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Properties

Label 784.6.a.n.1.1
Level $784$
Weight $6$
Character 784.1
Self dual yes
Analytic conductor $125.741$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [784,6,Mod(1,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,30,0,-32,0,0,0,657] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(125.740914733\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 56)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 784.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+30.0000 q^{3} -32.0000 q^{5} +657.000 q^{9} +624.000 q^{11} +708.000 q^{13} -960.000 q^{15} -934.000 q^{17} +1858.00 q^{19} +1120.00 q^{23} -2101.00 q^{25} +12420.0 q^{27} -1174.00 q^{29} +2908.00 q^{31} +18720.0 q^{33} -12462.0 q^{37} +21240.0 q^{39} -2662.00 q^{41} +7144.00 q^{43} -21024.0 q^{45} -7468.00 q^{47} -28020.0 q^{51} -27274.0 q^{53} -19968.0 q^{55} +55740.0 q^{57} +2490.00 q^{59} +11096.0 q^{61} -22656.0 q^{65} -39756.0 q^{67} +33600.0 q^{69} +69888.0 q^{71} -16450.0 q^{73} -63030.0 q^{75} -78376.0 q^{79} +212949. q^{81} +109818. q^{83} +29888.0 q^{85} -35220.0 q^{87} +56966.0 q^{89} +87240.0 q^{93} -59456.0 q^{95} +115946. q^{97} +409968. q^{99} +O(q^{100})\)

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 0
\(3\) 30.0000 1.92450 0.962250 − 0.272166i \(-0.0877398\pi\)
0.962250 + 0.272166i \(0.0877398\pi\)
\(4\) 0 0
\(5\) −32.0000 −0.572433 −0.286217 − 0.958165i \(-0.592398\pi\)
−0.286217 + 0.958165i \(0.592398\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 657.000 2.70370
\(10\) 0 0
\(11\) 624.000 1.55490 0.777451 − 0.628944i \(-0.216512\pi\)
0.777451 + 0.628944i \(0.216512\pi\)
\(12\) 0 0
\(13\) 708.000 1.16192 0.580958 − 0.813933i \(-0.302678\pi\)
0.580958 + 0.813933i \(0.302678\pi\)
\(14\) 0 0
\(15\) −960.000 −1.10165
\(16\) 0 0
\(17\) −934.000 −0.783835 −0.391917 − 0.920000i \(-0.628188\pi\)
−0.391917 + 0.920000i \(0.628188\pi\)
\(18\) 0 0
\(19\) 1858.00 1.18076 0.590380 − 0.807125i \(-0.298978\pi\)
0.590380 + 0.807125i \(0.298978\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1120.00 0.441467 0.220734 − 0.975334i \(-0.429155\pi\)
0.220734 + 0.975334i \(0.429155\pi\)
\(24\) 0 0
\(25\) −2101.00 −0.672320
\(26\) 0 0
\(27\) 12420.0 3.27878
\(28\) 0 0
\(29\) −1174.00 −0.259223 −0.129611 − 0.991565i \(-0.541373\pi\)
−0.129611 + 0.991565i \(0.541373\pi\)
\(30\) 0 0
\(31\) 2908.00 0.543488 0.271744 − 0.962370i \(-0.412400\pi\)
0.271744 + 0.962370i \(0.412400\pi\)
\(32\) 0 0
\(33\) 18720.0 2.99241
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −12462.0 −1.49652 −0.748262 − 0.663404i \(-0.769111\pi\)
−0.748262 + 0.663404i \(0.769111\pi\)
\(38\) 0 0
\(39\) 21240.0 2.23611
\(40\) 0 0
\(41\) −2662.00 −0.247314 −0.123657 − 0.992325i \(-0.539462\pi\)
−0.123657 + 0.992325i \(0.539462\pi\)
\(42\) 0 0
\(43\) 7144.00 0.589210 0.294605 − 0.955619i \(-0.404812\pi\)
0.294605 + 0.955619i \(0.404812\pi\)
\(44\) 0 0
\(45\) −21024.0 −1.54769
\(46\) 0 0
\(47\) −7468.00 −0.493128 −0.246564 − 0.969127i \(-0.579302\pi\)
−0.246564 + 0.969127i \(0.579302\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −28020.0 −1.50849
\(52\) 0 0
\(53\) −27274.0 −1.33370 −0.666852 − 0.745191i \(-0.732358\pi\)
−0.666852 + 0.745191i \(0.732358\pi\)
\(54\) 0 0
\(55\) −19968.0 −0.890078
\(56\) 0 0
\(57\) 55740.0 2.27237
\(58\) 0 0
\(59\) 2490.00 0.0931257 0.0465628 − 0.998915i \(-0.485173\pi\)
0.0465628 + 0.998915i \(0.485173\pi\)
\(60\) 0 0
\(61\) 11096.0 0.381805 0.190903 − 0.981609i \(-0.438859\pi\)
0.190903 + 0.981609i \(0.438859\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −22656.0 −0.665120
\(66\) 0 0
\(67\) −39756.0 −1.08197 −0.540986 − 0.841032i \(-0.681949\pi\)
−0.540986 + 0.841032i \(0.681949\pi\)
\(68\) 0 0
\(69\) 33600.0 0.849604
\(70\) 0 0
\(71\) 69888.0 1.64534 0.822672 − 0.568516i \(-0.192482\pi\)
0.822672 + 0.568516i \(0.192482\pi\)
\(72\) 0 0
\(73\) −16450.0 −0.361292 −0.180646 − 0.983548i \(-0.557819\pi\)
−0.180646 + 0.983548i \(0.557819\pi\)
\(74\) 0 0
\(75\) −63030.0 −1.29388
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −78376.0 −1.41291 −0.706456 − 0.707757i \(-0.749707\pi\)
−0.706456 + 0.707757i \(0.749707\pi\)
\(80\) 0 0
\(81\) 212949. 3.60631
\(82\) 0 0
\(83\) 109818. 1.74976 0.874880 − 0.484340i \(-0.160940\pi\)
0.874880 + 0.484340i \(0.160940\pi\)
\(84\) 0 0
\(85\) 29888.0 0.448693
\(86\) 0 0
\(87\) −35220.0 −0.498874
\(88\) 0 0
\(89\) 56966.0 0.762326 0.381163 − 0.924508i \(-0.375524\pi\)
0.381163 + 0.924508i \(0.375524\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 87240.0 1.04594
\(94\) 0 0
\(95\) −59456.0 −0.675907
\(96\) 0 0
\(97\) 115946. 1.25120 0.625600 − 0.780144i \(-0.284854\pi\)
0.625600 + 0.780144i \(0.284854\pi\)
\(98\) 0 0
\(99\) 409968. 4.20399
\(100\) 0 0
\(101\) −8352.00 −0.0814680 −0.0407340 − 0.999170i \(-0.512970\pi\)
−0.0407340 + 0.999170i \(0.512970\pi\)
\(102\) 0 0
\(103\) 179484. 1.66699 0.833494 − 0.552528i \(-0.186337\pi\)
0.833494 + 0.552528i \(0.186337\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 53892.0 0.455056 0.227528 − 0.973772i \(-0.426936\pi\)
0.227528 + 0.973772i \(0.426936\pi\)
\(108\) 0 0
\(109\) 105970. 0.854312 0.427156 − 0.904178i \(-0.359515\pi\)
0.427156 + 0.904178i \(0.359515\pi\)
\(110\) 0 0
\(111\) −373860. −2.88006
\(112\) 0 0
\(113\) 2502.00 0.0184328 0.00921640 − 0.999958i \(-0.497066\pi\)
0.00921640 + 0.999958i \(0.497066\pi\)
\(114\) 0 0
\(115\) −35840.0 −0.252711
\(116\) 0 0
\(117\) 465156. 3.14148
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 228325. 1.41772
\(122\) 0 0
\(123\) −79860.0 −0.475956
\(124\) 0 0
\(125\) 167232. 0.957292
\(126\) 0 0
\(127\) −287792. −1.58332 −0.791661 − 0.610960i \(-0.790784\pi\)
−0.791661 + 0.610960i \(0.790784\pi\)
\(128\) 0 0
\(129\) 214320. 1.13394
\(130\) 0 0
\(131\) −47662.0 −0.242658 −0.121329 − 0.992612i \(-0.538716\pi\)
−0.121329 + 0.992612i \(0.538716\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −397440. −1.87688
\(136\) 0 0
\(137\) 223154. 1.01579 0.507894 − 0.861419i \(-0.330424\pi\)
0.507894 + 0.861419i \(0.330424\pi\)
\(138\) 0 0
\(139\) 250542. 1.09988 0.549938 − 0.835206i \(-0.314651\pi\)
0.549938 + 0.835206i \(0.314651\pi\)
\(140\) 0 0
\(141\) −224040. −0.949025
\(142\) 0 0
\(143\) 441792. 1.80667
\(144\) 0 0
\(145\) 37568.0 0.148388
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −487394. −1.79852 −0.899258 − 0.437418i \(-0.855893\pi\)
−0.899258 + 0.437418i \(0.855893\pi\)
\(150\) 0 0
\(151\) 54680.0 0.195158 0.0975790 − 0.995228i \(-0.468890\pi\)
0.0975790 + 0.995228i \(0.468890\pi\)
\(152\) 0 0
\(153\) −613638. −2.11926
\(154\) 0 0
\(155\) −93056.0 −0.311111
\(156\) 0 0
\(157\) −211068. −0.683397 −0.341699 − 0.939810i \(-0.611002\pi\)
−0.341699 + 0.939810i \(0.611002\pi\)
\(158\) 0 0
\(159\) −818220. −2.56671
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 20192.0 0.0595265 0.0297632 − 0.999557i \(-0.490525\pi\)
0.0297632 + 0.999557i \(0.490525\pi\)
\(164\) 0 0
\(165\) −599040. −1.71296
\(166\) 0 0
\(167\) −4524.00 −0.0125525 −0.00627627 − 0.999980i \(-0.501998\pi\)
−0.00627627 + 0.999980i \(0.501998\pi\)
\(168\) 0 0
\(169\) 129971. 0.350050
\(170\) 0 0
\(171\) 1.22071e6 3.19243
\(172\) 0 0
\(173\) 104332. 0.265034 0.132517 − 0.991181i \(-0.457694\pi\)
0.132517 + 0.991181i \(0.457694\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 74700.0 0.179220
\(178\) 0 0
\(179\) 201724. 0.470571 0.235285 − 0.971926i \(-0.424398\pi\)
0.235285 + 0.971926i \(0.424398\pi\)
\(180\) 0 0
\(181\) −655700. −1.48768 −0.743839 − 0.668359i \(-0.766997\pi\)
−0.743839 + 0.668359i \(0.766997\pi\)
\(182\) 0 0
\(183\) 332880. 0.734784
\(184\) 0 0
\(185\) 398784. 0.856660
\(186\) 0 0
\(187\) −582816. −1.21879
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 151496. 0.300482 0.150241 − 0.988649i \(-0.451995\pi\)
0.150241 + 0.988649i \(0.451995\pi\)
\(192\) 0 0
\(193\) 229326. 0.443159 0.221580 − 0.975142i \(-0.428879\pi\)
0.221580 + 0.975142i \(0.428879\pi\)
\(194\) 0 0
\(195\) −679680. −1.28002
\(196\) 0 0
\(197\) 421086. 0.773046 0.386523 − 0.922280i \(-0.373676\pi\)
0.386523 + 0.922280i \(0.373676\pi\)
\(198\) 0 0
\(199\) 197300. 0.353179 0.176589 − 0.984285i \(-0.443494\pi\)
0.176589 + 0.984285i \(0.443494\pi\)
\(200\) 0 0
\(201\) −1.19268e6 −2.08225
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 85184.0 0.141571
\(206\) 0 0
\(207\) 735840. 1.19360
\(208\) 0 0
\(209\) 1.15939e6 1.83597
\(210\) 0 0
\(211\) −679052. −1.05002 −0.525009 − 0.851097i \(-0.675938\pi\)
−0.525009 + 0.851097i \(0.675938\pi\)
\(212\) 0 0
\(213\) 2.09664e6 3.16647
\(214\) 0 0
\(215\) −228608. −0.337284
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −493500. −0.695307
\(220\) 0 0
\(221\) −661272. −0.910751
\(222\) 0 0
\(223\) −184440. −0.248366 −0.124183 − 0.992259i \(-0.539631\pi\)
−0.124183 + 0.992259i \(0.539631\pi\)
\(224\) 0 0
\(225\) −1.38036e6 −1.81775
\(226\) 0 0
\(227\) −868078. −1.11813 −0.559067 − 0.829122i \(-0.688841\pi\)
−0.559067 + 0.829122i \(0.688841\pi\)
\(228\) 0 0
\(229\) 593860. 0.748334 0.374167 − 0.927361i \(-0.377929\pi\)
0.374167 + 0.927361i \(0.377929\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 48218.0 0.0581861 0.0290931 − 0.999577i \(-0.490738\pi\)
0.0290931 + 0.999577i \(0.490738\pi\)
\(234\) 0 0
\(235\) 238976. 0.282283
\(236\) 0 0
\(237\) −2.35128e6 −2.71915
\(238\) 0 0
\(239\) −241688. −0.273691 −0.136845 − 0.990592i \(-0.543696\pi\)
−0.136845 + 0.990592i \(0.543696\pi\)
\(240\) 0 0
\(241\) −565270. −0.626922 −0.313461 − 0.949601i \(-0.601488\pi\)
−0.313461 + 0.949601i \(0.601488\pi\)
\(242\) 0 0
\(243\) 3.37041e6 3.66157
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1.31546e6 1.37194
\(248\) 0 0
\(249\) 3.29454e6 3.36741
\(250\) 0 0
\(251\) −1.43775e6 −1.44045 −0.720224 − 0.693741i \(-0.755961\pi\)
−0.720224 + 0.693741i \(0.755961\pi\)
\(252\) 0 0
\(253\) 698880. 0.686438
\(254\) 0 0
\(255\) 896640. 0.863511
\(256\) 0 0
\(257\) −494802. −0.467303 −0.233652 − 0.972320i \(-0.575067\pi\)
−0.233652 + 0.972320i \(0.575067\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −771318. −0.700861
\(262\) 0 0
\(263\) −1.55654e6 −1.38762 −0.693812 − 0.720156i \(-0.744070\pi\)
−0.693812 + 0.720156i \(0.744070\pi\)
\(264\) 0 0
\(265\) 872768. 0.763456
\(266\) 0 0
\(267\) 1.70898e6 1.46710
\(268\) 0 0
\(269\) −1.36204e6 −1.14765 −0.573823 − 0.818979i \(-0.694540\pi\)
−0.573823 + 0.818979i \(0.694540\pi\)
\(270\) 0 0
\(271\) 558320. 0.461806 0.230903 − 0.972977i \(-0.425832\pi\)
0.230903 + 0.972977i \(0.425832\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1.31102e6 −1.04539
\(276\) 0 0
\(277\) 586342. 0.459147 0.229573 − 0.973291i \(-0.426267\pi\)
0.229573 + 0.973291i \(0.426267\pi\)
\(278\) 0 0
\(279\) 1.91056e6 1.46943
\(280\) 0 0
\(281\) 606234. 0.458010 0.229005 − 0.973425i \(-0.426453\pi\)
0.229005 + 0.973425i \(0.426453\pi\)
\(282\) 0 0
\(283\) −865174. −0.642151 −0.321076 − 0.947054i \(-0.604044\pi\)
−0.321076 + 0.947054i \(0.604044\pi\)
\(284\) 0 0
\(285\) −1.78368e6 −1.30078
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −547501. −0.385603
\(290\) 0 0
\(291\) 3.47838e6 2.40793
\(292\) 0 0
\(293\) 353352. 0.240458 0.120229 − 0.992746i \(-0.461637\pi\)
0.120229 + 0.992746i \(0.461637\pi\)
\(294\) 0 0
\(295\) −79680.0 −0.0533082
\(296\) 0 0
\(297\) 7.75008e6 5.09818
\(298\) 0 0
\(299\) 792960. 0.512948
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −250560. −0.156785
\(304\) 0 0
\(305\) −355072. −0.218558
\(306\) 0 0
\(307\) −1.95904e6 −1.18631 −0.593153 − 0.805090i \(-0.702117\pi\)
−0.593153 + 0.805090i \(0.702117\pi\)
\(308\) 0 0
\(309\) 5.38452e6 3.20812
\(310\) 0 0
\(311\) 3.06257e6 1.79550 0.897749 − 0.440508i \(-0.145202\pi\)
0.897749 + 0.440508i \(0.145202\pi\)
\(312\) 0 0
\(313\) 582634. 0.336151 0.168076 − 0.985774i \(-0.446245\pi\)
0.168076 + 0.985774i \(0.446245\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.09585e6 −1.73034 −0.865171 − 0.501478i \(-0.832790\pi\)
−0.865171 + 0.501478i \(0.832790\pi\)
\(318\) 0 0
\(319\) −732576. −0.403066
\(320\) 0 0
\(321\) 1.61676e6 0.875756
\(322\) 0 0
\(323\) −1.73537e6 −0.925521
\(324\) 0 0
\(325\) −1.48751e6 −0.781180
\(326\) 0 0
\(327\) 3.17910e6 1.64412
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −625496. −0.313801 −0.156901 − 0.987614i \(-0.550150\pi\)
−0.156901 + 0.987614i \(0.550150\pi\)
\(332\) 0 0
\(333\) −8.18753e6 −4.04616
\(334\) 0 0
\(335\) 1.27219e6 0.619356
\(336\) 0 0
\(337\) 2.32494e6 1.11516 0.557580 − 0.830123i \(-0.311730\pi\)
0.557580 + 0.830123i \(0.311730\pi\)
\(338\) 0 0
\(339\) 75060.0 0.0354739
\(340\) 0 0
\(341\) 1.81459e6 0.845071
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −1.07520e6 −0.486342
\(346\) 0 0
\(347\) 781128. 0.348256 0.174128 − 0.984723i \(-0.444289\pi\)
0.174128 + 0.984723i \(0.444289\pi\)
\(348\) 0 0
\(349\) −1.48586e6 −0.653002 −0.326501 − 0.945197i \(-0.605870\pi\)
−0.326501 + 0.945197i \(0.605870\pi\)
\(350\) 0 0
\(351\) 8.79336e6 3.80967
\(352\) 0 0
\(353\) −1.44463e6 −0.617048 −0.308524 − 0.951217i \(-0.599835\pi\)
−0.308524 + 0.951217i \(0.599835\pi\)
\(354\) 0 0
\(355\) −2.23642e6 −0.941850
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −404040. −0.165458 −0.0827291 − 0.996572i \(-0.526364\pi\)
−0.0827291 + 0.996572i \(0.526364\pi\)
\(360\) 0 0
\(361\) 976065. 0.394195
\(362\) 0 0
\(363\) 6.84975e6 2.72840
\(364\) 0 0
\(365\) 526400. 0.206816
\(366\) 0 0
\(367\) −2.71698e6 −1.05298 −0.526492 − 0.850180i \(-0.676493\pi\)
−0.526492 + 0.850180i \(0.676493\pi\)
\(368\) 0 0
\(369\) −1.74893e6 −0.668663
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.79399e6 −0.667647 −0.333824 − 0.942636i \(-0.608339\pi\)
−0.333824 + 0.942636i \(0.608339\pi\)
\(374\) 0 0
\(375\) 5.01696e6 1.84231
\(376\) 0 0
\(377\) −831192. −0.301195
\(378\) 0 0
\(379\) 18624.0 0.00666001 0.00333001 − 0.999994i \(-0.498940\pi\)
0.00333001 + 0.999994i \(0.498940\pi\)
\(380\) 0 0
\(381\) −8.63376e6 −3.04711
\(382\) 0 0
\(383\) −1.33004e6 −0.463307 −0.231654 − 0.972798i \(-0.574414\pi\)
−0.231654 + 0.972798i \(0.574414\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.69361e6 1.59305
\(388\) 0 0
\(389\) 2.26506e6 0.758936 0.379468 − 0.925205i \(-0.376107\pi\)
0.379468 + 0.925205i \(0.376107\pi\)
\(390\) 0 0
\(391\) −1.04608e6 −0.346037
\(392\) 0 0
\(393\) −1.42986e6 −0.466995
\(394\) 0 0
\(395\) 2.50803e6 0.808798
\(396\) 0 0
\(397\) 4.48900e6 1.42947 0.714733 − 0.699398i \(-0.246548\pi\)
0.714733 + 0.699398i \(0.246548\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −95442.0 −0.0296400 −0.0148200 − 0.999890i \(-0.504718\pi\)
−0.0148200 + 0.999890i \(0.504718\pi\)
\(402\) 0 0
\(403\) 2.05886e6 0.631488
\(404\) 0 0
\(405\) −6.81437e6 −2.06437
\(406\) 0 0
\(407\) −7.77629e6 −2.32695
\(408\) 0 0
\(409\) 2.99003e6 0.883828 0.441914 − 0.897057i \(-0.354300\pi\)
0.441914 + 0.897057i \(0.354300\pi\)
\(410\) 0 0
\(411\) 6.69462e6 1.95489
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3.51418e6 −1.00162
\(416\) 0 0
\(417\) 7.51626e6 2.11671
\(418\) 0 0
\(419\) 3.39037e6 0.943436 0.471718 − 0.881749i \(-0.343634\pi\)
0.471718 + 0.881749i \(0.343634\pi\)
\(420\) 0 0
\(421\) 3.38397e6 0.930512 0.465256 − 0.885176i \(-0.345962\pi\)
0.465256 + 0.885176i \(0.345962\pi\)
\(422\) 0 0
\(423\) −4.90648e6 −1.33327
\(424\) 0 0
\(425\) 1.96233e6 0.526988
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 1.32538e7 3.47693
\(430\) 0 0
\(431\) 1.98353e6 0.514334 0.257167 − 0.966367i \(-0.417211\pi\)
0.257167 + 0.966367i \(0.417211\pi\)
\(432\) 0 0
\(433\) 7.17581e6 1.83929 0.919647 − 0.392746i \(-0.128475\pi\)
0.919647 + 0.392746i \(0.128475\pi\)
\(434\) 0 0
\(435\) 1.12704e6 0.285572
\(436\) 0 0
\(437\) 2.08096e6 0.521267
\(438\) 0 0
\(439\) −2.44390e6 −0.605231 −0.302616 − 0.953113i \(-0.597860\pi\)
−0.302616 + 0.953113i \(0.597860\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −231716. −0.0560979 −0.0280490 − 0.999607i \(-0.508929\pi\)
−0.0280490 + 0.999607i \(0.508929\pi\)
\(444\) 0 0
\(445\) −1.82291e6 −0.436381
\(446\) 0 0
\(447\) −1.46218e7 −3.46125
\(448\) 0 0
\(449\) −4.73637e6 −1.10874 −0.554370 − 0.832271i \(-0.687041\pi\)
−0.554370 + 0.832271i \(0.687041\pi\)
\(450\) 0 0
\(451\) −1.66109e6 −0.384549
\(452\) 0 0
\(453\) 1.64040e6 0.375582
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −7.87486e6 −1.76381 −0.881906 − 0.471426i \(-0.843740\pi\)
−0.881906 + 0.471426i \(0.843740\pi\)
\(458\) 0 0
\(459\) −1.16003e7 −2.57002
\(460\) 0 0
\(461\) 8.23218e6 1.80411 0.902054 − 0.431623i \(-0.142059\pi\)
0.902054 + 0.431623i \(0.142059\pi\)
\(462\) 0 0
\(463\) −2.36038e6 −0.511717 −0.255859 − 0.966714i \(-0.582358\pi\)
−0.255859 + 0.966714i \(0.582358\pi\)
\(464\) 0 0
\(465\) −2.79168e6 −0.598733
\(466\) 0 0
\(467\) −6.31700e6 −1.34035 −0.670175 − 0.742203i \(-0.733781\pi\)
−0.670175 + 0.742203i \(0.733781\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −6.33204e6 −1.31520
\(472\) 0 0
\(473\) 4.45786e6 0.916164
\(474\) 0 0
\(475\) −3.90366e6 −0.793849
\(476\) 0 0
\(477\) −1.79190e7 −3.60594
\(478\) 0 0
\(479\) 1.45856e6 0.290459 0.145229 − 0.989398i \(-0.453608\pi\)
0.145229 + 0.989398i \(0.453608\pi\)
\(480\) 0 0
\(481\) −8.82310e6 −1.73883
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −3.71027e6 −0.716228
\(486\) 0 0
\(487\) −9.28782e6 −1.77456 −0.887282 − 0.461228i \(-0.847409\pi\)
−0.887282 + 0.461228i \(0.847409\pi\)
\(488\) 0 0
\(489\) 605760. 0.114559
\(490\) 0 0
\(491\) 234972. 0.0439858 0.0219929 − 0.999758i \(-0.492999\pi\)
0.0219929 + 0.999758i \(0.492999\pi\)
\(492\) 0 0
\(493\) 1.09652e6 0.203188
\(494\) 0 0
\(495\) −1.31190e7 −2.40651
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 7.00792e6 1.25991 0.629953 − 0.776633i \(-0.283074\pi\)
0.629953 + 0.776633i \(0.283074\pi\)
\(500\) 0 0
\(501\) −135720. −0.0241574
\(502\) 0 0
\(503\) 4.94752e6 0.871902 0.435951 − 0.899970i \(-0.356412\pi\)
0.435951 + 0.899970i \(0.356412\pi\)
\(504\) 0 0
\(505\) 267264. 0.0466350
\(506\) 0 0
\(507\) 3.89913e6 0.673671
\(508\) 0 0
\(509\) −5.50640e6 −0.942049 −0.471025 − 0.882120i \(-0.656116\pi\)
−0.471025 + 0.882120i \(0.656116\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 2.30764e7 3.87145
\(514\) 0 0
\(515\) −5.74349e6 −0.954240
\(516\) 0 0
\(517\) −4.66003e6 −0.766765
\(518\) 0 0
\(519\) 3.12996e6 0.510059
\(520\) 0 0
\(521\) −1.63076e6 −0.263206 −0.131603 − 0.991303i \(-0.542012\pi\)
−0.131603 + 0.991303i \(0.542012\pi\)
\(522\) 0 0
\(523\) −1.00765e7 −1.61086 −0.805429 − 0.592692i \(-0.798065\pi\)
−0.805429 + 0.592692i \(0.798065\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −2.71607e6 −0.426005
\(528\) 0 0
\(529\) −5.18194e6 −0.805107
\(530\) 0 0
\(531\) 1.63593e6 0.251784
\(532\) 0 0
\(533\) −1.88470e6 −0.287358
\(534\) 0 0
\(535\) −1.72454e6 −0.260489
\(536\) 0 0
\(537\) 6.05172e6 0.905614
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.25225e7 −1.83949 −0.919746 − 0.392513i \(-0.871606\pi\)
−0.919746 + 0.392513i \(0.871606\pi\)
\(542\) 0 0
\(543\) −1.96710e7 −2.86304
\(544\) 0 0
\(545\) −3.39104e6 −0.489037
\(546\) 0 0
\(547\) −6.67430e6 −0.953756 −0.476878 − 0.878970i \(-0.658232\pi\)
−0.476878 + 0.878970i \(0.658232\pi\)
\(548\) 0 0
\(549\) 7.29007e6 1.03229
\(550\) 0 0
\(551\) −2.18129e6 −0.306080
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1.19635e7 1.64864
\(556\) 0 0
\(557\) −4.61643e6 −0.630475 −0.315238 − 0.949013i \(-0.602084\pi\)
−0.315238 + 0.949013i \(0.602084\pi\)
\(558\) 0 0
\(559\) 5.05795e6 0.684613
\(560\) 0 0
\(561\) −1.74845e7 −2.34555
\(562\) 0 0
\(563\) −8.84218e6 −1.17568 −0.587839 − 0.808978i \(-0.700021\pi\)
−0.587839 + 0.808978i \(0.700021\pi\)
\(564\) 0 0
\(565\) −80064.0 −0.0105515
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.15771e7 1.49906 0.749532 − 0.661968i \(-0.230279\pi\)
0.749532 + 0.661968i \(0.230279\pi\)
\(570\) 0 0
\(571\) −4.48069e6 −0.575115 −0.287557 − 0.957763i \(-0.592843\pi\)
−0.287557 + 0.957763i \(0.592843\pi\)
\(572\) 0 0
\(573\) 4.54488e6 0.578277
\(574\) 0 0
\(575\) −2.35312e6 −0.296807
\(576\) 0 0
\(577\) −1.32788e7 −1.66042 −0.830212 − 0.557448i \(-0.811780\pi\)
−0.830212 + 0.557448i \(0.811780\pi\)
\(578\) 0 0
\(579\) 6.87978e6 0.852861
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.70190e7 −2.07378
\(584\) 0 0
\(585\) −1.48850e7 −1.79829
\(586\) 0 0
\(587\) −1.11188e7 −1.33187 −0.665936 − 0.746009i \(-0.731968\pi\)
−0.665936 + 0.746009i \(0.731968\pi\)
\(588\) 0 0
\(589\) 5.40306e6 0.641729
\(590\) 0 0
\(591\) 1.26326e7 1.48773
\(592\) 0 0
\(593\) 3.92737e6 0.458632 0.229316 − 0.973352i \(-0.426351\pi\)
0.229316 + 0.973352i \(0.426351\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 5.91900e6 0.679693
\(598\) 0 0
\(599\) 1.74099e7 1.98258 0.991289 − 0.131704i \(-0.0420449\pi\)
0.991289 + 0.131704i \(0.0420449\pi\)
\(600\) 0 0
\(601\) −7.46243e6 −0.842740 −0.421370 − 0.906889i \(-0.638451\pi\)
−0.421370 + 0.906889i \(0.638451\pi\)
\(602\) 0 0
\(603\) −2.61197e7 −2.92533
\(604\) 0 0
\(605\) −7.30640e6 −0.811549
\(606\) 0 0
\(607\) 701152. 0.0772397 0.0386198 − 0.999254i \(-0.487704\pi\)
0.0386198 + 0.999254i \(0.487704\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.28734e6 −0.572974
\(612\) 0 0
\(613\) 1.09575e7 1.17777 0.588886 − 0.808216i \(-0.299567\pi\)
0.588886 + 0.808216i \(0.299567\pi\)
\(614\) 0 0
\(615\) 2.55552e6 0.272453
\(616\) 0 0
\(617\) 1.90666e6 0.201633 0.100816 − 0.994905i \(-0.467855\pi\)
0.100816 + 0.994905i \(0.467855\pi\)
\(618\) 0 0
\(619\) −2.22346e6 −0.233240 −0.116620 − 0.993177i \(-0.537206\pi\)
−0.116620 + 0.993177i \(0.537206\pi\)
\(620\) 0 0
\(621\) 1.39104e7 1.44747
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.21420e6 0.124334
\(626\) 0 0
\(627\) 3.47818e7 3.53332
\(628\) 0 0
\(629\) 1.16395e7 1.17303
\(630\) 0 0
\(631\) −752624. −0.0752497 −0.0376248 − 0.999292i \(-0.511979\pi\)
−0.0376248 + 0.999292i \(0.511979\pi\)
\(632\) 0 0
\(633\) −2.03716e7 −2.02076
\(634\) 0 0
\(635\) 9.20934e6 0.906347
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 4.59164e7 4.44852
\(640\) 0 0
\(641\) −2.45429e6 −0.235929 −0.117964 − 0.993018i \(-0.537637\pi\)
−0.117964 + 0.993018i \(0.537637\pi\)
\(642\) 0 0
\(643\) 1.58237e7 1.50932 0.754660 − 0.656116i \(-0.227802\pi\)
0.754660 + 0.656116i \(0.227802\pi\)
\(644\) 0 0
\(645\) −6.85824e6 −0.649103
\(646\) 0 0
\(647\) −2.65489e6 −0.249337 −0.124668 − 0.992198i \(-0.539787\pi\)
−0.124668 + 0.992198i \(0.539787\pi\)
\(648\) 0 0
\(649\) 1.55376e6 0.144801
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 9.26899e6 0.850648 0.425324 − 0.905041i \(-0.360160\pi\)
0.425324 + 0.905041i \(0.360160\pi\)
\(654\) 0 0
\(655\) 1.52518e6 0.138905
\(656\) 0 0
\(657\) −1.08076e7 −0.976827
\(658\) 0 0
\(659\) 1.68242e7 1.50911 0.754556 − 0.656235i \(-0.227852\pi\)
0.754556 + 0.656235i \(0.227852\pi\)
\(660\) 0 0
\(661\) 6.77217e6 0.602871 0.301435 − 0.953487i \(-0.402534\pi\)
0.301435 + 0.953487i \(0.402534\pi\)
\(662\) 0 0
\(663\) −1.98382e7 −1.75274
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.31488e6 −0.114438
\(668\) 0 0
\(669\) −5.53320e6 −0.477981
\(670\) 0 0
\(671\) 6.92390e6 0.593669
\(672\) 0 0
\(673\) 7.61315e6 0.647928 0.323964 − 0.946069i \(-0.394984\pi\)
0.323964 + 0.946069i \(0.394984\pi\)
\(674\) 0 0
\(675\) −2.60944e7 −2.20439
\(676\) 0 0
\(677\) −6.11672e6 −0.512917 −0.256459 − 0.966555i \(-0.582556\pi\)
−0.256459 + 0.966555i \(0.582556\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −2.60423e7 −2.15185
\(682\) 0 0
\(683\) 1.12588e7 0.923511 0.461755 − 0.887007i \(-0.347220\pi\)
0.461755 + 0.887007i \(0.347220\pi\)
\(684\) 0 0
\(685\) −7.14093e6 −0.581471
\(686\) 0 0
\(687\) 1.78158e7 1.44017
\(688\) 0 0
\(689\) −1.93100e7 −1.54965
\(690\) 0 0
\(691\) 9.50952e6 0.757641 0.378821 − 0.925470i \(-0.376330\pi\)
0.378821 + 0.925470i \(0.376330\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −8.01734e6 −0.629605
\(696\) 0 0
\(697\) 2.48631e6 0.193853
\(698\) 0 0
\(699\) 1.44654e6 0.111979
\(700\) 0 0
\(701\) 1.53868e7 1.18264 0.591322 − 0.806436i \(-0.298606\pi\)
0.591322 + 0.806436i \(0.298606\pi\)
\(702\) 0 0
\(703\) −2.31544e7 −1.76703
\(704\) 0 0
\(705\) 7.16928e6 0.543254
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.21379e6 −0.0906834 −0.0453417 − 0.998972i \(-0.514438\pi\)
−0.0453417 + 0.998972i \(0.514438\pi\)
\(710\) 0 0
\(711\) −5.14930e7 −3.82010
\(712\) 0 0
\(713\) 3.25696e6 0.239932
\(714\) 0 0
\(715\) −1.41373e7 −1.03420
\(716\) 0 0
\(717\) −7.25064e6 −0.526718
\(718\) 0 0
\(719\) −1.00002e7 −0.721418 −0.360709 − 0.932678i \(-0.617465\pi\)
−0.360709 + 0.932678i \(0.617465\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −1.69581e7 −1.20651
\(724\) 0 0
\(725\) 2.46657e6 0.174281
\(726\) 0 0
\(727\) 1.33745e7 0.938514 0.469257 − 0.883062i \(-0.344522\pi\)
0.469257 + 0.883062i \(0.344522\pi\)
\(728\) 0 0
\(729\) 4.93657e7 3.44038
\(730\) 0 0
\(731\) −6.67250e6 −0.461844
\(732\) 0 0
\(733\) −1.61380e7 −1.10940 −0.554701 − 0.832050i \(-0.687167\pi\)
−0.554701 + 0.832050i \(0.687167\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.48077e7 −1.68236
\(738\) 0 0
\(739\) −8.61059e6 −0.579992 −0.289996 − 0.957028i \(-0.593654\pi\)
−0.289996 + 0.957028i \(0.593654\pi\)
\(740\) 0 0
\(741\) 3.94639e7 2.64031
\(742\) 0 0
\(743\) 2.85027e7 1.89415 0.947075 − 0.321012i \(-0.104023\pi\)
0.947075 + 0.321012i \(0.104023\pi\)
\(744\) 0 0
\(745\) 1.55966e7 1.02953
\(746\) 0 0
\(747\) 7.21504e7 4.73083
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 7.28721e6 0.471478 0.235739 − 0.971816i \(-0.424249\pi\)
0.235739 + 0.971816i \(0.424249\pi\)
\(752\) 0 0
\(753\) −4.31324e7 −2.77215
\(754\) 0 0
\(755\) −1.74976e6 −0.111715
\(756\) 0 0
\(757\) −2.77165e7 −1.75792 −0.878958 − 0.476899i \(-0.841761\pi\)
−0.878958 + 0.476899i \(0.841761\pi\)
\(758\) 0 0
\(759\) 2.09664e7 1.32105
\(760\) 0 0
\(761\) −3.07625e7 −1.92557 −0.962787 − 0.270262i \(-0.912890\pi\)
−0.962787 + 0.270262i \(0.912890\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 1.96364e7 1.21313
\(766\) 0 0
\(767\) 1.76292e6 0.108204
\(768\) 0 0
\(769\) 1.83665e7 1.11998 0.559990 − 0.828499i \(-0.310805\pi\)
0.559990 + 0.828499i \(0.310805\pi\)
\(770\) 0 0
\(771\) −1.48441e7 −0.899325
\(772\) 0 0
\(773\) 1.01397e7 0.610348 0.305174 − 0.952297i \(-0.401285\pi\)
0.305174 + 0.952297i \(0.401285\pi\)
\(774\) 0 0
\(775\) −6.10971e6 −0.365398
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −4.94600e6 −0.292018
\(780\) 0 0
\(781\) 4.36101e7 2.55835
\(782\) 0 0
\(783\) −1.45811e7 −0.849934
\(784\) 0 0
\(785\) 6.75418e6 0.391199
\(786\) 0 0
\(787\) −1.89442e7 −1.09028 −0.545140 − 0.838345i \(-0.683524\pi\)
−0.545140 + 0.838345i \(0.683524\pi\)
\(788\) 0 0
\(789\) −4.66963e7 −2.67049
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7.85597e6 0.443626
\(794\) 0 0
\(795\) 2.61830e7 1.46927
\(796\) 0 0
\(797\) 1.19835e7 0.668248 0.334124 − 0.942529i \(-0.391560\pi\)
0.334124 + 0.942529i \(0.391560\pi\)
\(798\) 0 0
\(799\) 6.97511e6 0.386531
\(800\) 0 0
\(801\) 3.74267e7 2.06110
\(802\) 0 0
\(803\) −1.02648e7 −0.561774
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.08611e7 −2.20865
\(808\) 0 0
\(809\) −6.31823e6 −0.339410 −0.169705 − 0.985495i \(-0.554281\pi\)
−0.169705 + 0.985495i \(0.554281\pi\)
\(810\) 0 0
\(811\) 1.47079e6 0.0785231 0.0392615 − 0.999229i \(-0.487499\pi\)
0.0392615 + 0.999229i \(0.487499\pi\)
\(812\) 0 0
\(813\) 1.67496e7 0.888747
\(814\) 0 0
\(815\) −646144. −0.0340750
\(816\) 0 0
\(817\) 1.32736e7 0.695716
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.10138e7 −1.08804 −0.544022 − 0.839071i \(-0.683099\pi\)
−0.544022 + 0.839071i \(0.683099\pi\)
\(822\) 0 0
\(823\) 1.35856e7 0.699163 0.349582 − 0.936906i \(-0.386324\pi\)
0.349582 + 0.936906i \(0.386324\pi\)
\(824\) 0 0
\(825\) −3.93307e7 −2.01186
\(826\) 0 0
\(827\) 2.62070e7 1.33246 0.666230 − 0.745747i \(-0.267907\pi\)
0.666230 + 0.745747i \(0.267907\pi\)
\(828\) 0 0
\(829\) 1.17710e7 0.594876 0.297438 − 0.954741i \(-0.403868\pi\)
0.297438 + 0.954741i \(0.403868\pi\)
\(830\) 0 0
\(831\) 1.75903e7 0.883628
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 144768. 0.00718549
\(836\) 0 0
\(837\) 3.61174e7 1.78198
\(838\) 0 0
\(839\) 1.44898e7 0.710651 0.355326 − 0.934743i \(-0.384370\pi\)
0.355326 + 0.934743i \(0.384370\pi\)
\(840\) 0 0
\(841\) −1.91329e7 −0.932804
\(842\) 0 0
\(843\) 1.81870e7 0.881440
\(844\) 0 0
\(845\) −4.15907e6 −0.200380
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −2.59552e7 −1.23582
\(850\) 0 0
\(851\) −1.39574e7 −0.660666
\(852\) 0 0
\(853\) −1.77865e7 −0.836984 −0.418492 − 0.908220i \(-0.637441\pi\)
−0.418492 + 0.908220i \(0.637441\pi\)
\(854\) 0 0
\(855\) −3.90626e7 −1.82745
\(856\) 0 0
\(857\) −3.79124e7 −1.76331 −0.881656 − 0.471893i \(-0.843571\pi\)
−0.881656 + 0.471893i \(0.843571\pi\)
\(858\) 0 0
\(859\) 3.32376e7 1.53690 0.768451 − 0.639909i \(-0.221028\pi\)
0.768451 + 0.639909i \(0.221028\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.33166e6 −0.106571 −0.0532853 − 0.998579i \(-0.516969\pi\)
−0.0532853 + 0.998579i \(0.516969\pi\)
\(864\) 0 0
\(865\) −3.33862e6 −0.151715
\(866\) 0 0
\(867\) −1.64250e7 −0.742093
\(868\) 0 0
\(869\) −4.89066e7 −2.19694
\(870\) 0 0
\(871\) −2.81472e7 −1.25716
\(872\) 0 0
\(873\) 7.61765e7 3.38287
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.38189e7 −1.48477 −0.742386 − 0.669972i \(-0.766306\pi\)
−0.742386 + 0.669972i \(0.766306\pi\)
\(878\) 0 0
\(879\) 1.06006e7 0.462761
\(880\) 0 0
\(881\) −2.65707e7 −1.15336 −0.576678 − 0.816972i \(-0.695651\pi\)
−0.576678 + 0.816972i \(0.695651\pi\)
\(882\) 0 0
\(883\) −1.74913e7 −0.754954 −0.377477 − 0.926019i \(-0.623208\pi\)
−0.377477 + 0.926019i \(0.623208\pi\)
\(884\) 0 0
\(885\) −2.39040e6 −0.102592
\(886\) 0 0
\(887\) −1.77452e7 −0.757305 −0.378652 − 0.925539i \(-0.623612\pi\)
−0.378652 + 0.925539i \(0.623612\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.32880e8 5.60746
\(892\) 0 0
\(893\) −1.38755e7 −0.582266
\(894\) 0 0
\(895\) −6.45517e6 −0.269370
\(896\) 0 0
\(897\) 2.37888e7 0.987169
\(898\) 0 0
\(899\) −3.41399e6 −0.140885
\(900\) 0 0
\(901\) 2.54739e7 1.04540
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.09824e7 0.851596
\(906\) 0 0
\(907\) 2.82335e7 1.13958 0.569792 − 0.821789i \(-0.307024\pi\)
0.569792 + 0.821789i \(0.307024\pi\)
\(908\) 0 0
\(909\) −5.48726e6 −0.220265
\(910\) 0 0
\(911\) −1.71757e7 −0.685674 −0.342837 − 0.939395i \(-0.611388\pi\)
−0.342837 + 0.939395i \(0.611388\pi\)
\(912\) 0 0
\(913\) 6.85264e7 2.72070
\(914\) 0 0
\(915\) −1.06522e7 −0.420615
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.42273e7 −0.946273 −0.473137 − 0.880989i \(-0.656878\pi\)
−0.473137 + 0.880989i \(0.656878\pi\)
\(920\) 0 0
\(921\) −5.87711e7 −2.28305
\(922\) 0 0
\(923\) 4.94807e7 1.91175
\(924\) 0 0
\(925\) 2.61827e7 1.00614
\(926\) 0 0
\(927\) 1.17921e8 4.50704
\(928\) 0 0
\(929\) −2.45919e7 −0.934874 −0.467437 − 0.884026i \(-0.654822\pi\)
−0.467437 + 0.884026i \(0.654822\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.18770e7 3.45544
\(934\) 0 0
\(935\) 1.86501e7 0.697674
\(936\) 0 0
\(937\) 1.31199e7 0.488181 0.244090 − 0.969753i \(-0.421511\pi\)
0.244090 + 0.969753i \(0.421511\pi\)
\(938\) 0 0
\(939\) 1.74790e7 0.646924
\(940\) 0 0
\(941\) −640776. −0.0235902 −0.0117951 − 0.999930i \(-0.503755\pi\)
−0.0117951 + 0.999930i \(0.503755\pi\)
\(942\) 0 0
\(943\) −2.98144e6 −0.109181
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.01502e7 −0.730135 −0.365068 − 0.930981i \(-0.618954\pi\)
−0.365068 + 0.930981i \(0.618954\pi\)
\(948\) 0 0
\(949\) −1.16466e7 −0.419791
\(950\) 0 0
\(951\) −9.28755e7 −3.33004
\(952\) 0 0
\(953\) −3.33217e7 −1.18849 −0.594244 − 0.804285i \(-0.702549\pi\)
−0.594244 + 0.804285i \(0.702549\pi\)
\(954\) 0 0
\(955\) −4.84787e6 −0.172006
\(956\) 0 0
\(957\) −2.19773e7 −0.775701
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.01727e7 −0.704621
\(962\) 0 0
\(963\) 3.54070e7 1.23034
\(964\) 0 0
\(965\) −7.33843e6 −0.253679
\(966\) 0 0
\(967\) −4.51857e7 −1.55394 −0.776970 − 0.629537i \(-0.783245\pi\)
−0.776970 + 0.629537i \(0.783245\pi\)
\(968\) 0 0
\(969\) −5.20612e7 −1.78117
\(970\) 0 0
\(971\) 2.62935e7 0.894953 0.447477 − 0.894296i \(-0.352323\pi\)
0.447477 + 0.894296i \(0.352323\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4.46252e7 −1.50338
\(976\) 0 0
\(977\) 2.02548e7 0.678879 0.339440 − 0.940628i \(-0.389763\pi\)
0.339440 + 0.940628i \(0.389763\pi\)
\(978\) 0 0
\(979\) 3.55468e7 1.18534
\(980\) 0 0
\(981\) 6.96223e7 2.30981
\(982\) 0 0
\(983\) 9.43782e6 0.311521 0.155761 − 0.987795i \(-0.450217\pi\)
0.155761 + 0.987795i \(0.450217\pi\)
\(984\) 0 0
\(985\) −1.34748e7 −0.442517
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.00128e6 0.260117
\(990\) 0 0
\(991\) −1.53265e7 −0.495747 −0.247874 − 0.968792i \(-0.579732\pi\)
−0.247874 + 0.968792i \(0.579732\pi\)
\(992\) 0 0
\(993\) −1.87649e7 −0.603911
\(994\) 0 0
\(995\) −6.31360e6 −0.202171
\(996\) 0 0
\(997\) 2.16514e6 0.0689841 0.0344920 − 0.999405i \(-0.489019\pi\)
0.0344920 + 0.999405i \(0.489019\pi\)
\(998\) 0 0
\(999\) −1.54778e8 −4.90677
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 784.6.a.n.1.1 1
4.3 odd 2 392.6.a.a.1.1 1
7.6 odd 2 112.6.a.a.1.1 1
21.20 even 2 1008.6.a.h.1.1 1
28.3 even 6 392.6.i.a.177.1 2
28.11 odd 6 392.6.i.f.177.1 2
28.19 even 6 392.6.i.a.361.1 2
28.23 odd 6 392.6.i.f.361.1 2
28.27 even 2 56.6.a.b.1.1 ✓ 1
56.13 odd 2 448.6.a.p.1.1 1
56.27 even 2 448.6.a.a.1.1 1
84.83 odd 2 504.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
56.6.a.b.1.1 ✓ 1 28.27 even 2
112.6.a.a.1.1 1 7.6 odd 2
392.6.a.a.1.1 1 4.3 odd 2
392.6.i.a.177.1 2 28.3 even 6
392.6.i.a.361.1 2 28.19 even 6
392.6.i.f.177.1 2 28.11 odd 6
392.6.i.f.361.1 2 28.23 odd 6
448.6.a.a.1.1 1 56.27 even 2
448.6.a.p.1.1 1 56.13 odd 2
504.6.a.b.1.1 1 84.83 odd 2
784.6.a.n.1.1 1 1.1 even 1 trivial
1008.6.a.h.1.1 1 21.20 even 2