Properties

Label 600.3.l.f.401.8
Level $600$
Weight $3$
Character 600.401
Analytic conductor $16.349$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [600,3,Mod(401,600)]
 
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, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("600.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 600 = 2^{3} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 600.l (of order \(2\), degree \(1\), 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: \(8\)
Coefficient field: 8.0.681615360000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4x^{7} - 2x^{6} + 20x^{5} + 49x^{4} - 136x^{3} + 168x^{2} - 96x + 864 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7}\cdot 3 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 401.8
Root \(3.22255 - 1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 600.401
Dual form 600.3.l.f.401.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(2.87275 + 0.864473i) q^{3} -9.02416 q^{7} +(7.50537 + 4.96683i) q^{9} +O(q^{10})\) \(q+(2.87275 + 0.864473i) q^{3} -9.02416 q^{7} +(7.50537 + 4.96683i) q^{9} -21.8827i q^{11} -21.6599 q^{13} -12.1078i q^{17} +3.03757 q^{19} +(-25.9241 - 7.80114i) q^{21} -28.5735i q^{23} +(17.2674 + 20.7566i) q^{27} -12.0364i q^{29} +2.19085 q^{31} +(18.9170 - 62.8636i) q^{33} -0.839959 q^{37} +(-62.2233 - 18.7244i) q^{39} -35.5690i q^{41} +12.7152 q^{43} +22.5481i q^{47} +32.4354 q^{49} +(10.4668 - 34.7826i) q^{51} +9.13775i q^{53} +(8.72618 + 2.62590i) q^{57} +80.4459i q^{59} -57.8816 q^{61} +(-67.7297 - 44.8214i) q^{63} +63.0560 q^{67} +(24.7011 - 82.0846i) q^{69} -17.0218i q^{71} -52.1181 q^{73} +197.473i q^{77} -7.46224 q^{79} +(31.6612 + 74.5558i) q^{81} -82.3758i q^{83} +(10.4051 - 34.5774i) q^{87} +27.5850i q^{89} +195.462 q^{91} +(6.29376 + 1.89393i) q^{93} -114.989 q^{97} +(108.688 - 164.238i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{3} - 16 q^{7} + 20 q^{9} + 8 q^{13} - 8 q^{19} + 28 q^{21} - 20 q^{27} + 120 q^{31} + 112 q^{33} - 8 q^{37} - 72 q^{39} + 328 q^{43} + 64 q^{49} + 64 q^{51} - 72 q^{57} + 8 q^{61} - 88 q^{63} - 152 q^{67} + 100 q^{69} - 32 q^{73} + 88 q^{79} + 224 q^{81} + 152 q^{87} + 560 q^{91} + 368 q^{93} - 144 q^{97} + 32 q^{99}+O(q^{100}) \) 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
\(3\) 2.87275 + 0.864473i 0.957583 + 0.288158i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −9.02416 −1.28917 −0.644583 0.764535i \(-0.722969\pi\)
−0.644583 + 0.764535i \(0.722969\pi\)
\(8\) 0 0
\(9\) 7.50537 + 4.96683i 0.833930 + 0.551870i
\(10\) 0 0
\(11\) 21.8827i 1.98934i −0.103119 0.994669i \(-0.532882\pi\)
0.103119 0.994669i \(-0.467118\pi\)
\(12\) 0 0
\(13\) −21.6599 −1.66614 −0.833071 0.553166i \(-0.813420\pi\)
−0.833071 + 0.553166i \(0.813420\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 12.1078i 0.712221i −0.934444 0.356111i \(-0.884103\pi\)
0.934444 0.356111i \(-0.115897\pi\)
\(18\) 0 0
\(19\) 3.03757 0.159872 0.0799361 0.996800i \(-0.474528\pi\)
0.0799361 + 0.996800i \(0.474528\pi\)
\(20\) 0 0
\(21\) −25.9241 7.80114i −1.23448 0.371483i
\(22\) 0 0
\(23\) 28.5735i 1.24233i −0.783681 0.621164i \(-0.786660\pi\)
0.783681 0.621164i \(-0.213340\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 17.2674 + 20.7566i 0.639532 + 0.768765i
\(28\) 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
\(33\) 18.9170 62.8636i 0.573243 1.90496i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −0.839959 −0.0227016 −0.0113508 0.999936i \(-0.503613\pi\)
−0.0113508 + 0.999936i \(0.503613\pi\)
\(38\) 0 0
\(39\) −62.2233 18.7244i −1.59547 0.480112i
\(40\) 0 0
\(41\) 35.5690i 0.867537i −0.901024 0.433769i \(-0.857184\pi\)
0.901024 0.433769i \(-0.142816\pi\)
\(42\) 0 0
\(43\) 12.7152 0.295702 0.147851 0.989010i \(-0.452764\pi\)
0.147851 + 0.989010i \(0.452764\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 22.5481i 0.479746i 0.970804 + 0.239873i \(0.0771058\pi\)
−0.970804 + 0.239873i \(0.922894\pi\)
\(48\) 0 0
\(49\) 32.4354 0.661947
\(50\) 0 0
\(51\) 10.4668 34.7826i 0.205232 0.682011i
\(52\) 0 0
\(53\) 9.13775i 0.172410i 0.996277 + 0.0862052i \(0.0274741\pi\)
−0.996277 + 0.0862052i \(0.972526\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.72618 + 2.62590i 0.153091 + 0.0460684i
\(58\) 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
\(63\) −67.7297 44.8214i −1.07507 0.711452i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 63.0560 0.941135 0.470567 0.882364i \(-0.344049\pi\)
0.470567 + 0.882364i \(0.344049\pi\)
\(68\) 0 0
\(69\) 24.7011 82.0846i 0.357986 1.18963i
\(70\) 0 0
\(71\) 17.0218i 0.239743i −0.992789 0.119872i \(-0.961752\pi\)
0.992789 0.119872i \(-0.0382483\pi\)
\(72\) 0 0
\(73\) −52.1181 −0.713947 −0.356973 0.934115i \(-0.616191\pi\)
−0.356973 + 0.934115i \(0.616191\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 197.473i 2.56459i
\(78\) 0 0
\(79\) −7.46224 −0.0944588 −0.0472294 0.998884i \(-0.515039\pi\)
−0.0472294 + 0.998884i \(0.515039\pi\)
\(80\) 0 0
\(81\) 31.6612 + 74.5558i 0.390879 + 0.920442i
\(82\) 0 0
\(83\) 82.3758i 0.992480i −0.868185 0.496240i \(-0.834714\pi\)
0.868185 0.496240i \(-0.165286\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 10.4051 34.5774i 0.119599 0.397442i
\(88\) 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
\(93\) 6.29376 + 1.89393i 0.0676748 + 0.0203648i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −114.989 −1.18545 −0.592727 0.805404i \(-0.701949\pi\)
−0.592727 + 0.805404i \(0.701949\pi\)
\(98\) 0 0
\(99\) 108.688 164.238i 1.09786 1.65897i
\(100\) 0 0
\(101\) 122.804i 1.21588i −0.793982 0.607941i \(-0.791996\pi\)
0.793982 0.607941i \(-0.208004\pi\)
\(102\) 0 0
\(103\) 46.8275 0.454636 0.227318 0.973821i \(-0.427004\pi\)
0.227318 + 0.973821i \(0.427004\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 105.086i 0.982113i −0.871128 0.491056i \(-0.836611\pi\)
0.871128 0.491056i \(-0.163389\pi\)
\(108\) 0 0
\(109\) −116.777 −1.07135 −0.535673 0.844426i \(-0.679942\pi\)
−0.535673 + 0.844426i \(0.679942\pi\)
\(110\) 0 0
\(111\) −2.41299 0.726122i −0.0217387 0.00654164i
\(112\) 0 0
\(113\) 10.8116i 0.0956779i 0.998855 + 0.0478389i \(0.0152334\pi\)
−0.998855 + 0.0478389i \(0.984767\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −162.565 107.581i −1.38945 0.919494i
\(118\) 0 0
\(119\) 109.262i 0.918171i
\(120\) 0 0
\(121\) −357.853 −2.95747
\(122\) 0 0
\(123\) 30.7485 102.181i 0.249988 0.830739i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 192.459 1.51543 0.757714 0.652587i \(-0.226316\pi\)
0.757714 + 0.652587i \(0.226316\pi\)
\(128\) 0 0
\(129\) 36.5276 + 10.9919i 0.283159 + 0.0852089i
\(130\) 0 0
\(131\) 48.6360i 0.371267i −0.982619 0.185633i \(-0.940566\pi\)
0.982619 0.185633i \(-0.0594337\pi\)
\(132\) 0 0
\(133\) −27.4115 −0.206102
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 157.869i 1.15233i −0.817335 0.576163i \(-0.804549\pi\)
0.817335 0.576163i \(-0.195451\pi\)
\(138\) 0 0
\(139\) −164.752 −1.18526 −0.592632 0.805473i \(-0.701911\pi\)
−0.592632 + 0.805473i \(0.701911\pi\)
\(140\) 0 0
\(141\) −19.4922 + 64.7749i −0.138243 + 0.459397i
\(142\) 0 0
\(143\) 473.977i 3.31452i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 93.1788 + 28.0396i 0.633869 + 0.190745i
\(148\) 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
\(153\) 60.1372 90.8733i 0.393054 0.593943i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.11941 0.0389771 0.0194886 0.999810i \(-0.493796\pi\)
0.0194886 + 0.999810i \(0.493796\pi\)
\(158\) 0 0
\(159\) −7.89934 + 26.2505i −0.0496814 + 0.165097i
\(160\) 0 0
\(161\) 257.852i 1.60157i
\(162\) 0 0
\(163\) 170.444 1.04567 0.522833 0.852435i \(-0.324875\pi\)
0.522833 + 0.852435i \(0.324875\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 61.2668i 0.366867i −0.983032 0.183434i \(-0.941279\pi\)
0.983032 0.183434i \(-0.0587212\pi\)
\(168\) 0 0
\(169\) 300.149 1.77603
\(170\) 0 0
\(171\) 22.7981 + 15.0871i 0.133322 + 0.0882286i
\(172\) 0 0
\(173\) 262.548i 1.51762i 0.651312 + 0.758810i \(0.274219\pi\)
−0.651312 + 0.758810i \(0.725781\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −69.5434 + 231.101i −0.392900 + 1.30566i
\(178\) 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
\(183\) −166.279 50.0371i −0.908630 0.273427i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −264.951 −1.41685
\(188\) 0 0
\(189\) −155.823 187.311i −0.824462 0.991065i
\(190\) 0 0
\(191\) 75.2506i 0.393982i −0.980405 0.196991i \(-0.936883\pi\)
0.980405 0.196991i \(-0.0631170\pi\)
\(192\) 0 0
\(193\) 212.587 1.10149 0.550744 0.834674i \(-0.314344\pi\)
0.550744 + 0.834674i \(0.314344\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 190.640i 0.967718i −0.875146 0.483859i \(-0.839235\pi\)
0.875146 0.483859i \(-0.160765\pi\)
\(198\) 0 0
\(199\) 209.996 1.05526 0.527629 0.849475i \(-0.323081\pi\)
0.527629 + 0.849475i \(0.323081\pi\)
\(200\) 0 0
\(201\) 181.144 + 54.5102i 0.901214 + 0.271195i
\(202\) 0 0
\(203\) 108.618i 0.535064i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 141.920 214.455i 0.685603 1.03601i
\(208\) 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
\(213\) 14.7149 48.8993i 0.0690839 0.229574i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −19.7706 −0.0911086
\(218\) 0 0
\(219\) −149.722 45.0547i −0.683663 0.205729i
\(220\) 0 0
\(221\) 262.252i 1.18666i
\(222\) 0 0
\(223\) −132.362 −0.593552 −0.296776 0.954947i \(-0.595911\pi\)
−0.296776 + 0.954947i \(0.595911\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 187.624i 0.826537i 0.910609 + 0.413268i \(0.135613\pi\)
−0.910609 + 0.413268i \(0.864387\pi\)
\(228\) 0 0
\(229\) −178.571 −0.779788 −0.389894 0.920860i \(-0.627488\pi\)
−0.389894 + 0.920860i \(0.627488\pi\)
\(230\) 0 0
\(231\) −170.710 + 567.291i −0.739005 + 2.45580i
\(232\) 0 0
\(233\) 296.711i 1.27344i 0.771097 + 0.636718i \(0.219708\pi\)
−0.771097 + 0.636718i \(0.780292\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −21.4371 6.45091i −0.0904521 0.0272190i
\(238\) 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
\(243\) 26.5032 + 241.550i 0.109067 + 0.994034i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −65.7933 −0.266370
\(248\) 0 0
\(249\) 71.2117 236.645i 0.285991 0.950382i
\(250\) 0 0
\(251\) 205.885i 0.820259i 0.912027 + 0.410130i \(0.134517\pi\)
−0.912027 + 0.410130i \(0.865483\pi\)
\(252\) 0 0
\(253\) −625.267 −2.47141
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 188.382i 0.733004i 0.930417 + 0.366502i \(0.119445\pi\)
−0.930417 + 0.366502i \(0.880555\pi\)
\(258\) 0 0
\(259\) 7.57992 0.0292661
\(260\) 0 0
\(261\) 59.7825 90.3373i 0.229052 0.346120i
\(262\) 0 0
\(263\) 188.745i 0.717660i −0.933403 0.358830i \(-0.883176\pi\)
0.933403 0.358830i \(-0.116824\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −23.8465 + 79.2447i −0.0893127 + 0.296797i
\(268\) 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
\(273\) 561.513 + 168.972i 2.05682 + 0.618944i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 400.100 1.44440 0.722202 0.691682i \(-0.243130\pi\)
0.722202 + 0.691682i \(0.243130\pi\)
\(278\) 0 0
\(279\) 16.4431 + 10.8816i 0.0589360 + 0.0390020i
\(280\) 0 0
\(281\) 350.698i 1.24804i −0.781410 0.624018i \(-0.785499\pi\)
0.781410 0.624018i \(-0.214501\pi\)
\(282\) 0 0
\(283\) 464.015 1.63963 0.819814 0.572630i \(-0.194077\pi\)
0.819814 + 0.572630i \(0.194077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 320.981i 1.11840i
\(288\) 0 0
\(289\) 142.402 0.492741
\(290\) 0 0
\(291\) −330.334 99.4049i −1.13517 0.341598i
\(292\) 0 0
\(293\) 47.4080i 0.161802i −0.996722 0.0809009i \(-0.974220\pi\)
0.996722 0.0809009i \(-0.0257797\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 454.212 377.857i 1.52933 1.27224i
\(298\) 0 0
\(299\) 618.899i 2.06990i
\(300\) 0 0
\(301\) −114.744 −0.381209
\(302\) 0 0
\(303\) 106.161 352.785i 0.350366 1.16431i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −461.894 −1.50454 −0.752270 0.658855i \(-0.771041\pi\)
−0.752270 + 0.658855i \(0.771041\pi\)
\(308\) 0 0
\(309\) 134.524 + 40.4812i 0.435352 + 0.131007i
\(310\) 0 0
\(311\) 123.057i 0.395681i −0.980234 0.197841i \(-0.936607\pi\)
0.980234 0.197841i \(-0.0633929\pi\)
\(312\) 0 0
\(313\) −97.4353 −0.311295 −0.155647 0.987813i \(-0.549746\pi\)
−0.155647 + 0.987813i \(0.549746\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 252.388i 0.796175i −0.917347 0.398088i \(-0.869674\pi\)
0.917347 0.398088i \(-0.130326\pi\)
\(318\) 0 0
\(319\) −263.388 −0.825668
\(320\) 0 0
\(321\) 90.8441 301.886i 0.283003 0.940454i
\(322\) 0 0
\(323\) 36.7782i 0.113864i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −335.470 100.950i −1.02590 0.308716i
\(328\) 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
\(333\) −6.30420 4.17193i −0.0189315 0.0125283i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 352.738 1.04670 0.523350 0.852118i \(-0.324682\pi\)
0.523350 + 0.852118i \(0.324682\pi\)
\(338\) 0 0
\(339\) −9.34634 + 31.0590i −0.0275703 + 0.0916195i
\(340\) 0 0
\(341\) 47.9417i 0.140592i
\(342\) 0 0
\(343\) 149.481 0.435806
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 280.382i 0.808018i −0.914755 0.404009i \(-0.867616\pi\)
0.914755 0.404009i \(-0.132384\pi\)
\(348\) 0 0
\(349\) −586.721 −1.68115 −0.840575 0.541696i \(-0.817783\pi\)
−0.840575 + 0.541696i \(0.817783\pi\)
\(350\) 0 0
\(351\) −374.008 449.586i −1.06555 1.28087i
\(352\) 0 0
\(353\) 558.927i 1.58336i −0.610935 0.791681i \(-0.709206\pi\)
0.610935 0.791681i \(-0.290794\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −94.4544 + 313.883i −0.264578 + 0.879225i
\(358\) 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
\(363\) −1028.02 309.355i −2.83202 0.852217i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −176.636 −0.481296 −0.240648 0.970612i \(-0.577360\pi\)
−0.240648 + 0.970612i \(0.577360\pi\)
\(368\) 0 0
\(369\) 176.665 266.959i 0.478768 0.723466i
\(370\) 0 0
\(371\) 82.4605i 0.222265i
\(372\) 0 0
\(373\) −367.327 −0.984790 −0.492395 0.870372i \(-0.663878\pi\)
−0.492395 + 0.870372i \(0.663878\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 260.706i 0.691527i
\(378\) 0 0
\(379\) 611.014 1.61217 0.806087 0.591798i \(-0.201582\pi\)
0.806087 + 0.591798i \(0.201582\pi\)
\(380\) 0 0
\(381\) 552.887 + 166.376i 1.45115 + 0.436682i
\(382\) 0 0
\(383\) 13.6994i 0.0357687i 0.999840 + 0.0178844i \(0.00569307\pi\)
−0.999840 + 0.0178844i \(0.994307\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 95.4322 + 63.1542i 0.246595 + 0.163189i
\(388\) 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
\(393\) 42.0445 139.719i 0.106983 0.355519i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −450.560 −1.13491 −0.567456 0.823404i \(-0.692072\pi\)
−0.567456 + 0.823404i \(0.692072\pi\)
\(398\) 0 0
\(399\) −78.7464 23.6965i −0.197359 0.0593898i
\(400\) 0 0
\(401\) 503.683i 1.25607i −0.778186 0.628034i \(-0.783860\pi\)
0.778186 0.628034i \(-0.216140\pi\)
\(402\) 0 0
\(403\) −47.4535 −0.117751
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 18.3806i 0.0451611i
\(408\) 0 0
\(409\) 240.726 0.588573 0.294286 0.955717i \(-0.404918\pi\)
0.294286 + 0.955717i \(0.404918\pi\)
\(410\) 0 0
\(411\) 136.473 453.517i 0.332051 1.10345i
\(412\) 0 0
\(413\) 725.957i 1.75776i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −473.290 142.423i −1.13499 0.341543i
\(418\) 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
\(423\) −111.992 + 169.232i −0.264757 + 0.400075i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 522.332 1.22326
\(428\) 0 0
\(429\) −409.740 + 1361.62i −0.955105 + 3.17393i
\(430\) 0 0
\(431\) 213.608i 0.495610i −0.968810 0.247805i \(-0.920291\pi\)
0.968810 0.247805i \(-0.0797092\pi\)
\(432\) 0 0
\(433\) 383.579 0.885864 0.442932 0.896555i \(-0.353938\pi\)
0.442932 + 0.896555i \(0.353938\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 86.7941i 0.198614i
\(438\) 0 0
\(439\) 523.900 1.19340 0.596698 0.802466i \(-0.296479\pi\)
0.596698 + 0.802466i \(0.296479\pi\)
\(440\) 0 0
\(441\) 243.440 + 161.101i 0.552018 + 0.365309i
\(442\) 0 0
\(443\) 391.277i 0.883243i −0.897201 0.441622i \(-0.854403\pi\)
0.897201 0.441622i \(-0.145597\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −227.301 + 755.347i −0.508502 + 1.68981i
\(448\) 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
\(453\) 45.4386 + 13.6735i 0.100306 + 0.0301843i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 734.032 1.60620 0.803098 0.595847i \(-0.203183\pi\)
0.803098 + 0.595847i \(0.203183\pi\)
\(458\) 0 0
\(459\) 251.317 209.069i 0.547531 0.455488i
\(460\) 0 0
\(461\) 775.239i 1.68165i −0.541310 0.840823i \(-0.682072\pi\)
0.541310 0.840823i \(-0.317928\pi\)
\(462\) 0 0
\(463\) 323.967 0.699714 0.349857 0.936803i \(-0.386230\pi\)
0.349857 + 0.936803i \(0.386230\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 160.785i 0.344294i 0.985071 + 0.172147i \(0.0550704\pi\)
−0.985071 + 0.172147i \(0.944930\pi\)
\(468\) 0 0
\(469\) −569.027 −1.21328
\(470\) 0 0
\(471\) 17.5795 + 5.29007i 0.0373239 + 0.0112316i
\(472\) 0 0
\(473\) 278.243i 0.588252i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −45.3856 + 68.5822i −0.0951481 + 0.143778i
\(478\) 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
\(483\) −222.906 + 740.744i −0.461504 + 1.53363i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −129.969 −0.266876 −0.133438 0.991057i \(-0.542602\pi\)
−0.133438 + 0.991057i \(0.542602\pi\)
\(488\) 0 0
\(489\) 489.642 + 147.344i 1.00131 + 0.301317i
\(490\) 0 0
\(491\) 810.511i 1.65074i 0.564595 + 0.825368i \(0.309032\pi\)
−0.564595 + 0.825368i \(0.690968\pi\)
\(492\) 0 0
\(493\) −145.733 −0.295605
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 153.607i 0.309069i
\(498\) 0 0
\(499\) −600.897 −1.20420 −0.602101 0.798420i \(-0.705670\pi\)
−0.602101 + 0.798420i \(0.705670\pi\)
\(500\) 0 0
\(501\) 52.9635 176.004i 0.105716 0.351306i
\(502\) 0 0
\(503\) 688.332i 1.36845i −0.729270 0.684226i \(-0.760140\pi\)
0.729270 0.684226i \(-0.239860\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 862.254 + 259.471i 1.70070 + 0.511777i
\(508\) 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
\(513\) 52.4508 + 63.0498i 0.102243 + 0.122904i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 493.413 0.954377
\(518\) 0 0
\(519\) −226.966 + 754.235i −0.437314 + 1.45325i
\(520\) 0 0
\(521\) 964.525i 1.85130i 0.378386 + 0.925648i \(0.376479\pi\)
−0.378386 + 0.925648i \(0.623521\pi\)
\(522\) 0 0
\(523\) −462.679 −0.884664 −0.442332 0.896851i \(-0.645849\pi\)
−0.442332 + 0.896851i \(0.645849\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 26.5263i 0.0503345i
\(528\) 0 0
\(529\) −287.447 −0.543378
\(530\) 0 0
\(531\) −399.561 + 603.777i −0.752469 + 1.13706i
\(532\) 0 0
\(533\) 770.420i 1.44544i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −5.95408 + 19.7861i −0.0110877 + 0.0368457i
\(538\) 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
\(543\) 627.800 + 188.919i 1.15617 + 0.347917i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −478.953 −0.875599 −0.437799 0.899073i \(-0.644242\pi\)
−0.437799 + 0.899073i \(0.644242\pi\)
\(548\) 0 0
\(549\) −434.423 287.488i −0.791298 0.523657i
\(550\) 0 0
\(551\) 36.5613i 0.0663544i
\(552\) 0 0
\(553\) 67.3404 0.121773
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 343.982i 0.617561i 0.951133 + 0.308781i \(0.0999209\pi\)
−0.951133 + 0.308781i \(0.900079\pi\)
\(558\) 0 0
\(559\) −275.409 −0.492682
\(560\) 0 0
\(561\) −761.137 229.043i −1.35675 0.408276i
\(562\) 0 0
\(563\) 394.056i 0.699922i −0.936764 0.349961i \(-0.886195\pi\)
0.936764 0.349961i \(-0.113805\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −285.716 672.803i −0.503908 1.18660i
\(568\) 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
\(573\) 65.0521 216.176i 0.113529 0.377271i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −452.846 −0.784828 −0.392414 0.919789i \(-0.628360\pi\)
−0.392414 + 0.919789i \(0.628360\pi\)
\(578\) 0 0
\(579\) 610.709 + 183.776i 1.05477 + 0.317402i
\(580\) 0 0
\(581\) 743.372i 1.27947i
\(582\) 0 0
\(583\) 199.959 0.342983
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1103.34i 1.87963i −0.341690 0.939813i \(-0.610999\pi\)
0.341690 0.939813i \(-0.389001\pi\)
\(588\) 0 0
\(589\) 6.65486 0.0112986
\(590\) 0 0
\(591\) 164.804 547.662i 0.278855 0.926670i
\(592\) 0 0
\(593\) 249.474i 0.420698i −0.977626 0.210349i \(-0.932540\pi\)
0.977626 0.210349i \(-0.0674601\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 603.267 + 181.536i 1.01050 + 0.304081i
\(598\) 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
\(603\) 473.259 + 313.188i 0.784841 + 0.519384i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −27.8813 −0.0459329 −0.0229664 0.999736i \(-0.507311\pi\)
−0.0229664 + 0.999736i \(0.507311\pi\)
\(608\) 0 0
\(609\) −93.8973 + 312.032i −0.154183 + 0.512368i
\(610\) 0 0
\(611\) 488.388i 0.799325i
\(612\) 0 0
\(613\) −465.472 −0.759334 −0.379667 0.925123i \(-0.623961\pi\)
−0.379667 + 0.925123i \(0.623961\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 48.6709i 0.0788832i 0.999222 + 0.0394416i \(0.0125579\pi\)
−0.999222 + 0.0394416i \(0.987442\pi\)
\(618\) 0 0
\(619\) −213.318 −0.344617 −0.172309 0.985043i \(-0.555123\pi\)
−0.172309 + 0.985043i \(0.555123\pi\)
\(620\) 0 0
\(621\) 593.091 493.389i 0.955058 0.794508i
\(622\) 0 0
\(623\) 248.931i 0.399569i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 57.4618 190.953i 0.0916456 0.304549i
\(628\) 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
\(633\) 506.808 + 152.510i 0.800644 + 0.240931i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −702.546 −1.10290
\(638\) 0 0
\(639\) 84.5442 127.755i 0.132307 0.199929i
\(640\) 0 0
\(641\) 319.635i 0.498650i 0.968420 + 0.249325i \(0.0802088\pi\)
−0.968420 + 0.249325i \(0.919791\pi\)
\(642\) 0 0
\(643\) −458.627 −0.713261 −0.356630 0.934246i \(-0.616074\pi\)
−0.356630 + 0.934246i \(0.616074\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 507.599i 0.784543i 0.919850 + 0.392271i \(0.128311\pi\)
−0.919850 + 0.392271i \(0.871689\pi\)
\(648\) 0 0
\(649\) 1760.38 2.71244
\(650\) 0 0
\(651\) −56.7959 17.0911i −0.0872440 0.0262536i
\(652\) 0 0
\(653\) 1197.20i 1.83339i 0.399591 + 0.916694i \(0.369152\pi\)
−0.399591 + 0.916694i \(0.630848\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −391.166 258.862i −0.595382 0.394006i
\(658\) 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
\(663\) −226.710 + 753.385i −0.341946 + 1.13633i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −343.921 −0.515624
\(668\) 0 0
\(669\) −380.243 114.423i −0.568375 0.171036i
\(670\) 0 0
\(671\) 1266.61i 1.88764i
\(672\) 0 0
\(673\) 306.607 0.455582 0.227791 0.973710i \(-0.426850\pi\)
0.227791 + 0.973710i \(0.426850\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 546.672i 0.807491i −0.914871 0.403746i \(-0.867708\pi\)
0.914871 0.403746i \(-0.132292\pi\)
\(678\) 0 0
\(679\) 1037.68 1.52824
\(680\) 0 0
\(681\) −162.196 + 538.996i −0.238173 + 0.791478i
\(682\) 0 0
\(683\) 812.204i 1.18917i 0.804032 + 0.594586i \(0.202684\pi\)
−0.804032 + 0.594586i \(0.797316\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −512.991 154.370i −0.746712 0.224702i
\(688\) 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
\(693\) −980.815 + 1482.11i −1.41532 + 2.13869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −430.661 −0.617879
\(698\) 0 0
\(699\) −256.498 + 852.375i −0.366950 + 1.21942i
\(700\) 0 0
\(701\) 674.615i 0.962360i −0.876622 0.481180i \(-0.840208\pi\)
0.876622 0.481180i \(-0.159792\pi\)
\(702\) 0 0
\(703\) −2.55143 −0.00362935
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1108.20i 1.56747i
\(708\) 0 0
\(709\) 656.626 0.926129 0.463065 0.886324i \(-0.346750\pi\)
0.463065 + 0.886324i \(0.346750\pi\)
\(710\) 0 0
\(711\) −56.0069 37.0637i −0.0787720 0.0521290i
\(712\) 0 0
\(713\) 62.6003i 0.0877984i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 119.277 396.371i 0.166355 0.552819i
\(718\) 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
\(723\) 122.289 + 36.7995i 0.169141 + 0.0508983i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 981.269 1.34975 0.674875 0.737932i \(-0.264197\pi\)
0.674875 + 0.737932i \(0.264197\pi\)
\(728\) 0 0
\(729\) −132.677 + 716.825i −0.181998 + 0.983299i
\(730\) 0 0
\(731\) 153.953i 0.210605i
\(732\) 0 0
\(733\) 127.756 0.174292 0.0871460 0.996196i \(-0.472225\pi\)
0.0871460 + 0.996196i \(0.472225\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1379.84i 1.87223i
\(738\) 0 0
\(739\) −1401.59 −1.89660 −0.948302 0.317368i \(-0.897201\pi\)
−0.948302 + 0.317368i \(0.897201\pi\)
\(740\) 0 0
\(741\) −189.008 56.8766i −0.255071 0.0767565i
\(742\) 0 0
\(743\) 11.6734i 0.0157112i 0.999969 + 0.00785561i \(0.00250055\pi\)
−0.999969 + 0.00785561i \(0.997499\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 409.147 618.261i 0.547720 0.827659i
\(748\) 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
\(753\) −177.982 + 591.456i −0.236364 + 0.785466i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −63.6621 −0.0840979 −0.0420490 0.999116i \(-0.513389\pi\)
−0.0420490 + 0.999116i \(0.513389\pi\)
\(758\) 0 0
\(759\) −1796.23 540.526i −2.36658 0.712156i
\(760\) 0 0
\(761\) 377.891i 0.496571i 0.968687 + 0.248286i \(0.0798671\pi\)
−0.968687 + 0.248286i \(0.920133\pi\)
\(762\) 0 0
\(763\) 1053.81 1.38114
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1742.45i 2.27177i
\(768\) 0 0
\(769\) 231.920 0.301586 0.150793 0.988565i \(-0.451817\pi\)
0.150793 + 0.988565i \(0.451817\pi\)
\(770\) 0 0
\(771\) −162.851 + 541.174i −0.211221 + 0.701912i
\(772\) 0 0
\(773\) 1509.05i 1.95220i −0.217320 0.976101i \(-0.569731\pi\)
0.217320 0.976101i \(-0.430269\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 21.7752 + 6.55264i 0.0280247 + 0.00843326i
\(778\) 0 0
\(779\) 108.043i 0.138695i
\(780\) 0 0
\(781\) −372.483 −0.476930
\(782\) 0 0
\(783\) 249.834 207.836i 0.319073 0.265436i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1171.43 −1.48848 −0.744240 0.667912i \(-0.767188\pi\)
−0.744240 + 0.667912i \(0.767188\pi\)
\(788\) 0 0
\(789\) 163.165 542.216i 0.206799 0.687219i
\(790\) 0 0
\(791\) 97.5656i 0.123345i
\(792\) 0 0
\(793\) 1253.71 1.58097
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 988.589i 1.24039i 0.784448 + 0.620194i \(0.212946\pi\)
−0.784448 + 0.620194i \(0.787054\pi\)
\(798\) 0 0
\(799\) 273.007 0.341685
\(800\) 0 0
\(801\) −137.010 + 207.036i −0.171049 + 0.258471i
\(802\) 0 0
\(803\) 1140.49i 1.42028i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −288.191 + 957.694i −0.357114 + 1.18673i
\(808\) 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
\(813\) 752.796 + 226.533i 0.925949 + 0.278638i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 38.6233 0.0472745
\(818\) 0 0
\(819\) 1467.01 + 970.826i 1.79123 + 1.18538i
\(820\) 0 0
\(821\) 877.538i 1.06887i 0.845211 + 0.534433i \(0.179475\pi\)
−0.845211 + 0.534433i \(0.820525\pi\)
\(822\) 0 0
\(823\) −230.855 −0.280504 −0.140252 0.990116i \(-0.544791\pi\)
−0.140252 + 0.990116i \(0.544791\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 969.434i 1.17223i 0.810228 + 0.586115i \(0.199343\pi\)
−0.810228 + 0.586115i \(0.800657\pi\)
\(828\) 0 0
\(829\) −1049.09 −1.26549 −0.632746 0.774360i \(-0.718072\pi\)
−0.632746 + 0.774360i \(0.718072\pi\)
\(830\) 0 0
\(831\) 1149.39 + 345.876i 1.38314 + 0.416216i
\(832\) 0 0
\(833\) 392.720i 0.471453i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 37.8302 + 45.4747i 0.0451973 + 0.0543305i
\(838\) 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
\(843\) 303.169 1007.47i 0.359631 1.19510i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 3229.33 3.81266
\(848\) 0 0
\(849\) 1333.00 + 401.128i 1.57008 + 0.472472i
\(850\) 0 0
\(851\) 24.0006i 0.0282028i
\(852\) 0 0
\(853\) −190.704 −0.223569 −0.111784 0.993732i \(-0.535657\pi\)
−0.111784 + 0.993732i \(0.535657\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 351.238i 0.409846i −0.978778 0.204923i \(-0.934306\pi\)
0.978778 0.204923i \(-0.0656944\pi\)
\(858\) 0 0
\(859\) −1233.74 −1.43625 −0.718123 0.695916i \(-0.754998\pi\)
−0.718123 + 0.695916i \(0.754998\pi\)
\(860\) 0 0
\(861\) −277.479 + 922.096i −0.322275 + 1.07096i
\(862\) 0 0
\(863\) 463.404i 0.536968i 0.963284 + 0.268484i \(0.0865227\pi\)
−0.963284 + 0.268484i \(0.913477\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 409.085 + 123.103i 0.471840 + 0.141987i
\(868\) 0 0
\(869\) 163.294i 0.187910i
\(870\) 0 0
\(871\) −1365.78 −1.56806
\(872\) 0 0
\(873\) −863.035 571.130i −0.988585 0.654216i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −777.195 −0.886197 −0.443099 0.896473i \(-0.646121\pi\)
−0.443099 + 0.896473i \(0.646121\pi\)
\(878\) 0 0
\(879\) 40.9829 136.191i 0.0466245 0.154939i
\(880\) 0 0
\(881\) 1609.50i 1.82690i 0.406951 + 0.913450i \(0.366592\pi\)
−0.406951 + 0.913450i \(0.633408\pi\)
\(882\) 0 0
\(883\) −331.153 −0.375032 −0.187516 0.982262i \(-0.560044\pi\)
−0.187516 + 0.982262i \(0.560044\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 365.778i 0.412377i 0.978512 + 0.206188i \(0.0661060\pi\)
−0.978512 + 0.206188i \(0.933894\pi\)
\(888\) 0 0
\(889\) −1736.78 −1.95364
\(890\) 0 0
\(891\) 1631.48 692.833i 1.83107 0.777591i
\(892\) 0 0
\(893\) 68.4913i 0.0766980i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −535.021 + 1777.94i −0.596456 + 1.98210i
\(898\) 0 0
\(899\) 26.3698i 0.0293324i
\(900\) 0 0
\(901\) 110.638 0.122794
\(902\) 0 0
\(903\) −329.630 99.1930i −0.365039 0.109848i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 270.791 0.298556 0.149278 0.988795i \(-0.452305\pi\)
0.149278 + 0.988795i \(0.452305\pi\)
\(908\) 0 0
\(909\) 609.947 921.690i 0.671008 1.01396i
\(910\) 0 0
\(911\) 1238.74i 1.35976i −0.733324 0.679879i \(-0.762032\pi\)
0.733324 0.679879i \(-0.237968\pi\)
\(912\) 0 0
\(913\) −1802.61 −1.97438
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 438.898i 0.478624i
\(918\) 0 0
\(919\) 578.894 0.629918 0.314959 0.949105i \(-0.398009\pi\)
0.314959 + 0.949105i \(0.398009\pi\)
\(920\) 0 0
\(921\) −1326.91 399.295i −1.44072 0.433545i
\(922\) 0 0
\(923\) 368.689i 0.399447i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 351.458 + 232.584i 0.379135 + 0.250900i
\(928\) 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
\(933\) 106.379 353.511i 0.114019 0.378898i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 693.314 0.739929 0.369965 0.929046i \(-0.379370\pi\)
0.369965 + 0.929046i \(0.379370\pi\)
\(938\) 0 0
\(939\) −279.907 84.2302i −0.298091 0.0897020i
\(940\) 0 0
\(941\) 783.592i 0.832723i 0.909199 + 0.416362i \(0.136695\pi\)
−0.909199 + 0.416362i \(0.863305\pi\)
\(942\) 0 0
\(943\) −1016.33 −1.07777
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1621.44i 1.71219i −0.516820 0.856094i \(-0.672884\pi\)
0.516820 0.856094i \(-0.327116\pi\)
\(948\) 0 0
\(949\) 1128.87 1.18954
\(950\) 0 0
\(951\) 218.182 725.046i 0.229424 0.762404i
\(952\) 0 0
\(953\) 1105.90i 1.16044i 0.814461 + 0.580218i \(0.197033\pi\)
−0.814461 + 0.580218i \(0.802967\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −756.648 227.692i −0.790646 0.237923i
\(958\) 0 0
\(959\) 1424.63i 1.48554i
\(960\) 0 0
\(961\) −956.200 −0.995005
\(962\) 0 0
\(963\) 521.945 788.710i 0.541998 0.819013i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1307.89 −1.35253 −0.676264 0.736659i \(-0.736402\pi\)
−0.676264 + 0.736659i \(0.736402\pi\)
\(968\) 0 0
\(969\) 31.7938 105.655i 0.0328109 0.109035i
\(970\) 0 0
\(971\) 749.988i 0.772387i −0.922418 0.386194i \(-0.873790\pi\)
0.922418 0.386194i \(-0.126210\pi\)
\(972\) 0 0
\(973\) 1486.75 1.52800
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 373.792i 0.382591i 0.981532 + 0.191296i \(0.0612690\pi\)
−0.981532 + 0.191296i \(0.938731\pi\)
\(978\) 0 0
\(979\) 603.634 0.616583
\(980\) 0 0
\(981\) −876.452 580.009i −0.893427 0.591243i
\(982\) 0 0
\(983\) 27.7025i 0.0281815i −0.999901 0.0140908i \(-0.995515\pi\)
0.999901 0.0140908i \(-0.00448538\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 175.901 584.539i 0.178218 0.592238i
\(988\) 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
\(993\) −871.226 262.171i −0.877367 0.264019i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1217.62 −1.22128 −0.610640 0.791909i \(-0.709088\pi\)
−0.610640 + 0.791909i \(0.709088\pi\)
\(998\) 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.l.f.401.8 8
3.2 odd 2 inner 600.3.l.f.401.7 8
4.3 odd 2 1200.3.l.x.401.1 8
5.2 odd 4 600.3.c.d.449.11 16
5.3 odd 4 600.3.c.d.449.6 16
5.4 even 2 120.3.l.a.41.1 8
12.11 even 2 1200.3.l.x.401.2 8
15.2 even 4 600.3.c.d.449.5 16
15.8 even 4 600.3.c.d.449.12 16
15.14 odd 2 120.3.l.a.41.2 yes 8
20.3 even 4 1200.3.c.m.449.11 16
20.7 even 4 1200.3.c.m.449.6 16
20.19 odd 2 240.3.l.d.161.8 8
40.19 odd 2 960.3.l.g.641.1 8
40.29 even 2 960.3.l.h.641.8 8
60.23 odd 4 1200.3.c.m.449.5 16
60.47 odd 4 1200.3.c.m.449.12 16
60.59 even 2 240.3.l.d.161.7 8
120.29 odd 2 960.3.l.h.641.7 8
120.59 even 2 960.3.l.g.641.2 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.3.l.a.41.1 8 5.4 even 2
120.3.l.a.41.2 yes 8 15.14 odd 2
240.3.l.d.161.7 8 60.59 even 2
240.3.l.d.161.8 8 20.19 odd 2
600.3.c.d.449.5 16 15.2 even 4
600.3.c.d.449.6 16 5.3 odd 4
600.3.c.d.449.11 16 5.2 odd 4
600.3.c.d.449.12 16 15.8 even 4
600.3.l.f.401.7 8 3.2 odd 2 inner
600.3.l.f.401.8 8 1.1 even 1 trivial
960.3.l.g.641.1 8 40.19 odd 2
960.3.l.g.641.2 8 120.59 even 2
960.3.l.h.641.7 8 120.29 odd 2
960.3.l.h.641.8 8 40.29 even 2
1200.3.c.m.449.5 16 60.23 odd 4
1200.3.c.m.449.6 16 20.7 even 4
1200.3.c.m.449.11 16 20.3 even 4
1200.3.c.m.449.12 16 60.47 odd 4
1200.3.l.x.401.1 8 4.3 odd 2
1200.3.l.x.401.2 8 12.11 even 2