Properties

Label 315.4.g.a.314.30
Level $315$
Weight $4$
Character 315.314
Analytic conductor $18.586$
Analytic rank $0$
Dimension $48$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [315,4,Mod(314,315)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(315, 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("315.314");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 315.g (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(18.5856016518\)
Analytic rank: \(0\)
Dimension: \(48\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 314.30
Character \(\chi\) \(=\) 315.314
Dual form 315.4.g.a.314.31

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41590 q^{2} -5.99524 q^{4} +(-2.48890 - 10.8998i) q^{5} +(5.69164 + 17.6240i) q^{7} -19.8158 q^{8} +(-3.52402 - 15.4330i) q^{10} -30.5871i q^{11} +16.5994 q^{13} +(8.05877 + 24.9538i) q^{14} +19.9047 q^{16} +81.3047i q^{17} +87.9721i q^{19} +(14.9215 + 65.3468i) q^{20} -43.3082i q^{22} -51.9402 q^{23} +(-112.611 + 54.2569i) q^{25} +23.5030 q^{26} +(-34.1227 - 105.660i) q^{28} +160.205i q^{29} +306.111i q^{31} +186.710 q^{32} +115.119i q^{34} +(177.932 - 105.902i) q^{35} -104.929i q^{37} +124.559i q^{38} +(49.3195 + 215.988i) q^{40} +477.920 q^{41} +265.819i q^{43} +183.377i q^{44} -73.5419 q^{46} -140.739i q^{47} +(-278.210 + 200.619i) q^{49} +(-159.445 + 76.8222i) q^{50} -99.5173 q^{52} -305.963 q^{53} +(-333.393 + 76.1282i) q^{55} +(-112.784 - 349.234i) q^{56} +226.833i q^{58} +134.064 q^{59} +303.805i q^{61} +433.422i q^{62} +105.123 q^{64} +(-41.3142 - 180.930i) q^{65} -671.592i q^{67} -487.441i q^{68} +(251.933 - 149.946i) q^{70} +616.486i q^{71} -1021.48 q^{73} -148.568i q^{74} -527.413i q^{76} +(539.067 - 174.091i) q^{77} -558.731 q^{79} +(-49.5409 - 216.958i) q^{80} +676.686 q^{82} -335.471i q^{83} +(886.204 - 202.359i) q^{85} +376.372i q^{86} +606.108i q^{88} +931.364 q^{89} +(94.4778 + 292.548i) q^{91} +311.394 q^{92} -199.272i q^{94} +(958.877 - 218.953i) q^{95} -772.248 q^{97} +(-393.917 + 284.056i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 48 q + 192 q^{4} + 768 q^{16} - 432 q^{25} + 816 q^{46} + 456 q^{49} + 7968 q^{64} + 1464 q^{70} + 4368 q^{79} - 1440 q^{85} - 4392 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/315\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(136\) \(281\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41590 0.500595 0.250298 0.968169i \(-0.419472\pi\)
0.250298 + 0.968169i \(0.419472\pi\)
\(3\) 0 0
\(4\) −5.99524 −0.749405
\(5\) −2.48890 10.8998i −0.222614 0.974907i
\(6\) 0 0
\(7\) 5.69164 + 17.6240i 0.307320 + 0.951606i
\(8\) −19.8158 −0.875743
\(9\) 0 0
\(10\) −3.52402 15.4330i −0.111439 0.488034i
\(11\) 30.5871i 0.838397i −0.907895 0.419198i \(-0.862311\pi\)
0.907895 0.419198i \(-0.137689\pi\)
\(12\) 0 0
\(13\) 16.5994 0.354142 0.177071 0.984198i \(-0.443338\pi\)
0.177071 + 0.984198i \(0.443338\pi\)
\(14\) 8.05877 + 24.9538i 0.153843 + 0.476369i
\(15\) 0 0
\(16\) 19.9047 0.311012
\(17\) 81.3047i 1.15996i 0.814632 + 0.579979i \(0.196939\pi\)
−0.814632 + 0.579979i \(0.803061\pi\)
\(18\) 0 0
\(19\) 87.9721i 1.06222i 0.847303 + 0.531110i \(0.178225\pi\)
−0.847303 + 0.531110i \(0.821775\pi\)
\(20\) 14.9215 + 65.3468i 0.166828 + 0.730600i
\(21\) 0 0
\(22\) 43.3082i 0.419697i
\(23\) −51.9402 −0.470882 −0.235441 0.971889i \(-0.575653\pi\)
−0.235441 + 0.971889i \(0.575653\pi\)
\(24\) 0 0
\(25\) −112.611 + 54.2569i −0.900886 + 0.434055i
\(26\) 23.5030 0.177282
\(27\) 0 0
\(28\) −34.1227 105.660i −0.230307 0.713138i
\(29\) 160.205i 1.02584i 0.858437 + 0.512919i \(0.171436\pi\)
−0.858437 + 0.512919i \(0.828564\pi\)
\(30\) 0 0
\(31\) 306.111i 1.77352i 0.462226 + 0.886762i \(0.347051\pi\)
−0.462226 + 0.886762i \(0.652949\pi\)
\(32\) 186.710 1.03143
\(33\) 0 0
\(34\) 115.119i 0.580669i
\(35\) 177.932 105.902i 0.859314 0.511449i
\(36\) 0 0
\(37\) 104.929i 0.466221i −0.972450 0.233111i \(-0.925110\pi\)
0.972450 0.233111i \(-0.0748904\pi\)
\(38\) 124.559i 0.531742i
\(39\) 0 0
\(40\) 49.3195 + 215.988i 0.194952 + 0.853768i
\(41\) 477.920 1.82045 0.910227 0.414110i \(-0.135907\pi\)
0.910227 + 0.414110i \(0.135907\pi\)
\(42\) 0 0
\(43\) 265.819i 0.942720i 0.881941 + 0.471360i \(0.156237\pi\)
−0.881941 + 0.471360i \(0.843763\pi\)
\(44\) 183.377i 0.628298i
\(45\) 0 0
\(46\) −73.5419 −0.235721
\(47\) 140.739i 0.436785i −0.975861 0.218393i \(-0.929919\pi\)
0.975861 0.218393i \(-0.0700813\pi\)
\(48\) 0 0
\(49\) −278.210 + 200.619i −0.811109 + 0.584895i
\(50\) −159.445 + 76.8222i −0.450979 + 0.217286i
\(51\) 0 0
\(52\) −99.5173 −0.265396
\(53\) −305.963 −0.792966 −0.396483 0.918042i \(-0.629769\pi\)
−0.396483 + 0.918042i \(0.629769\pi\)
\(54\) 0 0
\(55\) −333.393 + 76.1282i −0.817358 + 0.186639i
\(56\) −112.784 349.234i −0.269133 0.833363i
\(57\) 0 0
\(58\) 226.833i 0.513529i
\(59\) 134.064 0.295825 0.147912 0.989000i \(-0.452745\pi\)
0.147912 + 0.989000i \(0.452745\pi\)
\(60\) 0 0
\(61\) 303.805i 0.637676i 0.947809 + 0.318838i \(0.103293\pi\)
−0.947809 + 0.318838i \(0.896707\pi\)
\(62\) 433.422i 0.887817i
\(63\) 0 0
\(64\) 105.123 0.205319
\(65\) −41.3142 180.930i −0.0788368 0.345255i
\(66\) 0 0
\(67\) 671.592i 1.22460i −0.790626 0.612299i \(-0.790245\pi\)
0.790626 0.612299i \(-0.209755\pi\)
\(68\) 487.441i 0.869277i
\(69\) 0 0
\(70\) 251.933 149.946i 0.430168 0.256029i
\(71\) 616.486i 1.03047i 0.857049 + 0.515236i \(0.172296\pi\)
−0.857049 + 0.515236i \(0.827704\pi\)
\(72\) 0 0
\(73\) −1021.48 −1.63774 −0.818871 0.573977i \(-0.805400\pi\)
−0.818871 + 0.573977i \(0.805400\pi\)
\(74\) 148.568i 0.233388i
\(75\) 0 0
\(76\) 527.413i 0.796033i
\(77\) 539.067 174.091i 0.797823 0.257656i
\(78\) 0 0
\(79\) −558.731 −0.795723 −0.397862 0.917445i \(-0.630248\pi\)
−0.397862 + 0.917445i \(0.630248\pi\)
\(80\) −49.5409 216.958i −0.0692355 0.303207i
\(81\) 0 0
\(82\) 676.686 0.911310
\(83\) 335.471i 0.443648i −0.975087 0.221824i \(-0.928799\pi\)
0.975087 0.221824i \(-0.0712010\pi\)
\(84\) 0 0
\(85\) 886.204 202.359i 1.13085 0.258222i
\(86\) 376.372i 0.471921i
\(87\) 0 0
\(88\) 606.108i 0.734220i
\(89\) 931.364 1.10926 0.554632 0.832096i \(-0.312859\pi\)
0.554632 + 0.832096i \(0.312859\pi\)
\(90\) 0 0
\(91\) 94.4778 + 292.548i 0.108835 + 0.337004i
\(92\) 311.394 0.352881
\(93\) 0 0
\(94\) 199.272i 0.218653i
\(95\) 958.877 218.953i 1.03557 0.236465i
\(96\) 0 0
\(97\) −772.248 −0.808350 −0.404175 0.914682i \(-0.632441\pi\)
−0.404175 + 0.914682i \(0.632441\pi\)
\(98\) −393.917 + 284.056i −0.406037 + 0.292795i
\(99\) 0 0
\(100\) 675.128 325.283i 0.675128 0.325283i
\(101\) −889.653 −0.876473 −0.438237 0.898860i \(-0.644397\pi\)
−0.438237 + 0.898860i \(0.644397\pi\)
\(102\) 0 0
\(103\) 1586.47 1.51766 0.758831 0.651288i \(-0.225771\pi\)
0.758831 + 0.651288i \(0.225771\pi\)
\(104\) −328.931 −0.310137
\(105\) 0 0
\(106\) −433.211 −0.396955
\(107\) 767.328 0.693275 0.346638 0.937999i \(-0.387323\pi\)
0.346638 + 0.937999i \(0.387323\pi\)
\(108\) 0 0
\(109\) −814.720 −0.715927 −0.357963 0.933736i \(-0.616529\pi\)
−0.357963 + 0.933736i \(0.616529\pi\)
\(110\) −472.050 + 107.790i −0.409166 + 0.0934303i
\(111\) 0 0
\(112\) 113.291 + 350.801i 0.0955800 + 0.295961i
\(113\) −1649.68 −1.37335 −0.686676 0.726964i \(-0.740931\pi\)
−0.686676 + 0.726964i \(0.740931\pi\)
\(114\) 0 0
\(115\) 129.274 + 566.137i 0.104825 + 0.459066i
\(116\) 960.466i 0.768767i
\(117\) 0 0
\(118\) 189.821 0.148088
\(119\) −1432.91 + 462.757i −1.10382 + 0.356478i
\(120\) 0 0
\(121\) 395.428 0.297091
\(122\) 430.156i 0.319217i
\(123\) 0 0
\(124\) 1835.21i 1.32909i
\(125\) 871.665 + 1092.39i 0.623713 + 0.781654i
\(126\) 0 0
\(127\) 173.846i 0.121467i −0.998154 0.0607337i \(-0.980656\pi\)
0.998154 0.0607337i \(-0.0193440\pi\)
\(128\) −1344.83 −0.928652
\(129\) 0 0
\(130\) −58.4966 256.178i −0.0394653 0.172833i
\(131\) 468.800 0.312666 0.156333 0.987704i \(-0.450033\pi\)
0.156333 + 0.987704i \(0.450033\pi\)
\(132\) 0 0
\(133\) −1550.42 + 500.705i −1.01082 + 0.326441i
\(134\) 950.906i 0.613028i
\(135\) 0 0
\(136\) 1611.12i 1.01583i
\(137\) −1374.98 −0.857463 −0.428731 0.903432i \(-0.641039\pi\)
−0.428731 + 0.903432i \(0.641039\pi\)
\(138\) 0 0
\(139\) 1723.01i 1.05140i −0.850671 0.525698i \(-0.823804\pi\)
0.850671 0.525698i \(-0.176196\pi\)
\(140\) −1066.74 + 634.907i −0.643974 + 0.383282i
\(141\) 0 0
\(142\) 872.881i 0.515849i
\(143\) 507.728i 0.296911i
\(144\) 0 0
\(145\) 1746.20 398.733i 1.00010 0.228365i
\(146\) −1446.31 −0.819846
\(147\) 0 0
\(148\) 629.073i 0.349388i
\(149\) 621.643i 0.341792i −0.985289 0.170896i \(-0.945334\pi\)
0.985289 0.170896i \(-0.0546662\pi\)
\(150\) 0 0
\(151\) 2791.64 1.50451 0.752253 0.658874i \(-0.228967\pi\)
0.752253 + 0.658874i \(0.228967\pi\)
\(152\) 1743.24i 0.930232i
\(153\) 0 0
\(154\) 763.263 246.495i 0.399387 0.128981i
\(155\) 3336.55 761.880i 1.72902 0.394811i
\(156\) 0 0
\(157\) 1735.14 0.882034 0.441017 0.897499i \(-0.354618\pi\)
0.441017 + 0.897499i \(0.354618\pi\)
\(158\) −791.105 −0.398335
\(159\) 0 0
\(160\) −464.701 2035.09i −0.229611 1.00555i
\(161\) −295.625 915.393i −0.144711 0.448094i
\(162\) 0 0
\(163\) 995.750i 0.478486i 0.970960 + 0.239243i \(0.0768992\pi\)
−0.970960 + 0.239243i \(0.923101\pi\)
\(164\) −2865.25 −1.36426
\(165\) 0 0
\(166\) 474.993i 0.222088i
\(167\) 1877.32i 0.869888i 0.900457 + 0.434944i \(0.143232\pi\)
−0.900457 + 0.434944i \(0.856768\pi\)
\(168\) 0 0
\(169\) −1921.46 −0.874584
\(170\) 1254.77 286.519i 0.566098 0.129265i
\(171\) 0 0
\(172\) 1593.64i 0.706478i
\(173\) 2654.34i 1.16651i −0.812291 0.583253i \(-0.801780\pi\)
0.812291 0.583253i \(-0.198220\pi\)
\(174\) 0 0
\(175\) −1597.16 1675.84i −0.689910 0.723895i
\(176\) 608.829i 0.260751i
\(177\) 0 0
\(178\) 1318.72 0.555292
\(179\) 4472.92i 1.86772i 0.357642 + 0.933859i \(0.383581\pi\)
−0.357642 + 0.933859i \(0.616419\pi\)
\(180\) 0 0
\(181\) 396.033i 0.162635i −0.996688 0.0813173i \(-0.974087\pi\)
0.996688 0.0813173i \(-0.0259127\pi\)
\(182\) 133.771 + 414.217i 0.0544821 + 0.168702i
\(183\) 0 0
\(184\) 1029.24 0.412371
\(185\) −1143.70 + 261.157i −0.454522 + 0.103787i
\(186\) 0 0
\(187\) 2486.87 0.972504
\(188\) 843.764i 0.327329i
\(189\) 0 0
\(190\) 1357.67 310.015i 0.518399 0.118373i
\(191\) 1297.93i 0.491702i −0.969308 0.245851i \(-0.920933\pi\)
0.969308 0.245851i \(-0.0790673\pi\)
\(192\) 0 0
\(193\) 4471.94i 1.66786i 0.551868 + 0.833931i \(0.313915\pi\)
−0.551868 + 0.833931i \(0.686085\pi\)
\(194\) −1093.42 −0.404656
\(195\) 0 0
\(196\) 1667.94 1202.76i 0.607849 0.438323i
\(197\) −21.6839 −0.00784220 −0.00392110 0.999992i \(-0.501248\pi\)
−0.00392110 + 0.999992i \(0.501248\pi\)
\(198\) 0 0
\(199\) 1699.27i 0.605316i 0.953099 + 0.302658i \(0.0978740\pi\)
−0.953099 + 0.302658i \(0.902126\pi\)
\(200\) 2231.47 1075.14i 0.788945 0.380121i
\(201\) 0 0
\(202\) −1259.66 −0.438758
\(203\) −2823.45 + 911.828i −0.976193 + 0.315260i
\(204\) 0 0
\(205\) −1189.49 5209.23i −0.405258 1.77477i
\(206\) 2246.27 0.759734
\(207\) 0 0
\(208\) 330.407 0.110142
\(209\) 2690.81 0.890562
\(210\) 0 0
\(211\) −1882.71 −0.614269 −0.307135 0.951666i \(-0.599370\pi\)
−0.307135 + 0.951666i \(0.599370\pi\)
\(212\) 1834.32 0.594252
\(213\) 0 0
\(214\) 1086.46 0.347050
\(215\) 2897.37 661.595i 0.919064 0.209862i
\(216\) 0 0
\(217\) −5394.91 + 1742.28i −1.68770 + 0.545039i
\(218\) −1153.56 −0.358389
\(219\) 0 0
\(220\) 1998.77 456.406i 0.612532 0.139868i
\(221\) 1349.61i 0.410790i
\(222\) 0 0
\(223\) −5818.27 −1.74717 −0.873587 0.486668i \(-0.838212\pi\)
−0.873587 + 0.486668i \(0.838212\pi\)
\(224\) 1062.68 + 3290.57i 0.316980 + 0.981519i
\(225\) 0 0
\(226\) −2335.77 −0.687493
\(227\) 2706.94i 0.791480i −0.918363 0.395740i \(-0.870488\pi\)
0.918363 0.395740i \(-0.129512\pi\)
\(228\) 0 0
\(229\) 6228.53i 1.79735i 0.438616 + 0.898674i \(0.355469\pi\)
−0.438616 + 0.898674i \(0.644531\pi\)
\(230\) 183.038 + 801.592i 0.0524747 + 0.229806i
\(231\) 0 0
\(232\) 3174.59i 0.898370i
\(233\) −4128.59 −1.16083 −0.580414 0.814322i \(-0.697109\pi\)
−0.580414 + 0.814322i \(0.697109\pi\)
\(234\) 0 0
\(235\) −1534.03 + 350.285i −0.425825 + 0.0972344i
\(236\) −803.746 −0.221692
\(237\) 0 0
\(238\) −2028.86 + 655.216i −0.552568 + 0.178451i
\(239\) 4645.24i 1.25722i −0.777721 0.628610i \(-0.783624\pi\)
0.777721 0.628610i \(-0.216376\pi\)
\(240\) 0 0
\(241\) 6926.50i 1.85135i 0.378320 + 0.925675i \(0.376502\pi\)
−0.378320 + 0.925675i \(0.623498\pi\)
\(242\) 559.886 0.148722
\(243\) 0 0
\(244\) 1821.38i 0.477877i
\(245\) 2879.14 + 2533.12i 0.750782 + 0.660550i
\(246\) 0 0
\(247\) 1460.28i 0.376177i
\(248\) 6065.85i 1.55315i
\(249\) 0 0
\(250\) 1234.19 + 1546.72i 0.312228 + 0.391292i
\(251\) 5572.63 1.40136 0.700680 0.713476i \(-0.252880\pi\)
0.700680 + 0.713476i \(0.252880\pi\)
\(252\) 0 0
\(253\) 1588.70i 0.394785i
\(254\) 246.148i 0.0608060i
\(255\) 0 0
\(256\) −2745.13 −0.670198
\(257\) 2220.94i 0.539060i −0.962992 0.269530i \(-0.913132\pi\)
0.962992 0.269530i \(-0.0868683\pi\)
\(258\) 0 0
\(259\) 1849.26 597.217i 0.443659 0.143279i
\(260\) 247.688 + 1084.72i 0.0590807 + 0.258736i
\(261\) 0 0
\(262\) 663.772 0.156519
\(263\) −4665.04 −1.09376 −0.546880 0.837211i \(-0.684184\pi\)
−0.546880 + 0.837211i \(0.684184\pi\)
\(264\) 0 0
\(265\) 761.509 + 3334.93i 0.176525 + 0.773068i
\(266\) −2195.23 + 708.947i −0.506009 + 0.163415i
\(267\) 0 0
\(268\) 4026.36i 0.917719i
\(269\) 4373.91 0.991384 0.495692 0.868498i \(-0.334915\pi\)
0.495692 + 0.868498i \(0.334915\pi\)
\(270\) 0 0
\(271\) 1610.82i 0.361071i 0.983568 + 0.180535i \(0.0577830\pi\)
−0.983568 + 0.180535i \(0.942217\pi\)
\(272\) 1618.35i 0.360760i
\(273\) 0 0
\(274\) −1946.83 −0.429242
\(275\) 1659.56 + 3444.44i 0.363910 + 0.755300i
\(276\) 0 0
\(277\) 4785.35i 1.03799i −0.854777 0.518996i \(-0.826306\pi\)
0.854777 0.518996i \(-0.173694\pi\)
\(278\) 2439.61i 0.526324i
\(279\) 0 0
\(280\) −3525.87 + 2098.53i −0.752538 + 0.447898i
\(281\) 4559.34i 0.967927i 0.875088 + 0.483964i \(0.160803\pi\)
−0.875088 + 0.483964i \(0.839197\pi\)
\(282\) 0 0
\(283\) 3503.70 0.735949 0.367975 0.929836i \(-0.380051\pi\)
0.367975 + 0.929836i \(0.380051\pi\)
\(284\) 3695.98i 0.772240i
\(285\) 0 0
\(286\) 718.890i 0.148632i
\(287\) 2720.15 + 8422.87i 0.559461 + 1.73236i
\(288\) 0 0
\(289\) −1697.45 −0.345501
\(290\) 2472.44 564.565i 0.500643 0.114319i
\(291\) 0 0
\(292\) 6124.02 1.22733
\(293\) 5727.17i 1.14193i 0.820975 + 0.570964i \(0.193431\pi\)
−0.820975 + 0.570964i \(0.806569\pi\)
\(294\) 0 0
\(295\) −333.672 1461.27i −0.0658546 0.288401i
\(296\) 2079.25i 0.408290i
\(297\) 0 0
\(298\) 880.182i 0.171099i
\(299\) −862.176 −0.166759
\(300\) 0 0
\(301\) −4684.78 + 1512.94i −0.897098 + 0.289716i
\(302\) 3952.67 0.753149
\(303\) 0 0
\(304\) 1751.06i 0.330363i
\(305\) 3311.41 756.139i 0.621674 0.141955i
\(306\) 0 0
\(307\) −1876.50 −0.348852 −0.174426 0.984670i \(-0.555807\pi\)
−0.174426 + 0.984670i \(0.555807\pi\)
\(308\) −3231.84 + 1043.72i −0.597893 + 0.193088i
\(309\) 0 0
\(310\) 4724.21 1078.74i 0.865539 0.197640i
\(311\) −4517.34 −0.823650 −0.411825 0.911263i \(-0.635108\pi\)
−0.411825 + 0.911263i \(0.635108\pi\)
\(312\) 0 0
\(313\) 7150.40 1.29126 0.645630 0.763650i \(-0.276595\pi\)
0.645630 + 0.763650i \(0.276595\pi\)
\(314\) 2456.78 0.441542
\(315\) 0 0
\(316\) 3349.72 0.596318
\(317\) 4062.32 0.719756 0.359878 0.932999i \(-0.382818\pi\)
0.359878 + 0.932999i \(0.382818\pi\)
\(318\) 0 0
\(319\) 4900.20 0.860059
\(320\) −261.641 1145.82i −0.0457069 0.200167i
\(321\) 0 0
\(322\) −418.574 1296.10i −0.0724417 0.224314i
\(323\) −7152.54 −1.23213
\(324\) 0 0
\(325\) −1869.27 + 900.632i −0.319042 + 0.153717i
\(326\) 1409.88i 0.239528i
\(327\) 0 0
\(328\) −9470.38 −1.59425
\(329\) 2480.39 801.036i 0.415648 0.134233i
\(330\) 0 0
\(331\) −868.986 −0.144301 −0.0721507 0.997394i \(-0.522986\pi\)
−0.0721507 + 0.997394i \(0.522986\pi\)
\(332\) 2011.23i 0.332472i
\(333\) 0 0
\(334\) 2658.09i 0.435462i
\(335\) −7320.22 + 1671.52i −1.19387 + 0.272612i
\(336\) 0 0
\(337\) 1287.31i 0.208083i 0.994573 + 0.104042i \(0.0331775\pi\)
−0.994573 + 0.104042i \(0.966823\pi\)
\(338\) −2720.59 −0.437812
\(339\) 0 0
\(340\) −5313.00 + 1213.19i −0.847464 + 0.193513i
\(341\) 9363.07 1.48692
\(342\) 0 0
\(343\) −5119.18 3761.33i −0.805859 0.592107i
\(344\) 5267.41i 0.825580i
\(345\) 0 0
\(346\) 3758.27i 0.583947i
\(347\) −422.265 −0.0653268 −0.0326634 0.999466i \(-0.510399\pi\)
−0.0326634 + 0.999466i \(0.510399\pi\)
\(348\) 0 0
\(349\) 4640.81i 0.711797i 0.934525 + 0.355898i \(0.115825\pi\)
−0.934525 + 0.355898i \(0.884175\pi\)
\(350\) −2261.42 2372.82i −0.345365 0.362379i
\(351\) 0 0
\(352\) 5710.91i 0.864751i
\(353\) 1539.01i 0.232048i −0.993246 0.116024i \(-0.962985\pi\)
0.993246 0.116024i \(-0.0370150\pi\)
\(354\) 0 0
\(355\) 6719.57 1534.37i 1.00461 0.229397i
\(356\) −5583.75 −0.831287
\(357\) 0 0
\(358\) 6333.19i 0.934970i
\(359\) 1035.99i 0.152305i −0.997096 0.0761525i \(-0.975736\pi\)
0.997096 0.0761525i \(-0.0242636\pi\)
\(360\) 0 0
\(361\) −880.088 −0.128311
\(362\) 560.741i 0.0814141i
\(363\) 0 0
\(364\) −566.417 1753.89i −0.0815613 0.252552i
\(365\) 2542.36 + 11133.9i 0.364584 + 1.59665i
\(366\) 0 0
\(367\) 2278.61 0.324094 0.162047 0.986783i \(-0.448190\pi\)
0.162047 + 0.986783i \(0.448190\pi\)
\(368\) −1033.86 −0.146450
\(369\) 0 0
\(370\) −1619.36 + 369.771i −0.227532 + 0.0519554i
\(371\) −1741.43 5392.28i −0.243694 0.754591i
\(372\) 0 0
\(373\) 8610.23i 1.19523i 0.801783 + 0.597615i \(0.203885\pi\)
−0.801783 + 0.597615i \(0.796115\pi\)
\(374\) 3521.16 0.486831
\(375\) 0 0
\(376\) 2788.86i 0.382512i
\(377\) 2659.30i 0.363292i
\(378\) 0 0
\(379\) −16.5038 −0.00223679 −0.00111840 0.999999i \(-0.500356\pi\)
−0.00111840 + 0.999999i \(0.500356\pi\)
\(380\) −5748.70 + 1312.68i −0.776057 + 0.177208i
\(381\) 0 0
\(382\) 1837.74i 0.246143i
\(383\) 9887.59i 1.31914i −0.751641 0.659572i \(-0.770737\pi\)
0.751641 0.659572i \(-0.229263\pi\)
\(384\) 0 0
\(385\) −3239.24 5442.42i −0.428797 0.720446i
\(386\) 6331.81i 0.834924i
\(387\) 0 0
\(388\) 4629.81 0.605781
\(389\) 279.292i 0.0364028i −0.999834 0.0182014i \(-0.994206\pi\)
0.999834 0.0182014i \(-0.00579400\pi\)
\(390\) 0 0
\(391\) 4222.98i 0.546203i
\(392\) 5512.97 3975.43i 0.710324 0.512218i
\(393\) 0 0
\(394\) −30.7021 −0.00392577
\(395\) 1390.62 + 6090.05i 0.177139 + 0.775756i
\(396\) 0 0
\(397\) 10894.0 1.37721 0.688607 0.725135i \(-0.258223\pi\)
0.688607 + 0.725135i \(0.258223\pi\)
\(398\) 2405.99i 0.303018i
\(399\) 0 0
\(400\) −2241.49 + 1079.97i −0.280186 + 0.134996i
\(401\) 9631.75i 1.19947i −0.800200 0.599734i \(-0.795273\pi\)
0.800200 0.599734i \(-0.204727\pi\)
\(402\) 0 0
\(403\) 5081.27i 0.628079i
\(404\) 5333.68 0.656833
\(405\) 0 0
\(406\) −3997.71 + 1291.05i −0.488678 + 0.157818i
\(407\) −3209.47 −0.390878
\(408\) 0 0
\(409\) 4082.30i 0.493538i −0.969074 0.246769i \(-0.920631\pi\)
0.969074 0.246769i \(-0.0793689\pi\)
\(410\) −1684.20 7375.73i −0.202870 0.888443i
\(411\) 0 0
\(412\) −9511.24 −1.13734
\(413\) 763.044 + 2362.74i 0.0909127 + 0.281509i
\(414\) 0 0
\(415\) −3656.57 + 834.953i −0.432515 + 0.0987620i
\(416\) 3099.27 0.365274
\(417\) 0 0
\(418\) 3809.91 0.445811
\(419\) −15952.5 −1.85998 −0.929988 0.367589i \(-0.880183\pi\)
−0.929988 + 0.367589i \(0.880183\pi\)
\(420\) 0 0
\(421\) 16707.5 1.93415 0.967073 0.254499i \(-0.0819104\pi\)
0.967073 + 0.254499i \(0.0819104\pi\)
\(422\) −2665.72 −0.307500
\(423\) 0 0
\(424\) 6062.90 0.694435
\(425\) −4411.34 9155.78i −0.503486 1.04499i
\(426\) 0 0
\(427\) −5354.26 + 1729.15i −0.606816 + 0.195970i
\(428\) −4600.31 −0.519543
\(429\) 0 0
\(430\) 4102.37 936.750i 0.460079 0.105056i
\(431\) 1021.69i 0.114183i 0.998369 + 0.0570916i \(0.0181827\pi\)
−0.998369 + 0.0570916i \(0.981817\pi\)
\(432\) 0 0
\(433\) −1052.40 −0.116802 −0.0584008 0.998293i \(-0.518600\pi\)
−0.0584008 + 0.998293i \(0.518600\pi\)
\(434\) −7638.63 + 2466.88i −0.844853 + 0.272844i
\(435\) 0 0
\(436\) 4884.44 0.536519
\(437\) 4569.29i 0.500180i
\(438\) 0 0
\(439\) 13540.8i 1.47213i −0.676909 0.736066i \(-0.736681\pi\)
0.676909 0.736066i \(-0.263319\pi\)
\(440\) 6606.45 1508.54i 0.715796 0.163447i
\(441\) 0 0
\(442\) 1910.91i 0.205639i
\(443\) 12799.7 1.37276 0.686380 0.727243i \(-0.259199\pi\)
0.686380 + 0.727243i \(0.259199\pi\)
\(444\) 0 0
\(445\) −2318.07 10151.7i −0.246937 1.08143i
\(446\) −8238.06 −0.874627
\(447\) 0 0
\(448\) 598.325 + 1852.70i 0.0630986 + 0.195383i
\(449\) 523.887i 0.0550641i −0.999621 0.0275320i \(-0.991235\pi\)
0.999621 0.0275320i \(-0.00876482\pi\)
\(450\) 0 0
\(451\) 14618.2i 1.52626i
\(452\) 9890.21 1.02920
\(453\) 0 0
\(454\) 3832.75i 0.396211i
\(455\) 2953.56 1757.91i 0.304319 0.181125i
\(456\) 0 0
\(457\) 5749.64i 0.588527i −0.955724 0.294263i \(-0.904926\pi\)
0.955724 0.294263i \(-0.0950743\pi\)
\(458\) 8818.96i 0.899744i
\(459\) 0 0
\(460\) −775.027 3394.12i −0.0785561 0.344026i
\(461\) −14278.7 −1.44257 −0.721284 0.692640i \(-0.756448\pi\)
−0.721284 + 0.692640i \(0.756448\pi\)
\(462\) 0 0
\(463\) 16272.2i 1.63333i −0.577113 0.816664i \(-0.695821\pi\)
0.577113 0.816664i \(-0.304179\pi\)
\(464\) 3188.84i 0.319047i
\(465\) 0 0
\(466\) −5845.66 −0.581105
\(467\) 15119.6i 1.49819i −0.662464 0.749094i \(-0.730489\pi\)
0.662464 0.749094i \(-0.269511\pi\)
\(468\) 0 0
\(469\) 11836.1 3822.46i 1.16534 0.376343i
\(470\) −2172.02 + 495.967i −0.213166 + 0.0486751i
\(471\) 0 0
\(472\) −2656.59 −0.259066
\(473\) 8130.62 0.790373
\(474\) 0 0
\(475\) −4773.09 9906.61i −0.461062 0.956940i
\(476\) 8590.65 2774.34i 0.827210 0.267146i
\(477\) 0 0
\(478\) 6577.18i 0.629358i
\(479\) −13040.7 −1.24394 −0.621970 0.783041i \(-0.713667\pi\)
−0.621970 + 0.783041i \(0.713667\pi\)
\(480\) 0 0
\(481\) 1741.75i 0.165108i
\(482\) 9807.21i 0.926777i
\(483\) 0 0
\(484\) −2370.69 −0.222642
\(485\) 1922.05 + 8417.34i 0.179950 + 0.788066i
\(486\) 0 0
\(487\) 12919.7i 1.20215i 0.799192 + 0.601075i \(0.205261\pi\)
−0.799192 + 0.601075i \(0.794739\pi\)
\(488\) 6020.14i 0.558440i
\(489\) 0 0
\(490\) 4076.57 + 3586.63i 0.375838 + 0.330668i
\(491\) 10382.4i 0.954275i −0.878829 0.477138i \(-0.841674\pi\)
0.878829 0.477138i \(-0.158326\pi\)
\(492\) 0 0
\(493\) −13025.4 −1.18993
\(494\) 2067.61i 0.188312i
\(495\) 0 0
\(496\) 6093.07i 0.551587i
\(497\) −10865.0 + 3508.82i −0.980603 + 0.316684i
\(498\) 0 0
\(499\) 13113.5 1.17643 0.588217 0.808703i \(-0.299830\pi\)
0.588217 + 0.808703i \(0.299830\pi\)
\(500\) −5225.84 6549.16i −0.467413 0.585775i
\(501\) 0 0
\(502\) 7890.27 0.701514
\(503\) 9544.92i 0.846097i −0.906107 0.423049i \(-0.860960\pi\)
0.906107 0.423049i \(-0.139040\pi\)
\(504\) 0 0
\(505\) 2214.25 + 9697.03i 0.195115 + 0.854480i
\(506\) 2249.44i 0.197628i
\(507\) 0 0
\(508\) 1042.25i 0.0910282i
\(509\) 3806.73 0.331493 0.165747 0.986168i \(-0.446997\pi\)
0.165747 + 0.986168i \(0.446997\pi\)
\(510\) 0 0
\(511\) −5813.90 18002.6i −0.503311 1.55849i
\(512\) 6871.84 0.593155
\(513\) 0 0
\(514\) 3144.62i 0.269851i
\(515\) −3948.55 17292.2i −0.337852 1.47958i
\(516\) 0 0
\(517\) −4304.80 −0.366199
\(518\) 2618.37 845.597i 0.222094 0.0717247i
\(519\) 0 0
\(520\) 818.674 + 3585.27i 0.0690408 + 0.302355i
\(521\) 7738.68 0.650744 0.325372 0.945586i \(-0.394510\pi\)
0.325372 + 0.945586i \(0.394510\pi\)
\(522\) 0 0
\(523\) 19441.5 1.62547 0.812734 0.582635i \(-0.197978\pi\)
0.812734 + 0.582635i \(0.197978\pi\)
\(524\) −2810.57 −0.234313
\(525\) 0 0
\(526\) −6605.21 −0.547530
\(527\) −24888.3 −2.05721
\(528\) 0 0
\(529\) −9469.22 −0.778271
\(530\) 1078.22 + 4721.91i 0.0883676 + 0.386994i
\(531\) 0 0
\(532\) 9295.13 3001.85i 0.757510 0.244636i
\(533\) 7933.19 0.644699
\(534\) 0 0
\(535\) −1909.80 8363.72i −0.154333 0.675879i
\(536\) 13308.1i 1.07243i
\(537\) 0 0
\(538\) 6193.01 0.496282
\(539\) 6136.35 + 8509.66i 0.490374 + 0.680031i
\(540\) 0 0
\(541\) −10379.0 −0.824817 −0.412409 0.910999i \(-0.635312\pi\)
−0.412409 + 0.910999i \(0.635312\pi\)
\(542\) 2280.75i 0.180750i
\(543\) 0 0
\(544\) 15180.4i 1.19642i
\(545\) 2027.75 + 8880.28i 0.159375 + 0.697962i
\(546\) 0 0
\(547\) 16009.6i 1.25141i 0.780061 + 0.625703i \(0.215188\pi\)
−0.780061 + 0.625703i \(0.784812\pi\)
\(548\) 8243.33 0.642587
\(549\) 0 0
\(550\) 2349.77 + 4876.97i 0.182172 + 0.378099i
\(551\) −14093.6 −1.08967
\(552\) 0 0
\(553\) −3180.09 9847.07i −0.244541 0.757215i
\(554\) 6775.56i 0.519614i
\(555\) 0 0
\(556\) 10329.9i 0.787921i
\(557\) −11746.0 −0.893526 −0.446763 0.894652i \(-0.647423\pi\)
−0.446763 + 0.894652i \(0.647423\pi\)
\(558\) 0 0
\(559\) 4412.43i 0.333857i
\(560\) 3541.69 2107.95i 0.267257 0.159066i
\(561\) 0 0
\(562\) 6455.56i 0.484540i
\(563\) 9594.88i 0.718252i 0.933289 + 0.359126i \(0.116925\pi\)
−0.933289 + 0.359126i \(0.883075\pi\)
\(564\) 0 0
\(565\) 4105.88 + 17981.1i 0.305727 + 1.33889i
\(566\) 4960.88 0.368412
\(567\) 0 0
\(568\) 12216.2i 0.902428i
\(569\) 8438.18i 0.621700i −0.950459 0.310850i \(-0.899386\pi\)
0.950459 0.310850i \(-0.100614\pi\)
\(570\) 0 0
\(571\) 18415.6 1.34968 0.674841 0.737963i \(-0.264212\pi\)
0.674841 + 0.737963i \(0.264212\pi\)
\(572\) 3043.95i 0.222507i
\(573\) 0 0
\(574\) 3851.45 + 11925.9i 0.280064 + 0.867209i
\(575\) 5849.02 2818.11i 0.424211 0.204389i
\(576\) 0 0
\(577\) −5724.61 −0.413030 −0.206515 0.978443i \(-0.566212\pi\)
−0.206515 + 0.978443i \(0.566212\pi\)
\(578\) −2403.41 −0.172956
\(579\) 0 0
\(580\) −10468.9 + 2390.50i −0.749476 + 0.171138i
\(581\) 5912.35 1909.38i 0.422178 0.136342i
\(582\) 0 0
\(583\) 9358.51i 0.664820i
\(584\) 20241.5 1.43424
\(585\) 0 0
\(586\) 8109.09i 0.571644i
\(587\) 20033.2i 1.40862i −0.709894 0.704309i \(-0.751257\pi\)
0.709894 0.704309i \(-0.248743\pi\)
\(588\) 0 0
\(589\) −26929.3 −1.88387
\(590\) −472.444 2069.01i −0.0329665 0.144372i
\(591\) 0 0
\(592\) 2088.58i 0.145000i
\(593\) 10763.1i 0.745341i 0.927964 + 0.372670i \(0.121558\pi\)
−0.927964 + 0.372670i \(0.878442\pi\)
\(594\) 0 0
\(595\) 8610.32 + 14466.7i 0.593259 + 0.996768i
\(596\) 3726.90i 0.256140i
\(597\) 0 0
\(598\) −1220.75 −0.0834787
\(599\) 12751.4i 0.869799i 0.900479 + 0.434899i \(0.143216\pi\)
−0.900479 + 0.434899i \(0.856784\pi\)
\(600\) 0 0
\(601\) 8319.06i 0.564628i 0.959322 + 0.282314i \(0.0911020\pi\)
−0.959322 + 0.282314i \(0.908898\pi\)
\(602\) −6633.17 + 2142.17i −0.449083 + 0.145031i
\(603\) 0 0
\(604\) −16736.5 −1.12748
\(605\) −984.181 4310.09i −0.0661366 0.289636i
\(606\) 0 0
\(607\) −12259.8 −0.819786 −0.409893 0.912134i \(-0.634434\pi\)
−0.409893 + 0.912134i \(0.634434\pi\)
\(608\) 16425.2i 1.09561i
\(609\) 0 0
\(610\) 4688.61 1070.61i 0.311207 0.0710622i
\(611\) 2336.18i 0.154684i
\(612\) 0 0
\(613\) 23691.3i 1.56098i −0.625167 0.780491i \(-0.714969\pi\)
0.625167 0.780491i \(-0.285031\pi\)
\(614\) −2656.93 −0.174634
\(615\) 0 0
\(616\) −10682.1 + 3449.75i −0.698689 + 0.225640i
\(617\) −8805.60 −0.574554 −0.287277 0.957848i \(-0.592750\pi\)
−0.287277 + 0.957848i \(0.592750\pi\)
\(618\) 0 0
\(619\) 1540.85i 0.100051i −0.998748 0.0500257i \(-0.984070\pi\)
0.998748 0.0500257i \(-0.0159303\pi\)
\(620\) −20003.4 + 4567.65i −1.29574 + 0.295873i
\(621\) 0 0
\(622\) −6396.09 −0.412315
\(623\) 5300.99 + 16414.4i 0.340898 + 1.05558i
\(624\) 0 0
\(625\) 9737.38 12219.8i 0.623192 0.782069i
\(626\) 10124.2 0.646399
\(627\) 0 0
\(628\) −10402.6 −0.661000
\(629\) 8531.20 0.540797
\(630\) 0 0
\(631\) 4818.83 0.304017 0.152008 0.988379i \(-0.451426\pi\)
0.152008 + 0.988379i \(0.451426\pi\)
\(632\) 11071.7 0.696849
\(633\) 0 0
\(634\) 5751.83 0.360306
\(635\) −1894.89 + 432.685i −0.118419 + 0.0270403i
\(636\) 0 0
\(637\) −4618.13 + 3330.15i −0.287248 + 0.207136i
\(638\) 6938.18 0.430541
\(639\) 0 0
\(640\) 3347.15 + 14658.4i 0.206731 + 0.905350i
\(641\) 18206.7i 1.12187i 0.827858 + 0.560937i \(0.189559\pi\)
−0.827858 + 0.560937i \(0.810441\pi\)
\(642\) 0 0
\(643\) 18717.3 1.14796 0.573981 0.818868i \(-0.305398\pi\)
0.573981 + 0.818868i \(0.305398\pi\)
\(644\) 1772.34 + 5488.00i 0.108447 + 0.335804i
\(645\) 0 0
\(646\) −10127.3 −0.616798
\(647\) 1556.10i 0.0945542i 0.998882 + 0.0472771i \(0.0150544\pi\)
−0.998882 + 0.0472771i \(0.984946\pi\)
\(648\) 0 0
\(649\) 4100.63i 0.248018i
\(650\) −2646.70 + 1275.20i −0.159711 + 0.0769500i
\(651\) 0 0
\(652\) 5969.76i 0.358579i
\(653\) 29408.2 1.76238 0.881189 0.472765i \(-0.156744\pi\)
0.881189 + 0.472765i \(0.156744\pi\)
\(654\) 0 0
\(655\) −1166.79 5109.82i −0.0696037 0.304820i
\(656\) 9512.88 0.566182
\(657\) 0 0
\(658\) 3511.97 1134.18i 0.208071 0.0671962i
\(659\) 15993.9i 0.945425i −0.881217 0.472713i \(-0.843275\pi\)
0.881217 0.472713i \(-0.156725\pi\)
\(660\) 0 0
\(661\) 17299.8i 1.01798i 0.860773 + 0.508988i \(0.169980\pi\)
−0.860773 + 0.508988i \(0.830020\pi\)
\(662\) −1230.39 −0.0722366
\(663\) 0 0
\(664\) 6647.64i 0.388522i
\(665\) 9316.42 + 15653.0i 0.543271 + 0.912780i
\(666\) 0 0
\(667\) 8321.07i 0.483048i
\(668\) 11255.0i 0.651898i
\(669\) 0 0
\(670\) −10364.7 + 2366.71i −0.597645 + 0.136468i
\(671\) 9292.51 0.534625
\(672\) 0 0
\(673\) 17783.4i 1.01857i −0.860597 0.509287i \(-0.829909\pi\)
0.860597 0.509287i \(-0.170091\pi\)
\(674\) 1822.69i 0.104165i
\(675\) 0 0
\(676\) 11519.6 0.655417
\(677\) 12701.0i 0.721032i 0.932753 + 0.360516i \(0.117399\pi\)
−0.932753 + 0.360516i \(0.882601\pi\)
\(678\) 0 0
\(679\) −4395.36 13610.1i −0.248422 0.769231i
\(680\) −17560.8 + 4009.91i −0.990335 + 0.226137i
\(681\) 0 0
\(682\) 13257.1 0.744343
\(683\) 27112.0 1.51891 0.759453 0.650562i \(-0.225467\pi\)
0.759453 + 0.650562i \(0.225467\pi\)
\(684\) 0 0
\(685\) 3422.18 + 14987.0i 0.190883 + 0.835946i
\(686\) −7248.23 5325.66i −0.403409 0.296406i
\(687\) 0 0
\(688\) 5291.05i 0.293197i
\(689\) −5078.80 −0.280822
\(690\) 0 0
\(691\) 521.160i 0.0286916i −0.999897 0.0143458i \(-0.995433\pi\)
0.999897 0.0143458i \(-0.00456656\pi\)
\(692\) 15913.4i 0.874185i
\(693\) 0 0
\(694\) −597.884 −0.0327023
\(695\) −18780.5 + 4288.40i −1.02501 + 0.234055i
\(696\) 0 0
\(697\) 38857.1i 2.11165i
\(698\) 6570.91i 0.356322i
\(699\) 0 0
\(700\) 9575.37 + 10047.1i 0.517021 + 0.542491i
\(701\) 3411.27i 0.183797i 0.995768 + 0.0918987i \(0.0292936\pi\)
−0.995768 + 0.0918987i \(0.970706\pi\)
\(702\) 0 0
\(703\) 9230.80 0.495230
\(704\) 3215.42i 0.172139i
\(705\) 0 0
\(706\) 2179.07i 0.116162i
\(707\) −5063.58 15679.2i −0.269357 0.834057i
\(708\) 0 0
\(709\) −6386.93 −0.338316 −0.169158 0.985589i \(-0.554105\pi\)
−0.169158 + 0.985589i \(0.554105\pi\)
\(710\) 9514.22 2172.51i 0.502905 0.114835i
\(711\) 0 0
\(712\) −18455.7 −0.971430
\(713\) 15899.5i 0.835120i
\(714\) 0 0
\(715\) −5534.12 + 1263.68i −0.289461 + 0.0660965i
\(716\) 26816.2i 1.39968i
\(717\) 0 0
\(718\) 1466.86i 0.0762431i
\(719\) 1659.01 0.0860510 0.0430255 0.999074i \(-0.486300\pi\)
0.0430255 + 0.999074i \(0.486300\pi\)
\(720\) 0 0
\(721\) 9029.60 + 27959.9i 0.466407 + 1.44422i
\(722\) −1246.11 −0.0642321
\(723\) 0 0
\(724\) 2374.31i 0.121879i
\(725\) −8692.21 18040.8i −0.445270 0.924163i
\(726\) 0 0
\(727\) 25833.8 1.31791 0.658955 0.752182i \(-0.270999\pi\)
0.658955 + 0.752182i \(0.270999\pi\)
\(728\) −1872.15 5797.07i −0.0953113 0.295129i
\(729\) 0 0
\(730\) 3599.72 + 15764.5i 0.182509 + 0.799273i
\(731\) −21612.3 −1.09351
\(732\) 0 0
\(733\) −17410.3 −0.877304 −0.438652 0.898657i \(-0.644544\pi\)
−0.438652 + 0.898657i \(0.644544\pi\)
\(734\) 3226.28 0.162240
\(735\) 0 0
\(736\) −9697.73 −0.485683
\(737\) −20542.1 −1.02670
\(738\) 0 0
\(739\) −26348.6 −1.31157 −0.655784 0.754949i \(-0.727662\pi\)
−0.655784 + 0.754949i \(0.727662\pi\)
\(740\) 6856.76 1565.70i 0.340621 0.0777786i
\(741\) 0 0
\(742\) −2465.68 7634.92i −0.121992 0.377745i
\(743\) 27257.0 1.34585 0.672923 0.739713i \(-0.265039\pi\)
0.672923 + 0.739713i \(0.265039\pi\)
\(744\) 0 0
\(745\) −6775.78 + 1547.20i −0.333215 + 0.0760875i
\(746\) 12191.2i 0.598326i
\(747\) 0 0
\(748\) −14909.4 −0.728799
\(749\) 4367.36 + 13523.4i 0.213057 + 0.659725i
\(750\) 0 0
\(751\) 12085.2 0.587210 0.293605 0.955927i \(-0.405145\pi\)
0.293605 + 0.955927i \(0.405145\pi\)
\(752\) 2801.38i 0.135845i
\(753\) 0 0
\(754\) 3765.30i 0.181862i
\(755\) −6948.10 30428.3i −0.334924 1.46675i
\(756\) 0 0
\(757\) 1931.31i 0.0927275i 0.998925 + 0.0463638i \(0.0147633\pi\)
−0.998925 + 0.0463638i \(0.985237\pi\)
\(758\) −23.3677 −0.00111973
\(759\) 0 0
\(760\) −19000.9 + 4338.74i −0.906890 + 0.207082i
\(761\) 26856.6 1.27931 0.639653 0.768664i \(-0.279078\pi\)
0.639653 + 0.768664i \(0.279078\pi\)
\(762\) 0 0
\(763\) −4637.09 14358.6i −0.220018 0.681280i
\(764\) 7781.40i 0.368483i
\(765\) 0 0
\(766\) 13999.8i 0.660357i
\(767\) 2225.38 0.104764
\(768\) 0 0
\(769\) 2772.05i 0.129990i −0.997886 0.0649952i \(-0.979297\pi\)
0.997886 0.0649952i \(-0.0207032\pi\)
\(770\) −4586.42 7705.91i −0.214654 0.360652i
\(771\) 0 0
\(772\) 26810.3i 1.24990i
\(773\) 18591.8i 0.865071i 0.901617 + 0.432535i \(0.142381\pi\)
−0.901617 + 0.432535i \(0.857619\pi\)
\(774\) 0 0
\(775\) −16608.7 34471.5i −0.769807 1.59774i
\(776\) 15302.7 0.707907
\(777\) 0 0
\(778\) 395.449i 0.0182230i
\(779\) 42043.6i 1.93372i
\(780\) 0 0
\(781\) 18856.5 0.863944
\(782\) 5979.30i 0.273426i
\(783\) 0 0
\(784\) −5537.71 + 3993.27i −0.252264 + 0.181909i
\(785\) −4318.59 18912.7i −0.196353 0.859901i
\(786\) 0 0
\(787\) −10211.2 −0.462504 −0.231252 0.972894i \(-0.574282\pi\)
−0.231252 + 0.972894i \(0.574282\pi\)
\(788\) 130.000 0.00587698
\(789\) 0 0
\(790\) 1968.98 + 8622.88i 0.0886748 + 0.388340i
\(791\) −9389.37 29073.9i −0.422058 1.30689i
\(792\) 0 0
\(793\) 5042.98i 0.225828i
\(794\) 15424.8 0.689427
\(795\) 0 0
\(796\) 10187.5i 0.453627i
\(797\) 15129.4i 0.672412i −0.941788 0.336206i \(-0.890856\pi\)
0.941788 0.336206i \(-0.109144\pi\)
\(798\) 0 0
\(799\) 11442.7 0.506652
\(800\) −21025.5 + 10130.3i −0.929205 + 0.447699i
\(801\) 0 0
\(802\) 13637.6i 0.600448i
\(803\) 31244.1i 1.37308i
\(804\) 0 0
\(805\) −9241.82 + 5500.57i −0.404635 + 0.240832i
\(806\) 7194.55i 0.314413i
\(807\) 0 0
\(808\) 17629.2 0.767565
\(809\) 27312.2i 1.18695i −0.804851 0.593477i \(-0.797755\pi\)
0.804851 0.593477i \(-0.202245\pi\)
\(810\) 0 0
\(811\) 955.054i 0.0413520i −0.999786 0.0206760i \(-0.993418\pi\)
0.999786 0.0206760i \(-0.00658184\pi\)
\(812\) 16927.2 5466.62i 0.731564 0.236257i
\(813\) 0 0
\(814\) −4544.28 −0.195672
\(815\) 10853.5 2478.32i 0.466479 0.106517i
\(816\) 0 0
\(817\) −23384.6 −1.00138
\(818\) 5780.12i 0.247063i
\(819\) 0 0
\(820\) 7131.30 + 31230.6i 0.303702 + 1.33002i
\(821\) 4863.55i 0.206747i −0.994643 0.103373i \(-0.967036\pi\)
0.994643 0.103373i \(-0.0329636\pi\)
\(822\) 0 0
\(823\) 24899.9i 1.05463i −0.849671 0.527313i \(-0.823200\pi\)
0.849671 0.527313i \(-0.176800\pi\)
\(824\) −31437.1 −1.32908
\(825\) 0 0
\(826\) 1080.39 + 3345.40i 0.0455105 + 0.140922i
\(827\) −4217.13 −0.177321 −0.0886603 0.996062i \(-0.528259\pi\)
−0.0886603 + 0.996062i \(0.528259\pi\)
\(828\) 0 0
\(829\) 15003.0i 0.628561i 0.949330 + 0.314280i \(0.101763\pi\)
−0.949330 + 0.314280i \(0.898237\pi\)
\(830\) −5177.32 + 1182.21i −0.216515 + 0.0494398i
\(831\) 0 0
\(832\) 1744.99 0.0727122
\(833\) −16311.2 22619.8i −0.678453 0.940852i
\(834\) 0 0
\(835\) 20462.4 4672.45i 0.848060 0.193649i
\(836\) −16132.1 −0.667391
\(837\) 0 0
\(838\) −22587.1 −0.931095
\(839\) 2205.65 0.0907597 0.0453798 0.998970i \(-0.485550\pi\)
0.0453798 + 0.998970i \(0.485550\pi\)
\(840\) 0 0
\(841\) −1276.58 −0.0523425
\(842\) 23656.1 0.968224
\(843\) 0 0
\(844\) 11287.3 0.460336
\(845\) 4782.32 + 20943.5i 0.194694 + 0.852637i
\(846\) 0 0
\(847\) 2250.64 + 6969.03i 0.0913020 + 0.282714i
\(848\) −6090.11 −0.246622
\(849\) 0 0
\(850\) −6246.00 12963.6i −0.252042 0.523117i
\(851\) 5450.02i 0.219535i
\(852\) 0 0
\(853\) 41922.9 1.68278 0.841391 0.540426i \(-0.181737\pi\)
0.841391 + 0.540426i \(0.181737\pi\)
\(854\) −7581.07 + 2448.29i −0.303769 + 0.0981018i
\(855\) 0 0
\(856\) −15205.2 −0.607131
\(857\) 5480.18i 0.218436i 0.994018 + 0.109218i \(0.0348346\pi\)
−0.994018 + 0.109218i \(0.965165\pi\)
\(858\) 0 0
\(859\) 13807.4i 0.548431i −0.961668 0.274216i \(-0.911582\pi\)
0.961668 0.274216i \(-0.0884182\pi\)
\(860\) −17370.4 + 3966.42i −0.688751 + 0.157272i
\(861\) 0 0
\(862\) 1446.60i 0.0571595i
\(863\) 36042.1 1.42165 0.710827 0.703367i \(-0.248321\pi\)
0.710827 + 0.703367i \(0.248321\pi\)
\(864\) 0 0
\(865\) −28931.7 + 6606.37i −1.13723 + 0.259680i
\(866\) −1490.09 −0.0584703
\(867\) 0 0
\(868\) 32343.7 10445.4i 1.26477 0.408455i
\(869\) 17090.0i 0.667131i
\(870\) 0 0
\(871\) 11148.0i 0.433681i
\(872\) 16144.3 0.626968
\(873\) 0 0
\(874\) 6469.64i 0.250388i
\(875\) −14291.1 + 21579.7i −0.552147 + 0.833747i
\(876\) 0 0
\(877\) 25249.1i 0.972180i 0.873909 + 0.486090i \(0.161577\pi\)
−0.873909 + 0.486090i \(0.838423\pi\)
\(878\) 19172.4i 0.736943i
\(879\) 0 0
\(880\) −6636.11 + 1515.31i −0.254208 + 0.0580468i
\(881\) −7038.44 −0.269161 −0.134581 0.990903i \(-0.542969\pi\)
−0.134581 + 0.990903i \(0.542969\pi\)
\(882\) 0 0
\(883\) 17328.3i 0.660413i −0.943909 0.330207i \(-0.892882\pi\)
0.943909 0.330207i \(-0.107118\pi\)
\(884\) 8091.22i 0.307848i
\(885\) 0 0
\(886\) 18123.1 0.687197
\(887\) 23764.0i 0.899570i 0.893137 + 0.449785i \(0.148499\pi\)
−0.893137 + 0.449785i \(0.851501\pi\)
\(888\) 0 0
\(889\) 3063.86 989.470i 0.115589 0.0373293i
\(890\) −3282.15 14373.7i −0.123616 0.541358i
\(891\) 0 0
\(892\) 34881.9 1.30934
\(893\) 12381.1 0.463962
\(894\) 0 0
\(895\) 48753.8 11132.6i 1.82085 0.415779i
\(896\) −7654.30 23701.3i −0.285393 0.883712i
\(897\) 0 0
\(898\) 741.770i 0.0275648i
\(899\) −49040.5 −1.81935
\(900\) 0 0
\(901\) 24876.2i 0.919807i
\(902\) 20697.9i 0.764039i
\(903\) 0 0
\(904\) 32689.7 1.20270
\(905\) −4316.67 + 985.684i −0.158554 + 0.0362047i
\(906\) 0 0
\(907\) 16998.5i 0.622301i −0.950361 0.311151i \(-0.899286\pi\)
0.950361 0.311151i \(-0.100714\pi\)
\(908\) 16228.8i 0.593139i
\(909\) 0 0
\(910\) 4181.94 2489.02i 0.152341 0.0906705i
\(911\) 40955.7i 1.48949i 0.667351 + 0.744743i \(0.267428\pi\)
−0.667351 + 0.744743i \(0.732572\pi\)
\(912\) 0 0
\(913\) −10261.1 −0.371953
\(914\) 8140.90i 0.294614i
\(915\) 0 0
\(916\) 37341.5i 1.34694i
\(917\) 2668.24 + 8262.13i 0.0960884 + 0.297535i
\(918\) 0 0
\(919\) 42060.9 1.50975 0.754876 0.655867i \(-0.227697\pi\)
0.754876 + 0.655867i \(0.227697\pi\)
\(920\) −2561.66 11218.5i −0.0917995 0.402024i
\(921\) 0 0
\(922\) −20217.1 −0.722142
\(923\) 10233.3i 0.364933i
\(924\) 0 0
\(925\) 5693.11 + 11816.1i 0.202366 + 0.420012i
\(926\) 23039.7i 0.817636i
\(927\) 0 0
\(928\) 29911.8i 1.05808i
\(929\) −1206.74 −0.0426177 −0.0213088 0.999773i \(-0.506783\pi\)
−0.0213088 + 0.999773i \(0.506783\pi\)
\(930\) 0 0
\(931\) −17648.9 24474.8i −0.621287 0.861577i
\(932\) 24751.9 0.869930
\(933\) 0 0
\(934\) 21407.9i 0.749985i
\(935\) −6189.57 27106.4i −0.216493 0.948101i
\(936\) 0 0
\(937\) 40024.3 1.39545 0.697725 0.716366i \(-0.254196\pi\)
0.697725 + 0.716366i \(0.254196\pi\)
\(938\) 16758.8 5412.21i 0.583361 0.188395i
\(939\) 0 0
\(940\) 9196.85 2100.04i 0.319115 0.0728679i
\(941\) 7112.58 0.246401 0.123201 0.992382i \(-0.460684\pi\)
0.123201 + 0.992382i \(0.460684\pi\)
\(942\) 0 0
\(943\) −24823.3 −0.857218
\(944\) 2668.51 0.0920049
\(945\) 0 0
\(946\) 11512.1 0.395657
\(947\) −52031.5 −1.78543 −0.892713 0.450626i \(-0.851201\pi\)
−0.892713 + 0.450626i \(0.851201\pi\)
\(948\) 0 0
\(949\) −16956.0 −0.579993
\(950\) −6758.21 14026.7i −0.230805 0.479039i
\(951\) 0 0
\(952\) 28394.3 9169.90i 0.966666 0.312183i
\(953\) −50850.3 −1.72844 −0.864220 0.503114i \(-0.832188\pi\)
−0.864220 + 0.503114i \(0.832188\pi\)
\(954\) 0 0
\(955\) −14147.2 + 3230.42i −0.479363 + 0.109459i
\(956\) 27849.3i 0.942166i
\(957\) 0 0
\(958\) −18464.3 −0.622710
\(959\) −7825.89 24232.6i −0.263515 0.815967i
\(960\) 0 0
\(961\) −63913.2 −2.14539
\(962\) 2466.14i 0.0826525i
\(963\) 0 0
\(964\) 41526.0i 1.38741i
\(965\) 48743.2 11130.2i 1.62601 0.371289i
\(966\) 0 0
\(967\) 35434.8i 1.17839i 0.807990 + 0.589197i \(0.200556\pi\)
−0.807990 + 0.589197i \(0.799444\pi\)
\(968\) −7835.74 −0.260176
\(969\) 0 0
\(970\) 2721.42 + 11918.1i 0.0900820 + 0.394502i
\(971\) −46207.8 −1.52717 −0.763584 0.645709i \(-0.776562\pi\)
−0.763584 + 0.645709i \(0.776562\pi\)
\(972\) 0 0
\(973\) 30366.4 9806.77i 1.00052 0.323115i
\(974\) 18293.0i 0.601791i
\(975\) 0 0
\(976\) 6047.16i 0.198325i
\(977\) −14154.5 −0.463503 −0.231752 0.972775i \(-0.574446\pi\)
−0.231752 + 0.972775i \(0.574446\pi\)
\(978\) 0 0
\(979\) 28487.7i 0.930002i
\(980\) −17261.1 15186.6i −0.562639 0.495019i
\(981\) 0 0
\(982\) 14700.3i 0.477705i
\(983\) 12680.8i 0.411451i −0.978610 0.205725i \(-0.934045\pi\)
0.978610 0.205725i \(-0.0659554\pi\)
\(984\) 0 0
\(985\) 53.9689 + 236.350i 0.00174578 + 0.00764541i
\(986\) −18442.6 −0.595672
\(987\) 0 0
\(988\) 8754.75i 0.281908i
\(989\) 13806.7i 0.443909i
\(990\) 0 0
\(991\) 36020.9 1.15463 0.577317 0.816520i \(-0.304100\pi\)
0.577317 + 0.816520i \(0.304100\pi\)
\(992\) 57153.9i 1.82927i
\(993\) 0 0
\(994\) −15383.7 + 4968.12i −0.490885 + 0.158531i
\(995\) 18521.7 4229.30i 0.590127 0.134752i
\(996\) 0 0
\(997\) −61180.4 −1.94343 −0.971716 0.236152i \(-0.924114\pi\)
−0.971716 + 0.236152i \(0.924114\pi\)
\(998\) 18567.3 0.588917
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 315.4.g.a.314.30 yes 48
3.2 odd 2 inner 315.4.g.a.314.20 yes 48
5.4 even 2 inner 315.4.g.a.314.18 yes 48
7.6 odd 2 inner 315.4.g.a.314.29 yes 48
15.14 odd 2 inner 315.4.g.a.314.32 yes 48
21.20 even 2 inner 315.4.g.a.314.19 yes 48
35.34 odd 2 inner 315.4.g.a.314.17 48
105.104 even 2 inner 315.4.g.a.314.31 yes 48
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
315.4.g.a.314.17 48 35.34 odd 2 inner
315.4.g.a.314.18 yes 48 5.4 even 2 inner
315.4.g.a.314.19 yes 48 21.20 even 2 inner
315.4.g.a.314.20 yes 48 3.2 odd 2 inner
315.4.g.a.314.29 yes 48 7.6 odd 2 inner
315.4.g.a.314.30 yes 48 1.1 even 1 trivial
315.4.g.a.314.31 yes 48 105.104 even 2 inner
315.4.g.a.314.32 yes 48 15.14 odd 2 inner