Properties

Label 128.4.g.a.81.6
Level $128$
Weight $4$
Character 128.81
Analytic conductor $7.552$
Analytic rank $0$
Dimension $44$
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] = [128,4,Mod(17,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([0, 7]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("128.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 128.g (of order \(8\), degree \(4\), 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: \(7.55224448073\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(11\) over \(\Q(\zeta_{8})\)
Twist minimal: no (minimal twist has level 32)
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

Embedding invariants

Embedding label 81.6
Character \(\chi\) \(=\) 128.81
Dual form 128.4.g.a.49.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.36212 + 0.564209i) q^{3} +(6.58151 + 15.8892i) q^{5} +(-14.5517 + 14.5517i) q^{7} +(-17.5548 - 17.5548i) q^{9} +O(q^{10})\) \(q+(1.36212 + 0.564209i) q^{3} +(6.58151 + 15.8892i) q^{5} +(-14.5517 + 14.5517i) q^{7} +(-17.5548 - 17.5548i) q^{9} +(-34.4676 + 14.2769i) q^{11} +(15.3924 - 37.1606i) q^{13} +25.3563i q^{15} +103.310i q^{17} +(-12.4092 + 29.9584i) q^{19} +(-28.0313 + 11.6109i) q^{21} +(72.3950 + 72.3950i) q^{23} +(-120.761 + 120.761i) q^{25} +(-29.2409 - 70.5937i) q^{27} +(23.9061 + 9.90225i) q^{29} +124.769 q^{31} -55.0042 q^{33} +(-326.986 - 135.442i) q^{35} +(18.0425 + 43.5584i) q^{37} +(41.9327 - 41.9327i) q^{39} +(-45.1360 - 45.1360i) q^{41} +(457.470 - 189.490i) q^{43} +(163.395 - 394.469i) q^{45} +582.766i q^{47} -80.5012i q^{49} +(-58.2884 + 140.721i) q^{51} +(395.810 - 163.950i) q^{53} +(-453.698 - 453.698i) q^{55} +(-33.8056 + 33.8056i) q^{57} +(-142.805 - 344.762i) q^{59} +(-34.8958 - 14.4543i) q^{61} +510.904 q^{63} +691.757 q^{65} +(-196.686 - 81.4699i) q^{67} +(57.7649 + 139.457i) q^{69} +(520.831 - 520.831i) q^{71} +(-582.329 - 582.329i) q^{73} +(-232.626 + 96.3570i) q^{75} +(293.807 - 709.314i) q^{77} -157.779i q^{79} +557.655i q^{81} +(-54.5324 + 131.653i) q^{83} +(-1641.51 + 679.936i) q^{85} +(26.9761 + 26.9761i) q^{87} +(-272.884 + 272.884i) q^{89} +(316.763 + 764.733i) q^{91} +(169.951 + 70.3959i) q^{93} -557.686 q^{95} +788.873 q^{97} +(855.703 + 354.444i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 44 q + 4 q^{3} - 4 q^{5} + 4 q^{7} - 4 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{19} - 4 q^{21} - 324 q^{23} - 4 q^{25} + 268 q^{27} - 4 q^{29} + 752 q^{31} - 8 q^{33} + 460 q^{35} - 4 q^{37} - 596 q^{39} - 4 q^{41} - 804 q^{43} + 104 q^{45} + 1384 q^{51} + 748 q^{53} + 292 q^{55} - 4 q^{57} - 1372 q^{59} - 1828 q^{61} - 2512 q^{63} - 8 q^{65} - 2036 q^{67} - 1060 q^{69} - 220 q^{71} - 4 q^{73} + 1712 q^{75} + 1900 q^{77} - 2436 q^{83} + 496 q^{85} + 1292 q^{87} - 4 q^{89} + 3604 q^{91} - 112 q^{93} + 6088 q^{95} - 8 q^{97} + 5424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{8}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.36212 + 0.564209i 0.262140 + 0.108582i 0.509883 0.860244i \(-0.329689\pi\)
−0.247743 + 0.968826i \(0.579689\pi\)
\(4\) 0 0
\(5\) 6.58151 + 15.8892i 0.588669 + 1.42117i 0.884776 + 0.466017i \(0.154311\pi\)
−0.296107 + 0.955155i \(0.595689\pi\)
\(6\) 0 0
\(7\) −14.5517 + 14.5517i −0.785715 + 0.785715i −0.980789 0.195073i \(-0.937506\pi\)
0.195073 + 0.980789i \(0.437506\pi\)
\(8\) 0 0
\(9\) −17.5548 17.5548i −0.650179 0.650179i
\(10\) 0 0
\(11\) −34.4676 + 14.2769i −0.944761 + 0.391333i −0.801259 0.598317i \(-0.795836\pi\)
−0.143502 + 0.989650i \(0.545836\pi\)
\(12\) 0 0
\(13\) 15.3924 37.1606i 0.328391 0.792807i −0.670321 0.742071i \(-0.733843\pi\)
0.998712 0.0507354i \(-0.0161565\pi\)
\(14\) 0 0
\(15\) 25.3563i 0.436465i
\(16\) 0 0
\(17\) 103.310i 1.47390i 0.675946 + 0.736951i \(0.263735\pi\)
−0.675946 + 0.736951i \(0.736265\pi\)
\(18\) 0 0
\(19\) −12.4092 + 29.9584i −0.149835 + 0.361733i −0.980920 0.194412i \(-0.937720\pi\)
0.831085 + 0.556145i \(0.187720\pi\)
\(20\) 0 0
\(21\) −28.0313 + 11.6109i −0.291282 + 0.120653i
\(22\) 0 0
\(23\) 72.3950 + 72.3950i 0.656322 + 0.656322i 0.954508 0.298186i \(-0.0963815\pi\)
−0.298186 + 0.954508i \(0.596381\pi\)
\(24\) 0 0
\(25\) −120.761 + 120.761i −0.966091 + 0.966091i
\(26\) 0 0
\(27\) −29.2409 70.5937i −0.208422 0.503176i
\(28\) 0 0
\(29\) 23.9061 + 9.90225i 0.153078 + 0.0634069i 0.457907 0.889000i \(-0.348599\pi\)
−0.304829 + 0.952407i \(0.598599\pi\)
\(30\) 0 0
\(31\) 124.769 0.722877 0.361439 0.932396i \(-0.382286\pi\)
0.361439 + 0.932396i \(0.382286\pi\)
\(32\) 0 0
\(33\) −55.0042 −0.290152
\(34\) 0 0
\(35\) −326.986 135.442i −1.57916 0.654110i
\(36\) 0 0
\(37\) 18.0425 + 43.5584i 0.0801667 + 0.193540i 0.958881 0.283809i \(-0.0915980\pi\)
−0.878714 + 0.477348i \(0.841598\pi\)
\(38\) 0 0
\(39\) 41.9327 41.9327i 0.172169 0.172169i
\(40\) 0 0
\(41\) −45.1360 45.1360i −0.171928 0.171928i 0.615898 0.787826i \(-0.288793\pi\)
−0.787826 + 0.615898i \(0.788793\pi\)
\(42\) 0 0
\(43\) 457.470 189.490i 1.62241 0.672023i 0.628057 0.778168i \(-0.283851\pi\)
0.994351 + 0.106144i \(0.0338506\pi\)
\(44\) 0 0
\(45\) 163.395 394.469i 0.541276 1.30676i
\(46\) 0 0
\(47\) 582.766i 1.80862i 0.426877 + 0.904310i \(0.359614\pi\)
−0.426877 + 0.904310i \(0.640386\pi\)
\(48\) 0 0
\(49\) 80.5012i 0.234697i
\(50\) 0 0
\(51\) −58.2884 + 140.721i −0.160039 + 0.386369i
\(52\) 0 0
\(53\) 395.810 163.950i 1.02582 0.424910i 0.194620 0.980879i \(-0.437653\pi\)
0.831203 + 0.555969i \(0.187653\pi\)
\(54\) 0 0
\(55\) −453.698 453.698i −1.11230 1.11230i
\(56\) 0 0
\(57\) −33.8056 + 33.8056i −0.0785555 + 0.0785555i
\(58\) 0 0
\(59\) −142.805 344.762i −0.315113 0.760750i −0.999500 0.0316306i \(-0.989930\pi\)
0.684387 0.729119i \(-0.260070\pi\)
\(60\) 0 0
\(61\) −34.8958 14.4543i −0.0732450 0.0303391i 0.345760 0.938323i \(-0.387621\pi\)
−0.419005 + 0.907984i \(0.637621\pi\)
\(62\) 0 0
\(63\) 510.904 1.02171
\(64\) 0 0
\(65\) 691.757 1.32003
\(66\) 0 0
\(67\) −196.686 81.4699i −0.358642 0.148554i 0.196085 0.980587i \(-0.437177\pi\)
−0.554727 + 0.832033i \(0.687177\pi\)
\(68\) 0 0
\(69\) 57.7649 + 139.457i 0.100784 + 0.243313i
\(70\) 0 0
\(71\) 520.831 520.831i 0.870582 0.870582i −0.121954 0.992536i \(-0.538916\pi\)
0.992536 + 0.121954i \(0.0389161\pi\)
\(72\) 0 0
\(73\) −582.329 582.329i −0.933649 0.933649i 0.0642823 0.997932i \(-0.479524\pi\)
−0.997932 + 0.0642823i \(0.979524\pi\)
\(74\) 0 0
\(75\) −232.626 + 96.3570i −0.358152 + 0.148351i
\(76\) 0 0
\(77\) 293.807 709.314i 0.434837 1.04979i
\(78\) 0 0
\(79\) 157.779i 0.224702i −0.993669 0.112351i \(-0.964162\pi\)
0.993669 0.112351i \(-0.0358381\pi\)
\(80\) 0 0
\(81\) 557.655i 0.764959i
\(82\) 0 0
\(83\) −54.5324 + 131.653i −0.0721170 + 0.174106i −0.955828 0.293928i \(-0.905037\pi\)
0.883711 + 0.468034i \(0.155037\pi\)
\(84\) 0 0
\(85\) −1641.51 + 679.936i −2.09467 + 0.867640i
\(86\) 0 0
\(87\) 26.9761 + 26.9761i 0.0332430 + 0.0332430i
\(88\) 0 0
\(89\) −272.884 + 272.884i −0.325007 + 0.325007i −0.850684 0.525677i \(-0.823812\pi\)
0.525677 + 0.850684i \(0.323812\pi\)
\(90\) 0 0
\(91\) 316.763 + 764.733i 0.364898 + 0.880943i
\(92\) 0 0
\(93\) 169.951 + 70.3959i 0.189495 + 0.0784915i
\(94\) 0 0
\(95\) −557.686 −0.602288
\(96\) 0 0
\(97\) 788.873 0.825752 0.412876 0.910787i \(-0.364524\pi\)
0.412876 + 0.910787i \(0.364524\pi\)
\(98\) 0 0
\(99\) 855.703 + 354.444i 0.868701 + 0.359828i
\(100\) 0 0
\(101\) 585.257 + 1412.93i 0.576586 + 1.39200i 0.895858 + 0.444339i \(0.146562\pi\)
−0.319272 + 0.947663i \(0.603438\pi\)
\(102\) 0 0
\(103\) −1120.20 + 1120.20i −1.07161 + 1.07161i −0.0743853 + 0.997230i \(0.523699\pi\)
−0.997230 + 0.0743853i \(0.976301\pi\)
\(104\) 0 0
\(105\) −368.977 368.977i −0.342937 0.342937i
\(106\) 0 0
\(107\) 1665.63 689.926i 1.50488 0.623343i 0.530388 0.847755i \(-0.322046\pi\)
0.974494 + 0.224413i \(0.0720463\pi\)
\(108\) 0 0
\(109\) −342.786 + 827.559i −0.301220 + 0.727209i 0.698710 + 0.715405i \(0.253758\pi\)
−0.999930 + 0.0118047i \(0.996242\pi\)
\(110\) 0 0
\(111\) 69.5116i 0.0594392i
\(112\) 0 0
\(113\) 924.353i 0.769521i −0.923016 0.384760i \(-0.874284\pi\)
0.923016 0.384760i \(-0.125716\pi\)
\(114\) 0 0
\(115\) −673.829 + 1626.77i −0.546390 + 1.31910i
\(116\) 0 0
\(117\) −922.559 + 382.137i −0.728980 + 0.301953i
\(118\) 0 0
\(119\) −1503.33 1503.33i −1.15807 1.15807i
\(120\) 0 0
\(121\) 43.0249 43.0249i 0.0323253 0.0323253i
\(122\) 0 0
\(123\) −36.0145 86.9467i −0.0264010 0.0637376i
\(124\) 0 0
\(125\) −727.445 301.318i −0.520517 0.215605i
\(126\) 0 0
\(127\) −569.829 −0.398143 −0.199072 0.979985i \(-0.563793\pi\)
−0.199072 + 0.979985i \(0.563793\pi\)
\(128\) 0 0
\(129\) 730.042 0.498268
\(130\) 0 0
\(131\) −1171.87 485.406i −0.781581 0.323742i −0.0440278 0.999030i \(-0.514019\pi\)
−0.737553 + 0.675289i \(0.764019\pi\)
\(132\) 0 0
\(133\) −255.370 616.518i −0.166492 0.401947i
\(134\) 0 0
\(135\) 929.227 929.227i 0.592408 0.592408i
\(136\) 0 0
\(137\) 946.041 + 946.041i 0.589969 + 0.589969i 0.937623 0.347654i \(-0.113022\pi\)
−0.347654 + 0.937623i \(0.613022\pi\)
\(138\) 0 0
\(139\) −1313.93 + 544.249i −0.801772 + 0.332105i −0.745666 0.666320i \(-0.767869\pi\)
−0.0561063 + 0.998425i \(0.517869\pi\)
\(140\) 0 0
\(141\) −328.802 + 793.797i −0.196384 + 0.474112i
\(142\) 0 0
\(143\) 1500.59i 0.877523i
\(144\) 0 0
\(145\) 445.021i 0.254876i
\(146\) 0 0
\(147\) 45.4195 109.652i 0.0254839 0.0615236i
\(148\) 0 0
\(149\) 2438.15 1009.92i 1.34054 0.555272i 0.406900 0.913473i \(-0.366610\pi\)
0.933644 + 0.358201i \(0.116610\pi\)
\(150\) 0 0
\(151\) 872.975 + 872.975i 0.470475 + 0.470475i 0.902068 0.431594i \(-0.142049\pi\)
−0.431594 + 0.902068i \(0.642049\pi\)
\(152\) 0 0
\(153\) 1813.59 1813.59i 0.958301 0.958301i
\(154\) 0 0
\(155\) 821.170 + 1982.48i 0.425535 + 1.02733i
\(156\) 0 0
\(157\) −866.489 358.912i −0.440467 0.182447i 0.151418 0.988470i \(-0.451616\pi\)
−0.591885 + 0.806022i \(0.701616\pi\)
\(158\) 0 0
\(159\) 631.643 0.315047
\(160\) 0 0
\(161\) −2106.94 −1.03136
\(162\) 0 0
\(163\) 745.455 + 308.777i 0.358212 + 0.148376i 0.554529 0.832164i \(-0.312898\pi\)
−0.196318 + 0.980540i \(0.562898\pi\)
\(164\) 0 0
\(165\) −362.011 873.972i −0.170803 0.412355i
\(166\) 0 0
\(167\) 741.751 741.751i 0.343703 0.343703i −0.514055 0.857758i \(-0.671857\pi\)
0.857758 + 0.514055i \(0.171857\pi\)
\(168\) 0 0
\(169\) 409.532 + 409.532i 0.186405 + 0.186405i
\(170\) 0 0
\(171\) 743.756 308.074i 0.332611 0.137772i
\(172\) 0 0
\(173\) −34.3941 + 83.0347i −0.0151152 + 0.0364914i −0.931257 0.364363i \(-0.881287\pi\)
0.916142 + 0.400854i \(0.131287\pi\)
\(174\) 0 0
\(175\) 3514.56i 1.51815i
\(176\) 0 0
\(177\) 550.180i 0.233639i
\(178\) 0 0
\(179\) 1402.83 3386.73i 0.585768 1.41417i −0.301746 0.953388i \(-0.597570\pi\)
0.887514 0.460781i \(-0.152430\pi\)
\(180\) 0 0
\(181\) 504.860 209.120i 0.207326 0.0858771i −0.276604 0.960984i \(-0.589209\pi\)
0.483930 + 0.875107i \(0.339209\pi\)
\(182\) 0 0
\(183\) −39.3770 39.3770i −0.0159062 0.0159062i
\(184\) 0 0
\(185\) −573.361 + 573.361i −0.227861 + 0.227861i
\(186\) 0 0
\(187\) −1474.95 3560.85i −0.576786 1.39249i
\(188\) 0 0
\(189\) 1452.76 + 601.752i 0.559114 + 0.231593i
\(190\) 0 0
\(191\) 17.6918 0.00670226 0.00335113 0.999994i \(-0.498933\pi\)
0.00335113 + 0.999994i \(0.498933\pi\)
\(192\) 0 0
\(193\) −3321.63 −1.23884 −0.619420 0.785059i \(-0.712632\pi\)
−0.619420 + 0.785059i \(0.712632\pi\)
\(194\) 0 0
\(195\) 942.256 + 390.295i 0.346033 + 0.143331i
\(196\) 0 0
\(197\) −310.402 749.376i −0.112260 0.271020i 0.857758 0.514054i \(-0.171857\pi\)
−0.970018 + 0.243035i \(0.921857\pi\)
\(198\) 0 0
\(199\) −1908.35 + 1908.35i −0.679795 + 0.679795i −0.959954 0.280158i \(-0.909613\pi\)
0.280158 + 0.959954i \(0.409613\pi\)
\(200\) 0 0
\(201\) −221.944 221.944i −0.0778841 0.0778841i
\(202\) 0 0
\(203\) −491.968 + 203.780i −0.170095 + 0.0704559i
\(204\) 0 0
\(205\) 420.110 1014.24i 0.143131 0.345548i
\(206\) 0 0
\(207\) 2541.77i 0.853454i
\(208\) 0 0
\(209\) 1209.76i 0.400387i
\(210\) 0 0
\(211\) 969.860 2341.45i 0.316435 0.763943i −0.683002 0.730416i \(-0.739326\pi\)
0.999438 0.0335266i \(-0.0106739\pi\)
\(212\) 0 0
\(213\) 1003.29 415.578i 0.322744 0.133685i
\(214\) 0 0
\(215\) 6021.69 + 6021.69i 1.91012 + 1.91012i
\(216\) 0 0
\(217\) −1815.60 + 1815.60i −0.567976 + 0.567976i
\(218\) 0 0
\(219\) −464.647 1121.76i −0.143370 0.346125i
\(220\) 0 0
\(221\) 3839.06 + 1590.19i 1.16852 + 0.484017i
\(222\) 0 0
\(223\) 267.474 0.0803200 0.0401600 0.999193i \(-0.487213\pi\)
0.0401600 + 0.999193i \(0.487213\pi\)
\(224\) 0 0
\(225\) 4239.89 1.25627
\(226\) 0 0
\(227\) 721.748 + 298.958i 0.211031 + 0.0874120i 0.485695 0.874128i \(-0.338567\pi\)
−0.274664 + 0.961540i \(0.588567\pi\)
\(228\) 0 0
\(229\) −1403.35 3387.98i −0.404960 0.977660i −0.986443 0.164101i \(-0.947528\pi\)
0.581484 0.813558i \(-0.302472\pi\)
\(230\) 0 0
\(231\) 800.402 800.402i 0.227977 0.227977i
\(232\) 0 0
\(233\) 95.4522 + 95.4522i 0.0268381 + 0.0268381i 0.720399 0.693560i \(-0.243959\pi\)
−0.693560 + 0.720399i \(0.743959\pi\)
\(234\) 0 0
\(235\) −9259.67 + 3835.48i −2.57036 + 1.06468i
\(236\) 0 0
\(237\) 89.0201 214.913i 0.0243986 0.0589035i
\(238\) 0 0
\(239\) 2210.32i 0.598216i 0.954219 + 0.299108i \(0.0966890\pi\)
−0.954219 + 0.299108i \(0.903311\pi\)
\(240\) 0 0
\(241\) 3029.15i 0.809645i 0.914395 + 0.404823i \(0.132667\pi\)
−0.914395 + 0.404823i \(0.867333\pi\)
\(242\) 0 0
\(243\) −1104.14 + 2665.62i −0.291483 + 0.703703i
\(244\) 0 0
\(245\) 1279.10 529.820i 0.333545 0.138159i
\(246\) 0 0
\(247\) 922.264 + 922.264i 0.237580 + 0.237580i
\(248\) 0 0
\(249\) −148.560 + 148.560i −0.0378095 + 0.0378095i
\(250\) 0 0
\(251\) 134.300 + 324.230i 0.0337727 + 0.0815346i 0.939866 0.341543i \(-0.110950\pi\)
−0.906093 + 0.423078i \(0.860950\pi\)
\(252\) 0 0
\(253\) −3528.86 1461.70i −0.876908 0.363227i
\(254\) 0 0
\(255\) −2619.56 −0.643307
\(256\) 0 0
\(257\) −7459.69 −1.81059 −0.905297 0.424779i \(-0.860352\pi\)
−0.905297 + 0.424779i \(0.860352\pi\)
\(258\) 0 0
\(259\) −896.396 371.299i −0.215055 0.0890788i
\(260\) 0 0
\(261\) −245.836 593.501i −0.0583022 0.140754i
\(262\) 0 0
\(263\) −4192.29 + 4192.29i −0.982919 + 0.982919i −0.999857 0.0169374i \(-0.994608\pi\)
0.0169374 + 0.999857i \(0.494608\pi\)
\(264\) 0 0
\(265\) 5210.05 + 5210.05i 1.20774 + 1.20774i
\(266\) 0 0
\(267\) −525.664 + 217.737i −0.120487 + 0.0499075i
\(268\) 0 0
\(269\) −1275.44 + 3079.18i −0.289088 + 0.697921i −0.999986 0.00537488i \(-0.998289\pi\)
0.710897 + 0.703296i \(0.248289\pi\)
\(270\) 0 0
\(271\) 1188.89i 0.266494i 0.991083 + 0.133247i \(0.0425404\pi\)
−0.991083 + 0.133247i \(0.957460\pi\)
\(272\) 0 0
\(273\) 1220.38i 0.270552i
\(274\) 0 0
\(275\) 2438.25 5886.46i 0.534662 1.29079i
\(276\) 0 0
\(277\) 3832.51 1587.48i 0.831310 0.344340i 0.0738890 0.997266i \(-0.476459\pi\)
0.757421 + 0.652926i \(0.226459\pi\)
\(278\) 0 0
\(279\) −2190.30 2190.30i −0.470000 0.470000i
\(280\) 0 0
\(281\) 230.048 230.048i 0.0488382 0.0488382i −0.682266 0.731104i \(-0.739005\pi\)
0.731104 + 0.682266i \(0.239005\pi\)
\(282\) 0 0
\(283\) 405.744 + 979.554i 0.0852261 + 0.205754i 0.960747 0.277426i \(-0.0894815\pi\)
−0.875521 + 0.483181i \(0.839481\pi\)
\(284\) 0 0
\(285\) −759.635 314.651i −0.157884 0.0653977i
\(286\) 0 0
\(287\) 1313.61 0.270173
\(288\) 0 0
\(289\) −5759.94 −1.17239
\(290\) 0 0
\(291\) 1074.54 + 445.089i 0.216463 + 0.0896619i
\(292\) 0 0
\(293\) −614.388 1483.26i −0.122501 0.295744i 0.850718 0.525622i \(-0.176167\pi\)
−0.973220 + 0.229878i \(0.926167\pi\)
\(294\) 0 0
\(295\) 4538.12 4538.12i 0.895659 0.895659i
\(296\) 0 0
\(297\) 2015.72 + 2015.72i 0.393819 + 0.393819i
\(298\) 0 0
\(299\) 3804.58 1575.91i 0.735867 0.304806i
\(300\) 0 0
\(301\) −3899.55 + 9414.34i −0.746732 + 1.80277i
\(302\) 0 0
\(303\) 2254.80i 0.427507i
\(304\) 0 0
\(305\) 649.596i 0.121953i
\(306\) 0 0
\(307\) −2158.05 + 5210.00i −0.401194 + 0.968569i 0.586182 + 0.810179i \(0.300630\pi\)
−0.987377 + 0.158390i \(0.949370\pi\)
\(308\) 0 0
\(309\) −2157.87 + 893.819i −0.397272 + 0.164555i
\(310\) 0 0
\(311\) −182.317 182.317i −0.0332420 0.0332420i 0.690290 0.723532i \(-0.257483\pi\)
−0.723532 + 0.690290i \(0.757483\pi\)
\(312\) 0 0
\(313\) 3731.88 3731.88i 0.673924 0.673924i −0.284694 0.958618i \(-0.591892\pi\)
0.958618 + 0.284694i \(0.0918921\pi\)
\(314\) 0 0
\(315\) 3362.52 + 8117.84i 0.601450 + 1.45203i
\(316\) 0 0
\(317\) 2940.58 + 1218.03i 0.521007 + 0.215808i 0.627659 0.778488i \(-0.284013\pi\)
−0.106652 + 0.994296i \(0.534013\pi\)
\(318\) 0 0
\(319\) −965.361 −0.169435
\(320\) 0 0
\(321\) 2658.05 0.462174
\(322\) 0 0
\(323\) −3095.00 1281.99i −0.533159 0.220842i
\(324\) 0 0
\(325\) 2628.75 + 6346.37i 0.448668 + 1.08318i
\(326\) 0 0
\(327\) −933.833 + 933.833i −0.157924 + 0.157924i
\(328\) 0 0
\(329\) −8480.20 8480.20i −1.42106 1.42106i
\(330\) 0 0
\(331\) 5918.75 2451.63i 0.982852 0.407111i 0.167371 0.985894i \(-0.446472\pi\)
0.815481 + 0.578783i \(0.196472\pi\)
\(332\) 0 0
\(333\) 447.928 1081.39i 0.0737127 0.177958i
\(334\) 0 0
\(335\) 3661.37i 0.597141i
\(336\) 0 0
\(337\) 11283.3i 1.82385i −0.410355 0.911926i \(-0.634595\pi\)
0.410355 0.911926i \(-0.365405\pi\)
\(338\) 0 0
\(339\) 521.529 1259.08i 0.0835562 0.201722i
\(340\) 0 0
\(341\) −4300.49 + 1781.32i −0.682946 + 0.282886i
\(342\) 0 0
\(343\) −3819.79 3819.79i −0.601310 0.601310i
\(344\) 0 0
\(345\) −1835.67 + 1835.67i −0.286462 + 0.286462i
\(346\) 0 0
\(347\) −4803.69 11597.1i −0.743157 1.79414i −0.592527 0.805550i \(-0.701870\pi\)
−0.150630 0.988590i \(-0.548130\pi\)
\(348\) 0 0
\(349\) −5191.29 2150.30i −0.796228 0.329808i −0.0527834 0.998606i \(-0.516809\pi\)
−0.743445 + 0.668798i \(0.766809\pi\)
\(350\) 0 0
\(351\) −3073.39 −0.467366
\(352\) 0 0
\(353\) 9949.61 1.50018 0.750091 0.661335i \(-0.230010\pi\)
0.750091 + 0.661335i \(0.230010\pi\)
\(354\) 0 0
\(355\) 11703.4 + 4847.72i 1.74973 + 0.724762i
\(356\) 0 0
\(357\) −1199.53 2895.91i −0.177831 0.429322i
\(358\) 0 0
\(359\) 3301.10 3301.10i 0.485308 0.485308i −0.421514 0.906822i \(-0.638501\pi\)
0.906822 + 0.421514i \(0.138501\pi\)
\(360\) 0 0
\(361\) 4106.53 + 4106.53i 0.598706 + 0.598706i
\(362\) 0 0
\(363\) 82.8802 34.3301i 0.0119837 0.00496381i
\(364\) 0 0
\(365\) 5420.12 13085.3i 0.777266 1.87649i
\(366\) 0 0
\(367\) 10621.1i 1.51067i −0.655340 0.755334i \(-0.727474\pi\)
0.655340 0.755334i \(-0.272526\pi\)
\(368\) 0 0
\(369\) 1584.71i 0.223568i
\(370\) 0 0
\(371\) −3373.95 + 8145.43i −0.472147 + 1.13986i
\(372\) 0 0
\(373\) −527.627 + 218.550i −0.0732426 + 0.0303381i −0.419004 0.907984i \(-0.637621\pi\)
0.345761 + 0.938322i \(0.387621\pi\)
\(374\) 0 0
\(375\) −820.862 820.862i −0.113038 0.113038i
\(376\) 0 0
\(377\) 735.947 735.947i 0.100539 0.100539i
\(378\) 0 0
\(379\) 3589.96 + 8666.94i 0.486554 + 1.17465i 0.956443 + 0.291920i \(0.0942941\pi\)
−0.469888 + 0.882726i \(0.655706\pi\)
\(380\) 0 0
\(381\) −776.177 321.503i −0.104369 0.0432312i
\(382\) 0 0
\(383\) 4350.16 0.580372 0.290186 0.956970i \(-0.406283\pi\)
0.290186 + 0.956970i \(0.406283\pi\)
\(384\) 0 0
\(385\) 13204.1 1.74791
\(386\) 0 0
\(387\) −11357.3 4704.34i −1.49179 0.617920i
\(388\) 0 0
\(389\) 763.322 + 1842.82i 0.0994909 + 0.240192i 0.965786 0.259340i \(-0.0835051\pi\)
−0.866295 + 0.499533i \(0.833505\pi\)
\(390\) 0 0
\(391\) −7479.13 + 7479.13i −0.967355 + 0.967355i
\(392\) 0 0
\(393\) −1322.36 1322.36i −0.169731 0.169731i
\(394\) 0 0
\(395\) 2506.97 1038.42i 0.319340 0.132275i
\(396\) 0 0
\(397\) 1876.19 4529.53i 0.237187 0.572621i −0.759802 0.650154i \(-0.774704\pi\)
0.996990 + 0.0775330i \(0.0247043\pi\)
\(398\) 0 0
\(399\) 983.855i 0.123444i
\(400\) 0 0
\(401\) 14992.8i 1.86710i 0.358450 + 0.933549i \(0.383305\pi\)
−0.358450 + 0.933549i \(0.616695\pi\)
\(402\) 0 0
\(403\) 1920.50 4636.49i 0.237387 0.573102i
\(404\) 0 0
\(405\) −8860.68 + 3670.21i −1.08714 + 0.450307i
\(406\) 0 0
\(407\) −1243.76 1243.76i −0.151477 0.151477i
\(408\) 0 0
\(409\) 8450.62 8450.62i 1.02165 1.02165i 0.0218936 0.999760i \(-0.493031\pi\)
0.999760 0.0218936i \(-0.00696949\pi\)
\(410\) 0 0
\(411\) 754.857 + 1822.39i 0.0905946 + 0.218715i
\(412\) 0 0
\(413\) 7094.91 + 2938.81i 0.845322 + 0.350144i
\(414\) 0 0
\(415\) −2450.76 −0.289887
\(416\) 0 0
\(417\) −2096.81 −0.246238
\(418\) 0 0
\(419\) 3136.96 + 1299.37i 0.365753 + 0.151500i 0.557988 0.829849i \(-0.311574\pi\)
−0.192235 + 0.981349i \(0.561574\pi\)
\(420\) 0 0
\(421\) 3434.08 + 8290.60i 0.397546 + 0.959760i 0.988246 + 0.152870i \(0.0488514\pi\)
−0.590701 + 0.806891i \(0.701149\pi\)
\(422\) 0 0
\(423\) 10230.4 10230.4i 1.17593 1.17593i
\(424\) 0 0
\(425\) −12475.9 12475.9i −1.42392 1.42392i
\(426\) 0 0
\(427\) 718.125 297.457i 0.0813876 0.0337118i
\(428\) 0 0
\(429\) −846.648 + 2043.99i −0.0952833 + 0.230034i
\(430\) 0 0
\(431\) 15375.3i 1.71833i 0.511699 + 0.859165i \(0.329016\pi\)
−0.511699 + 0.859165i \(0.670984\pi\)
\(432\) 0 0
\(433\) 12522.9i 1.38987i −0.719073 0.694934i \(-0.755433\pi\)
0.719073 0.694934i \(-0.244567\pi\)
\(434\) 0 0
\(435\) −251.085 + 606.172i −0.0276749 + 0.0668132i
\(436\) 0 0
\(437\) −3067.20 + 1270.48i −0.335753 + 0.139074i
\(438\) 0 0
\(439\) −6689.38 6689.38i −0.727259 0.727259i 0.242814 0.970073i \(-0.421929\pi\)
−0.970073 + 0.242814i \(0.921929\pi\)
\(440\) 0 0
\(441\) −1413.19 + 1413.19i −0.152595 + 0.152595i
\(442\) 0 0
\(443\) −1232.30 2975.03i −0.132163 0.319070i 0.843920 0.536470i \(-0.180242\pi\)
−0.976083 + 0.217400i \(0.930242\pi\)
\(444\) 0 0
\(445\) −6131.88 2539.91i −0.653212 0.270569i
\(446\) 0 0
\(447\) 3890.86 0.411703
\(448\) 0 0
\(449\) −2465.75 −0.259167 −0.129584 0.991569i \(-0.541364\pi\)
−0.129584 + 0.991569i \(0.541364\pi\)
\(450\) 0 0
\(451\) 2200.13 + 911.324i 0.229712 + 0.0951498i
\(452\) 0 0
\(453\) 696.557 + 1681.64i 0.0722453 + 0.174415i
\(454\) 0 0
\(455\) −10066.2 + 10066.2i −1.03717 + 1.03717i
\(456\) 0 0
\(457\) 720.641 + 720.641i 0.0737640 + 0.0737640i 0.743026 0.669262i \(-0.233390\pi\)
−0.669262 + 0.743026i \(0.733390\pi\)
\(458\) 0 0
\(459\) 7293.03 3020.87i 0.741633 0.307194i
\(460\) 0 0
\(461\) 94.7149 228.662i 0.00956900 0.0231016i −0.919023 0.394204i \(-0.871020\pi\)
0.928592 + 0.371103i \(0.121020\pi\)
\(462\) 0 0
\(463\) 6979.44i 0.700566i 0.936644 + 0.350283i \(0.113915\pi\)
−0.936644 + 0.350283i \(0.886085\pi\)
\(464\) 0 0
\(465\) 3163.69i 0.315511i
\(466\) 0 0
\(467\) −795.725 + 1921.05i −0.0788475 + 0.190355i −0.958388 0.285470i \(-0.907850\pi\)
0.879540 + 0.475825i \(0.157850\pi\)
\(468\) 0 0
\(469\) 4047.63 1676.58i 0.398512 0.165069i
\(470\) 0 0
\(471\) −977.762 977.762i −0.0956537 0.0956537i
\(472\) 0 0
\(473\) −13062.5 + 13062.5i −1.26980 + 1.26980i
\(474\) 0 0
\(475\) −2119.27 5116.37i −0.204713 0.494221i
\(476\) 0 0
\(477\) −9826.49 4070.26i −0.943237 0.390702i
\(478\) 0 0
\(479\) −973.751 −0.0928848 −0.0464424 0.998921i \(-0.514788\pi\)
−0.0464424 + 0.998921i \(0.514788\pi\)
\(480\) 0 0
\(481\) 1896.37 0.179766
\(482\) 0 0
\(483\) −2869.90 1188.75i −0.270362 0.111988i
\(484\) 0 0
\(485\) 5191.98 + 12534.6i 0.486094 + 1.17354i
\(486\) 0 0
\(487\) 11379.8 11379.8i 1.05887 1.05887i 0.0607112 0.998155i \(-0.480663\pi\)
0.998155 0.0607112i \(-0.0193369\pi\)
\(488\) 0 0
\(489\) 841.184 + 841.184i 0.0777907 + 0.0777907i
\(490\) 0 0
\(491\) 1394.56 577.645i 0.128178 0.0530932i −0.317672 0.948201i \(-0.602901\pi\)
0.445851 + 0.895107i \(0.352901\pi\)
\(492\) 0 0
\(493\) −1023.00 + 2469.74i −0.0934557 + 0.225622i
\(494\) 0 0
\(495\) 15929.2i 1.44639i
\(496\) 0 0
\(497\) 15157.9i 1.36806i
\(498\) 0 0
\(499\) −5429.65 + 13108.3i −0.487103 + 1.17597i 0.469068 + 0.883162i \(0.344590\pi\)
−0.956171 + 0.292809i \(0.905410\pi\)
\(500\) 0 0
\(501\) 1428.86 591.852i 0.127418 0.0527784i
\(502\) 0 0
\(503\) −11037.0 11037.0i −0.978362 0.978362i 0.0214085 0.999771i \(-0.493185\pi\)
−0.999771 + 0.0214085i \(0.993185\pi\)
\(504\) 0 0
\(505\) −18598.5 + 18598.5i −1.63886 + 1.63886i
\(506\) 0 0
\(507\) 326.770 + 788.893i 0.0286240 + 0.0691045i
\(508\) 0 0
\(509\) 13984.8 + 5792.71i 1.21781 + 0.504435i 0.896714 0.442611i \(-0.145948\pi\)
0.321099 + 0.947046i \(0.395948\pi\)
\(510\) 0 0
\(511\) 16947.7 1.46717
\(512\) 0 0
\(513\) 2477.73 0.213244
\(514\) 0 0
\(515\) −25171.6 10426.4i −2.15377 0.892123i
\(516\) 0 0
\(517\) −8320.11 20086.5i −0.707772 1.70871i
\(518\) 0 0
\(519\) −93.6978 + 93.6978i −0.00792462 + 0.00792462i
\(520\) 0 0
\(521\) 8504.46 + 8504.46i 0.715139 + 0.715139i 0.967606 0.252467i \(-0.0812418\pi\)
−0.252467 + 0.967606i \(0.581242\pi\)
\(522\) 0 0
\(523\) 2991.17 1238.98i 0.250086 0.103589i −0.254119 0.967173i \(-0.581786\pi\)
0.504205 + 0.863584i \(0.331786\pi\)
\(524\) 0 0
\(525\) 1982.94 4787.25i 0.164843 0.397967i
\(526\) 0 0
\(527\) 12889.9i 1.06545i
\(528\) 0 0
\(529\) 1684.91i 0.138482i
\(530\) 0 0
\(531\) −3545.33 + 8559.17i −0.289744 + 0.699504i
\(532\) 0 0
\(533\) −2372.03 + 982.527i −0.192765 + 0.0798461i
\(534\) 0 0
\(535\) 21924.7 + 21924.7i 1.77175 + 1.77175i
\(536\) 0 0
\(537\) 3821.65 3821.65i 0.307107 0.307107i
\(538\) 0 0
\(539\) 1149.31 + 2774.68i 0.0918448 + 0.221733i
\(540\) 0 0
\(541\) 4615.58 + 1911.84i 0.366801 + 0.151934i 0.558468 0.829526i \(-0.311389\pi\)
−0.191667 + 0.981460i \(0.561389\pi\)
\(542\) 0 0
\(543\) 805.668 0.0636732
\(544\) 0 0
\(545\) −15405.3 −1.21081
\(546\) 0 0
\(547\) −1759.02 728.609i −0.137496 0.0569526i 0.312875 0.949794i \(-0.398708\pi\)
−0.450370 + 0.892842i \(0.648708\pi\)
\(548\) 0 0
\(549\) 358.847 + 866.332i 0.0278965 + 0.0673482i
\(550\) 0 0
\(551\) −593.311 + 593.311i −0.0458728 + 0.0458728i
\(552\) 0 0
\(553\) 2295.94 + 2295.94i 0.176552 + 0.176552i
\(554\) 0 0
\(555\) −1104.48 + 457.492i −0.0844733 + 0.0349900i
\(556\) 0 0
\(557\) 7305.07 17636.0i 0.555702 1.34158i −0.357438 0.933937i \(-0.616350\pi\)
0.913140 0.407646i \(-0.133650\pi\)
\(558\) 0 0
\(559\) 19916.6i 1.50694i
\(560\) 0 0
\(561\) 5682.48i 0.427655i
\(562\) 0 0
\(563\) −498.470 + 1203.41i −0.0373144 + 0.0900849i −0.941438 0.337186i \(-0.890525\pi\)
0.904124 + 0.427271i \(0.140525\pi\)
\(564\) 0 0
\(565\) 14687.2 6083.65i 1.09362 0.452993i
\(566\) 0 0
\(567\) −8114.80 8114.80i −0.601040 0.601040i
\(568\) 0 0
\(569\) 13188.4 13188.4i 0.971684 0.971684i −0.0279264 0.999610i \(-0.508890\pi\)
0.999610 + 0.0279264i \(0.00889041\pi\)
\(570\) 0 0
\(571\) −5719.51 13808.1i −0.419184 1.01200i −0.982585 0.185815i \(-0.940507\pi\)
0.563401 0.826184i \(-0.309493\pi\)
\(572\) 0 0
\(573\) 24.0983 + 9.98186i 0.00175693 + 0.000727746i
\(574\) 0 0
\(575\) −17485.1 −1.26813
\(576\) 0 0
\(577\) 7866.08 0.567538 0.283769 0.958893i \(-0.408415\pi\)
0.283769 + 0.958893i \(0.408415\pi\)
\(578\) 0 0
\(579\) −4524.46 1874.09i −0.324750 0.134516i
\(580\) 0 0
\(581\) −1122.23 2709.30i −0.0801342 0.193461i
\(582\) 0 0
\(583\) −11301.9 + 11301.9i −0.802877 + 0.802877i
\(584\) 0 0
\(585\) −12143.7 12143.7i −0.858255 0.858255i
\(586\) 0 0
\(587\) −13370.3 + 5538.16i −0.940122 + 0.389411i −0.799510 0.600653i \(-0.794907\pi\)
−0.140613 + 0.990065i \(0.544907\pi\)
\(588\) 0 0
\(589\) −1548.28 + 3737.88i −0.108312 + 0.261489i
\(590\) 0 0
\(591\) 1195.87i 0.0832346i
\(592\) 0 0
\(593\) 7176.23i 0.496952i −0.968638 0.248476i \(-0.920070\pi\)
0.968638 0.248476i \(-0.0799297\pi\)
\(594\) 0 0
\(595\) 13992.5 33780.9i 0.964095 2.32753i
\(596\) 0 0
\(597\) −3676.11 + 1522.69i −0.252015 + 0.104388i
\(598\) 0 0
\(599\) 11813.4 + 11813.4i 0.805815 + 0.805815i 0.983997 0.178183i \(-0.0570218\pi\)
−0.178183 + 0.983997i \(0.557022\pi\)
\(600\) 0 0
\(601\) −1438.92 + 1438.92i −0.0976618 + 0.0976618i −0.754250 0.656588i \(-0.771999\pi\)
0.656588 + 0.754250i \(0.271999\pi\)
\(602\) 0 0
\(603\) 2022.60 + 4882.98i 0.136595 + 0.329768i
\(604\) 0 0
\(605\) 966.800 + 400.462i 0.0649686 + 0.0269109i
\(606\) 0 0
\(607\) 19331.1 1.29263 0.646314 0.763071i \(-0.276309\pi\)
0.646314 + 0.763071i \(0.276309\pi\)
\(608\) 0 0
\(609\) −785.094 −0.0522391
\(610\) 0 0
\(611\) 21655.9 + 8970.17i 1.43389 + 0.593935i
\(612\) 0 0
\(613\) 2013.56 + 4861.15i 0.132670 + 0.320294i 0.976229 0.216743i \(-0.0695435\pi\)
−0.843559 + 0.537037i \(0.819543\pi\)
\(614\) 0 0
\(615\) 1144.48 1144.48i 0.0750406 0.0750406i
\(616\) 0 0
\(617\) −6574.52 6574.52i −0.428979 0.428979i 0.459301 0.888281i \(-0.348100\pi\)
−0.888281 + 0.459301i \(0.848100\pi\)
\(618\) 0 0
\(619\) −2395.09 + 992.077i −0.155520 + 0.0644183i −0.459085 0.888392i \(-0.651823\pi\)
0.303566 + 0.952810i \(0.401823\pi\)
\(620\) 0 0
\(621\) 2993.74 7227.53i 0.193454 0.467038i
\(622\) 0 0
\(623\) 7941.81i 0.510726i
\(624\) 0 0
\(625\) 7806.17i 0.499595i
\(626\) 0 0
\(627\) 682.557 1647.84i 0.0434748 0.104957i
\(628\) 0 0
\(629\) −4500.02 + 1863.97i −0.285258 + 0.118158i
\(630\) 0 0
\(631\) 2820.94 + 2820.94i 0.177971 + 0.177971i 0.790471 0.612500i \(-0.209836\pi\)
−0.612500 + 0.790471i \(0.709836\pi\)
\(632\) 0 0
\(633\) 2642.13 2642.13i 0.165901 0.165901i
\(634\) 0 0
\(635\) −3750.34 9054.12i −0.234374 0.565830i
\(636\) 0 0
\(637\) −2991.47 1239.11i −0.186070 0.0770726i
\(638\) 0 0
\(639\) −18286.2 −1.13207
\(640\) 0 0
\(641\) 8220.77 0.506553 0.253277 0.967394i \(-0.418492\pi\)
0.253277 + 0.967394i \(0.418492\pi\)
\(642\) 0 0
\(643\) 21031.3 + 8711.45i 1.28988 + 0.534286i 0.918950 0.394374i \(-0.129038\pi\)
0.370932 + 0.928660i \(0.379038\pi\)
\(644\) 0 0
\(645\) 4804.78 + 11599.8i 0.293315 + 0.708124i
\(646\) 0 0
\(647\) 14749.1 14749.1i 0.896208 0.896208i −0.0988901 0.995098i \(-0.531529\pi\)
0.995098 + 0.0988901i \(0.0315292\pi\)
\(648\) 0 0
\(649\) 9844.31 + 9844.31i 0.595413 + 0.595413i
\(650\) 0 0
\(651\) −3497.44 + 1448.69i −0.210561 + 0.0872174i
\(652\) 0 0
\(653\) −4279.06 + 10330.6i −0.256436 + 0.619091i −0.998698 0.0510191i \(-0.983753\pi\)
0.742262 + 0.670110i \(0.233753\pi\)
\(654\) 0 0
\(655\) 21814.8i 1.30134i
\(656\) 0 0
\(657\) 20445.4i 1.21408i
\(658\) 0 0
\(659\) 11437.1 27611.6i 0.676063 1.63216i −0.0950580 0.995472i \(-0.530304\pi\)
0.771121 0.636689i \(-0.219696\pi\)
\(660\) 0 0
\(661\) 19843.3 8219.36i 1.16765 0.483655i 0.287233 0.957861i \(-0.407265\pi\)
0.880414 + 0.474205i \(0.157265\pi\)
\(662\) 0 0
\(663\) 4332.06 + 4332.06i 0.253761 + 0.253761i
\(664\) 0 0
\(665\) 8115.25 8115.25i 0.473227 0.473227i
\(666\) 0 0
\(667\) 1013.81 + 2447.56i 0.0588530 + 0.142084i
\(668\) 0 0
\(669\) 364.332 + 150.911i 0.0210551 + 0.00872132i
\(670\) 0 0
\(671\) 1409.14 0.0810717
\(672\) 0 0
\(673\) −13639.3 −0.781216 −0.390608 0.920557i \(-0.627735\pi\)
−0.390608 + 0.920557i \(0.627735\pi\)
\(674\) 0 0
\(675\) 12056.2 + 4993.82i 0.687469 + 0.284759i
\(676\) 0 0
\(677\) −10163.0 24535.6i −0.576951 1.39288i −0.895537 0.444988i \(-0.853208\pi\)
0.318586 0.947894i \(-0.396792\pi\)
\(678\) 0 0
\(679\) −11479.4 + 11479.4i −0.648806 + 0.648806i
\(680\) 0 0
\(681\) 814.433 + 814.433i 0.0458284 + 0.0458284i
\(682\) 0 0
\(683\) 12279.3 5086.25i 0.687928 0.284949i −0.0112089 0.999937i \(-0.503568\pi\)
0.699136 + 0.714988i \(0.253568\pi\)
\(684\) 0 0
\(685\) −8805.43 + 21258.2i −0.491151 + 1.18574i
\(686\) 0 0
\(687\) 5406.62i 0.300255i
\(688\) 0 0
\(689\) 17232.1i 0.952817i
\(690\) 0 0
\(691\) −12427.1 + 30001.7i −0.684153 + 1.65169i 0.0720884 + 0.997398i \(0.477034\pi\)
−0.756241 + 0.654293i \(0.772966\pi\)
\(692\) 0 0
\(693\) −17609.6 + 7294.15i −0.965273 + 0.399829i
\(694\) 0 0
\(695\) −17295.3 17295.3i −0.943956 0.943956i
\(696\) 0 0
\(697\) 4662.99 4662.99i 0.253405 0.253405i
\(698\) 0 0
\(699\) 76.1625 + 183.873i 0.00412122 + 0.00994950i
\(700\) 0 0
\(701\) −24563.9 10174.7i −1.32349 0.548207i −0.394697 0.918811i \(-0.629150\pi\)
−0.928791 + 0.370605i \(0.879150\pi\)
\(702\) 0 0
\(703\) −1528.83 −0.0820214
\(704\) 0 0
\(705\) −14776.8 −0.789399
\(706\) 0 0
\(707\) −29077.0 12044.1i −1.54675 0.640685i
\(708\) 0 0
\(709\) −4032.81 9736.05i −0.213618 0.515720i 0.780356 0.625336i \(-0.215038\pi\)
−0.993974 + 0.109616i \(0.965038\pi\)
\(710\) 0 0
\(711\) −2769.78 + 2769.78i −0.146097 + 0.146097i
\(712\) 0 0
\(713\) 9032.67 + 9032.67i 0.474440 + 0.474440i
\(714\) 0 0
\(715\) −23843.2 + 9876.17i −1.24711 + 0.516570i
\(716\) 0 0
\(717\) −1247.08 + 3010.72i −0.0649555 + 0.156816i
\(718\) 0 0
\(719\) 2159.37i 0.112004i −0.998431 0.0560021i \(-0.982165\pi\)
0.998431 0.0560021i \(-0.0178353\pi\)
\(720\) 0 0
\(721\) 32601.4i 1.68397i
\(722\) 0 0
\(723\) −1709.07 + 4126.06i −0.0879129 + 0.212241i
\(724\) 0 0
\(725\) −4082.75 + 1691.13i −0.209144 + 0.0866303i
\(726\) 0 0
\(727\) 15999.3 + 15999.3i 0.816204 + 0.816204i 0.985556 0.169352i \(-0.0541674\pi\)
−0.169352 + 0.985556i \(0.554167\pi\)
\(728\) 0 0
\(729\) 7638.74 7638.74i 0.388088 0.388088i
\(730\) 0 0
\(731\) 19576.2 + 47261.2i 0.990497 + 2.39127i
\(732\) 0 0
\(733\) −2232.08 924.559i −0.112475 0.0465885i 0.325737 0.945461i \(-0.394388\pi\)
−0.438211 + 0.898872i \(0.644388\pi\)
\(734\) 0 0
\(735\) 2041.22 0.102437
\(736\) 0 0
\(737\) 7942.43 0.396965
\(738\) 0 0
\(739\) 15900.0 + 6585.99i 0.791462 + 0.327834i 0.741531 0.670919i \(-0.234100\pi\)
0.0499305 + 0.998753i \(0.484100\pi\)
\(740\) 0 0
\(741\) 735.886 + 1776.59i 0.0364824 + 0.0880762i
\(742\) 0 0
\(743\) 6184.20 6184.20i 0.305352 0.305352i −0.537751 0.843103i \(-0.680726\pi\)
0.843103 + 0.537751i \(0.180726\pi\)
\(744\) 0 0
\(745\) 32093.4 + 32093.4i 1.57827 + 1.57827i
\(746\) 0 0
\(747\) 3268.45 1353.84i 0.160089 0.0663110i
\(748\) 0 0
\(749\) −14198.1 + 34277.2i −0.692639 + 1.67218i
\(750\) 0 0
\(751\) 11095.5i 0.539122i −0.962983 0.269561i \(-0.913121\pi\)
0.962983 0.269561i \(-0.0868786\pi\)
\(752\) 0 0
\(753\) 517.413i 0.0250406i
\(754\) 0 0
\(755\) −8125.36 + 19616.3i −0.391672 + 0.945579i
\(756\) 0 0
\(757\) 25020.9 10364.0i 1.20132 0.497604i 0.309896 0.950771i \(-0.399706\pi\)
0.891426 + 0.453167i \(0.149706\pi\)
\(758\) 0 0
\(759\) −3982.03 3982.03i −0.190433 0.190433i
\(760\) 0 0
\(761\) 2393.12 2393.12i 0.113995 0.113995i −0.647808 0.761804i \(-0.724314\pi\)
0.761804 + 0.647808i \(0.224314\pi\)
\(762\) 0 0
\(763\) −7054.25 17030.5i −0.334706 0.808053i
\(764\) 0 0
\(765\) 40752.6 + 16880.3i 1.92603 + 0.797788i
\(766\) 0 0
\(767\) −15009.7 −0.706608
\(768\) 0 0
\(769\) −33544.1 −1.57299 −0.786496 0.617595i \(-0.788107\pi\)
−0.786496 + 0.617595i \(0.788107\pi\)
\(770\) 0 0
\(771\) −10161.0 4208.82i −0.474630 0.196598i
\(772\) 0 0
\(773\) 9560.71 + 23081.6i 0.444857 + 1.07398i 0.974223 + 0.225587i \(0.0724300\pi\)
−0.529366 + 0.848394i \(0.677570\pi\)
\(774\) 0 0
\(775\) −15067.3 + 15067.3i −0.698365 + 0.698365i
\(776\) 0 0
\(777\) −1011.51 1011.51i −0.0467023 0.0467023i
\(778\) 0 0
\(779\) 1912.30 792.101i 0.0879529 0.0364313i
\(780\) 0 0
\(781\) −10515.9 + 25387.7i −0.481804 + 1.16318i
\(782\) 0 0
\(783\) 1977.17i 0.0902406i
\(784\) 0 0
\(785\) 16130.0i 0.733381i
\(786\) 0 0
\(787\) 6144.82 14834.9i 0.278321 0.671927i −0.721468 0.692448i \(-0.756532\pi\)
0.999789 + 0.0205203i \(0.00653228\pi\)
\(788\) 0 0
\(789\) −8075.74 + 3345.08i −0.364390 + 0.150935i
\(790\) 0 0
\(791\) 13450.9 + 13450.9i 0.604624 + 0.604624i
\(792\) 0 0
\(793\) −1074.26 + 1074.26i −0.0481060 + 0.0481060i
\(794\) 0 0
\(795\) 4157.17 + 10036.3i 0.185458 + 0.447736i
\(796\) 0 0
\(797\) −15066.3 6240.66i −0.669605 0.277359i 0.0218691 0.999761i \(-0.493038\pi\)
−0.691474 + 0.722401i \(0.743038\pi\)
\(798\) 0 0
\(799\) −60205.5 −2.66573
\(800\) 0 0
\(801\) 9580.85 0.422625
\(802\) 0 0
\(803\) 28385.3 + 11757.6i 1.24744 + 0.516708i
\(804\) 0 0
\(805\) −13866.8 33477.5i −0.607132 1.46575i
\(806\) 0 0
\(807\) −3474.60 + 3474.60i −0.151563 + 0.151563i
\(808\) 0 0
\(809\) 2211.65 + 2211.65i 0.0961154 + 0.0961154i 0.753529 0.657414i \(-0.228350\pi\)
−0.657414 + 0.753529i \(0.728350\pi\)
\(810\) 0 0
\(811\) −19941.1 + 8259.86i −0.863410 + 0.357636i −0.770040 0.637995i \(-0.779764\pi\)
−0.0933697 + 0.995632i \(0.529764\pi\)
\(812\) 0 0
\(813\) −670.782 + 1619.41i −0.0289365 + 0.0698588i
\(814\) 0 0
\(815\) 13876.9i 0.596425i
\(816\) 0 0
\(817\) 16056.5i 0.687571i
\(818\) 0 0
\(819\) 7864.04 18985.5i 0.335521 0.810020i
\(820\) 0 0
\(821\) −13446.8 + 5569.86i −0.571617 + 0.236772i −0.649720 0.760174i \(-0.725114\pi\)
0.0781027 + 0.996945i \(0.475114\pi\)
\(822\) 0 0
\(823\) −23548.4 23548.4i −0.997383 0.997383i 0.00261392 0.999997i \(-0.499168\pi\)
−0.999997 + 0.00261392i \(0.999168\pi\)
\(824\) 0 0
\(825\) 6642.39 6642.39i 0.280313 0.280313i
\(826\) 0 0
\(827\) 6650.02 + 16054.6i 0.279618 + 0.675057i 0.999825 0.0187017i \(-0.00595330\pi\)
−0.720207 + 0.693759i \(0.755953\pi\)
\(828\) 0 0
\(829\) 1367.19 + 566.309i 0.0572792 + 0.0237258i 0.411139 0.911573i \(-0.365131\pi\)
−0.353860 + 0.935298i \(0.615131\pi\)
\(830\) 0 0
\(831\) 6116.00 0.255309
\(832\) 0 0
\(833\) 8316.57 0.345921
\(834\) 0 0
\(835\) 16667.7 + 6903.97i 0.690788 + 0.286134i
\(836\) 0 0
\(837\) −3648.36 8807.91i −0.150664 0.363735i
\(838\) 0 0
\(839\) 10926.4 10926.4i 0.449609 0.449609i −0.445616 0.895224i \(-0.647015\pi\)
0.895224 + 0.445616i \(0.147015\pi\)
\(840\) 0 0
\(841\) −16772.2 16772.2i −0.687694 0.687694i
\(842\) 0 0
\(843\) 443.149 183.558i 0.0181054 0.00749951i
\(844\) 0 0
\(845\) −3811.78 + 9202.46i −0.155183 + 0.374644i
\(846\) 0 0
\(847\) 1252.17i 0.0507969i
\(848\) 0 0
\(849\) 1563.20i 0.0631905i
\(850\) 0 0
\(851\) −1847.23 + 4459.60i −0.0744091 + 0.179640i
\(852\) 0 0
\(853\) −8874.25 + 3675.84i −0.356212 + 0.147548i −0.553611 0.832775i \(-0.686751\pi\)
0.197399 + 0.980323i \(0.436751\pi\)
\(854\) 0 0
\(855\) 9790.08 + 9790.08i 0.391595 + 0.391595i
\(856\) 0 0
\(857\) −16585.9 + 16585.9i −0.661100 + 0.661100i −0.955639 0.294539i \(-0.904834\pi\)
0.294539 + 0.955639i \(0.404834\pi\)
\(858\) 0 0
\(859\) −8809.72 21268.5i −0.349923 0.844788i −0.996628 0.0820486i \(-0.973854\pi\)
0.646706 0.762740i \(-0.276146\pi\)
\(860\) 0 0
\(861\) 1789.29 + 741.148i 0.0708233 + 0.0293360i
\(862\) 0 0
\(863\) 19940.2 0.786527 0.393263 0.919426i \(-0.371346\pi\)
0.393263 + 0.919426i \(0.371346\pi\)
\(864\) 0 0
\(865\) −1545.72 −0.0607584
\(866\) 0 0
\(867\) −7845.74 3249.81i −0.307330 0.127300i
\(868\) 0 0
\(869\) 2252.60 + 5438.25i 0.0879333 + 0.212290i
\(870\) 0 0
\(871\) −6054.94 + 6054.94i −0.235550 + 0.235550i
\(872\) 0 0
\(873\) −13848.5 13848.5i −0.536887 0.536887i
\(874\) 0 0
\(875\) 14970.2 6200.86i 0.578383 0.239574i
\(876\) 0 0
\(877\) 11501.0 27765.8i 0.442828 1.06908i −0.532125 0.846666i \(-0.678606\pi\)
0.974952 0.222414i \(-0.0713938\pi\)
\(878\) 0 0
\(879\) 2367.03i 0.0908280i
\(880\) 0 0
\(881\) 18012.3i 0.688821i 0.938819 + 0.344410i \(0.111921\pi\)
−0.938819 + 0.344410i \(0.888079\pi\)
\(882\) 0 0
\(883\) 6581.16 15888.3i 0.250819 0.605532i −0.747451 0.664317i \(-0.768723\pi\)
0.998271 + 0.0587851i \(0.0187227\pi\)
\(884\) 0 0
\(885\) 8741.91 3621.02i 0.332041 0.137536i
\(886\) 0 0
\(887\) 2226.45 + 2226.45i 0.0842806 + 0.0842806i 0.747990 0.663710i \(-0.231019\pi\)
−0.663710 + 0.747990i \(0.731019\pi\)
\(888\) 0 0
\(889\) 8291.96 8291.96i 0.312827 0.312827i
\(890\) 0 0
\(891\) −7961.61 19221.0i −0.299353 0.722703i
\(892\) 0 0
\(893\) −17458.7 7231.64i −0.654237 0.270994i
\(894\) 0 0
\(895\) 63045.1 2.35460
\(896\) 0 0
\(897\) 6071.43 0.225997
\(898\) 0 0
\(899\) 2982.75 + 1235.49i 0.110657 + 0.0458354i
\(900\) 0 0
\(901\) 16937.6 + 40891.1i 0.626276 + 1.51196i
\(902\) 0 0
\(903\) −10623.3 + 10623.3i −0.391497 + 0.391497i
\(904\) 0 0
\(905\) 6645.49 + 6645.49i 0.244092 + 0.244092i
\(906\) 0 0
\(907\) −43002.0 + 17812.0i −1.57427 + 0.652082i −0.987492 0.157668i \(-0.949602\pi\)
−0.586774 + 0.809751i \(0.699602\pi\)
\(908\) 0 0
\(909\) 14529.8 35077.9i 0.530167 1.27994i
\(910\) 0 0
\(911\) 13008.4i 0.473094i −0.971620 0.236547i \(-0.923984\pi\)
0.971620 0.236547i \(-0.0760156\pi\)
\(912\) 0 0
\(913\) 5316.32i 0.192710i
\(914\) 0 0
\(915\) 366.508 884.828i 0.0132419 0.0319689i
\(916\) 0 0
\(917\) 24116.2 9989.25i 0.868469 0.359732i
\(918\) 0 0
\(919\) 16677.5 + 16677.5i 0.598628 + 0.598628i 0.939947 0.341319i \(-0.110874\pi\)
−0.341319 + 0.939947i \(0.610874\pi\)
\(920\) 0 0
\(921\) −5879.06 + 5879.06i −0.210338 + 0.210338i
\(922\) 0 0
\(923\) −11337.5 27371.2i −0.404312 0.976095i
\(924\) 0 0
\(925\) −7439.02 3081.34i −0.264425 0.109529i
\(926\) 0 0
\(927\) 39329.8 1.39348
\(928\) 0 0
\(929\) 24373.6 0.860786 0.430393 0.902642i \(-0.358375\pi\)
0.430393 + 0.902642i \(0.358375\pi\)
\(930\) 0 0
\(931\) 2411.69 + 998.953i 0.0848978 + 0.0351658i
\(932\) 0 0
\(933\) −145.473 351.203i −0.00510458 0.0123235i
\(934\) 0 0
\(935\) 46871.5 46871.5i 1.63942 1.63942i
\(936\) 0 0
\(937\) −3495.17 3495.17i −0.121859 0.121859i 0.643547 0.765406i \(-0.277462\pi\)
−0.765406 + 0.643547i \(0.777462\pi\)
\(938\) 0 0
\(939\) 7188.83 2977.71i 0.249839 0.103487i
\(940\) 0 0
\(941\) 15711.1 37929.9i 0.544279 1.31401i −0.377400 0.926050i \(-0.623182\pi\)
0.921678 0.387955i \(-0.126818\pi\)
\(942\) 0 0
\(943\) 6535.24i 0.225680i
\(944\) 0 0
\(945\) 27043.6i 0.930928i
\(946\) 0 0
\(947\) −3104.11 + 7493.99i −0.106515 + 0.257151i −0.968147 0.250384i \(-0.919443\pi\)
0.861631 + 0.507535i \(0.169443\pi\)
\(948\) 0 0
\(949\) −30603.1 + 12676.2i −1.04681 + 0.433601i
\(950\) 0 0
\(951\) 3318.20 + 3318.20i 0.113144 + 0.113144i
\(952\) 0 0
\(953\) −16440.0 + 16440.0i −0.558808 + 0.558808i −0.928968 0.370160i \(-0.879303\pi\)
0.370160 + 0.928968i \(0.379303\pi\)
\(954\) 0 0
\(955\) 116.439 + 281.108i 0.00394541 + 0.00952506i
\(956\) 0 0
\(957\) −1314.94 544.665i −0.0444158 0.0183976i
\(958\) 0 0
\(959\) −27532.9 −0.927095
\(960\) 0 0
\(961\) −14223.7 −0.477449
\(962\) 0 0
\(963\) −41351.4 17128.3i −1.38373 0.573159i
\(964\) 0 0
\(965\) −21861.4 52778.0i −0.729267 1.76061i
\(966\) 0 0
\(967\) −25408.6 + 25408.6i −0.844968 + 0.844968i −0.989500 0.144532i \(-0.953832\pi\)
0.144532 + 0.989500i \(0.453832\pi\)
\(968\) 0 0
\(969\) −3492.45 3492.45i −0.115783 0.115783i
\(970\) 0 0
\(971\) 33005.6 13671.4i 1.09083 0.451838i 0.236536 0.971623i \(-0.423988\pi\)
0.854297 + 0.519785i \(0.173988\pi\)
\(972\) 0 0
\(973\) 11200.2 27039.6i 0.369025 0.890905i
\(974\) 0 0
\(975\) 10127.7i 0.332662i
\(976\) 0 0
\(977\) 23994.9i 0.785738i 0.919594 + 0.392869i \(0.128517\pi\)
−0.919594 + 0.392869i \(0.871483\pi\)
\(978\) 0 0
\(979\) 5509.70 13301.6i 0.179868 0.434239i
\(980\) 0 0
\(981\) 20545.2 8510.11i 0.668663 0.276969i
\(982\) 0 0
\(983\) −11434.9 11434.9i −0.371023 0.371023i 0.496827 0.867850i \(-0.334498\pi\)
−0.867850 + 0.496827i \(0.834498\pi\)
\(984\) 0 0
\(985\) 9864.06 9864.06i 0.319081 0.319081i
\(986\) 0 0
\(987\) −6766.46 16335.7i −0.218215 0.526819i
\(988\) 0 0
\(989\) 46836.7 + 19400.4i 1.50589 + 0.623758i
\(990\) 0 0
\(991\) −49651.4 −1.59155 −0.795777 0.605590i \(-0.792937\pi\)
−0.795777 + 0.605590i \(0.792937\pi\)
\(992\) 0 0
\(993\) 9445.29 0.301850
\(994\) 0 0
\(995\) −42881.9 17762.3i −1.36628 0.565932i
\(996\) 0 0
\(997\) 13807.6 + 33334.5i 0.438607 + 1.05889i 0.976430 + 0.215833i \(0.0692468\pi\)
−0.537823 + 0.843058i \(0.680753\pi\)
\(998\) 0 0
\(999\) 2547.37 2547.37i 0.0806760 0.0806760i
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 128.4.g.a.81.6 44
4.3 odd 2 32.4.g.a.29.8 yes 44
8.3 odd 2 256.4.g.b.161.6 44
8.5 even 2 256.4.g.a.161.6 44
32.5 even 8 256.4.g.a.97.6 44
32.11 odd 8 32.4.g.a.21.8 44
32.21 even 8 inner 128.4.g.a.49.6 44
32.27 odd 8 256.4.g.b.97.6 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.4.g.a.21.8 44 32.11 odd 8
32.4.g.a.29.8 yes 44 4.3 odd 2
128.4.g.a.49.6 44 32.21 even 8 inner
128.4.g.a.81.6 44 1.1 even 1 trivial
256.4.g.a.97.6 44 32.5 even 8
256.4.g.a.161.6 44 8.5 even 2
256.4.g.b.97.6 44 32.27 odd 8
256.4.g.b.161.6 44 8.3 odd 2