Properties

Label 50.6.b.d.49.1
Level $50$
Weight $6$
Character 50.49
Analytic conductor $8.019$
Analytic rank $0$
Dimension $2$
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] = [50,6,Mod(49,50)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(50, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("50.49");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.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: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 10)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 50.49
Dual form 50.6.b.d.49.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} +26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} -22.0000i q^{7} +64.0000i q^{8} -433.000 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} +26.0000i q^{3} -16.0000 q^{4} +104.000 q^{6} -22.0000i q^{7} +64.0000i q^{8} -433.000 q^{9} -768.000 q^{11} -416.000i q^{12} +46.0000i q^{13} -88.0000 q^{14} +256.000 q^{16} +378.000i q^{17} +1732.00i q^{18} -1100.00 q^{19} +572.000 q^{21} +3072.00i q^{22} +1986.00i q^{23} -1664.00 q^{24} +184.000 q^{26} -4940.00i q^{27} +352.000i q^{28} +5610.00 q^{29} -3988.00 q^{31} -1024.00i q^{32} -19968.0i q^{33} +1512.00 q^{34} +6928.00 q^{36} -142.000i q^{37} +4400.00i q^{38} -1196.00 q^{39} +1542.00 q^{41} -2288.00i q^{42} +5026.00i q^{43} +12288.0 q^{44} +7944.00 q^{46} +24738.0i q^{47} +6656.00i q^{48} +16323.0 q^{49} -9828.00 q^{51} -736.000i q^{52} +14166.0i q^{53} -19760.0 q^{54} +1408.00 q^{56} -28600.0i q^{57} -22440.0i q^{58} -28380.0 q^{59} +5522.00 q^{61} +15952.0i q^{62} +9526.00i q^{63} -4096.00 q^{64} -79872.0 q^{66} -24742.0i q^{67} -6048.00i q^{68} -51636.0 q^{69} +42372.0 q^{71} -27712.0i q^{72} +52126.0i q^{73} -568.000 q^{74} +17600.0 q^{76} +16896.0i q^{77} +4784.00i q^{78} +39640.0 q^{79} +23221.0 q^{81} -6168.00i q^{82} +59826.0i q^{83} -9152.00 q^{84} +20104.0 q^{86} +145860. i q^{87} -49152.0i q^{88} -57690.0 q^{89} +1012.00 q^{91} -31776.0i q^{92} -103688. i q^{93} +98952.0 q^{94} +26624.0 q^{96} -144382. i q^{97} -65292.0i q^{98} +332544. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 32 q^{4} + 208 q^{6} - 866 q^{9} - 1536 q^{11} - 176 q^{14} + 512 q^{16} - 2200 q^{19} + 1144 q^{21} - 3328 q^{24} + 368 q^{26} + 11220 q^{29} - 7976 q^{31} + 3024 q^{34} + 13856 q^{36} - 2392 q^{39} + 3084 q^{41} + 24576 q^{44} + 15888 q^{46} + 32646 q^{49} - 19656 q^{51} - 39520 q^{54} + 2816 q^{56} - 56760 q^{59} + 11044 q^{61} - 8192 q^{64} - 159744 q^{66} - 103272 q^{69} + 84744 q^{71} - 1136 q^{74} + 35200 q^{76} + 79280 q^{79} + 46442 q^{81} - 18304 q^{84} + 40208 q^{86} - 115380 q^{89} + 2024 q^{91} + 197904 q^{94} + 53248 q^{96} + 665088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/50\mathbb{Z}\right)^\times\).

\(n\) \(27\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 26.0000i 1.66790i 0.551839 + 0.833950i \(0.313926\pi\)
−0.551839 + 0.833950i \(0.686074\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 104.000 1.17938
\(7\) − 22.0000i − 0.169698i −0.996394 0.0848492i \(-0.972959\pi\)
0.996394 0.0848492i \(-0.0270408\pi\)
\(8\) 64.0000i 0.353553i
\(9\) −433.000 −1.78189
\(10\) 0 0
\(11\) −768.000 −1.91372 −0.956862 0.290541i \(-0.906165\pi\)
−0.956862 + 0.290541i \(0.906165\pi\)
\(12\) − 416.000i − 0.833950i
\(13\) 46.0000i 0.0754917i 0.999287 + 0.0377459i \(0.0120177\pi\)
−0.999287 + 0.0377459i \(0.987982\pi\)
\(14\) −88.0000 −0.119995
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 378.000i 0.317227i 0.987341 + 0.158613i \(0.0507023\pi\)
−0.987341 + 0.158613i \(0.949298\pi\)
\(18\) 1732.00i 1.25999i
\(19\) −1100.00 −0.699051 −0.349525 0.936927i \(-0.613657\pi\)
−0.349525 + 0.936927i \(0.613657\pi\)
\(20\) 0 0
\(21\) 572.000 0.283040
\(22\) 3072.00i 1.35321i
\(23\) 1986.00i 0.782816i 0.920217 + 0.391408i \(0.128012\pi\)
−0.920217 + 0.391408i \(0.871988\pi\)
\(24\) −1664.00 −0.589692
\(25\) 0 0
\(26\) 184.000 0.0533807
\(27\) − 4940.00i − 1.30412i
\(28\) 352.000i 0.0848492i
\(29\) 5610.00 1.23870 0.619352 0.785113i \(-0.287395\pi\)
0.619352 + 0.785113i \(0.287395\pi\)
\(30\) 0 0
\(31\) −3988.00 −0.745334 −0.372667 0.927965i \(-0.621557\pi\)
−0.372667 + 0.927965i \(0.621557\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) − 19968.0i − 3.19190i
\(34\) 1512.00 0.224313
\(35\) 0 0
\(36\) 6928.00 0.890947
\(37\) − 142.000i − 0.0170523i −0.999964 0.00852617i \(-0.997286\pi\)
0.999964 0.00852617i \(-0.00271400\pi\)
\(38\) 4400.00i 0.494303i
\(39\) −1196.00 −0.125913
\(40\) 0 0
\(41\) 1542.00 0.143260 0.0716300 0.997431i \(-0.477180\pi\)
0.0716300 + 0.997431i \(0.477180\pi\)
\(42\) − 2288.00i − 0.200139i
\(43\) 5026.00i 0.414526i 0.978285 + 0.207263i \(0.0664555\pi\)
−0.978285 + 0.207263i \(0.933544\pi\)
\(44\) 12288.0 0.956862
\(45\) 0 0
\(46\) 7944.00 0.553534
\(47\) 24738.0i 1.63350i 0.576990 + 0.816752i \(0.304227\pi\)
−0.576990 + 0.816752i \(0.695773\pi\)
\(48\) 6656.00i 0.416975i
\(49\) 16323.0 0.971202
\(50\) 0 0
\(51\) −9828.00 −0.529102
\(52\) − 736.000i − 0.0377459i
\(53\) 14166.0i 0.692720i 0.938102 + 0.346360i \(0.112582\pi\)
−0.938102 + 0.346360i \(0.887418\pi\)
\(54\) −19760.0 −0.922152
\(55\) 0 0
\(56\) 1408.00 0.0599974
\(57\) − 28600.0i − 1.16595i
\(58\) − 22440.0i − 0.875897i
\(59\) −28380.0 −1.06141 −0.530704 0.847557i \(-0.678072\pi\)
−0.530704 + 0.847557i \(0.678072\pi\)
\(60\) 0 0
\(61\) 5522.00 0.190008 0.0950040 0.995477i \(-0.469714\pi\)
0.0950040 + 0.995477i \(0.469714\pi\)
\(62\) 15952.0i 0.527031i
\(63\) 9526.00i 0.302384i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) −79872.0 −2.25702
\(67\) − 24742.0i − 0.673361i −0.941619 0.336680i \(-0.890696\pi\)
0.941619 0.336680i \(-0.109304\pi\)
\(68\) − 6048.00i − 0.158613i
\(69\) −51636.0 −1.30566
\(70\) 0 0
\(71\) 42372.0 0.997546 0.498773 0.866733i \(-0.333784\pi\)
0.498773 + 0.866733i \(0.333784\pi\)
\(72\) − 27712.0i − 0.629994i
\(73\) 52126.0i 1.14485i 0.819958 + 0.572423i \(0.193997\pi\)
−0.819958 + 0.572423i \(0.806003\pi\)
\(74\) −568.000 −0.0120578
\(75\) 0 0
\(76\) 17600.0 0.349525
\(77\) 16896.0i 0.324756i
\(78\) 4784.00i 0.0890337i
\(79\) 39640.0 0.714605 0.357302 0.933989i \(-0.383697\pi\)
0.357302 + 0.933989i \(0.383697\pi\)
\(80\) 0 0
\(81\) 23221.0 0.393250
\(82\) − 6168.00i − 0.101300i
\(83\) 59826.0i 0.953223i 0.879114 + 0.476612i \(0.158135\pi\)
−0.879114 + 0.476612i \(0.841865\pi\)
\(84\) −9152.00 −0.141520
\(85\) 0 0
\(86\) 20104.0 0.293114
\(87\) 145860.i 2.06604i
\(88\) − 49152.0i − 0.676604i
\(89\) −57690.0 −0.772015 −0.386007 0.922496i \(-0.626146\pi\)
−0.386007 + 0.922496i \(0.626146\pi\)
\(90\) 0 0
\(91\) 1012.00 0.0128108
\(92\) − 31776.0i − 0.391408i
\(93\) − 103688.i − 1.24314i
\(94\) 98952.0 1.15506
\(95\) 0 0
\(96\) 26624.0 0.294846
\(97\) − 144382.i − 1.55806i −0.626988 0.779029i \(-0.715712\pi\)
0.626988 0.779029i \(-0.284288\pi\)
\(98\) − 65292.0i − 0.686744i
\(99\) 332544. 3.41005
\(100\) 0 0
\(101\) −141258. −1.37787 −0.688937 0.724821i \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(102\) 39312.0i 0.374132i
\(103\) − 139814.i − 1.29855i −0.760555 0.649273i \(-0.775073\pi\)
0.760555 0.649273i \(-0.224927\pi\)
\(104\) −2944.00 −0.0266904
\(105\) 0 0
\(106\) 56664.0 0.489827
\(107\) 86418.0i 0.729701i 0.931066 + 0.364850i \(0.118880\pi\)
−0.931066 + 0.364850i \(0.881120\pi\)
\(108\) 79040.0i 0.652060i
\(109\) −218450. −1.76111 −0.880554 0.473947i \(-0.842829\pi\)
−0.880554 + 0.473947i \(0.842829\pi\)
\(110\) 0 0
\(111\) 3692.00 0.0284416
\(112\) − 5632.00i − 0.0424246i
\(113\) 28806.0i 0.212220i 0.994354 + 0.106110i \(0.0338396\pi\)
−0.994354 + 0.106110i \(0.966160\pi\)
\(114\) −114400. −0.824449
\(115\) 0 0
\(116\) −89760.0 −0.619352
\(117\) − 19918.0i − 0.134518i
\(118\) 113520.i 0.750529i
\(119\) 8316.00 0.0538328
\(120\) 0 0
\(121\) 428773. 2.66234
\(122\) − 22088.0i − 0.134356i
\(123\) 40092.0i 0.238943i
\(124\) 63808.0 0.372667
\(125\) 0 0
\(126\) 38104.0 0.213818
\(127\) − 216502.i − 1.19111i −0.803314 0.595556i \(-0.796932\pi\)
0.803314 0.595556i \(-0.203068\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) −130676. −0.691388
\(130\) 0 0
\(131\) −244608. −1.24535 −0.622676 0.782479i \(-0.713955\pi\)
−0.622676 + 0.782479i \(0.713955\pi\)
\(132\) 319488.i 1.59595i
\(133\) 24200.0i 0.118628i
\(134\) −98968.0 −0.476138
\(135\) 0 0
\(136\) −24192.0 −0.112157
\(137\) − 239502.i − 1.09020i −0.838370 0.545102i \(-0.816491\pi\)
0.838370 0.545102i \(-0.183509\pi\)
\(138\) 206544.i 0.923241i
\(139\) −30860.0 −0.135475 −0.0677375 0.997703i \(-0.521578\pi\)
−0.0677375 + 0.997703i \(0.521578\pi\)
\(140\) 0 0
\(141\) −643188. −2.72452
\(142\) − 169488.i − 0.705372i
\(143\) − 35328.0i − 0.144470i
\(144\) −110848. −0.445473
\(145\) 0 0
\(146\) 208504. 0.809529
\(147\) 424398.i 1.61987i
\(148\) 2272.00i 0.00852617i
\(149\) 100950. 0.372512 0.186256 0.982501i \(-0.440365\pi\)
0.186256 + 0.982501i \(0.440365\pi\)
\(150\) 0 0
\(151\) 12452.0 0.0444423 0.0222212 0.999753i \(-0.492926\pi\)
0.0222212 + 0.999753i \(0.492926\pi\)
\(152\) − 70400.0i − 0.247152i
\(153\) − 163674.i − 0.565264i
\(154\) 67584.0 0.229637
\(155\) 0 0
\(156\) 19136.0 0.0629564
\(157\) − 6022.00i − 0.0194981i −0.999952 0.00974903i \(-0.996897\pi\)
0.999952 0.00974903i \(-0.00310326\pi\)
\(158\) − 158560.i − 0.505302i
\(159\) −368316. −1.15539
\(160\) 0 0
\(161\) 43692.0 0.132843
\(162\) − 92884.0i − 0.278070i
\(163\) 500866.i 1.47656i 0.674492 + 0.738282i \(0.264363\pi\)
−0.674492 + 0.738282i \(0.735637\pi\)
\(164\) −24672.0 −0.0716300
\(165\) 0 0
\(166\) 239304. 0.674031
\(167\) 555258.i 1.54065i 0.637652 + 0.770324i \(0.279906\pi\)
−0.637652 + 0.770324i \(0.720094\pi\)
\(168\) 36608.0i 0.100070i
\(169\) 369177. 0.994301
\(170\) 0 0
\(171\) 476300. 1.24563
\(172\) − 80416.0i − 0.207263i
\(173\) − 417354.i − 1.06020i −0.847934 0.530102i \(-0.822154\pi\)
0.847934 0.530102i \(-0.177846\pi\)
\(174\) 583440. 1.46091
\(175\) 0 0
\(176\) −196608. −0.478431
\(177\) − 737880.i − 1.77032i
\(178\) 230760.i 0.545897i
\(179\) 52380.0 0.122189 0.0610946 0.998132i \(-0.480541\pi\)
0.0610946 + 0.998132i \(0.480541\pi\)
\(180\) 0 0
\(181\) 546662. 1.24029 0.620144 0.784488i \(-0.287074\pi\)
0.620144 + 0.784488i \(0.287074\pi\)
\(182\) − 4048.00i − 0.00905862i
\(183\) 143572.i 0.316914i
\(184\) −127104. −0.276767
\(185\) 0 0
\(186\) −414752. −0.879035
\(187\) − 290304.i − 0.607084i
\(188\) − 395808.i − 0.816752i
\(189\) −108680. −0.221307
\(190\) 0 0
\(191\) −452028. −0.896565 −0.448283 0.893892i \(-0.647964\pi\)
−0.448283 + 0.893892i \(0.647964\pi\)
\(192\) − 106496.i − 0.208488i
\(193\) − 485594.i − 0.938383i −0.883097 0.469191i \(-0.844545\pi\)
0.883097 0.469191i \(-0.155455\pi\)
\(194\) −577528. −1.10171
\(195\) 0 0
\(196\) −261168. −0.485601
\(197\) 1.01018e6i 1.85452i 0.374414 + 0.927262i \(0.377844\pi\)
−0.374414 + 0.927262i \(0.622156\pi\)
\(198\) − 1.33018e6i − 2.41127i
\(199\) 807640. 1.44572 0.722862 0.690993i \(-0.242826\pi\)
0.722862 + 0.690993i \(0.242826\pi\)
\(200\) 0 0
\(201\) 643292. 1.12310
\(202\) 565032.i 0.974304i
\(203\) − 123420.i − 0.210206i
\(204\) 157248. 0.264551
\(205\) 0 0
\(206\) −559256. −0.918211
\(207\) − 859938.i − 1.39489i
\(208\) 11776.0i 0.0188729i
\(209\) 844800. 1.33779
\(210\) 0 0
\(211\) 149552. 0.231252 0.115626 0.993293i \(-0.463113\pi\)
0.115626 + 0.993293i \(0.463113\pi\)
\(212\) − 226656.i − 0.346360i
\(213\) 1.10167e6i 1.66381i
\(214\) 345672. 0.515976
\(215\) 0 0
\(216\) 316160. 0.461076
\(217\) 87736.0i 0.126482i
\(218\) 873800.i 1.24529i
\(219\) −1.35528e6 −1.90949
\(220\) 0 0
\(221\) −17388.0 −0.0239480
\(222\) − 14768.0i − 0.0201113i
\(223\) 443506.i 0.597224i 0.954375 + 0.298612i \(0.0965237\pi\)
−0.954375 + 0.298612i \(0.903476\pi\)
\(224\) −22528.0 −0.0299987
\(225\) 0 0
\(226\) 115224. 0.150062
\(227\) 420018.i 0.541007i 0.962719 + 0.270504i \(0.0871902\pi\)
−0.962719 + 0.270504i \(0.912810\pi\)
\(228\) 457600.i 0.582974i
\(229\) −1.05875e6 −1.33415 −0.667075 0.744990i \(-0.732454\pi\)
−0.667075 + 0.744990i \(0.732454\pi\)
\(230\) 0 0
\(231\) −439296. −0.541661
\(232\) 359040.i 0.437948i
\(233\) 1.27345e6i 1.53671i 0.640026 + 0.768353i \(0.278923\pi\)
−0.640026 + 0.768353i \(0.721077\pi\)
\(234\) −79672.0 −0.0951187
\(235\) 0 0
\(236\) 454080. 0.530704
\(237\) 1.03064e6i 1.19189i
\(238\) − 33264.0i − 0.0380655i
\(239\) 370680. 0.419763 0.209882 0.977727i \(-0.432692\pi\)
0.209882 + 0.977727i \(0.432692\pi\)
\(240\) 0 0
\(241\) −561298. −0.622517 −0.311258 0.950325i \(-0.600750\pi\)
−0.311258 + 0.950325i \(0.600750\pi\)
\(242\) − 1.71509e6i − 1.88256i
\(243\) − 596674.i − 0.648219i
\(244\) −88352.0 −0.0950040
\(245\) 0 0
\(246\) 160368. 0.168958
\(247\) − 50600.0i − 0.0527726i
\(248\) − 255232.i − 0.263515i
\(249\) −1.55548e6 −1.58988
\(250\) 0 0
\(251\) 577152. 0.578237 0.289119 0.957293i \(-0.406638\pi\)
0.289119 + 0.957293i \(0.406638\pi\)
\(252\) − 152416.i − 0.151192i
\(253\) − 1.52525e6i − 1.49809i
\(254\) −866008. −0.842243
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 651462.i − 0.615257i −0.951507 0.307628i \(-0.900465\pi\)
0.951507 0.307628i \(-0.0995353\pi\)
\(258\) 522704.i 0.488885i
\(259\) −3124.00 −0.00289375
\(260\) 0 0
\(261\) −2.42913e6 −2.20724
\(262\) 978432.i 0.880597i
\(263\) − 917574.i − 0.817997i −0.912535 0.408999i \(-0.865878\pi\)
0.912535 0.408999i \(-0.134122\pi\)
\(264\) 1.27795e6 1.12851
\(265\) 0 0
\(266\) 96800.0 0.0838825
\(267\) − 1.49994e6i − 1.28764i
\(268\) 395872.i 0.336680i
\(269\) 735390. 0.619637 0.309818 0.950796i \(-0.399732\pi\)
0.309818 + 0.950796i \(0.399732\pi\)
\(270\) 0 0
\(271\) −1.12131e6 −0.927474 −0.463737 0.885973i \(-0.653492\pi\)
−0.463737 + 0.885973i \(0.653492\pi\)
\(272\) 96768.0i 0.0793066i
\(273\) 26312.0i 0.0213672i
\(274\) −958008. −0.770891
\(275\) 0 0
\(276\) 826176. 0.652830
\(277\) − 1.66034e6i − 1.30016i −0.759864 0.650082i \(-0.774735\pi\)
0.759864 0.650082i \(-0.225265\pi\)
\(278\) 123440.i 0.0957952i
\(279\) 1.72680e6 1.32811
\(280\) 0 0
\(281\) 1.45210e6 1.09706 0.548531 0.836130i \(-0.315187\pi\)
0.548531 + 0.836130i \(0.315187\pi\)
\(282\) 2.57275e6i 1.92653i
\(283\) − 309014.i − 0.229357i −0.993403 0.114679i \(-0.963416\pi\)
0.993403 0.114679i \(-0.0365838\pi\)
\(284\) −677952. −0.498773
\(285\) 0 0
\(286\) −141312. −0.102156
\(287\) − 33924.0i − 0.0243110i
\(288\) 443392.i 0.314997i
\(289\) 1.27697e6 0.899367
\(290\) 0 0
\(291\) 3.75393e6 2.59869
\(292\) − 834016.i − 0.572423i
\(293\) 1.59301e6i 1.08405i 0.840363 + 0.542024i \(0.182342\pi\)
−0.840363 + 0.542024i \(0.817658\pi\)
\(294\) 1.69759e6 1.14542
\(295\) 0 0
\(296\) 9088.00 0.00602891
\(297\) 3.79392e6i 2.49573i
\(298\) − 403800.i − 0.263406i
\(299\) −91356.0 −0.0590961
\(300\) 0 0
\(301\) 110572. 0.0703443
\(302\) − 49808.0i − 0.0314255i
\(303\) − 3.67271e6i − 2.29816i
\(304\) −281600. −0.174763
\(305\) 0 0
\(306\) −654696. −0.399702
\(307\) 1.24726e6i 0.755284i 0.925952 + 0.377642i \(0.123265\pi\)
−0.925952 + 0.377642i \(0.876735\pi\)
\(308\) − 270336.i − 0.162378i
\(309\) 3.63516e6 2.16585
\(310\) 0 0
\(311\) −665988. −0.390450 −0.195225 0.980758i \(-0.562544\pi\)
−0.195225 + 0.980758i \(0.562544\pi\)
\(312\) − 76544.0i − 0.0445169i
\(313\) 591286.i 0.341143i 0.985345 + 0.170572i \(0.0545614\pi\)
−0.985345 + 0.170572i \(0.945439\pi\)
\(314\) −24088.0 −0.0137872
\(315\) 0 0
\(316\) −634240. −0.357302
\(317\) − 516342.i − 0.288595i −0.989534 0.144298i \(-0.953908\pi\)
0.989534 0.144298i \(-0.0460923\pi\)
\(318\) 1.47326e6i 0.816983i
\(319\) −4.30848e6 −2.37054
\(320\) 0 0
\(321\) −2.24687e6 −1.21707
\(322\) − 174768.i − 0.0939339i
\(323\) − 415800.i − 0.221757i
\(324\) −371536. −0.196625
\(325\) 0 0
\(326\) 2.00346e6 1.04409
\(327\) − 5.67970e6i − 2.93735i
\(328\) 98688.0i 0.0506500i
\(329\) 544236. 0.277203
\(330\) 0 0
\(331\) −3.29577e6 −1.65343 −0.826717 0.562619i \(-0.809794\pi\)
−0.826717 + 0.562619i \(0.809794\pi\)
\(332\) − 957216.i − 0.476612i
\(333\) 61486.0i 0.0303854i
\(334\) 2.22103e6 1.08940
\(335\) 0 0
\(336\) 146432. 0.0707600
\(337\) 1.91098e6i 0.916602i 0.888797 + 0.458301i \(0.151542\pi\)
−0.888797 + 0.458301i \(0.848458\pi\)
\(338\) − 1.47671e6i − 0.703077i
\(339\) −748956. −0.353962
\(340\) 0 0
\(341\) 3.06278e6 1.42636
\(342\) − 1.90520e6i − 0.880796i
\(343\) − 728860.i − 0.334510i
\(344\) −321664. −0.146557
\(345\) 0 0
\(346\) −1.66942e6 −0.749677
\(347\) 2.42006e6i 1.07895i 0.842001 + 0.539476i \(0.181378\pi\)
−0.842001 + 0.539476i \(0.818622\pi\)
\(348\) − 2.33376e6i − 1.03302i
\(349\) −2.50727e6 −1.10189 −0.550944 0.834542i \(-0.685732\pi\)
−0.550944 + 0.834542i \(0.685732\pi\)
\(350\) 0 0
\(351\) 227240. 0.0984503
\(352\) 786432.i 0.338302i
\(353\) 413166.i 0.176477i 0.996099 + 0.0882384i \(0.0281237\pi\)
−0.996099 + 0.0882384i \(0.971876\pi\)
\(354\) −2.95152e6 −1.25181
\(355\) 0 0
\(356\) 923040. 0.386007
\(357\) 216216.i 0.0897878i
\(358\) − 209520.i − 0.0864008i
\(359\) −1.73772e6 −0.711613 −0.355806 0.934560i \(-0.615794\pi\)
−0.355806 + 0.934560i \(0.615794\pi\)
\(360\) 0 0
\(361\) −1.26610e6 −0.511328
\(362\) − 2.18665e6i − 0.877016i
\(363\) 1.11481e7i 4.44052i
\(364\) −16192.0 −0.00640541
\(365\) 0 0
\(366\) 574288. 0.224092
\(367\) 1.16098e6i 0.449944i 0.974365 + 0.224972i \(0.0722291\pi\)
−0.974365 + 0.224972i \(0.927771\pi\)
\(368\) 508416.i 0.195704i
\(369\) −667686. −0.255274
\(370\) 0 0
\(371\) 311652. 0.117553
\(372\) 1.65901e6i 0.621572i
\(373\) − 343754.i − 0.127931i −0.997952 0.0639655i \(-0.979625\pi\)
0.997952 0.0639655i \(-0.0203748\pi\)
\(374\) −1.16122e6 −0.429273
\(375\) 0 0
\(376\) −1.58323e6 −0.577531
\(377\) 258060.i 0.0935120i
\(378\) 434720.i 0.156488i
\(379\) −573140. −0.204957 −0.102478 0.994735i \(-0.532677\pi\)
−0.102478 + 0.994735i \(0.532677\pi\)
\(380\) 0 0
\(381\) 5.62905e6 1.98666
\(382\) 1.80811e6i 0.633967i
\(383\) 2.88055e6i 1.00341i 0.865039 + 0.501704i \(0.167293\pi\)
−0.865039 + 0.501704i \(0.832707\pi\)
\(384\) −425984. −0.147423
\(385\) 0 0
\(386\) −1.94238e6 −0.663537
\(387\) − 2.17626e6i − 0.738640i
\(388\) 2.31011e6i 0.779029i
\(389\) 3.08559e6 1.03387 0.516933 0.856026i \(-0.327074\pi\)
0.516933 + 0.856026i \(0.327074\pi\)
\(390\) 0 0
\(391\) −750708. −0.248330
\(392\) 1.04467e6i 0.343372i
\(393\) − 6.35981e6i − 2.07712i
\(394\) 4.04071e6 1.31135
\(395\) 0 0
\(396\) −5.32070e6 −1.70503
\(397\) 885458.i 0.281963i 0.990012 + 0.140981i \(0.0450258\pi\)
−0.990012 + 0.140981i \(0.954974\pi\)
\(398\) − 3.23056e6i − 1.02228i
\(399\) −629200. −0.197859
\(400\) 0 0
\(401\) −3.75344e6 −1.16565 −0.582825 0.812598i \(-0.698053\pi\)
−0.582825 + 0.812598i \(0.698053\pi\)
\(402\) − 2.57317e6i − 0.794151i
\(403\) − 183448.i − 0.0562666i
\(404\) 2.26013e6 0.688937
\(405\) 0 0
\(406\) −493680. −0.148638
\(407\) 109056.i 0.0326335i
\(408\) − 628992.i − 0.187066i
\(409\) 1.94653e6 0.575377 0.287689 0.957724i \(-0.407113\pi\)
0.287689 + 0.957724i \(0.407113\pi\)
\(410\) 0 0
\(411\) 6.22705e6 1.81835
\(412\) 2.23702e6i 0.649273i
\(413\) 624360.i 0.180119i
\(414\) −3.43975e6 −0.986339
\(415\) 0 0
\(416\) 47104.0 0.0133452
\(417\) − 802360.i − 0.225959i
\(418\) − 3.37920e6i − 0.945961i
\(419\) 2.99166e6 0.832486 0.416243 0.909253i \(-0.363346\pi\)
0.416243 + 0.909253i \(0.363346\pi\)
\(420\) 0 0
\(421\) 3.96660e6 1.09072 0.545360 0.838202i \(-0.316393\pi\)
0.545360 + 0.838202i \(0.316393\pi\)
\(422\) − 598208.i − 0.163520i
\(423\) − 1.07116e7i − 2.91073i
\(424\) −906624. −0.244913
\(425\) 0 0
\(426\) 4.40669e6 1.17649
\(427\) − 121484.i − 0.0322440i
\(428\) − 1.38269e6i − 0.364850i
\(429\) 918528. 0.240962
\(430\) 0 0
\(431\) −5.17115e6 −1.34089 −0.670446 0.741958i \(-0.733897\pi\)
−0.670446 + 0.741958i \(0.733897\pi\)
\(432\) − 1.26464e6i − 0.326030i
\(433\) 4.53485e6i 1.16237i 0.813773 + 0.581183i \(0.197410\pi\)
−0.813773 + 0.581183i \(0.802590\pi\)
\(434\) 350944. 0.0894362
\(435\) 0 0
\(436\) 3.49520e6 0.880554
\(437\) − 2.18460e6i − 0.547228i
\(438\) 5.42110e6i 1.35021i
\(439\) 1.08220e6 0.268007 0.134004 0.990981i \(-0.457217\pi\)
0.134004 + 0.990981i \(0.457217\pi\)
\(440\) 0 0
\(441\) −7.06786e6 −1.73058
\(442\) 69552.0i 0.0169338i
\(443\) 1.08079e6i 0.261656i 0.991405 + 0.130828i \(0.0417635\pi\)
−0.991405 + 0.130828i \(0.958236\pi\)
\(444\) −59072.0 −0.0142208
\(445\) 0 0
\(446\) 1.77402e6 0.422301
\(447\) 2.62470e6i 0.621314i
\(448\) 90112.0i 0.0212123i
\(449\) −2.61783e6 −0.612810 −0.306405 0.951901i \(-0.599126\pi\)
−0.306405 + 0.951901i \(0.599126\pi\)
\(450\) 0 0
\(451\) −1.18426e6 −0.274160
\(452\) − 460896.i − 0.106110i
\(453\) 323752.i 0.0741254i
\(454\) 1.68007e6 0.382550
\(455\) 0 0
\(456\) 1.83040e6 0.412225
\(457\) 1.59046e6i 0.356231i 0.984010 + 0.178115i \(0.0570001\pi\)
−0.984010 + 0.178115i \(0.943000\pi\)
\(458\) 4.23500e6i 0.943387i
\(459\) 1.86732e6 0.413701
\(460\) 0 0
\(461\) 4.25470e6 0.932431 0.466216 0.884671i \(-0.345617\pi\)
0.466216 + 0.884671i \(0.345617\pi\)
\(462\) 1.75718e6i 0.383012i
\(463\) − 3.26605e6i − 0.708061i −0.935234 0.354031i \(-0.884811\pi\)
0.935234 0.354031i \(-0.115189\pi\)
\(464\) 1.43616e6 0.309676
\(465\) 0 0
\(466\) 5.09378e6 1.08662
\(467\) − 601542.i − 0.127636i −0.997962 0.0638181i \(-0.979672\pi\)
0.997962 0.0638181i \(-0.0203277\pi\)
\(468\) 318688.i 0.0672591i
\(469\) −544324. −0.114268
\(470\) 0 0
\(471\) 156572. 0.0325208
\(472\) − 1.81632e6i − 0.375264i
\(473\) − 3.85997e6i − 0.793288i
\(474\) 4.12256e6 0.842793
\(475\) 0 0
\(476\) −133056. −0.0269164
\(477\) − 6.13388e6i − 1.23435i
\(478\) − 1.48272e6i − 0.296817i
\(479\) 4.57932e6 0.911931 0.455966 0.889997i \(-0.349294\pi\)
0.455966 + 0.889997i \(0.349294\pi\)
\(480\) 0 0
\(481\) 6532.00 0.00128731
\(482\) 2.24519e6i 0.440186i
\(483\) 1.13599e6i 0.221568i
\(484\) −6.86037e6 −1.33117
\(485\) 0 0
\(486\) −2.38670e6 −0.458360
\(487\) 7.05226e6i 1.34743i 0.738992 + 0.673714i \(0.235302\pi\)
−0.738992 + 0.673714i \(0.764698\pi\)
\(488\) 353408.i 0.0671780i
\(489\) −1.30225e7 −2.46276
\(490\) 0 0
\(491\) −2.62349e6 −0.491106 −0.245553 0.969383i \(-0.578970\pi\)
−0.245553 + 0.969383i \(0.578970\pi\)
\(492\) − 641472.i − 0.119472i
\(493\) 2.12058e6i 0.392950i
\(494\) −202400. −0.0373158
\(495\) 0 0
\(496\) −1.02093e6 −0.186333
\(497\) − 932184.i − 0.169282i
\(498\) 6.22190e6i 1.12422i
\(499\) 3.61234e6 0.649437 0.324719 0.945811i \(-0.394730\pi\)
0.324719 + 0.945811i \(0.394730\pi\)
\(500\) 0 0
\(501\) −1.44367e7 −2.56965
\(502\) − 2.30861e6i − 0.408875i
\(503\) − 9.15629e6i − 1.61361i −0.590815 0.806807i \(-0.701194\pi\)
0.590815 0.806807i \(-0.298806\pi\)
\(504\) −609664. −0.106909
\(505\) 0 0
\(506\) −6.10099e6 −1.05931
\(507\) 9.59860e6i 1.65840i
\(508\) 3.46403e6i 0.595556i
\(509\) −7.26159e6 −1.24233 −0.621165 0.783679i \(-0.713340\pi\)
−0.621165 + 0.783679i \(0.713340\pi\)
\(510\) 0 0
\(511\) 1.14677e6 0.194279
\(512\) − 262144.i − 0.0441942i
\(513\) 5.43400e6i 0.911646i
\(514\) −2.60585e6 −0.435052
\(515\) 0 0
\(516\) 2.09082e6 0.345694
\(517\) − 1.89988e7i − 3.12608i
\(518\) 12496.0i 0.00204619i
\(519\) 1.08512e7 1.76831
\(520\) 0 0
\(521\) 5.81020e6 0.937771 0.468886 0.883259i \(-0.344656\pi\)
0.468886 + 0.883259i \(0.344656\pi\)
\(522\) 9.71652e6i 1.56075i
\(523\) 8.17067e6i 1.30618i 0.757280 + 0.653090i \(0.226528\pi\)
−0.757280 + 0.653090i \(0.773472\pi\)
\(524\) 3.91373e6 0.622676
\(525\) 0 0
\(526\) −3.67030e6 −0.578411
\(527\) − 1.50746e6i − 0.236440i
\(528\) − 5.11181e6i − 0.797976i
\(529\) 2.49215e6 0.387199
\(530\) 0 0
\(531\) 1.22885e7 1.89132
\(532\) − 387200.i − 0.0593139i
\(533\) 70932.0i 0.0108149i
\(534\) −5.99976e6 −0.910502
\(535\) 0 0
\(536\) 1.58349e6 0.238069
\(537\) 1.36188e6i 0.203800i
\(538\) − 2.94156e6i − 0.438149i
\(539\) −1.25361e7 −1.85861
\(540\) 0 0
\(541\) −817378. −0.120069 −0.0600343 0.998196i \(-0.519121\pi\)
−0.0600343 + 0.998196i \(0.519121\pi\)
\(542\) 4.48523e6i 0.655823i
\(543\) 1.42132e7i 2.06868i
\(544\) 387072. 0.0560783
\(545\) 0 0
\(546\) 105248. 0.0151089
\(547\) − 3.50750e6i − 0.501221i −0.968088 0.250611i \(-0.919369\pi\)
0.968088 0.250611i \(-0.0806314\pi\)
\(548\) 3.83203e6i 0.545102i
\(549\) −2.39103e6 −0.338574
\(550\) 0 0
\(551\) −6.17100e6 −0.865918
\(552\) − 3.30470e6i − 0.461620i
\(553\) − 872080.i − 0.121267i
\(554\) −6.64137e6 −0.919355
\(555\) 0 0
\(556\) 493760. 0.0677375
\(557\) 9.61490e6i 1.31313i 0.754271 + 0.656563i \(0.227991\pi\)
−0.754271 + 0.656563i \(0.772009\pi\)
\(558\) − 6.90722e6i − 0.939112i
\(559\) −231196. −0.0312933
\(560\) 0 0
\(561\) 7.54790e6 1.01256
\(562\) − 5.80841e6i − 0.775740i
\(563\) − 2.01941e6i − 0.268506i −0.990947 0.134253i \(-0.957136\pi\)
0.990947 0.134253i \(-0.0428635\pi\)
\(564\) 1.02910e7 1.36226
\(565\) 0 0
\(566\) −1.23606e6 −0.162180
\(567\) − 510862.i − 0.0667338i
\(568\) 2.71181e6i 0.352686i
\(569\) −1.37859e6 −0.178507 −0.0892533 0.996009i \(-0.528448\pi\)
−0.0892533 + 0.996009i \(0.528448\pi\)
\(570\) 0 0
\(571\) 8.54295e6 1.09652 0.548261 0.836307i \(-0.315290\pi\)
0.548261 + 0.836307i \(0.315290\pi\)
\(572\) 565248.i 0.0722352i
\(573\) − 1.17527e7i − 1.49538i
\(574\) −135696. −0.0171905
\(575\) 0 0
\(576\) 1.77357e6 0.222737
\(577\) − 2.31458e6i − 0.289423i −0.989474 0.144711i \(-0.953775\pi\)
0.989474 0.144711i \(-0.0462254\pi\)
\(578\) − 5.10789e6i − 0.635949i
\(579\) 1.26254e7 1.56513
\(580\) 0 0
\(581\) 1.31617e6 0.161760
\(582\) − 1.50157e7i − 1.83755i
\(583\) − 1.08795e7i − 1.32568i
\(584\) −3.33606e6 −0.404764
\(585\) 0 0
\(586\) 6.37202e6 0.766537
\(587\) 928338.i 0.111202i 0.998453 + 0.0556008i \(0.0177074\pi\)
−0.998453 + 0.0556008i \(0.982293\pi\)
\(588\) − 6.79037e6i − 0.809935i
\(589\) 4.38680e6 0.521026
\(590\) 0 0
\(591\) −2.62646e7 −3.09316
\(592\) − 36352.0i − 0.00426309i
\(593\) 909486.i 0.106209i 0.998589 + 0.0531043i \(0.0169116\pi\)
−0.998589 + 0.0531043i \(0.983088\pi\)
\(594\) 1.51757e7 1.76475
\(595\) 0 0
\(596\) −1.61520e6 −0.186256
\(597\) 2.09986e7i 2.41132i
\(598\) 365424.i 0.0417873i
\(599\) 8.51136e6 0.969241 0.484621 0.874724i \(-0.338958\pi\)
0.484621 + 0.874724i \(0.338958\pi\)
\(600\) 0 0
\(601\) 6.12498e6 0.691701 0.345851 0.938290i \(-0.387590\pi\)
0.345851 + 0.938290i \(0.387590\pi\)
\(602\) − 442288.i − 0.0497409i
\(603\) 1.07133e7i 1.19986i
\(604\) −199232. −0.0222212
\(605\) 0 0
\(606\) −1.46908e7 −1.62504
\(607\) − 4.51646e6i − 0.497538i −0.968563 0.248769i \(-0.919974\pi\)
0.968563 0.248769i \(-0.0800261\pi\)
\(608\) 1.12640e6i 0.123576i
\(609\) 3.20892e6 0.350603
\(610\) 0 0
\(611\) −1.13795e6 −0.123316
\(612\) 2.61878e6i 0.282632i
\(613\) − 9.63979e6i − 1.03614i −0.855340 0.518068i \(-0.826651\pi\)
0.855340 0.518068i \(-0.173349\pi\)
\(614\) 4.98903e6 0.534067
\(615\) 0 0
\(616\) −1.08134e6 −0.114819
\(617\) − 9.92650e6i − 1.04974i −0.851181 0.524872i \(-0.824113\pi\)
0.851181 0.524872i \(-0.175887\pi\)
\(618\) − 1.45407e7i − 1.53149i
\(619\) −7.63322e6 −0.800721 −0.400360 0.916358i \(-0.631115\pi\)
−0.400360 + 0.916358i \(0.631115\pi\)
\(620\) 0 0
\(621\) 9.81084e6 1.02089
\(622\) 2.66395e6i 0.276090i
\(623\) 1.26918e6i 0.131010i
\(624\) −306176. −0.0314782
\(625\) 0 0
\(626\) 2.36514e6 0.241225
\(627\) 2.19648e7i 2.23130i
\(628\) 96352.0i 0.00974903i
\(629\) 53676.0 0.00540946
\(630\) 0 0
\(631\) 1.80314e7 1.80284 0.901418 0.432949i \(-0.142527\pi\)
0.901418 + 0.432949i \(0.142527\pi\)
\(632\) 2.53696e6i 0.252651i
\(633\) 3.88835e6i 0.385706i
\(634\) −2.06537e6 −0.204068
\(635\) 0 0
\(636\) 5.89306e6 0.577694
\(637\) 750858.i 0.0733178i
\(638\) 1.72339e7i 1.67623i
\(639\) −1.83471e7 −1.77752
\(640\) 0 0
\(641\) 9.30190e6 0.894184 0.447092 0.894488i \(-0.352460\pi\)
0.447092 + 0.894488i \(0.352460\pi\)
\(642\) 8.98747e6i 0.860597i
\(643\) 1.38332e7i 1.31946i 0.751503 + 0.659730i \(0.229329\pi\)
−0.751503 + 0.659730i \(0.770671\pi\)
\(644\) −699072. −0.0664213
\(645\) 0 0
\(646\) −1.66320e6 −0.156806
\(647\) − 1.48997e7i − 1.39932i −0.714478 0.699658i \(-0.753336\pi\)
0.714478 0.699658i \(-0.246664\pi\)
\(648\) 1.48614e6i 0.139035i
\(649\) 2.17958e7 2.03124
\(650\) 0 0
\(651\) −2.28114e6 −0.210959
\(652\) − 8.01386e6i − 0.738282i
\(653\) 1.93306e7i 1.77403i 0.461738 + 0.887016i \(0.347226\pi\)
−0.461738 + 0.887016i \(0.652774\pi\)
\(654\) −2.27188e7 −2.07702
\(655\) 0 0
\(656\) 394752. 0.0358150
\(657\) − 2.25706e7i − 2.03999i
\(658\) − 2.17694e6i − 0.196012i
\(659\) 4.06110e6 0.364276 0.182138 0.983273i \(-0.441698\pi\)
0.182138 + 0.983273i \(0.441698\pi\)
\(660\) 0 0
\(661\) −1.35152e7 −1.20315 −0.601575 0.798816i \(-0.705460\pi\)
−0.601575 + 0.798816i \(0.705460\pi\)
\(662\) 1.31831e7i 1.16915i
\(663\) − 452088.i − 0.0399429i
\(664\) −3.82886e6 −0.337015
\(665\) 0 0
\(666\) 245944. 0.0214858
\(667\) 1.11415e7i 0.969678i
\(668\) − 8.88413e6i − 0.770324i
\(669\) −1.15312e7 −0.996111
\(670\) 0 0
\(671\) −4.24090e6 −0.363623
\(672\) − 585728.i − 0.0500349i
\(673\) − 1.43520e7i − 1.22144i −0.791845 0.610722i \(-0.790879\pi\)
0.791845 0.610722i \(-0.209121\pi\)
\(674\) 7.64391e6 0.648136
\(675\) 0 0
\(676\) −5.90683e6 −0.497150
\(677\) 1.89530e6i 0.158930i 0.996838 + 0.0794650i \(0.0253212\pi\)
−0.996838 + 0.0794650i \(0.974679\pi\)
\(678\) 2.99582e6i 0.250289i
\(679\) −3.17640e6 −0.264400
\(680\) 0 0
\(681\) −1.09205e7 −0.902347
\(682\) − 1.22511e7i − 1.00859i
\(683\) − 2.91641e6i − 0.239220i −0.992821 0.119610i \(-0.961836\pi\)
0.992821 0.119610i \(-0.0381644\pi\)
\(684\) −7.62080e6 −0.622817
\(685\) 0 0
\(686\) −2.91544e6 −0.236534
\(687\) − 2.75275e7i − 2.22523i
\(688\) 1.28666e6i 0.103631i
\(689\) −651636. −0.0522946
\(690\) 0 0
\(691\) 1.44278e7 1.14949 0.574743 0.818334i \(-0.305102\pi\)
0.574743 + 0.818334i \(0.305102\pi\)
\(692\) 6.67766e6i 0.530102i
\(693\) − 7.31597e6i − 0.578680i
\(694\) 9.68023e6 0.762934
\(695\) 0 0
\(696\) −9.33504e6 −0.730454
\(697\) 582876.i 0.0454458i
\(698\) 1.00291e7i 0.779153i
\(699\) −3.31096e7 −2.56307
\(700\) 0 0
\(701\) −1.58679e7 −1.21962 −0.609811 0.792547i \(-0.708754\pi\)
−0.609811 + 0.792547i \(0.708754\pi\)
\(702\) − 908960.i − 0.0696149i
\(703\) 156200.i 0.0119205i
\(704\) 3.14573e6 0.239216
\(705\) 0 0
\(706\) 1.65266e6 0.124788
\(707\) 3.10768e6i 0.233823i
\(708\) 1.18061e7i 0.885162i
\(709\) 301810. 0.0225485 0.0112743 0.999936i \(-0.496411\pi\)
0.0112743 + 0.999936i \(0.496411\pi\)
\(710\) 0 0
\(711\) −1.71641e7 −1.27335
\(712\) − 3.69216e6i − 0.272948i
\(713\) − 7.92017e6i − 0.583459i
\(714\) 864864. 0.0634896
\(715\) 0 0
\(716\) −838080. −0.0610946
\(717\) 9.63768e6i 0.700123i
\(718\) 6.95088e6i 0.503186i
\(719\) −2.12677e7 −1.53426 −0.767130 0.641492i \(-0.778316\pi\)
−0.767130 + 0.641492i \(0.778316\pi\)
\(720\) 0 0
\(721\) −3.07591e6 −0.220361
\(722\) 5.06440e6i 0.361564i
\(723\) − 1.45937e7i − 1.03830i
\(724\) −8.74659e6 −0.620144
\(725\) 0 0
\(726\) 4.45924e7 3.13992
\(727\) 1.55009e7i 1.08773i 0.839174 + 0.543863i \(0.183039\pi\)
−0.839174 + 0.543863i \(0.816961\pi\)
\(728\) 64768.0i 0.00452931i
\(729\) 2.11562e7 1.47441
\(730\) 0 0
\(731\) −1.89983e6 −0.131499
\(732\) − 2.29715e6i − 0.158457i
\(733\) 1.21850e7i 0.837653i 0.908066 + 0.418827i \(0.137559\pi\)
−0.908066 + 0.418827i \(0.862441\pi\)
\(734\) 4.64391e6 0.318159
\(735\) 0 0
\(736\) 2.03366e6 0.138384
\(737\) 1.90019e7i 1.28863i
\(738\) 2.67074e6i 0.180506i
\(739\) 2.90282e7 1.95528 0.977641 0.210282i \(-0.0674382\pi\)
0.977641 + 0.210282i \(0.0674382\pi\)
\(740\) 0 0
\(741\) 1.31560e6 0.0880194
\(742\) − 1.24661e6i − 0.0831228i
\(743\) − 1.61145e7i − 1.07089i −0.844570 0.535445i \(-0.820144\pi\)
0.844570 0.535445i \(-0.179856\pi\)
\(744\) 6.63603e6 0.439517
\(745\) 0 0
\(746\) −1.37502e6 −0.0904609
\(747\) − 2.59047e7i − 1.69854i
\(748\) 4.64486e6i 0.303542i
\(749\) 1.90120e6 0.123829
\(750\) 0 0
\(751\) −2.92431e6 −0.189201 −0.0946005 0.995515i \(-0.530157\pi\)
−0.0946005 + 0.995515i \(0.530157\pi\)
\(752\) 6.33293e6i 0.408376i
\(753\) 1.50060e7i 0.964442i
\(754\) 1.03224e6 0.0661230
\(755\) 0 0
\(756\) 1.73888e6 0.110653
\(757\) 2.60325e7i 1.65111i 0.564319 + 0.825557i \(0.309139\pi\)
−0.564319 + 0.825557i \(0.690861\pi\)
\(758\) 2.29256e6i 0.144926i
\(759\) 3.96564e7 2.49867
\(760\) 0 0
\(761\) 1.63263e7 1.02194 0.510970 0.859598i \(-0.329286\pi\)
0.510970 + 0.859598i \(0.329286\pi\)
\(762\) − 2.25162e7i − 1.40478i
\(763\) 4.80590e6i 0.298857i
\(764\) 7.23245e6 0.448283
\(765\) 0 0
\(766\) 1.15222e7 0.709517
\(767\) − 1.30548e6i − 0.0801275i
\(768\) 1.70394e6i 0.104244i
\(769\) −2.58132e7 −1.57408 −0.787040 0.616902i \(-0.788388\pi\)
−0.787040 + 0.616902i \(0.788388\pi\)
\(770\) 0 0
\(771\) 1.69380e7 1.02619
\(772\) 7.76950e6i 0.469191i
\(773\) 1.90592e7i 1.14725i 0.819119 + 0.573624i \(0.194463\pi\)
−0.819119 + 0.573624i \(0.805537\pi\)
\(774\) −8.70503e6 −0.522298
\(775\) 0 0
\(776\) 9.24045e6 0.550857
\(777\) − 81224.0i − 0.00482649i
\(778\) − 1.23424e7i − 0.731054i
\(779\) −1.69620e6 −0.100146
\(780\) 0 0
\(781\) −3.25417e7 −1.90903
\(782\) 3.00283e6i 0.175596i
\(783\) − 2.77134e7i − 1.61542i
\(784\) 4.17869e6 0.242801
\(785\) 0 0
\(786\) −2.54392e7 −1.46875
\(787\) − 1.73411e7i − 0.998021i −0.866596 0.499011i \(-0.833697\pi\)
0.866596 0.499011i \(-0.166303\pi\)
\(788\) − 1.61628e7i − 0.927262i
\(789\) 2.38569e7 1.36434
\(790\) 0 0
\(791\) 633732. 0.0360134
\(792\) 2.12828e7i 1.20564i
\(793\) 254012.i 0.0143440i
\(794\) 3.54183e6 0.199378
\(795\) 0 0
\(796\) −1.29222e7 −0.722862
\(797\) − 2.58169e7i − 1.43965i −0.694153 0.719827i \(-0.744221\pi\)
0.694153 0.719827i \(-0.255779\pi\)
\(798\) 2.51680e6i 0.139908i
\(799\) −9.35096e6 −0.518190
\(800\) 0 0
\(801\) 2.49798e7 1.37565
\(802\) 1.50138e7i 0.824239i
\(803\) − 4.00328e7i − 2.19092i
\(804\) −1.02927e7 −0.561549
\(805\) 0 0
\(806\) −733792. −0.0397865
\(807\) 1.91201e7i 1.03349i
\(808\) − 9.04051e6i − 0.487152i
\(809\) −8.88489e6 −0.477288 −0.238644 0.971107i \(-0.576703\pi\)
−0.238644 + 0.971107i \(0.576703\pi\)
\(810\) 0 0
\(811\) −2.46396e7 −1.31547 −0.657735 0.753249i \(-0.728485\pi\)
−0.657735 + 0.753249i \(0.728485\pi\)
\(812\) 1.97472e6i 0.105103i
\(813\) − 2.91540e7i − 1.54693i
\(814\) 436224. 0.0230754
\(815\) 0 0
\(816\) −2.51597e6 −0.132276
\(817\) − 5.52860e6i − 0.289774i
\(818\) − 7.78612e6i − 0.406853i
\(819\) −438196. −0.0228275
\(820\) 0 0
\(821\) 1.13768e7 0.589062 0.294531 0.955642i \(-0.404837\pi\)
0.294531 + 0.955642i \(0.404837\pi\)
\(822\) − 2.49082e7i − 1.28577i
\(823\) 1.30783e7i 0.673057i 0.941673 + 0.336529i \(0.109253\pi\)
−0.941673 + 0.336529i \(0.890747\pi\)
\(824\) 8.94810e6 0.459106
\(825\) 0 0
\(826\) 2.49744e6 0.127363
\(827\) − 3.57188e7i − 1.81607i −0.418891 0.908037i \(-0.637581\pi\)
0.418891 0.908037i \(-0.362419\pi\)
\(828\) 1.37590e7i 0.697447i
\(829\) −1.61880e7 −0.818103 −0.409052 0.912511i \(-0.634140\pi\)
−0.409052 + 0.912511i \(0.634140\pi\)
\(830\) 0 0
\(831\) 4.31689e7 2.16854
\(832\) − 188416.i − 0.00943647i
\(833\) 6.17009e6i 0.308091i
\(834\) −3.20944e6 −0.159777
\(835\) 0 0
\(836\) −1.35168e7 −0.668895
\(837\) 1.97007e7i 0.972005i
\(838\) − 1.19666e7i − 0.588657i
\(839\) 2.55497e7 1.25309 0.626543 0.779387i \(-0.284469\pi\)
0.626543 + 0.779387i \(0.284469\pi\)
\(840\) 0 0
\(841\) 1.09610e7 0.534390
\(842\) − 1.58664e7i − 0.771256i
\(843\) 3.77547e7i 1.82979i
\(844\) −2.39283e6 −0.115626
\(845\) 0 0
\(846\) −4.28462e7 −2.05820
\(847\) − 9.43301e6i − 0.451795i
\(848\) 3.62650e6i 0.173180i
\(849\) 8.03436e6 0.382545
\(850\) 0 0
\(851\) 282012. 0.0133488
\(852\) − 1.76268e7i − 0.831904i
\(853\) 2.22953e7i 1.04916i 0.851362 + 0.524579i \(0.175777\pi\)
−0.851362 + 0.524579i \(0.824223\pi\)
\(854\) −485936. −0.0228000
\(855\) 0 0
\(856\) −5.53075e6 −0.257988
\(857\) 1.96872e7i 0.915656i 0.889041 + 0.457828i \(0.151372\pi\)
−0.889041 + 0.457828i \(0.848628\pi\)
\(858\) − 3.67411e6i − 0.170386i
\(859\) −6.77582e6 −0.313313 −0.156657 0.987653i \(-0.550072\pi\)
−0.156657 + 0.987653i \(0.550072\pi\)
\(860\) 0 0
\(861\) 882024. 0.0405483
\(862\) 2.06846e7i 0.948154i
\(863\) 2.63804e7i 1.20574i 0.797839 + 0.602871i \(0.205977\pi\)
−0.797839 + 0.602871i \(0.794023\pi\)
\(864\) −5.05856e6 −0.230538
\(865\) 0 0
\(866\) 1.81394e7 0.821917
\(867\) 3.32013e7i 1.50006i
\(868\) − 1.40378e6i − 0.0632410i
\(869\) −3.04435e7 −1.36756
\(870\) 0 0
\(871\) 1.13813e6 0.0508332
\(872\) − 1.39808e7i − 0.622645i
\(873\) 6.25174e7i 2.77629i
\(874\) −8.73840e6 −0.386949
\(875\) 0 0
\(876\) 2.16844e7 0.954745
\(877\) 2.95161e7i 1.29587i 0.761697 + 0.647934i \(0.224367\pi\)
−0.761697 + 0.647934i \(0.775633\pi\)
\(878\) − 4.32880e6i − 0.189510i
\(879\) −4.14182e7 −1.80808
\(880\) 0 0
\(881\) −1.48565e7 −0.644877 −0.322438 0.946590i \(-0.604502\pi\)
−0.322438 + 0.946590i \(0.604502\pi\)
\(882\) 2.82714e7i 1.22370i
\(883\) 1.45340e7i 0.627313i 0.949537 + 0.313656i \(0.101554\pi\)
−0.949537 + 0.313656i \(0.898446\pi\)
\(884\) 278208. 0.0119740
\(885\) 0 0
\(886\) 4.32314e6 0.185019
\(887\) − 1.72028e7i − 0.734160i −0.930189 0.367080i \(-0.880358\pi\)
0.930189 0.367080i \(-0.119642\pi\)
\(888\) 236288.i 0.0100556i
\(889\) −4.76304e6 −0.202130
\(890\) 0 0
\(891\) −1.78337e7 −0.752572
\(892\) − 7.09610e6i − 0.298612i
\(893\) − 2.72118e7i − 1.14190i
\(894\) 1.04988e7 0.439335
\(895\) 0 0
\(896\) 360448. 0.0149994
\(897\) − 2.37526e6i − 0.0985665i
\(898\) 1.04713e7i 0.433322i
\(899\) −2.23727e7 −0.923249
\(900\) 0 0
\(901\) −5.35475e6 −0.219749
\(902\) 4.73702e6i 0.193860i
\(903\) 2.87487e6i 0.117327i
\(904\) −1.84358e6 −0.0750312
\(905\) 0 0
\(906\) 1.29501e6 0.0524146
\(907\) − 3.44434e7i − 1.39023i −0.718897 0.695116i \(-0.755353\pi\)
0.718897 0.695116i \(-0.244647\pi\)
\(908\) − 6.72029e6i − 0.270504i
\(909\) 6.11647e7 2.45522
\(910\) 0 0
\(911\) −983748. −0.0392724 −0.0196362 0.999807i \(-0.506251\pi\)
−0.0196362 + 0.999807i \(0.506251\pi\)
\(912\) − 7.32160e6i − 0.291487i
\(913\) − 4.59464e7i − 1.82421i
\(914\) 6.36183e6 0.251893
\(915\) 0 0
\(916\) 1.69400e7 0.667075
\(917\) 5.38138e6i 0.211334i
\(918\) − 7.46928e6i − 0.292531i
\(919\) −3.08857e7 −1.20634 −0.603168 0.797614i \(-0.706095\pi\)
−0.603168 + 0.797614i \(0.706095\pi\)
\(920\) 0 0
\(921\) −3.24287e7 −1.25974
\(922\) − 1.70188e7i − 0.659328i
\(923\) 1.94911e6i 0.0753065i
\(924\) 7.02874e6 0.270830
\(925\) 0 0
\(926\) −1.30642e7 −0.500675
\(927\) 6.05395e7i 2.31387i
\(928\) − 5.74464e6i − 0.218974i
\(929\) 3.20874e7 1.21982 0.609909 0.792472i \(-0.291206\pi\)
0.609909 + 0.792472i \(0.291206\pi\)
\(930\) 0 0
\(931\) −1.79553e7 −0.678920
\(932\) − 2.03751e7i − 0.768353i
\(933\) − 1.73157e7i − 0.651232i
\(934\) −2.40617e6 −0.0902524
\(935\) 0 0
\(936\) 1.27475e6 0.0475594
\(937\) 1.52520e7i 0.567515i 0.958896 + 0.283757i \(0.0915810\pi\)
−0.958896 + 0.283757i \(0.908419\pi\)
\(938\) 2.17730e6i 0.0807998i
\(939\) −1.53734e7 −0.568993
\(940\) 0 0
\(941\) 3.48166e6 0.128178 0.0640889 0.997944i \(-0.479586\pi\)
0.0640889 + 0.997944i \(0.479586\pi\)
\(942\) − 626288.i − 0.0229957i
\(943\) 3.06241e6i 0.112146i
\(944\) −7.26528e6 −0.265352
\(945\) 0 0
\(946\) −1.54399e7 −0.560939
\(947\) − 2.54010e7i − 0.920398i −0.887816 0.460199i \(-0.847778\pi\)
0.887816 0.460199i \(-0.152222\pi\)
\(948\) − 1.64902e7i − 0.595945i
\(949\) −2.39780e6 −0.0864265
\(950\) 0 0
\(951\) 1.34249e7 0.481348
\(952\) 532224.i 0.0190328i
\(953\) 4.97352e7i 1.77391i 0.461856 + 0.886955i \(0.347184\pi\)
−0.461856 + 0.886955i \(0.652816\pi\)
\(954\) −2.45355e7 −0.872819
\(955\) 0 0
\(956\) −5.93088e6 −0.209882
\(957\) − 1.12020e8i − 3.95383i
\(958\) − 1.83173e7i − 0.644833i
\(959\) −5.26904e6 −0.185006
\(960\) 0 0
\(961\) −1.27250e7 −0.444477
\(962\) − 26128.0i 0 0.000910266i
\(963\) − 3.74190e7i − 1.30025i
\(964\) 8.98077e6 0.311258
\(965\) 0 0
\(966\) 4.54397e6 0.156672
\(967\) 3.05173e7i 1.04949i 0.851258 + 0.524747i \(0.175840\pi\)
−0.851258 + 0.524747i \(0.824160\pi\)
\(968\) 2.74415e7i 0.941280i
\(969\) 1.08108e7 0.369869
\(970\) 0 0
\(971\) 3.19854e7 1.08869 0.544344 0.838862i \(-0.316779\pi\)
0.544344 + 0.838862i \(0.316779\pi\)
\(972\) 9.54678e6i 0.324109i
\(973\) 678920.i 0.0229899i
\(974\) 2.82090e7 0.952776
\(975\) 0 0
\(976\) 1.41363e6 0.0475020
\(977\) 2.90786e6i 0.0974623i 0.998812 + 0.0487312i \(0.0155178\pi\)
−0.998812 + 0.0487312i \(0.984482\pi\)
\(978\) 5.20901e7i 1.74144i
\(979\) 4.43059e7 1.47742
\(980\) 0 0
\(981\) 9.45888e7 3.13810
\(982\) 1.04940e7i 0.347264i
\(983\) − 3.49621e7i − 1.15402i −0.816737 0.577010i \(-0.804219\pi\)
0.816737 0.577010i \(-0.195781\pi\)
\(984\) −2.56589e6 −0.0844792
\(985\) 0 0
\(986\) 8.48232e6 0.277858
\(987\) 1.41501e7i 0.462347i
\(988\) 809600.i 0.0263863i
\(989\) −9.98164e6 −0.324497
\(990\) 0 0
\(991\) 3.00465e6 0.0971874 0.0485937 0.998819i \(-0.484526\pi\)
0.0485937 + 0.998819i \(0.484526\pi\)
\(992\) 4.08371e6i 0.131758i
\(993\) − 8.56900e7i − 2.75776i
\(994\) −3.72874e6 −0.119700
\(995\) 0 0
\(996\) 2.48876e7 0.794941
\(997\) 3.20789e7i 1.02207i 0.859560 + 0.511035i \(0.170738\pi\)
−0.859560 + 0.511035i \(0.829262\pi\)
\(998\) − 1.44494e7i − 0.459222i
\(999\) −701480. −0.0222383
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 50.6.b.d.49.1 2
3.2 odd 2 450.6.c.o.199.2 2
4.3 odd 2 400.6.c.a.49.1 2
5.2 odd 4 50.6.a.g.1.1 1
5.3 odd 4 10.6.a.a.1.1 1
5.4 even 2 inner 50.6.b.d.49.2 2
15.2 even 4 450.6.a.h.1.1 1
15.8 even 4 90.6.a.f.1.1 1
15.14 odd 2 450.6.c.o.199.1 2
20.3 even 4 80.6.a.h.1.1 1
20.7 even 4 400.6.a.a.1.1 1
20.19 odd 2 400.6.c.a.49.2 2
35.13 even 4 490.6.a.j.1.1 1
40.3 even 4 320.6.a.a.1.1 1
40.13 odd 4 320.6.a.p.1.1 1
60.23 odd 4 720.6.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.a.a.1.1 1 5.3 odd 4
50.6.a.g.1.1 1 5.2 odd 4
50.6.b.d.49.1 2 1.1 even 1 trivial
50.6.b.d.49.2 2 5.4 even 2 inner
80.6.a.h.1.1 1 20.3 even 4
90.6.a.f.1.1 1 15.8 even 4
320.6.a.a.1.1 1 40.3 even 4
320.6.a.p.1.1 1 40.13 odd 4
400.6.a.a.1.1 1 20.7 even 4
400.6.c.a.49.1 2 4.3 odd 2
400.6.c.a.49.2 2 20.19 odd 2
450.6.a.h.1.1 1 15.2 even 4
450.6.c.o.199.1 2 15.14 odd 2
450.6.c.o.199.2 2 3.2 odd 2
490.6.a.j.1.1 1 35.13 even 4
720.6.a.r.1.1 1 60.23 odd 4