Properties

Label 825.2.c.d.199.1
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.d.199.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.414214i q^{7} +4.41421i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-2.41421i q^{2} +1.00000i q^{3} -3.82843 q^{4} +2.41421 q^{6} -0.414214i q^{7} +4.41421i q^{8} -1.00000 q^{9} -1.00000 q^{11} -3.82843i q^{12} -2.82843i q^{13} -1.00000 q^{14} +3.00000 q^{16} -2.41421i q^{17} +2.41421i q^{18} -6.41421 q^{19} +0.414214 q^{21} +2.41421i q^{22} +1.00000i q^{23} -4.41421 q^{24} -6.82843 q^{26} -1.00000i q^{27} +1.58579i q^{28} -1.17157 q^{29} -8.48528 q^{31} +1.58579i q^{32} -1.00000i q^{33} -5.82843 q^{34} +3.82843 q^{36} +0.171573i q^{37} +15.4853i q^{38} +2.82843 q^{39} -10.8995 q^{41} -1.00000i q^{42} -11.6569i q^{43} +3.82843 q^{44} +2.41421 q^{46} +7.48528i q^{47} +3.00000i q^{48} +6.82843 q^{49} +2.41421 q^{51} +10.8284i q^{52} +7.65685i q^{53} -2.41421 q^{54} +1.82843 q^{56} -6.41421i q^{57} +2.82843i q^{58} -11.0000 q^{59} +8.82843 q^{61} +20.4853i q^{62} +0.414214i q^{63} +9.82843 q^{64} -2.41421 q^{66} +0.343146i q^{67} +9.24264i q^{68} -1.00000 q^{69} +7.82843 q^{71} -4.41421i q^{72} -8.82843i q^{73} +0.414214 q^{74} +24.5563 q^{76} +0.414214i q^{77} -6.82843i q^{78} -13.2426 q^{79} +1.00000 q^{81} +26.3137i q^{82} -4.48528i q^{83} -1.58579 q^{84} -28.1421 q^{86} -1.17157i q^{87} -4.41421i q^{88} -3.65685 q^{89} -1.17157 q^{91} -3.82843i q^{92} -8.48528i q^{93} +18.0711 q^{94} -1.58579 q^{96} -5.82843i q^{97} -16.4853i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} + O(q^{10}) \) \( 4 q - 4 q^{4} + 4 q^{6} - 4 q^{9} - 4 q^{11} - 4 q^{14} + 12 q^{16} - 20 q^{19} - 4 q^{21} - 12 q^{24} - 16 q^{26} - 16 q^{29} - 12 q^{34} + 4 q^{36} - 4 q^{41} + 4 q^{44} + 4 q^{46} + 16 q^{49} + 4 q^{51} - 4 q^{54} - 4 q^{56} - 44 q^{59} + 24 q^{61} + 28 q^{64} - 4 q^{66} - 4 q^{69} + 20 q^{71} - 4 q^{74} + 36 q^{76} - 36 q^{79} + 4 q^{81} - 12 q^{84} - 56 q^{86} + 8 q^{89} - 16 q^{91} + 44 q^{94} - 12 q^{96} + 4 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.41421i − 1.70711i −0.521005 0.853553i \(-0.674443\pi\)
0.521005 0.853553i \(-0.325557\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) 2.41421 0.985599
\(7\) − 0.414214i − 0.156558i −0.996931 0.0782790i \(-0.975058\pi\)
0.996931 0.0782790i \(-0.0249425\pi\)
\(8\) 4.41421i 1.56066i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 3.82843i − 1.10517i
\(13\) − 2.82843i − 0.784465i −0.919866 0.392232i \(-0.871703\pi\)
0.919866 0.392232i \(-0.128297\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) − 2.41421i − 0.585533i −0.956184 0.292766i \(-0.905424\pi\)
0.956184 0.292766i \(-0.0945758\pi\)
\(18\) 2.41421i 0.569036i
\(19\) −6.41421 −1.47152 −0.735761 0.677242i \(-0.763175\pi\)
−0.735761 + 0.677242i \(0.763175\pi\)
\(20\) 0 0
\(21\) 0.414214 0.0903888
\(22\) 2.41421i 0.514712i
\(23\) 1.00000i 0.208514i 0.994550 + 0.104257i \(0.0332465\pi\)
−0.994550 + 0.104257i \(0.966753\pi\)
\(24\) −4.41421 −0.901048
\(25\) 0 0
\(26\) −6.82843 −1.33916
\(27\) − 1.00000i − 0.192450i
\(28\) 1.58579i 0.299685i
\(29\) −1.17157 −0.217556 −0.108778 0.994066i \(-0.534694\pi\)
−0.108778 + 0.994066i \(0.534694\pi\)
\(30\) 0 0
\(31\) −8.48528 −1.52400 −0.762001 0.647576i \(-0.775783\pi\)
−0.762001 + 0.647576i \(0.775783\pi\)
\(32\) 1.58579i 0.280330i
\(33\) − 1.00000i − 0.174078i
\(34\) −5.82843 −0.999567
\(35\) 0 0
\(36\) 3.82843 0.638071
\(37\) 0.171573i 0.0282064i 0.999901 + 0.0141032i \(0.00448934\pi\)
−0.999901 + 0.0141032i \(0.995511\pi\)
\(38\) 15.4853i 2.51204i
\(39\) 2.82843 0.452911
\(40\) 0 0
\(41\) −10.8995 −1.70222 −0.851108 0.524991i \(-0.824069\pi\)
−0.851108 + 0.524991i \(0.824069\pi\)
\(42\) − 1.00000i − 0.154303i
\(43\) − 11.6569i − 1.77765i −0.458243 0.888827i \(-0.651521\pi\)
0.458243 0.888827i \(-0.348479\pi\)
\(44\) 3.82843 0.577157
\(45\) 0 0
\(46\) 2.41421 0.355956
\(47\) 7.48528i 1.09184i 0.837837 + 0.545920i \(0.183820\pi\)
−0.837837 + 0.545920i \(0.816180\pi\)
\(48\) 3.00000i 0.433013i
\(49\) 6.82843 0.975490
\(50\) 0 0
\(51\) 2.41421 0.338058
\(52\) 10.8284i 1.50163i
\(53\) 7.65685i 1.05175i 0.850562 + 0.525875i \(0.176262\pi\)
−0.850562 + 0.525875i \(0.823738\pi\)
\(54\) −2.41421 −0.328533
\(55\) 0 0
\(56\) 1.82843 0.244334
\(57\) − 6.41421i − 0.849583i
\(58\) 2.82843i 0.371391i
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) 0 0
\(61\) 8.82843 1.13036 0.565182 0.824966i \(-0.308806\pi\)
0.565182 + 0.824966i \(0.308806\pi\)
\(62\) 20.4853i 2.60163i
\(63\) 0.414214i 0.0521860i
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) −2.41421 −0.297169
\(67\) 0.343146i 0.0419219i 0.999780 + 0.0209610i \(0.00667257\pi\)
−0.999780 + 0.0209610i \(0.993327\pi\)
\(68\) 9.24264i 1.12083i
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 7.82843 0.929063 0.464532 0.885556i \(-0.346223\pi\)
0.464532 + 0.885556i \(0.346223\pi\)
\(72\) − 4.41421i − 0.520220i
\(73\) − 8.82843i − 1.03329i −0.856200 0.516645i \(-0.827181\pi\)
0.856200 0.516645i \(-0.172819\pi\)
\(74\) 0.414214 0.0481513
\(75\) 0 0
\(76\) 24.5563 2.81681
\(77\) 0.414214i 0.0472040i
\(78\) − 6.82843i − 0.773167i
\(79\) −13.2426 −1.48991 −0.744957 0.667113i \(-0.767530\pi\)
−0.744957 + 0.667113i \(0.767530\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 26.3137i 2.90586i
\(83\) − 4.48528i − 0.492324i −0.969229 0.246162i \(-0.920831\pi\)
0.969229 0.246162i \(-0.0791695\pi\)
\(84\) −1.58579 −0.173023
\(85\) 0 0
\(86\) −28.1421 −3.03464
\(87\) − 1.17157i − 0.125606i
\(88\) − 4.41421i − 0.470557i
\(89\) −3.65685 −0.387626 −0.193813 0.981039i \(-0.562085\pi\)
−0.193813 + 0.981039i \(0.562085\pi\)
\(90\) 0 0
\(91\) −1.17157 −0.122814
\(92\) − 3.82843i − 0.399141i
\(93\) − 8.48528i − 0.879883i
\(94\) 18.0711 1.86389
\(95\) 0 0
\(96\) −1.58579 −0.161849
\(97\) − 5.82843i − 0.591787i −0.955221 0.295894i \(-0.904383\pi\)
0.955221 0.295894i \(-0.0956174\pi\)
\(98\) − 16.4853i − 1.66526i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 14.8995 1.48256 0.741278 0.671199i \(-0.234220\pi\)
0.741278 + 0.671199i \(0.234220\pi\)
\(102\) − 5.82843i − 0.577100i
\(103\) − 13.6569i − 1.34565i −0.739802 0.672825i \(-0.765081\pi\)
0.739802 0.672825i \(-0.234919\pi\)
\(104\) 12.4853 1.22428
\(105\) 0 0
\(106\) 18.4853 1.79545
\(107\) 5.31371i 0.513696i 0.966452 + 0.256848i \(0.0826839\pi\)
−0.966452 + 0.256848i \(0.917316\pi\)
\(108\) 3.82843i 0.368391i
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) −0.171573 −0.0162850
\(112\) − 1.24264i − 0.117419i
\(113\) − 10.0000i − 0.940721i −0.882474 0.470360i \(-0.844124\pi\)
0.882474 0.470360i \(-0.155876\pi\)
\(114\) −15.4853 −1.45033
\(115\) 0 0
\(116\) 4.48528 0.416448
\(117\) 2.82843i 0.261488i
\(118\) 26.5563i 2.44471i
\(119\) −1.00000 −0.0916698
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 21.3137i − 1.92965i
\(123\) − 10.8995i − 0.982774i
\(124\) 32.4853 2.91726
\(125\) 0 0
\(126\) 1.00000 0.0890871
\(127\) − 7.24264i − 0.642680i −0.946964 0.321340i \(-0.895867\pi\)
0.946964 0.321340i \(-0.104133\pi\)
\(128\) − 20.5563i − 1.81694i
\(129\) 11.6569 1.02633
\(130\) 0 0
\(131\) 6.82843 0.596602 0.298301 0.954472i \(-0.403580\pi\)
0.298301 + 0.954472i \(0.403580\pi\)
\(132\) 3.82843i 0.333222i
\(133\) 2.65685i 0.230378i
\(134\) 0.828427 0.0715652
\(135\) 0 0
\(136\) 10.6569 0.913818
\(137\) 12.1421i 1.03737i 0.854965 + 0.518686i \(0.173579\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(138\) 2.41421i 0.205512i
\(139\) 18.9706 1.60906 0.804531 0.593911i \(-0.202417\pi\)
0.804531 + 0.593911i \(0.202417\pi\)
\(140\) 0 0
\(141\) −7.48528 −0.630374
\(142\) − 18.8995i − 1.58601i
\(143\) 2.82843i 0.236525i
\(144\) −3.00000 −0.250000
\(145\) 0 0
\(146\) −21.3137 −1.76394
\(147\) 6.82843i 0.563199i
\(148\) − 0.656854i − 0.0539931i
\(149\) −7.72792 −0.633096 −0.316548 0.948576i \(-0.602524\pi\)
−0.316548 + 0.948576i \(0.602524\pi\)
\(150\) 0 0
\(151\) 14.0000 1.13930 0.569652 0.821886i \(-0.307078\pi\)
0.569652 + 0.821886i \(0.307078\pi\)
\(152\) − 28.3137i − 2.29655i
\(153\) 2.41421i 0.195178i
\(154\) 1.00000 0.0805823
\(155\) 0 0
\(156\) −10.8284 −0.866968
\(157\) − 6.00000i − 0.478852i −0.970915 0.239426i \(-0.923041\pi\)
0.970915 0.239426i \(-0.0769593\pi\)
\(158\) 31.9706i 2.54344i
\(159\) −7.65685 −0.607228
\(160\) 0 0
\(161\) 0.414214 0.0326446
\(162\) − 2.41421i − 0.189679i
\(163\) 15.7990i 1.23747i 0.785599 + 0.618736i \(0.212355\pi\)
−0.785599 + 0.618736i \(0.787645\pi\)
\(164\) 41.7279 3.25840
\(165\) 0 0
\(166\) −10.8284 −0.840449
\(167\) − 21.7990i − 1.68686i −0.537242 0.843428i \(-0.680534\pi\)
0.537242 0.843428i \(-0.319466\pi\)
\(168\) 1.82843i 0.141066i
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 6.41421 0.490507
\(172\) 44.6274i 3.40281i
\(173\) − 12.5563i − 0.954642i −0.878729 0.477321i \(-0.841608\pi\)
0.878729 0.477321i \(-0.158392\pi\)
\(174\) −2.82843 −0.214423
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) − 11.0000i − 0.826811i
\(178\) 8.82843i 0.661719i
\(179\) −16.7990 −1.25562 −0.627808 0.778368i \(-0.716048\pi\)
−0.627808 + 0.778368i \(0.716048\pi\)
\(180\) 0 0
\(181\) −21.9706 −1.63306 −0.816530 0.577304i \(-0.804105\pi\)
−0.816530 + 0.577304i \(0.804105\pi\)
\(182\) 2.82843i 0.209657i
\(183\) 8.82843i 0.652616i
\(184\) −4.41421 −0.325420
\(185\) 0 0
\(186\) −20.4853 −1.50205
\(187\) 2.41421i 0.176545i
\(188\) − 28.6569i − 2.09002i
\(189\) −0.414214 −0.0301296
\(190\) 0 0
\(191\) 6.17157 0.446559 0.223280 0.974754i \(-0.428324\pi\)
0.223280 + 0.974754i \(0.428324\pi\)
\(192\) 9.82843i 0.709306i
\(193\) 3.31371i 0.238526i 0.992863 + 0.119263i \(0.0380532\pi\)
−0.992863 + 0.119263i \(0.961947\pi\)
\(194\) −14.0711 −1.01024
\(195\) 0 0
\(196\) −26.1421 −1.86730
\(197\) − 4.75736i − 0.338948i −0.985535 0.169474i \(-0.945793\pi\)
0.985535 0.169474i \(-0.0542068\pi\)
\(198\) − 2.41421i − 0.171571i
\(199\) −10.8284 −0.767607 −0.383803 0.923415i \(-0.625386\pi\)
−0.383803 + 0.923415i \(0.625386\pi\)
\(200\) 0 0
\(201\) −0.343146 −0.0242036
\(202\) − 35.9706i − 2.53088i
\(203\) 0.485281i 0.0340601i
\(204\) −9.24264 −0.647114
\(205\) 0 0
\(206\) −32.9706 −2.29717
\(207\) − 1.00000i − 0.0695048i
\(208\) − 8.48528i − 0.588348i
\(209\) 6.41421 0.443680
\(210\) 0 0
\(211\) −13.3137 −0.916553 −0.458277 0.888810i \(-0.651533\pi\)
−0.458277 + 0.888810i \(0.651533\pi\)
\(212\) − 29.3137i − 2.01327i
\(213\) 7.82843i 0.536395i
\(214\) 12.8284 0.876933
\(215\) 0 0
\(216\) 4.41421 0.300349
\(217\) 3.51472i 0.238595i
\(218\) 12.8284i 0.868851i
\(219\) 8.82843 0.596570
\(220\) 0 0
\(221\) −6.82843 −0.459330
\(222\) 0.414214i 0.0278002i
\(223\) − 21.1716i − 1.41775i −0.705332 0.708877i \(-0.749202\pi\)
0.705332 0.708877i \(-0.250798\pi\)
\(224\) 0.656854 0.0438879
\(225\) 0 0
\(226\) −24.1421 −1.60591
\(227\) 18.4853i 1.22691i 0.789729 + 0.613456i \(0.210221\pi\)
−0.789729 + 0.613456i \(0.789779\pi\)
\(228\) 24.5563i 1.62628i
\(229\) 2.51472 0.166177 0.0830886 0.996542i \(-0.473522\pi\)
0.0830886 + 0.996542i \(0.473522\pi\)
\(230\) 0 0
\(231\) −0.414214 −0.0272533
\(232\) − 5.17157i − 0.339530i
\(233\) 16.5563i 1.08464i 0.840171 + 0.542321i \(0.182454\pi\)
−0.840171 + 0.542321i \(0.817546\pi\)
\(234\) 6.82843 0.446388
\(235\) 0 0
\(236\) 42.1127 2.74130
\(237\) − 13.2426i − 0.860202i
\(238\) 2.41421i 0.156490i
\(239\) −23.6569 −1.53023 −0.765117 0.643891i \(-0.777319\pi\)
−0.765117 + 0.643891i \(0.777319\pi\)
\(240\) 0 0
\(241\) 14.1421 0.910975 0.455488 0.890242i \(-0.349465\pi\)
0.455488 + 0.890242i \(0.349465\pi\)
\(242\) − 2.41421i − 0.155192i
\(243\) 1.00000i 0.0641500i
\(244\) −33.7990 −2.16376
\(245\) 0 0
\(246\) −26.3137 −1.67770
\(247\) 18.1421i 1.15436i
\(248\) − 37.4558i − 2.37845i
\(249\) 4.48528 0.284243
\(250\) 0 0
\(251\) 8.97056 0.566217 0.283108 0.959088i \(-0.408634\pi\)
0.283108 + 0.959088i \(0.408634\pi\)
\(252\) − 1.58579i − 0.0998952i
\(253\) − 1.00000i − 0.0628695i
\(254\) −17.4853 −1.09712
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) − 9.31371i − 0.580973i −0.956879 0.290487i \(-0.906183\pi\)
0.956879 0.290487i \(-0.0938172\pi\)
\(258\) − 28.1421i − 1.75205i
\(259\) 0.0710678 0.00441594
\(260\) 0 0
\(261\) 1.17157 0.0725185
\(262\) − 16.4853i − 1.01846i
\(263\) 22.9706i 1.41643i 0.705999 + 0.708213i \(0.250498\pi\)
−0.705999 + 0.708213i \(0.749502\pi\)
\(264\) 4.41421 0.271676
\(265\) 0 0
\(266\) 6.41421 0.393281
\(267\) − 3.65685i − 0.223796i
\(268\) − 1.31371i − 0.0802475i
\(269\) 15.7990 0.963281 0.481641 0.876369i \(-0.340041\pi\)
0.481641 + 0.876369i \(0.340041\pi\)
\(270\) 0 0
\(271\) 8.89949 0.540606 0.270303 0.962775i \(-0.412876\pi\)
0.270303 + 0.962775i \(0.412876\pi\)
\(272\) − 7.24264i − 0.439150i
\(273\) − 1.17157i − 0.0709068i
\(274\) 29.3137 1.77091
\(275\) 0 0
\(276\) 3.82843 0.230444
\(277\) − 4.82843i − 0.290112i −0.989423 0.145056i \(-0.953664\pi\)
0.989423 0.145056i \(-0.0463362\pi\)
\(278\) − 45.7990i − 2.74684i
\(279\) 8.48528 0.508001
\(280\) 0 0
\(281\) 32.0711 1.91320 0.956600 0.291405i \(-0.0941227\pi\)
0.956600 + 0.291405i \(0.0941227\pi\)
\(282\) 18.0711i 1.07612i
\(283\) − 0.899495i − 0.0534694i −0.999643 0.0267347i \(-0.991489\pi\)
0.999643 0.0267347i \(-0.00851094\pi\)
\(284\) −29.9706 −1.77843
\(285\) 0 0
\(286\) 6.82843 0.403773
\(287\) 4.51472i 0.266495i
\(288\) − 1.58579i − 0.0934434i
\(289\) 11.1716 0.657151
\(290\) 0 0
\(291\) 5.82843 0.341668
\(292\) 33.7990i 1.97794i
\(293\) 17.5858i 1.02737i 0.857978 + 0.513686i \(0.171720\pi\)
−0.857978 + 0.513686i \(0.828280\pi\)
\(294\) 16.4853 0.961441
\(295\) 0 0
\(296\) −0.757359 −0.0440206
\(297\) 1.00000i 0.0580259i
\(298\) 18.6569i 1.08076i
\(299\) 2.82843 0.163572
\(300\) 0 0
\(301\) −4.82843 −0.278306
\(302\) − 33.7990i − 1.94491i
\(303\) 14.8995i 0.855954i
\(304\) −19.2426 −1.10364
\(305\) 0 0
\(306\) 5.82843 0.333189
\(307\) − 6.68629i − 0.381607i −0.981628 0.190803i \(-0.938891\pi\)
0.981628 0.190803i \(-0.0611093\pi\)
\(308\) − 1.58579i − 0.0903586i
\(309\) 13.6569 0.776911
\(310\) 0 0
\(311\) 13.6569 0.774409 0.387205 0.921994i \(-0.373441\pi\)
0.387205 + 0.921994i \(0.373441\pi\)
\(312\) 12.4853i 0.706840i
\(313\) − 27.1421i − 1.53416i −0.641549 0.767082i \(-0.721708\pi\)
0.641549 0.767082i \(-0.278292\pi\)
\(314\) −14.4853 −0.817452
\(315\) 0 0
\(316\) 50.6985 2.85201
\(317\) − 30.8284i − 1.73150i −0.500479 0.865748i \(-0.666843\pi\)
0.500479 0.865748i \(-0.333157\pi\)
\(318\) 18.4853i 1.03660i
\(319\) 1.17157 0.0655955
\(320\) 0 0
\(321\) −5.31371 −0.296582
\(322\) − 1.00000i − 0.0557278i
\(323\) 15.4853i 0.861624i
\(324\) −3.82843 −0.212690
\(325\) 0 0
\(326\) 38.1421 2.11250
\(327\) − 5.31371i − 0.293849i
\(328\) − 48.1127i − 2.65658i
\(329\) 3.10051 0.170936
\(330\) 0 0
\(331\) −32.1421 −1.76669 −0.883346 0.468722i \(-0.844715\pi\)
−0.883346 + 0.468722i \(0.844715\pi\)
\(332\) 17.1716i 0.942412i
\(333\) − 0.171573i − 0.00940214i
\(334\) −52.6274 −2.87964
\(335\) 0 0
\(336\) 1.24264 0.0677916
\(337\) − 4.14214i − 0.225637i −0.993616 0.112818i \(-0.964012\pi\)
0.993616 0.112818i \(-0.0359878\pi\)
\(338\) − 12.0711i − 0.656580i
\(339\) 10.0000 0.543125
\(340\) 0 0
\(341\) 8.48528 0.459504
\(342\) − 15.4853i − 0.837348i
\(343\) − 5.72792i − 0.309279i
\(344\) 51.4558 2.77431
\(345\) 0 0
\(346\) −30.3137 −1.62968
\(347\) 21.1716i 1.13655i 0.822839 + 0.568275i \(0.192389\pi\)
−0.822839 + 0.568275i \(0.807611\pi\)
\(348\) 4.48528i 0.240436i
\(349\) 2.48528 0.133034 0.0665170 0.997785i \(-0.478811\pi\)
0.0665170 + 0.997785i \(0.478811\pi\)
\(350\) 0 0
\(351\) −2.82843 −0.150970
\(352\) − 1.58579i − 0.0845227i
\(353\) − 4.48528i − 0.238727i −0.992851 0.119364i \(-0.961915\pi\)
0.992851 0.119364i \(-0.0380855\pi\)
\(354\) −26.5563 −1.41145
\(355\) 0 0
\(356\) 14.0000 0.741999
\(357\) − 1.00000i − 0.0529256i
\(358\) 40.5563i 2.14347i
\(359\) −15.5147 −0.818836 −0.409418 0.912347i \(-0.634268\pi\)
−0.409418 + 0.912347i \(0.634268\pi\)
\(360\) 0 0
\(361\) 22.1421 1.16538
\(362\) 53.0416i 2.78781i
\(363\) 1.00000i 0.0524864i
\(364\) 4.48528 0.235093
\(365\) 0 0
\(366\) 21.3137 1.11409
\(367\) − 1.31371i − 0.0685750i −0.999412 0.0342875i \(-0.989084\pi\)
0.999412 0.0342875i \(-0.0109162\pi\)
\(368\) 3.00000i 0.156386i
\(369\) 10.8995 0.567405
\(370\) 0 0
\(371\) 3.17157 0.164660
\(372\) 32.4853i 1.68428i
\(373\) − 23.6569i − 1.22491i −0.790507 0.612453i \(-0.790183\pi\)
0.790507 0.612453i \(-0.209817\pi\)
\(374\) 5.82843 0.301381
\(375\) 0 0
\(376\) −33.0416 −1.70399
\(377\) 3.31371i 0.170665i
\(378\) 1.00000i 0.0514344i
\(379\) −9.17157 −0.471112 −0.235556 0.971861i \(-0.575691\pi\)
−0.235556 + 0.971861i \(0.575691\pi\)
\(380\) 0 0
\(381\) 7.24264 0.371052
\(382\) − 14.8995i − 0.762324i
\(383\) 20.0000i 1.02195i 0.859595 + 0.510976i \(0.170716\pi\)
−0.859595 + 0.510976i \(0.829284\pi\)
\(384\) 20.5563 1.04901
\(385\) 0 0
\(386\) 8.00000 0.407189
\(387\) 11.6569i 0.592551i
\(388\) 22.3137i 1.13281i
\(389\) 17.6569 0.895238 0.447619 0.894224i \(-0.352272\pi\)
0.447619 + 0.894224i \(0.352272\pi\)
\(390\) 0 0
\(391\) 2.41421 0.122092
\(392\) 30.1421i 1.52241i
\(393\) 6.82843i 0.344449i
\(394\) −11.4853 −0.578620
\(395\) 0 0
\(396\) −3.82843 −0.192386
\(397\) 35.9411i 1.80383i 0.431910 + 0.901917i \(0.357840\pi\)
−0.431910 + 0.901917i \(0.642160\pi\)
\(398\) 26.1421i 1.31039i
\(399\) −2.65685 −0.133009
\(400\) 0 0
\(401\) 31.7990 1.58797 0.793983 0.607940i \(-0.208004\pi\)
0.793983 + 0.607940i \(0.208004\pi\)
\(402\) 0.828427i 0.0413182i
\(403\) 24.0000i 1.19553i
\(404\) −57.0416 −2.83793
\(405\) 0 0
\(406\) 1.17157 0.0581442
\(407\) − 0.171573i − 0.00850455i
\(408\) 10.6569i 0.527593i
\(409\) 4.14214 0.204815 0.102408 0.994743i \(-0.467345\pi\)
0.102408 + 0.994743i \(0.467345\pi\)
\(410\) 0 0
\(411\) −12.1421 −0.598927
\(412\) 52.2843i 2.57586i
\(413\) 4.55635i 0.224203i
\(414\) −2.41421 −0.118652
\(415\) 0 0
\(416\) 4.48528 0.219909
\(417\) 18.9706i 0.928992i
\(418\) − 15.4853i − 0.757410i
\(419\) 25.4853 1.24504 0.622519 0.782605i \(-0.286109\pi\)
0.622519 + 0.782605i \(0.286109\pi\)
\(420\) 0 0
\(421\) −27.0000 −1.31590 −0.657950 0.753062i \(-0.728576\pi\)
−0.657950 + 0.753062i \(0.728576\pi\)
\(422\) 32.1421i 1.56465i
\(423\) − 7.48528i − 0.363947i
\(424\) −33.7990 −1.64142
\(425\) 0 0
\(426\) 18.8995 0.915684
\(427\) − 3.65685i − 0.176968i
\(428\) − 20.3431i − 0.983323i
\(429\) −2.82843 −0.136558
\(430\) 0 0
\(431\) 1.17157 0.0564327 0.0282163 0.999602i \(-0.491017\pi\)
0.0282163 + 0.999602i \(0.491017\pi\)
\(432\) − 3.00000i − 0.144338i
\(433\) − 13.3137i − 0.639816i −0.947449 0.319908i \(-0.896348\pi\)
0.947449 0.319908i \(-0.103652\pi\)
\(434\) 8.48528 0.407307
\(435\) 0 0
\(436\) 20.3431 0.974260
\(437\) − 6.41421i − 0.306833i
\(438\) − 21.3137i − 1.01841i
\(439\) −2.27208 −0.108440 −0.0542202 0.998529i \(-0.517267\pi\)
−0.0542202 + 0.998529i \(0.517267\pi\)
\(440\) 0 0
\(441\) −6.82843 −0.325163
\(442\) 16.4853i 0.784125i
\(443\) − 7.97056i − 0.378693i −0.981910 0.189346i \(-0.939363\pi\)
0.981910 0.189346i \(-0.0606369\pi\)
\(444\) 0.656854 0.0311729
\(445\) 0 0
\(446\) −51.1127 −2.42026
\(447\) − 7.72792i − 0.365518i
\(448\) − 4.07107i − 0.192340i
\(449\) 6.48528 0.306059 0.153030 0.988222i \(-0.451097\pi\)
0.153030 + 0.988222i \(0.451097\pi\)
\(450\) 0 0
\(451\) 10.8995 0.513237
\(452\) 38.2843i 1.80074i
\(453\) 14.0000i 0.657777i
\(454\) 44.6274 2.09447
\(455\) 0 0
\(456\) 28.3137 1.32591
\(457\) − 3.85786i − 0.180463i −0.995921 0.0902316i \(-0.971239\pi\)
0.995921 0.0902316i \(-0.0287607\pi\)
\(458\) − 6.07107i − 0.283682i
\(459\) −2.41421 −0.112686
\(460\) 0 0
\(461\) −32.7696 −1.52623 −0.763115 0.646263i \(-0.776331\pi\)
−0.763115 + 0.646263i \(0.776331\pi\)
\(462\) 1.00000i 0.0465242i
\(463\) 34.9706i 1.62522i 0.582808 + 0.812610i \(0.301954\pi\)
−0.582808 + 0.812610i \(0.698046\pi\)
\(464\) −3.51472 −0.163167
\(465\) 0 0
\(466\) 39.9706 1.85160
\(467\) − 10.6274i − 0.491778i −0.969298 0.245889i \(-0.920920\pi\)
0.969298 0.245889i \(-0.0790799\pi\)
\(468\) − 10.8284i − 0.500544i
\(469\) 0.142136 0.00656321
\(470\) 0 0
\(471\) 6.00000 0.276465
\(472\) − 48.5563i − 2.23499i
\(473\) 11.6569i 0.535983i
\(474\) −31.9706 −1.46846
\(475\) 0 0
\(476\) 3.82843 0.175476
\(477\) − 7.65685i − 0.350583i
\(478\) 57.1127i 2.61227i
\(479\) 24.4853 1.11876 0.559381 0.828911i \(-0.311039\pi\)
0.559381 + 0.828911i \(0.311039\pi\)
\(480\) 0 0
\(481\) 0.485281 0.0221269
\(482\) − 34.1421i − 1.55513i
\(483\) 0.414214i 0.0188474i
\(484\) −3.82843 −0.174019
\(485\) 0 0
\(486\) 2.41421 0.109511
\(487\) 6.48528i 0.293876i 0.989146 + 0.146938i \(0.0469418\pi\)
−0.989146 + 0.146938i \(0.953058\pi\)
\(488\) 38.9706i 1.76411i
\(489\) −15.7990 −0.714455
\(490\) 0 0
\(491\) −24.1421 −1.08952 −0.544760 0.838592i \(-0.683379\pi\)
−0.544760 + 0.838592i \(0.683379\pi\)
\(492\) 41.7279i 1.88124i
\(493\) 2.82843i 0.127386i
\(494\) 43.7990 1.97061
\(495\) 0 0
\(496\) −25.4558 −1.14300
\(497\) − 3.24264i − 0.145452i
\(498\) − 10.8284i − 0.485233i
\(499\) −35.1716 −1.57450 −0.787248 0.616637i \(-0.788495\pi\)
−0.787248 + 0.616637i \(0.788495\pi\)
\(500\) 0 0
\(501\) 21.7990 0.973907
\(502\) − 21.6569i − 0.966593i
\(503\) − 34.2843i − 1.52866i −0.644825 0.764330i \(-0.723070\pi\)
0.644825 0.764330i \(-0.276930\pi\)
\(504\) −1.82843 −0.0814446
\(505\) 0 0
\(506\) −2.41421 −0.107325
\(507\) 5.00000i 0.222058i
\(508\) 27.7279i 1.23023i
\(509\) −4.62742 −0.205107 −0.102553 0.994728i \(-0.532701\pi\)
−0.102553 + 0.994728i \(0.532701\pi\)
\(510\) 0 0
\(511\) −3.65685 −0.161770
\(512\) 31.2426i 1.38074i
\(513\) 6.41421i 0.283194i
\(514\) −22.4853 −0.991783
\(515\) 0 0
\(516\) −44.6274 −1.96461
\(517\) − 7.48528i − 0.329202i
\(518\) − 0.171573i − 0.00753848i
\(519\) 12.5563 0.551163
\(520\) 0 0
\(521\) −36.1421 −1.58342 −0.791708 0.610900i \(-0.790808\pi\)
−0.791708 + 0.610900i \(0.790808\pi\)
\(522\) − 2.82843i − 0.123797i
\(523\) 42.2132i 1.84585i 0.384974 + 0.922927i \(0.374210\pi\)
−0.384974 + 0.922927i \(0.625790\pi\)
\(524\) −26.1421 −1.14202
\(525\) 0 0
\(526\) 55.4558 2.41799
\(527\) 20.4853i 0.892353i
\(528\) − 3.00000i − 0.130558i
\(529\) 22.0000 0.956522
\(530\) 0 0
\(531\) 11.0000 0.477359
\(532\) − 10.1716i − 0.440994i
\(533\) 30.8284i 1.33533i
\(534\) −8.82843 −0.382043
\(535\) 0 0
\(536\) −1.51472 −0.0654259
\(537\) − 16.7990i − 0.724930i
\(538\) − 38.1421i − 1.64442i
\(539\) −6.82843 −0.294121
\(540\) 0 0
\(541\) 11.3137 0.486414 0.243207 0.969974i \(-0.421801\pi\)
0.243207 + 0.969974i \(0.421801\pi\)
\(542\) − 21.4853i − 0.922872i
\(543\) − 21.9706i − 0.942847i
\(544\) 3.82843 0.164142
\(545\) 0 0
\(546\) −2.82843 −0.121046
\(547\) − 35.8701i − 1.53369i −0.641831 0.766846i \(-0.721825\pi\)
0.641831 0.766846i \(-0.278175\pi\)
\(548\) − 46.4853i − 1.98575i
\(549\) −8.82843 −0.376788
\(550\) 0 0
\(551\) 7.51472 0.320138
\(552\) − 4.41421i − 0.187881i
\(553\) 5.48528i 0.233258i
\(554\) −11.6569 −0.495252
\(555\) 0 0
\(556\) −72.6274 −3.08009
\(557\) 5.17157i 0.219127i 0.993980 + 0.109563i \(0.0349452\pi\)
−0.993980 + 0.109563i \(0.965055\pi\)
\(558\) − 20.4853i − 0.867211i
\(559\) −32.9706 −1.39451
\(560\) 0 0
\(561\) −2.41421 −0.101928
\(562\) − 77.4264i − 3.26604i
\(563\) 15.3137i 0.645396i 0.946502 + 0.322698i \(0.104590\pi\)
−0.946502 + 0.322698i \(0.895410\pi\)
\(564\) 28.6569 1.20667
\(565\) 0 0
\(566\) −2.17157 −0.0912780
\(567\) − 0.414214i − 0.0173953i
\(568\) 34.5563i 1.44995i
\(569\) −15.2426 −0.639005 −0.319502 0.947585i \(-0.603516\pi\)
−0.319502 + 0.947585i \(0.603516\pi\)
\(570\) 0 0
\(571\) −9.02944 −0.377870 −0.188935 0.981990i \(-0.560504\pi\)
−0.188935 + 0.981990i \(0.560504\pi\)
\(572\) − 10.8284i − 0.452759i
\(573\) 6.17157i 0.257821i
\(574\) 10.8995 0.454936
\(575\) 0 0
\(576\) −9.82843 −0.409518
\(577\) − 23.9706i − 0.997908i −0.866629 0.498954i \(-0.833718\pi\)
0.866629 0.498954i \(-0.166282\pi\)
\(578\) − 26.9706i − 1.12183i
\(579\) −3.31371 −0.137713
\(580\) 0 0
\(581\) −1.85786 −0.0770772
\(582\) − 14.0711i − 0.583265i
\(583\) − 7.65685i − 0.317115i
\(584\) 38.9706 1.61261
\(585\) 0 0
\(586\) 42.4558 1.75383
\(587\) 36.6569i 1.51299i 0.653999 + 0.756495i \(0.273090\pi\)
−0.653999 + 0.756495i \(0.726910\pi\)
\(588\) − 26.1421i − 1.07808i
\(589\) 54.4264 2.24260
\(590\) 0 0
\(591\) 4.75736 0.195692
\(592\) 0.514719i 0.0211548i
\(593\) 3.79899i 0.156006i 0.996953 + 0.0780029i \(0.0248543\pi\)
−0.996953 + 0.0780029i \(0.975146\pi\)
\(594\) 2.41421 0.0990564
\(595\) 0 0
\(596\) 29.5858 1.21188
\(597\) − 10.8284i − 0.443178i
\(598\) − 6.82843i − 0.279235i
\(599\) −36.3137 −1.48374 −0.741869 0.670545i \(-0.766060\pi\)
−0.741869 + 0.670545i \(0.766060\pi\)
\(600\) 0 0
\(601\) −14.8284 −0.604864 −0.302432 0.953171i \(-0.597799\pi\)
−0.302432 + 0.953171i \(0.597799\pi\)
\(602\) 11.6569i 0.475098i
\(603\) − 0.343146i − 0.0139740i
\(604\) −53.5980 −2.18087
\(605\) 0 0
\(606\) 35.9706 1.46120
\(607\) 2.97056i 0.120571i 0.998181 + 0.0602857i \(0.0192012\pi\)
−0.998181 + 0.0602857i \(0.980799\pi\)
\(608\) − 10.1716i − 0.412512i
\(609\) −0.485281 −0.0196646
\(610\) 0 0
\(611\) 21.1716 0.856510
\(612\) − 9.24264i − 0.373612i
\(613\) − 42.0000i − 1.69636i −0.529705 0.848182i \(-0.677697\pi\)
0.529705 0.848182i \(-0.322303\pi\)
\(614\) −16.1421 −0.651444
\(615\) 0 0
\(616\) −1.82843 −0.0736694
\(617\) 9.85786i 0.396863i 0.980115 + 0.198431i \(0.0635847\pi\)
−0.980115 + 0.198431i \(0.936415\pi\)
\(618\) − 32.9706i − 1.32627i
\(619\) −38.6274 −1.55257 −0.776283 0.630384i \(-0.782897\pi\)
−0.776283 + 0.630384i \(0.782897\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) − 32.9706i − 1.32200i
\(623\) 1.51472i 0.0606859i
\(624\) 8.48528 0.339683
\(625\) 0 0
\(626\) −65.5269 −2.61898
\(627\) 6.41421i 0.256159i
\(628\) 22.9706i 0.916625i
\(629\) 0.414214 0.0165158
\(630\) 0 0
\(631\) 26.6274 1.06002 0.530010 0.847991i \(-0.322188\pi\)
0.530010 + 0.847991i \(0.322188\pi\)
\(632\) − 58.4558i − 2.32525i
\(633\) − 13.3137i − 0.529172i
\(634\) −74.4264 −2.95585
\(635\) 0 0
\(636\) 29.3137 1.16236
\(637\) − 19.3137i − 0.765237i
\(638\) − 2.82843i − 0.111979i
\(639\) −7.82843 −0.309688
\(640\) 0 0
\(641\) −42.4853 −1.67807 −0.839034 0.544079i \(-0.816879\pi\)
−0.839034 + 0.544079i \(0.816879\pi\)
\(642\) 12.8284i 0.506298i
\(643\) − 32.9706i − 1.30023i −0.759835 0.650116i \(-0.774720\pi\)
0.759835 0.650116i \(-0.225280\pi\)
\(644\) −1.58579 −0.0624887
\(645\) 0 0
\(646\) 37.3848 1.47088
\(647\) − 17.3431i − 0.681829i −0.940094 0.340915i \(-0.889263\pi\)
0.940094 0.340915i \(-0.110737\pi\)
\(648\) 4.41421i 0.173407i
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) −3.51472 −0.137753
\(652\) − 60.4853i − 2.36879i
\(653\) − 40.4853i − 1.58431i −0.610319 0.792156i \(-0.708959\pi\)
0.610319 0.792156i \(-0.291041\pi\)
\(654\) −12.8284 −0.501631
\(655\) 0 0
\(656\) −32.6985 −1.27666
\(657\) 8.82843i 0.344430i
\(658\) − 7.48528i − 0.291807i
\(659\) −15.1127 −0.588707 −0.294354 0.955697i \(-0.595104\pi\)
−0.294354 + 0.955697i \(0.595104\pi\)
\(660\) 0 0
\(661\) −34.6569 −1.34800 −0.673998 0.738733i \(-0.735424\pi\)
−0.673998 + 0.738733i \(0.735424\pi\)
\(662\) 77.5980i 3.01593i
\(663\) − 6.82843i − 0.265194i
\(664\) 19.7990 0.768350
\(665\) 0 0
\(666\) −0.414214 −0.0160504
\(667\) − 1.17157i − 0.0453635i
\(668\) 83.4558i 3.22900i
\(669\) 21.1716 0.818540
\(670\) 0 0
\(671\) −8.82843 −0.340818
\(672\) 0.656854i 0.0253387i
\(673\) 11.6569i 0.449339i 0.974435 + 0.224669i \(0.0721302\pi\)
−0.974435 + 0.224669i \(0.927870\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −19.1421 −0.736236
\(677\) − 40.4853i − 1.55598i −0.628279 0.777988i \(-0.716240\pi\)
0.628279 0.777988i \(-0.283760\pi\)
\(678\) − 24.1421i − 0.927173i
\(679\) −2.41421 −0.0926490
\(680\) 0 0
\(681\) −18.4853 −0.708358
\(682\) − 20.4853i − 0.784422i
\(683\) 29.4853i 1.12822i 0.825699 + 0.564111i \(0.190781\pi\)
−0.825699 + 0.564111i \(0.809219\pi\)
\(684\) −24.5563 −0.938935
\(685\) 0 0
\(686\) −13.8284 −0.527972
\(687\) 2.51472i 0.0959425i
\(688\) − 34.9706i − 1.33324i
\(689\) 21.6569 0.825060
\(690\) 0 0
\(691\) 15.4558 0.587968 0.293984 0.955810i \(-0.405019\pi\)
0.293984 + 0.955810i \(0.405019\pi\)
\(692\) 48.0711i 1.82739i
\(693\) − 0.414214i − 0.0157347i
\(694\) 51.1127 1.94021
\(695\) 0 0
\(696\) 5.17157 0.196028
\(697\) 26.3137i 0.996703i
\(698\) − 6.00000i − 0.227103i
\(699\) −16.5563 −0.626219
\(700\) 0 0
\(701\) −4.61522 −0.174315 −0.0871573 0.996195i \(-0.527778\pi\)
−0.0871573 + 0.996195i \(0.527778\pi\)
\(702\) 6.82843i 0.257722i
\(703\) − 1.10051i − 0.0415063i
\(704\) −9.82843 −0.370423
\(705\) 0 0
\(706\) −10.8284 −0.407533
\(707\) − 6.17157i − 0.232106i
\(708\) 42.1127i 1.58269i
\(709\) 0.857864 0.0322178 0.0161089 0.999870i \(-0.494872\pi\)
0.0161089 + 0.999870i \(0.494872\pi\)
\(710\) 0 0
\(711\) 13.2426 0.496638
\(712\) − 16.1421i − 0.604952i
\(713\) − 8.48528i − 0.317776i
\(714\) −2.41421 −0.0903497
\(715\) 0 0
\(716\) 64.3137 2.40352
\(717\) − 23.6569i − 0.883481i
\(718\) 37.4558i 1.39784i
\(719\) 1.65685 0.0617902 0.0308951 0.999523i \(-0.490164\pi\)
0.0308951 + 0.999523i \(0.490164\pi\)
\(720\) 0 0
\(721\) −5.65685 −0.210672
\(722\) − 53.4558i − 1.98942i
\(723\) 14.1421i 0.525952i
\(724\) 84.1127 3.12602
\(725\) 0 0
\(726\) 2.41421 0.0895999
\(727\) 16.9706i 0.629403i 0.949191 + 0.314702i \(0.101904\pi\)
−0.949191 + 0.314702i \(0.898096\pi\)
\(728\) − 5.17157i − 0.191671i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −28.1421 −1.04087
\(732\) − 33.7990i − 1.24925i
\(733\) 3.85786i 0.142493i 0.997459 + 0.0712467i \(0.0226978\pi\)
−0.997459 + 0.0712467i \(0.977302\pi\)
\(734\) −3.17157 −0.117065
\(735\) 0 0
\(736\) −1.58579 −0.0584529
\(737\) − 0.343146i − 0.0126399i
\(738\) − 26.3137i − 0.968621i
\(739\) −17.5858 −0.646904 −0.323452 0.946245i \(-0.604843\pi\)
−0.323452 + 0.946245i \(0.604843\pi\)
\(740\) 0 0
\(741\) −18.1421 −0.666468
\(742\) − 7.65685i − 0.281092i
\(743\) 31.1127i 1.14141i 0.821154 + 0.570707i \(0.193331\pi\)
−0.821154 + 0.570707i \(0.806669\pi\)
\(744\) 37.4558 1.37320
\(745\) 0 0
\(746\) −57.1127 −2.09104
\(747\) 4.48528i 0.164108i
\(748\) − 9.24264i − 0.337944i
\(749\) 2.20101 0.0804232
\(750\) 0 0
\(751\) −47.5980 −1.73687 −0.868437 0.495799i \(-0.834875\pi\)
−0.868437 + 0.495799i \(0.834875\pi\)
\(752\) 22.4558i 0.818880i
\(753\) 8.97056i 0.326905i
\(754\) 8.00000 0.291343
\(755\) 0 0
\(756\) 1.58579 0.0576745
\(757\) 33.3137i 1.21081i 0.795919 + 0.605404i \(0.206988\pi\)
−0.795919 + 0.605404i \(0.793012\pi\)
\(758\) 22.1421i 0.804239i
\(759\) 1.00000 0.0362977
\(760\) 0 0
\(761\) −10.8284 −0.392530 −0.196265 0.980551i \(-0.562881\pi\)
−0.196265 + 0.980551i \(0.562881\pi\)
\(762\) − 17.4853i − 0.633425i
\(763\) 2.20101i 0.0796819i
\(764\) −23.6274 −0.854810
\(765\) 0 0
\(766\) 48.2843 1.74458
\(767\) 31.1127i 1.12341i
\(768\) − 29.9706i − 1.08147i
\(769\) −3.65685 −0.131870 −0.0659348 0.997824i \(-0.521003\pi\)
−0.0659348 + 0.997824i \(0.521003\pi\)
\(770\) 0 0
\(771\) 9.31371 0.335425
\(772\) − 12.6863i − 0.456590i
\(773\) 4.82843i 0.173666i 0.996223 + 0.0868332i \(0.0276747\pi\)
−0.996223 + 0.0868332i \(0.972325\pi\)
\(774\) 28.1421 1.01155
\(775\) 0 0
\(776\) 25.7279 0.923579
\(777\) 0.0710678i 0.00254954i
\(778\) − 42.6274i − 1.52827i
\(779\) 69.9117 2.50485
\(780\) 0 0
\(781\) −7.82843 −0.280123
\(782\) − 5.82843i − 0.208424i
\(783\) 1.17157i 0.0418686i
\(784\) 20.4853 0.731617
\(785\) 0 0
\(786\) 16.4853 0.588011
\(787\) 22.0711i 0.786749i 0.919378 + 0.393374i \(0.128692\pi\)
−0.919378 + 0.393374i \(0.871308\pi\)
\(788\) 18.2132i 0.648819i
\(789\) −22.9706 −0.817774
\(790\) 0 0
\(791\) −4.14214 −0.147277
\(792\) 4.41421i 0.156852i
\(793\) − 24.9706i − 0.886731i
\(794\) 86.7696 3.07934
\(795\) 0 0
\(796\) 41.4558 1.46936
\(797\) − 7.02944i − 0.248995i −0.992220 0.124498i \(-0.960268\pi\)
0.992220 0.124498i \(-0.0397319\pi\)
\(798\) 6.41421i 0.227061i
\(799\) 18.0711 0.639308
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) − 76.7696i − 2.71083i
\(803\) 8.82843i 0.311548i
\(804\) 1.31371 0.0463309
\(805\) 0 0
\(806\) 57.9411 2.04089
\(807\) 15.7990i 0.556151i
\(808\) 65.7696i 2.31376i
\(809\) 17.7279 0.623281 0.311640 0.950200i \(-0.399122\pi\)
0.311640 + 0.950200i \(0.399122\pi\)
\(810\) 0 0
\(811\) −32.2132 −1.13116 −0.565579 0.824694i \(-0.691347\pi\)
−0.565579 + 0.824694i \(0.691347\pi\)
\(812\) − 1.85786i − 0.0651983i
\(813\) 8.89949i 0.312119i
\(814\) −0.414214 −0.0145182
\(815\) 0 0
\(816\) 7.24264 0.253543
\(817\) 74.7696i 2.61586i
\(818\) − 10.0000i − 0.349642i
\(819\) 1.17157 0.0409381
\(820\) 0 0
\(821\) −23.7990 −0.830590 −0.415295 0.909687i \(-0.636322\pi\)
−0.415295 + 0.909687i \(0.636322\pi\)
\(822\) 29.3137i 1.02243i
\(823\) 18.9706i 0.661272i 0.943758 + 0.330636i \(0.107263\pi\)
−0.943758 + 0.330636i \(0.892737\pi\)
\(824\) 60.2843 2.10010
\(825\) 0 0
\(826\) 11.0000 0.382739
\(827\) 35.3137i 1.22798i 0.789315 + 0.613989i \(0.210436\pi\)
−0.789315 + 0.613989i \(0.789564\pi\)
\(828\) 3.82843i 0.133047i
\(829\) −19.9411 −0.692584 −0.346292 0.938127i \(-0.612559\pi\)
−0.346292 + 0.938127i \(0.612559\pi\)
\(830\) 0 0
\(831\) 4.82843 0.167496
\(832\) − 27.7990i − 0.963757i
\(833\) − 16.4853i − 0.571181i
\(834\) 45.7990 1.58589
\(835\) 0 0
\(836\) −24.5563 −0.849299
\(837\) 8.48528i 0.293294i
\(838\) − 61.5269i − 2.12541i
\(839\) 24.6863 0.852265 0.426133 0.904661i \(-0.359876\pi\)
0.426133 + 0.904661i \(0.359876\pi\)
\(840\) 0 0
\(841\) −27.6274 −0.952670
\(842\) 65.1838i 2.24638i
\(843\) 32.0711i 1.10459i
\(844\) 50.9706 1.75448
\(845\) 0 0
\(846\) −18.0711 −0.621296
\(847\) − 0.414214i − 0.0142325i
\(848\) 22.9706i 0.788812i
\(849\) 0.899495 0.0308706
\(850\) 0 0
\(851\) −0.171573 −0.00588144
\(852\) − 29.9706i − 1.02677i
\(853\) − 24.8284i − 0.850109i −0.905168 0.425055i \(-0.860255\pi\)
0.905168 0.425055i \(-0.139745\pi\)
\(854\) −8.82843 −0.302103
\(855\) 0 0
\(856\) −23.4558 −0.801704
\(857\) − 22.6985i − 0.775365i −0.921793 0.387683i \(-0.873276\pi\)
0.921793 0.387683i \(-0.126724\pi\)
\(858\) 6.82843i 0.233119i
\(859\) 3.51472 0.119921 0.0599603 0.998201i \(-0.480903\pi\)
0.0599603 + 0.998201i \(0.480903\pi\)
\(860\) 0 0
\(861\) −4.51472 −0.153861
\(862\) − 2.82843i − 0.0963366i
\(863\) − 51.3137i − 1.74674i −0.487058 0.873369i \(-0.661930\pi\)
0.487058 0.873369i \(-0.338070\pi\)
\(864\) 1.58579 0.0539496
\(865\) 0 0
\(866\) −32.1421 −1.09223
\(867\) 11.1716i 0.379407i
\(868\) − 13.4558i − 0.456721i
\(869\) 13.2426 0.449226
\(870\) 0 0
\(871\) 0.970563 0.0328863
\(872\) − 23.4558i − 0.794315i
\(873\) 5.82843i 0.197262i
\(874\) −15.4853 −0.523797
\(875\) 0 0
\(876\) −33.7990 −1.14196
\(877\) 47.1127i 1.59088i 0.606031 + 0.795441i \(0.292761\pi\)
−0.606031 + 0.795441i \(0.707239\pi\)
\(878\) 5.48528i 0.185119i
\(879\) −17.5858 −0.593154
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 16.4853i 0.555088i
\(883\) − 6.62742i − 0.223030i −0.993763 0.111515i \(-0.964430\pi\)
0.993763 0.111515i \(-0.0355704\pi\)
\(884\) 26.1421 0.879255
\(885\) 0 0
\(886\) −19.2426 −0.646469
\(887\) − 6.14214i − 0.206233i −0.994669 0.103116i \(-0.967119\pi\)
0.994669 0.103116i \(-0.0328814\pi\)
\(888\) − 0.757359i − 0.0254153i
\(889\) −3.00000 −0.100617
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 81.0538i 2.71388i
\(893\) − 48.0122i − 1.60667i
\(894\) −18.6569 −0.623979
\(895\) 0 0
\(896\) −8.51472 −0.284457
\(897\) 2.82843i 0.0944384i
\(898\) − 15.6569i − 0.522476i
\(899\) 9.94113 0.331555
\(900\) 0 0
\(901\) 18.4853 0.615834
\(902\) − 26.3137i − 0.876151i
\(903\) − 4.82843i − 0.160680i
\(904\) 44.1421 1.46815
\(905\) 0 0
\(906\) 33.7990 1.12290
\(907\) − 14.4853i − 0.480976i −0.970652 0.240488i \(-0.922693\pi\)
0.970652 0.240488i \(-0.0773074\pi\)
\(908\) − 70.7696i − 2.34857i
\(909\) −14.8995 −0.494185
\(910\) 0 0
\(911\) 45.4853 1.50699 0.753497 0.657451i \(-0.228365\pi\)
0.753497 + 0.657451i \(0.228365\pi\)
\(912\) − 19.2426i − 0.637188i
\(913\) 4.48528i 0.148441i
\(914\) −9.31371 −0.308070
\(915\) 0 0
\(916\) −9.62742 −0.318099
\(917\) − 2.82843i − 0.0934029i
\(918\) 5.82843i 0.192367i
\(919\) −44.2132 −1.45846 −0.729230 0.684269i \(-0.760121\pi\)
−0.729230 + 0.684269i \(0.760121\pi\)
\(920\) 0 0
\(921\) 6.68629 0.220321
\(922\) 79.1127i 2.60544i
\(923\) − 22.1421i − 0.728817i
\(924\) 1.58579 0.0521685
\(925\) 0 0
\(926\) 84.4264 2.77442
\(927\) 13.6569i 0.448550i
\(928\) − 1.85786i − 0.0609874i
\(929\) −25.7990 −0.846437 −0.423219 0.906028i \(-0.639100\pi\)
−0.423219 + 0.906028i \(0.639100\pi\)
\(930\) 0 0
\(931\) −43.7990 −1.43545
\(932\) − 63.3848i − 2.07624i
\(933\) 13.6569i 0.447105i
\(934\) −25.6569 −0.839518
\(935\) 0 0
\(936\) −12.4853 −0.408094
\(937\) 16.0000i 0.522697i 0.965244 + 0.261349i \(0.0841672\pi\)
−0.965244 + 0.261349i \(0.915833\pi\)
\(938\) − 0.343146i − 0.0112041i
\(939\) 27.1421 0.885750
\(940\) 0 0
\(941\) −3.10051 −0.101074 −0.0505368 0.998722i \(-0.516093\pi\)
−0.0505368 + 0.998722i \(0.516093\pi\)
\(942\) − 14.4853i − 0.471956i
\(943\) − 10.8995i − 0.354936i
\(944\) −33.0000 −1.07406
\(945\) 0 0
\(946\) 28.1421 0.914980
\(947\) − 2.79899i − 0.0909549i −0.998965 0.0454775i \(-0.985519\pi\)
0.998965 0.0454775i \(-0.0144809\pi\)
\(948\) 50.6985i 1.64661i
\(949\) −24.9706 −0.810579
\(950\) 0 0
\(951\) 30.8284 0.999680
\(952\) − 4.41421i − 0.143065i
\(953\) 19.0416i 0.616819i 0.951254 + 0.308409i \(0.0997967\pi\)
−0.951254 + 0.308409i \(0.900203\pi\)
\(954\) −18.4853 −0.598483
\(955\) 0 0
\(956\) 90.5685 2.92920
\(957\) 1.17157i 0.0378716i
\(958\) − 59.1127i − 1.90984i
\(959\) 5.02944 0.162409
\(960\) 0 0
\(961\) 41.0000 1.32258
\(962\) − 1.17157i − 0.0377730i
\(963\) − 5.31371i − 0.171232i
\(964\) −54.1421 −1.74380
\(965\) 0 0
\(966\) 1.00000 0.0321745
\(967\) − 38.0000i − 1.22200i −0.791632 0.610999i \(-0.790768\pi\)
0.791632 0.610999i \(-0.209232\pi\)
\(968\) 4.41421i 0.141878i
\(969\) −15.4853 −0.497459
\(970\) 0 0
\(971\) −27.6863 −0.888495 −0.444248 0.895904i \(-0.646529\pi\)
−0.444248 + 0.895904i \(0.646529\pi\)
\(972\) − 3.82843i − 0.122797i
\(973\) − 7.85786i − 0.251912i
\(974\) 15.6569 0.501678
\(975\) 0 0
\(976\) 26.4853 0.847773
\(977\) 52.5685i 1.68182i 0.541178 + 0.840908i \(0.317979\pi\)
−0.541178 + 0.840908i \(0.682021\pi\)
\(978\) 38.1421i 1.21965i
\(979\) 3.65685 0.116874
\(980\) 0 0
\(981\) 5.31371 0.169654
\(982\) 58.2843i 1.85993i
\(983\) − 5.28427i − 0.168542i −0.996443 0.0842710i \(-0.973144\pi\)
0.996443 0.0842710i \(-0.0268562\pi\)
\(984\) 48.1127 1.53378
\(985\) 0 0
\(986\) 6.82843 0.217461
\(987\) 3.10051i 0.0986902i
\(988\) − 69.4558i − 2.20968i
\(989\) 11.6569 0.370666
\(990\) 0 0
\(991\) 14.2843 0.453755 0.226877 0.973923i \(-0.427148\pi\)
0.226877 + 0.973923i \(0.427148\pi\)
\(992\) − 13.4558i − 0.427223i
\(993\) − 32.1421i − 1.02000i
\(994\) −7.82843 −0.248303
\(995\) 0 0
\(996\) −17.1716 −0.544102
\(997\) − 43.2548i − 1.36989i −0.728593 0.684947i \(-0.759825\pi\)
0.728593 0.684947i \(-0.240175\pi\)
\(998\) 84.9117i 2.68783i
\(999\) 0.171573 0.00542833
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 825.2.c.d.199.1 4
3.2 odd 2 2475.2.c.o.199.4 4
5.2 odd 4 825.2.a.f.1.2 yes 2
5.3 odd 4 825.2.a.d.1.1 2
5.4 even 2 inner 825.2.c.d.199.4 4
15.2 even 4 2475.2.a.l.1.1 2
15.8 even 4 2475.2.a.w.1.2 2
15.14 odd 2 2475.2.c.o.199.1 4
55.32 even 4 9075.2.a.w.1.1 2
55.43 even 4 9075.2.a.ca.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.2.a.d.1.1 2 5.3 odd 4
825.2.a.f.1.2 yes 2 5.2 odd 4
825.2.c.d.199.1 4 1.1 even 1 trivial
825.2.c.d.199.4 4 5.4 even 2 inner
2475.2.a.l.1.1 2 15.2 even 4
2475.2.a.w.1.2 2 15.8 even 4
2475.2.c.o.199.1 4 15.14 odd 2
2475.2.c.o.199.4 4 3.2 odd 2
9075.2.a.w.1.1 2 55.32 even 4
9075.2.a.ca.1.2 2 55.43 even 4