Properties

Label 456.6.a.f.1.5
Level $456$
Weight $6$
Character 456.1
Self dual yes
Analytic conductor $73.135$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 456 = 2^{3} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 456.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(73.1350218347\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 4725x^{4} + 92430x^{3} + 1610577x^{2} - 16081740x - 24661341 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-15.6548\) of defining polynomial
Character \(\chi\) \(=\) 456.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q-9.00000 q^{3} +46.6057 q^{5} -58.6437 q^{7} +81.0000 q^{9} -125.529 q^{11} +351.083 q^{13} -419.452 q^{15} -885.933 q^{17} +361.000 q^{19} +527.793 q^{21} +936.962 q^{23} -952.906 q^{25} -729.000 q^{27} +3117.41 q^{29} -10558.4 q^{31} +1129.76 q^{33} -2733.13 q^{35} +10653.1 q^{37} -3159.75 q^{39} +11247.7 q^{41} +5845.74 q^{43} +3775.06 q^{45} -4849.81 q^{47} -13367.9 q^{49} +7973.40 q^{51} -36611.8 q^{53} -5850.36 q^{55} -3249.00 q^{57} -7920.92 q^{59} +49482.3 q^{61} -4750.14 q^{63} +16362.5 q^{65} +15138.3 q^{67} -8432.66 q^{69} -39065.7 q^{71} +36875.3 q^{73} +8576.15 q^{75} +7361.46 q^{77} +8875.46 q^{79} +6561.00 q^{81} -61884.5 q^{83} -41289.5 q^{85} -28056.7 q^{87} -25935.2 q^{89} -20588.8 q^{91} +95025.4 q^{93} +16824.7 q^{95} -88399.1 q^{97} -10167.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 54 q^{3} - 65 q^{5} - 149 q^{7} + 486 q^{9} - 203 q^{11} - 298 q^{13} + 585 q^{15} + 1319 q^{17} + 2166 q^{19} + 1341 q^{21} + 1234 q^{23} + 4589 q^{25} - 4374 q^{27} + 7356 q^{29} + 1632 q^{31} + 1827 q^{33} + 4383 q^{35} + 14204 q^{37} + 2682 q^{39} + 14734 q^{41} - 4693 q^{43} - 5265 q^{45} - 10955 q^{47} + 38561 q^{49} - 11871 q^{51} + 47500 q^{53} + 769 q^{55} - 19494 q^{57} - 61744 q^{59} - 81581 q^{61} - 12069 q^{63} - 59686 q^{65} - 45756 q^{67} - 11106 q^{69} - 10416 q^{71} - 54615 q^{73} - 41301 q^{75} - 29515 q^{77} - 145594 q^{79} + 39366 q^{81} - 160548 q^{83} - 53947 q^{85} - 66204 q^{87} - 97728 q^{89} - 418294 q^{91} - 14688 q^{93} - 23465 q^{95} - 760 q^{97} - 16443 q^{99}+O(q^{100}) \) 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 0
\(3\) −9.00000 −0.577350
\(4\) 0 0
\(5\) 46.6057 0.833709 0.416854 0.908973i \(-0.363133\pi\)
0.416854 + 0.908973i \(0.363133\pi\)
\(6\) 0 0
\(7\) −58.6437 −0.452351 −0.226176 0.974087i \(-0.572622\pi\)
−0.226176 + 0.974087i \(0.572622\pi\)
\(8\) 0 0
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −125.529 −0.312796 −0.156398 0.987694i \(-0.549988\pi\)
−0.156398 + 0.987694i \(0.549988\pi\)
\(12\) 0 0
\(13\) 351.083 0.576171 0.288085 0.957605i \(-0.406981\pi\)
0.288085 + 0.957605i \(0.406981\pi\)
\(14\) 0 0
\(15\) −419.452 −0.481342
\(16\) 0 0
\(17\) −885.933 −0.743496 −0.371748 0.928334i \(-0.621241\pi\)
−0.371748 + 0.928334i \(0.621241\pi\)
\(18\) 0 0
\(19\) 361.000 0.229416
\(20\) 0 0
\(21\) 527.793 0.261165
\(22\) 0 0
\(23\) 936.962 0.369320 0.184660 0.982803i \(-0.440882\pi\)
0.184660 + 0.982803i \(0.440882\pi\)
\(24\) 0 0
\(25\) −952.906 −0.304930
\(26\) 0 0
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 3117.41 0.688333 0.344166 0.938909i \(-0.388162\pi\)
0.344166 + 0.938909i \(0.388162\pi\)
\(30\) 0 0
\(31\) −10558.4 −1.97330 −0.986649 0.162859i \(-0.947929\pi\)
−0.986649 + 0.162859i \(0.947929\pi\)
\(32\) 0 0
\(33\) 1129.76 0.180593
\(34\) 0 0
\(35\) −2733.13 −0.377129
\(36\) 0 0
\(37\) 10653.1 1.27929 0.639646 0.768670i \(-0.279081\pi\)
0.639646 + 0.768670i \(0.279081\pi\)
\(38\) 0 0
\(39\) −3159.75 −0.332652
\(40\) 0 0
\(41\) 11247.7 1.04497 0.522486 0.852648i \(-0.325005\pi\)
0.522486 + 0.852648i \(0.325005\pi\)
\(42\) 0 0
\(43\) 5845.74 0.482134 0.241067 0.970508i \(-0.422503\pi\)
0.241067 + 0.970508i \(0.422503\pi\)
\(44\) 0 0
\(45\) 3775.06 0.277903
\(46\) 0 0
\(47\) −4849.81 −0.320244 −0.160122 0.987097i \(-0.551189\pi\)
−0.160122 + 0.987097i \(0.551189\pi\)
\(48\) 0 0
\(49\) −13367.9 −0.795378
\(50\) 0 0
\(51\) 7973.40 0.429257
\(52\) 0 0
\(53\) −36611.8 −1.79033 −0.895163 0.445740i \(-0.852941\pi\)
−0.895163 + 0.445740i \(0.852941\pi\)
\(54\) 0 0
\(55\) −5850.36 −0.260781
\(56\) 0 0
\(57\) −3249.00 −0.132453
\(58\) 0 0
\(59\) −7920.92 −0.296241 −0.148121 0.988969i \(-0.547322\pi\)
−0.148121 + 0.988969i \(0.547322\pi\)
\(60\) 0 0
\(61\) 49482.3 1.70265 0.851325 0.524639i \(-0.175800\pi\)
0.851325 + 0.524639i \(0.175800\pi\)
\(62\) 0 0
\(63\) −4750.14 −0.150784
\(64\) 0 0
\(65\) 16362.5 0.480359
\(66\) 0 0
\(67\) 15138.3 0.411994 0.205997 0.978553i \(-0.433956\pi\)
0.205997 + 0.978553i \(0.433956\pi\)
\(68\) 0 0
\(69\) −8432.66 −0.213227
\(70\) 0 0
\(71\) −39065.7 −0.919708 −0.459854 0.887995i \(-0.652098\pi\)
−0.459854 + 0.887995i \(0.652098\pi\)
\(72\) 0 0
\(73\) 36875.3 0.809895 0.404947 0.914340i \(-0.367290\pi\)
0.404947 + 0.914340i \(0.367290\pi\)
\(74\) 0 0
\(75\) 8576.15 0.176051
\(76\) 0 0
\(77\) 7361.46 0.141494
\(78\) 0 0
\(79\) 8875.46 0.160001 0.0800006 0.996795i \(-0.474508\pi\)
0.0800006 + 0.996795i \(0.474508\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 0 0
\(83\) −61884.5 −0.986021 −0.493011 0.870023i \(-0.664104\pi\)
−0.493011 + 0.870023i \(0.664104\pi\)
\(84\) 0 0
\(85\) −41289.5 −0.619859
\(86\) 0 0
\(87\) −28056.7 −0.397409
\(88\) 0 0
\(89\) −25935.2 −0.347068 −0.173534 0.984828i \(-0.555519\pi\)
−0.173534 + 0.984828i \(0.555519\pi\)
\(90\) 0 0
\(91\) −20588.8 −0.260632
\(92\) 0 0
\(93\) 95025.4 1.13928
\(94\) 0 0
\(95\) 16824.7 0.191266
\(96\) 0 0
\(97\) −88399.1 −0.953934 −0.476967 0.878921i \(-0.658264\pi\)
−0.476967 + 0.878921i \(0.658264\pi\)
\(98\) 0 0
\(99\) −10167.8 −0.104265
\(100\) 0 0
\(101\) −97916.9 −0.955112 −0.477556 0.878601i \(-0.658477\pi\)
−0.477556 + 0.878601i \(0.658477\pi\)
\(102\) 0 0
\(103\) −158754. −1.47445 −0.737225 0.675647i \(-0.763864\pi\)
−0.737225 + 0.675647i \(0.763864\pi\)
\(104\) 0 0
\(105\) 24598.2 0.217736
\(106\) 0 0
\(107\) −147837. −1.24831 −0.624156 0.781300i \(-0.714557\pi\)
−0.624156 + 0.781300i \(0.714557\pi\)
\(108\) 0 0
\(109\) −81578.7 −0.657674 −0.328837 0.944387i \(-0.606657\pi\)
−0.328837 + 0.944387i \(0.606657\pi\)
\(110\) 0 0
\(111\) −95877.5 −0.738600
\(112\) 0 0
\(113\) 29397.4 0.216577 0.108289 0.994119i \(-0.465463\pi\)
0.108289 + 0.994119i \(0.465463\pi\)
\(114\) 0 0
\(115\) 43667.8 0.307905
\(116\) 0 0
\(117\) 28437.7 0.192057
\(118\) 0 0
\(119\) 51954.3 0.336321
\(120\) 0 0
\(121\) −145294. −0.902159
\(122\) 0 0
\(123\) −101229. −0.603315
\(124\) 0 0
\(125\) −190054. −1.08793
\(126\) 0 0
\(127\) −115145. −0.633482 −0.316741 0.948512i \(-0.602589\pi\)
−0.316741 + 0.948512i \(0.602589\pi\)
\(128\) 0 0
\(129\) −52611.6 −0.278360
\(130\) 0 0
\(131\) −60304.8 −0.307025 −0.153512 0.988147i \(-0.549059\pi\)
−0.153512 + 0.988147i \(0.549059\pi\)
\(132\) 0 0
\(133\) −21170.4 −0.103777
\(134\) 0 0
\(135\) −33975.6 −0.160447
\(136\) 0 0
\(137\) −231240. −1.05259 −0.526297 0.850301i \(-0.676420\pi\)
−0.526297 + 0.850301i \(0.676420\pi\)
\(138\) 0 0
\(139\) −50266.0 −0.220667 −0.110333 0.993895i \(-0.535192\pi\)
−0.110333 + 0.993895i \(0.535192\pi\)
\(140\) 0 0
\(141\) 43648.3 0.184893
\(142\) 0 0
\(143\) −44071.0 −0.180224
\(144\) 0 0
\(145\) 145289. 0.573869
\(146\) 0 0
\(147\) 120311. 0.459212
\(148\) 0 0
\(149\) 353949. 1.30610 0.653048 0.757317i \(-0.273490\pi\)
0.653048 + 0.757317i \(0.273490\pi\)
\(150\) 0 0
\(151\) −337116. −1.20320 −0.601599 0.798798i \(-0.705469\pi\)
−0.601599 + 0.798798i \(0.705469\pi\)
\(152\) 0 0
\(153\) −71760.6 −0.247832
\(154\) 0 0
\(155\) −492081. −1.64516
\(156\) 0 0
\(157\) −494620. −1.60148 −0.800741 0.599010i \(-0.795561\pi\)
−0.800741 + 0.599010i \(0.795561\pi\)
\(158\) 0 0
\(159\) 329507. 1.03364
\(160\) 0 0
\(161\) −54946.9 −0.167062
\(162\) 0 0
\(163\) −117016. −0.344966 −0.172483 0.985013i \(-0.555179\pi\)
−0.172483 + 0.985013i \(0.555179\pi\)
\(164\) 0 0
\(165\) 52653.2 0.150562
\(166\) 0 0
\(167\) −300735. −0.834435 −0.417217 0.908807i \(-0.636995\pi\)
−0.417217 + 0.908807i \(0.636995\pi\)
\(168\) 0 0
\(169\) −248034. −0.668027
\(170\) 0 0
\(171\) 29241.0 0.0764719
\(172\) 0 0
\(173\) 468620. 1.19043 0.595217 0.803565i \(-0.297066\pi\)
0.595217 + 0.803565i \(0.297066\pi\)
\(174\) 0 0
\(175\) 55881.9 0.137936
\(176\) 0 0
\(177\) 71288.2 0.171035
\(178\) 0 0
\(179\) 695221. 1.62177 0.810887 0.585203i \(-0.198985\pi\)
0.810887 + 0.585203i \(0.198985\pi\)
\(180\) 0 0
\(181\) −606339. −1.37568 −0.687842 0.725860i \(-0.741442\pi\)
−0.687842 + 0.725860i \(0.741442\pi\)
\(182\) 0 0
\(183\) −445341. −0.983025
\(184\) 0 0
\(185\) 496493. 1.06656
\(186\) 0 0
\(187\) 111210. 0.232563
\(188\) 0 0
\(189\) 42751.2 0.0870551
\(190\) 0 0
\(191\) 84147.2 0.166900 0.0834500 0.996512i \(-0.473406\pi\)
0.0834500 + 0.996512i \(0.473406\pi\)
\(192\) 0 0
\(193\) 757920. 1.46464 0.732319 0.680962i \(-0.238438\pi\)
0.732319 + 0.680962i \(0.238438\pi\)
\(194\) 0 0
\(195\) −147262. −0.277335
\(196\) 0 0
\(197\) 353050. 0.648143 0.324072 0.946033i \(-0.394948\pi\)
0.324072 + 0.946033i \(0.394948\pi\)
\(198\) 0 0
\(199\) 90967.9 0.162838 0.0814189 0.996680i \(-0.474055\pi\)
0.0814189 + 0.996680i \(0.474055\pi\)
\(200\) 0 0
\(201\) −136245. −0.237865
\(202\) 0 0
\(203\) −182816. −0.311368
\(204\) 0 0
\(205\) 524208. 0.871203
\(206\) 0 0
\(207\) 75893.9 0.123107
\(208\) 0 0
\(209\) −45315.9 −0.0717603
\(210\) 0 0
\(211\) −612292. −0.946788 −0.473394 0.880851i \(-0.656971\pi\)
−0.473394 + 0.880851i \(0.656971\pi\)
\(212\) 0 0
\(213\) 351591. 0.530993
\(214\) 0 0
\(215\) 272445. 0.401960
\(216\) 0 0
\(217\) 619182. 0.892625
\(218\) 0 0
\(219\) −331878. −0.467593
\(220\) 0 0
\(221\) −311036. −0.428381
\(222\) 0 0
\(223\) −904520. −1.21802 −0.609012 0.793161i \(-0.708434\pi\)
−0.609012 + 0.793161i \(0.708434\pi\)
\(224\) 0 0
\(225\) −77185.4 −0.101643
\(226\) 0 0
\(227\) −10704.5 −0.0137881 −0.00689403 0.999976i \(-0.502194\pi\)
−0.00689403 + 0.999976i \(0.502194\pi\)
\(228\) 0 0
\(229\) −536171. −0.675639 −0.337819 0.941211i \(-0.609689\pi\)
−0.337819 + 0.941211i \(0.609689\pi\)
\(230\) 0 0
\(231\) −66253.2 −0.0816915
\(232\) 0 0
\(233\) 124720. 0.150504 0.0752519 0.997165i \(-0.476024\pi\)
0.0752519 + 0.997165i \(0.476024\pi\)
\(234\) 0 0
\(235\) −226029. −0.266990
\(236\) 0 0
\(237\) −79879.1 −0.0923767
\(238\) 0 0
\(239\) −580116. −0.656931 −0.328466 0.944516i \(-0.606532\pi\)
−0.328466 + 0.944516i \(0.606532\pi\)
\(240\) 0 0
\(241\) −381173. −0.422746 −0.211373 0.977406i \(-0.567793\pi\)
−0.211373 + 0.977406i \(0.567793\pi\)
\(242\) 0 0
\(243\) −59049.0 −0.0641500
\(244\) 0 0
\(245\) −623022. −0.663114
\(246\) 0 0
\(247\) 126741. 0.132183
\(248\) 0 0
\(249\) 556960. 0.569280
\(250\) 0 0
\(251\) 710860. 0.712197 0.356098 0.934448i \(-0.384107\pi\)
0.356098 + 0.934448i \(0.384107\pi\)
\(252\) 0 0
\(253\) −117616. −0.115522
\(254\) 0 0
\(255\) 371606. 0.357876
\(256\) 0 0
\(257\) 143727. 0.135739 0.0678695 0.997694i \(-0.478380\pi\)
0.0678695 + 0.997694i \(0.478380\pi\)
\(258\) 0 0
\(259\) −624734. −0.578690
\(260\) 0 0
\(261\) 252510. 0.229444
\(262\) 0 0
\(263\) 20933.3 0.0186616 0.00933079 0.999956i \(-0.497030\pi\)
0.00933079 + 0.999956i \(0.497030\pi\)
\(264\) 0 0
\(265\) −1.70632e6 −1.49261
\(266\) 0 0
\(267\) 233417. 0.200380
\(268\) 0 0
\(269\) 899788. 0.758158 0.379079 0.925364i \(-0.376241\pi\)
0.379079 + 0.925364i \(0.376241\pi\)
\(270\) 0 0
\(271\) −1.38128e6 −1.14251 −0.571255 0.820773i \(-0.693543\pi\)
−0.571255 + 0.820773i \(0.693543\pi\)
\(272\) 0 0
\(273\) 185299. 0.150476
\(274\) 0 0
\(275\) 119617. 0.0953809
\(276\) 0 0
\(277\) 1.93050e6 1.51172 0.755858 0.654735i \(-0.227220\pi\)
0.755858 + 0.654735i \(0.227220\pi\)
\(278\) 0 0
\(279\) −855228. −0.657766
\(280\) 0 0
\(281\) 1.99112e6 1.50429 0.752146 0.658996i \(-0.229019\pi\)
0.752146 + 0.658996i \(0.229019\pi\)
\(282\) 0 0
\(283\) 240309. 0.178363 0.0891813 0.996015i \(-0.471575\pi\)
0.0891813 + 0.996015i \(0.471575\pi\)
\(284\) 0 0
\(285\) −151422. −0.110427
\(286\) 0 0
\(287\) −659607. −0.472695
\(288\) 0 0
\(289\) −634980. −0.447214
\(290\) 0 0
\(291\) 795592. 0.550754
\(292\) 0 0
\(293\) −1.75213e6 −1.19234 −0.596168 0.802860i \(-0.703311\pi\)
−0.596168 + 0.802860i \(0.703311\pi\)
\(294\) 0 0
\(295\) −369160. −0.246979
\(296\) 0 0
\(297\) 91510.4 0.0601976
\(298\) 0 0
\(299\) 328951. 0.212791
\(300\) 0 0
\(301\) −342815. −0.218094
\(302\) 0 0
\(303\) 881252. 0.551434
\(304\) 0 0
\(305\) 2.30616e6 1.41951
\(306\) 0 0
\(307\) 2.56334e6 1.55224 0.776122 0.630583i \(-0.217184\pi\)
0.776122 + 0.630583i \(0.217184\pi\)
\(308\) 0 0
\(309\) 1.42878e6 0.851275
\(310\) 0 0
\(311\) −1.22333e6 −0.717203 −0.358601 0.933491i \(-0.616746\pi\)
−0.358601 + 0.933491i \(0.616746\pi\)
\(312\) 0 0
\(313\) 2.36852e6 1.36652 0.683262 0.730174i \(-0.260561\pi\)
0.683262 + 0.730174i \(0.260561\pi\)
\(314\) 0 0
\(315\) −221384. −0.125710
\(316\) 0 0
\(317\) 2.41136e6 1.34776 0.673882 0.738839i \(-0.264626\pi\)
0.673882 + 0.738839i \(0.264626\pi\)
\(318\) 0 0
\(319\) −391324. −0.215308
\(320\) 0 0
\(321\) 1.33053e6 0.720713
\(322\) 0 0
\(323\) −319822. −0.170570
\(324\) 0 0
\(325\) −334549. −0.175692
\(326\) 0 0
\(327\) 734208. 0.379708
\(328\) 0 0
\(329\) 284411. 0.144863
\(330\) 0 0
\(331\) −2.96614e6 −1.48806 −0.744032 0.668144i \(-0.767089\pi\)
−0.744032 + 0.668144i \(0.767089\pi\)
\(332\) 0 0
\(333\) 862897. 0.426431
\(334\) 0 0
\(335\) 705533. 0.343483
\(336\) 0 0
\(337\) 1.06940e6 0.512937 0.256469 0.966553i \(-0.417441\pi\)
0.256469 + 0.966553i \(0.417441\pi\)
\(338\) 0 0
\(339\) −264577. −0.125041
\(340\) 0 0
\(341\) 1.32538e6 0.617240
\(342\) 0 0
\(343\) 1.76957e6 0.812142
\(344\) 0 0
\(345\) −393010. −0.177769
\(346\) 0 0
\(347\) −146634. −0.0653748 −0.0326874 0.999466i \(-0.510407\pi\)
−0.0326874 + 0.999466i \(0.510407\pi\)
\(348\) 0 0
\(349\) 398157. 0.174981 0.0874905 0.996165i \(-0.472115\pi\)
0.0874905 + 0.996165i \(0.472115\pi\)
\(350\) 0 0
\(351\) −255939. −0.110884
\(352\) 0 0
\(353\) −3.57018e6 −1.52494 −0.762470 0.647023i \(-0.776014\pi\)
−0.762470 + 0.647023i \(0.776014\pi\)
\(354\) 0 0
\(355\) −1.82069e6 −0.766768
\(356\) 0 0
\(357\) −467589. −0.194175
\(358\) 0 0
\(359\) 2.51582e6 1.03025 0.515125 0.857115i \(-0.327745\pi\)
0.515125 + 0.857115i \(0.327745\pi\)
\(360\) 0 0
\(361\) 130321. 0.0526316
\(362\) 0 0
\(363\) 1.30764e6 0.520862
\(364\) 0 0
\(365\) 1.71860e6 0.675216
\(366\) 0 0
\(367\) −4.82115e6 −1.86847 −0.934233 0.356663i \(-0.883914\pi\)
−0.934233 + 0.356663i \(0.883914\pi\)
\(368\) 0 0
\(369\) 911065. 0.348324
\(370\) 0 0
\(371\) 2.14705e6 0.809856
\(372\) 0 0
\(373\) −1.29172e6 −0.480723 −0.240362 0.970683i \(-0.577266\pi\)
−0.240362 + 0.970683i \(0.577266\pi\)
\(374\) 0 0
\(375\) 1.71048e6 0.628117
\(376\) 0 0
\(377\) 1.09447e6 0.396597
\(378\) 0 0
\(379\) 2.36358e6 0.845225 0.422613 0.906310i \(-0.361113\pi\)
0.422613 + 0.906310i \(0.361113\pi\)
\(380\) 0 0
\(381\) 1.03630e6 0.365741
\(382\) 0 0
\(383\) 3.86944e6 1.34788 0.673939 0.738787i \(-0.264601\pi\)
0.673939 + 0.738787i \(0.264601\pi\)
\(384\) 0 0
\(385\) 343086. 0.117965
\(386\) 0 0
\(387\) 473505. 0.160711
\(388\) 0 0
\(389\) −3.20371e6 −1.07344 −0.536722 0.843759i \(-0.680338\pi\)
−0.536722 + 0.843759i \(0.680338\pi\)
\(390\) 0 0
\(391\) −830085. −0.274588
\(392\) 0 0
\(393\) 542743. 0.177261
\(394\) 0 0
\(395\) 413647. 0.133394
\(396\) 0 0
\(397\) 861302. 0.274270 0.137135 0.990552i \(-0.456211\pi\)
0.137135 + 0.990552i \(0.456211\pi\)
\(398\) 0 0
\(399\) 190533. 0.0599154
\(400\) 0 0
\(401\) 5.75486e6 1.78720 0.893601 0.448862i \(-0.148171\pi\)
0.893601 + 0.448862i \(0.148171\pi\)
\(402\) 0 0
\(403\) −3.70687e6 −1.13696
\(404\) 0 0
\(405\) 305780. 0.0926343
\(406\) 0 0
\(407\) −1.33726e6 −0.400158
\(408\) 0 0
\(409\) 3.25999e6 0.963625 0.481813 0.876274i \(-0.339979\pi\)
0.481813 + 0.876274i \(0.339979\pi\)
\(410\) 0 0
\(411\) 2.08116e6 0.607715
\(412\) 0 0
\(413\) 464511. 0.134005
\(414\) 0 0
\(415\) −2.88417e6 −0.822055
\(416\) 0 0
\(417\) 452394. 0.127402
\(418\) 0 0
\(419\) −1.10038e6 −0.306203 −0.153101 0.988210i \(-0.548926\pi\)
−0.153101 + 0.988210i \(0.548926\pi\)
\(420\) 0 0
\(421\) −4.45235e6 −1.22429 −0.612145 0.790746i \(-0.709693\pi\)
−0.612145 + 0.790746i \(0.709693\pi\)
\(422\) 0 0
\(423\) −392835. −0.106748
\(424\) 0 0
\(425\) 844211. 0.226714
\(426\) 0 0
\(427\) −2.90182e6 −0.770196
\(428\) 0 0
\(429\) 396639. 0.104052
\(430\) 0 0
\(431\) 2.46438e6 0.639020 0.319510 0.947583i \(-0.396482\pi\)
0.319510 + 0.947583i \(0.396482\pi\)
\(432\) 0 0
\(433\) 6.18726e6 1.58591 0.792955 0.609280i \(-0.208542\pi\)
0.792955 + 0.609280i \(0.208542\pi\)
\(434\) 0 0
\(435\) −1.30760e6 −0.331323
\(436\) 0 0
\(437\) 338243. 0.0847277
\(438\) 0 0
\(439\) 5.72940e6 1.41889 0.709443 0.704762i \(-0.248946\pi\)
0.709443 + 0.704762i \(0.248946\pi\)
\(440\) 0 0
\(441\) −1.08280e6 −0.265126
\(442\) 0 0
\(443\) −820524. −0.198647 −0.0993234 0.995055i \(-0.531668\pi\)
−0.0993234 + 0.995055i \(0.531668\pi\)
\(444\) 0 0
\(445\) −1.20873e6 −0.289354
\(446\) 0 0
\(447\) −3.18554e6 −0.754075
\(448\) 0 0
\(449\) −1.12512e6 −0.263380 −0.131690 0.991291i \(-0.542040\pi\)
−0.131690 + 0.991291i \(0.542040\pi\)
\(450\) 0 0
\(451\) −1.41191e6 −0.326863
\(452\) 0 0
\(453\) 3.03404e6 0.694667
\(454\) 0 0
\(455\) −959555. −0.217291
\(456\) 0 0
\(457\) −1.66587e6 −0.373121 −0.186561 0.982443i \(-0.559734\pi\)
−0.186561 + 0.982443i \(0.559734\pi\)
\(458\) 0 0
\(459\) 645845. 0.143086
\(460\) 0 0
\(461\) −3.75543e6 −0.823014 −0.411507 0.911407i \(-0.634997\pi\)
−0.411507 + 0.911407i \(0.634997\pi\)
\(462\) 0 0
\(463\) −641032. −0.138972 −0.0694860 0.997583i \(-0.522136\pi\)
−0.0694860 + 0.997583i \(0.522136\pi\)
\(464\) 0 0
\(465\) 4.42873e6 0.949831
\(466\) 0 0
\(467\) −2.90723e6 −0.616862 −0.308431 0.951247i \(-0.599804\pi\)
−0.308431 + 0.951247i \(0.599804\pi\)
\(468\) 0 0
\(469\) −887767. −0.186366
\(470\) 0 0
\(471\) 4.45158e6 0.924616
\(472\) 0 0
\(473\) −733808. −0.150810
\(474\) 0 0
\(475\) −343999. −0.0699557
\(476\) 0 0
\(477\) −2.96556e6 −0.596775
\(478\) 0 0
\(479\) 8.55294e6 1.70324 0.851622 0.524157i \(-0.175619\pi\)
0.851622 + 0.524157i \(0.175619\pi\)
\(480\) 0 0
\(481\) 3.74010e6 0.737091
\(482\) 0 0
\(483\) 494522. 0.0964534
\(484\) 0 0
\(485\) −4.11990e6 −0.795303
\(486\) 0 0
\(487\) −6.51187e6 −1.24418 −0.622090 0.782945i \(-0.713716\pi\)
−0.622090 + 0.782945i \(0.713716\pi\)
\(488\) 0 0
\(489\) 1.05314e6 0.199166
\(490\) 0 0
\(491\) −1.47134e6 −0.275429 −0.137714 0.990472i \(-0.543976\pi\)
−0.137714 + 0.990472i \(0.543976\pi\)
\(492\) 0 0
\(493\) −2.76181e6 −0.511772
\(494\) 0 0
\(495\) −473879. −0.0869269
\(496\) 0 0
\(497\) 2.29096e6 0.416031
\(498\) 0 0
\(499\) −8.75236e6 −1.57352 −0.786762 0.617256i \(-0.788244\pi\)
−0.786762 + 0.617256i \(0.788244\pi\)
\(500\) 0 0
\(501\) 2.70661e6 0.481761
\(502\) 0 0
\(503\) 9.02461e6 1.59041 0.795204 0.606342i \(-0.207364\pi\)
0.795204 + 0.606342i \(0.207364\pi\)
\(504\) 0 0
\(505\) −4.56349e6 −0.796285
\(506\) 0 0
\(507\) 2.23230e6 0.385686
\(508\) 0 0
\(509\) 890437. 0.152338 0.0761691 0.997095i \(-0.475731\pi\)
0.0761691 + 0.997095i \(0.475731\pi\)
\(510\) 0 0
\(511\) −2.16250e6 −0.366357
\(512\) 0 0
\(513\) −263169. −0.0441511
\(514\) 0 0
\(515\) −7.39882e6 −1.22926
\(516\) 0 0
\(517\) 608791. 0.100171
\(518\) 0 0
\(519\) −4.21758e6 −0.687298
\(520\) 0 0
\(521\) −3.15267e6 −0.508844 −0.254422 0.967093i \(-0.581885\pi\)
−0.254422 + 0.967093i \(0.581885\pi\)
\(522\) 0 0
\(523\) −4.92318e6 −0.787030 −0.393515 0.919318i \(-0.628741\pi\)
−0.393515 + 0.919318i \(0.628741\pi\)
\(524\) 0 0
\(525\) −502937. −0.0796371
\(526\) 0 0
\(527\) 9.35401e6 1.46714
\(528\) 0 0
\(529\) −5.55845e6 −0.863603
\(530\) 0 0
\(531\) −641594. −0.0987470
\(532\) 0 0
\(533\) 3.94888e6 0.602083
\(534\) 0 0
\(535\) −6.89004e6 −1.04073
\(536\) 0 0
\(537\) −6.25699e6 −0.936331
\(538\) 0 0
\(539\) 1.67806e6 0.248791
\(540\) 0 0
\(541\) −7.78529e6 −1.14362 −0.571810 0.820386i \(-0.693758\pi\)
−0.571810 + 0.820386i \(0.693758\pi\)
\(542\) 0 0
\(543\) 5.45705e6 0.794252
\(544\) 0 0
\(545\) −3.80204e6 −0.548308
\(546\) 0 0
\(547\) 2.10139e6 0.300289 0.150144 0.988664i \(-0.452026\pi\)
0.150144 + 0.988664i \(0.452026\pi\)
\(548\) 0 0
\(549\) 4.00807e6 0.567550
\(550\) 0 0
\(551\) 1.12538e6 0.157914
\(552\) 0 0
\(553\) −520489. −0.0723768
\(554\) 0 0
\(555\) −4.46844e6 −0.615777
\(556\) 0 0
\(557\) −2.52411e6 −0.344722 −0.172361 0.985034i \(-0.555140\pi\)
−0.172361 + 0.985034i \(0.555140\pi\)
\(558\) 0 0
\(559\) 2.05234e6 0.277792
\(560\) 0 0
\(561\) −1.00089e6 −0.134270
\(562\) 0 0
\(563\) 7.21244e6 0.958984 0.479492 0.877546i \(-0.340821\pi\)
0.479492 + 0.877546i \(0.340821\pi\)
\(564\) 0 0
\(565\) 1.37009e6 0.180562
\(566\) 0 0
\(567\) −384761. −0.0502613
\(568\) 0 0
\(569\) −5.98706e6 −0.775234 −0.387617 0.921821i \(-0.626702\pi\)
−0.387617 + 0.921821i \(0.626702\pi\)
\(570\) 0 0
\(571\) −3.41302e6 −0.438075 −0.219037 0.975716i \(-0.570292\pi\)
−0.219037 + 0.975716i \(0.570292\pi\)
\(572\) 0 0
\(573\) −757325. −0.0963598
\(574\) 0 0
\(575\) −892837. −0.112617
\(576\) 0 0
\(577\) 956178. 0.119564 0.0597818 0.998211i \(-0.480960\pi\)
0.0597818 + 0.998211i \(0.480960\pi\)
\(578\) 0 0
\(579\) −6.82128e6 −0.845609
\(580\) 0 0
\(581\) 3.62913e6 0.446028
\(582\) 0 0
\(583\) 4.59584e6 0.560007
\(584\) 0 0
\(585\) 1.32536e6 0.160120
\(586\) 0 0
\(587\) 5.95696e6 0.713559 0.356779 0.934189i \(-0.383875\pi\)
0.356779 + 0.934189i \(0.383875\pi\)
\(588\) 0 0
\(589\) −3.81157e6 −0.452706
\(590\) 0 0
\(591\) −3.17745e6 −0.374206
\(592\) 0 0
\(593\) −1.17783e7 −1.37545 −0.687725 0.725971i \(-0.741391\pi\)
−0.687725 + 0.725971i \(0.741391\pi\)
\(594\) 0 0
\(595\) 2.42137e6 0.280394
\(596\) 0 0
\(597\) −818711. −0.0940145
\(598\) 0 0
\(599\) 131291. 0.0149509 0.00747544 0.999972i \(-0.497620\pi\)
0.00747544 + 0.999972i \(0.497620\pi\)
\(600\) 0 0
\(601\) −5.73312e6 −0.647448 −0.323724 0.946152i \(-0.604935\pi\)
−0.323724 + 0.946152i \(0.604935\pi\)
\(602\) 0 0
\(603\) 1.22620e6 0.137331
\(604\) 0 0
\(605\) −6.77151e6 −0.752137
\(606\) 0 0
\(607\) 4.70647e6 0.518470 0.259235 0.965814i \(-0.416530\pi\)
0.259235 + 0.965814i \(0.416530\pi\)
\(608\) 0 0
\(609\) 1.64534e6 0.179769
\(610\) 0 0
\(611\) −1.70269e6 −0.184515
\(612\) 0 0
\(613\) −5.56498e6 −0.598153 −0.299077 0.954229i \(-0.596679\pi\)
−0.299077 + 0.954229i \(0.596679\pi\)
\(614\) 0 0
\(615\) −4.71787e6 −0.502989
\(616\) 0 0
\(617\) −8.82742e6 −0.933514 −0.466757 0.884386i \(-0.654578\pi\)
−0.466757 + 0.884386i \(0.654578\pi\)
\(618\) 0 0
\(619\) −1.31806e7 −1.38264 −0.691319 0.722550i \(-0.742970\pi\)
−0.691319 + 0.722550i \(0.742970\pi\)
\(620\) 0 0
\(621\) −683045. −0.0710756
\(622\) 0 0
\(623\) 1.52094e6 0.156997
\(624\) 0 0
\(625\) −5.87976e6 −0.602088
\(626\) 0 0
\(627\) 407843. 0.0414309
\(628\) 0 0
\(629\) −9.43789e6 −0.951148
\(630\) 0 0
\(631\) 1.16412e7 1.16392 0.581959 0.813218i \(-0.302286\pi\)
0.581959 + 0.813218i \(0.302286\pi\)
\(632\) 0 0
\(633\) 5.51063e6 0.546628
\(634\) 0 0
\(635\) −5.36640e6 −0.528139
\(636\) 0 0
\(637\) −4.69325e6 −0.458274
\(638\) 0 0
\(639\) −3.16432e6 −0.306569
\(640\) 0 0
\(641\) −9.10465e6 −0.875222 −0.437611 0.899164i \(-0.644175\pi\)
−0.437611 + 0.899164i \(0.644175\pi\)
\(642\) 0 0
\(643\) −1.08215e7 −1.03219 −0.516096 0.856531i \(-0.672615\pi\)
−0.516096 + 0.856531i \(0.672615\pi\)
\(644\) 0 0
\(645\) −2.45200e6 −0.232071
\(646\) 0 0
\(647\) 1.42375e7 1.33713 0.668564 0.743655i \(-0.266909\pi\)
0.668564 + 0.743655i \(0.266909\pi\)
\(648\) 0 0
\(649\) 994302. 0.0926631
\(650\) 0 0
\(651\) −5.57264e6 −0.515357
\(652\) 0 0
\(653\) −919718. −0.0844057 −0.0422029 0.999109i \(-0.513438\pi\)
−0.0422029 + 0.999109i \(0.513438\pi\)
\(654\) 0 0
\(655\) −2.81055e6 −0.255969
\(656\) 0 0
\(657\) 2.98690e6 0.269965
\(658\) 0 0
\(659\) −2.87830e6 −0.258180 −0.129090 0.991633i \(-0.541206\pi\)
−0.129090 + 0.991633i \(0.541206\pi\)
\(660\) 0 0
\(661\) 477439. 0.0425024 0.0212512 0.999774i \(-0.493235\pi\)
0.0212512 + 0.999774i \(0.493235\pi\)
\(662\) 0 0
\(663\) 2.79932e6 0.247326
\(664\) 0 0
\(665\) −986660. −0.0865194
\(666\) 0 0
\(667\) 2.92089e6 0.254215
\(668\) 0 0
\(669\) 8.14068e6 0.703227
\(670\) 0 0
\(671\) −6.21145e6 −0.532582
\(672\) 0 0
\(673\) −5.74481e6 −0.488920 −0.244460 0.969659i \(-0.578611\pi\)
−0.244460 + 0.969659i \(0.578611\pi\)
\(674\) 0 0
\(675\) 694669. 0.0586838
\(676\) 0 0
\(677\) 1.47442e7 1.23637 0.618187 0.786031i \(-0.287867\pi\)
0.618187 + 0.786031i \(0.287867\pi\)
\(678\) 0 0
\(679\) 5.18404e6 0.431513
\(680\) 0 0
\(681\) 96340.9 0.00796054
\(682\) 0 0
\(683\) 1.47287e7 1.20813 0.604065 0.796935i \(-0.293547\pi\)
0.604065 + 0.796935i \(0.293547\pi\)
\(684\) 0 0
\(685\) −1.07771e7 −0.877556
\(686\) 0 0
\(687\) 4.82554e6 0.390080
\(688\) 0 0
\(689\) −1.28538e7 −1.03153
\(690\) 0 0
\(691\) 1.51588e7 1.20773 0.603863 0.797088i \(-0.293627\pi\)
0.603863 + 0.797088i \(0.293627\pi\)
\(692\) 0 0
\(693\) 596279. 0.0471646
\(694\) 0 0
\(695\) −2.34268e6 −0.183972
\(696\) 0 0
\(697\) −9.96472e6 −0.776933
\(698\) 0 0
\(699\) −1.12248e6 −0.0868934
\(700\) 0 0
\(701\) −1.08937e7 −0.837295 −0.418647 0.908149i \(-0.637496\pi\)
−0.418647 + 0.908149i \(0.637496\pi\)
\(702\) 0 0
\(703\) 3.84575e6 0.293490
\(704\) 0 0
\(705\) 2.03426e6 0.154147
\(706\) 0 0
\(707\) 5.74221e6 0.432046
\(708\) 0 0
\(709\) −1.52402e7 −1.13861 −0.569304 0.822127i \(-0.692787\pi\)
−0.569304 + 0.822127i \(0.692787\pi\)
\(710\) 0 0
\(711\) 718912. 0.0533337
\(712\) 0 0
\(713\) −9.89280e6 −0.728778
\(714\) 0 0
\(715\) −2.05396e6 −0.150254
\(716\) 0 0
\(717\) 5.22104e6 0.379279
\(718\) 0 0
\(719\) 1.55304e7 1.12037 0.560183 0.828369i \(-0.310731\pi\)
0.560183 + 0.828369i \(0.310731\pi\)
\(720\) 0 0
\(721\) 9.30989e6 0.666970
\(722\) 0 0
\(723\) 3.43055e6 0.244072
\(724\) 0 0
\(725\) −2.97059e6 −0.209893
\(726\) 0 0
\(727\) 4.95054e6 0.347389 0.173695 0.984800i \(-0.444429\pi\)
0.173695 + 0.984800i \(0.444429\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −5.17893e6 −0.358465
\(732\) 0 0
\(733\) −7.63011e6 −0.524531 −0.262265 0.964996i \(-0.584470\pi\)
−0.262265 + 0.964996i \(0.584470\pi\)
\(734\) 0 0
\(735\) 5.60720e6 0.382849
\(736\) 0 0
\(737\) −1.90029e6 −0.128870
\(738\) 0 0
\(739\) −7.81572e6 −0.526451 −0.263226 0.964734i \(-0.584786\pi\)
−0.263226 + 0.964734i \(0.584786\pi\)
\(740\) 0 0
\(741\) −1.14067e6 −0.0763157
\(742\) 0 0
\(743\) 8.78448e6 0.583773 0.291887 0.956453i \(-0.405717\pi\)
0.291887 + 0.956453i \(0.405717\pi\)
\(744\) 0 0
\(745\) 1.64960e7 1.08890
\(746\) 0 0
\(747\) −5.01264e6 −0.328674
\(748\) 0 0
\(749\) 8.66969e6 0.564676
\(750\) 0 0
\(751\) 1.84385e7 1.19296 0.596479 0.802628i \(-0.296566\pi\)
0.596479 + 0.802628i \(0.296566\pi\)
\(752\) 0 0
\(753\) −6.39774e6 −0.411187
\(754\) 0 0
\(755\) −1.57115e7 −1.00312
\(756\) 0 0
\(757\) 2.01886e7 1.28046 0.640232 0.768181i \(-0.278838\pi\)
0.640232 + 0.768181i \(0.278838\pi\)
\(758\) 0 0
\(759\) 1.05854e6 0.0666965
\(760\) 0 0
\(761\) 5.59816e6 0.350416 0.175208 0.984531i \(-0.443940\pi\)
0.175208 + 0.984531i \(0.443940\pi\)
\(762\) 0 0
\(763\) 4.78407e6 0.297500
\(764\) 0 0
\(765\) −3.34445e6 −0.206620
\(766\) 0 0
\(767\) −2.78090e6 −0.170686
\(768\) 0 0
\(769\) −3.22353e6 −0.196569 −0.0982846 0.995158i \(-0.531336\pi\)
−0.0982846 + 0.995158i \(0.531336\pi\)
\(770\) 0 0
\(771\) −1.29354e6 −0.0783689
\(772\) 0 0
\(773\) −3.00281e6 −0.180750 −0.0903752 0.995908i \(-0.528807\pi\)
−0.0903752 + 0.995908i \(0.528807\pi\)
\(774\) 0 0
\(775\) 1.00611e7 0.601718
\(776\) 0 0
\(777\) 5.62261e6 0.334107
\(778\) 0 0
\(779\) 4.06043e6 0.239733
\(780\) 0 0
\(781\) 4.90387e6 0.287681
\(782\) 0 0
\(783\) −2.27259e6 −0.132470
\(784\) 0 0
\(785\) −2.30521e7 −1.33517
\(786\) 0 0
\(787\) 9.05219e6 0.520975 0.260487 0.965477i \(-0.416117\pi\)
0.260487 + 0.965477i \(0.416117\pi\)
\(788\) 0 0
\(789\) −188400. −0.0107743
\(790\) 0 0
\(791\) −1.72397e6 −0.0979690
\(792\) 0 0
\(793\) 1.73724e7 0.981017
\(794\) 0 0
\(795\) 1.53569e7 0.861759
\(796\) 0 0
\(797\) 77865.6 0.00434210 0.00217105 0.999998i \(-0.499309\pi\)
0.00217105 + 0.999998i \(0.499309\pi\)
\(798\) 0 0
\(799\) 4.29661e6 0.238100
\(800\) 0 0
\(801\) −2.10075e6 −0.115689
\(802\) 0 0
\(803\) −4.62891e6 −0.253332
\(804\) 0 0
\(805\) −2.56084e6 −0.139281
\(806\) 0 0
\(807\) −8.09809e6 −0.437722
\(808\) 0 0
\(809\) 2.45959e6 0.132127 0.0660634 0.997815i \(-0.478956\pi\)
0.0660634 + 0.997815i \(0.478956\pi\)
\(810\) 0 0
\(811\) 1.57822e7 0.842587 0.421294 0.906924i \(-0.361576\pi\)
0.421294 + 0.906924i \(0.361576\pi\)
\(812\) 0 0
\(813\) 1.24316e7 0.659628
\(814\) 0 0
\(815\) −5.45362e6 −0.287601
\(816\) 0 0
\(817\) 2.11031e6 0.110609
\(818\) 0 0
\(819\) −1.66769e6 −0.0868772
\(820\) 0 0
\(821\) 1.10300e7 0.571106 0.285553 0.958363i \(-0.407823\pi\)
0.285553 + 0.958363i \(0.407823\pi\)
\(822\) 0 0
\(823\) −1.90977e7 −0.982839 −0.491420 0.870923i \(-0.663522\pi\)
−0.491420 + 0.870923i \(0.663522\pi\)
\(824\) 0 0
\(825\) −1.07655e6 −0.0550682
\(826\) 0 0
\(827\) 2.63862e7 1.34157 0.670784 0.741652i \(-0.265958\pi\)
0.670784 + 0.741652i \(0.265958\pi\)
\(828\) 0 0
\(829\) 1.35815e7 0.686374 0.343187 0.939267i \(-0.388494\pi\)
0.343187 + 0.939267i \(0.388494\pi\)
\(830\) 0 0
\(831\) −1.73745e7 −0.872790
\(832\) 0 0
\(833\) 1.18431e7 0.591360
\(834\) 0 0
\(835\) −1.40160e7 −0.695675
\(836\) 0 0
\(837\) 7.69706e6 0.379762
\(838\) 0 0
\(839\) 4.73494e6 0.232225 0.116113 0.993236i \(-0.462957\pi\)
0.116113 + 0.993236i \(0.462957\pi\)
\(840\) 0 0
\(841\) −1.07929e7 −0.526198
\(842\) 0 0
\(843\) −1.79201e7 −0.868503
\(844\) 0 0
\(845\) −1.15598e7 −0.556940
\(846\) 0 0
\(847\) 8.52055e6 0.408093
\(848\) 0 0
\(849\) −2.16278e6 −0.102978
\(850\) 0 0
\(851\) 9.98150e6 0.472468
\(852\) 0 0
\(853\) 7.94247e6 0.373751 0.186876 0.982384i \(-0.440164\pi\)
0.186876 + 0.982384i \(0.440164\pi\)
\(854\) 0 0
\(855\) 1.36280e6 0.0637553
\(856\) 0 0
\(857\) −1.50658e7 −0.700713 −0.350356 0.936616i \(-0.613940\pi\)
−0.350356 + 0.936616i \(0.613940\pi\)
\(858\) 0 0
\(859\) −2.28464e7 −1.05641 −0.528207 0.849116i \(-0.677135\pi\)
−0.528207 + 0.849116i \(0.677135\pi\)
\(860\) 0 0
\(861\) 5.93647e6 0.272910
\(862\) 0 0
\(863\) 7.03849e6 0.321701 0.160851 0.986979i \(-0.448576\pi\)
0.160851 + 0.986979i \(0.448576\pi\)
\(864\) 0 0
\(865\) 2.18404e7 0.992475
\(866\) 0 0
\(867\) 5.71482e6 0.258199
\(868\) 0 0
\(869\) −1.11412e6 −0.0500477
\(870\) 0 0
\(871\) 5.31481e6 0.237379
\(872\) 0 0
\(873\) −7.16032e6 −0.317978
\(874\) 0 0
\(875\) 1.11455e7 0.492127
\(876\) 0 0
\(877\) 2.02855e7 0.890608 0.445304 0.895379i \(-0.353096\pi\)
0.445304 + 0.895379i \(0.353096\pi\)
\(878\) 0 0
\(879\) 1.57692e7 0.688395
\(880\) 0 0
\(881\) 3.70861e7 1.60980 0.804898 0.593413i \(-0.202220\pi\)
0.804898 + 0.593413i \(0.202220\pi\)
\(882\) 0 0
\(883\) 1.26049e7 0.544048 0.272024 0.962290i \(-0.412307\pi\)
0.272024 + 0.962290i \(0.412307\pi\)
\(884\) 0 0
\(885\) 3.32244e6 0.142593
\(886\) 0 0
\(887\) 3.69844e7 1.57837 0.789187 0.614153i \(-0.210502\pi\)
0.789187 + 0.614153i \(0.210502\pi\)
\(888\) 0 0
\(889\) 6.75250e6 0.286556
\(890\) 0 0
\(891\) −823594. −0.0347551
\(892\) 0 0
\(893\) −1.75078e6 −0.0734689
\(894\) 0 0
\(895\) 3.24013e7 1.35209
\(896\) 0 0
\(897\) −2.96056e6 −0.122855
\(898\) 0 0
\(899\) −3.29147e7 −1.35829
\(900\) 0 0
\(901\) 3.24356e7 1.33110
\(902\) 0 0
\(903\) 3.08534e6 0.125917
\(904\) 0 0
\(905\) −2.82589e7 −1.14692
\(906\) 0 0
\(907\) −1.95026e6 −0.0787180 −0.0393590 0.999225i \(-0.512532\pi\)
−0.0393590 + 0.999225i \(0.512532\pi\)
\(908\) 0 0
\(909\) −7.93127e6 −0.318371
\(910\) 0 0
\(911\) −1.72654e7 −0.689258 −0.344629 0.938739i \(-0.611995\pi\)
−0.344629 + 0.938739i \(0.611995\pi\)
\(912\) 0 0
\(913\) 7.76828e6 0.308424
\(914\) 0 0
\(915\) −2.07554e7 −0.819556
\(916\) 0 0
\(917\) 3.53649e6 0.138883
\(918\) 0 0
\(919\) −1.36353e7 −0.532570 −0.266285 0.963894i \(-0.585796\pi\)
−0.266285 + 0.963894i \(0.585796\pi\)
\(920\) 0 0
\(921\) −2.30700e7 −0.896188
\(922\) 0 0
\(923\) −1.37153e7 −0.529909
\(924\) 0 0
\(925\) −1.01514e7 −0.390094
\(926\) 0 0
\(927\) −1.28590e7 −0.491484
\(928\) 0 0
\(929\) −1.31495e7 −0.499885 −0.249943 0.968261i \(-0.580412\pi\)
−0.249943 + 0.968261i \(0.580412\pi\)
\(930\) 0 0
\(931\) −4.82582e6 −0.182472
\(932\) 0 0
\(933\) 1.10100e7 0.414077
\(934\) 0 0
\(935\) 5.18302e6 0.193889
\(936\) 0 0
\(937\) −1.19341e7 −0.444058 −0.222029 0.975040i \(-0.571268\pi\)
−0.222029 + 0.975040i \(0.571268\pi\)
\(938\) 0 0
\(939\) −2.13167e7 −0.788963
\(940\) 0 0
\(941\) −4.51597e7 −1.66256 −0.831279 0.555856i \(-0.812391\pi\)
−0.831279 + 0.555856i \(0.812391\pi\)
\(942\) 0 0
\(943\) 1.05387e7 0.385929
\(944\) 0 0
\(945\) 1.99245e6 0.0725786
\(946\) 0 0
\(947\) −4.17275e7 −1.51198 −0.755992 0.654581i \(-0.772845\pi\)
−0.755992 + 0.654581i \(0.772845\pi\)
\(948\) 0 0
\(949\) 1.29463e7 0.466638
\(950\) 0 0
\(951\) −2.17022e7 −0.778132
\(952\) 0 0
\(953\) −4.35052e7 −1.55170 −0.775851 0.630916i \(-0.782679\pi\)
−0.775851 + 0.630916i \(0.782679\pi\)
\(954\) 0 0
\(955\) 3.92174e6 0.139146
\(956\) 0 0
\(957\) 3.52192e6 0.124308
\(958\) 0 0
\(959\) 1.35607e7 0.476142
\(960\) 0 0
\(961\) 8.28501e7 2.89391
\(962\) 0 0
\(963\) −1.19748e7 −0.416104
\(964\) 0 0
\(965\) 3.53234e7 1.22108
\(966\) 0 0
\(967\) −164114. −0.00564391 −0.00282195 0.999996i \(-0.500898\pi\)
−0.00282195 + 0.999996i \(0.500898\pi\)
\(968\) 0 0
\(969\) 2.87840e6 0.0984784
\(970\) 0 0
\(971\) 3.59222e7 1.22268 0.611342 0.791366i \(-0.290630\pi\)
0.611342 + 0.791366i \(0.290630\pi\)
\(972\) 0 0
\(973\) 2.94778e6 0.0998190
\(974\) 0 0
\(975\) 3.01094e6 0.101436
\(976\) 0 0
\(977\) 1.59785e7 0.535550 0.267775 0.963481i \(-0.413712\pi\)
0.267775 + 0.963481i \(0.413712\pi\)
\(978\) 0 0
\(979\) 3.25561e6 0.108562
\(980\) 0 0
\(981\) −6.60788e6 −0.219225
\(982\) 0 0
\(983\) 5.38598e7 1.77779 0.888896 0.458109i \(-0.151473\pi\)
0.888896 + 0.458109i \(0.151473\pi\)
\(984\) 0 0
\(985\) 1.64542e7 0.540363
\(986\) 0 0
\(987\) −2.55970e6 −0.0836365
\(988\) 0 0
\(989\) 5.47723e6 0.178062
\(990\) 0 0
\(991\) −7.75820e6 −0.250944 −0.125472 0.992097i \(-0.540045\pi\)
−0.125472 + 0.992097i \(0.540045\pi\)
\(992\) 0 0
\(993\) 2.66953e7 0.859134
\(994\) 0 0
\(995\) 4.23962e6 0.135759
\(996\) 0 0
\(997\) 967884. 0.0308380 0.0154190 0.999881i \(-0.495092\pi\)
0.0154190 + 0.999881i \(0.495092\pi\)
\(998\) 0 0
\(999\) −7.76607e6 −0.246200
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 456.6.a.f.1.5 6
4.3 odd 2 912.6.a.y.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
456.6.a.f.1.5 6 1.1 even 1 trivial
912.6.a.y.1.5 6 4.3 odd 2