Properties

Label 147.6.a.i.1.1
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

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: yes
Analytic conductor: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{249}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 62 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(8.38987\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-9.38987 q^{2} -9.00000 q^{3} +56.1696 q^{4} -71.7291 q^{5} +84.5088 q^{6} -226.949 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-9.38987 q^{2} -9.00000 q^{3} +56.1696 q^{4} -71.7291 q^{5} +84.5088 q^{6} -226.949 q^{8} +81.0000 q^{9} +673.526 q^{10} -560.610 q^{11} -505.526 q^{12} +533.509 q^{13} +645.562 q^{15} +333.597 q^{16} +1005.70 q^{17} -760.579 q^{18} +1368.53 q^{19} -4028.99 q^{20} +5264.05 q^{22} +3228.08 q^{23} +2042.54 q^{24} +2020.06 q^{25} -5009.58 q^{26} -729.000 q^{27} -753.456 q^{29} -6061.74 q^{30} +8206.42 q^{31} +4129.95 q^{32} +5045.49 q^{33} -9443.39 q^{34} +4549.74 q^{36} -2808.66 q^{37} -12850.3 q^{38} -4801.58 q^{39} +16278.9 q^{40} +245.827 q^{41} -17504.5 q^{43} -31489.2 q^{44} -5810.05 q^{45} -30311.2 q^{46} -16345.5 q^{47} -3002.37 q^{48} -18968.1 q^{50} -9051.30 q^{51} +29967.0 q^{52} -29641.7 q^{53} +6845.21 q^{54} +40212.0 q^{55} -12316.7 q^{57} +7074.85 q^{58} -10356.1 q^{59} +36260.9 q^{60} +954.179 q^{61} -77057.2 q^{62} -49454.8 q^{64} -38268.1 q^{65} -47376.5 q^{66} -19815.2 q^{67} +56489.8 q^{68} -29052.7 q^{69} +62125.4 q^{71} -18382.9 q^{72} +27109.6 q^{73} +26373.0 q^{74} -18180.5 q^{75} +76869.6 q^{76} +45086.2 q^{78} +44687.4 q^{79} -23928.6 q^{80} +6561.00 q^{81} -2308.29 q^{82} +15606.6 q^{83} -72137.9 q^{85} +164365. q^{86} +6781.10 q^{87} +127230. q^{88} +13635.3 q^{89} +54555.6 q^{90} +181320. q^{92} -73857.8 q^{93} +153482. q^{94} -98163.1 q^{95} -37169.5 q^{96} -12919.5 q^{97} -45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{2} - 18 q^{3} + 65 q^{4} - 33 q^{5} + 27 q^{6} - 375 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 3 q^{2} - 18 q^{3} + 65 q^{4} - 33 q^{5} + 27 q^{6} - 375 q^{8} + 162 q^{9} + 921 q^{10} - 1137 q^{11} - 585 q^{12} + 925 q^{13} + 297 q^{15} - 895 q^{16} - 324 q^{17} - 243 q^{18} + 2311 q^{19} - 3687 q^{20} + 1581 q^{22} + 1596 q^{23} + 3375 q^{24} + 395 q^{25} - 2508 q^{26} - 1458 q^{27} - 2217 q^{29} - 8289 q^{30} + 4294 q^{31} + 1017 q^{32} + 10233 q^{33} - 17940 q^{34} + 5265 q^{36} - 19109 q^{37} - 6828 q^{38} - 8325 q^{39} + 10545 q^{40} - 12858 q^{41} - 2771 q^{43} - 36579 q^{44} - 2673 q^{45} - 40740 q^{46} - 23160 q^{47} + 8055 q^{48} - 29352 q^{50} + 2916 q^{51} + 33424 q^{52} - 31653 q^{53} + 2187 q^{54} + 17889 q^{55} - 20799 q^{57} - 2277 q^{58} + 41097 q^{59} + 33183 q^{60} + 42052 q^{61} - 102057 q^{62} - 30031 q^{64} - 23106 q^{65} - 14229 q^{66} + 30763 q^{67} + 44748 q^{68} - 14364 q^{69} + 102096 q^{71} - 30375 q^{72} - 28577 q^{73} - 77784 q^{74} - 3555 q^{75} + 85192 q^{76} + 22572 q^{78} - 18464 q^{79} - 71511 q^{80} + 13122 q^{81} - 86040 q^{82} + 61179 q^{83} - 123636 q^{85} + 258510 q^{86} + 19953 q^{87} + 212565 q^{88} + 29322 q^{89} + 74601 q^{90} + 166908 q^{92} - 38646 q^{93} + 109938 q^{94} - 61662 q^{95} - 9153 q^{96} - 9791 q^{97} - 92097 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.38987 −1.65991 −0.829955 0.557831i \(-0.811634\pi\)
−0.829955 + 0.557831i \(0.811634\pi\)
\(3\) −9.00000 −0.577350
\(4\) 56.1696 1.75530
\(5\) −71.7291 −1.28313 −0.641564 0.767069i \(-0.721714\pi\)
−0.641564 + 0.767069i \(0.721714\pi\)
\(6\) 84.5088 0.958349
\(7\) 0 0
\(8\) −226.949 −1.25373
\(9\) 81.0000 0.333333
\(10\) 673.526 2.12988
\(11\) −560.610 −1.39694 −0.698472 0.715637i \(-0.746137\pi\)
−0.698472 + 0.715637i \(0.746137\pi\)
\(12\) −505.526 −1.01342
\(13\) 533.509 0.875555 0.437777 0.899083i \(-0.355766\pi\)
0.437777 + 0.899083i \(0.355766\pi\)
\(14\) 0 0
\(15\) 645.562 0.740815
\(16\) 333.597 0.325778
\(17\) 1005.70 0.844007 0.422004 0.906594i \(-0.361327\pi\)
0.422004 + 0.906594i \(0.361327\pi\)
\(18\) −760.579 −0.553303
\(19\) 1368.53 0.869699 0.434850 0.900503i \(-0.356801\pi\)
0.434850 + 0.900503i \(0.356801\pi\)
\(20\) −4028.99 −2.25228
\(21\) 0 0
\(22\) 5264.05 2.31880
\(23\) 3228.08 1.27240 0.636201 0.771523i \(-0.280505\pi\)
0.636201 + 0.771523i \(0.280505\pi\)
\(24\) 2042.54 0.723841
\(25\) 2020.06 0.646419
\(26\) −5009.58 −1.45334
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −753.456 −0.166365 −0.0831827 0.996534i \(-0.526508\pi\)
−0.0831827 + 0.996534i \(0.526508\pi\)
\(30\) −6061.74 −1.22969
\(31\) 8206.42 1.53373 0.766866 0.641807i \(-0.221815\pi\)
0.766866 + 0.641807i \(0.221815\pi\)
\(32\) 4129.95 0.712968
\(33\) 5045.49 0.806526
\(34\) −9443.39 −1.40098
\(35\) 0 0
\(36\) 4549.74 0.585100
\(37\) −2808.66 −0.337284 −0.168642 0.985677i \(-0.553938\pi\)
−0.168642 + 0.985677i \(0.553938\pi\)
\(38\) −12850.3 −1.44362
\(39\) −4801.58 −0.505502
\(40\) 16278.9 1.60870
\(41\) 245.827 0.0228387 0.0114193 0.999935i \(-0.496365\pi\)
0.0114193 + 0.999935i \(0.496365\pi\)
\(42\) 0 0
\(43\) −17504.5 −1.44371 −0.721853 0.692047i \(-0.756709\pi\)
−0.721853 + 0.692047i \(0.756709\pi\)
\(44\) −31489.2 −2.45206
\(45\) −5810.05 −0.427710
\(46\) −30311.2 −2.11207
\(47\) −16345.5 −1.07933 −0.539663 0.841881i \(-0.681449\pi\)
−0.539663 + 0.841881i \(0.681449\pi\)
\(48\) −3002.37 −0.188088
\(49\) 0 0
\(50\) −18968.1 −1.07300
\(51\) −9051.30 −0.487288
\(52\) 29967.0 1.53686
\(53\) −29641.7 −1.44948 −0.724741 0.689021i \(-0.758041\pi\)
−0.724741 + 0.689021i \(0.758041\pi\)
\(54\) 6845.21 0.319450
\(55\) 40212.0 1.79246
\(56\) 0 0
\(57\) −12316.7 −0.502121
\(58\) 7074.85 0.276151
\(59\) −10356.1 −0.387317 −0.193659 0.981069i \(-0.562035\pi\)
−0.193659 + 0.981069i \(0.562035\pi\)
\(60\) 36260.9 1.30035
\(61\) 954.179 0.0328326 0.0164163 0.999865i \(-0.494774\pi\)
0.0164163 + 0.999865i \(0.494774\pi\)
\(62\) −77057.2 −2.54586
\(63\) 0 0
\(64\) −49454.8 −1.50924
\(65\) −38268.1 −1.12345
\(66\) −47376.5 −1.33876
\(67\) −19815.2 −0.539276 −0.269638 0.962962i \(-0.586904\pi\)
−0.269638 + 0.962962i \(0.586904\pi\)
\(68\) 56489.8 1.48149
\(69\) −29052.7 −0.734622
\(70\) 0 0
\(71\) 62125.4 1.46259 0.731296 0.682060i \(-0.238916\pi\)
0.731296 + 0.682060i \(0.238916\pi\)
\(72\) −18382.9 −0.417910
\(73\) 27109.6 0.595410 0.297705 0.954658i \(-0.403779\pi\)
0.297705 + 0.954658i \(0.403779\pi\)
\(74\) 26373.0 0.559861
\(75\) −18180.5 −0.373210
\(76\) 76869.6 1.52658
\(77\) 0 0
\(78\) 45086.2 0.839087
\(79\) 44687.4 0.805595 0.402798 0.915289i \(-0.368038\pi\)
0.402798 + 0.915289i \(0.368038\pi\)
\(80\) −23928.6 −0.418015
\(81\) 6561.00 0.111111
\(82\) −2308.29 −0.0379101
\(83\) 15606.6 0.248665 0.124332 0.992241i \(-0.460321\pi\)
0.124332 + 0.992241i \(0.460321\pi\)
\(84\) 0 0
\(85\) −72137.9 −1.08297
\(86\) 164365. 2.39642
\(87\) 6781.10 0.0960511
\(88\) 127230. 1.75139
\(89\) 13635.3 0.182469 0.0912347 0.995829i \(-0.470919\pi\)
0.0912347 + 0.995829i \(0.470919\pi\)
\(90\) 54555.6 0.709959
\(91\) 0 0
\(92\) 181320. 2.23345
\(93\) −73857.8 −0.885500
\(94\) 153482. 1.79159
\(95\) −98163.1 −1.11594
\(96\) −37169.5 −0.411632
\(97\) −12919.5 −0.139417 −0.0697086 0.997567i \(-0.522207\pi\)
−0.0697086 + 0.997567i \(0.522207\pi\)
\(98\) 0 0
\(99\) −45409.4 −0.465648
\(100\) 113466. 1.13466
\(101\) 25142.9 0.245252 0.122626 0.992453i \(-0.460868\pi\)
0.122626 + 0.992453i \(0.460868\pi\)
\(102\) 84990.5 0.808854
\(103\) −160753. −1.49302 −0.746511 0.665373i \(-0.768273\pi\)
−0.746511 + 0.665373i \(0.768273\pi\)
\(104\) −121079. −1.09771
\(105\) 0 0
\(106\) 278331. 2.40601
\(107\) −94375.5 −0.796893 −0.398446 0.917192i \(-0.630451\pi\)
−0.398446 + 0.917192i \(0.630451\pi\)
\(108\) −40947.6 −0.337808
\(109\) −83393.5 −0.672304 −0.336152 0.941808i \(-0.609126\pi\)
−0.336152 + 0.941808i \(0.609126\pi\)
\(110\) −377586. −2.97532
\(111\) 25278.0 0.194731
\(112\) 0 0
\(113\) −179254. −1.32060 −0.660301 0.751001i \(-0.729571\pi\)
−0.660301 + 0.751001i \(0.729571\pi\)
\(114\) 115653. 0.833476
\(115\) −231547. −1.63266
\(116\) −42321.3 −0.292021
\(117\) 43214.2 0.291852
\(118\) 97242.5 0.642911
\(119\) 0 0
\(120\) −146510. −0.928781
\(121\) 153233. 0.951455
\(122\) −8959.61 −0.0544991
\(123\) −2212.45 −0.0131859
\(124\) 460951. 2.69216
\(125\) 79256.4 0.453690
\(126\) 0 0
\(127\) −143674. −0.790440 −0.395220 0.918586i \(-0.629332\pi\)
−0.395220 + 0.918586i \(0.629332\pi\)
\(128\) 332215. 1.79223
\(129\) 157540. 0.833524
\(130\) 359332. 1.86482
\(131\) 52289.9 0.266219 0.133110 0.991101i \(-0.457504\pi\)
0.133110 + 0.991101i \(0.457504\pi\)
\(132\) 283403. 1.41570
\(133\) 0 0
\(134\) 186062. 0.895150
\(135\) 52290.5 0.246938
\(136\) −228243. −1.05816
\(137\) 9410.10 0.0428344 0.0214172 0.999771i \(-0.493182\pi\)
0.0214172 + 0.999771i \(0.493182\pi\)
\(138\) 272801. 1.21941
\(139\) 183094. 0.803781 0.401890 0.915688i \(-0.368353\pi\)
0.401890 + 0.915688i \(0.368353\pi\)
\(140\) 0 0
\(141\) 147109. 0.623150
\(142\) −583349. −2.42777
\(143\) −299090. −1.22310
\(144\) 27021.3 0.108593
\(145\) 54044.7 0.213468
\(146\) −254556. −0.988328
\(147\) 0 0
\(148\) −157762. −0.592034
\(149\) −167002. −0.616247 −0.308123 0.951346i \(-0.599701\pi\)
−0.308123 + 0.951346i \(0.599701\pi\)
\(150\) 170713. 0.619495
\(151\) 376264. 1.34292 0.671461 0.741040i \(-0.265667\pi\)
0.671461 + 0.741040i \(0.265667\pi\)
\(152\) −310586. −1.09037
\(153\) 81461.7 0.281336
\(154\) 0 0
\(155\) −588639. −1.96797
\(156\) −269703. −0.887307
\(157\) 39075.0 0.126517 0.0632587 0.997997i \(-0.479851\pi\)
0.0632587 + 0.997997i \(0.479851\pi\)
\(158\) −419608. −1.33722
\(159\) 266775. 0.836859
\(160\) −296237. −0.914829
\(161\) 0 0
\(162\) −61606.9 −0.184434
\(163\) −477919. −1.40892 −0.704458 0.709745i \(-0.748810\pi\)
−0.704458 + 0.709745i \(0.748810\pi\)
\(164\) 13808.0 0.0400887
\(165\) −361908. −1.03488
\(166\) −146544. −0.412761
\(167\) 39793.4 0.110413 0.0552064 0.998475i \(-0.482418\pi\)
0.0552064 + 0.998475i \(0.482418\pi\)
\(168\) 0 0
\(169\) −86661.4 −0.233404
\(170\) 677366. 1.79763
\(171\) 110851. 0.289900
\(172\) −983221. −2.53414
\(173\) 48338.2 0.122794 0.0613968 0.998113i \(-0.480445\pi\)
0.0613968 + 0.998113i \(0.480445\pi\)
\(174\) −63673.7 −0.159436
\(175\) 0 0
\(176\) −187018. −0.455094
\(177\) 93205.0 0.223618
\(178\) −128034. −0.302883
\(179\) −142911. −0.333374 −0.166687 0.986010i \(-0.553307\pi\)
−0.166687 + 0.986010i \(0.553307\pi\)
\(180\) −326348. −0.750759
\(181\) −77245.3 −0.175257 −0.0876285 0.996153i \(-0.527929\pi\)
−0.0876285 + 0.996153i \(0.527929\pi\)
\(182\) 0 0
\(183\) −8587.61 −0.0189559
\(184\) −732610. −1.59525
\(185\) 201463. 0.432778
\(186\) 693515. 1.46985
\(187\) −563806. −1.17903
\(188\) −918119. −1.89454
\(189\) 0 0
\(190\) 921739. 1.85235
\(191\) 272054. 0.539600 0.269800 0.962916i \(-0.413042\pi\)
0.269800 + 0.962916i \(0.413042\pi\)
\(192\) 445093. 0.871360
\(193\) −16033.8 −0.0309844 −0.0154922 0.999880i \(-0.504932\pi\)
−0.0154922 + 0.999880i \(0.504932\pi\)
\(194\) 121312. 0.231420
\(195\) 344413. 0.648624
\(196\) 0 0
\(197\) 1.03228e6 1.89510 0.947552 0.319603i \(-0.103549\pi\)
0.947552 + 0.319603i \(0.103549\pi\)
\(198\) 426388. 0.772934
\(199\) −881736. −1.57836 −0.789180 0.614162i \(-0.789494\pi\)
−0.789180 + 0.614162i \(0.789494\pi\)
\(200\) −458451. −0.810435
\(201\) 178337. 0.311351
\(202\) −236089. −0.407096
\(203\) 0 0
\(204\) −508408. −0.855337
\(205\) −17633.0 −0.0293049
\(206\) 1.50945e6 2.47828
\(207\) 261474. 0.424134
\(208\) 177977. 0.285237
\(209\) −767210. −1.21492
\(210\) 0 0
\(211\) −372813. −0.576480 −0.288240 0.957558i \(-0.593070\pi\)
−0.288240 + 0.957558i \(0.593070\pi\)
\(212\) −1.66496e6 −2.54428
\(213\) −559128. −0.844428
\(214\) 886174. 1.32277
\(215\) 1.25558e6 1.85246
\(216\) 165446. 0.241280
\(217\) 0 0
\(218\) 783054. 1.11596
\(219\) −243987. −0.343760
\(220\) 2.25869e6 3.14630
\(221\) 536550. 0.738975
\(222\) −237357. −0.323236
\(223\) −1.08205e6 −1.45708 −0.728541 0.685002i \(-0.759801\pi\)
−0.728541 + 0.685002i \(0.759801\pi\)
\(224\) 0 0
\(225\) 163625. 0.215473
\(226\) 1.68317e6 2.19208
\(227\) 553049. 0.712359 0.356179 0.934418i \(-0.384079\pi\)
0.356179 + 0.934418i \(0.384079\pi\)
\(228\) −691826. −0.881373
\(229\) 523024. 0.659072 0.329536 0.944143i \(-0.393108\pi\)
0.329536 + 0.944143i \(0.393108\pi\)
\(230\) 2.17420e6 2.71006
\(231\) 0 0
\(232\) 170996. 0.208577
\(233\) −364181. −0.439468 −0.219734 0.975560i \(-0.570519\pi\)
−0.219734 + 0.975560i \(0.570519\pi\)
\(234\) −405776. −0.484447
\(235\) 1.17245e6 1.38492
\(236\) −581698. −0.679858
\(237\) −402186. −0.465111
\(238\) 0 0
\(239\) 371841. 0.421078 0.210539 0.977585i \(-0.432478\pi\)
0.210539 + 0.977585i \(0.432478\pi\)
\(240\) 215357. 0.241341
\(241\) −1.71147e6 −1.89814 −0.949069 0.315067i \(-0.897973\pi\)
−0.949069 + 0.315067i \(0.897973\pi\)
\(242\) −1.43883e6 −1.57933
\(243\) −59049.0 −0.0641500
\(244\) 53595.8 0.0576310
\(245\) 0 0
\(246\) 20774.6 0.0218874
\(247\) 730121. 0.761469
\(248\) −1.86244e6 −1.92289
\(249\) −140460. −0.143567
\(250\) −744207. −0.753084
\(251\) 58134.1 0.0582434 0.0291217 0.999576i \(-0.490729\pi\)
0.0291217 + 0.999576i \(0.490729\pi\)
\(252\) 0 0
\(253\) −1.80969e6 −1.77748
\(254\) 1.34908e6 1.31206
\(255\) 649242. 0.625253
\(256\) −1.53691e6 −1.46571
\(257\) −311839. −0.294509 −0.147254 0.989099i \(-0.547044\pi\)
−0.147254 + 0.989099i \(0.547044\pi\)
\(258\) −1.47928e6 −1.38357
\(259\) 0 0
\(260\) −2.14950e6 −1.97199
\(261\) −61029.9 −0.0554551
\(262\) −490995. −0.441900
\(263\) 863965. 0.770206 0.385103 0.922874i \(-0.374166\pi\)
0.385103 + 0.922874i \(0.374166\pi\)
\(264\) −1.14507e6 −1.01117
\(265\) 2.12617e6 1.85987
\(266\) 0 0
\(267\) −122718. −0.105349
\(268\) −1.11301e6 −0.946592
\(269\) −1.12069e6 −0.944290 −0.472145 0.881521i \(-0.656520\pi\)
−0.472145 + 0.881521i \(0.656520\pi\)
\(270\) −491001. −0.409895
\(271\) 1.14012e6 0.943030 0.471515 0.881858i \(-0.343707\pi\)
0.471515 + 0.881858i \(0.343707\pi\)
\(272\) 335498. 0.274959
\(273\) 0 0
\(274\) −88359.6 −0.0711013
\(275\) −1.13247e6 −0.903012
\(276\) −1.63188e6 −1.28948
\(277\) −1.98801e6 −1.55675 −0.778375 0.627799i \(-0.783956\pi\)
−0.778375 + 0.627799i \(0.783956\pi\)
\(278\) −1.71923e6 −1.33420
\(279\) 664720. 0.511244
\(280\) 0 0
\(281\) 532321. 0.402168 0.201084 0.979574i \(-0.435554\pi\)
0.201084 + 0.979574i \(0.435554\pi\)
\(282\) −1.38134e6 −1.03437
\(283\) 2.62473e6 1.94813 0.974067 0.226259i \(-0.0726497\pi\)
0.974067 + 0.226259i \(0.0726497\pi\)
\(284\) 3.48956e6 2.56729
\(285\) 883468. 0.644286
\(286\) 2.80842e6 2.03024
\(287\) 0 0
\(288\) 334526. 0.237656
\(289\) −408424. −0.287651
\(290\) −507473. −0.354338
\(291\) 116275. 0.0804925
\(292\) 1.52274e6 1.04512
\(293\) 609962. 0.415082 0.207541 0.978226i \(-0.433454\pi\)
0.207541 + 0.978226i \(0.433454\pi\)
\(294\) 0 0
\(295\) 742834. 0.496978
\(296\) 637424. 0.422863
\(297\) 408685. 0.268842
\(298\) 1.56812e6 1.02291
\(299\) 1.72221e6 1.11406
\(300\) −1.02119e6 −0.655096
\(301\) 0 0
\(302\) −3.53307e6 −2.22913
\(303\) −226286. −0.141596
\(304\) 456536. 0.283329
\(305\) −68442.3 −0.0421284
\(306\) −764915. −0.466992
\(307\) −1.34843e6 −0.816551 −0.408275 0.912859i \(-0.633870\pi\)
−0.408275 + 0.912859i \(0.633870\pi\)
\(308\) 0 0
\(309\) 1.44678e6 0.861997
\(310\) 5.52724e6 3.26666
\(311\) −3.33061e6 −1.95264 −0.976320 0.216330i \(-0.930591\pi\)
−0.976320 + 0.216330i \(0.930591\pi\)
\(312\) 1.08972e6 0.633762
\(313\) −2.83670e6 −1.63664 −0.818320 0.574763i \(-0.805094\pi\)
−0.818320 + 0.574763i \(0.805094\pi\)
\(314\) −366909. −0.210007
\(315\) 0 0
\(316\) 2.51007e6 1.41406
\(317\) −1.08839e6 −0.608326 −0.304163 0.952620i \(-0.598377\pi\)
−0.304163 + 0.952620i \(0.598377\pi\)
\(318\) −2.50498e6 −1.38911
\(319\) 422395. 0.232403
\(320\) 3.54734e6 1.93655
\(321\) 849380. 0.460086
\(322\) 0 0
\(323\) 1.37633e6 0.734033
\(324\) 368529. 0.195033
\(325\) 1.07772e6 0.565975
\(326\) 4.48760e6 2.33867
\(327\) 750541. 0.388155
\(328\) −55790.4 −0.0286335
\(329\) 0 0
\(330\) 3.39827e6 1.71780
\(331\) 1.30555e6 0.654971 0.327485 0.944856i \(-0.393799\pi\)
0.327485 + 0.944856i \(0.393799\pi\)
\(332\) 876619. 0.436481
\(333\) −227502. −0.112428
\(334\) −373654. −0.183275
\(335\) 1.42133e6 0.691961
\(336\) 0 0
\(337\) −3.17016e6 −1.52057 −0.760285 0.649590i \(-0.774941\pi\)
−0.760285 + 0.649590i \(0.774941\pi\)
\(338\) 813739. 0.387430
\(339\) 1.61328e6 0.762450
\(340\) −4.05196e6 −1.90094
\(341\) −4.60060e6 −2.14254
\(342\) −1.04087e6 −0.481207
\(343\) 0 0
\(344\) 3.97263e6 1.81002
\(345\) 2.08392e6 0.942615
\(346\) −453889. −0.203826
\(347\) −1.71592e6 −0.765019 −0.382510 0.923951i \(-0.624940\pi\)
−0.382510 + 0.923951i \(0.624940\pi\)
\(348\) 380892. 0.168598
\(349\) −2.95822e6 −1.30007 −0.650034 0.759905i \(-0.725245\pi\)
−0.650034 + 0.759905i \(0.725245\pi\)
\(350\) 0 0
\(351\) −388928. −0.168501
\(352\) −2.31529e6 −0.995976
\(353\) 3.76980e6 1.61021 0.805103 0.593135i \(-0.202110\pi\)
0.805103 + 0.593135i \(0.202110\pi\)
\(354\) −875182. −0.371185
\(355\) −4.45620e6 −1.87669
\(356\) 765890. 0.320289
\(357\) 0 0
\(358\) 1.34191e6 0.553371
\(359\) 1.92987e6 0.790300 0.395150 0.918617i \(-0.370693\pi\)
0.395150 + 0.918617i \(0.370693\pi\)
\(360\) 1.31859e6 0.536232
\(361\) −603234. −0.243623
\(362\) 725323. 0.290911
\(363\) −1.37909e6 −0.549323
\(364\) 0 0
\(365\) −1.94455e6 −0.763988
\(366\) 80636.5 0.0314651
\(367\) 2.36742e6 0.917509 0.458754 0.888563i \(-0.348296\pi\)
0.458754 + 0.888563i \(0.348296\pi\)
\(368\) 1.07688e6 0.414521
\(369\) 19912.0 0.00761289
\(370\) −1.89171e6 −0.718373
\(371\) 0 0
\(372\) −4.14856e6 −1.55432
\(373\) 3.53829e6 1.31680 0.658402 0.752666i \(-0.271233\pi\)
0.658402 + 0.752666i \(0.271233\pi\)
\(374\) 5.29406e6 1.95709
\(375\) −713307. −0.261938
\(376\) 3.70960e6 1.35318
\(377\) −401975. −0.145662
\(378\) 0 0
\(379\) 1.79847e6 0.643139 0.321569 0.946886i \(-0.395790\pi\)
0.321569 + 0.946886i \(0.395790\pi\)
\(380\) −5.51378e6 −1.95880
\(381\) 1.29307e6 0.456361
\(382\) −2.55455e6 −0.895686
\(383\) 2.60815e6 0.908521 0.454261 0.890869i \(-0.349904\pi\)
0.454261 + 0.890869i \(0.349904\pi\)
\(384\) −2.98994e6 −1.03475
\(385\) 0 0
\(386\) 150555. 0.0514313
\(387\) −1.41786e6 −0.481235
\(388\) −725683. −0.244719
\(389\) −2.82995e6 −0.948211 −0.474106 0.880468i \(-0.657228\pi\)
−0.474106 + 0.880468i \(0.657228\pi\)
\(390\) −3.23399e6 −1.07666
\(391\) 3.24648e6 1.07392
\(392\) 0 0
\(393\) −470609. −0.153702
\(394\) −9.69299e6 −3.14570
\(395\) −3.20538e6 −1.03368
\(396\) −2.55063e6 −0.817352
\(397\) −2.43062e6 −0.773999 −0.387000 0.922080i \(-0.626489\pi\)
−0.387000 + 0.922080i \(0.626489\pi\)
\(398\) 8.27939e6 2.61993
\(399\) 0 0
\(400\) 673885. 0.210589
\(401\) −2.43184e6 −0.755222 −0.377611 0.925964i \(-0.623254\pi\)
−0.377611 + 0.925964i \(0.623254\pi\)
\(402\) −1.67456e6 −0.516815
\(403\) 4.37820e6 1.34287
\(404\) 1.41227e6 0.430491
\(405\) −470614. −0.142570
\(406\) 0 0
\(407\) 1.57457e6 0.471167
\(408\) 2.05419e6 0.610927
\(409\) −4.77466e6 −1.41135 −0.705674 0.708537i \(-0.749356\pi\)
−0.705674 + 0.708537i \(0.749356\pi\)
\(410\) 165571. 0.0486436
\(411\) −84690.9 −0.0247305
\(412\) −9.02944e6 −2.62070
\(413\) 0 0
\(414\) −2.45521e6 −0.704024
\(415\) −1.11945e6 −0.319069
\(416\) 2.20336e6 0.624242
\(417\) −1.64785e6 −0.464063
\(418\) 7.20400e6 2.01666
\(419\) −457181. −0.127219 −0.0636097 0.997975i \(-0.520261\pi\)
−0.0636097 + 0.997975i \(0.520261\pi\)
\(420\) 0 0
\(421\) −1.82396e6 −0.501545 −0.250773 0.968046i \(-0.580685\pi\)
−0.250773 + 0.968046i \(0.580685\pi\)
\(422\) 3.50066e6 0.956905
\(423\) −1.32398e6 −0.359776
\(424\) 6.72715e6 1.81726
\(425\) 2.03157e6 0.545582
\(426\) 5.25014e6 1.40167
\(427\) 0 0
\(428\) −5.30104e6 −1.39879
\(429\) 2.69181e6 0.706158
\(430\) −1.17897e7 −3.07492
\(431\) −3.38249e6 −0.877087 −0.438544 0.898710i \(-0.644506\pi\)
−0.438544 + 0.898710i \(0.644506\pi\)
\(432\) −243192. −0.0626960
\(433\) −285266. −0.0731190 −0.0365595 0.999331i \(-0.511640\pi\)
−0.0365595 + 0.999331i \(0.511640\pi\)
\(434\) 0 0
\(435\) −486402. −0.123246
\(436\) −4.68418e6 −1.18010
\(437\) 4.41771e6 1.10661
\(438\) 2.29100e6 0.570611
\(439\) 4.35220e6 1.07782 0.538911 0.842363i \(-0.318836\pi\)
0.538911 + 0.842363i \(0.318836\pi\)
\(440\) −9.12610e6 −2.24726
\(441\) 0 0
\(442\) −5.03813e6 −1.22663
\(443\) −5.10560e6 −1.23605 −0.618027 0.786157i \(-0.712068\pi\)
−0.618027 + 0.786157i \(0.712068\pi\)
\(444\) 1.41985e6 0.341811
\(445\) −978049. −0.234132
\(446\) 1.01603e7 2.41862
\(447\) 1.50301e6 0.355790
\(448\) 0 0
\(449\) 3.04163e6 0.712016 0.356008 0.934483i \(-0.384138\pi\)
0.356008 + 0.934483i \(0.384138\pi\)
\(450\) −1.53642e6 −0.357666
\(451\) −137813. −0.0319043
\(452\) −1.00686e7 −2.31805
\(453\) −3.38638e6 −0.775336
\(454\) −5.19305e6 −1.18245
\(455\) 0 0
\(456\) 2.79528e6 0.629524
\(457\) −1.74432e6 −0.390694 −0.195347 0.980734i \(-0.562583\pi\)
−0.195347 + 0.980734i \(0.562583\pi\)
\(458\) −4.91113e6 −1.09400
\(459\) −733156. −0.162429
\(460\) −1.30059e7 −2.86580
\(461\) −6.85701e6 −1.50273 −0.751367 0.659884i \(-0.770605\pi\)
−0.751367 + 0.659884i \(0.770605\pi\)
\(462\) 0 0
\(463\) 5.13844e6 1.11398 0.556992 0.830518i \(-0.311955\pi\)
0.556992 + 0.830518i \(0.311955\pi\)
\(464\) −251351. −0.0541982
\(465\) 5.29775e6 1.13621
\(466\) 3.41961e6 0.729477
\(467\) −4.58171e6 −0.972154 −0.486077 0.873916i \(-0.661573\pi\)
−0.486077 + 0.873916i \(0.661573\pi\)
\(468\) 2.42733e6 0.512287
\(469\) 0 0
\(470\) −1.10091e7 −2.29883
\(471\) −351675. −0.0730448
\(472\) 2.35031e6 0.485591
\(473\) 9.81320e6 2.01678
\(474\) 3.77647e6 0.772042
\(475\) 2.76450e6 0.562190
\(476\) 0 0
\(477\) −2.40097e6 −0.483161
\(478\) −3.49154e6 −0.698952
\(479\) −288523. −0.0574568 −0.0287284 0.999587i \(-0.509146\pi\)
−0.0287284 + 0.999587i \(0.509146\pi\)
\(480\) 2.66614e6 0.528177
\(481\) −1.49845e6 −0.295310
\(482\) 1.60705e7 3.15074
\(483\) 0 0
\(484\) 8.60702e6 1.67009
\(485\) 926703. 0.178890
\(486\) 554462. 0.106483
\(487\) −7.81685e6 −1.49351 −0.746757 0.665097i \(-0.768390\pi\)
−0.746757 + 0.665097i \(0.768390\pi\)
\(488\) −216550. −0.0411632
\(489\) 4.30127e6 0.813438
\(490\) 0 0
\(491\) 3.14467e6 0.588669 0.294335 0.955702i \(-0.404902\pi\)
0.294335 + 0.955702i \(0.404902\pi\)
\(492\) −124272. −0.0231452
\(493\) −757751. −0.140414
\(494\) −6.85574e6 −1.26397
\(495\) 3.25718e6 0.597487
\(496\) 2.73763e6 0.499656
\(497\) 0 0
\(498\) 1.31890e6 0.238308
\(499\) 6.72563e6 1.20915 0.604577 0.796547i \(-0.293342\pi\)
0.604577 + 0.796547i \(0.293342\pi\)
\(500\) 4.45180e6 0.796362
\(501\) −358140. −0.0637469
\(502\) −545871. −0.0966787
\(503\) −9.45056e6 −1.66547 −0.832737 0.553669i \(-0.813227\pi\)
−0.832737 + 0.553669i \(0.813227\pi\)
\(504\) 0 0
\(505\) −1.80348e6 −0.314690
\(506\) 1.69928e7 2.95045
\(507\) 779952. 0.134756
\(508\) −8.07011e6 −1.38746
\(509\) −8.83702e6 −1.51186 −0.755930 0.654653i \(-0.772815\pi\)
−0.755930 + 0.654653i \(0.772815\pi\)
\(510\) −6.09629e6 −1.03786
\(511\) 0 0
\(512\) 3.80044e6 0.640707
\(513\) −997656. −0.167374
\(514\) 2.92813e6 0.488858
\(515\) 1.15307e7 1.91574
\(516\) 8.84899e6 1.46308
\(517\) 9.16344e6 1.50776
\(518\) 0 0
\(519\) −435044. −0.0708949
\(520\) 8.68492e6 1.40850
\(521\) 694985. 0.112171 0.0560855 0.998426i \(-0.482138\pi\)
0.0560855 + 0.998426i \(0.482138\pi\)
\(522\) 573063. 0.0920505
\(523\) 3.58210e6 0.572643 0.286321 0.958134i \(-0.407568\pi\)
0.286321 + 0.958134i \(0.407568\pi\)
\(524\) 2.93710e6 0.467294
\(525\) 0 0
\(526\) −8.11252e6 −1.27847
\(527\) 8.25320e6 1.29448
\(528\) 1.68316e6 0.262749
\(529\) 3.98415e6 0.619009
\(530\) −1.99644e7 −3.08722
\(531\) −838845. −0.129106
\(532\) 0 0
\(533\) 131151. 0.0199965
\(534\) 1.15230e6 0.174869
\(535\) 6.76947e6 1.02252
\(536\) 4.49705e6 0.676107
\(537\) 1.28620e6 0.192474
\(538\) 1.05231e7 1.56744
\(539\) 0 0
\(540\) 2.93714e6 0.433451
\(541\) −5.16846e6 −0.759220 −0.379610 0.925147i \(-0.623942\pi\)
−0.379610 + 0.925147i \(0.623942\pi\)
\(542\) −1.07055e7 −1.56535
\(543\) 695208. 0.101185
\(544\) 4.15349e6 0.601750
\(545\) 5.98174e6 0.862653
\(546\) 0 0
\(547\) 8.47489e6 1.21106 0.605530 0.795822i \(-0.292961\pi\)
0.605530 + 0.795822i \(0.292961\pi\)
\(548\) 528562. 0.0751873
\(549\) 77288.5 0.0109442
\(550\) 1.06337e7 1.49892
\(551\) −1.03112e6 −0.144688
\(552\) 6.59349e6 0.921018
\(553\) 0 0
\(554\) 1.86671e7 2.58407
\(555\) −1.81317e6 −0.249865
\(556\) 1.02843e7 1.41088
\(557\) −1.04213e7 −1.42325 −0.711627 0.702557i \(-0.752041\pi\)
−0.711627 + 0.702557i \(0.752041\pi\)
\(558\) −6.24163e6 −0.848619
\(559\) −9.33880e6 −1.26404
\(560\) 0 0
\(561\) 5.07425e6 0.680714
\(562\) −4.99842e6 −0.667562
\(563\) −7.24077e6 −0.962751 −0.481375 0.876515i \(-0.659863\pi\)
−0.481375 + 0.876515i \(0.659863\pi\)
\(564\) 8.26307e6 1.09381
\(565\) 1.28577e7 1.69450
\(566\) −2.46459e7 −3.23373
\(567\) 0 0
\(568\) −1.40993e7 −1.83369
\(569\) 916536. 0.118678 0.0593388 0.998238i \(-0.481101\pi\)
0.0593388 + 0.998238i \(0.481101\pi\)
\(570\) −8.29565e6 −1.06946
\(571\) 1.00708e7 1.29262 0.646312 0.763073i \(-0.276311\pi\)
0.646312 + 0.763073i \(0.276311\pi\)
\(572\) −1.67998e7 −2.14691
\(573\) −2.44849e6 −0.311538
\(574\) 0 0
\(575\) 6.52091e6 0.822505
\(576\) −4.00584e6 −0.503080
\(577\) 1.16997e7 1.46298 0.731488 0.681855i \(-0.238826\pi\)
0.731488 + 0.681855i \(0.238826\pi\)
\(578\) 3.83505e6 0.477475
\(579\) 144304. 0.0178888
\(580\) 3.03567e6 0.374701
\(581\) 0 0
\(582\) −1.09181e6 −0.133610
\(583\) 1.66174e7 2.02485
\(584\) −6.15251e6 −0.746484
\(585\) −3.09972e6 −0.374483
\(586\) −5.72747e6 −0.688999
\(587\) −6.92367e6 −0.829357 −0.414678 0.909968i \(-0.636106\pi\)
−0.414678 + 0.909968i \(0.636106\pi\)
\(588\) 0 0
\(589\) 1.12307e7 1.33389
\(590\) −6.97511e6 −0.824938
\(591\) −9.29054e6 −1.09414
\(592\) −936961. −0.109880
\(593\) 1.57770e7 1.84242 0.921208 0.389069i \(-0.127203\pi\)
0.921208 + 0.389069i \(0.127203\pi\)
\(594\) −3.83750e6 −0.446254
\(595\) 0 0
\(596\) −9.38041e6 −1.08170
\(597\) 7.93563e6 0.911266
\(598\) −1.61713e7 −1.84924
\(599\) 1.42708e7 1.62511 0.812553 0.582887i \(-0.198077\pi\)
0.812553 + 0.582887i \(0.198077\pi\)
\(600\) 4.12606e6 0.467905
\(601\) 7.63222e6 0.861916 0.430958 0.902372i \(-0.358176\pi\)
0.430958 + 0.902372i \(0.358176\pi\)
\(602\) 0 0
\(603\) −1.60503e6 −0.179759
\(604\) 2.11346e7 2.35723
\(605\) −1.09912e7 −1.22084
\(606\) 2.12480e6 0.235037
\(607\) −3.56035e6 −0.392212 −0.196106 0.980583i \(-0.562830\pi\)
−0.196106 + 0.980583i \(0.562830\pi\)
\(608\) 5.65194e6 0.620067
\(609\) 0 0
\(610\) 642664. 0.0699294
\(611\) −8.72046e6 −0.945010
\(612\) 4.57567e6 0.493829
\(613\) −1.37284e7 −1.47560 −0.737800 0.675019i \(-0.764135\pi\)
−0.737800 + 0.675019i \(0.764135\pi\)
\(614\) 1.26616e7 1.35540
\(615\) 158697. 0.0169192
\(616\) 0 0
\(617\) −6.51173e6 −0.688626 −0.344313 0.938855i \(-0.611888\pi\)
−0.344313 + 0.938855i \(0.611888\pi\)
\(618\) −1.35851e7 −1.43084
\(619\) 8.85110e6 0.928476 0.464238 0.885711i \(-0.346328\pi\)
0.464238 + 0.885711i \(0.346328\pi\)
\(620\) −3.30636e7 −3.45439
\(621\) −2.35327e6 −0.244874
\(622\) 3.12739e7 3.24121
\(623\) 0 0
\(624\) −1.60179e6 −0.164681
\(625\) −1.19977e7 −1.22856
\(626\) 2.66363e7 2.71667
\(627\) 6.90489e6 0.701436
\(628\) 2.19483e6 0.222076
\(629\) −2.82467e6 −0.284670
\(630\) 0 0
\(631\) −6.89663e6 −0.689546 −0.344773 0.938686i \(-0.612044\pi\)
−0.344773 + 0.938686i \(0.612044\pi\)
\(632\) −1.01418e7 −1.01000
\(633\) 3.35531e6 0.332831
\(634\) 1.02198e7 1.00977
\(635\) 1.03056e7 1.01424
\(636\) 1.49846e7 1.46894
\(637\) 0 0
\(638\) −3.96623e6 −0.385768
\(639\) 5.03216e6 0.487531
\(640\) −2.38295e7 −2.29967
\(641\) −1.66695e7 −1.60242 −0.801210 0.598383i \(-0.795810\pi\)
−0.801210 + 0.598383i \(0.795810\pi\)
\(642\) −7.97556e6 −0.763702
\(643\) −1.28697e7 −1.22756 −0.613779 0.789478i \(-0.710351\pi\)
−0.613779 + 0.789478i \(0.710351\pi\)
\(644\) 0 0
\(645\) −1.13002e7 −1.06952
\(646\) −1.29235e7 −1.21843
\(647\) −1.14731e7 −1.07751 −0.538754 0.842463i \(-0.681105\pi\)
−0.538754 + 0.842463i \(0.681105\pi\)
\(648\) −1.48901e6 −0.139303
\(649\) 5.80574e6 0.541060
\(650\) −1.01196e7 −0.939467
\(651\) 0 0
\(652\) −2.68445e7 −2.47307
\(653\) −1.31801e7 −1.20958 −0.604791 0.796384i \(-0.706743\pi\)
−0.604791 + 0.796384i \(0.706743\pi\)
\(654\) −7.04748e6 −0.644302
\(655\) −3.75070e6 −0.341593
\(656\) 82007.2 0.00744034
\(657\) 2.19588e6 0.198470
\(658\) 0 0
\(659\) −1.13068e7 −1.01421 −0.507104 0.861885i \(-0.669284\pi\)
−0.507104 + 0.861885i \(0.669284\pi\)
\(660\) −2.03282e7 −1.81652
\(661\) 2.20319e6 0.196132 0.0980661 0.995180i \(-0.468734\pi\)
0.0980661 + 0.995180i \(0.468734\pi\)
\(662\) −1.22589e7 −1.08719
\(663\) −4.82895e6 −0.426647
\(664\) −3.54192e6 −0.311758
\(665\) 0 0
\(666\) 2.13621e6 0.186620
\(667\) −2.43222e6 −0.211684
\(668\) 2.23518e6 0.193808
\(669\) 9.73843e6 0.841247
\(670\) −1.33461e7 −1.14859
\(671\) −534922. −0.0458653
\(672\) 0 0
\(673\) −1.89787e7 −1.61521 −0.807606 0.589723i \(-0.799237\pi\)
−0.807606 + 0.589723i \(0.799237\pi\)
\(674\) 2.97674e7 2.52401
\(675\) −1.47262e6 −0.124403
\(676\) −4.86773e6 −0.409694
\(677\) −1.96475e7 −1.64754 −0.823771 0.566923i \(-0.808134\pi\)
−0.823771 + 0.566923i \(0.808134\pi\)
\(678\) −1.51485e7 −1.26560
\(679\) 0 0
\(680\) 1.63717e7 1.35775
\(681\) −4.97744e6 −0.411280
\(682\) 4.31990e7 3.55642
\(683\) 1.62705e7 1.33459 0.667295 0.744793i \(-0.267452\pi\)
0.667295 + 0.744793i \(0.267452\pi\)
\(684\) 6.22644e6 0.508861
\(685\) −674978. −0.0549621
\(686\) 0 0
\(687\) −4.70722e6 −0.380515
\(688\) −5.83944e6 −0.470328
\(689\) −1.58141e7 −1.26910
\(690\) −1.95678e7 −1.56465
\(691\) 1.94467e7 1.54935 0.774677 0.632357i \(-0.217912\pi\)
0.774677 + 0.632357i \(0.217912\pi\)
\(692\) 2.71514e6 0.215539
\(693\) 0 0
\(694\) 1.61122e7 1.26986
\(695\) −1.31332e7 −1.03135
\(696\) −1.53897e6 −0.120422
\(697\) 247229. 0.0192760
\(698\) 2.77772e7 2.15800
\(699\) 3.27763e6 0.253727
\(700\) 0 0
\(701\) 1.49625e7 1.15003 0.575014 0.818144i \(-0.304997\pi\)
0.575014 + 0.818144i \(0.304997\pi\)
\(702\) 3.65198e6 0.279696
\(703\) −3.84373e6 −0.293335
\(704\) 2.77248e7 2.10832
\(705\) −1.05520e7 −0.799581
\(706\) −3.53979e7 −2.67280
\(707\) 0 0
\(708\) 5.23529e6 0.392516
\(709\) 1.58639e6 0.118521 0.0592603 0.998243i \(-0.481126\pi\)
0.0592603 + 0.998243i \(0.481126\pi\)
\(710\) 4.18431e7 3.11514
\(711\) 3.61968e6 0.268532
\(712\) −3.09453e6 −0.228767
\(713\) 2.64910e7 1.95152
\(714\) 0 0
\(715\) 2.14535e7 1.56940
\(716\) −8.02723e6 −0.585172
\(717\) −3.34657e6 −0.243110
\(718\) −1.81212e7 −1.31183
\(719\) −1.81675e7 −1.31061 −0.655305 0.755364i \(-0.727460\pi\)
−0.655305 + 0.755364i \(0.727460\pi\)
\(720\) −1.93822e6 −0.139338
\(721\) 0 0
\(722\) 5.66429e6 0.404392
\(723\) 1.54033e7 1.09589
\(724\) −4.33884e6 −0.307629
\(725\) −1.52203e6 −0.107542
\(726\) 1.29495e7 0.911826
\(727\) 1.26903e7 0.890506 0.445253 0.895405i \(-0.353114\pi\)
0.445253 + 0.895405i \(0.353114\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 1.82591e7 1.26815
\(731\) −1.76043e7 −1.21850
\(732\) −482362. −0.0332733
\(733\) −3.11401e6 −0.214072 −0.107036 0.994255i \(-0.534136\pi\)
−0.107036 + 0.994255i \(0.534136\pi\)
\(734\) −2.22298e7 −1.52298
\(735\) 0 0
\(736\) 1.33318e7 0.907182
\(737\) 1.11086e7 0.753339
\(738\) −186971. −0.0126367
\(739\) 9.09899e6 0.612890 0.306445 0.951888i \(-0.400861\pi\)
0.306445 + 0.951888i \(0.400861\pi\)
\(740\) 1.13161e7 0.759656
\(741\) −6.57109e6 −0.439634
\(742\) 0 0
\(743\) −8.79218e6 −0.584285 −0.292142 0.956375i \(-0.594368\pi\)
−0.292142 + 0.956375i \(0.594368\pi\)
\(744\) 1.67620e7 1.11018
\(745\) 1.19789e7 0.790724
\(746\) −3.32241e7 −2.18578
\(747\) 1.26414e6 0.0828883
\(748\) −3.16687e7 −2.06955
\(749\) 0 0
\(750\) 6.69786e6 0.434793
\(751\) −2.80918e7 −1.81752 −0.908759 0.417320i \(-0.862969\pi\)
−0.908759 + 0.417320i \(0.862969\pi\)
\(752\) −5.45280e6 −0.351621
\(753\) −523207. −0.0336268
\(754\) 3.77450e6 0.241786
\(755\) −2.69891e7 −1.72314
\(756\) 0 0
\(757\) −4.18815e6 −0.265634 −0.132817 0.991141i \(-0.542402\pi\)
−0.132817 + 0.991141i \(0.542402\pi\)
\(758\) −1.68874e7 −1.06755
\(759\) 1.62872e7 1.02623
\(760\) 2.22781e7 1.39908
\(761\) −393182. −0.0246111 −0.0123056 0.999924i \(-0.503917\pi\)
−0.0123056 + 0.999924i \(0.503917\pi\)
\(762\) −1.21417e7 −0.757518
\(763\) 0 0
\(764\) 1.52812e7 0.947159
\(765\) −5.84317e6 −0.360990
\(766\) −2.44901e7 −1.50806
\(767\) −5.52508e6 −0.339117
\(768\) 1.38321e7 0.846226
\(769\) 1.57517e7 0.960530 0.480265 0.877124i \(-0.340541\pi\)
0.480265 + 0.877124i \(0.340541\pi\)
\(770\) 0 0
\(771\) 2.80655e6 0.170035
\(772\) −900611. −0.0543869
\(773\) 2.88472e6 0.173642 0.0868210 0.996224i \(-0.472329\pi\)
0.0868210 + 0.996224i \(0.472329\pi\)
\(774\) 1.33136e7 0.798807
\(775\) 1.65775e7 0.991433
\(776\) 2.93207e6 0.174791
\(777\) 0 0
\(778\) 2.65729e7 1.57394
\(779\) 336421. 0.0198628
\(780\) 1.93455e7 1.13853
\(781\) −3.48281e7 −2.04316
\(782\) −3.04840e7 −1.78261
\(783\) 549269. 0.0320170
\(784\) 0 0
\(785\) −2.80281e6 −0.162338
\(786\) 4.41895e6 0.255131
\(787\) −2.97866e7 −1.71429 −0.857145 0.515076i \(-0.827764\pi\)
−0.857145 + 0.515076i \(0.827764\pi\)
\(788\) 5.79829e7 3.32647
\(789\) −7.77569e6 −0.444679
\(790\) 3.00981e7 1.71582
\(791\) 0 0
\(792\) 1.03056e7 0.583797
\(793\) 509063. 0.0287467
\(794\) 2.28232e7 1.28477
\(795\) −1.91355e7 −1.07380
\(796\) −4.95268e7 −2.77049
\(797\) 8.11321e6 0.452425 0.226213 0.974078i \(-0.427366\pi\)
0.226213 + 0.974078i \(0.427366\pi\)
\(798\) 0 0
\(799\) −1.64387e7 −0.910960
\(800\) 8.34274e6 0.460876
\(801\) 1.10446e6 0.0608232
\(802\) 2.28347e7 1.25360
\(803\) −1.51979e7 −0.831756
\(804\) 1.00171e7 0.546515
\(805\) 0 0
\(806\) −4.11107e7 −2.22904
\(807\) 1.00862e7 0.545186
\(808\) −5.70617e6 −0.307480
\(809\) 111597. 0.00599489 0.00299744 0.999996i \(-0.499046\pi\)
0.00299744 + 0.999996i \(0.499046\pi\)
\(810\) 4.41901e6 0.236653
\(811\) −209250. −0.0111716 −0.00558578 0.999984i \(-0.501778\pi\)
−0.00558578 + 0.999984i \(0.501778\pi\)
\(812\) 0 0
\(813\) −1.02610e7 −0.544459
\(814\) −1.47850e7 −0.782094
\(815\) 3.42807e7 1.80782
\(816\) −3.01949e6 −0.158748
\(817\) −2.39554e7 −1.25559
\(818\) 4.48334e7 2.34271
\(819\) 0 0
\(820\) −990437. −0.0514390
\(821\) 6.84614e6 0.354477 0.177238 0.984168i \(-0.443284\pi\)
0.177238 + 0.984168i \(0.443284\pi\)
\(822\) 795237. 0.0410503
\(823\) −5.61210e6 −0.288819 −0.144409 0.989518i \(-0.546128\pi\)
−0.144409 + 0.989518i \(0.546128\pi\)
\(824\) 3.64828e7 1.87185
\(825\) 1.01922e7 0.521354
\(826\) 0 0
\(827\) −2.15868e7 −1.09755 −0.548776 0.835969i \(-0.684906\pi\)
−0.548776 + 0.835969i \(0.684906\pi\)
\(828\) 1.46869e7 0.744483
\(829\) 1.92427e7 0.972478 0.486239 0.873826i \(-0.338368\pi\)
0.486239 + 0.873826i \(0.338368\pi\)
\(830\) 1.05115e7 0.529625
\(831\) 1.78921e7 0.898790
\(832\) −2.63846e7 −1.32142
\(833\) 0 0
\(834\) 1.54731e7 0.770303
\(835\) −2.85434e6 −0.141674
\(836\) −4.30939e7 −2.13255
\(837\) −5.98248e6 −0.295167
\(838\) 4.29287e6 0.211173
\(839\) 2.47678e7 1.21474 0.607369 0.794420i \(-0.292225\pi\)
0.607369 + 0.794420i \(0.292225\pi\)
\(840\) 0 0
\(841\) −1.99435e7 −0.972323
\(842\) 1.71267e7 0.832520
\(843\) −4.79089e6 −0.232192
\(844\) −2.09407e7 −1.01190
\(845\) 6.21614e6 0.299488
\(846\) 1.24320e7 0.597195
\(847\) 0 0
\(848\) −9.88836e6 −0.472210
\(849\) −2.36226e7 −1.12476
\(850\) −1.90762e7 −0.905618
\(851\) −9.06659e6 −0.429161
\(852\) −3.14060e7 −1.48222
\(853\) −2.91267e7 −1.37062 −0.685312 0.728250i \(-0.740334\pi\)
−0.685312 + 0.728250i \(0.740334\pi\)
\(854\) 0 0
\(855\) −7.95121e6 −0.371979
\(856\) 2.14185e7 0.999088
\(857\) −3.22586e6 −0.150035 −0.0750175 0.997182i \(-0.523901\pi\)
−0.0750175 + 0.997182i \(0.523901\pi\)
\(858\) −2.52758e7 −1.17216
\(859\) 2.64671e7 1.22384 0.611918 0.790921i \(-0.290398\pi\)
0.611918 + 0.790921i \(0.290398\pi\)
\(860\) 7.05255e7 3.25162
\(861\) 0 0
\(862\) 3.17611e7 1.45589
\(863\) 1.58510e7 0.724487 0.362243 0.932084i \(-0.382011\pi\)
0.362243 + 0.932084i \(0.382011\pi\)
\(864\) −3.01073e6 −0.137211
\(865\) −3.46726e6 −0.157560
\(866\) 2.67861e6 0.121371
\(867\) 3.67582e6 0.166076
\(868\) 0 0
\(869\) −2.50522e7 −1.12537
\(870\) 4.56725e6 0.204577
\(871\) −1.05716e7 −0.472166
\(872\) 1.89261e7 0.842888
\(873\) −1.04648e6 −0.0464724
\(874\) −4.14817e7 −1.83687
\(875\) 0 0
\(876\) −1.37046e7 −0.603403
\(877\) −1.49132e7 −0.654743 −0.327372 0.944896i \(-0.606163\pi\)
−0.327372 + 0.944896i \(0.606163\pi\)
\(878\) −4.08665e7 −1.78909
\(879\) −5.48966e6 −0.239648
\(880\) 1.34146e7 0.583944
\(881\) −3.70679e7 −1.60901 −0.804504 0.593947i \(-0.797569\pi\)
−0.804504 + 0.593947i \(0.797569\pi\)
\(882\) 0 0
\(883\) 1.50440e7 0.649325 0.324663 0.945830i \(-0.394749\pi\)
0.324663 + 0.945830i \(0.394749\pi\)
\(884\) 3.01378e7 1.29712
\(885\) −6.68551e6 −0.286930
\(886\) 4.79409e7 2.05174
\(887\) 1.78101e7 0.760075 0.380038 0.924971i \(-0.375911\pi\)
0.380038 + 0.924971i \(0.375911\pi\)
\(888\) −5.73682e6 −0.244140
\(889\) 0 0
\(890\) 9.18375e6 0.388638
\(891\) −3.67816e6 −0.155216
\(892\) −6.07782e7 −2.55762
\(893\) −2.23692e7 −0.938690
\(894\) −1.41131e7 −0.590580
\(895\) 1.02508e7 0.427762
\(896\) 0 0
\(897\) −1.54999e7 −0.643202
\(898\) −2.85605e7 −1.18188
\(899\) −6.18317e6 −0.255160
\(900\) 9.19074e6 0.378220
\(901\) −2.98106e7 −1.22337
\(902\) 1.29405e6 0.0529583
\(903\) 0 0
\(904\) 4.06815e7 1.65568
\(905\) 5.54073e6 0.224877
\(906\) 3.17977e7 1.28699
\(907\) −1.16936e6 −0.0471987 −0.0235993 0.999721i \(-0.507513\pi\)
−0.0235993 + 0.999721i \(0.507513\pi\)
\(908\) 3.10645e7 1.25040
\(909\) 2.03658e6 0.0817506
\(910\) 0 0
\(911\) 2.74389e7 1.09539 0.547697 0.836677i \(-0.315505\pi\)
0.547697 + 0.836677i \(0.315505\pi\)
\(912\) −4.10882e6 −0.163580
\(913\) −8.74924e6 −0.347371
\(914\) 1.63790e7 0.648516
\(915\) 615981. 0.0243229
\(916\) 2.93781e7 1.15687
\(917\) 0 0
\(918\) 6.88423e6 0.269618
\(919\) 1.10218e7 0.430492 0.215246 0.976560i \(-0.430945\pi\)
0.215246 + 0.976560i \(0.430945\pi\)
\(920\) 5.25495e7 2.04691
\(921\) 1.21359e7 0.471436
\(922\) 6.43864e7 2.49440
\(923\) 3.31444e7 1.28058
\(924\) 0 0
\(925\) −5.67367e6 −0.218027
\(926\) −4.82493e7 −1.84911
\(927\) −1.30210e7 −0.497674
\(928\) −3.11173e6 −0.118613
\(929\) 4.25051e6 0.161585 0.0807927 0.996731i \(-0.474255\pi\)
0.0807927 + 0.996731i \(0.474255\pi\)
\(930\) −4.97452e7 −1.88601
\(931\) 0 0
\(932\) −2.04559e7 −0.771398
\(933\) 2.99755e7 1.12736
\(934\) 4.30216e7 1.61369
\(935\) 4.04413e7 1.51285
\(936\) −9.80744e6 −0.365903
\(937\) 3.75738e7 1.39809 0.699046 0.715077i \(-0.253608\pi\)
0.699046 + 0.715077i \(0.253608\pi\)
\(938\) 0 0
\(939\) 2.55303e7 0.944915
\(940\) 6.58558e7 2.43094
\(941\) 6.87043e6 0.252936 0.126468 0.991971i \(-0.459636\pi\)
0.126468 + 0.991971i \(0.459636\pi\)
\(942\) 3.30218e6 0.121248
\(943\) 793550. 0.0290600
\(944\) −3.45476e6 −0.126179
\(945\) 0 0
\(946\) −9.21446e7 −3.34767
\(947\) −2.30923e7 −0.836744 −0.418372 0.908276i \(-0.637399\pi\)
−0.418372 + 0.908276i \(0.637399\pi\)
\(948\) −2.25906e7 −0.816409
\(949\) 1.44632e7 0.521314
\(950\) −2.59583e7 −0.933185
\(951\) 9.79551e6 0.351217
\(952\) 0 0
\(953\) −2.95399e7 −1.05360 −0.526802 0.849988i \(-0.676609\pi\)
−0.526802 + 0.849988i \(0.676609\pi\)
\(954\) 2.25448e7 0.802003
\(955\) −1.95142e7 −0.692376
\(956\) 2.08862e7 0.739119
\(957\) −3.80156e6 −0.134178
\(958\) 2.70919e6 0.0953730
\(959\) 0 0
\(960\) −3.19261e7 −1.11807
\(961\) 3.87161e7 1.35233
\(962\) 1.40702e7 0.490188
\(963\) −7.64442e6 −0.265631
\(964\) −9.61329e7 −3.33180
\(965\) 1.15009e6 0.0397569
\(966\) 0 0
\(967\) 1.49151e7 0.512931 0.256466 0.966553i \(-0.417442\pi\)
0.256466 + 0.966553i \(0.417442\pi\)
\(968\) −3.47761e7 −1.19287
\(969\) −1.23869e7 −0.423794
\(970\) −8.70162e6 −0.296941
\(971\) 1.54830e7 0.526995 0.263498 0.964660i \(-0.415124\pi\)
0.263498 + 0.964660i \(0.415124\pi\)
\(972\) −3.31676e6 −0.112603
\(973\) 0 0
\(974\) 7.33992e7 2.47910
\(975\) −9.69947e6 −0.326766
\(976\) 318311. 0.0106961
\(977\) −2.83428e7 −0.949961 −0.474980 0.879996i \(-0.657545\pi\)
−0.474980 + 0.879996i \(0.657545\pi\)
\(978\) −4.03884e7 −1.35023
\(979\) −7.64410e6 −0.254900
\(980\) 0 0
\(981\) −6.75487e6 −0.224101
\(982\) −2.95280e7 −0.977138
\(983\) 2.56993e7 0.848278 0.424139 0.905597i \(-0.360577\pi\)
0.424139 + 0.905597i \(0.360577\pi\)
\(984\) 502113. 0.0165316
\(985\) −7.40446e7 −2.43166
\(986\) 7.11518e6 0.233074
\(987\) 0 0
\(988\) 4.10106e7 1.33661
\(989\) −5.65059e7 −1.83697
\(990\) −3.05844e7 −0.991774
\(991\) 4.46405e7 1.44392 0.721962 0.691933i \(-0.243240\pi\)
0.721962 + 0.691933i \(0.243240\pi\)
\(992\) 3.38921e7 1.09350
\(993\) −1.17499e7 −0.378148
\(994\) 0 0
\(995\) 6.32461e7 2.02524
\(996\) −7.88957e6 −0.252003
\(997\) 2.30554e7 0.734572 0.367286 0.930108i \(-0.380287\pi\)
0.367286 + 0.930108i \(0.380287\pi\)
\(998\) −6.31528e7 −2.00709
\(999\) 2.04752e6 0.0649103
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 147.6.a.i.1.1 2
3.2 odd 2 441.6.a.t.1.2 2
7.2 even 3 21.6.e.b.4.2 4
7.3 odd 6 147.6.e.l.79.2 4
7.4 even 3 21.6.e.b.16.2 yes 4
7.5 odd 6 147.6.e.l.67.2 4
7.6 odd 2 147.6.a.k.1.1 2
21.2 odd 6 63.6.e.c.46.1 4
21.11 odd 6 63.6.e.c.37.1 4
21.20 even 2 441.6.a.s.1.2 2
28.11 odd 6 336.6.q.e.289.2 4
28.23 odd 6 336.6.q.e.193.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.b.4.2 4 7.2 even 3
21.6.e.b.16.2 yes 4 7.4 even 3
63.6.e.c.37.1 4 21.11 odd 6
63.6.e.c.46.1 4 21.2 odd 6
147.6.a.i.1.1 2 1.1 even 1 trivial
147.6.a.k.1.1 2 7.6 odd 2
147.6.e.l.67.2 4 7.5 odd 6
147.6.e.l.79.2 4 7.3 odd 6
336.6.q.e.193.2 4 28.23 odd 6
336.6.q.e.289.2 4 28.11 odd 6
441.6.a.s.1.2 2 21.20 even 2
441.6.a.t.1.2 2 3.2 odd 2