Properties

Label 441.6.a.bb.1.5
Level $441$
Weight $6$
Character 441.1
Self dual yes
Analytic conductor $70.729$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(70.7292645375\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: no (minimal twist has level 147)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-0.910122\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +O(q^{10})\) \(q+5.31815 q^{2} -3.71724 q^{4} +103.471 q^{5} -189.950 q^{8} +550.272 q^{10} -653.308 q^{11} +138.055 q^{13} -891.231 q^{16} +1174.38 q^{17} +1710.77 q^{19} -384.625 q^{20} -3474.39 q^{22} +4020.28 q^{23} +7581.15 q^{25} +734.196 q^{26} -2649.46 q^{29} +2874.65 q^{31} +1338.69 q^{32} +6245.51 q^{34} +2856.87 q^{37} +9098.15 q^{38} -19654.2 q^{40} +216.487 q^{41} +2928.29 q^{43} +2428.50 q^{44} +21380.5 q^{46} +14816.8 q^{47} +40317.7 q^{50} -513.182 q^{52} -21167.7 q^{53} -67598.1 q^{55} -14090.3 q^{58} +34689.1 q^{59} +8753.12 q^{61} +15287.8 q^{62} +35638.7 q^{64} +14284.6 q^{65} -12068.5 q^{67} -4365.43 q^{68} +35541.5 q^{71} -33485.8 q^{73} +15193.3 q^{74} -6359.35 q^{76} +43134.8 q^{79} -92216.1 q^{80} +1151.31 q^{82} -43338.8 q^{83} +121513. q^{85} +15573.1 q^{86} +124096. q^{88} -103533. q^{89} -14944.3 q^{92} +78798.1 q^{94} +177015. q^{95} +86294.4 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{2} + 150 q^{4} + 100 q^{5} + 114 q^{8} - 864 q^{10} - 604 q^{11} - 1352 q^{13} + 4578 q^{16} + 3028 q^{17} - 1728 q^{19} + 452 q^{20} - 4116 q^{22} + 4484 q^{23} + 4806 q^{25} + 14172 q^{26} + 5320 q^{29} - 3976 q^{31} + 37326 q^{32} + 16336 q^{34} + 22680 q^{37} + 52744 q^{38} - 100600 q^{40} + 28756 q^{41} - 6768 q^{43} + 64940 q^{44} + 540 q^{46} + 51552 q^{47} + 40622 q^{50} - 119296 q^{52} - 80884 q^{53} - 11656 q^{55} - 70464 q^{58} + 8872 q^{59} - 50896 q^{61} + 11824 q^{62} + 199590 q^{64} - 3492 q^{65} + 6480 q^{67} + 37348 q^{68} + 110852 q^{71} - 64232 q^{73} + 27464 q^{74} + 194864 q^{76} + 111696 q^{79} - 308940 q^{80} + 189640 q^{82} + 101128 q^{83} - 23292 q^{85} - 3824 q^{86} - 97788 q^{88} - 35012 q^{89} + 449260 q^{92} + 121016 q^{94} + 119080 q^{95} - 70952 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.31815 0.940126 0.470063 0.882633i \(-0.344231\pi\)
0.470063 + 0.882633i \(0.344231\pi\)
\(3\) 0 0
\(4\) −3.71724 −0.116164
\(5\) 103.471 1.85094 0.925468 0.378825i \(-0.123672\pi\)
0.925468 + 0.378825i \(0.123672\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −189.950 −1.04933
\(9\) 0 0
\(10\) 550.272 1.74011
\(11\) −653.308 −1.62793 −0.813966 0.580912i \(-0.802696\pi\)
−0.813966 + 0.580912i \(0.802696\pi\)
\(12\) 0 0
\(13\) 138.055 0.226565 0.113282 0.993563i \(-0.463863\pi\)
0.113282 + 0.993563i \(0.463863\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −891.231 −0.870342
\(17\) 1174.38 0.985564 0.492782 0.870153i \(-0.335980\pi\)
0.492782 + 0.870153i \(0.335980\pi\)
\(18\) 0 0
\(19\) 1710.77 1.08720 0.543599 0.839345i \(-0.317061\pi\)
0.543599 + 0.839345i \(0.317061\pi\)
\(20\) −384.625 −0.215012
\(21\) 0 0
\(22\) −3474.39 −1.53046
\(23\) 4020.28 1.58466 0.792332 0.610091i \(-0.208867\pi\)
0.792332 + 0.610091i \(0.208867\pi\)
\(24\) 0 0
\(25\) 7581.15 2.42597
\(26\) 734.196 0.213000
\(27\) 0 0
\(28\) 0 0
\(29\) −2649.46 −0.585009 −0.292505 0.956264i \(-0.594489\pi\)
−0.292505 + 0.956264i \(0.594489\pi\)
\(30\) 0 0
\(31\) 2874.65 0.537255 0.268628 0.963244i \(-0.413430\pi\)
0.268628 + 0.963244i \(0.413430\pi\)
\(32\) 1338.69 0.231103
\(33\) 0 0
\(34\) 6245.51 0.926554
\(35\) 0 0
\(36\) 0 0
\(37\) 2856.87 0.343073 0.171536 0.985178i \(-0.445127\pi\)
0.171536 + 0.985178i \(0.445127\pi\)
\(38\) 9098.15 1.02210
\(39\) 0 0
\(40\) −19654.2 −1.94225
\(41\) 216.487 0.0201128 0.0100564 0.999949i \(-0.496799\pi\)
0.0100564 + 0.999949i \(0.496799\pi\)
\(42\) 0 0
\(43\) 2928.29 0.241514 0.120757 0.992682i \(-0.461468\pi\)
0.120757 + 0.992682i \(0.461468\pi\)
\(44\) 2428.50 0.189107
\(45\) 0 0
\(46\) 21380.5 1.48978
\(47\) 14816.8 0.978386 0.489193 0.872176i \(-0.337291\pi\)
0.489193 + 0.872176i \(0.337291\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 40317.7 2.28071
\(51\) 0 0
\(52\) −513.182 −0.0263186
\(53\) −21167.7 −1.03510 −0.517552 0.855652i \(-0.673157\pi\)
−0.517552 + 0.855652i \(0.673157\pi\)
\(54\) 0 0
\(55\) −67598.1 −3.01320
\(56\) 0 0
\(57\) 0 0
\(58\) −14090.3 −0.549982
\(59\) 34689.1 1.29737 0.648684 0.761058i \(-0.275320\pi\)
0.648684 + 0.761058i \(0.275320\pi\)
\(60\) 0 0
\(61\) 8753.12 0.301189 0.150594 0.988596i \(-0.451881\pi\)
0.150594 + 0.988596i \(0.451881\pi\)
\(62\) 15287.8 0.505087
\(63\) 0 0
\(64\) 35638.7 1.08761
\(65\) 14284.6 0.419357
\(66\) 0 0
\(67\) −12068.5 −0.328448 −0.164224 0.986423i \(-0.552512\pi\)
−0.164224 + 0.986423i \(0.552512\pi\)
\(68\) −4365.43 −0.114487
\(69\) 0 0
\(70\) 0 0
\(71\) 35541.5 0.836738 0.418369 0.908277i \(-0.362602\pi\)
0.418369 + 0.908277i \(0.362602\pi\)
\(72\) 0 0
\(73\) −33485.8 −0.735450 −0.367725 0.929935i \(-0.619863\pi\)
−0.367725 + 0.929935i \(0.619863\pi\)
\(74\) 15193.3 0.322531
\(75\) 0 0
\(76\) −6359.35 −0.126293
\(77\) 0 0
\(78\) 0 0
\(79\) 43134.8 0.777607 0.388804 0.921321i \(-0.372888\pi\)
0.388804 + 0.921321i \(0.372888\pi\)
\(80\) −92216.1 −1.61095
\(81\) 0 0
\(82\) 1151.31 0.0189086
\(83\) −43338.8 −0.690529 −0.345265 0.938505i \(-0.612211\pi\)
−0.345265 + 0.938505i \(0.612211\pi\)
\(84\) 0 0
\(85\) 121513. 1.82422
\(86\) 15573.1 0.227054
\(87\) 0 0
\(88\) 124096. 1.70825
\(89\) −103533. −1.38550 −0.692748 0.721180i \(-0.743600\pi\)
−0.692748 + 0.721180i \(0.743600\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −14944.3 −0.184080
\(93\) 0 0
\(94\) 78798.1 0.919806
\(95\) 177015. 2.01233
\(96\) 0 0
\(97\) 86294.4 0.931223 0.465611 0.884989i \(-0.345835\pi\)
0.465611 + 0.884989i \(0.345835\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −28180.9 −0.281809
\(101\) 182677. 1.78189 0.890944 0.454114i \(-0.150044\pi\)
0.890944 + 0.454114i \(0.150044\pi\)
\(102\) 0 0
\(103\) 40088.2 0.372326 0.186163 0.982519i \(-0.440395\pi\)
0.186163 + 0.982519i \(0.440395\pi\)
\(104\) −26223.5 −0.237742
\(105\) 0 0
\(106\) −112573. −0.973129
\(107\) 110207. 0.930575 0.465287 0.885160i \(-0.345951\pi\)
0.465287 + 0.885160i \(0.345951\pi\)
\(108\) 0 0
\(109\) −222354. −1.79258 −0.896289 0.443471i \(-0.853747\pi\)
−0.896289 + 0.443471i \(0.853747\pi\)
\(110\) −359497. −2.83279
\(111\) 0 0
\(112\) 0 0
\(113\) 153473. 1.13067 0.565335 0.824861i \(-0.308747\pi\)
0.565335 + 0.824861i \(0.308747\pi\)
\(114\) 0 0
\(115\) 415981. 2.93311
\(116\) 9848.68 0.0679568
\(117\) 0 0
\(118\) 184482. 1.21969
\(119\) 0 0
\(120\) 0 0
\(121\) 265761. 1.65016
\(122\) 46550.5 0.283155
\(123\) 0 0
\(124\) −10685.8 −0.0624095
\(125\) 461080. 2.63938
\(126\) 0 0
\(127\) 142247. 0.782591 0.391296 0.920265i \(-0.372027\pi\)
0.391296 + 0.920265i \(0.372027\pi\)
\(128\) 146694. 0.791385
\(129\) 0 0
\(130\) 75967.6 0.394249
\(131\) 127000. 0.646587 0.323293 0.946299i \(-0.395210\pi\)
0.323293 + 0.946299i \(0.395210\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −64182.2 −0.308782
\(135\) 0 0
\(136\) −223072. −1.03419
\(137\) −99654.8 −0.453625 −0.226812 0.973938i \(-0.572830\pi\)
−0.226812 + 0.973938i \(0.572830\pi\)
\(138\) 0 0
\(139\) −23441.9 −0.102909 −0.0514547 0.998675i \(-0.516386\pi\)
−0.0514547 + 0.998675i \(0.516386\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 189015. 0.786639
\(143\) −90192.2 −0.368832
\(144\) 0 0
\(145\) −274141. −1.08282
\(146\) −178082. −0.691415
\(147\) 0 0
\(148\) −10619.7 −0.0398526
\(149\) −498705. −1.84025 −0.920127 0.391620i \(-0.871915\pi\)
−0.920127 + 0.391620i \(0.871915\pi\)
\(150\) 0 0
\(151\) −216338. −0.772130 −0.386065 0.922472i \(-0.626166\pi\)
−0.386065 + 0.922472i \(0.626166\pi\)
\(152\) −324961. −1.14083
\(153\) 0 0
\(154\) 0 0
\(155\) 297441. 0.994426
\(156\) 0 0
\(157\) −426837. −1.38201 −0.691007 0.722848i \(-0.742833\pi\)
−0.691007 + 0.722848i \(0.742833\pi\)
\(158\) 229398. 0.731049
\(159\) 0 0
\(160\) 138515. 0.427757
\(161\) 0 0
\(162\) 0 0
\(163\) −273196. −0.805387 −0.402694 0.915335i \(-0.631926\pi\)
−0.402694 + 0.915335i \(0.631926\pi\)
\(164\) −804.735 −0.00233638
\(165\) 0 0
\(166\) −230483. −0.649184
\(167\) 600365. 1.66581 0.832903 0.553420i \(-0.186677\pi\)
0.832903 + 0.553420i \(0.186677\pi\)
\(168\) 0 0
\(169\) −352234. −0.948668
\(170\) 646226. 1.71499
\(171\) 0 0
\(172\) −10885.1 −0.0280552
\(173\) −500188. −1.27063 −0.635313 0.772254i \(-0.719129\pi\)
−0.635313 + 0.772254i \(0.719129\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 582248. 1.41686
\(177\) 0 0
\(178\) −550606. −1.30254
\(179\) −289563. −0.675477 −0.337739 0.941240i \(-0.609662\pi\)
−0.337739 + 0.941240i \(0.609662\pi\)
\(180\) 0 0
\(181\) −168243. −0.381715 −0.190858 0.981618i \(-0.561127\pi\)
−0.190858 + 0.981618i \(0.561127\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −763652. −1.66284
\(185\) 295602. 0.635006
\(186\) 0 0
\(187\) −767229. −1.60443
\(188\) −55077.6 −0.113653
\(189\) 0 0
\(190\) 941391. 1.89185
\(191\) 386110. 0.765822 0.382911 0.923785i \(-0.374922\pi\)
0.382911 + 0.923785i \(0.374922\pi\)
\(192\) 0 0
\(193\) −339682. −0.656416 −0.328208 0.944606i \(-0.606445\pi\)
−0.328208 + 0.944606i \(0.606445\pi\)
\(194\) 458927. 0.875466
\(195\) 0 0
\(196\) 0 0
\(197\) 460915. 0.846166 0.423083 0.906091i \(-0.360948\pi\)
0.423083 + 0.906091i \(0.360948\pi\)
\(198\) 0 0
\(199\) 392888. 0.703292 0.351646 0.936133i \(-0.385622\pi\)
0.351646 + 0.936133i \(0.385622\pi\)
\(200\) −1.44004e6 −2.54565
\(201\) 0 0
\(202\) 971504. 1.67520
\(203\) 0 0
\(204\) 0 0
\(205\) 22400.0 0.0372275
\(206\) 213195. 0.350033
\(207\) 0 0
\(208\) −123039. −0.197189
\(209\) −1.11766e6 −1.76988
\(210\) 0 0
\(211\) −469348. −0.725752 −0.362876 0.931837i \(-0.618205\pi\)
−0.362876 + 0.931837i \(0.618205\pi\)
\(212\) 78685.4 0.120242
\(213\) 0 0
\(214\) 586100. 0.874857
\(215\) 302991. 0.447027
\(216\) 0 0
\(217\) 0 0
\(218\) −1.18251e6 −1.68525
\(219\) 0 0
\(220\) 251278. 0.350024
\(221\) 162128. 0.223294
\(222\) 0 0
\(223\) −83059.4 −0.111848 −0.0559238 0.998435i \(-0.517810\pi\)
−0.0559238 + 0.998435i \(0.517810\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 816194. 1.06297
\(227\) 139769. 0.180030 0.0900152 0.995940i \(-0.471308\pi\)
0.0900152 + 0.995940i \(0.471308\pi\)
\(228\) 0 0
\(229\) −143847. −0.181264 −0.0906321 0.995884i \(-0.528889\pi\)
−0.0906321 + 0.995884i \(0.528889\pi\)
\(230\) 2.21225e6 2.75749
\(231\) 0 0
\(232\) 503265. 0.613870
\(233\) 372492. 0.449498 0.224749 0.974417i \(-0.427844\pi\)
0.224749 + 0.974417i \(0.427844\pi\)
\(234\) 0 0
\(235\) 1.53310e6 1.81093
\(236\) −128948. −0.150707
\(237\) 0 0
\(238\) 0 0
\(239\) 201182. 0.227822 0.113911 0.993491i \(-0.463662\pi\)
0.113911 + 0.993491i \(0.463662\pi\)
\(240\) 0 0
\(241\) 42020.6 0.0466036 0.0233018 0.999728i \(-0.492582\pi\)
0.0233018 + 0.999728i \(0.492582\pi\)
\(242\) 1.41336e6 1.55136
\(243\) 0 0
\(244\) −32537.4 −0.0349872
\(245\) 0 0
\(246\) 0 0
\(247\) 236180. 0.246321
\(248\) −546039. −0.563760
\(249\) 0 0
\(250\) 2.45209e6 2.48135
\(251\) −1.23330e6 −1.23562 −0.617809 0.786329i \(-0.711979\pi\)
−0.617809 + 0.786329i \(0.711979\pi\)
\(252\) 0 0
\(253\) −2.62648e6 −2.57972
\(254\) 756493. 0.735734
\(255\) 0 0
\(256\) −360297. −0.343606
\(257\) 560688. 0.529528 0.264764 0.964313i \(-0.414706\pi\)
0.264764 + 0.964313i \(0.414706\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −53099.2 −0.0487141
\(261\) 0 0
\(262\) 675408. 0.607873
\(263\) 1.44524e6 1.28840 0.644198 0.764859i \(-0.277191\pi\)
0.644198 + 0.764859i \(0.277191\pi\)
\(264\) 0 0
\(265\) −2.19023e6 −1.91591
\(266\) 0 0
\(267\) 0 0
\(268\) 44861.5 0.0381537
\(269\) 417542. 0.351819 0.175910 0.984406i \(-0.443713\pi\)
0.175910 + 0.984406i \(0.443713\pi\)
\(270\) 0 0
\(271\) 428512. 0.354437 0.177219 0.984172i \(-0.443290\pi\)
0.177219 + 0.984172i \(0.443290\pi\)
\(272\) −1.04664e6 −0.857778
\(273\) 0 0
\(274\) −529980. −0.426465
\(275\) −4.95283e6 −3.94931
\(276\) 0 0
\(277\) −690887. −0.541013 −0.270506 0.962718i \(-0.587191\pi\)
−0.270506 + 0.962718i \(0.587191\pi\)
\(278\) −124667. −0.0967478
\(279\) 0 0
\(280\) 0 0
\(281\) −1.91606e6 −1.44758 −0.723792 0.690018i \(-0.757603\pi\)
−0.723792 + 0.690018i \(0.757603\pi\)
\(282\) 0 0
\(283\) 1.62417e6 1.20550 0.602748 0.797931i \(-0.294072\pi\)
0.602748 + 0.797931i \(0.294072\pi\)
\(284\) −132116. −0.0971986
\(285\) 0 0
\(286\) −479656. −0.346749
\(287\) 0 0
\(288\) 0 0
\(289\) −40699.3 −0.0286644
\(290\) −1.45793e6 −1.01798
\(291\) 0 0
\(292\) 124475. 0.0854326
\(293\) −531272. −0.361533 −0.180767 0.983526i \(-0.557858\pi\)
−0.180767 + 0.983526i \(0.557858\pi\)
\(294\) 0 0
\(295\) 3.58930e6 2.40135
\(296\) −542661. −0.359998
\(297\) 0 0
\(298\) −2.65219e6 −1.73007
\(299\) 555019. 0.359029
\(300\) 0 0
\(301\) 0 0
\(302\) −1.15052e6 −0.725899
\(303\) 0 0
\(304\) −1.52469e6 −0.946234
\(305\) 905690. 0.557481
\(306\) 0 0
\(307\) −409059. −0.247708 −0.123854 0.992300i \(-0.539525\pi\)
−0.123854 + 0.992300i \(0.539525\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.58184e6 0.934885
\(311\) −308704. −0.180984 −0.0904922 0.995897i \(-0.528844\pi\)
−0.0904922 + 0.995897i \(0.528844\pi\)
\(312\) 0 0
\(313\) −637733. −0.367941 −0.183970 0.982932i \(-0.558895\pi\)
−0.183970 + 0.982932i \(0.558895\pi\)
\(314\) −2.26998e6 −1.29927
\(315\) 0 0
\(316\) −160342. −0.0903297
\(317\) 2.94325e6 1.64505 0.822525 0.568729i \(-0.192565\pi\)
0.822525 + 0.568729i \(0.192565\pi\)
\(318\) 0 0
\(319\) 1.73092e6 0.952356
\(320\) 3.68756e6 2.01309
\(321\) 0 0
\(322\) 0 0
\(323\) 2.00909e6 1.07150
\(324\) 0 0
\(325\) 1.04661e6 0.549639
\(326\) −1.45290e6 −0.757165
\(327\) 0 0
\(328\) −41121.7 −0.0211051
\(329\) 0 0
\(330\) 0 0
\(331\) −2.84991e6 −1.42976 −0.714878 0.699249i \(-0.753518\pi\)
−0.714878 + 0.699249i \(0.753518\pi\)
\(332\) 161101. 0.0802144
\(333\) 0 0
\(334\) 3.19283e6 1.56607
\(335\) −1.24874e6 −0.607937
\(336\) 0 0
\(337\) 3.46617e6 1.66255 0.831277 0.555859i \(-0.187611\pi\)
0.831277 + 0.555859i \(0.187611\pi\)
\(338\) −1.87323e6 −0.891867
\(339\) 0 0
\(340\) −451694. −0.211908
\(341\) −1.87803e6 −0.874615
\(342\) 0 0
\(343\) 0 0
\(344\) −556228. −0.253429
\(345\) 0 0
\(346\) −2.66008e6 −1.19455
\(347\) −15115.3 −0.00673897 −0.00336949 0.999994i \(-0.501073\pi\)
−0.00336949 + 0.999994i \(0.501073\pi\)
\(348\) 0 0
\(349\) −3.24698e6 −1.42698 −0.713488 0.700667i \(-0.752886\pi\)
−0.713488 + 0.700667i \(0.752886\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −874578. −0.376220
\(353\) −1.76786e6 −0.755110 −0.377555 0.925987i \(-0.623235\pi\)
−0.377555 + 0.925987i \(0.623235\pi\)
\(354\) 0 0
\(355\) 3.67749e6 1.54875
\(356\) 384858. 0.160944
\(357\) 0 0
\(358\) −1.53994e6 −0.635034
\(359\) 121581. 0.0497885 0.0248942 0.999690i \(-0.492075\pi\)
0.0248942 + 0.999690i \(0.492075\pi\)
\(360\) 0 0
\(361\) 450645. 0.181998
\(362\) −894740. −0.358860
\(363\) 0 0
\(364\) 0 0
\(365\) −3.46479e6 −1.36127
\(366\) 0 0
\(367\) −2.08841e6 −0.809378 −0.404689 0.914454i \(-0.632620\pi\)
−0.404689 + 0.914454i \(0.632620\pi\)
\(368\) −3.58300e6 −1.37920
\(369\) 0 0
\(370\) 1.57206e6 0.596985
\(371\) 0 0
\(372\) 0 0
\(373\) −2.29355e6 −0.853563 −0.426782 0.904355i \(-0.640353\pi\)
−0.426782 + 0.904355i \(0.640353\pi\)
\(374\) −4.08024e6 −1.50837
\(375\) 0 0
\(376\) −2.81445e6 −1.02665
\(377\) −365771. −0.132543
\(378\) 0 0
\(379\) −2.93812e6 −1.05068 −0.525341 0.850892i \(-0.676062\pi\)
−0.525341 + 0.850892i \(0.676062\pi\)
\(380\) −658005. −0.233760
\(381\) 0 0
\(382\) 2.05339e6 0.719969
\(383\) 5.04605e6 1.75774 0.878871 0.477060i \(-0.158298\pi\)
0.878871 + 0.477060i \(0.158298\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.80648e6 −0.617113
\(387\) 0 0
\(388\) −320777. −0.108174
\(389\) 1.07621e6 0.360599 0.180300 0.983612i \(-0.442293\pi\)
0.180300 + 0.983612i \(0.442293\pi\)
\(390\) 0 0
\(391\) 4.72132e6 1.56179
\(392\) 0 0
\(393\) 0 0
\(394\) 2.45122e6 0.795502
\(395\) 4.46318e6 1.43930
\(396\) 0 0
\(397\) −4.34345e6 −1.38312 −0.691559 0.722320i \(-0.743076\pi\)
−0.691559 + 0.722320i \(0.743076\pi\)
\(398\) 2.08944e6 0.661183
\(399\) 0 0
\(400\) −6.75655e6 −2.11142
\(401\) 704083. 0.218657 0.109328 0.994006i \(-0.465130\pi\)
0.109328 + 0.994006i \(0.465130\pi\)
\(402\) 0 0
\(403\) 396859. 0.121723
\(404\) −679054. −0.206991
\(405\) 0 0
\(406\) 0 0
\(407\) −1.86642e6 −0.558499
\(408\) 0 0
\(409\) 1.63500e6 0.483292 0.241646 0.970365i \(-0.422313\pi\)
0.241646 + 0.970365i \(0.422313\pi\)
\(410\) 119127. 0.0349986
\(411\) 0 0
\(412\) −149017. −0.0432508
\(413\) 0 0
\(414\) 0 0
\(415\) −4.48429e6 −1.27813
\(416\) 184813. 0.0523598
\(417\) 0 0
\(418\) −5.94390e6 −1.66391
\(419\) −5.92839e6 −1.64969 −0.824844 0.565360i \(-0.808737\pi\)
−0.824844 + 0.565360i \(0.808737\pi\)
\(420\) 0 0
\(421\) −5.46170e6 −1.50184 −0.750918 0.660395i \(-0.770389\pi\)
−0.750918 + 0.660395i \(0.770389\pi\)
\(422\) −2.49606e6 −0.682299
\(423\) 0 0
\(424\) 4.02080e6 1.08617
\(425\) 8.90311e6 2.39095
\(426\) 0 0
\(427\) 0 0
\(428\) −409667. −0.108099
\(429\) 0 0
\(430\) 1.61136e6 0.420262
\(431\) −3.44526e6 −0.893366 −0.446683 0.894692i \(-0.647395\pi\)
−0.446683 + 0.894692i \(0.647395\pi\)
\(432\) 0 0
\(433\) −2.46833e6 −0.632680 −0.316340 0.948646i \(-0.602454\pi\)
−0.316340 + 0.948646i \(0.602454\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 826541. 0.208232
\(437\) 6.87779e6 1.72284
\(438\) 0 0
\(439\) 6.13780e6 1.52003 0.760014 0.649907i \(-0.225192\pi\)
0.760014 + 0.649907i \(0.225192\pi\)
\(440\) 1.28403e7 3.16185
\(441\) 0 0
\(442\) 862222. 0.209925
\(443\) 1.79219e6 0.433885 0.216942 0.976184i \(-0.430392\pi\)
0.216942 + 0.976184i \(0.430392\pi\)
\(444\) 0 0
\(445\) −1.07126e7 −2.56446
\(446\) −441723. −0.105151
\(447\) 0 0
\(448\) 0 0
\(449\) 3.76014e6 0.880213 0.440107 0.897945i \(-0.354941\pi\)
0.440107 + 0.897945i \(0.354941\pi\)
\(450\) 0 0
\(451\) −141433. −0.0327423
\(452\) −570496. −0.131343
\(453\) 0 0
\(454\) 743312. 0.169251
\(455\) 0 0
\(456\) 0 0
\(457\) 7.54502e6 1.68994 0.844968 0.534817i \(-0.179620\pi\)
0.844968 + 0.534817i \(0.179620\pi\)
\(458\) −765000. −0.170411
\(459\) 0 0
\(460\) −1.54630e6 −0.340721
\(461\) 359098. 0.0786975 0.0393487 0.999226i \(-0.487472\pi\)
0.0393487 + 0.999226i \(0.487472\pi\)
\(462\) 0 0
\(463\) −890950. −0.193153 −0.0965764 0.995326i \(-0.530789\pi\)
−0.0965764 + 0.995326i \(0.530789\pi\)
\(464\) 2.36128e6 0.509158
\(465\) 0 0
\(466\) 1.98097e6 0.422584
\(467\) −7.61319e6 −1.61538 −0.807690 0.589608i \(-0.799282\pi\)
−0.807690 + 0.589608i \(0.799282\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 8.15328e6 1.70250
\(471\) 0 0
\(472\) −6.58919e6 −1.36137
\(473\) −1.91307e6 −0.393169
\(474\) 0 0
\(475\) 1.29696e7 2.63751
\(476\) 0 0
\(477\) 0 0
\(478\) 1.06992e6 0.214181
\(479\) −5.96007e6 −1.18690 −0.593448 0.804872i \(-0.702234\pi\)
−0.593448 + 0.804872i \(0.702234\pi\)
\(480\) 0 0
\(481\) 394404. 0.0777282
\(482\) 223472. 0.0438132
\(483\) 0 0
\(484\) −987895. −0.191689
\(485\) 8.92893e6 1.72363
\(486\) 0 0
\(487\) −257827. −0.0492612 −0.0246306 0.999697i \(-0.507841\pi\)
−0.0246306 + 0.999697i \(0.507841\pi\)
\(488\) −1.66265e6 −0.316047
\(489\) 0 0
\(490\) 0 0
\(491\) −592660. −0.110944 −0.0554718 0.998460i \(-0.517666\pi\)
−0.0554718 + 0.998460i \(0.517666\pi\)
\(492\) 0 0
\(493\) −3.11146e6 −0.576564
\(494\) 1.25604e6 0.231573
\(495\) 0 0
\(496\) −2.56198e6 −0.467596
\(497\) 0 0
\(498\) 0 0
\(499\) −2.42242e6 −0.435511 −0.217755 0.976003i \(-0.569874\pi\)
−0.217755 + 0.976003i \(0.569874\pi\)
\(500\) −1.71394e6 −0.306600
\(501\) 0 0
\(502\) −6.55887e6 −1.16164
\(503\) 2.31849e6 0.408588 0.204294 0.978910i \(-0.434510\pi\)
0.204294 + 0.978910i \(0.434510\pi\)
\(504\) 0 0
\(505\) 1.89017e7 3.29816
\(506\) −1.39680e7 −2.42527
\(507\) 0 0
\(508\) −528767. −0.0909087
\(509\) −8.49886e6 −1.45401 −0.727003 0.686634i \(-0.759087\pi\)
−0.727003 + 0.686634i \(0.759087\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.61033e6 −1.11442
\(513\) 0 0
\(514\) 2.98183e6 0.497822
\(515\) 4.14795e6 0.689152
\(516\) 0 0
\(517\) −9.67995e6 −1.59275
\(518\) 0 0
\(519\) 0 0
\(520\) −2.71335e6 −0.440046
\(521\) −1.16092e6 −0.187374 −0.0936868 0.995602i \(-0.529865\pi\)
−0.0936868 + 0.995602i \(0.529865\pi\)
\(522\) 0 0
\(523\) −1.22907e6 −0.196482 −0.0982411 0.995163i \(-0.531322\pi\)
−0.0982411 + 0.995163i \(0.531322\pi\)
\(524\) −472091. −0.0751099
\(525\) 0 0
\(526\) 7.68598e6 1.21125
\(527\) 3.37592e6 0.529499
\(528\) 0 0
\(529\) 9.72633e6 1.51116
\(530\) −1.16480e7 −1.80120
\(531\) 0 0
\(532\) 0 0
\(533\) 29887.1 0.00455686
\(534\) 0 0
\(535\) 1.14032e7 1.72244
\(536\) 2.29241e6 0.344652
\(537\) 0 0
\(538\) 2.22055e6 0.330754
\(539\) 0 0
\(540\) 0 0
\(541\) −790279. −0.116088 −0.0580440 0.998314i \(-0.518486\pi\)
−0.0580440 + 0.998314i \(0.518486\pi\)
\(542\) 2.27889e6 0.333216
\(543\) 0 0
\(544\) 1.57213e6 0.227767
\(545\) −2.30070e7 −3.31795
\(546\) 0 0
\(547\) 1.74335e6 0.249124 0.124562 0.992212i \(-0.460247\pi\)
0.124562 + 0.992212i \(0.460247\pi\)
\(548\) 370441. 0.0526947
\(549\) 0 0
\(550\) −2.63399e7 −3.71285
\(551\) −4.53263e6 −0.636021
\(552\) 0 0
\(553\) 0 0
\(554\) −3.67424e6 −0.508620
\(555\) 0 0
\(556\) 87139.0 0.0119543
\(557\) 1.01834e7 1.39077 0.695385 0.718637i \(-0.255234\pi\)
0.695385 + 0.718637i \(0.255234\pi\)
\(558\) 0 0
\(559\) 404264. 0.0547186
\(560\) 0 0
\(561\) 0 0
\(562\) −1.01899e7 −1.36091
\(563\) −8.57420e6 −1.14005 −0.570023 0.821629i \(-0.693066\pi\)
−0.570023 + 0.821629i \(0.693066\pi\)
\(564\) 0 0
\(565\) 1.58799e7 2.09280
\(566\) 8.63760e6 1.13332
\(567\) 0 0
\(568\) −6.75109e6 −0.878018
\(569\) −2.67960e6 −0.346968 −0.173484 0.984837i \(-0.555502\pi\)
−0.173484 + 0.984837i \(0.555502\pi\)
\(570\) 0 0
\(571\) 7.45017e6 0.956260 0.478130 0.878289i \(-0.341315\pi\)
0.478130 + 0.878289i \(0.341315\pi\)
\(572\) 335266. 0.0428449
\(573\) 0 0
\(574\) 0 0
\(575\) 3.04784e7 3.84434
\(576\) 0 0
\(577\) −5.72891e6 −0.716362 −0.358181 0.933652i \(-0.616603\pi\)
−0.358181 + 0.933652i \(0.616603\pi\)
\(578\) −216445. −0.0269481
\(579\) 0 0
\(580\) 1.01905e6 0.125784
\(581\) 0 0
\(582\) 0 0
\(583\) 1.38290e7 1.68508
\(584\) 6.36061e6 0.771733
\(585\) 0 0
\(586\) −2.82539e6 −0.339887
\(587\) 1.41744e7 1.69788 0.848942 0.528486i \(-0.177240\pi\)
0.848942 + 0.528486i \(0.177240\pi\)
\(588\) 0 0
\(589\) 4.91787e6 0.584102
\(590\) 1.90885e7 2.25757
\(591\) 0 0
\(592\) −2.54613e6 −0.298591
\(593\) −6.09494e6 −0.711758 −0.355879 0.934532i \(-0.615818\pi\)
−0.355879 + 0.934532i \(0.615818\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.85380e6 0.213771
\(597\) 0 0
\(598\) 2.95168e6 0.337533
\(599\) −1.97464e6 −0.224865 −0.112432 0.993659i \(-0.535864\pi\)
−0.112432 + 0.993659i \(0.535864\pi\)
\(600\) 0 0
\(601\) −5.14078e6 −0.580554 −0.290277 0.956943i \(-0.593747\pi\)
−0.290277 + 0.956943i \(0.593747\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 804180. 0.0896935
\(605\) 2.74984e7 3.05435
\(606\) 0 0
\(607\) 1.31482e6 0.144843 0.0724213 0.997374i \(-0.476927\pi\)
0.0724213 + 0.997374i \(0.476927\pi\)
\(608\) 2.29020e6 0.251255
\(609\) 0 0
\(610\) 4.81660e6 0.524102
\(611\) 2.04553e6 0.221668
\(612\) 0 0
\(613\) 1.34181e7 1.44224 0.721122 0.692808i \(-0.243627\pi\)
0.721122 + 0.692808i \(0.243627\pi\)
\(614\) −2.17544e6 −0.232877
\(615\) 0 0
\(616\) 0 0
\(617\) −1.48394e7 −1.56929 −0.784646 0.619944i \(-0.787155\pi\)
−0.784646 + 0.619944i \(0.787155\pi\)
\(618\) 0 0
\(619\) 1.14468e7 1.20077 0.600383 0.799713i \(-0.295015\pi\)
0.600383 + 0.799713i \(0.295015\pi\)
\(620\) −1.10566e6 −0.115516
\(621\) 0 0
\(622\) −1.64173e6 −0.170148
\(623\) 0 0
\(624\) 0 0
\(625\) 2.40171e7 2.45935
\(626\) −3.39156e6 −0.345910
\(627\) 0 0
\(628\) 1.58665e6 0.160540
\(629\) 3.35504e6 0.338120
\(630\) 0 0
\(631\) −1.11608e7 −1.11589 −0.557944 0.829878i \(-0.688410\pi\)
−0.557944 + 0.829878i \(0.688410\pi\)
\(632\) −8.19345e6 −0.815970
\(633\) 0 0
\(634\) 1.56527e7 1.54655
\(635\) 1.47184e7 1.44853
\(636\) 0 0
\(637\) 0 0
\(638\) 9.20528e6 0.895334
\(639\) 0 0
\(640\) 1.51785e7 1.46480
\(641\) 1.72150e7 1.65486 0.827431 0.561567i \(-0.189801\pi\)
0.827431 + 0.561567i \(0.189801\pi\)
\(642\) 0 0
\(643\) 2.73062e6 0.260456 0.130228 0.991484i \(-0.458429\pi\)
0.130228 + 0.991484i \(0.458429\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 1.06846e7 1.00735
\(647\) −4.39969e6 −0.413201 −0.206600 0.978425i \(-0.566240\pi\)
−0.206600 + 0.978425i \(0.566240\pi\)
\(648\) 0 0
\(649\) −2.26627e7 −2.11203
\(650\) 5.56605e6 0.516730
\(651\) 0 0
\(652\) 1.01553e6 0.0935567
\(653\) −5.60500e6 −0.514390 −0.257195 0.966360i \(-0.582798\pi\)
−0.257195 + 0.966360i \(0.582798\pi\)
\(654\) 0 0
\(655\) 1.31408e7 1.19679
\(656\) −192940. −0.0175050
\(657\) 0 0
\(658\) 0 0
\(659\) 1.33959e7 1.20159 0.600796 0.799402i \(-0.294850\pi\)
0.600796 + 0.799402i \(0.294850\pi\)
\(660\) 0 0
\(661\) −1.87657e6 −0.167056 −0.0835279 0.996505i \(-0.526619\pi\)
−0.0835279 + 0.996505i \(0.526619\pi\)
\(662\) −1.51563e7 −1.34415
\(663\) 0 0
\(664\) 8.23220e6 0.724596
\(665\) 0 0
\(666\) 0 0
\(667\) −1.06516e7 −0.927043
\(668\) −2.23170e6 −0.193506
\(669\) 0 0
\(670\) −6.64097e6 −0.571537
\(671\) −5.71849e6 −0.490315
\(672\) 0 0
\(673\) 2.82367e6 0.240313 0.120156 0.992755i \(-0.461660\pi\)
0.120156 + 0.992755i \(0.461660\pi\)
\(674\) 1.84336e7 1.56301
\(675\) 0 0
\(676\) 1.30934e6 0.110201
\(677\) 9.31849e6 0.781401 0.390700 0.920518i \(-0.372233\pi\)
0.390700 + 0.920518i \(0.372233\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −2.30814e7 −1.91421
\(681\) 0 0
\(682\) −9.98766e6 −0.822248
\(683\) 2.13619e7 1.75222 0.876108 0.482114i \(-0.160131\pi\)
0.876108 + 0.482114i \(0.160131\pi\)
\(684\) 0 0
\(685\) −1.03113e7 −0.839631
\(686\) 0 0
\(687\) 0 0
\(688\) −2.60978e6 −0.210200
\(689\) −2.92230e6 −0.234518
\(690\) 0 0
\(691\) −1.10885e7 −0.883439 −0.441720 0.897153i \(-0.645631\pi\)
−0.441720 + 0.897153i \(0.645631\pi\)
\(692\) 1.85932e6 0.147601
\(693\) 0 0
\(694\) −80385.6 −0.00633548
\(695\) −2.42554e6 −0.190479
\(696\) 0 0
\(697\) 254237. 0.0198224
\(698\) −1.72680e7 −1.34154
\(699\) 0 0
\(700\) 0 0
\(701\) 2.02256e7 1.55456 0.777279 0.629156i \(-0.216599\pi\)
0.777279 + 0.629156i \(0.216599\pi\)
\(702\) 0 0
\(703\) 4.88745e6 0.372988
\(704\) −2.32831e7 −1.77055
\(705\) 0 0
\(706\) −9.40174e6 −0.709899
\(707\) 0 0
\(708\) 0 0
\(709\) −2.10834e7 −1.57516 −0.787580 0.616213i \(-0.788666\pi\)
−0.787580 + 0.616213i \(0.788666\pi\)
\(710\) 1.95575e7 1.45602
\(711\) 0 0
\(712\) 1.96661e7 1.45385
\(713\) 1.15569e7 0.851369
\(714\) 0 0
\(715\) −9.33224e6 −0.682686
\(716\) 1.07638e6 0.0784659
\(717\) 0 0
\(718\) 646585. 0.0468074
\(719\) −5.88060e6 −0.424228 −0.212114 0.977245i \(-0.568035\pi\)
−0.212114 + 0.977245i \(0.568035\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 2.39660e6 0.171101
\(723\) 0 0
\(724\) 625398. 0.0443414
\(725\) −2.00860e7 −1.41921
\(726\) 0 0
\(727\) −9.23778e6 −0.648233 −0.324117 0.946017i \(-0.605067\pi\)
−0.324117 + 0.946017i \(0.605067\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −1.84263e7 −1.27977
\(731\) 3.43891e6 0.238028
\(732\) 0 0
\(733\) −7.75113e6 −0.532850 −0.266425 0.963856i \(-0.585842\pi\)
−0.266425 + 0.963856i \(0.585842\pi\)
\(734\) −1.11065e7 −0.760917
\(735\) 0 0
\(736\) 5.38192e6 0.366220
\(737\) 7.88446e6 0.534691
\(738\) 0 0
\(739\) 1.02902e7 0.693125 0.346563 0.938027i \(-0.387349\pi\)
0.346563 + 0.938027i \(0.387349\pi\)
\(740\) −1.09882e6 −0.0737646
\(741\) 0 0
\(742\) 0 0
\(743\) 2.04494e6 0.135897 0.0679483 0.997689i \(-0.478355\pi\)
0.0679483 + 0.997689i \(0.478355\pi\)
\(744\) 0 0
\(745\) −5.16012e7 −3.40619
\(746\) −1.21974e7 −0.802457
\(747\) 0 0
\(748\) 2.85197e6 0.186377
\(749\) 0 0
\(750\) 0 0
\(751\) −1.18742e7 −0.768251 −0.384125 0.923281i \(-0.625497\pi\)
−0.384125 + 0.923281i \(0.625497\pi\)
\(752\) −1.32052e7 −0.851531
\(753\) 0 0
\(754\) −1.94523e6 −0.124607
\(755\) −2.23846e7 −1.42916
\(756\) 0 0
\(757\) −8.48702e6 −0.538289 −0.269145 0.963100i \(-0.586741\pi\)
−0.269145 + 0.963100i \(0.586741\pi\)
\(758\) −1.56254e7 −0.987772
\(759\) 0 0
\(760\) −3.36239e7 −2.11161
\(761\) 2.39932e7 1.50185 0.750925 0.660388i \(-0.229608\pi\)
0.750925 + 0.660388i \(0.229608\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.43526e6 −0.0889607
\(765\) 0 0
\(766\) 2.68357e7 1.65250
\(767\) 4.78899e6 0.293938
\(768\) 0 0
\(769\) 2.43194e7 1.48299 0.741494 0.670959i \(-0.234117\pi\)
0.741494 + 0.670959i \(0.234117\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.26268e6 0.0762517
\(773\) −5.38149e6 −0.323932 −0.161966 0.986796i \(-0.551784\pi\)
−0.161966 + 0.986796i \(0.551784\pi\)
\(774\) 0 0
\(775\) 2.17931e7 1.30336
\(776\) −1.63916e7 −0.977164
\(777\) 0 0
\(778\) 5.72347e6 0.339009
\(779\) 370361. 0.0218666
\(780\) 0 0
\(781\) −2.32195e7 −1.36215
\(782\) 2.51087e7 1.46828
\(783\) 0 0
\(784\) 0 0
\(785\) −4.41650e7 −2.55802
\(786\) 0 0
\(787\) −2.80718e7 −1.61560 −0.807800 0.589456i \(-0.799342\pi\)
−0.807800 + 0.589456i \(0.799342\pi\)
\(788\) −1.71333e6 −0.0982938
\(789\) 0 0
\(790\) 2.37359e7 1.35312
\(791\) 0 0
\(792\) 0 0
\(793\) 1.20841e6 0.0682388
\(794\) −2.30992e7 −1.30030
\(795\) 0 0
\(796\) −1.46046e6 −0.0816970
\(797\) 1.97055e7 1.09886 0.549428 0.835541i \(-0.314846\pi\)
0.549428 + 0.835541i \(0.314846\pi\)
\(798\) 0 0
\(799\) 1.74005e7 0.964262
\(800\) 1.01488e7 0.560648
\(801\) 0 0
\(802\) 3.74442e6 0.205565
\(803\) 2.18765e7 1.19726
\(804\) 0 0
\(805\) 0 0
\(806\) 2.11056e6 0.114435
\(807\) 0 0
\(808\) −3.46994e7 −1.86980
\(809\) 3.39408e6 0.182327 0.0911633 0.995836i \(-0.470941\pi\)
0.0911633 + 0.995836i \(0.470941\pi\)
\(810\) 0 0
\(811\) −1.78837e7 −0.954786 −0.477393 0.878690i \(-0.658418\pi\)
−0.477393 + 0.878690i \(0.658418\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −9.92588e6 −0.525059
\(815\) −2.82677e7 −1.49072
\(816\) 0 0
\(817\) 5.00964e6 0.262574
\(818\) 8.69518e6 0.454355
\(819\) 0 0
\(820\) −83266.3 −0.00432449
\(821\) −9.26949e6 −0.479952 −0.239976 0.970779i \(-0.577140\pi\)
−0.239976 + 0.970779i \(0.577140\pi\)
\(822\) 0 0
\(823\) −1.52579e6 −0.0785228 −0.0392614 0.999229i \(-0.512501\pi\)
−0.0392614 + 0.999229i \(0.512501\pi\)
\(824\) −7.61474e6 −0.390694
\(825\) 0 0
\(826\) 0 0
\(827\) −806595. −0.0410102 −0.0205051 0.999790i \(-0.506527\pi\)
−0.0205051 + 0.999790i \(0.506527\pi\)
\(828\) 0 0
\(829\) 2.19275e7 1.10816 0.554080 0.832463i \(-0.313070\pi\)
0.554080 + 0.832463i \(0.313070\pi\)
\(830\) −2.38482e7 −1.20160
\(831\) 0 0
\(832\) 4.92010e6 0.246414
\(833\) 0 0
\(834\) 0 0
\(835\) 6.21201e7 3.08330
\(836\) 4.15462e6 0.205596
\(837\) 0 0
\(838\) −3.15281e7 −1.55091
\(839\) −2.40115e7 −1.17765 −0.588823 0.808262i \(-0.700409\pi\)
−0.588823 + 0.808262i \(0.700409\pi\)
\(840\) 0 0
\(841\) −1.34915e7 −0.657764
\(842\) −2.90462e7 −1.41192
\(843\) 0 0
\(844\) 1.74468e6 0.0843061
\(845\) −3.64458e7 −1.75593
\(846\) 0 0
\(847\) 0 0
\(848\) 1.88653e7 0.900895
\(849\) 0 0
\(850\) 4.73481e7 2.24779
\(851\) 1.14854e7 0.543654
\(852\) 0 0
\(853\) −2.26785e7 −1.06719 −0.533594 0.845741i \(-0.679159\pi\)
−0.533594 + 0.845741i \(0.679159\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −2.09339e7 −0.976484
\(857\) 5.07518e6 0.236048 0.118024 0.993011i \(-0.462344\pi\)
0.118024 + 0.993011i \(0.462344\pi\)
\(858\) 0 0
\(859\) 1.71405e7 0.792577 0.396289 0.918126i \(-0.370298\pi\)
0.396289 + 0.918126i \(0.370298\pi\)
\(860\) −1.12629e6 −0.0519284
\(861\) 0 0
\(862\) −1.83224e7 −0.839876
\(863\) −3.84686e7 −1.75824 −0.879122 0.476597i \(-0.841870\pi\)
−0.879122 + 0.476597i \(0.841870\pi\)
\(864\) 0 0
\(865\) −5.17547e7 −2.35185
\(866\) −1.31270e7 −0.594799
\(867\) 0 0
\(868\) 0 0
\(869\) −2.81803e7 −1.26589
\(870\) 0 0
\(871\) −1.66611e6 −0.0744148
\(872\) 4.22360e7 1.88101
\(873\) 0 0
\(874\) 3.65772e7 1.61969
\(875\) 0 0
\(876\) 0 0
\(877\) −1.91671e7 −0.841504 −0.420752 0.907176i \(-0.638234\pi\)
−0.420752 + 0.907176i \(0.638234\pi\)
\(878\) 3.26418e7 1.42902
\(879\) 0 0
\(880\) 6.02455e7 2.62252
\(881\) 4.15340e7 1.80287 0.901434 0.432916i \(-0.142515\pi\)
0.901434 + 0.432916i \(0.142515\pi\)
\(882\) 0 0
\(883\) −4.31950e7 −1.86437 −0.932183 0.361987i \(-0.882099\pi\)
−0.932183 + 0.361987i \(0.882099\pi\)
\(884\) −602668. −0.0259387
\(885\) 0 0
\(886\) 9.53113e6 0.407906
\(887\) 6.87508e6 0.293406 0.146703 0.989181i \(-0.453134\pi\)
0.146703 + 0.989181i \(0.453134\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −5.69715e7 −2.41092
\(891\) 0 0
\(892\) 308752. 0.0129926
\(893\) 2.53482e7 1.06370
\(894\) 0 0
\(895\) −2.99613e7 −1.25027
\(896\) 0 0
\(897\) 0 0
\(898\) 1.99970e7 0.827511
\(899\) −7.61628e6 −0.314299
\(900\) 0 0
\(901\) −2.48588e7 −1.02016
\(902\) −752162. −0.0307819
\(903\) 0 0
\(904\) −2.91522e7 −1.18645
\(905\) −1.74081e7 −0.706531
\(906\) 0 0
\(907\) 1.49747e7 0.604423 0.302212 0.953241i \(-0.402275\pi\)
0.302212 + 0.953241i \(0.402275\pi\)
\(908\) −519554. −0.0209130
\(909\) 0 0
\(910\) 0 0
\(911\) −3.13061e7 −1.24978 −0.624889 0.780714i \(-0.714856\pi\)
−0.624889 + 0.780714i \(0.714856\pi\)
\(912\) 0 0
\(913\) 2.83136e7 1.12414
\(914\) 4.01256e7 1.58875
\(915\) 0 0
\(916\) 534713. 0.0210563
\(917\) 0 0
\(918\) 0 0
\(919\) 1.46990e7 0.574117 0.287058 0.957913i \(-0.407323\pi\)
0.287058 + 0.957913i \(0.407323\pi\)
\(920\) −7.90154e7 −3.07781
\(921\) 0 0
\(922\) 1.90974e6 0.0739855
\(923\) 4.90666e6 0.189575
\(924\) 0 0
\(925\) 2.16583e7 0.832283
\(926\) −4.73821e6 −0.181588
\(927\) 0 0
\(928\) −3.54681e6 −0.135197
\(929\) 1.68420e7 0.640256 0.320128 0.947374i \(-0.396274\pi\)
0.320128 + 0.947374i \(0.396274\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1.38464e6 −0.0522153
\(933\) 0 0
\(934\) −4.04881e7 −1.51866
\(935\) −7.93856e7 −2.96970
\(936\) 0 0
\(937\) 3.22026e7 1.19824 0.599118 0.800661i \(-0.295518\pi\)
0.599118 + 0.800661i \(0.295518\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −5.69891e6 −0.210364
\(941\) 2.65064e7 0.975837 0.487919 0.872889i \(-0.337756\pi\)
0.487919 + 0.872889i \(0.337756\pi\)
\(942\) 0 0
\(943\) 870340. 0.0318720
\(944\) −3.09160e7 −1.12915
\(945\) 0 0
\(946\) −1.01740e7 −0.369628
\(947\) 4.98682e7 1.80696 0.903481 0.428629i \(-0.141003\pi\)
0.903481 + 0.428629i \(0.141003\pi\)
\(948\) 0 0
\(949\) −4.62287e6 −0.166627
\(950\) 6.89745e7 2.47959
\(951\) 0 0
\(952\) 0 0
\(953\) 2.46027e7 0.877508 0.438754 0.898607i \(-0.355420\pi\)
0.438754 + 0.898607i \(0.355420\pi\)
\(954\) 0 0
\(955\) 3.99510e7 1.41749
\(956\) −747843. −0.0264646
\(957\) 0 0
\(958\) −3.16966e7 −1.11583
\(959\) 0 0
\(960\) 0 0
\(961\) −2.03655e7 −0.711357
\(962\) 2.09750e6 0.0730743
\(963\) 0 0
\(964\) −156200. −0.00541364
\(965\) −3.51471e7 −1.21498
\(966\) 0 0
\(967\) 1.59962e7 0.550111 0.275056 0.961428i \(-0.411304\pi\)
0.275056 + 0.961428i \(0.411304\pi\)
\(968\) −5.04812e7 −1.73157
\(969\) 0 0
\(970\) 4.74854e7 1.62043
\(971\) 9.94424e6 0.338473 0.169236 0.985576i \(-0.445870\pi\)
0.169236 + 0.985576i \(0.445870\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1.37116e6 −0.0463118
\(975\) 0 0
\(976\) −7.80105e6 −0.262137
\(977\) 1.20871e7 0.405121 0.202561 0.979270i \(-0.435074\pi\)
0.202561 + 0.979270i \(0.435074\pi\)
\(978\) 0 0
\(979\) 6.76392e7 2.25549
\(980\) 0 0
\(981\) 0 0
\(982\) −3.15186e6 −0.104301
\(983\) 3.42776e7 1.13143 0.565714 0.824601i \(-0.308601\pi\)
0.565714 + 0.824601i \(0.308601\pi\)
\(984\) 0 0
\(985\) 4.76911e7 1.56620
\(986\) −1.65472e7 −0.542043
\(987\) 0 0
\(988\) −877938. −0.0286135
\(989\) 1.17725e7 0.382719
\(990\) 0 0
\(991\) −3.50099e7 −1.13242 −0.566209 0.824262i \(-0.691590\pi\)
−0.566209 + 0.824262i \(0.691590\pi\)
\(992\) 3.84827e6 0.124161
\(993\) 0 0
\(994\) 0 0
\(995\) 4.06523e7 1.30175
\(996\) 0 0
\(997\) −9.99230e6 −0.318367 −0.159183 0.987249i \(-0.550886\pi\)
−0.159183 + 0.987249i \(0.550886\pi\)
\(998\) −1.28828e7 −0.409435
\(999\) 0 0
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 441.6.a.bb.1.5 6
3.2 odd 2 147.6.a.n.1.2 6
7.6 odd 2 441.6.a.ba.1.5 6
21.2 odd 6 147.6.e.q.67.5 12
21.5 even 6 147.6.e.p.67.5 12
21.11 odd 6 147.6.e.q.79.5 12
21.17 even 6 147.6.e.p.79.5 12
21.20 even 2 147.6.a.o.1.2 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.2 6 3.2 odd 2
147.6.a.o.1.2 yes 6 21.20 even 2
147.6.e.p.67.5 12 21.5 even 6
147.6.e.p.79.5 12 21.17 even 6
147.6.e.q.67.5 12 21.2 odd 6
147.6.e.q.79.5 12 21.11 odd 6
441.6.a.ba.1.5 6 7.6 odd 2
441.6.a.bb.1.5 6 1.1 even 1 trivial