Properties

Label 126.3.c.a.55.2
Level $126$
Weight $3$
Character 126.55
Analytic conductor $3.433$
Analytic rank $0$
Dimension $4$
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] = [126,3,Mod(55,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.55");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.c (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: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-33})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 32x^{2} + 289 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 55.2
Root \(0.707107 - 4.06202i\) of defining polynomial
Character \(\chi\) \(=\) 126.55
Dual form 126.3.c.a.55.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.41421 q^{2} +2.00000 q^{4} +8.12404i q^{5} +(-4.00000 - 5.74456i) q^{7} -2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +2.00000 q^{4} +8.12404i q^{5} +(-4.00000 - 5.74456i) q^{7} -2.82843 q^{8} -11.4891i q^{10} -12.7279 q^{11} +22.9783i q^{13} +(5.65685 + 8.12404i) q^{14} +4.00000 q^{16} +8.12404i q^{17} +11.4891i q^{19} +16.2481i q^{20} +18.0000 q^{22} -21.2132 q^{23} -41.0000 q^{25} -32.4962i q^{26} +(-8.00000 - 11.4891i) q^{28} +33.9411 q^{29} -5.65685 q^{32} -11.4891i q^{34} +(46.6690 - 32.4962i) q^{35} +16.0000 q^{37} -16.2481i q^{38} -22.9783i q^{40} -56.8683i q^{41} +52.0000 q^{43} -25.4558 q^{44} +30.0000 q^{46} +32.4962i q^{47} +(-17.0000 + 45.9565i) q^{49} +57.9828 q^{50} +45.9565i q^{52} +16.9706 q^{53} -103.402i q^{55} +(11.3137 + 16.2481i) q^{56} -48.0000 q^{58} -32.4962i q^{59} -22.9783i q^{61} +8.00000 q^{64} -186.676 q^{65} -52.0000 q^{67} +16.2481i q^{68} +(-66.0000 + 45.9565i) q^{70} +89.0955 q^{71} +45.9565i q^{73} -22.6274 q^{74} +22.9783i q^{76} +(50.9117 + 73.1163i) q^{77} +104.000 q^{79} +32.4962i q^{80} +80.4239i q^{82} +162.481i q^{83} -66.0000 q^{85} -73.5391 q^{86} +36.0000 q^{88} -73.1163i q^{89} +(132.000 - 91.9130i) q^{91} -42.4264 q^{92} -45.9565i q^{94} -93.3381 q^{95} +91.9130i q^{97} +(24.0416 - 64.9923i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{4} - 16 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{4} - 16 q^{7} + 16 q^{16} + 72 q^{22} - 164 q^{25} - 32 q^{28} + 64 q^{37} + 208 q^{43} + 120 q^{46} - 68 q^{49} - 192 q^{58} + 32 q^{64} - 208 q^{67} - 264 q^{70} + 416 q^{79} - 264 q^{85} + 144 q^{88} + 528 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −0.707107
\(3\) 0 0
\(4\) 2.00000 0.500000
\(5\) 8.12404i 1.62481i 0.583095 + 0.812404i \(0.301841\pi\)
−0.583095 + 0.812404i \(0.698159\pi\)
\(6\) 0 0
\(7\) −4.00000 5.74456i −0.571429 0.820652i
\(8\) −2.82843 −0.353553
\(9\) 0 0
\(10\) 11.4891i 1.14891i
\(11\) −12.7279 −1.15708 −0.578542 0.815653i \(-0.696378\pi\)
−0.578542 + 0.815653i \(0.696378\pi\)
\(12\) 0 0
\(13\) 22.9783i 1.76756i 0.467905 + 0.883779i \(0.345009\pi\)
−0.467905 + 0.883779i \(0.654991\pi\)
\(14\) 5.65685 + 8.12404i 0.404061 + 0.580288i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 8.12404i 0.477885i 0.971034 + 0.238942i \(0.0768007\pi\)
−0.971034 + 0.238942i \(0.923199\pi\)
\(18\) 0 0
\(19\) 11.4891i 0.604691i 0.953198 + 0.302345i \(0.0977696\pi\)
−0.953198 + 0.302345i \(0.902230\pi\)
\(20\) 16.2481i 0.812404i
\(21\) 0 0
\(22\) 18.0000 0.818182
\(23\) −21.2132 −0.922313 −0.461157 0.887319i \(-0.652565\pi\)
−0.461157 + 0.887319i \(0.652565\pi\)
\(24\) 0 0
\(25\) −41.0000 −1.64000
\(26\) 32.4962i 1.24985i
\(27\) 0 0
\(28\) −8.00000 11.4891i −0.285714 0.410326i
\(29\) 33.9411 1.17038 0.585192 0.810895i \(-0.301019\pi\)
0.585192 + 0.810895i \(0.301019\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) −5.65685 −0.176777
\(33\) 0 0
\(34\) 11.4891i 0.337915i
\(35\) 46.6690 32.4962i 1.33340 0.928462i
\(36\) 0 0
\(37\) 16.0000 0.432432 0.216216 0.976346i \(-0.430628\pi\)
0.216216 + 0.976346i \(0.430628\pi\)
\(38\) 16.2481i 0.427581i
\(39\) 0 0
\(40\) 22.9783i 0.574456i
\(41\) 56.8683i 1.38703i −0.720442 0.693515i \(-0.756061\pi\)
0.720442 0.693515i \(-0.243939\pi\)
\(42\) 0 0
\(43\) 52.0000 1.20930 0.604651 0.796490i \(-0.293313\pi\)
0.604651 + 0.796490i \(0.293313\pi\)
\(44\) −25.4558 −0.578542
\(45\) 0 0
\(46\) 30.0000 0.652174
\(47\) 32.4962i 0.691408i 0.938344 + 0.345704i \(0.112360\pi\)
−0.938344 + 0.345704i \(0.887640\pi\)
\(48\) 0 0
\(49\) −17.0000 + 45.9565i −0.346939 + 0.937888i
\(50\) 57.9828 1.15966
\(51\) 0 0
\(52\) 45.9565i 0.883779i
\(53\) 16.9706 0.320199 0.160100 0.987101i \(-0.448818\pi\)
0.160100 + 0.987101i \(0.448818\pi\)
\(54\) 0 0
\(55\) 103.402i 1.88004i
\(56\) 11.3137 + 16.2481i 0.202031 + 0.290144i
\(57\) 0 0
\(58\) −48.0000 −0.827586
\(59\) 32.4962i 0.550782i −0.961332 0.275391i \(-0.911193\pi\)
0.961332 0.275391i \(-0.0888074\pi\)
\(60\) 0 0
\(61\) 22.9783i 0.376693i −0.982103 0.188346i \(-0.939687\pi\)
0.982103 0.188346i \(-0.0603127\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 8.00000 0.125000
\(65\) −186.676 −2.87194
\(66\) 0 0
\(67\) −52.0000 −0.776119 −0.388060 0.921634i \(-0.626855\pi\)
−0.388060 + 0.921634i \(0.626855\pi\)
\(68\) 16.2481i 0.238942i
\(69\) 0 0
\(70\) −66.0000 + 45.9565i −0.942857 + 0.656521i
\(71\) 89.0955 1.25487 0.627433 0.778671i \(-0.284106\pi\)
0.627433 + 0.778671i \(0.284106\pi\)
\(72\) 0 0
\(73\) 45.9565i 0.629541i 0.949168 + 0.314771i \(0.101928\pi\)
−0.949168 + 0.314771i \(0.898072\pi\)
\(74\) −22.6274 −0.305776
\(75\) 0 0
\(76\) 22.9783i 0.302345i
\(77\) 50.9117 + 73.1163i 0.661191 + 0.949563i
\(78\) 0 0
\(79\) 104.000 1.31646 0.658228 0.752819i \(-0.271306\pi\)
0.658228 + 0.752819i \(0.271306\pi\)
\(80\) 32.4962i 0.406202i
\(81\) 0 0
\(82\) 80.4239i 0.980779i
\(83\) 162.481i 1.95760i 0.204819 + 0.978800i \(0.434339\pi\)
−0.204819 + 0.978800i \(0.565661\pi\)
\(84\) 0 0
\(85\) −66.0000 −0.776471
\(86\) −73.5391 −0.855106
\(87\) 0 0
\(88\) 36.0000 0.409091
\(89\) 73.1163i 0.821532i −0.911741 0.410766i \(-0.865261\pi\)
0.911741 0.410766i \(-0.134739\pi\)
\(90\) 0 0
\(91\) 132.000 91.9130i 1.45055 1.01003i
\(92\) −42.4264 −0.461157
\(93\) 0 0
\(94\) 45.9565i 0.488899i
\(95\) −93.3381 −0.982506
\(96\) 0 0
\(97\) 91.9130i 0.947557i 0.880644 + 0.473778i \(0.157110\pi\)
−0.880644 + 0.473778i \(0.842890\pi\)
\(98\) 24.0416 64.9923i 0.245323 0.663187i
\(99\) 0 0
\(100\) −82.0000 −0.820000
\(101\) 121.861i 1.20654i 0.797537 + 0.603270i \(0.206136\pi\)
−0.797537 + 0.603270i \(0.793864\pi\)
\(102\) 0 0
\(103\) 91.9130i 0.892359i −0.894943 0.446180i \(-0.852784\pi\)
0.894943 0.446180i \(-0.147216\pi\)
\(104\) 64.9923i 0.624926i
\(105\) 0 0
\(106\) −24.0000 −0.226415
\(107\) −123.037 −1.14987 −0.574937 0.818197i \(-0.694974\pi\)
−0.574937 + 0.818197i \(0.694974\pi\)
\(108\) 0 0
\(109\) −62.0000 −0.568807 −0.284404 0.958705i \(-0.591796\pi\)
−0.284404 + 0.958705i \(0.591796\pi\)
\(110\) 146.233i 1.32939i
\(111\) 0 0
\(112\) −16.0000 22.9783i −0.142857 0.205163i
\(113\) 118.794 1.05127 0.525637 0.850709i \(-0.323827\pi\)
0.525637 + 0.850709i \(0.323827\pi\)
\(114\) 0 0
\(115\) 172.337i 1.49858i
\(116\) 67.8823 0.585192
\(117\) 0 0
\(118\) 45.9565i 0.389462i
\(119\) 46.6690 32.4962i 0.392177 0.273077i
\(120\) 0 0
\(121\) 41.0000 0.338843
\(122\) 32.4962i 0.266362i
\(123\) 0 0
\(124\) 0 0
\(125\) 129.985i 1.03988i
\(126\) 0 0
\(127\) −64.0000 −0.503937 −0.251969 0.967735i \(-0.581078\pi\)
−0.251969 + 0.967735i \(0.581078\pi\)
\(128\) −11.3137 −0.0883883
\(129\) 0 0
\(130\) 264.000 2.03077
\(131\) 97.4885i 0.744187i −0.928195 0.372093i \(-0.878640\pi\)
0.928195 0.372093i \(-0.121360\pi\)
\(132\) 0 0
\(133\) 66.0000 45.9565i 0.496241 0.345538i
\(134\) 73.5391 0.548799
\(135\) 0 0
\(136\) 22.9783i 0.168958i
\(137\) −33.9411 −0.247745 −0.123873 0.992298i \(-0.539531\pi\)
−0.123873 + 0.992298i \(0.539531\pi\)
\(138\) 0 0
\(139\) 183.826i 1.32249i 0.750170 + 0.661245i \(0.229971\pi\)
−0.750170 + 0.661245i \(0.770029\pi\)
\(140\) 93.3381 64.9923i 0.666701 0.464231i
\(141\) 0 0
\(142\) −126.000 −0.887324
\(143\) 292.465i 2.04521i
\(144\) 0 0
\(145\) 275.739i 1.90165i
\(146\) 64.9923i 0.445153i
\(147\) 0 0
\(148\) 32.0000 0.216216
\(149\) 186.676 1.25286 0.626430 0.779478i \(-0.284515\pi\)
0.626430 + 0.779478i \(0.284515\pi\)
\(150\) 0 0
\(151\) −32.0000 −0.211921 −0.105960 0.994370i \(-0.533792\pi\)
−0.105960 + 0.994370i \(0.533792\pi\)
\(152\) 32.4962i 0.213790i
\(153\) 0 0
\(154\) −72.0000 103.402i −0.467532 0.671442i
\(155\) 0 0
\(156\) 0 0
\(157\) 68.9348i 0.439075i 0.975604 + 0.219537i \(0.0704548\pi\)
−0.975604 + 0.219537i \(0.929545\pi\)
\(158\) −147.078 −0.930875
\(159\) 0 0
\(160\) 45.9565i 0.287228i
\(161\) 84.8528 + 121.861i 0.527036 + 0.756898i
\(162\) 0 0
\(163\) −124.000 −0.760736 −0.380368 0.924835i \(-0.624203\pi\)
−0.380368 + 0.924835i \(0.624203\pi\)
\(164\) 113.737i 0.693515i
\(165\) 0 0
\(166\) 229.783i 1.38423i
\(167\) 129.985i 0.778351i 0.921164 + 0.389175i \(0.127240\pi\)
−0.921164 + 0.389175i \(0.872760\pi\)
\(168\) 0 0
\(169\) −359.000 −2.12426
\(170\) 93.3381 0.549048
\(171\) 0 0
\(172\) 104.000 0.604651
\(173\) 73.1163i 0.422638i −0.977417 0.211319i \(-0.932224\pi\)
0.977417 0.211319i \(-0.0677759\pi\)
\(174\) 0 0
\(175\) 164.000 + 235.527i 0.937143 + 1.34587i
\(176\) −50.9117 −0.289271
\(177\) 0 0
\(178\) 103.402i 0.580911i
\(179\) 156.978 0.876970 0.438485 0.898738i \(-0.355515\pi\)
0.438485 + 0.898738i \(0.355515\pi\)
\(180\) 0 0
\(181\) 252.761i 1.39647i −0.715869 0.698234i \(-0.753969\pi\)
0.715869 0.698234i \(-0.246031\pi\)
\(182\) −186.676 + 129.985i −1.02569 + 0.714201i
\(183\) 0 0
\(184\) 60.0000 0.326087
\(185\) 129.985i 0.702620i
\(186\) 0 0
\(187\) 103.402i 0.552953i
\(188\) 64.9923i 0.345704i
\(189\) 0 0
\(190\) 132.000 0.694737
\(191\) 12.7279 0.0666383 0.0333192 0.999445i \(-0.489392\pi\)
0.0333192 + 0.999445i \(0.489392\pi\)
\(192\) 0 0
\(193\) −224.000 −1.16062 −0.580311 0.814395i \(-0.697069\pi\)
−0.580311 + 0.814395i \(0.697069\pi\)
\(194\) 129.985i 0.670024i
\(195\) 0 0
\(196\) −34.0000 + 91.9130i −0.173469 + 0.468944i
\(197\) −305.470 −1.55061 −0.775305 0.631587i \(-0.782404\pi\)
−0.775305 + 0.631587i \(0.782404\pi\)
\(198\) 0 0
\(199\) 264.250i 1.32789i 0.747782 + 0.663944i \(0.231119\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(200\) 115.966 0.579828
\(201\) 0 0
\(202\) 172.337i 0.853153i
\(203\) −135.765 194.977i −0.668791 0.960477i
\(204\) 0 0
\(205\) 462.000 2.25366
\(206\) 129.985i 0.630993i
\(207\) 0 0
\(208\) 91.9130i 0.441889i
\(209\) 146.233i 0.699678i
\(210\) 0 0
\(211\) 124.000 0.587678 0.293839 0.955855i \(-0.405067\pi\)
0.293839 + 0.955855i \(0.405067\pi\)
\(212\) 33.9411 0.160100
\(213\) 0 0
\(214\) 174.000 0.813084
\(215\) 422.450i 1.96488i
\(216\) 0 0
\(217\) 0 0
\(218\) 87.6812 0.402208
\(219\) 0 0
\(220\) 206.804i 0.940019i
\(221\) −186.676 −0.844689
\(222\) 0 0
\(223\) 172.337i 0.772811i −0.922329 0.386406i \(-0.873717\pi\)
0.922329 0.386406i \(-0.126283\pi\)
\(224\) 22.6274 + 32.4962i 0.101015 + 0.145072i
\(225\) 0 0
\(226\) −168.000 −0.743363
\(227\) 389.954i 1.71786i 0.512094 + 0.858929i \(0.328870\pi\)
−0.512094 + 0.858929i \(0.671130\pi\)
\(228\) 0 0
\(229\) 206.804i 0.903075i −0.892252 0.451538i \(-0.850876\pi\)
0.892252 0.451538i \(-0.149124\pi\)
\(230\) 243.721i 1.05966i
\(231\) 0 0
\(232\) −96.0000 −0.413793
\(233\) 186.676 0.801185 0.400593 0.916256i \(-0.368804\pi\)
0.400593 + 0.916256i \(0.368804\pi\)
\(234\) 0 0
\(235\) −264.000 −1.12340
\(236\) 64.9923i 0.275391i
\(237\) 0 0
\(238\) −66.0000 + 45.9565i −0.277311 + 0.193095i
\(239\) −258.801 −1.08285 −0.541425 0.840749i \(-0.682115\pi\)
−0.541425 + 0.840749i \(0.682115\pi\)
\(240\) 0 0
\(241\) 45.9565i 0.190691i 0.995444 + 0.0953454i \(0.0303956\pi\)
−0.995444 + 0.0953454i \(0.969604\pi\)
\(242\) −57.9828 −0.239598
\(243\) 0 0
\(244\) 45.9565i 0.188346i
\(245\) −373.352 138.109i −1.52389 0.563709i
\(246\) 0 0
\(247\) −264.000 −1.06883
\(248\) 0 0
\(249\) 0 0
\(250\) 183.826i 0.735304i
\(251\) 64.9923i 0.258933i −0.991584 0.129467i \(-0.958673\pi\)
0.991584 0.129467i \(-0.0413265\pi\)
\(252\) 0 0
\(253\) 270.000 1.06719
\(254\) 90.5097 0.356337
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 268.093i 1.04316i 0.853201 + 0.521582i \(0.174658\pi\)
−0.853201 + 0.521582i \(0.825342\pi\)
\(258\) 0 0
\(259\) −64.0000 91.9130i −0.247104 0.354876i
\(260\) −373.352 −1.43597
\(261\) 0 0
\(262\) 137.870i 0.526219i
\(263\) 428.507 1.62930 0.814652 0.579951i \(-0.196928\pi\)
0.814652 + 0.579951i \(0.196928\pi\)
\(264\) 0 0
\(265\) 137.870i 0.520262i
\(266\) −93.3381 + 64.9923i −0.350895 + 0.244332i
\(267\) 0 0
\(268\) −104.000 −0.388060
\(269\) 203.101i 0.755022i −0.926005 0.377511i \(-0.876780\pi\)
0.926005 0.377511i \(-0.123220\pi\)
\(270\) 0 0
\(271\) 91.9130i 0.339162i 0.985516 + 0.169581i \(0.0542415\pi\)
−0.985516 + 0.169581i \(0.945759\pi\)
\(272\) 32.4962i 0.119471i
\(273\) 0 0
\(274\) 48.0000 0.175182
\(275\) 521.845 1.89762
\(276\) 0 0
\(277\) 272.000 0.981949 0.490975 0.871174i \(-0.336641\pi\)
0.490975 + 0.871174i \(0.336641\pi\)
\(278\) 259.969i 0.935141i
\(279\) 0 0
\(280\) −132.000 + 91.9130i −0.471429 + 0.328261i
\(281\) 50.9117 0.181180 0.0905902 0.995888i \(-0.471125\pi\)
0.0905902 + 0.995888i \(0.471125\pi\)
\(282\) 0 0
\(283\) 103.402i 0.365379i −0.983171 0.182689i \(-0.941520\pi\)
0.983171 0.182689i \(-0.0584802\pi\)
\(284\) 178.191 0.627433
\(285\) 0 0
\(286\) 413.609i 1.44618i
\(287\) −326.683 + 227.473i −1.13827 + 0.792589i
\(288\) 0 0
\(289\) 223.000 0.771626
\(290\) 389.954i 1.34467i
\(291\) 0 0
\(292\) 91.9130i 0.314771i
\(293\) 8.12404i 0.0277271i −0.999904 0.0138635i \(-0.995587\pi\)
0.999904 0.0138635i \(-0.00441305\pi\)
\(294\) 0 0
\(295\) 264.000 0.894915
\(296\) −45.2548 −0.152888
\(297\) 0 0
\(298\) −264.000 −0.885906
\(299\) 487.442i 1.63024i
\(300\) 0 0
\(301\) −208.000 298.717i −0.691030 0.992416i
\(302\) 45.2548 0.149850
\(303\) 0 0
\(304\) 45.9565i 0.151173i
\(305\) 186.676 0.612053
\(306\) 0 0
\(307\) 379.141i 1.23499i 0.786576 + 0.617494i \(0.211852\pi\)
−0.786576 + 0.617494i \(0.788148\pi\)
\(308\) 101.823 + 146.233i 0.330595 + 0.474781i
\(309\) 0 0
\(310\) 0 0
\(311\) 357.458i 1.14938i −0.818371 0.574691i \(-0.805122\pi\)
0.818371 0.574691i \(-0.194878\pi\)
\(312\) 0 0
\(313\) 183.826i 0.587304i −0.955912 0.293652i \(-0.905129\pi\)
0.955912 0.293652i \(-0.0948706\pi\)
\(314\) 97.4885i 0.310473i
\(315\) 0 0
\(316\) 208.000 0.658228
\(317\) 356.382 1.12423 0.562116 0.827058i \(-0.309987\pi\)
0.562116 + 0.827058i \(0.309987\pi\)
\(318\) 0 0
\(319\) −432.000 −1.35423
\(320\) 64.9923i 0.203101i
\(321\) 0 0
\(322\) −120.000 172.337i −0.372671 0.535208i
\(323\) −93.3381 −0.288972
\(324\) 0 0
\(325\) 942.108i 2.89879i
\(326\) 175.362 0.537922
\(327\) 0 0
\(328\) 160.848i 0.490389i
\(329\) 186.676 129.985i 0.567405 0.395090i
\(330\) 0 0
\(331\) 4.00000 0.0120846 0.00604230 0.999982i \(-0.498077\pi\)
0.00604230 + 0.999982i \(0.498077\pi\)
\(332\) 324.962i 0.978800i
\(333\) 0 0
\(334\) 183.826i 0.550377i
\(335\) 422.450i 1.26104i
\(336\) 0 0
\(337\) 82.0000 0.243323 0.121662 0.992572i \(-0.461178\pi\)
0.121662 + 0.992572i \(0.461178\pi\)
\(338\) 507.703 1.50208
\(339\) 0 0
\(340\) −132.000 −0.388235
\(341\) 0 0
\(342\) 0 0
\(343\) 332.000 86.1684i 0.967930 0.251220i
\(344\) −147.078 −0.427553
\(345\) 0 0
\(346\) 103.402i 0.298850i
\(347\) −555.786 −1.60169 −0.800844 0.598873i \(-0.795616\pi\)
−0.800844 + 0.598873i \(0.795616\pi\)
\(348\) 0 0
\(349\) 160.848i 0.460882i −0.973086 0.230441i \(-0.925983\pi\)
0.973086 0.230441i \(-0.0740168\pi\)
\(350\) −231.931 333.086i −0.662660 0.951673i
\(351\) 0 0
\(352\) 72.0000 0.204545
\(353\) 138.109i 0.391243i −0.980680 0.195621i \(-0.937328\pi\)
0.980680 0.195621i \(-0.0626723\pi\)
\(354\) 0 0
\(355\) 723.815i 2.03892i
\(356\) 146.233i 0.410766i
\(357\) 0 0
\(358\) −222.000 −0.620112
\(359\) 224.860 0.626351 0.313175 0.949695i \(-0.398607\pi\)
0.313175 + 0.949695i \(0.398607\pi\)
\(360\) 0 0
\(361\) 229.000 0.634349
\(362\) 357.458i 0.987452i
\(363\) 0 0
\(364\) 264.000 183.826i 0.725275 0.505016i
\(365\) −373.352 −1.02288
\(366\) 0 0
\(367\) 195.315i 0.532194i 0.963946 + 0.266097i \(0.0857341\pi\)
−0.963946 + 0.266097i \(0.914266\pi\)
\(368\) −84.8528 −0.230578
\(369\) 0 0
\(370\) 183.826i 0.496827i
\(371\) −67.8823 97.4885i −0.182971 0.262772i
\(372\) 0 0
\(373\) 530.000 1.42091 0.710456 0.703742i \(-0.248489\pi\)
0.710456 + 0.703742i \(0.248489\pi\)
\(374\) 146.233i 0.390997i
\(375\) 0 0
\(376\) 91.9130i 0.244449i
\(377\) 779.908i 2.06872i
\(378\) 0 0
\(379\) 196.000 0.517150 0.258575 0.965991i \(-0.416747\pi\)
0.258575 + 0.965991i \(0.416747\pi\)
\(380\) −186.676 −0.491253
\(381\) 0 0
\(382\) −18.0000 −0.0471204
\(383\) 32.4962i 0.0848464i −0.999100 0.0424232i \(-0.986492\pi\)
0.999100 0.0424232i \(-0.0135078\pi\)
\(384\) 0 0
\(385\) −594.000 + 413.609i −1.54286 + 1.07431i
\(386\) 316.784 0.820684
\(387\) 0 0
\(388\) 183.826i 0.473778i
\(389\) 644.881 1.65779 0.828896 0.559402i \(-0.188969\pi\)
0.828896 + 0.559402i \(0.188969\pi\)
\(390\) 0 0
\(391\) 172.337i 0.440759i
\(392\) 48.0833 129.985i 0.122661 0.331593i
\(393\) 0 0
\(394\) 432.000 1.09645
\(395\) 844.900i 2.13899i
\(396\) 0 0
\(397\) 758.282i 1.91003i −0.296556 0.955015i \(-0.595838\pi\)
0.296556 0.955015i \(-0.404162\pi\)
\(398\) 373.706i 0.938959i
\(399\) 0 0
\(400\) −164.000 −0.410000
\(401\) 33.9411 0.0846412 0.0423206 0.999104i \(-0.486525\pi\)
0.0423206 + 0.999104i \(0.486525\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 243.721i 0.603270i
\(405\) 0 0
\(406\) 192.000 + 275.739i 0.472906 + 0.679160i
\(407\) −203.647 −0.500361
\(408\) 0 0
\(409\) 735.304i 1.79781i 0.438144 + 0.898905i \(0.355636\pi\)
−0.438144 + 0.898905i \(0.644364\pi\)
\(410\) −653.367 −1.59358
\(411\) 0 0
\(412\) 183.826i 0.446180i
\(413\) −186.676 + 129.985i −0.452000 + 0.314733i
\(414\) 0 0
\(415\) −1320.00 −3.18072
\(416\) 129.985i 0.312463i
\(417\) 0 0
\(418\) 206.804i 0.494747i
\(419\) 32.4962i 0.0775565i 0.999248 + 0.0387782i \(0.0123466\pi\)
−0.999248 + 0.0387782i \(0.987653\pi\)
\(420\) 0 0
\(421\) −496.000 −1.17815 −0.589074 0.808079i \(-0.700507\pi\)
−0.589074 + 0.808079i \(0.700507\pi\)
\(422\) −175.362 −0.415551
\(423\) 0 0
\(424\) −48.0000 −0.113208
\(425\) 333.086i 0.783731i
\(426\) 0 0
\(427\) −132.000 + 91.9130i −0.309133 + 0.215253i
\(428\) −246.073 −0.574937
\(429\) 0 0
\(430\) 597.435i 1.38938i
\(431\) −462.448 −1.07296 −0.536482 0.843912i \(-0.680247\pi\)
−0.536482 + 0.843912i \(0.680247\pi\)
\(432\) 0 0
\(433\) 137.870i 0.318405i 0.987246 + 0.159203i \(0.0508923\pi\)
−0.987246 + 0.159203i \(0.949108\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −124.000 −0.284404
\(437\) 243.721i 0.557714i
\(438\) 0 0
\(439\) 172.337i 0.392567i 0.980547 + 0.196283i \(0.0628873\pi\)
−0.980547 + 0.196283i \(0.937113\pi\)
\(440\) 292.465i 0.664694i
\(441\) 0 0
\(442\) 264.000 0.597285
\(443\) −394.566 −0.890667 −0.445334 0.895365i \(-0.646915\pi\)
−0.445334 + 0.895365i \(0.646915\pi\)
\(444\) 0 0
\(445\) 594.000 1.33483
\(446\) 243.721i 0.546460i
\(447\) 0 0
\(448\) −32.0000 45.9565i −0.0714286 0.102581i
\(449\) −50.9117 −0.113389 −0.0566945 0.998392i \(-0.518056\pi\)
−0.0566945 + 0.998392i \(0.518056\pi\)
\(450\) 0 0
\(451\) 723.815i 1.60491i
\(452\) 237.588 0.525637
\(453\) 0 0
\(454\) 551.478i 1.21471i
\(455\) 746.705 + 1072.37i 1.64111 + 2.35686i
\(456\) 0 0
\(457\) 130.000 0.284464 0.142232 0.989833i \(-0.454572\pi\)
0.142232 + 0.989833i \(0.454572\pi\)
\(458\) 292.465i 0.638571i
\(459\) 0 0
\(460\) 344.674i 0.749291i
\(461\) 723.039i 1.56842i 0.620499 + 0.784208i \(0.286930\pi\)
−0.620499 + 0.784208i \(0.713070\pi\)
\(462\) 0 0
\(463\) 80.0000 0.172786 0.0863931 0.996261i \(-0.472466\pi\)
0.0863931 + 0.996261i \(0.472466\pi\)
\(464\) 135.765 0.292596
\(465\) 0 0
\(466\) −264.000 −0.566524
\(467\) 194.977i 0.417509i −0.977968 0.208755i \(-0.933059\pi\)
0.977968 0.208755i \(-0.0669410\pi\)
\(468\) 0 0
\(469\) 208.000 + 298.717i 0.443497 + 0.636924i
\(470\) 373.352 0.794367
\(471\) 0 0
\(472\) 91.9130i 0.194731i
\(473\) −661.852 −1.39926
\(474\) 0 0
\(475\) 471.054i 0.991693i
\(476\) 93.3381 64.9923i 0.196088 0.136538i
\(477\) 0 0
\(478\) 366.000 0.765690
\(479\) 422.450i 0.881942i −0.897521 0.440971i \(-0.854634\pi\)
0.897521 0.440971i \(-0.145366\pi\)
\(480\) 0 0
\(481\) 367.652i 0.764349i
\(482\) 64.9923i 0.134839i
\(483\) 0 0
\(484\) 82.0000 0.169421
\(485\) −746.705 −1.53960
\(486\) 0 0
\(487\) 904.000 1.85626 0.928131 0.372253i \(-0.121415\pi\)
0.928131 + 0.372253i \(0.121415\pi\)
\(488\) 64.9923i 0.133181i
\(489\) 0 0
\(490\) 528.000 + 195.315i 1.07755 + 0.398602i
\(491\) 420.021 0.855441 0.427720 0.903911i \(-0.359317\pi\)
0.427720 + 0.903911i \(0.359317\pi\)
\(492\) 0 0
\(493\) 275.739i 0.559308i
\(494\) 373.352 0.755774
\(495\) 0 0
\(496\) 0 0
\(497\) −356.382 511.814i −0.717066 1.02981i
\(498\) 0 0
\(499\) −716.000 −1.43487 −0.717435 0.696626i \(-0.754684\pi\)
−0.717435 + 0.696626i \(0.754684\pi\)
\(500\) 259.969i 0.519938i
\(501\) 0 0
\(502\) 91.9130i 0.183094i
\(503\) 812.404i 1.61512i −0.589787 0.807558i \(-0.700788\pi\)
0.589787 0.807558i \(-0.299212\pi\)
\(504\) 0 0
\(505\) −990.000 −1.96040
\(506\) −381.838 −0.754620
\(507\) 0 0
\(508\) −128.000 −0.251969
\(509\) 186.853i 0.367098i 0.983011 + 0.183549i \(0.0587586\pi\)
−0.983011 + 0.183549i \(0.941241\pi\)
\(510\) 0 0
\(511\) 264.000 183.826i 0.516634 0.359738i
\(512\) −22.6274 −0.0441942
\(513\) 0 0
\(514\) 379.141i 0.737629i
\(515\) 746.705 1.44991
\(516\) 0 0
\(517\) 413.609i 0.800016i
\(518\) 90.5097 + 129.985i 0.174729 + 0.250936i
\(519\) 0 0
\(520\) 528.000 1.01538
\(521\) 593.055i 1.13830i −0.822233 0.569150i \(-0.807272\pi\)
0.822233 0.569150i \(-0.192728\pi\)
\(522\) 0 0
\(523\) 735.304i 1.40594i −0.711222 0.702968i \(-0.751858\pi\)
0.711222 0.702968i \(-0.248142\pi\)
\(524\) 194.977i 0.372093i
\(525\) 0 0
\(526\) −606.000 −1.15209
\(527\) 0 0
\(528\) 0 0
\(529\) −79.0000 −0.149338
\(530\) 194.977i 0.367881i
\(531\) 0 0
\(532\) 132.000 91.9130i 0.248120 0.172769i
\(533\) 1306.73 2.45166
\(534\) 0 0
\(535\) 999.554i 1.86833i
\(536\) 147.078 0.274400
\(537\) 0 0
\(538\) 287.228i 0.533881i
\(539\) 216.375 584.931i 0.401437 1.08521i
\(540\) 0 0
\(541\) 272.000 0.502773 0.251386 0.967887i \(-0.419114\pi\)
0.251386 + 0.967887i \(0.419114\pi\)
\(542\) 129.985i 0.239824i
\(543\) 0 0
\(544\) 45.9565i 0.0844789i
\(545\) 503.690i 0.924203i
\(546\) 0 0
\(547\) 644.000 1.17733 0.588665 0.808377i \(-0.299654\pi\)
0.588665 + 0.808377i \(0.299654\pi\)
\(548\) −67.8823 −0.123873
\(549\) 0 0
\(550\) −738.000 −1.34182
\(551\) 389.954i 0.707720i
\(552\) 0 0
\(553\) −416.000 597.435i −0.752260 1.08035i
\(554\) −384.666 −0.694343
\(555\) 0 0
\(556\) 367.652i 0.661245i
\(557\) 152.735 0.274210 0.137105 0.990557i \(-0.456220\pi\)
0.137105 + 0.990557i \(0.456220\pi\)
\(558\) 0 0
\(559\) 1194.87i 2.13751i
\(560\) 186.676 129.985i 0.333350 0.232115i
\(561\) 0 0
\(562\) −72.0000 −0.128114
\(563\) 877.396i 1.55843i 0.626757 + 0.779215i \(0.284382\pi\)
−0.626757 + 0.779215i \(0.715618\pi\)
\(564\) 0 0
\(565\) 965.087i 1.70812i
\(566\) 146.233i 0.258362i
\(567\) 0 0
\(568\) −252.000 −0.443662
\(569\) −933.381 −1.64039 −0.820194 0.572085i \(-0.806135\pi\)
−0.820194 + 0.572085i \(0.806135\pi\)
\(570\) 0 0
\(571\) 260.000 0.455342 0.227671 0.973738i \(-0.426889\pi\)
0.227671 + 0.973738i \(0.426889\pi\)
\(572\) 584.931i 1.02261i
\(573\) 0 0
\(574\) 462.000 321.696i 0.804878 0.560445i
\(575\) 869.741 1.51259
\(576\) 0 0
\(577\) 505.522i 0.876120i −0.898946 0.438060i \(-0.855666\pi\)
0.898946 0.438060i \(-0.144334\pi\)
\(578\) −315.370 −0.545622
\(579\) 0 0
\(580\) 551.478i 0.950824i
\(581\) 933.381 649.923i 1.60651 1.11863i
\(582\) 0 0
\(583\) −216.000 −0.370497
\(584\) 129.985i 0.222576i
\(585\) 0 0
\(586\) 11.4891i 0.0196060i
\(587\) 194.977i 0.332158i −0.986112 0.166079i \(-0.946889\pi\)
0.986112 0.166079i \(-0.0531107\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −373.352 −0.632801
\(591\) 0 0
\(592\) 64.0000 0.108108
\(593\) 138.109i 0.232898i 0.993197 + 0.116449i \(0.0371512\pi\)
−0.993197 + 0.116449i \(0.962849\pi\)
\(594\) 0 0
\(595\) 264.000 + 379.141i 0.443697 + 0.637212i
\(596\) 373.352 0.626430
\(597\) 0 0
\(598\) 689.348i 1.15276i
\(599\) −453.963 −0.757867 −0.378934 0.925424i \(-0.623709\pi\)
−0.378934 + 0.925424i \(0.623709\pi\)
\(600\) 0 0
\(601\) 827.217i 1.37640i 0.725521 + 0.688201i \(0.241599\pi\)
−0.725521 + 0.688201i \(0.758401\pi\)
\(602\) 294.156 + 422.450i 0.488632 + 0.701744i
\(603\) 0 0
\(604\) −64.0000 −0.105960
\(605\) 333.086i 0.550555i
\(606\) 0 0
\(607\) 356.163i 0.586759i −0.955996 0.293380i \(-0.905220\pi\)
0.955996 0.293380i \(-0.0947800\pi\)
\(608\) 64.9923i 0.106895i
\(609\) 0 0
\(610\) −264.000 −0.432787
\(611\) −746.705 −1.22210
\(612\) 0 0
\(613\) 878.000 1.43230 0.716150 0.697946i \(-0.245903\pi\)
0.716150 + 0.697946i \(0.245903\pi\)
\(614\) 536.187i 0.873268i
\(615\) 0 0
\(616\) −144.000 206.804i −0.233766 0.335721i
\(617\) −101.823 −0.165030 −0.0825149 0.996590i \(-0.526295\pi\)
−0.0825149 + 0.996590i \(0.526295\pi\)
\(618\) 0 0
\(619\) 735.304i 1.18789i −0.804506 0.593945i \(-0.797570\pi\)
0.804506 0.593945i \(-0.202430\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 505.522i 0.812736i
\(623\) −420.021 + 292.465i −0.674192 + 0.469447i
\(624\) 0 0
\(625\) 31.0000 0.0496000
\(626\) 259.969i 0.415286i
\(627\) 0 0
\(628\) 137.870i 0.219537i
\(629\) 129.985i 0.206653i
\(630\) 0 0
\(631\) −16.0000 −0.0253566 −0.0126783 0.999920i \(-0.504036\pi\)
−0.0126783 + 0.999920i \(0.504036\pi\)
\(632\) −294.156 −0.465437
\(633\) 0 0
\(634\) −504.000 −0.794953
\(635\) 519.938i 0.818801i
\(636\) 0 0
\(637\) −1056.00 390.630i −1.65777 0.613234i
\(638\) 610.940 0.957587
\(639\) 0 0
\(640\) 91.9130i 0.143614i
\(641\) 67.8823 0.105901 0.0529503 0.998597i \(-0.483138\pi\)
0.0529503 + 0.998597i \(0.483138\pi\)
\(642\) 0 0
\(643\) 930.619i 1.44731i −0.690163 0.723654i \(-0.742461\pi\)
0.690163 0.723654i \(-0.257539\pi\)
\(644\) 169.706 + 243.721i 0.263518 + 0.378449i
\(645\) 0 0
\(646\) 132.000 0.204334
\(647\) 584.931i 0.904066i −0.892001 0.452033i \(-0.850699\pi\)
0.892001 0.452033i \(-0.149301\pi\)
\(648\) 0 0
\(649\) 413.609i 0.637301i
\(650\) 1332.34i 2.04976i
\(651\) 0 0
\(652\) −248.000 −0.380368
\(653\) −84.8528 −0.129943 −0.0649715 0.997887i \(-0.520696\pi\)
−0.0649715 + 0.997887i \(0.520696\pi\)
\(654\) 0 0
\(655\) 792.000 1.20916
\(656\) 227.473i 0.346758i
\(657\) 0 0
\(658\) −264.000 + 183.826i −0.401216 + 0.279371i
\(659\) −700.036 −1.06227 −0.531135 0.847287i \(-0.678234\pi\)
−0.531135 + 0.847287i \(0.678234\pi\)
\(660\) 0 0
\(661\) 114.891i 0.173814i −0.996216 0.0869072i \(-0.972302\pi\)
0.996216 0.0869072i \(-0.0276983\pi\)
\(662\) −5.65685 −0.00854510
\(663\) 0 0
\(664\) 459.565i 0.692116i
\(665\) 373.352 + 536.187i 0.561432 + 0.806296i
\(666\) 0 0
\(667\) −720.000 −1.07946
\(668\) 259.969i 0.389175i
\(669\) 0 0
\(670\) 597.435i 0.891693i
\(671\) 292.465i 0.435865i
\(672\) 0 0
\(673\) −368.000 −0.546805 −0.273403 0.961900i \(-0.588149\pi\)
−0.273403 + 0.961900i \(0.588149\pi\)
\(674\) −115.966 −0.172056
\(675\) 0 0
\(676\) −718.000 −1.06213
\(677\) 1048.00i 1.54801i 0.633181 + 0.774004i \(0.281749\pi\)
−0.633181 + 0.774004i \(0.718251\pi\)
\(678\) 0 0
\(679\) 528.000 367.652i 0.777614 0.541461i
\(680\) 186.676 0.274524
\(681\) 0 0
\(682\) 0 0
\(683\) −258.801 −0.378918 −0.189459 0.981889i \(-0.560673\pi\)
−0.189459 + 0.981889i \(0.560673\pi\)
\(684\) 0 0
\(685\) 275.739i 0.402539i
\(686\) −469.519 + 121.861i −0.684430 + 0.177639i
\(687\) 0 0
\(688\) 208.000 0.302326
\(689\) 389.954i 0.565971i
\(690\) 0 0
\(691\) 1102.96i 1.59617i −0.602542 0.798087i \(-0.705846\pi\)
0.602542 0.798087i \(-0.294154\pi\)
\(692\) 146.233i 0.211319i
\(693\) 0 0
\(694\) 786.000 1.13256
\(695\) −1493.41 −2.14879
\(696\) 0 0
\(697\) 462.000 0.662841
\(698\) 227.473i 0.325893i
\(699\) 0 0
\(700\) 328.000 + 471.054i 0.468571 + 0.672934i
\(701\) 390.323 0.556809 0.278404 0.960464i \(-0.410194\pi\)
0.278404 + 0.960464i \(0.410194\pi\)
\(702\) 0 0
\(703\) 183.826i 0.261488i
\(704\) −101.823 −0.144635
\(705\) 0 0
\(706\) 195.315i 0.276650i
\(707\) 700.036 487.442i 0.990150 0.689452i
\(708\) 0 0
\(709\) 238.000 0.335684 0.167842 0.985814i \(-0.446320\pi\)
0.167842 + 0.985814i \(0.446320\pi\)
\(710\) 1023.63i 1.44173i
\(711\) 0 0
\(712\) 206.804i 0.290455i
\(713\) 0 0
\(714\) 0 0
\(715\) 2376.00 3.32308
\(716\) 313.955 0.438485
\(717\) 0 0
\(718\) −318.000 −0.442897
\(719\) 1137.37i 1.58187i 0.611899 + 0.790936i \(0.290406\pi\)
−0.611899 + 0.790936i \(0.709594\pi\)
\(720\) 0 0
\(721\) −528.000 + 367.652i −0.732316 + 0.509920i
\(722\) −323.855 −0.448553
\(723\) 0 0
\(724\) 505.522i 0.698234i
\(725\) −1391.59 −1.91943
\(726\) 0 0
\(727\) 275.739i 0.379283i −0.981853 0.189642i \(-0.939267\pi\)
0.981853 0.189642i \(-0.0607326\pi\)
\(728\) −373.352 + 259.969i −0.512847 + 0.357101i
\(729\) 0 0
\(730\) 528.000 0.723288
\(731\) 422.450i 0.577907i
\(732\) 0 0
\(733\) 206.804i 0.282134i −0.990000 0.141067i \(-0.954947\pi\)
0.990000 0.141067i \(-0.0450533\pi\)
\(734\) 276.217i 0.376318i
\(735\) 0 0
\(736\) 120.000 0.163043
\(737\) 661.852 0.898035
\(738\) 0 0
\(739\) −1348.00 −1.82409 −0.912043 0.410094i \(-0.865496\pi\)
−0.912043 + 0.410094i \(0.865496\pi\)
\(740\) 259.969i 0.351310i
\(741\) 0 0
\(742\) 96.0000 + 137.870i 0.129380 + 0.185808i
\(743\) 182.434 0.245536 0.122768 0.992435i \(-0.460823\pi\)
0.122768 + 0.992435i \(0.460823\pi\)
\(744\) 0 0
\(745\) 1516.56i 2.03566i
\(746\) −749.533 −1.00474
\(747\) 0 0
\(748\) 206.804i 0.276476i
\(749\) 492.146 + 706.791i 0.657071 + 0.943647i
\(750\) 0 0
\(751\) 232.000 0.308921 0.154461 0.987999i \(-0.450636\pi\)
0.154461 + 0.987999i \(0.450636\pi\)
\(752\) 129.985i 0.172852i
\(753\) 0 0
\(754\) 1102.96i 1.46281i
\(755\) 259.969i 0.344330i
\(756\) 0 0
\(757\) −434.000 −0.573316 −0.286658 0.958033i \(-0.592544\pi\)
−0.286658 + 0.958033i \(0.592544\pi\)
\(758\) −277.186 −0.365681
\(759\) 0 0
\(760\) 264.000 0.347368
\(761\) 56.8683i 0.0747283i −0.999302 0.0373642i \(-0.988104\pi\)
0.999302 0.0373642i \(-0.0118962\pi\)
\(762\) 0 0
\(763\) 248.000 + 356.163i 0.325033 + 0.466793i
\(764\) 25.4558 0.0333192
\(765\) 0 0
\(766\) 45.9565i 0.0599954i
\(767\) 746.705 0.973539
\(768\) 0 0
\(769\) 505.522i 0.657375i −0.944439 0.328688i \(-0.893394\pi\)
0.944439 0.328688i \(-0.106606\pi\)
\(770\) 840.043 584.931i 1.09096 0.759650i
\(771\) 0 0
\(772\) −448.000 −0.580311
\(773\) 8.12404i 0.0105098i 0.999986 + 0.00525488i \(0.00167269\pi\)
−0.999986 + 0.00525488i \(0.998327\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 259.969i 0.335012i
\(777\) 0 0
\(778\) −912.000 −1.17224
\(779\) 653.367 0.838725
\(780\) 0 0
\(781\) −1134.00 −1.45198
\(782\) 243.721i 0.311664i
\(783\) 0 0
\(784\) −68.0000 + 183.826i −0.0867347 + 0.234472i
\(785\) −560.029 −0.713412
\(786\) 0 0
\(787\) 919.130i 1.16789i 0.811793 + 0.583945i \(0.198492\pi\)
−0.811793 + 0.583945i \(0.801508\pi\)
\(788\) −610.940 −0.775305
\(789\) 0 0
\(790\) 1194.87i 1.51249i
\(791\) −475.176 682.419i −0.600728 0.862730i
\(792\) 0 0
\(793\) 528.000 0.665826
\(794\) 1072.37i 1.35060i
\(795\) 0 0
\(796\) 528.500i 0.663944i
\(797\) 706.791i 0.886815i −0.896320 0.443407i \(-0.853770\pi\)
0.896320 0.443407i \(-0.146230\pi\)
\(798\) 0 0
\(799\) −264.000 −0.330413
\(800\) 231.931 0.289914
\(801\) 0 0
\(802\) −48.0000 −0.0598504
\(803\) 584.931i 0.728432i
\(804\) 0 0
\(805\) −990.000 + 689.348i −1.22981 + 0.856332i
\(806\) 0 0
\(807\) 0 0
\(808\) 344.674i 0.426576i
\(809\) 899.440 1.11179 0.555896 0.831252i \(-0.312375\pi\)
0.555896 + 0.831252i \(0.312375\pi\)
\(810\) 0 0
\(811\) 1102.96i 1.36000i −0.733214 0.679998i \(-0.761981\pi\)
0.733214 0.679998i \(-0.238019\pi\)
\(812\) −271.529 389.954i −0.334395 0.480239i
\(813\) 0 0
\(814\) 288.000 0.353808
\(815\) 1007.38i 1.23605i
\(816\) 0 0
\(817\) 597.435i 0.731254i
\(818\) 1039.88i 1.27124i
\(819\) 0 0
\(820\) 924.000 1.12683
\(821\) −186.676 −0.227377 −0.113688 0.993516i \(-0.536267\pi\)
−0.113688 + 0.993516i \(0.536267\pi\)
\(822\) 0 0
\(823\) −424.000 −0.515188 −0.257594 0.966253i \(-0.582930\pi\)
−0.257594 + 0.966253i \(0.582930\pi\)
\(824\) 259.969i 0.315497i
\(825\) 0 0
\(826\) 264.000 183.826i 0.319613 0.222550i
\(827\) 1073.39 1.29793 0.648965 0.760818i \(-0.275202\pi\)
0.648965 + 0.760818i \(0.275202\pi\)
\(828\) 0 0
\(829\) 850.195i 1.02557i 0.858518 + 0.512784i \(0.171386\pi\)
−0.858518 + 0.512784i \(0.828614\pi\)
\(830\) 1866.76 2.24911
\(831\) 0 0
\(832\) 183.826i 0.220945i
\(833\) −373.352 138.109i −0.448202 0.165797i
\(834\) 0 0
\(835\) −1056.00 −1.26467
\(836\) 292.465i 0.349839i
\(837\) 0 0
\(838\) 45.9565i 0.0548407i
\(839\) 1299.85i 1.54928i 0.632402 + 0.774640i \(0.282069\pi\)
−0.632402 + 0.774640i \(0.717931\pi\)
\(840\) 0 0
\(841\) 311.000 0.369798
\(842\) 701.450 0.833076
\(843\) 0 0
\(844\) 248.000 0.293839
\(845\) 2916.53i 3.45151i
\(846\) 0 0
\(847\) −164.000 235.527i −0.193625 0.278072i
\(848\) 67.8823 0.0800498
\(849\) 0 0
\(850\) 471.054i 0.554181i
\(851\) −339.411 −0.398838
\(852\) 0 0
\(853\) 1401.67i 1.64323i 0.570044 + 0.821614i \(0.306926\pi\)
−0.570044 + 0.821614i \(0.693074\pi\)
\(854\) 186.676 129.985i 0.218590 0.152207i
\(855\) 0 0
\(856\) 348.000 0.406542
\(857\) 186.853i 0.218031i −0.994040 0.109016i \(-0.965230\pi\)
0.994040 0.109016i \(-0.0347699\pi\)
\(858\) 0 0
\(859\) 471.054i 0.548375i 0.961676 + 0.274188i \(0.0884089\pi\)
−0.961676 + 0.274188i \(0.911591\pi\)
\(860\) 844.900i 0.982442i
\(861\) 0 0
\(862\) 654.000 0.758701
\(863\) 1166.73 1.35194 0.675971 0.736928i \(-0.263724\pi\)
0.675971 + 0.736928i \(0.263724\pi\)
\(864\) 0 0
\(865\) 594.000 0.686705
\(866\) 194.977i 0.225147i
\(867\) 0 0
\(868\) 0 0
\(869\) −1323.70 −1.52325
\(870\) 0 0
\(871\) 1194.87i 1.37184i
\(872\) 175.362 0.201104
\(873\) 0 0
\(874\) 344.674i 0.394364i
\(875\) −746.705 + 519.938i −0.853377 + 0.594215i
\(876\) 0 0
\(877\) 350.000 0.399088 0.199544 0.979889i \(-0.436054\pi\)
0.199544 + 0.979889i \(0.436054\pi\)
\(878\) 243.721i 0.277587i
\(879\) 0 0
\(880\) 413.609i 0.470010i
\(881\) 268.093i 0.304306i −0.988357 0.152153i \(-0.951379\pi\)
0.988357 0.152153i \(-0.0486206\pi\)
\(882\) 0 0
\(883\) 412.000 0.466591 0.233296 0.972406i \(-0.425049\pi\)
0.233296 + 0.972406i \(0.425049\pi\)
\(884\) −373.352 −0.422344
\(885\) 0 0
\(886\) 558.000 0.629797
\(887\) 1039.88i 1.17235i −0.810183 0.586176i \(-0.800633\pi\)
0.810183 0.586176i \(-0.199367\pi\)
\(888\) 0 0
\(889\) 256.000 + 367.652i 0.287964 + 0.413557i
\(890\) −840.043 −0.943868
\(891\) 0 0
\(892\) 344.674i 0.386406i
\(893\) −373.352 −0.418088
\(894\) 0 0
\(895\) 1275.29i 1.42491i
\(896\) 45.2548 + 64.9923i 0.0505076 + 0.0725361i
\(897\) 0 0
\(898\) 72.0000 0.0801782
\(899\) 0 0
\(900\) 0 0
\(901\) 137.870i 0.153018i
\(902\) 1023.63i 1.13484i
\(903\) 0 0
\(904\) −336.000 −0.371681
\(905\) 2053.44 2.26899
\(906\) 0 0
\(907\) 436.000 0.480706 0.240353 0.970686i \(-0.422737\pi\)
0.240353 + 0.970686i \(0.422737\pi\)
\(908\) 779.908i 0.858929i
\(909\) 0 0
\(910\) −1056.00 1516.56i −1.16044 1.66655i
\(911\) 326.683 0.358599 0.179299 0.983795i \(-0.442617\pi\)
0.179299 + 0.983795i \(0.442617\pi\)
\(912\) 0 0
\(913\) 2068.04i 2.26511i
\(914\) −183.848 −0.201146
\(915\) 0 0
\(916\) 413.609i 0.451538i
\(917\) −560.029 + 389.954i −0.610718 + 0.425250i
\(918\) 0 0
\(919\) −224.000 −0.243743 −0.121872 0.992546i \(-0.538890\pi\)
−0.121872 + 0.992546i \(0.538890\pi\)
\(920\) 487.442i 0.529829i
\(921\) 0 0
\(922\) 1022.53i 1.10904i
\(923\) 2047.26i 2.21805i
\(924\) 0 0
\(925\) −656.000 −0.709189
\(926\) −113.137 −0.122178
\(927\) 0 0
\(928\) −192.000 −0.206897
\(929\) 788.032i 0.848258i 0.905602 + 0.424129i \(0.139420\pi\)
−0.905602 + 0.424129i \(0.860580\pi\)
\(930\) 0 0
\(931\) −528.000 195.315i −0.567132 0.209791i
\(932\) 373.352 0.400593
\(933\) 0 0
\(934\) 275.739i 0.295224i
\(935\) 840.043 0.898442
\(936\) 0 0
\(937\) 505.522i 0.539511i −0.962929 0.269755i \(-0.913057\pi\)
0.962929 0.269755i \(-0.0869428\pi\)
\(938\) −294.156 422.450i −0.313600 0.450373i
\(939\) 0 0
\(940\) −528.000 −0.561702
\(941\) 333.086i 0.353970i 0.984214 + 0.176985i \(0.0566344\pi\)
−0.984214 + 0.176985i \(0.943366\pi\)
\(942\) 0 0
\(943\) 1206.36i 1.27928i
\(944\) 129.985i 0.137696i
\(945\) 0 0
\(946\) 936.000 0.989429
\(947\) −46.6690 −0.0492809 −0.0246405 0.999696i \(-0.507844\pi\)
−0.0246405 + 0.999696i \(0.507844\pi\)
\(948\) 0 0
\(949\) −1056.00 −1.11275
\(950\) 666.171i 0.701233i
\(951\) 0 0
\(952\) −132.000 + 91.9130i −0.138655 + 0.0965473i
\(953\) 101.823 0.106845 0.0534225 0.998572i \(-0.482987\pi\)
0.0534225 + 0.998572i \(0.482987\pi\)
\(954\) 0 0
\(955\) 103.402i 0.108274i
\(956\) −517.602 −0.541425
\(957\) 0 0
\(958\) 597.435i 0.623627i
\(959\) 135.765 + 194.977i 0.141569 + 0.203313i
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 519.938i 0.540477i
\(963\) 0 0
\(964\) 91.9130i 0.0953454i
\(965\) 1819.78i 1.88579i
\(966\) 0 0
\(967\) −664.000 −0.686660 −0.343330 0.939215i \(-0.611555\pi\)
−0.343330 + 0.939215i \(0.611555\pi\)
\(968\) −115.966 −0.119799
\(969\) 0 0
\(970\) 1056.00 1.08866
\(971\) 844.900i 0.870134i 0.900398 + 0.435067i \(0.143275\pi\)
−0.900398 + 0.435067i \(0.856725\pi\)
\(972\) 0 0
\(973\) 1056.00 735.304i 1.08530 0.755708i
\(974\) −1278.45 −1.31258
\(975\) 0 0
\(976\) 91.9130i 0.0941732i
\(977\) 424.264 0.434252 0.217126 0.976144i \(-0.430332\pi\)
0.217126 + 0.976144i \(0.430332\pi\)
\(978\) 0 0
\(979\) 930.619i 0.950581i
\(980\) −746.705 276.217i −0.761944 0.281854i
\(981\) 0 0
\(982\) −594.000 −0.604888
\(983\) 1104.87i 1.12398i −0.827145 0.561988i \(-0.810037\pi\)
0.827145 0.561988i \(-0.189963\pi\)
\(984\) 0 0
\(985\) 2481.65i 2.51944i
\(986\) 389.954i 0.395491i
\(987\) 0 0
\(988\) −528.000 −0.534413
\(989\) −1103.09 −1.11536
\(990\) 0 0
\(991\) 592.000 0.597376 0.298688 0.954351i \(-0.403451\pi\)
0.298688 + 0.954351i \(0.403451\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 504.000 + 723.815i 0.507042 + 0.728184i
\(995\) −2146.78 −2.15756
\(996\) 0 0
\(997\) 344.674i 0.345711i −0.984947 0.172855i \(-0.944701\pi\)
0.984947 0.172855i \(-0.0552993\pi\)
\(998\) 1012.58 1.01461
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 126.3.c.a.55.2 yes 4
3.2 odd 2 inner 126.3.c.a.55.3 yes 4
4.3 odd 2 1008.3.f.i.433.4 4
7.2 even 3 882.3.n.i.325.4 8
7.3 odd 6 882.3.n.i.19.4 8
7.4 even 3 882.3.n.i.19.3 8
7.5 odd 6 882.3.n.i.325.3 8
7.6 odd 2 inner 126.3.c.a.55.1 4
12.11 even 2 1008.3.f.i.433.2 4
21.2 odd 6 882.3.n.i.325.1 8
21.5 even 6 882.3.n.i.325.2 8
21.11 odd 6 882.3.n.i.19.2 8
21.17 even 6 882.3.n.i.19.1 8
21.20 even 2 inner 126.3.c.a.55.4 yes 4
28.27 even 2 1008.3.f.i.433.1 4
84.83 odd 2 1008.3.f.i.433.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.c.a.55.1 4 7.6 odd 2 inner
126.3.c.a.55.2 yes 4 1.1 even 1 trivial
126.3.c.a.55.3 yes 4 3.2 odd 2 inner
126.3.c.a.55.4 yes 4 21.20 even 2 inner
882.3.n.i.19.1 8 21.17 even 6
882.3.n.i.19.2 8 21.11 odd 6
882.3.n.i.19.3 8 7.4 even 3
882.3.n.i.19.4 8 7.3 odd 6
882.3.n.i.325.1 8 21.2 odd 6
882.3.n.i.325.2 8 21.5 even 6
882.3.n.i.325.3 8 7.5 odd 6
882.3.n.i.325.4 8 7.2 even 3
1008.3.f.i.433.1 4 28.27 even 2
1008.3.f.i.433.2 4 12.11 even 2
1008.3.f.i.433.3 4 84.83 odd 2
1008.3.f.i.433.4 4 4.3 odd 2