Properties

Label 650.6.b.j.599.3
Level $650$
Weight $6$
Character 650.599
Analytic conductor $104.249$
Analytic rank $0$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,6,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("650.599");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 650.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: \(104.249482878\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} - 390x^{3} + 32400x^{2} - 135000x + 281250 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 130)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 599.3
Root \(-9.94525 + 9.94525i\) of defining polynomial
Character \(\chi\) \(=\) 650.599
Dual form 650.6.b.j.599.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000i q^{2} +27.8905i q^{3} -16.0000 q^{4} +111.562 q^{6} -240.426i q^{7} +64.0000i q^{8} -534.880 q^{9} +O(q^{10})\) \(q-4.00000i q^{2} +27.8905i q^{3} -16.0000 q^{4} +111.562 q^{6} -240.426i q^{7} +64.0000i q^{8} -534.880 q^{9} +544.151 q^{11} -446.248i q^{12} -169.000i q^{13} -961.702 q^{14} +256.000 q^{16} +1629.30i q^{17} +2139.52i q^{18} +805.920 q^{19} +6705.59 q^{21} -2176.61i q^{22} -373.152i q^{23} -1784.99 q^{24} -676.000 q^{26} -8140.67i q^{27} +3846.81i q^{28} -1503.62 q^{29} -2200.08 q^{31} -1024.00i q^{32} +15176.7i q^{33} +6517.20 q^{34} +8558.08 q^{36} -13109.1i q^{37} -3223.68i q^{38} +4713.49 q^{39} -17099.9 q^{41} -26822.4i q^{42} +8935.58i q^{43} -8706.42 q^{44} -1492.61 q^{46} +15749.7i q^{47} +7139.97i q^{48} -40997.5 q^{49} -45442.0 q^{51} +2704.00i q^{52} -40379.7i q^{53} -32562.7 q^{54} +15387.2 q^{56} +22477.5i q^{57} +6014.46i q^{58} +47562.1 q^{59} -30280.0 q^{61} +8800.31i q^{62} +128599. i q^{63} -4096.00 q^{64} +60706.6 q^{66} +38769.4i q^{67} -26068.8i q^{68} +10407.4 q^{69} -10519.7 q^{71} -34232.3i q^{72} -1582.12i q^{73} -52436.5 q^{74} -12894.7 q^{76} -130828. i q^{77} -18854.0i q^{78} +6191.23 q^{79} +97071.7 q^{81} +68399.8i q^{82} -37849.2i q^{83} -107289. q^{84} +35742.3 q^{86} -41936.6i q^{87} +34825.7i q^{88} -49151.0 q^{89} -40631.9 q^{91} +5970.43i q^{92} -61361.3i q^{93} +62999.0 q^{94} +28559.9 q^{96} -15654.2i q^{97} +163990. i q^{98} -291056. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 96 q^{4} + 176 q^{6} - 310 q^{9} + 96 q^{11} - 1872 q^{14} + 1536 q^{16} + 720 q^{19} + 13808 q^{21} - 2816 q^{24} - 4056 q^{26} + 6156 q^{29} - 10776 q^{31} + 12048 q^{34} + 4960 q^{36} + 7436 q^{39} - 31812 q^{41} - 1536 q^{44} + 18960 q^{46} - 15534 q^{49} - 67944 q^{51} - 81344 q^{54} + 29952 q^{56} + 155520 q^{59} + 35964 q^{61} - 24576 q^{64} + 180576 q^{66} + 63720 q^{69} - 9888 q^{71} - 202896 q^{74} - 11520 q^{76} - 84240 q^{79} + 293038 q^{81} - 220928 q^{84} + 314448 q^{86} - 39228 q^{89} - 79092 q^{91} - 142224 q^{94} + 45056 q^{96} - 700680 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.00000i − 0.707107i
\(3\) 27.8905i 1.78918i 0.446892 + 0.894588i \(0.352531\pi\)
−0.446892 + 0.894588i \(0.647469\pi\)
\(4\) −16.0000 −0.500000
\(5\) 0 0
\(6\) 111.562 1.26514
\(7\) − 240.426i − 1.85454i −0.374397 0.927269i \(-0.622150\pi\)
0.374397 0.927269i \(-0.377850\pi\)
\(8\) 64.0000i 0.353553i
\(9\) −534.880 −2.20115
\(10\) 0 0
\(11\) 544.151 1.35593 0.677966 0.735093i \(-0.262862\pi\)
0.677966 + 0.735093i \(0.262862\pi\)
\(12\) − 446.248i − 0.894588i
\(13\) − 169.000i − 0.277350i
\(14\) −961.702 −1.31136
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 1629.30i 1.36735i 0.729788 + 0.683673i \(0.239619\pi\)
−0.729788 + 0.683673i \(0.760381\pi\)
\(18\) 2139.52i 1.55645i
\(19\) 805.920 0.512162 0.256081 0.966655i \(-0.417569\pi\)
0.256081 + 0.966655i \(0.417569\pi\)
\(20\) 0 0
\(21\) 6705.59 3.31809
\(22\) − 2176.61i − 0.958789i
\(23\) − 373.152i − 0.147084i −0.997292 0.0735421i \(-0.976570\pi\)
0.997292 0.0735421i \(-0.0234303\pi\)
\(24\) −1784.99 −0.632569
\(25\) 0 0
\(26\) −676.000 −0.196116
\(27\) − 8140.67i − 2.14907i
\(28\) 3846.81i 0.927269i
\(29\) −1503.62 −0.332003 −0.166001 0.986126i \(-0.553086\pi\)
−0.166001 + 0.986126i \(0.553086\pi\)
\(30\) 0 0
\(31\) −2200.08 −0.411182 −0.205591 0.978638i \(-0.565912\pi\)
−0.205591 + 0.978638i \(0.565912\pi\)
\(32\) − 1024.00i − 0.176777i
\(33\) 15176.7i 2.42600i
\(34\) 6517.20 0.966860
\(35\) 0 0
\(36\) 8558.08 1.10058
\(37\) − 13109.1i − 1.57423i −0.616804 0.787117i \(-0.711573\pi\)
0.616804 0.787117i \(-0.288427\pi\)
\(38\) − 3223.68i − 0.362154i
\(39\) 4713.49 0.496228
\(40\) 0 0
\(41\) −17099.9 −1.58867 −0.794337 0.607477i \(-0.792182\pi\)
−0.794337 + 0.607477i \(0.792182\pi\)
\(42\) − 26822.4i − 2.34625i
\(43\) 8935.58i 0.736973i 0.929633 + 0.368487i \(0.120124\pi\)
−0.929633 + 0.368487i \(0.879876\pi\)
\(44\) −8706.42 −0.677966
\(45\) 0 0
\(46\) −1492.61 −0.104004
\(47\) 15749.7i 1.03999i 0.854170 + 0.519995i \(0.174066\pi\)
−0.854170 + 0.519995i \(0.825934\pi\)
\(48\) 7139.97i 0.447294i
\(49\) −40997.5 −2.43931
\(50\) 0 0
\(51\) −45442.0 −2.44642
\(52\) 2704.00i 0.138675i
\(53\) − 40379.7i − 1.97457i −0.158951 0.987286i \(-0.550811\pi\)
0.158951 0.987286i \(-0.449189\pi\)
\(54\) −32562.7 −1.51962
\(55\) 0 0
\(56\) 15387.2 0.655678
\(57\) 22477.5i 0.916349i
\(58\) 6014.46i 0.234761i
\(59\) 47562.1 1.77881 0.889407 0.457116i \(-0.151118\pi\)
0.889407 + 0.457116i \(0.151118\pi\)
\(60\) 0 0
\(61\) −30280.0 −1.04191 −0.520956 0.853584i \(-0.674424\pi\)
−0.520956 + 0.853584i \(0.674424\pi\)
\(62\) 8800.31i 0.290749i
\(63\) 128599.i 4.08212i
\(64\) −4096.00 −0.125000
\(65\) 0 0
\(66\) 60706.6 1.71544
\(67\) 38769.4i 1.05512i 0.849517 + 0.527561i \(0.176893\pi\)
−0.849517 + 0.527561i \(0.823107\pi\)
\(68\) − 26068.8i − 0.683673i
\(69\) 10407.4 0.263160
\(70\) 0 0
\(71\) −10519.7 −0.247660 −0.123830 0.992303i \(-0.539518\pi\)
−0.123830 + 0.992303i \(0.539518\pi\)
\(72\) − 34232.3i − 0.778225i
\(73\) − 1582.12i − 0.0347483i −0.999849 0.0173742i \(-0.994469\pi\)
0.999849 0.0173742i \(-0.00553064\pi\)
\(74\) −52436.5 −1.11315
\(75\) 0 0
\(76\) −12894.7 −0.256081
\(77\) − 130828.i − 2.51463i
\(78\) − 18854.0i − 0.350886i
\(79\) 6191.23 0.111612 0.0558058 0.998442i \(-0.482227\pi\)
0.0558058 + 0.998442i \(0.482227\pi\)
\(80\) 0 0
\(81\) 97071.7 1.64392
\(82\) 68399.8i 1.12336i
\(83\) − 37849.2i − 0.603062i −0.953456 0.301531i \(-0.902502\pi\)
0.953456 0.301531i \(-0.0974977\pi\)
\(84\) −107289. −1.65905
\(85\) 0 0
\(86\) 35742.3 0.521119
\(87\) − 41936.6i − 0.594011i
\(88\) 34825.7i 0.479394i
\(89\) −49151.0 −0.657744 −0.328872 0.944374i \(-0.606669\pi\)
−0.328872 + 0.944374i \(0.606669\pi\)
\(90\) 0 0
\(91\) −40631.9 −0.514356
\(92\) 5970.43i 0.0735421i
\(93\) − 61361.3i − 0.735677i
\(94\) 62999.0 0.735383
\(95\) 0 0
\(96\) 28559.9 0.316285
\(97\) − 15654.2i − 0.168928i −0.996427 0.0844641i \(-0.973082\pi\)
0.996427 0.0844641i \(-0.0269178\pi\)
\(98\) 163990.i 1.72485i
\(99\) −291056. −2.98461
\(100\) 0 0
\(101\) −138108. −1.34714 −0.673572 0.739122i \(-0.735241\pi\)
−0.673572 + 0.739122i \(0.735241\pi\)
\(102\) 181768.i 1.72988i
\(103\) 47642.1i 0.442484i 0.975219 + 0.221242i \(0.0710111\pi\)
−0.975219 + 0.221242i \(0.928989\pi\)
\(104\) 10816.0 0.0980581
\(105\) 0 0
\(106\) −161519. −1.39623
\(107\) − 186527.i − 1.57500i −0.616313 0.787501i \(-0.711374\pi\)
0.616313 0.787501i \(-0.288626\pi\)
\(108\) 130251.i 1.07454i
\(109\) −14407.1 −0.116147 −0.0580736 0.998312i \(-0.518496\pi\)
−0.0580736 + 0.998312i \(0.518496\pi\)
\(110\) 0 0
\(111\) 365620. 2.81658
\(112\) − 61548.9i − 0.463634i
\(113\) 50802.0i 0.374270i 0.982334 + 0.187135i \(0.0599201\pi\)
−0.982334 + 0.187135i \(0.940080\pi\)
\(114\) 89910.0 0.647957
\(115\) 0 0
\(116\) 24057.8 0.166001
\(117\) 90394.7i 0.610490i
\(118\) − 190248.i − 1.25781i
\(119\) 391725. 2.53580
\(120\) 0 0
\(121\) 135050. 0.838552
\(122\) 121120.i 0.736742i
\(123\) − 476926.i − 2.84242i
\(124\) 35201.3 0.205591
\(125\) 0 0
\(126\) 514395. 2.88649
\(127\) − 182278.i − 1.00282i −0.865209 0.501411i \(-0.832814\pi\)
0.865209 0.501411i \(-0.167186\pi\)
\(128\) 16384.0i 0.0883883i
\(129\) −249218. −1.31858
\(130\) 0 0
\(131\) 199714. 1.01679 0.508394 0.861124i \(-0.330239\pi\)
0.508394 + 0.861124i \(0.330239\pi\)
\(132\) − 242826.i − 1.21300i
\(133\) − 193764.i − 0.949824i
\(134\) 155078. 0.746083
\(135\) 0 0
\(136\) −104275. −0.483430
\(137\) − 345418.i − 1.57233i −0.618017 0.786165i \(-0.712064\pi\)
0.618017 0.786165i \(-0.287936\pi\)
\(138\) − 41629.6i − 0.186082i
\(139\) −183088. −0.803754 −0.401877 0.915694i \(-0.631642\pi\)
−0.401877 + 0.915694i \(0.631642\pi\)
\(140\) 0 0
\(141\) −439268. −1.86072
\(142\) 42078.7i 0.175122i
\(143\) − 91961.6i − 0.376068i
\(144\) −136929. −0.550288
\(145\) 0 0
\(146\) −6328.50 −0.0245708
\(147\) − 1.14344e6i − 4.36435i
\(148\) 209746.i 0.787117i
\(149\) 187082. 0.690344 0.345172 0.938540i \(-0.387821\pi\)
0.345172 + 0.938540i \(0.387821\pi\)
\(150\) 0 0
\(151\) −10616.3 −0.0378907 −0.0189454 0.999821i \(-0.506031\pi\)
−0.0189454 + 0.999821i \(0.506031\pi\)
\(152\) 51578.9i 0.181077i
\(153\) − 871480.i − 3.00974i
\(154\) −523312. −1.77811
\(155\) 0 0
\(156\) −75415.9 −0.248114
\(157\) − 335884.i − 1.08753i −0.839238 0.543764i \(-0.816999\pi\)
0.839238 0.543764i \(-0.183001\pi\)
\(158\) − 24764.9i − 0.0789213i
\(159\) 1.12621e6 3.53286
\(160\) 0 0
\(161\) −89715.3 −0.272773
\(162\) − 388287.i − 1.16242i
\(163\) − 201881.i − 0.595150i −0.954698 0.297575i \(-0.903822\pi\)
0.954698 0.297575i \(-0.0961778\pi\)
\(164\) 273599. 0.794337
\(165\) 0 0
\(166\) −151397. −0.426429
\(167\) − 446050.i − 1.23764i −0.785535 0.618818i \(-0.787612\pi\)
0.785535 0.618818i \(-0.212388\pi\)
\(168\) 429158.i 1.17312i
\(169\) −28561.0 −0.0769231
\(170\) 0 0
\(171\) −431070. −1.12735
\(172\) − 142969.i − 0.368487i
\(173\) − 57408.8i − 0.145835i −0.997338 0.0729177i \(-0.976769\pi\)
0.997338 0.0729177i \(-0.0232311\pi\)
\(174\) −167746. −0.420030
\(175\) 0 0
\(176\) 139303. 0.338983
\(177\) 1.32653e6i 3.18261i
\(178\) 196604.i 0.465095i
\(179\) −87439.4 −0.203974 −0.101987 0.994786i \(-0.532520\pi\)
−0.101987 + 0.994786i \(0.532520\pi\)
\(180\) 0 0
\(181\) 21341.6 0.0484206 0.0242103 0.999707i \(-0.492293\pi\)
0.0242103 + 0.999707i \(0.492293\pi\)
\(182\) 162528.i 0.363705i
\(183\) − 844523.i − 1.86416i
\(184\) 23881.7 0.0520021
\(185\) 0 0
\(186\) −245445. −0.520202
\(187\) 886586.i 1.85403i
\(188\) − 251996.i − 0.519995i
\(189\) −1.95723e6 −3.98553
\(190\) 0 0
\(191\) 307517. 0.609938 0.304969 0.952362i \(-0.401354\pi\)
0.304969 + 0.952362i \(0.401354\pi\)
\(192\) − 114239.i − 0.223647i
\(193\) − 182409.i − 0.352496i −0.984346 0.176248i \(-0.943604\pi\)
0.984346 0.176248i \(-0.0563960\pi\)
\(194\) −62616.9 −0.119450
\(195\) 0 0
\(196\) 655959. 1.21965
\(197\) − 152508.i − 0.279980i −0.990153 0.139990i \(-0.955293\pi\)
0.990153 0.139990i \(-0.0447070\pi\)
\(198\) 1.16422e6i 2.11044i
\(199\) 586606. 1.05006 0.525029 0.851084i \(-0.324054\pi\)
0.525029 + 0.851084i \(0.324054\pi\)
\(200\) 0 0
\(201\) −1.08130e6 −1.88780
\(202\) 552430.i 0.952574i
\(203\) 361508.i 0.615711i
\(204\) 727072. 1.22321
\(205\) 0 0
\(206\) 190568. 0.312884
\(207\) 199592.i 0.323755i
\(208\) − 43264.0i − 0.0693375i
\(209\) 438542. 0.694458
\(210\) 0 0
\(211\) −207041. −0.320147 −0.160073 0.987105i \(-0.551173\pi\)
−0.160073 + 0.987105i \(0.551173\pi\)
\(212\) 646075.i 0.987286i
\(213\) − 293399.i − 0.443108i
\(214\) −746106. −1.11369
\(215\) 0 0
\(216\) 521003. 0.759812
\(217\) 528955.i 0.762552i
\(218\) 57628.2i 0.0821285i
\(219\) 44126.3 0.0621708
\(220\) 0 0
\(221\) 275352. 0.379234
\(222\) − 1.46248e6i − 1.99162i
\(223\) 1.04889e6i 1.41243i 0.707998 + 0.706214i \(0.249599\pi\)
−0.707998 + 0.706214i \(0.750401\pi\)
\(224\) −246196. −0.327839
\(225\) 0 0
\(226\) 203208. 0.264649
\(227\) − 100709.i − 0.129719i −0.997894 0.0648593i \(-0.979340\pi\)
0.997894 0.0648593i \(-0.0206599\pi\)
\(228\) − 359640.i − 0.458174i
\(229\) −487174. −0.613896 −0.306948 0.951726i \(-0.599308\pi\)
−0.306948 + 0.951726i \(0.599308\pi\)
\(230\) 0 0
\(231\) 3.64886e6 4.49911
\(232\) − 96231.4i − 0.117381i
\(233\) 832844.i 1.00502i 0.864572 + 0.502509i \(0.167590\pi\)
−0.864572 + 0.502509i \(0.832410\pi\)
\(234\) 361579. 0.431681
\(235\) 0 0
\(236\) −760993. −0.889407
\(237\) 172677.i 0.199693i
\(238\) − 1.56690e6i − 1.79308i
\(239\) −1.34919e6 −1.52785 −0.763923 0.645308i \(-0.776729\pi\)
−0.763923 + 0.645308i \(0.776729\pi\)
\(240\) 0 0
\(241\) −1.66283e6 −1.84419 −0.922096 0.386962i \(-0.873524\pi\)
−0.922096 + 0.386962i \(0.873524\pi\)
\(242\) − 540199.i − 0.592946i
\(243\) 729193.i 0.792185i
\(244\) 484479. 0.520956
\(245\) 0 0
\(246\) −1.90770e6 −2.00989
\(247\) − 136200.i − 0.142048i
\(248\) − 140805.i − 0.145375i
\(249\) 1.05563e6 1.07898
\(250\) 0 0
\(251\) −291273. −0.291820 −0.145910 0.989298i \(-0.546611\pi\)
−0.145910 + 0.989298i \(0.546611\pi\)
\(252\) − 2.05758e6i − 2.04106i
\(253\) − 203051.i − 0.199436i
\(254\) −729110. −0.709102
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) − 548204.i − 0.517737i −0.965913 0.258868i \(-0.916650\pi\)
0.965913 0.258868i \(-0.0833496\pi\)
\(258\) 996872.i 0.932374i
\(259\) −3.15177e6 −2.91947
\(260\) 0 0
\(261\) 804253. 0.730788
\(262\) − 798857.i − 0.718978i
\(263\) − 818412.i − 0.729596i −0.931087 0.364798i \(-0.881138\pi\)
0.931087 0.364798i \(-0.118862\pi\)
\(264\) −971306. −0.857721
\(265\) 0 0
\(266\) −775055. −0.671627
\(267\) − 1.37085e6i − 1.17682i
\(268\) − 620311.i − 0.527561i
\(269\) 939455. 0.791581 0.395790 0.918341i \(-0.370471\pi\)
0.395790 + 0.918341i \(0.370471\pi\)
\(270\) 0 0
\(271\) −1.52416e6 −1.26069 −0.630345 0.776315i \(-0.717087\pi\)
−0.630345 + 0.776315i \(0.717087\pi\)
\(272\) 417101.i 0.341837i
\(273\) − 1.13324e6i − 0.920274i
\(274\) −1.38167e6 −1.11180
\(275\) 0 0
\(276\) −166518. −0.131580
\(277\) − 439704.i − 0.344319i −0.985069 0.172159i \(-0.944926\pi\)
0.985069 0.172159i \(-0.0550744\pi\)
\(278\) 732352.i 0.568340i
\(279\) 1.17678e6 0.905074
\(280\) 0 0
\(281\) −592033. −0.447280 −0.223640 0.974672i \(-0.571794\pi\)
−0.223640 + 0.974672i \(0.571794\pi\)
\(282\) 1.75707e6i 1.31573i
\(283\) 777226.i 0.576875i 0.957499 + 0.288437i \(0.0931357\pi\)
−0.957499 + 0.288437i \(0.906864\pi\)
\(284\) 168315. 0.123830
\(285\) 0 0
\(286\) −367846. −0.265920
\(287\) 4.11126e6i 2.94626i
\(288\) 547717.i 0.389112i
\(289\) −1.23476e6 −0.869637
\(290\) 0 0
\(291\) 436604. 0.302242
\(292\) 25314.0i 0.0173742i
\(293\) − 609025.i − 0.414444i −0.978294 0.207222i \(-0.933558\pi\)
0.978294 0.207222i \(-0.0664423\pi\)
\(294\) −4.57376e6 −3.08606
\(295\) 0 0
\(296\) 838984. 0.556576
\(297\) − 4.42976e6i − 2.91400i
\(298\) − 748326.i − 0.488147i
\(299\) −63062.7 −0.0407938
\(300\) 0 0
\(301\) 2.14834e6 1.36674
\(302\) 42465.4i 0.0267928i
\(303\) − 3.85189e6i − 2.41028i
\(304\) 206315. 0.128041
\(305\) 0 0
\(306\) −3.48592e6 −2.12821
\(307\) − 2.66761e6i − 1.61538i −0.589604 0.807692i \(-0.700716\pi\)
0.589604 0.807692i \(-0.299284\pi\)
\(308\) 2.09325e6i 1.25731i
\(309\) −1.32876e6 −0.791682
\(310\) 0 0
\(311\) −1.62320e6 −0.951634 −0.475817 0.879544i \(-0.657847\pi\)
−0.475817 + 0.879544i \(0.657847\pi\)
\(312\) 301664.i 0.175443i
\(313\) − 592115.i − 0.341622i −0.985304 0.170811i \(-0.945361\pi\)
0.985304 0.170811i \(-0.0546387\pi\)
\(314\) −1.34354e6 −0.768998
\(315\) 0 0
\(316\) −99059.7 −0.0558058
\(317\) − 2.61036e6i − 1.45899i −0.683987 0.729494i \(-0.739756\pi\)
0.683987 0.729494i \(-0.260244\pi\)
\(318\) − 4.50484e6i − 2.49811i
\(319\) −818194. −0.450173
\(320\) 0 0
\(321\) 5.20232e6 2.81796
\(322\) 358861.i 0.192880i
\(323\) 1.31308e6i 0.700304i
\(324\) −1.55315e6 −0.821959
\(325\) 0 0
\(326\) −807524. −0.420834
\(327\) − 401820.i − 0.207808i
\(328\) − 1.09440e6i − 0.561681i
\(329\) 3.78664e6 1.92870
\(330\) 0 0
\(331\) 818785. 0.410771 0.205386 0.978681i \(-0.434155\pi\)
0.205386 + 0.978681i \(0.434155\pi\)
\(332\) 605588.i 0.301531i
\(333\) 7.01180e6i 3.46513i
\(334\) −1.78420e6 −0.875140
\(335\) 0 0
\(336\) 1.71663e6 0.829523
\(337\) 3.28910e6i 1.57762i 0.614638 + 0.788809i \(0.289302\pi\)
−0.614638 + 0.788809i \(0.710698\pi\)
\(338\) 114244.i 0.0543928i
\(339\) −1.41689e6 −0.669634
\(340\) 0 0
\(341\) −1.19718e6 −0.557535
\(342\) 1.72428e6i 0.797155i
\(343\) 5.81600e6i 2.66925i
\(344\) −571877. −0.260559
\(345\) 0 0
\(346\) −229635. −0.103121
\(347\) 139108.i 0.0620195i 0.999519 + 0.0310098i \(0.00987229\pi\)
−0.999519 + 0.0310098i \(0.990128\pi\)
\(348\) 670985.i 0.297006i
\(349\) −1.41781e6 −0.623097 −0.311549 0.950230i \(-0.600848\pi\)
−0.311549 + 0.950230i \(0.600848\pi\)
\(350\) 0 0
\(351\) −1.37577e6 −0.596045
\(352\) − 557211.i − 0.239697i
\(353\) − 2.45675e6i − 1.04936i −0.851300 0.524679i \(-0.824185\pi\)
0.851300 0.524679i \(-0.175815\pi\)
\(354\) 5.30612e6 2.25045
\(355\) 0 0
\(356\) 786416. 0.328872
\(357\) 1.09254e7i 4.53698i
\(358\) 349758.i 0.144231i
\(359\) 2.88601e6 1.18185 0.590924 0.806727i \(-0.298763\pi\)
0.590924 + 0.806727i \(0.298763\pi\)
\(360\) 0 0
\(361\) −1.82659e6 −0.737690
\(362\) − 85366.4i − 0.0342386i
\(363\) 3.76660e6i 1.50032i
\(364\) 650111. 0.257178
\(365\) 0 0
\(366\) −3.37809e6 −1.31816
\(367\) − 2.83587e6i − 1.09906i −0.835474 0.549530i \(-0.814807\pi\)
0.835474 0.549530i \(-0.185193\pi\)
\(368\) − 95526.9i − 0.0367711i
\(369\) 9.14641e6 3.49691
\(370\) 0 0
\(371\) −9.70831e6 −3.66192
\(372\) 981781.i 0.367838i
\(373\) − 5.02691e6i − 1.87081i −0.353583 0.935403i \(-0.615037\pi\)
0.353583 0.935403i \(-0.384963\pi\)
\(374\) 3.54634e6 1.31100
\(375\) 0 0
\(376\) −1.00798e6 −0.367692
\(377\) 254111.i 0.0920810i
\(378\) 7.82891e6i 2.81820i
\(379\) −2.14892e6 −0.768462 −0.384231 0.923237i \(-0.625533\pi\)
−0.384231 + 0.923237i \(0.625533\pi\)
\(380\) 0 0
\(381\) 5.08381e6 1.79423
\(382\) − 1.23007e6i − 0.431291i
\(383\) − 832971.i − 0.290157i −0.989420 0.145078i \(-0.953657\pi\)
0.989420 0.145078i \(-0.0463434\pi\)
\(384\) −456958. −0.158142
\(385\) 0 0
\(386\) −729637. −0.249252
\(387\) − 4.77946e6i − 1.62219i
\(388\) 250468.i 0.0844641i
\(389\) 725243. 0.243002 0.121501 0.992591i \(-0.461229\pi\)
0.121501 + 0.992591i \(0.461229\pi\)
\(390\) 0 0
\(391\) 607976. 0.201115
\(392\) − 2.62384e6i − 0.862426i
\(393\) 5.57013e6i 1.81921i
\(394\) −610032. −0.197976
\(395\) 0 0
\(396\) 4.65689e6 1.49231
\(397\) 2.42654e6i 0.772701i 0.922352 + 0.386351i \(0.126265\pi\)
−0.922352 + 0.386351i \(0.873735\pi\)
\(398\) − 2.34642e6i − 0.742504i
\(399\) 5.40417e6 1.69940
\(400\) 0 0
\(401\) −2.14376e6 −0.665758 −0.332879 0.942970i \(-0.608020\pi\)
−0.332879 + 0.942970i \(0.608020\pi\)
\(402\) 4.32519e6i 1.33487i
\(403\) 371813.i 0.114041i
\(404\) 2.20972e6 0.673572
\(405\) 0 0
\(406\) 1.44603e6 0.435374
\(407\) − 7.13335e6i − 2.13455i
\(408\) − 2.90829e6i − 0.864942i
\(409\) 3.66144e6 1.08229 0.541145 0.840929i \(-0.317991\pi\)
0.541145 + 0.840929i \(0.317991\pi\)
\(410\) 0 0
\(411\) 9.63388e6 2.81317
\(412\) − 762273.i − 0.221242i
\(413\) − 1.14351e7i − 3.29888i
\(414\) 798366. 0.228929
\(415\) 0 0
\(416\) −173056. −0.0490290
\(417\) − 5.10642e6i − 1.43806i
\(418\) − 1.75417e6i − 0.491056i
\(419\) −3.03786e6 −0.845341 −0.422671 0.906283i \(-0.638907\pi\)
−0.422671 + 0.906283i \(0.638907\pi\)
\(420\) 0 0
\(421\) −1.88663e6 −0.518777 −0.259389 0.965773i \(-0.583521\pi\)
−0.259389 + 0.965773i \(0.583521\pi\)
\(422\) 828162.i 0.226378i
\(423\) − 8.42422e6i − 2.28917i
\(424\) 2.58430e6 0.698117
\(425\) 0 0
\(426\) −1.17360e6 −0.313325
\(427\) 7.28008e6i 1.93226i
\(428\) 2.98442e6i 0.787501i
\(429\) 2.56485e6 0.672852
\(430\) 0 0
\(431\) 717193. 0.185970 0.0929850 0.995668i \(-0.470359\pi\)
0.0929850 + 0.995668i \(0.470359\pi\)
\(432\) − 2.08401e6i − 0.537268i
\(433\) − 2.64569e6i − 0.678140i −0.940761 0.339070i \(-0.889888\pi\)
0.940761 0.339070i \(-0.110112\pi\)
\(434\) 2.11582e6 0.539206
\(435\) 0 0
\(436\) 230513. 0.0580736
\(437\) − 300731.i − 0.0753310i
\(438\) − 176505.i − 0.0439614i
\(439\) −4.47973e6 −1.10941 −0.554704 0.832048i \(-0.687168\pi\)
−0.554704 + 0.832048i \(0.687168\pi\)
\(440\) 0 0
\(441\) 2.19287e7 5.36929
\(442\) − 1.10141e6i − 0.268159i
\(443\) − 4.30731e6i − 1.04279i −0.853315 0.521395i \(-0.825412\pi\)
0.853315 0.521395i \(-0.174588\pi\)
\(444\) −5.84992e6 −1.40829
\(445\) 0 0
\(446\) 4.19555e6 0.998738
\(447\) 5.21780e6i 1.23515i
\(448\) 984783.i 0.231817i
\(449\) 25272.9 0.00591615 0.00295808 0.999996i \(-0.499058\pi\)
0.00295808 + 0.999996i \(0.499058\pi\)
\(450\) 0 0
\(451\) −9.30495e6 −2.15413
\(452\) − 812832.i − 0.187135i
\(453\) − 296095.i − 0.0677932i
\(454\) −402835. −0.0917250
\(455\) 0 0
\(456\) −1.43856e6 −0.323978
\(457\) 1.14360e6i 0.256143i 0.991765 + 0.128072i \(0.0408787\pi\)
−0.991765 + 0.128072i \(0.959121\pi\)
\(458\) 1.94869e6i 0.434090i
\(459\) 1.32636e7 2.93853
\(460\) 0 0
\(461\) −8.34911e6 −1.82973 −0.914867 0.403755i \(-0.867705\pi\)
−0.914867 + 0.403755i \(0.867705\pi\)
\(462\) − 1.45954e7i − 3.18135i
\(463\) 3.41596e6i 0.740561i 0.928920 + 0.370280i \(0.120738\pi\)
−0.928920 + 0.370280i \(0.879262\pi\)
\(464\) −384925. −0.0830007
\(465\) 0 0
\(466\) 3.33138e6 0.710655
\(467\) 5.67105e6i 1.20329i 0.798763 + 0.601646i \(0.205488\pi\)
−0.798763 + 0.601646i \(0.794512\pi\)
\(468\) − 1.44632e6i − 0.305245i
\(469\) 9.32116e6 1.95676
\(470\) 0 0
\(471\) 9.36797e6 1.94578
\(472\) 3.04397e6i 0.628906i
\(473\) 4.86231e6i 0.999286i
\(474\) 690706. 0.141204
\(475\) 0 0
\(476\) −6.26760e6 −1.26790
\(477\) 2.15983e7i 4.34633i
\(478\) 5.39677e6i 1.08035i
\(479\) 6.16340e6 1.22739 0.613694 0.789544i \(-0.289683\pi\)
0.613694 + 0.789544i \(0.289683\pi\)
\(480\) 0 0
\(481\) −2.21544e6 −0.436614
\(482\) 6.65133e6i 1.30404i
\(483\) − 2.50220e6i − 0.488039i
\(484\) −2.16080e6 −0.419276
\(485\) 0 0
\(486\) 2.91677e6 0.560160
\(487\) − 7.32701e6i − 1.39992i −0.714180 0.699962i \(-0.753200\pi\)
0.714180 0.699962i \(-0.246800\pi\)
\(488\) − 1.93792e6i − 0.368371i
\(489\) 5.63056e6 1.06483
\(490\) 0 0
\(491\) 8.89739e6 1.66555 0.832777 0.553608i \(-0.186750\pi\)
0.832777 + 0.553608i \(0.186750\pi\)
\(492\) 7.63081e6i 1.42121i
\(493\) − 2.44984e6i − 0.453963i
\(494\) −544802. −0.100443
\(495\) 0 0
\(496\) −563220. −0.102795
\(497\) 2.52920e6i 0.459295i
\(498\) − 4.22254e6i − 0.762957i
\(499\) −3.02237e6 −0.543371 −0.271686 0.962386i \(-0.587581\pi\)
−0.271686 + 0.962386i \(0.587581\pi\)
\(500\) 0 0
\(501\) 1.24406e7 2.21435
\(502\) 1.16509e6i 0.206348i
\(503\) 7.48799e6i 1.31961i 0.751437 + 0.659805i \(0.229361\pi\)
−0.751437 + 0.659805i \(0.770639\pi\)
\(504\) −8.23032e6 −1.44325
\(505\) 0 0
\(506\) −812205. −0.141023
\(507\) − 796581.i − 0.137629i
\(508\) 2.91644e6i 0.501411i
\(509\) −4.87997e6 −0.834878 −0.417439 0.908705i \(-0.637072\pi\)
−0.417439 + 0.908705i \(0.637072\pi\)
\(510\) 0 0
\(511\) −380383. −0.0644420
\(512\) − 262144.i − 0.0441942i
\(513\) − 6.56073e6i − 1.10067i
\(514\) −2.19281e6 −0.366095
\(515\) 0 0
\(516\) 3.98749e6 0.659288
\(517\) 8.57024e6i 1.41015i
\(518\) 1.26071e7i 2.06438i
\(519\) 1.60116e6 0.260925
\(520\) 0 0
\(521\) −3.79679e6 −0.612805 −0.306403 0.951902i \(-0.599125\pi\)
−0.306403 + 0.951902i \(0.599125\pi\)
\(522\) − 3.21701e6i − 0.516745i
\(523\) 5.66643e6i 0.905848i 0.891549 + 0.452924i \(0.149619\pi\)
−0.891549 + 0.452924i \(0.850381\pi\)
\(524\) −3.19543e6 −0.508394
\(525\) 0 0
\(526\) −3.27365e6 −0.515902
\(527\) − 3.58459e6i − 0.562228i
\(528\) 3.88522e6i 0.606500i
\(529\) 6.29710e6 0.978366
\(530\) 0 0
\(531\) −2.54400e7 −3.91544
\(532\) 3.10022e6i 0.474912i
\(533\) 2.88989e6i 0.440619i
\(534\) −5.48338e6 −0.832138
\(535\) 0 0
\(536\) −2.48124e6 −0.373042
\(537\) − 2.43873e6i − 0.364945i
\(538\) − 3.75782e6i − 0.559732i
\(539\) −2.23088e7 −3.30754
\(540\) 0 0
\(541\) 1.15648e7 1.69881 0.849403 0.527745i \(-0.176962\pi\)
0.849403 + 0.527745i \(0.176962\pi\)
\(542\) 6.09665e6i 0.891442i
\(543\) 595228.i 0.0866330i
\(544\) 1.66840e6 0.241715
\(545\) 0 0
\(546\) −4.53298e6 −0.650732
\(547\) − 1.28104e7i − 1.83060i −0.402776 0.915299i \(-0.631955\pi\)
0.402776 0.915299i \(-0.368045\pi\)
\(548\) 5.52669e6i 0.786165i
\(549\) 1.61961e7 2.29340
\(550\) 0 0
\(551\) −1.21179e6 −0.170039
\(552\) 666073.i 0.0930410i
\(553\) − 1.48853e6i − 0.206988i
\(554\) −1.75881e6 −0.243470
\(555\) 0 0
\(556\) 2.92941e6 0.401877
\(557\) 3.51737e6i 0.480374i 0.970727 + 0.240187i \(0.0772088\pi\)
−0.970727 + 0.240187i \(0.922791\pi\)
\(558\) − 4.70711e6i − 0.639984i
\(559\) 1.51011e6 0.204400
\(560\) 0 0
\(561\) −2.47273e7 −3.31719
\(562\) 2.36813e6i 0.316275i
\(563\) − 9.45258e6i − 1.25684i −0.777875 0.628419i \(-0.783702\pi\)
0.777875 0.628419i \(-0.216298\pi\)
\(564\) 7.02829e6 0.930362
\(565\) 0 0
\(566\) 3.10891e6 0.407912
\(567\) − 2.33385e7i − 3.04871i
\(568\) − 673259.i − 0.0875611i
\(569\) −1.11315e7 −1.44136 −0.720679 0.693269i \(-0.756170\pi\)
−0.720679 + 0.693269i \(0.756170\pi\)
\(570\) 0 0
\(571\) 4.92935e6 0.632702 0.316351 0.948642i \(-0.397542\pi\)
0.316351 + 0.948642i \(0.397542\pi\)
\(572\) 1.47139e6i 0.188034i
\(573\) 8.57680e6i 1.09129i
\(574\) 1.64450e7 2.08332
\(575\) 0 0
\(576\) 2.19087e6 0.275144
\(577\) 1.09437e7i 1.36843i 0.729279 + 0.684217i \(0.239856\pi\)
−0.729279 + 0.684217i \(0.760144\pi\)
\(578\) 4.93904e6i 0.614926i
\(579\) 5.08749e6 0.630677
\(580\) 0 0
\(581\) −9.09993e6 −1.11840
\(582\) − 1.74642e6i − 0.213718i
\(583\) − 2.19727e7i − 2.67739i
\(584\) 101256. 0.0122854
\(585\) 0 0
\(586\) −2.43610e6 −0.293056
\(587\) 7.96444e6i 0.954025i 0.878896 + 0.477013i \(0.158280\pi\)
−0.878896 + 0.477013i \(0.841720\pi\)
\(588\) 1.82950e7i 2.18218i
\(589\) −1.77309e6 −0.210592
\(590\) 0 0
\(591\) 4.25352e6 0.500934
\(592\) − 3.35593e6i − 0.393558i
\(593\) 1.19793e7i 1.39892i 0.714671 + 0.699460i \(0.246576\pi\)
−0.714671 + 0.699460i \(0.753424\pi\)
\(594\) −1.77190e7 −2.06051
\(595\) 0 0
\(596\) −2.99330e6 −0.345172
\(597\) 1.63607e7i 1.87874i
\(598\) 252251.i 0.0288456i
\(599\) −7.45663e6 −0.849132 −0.424566 0.905397i \(-0.639573\pi\)
−0.424566 + 0.905397i \(0.639573\pi\)
\(600\) 0 0
\(601\) −1.04196e7 −1.17670 −0.588351 0.808606i \(-0.700223\pi\)
−0.588351 + 0.808606i \(0.700223\pi\)
\(602\) − 8.59337e6i − 0.966434i
\(603\) − 2.07370e7i − 2.32248i
\(604\) 169862. 0.0189454
\(605\) 0 0
\(606\) −1.54075e7 −1.70432
\(607\) 571939.i 0.0630054i 0.999504 + 0.0315027i \(0.0100293\pi\)
−0.999504 + 0.0315027i \(0.989971\pi\)
\(608\) − 825262.i − 0.0905384i
\(609\) −1.00826e7 −1.10162
\(610\) 0 0
\(611\) 2.66171e6 0.288441
\(612\) 1.39437e7i 1.50487i
\(613\) 1.61323e7i 1.73398i 0.498323 + 0.866991i \(0.333949\pi\)
−0.498323 + 0.866991i \(0.666051\pi\)
\(614\) −1.06704e7 −1.14225
\(615\) 0 0
\(616\) 8.37299e6 0.889055
\(617\) − 6.39133e6i − 0.675894i −0.941165 0.337947i \(-0.890268\pi\)
0.941165 0.337947i \(-0.109732\pi\)
\(618\) 5.31505e6i 0.559804i
\(619\) −1.27062e7 −1.33287 −0.666435 0.745563i \(-0.732181\pi\)
−0.666435 + 0.745563i \(0.732181\pi\)
\(620\) 0 0
\(621\) −3.03771e6 −0.316095
\(622\) 6.49278e6i 0.672907i
\(623\) 1.18172e7i 1.21981i
\(624\) 1.20665e6 0.124057
\(625\) 0 0
\(626\) −2.36846e6 −0.241563
\(627\) 1.22312e7i 1.24251i
\(628\) 5.37414e6i 0.543764i
\(629\) 2.13587e7 2.15252
\(630\) 0 0
\(631\) −694354. −0.0694237 −0.0347118 0.999397i \(-0.511051\pi\)
−0.0347118 + 0.999397i \(0.511051\pi\)
\(632\) 396239.i 0.0394607i
\(633\) − 5.77447e6i − 0.572799i
\(634\) −1.04414e7 −1.03166
\(635\) 0 0
\(636\) −1.80193e7 −1.76643
\(637\) 6.92857e6i 0.676542i
\(638\) 3.27278e6i 0.318321i
\(639\) 5.62676e6 0.545138
\(640\) 0 0
\(641\) −8.71863e6 −0.838114 −0.419057 0.907960i \(-0.637639\pi\)
−0.419057 + 0.907960i \(0.637639\pi\)
\(642\) − 2.08093e7i − 1.99260i
\(643\) − 5.98737e6i − 0.571096i −0.958364 0.285548i \(-0.907824\pi\)
0.958364 0.285548i \(-0.0921756\pi\)
\(644\) 1.43544e6 0.136387
\(645\) 0 0
\(646\) 5.25234e6 0.495189
\(647\) − 1.09581e7i − 1.02914i −0.857448 0.514571i \(-0.827951\pi\)
0.857448 0.514571i \(-0.172049\pi\)
\(648\) 6.21259e6i 0.581212i
\(649\) 2.58810e7 2.41195
\(650\) 0 0
\(651\) −1.47528e7 −1.36434
\(652\) 3.23009e6i 0.297575i
\(653\) − 6.47503e6i − 0.594236i −0.954841 0.297118i \(-0.903975\pi\)
0.954841 0.297118i \(-0.0960255\pi\)
\(654\) −1.60728e6 −0.146942
\(655\) 0 0
\(656\) −4.37758e6 −0.397169
\(657\) 846247.i 0.0764863i
\(658\) − 1.51466e7i − 1.36380i
\(659\) −501723. −0.0450039 −0.0225020 0.999747i \(-0.507163\pi\)
−0.0225020 + 0.999747i \(0.507163\pi\)
\(660\) 0 0
\(661\) 4.00659e6 0.356674 0.178337 0.983969i \(-0.442928\pi\)
0.178337 + 0.983969i \(0.442928\pi\)
\(662\) − 3.27514e6i − 0.290459i
\(663\) 7.67969e6i 0.678516i
\(664\) 2.42235e6 0.213215
\(665\) 0 0
\(666\) 2.80472e7 2.45021
\(667\) 561077.i 0.0488324i
\(668\) 7.13681e6i 0.618818i
\(669\) −2.92540e7 −2.52708
\(670\) 0 0
\(671\) −1.64769e7 −1.41276
\(672\) − 6.86652e6i − 0.586562i
\(673\) − 8.17473e6i − 0.695722i −0.937546 0.347861i \(-0.886908\pi\)
0.937546 0.347861i \(-0.113092\pi\)
\(674\) 1.31564e7 1.11554
\(675\) 0 0
\(676\) 456976. 0.0384615
\(677\) 1.00935e6i 0.0846392i 0.999104 + 0.0423196i \(0.0134748\pi\)
−0.999104 + 0.0423196i \(0.986525\pi\)
\(678\) 5.66757e6i 0.473503i
\(679\) −3.76368e6 −0.313284
\(680\) 0 0
\(681\) 2.80882e6 0.232090
\(682\) 4.78870e6i 0.394237i
\(683\) − 1.99189e7i − 1.63386i −0.576739 0.816928i \(-0.695675\pi\)
0.576739 0.816928i \(-0.304325\pi\)
\(684\) 6.89712e6 0.563674
\(685\) 0 0
\(686\) 2.32640e7 1.88745
\(687\) − 1.35875e7i − 1.09837i
\(688\) 2.28751e6i 0.184243i
\(689\) −6.82417e6 −0.547648
\(690\) 0 0
\(691\) −677407. −0.0539703 −0.0269851 0.999636i \(-0.508591\pi\)
−0.0269851 + 0.999636i \(0.508591\pi\)
\(692\) 918541.i 0.0729177i
\(693\) 6.99772e7i 5.53508i
\(694\) 556432. 0.0438544
\(695\) 0 0
\(696\) 2.68394e6 0.210015
\(697\) − 2.78609e7i − 2.17227i
\(698\) 5.67126e6i 0.440596i
\(699\) −2.32284e7 −1.79815
\(700\) 0 0
\(701\) 1.04542e7 0.803518 0.401759 0.915745i \(-0.368399\pi\)
0.401759 + 0.915745i \(0.368399\pi\)
\(702\) 5.50310e6i 0.421468i
\(703\) − 1.05649e7i − 0.806263i
\(704\) −2.22884e6 −0.169492
\(705\) 0 0
\(706\) −9.82700e6 −0.742009
\(707\) 3.32046e7i 2.49833i
\(708\) − 2.12245e7i − 1.59131i
\(709\) 2.25125e7 1.68193 0.840964 0.541091i \(-0.181989\pi\)
0.840964 + 0.541091i \(0.181989\pi\)
\(710\) 0 0
\(711\) −3.31157e6 −0.245674
\(712\) − 3.14566e6i − 0.232548i
\(713\) 820964.i 0.0604784i
\(714\) 4.37017e7 3.20813
\(715\) 0 0
\(716\) 1.39903e6 0.101987
\(717\) − 3.76297e7i − 2.73358i
\(718\) − 1.15440e7i − 0.835693i
\(719\) 2.77678e6 0.200317 0.100159 0.994971i \(-0.468065\pi\)
0.100159 + 0.994971i \(0.468065\pi\)
\(720\) 0 0
\(721\) 1.14544e7 0.820603
\(722\) 7.30637e6i 0.521625i
\(723\) − 4.63772e7i − 3.29958i
\(724\) −341466. −0.0242103
\(725\) 0 0
\(726\) 1.50664e7 1.06089
\(727\) 2.76872e6i 0.194287i 0.995270 + 0.0971434i \(0.0309705\pi\)
−0.995270 + 0.0971434i \(0.969029\pi\)
\(728\) − 2.60044e6i − 0.181852i
\(729\) 3.25086e6 0.226558
\(730\) 0 0
\(731\) −1.45587e7 −1.00770
\(732\) 1.35124e7i 0.932081i
\(733\) 1.80998e7i 1.24427i 0.782911 + 0.622133i \(0.213734\pi\)
−0.782911 + 0.622133i \(0.786266\pi\)
\(734\) −1.13435e7 −0.777152
\(735\) 0 0
\(736\) −382108. −0.0260011
\(737\) 2.10964e7i 1.43067i
\(738\) − 3.65857e7i − 2.47269i
\(739\) 3.57016e6 0.240479 0.120239 0.992745i \(-0.461634\pi\)
0.120239 + 0.992745i \(0.461634\pi\)
\(740\) 0 0
\(741\) 3.79870e6 0.254149
\(742\) 3.88332e7i 2.58937i
\(743\) 1.73285e7i 1.15157i 0.817603 + 0.575783i \(0.195303\pi\)
−0.817603 + 0.575783i \(0.804697\pi\)
\(744\) 3.92712e6 0.260101
\(745\) 0 0
\(746\) −2.01076e7 −1.32286
\(747\) 2.02448e7i 1.32743i
\(748\) − 1.41854e7i − 0.927015i
\(749\) −4.48457e7 −2.92090
\(750\) 0 0
\(751\) 3.04634e6 0.197096 0.0985480 0.995132i \(-0.468580\pi\)
0.0985480 + 0.995132i \(0.468580\pi\)
\(752\) 4.03193e6i 0.259997i
\(753\) − 8.12374e6i − 0.522118i
\(754\) 1.01644e6 0.0651111
\(755\) 0 0
\(756\) 3.13156e7 1.99277
\(757\) − 8.98398e6i − 0.569808i −0.958556 0.284904i \(-0.908038\pi\)
0.958556 0.284904i \(-0.0919618\pi\)
\(758\) 8.59568e6i 0.543384i
\(759\) 5.66320e6 0.356827
\(760\) 0 0
\(761\) 1.31757e7 0.824731 0.412366 0.911018i \(-0.364703\pi\)
0.412366 + 0.911018i \(0.364703\pi\)
\(762\) − 2.03352e7i − 1.26871i
\(763\) 3.46382e6i 0.215399i
\(764\) −4.92027e6 −0.304969
\(765\) 0 0
\(766\) −3.33188e6 −0.205172
\(767\) − 8.03799e6i − 0.493354i
\(768\) 1.82783e6i 0.111824i
\(769\) −3.53661e6 −0.215661 −0.107830 0.994169i \(-0.534390\pi\)
−0.107830 + 0.994169i \(0.534390\pi\)
\(770\) 0 0
\(771\) 1.52897e7 0.926322
\(772\) 2.91855e6i 0.176248i
\(773\) 3.14397e7i 1.89247i 0.323479 + 0.946235i \(0.395147\pi\)
−0.323479 + 0.946235i \(0.604853\pi\)
\(774\) −1.91179e7 −1.14706
\(775\) 0 0
\(776\) 1.00187e6 0.0597251
\(777\) − 8.79044e7i − 5.22346i
\(778\) − 2.90097e6i − 0.171828i
\(779\) −1.37812e7 −0.813659
\(780\) 0 0
\(781\) −5.72429e6 −0.335810
\(782\) − 2.43191e6i − 0.142210i
\(783\) 1.22404e7i 0.713498i
\(784\) −1.04953e7 −0.609827
\(785\) 0 0
\(786\) 2.22805e7 1.28638
\(787\) 1.27213e7i 0.732139i 0.930588 + 0.366069i \(0.119297\pi\)
−0.930588 + 0.366069i \(0.880703\pi\)
\(788\) 2.44013e6i 0.139990i
\(789\) 2.28259e7 1.30538
\(790\) 0 0
\(791\) 1.22141e7 0.694097
\(792\) − 1.86276e7i − 1.05522i
\(793\) 5.11731e6i 0.288974i
\(794\) 9.70617e6 0.546382
\(795\) 0 0
\(796\) −9.38569e6 −0.525029
\(797\) 1.84237e7i 1.02738i 0.857977 + 0.513689i \(0.171721\pi\)
−0.857977 + 0.513689i \(0.828279\pi\)
\(798\) − 2.16167e7i − 1.20166i
\(799\) −2.56610e7 −1.42203
\(800\) 0 0
\(801\) 2.62899e7 1.44780
\(802\) 8.57506e6i 0.470762i
\(803\) − 860915.i − 0.0471164i
\(804\) 1.73008e7 0.943899
\(805\) 0 0
\(806\) 1.48725e6 0.0806394
\(807\) 2.62019e7i 1.41628i
\(808\) − 8.83888e6i − 0.476287i
\(809\) −2.10360e7 −1.13004 −0.565018 0.825079i \(-0.691131\pi\)
−0.565018 + 0.825079i \(0.691131\pi\)
\(810\) 0 0
\(811\) 3.53571e7 1.88767 0.943833 0.330424i \(-0.107192\pi\)
0.943833 + 0.330424i \(0.107192\pi\)
\(812\) − 5.78412e6i − 0.307856i
\(813\) − 4.25097e7i − 2.25560i
\(814\) −2.85334e7 −1.50936
\(815\) 0 0
\(816\) −1.16331e7 −0.611606
\(817\) 7.20136e6i 0.377450i
\(818\) − 1.46458e7i − 0.765295i
\(819\) 2.17332e7 1.13218
\(820\) 0 0
\(821\) 1.72038e7 0.890770 0.445385 0.895339i \(-0.353067\pi\)
0.445385 + 0.895339i \(0.353067\pi\)
\(822\) − 3.85355e7i − 1.98921i
\(823\) 947098.i 0.0487411i 0.999703 + 0.0243705i \(0.00775815\pi\)
−0.999703 + 0.0243705i \(0.992242\pi\)
\(824\) −3.04909e6 −0.156442
\(825\) 0 0
\(826\) −4.57405e7 −2.33266
\(827\) − 3.05385e7i − 1.55269i −0.630308 0.776345i \(-0.717072\pi\)
0.630308 0.776345i \(-0.282928\pi\)
\(828\) − 3.19346e6i − 0.161877i
\(829\) 1.57415e7 0.795538 0.397769 0.917486i \(-0.369785\pi\)
0.397769 + 0.917486i \(0.369785\pi\)
\(830\) 0 0
\(831\) 1.22636e7 0.616047
\(832\) 692224.i 0.0346688i
\(833\) − 6.67971e7i − 3.33538i
\(834\) −2.04257e7 −1.01686
\(835\) 0 0
\(836\) −7.01668e6 −0.347229
\(837\) 1.79101e7i 0.883659i
\(838\) 1.21514e7i 0.597746i
\(839\) −2.07700e7 −1.01867 −0.509333 0.860570i \(-0.670108\pi\)
−0.509333 + 0.860570i \(0.670108\pi\)
\(840\) 0 0
\(841\) −1.82503e7 −0.889774
\(842\) 7.54651e6i 0.366831i
\(843\) − 1.65121e7i − 0.800263i
\(844\) 3.31265e6 0.160073
\(845\) 0 0
\(846\) −3.36969e7 −1.61869
\(847\) − 3.24694e7i − 1.55513i
\(848\) − 1.03372e7i − 0.493643i
\(849\) −2.16772e7 −1.03213
\(850\) 0 0
\(851\) −4.89169e6 −0.231545
\(852\) 4.69438e6i 0.221554i
\(853\) 1.91838e7i 0.902740i 0.892337 + 0.451370i \(0.149064\pi\)
−0.892337 + 0.451370i \(0.850936\pi\)
\(854\) 2.91203e7 1.36632
\(855\) 0 0
\(856\) 1.19377e7 0.556847
\(857\) 2.64211e7i 1.22885i 0.788975 + 0.614425i \(0.210612\pi\)
−0.788975 + 0.614425i \(0.789388\pi\)
\(858\) − 1.02594e7i − 0.475778i
\(859\) 3.30576e7 1.52858 0.764289 0.644873i \(-0.223090\pi\)
0.764289 + 0.644873i \(0.223090\pi\)
\(860\) 0 0
\(861\) −1.14665e8 −5.27137
\(862\) − 2.86877e6i − 0.131501i
\(863\) − 1.19324e7i − 0.545380i −0.962102 0.272690i \(-0.912087\pi\)
0.962102 0.272690i \(-0.0879133\pi\)
\(864\) −8.33605e6 −0.379906
\(865\) 0 0
\(866\) −1.05828e7 −0.479518
\(867\) − 3.44381e7i − 1.55593i
\(868\) − 8.46328e6i − 0.381276i
\(869\) 3.36897e6 0.151338
\(870\) 0 0
\(871\) 6.55203e6 0.292638
\(872\) − 922051.i − 0.0410643i
\(873\) 8.37313e6i 0.371837i
\(874\) −1.20292e6 −0.0532671
\(875\) 0 0
\(876\) −706020. −0.0310854
\(877\) 3.88434e6i 0.170537i 0.996358 + 0.0852684i \(0.0271748\pi\)
−0.996358 + 0.0852684i \(0.972825\pi\)
\(878\) 1.79189e7i 0.784469i
\(879\) 1.69860e7 0.741513
\(880\) 0 0
\(881\) −8.93397e6 −0.387797 −0.193899 0.981022i \(-0.562113\pi\)
−0.193899 + 0.981022i \(0.562113\pi\)
\(882\) − 8.77148e7i − 3.79666i
\(883\) 4.32552e6i 0.186697i 0.995634 + 0.0933484i \(0.0297570\pi\)
−0.995634 + 0.0933484i \(0.970243\pi\)
\(884\) −4.40563e6 −0.189617
\(885\) 0 0
\(886\) −1.72292e7 −0.737364
\(887\) 3.55552e7i 1.51738i 0.651453 + 0.758689i \(0.274160\pi\)
−0.651453 + 0.758689i \(0.725840\pi\)
\(888\) 2.33997e7i 0.995812i
\(889\) −4.38242e7 −1.85977
\(890\) 0 0
\(891\) 5.28217e7 2.22904
\(892\) − 1.67822e7i − 0.706214i
\(893\) 1.26930e7i 0.532643i
\(894\) 2.08712e7 0.873380
\(895\) 0 0
\(896\) 3.93913e6 0.163919
\(897\) − 1.75885e6i − 0.0729874i
\(898\) − 101092.i − 0.00418335i
\(899\) 3.30807e6 0.136514
\(900\) 0 0
\(901\) 6.57906e7 2.69993
\(902\) 3.72198e7i 1.52320i
\(903\) 5.99184e7i 2.44535i
\(904\) −3.25133e6 −0.132324
\(905\) 0 0
\(906\) −1.18438e6 −0.0479370
\(907\) 2.44157e7i 0.985486i 0.870175 + 0.492743i \(0.164006\pi\)
−0.870175 + 0.492743i \(0.835994\pi\)
\(908\) 1.61134e6i 0.0648593i
\(909\) 7.38709e7 2.96527
\(910\) 0 0
\(911\) 3.00267e7 1.19870 0.599352 0.800486i \(-0.295425\pi\)
0.599352 + 0.800486i \(0.295425\pi\)
\(912\) 5.75424e6i 0.229087i
\(913\) − 2.05957e7i − 0.817711i
\(914\) 4.57439e6 0.181120
\(915\) 0 0
\(916\) 7.79478e6 0.306948
\(917\) − 4.80164e7i − 1.88567i
\(918\) − 5.30544e7i − 2.07785i
\(919\) 1.30118e7 0.508216 0.254108 0.967176i \(-0.418218\pi\)
0.254108 + 0.967176i \(0.418218\pi\)
\(920\) 0 0
\(921\) 7.44009e7 2.89021
\(922\) 3.33964e7i 1.29382i
\(923\) 1.77782e6i 0.0686886i
\(924\) −5.83817e7 −2.24956
\(925\) 0 0
\(926\) 1.36639e7 0.523656
\(927\) − 2.54828e7i − 0.973975i
\(928\) 1.53970e6i 0.0586904i
\(929\) 5.16678e7 1.96418 0.982089 0.188419i \(-0.0603362\pi\)
0.982089 + 0.188419i \(0.0603362\pi\)
\(930\) 0 0
\(931\) −3.30407e7 −1.24932
\(932\) − 1.33255e7i − 0.502509i
\(933\) − 4.52717e7i − 1.70264i
\(934\) 2.26842e7 0.850857
\(935\) 0 0
\(936\) −5.78526e6 −0.215841
\(937\) − 4.04703e6i − 0.150587i −0.997161 0.0752936i \(-0.976011\pi\)
0.997161 0.0752936i \(-0.0239894\pi\)
\(938\) − 3.72846e7i − 1.38364i
\(939\) 1.65144e7 0.611221
\(940\) 0 0
\(941\) −4.43008e7 −1.63094 −0.815470 0.578799i \(-0.803521\pi\)
−0.815470 + 0.578799i \(0.803521\pi\)
\(942\) − 3.74719e7i − 1.37587i
\(943\) 6.38088e6i 0.233669i
\(944\) 1.21759e7 0.444704
\(945\) 0 0
\(946\) 1.94492e7 0.706602
\(947\) − 3.60463e7i − 1.30613i −0.757303 0.653064i \(-0.773483\pi\)
0.757303 0.653064i \(-0.226517\pi\)
\(948\) − 2.76283e6i − 0.0998464i
\(949\) −267379. −0.00963745
\(950\) 0 0
\(951\) 7.28042e7 2.61039
\(952\) 2.50704e7i 0.896539i
\(953\) 1.31248e7i 0.468123i 0.972222 + 0.234061i \(0.0752017\pi\)
−0.972222 + 0.234061i \(0.924798\pi\)
\(954\) 8.63931e7 3.07332
\(955\) 0 0
\(956\) 2.15871e7 0.763923
\(957\) − 2.28198e7i − 0.805439i
\(958\) − 2.46536e7i − 0.867894i
\(959\) −8.30473e7 −2.91594
\(960\) 0 0
\(961\) −2.37888e7 −0.830929
\(962\) 8.86176e6i 0.308733i
\(963\) 9.97693e7i 3.46682i
\(964\) 2.66053e7 0.922096
\(965\) 0 0
\(966\) −1.00088e7 −0.345096
\(967\) 2.48150e7i 0.853390i 0.904396 + 0.426695i \(0.140322\pi\)
−0.904396 + 0.426695i \(0.859678\pi\)
\(968\) 8.64318e6i 0.296473i
\(969\) −3.66226e7 −1.25297
\(970\) 0 0
\(971\) −3.54227e6 −0.120569 −0.0602843 0.998181i \(-0.519201\pi\)
−0.0602843 + 0.998181i \(0.519201\pi\)
\(972\) − 1.16671e7i − 0.396093i
\(973\) 4.40191e7i 1.49059i
\(974\) −2.93080e7 −0.989896
\(975\) 0 0
\(976\) −7.75167e6 −0.260478
\(977\) 9.16916e6i 0.307322i 0.988124 + 0.153661i \(0.0491063\pi\)
−0.988124 + 0.153661i \(0.950894\pi\)
\(978\) − 2.25222e7i − 0.752947i
\(979\) −2.67456e7 −0.891857
\(980\) 0 0
\(981\) 7.70604e6 0.255658
\(982\) − 3.55895e7i − 1.17772i
\(983\) 3.05912e7i 1.00975i 0.863193 + 0.504874i \(0.168461\pi\)
−0.863193 + 0.504874i \(0.831539\pi\)
\(984\) 3.05233e7 1.00495
\(985\) 0 0
\(986\) −9.79936e6 −0.321000
\(987\) 1.05611e8i 3.45078i
\(988\) 2.17921e6i 0.0710242i
\(989\) 3.33433e6 0.108397
\(990\) 0 0
\(991\) 3.47852e7 1.12515 0.562575 0.826746i \(-0.309811\pi\)
0.562575 + 0.826746i \(0.309811\pi\)
\(992\) 2.25288e6i 0.0726874i
\(993\) 2.28363e7i 0.734942i
\(994\) 1.01168e7 0.324771
\(995\) 0 0
\(996\) −1.68902e7 −0.539492
\(997\) − 4.21812e7i − 1.34394i −0.740577 0.671972i \(-0.765448\pi\)
0.740577 0.671972i \(-0.234552\pi\)
\(998\) 1.20895e7i 0.384222i
\(999\) −1.06717e8 −3.38314
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 650.6.b.j.599.3 6
5.2 odd 4 650.6.a.j.1.3 3
5.3 odd 4 130.6.a.f.1.1 3
5.4 even 2 inner 650.6.b.j.599.4 6
20.3 even 4 1040.6.a.l.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
130.6.a.f.1.1 3 5.3 odd 4
650.6.a.j.1.3 3 5.2 odd 4
650.6.b.j.599.3 6 1.1 even 1 trivial
650.6.b.j.599.4 6 5.4 even 2 inner
1040.6.a.l.1.3 3 20.3 even 4