Properties

Label 150.6.a.n.1.2
Level $150$
Weight $6$
Character 150.1
Self dual yes
Analytic conductor $24.058$
Analytic rank $0$
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] = [150,6,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, 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("150.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(24.0575729719\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{1249}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 312 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 5 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-17.1706\) of defining polynomial
Character \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +119.706 q^{7} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +36.0000 q^{6} +119.706 q^{7} -64.0000 q^{8} +81.0000 q^{9} +263.706 q^{11} -144.000 q^{12} -851.118 q^{13} -478.824 q^{14} +256.000 q^{16} -1287.12 q^{17} -324.000 q^{18} +2060.47 q^{19} -1077.35 q^{21} -1054.82 q^{22} -55.5284 q^{23} +576.000 q^{24} +3404.47 q^{26} -729.000 q^{27} +1915.30 q^{28} -5986.06 q^{29} +4781.76 q^{31} -1024.00 q^{32} -2373.35 q^{33} +5148.47 q^{34} +1296.00 q^{36} +12150.4 q^{37} -8241.89 q^{38} +7660.06 q^{39} +18500.0 q^{41} +4309.41 q^{42} +2188.47 q^{43} +4219.30 q^{44} +222.113 q^{46} +5597.76 q^{47} -2304.00 q^{48} -2477.48 q^{49} +11584.1 q^{51} -13617.9 q^{52} +26463.4 q^{53} +2916.00 q^{54} -7661.18 q^{56} -18544.2 q^{57} +23944.2 q^{58} +20825.6 q^{59} +45525.8 q^{61} -19127.1 q^{62} +9696.18 q^{63} +4096.00 q^{64} +9493.41 q^{66} +34354.7 q^{67} -20593.9 q^{68} +499.755 q^{69} -57489.6 q^{71} -5184.00 q^{72} +26956.8 q^{73} -48601.7 q^{74} +32967.5 q^{76} +31567.2 q^{77} -30640.2 q^{78} +42097.8 q^{79} +6561.00 q^{81} -74000.0 q^{82} -101733. q^{83} -17237.7 q^{84} -8753.86 q^{86} +53874.6 q^{87} -16877.2 q^{88} -65551.2 q^{89} -101884. q^{91} -888.454 q^{92} -43035.9 q^{93} -22391.1 q^{94} +9216.00 q^{96} +82780.7 q^{97} +9909.92 q^{98} +21360.2 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} - 18 q^{3} + 32 q^{4} + 72 q^{6} - 114 q^{7} - 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} - 18 q^{3} + 32 q^{4} + 72 q^{6} - 114 q^{7} - 128 q^{8} + 162 q^{9} + 174 q^{11} - 288 q^{12} - 642 q^{13} + 456 q^{14} + 512 q^{16} - 1514 q^{17} - 648 q^{18} - 120 q^{19} + 1026 q^{21} - 696 q^{22} - 4352 q^{23} + 1152 q^{24} + 2568 q^{26} - 1458 q^{27} - 1824 q^{28} - 2430 q^{29} + 11684 q^{31} - 2048 q^{32} - 1566 q^{33} + 6056 q^{34} + 2592 q^{36} + 17586 q^{37} + 480 q^{38} + 5778 q^{39} + 24984 q^{41} - 4104 q^{42} + 24168 q^{43} + 2784 q^{44} + 17408 q^{46} + 13316 q^{47} - 4608 q^{48} + 35334 q^{49} + 13626 q^{51} - 10272 q^{52} + 13698 q^{53} + 5832 q^{54} + 7296 q^{56} + 1080 q^{57} + 9720 q^{58} - 23730 q^{59} + 57124 q^{61} - 46736 q^{62} - 9234 q^{63} + 8192 q^{64} + 6264 q^{66} + 38316 q^{67} - 24224 q^{68} + 39168 q^{69} - 11076 q^{71} - 10368 q^{72} + 88548 q^{73} - 70344 q^{74} - 1920 q^{76} + 52532 q^{77} - 23112 q^{78} + 14220 q^{79} + 13122 q^{81} - 99936 q^{82} - 50792 q^{83} + 16416 q^{84} - 96672 q^{86} + 21870 q^{87} - 11136 q^{88} - 60420 q^{89} - 150756 q^{91} - 69632 q^{92} - 105156 q^{93} - 53264 q^{94} + 18432 q^{96} + 171216 q^{97} - 141336 q^{98} + 14094 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) 36.0000 0.408248
\(7\) 119.706 0.923359 0.461680 0.887047i \(-0.347247\pi\)
0.461680 + 0.887047i \(0.347247\pi\)
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 263.706 0.657110 0.328555 0.944485i \(-0.393438\pi\)
0.328555 + 0.944485i \(0.393438\pi\)
\(12\) −144.000 −0.288675
\(13\) −851.118 −1.39679 −0.698395 0.715712i \(-0.746102\pi\)
−0.698395 + 0.715712i \(0.746102\pi\)
\(14\) −478.824 −0.652914
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −1287.12 −1.08018 −0.540090 0.841607i \(-0.681610\pi\)
−0.540090 + 0.841607i \(0.681610\pi\)
\(18\) −324.000 −0.235702
\(19\) 2060.47 1.30943 0.654716 0.755875i \(-0.272788\pi\)
0.654716 + 0.755875i \(0.272788\pi\)
\(20\) 0 0
\(21\) −1077.35 −0.533102
\(22\) −1054.82 −0.464647
\(23\) −55.5284 −0.0218875 −0.0109437 0.999940i \(-0.503484\pi\)
−0.0109437 + 0.999940i \(0.503484\pi\)
\(24\) 576.000 0.204124
\(25\) 0 0
\(26\) 3404.47 0.987680
\(27\) −729.000 −0.192450
\(28\) 1915.30 0.461680
\(29\) −5986.06 −1.32174 −0.660870 0.750500i \(-0.729813\pi\)
−0.660870 + 0.750500i \(0.729813\pi\)
\(30\) 0 0
\(31\) 4781.76 0.893684 0.446842 0.894613i \(-0.352549\pi\)
0.446842 + 0.894613i \(0.352549\pi\)
\(32\) −1024.00 −0.176777
\(33\) −2373.35 −0.379383
\(34\) 5148.47 0.763802
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 12150.4 1.45911 0.729553 0.683924i \(-0.239728\pi\)
0.729553 + 0.683924i \(0.239728\pi\)
\(38\) −8241.89 −0.925908
\(39\) 7660.06 0.806438
\(40\) 0 0
\(41\) 18500.0 1.71875 0.859374 0.511348i \(-0.170854\pi\)
0.859374 + 0.511348i \(0.170854\pi\)
\(42\) 4309.41 0.376960
\(43\) 2188.47 0.180496 0.0902482 0.995919i \(-0.471234\pi\)
0.0902482 + 0.995919i \(0.471234\pi\)
\(44\) 4219.30 0.328555
\(45\) 0 0
\(46\) 222.113 0.0154768
\(47\) 5597.76 0.369632 0.184816 0.982773i \(-0.440831\pi\)
0.184816 + 0.982773i \(0.440831\pi\)
\(48\) −2304.00 −0.144338
\(49\) −2477.48 −0.147408
\(50\) 0 0
\(51\) 11584.1 0.623642
\(52\) −13617.9 −0.698395
\(53\) 26463.4 1.29406 0.647031 0.762463i \(-0.276010\pi\)
0.647031 + 0.762463i \(0.276010\pi\)
\(54\) 2916.00 0.136083
\(55\) 0 0
\(56\) −7661.18 −0.326457
\(57\) −18544.2 −0.756000
\(58\) 23944.2 0.934612
\(59\) 20825.6 0.778875 0.389437 0.921053i \(-0.372669\pi\)
0.389437 + 0.921053i \(0.372669\pi\)
\(60\) 0 0
\(61\) 45525.8 1.56651 0.783254 0.621702i \(-0.213558\pi\)
0.783254 + 0.621702i \(0.213558\pi\)
\(62\) −19127.1 −0.631930
\(63\) 9696.18 0.307786
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 9493.41 0.268264
\(67\) 34354.7 0.934974 0.467487 0.884000i \(-0.345160\pi\)
0.467487 + 0.884000i \(0.345160\pi\)
\(68\) −20593.9 −0.540090
\(69\) 499.755 0.0126367
\(70\) 0 0
\(71\) −57489.6 −1.35345 −0.676726 0.736235i \(-0.736602\pi\)
−0.676726 + 0.736235i \(0.736602\pi\)
\(72\) −5184.00 −0.117851
\(73\) 26956.8 0.592054 0.296027 0.955180i \(-0.404338\pi\)
0.296027 + 0.955180i \(0.404338\pi\)
\(74\) −48601.7 −1.03174
\(75\) 0 0
\(76\) 32967.5 0.654716
\(77\) 31567.2 0.606749
\(78\) −30640.2 −0.570237
\(79\) 42097.8 0.758912 0.379456 0.925210i \(-0.376111\pi\)
0.379456 + 0.925210i \(0.376111\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −74000.0 −1.21534
\(83\) −101733. −1.62094 −0.810469 0.585781i \(-0.800788\pi\)
−0.810469 + 0.585781i \(0.800788\pi\)
\(84\) −17237.7 −0.266551
\(85\) 0 0
\(86\) −8753.86 −0.127630
\(87\) 53874.6 0.763107
\(88\) −16877.2 −0.232324
\(89\) −65551.2 −0.877214 −0.438607 0.898679i \(-0.644528\pi\)
−0.438607 + 0.898679i \(0.644528\pi\)
\(90\) 0 0
\(91\) −101884. −1.28974
\(92\) −888.454 −0.0109437
\(93\) −43035.9 −0.515969
\(94\) −22391.1 −0.261370
\(95\) 0 0
\(96\) 9216.00 0.102062
\(97\) 82780.7 0.893305 0.446653 0.894708i \(-0.352616\pi\)
0.446653 + 0.894708i \(0.352616\pi\)
\(98\) 9909.92 0.104233
\(99\) 21360.2 0.219037
\(100\) 0 0
\(101\) 14644.9 0.142851 0.0714255 0.997446i \(-0.477245\pi\)
0.0714255 + 0.997446i \(0.477245\pi\)
\(102\) −46336.2 −0.440982
\(103\) 199927. 1.85686 0.928429 0.371511i \(-0.121160\pi\)
0.928429 + 0.371511i \(0.121160\pi\)
\(104\) 54471.5 0.493840
\(105\) 0 0
\(106\) −105853. −0.915041
\(107\) −34770.4 −0.293596 −0.146798 0.989166i \(-0.546897\pi\)
−0.146798 + 0.989166i \(0.546897\pi\)
\(108\) −11664.0 −0.0962250
\(109\) −19636.5 −0.158306 −0.0791529 0.996862i \(-0.525222\pi\)
−0.0791529 + 0.996862i \(0.525222\pi\)
\(110\) 0 0
\(111\) −109354. −0.842415
\(112\) 30644.7 0.230840
\(113\) 19716.6 0.145256 0.0726282 0.997359i \(-0.476861\pi\)
0.0726282 + 0.997359i \(0.476861\pi\)
\(114\) 74177.0 0.534573
\(115\) 0 0
\(116\) −95777.0 −0.660870
\(117\) −68940.6 −0.465597
\(118\) −83302.4 −0.550748
\(119\) −154076. −0.997394
\(120\) 0 0
\(121\) −91510.2 −0.568206
\(122\) −182103. −1.10769
\(123\) −166500. −0.992320
\(124\) 76508.2 0.446842
\(125\) 0 0
\(126\) −38784.7 −0.217638
\(127\) 55823.7 0.307121 0.153560 0.988139i \(-0.450926\pi\)
0.153560 + 0.988139i \(0.450926\pi\)
\(128\) −16384.0 −0.0883883
\(129\) −19696.2 −0.104210
\(130\) 0 0
\(131\) 136377. 0.694326 0.347163 0.937805i \(-0.387145\pi\)
0.347163 + 0.937805i \(0.387145\pi\)
\(132\) −37973.7 −0.189691
\(133\) 246651. 1.20908
\(134\) −137419. −0.661126
\(135\) 0 0
\(136\) 82375.5 0.381901
\(137\) 134387. 0.611722 0.305861 0.952076i \(-0.401056\pi\)
0.305861 + 0.952076i \(0.401056\pi\)
\(138\) −1999.02 −0.00893551
\(139\) −305523. −1.34124 −0.670620 0.741801i \(-0.733972\pi\)
−0.670620 + 0.741801i \(0.733972\pi\)
\(140\) 0 0
\(141\) −50379.9 −0.213407
\(142\) 229958. 0.957036
\(143\) −224445. −0.917846
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) −107827. −0.418646
\(147\) 22297.3 0.0851059
\(148\) 194407. 0.729553
\(149\) 349387. 1.28926 0.644630 0.764494i \(-0.277011\pi\)
0.644630 + 0.764494i \(0.277011\pi\)
\(150\) 0 0
\(151\) −314525. −1.12257 −0.561284 0.827623i \(-0.689692\pi\)
−0.561284 + 0.827623i \(0.689692\pi\)
\(152\) −131870. −0.462954
\(153\) −104257. −0.360060
\(154\) −126269. −0.429036
\(155\) 0 0
\(156\) 122561. 0.403219
\(157\) 131302. 0.425132 0.212566 0.977147i \(-0.431818\pi\)
0.212566 + 0.977147i \(0.431818\pi\)
\(158\) −168391. −0.536632
\(159\) −238170. −0.747128
\(160\) 0 0
\(161\) −6647.08 −0.0202100
\(162\) −26244.0 −0.0785674
\(163\) 633204. 1.86670 0.933351 0.358966i \(-0.116871\pi\)
0.933351 + 0.358966i \(0.116871\pi\)
\(164\) 296000. 0.859374
\(165\) 0 0
\(166\) 406932. 1.14618
\(167\) −137690. −0.382042 −0.191021 0.981586i \(-0.561180\pi\)
−0.191021 + 0.981586i \(0.561180\pi\)
\(168\) 68950.6 0.188480
\(169\) 353109. 0.951024
\(170\) 0 0
\(171\) 166898. 0.436477
\(172\) 35015.5 0.0902482
\(173\) 90278.1 0.229333 0.114667 0.993404i \(-0.463420\pi\)
0.114667 + 0.993404i \(0.463420\pi\)
\(174\) −215498. −0.539598
\(175\) 0 0
\(176\) 67508.7 0.164278
\(177\) −187430. −0.449684
\(178\) 262205. 0.620284
\(179\) 601579. 1.40333 0.701665 0.712507i \(-0.252440\pi\)
0.701665 + 0.712507i \(0.252440\pi\)
\(180\) 0 0
\(181\) 447746. 1.01586 0.507931 0.861398i \(-0.330410\pi\)
0.507931 + 0.861398i \(0.330410\pi\)
\(182\) 407536. 0.911984
\(183\) −409732. −0.904424
\(184\) 3553.81 0.00773838
\(185\) 0 0
\(186\) 172144. 0.364845
\(187\) −339421. −0.709797
\(188\) 89564.2 0.184816
\(189\) −87265.7 −0.177701
\(190\) 0 0
\(191\) 673846. 1.33653 0.668263 0.743925i \(-0.267038\pi\)
0.668263 + 0.743925i \(0.267038\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −255962. −0.494632 −0.247316 0.968935i \(-0.579549\pi\)
−0.247316 + 0.968935i \(0.579549\pi\)
\(194\) −331123. −0.631662
\(195\) 0 0
\(196\) −39639.7 −0.0737038
\(197\) −196202. −0.360195 −0.180097 0.983649i \(-0.557641\pi\)
−0.180097 + 0.983649i \(0.557641\pi\)
\(198\) −85440.7 −0.154882
\(199\) 118265. 0.211702 0.105851 0.994382i \(-0.466243\pi\)
0.105851 + 0.994382i \(0.466243\pi\)
\(200\) 0 0
\(201\) −309192. −0.539807
\(202\) −58579.7 −0.101011
\(203\) −716567. −1.22044
\(204\) 185345. 0.311821
\(205\) 0 0
\(206\) −799708. −1.31300
\(207\) −4497.80 −0.00729582
\(208\) −217886. −0.349198
\(209\) 543359. 0.860441
\(210\) 0 0
\(211\) 910278. 1.40756 0.703781 0.710417i \(-0.251493\pi\)
0.703781 + 0.710417i \(0.251493\pi\)
\(212\) 423414. 0.647031
\(213\) 517406. 0.781416
\(214\) 139082. 0.207604
\(215\) 0 0
\(216\) 46656.0 0.0680414
\(217\) 572406. 0.825191
\(218\) 78545.9 0.111939
\(219\) −242611. −0.341823
\(220\) 0 0
\(221\) 1.09549e6 1.50879
\(222\) 437415. 0.595677
\(223\) −1.04513e6 −1.40737 −0.703683 0.710514i \(-0.748462\pi\)
−0.703683 + 0.710514i \(0.748462\pi\)
\(224\) −122579. −0.163228
\(225\) 0 0
\(226\) −78866.3 −0.102712
\(227\) −869120. −1.11948 −0.559739 0.828669i \(-0.689098\pi\)
−0.559739 + 0.828669i \(0.689098\pi\)
\(228\) −296708. −0.378000
\(229\) −1.39907e6 −1.76299 −0.881494 0.472195i \(-0.843462\pi\)
−0.881494 + 0.472195i \(0.843462\pi\)
\(230\) 0 0
\(231\) −284105. −0.350307
\(232\) 383108. 0.467306
\(233\) 403494. 0.486908 0.243454 0.969912i \(-0.421720\pi\)
0.243454 + 0.969912i \(0.421720\pi\)
\(234\) 275762. 0.329227
\(235\) 0 0
\(236\) 333210. 0.389437
\(237\) −378880. −0.438158
\(238\) 616303. 0.705264
\(239\) −611076. −0.691991 −0.345995 0.938236i \(-0.612459\pi\)
−0.345995 + 0.938236i \(0.612459\pi\)
\(240\) 0 0
\(241\) −1.50639e6 −1.67068 −0.835342 0.549730i \(-0.814730\pi\)
−0.835342 + 0.549730i \(0.814730\pi\)
\(242\) 366041. 0.401782
\(243\) −59049.0 −0.0641500
\(244\) 728412. 0.783254
\(245\) 0 0
\(246\) 666000. 0.701676
\(247\) −1.75370e6 −1.82900
\(248\) −306033. −0.315965
\(249\) 915597. 0.935849
\(250\) 0 0
\(251\) 558727. 0.559777 0.279889 0.960032i \(-0.409702\pi\)
0.279889 + 0.960032i \(0.409702\pi\)
\(252\) 155139. 0.153893
\(253\) −14643.2 −0.0143825
\(254\) −223295. −0.217167
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) −680108. −0.642310 −0.321155 0.947027i \(-0.604071\pi\)
−0.321155 + 0.947027i \(0.604071\pi\)
\(258\) 78784.8 0.0736874
\(259\) 1.45448e6 1.34728
\(260\) 0 0
\(261\) −484871. −0.440580
\(262\) −545508. −0.490962
\(263\) 428668. 0.382148 0.191074 0.981576i \(-0.438803\pi\)
0.191074 + 0.981576i \(0.438803\pi\)
\(264\) 151895. 0.134132
\(265\) 0 0
\(266\) −986603. −0.854945
\(267\) 589961. 0.506460
\(268\) 549675. 0.467487
\(269\) −1.51936e6 −1.28020 −0.640101 0.768291i \(-0.721108\pi\)
−0.640101 + 0.768291i \(0.721108\pi\)
\(270\) 0 0
\(271\) −1.34595e6 −1.11329 −0.556643 0.830752i \(-0.687911\pi\)
−0.556643 + 0.830752i \(0.687911\pi\)
\(272\) −329502. −0.270045
\(273\) 916955. 0.744632
\(274\) −537546. −0.432553
\(275\) 0 0
\(276\) 7996.08 0.00631836
\(277\) 885541. 0.693440 0.346720 0.937969i \(-0.387295\pi\)
0.346720 + 0.937969i \(0.387295\pi\)
\(278\) 1.22209e6 0.948400
\(279\) 387323. 0.297895
\(280\) 0 0
\(281\) −1.26803e6 −0.957999 −0.478999 0.877815i \(-0.659000\pi\)
−0.478999 + 0.877815i \(0.659000\pi\)
\(282\) 201520. 0.150902
\(283\) −685833. −0.509040 −0.254520 0.967067i \(-0.581918\pi\)
−0.254520 + 0.967067i \(0.581918\pi\)
\(284\) −919833. −0.676726
\(285\) 0 0
\(286\) 897779. 0.649015
\(287\) 2.21456e6 1.58702
\(288\) −82944.0 −0.0589256
\(289\) 236816. 0.166788
\(290\) 0 0
\(291\) −745026. −0.515750
\(292\) 431309. 0.296027
\(293\) 1.66857e6 1.13547 0.567736 0.823211i \(-0.307819\pi\)
0.567736 + 0.823211i \(0.307819\pi\)
\(294\) −89189.3 −0.0601789
\(295\) 0 0
\(296\) −777626. −0.515872
\(297\) −192242. −0.126461
\(298\) −1.39755e6 −0.911645
\(299\) 47261.2 0.0305722
\(300\) 0 0
\(301\) 261972. 0.166663
\(302\) 1.25810e6 0.793776
\(303\) −131804. −0.0824751
\(304\) 527481. 0.327358
\(305\) 0 0
\(306\) 417026. 0.254601
\(307\) 560776. 0.339581 0.169790 0.985480i \(-0.445691\pi\)
0.169790 + 0.985480i \(0.445691\pi\)
\(308\) 505075. 0.303374
\(309\) −1.79934e6 −1.07206
\(310\) 0 0
\(311\) −1.33943e6 −0.785268 −0.392634 0.919695i \(-0.628436\pi\)
−0.392634 + 0.919695i \(0.628436\pi\)
\(312\) −490244. −0.285119
\(313\) −621225. −0.358417 −0.179208 0.983811i \(-0.557354\pi\)
−0.179208 + 0.983811i \(0.557354\pi\)
\(314\) −525210. −0.300614
\(315\) 0 0
\(316\) 673565. 0.379456
\(317\) −300282. −0.167834 −0.0839172 0.996473i \(-0.526743\pi\)
−0.0839172 + 0.996473i \(0.526743\pi\)
\(318\) 952681. 0.528299
\(319\) −1.57856e6 −0.868529
\(320\) 0 0
\(321\) 312934. 0.169508
\(322\) 26588.3 0.0142906
\(323\) −2.65207e6 −1.41442
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) −2.53282e6 −1.31996
\(327\) 176728. 0.0913979
\(328\) −1.18400e6 −0.607669
\(329\) 670086. 0.341303
\(330\) 0 0
\(331\) −1.94914e6 −0.977851 −0.488925 0.872326i \(-0.662611\pi\)
−0.488925 + 0.872326i \(0.662611\pi\)
\(332\) −1.62773e6 −0.810469
\(333\) 984183. 0.486369
\(334\) 550760. 0.270145
\(335\) 0 0
\(336\) −275803. −0.133275
\(337\) −253373. −0.121530 −0.0607652 0.998152i \(-0.519354\pi\)
−0.0607652 + 0.998152i \(0.519354\pi\)
\(338\) −1.41243e6 −0.672476
\(339\) −177449. −0.0838638
\(340\) 0 0
\(341\) 1.26098e6 0.587249
\(342\) −667593. −0.308636
\(343\) −2.30847e6 −1.05947
\(344\) −140062. −0.0638151
\(345\) 0 0
\(346\) −361112. −0.162163
\(347\) −1.60688e6 −0.716406 −0.358203 0.933644i \(-0.616610\pi\)
−0.358203 + 0.933644i \(0.616610\pi\)
\(348\) 861993. 0.381554
\(349\) −608374. −0.267367 −0.133683 0.991024i \(-0.542680\pi\)
−0.133683 + 0.991024i \(0.542680\pi\)
\(350\) 0 0
\(351\) 620465. 0.268813
\(352\) −270035. −0.116162
\(353\) 3.83773e6 1.63922 0.819612 0.572919i \(-0.194189\pi\)
0.819612 + 0.572919i \(0.194189\pi\)
\(354\) 749722. 0.317974
\(355\) 0 0
\(356\) −1.04882e6 −0.438607
\(357\) 1.38668e6 0.575846
\(358\) −2.40632e6 −0.992305
\(359\) 2.13850e6 0.875738 0.437869 0.899039i \(-0.355733\pi\)
0.437869 + 0.899039i \(0.355733\pi\)
\(360\) 0 0
\(361\) 1.76944e6 0.714610
\(362\) −1.79098e6 −0.718323
\(363\) 823591. 0.328054
\(364\) −1.63014e6 −0.644870
\(365\) 0 0
\(366\) 1.63893e6 0.639524
\(367\) −4.50396e6 −1.74554 −0.872769 0.488133i \(-0.837678\pi\)
−0.872769 + 0.488133i \(0.837678\pi\)
\(368\) −14215.3 −0.00547186
\(369\) 1.49850e6 0.572916
\(370\) 0 0
\(371\) 3.16782e6 1.19488
\(372\) −688574. −0.257984
\(373\) 1.15223e6 0.428811 0.214406 0.976745i \(-0.431219\pi\)
0.214406 + 0.976745i \(0.431219\pi\)
\(374\) 1.35768e6 0.501902
\(375\) 0 0
\(376\) −358257. −0.130685
\(377\) 5.09484e6 1.84619
\(378\) 349063. 0.125653
\(379\) 797740. 0.285275 0.142637 0.989775i \(-0.454442\pi\)
0.142637 + 0.989775i \(0.454442\pi\)
\(380\) 0 0
\(381\) −502413. −0.177316
\(382\) −2.69538e6 −0.945066
\(383\) −1.02124e6 −0.355737 −0.177869 0.984054i \(-0.556920\pi\)
−0.177869 + 0.984054i \(0.556920\pi\)
\(384\) 147456. 0.0510310
\(385\) 0 0
\(386\) 1.02385e6 0.349757
\(387\) 177266. 0.0601655
\(388\) 1.32449e6 0.446653
\(389\) −1.06820e6 −0.357914 −0.178957 0.983857i \(-0.557272\pi\)
−0.178957 + 0.983857i \(0.557272\pi\)
\(390\) 0 0
\(391\) 71471.5 0.0236424
\(392\) 158559. 0.0521165
\(393\) −1.22739e6 −0.400869
\(394\) 784807. 0.254696
\(395\) 0 0
\(396\) 341763. 0.109518
\(397\) −109330. −0.0348148 −0.0174074 0.999848i \(-0.505541\pi\)
−0.0174074 + 0.999848i \(0.505541\pi\)
\(398\) −473061. −0.149696
\(399\) −2.21986e6 −0.698060
\(400\) 0 0
\(401\) 1.42131e6 0.441395 0.220697 0.975342i \(-0.429167\pi\)
0.220697 + 0.975342i \(0.429167\pi\)
\(402\) 1.23677e6 0.381701
\(403\) −4.06985e6 −1.24829
\(404\) 234319. 0.0714255
\(405\) 0 0
\(406\) 2.86627e6 0.862982
\(407\) 3.20414e6 0.958793
\(408\) −741380. −0.220491
\(409\) 3.07511e6 0.908975 0.454487 0.890753i \(-0.349823\pi\)
0.454487 + 0.890753i \(0.349823\pi\)
\(410\) 0 0
\(411\) −1.20948e6 −0.353178
\(412\) 3.19883e6 0.928429
\(413\) 2.49295e6 0.719181
\(414\) 17991.2 0.00515892
\(415\) 0 0
\(416\) 871545. 0.246920
\(417\) 2.74971e6 0.774365
\(418\) −2.17343e6 −0.608423
\(419\) 3.06462e6 0.852789 0.426395 0.904537i \(-0.359783\pi\)
0.426395 + 0.904537i \(0.359783\pi\)
\(420\) 0 0
\(421\) 837782. 0.230370 0.115185 0.993344i \(-0.463254\pi\)
0.115185 + 0.993344i \(0.463254\pi\)
\(422\) −3.64111e6 −0.995297
\(423\) 453419. 0.123211
\(424\) −1.69366e6 −0.457520
\(425\) 0 0
\(426\) −2.06962e6 −0.552545
\(427\) 5.44971e6 1.44645
\(428\) −556326. −0.146798
\(429\) 2.02000e6 0.529918
\(430\) 0 0
\(431\) 5.90720e6 1.53175 0.765876 0.642988i \(-0.222306\pi\)
0.765876 + 0.642988i \(0.222306\pi\)
\(432\) −186624. −0.0481125
\(433\) 1.84995e6 0.474177 0.237089 0.971488i \(-0.423807\pi\)
0.237089 + 0.971488i \(0.423807\pi\)
\(434\) −2.28962e6 −0.583498
\(435\) 0 0
\(436\) −314183. −0.0791529
\(437\) −114415. −0.0286601
\(438\) 970445. 0.241705
\(439\) −2.81172e6 −0.696323 −0.348161 0.937435i \(-0.613194\pi\)
−0.348161 + 0.937435i \(0.613194\pi\)
\(440\) 0 0
\(441\) −200676. −0.0491359
\(442\) −4.38196e6 −1.06687
\(443\) 2.41386e6 0.584391 0.292196 0.956359i \(-0.405614\pi\)
0.292196 + 0.956359i \(0.405614\pi\)
\(444\) −1.74966e6 −0.421208
\(445\) 0 0
\(446\) 4.18051e6 0.995158
\(447\) −3.14448e6 −0.744355
\(448\) 490316. 0.115420
\(449\) −507124. −0.118713 −0.0593565 0.998237i \(-0.518905\pi\)
−0.0593565 + 0.998237i \(0.518905\pi\)
\(450\) 0 0
\(451\) 4.87856e6 1.12941
\(452\) 315465. 0.0726282
\(453\) 2.83072e6 0.648115
\(454\) 3.47648e6 0.791590
\(455\) 0 0
\(456\) 1.18683e6 0.267286
\(457\) 1.09879e6 0.246106 0.123053 0.992400i \(-0.460731\pi\)
0.123053 + 0.992400i \(0.460731\pi\)
\(458\) 5.59626e6 1.24662
\(459\) 938309. 0.207881
\(460\) 0 0
\(461\) 106066. 0.0232446 0.0116223 0.999932i \(-0.496300\pi\)
0.0116223 + 0.999932i \(0.496300\pi\)
\(462\) 1.13642e6 0.247704
\(463\) 1.36176e6 0.295222 0.147611 0.989045i \(-0.452842\pi\)
0.147611 + 0.989045i \(0.452842\pi\)
\(464\) −1.53243e6 −0.330435
\(465\) 0 0
\(466\) −1.61397e6 −0.344296
\(467\) −2.18710e6 −0.464062 −0.232031 0.972708i \(-0.574537\pi\)
−0.232031 + 0.972708i \(0.574537\pi\)
\(468\) −1.10305e6 −0.232798
\(469\) 4.11246e6 0.863316
\(470\) 0 0
\(471\) −1.18172e6 −0.245450
\(472\) −1.33284e6 −0.275374
\(473\) 577111. 0.118606
\(474\) 1.51552e6 0.309825
\(475\) 0 0
\(476\) −2.46521e6 −0.498697
\(477\) 2.14353e6 0.431354
\(478\) 2.44430e6 0.489311
\(479\) 1.01595e6 0.202318 0.101159 0.994870i \(-0.467745\pi\)
0.101159 + 0.994870i \(0.467745\pi\)
\(480\) 0 0
\(481\) −1.03414e7 −2.03807
\(482\) 6.02555e6 1.18135
\(483\) 59823.7 0.0116682
\(484\) −1.46416e6 −0.284103
\(485\) 0 0
\(486\) 236196. 0.0453609
\(487\) 2.02339e6 0.386596 0.193298 0.981140i \(-0.438082\pi\)
0.193298 + 0.981140i \(0.438082\pi\)
\(488\) −2.91365e6 −0.553844
\(489\) −5.69884e6 −1.07774
\(490\) 0 0
\(491\) −4.22487e6 −0.790878 −0.395439 0.918492i \(-0.629408\pi\)
−0.395439 + 0.918492i \(0.629408\pi\)
\(492\) −2.66400e6 −0.496160
\(493\) 7.70477e6 1.42772
\(494\) 7.01482e6 1.29330
\(495\) 0 0
\(496\) 1.22413e6 0.223421
\(497\) −6.88184e6 −1.24972
\(498\) −3.66239e6 −0.661745
\(499\) −6.71381e6 −1.20703 −0.603515 0.797352i \(-0.706233\pi\)
−0.603515 + 0.797352i \(0.706233\pi\)
\(500\) 0 0
\(501\) 1.23921e6 0.220572
\(502\) −2.23491e6 −0.395822
\(503\) −5.36486e6 −0.945450 −0.472725 0.881210i \(-0.656730\pi\)
−0.472725 + 0.881210i \(0.656730\pi\)
\(504\) −620556. −0.108819
\(505\) 0 0
\(506\) 58572.6 0.0101699
\(507\) −3.17798e6 −0.549074
\(508\) 893179. 0.153560
\(509\) 1.10627e6 0.189264 0.0946318 0.995512i \(-0.469833\pi\)
0.0946318 + 0.995512i \(0.469833\pi\)
\(510\) 0 0
\(511\) 3.22689e6 0.546679
\(512\) −262144. −0.0441942
\(513\) −1.50208e6 −0.252000
\(514\) 2.72043e6 0.454182
\(515\) 0 0
\(516\) −315139. −0.0521048
\(517\) 1.47616e6 0.242889
\(518\) −5.81791e6 −0.952670
\(519\) −812503. −0.132406
\(520\) 0 0
\(521\) −3.11641e6 −0.502991 −0.251495 0.967859i \(-0.580922\pi\)
−0.251495 + 0.967859i \(0.580922\pi\)
\(522\) 1.93948e6 0.311537
\(523\) −9.29324e6 −1.48564 −0.742819 0.669492i \(-0.766512\pi\)
−0.742819 + 0.669492i \(0.766512\pi\)
\(524\) 2.18203e6 0.347163
\(525\) 0 0
\(526\) −1.71467e6 −0.270219
\(527\) −6.15469e6 −0.965339
\(528\) −607579. −0.0948457
\(529\) −6.43326e6 −0.999521
\(530\) 0 0
\(531\) 1.68687e6 0.259625
\(532\) 3.94641e6 0.604538
\(533\) −1.57457e7 −2.40073
\(534\) −2.35984e6 −0.358121
\(535\) 0 0
\(536\) −2.19870e6 −0.330563
\(537\) −5.41421e6 −0.810213
\(538\) 6.07742e6 0.905240
\(539\) −653326. −0.0968631
\(540\) 0 0
\(541\) 1.00152e7 1.47118 0.735592 0.677425i \(-0.236904\pi\)
0.735592 + 0.677425i \(0.236904\pi\)
\(542\) 5.38382e6 0.787213
\(543\) −4.02971e6 −0.586509
\(544\) 1.31801e6 0.190951
\(545\) 0 0
\(546\) −3.66782e6 −0.526534
\(547\) 605416. 0.0865138 0.0432569 0.999064i \(-0.486227\pi\)
0.0432569 + 0.999064i \(0.486227\pi\)
\(548\) 2.15018e6 0.305861
\(549\) 3.68759e6 0.522169
\(550\) 0 0
\(551\) −1.23341e7 −1.73073
\(552\) −31984.3 −0.00446776
\(553\) 5.03936e6 0.700749
\(554\) −3.54216e6 −0.490336
\(555\) 0 0
\(556\) −4.88836e6 −0.670620
\(557\) 9.21693e6 1.25878 0.629388 0.777091i \(-0.283306\pi\)
0.629388 + 0.777091i \(0.283306\pi\)
\(558\) −1.54929e6 −0.210643
\(559\) −1.86264e6 −0.252116
\(560\) 0 0
\(561\) 3.05479e6 0.409802
\(562\) 5.07213e6 0.677407
\(563\) 9.44213e6 1.25545 0.627724 0.778436i \(-0.283986\pi\)
0.627724 + 0.778436i \(0.283986\pi\)
\(564\) −806078. −0.106704
\(565\) 0 0
\(566\) 2.74333e6 0.359946
\(567\) 785391. 0.102595
\(568\) 3.67933e6 0.478518
\(569\) −1.24216e6 −0.160841 −0.0804206 0.996761i \(-0.525626\pi\)
−0.0804206 + 0.996761i \(0.525626\pi\)
\(570\) 0 0
\(571\) −1.35633e7 −1.74090 −0.870450 0.492256i \(-0.836172\pi\)
−0.870450 + 0.492256i \(0.836172\pi\)
\(572\) −3.59112e6 −0.458923
\(573\) −6.06462e6 −0.771644
\(574\) −8.85824e6 −1.12219
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) −6.75669e6 −0.844879 −0.422440 0.906391i \(-0.638826\pi\)
−0.422440 + 0.906391i \(0.638826\pi\)
\(578\) −947262. −0.117937
\(579\) 2.30366e6 0.285576
\(580\) 0 0
\(581\) −1.21780e7 −1.49671
\(582\) 2.98011e6 0.364690
\(583\) 6.97855e6 0.850342
\(584\) −1.72524e6 −0.209323
\(585\) 0 0
\(586\) −6.67430e6 −0.802900
\(587\) −1.08655e7 −1.30153 −0.650764 0.759280i \(-0.725551\pi\)
−0.650764 + 0.759280i \(0.725551\pi\)
\(588\) 356757. 0.0425529
\(589\) 9.85269e6 1.17022
\(590\) 0 0
\(591\) 1.76582e6 0.207959
\(592\) 3.11051e6 0.364776
\(593\) 1.20061e6 0.140205 0.0701027 0.997540i \(-0.477667\pi\)
0.0701027 + 0.997540i \(0.477667\pi\)
\(594\) 768967. 0.0894214
\(595\) 0 0
\(596\) 5.59019e6 0.644630
\(597\) −1.06439e6 −0.122226
\(598\) −189045. −0.0216178
\(599\) −6.77826e6 −0.771882 −0.385941 0.922523i \(-0.626123\pi\)
−0.385941 + 0.922523i \(0.626123\pi\)
\(600\) 0 0
\(601\) 6.32426e6 0.714207 0.357103 0.934065i \(-0.383764\pi\)
0.357103 + 0.934065i \(0.383764\pi\)
\(602\) −1.04789e6 −0.117849
\(603\) 2.78273e6 0.311658
\(604\) −5.03240e6 −0.561284
\(605\) 0 0
\(606\) 527217. 0.0583187
\(607\) 2.63732e6 0.290530 0.145265 0.989393i \(-0.453596\pi\)
0.145265 + 0.989393i \(0.453596\pi\)
\(608\) −2.10992e6 −0.231477
\(609\) 6.44911e6 0.704622
\(610\) 0 0
\(611\) −4.76436e6 −0.516299
\(612\) −1.66810e6 −0.180030
\(613\) 1.43305e7 1.54031 0.770157 0.637854i \(-0.220178\pi\)
0.770157 + 0.637854i \(0.220178\pi\)
\(614\) −2.24310e6 −0.240120
\(615\) 0 0
\(616\) −2.02030e6 −0.214518
\(617\) 29265.6 0.00309489 0.00154744 0.999999i \(-0.499507\pi\)
0.00154744 + 0.999999i \(0.499507\pi\)
\(618\) 7.19737e6 0.758059
\(619\) 7.84138e6 0.822557 0.411279 0.911510i \(-0.365082\pi\)
0.411279 + 0.911510i \(0.365082\pi\)
\(620\) 0 0
\(621\) 40480.2 0.00421224
\(622\) 5.35771e6 0.555269
\(623\) −7.84687e6 −0.809984
\(624\) 1.96098e6 0.201609
\(625\) 0 0
\(626\) 2.48490e6 0.253439
\(627\) −4.89023e6 −0.496776
\(628\) 2.10084e6 0.212566
\(629\) −1.56390e7 −1.57610
\(630\) 0 0
\(631\) −668942. −0.0668829 −0.0334414 0.999441i \(-0.510647\pi\)
−0.0334414 + 0.999441i \(0.510647\pi\)
\(632\) −2.69426e6 −0.268316
\(633\) −8.19250e6 −0.812657
\(634\) 1.20113e6 0.118677
\(635\) 0 0
\(636\) −3.81072e6 −0.373564
\(637\) 2.10863e6 0.205898
\(638\) 6.31424e6 0.614143
\(639\) −4.65665e6 −0.451151
\(640\) 0 0
\(641\) 1.26732e7 1.21826 0.609130 0.793070i \(-0.291519\pi\)
0.609130 + 0.793070i \(0.291519\pi\)
\(642\) −1.25173e6 −0.119860
\(643\) −6.26256e6 −0.597344 −0.298672 0.954356i \(-0.596544\pi\)
−0.298672 + 0.954356i \(0.596544\pi\)
\(644\) −106353. −0.0101050
\(645\) 0 0
\(646\) 1.06083e7 1.00015
\(647\) 8.69452e6 0.816554 0.408277 0.912858i \(-0.366130\pi\)
0.408277 + 0.912858i \(0.366130\pi\)
\(648\) −419904. −0.0392837
\(649\) 5.49184e6 0.511807
\(650\) 0 0
\(651\) −5.15165e6 −0.476424
\(652\) 1.01313e7 0.933351
\(653\) 4.56532e6 0.418976 0.209488 0.977811i \(-0.432820\pi\)
0.209488 + 0.977811i \(0.432820\pi\)
\(654\) −706913. −0.0646281
\(655\) 0 0
\(656\) 4.73600e6 0.429687
\(657\) 2.18350e6 0.197351
\(658\) −2.68034e6 −0.241338
\(659\) 1.00273e7 0.899437 0.449718 0.893170i \(-0.351524\pi\)
0.449718 + 0.893170i \(0.351524\pi\)
\(660\) 0 0
\(661\) −8.14715e6 −0.725274 −0.362637 0.931930i \(-0.618123\pi\)
−0.362637 + 0.931930i \(0.618123\pi\)
\(662\) 7.79655e6 0.691445
\(663\) −9.85940e6 −0.871097
\(664\) 6.51091e6 0.573088
\(665\) 0 0
\(666\) −3.93673e6 −0.343915
\(667\) 332396. 0.0289295
\(668\) −2.20304e6 −0.191021
\(669\) 9.40614e6 0.812543
\(670\) 0 0
\(671\) 1.20054e7 1.02937
\(672\) 1.10321e6 0.0942400
\(673\) 1.16421e6 0.0990818 0.0495409 0.998772i \(-0.484224\pi\)
0.0495409 + 0.998772i \(0.484224\pi\)
\(674\) 1.01349e6 0.0859350
\(675\) 0 0
\(676\) 5.64974e6 0.475512
\(677\) −1.20869e7 −1.01355 −0.506775 0.862079i \(-0.669162\pi\)
−0.506775 + 0.862079i \(0.669162\pi\)
\(678\) 709796. 0.0593007
\(679\) 9.90934e6 0.824841
\(680\) 0 0
\(681\) 7.82208e6 0.646331
\(682\) −5.04392e6 −0.415248
\(683\) −7.57524e6 −0.621362 −0.310681 0.950514i \(-0.600557\pi\)
−0.310681 + 0.950514i \(0.600557\pi\)
\(684\) 2.67037e6 0.218239
\(685\) 0 0
\(686\) 9.23387e6 0.749158
\(687\) 1.25916e7 1.01786
\(688\) 560247. 0.0451241
\(689\) −2.25234e7 −1.80753
\(690\) 0 0
\(691\) 2.32589e7 1.85308 0.926539 0.376198i \(-0.122769\pi\)
0.926539 + 0.376198i \(0.122769\pi\)
\(692\) 1.44445e6 0.114667
\(693\) 2.55694e6 0.202250
\(694\) 6.42751e6 0.506575
\(695\) 0 0
\(696\) −3.44797e6 −0.269799
\(697\) −2.38117e7 −1.85656
\(698\) 2.43350e6 0.189057
\(699\) −3.63144e6 −0.281116
\(700\) 0 0
\(701\) −1.11461e7 −0.856700 −0.428350 0.903613i \(-0.640905\pi\)
−0.428350 + 0.903613i \(0.640905\pi\)
\(702\) −2.48186e6 −0.190079
\(703\) 2.50356e7 1.91060
\(704\) 1.08014e6 0.0821388
\(705\) 0 0
\(706\) −1.53509e7 −1.15911
\(707\) 1.75308e6 0.131903
\(708\) −2.99889e6 −0.224842
\(709\) 264391. 0.0197529 0.00987646 0.999951i \(-0.496856\pi\)
0.00987646 + 0.999951i \(0.496856\pi\)
\(710\) 0 0
\(711\) 3.40992e6 0.252971
\(712\) 4.19528e6 0.310142
\(713\) −265523. −0.0195605
\(714\) −5.54673e6 −0.407184
\(715\) 0 0
\(716\) 9.62526e6 0.701665
\(717\) 5.49968e6 0.399521
\(718\) −8.55402e6 −0.619240
\(719\) 7.09616e6 0.511919 0.255959 0.966688i \(-0.417609\pi\)
0.255959 + 0.966688i \(0.417609\pi\)
\(720\) 0 0
\(721\) 2.39325e7 1.71455
\(722\) −7.07778e6 −0.505305
\(723\) 1.35575e7 0.964570
\(724\) 7.16393e6 0.507931
\(725\) 0 0
\(726\) −3.29437e6 −0.231969
\(727\) −3.83492e6 −0.269104 −0.134552 0.990907i \(-0.542960\pi\)
−0.134552 + 0.990907i \(0.542960\pi\)
\(728\) 6.52057e6 0.455992
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.81681e6 −0.194969
\(732\) −6.55571e6 −0.452212
\(733\) −1.25850e7 −0.865151 −0.432575 0.901598i \(-0.642395\pi\)
−0.432575 + 0.901598i \(0.642395\pi\)
\(734\) 1.80158e7 1.23428
\(735\) 0 0
\(736\) 56861.0 0.00386919
\(737\) 9.05954e6 0.614381
\(738\) −5.99400e6 −0.405113
\(739\) −1.48983e7 −1.00352 −0.501761 0.865006i \(-0.667314\pi\)
−0.501761 + 0.865006i \(0.667314\pi\)
\(740\) 0 0
\(741\) 1.57833e7 1.05597
\(742\) −1.26713e7 −0.844911
\(743\) 921340. 0.0612277 0.0306139 0.999531i \(-0.490254\pi\)
0.0306139 + 0.999531i \(0.490254\pi\)
\(744\) 2.75430e6 0.182422
\(745\) 0 0
\(746\) −4.60891e6 −0.303215
\(747\) −8.24037e6 −0.540313
\(748\) −5.43073e6 −0.354899
\(749\) −4.16222e6 −0.271095
\(750\) 0 0
\(751\) −1.19124e7 −0.770726 −0.385363 0.922765i \(-0.625924\pi\)
−0.385363 + 0.922765i \(0.625924\pi\)
\(752\) 1.43303e6 0.0924081
\(753\) −5.02854e6 −0.323188
\(754\) −2.03794e7 −1.30546
\(755\) 0 0
\(756\) −1.39625e6 −0.0888503
\(757\) 1.43092e7 0.907558 0.453779 0.891114i \(-0.350076\pi\)
0.453779 + 0.891114i \(0.350076\pi\)
\(758\) −3.19096e6 −0.201720
\(759\) 131788. 0.00830372
\(760\) 0 0
\(761\) −1.78549e6 −0.111762 −0.0558812 0.998437i \(-0.517797\pi\)
−0.0558812 + 0.998437i \(0.517797\pi\)
\(762\) 2.00965e6 0.125381
\(763\) −2.35060e6 −0.146173
\(764\) 1.07815e7 0.668263
\(765\) 0 0
\(766\) 4.08494e6 0.251544
\(767\) −1.77250e7 −1.08792
\(768\) −589824. −0.0360844
\(769\) −1.03207e7 −0.629350 −0.314675 0.949200i \(-0.601895\pi\)
−0.314675 + 0.949200i \(0.601895\pi\)
\(770\) 0 0
\(771\) 6.12097e6 0.370838
\(772\) −4.09539e6 −0.247316
\(773\) −6.93308e6 −0.417328 −0.208664 0.977987i \(-0.566911\pi\)
−0.208664 + 0.977987i \(0.566911\pi\)
\(774\) −709063. −0.0425434
\(775\) 0 0
\(776\) −5.29797e6 −0.315831
\(777\) −1.30903e7 −0.777852
\(778\) 4.27281e6 0.253084
\(779\) 3.81187e7 2.25058
\(780\) 0 0
\(781\) −1.51603e7 −0.889368
\(782\) −285886. −0.0167177
\(783\) 4.36384e6 0.254369
\(784\) −634235. −0.0368519
\(785\) 0 0
\(786\) 4.90958e6 0.283457
\(787\) 3.05516e7 1.75831 0.879157 0.476532i \(-0.158106\pi\)
0.879157 + 0.476532i \(0.158106\pi\)
\(788\) −3.13923e6 −0.180097
\(789\) −3.85801e6 −0.220633
\(790\) 0 0
\(791\) 2.36019e6 0.134124
\(792\) −1.36705e6 −0.0774412
\(793\) −3.87478e7 −2.18808
\(794\) 437321. 0.0246178
\(795\) 0 0
\(796\) 1.89225e6 0.105851
\(797\) 2.76107e7 1.53968 0.769841 0.638236i \(-0.220336\pi\)
0.769841 + 0.638236i \(0.220336\pi\)
\(798\) 8.87943e6 0.493603
\(799\) −7.20498e6 −0.399269
\(800\) 0 0
\(801\) −5.30965e6 −0.292405
\(802\) −5.68523e6 −0.312113
\(803\) 7.10867e6 0.389045
\(804\) −4.94708e6 −0.269904
\(805\) 0 0
\(806\) 1.62794e7 0.882674
\(807\) 1.36742e7 0.739125
\(808\) −937275. −0.0505055
\(809\) 9.60377e6 0.515906 0.257953 0.966157i \(-0.416952\pi\)
0.257953 + 0.966157i \(0.416952\pi\)
\(810\) 0 0
\(811\) 1.77352e7 0.946856 0.473428 0.880832i \(-0.343016\pi\)
0.473428 + 0.880832i \(0.343016\pi\)
\(812\) −1.14651e7 −0.610221
\(813\) 1.21136e7 0.642756
\(814\) −1.28165e7 −0.677969
\(815\) 0 0
\(816\) 2.96552e6 0.155911
\(817\) 4.50927e6 0.236348
\(818\) −1.23004e7 −0.642742
\(819\) −8.25260e6 −0.429913
\(820\) 0 0
\(821\) −2.66186e7 −1.37825 −0.689125 0.724643i \(-0.742005\pi\)
−0.689125 + 0.724643i \(0.742005\pi\)
\(822\) 4.83791e6 0.249735
\(823\) 1.38059e7 0.710504 0.355252 0.934771i \(-0.384395\pi\)
0.355252 + 0.934771i \(0.384395\pi\)
\(824\) −1.27953e7 −0.656498
\(825\) 0 0
\(826\) −9.97180e6 −0.508538
\(827\) 3.25738e7 1.65617 0.828084 0.560604i \(-0.189431\pi\)
0.828084 + 0.560604i \(0.189431\pi\)
\(828\) −71964.7 −0.00364791
\(829\) 1.67628e7 0.847150 0.423575 0.905861i \(-0.360775\pi\)
0.423575 + 0.905861i \(0.360775\pi\)
\(830\) 0 0
\(831\) −7.96987e6 −0.400358
\(832\) −3.48618e6 −0.174599
\(833\) 3.18881e6 0.159227
\(834\) −1.09988e7 −0.547559
\(835\) 0 0
\(836\) 8.69374e6 0.430220
\(837\) −3.48591e6 −0.171990
\(838\) −1.22585e7 −0.603013
\(839\) −1.54475e6 −0.0757625 −0.0378812 0.999282i \(-0.512061\pi\)
−0.0378812 + 0.999282i \(0.512061\pi\)
\(840\) 0 0
\(841\) 1.53218e7 0.746998
\(842\) −3.35113e6 −0.162896
\(843\) 1.14123e7 0.553101
\(844\) 1.45644e7 0.703781
\(845\) 0 0
\(846\) −1.81368e6 −0.0871232
\(847\) −1.09543e7 −0.524658
\(848\) 6.77462e6 0.323516
\(849\) 6.17249e6 0.293895
\(850\) 0 0
\(851\) −674692. −0.0319361
\(852\) 8.27850e6 0.390708
\(853\) −2.39801e7 −1.12844 −0.564220 0.825625i \(-0.690823\pi\)
−0.564220 + 0.825625i \(0.690823\pi\)
\(854\) −2.17988e7 −1.02279
\(855\) 0 0
\(856\) 2.22531e6 0.103802
\(857\) −3.48306e7 −1.61998 −0.809989 0.586445i \(-0.800527\pi\)
−0.809989 + 0.586445i \(0.800527\pi\)
\(858\) −8.08002e6 −0.374709
\(859\) −1.01532e7 −0.469482 −0.234741 0.972058i \(-0.575424\pi\)
−0.234741 + 0.972058i \(0.575424\pi\)
\(860\) 0 0
\(861\) −1.99310e7 −0.916267
\(862\) −2.36288e7 −1.08311
\(863\) 2.71085e7 1.23902 0.619510 0.784989i \(-0.287331\pi\)
0.619510 + 0.784989i \(0.287331\pi\)
\(864\) 746496. 0.0340207
\(865\) 0 0
\(866\) −7.39981e6 −0.335294
\(867\) −2.13134e6 −0.0962953
\(868\) 9.15849e6 0.412596
\(869\) 1.11014e7 0.498689
\(870\) 0 0
\(871\) −2.92399e7 −1.30596
\(872\) 1.25673e6 0.0559696
\(873\) 6.70524e6 0.297768
\(874\) 457658. 0.0202658
\(875\) 0 0
\(876\) −3.88178e6 −0.170911
\(877\) 5.76852e6 0.253259 0.126630 0.991950i \(-0.459584\pi\)
0.126630 + 0.991950i \(0.459584\pi\)
\(878\) 1.12469e7 0.492374
\(879\) −1.50172e7 −0.655565
\(880\) 0 0
\(881\) −3.56949e7 −1.54941 −0.774705 0.632322i \(-0.782102\pi\)
−0.774705 + 0.632322i \(0.782102\pi\)
\(882\) 802704. 0.0347443
\(883\) 4.55843e6 0.196749 0.0983747 0.995149i \(-0.468636\pi\)
0.0983747 + 0.995149i \(0.468636\pi\)
\(884\) 1.75278e7 0.754393
\(885\) 0 0
\(886\) −9.65546e6 −0.413227
\(887\) 3.23885e6 0.138223 0.0691117 0.997609i \(-0.477984\pi\)
0.0691117 + 0.997609i \(0.477984\pi\)
\(888\) 6.99864e6 0.297839
\(889\) 6.68243e6 0.283583
\(890\) 0 0
\(891\) 1.73017e6 0.0730123
\(892\) −1.67220e7 −0.703683
\(893\) 1.15340e7 0.484008
\(894\) 1.25779e7 0.526338
\(895\) 0 0
\(896\) −1.96126e6 −0.0816142
\(897\) −425351. −0.0176509
\(898\) 2.02850e6 0.0839428
\(899\) −2.86239e7 −1.18122
\(900\) 0 0
\(901\) −3.40615e7 −1.39782
\(902\) −1.95142e7 −0.798611
\(903\) −2.35775e6 −0.0962230
\(904\) −1.26186e6 −0.0513559
\(905\) 0 0
\(906\) −1.13229e7 −0.458287
\(907\) −168518. −0.00680186 −0.00340093 0.999994i \(-0.501083\pi\)
−0.00340093 + 0.999994i \(0.501083\pi\)
\(908\) −1.39059e7 −0.559739
\(909\) 1.18624e6 0.0476170
\(910\) 0 0
\(911\) 3.36451e7 1.34315 0.671577 0.740935i \(-0.265617\pi\)
0.671577 + 0.740935i \(0.265617\pi\)
\(912\) −4.74733e6 −0.189000
\(913\) −2.68276e7 −1.06514
\(914\) −4.39514e6 −0.174023
\(915\) 0 0
\(916\) −2.23851e7 −0.881494
\(917\) 1.63252e7 0.641112
\(918\) −3.75324e6 −0.146994
\(919\) 6.60944e6 0.258152 0.129076 0.991635i \(-0.458799\pi\)
0.129076 + 0.991635i \(0.458799\pi\)
\(920\) 0 0
\(921\) −5.04698e6 −0.196057
\(922\) −424262. −0.0164364
\(923\) 4.89304e7 1.89049
\(924\) −4.54567e6 −0.175153
\(925\) 0 0
\(926\) −5.44706e6 −0.208754
\(927\) 1.61941e7 0.618952
\(928\) 6.12973e6 0.233653
\(929\) −4.55946e7 −1.73330 −0.866651 0.498915i \(-0.833732\pi\)
−0.866651 + 0.498915i \(0.833732\pi\)
\(930\) 0 0
\(931\) −5.10478e6 −0.193020
\(932\) 6.45590e6 0.243454
\(933\) 1.20548e7 0.453375
\(934\) 8.74839e6 0.328141
\(935\) 0 0
\(936\) 4.41220e6 0.164613
\(937\) −3.05380e6 −0.113630 −0.0568148 0.998385i \(-0.518094\pi\)
−0.0568148 + 0.998385i \(0.518094\pi\)
\(938\) −1.64499e7 −0.610457
\(939\) 5.59103e6 0.206932
\(940\) 0 0
\(941\) 3.36840e7 1.24008 0.620040 0.784570i \(-0.287116\pi\)
0.620040 + 0.784570i \(0.287116\pi\)
\(942\) 4.72689e6 0.173559
\(943\) −1.02727e6 −0.0376190
\(944\) 5.33135e6 0.194719
\(945\) 0 0
\(946\) −2.30845e6 −0.0838671
\(947\) 5.69161e6 0.206234 0.103117 0.994669i \(-0.467118\pi\)
0.103117 + 0.994669i \(0.467118\pi\)
\(948\) −6.06208e6 −0.219079
\(949\) −2.29434e7 −0.826976
\(950\) 0 0
\(951\) 2.70254e6 0.0968992
\(952\) 9.86084e6 0.352632
\(953\) 1.73627e7 0.619278 0.309639 0.950854i \(-0.399792\pi\)
0.309639 + 0.950854i \(0.399792\pi\)
\(954\) −8.57413e6 −0.305014
\(955\) 0 0
\(956\) −9.77721e6 −0.345995
\(957\) 1.42070e7 0.501446
\(958\) −4.06380e6 −0.143060
\(959\) 1.60869e7 0.564839
\(960\) 0 0
\(961\) −5.76388e6 −0.201329
\(962\) 4.13657e7 1.44113
\(963\) −2.81640e6 −0.0978653
\(964\) −2.41022e7 −0.835342
\(965\) 0 0
\(966\) −239295. −0.00825069
\(967\) −4.93082e7 −1.69572 −0.847858 0.530224i \(-0.822108\pi\)
−0.847858 + 0.530224i \(0.822108\pi\)
\(968\) 5.85665e6 0.200891
\(969\) 2.38686e7 0.816616
\(970\) 0 0
\(971\) 3.23520e7 1.10117 0.550584 0.834780i \(-0.314405\pi\)
0.550584 + 0.834780i \(0.314405\pi\)
\(972\) −944784. −0.0320750
\(973\) −3.65729e7 −1.23845
\(974\) −8.09357e6 −0.273365
\(975\) 0 0
\(976\) 1.16546e7 0.391627
\(977\) −2.61290e7 −0.875763 −0.437882 0.899033i \(-0.644271\pi\)
−0.437882 + 0.899033i \(0.644271\pi\)
\(978\) 2.27954e7 0.762078
\(979\) −1.72862e7 −0.576426
\(980\) 0 0
\(981\) −1.59055e6 −0.0527686
\(982\) 1.68995e7 0.559235
\(983\) −3.19496e7 −1.05458 −0.527292 0.849684i \(-0.676793\pi\)
−0.527292 + 0.849684i \(0.676793\pi\)
\(984\) 1.06560e7 0.350838
\(985\) 0 0
\(986\) −3.08191e7 −1.00955
\(987\) −6.03077e6 −0.197052
\(988\) −2.80593e7 −0.914501
\(989\) −121522. −0.00395061
\(990\) 0 0
\(991\) 1.14783e7 0.371272 0.185636 0.982619i \(-0.440565\pi\)
0.185636 + 0.982619i \(0.440565\pi\)
\(992\) −4.89653e6 −0.157982
\(993\) 1.75422e7 0.564563
\(994\) 2.75274e7 0.883688
\(995\) 0 0
\(996\) 1.46495e7 0.467925
\(997\) 5.48929e7 1.74895 0.874476 0.485068i \(-0.161205\pi\)
0.874476 + 0.485068i \(0.161205\pi\)
\(998\) 2.68552e7 0.853499
\(999\) −8.85765e6 −0.280805
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 150.6.a.n.1.2 2
3.2 odd 2 450.6.a.bc.1.2 2
5.2 odd 4 30.6.c.b.19.1 4
5.3 odd 4 30.6.c.b.19.3 yes 4
5.4 even 2 150.6.a.o.1.1 2
15.2 even 4 90.6.c.c.19.4 4
15.8 even 4 90.6.c.c.19.2 4
15.14 odd 2 450.6.a.bb.1.1 2
20.3 even 4 240.6.f.b.49.3 4
20.7 even 4 240.6.f.b.49.1 4
60.23 odd 4 720.6.f.i.289.4 4
60.47 odd 4 720.6.f.i.289.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
30.6.c.b.19.1 4 5.2 odd 4
30.6.c.b.19.3 yes 4 5.3 odd 4
90.6.c.c.19.2 4 15.8 even 4
90.6.c.c.19.4 4 15.2 even 4
150.6.a.n.1.2 2 1.1 even 1 trivial
150.6.a.o.1.1 2 5.4 even 2
240.6.f.b.49.1 4 20.7 even 4
240.6.f.b.49.3 4 20.3 even 4
450.6.a.bb.1.1 2 15.14 odd 2
450.6.a.bc.1.2 2 3.2 odd 2
720.6.f.i.289.3 4 60.47 odd 4
720.6.f.i.289.4 4 60.23 odd 4