Properties

Label 378.5.b.a.323.8
Level $378$
Weight $5$
Character 378.323
Analytic conductor $39.074$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 378.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(39.0738460457\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.5443747577856.29
Defining polynomial: \( x^{8} + 24x^{6} + 180x^{4} + 488x^{2} + 324 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.8
Root \(-0.982345i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.a.323.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{2} -8.00000 q^{4} +42.2558i q^{5} -18.5203 q^{7} -22.6274i q^{8} +O(q^{10})\) \(q+2.82843i q^{2} -8.00000 q^{4} +42.2558i q^{5} -18.5203 q^{7} -22.6274i q^{8} -119.518 q^{10} +63.4392i q^{11} -99.9091 q^{13} -52.3832i q^{14} +64.0000 q^{16} +450.455i q^{17} -262.252 q^{19} -338.047i q^{20} -179.433 q^{22} -673.509i q^{23} -1160.56 q^{25} -282.586i q^{26} +148.162 q^{28} +498.837i q^{29} -201.115 q^{31} +181.019i q^{32} -1274.08 q^{34} -782.589i q^{35} +2517.87 q^{37} -741.761i q^{38} +956.140 q^{40} +1698.04i q^{41} -1359.85 q^{43} -507.513i q^{44} +1904.97 q^{46} -685.838i q^{47} +343.000 q^{49} -3282.55i q^{50} +799.273 q^{52} -3969.53i q^{53} -2680.68 q^{55} +419.066i q^{56} -1410.93 q^{58} -741.755i q^{59} -4675.57 q^{61} -568.838i q^{62} -512.000 q^{64} -4221.74i q^{65} +7531.63 q^{67} -3603.64i q^{68} +2213.50 q^{70} -8956.91i q^{71} -1465.30 q^{73} +7121.61i q^{74} +2098.02 q^{76} -1174.91i q^{77} -5557.36 q^{79} +2704.37i q^{80} -4802.77 q^{82} -2588.64i q^{83} -19034.3 q^{85} -3846.24i q^{86} +1435.46 q^{88} -8753.43i q^{89} +1850.34 q^{91} +5388.07i q^{92} +1939.84 q^{94} -11081.7i q^{95} +4336.45 q^{97} +970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 224 q^{10} - 376 q^{13} + 512 q^{16} + 1120 q^{19} - 1120 q^{22} + 792 q^{25} - 880 q^{31} - 1792 q^{34} + 1576 q^{37} + 1792 q^{40} - 5768 q^{43} - 160 q^{46} + 2744 q^{49} + 3008 q^{52} + 488 q^{55} + 7552 q^{58} - 2560 q^{61} - 4096 q^{64} + 23784 q^{67} + 1568 q^{70} - 13176 q^{73} - 8960 q^{76} - 9592 q^{79} - 3360 q^{82} - 39880 q^{85} + 8960 q^{88} + 5096 q^{91} + 4608 q^{94} - 14016 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.82843i 0.707107i
\(3\) 0 0
\(4\) −8.00000 −0.500000
\(5\) 42.2558i 1.69023i 0.534582 + 0.845117i \(0.320469\pi\)
−0.534582 + 0.845117i \(0.679531\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) − 22.6274i − 0.353553i
\(9\) 0 0
\(10\) −119.518 −1.19518
\(11\) 63.4392i 0.524291i 0.965028 + 0.262145i \(0.0844300\pi\)
−0.965028 + 0.262145i \(0.915570\pi\)
\(12\) 0 0
\(13\) −99.9091 −0.591178 −0.295589 0.955315i \(-0.595516\pi\)
−0.295589 + 0.955315i \(0.595516\pi\)
\(14\) − 52.3832i − 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 450.455i 1.55867i 0.626609 + 0.779334i \(0.284442\pi\)
−0.626609 + 0.779334i \(0.715558\pi\)
\(18\) 0 0
\(19\) −262.252 −0.726460 −0.363230 0.931700i \(-0.618326\pi\)
−0.363230 + 0.931700i \(0.618326\pi\)
\(20\) − 338.047i − 0.845117i
\(21\) 0 0
\(22\) −179.433 −0.370730
\(23\) − 673.509i − 1.27317i −0.771205 0.636587i \(-0.780346\pi\)
0.771205 0.636587i \(-0.219654\pi\)
\(24\) 0 0
\(25\) −1160.56 −1.85689
\(26\) − 282.586i − 0.418026i
\(27\) 0 0
\(28\) 148.162 0.188982
\(29\) 498.837i 0.593148i 0.955010 + 0.296574i \(0.0958441\pi\)
−0.955010 + 0.296574i \(0.904156\pi\)
\(30\) 0 0
\(31\) −201.115 −0.209276 −0.104638 0.994510i \(-0.533368\pi\)
−0.104638 + 0.994510i \(0.533368\pi\)
\(32\) 181.019i 0.176777i
\(33\) 0 0
\(34\) −1274.08 −1.10214
\(35\) − 782.589i − 0.638848i
\(36\) 0 0
\(37\) 2517.87 1.83920 0.919602 0.392852i \(-0.128512\pi\)
0.919602 + 0.392852i \(0.128512\pi\)
\(38\) − 741.761i − 0.513685i
\(39\) 0 0
\(40\) 956.140 0.597588
\(41\) 1698.04i 1.01013i 0.863080 + 0.505067i \(0.168532\pi\)
−0.863080 + 0.505067i \(0.831468\pi\)
\(42\) 0 0
\(43\) −1359.85 −0.735452 −0.367726 0.929934i \(-0.619864\pi\)
−0.367726 + 0.929934i \(0.619864\pi\)
\(44\) − 507.513i − 0.262145i
\(45\) 0 0
\(46\) 1904.97 0.900270
\(47\) − 685.838i − 0.310474i −0.987877 0.155237i \(-0.950386\pi\)
0.987877 0.155237i \(-0.0496142\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) − 3282.55i − 1.31302i
\(51\) 0 0
\(52\) 799.273 0.295589
\(53\) − 3969.53i − 1.41315i −0.707640 0.706573i \(-0.750240\pi\)
0.707640 0.706573i \(-0.249760\pi\)
\(54\) 0 0
\(55\) −2680.68 −0.886174
\(56\) 419.066i 0.133631i
\(57\) 0 0
\(58\) −1410.93 −0.419419
\(59\) − 741.755i − 0.213087i −0.994308 0.106543i \(-0.966022\pi\)
0.994308 0.106543i \(-0.0339783\pi\)
\(60\) 0 0
\(61\) −4675.57 −1.25654 −0.628268 0.777997i \(-0.716236\pi\)
−0.628268 + 0.777997i \(0.716236\pi\)
\(62\) − 568.838i − 0.147981i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 4221.74i − 0.999229i
\(66\) 0 0
\(67\) 7531.63 1.67780 0.838899 0.544288i \(-0.183200\pi\)
0.838899 + 0.544288i \(0.183200\pi\)
\(68\) − 3603.64i − 0.779334i
\(69\) 0 0
\(70\) 2213.50 0.451734
\(71\) − 8956.91i − 1.77681i −0.459059 0.888406i \(-0.651813\pi\)
0.459059 0.888406i \(-0.348187\pi\)
\(72\) 0 0
\(73\) −1465.30 −0.274967 −0.137484 0.990504i \(-0.543901\pi\)
−0.137484 + 0.990504i \(0.543901\pi\)
\(74\) 7121.61i 1.30051i
\(75\) 0 0
\(76\) 2098.02 0.363230
\(77\) − 1174.91i − 0.198163i
\(78\) 0 0
\(79\) −5557.36 −0.890460 −0.445230 0.895416i \(-0.646878\pi\)
−0.445230 + 0.895416i \(0.646878\pi\)
\(80\) 2704.37i 0.422558i
\(81\) 0 0
\(82\) −4802.77 −0.714273
\(83\) − 2588.64i − 0.375765i −0.982192 0.187882i \(-0.939838\pi\)
0.982192 0.187882i \(-0.0601623\pi\)
\(84\) 0 0
\(85\) −19034.3 −2.63451
\(86\) − 3846.24i − 0.520043i
\(87\) 0 0
\(88\) 1435.46 0.185365
\(89\) − 8753.43i − 1.10509i −0.833483 0.552545i \(-0.813657\pi\)
0.833483 0.552545i \(-0.186343\pi\)
\(90\) 0 0
\(91\) 1850.34 0.223444
\(92\) 5388.07i 0.636587i
\(93\) 0 0
\(94\) 1939.84 0.219538
\(95\) − 11081.7i − 1.22789i
\(96\) 0 0
\(97\) 4336.45 0.460883 0.230442 0.973086i \(-0.425983\pi\)
0.230442 + 0.973086i \(0.425983\pi\)
\(98\) 970.151i 0.101015i
\(99\) 0 0
\(100\) 9284.44 0.928444
\(101\) 1236.26i 0.121190i 0.998162 + 0.0605950i \(0.0192998\pi\)
−0.998162 + 0.0605950i \(0.980700\pi\)
\(102\) 0 0
\(103\) −16956.4 −1.59831 −0.799153 0.601128i \(-0.794718\pi\)
−0.799153 + 0.601128i \(0.794718\pi\)
\(104\) 2260.68i 0.209013i
\(105\) 0 0
\(106\) 11227.5 0.999246
\(107\) − 4735.53i − 0.413619i −0.978381 0.206810i \(-0.933692\pi\)
0.978381 0.206810i \(-0.0663081\pi\)
\(108\) 0 0
\(109\) 18182.9 1.53042 0.765209 0.643782i \(-0.222636\pi\)
0.765209 + 0.643782i \(0.222636\pi\)
\(110\) − 7582.10i − 0.626619i
\(111\) 0 0
\(112\) −1185.30 −0.0944911
\(113\) 19988.6i 1.56540i 0.622398 + 0.782701i \(0.286159\pi\)
−0.622398 + 0.782701i \(0.713841\pi\)
\(114\) 0 0
\(115\) 28459.7 2.15196
\(116\) − 3990.70i − 0.296574i
\(117\) 0 0
\(118\) 2098.00 0.150675
\(119\) − 8342.54i − 0.589121i
\(120\) 0 0
\(121\) 10616.5 0.725119
\(122\) − 13224.5i − 0.888504i
\(123\) 0 0
\(124\) 1608.92 0.104638
\(125\) − 22630.3i − 1.44834i
\(126\) 0 0
\(127\) 9159.59 0.567896 0.283948 0.958840i \(-0.408356\pi\)
0.283948 + 0.958840i \(0.408356\pi\)
\(128\) − 1448.15i − 0.0883883i
\(129\) 0 0
\(130\) 11940.9 0.706561
\(131\) − 8015.64i − 0.467084i −0.972347 0.233542i \(-0.924968\pi\)
0.972347 0.233542i \(-0.0750317\pi\)
\(132\) 0 0
\(133\) 4856.98 0.274576
\(134\) 21302.7i 1.18638i
\(135\) 0 0
\(136\) 10192.6 0.551072
\(137\) 13447.2i 0.716459i 0.933634 + 0.358229i \(0.116619\pi\)
−0.933634 + 0.358229i \(0.883381\pi\)
\(138\) 0 0
\(139\) −27654.5 −1.43132 −0.715658 0.698451i \(-0.753873\pi\)
−0.715658 + 0.698451i \(0.753873\pi\)
\(140\) 6260.71i 0.319424i
\(141\) 0 0
\(142\) 25334.0 1.25640
\(143\) − 6338.15i − 0.309949i
\(144\) 0 0
\(145\) −21078.8 −1.00256
\(146\) − 4144.49i − 0.194431i
\(147\) 0 0
\(148\) −20143.0 −0.919602
\(149\) 23453.9i 1.05643i 0.849110 + 0.528217i \(0.177139\pi\)
−0.849110 + 0.528217i \(0.822861\pi\)
\(150\) 0 0
\(151\) 2549.51 0.111816 0.0559078 0.998436i \(-0.482195\pi\)
0.0559078 + 0.998436i \(0.482195\pi\)
\(152\) 5934.09i 0.256842i
\(153\) 0 0
\(154\) 3323.15 0.140123
\(155\) − 8498.27i − 0.353726i
\(156\) 0 0
\(157\) −17969.8 −0.729029 −0.364514 0.931198i \(-0.618765\pi\)
−0.364514 + 0.931198i \(0.618765\pi\)
\(158\) − 15718.6i − 0.629650i
\(159\) 0 0
\(160\) −7649.12 −0.298794
\(161\) 12473.6i 0.481214i
\(162\) 0 0
\(163\) −25505.1 −0.959956 −0.479978 0.877281i \(-0.659355\pi\)
−0.479978 + 0.877281i \(0.659355\pi\)
\(164\) − 13584.3i − 0.505067i
\(165\) 0 0
\(166\) 7321.79 0.265706
\(167\) 1848.54i 0.0662821i 0.999451 + 0.0331410i \(0.0105510\pi\)
−0.999451 + 0.0331410i \(0.989449\pi\)
\(168\) 0 0
\(169\) −18579.2 −0.650509
\(170\) − 53837.3i − 1.86288i
\(171\) 0 0
\(172\) 10878.8 0.367726
\(173\) 40682.6i 1.35930i 0.733535 + 0.679652i \(0.237869\pi\)
−0.733535 + 0.679652i \(0.762131\pi\)
\(174\) 0 0
\(175\) 21493.8 0.701838
\(176\) 4060.11i 0.131073i
\(177\) 0 0
\(178\) 24758.4 0.781417
\(179\) − 38271.8i − 1.19446i −0.802069 0.597232i \(-0.796267\pi\)
0.802069 0.597232i \(-0.203733\pi\)
\(180\) 0 0
\(181\) −46467.3 −1.41837 −0.709186 0.705022i \(-0.750937\pi\)
−0.709186 + 0.705022i \(0.750937\pi\)
\(182\) 5233.56i 0.157999i
\(183\) 0 0
\(184\) −15239.8 −0.450135
\(185\) 106395.i 3.10868i
\(186\) 0 0
\(187\) −28576.5 −0.817195
\(188\) 5486.70i 0.155237i
\(189\) 0 0
\(190\) 31343.7 0.868247
\(191\) 48153.9i 1.31997i 0.751278 + 0.659986i \(0.229438\pi\)
−0.751278 + 0.659986i \(0.770562\pi\)
\(192\) 0 0
\(193\) −51645.0 −1.38648 −0.693240 0.720706i \(-0.743818\pi\)
−0.693240 + 0.720706i \(0.743818\pi\)
\(194\) 12265.3i 0.325894i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 4002.20i 0.103126i 0.998670 + 0.0515628i \(0.0164202\pi\)
−0.998670 + 0.0515628i \(0.983580\pi\)
\(198\) 0 0
\(199\) 22970.5 0.580048 0.290024 0.957019i \(-0.406337\pi\)
0.290024 + 0.957019i \(0.406337\pi\)
\(200\) 26260.4i 0.656509i
\(201\) 0 0
\(202\) −3496.67 −0.0856942
\(203\) − 9238.60i − 0.224189i
\(204\) 0 0
\(205\) −71751.9 −1.70736
\(206\) − 47960.0i − 1.13017i
\(207\) 0 0
\(208\) −6394.18 −0.147794
\(209\) − 16637.1i − 0.380876i
\(210\) 0 0
\(211\) −51005.4 −1.14565 −0.572824 0.819679i \(-0.694152\pi\)
−0.572824 + 0.819679i \(0.694152\pi\)
\(212\) 31756.2i 0.706573i
\(213\) 0 0
\(214\) 13394.1 0.292473
\(215\) − 57461.6i − 1.24309i
\(216\) 0 0
\(217\) 3724.70 0.0790991
\(218\) 51429.0i 1.08217i
\(219\) 0 0
\(220\) 21445.4 0.443087
\(221\) − 45004.5i − 0.921450i
\(222\) 0 0
\(223\) −16479.0 −0.331375 −0.165688 0.986178i \(-0.552984\pi\)
−0.165688 + 0.986178i \(0.552984\pi\)
\(224\) − 3352.53i − 0.0668153i
\(225\) 0 0
\(226\) −56536.3 −1.10691
\(227\) 101140.i 1.96278i 0.192026 + 0.981390i \(0.438494\pi\)
−0.192026 + 0.981390i \(0.561506\pi\)
\(228\) 0 0
\(229\) 98918.5 1.88628 0.943140 0.332394i \(-0.107857\pi\)
0.943140 + 0.332394i \(0.107857\pi\)
\(230\) 80496.1i 1.52167i
\(231\) 0 0
\(232\) 11287.4 0.209709
\(233\) 8359.71i 0.153985i 0.997032 + 0.0769926i \(0.0245318\pi\)
−0.997032 + 0.0769926i \(0.975468\pi\)
\(234\) 0 0
\(235\) 28980.6 0.524774
\(236\) 5934.04i 0.106543i
\(237\) 0 0
\(238\) 23596.3 0.416571
\(239\) 29969.3i 0.524663i 0.964978 + 0.262331i \(0.0844913\pi\)
−0.964978 + 0.262331i \(0.915509\pi\)
\(240\) 0 0
\(241\) −74966.2 −1.29072 −0.645359 0.763879i \(-0.723292\pi\)
−0.645359 + 0.763879i \(0.723292\pi\)
\(242\) 30027.9i 0.512737i
\(243\) 0 0
\(244\) 37404.5 0.628268
\(245\) 14493.8i 0.241462i
\(246\) 0 0
\(247\) 26201.4 0.429467
\(248\) 4550.71i 0.0739904i
\(249\) 0 0
\(250\) 64008.3 1.02413
\(251\) 92012.3i 1.46049i 0.683186 + 0.730244i \(0.260594\pi\)
−0.683186 + 0.730244i \(0.739406\pi\)
\(252\) 0 0
\(253\) 42726.8 0.667513
\(254\) 25907.2i 0.401563i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 30757.4i − 0.465676i −0.972516 0.232838i \(-0.925199\pi\)
0.972516 0.232838i \(-0.0748011\pi\)
\(258\) 0 0
\(259\) −46631.6 −0.695153
\(260\) 33773.9i 0.499614i
\(261\) 0 0
\(262\) 22671.6 0.330279
\(263\) 37491.9i 0.542034i 0.962575 + 0.271017i \(0.0873599\pi\)
−0.962575 + 0.271017i \(0.912640\pi\)
\(264\) 0 0
\(265\) 167736. 2.38855
\(266\) 13737.6i 0.194155i
\(267\) 0 0
\(268\) −60253.1 −0.838899
\(269\) 37506.9i 0.518330i 0.965833 + 0.259165i \(0.0834473\pi\)
−0.965833 + 0.259165i \(0.916553\pi\)
\(270\) 0 0
\(271\) 53171.4 0.724002 0.362001 0.932178i \(-0.382094\pi\)
0.362001 + 0.932178i \(0.382094\pi\)
\(272\) 28829.1i 0.389667i
\(273\) 0 0
\(274\) −38034.5 −0.506613
\(275\) − 73624.7i − 0.973550i
\(276\) 0 0
\(277\) −30111.4 −0.392438 −0.196219 0.980560i \(-0.562866\pi\)
−0.196219 + 0.980560i \(0.562866\pi\)
\(278\) − 78218.6i − 1.01209i
\(279\) 0 0
\(280\) −17708.0 −0.225867
\(281\) − 28226.0i − 0.357467i −0.983898 0.178734i \(-0.942800\pi\)
0.983898 0.178734i \(-0.0572000\pi\)
\(282\) 0 0
\(283\) 11670.3 0.145716 0.0728581 0.997342i \(-0.476788\pi\)
0.0728581 + 0.997342i \(0.476788\pi\)
\(284\) 71655.3i 0.888406i
\(285\) 0 0
\(286\) 17927.0 0.219167
\(287\) − 31448.1i − 0.381795i
\(288\) 0 0
\(289\) −119389. −1.42944
\(290\) − 59619.8i − 0.708916i
\(291\) 0 0
\(292\) 11722.4 0.137484
\(293\) − 29595.2i − 0.344735i −0.985033 0.172368i \(-0.944858\pi\)
0.985033 0.172368i \(-0.0551417\pi\)
\(294\) 0 0
\(295\) 31343.5 0.360166
\(296\) − 56972.9i − 0.650257i
\(297\) 0 0
\(298\) −66337.6 −0.747011
\(299\) 67289.6i 0.752672i
\(300\) 0 0
\(301\) 25184.8 0.277975
\(302\) 7211.09i 0.0790655i
\(303\) 0 0
\(304\) −16784.1 −0.181615
\(305\) − 197570.i − 2.12384i
\(306\) 0 0
\(307\) −96742.0 −1.02645 −0.513225 0.858254i \(-0.671550\pi\)
−0.513225 + 0.858254i \(0.671550\pi\)
\(308\) 9399.28i 0.0990816i
\(309\) 0 0
\(310\) 24036.7 0.250122
\(311\) 83000.7i 0.858146i 0.903270 + 0.429073i \(0.141160\pi\)
−0.903270 + 0.429073i \(0.858840\pi\)
\(312\) 0 0
\(313\) 148025. 1.51093 0.755467 0.655186i \(-0.227410\pi\)
0.755467 + 0.655186i \(0.227410\pi\)
\(314\) − 50826.3i − 0.515501i
\(315\) 0 0
\(316\) 44458.9 0.445230
\(317\) − 61615.0i − 0.613152i −0.951846 0.306576i \(-0.900817\pi\)
0.951846 0.306576i \(-0.0991833\pi\)
\(318\) 0 0
\(319\) −31645.8 −0.310982
\(320\) − 21635.0i − 0.211279i
\(321\) 0 0
\(322\) −35280.5 −0.340270
\(323\) − 118133.i − 1.13231i
\(324\) 0 0
\(325\) 115950. 1.09775
\(326\) − 72139.2i − 0.678791i
\(327\) 0 0
\(328\) 38422.2 0.357136
\(329\) 12701.9i 0.117348i
\(330\) 0 0
\(331\) −100037. −0.913067 −0.456534 0.889706i \(-0.650909\pi\)
−0.456534 + 0.889706i \(0.650909\pi\)
\(332\) 20709.1i 0.187882i
\(333\) 0 0
\(334\) −5228.46 −0.0468685
\(335\) 318255.i 2.83587i
\(336\) 0 0
\(337\) −170263. −1.49920 −0.749600 0.661891i \(-0.769754\pi\)
−0.749600 + 0.661891i \(0.769754\pi\)
\(338\) − 52549.8i − 0.459979i
\(339\) 0 0
\(340\) 152275. 1.31726
\(341\) − 12758.5i − 0.109722i
\(342\) 0 0
\(343\) −6352.45 −0.0539949
\(344\) 30769.9i 0.260022i
\(345\) 0 0
\(346\) −115068. −0.961173
\(347\) − 186399.i − 1.54805i −0.633155 0.774025i \(-0.718241\pi\)
0.633155 0.774025i \(-0.281759\pi\)
\(348\) 0 0
\(349\) −230470. −1.89218 −0.946091 0.323901i \(-0.895005\pi\)
−0.946091 + 0.323901i \(0.895005\pi\)
\(350\) 60793.6i 0.496274i
\(351\) 0 0
\(352\) −11483.7 −0.0926824
\(353\) − 6167.83i − 0.0494975i −0.999694 0.0247488i \(-0.992121\pi\)
0.999694 0.0247488i \(-0.00787858\pi\)
\(354\) 0 0
\(355\) 378482. 3.00323
\(356\) 70027.4i 0.552545i
\(357\) 0 0
\(358\) 108249. 0.844613
\(359\) − 8588.08i − 0.0666357i −0.999445 0.0333179i \(-0.989393\pi\)
0.999445 0.0333179i \(-0.0106074\pi\)
\(360\) 0 0
\(361\) −61544.9 −0.472256
\(362\) − 131429.i − 1.00294i
\(363\) 0 0
\(364\) −14802.7 −0.111722
\(365\) − 61917.5i − 0.464759i
\(366\) 0 0
\(367\) 147820. 1.09749 0.548747 0.835988i \(-0.315105\pi\)
0.548747 + 0.835988i \(0.315105\pi\)
\(368\) − 43104.6i − 0.318293i
\(369\) 0 0
\(370\) −300930. −2.19817
\(371\) 73516.7i 0.534119i
\(372\) 0 0
\(373\) −142961. −1.02754 −0.513771 0.857927i \(-0.671752\pi\)
−0.513771 + 0.857927i \(0.671752\pi\)
\(374\) − 80826.5i − 0.577844i
\(375\) 0 0
\(376\) −15518.7 −0.109769
\(377\) − 49838.4i − 0.350656i
\(378\) 0 0
\(379\) 252021. 1.75452 0.877260 0.480016i \(-0.159369\pi\)
0.877260 + 0.480016i \(0.159369\pi\)
\(380\) 88653.4i 0.613943i
\(381\) 0 0
\(382\) −136200. −0.933362
\(383\) 246474.i 1.68025i 0.542394 + 0.840125i \(0.317518\pi\)
−0.542394 + 0.840125i \(0.682482\pi\)
\(384\) 0 0
\(385\) 49646.8 0.334942
\(386\) − 146074.i − 0.980390i
\(387\) 0 0
\(388\) −34691.6 −0.230442
\(389\) − 113655.i − 0.751083i −0.926805 0.375542i \(-0.877457\pi\)
0.926805 0.375542i \(-0.122543\pi\)
\(390\) 0 0
\(391\) 303385. 1.98445
\(392\) − 7761.20i − 0.0505076i
\(393\) 0 0
\(394\) −11319.9 −0.0729208
\(395\) − 234831.i − 1.50509i
\(396\) 0 0
\(397\) 48018.2 0.304667 0.152333 0.988329i \(-0.451321\pi\)
0.152333 + 0.988329i \(0.451321\pi\)
\(398\) 64970.3i 0.410156i
\(399\) 0 0
\(400\) −74275.5 −0.464222
\(401\) 78669.6i 0.489236i 0.969620 + 0.244618i \(0.0786625\pi\)
−0.969620 + 0.244618i \(0.921337\pi\)
\(402\) 0 0
\(403\) 20093.2 0.123720
\(404\) − 9890.07i − 0.0605950i
\(405\) 0 0
\(406\) 26130.7 0.158525
\(407\) 159732.i 0.964277i
\(408\) 0 0
\(409\) −89356.3 −0.534169 −0.267085 0.963673i \(-0.586060\pi\)
−0.267085 + 0.963673i \(0.586060\pi\)
\(410\) − 202945.i − 1.20729i
\(411\) 0 0
\(412\) 135651. 0.799153
\(413\) 13737.5i 0.0805393i
\(414\) 0 0
\(415\) 109385. 0.635130
\(416\) − 18085.5i − 0.104506i
\(417\) 0 0
\(418\) 47056.7 0.269320
\(419\) − 109718.i − 0.624955i −0.949925 0.312477i \(-0.898841\pi\)
0.949925 0.312477i \(-0.101159\pi\)
\(420\) 0 0
\(421\) 146241. 0.825096 0.412548 0.910936i \(-0.364639\pi\)
0.412548 + 0.910936i \(0.364639\pi\)
\(422\) − 144265.i − 0.810095i
\(423\) 0 0
\(424\) −89820.2 −0.499623
\(425\) − 522778.i − 2.89427i
\(426\) 0 0
\(427\) 86592.7 0.474926
\(428\) 37884.2i 0.206810i
\(429\) 0 0
\(430\) 162526. 0.878994
\(431\) − 215583.i − 1.16054i −0.814425 0.580269i \(-0.802947\pi\)
0.814425 0.580269i \(-0.197053\pi\)
\(432\) 0 0
\(433\) −171626. −0.915390 −0.457695 0.889109i \(-0.651325\pi\)
−0.457695 + 0.889109i \(0.651325\pi\)
\(434\) 10535.0i 0.0559315i
\(435\) 0 0
\(436\) −145463. −0.765209
\(437\) 176629.i 0.924909i
\(438\) 0 0
\(439\) −148476. −0.770420 −0.385210 0.922829i \(-0.625871\pi\)
−0.385210 + 0.922829i \(0.625871\pi\)
\(440\) 60656.8i 0.313310i
\(441\) 0 0
\(442\) 127292. 0.651564
\(443\) − 282139.i − 1.43766i −0.695186 0.718830i \(-0.744678\pi\)
0.695186 0.718830i \(-0.255322\pi\)
\(444\) 0 0
\(445\) 369883. 1.86786
\(446\) − 46609.5i − 0.234318i
\(447\) 0 0
\(448\) 9482.37 0.0472456
\(449\) 302834.i 1.50214i 0.660220 + 0.751072i \(0.270463\pi\)
−0.660220 + 0.751072i \(0.729537\pi\)
\(450\) 0 0
\(451\) −107722. −0.529604
\(452\) − 159909.i − 0.782701i
\(453\) 0 0
\(454\) −286067. −1.38789
\(455\) 78187.7i 0.377673i
\(456\) 0 0
\(457\) −216256. −1.03547 −0.517734 0.855542i \(-0.673224\pi\)
−0.517734 + 0.855542i \(0.673224\pi\)
\(458\) 279784.i 1.33380i
\(459\) 0 0
\(460\) −227677. −1.07598
\(461\) − 338541.i − 1.59298i −0.604653 0.796489i \(-0.706688\pi\)
0.604653 0.796489i \(-0.293312\pi\)
\(462\) 0 0
\(463\) −406862. −1.89795 −0.948975 0.315351i \(-0.897878\pi\)
−0.948975 + 0.315351i \(0.897878\pi\)
\(464\) 31925.6i 0.148287i
\(465\) 0 0
\(466\) −23644.8 −0.108884
\(467\) − 143548.i − 0.658208i −0.944294 0.329104i \(-0.893253\pi\)
0.944294 0.329104i \(-0.106747\pi\)
\(468\) 0 0
\(469\) −139488. −0.634148
\(470\) 81969.6i 0.371071i
\(471\) 0 0
\(472\) −16784.0 −0.0753376
\(473\) − 86267.8i − 0.385591i
\(474\) 0 0
\(475\) 304358. 1.34896
\(476\) 66740.3i 0.294560i
\(477\) 0 0
\(478\) −84765.8 −0.370992
\(479\) 329574.i 1.43642i 0.695825 + 0.718212i \(0.255039\pi\)
−0.695825 + 0.718212i \(0.744961\pi\)
\(480\) 0 0
\(481\) −251558. −1.08730
\(482\) − 212037.i − 0.912676i
\(483\) 0 0
\(484\) −84931.8 −0.362560
\(485\) 183240.i 0.779000i
\(486\) 0 0
\(487\) 119437. 0.503594 0.251797 0.967780i \(-0.418978\pi\)
0.251797 + 0.967780i \(0.418978\pi\)
\(488\) 105796.i 0.444252i
\(489\) 0 0
\(490\) −40994.5 −0.170739
\(491\) − 277082.i − 1.14933i −0.818388 0.574666i \(-0.805132\pi\)
0.818388 0.574666i \(-0.194868\pi\)
\(492\) 0 0
\(493\) −224704. −0.924520
\(494\) 74108.6i 0.303679i
\(495\) 0 0
\(496\) −12871.3 −0.0523191
\(497\) 165884.i 0.671572i
\(498\) 0 0
\(499\) 384780. 1.54529 0.772647 0.634836i \(-0.218932\pi\)
0.772647 + 0.634836i \(0.218932\pi\)
\(500\) 181043.i 0.724171i
\(501\) 0 0
\(502\) −260250. −1.03272
\(503\) 288691.i 1.14103i 0.821286 + 0.570516i \(0.193257\pi\)
−0.821286 + 0.570516i \(0.806743\pi\)
\(504\) 0 0
\(505\) −52239.1 −0.204839
\(506\) 120850.i 0.472003i
\(507\) 0 0
\(508\) −73276.8 −0.283948
\(509\) − 181063.i − 0.698866i −0.936961 0.349433i \(-0.886374\pi\)
0.936961 0.349433i \(-0.113626\pi\)
\(510\) 0 0
\(511\) 27137.7 0.103928
\(512\) 11585.2i 0.0441942i
\(513\) 0 0
\(514\) 86995.1 0.329282
\(515\) − 716508.i − 2.70151i
\(516\) 0 0
\(517\) 43509.0 0.162779
\(518\) − 131894.i − 0.491548i
\(519\) 0 0
\(520\) −95527.1 −0.353281
\(521\) − 313770.i − 1.15594i −0.816057 0.577971i \(-0.803845\pi\)
0.816057 0.577971i \(-0.196155\pi\)
\(522\) 0 0
\(523\) 10145.3 0.0370903 0.0185452 0.999828i \(-0.494097\pi\)
0.0185452 + 0.999828i \(0.494097\pi\)
\(524\) 64125.1i 0.233542i
\(525\) 0 0
\(526\) −106043. −0.383276
\(527\) − 90593.1i − 0.326192i
\(528\) 0 0
\(529\) −173773. −0.620970
\(530\) 474428.i 1.68896i
\(531\) 0 0
\(532\) −38855.8 −0.137288
\(533\) − 169649.i − 0.597169i
\(534\) 0 0
\(535\) 200104. 0.699113
\(536\) − 170421.i − 0.593191i
\(537\) 0 0
\(538\) −106085. −0.366515
\(539\) 21759.6i 0.0748987i
\(540\) 0 0
\(541\) 496514. 1.69643 0.848217 0.529649i \(-0.177676\pi\)
0.848217 + 0.529649i \(0.177676\pi\)
\(542\) 150392.i 0.511947i
\(543\) 0 0
\(544\) −81541.1 −0.275536
\(545\) 768333.i 2.58676i
\(546\) 0 0
\(547\) −557220. −1.86231 −0.931155 0.364623i \(-0.881198\pi\)
−0.931155 + 0.364623i \(0.881198\pi\)
\(548\) − 107578.i − 0.358229i
\(549\) 0 0
\(550\) 208242. 0.688404
\(551\) − 130821.i − 0.430898i
\(552\) 0 0
\(553\) 102924. 0.336562
\(554\) − 85167.9i − 0.277496i
\(555\) 0 0
\(556\) 221236. 0.715658
\(557\) 67362.8i 0.217125i 0.994090 + 0.108562i \(0.0346247\pi\)
−0.994090 + 0.108562i \(0.965375\pi\)
\(558\) 0 0
\(559\) 135861. 0.434783
\(560\) − 50085.7i − 0.159712i
\(561\) 0 0
\(562\) 79835.1 0.252767
\(563\) − 171298.i − 0.540425i −0.962801 0.270212i \(-0.912906\pi\)
0.962801 0.270212i \(-0.0870940\pi\)
\(564\) 0 0
\(565\) −844636. −2.64589
\(566\) 33008.5i 0.103037i
\(567\) 0 0
\(568\) −202672. −0.628198
\(569\) 23818.0i 0.0735666i 0.999323 + 0.0367833i \(0.0117111\pi\)
−0.999323 + 0.0367833i \(0.988289\pi\)
\(570\) 0 0
\(571\) 58622.1 0.179800 0.0898999 0.995951i \(-0.471345\pi\)
0.0898999 + 0.995951i \(0.471345\pi\)
\(572\) 50705.2i 0.154975i
\(573\) 0 0
\(574\) 88948.6 0.269970
\(575\) 781644.i 2.36414i
\(576\) 0 0
\(577\) 421423. 1.26580 0.632901 0.774232i \(-0.281864\pi\)
0.632901 + 0.774232i \(0.281864\pi\)
\(578\) − 337682.i − 1.01077i
\(579\) 0 0
\(580\) 168630. 0.501279
\(581\) 47942.3i 0.142026i
\(582\) 0 0
\(583\) 251824. 0.740900
\(584\) 33156.0i 0.0972156i
\(585\) 0 0
\(586\) 83707.8 0.243765
\(587\) 404875.i 1.17502i 0.809218 + 0.587509i \(0.199891\pi\)
−0.809218 + 0.587509i \(0.800109\pi\)
\(588\) 0 0
\(589\) 52742.7 0.152031
\(590\) 88652.8i 0.254676i
\(591\) 0 0
\(592\) 161144. 0.459801
\(593\) − 541601.i − 1.54018i −0.637938 0.770088i \(-0.720212\pi\)
0.637938 0.770088i \(-0.279788\pi\)
\(594\) 0 0
\(595\) 352521. 0.995752
\(596\) − 187631.i − 0.528217i
\(597\) 0 0
\(598\) −190324. −0.532219
\(599\) − 236556.i − 0.659295i −0.944104 0.329648i \(-0.893070\pi\)
0.944104 0.329648i \(-0.106930\pi\)
\(600\) 0 0
\(601\) 382066. 1.05776 0.528882 0.848695i \(-0.322611\pi\)
0.528882 + 0.848695i \(0.322611\pi\)
\(602\) 71233.3i 0.196558i
\(603\) 0 0
\(604\) −20396.1 −0.0559078
\(605\) 448608.i 1.22562i
\(606\) 0 0
\(607\) 376071. 1.02069 0.510344 0.859970i \(-0.329518\pi\)
0.510344 + 0.859970i \(0.329518\pi\)
\(608\) − 47472.7i − 0.128421i
\(609\) 0 0
\(610\) 558812. 1.50178
\(611\) 68521.4i 0.183546i
\(612\) 0 0
\(613\) −21902.9 −0.0582882 −0.0291441 0.999575i \(-0.509278\pi\)
−0.0291441 + 0.999575i \(0.509278\pi\)
\(614\) − 273628.i − 0.725810i
\(615\) 0 0
\(616\) −26585.2 −0.0700613
\(617\) 358506.i 0.941729i 0.882206 + 0.470864i \(0.156058\pi\)
−0.882206 + 0.470864i \(0.843942\pi\)
\(618\) 0 0
\(619\) 448702. 1.17105 0.585526 0.810653i \(-0.300888\pi\)
0.585526 + 0.810653i \(0.300888\pi\)
\(620\) 67986.1i 0.176863i
\(621\) 0 0
\(622\) −234761. −0.606801
\(623\) 162116.i 0.417685i
\(624\) 0 0
\(625\) 230917. 0.591147
\(626\) 418677.i 1.06839i
\(627\) 0 0
\(628\) 143759. 0.364514
\(629\) 1.13419e6i 2.86671i
\(630\) 0 0
\(631\) 302074. 0.758672 0.379336 0.925259i \(-0.376152\pi\)
0.379336 + 0.925259i \(0.376152\pi\)
\(632\) 125749.i 0.314825i
\(633\) 0 0
\(634\) 174274. 0.433564
\(635\) 387046.i 0.959877i
\(636\) 0 0
\(637\) −34268.8 −0.0844540
\(638\) − 89507.9i − 0.219897i
\(639\) 0 0
\(640\) 61193.0 0.149397
\(641\) 757749.i 1.84421i 0.386944 + 0.922103i \(0.373531\pi\)
−0.386944 + 0.922103i \(0.626469\pi\)
\(642\) 0 0
\(643\) −211423. −0.511364 −0.255682 0.966761i \(-0.582300\pi\)
−0.255682 + 0.966761i \(0.582300\pi\)
\(644\) − 99788.4i − 0.240607i
\(645\) 0 0
\(646\) 334130. 0.800664
\(647\) 361157.i 0.862756i 0.902171 + 0.431378i \(0.141972\pi\)
−0.902171 + 0.431378i \(0.858028\pi\)
\(648\) 0 0
\(649\) 47056.3 0.111719
\(650\) 327956.i 0.776228i
\(651\) 0 0
\(652\) 204041. 0.479978
\(653\) − 472486.i − 1.10806i −0.832497 0.554030i \(-0.813089\pi\)
0.832497 0.554030i \(-0.186911\pi\)
\(654\) 0 0
\(655\) 338707. 0.789482
\(656\) 108674.i 0.252534i
\(657\) 0 0
\(658\) −35926.4 −0.0829777
\(659\) 479228.i 1.10350i 0.834010 + 0.551749i \(0.186039\pi\)
−0.834010 + 0.551749i \(0.813961\pi\)
\(660\) 0 0
\(661\) −653224. −1.49506 −0.747531 0.664227i \(-0.768761\pi\)
−0.747531 + 0.664227i \(0.768761\pi\)
\(662\) − 282946.i − 0.645636i
\(663\) 0 0
\(664\) −58574.3 −0.132853
\(665\) 205236.i 0.464098i
\(666\) 0 0
\(667\) 335971. 0.755180
\(668\) − 14788.3i − 0.0331410i
\(669\) 0 0
\(670\) −900162. −2.00526
\(671\) − 296614.i − 0.658790i
\(672\) 0 0
\(673\) −852808. −1.88287 −0.941436 0.337190i \(-0.890523\pi\)
−0.941436 + 0.337190i \(0.890523\pi\)
\(674\) − 481575.i − 1.06009i
\(675\) 0 0
\(676\) 148633. 0.325254
\(677\) − 344306.i − 0.751220i −0.926778 0.375610i \(-0.877433\pi\)
0.926778 0.375610i \(-0.122567\pi\)
\(678\) 0 0
\(679\) −80312.2 −0.174198
\(680\) 430698.i 0.931441i
\(681\) 0 0
\(682\) 36086.6 0.0775850
\(683\) − 189722.i − 0.406701i −0.979106 0.203350i \(-0.934817\pi\)
0.979106 0.203350i \(-0.0651831\pi\)
\(684\) 0 0
\(685\) −568223. −1.21098
\(686\) − 17967.4i − 0.0381802i
\(687\) 0 0
\(688\) −87030.4 −0.183863
\(689\) 396592.i 0.835421i
\(690\) 0 0
\(691\) 323591. 0.677704 0.338852 0.940840i \(-0.389961\pi\)
0.338852 + 0.940840i \(0.389961\pi\)
\(692\) − 325461.i − 0.679652i
\(693\) 0 0
\(694\) 527216. 1.09464
\(695\) − 1.16856e6i − 2.41926i
\(696\) 0 0
\(697\) −764889. −1.57446
\(698\) − 651867.i − 1.33797i
\(699\) 0 0
\(700\) −171950. −0.350919
\(701\) 705697.i 1.43609i 0.695996 + 0.718046i \(0.254963\pi\)
−0.695996 + 0.718046i \(0.745037\pi\)
\(702\) 0 0
\(703\) −660316. −1.33611
\(704\) − 32480.9i − 0.0655363i
\(705\) 0 0
\(706\) 17445.3 0.0350000
\(707\) − 22895.8i − 0.0458055i
\(708\) 0 0
\(709\) 278238. 0.553509 0.276755 0.960941i \(-0.410741\pi\)
0.276755 + 0.960941i \(0.410741\pi\)
\(710\) 1.07051e6i 2.12360i
\(711\) 0 0
\(712\) −198067. −0.390709
\(713\) 135452.i 0.266445i
\(714\) 0 0
\(715\) 267824. 0.523886
\(716\) 306174.i 0.597232i
\(717\) 0 0
\(718\) 24290.8 0.0471186
\(719\) 482776.i 0.933873i 0.884291 + 0.466937i \(0.154642\pi\)
−0.884291 + 0.466937i \(0.845358\pi\)
\(720\) 0 0
\(721\) 314037. 0.604103
\(722\) − 174075.i − 0.333935i
\(723\) 0 0
\(724\) 371738. 0.709186
\(725\) − 578928.i − 1.10141i
\(726\) 0 0
\(727\) 130521. 0.246952 0.123476 0.992348i \(-0.460596\pi\)
0.123476 + 0.992348i \(0.460596\pi\)
\(728\) − 41868.5i − 0.0789995i
\(729\) 0 0
\(730\) 175129. 0.328634
\(731\) − 612551.i − 1.14633i
\(732\) 0 0
\(733\) −342877. −0.638161 −0.319081 0.947728i \(-0.603374\pi\)
−0.319081 + 0.947728i \(0.603374\pi\)
\(734\) 418099.i 0.776046i
\(735\) 0 0
\(736\) 121918. 0.225067
\(737\) 477801.i 0.879654i
\(738\) 0 0
\(739\) −693065. −1.26907 −0.634534 0.772895i \(-0.718808\pi\)
−0.634534 + 0.772895i \(0.718808\pi\)
\(740\) − 851157.i − 1.55434i
\(741\) 0 0
\(742\) −207937. −0.377679
\(743\) 788374.i 1.42809i 0.700102 + 0.714043i \(0.253138\pi\)
−0.700102 + 0.714043i \(0.746862\pi\)
\(744\) 0 0
\(745\) −991063. −1.78562
\(746\) − 404354.i − 0.726582i
\(747\) 0 0
\(748\) 228612. 0.408598
\(749\) 87703.2i 0.156333i
\(750\) 0 0
\(751\) −346405. −0.614192 −0.307096 0.951678i \(-0.599357\pi\)
−0.307096 + 0.951678i \(0.599357\pi\)
\(752\) − 43893.6i − 0.0776186i
\(753\) 0 0
\(754\) 140964. 0.247951
\(755\) 107732.i 0.188994i
\(756\) 0 0
\(757\) −774595. −1.35171 −0.675854 0.737036i \(-0.736225\pi\)
−0.675854 + 0.737036i \(0.736225\pi\)
\(758\) 712823.i 1.24063i
\(759\) 0 0
\(760\) −250750. −0.434124
\(761\) − 956099.i − 1.65095i −0.564439 0.825475i \(-0.690908\pi\)
0.564439 0.825475i \(-0.309092\pi\)
\(762\) 0 0
\(763\) −336752. −0.578443
\(764\) − 385231.i − 0.659986i
\(765\) 0 0
\(766\) −697134. −1.18812
\(767\) 74108.1i 0.125972i
\(768\) 0 0
\(769\) 62358.6 0.105449 0.0527247 0.998609i \(-0.483209\pi\)
0.0527247 + 0.998609i \(0.483209\pi\)
\(770\) 140422.i 0.236840i
\(771\) 0 0
\(772\) 413160. 0.693240
\(773\) 664455.i 1.11201i 0.831181 + 0.556003i \(0.187666\pi\)
−0.831181 + 0.556003i \(0.812334\pi\)
\(774\) 0 0
\(775\) 233405. 0.388603
\(776\) − 98122.7i − 0.162947i
\(777\) 0 0
\(778\) 321464. 0.531096
\(779\) − 445313.i − 0.733822i
\(780\) 0 0
\(781\) 568219. 0.931566
\(782\) 858103.i 1.40322i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) − 759330.i − 1.23223i
\(786\) 0 0
\(787\) 206026. 0.332639 0.166320 0.986072i \(-0.446812\pi\)
0.166320 + 0.986072i \(0.446812\pi\)
\(788\) − 32017.6i − 0.0515628i
\(789\) 0 0
\(790\) 664202. 1.06426
\(791\) − 370194.i − 0.591666i
\(792\) 0 0
\(793\) 467132. 0.742836
\(794\) 135816.i 0.215432i
\(795\) 0 0
\(796\) −183764. −0.290024
\(797\) − 44688.5i − 0.0703524i −0.999381 0.0351762i \(-0.988801\pi\)
0.999381 0.0351762i \(-0.0111992\pi\)
\(798\) 0 0
\(799\) 308939. 0.483926
\(800\) − 210083.i − 0.328255i
\(801\) 0 0
\(802\) −222511. −0.345942
\(803\) − 92957.5i − 0.144163i
\(804\) 0 0
\(805\) −527080. −0.813364
\(806\) 56832.1i 0.0874830i
\(807\) 0 0
\(808\) 27973.3 0.0428471
\(809\) − 377340.i − 0.576549i −0.957548 0.288274i \(-0.906918\pi\)
0.957548 0.288274i \(-0.0930815\pi\)
\(810\) 0 0
\(811\) 289441. 0.440067 0.220034 0.975492i \(-0.429383\pi\)
0.220034 + 0.975492i \(0.429383\pi\)
\(812\) 73908.8i 0.112094i
\(813\) 0 0
\(814\) −451789. −0.681847
\(815\) − 1.07774e6i − 1.62255i
\(816\) 0 0
\(817\) 356624. 0.534276
\(818\) − 252738.i − 0.377715i
\(819\) 0 0
\(820\) 574015. 0.853681
\(821\) 549071.i 0.814596i 0.913295 + 0.407298i \(0.133529\pi\)
−0.913295 + 0.407298i \(0.866471\pi\)
\(822\) 0 0
\(823\) 262600. 0.387700 0.193850 0.981031i \(-0.437903\pi\)
0.193850 + 0.981031i \(0.437903\pi\)
\(824\) 383680.i 0.565086i
\(825\) 0 0
\(826\) −38855.5 −0.0569499
\(827\) 49767.0i 0.0727664i 0.999338 + 0.0363832i \(0.0115837\pi\)
−0.999338 + 0.0363832i \(0.988416\pi\)
\(828\) 0 0
\(829\) −738419. −1.07447 −0.537234 0.843433i \(-0.680531\pi\)
−0.537234 + 0.843433i \(0.680531\pi\)
\(830\) 309388.i 0.449105i
\(831\) 0 0
\(832\) 51153.4 0.0738972
\(833\) 154506.i 0.222667i
\(834\) 0 0
\(835\) −78111.6 −0.112032
\(836\) 133096.i 0.190438i
\(837\) 0 0
\(838\) 310329. 0.441910
\(839\) − 1.18604e6i − 1.68490i −0.538776 0.842449i \(-0.681113\pi\)
0.538776 0.842449i \(-0.318887\pi\)
\(840\) 0 0
\(841\) 458442. 0.648176
\(842\) 413632.i 0.583431i
\(843\) 0 0
\(844\) 408043. 0.572824
\(845\) − 785079.i − 1.09951i
\(846\) 0 0
\(847\) −196620. −0.274069
\(848\) − 254050.i − 0.353287i
\(849\) 0 0
\(850\) 1.47864e6 2.04656
\(851\) − 1.69581e6i − 2.34162i
\(852\) 0 0
\(853\) 470813. 0.647069 0.323534 0.946216i \(-0.395129\pi\)
0.323534 + 0.946216i \(0.395129\pi\)
\(854\) 244921.i 0.335823i
\(855\) 0 0
\(856\) −107153. −0.146236
\(857\) 27792.1i 0.0378408i 0.999821 + 0.0189204i \(0.00602291\pi\)
−0.999821 + 0.0189204i \(0.993977\pi\)
\(858\) 0 0
\(859\) −897338. −1.21610 −0.608051 0.793898i \(-0.708048\pi\)
−0.608051 + 0.793898i \(0.708048\pi\)
\(860\) 459693.i 0.621543i
\(861\) 0 0
\(862\) 609760. 0.820624
\(863\) 327757.i 0.440078i 0.975491 + 0.220039i \(0.0706185\pi\)
−0.975491 + 0.220039i \(0.929382\pi\)
\(864\) 0 0
\(865\) −1.71908e6 −2.29754
\(866\) − 485430.i − 0.647278i
\(867\) 0 0
\(868\) −29797.6 −0.0395495
\(869\) − 352555.i − 0.466860i
\(870\) 0 0
\(871\) −752478. −0.991877
\(872\) − 411432.i − 0.541084i
\(873\) 0 0
\(874\) −499582. −0.654010
\(875\) 419120.i 0.547422i
\(876\) 0 0
\(877\) −675535. −0.878312 −0.439156 0.898411i \(-0.644722\pi\)
−0.439156 + 0.898411i \(0.644722\pi\)
\(878\) − 419954.i − 0.544770i
\(879\) 0 0
\(880\) −171563. −0.221543
\(881\) 105642.i 0.136108i 0.997682 + 0.0680540i \(0.0216790\pi\)
−0.997682 + 0.0680540i \(0.978321\pi\)
\(882\) 0 0
\(883\) −214939. −0.275672 −0.137836 0.990455i \(-0.544015\pi\)
−0.137836 + 0.990455i \(0.544015\pi\)
\(884\) 360036.i 0.460725i
\(885\) 0 0
\(886\) 798011. 1.01658
\(887\) 1.38541e6i 1.76089i 0.474152 + 0.880443i \(0.342755\pi\)
−0.474152 + 0.880443i \(0.657245\pi\)
\(888\) 0 0
\(889\) −169638. −0.214645
\(890\) 1.04619e6i 1.32078i
\(891\) 0 0
\(892\) 131832. 0.165688
\(893\) 179862.i 0.225547i
\(894\) 0 0
\(895\) 1.61721e6 2.01892
\(896\) 26820.2i 0.0334077i
\(897\) 0 0
\(898\) −856544. −1.06218
\(899\) − 100324.i − 0.124132i
\(900\) 0 0
\(901\) 1.78809e6 2.20263
\(902\) − 304684.i − 0.374487i
\(903\) 0 0
\(904\) 452291. 0.553453
\(905\) − 1.96351e6i − 2.39738i
\(906\) 0 0
\(907\) −1.24618e6 −1.51483 −0.757417 0.652932i \(-0.773539\pi\)
−0.757417 + 0.652932i \(0.773539\pi\)
\(908\) − 809121.i − 0.981390i
\(909\) 0 0
\(910\) −221148. −0.267055
\(911\) 289663.i 0.349025i 0.984655 + 0.174513i \(0.0558350\pi\)
−0.984655 + 0.174513i \(0.944165\pi\)
\(912\) 0 0
\(913\) 164221. 0.197010
\(914\) − 611665.i − 0.732186i
\(915\) 0 0
\(916\) −791348. −0.943140
\(917\) 148452.i 0.176541i
\(918\) 0 0
\(919\) 781183. 0.924958 0.462479 0.886630i \(-0.346960\pi\)
0.462479 + 0.886630i \(0.346960\pi\)
\(920\) − 643969.i − 0.760833i
\(921\) 0 0
\(922\) 957539. 1.12641
\(923\) 894876.i 1.05041i
\(924\) 0 0
\(925\) −2.92213e6 −3.41520
\(926\) − 1.15078e6i − 1.34205i
\(927\) 0 0
\(928\) −90299.2 −0.104855
\(929\) − 112122.i − 0.129915i −0.997888 0.0649576i \(-0.979309\pi\)
0.997888 0.0649576i \(-0.0206912\pi\)
\(930\) 0 0
\(931\) −89952.4 −0.103780
\(932\) − 66877.6i − 0.0769926i
\(933\) 0 0
\(934\) 406015. 0.465424
\(935\) − 1.20752e6i − 1.38125i
\(936\) 0 0
\(937\) 405756. 0.462152 0.231076 0.972936i \(-0.425775\pi\)
0.231076 + 0.972936i \(0.425775\pi\)
\(938\) − 394531.i − 0.448410i
\(939\) 0 0
\(940\) −231845. −0.262387
\(941\) 1.62677e6i 1.83716i 0.395239 + 0.918578i \(0.370662\pi\)
−0.395239 + 0.918578i \(0.629338\pi\)
\(942\) 0 0
\(943\) 1.14364e6 1.28608
\(944\) − 47472.3i − 0.0532717i
\(945\) 0 0
\(946\) 244002. 0.272654
\(947\) 641839.i 0.715692i 0.933781 + 0.357846i \(0.116489\pi\)
−0.933781 + 0.357846i \(0.883511\pi\)
\(948\) 0 0
\(949\) 146397. 0.162555
\(950\) 860854.i 0.953855i
\(951\) 0 0
\(952\) −188770. −0.208286
\(953\) − 1.18936e6i − 1.30957i −0.755816 0.654785i \(-0.772759\pi\)
0.755816 0.654785i \(-0.227241\pi\)
\(954\) 0 0
\(955\) −2.03478e6 −2.23106
\(956\) − 239754.i − 0.262331i
\(957\) 0 0
\(958\) −932177. −1.01570
\(959\) − 249046.i − 0.270796i
\(960\) 0 0
\(961\) −883074. −0.956203
\(962\) − 711513.i − 0.768835i
\(963\) 0 0
\(964\) 599730. 0.645359
\(965\) − 2.18230e6i − 2.34348i
\(966\) 0 0
\(967\) 1.68998e6 1.80730 0.903648 0.428275i \(-0.140879\pi\)
0.903648 + 0.428275i \(0.140879\pi\)
\(968\) − 240223.i − 0.256368i
\(969\) 0 0
\(970\) −518282. −0.550836
\(971\) 400896.i 0.425200i 0.977139 + 0.212600i \(0.0681932\pi\)
−0.977139 + 0.212600i \(0.931807\pi\)
\(972\) 0 0
\(973\) 512168. 0.540987
\(974\) 337819.i 0.356095i
\(975\) 0 0
\(976\) −299236. −0.314134
\(977\) 547999.i 0.574104i 0.957915 + 0.287052i \(0.0926753\pi\)
−0.957915 + 0.287052i \(0.907325\pi\)
\(978\) 0 0
\(979\) 555310. 0.579389
\(980\) − 115950.i − 0.120731i
\(981\) 0 0
\(982\) 783707. 0.812701
\(983\) 1.36963e6i 1.41742i 0.705502 + 0.708708i \(0.250721\pi\)
−0.705502 + 0.708708i \(0.749279\pi\)
\(984\) 0 0
\(985\) −169116. −0.174306
\(986\) − 635558.i − 0.653735i
\(987\) 0 0
\(988\) −209611. −0.214734
\(989\) 915871.i 0.936358i
\(990\) 0 0
\(991\) −406038. −0.413446 −0.206723 0.978399i \(-0.566280\pi\)
−0.206723 + 0.978399i \(0.566280\pi\)
\(992\) − 36405.6i − 0.0369952i
\(993\) 0 0
\(994\) −469192. −0.474873
\(995\) 970637.i 0.980417i
\(996\) 0 0
\(997\) 301551. 0.303368 0.151684 0.988429i \(-0.451530\pi\)
0.151684 + 0.988429i \(0.451530\pi\)
\(998\) 1.08832e6i 1.09269i
\(999\) 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 378.5.b.a.323.8 yes 8
3.2 odd 2 inner 378.5.b.a.323.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.a.323.1 8 3.2 odd 2 inner
378.5.b.a.323.8 yes 8 1.1 even 1 trivial