Properties

Label 378.5.b.b.323.2
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: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 92x^{6} + 2949x^{4} + 37548x^{2} + 142884 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 323.2
Root \(5.12618i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.b.323.7

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{2} -8.00000 q^{4} -28.1791i q^{5} -18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\) \(q-2.82843i q^{2} -8.00000 q^{4} -28.1791i q^{5} -18.5203 q^{7} +22.6274i q^{8} -79.7026 q^{10} +168.507i q^{11} +319.388 q^{13} +52.3832i q^{14} +64.0000 q^{16} +39.1240i q^{17} +457.590 q^{19} +225.433i q^{20} +476.611 q^{22} -435.106i q^{23} -169.064 q^{25} -903.365i q^{26} +148.162 q^{28} +135.719i q^{29} -267.371 q^{31} -181.019i q^{32} +110.659 q^{34} +521.885i q^{35} +812.186 q^{37} -1294.26i q^{38} +637.621 q^{40} +737.686i q^{41} -2092.00 q^{43} -1348.06i q^{44} -1230.66 q^{46} -2962.37i q^{47} +343.000 q^{49} +478.184i q^{50} -2555.10 q^{52} +3190.68i q^{53} +4748.39 q^{55} -419.066i q^{56} +383.871 q^{58} -5620.52i q^{59} +4862.90 q^{61} +756.240i q^{62} -512.000 q^{64} -9000.07i q^{65} -2008.97 q^{67} -312.992i q^{68} +1476.11 q^{70} -4306.53i q^{71} +6314.60 q^{73} -2297.21i q^{74} -3660.72 q^{76} -3120.80i q^{77} -3703.76 q^{79} -1803.46i q^{80} +2086.49 q^{82} -9033.39i q^{83} +1102.48 q^{85} +5917.06i q^{86} -3812.88 q^{88} +3711.45i q^{89} -5915.14 q^{91} +3480.84i q^{92} -8378.85 q^{94} -12894.5i q^{95} +13748.8 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 64 q^{4} - 32 q^{10} + 296 q^{13} + 512 q^{16} - 80 q^{19} + 800 q^{22} - 1464 q^{25} - 1024 q^{31} + 896 q^{34} + 904 q^{37} + 256 q^{40} - 3224 q^{43} - 4384 q^{46} + 2744 q^{49} - 2368 q^{52} + 8312 q^{55} + 2944 q^{58} + 128 q^{61} - 4096 q^{64} - 10152 q^{67} + 1568 q^{70} + 10632 q^{73} + 640 q^{76} - 4072 q^{79} - 3552 q^{82} + 13448 q^{85} - 6400 q^{88} - 6664 q^{91} - 384 q^{94} + 35616 q^{97}+O(q^{100}) \) 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
\(5\) − 28.1791i − 1.12717i −0.826060 0.563583i \(-0.809423\pi\)
0.826060 0.563583i \(-0.190577\pi\)
\(6\) 0 0
\(7\) −18.5203 −0.377964
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) −79.7026 −0.797026
\(11\) 168.507i 1.39262i 0.717740 + 0.696311i \(0.245177\pi\)
−0.717740 + 0.696311i \(0.754823\pi\)
\(12\) 0 0
\(13\) 319.388 1.88987 0.944934 0.327261i \(-0.106126\pi\)
0.944934 + 0.327261i \(0.106126\pi\)
\(14\) 52.3832i 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 39.1240i 0.135377i 0.997706 + 0.0676886i \(0.0215624\pi\)
−0.997706 + 0.0676886i \(0.978438\pi\)
\(18\) 0 0
\(19\) 457.590 1.26756 0.633782 0.773512i \(-0.281502\pi\)
0.633782 + 0.773512i \(0.281502\pi\)
\(20\) 225.433i 0.563583i
\(21\) 0 0
\(22\) 476.611 0.984733
\(23\) − 435.106i − 0.822506i −0.911521 0.411253i \(-0.865091\pi\)
0.911521 0.411253i \(-0.134909\pi\)
\(24\) 0 0
\(25\) −169.064 −0.270502
\(26\) − 903.365i − 1.33634i
\(27\) 0 0
\(28\) 148.162 0.188982
\(29\) 135.719i 0.161378i 0.996739 + 0.0806890i \(0.0257121\pi\)
−0.996739 + 0.0806890i \(0.974288\pi\)
\(30\) 0 0
\(31\) −267.371 −0.278222 −0.139111 0.990277i \(-0.544424\pi\)
−0.139111 + 0.990277i \(0.544424\pi\)
\(32\) − 181.019i − 0.176777i
\(33\) 0 0
\(34\) 110.659 0.0957261
\(35\) 521.885i 0.426028i
\(36\) 0 0
\(37\) 812.186 0.593270 0.296635 0.954991i \(-0.404136\pi\)
0.296635 + 0.954991i \(0.404136\pi\)
\(38\) − 1294.26i − 0.896303i
\(39\) 0 0
\(40\) 637.621 0.398513
\(41\) 737.686i 0.438837i 0.975631 + 0.219419i \(0.0704161\pi\)
−0.975631 + 0.219419i \(0.929584\pi\)
\(42\) 0 0
\(43\) −2092.00 −1.13142 −0.565710 0.824604i \(-0.691398\pi\)
−0.565710 + 0.824604i \(0.691398\pi\)
\(44\) − 1348.06i − 0.696311i
\(45\) 0 0
\(46\) −1230.66 −0.581599
\(47\) − 2962.37i − 1.34105i −0.741889 0.670523i \(-0.766070\pi\)
0.741889 0.670523i \(-0.233930\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 478.184i 0.191274i
\(51\) 0 0
\(52\) −2555.10 −0.944934
\(53\) 3190.68i 1.13588i 0.823071 + 0.567939i \(0.192259\pi\)
−0.823071 + 0.567939i \(0.807741\pi\)
\(54\) 0 0
\(55\) 4748.39 1.56972
\(56\) − 419.066i − 0.133631i
\(57\) 0 0
\(58\) 383.871 0.114111
\(59\) − 5620.52i − 1.61463i −0.590122 0.807314i \(-0.700920\pi\)
0.590122 0.807314i \(-0.299080\pi\)
\(60\) 0 0
\(61\) 4862.90 1.30688 0.653439 0.756979i \(-0.273325\pi\)
0.653439 + 0.756979i \(0.273325\pi\)
\(62\) 756.240i 0.196733i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) − 9000.07i − 2.13019i
\(66\) 0 0
\(67\) −2008.97 −0.447533 −0.223766 0.974643i \(-0.571835\pi\)
−0.223766 + 0.974643i \(0.571835\pi\)
\(68\) − 312.992i − 0.0676886i
\(69\) 0 0
\(70\) 1476.11 0.301248
\(71\) − 4306.53i − 0.854302i −0.904180 0.427151i \(-0.859517\pi\)
0.904180 0.427151i \(-0.140483\pi\)
\(72\) 0 0
\(73\) 6314.60 1.18495 0.592475 0.805589i \(-0.298151\pi\)
0.592475 + 0.805589i \(0.298151\pi\)
\(74\) − 2297.21i − 0.419505i
\(75\) 0 0
\(76\) −3660.72 −0.633782
\(77\) − 3120.80i − 0.526362i
\(78\) 0 0
\(79\) −3703.76 −0.593457 −0.296728 0.954962i \(-0.595896\pi\)
−0.296728 + 0.954962i \(0.595896\pi\)
\(80\) − 1803.46i − 0.281791i
\(81\) 0 0
\(82\) 2086.49 0.310305
\(83\) − 9033.39i − 1.31128i −0.755075 0.655639i \(-0.772399\pi\)
0.755075 0.655639i \(-0.227601\pi\)
\(84\) 0 0
\(85\) 1102.48 0.152592
\(86\) 5917.06i 0.800035i
\(87\) 0 0
\(88\) −3812.88 −0.492366
\(89\) 3711.45i 0.468559i 0.972169 + 0.234279i \(0.0752730\pi\)
−0.972169 + 0.234279i \(0.924727\pi\)
\(90\) 0 0
\(91\) −5915.14 −0.714303
\(92\) 3480.84i 0.411253i
\(93\) 0 0
\(94\) −8378.85 −0.948263
\(95\) − 12894.5i − 1.42875i
\(96\) 0 0
\(97\) 13748.8 1.46124 0.730620 0.682784i \(-0.239231\pi\)
0.730620 + 0.682784i \(0.239231\pi\)
\(98\) − 970.151i − 0.101015i
\(99\) 0 0
\(100\) 1352.51 0.135251
\(101\) 17899.9i 1.75472i 0.479835 + 0.877359i \(0.340697\pi\)
−0.479835 + 0.877359i \(0.659303\pi\)
\(102\) 0 0
\(103\) 3979.89 0.375143 0.187571 0.982251i \(-0.439938\pi\)
0.187571 + 0.982251i \(0.439938\pi\)
\(104\) 7226.92i 0.668169i
\(105\) 0 0
\(106\) 9024.61 0.803187
\(107\) − 8914.65i − 0.778640i −0.921103 0.389320i \(-0.872710\pi\)
0.921103 0.389320i \(-0.127290\pi\)
\(108\) 0 0
\(109\) −13486.3 −1.13511 −0.567557 0.823334i \(-0.692111\pi\)
−0.567557 + 0.823334i \(0.692111\pi\)
\(110\) − 13430.5i − 1.10996i
\(111\) 0 0
\(112\) −1185.30 −0.0944911
\(113\) − 4042.36i − 0.316576i −0.987393 0.158288i \(-0.949403\pi\)
0.987393 0.158288i \(-0.0505974\pi\)
\(114\) 0 0
\(115\) −12260.9 −0.927100
\(116\) − 1085.75i − 0.0806890i
\(117\) 0 0
\(118\) −15897.2 −1.14171
\(119\) − 724.586i − 0.0511677i
\(120\) 0 0
\(121\) −13753.7 −0.939397
\(122\) − 13754.3i − 0.924103i
\(123\) 0 0
\(124\) 2138.97 0.139111
\(125\) − 12847.9i − 0.822265i
\(126\) 0 0
\(127\) 18650.2 1.15631 0.578157 0.815926i \(-0.303772\pi\)
0.578157 + 0.815926i \(0.303772\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) −25456.0 −1.50627
\(131\) − 14164.3i − 0.825375i −0.910873 0.412688i \(-0.864590\pi\)
0.910873 0.412688i \(-0.135410\pi\)
\(132\) 0 0
\(133\) −8474.69 −0.479094
\(134\) 5682.24i 0.316453i
\(135\) 0 0
\(136\) −885.275 −0.0478630
\(137\) − 27896.5i − 1.48631i −0.669120 0.743154i \(-0.733329\pi\)
0.669120 0.743154i \(-0.266671\pi\)
\(138\) 0 0
\(139\) 6374.52 0.329927 0.164963 0.986300i \(-0.447249\pi\)
0.164963 + 0.986300i \(0.447249\pi\)
\(140\) − 4175.08i − 0.213014i
\(141\) 0 0
\(142\) −12180.7 −0.604082
\(143\) 53819.2i 2.63187i
\(144\) 0 0
\(145\) 3824.44 0.181900
\(146\) − 17860.4i − 0.837887i
\(147\) 0 0
\(148\) −6497.49 −0.296635
\(149\) 32359.8i 1.45758i 0.684736 + 0.728791i \(0.259918\pi\)
−0.684736 + 0.728791i \(0.740082\pi\)
\(150\) 0 0
\(151\) 19569.8 0.858288 0.429144 0.903236i \(-0.358815\pi\)
0.429144 + 0.903236i \(0.358815\pi\)
\(152\) 10354.1i 0.448151i
\(153\) 0 0
\(154\) −8826.95 −0.372194
\(155\) 7534.29i 0.313602i
\(156\) 0 0
\(157\) −29337.2 −1.19020 −0.595100 0.803652i \(-0.702888\pi\)
−0.595100 + 0.803652i \(0.702888\pi\)
\(158\) 10475.8i 0.419637i
\(159\) 0 0
\(160\) −5100.97 −0.199257
\(161\) 8058.27i 0.310878i
\(162\) 0 0
\(163\) 39562.6 1.48905 0.744526 0.667593i \(-0.232675\pi\)
0.744526 + 0.667593i \(0.232675\pi\)
\(164\) − 5901.49i − 0.219419i
\(165\) 0 0
\(166\) −25550.3 −0.927214
\(167\) − 26576.6i − 0.952943i −0.879190 0.476471i \(-0.841916\pi\)
0.879190 0.476471i \(-0.158084\pi\)
\(168\) 0 0
\(169\) 73447.5 2.57160
\(170\) − 3118.28i − 0.107899i
\(171\) 0 0
\(172\) 16736.0 0.565710
\(173\) − 2650.12i − 0.0885468i −0.999019 0.0442734i \(-0.985903\pi\)
0.999019 0.0442734i \(-0.0140973\pi\)
\(174\) 0 0
\(175\) 3131.10 0.102240
\(176\) 10784.5i 0.348156i
\(177\) 0 0
\(178\) 10497.6 0.331321
\(179\) 28123.7i 0.877741i 0.898550 + 0.438871i \(0.144621\pi\)
−0.898550 + 0.438871i \(0.855379\pi\)
\(180\) 0 0
\(181\) 6956.01 0.212326 0.106163 0.994349i \(-0.466143\pi\)
0.106163 + 0.994349i \(0.466143\pi\)
\(182\) 16730.6i 0.505089i
\(183\) 0 0
\(184\) 9845.31 0.290800
\(185\) − 22886.7i − 0.668713i
\(186\) 0 0
\(187\) −6592.68 −0.188529
\(188\) 23699.0i 0.670523i
\(189\) 0 0
\(190\) −36471.1 −1.01028
\(191\) − 35193.7i − 0.964713i −0.875975 0.482356i \(-0.839781\pi\)
0.875975 0.482356i \(-0.160219\pi\)
\(192\) 0 0
\(193\) 37582.9 1.00897 0.504483 0.863422i \(-0.331683\pi\)
0.504483 + 0.863422i \(0.331683\pi\)
\(194\) − 38887.5i − 1.03325i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) 40590.7i 1.04591i 0.852361 + 0.522954i \(0.175170\pi\)
−0.852361 + 0.522954i \(0.824830\pi\)
\(198\) 0 0
\(199\) 71304.5 1.80057 0.900286 0.435298i \(-0.143357\pi\)
0.900286 + 0.435298i \(0.143357\pi\)
\(200\) − 3825.47i − 0.0956368i
\(201\) 0 0
\(202\) 50628.5 1.24077
\(203\) − 2513.55i − 0.0609951i
\(204\) 0 0
\(205\) 20787.3 0.494642
\(206\) − 11256.8i − 0.265266i
\(207\) 0 0
\(208\) 20440.8 0.472467
\(209\) 77107.3i 1.76524i
\(210\) 0 0
\(211\) −58607.1 −1.31639 −0.658196 0.752847i \(-0.728680\pi\)
−0.658196 + 0.752847i \(0.728680\pi\)
\(212\) − 25525.5i − 0.567939i
\(213\) 0 0
\(214\) −25214.4 −0.550582
\(215\) 58950.6i 1.27530i
\(216\) 0 0
\(217\) 4951.79 0.105158
\(218\) 38145.0i 0.802646i
\(219\) 0 0
\(220\) −37987.1 −0.784858
\(221\) 12495.7i 0.255845i
\(222\) 0 0
\(223\) −31333.8 −0.630091 −0.315045 0.949077i \(-0.602020\pi\)
−0.315045 + 0.949077i \(0.602020\pi\)
\(224\) 3352.53i 0.0668153i
\(225\) 0 0
\(226\) −11433.5 −0.223853
\(227\) 3437.19i 0.0667040i 0.999444 + 0.0333520i \(0.0106182\pi\)
−0.999444 + 0.0333520i \(0.989382\pi\)
\(228\) 0 0
\(229\) 67814.1 1.29315 0.646576 0.762850i \(-0.276201\pi\)
0.646576 + 0.762850i \(0.276201\pi\)
\(230\) 34679.1i 0.655559i
\(231\) 0 0
\(232\) −3070.97 −0.0570557
\(233\) 62389.0i 1.14920i 0.818434 + 0.574601i \(0.194843\pi\)
−0.818434 + 0.574601i \(0.805157\pi\)
\(234\) 0 0
\(235\) −83477.0 −1.51158
\(236\) 44964.2i 0.807314i
\(237\) 0 0
\(238\) −2049.44 −0.0361811
\(239\) 93904.5i 1.64396i 0.569518 + 0.821979i \(0.307130\pi\)
−0.569518 + 0.821979i \(0.692870\pi\)
\(240\) 0 0
\(241\) −107318. −1.84773 −0.923864 0.382722i \(-0.874987\pi\)
−0.923864 + 0.382722i \(0.874987\pi\)
\(242\) 38901.4i 0.664254i
\(243\) 0 0
\(244\) −38903.2 −0.653439
\(245\) − 9665.44i − 0.161024i
\(246\) 0 0
\(247\) 146149. 2.39553
\(248\) − 6049.92i − 0.0983663i
\(249\) 0 0
\(250\) −36339.3 −0.581429
\(251\) 56349.2i 0.894417i 0.894430 + 0.447209i \(0.147582\pi\)
−0.894430 + 0.447209i \(0.852418\pi\)
\(252\) 0 0
\(253\) 73318.5 1.14544
\(254\) − 52750.7i − 0.817638i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 34716.3i − 0.525614i −0.964848 0.262807i \(-0.915352\pi\)
0.964848 0.262807i \(-0.0846482\pi\)
\(258\) 0 0
\(259\) −15041.9 −0.224235
\(260\) 72000.6i 1.06510i
\(261\) 0 0
\(262\) −40062.6 −0.583628
\(263\) − 104358.i − 1.50874i −0.656449 0.754371i \(-0.727942\pi\)
0.656449 0.754371i \(-0.272058\pi\)
\(264\) 0 0
\(265\) 89910.7 1.28032
\(266\) 23970.0i 0.338771i
\(267\) 0 0
\(268\) 16071.8 0.223766
\(269\) 81356.1i 1.12431i 0.827032 + 0.562155i \(0.190027\pi\)
−0.827032 + 0.562155i \(0.809973\pi\)
\(270\) 0 0
\(271\) −125208. −1.70488 −0.852439 0.522827i \(-0.824878\pi\)
−0.852439 + 0.522827i \(0.824878\pi\)
\(272\) 2503.94i 0.0338443i
\(273\) 0 0
\(274\) −78903.3 −1.05098
\(275\) − 28488.4i − 0.376707i
\(276\) 0 0
\(277\) 3328.93 0.0433855 0.0216928 0.999765i \(-0.493094\pi\)
0.0216928 + 0.999765i \(0.493094\pi\)
\(278\) − 18029.9i − 0.233294i
\(279\) 0 0
\(280\) −11808.9 −0.150624
\(281\) − 151897.i − 1.92370i −0.273584 0.961848i \(-0.588209\pi\)
0.273584 0.961848i \(-0.411791\pi\)
\(282\) 0 0
\(283\) 4034.61 0.0503765 0.0251883 0.999683i \(-0.491981\pi\)
0.0251883 + 0.999683i \(0.491981\pi\)
\(284\) 34452.3i 0.427151i
\(285\) 0 0
\(286\) 152224. 1.86101
\(287\) − 13662.1i − 0.165865i
\(288\) 0 0
\(289\) 81990.3 0.981673
\(290\) − 10817.1i − 0.128622i
\(291\) 0 0
\(292\) −50516.8 −0.592475
\(293\) 39529.1i 0.460449i 0.973138 + 0.230225i \(0.0739461\pi\)
−0.973138 + 0.230225i \(0.926054\pi\)
\(294\) 0 0
\(295\) −158381. −1.81995
\(296\) 18377.7i 0.209752i
\(297\) 0 0
\(298\) 91527.3 1.03067
\(299\) − 138967.i − 1.55443i
\(300\) 0 0
\(301\) 38744.3 0.427637
\(302\) − 55351.8i − 0.606902i
\(303\) 0 0
\(304\) 29285.8 0.316891
\(305\) − 137032.i − 1.47307i
\(306\) 0 0
\(307\) −87173.0 −0.924922 −0.462461 0.886640i \(-0.653034\pi\)
−0.462461 + 0.886640i \(0.653034\pi\)
\(308\) 24966.4i 0.263181i
\(309\) 0 0
\(310\) 21310.2 0.221750
\(311\) 130723.i 1.35154i 0.737110 + 0.675772i \(0.236190\pi\)
−0.737110 + 0.675772i \(0.763810\pi\)
\(312\) 0 0
\(313\) −129850. −1.32542 −0.662711 0.748876i \(-0.730594\pi\)
−0.662711 + 0.748876i \(0.730594\pi\)
\(314\) 82978.2i 0.841598i
\(315\) 0 0
\(316\) 29630.1 0.296728
\(317\) 80859.0i 0.804656i 0.915496 + 0.402328i \(0.131799\pi\)
−0.915496 + 0.402328i \(0.868201\pi\)
\(318\) 0 0
\(319\) −22869.6 −0.224739
\(320\) 14427.7i 0.140896i
\(321\) 0 0
\(322\) 22792.2 0.219824
\(323\) 17902.8i 0.171599i
\(324\) 0 0
\(325\) −53996.8 −0.511213
\(326\) − 111900.i − 1.05292i
\(327\) 0 0
\(328\) −16691.9 −0.155152
\(329\) 54863.9i 0.506868i
\(330\) 0 0
\(331\) 44166.4 0.403122 0.201561 0.979476i \(-0.435399\pi\)
0.201561 + 0.979476i \(0.435399\pi\)
\(332\) 72267.1i 0.655639i
\(333\) 0 0
\(334\) −75170.0 −0.673832
\(335\) 56611.2i 0.504443i
\(336\) 0 0
\(337\) −19903.7 −0.175256 −0.0876280 0.996153i \(-0.527929\pi\)
−0.0876280 + 0.996153i \(0.527929\pi\)
\(338\) − 207741.i − 1.81840i
\(339\) 0 0
\(340\) −8819.84 −0.0762962
\(341\) − 45054.0i − 0.387458i
\(342\) 0 0
\(343\) −6352.45 −0.0539949
\(344\) − 47336.5i − 0.400017i
\(345\) 0 0
\(346\) −7495.66 −0.0626120
\(347\) − 151074.i − 1.25467i −0.778749 0.627335i \(-0.784146\pi\)
0.778749 0.627335i \(-0.215854\pi\)
\(348\) 0 0
\(349\) −67141.3 −0.551237 −0.275619 0.961267i \(-0.588883\pi\)
−0.275619 + 0.961267i \(0.588883\pi\)
\(350\) − 8856.09i − 0.0722946i
\(351\) 0 0
\(352\) 30503.1 0.246183
\(353\) 48713.4i 0.390930i 0.980711 + 0.195465i \(0.0626216\pi\)
−0.980711 + 0.195465i \(0.937378\pi\)
\(354\) 0 0
\(355\) −121354. −0.962939
\(356\) − 29691.6i − 0.234279i
\(357\) 0 0
\(358\) 79545.9 0.620657
\(359\) 149704.i 1.16157i 0.814057 + 0.580784i \(0.197254\pi\)
−0.814057 + 0.580784i \(0.802746\pi\)
\(360\) 0 0
\(361\) 79067.9 0.606716
\(362\) − 19674.6i − 0.150137i
\(363\) 0 0
\(364\) 47321.1 0.357152
\(365\) − 177940.i − 1.33564i
\(366\) 0 0
\(367\) 170863. 1.26857 0.634286 0.773098i \(-0.281294\pi\)
0.634286 + 0.773098i \(0.281294\pi\)
\(368\) − 27846.8i − 0.205626i
\(369\) 0 0
\(370\) −64733.4 −0.472851
\(371\) − 59092.3i − 0.429322i
\(372\) 0 0
\(373\) 75344.3 0.541542 0.270771 0.962644i \(-0.412721\pi\)
0.270771 + 0.962644i \(0.412721\pi\)
\(374\) 18646.9i 0.133310i
\(375\) 0 0
\(376\) 67030.8 0.474131
\(377\) 43346.9i 0.304983i
\(378\) 0 0
\(379\) 213599. 1.48703 0.743515 0.668719i \(-0.233157\pi\)
0.743515 + 0.668719i \(0.233157\pi\)
\(380\) 103156.i 0.714377i
\(381\) 0 0
\(382\) −99542.8 −0.682155
\(383\) 37077.1i 0.252760i 0.991982 + 0.126380i \(0.0403359\pi\)
−0.991982 + 0.126380i \(0.959664\pi\)
\(384\) 0 0
\(385\) −87941.4 −0.593297
\(386\) − 106301.i − 0.713446i
\(387\) 0 0
\(388\) −109991. −0.730620
\(389\) − 152168.i − 1.00560i −0.864403 0.502799i \(-0.832303\pi\)
0.864403 0.502799i \(-0.167697\pi\)
\(390\) 0 0
\(391\) 17023.1 0.111348
\(392\) 7761.20i 0.0505076i
\(393\) 0 0
\(394\) 114808. 0.739569
\(395\) 104369.i 0.668924i
\(396\) 0 0
\(397\) −273698. −1.73656 −0.868281 0.496073i \(-0.834775\pi\)
−0.868281 + 0.496073i \(0.834775\pi\)
\(398\) − 201680.i − 1.27320i
\(399\) 0 0
\(400\) −10820.1 −0.0676254
\(401\) − 206734.i − 1.28565i −0.766014 0.642824i \(-0.777763\pi\)
0.766014 0.642824i \(-0.222237\pi\)
\(402\) 0 0
\(403\) −85395.1 −0.525803
\(404\) − 143199.i − 0.877359i
\(405\) 0 0
\(406\) −7109.39 −0.0431301
\(407\) 136859.i 0.826200i
\(408\) 0 0
\(409\) −18110.7 −0.108265 −0.0541327 0.998534i \(-0.517239\pi\)
−0.0541327 + 0.998534i \(0.517239\pi\)
\(410\) − 58795.5i − 0.349765i
\(411\) 0 0
\(412\) −31839.1 −0.187571
\(413\) 104094.i 0.610272i
\(414\) 0 0
\(415\) −254553. −1.47803
\(416\) − 57815.4i − 0.334085i
\(417\) 0 0
\(418\) 218092. 1.24821
\(419\) − 31032.7i − 0.176763i −0.996087 0.0883816i \(-0.971830\pi\)
0.996087 0.0883816i \(-0.0281695\pi\)
\(420\) 0 0
\(421\) 104982. 0.592314 0.296157 0.955139i \(-0.404295\pi\)
0.296157 + 0.955139i \(0.404295\pi\)
\(422\) 165766.i 0.930830i
\(423\) 0 0
\(424\) −72196.9 −0.401594
\(425\) − 6614.44i − 0.0366197i
\(426\) 0 0
\(427\) −90062.1 −0.493954
\(428\) 71317.2i 0.389320i
\(429\) 0 0
\(430\) 166738. 0.901771
\(431\) − 279476.i − 1.50449i −0.658882 0.752246i \(-0.728971\pi\)
0.658882 0.752246i \(-0.271029\pi\)
\(432\) 0 0
\(433\) 147044. 0.784280 0.392140 0.919905i \(-0.371735\pi\)
0.392140 + 0.919905i \(0.371735\pi\)
\(434\) − 14005.8i − 0.0743580i
\(435\) 0 0
\(436\) 107890. 0.567557
\(437\) − 199100.i − 1.04258i
\(438\) 0 0
\(439\) −191301. −0.992631 −0.496316 0.868142i \(-0.665314\pi\)
−0.496316 + 0.868142i \(0.665314\pi\)
\(440\) 107444.i 0.554978i
\(441\) 0 0
\(442\) 35343.2 0.180910
\(443\) − 268170.i − 1.36648i −0.730195 0.683239i \(-0.760571\pi\)
0.730195 0.683239i \(-0.239429\pi\)
\(444\) 0 0
\(445\) 104586. 0.528143
\(446\) 88625.3i 0.445541i
\(447\) 0 0
\(448\) 9482.37 0.0472456
\(449\) − 71968.8i − 0.356986i −0.983941 0.178493i \(-0.942878\pi\)
0.983941 0.178493i \(-0.0571222\pi\)
\(450\) 0 0
\(451\) −124305. −0.611135
\(452\) 32338.9i 0.158288i
\(453\) 0 0
\(454\) 9721.85 0.0471669
\(455\) 166684.i 0.805138i
\(456\) 0 0
\(457\) −261174. −1.25054 −0.625270 0.780408i \(-0.715011\pi\)
−0.625270 + 0.780408i \(0.715011\pi\)
\(458\) − 191807.i − 0.914396i
\(459\) 0 0
\(460\) 98087.2 0.463550
\(461\) 148535.i 0.698917i 0.936952 + 0.349458i \(0.113634\pi\)
−0.936952 + 0.349458i \(0.886366\pi\)
\(462\) 0 0
\(463\) −179727. −0.838401 −0.419200 0.907894i \(-0.637689\pi\)
−0.419200 + 0.907894i \(0.637689\pi\)
\(464\) 8686.01i 0.0403445i
\(465\) 0 0
\(466\) 176463. 0.812609
\(467\) − 86089.6i − 0.394745i −0.980329 0.197373i \(-0.936759\pi\)
0.980329 0.197373i \(-0.0632409\pi\)
\(468\) 0 0
\(469\) 37206.7 0.169152
\(470\) 236109.i 1.06885i
\(471\) 0 0
\(472\) 127178. 0.570857
\(473\) − 352517.i − 1.57564i
\(474\) 0 0
\(475\) −77361.8 −0.342878
\(476\) 5796.69i 0.0255839i
\(477\) 0 0
\(478\) 265602. 1.16245
\(479\) − 100240.i − 0.436890i −0.975849 0.218445i \(-0.929902\pi\)
0.975849 0.218445i \(-0.0700984\pi\)
\(480\) 0 0
\(481\) 259402. 1.12120
\(482\) 303541.i 1.30654i
\(483\) 0 0
\(484\) 110030. 0.469698
\(485\) − 387430.i − 1.64706i
\(486\) 0 0
\(487\) −79918.4 −0.336968 −0.168484 0.985704i \(-0.553887\pi\)
−0.168484 + 0.985704i \(0.553887\pi\)
\(488\) 110035.i 0.462051i
\(489\) 0 0
\(490\) −27338.0 −0.113861
\(491\) 376135.i 1.56020i 0.625654 + 0.780101i \(0.284832\pi\)
−0.625654 + 0.780101i \(0.715168\pi\)
\(492\) 0 0
\(493\) −5309.86 −0.0218469
\(494\) − 413371.i − 1.69389i
\(495\) 0 0
\(496\) −17111.8 −0.0695555
\(497\) 79758.1i 0.322896i
\(498\) 0 0
\(499\) 104632. 0.420205 0.210103 0.977679i \(-0.432620\pi\)
0.210103 + 0.977679i \(0.432620\pi\)
\(500\) 102783.i 0.411133i
\(501\) 0 0
\(502\) 159380. 0.632449
\(503\) 8060.64i 0.0318591i 0.999873 + 0.0159295i \(0.00507075\pi\)
−0.999873 + 0.0159295i \(0.994929\pi\)
\(504\) 0 0
\(505\) 504403. 1.97786
\(506\) − 207376.i − 0.809948i
\(507\) 0 0
\(508\) −149202. −0.578157
\(509\) − 383177.i − 1.47898i −0.673165 0.739492i \(-0.735066\pi\)
0.673165 0.739492i \(-0.264934\pi\)
\(510\) 0 0
\(511\) −116948. −0.447869
\(512\) − 11585.2i − 0.0441942i
\(513\) 0 0
\(514\) −98192.4 −0.371665
\(515\) − 112150.i − 0.422848i
\(516\) 0 0
\(517\) 499181. 1.86757
\(518\) 42544.9i 0.158558i
\(519\) 0 0
\(520\) 203648. 0.753137
\(521\) 165297.i 0.608962i 0.952518 + 0.304481i \(0.0984831\pi\)
−0.952518 + 0.304481i \(0.901517\pi\)
\(522\) 0 0
\(523\) 346312. 1.26609 0.633044 0.774116i \(-0.281805\pi\)
0.633044 + 0.774116i \(0.281805\pi\)
\(524\) 113314.i 0.412688i
\(525\) 0 0
\(526\) −295169. −1.06684
\(527\) − 10460.6i − 0.0376649i
\(528\) 0 0
\(529\) 90524.2 0.323484
\(530\) − 254306.i − 0.905325i
\(531\) 0 0
\(532\) 67797.5 0.239547
\(533\) 235608.i 0.829345i
\(534\) 0 0
\(535\) −251207. −0.877656
\(536\) − 45457.9i − 0.158227i
\(537\) 0 0
\(538\) 230110. 0.795007
\(539\) 57798.0i 0.198946i
\(540\) 0 0
\(541\) 138669. 0.473787 0.236894 0.971536i \(-0.423871\pi\)
0.236894 + 0.971536i \(0.423871\pi\)
\(542\) 354142.i 1.20553i
\(543\) 0 0
\(544\) 7082.20 0.0239315
\(545\) 380032.i 1.27946i
\(546\) 0 0
\(547\) −284471. −0.950742 −0.475371 0.879785i \(-0.657686\pi\)
−0.475371 + 0.879785i \(0.657686\pi\)
\(548\) 223172.i 0.743154i
\(549\) 0 0
\(550\) −80577.5 −0.266372
\(551\) 62103.6i 0.204557i
\(552\) 0 0
\(553\) 68594.6 0.224305
\(554\) − 9415.63i − 0.0306782i
\(555\) 0 0
\(556\) −50996.2 −0.164963
\(557\) 37910.6i 0.122194i 0.998132 + 0.0610971i \(0.0194599\pi\)
−0.998132 + 0.0610971i \(0.980540\pi\)
\(558\) 0 0
\(559\) −668158. −2.13823
\(560\) 33400.6i 0.106507i
\(561\) 0 0
\(562\) −429629. −1.36026
\(563\) − 164337.i − 0.518465i −0.965815 0.259233i \(-0.916530\pi\)
0.965815 0.259233i \(-0.0834696\pi\)
\(564\) 0 0
\(565\) −113910. −0.356833
\(566\) − 11411.6i − 0.0356216i
\(567\) 0 0
\(568\) 97445.7 0.302041
\(569\) 394027.i 1.21703i 0.793542 + 0.608516i \(0.208235\pi\)
−0.793542 + 0.608516i \(0.791765\pi\)
\(570\) 0 0
\(571\) −269585. −0.826844 −0.413422 0.910540i \(-0.635667\pi\)
−0.413422 + 0.910540i \(0.635667\pi\)
\(572\) − 430553.i − 1.31594i
\(573\) 0 0
\(574\) −38642.3 −0.117284
\(575\) 73560.5i 0.222489i
\(576\) 0 0
\(577\) 11879.8 0.0356826 0.0178413 0.999841i \(-0.494321\pi\)
0.0178413 + 0.999841i \(0.494321\pi\)
\(578\) − 231904.i − 0.694148i
\(579\) 0 0
\(580\) −30595.5 −0.0909498
\(581\) 167301.i 0.495616i
\(582\) 0 0
\(583\) −537653. −1.58185
\(584\) 142883.i 0.418943i
\(585\) 0 0
\(586\) 111805. 0.325587
\(587\) 433502.i 1.25810i 0.777365 + 0.629050i \(0.216556\pi\)
−0.777365 + 0.629050i \(0.783444\pi\)
\(588\) 0 0
\(589\) −122347. −0.352664
\(590\) 447970.i 1.28690i
\(591\) 0 0
\(592\) 51979.9 0.148317
\(593\) 65763.0i 0.187013i 0.995619 + 0.0935066i \(0.0298076\pi\)
−0.995619 + 0.0935066i \(0.970192\pi\)
\(594\) 0 0
\(595\) −20418.2 −0.0576745
\(596\) − 258878.i − 0.728791i
\(597\) 0 0
\(598\) −393059. −1.09915
\(599\) 232771.i 0.648746i 0.945929 + 0.324373i \(0.105153\pi\)
−0.945929 + 0.324373i \(0.894847\pi\)
\(600\) 0 0
\(601\) −98549.6 −0.272839 −0.136419 0.990651i \(-0.543559\pi\)
−0.136419 + 0.990651i \(0.543559\pi\)
\(602\) − 109585.i − 0.302385i
\(603\) 0 0
\(604\) −156559. −0.429144
\(605\) 387568.i 1.05886i
\(606\) 0 0
\(607\) 58943.3 0.159977 0.0799884 0.996796i \(-0.474512\pi\)
0.0799884 + 0.996796i \(0.474512\pi\)
\(608\) − 82832.7i − 0.224076i
\(609\) 0 0
\(610\) −387586. −1.04162
\(611\) − 946145.i − 2.53440i
\(612\) 0 0
\(613\) 127556. 0.339453 0.169727 0.985491i \(-0.445712\pi\)
0.169727 + 0.985491i \(0.445712\pi\)
\(614\) 246562.i 0.654019i
\(615\) 0 0
\(616\) 70615.6 0.186097
\(617\) 183225.i 0.481299i 0.970612 + 0.240649i \(0.0773604\pi\)
−0.970612 + 0.240649i \(0.922640\pi\)
\(618\) 0 0
\(619\) −171376. −0.447268 −0.223634 0.974673i \(-0.571792\pi\)
−0.223634 + 0.974673i \(0.571792\pi\)
\(620\) − 60274.3i − 0.156801i
\(621\) 0 0
\(622\) 369740. 0.955686
\(623\) − 68737.1i − 0.177099i
\(624\) 0 0
\(625\) −467707. −1.19733
\(626\) 367272.i 0.937214i
\(627\) 0 0
\(628\) 234698. 0.595100
\(629\) 31776.0i 0.0803151i
\(630\) 0 0
\(631\) −195310. −0.490530 −0.245265 0.969456i \(-0.578875\pi\)
−0.245265 + 0.969456i \(0.578875\pi\)
\(632\) − 83806.6i − 0.209819i
\(633\) 0 0
\(634\) 228704. 0.568977
\(635\) − 525546.i − 1.30336i
\(636\) 0 0
\(637\) 109550. 0.269981
\(638\) 64685.0i 0.158914i
\(639\) 0 0
\(640\) 40807.7 0.0996283
\(641\) 10697.3i 0.0260351i 0.999915 + 0.0130175i \(0.00414373\pi\)
−0.999915 + 0.0130175i \(0.995856\pi\)
\(642\) 0 0
\(643\) −171673. −0.415221 −0.207611 0.978212i \(-0.566569\pi\)
−0.207611 + 0.978212i \(0.566569\pi\)
\(644\) − 64466.1i − 0.155439i
\(645\) 0 0
\(646\) 50636.6 0.121339
\(647\) 814856.i 1.94658i 0.229580 + 0.973290i \(0.426265\pi\)
−0.229580 + 0.973290i \(0.573735\pi\)
\(648\) 0 0
\(649\) 947099. 2.24857
\(650\) 152726.i 0.361482i
\(651\) 0 0
\(652\) −316501. −0.744526
\(653\) 741201.i 1.73824i 0.494601 + 0.869120i \(0.335314\pi\)
−0.494601 + 0.869120i \(0.664686\pi\)
\(654\) 0 0
\(655\) −399137. −0.930334
\(656\) 47211.9i 0.109709i
\(657\) 0 0
\(658\) 155178. 0.358410
\(659\) 548473.i 1.26294i 0.775398 + 0.631472i \(0.217549\pi\)
−0.775398 + 0.631472i \(0.782451\pi\)
\(660\) 0 0
\(661\) 426168. 0.975389 0.487695 0.873014i \(-0.337838\pi\)
0.487695 + 0.873014i \(0.337838\pi\)
\(662\) − 124922.i − 0.285050i
\(663\) 0 0
\(664\) 204402. 0.463607
\(665\) 238809.i 0.540018i
\(666\) 0 0
\(667\) 59052.0 0.132734
\(668\) 212613.i 0.476471i
\(669\) 0 0
\(670\) 160121. 0.356695
\(671\) 819433.i 1.81999i
\(672\) 0 0
\(673\) 332282. 0.733630 0.366815 0.930294i \(-0.380448\pi\)
0.366815 + 0.930294i \(0.380448\pi\)
\(674\) 56296.0i 0.123925i
\(675\) 0 0
\(676\) −587580. −1.28580
\(677\) 371163.i 0.809818i 0.914357 + 0.404909i \(0.132697\pi\)
−0.914357 + 0.404909i \(0.867303\pi\)
\(678\) 0 0
\(679\) −254632. −0.552297
\(680\) 24946.3i 0.0539496i
\(681\) 0 0
\(682\) −127432. −0.273974
\(683\) − 688730.i − 1.47641i −0.674575 0.738206i \(-0.735673\pi\)
0.674575 0.738206i \(-0.264327\pi\)
\(684\) 0 0
\(685\) −786100. −1.67532
\(686\) 17967.4i 0.0381802i
\(687\) 0 0
\(688\) −133888. −0.282855
\(689\) 1.01906e6i 2.14666i
\(690\) 0 0
\(691\) 28665.8 0.0600354 0.0300177 0.999549i \(-0.490444\pi\)
0.0300177 + 0.999549i \(0.490444\pi\)
\(692\) 21200.9i 0.0442734i
\(693\) 0 0
\(694\) −427301. −0.887186
\(695\) − 179628.i − 0.371882i
\(696\) 0 0
\(697\) −28861.2 −0.0594085
\(698\) 189904.i 0.389784i
\(699\) 0 0
\(700\) −25048.8 −0.0511200
\(701\) − 347036.i − 0.706218i −0.935582 0.353109i \(-0.885124\pi\)
0.935582 0.353109i \(-0.114876\pi\)
\(702\) 0 0
\(703\) 371648. 0.752007
\(704\) − 86275.7i − 0.174078i
\(705\) 0 0
\(706\) 137782. 0.276429
\(707\) − 331510.i − 0.663221i
\(708\) 0 0
\(709\) 117651. 0.234047 0.117024 0.993129i \(-0.462665\pi\)
0.117024 + 0.993129i \(0.462665\pi\)
\(710\) 343242.i 0.680901i
\(711\) 0 0
\(712\) −83980.6 −0.165661
\(713\) 116335.i 0.228839i
\(714\) 0 0
\(715\) 1.51658e6 2.96656
\(716\) − 224990.i − 0.438871i
\(717\) 0 0
\(718\) 423427. 0.821353
\(719\) − 376532.i − 0.728356i −0.931329 0.364178i \(-0.881350\pi\)
0.931329 0.364178i \(-0.118650\pi\)
\(720\) 0 0
\(721\) −73708.6 −0.141791
\(722\) − 223638.i − 0.429013i
\(723\) 0 0
\(724\) −55648.0 −0.106163
\(725\) − 22945.1i − 0.0436530i
\(726\) 0 0
\(727\) 618700. 1.17061 0.585304 0.810814i \(-0.300975\pi\)
0.585304 + 0.810814i \(0.300975\pi\)
\(728\) − 133844.i − 0.252544i
\(729\) 0 0
\(730\) −503290. −0.944437
\(731\) − 81847.2i − 0.153168i
\(732\) 0 0
\(733\) −454163. −0.845287 −0.422643 0.906296i \(-0.638898\pi\)
−0.422643 + 0.906296i \(0.638898\pi\)
\(734\) − 483273.i − 0.897016i
\(735\) 0 0
\(736\) −78762.5 −0.145400
\(737\) − 338527.i − 0.623244i
\(738\) 0 0
\(739\) 200302. 0.366771 0.183386 0.983041i \(-0.441294\pi\)
0.183386 + 0.983041i \(0.441294\pi\)
\(740\) 183094.i 0.334356i
\(741\) 0 0
\(742\) −167138. −0.303576
\(743\) 854018.i 1.54700i 0.633799 + 0.773498i \(0.281495\pi\)
−0.633799 + 0.773498i \(0.718505\pi\)
\(744\) 0 0
\(745\) 911871. 1.64294
\(746\) − 213106.i − 0.382928i
\(747\) 0 0
\(748\) 52741.4 0.0942646
\(749\) 165102.i 0.294298i
\(750\) 0 0
\(751\) −692616. −1.22804 −0.614020 0.789290i \(-0.710449\pi\)
−0.614020 + 0.789290i \(0.710449\pi\)
\(752\) − 189592.i − 0.335261i
\(753\) 0 0
\(754\) 122604. 0.215656
\(755\) − 551461.i − 0.967433i
\(756\) 0 0
\(757\) 154839. 0.270202 0.135101 0.990832i \(-0.456864\pi\)
0.135101 + 0.990832i \(0.456864\pi\)
\(758\) − 604148.i − 1.05149i
\(759\) 0 0
\(760\) 291769. 0.505141
\(761\) − 913167.i − 1.57682i −0.615153 0.788408i \(-0.710906\pi\)
0.615153 0.788408i \(-0.289094\pi\)
\(762\) 0 0
\(763\) 249769. 0.429032
\(764\) 281549.i 0.482356i
\(765\) 0 0
\(766\) 104870. 0.178728
\(767\) − 1.79513e6i − 3.05144i
\(768\) 0 0
\(769\) −1.00329e6 −1.69657 −0.848286 0.529538i \(-0.822365\pi\)
−0.848286 + 0.529538i \(0.822365\pi\)
\(770\) 248736.i 0.419524i
\(771\) 0 0
\(772\) −300664. −0.504483
\(773\) − 474439.i − 0.794001i −0.917818 0.397001i \(-0.870051\pi\)
0.917818 0.397001i \(-0.129949\pi\)
\(774\) 0 0
\(775\) 45202.8 0.0752595
\(776\) 311100.i 0.516627i
\(777\) 0 0
\(778\) −430397. −0.711066
\(779\) 337558.i 0.556254i
\(780\) 0 0
\(781\) 725682. 1.18972
\(782\) − 48148.5i − 0.0787353i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) 826698.i 1.34155i
\(786\) 0 0
\(787\) 590744. 0.953783 0.476892 0.878962i \(-0.341763\pi\)
0.476892 + 0.878962i \(0.341763\pi\)
\(788\) − 324725.i − 0.522954i
\(789\) 0 0
\(790\) 295200. 0.473000
\(791\) 74865.5i 0.119654i
\(792\) 0 0
\(793\) 1.55315e6 2.46983
\(794\) 774134.i 1.22793i
\(795\) 0 0
\(796\) −570436. −0.900286
\(797\) 600262.i 0.944984i 0.881335 + 0.472492i \(0.156645\pi\)
−0.881335 + 0.472492i \(0.843355\pi\)
\(798\) 0 0
\(799\) 115900. 0.181547
\(800\) 30603.8i 0.0478184i
\(801\) 0 0
\(802\) −584731. −0.909091
\(803\) 1.06406e6i 1.65019i
\(804\) 0 0
\(805\) 227075. 0.350411
\(806\) 241534.i 0.371799i
\(807\) 0 0
\(808\) −405028. −0.620386
\(809\) 649360.i 0.992176i 0.868272 + 0.496088i \(0.165231\pi\)
−0.868272 + 0.496088i \(0.834769\pi\)
\(810\) 0 0
\(811\) −1.13293e6 −1.72250 −0.861251 0.508179i \(-0.830319\pi\)
−0.861251 + 0.508179i \(0.830319\pi\)
\(812\) 20108.4i 0.0304976i
\(813\) 0 0
\(814\) 387096. 0.584212
\(815\) − 1.11484e6i − 1.67841i
\(816\) 0 0
\(817\) −957277. −1.43415
\(818\) 51224.9i 0.0765552i
\(819\) 0 0
\(820\) −166299. −0.247321
\(821\) − 463203.i − 0.687204i −0.939115 0.343602i \(-0.888353\pi\)
0.939115 0.343602i \(-0.111647\pi\)
\(822\) 0 0
\(823\) 1.00476e6 1.48341 0.741705 0.670726i \(-0.234017\pi\)
0.741705 + 0.670726i \(0.234017\pi\)
\(824\) 90054.6i 0.132633i
\(825\) 0 0
\(826\) 294421. 0.431528
\(827\) − 57967.0i − 0.0847559i −0.999102 0.0423779i \(-0.986507\pi\)
0.999102 0.0423779i \(-0.0134934\pi\)
\(828\) 0 0
\(829\) 1.08695e6 1.58162 0.790809 0.612063i \(-0.209660\pi\)
0.790809 + 0.612063i \(0.209660\pi\)
\(830\) 719985.i 1.04512i
\(831\) 0 0
\(832\) −163527. −0.236234
\(833\) 13419.5i 0.0193396i
\(834\) 0 0
\(835\) −748906. −1.07412
\(836\) − 616858.i − 0.882618i
\(837\) 0 0
\(838\) −87773.8 −0.124990
\(839\) 1.04936e6i 1.49073i 0.666655 + 0.745367i \(0.267726\pi\)
−0.666655 + 0.745367i \(0.732274\pi\)
\(840\) 0 0
\(841\) 688861. 0.973957
\(842\) − 296935.i − 0.418829i
\(843\) 0 0
\(844\) 468857. 0.658196
\(845\) − 2.06969e6i − 2.89862i
\(846\) 0 0
\(847\) 254722. 0.355059
\(848\) 204204.i 0.283970i
\(849\) 0 0
\(850\) −18708.5 −0.0258941
\(851\) − 353387.i − 0.487968i
\(852\) 0 0
\(853\) 731878. 1.00587 0.502933 0.864325i \(-0.332254\pi\)
0.502933 + 0.864325i \(0.332254\pi\)
\(854\) 254734.i 0.349278i
\(855\) 0 0
\(856\) 201715. 0.275291
\(857\) − 704349.i − 0.959017i −0.877537 0.479509i \(-0.840815\pi\)
0.877537 0.479509i \(-0.159185\pi\)
\(858\) 0 0
\(859\) −1.16702e6 −1.58159 −0.790793 0.612084i \(-0.790331\pi\)
−0.790793 + 0.612084i \(0.790331\pi\)
\(860\) − 471605.i − 0.637649i
\(861\) 0 0
\(862\) −790477. −1.06384
\(863\) − 126671.i − 0.170081i −0.996377 0.0850405i \(-0.972898\pi\)
0.996377 0.0850405i \(-0.0271020\pi\)
\(864\) 0 0
\(865\) −74678.0 −0.0998069
\(866\) − 415903.i − 0.554570i
\(867\) 0 0
\(868\) −39614.3 −0.0525790
\(869\) − 624111.i − 0.826461i
\(870\) 0 0
\(871\) −641642. −0.845778
\(872\) − 305160.i − 0.401323i
\(873\) 0 0
\(874\) −563140. −0.737214
\(875\) 237946.i 0.310787i
\(876\) 0 0
\(877\) −891593. −1.15922 −0.579612 0.814892i \(-0.696796\pi\)
−0.579612 + 0.814892i \(0.696796\pi\)
\(878\) 541081.i 0.701896i
\(879\) 0 0
\(880\) 303897. 0.392429
\(881\) − 494760.i − 0.637445i −0.947848 0.318723i \(-0.896746\pi\)
0.947848 0.318723i \(-0.103254\pi\)
\(882\) 0 0
\(883\) −104454. −0.133969 −0.0669847 0.997754i \(-0.521338\pi\)
−0.0669847 + 0.997754i \(0.521338\pi\)
\(884\) − 99965.8i − 0.127922i
\(885\) 0 0
\(886\) −758499. −0.966246
\(887\) 176125.i 0.223859i 0.993716 + 0.111929i \(0.0357031\pi\)
−0.993716 + 0.111929i \(0.964297\pi\)
\(888\) 0 0
\(889\) −345406. −0.437046
\(890\) − 295813.i − 0.373454i
\(891\) 0 0
\(892\) 250670. 0.315045
\(893\) − 1.35555e6i − 1.69986i
\(894\) 0 0
\(895\) 792502. 0.989360
\(896\) − 26820.2i − 0.0334077i
\(897\) 0 0
\(898\) −203559. −0.252427
\(899\) − 36287.3i − 0.0448989i
\(900\) 0 0
\(901\) −124832. −0.153772
\(902\) 351589.i 0.432138i
\(903\) 0 0
\(904\) 91468.1 0.111926
\(905\) − 196014.i − 0.239326i
\(906\) 0 0
\(907\) 734712. 0.893105 0.446553 0.894757i \(-0.352652\pi\)
0.446553 + 0.894757i \(0.352652\pi\)
\(908\) − 27497.5i − 0.0333520i
\(909\) 0 0
\(910\) 471452. 0.569318
\(911\) 628633.i 0.757462i 0.925507 + 0.378731i \(0.123639\pi\)
−0.925507 + 0.378731i \(0.876361\pi\)
\(912\) 0 0
\(913\) 1.52219e6 1.82611
\(914\) 738712.i 0.884266i
\(915\) 0 0
\(916\) −542513. −0.646576
\(917\) 262326.i 0.311963i
\(918\) 0 0
\(919\) −1.34955e6 −1.59793 −0.798965 0.601377i \(-0.794619\pi\)
−0.798965 + 0.601377i \(0.794619\pi\)
\(920\) − 277432.i − 0.327779i
\(921\) 0 0
\(922\) 420119. 0.494209
\(923\) − 1.37545e6i − 1.61452i
\(924\) 0 0
\(925\) −137311. −0.160480
\(926\) 508345.i 0.592839i
\(927\) 0 0
\(928\) 24567.7 0.0285279
\(929\) − 1.52629e6i − 1.76851i −0.467007 0.884254i \(-0.654668\pi\)
0.467007 0.884254i \(-0.345332\pi\)
\(930\) 0 0
\(931\) 156953. 0.181080
\(932\) − 499112.i − 0.574601i
\(933\) 0 0
\(934\) −243498. −0.279127
\(935\) 185776.i 0.212504i
\(936\) 0 0
\(937\) 542424. 0.617817 0.308908 0.951092i \(-0.400036\pi\)
0.308908 + 0.951092i \(0.400036\pi\)
\(938\) − 105237.i − 0.119608i
\(939\) 0 0
\(940\) 667816. 0.755790
\(941\) 620071.i 0.700265i 0.936700 + 0.350132i \(0.113863\pi\)
−0.936700 + 0.350132i \(0.886137\pi\)
\(942\) 0 0
\(943\) 320971. 0.360946
\(944\) − 359713.i − 0.403657i
\(945\) 0 0
\(946\) −997067. −1.11415
\(947\) 620123.i 0.691477i 0.938331 + 0.345738i \(0.112372\pi\)
−0.938331 + 0.345738i \(0.887628\pi\)
\(948\) 0 0
\(949\) 2.01681e6 2.23940
\(950\) 218812.i 0.242451i
\(951\) 0 0
\(952\) 16395.5 0.0180905
\(953\) 568767.i 0.626251i 0.949712 + 0.313126i \(0.101376\pi\)
−0.949712 + 0.313126i \(0.898624\pi\)
\(954\) 0 0
\(955\) −991727. −1.08739
\(956\) − 751236.i − 0.821979i
\(957\) 0 0
\(958\) −283523. −0.308928
\(959\) 516651.i 0.561772i
\(960\) 0 0
\(961\) −852034. −0.922593
\(962\) − 733700.i − 0.792809i
\(963\) 0 0
\(964\) 858543. 0.923864
\(965\) − 1.05905e6i − 1.13727i
\(966\) 0 0
\(967\) 747986. 0.799909 0.399954 0.916535i \(-0.369026\pi\)
0.399954 + 0.916535i \(0.369026\pi\)
\(968\) − 311211.i − 0.332127i
\(969\) 0 0
\(970\) −1.09582e6 −1.16465
\(971\) 1.50806e6i 1.59949i 0.600343 + 0.799743i \(0.295031\pi\)
−0.600343 + 0.799743i \(0.704969\pi\)
\(972\) 0 0
\(973\) −118058. −0.124701
\(974\) 226043.i 0.238272i
\(975\) 0 0
\(976\) 311225. 0.326720
\(977\) 1.24528e6i 1.30460i 0.757960 + 0.652301i \(0.226196\pi\)
−0.757960 + 0.652301i \(0.773804\pi\)
\(978\) 0 0
\(979\) −625407. −0.652525
\(980\) 77323.5i 0.0805118i
\(981\) 0 0
\(982\) 1.06387e6 1.10323
\(983\) − 753418.i − 0.779703i −0.920878 0.389851i \(-0.872526\pi\)
0.920878 0.389851i \(-0.127474\pi\)
\(984\) 0 0
\(985\) 1.14381e6 1.17891
\(986\) 15018.6i 0.0154481i
\(987\) 0 0
\(988\) −1.16919e6 −1.19776
\(989\) 910239.i 0.930599i
\(990\) 0 0
\(991\) −77288.4 −0.0786986 −0.0393493 0.999226i \(-0.512528\pi\)
−0.0393493 + 0.999226i \(0.512528\pi\)
\(992\) 48399.4i 0.0491832i
\(993\) 0 0
\(994\) 225590. 0.228322
\(995\) − 2.00930e6i − 2.02954i
\(996\) 0 0
\(997\) 1.32860e6 1.33661 0.668305 0.743888i \(-0.267020\pi\)
0.668305 + 0.743888i \(0.267020\pi\)
\(998\) − 295943.i − 0.297130i
\(999\) 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.b.323.2 8
3.2 odd 2 inner 378.5.b.b.323.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.b.323.2 8 1.1 even 1 trivial
378.5.b.b.323.7 yes 8 3.2 odd 2 inner