Properties

Label 294.6.a.u.1.1
Level $294$
Weight $6$
Character 294.1
Self dual yes
Analytic conductor $47.153$
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] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, 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("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.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: \(47.1528430250\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 294.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -61.0711 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +9.00000 q^{3} +16.0000 q^{4} -61.0711 q^{5} +36.0000 q^{6} +64.0000 q^{8} +81.0000 q^{9} -244.284 q^{10} -36.5442 q^{11} +144.000 q^{12} +34.5656 q^{13} -549.640 q^{15} +256.000 q^{16} -2061.04 q^{17} +324.000 q^{18} +452.238 q^{19} -977.137 q^{20} -146.177 q^{22} +1684.25 q^{23} +576.000 q^{24} +604.675 q^{25} +138.262 q^{26} +729.000 q^{27} -4765.22 q^{29} -2198.56 q^{30} -5260.87 q^{31} +1024.00 q^{32} -328.897 q^{33} -8244.16 q^{34} +1296.00 q^{36} -12822.0 q^{37} +1808.95 q^{38} +311.090 q^{39} -3908.55 q^{40} -7126.74 q^{41} +11141.7 q^{43} -584.706 q^{44} -4946.76 q^{45} +6736.99 q^{46} -23443.2 q^{47} +2304.00 q^{48} +2418.70 q^{50} -18549.4 q^{51} +553.049 q^{52} -7030.30 q^{53} +2916.00 q^{54} +2231.79 q^{55} +4070.14 q^{57} -19060.9 q^{58} -44222.8 q^{59} -8794.23 q^{60} +19380.5 q^{61} -21043.5 q^{62} +4096.00 q^{64} -2110.96 q^{65} -1315.59 q^{66} +20944.1 q^{67} -32976.6 q^{68} +15158.2 q^{69} +79843.4 q^{71} +5184.00 q^{72} -37065.8 q^{73} -51287.8 q^{74} +5442.08 q^{75} +7235.81 q^{76} +1244.36 q^{78} +42072.0 q^{79} -15634.2 q^{80} +6561.00 q^{81} -28506.9 q^{82} -6311.34 q^{83} +125870. q^{85} +44566.8 q^{86} -42887.0 q^{87} -2338.83 q^{88} -51495.8 q^{89} -19787.0 q^{90} +26948.0 q^{92} -47347.8 q^{93} -93772.8 q^{94} -27618.7 q^{95} +9216.00 q^{96} -127357. q^{97} -2960.08 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} - 108 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{2} + 18 q^{3} + 32 q^{4} - 108 q^{5} + 72 q^{6} + 128 q^{8} + 162 q^{9} - 432 q^{10} - 124 q^{11} + 288 q^{12} - 720 q^{13} - 972 q^{15} + 512 q^{16} - 612 q^{17} + 648 q^{18} - 2088 q^{19} - 1728 q^{20} - 496 q^{22} + 772 q^{23} + 1152 q^{24} - 318 q^{25} - 2880 q^{26} + 1458 q^{27} - 4592 q^{29} - 3888 q^{30} - 9792 q^{31} + 2048 q^{32} - 1116 q^{33} - 2448 q^{34} + 2592 q^{36} - 5992 q^{37} - 8352 q^{38} - 6480 q^{39} - 6912 q^{40} - 20196 q^{41} - 1136 q^{43} - 1984 q^{44} - 8748 q^{45} + 3088 q^{46} - 36936 q^{47} + 4608 q^{48} - 1272 q^{50} - 5508 q^{51} - 11520 q^{52} - 16708 q^{53} + 5832 q^{54} + 6336 q^{55} - 18792 q^{57} - 18368 q^{58} - 74592 q^{59} - 15552 q^{60} + 18648 q^{61} - 39168 q^{62} + 8192 q^{64} + 33300 q^{65} - 4464 q^{66} + 67344 q^{67} - 9792 q^{68} + 6948 q^{69} + 76548 q^{71} + 10368 q^{72} - 47304 q^{73} - 23968 q^{74} - 2862 q^{75} - 33408 q^{76} - 25920 q^{78} + 140656 q^{79} - 27648 q^{80} + 13122 q^{81} - 80784 q^{82} - 94104 q^{83} + 57868 q^{85} - 4544 q^{86} - 41328 q^{87} - 7936 q^{88} + 17604 q^{89} - 34992 q^{90} + 12352 q^{92} - 88128 q^{93} - 147744 q^{94} + 91592 q^{95} + 18432 q^{96} - 85176 q^{97} - 10044 q^{99}+O(q^{100}) \) 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
\(5\) −61.0711 −1.09247 −0.546236 0.837631i \(-0.683940\pi\)
−0.546236 + 0.837631i \(0.683940\pi\)
\(6\) 36.0000 0.408248
\(7\) 0 0
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) −244.284 −0.772495
\(11\) −36.5442 −0.0910618 −0.0455309 0.998963i \(-0.514498\pi\)
−0.0455309 + 0.998963i \(0.514498\pi\)
\(12\) 144.000 0.288675
\(13\) 34.5656 0.0567264 0.0283632 0.999598i \(-0.490970\pi\)
0.0283632 + 0.999598i \(0.490970\pi\)
\(14\) 0 0
\(15\) −549.640 −0.630739
\(16\) 256.000 0.250000
\(17\) −2061.04 −1.72967 −0.864836 0.502054i \(-0.832578\pi\)
−0.864836 + 0.502054i \(0.832578\pi\)
\(18\) 324.000 0.235702
\(19\) 452.238 0.287398 0.143699 0.989621i \(-0.454100\pi\)
0.143699 + 0.989621i \(0.454100\pi\)
\(20\) −977.137 −0.546236
\(21\) 0 0
\(22\) −146.177 −0.0643904
\(23\) 1684.25 0.663875 0.331938 0.943301i \(-0.392298\pi\)
0.331938 + 0.943301i \(0.392298\pi\)
\(24\) 576.000 0.204124
\(25\) 604.675 0.193496
\(26\) 138.262 0.0401116
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −4765.22 −1.05217 −0.526087 0.850431i \(-0.676341\pi\)
−0.526087 + 0.850431i \(0.676341\pi\)
\(30\) −2198.56 −0.446000
\(31\) −5260.87 −0.983225 −0.491613 0.870814i \(-0.663592\pi\)
−0.491613 + 0.870814i \(0.663592\pi\)
\(32\) 1024.00 0.176777
\(33\) −328.897 −0.0525746
\(34\) −8244.16 −1.22306
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −12822.0 −1.53975 −0.769875 0.638195i \(-0.779681\pi\)
−0.769875 + 0.638195i \(0.779681\pi\)
\(38\) 1808.95 0.203221
\(39\) 311.090 0.0327510
\(40\) −3908.55 −0.386247
\(41\) −7126.74 −0.662111 −0.331056 0.943611i \(-0.607405\pi\)
−0.331056 + 0.943611i \(0.607405\pi\)
\(42\) 0 0
\(43\) 11141.7 0.918925 0.459462 0.888197i \(-0.348042\pi\)
0.459462 + 0.888197i \(0.348042\pi\)
\(44\) −584.706 −0.0455309
\(45\) −4946.76 −0.364157
\(46\) 6736.99 0.469431
\(47\) −23443.2 −1.54800 −0.774002 0.633183i \(-0.781748\pi\)
−0.774002 + 0.633183i \(0.781748\pi\)
\(48\) 2304.00 0.144338
\(49\) 0 0
\(50\) 2418.70 0.136822
\(51\) −18549.4 −0.998627
\(52\) 553.049 0.0283632
\(53\) −7030.30 −0.343783 −0.171891 0.985116i \(-0.554988\pi\)
−0.171891 + 0.985116i \(0.554988\pi\)
\(54\) 2916.00 0.136083
\(55\) 2231.79 0.0994825
\(56\) 0 0
\(57\) 4070.14 0.165929
\(58\) −19060.9 −0.744000
\(59\) −44222.8 −1.65393 −0.826964 0.562255i \(-0.809934\pi\)
−0.826964 + 0.562255i \(0.809934\pi\)
\(60\) −8794.23 −0.315370
\(61\) 19380.5 0.666868 0.333434 0.942773i \(-0.391793\pi\)
0.333434 + 0.942773i \(0.391793\pi\)
\(62\) −21043.5 −0.695245
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) −2110.96 −0.0619721
\(66\) −1315.59 −0.0371758
\(67\) 20944.1 0.569999 0.285000 0.958528i \(-0.408007\pi\)
0.285000 + 0.958528i \(0.408007\pi\)
\(68\) −32976.6 −0.864836
\(69\) 15158.2 0.383289
\(70\) 0 0
\(71\) 79843.4 1.87972 0.939860 0.341560i \(-0.110955\pi\)
0.939860 + 0.341560i \(0.110955\pi\)
\(72\) 5184.00 0.117851
\(73\) −37065.8 −0.814079 −0.407039 0.913411i \(-0.633439\pi\)
−0.407039 + 0.913411i \(0.633439\pi\)
\(74\) −51287.8 −1.08877
\(75\) 5442.08 0.111715
\(76\) 7235.81 0.143699
\(77\) 0 0
\(78\) 1244.36 0.0231585
\(79\) 42072.0 0.758448 0.379224 0.925305i \(-0.376191\pi\)
0.379224 + 0.925305i \(0.376191\pi\)
\(80\) −15634.2 −0.273118
\(81\) 6561.00 0.111111
\(82\) −28506.9 −0.468184
\(83\) −6311.34 −0.100560 −0.0502801 0.998735i \(-0.516011\pi\)
−0.0502801 + 0.998735i \(0.516011\pi\)
\(84\) 0 0
\(85\) 125870. 1.88962
\(86\) 44566.8 0.649778
\(87\) −42887.0 −0.607473
\(88\) −2338.83 −0.0321952
\(89\) −51495.8 −0.689123 −0.344562 0.938764i \(-0.611972\pi\)
−0.344562 + 0.938764i \(0.611972\pi\)
\(90\) −19787.0 −0.257498
\(91\) 0 0
\(92\) 26948.0 0.331938
\(93\) −47347.8 −0.567665
\(94\) −93772.8 −1.09460
\(95\) −27618.7 −0.313974
\(96\) 9216.00 0.102062
\(97\) −127357. −1.37434 −0.687171 0.726496i \(-0.741148\pi\)
−0.687171 + 0.726496i \(0.741148\pi\)
\(98\) 0 0
\(99\) −2960.08 −0.0303539
\(100\) 9674.81 0.0967481
\(101\) 65757.9 0.641423 0.320712 0.947177i \(-0.396078\pi\)
0.320712 + 0.947177i \(0.396078\pi\)
\(102\) −74197.4 −0.706136
\(103\) 170010. 1.57900 0.789499 0.613751i \(-0.210340\pi\)
0.789499 + 0.613751i \(0.210340\pi\)
\(104\) 2212.20 0.0200558
\(105\) 0 0
\(106\) −28121.2 −0.243091
\(107\) −107933. −0.911369 −0.455685 0.890141i \(-0.650605\pi\)
−0.455685 + 0.890141i \(0.650605\pi\)
\(108\) 11664.0 0.0962250
\(109\) −222698. −1.79536 −0.897679 0.440651i \(-0.854748\pi\)
−0.897679 + 0.440651i \(0.854748\pi\)
\(110\) 8927.16 0.0703448
\(111\) −115398. −0.888975
\(112\) 0 0
\(113\) 181055. 1.33387 0.666936 0.745115i \(-0.267605\pi\)
0.666936 + 0.745115i \(0.267605\pi\)
\(114\) 16280.6 0.117330
\(115\) −102859. −0.725265
\(116\) −76243.5 −0.526087
\(117\) 2799.81 0.0189088
\(118\) −176891. −1.16950
\(119\) 0 0
\(120\) −35176.9 −0.223000
\(121\) −159716. −0.991708
\(122\) 77521.9 0.471547
\(123\) −64140.6 −0.382270
\(124\) −84173.9 −0.491613
\(125\) 153919. 0.881083
\(126\) 0 0
\(127\) 91474.7 0.503259 0.251630 0.967824i \(-0.419034\pi\)
0.251630 + 0.967824i \(0.419034\pi\)
\(128\) 16384.0 0.0883883
\(129\) 100275. 0.530541
\(130\) −8443.83 −0.0438209
\(131\) −19343.9 −0.0984840 −0.0492420 0.998787i \(-0.515681\pi\)
−0.0492420 + 0.998787i \(0.515681\pi\)
\(132\) −5262.36 −0.0262873
\(133\) 0 0
\(134\) 83776.3 0.403050
\(135\) −44520.8 −0.210246
\(136\) −131906. −0.611532
\(137\) −3077.04 −0.0140065 −0.00700327 0.999975i \(-0.502229\pi\)
−0.00700327 + 0.999975i \(0.502229\pi\)
\(138\) 60632.9 0.271026
\(139\) 370001. 1.62430 0.812149 0.583450i \(-0.198298\pi\)
0.812149 + 0.583450i \(0.198298\pi\)
\(140\) 0 0
\(141\) −210989. −0.893741
\(142\) 319374. 1.32916
\(143\) −1263.17 −0.00516561
\(144\) 20736.0 0.0833333
\(145\) 291017. 1.14947
\(146\) −148263. −0.575641
\(147\) 0 0
\(148\) −205151. −0.769875
\(149\) −454120. −1.67573 −0.837866 0.545875i \(-0.816197\pi\)
−0.837866 + 0.545875i \(0.816197\pi\)
\(150\) 21768.3 0.0789945
\(151\) 177592. 0.633841 0.316921 0.948452i \(-0.397351\pi\)
0.316921 + 0.948452i \(0.397351\pi\)
\(152\) 28943.2 0.101610
\(153\) −166944. −0.576558
\(154\) 0 0
\(155\) 321287. 1.07415
\(156\) 4977.44 0.0163755
\(157\) −248981. −0.806153 −0.403076 0.915166i \(-0.632059\pi\)
−0.403076 + 0.915166i \(0.632059\pi\)
\(158\) 168288. 0.536303
\(159\) −63272.7 −0.198483
\(160\) −62536.8 −0.193124
\(161\) 0 0
\(162\) 26244.0 0.0785674
\(163\) 202248. 0.596231 0.298116 0.954530i \(-0.403642\pi\)
0.298116 + 0.954530i \(0.403642\pi\)
\(164\) −114028. −0.331056
\(165\) 20086.1 0.0574363
\(166\) −25245.3 −0.0711068
\(167\) 475790. 1.32015 0.660076 0.751199i \(-0.270524\pi\)
0.660076 + 0.751199i \(0.270524\pi\)
\(168\) 0 0
\(169\) −370098. −0.996782
\(170\) 503479. 1.33616
\(171\) 36631.3 0.0957992
\(172\) 178267. 0.459462
\(173\) −284703. −0.723231 −0.361616 0.932327i \(-0.617775\pi\)
−0.361616 + 0.932327i \(0.617775\pi\)
\(174\) −171548. −0.429548
\(175\) 0 0
\(176\) −9355.30 −0.0227654
\(177\) −398005. −0.954895
\(178\) −205983. −0.487284
\(179\) 347958. 0.811699 0.405849 0.913940i \(-0.366976\pi\)
0.405849 + 0.913940i \(0.366976\pi\)
\(180\) −79148.1 −0.182079
\(181\) 379706. 0.861492 0.430746 0.902473i \(-0.358250\pi\)
0.430746 + 0.902473i \(0.358250\pi\)
\(182\) 0 0
\(183\) 174424. 0.385016
\(184\) 107792. 0.234715
\(185\) 783051. 1.68213
\(186\) −189391. −0.401400
\(187\) 75318.9 0.157507
\(188\) −375091. −0.774002
\(189\) 0 0
\(190\) −110475. −0.222013
\(191\) 194492. 0.385762 0.192881 0.981222i \(-0.438217\pi\)
0.192881 + 0.981222i \(0.438217\pi\)
\(192\) 36864.0 0.0721688
\(193\) 877401. 1.69553 0.847763 0.530375i \(-0.177949\pi\)
0.847763 + 0.530375i \(0.177949\pi\)
\(194\) −509430. −0.971806
\(195\) −18998.6 −0.0357796
\(196\) 0 0
\(197\) −752368. −1.38123 −0.690613 0.723225i \(-0.742659\pi\)
−0.690613 + 0.723225i \(0.742659\pi\)
\(198\) −11840.3 −0.0214635
\(199\) −155453. −0.278270 −0.139135 0.990273i \(-0.544432\pi\)
−0.139135 + 0.990273i \(0.544432\pi\)
\(200\) 38699.2 0.0684112
\(201\) 188497. 0.329089
\(202\) 263032. 0.453555
\(203\) 0 0
\(204\) −296790. −0.499313
\(205\) 435237. 0.723339
\(206\) 680041. 1.11652
\(207\) 136424. 0.221292
\(208\) 8848.79 0.0141816
\(209\) −16526.7 −0.0261709
\(210\) 0 0
\(211\) 3898.23 0.00602783 0.00301391 0.999995i \(-0.499041\pi\)
0.00301391 + 0.999995i \(0.499041\pi\)
\(212\) −112485. −0.171891
\(213\) 718591. 1.08526
\(214\) −431731. −0.644435
\(215\) −680435. −1.00390
\(216\) 46656.0 0.0680414
\(217\) 0 0
\(218\) −890794. −1.26951
\(219\) −333592. −0.470009
\(220\) 35708.6 0.0497413
\(221\) −71241.0 −0.0981182
\(222\) −461590. −0.628600
\(223\) 782574. 1.05381 0.526906 0.849924i \(-0.323352\pi\)
0.526906 + 0.849924i \(0.323352\pi\)
\(224\) 0 0
\(225\) 48978.7 0.0644987
\(226\) 724220. 0.943190
\(227\) 796543. 1.02599 0.512997 0.858391i \(-0.328535\pi\)
0.512997 + 0.858391i \(0.328535\pi\)
\(228\) 65122.3 0.0829645
\(229\) 387045. 0.487722 0.243861 0.969810i \(-0.421586\pi\)
0.243861 + 0.969810i \(0.421586\pi\)
\(230\) −411435. −0.512840
\(231\) 0 0
\(232\) −304974. −0.372000
\(233\) −383759. −0.463093 −0.231547 0.972824i \(-0.574379\pi\)
−0.231547 + 0.972824i \(0.574379\pi\)
\(234\) 11199.2 0.0133705
\(235\) 1.43170e6 1.69115
\(236\) −707565. −0.826964
\(237\) 378648. 0.437890
\(238\) 0 0
\(239\) −465409. −0.527036 −0.263518 0.964654i \(-0.584883\pi\)
−0.263518 + 0.964654i \(0.584883\pi\)
\(240\) −140708. −0.157685
\(241\) −348744. −0.386781 −0.193390 0.981122i \(-0.561948\pi\)
−0.193390 + 0.981122i \(0.561948\pi\)
\(242\) −638862. −0.701243
\(243\) 59049.0 0.0641500
\(244\) 310088. 0.333434
\(245\) 0 0
\(246\) −256563. −0.270306
\(247\) 15631.9 0.0163030
\(248\) −336695. −0.347623
\(249\) −56802.0 −0.0580584
\(250\) 615676. 0.623020
\(251\) −186543. −0.186894 −0.0934468 0.995624i \(-0.529789\pi\)
−0.0934468 + 0.995624i \(0.529789\pi\)
\(252\) 0 0
\(253\) −61549.4 −0.0604537
\(254\) 365899. 0.355858
\(255\) 1.13283e6 1.09097
\(256\) 65536.0 0.0625000
\(257\) 1.55355e6 1.46721 0.733605 0.679577i \(-0.237836\pi\)
0.733605 + 0.679577i \(0.237836\pi\)
\(258\) 401101. 0.375149
\(259\) 0 0
\(260\) −33775.3 −0.0309860
\(261\) −385983. −0.350725
\(262\) −77375.6 −0.0696387
\(263\) −1.25960e6 −1.12290 −0.561451 0.827510i \(-0.689757\pi\)
−0.561451 + 0.827510i \(0.689757\pi\)
\(264\) −21049.4 −0.0185879
\(265\) 429348. 0.375573
\(266\) 0 0
\(267\) −463462. −0.397866
\(268\) 335105. 0.285000
\(269\) −170461. −0.143630 −0.0718149 0.997418i \(-0.522879\pi\)
−0.0718149 + 0.997418i \(0.522879\pi\)
\(270\) −178083. −0.148667
\(271\) 2.22743e6 1.84239 0.921194 0.389104i \(-0.127215\pi\)
0.921194 + 0.389104i \(0.127215\pi\)
\(272\) −527626. −0.432418
\(273\) 0 0
\(274\) −12308.1 −0.00990412
\(275\) −22097.3 −0.0176201
\(276\) 242532. 0.191644
\(277\) 1.98339e6 1.55314 0.776568 0.630034i \(-0.216959\pi\)
0.776568 + 0.630034i \(0.216959\pi\)
\(278\) 1.48000e6 1.14855
\(279\) −426130. −0.327742
\(280\) 0 0
\(281\) −1.16687e6 −0.881573 −0.440786 0.897612i \(-0.645300\pi\)
−0.440786 + 0.897612i \(0.645300\pi\)
\(282\) −843955. −0.631970
\(283\) −1.90090e6 −1.41089 −0.705445 0.708765i \(-0.749253\pi\)
−0.705445 + 0.708765i \(0.749253\pi\)
\(284\) 1.27749e6 0.939860
\(285\) −248568. −0.181273
\(286\) −5052.68 −0.00365264
\(287\) 0 0
\(288\) 82944.0 0.0589256
\(289\) 2.82802e6 1.99177
\(290\) 1.16407e6 0.812799
\(291\) −1.14622e6 −0.793477
\(292\) −593053. −0.407039
\(293\) 1.65009e6 1.12289 0.561445 0.827514i \(-0.310245\pi\)
0.561445 + 0.827514i \(0.310245\pi\)
\(294\) 0 0
\(295\) 2.70073e6 1.80687
\(296\) −820605. −0.544384
\(297\) −26640.7 −0.0175249
\(298\) −1.81648e6 −1.18492
\(299\) 58217.0 0.0376593
\(300\) 87073.2 0.0558575
\(301\) 0 0
\(302\) 710367. 0.448194
\(303\) 591821. 0.370326
\(304\) 115773. 0.0718494
\(305\) −1.18359e6 −0.728535
\(306\) −667777. −0.407688
\(307\) 597936. 0.362084 0.181042 0.983475i \(-0.442053\pi\)
0.181042 + 0.983475i \(0.442053\pi\)
\(308\) 0 0
\(309\) 1.53009e6 0.911635
\(310\) 1.28515e6 0.759536
\(311\) 230480. 0.135124 0.0675620 0.997715i \(-0.478478\pi\)
0.0675620 + 0.997715i \(0.478478\pi\)
\(312\) 19909.8 0.0115792
\(313\) 442577. 0.255346 0.127673 0.991816i \(-0.459249\pi\)
0.127673 + 0.991816i \(0.459249\pi\)
\(314\) −995925. −0.570036
\(315\) 0 0
\(316\) 673152. 0.379224
\(317\) −1.26248e6 −0.705627 −0.352813 0.935694i \(-0.614775\pi\)
−0.352813 + 0.935694i \(0.614775\pi\)
\(318\) −253091. −0.140349
\(319\) 174141. 0.0958129
\(320\) −250147. −0.136559
\(321\) −971396. −0.526179
\(322\) 0 0
\(323\) −932080. −0.497104
\(324\) 104976. 0.0555556
\(325\) 20901.0 0.0109763
\(326\) 808991. 0.421599
\(327\) −2.00429e6 −1.03655
\(328\) −456111. −0.234092
\(329\) 0 0
\(330\) 80344.5 0.0406136
\(331\) −468936. −0.235257 −0.117629 0.993058i \(-0.537529\pi\)
−0.117629 + 0.993058i \(0.537529\pi\)
\(332\) −100981. −0.0502801
\(333\) −1.03858e6 −0.513250
\(334\) 1.90316e6 0.933489
\(335\) −1.27908e6 −0.622708
\(336\) 0 0
\(337\) −2.46798e6 −1.18377 −0.591884 0.806023i \(-0.701616\pi\)
−0.591884 + 0.806023i \(0.701616\pi\)
\(338\) −1.48039e6 −0.704831
\(339\) 1.62949e6 0.770112
\(340\) 2.01392e6 0.944810
\(341\) 192254. 0.0895343
\(342\) 146525. 0.0677402
\(343\) 0 0
\(344\) 713068. 0.324889
\(345\) −925729. −0.418732
\(346\) −1.13881e6 −0.511402
\(347\) −4.28250e6 −1.90930 −0.954648 0.297736i \(-0.903769\pi\)
−0.954648 + 0.297736i \(0.903769\pi\)
\(348\) −686191. −0.303737
\(349\) −1.84999e6 −0.813028 −0.406514 0.913645i \(-0.633256\pi\)
−0.406514 + 0.913645i \(0.633256\pi\)
\(350\) 0 0
\(351\) 25198.3 0.0109170
\(352\) −37421.2 −0.0160976
\(353\) −2.27380e6 −0.971213 −0.485607 0.874177i \(-0.661401\pi\)
−0.485607 + 0.874177i \(0.661401\pi\)
\(354\) −1.59202e6 −0.675213
\(355\) −4.87612e6 −2.05354
\(356\) −823933. −0.344562
\(357\) 0 0
\(358\) 1.39183e6 0.573958
\(359\) −4.39992e6 −1.80181 −0.900905 0.434017i \(-0.857096\pi\)
−0.900905 + 0.434017i \(0.857096\pi\)
\(360\) −316592. −0.128749
\(361\) −2.27158e6 −0.917403
\(362\) 1.51882e6 0.609167
\(363\) −1.43744e6 −0.572563
\(364\) 0 0
\(365\) 2.26365e6 0.889359
\(366\) 697697. 0.272248
\(367\) −1.89743e6 −0.735362 −0.367681 0.929952i \(-0.619848\pi\)
−0.367681 + 0.929952i \(0.619848\pi\)
\(368\) 431168. 0.165969
\(369\) −577266. −0.220704
\(370\) 3.13220e6 1.18945
\(371\) 0 0
\(372\) −757565. −0.283833
\(373\) 4.56260e6 1.69801 0.849005 0.528385i \(-0.177202\pi\)
0.849005 + 0.528385i \(0.177202\pi\)
\(374\) 301276. 0.111374
\(375\) 1.38527e6 0.508694
\(376\) −1.50037e6 −0.547302
\(377\) −164713. −0.0596861
\(378\) 0 0
\(379\) 4.99400e6 1.78587 0.892936 0.450184i \(-0.148642\pi\)
0.892936 + 0.450184i \(0.148642\pi\)
\(380\) −441898. −0.156987
\(381\) 823273. 0.290557
\(382\) 777970. 0.272775
\(383\) −3.67907e6 −1.28157 −0.640783 0.767722i \(-0.721390\pi\)
−0.640783 + 0.767722i \(0.721390\pi\)
\(384\) 147456. 0.0510310
\(385\) 0 0
\(386\) 3.50960e6 1.19892
\(387\) 902477. 0.306308
\(388\) −2.03772e6 −0.687171
\(389\) 257234. 0.0861894 0.0430947 0.999071i \(-0.486278\pi\)
0.0430947 + 0.999071i \(0.486278\pi\)
\(390\) −75994.5 −0.0253000
\(391\) −3.47130e6 −1.14829
\(392\) 0 0
\(393\) −174095. −0.0568598
\(394\) −3.00947e6 −0.976674
\(395\) −2.56938e6 −0.828583
\(396\) −47361.2 −0.0151770
\(397\) 433280. 0.137972 0.0689862 0.997618i \(-0.478024\pi\)
0.0689862 + 0.997618i \(0.478024\pi\)
\(398\) −621811. −0.196766
\(399\) 0 0
\(400\) 154797. 0.0483740
\(401\) −1.13824e6 −0.353488 −0.176744 0.984257i \(-0.556556\pi\)
−0.176744 + 0.984257i \(0.556556\pi\)
\(402\) 753987. 0.232701
\(403\) −181845. −0.0557749
\(404\) 1.05213e6 0.320712
\(405\) −400687. −0.121386
\(406\) 0 0
\(407\) 468568. 0.140212
\(408\) −1.18716e6 −0.353068
\(409\) −5.02438e6 −1.48516 −0.742581 0.669756i \(-0.766399\pi\)
−0.742581 + 0.669756i \(0.766399\pi\)
\(410\) 1.74095e6 0.511478
\(411\) −27693.3 −0.00808668
\(412\) 2.72016e6 0.789499
\(413\) 0 0
\(414\) 545696. 0.156477
\(415\) 385440. 0.109859
\(416\) 35395.2 0.0100279
\(417\) 3.33001e6 0.937789
\(418\) −66106.6 −0.0185056
\(419\) −2.57295e6 −0.715974 −0.357987 0.933727i \(-0.616537\pi\)
−0.357987 + 0.933727i \(0.616537\pi\)
\(420\) 0 0
\(421\) 336425. 0.0925089 0.0462545 0.998930i \(-0.485271\pi\)
0.0462545 + 0.998930i \(0.485271\pi\)
\(422\) 15592.9 0.00426232
\(423\) −1.89890e6 −0.516002
\(424\) −449939. −0.121546
\(425\) −1.24626e6 −0.334685
\(426\) 2.87436e6 0.767392
\(427\) 0 0
\(428\) −1.72693e6 −0.455685
\(429\) −11368.5 −0.00298237
\(430\) −2.72174e6 −0.709864
\(431\) 236410. 0.0613018 0.0306509 0.999530i \(-0.490242\pi\)
0.0306509 + 0.999530i \(0.490242\pi\)
\(432\) 186624. 0.0481125
\(433\) −2.33004e6 −0.597232 −0.298616 0.954373i \(-0.596525\pi\)
−0.298616 + 0.954373i \(0.596525\pi\)
\(434\) 0 0
\(435\) 2.61915e6 0.663648
\(436\) −3.56317e6 −0.897679
\(437\) 761681. 0.190796
\(438\) −1.33437e6 −0.332346
\(439\) 3.19962e6 0.792387 0.396193 0.918167i \(-0.370331\pi\)
0.396193 + 0.918167i \(0.370331\pi\)
\(440\) 142835. 0.0351724
\(441\) 0 0
\(442\) −284964. −0.0693800
\(443\) 1.13240e6 0.274152 0.137076 0.990561i \(-0.456230\pi\)
0.137076 + 0.990561i \(0.456230\pi\)
\(444\) −1.84636e6 −0.444487
\(445\) 3.14490e6 0.752848
\(446\) 3.13029e6 0.745157
\(447\) −4.08708e6 −0.967485
\(448\) 0 0
\(449\) −8.23687e6 −1.92817 −0.964087 0.265585i \(-0.914435\pi\)
−0.964087 + 0.265585i \(0.914435\pi\)
\(450\) 195915. 0.0456075
\(451\) 260441. 0.0602931
\(452\) 2.89688e6 0.666936
\(453\) 1.59833e6 0.365949
\(454\) 3.18617e6 0.725487
\(455\) 0 0
\(456\) 260489. 0.0586648
\(457\) −365132. −0.0817824 −0.0408912 0.999164i \(-0.513020\pi\)
−0.0408912 + 0.999164i \(0.513020\pi\)
\(458\) 1.54818e6 0.344872
\(459\) −1.50250e6 −0.332876
\(460\) −1.64574e6 −0.362633
\(461\) 332567. 0.0728832 0.0364416 0.999336i \(-0.488398\pi\)
0.0364416 + 0.999336i \(0.488398\pi\)
\(462\) 0 0
\(463\) 1.69992e6 0.368533 0.184266 0.982876i \(-0.441009\pi\)
0.184266 + 0.982876i \(0.441009\pi\)
\(464\) −1.21990e6 −0.263044
\(465\) 2.89158e6 0.620159
\(466\) −1.53504e6 −0.327456
\(467\) −990465. −0.210159 −0.105079 0.994464i \(-0.533510\pi\)
−0.105079 + 0.994464i \(0.533510\pi\)
\(468\) 44797.0 0.00945441
\(469\) 0 0
\(470\) 5.72681e6 1.19583
\(471\) −2.24083e6 −0.465432
\(472\) −2.83026e6 −0.584752
\(473\) −407164. −0.0836789
\(474\) 1.51459e6 0.309635
\(475\) 273457. 0.0556103
\(476\) 0 0
\(477\) −569454. −0.114594
\(478\) −1.86164e6 −0.372671
\(479\) −6.08065e6 −1.21091 −0.605454 0.795880i \(-0.707009\pi\)
−0.605454 + 0.795880i \(0.707009\pi\)
\(480\) −562831. −0.111500
\(481\) −443198. −0.0873445
\(482\) −1.39498e6 −0.273495
\(483\) 0 0
\(484\) −2.55545e6 −0.495854
\(485\) 7.77785e6 1.50143
\(486\) 236196. 0.0453609
\(487\) −6.71098e6 −1.28222 −0.641112 0.767448i \(-0.721526\pi\)
−0.641112 + 0.767448i \(0.721526\pi\)
\(488\) 1.24035e6 0.235773
\(489\) 1.82023e6 0.344234
\(490\) 0 0
\(491\) −914042. −0.171105 −0.0855525 0.996334i \(-0.527266\pi\)
−0.0855525 + 0.996334i \(0.527266\pi\)
\(492\) −1.02625e6 −0.191135
\(493\) 9.82130e6 1.81992
\(494\) 62527.5 0.0115280
\(495\) 180775. 0.0331608
\(496\) −1.34678e6 −0.245806
\(497\) 0 0
\(498\) −227208. −0.0410535
\(499\) −7.75492e6 −1.39420 −0.697101 0.716972i \(-0.745527\pi\)
−0.697101 + 0.716972i \(0.745527\pi\)
\(500\) 2.46270e6 0.440542
\(501\) 4.28211e6 0.762190
\(502\) −746172. −0.132154
\(503\) −3.79381e6 −0.668584 −0.334292 0.942470i \(-0.608497\pi\)
−0.334292 + 0.942470i \(0.608497\pi\)
\(504\) 0 0
\(505\) −4.01591e6 −0.700737
\(506\) −246198. −0.0427472
\(507\) −3.33088e6 −0.575492
\(508\) 1.46360e6 0.251630
\(509\) −8.53160e6 −1.45961 −0.729804 0.683657i \(-0.760389\pi\)
−0.729804 + 0.683657i \(0.760389\pi\)
\(510\) 4.53131e6 0.771434
\(511\) 0 0
\(512\) 262144. 0.0441942
\(513\) 329681. 0.0553097
\(514\) 6.21419e6 1.03747
\(515\) −1.03827e7 −1.72501
\(516\) 1.60440e6 0.265271
\(517\) 856712. 0.140964
\(518\) 0 0
\(519\) −2.56233e6 −0.417558
\(520\) −135101. −0.0219104
\(521\) 3.19066e6 0.514976 0.257488 0.966282i \(-0.417105\pi\)
0.257488 + 0.966282i \(0.417105\pi\)
\(522\) −1.54393e6 −0.248000
\(523\) 9.60678e6 1.53576 0.767880 0.640593i \(-0.221311\pi\)
0.767880 + 0.640593i \(0.221311\pi\)
\(524\) −309502. −0.0492420
\(525\) 0 0
\(526\) −5.03838e6 −0.794012
\(527\) 1.08429e7 1.70066
\(528\) −84197.7 −0.0131436
\(529\) −3.59965e6 −0.559270
\(530\) 1.71739e6 0.265570
\(531\) −3.58205e6 −0.551309
\(532\) 0 0
\(533\) −246340. −0.0375592
\(534\) −1.85385e6 −0.281333
\(535\) 6.59158e6 0.995646
\(536\) 1.34042e6 0.201525
\(537\) 3.13163e6 0.468634
\(538\) −681844. −0.101562
\(539\) 0 0
\(540\) −712333. −0.105123
\(541\) 1.02366e7 1.50371 0.751855 0.659328i \(-0.229159\pi\)
0.751855 + 0.659328i \(0.229159\pi\)
\(542\) 8.90972e6 1.30276
\(543\) 3.41736e6 0.497383
\(544\) −2.11050e6 −0.305766
\(545\) 1.36004e7 1.96138
\(546\) 0 0
\(547\) −9.27757e6 −1.32576 −0.662882 0.748724i \(-0.730667\pi\)
−0.662882 + 0.748724i \(0.730667\pi\)
\(548\) −49232.6 −0.00700327
\(549\) 1.56982e6 0.222289
\(550\) −88389.4 −0.0124593
\(551\) −2.15501e6 −0.302392
\(552\) 970127. 0.135513
\(553\) 0 0
\(554\) 7.93357e6 1.09823
\(555\) 7.04745e6 0.971180
\(556\) 5.92001e6 0.812149
\(557\) 620098. 0.0846881 0.0423441 0.999103i \(-0.486517\pi\)
0.0423441 + 0.999103i \(0.486517\pi\)
\(558\) −1.70452e6 −0.231748
\(559\) 385119. 0.0521273
\(560\) 0 0
\(561\) 677870. 0.0909368
\(562\) −4.66750e6 −0.623366
\(563\) −7.36664e6 −0.979487 −0.489744 0.871867i \(-0.662910\pi\)
−0.489744 + 0.871867i \(0.662910\pi\)
\(564\) −3.37582e6 −0.446871
\(565\) −1.10572e7 −1.45722
\(566\) −7.60360e6 −0.997650
\(567\) 0 0
\(568\) 5.10998e6 0.664581
\(569\) 3.68222e6 0.476792 0.238396 0.971168i \(-0.423378\pi\)
0.238396 + 0.971168i \(0.423378\pi\)
\(570\) −994272. −0.128179
\(571\) 1.03320e7 1.32615 0.663077 0.748551i \(-0.269250\pi\)
0.663077 + 0.748551i \(0.269250\pi\)
\(572\) −20210.7 −0.00258281
\(573\) 1.75043e6 0.222720
\(574\) 0 0
\(575\) 1.01842e6 0.128457
\(576\) 331776. 0.0416667
\(577\) 1.25123e7 1.56458 0.782292 0.622911i \(-0.214050\pi\)
0.782292 + 0.622911i \(0.214050\pi\)
\(578\) 1.13121e7 1.40839
\(579\) 7.89661e6 0.978913
\(580\) 4.65627e6 0.574736
\(581\) 0 0
\(582\) −4.58487e6 −0.561073
\(583\) 256916. 0.0313055
\(584\) −2.37221e6 −0.287820
\(585\) −170988. −0.0206574
\(586\) 6.60034e6 0.794004
\(587\) 1.60865e7 1.92694 0.963468 0.267823i \(-0.0863042\pi\)
0.963468 + 0.267823i \(0.0863042\pi\)
\(588\) 0 0
\(589\) −2.37916e6 −0.282577
\(590\) 1.08029e7 1.27765
\(591\) −6.77131e6 −0.797451
\(592\) −3.28242e6 −0.384937
\(593\) 9.46330e6 1.10511 0.552556 0.833476i \(-0.313653\pi\)
0.552556 + 0.833476i \(0.313653\pi\)
\(594\) −106563. −0.0123919
\(595\) 0 0
\(596\) −7.26592e6 −0.837866
\(597\) −1.39908e6 −0.160659
\(598\) 232868. 0.0266291
\(599\) 5.01498e6 0.571087 0.285544 0.958366i \(-0.407826\pi\)
0.285544 + 0.958366i \(0.407826\pi\)
\(600\) 348293. 0.0394972
\(601\) −1.26473e7 −1.42828 −0.714139 0.700004i \(-0.753182\pi\)
−0.714139 + 0.700004i \(0.753182\pi\)
\(602\) 0 0
\(603\) 1.69647e6 0.190000
\(604\) 2.84147e6 0.316921
\(605\) 9.75400e6 1.08341
\(606\) 2.36729e6 0.261860
\(607\) 2.87702e6 0.316936 0.158468 0.987364i \(-0.449345\pi\)
0.158468 + 0.987364i \(0.449345\pi\)
\(608\) 463092. 0.0508052
\(609\) 0 0
\(610\) −4.73434e6 −0.515152
\(611\) −810328. −0.0878128
\(612\) −2.67111e6 −0.288279
\(613\) −3.72582e6 −0.400471 −0.200236 0.979748i \(-0.564171\pi\)
−0.200236 + 0.979748i \(0.564171\pi\)
\(614\) 2.39174e6 0.256032
\(615\) 3.91714e6 0.417620
\(616\) 0 0
\(617\) −1.15861e7 −1.22525 −0.612625 0.790374i \(-0.709886\pi\)
−0.612625 + 0.790374i \(0.709886\pi\)
\(618\) 6.12037e6 0.644624
\(619\) −1.85608e6 −0.194702 −0.0973508 0.995250i \(-0.531037\pi\)
−0.0973508 + 0.995250i \(0.531037\pi\)
\(620\) 5.14059e6 0.537073
\(621\) 1.22782e6 0.127763
\(622\) 921921. 0.0955471
\(623\) 0 0
\(624\) 79639.1 0.00818776
\(625\) −1.12896e7 −1.15606
\(626\) 1.77031e6 0.180557
\(627\) −148740. −0.0151098
\(628\) −3.98370e6 −0.403076
\(629\) 2.64266e7 2.66326
\(630\) 0 0
\(631\) 8.92135e6 0.891984 0.445992 0.895037i \(-0.352851\pi\)
0.445992 + 0.895037i \(0.352851\pi\)
\(632\) 2.69261e6 0.268152
\(633\) 35084.0 0.00348017
\(634\) −5.04990e6 −0.498953
\(635\) −5.58646e6 −0.549797
\(636\) −1.01236e6 −0.0992415
\(637\) 0 0
\(638\) 696563. 0.0677499
\(639\) 6.46731e6 0.626573
\(640\) −1.00059e6 −0.0965618
\(641\) 1.31866e7 1.26762 0.633810 0.773489i \(-0.281490\pi\)
0.633810 + 0.773489i \(0.281490\pi\)
\(642\) −3.88558e6 −0.372065
\(643\) −9.67813e6 −0.923132 −0.461566 0.887106i \(-0.652712\pi\)
−0.461566 + 0.887106i \(0.652712\pi\)
\(644\) 0 0
\(645\) −6.12391e6 −0.579602
\(646\) −3.72832e6 −0.351505
\(647\) −1.72491e6 −0.161996 −0.0809982 0.996714i \(-0.525811\pi\)
−0.0809982 + 0.996714i \(0.525811\pi\)
\(648\) 419904. 0.0392837
\(649\) 1.61609e6 0.150610
\(650\) 83603.8 0.00776145
\(651\) 0 0
\(652\) 3.23596e6 0.298116
\(653\) 597510. 0.0548356 0.0274178 0.999624i \(-0.491272\pi\)
0.0274178 + 0.999624i \(0.491272\pi\)
\(654\) −8.01714e6 −0.732951
\(655\) 1.18135e6 0.107591
\(656\) −1.82444e6 −0.165528
\(657\) −3.00233e6 −0.271360
\(658\) 0 0
\(659\) 1.47605e7 1.32400 0.662001 0.749503i \(-0.269707\pi\)
0.662001 + 0.749503i \(0.269707\pi\)
\(660\) 321378. 0.0287181
\(661\) 1.69041e7 1.50484 0.752418 0.658686i \(-0.228888\pi\)
0.752418 + 0.658686i \(0.228888\pi\)
\(662\) −1.87574e6 −0.166352
\(663\) −641169. −0.0566486
\(664\) −403925. −0.0355534
\(665\) 0 0
\(666\) −4.15431e6 −0.362922
\(667\) −8.02581e6 −0.698512
\(668\) 7.61264e6 0.660076
\(669\) 7.04316e6 0.608418
\(670\) −5.11631e6 −0.440321
\(671\) −708243. −0.0607262
\(672\) 0 0
\(673\) −1.58960e7 −1.35285 −0.676425 0.736512i \(-0.736472\pi\)
−0.676425 + 0.736512i \(0.736472\pi\)
\(674\) −9.87191e6 −0.837050
\(675\) 440808. 0.0372383
\(676\) −5.92157e6 −0.498391
\(677\) −1.95361e6 −0.163820 −0.0819099 0.996640i \(-0.526102\pi\)
−0.0819099 + 0.996640i \(0.526102\pi\)
\(678\) 6.51798e6 0.544551
\(679\) 0 0
\(680\) 8.05567e6 0.668081
\(681\) 7.16889e6 0.592357
\(682\) 769016. 0.0633103
\(683\) −9.23970e6 −0.757890 −0.378945 0.925419i \(-0.623713\pi\)
−0.378945 + 0.925419i \(0.623713\pi\)
\(684\) 586100. 0.0478996
\(685\) 187918. 0.0153018
\(686\) 0 0
\(687\) 3.48340e6 0.281587
\(688\) 2.85227e6 0.229731
\(689\) −243006. −0.0195016
\(690\) −3.70292e6 −0.296088
\(691\) −1.45698e7 −1.16080 −0.580401 0.814331i \(-0.697104\pi\)
−0.580401 + 0.814331i \(0.697104\pi\)
\(692\) −4.55525e6 −0.361616
\(693\) 0 0
\(694\) −1.71300e7 −1.35008
\(695\) −2.25964e7 −1.77450
\(696\) −2.74476e6 −0.214774
\(697\) 1.46885e7 1.14524
\(698\) −7.39995e6 −0.574898
\(699\) −3.45383e6 −0.267367
\(700\) 0 0
\(701\) −3.70190e6 −0.284531 −0.142266 0.989829i \(-0.545439\pi\)
−0.142266 + 0.989829i \(0.545439\pi\)
\(702\) 100793. 0.00771949
\(703\) −5.79858e6 −0.442520
\(704\) −149685. −0.0113827
\(705\) 1.28853e7 0.976387
\(706\) −9.09518e6 −0.686751
\(707\) 0 0
\(708\) −6.36809e6 −0.477448
\(709\) −2.52937e7 −1.88971 −0.944857 0.327484i \(-0.893799\pi\)
−0.944857 + 0.327484i \(0.893799\pi\)
\(710\) −1.95045e7 −1.45207
\(711\) 3.40783e6 0.252816
\(712\) −3.29573e6 −0.243642
\(713\) −8.86061e6 −0.652739
\(714\) 0 0
\(715\) 77143.1 0.00564329
\(716\) 5.56734e6 0.405849
\(717\) −4.18868e6 −0.304284
\(718\) −1.75997e7 −1.27407
\(719\) 1.45185e7 1.04737 0.523683 0.851913i \(-0.324557\pi\)
0.523683 + 0.851913i \(0.324557\pi\)
\(720\) −1.26637e6 −0.0910394
\(721\) 0 0
\(722\) −9.08632e6 −0.648702
\(723\) −3.13870e6 −0.223308
\(724\) 6.07530e6 0.430746
\(725\) −2.88141e6 −0.203592
\(726\) −5.74976e6 −0.404863
\(727\) −1.42855e7 −1.00244 −0.501222 0.865319i \(-0.667116\pi\)
−0.501222 + 0.865319i \(0.667116\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 9.05460e6 0.628872
\(731\) −2.29635e7 −1.58944
\(732\) 2.79079e6 0.192508
\(733\) −8.21556e6 −0.564777 −0.282389 0.959300i \(-0.591127\pi\)
−0.282389 + 0.959300i \(0.591127\pi\)
\(734\) −7.58973e6 −0.519979
\(735\) 0 0
\(736\) 1.72467e6 0.117358
\(737\) −765384. −0.0519052
\(738\) −2.30906e6 −0.156061
\(739\) −3.43896e6 −0.231641 −0.115821 0.993270i \(-0.536950\pi\)
−0.115821 + 0.993270i \(0.536950\pi\)
\(740\) 1.25288e7 0.841067
\(741\) 140687. 0.00941256
\(742\) 0 0
\(743\) 1.53588e6 0.102067 0.0510334 0.998697i \(-0.483748\pi\)
0.0510334 + 0.998697i \(0.483748\pi\)
\(744\) −3.03026e6 −0.200700
\(745\) 2.77336e7 1.83069
\(746\) 1.82504e7 1.20067
\(747\) −511218. −0.0335201
\(748\) 1.20510e6 0.0787535
\(749\) 0 0
\(750\) 5.54108e6 0.359701
\(751\) −2.01146e7 −1.30140 −0.650700 0.759335i \(-0.725524\pi\)
−0.650700 + 0.759335i \(0.725524\pi\)
\(752\) −6.00146e6 −0.387001
\(753\) −1.67889e6 −0.107903
\(754\) −658850. −0.0422044
\(755\) −1.08457e7 −0.692454
\(756\) 0 0
\(757\) 202045. 0.0128147 0.00640733 0.999979i \(-0.497960\pi\)
0.00640733 + 0.999979i \(0.497960\pi\)
\(758\) 1.99760e7 1.26280
\(759\) −553945. −0.0349029
\(760\) −1.76759e6 −0.111007
\(761\) −1.08202e6 −0.0677290 −0.0338645 0.999426i \(-0.510781\pi\)
−0.0338645 + 0.999426i \(0.510781\pi\)
\(762\) 3.29309e6 0.205455
\(763\) 0 0
\(764\) 3.11188e6 0.192881
\(765\) 1.01955e7 0.629873
\(766\) −1.47163e7 −0.906205
\(767\) −1.52859e6 −0.0938214
\(768\) 589824. 0.0360844
\(769\) 1.19305e7 0.727517 0.363759 0.931493i \(-0.381493\pi\)
0.363759 + 0.931493i \(0.381493\pi\)
\(770\) 0 0
\(771\) 1.39819e7 0.847094
\(772\) 1.40384e7 0.847763
\(773\) −6.58657e6 −0.396470 −0.198235 0.980155i \(-0.563521\pi\)
−0.198235 + 0.980155i \(0.563521\pi\)
\(774\) 3.60991e6 0.216593
\(775\) −3.18112e6 −0.190250
\(776\) −8.15087e6 −0.485903
\(777\) 0 0
\(778\) 1.02894e6 0.0609451
\(779\) −3.22298e6 −0.190289
\(780\) −303978. −0.0178898
\(781\) −2.91781e6 −0.171171
\(782\) −1.38852e7 −0.811961
\(783\) −3.47384e6 −0.202491
\(784\) 0 0
\(785\) 1.52055e7 0.880700
\(786\) −696380. −0.0402059
\(787\) −1.92720e7 −1.10915 −0.554575 0.832134i \(-0.687119\pi\)
−0.554575 + 0.832134i \(0.687119\pi\)
\(788\) −1.20379e7 −0.690613
\(789\) −1.13364e7 −0.648308
\(790\) −1.02775e7 −0.585897
\(791\) 0 0
\(792\) −189445. −0.0107317
\(793\) 669897. 0.0378290
\(794\) 1.73312e6 0.0975613
\(795\) 3.86413e6 0.216837
\(796\) −2.48724e6 −0.139135
\(797\) 1.54906e7 0.863816 0.431908 0.901918i \(-0.357841\pi\)
0.431908 + 0.901918i \(0.357841\pi\)
\(798\) 0 0
\(799\) 4.83174e7 2.67754
\(800\) 619188. 0.0342056
\(801\) −4.17116e6 −0.229708
\(802\) −4.55298e6 −0.249954
\(803\) 1.35454e6 0.0741315
\(804\) 3.01595e6 0.164545
\(805\) 0 0
\(806\) −727380. −0.0394388
\(807\) −1.53415e6 −0.0829247
\(808\) 4.20851e6 0.226777
\(809\) −1.77071e6 −0.0951210 −0.0475605 0.998868i \(-0.515145\pi\)
−0.0475605 + 0.998868i \(0.515145\pi\)
\(810\) −1.60275e6 −0.0858327
\(811\) 6.80547e6 0.363334 0.181667 0.983360i \(-0.441851\pi\)
0.181667 + 0.983360i \(0.441851\pi\)
\(812\) 0 0
\(813\) 2.00469e7 1.06370
\(814\) 1.87427e6 0.0991451
\(815\) −1.23515e7 −0.651366
\(816\) −4.74863e6 −0.249657
\(817\) 5.03869e6 0.264097
\(818\) −2.00975e7 −1.05017
\(819\) 0 0
\(820\) 6.96380e6 0.361669
\(821\) 2.27127e7 1.17601 0.588006 0.808857i \(-0.299913\pi\)
0.588006 + 0.808857i \(0.299913\pi\)
\(822\) −110773. −0.00571815
\(823\) −3.03576e7 −1.56231 −0.781156 0.624336i \(-0.785370\pi\)
−0.781156 + 0.624336i \(0.785370\pi\)
\(824\) 1.08807e7 0.558260
\(825\) −198876. −0.0101730
\(826\) 0 0
\(827\) −5.80776e6 −0.295287 −0.147644 0.989041i \(-0.547169\pi\)
−0.147644 + 0.989041i \(0.547169\pi\)
\(828\) 2.18279e6 0.110646
\(829\) −9.04916e6 −0.457322 −0.228661 0.973506i \(-0.573435\pi\)
−0.228661 + 0.973506i \(0.573435\pi\)
\(830\) 1.54176e6 0.0776822
\(831\) 1.78505e7 0.896703
\(832\) 141581. 0.00709080
\(833\) 0 0
\(834\) 1.33200e7 0.663117
\(835\) −2.90570e7 −1.44223
\(836\) −264426. −0.0130855
\(837\) −3.83517e6 −0.189222
\(838\) −1.02918e7 −0.506270
\(839\) −8.17101e6 −0.400747 −0.200374 0.979720i \(-0.564216\pi\)
−0.200374 + 0.979720i \(0.564216\pi\)
\(840\) 0 0
\(841\) 2.19614e6 0.107071
\(842\) 1.34570e6 0.0654137
\(843\) −1.05019e7 −0.508976
\(844\) 62371.6 0.00301391
\(845\) 2.26023e7 1.08896
\(846\) −7.59560e6 −0.364868
\(847\) 0 0
\(848\) −1.79976e6 −0.0859457
\(849\) −1.71081e7 −0.814578
\(850\) −4.98504e6 −0.236658
\(851\) −2.15954e7 −1.02220
\(852\) 1.14974e7 0.542628
\(853\) 2.88437e7 1.35731 0.678654 0.734458i \(-0.262564\pi\)
0.678654 + 0.734458i \(0.262564\pi\)
\(854\) 0 0
\(855\) −2.23711e6 −0.104658
\(856\) −6.90770e6 −0.322218
\(857\) −2.69462e7 −1.25327 −0.626637 0.779311i \(-0.715569\pi\)
−0.626637 + 0.779311i \(0.715569\pi\)
\(858\) −45474.1 −0.00210885
\(859\) −7.49124e6 −0.346394 −0.173197 0.984887i \(-0.555410\pi\)
−0.173197 + 0.984887i \(0.555410\pi\)
\(860\) −1.08870e7 −0.501950
\(861\) 0 0
\(862\) 945641. 0.0433469
\(863\) −1.41552e6 −0.0646977 −0.0323489 0.999477i \(-0.510299\pi\)
−0.0323489 + 0.999477i \(0.510299\pi\)
\(864\) 746496. 0.0340207
\(865\) 1.73871e7 0.790110
\(866\) −9.32015e6 −0.422307
\(867\) 2.54522e7 1.14995
\(868\) 0 0
\(869\) −1.53749e6 −0.0690656
\(870\) 1.04766e7 0.469270
\(871\) 723944. 0.0323340
\(872\) −1.42527e7 −0.634755
\(873\) −1.03159e7 −0.458114
\(874\) 3.04672e6 0.134913
\(875\) 0 0
\(876\) −5.33748e6 −0.235004
\(877\) 2.55232e7 1.12056 0.560281 0.828303i \(-0.310693\pi\)
0.560281 + 0.828303i \(0.310693\pi\)
\(878\) 1.27985e7 0.560302
\(879\) 1.48508e7 0.648301
\(880\) 571338. 0.0248706
\(881\) −2.07516e6 −0.0900764 −0.0450382 0.998985i \(-0.514341\pi\)
−0.0450382 + 0.998985i \(0.514341\pi\)
\(882\) 0 0
\(883\) 1.80641e7 0.779677 0.389839 0.920883i \(-0.372531\pi\)
0.389839 + 0.920883i \(0.372531\pi\)
\(884\) −1.13986e6 −0.0490591
\(885\) 2.43066e7 1.04320
\(886\) 4.52961e6 0.193855
\(887\) −9.28389e6 −0.396206 −0.198103 0.980181i \(-0.563478\pi\)
−0.198103 + 0.980181i \(0.563478\pi\)
\(888\) −7.38545e6 −0.314300
\(889\) 0 0
\(890\) 1.25796e7 0.532344
\(891\) −239766. −0.0101180
\(892\) 1.25212e7 0.526906
\(893\) −1.06019e7 −0.444893
\(894\) −1.63483e7 −0.684115
\(895\) −2.12502e7 −0.886758
\(896\) 0 0
\(897\) 523953. 0.0217426
\(898\) −3.29475e7 −1.36343
\(899\) 2.50692e7 1.03452
\(900\) 783659. 0.0322494
\(901\) 1.44897e7 0.594631
\(902\) 1.04176e6 0.0426336
\(903\) 0 0
\(904\) 1.15875e7 0.471595
\(905\) −2.31891e7 −0.941156
\(906\) 6.39331e6 0.258765
\(907\) −4.20818e7 −1.69854 −0.849270 0.527958i \(-0.822958\pi\)
−0.849270 + 0.527958i \(0.822958\pi\)
\(908\) 1.27447e7 0.512997
\(909\) 5.32639e6 0.213808
\(910\) 0 0
\(911\) 2.33510e7 0.932199 0.466100 0.884732i \(-0.345659\pi\)
0.466100 + 0.884732i \(0.345659\pi\)
\(912\) 1.04196e6 0.0414823
\(913\) 230642. 0.00915719
\(914\) −1.46053e6 −0.0578289
\(915\) −1.06523e7 −0.420620
\(916\) 6.19272e6 0.243861
\(917\) 0 0
\(918\) −6.00999e6 −0.235379
\(919\) 1.54594e7 0.603817 0.301908 0.953337i \(-0.402376\pi\)
0.301908 + 0.953337i \(0.402376\pi\)
\(920\) −6.58296e6 −0.256420
\(921\) 5.38142e6 0.209049
\(922\) 1.33027e6 0.0515362
\(923\) 2.75983e6 0.106630
\(924\) 0 0
\(925\) −7.75312e6 −0.297935
\(926\) 6.79968e6 0.260592
\(927\) 1.37708e7 0.526333
\(928\) −4.87958e6 −0.186000
\(929\) 1.49993e7 0.570204 0.285102 0.958497i \(-0.407972\pi\)
0.285102 + 0.958497i \(0.407972\pi\)
\(930\) 1.15663e7 0.438519
\(931\) 0 0
\(932\) −6.14014e6 −0.231547
\(933\) 2.07432e6 0.0780139
\(934\) −3.96186e6 −0.148605
\(935\) −4.59981e6 −0.172072
\(936\) 179188. 0.00668527
\(937\) 1.52742e7 0.568343 0.284172 0.958773i \(-0.408281\pi\)
0.284172 + 0.958773i \(0.408281\pi\)
\(938\) 0 0
\(939\) 3.98320e6 0.147424
\(940\) 2.29072e7 0.845576
\(941\) −5.13600e6 −0.189082 −0.0945412 0.995521i \(-0.530138\pi\)
−0.0945412 + 0.995521i \(0.530138\pi\)
\(942\) −8.96332e6 −0.329110
\(943\) −1.20032e7 −0.439559
\(944\) −1.13210e7 −0.413482
\(945\) 0 0
\(946\) −1.62865e6 −0.0591699
\(947\) −4.50054e7 −1.63076 −0.815379 0.578928i \(-0.803471\pi\)
−0.815379 + 0.578928i \(0.803471\pi\)
\(948\) 6.05837e6 0.218945
\(949\) −1.28120e6 −0.0461798
\(950\) 1.09383e6 0.0393224
\(951\) −1.13623e7 −0.407394
\(952\) 0 0
\(953\) −4.49596e7 −1.60358 −0.801789 0.597607i \(-0.796118\pi\)
−0.801789 + 0.597607i \(0.796118\pi\)
\(954\) −2.27782e6 −0.0810303
\(955\) −1.18779e7 −0.421434
\(956\) −7.44655e6 −0.263518
\(957\) 1.56727e6 0.0553176
\(958\) −2.43226e7 −0.856242
\(959\) 0 0
\(960\) −2.25132e6 −0.0788424
\(961\) −952428. −0.0332678
\(962\) −1.77279e6 −0.0617619
\(963\) −8.74256e6 −0.303790
\(964\) −5.57991e6 −0.193390
\(965\) −5.35838e7 −1.85232
\(966\) 0 0
\(967\) 1.04364e7 0.358907 0.179454 0.983766i \(-0.442567\pi\)
0.179454 + 0.983766i \(0.442567\pi\)
\(968\) −1.02218e7 −0.350622
\(969\) −8.38872e6 −0.287003
\(970\) 3.11114e7 1.06167
\(971\) −1.72207e7 −0.586142 −0.293071 0.956091i \(-0.594677\pi\)
−0.293071 + 0.956091i \(0.594677\pi\)
\(972\) 944784. 0.0320750
\(973\) 0 0
\(974\) −2.68439e7 −0.906669
\(975\) 188109. 0.00633720
\(976\) 4.96140e6 0.166717
\(977\) 3.24376e7 1.08721 0.543603 0.839343i \(-0.317060\pi\)
0.543603 + 0.839343i \(0.317060\pi\)
\(978\) 7.28092e6 0.243410
\(979\) 1.88187e6 0.0627528
\(980\) 0 0
\(981\) −1.80386e7 −0.598452
\(982\) −3.65617e6 −0.120989
\(983\) 3.19804e7 1.05560 0.527801 0.849368i \(-0.323017\pi\)
0.527801 + 0.849368i \(0.323017\pi\)
\(984\) −4.10500e6 −0.135153
\(985\) 4.59479e7 1.50895
\(986\) 3.92852e7 1.28688
\(987\) 0 0
\(988\) 250110. 0.00815152
\(989\) 1.87654e7 0.610051
\(990\) 723100. 0.0234483
\(991\) 5.49614e7 1.77776 0.888880 0.458139i \(-0.151484\pi\)
0.888880 + 0.458139i \(0.151484\pi\)
\(992\) −5.38713e6 −0.173811
\(993\) −4.22042e6 −0.135826
\(994\) 0 0
\(995\) 9.49367e6 0.304002
\(996\) −908832. −0.0290292
\(997\) 7.51010e6 0.239281 0.119640 0.992817i \(-0.461826\pi\)
0.119640 + 0.992817i \(0.461826\pi\)
\(998\) −3.10197e7 −0.985850
\(999\) −9.34721e6 −0.296325
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 294.6.a.u.1.1 yes 2
3.2 odd 2 882.6.a.bj.1.2 2
7.2 even 3 294.6.e.u.67.2 4
7.3 odd 6 294.6.e.v.79.1 4
7.4 even 3 294.6.e.u.79.2 4
7.5 odd 6 294.6.e.v.67.1 4
7.6 odd 2 294.6.a.t.1.2 2
21.20 even 2 882.6.a.z.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.6.a.t.1.2 2 7.6 odd 2
294.6.a.u.1.1 yes 2 1.1 even 1 trivial
294.6.e.u.67.2 4 7.2 even 3
294.6.e.u.79.2 4 7.4 even 3
294.6.e.v.67.1 4 7.5 odd 6
294.6.e.v.79.1 4 7.3 odd 6
882.6.a.z.1.1 2 21.20 even 2
882.6.a.bj.1.2 2 3.2 odd 2