 6.30420i 0.0125283 + 0.0189315i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 352.738i 1.04670i 0.852118 + 0.523350i \(0.175318\pi\)
−0.852118 + 0.523350i \(0.824682\pi\)
\(338\) 0 0
\(339\) 9.34634 + 31.0590i 0.0275703 + 0.0916195i
\(340\) 0 0
\(341\) 47.9417i 0.140592i
\(342\) 0 0
\(343\) 149.481i 0.435806i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −280.382 −0.808018 −0.404009 0.914755i \(-0.632384\pi\)
−0.404009 + 0.914755i \(0.632384\pi\)
\(348\) 0 0
\(349\) 586.721 1.68115 0.840575 0.541696i \(-0.182217\pi\)
0.840575 + 0.541696i \(0.182217\pi\)
\(350\) 0 0
\(351\) −374.008 + 449.586i −1.06555 + 1.28087i
\(352\) 0 0
\(353\) 558.927 1.58336 0.791681 0.610935i \(-0.209206\pi\)
0.791681 + 0.610935i \(0.209206\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 313.883 94.4544i 0.879225 0.264578i
\(358\) 0 0
\(359\) 323.554i 0.901264i −0.892710 0.450632i \(-0.851199\pi\)
0.892710 0.450632i \(-0.148801\pi\)
\(360\) 0 0
\(361\) −351.773 −0.974441
\(362\) 0 0
\(363\) 309.355 + 1028.02i 0.852217 + 2.83202i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 176.636i 0.481296i −0.970612 0.240648i \(-0.922640\pi\)
0.970612 0.240648i \(-0.0773600\pi\)
\(368\) 0 0
\(369\) −176.665 266.959i −0.478768 0.723466i
\(370\) 0 0
\(371\) 82.4605i 0.222265i
\(372\) 0 0
\(373\) 367.327i 0.984790i 0.870372 + 0.492395i \(0.163878\pi\)
−0.870372 + 0.492395i \(0.836122\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 260.706 0.691527
\(378\) 0 0
\(379\) −611.014 −1.61217 −0.806087 0.591798i \(-0.798418\pi\)
−0.806087 + 0.591798i \(0.798418\pi\)
\(380\) 0 0
\(381\) 552.887 166.376i 1.45115 0.436682i
\(382\) 0 0
\(383\) −13.6994 −0.0357687 −0.0178844 0.999840i \(-0.505693\pi\)
−0.0178844 + 0.999840i \(0.505693\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 63.1542 + 95.4322i 0.163189 + 0.246595i
\(388\) 0 0
\(389\) 379.601i 0.975838i −0.872889 0.487919i \(-0.837756\pi\)
0.872889 0.487919i \(-0.162244\pi\)
\(390\) 0 0
\(391\) −345.962 −0.884812
\(392\) 0 0
\(393\) 139.719 42.0445i 0.355519 0.106983i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 450.560i 1.13491i −0.823404 0.567456i \(-0.807928\pi\)
0.823404 0.567456i \(-0.192072\pi\)
\(398\) 0 0
\(399\) 78.7464 23.6965i 0.197359 0.0593898i
\(400\) 0 0
\(401\) 503.683i 1.25607i 0.778186 + 0.628034i \(0.216140\pi\)
−0.778186 + 0.628034i \(0.783860\pi\)
\(402\) 0 0
\(403\) 47.4535i 0.117751i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.3806 0.0451611
\(408\) 0 0
\(409\) −240.726 −0.588573 −0.294286 0.955717i \(-0.595082\pi\)
−0.294286 + 0.955717i \(0.595082\pi\)
\(410\) 0 0
\(411\) 136.473 + 453.517i 0.332051 + 1.10345i
\(412\) 0 0
\(413\) 725.957 1.75776
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −142.423 473.290i −0.341543 1.13499i
\(418\) 0 0
\(419\) 676.873i 1.61545i −0.589561 0.807724i \(-0.700699\pi\)
0.589561 0.807724i \(-0.299301\pi\)
\(420\) 0 0
\(421\) 683.755 1.62412 0.812061 0.583572i \(-0.198346\pi\)
0.812061 + 0.583572i \(0.198346\pi\)
\(422\) 0 0
\(423\) −169.232 + 111.992i −0.400075 + 0.264757i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 522.332i 1.22326i
\(428\) 0 0
\(429\) 409.740 + 1361.62i 0.955105 + 3.17393i
\(430\) 0 0
\(431\) 213.608i 0.495610i 0.968810 + 0.247805i \(0.0797092\pi\)
−0.968810 + 0.247805i \(0.920291\pi\)
\(432\) 0 0
\(433\) 383.579i 0.885864i −0.896555 0.442932i \(-0.853938\pi\)
0.896555 0.442932i \(-0.146062\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −86.7941 −0.198614
\(438\) 0 0
\(439\) −523.900 −1.19340 −0.596698 0.802466i \(-0.703521\pi\)
−0.596698 + 0.802466i \(0.703521\pi\)
\(440\) 0 0
\(441\) 243.440 161.101i 0.552018 0.365309i
\(442\) 0 0
\(443\) 391.277 0.883243 0.441622 0.897201i \(-0.354403\pi\)
0.441622 + 0.897201i \(0.354403\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 755.347 227.301i 1.68981 0.508502i
\(448\) 0 0
\(449\) 375.014i 0.835220i 0.908626 + 0.417610i \(0.137132\pi\)
−0.908626 + 0.417610i \(0.862868\pi\)
\(450\) 0 0
\(451\) −778.347 −1.72582
\(452\) 0 0
\(453\) −13.6735 45.4386i −0.0301843 0.100306i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 734.032i 1.60620i 0.595847 + 0.803098i \(0.296817\pi\)
−0.595847 + 0.803098i \(0.703183\pi\)
\(458\) 0 0
\(459\) −251.317 209.069i −0.547531 0.455488i
\(460\) 0 0
\(461\) 775.239i 1.68165i 0.541310 + 0.840823i \(0.317928\pi\)
−0.541310 + 0.840823i \(0.682072\pi\)
\(462\) 0 0
\(463\) 323.967i 0.699714i −0.936803 0.349857i \(-0.886230\pi\)
0.936803 0.349857i \(-0.113770\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 160.785 0.344294 0.172147 0.985071i \(-0.444930\pi\)
0.172147 + 0.985071i \(0.444930\pi\)
\(468\) 0 0
\(469\) 569.027 1.21328
\(470\) 0 0
\(471\) 17.5795 5.29007i 0.0373239 0.0112316i
\(472\) 0 0
\(473\) 278.243 0.588252
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 68.5822 45.3856i 0.143778 0.0951481i
\(478\) 0 0
\(479\) 61.7565i 0.128928i −0.997920 0.0644640i \(-0.979466\pi\)
0.997920 0.0644640i \(-0.0205338\pi\)
\(480\) 0 0
\(481\) 18.1934 0.0378241
\(482\) 0 0
\(483\) −740.744 + 222.906i −1.53363 + 0.461504i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 129.969i 0.266876i −0.991057 0.133438i \(-0.957398\pi\)
0.991057 0.133438i \(-0.0426017\pi\)
\(488\) 0 0
\(489\) −489.642 + 147.344i −1.00131 + 0.301317i
\(490\) 0 0
\(491\) 810.511i 1.65074i −0.564595 0.825368i \(-0.690968\pi\)
0.564595 0.825368i \(-0.309032\pi\)
\(492\) 0 0
\(493\) 145.733i 0.295605i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 153.607 0.309069
\(498\) 0 0
\(499\) 600.897 1.20420 0.602101 0.798420i \(-0.294330\pi\)
0.602101 + 0.798420i \(0.294330\pi\)
\(500\) 0 0
\(501\) 52.9635 + 176.004i 0.105716 + 0.351306i
\(502\) 0 0
\(503\) 688.332 1.36845 0.684226 0.729270i \(-0.260140\pi\)
0.684226 + 0.729270i \(0.260140\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 259.471 + 862.254i 0.511777 + 1.70070i
\(508\) 0 0
\(509\) 823.791i 1.61845i −0.587498 0.809225i \(-0.699887\pi\)
0.587498 0.809225i \(-0.300113\pi\)
\(510\) 0 0
\(511\) 470.322 0.920395
\(512\) 0 0
\(513\) −63.0498 52.4508i −0.122904 0.102243i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 493.413i 0.954377i
\(518\) 0 0
\(519\) 226.966 + 754.235i 0.437314 + 1.45325i
\(520\) 0 0
\(521\) 964.525i 1.85130i −0.378386 0.925648i \(-0.623521\pi\)
0.378386 0.925648i \(-0.376479\pi\)
\(522\) 0 0
\(523\) 462.679i 0.884664i 0.896851 + 0.442332i \(0.145849\pi\)
−0.896851 + 0.442332i \(0.854151\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −26.5263 −0.0503345
\(528\) 0 0
\(529\) 287.447 0.543378
\(530\) 0 0
\(531\) −399.561 603.777i −0.752469 1.13706i
\(532\) 0 0
\(533\) −770.420 −1.44544
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 19.7861 5.95408i 0.0368457 0.0110877i
\(538\) 0 0
\(539\) 709.775i 1.31684i
\(540\) 0 0
\(541\) −516.752 −0.955180 −0.477590 0.878583i \(-0.658490\pi\)
−0.477590 + 0.878583i \(0.658490\pi\)
\(542\) 0 0
\(543\) −188.919 627.800i −0.347917 1.15617i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 478.953i 0.875599i −0.899073 0.437799i \(-0.855758\pi\)
0.899073 0.437799i \(-0.144242\pi\)
\(548\) 0 0
\(549\) 434.423 287.488i 0.791298 0.523657i
\(550\) 0 0
\(551\) 36.5613i 0.0663544i
\(552\) 0 0
\(553\) 67.3404i 0.121773i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 343.982 0.617561 0.308781 0.951133i \(-0.400079\pi\)
0.308781 + 0.951133i \(0.400079\pi\)
\(558\) 0 0
\(559\) 275.409 0.492682
\(560\) 0 0
\(561\) −761.137 + 229.043i −1.35675 + 0.408276i
\(562\) 0 0
\(563\) 394.056 0.699922 0.349961 0.936764i \(-0.386195\pi\)
0.349961 + 0.936764i \(0.386195\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −672.803 285.716i −1.18660 0.503908i
\(568\) 0 0
\(569\) 537.452i 0.944556i −0.881450 0.472278i \(-0.843432\pi\)
0.881450 0.472278i \(-0.156568\pi\)
\(570\) 0 0
\(571\) 710.555 1.24440 0.622202 0.782856i \(-0.286238\pi\)
0.622202 + 0.782856i \(0.286238\pi\)
\(572\) 0 0
\(573\) 216.176 65.0521i 0.377271 0.113529i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 452.846i 0.784828i −0.919789 0.392414i \(-0.871640\pi\)
0.919789 0.392414i \(-0.128360\pi\)
\(578\) 0 0
\(579\) −610.709 + 183.776i −1.05477 + 0.317402i
\(580\) 0 0
\(581\) 743.372i 1.27947i
\(582\) 0 0
\(583\) 199.959i 0.342983i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1103.34 −1.87963 −0.939813 0.341690i \(-0.889001\pi\)
−0.939813 + 0.341690i \(0.889001\pi\)
\(588\) 0 0
\(589\) −6.65486 −0.0112986
\(590\) 0 0
\(591\) 164.804 + 547.662i 0.278855 + 0.926670i
\(592\) 0 0
\(593\) 249.474 0.420698 0.210349 0.977626i \(-0.432540\pi\)
0.210349 + 0.977626i \(0.432540\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 181.536 + 603.267i 0.304081 + 1.01050i
\(598\) 0 0
\(599\) 596.120i 0.995191i 0.867409 + 0.497596i \(0.165784\pi\)
−0.867409 + 0.497596i \(0.834216\pi\)
\(600\) 0 0
\(601\) 476.515 0.792871 0.396435 0.918063i \(-0.370247\pi\)
0.396435 + 0.918063i \(0.370247\pi\)
\(602\) 0 0
\(603\) −313.188 473.259i −0.519384 0.784841i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 27.8813i 0.0459329i −0.999736 0.0229664i \(-0.992689\pi\)
0.999736 0.0229664i \(-0.00731109\pi\)
\(608\) 0 0
\(609\) 93.8973 + 312.032i 0.154183 + 0.512368i
\(610\) 0 0
\(611\) 488.388i 0.799325i
\(612\) 0 0
\(613\) 465.472i 0.759334i 0.925123 + 0.379667i \(0.123961\pi\)
−0.925123 + 0.379667i \(0.876039\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.6709 0.0788832 0.0394416 0.999222i \(-0.487442\pi\)
0.0394416 + 0.999222i \(0.487442\pi\)
\(618\) 0 0
\(619\) 213.318 0.344617 0.172309 0.985043i \(-0.444877\pi\)
0.172309 + 0.985043i \(0.444877\pi\)
\(620\) 0 0
\(621\) 593.091 + 493.389i 0.955058 + 0.794508i
\(622\) 0 0
\(623\) 248.931 0.399569
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −190.953 + 57.4618i −0.304549 + 0.0916456i
\(628\) 0 0
\(629\) 10.1700i 0.0161686i
\(630\) 0 0
\(631\) 582.489 0.923121 0.461560 0.887109i \(-0.347290\pi\)
0.461560 + 0.887109i \(0.347290\pi\)
\(632\) 0 0
\(633\) −152.510 506.808i −0.240931 0.800644i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 702.546i 1.10290i
\(638\) 0 0
\(639\) −84.5442 127.755i −0.132307 0.199929i
\(640\) 0 0
\(641\) 319.635i 0.498650i −0.968420 0.249325i \(-0.919791\pi\)
0.968420 0.249325i \(-0.0802088\pi\)
\(642\) 0 0
\(643\) 458.627i 0.713261i 0.934246 + 0.356630i \(0.116074\pi\)
−0.934246 + 0.356630i \(0.883926\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 507.599 0.784543 0.392271 0.919850i \(-0.371689\pi\)
0.392271 + 0.919850i \(0.371689\pi\)
\(648\) 0 0
\(649\) −1760.38 −2.71244
\(650\) 0 0
\(651\) −56.7959 + 17.0911i −0.0872440 + 0.0262536i
\(652\) 0 0
\(653\) −1197.20 −1.83339 −0.916694 0.399591i \(-0.869152\pi\)
−0.916694 + 0.399591i \(0.869152\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −258.862 391.166i −0.394006 0.595382i
\(658\) 0 0
\(659\) 632.173i 0.959291i 0.877462 + 0.479645i \(0.159235\pi\)
−0.877462 + 0.479645i \(0.840765\pi\)
\(660\) 0 0
\(661\) −565.316 −0.855243 −0.427622 0.903958i \(-0.640648\pi\)
−0.427622 + 0.903958i \(0.640648\pi\)
\(662\) 0 0
\(663\) −753.385 + 226.710i −1.13633 + 0.341946i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 343.921i 0.515624i
\(668\) 0 0
\(669\) 380.243 114.423i 0.568375 0.171036i
\(670\) 0 0
\(671\) 1266.61i 1.88764i
\(672\) 0 0
\(673\) 306.607i 0.455582i −0.973710 0.227791i \(-0.926850\pi\)
0.973710 0.227791i \(-0.0731503\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −546.672 −0.807491 −0.403746 0.914871i \(-0.632292\pi\)
−0.403746 + 0.914871i \(0.632292\pi\)
\(678\) 0 0
\(679\) −1037.68 −1.52824
\(680\) 0 0
\(681\) −162.196 538.996i −0.238173 0.791478i
\(682\) 0 0
\(683\) −812.204 −1.18917 −0.594586 0.804032i \(-0.702684\pi\)
−0.594586 + 0.804032i \(0.702684\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −154.370 512.991i −0.224702 0.746712i
\(688\) 0 0
\(689\) 197.922i 0.287260i
\(690\) 0 0
\(691\) −45.9358 −0.0664773 −0.0332387 0.999447i \(-0.510582\pi\)
−0.0332387 + 0.999447i \(0.510582\pi\)
\(692\) 0 0
\(693\) −1482.11 + 980.815i −2.13869 + 1.41532i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 430.661i 0.617879i
\(698\) 0 0
\(699\) 256.498 + 852.375i 0.366950 + 1.21942i
\(700\) 0 0
\(701\) 674.615i 0.962360i 0.876622 + 0.481180i \(0.159792\pi\)
−0.876622 + 0.481180i \(0.840208\pi\)
\(702\) 0 0
\(703\) 2.55143i 0.00362935i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1108.20 1.56747
\(708\) 0 0
\(709\) −656.626 −0.926129 −0.463065 0.886324i \(-0.653250\pi\)
−0.463065 + 0.886324i \(0.653250\pi\)
\(710\) 0 0
\(711\) −56.0069 + 37.0637i −0.0787720 + 0.0521290i
\(712\) 0 0
\(713\) 62.6003 0.0877984
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −396.371 + 119.277i −0.552819 + 0.166355i
\(718\) 0 0
\(719\) 644.279i 0.896076i −0.894014 0.448038i \(-0.852123\pi\)
0.894014 0.448038i \(-0.147877\pi\)
\(720\) 0 0
\(721\) −422.579 −0.586101
\(722\) 0 0
\(723\) −36.7995 122.289i −0.0508983 0.169141i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 981.269i 1.34975i 0.737932 + 0.674875i \(0.235803\pi\)
−0.737932 + 0.674875i \(0.764197\pi\)
\(728\) 0 0
\(729\) 132.677 + 716.825i 0.181998 + 0.983299i
\(730\) 0 0
\(731\) 153.953i 0.210605i
\(732\) 0 0
\(733\) 127.756i 0.174292i −0.996196 0.0871460i \(-0.972225\pi\)
0.996196 0.0871460i \(-0.0277747\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1379.84 −1.87223
\(738\) 0 0
\(739\) 1401.59 1.89660 0.948302 0.317368i \(-0.102799\pi\)
0.948302 + 0.317368i \(0.102799\pi\)
\(740\) 0 0
\(741\) −189.008 + 56.8766i −0.255071 + 0.0767565i
\(742\) 0 0
\(743\) −11.6734 −0.0157112 −0.00785561 0.999969i \(-0.502501\pi\)
−0.00785561 + 0.999969i \(0.502501\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −618.261 + 409.147i −0.827659 + 0.547720i
\(748\) 0 0
\(749\) 948.313i 1.26611i
\(750\) 0 0
\(751\) 380.403 0.506529 0.253264 0.967397i \(-0.418496\pi\)
0.253264 + 0.967397i \(0.418496\pi\)
\(752\) 0 0
\(753\) −591.456 + 177.982i −0.785466 + 0.236364i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 63.6621i 0.0840979i −0.999116 0.0420490i \(-0.986611\pi\)
0.999116 0.0420490i \(-0.0133885\pi\)
\(758\) 0 0
\(759\) 1796.23 540.526i 2.36658 0.712156i
\(760\) 0 0
\(761\) 377.891i 0.496571i −0.968687 0.248286i \(-0.920133\pi\)
0.968687 0.248286i \(-0.0798671\pi\)
\(762\) 0 0
\(763\) 1053.81i 1.38114i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1742.45 −2.27177
\(768\) 0 0
\(769\) −231.920 −0.301586 −0.150793 0.988565i \(-0.548183\pi\)
−0.150793 + 0.988565i \(0.548183\pi\)
\(770\) 0 0
\(771\) −162.851 541.174i −0.211221 0.701912i
\(772\) 0 0
\(773\) 1509.05 1.95220 0.976101 0.217320i \(-0.0697314\pi\)
0.976101 + 0.217320i \(0.0697314\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.55264 + 21.7752i 0.00843326 + 0.0280247i
\(778\) 0 0
\(779\) 108.043i 0.138695i
\(780\) 0 0
\(781\) −372.483 −0.476930
\(782\) 0 0
\(783\) 207.836 249.834i 0.265436 0.319073i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1171.43i 1.48848i −0.667912 0.744240i \(-0.732812\pi\)
0.667912 0.744240i \(-0.267188\pi\)
\(788\) 0 0
\(789\) −163.165 542.216i −0.206799 0.687219i
\(790\) 0 0
\(791\) 97.5656i 0.123345i
\(792\) 0 0
\(793\) 1253.71i 1.58097i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 988.589 1.24039 0.620194 0.784448i \(-0.287054\pi\)
0.620194 + 0.784448i \(0.287054\pi\)
\(798\) 0 0
\(799\) −273.007 −0.341685
\(800\) 0 0
\(801\) −137.010 207.036i −0.171049 0.258471i
\(802\) 0 0
\(803\) −1140.49 −1.42028
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 957.694 288.191i 1.18673 0.357114i
\(808\) 0 0
\(809\) 1334.53i 1.64961i −0.565419 0.824804i \(-0.691286\pi\)
0.565419 0.824804i \(-0.308714\pi\)
\(810\) 0 0
\(811\) 610.590 0.752885 0.376443 0.926440i \(-0.377147\pi\)
0.376443 + 0.926440i \(0.377147\pi\)
\(812\) 0 0
\(813\) −226.533 752.796i −0.278638 0.925949i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 38.6233i 0.0472745i
\(818\) 0 0
\(819\) −1467.01 + 970.826i −1.79123 + 1.18538i
\(820\) 0 0
\(821\) 877.538i 1.06887i −0.845211 0.534433i \(-0.820525\pi\)
0.845211 0.534433i \(-0.179475\pi\)
\(822\) 0 0
\(823\) 230.855i 0.280504i 0.990116 + 0.140252i \(0.0447913\pi\)
−0.990116 + 0.140252i \(0.955209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 969.434 1.17223 0.586115 0.810228i \(-0.300657\pi\)
0.586115 + 0.810228i \(0.300657\pi\)
\(828\) 0 0
\(829\) 1049.09 1.26549 0.632746 0.774360i \(-0.281928\pi\)
0.632746 + 0.774360i \(0.281928\pi\)
\(830\) 0 0
\(831\) 1149.39 345.876i 1.38314 0.416216i
\(832\) 0 0
\(833\) 392.720 0.471453
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 45.4747 + 37.8302i 0.0543305 + 0.0451973i
\(838\) 0 0
\(839\) 1214.62i 1.44770i −0.689959 0.723849i \(-0.742371\pi\)
0.689959 0.723849i \(-0.257629\pi\)
\(840\) 0 0
\(841\) 696.126 0.827736
\(842\) 0 0
\(843\) 1007.47 303.169i 1.19510 0.359631i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3229.33i 3.81266i
\(848\) 0 0
\(849\) −1333.00 + 401.128i −1.57008 + 0.472472i
\(850\) 0 0
\(851\) 24.0006i 0.0282028i
\(852\) 0 0
\(853\) 190.704i 0.223569i 0.993732 + 0.111784i \(0.0356566\pi\)
−0.993732 + 0.111784i \(0.964343\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −351.238 −0.409846 −0.204923 0.978778i \(-0.565694\pi\)
−0.204923 + 0.978778i \(0.565694\pi\)
\(858\) 0 0
\(859\) 1233.74 1.43625 0.718123 0.695916i \(-0.245002\pi\)
0.718123 + 0.695916i \(0.245002\pi\)
\(860\) 0 0
\(861\) −277.479 922.096i −0.322275 1.07096i
\(862\) 0 0
\(863\) −463.404 −0.536968 −0.268484 0.963284i \(-0.586523\pi\)
−0.268484 + 0.963284i \(0.586523\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 123.103 + 409.085i 0.141987 + 0.471840i
\(868\) 0 0
\(869\) 163.294i 0.187910i
\(870\) 0 0
\(871\) −1365.78 −1.56806
\(872\) 0 0
\(873\) 571.130 + 863.035i 0.654216 + 0.988585i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 777.195i 0.886197i −0.896473 0.443099i \(-0.853879\pi\)
0.896473 0.443099i \(-0.146121\pi\)
\(878\) 0 0
\(879\) −40.9829 136.191i −0.0466245 0.154939i
\(880\) 0 0
\(881\) 1609.50i 1.82690i −0.406951 0.913450i \(-0.633408\pi\)
0.406951 0.913450i \(-0.366592\pi\)
\(882\) 0 0
\(883\) 331.153i 0.375032i 0.982262 + 0.187516i \(0.0600437\pi\)
−0.982262 + 0.187516i \(0.939956\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 365.778 0.412377 0.206188 0.978512i \(-0.433894\pi\)
0.206188 + 0.978512i \(0.433894\pi\)
\(888\) 0 0
\(889\) 1736.78 1.95364
\(890\) 0 0
\(891\) 1631.48 + 692.833i 1.83107 + 0.777591i
\(892\) 0 0
\(893\) −68.4913 −0.0766980
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1777.94 535.021i 1.98210 0.596456i
\(898\) 0 0
\(899\) 26.3698i 0.0293324i
\(900\) 0 0
\(901\) 110.638 0.122794
\(902\) 0 0
\(903\) 99.1930 + 329.630i 0.109848 + 0.365039i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 270.791i 0.298556i 0.988795 + 0.149278i \(0.0476950\pi\)
−0.988795 + 0.149278i \(0.952305\pi\)
\(908\) 0 0
\(909\) −609.947 921.690i −0.671008 1.01396i
\(910\) 0 0
\(911\) 1238.74i 1.35976i 0.733324 + 0.679879i \(0.237968\pi\)
−0.733324 + 0.679879i \(0.762032\pi\)
\(912\) 0 0
\(913\) 1802.61i 1.97438i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 438.898 0.478624
\(918\) 0 0
\(919\) −578.894 −0.629918 −0.314959 0.949105i \(-0.601991\pi\)
−0.314959 + 0.949105i \(0.601991\pi\)
\(920\) 0 0
\(921\) −1326.91 + 399.295i −1.44072 + 0.433545i
\(922\) 0 0
\(923\) −368.689 −0.399447
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 232.584 + 351.458i 0.250900 + 0.379135i
\(928\) 0 0
\(929\) 1307.58i 1.40752i 0.710440 + 0.703758i \(0.248496\pi\)
−0.710440 + 0.703758i \(0.751504\pi\)
\(930\) 0 0
\(931\) 98.5249 0.105827
\(932\) 0 0
\(933\) 353.511 106.379i 0.378898 0.114019i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 693.314i 0.739929i 0.929046 + 0.369965i \(0.120630\pi\)
−0.929046 + 0.369965i \(0.879370\pi\)
\(938\) 0 0
\(939\) 279.907 84.2302i 0.298091 0.0897020i
\(940\) 0 0
\(941\) 783.592i 0.832723i −0.909199 0.416362i \(-0.863305\pi\)
0.909199 0.416362i \(-0.136695\pi\)
\(942\) 0 0
\(943\) 1016.33i 1.07777i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1621.44 −1.71219 −0.856094 0.516820i \(-0.827116\pi\)
−0.856094 + 0.516820i \(0.827116\pi\)
\(948\) 0 0
\(949\) −1128.87 −1.18954
\(950\) 0 0
\(951\) 218.182 + 725.046i 0.229424 + 0.762404i
\(952\) 0 0
\(953\) −1105.90 −1.16044 −0.580218 0.814461i \(-0.697033\pi\)
−0.580218 + 0.814461i \(0.697033\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −227.692 756.648i −0.237923 0.790646i
\(958\) 0 0
\(959\) 1424.63i 1.48554i
\(960\) 0 0
\(961\) −956.200 −0.995005
\(962\) 0 0
\(963\) 788.710 521.945i 0.819013 0.541998i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1307.89i 1.35253i −0.736659 0.676264i \(-0.763598\pi\)
0.736659 0.676264i \(-0.236402\pi\)
\(968\) 0 0
\(969\) −31.7938 105.655i −0.0328109 0.109035i
\(970\) 0 0
\(971\) 749.988i 0.772387i 0.922418 + 0.386194i \(0.126210\pi\)
−0.922418 + 0.386194i \(0.873790\pi\)
\(972\) 0 0
\(973\) 1486.75i 1.52800i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 373.792 0.382591 0.191296 0.981532i \(-0.438731\pi\)
0.191296 + 0.981532i \(0.438731\pi\)
\(978\) 0 0
\(979\) −603.634 −0.616583
\(980\) 0 0
\(981\) −876.452 + 580.009i −0.893427 + 0.591243i
\(982\) 0 0
\(983\) 27.7025 0.0281815 0.0140908 0.999901i \(-0.495515\pi\)
0.0140908 + 0.999901i \(0.495515\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −584.539 + 175.901i −0.592238 + 0.178218i
\(988\) 0 0
\(989\) 363.318i 0.367359i
\(990\) 0 0
\(991\) 1030.34 1.03970 0.519850 0.854258i \(-0.325988\pi\)
0.519850 + 0.854258i \(0.325988\pi\)
\(992\) 0 0
\(993\) 262.171 + 871.226i 0.264019 + 0.877367i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1217.62i 1.22128i −0.791909 0.610640i \(-0.790912\pi\)
0.791909 0.610640i \(-0.209088\pi\)
\(998\) 0 0
\(999\) 14.5039 17.4347i 0.0145184 0.0174522i
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.449.5 16
3.2 odd 2 inner 600.3.c.d.449.11 16
4.3 odd 2 1200.3.c.m.449.12 16
5.2 odd 4 120.3.l.a.41.2 yes 8
5.3 odd 4 600.3.l.f.401.7 8
5.4 even 2 inner 600.3.c.d.449.12 16
12.11 even 2 1200.3.c.m.449.6 16
15.2 even 4 120.3.l.a.41.1 8
15.8 even 4 600.3.l.f.401.8 8
15.14 odd 2 inner 600.3.c.d.449.6 16
20.3 even 4 1200.3.l.x.401.2 8
20.7 even 4 240.3.l.d.161.7 8
20.19 odd 2 1200.3.c.m.449.5 16
40.27 even 4 960.3.l.g.641.2 8
40.37 odd 4 960.3.l.h.641.7 8
60.23 odd 4 1200.3.l.x.401.1 8
60.47 odd 4 240.3.l.d.161.8 8
60.59 even 2 1200.3.c.m.449.11 16
120.77 even 4 960.3.l.h.641.8 8
120.107 odd 4 960.3.l.g.641.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.1 8 15.2 even 4
120.3.l.a.41.2 yes 8 5.2 odd 4
240.3.l.d.161.7 8 20.7 even 4
240.3.l.d.161.8 8 60.47 odd 4
600.3.c.d.449.5 16 1.1 even 1 trivial
600.3.c.d.449.6 16 15.14 odd 2 inner
600.3.c.d.449.11 16 3.2 odd 2 inner
600.3.c.d.449.12 16 5.4 even 2 inner
600.3.l.f.401.7 8 5.3 odd 4
600.3.l.f.401.8 8 15.8 even 4
960.3.l.g.641.1 8 120.107 odd 4
960.3.l.g.641.2 8 40.27 even 4
960.3.l.h.641.7 8 40.37 odd 4
960.3.l.h.641.8 8 120.77 even 4
1200.3.c.m.449.5 16 20.19 odd 2
1200.3.c.m.449.6 16 12.11 even 2
1200.3.c.m.449.11 16 60.59 even 2
1200.3.c.m.449.12 16 4.3 odd 2
1200.3.l.x.401.1 8 60.23 odd 4
1200.3.l.x.401.2 8 20.3 even 4