Properties

Label 825.6.a.g.1.2
Level $825$
Weight $6$
Character 825.1
Self dual yes
Analytic conductor $132.317$
Analytic rank $1$
Dimension $3$
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] = [825,6,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("825.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 825.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: \(132.316651346\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.18257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 26x + 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.305203\) of defining polynomial
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-0.694797 q^{2} +9.00000 q^{3} -31.5173 q^{4} -6.25317 q^{6} -83.1683 q^{7} +44.1316 q^{8} +81.0000 q^{9} +121.000 q^{11} -283.655 q^{12} +674.655 q^{13} +57.7851 q^{14} +977.890 q^{16} -1927.66 q^{17} -56.2785 q^{18} -149.751 q^{19} -748.514 q^{21} -84.0704 q^{22} +1355.59 q^{23} +397.184 q^{24} -468.748 q^{26} +729.000 q^{27} +2621.24 q^{28} -7320.99 q^{29} -4215.76 q^{31} -2091.65 q^{32} +1089.00 q^{33} +1339.33 q^{34} -2552.90 q^{36} +13420.1 q^{37} +104.046 q^{38} +6071.90 q^{39} +2865.39 q^{41} +520.065 q^{42} +22078.1 q^{43} -3813.59 q^{44} -941.861 q^{46} +14556.4 q^{47} +8801.01 q^{48} -9890.04 q^{49} -17348.9 q^{51} -21263.3 q^{52} -13349.7 q^{53} -506.507 q^{54} -3670.35 q^{56} -1347.76 q^{57} +5086.60 q^{58} +45803.3 q^{59} +18996.5 q^{61} +2929.10 q^{62} -6736.63 q^{63} -29839.2 q^{64} -756.634 q^{66} -6651.05 q^{67} +60754.5 q^{68} +12200.3 q^{69} -61028.7 q^{71} +3574.66 q^{72} +17353.4 q^{73} -9324.27 q^{74} +4719.73 q^{76} -10063.4 q^{77} -4218.74 q^{78} +61676.5 q^{79} +6561.00 q^{81} -1990.87 q^{82} +65230.8 q^{83} +23591.1 q^{84} -15339.8 q^{86} -65888.9 q^{87} +5339.92 q^{88} -109563. q^{89} -56109.9 q^{91} -42724.5 q^{92} -37941.8 q^{93} -10113.8 q^{94} -18824.8 q^{96} -83736.1 q^{97} +6871.57 q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} - 42 q^{4} - 18 q^{6} + 68 q^{7} + 24 q^{8} + 243 q^{9} + 363 q^{11} - 378 q^{12} - 290 q^{13} + 916 q^{14} - 590 q^{16} - 434 q^{17} - 162 q^{18} - 2856 q^{19} + 612 q^{21} - 242 q^{22} + 640 q^{23} + 216 q^{24} + 2132 q^{26} + 2187 q^{27} + 580 q^{28} - 4538 q^{29} - 14968 q^{31} + 2496 q^{32} + 3267 q^{33} - 13704 q^{34} - 3402 q^{36} + 6190 q^{37} + 11668 q^{38} - 2610 q^{39} - 8926 q^{41} + 8244 q^{42} + 33592 q^{43} - 5082 q^{44} - 35680 q^{46} + 24640 q^{47} - 5310 q^{48} - 14693 q^{49} - 3906 q^{51} - 18780 q^{52} + 22934 q^{53} - 1458 q^{54} - 40012 q^{56} - 25704 q^{57} + 32304 q^{58} - 13756 q^{59} + 24602 q^{61} + 7704 q^{62} + 5508 q^{63} + 35474 q^{64} - 2178 q^{66} - 16868 q^{67} + 71288 q^{68} + 5760 q^{69} + 4856 q^{71} + 1944 q^{72} - 1910 q^{73} + 29404 q^{74} + 6116 q^{76} + 8228 q^{77} + 19188 q^{78} - 36844 q^{79} + 19683 q^{81} - 84000 q^{82} + 48796 q^{83} + 5220 q^{84} - 83492 q^{86} - 40842 q^{87} + 2904 q^{88} - 188978 q^{89} - 93208 q^{91} + 6976 q^{92} - 134712 q^{93} + 70472 q^{94} + 22464 q^{96} - 247526 q^{97} + 154654 q^{98} + 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.694797 −0.122824 −0.0614120 0.998113i \(-0.519560\pi\)
−0.0614120 + 0.998113i \(0.519560\pi\)
\(3\) 9.00000 0.577350
\(4\) −31.5173 −0.984914
\(5\) 0 0
\(6\) −6.25317 −0.0709124
\(7\) −83.1683 −0.641523 −0.320762 0.947160i \(-0.603939\pi\)
−0.320762 + 0.947160i \(0.603939\pi\)
\(8\) 44.1316 0.243795
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 121.000 0.301511
\(12\) −283.655 −0.568641
\(13\) 674.655 1.10719 0.553597 0.832785i \(-0.313255\pi\)
0.553597 + 0.832785i \(0.313255\pi\)
\(14\) 57.7851 0.0787944
\(15\) 0 0
\(16\) 977.890 0.954970
\(17\) −1927.66 −1.61774 −0.808868 0.587991i \(-0.799919\pi\)
−0.808868 + 0.587991i \(0.799919\pi\)
\(18\) −56.2785 −0.0409413
\(19\) −149.751 −0.0951666 −0.0475833 0.998867i \(-0.515152\pi\)
−0.0475833 + 0.998867i \(0.515152\pi\)
\(20\) 0 0
\(21\) −748.514 −0.370384
\(22\) −84.0704 −0.0370328
\(23\) 1355.59 0.534330 0.267165 0.963651i \(-0.413913\pi\)
0.267165 + 0.963651i \(0.413913\pi\)
\(24\) 397.184 0.140755
\(25\) 0 0
\(26\) −468.748 −0.135990
\(27\) 729.000 0.192450
\(28\) 2621.24 0.631846
\(29\) −7320.99 −1.61650 −0.808248 0.588842i \(-0.799584\pi\)
−0.808248 + 0.588842i \(0.799584\pi\)
\(30\) 0 0
\(31\) −4215.76 −0.787901 −0.393951 0.919132i \(-0.628892\pi\)
−0.393951 + 0.919132i \(0.628892\pi\)
\(32\) −2091.65 −0.361088
\(33\) 1089.00 0.174078
\(34\) 1339.33 0.198697
\(35\) 0 0
\(36\) −2552.90 −0.328305
\(37\) 13420.1 1.61158 0.805792 0.592199i \(-0.201740\pi\)
0.805792 + 0.592199i \(0.201740\pi\)
\(38\) 104.046 0.0116887
\(39\) 6071.90 0.639238
\(40\) 0 0
\(41\) 2865.39 0.266210 0.133105 0.991102i \(-0.457505\pi\)
0.133105 + 0.991102i \(0.457505\pi\)
\(42\) 520.065 0.0454920
\(43\) 22078.1 1.82092 0.910458 0.413601i \(-0.135729\pi\)
0.910458 + 0.413601i \(0.135729\pi\)
\(44\) −3813.59 −0.296963
\(45\) 0 0
\(46\) −941.861 −0.0656285
\(47\) 14556.4 0.961192 0.480596 0.876942i \(-0.340420\pi\)
0.480596 + 0.876942i \(0.340420\pi\)
\(48\) 8801.01 0.551352
\(49\) −9890.04 −0.588448
\(50\) 0 0
\(51\) −17348.9 −0.934000
\(52\) −21263.3 −1.09049
\(53\) −13349.7 −0.652805 −0.326402 0.945231i \(-0.605836\pi\)
−0.326402 + 0.945231i \(0.605836\pi\)
\(54\) −506.507 −0.0236375
\(55\) 0 0
\(56\) −3670.35 −0.156400
\(57\) −1347.76 −0.0549445
\(58\) 5086.60 0.198544
\(59\) 45803.3 1.71304 0.856518 0.516117i \(-0.172623\pi\)
0.856518 + 0.516117i \(0.172623\pi\)
\(60\) 0 0
\(61\) 18996.5 0.653655 0.326827 0.945084i \(-0.394020\pi\)
0.326827 + 0.945084i \(0.394020\pi\)
\(62\) 2929.10 0.0967731
\(63\) −6736.63 −0.213841
\(64\) −29839.2 −0.910620
\(65\) 0 0
\(66\) −756.634 −0.0213809
\(67\) −6651.05 −0.181010 −0.0905052 0.995896i \(-0.528848\pi\)
−0.0905052 + 0.995896i \(0.528848\pi\)
\(68\) 60754.5 1.59333
\(69\) 12200.3 0.308495
\(70\) 0 0
\(71\) −61028.7 −1.43677 −0.718386 0.695644i \(-0.755119\pi\)
−0.718386 + 0.695644i \(0.755119\pi\)
\(72\) 3574.66 0.0812650
\(73\) 17353.4 0.381134 0.190567 0.981674i \(-0.438967\pi\)
0.190567 + 0.981674i \(0.438967\pi\)
\(74\) −9324.27 −0.197941
\(75\) 0 0
\(76\) 4719.73 0.0937309
\(77\) −10063.4 −0.193427
\(78\) −4218.74 −0.0785138
\(79\) 61676.5 1.11187 0.555933 0.831227i \(-0.312361\pi\)
0.555933 + 0.831227i \(0.312361\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −1990.87 −0.0326969
\(83\) 65230.8 1.03934 0.519670 0.854367i \(-0.326055\pi\)
0.519670 + 0.854367i \(0.326055\pi\)
\(84\) 23591.1 0.364796
\(85\) 0 0
\(86\) −15339.8 −0.223652
\(87\) −65888.9 −0.933284
\(88\) 5339.92 0.0735069
\(89\) −109563. −1.46618 −0.733091 0.680131i \(-0.761923\pi\)
−0.733091 + 0.680131i \(0.761923\pi\)
\(90\) 0 0
\(91\) −56109.9 −0.710291
\(92\) −42724.5 −0.526269
\(93\) −37941.8 −0.454895
\(94\) −10113.8 −0.118057
\(95\) 0 0
\(96\) −18824.8 −0.208474
\(97\) −83736.1 −0.903615 −0.451808 0.892115i \(-0.649221\pi\)
−0.451808 + 0.892115i \(0.649221\pi\)
\(98\) 6871.57 0.0722754
\(99\) 9801.00 0.100504
\(100\) 0 0
\(101\) −129877. −1.26686 −0.633428 0.773801i \(-0.718353\pi\)
−0.633428 + 0.773801i \(0.718353\pi\)
\(102\) 12054.0 0.114718
\(103\) −201851. −1.87472 −0.937362 0.348358i \(-0.886739\pi\)
−0.937362 + 0.348358i \(0.886739\pi\)
\(104\) 29773.6 0.269928
\(105\) 0 0
\(106\) 9275.36 0.0801800
\(107\) 132285. 1.11699 0.558496 0.829507i \(-0.311378\pi\)
0.558496 + 0.829507i \(0.311378\pi\)
\(108\) −22976.1 −0.189547
\(109\) −144044. −1.16126 −0.580629 0.814168i \(-0.697193\pi\)
−0.580629 + 0.814168i \(0.697193\pi\)
\(110\) 0 0
\(111\) 120781. 0.930448
\(112\) −81329.4 −0.612636
\(113\) 1095.62 0.00807165 0.00403582 0.999992i \(-0.498715\pi\)
0.00403582 + 0.999992i \(0.498715\pi\)
\(114\) 936.416 0.00674849
\(115\) 0 0
\(116\) 230737. 1.59211
\(117\) 54647.1 0.369065
\(118\) −31824.0 −0.210402
\(119\) 160320. 1.03782
\(120\) 0 0
\(121\) 14641.0 0.0909091
\(122\) −13198.7 −0.0802844
\(123\) 25788.5 0.153696
\(124\) 132869. 0.776015
\(125\) 0 0
\(126\) 4680.59 0.0262648
\(127\) −315758. −1.73718 −0.868591 0.495529i \(-0.834974\pi\)
−0.868591 + 0.495529i \(0.834974\pi\)
\(128\) 87664.9 0.472934
\(129\) 198703. 1.05131
\(130\) 0 0
\(131\) −187001. −0.952064 −0.476032 0.879428i \(-0.657925\pi\)
−0.476032 + 0.879428i \(0.657925\pi\)
\(132\) −34322.3 −0.171452
\(133\) 12454.5 0.0610516
\(134\) 4621.13 0.0222324
\(135\) 0 0
\(136\) −85070.6 −0.394396
\(137\) −232301. −1.05743 −0.528713 0.848801i \(-0.677325\pi\)
−0.528713 + 0.848801i \(0.677325\pi\)
\(138\) −8476.75 −0.0378906
\(139\) −311665. −1.36820 −0.684102 0.729386i \(-0.739806\pi\)
−0.684102 + 0.729386i \(0.739806\pi\)
\(140\) 0 0
\(141\) 131008. 0.554944
\(142\) 42402.5 0.176470
\(143\) 81633.3 0.333831
\(144\) 79209.1 0.318323
\(145\) 0 0
\(146\) −12057.1 −0.0468124
\(147\) −89010.4 −0.339740
\(148\) −422966. −1.58727
\(149\) 30354.5 0.112010 0.0560050 0.998430i \(-0.482164\pi\)
0.0560050 + 0.998430i \(0.482164\pi\)
\(150\) 0 0
\(151\) 131015. 0.467604 0.233802 0.972284i \(-0.424883\pi\)
0.233802 + 0.972284i \(0.424883\pi\)
\(152\) −6608.73 −0.0232011
\(153\) −156140. −0.539245
\(154\) 6991.99 0.0237574
\(155\) 0 0
\(156\) −191370. −0.629595
\(157\) −326922. −1.05851 −0.529255 0.848463i \(-0.677528\pi\)
−0.529255 + 0.848463i \(0.677528\pi\)
\(158\) −42852.7 −0.136564
\(159\) −120148. −0.376897
\(160\) 0 0
\(161\) −112742. −0.342785
\(162\) −4558.56 −0.0136471
\(163\) 501854. 1.47948 0.739739 0.672894i \(-0.234949\pi\)
0.739739 + 0.672894i \(0.234949\pi\)
\(164\) −90309.3 −0.262194
\(165\) 0 0
\(166\) −45322.2 −0.127656
\(167\) 590462. 1.63833 0.819164 0.573559i \(-0.194438\pi\)
0.819164 + 0.573559i \(0.194438\pi\)
\(168\) −33033.1 −0.0902977
\(169\) 83866.7 0.225877
\(170\) 0 0
\(171\) −12129.8 −0.0317222
\(172\) −695840. −1.79345
\(173\) −570934. −1.45034 −0.725172 0.688568i \(-0.758240\pi\)
−0.725172 + 0.688568i \(0.758240\pi\)
\(174\) 45779.4 0.114630
\(175\) 0 0
\(176\) 118325. 0.287934
\(177\) 412230. 0.989022
\(178\) 76123.9 0.180082
\(179\) −464402. −1.08333 −0.541666 0.840594i \(-0.682206\pi\)
−0.541666 + 0.840594i \(0.682206\pi\)
\(180\) 0 0
\(181\) −409792. −0.929751 −0.464876 0.885376i \(-0.653901\pi\)
−0.464876 + 0.885376i \(0.653901\pi\)
\(182\) 38985.0 0.0872407
\(183\) 170968. 0.377388
\(184\) 59824.4 0.130267
\(185\) 0 0
\(186\) 26361.9 0.0558720
\(187\) −233247. −0.487766
\(188\) −458778. −0.946691
\(189\) −60629.7 −0.123461
\(190\) 0 0
\(191\) −591029. −1.17226 −0.586132 0.810216i \(-0.699350\pi\)
−0.586132 + 0.810216i \(0.699350\pi\)
\(192\) −268553. −0.525747
\(193\) 310907. 0.600811 0.300405 0.953812i \(-0.402878\pi\)
0.300405 + 0.953812i \(0.402878\pi\)
\(194\) 58179.6 0.110986
\(195\) 0 0
\(196\) 311707. 0.579571
\(197\) −161613. −0.296695 −0.148348 0.988935i \(-0.547395\pi\)
−0.148348 + 0.988935i \(0.547395\pi\)
\(198\) −6809.70 −0.0123443
\(199\) 435622. 0.779790 0.389895 0.920859i \(-0.372511\pi\)
0.389895 + 0.920859i \(0.372511\pi\)
\(200\) 0 0
\(201\) −59859.5 −0.104506
\(202\) 90237.8 0.155600
\(203\) 608874. 1.03702
\(204\) 546790. 0.919910
\(205\) 0 0
\(206\) 140245. 0.230261
\(207\) 109803. 0.178110
\(208\) 659738. 1.05734
\(209\) −18119.8 −0.0286938
\(210\) 0 0
\(211\) −539434. −0.834127 −0.417063 0.908877i \(-0.636941\pi\)
−0.417063 + 0.908877i \(0.636941\pi\)
\(212\) 420747. 0.642957
\(213\) −549258. −0.829521
\(214\) −91911.1 −0.137193
\(215\) 0 0
\(216\) 32171.9 0.0469184
\(217\) 350618. 0.505457
\(218\) 100081. 0.142630
\(219\) 156181. 0.220048
\(220\) 0 0
\(221\) −1.30050e6 −1.79115
\(222\) −83918.5 −0.114281
\(223\) −452834. −0.609786 −0.304893 0.952387i \(-0.598621\pi\)
−0.304893 + 0.952387i \(0.598621\pi\)
\(224\) 173959. 0.231646
\(225\) 0 0
\(226\) −761.230 −0.000991391 0
\(227\) 40608.6 0.0523062 0.0261531 0.999658i \(-0.491674\pi\)
0.0261531 + 0.999658i \(0.491674\pi\)
\(228\) 42477.5 0.0541156
\(229\) 988410. 1.24551 0.622757 0.782415i \(-0.286013\pi\)
0.622757 + 0.782415i \(0.286013\pi\)
\(230\) 0 0
\(231\) −90570.2 −0.111675
\(232\) −323087. −0.394093
\(233\) 47673.4 0.0575289 0.0287645 0.999586i \(-0.490843\pi\)
0.0287645 + 0.999586i \(0.490843\pi\)
\(234\) −37968.6 −0.0453299
\(235\) 0 0
\(236\) −1.44359e6 −1.68719
\(237\) 555089. 0.641936
\(238\) −111390. −0.127469
\(239\) −165174. −0.187045 −0.0935224 0.995617i \(-0.529813\pi\)
−0.0935224 + 0.995617i \(0.529813\pi\)
\(240\) 0 0
\(241\) −1.13302e6 −1.25659 −0.628294 0.777976i \(-0.716247\pi\)
−0.628294 + 0.777976i \(0.716247\pi\)
\(242\) −10172.5 −0.0111658
\(243\) 59049.0 0.0641500
\(244\) −598717. −0.643794
\(245\) 0 0
\(246\) −17917.8 −0.0188776
\(247\) −101030. −0.105368
\(248\) −186048. −0.192086
\(249\) 587078. 0.600063
\(250\) 0 0
\(251\) −1.08030e6 −1.08233 −0.541165 0.840917i \(-0.682016\pi\)
−0.541165 + 0.840917i \(0.682016\pi\)
\(252\) 212320. 0.210615
\(253\) 164027. 0.161106
\(254\) 219388. 0.213368
\(255\) 0 0
\(256\) 893945. 0.852533
\(257\) −301460. −0.284707 −0.142353 0.989816i \(-0.545467\pi\)
−0.142353 + 0.989816i \(0.545467\pi\)
\(258\) −138058. −0.129126
\(259\) −1.11613e6 −1.03387
\(260\) 0 0
\(261\) −593000. −0.538832
\(262\) 129928. 0.116936
\(263\) 347755. 0.310016 0.155008 0.987913i \(-0.450460\pi\)
0.155008 + 0.987913i \(0.450460\pi\)
\(264\) 48059.3 0.0424392
\(265\) 0 0
\(266\) −8653.34 −0.00749860
\(267\) −986065. −0.846501
\(268\) 209623. 0.178280
\(269\) 980953. 0.826547 0.413274 0.910607i \(-0.364385\pi\)
0.413274 + 0.910607i \(0.364385\pi\)
\(270\) 0 0
\(271\) 9630.73 0.00796592 0.00398296 0.999992i \(-0.498732\pi\)
0.00398296 + 0.999992i \(0.498732\pi\)
\(272\) −1.88504e6 −1.54489
\(273\) −504989. −0.410086
\(274\) 161402. 0.129877
\(275\) 0 0
\(276\) −384521. −0.303842
\(277\) −324368. −0.254003 −0.127001 0.991903i \(-0.540535\pi\)
−0.127001 + 0.991903i \(0.540535\pi\)
\(278\) 216544. 0.168048
\(279\) −341477. −0.262634
\(280\) 0 0
\(281\) 866856. 0.654909 0.327455 0.944867i \(-0.393809\pi\)
0.327455 + 0.944867i \(0.393809\pi\)
\(282\) −91023.8 −0.0681604
\(283\) 575157. 0.426895 0.213447 0.976955i \(-0.431531\pi\)
0.213447 + 0.976955i \(0.431531\pi\)
\(284\) 1.92346e6 1.41510
\(285\) 0 0
\(286\) −56718.6 −0.0410025
\(287\) −238310. −0.170780
\(288\) −169423. −0.120363
\(289\) 2.29601e6 1.61707
\(290\) 0 0
\(291\) −753625. −0.521702
\(292\) −546932. −0.375384
\(293\) −1.96073e6 −1.33429 −0.667144 0.744929i \(-0.732483\pi\)
−0.667144 + 0.744929i \(0.732483\pi\)
\(294\) 61844.1 0.0417282
\(295\) 0 0
\(296\) 592252. 0.392896
\(297\) 88209.0 0.0580259
\(298\) −21090.2 −0.0137575
\(299\) 914557. 0.591606
\(300\) 0 0
\(301\) −1.83620e6 −1.16816
\(302\) −91028.8 −0.0574330
\(303\) −1.16889e6 −0.731420
\(304\) −146440. −0.0908813
\(305\) 0 0
\(306\) 108486. 0.0662322
\(307\) −1.60549e6 −0.972215 −0.486108 0.873899i \(-0.661584\pi\)
−0.486108 + 0.873899i \(0.661584\pi\)
\(308\) 317170. 0.190509
\(309\) −1.81666e6 −1.08237
\(310\) 0 0
\(311\) 6622.52 0.00388260 0.00194130 0.999998i \(-0.499382\pi\)
0.00194130 + 0.999998i \(0.499382\pi\)
\(312\) 267963. 0.155843
\(313\) 1.08764e6 0.627516 0.313758 0.949503i \(-0.398412\pi\)
0.313758 + 0.949503i \(0.398412\pi\)
\(314\) 227144. 0.130010
\(315\) 0 0
\(316\) −1.94387e6 −1.09509
\(317\) 456183. 0.254971 0.127486 0.991840i \(-0.459309\pi\)
0.127486 + 0.991840i \(0.459309\pi\)
\(318\) 83478.2 0.0462920
\(319\) −885839. −0.487392
\(320\) 0 0
\(321\) 1.19056e6 0.644896
\(322\) 78332.9 0.0421022
\(323\) 288668. 0.153954
\(324\) −206785. −0.109435
\(325\) 0 0
\(326\) −348687. −0.181715
\(327\) −1.29640e6 −0.670452
\(328\) 126454. 0.0649006
\(329\) −1.21063e6 −0.616627
\(330\) 0 0
\(331\) 1.73077e6 0.868297 0.434149 0.900841i \(-0.357049\pi\)
0.434149 + 0.900841i \(0.357049\pi\)
\(332\) −2.05590e6 −1.02366
\(333\) 1.08703e6 0.537194
\(334\) −410251. −0.201226
\(335\) 0 0
\(336\) −731965. −0.353706
\(337\) −3.01707e6 −1.44714 −0.723571 0.690250i \(-0.757501\pi\)
−0.723571 + 0.690250i \(0.757501\pi\)
\(338\) −58270.3 −0.0277431
\(339\) 9860.54 0.00466017
\(340\) 0 0
\(341\) −510107. −0.237561
\(342\) 8427.75 0.00389624
\(343\) 2.22035e6 1.01903
\(344\) 974341. 0.443930
\(345\) 0 0
\(346\) 396683. 0.178137
\(347\) −1.26947e6 −0.565978 −0.282989 0.959123i \(-0.591326\pi\)
−0.282989 + 0.959123i \(0.591326\pi\)
\(348\) 2.07664e6 0.919205
\(349\) 3.26420e6 1.43454 0.717270 0.696795i \(-0.245391\pi\)
0.717270 + 0.696795i \(0.245391\pi\)
\(350\) 0 0
\(351\) 491824. 0.213079
\(352\) −253089. −0.108872
\(353\) −1.70741e6 −0.729290 −0.364645 0.931147i \(-0.618810\pi\)
−0.364645 + 0.931147i \(0.618810\pi\)
\(354\) −286416. −0.121476
\(355\) 0 0
\(356\) 3.45312e6 1.44406
\(357\) 1.44288e6 0.599183
\(358\) 322665. 0.133059
\(359\) 257688. 0.105526 0.0527628 0.998607i \(-0.483197\pi\)
0.0527628 + 0.998607i \(0.483197\pi\)
\(360\) 0 0
\(361\) −2.45367e6 −0.990943
\(362\) 284722. 0.114196
\(363\) 131769. 0.0524864
\(364\) 1.76843e6 0.699575
\(365\) 0 0
\(366\) −118788. −0.0463522
\(367\) 683489. 0.264891 0.132445 0.991190i \(-0.457717\pi\)
0.132445 + 0.991190i \(0.457717\pi\)
\(368\) 1.32562e6 0.510269
\(369\) 232097. 0.0887367
\(370\) 0 0
\(371\) 1.11027e6 0.418790
\(372\) 1.19582e6 0.448033
\(373\) −3.58176e6 −1.33298 −0.666492 0.745512i \(-0.732205\pi\)
−0.666492 + 0.745512i \(0.732205\pi\)
\(374\) 162059. 0.0599093
\(375\) 0 0
\(376\) 642398. 0.234334
\(377\) −4.93914e6 −1.78977
\(378\) 42125.3 0.0151640
\(379\) −4.87955e6 −1.74494 −0.872472 0.488665i \(-0.837484\pi\)
−0.872472 + 0.488665i \(0.837484\pi\)
\(380\) 0 0
\(381\) −2.84182e6 −1.00296
\(382\) 410645. 0.143982
\(383\) −4.97559e6 −1.73319 −0.866597 0.499008i \(-0.833698\pi\)
−0.866597 + 0.499008i \(0.833698\pi\)
\(384\) 788984. 0.273049
\(385\) 0 0
\(386\) −216017. −0.0737939
\(387\) 1.78832e6 0.606972
\(388\) 2.63913e6 0.889983
\(389\) −4.09866e6 −1.37331 −0.686654 0.726984i \(-0.740921\pi\)
−0.686654 + 0.726984i \(0.740921\pi\)
\(390\) 0 0
\(391\) −2.61312e6 −0.864404
\(392\) −436463. −0.143461
\(393\) −1.68301e6 −0.549674
\(394\) 112288. 0.0364413
\(395\) 0 0
\(396\) −308901. −0.0989876
\(397\) −1.61215e6 −0.513369 −0.256684 0.966495i \(-0.582630\pi\)
−0.256684 + 0.966495i \(0.582630\pi\)
\(398\) −302669. −0.0957768
\(399\) 112090. 0.0352482
\(400\) 0 0
\(401\) −648362. −0.201352 −0.100676 0.994919i \(-0.532101\pi\)
−0.100676 + 0.994919i \(0.532101\pi\)
\(402\) 41590.2 0.0128359
\(403\) −2.84419e6 −0.872359
\(404\) 4.09335e6 1.24775
\(405\) 0 0
\(406\) −423044. −0.127371
\(407\) 1.62384e6 0.485911
\(408\) −765635. −0.227704
\(409\) 1.88327e6 0.556677 0.278338 0.960483i \(-0.410216\pi\)
0.278338 + 0.960483i \(0.410216\pi\)
\(410\) 0 0
\(411\) −2.09071e6 −0.610505
\(412\) 6.36178e6 1.84644
\(413\) −3.80938e6 −1.09895
\(414\) −76290.7 −0.0218762
\(415\) 0 0
\(416\) −1.41114e6 −0.399794
\(417\) −2.80499e6 −0.789933
\(418\) 12589.6 0.00352429
\(419\) −5.10510e6 −1.42059 −0.710296 0.703903i \(-0.751439\pi\)
−0.710296 + 0.703903i \(0.751439\pi\)
\(420\) 0 0
\(421\) 5.48811e6 1.50910 0.754550 0.656243i \(-0.227855\pi\)
0.754550 + 0.656243i \(0.227855\pi\)
\(422\) 374797. 0.102451
\(423\) 1.17907e6 0.320397
\(424\) −589145. −0.159150
\(425\) 0 0
\(426\) 381623. 0.101885
\(427\) −1.57990e6 −0.419335
\(428\) −4.16925e6 −1.10014
\(429\) 734700. 0.192738
\(430\) 0 0
\(431\) −3.59589e6 −0.932424 −0.466212 0.884673i \(-0.654382\pi\)
−0.466212 + 0.884673i \(0.654382\pi\)
\(432\) 712882. 0.183784
\(433\) 5.03603e6 1.29083 0.645414 0.763833i \(-0.276685\pi\)
0.645414 + 0.763833i \(0.276685\pi\)
\(434\) −243608. −0.0620822
\(435\) 0 0
\(436\) 4.53987e6 1.14374
\(437\) −203001. −0.0508503
\(438\) −108514. −0.0270271
\(439\) −3.25961e6 −0.807244 −0.403622 0.914926i \(-0.632249\pi\)
−0.403622 + 0.914926i \(0.632249\pi\)
\(440\) 0 0
\(441\) −801093. −0.196149
\(442\) 903586. 0.219996
\(443\) 1.35658e6 0.328424 0.164212 0.986425i \(-0.447492\pi\)
0.164212 + 0.986425i \(0.447492\pi\)
\(444\) −3.80669e6 −0.916412
\(445\) 0 0
\(446\) 314628. 0.0748962
\(447\) 273190. 0.0646690
\(448\) 2.48167e6 0.584184
\(449\) −6.80906e6 −1.59394 −0.796969 0.604020i \(-0.793565\pi\)
−0.796969 + 0.604020i \(0.793565\pi\)
\(450\) 0 0
\(451\) 346712. 0.0802653
\(452\) −34530.8 −0.00794988
\(453\) 1.17913e6 0.269972
\(454\) −28214.7 −0.00642445
\(455\) 0 0
\(456\) −59478.6 −0.0133952
\(457\) 3.03977e6 0.680848 0.340424 0.940272i \(-0.389429\pi\)
0.340424 + 0.940272i \(0.389429\pi\)
\(458\) −686744. −0.152979
\(459\) −1.40526e6 −0.311333
\(460\) 0 0
\(461\) 3.27858e6 0.718511 0.359256 0.933239i \(-0.383031\pi\)
0.359256 + 0.933239i \(0.383031\pi\)
\(462\) 62927.9 0.0137163
\(463\) −5.51970e6 −1.19664 −0.598319 0.801258i \(-0.704164\pi\)
−0.598319 + 0.801258i \(0.704164\pi\)
\(464\) −7.15912e6 −1.54371
\(465\) 0 0
\(466\) −33123.3 −0.00706593
\(467\) 4.07534e6 0.864711 0.432356 0.901703i \(-0.357683\pi\)
0.432356 + 0.901703i \(0.357683\pi\)
\(468\) −1.72233e6 −0.363497
\(469\) 553157. 0.116122
\(470\) 0 0
\(471\) −2.94230e6 −0.611131
\(472\) 2.02137e6 0.417630
\(473\) 2.67145e6 0.549027
\(474\) −385674. −0.0788450
\(475\) 0 0
\(476\) −5.05284e6 −1.02216
\(477\) −1.08133e6 −0.217602
\(478\) 114762. 0.0229736
\(479\) 4.77799e6 0.951495 0.475748 0.879582i \(-0.342178\pi\)
0.475748 + 0.879582i \(0.342178\pi\)
\(480\) 0 0
\(481\) 9.05397e6 1.78433
\(482\) 787216. 0.154339
\(483\) −1.01468e6 −0.197907
\(484\) −461444. −0.0895377
\(485\) 0 0
\(486\) −41027.1 −0.00787916
\(487\) −1.76923e6 −0.338035 −0.169017 0.985613i \(-0.554059\pi\)
−0.169017 + 0.985613i \(0.554059\pi\)
\(488\) 838345. 0.159358
\(489\) 4.51669e6 0.854177
\(490\) 0 0
\(491\) −1.04139e6 −0.194943 −0.0974716 0.995238i \(-0.531076\pi\)
−0.0974716 + 0.995238i \(0.531076\pi\)
\(492\) −812784. −0.151378
\(493\) 1.41124e7 2.61506
\(494\) 70195.3 0.0129417
\(495\) 0 0
\(496\) −4.12255e6 −0.752422
\(497\) 5.07565e6 0.921723
\(498\) −407900. −0.0737021
\(499\) 2.03068e6 0.365082 0.182541 0.983198i \(-0.441568\pi\)
0.182541 + 0.983198i \(0.441568\pi\)
\(500\) 0 0
\(501\) 5.31416e6 0.945890
\(502\) 750588. 0.132936
\(503\) 5.06008e6 0.891739 0.445869 0.895098i \(-0.352895\pi\)
0.445869 + 0.895098i \(0.352895\pi\)
\(504\) −297298. −0.0521334
\(505\) 0 0
\(506\) −113965. −0.0197877
\(507\) 754800. 0.130410
\(508\) 9.95183e6 1.71098
\(509\) 5.64302e6 0.965421 0.482711 0.875780i \(-0.339652\pi\)
0.482711 + 0.875780i \(0.339652\pi\)
\(510\) 0 0
\(511\) −1.44325e6 −0.244506
\(512\) −3.42639e6 −0.577645
\(513\) −109168. −0.0183148
\(514\) 209454. 0.0349688
\(515\) 0 0
\(516\) −6.26256e6 −1.03545
\(517\) 1.76133e6 0.289810
\(518\) 775484. 0.126984
\(519\) −5.13841e6 −0.837356
\(520\) 0 0
\(521\) 5.37673e6 0.867809 0.433905 0.900959i \(-0.357136\pi\)
0.433905 + 0.900959i \(0.357136\pi\)
\(522\) 412014. 0.0661814
\(523\) −8.42403e6 −1.34668 −0.673342 0.739331i \(-0.735142\pi\)
−0.673342 + 0.739331i \(0.735142\pi\)
\(524\) 5.89376e6 0.937701
\(525\) 0 0
\(526\) −241619. −0.0380773
\(527\) 8.12654e6 1.27462
\(528\) 1.06492e6 0.166239
\(529\) −4.59871e6 −0.714492
\(530\) 0 0
\(531\) 3.71007e6 0.571012
\(532\) −392532. −0.0601306
\(533\) 1.93315e6 0.294746
\(534\) 685115. 0.103970
\(535\) 0 0
\(536\) −293522. −0.0441294
\(537\) −4.17962e6 −0.625462
\(538\) −681563. −0.101520
\(539\) −1.19669e6 −0.177424
\(540\) 0 0
\(541\) 8.85130e6 1.30021 0.650106 0.759844i \(-0.274725\pi\)
0.650106 + 0.759844i \(0.274725\pi\)
\(542\) −6691.40 −0.000978405 0
\(543\) −3.68813e6 −0.536792
\(544\) 4.03198e6 0.584145
\(545\) 0 0
\(546\) 350865. 0.0503684
\(547\) −3.13327e6 −0.447744 −0.223872 0.974619i \(-0.571870\pi\)
−0.223872 + 0.974619i \(0.571870\pi\)
\(548\) 7.32150e6 1.04147
\(549\) 1.53871e6 0.217885
\(550\) 0 0
\(551\) 1.09632e6 0.153836
\(552\) 538420. 0.0752096
\(553\) −5.12953e6 −0.713288
\(554\) 225370. 0.0311976
\(555\) 0 0
\(556\) 9.82283e6 1.34756
\(557\) 5.46691e6 0.746627 0.373313 0.927705i \(-0.378222\pi\)
0.373313 + 0.927705i \(0.378222\pi\)
\(558\) 237257. 0.0322577
\(559\) 1.48951e7 2.01611
\(560\) 0 0
\(561\) −2.09922e6 −0.281612
\(562\) −602289. −0.0804385
\(563\) −5.19043e6 −0.690132 −0.345066 0.938578i \(-0.612143\pi\)
−0.345066 + 0.938578i \(0.612143\pi\)
\(564\) −4.12901e6 −0.546573
\(565\) 0 0
\(566\) −399618. −0.0524329
\(567\) −545667. −0.0712804
\(568\) −2.69329e6 −0.350278
\(569\) 1.24058e7 1.60636 0.803180 0.595737i \(-0.203140\pi\)
0.803180 + 0.595737i \(0.203140\pi\)
\(570\) 0 0
\(571\) 8.79234e6 1.12853 0.564266 0.825593i \(-0.309159\pi\)
0.564266 + 0.825593i \(0.309159\pi\)
\(572\) −2.57286e6 −0.328795
\(573\) −5.31926e6 −0.676807
\(574\) 165577. 0.0209759
\(575\) 0 0
\(576\) −2.41698e6 −0.303540
\(577\) 1.14336e7 1.42969 0.714845 0.699283i \(-0.246497\pi\)
0.714845 + 0.699283i \(0.246497\pi\)
\(578\) −1.59526e6 −0.198615
\(579\) 2.79817e6 0.346878
\(580\) 0 0
\(581\) −5.42514e6 −0.666761
\(582\) 523616. 0.0640775
\(583\) −1.61532e6 −0.196828
\(584\) 765834. 0.0929185
\(585\) 0 0
\(586\) 1.36231e6 0.163882
\(587\) 2.32244e6 0.278195 0.139097 0.990279i \(-0.455580\pi\)
0.139097 + 0.990279i \(0.455580\pi\)
\(588\) 2.80536e6 0.334615
\(589\) 631313. 0.0749819
\(590\) 0 0
\(591\) −1.45452e6 −0.171297
\(592\) 1.31234e7 1.53901
\(593\) −7.98435e6 −0.932401 −0.466201 0.884679i \(-0.654377\pi\)
−0.466201 + 0.884679i \(0.654377\pi\)
\(594\) −61287.3 −0.00712697
\(595\) 0 0
\(596\) −956690. −0.110320
\(597\) 3.92060e6 0.450212
\(598\) −635431. −0.0726634
\(599\) −139225. −0.0158544 −0.00792721 0.999969i \(-0.502523\pi\)
−0.00792721 + 0.999969i \(0.502523\pi\)
\(600\) 0 0
\(601\) 1.55602e7 1.75723 0.878616 0.477529i \(-0.158467\pi\)
0.878616 + 0.477529i \(0.158467\pi\)
\(602\) 1.27578e6 0.143478
\(603\) −538735. −0.0603368
\(604\) −4.12923e6 −0.460550
\(605\) 0 0
\(606\) 812141. 0.0898358
\(607\) 6.00544e6 0.661566 0.330783 0.943707i \(-0.392687\pi\)
0.330783 + 0.943707i \(0.392687\pi\)
\(608\) 313225. 0.0343635
\(609\) 5.47986e6 0.598724
\(610\) 0 0
\(611\) 9.82057e6 1.06423
\(612\) 4.92111e6 0.531110
\(613\) 2.32775e6 0.250199 0.125099 0.992144i \(-0.460075\pi\)
0.125099 + 0.992144i \(0.460075\pi\)
\(614\) 1.11549e6 0.119411
\(615\) 0 0
\(616\) −444112. −0.0471564
\(617\) −1.30284e7 −1.37777 −0.688886 0.724870i \(-0.741900\pi\)
−0.688886 + 0.724870i \(0.741900\pi\)
\(618\) 1.26221e6 0.132941
\(619\) −2.53159e6 −0.265562 −0.132781 0.991145i \(-0.542391\pi\)
−0.132781 + 0.991145i \(0.542391\pi\)
\(620\) 0 0
\(621\) 988226. 0.102832
\(622\) −4601.30 −0.000476876 0
\(623\) 9.11214e6 0.940590
\(624\) 5.93765e6 0.610454
\(625\) 0 0
\(626\) −755689. −0.0770739
\(627\) −163078. −0.0165664
\(628\) 1.03037e7 1.04254
\(629\) −2.58694e7 −2.60712
\(630\) 0 0
\(631\) −2.17733e6 −0.217696 −0.108848 0.994058i \(-0.534716\pi\)
−0.108848 + 0.994058i \(0.534716\pi\)
\(632\) 2.72188e6 0.271067
\(633\) −4.85491e6 −0.481583
\(634\) −316955. −0.0313166
\(635\) 0 0
\(636\) 3.78673e6 0.371211
\(637\) −6.67237e6 −0.651525
\(638\) 615478. 0.0598634
\(639\) −4.94332e6 −0.478924
\(640\) 0 0
\(641\) −2.58180e6 −0.248187 −0.124093 0.992271i \(-0.539602\pi\)
−0.124093 + 0.992271i \(0.539602\pi\)
\(642\) −827200. −0.0792087
\(643\) 1.67813e7 1.60066 0.800329 0.599561i \(-0.204658\pi\)
0.800329 + 0.599561i \(0.204658\pi\)
\(644\) 3.55333e6 0.337614
\(645\) 0 0
\(646\) −200566. −0.0189093
\(647\) 1.01691e7 0.955037 0.477518 0.878622i \(-0.341536\pi\)
0.477518 + 0.878622i \(0.341536\pi\)
\(648\) 289547. 0.0270883
\(649\) 5.54220e6 0.516500
\(650\) 0 0
\(651\) 3.15556e6 0.291826
\(652\) −1.58171e7 −1.45716
\(653\) 8.99282e6 0.825302 0.412651 0.910889i \(-0.364603\pi\)
0.412651 + 0.910889i \(0.364603\pi\)
\(654\) 900731. 0.0823476
\(655\) 0 0
\(656\) 2.80204e6 0.254223
\(657\) 1.40563e6 0.127045
\(658\) 841144. 0.0757365
\(659\) −1.03480e7 −0.928203 −0.464102 0.885782i \(-0.653623\pi\)
−0.464102 + 0.885782i \(0.653623\pi\)
\(660\) 0 0
\(661\) 1.48595e7 1.32282 0.661409 0.750025i \(-0.269959\pi\)
0.661409 + 0.750025i \(0.269959\pi\)
\(662\) −1.20253e6 −0.106648
\(663\) −1.17045e7 −1.03412
\(664\) 2.87874e6 0.253386
\(665\) 0 0
\(666\) −755266. −0.0659803
\(667\) −9.92427e6 −0.863742
\(668\) −1.86098e7 −1.61361
\(669\) −4.07551e6 −0.352060
\(670\) 0 0
\(671\) 2.29857e6 0.197084
\(672\) 1.56563e6 0.133741
\(673\) −217545. −0.0185145 −0.00925725 0.999957i \(-0.502947\pi\)
−0.00925725 + 0.999957i \(0.502947\pi\)
\(674\) 2.09625e6 0.177744
\(675\) 0 0
\(676\) −2.64325e6 −0.222470
\(677\) 2.64314e6 0.221640 0.110820 0.993840i \(-0.464652\pi\)
0.110820 + 0.993840i \(0.464652\pi\)
\(678\) −6851.07 −0.000572380 0
\(679\) 6.96419e6 0.579690
\(680\) 0 0
\(681\) 365477. 0.0301990
\(682\) 354421. 0.0291782
\(683\) −4.66785e6 −0.382882 −0.191441 0.981504i \(-0.561316\pi\)
−0.191441 + 0.981504i \(0.561316\pi\)
\(684\) 382298. 0.0312436
\(685\) 0 0
\(686\) −1.54269e6 −0.125161
\(687\) 8.89569e6 0.719098
\(688\) 2.15899e7 1.73892
\(689\) −9.00647e6 −0.722781
\(690\) 0 0
\(691\) 3.11273e6 0.247997 0.123998 0.992282i \(-0.460428\pi\)
0.123998 + 0.992282i \(0.460428\pi\)
\(692\) 1.79943e7 1.42846
\(693\) −815132. −0.0644755
\(694\) 882026. 0.0695156
\(695\) 0 0
\(696\) −2.90778e6 −0.227530
\(697\) −5.52349e6 −0.430657
\(698\) −2.26795e6 −0.176196
\(699\) 429061. 0.0332143
\(700\) 0 0
\(701\) 1.16786e7 0.897629 0.448815 0.893625i \(-0.351846\pi\)
0.448815 + 0.893625i \(0.351846\pi\)
\(702\) −341718. −0.0261713
\(703\) −2.00967e6 −0.153369
\(704\) −3.61054e6 −0.274562
\(705\) 0 0
\(706\) 1.18630e6 0.0895743
\(707\) 1.08016e7 0.812718
\(708\) −1.29923e7 −0.974102
\(709\) −1.96571e7 −1.46860 −0.734302 0.678823i \(-0.762490\pi\)
−0.734302 + 0.678823i \(0.762490\pi\)
\(710\) 0 0
\(711\) 4.99580e6 0.370622
\(712\) −4.83518e6 −0.357448
\(713\) −5.71485e6 −0.420999
\(714\) −1.00251e6 −0.0735940
\(715\) 0 0
\(716\) 1.46367e7 1.06699
\(717\) −1.48656e6 −0.107990
\(718\) −179041. −0.0129611
\(719\) −1.89013e7 −1.36354 −0.681772 0.731565i \(-0.738790\pi\)
−0.681772 + 0.731565i \(0.738790\pi\)
\(720\) 0 0
\(721\) 1.67876e7 1.20268
\(722\) 1.70480e6 0.121712
\(723\) −1.01971e7 −0.725492
\(724\) 1.29155e7 0.915725
\(725\) 0 0
\(726\) −91552.7 −0.00644658
\(727\) −6.78994e6 −0.476463 −0.238232 0.971208i \(-0.576568\pi\)
−0.238232 + 0.971208i \(0.576568\pi\)
\(728\) −2.47622e6 −0.173165
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −4.25590e7 −2.94576
\(732\) −5.38845e6 −0.371695
\(733\) 3.60300e6 0.247688 0.123844 0.992302i \(-0.460478\pi\)
0.123844 + 0.992302i \(0.460478\pi\)
\(734\) −474886. −0.0325349
\(735\) 0 0
\(736\) −2.83542e6 −0.192940
\(737\) −804778. −0.0545767
\(738\) −161260. −0.0108990
\(739\) −7.58466e6 −0.510887 −0.255443 0.966824i \(-0.582221\pi\)
−0.255443 + 0.966824i \(0.582221\pi\)
\(740\) 0 0
\(741\) −909270. −0.0608341
\(742\) −771416. −0.0514374
\(743\) −1.17473e7 −0.780669 −0.390335 0.920673i \(-0.627641\pi\)
−0.390335 + 0.920673i \(0.627641\pi\)
\(744\) −1.67443e6 −0.110901
\(745\) 0 0
\(746\) 2.48860e6 0.163722
\(747\) 5.28370e6 0.346447
\(748\) 7.35129e6 0.480407
\(749\) −1.10019e7 −0.716577
\(750\) 0 0
\(751\) 4.05057e6 0.262070 0.131035 0.991378i \(-0.458170\pi\)
0.131035 + 0.991378i \(0.458170\pi\)
\(752\) 1.42346e7 0.917910
\(753\) −9.72269e6 −0.624883
\(754\) 3.43170e6 0.219827
\(755\) 0 0
\(756\) 1.91088e6 0.121599
\(757\) −2.12290e7 −1.34645 −0.673225 0.739438i \(-0.735091\pi\)
−0.673225 + 0.739438i \(0.735091\pi\)
\(758\) 3.39029e6 0.214321
\(759\) 1.47624e6 0.0930149
\(760\) 0 0
\(761\) −1.06438e7 −0.666247 −0.333123 0.942883i \(-0.608103\pi\)
−0.333123 + 0.942883i \(0.608103\pi\)
\(762\) 1.97449e6 0.123188
\(763\) 1.19799e7 0.744974
\(764\) 1.86276e7 1.15458
\(765\) 0 0
\(766\) 3.45702e6 0.212878
\(767\) 3.09014e7 1.89666
\(768\) 8.04551e6 0.492210
\(769\) 3.17109e6 0.193372 0.0966859 0.995315i \(-0.469176\pi\)
0.0966859 + 0.995315i \(0.469176\pi\)
\(770\) 0 0
\(771\) −2.71314e6 −0.164375
\(772\) −9.79894e6 −0.591747
\(773\) −1.80501e7 −1.08650 −0.543252 0.839570i \(-0.682807\pi\)
−0.543252 + 0.839570i \(0.682807\pi\)
\(774\) −1.24252e6 −0.0745507
\(775\) 0 0
\(776\) −3.69541e6 −0.220297
\(777\) −1.00452e7 −0.596904
\(778\) 2.84774e6 0.168675
\(779\) −429094. −0.0253343
\(780\) 0 0
\(781\) −7.38447e6 −0.433203
\(782\) 1.81559e6 0.106169
\(783\) −5.33700e6 −0.311095
\(784\) −9.67137e6 −0.561950
\(785\) 0 0
\(786\) 1.16935e6 0.0675132
\(787\) 2.58223e7 1.48614 0.743068 0.669216i \(-0.233370\pi\)
0.743068 + 0.669216i \(0.233370\pi\)
\(788\) 5.09360e6 0.292219
\(789\) 3.12979e6 0.178988
\(790\) 0 0
\(791\) −91120.5 −0.00517815
\(792\) 432534. 0.0245023
\(793\) 1.28161e7 0.723722
\(794\) 1.12012e6 0.0630540
\(795\) 0 0
\(796\) −1.37296e7 −0.768026
\(797\) −1.50658e7 −0.840129 −0.420065 0.907494i \(-0.637993\pi\)
−0.420065 + 0.907494i \(0.637993\pi\)
\(798\) −77880.1 −0.00432932
\(799\) −2.80598e7 −1.55495
\(800\) 0 0
\(801\) −8.87458e6 −0.488727
\(802\) 450480. 0.0247309
\(803\) 2.09976e6 0.114916
\(804\) 1.88661e6 0.102930
\(805\) 0 0
\(806\) 1.97613e6 0.107147
\(807\) 8.82858e6 0.477207
\(808\) −5.73166e6 −0.308853
\(809\) −2.08560e7 −1.12037 −0.560183 0.828369i \(-0.689269\pi\)
−0.560183 + 0.828369i \(0.689269\pi\)
\(810\) 0 0
\(811\) 8.46233e6 0.451791 0.225896 0.974152i \(-0.427469\pi\)
0.225896 + 0.974152i \(0.427469\pi\)
\(812\) −1.91900e7 −1.02138
\(813\) 86676.5 0.00459912
\(814\) −1.12824e6 −0.0596814
\(815\) 0 0
\(816\) −1.69653e7 −0.891943
\(817\) −3.30620e6 −0.173290
\(818\) −1.30849e6 −0.0683732
\(819\) −4.54490e6 −0.236764
\(820\) 0 0
\(821\) 8.67359e6 0.449098 0.224549 0.974463i \(-0.427909\pi\)
0.224549 + 0.974463i \(0.427909\pi\)
\(822\) 1.45262e6 0.0749846
\(823\) 1.24194e7 0.639148 0.319574 0.947561i \(-0.396460\pi\)
0.319574 + 0.947561i \(0.396460\pi\)
\(824\) −8.90799e6 −0.457048
\(825\) 0 0
\(826\) 2.64675e6 0.134978
\(827\) 2.38473e7 1.21248 0.606242 0.795280i \(-0.292676\pi\)
0.606242 + 0.795280i \(0.292676\pi\)
\(828\) −3.46069e6 −0.175423
\(829\) −3.26873e7 −1.65193 −0.825967 0.563719i \(-0.809370\pi\)
−0.825967 + 0.563719i \(0.809370\pi\)
\(830\) 0 0
\(831\) −2.91931e6 −0.146649
\(832\) −2.01312e7 −1.00823
\(833\) 1.90646e7 0.951953
\(834\) 1.94890e6 0.0970227
\(835\) 0 0
\(836\) 571087. 0.0282609
\(837\) −3.07329e6 −0.151632
\(838\) 3.54701e6 0.174483
\(839\) 8.09423e6 0.396982 0.198491 0.980103i \(-0.436396\pi\)
0.198491 + 0.980103i \(0.436396\pi\)
\(840\) 0 0
\(841\) 3.30857e7 1.61306
\(842\) −3.81313e6 −0.185354
\(843\) 7.80171e6 0.378112
\(844\) 1.70015e7 0.821544
\(845\) 0 0
\(846\) −819214. −0.0393524
\(847\) −1.21767e6 −0.0583203
\(848\) −1.30546e7 −0.623409
\(849\) 5.17642e6 0.246468
\(850\) 0 0
\(851\) 1.81922e7 0.861117
\(852\) 1.73111e7 0.817007
\(853\) −3.67970e6 −0.173157 −0.0865785 0.996245i \(-0.527593\pi\)
−0.0865785 + 0.996245i \(0.527593\pi\)
\(854\) 1.09771e6 0.0515043
\(855\) 0 0
\(856\) 5.83794e6 0.272317
\(857\) −2.17589e7 −1.01201 −0.506004 0.862531i \(-0.668878\pi\)
−0.506004 + 0.862531i \(0.668878\pi\)
\(858\) −510467. −0.0236728
\(859\) −3.76305e7 −1.74003 −0.870015 0.493025i \(-0.835891\pi\)
−0.870015 + 0.493025i \(0.835891\pi\)
\(860\) 0 0
\(861\) −2.14479e6 −0.0985998
\(862\) 2.49841e6 0.114524
\(863\) 2.50264e7 1.14386 0.571928 0.820304i \(-0.306196\pi\)
0.571928 + 0.820304i \(0.306196\pi\)
\(864\) −1.52481e6 −0.0694914
\(865\) 0 0
\(866\) −3.49902e6 −0.158545
\(867\) 2.06641e7 0.933615
\(868\) −1.10505e7 −0.497832
\(869\) 7.46286e6 0.335240
\(870\) 0 0
\(871\) −4.48717e6 −0.200414
\(872\) −6.35689e6 −0.283109
\(873\) −6.78263e6 −0.301205
\(874\) 141044. 0.00624564
\(875\) 0 0
\(876\) −4.92239e6 −0.216728
\(877\) −1.12677e7 −0.494692 −0.247346 0.968927i \(-0.579558\pi\)
−0.247346 + 0.968927i \(0.579558\pi\)
\(878\) 2.26477e6 0.0991488
\(879\) −1.76466e7 −0.770351
\(880\) 0 0
\(881\) −4.36223e7 −1.89352 −0.946758 0.321947i \(-0.895663\pi\)
−0.946758 + 0.321947i \(0.895663\pi\)
\(882\) 556597. 0.0240918
\(883\) 3.14744e7 1.35849 0.679244 0.733912i \(-0.262308\pi\)
0.679244 + 0.733912i \(0.262308\pi\)
\(884\) 4.09883e7 1.76413
\(885\) 0 0
\(886\) −942545. −0.0403383
\(887\) −2.33777e7 −0.997684 −0.498842 0.866693i \(-0.666241\pi\)
−0.498842 + 0.866693i \(0.666241\pi\)
\(888\) 5.33027e6 0.226839
\(889\) 2.62611e7 1.11444
\(890\) 0 0
\(891\) 793881. 0.0335013
\(892\) 1.42721e7 0.600587
\(893\) −2.17983e6 −0.0914733
\(894\) −189812. −0.00794290
\(895\) 0 0
\(896\) −7.29093e6 −0.303398
\(897\) 8.23101e6 0.341564
\(898\) 4.73091e6 0.195774
\(899\) 3.08635e7 1.27364
\(900\) 0 0
\(901\) 2.57337e7 1.05607
\(902\) −240895. −0.00985850
\(903\) −1.65258e7 −0.674438
\(904\) 48351.3 0.00196783
\(905\) 0 0
\(906\) −819259. −0.0331590
\(907\) −2.58345e7 −1.04275 −0.521377 0.853326i \(-0.674582\pi\)
−0.521377 + 0.853326i \(0.674582\pi\)
\(908\) −1.27987e6 −0.0515171
\(909\) −1.05200e7 −0.422285
\(910\) 0 0
\(911\) 1.24995e7 0.498996 0.249498 0.968375i \(-0.419734\pi\)
0.249498 + 0.968375i \(0.419734\pi\)
\(912\) −1.31796e6 −0.0524703
\(913\) 7.89293e6 0.313373
\(914\) −2.11202e6 −0.0836244
\(915\) 0 0
\(916\) −3.11520e7 −1.22672
\(917\) 1.55526e7 0.610771
\(918\) 976372. 0.0382392
\(919\) 1.24766e7 0.487313 0.243657 0.969862i \(-0.421653\pi\)
0.243657 + 0.969862i \(0.421653\pi\)
\(920\) 0 0
\(921\) −1.44494e7 −0.561309
\(922\) −2.27795e6 −0.0882503
\(923\) −4.11733e7 −1.59079
\(924\) 2.85453e6 0.109990
\(925\) 0 0
\(926\) 3.83507e6 0.146976
\(927\) −1.63499e7 −0.624908
\(928\) 1.53129e7 0.583697
\(929\) −1.63852e6 −0.0622891 −0.0311445 0.999515i \(-0.509915\pi\)
−0.0311445 + 0.999515i \(0.509915\pi\)
\(930\) 0 0
\(931\) 1.48104e6 0.0560006
\(932\) −1.50253e6 −0.0566611
\(933\) 59602.7 0.00224162
\(934\) −2.83153e6 −0.106207
\(935\) 0 0
\(936\) 2.41166e6 0.0899760
\(937\) −1.03137e7 −0.383766 −0.191883 0.981418i \(-0.561459\pi\)
−0.191883 + 0.981418i \(0.561459\pi\)
\(938\) −384331. −0.0142626
\(939\) 9.78876e6 0.362296
\(940\) 0 0
\(941\) −1.30200e7 −0.479332 −0.239666 0.970855i \(-0.577038\pi\)
−0.239666 + 0.970855i \(0.577038\pi\)
\(942\) 2.04430e6 0.0750615
\(943\) 3.88430e6 0.142244
\(944\) 4.47906e7 1.63590
\(945\) 0 0
\(946\) −1.85611e6 −0.0674336
\(947\) −4.07950e7 −1.47820 −0.739098 0.673598i \(-0.764748\pi\)
−0.739098 + 0.673598i \(0.764748\pi\)
\(948\) −1.74949e7 −0.632252
\(949\) 1.17076e7 0.421989
\(950\) 0 0
\(951\) 4.10565e6 0.147208
\(952\) 7.07517e6 0.253014
\(953\) −4.68439e7 −1.67078 −0.835392 0.549654i \(-0.814760\pi\)
−0.835392 + 0.549654i \(0.814760\pi\)
\(954\) 751304. 0.0267267
\(955\) 0 0
\(956\) 5.20582e6 0.184223
\(957\) −7.97255e6 −0.281396
\(958\) −3.31973e6 −0.116866
\(959\) 1.93201e7 0.678364
\(960\) 0 0
\(961\) −1.08565e7 −0.379212
\(962\) −6.29067e6 −0.219159
\(963\) 1.07151e7 0.372331
\(964\) 3.57095e7 1.23763
\(965\) 0 0
\(966\) 704996. 0.0243077
\(967\) −1.63484e7 −0.562223 −0.281111 0.959675i \(-0.590703\pi\)
−0.281111 + 0.959675i \(0.590703\pi\)
\(968\) 646131. 0.0221632
\(969\) 2.59801e6 0.0888856
\(970\) 0 0
\(971\) −1.62139e6 −0.0551875 −0.0275937 0.999619i \(-0.508784\pi\)
−0.0275937 + 0.999619i \(0.508784\pi\)
\(972\) −1.86106e6 −0.0631823
\(973\) 2.59206e7 0.877735
\(974\) 1.22925e6 0.0415187
\(975\) 0 0
\(976\) 1.85765e7 0.624221
\(977\) 2.33840e7 0.783759 0.391880 0.920017i \(-0.371825\pi\)
0.391880 + 0.920017i \(0.371825\pi\)
\(978\) −3.13818e6 −0.104913
\(979\) −1.32571e7 −0.442070
\(980\) 0 0
\(981\) −1.16676e7 −0.387086
\(982\) 723552. 0.0239437
\(983\) −1.96781e7 −0.649529 −0.324764 0.945795i \(-0.605285\pi\)
−0.324764 + 0.945795i \(0.605285\pi\)
\(984\) 1.13809e6 0.0374704
\(985\) 0 0
\(986\) −9.80522e6 −0.321192
\(987\) −1.08957e7 −0.356010
\(988\) 3.18419e6 0.103778
\(989\) 2.99289e7 0.972970
\(990\) 0 0
\(991\) 1.56015e7 0.504642 0.252321 0.967644i \(-0.418806\pi\)
0.252321 + 0.967644i \(0.418806\pi\)
\(992\) 8.81788e6 0.284502
\(993\) 1.55769e7 0.501312
\(994\) −3.52654e6 −0.113210
\(995\) 0 0
\(996\) −1.85031e7 −0.591011
\(997\) 5.71913e6 0.182218 0.0911091 0.995841i \(-0.470959\pi\)
0.0911091 + 0.995841i \(0.470959\pi\)
\(998\) −1.41091e6 −0.0448408
\(999\) 9.78328e6 0.310149
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 825.6.a.g.1.2 3
5.4 even 2 165.6.a.c.1.2 3
15.14 odd 2 495.6.a.b.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.c.1.2 3 5.4 even 2
495.6.a.b.1.2 3 15.14 odd 2
825.6.a.g.1.2 3 1.1 even 1 trivial