Properties

Label 768.4.f.g.383.9
Level $768$
Weight $4$
Character 768.383
Analytic conductor $45.313$
Analytic rank $0$
Dimension $12$
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] = [768,4,Mod(383,768)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(768, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("768.383");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 768 = 2^{8} \cdot 3 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 768.f (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: \(45.3134668844\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 384)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 383.9
Root \(1.08600i\) of defining polynomial
Character \(\chi\) \(=\) 768.383
Dual form 768.4.f.g.383.10

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.21320 - 3.04120i) q^{3} -9.33303 q^{5} +36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} +O(q^{10})\) \(q+(4.21320 - 3.04120i) q^{3} -9.33303 q^{5} +36.3792i q^{7} +(8.50216 - 25.6264i) q^{9} +48.4408i q^{11} -25.8988i q^{13} +(-39.3219 + 28.3836i) q^{15} -74.2219i q^{17} +82.9826 q^{19} +(110.637 + 153.273i) q^{21} -179.084 q^{23} -37.8946 q^{25} +(-42.1138 - 133.826i) q^{27} -122.309 q^{29} +64.1893i q^{31} +(147.318 + 204.091i) q^{33} -339.528i q^{35} +5.01487i q^{37} +(-78.7635 - 109.117i) q^{39} -325.933i q^{41} -321.853 q^{43} +(-79.3509 + 239.172i) q^{45} -95.9780 q^{47} -980.446 q^{49} +(-225.724 - 312.712i) q^{51} +185.069 q^{53} -452.099i q^{55} +(349.623 - 252.367i) q^{57} +226.316i q^{59} -198.851i q^{61} +(932.268 + 309.302i) q^{63} +241.714i q^{65} -23.9524 q^{67} +(-754.518 + 544.632i) q^{69} -399.741 q^{71} -669.631 q^{73} +(-159.658 + 115.245i) q^{75} -1762.24 q^{77} +229.453i q^{79} +(-584.427 - 435.760i) q^{81} +321.150i q^{83} +692.715i q^{85} +(-515.313 + 371.967i) q^{87} -131.446i q^{89} +942.177 q^{91} +(195.213 + 270.443i) q^{93} -774.479 q^{95} +136.371 q^{97} +(1241.36 + 411.851i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{3} - 12 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{3} - 12 q^{5} + 84 q^{15} + 180 q^{19} - 156 q^{21} + 120 q^{23} + 300 q^{25} - 130 q^{27} - 588 q^{29} - 116 q^{33} - 620 q^{39} - 372 q^{43} + 740 q^{45} - 1248 q^{47} - 948 q^{49} - 360 q^{51} + 948 q^{53} + 172 q^{57} + 2744 q^{63} + 2292 q^{67} - 3280 q^{69} + 2040 q^{71} + 216 q^{73} - 2522 q^{75} - 4824 q^{77} - 1076 q^{81} - 4156 q^{87} + 3480 q^{91} + 4180 q^{93} - 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/768\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(511\) \(517\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 4.21320 3.04120i 0.810831 0.585280i
\(4\) 0 0
\(5\) −9.33303 −0.834771 −0.417386 0.908729i \(-0.637054\pi\)
−0.417386 + 0.908729i \(0.637054\pi\)
\(6\) 0 0
\(7\) 36.3792i 1.96429i 0.188120 + 0.982146i \(0.439760\pi\)
−0.188120 + 0.982146i \(0.560240\pi\)
\(8\) 0 0
\(9\) 8.50216 25.6264i 0.314895 0.949127i
\(10\) 0 0
\(11\) 48.4408i 1.32777i 0.747836 + 0.663884i \(0.231093\pi\)
−0.747836 + 0.663884i \(0.768907\pi\)
\(12\) 0 0
\(13\) 25.8988i 0.552541i −0.961080 0.276270i \(-0.910901\pi\)
0.961080 0.276270i \(-0.0890985\pi\)
\(14\) 0 0
\(15\) −39.3219 + 28.3836i −0.676859 + 0.488575i
\(16\) 0 0
\(17\) 74.2219i 1.05891i −0.848338 0.529455i \(-0.822397\pi\)
0.848338 0.529455i \(-0.177603\pi\)
\(18\) 0 0
\(19\) 82.9826 1.00197 0.500987 0.865455i \(-0.332970\pi\)
0.500987 + 0.865455i \(0.332970\pi\)
\(20\) 0 0
\(21\) 110.637 + 153.273i 1.14966 + 1.59271i
\(22\) 0 0
\(23\) −179.084 −1.62355 −0.811775 0.583971i \(-0.801498\pi\)
−0.811775 + 0.583971i \(0.801498\pi\)
\(24\) 0 0
\(25\) −37.8946 −0.303157
\(26\) 0 0
\(27\) −42.1138 133.826i −0.300178 0.953883i
\(28\) 0 0
\(29\) −122.309 −0.783180 −0.391590 0.920140i \(-0.628075\pi\)
−0.391590 + 0.920140i \(0.628075\pi\)
\(30\) 0 0
\(31\) 64.1893i 0.371895i 0.982560 + 0.185947i \(0.0595354\pi\)
−0.982560 + 0.185947i \(0.940465\pi\)
\(32\) 0 0
\(33\) 147.318 + 204.091i 0.777116 + 1.07660i
\(34\) 0 0
\(35\) 339.528i 1.63973i
\(36\) 0 0
\(37\) 5.01487i 0.0222822i 0.999938 + 0.0111411i \(0.00354639\pi\)
−0.999938 + 0.0111411i \(0.996454\pi\)
\(38\) 0 0
\(39\) −78.7635 109.117i −0.323391 0.448017i
\(40\) 0 0
\(41\) 325.933i 1.24152i −0.784003 0.620758i \(-0.786825\pi\)
0.784003 0.620758i \(-0.213175\pi\)
\(42\) 0 0
\(43\) −321.853 −1.14144 −0.570722 0.821143i \(-0.693337\pi\)
−0.570722 + 0.821143i \(0.693337\pi\)
\(44\) 0 0
\(45\) −79.3509 + 239.172i −0.262865 + 0.792304i
\(46\) 0 0
\(47\) −95.9780 −0.297869 −0.148934 0.988847i \(-0.547584\pi\)
−0.148934 + 0.988847i \(0.547584\pi\)
\(48\) 0 0
\(49\) −980.446 −2.85844
\(50\) 0 0
\(51\) −225.724 312.712i −0.619758 0.858597i
\(52\) 0 0
\(53\) 185.069 0.479645 0.239822 0.970817i \(-0.422911\pi\)
0.239822 + 0.970817i \(0.422911\pi\)
\(54\) 0 0
\(55\) 452.099i 1.10838i
\(56\) 0 0
\(57\) 349.623 252.367i 0.812432 0.586436i
\(58\) 0 0
\(59\) 226.316i 0.499386i 0.968325 + 0.249693i \(0.0803297\pi\)
−0.968325 + 0.249693i \(0.919670\pi\)
\(60\) 0 0
\(61\) 198.851i 0.417382i −0.977982 0.208691i \(-0.933080\pi\)
0.977982 0.208691i \(-0.0669203\pi\)
\(62\) 0 0
\(63\) 932.268 + 309.302i 1.86436 + 0.618545i
\(64\) 0 0
\(65\) 241.714i 0.461245i
\(66\) 0 0
\(67\) −23.9524 −0.0436753 −0.0218377 0.999762i \(-0.506952\pi\)
−0.0218377 + 0.999762i \(0.506952\pi\)
\(68\) 0 0
\(69\) −754.518 + 544.632i −1.31642 + 0.950231i
\(70\) 0 0
\(71\) −399.741 −0.668176 −0.334088 0.942542i \(-0.608428\pi\)
−0.334088 + 0.942542i \(0.608428\pi\)
\(72\) 0 0
\(73\) −669.631 −1.07362 −0.536811 0.843703i \(-0.680371\pi\)
−0.536811 + 0.843703i \(0.680371\pi\)
\(74\) 0 0
\(75\) −159.658 + 115.245i −0.245809 + 0.177432i
\(76\) 0 0
\(77\) −1762.24 −2.60812
\(78\) 0 0
\(79\) 229.453i 0.326778i 0.986562 + 0.163389i \(0.0522426\pi\)
−0.986562 + 0.163389i \(0.947757\pi\)
\(80\) 0 0
\(81\) −584.427 435.760i −0.801683 0.597750i
\(82\) 0 0
\(83\) 321.150i 0.424709i 0.977193 + 0.212354i \(0.0681131\pi\)
−0.977193 + 0.212354i \(0.931887\pi\)
\(84\) 0 0
\(85\) 692.715i 0.883947i
\(86\) 0 0
\(87\) −515.313 + 371.967i −0.635027 + 0.458380i
\(88\) 0 0
\(89\) 131.446i 0.156554i −0.996932 0.0782768i \(-0.975058\pi\)
0.996932 0.0782768i \(-0.0249418\pi\)
\(90\) 0 0
\(91\) 942.177 1.08535
\(92\) 0 0
\(93\) 195.213 + 270.443i 0.217663 + 0.301544i
\(94\) 0 0
\(95\) −774.479 −0.836420
\(96\) 0 0
\(97\) 136.371 0.142746 0.0713732 0.997450i \(-0.477262\pi\)
0.0713732 + 0.997450i \(0.477262\pi\)
\(98\) 0 0
\(99\) 1241.36 + 411.851i 1.26022 + 0.418107i
\(100\) 0 0
\(101\) 411.095 0.405005 0.202502 0.979282i \(-0.435093\pi\)
0.202502 + 0.979282i \(0.435093\pi\)
\(102\) 0 0
\(103\) 137.507i 0.131543i 0.997835 + 0.0657716i \(0.0209509\pi\)
−0.997835 + 0.0657716i \(0.979049\pi\)
\(104\) 0 0
\(105\) −1032.57 1430.50i −0.959704 1.32955i
\(106\) 0 0
\(107\) 2113.08i 1.90915i 0.297964 + 0.954577i \(0.403692\pi\)
−0.297964 + 0.954577i \(0.596308\pi\)
\(108\) 0 0
\(109\) 1310.07i 1.15121i 0.817727 + 0.575606i \(0.195234\pi\)
−0.817727 + 0.575606i \(0.804766\pi\)
\(110\) 0 0
\(111\) 15.2512 + 21.1287i 0.0130413 + 0.0180671i
\(112\) 0 0
\(113\) 1403.62i 1.16851i −0.811571 0.584254i \(-0.801387\pi\)
0.811571 0.584254i \(-0.198613\pi\)
\(114\) 0 0
\(115\) 1671.40 1.35529
\(116\) 0 0
\(117\) −663.693 220.196i −0.524431 0.173992i
\(118\) 0 0
\(119\) 2700.13 2.08001
\(120\) 0 0
\(121\) −1015.51 −0.762967
\(122\) 0 0
\(123\) −991.227 1373.22i −0.726634 1.00666i
\(124\) 0 0
\(125\) 1520.30 1.08784
\(126\) 0 0
\(127\) 1583.69i 1.10653i −0.833004 0.553267i \(-0.813381\pi\)
0.833004 0.553267i \(-0.186619\pi\)
\(128\) 0 0
\(129\) −1356.03 + 978.820i −0.925519 + 0.668065i
\(130\) 0 0
\(131\) 160.326i 0.106929i −0.998570 0.0534646i \(-0.982974\pi\)
0.998570 0.0534646i \(-0.0170264\pi\)
\(132\) 0 0
\(133\) 3018.84i 1.96817i
\(134\) 0 0
\(135\) 393.050 + 1249.00i 0.250580 + 0.796274i
\(136\) 0 0
\(137\) 1487.47i 0.927614i 0.885936 + 0.463807i \(0.153517\pi\)
−0.885936 + 0.463807i \(0.846483\pi\)
\(138\) 0 0
\(139\) 58.7660 0.0358595 0.0179298 0.999839i \(-0.494292\pi\)
0.0179298 + 0.999839i \(0.494292\pi\)
\(140\) 0 0
\(141\) −404.375 + 291.889i −0.241521 + 0.174337i
\(142\) 0 0
\(143\) 1254.56 0.733646
\(144\) 0 0
\(145\) 1141.51 0.653776
\(146\) 0 0
\(147\) −4130.82 + 2981.74i −2.31771 + 1.67299i
\(148\) 0 0
\(149\) −940.146 −0.516911 −0.258456 0.966023i \(-0.583214\pi\)
−0.258456 + 0.966023i \(0.583214\pi\)
\(150\) 0 0
\(151\) 1800.43i 0.970308i −0.874429 0.485154i \(-0.838763\pi\)
0.874429 0.485154i \(-0.161237\pi\)
\(152\) 0 0
\(153\) −1902.04 631.046i −1.00504 0.333445i
\(154\) 0 0
\(155\) 599.081i 0.310447i
\(156\) 0 0
\(157\) 3024.10i 1.53726i 0.639695 + 0.768629i \(0.279061\pi\)
−0.639695 + 0.768629i \(0.720939\pi\)
\(158\) 0 0
\(159\) 779.733 562.832i 0.388911 0.280726i
\(160\) 0 0
\(161\) 6514.94i 3.18913i
\(162\) 0 0
\(163\) −3124.46 −1.50139 −0.750694 0.660650i \(-0.770281\pi\)
−0.750694 + 0.660650i \(0.770281\pi\)
\(164\) 0 0
\(165\) −1374.93 1904.79i −0.648714 0.898711i
\(166\) 0 0
\(167\) 371.384 0.172087 0.0860435 0.996291i \(-0.472578\pi\)
0.0860435 + 0.996291i \(0.472578\pi\)
\(168\) 0 0
\(169\) 1526.25 0.694699
\(170\) 0 0
\(171\) 705.531 2126.55i 0.315517 0.951001i
\(172\) 0 0
\(173\) −2931.24 −1.28820 −0.644099 0.764943i \(-0.722767\pi\)
−0.644099 + 0.764943i \(0.722767\pi\)
\(174\) 0 0
\(175\) 1378.58i 0.595489i
\(176\) 0 0
\(177\) 688.272 + 953.513i 0.292281 + 0.404918i
\(178\) 0 0
\(179\) 2834.46i 1.18356i −0.806098 0.591781i \(-0.798425\pi\)
0.806098 0.591781i \(-0.201575\pi\)
\(180\) 0 0
\(181\) 1819.98i 0.747392i −0.927551 0.373696i \(-0.878090\pi\)
0.927551 0.373696i \(-0.121910\pi\)
\(182\) 0 0
\(183\) −604.747 837.800i −0.244285 0.338426i
\(184\) 0 0
\(185\) 46.8039i 0.0186005i
\(186\) 0 0
\(187\) 3595.37 1.40599
\(188\) 0 0
\(189\) 4868.49 1532.07i 1.87370 0.589638i
\(190\) 0 0
\(191\) −4627.06 −1.75289 −0.876446 0.481500i \(-0.840092\pi\)
−0.876446 + 0.481500i \(0.840092\pi\)
\(192\) 0 0
\(193\) −2770.29 −1.03321 −0.516606 0.856223i \(-0.672805\pi\)
−0.516606 + 0.856223i \(0.672805\pi\)
\(194\) 0 0
\(195\) 735.102 + 1018.39i 0.269958 + 0.373992i
\(196\) 0 0
\(197\) −3880.01 −1.40324 −0.701622 0.712550i \(-0.747540\pi\)
−0.701622 + 0.712550i \(0.747540\pi\)
\(198\) 0 0
\(199\) 2015.00i 0.717785i 0.933379 + 0.358893i \(0.116846\pi\)
−0.933379 + 0.358893i \(0.883154\pi\)
\(200\) 0 0
\(201\) −100.916 + 72.8440i −0.0354133 + 0.0255623i
\(202\) 0 0
\(203\) 4449.51i 1.53839i
\(204\) 0 0
\(205\) 3041.94i 1.03638i
\(206\) 0 0
\(207\) −1522.60 + 4589.29i −0.511247 + 1.54095i
\(208\) 0 0
\(209\) 4019.74i 1.33039i
\(210\) 0 0
\(211\) 5611.42 1.83084 0.915418 0.402505i \(-0.131860\pi\)
0.915418 + 0.402505i \(0.131860\pi\)
\(212\) 0 0
\(213\) −1684.19 + 1215.69i −0.541778 + 0.391070i
\(214\) 0 0
\(215\) 3003.86 0.952845
\(216\) 0 0
\(217\) −2335.16 −0.730510
\(218\) 0 0
\(219\) −2821.29 + 2036.48i −0.870526 + 0.628369i
\(220\) 0 0
\(221\) −1922.26 −0.585091
\(222\) 0 0
\(223\) 2025.65i 0.608285i −0.952627 0.304142i \(-0.901630\pi\)
0.952627 0.304142i \(-0.0983699\pi\)
\(224\) 0 0
\(225\) −322.186 + 971.103i −0.0954625 + 0.287734i
\(226\) 0 0
\(227\) 404.049i 0.118140i 0.998254 + 0.0590698i \(0.0188135\pi\)
−0.998254 + 0.0590698i \(0.981187\pi\)
\(228\) 0 0
\(229\) 4735.42i 1.36649i −0.730191 0.683243i \(-0.760569\pi\)
0.730191 0.683243i \(-0.239431\pi\)
\(230\) 0 0
\(231\) −7424.66 + 5359.32i −2.11475 + 1.52648i
\(232\) 0 0
\(233\) 3703.79i 1.04139i 0.853744 + 0.520694i \(0.174327\pi\)
−0.853744 + 0.520694i \(0.825673\pi\)
\(234\) 0 0
\(235\) 895.765 0.248652
\(236\) 0 0
\(237\) 697.814 + 966.733i 0.191257 + 0.264962i
\(238\) 0 0
\(239\) −6398.83 −1.73182 −0.865912 0.500197i \(-0.833261\pi\)
−0.865912 + 0.500197i \(0.833261\pi\)
\(240\) 0 0
\(241\) 3766.48 1.00672 0.503362 0.864076i \(-0.332096\pi\)
0.503362 + 0.864076i \(0.332096\pi\)
\(242\) 0 0
\(243\) −3787.54 58.5842i −0.999880 0.0154658i
\(244\) 0 0
\(245\) 9150.53 2.38615
\(246\) 0 0
\(247\) 2149.15i 0.553632i
\(248\) 0 0
\(249\) 976.683 + 1353.07i 0.248573 + 0.344367i
\(250\) 0 0
\(251\) 1979.71i 0.497841i 0.968524 + 0.248921i \(0.0800758\pi\)
−0.968524 + 0.248921i \(0.919924\pi\)
\(252\) 0 0
\(253\) 8674.98i 2.15570i
\(254\) 0 0
\(255\) 2106.69 + 2918.55i 0.517356 + 0.716732i
\(256\) 0 0
\(257\) 1081.54i 0.262508i 0.991349 + 0.131254i \(0.0419004\pi\)
−0.991349 + 0.131254i \(0.958100\pi\)
\(258\) 0 0
\(259\) −182.437 −0.0437687
\(260\) 0 0
\(261\) −1039.89 + 3134.34i −0.246619 + 0.743337i
\(262\) 0 0
\(263\) −1121.20 −0.262875 −0.131438 0.991324i \(-0.541959\pi\)
−0.131438 + 0.991324i \(0.541959\pi\)
\(264\) 0 0
\(265\) −1727.25 −0.400394
\(266\) 0 0
\(267\) −399.755 553.810i −0.0916277 0.126939i
\(268\) 0 0
\(269\) 1844.86 0.418153 0.209076 0.977899i \(-0.432954\pi\)
0.209076 + 0.977899i \(0.432954\pi\)
\(270\) 0 0
\(271\) 938.970i 0.210474i −0.994447 0.105237i \(-0.966440\pi\)
0.994447 0.105237i \(-0.0335601\pi\)
\(272\) 0 0
\(273\) 3969.58 2865.35i 0.880037 0.635234i
\(274\) 0 0
\(275\) 1835.64i 0.402522i
\(276\) 0 0
\(277\) 1100.93i 0.238803i 0.992846 + 0.119401i \(0.0380975\pi\)
−0.992846 + 0.119401i \(0.961902\pi\)
\(278\) 0 0
\(279\) 1644.94 + 545.748i 0.352975 + 0.117108i
\(280\) 0 0
\(281\) 748.208i 0.158841i 0.996841 + 0.0794205i \(0.0253070\pi\)
−0.996841 + 0.0794205i \(0.974693\pi\)
\(282\) 0 0
\(283\) −2150.15 −0.451636 −0.225818 0.974170i \(-0.572505\pi\)
−0.225818 + 0.974170i \(0.572505\pi\)
\(284\) 0 0
\(285\) −3263.04 + 2355.35i −0.678195 + 0.489540i
\(286\) 0 0
\(287\) 11857.2 2.43870
\(288\) 0 0
\(289\) −595.891 −0.121289
\(290\) 0 0
\(291\) 574.559 414.733i 0.115743 0.0835466i
\(292\) 0 0
\(293\) 5252.83 1.04735 0.523675 0.851918i \(-0.324561\pi\)
0.523675 + 0.851918i \(0.324561\pi\)
\(294\) 0 0
\(295\) 2112.21i 0.416873i
\(296\) 0 0
\(297\) 6482.64 2040.03i 1.26654 0.398567i
\(298\) 0 0
\(299\) 4638.06i 0.897077i
\(300\) 0 0
\(301\) 11708.8i 2.24213i
\(302\) 0 0
\(303\) 1732.03 1250.22i 0.328390 0.237041i
\(304\) 0 0
\(305\) 1855.88i 0.348418i
\(306\) 0 0
\(307\) 5337.06 0.992190 0.496095 0.868268i \(-0.334767\pi\)
0.496095 + 0.868268i \(0.334767\pi\)
\(308\) 0 0
\(309\) 418.186 + 579.344i 0.0769896 + 0.106659i
\(310\) 0 0
\(311\) 1034.85 0.188685 0.0943424 0.995540i \(-0.469925\pi\)
0.0943424 + 0.995540i \(0.469925\pi\)
\(312\) 0 0
\(313\) 9570.33 1.72826 0.864132 0.503265i \(-0.167868\pi\)
0.864132 + 0.503265i \(0.167868\pi\)
\(314\) 0 0
\(315\) −8700.89 2886.72i −1.55632 0.516344i
\(316\) 0 0
\(317\) 3593.05 0.636612 0.318306 0.947988i \(-0.396886\pi\)
0.318306 + 0.947988i \(0.396886\pi\)
\(318\) 0 0
\(319\) 5924.75i 1.03988i
\(320\) 0 0
\(321\) 6426.32 + 8902.85i 1.11739 + 1.54800i
\(322\) 0 0
\(323\) 6159.13i 1.06100i
\(324\) 0 0
\(325\) 981.425i 0.167507i
\(326\) 0 0
\(327\) 3984.19 + 5519.60i 0.673781 + 0.933438i
\(328\) 0 0
\(329\) 3491.60i 0.585101i
\(330\) 0 0
\(331\) 4119.28 0.684037 0.342018 0.939693i \(-0.388890\pi\)
0.342018 + 0.939693i \(0.388890\pi\)
\(332\) 0 0
\(333\) 128.513 + 42.6372i 0.0211486 + 0.00701654i
\(334\) 0 0
\(335\) 223.548 0.0364589
\(336\) 0 0
\(337\) −6488.93 −1.04889 −0.524443 0.851446i \(-0.675726\pi\)
−0.524443 + 0.851446i \(0.675726\pi\)
\(338\) 0 0
\(339\) −4268.69 5913.73i −0.683904 0.947462i
\(340\) 0 0
\(341\) −3109.38 −0.493790
\(342\) 0 0
\(343\) 23189.8i 3.65052i
\(344\) 0 0
\(345\) 7041.94 5083.06i 1.09891 0.793225i
\(346\) 0 0
\(347\) 7085.97i 1.09624i 0.836400 + 0.548119i \(0.184656\pi\)
−0.836400 + 0.548119i \(0.815344\pi\)
\(348\) 0 0
\(349\) 6075.79i 0.931891i 0.884814 + 0.465945i \(0.154286\pi\)
−0.884814 + 0.465945i \(0.845714\pi\)
\(350\) 0 0
\(351\) −3465.93 + 1090.70i −0.527059 + 0.165861i
\(352\) 0 0
\(353\) 4451.86i 0.671242i 0.941997 + 0.335621i \(0.108946\pi\)
−0.941997 + 0.335621i \(0.891054\pi\)
\(354\) 0 0
\(355\) 3730.79 0.557774
\(356\) 0 0
\(357\) 11376.2 8211.66i 1.68653 1.21739i
\(358\) 0 0
\(359\) −10771.6 −1.58357 −0.791785 0.610800i \(-0.790848\pi\)
−0.791785 + 0.610800i \(0.790848\pi\)
\(360\) 0 0
\(361\) 27.1131 0.00395293
\(362\) 0 0
\(363\) −4278.54 + 3088.37i −0.618637 + 0.446549i
\(364\) 0 0
\(365\) 6249.68 0.896228
\(366\) 0 0
\(367\) 6671.95i 0.948973i 0.880263 + 0.474486i \(0.157366\pi\)
−0.880263 + 0.474486i \(0.842634\pi\)
\(368\) 0 0
\(369\) −8352.48 2771.13i −1.17835 0.390947i
\(370\) 0 0
\(371\) 6732.66i 0.942162i
\(372\) 0 0
\(373\) 12618.5i 1.75164i −0.482641 0.875819i \(-0.660322\pi\)
0.482641 0.875819i \(-0.339678\pi\)
\(374\) 0 0
\(375\) 6405.33 4623.54i 0.882053 0.636690i
\(376\) 0 0
\(377\) 3167.66i 0.432739i
\(378\) 0 0
\(379\) 7915.38 1.07279 0.536393 0.843968i \(-0.319787\pi\)
0.536393 + 0.843968i \(0.319787\pi\)
\(380\) 0 0
\(381\) −4816.32 6672.40i −0.647632 0.897212i
\(382\) 0 0
\(383\) 10820.5 1.44360 0.721802 0.692099i \(-0.243314\pi\)
0.721802 + 0.692099i \(0.243314\pi\)
\(384\) 0 0
\(385\) 16447.0 2.17719
\(386\) 0 0
\(387\) −2736.45 + 8247.94i −0.359435 + 1.08338i
\(388\) 0 0
\(389\) 7990.57 1.04149 0.520743 0.853714i \(-0.325655\pi\)
0.520743 + 0.853714i \(0.325655\pi\)
\(390\) 0 0
\(391\) 13292.0i 1.71919i
\(392\) 0 0
\(393\) −487.583 675.485i −0.0625835 0.0867016i
\(394\) 0 0
\(395\) 2141.49i 0.272785i
\(396\) 0 0
\(397\) 5526.21i 0.698620i −0.937007 0.349310i \(-0.886416\pi\)
0.937007 0.349310i \(-0.113584\pi\)
\(398\) 0 0
\(399\) 9180.91 + 12719.0i 1.15193 + 1.59585i
\(400\) 0 0
\(401\) 7844.55i 0.976903i 0.872591 + 0.488451i \(0.162438\pi\)
−0.872591 + 0.488451i \(0.837562\pi\)
\(402\) 0 0
\(403\) 1662.43 0.205487
\(404\) 0 0
\(405\) 5454.47 + 4066.96i 0.669222 + 0.498985i
\(406\) 0 0
\(407\) −242.924 −0.0295855
\(408\) 0 0
\(409\) 4407.65 0.532871 0.266435 0.963853i \(-0.414154\pi\)
0.266435 + 0.963853i \(0.414154\pi\)
\(410\) 0 0
\(411\) 4523.70 + 6267.01i 0.542914 + 0.752138i
\(412\) 0 0
\(413\) −8233.18 −0.980940
\(414\) 0 0
\(415\) 2997.30i 0.354534i
\(416\) 0 0
\(417\) 247.593 178.720i 0.0290760 0.0209878i
\(418\) 0 0
\(419\) 12032.8i 1.40296i 0.712688 + 0.701481i \(0.247477\pi\)
−0.712688 + 0.701481i \(0.752523\pi\)
\(420\) 0 0
\(421\) 10320.3i 1.19473i 0.801969 + 0.597365i \(0.203786\pi\)
−0.801969 + 0.597365i \(0.796214\pi\)
\(422\) 0 0
\(423\) −816.021 + 2459.57i −0.0937973 + 0.282715i
\(424\) 0 0
\(425\) 2812.61i 0.321016i
\(426\) 0 0
\(427\) 7234.05 0.819860
\(428\) 0 0
\(429\) 5285.70 3815.36i 0.594863 0.429388i
\(430\) 0 0
\(431\) −8068.52 −0.901733 −0.450866 0.892591i \(-0.648885\pi\)
−0.450866 + 0.892591i \(0.648885\pi\)
\(432\) 0 0
\(433\) −3236.75 −0.359234 −0.179617 0.983737i \(-0.557486\pi\)
−0.179617 + 0.983737i \(0.557486\pi\)
\(434\) 0 0
\(435\) 4809.43 3471.58i 0.530102 0.382642i
\(436\) 0 0
\(437\) −14860.9 −1.62676
\(438\) 0 0
\(439\) 2207.71i 0.240019i −0.992773 0.120009i \(-0.961708\pi\)
0.992773 0.120009i \(-0.0382924\pi\)
\(440\) 0 0
\(441\) −8335.91 + 25125.3i −0.900109 + 2.71302i
\(442\) 0 0
\(443\) 1058.18i 0.113489i 0.998389 + 0.0567447i \(0.0180721\pi\)
−0.998389 + 0.0567447i \(0.981928\pi\)
\(444\) 0 0
\(445\) 1226.79i 0.130686i
\(446\) 0 0
\(447\) −3961.03 + 2859.18i −0.419128 + 0.302538i
\(448\) 0 0
\(449\) 10836.4i 1.13898i 0.822000 + 0.569488i \(0.192858\pi\)
−0.822000 + 0.569488i \(0.807142\pi\)
\(450\) 0 0
\(451\) 15788.4 1.64844
\(452\) 0 0
\(453\) −5475.46 7585.56i −0.567902 0.786756i
\(454\) 0 0
\(455\) −8793.36 −0.906020
\(456\) 0 0
\(457\) 9827.12 1.00589 0.502947 0.864317i \(-0.332249\pi\)
0.502947 + 0.864317i \(0.332249\pi\)
\(458\) 0 0
\(459\) −9932.83 + 3125.77i −1.01008 + 0.317861i
\(460\) 0 0
\(461\) 9311.74 0.940761 0.470381 0.882464i \(-0.344117\pi\)
0.470381 + 0.882464i \(0.344117\pi\)
\(462\) 0 0
\(463\) 7601.37i 0.762993i −0.924370 0.381497i \(-0.875409\pi\)
0.924370 0.381497i \(-0.124591\pi\)
\(464\) 0 0
\(465\) −1821.93 2524.05i −0.181699 0.251720i
\(466\) 0 0
\(467\) 4446.20i 0.440569i 0.975436 + 0.220284i \(0.0706985\pi\)
−0.975436 + 0.220284i \(0.929301\pi\)
\(468\) 0 0
\(469\) 871.368i 0.0857911i
\(470\) 0 0
\(471\) 9196.90 + 12741.1i 0.899726 + 1.24646i
\(472\) 0 0
\(473\) 15590.8i 1.51557i
\(474\) 0 0
\(475\) −3144.59 −0.303756
\(476\) 0 0
\(477\) 1573.49 4742.65i 0.151038 0.455244i
\(478\) 0 0
\(479\) −12285.9 −1.17194 −0.585970 0.810333i \(-0.699286\pi\)
−0.585970 + 0.810333i \(0.699286\pi\)
\(480\) 0 0
\(481\) 129.879 0.0123118
\(482\) 0 0
\(483\) −19813.3 27448.8i −1.86653 2.58584i
\(484\) 0 0
\(485\) −1272.76 −0.119161
\(486\) 0 0
\(487\) 8464.78i 0.787630i 0.919190 + 0.393815i \(0.128845\pi\)
−0.919190 + 0.393815i \(0.871155\pi\)
\(488\) 0 0
\(489\) −13164.0 + 9502.11i −1.21737 + 0.878732i
\(490\) 0 0
\(491\) 6356.37i 0.584234i −0.956383 0.292117i \(-0.905640\pi\)
0.956383 0.292117i \(-0.0943596\pi\)
\(492\) 0 0
\(493\) 9078.02i 0.829317i
\(494\) 0 0
\(495\) −11585.7 3843.82i −1.05200 0.349024i
\(496\) 0 0
\(497\) 14542.2i 1.31249i
\(498\) 0 0
\(499\) −11856.6 −1.06368 −0.531840 0.846845i \(-0.678499\pi\)
−0.531840 + 0.846845i \(0.678499\pi\)
\(500\) 0 0
\(501\) 1564.72 1129.45i 0.139534 0.100719i
\(502\) 0 0
\(503\) 17032.5 1.50982 0.754912 0.655826i \(-0.227679\pi\)
0.754912 + 0.655826i \(0.227679\pi\)
\(504\) 0 0
\(505\) −3836.76 −0.338086
\(506\) 0 0
\(507\) 6430.41 4641.65i 0.563283 0.406593i
\(508\) 0 0
\(509\) −1563.18 −0.136123 −0.0680617 0.997681i \(-0.521681\pi\)
−0.0680617 + 0.997681i \(0.521681\pi\)
\(510\) 0 0
\(511\) 24360.6i 2.10891i
\(512\) 0 0
\(513\) −3494.72 11105.2i −0.300771 0.955767i
\(514\) 0 0
\(515\) 1283.36i 0.109809i
\(516\) 0 0
\(517\) 4649.25i 0.395501i
\(518\) 0 0
\(519\) −12349.9 + 8914.50i −1.04451 + 0.753956i
\(520\) 0 0
\(521\) 9096.23i 0.764900i −0.923976 0.382450i \(-0.875080\pi\)
0.923976 0.382450i \(-0.124920\pi\)
\(522\) 0 0
\(523\) −10108.4 −0.845143 −0.422572 0.906330i \(-0.638873\pi\)
−0.422572 + 0.906330i \(0.638873\pi\)
\(524\) 0 0
\(525\) −4192.53 5808.22i −0.348528 0.482841i
\(526\) 0 0
\(527\) 4764.25 0.393803
\(528\) 0 0
\(529\) 19904.2 1.63591
\(530\) 0 0
\(531\) 5799.66 + 1924.17i 0.473980 + 0.157254i
\(532\) 0 0
\(533\) −8441.26 −0.685988
\(534\) 0 0
\(535\) 19721.5i 1.59371i
\(536\) 0 0
\(537\) −8620.18 11942.2i −0.692716 0.959670i
\(538\) 0 0
\(539\) 47493.6i 3.79535i
\(540\) 0 0
\(541\) 10097.6i 0.802462i 0.915977 + 0.401231i \(0.131417\pi\)
−0.915977 + 0.401231i \(0.868583\pi\)
\(542\) 0 0
\(543\) −5534.93 7667.94i −0.437434 0.606009i
\(544\) 0 0
\(545\) 12226.9i 0.960998i
\(546\) 0 0
\(547\) 2064.83 0.161400 0.0806998 0.996738i \(-0.474284\pi\)
0.0806998 + 0.996738i \(0.474284\pi\)
\(548\) 0 0
\(549\) −5095.84 1690.66i −0.396148 0.131431i
\(550\) 0 0
\(551\) −10149.5 −0.784727
\(552\) 0 0
\(553\) −8347.32 −0.641888
\(554\) 0 0
\(555\) −142.340 197.194i −0.0108865 0.0150819i
\(556\) 0 0
\(557\) 12125.7 0.922409 0.461204 0.887294i \(-0.347417\pi\)
0.461204 + 0.887294i \(0.347417\pi\)
\(558\) 0 0
\(559\) 8335.60i 0.630695i
\(560\) 0 0
\(561\) 15148.0 10934.2i 1.14002 0.822895i
\(562\) 0 0
\(563\) 7278.58i 0.544859i 0.962176 + 0.272429i \(0.0878271\pi\)
−0.962176 + 0.272429i \(0.912173\pi\)
\(564\) 0 0
\(565\) 13100.0i 0.975436i
\(566\) 0 0
\(567\) 15852.6 21261.0i 1.17416 1.57474i
\(568\) 0 0
\(569\) 17428.1i 1.28405i 0.766685 + 0.642024i \(0.221905\pi\)
−0.766685 + 0.642024i \(0.778095\pi\)
\(570\) 0 0
\(571\) −2423.73 −0.177635 −0.0888177 0.996048i \(-0.528309\pi\)
−0.0888177 + 0.996048i \(0.528309\pi\)
\(572\) 0 0
\(573\) −19494.7 + 14071.8i −1.42130 + 1.02593i
\(574\) 0 0
\(575\) 6786.33 0.492190
\(576\) 0 0
\(577\) −17471.1 −1.26054 −0.630268 0.776377i \(-0.717055\pi\)
−0.630268 + 0.776377i \(0.717055\pi\)
\(578\) 0 0
\(579\) −11671.8 + 8425.03i −0.837761 + 0.604719i
\(580\) 0 0
\(581\) −11683.2 −0.834252
\(582\) 0 0
\(583\) 8964.88i 0.636857i
\(584\) 0 0
\(585\) 6194.27 + 2055.09i 0.437780 + 0.145244i
\(586\) 0 0
\(587\) 7273.06i 0.511399i 0.966756 + 0.255700i \(0.0823058\pi\)
−0.966756 + 0.255700i \(0.917694\pi\)
\(588\) 0 0
\(589\) 5326.60i 0.372629i
\(590\) 0 0
\(591\) −16347.3 + 11799.9i −1.13779 + 0.821290i
\(592\) 0 0
\(593\) 19957.9i 1.38208i 0.722818 + 0.691038i \(0.242846\pi\)
−0.722818 + 0.691038i \(0.757154\pi\)
\(594\) 0 0
\(595\) −25200.4 −1.73633
\(596\) 0 0
\(597\) 6128.01 + 8489.59i 0.420105 + 0.582003i
\(598\) 0 0
\(599\) −6101.48 −0.416193 −0.208097 0.978108i \(-0.566727\pi\)
−0.208097 + 0.978108i \(0.566727\pi\)
\(600\) 0 0
\(601\) −16659.6 −1.13071 −0.565357 0.824846i \(-0.691262\pi\)
−0.565357 + 0.824846i \(0.691262\pi\)
\(602\) 0 0
\(603\) −203.647 + 613.813i −0.0137531 + 0.0414534i
\(604\) 0 0
\(605\) 9477.77 0.636903
\(606\) 0 0
\(607\) 13728.6i 0.917999i 0.888437 + 0.458999i \(0.151792\pi\)
−0.888437 + 0.458999i \(0.848208\pi\)
\(608\) 0 0
\(609\) −13531.9 18746.7i −0.900392 1.24738i
\(610\) 0 0
\(611\) 2485.71i 0.164585i
\(612\) 0 0
\(613\) 7514.10i 0.495093i 0.968876 + 0.247546i \(0.0796242\pi\)
−0.968876 + 0.247546i \(0.920376\pi\)
\(614\) 0 0
\(615\) 9251.15 + 12816.3i 0.606573 + 0.840330i
\(616\) 0 0
\(617\) 8052.78i 0.525434i 0.964873 + 0.262717i \(0.0846186\pi\)
−0.964873 + 0.262717i \(0.915381\pi\)
\(618\) 0 0
\(619\) 25194.4 1.63594 0.817970 0.575261i \(-0.195100\pi\)
0.817970 + 0.575261i \(0.195100\pi\)
\(620\) 0 0
\(621\) 7541.92 + 23966.1i 0.487354 + 1.54868i
\(622\) 0 0
\(623\) 4781.91 0.307517
\(624\) 0 0
\(625\) −9452.17 −0.604939
\(626\) 0 0
\(627\) 12224.9 + 16936.0i 0.778650 + 1.07872i
\(628\) 0 0
\(629\) 372.213 0.0235948
\(630\) 0 0
\(631\) 14945.1i 0.942876i 0.881899 + 0.471438i \(0.156265\pi\)
−0.881899 + 0.471438i \(0.843735\pi\)
\(632\) 0 0
\(633\) 23642.1 17065.5i 1.48450 1.07155i
\(634\) 0 0
\(635\) 14780.6i 0.923702i
\(636\) 0 0
\(637\) 25392.4i 1.57941i
\(638\) 0 0
\(639\) −3398.66 + 10243.9i −0.210405 + 0.634184i
\(640\) 0 0
\(641\) 13909.9i 0.857109i −0.903516 0.428555i \(-0.859023\pi\)
0.903516 0.428555i \(-0.140977\pi\)
\(642\) 0 0
\(643\) 13338.7 0.818084 0.409042 0.912516i \(-0.365863\pi\)
0.409042 + 0.912516i \(0.365863\pi\)
\(644\) 0 0
\(645\) 12655.9 9135.36i 0.772597 0.557681i
\(646\) 0 0
\(647\) −719.190 −0.0437006 −0.0218503 0.999761i \(-0.506956\pi\)
−0.0218503 + 0.999761i \(0.506956\pi\)
\(648\) 0 0
\(649\) −10962.9 −0.663069
\(650\) 0 0
\(651\) −9838.49 + 7101.69i −0.592321 + 0.427553i
\(652\) 0 0
\(653\) 19019.4 1.13980 0.569899 0.821714i \(-0.306982\pi\)
0.569899 + 0.821714i \(0.306982\pi\)
\(654\) 0 0
\(655\) 1496.32i 0.0892615i
\(656\) 0 0
\(657\) −5693.31 + 17160.2i −0.338078 + 1.01900i
\(658\) 0 0
\(659\) 22748.7i 1.34471i −0.740230 0.672354i \(-0.765283\pi\)
0.740230 0.672354i \(-0.234717\pi\)
\(660\) 0 0
\(661\) 23275.3i 1.36960i −0.728733 0.684798i \(-0.759890\pi\)
0.728733 0.684798i \(-0.240110\pi\)
\(662\) 0 0
\(663\) −8098.86 + 5845.98i −0.474410 + 0.342442i
\(664\) 0 0
\(665\) 28174.9i 1.64297i
\(666\) 0 0
\(667\) 21903.6 1.27153
\(668\) 0 0
\(669\) −6160.41 8534.47i −0.356017 0.493216i
\(670\) 0 0
\(671\) 9632.51 0.554186
\(672\) 0 0
\(673\) −28195.5 −1.61495 −0.807473 0.589905i \(-0.799165\pi\)
−0.807473 + 0.589905i \(0.799165\pi\)
\(674\) 0 0
\(675\) 1595.89 + 5071.29i 0.0910011 + 0.289176i
\(676\) 0 0
\(677\) −32369.7 −1.83762 −0.918809 0.394701i \(-0.870848\pi\)
−0.918809 + 0.394701i \(0.870848\pi\)
\(678\) 0 0
\(679\) 4961.07i 0.280396i
\(680\) 0 0
\(681\) 1228.80 + 1702.34i 0.0691447 + 0.0957913i
\(682\) 0 0
\(683\) 1227.33i 0.0687592i −0.999409 0.0343796i \(-0.989054\pi\)
0.999409 0.0343796i \(-0.0109455\pi\)
\(684\) 0 0
\(685\) 13882.6i 0.774345i
\(686\) 0 0
\(687\) −14401.4 19951.3i −0.799776 1.10799i
\(688\) 0 0
\(689\) 4793.06i 0.265023i
\(690\) 0 0
\(691\) 3563.29 0.196170 0.0980852 0.995178i \(-0.468728\pi\)
0.0980852 + 0.995178i \(0.468728\pi\)
\(692\) 0 0
\(693\) −14982.8 + 45159.8i −0.821284 + 2.47544i
\(694\) 0 0
\(695\) −548.465 −0.0299345
\(696\) 0 0
\(697\) −24191.3 −1.31465
\(698\) 0 0
\(699\) 11264.0 + 15604.8i 0.609503 + 0.844389i
\(700\) 0 0
\(701\) −11942.4 −0.643448 −0.321724 0.946834i \(-0.604262\pi\)
−0.321724 + 0.946834i \(0.604262\pi\)
\(702\) 0 0
\(703\) 416.147i 0.0223262i
\(704\) 0 0
\(705\) 3774.04 2724.21i 0.201615 0.145531i
\(706\) 0 0
\(707\) 14955.3i 0.795547i
\(708\) 0 0
\(709\) 22547.9i 1.19436i 0.802105 + 0.597182i \(0.203713\pi\)
−0.802105 + 0.597182i \(0.796287\pi\)
\(710\) 0 0
\(711\) 5880.06 + 1950.85i 0.310154 + 0.102901i
\(712\) 0 0
\(713\) 11495.3i 0.603790i
\(714\) 0 0
\(715\) −11708.8 −0.612426
\(716\) 0 0
\(717\) −26959.6 + 19460.1i −1.40422 + 1.01360i
\(718\) 0 0
\(719\) 2034.54 0.105529 0.0527647 0.998607i \(-0.483197\pi\)
0.0527647 + 0.998607i \(0.483197\pi\)
\(720\) 0 0
\(721\) −5002.39 −0.258389
\(722\) 0 0
\(723\) 15869.0 11454.6i 0.816283 0.589215i
\(724\) 0 0
\(725\) 4634.86 0.237427
\(726\) 0 0
\(727\) 26487.5i 1.35126i −0.737240 0.675631i \(-0.763871\pi\)
0.737240 0.675631i \(-0.236129\pi\)
\(728\) 0 0
\(729\) −16135.9 + 11271.9i −0.819786 + 0.572670i
\(730\) 0 0
\(731\) 23888.5i 1.20869i
\(732\) 0 0
\(733\) 38967.7i 1.96358i −0.189964 0.981791i \(-0.560837\pi\)
0.189964 0.981791i \(-0.439163\pi\)
\(734\) 0 0
\(735\) 38553.0 27828.6i 1.93476 1.39656i
\(736\) 0 0
\(737\) 1160.27i 0.0579907i
\(738\) 0 0
\(739\) 14312.2 0.712425 0.356212 0.934405i \(-0.384068\pi\)
0.356212 + 0.934405i \(0.384068\pi\)
\(740\) 0 0
\(741\) −6536.00 9054.80i −0.324030 0.448902i
\(742\) 0 0
\(743\) −12790.0 −0.631520 −0.315760 0.948839i \(-0.602259\pi\)
−0.315760 + 0.948839i \(0.602259\pi\)
\(744\) 0 0
\(745\) 8774.41 0.431503
\(746\) 0 0
\(747\) 8229.93 + 2730.47i 0.403102 + 0.133739i
\(748\) 0 0
\(749\) −76872.3 −3.75014
\(750\) 0 0
\(751\) 19441.6i 0.944653i 0.881424 + 0.472327i \(0.156586\pi\)
−0.881424 + 0.472327i \(0.843414\pi\)
\(752\) 0 0
\(753\) 6020.70 + 8340.92i 0.291376 + 0.403665i
\(754\) 0 0
\(755\) 16803.4i 0.809985i
\(756\) 0 0
\(757\) 13901.0i 0.667424i −0.942675 0.333712i \(-0.891699\pi\)
0.942675 0.333712i \(-0.108301\pi\)
\(758\) 0 0
\(759\) −26382.4 36549.4i −1.26169 1.74791i
\(760\) 0 0
\(761\) 8818.58i 0.420070i 0.977694 + 0.210035i \(0.0673578\pi\)
−0.977694 + 0.210035i \(0.932642\pi\)
\(762\) 0 0
\(763\) −47659.3 −2.26132
\(764\) 0 0
\(765\) 17751.8 + 5889.57i 0.838978 + 0.278350i
\(766\) 0 0
\(767\) 5861.30 0.275931
\(768\) 0 0
\(769\) 1323.39 0.0620582 0.0310291 0.999518i \(-0.490122\pi\)
0.0310291 + 0.999518i \(0.490122\pi\)
\(770\) 0 0
\(771\) 3289.18 + 4556.75i 0.153641 + 0.212850i
\(772\) 0 0
\(773\) 34327.1 1.59723 0.798617 0.601840i \(-0.205565\pi\)
0.798617 + 0.601840i \(0.205565\pi\)
\(774\) 0 0
\(775\) 2432.43i 0.112743i
\(776\) 0 0
\(777\) −768.644 + 554.828i −0.0354890 + 0.0256169i
\(778\) 0 0
\(779\) 27046.7i 1.24397i
\(780\) 0 0
\(781\) 19363.8i 0.887183i
\(782\) 0 0
\(783\) 5150.91 + 16368.2i 0.235094 + 0.747063i
\(784\) 0 0
\(785\) 28224.0i 1.28326i
\(786\) 0 0
\(787\) −15375.3 −0.696405 −0.348203 0.937419i \(-0.613208\pi\)
−0.348203 + 0.937419i \(0.613208\pi\)
\(788\) 0 0
\(789\) −4723.84 + 3409.80i −0.213147 + 0.153856i
\(790\) 0 0
\(791\) 51062.5 2.29529
\(792\) 0 0
\(793\) −5150.00 −0.230620
\(794\) 0 0
\(795\) −7277.27 + 5252.93i −0.324652 + 0.234342i
\(796\) 0 0
\(797\) 3347.25 0.148765 0.0743826 0.997230i \(-0.476301\pi\)
0.0743826 + 0.997230i \(0.476301\pi\)
\(798\) 0 0
\(799\) 7123.67i 0.315416i
\(800\) 0 0
\(801\) −3368.50 1117.58i −0.148589 0.0492979i
\(802\) 0 0
\(803\) 32437.4i 1.42552i
\(804\) 0 0
\(805\) 60804.1i 2.66219i
\(806\) 0 0
\(807\) 7772.76 5610.59i 0.339051 0.244736i
\(808\) 0 0
\(809\) 39577.3i 1.71998i −0.510309 0.859991i \(-0.670469\pi\)
0.510309 0.859991i \(-0.329531\pi\)
\(810\) 0 0
\(811\) −12614.4 −0.546180 −0.273090 0.961989i \(-0.588046\pi\)
−0.273090 + 0.961989i \(0.588046\pi\)
\(812\) 0 0
\(813\) −2855.60 3956.07i −0.123186 0.170659i
\(814\) 0 0
\(815\) 29160.6 1.25332
\(816\) 0 0
\(817\) −26708.2 −1.14370
\(818\) 0 0
\(819\) 8010.54 24144.6i 0.341772 1.03014i
\(820\) 0 0
\(821\) 18243.6 0.775526 0.387763 0.921759i \(-0.373248\pi\)
0.387763 + 0.921759i \(0.373248\pi\)
\(822\) 0 0
\(823\) 18350.6i 0.777230i 0.921400 + 0.388615i \(0.127046\pi\)
−0.921400 + 0.388615i \(0.872954\pi\)
\(824\) 0 0
\(825\) −5582.57 7733.94i −0.235588 0.326377i
\(826\) 0 0
\(827\) 15785.4i 0.663739i 0.943325 + 0.331869i \(0.107679\pi\)
−0.943325 + 0.331869i \(0.892321\pi\)
\(828\) 0 0
\(829\) 36102.8i 1.51255i 0.654255 + 0.756274i \(0.272982\pi\)
−0.654255 + 0.756274i \(0.727018\pi\)
\(830\) 0 0
\(831\) 3348.15 + 4638.43i 0.139766 + 0.193629i
\(832\) 0 0
\(833\) 72770.6i 3.02683i
\(834\) 0 0
\(835\) −3466.13 −0.143653
\(836\) 0 0
\(837\) 8590.21 2703.26i 0.354744 0.111635i
\(838\) 0 0
\(839\) 18961.8 0.780255 0.390127 0.920761i \(-0.372431\pi\)
0.390127 + 0.920761i \(0.372431\pi\)
\(840\) 0 0
\(841\) −9429.48 −0.386628
\(842\) 0 0
\(843\) 2275.45 + 3152.35i 0.0929665 + 0.128793i
\(844\) 0 0
\(845\) −14244.6 −0.579914
\(846\) 0 0
\(847\) 36943.4i 1.49869i
\(848\) 0 0
\(849\) −9059.00 + 6539.03i −0.366201 + 0.264333i
\(850\) 0 0
\(851\) 898.084i 0.0361762i
\(852\) 0 0
\(853\) 27761.8i 1.11435i −0.830394 0.557177i \(-0.811884\pi\)
0.830394 0.557177i \(-0.188116\pi\)
\(854\) 0 0
\(855\) −6584.74 + 19847.1i −0.263384 + 0.793868i
\(856\) 0 0
\(857\) 2082.01i 0.0829874i 0.999139 + 0.0414937i \(0.0132117\pi\)
−0.999139 + 0.0414937i \(0.986788\pi\)
\(858\) 0 0
\(859\) −34995.2 −1.39001 −0.695005 0.719004i \(-0.744598\pi\)
−0.695005 + 0.719004i \(0.744598\pi\)
\(860\) 0 0
\(861\) 49956.6 36060.1i 1.97737 1.42732i
\(862\) 0 0
\(863\) −6045.46 −0.238459 −0.119229 0.992867i \(-0.538042\pi\)
−0.119229 + 0.992867i \(0.538042\pi\)
\(864\) 0 0
\(865\) 27357.3 1.07535
\(866\) 0 0
\(867\) −2510.61 + 1812.23i −0.0983446 + 0.0709878i
\(868\) 0 0
\(869\) −11114.9 −0.433886
\(870\) 0 0
\(871\) 620.337i 0.0241324i
\(872\) 0 0
\(873\) 1159.45 3494.70i 0.0449501 0.135484i
\(874\) 0 0
\(875\) 55307.3i 2.13683i
\(876\) 0 0
\(877\) 30101.8i 1.15903i 0.814963 + 0.579514i \(0.196757\pi\)
−0.814963 + 0.579514i \(0.803243\pi\)
\(878\) 0 0
\(879\) 22131.2 15974.9i 0.849224 0.612993i
\(880\) 0 0
\(881\) 16655.3i 0.636926i −0.947935 0.318463i \(-0.896833\pi\)
0.947935 0.318463i \(-0.103167\pi\)
\(882\) 0 0
\(883\) 24940.5 0.950528 0.475264 0.879843i \(-0.342353\pi\)
0.475264 + 0.879843i \(0.342353\pi\)
\(884\) 0 0
\(885\) −6423.66 8899.16i −0.243987 0.338014i
\(886\) 0 0
\(887\) −32143.9 −1.21678 −0.608391 0.793637i \(-0.708185\pi\)
−0.608391 + 0.793637i \(0.708185\pi\)
\(888\) 0 0
\(889\) 57613.3 2.17355
\(890\) 0 0
\(891\) 21108.5 28310.1i 0.793673 1.06445i
\(892\) 0 0
\(893\) −7964.51 −0.298457
\(894\) 0 0
\(895\) 26454.1i 0.988004i
\(896\) 0 0
\(897\) 14105.3 + 19541.1i 0.525041 + 0.727378i
\(898\) 0 0
\(899\) 7850.94i 0.291261i
\(900\) 0 0
\(901\) 13736.2i 0.507900i
\(902\) 0 0
\(903\) −35608.7 49331.3i −1.31227 1.81799i
\(904\) 0 0
\(905\) 16985.9i 0.623902i
\(906\) 0 0
\(907\) −3376.58 −0.123613 −0.0618067 0.998088i \(-0.519686\pi\)
−0.0618067 + 0.998088i \(0.519686\pi\)
\(908\) 0 0
\(909\) 3495.19 10534.9i 0.127534 0.384401i
\(910\) 0 0
\(911\) −45248.4 −1.64561 −0.822803 0.568326i \(-0.807591\pi\)
−0.822803 + 0.568326i \(0.807591\pi\)
\(912\) 0 0
\(913\) −15556.8 −0.563914
\(914\) 0 0
\(915\) 5644.12 + 7819.21i 0.203922 + 0.282508i
\(916\) 0 0
\(917\) 5832.52 0.210040
\(918\) 0 0
\(919\) 18545.2i 0.665668i 0.942985 + 0.332834i \(0.108005\pi\)
−0.942985 + 0.332834i \(0.891995\pi\)
\(920\) 0 0
\(921\) 22486.1 16231.1i 0.804498 0.580709i
\(922\) 0 0
\(923\) 10352.8i 0.369195i
\(924\) 0 0
\(925\) 190.037i 0.00675499i
\(926\) 0 0
\(927\) 3523.81 + 1169.11i 0.124851 + 0.0414223i
\(928\) 0 0
\(929\) 33976.7i 1.19993i −0.800024 0.599967i \(-0.795180\pi\)
0.800024 0.599967i \(-0.204820\pi\)
\(930\) 0 0
\(931\) −81360.0 −2.86409
\(932\) 0 0
\(933\) 4360.03 3147.19i 0.152992 0.110433i
\(934\) 0 0
\(935\) −33555.7 −1.17368
\(936\) 0 0
\(937\) 22803.1 0.795031 0.397516 0.917595i \(-0.369872\pi\)
0.397516 + 0.917595i \(0.369872\pi\)
\(938\) 0 0
\(939\) 40321.7 29105.3i 1.40133 1.01152i
\(940\) 0 0
\(941\) 24174.7 0.837484 0.418742 0.908105i \(-0.362471\pi\)
0.418742 + 0.908105i \(0.362471\pi\)
\(942\) 0 0
\(943\) 58369.4i 2.01566i
\(944\) 0 0
\(945\) −45437.7 + 14298.8i −1.56412 + 0.492213i
\(946\) 0 0
\(947\) 8044.99i 0.276058i −0.990428 0.138029i \(-0.955923\pi\)
0.990428 0.138029i \(-0.0440767\pi\)
\(948\) 0 0
\(949\) 17342.6i 0.593220i
\(950\) 0 0
\(951\) 15138.3 10927.2i 0.516185 0.372596i
\(952\) 0 0
\(953\) 39730.8i 1.35048i 0.737598 + 0.675240i \(0.235960\pi\)
−0.737598 + 0.675240i \(0.764040\pi\)
\(954\) 0 0
\(955\) 43184.5 1.46326
\(956\) 0 0
\(957\) −18018.4 24962.2i −0.608622 0.843169i
\(958\) 0 0
\(959\) −54112.9 −1.82210
\(960\) 0 0
\(961\) 25670.7 0.861694
\(962\) 0 0
\(963\) 54150.8 + 17965.8i 1.81203 + 0.601183i
\(964\) 0 0
\(965\) 25855.2 0.862496
\(966\) 0 0
\(967\) 17685.2i 0.588126i 0.955786 + 0.294063i \(0.0950076\pi\)
−0.955786 + 0.294063i \(0.904992\pi\)
\(968\) 0 0
\(969\) −18731.2 25949.7i −0.620982 0.860292i
\(970\) 0 0
\(971\) 44010.1i 1.45453i −0.686355 0.727267i \(-0.740790\pi\)
0.686355 0.727267i \(-0.259210\pi\)
\(972\) 0 0
\(973\) 2137.86i 0.0704385i
\(974\) 0 0
\(975\) 2984.71 + 4134.94i 0.0980382 + 0.135820i
\(976\) 0 0
\(977\) 33518.4i 1.09759i −0.835956 0.548796i \(-0.815086\pi\)
0.835956 0.548796i \(-0.184914\pi\)
\(978\) 0 0
\(979\) 6367.36 0.207867
\(980\) 0 0
\(981\) 33572.4 + 11138.4i 1.09265 + 0.362511i
\(982\) 0 0
\(983\) −43740.3 −1.41923 −0.709613 0.704592i \(-0.751130\pi\)
−0.709613 + 0.704592i \(0.751130\pi\)
\(984\) 0 0
\(985\) 36212.2 1.17139
\(986\) 0 0
\(987\) −10618.7 14710.8i −0.342448 0.474418i
\(988\) 0 0
\(989\) 57638.8 1.85319
\(990\) 0 0
\(991\) 10468.7i 0.335568i 0.985824 + 0.167784i \(0.0536611\pi\)
−0.985824 + 0.167784i \(0.946339\pi\)
\(992\) 0 0
\(993\) 17355.4 12527.6i 0.554638 0.400353i
\(994\) 0 0
\(995\) 18806.0i 0.599186i
\(996\) 0 0
\(997\) 9561.56i 0.303729i −0.988401 0.151864i \(-0.951472\pi\)
0.988401 0.151864i \(-0.0485277\pi\)
\(998\) 0 0
\(999\) 671.121 211.195i 0.0212546 0.00668862i
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 768.4.f.g.383.9 12
3.2 odd 2 768.4.f.h.383.10 12
4.3 odd 2 768.4.f.e.383.4 12
8.3 odd 2 768.4.f.h.383.9 12
8.5 even 2 768.4.f.f.383.4 12
12.11 even 2 768.4.f.f.383.3 12
16.3 odd 4 384.4.c.a.383.5 12
16.5 even 4 384.4.c.b.383.5 yes 12
16.11 odd 4 384.4.c.c.383.8 yes 12
16.13 even 4 384.4.c.d.383.8 yes 12
24.5 odd 2 768.4.f.e.383.3 12
24.11 even 2 inner 768.4.f.g.383.10 12
48.5 odd 4 384.4.c.c.383.7 yes 12
48.11 even 4 384.4.c.b.383.6 yes 12
48.29 odd 4 384.4.c.a.383.6 yes 12
48.35 even 4 384.4.c.d.383.7 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.4.c.a.383.5 12 16.3 odd 4
384.4.c.a.383.6 yes 12 48.29 odd 4
384.4.c.b.383.5 yes 12 16.5 even 4
384.4.c.b.383.6 yes 12 48.11 even 4
384.4.c.c.383.7 yes 12 48.5 odd 4
384.4.c.c.383.8 yes 12 16.11 odd 4
384.4.c.d.383.7 yes 12 48.35 even 4
384.4.c.d.383.8 yes 12 16.13 even 4
768.4.f.e.383.3 12 24.5 odd 2
768.4.f.e.383.4 12 4.3 odd 2
768.4.f.f.383.3 12 12.11 even 2
768.4.f.f.383.4 12 8.5 even 2
768.4.f.g.383.9 12 1.1 even 1 trivial
768.4.f.g.383.10 12 24.11 even 2 inner
768.4.f.h.383.9 12 8.3 odd 2
768.4.f.h.383.10 12 3.2 odd 2