Properties

Label 375.2.b.c.124.5
Level $375$
Weight $2$
Character 375.124
Analytic conductor $2.994$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [375,2,Mod(124,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.b (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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1632160000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 124.5
Root \(1.46673i\) of defining polynomial
Character \(\chi\) \(=\) 375.124
Dual form 375.2.b.c.124.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.0935099i q^{2} -1.00000i q^{3} +1.99126 q^{4} +0.0935099 q^{6} +2.93346i q^{7} +0.373222i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+0.0935099i q^{2} -1.00000i q^{3} +1.99126 q^{4} +0.0935099 q^{6} +2.93346i q^{7} +0.373222i q^{8} -1.00000 q^{9} +5.04905 q^{11} -1.99126i q^{12} +3.69740i q^{13} -0.274308 q^{14} +3.94761 q^{16} -2.85410i q^{17} -0.0935099i q^{18} -4.43101 q^{19} +2.93346 q^{21} +0.472136i q^{22} -8.36448i q^{23} +0.373222 q^{24} -0.345743 q^{26} +1.00000i q^{27} +5.84128i q^{28} -1.88442 q^{29} +0.266381 q^{31} +1.11558i q^{32} -5.04905i q^{33} +0.266887 q^{34} -1.99126 q^{36} -8.35655i q^{37} -0.414344i q^{38} +3.69740 q^{39} -0.169532 q^{41} +0.274308i q^{42} +9.59262i q^{43} +10.0539 q^{44} +0.782161 q^{46} +2.43101i q^{47} -3.94761i q^{48} -1.60521 q^{49} -2.85410 q^{51} +7.36246i q^{52} +7.24889i q^{53} -0.0935099 q^{54} -1.09483 q^{56} +4.43101i q^{57} -0.176211i q^{58} -5.89522 q^{59} -6.48496 q^{61} +0.0249093i q^{62} -2.93346i q^{63} +7.79091 q^{64} +0.472136 q^{66} -8.55942i q^{67} -5.68325i q^{68} -8.36448 q^{69} -16.2676 q^{71} -0.373222i q^{72} +1.81298i q^{73} +0.781420 q^{74} -8.82328 q^{76} +14.8112i q^{77} +0.345743i q^{78} -2.67198 q^{79} +1.00000 q^{81} -0.0158529i q^{82} -12.9032i q^{83} +5.84128 q^{84} -0.897005 q^{86} +1.88442i q^{87} +1.88442i q^{88} -12.6309 q^{89} -10.8462 q^{91} -16.6558i q^{92} -0.266381i q^{93} -0.227324 q^{94} +1.11558 q^{96} -6.89032i q^{97} -0.150103i q^{98} -5.04905 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9} + 12 q^{11} + 4 q^{14} + 10 q^{16} - 16 q^{19} - 8 q^{21} - 18 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 12 q^{34} + 14 q^{36} + 16 q^{39} + 48 q^{41} + 28 q^{44} - 32 q^{46} - 40 q^{49} + 4 q^{51} - 6 q^{54} - 60 q^{56} - 4 q^{59} + 20 q^{61} + 54 q^{64} - 32 q^{66} - 16 q^{69} - 24 q^{71} + 84 q^{74} - 16 q^{76} + 40 q^{79} + 8 q^{81} + 56 q^{84} - 88 q^{86} - 56 q^{89} - 72 q^{91} + 46 q^{94} + 12 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.0935099i 0.0661215i 0.999453 + 0.0330607i \(0.0105255\pi\)
−0.999453 + 0.0330607i \(0.989475\pi\)
\(3\) − 1.00000i − 0.577350i
\(4\) 1.99126 0.995628
\(5\) 0 0
\(6\) 0.0935099 0.0381753
\(7\) 2.93346i 1.10875i 0.832269 + 0.554373i \(0.187042\pi\)
−0.832269 + 0.554373i \(0.812958\pi\)
\(8\) 0.373222i 0.131954i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) 5.04905 1.52235 0.761173 0.648549i \(-0.224624\pi\)
0.761173 + 0.648549i \(0.224624\pi\)
\(12\) − 1.99126i − 0.574826i
\(13\) 3.69740i 1.02547i 0.858546 + 0.512737i \(0.171368\pi\)
−0.858546 + 0.512737i \(0.828632\pi\)
\(14\) −0.274308 −0.0733119
\(15\) 0 0
\(16\) 3.94761 0.986903
\(17\) − 2.85410i − 0.692221i −0.938194 0.346111i \(-0.887502\pi\)
0.938194 0.346111i \(-0.112498\pi\)
\(18\) − 0.0935099i − 0.0220405i
\(19\) −4.43101 −1.01654 −0.508272 0.861196i \(-0.669716\pi\)
−0.508272 + 0.861196i \(0.669716\pi\)
\(20\) 0 0
\(21\) 2.93346 0.640134
\(22\) 0.472136i 0.100660i
\(23\) − 8.36448i − 1.74411i −0.489404 0.872057i \(-0.662786\pi\)
0.489404 0.872057i \(-0.337214\pi\)
\(24\) 0.373222 0.0761836
\(25\) 0 0
\(26\) −0.345743 −0.0678058
\(27\) 1.00000i 0.192450i
\(28\) 5.84128i 1.10390i
\(29\) −1.88442 −0.349927 −0.174964 0.984575i \(-0.555981\pi\)
−0.174964 + 0.984575i \(0.555981\pi\)
\(30\) 0 0
\(31\) 0.266381 0.0478435 0.0239218 0.999714i \(-0.492385\pi\)
0.0239218 + 0.999714i \(0.492385\pi\)
\(32\) 1.11558i 0.197209i
\(33\) − 5.04905i − 0.878926i
\(34\) 0.266887 0.0457707
\(35\) 0 0
\(36\) −1.99126 −0.331876
\(37\) − 8.35655i − 1.37381i −0.726748 0.686904i \(-0.758969\pi\)
0.726748 0.686904i \(-0.241031\pi\)
\(38\) − 0.414344i − 0.0672154i
\(39\) 3.69740 0.592057
\(40\) 0 0
\(41\) −0.169532 −0.0264764 −0.0132382 0.999912i \(-0.504214\pi\)
−0.0132382 + 0.999912i \(0.504214\pi\)
\(42\) 0.274308i 0.0423266i
\(43\) 9.59262i 1.46286i 0.681916 + 0.731430i \(0.261147\pi\)
−0.681916 + 0.731430i \(0.738853\pi\)
\(44\) 10.0539 1.51569
\(45\) 0 0
\(46\) 0.782161 0.115323
\(47\) 2.43101i 0.354600i 0.984157 + 0.177300i \(0.0567363\pi\)
−0.984157 + 0.177300i \(0.943264\pi\)
\(48\) − 3.94761i − 0.569789i
\(49\) −1.60521 −0.229316
\(50\) 0 0
\(51\) −2.85410 −0.399654
\(52\) 7.36246i 1.02099i
\(53\) 7.24889i 0.995712i 0.867260 + 0.497856i \(0.165879\pi\)
−0.867260 + 0.497856i \(0.834121\pi\)
\(54\) −0.0935099 −0.0127251
\(55\) 0 0
\(56\) −1.09483 −0.146303
\(57\) 4.43101i 0.586902i
\(58\) − 0.176211i − 0.0231377i
\(59\) −5.89522 −0.767493 −0.383746 0.923439i \(-0.625366\pi\)
−0.383746 + 0.923439i \(0.625366\pi\)
\(60\) 0 0
\(61\) −6.48496 −0.830314 −0.415157 0.909750i \(-0.636273\pi\)
−0.415157 + 0.909750i \(0.636273\pi\)
\(62\) 0.0249093i 0.00316348i
\(63\) − 2.93346i − 0.369582i
\(64\) 7.79091 0.973863
\(65\) 0 0
\(66\) 0.472136 0.0581159
\(67\) − 8.55942i − 1.04570i −0.852425 0.522850i \(-0.824869\pi\)
0.852425 0.522850i \(-0.175131\pi\)
\(68\) − 5.68325i − 0.689195i
\(69\) −8.36448 −1.00696
\(70\) 0 0
\(71\) −16.2676 −1.93061 −0.965306 0.261121i \(-0.915908\pi\)
−0.965306 + 0.261121i \(0.915908\pi\)
\(72\) − 0.373222i − 0.0439846i
\(73\) 1.81298i 0.212193i 0.994356 + 0.106097i \(0.0338353\pi\)
−0.994356 + 0.106097i \(0.966165\pi\)
\(74\) 0.781420 0.0908383
\(75\) 0 0
\(76\) −8.82328 −1.01210
\(77\) 14.8112i 1.68789i
\(78\) 0.345743i 0.0391477i
\(79\) −2.67198 −0.300621 −0.150311 0.988639i \(-0.548027\pi\)
−0.150311 + 0.988639i \(0.548027\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 0.0158529i − 0.00175066i
\(83\) − 12.9032i − 1.41630i −0.706060 0.708152i \(-0.749529\pi\)
0.706060 0.708152i \(-0.250471\pi\)
\(84\) 5.84128 0.637336
\(85\) 0 0
\(86\) −0.897005 −0.0967265
\(87\) 1.88442i 0.202031i
\(88\) 1.88442i 0.200879i
\(89\) −12.6309 −1.33887 −0.669434 0.742871i \(-0.733463\pi\)
−0.669434 + 0.742871i \(0.733463\pi\)
\(90\) 0 0
\(91\) −10.8462 −1.13699
\(92\) − 16.6558i − 1.73649i
\(93\) − 0.266381i − 0.0276225i
\(94\) −0.227324 −0.0234467
\(95\) 0 0
\(96\) 1.11558 0.113859
\(97\) − 6.89032i − 0.699606i −0.936823 0.349803i \(-0.886248\pi\)
0.936823 0.349803i \(-0.113752\pi\)
\(98\) − 0.150103i − 0.0151627i
\(99\) −5.04905 −0.507448
\(100\) 0 0
\(101\) 9.77964 0.973110 0.486555 0.873650i \(-0.338253\pi\)
0.486555 + 0.873650i \(0.338253\pi\)
\(102\) − 0.266887i − 0.0264257i
\(103\) 14.8620i 1.46440i 0.681090 + 0.732200i \(0.261506\pi\)
−0.681090 + 0.732200i \(0.738494\pi\)
\(104\) −1.37995 −0.135315
\(105\) 0 0
\(106\) −0.677843 −0.0658380
\(107\) 3.63552i 0.351459i 0.984438 + 0.175730i \(0.0562284\pi\)
−0.984438 + 0.175730i \(0.943772\pi\)
\(108\) 1.99126i 0.191609i
\(109\) 4.62293 0.442797 0.221398 0.975183i \(-0.428938\pi\)
0.221398 + 0.975183i \(0.428938\pi\)
\(110\) 0 0
\(111\) −8.35655 −0.793169
\(112\) 11.5802i 1.09422i
\(113\) − 0.776757i − 0.0730712i −0.999332 0.0365356i \(-0.988368\pi\)
0.999332 0.0365356i \(-0.0116322\pi\)
\(114\) −0.414344 −0.0388068
\(115\) 0 0
\(116\) −3.75235 −0.348397
\(117\) − 3.69740i − 0.341824i
\(118\) − 0.551262i − 0.0507478i
\(119\) 8.37240 0.767497
\(120\) 0 0
\(121\) 14.4929 1.31754
\(122\) − 0.606408i − 0.0549016i
\(123\) 0.169532i 0.0152861i
\(124\) 0.530434 0.0476343
\(125\) 0 0
\(126\) 0.274308 0.0244373
\(127\) 6.26763i 0.556162i 0.960558 + 0.278081i \(0.0896984\pi\)
−0.960558 + 0.278081i \(0.910302\pi\)
\(128\) 2.95970i 0.261603i
\(129\) 9.59262 0.844583
\(130\) 0 0
\(131\) −16.1346 −1.40968 −0.704841 0.709365i \(-0.748982\pi\)
−0.704841 + 0.709365i \(0.748982\pi\)
\(132\) − 10.0539i − 0.875084i
\(133\) − 12.9982i − 1.12709i
\(134\) 0.800391 0.0691432
\(135\) 0 0
\(136\) 1.06521 0.0913413
\(137\) − 17.9650i − 1.53486i −0.641135 0.767428i \(-0.721536\pi\)
0.641135 0.767428i \(-0.278464\pi\)
\(138\) − 0.782161i − 0.0665820i
\(139\) −8.83661 −0.749512 −0.374756 0.927124i \(-0.622273\pi\)
−0.374756 + 0.927124i \(0.622273\pi\)
\(140\) 0 0
\(141\) 2.43101 0.204728
\(142\) − 1.52118i − 0.127655i
\(143\) 18.6683i 1.56112i
\(144\) −3.94761 −0.328968
\(145\) 0 0
\(146\) −0.169532 −0.0140305
\(147\) 1.60521i 0.132395i
\(148\) − 16.6400i − 1.36780i
\(149\) 17.3881 1.42449 0.712245 0.701931i \(-0.247679\pi\)
0.712245 + 0.701931i \(0.247679\pi\)
\(150\) 0 0
\(151\) 2.96378 0.241189 0.120594 0.992702i \(-0.461520\pi\)
0.120594 + 0.992702i \(0.461520\pi\)
\(152\) − 1.65375i − 0.134137i
\(153\) 2.85410i 0.230740i
\(154\) −1.38499 −0.111606
\(155\) 0 0
\(156\) 7.36246 0.589469
\(157\) 3.06654i 0.244736i 0.992485 + 0.122368i \(0.0390489\pi\)
−0.992485 + 0.122368i \(0.960951\pi\)
\(158\) − 0.249857i − 0.0198775i
\(159\) 7.24889 0.574875
\(160\) 0 0
\(161\) 24.5369 1.93378
\(162\) 0.0935099i 0.00734683i
\(163\) − 10.1803i − 0.797386i −0.917085 0.398693i \(-0.869464\pi\)
0.917085 0.398693i \(-0.130536\pi\)
\(164\) −0.337581 −0.0263606
\(165\) 0 0
\(166\) 1.20657 0.0936482
\(167\) 20.2373i 1.56601i 0.622015 + 0.783005i \(0.286314\pi\)
−0.622015 + 0.783005i \(0.713686\pi\)
\(168\) 1.09483i 0.0844682i
\(169\) −0.670734 −0.0515950
\(170\) 0 0
\(171\) 4.43101 0.338848
\(172\) 19.1014i 1.45647i
\(173\) 5.07734i 0.386023i 0.981196 + 0.193012i \(0.0618255\pi\)
−0.981196 + 0.193012i \(0.938174\pi\)
\(174\) −0.176211 −0.0133586
\(175\) 0 0
\(176\) 19.9317 1.50241
\(177\) 5.89522i 0.443112i
\(178\) − 1.18111i − 0.0885280i
\(179\) −12.3216 −0.920958 −0.460479 0.887671i \(-0.652322\pi\)
−0.460479 + 0.887671i \(0.652322\pi\)
\(180\) 0 0
\(181\) 18.4243 1.36947 0.684735 0.728792i \(-0.259918\pi\)
0.684735 + 0.728792i \(0.259918\pi\)
\(182\) − 1.01422i − 0.0751793i
\(183\) 6.48496i 0.479382i
\(184\) 3.12181 0.230143
\(185\) 0 0
\(186\) 0.0249093 0.00182644
\(187\) − 14.4105i − 1.05380i
\(188\) 4.84077i 0.353050i
\(189\) −2.93346 −0.213378
\(190\) 0 0
\(191\) 1.60847 0.116385 0.0581925 0.998305i \(-0.481466\pi\)
0.0581925 + 0.998305i \(0.481466\pi\)
\(192\) − 7.79091i − 0.562260i
\(193\) − 15.3167i − 1.10252i −0.834334 0.551259i \(-0.814148\pi\)
0.834334 0.551259i \(-0.185852\pi\)
\(194\) 0.644314 0.0462590
\(195\) 0 0
\(196\) −3.19638 −0.228313
\(197\) − 5.48496i − 0.390787i −0.980725 0.195394i \(-0.937402\pi\)
0.980725 0.195394i \(-0.0625984\pi\)
\(198\) − 0.472136i − 0.0335532i
\(199\) −12.0885 −0.856933 −0.428467 0.903558i \(-0.640946\pi\)
−0.428467 + 0.903558i \(0.640946\pi\)
\(200\) 0 0
\(201\) −8.55942 −0.603735
\(202\) 0.914493i 0.0643435i
\(203\) − 5.52786i − 0.387980i
\(204\) −5.68325 −0.397907
\(205\) 0 0
\(206\) −1.38975 −0.0968282
\(207\) 8.36448i 0.581371i
\(208\) 14.5959i 1.01204i
\(209\) −22.3724 −1.54753
\(210\) 0 0
\(211\) −5.26148 −0.362215 −0.181108 0.983463i \(-0.557968\pi\)
−0.181108 + 0.983463i \(0.557968\pi\)
\(212\) 14.4344i 0.991359i
\(213\) 16.2676i 1.11464i
\(214\) −0.339957 −0.0232390
\(215\) 0 0
\(216\) −0.373222 −0.0253945
\(217\) 0.781420i 0.0530463i
\(218\) 0.432290i 0.0292784i
\(219\) 1.81298 0.122510
\(220\) 0 0
\(221\) 10.5527 0.709854
\(222\) − 0.781420i − 0.0524455i
\(223\) − 1.41228i − 0.0945732i −0.998881 0.0472866i \(-0.984943\pi\)
0.998881 0.0472866i \(-0.0150574\pi\)
\(224\) −3.27253 −0.218655
\(225\) 0 0
\(226\) 0.0726345 0.00483157
\(227\) − 19.5656i − 1.29861i −0.760527 0.649306i \(-0.775059\pi\)
0.760527 0.649306i \(-0.224941\pi\)
\(228\) 8.82328i 0.584336i
\(229\) 21.3735 1.41240 0.706200 0.708012i \(-0.250408\pi\)
0.706200 + 0.708012i \(0.250408\pi\)
\(230\) 0 0
\(231\) 14.8112 0.974505
\(232\) − 0.703305i − 0.0461742i
\(233\) 3.15382i 0.206614i 0.994650 + 0.103307i \(0.0329424\pi\)
−0.994650 + 0.103307i \(0.967058\pi\)
\(234\) 0.345743 0.0226019
\(235\) 0 0
\(236\) −11.7389 −0.764137
\(237\) 2.67198i 0.173564i
\(238\) 0.782903i 0.0507480i
\(239\) 19.2725 1.24664 0.623318 0.781968i \(-0.285784\pi\)
0.623318 + 0.781968i \(0.285784\pi\)
\(240\) 0 0
\(241\) 22.7210 1.46359 0.731795 0.681525i \(-0.238683\pi\)
0.731795 + 0.681525i \(0.238683\pi\)
\(242\) 1.35523i 0.0871174i
\(243\) − 1.00000i − 0.0641500i
\(244\) −12.9132 −0.826684
\(245\) 0 0
\(246\) −0.0158529 −0.00101074
\(247\) − 16.3832i − 1.04244i
\(248\) 0.0994194i 0.00631314i
\(249\) −12.9032 −0.817704
\(250\) 0 0
\(251\) 8.14124 0.513870 0.256935 0.966429i \(-0.417287\pi\)
0.256935 + 0.966429i \(0.417287\pi\)
\(252\) − 5.84128i − 0.367966i
\(253\) − 42.2327i − 2.65514i
\(254\) −0.586085 −0.0367743
\(255\) 0 0
\(256\) 15.3051 0.956566
\(257\) 22.6653i 1.41382i 0.707302 + 0.706911i \(0.249912\pi\)
−0.707302 + 0.706911i \(0.750088\pi\)
\(258\) 0.897005i 0.0558451i
\(259\) 24.5136 1.52320
\(260\) 0 0
\(261\) 1.88442 0.116642
\(262\) − 1.50874i − 0.0932103i
\(263\) 6.81500i 0.420231i 0.977677 + 0.210115i \(0.0673840\pi\)
−0.977677 + 0.210115i \(0.932616\pi\)
\(264\) 1.88442 0.115978
\(265\) 0 0
\(266\) 1.21546 0.0745248
\(267\) 12.6309i 0.772996i
\(268\) − 17.0440i − 1.04113i
\(269\) 20.1089 1.22606 0.613031 0.790059i \(-0.289950\pi\)
0.613031 + 0.790059i \(0.289950\pi\)
\(270\) 0 0
\(271\) −16.8366 −1.02275 −0.511376 0.859357i \(-0.670864\pi\)
−0.511376 + 0.859357i \(0.670864\pi\)
\(272\) − 11.2669i − 0.683155i
\(273\) 10.8462i 0.656440i
\(274\) 1.67991 0.101487
\(275\) 0 0
\(276\) −16.6558 −1.00256
\(277\) − 25.7015i − 1.54425i −0.635468 0.772127i \(-0.719193\pi\)
0.635468 0.772127i \(-0.280807\pi\)
\(278\) − 0.826311i − 0.0495588i
\(279\) −0.266381 −0.0159478
\(280\) 0 0
\(281\) −27.7097 −1.65302 −0.826511 0.562921i \(-0.809677\pi\)
−0.826511 + 0.562921i \(0.809677\pi\)
\(282\) 0.227324i 0.0135369i
\(283\) − 11.6466i − 0.692317i −0.938176 0.346158i \(-0.887486\pi\)
0.938176 0.346158i \(-0.112514\pi\)
\(284\) −32.3930 −1.92217
\(285\) 0 0
\(286\) −1.74567 −0.103224
\(287\) − 0.497315i − 0.0293556i
\(288\) − 1.11558i − 0.0657365i
\(289\) 8.85410 0.520830
\(290\) 0 0
\(291\) −6.89032 −0.403918
\(292\) 3.61011i 0.211266i
\(293\) 7.72593i 0.451354i 0.974202 + 0.225677i \(0.0724593\pi\)
−0.974202 + 0.225677i \(0.927541\pi\)
\(294\) −0.150103 −0.00875418
\(295\) 0 0
\(296\) 3.11885 0.181279
\(297\) 5.04905i 0.292975i
\(298\) 1.62596i 0.0941894i
\(299\) 30.9268 1.78854
\(300\) 0 0
\(301\) −28.1396 −1.62194
\(302\) 0.277142i 0.0159478i
\(303\) − 9.77964i − 0.561826i
\(304\) −17.4919 −1.00323
\(305\) 0 0
\(306\) −0.266887 −0.0152569
\(307\) − 12.3674i − 0.705842i −0.935653 0.352921i \(-0.885188\pi\)
0.935653 0.352921i \(-0.114812\pi\)
\(308\) 29.4929i 1.68051i
\(309\) 14.8620 0.845471
\(310\) 0 0
\(311\) −4.53276 −0.257029 −0.128515 0.991708i \(-0.541021\pi\)
−0.128515 + 0.991708i \(0.541021\pi\)
\(312\) 1.37995i 0.0781242i
\(313\) − 2.33090i − 0.131750i −0.997828 0.0658752i \(-0.979016\pi\)
0.997828 0.0658752i \(-0.0209839\pi\)
\(314\) −0.286751 −0.0161823
\(315\) 0 0
\(316\) −5.32060 −0.299307
\(317\) 7.86693i 0.441851i 0.975291 + 0.220925i \(0.0709077\pi\)
−0.975291 + 0.220925i \(0.929092\pi\)
\(318\) 0.677843i 0.0380116i
\(319\) −9.51450 −0.532710
\(320\) 0 0
\(321\) 3.63552 0.202915
\(322\) 2.29444i 0.127864i
\(323\) 12.6466i 0.703674i
\(324\) 1.99126 0.110625
\(325\) 0 0
\(326\) 0.951962 0.0527243
\(327\) − 4.62293i − 0.255649i
\(328\) − 0.0632729i − 0.00349366i
\(329\) −7.13129 −0.393161
\(330\) 0 0
\(331\) −20.3744 −1.11988 −0.559940 0.828533i \(-0.689176\pi\)
−0.559940 + 0.828533i \(0.689176\pi\)
\(332\) − 25.6935i − 1.41011i
\(333\) 8.35655i 0.457936i
\(334\) −1.89239 −0.103547
\(335\) 0 0
\(336\) 11.5802 0.631750
\(337\) 17.0141i 0.926816i 0.886145 + 0.463408i \(0.153373\pi\)
−0.886145 + 0.463408i \(0.846627\pi\)
\(338\) − 0.0627203i − 0.00341153i
\(339\) −0.776757 −0.0421877
\(340\) 0 0
\(341\) 1.34497 0.0728344
\(342\) 0.414344i 0.0224051i
\(343\) 15.8254i 0.854493i
\(344\) −3.58018 −0.193030
\(345\) 0 0
\(346\) −0.474782 −0.0255244
\(347\) 33.7908i 1.81399i 0.421145 + 0.906993i \(0.361628\pi\)
−0.421145 + 0.906993i \(0.638372\pi\)
\(348\) 3.75235i 0.201147i
\(349\) −18.8798 −1.01061 −0.505305 0.862941i \(-0.668620\pi\)
−0.505305 + 0.862941i \(0.668620\pi\)
\(350\) 0 0
\(351\) −3.69740 −0.197352
\(352\) 5.63264i 0.300221i
\(353\) − 14.2330i − 0.757548i −0.925489 0.378774i \(-0.876346\pi\)
0.925489 0.378774i \(-0.123654\pi\)
\(354\) −0.551262 −0.0292992
\(355\) 0 0
\(356\) −25.1513 −1.33301
\(357\) − 8.37240i − 0.443115i
\(358\) − 1.15219i − 0.0608951i
\(359\) −11.1014 −0.585907 −0.292954 0.956127i \(-0.594638\pi\)
−0.292954 + 0.956127i \(0.594638\pi\)
\(360\) 0 0
\(361\) 0.633887 0.0333625
\(362\) 1.72286i 0.0905514i
\(363\) − 14.4929i − 0.760679i
\(364\) −21.5975 −1.13202
\(365\) 0 0
\(366\) −0.606408 −0.0316975
\(367\) 32.1387i 1.67763i 0.544420 + 0.838813i \(0.316750\pi\)
−0.544420 + 0.838813i \(0.683250\pi\)
\(368\) − 33.0197i − 1.72127i
\(369\) 0.169532 0.00882546
\(370\) 0 0
\(371\) −21.2644 −1.10399
\(372\) − 0.530434i − 0.0275017i
\(373\) 15.4754i 0.801286i 0.916234 + 0.400643i \(0.131213\pi\)
−0.916234 + 0.400643i \(0.868787\pi\)
\(374\) 1.34752 0.0696788
\(375\) 0 0
\(376\) −0.907308 −0.0467908
\(377\) − 6.96743i − 0.358841i
\(378\) − 0.274308i − 0.0141089i
\(379\) 23.4164 1.20282 0.601410 0.798941i \(-0.294606\pi\)
0.601410 + 0.798941i \(0.294606\pi\)
\(380\) 0 0
\(381\) 6.26763 0.321100
\(382\) 0.150408i 0.00769555i
\(383\) 2.64042i 0.134919i 0.997722 + 0.0674596i \(0.0214894\pi\)
−0.997722 + 0.0674596i \(0.978511\pi\)
\(384\) 2.95970 0.151036
\(385\) 0 0
\(386\) 1.43226 0.0729002
\(387\) − 9.59262i − 0.487620i
\(388\) − 13.7204i − 0.696548i
\(389\) 3.98415 0.202004 0.101002 0.994886i \(-0.467795\pi\)
0.101002 + 0.994886i \(0.467795\pi\)
\(390\) 0 0
\(391\) −23.8731 −1.20731
\(392\) − 0.599099i − 0.0302591i
\(393\) 16.1346i 0.813881i
\(394\) 0.512898 0.0258394
\(395\) 0 0
\(396\) −10.0539 −0.505230
\(397\) − 7.25768i − 0.364253i −0.983275 0.182126i \(-0.941702\pi\)
0.983275 0.182126i \(-0.0582980\pi\)
\(398\) − 1.13040i − 0.0566617i
\(399\) −12.9982 −0.650725
\(400\) 0 0
\(401\) 4.83373 0.241385 0.120693 0.992690i \(-0.461488\pi\)
0.120693 + 0.992690i \(0.461488\pi\)
\(402\) − 0.800391i − 0.0399199i
\(403\) 0.984918i 0.0490622i
\(404\) 19.4738 0.968856
\(405\) 0 0
\(406\) 0.516910 0.0256538
\(407\) − 42.1926i − 2.09141i
\(408\) − 1.06521i − 0.0527359i
\(409\) −15.1508 −0.749159 −0.374579 0.927195i \(-0.622213\pi\)
−0.374579 + 0.927195i \(0.622213\pi\)
\(410\) 0 0
\(411\) −17.9650 −0.886149
\(412\) 29.5941i 1.45800i
\(413\) − 17.2934i − 0.850954i
\(414\) −0.782161 −0.0384411
\(415\) 0 0
\(416\) −4.12476 −0.202233
\(417\) 8.83661i 0.432731i
\(418\) − 2.09204i − 0.102325i
\(419\) −16.2070 −0.791764 −0.395882 0.918301i \(-0.629561\pi\)
−0.395882 + 0.918301i \(0.629561\pi\)
\(420\) 0 0
\(421\) 12.4085 0.604752 0.302376 0.953189i \(-0.402220\pi\)
0.302376 + 0.953189i \(0.402220\pi\)
\(422\) − 0.492001i − 0.0239502i
\(423\) − 2.43101i − 0.118200i
\(424\) −2.70545 −0.131388
\(425\) 0 0
\(426\) −1.52118 −0.0737016
\(427\) − 19.0234i − 0.920607i
\(428\) 7.23926i 0.349923i
\(429\) 18.6683 0.901315
\(430\) 0 0
\(431\) −1.68255 −0.0810457 −0.0405229 0.999179i \(-0.512902\pi\)
−0.0405229 + 0.999179i \(0.512902\pi\)
\(432\) 3.94761i 0.189930i
\(433\) 14.9052i 0.716297i 0.933665 + 0.358148i \(0.116592\pi\)
−0.933665 + 0.358148i \(0.883408\pi\)
\(434\) −0.0730705 −0.00350750
\(435\) 0 0
\(436\) 9.20544 0.440861
\(437\) 37.0631i 1.77297i
\(438\) 0.169532i 0.00810053i
\(439\) 22.0631 1.05302 0.526508 0.850170i \(-0.323501\pi\)
0.526508 + 0.850170i \(0.323501\pi\)
\(440\) 0 0
\(441\) 1.60521 0.0764385
\(442\) 0.986786i 0.0469366i
\(443\) − 20.0061i − 0.950521i −0.879845 0.475260i \(-0.842354\pi\)
0.879845 0.475260i \(-0.157646\pi\)
\(444\) −16.6400 −0.789701
\(445\) 0 0
\(446\) 0.132062 0.00625332
\(447\) − 17.3881i − 0.822429i
\(448\) 22.8543i 1.07977i
\(449\) 16.5517 0.781125 0.390562 0.920576i \(-0.372281\pi\)
0.390562 + 0.920576i \(0.372281\pi\)
\(450\) 0 0
\(451\) −0.855973 −0.0403062
\(452\) − 1.54672i − 0.0727517i
\(453\) − 2.96378i − 0.139250i
\(454\) 1.82957 0.0858662
\(455\) 0 0
\(456\) −1.65375 −0.0774440
\(457\) 9.29771i 0.434928i 0.976068 + 0.217464i \(0.0697785\pi\)
−0.976068 + 0.217464i \(0.930221\pi\)
\(458\) 1.99863i 0.0933900i
\(459\) 2.85410 0.133218
\(460\) 0 0
\(461\) 18.8179 0.876436 0.438218 0.898869i \(-0.355610\pi\)
0.438218 + 0.898869i \(0.355610\pi\)
\(462\) 1.38499i 0.0644357i
\(463\) 7.50533i 0.348802i 0.984675 + 0.174401i \(0.0557990\pi\)
−0.984675 + 0.174401i \(0.944201\pi\)
\(464\) −7.43894 −0.345344
\(465\) 0 0
\(466\) −0.294914 −0.0136616
\(467\) − 4.50749i − 0.208582i −0.994547 0.104291i \(-0.966743\pi\)
0.994547 0.104291i \(-0.0332573\pi\)
\(468\) − 7.36246i − 0.340330i
\(469\) 25.1088 1.15941
\(470\) 0 0
\(471\) 3.06654 0.141299
\(472\) − 2.20023i − 0.101274i
\(473\) 48.4336i 2.22698i
\(474\) −0.249857 −0.0114763
\(475\) 0 0
\(476\) 16.6716 0.764142
\(477\) − 7.24889i − 0.331904i
\(478\) 1.80217i 0.0824294i
\(479\) −21.3291 −0.974552 −0.487276 0.873248i \(-0.662009\pi\)
−0.487276 + 0.873248i \(0.662009\pi\)
\(480\) 0 0
\(481\) 30.8975 1.40880
\(482\) 2.12464i 0.0967747i
\(483\) − 24.5369i − 1.11647i
\(484\) 28.8590 1.31177
\(485\) 0 0
\(486\) 0.0935099 0.00424169
\(487\) − 18.5203i − 0.839236i −0.907701 0.419618i \(-0.862164\pi\)
0.907701 0.419618i \(-0.137836\pi\)
\(488\) − 2.42033i − 0.109563i
\(489\) −10.1803 −0.460371
\(490\) 0 0
\(491\) 9.44369 0.426188 0.213094 0.977032i \(-0.431646\pi\)
0.213094 + 0.977032i \(0.431646\pi\)
\(492\) 0.337581i 0.0152193i
\(493\) 5.37831i 0.242227i
\(494\) 1.53199 0.0689276
\(495\) 0 0
\(496\) 1.05157 0.0472169
\(497\) − 47.7205i − 2.14056i
\(498\) − 1.20657i − 0.0540678i
\(499\) 24.5706 1.09993 0.549966 0.835187i \(-0.314641\pi\)
0.549966 + 0.835187i \(0.314641\pi\)
\(500\) 0 0
\(501\) 20.2373 0.904137
\(502\) 0.761286i 0.0339779i
\(503\) − 25.7970i − 1.15023i −0.818072 0.575115i \(-0.804957\pi\)
0.818072 0.575115i \(-0.195043\pi\)
\(504\) 1.09483 0.0487677
\(505\) 0 0
\(506\) 3.94917 0.175562
\(507\) 0.670734i 0.0297884i
\(508\) 12.4805i 0.553730i
\(509\) 27.4970 1.21878 0.609392 0.792869i \(-0.291414\pi\)
0.609392 + 0.792869i \(0.291414\pi\)
\(510\) 0 0
\(511\) −5.31831 −0.235268
\(512\) 7.35057i 0.324852i
\(513\) − 4.43101i − 0.195634i
\(514\) −2.11943 −0.0934840
\(515\) 0 0
\(516\) 19.1014 0.840890
\(517\) 12.2743i 0.539823i
\(518\) 2.29227i 0.100716i
\(519\) 5.07734 0.222871
\(520\) 0 0
\(521\) −9.23443 −0.404568 −0.202284 0.979327i \(-0.564836\pi\)
−0.202284 + 0.979327i \(0.564836\pi\)
\(522\) 0.176211i 0.00771257i
\(523\) 34.1612i 1.49377i 0.664956 + 0.746883i \(0.268450\pi\)
−0.664956 + 0.746883i \(0.731550\pi\)
\(524\) −32.1280 −1.40352
\(525\) 0 0
\(526\) −0.637270 −0.0277863
\(527\) − 0.760280i − 0.0331183i
\(528\) − 19.9317i − 0.867415i
\(529\) −46.9645 −2.04193
\(530\) 0 0
\(531\) 5.89522 0.255831
\(532\) − 25.8828i − 1.12216i
\(533\) − 0.626825i − 0.0271508i
\(534\) −1.18111 −0.0511116
\(535\) 0 0
\(536\) 3.19456 0.137984
\(537\) 12.3216i 0.531715i
\(538\) 1.88038i 0.0810690i
\(539\) −8.10478 −0.349097
\(540\) 0 0
\(541\) 16.4858 0.708781 0.354391 0.935098i \(-0.384688\pi\)
0.354391 + 0.935098i \(0.384688\pi\)
\(542\) − 1.57439i − 0.0676258i
\(543\) − 18.4243i − 0.790664i
\(544\) 3.18399 0.136513
\(545\) 0 0
\(546\) −1.01422 −0.0434048
\(547\) 17.6399i 0.754227i 0.926167 + 0.377114i \(0.123083\pi\)
−0.926167 + 0.377114i \(0.876917\pi\)
\(548\) − 35.7730i − 1.52815i
\(549\) 6.48496 0.276771
\(550\) 0 0
\(551\) 8.34987 0.355716
\(552\) − 3.12181i − 0.132873i
\(553\) − 7.83816i − 0.333312i
\(554\) 2.40335 0.102108
\(555\) 0 0
\(556\) −17.5960 −0.746235
\(557\) 23.9964i 1.01676i 0.861132 + 0.508381i \(0.169756\pi\)
−0.861132 + 0.508381i \(0.830244\pi\)
\(558\) − 0.0249093i − 0.00105449i
\(559\) −35.4677 −1.50012
\(560\) 0 0
\(561\) −14.4105 −0.608412
\(562\) − 2.59113i − 0.109300i
\(563\) − 3.56884i − 0.150409i −0.997168 0.0752043i \(-0.976039\pi\)
0.997168 0.0752043i \(-0.0239609\pi\)
\(564\) 4.84077 0.203833
\(565\) 0 0
\(566\) 1.08907 0.0457770
\(567\) 2.93346i 0.123194i
\(568\) − 6.07144i − 0.254752i
\(569\) 11.4585 0.480367 0.240183 0.970728i \(-0.422792\pi\)
0.240183 + 0.970728i \(0.422792\pi\)
\(570\) 0 0
\(571\) 35.5027 1.48574 0.742871 0.669435i \(-0.233464\pi\)
0.742871 + 0.669435i \(0.233464\pi\)
\(572\) 37.1734i 1.55430i
\(573\) − 1.60847i − 0.0671949i
\(574\) 0.0465038 0.00194103
\(575\) 0 0
\(576\) −7.79091 −0.324621
\(577\) 0.225114i 0.00937161i 0.999989 + 0.00468581i \(0.00149154\pi\)
−0.999989 + 0.00468581i \(0.998508\pi\)
\(578\) 0.827946i 0.0344380i
\(579\) −15.3167 −0.636539
\(580\) 0 0
\(581\) 37.8509 1.57032
\(582\) − 0.644314i − 0.0267077i
\(583\) 36.6000i 1.51582i
\(584\) −0.676644 −0.0279997
\(585\) 0 0
\(586\) −0.722451 −0.0298442
\(587\) − 13.8981i − 0.573636i −0.957985 0.286818i \(-0.907402\pi\)
0.957985 0.286818i \(-0.0925975\pi\)
\(588\) 3.19638i 0.131817i
\(589\) −1.18034 −0.0486351
\(590\) 0 0
\(591\) −5.48496 −0.225621
\(592\) − 32.9884i − 1.35582i
\(593\) − 9.45488i − 0.388266i −0.980975 0.194133i \(-0.937811\pi\)
0.980975 0.194133i \(-0.0621893\pi\)
\(594\) −0.472136 −0.0193720
\(595\) 0 0
\(596\) 34.6242 1.41826
\(597\) 12.0885i 0.494751i
\(598\) 2.89196i 0.118261i
\(599\) −8.07648 −0.329996 −0.164998 0.986294i \(-0.552762\pi\)
−0.164998 + 0.986294i \(0.552762\pi\)
\(600\) 0 0
\(601\) 28.6038 1.16677 0.583387 0.812194i \(-0.301727\pi\)
0.583387 + 0.812194i \(0.301727\pi\)
\(602\) − 2.63133i − 0.107245i
\(603\) 8.55942i 0.348567i
\(604\) 5.90164 0.240134
\(605\) 0 0
\(606\) 0.914493 0.0371487
\(607\) − 1.42208i − 0.0577203i −0.999583 0.0288602i \(-0.990812\pi\)
0.999583 0.0288602i \(-0.00918775\pi\)
\(608\) − 4.94317i − 0.200472i
\(609\) −5.52786 −0.224000
\(610\) 0 0
\(611\) −8.98842 −0.363633
\(612\) 5.68325i 0.229732i
\(613\) 14.7556i 0.595974i 0.954570 + 0.297987i \(0.0963152\pi\)
−0.954570 + 0.297987i \(0.903685\pi\)
\(614\) 1.15647 0.0466714
\(615\) 0 0
\(616\) −5.52786 −0.222724
\(617\) 31.3519i 1.26218i 0.775710 + 0.631090i \(0.217392\pi\)
−0.775710 + 0.631090i \(0.782608\pi\)
\(618\) 1.38975i 0.0559038i
\(619\) −12.6563 −0.508699 −0.254349 0.967112i \(-0.581861\pi\)
−0.254349 + 0.967112i \(0.581861\pi\)
\(620\) 0 0
\(621\) 8.36448 0.335655
\(622\) − 0.423858i − 0.0169952i
\(623\) − 37.0522i − 1.48446i
\(624\) 14.5959 0.584303
\(625\) 0 0
\(626\) 0.217962 0.00871153
\(627\) 22.3724i 0.893468i
\(628\) 6.10626i 0.243666i
\(629\) −23.8504 −0.950980
\(630\) 0 0
\(631\) 19.0741 0.759327 0.379663 0.925125i \(-0.376040\pi\)
0.379663 + 0.925125i \(0.376040\pi\)
\(632\) − 0.997242i − 0.0396682i
\(633\) 5.26148i 0.209125i
\(634\) −0.735636 −0.0292158
\(635\) 0 0
\(636\) 14.4344 0.572361
\(637\) − 5.93509i − 0.235157i
\(638\) − 0.889700i − 0.0352236i
\(639\) 16.2676 0.643537
\(640\) 0 0
\(641\) −3.58594 −0.141636 −0.0708180 0.997489i \(-0.522561\pi\)
−0.0708180 + 0.997489i \(0.522561\pi\)
\(642\) 0.339957i 0.0134170i
\(643\) 37.6409i 1.48441i 0.670172 + 0.742206i \(0.266220\pi\)
−0.670172 + 0.742206i \(0.733780\pi\)
\(644\) 48.8592 1.92532
\(645\) 0 0
\(646\) −1.18258 −0.0465280
\(647\) − 2.21935i − 0.0872517i −0.999048 0.0436258i \(-0.986109\pi\)
0.999048 0.0436258i \(-0.0138909\pi\)
\(648\) 0.373222i 0.0146615i
\(649\) −29.7653 −1.16839
\(650\) 0 0
\(651\) 0.781420 0.0306263
\(652\) − 20.2717i − 0.793900i
\(653\) − 3.13494i − 0.122680i −0.998117 0.0613399i \(-0.980463\pi\)
0.998117 0.0613399i \(-0.0195374\pi\)
\(654\) 0.432290 0.0169039
\(655\) 0 0
\(656\) −0.669245 −0.0261296
\(657\) − 1.81298i − 0.0707311i
\(658\) − 0.666846i − 0.0259964i
\(659\) 21.6691 0.844108 0.422054 0.906571i \(-0.361309\pi\)
0.422054 + 0.906571i \(0.361309\pi\)
\(660\) 0 0
\(661\) −21.9730 −0.854649 −0.427324 0.904098i \(-0.640544\pi\)
−0.427324 + 0.904098i \(0.640544\pi\)
\(662\) − 1.90521i − 0.0740481i
\(663\) − 10.5527i − 0.409835i
\(664\) 4.81574 0.186887
\(665\) 0 0
\(666\) −0.781420 −0.0302794
\(667\) 15.7622i 0.610313i
\(668\) 40.2977i 1.55916i
\(669\) −1.41228 −0.0546019
\(670\) 0 0
\(671\) −32.7429 −1.26402
\(672\) 3.27253i 0.126240i
\(673\) − 32.7911i − 1.26400i −0.774967 0.632001i \(-0.782234\pi\)
0.774967 0.632001i \(-0.217766\pi\)
\(674\) −1.59098 −0.0612824
\(675\) 0 0
\(676\) −1.33560 −0.0513694
\(677\) − 25.8289i − 0.992686i −0.868126 0.496343i \(-0.834676\pi\)
0.868126 0.496343i \(-0.165324\pi\)
\(678\) − 0.0726345i − 0.00278951i
\(679\) 20.2125 0.775685
\(680\) 0 0
\(681\) −19.5656 −0.749754
\(682\) 0.125768i 0.00481592i
\(683\) − 23.4044i − 0.895543i −0.894148 0.447771i \(-0.852218\pi\)
0.894148 0.447771i \(-0.147782\pi\)
\(684\) 8.82328 0.337367
\(685\) 0 0
\(686\) −1.47983 −0.0565003
\(687\) − 21.3735i − 0.815450i
\(688\) 37.8679i 1.44370i
\(689\) −26.8020 −1.02108
\(690\) 0 0
\(691\) 9.91799 0.377298 0.188649 0.982045i \(-0.439589\pi\)
0.188649 + 0.982045i \(0.439589\pi\)
\(692\) 10.1103i 0.384336i
\(693\) − 14.8112i − 0.562631i
\(694\) −3.15978 −0.119943
\(695\) 0 0
\(696\) −0.703305 −0.0266587
\(697\) 0.483860i 0.0183275i
\(698\) − 1.76544i − 0.0668230i
\(699\) 3.15382 0.119289
\(700\) 0 0
\(701\) −36.6906 −1.38578 −0.692892 0.721042i \(-0.743664\pi\)
−0.692892 + 0.721042i \(0.743664\pi\)
\(702\) − 0.345743i − 0.0130492i
\(703\) 37.0280i 1.39654i
\(704\) 39.3367 1.48256
\(705\) 0 0
\(706\) 1.33093 0.0500902
\(707\) 28.6882i 1.07893i
\(708\) 11.7389i 0.441175i
\(709\) 33.0393 1.24082 0.620409 0.784278i \(-0.286967\pi\)
0.620409 + 0.784278i \(0.286967\pi\)
\(710\) 0 0
\(711\) 2.67198 0.100207
\(712\) − 4.71411i − 0.176669i
\(713\) − 2.22814i − 0.0834446i
\(714\) 0.782903 0.0292994
\(715\) 0 0
\(716\) −24.5354 −0.916931
\(717\) − 19.2725i − 0.719746i
\(718\) − 1.03809i − 0.0387411i
\(719\) −30.0339 −1.12008 −0.560038 0.828467i \(-0.689213\pi\)
−0.560038 + 0.828467i \(0.689213\pi\)
\(720\) 0 0
\(721\) −43.5972 −1.62365
\(722\) 0.0592747i 0.00220598i
\(723\) − 22.7210i − 0.845004i
\(724\) 36.6876 1.36348
\(725\) 0 0
\(726\) 1.35523 0.0502972
\(727\) 29.5053i 1.09429i 0.837037 + 0.547146i \(0.184286\pi\)
−0.837037 + 0.547146i \(0.815714\pi\)
\(728\) − 4.04803i − 0.150030i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 27.3783 1.01262
\(732\) 12.9132i 0.477286i
\(733\) − 39.2642i − 1.45026i −0.688614 0.725128i \(-0.741780\pi\)
0.688614 0.725128i \(-0.258220\pi\)
\(734\) −3.00528 −0.110927
\(735\) 0 0
\(736\) 9.33128 0.343956
\(737\) − 43.2169i − 1.59192i
\(738\) 0.0158529i 0 0.000583553i
\(739\) −7.12841 −0.262223 −0.131111 0.991368i \(-0.541855\pi\)
−0.131111 + 0.991368i \(0.541855\pi\)
\(740\) 0 0
\(741\) −16.3832 −0.601852
\(742\) − 1.98843i − 0.0729975i
\(743\) 10.2060i 0.374421i 0.982320 + 0.187211i \(0.0599447\pi\)
−0.982320 + 0.187211i \(0.940055\pi\)
\(744\) 0.0994194 0.00364489
\(745\) 0 0
\(746\) −1.44710 −0.0529822
\(747\) 12.9032i 0.472102i
\(748\) − 28.6950i − 1.04919i
\(749\) −10.6647 −0.389679
\(750\) 0 0
\(751\) 5.89220 0.215009 0.107505 0.994205i \(-0.465714\pi\)
0.107505 + 0.994205i \(0.465714\pi\)
\(752\) 9.59670i 0.349956i
\(753\) − 8.14124i − 0.296683i
\(754\) 0.651524 0.0237271
\(755\) 0 0
\(756\) −5.84128 −0.212445
\(757\) 28.9643i 1.05272i 0.850261 + 0.526362i \(0.176444\pi\)
−0.850261 + 0.526362i \(0.823556\pi\)
\(758\) 2.18967i 0.0795322i
\(759\) −42.2327 −1.53295
\(760\) 0 0
\(761\) 23.8720 0.865358 0.432679 0.901548i \(-0.357568\pi\)
0.432679 + 0.901548i \(0.357568\pi\)
\(762\) 0.586085i 0.0212316i
\(763\) 13.5612i 0.490949i
\(764\) 3.20288 0.115876
\(765\) 0 0
\(766\) −0.246905 −0.00892106
\(767\) − 21.7970i − 0.787043i
\(768\) − 15.3051i − 0.552273i
\(769\) −32.1167 −1.15816 −0.579079 0.815272i \(-0.696588\pi\)
−0.579079 + 0.815272i \(0.696588\pi\)
\(770\) 0 0
\(771\) 22.6653 0.816271
\(772\) − 30.4994i − 1.09770i
\(773\) 22.9098i 0.824009i 0.911182 + 0.412005i \(0.135171\pi\)
−0.911182 + 0.412005i \(0.864829\pi\)
\(774\) 0.897005 0.0322422
\(775\) 0 0
\(776\) 2.57162 0.0923158
\(777\) − 24.5136i − 0.879422i
\(778\) 0.372557i 0.0133568i
\(779\) 0.751197 0.0269144
\(780\) 0 0
\(781\) −82.1360 −2.93906
\(782\) − 2.23237i − 0.0798293i
\(783\) − 1.88442i − 0.0673435i
\(784\) −6.33674 −0.226312
\(785\) 0 0
\(786\) −1.50874 −0.0538150
\(787\) 0.790593i 0.0281816i 0.999901 + 0.0140908i \(0.00448539\pi\)
−0.999901 + 0.0140908i \(0.995515\pi\)
\(788\) − 10.9220i − 0.389079i
\(789\) 6.81500 0.242620
\(790\) 0 0
\(791\) 2.27859 0.0810173
\(792\) − 1.88442i − 0.0669598i
\(793\) − 23.9775i − 0.851465i
\(794\) 0.678665 0.0240849
\(795\) 0 0
\(796\) −24.0714 −0.853187
\(797\) − 31.1524i − 1.10348i −0.834018 0.551738i \(-0.813965\pi\)
0.834018 0.551738i \(-0.186035\pi\)
\(798\) − 1.21546i − 0.0430269i
\(799\) 6.93836 0.245462
\(800\) 0 0
\(801\) 12.6309 0.446289
\(802\) 0.452002i 0.0159607i
\(803\) 9.15382i 0.323031i
\(804\) −17.0440 −0.601096
\(805\) 0 0
\(806\) −0.0920995 −0.00324407
\(807\) − 20.1089i − 0.707867i
\(808\) 3.64998i 0.128406i
\(809\) 0.120629 0.00424110 0.00212055 0.999998i \(-0.499325\pi\)
0.00212055 + 0.999998i \(0.499325\pi\)
\(810\) 0 0
\(811\) 1.55313 0.0545379 0.0272689 0.999628i \(-0.491319\pi\)
0.0272689 + 0.999628i \(0.491319\pi\)
\(812\) − 11.0074i − 0.386284i
\(813\) 16.8366i 0.590486i
\(814\) 3.94543 0.138287
\(815\) 0 0
\(816\) −11.2669 −0.394420
\(817\) − 42.5050i − 1.48706i
\(818\) − 1.41675i − 0.0495355i
\(819\) 10.8462 0.378996
\(820\) 0 0
\(821\) −14.6965 −0.512912 −0.256456 0.966556i \(-0.582555\pi\)
−0.256456 + 0.966556i \(0.582555\pi\)
\(822\) − 1.67991i − 0.0585935i
\(823\) − 9.71652i − 0.338697i −0.985556 0.169348i \(-0.945834\pi\)
0.985556 0.169348i \(-0.0541663\pi\)
\(824\) −5.54683 −0.193233
\(825\) 0 0
\(826\) 1.61711 0.0562663
\(827\) 32.6997i 1.13708i 0.822656 + 0.568539i \(0.192491\pi\)
−0.822656 + 0.568539i \(0.807509\pi\)
\(828\) 16.6558i 0.578830i
\(829\) 37.5623 1.30459 0.652296 0.757964i \(-0.273806\pi\)
0.652296 + 0.757964i \(0.273806\pi\)
\(830\) 0 0
\(831\) −25.7015 −0.891576
\(832\) 28.8061i 0.998670i
\(833\) 4.58143i 0.158737i
\(834\) −0.826311 −0.0286128
\(835\) 0 0
\(836\) −44.5492 −1.54077
\(837\) 0.266381i 0.00920749i
\(838\) − 1.51551i − 0.0523526i
\(839\) −23.6116 −0.815163 −0.407581 0.913169i \(-0.633628\pi\)
−0.407581 + 0.913169i \(0.633628\pi\)
\(840\) 0 0
\(841\) −25.4490 −0.877551
\(842\) 1.16032i 0.0399871i
\(843\) 27.7097i 0.954372i
\(844\) −10.4770 −0.360632
\(845\) 0 0
\(846\) 0.227324 0.00781556
\(847\) 42.5144i 1.46081i
\(848\) 28.6158i 0.982671i
\(849\) −11.6466 −0.399709
\(850\) 0 0
\(851\) −69.8982 −2.39608
\(852\) 32.3930i 1.10977i
\(853\) 1.72305i 0.0589960i 0.999565 + 0.0294980i \(0.00939086\pi\)
−0.999565 + 0.0294980i \(0.990609\pi\)
\(854\) 1.77888 0.0608719
\(855\) 0 0
\(856\) −1.35686 −0.0463764
\(857\) 25.9461i 0.886303i 0.896447 + 0.443152i \(0.146140\pi\)
−0.896447 + 0.443152i \(0.853860\pi\)
\(858\) 1.74567i 0.0595963i
\(859\) −33.1188 −1.13000 −0.565000 0.825091i \(-0.691124\pi\)
−0.565000 + 0.825091i \(0.691124\pi\)
\(860\) 0 0
\(861\) −0.497315 −0.0169484
\(862\) − 0.157335i − 0.00535886i
\(863\) 29.3696i 0.999753i 0.866097 + 0.499877i \(0.166621\pi\)
−0.866097 + 0.499877i \(0.833379\pi\)
\(864\) −1.11558 −0.0379530
\(865\) 0 0
\(866\) −1.39378 −0.0473626
\(867\) − 8.85410i − 0.300701i
\(868\) 1.55601i 0.0528143i
\(869\) −13.4910 −0.457649
\(870\) 0 0
\(871\) 31.6476 1.07234
\(872\) 1.72538i 0.0584287i
\(873\) 6.89032i 0.233202i
\(874\) −3.46577 −0.117231
\(875\) 0 0
\(876\) 3.61011 0.121974
\(877\) 51.6306i 1.74344i 0.490004 + 0.871720i \(0.336995\pi\)
−0.490004 + 0.871720i \(0.663005\pi\)
\(878\) 2.06312i 0.0696269i
\(879\) 7.72593 0.260589
\(880\) 0 0
\(881\) −17.6132 −0.593405 −0.296702 0.954970i \(-0.595887\pi\)
−0.296702 + 0.954970i \(0.595887\pi\)
\(882\) 0.150103i 0.00505423i
\(883\) − 1.65829i − 0.0558059i −0.999611 0.0279030i \(-0.991117\pi\)
0.999611 0.0279030i \(-0.00888294\pi\)
\(884\) 21.0132 0.706751
\(885\) 0 0
\(886\) 1.87077 0.0628498
\(887\) − 20.1490i − 0.676538i −0.941050 0.338269i \(-0.890159\pi\)
0.941050 0.338269i \(-0.109841\pi\)
\(888\) − 3.11885i − 0.104662i
\(889\) −18.3859 −0.616642
\(890\) 0 0
\(891\) 5.04905 0.169149
\(892\) − 2.81221i − 0.0941597i
\(893\) − 10.7719i − 0.360466i
\(894\) 1.62596 0.0543803
\(895\) 0 0
\(896\) −8.68216 −0.290051
\(897\) − 30.9268i − 1.03262i
\(898\) 1.54775i 0.0516491i
\(899\) −0.501973 −0.0167417
\(900\) 0 0
\(901\) 20.6891 0.689253
\(902\) − 0.0800420i − 0.00266511i
\(903\) 28.1396i 0.936427i
\(904\) 0.289903 0.00964202
\(905\) 0 0
\(906\) 0.277142 0.00920744
\(907\) − 31.8504i − 1.05758i −0.848754 0.528788i \(-0.822647\pi\)
0.848754 0.528788i \(-0.177353\pi\)
\(908\) − 38.9601i − 1.29293i
\(909\) −9.77964 −0.324370
\(910\) 0 0
\(911\) 30.7213 1.01784 0.508920 0.860814i \(-0.330045\pi\)
0.508920 + 0.860814i \(0.330045\pi\)
\(912\) 17.4919i 0.579215i
\(913\) − 65.1486i − 2.15610i
\(914\) −0.869427 −0.0287581
\(915\) 0 0
\(916\) 42.5601 1.40623
\(917\) − 47.3301i − 1.56298i
\(918\) 0.266887i 0.00880858i
\(919\) 36.0752 1.19001 0.595005 0.803722i \(-0.297150\pi\)
0.595005 + 0.803722i \(0.297150\pi\)
\(920\) 0 0
\(921\) −12.3674 −0.407518
\(922\) 1.75966i 0.0579512i
\(923\) − 60.1479i − 1.97979i
\(924\) 29.4929 0.970245
\(925\) 0 0
\(926\) −0.701823 −0.0230633
\(927\) − 14.8620i − 0.488133i
\(928\) − 2.10222i − 0.0690089i
\(929\) −13.7648 −0.451608 −0.225804 0.974173i \(-0.572501\pi\)
−0.225804 + 0.974173i \(0.572501\pi\)
\(930\) 0 0
\(931\) 7.11270 0.233109
\(932\) 6.28007i 0.205711i
\(933\) 4.53276i 0.148396i
\(934\) 0.421495 0.0137917
\(935\) 0 0
\(936\) 1.37995 0.0451050
\(937\) 40.1571i 1.31187i 0.754815 + 0.655937i \(0.227726\pi\)
−0.754815 + 0.655937i \(0.772274\pi\)
\(938\) 2.34792i 0.0766622i
\(939\) −2.33090 −0.0760661
\(940\) 0 0
\(941\) 13.7005 0.446624 0.223312 0.974747i \(-0.428313\pi\)
0.223312 + 0.974747i \(0.428313\pi\)
\(942\) 0.286751i 0.00934287i
\(943\) 1.41804i 0.0461778i
\(944\) −23.2721 −0.757441
\(945\) 0 0
\(946\) −4.52902 −0.147251
\(947\) 27.1382i 0.881873i 0.897538 + 0.440937i \(0.145354\pi\)
−0.897538 + 0.440937i \(0.854646\pi\)
\(948\) 5.32060i 0.172805i
\(949\) −6.70331 −0.217598
\(950\) 0 0
\(951\) 7.86693 0.255103
\(952\) 3.12476i 0.101274i
\(953\) 47.9437i 1.55305i 0.630086 + 0.776525i \(0.283019\pi\)
−0.630086 + 0.776525i \(0.716981\pi\)
\(954\) 0.677843 0.0219460
\(955\) 0 0
\(956\) 38.3765 1.24119
\(957\) 9.51450i 0.307560i
\(958\) − 1.99448i − 0.0644388i
\(959\) 52.6997 1.70176
\(960\) 0 0
\(961\) −30.9290 −0.997711
\(962\) 2.88922i 0.0931522i
\(963\) − 3.63552i − 0.117153i
\(964\) 45.2434 1.45719
\(965\) 0 0
\(966\) 2.29444 0.0738225
\(967\) − 2.33998i − 0.0752487i −0.999292 0.0376243i \(-0.988021\pi\)
0.999292 0.0376243i \(-0.0119790\pi\)
\(968\) 5.40906i 0.173854i
\(969\) 12.6466 0.406266
\(970\) 0 0
\(971\) 50.9343 1.63456 0.817280 0.576241i \(-0.195481\pi\)
0.817280 + 0.576241i \(0.195481\pi\)
\(972\) − 1.99126i − 0.0638696i
\(973\) − 25.9219i − 0.831017i
\(974\) 1.73183 0.0554915
\(975\) 0 0
\(976\) −25.6001 −0.819440
\(977\) 58.0048i 1.85574i 0.372905 + 0.927870i \(0.378362\pi\)
−0.372905 + 0.927870i \(0.621638\pi\)
\(978\) − 0.951962i − 0.0304404i
\(979\) −63.7738 −2.03822
\(980\) 0 0
\(981\) −4.62293 −0.147599
\(982\) 0.883079i 0.0281802i
\(983\) − 35.2324i − 1.12374i −0.827226 0.561870i \(-0.810082\pi\)
0.827226 0.561870i \(-0.189918\pi\)
\(984\) −0.0632729 −0.00201707
\(985\) 0 0
\(986\) −0.502926 −0.0160164
\(987\) 7.13129i 0.226992i
\(988\) − 32.6232i − 1.03788i
\(989\) 80.2373 2.55140
\(990\) 0 0
\(991\) −23.7908 −0.755741 −0.377870 0.925859i \(-0.623343\pi\)
−0.377870 + 0.925859i \(0.623343\pi\)
\(992\) 0.297171i 0.00943519i
\(993\) 20.3744i 0.646563i
\(994\) 4.46234 0.141537
\(995\) 0 0
\(996\) −25.6935 −0.814129
\(997\) 8.78958i 0.278369i 0.990266 + 0.139184i \(0.0444481\pi\)
−0.990266 + 0.139184i \(0.955552\pi\)
\(998\) 2.29760i 0.0727291i
\(999\) 8.35655 0.264390
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 375.2.b.c.124.5 8
3.2 odd 2 1125.2.b.g.874.4 8
4.3 odd 2 6000.2.f.o.1249.5 8
5.2 odd 4 375.2.a.e.1.3 4
5.3 odd 4 375.2.a.f.1.2 yes 4
5.4 even 2 inner 375.2.b.c.124.4 8
15.2 even 4 1125.2.a.l.1.2 4
15.8 even 4 1125.2.a.h.1.3 4
15.14 odd 2 1125.2.b.g.874.5 8
20.3 even 4 6000.2.a.bg.1.1 4
20.7 even 4 6000.2.a.bh.1.4 4
20.19 odd 2 6000.2.f.o.1249.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
375.2.a.e.1.3 4 5.2 odd 4
375.2.a.f.1.2 yes 4 5.3 odd 4
375.2.b.c.124.4 8 5.4 even 2 inner
375.2.b.c.124.5 8 1.1 even 1 trivial
1125.2.a.h.1.3 4 15.8 even 4
1125.2.a.l.1.2 4 15.2 even 4
1125.2.b.g.874.4 8 3.2 odd 2
1125.2.b.g.874.5 8 15.14 odd 2
6000.2.a.bg.1.1 4 20.3 even 4
6000.2.a.bh.1.4 4 20.7 even 4
6000.2.f.o.1249.4 8 20.19 odd 2
6000.2.f.o.1249.5 8 4.3 odd 2