Properties

Label 18.7.b.a.17.1
Level $18$
Weight $7$
Character 18.17
Analytic conductor $4.141$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 18.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(4.14097350516\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 18.17
Dual form 18.7.b.a.17.2

$q$-expansion

\(f(q)\) \(=\) \(q-5.65685i q^{2} -32.0000 q^{4} -173.948i q^{5} -484.000 q^{7} +181.019i q^{8} +O(q^{10})\) \(q-5.65685i q^{2} -32.0000 q^{4} -173.948i q^{5} -484.000 q^{7} +181.019i q^{8} -984.000 q^{10} -1340.67i q^{11} +3368.00 q^{13} +2737.92i q^{14} +1024.00 q^{16} -12.7279i q^{17} +5744.00 q^{19} +5566.34i q^{20} -7584.00 q^{22} +3377.14i q^{23} -14633.0 q^{25} -19052.3i q^{26} +15488.0 q^{28} -29354.8i q^{29} -39796.0 q^{31} -5792.62i q^{32} -72.0000 q^{34} +84191.0i q^{35} +52526.0 q^{37} -32493.0i q^{38} +31488.0 q^{40} +37042.5i q^{41} +3800.00 q^{43} +42901.6i q^{44} +19104.0 q^{46} -76791.8i q^{47} +116607. q^{49} +82776.7i q^{50} -107776. q^{52} -238738. i q^{53} -233208. q^{55} -87613.4i q^{56} -166056. q^{58} +249841. i q^{59} +13250.0 q^{61} +225120. i q^{62} -32768.0 q^{64} -585858. i q^{65} +168968. q^{67} +407.294i q^{68} +476256. q^{70} +531467. i q^{71} +236144. q^{73} -297132. i q^{74} -183808. q^{76} +648886. i q^{77} -35116.0 q^{79} -178123. i q^{80} +209544. q^{82} +10980.0i q^{83} -2214.00 q^{85} -21496.0i q^{86} +242688. q^{88} +129328. i q^{89} -1.63011e6 q^{91} -108069. i q^{92} -434400. q^{94} -999159. i q^{95} -321424. q^{97} -659629. i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 64 q^{4} - 968 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 64 q^{4} - 968 q^{7} - 1968 q^{10} + 6736 q^{13} + 2048 q^{16} + 11488 q^{19} - 15168 q^{22} - 29266 q^{25} + 30976 q^{28} - 79592 q^{31} - 144 q^{34} + 105052 q^{37} + 62976 q^{40} + 7600 q^{43} + 38208 q^{46} + 233214 q^{49} - 215552 q^{52} - 466416 q^{55} - 332112 q^{58} + 26500 q^{61} - 65536 q^{64} + 337936 q^{67} + 952512 q^{70} + 472288 q^{73} - 367616 q^{76} - 70232 q^{79} + 419088 q^{82} - 4428 q^{85} + 485376 q^{88} - 3260224 q^{91} - 868800 q^{94} - 642848 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 5.65685i − 0.707107i
\(3\) 0 0
\(4\) −32.0000 −0.500000
\(5\) − 173.948i − 1.39159i −0.718242 0.695793i \(-0.755053\pi\)
0.718242 0.695793i \(-0.244947\pi\)
\(6\) 0 0
\(7\) −484.000 −1.41108 −0.705539 0.708671i \(-0.749295\pi\)
−0.705539 + 0.708671i \(0.749295\pi\)
\(8\) 181.019i 0.353553i
\(9\) 0 0
\(10\) −984.000 −0.984000
\(11\) − 1340.67i − 1.00727i −0.863917 0.503634i \(-0.831996\pi\)
0.863917 0.503634i \(-0.168004\pi\)
\(12\) 0 0
\(13\) 3368.00 1.53300 0.766500 0.642245i \(-0.221997\pi\)
0.766500 + 0.642245i \(0.221997\pi\)
\(14\) 2737.92i 0.997783i
\(15\) 0 0
\(16\) 1024.00 0.250000
\(17\) − 12.7279i − 0.00259066i −0.999999 0.00129533i \(-0.999588\pi\)
0.999999 0.00129533i \(-0.000412317\pi\)
\(18\) 0 0
\(19\) 5744.00 0.837440 0.418720 0.908115i \(-0.362479\pi\)
0.418720 + 0.908115i \(0.362479\pi\)
\(20\) 5566.34i 0.695793i
\(21\) 0 0
\(22\) −7584.00 −0.712246
\(23\) 3377.14i 0.277566i 0.990323 + 0.138783i \(0.0443190\pi\)
−0.990323 + 0.138783i \(0.955681\pi\)
\(24\) 0 0
\(25\) −14633.0 −0.936512
\(26\) − 19052.3i − 1.08399i
\(27\) 0 0
\(28\) 15488.0 0.705539
\(29\) − 29354.8i − 1.20361i −0.798643 0.601805i \(-0.794449\pi\)
0.798643 0.601805i \(-0.205551\pi\)
\(30\) 0 0
\(31\) −39796.0 −1.33584 −0.667920 0.744233i \(-0.732815\pi\)
−0.667920 + 0.744233i \(0.732815\pi\)
\(32\) − 5792.62i − 0.176777i
\(33\) 0 0
\(34\) −72.0000 −0.00183187
\(35\) 84191.0i 1.96364i
\(36\) 0 0
\(37\) 52526.0 1.03698 0.518489 0.855085i \(-0.326495\pi\)
0.518489 + 0.855085i \(0.326495\pi\)
\(38\) − 32493.0i − 0.592159i
\(39\) 0 0
\(40\) 31488.0 0.492000
\(41\) 37042.5i 0.537463i 0.963215 + 0.268732i \(0.0866045\pi\)
−0.963215 + 0.268732i \(0.913396\pi\)
\(42\) 0 0
\(43\) 3800.00 0.0477945 0.0238973 0.999714i \(-0.492393\pi\)
0.0238973 + 0.999714i \(0.492393\pi\)
\(44\) 42901.6i 0.503634i
\(45\) 0 0
\(46\) 19104.0 0.196269
\(47\) − 76791.8i − 0.739641i −0.929103 0.369821i \(-0.879419\pi\)
0.929103 0.369821i \(-0.120581\pi\)
\(48\) 0 0
\(49\) 116607. 0.991143
\(50\) 82776.7i 0.662214i
\(51\) 0 0
\(52\) −107776. −0.766500
\(53\) − 238738.i − 1.60359i −0.597599 0.801795i \(-0.703879\pi\)
0.597599 0.801795i \(-0.296121\pi\)
\(54\) 0 0
\(55\) −233208. −1.40170
\(56\) − 87613.4i − 0.498892i
\(57\) 0 0
\(58\) −166056. −0.851080
\(59\) 249841.i 1.21649i 0.793751 + 0.608243i \(0.208125\pi\)
−0.793751 + 0.608243i \(0.791875\pi\)
\(60\) 0 0
\(61\) 13250.0 0.0583749 0.0291875 0.999574i \(-0.490708\pi\)
0.0291875 + 0.999574i \(0.490708\pi\)
\(62\) 225120.i 0.944581i
\(63\) 0 0
\(64\) −32768.0 −0.125000
\(65\) − 585858.i − 2.13330i
\(66\) 0 0
\(67\) 168968. 0.561798 0.280899 0.959737i \(-0.409367\pi\)
0.280899 + 0.959737i \(0.409367\pi\)
\(68\) 407.294i 0.00129533i
\(69\) 0 0
\(70\) 476256. 1.38850
\(71\) 531467.i 1.48491i 0.669894 + 0.742457i \(0.266340\pi\)
−0.669894 + 0.742457i \(0.733660\pi\)
\(72\) 0 0
\(73\) 236144. 0.607027 0.303514 0.952827i \(-0.401840\pi\)
0.303514 + 0.952827i \(0.401840\pi\)
\(74\) − 297132.i − 0.733254i
\(75\) 0 0
\(76\) −183808. −0.418720
\(77\) 648886.i 1.42134i
\(78\) 0 0
\(79\) −35116.0 −0.0712236 −0.0356118 0.999366i \(-0.511338\pi\)
−0.0356118 + 0.999366i \(0.511338\pi\)
\(80\) − 178123.i − 0.347897i
\(81\) 0 0
\(82\) 209544. 0.380044
\(83\) 10980.0i 0.0192029i 0.999954 + 0.00960144i \(0.00305628\pi\)
−0.999954 + 0.00960144i \(0.996944\pi\)
\(84\) 0 0
\(85\) −2214.00 −0.00360513
\(86\) − 21496.0i − 0.0337958i
\(87\) 0 0
\(88\) 242688. 0.356123
\(89\) 129328.i 0.183453i 0.995784 + 0.0917263i \(0.0292385\pi\)
−0.995784 + 0.0917263i \(0.970762\pi\)
\(90\) 0 0
\(91\) −1.63011e6 −2.16318
\(92\) − 108069.i − 0.138783i
\(93\) 0 0
\(94\) −434400. −0.523005
\(95\) − 999159.i − 1.16537i
\(96\) 0 0
\(97\) −321424. −0.352179 −0.176089 0.984374i \(-0.556345\pi\)
−0.176089 + 0.984374i \(0.556345\pi\)
\(98\) − 659629.i − 0.700844i
\(99\) 0 0
\(100\) 468256. 0.468256
\(101\) − 668780.i − 0.649111i −0.945867 0.324556i \(-0.894785\pi\)
0.945867 0.324556i \(-0.105215\pi\)
\(102\) 0 0
\(103\) 1.99341e6 1.82425 0.912127 0.409907i \(-0.134439\pi\)
0.912127 + 0.409907i \(0.134439\pi\)
\(104\) 609673.i 0.541997i
\(105\) 0 0
\(106\) −1.35050e6 −1.13391
\(107\) 260668.i 0.212783i 0.994324 + 0.106391i \(0.0339296\pi\)
−0.994324 + 0.106391i \(0.966070\pi\)
\(108\) 0 0
\(109\) 194456. 0.150156 0.0750779 0.997178i \(-0.476079\pi\)
0.0750779 + 0.997178i \(0.476079\pi\)
\(110\) 1.31922e6i 0.991152i
\(111\) 0 0
\(112\) −495616. −0.352770
\(113\) 821897.i 0.569616i 0.958585 + 0.284808i \(0.0919298\pi\)
−0.958585 + 0.284808i \(0.908070\pi\)
\(114\) 0 0
\(115\) 587448. 0.386257
\(116\) 939355.i 0.601805i
\(117\) 0 0
\(118\) 1.41331e6 0.860185
\(119\) 6160.31i 0.00365563i
\(120\) 0 0
\(121\) −25847.0 −0.0145900
\(122\) − 74953.3i − 0.0412773i
\(123\) 0 0
\(124\) 1.27347e6 0.667920
\(125\) − 172557.i − 0.0883490i
\(126\) 0 0
\(127\) 3.05721e6 1.49250 0.746250 0.665666i \(-0.231852\pi\)
0.746250 + 0.665666i \(0.231852\pi\)
\(128\) 185364.i 0.0883883i
\(129\) 0 0
\(130\) −3.31411e6 −1.50847
\(131\) − 3.07388e6i − 1.36733i −0.729797 0.683664i \(-0.760385\pi\)
0.729797 0.683664i \(-0.239615\pi\)
\(132\) 0 0
\(133\) −2.78010e6 −1.18169
\(134\) − 955827.i − 0.397251i
\(135\) 0 0
\(136\) 2304.00 0.000915937 0
\(137\) 4.48412e6i 1.74388i 0.489617 + 0.871938i \(0.337137\pi\)
−0.489617 + 0.871938i \(0.662863\pi\)
\(138\) 0 0
\(139\) −1.09233e6 −0.406732 −0.203366 0.979103i \(-0.565188\pi\)
−0.203366 + 0.979103i \(0.565188\pi\)
\(140\) − 2.69411e6i − 0.981819i
\(141\) 0 0
\(142\) 3.00643e6 1.04999
\(143\) − 4.51539e6i − 1.54414i
\(144\) 0 0
\(145\) −5.10622e6 −1.67493
\(146\) − 1.33583e6i − 0.429233i
\(147\) 0 0
\(148\) −1.68083e6 −0.518489
\(149\) 2.22087e6i 0.671375i 0.941973 + 0.335687i \(0.108969\pi\)
−0.941973 + 0.335687i \(0.891031\pi\)
\(150\) 0 0
\(151\) −4.07871e6 −1.18465 −0.592327 0.805697i \(-0.701791\pi\)
−0.592327 + 0.805697i \(0.701791\pi\)
\(152\) 1.03978e6i 0.296080i
\(153\) 0 0
\(154\) 3.67066e6 1.00504
\(155\) 6.92245e6i 1.85894i
\(156\) 0 0
\(157\) 6.15568e6 1.59066 0.795329 0.606178i \(-0.207298\pi\)
0.795329 + 0.606178i \(0.207298\pi\)
\(158\) 198646.i 0.0503627i
\(159\) 0 0
\(160\) −1.00762e6 −0.246000
\(161\) − 1.63454e6i − 0.391667i
\(162\) 0 0
\(163\) 800696. 0.184886 0.0924432 0.995718i \(-0.470532\pi\)
0.0924432 + 0.995718i \(0.470532\pi\)
\(164\) − 1.18536e6i − 0.268732i
\(165\) 0 0
\(166\) 62112.0 0.0135785
\(167\) − 4.80467e6i − 1.03161i −0.856707 0.515804i \(-0.827493\pi\)
0.856707 0.515804i \(-0.172507\pi\)
\(168\) 0 0
\(169\) 6.51661e6 1.35009
\(170\) 12524.3i 0.00254921i
\(171\) 0 0
\(172\) −121600. −0.0238973
\(173\) 3.56992e6i 0.689478i 0.938699 + 0.344739i \(0.112033\pi\)
−0.938699 + 0.344739i \(0.887967\pi\)
\(174\) 0 0
\(175\) 7.08237e6 1.32149
\(176\) − 1.37285e6i − 0.251817i
\(177\) 0 0
\(178\) 731592. 0.129721
\(179\) 7.43698e6i 1.29669i 0.761345 + 0.648347i \(0.224539\pi\)
−0.761345 + 0.648347i \(0.775461\pi\)
\(180\) 0 0
\(181\) −1.03812e7 −1.75070 −0.875350 0.483491i \(-0.839369\pi\)
−0.875350 + 0.483491i \(0.839369\pi\)
\(182\) 9.22131e6i 1.52960i
\(183\) 0 0
\(184\) −611328. −0.0981343
\(185\) − 9.13681e6i − 1.44304i
\(186\) 0 0
\(187\) −17064.0 −0.00260949
\(188\) 2.45734e6i 0.369821i
\(189\) 0 0
\(190\) −5.65210e6 −0.824041
\(191\) − 1.29941e7i − 1.86485i −0.361360 0.932426i \(-0.617687\pi\)
0.361360 0.932426i \(-0.382313\pi\)
\(192\) 0 0
\(193\) −3.93195e6 −0.546936 −0.273468 0.961881i \(-0.588171\pi\)
−0.273468 + 0.961881i \(0.588171\pi\)
\(194\) 1.81825e6i 0.249028i
\(195\) 0 0
\(196\) −3.73142e6 −0.495572
\(197\) − 5.37967e6i − 0.703651i −0.936066 0.351825i \(-0.885561\pi\)
0.936066 0.351825i \(-0.114439\pi\)
\(198\) 0 0
\(199\) −565900. −0.0718093 −0.0359046 0.999355i \(-0.511431\pi\)
−0.0359046 + 0.999355i \(0.511431\pi\)
\(200\) − 2.64886e6i − 0.331107i
\(201\) 0 0
\(202\) −3.78319e6 −0.458991
\(203\) 1.42077e7i 1.69839i
\(204\) 0 0
\(205\) 6.44348e6 0.747926
\(206\) − 1.12764e7i − 1.28994i
\(207\) 0 0
\(208\) 3.44883e6 0.383250
\(209\) − 7.70083e6i − 0.843527i
\(210\) 0 0
\(211\) −1.35165e7 −1.43885 −0.719427 0.694568i \(-0.755596\pi\)
−0.719427 + 0.694568i \(0.755596\pi\)
\(212\) 7.63960e6i 0.801795i
\(213\) 0 0
\(214\) 1.47456e6 0.150460
\(215\) − 661003.i − 0.0665102i
\(216\) 0 0
\(217\) 1.92613e7 1.88497
\(218\) − 1.10001e6i − 0.106176i
\(219\) 0 0
\(220\) 7.46266e6 0.700850
\(221\) − 42867.6i − 0.00397148i
\(222\) 0 0
\(223\) −5.35484e6 −0.482872 −0.241436 0.970417i \(-0.577618\pi\)
−0.241436 + 0.970417i \(0.577618\pi\)
\(224\) 2.80363e6i 0.249446i
\(225\) 0 0
\(226\) 4.64935e6 0.402779
\(227\) 1.36063e7i 1.16322i 0.813466 + 0.581612i \(0.197578\pi\)
−0.813466 + 0.581612i \(0.802422\pi\)
\(228\) 0 0
\(229\) 4.34641e6 0.361930 0.180965 0.983490i \(-0.442078\pi\)
0.180965 + 0.983490i \(0.442078\pi\)
\(230\) − 3.32311e6i − 0.273125i
\(231\) 0 0
\(232\) 5.31379e6 0.425540
\(233\) 2.02333e7i 1.59956i 0.600297 + 0.799778i \(0.295049\pi\)
−0.600297 + 0.799778i \(0.704951\pi\)
\(234\) 0 0
\(235\) −1.33578e7 −1.02927
\(236\) − 7.99490e6i − 0.608243i
\(237\) 0 0
\(238\) 34848.0 0.00258492
\(239\) 2.03947e7i 1.49391i 0.664877 + 0.746953i \(0.268484\pi\)
−0.664877 + 0.746953i \(0.731516\pi\)
\(240\) 0 0
\(241\) −3.12093e6 −0.222963 −0.111481 0.993767i \(-0.535560\pi\)
−0.111481 + 0.993767i \(0.535560\pi\)
\(242\) 146213.i 0.0103167i
\(243\) 0 0
\(244\) −424000. −0.0291875
\(245\) − 2.02836e7i − 1.37926i
\(246\) 0 0
\(247\) 1.93458e7 1.28379
\(248\) − 7.20385e6i − 0.472291i
\(249\) 0 0
\(250\) −976128. −0.0624722
\(251\) − 5.09519e6i − 0.322210i −0.986937 0.161105i \(-0.948494\pi\)
0.986937 0.161105i \(-0.0515058\pi\)
\(252\) 0 0
\(253\) 4.52765e6 0.279583
\(254\) − 1.72942e7i − 1.05536i
\(255\) 0 0
\(256\) 1.04858e6 0.0625000
\(257\) − 1.44374e7i − 0.850529i −0.905069 0.425264i \(-0.860181\pi\)
0.905069 0.425264i \(-0.139819\pi\)
\(258\) 0 0
\(259\) −2.54226e7 −1.46326
\(260\) 1.87474e7i 1.06665i
\(261\) 0 0
\(262\) −1.73885e7 −0.966847
\(263\) − 3.12567e7i − 1.71821i −0.511801 0.859104i \(-0.671022\pi\)
0.511801 0.859104i \(-0.328978\pi\)
\(264\) 0 0
\(265\) −4.15280e7 −2.23153
\(266\) 1.57266e7i 0.835584i
\(267\) 0 0
\(268\) −5.40698e6 −0.280899
\(269\) − 251338.i − 0.0129122i −0.999979 0.00645612i \(-0.997945\pi\)
0.999979 0.00645612i \(-0.00205506\pi\)
\(270\) 0 0
\(271\) 2.96399e7 1.48925 0.744627 0.667481i \(-0.232627\pi\)
0.744627 + 0.667481i \(0.232627\pi\)
\(272\) − 13033.4i 0 0.000647665i
\(273\) 0 0
\(274\) 2.53660e7 1.23311
\(275\) 1.96181e7i 0.943319i
\(276\) 0 0
\(277\) 1.32213e7 0.622062 0.311031 0.950400i \(-0.399326\pi\)
0.311031 + 0.950400i \(0.399326\pi\)
\(278\) 6.17914e6i 0.287603i
\(279\) 0 0
\(280\) −1.52402e7 −0.694251
\(281\) 6.12360e6i 0.275987i 0.990433 + 0.137993i \(0.0440652\pi\)
−0.990433 + 0.137993i \(0.955935\pi\)
\(282\) 0 0
\(283\) −6.74325e6 −0.297516 −0.148758 0.988874i \(-0.547527\pi\)
−0.148758 + 0.988874i \(0.547527\pi\)
\(284\) − 1.70069e7i − 0.742457i
\(285\) 0 0
\(286\) −2.55429e7 −1.09187
\(287\) − 1.79286e7i − 0.758403i
\(288\) 0 0
\(289\) 2.41374e7 0.999993
\(290\) 2.88852e7i 1.18435i
\(291\) 0 0
\(292\) −7.55661e6 −0.303514
\(293\) 1.00239e7i 0.398505i 0.979948 + 0.199253i \(0.0638514\pi\)
−0.979948 + 0.199253i \(0.936149\pi\)
\(294\) 0 0
\(295\) 4.34593e7 1.69284
\(296\) 9.50822e6i 0.366627i
\(297\) 0 0
\(298\) 1.25632e7 0.474734
\(299\) 1.13742e7i 0.425508i
\(300\) 0 0
\(301\) −1.83920e6 −0.0674418
\(302\) 2.30727e7i 0.837677i
\(303\) 0 0
\(304\) 5.88186e6 0.209360
\(305\) − 2.30481e6i − 0.0812337i
\(306\) 0 0
\(307\) −5.23060e6 −0.180774 −0.0903871 0.995907i \(-0.528810\pi\)
−0.0903871 + 0.995907i \(0.528810\pi\)
\(308\) − 2.07644e7i − 0.710668i
\(309\) 0 0
\(310\) 3.91593e7 1.31447
\(311\) − 3.12221e7i − 1.03796i −0.854786 0.518981i \(-0.826312\pi\)
0.854786 0.518981i \(-0.173688\pi\)
\(312\) 0 0
\(313\) 2.24778e7 0.733029 0.366515 0.930412i \(-0.380551\pi\)
0.366515 + 0.930412i \(0.380551\pi\)
\(314\) − 3.48218e7i − 1.12477i
\(315\) 0 0
\(316\) 1.12371e6 0.0356118
\(317\) 2.76211e7i 0.867088i 0.901132 + 0.433544i \(0.142737\pi\)
−0.901132 + 0.433544i \(0.857263\pi\)
\(318\) 0 0
\(319\) −3.93553e7 −1.21236
\(320\) 5.69994e6i 0.173948i
\(321\) 0 0
\(322\) −9.24634e6 −0.276950
\(323\) − 73109.2i − 0.00216952i
\(324\) 0 0
\(325\) −4.92839e7 −1.43567
\(326\) − 4.52942e6i − 0.130734i
\(327\) 0 0
\(328\) −6.70541e6 −0.190022
\(329\) 3.71672e7i 1.04369i
\(330\) 0 0
\(331\) −5.76138e6 −0.158870 −0.0794352 0.996840i \(-0.525312\pi\)
−0.0794352 + 0.996840i \(0.525312\pi\)
\(332\) − 351359.i − 0.00960144i
\(333\) 0 0
\(334\) −2.71793e7 −0.729456
\(335\) − 2.93917e7i − 0.781790i
\(336\) 0 0
\(337\) −4.01052e7 −1.04788 −0.523939 0.851756i \(-0.675538\pi\)
−0.523939 + 0.851756i \(0.675538\pi\)
\(338\) − 3.68635e7i − 0.954656i
\(339\) 0 0
\(340\) 70848.0 0.00180256
\(341\) 5.33535e7i 1.34555i
\(342\) 0 0
\(343\) 504328. 0.0124977
\(344\) 687873.i 0.0168979i
\(345\) 0 0
\(346\) 2.01945e7 0.487535
\(347\) − 6.78127e7i − 1.62302i −0.584341 0.811508i \(-0.698647\pi\)
0.584341 0.811508i \(-0.301353\pi\)
\(348\) 0 0
\(349\) −4.20638e7 −0.989538 −0.494769 0.869024i \(-0.664747\pi\)
−0.494769 + 0.869024i \(0.664747\pi\)
\(350\) − 4.00639e7i − 0.934436i
\(351\) 0 0
\(352\) −7.76602e6 −0.178062
\(353\) 1.75976e7i 0.400063i 0.979789 + 0.200032i \(0.0641045\pi\)
−0.979789 + 0.200032i \(0.935896\pi\)
\(354\) 0 0
\(355\) 9.24478e7 2.06639
\(356\) − 4.13851e6i − 0.0917263i
\(357\) 0 0
\(358\) 4.20699e7 0.916901
\(359\) 1.39920e7i 0.302410i 0.988502 + 0.151205i \(0.0483154\pi\)
−0.988502 + 0.151205i \(0.951685\pi\)
\(360\) 0 0
\(361\) −1.40523e7 −0.298694
\(362\) 5.87249e7i 1.23793i
\(363\) 0 0
\(364\) 5.21636e7 1.08159
\(365\) − 4.10768e7i − 0.844731i
\(366\) 0 0
\(367\) −2.65855e7 −0.537832 −0.268916 0.963164i \(-0.586665\pi\)
−0.268916 + 0.963164i \(0.586665\pi\)
\(368\) 3.45819e6i 0.0693914i
\(369\) 0 0
\(370\) −5.16856e7 −1.02039
\(371\) 1.15549e8i 2.26279i
\(372\) 0 0
\(373\) 1.78829e7 0.344598 0.172299 0.985045i \(-0.444881\pi\)
0.172299 + 0.985045i \(0.444881\pi\)
\(374\) 96528.6i 0.00184519i
\(375\) 0 0
\(376\) 1.39008e7 0.261503
\(377\) − 9.88671e7i − 1.84513i
\(378\) 0 0
\(379\) 7.20978e7 1.32435 0.662177 0.749347i \(-0.269633\pi\)
0.662177 + 0.749347i \(0.269633\pi\)
\(380\) 3.19731e7i 0.582685i
\(381\) 0 0
\(382\) −7.35055e7 −1.31865
\(383\) − 8.68648e6i − 0.154614i −0.997007 0.0773068i \(-0.975368\pi\)
0.997007 0.0773068i \(-0.0246321\pi\)
\(384\) 0 0
\(385\) 1.12873e8 1.97791
\(386\) 2.22425e7i 0.386742i
\(387\) 0 0
\(388\) 1.02856e7 0.176089
\(389\) 4.94411e7i 0.839923i 0.907542 + 0.419962i \(0.137956\pi\)
−0.907542 + 0.419962i \(0.862044\pi\)
\(390\) 0 0
\(391\) 42984.0 0.000719079 0
\(392\) 2.11081e7i 0.350422i
\(393\) 0 0
\(394\) −3.04320e7 −0.497556
\(395\) 6.10837e6i 0.0991137i
\(396\) 0 0
\(397\) 1.56911e7 0.250774 0.125387 0.992108i \(-0.459983\pi\)
0.125387 + 0.992108i \(0.459983\pi\)
\(398\) 3.20121e6i 0.0507768i
\(399\) 0 0
\(400\) −1.49842e7 −0.234128
\(401\) 4.74514e7i 0.735895i 0.929847 + 0.367947i \(0.119939\pi\)
−0.929847 + 0.367947i \(0.880061\pi\)
\(402\) 0 0
\(403\) −1.34033e8 −2.04784
\(404\) 2.14010e7i 0.324556i
\(405\) 0 0
\(406\) 8.03711e7 1.20094
\(407\) − 7.04203e7i − 1.04451i
\(408\) 0 0
\(409\) −1.15512e8 −1.68832 −0.844162 0.536088i \(-0.819901\pi\)
−0.844162 + 0.536088i \(0.819901\pi\)
\(410\) − 3.64498e7i − 0.528864i
\(411\) 0 0
\(412\) −6.37892e7 −0.912127
\(413\) − 1.20923e8i − 1.71656i
\(414\) 0 0
\(415\) 1.90994e6 0.0267225
\(416\) − 1.95095e7i − 0.270999i
\(417\) 0 0
\(418\) −4.35625e7 −0.596464
\(419\) − 1.46693e8i − 1.99420i −0.0761306 0.997098i \(-0.524257\pi\)
0.0761306 0.997098i \(-0.475743\pi\)
\(420\) 0 0
\(421\) 1.39239e8 1.86601 0.933005 0.359863i \(-0.117176\pi\)
0.933005 + 0.359863i \(0.117176\pi\)
\(422\) 7.64609e7i 1.01742i
\(423\) 0 0
\(424\) 4.32161e7 0.566955
\(425\) 186248.i 0.00242619i
\(426\) 0 0
\(427\) −6.41300e6 −0.0823716
\(428\) − 8.34137e6i − 0.106391i
\(429\) 0 0
\(430\) −3.73920e6 −0.0470298
\(431\) 1.00392e8i 1.25391i 0.779056 + 0.626954i \(0.215699\pi\)
−0.779056 + 0.626954i \(0.784301\pi\)
\(432\) 0 0
\(433\) −4.00631e7 −0.493493 −0.246747 0.969080i \(-0.579362\pi\)
−0.246747 + 0.969080i \(0.579362\pi\)
\(434\) − 1.08958e8i − 1.33288i
\(435\) 0 0
\(436\) −6.22259e6 −0.0750779
\(437\) 1.93983e7i 0.232445i
\(438\) 0 0
\(439\) −1.38592e8 −1.63811 −0.819057 0.573712i \(-0.805503\pi\)
−0.819057 + 0.573712i \(0.805503\pi\)
\(440\) − 4.22152e7i − 0.495576i
\(441\) 0 0
\(442\) −242496. −0.00280826
\(443\) 1.11443e8i 1.28186i 0.767600 + 0.640929i \(0.221451\pi\)
−0.767600 + 0.640929i \(0.778549\pi\)
\(444\) 0 0
\(445\) 2.24965e7 0.255290
\(446\) 3.02915e7i 0.341442i
\(447\) 0 0
\(448\) 1.58597e7 0.176385
\(449\) 6.11166e7i 0.675181i 0.941293 + 0.337591i \(0.109612\pi\)
−0.941293 + 0.337591i \(0.890388\pi\)
\(450\) 0 0
\(451\) 4.96619e7 0.541370
\(452\) − 2.63007e7i − 0.284808i
\(453\) 0 0
\(454\) 7.69691e7 0.822524
\(455\) 2.83555e8i 3.01026i
\(456\) 0 0
\(457\) 3.56665e7 0.373690 0.186845 0.982389i \(-0.440174\pi\)
0.186845 + 0.982389i \(0.440174\pi\)
\(458\) − 2.45870e7i − 0.255923i
\(459\) 0 0
\(460\) −1.87983e7 −0.193128
\(461\) − 1.51983e8i − 1.55128i −0.631173 0.775642i \(-0.717426\pi\)
0.631173 0.775642i \(-0.282574\pi\)
\(462\) 0 0
\(463\) 1.14978e8 1.15844 0.579218 0.815173i \(-0.303358\pi\)
0.579218 + 0.815173i \(0.303358\pi\)
\(464\) − 3.00593e7i − 0.300902i
\(465\) 0 0
\(466\) 1.14457e8 1.13106
\(467\) − 8.81705e7i − 0.865711i −0.901463 0.432855i \(-0.857506\pi\)
0.901463 0.432855i \(-0.142494\pi\)
\(468\) 0 0
\(469\) −8.17805e7 −0.792741
\(470\) 7.55631e7i 0.727807i
\(471\) 0 0
\(472\) −4.52260e7 −0.430093
\(473\) − 5.09456e6i − 0.0481419i
\(474\) 0 0
\(475\) −8.40520e7 −0.784272
\(476\) − 197130.i − 0.00182781i
\(477\) 0 0
\(478\) 1.15370e8 1.05635
\(479\) 8.94388e7i 0.813803i 0.913472 + 0.406902i \(0.133391\pi\)
−0.913472 + 0.406902i \(0.866609\pi\)
\(480\) 0 0
\(481\) 1.76908e8 1.58969
\(482\) 1.76546e7i 0.157659i
\(483\) 0 0
\(484\) 827104. 0.00729498
\(485\) 5.59111e7i 0.490087i
\(486\) 0 0
\(487\) −7.51688e7 −0.650805 −0.325403 0.945576i \(-0.605500\pi\)
−0.325403 + 0.945576i \(0.605500\pi\)
\(488\) 2.39851e6i 0.0206387i
\(489\) 0 0
\(490\) −1.14741e8 −0.975285
\(491\) − 4.50822e7i − 0.380856i −0.981701 0.190428i \(-0.939012\pi\)
0.981701 0.190428i \(-0.0609876\pi\)
\(492\) 0 0
\(493\) −373626. −0.00311815
\(494\) − 1.09436e8i − 0.907780i
\(495\) 0 0
\(496\) −4.07511e7 −0.333960
\(497\) − 2.57230e8i − 2.09533i
\(498\) 0 0
\(499\) 9.15458e7 0.736778 0.368389 0.929672i \(-0.379909\pi\)
0.368389 + 0.929672i \(0.379909\pi\)
\(500\) 5.52181e6i 0.0441745i
\(501\) 0 0
\(502\) −2.88228e7 −0.227837
\(503\) 1.61043e8i 1.26543i 0.774386 + 0.632713i \(0.218059\pi\)
−0.774386 + 0.632713i \(0.781941\pi\)
\(504\) 0 0
\(505\) −1.16333e8 −0.903295
\(506\) − 2.56122e7i − 0.197695i
\(507\) 0 0
\(508\) −9.78308e7 −0.746250
\(509\) − 2.39995e7i − 0.181990i −0.995851 0.0909951i \(-0.970995\pi\)
0.995851 0.0909951i \(-0.0290048\pi\)
\(510\) 0 0
\(511\) −1.14294e8 −0.856564
\(512\) − 5.93164e6i − 0.0441942i
\(513\) 0 0
\(514\) −8.16702e7 −0.601415
\(515\) − 3.46751e8i − 2.53861i
\(516\) 0 0
\(517\) −1.02953e8 −0.745018
\(518\) 1.43812e8i 1.03468i
\(519\) 0 0
\(520\) 1.06052e8 0.754236
\(521\) 9.00897e7i 0.637033i 0.947917 + 0.318517i \(0.103185\pi\)
−0.947917 + 0.318517i \(0.896815\pi\)
\(522\) 0 0
\(523\) −3.77691e7 −0.264016 −0.132008 0.991249i \(-0.542143\pi\)
−0.132008 + 0.991249i \(0.542143\pi\)
\(524\) 9.83641e7i 0.683664i
\(525\) 0 0
\(526\) −1.76815e8 −1.21496
\(527\) 506520.i 0.00346071i
\(528\) 0 0
\(529\) 1.36631e8 0.922957
\(530\) 2.34918e8i 1.57793i
\(531\) 0 0
\(532\) 8.89631e7 0.590847
\(533\) 1.24759e8i 0.823931i
\(534\) 0 0
\(535\) 4.53427e7 0.296105
\(536\) 3.05865e7i 0.198626i
\(537\) 0 0
\(538\) −1.42178e6 −0.00913034
\(539\) − 1.56332e8i − 0.998347i
\(540\) 0 0
\(541\) 2.54800e7 0.160919 0.0804595 0.996758i \(-0.474361\pi\)
0.0804595 + 0.996758i \(0.474361\pi\)
\(542\) − 1.67669e8i − 1.05306i
\(543\) 0 0
\(544\) −73728.0 −0.000457969 0
\(545\) − 3.38253e7i − 0.208955i
\(546\) 0 0
\(547\) 2.05216e8 1.25386 0.626930 0.779076i \(-0.284311\pi\)
0.626930 + 0.779076i \(0.284311\pi\)
\(548\) − 1.43492e8i − 0.871938i
\(549\) 0 0
\(550\) 1.10977e8 0.667027
\(551\) − 1.68614e8i − 1.00795i
\(552\) 0 0
\(553\) 1.69961e7 0.100502
\(554\) − 7.47908e7i − 0.439865i
\(555\) 0 0
\(556\) 3.49545e7 0.203366
\(557\) 2.41143e8i 1.39543i 0.716375 + 0.697715i \(0.245800\pi\)
−0.716375 + 0.697715i \(0.754200\pi\)
\(558\) 0 0
\(559\) 1.27984e7 0.0732690
\(560\) 8.62115e7i 0.490909i
\(561\) 0 0
\(562\) 3.46403e7 0.195152
\(563\) 1.68877e8i 0.946337i 0.880972 + 0.473168i \(0.156890\pi\)
−0.880972 + 0.473168i \(0.843110\pi\)
\(564\) 0 0
\(565\) 1.42968e8 0.792670
\(566\) 3.81456e7i 0.210375i
\(567\) 0 0
\(568\) −9.62058e7 −0.524996
\(569\) 2.43995e8i 1.32448i 0.749293 + 0.662238i \(0.230393\pi\)
−0.749293 + 0.662238i \(0.769607\pi\)
\(570\) 0 0
\(571\) 2.41502e8 1.29722 0.648608 0.761123i \(-0.275352\pi\)
0.648608 + 0.761123i \(0.275352\pi\)
\(572\) 1.44493e8i 0.772071i
\(573\) 0 0
\(574\) −1.01419e8 −0.536272
\(575\) − 4.94177e7i − 0.259944i
\(576\) 0 0
\(577\) −4.93979e7 −0.257147 −0.128573 0.991700i \(-0.541040\pi\)
−0.128573 + 0.991700i \(0.541040\pi\)
\(578\) − 1.36542e8i − 0.707102i
\(579\) 0 0
\(580\) 1.63399e8 0.837463
\(581\) − 5.31430e6i − 0.0270968i
\(582\) 0 0
\(583\) −3.20069e8 −1.61525
\(584\) 4.27466e7i 0.214617i
\(585\) 0 0
\(586\) 5.67038e7 0.281786
\(587\) 1.72052e8i 0.850639i 0.905043 + 0.425320i \(0.139838\pi\)
−0.905043 + 0.425320i \(0.860162\pi\)
\(588\) 0 0
\(589\) −2.28588e8 −1.11869
\(590\) − 2.45843e8i − 1.19702i
\(591\) 0 0
\(592\) 5.37866e7 0.259244
\(593\) − 2.70643e8i − 1.29788i −0.760841 0.648938i \(-0.775213\pi\)
0.760841 0.648938i \(-0.224787\pi\)
\(594\) 0 0
\(595\) 1.07158e6 0.00508712
\(596\) − 7.10680e7i − 0.335687i
\(597\) 0 0
\(598\) 6.43423e7 0.300880
\(599\) 1.73299e8i 0.806337i 0.915126 + 0.403169i \(0.132091\pi\)
−0.915126 + 0.403169i \(0.867909\pi\)
\(600\) 0 0
\(601\) −4.31090e8 −1.98584 −0.992921 0.118775i \(-0.962103\pi\)
−0.992921 + 0.118775i \(0.962103\pi\)
\(602\) 1.04041e7i 0.0476886i
\(603\) 0 0
\(604\) 1.30519e8 0.592327
\(605\) 4.49604e6i 0.0203032i
\(606\) 0 0
\(607\) 1.66991e7 0.0746665 0.0373332 0.999303i \(-0.488114\pi\)
0.0373332 + 0.999303i \(0.488114\pi\)
\(608\) − 3.32728e7i − 0.148040i
\(609\) 0 0
\(610\) −1.30380e7 −0.0574409
\(611\) − 2.58635e8i − 1.13387i
\(612\) 0 0
\(613\) −1.92321e8 −0.834920 −0.417460 0.908695i \(-0.637080\pi\)
−0.417460 + 0.908695i \(0.637080\pi\)
\(614\) 2.95887e7i 0.127827i
\(615\) 0 0
\(616\) −1.17461e8 −0.502518
\(617\) 1.87023e8i 0.796233i 0.917335 + 0.398117i \(0.130336\pi\)
−0.917335 + 0.398117i \(0.869664\pi\)
\(618\) 0 0
\(619\) 2.54873e8 1.07461 0.537307 0.843387i \(-0.319442\pi\)
0.537307 + 0.843387i \(0.319442\pi\)
\(620\) − 2.21518e8i − 0.929468i
\(621\) 0 0
\(622\) −1.76619e8 −0.733950
\(623\) − 6.25950e7i − 0.258866i
\(624\) 0 0
\(625\) −2.58657e8 −1.05946
\(626\) − 1.27154e8i − 0.518330i
\(627\) 0 0
\(628\) −1.96982e8 −0.795329
\(629\) − 668547.i − 0.00268646i
\(630\) 0 0
\(631\) 9.23602e7 0.367618 0.183809 0.982962i \(-0.441157\pi\)
0.183809 + 0.982962i \(0.441157\pi\)
\(632\) − 6.35668e6i − 0.0251813i
\(633\) 0 0
\(634\) 1.56249e8 0.613124
\(635\) − 5.31797e8i − 2.07694i
\(636\) 0 0
\(637\) 3.92732e8 1.51942
\(638\) 2.22627e8i 0.857267i
\(639\) 0 0
\(640\) 3.22437e7 0.123000
\(641\) 4.24666e8i 1.61240i 0.591643 + 0.806200i \(0.298480\pi\)
−0.591643 + 0.806200i \(0.701520\pi\)
\(642\) 0 0
\(643\) 3.75946e8 1.41414 0.707071 0.707143i \(-0.250016\pi\)
0.707071 + 0.707143i \(0.250016\pi\)
\(644\) 5.23052e7i 0.195834i
\(645\) 0 0
\(646\) −413568. −0.00153408
\(647\) − 2.63747e7i − 0.0973813i −0.998814 0.0486906i \(-0.984495\pi\)
0.998814 0.0486906i \(-0.0155048\pi\)
\(648\) 0 0
\(649\) 3.34955e8 1.22533
\(650\) 2.78792e8i 1.01517i
\(651\) 0 0
\(652\) −2.56223e7 −0.0924432
\(653\) − 2.58756e8i − 0.929291i −0.885497 0.464645i \(-0.846182\pi\)
0.885497 0.464645i \(-0.153818\pi\)
\(654\) 0 0
\(655\) −5.34696e8 −1.90275
\(656\) 3.79315e7i 0.134366i
\(657\) 0 0
\(658\) 2.10250e8 0.738002
\(659\) 1.39345e8i 0.486895i 0.969914 + 0.243447i \(0.0782783\pi\)
−0.969914 + 0.243447i \(0.921722\pi\)
\(660\) 0 0
\(661\) −4.72545e8 −1.63621 −0.818104 0.575070i \(-0.804975\pi\)
−0.818104 + 0.575070i \(0.804975\pi\)
\(662\) 3.25913e7i 0.112338i
\(663\) 0 0
\(664\) −1.98758e6 −0.00678924
\(665\) 4.83593e8i 1.64443i
\(666\) 0 0
\(667\) 9.91354e7 0.334081
\(668\) 1.53749e8i 0.515804i
\(669\) 0 0
\(670\) −1.66265e8 −0.552809
\(671\) − 1.77639e7i − 0.0587992i
\(672\) 0 0
\(673\) 5.48833e8 1.80051 0.900254 0.435364i \(-0.143380\pi\)
0.900254 + 0.435364i \(0.143380\pi\)
\(674\) 2.26869e8i 0.740961i
\(675\) 0 0
\(676\) −2.08532e8 −0.675044
\(677\) 1.00760e8i 0.324731i 0.986731 + 0.162365i \(0.0519123\pi\)
−0.986731 + 0.162365i \(0.948088\pi\)
\(678\) 0 0
\(679\) 1.55569e8 0.496952
\(680\) − 400777.i − 0.00127461i
\(681\) 0 0
\(682\) 3.01813e8 0.951447
\(683\) 313056.i 0 0.000982562i 1.00000 0.000491281i \(0.000156380\pi\)
−1.00000 0.000491281i \(0.999844\pi\)
\(684\) 0 0
\(685\) 7.80005e8 2.42675
\(686\) − 2.85291e6i − 0.00883722i
\(687\) 0 0
\(688\) 3.89120e6 0.0119486
\(689\) − 8.04068e8i − 2.45830i
\(690\) 0 0
\(691\) −3.72812e8 −1.12994 −0.564971 0.825111i \(-0.691113\pi\)
−0.564971 + 0.825111i \(0.691113\pi\)
\(692\) − 1.14238e8i − 0.344739i
\(693\) 0 0
\(694\) −3.83607e8 −1.14765
\(695\) 1.90009e8i 0.566003i
\(696\) 0 0
\(697\) 471474. 0.00139239
\(698\) 2.37949e8i 0.699709i
\(699\) 0 0
\(700\) −2.26636e8 −0.660746
\(701\) − 6.21170e8i − 1.80325i −0.432517 0.901626i \(-0.642374\pi\)
0.432517 0.901626i \(-0.357626\pi\)
\(702\) 0 0
\(703\) 3.01709e8 0.868406
\(704\) 4.39312e7i 0.125909i
\(705\) 0 0
\(706\) 9.95469e7 0.282887
\(707\) 3.23690e8i 0.915947i
\(708\) 0 0
\(709\) −2.46510e8 −0.691666 −0.345833 0.938296i \(-0.612404\pi\)
−0.345833 + 0.938296i \(0.612404\pi\)
\(710\) − 5.22964e8i − 1.46116i
\(711\) 0 0
\(712\) −2.34109e7 −0.0648603
\(713\) − 1.34397e8i − 0.370783i
\(714\) 0 0
\(715\) −7.85445e8 −2.14881
\(716\) − 2.37983e8i − 0.648347i
\(717\) 0 0
\(718\) 7.91508e7 0.213836
\(719\) 9.60389e7i 0.258381i 0.991620 + 0.129191i \(0.0412379\pi\)
−0.991620 + 0.129191i \(0.958762\pi\)
\(720\) 0 0
\(721\) −9.64811e8 −2.57417
\(722\) 7.94921e7i 0.211209i
\(723\) 0 0
\(724\) 3.32198e8 0.875350
\(725\) 4.29549e8i 1.12719i
\(726\) 0 0
\(727\) 3.91371e8 1.01856 0.509278 0.860602i \(-0.329912\pi\)
0.509278 + 0.860602i \(0.329912\pi\)
\(728\) − 2.95082e8i − 0.764801i
\(729\) 0 0
\(730\) −2.32366e8 −0.597315
\(731\) − 48366.1i 0 0.000123819i
\(732\) 0 0
\(733\) 3.49078e7 0.0886361 0.0443181 0.999017i \(-0.485889\pi\)
0.0443181 + 0.999017i \(0.485889\pi\)
\(734\) 1.50390e8i 0.380304i
\(735\) 0 0
\(736\) 1.95625e7 0.0490671
\(737\) − 2.26531e8i − 0.565881i
\(738\) 0 0
\(739\) −3.02999e8 −0.750773 −0.375386 0.926868i \(-0.622490\pi\)
−0.375386 + 0.926868i \(0.622490\pi\)
\(740\) 2.92378e8i 0.721521i
\(741\) 0 0
\(742\) 6.53644e8 1.60004
\(743\) − 2.45628e8i − 0.598842i −0.954121 0.299421i \(-0.903207\pi\)
0.954121 0.299421i \(-0.0967935\pi\)
\(744\) 0 0
\(745\) 3.86317e8 0.934276
\(746\) − 1.01161e8i − 0.243667i
\(747\) 0 0
\(748\) 546048. 0.00130475
\(749\) − 1.26163e8i − 0.300253i
\(750\) 0 0
\(751\) −8.23270e7 −0.194367 −0.0971835 0.995266i \(-0.530983\pi\)
−0.0971835 + 0.995266i \(0.530983\pi\)
\(752\) − 7.86348e7i − 0.184910i
\(753\) 0 0
\(754\) −5.59277e8 −1.30471
\(755\) 7.09484e8i 1.64855i
\(756\) 0 0
\(757\) −6.03579e8 −1.39138 −0.695691 0.718341i \(-0.744902\pi\)
−0.695691 + 0.718341i \(0.744902\pi\)
\(758\) − 4.07847e8i − 0.936460i
\(759\) 0 0
\(760\) 1.80867e8 0.412020
\(761\) 2.32982e8i 0.528651i 0.964434 + 0.264325i \(0.0851493\pi\)
−0.964434 + 0.264325i \(0.914851\pi\)
\(762\) 0 0
\(763\) −9.41167e7 −0.211882
\(764\) 4.15810e8i 0.932426i
\(765\) 0 0
\(766\) −4.91382e7 −0.109328
\(767\) 8.41463e8i 1.86487i
\(768\) 0 0
\(769\) 8.15796e8 1.79392 0.896958 0.442115i \(-0.145772\pi\)
0.896958 + 0.442115i \(0.145772\pi\)
\(770\) − 6.38504e8i − 1.39859i
\(771\) 0 0
\(772\) 1.25823e8 0.273468
\(773\) 3.66587e8i 0.793667i 0.917891 + 0.396833i \(0.129891\pi\)
−0.917891 + 0.396833i \(0.870109\pi\)
\(774\) 0 0
\(775\) 5.82335e8 1.25103
\(776\) − 5.81840e7i − 0.124514i
\(777\) 0 0
\(778\) 2.79681e8 0.593915
\(779\) 2.12772e8i 0.450093i
\(780\) 0 0
\(781\) 7.12524e8 1.49571
\(782\) − 243154.i 0 0.000508466i
\(783\) 0 0
\(784\) 1.19406e8 0.247786
\(785\) − 1.07077e9i − 2.21354i
\(786\) 0 0
\(787\) −4.02462e8 −0.825659 −0.412830 0.910808i \(-0.635460\pi\)
−0.412830 + 0.910808i \(0.635460\pi\)
\(788\) 1.72150e8i 0.351825i
\(789\) 0 0
\(790\) 3.45541e7 0.0700840
\(791\) − 3.97798e8i − 0.803773i
\(792\) 0 0
\(793\) 4.46260e7 0.0894887
\(794\) − 8.87623e7i − 0.177324i
\(795\) 0 0
\(796\) 1.81088e7 0.0359046
\(797\) 5.18940e8i 1.02504i 0.858675 + 0.512521i \(0.171288\pi\)
−0.858675 + 0.512521i \(0.828712\pi\)
\(798\) 0 0
\(799\) −977400. −0.00191616
\(800\) 8.47634e7i 0.165553i
\(801\) 0 0
\(802\) 2.68426e8 0.520356
\(803\) − 3.16592e8i − 0.611440i
\(804\) 0 0
\(805\) −2.84325e8 −0.545038
\(806\) 7.58205e8i 1.44804i
\(807\) 0 0
\(808\) 1.21062e8 0.229496
\(809\) 3.04036e6i 0.00574221i 0.999996 + 0.00287110i \(0.000913902\pi\)
−0.999996 + 0.00287110i \(0.999086\pi\)
\(810\) 0 0
\(811\) 2.25521e8 0.422790 0.211395 0.977401i \(-0.432199\pi\)
0.211395 + 0.977401i \(0.432199\pi\)
\(812\) − 4.54648e8i − 0.849194i
\(813\) 0 0
\(814\) −3.98357e8 −0.738583
\(815\) − 1.39280e8i − 0.257285i
\(816\) 0 0
\(817\) 2.18272e7 0.0400250
\(818\) 6.53432e8i 1.19383i
\(819\) 0 0
\(820\) −2.06191e8 −0.373963
\(821\) − 2.77035e8i − 0.500617i −0.968166 0.250309i \(-0.919468\pi\)
0.968166 0.250309i \(-0.0805321\pi\)
\(822\) 0 0
\(823\) −7.07336e8 −1.26890 −0.634448 0.772965i \(-0.718773\pi\)
−0.634448 + 0.772965i \(0.718773\pi\)
\(824\) 3.60846e8i 0.644971i
\(825\) 0 0
\(826\) −6.84043e8 −1.21379
\(827\) 2.66346e8i 0.470900i 0.971886 + 0.235450i \(0.0756564\pi\)
−0.971886 + 0.235450i \(0.924344\pi\)
\(828\) 0 0
\(829\) 5.03826e8 0.884336 0.442168 0.896932i \(-0.354209\pi\)
0.442168 + 0.896932i \(0.354209\pi\)
\(830\) − 1.08043e7i − 0.0188956i
\(831\) 0 0
\(832\) −1.10363e8 −0.191625
\(833\) − 1.48416e6i − 0.00256772i
\(834\) 0 0
\(835\) −8.35764e8 −1.43557
\(836\) 2.46427e8i 0.421763i
\(837\) 0 0
\(838\) −8.29822e8 −1.41011
\(839\) − 7.63364e8i − 1.29255i −0.763106 0.646273i \(-0.776327\pi\)
0.763106 0.646273i \(-0.223673\pi\)
\(840\) 0 0
\(841\) −2.66883e8 −0.448676
\(842\) − 7.87654e8i − 1.31947i
\(843\) 0 0
\(844\) 4.32528e8 0.719427
\(845\) − 1.13355e9i − 1.87876i
\(846\) 0 0
\(847\) 1.25099e7 0.0205876
\(848\) − 2.44467e8i − 0.400897i
\(849\) 0 0
\(850\) 1.05358e6 0.00171557
\(851\) 1.77388e8i 0.287829i
\(852\) 0 0
\(853\) −1.87985e7 −0.0302884 −0.0151442 0.999885i \(-0.504821\pi\)
−0.0151442 + 0.999885i \(0.504821\pi\)
\(854\) 3.62774e7i 0.0582455i
\(855\) 0 0
\(856\) −4.71859e7 −0.0752300
\(857\) 6.86427e8i 1.09057i 0.838252 + 0.545283i \(0.183578\pi\)
−0.838252 + 0.545283i \(0.816422\pi\)
\(858\) 0 0
\(859\) 5.51932e8 0.870775 0.435387 0.900243i \(-0.356611\pi\)
0.435387 + 0.900243i \(0.356611\pi\)
\(860\) 2.11521e7i 0.0332551i
\(861\) 0 0
\(862\) 5.67901e8 0.886647
\(863\) − 3.65665e8i − 0.568920i −0.958688 0.284460i \(-0.908186\pi\)
0.958688 0.284460i \(-0.0918142\pi\)
\(864\) 0 0
\(865\) 6.20982e8 0.959468
\(866\) 2.26631e8i 0.348952i
\(867\) 0 0
\(868\) −6.16360e8 −0.942487
\(869\) 4.70791e7i 0.0717413i
\(870\) 0 0
\(871\) 5.69084e8 0.861236
\(872\) 3.52003e7i 0.0530881i
\(873\) 0 0
\(874\) 1.09733e8 0.164363
\(875\) 8.35174e7i 0.124667i
\(876\) 0 0
\(877\) −5.85387e8 −0.867849 −0.433925 0.900949i \(-0.642872\pi\)
−0.433925 + 0.900949i \(0.642872\pi\)
\(878\) 7.83994e8i 1.15832i
\(879\) 0 0
\(880\) −2.38805e8 −0.350425
\(881\) 4.29761e8i 0.628491i 0.949342 + 0.314246i \(0.101752\pi\)
−0.949342 + 0.314246i \(0.898248\pi\)
\(882\) 0 0
\(883\) 2.20085e8 0.319675 0.159837 0.987143i \(-0.448903\pi\)
0.159837 + 0.987143i \(0.448903\pi\)
\(884\) 1.37176e6i 0.00198574i
\(885\) 0 0
\(886\) 6.30415e8 0.906411
\(887\) − 1.17196e9i − 1.67936i −0.543084 0.839678i \(-0.682744\pi\)
0.543084 0.839678i \(-0.317256\pi\)
\(888\) 0 0
\(889\) −1.47969e9 −2.10604
\(890\) − 1.27259e8i − 0.180517i
\(891\) 0 0
\(892\) 1.71355e8 0.241436
\(893\) − 4.41092e8i − 0.619405i
\(894\) 0 0
\(895\) 1.29365e9 1.80446
\(896\) − 8.97161e7i − 0.124723i
\(897\) 0 0
\(898\) 3.45728e8 0.477425
\(899\) 1.16820e9i 1.60783i
\(900\) 0 0
\(901\) −3.03863e6 −0.00415436
\(902\) − 2.80930e8i − 0.382806i
\(903\) 0 0
\(904\) −1.48779e8 −0.201390
\(905\) 1.80579e9i 2.43625i
\(906\) 0 0
\(907\) 7.31614e8 0.980529 0.490264 0.871574i \(-0.336900\pi\)
0.490264 + 0.871574i \(0.336900\pi\)
\(908\) − 4.35403e8i − 0.581612i
\(909\) 0 0
\(910\) 1.60403e9 2.12857
\(911\) − 9.18595e8i − 1.21498i −0.794327 0.607490i \(-0.792177\pi\)
0.794327 0.607490i \(-0.207823\pi\)
\(912\) 0 0
\(913\) 1.47205e7 0.0193425
\(914\) − 2.01760e8i − 0.264239i
\(915\) 0 0
\(916\) −1.39085e8 −0.180965
\(917\) 1.48776e9i 1.92941i
\(918\) 0 0
\(919\) −2.15987e8 −0.278279 −0.139139 0.990273i \(-0.544434\pi\)
−0.139139 + 0.990273i \(0.544434\pi\)
\(920\) 1.06339e8i 0.136562i
\(921\) 0 0
\(922\) −8.59744e8 −1.09692
\(923\) 1.78998e9i 2.27637i
\(924\) 0 0
\(925\) −7.68613e8 −0.971141
\(926\) − 6.50414e8i − 0.819138i
\(927\) 0 0
\(928\) −1.70041e8 −0.212770
\(929\) − 3.10124e8i − 0.386802i −0.981120 0.193401i \(-0.938048\pi\)
0.981120 0.193401i \(-0.0619518\pi\)
\(930\) 0 0
\(931\) 6.69791e8 0.830023
\(932\) − 6.47466e8i − 0.799778i
\(933\) 0 0
\(934\) −4.98768e8 −0.612150
\(935\) 2.96825e6i 0.00363133i
\(936\) 0 0
\(937\) −7.42448e8 −0.902501 −0.451250 0.892397i \(-0.649022\pi\)
−0.451250 + 0.892397i \(0.649022\pi\)
\(938\) 4.62620e8i 0.560553i
\(939\) 0 0
\(940\) 4.27450e8 0.514637
\(941\) − 1.81766e8i − 0.218144i −0.994034 0.109072i \(-0.965212\pi\)
0.994034 0.109072i \(-0.0347879\pi\)
\(942\) 0 0
\(943\) −1.25098e8 −0.149181
\(944\) 2.55837e8i 0.304121i
\(945\) 0 0
\(946\) −2.88192e7 −0.0340415
\(947\) − 8.59189e8i − 1.01167i −0.862630 0.505835i \(-0.831184\pi\)
0.862630 0.505835i \(-0.168816\pi\)
\(948\) 0 0
\(949\) 7.95333e8 0.930573
\(950\) 4.75470e8i 0.554564i
\(951\) 0 0
\(952\) −1.11514e6 −0.00129246
\(953\) − 6.86819e8i − 0.793530i −0.917920 0.396765i \(-0.870133\pi\)
0.917920 0.396765i \(-0.129867\pi\)
\(954\) 0 0
\(955\) −2.26029e9 −2.59510
\(956\) − 6.52630e8i − 0.746953i
\(957\) 0 0
\(958\) 5.05942e8 0.575446
\(959\) − 2.17031e9i − 2.46075i
\(960\) 0 0
\(961\) 6.96218e8 0.784468
\(962\) − 1.00074e9i − 1.12408i
\(963\) 0 0
\(964\) 9.98697e7 0.111481
\(965\) 6.83957e8i 0.761109i
\(966\) 0 0
\(967\) −1.09411e9 −1.20999 −0.604995 0.796230i \(-0.706825\pi\)
−0.604995 + 0.796230i \(0.706825\pi\)
\(968\) − 4.67881e6i − 0.00515833i
\(969\) 0 0
\(970\) 3.16281e8 0.346544
\(971\) − 4.43115e8i − 0.484014i −0.970274 0.242007i \(-0.922194\pi\)
0.970274 0.242007i \(-0.0778058\pi\)
\(972\) 0 0
\(973\) 5.28687e8 0.573931
\(974\) 4.25219e8i 0.460189i
\(975\) 0 0
\(976\) 1.35680e7 0.0145937
\(977\) − 1.19004e9i − 1.27608i −0.770001 0.638042i \(-0.779744\pi\)
0.770001 0.638042i \(-0.220256\pi\)
\(978\) 0 0
\(979\) 1.73387e8 0.184786
\(980\) 6.49075e8i 0.689631i
\(981\) 0 0
\(982\) −2.55024e8 −0.269306
\(983\) 1.18187e9i 1.24425i 0.782918 + 0.622125i \(0.213730\pi\)
−0.782918 + 0.622125i \(0.786270\pi\)
\(984\) 0 0
\(985\) −9.35785e8 −0.979191
\(986\) 2.11355e6i 0.00220486i
\(987\) 0 0
\(988\) −6.19065e8 −0.641897
\(989\) 1.28331e7i 0.0132661i
\(990\) 0 0
\(991\) 5.09602e8 0.523613 0.261806 0.965120i \(-0.415682\pi\)
0.261806 + 0.965120i \(0.415682\pi\)
\(992\) 2.30523e8i 0.236145i
\(993\) 0 0
\(994\) −1.45511e9 −1.48162
\(995\) 9.84373e7i 0.0999288i
\(996\) 0 0
\(997\) −9.90780e8 −0.999751 −0.499875 0.866097i \(-0.666621\pi\)
−0.499875 + 0.866097i \(0.666621\pi\)
\(998\) − 5.17861e8i − 0.520981i
\(999\) 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 18.7.b.a.17.1 2
3.2 odd 2 inner 18.7.b.a.17.2 yes 2
4.3 odd 2 144.7.e.d.17.1 2
5.2 odd 4 450.7.b.a.449.3 4
5.3 odd 4 450.7.b.a.449.2 4
5.4 even 2 450.7.d.a.251.2 2
8.3 odd 2 576.7.e.k.449.2 2
8.5 even 2 576.7.e.b.449.2 2
9.2 odd 6 162.7.d.d.53.2 4
9.4 even 3 162.7.d.d.107.2 4
9.5 odd 6 162.7.d.d.107.1 4
9.7 even 3 162.7.d.d.53.1 4
12.11 even 2 144.7.e.d.17.2 2
15.2 even 4 450.7.b.a.449.1 4
15.8 even 4 450.7.b.a.449.4 4
15.14 odd 2 450.7.d.a.251.1 2
24.5 odd 2 576.7.e.b.449.1 2
24.11 even 2 576.7.e.k.449.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.7.b.a.17.1 2 1.1 even 1 trivial
18.7.b.a.17.2 yes 2 3.2 odd 2 inner
144.7.e.d.17.1 2 4.3 odd 2
144.7.e.d.17.2 2 12.11 even 2
162.7.d.d.53.1 4 9.7 even 3
162.7.d.d.53.2 4 9.2 odd 6
162.7.d.d.107.1 4 9.5 odd 6
162.7.d.d.107.2 4 9.4 even 3
450.7.b.a.449.1 4 15.2 even 4
450.7.b.a.449.2 4 5.3 odd 4
450.7.b.a.449.3 4 5.2 odd 4
450.7.b.a.449.4 4 15.8 even 4
450.7.d.a.251.1 2 15.14 odd 2
450.7.d.a.251.2 2 5.4 even 2
576.7.e.b.449.1 2 24.5 odd 2
576.7.e.b.449.2 2 8.5 even 2
576.7.e.k.449.1 2 24.11 even 2
576.7.e.k.449.2 2 8.3 odd 2