Properties

Label 378.5.b.b.323.1
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 323.1
Root \(4.82885i\) of defining polynomial
Character \(\chi\) \(=\) 378.323
Dual form 378.5.b.b.323.8

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{2} -8.00000 q^{4} -30.1368i q^{5} +18.5203 q^{7} +22.6274i q^{8} +O(q^{10})\) \(q-2.82843i q^{2} -8.00000 q^{4} -30.1368i q^{5} +18.5203 q^{7} +22.6274i q^{8} -85.2398 q^{10} -11.8001i q^{11} -159.095 q^{13} -52.3832i q^{14} +64.0000 q^{16} +120.538i q^{17} -391.298 q^{19} +241.095i q^{20} -33.3758 q^{22} -371.311i q^{23} -283.229 q^{25} +449.990i q^{26} -148.162 q^{28} -216.816i q^{29} -930.283 q^{31} -181.019i q^{32} +340.933 q^{34} -558.142i q^{35} -672.478 q^{37} +1106.76i q^{38} +681.919 q^{40} +368.730i q^{41} +1278.43 q^{43} +94.4009i q^{44} -1050.23 q^{46} +1899.14i q^{47} +343.000 q^{49} +801.092i q^{50} +1272.76 q^{52} +1723.71i q^{53} -355.618 q^{55} +419.066i q^{56} -613.249 q^{58} +6318.71i q^{59} +2529.76 q^{61} +2631.24i q^{62} -512.000 q^{64} +4794.63i q^{65} -6171.39 q^{67} -964.305i q^{68} -1578.66 q^{70} +907.855i q^{71} -3649.04 q^{73} +1902.06i q^{74} +3130.38 q^{76} -218.541i q^{77} -6817.08 q^{79} -1928.76i q^{80} +1042.93 q^{82} +9100.53i q^{83} +3632.64 q^{85} -3615.96i q^{86} +267.006 q^{88} -8432.60i q^{89} -2946.49 q^{91} +2970.49i q^{92} +5371.59 q^{94} +11792.5i q^{95} -545.318 q^{97} -970.151i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 64 q^{4}+O(q^{10}) \) 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
\(5\) − 30.1368i − 1.20547i −0.797940 0.602737i \(-0.794077\pi\)
0.797940 0.602737i \(-0.205923\pi\)
\(6\) 0 0
\(7\) 18.5203 0.377964
\(8\) 22.6274i 0.353553i
\(9\) 0 0
\(10\) −85.2398 −0.852398
\(11\) − 11.8001i − 0.0975216i −0.998810 0.0487608i \(-0.984473\pi\)
0.998810 0.0487608i \(-0.0155272\pi\)
\(12\) 0 0
\(13\) −159.095 −0.941393 −0.470696 0.882295i \(-0.655997\pi\)
−0.470696 + 0.882295i \(0.655997\pi\)
\(14\) − 52.3832i − 0.267261i
\(15\) 0 0
\(16\) 64.0000 0.250000
\(17\) 120.538i 0.417087i 0.978013 + 0.208543i \(0.0668723\pi\)
−0.978013 + 0.208543i \(0.933128\pi\)
\(18\) 0 0
\(19\) −391.298 −1.08393 −0.541964 0.840402i \(-0.682319\pi\)
−0.541964 + 0.840402i \(0.682319\pi\)
\(20\) 241.095i 0.602737i
\(21\) 0 0
\(22\) −33.3758 −0.0689582
\(23\) − 371.311i − 0.701911i −0.936392 0.350955i \(-0.885857\pi\)
0.936392 0.350955i \(-0.114143\pi\)
\(24\) 0 0
\(25\) −283.229 −0.453166
\(26\) 449.990i 0.665665i
\(27\) 0 0
\(28\) −148.162 −0.188982
\(29\) − 216.816i − 0.257808i −0.991657 0.128904i \(-0.958854\pi\)
0.991657 0.128904i \(-0.0411459\pi\)
\(30\) 0 0
\(31\) −930.283 −0.968037 −0.484018 0.875058i \(-0.660823\pi\)
−0.484018 + 0.875058i \(0.660823\pi\)
\(32\) − 181.019i − 0.176777i
\(33\) 0 0
\(34\) 340.933 0.294925
\(35\) − 558.142i − 0.455626i
\(36\) 0 0
\(37\) −672.478 −0.491219 −0.245609 0.969369i \(-0.578988\pi\)
−0.245609 + 0.969369i \(0.578988\pi\)
\(38\) 1106.76i 0.766453i
\(39\) 0 0
\(40\) 681.919 0.426199
\(41\) 368.730i 0.219351i 0.993967 + 0.109676i \(0.0349812\pi\)
−0.993967 + 0.109676i \(0.965019\pi\)
\(42\) 0 0
\(43\) 1278.43 0.691419 0.345710 0.938342i \(-0.387638\pi\)
0.345710 + 0.938342i \(0.387638\pi\)
\(44\) 94.4009i 0.0487608i
\(45\) 0 0
\(46\) −1050.23 −0.496326
\(47\) 1899.14i 0.859730i 0.902893 + 0.429865i \(0.141439\pi\)
−0.902893 + 0.429865i \(0.858561\pi\)
\(48\) 0 0
\(49\) 343.000 0.142857
\(50\) 801.092i 0.320437i
\(51\) 0 0
\(52\) 1272.76 0.470696
\(53\) 1723.71i 0.613638i 0.951768 + 0.306819i \(0.0992647\pi\)
−0.951768 + 0.306819i \(0.900735\pi\)
\(54\) 0 0
\(55\) −355.618 −0.117560
\(56\) 419.066i 0.133631i
\(57\) 0 0
\(58\) −613.249 −0.182298
\(59\) 6318.71i 1.81520i 0.419837 + 0.907600i \(0.362087\pi\)
−0.419837 + 0.907600i \(0.637913\pi\)
\(60\) 0 0
\(61\) 2529.76 0.679860 0.339930 0.940451i \(-0.389597\pi\)
0.339930 + 0.940451i \(0.389597\pi\)
\(62\) 2631.24i 0.684505i
\(63\) 0 0
\(64\) −512.000 −0.125000
\(65\) 4794.63i 1.13482i
\(66\) 0 0
\(67\) −6171.39 −1.37478 −0.687391 0.726288i \(-0.741244\pi\)
−0.687391 + 0.726288i \(0.741244\pi\)
\(68\) − 964.305i − 0.208543i
\(69\) 0 0
\(70\) −1578.66 −0.322176
\(71\) 907.855i 0.180094i 0.995938 + 0.0900471i \(0.0287017\pi\)
−0.995938 + 0.0900471i \(0.971298\pi\)
\(72\) 0 0
\(73\) −3649.04 −0.684751 −0.342376 0.939563i \(-0.611232\pi\)
−0.342376 + 0.939563i \(0.611232\pi\)
\(74\) 1902.06i 0.347344i
\(75\) 0 0
\(76\) 3130.38 0.541964
\(77\) − 218.541i − 0.0368597i
\(78\) 0 0
\(79\) −6817.08 −1.09231 −0.546153 0.837686i \(-0.683908\pi\)
−0.546153 + 0.837686i \(0.683908\pi\)
\(80\) − 1928.76i − 0.301368i
\(81\) 0 0
\(82\) 1042.93 0.155105
\(83\) 9100.53i 1.32102i 0.750816 + 0.660512i \(0.229661\pi\)
−0.750816 + 0.660512i \(0.770339\pi\)
\(84\) 0 0
\(85\) 3632.64 0.502787
\(86\) − 3615.96i − 0.488907i
\(87\) 0 0
\(88\) 267.006 0.0344791
\(89\) − 8432.60i − 1.06459i −0.846560 0.532294i \(-0.821330\pi\)
0.846560 0.532294i \(-0.178670\pi\)
\(90\) 0 0
\(91\) −2946.49 −0.355813
\(92\) 2970.49i 0.350955i
\(93\) 0 0
\(94\) 5371.59 0.607921
\(95\) 11792.5i 1.30665i
\(96\) 0 0
\(97\) −545.318 −0.0579571 −0.0289785 0.999580i \(-0.509225\pi\)
−0.0289785 + 0.999580i \(0.509225\pi\)
\(98\) − 970.151i − 0.101015i
\(99\) 0 0
\(100\) 2265.83 0.226583
\(101\) 7298.60i 0.715479i 0.933822 + 0.357739i \(0.116452\pi\)
−0.933822 + 0.357739i \(0.883548\pi\)
\(102\) 0 0
\(103\) 12988.8 1.22432 0.612159 0.790735i \(-0.290301\pi\)
0.612159 + 0.790735i \(0.290301\pi\)
\(104\) − 3599.92i − 0.332833i
\(105\) 0 0
\(106\) 4875.39 0.433908
\(107\) 1506.48i 0.131582i 0.997833 + 0.0657910i \(0.0209571\pi\)
−0.997833 + 0.0657910i \(0.979043\pi\)
\(108\) 0 0
\(109\) −11205.6 −0.943149 −0.471575 0.881826i \(-0.656314\pi\)
−0.471575 + 0.881826i \(0.656314\pi\)
\(110\) 1005.84i 0.0831273i
\(111\) 0 0
\(112\) 1185.30 0.0944911
\(113\) 15734.3i 1.23223i 0.787658 + 0.616113i \(0.211294\pi\)
−0.787658 + 0.616113i \(0.788706\pi\)
\(114\) 0 0
\(115\) −11190.1 −0.846135
\(116\) 1734.53i 0.128904i
\(117\) 0 0
\(118\) 17872.0 1.28354
\(119\) 2232.40i 0.157644i
\(120\) 0 0
\(121\) 14501.8 0.990490
\(122\) − 7155.24i − 0.480733i
\(123\) 0 0
\(124\) 7442.27 0.484018
\(125\) − 10299.9i − 0.659194i
\(126\) 0 0
\(127\) 14352.6 0.889864 0.444932 0.895564i \(-0.353228\pi\)
0.444932 + 0.895564i \(0.353228\pi\)
\(128\) 1448.15i 0.0883883i
\(129\) 0 0
\(130\) 13561.3 0.802442
\(131\) 15941.9i 0.928963i 0.885583 + 0.464481i \(0.153759\pi\)
−0.885583 + 0.464481i \(0.846241\pi\)
\(132\) 0 0
\(133\) −7246.94 −0.409686
\(134\) 17455.3i 0.972117i
\(135\) 0 0
\(136\) −2727.47 −0.147462
\(137\) − 27139.8i − 1.44599i −0.690852 0.722996i \(-0.742764\pi\)
0.690852 0.722996i \(-0.257236\pi\)
\(138\) 0 0
\(139\) 20223.3 1.04670 0.523351 0.852117i \(-0.324681\pi\)
0.523351 + 0.852117i \(0.324681\pi\)
\(140\) 4465.14i 0.227813i
\(141\) 0 0
\(142\) 2567.80 0.127346
\(143\) 1877.34i 0.0918061i
\(144\) 0 0
\(145\) −6534.16 −0.310780
\(146\) 10321.0i 0.484192i
\(147\) 0 0
\(148\) 5379.83 0.245609
\(149\) − 6130.36i − 0.276130i −0.990423 0.138065i \(-0.955912\pi\)
0.990423 0.138065i \(-0.0440882\pi\)
\(150\) 0 0
\(151\) −24047.1 −1.05465 −0.527325 0.849664i \(-0.676805\pi\)
−0.527325 + 0.849664i \(0.676805\pi\)
\(152\) − 8854.06i − 0.383226i
\(153\) 0 0
\(154\) −618.128 −0.0260637
\(155\) 28035.8i 1.16694i
\(156\) 0 0
\(157\) −21025.6 −0.852999 −0.426499 0.904488i \(-0.640253\pi\)
−0.426499 + 0.904488i \(0.640253\pi\)
\(158\) 19281.6i 0.772377i
\(159\) 0 0
\(160\) −5455.35 −0.213100
\(161\) − 6876.77i − 0.265297i
\(162\) 0 0
\(163\) −16709.0 −0.628890 −0.314445 0.949276i \(-0.601818\pi\)
−0.314445 + 0.949276i \(0.601818\pi\)
\(164\) − 2949.84i − 0.109676i
\(165\) 0 0
\(166\) 25740.2 0.934105
\(167\) − 28204.0i − 1.01129i −0.862741 0.505647i \(-0.831254\pi\)
0.862741 0.505647i \(-0.168746\pi\)
\(168\) 0 0
\(169\) −3249.67 −0.113780
\(170\) − 10274.6i − 0.355524i
\(171\) 0 0
\(172\) −10227.5 −0.345710
\(173\) − 14376.2i − 0.480343i −0.970731 0.240171i \(-0.922796\pi\)
0.970731 0.240171i \(-0.0772036\pi\)
\(174\) 0 0
\(175\) −5245.47 −0.171281
\(176\) − 755.207i − 0.0243804i
\(177\) 0 0
\(178\) −23851.0 −0.752777
\(179\) − 55016.8i − 1.71708i −0.512750 0.858538i \(-0.671373\pi\)
0.512750 0.858538i \(-0.328627\pi\)
\(180\) 0 0
\(181\) −59154.3 −1.80563 −0.902816 0.430027i \(-0.858504\pi\)
−0.902816 + 0.430027i \(0.858504\pi\)
\(182\) 8333.92i 0.251598i
\(183\) 0 0
\(184\) 8401.80 0.248163
\(185\) 20266.4i 0.592151i
\(186\) 0 0
\(187\) 1422.36 0.0406750
\(188\) − 15193.2i − 0.429865i
\(189\) 0 0
\(190\) 33354.2 0.923938
\(191\) 1660.41i 0.0455143i 0.999741 + 0.0227572i \(0.00724446\pi\)
−0.999741 + 0.0227572i \(0.992756\pi\)
\(192\) 0 0
\(193\) −28950.8 −0.777224 −0.388612 0.921402i \(-0.627045\pi\)
−0.388612 + 0.921402i \(0.627045\pi\)
\(194\) 1542.39i 0.0409818i
\(195\) 0 0
\(196\) −2744.00 −0.0714286
\(197\) − 66964.1i − 1.72548i −0.505650 0.862739i \(-0.668747\pi\)
0.505650 0.862739i \(-0.331253\pi\)
\(198\) 0 0
\(199\) −25164.4 −0.635449 −0.317725 0.948183i \(-0.602919\pi\)
−0.317725 + 0.948183i \(0.602919\pi\)
\(200\) − 6408.74i − 0.160218i
\(201\) 0 0
\(202\) 20643.5 0.505920
\(203\) − 4015.49i − 0.0974421i
\(204\) 0 0
\(205\) 11112.3 0.264422
\(206\) − 36737.8i − 0.865724i
\(207\) 0 0
\(208\) −10182.1 −0.235348
\(209\) 4617.36i 0.105706i
\(210\) 0 0
\(211\) −68772.5 −1.54472 −0.772361 0.635184i \(-0.780924\pi\)
−0.772361 + 0.635184i \(0.780924\pi\)
\(212\) − 13789.7i − 0.306819i
\(213\) 0 0
\(214\) 4260.97 0.0930425
\(215\) − 38528.0i − 0.833488i
\(216\) 0 0
\(217\) −17229.1 −0.365884
\(218\) 31694.1i 0.666907i
\(219\) 0 0
\(220\) 2844.94 0.0587798
\(221\) − 19177.0i − 0.392642i
\(222\) 0 0
\(223\) 41818.0 0.840918 0.420459 0.907312i \(-0.361869\pi\)
0.420459 + 0.907312i \(0.361869\pi\)
\(224\) − 3352.53i − 0.0668153i
\(225\) 0 0
\(226\) 44503.3 0.871315
\(227\) 88775.6i 1.72283i 0.507903 + 0.861414i \(0.330421\pi\)
−0.507903 + 0.861414i \(0.669579\pi\)
\(228\) 0 0
\(229\) −88748.9 −1.69236 −0.846179 0.532899i \(-0.821102\pi\)
−0.846179 + 0.532899i \(0.821102\pi\)
\(230\) 31650.5i 0.598308i
\(231\) 0 0
\(232\) 4905.99 0.0911488
\(233\) − 27765.9i − 0.511447i −0.966750 0.255723i \(-0.917686\pi\)
0.966750 0.255723i \(-0.0823136\pi\)
\(234\) 0 0
\(235\) 57234.2 1.03638
\(236\) − 50549.7i − 0.907600i
\(237\) 0 0
\(238\) 6314.17 0.111471
\(239\) − 30454.6i − 0.533159i −0.963813 0.266579i \(-0.914107\pi\)
0.963813 0.266579i \(-0.0858934\pi\)
\(240\) 0 0
\(241\) 41178.6 0.708985 0.354493 0.935059i \(-0.384654\pi\)
0.354493 + 0.935059i \(0.384654\pi\)
\(242\) − 41017.2i − 0.700382i
\(243\) 0 0
\(244\) −20238.1 −0.339930
\(245\) − 10336.9i − 0.172210i
\(246\) 0 0
\(247\) 62253.7 1.02040
\(248\) − 21049.9i − 0.342253i
\(249\) 0 0
\(250\) −29132.5 −0.466120
\(251\) − 66882.6i − 1.06161i −0.847493 0.530806i \(-0.821889\pi\)
0.847493 0.530806i \(-0.178111\pi\)
\(252\) 0 0
\(253\) −4381.51 −0.0684514
\(254\) − 40595.4i − 0.629229i
\(255\) 0 0
\(256\) 4096.00 0.0625000
\(257\) − 64609.6i − 0.978207i −0.872226 0.489104i \(-0.837324\pi\)
0.872226 0.489104i \(-0.162676\pi\)
\(258\) 0 0
\(259\) −12454.5 −0.185663
\(260\) − 38357.0i − 0.567412i
\(261\) 0 0
\(262\) 45090.6 0.656876
\(263\) − 57042.8i − 0.824688i −0.911028 0.412344i \(-0.864710\pi\)
0.911028 0.412344i \(-0.135290\pi\)
\(264\) 0 0
\(265\) 51947.2 0.739725
\(266\) 20497.4i 0.289692i
\(267\) 0 0
\(268\) 49371.1 0.687391
\(269\) 41750.8i 0.576979i 0.957483 + 0.288490i \(0.0931531\pi\)
−0.957483 + 0.288490i \(0.906847\pi\)
\(270\) 0 0
\(271\) −76655.5 −1.04377 −0.521885 0.853016i \(-0.674771\pi\)
−0.521885 + 0.853016i \(0.674771\pi\)
\(272\) 7714.44i 0.104272i
\(273\) 0 0
\(274\) −76763.0 −1.02247
\(275\) 3342.13i 0.0441935i
\(276\) 0 0
\(277\) 19904.0 0.259406 0.129703 0.991553i \(-0.458598\pi\)
0.129703 + 0.991553i \(0.458598\pi\)
\(278\) − 57200.3i − 0.740131i
\(279\) 0 0
\(280\) 12629.3 0.161088
\(281\) 23149.0i 0.293170i 0.989198 + 0.146585i \(0.0468282\pi\)
−0.989198 + 0.146585i \(0.953172\pi\)
\(282\) 0 0
\(283\) −125867. −1.57158 −0.785792 0.618491i \(-0.787745\pi\)
−0.785792 + 0.618491i \(0.787745\pi\)
\(284\) − 7262.84i − 0.0900471i
\(285\) 0 0
\(286\) 5309.93 0.0649167
\(287\) 6828.97i 0.0829070i
\(288\) 0 0
\(289\) 68991.6 0.826039
\(290\) 18481.4i 0.219755i
\(291\) 0 0
\(292\) 29192.3 0.342376
\(293\) 150189.i 1.74945i 0.484616 + 0.874727i \(0.338959\pi\)
−0.484616 + 0.874727i \(0.661041\pi\)
\(294\) 0 0
\(295\) 190426. 2.18817
\(296\) − 15216.4i − 0.173672i
\(297\) 0 0
\(298\) −17339.3 −0.195253
\(299\) 59073.8i 0.660773i
\(300\) 0 0
\(301\) 23676.9 0.261332
\(302\) 68015.4i 0.745750i
\(303\) 0 0
\(304\) −25043.1 −0.270982
\(305\) − 76238.9i − 0.819553i
\(306\) 0 0
\(307\) 8930.45 0.0947538 0.0473769 0.998877i \(-0.484914\pi\)
0.0473769 + 0.998877i \(0.484914\pi\)
\(308\) 1748.33i 0.0184298i
\(309\) 0 0
\(310\) 79297.2 0.825153
\(311\) − 60394.3i − 0.624418i −0.950013 0.312209i \(-0.898931\pi\)
0.950013 0.312209i \(-0.101069\pi\)
\(312\) 0 0
\(313\) −149441. −1.52540 −0.762698 0.646755i \(-0.776126\pi\)
−0.762698 + 0.646755i \(0.776126\pi\)
\(314\) 59469.3i 0.603161i
\(315\) 0 0
\(316\) 54536.6 0.546153
\(317\) 55635.4i 0.553647i 0.960921 + 0.276823i \(0.0892817\pi\)
−0.960921 + 0.276823i \(0.910718\pi\)
\(318\) 0 0
\(319\) −2558.46 −0.0251418
\(320\) 15430.1i 0.150684i
\(321\) 0 0
\(322\) −19450.4 −0.187594
\(323\) − 47166.3i − 0.452092i
\(324\) 0 0
\(325\) 45060.4 0.426607
\(326\) 47260.2i 0.444693i
\(327\) 0 0
\(328\) −8343.40 −0.0775524
\(329\) 35172.6i 0.324948i
\(330\) 0 0
\(331\) 76613.6 0.699278 0.349639 0.936884i \(-0.386304\pi\)
0.349639 + 0.936884i \(0.386304\pi\)
\(332\) − 72804.2i − 0.660512i
\(333\) 0 0
\(334\) −79772.8 −0.715092
\(335\) 185986.i 1.65726i
\(336\) 0 0
\(337\) −177974. −1.56710 −0.783549 0.621330i \(-0.786593\pi\)
−0.783549 + 0.621330i \(0.786593\pi\)
\(338\) 9191.46i 0.0804547i
\(339\) 0 0
\(340\) −29061.1 −0.251394
\(341\) 10977.4i 0.0944045i
\(342\) 0 0
\(343\) 6352.45 0.0539949
\(344\) 28927.7i 0.244454i
\(345\) 0 0
\(346\) −40662.0 −0.339653
\(347\) − 122160.i − 1.01454i −0.861787 0.507271i \(-0.830654\pi\)
0.861787 0.507271i \(-0.169346\pi\)
\(348\) 0 0
\(349\) 60823.0 0.499363 0.249682 0.968328i \(-0.419674\pi\)
0.249682 + 0.968328i \(0.419674\pi\)
\(350\) 14836.4i 0.121114i
\(351\) 0 0
\(352\) −2136.05 −0.0172395
\(353\) 96167.4i 0.771753i 0.922550 + 0.385877i \(0.126101\pi\)
−0.922550 + 0.385877i \(0.873899\pi\)
\(354\) 0 0
\(355\) 27359.9 0.217099
\(356\) 67460.8i 0.532294i
\(357\) 0 0
\(358\) −155611. −1.21416
\(359\) − 200553.i − 1.55611i −0.628196 0.778055i \(-0.716206\pi\)
0.628196 0.778055i \(-0.283794\pi\)
\(360\) 0 0
\(361\) 22793.1 0.174899
\(362\) 167314.i 1.27677i
\(363\) 0 0
\(364\) 23571.9 0.177906
\(365\) 109971.i 0.825450i
\(366\) 0 0
\(367\) 114564. 0.850581 0.425291 0.905057i \(-0.360172\pi\)
0.425291 + 0.905057i \(0.360172\pi\)
\(368\) − 23763.9i − 0.175478i
\(369\) 0 0
\(370\) 57322.0 0.418714
\(371\) 31923.6i 0.231933i
\(372\) 0 0
\(373\) 87845.8 0.631398 0.315699 0.948859i \(-0.397761\pi\)
0.315699 + 0.948859i \(0.397761\pi\)
\(374\) − 4023.05i − 0.0287615i
\(375\) 0 0
\(376\) −42972.7 −0.303961
\(377\) 34494.5i 0.242698i
\(378\) 0 0
\(379\) 277656. 1.93298 0.966492 0.256696i \(-0.0826339\pi\)
0.966492 + 0.256696i \(0.0826339\pi\)
\(380\) − 94339.8i − 0.653323i
\(381\) 0 0
\(382\) 4696.34 0.0321835
\(383\) − 92652.4i − 0.631625i −0.948822 0.315812i \(-0.897723\pi\)
0.948822 0.315812i \(-0.102277\pi\)
\(384\) 0 0
\(385\) −6586.14 −0.0444334
\(386\) 81885.2i 0.549580i
\(387\) 0 0
\(388\) 4362.55 0.0289785
\(389\) − 77310.8i − 0.510906i −0.966821 0.255453i \(-0.917775\pi\)
0.966821 0.255453i \(-0.0822246\pi\)
\(390\) 0 0
\(391\) 44757.1 0.292758
\(392\) 7761.20i 0.0505076i
\(393\) 0 0
\(394\) −189403. −1.22010
\(395\) 205445.i 1.31675i
\(396\) 0 0
\(397\) −107227. −0.680333 −0.340167 0.940365i \(-0.610483\pi\)
−0.340167 + 0.940365i \(0.610483\pi\)
\(398\) 71175.8i 0.449331i
\(399\) 0 0
\(400\) −18126.6 −0.113292
\(401\) 140836.i 0.875839i 0.899014 + 0.437920i \(0.144285\pi\)
−0.899014 + 0.437920i \(0.855715\pi\)
\(402\) 0 0
\(403\) 148004. 0.911303
\(404\) − 58388.8i − 0.357739i
\(405\) 0 0
\(406\) −11357.5 −0.0689020
\(407\) 7935.32i 0.0479044i
\(408\) 0 0
\(409\) 2186.60 0.0130714 0.00653572 0.999979i \(-0.497920\pi\)
0.00653572 + 0.999979i \(0.497920\pi\)
\(410\) − 31430.5i − 0.186975i
\(411\) 0 0
\(412\) −103910. −0.612159
\(413\) 117024.i 0.686081i
\(414\) 0 0
\(415\) 274261. 1.59246
\(416\) 28799.3i 0.166416i
\(417\) 0 0
\(418\) 13059.9 0.0747457
\(419\) 147204.i 0.838480i 0.907875 + 0.419240i \(0.137703\pi\)
−0.907875 + 0.419240i \(0.862297\pi\)
\(420\) 0 0
\(421\) −3618.79 −0.0204173 −0.0102087 0.999948i \(-0.503250\pi\)
−0.0102087 + 0.999948i \(0.503250\pi\)
\(422\) 194518.i 1.09228i
\(423\) 0 0
\(424\) −39003.1 −0.216954
\(425\) − 34139.9i − 0.189010i
\(426\) 0 0
\(427\) 46851.8 0.256963
\(428\) − 12051.9i − 0.0657910i
\(429\) 0 0
\(430\) −108974. −0.589365
\(431\) − 269119.i − 1.44874i −0.689411 0.724370i \(-0.742131\pi\)
0.689411 0.724370i \(-0.257869\pi\)
\(432\) 0 0
\(433\) −278335. −1.48454 −0.742271 0.670100i \(-0.766251\pi\)
−0.742271 + 0.670100i \(0.766251\pi\)
\(434\) 48731.2i 0.258719i
\(435\) 0 0
\(436\) 89644.5 0.471575
\(437\) 145293.i 0.760820i
\(438\) 0 0
\(439\) 12385.3 0.0642656 0.0321328 0.999484i \(-0.489770\pi\)
0.0321328 + 0.999484i \(0.489770\pi\)
\(440\) − 8046.72i − 0.0415636i
\(441\) 0 0
\(442\) −54240.9 −0.277640
\(443\) 190001.i 0.968164i 0.875023 + 0.484082i \(0.160846\pi\)
−0.875023 + 0.484082i \(0.839154\pi\)
\(444\) 0 0
\(445\) −254132. −1.28333
\(446\) − 118279.i − 0.594619i
\(447\) 0 0
\(448\) −9482.37 −0.0472456
\(449\) 75081.0i 0.372424i 0.982510 + 0.186212i \(0.0596211\pi\)
−0.982510 + 0.186212i \(0.940379\pi\)
\(450\) 0 0
\(451\) 4351.05 0.0213915
\(452\) − 125874.i − 0.616113i
\(453\) 0 0
\(454\) 251095. 1.21822
\(455\) 88797.8i 0.428923i
\(456\) 0 0
\(457\) 58972.2 0.282367 0.141184 0.989983i \(-0.454909\pi\)
0.141184 + 0.989983i \(0.454909\pi\)
\(458\) 251020.i 1.19668i
\(459\) 0 0
\(460\) 89521.0 0.423067
\(461\) − 69046.3i − 0.324892i −0.986717 0.162446i \(-0.948062\pi\)
0.986717 0.162446i \(-0.0519383\pi\)
\(462\) 0 0
\(463\) −112506. −0.524824 −0.262412 0.964956i \(-0.584518\pi\)
−0.262412 + 0.964956i \(0.584518\pi\)
\(464\) − 13876.2i − 0.0644519i
\(465\) 0 0
\(466\) −78533.9 −0.361648
\(467\) 154153.i 0.706836i 0.935465 + 0.353418i \(0.114981\pi\)
−0.935465 + 0.353418i \(0.885019\pi\)
\(468\) 0 0
\(469\) −114296. −0.519618
\(470\) − 161883.i − 0.732833i
\(471\) 0 0
\(472\) −142976. −0.641770
\(473\) − 15085.7i − 0.0674283i
\(474\) 0 0
\(475\) 110827. 0.491199
\(476\) − 17859.2i − 0.0788220i
\(477\) 0 0
\(478\) −86138.5 −0.377000
\(479\) − 165690.i − 0.722144i −0.932538 0.361072i \(-0.882411\pi\)
0.932538 0.361072i \(-0.117589\pi\)
\(480\) 0 0
\(481\) 106988. 0.462430
\(482\) − 116471.i − 0.501328i
\(483\) 0 0
\(484\) −116014. −0.495245
\(485\) 16434.2i 0.0698657i
\(486\) 0 0
\(487\) −154280. −0.650508 −0.325254 0.945627i \(-0.605450\pi\)
−0.325254 + 0.945627i \(0.605450\pi\)
\(488\) 57241.9i 0.240367i
\(489\) 0 0
\(490\) −29237.3 −0.121771
\(491\) 433288.i 1.79727i 0.438695 + 0.898636i \(0.355441\pi\)
−0.438695 + 0.898636i \(0.644559\pi\)
\(492\) 0 0
\(493\) 26134.6 0.107528
\(494\) − 176080.i − 0.721533i
\(495\) 0 0
\(496\) −59538.1 −0.242009
\(497\) 16813.7i 0.0680692i
\(498\) 0 0
\(499\) −298370. −1.19827 −0.599134 0.800649i \(-0.704488\pi\)
−0.599134 + 0.800649i \(0.704488\pi\)
\(500\) 82399.2i 0.329597i
\(501\) 0 0
\(502\) −189173. −0.750673
\(503\) 284631.i 1.12498i 0.826803 + 0.562491i \(0.190157\pi\)
−0.826803 + 0.562491i \(0.809843\pi\)
\(504\) 0 0
\(505\) 219957. 0.862490
\(506\) 12392.8i 0.0484025i
\(507\) 0 0
\(508\) −114821. −0.444932
\(509\) − 287223.i − 1.10862i −0.832309 0.554312i \(-0.812981\pi\)
0.832309 0.554312i \(-0.187019\pi\)
\(510\) 0 0
\(511\) −67581.2 −0.258812
\(512\) − 11585.2i − 0.0441942i
\(513\) 0 0
\(514\) −182744. −0.691697
\(515\) − 391441.i − 1.47588i
\(516\) 0 0
\(517\) 22410.1 0.0838423
\(518\) 35226.6i 0.131284i
\(519\) 0 0
\(520\) −108490. −0.401221
\(521\) 458107.i 1.68769i 0.536591 + 0.843843i \(0.319712\pi\)
−0.536591 + 0.843843i \(0.680288\pi\)
\(522\) 0 0
\(523\) 99411.5 0.363441 0.181720 0.983350i \(-0.441833\pi\)
0.181720 + 0.983350i \(0.441833\pi\)
\(524\) − 127535.i − 0.464481i
\(525\) 0 0
\(526\) −161341. −0.583142
\(527\) − 112135.i − 0.403755i
\(528\) 0 0
\(529\) 141969. 0.507321
\(530\) − 146929.i − 0.523064i
\(531\) 0 0
\(532\) 57975.5 0.204843
\(533\) − 58663.2i − 0.206496i
\(534\) 0 0
\(535\) 45400.6 0.158619
\(536\) − 139643.i − 0.486059i
\(537\) 0 0
\(538\) 118089. 0.407986
\(539\) − 4047.44i − 0.0139317i
\(540\) 0 0
\(541\) −355697. −1.21531 −0.607653 0.794202i \(-0.707889\pi\)
−0.607653 + 0.794202i \(0.707889\pi\)
\(542\) 216815.i 0.738057i
\(543\) 0 0
\(544\) 21819.7 0.0737312
\(545\) 337700.i 1.13694i
\(546\) 0 0
\(547\) 356592. 1.19178 0.595892 0.803065i \(-0.296799\pi\)
0.595892 + 0.803065i \(0.296799\pi\)
\(548\) 217119.i 0.722996i
\(549\) 0 0
\(550\) 9452.98 0.0312495
\(551\) 84839.7i 0.279445i
\(552\) 0 0
\(553\) −126254. −0.412853
\(554\) − 56296.9i − 0.183428i
\(555\) 0 0
\(556\) −161787. −0.523351
\(557\) − 89649.4i − 0.288960i −0.989508 0.144480i \(-0.953849\pi\)
0.989508 0.144480i \(-0.0461509\pi\)
\(558\) 0 0
\(559\) −203393. −0.650897
\(560\) − 35721.1i − 0.113907i
\(561\) 0 0
\(562\) 65475.3 0.207303
\(563\) − 146102.i − 0.460934i −0.973080 0.230467i \(-0.925975\pi\)
0.973080 0.230467i \(-0.0740254\pi\)
\(564\) 0 0
\(565\) 474182. 1.48542
\(566\) 356005.i 1.11128i
\(567\) 0 0
\(568\) −20542.4 −0.0636729
\(569\) − 437298.i − 1.35068i −0.737506 0.675341i \(-0.763997\pi\)
0.737506 0.675341i \(-0.236003\pi\)
\(570\) 0 0
\(571\) 429334. 1.31681 0.658405 0.752664i \(-0.271232\pi\)
0.658405 + 0.752664i \(0.271232\pi\)
\(572\) − 15018.7i − 0.0459031i
\(573\) 0 0
\(574\) 19315.2 0.0586241
\(575\) 105166.i 0.318082i
\(576\) 0 0
\(577\) −140587. −0.422272 −0.211136 0.977457i \(-0.567716\pi\)
−0.211136 + 0.977457i \(0.567716\pi\)
\(578\) − 195138.i − 0.584097i
\(579\) 0 0
\(580\) 52273.2 0.155390
\(581\) 168544.i 0.499300i
\(582\) 0 0
\(583\) 20340.0 0.0598430
\(584\) − 82568.4i − 0.242096i
\(585\) 0 0
\(586\) 424798. 1.23705
\(587\) 264968.i 0.768985i 0.923128 + 0.384492i \(0.125623\pi\)
−0.923128 + 0.384492i \(0.874377\pi\)
\(588\) 0 0
\(589\) 364018. 1.04928
\(590\) − 538606.i − 1.54727i
\(591\) 0 0
\(592\) −43038.6 −0.122805
\(593\) − 510014.i − 1.45035i −0.688565 0.725175i \(-0.741759\pi\)
0.688565 0.725175i \(-0.258241\pi\)
\(594\) 0 0
\(595\) 67277.4 0.190036
\(596\) 49042.8i 0.138065i
\(597\) 0 0
\(598\) 167086. 0.467237
\(599\) − 590648.i − 1.64617i −0.567917 0.823086i \(-0.692251\pi\)
0.567917 0.823086i \(-0.307749\pi\)
\(600\) 0 0
\(601\) 260749. 0.721896 0.360948 0.932586i \(-0.382453\pi\)
0.360948 + 0.932586i \(0.382453\pi\)
\(602\) − 66968.5i − 0.184790i
\(603\) 0 0
\(604\) 192377. 0.527325
\(605\) − 437037.i − 1.19401i
\(606\) 0 0
\(607\) 432739. 1.17449 0.587245 0.809409i \(-0.300213\pi\)
0.587245 + 0.809409i \(0.300213\pi\)
\(608\) 70832.5i 0.191613i
\(609\) 0 0
\(610\) −215636. −0.579511
\(611\) − 302145.i − 0.809344i
\(612\) 0 0
\(613\) −509866. −1.35686 −0.678430 0.734665i \(-0.737339\pi\)
−0.678430 + 0.734665i \(0.737339\pi\)
\(614\) − 25259.1i − 0.0670010i
\(615\) 0 0
\(616\) 4945.02 0.0130319
\(617\) − 400846.i − 1.05295i −0.850191 0.526474i \(-0.823514\pi\)
0.850191 0.526474i \(-0.176486\pi\)
\(618\) 0 0
\(619\) 435350. 1.13621 0.568103 0.822957i \(-0.307678\pi\)
0.568103 + 0.822957i \(0.307678\pi\)
\(620\) − 224286.i − 0.583471i
\(621\) 0 0
\(622\) −170821. −0.441530
\(623\) − 156174.i − 0.402376i
\(624\) 0 0
\(625\) −487424. −1.24781
\(626\) 422684.i 1.07862i
\(627\) 0 0
\(628\) 168205. 0.426499
\(629\) − 81059.3i − 0.204881i
\(630\) 0 0
\(631\) 396490. 0.995803 0.497901 0.867234i \(-0.334104\pi\)
0.497901 + 0.867234i \(0.334104\pi\)
\(632\) − 154253.i − 0.386188i
\(633\) 0 0
\(634\) 157361. 0.391487
\(635\) − 432543.i − 1.07271i
\(636\) 0 0
\(637\) −54569.7 −0.134485
\(638\) 7236.41i 0.0177779i
\(639\) 0 0
\(640\) 43642.8 0.106550
\(641\) − 568080.i − 1.38259i −0.722572 0.691295i \(-0.757040\pi\)
0.722572 0.691295i \(-0.242960\pi\)
\(642\) 0 0
\(643\) 296340. 0.716751 0.358376 0.933577i \(-0.383331\pi\)
0.358376 + 0.933577i \(0.383331\pi\)
\(644\) 55014.2i 0.132649i
\(645\) 0 0
\(646\) −133406. −0.319677
\(647\) 36510.4i 0.0872184i 0.999049 + 0.0436092i \(0.0138856\pi\)
−0.999049 + 0.0436092i \(0.986114\pi\)
\(648\) 0 0
\(649\) 74561.5 0.177021
\(650\) − 127450.i − 0.301657i
\(651\) 0 0
\(652\) 133672. 0.314445
\(653\) − 596493.i − 1.39888i −0.714693 0.699438i \(-0.753434\pi\)
0.714693 0.699438i \(-0.246566\pi\)
\(654\) 0 0
\(655\) 480439. 1.11984
\(656\) 23598.7i 0.0548378i
\(657\) 0 0
\(658\) 99483.3 0.229773
\(659\) 746970.i 1.72002i 0.510280 + 0.860008i \(0.329542\pi\)
−0.510280 + 0.860008i \(0.670458\pi\)
\(660\) 0 0
\(661\) −147076. −0.336619 −0.168309 0.985734i \(-0.553831\pi\)
−0.168309 + 0.985734i \(0.553831\pi\)
\(662\) − 216696.i − 0.494464i
\(663\) 0 0
\(664\) −205921. −0.467052
\(665\) 218400.i 0.493866i
\(666\) 0 0
\(667\) −80506.2 −0.180958
\(668\) 225632.i 0.505647i
\(669\) 0 0
\(670\) 526049. 1.17186
\(671\) − 29851.4i − 0.0663010i
\(672\) 0 0
\(673\) −452502. −0.999058 −0.499529 0.866297i \(-0.666494\pi\)
−0.499529 + 0.866297i \(0.666494\pi\)
\(674\) 503386.i 1.10811i
\(675\) 0 0
\(676\) 25997.4 0.0568900
\(677\) − 772073.i − 1.68454i −0.539056 0.842270i \(-0.681219\pi\)
0.539056 0.842270i \(-0.318781\pi\)
\(678\) 0 0
\(679\) −10099.4 −0.0219057
\(680\) 82197.2i 0.177762i
\(681\) 0 0
\(682\) 31048.9 0.0667541
\(683\) 404022.i 0.866092i 0.901372 + 0.433046i \(0.142561\pi\)
−0.901372 + 0.433046i \(0.857439\pi\)
\(684\) 0 0
\(685\) −817909. −1.74311
\(686\) − 17967.4i − 0.0381802i
\(687\) 0 0
\(688\) 81819.8 0.172855
\(689\) − 274234.i − 0.577674i
\(690\) 0 0
\(691\) −615946. −1.28999 −0.644995 0.764187i \(-0.723141\pi\)
−0.644995 + 0.764187i \(0.723141\pi\)
\(692\) 115009.i 0.240171i
\(693\) 0 0
\(694\) −345520. −0.717389
\(695\) − 609468.i − 1.26177i
\(696\) 0 0
\(697\) −44446.0 −0.0914886
\(698\) − 172033.i − 0.353103i
\(699\) 0 0
\(700\) 41963.8 0.0856403
\(701\) 510387.i 1.03864i 0.854581 + 0.519319i \(0.173814\pi\)
−0.854581 + 0.519319i \(0.826186\pi\)
\(702\) 0 0
\(703\) 263139. 0.532446
\(704\) 6041.66i 0.0121902i
\(705\) 0 0
\(706\) 272003. 0.545712
\(707\) 135172.i 0.270425i
\(708\) 0 0
\(709\) −682535. −1.35779 −0.678895 0.734235i \(-0.737541\pi\)
−0.678895 + 0.734235i \(0.737541\pi\)
\(710\) − 77385.4i − 0.153512i
\(711\) 0 0
\(712\) 190808. 0.376389
\(713\) 345424.i 0.679475i
\(714\) 0 0
\(715\) 56577.2 0.110670
\(716\) 440135.i 0.858538i
\(717\) 0 0
\(718\) −567250. −1.10034
\(719\) 108955.i 0.210760i 0.994432 + 0.105380i \(0.0336059\pi\)
−0.994432 + 0.105380i \(0.966394\pi\)
\(720\) 0 0
\(721\) 240556. 0.462749
\(722\) − 64468.5i − 0.123673i
\(723\) 0 0
\(724\) 473235. 0.902816
\(725\) 61408.6i 0.116830i
\(726\) 0 0
\(727\) −146049. −0.276331 −0.138165 0.990409i \(-0.544121\pi\)
−0.138165 + 0.990409i \(0.544121\pi\)
\(728\) − 66671.4i − 0.125799i
\(729\) 0 0
\(730\) 311044. 0.583681
\(731\) 154100.i 0.288382i
\(732\) 0 0
\(733\) −925797. −1.72309 −0.861544 0.507682i \(-0.830502\pi\)
−0.861544 + 0.507682i \(0.830502\pi\)
\(734\) − 324036.i − 0.601452i
\(735\) 0 0
\(736\) −67214.4 −0.124081
\(737\) 72823.1i 0.134071i
\(738\) 0 0
\(739\) 40824.3 0.0747532 0.0373766 0.999301i \(-0.488100\pi\)
0.0373766 + 0.999301i \(0.488100\pi\)
\(740\) − 162131.i − 0.296076i
\(741\) 0 0
\(742\) 90293.4 0.164002
\(743\) − 106961.i − 0.193753i −0.995296 0.0968766i \(-0.969115\pi\)
0.995296 0.0968766i \(-0.0308852\pi\)
\(744\) 0 0
\(745\) −184750. −0.332867
\(746\) − 248465.i − 0.446466i
\(747\) 0 0
\(748\) −11378.9 −0.0203375
\(749\) 27900.4i 0.0497333i
\(750\) 0 0
\(751\) −111510. −0.197713 −0.0988564 0.995102i \(-0.531518\pi\)
−0.0988564 + 0.995102i \(0.531518\pi\)
\(752\) 121545.i 0.214933i
\(753\) 0 0
\(754\) 97565.1 0.171614
\(755\) 724703.i 1.27135i
\(756\) 0 0
\(757\) −78268.3 −0.136582 −0.0682911 0.997665i \(-0.521755\pi\)
−0.0682911 + 0.997665i \(0.521755\pi\)
\(758\) − 785329.i − 1.36683i
\(759\) 0 0
\(760\) −266833. −0.461969
\(761\) − 799900.i − 1.38123i −0.723222 0.690616i \(-0.757340\pi\)
0.723222 0.690616i \(-0.242660\pi\)
\(762\) 0 0
\(763\) −207530. −0.356477
\(764\) − 13283.3i − 0.0227572i
\(765\) 0 0
\(766\) −262061. −0.446626
\(767\) − 1.00528e6i − 1.70882i
\(768\) 0 0
\(769\) −696974. −1.17859 −0.589296 0.807917i \(-0.700595\pi\)
−0.589296 + 0.807917i \(0.700595\pi\)
\(770\) 18628.4i 0.0314191i
\(771\) 0 0
\(772\) 231606. 0.388612
\(773\) 946901.i 1.58469i 0.610071 + 0.792347i \(0.291141\pi\)
−0.610071 + 0.792347i \(0.708859\pi\)
\(774\) 0 0
\(775\) 263483. 0.438681
\(776\) − 12339.1i − 0.0204909i
\(777\) 0 0
\(778\) −218668. −0.361265
\(779\) − 144283.i − 0.237761i
\(780\) 0 0
\(781\) 10712.8 0.0175631
\(782\) − 126592.i − 0.207011i
\(783\) 0 0
\(784\) 21952.0 0.0357143
\(785\) 633644.i 1.02827i
\(786\) 0 0
\(787\) 693165. 1.11915 0.559574 0.828780i \(-0.310965\pi\)
0.559574 + 0.828780i \(0.310965\pi\)
\(788\) 535712.i 0.862739i
\(789\) 0 0
\(790\) 581087. 0.931079
\(791\) 291403.i 0.465738i
\(792\) 0 0
\(793\) −402473. −0.640015
\(794\) 303283.i 0.481068i
\(795\) 0 0
\(796\) 201315. 0.317725
\(797\) − 366791.i − 0.577434i −0.957414 0.288717i \(-0.906771\pi\)
0.957414 0.288717i \(-0.0932287\pi\)
\(798\) 0 0
\(799\) −228919. −0.358582
\(800\) 51269.9i 0.0801092i
\(801\) 0 0
\(802\) 398344. 0.619312
\(803\) 43059.1i 0.0667781i
\(804\) 0 0
\(805\) −207244. −0.319809
\(806\) − 418618.i − 0.644388i
\(807\) 0 0
\(808\) −165148. −0.252960
\(809\) − 17053.8i − 0.0260569i −0.999915 0.0130285i \(-0.995853\pi\)
0.999915 0.0130285i \(-0.00414721\pi\)
\(810\) 0 0
\(811\) 127087. 0.193223 0.0966116 0.995322i \(-0.469200\pi\)
0.0966116 + 0.995322i \(0.469200\pi\)
\(812\) 32123.9i 0.0487211i
\(813\) 0 0
\(814\) 22444.5 0.0338735
\(815\) 503556.i 0.758111i
\(816\) 0 0
\(817\) −500249. −0.749449
\(818\) − 6184.64i − 0.00924290i
\(819\) 0 0
\(820\) −88898.8 −0.132211
\(821\) 870988.i 1.29219i 0.763258 + 0.646094i \(0.223599\pi\)
−0.763258 + 0.646094i \(0.776401\pi\)
\(822\) 0 0
\(823\) 590520. 0.871836 0.435918 0.899986i \(-0.356424\pi\)
0.435918 + 0.899986i \(0.356424\pi\)
\(824\) 293903.i 0.432862i
\(825\) 0 0
\(826\) 330994. 0.485132
\(827\) 254863.i 0.372645i 0.982489 + 0.186323i \(0.0596570\pi\)
−0.982489 + 0.186323i \(0.940343\pi\)
\(828\) 0 0
\(829\) 102666. 0.149388 0.0746942 0.997206i \(-0.476202\pi\)
0.0746942 + 0.997206i \(0.476202\pi\)
\(830\) − 775728.i − 1.12604i
\(831\) 0 0
\(832\) 81456.8 0.117674
\(833\) 41344.6i 0.0595838i
\(834\) 0 0
\(835\) −849978. −1.21909
\(836\) − 36938.9i − 0.0528532i
\(837\) 0 0
\(838\) 416357. 0.592895
\(839\) 428504.i 0.608739i 0.952554 + 0.304369i \(0.0984458\pi\)
−0.952554 + 0.304369i \(0.901554\pi\)
\(840\) 0 0
\(841\) 660272. 0.933535
\(842\) 10235.5i 0.0144372i
\(843\) 0 0
\(844\) 550180. 0.772361
\(845\) 97934.8i 0.137159i
\(846\) 0 0
\(847\) 268576. 0.374370
\(848\) 110317.i 0.153410i
\(849\) 0 0
\(850\) −96562.1 −0.133650
\(851\) 249698.i 0.344792i
\(852\) 0 0
\(853\) 248624. 0.341700 0.170850 0.985297i \(-0.445349\pi\)
0.170850 + 0.985297i \(0.445349\pi\)
\(854\) − 132517.i − 0.181700i
\(855\) 0 0
\(856\) −34087.8 −0.0465212
\(857\) 222103.i 0.302407i 0.988503 + 0.151204i \(0.0483149\pi\)
−0.988503 + 0.151204i \(0.951685\pi\)
\(858\) 0 0
\(859\) −893111. −1.21037 −0.605187 0.796084i \(-0.706902\pi\)
−0.605187 + 0.796084i \(0.706902\pi\)
\(860\) 308224.i 0.416744i
\(861\) 0 0
\(862\) −761185. −1.02441
\(863\) 153799.i 0.206505i 0.994655 + 0.103253i \(0.0329250\pi\)
−0.994655 + 0.103253i \(0.967075\pi\)
\(864\) 0 0
\(865\) −433252. −0.579040
\(866\) 787251.i 1.04973i
\(867\) 0 0
\(868\) 137833. 0.182942
\(869\) 80442.3i 0.106523i
\(870\) 0 0
\(871\) 981840. 1.29421
\(872\) − 253553.i − 0.333454i
\(873\) 0 0
\(874\) 410951. 0.537981
\(875\) − 190757.i − 0.249152i
\(876\) 0 0
\(877\) −845361. −1.09912 −0.549558 0.835456i \(-0.685204\pi\)
−0.549558 + 0.835456i \(0.685204\pi\)
\(878\) − 35031.0i − 0.0454426i
\(879\) 0 0
\(880\) −22759.6 −0.0293899
\(881\) − 603302.i − 0.777290i −0.921388 0.388645i \(-0.872943\pi\)
0.921388 0.388645i \(-0.127057\pi\)
\(882\) 0 0
\(883\) −642582. −0.824152 −0.412076 0.911149i \(-0.635196\pi\)
−0.412076 + 0.911149i \(0.635196\pi\)
\(884\) 153416.i 0.196321i
\(885\) 0 0
\(886\) 537404. 0.684595
\(887\) 516432.i 0.656396i 0.944609 + 0.328198i \(0.106441\pi\)
−0.944609 + 0.328198i \(0.893559\pi\)
\(888\) 0 0
\(889\) 265814. 0.336337
\(890\) 718793.i 0.907453i
\(891\) 0 0
\(892\) −334544. −0.420459
\(893\) − 743131.i − 0.931886i
\(894\) 0 0
\(895\) −1.65803e6 −2.06989
\(896\) 26820.2i 0.0334077i
\(897\) 0 0
\(898\) 212361. 0.263343
\(899\) 201701.i 0.249567i
\(900\) 0 0
\(901\) −207773. −0.255940
\(902\) − 12306.6i − 0.0151261i
\(903\) 0 0
\(904\) −356026. −0.435658
\(905\) 1.78272e6i 2.17664i
\(906\) 0 0
\(907\) 332852. 0.404610 0.202305 0.979323i \(-0.435157\pi\)
0.202305 + 0.979323i \(0.435157\pi\)
\(908\) − 710205.i − 0.861414i
\(909\) 0 0
\(910\) 251158. 0.303294
\(911\) 128205.i 0.154478i 0.997013 + 0.0772390i \(0.0246104\pi\)
−0.997013 + 0.0772390i \(0.975390\pi\)
\(912\) 0 0
\(913\) 107387. 0.128828
\(914\) − 166798.i − 0.199664i
\(915\) 0 0
\(916\) 709991. 0.846179
\(917\) 295249.i 0.351115i
\(918\) 0 0
\(919\) 1.30265e6 1.54240 0.771201 0.636592i \(-0.219656\pi\)
0.771201 + 0.636592i \(0.219656\pi\)
\(920\) − 253204.i − 0.299154i
\(921\) 0 0
\(922\) −195292. −0.229733
\(923\) − 144435.i − 0.169539i
\(924\) 0 0
\(925\) 190465. 0.222604
\(926\) 318215.i 0.371106i
\(927\) 0 0
\(928\) −39247.9 −0.0455744
\(929\) 1.34170e6i 1.55462i 0.629117 + 0.777311i \(0.283417\pi\)
−0.629117 + 0.777311i \(0.716583\pi\)
\(930\) 0 0
\(931\) −134215. −0.154847
\(932\) 222127.i 0.255723i
\(933\) 0 0
\(934\) 436011. 0.499809
\(935\) − 42865.5i − 0.0490326i
\(936\) 0 0
\(937\) −1.49619e6 −1.70415 −0.852074 0.523421i \(-0.824656\pi\)
−0.852074 + 0.523421i \(0.824656\pi\)
\(938\) 323277.i 0.367426i
\(939\) 0 0
\(940\) −457874. −0.518191
\(941\) 778807.i 0.879530i 0.898113 + 0.439765i \(0.144938\pi\)
−0.898113 + 0.439765i \(0.855062\pi\)
\(942\) 0 0
\(943\) 136913. 0.153965
\(944\) 404397.i 0.453800i
\(945\) 0 0
\(946\) −42668.7 −0.0476790
\(947\) − 1.36722e6i − 1.52454i −0.647258 0.762271i \(-0.724084\pi\)
0.647258 0.762271i \(-0.275916\pi\)
\(948\) 0 0
\(949\) 580545. 0.644620
\(950\) − 313466.i − 0.347330i
\(951\) 0 0
\(952\) −50513.4 −0.0557356
\(953\) − 656858.i − 0.723245i −0.932325 0.361623i \(-0.882223\pi\)
0.932325 0.361623i \(-0.117777\pi\)
\(954\) 0 0
\(955\) 50039.5 0.0548663
\(956\) 243636.i 0.266579i
\(957\) 0 0
\(958\) −468641. −0.510633
\(959\) − 502637.i − 0.546534i
\(960\) 0 0
\(961\) −58093.7 −0.0629046
\(962\) − 302608.i − 0.326987i
\(963\) 0 0
\(964\) −329429. −0.354493
\(965\) 872486.i 0.936923i
\(966\) 0 0
\(967\) 817206. 0.873934 0.436967 0.899478i \(-0.356053\pi\)
0.436967 + 0.899478i \(0.356053\pi\)
\(968\) 328137.i 0.350191i
\(969\) 0 0
\(970\) 46482.8 0.0494025
\(971\) 1.02544e6i 1.08761i 0.839211 + 0.543805i \(0.183017\pi\)
−0.839211 + 0.543805i \(0.816983\pi\)
\(972\) 0 0
\(973\) 374542. 0.395617
\(974\) 436370.i 0.459978i
\(975\) 0 0
\(976\) 161905. 0.169965
\(977\) 233580.i 0.244707i 0.992487 + 0.122354i \(0.0390442\pi\)
−0.992487 + 0.122354i \(0.960956\pi\)
\(978\) 0 0
\(979\) −99505.6 −0.103820
\(980\) 82695.5i 0.0861052i
\(981\) 0 0
\(982\) 1.22552e6 1.27086
\(983\) − 512053.i − 0.529917i −0.964260 0.264958i \(-0.914642\pi\)
0.964260 0.264958i \(-0.0853582\pi\)
\(984\) 0 0
\(985\) −2.01808e6 −2.08002
\(986\) − 73919.9i − 0.0760339i
\(987\) 0 0
\(988\) −498029. −0.510201
\(989\) − 474696.i − 0.485315i
\(990\) 0 0
\(991\) 153330. 0.156128 0.0780639 0.996948i \(-0.475126\pi\)
0.0780639 + 0.996948i \(0.475126\pi\)
\(992\) 168399.i 0.171126i
\(993\) 0 0
\(994\) 47556.3 0.0481322
\(995\) 758376.i 0.766017i
\(996\) 0 0
\(997\) 731704. 0.736114 0.368057 0.929803i \(-0.380023\pi\)
0.368057 + 0.929803i \(0.380023\pi\)
\(998\) 843917.i 0.847303i
\(999\) 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.323.1 8
3.2 odd 2 inner 378.5.b.b.323.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.5.b.b.323.1 8 1.1 even 1 trivial
378.5.b.b.323.8 yes 8 3.2 odd 2 inner