Properties

Label 525.4.d.k.274.4
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $4$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [525,4,Mod(274,525)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(525, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("525.274");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (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: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 105)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.k.274.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.23607i q^{2} +3.00000i q^{3} -9.94427 q^{4} -12.7082 q^{6} +7.00000i q^{7} -8.23607i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+4.23607i q^{2} +3.00000i q^{3} -9.94427 q^{4} -12.7082 q^{6} +7.00000i q^{7} -8.23607i q^{8} -9.00000 q^{9} -41.5279 q^{11} -29.8328i q^{12} +88.9706i q^{13} -29.6525 q^{14} -44.6656 q^{16} +120.387i q^{17} -38.1246i q^{18} +112.138 q^{19} -21.0000 q^{21} -175.915i q^{22} -115.279i q^{23} +24.7082 q^{24} -376.885 q^{26} -27.0000i q^{27} -69.6099i q^{28} +144.833 q^{29} -258.079 q^{31} -255.095i q^{32} -124.584i q^{33} -509.967 q^{34} +89.4984 q^{36} -48.3344i q^{37} +475.023i q^{38} -266.912 q^{39} +200.885 q^{41} -88.9574i q^{42} -218.217i q^{43} +412.964 q^{44} +488.328 q^{46} -575.659i q^{47} -133.997i q^{48} -49.0000 q^{49} -361.161 q^{51} -884.748i q^{52} -184.302i q^{53} +114.374 q^{54} +57.6525 q^{56} +336.413i q^{57} +613.522i q^{58} +151.502 q^{59} -529.830 q^{61} -1093.24i q^{62} -63.0000i q^{63} +723.276 q^{64} +527.745 q^{66} -1.28485i q^{67} -1197.16i q^{68} +345.836 q^{69} -61.4226 q^{71} +74.1246i q^{72} +484.800i q^{73} +204.748 q^{74} -1115.13 q^{76} -290.695i q^{77} -1130.66i q^{78} -878.257 q^{79} +81.0000 q^{81} +850.964i q^{82} +491.830i q^{83} +208.830 q^{84} +924.381 q^{86} +434.498i q^{87} +342.026i q^{88} +415.560 q^{89} -622.794 q^{91} +1146.36i q^{92} -774.237i q^{93} +2438.53 q^{94} +765.286 q^{96} +1031.70i q^{97} -207.567i q^{98} +373.751 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 24 q^{6} - 36 q^{9} - 184 q^{11} - 56 q^{14} + 36 q^{16} + 216 q^{19} - 84 q^{21} + 72 q^{24} - 792 q^{26} + 472 q^{29} - 120 q^{31} - 1056 q^{34} + 36 q^{36} - 48 q^{39} + 88 q^{41} + 24 q^{44} + 880 q^{46} - 196 q^{49} - 264 q^{51} + 216 q^{54} + 168 q^{56} + 928 q^{59} - 1368 q^{61} + 2428 q^{64} + 984 q^{66} + 1920 q^{69} - 1480 q^{71} + 336 q^{74} - 2296 q^{76} + 816 q^{79} + 324 q^{81} + 84 q^{84} + 2016 q^{86} + 2664 q^{89} - 112 q^{91} + 4960 q^{94} + 1800 q^{96} + 1656 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.23607i 1.49768i 0.662753 + 0.748838i \(0.269388\pi\)
−0.662753 + 0.748838i \(0.730612\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −9.94427 −1.24303
\(5\) 0 0
\(6\) −12.7082 −0.864684
\(7\) 7.00000i 0.377964i
\(8\) − 8.23607i − 0.363986i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −41.5279 −1.13828 −0.569142 0.822239i \(-0.692725\pi\)
−0.569142 + 0.822239i \(0.692725\pi\)
\(12\) − 29.8328i − 0.717666i
\(13\) 88.9706i 1.89815i 0.315044 + 0.949077i \(0.397981\pi\)
−0.315044 + 0.949077i \(0.602019\pi\)
\(14\) −29.6525 −0.566068
\(15\) 0 0
\(16\) −44.6656 −0.697900
\(17\) 120.387i 1.71754i 0.512364 + 0.858769i \(0.328770\pi\)
−0.512364 + 0.858769i \(0.671230\pi\)
\(18\) − 38.1246i − 0.499225i
\(19\) 112.138 1.35401 0.677004 0.735979i \(-0.263278\pi\)
0.677004 + 0.735979i \(0.263278\pi\)
\(20\) 0 0
\(21\) −21.0000 −0.218218
\(22\) − 175.915i − 1.70478i
\(23\) − 115.279i − 1.04510i −0.852609 0.522549i \(-0.824981\pi\)
0.852609 0.522549i \(-0.175019\pi\)
\(24\) 24.7082 0.210148
\(25\) 0 0
\(26\) −376.885 −2.84282
\(27\) − 27.0000i − 0.192450i
\(28\) − 69.6099i − 0.469823i
\(29\) 144.833 0.927406 0.463703 0.885991i \(-0.346520\pi\)
0.463703 + 0.885991i \(0.346520\pi\)
\(30\) 0 0
\(31\) −258.079 −1.49524 −0.747618 0.664128i \(-0.768803\pi\)
−0.747618 + 0.664128i \(0.768803\pi\)
\(32\) − 255.095i − 1.40922i
\(33\) − 124.584i − 0.657188i
\(34\) −509.967 −2.57231
\(35\) 0 0
\(36\) 89.4984 0.414345
\(37\) − 48.3344i − 0.214760i −0.994218 0.107380i \(-0.965754\pi\)
0.994218 0.107380i \(-0.0342461\pi\)
\(38\) 475.023i 2.02787i
\(39\) −266.912 −1.09590
\(40\) 0 0
\(41\) 200.885 0.765196 0.382598 0.923915i \(-0.375029\pi\)
0.382598 + 0.923915i \(0.375029\pi\)
\(42\) − 88.9574i − 0.326820i
\(43\) − 218.217i − 0.773901i −0.922101 0.386950i \(-0.873528\pi\)
0.922101 0.386950i \(-0.126472\pi\)
\(44\) 412.964 1.41493
\(45\) 0 0
\(46\) 488.328 1.56522
\(47\) − 575.659i − 1.78657i −0.449496 0.893283i \(-0.648396\pi\)
0.449496 0.893283i \(-0.351604\pi\)
\(48\) − 133.997i − 0.402933i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −361.161 −0.991621
\(52\) − 884.748i − 2.35947i
\(53\) − 184.302i − 0.477657i −0.971062 0.238828i \(-0.923237\pi\)
0.971062 0.238828i \(-0.0767633\pi\)
\(54\) 114.374 0.288228
\(55\) 0 0
\(56\) 57.6525 0.137574
\(57\) 336.413i 0.781737i
\(58\) 613.522i 1.38895i
\(59\) 151.502 0.334302 0.167151 0.985931i \(-0.446543\pi\)
0.167151 + 0.985931i \(0.446543\pi\)
\(60\) 0 0
\(61\) −529.830 −1.11209 −0.556047 0.831151i \(-0.687683\pi\)
−0.556047 + 0.831151i \(0.687683\pi\)
\(62\) − 1093.24i − 2.23938i
\(63\) − 63.0000i − 0.125988i
\(64\) 723.276 1.41265
\(65\) 0 0
\(66\) 527.745 0.984256
\(67\) − 1.28485i − 0.00234283i −0.999999 0.00117142i \(-0.999627\pi\)
0.999999 0.00117142i \(-0.000372873\pi\)
\(68\) − 1197.16i − 2.13496i
\(69\) 345.836 0.603388
\(70\) 0 0
\(71\) −61.4226 −0.102669 −0.0513347 0.998682i \(-0.516348\pi\)
−0.0513347 + 0.998682i \(0.516348\pi\)
\(72\) 74.1246i 0.121329i
\(73\) 484.800i 0.777282i 0.921389 + 0.388641i \(0.127055\pi\)
−0.921389 + 0.388641i \(0.872945\pi\)
\(74\) 204.748 0.321641
\(75\) 0 0
\(76\) −1115.13 −1.68308
\(77\) − 290.695i − 0.430231i
\(78\) − 1130.66i − 1.64130i
\(79\) −878.257 −1.25078 −0.625390 0.780312i \(-0.715060\pi\)
−0.625390 + 0.780312i \(0.715060\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 850.964i 1.14602i
\(83\) 491.830i 0.650426i 0.945641 + 0.325213i \(0.105436\pi\)
−0.945641 + 0.325213i \(0.894564\pi\)
\(84\) 208.830 0.271252
\(85\) 0 0
\(86\) 924.381 1.15905
\(87\) 434.498i 0.535438i
\(88\) 342.026i 0.414320i
\(89\) 415.560 0.494936 0.247468 0.968896i \(-0.420401\pi\)
0.247468 + 0.968896i \(0.420401\pi\)
\(90\) 0 0
\(91\) −622.794 −0.717435
\(92\) 1146.36i 1.29909i
\(93\) − 774.237i − 0.863275i
\(94\) 2438.53 2.67570
\(95\) 0 0
\(96\) 765.286 0.813611
\(97\) 1031.70i 1.07993i 0.841688 + 0.539964i \(0.181562\pi\)
−0.841688 + 0.539964i \(0.818438\pi\)
\(98\) − 207.567i − 0.213954i
\(99\) 373.751 0.379428
\(100\) 0 0
\(101\) 1447.19 1.42576 0.712878 0.701288i \(-0.247391\pi\)
0.712878 + 0.701288i \(0.247391\pi\)
\(102\) − 1529.90i − 1.48513i
\(103\) − 163.567i − 0.156473i −0.996935 0.0782364i \(-0.975071\pi\)
0.996935 0.0782364i \(-0.0249289\pi\)
\(104\) 732.768 0.690902
\(105\) 0 0
\(106\) 780.715 0.715375
\(107\) 129.653i 0.117141i 0.998283 + 0.0585703i \(0.0186542\pi\)
−0.998283 + 0.0585703i \(0.981346\pi\)
\(108\) 268.495i 0.239222i
\(109\) −566.681 −0.497965 −0.248983 0.968508i \(-0.580096\pi\)
−0.248983 + 0.968508i \(0.580096\pi\)
\(110\) 0 0
\(111\) 145.003 0.123992
\(112\) − 312.659i − 0.263782i
\(113\) 809.890i 0.674230i 0.941463 + 0.337115i \(0.109451\pi\)
−0.941463 + 0.337115i \(0.890549\pi\)
\(114\) −1425.07 −1.17079
\(115\) 0 0
\(116\) −1440.26 −1.15280
\(117\) − 800.735i − 0.632718i
\(118\) 641.771i 0.500676i
\(119\) −842.709 −0.649168
\(120\) 0 0
\(121\) 393.563 0.295690
\(122\) − 2244.39i − 1.66556i
\(123\) 602.656i 0.441786i
\(124\) 2566.41 1.85863
\(125\) 0 0
\(126\) 266.872 0.188689
\(127\) 2584.25i 1.80563i 0.430028 + 0.902816i \(0.358504\pi\)
−0.430028 + 0.902816i \(0.641496\pi\)
\(128\) 1023.08i 0.706473i
\(129\) 654.650 0.446812
\(130\) 0 0
\(131\) −1421.10 −0.947804 −0.473902 0.880578i \(-0.657155\pi\)
−0.473902 + 0.880578i \(0.657155\pi\)
\(132\) 1238.89i 0.816908i
\(133\) 784.964i 0.511767i
\(134\) 5.44272 0.00350880
\(135\) 0 0
\(136\) 991.515 0.625160
\(137\) − 104.878i − 0.0654037i −0.999465 0.0327019i \(-0.989589\pi\)
0.999465 0.0327019i \(-0.0104112\pi\)
\(138\) 1464.98i 0.903679i
\(139\) 913.160 0.557217 0.278609 0.960405i \(-0.410127\pi\)
0.278609 + 0.960405i \(0.410127\pi\)
\(140\) 0 0
\(141\) 1726.98 1.03147
\(142\) − 260.190i − 0.153765i
\(143\) − 3694.76i − 2.16064i
\(144\) 401.991 0.232633
\(145\) 0 0
\(146\) −2053.65 −1.16412
\(147\) − 147.000i − 0.0824786i
\(148\) 480.650i 0.266954i
\(149\) −1781.45 −0.979476 −0.489738 0.871870i \(-0.662908\pi\)
−0.489738 + 0.871870i \(0.662908\pi\)
\(150\) 0 0
\(151\) 1407.53 0.758564 0.379282 0.925281i \(-0.376171\pi\)
0.379282 + 0.925281i \(0.376171\pi\)
\(152\) − 923.574i − 0.492841i
\(153\) − 1083.48i − 0.572512i
\(154\) 1231.40 0.644346
\(155\) 0 0
\(156\) 2654.24 1.36224
\(157\) 1598.94i 0.812798i 0.913696 + 0.406399i \(0.133216\pi\)
−0.913696 + 0.406399i \(0.866784\pi\)
\(158\) − 3720.36i − 1.87326i
\(159\) 552.906 0.275775
\(160\) 0 0
\(161\) 806.950 0.395010
\(162\) 343.122i 0.166408i
\(163\) − 204.892i − 0.0984562i −0.998788 0.0492281i \(-0.984324\pi\)
0.998788 0.0492281i \(-0.0156761\pi\)
\(164\) −1997.66 −0.951165
\(165\) 0 0
\(166\) −2083.42 −0.974127
\(167\) 1165.94i 0.540259i 0.962824 + 0.270129i \(0.0870665\pi\)
−0.962824 + 0.270129i \(0.912933\pi\)
\(168\) 172.957i 0.0794283i
\(169\) −5718.76 −2.60299
\(170\) 0 0
\(171\) −1009.24 −0.451336
\(172\) 2170.01i 0.961985i
\(173\) − 2538.00i − 1.11538i −0.830049 0.557690i \(-0.811688\pi\)
0.830049 0.557690i \(-0.188312\pi\)
\(174\) −1840.56 −0.801913
\(175\) 0 0
\(176\) 1854.87 0.794409
\(177\) 454.505i 0.193009i
\(178\) 1760.34i 0.741254i
\(179\) −392.255 −0.163791 −0.0818954 0.996641i \(-0.526097\pi\)
−0.0818954 + 0.996641i \(0.526097\pi\)
\(180\) 0 0
\(181\) −2978.08 −1.22298 −0.611489 0.791253i \(-0.709429\pi\)
−0.611489 + 0.791253i \(0.709429\pi\)
\(182\) − 2638.20i − 1.07448i
\(183\) − 1589.49i − 0.642068i
\(184\) −949.443 −0.380401
\(185\) 0 0
\(186\) 3279.72 1.29291
\(187\) − 4999.41i − 1.95504i
\(188\) 5724.51i 2.22076i
\(189\) 189.000 0.0727393
\(190\) 0 0
\(191\) −1097.37 −0.415722 −0.207861 0.978158i \(-0.566650\pi\)
−0.207861 + 0.978158i \(0.566650\pi\)
\(192\) 2169.83i 0.815592i
\(193\) 3500.31i 1.30548i 0.757582 + 0.652740i \(0.226381\pi\)
−0.757582 + 0.652740i \(0.773619\pi\)
\(194\) −4370.34 −1.61738
\(195\) 0 0
\(196\) 487.269 0.177576
\(197\) − 1573.96i − 0.569237i −0.958641 0.284618i \(-0.908133\pi\)
0.958641 0.284618i \(-0.0918669\pi\)
\(198\) 1583.23i 0.568260i
\(199\) 3396.62 1.20995 0.604976 0.796244i \(-0.293183\pi\)
0.604976 + 0.796244i \(0.293183\pi\)
\(200\) 0 0
\(201\) 3.85456 0.00135263
\(202\) 6130.42i 2.13532i
\(203\) 1013.83i 0.350527i
\(204\) 3591.48 1.23262
\(205\) 0 0
\(206\) 692.879 0.234346
\(207\) 1037.51i 0.348366i
\(208\) − 3973.93i − 1.32472i
\(209\) −4656.84 −1.54125
\(210\) 0 0
\(211\) 3337.81 1.08903 0.544513 0.838753i \(-0.316715\pi\)
0.544513 + 0.838753i \(0.316715\pi\)
\(212\) 1832.75i 0.593744i
\(213\) − 184.268i − 0.0592762i
\(214\) −549.220 −0.175439
\(215\) 0 0
\(216\) −222.374 −0.0700492
\(217\) − 1806.55i − 0.565146i
\(218\) − 2400.50i − 0.745791i
\(219\) −1454.40 −0.448764
\(220\) 0 0
\(221\) −10710.9 −3.26015
\(222\) 614.243i 0.185700i
\(223\) 127.328i 0.0382356i 0.999817 + 0.0191178i \(0.00608575\pi\)
−0.999817 + 0.0191178i \(0.993914\pi\)
\(224\) 1785.67 0.532633
\(225\) 0 0
\(226\) −3430.75 −1.00978
\(227\) − 3844.12i − 1.12398i −0.827144 0.561990i \(-0.810036\pi\)
0.827144 0.561990i \(-0.189964\pi\)
\(228\) − 3345.39i − 0.971726i
\(229\) −2536.95 −0.732080 −0.366040 0.930599i \(-0.619287\pi\)
−0.366040 + 0.930599i \(0.619287\pi\)
\(230\) 0 0
\(231\) 872.085 0.248394
\(232\) − 1192.85i − 0.337563i
\(233\) 3987.44i 1.12114i 0.828107 + 0.560570i \(0.189418\pi\)
−0.828107 + 0.560570i \(0.810582\pi\)
\(234\) 3391.97 0.947607
\(235\) 0 0
\(236\) −1506.57 −0.415549
\(237\) − 2634.77i − 0.722138i
\(238\) − 3569.77i − 0.972244i
\(239\) 3367.18 0.911317 0.455659 0.890155i \(-0.349404\pi\)
0.455659 + 0.890155i \(0.349404\pi\)
\(240\) 0 0
\(241\) −939.551 −0.251128 −0.125564 0.992086i \(-0.540074\pi\)
−0.125564 + 0.992086i \(0.540074\pi\)
\(242\) 1667.16i 0.442848i
\(243\) 243.000i 0.0641500i
\(244\) 5268.77 1.38237
\(245\) 0 0
\(246\) −2552.89 −0.661653
\(247\) 9976.96i 2.57012i
\(248\) 2125.56i 0.544246i
\(249\) −1475.49 −0.375523
\(250\) 0 0
\(251\) −1403.96 −0.353056 −0.176528 0.984296i \(-0.556487\pi\)
−0.176528 + 0.984296i \(0.556487\pi\)
\(252\) 626.489i 0.156608i
\(253\) 4787.28i 1.18962i
\(254\) −10947.1 −2.70425
\(255\) 0 0
\(256\) 1452.36 0.354579
\(257\) 1964.86i 0.476905i 0.971154 + 0.238453i \(0.0766402\pi\)
−0.971154 + 0.238453i \(0.923360\pi\)
\(258\) 2773.14i 0.669179i
\(259\) 338.341 0.0811717
\(260\) 0 0
\(261\) −1303.50 −0.309135
\(262\) − 6019.89i − 1.41950i
\(263\) − 393.821i − 0.0923347i −0.998934 0.0461673i \(-0.985299\pi\)
0.998934 0.0461673i \(-0.0147007\pi\)
\(264\) −1026.08 −0.239208
\(265\) 0 0
\(266\) −3325.16 −0.766462
\(267\) 1246.68i 0.285751i
\(268\) 12.7769i 0.00291222i
\(269\) 1877.03 0.425444 0.212722 0.977113i \(-0.431767\pi\)
0.212722 + 0.977113i \(0.431767\pi\)
\(270\) 0 0
\(271\) −689.909 −0.154646 −0.0773228 0.997006i \(-0.524637\pi\)
−0.0773228 + 0.997006i \(0.524637\pi\)
\(272\) − 5377.16i − 1.19867i
\(273\) − 1868.38i − 0.414211i
\(274\) 444.269 0.0979536
\(275\) 0 0
\(276\) −3439.09 −0.750031
\(277\) − 6289.13i − 1.36418i −0.731270 0.682088i \(-0.761072\pi\)
0.731270 0.682088i \(-0.238928\pi\)
\(278\) 3868.21i 0.834531i
\(279\) 2322.71 0.498412
\(280\) 0 0
\(281\) −1954.87 −0.415010 −0.207505 0.978234i \(-0.566534\pi\)
−0.207505 + 0.978234i \(0.566534\pi\)
\(282\) 7315.60i 1.54481i
\(283\) 5033.96i 1.05738i 0.848816 + 0.528688i \(0.177316\pi\)
−0.848816 + 0.528688i \(0.822684\pi\)
\(284\) 610.803 0.127621
\(285\) 0 0
\(286\) 15651.2 3.23594
\(287\) 1406.20i 0.289217i
\(288\) 2295.86i 0.469738i
\(289\) −9580.03 −1.94993
\(290\) 0 0
\(291\) −3095.09 −0.623497
\(292\) − 4820.99i − 0.966188i
\(293\) 6369.12i 1.26993i 0.772543 + 0.634963i \(0.218985\pi\)
−0.772543 + 0.634963i \(0.781015\pi\)
\(294\) 622.702 0.123526
\(295\) 0 0
\(296\) −398.085 −0.0781697
\(297\) 1121.25i 0.219063i
\(298\) − 7546.34i − 1.46694i
\(299\) 10256.4 1.98376
\(300\) 0 0
\(301\) 1527.52 0.292507
\(302\) 5962.39i 1.13608i
\(303\) 4341.58i 0.823160i
\(304\) −5008.70 −0.944963
\(305\) 0 0
\(306\) 4589.71 0.857438
\(307\) 6619.83i 1.23066i 0.788268 + 0.615332i \(0.210978\pi\)
−0.788268 + 0.615332i \(0.789022\pi\)
\(308\) 2890.75i 0.534792i
\(309\) 490.700 0.0903396
\(310\) 0 0
\(311\) −9909.22 −1.80675 −0.903377 0.428848i \(-0.858920\pi\)
−0.903377 + 0.428848i \(0.858920\pi\)
\(312\) 2198.30i 0.398892i
\(313\) − 422.336i − 0.0762678i −0.999273 0.0381339i \(-0.987859\pi\)
0.999273 0.0381339i \(-0.0121413\pi\)
\(314\) −6773.22 −1.21731
\(315\) 0 0
\(316\) 8733.63 1.55476
\(317\) 4902.78i 0.868668i 0.900752 + 0.434334i \(0.143016\pi\)
−0.900752 + 0.434334i \(0.856984\pi\)
\(318\) 2342.15i 0.413022i
\(319\) −6014.60 −1.05565
\(320\) 0 0
\(321\) −388.960 −0.0676312
\(322\) 3418.30i 0.591597i
\(323\) 13499.9i 2.32556i
\(324\) −805.486 −0.138115
\(325\) 0 0
\(326\) 867.935 0.147455
\(327\) − 1700.04i − 0.287500i
\(328\) − 1654.51i − 0.278521i
\(329\) 4029.62 0.675258
\(330\) 0 0
\(331\) −5281.74 −0.877071 −0.438535 0.898714i \(-0.644503\pi\)
−0.438535 + 0.898714i \(0.644503\pi\)
\(332\) − 4890.89i − 0.808501i
\(333\) 435.009i 0.0715867i
\(334\) −4939.01 −0.809133
\(335\) 0 0
\(336\) 937.978 0.152294
\(337\) − 4459.60i − 0.720860i −0.932786 0.360430i \(-0.882630\pi\)
0.932786 0.360430i \(-0.117370\pi\)
\(338\) − 24225.1i − 3.89843i
\(339\) −2429.67 −0.389267
\(340\) 0 0
\(341\) 10717.5 1.70200
\(342\) − 4275.21i − 0.675956i
\(343\) − 343.000i − 0.0539949i
\(344\) −1797.25 −0.281689
\(345\) 0 0
\(346\) 10751.2 1.67048
\(347\) − 5261.97i − 0.814056i −0.913416 0.407028i \(-0.866565\pi\)
0.913416 0.407028i \(-0.133435\pi\)
\(348\) − 4320.77i − 0.665568i
\(349\) −960.325 −0.147292 −0.0736461 0.997284i \(-0.523464\pi\)
−0.0736461 + 0.997284i \(0.523464\pi\)
\(350\) 0 0
\(351\) 2402.21 0.365300
\(352\) 10593.6i 1.60409i
\(353\) − 8925.80i − 1.34581i −0.739727 0.672907i \(-0.765045\pi\)
0.739727 0.672907i \(-0.234955\pi\)
\(354\) −1925.31 −0.289066
\(355\) 0 0
\(356\) −4132.45 −0.615222
\(357\) − 2528.13i − 0.374797i
\(358\) − 1661.62i − 0.245306i
\(359\) 3056.27 0.449314 0.224657 0.974438i \(-0.427874\pi\)
0.224657 + 0.974438i \(0.427874\pi\)
\(360\) 0 0
\(361\) 5715.88 0.833340
\(362\) − 12615.4i − 1.83162i
\(363\) 1180.69i 0.170717i
\(364\) 6193.23 0.891796
\(365\) 0 0
\(366\) 6733.18 0.961610
\(367\) 1813.52i 0.257943i 0.991648 + 0.128971i \(0.0411675\pi\)
−0.991648 + 0.128971i \(0.958832\pi\)
\(368\) 5148.99i 0.729375i
\(369\) −1807.97 −0.255065
\(370\) 0 0
\(371\) 1290.11 0.180537
\(372\) 7699.22i 1.07308i
\(373\) − 4517.48i − 0.627094i −0.949573 0.313547i \(-0.898483\pi\)
0.949573 0.313547i \(-0.101517\pi\)
\(374\) 21177.9 2.92802
\(375\) 0 0
\(376\) −4741.17 −0.650285
\(377\) 12885.9i 1.76036i
\(378\) 800.617i 0.108940i
\(379\) 4931.24 0.668340 0.334170 0.942513i \(-0.391544\pi\)
0.334170 + 0.942513i \(0.391544\pi\)
\(380\) 0 0
\(381\) −7752.75 −1.04248
\(382\) − 4648.53i − 0.622617i
\(383\) − 1482.37i − 0.197770i −0.995099 0.0988849i \(-0.968472\pi\)
0.995099 0.0988849i \(-0.0315276\pi\)
\(384\) −3069.25 −0.407883
\(385\) 0 0
\(386\) −14827.5 −1.95519
\(387\) 1963.95i 0.257967i
\(388\) − 10259.5i − 1.34239i
\(389\) 5448.98 0.710217 0.355109 0.934825i \(-0.384444\pi\)
0.355109 + 0.934825i \(0.384444\pi\)
\(390\) 0 0
\(391\) 13878.0 1.79500
\(392\) 403.567i 0.0519980i
\(393\) − 4263.31i − 0.547215i
\(394\) 6667.38 0.852532
\(395\) 0 0
\(396\) −3716.68 −0.471642
\(397\) 13675.9i 1.72891i 0.502713 + 0.864453i \(0.332335\pi\)
−0.502713 + 0.864453i \(0.667665\pi\)
\(398\) 14388.3i 1.81212i
\(399\) −2354.89 −0.295469
\(400\) 0 0
\(401\) 14109.9 1.75714 0.878570 0.477613i \(-0.158498\pi\)
0.878570 + 0.477613i \(0.158498\pi\)
\(402\) 16.3282i 0.00202581i
\(403\) − 22961.4i − 2.83819i
\(404\) −14391.3 −1.77226
\(405\) 0 0
\(406\) −4294.65 −0.524975
\(407\) 2007.22i 0.244458i
\(408\) 2974.55i 0.360936i
\(409\) 13995.6 1.69203 0.846015 0.533159i \(-0.178995\pi\)
0.846015 + 0.533159i \(0.178995\pi\)
\(410\) 0 0
\(411\) 314.633 0.0377609
\(412\) 1626.55i 0.194501i
\(413\) 1060.51i 0.126354i
\(414\) −4394.95 −0.521740
\(415\) 0 0
\(416\) 22696.0 2.67491
\(417\) 2739.48i 0.321709i
\(418\) − 19726.7i − 2.30829i
\(419\) 9840.61 1.14736 0.573682 0.819078i \(-0.305515\pi\)
0.573682 + 0.819078i \(0.305515\pi\)
\(420\) 0 0
\(421\) −12660.5 −1.46564 −0.732822 0.680420i \(-0.761797\pi\)
−0.732822 + 0.680420i \(0.761797\pi\)
\(422\) 14139.2i 1.63101i
\(423\) 5180.93i 0.595522i
\(424\) −1517.92 −0.173860
\(425\) 0 0
\(426\) 780.571 0.0887765
\(427\) − 3708.81i − 0.420332i
\(428\) − 1289.31i − 0.145610i
\(429\) 11084.3 1.24744
\(430\) 0 0
\(431\) −4578.91 −0.511736 −0.255868 0.966712i \(-0.582361\pi\)
−0.255868 + 0.966712i \(0.582361\pi\)
\(432\) 1205.97i 0.134311i
\(433\) − 3279.88i − 0.364020i −0.983297 0.182010i \(-0.941740\pi\)
0.983297 0.182010i \(-0.0582604\pi\)
\(434\) 7652.68 0.846406
\(435\) 0 0
\(436\) 5635.23 0.618988
\(437\) − 12927.1i − 1.41507i
\(438\) − 6160.94i − 0.672103i
\(439\) 427.807 0.0465105 0.0232552 0.999730i \(-0.492597\pi\)
0.0232552 + 0.999730i \(0.492597\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 45372.1i − 4.88265i
\(443\) 15441.2i 1.65605i 0.560688 + 0.828027i \(0.310537\pi\)
−0.560688 + 0.828027i \(0.689463\pi\)
\(444\) −1441.95 −0.154126
\(445\) 0 0
\(446\) −539.371 −0.0572645
\(447\) − 5344.35i − 0.565501i
\(448\) 5062.93i 0.533931i
\(449\) −9382.02 −0.986113 −0.493057 0.869997i \(-0.664120\pi\)
−0.493057 + 0.869997i \(0.664120\pi\)
\(450\) 0 0
\(451\) −8342.34 −0.871010
\(452\) − 8053.77i − 0.838091i
\(453\) 4222.59i 0.437957i
\(454\) 16284.0 1.68336
\(455\) 0 0
\(456\) 2770.72 0.284542
\(457\) − 13570.4i − 1.38905i −0.719469 0.694524i \(-0.755615\pi\)
0.719469 0.694524i \(-0.244385\pi\)
\(458\) − 10746.7i − 1.09642i
\(459\) 3250.45 0.330540
\(460\) 0 0
\(461\) 1251.88 0.126477 0.0632386 0.997998i \(-0.479857\pi\)
0.0632386 + 0.997998i \(0.479857\pi\)
\(462\) 3694.21i 0.372014i
\(463\) 7934.36i 0.796417i 0.917295 + 0.398209i \(0.130368\pi\)
−0.917295 + 0.398209i \(0.869632\pi\)
\(464\) −6469.05 −0.647237
\(465\) 0 0
\(466\) −16891.1 −1.67911
\(467\) − 7583.76i − 0.751466i −0.926728 0.375733i \(-0.877391\pi\)
0.926728 0.375733i \(-0.122609\pi\)
\(468\) 7962.73i 0.786490i
\(469\) 8.99396 0.000885507 0
\(470\) 0 0
\(471\) −4796.82 −0.469269
\(472\) − 1247.78i − 0.121681i
\(473\) 9062.07i 0.880919i
\(474\) 11161.1 1.08153
\(475\) 0 0
\(476\) 8380.13 0.806938
\(477\) 1658.72i 0.159219i
\(478\) 14263.6i 1.36486i
\(479\) 5829.34 0.556053 0.278027 0.960573i \(-0.410320\pi\)
0.278027 + 0.960573i \(0.410320\pi\)
\(480\) 0 0
\(481\) 4300.34 0.407648
\(482\) − 3980.00i − 0.376108i
\(483\) 2420.85i 0.228059i
\(484\) −3913.70 −0.367553
\(485\) 0 0
\(486\) −1029.36 −0.0960760
\(487\) 19902.1i 1.85185i 0.377708 + 0.925925i \(0.376712\pi\)
−0.377708 + 0.925925i \(0.623288\pi\)
\(488\) 4363.71i 0.404787i
\(489\) 614.675 0.0568437
\(490\) 0 0
\(491\) −16821.6 −1.54613 −0.773065 0.634327i \(-0.781277\pi\)
−0.773065 + 0.634327i \(0.781277\pi\)
\(492\) − 5992.98i − 0.549155i
\(493\) 17436.0i 1.59285i
\(494\) −42263.1 −3.84920
\(495\) 0 0
\(496\) 11527.3 1.04353
\(497\) − 429.958i − 0.0388054i
\(498\) − 6250.27i − 0.562412i
\(499\) −6031.83 −0.541126 −0.270563 0.962702i \(-0.587210\pi\)
−0.270563 + 0.962702i \(0.587210\pi\)
\(500\) 0 0
\(501\) −3497.82 −0.311919
\(502\) − 5947.27i − 0.528764i
\(503\) − 17176.4i − 1.52258i −0.648412 0.761290i \(-0.724566\pi\)
0.648412 0.761290i \(-0.275434\pi\)
\(504\) −518.872 −0.0458580
\(505\) 0 0
\(506\) −20279.2 −1.78166
\(507\) − 17156.3i − 1.50284i
\(508\) − 25698.5i − 2.24446i
\(509\) −4706.59 −0.409854 −0.204927 0.978777i \(-0.565696\pi\)
−0.204927 + 0.978777i \(0.565696\pi\)
\(510\) 0 0
\(511\) −3393.60 −0.293785
\(512\) 14336.9i 1.23752i
\(513\) − 3027.72i − 0.260579i
\(514\) −8323.28 −0.714250
\(515\) 0 0
\(516\) −6510.02 −0.555402
\(517\) 23905.9i 2.03362i
\(518\) 1433.23i 0.121569i
\(519\) 7614.01 0.643965
\(520\) 0 0
\(521\) −8557.18 −0.719572 −0.359786 0.933035i \(-0.617150\pi\)
−0.359786 + 0.933035i \(0.617150\pi\)
\(522\) − 5521.69i − 0.462985i
\(523\) 18248.5i 1.52572i 0.646566 + 0.762858i \(0.276204\pi\)
−0.646566 + 0.762858i \(0.723796\pi\)
\(524\) 14131.8 1.17815
\(525\) 0 0
\(526\) 1668.25 0.138287
\(527\) − 31069.3i − 2.56813i
\(528\) 5564.60i 0.458652i
\(529\) −1122.16 −0.0922302
\(530\) 0 0
\(531\) −1363.51 −0.111434
\(532\) − 7805.90i − 0.636144i
\(533\) 17872.9i 1.45246i
\(534\) −5281.03 −0.427963
\(535\) 0 0
\(536\) −10.5821 −0.000852758 0
\(537\) − 1176.77i − 0.0945646i
\(538\) 7951.22i 0.637177i
\(539\) 2034.87 0.162612
\(540\) 0 0
\(541\) −5734.17 −0.455696 −0.227848 0.973697i \(-0.573169\pi\)
−0.227848 + 0.973697i \(0.573169\pi\)
\(542\) − 2922.50i − 0.231609i
\(543\) − 8934.24i − 0.706087i
\(544\) 30710.1 2.42038
\(545\) 0 0
\(546\) 7914.59 0.620354
\(547\) 8002.52i 0.625527i 0.949831 + 0.312763i \(0.101255\pi\)
−0.949831 + 0.312763i \(0.898745\pi\)
\(548\) 1042.93i 0.0812991i
\(549\) 4768.47 0.370698
\(550\) 0 0
\(551\) 16241.2 1.25572
\(552\) − 2848.33i − 0.219625i
\(553\) − 6147.80i − 0.472750i
\(554\) 26641.2 2.04310
\(555\) 0 0
\(556\) −9080.71 −0.692640
\(557\) − 1276.82i − 0.0971289i −0.998820 0.0485644i \(-0.984535\pi\)
0.998820 0.0485644i \(-0.0154646\pi\)
\(558\) 9839.16i 0.746460i
\(559\) 19414.9 1.46898
\(560\) 0 0
\(561\) 14998.2 1.12875
\(562\) − 8280.96i − 0.621550i
\(563\) 11027.7i 0.825507i 0.910843 + 0.412753i \(0.135433\pi\)
−0.910843 + 0.412753i \(0.864567\pi\)
\(564\) −17173.5 −1.28216
\(565\) 0 0
\(566\) −21324.2 −1.58361
\(567\) 567.000i 0.0419961i
\(568\) 505.881i 0.0373702i
\(569\) 4519.03 0.332948 0.166474 0.986046i \(-0.446762\pi\)
0.166474 + 0.986046i \(0.446762\pi\)
\(570\) 0 0
\(571\) 3598.81 0.263758 0.131879 0.991266i \(-0.457899\pi\)
0.131879 + 0.991266i \(0.457899\pi\)
\(572\) 36741.7i 2.68575i
\(573\) − 3292.11i − 0.240017i
\(574\) −5956.75 −0.433153
\(575\) 0 0
\(576\) −6509.48 −0.470883
\(577\) 3439.23i 0.248140i 0.992273 + 0.124070i \(0.0395948\pi\)
−0.992273 + 0.124070i \(0.960405\pi\)
\(578\) − 40581.6i − 2.92037i
\(579\) −10500.9 −0.753720
\(580\) 0 0
\(581\) −3442.81 −0.245838
\(582\) − 13111.0i − 0.933797i
\(583\) 7653.66i 0.543709i
\(584\) 3992.85 0.282920
\(585\) 0 0
\(586\) −26980.0 −1.90194
\(587\) − 21285.2i − 1.49665i −0.663330 0.748327i \(-0.730857\pi\)
0.663330 0.748327i \(-0.269143\pi\)
\(588\) 1461.81i 0.102524i
\(589\) −28940.4 −2.02456
\(590\) 0 0
\(591\) 4721.87 0.328649
\(592\) 2158.89i 0.149881i
\(593\) − 14200.8i − 0.983404i −0.870764 0.491702i \(-0.836375\pi\)
0.870764 0.491702i \(-0.163625\pi\)
\(594\) −4749.70 −0.328085
\(595\) 0 0
\(596\) 17715.2 1.21752
\(597\) 10189.9i 0.698566i
\(598\) 43446.8i 2.97103i
\(599\) 8885.05 0.606065 0.303033 0.952980i \(-0.402001\pi\)
0.303033 + 0.952980i \(0.402001\pi\)
\(600\) 0 0
\(601\) −2052.89 −0.139333 −0.0696664 0.997570i \(-0.522193\pi\)
−0.0696664 + 0.997570i \(0.522193\pi\)
\(602\) 6470.67i 0.438081i
\(603\) 11.5637i 0 0.000780943i
\(604\) −13996.9 −0.942920
\(605\) 0 0
\(606\) −18391.2 −1.23283
\(607\) − 10280.0i − 0.687404i −0.939079 0.343702i \(-0.888319\pi\)
0.939079 0.343702i \(-0.111681\pi\)
\(608\) − 28605.8i − 1.90809i
\(609\) −3041.49 −0.202377
\(610\) 0 0
\(611\) 51216.8 3.39118
\(612\) 10774.4i 0.711652i
\(613\) 23409.5i 1.54242i 0.636584 + 0.771208i \(0.280347\pi\)
−0.636584 + 0.771208i \(0.719653\pi\)
\(614\) −28042.0 −1.84314
\(615\) 0 0
\(616\) −2394.18 −0.156598
\(617\) 6632.75i 0.432779i 0.976307 + 0.216389i \(0.0694281\pi\)
−0.976307 + 0.216389i \(0.930572\pi\)
\(618\) 2078.64i 0.135299i
\(619\) −10734.0 −0.696990 −0.348495 0.937311i \(-0.613307\pi\)
−0.348495 + 0.937311i \(0.613307\pi\)
\(620\) 0 0
\(621\) −3112.52 −0.201129
\(622\) − 41976.1i − 2.70593i
\(623\) 2908.92i 0.187068i
\(624\) 11921.8 0.764829
\(625\) 0 0
\(626\) 1789.04 0.114225
\(627\) − 13970.5i − 0.889839i
\(628\) − 15900.3i − 1.01034i
\(629\) 5818.83 0.368858
\(630\) 0 0
\(631\) −17071.0 −1.07700 −0.538499 0.842626i \(-0.681008\pi\)
−0.538499 + 0.842626i \(0.681008\pi\)
\(632\) 7233.38i 0.455267i
\(633\) 10013.4i 0.628749i
\(634\) −20768.5 −1.30098
\(635\) 0 0
\(636\) −5498.24 −0.342798
\(637\) − 4359.56i − 0.271165i
\(638\) − 25478.2i − 1.58102i
\(639\) 552.804 0.0342231
\(640\) 0 0
\(641\) −19389.7 −1.19477 −0.597386 0.801954i \(-0.703794\pi\)
−0.597386 + 0.801954i \(0.703794\pi\)
\(642\) − 1647.66i − 0.101290i
\(643\) − 25409.3i − 1.55839i −0.626780 0.779196i \(-0.715628\pi\)
0.626780 0.779196i \(-0.284372\pi\)
\(644\) −8024.54 −0.491011
\(645\) 0 0
\(646\) −57186.6 −3.48294
\(647\) 6039.08i 0.366956i 0.983024 + 0.183478i \(0.0587357\pi\)
−0.983024 + 0.183478i \(0.941264\pi\)
\(648\) − 667.122i − 0.0404429i
\(649\) −6291.54 −0.380531
\(650\) 0 0
\(651\) 5419.66 0.326287
\(652\) 2037.50i 0.122384i
\(653\) − 30666.2i − 1.83776i −0.394532 0.918882i \(-0.629093\pi\)
0.394532 0.918882i \(-0.370907\pi\)
\(654\) 7201.50 0.430582
\(655\) 0 0
\(656\) −8972.67 −0.534031
\(657\) − 4363.20i − 0.259094i
\(658\) 17069.7i 1.01132i
\(659\) 2765.96 0.163500 0.0817500 0.996653i \(-0.473949\pi\)
0.0817500 + 0.996653i \(0.473949\pi\)
\(660\) 0 0
\(661\) 27261.8 1.60418 0.802089 0.597204i \(-0.203722\pi\)
0.802089 + 0.597204i \(0.203722\pi\)
\(662\) − 22373.8i − 1.31357i
\(663\) − 32132.7i − 1.88225i
\(664\) 4050.74 0.236746
\(665\) 0 0
\(666\) −1842.73 −0.107214
\(667\) − 16696.1i − 0.969230i
\(668\) − 11594.4i − 0.671560i
\(669\) −381.985 −0.0220753
\(670\) 0 0
\(671\) 22002.7 1.26588
\(672\) 5357.00i 0.307516i
\(673\) − 1048.17i − 0.0600356i −0.999549 0.0300178i \(-0.990444\pi\)
0.999549 0.0300178i \(-0.00955640\pi\)
\(674\) 18891.2 1.07961
\(675\) 0 0
\(676\) 56869.0 3.23560
\(677\) 34554.7i 1.96166i 0.194860 + 0.980831i \(0.437575\pi\)
−0.194860 + 0.980831i \(0.562425\pi\)
\(678\) − 10292.2i − 0.582996i
\(679\) −7221.89 −0.408175
\(680\) 0 0
\(681\) 11532.4 0.648930
\(682\) 45399.9i 2.54905i
\(683\) 14711.6i 0.824192i 0.911140 + 0.412096i \(0.135203\pi\)
−0.911140 + 0.412096i \(0.864797\pi\)
\(684\) 10036.2 0.561026
\(685\) 0 0
\(686\) 1452.97 0.0808669
\(687\) − 7610.85i − 0.422667i
\(688\) 9746.79i 0.540106i
\(689\) 16397.4 0.906666
\(690\) 0 0
\(691\) −24522.6 −1.35005 −0.675024 0.737796i \(-0.735867\pi\)
−0.675024 + 0.737796i \(0.735867\pi\)
\(692\) 25238.6i 1.38646i
\(693\) 2616.26i 0.143410i
\(694\) 22290.1 1.21919
\(695\) 0 0
\(696\) 3578.56 0.194892
\(697\) 24184.0i 1.31425i
\(698\) − 4068.00i − 0.220596i
\(699\) −11962.3 −0.647291
\(700\) 0 0
\(701\) 19912.2 1.07286 0.536429 0.843946i \(-0.319773\pi\)
0.536429 + 0.843946i \(0.319773\pi\)
\(702\) 10175.9i 0.547101i
\(703\) − 5420.11i − 0.290787i
\(704\) −30036.1 −1.60799
\(705\) 0 0
\(706\) 37810.3 2.01559
\(707\) 10130.4i 0.538885i
\(708\) − 4519.72i − 0.239917i
\(709\) −6208.79 −0.328880 −0.164440 0.986387i \(-0.552582\pi\)
−0.164440 + 0.986387i \(0.552582\pi\)
\(710\) 0 0
\(711\) 7904.31 0.416927
\(712\) − 3422.58i − 0.180150i
\(713\) 29751.0i 1.56267i
\(714\) 10709.3 0.561325
\(715\) 0 0
\(716\) 3900.69 0.203597
\(717\) 10101.5i 0.526149i
\(718\) 12946.6i 0.672927i
\(719\) −13063.6 −0.677593 −0.338797 0.940860i \(-0.610020\pi\)
−0.338797 + 0.940860i \(0.610020\pi\)
\(720\) 0 0
\(721\) 1144.97 0.0591411
\(722\) 24212.9i 1.24807i
\(723\) − 2818.65i − 0.144989i
\(724\) 29614.8 1.52020
\(725\) 0 0
\(726\) −5001.49 −0.255678
\(727\) 12897.0i 0.657940i 0.944340 + 0.328970i \(0.106702\pi\)
−0.944340 + 0.328970i \(0.893298\pi\)
\(728\) 5129.37i 0.261136i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 26270.5 1.32920
\(732\) 15806.3i 0.798112i
\(733\) 11699.6i 0.589540i 0.955568 + 0.294770i \(0.0952430\pi\)
−0.955568 + 0.294770i \(0.904757\pi\)
\(734\) −7682.19 −0.386315
\(735\) 0 0
\(736\) −29407.0 −1.47277
\(737\) 53.3571i 0.00266681i
\(738\) − 7658.68i − 0.382005i
\(739\) −14974.0 −0.745368 −0.372684 0.927958i \(-0.621562\pi\)
−0.372684 + 0.927958i \(0.621562\pi\)
\(740\) 0 0
\(741\) −29930.9 −1.48386
\(742\) 5465.01i 0.270386i
\(743\) 18500.7i 0.913492i 0.889597 + 0.456746i \(0.150985\pi\)
−0.889597 + 0.456746i \(0.849015\pi\)
\(744\) −6376.67 −0.314220
\(745\) 0 0
\(746\) 19136.3 0.939184
\(747\) − 4426.47i − 0.216809i
\(748\) 49715.5i 2.43019i
\(749\) −907.572 −0.0442750
\(750\) 0 0
\(751\) −26348.4 −1.28025 −0.640125 0.768271i \(-0.721117\pi\)
−0.640125 + 0.768271i \(0.721117\pi\)
\(752\) 25712.2i 1.24684i
\(753\) − 4211.88i − 0.203837i
\(754\) −54585.4 −2.63645
\(755\) 0 0
\(756\) −1879.47 −0.0904174
\(757\) 28061.7i 1.34732i 0.739042 + 0.673659i \(0.235278\pi\)
−0.739042 + 0.673659i \(0.764722\pi\)
\(758\) 20889.1i 1.00096i
\(759\) −14361.8 −0.686826
\(760\) 0 0
\(761\) −3579.22 −0.170495 −0.0852476 0.996360i \(-0.527168\pi\)
−0.0852476 + 0.996360i \(0.527168\pi\)
\(762\) − 32841.2i − 1.56130i
\(763\) − 3966.77i − 0.188213i
\(764\) 10912.5 0.516757
\(765\) 0 0
\(766\) 6279.44 0.296195
\(767\) 13479.2i 0.634557i
\(768\) 4357.07i 0.204716i
\(769\) −4339.61 −0.203499 −0.101749 0.994810i \(-0.532444\pi\)
−0.101749 + 0.994810i \(0.532444\pi\)
\(770\) 0 0
\(771\) −5894.58 −0.275341
\(772\) − 34808.0i − 1.62276i
\(773\) 10005.1i 0.465537i 0.972532 + 0.232769i \(0.0747785\pi\)
−0.972532 + 0.232769i \(0.925222\pi\)
\(774\) −8319.43 −0.386351
\(775\) 0 0
\(776\) 8497.14 0.393079
\(777\) 1015.02i 0.0468645i
\(778\) 23082.3i 1.06368i
\(779\) 22526.8 1.03608
\(780\) 0 0
\(781\) 2550.75 0.116867
\(782\) 58788.4i 2.68832i
\(783\) − 3910.49i − 0.178479i
\(784\) 2188.62 0.0997001
\(785\) 0 0
\(786\) 18059.7 0.819550
\(787\) − 17826.8i − 0.807443i −0.914882 0.403721i \(-0.867716\pi\)
0.914882 0.403721i \(-0.132284\pi\)
\(788\) 15651.8i 0.707581i
\(789\) 1181.46 0.0533094
\(790\) 0 0
\(791\) −5669.23 −0.254835
\(792\) − 3078.24i − 0.138107i
\(793\) − 47139.3i − 2.11093i
\(794\) −57932.2 −2.58934
\(795\) 0 0
\(796\) −33777.0 −1.50401
\(797\) − 36723.0i − 1.63211i −0.577971 0.816057i \(-0.696155\pi\)
0.577971 0.816057i \(-0.303845\pi\)
\(798\) − 9975.49i − 0.442517i
\(799\) 69301.9 3.06849
\(800\) 0 0
\(801\) −3740.04 −0.164979
\(802\) 59770.4i 2.63163i
\(803\) − 20132.7i − 0.884767i
\(804\) −38.3307 −0.00168137
\(805\) 0 0
\(806\) 97266.2 4.25069
\(807\) 5631.08i 0.245630i
\(808\) − 11919.2i − 0.518955i
\(809\) −5657.55 −0.245870 −0.122935 0.992415i \(-0.539231\pi\)
−0.122935 + 0.992415i \(0.539231\pi\)
\(810\) 0 0
\(811\) 7532.41 0.326139 0.163070 0.986615i \(-0.447861\pi\)
0.163070 + 0.986615i \(0.447861\pi\)
\(812\) − 10081.8i − 0.435716i
\(813\) − 2069.73i − 0.0892847i
\(814\) −8502.73 −0.366119
\(815\) 0 0
\(816\) 16131.5 0.692053
\(817\) − 24470.3i − 1.04787i
\(818\) 59286.5i 2.53411i
\(819\) 5605.15 0.239145
\(820\) 0 0
\(821\) −6489.25 −0.275854 −0.137927 0.990442i \(-0.544044\pi\)
−0.137927 + 0.990442i \(0.544044\pi\)
\(822\) 1332.81i 0.0565535i
\(823\) 7901.57i 0.334668i 0.985900 + 0.167334i \(0.0535157\pi\)
−0.985900 + 0.167334i \(0.946484\pi\)
\(824\) −1347.15 −0.0569539
\(825\) 0 0
\(826\) −4492.40 −0.189238
\(827\) 37815.8i 1.59007i 0.606566 + 0.795033i \(0.292547\pi\)
−0.606566 + 0.795033i \(0.707453\pi\)
\(828\) − 10317.3i − 0.433031i
\(829\) −26073.5 −1.09236 −0.546182 0.837667i \(-0.683919\pi\)
−0.546182 + 0.837667i \(0.683919\pi\)
\(830\) 0 0
\(831\) 18867.4 0.787608
\(832\) 64350.2i 2.68142i
\(833\) − 5898.96i − 0.245362i
\(834\) −11604.6 −0.481817
\(835\) 0 0
\(836\) 46308.9 1.91582
\(837\) 6968.13i 0.287758i
\(838\) 41685.5i 1.71838i
\(839\) 15590.3 0.641523 0.320762 0.947160i \(-0.396061\pi\)
0.320762 + 0.947160i \(0.396061\pi\)
\(840\) 0 0
\(841\) −3412.46 −0.139918
\(842\) − 53630.8i − 2.19506i
\(843\) − 5864.61i − 0.239606i
\(844\) −33192.1 −1.35370
\(845\) 0 0
\(846\) −21946.8 −0.891899
\(847\) 2754.94i 0.111760i
\(848\) 8231.96i 0.333357i
\(849\) −15101.9 −0.610477
\(850\) 0 0
\(851\) −5571.92 −0.224445
\(852\) 1832.41i 0.0736823i
\(853\) − 17476.1i − 0.701488i −0.936471 0.350744i \(-0.885929\pi\)
0.936471 0.350744i \(-0.114071\pi\)
\(854\) 15710.8 0.629521
\(855\) 0 0
\(856\) 1067.83 0.0426376
\(857\) 5694.54i 0.226980i 0.993539 + 0.113490i \(0.0362030\pi\)
−0.993539 + 0.113490i \(0.963797\pi\)
\(858\) 46953.7i 1.86827i
\(859\) −27313.6 −1.08490 −0.542448 0.840089i \(-0.682503\pi\)
−0.542448 + 0.840089i \(0.682503\pi\)
\(860\) 0 0
\(861\) −4218.59 −0.166979
\(862\) − 19396.6i − 0.766415i
\(863\) − 9046.07i − 0.356815i −0.983957 0.178408i \(-0.942905\pi\)
0.983957 0.178408i \(-0.0570946\pi\)
\(864\) −6887.57 −0.271204
\(865\) 0 0
\(866\) 13893.8 0.545185
\(867\) − 28740.1i − 1.12580i
\(868\) 17964.8i 0.702496i
\(869\) 36472.1 1.42374
\(870\) 0 0
\(871\) 114.314 0.00444705
\(872\) 4667.22i 0.181252i
\(873\) − 9285.28i − 0.359976i
\(874\) 54760.0 2.11932
\(875\) 0 0
\(876\) 14463.0 0.557829
\(877\) 2104.29i 0.0810224i 0.999179 + 0.0405112i \(0.0128986\pi\)
−0.999179 + 0.0405112i \(0.987101\pi\)
\(878\) 1812.22i 0.0696576i
\(879\) −19107.4 −0.733192
\(880\) 0 0
\(881\) −22589.6 −0.863861 −0.431931 0.901907i \(-0.642167\pi\)
−0.431931 + 0.901907i \(0.642167\pi\)
\(882\) 1868.11i 0.0713179i
\(883\) − 2419.71i − 0.0922193i −0.998936 0.0461096i \(-0.985318\pi\)
0.998936 0.0461096i \(-0.0146824\pi\)
\(884\) 106512. 4.05248
\(885\) 0 0
\(886\) −65409.8 −2.48023
\(887\) − 13177.0i − 0.498806i −0.968400 0.249403i \(-0.919766\pi\)
0.968400 0.249403i \(-0.0802344\pi\)
\(888\) − 1194.26i − 0.0451313i
\(889\) −18089.8 −0.682464
\(890\) 0 0
\(891\) −3363.76 −0.126476
\(892\) − 1266.19i − 0.0475281i
\(893\) − 64553.2i − 2.41902i
\(894\) 22639.0 0.846937
\(895\) 0 0
\(896\) −7161.58 −0.267022
\(897\) 30769.2i 1.14532i
\(898\) − 39742.9i − 1.47688i
\(899\) −37378.3 −1.38669
\(900\) 0 0
\(901\) 22187.5 0.820393
\(902\) − 35338.7i − 1.30449i
\(903\) 4582.55i 0.168879i
\(904\) 6670.31 0.245411
\(905\) 0 0
\(906\) −17887.2 −0.655918
\(907\) − 9189.14i − 0.336406i −0.985752 0.168203i \(-0.946204\pi\)
0.985752 0.168203i \(-0.0537964\pi\)
\(908\) 38227.0i 1.39714i
\(909\) −13024.8 −0.475252
\(910\) 0 0
\(911\) −17045.8 −0.619928 −0.309964 0.950748i \(-0.600317\pi\)
−0.309964 + 0.950748i \(0.600317\pi\)
\(912\) − 15026.1i − 0.545575i
\(913\) − 20424.6i − 0.740369i
\(914\) 57485.0 2.08034
\(915\) 0 0
\(916\) 25228.1 0.910001
\(917\) − 9947.71i − 0.358236i
\(918\) 13769.1i 0.495042i
\(919\) 30825.0 1.10645 0.553223 0.833033i \(-0.313398\pi\)
0.553223 + 0.833033i \(0.313398\pi\)
\(920\) 0 0
\(921\) −19859.5 −0.710524
\(922\) 5303.06i 0.189422i
\(923\) − 5464.81i − 0.194882i
\(924\) −8672.25 −0.308762
\(925\) 0 0
\(926\) −33610.5 −1.19277
\(927\) 1472.10i 0.0521576i
\(928\) − 36946.2i − 1.30691i
\(929\) 5785.88 0.204336 0.102168 0.994767i \(-0.467422\pi\)
0.102168 + 0.994767i \(0.467422\pi\)
\(930\) 0 0
\(931\) −5494.75 −0.193430
\(932\) − 39652.2i − 1.39362i
\(933\) − 29727.7i − 1.04313i
\(934\) 32125.3 1.12545
\(935\) 0 0
\(936\) −6594.91 −0.230301
\(937\) 13680.9i 0.476986i 0.971144 + 0.238493i \(0.0766534\pi\)
−0.971144 + 0.238493i \(0.923347\pi\)
\(938\) 38.0990i 0.00132620i
\(939\) 1267.01 0.0440333
\(940\) 0 0
\(941\) −45448.8 −1.57448 −0.787242 0.616644i \(-0.788492\pi\)
−0.787242 + 0.616644i \(0.788492\pi\)
\(942\) − 20319.6i − 0.702813i
\(943\) − 23157.8i − 0.799705i
\(944\) −6766.91 −0.233310
\(945\) 0 0
\(946\) −38387.6 −1.31933
\(947\) 7788.45i 0.267255i 0.991032 + 0.133628i \(0.0426626\pi\)
−0.991032 + 0.133628i \(0.957337\pi\)
\(948\) 26200.9i 0.897642i
\(949\) −43133.0 −1.47540
\(950\) 0 0
\(951\) −14708.4 −0.501526
\(952\) 6940.61i 0.236288i
\(953\) 6149.43i 0.209024i 0.994524 + 0.104512i \(0.0333280\pi\)
−0.994524 + 0.104512i \(0.966672\pi\)
\(954\) −7026.44 −0.238458
\(955\) 0 0
\(956\) −33484.2 −1.13280
\(957\) − 18043.8i − 0.609481i
\(958\) 24693.5i 0.832788i
\(959\) 734.144 0.0247203
\(960\) 0 0
\(961\) 36813.7 1.23573
\(962\) 18216.5i 0.610524i
\(963\) − 1166.88i − 0.0390469i
\(964\) 9343.15 0.312160
\(965\) 0 0
\(966\) −10254.9 −0.341559
\(967\) − 23902.9i − 0.794896i −0.917625 0.397448i \(-0.869896\pi\)
0.917625 0.397448i \(-0.130104\pi\)
\(968\) − 3241.42i − 0.107627i
\(969\) −40499.8 −1.34266
\(970\) 0 0
\(971\) −8015.06 −0.264898 −0.132449 0.991190i \(-0.542284\pi\)
−0.132449 + 0.991190i \(0.542284\pi\)
\(972\) − 2416.46i − 0.0797407i
\(973\) 6392.12i 0.210608i
\(974\) −84306.7 −2.77347
\(975\) 0 0
\(976\) 23665.2 0.776131
\(977\) 34861.1i 1.14156i 0.821103 + 0.570780i \(0.193359\pi\)
−0.821103 + 0.570780i \(0.806641\pi\)
\(978\) 2603.80i 0.0851334i
\(979\) −17257.3 −0.563378
\(980\) 0 0
\(981\) 5100.13 0.165988
\(982\) − 71257.6i − 2.31560i
\(983\) 6620.83i 0.214824i 0.994215 + 0.107412i \(0.0342563\pi\)
−0.994215 + 0.107412i \(0.965744\pi\)
\(984\) 4963.52 0.160804
\(985\) 0 0
\(986\) −73860.0 −2.38558
\(987\) 12088.8i 0.389860i
\(988\) − 99213.6i − 3.19474i
\(989\) −25155.7 −0.808802
\(990\) 0 0
\(991\) 10360.1 0.332089 0.166045 0.986118i \(-0.446900\pi\)
0.166045 + 0.986118i \(0.446900\pi\)
\(992\) 65834.7i 2.10711i
\(993\) − 15845.2i − 0.506377i
\(994\) 1821.33 0.0581179
\(995\) 0 0
\(996\) 14672.7 0.466788
\(997\) 40309.3i 1.28045i 0.768188 + 0.640225i \(0.221159\pi\)
−0.768188 + 0.640225i \(0.778841\pi\)
\(998\) − 25551.3i − 0.810432i
\(999\) −1305.03 −0.0413306
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 525.4.d.k.274.4 4
5.2 odd 4 105.4.a.d.1.1 2
5.3 odd 4 525.4.a.o.1.2 2
5.4 even 2 inner 525.4.d.k.274.1 4
15.2 even 4 315.4.a.l.1.2 2
15.8 even 4 1575.4.a.n.1.1 2
20.7 even 4 1680.4.a.bd.1.1 2
35.27 even 4 735.4.a.m.1.1 2
105.62 odd 4 2205.4.a.be.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.d.1.1 2 5.2 odd 4
315.4.a.l.1.2 2 15.2 even 4
525.4.a.o.1.2 2 5.3 odd 4
525.4.d.k.274.1 4 5.4 even 2 inner
525.4.d.k.274.4 4 1.1 even 1 trivial
735.4.a.m.1.1 2 35.27 even 4
1575.4.a.n.1.1 2 15.8 even 4
1680.4.a.bd.1.1 2 20.7 even 4
2205.4.a.be.1.2 2 105.62 odd 4