Properties

Label 600.3.c.d.449.13
Level $600$
Weight $3$
Character 600.449
Analytic conductor $16.349$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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 449.13
Root \(1.09102 - 1.09102i\) of defining polynomial
Character \(\chi\) \(=\) 600.449
Dual form 600.3.c.d.449.14

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +O(q^{10})\) \(q+(1.79813 - 2.40140i) q^{3} +10.2132i q^{7} +(-2.53346 - 8.63606i) q^{9} +8.19300i q^{11} -13.5822i q^{13} +15.4710 q^{17} +25.4934 q^{19} +(24.5261 + 18.3647i) q^{21} +17.9156 q^{23} +(-25.2941 - 9.44491i) q^{27} +42.0022i q^{29} +38.4878 q^{31} +(19.6747 + 14.7321i) q^{33} -11.8387i q^{37} +(-32.6163 - 24.4225i) q^{39} +46.3781i q^{41} -54.0181i q^{43} +43.0955 q^{47} -55.3102 q^{49} +(27.8188 - 37.1521i) q^{51} +82.7421 q^{53} +(45.8404 - 61.2199i) q^{57} -45.8928i q^{59} -93.6873 q^{61} +(88.2022 - 25.8748i) q^{63} -34.4995i q^{67} +(32.2147 - 43.0227i) q^{69} +68.0061i q^{71} -44.7191i q^{73} -83.6770 q^{77} +11.7499 q^{79} +(-68.1632 + 43.7582i) q^{81} -144.436 q^{83} +(100.864 + 75.5255i) q^{87} +63.7094i q^{89} +138.718 q^{91} +(69.2061 - 92.4247i) q^{93} -63.9013i q^{97} +(70.7552 - 20.7566i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 40 q^{9}+O(q^{10}) \) 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
\(3\) 1.79813 2.40140i 0.599377 0.800467i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 10.2132i 1.45903i 0.683963 + 0.729517i \(0.260255\pi\)
−0.683963 + 0.729517i \(0.739745\pi\)
\(8\) 0 0
\(9\) −2.53346 8.63606i −0.281496 0.959563i
\(10\) 0 0
\(11\) 8.19300i 0.744818i 0.928069 + 0.372409i \(0.121468\pi\)
−0.928069 + 0.372409i \(0.878532\pi\)
\(12\) 0 0
\(13\) 13.5822i 1.04478i −0.852705 0.522392i \(-0.825040\pi\)
0.852705 0.522392i \(-0.174960\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15.4710 0.910058 0.455029 0.890477i \(-0.349629\pi\)
0.455029 + 0.890477i \(0.349629\pi\)
\(18\) 0 0
\(19\) 25.4934 1.34176 0.670879 0.741567i \(-0.265917\pi\)
0.670879 + 0.741567i \(0.265917\pi\)
\(20\) 0 0
\(21\) 24.5261 + 18.3647i 1.16791 + 0.874511i
\(22\) 0 0
\(23\) 17.9156 0.778941 0.389471 0.921039i \(-0.372658\pi\)
0.389471 + 0.921039i \(0.372658\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −25.2941 9.44491i −0.936820 0.349811i
\(28\) 0 0
\(29\) 42.0022i 1.44835i 0.689615 + 0.724177i \(0.257780\pi\)
−0.689615 + 0.724177i \(0.742220\pi\)
\(30\) 0 0
\(31\) 38.4878 1.24154 0.620771 0.783992i \(-0.286820\pi\)
0.620771 + 0.783992i \(0.286820\pi\)
\(32\) 0 0
\(33\) 19.6747 + 14.7321i 0.596202 + 0.446426i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 11.8387i 0.319965i −0.987120 0.159982i \(-0.948856\pi\)
0.987120 0.159982i \(-0.0511437\pi\)
\(38\) 0 0
\(39\) −32.6163 24.4225i −0.836315 0.626219i
\(40\) 0 0
\(41\) 46.3781i 1.13117i 0.824689 + 0.565587i \(0.191350\pi\)
−0.824689 + 0.565587i \(0.808650\pi\)
\(42\) 0 0
\(43\) 54.0181i 1.25623i −0.778119 0.628117i \(-0.783826\pi\)
0.778119 0.628117i \(-0.216174\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 43.0955 0.916925 0.458462 0.888714i \(-0.348400\pi\)
0.458462 + 0.888714i \(0.348400\pi\)
\(48\) 0 0
\(49\) −55.3102 −1.12878
\(50\) 0 0
\(51\) 27.8188 37.1521i 0.545468 0.728472i
\(52\) 0 0
\(53\) 82.7421 1.56117 0.780586 0.625049i \(-0.214921\pi\)
0.780586 + 0.625049i \(0.214921\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 45.8404 61.2199i 0.804218 1.07403i
\(58\) 0 0
\(59\) 45.8928i 0.777844i −0.921271 0.388922i \(-0.872848\pi\)
0.921271 0.388922i \(-0.127152\pi\)
\(60\) 0 0
\(61\) −93.6873 −1.53586 −0.767929 0.640535i \(-0.778713\pi\)
−0.767929 + 0.640535i \(0.778713\pi\)
\(62\) 0 0
\(63\) 88.2022 25.8748i 1.40003 0.410712i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 34.4995i 0.514917i −0.966289 0.257459i \(-0.917115\pi\)
0.966289 0.257459i \(-0.0828851\pi\)
\(68\) 0 0
\(69\) 32.2147 43.0227i 0.466879 0.623517i
\(70\) 0 0
\(71\) 68.0061i 0.957832i 0.877861 + 0.478916i \(0.158970\pi\)
−0.877861 + 0.478916i \(0.841030\pi\)
\(72\) 0 0
\(73\) 44.7191i 0.612591i −0.951937 0.306295i \(-0.900911\pi\)
0.951937 0.306295i \(-0.0990895\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −83.6770 −1.08671
\(78\) 0 0
\(79\) 11.7499 0.148733 0.0743665 0.997231i \(-0.476307\pi\)
0.0743665 + 0.997231i \(0.476307\pi\)
\(80\) 0 0
\(81\) −68.1632 + 43.7582i −0.841521 + 0.540225i
\(82\) 0 0
\(83\) −144.436 −1.74020 −0.870099 0.492877i \(-0.835945\pi\)
−0.870099 + 0.492877i \(0.835945\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 100.864 + 75.5255i 1.15936 + 0.868109i
\(88\) 0 0
\(89\) 63.7094i 0.715836i 0.933753 + 0.357918i \(0.116513\pi\)
−0.933753 + 0.357918i \(0.883487\pi\)
\(90\) 0 0
\(91\) 138.718 1.52438
\(92\) 0 0
\(93\) 69.2061 92.4247i 0.744151 0.993814i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 63.9013i 0.658776i −0.944195 0.329388i \(-0.893158\pi\)
0.944195 0.329388i \(-0.106842\pi\)
\(98\) 0 0
\(99\) 70.7552 20.7566i 0.714699 0.209663i
\(100\) 0 0
\(101\) 50.8769i 0.503731i 0.967762 + 0.251866i \(0.0810441\pi\)
−0.967762 + 0.251866i \(0.918956\pi\)
\(102\) 0 0
\(103\) 45.9491i 0.446108i 0.974806 + 0.223054i \(0.0716026\pi\)
−0.974806 + 0.223054i \(0.928397\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 110.249 1.03036 0.515181 0.857082i \(-0.327725\pi\)
0.515181 + 0.857082i \(0.327725\pi\)
\(108\) 0 0
\(109\) 49.2797 0.452107 0.226054 0.974115i \(-0.427418\pi\)
0.226054 + 0.974115i \(0.427418\pi\)
\(110\) 0 0
\(111\) −28.4295 21.2875i −0.256121 0.191779i
\(112\) 0 0
\(113\) −157.450 −1.39337 −0.696683 0.717379i \(-0.745342\pi\)
−0.696683 + 0.717379i \(0.745342\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −117.297 + 34.4099i −1.00254 + 0.294102i
\(118\) 0 0
\(119\) 158.009i 1.32781i
\(120\) 0 0
\(121\) 53.8748 0.445246
\(122\) 0 0
\(123\) 111.373 + 83.3939i 0.905468 + 0.677999i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 135.065i 1.06351i 0.846900 + 0.531753i \(0.178466\pi\)
−0.846900 + 0.531753i \(0.821534\pi\)
\(128\) 0 0
\(129\) −129.719 97.1315i −1.00557 0.752957i
\(130\) 0 0
\(131\) 150.653i 1.15002i −0.818146 0.575010i \(-0.804998\pi\)
0.818146 0.575010i \(-0.195002\pi\)
\(132\) 0 0
\(133\) 260.370i 1.95767i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −139.086 −1.01523 −0.507613 0.861585i \(-0.669472\pi\)
−0.507613 + 0.861585i \(0.669472\pi\)
\(138\) 0 0
\(139\) −110.296 −0.793496 −0.396748 0.917928i \(-0.629861\pi\)
−0.396748 + 0.917928i \(0.629861\pi\)
\(140\) 0 0
\(141\) 77.4912 103.490i 0.549583 0.733968i
\(142\) 0 0
\(143\) 111.279 0.778174
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −99.4549 + 132.822i −0.676564 + 0.903551i
\(148\) 0 0
\(149\) 90.8076i 0.609447i 0.952441 + 0.304723i \(0.0985640\pi\)
−0.952441 + 0.304723i \(0.901436\pi\)
\(150\) 0 0
\(151\) 111.286 0.736992 0.368496 0.929629i \(-0.379873\pi\)
0.368496 + 0.929629i \(0.379873\pi\)
\(152\) 0 0
\(153\) −39.1951 133.608i −0.256177 0.873258i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 142.755i 0.909269i −0.890678 0.454635i \(-0.849770\pi\)
0.890678 0.454635i \(-0.150230\pi\)
\(158\) 0 0
\(159\) 148.781 198.697i 0.935730 1.24967i
\(160\) 0 0
\(161\) 182.977i 1.13650i
\(162\) 0 0
\(163\) 170.179i 1.04405i −0.852932 0.522023i \(-0.825178\pi\)
0.852932 0.522023i \(-0.174822\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.38780 0.0382503 0.0191251 0.999817i \(-0.493912\pi\)
0.0191251 + 0.999817i \(0.493912\pi\)
\(168\) 0 0
\(169\) −15.4759 −0.0915734
\(170\) 0 0
\(171\) −64.5865 220.163i −0.377699 1.28750i
\(172\) 0 0
\(173\) −238.403 −1.37805 −0.689027 0.724736i \(-0.741962\pi\)
−0.689027 + 0.724736i \(0.741962\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −110.207 82.5212i −0.622638 0.466221i
\(178\) 0 0
\(179\) 243.541i 1.36057i 0.732950 + 0.680283i \(0.238143\pi\)
−0.732950 + 0.680283i \(0.761857\pi\)
\(180\) 0 0
\(181\) −325.449 −1.79806 −0.899030 0.437887i \(-0.855727\pi\)
−0.899030 + 0.437887i \(0.855727\pi\)
\(182\) 0 0
\(183\) −168.462 + 224.981i −0.920557 + 1.22940i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 126.754i 0.677828i
\(188\) 0 0
\(189\) 96.4631 258.335i 0.510387 1.36685i
\(190\) 0 0
\(191\) 166.001i 0.869113i −0.900645 0.434556i \(-0.856905\pi\)
0.900645 0.434556i \(-0.143095\pi\)
\(192\) 0 0
\(193\) 239.408i 1.24045i 0.784422 + 0.620227i \(0.212959\pi\)
−0.784422 + 0.620227i \(0.787041\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 257.518 1.30720 0.653600 0.756840i \(-0.273258\pi\)
0.653600 + 0.756840i \(0.273258\pi\)
\(198\) 0 0
\(199\) 78.4735 0.394339 0.197170 0.980369i \(-0.436825\pi\)
0.197170 + 0.980369i \(0.436825\pi\)
\(200\) 0 0
\(201\) −82.8471 62.0345i −0.412174 0.308629i
\(202\) 0 0
\(203\) −428.979 −2.11320
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −45.3886 154.721i −0.219268 0.747443i
\(208\) 0 0
\(209\) 208.867i 0.999365i
\(210\) 0 0
\(211\) −112.724 −0.534237 −0.267119 0.963664i \(-0.586072\pi\)
−0.267119 + 0.963664i \(0.586072\pi\)
\(212\) 0 0
\(213\) 163.310 + 122.284i 0.766713 + 0.574102i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 393.085i 1.81145i
\(218\) 0 0
\(219\) −107.389 80.4108i −0.490359 0.367172i
\(220\) 0 0
\(221\) 210.130i 0.950814i
\(222\) 0 0
\(223\) 204.686i 0.917872i −0.888469 0.458936i \(-0.848231\pi\)
0.888469 0.458936i \(-0.151769\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −115.071 −0.506920 −0.253460 0.967346i \(-0.581569\pi\)
−0.253460 + 0.967346i \(0.581569\pi\)
\(228\) 0 0
\(229\) 129.458 0.565317 0.282659 0.959221i \(-0.408784\pi\)
0.282659 + 0.959221i \(0.408784\pi\)
\(230\) 0 0
\(231\) −150.462 + 200.942i −0.651351 + 0.869879i
\(232\) 0 0
\(233\) −250.150 −1.07361 −0.536803 0.843708i \(-0.680368\pi\)
−0.536803 + 0.843708i \(0.680368\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 21.1278 28.2162i 0.0891470 0.119056i
\(238\) 0 0
\(239\) 49.2556i 0.206090i 0.994677 + 0.103045i \(0.0328586\pi\)
−0.994677 + 0.103045i \(0.967141\pi\)
\(240\) 0 0
\(241\) −457.672 −1.89905 −0.949526 0.313688i \(-0.898435\pi\)
−0.949526 + 0.313688i \(0.898435\pi\)
\(242\) 0 0
\(243\) −17.4851 + 242.370i −0.0719551 + 0.997408i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 346.256i 1.40185i
\(248\) 0 0
\(249\) −259.715 + 346.850i −1.04303 + 1.39297i
\(250\) 0 0
\(251\) 119.717i 0.476959i 0.971148 + 0.238479i \(0.0766489\pi\)
−0.971148 + 0.238479i \(0.923351\pi\)
\(252\) 0 0
\(253\) 146.783i 0.580169i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 111.805 0.435041 0.217520 0.976056i \(-0.430203\pi\)
0.217520 + 0.976056i \(0.430203\pi\)
\(258\) 0 0
\(259\) 120.911 0.466839
\(260\) 0 0
\(261\) 362.734 106.411i 1.38979 0.407705i
\(262\) 0 0
\(263\) −55.3652 −0.210514 −0.105257 0.994445i \(-0.533567\pi\)
−0.105257 + 0.994445i \(0.533567\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 152.992 + 114.558i 0.573003 + 0.429055i
\(268\) 0 0
\(269\) 309.427i 1.15028i −0.818053 0.575142i \(-0.804947\pi\)
0.818053 0.575142i \(-0.195053\pi\)
\(270\) 0 0
\(271\) −194.062 −0.716096 −0.358048 0.933703i \(-0.616558\pi\)
−0.358048 + 0.933703i \(0.616558\pi\)
\(272\) 0 0
\(273\) 249.433 333.118i 0.913675 1.22021i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.20416i 0.0187876i −0.999956 0.00939379i \(-0.997010\pi\)
0.999956 0.00939379i \(-0.00299018\pi\)
\(278\) 0 0
\(279\) −97.5073 332.383i −0.349489 1.19134i
\(280\) 0 0
\(281\) 240.004i 0.854105i −0.904227 0.427053i \(-0.859552\pi\)
0.904227 0.427053i \(-0.140448\pi\)
\(282\) 0 0
\(283\) 153.109i 0.541022i 0.962717 + 0.270511i \(0.0871927\pi\)
−0.962717 + 0.270511i \(0.912807\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −473.671 −1.65042
\(288\) 0 0
\(289\) −49.6485 −0.171794
\(290\) 0 0
\(291\) −153.453 114.903i −0.527329 0.394855i
\(292\) 0 0
\(293\) 169.252 0.577653 0.288827 0.957381i \(-0.406735\pi\)
0.288827 + 0.957381i \(0.406735\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 77.3821 207.235i 0.260546 0.697760i
\(298\) 0 0
\(299\) 243.334i 0.813825i
\(300\) 0 0
\(301\) 551.699 1.83289
\(302\) 0 0
\(303\) 122.176 + 91.4832i 0.403220 + 0.301925i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 330.401i 1.07623i −0.842873 0.538113i \(-0.819137\pi\)
0.842873 0.538113i \(-0.180863\pi\)
\(308\) 0 0
\(309\) 110.342 + 82.6224i 0.357095 + 0.267387i
\(310\) 0 0
\(311\) 133.596i 0.429570i 0.976661 + 0.214785i \(0.0689050\pi\)
−0.976661 + 0.214785i \(0.931095\pi\)
\(312\) 0 0
\(313\) 459.981i 1.46959i −0.678290 0.734794i \(-0.737279\pi\)
0.678290 0.734794i \(-0.262721\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −298.150 −0.940536 −0.470268 0.882524i \(-0.655843\pi\)
−0.470268 + 0.882524i \(0.655843\pi\)
\(318\) 0 0
\(319\) −344.124 −1.07876
\(320\) 0 0
\(321\) 198.241 264.751i 0.617574 0.824771i
\(322\) 0 0
\(323\) 394.408 1.22108
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 88.6112 118.340i 0.270982 0.361897i
\(328\) 0 0
\(329\) 440.144i 1.33782i
\(330\) 0 0
\(331\) 397.599 1.20120 0.600602 0.799548i \(-0.294927\pi\)
0.600602 + 0.799548i \(0.294927\pi\)
\(332\) 0 0
\(333\) −102.240 + 29.9929i −0.307026 + 0.0900687i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 427.942i 1.26986i 0.772571 + 0.634929i \(0.218970\pi\)
−0.772571 + 0.634929i \(0.781030\pi\)
\(338\) 0 0
\(339\) −283.116 + 378.101i −0.835151 + 1.11534i
\(340\) 0 0
\(341\) 315.331i 0.924723i
\(342\) 0 0
\(343\) 64.4477i 0.187894i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 138.137 0.398090 0.199045 0.979990i \(-0.436216\pi\)
0.199045 + 0.979990i \(0.436216\pi\)
\(348\) 0 0
\(349\) 301.421 0.863669 0.431835 0.901953i \(-0.357866\pi\)
0.431835 + 0.901953i \(0.357866\pi\)
\(350\) 0 0
\(351\) −128.283 + 343.550i −0.365477 + 0.978775i
\(352\) 0 0
\(353\) 142.622 0.404028 0.202014 0.979383i \(-0.435251\pi\)
0.202014 + 0.979383i \(0.435251\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 379.443 + 284.120i 1.06286 + 0.795856i
\(358\) 0 0
\(359\) 3.05663i 0.00851429i 0.999991 + 0.00425715i \(0.00135510\pi\)
−0.999991 + 0.00425715i \(0.998645\pi\)
\(360\) 0 0
\(361\) 288.913 0.800313
\(362\) 0 0
\(363\) 96.8739 129.375i 0.266870 0.356405i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 370.072i 1.00837i 0.863595 + 0.504186i \(0.168207\pi\)
−0.863595 + 0.504186i \(0.831793\pi\)
\(368\) 0 0
\(369\) 400.525 117.497i 1.08543 0.318420i
\(370\) 0 0
\(371\) 845.065i 2.27780i
\(372\) 0 0
\(373\) 455.556i 1.22133i 0.791889 + 0.610665i \(0.209098\pi\)
−0.791889 + 0.610665i \(0.790902\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 570.482 1.51322
\(378\) 0 0
\(379\) 380.051 1.00277 0.501387 0.865223i \(-0.332823\pi\)
0.501387 + 0.865223i \(0.332823\pi\)
\(380\) 0 0
\(381\) 324.346 + 242.865i 0.851301 + 0.637440i
\(382\) 0 0
\(383\) 164.078 0.428402 0.214201 0.976790i \(-0.431285\pi\)
0.214201 + 0.976790i \(0.431285\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −466.503 + 136.853i −1.20544 + 0.353624i
\(388\) 0 0
\(389\) 562.622i 1.44633i −0.690676 0.723164i \(-0.742687\pi\)
0.690676 0.723164i \(-0.257313\pi\)
\(390\) 0 0
\(391\) 277.173 0.708882
\(392\) 0 0
\(393\) −361.778 270.893i −0.920553 0.689295i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 93.4668i 0.235433i 0.993047 + 0.117716i \(0.0375574\pi\)
−0.993047 + 0.117716i \(0.962443\pi\)
\(398\) 0 0
\(399\) 625.253 + 468.179i 1.56705 + 1.17338i
\(400\) 0 0
\(401\) 23.4831i 0.0585613i 0.999571 + 0.0292807i \(0.00932166\pi\)
−0.999571 + 0.0292807i \(0.990678\pi\)
\(402\) 0 0
\(403\) 522.749i 1.29714i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 96.9944 0.238316
\(408\) 0 0
\(409\) −697.472 −1.70531 −0.852655 0.522474i \(-0.825009\pi\)
−0.852655 + 0.522474i \(0.825009\pi\)
\(410\) 0 0
\(411\) −250.095 + 334.001i −0.608503 + 0.812656i
\(412\) 0 0
\(413\) 468.714 1.13490
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −198.326 + 264.865i −0.475603 + 0.635167i
\(418\) 0 0
\(419\) 115.537i 0.275744i −0.990450 0.137872i \(-0.955974\pi\)
0.990450 0.137872i \(-0.0440262\pi\)
\(420\) 0 0
\(421\) 117.396 0.278850 0.139425 0.990233i \(-0.455475\pi\)
0.139425 + 0.990233i \(0.455475\pi\)
\(422\) 0 0
\(423\) −109.181 372.175i −0.258110 0.879847i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 956.851i 2.24087i
\(428\) 0 0
\(429\) 200.094 267.225i 0.466419 0.622903i
\(430\) 0 0
\(431\) 673.123i 1.56177i −0.624675 0.780885i \(-0.714768\pi\)
0.624675 0.780885i \(-0.285232\pi\)
\(432\) 0 0
\(433\) 392.655i 0.906824i −0.891301 0.453412i \(-0.850207\pi\)
0.891301 0.453412i \(-0.149793\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 456.731 1.04515
\(438\) 0 0
\(439\) −409.353 −0.932468 −0.466234 0.884661i \(-0.654389\pi\)
−0.466234 + 0.884661i \(0.654389\pi\)
\(440\) 0 0
\(441\) 140.126 + 477.662i 0.317746 + 1.08313i
\(442\) 0 0
\(443\) 355.935 0.803464 0.401732 0.915757i \(-0.368408\pi\)
0.401732 + 0.915757i \(0.368408\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 218.065 + 163.284i 0.487842 + 0.365288i
\(448\) 0 0
\(449\) 294.636i 0.656205i −0.944642 0.328103i \(-0.893591\pi\)
0.944642 0.328103i \(-0.106409\pi\)
\(450\) 0 0
\(451\) −379.976 −0.842519
\(452\) 0 0
\(453\) 200.106 267.242i 0.441736 0.589938i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 491.814i 1.07618i −0.842887 0.538090i \(-0.819146\pi\)
0.842887 0.538090i \(-0.180854\pi\)
\(458\) 0 0
\(459\) −391.325 146.122i −0.852561 0.318349i
\(460\) 0 0
\(461\) 42.6832i 0.0925883i −0.998928 0.0462942i \(-0.985259\pi\)
0.998928 0.0462942i \(-0.0147412\pi\)
\(462\) 0 0
\(463\) 422.582i 0.912705i −0.889799 0.456352i \(-0.849156\pi\)
0.889799 0.456352i \(-0.150844\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 203.497 0.435754 0.217877 0.975976i \(-0.430087\pi\)
0.217877 + 0.975976i \(0.430087\pi\)
\(468\) 0 0
\(469\) 352.351 0.751282
\(470\) 0 0
\(471\) −342.813 256.693i −0.727840 0.544995i
\(472\) 0 0
\(473\) 442.570 0.935666
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −209.624 714.566i −0.439463 1.49804i
\(478\) 0 0
\(479\) 485.253i 1.01305i 0.862224 + 0.506527i \(0.169071\pi\)
−0.862224 + 0.506527i \(0.830929\pi\)
\(480\) 0 0
\(481\) −160.795 −0.334294
\(482\) 0 0
\(483\) 439.401 + 329.016i 0.909732 + 0.681192i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 140.013i 0.287500i −0.989614 0.143750i \(-0.954084\pi\)
0.989614 0.143750i \(-0.0459162\pi\)
\(488\) 0 0
\(489\) −408.669 306.005i −0.835724 0.625776i
\(490\) 0 0
\(491\) 20.4450i 0.0416396i 0.999783 + 0.0208198i \(0.00662763\pi\)
−0.999783 + 0.0208198i \(0.993372\pi\)
\(492\) 0 0
\(493\) 649.816i 1.31809i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −694.562 −1.39751
\(498\) 0 0
\(499\) −49.2051 −0.0986074 −0.0493037 0.998784i \(-0.515700\pi\)
−0.0493037 + 0.998784i \(0.515700\pi\)
\(500\) 0 0
\(501\) 11.4861 15.3397i 0.0229263 0.0306181i
\(502\) 0 0
\(503\) 149.060 0.296343 0.148171 0.988962i \(-0.452661\pi\)
0.148171 + 0.988962i \(0.452661\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −27.8277 + 37.1639i −0.0548870 + 0.0733015i
\(508\) 0 0
\(509\) 655.375i 1.28757i −0.765205 0.643787i \(-0.777362\pi\)
0.765205 0.643787i \(-0.222638\pi\)
\(510\) 0 0
\(511\) 456.727 0.893790
\(512\) 0 0
\(513\) −644.834 240.783i −1.25699 0.469362i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 353.081i 0.682942i
\(518\) 0 0
\(519\) −428.680 + 572.502i −0.825973 + 1.10309i
\(520\) 0 0
\(521\) 419.868i 0.805890i −0.915224 0.402945i \(-0.867987\pi\)
0.915224 0.402945i \(-0.132013\pi\)
\(522\) 0 0
\(523\) 146.678i 0.280454i 0.990119 + 0.140227i \(0.0447833\pi\)
−0.990119 + 0.140227i \(0.955217\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 595.444 1.12988
\(528\) 0 0
\(529\) −208.030 −0.393251
\(530\) 0 0
\(531\) −396.333 + 116.268i −0.746390 + 0.218960i
\(532\) 0 0
\(533\) 629.917 1.18183
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 584.840 + 437.919i 1.08909 + 0.815491i
\(538\) 0 0
\(539\) 453.156i 0.840735i
\(540\) 0 0
\(541\) 7.07300 0.0130739 0.00653697 0.999979i \(-0.497919\pi\)
0.00653697 + 0.999979i \(0.497919\pi\)
\(542\) 0 0
\(543\) −585.199 + 781.534i −1.07772 + 1.43929i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.46995i 0.00634360i −0.999995 0.00317180i \(-0.998990\pi\)
0.999995 0.00317180i \(-0.00100962\pi\)
\(548\) 0 0
\(549\) 237.353 + 809.090i 0.432337 + 1.47375i
\(550\) 0 0
\(551\) 1070.78i 1.94334i
\(552\) 0 0
\(553\) 120.005i 0.217006i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −654.467 −1.17499 −0.587493 0.809230i \(-0.699885\pi\)
−0.587493 + 0.809230i \(0.699885\pi\)
\(558\) 0 0
\(559\) −733.684 −1.31249
\(560\) 0 0
\(561\) 304.387 + 227.920i 0.542579 + 0.406274i
\(562\) 0 0
\(563\) 546.529 0.970745 0.485373 0.874307i \(-0.338684\pi\)
0.485373 + 0.874307i \(0.338684\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −446.913 696.167i −0.788207 1.22781i
\(568\) 0 0
\(569\) 746.774i 1.31243i 0.754573 + 0.656216i \(0.227844\pi\)
−0.754573 + 0.656216i \(0.772156\pi\)
\(570\) 0 0
\(571\) −230.358 −0.403430 −0.201715 0.979444i \(-0.564651\pi\)
−0.201715 + 0.979444i \(0.564651\pi\)
\(572\) 0 0
\(573\) −398.634 298.491i −0.695696 0.520926i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 525.585i 0.910893i −0.890263 0.455446i \(-0.849480\pi\)
0.890263 0.455446i \(-0.150520\pi\)
\(578\) 0 0
\(579\) 574.914 + 430.486i 0.992943 + 0.743499i
\(580\) 0 0
\(581\) 1475.16i 2.53901i
\(582\) 0 0
\(583\) 677.906i 1.16279i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −75.3619 −0.128385 −0.0641924 0.997938i \(-0.520447\pi\)
−0.0641924 + 0.997938i \(0.520447\pi\)
\(588\) 0 0
\(589\) 981.185 1.66585
\(590\) 0 0
\(591\) 463.051 618.405i 0.783505 1.04637i
\(592\) 0 0
\(593\) −1142.87 −1.92727 −0.963633 0.267230i \(-0.913892\pi\)
−0.963633 + 0.267230i \(0.913892\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 141.106 188.446i 0.236358 0.315656i
\(598\) 0 0
\(599\) 1060.02i 1.76965i −0.465927 0.884823i \(-0.654279\pi\)
0.465927 0.884823i \(-0.345721\pi\)
\(600\) 0 0
\(601\) −530.900 −0.883361 −0.441681 0.897172i \(-0.645618\pi\)
−0.441681 + 0.897172i \(0.645618\pi\)
\(602\) 0 0
\(603\) −297.940 + 87.4030i −0.494095 + 0.144947i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 719.995i 1.18615i 0.805146 + 0.593077i \(0.202087\pi\)
−0.805146 + 0.593077i \(0.797913\pi\)
\(608\) 0 0
\(609\) −771.360 + 1030.15i −1.26660 + 1.69154i
\(610\) 0 0
\(611\) 585.331i 0.957988i
\(612\) 0 0
\(613\) 216.348i 0.352932i −0.984307 0.176466i \(-0.943533\pi\)
0.984307 0.176466i \(-0.0564666\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 231.190 0.374701 0.187350 0.982293i \(-0.440010\pi\)
0.187350 + 0.982293i \(0.440010\pi\)
\(618\) 0 0
\(619\) 689.633 1.11411 0.557054 0.830476i \(-0.311932\pi\)
0.557054 + 0.830476i \(0.311932\pi\)
\(620\) 0 0
\(621\) −453.161 169.212i −0.729728 0.272482i
\(622\) 0 0
\(623\) −650.679 −1.04443
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 501.574 + 375.570i 0.799959 + 0.598996i
\(628\) 0 0
\(629\) 183.156i 0.291187i
\(630\) 0 0
\(631\) −616.973 −0.977770 −0.488885 0.872348i \(-0.662596\pi\)
−0.488885 + 0.872348i \(0.662596\pi\)
\(632\) 0 0
\(633\) −202.692 + 270.696i −0.320209 + 0.427639i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 751.234i 1.17933i
\(638\) 0 0
\(639\) 587.305 172.291i 0.919100 0.269626i
\(640\) 0 0
\(641\) 1131.85i 1.76576i 0.469598 + 0.882880i \(0.344399\pi\)
−0.469598 + 0.882880i \(0.655601\pi\)
\(642\) 0 0
\(643\) 637.481i 0.991418i −0.868489 0.495709i \(-0.834908\pi\)
0.868489 0.495709i \(-0.165092\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −800.782 −1.23769 −0.618843 0.785515i \(-0.712398\pi\)
−0.618843 + 0.785515i \(0.712398\pi\)
\(648\) 0 0
\(649\) 375.999 0.579352
\(650\) 0 0
\(651\) 943.955 + 706.818i 1.45001 + 1.08574i
\(652\) 0 0
\(653\) 313.185 0.479610 0.239805 0.970821i \(-0.422916\pi\)
0.239805 + 0.970821i \(0.422916\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −386.197 + 113.294i −0.587819 + 0.172442i
\(658\) 0 0
\(659\) 878.311i 1.33279i −0.745597 0.666397i \(-0.767836\pi\)
0.745597 0.666397i \(-0.232164\pi\)
\(660\) 0 0
\(661\) 200.421 0.303208 0.151604 0.988441i \(-0.451556\pi\)
0.151604 + 0.988441i \(0.451556\pi\)
\(662\) 0 0
\(663\) −504.606 377.841i −0.761096 0.569896i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 752.497i 1.12818i
\(668\) 0 0
\(669\) −491.532 368.051i −0.734727 0.550151i
\(670\) 0 0
\(671\) 767.580i 1.14393i
\(672\) 0 0
\(673\) 60.8622i 0.0904341i −0.998977 0.0452171i \(-0.985602\pi\)
0.998977 0.0452171i \(-0.0143980\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −415.595 −0.613877 −0.306939 0.951729i \(-0.599305\pi\)
−0.306939 + 0.951729i \(0.599305\pi\)
\(678\) 0 0
\(679\) 652.639 0.961177
\(680\) 0 0
\(681\) −206.912 + 276.331i −0.303836 + 0.405772i
\(682\) 0 0
\(683\) −434.504 −0.636170 −0.318085 0.948062i \(-0.603040\pi\)
−0.318085 + 0.948062i \(0.603040\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 232.782 310.880i 0.338838 0.452518i
\(688\) 0 0
\(689\) 1123.82i 1.63109i
\(690\) 0 0
\(691\) −166.630 −0.241144 −0.120572 0.992705i \(-0.538473\pi\)
−0.120572 + 0.992705i \(0.538473\pi\)
\(692\) 0 0
\(693\) 211.992 + 722.640i 0.305905 + 1.04277i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 717.516i 1.02943i
\(698\) 0 0
\(699\) −449.802 + 600.711i −0.643494 + 0.859386i
\(700\) 0 0
\(701\) 248.345i 0.354272i −0.984186 0.177136i \(-0.943317\pi\)
0.984186 0.177136i \(-0.0566833\pi\)
\(702\) 0 0
\(703\) 301.809i 0.429315i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −519.617 −0.734961
\(708\) 0 0
\(709\) 24.1911 0.0341200 0.0170600 0.999854i \(-0.494569\pi\)
0.0170600 + 0.999854i \(0.494569\pi\)
\(710\) 0 0
\(711\) −29.7679 101.473i −0.0418677 0.142719i
\(712\) 0 0
\(713\) 689.534 0.967088
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 118.282 + 88.5679i 0.164969 + 0.123526i
\(718\) 0 0
\(719\) 323.584i 0.450048i −0.974353 0.225024i \(-0.927754\pi\)
0.974353 0.225024i \(-0.0722460\pi\)
\(720\) 0 0
\(721\) −469.289 −0.650886
\(722\) 0 0
\(723\) −822.953 + 1099.05i −1.13825 + 1.52013i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 454.949i 0.625789i 0.949788 + 0.312895i \(0.101299\pi\)
−0.949788 + 0.312895i \(0.898701\pi\)
\(728\) 0 0
\(729\) 550.588 + 477.802i 0.755264 + 0.655421i
\(730\) 0 0
\(731\) 835.713i 1.14325i
\(732\) 0 0
\(733\) 131.588i 0.179520i 0.995963 + 0.0897600i \(0.0286100\pi\)
−0.995963 + 0.0897600i \(0.971390\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 282.654 0.383520
\(738\) 0 0
\(739\) −1033.59 −1.39864 −0.699319 0.714810i \(-0.746513\pi\)
−0.699319 + 0.714810i \(0.746513\pi\)
\(740\) 0 0
\(741\) −831.500 622.613i −1.12213 0.840234i
\(742\) 0 0
\(743\) −439.535 −0.591569 −0.295784 0.955255i \(-0.595581\pi\)
−0.295784 + 0.955255i \(0.595581\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 365.924 + 1247.36i 0.489858 + 1.66983i
\(748\) 0 0
\(749\) 1126.00i 1.50333i
\(750\) 0 0
\(751\) 496.343 0.660909 0.330455 0.943822i \(-0.392798\pi\)
0.330455 + 0.943822i \(0.392798\pi\)
\(752\) 0 0
\(753\) 287.488 + 215.266i 0.381790 + 0.285878i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 468.694i 0.619147i −0.950875 0.309574i \(-0.899814\pi\)
0.950875 0.309574i \(-0.100186\pi\)
\(758\) 0 0
\(759\) 352.485 + 263.935i 0.464406 + 0.347740i
\(760\) 0 0
\(761\) 156.940i 0.206229i 0.994669 + 0.103115i \(0.0328808\pi\)
−0.994669 + 0.103115i \(0.967119\pi\)
\(762\) 0 0
\(763\) 503.305i 0.659640i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −623.325 −0.812679
\(768\) 0 0
\(769\) −1188.63 −1.54569 −0.772844 0.634596i \(-0.781166\pi\)
−0.772844 + 0.634596i \(0.781166\pi\)
\(770\) 0 0
\(771\) 201.041 268.490i 0.260753 0.348236i
\(772\) 0 0
\(773\) −950.291 −1.22935 −0.614677 0.788779i \(-0.710714\pi\)
−0.614677 + 0.788779i \(0.710714\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 217.414 290.357i 0.279813 0.373690i
\(778\) 0 0
\(779\) 1182.34i 1.51776i
\(780\) 0 0
\(781\) −557.174 −0.713411
\(782\) 0 0
\(783\) 396.707 1062.41i 0.506650 1.35685i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1056.60i 1.34257i −0.741200 0.671284i \(-0.765743\pi\)
0.741200 0.671284i \(-0.234257\pi\)
\(788\) 0 0
\(789\) −99.5538 + 132.954i −0.126177 + 0.168510i
\(790\) 0 0
\(791\) 1608.08i 2.03297i
\(792\) 0 0
\(793\) 1272.48i 1.60464i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −78.7126 −0.0987612 −0.0493806 0.998780i \(-0.515725\pi\)
−0.0493806 + 0.998780i \(0.515725\pi\)
\(798\) 0 0
\(799\) 666.730 0.834455
\(800\) 0 0
\(801\) 550.198 161.405i 0.686889 0.201505i
\(802\) 0 0
\(803\) 366.384 0.456268
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −743.058 556.389i −0.920765 0.689454i
\(808\) 0 0
\(809\) 176.199i 0.217799i −0.994053 0.108899i \(-0.965267\pi\)
0.994053 0.108899i \(-0.0347327\pi\)
\(810\) 0 0
\(811\) 1103.89 1.36115 0.680574 0.732679i \(-0.261730\pi\)
0.680574 + 0.732679i \(0.261730\pi\)
\(812\) 0 0
\(813\) −348.949 + 466.021i −0.429211 + 0.573211i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1377.10i 1.68556i
\(818\) 0 0
\(819\) −351.437 1197.98i −0.429105 1.46273i
\(820\) 0 0
\(821\) 1321.32i 1.60940i 0.593679 + 0.804702i \(0.297675\pi\)
−0.593679 + 0.804702i \(0.702325\pi\)
\(822\) 0 0
\(823\) 760.814i 0.924439i 0.886765 + 0.462220i \(0.152947\pi\)
−0.886765 + 0.462220i \(0.847053\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 610.599 0.738331 0.369165 0.929364i \(-0.379644\pi\)
0.369165 + 0.929364i \(0.379644\pi\)
\(828\) 0 0
\(829\) −265.766 −0.320586 −0.160293 0.987069i \(-0.551244\pi\)
−0.160293 + 0.987069i \(0.551244\pi\)
\(830\) 0 0
\(831\) −12.4973 9.35776i −0.0150388 0.0112608i
\(832\) 0 0
\(833\) −855.704 −1.02726
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −973.516 363.514i −1.16310 0.434306i
\(838\) 0 0
\(839\) 48.5299i 0.0578425i −0.999582 0.0289213i \(-0.990793\pi\)
0.999582 0.0289213i \(-0.00920721\pi\)
\(840\) 0 0
\(841\) −923.188 −1.09773
\(842\) 0 0
\(843\) −576.345 431.558i −0.683683 0.511931i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 550.236i 0.649629i
\(848\) 0 0
\(849\) 367.677 + 275.310i 0.433071 + 0.324276i
\(850\) 0 0
\(851\) 212.098i 0.249234i
\(852\) 0 0
\(853\) 331.180i 0.388253i −0.980976 0.194127i \(-0.937813\pi\)
0.980976 0.194127i \(-0.0621873\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 159.395 0.185992 0.0929961 0.995666i \(-0.470356\pi\)
0.0929961 + 0.995666i \(0.470356\pi\)
\(858\) 0 0
\(859\) 1635.01 1.90339 0.951694 0.307049i \(-0.0993416\pi\)
0.951694 + 0.307049i \(0.0993416\pi\)
\(860\) 0 0
\(861\) −851.722 + 1137.47i −0.989224 + 1.32111i
\(862\) 0 0
\(863\) 152.237 0.176405 0.0882025 0.996103i \(-0.471888\pi\)
0.0882025 + 0.996103i \(0.471888\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −89.2744 + 119.226i −0.102969 + 0.137516i
\(868\) 0 0
\(869\) 96.2669i 0.110779i
\(870\) 0 0
\(871\) −468.578 −0.537977
\(872\) 0 0
\(873\) −551.856 + 161.891i −0.632137 + 0.185443i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 521.588i 0.594742i −0.954762 0.297371i \(-0.903890\pi\)
0.954762 0.297371i \(-0.0961097\pi\)
\(878\) 0 0
\(879\) 304.338 406.443i 0.346232 0.462392i
\(880\) 0 0
\(881\) 821.701i 0.932692i −0.884603 0.466346i \(-0.845570\pi\)
0.884603 0.466346i \(-0.154430\pi\)
\(882\) 0 0
\(883\) 747.791i 0.846876i 0.905925 + 0.423438i \(0.139177\pi\)
−0.905925 + 0.423438i \(0.860823\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1160.21 1.30801 0.654006 0.756490i \(-0.273087\pi\)
0.654006 + 0.756490i \(0.273087\pi\)
\(888\) 0 0
\(889\) −1379.45 −1.55169
\(890\) 0 0
\(891\) −358.511 558.461i −0.402369 0.626780i
\(892\) 0 0
\(893\) 1098.65 1.23029
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −584.342 437.546i −0.651440 0.487788i
\(898\) 0 0
\(899\) 1616.57i 1.79819i
\(900\) 0 0
\(901\) 1280.10 1.42076
\(902\) 0 0
\(903\) 992.027 1324.85i 1.09859 1.46717i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 354.207i 0.390526i 0.980751 + 0.195263i \(0.0625560\pi\)
−0.980751 + 0.195263i \(0.937444\pi\)
\(908\) 0 0
\(909\) 439.376 128.894i 0.483362 0.141798i
\(910\) 0 0
\(911\) 1461.82i 1.60463i 0.596900 + 0.802316i \(0.296399\pi\)
−0.596900 + 0.802316i \(0.703601\pi\)
\(912\) 0 0
\(913\) 1183.37i 1.29613i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1538.65 1.67792
\(918\) 0 0
\(919\) −600.104 −0.652997 −0.326499 0.945198i \(-0.605869\pi\)
−0.326499 + 0.945198i \(0.605869\pi\)
\(920\) 0 0
\(921\) −793.427 594.105i −0.861484 0.645065i
\(922\) 0 0
\(923\) 923.672 1.00073
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 396.819 116.410i 0.428068 0.125577i
\(928\) 0 0
\(929\) 243.828i 0.262462i −0.991352 0.131231i \(-0.958107\pi\)
0.991352 0.131231i \(-0.0418930\pi\)
\(930\) 0 0
\(931\) −1410.05 −1.51455
\(932\) 0 0
\(933\) 320.818 + 240.223i 0.343856 + 0.257474i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 894.637i 0.954789i −0.878689 0.477395i \(-0.841581\pi\)
0.878689 0.477395i \(-0.158419\pi\)
\(938\) 0 0
\(939\) −1104.60 827.105i −1.17636 0.880837i
\(940\) 0 0
\(941\) 592.336i 0.629475i 0.949179 + 0.314737i \(0.101916\pi\)
−0.949179 + 0.314737i \(0.898084\pi\)
\(942\) 0 0
\(943\) 830.894i 0.881118i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −526.526 −0.555994 −0.277997 0.960582i \(-0.589670\pi\)
−0.277997 + 0.960582i \(0.589670\pi\)
\(948\) 0 0
\(949\) −607.383 −0.640025
\(950\) 0 0
\(951\) −536.112 + 715.977i −0.563735 + 0.752868i
\(952\) 0 0
\(953\) 117.158 0.122936 0.0614678 0.998109i \(-0.480422\pi\)
0.0614678 + 0.998109i \(0.480422\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −618.780 + 826.381i −0.646583 + 0.863512i
\(958\) 0 0
\(959\) 1420.52i 1.48125i
\(960\) 0 0
\(961\) 520.311 0.541427
\(962\) 0 0
\(963\) −279.311 952.114i −0.290042 0.988696i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 811.294i 0.838981i 0.907760 + 0.419490i \(0.137791\pi\)
−0.907760 + 0.419490i \(0.862209\pi\)
\(968\) 0 0
\(969\) 709.197 947.132i 0.731885 0.977432i
\(970\) 0 0
\(971\) 1052.66i 1.08410i −0.840346 0.542050i \(-0.817648\pi\)
0.840346 0.542050i \(-0.182352\pi\)
\(972\) 0 0
\(973\) 1126.48i 1.15774i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −830.832 −0.850391 −0.425195 0.905102i \(-0.639795\pi\)
−0.425195 + 0.905102i \(0.639795\pi\)
\(978\) 0 0
\(979\) −521.971 −0.533167
\(980\) 0 0
\(981\) −124.848 425.582i −0.127266 0.433825i
\(982\) 0 0
\(983\) 1056.30 1.07457 0.537283 0.843402i \(-0.319451\pi\)
0.537283 + 0.843402i \(0.319451\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 1056.96 + 791.436i 1.07088 + 0.801861i
\(988\) 0 0
\(989\) 967.768i 0.978532i
\(990\) 0 0
\(991\) 1592.65 1.60712 0.803559 0.595225i \(-0.202937\pi\)
0.803559 + 0.595225i \(0.202937\pi\)
\(992\) 0 0
\(993\) 714.934 954.795i 0.719974 0.961525i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 978.023i 0.980966i −0.871451 0.490483i \(-0.836820\pi\)
0.871451 0.490483i \(-0.163180\pi\)
\(998\) 0 0
\(999\) −111.815 + 299.450i −0.111927 + 0.299749i
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.449.13 16
3.2 odd 2 inner 600.3.c.d.449.3 16
4.3 odd 2 1200.3.c.m.449.4 16
5.2 odd 4 120.3.l.a.41.3 8
5.3 odd 4 600.3.l.f.401.6 8
5.4 even 2 inner 600.3.c.d.449.4 16
12.11 even 2 1200.3.c.m.449.14 16
15.2 even 4 120.3.l.a.41.4 yes 8
15.8 even 4 600.3.l.f.401.5 8
15.14 odd 2 inner 600.3.c.d.449.14 16
20.3 even 4 1200.3.l.x.401.3 8
20.7 even 4 240.3.l.d.161.6 8
20.19 odd 2 1200.3.c.m.449.13 16
40.27 even 4 960.3.l.g.641.3 8
40.37 odd 4 960.3.l.h.641.6 8
60.23 odd 4 1200.3.l.x.401.4 8
60.47 odd 4 240.3.l.d.161.5 8
60.59 even 2 1200.3.c.m.449.3 16
120.77 even 4 960.3.l.h.641.5 8
120.107 odd 4 960.3.l.g.641.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.3 8 5.2 odd 4
120.3.l.a.41.4 yes 8 15.2 even 4
240.3.l.d.161.5 8 60.47 odd 4
240.3.l.d.161.6 8 20.7 even 4
600.3.c.d.449.3 16 3.2 odd 2 inner
600.3.c.d.449.4 16 5.4 even 2 inner
600.3.c.d.449.13 16 1.1 even 1 trivial
600.3.c.d.449.14 16 15.14 odd 2 inner
600.3.l.f.401.5 8 15.8 even 4
600.3.l.f.401.6 8 5.3 odd 4
960.3.l.g.641.3 8 40.27 even 4
960.3.l.g.641.4 8 120.107 odd 4
960.3.l.h.641.5 8 120.77 even 4
960.3.l.h.641.6 8 40.37 odd 4
1200.3.c.m.449.3 16 60.59 even 2
1200.3.c.m.449.4 16 4.3 odd 2
1200.3.c.m.449.13 16 20.19 odd 2
1200.3.c.m.449.14 16 12.11 even 2
1200.3.l.x.401.3 8 20.3 even 4
1200.3.l.x.401.4 8 60.23 odd 4