Properties

Label 147.6.a.k.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 \(-0.278286\pi\)
0.641564 + 0.767069i \(0.278286\pi\)
\(6\) −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 \(-0.644234\pi\)
−0.437777 + 0.899083i \(0.644234\pi\)
\(14\) 0 0
\(15\) 645.562 0.740815
\(16\) 333.597 0.325778
\(17\) −1005.70 −0.844007 −0.422004 0.906594i \(-0.638673\pi\)
−0.422004 + 0.906594i \(0.638673\pi\)
\(18\) −760.579 −0.553303
\(19\) −1368.53 −0.869699 −0.434850 0.900503i \(-0.643199\pi\)
−0.434850 + 0.900503i \(0.643199\pi\)
\(20\) 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 \(-0.778185\pi\)
−0.766866 + 0.641807i \(0.778185\pi\)
\(32\) 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 \(-0.503635\pi\)
−0.0114193 + 0.999935i \(0.503635\pi\)
\(42\) 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 \(-0.318551\pi\)
0.539663 + 0.841881i \(0.318551\pi\)
\(48\) 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 \(-0.437965\pi\)
0.193659 + 0.981069i \(0.437965\pi\)
\(60\) 36260.9 1.30035
\(61\) −954.179 −0.0328326 −0.0164163 0.999865i \(-0.505226\pi\)
−0.0164163 + 0.999865i \(0.505226\pi\)
\(62\) 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 \(-0.596221\pi\)
−0.297705 + 0.954658i \(0.596221\pi\)
\(74\) 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 \(-0.539679\pi\)
−0.124332 + 0.992241i \(0.539679\pi\)
\(84\) 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 \(-0.529081\pi\)
−0.0912347 + 0.995829i \(0.529081\pi\)
\(90\) −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 \(-0.477793\pi\)
0.0697086 + 0.997567i \(0.477793\pi\)
\(98\) 0 0
\(99\) −45409.4 −0.465648
\(100\) 113466. 1.13466
\(101\) −25142.9 −0.245252 −0.122626 0.992453i \(-0.539132\pi\)
−0.122626 + 0.992453i \(0.539132\pi\)
\(102\) 84990.5 0.808854
\(103\) 160753. 1.49302 0.746511 0.665373i \(-0.231727\pi\)
0.746511 + 0.665373i \(0.231727\pi\)
\(104\) 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 \(-0.542496\pi\)
−0.133110 + 0.991101i \(0.542496\pi\)
\(132\) −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 \(-0.631647\pi\)
−0.401890 + 0.915688i \(0.631647\pi\)
\(140\) 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 \(-0.520149\pi\)
−0.0632587 + 0.997997i \(0.520149\pi\)
\(158\) −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 \(-0.517582\pi\)
−0.0552064 + 0.998475i \(0.517582\pi\)
\(168\) 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 \(-0.519555\pi\)
−0.0613968 + 0.998113i \(0.519555\pi\)
\(174\) 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 \(-0.472071\pi\)
0.0876285 + 0.996153i \(0.472071\pi\)
\(182\) 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 \(-0.210506\pi\)
0.789180 + 0.614162i \(0.210506\pi\)
\(200\) −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 \(-0.240199\pi\)
0.728541 + 0.685002i \(0.240199\pi\)
\(224\) 0 0
\(225\) 163625. 0.215473
\(226\) 1.68317e6 2.19208
\(227\) −553049. −0.712359 −0.356179 0.934418i \(-0.615921\pi\)
−0.356179 + 0.934418i \(0.615921\pi\)
\(228\) −691826. −0.881373
\(229\) −523024. −0.659072 −0.329536 0.944143i \(-0.606892\pi\)
−0.329536 + 0.944143i \(0.606892\pi\)
\(230\) −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 \(-0.102027\pi\)
0.949069 + 0.315067i \(0.102027\pi\)
\(242\) −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 \(-0.509271\pi\)
−0.0291217 + 0.999576i \(0.509271\pi\)
\(252\) 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 \(-0.452956\pi\)
0.147254 + 0.989099i \(0.452956\pi\)
\(258\) 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 \(-0.343480\pi\)
0.472145 + 0.881521i \(0.343480\pi\)
\(270\) −491001. −0.409895
\(271\) −1.14012e6 −0.943030 −0.471515 0.881858i \(-0.656293\pi\)
−0.471515 + 0.881858i \(0.656293\pi\)
\(272\) −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 \(-0.927350\pi\)
−0.974067 + 0.226259i \(0.927350\pi\)
\(284\) 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 \(-0.566546\pi\)
−0.207541 + 0.978226i \(0.566546\pi\)
\(294\) 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 \(-0.366130\pi\)
0.408275 + 0.912859i \(0.366130\pi\)
\(308\) 0 0
\(309\) 1.44678e6 0.861997
\(310\) 5.52724e6 3.26666
\(311\) 3.33061e6 1.95264 0.976320 0.216330i \(-0.0694087\pi\)
0.976320 + 0.216330i \(0.0694087\pi\)
\(312\) 1.08972e6 0.633762
\(313\) 2.83670e6 1.63664 0.818320 0.574763i \(-0.194906\pi\)
0.818320 + 0.574763i \(0.194906\pi\)
\(314\) 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 \(-0.274755\pi\)
0.650034 + 0.759905i \(0.274755\pi\)
\(350\) 0 0
\(351\) −388928. −0.168501
\(352\) −2.31529e6 −0.995976
\(353\) −3.76980e6 −1.61021 −0.805103 0.593135i \(-0.797890\pi\)
−0.805103 + 0.593135i \(0.797890\pi\)
\(354\) −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 \(-0.651704\pi\)
−0.458754 + 0.888563i \(0.651704\pi\)
\(368\) 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 \(-0.650096\pi\)
−0.454261 + 0.890869i \(0.650096\pi\)
\(384\) 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 \(-0.373511\pi\)
0.387000 + 0.922080i \(0.373511\pi\)
\(398\) −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 \(-0.250644\pi\)
0.705674 + 0.708537i \(0.250644\pi\)
\(410\) 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 \(-0.479739\pi\)
0.0636097 + 0.997975i \(0.479739\pi\)
\(420\) 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 \(-0.488360\pi\)
0.0365595 + 0.999331i \(0.488360\pi\)
\(434\) 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 \(-0.681164\pi\)
−0.538911 + 0.842363i \(0.681164\pi\)
\(440\) 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 \(-0.229395\pi\)
0.751367 + 0.659884i \(0.229395\pi\)
\(462\) 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 \(-0.338427\pi\)
0.486077 + 0.873916i \(0.338427\pi\)
\(468\) −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 \(-0.490854\pi\)
0.0287284 + 0.999587i \(0.490854\pi\)
\(480\) 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 \(-0.186773\pi\)
0.832737 + 0.553669i \(0.186773\pi\)
\(504\) 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 \(-0.227185\pi\)
0.755930 + 0.654653i \(0.227185\pi\)
\(510\) 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 \(-0.517862\pi\)
−0.0560855 + 0.998426i \(0.517862\pi\)
\(522\) 573063. 0.0920505
\(523\) −3.58210e6 −0.572643 −0.286321 0.958134i \(-0.592432\pi\)
−0.286321 + 0.958134i \(0.592432\pi\)
\(524\) −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 \(-0.340137\pi\)
0.481375 + 0.876515i \(0.340137\pi\)
\(564\) 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 \(-0.761174\pi\)
−0.731488 + 0.681855i \(0.761174\pi\)
\(578\) 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 \(-0.363894\pi\)
0.414678 + 0.909968i \(0.363894\pi\)
\(588\) 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 \(-0.872797\pi\)
−0.921208 + 0.389069i \(0.872797\pi\)
\(594\) 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 \(-0.641824\pi\)
−0.430958 + 0.902372i \(0.641824\pi\)
\(602\) 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 \(-0.437170\pi\)
0.196106 + 0.980583i \(0.437170\pi\)
\(608\) −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 \(-0.653672\pi\)
−0.464238 + 0.885711i \(0.653672\pi\)
\(620\) −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 \(-0.289649\pi\)
0.613779 + 0.789478i \(0.289649\pi\)
\(644\) 0 0
\(645\) −1.13002e7 −1.06952
\(646\) −1.29235e7 −1.21843
\(647\) 1.14731e7 1.07751 0.538754 0.842463i \(-0.318895\pi\)
0.538754 + 0.842463i \(0.318895\pi\)
\(648\) −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 \(-0.531266\pi\)
−0.0980661 + 0.995180i \(0.531266\pi\)
\(662\) −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 \(-0.191866\pi\)
0.823771 + 0.566923i \(0.191866\pi\)
\(678\) 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 \(-0.782088\pi\)
−0.774677 + 0.632357i \(0.782088\pi\)
\(692\) −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 \(-0.272540\pi\)
0.655305 + 0.755364i \(0.272540\pi\)
\(720\) 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 \(-0.646886\pi\)
−0.445253 + 0.895405i \(0.646886\pi\)
\(728\) 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 \(-0.465864\pi\)
0.107036 + 0.994255i \(0.465864\pi\)
\(734\) 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 \(-0.496083\pi\)
0.0123056 + 0.999924i \(0.496083\pi\)
\(762\) 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 \(-0.659459\pi\)
−0.480265 + 0.877124i \(0.659459\pi\)
\(770\) 0 0
\(771\) 2.80655e6 0.170035
\(772\) −900611. −0.0543869
\(773\) −2.88472e6 −0.173642 −0.0868210 0.996224i \(-0.527671\pi\)
−0.0868210 + 0.996224i \(0.527671\pi\)
\(774\) 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 \(-0.172236\pi\)
0.857145 + 0.515076i \(0.172236\pi\)
\(788\) 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 \(-0.572634\pi\)
−0.226213 + 0.974078i \(0.572634\pi\)
\(798\) 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 \(-0.498222\pi\)
0.00558578 + 0.999984i \(0.498222\pi\)
\(812\) 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 \(-0.661632\pi\)
−0.486239 + 0.873826i \(0.661632\pi\)
\(830\) 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 \(-0.707775\pi\)
−0.607369 + 0.794420i \(0.707775\pi\)
\(840\) 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 \(-0.259666\pi\)
0.685312 + 0.728250i \(0.259666\pi\)
\(854\) 0 0
\(855\) −7.95121e6 −0.371979
\(856\) 2.14185e7 0.999088
\(857\) 3.22586e6 0.150035 0.0750175 0.997182i \(-0.476099\pi\)
0.0750175 + 0.997182i \(0.476099\pi\)
\(858\) −2.52758e7 −1.17216
\(859\) −2.64671e7 −1.22384 −0.611918 0.790921i \(-0.709602\pi\)
−0.611918 + 0.790921i \(0.709602\pi\)
\(860\) −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 \(-0.202431\pi\)
0.804504 + 0.593947i \(0.202431\pi\)
\(882\) 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 \(-0.624089\pi\)
−0.380038 + 0.924971i \(0.624089\pi\)
\(888\) 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 \(-0.525745\pi\)
−0.0807927 + 0.996731i \(0.525745\pi\)
\(930\) 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 \(-0.746392\pi\)
−0.699046 + 0.715077i \(0.746392\pi\)
\(938\) 0 0
\(939\) 2.55303e7 0.944915
\(940\) 6.58558e7 2.43094
\(941\) −6.87043e6 −0.252936 −0.126468 0.991971i \(-0.540364\pi\)
−0.126468 + 0.991971i \(0.540364\pi\)
\(942\) 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 \(-0.584876\pi\)
−0.263498 + 0.964660i \(0.584876\pi\)
\(972\) 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 \(-0.639423\pi\)
−0.424139 + 0.905597i \(0.639423\pi\)
\(984\) 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 \(-0.619713\pi\)
−0.367286 + 0.930108i \(0.619713\pi\)
\(998\) −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.k.1.1 2
3.2 odd 2 441.6.a.s.1.2 2
7.2 even 3 147.6.e.l.67.2 4
7.3 odd 6 21.6.e.b.16.2 yes 4
7.4 even 3 147.6.e.l.79.2 4
7.5 odd 6 21.6.e.b.4.2 4
7.6 odd 2 147.6.a.i.1.1 2
21.5 even 6 63.6.e.c.46.1 4
21.17 even 6 63.6.e.c.37.1 4
21.20 even 2 441.6.a.t.1.2 2
28.3 even 6 336.6.q.e.289.2 4
28.19 even 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 7.5 odd 6
21.6.e.b.16.2 yes 4 7.3 odd 6
63.6.e.c.37.1 4 21.17 even 6
63.6.e.c.46.1 4 21.5 even 6
147.6.a.i.1.1 2 7.6 odd 2
147.6.a.k.1.1 2 1.1 even 1 trivial
147.6.e.l.67.2 4 7.2 even 3
147.6.e.l.79.2 4 7.4 even 3
336.6.q.e.193.2 4 28.19 even 6
336.6.q.e.289.2 4 28.3 even 6
441.6.a.s.1.2 2 3.2 odd 2
441.6.a.t.1.2 2 21.20 even 2