Properties

Label 350.10.c.b.99.1
Level $350$
Weight $10$
Character 350.99
Analytic conductor $180.263$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(180.262542657\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 14)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.1
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 350.99
Dual form 350.10.c.b.99.2

$q$-expansion

\(f(q)\) \(=\) \(q-16.0000i q^{2} +6.00000i q^{3} -256.000 q^{4} +96.0000 q^{6} -2401.00i q^{7} +4096.00i q^{8} +19647.0 q^{9} +O(q^{10})\) \(q-16.0000i q^{2} +6.00000i q^{3} -256.000 q^{4} +96.0000 q^{6} -2401.00i q^{7} +4096.00i q^{8} +19647.0 q^{9} -54152.0 q^{11} -1536.00i q^{12} +113172. i q^{13} -38416.0 q^{14} +65536.0 q^{16} +6262.00i q^{17} -314352. i q^{18} -257078. q^{19} +14406.0 q^{21} +866432. i q^{22} +266000. i q^{23} -24576.0 q^{24} +1.81075e6 q^{26} +235980. i q^{27} +614656. i q^{28} -1.57471e6 q^{29} -4.63748e6 q^{31} -1.04858e6i q^{32} -324912. i q^{33} +100192. q^{34} -5.02963e6 q^{36} -1.19462e7i q^{37} +4.11325e6i q^{38} -679032. q^{39} +2.19091e7 q^{41} -230496. i q^{42} -2.75206e7i q^{43} +1.38629e7 q^{44} +4.25600e6 q^{46} +5.29278e7i q^{47} +393216. i q^{48} -5.76480e6 q^{49} -37572.0 q^{51} -2.89720e7i q^{52} -1.62212e7i q^{53} +3.77568e6 q^{54} +9.83450e6 q^{56} -1.54247e6i q^{57} +2.51954e7i q^{58} +1.40510e8 q^{59} -2.02964e8 q^{61} +7.41997e7i q^{62} -4.71724e7i q^{63} -1.67772e7 q^{64} -5.19859e6 q^{66} +1.53735e8i q^{67} -1.60307e6i q^{68} -1.59600e6 q^{69} +2.79656e8 q^{71} +8.04741e7i q^{72} +4.04023e8i q^{73} -1.91140e8 q^{74} +6.58120e7 q^{76} +1.30019e8i q^{77} +1.08645e7i q^{78} +1.30690e8 q^{79} +3.85296e8 q^{81} -3.50546e8i q^{82} -4.20134e8i q^{83} -3.68794e6 q^{84} -4.40329e8 q^{86} -9.44828e6i q^{87} -2.21807e8i q^{88} +4.69542e8 q^{89} +2.71726e8 q^{91} -6.80960e7i q^{92} -2.78249e7i q^{93} +8.46845e8 q^{94} +6.29146e6 q^{96} -8.72502e8i q^{97} +9.22368e7i q^{98} -1.06392e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 512 q^{4} + 192 q^{6} + 39294 q^{9} + O(q^{10}) \) \( 2 q - 512 q^{4} + 192 q^{6} + 39294 q^{9} - 108304 q^{11} - 76832 q^{14} + 131072 q^{16} - 514156 q^{19} + 28812 q^{21} - 49152 q^{24} + 3621504 q^{26} - 3149428 q^{29} - 9274968 q^{31} + 200384 q^{34} - 10059264 q^{36} - 1358064 q^{39} + 43818252 q^{41} + 27725824 q^{44} + 8512000 q^{46} - 11529602 q^{49} - 75144 q^{51} + 7551360 q^{54} + 19668992 q^{56} + 281019236 q^{59} - 405927120 q^{61} - 33554432 q^{64} - 10397184 q^{66} - 3192000 q^{69} + 559311872 q^{71} - 382279616 q^{74} + 131623936 q^{76} + 261379632 q^{79} + 770592042 q^{81} - 7375872 q^{84} - 880658944 q^{86} + 939084780 q^{89} + 543451944 q^{91} + 1693690752 q^{94} + 12582912 q^{96} - 2127848688 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 16.0000i − 0.707107i
\(3\) 6.00000i 0.0427667i 0.999771 + 0.0213833i \(0.00680705\pi\)
−0.999771 + 0.0213833i \(0.993193\pi\)
\(4\) −256.000 −0.500000
\(5\) 0 0
\(6\) 96.0000 0.0302406
\(7\) − 2401.00i − 0.377964i
\(8\) 4096.00i 0.353553i
\(9\) 19647.0 0.998171
\(10\) 0 0
\(11\) −54152.0 −1.11519 −0.557593 0.830114i \(-0.688275\pi\)
−0.557593 + 0.830114i \(0.688275\pi\)
\(12\) − 1536.00i − 0.0213833i
\(13\) 113172.i 1.09899i 0.835497 + 0.549495i \(0.185180\pi\)
−0.835497 + 0.549495i \(0.814820\pi\)
\(14\) −38416.0 −0.267261
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 6262.00i 0.0181841i 0.999959 + 0.00909207i \(0.00289414\pi\)
−0.999959 + 0.00909207i \(0.997106\pi\)
\(18\) − 314352.i − 0.705813i
\(19\) −257078. −0.452557 −0.226279 0.974063i \(-0.572656\pi\)
−0.226279 + 0.974063i \(0.572656\pi\)
\(20\) 0 0
\(21\) 14406.0 0.0161643
\(22\) 866432.i 0.788556i
\(23\) 266000.i 0.198201i 0.995077 + 0.0991006i \(0.0315966\pi\)
−0.995077 + 0.0991006i \(0.968403\pi\)
\(24\) −24576.0 −0.0151203
\(25\) 0 0
\(26\) 1.81075e6 0.777104
\(27\) 235980.i 0.0854552i
\(28\) 614656.i 0.188982i
\(29\) −1.57471e6 −0.413438 −0.206719 0.978400i \(-0.566279\pi\)
−0.206719 + 0.978400i \(0.566279\pi\)
\(30\) 0 0
\(31\) −4.63748e6 −0.901893 −0.450946 0.892551i \(-0.648913\pi\)
−0.450946 + 0.892551i \(0.648913\pi\)
\(32\) − 1.04858e6i − 0.176777i
\(33\) − 324912.i − 0.0476928i
\(34\) 100192. 0.0128581
\(35\) 0 0
\(36\) −5.02963e6 −0.499086
\(37\) − 1.19462e7i − 1.04791i −0.851746 0.523954i \(-0.824456\pi\)
0.851746 0.523954i \(-0.175544\pi\)
\(38\) 4.11325e6i 0.320006i
\(39\) −679032. −0.0470002
\(40\) 0 0
\(41\) 2.19091e7 1.21087 0.605435 0.795895i \(-0.292999\pi\)
0.605435 + 0.795895i \(0.292999\pi\)
\(42\) − 230496.i − 0.0114299i
\(43\) − 2.75206e7i − 1.22758i −0.789469 0.613790i \(-0.789644\pi\)
0.789469 0.613790i \(-0.210356\pi\)
\(44\) 1.38629e7 0.557593
\(45\) 0 0
\(46\) 4.25600e6 0.140149
\(47\) 5.29278e7i 1.58214i 0.611728 + 0.791068i \(0.290475\pi\)
−0.611728 + 0.791068i \(0.709525\pi\)
\(48\) 393216.i 0.0106917i
\(49\) −5.76480e6 −0.142857
\(50\) 0 0
\(51\) −37572.0 −0.000777676 0
\(52\) − 2.89720e7i − 0.549495i
\(53\) − 1.62212e7i − 0.282385i −0.989982 0.141193i \(-0.954906\pi\)
0.989982 0.141193i \(-0.0450937\pi\)
\(54\) 3.77568e6 0.0604259
\(55\) 0 0
\(56\) 9.83450e6 0.133631
\(57\) − 1.54247e6i − 0.0193544i
\(58\) 2.51954e7i 0.292345i
\(59\) 1.40510e8 1.50964 0.754818 0.655935i \(-0.227725\pi\)
0.754818 + 0.655935i \(0.227725\pi\)
\(60\) 0 0
\(61\) −2.02964e8 −1.87687 −0.938434 0.345458i \(-0.887724\pi\)
−0.938434 + 0.345458i \(0.887724\pi\)
\(62\) 7.41997e7i 0.637734i
\(63\) − 4.71724e7i − 0.377273i
\(64\) −1.67772e7 −0.125000
\(65\) 0 0
\(66\) −5.19859e6 −0.0337239
\(67\) 1.53735e8i 0.932041i 0.884774 + 0.466020i \(0.154313\pi\)
−0.884774 + 0.466020i \(0.845687\pi\)
\(68\) − 1.60307e6i − 0.00909207i
\(69\) −1.59600e6 −0.00847641
\(70\) 0 0
\(71\) 2.79656e8 1.30606 0.653028 0.757334i \(-0.273499\pi\)
0.653028 + 0.757334i \(0.273499\pi\)
\(72\) 8.04741e7i 0.352907i
\(73\) 4.04023e8i 1.66515i 0.553913 + 0.832574i \(0.313134\pi\)
−0.553913 + 0.832574i \(0.686866\pi\)
\(74\) −1.91140e8 −0.740983
\(75\) 0 0
\(76\) 6.58120e7 0.226279
\(77\) 1.30019e8i 0.421501i
\(78\) 1.08645e7i 0.0332341i
\(79\) 1.30690e8 0.377503 0.188751 0.982025i \(-0.439556\pi\)
0.188751 + 0.982025i \(0.439556\pi\)
\(80\) 0 0
\(81\) 3.85296e8 0.994516
\(82\) − 3.50546e8i − 0.856215i
\(83\) − 4.20134e8i − 0.971709i −0.874040 0.485855i \(-0.838508\pi\)
0.874040 0.485855i \(-0.161492\pi\)
\(84\) −3.68794e6 −0.00808214
\(85\) 0 0
\(86\) −4.40329e8 −0.868030
\(87\) − 9.44828e6i − 0.0176814i
\(88\) − 2.21807e8i − 0.394278i
\(89\) 4.69542e8 0.793268 0.396634 0.917977i \(-0.370178\pi\)
0.396634 + 0.917977i \(0.370178\pi\)
\(90\) 0 0
\(91\) 2.71726e8 0.415379
\(92\) − 6.80960e7i − 0.0991006i
\(93\) − 2.78249e7i − 0.0385710i
\(94\) 8.46845e8 1.11874
\(95\) 0 0
\(96\) 6.29146e6 0.00756015
\(97\) − 8.72502e8i − 1.00068i −0.865830 0.500338i \(-0.833209\pi\)
0.865830 0.500338i \(-0.166791\pi\)
\(98\) 9.22368e7i 0.101015i
\(99\) −1.06392e9 −1.11315
\(100\) 0 0
\(101\) −1.20901e9 −1.15607 −0.578037 0.816011i \(-0.696181\pi\)
−0.578037 + 0.816011i \(0.696181\pi\)
\(102\) 601152.i 0 0.000549900i
\(103\) − 6.90563e8i − 0.604555i −0.953220 0.302277i \(-0.902253\pi\)
0.953220 0.302277i \(-0.0977469\pi\)
\(104\) −4.63553e8 −0.388552
\(105\) 0 0
\(106\) −2.59540e8 −0.199677
\(107\) 1.79499e8i 0.132384i 0.997807 + 0.0661921i \(0.0210850\pi\)
−0.997807 + 0.0661921i \(0.978915\pi\)
\(108\) − 6.04109e7i − 0.0427276i
\(109\) 1.60361e9 1.08813 0.544063 0.839044i \(-0.316885\pi\)
0.544063 + 0.839044i \(0.316885\pi\)
\(110\) 0 0
\(111\) 7.16774e7 0.0448156
\(112\) − 1.57352e8i − 0.0944911i
\(113\) − 1.42785e9i − 0.823815i −0.911226 0.411908i \(-0.864863\pi\)
0.911226 0.411908i \(-0.135137\pi\)
\(114\) −2.46795e7 −0.0136856
\(115\) 0 0
\(116\) 4.03127e8 0.206719
\(117\) 2.22349e9i 1.09698i
\(118\) − 2.24815e9i − 1.06747i
\(119\) 1.50351e7 0.00687296
\(120\) 0 0
\(121\) 5.74491e8 0.243640
\(122\) 3.24742e9i 1.32715i
\(123\) 1.31455e8i 0.0517849i
\(124\) 1.18720e9 0.450946
\(125\) 0 0
\(126\) −7.54759e8 −0.266772
\(127\) − 2.35873e9i − 0.804565i −0.915516 0.402282i \(-0.868217\pi\)
0.915516 0.402282i \(-0.131783\pi\)
\(128\) 2.68435e8i 0.0883883i
\(129\) 1.65124e8 0.0524995
\(130\) 0 0
\(131\) 6.01665e8 0.178498 0.0892492 0.996009i \(-0.471553\pi\)
0.0892492 + 0.996009i \(0.471553\pi\)
\(132\) 8.31775e7i 0.0238464i
\(133\) 6.17244e8i 0.171051i
\(134\) 2.45975e9 0.659052
\(135\) 0 0
\(136\) −2.56492e7 −0.00642907
\(137\) − 5.16009e9i − 1.25145i −0.780043 0.625726i \(-0.784803\pi\)
0.780043 0.625726i \(-0.215197\pi\)
\(138\) 2.55360e7i 0.00599373i
\(139\) 7.14356e9 1.62311 0.811556 0.584275i \(-0.198621\pi\)
0.811556 + 0.584275i \(0.198621\pi\)
\(140\) 0 0
\(141\) −3.17567e8 −0.0676627
\(142\) − 4.47449e9i − 0.923520i
\(143\) − 6.12849e9i − 1.22558i
\(144\) 1.28759e9 0.249543
\(145\) 0 0
\(146\) 6.46437e9 1.17744
\(147\) − 3.45888e7i − 0.00610953i
\(148\) 3.05824e9i 0.523954i
\(149\) −9.10424e9 −1.51323 −0.756616 0.653859i \(-0.773149\pi\)
−0.756616 + 0.653859i \(0.773149\pi\)
\(150\) 0 0
\(151\) −2.89432e8 −0.0453054 −0.0226527 0.999743i \(-0.507211\pi\)
−0.0226527 + 0.999743i \(0.507211\pi\)
\(152\) − 1.05299e9i − 0.160003i
\(153\) 1.23030e8i 0.0181509i
\(154\) 2.08030e9 0.298046
\(155\) 0 0
\(156\) 1.73832e8 0.0235001
\(157\) 1.39068e10i 1.82675i 0.407124 + 0.913373i \(0.366532\pi\)
−0.407124 + 0.913373i \(0.633468\pi\)
\(158\) − 2.09104e9i − 0.266935i
\(159\) 9.73273e7 0.0120767
\(160\) 0 0
\(161\) 6.38666e8 0.0749130
\(162\) − 6.16474e9i − 0.703229i
\(163\) − 1.66232e10i − 1.84447i −0.386629 0.922235i \(-0.626361\pi\)
0.386629 0.922235i \(-0.373639\pi\)
\(164\) −5.60874e9 −0.605435
\(165\) 0 0
\(166\) −6.72214e9 −0.687102
\(167\) − 1.58019e10i − 1.57212i −0.618149 0.786061i \(-0.712117\pi\)
0.618149 0.786061i \(-0.287883\pi\)
\(168\) 5.90070e7i 0.00571494i
\(169\) −2.20340e9 −0.207780
\(170\) 0 0
\(171\) −5.05081e9 −0.451730
\(172\) 7.04527e9i 0.613790i
\(173\) − 3.23125e9i − 0.274260i −0.990553 0.137130i \(-0.956212\pi\)
0.990553 0.137130i \(-0.0437879\pi\)
\(174\) −1.51173e8 −0.0125026
\(175\) 0 0
\(176\) −3.54891e9 −0.278797
\(177\) 8.43058e8i 0.0645621i
\(178\) − 7.51268e9i − 0.560925i
\(179\) 2.41408e10 1.75757 0.878785 0.477218i \(-0.158355\pi\)
0.878785 + 0.477218i \(0.158355\pi\)
\(180\) 0 0
\(181\) −3.89332e9 −0.269629 −0.134814 0.990871i \(-0.543044\pi\)
−0.134814 + 0.990871i \(0.543044\pi\)
\(182\) − 4.34762e9i − 0.293718i
\(183\) − 1.21778e9i − 0.0802674i
\(184\) −1.08954e9 −0.0700747
\(185\) 0 0
\(186\) −4.45198e8 −0.0272738
\(187\) − 3.39100e8i − 0.0202787i
\(188\) − 1.35495e10i − 0.791068i
\(189\) 5.66588e8 0.0322990
\(190\) 0 0
\(191\) −2.58988e10 −1.40809 −0.704043 0.710157i \(-0.748624\pi\)
−0.704043 + 0.710157i \(0.748624\pi\)
\(192\) − 1.00663e8i − 0.00534584i
\(193\) − 1.59367e10i − 0.826783i −0.910553 0.413391i \(-0.864344\pi\)
0.910553 0.413391i \(-0.135656\pi\)
\(194\) −1.39600e10 −0.707585
\(195\) 0 0
\(196\) 1.47579e9 0.0714286
\(197\) − 3.34685e9i − 0.158321i −0.996862 0.0791604i \(-0.974776\pi\)
0.996862 0.0791604i \(-0.0252239\pi\)
\(198\) 1.70228e10i 0.787114i
\(199\) 1.34261e10 0.606891 0.303445 0.952849i \(-0.401863\pi\)
0.303445 + 0.952849i \(0.401863\pi\)
\(200\) 0 0
\(201\) −9.22407e8 −0.0398603
\(202\) 1.93442e10i 0.817467i
\(203\) 3.78089e9i 0.156265i
\(204\) 9.61843e6 0.000388838 0
\(205\) 0 0
\(206\) −1.10490e10 −0.427485
\(207\) 5.22610e9i 0.197839i
\(208\) 7.41684e9i 0.274748i
\(209\) 1.39213e10 0.504686
\(210\) 0 0
\(211\) 3.01702e10 1.04787 0.523935 0.851759i \(-0.324464\pi\)
0.523935 + 0.851759i \(0.324464\pi\)
\(212\) 4.15263e9i 0.141193i
\(213\) 1.67794e9i 0.0558556i
\(214\) 2.87199e9 0.0936097
\(215\) 0 0
\(216\) −9.66574e8 −0.0302130
\(217\) 1.11346e10i 0.340883i
\(218\) − 2.56577e10i − 0.769421i
\(219\) −2.42414e9 −0.0712129
\(220\) 0 0
\(221\) −7.08683e8 −0.0199842
\(222\) − 1.14684e9i − 0.0316894i
\(223\) − 5.35030e10i − 1.44879i −0.689384 0.724396i \(-0.742119\pi\)
0.689384 0.724396i \(-0.257881\pi\)
\(224\) −2.51763e9 −0.0668153
\(225\) 0 0
\(226\) −2.28456e10 −0.582525
\(227\) − 4.02704e10i − 1.00663i −0.864103 0.503315i \(-0.832114\pi\)
0.864103 0.503315i \(-0.167886\pi\)
\(228\) 3.94872e8i 0.00967719i
\(229\) −1.90247e10 −0.457150 −0.228575 0.973526i \(-0.573407\pi\)
−0.228575 + 0.973526i \(0.573407\pi\)
\(230\) 0 0
\(231\) −7.80114e8 −0.0180262
\(232\) − 6.45003e9i − 0.146173i
\(233\) 3.67748e10i 0.817426i 0.912663 + 0.408713i \(0.134022\pi\)
−0.912663 + 0.408713i \(0.865978\pi\)
\(234\) 3.55758e10 0.775682
\(235\) 0 0
\(236\) −3.59705e10 −0.754818
\(237\) 7.84139e8i 0.0161445i
\(238\) − 2.40561e8i − 0.00485992i
\(239\) −6.56110e9 −0.130073 −0.0650363 0.997883i \(-0.520716\pi\)
−0.0650363 + 0.997883i \(0.520716\pi\)
\(240\) 0 0
\(241\) −8.96818e10 −1.71249 −0.856244 0.516572i \(-0.827208\pi\)
−0.856244 + 0.516572i \(0.827208\pi\)
\(242\) − 9.19186e9i − 0.172280i
\(243\) 6.95657e9i 0.127987i
\(244\) 5.19587e10 0.938434
\(245\) 0 0
\(246\) 2.10328e9 0.0366175
\(247\) − 2.90940e10i − 0.497356i
\(248\) − 1.89951e10i − 0.318867i
\(249\) 2.52080e9 0.0415568
\(250\) 0 0
\(251\) 5.33703e9 0.0848727 0.0424363 0.999099i \(-0.486488\pi\)
0.0424363 + 0.999099i \(0.486488\pi\)
\(252\) 1.20761e10i 0.188637i
\(253\) − 1.44044e10i − 0.221031i
\(254\) −3.77396e10 −0.568913
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) − 8.35575e10i − 1.19478i −0.801952 0.597388i \(-0.796205\pi\)
0.801952 0.597388i \(-0.203795\pi\)
\(258\) − 2.64198e9i − 0.0371228i
\(259\) −2.86829e10 −0.396072
\(260\) 0 0
\(261\) −3.09384e10 −0.412682
\(262\) − 9.62665e9i − 0.126217i
\(263\) − 1.08635e11i − 1.40013i −0.714079 0.700065i \(-0.753154\pi\)
0.714079 0.700065i \(-0.246846\pi\)
\(264\) 1.33084e9 0.0168620
\(265\) 0 0
\(266\) 9.87591e9 0.120951
\(267\) 2.81725e9i 0.0339254i
\(268\) − 3.93561e10i − 0.466020i
\(269\) −1.41401e11 −1.64652 −0.823258 0.567668i \(-0.807846\pi\)
−0.823258 + 0.567668i \(0.807846\pi\)
\(270\) 0 0
\(271\) −9.08353e10 −1.02304 −0.511520 0.859271i \(-0.670917\pi\)
−0.511520 + 0.859271i \(0.670917\pi\)
\(272\) 4.10386e8i 0.00454604i
\(273\) 1.63036e9i 0.0177644i
\(274\) −8.25614e10 −0.884911
\(275\) 0 0
\(276\) 4.08576e8 0.00423821
\(277\) − 2.65075e10i − 0.270527i −0.990810 0.135263i \(-0.956812\pi\)
0.990810 0.135263i \(-0.0431880\pi\)
\(278\) − 1.14297e11i − 1.14771i
\(279\) −9.11126e10 −0.900243
\(280\) 0 0
\(281\) −1.86968e11 −1.78891 −0.894455 0.447158i \(-0.852436\pi\)
−0.894455 + 0.447158i \(0.852436\pi\)
\(282\) 5.08107e9i 0.0478448i
\(283\) 5.33413e9i 0.0494338i 0.999694 + 0.0247169i \(0.00786844\pi\)
−0.999694 + 0.0247169i \(0.992132\pi\)
\(284\) −7.15919e10 −0.653028
\(285\) 0 0
\(286\) −9.80558e10 −0.866615
\(287\) − 5.26038e10i − 0.457666i
\(288\) − 2.06014e10i − 0.176453i
\(289\) 1.18549e11 0.999669
\(290\) 0 0
\(291\) 5.23501e9 0.0427956
\(292\) − 1.03430e11i − 0.832574i
\(293\) − 7.65433e10i − 0.606741i −0.952873 0.303370i \(-0.901888\pi\)
0.952873 0.303370i \(-0.0981119\pi\)
\(294\) −5.53421e8 −0.00432009
\(295\) 0 0
\(296\) 4.89318e10 0.370492
\(297\) − 1.27788e10i − 0.0952984i
\(298\) 1.45668e11i 1.07002i
\(299\) −3.01038e10 −0.217821
\(300\) 0 0
\(301\) −6.60769e10 −0.463982
\(302\) 4.63091e9i 0.0320358i
\(303\) − 7.25409e9i − 0.0494414i
\(304\) −1.68479e10 −0.113139
\(305\) 0 0
\(306\) 1.96847e9 0.0128346
\(307\) − 7.51944e10i − 0.483128i −0.970385 0.241564i \(-0.922340\pi\)
0.970385 0.241564i \(-0.0776605\pi\)
\(308\) − 3.32849e10i − 0.210750i
\(309\) 4.14338e9 0.0258548
\(310\) 0 0
\(311\) 2.15134e11 1.30403 0.652014 0.758207i \(-0.273924\pi\)
0.652014 + 0.758207i \(0.273924\pi\)
\(312\) − 2.78132e9i − 0.0166171i
\(313\) − 9.59075e10i − 0.564811i −0.959295 0.282405i \(-0.908868\pi\)
0.959295 0.282405i \(-0.0911323\pi\)
\(314\) 2.22508e11 1.29170
\(315\) 0 0
\(316\) −3.34566e10 −0.188751
\(317\) 1.70586e11i 0.948807i 0.880308 + 0.474403i \(0.157336\pi\)
−0.880308 + 0.474403i \(0.842664\pi\)
\(318\) − 1.55724e9i − 0.00853951i
\(319\) 8.52739e10 0.461061
\(320\) 0 0
\(321\) −1.07700e9 −0.00566163
\(322\) − 1.02187e10i − 0.0529715i
\(323\) − 1.60982e9i − 0.00822937i
\(324\) −9.86358e10 −0.497258
\(325\) 0 0
\(326\) −2.65972e11 −1.30424
\(327\) 9.62165e9i 0.0465355i
\(328\) 8.97398e10i 0.428107i
\(329\) 1.27080e11 0.597991
\(330\) 0 0
\(331\) −1.23992e11 −0.567762 −0.283881 0.958859i \(-0.591622\pi\)
−0.283881 + 0.958859i \(0.591622\pi\)
\(332\) 1.07554e11i 0.485855i
\(333\) − 2.34708e11i − 1.04599i
\(334\) −2.52831e11 −1.11166
\(335\) 0 0
\(336\) 9.44112e8 0.00404107
\(337\) − 7.29335e10i − 0.308030i −0.988069 0.154015i \(-0.950780\pi\)
0.988069 0.154015i \(-0.0492204\pi\)
\(338\) 3.52544e10i 0.146923i
\(339\) 8.56710e9 0.0352318
\(340\) 0 0
\(341\) 2.51129e11 1.00578
\(342\) 8.08130e10i 0.319421i
\(343\) 1.38413e10i 0.0539949i
\(344\) 1.12724e11 0.434015
\(345\) 0 0
\(346\) −5.17000e10 −0.193931
\(347\) − 1.55720e11i − 0.576584i −0.957542 0.288292i \(-0.906913\pi\)
0.957542 0.288292i \(-0.0930874\pi\)
\(348\) 2.41876e9i 0.00884069i
\(349\) −1.08728e11 −0.392310 −0.196155 0.980573i \(-0.562846\pi\)
−0.196155 + 0.980573i \(0.562846\pi\)
\(350\) 0 0
\(351\) −2.67063e10 −0.0939144
\(352\) 5.67825e10i 0.197139i
\(353\) − 3.25585e11i − 1.11604i −0.829829 0.558018i \(-0.811562\pi\)
0.829829 0.558018i \(-0.188438\pi\)
\(354\) 1.34889e10 0.0456523
\(355\) 0 0
\(356\) −1.20203e11 −0.396634
\(357\) 9.02104e7i 0 0.000293934i
\(358\) − 3.86252e11i − 1.24279i
\(359\) 2.27550e11 0.723022 0.361511 0.932368i \(-0.382261\pi\)
0.361511 + 0.932368i \(0.382261\pi\)
\(360\) 0 0
\(361\) −2.56599e11 −0.795192
\(362\) 6.22931e10i 0.190656i
\(363\) 3.44695e9i 0.0104197i
\(364\) −6.95618e10 −0.207690
\(365\) 0 0
\(366\) −1.94845e10 −0.0567577
\(367\) − 4.21993e11i − 1.21425i −0.794607 0.607125i \(-0.792323\pi\)
0.794607 0.607125i \(-0.207677\pi\)
\(368\) 1.74326e10i 0.0495503i
\(369\) 4.30449e11 1.20866
\(370\) 0 0
\(371\) −3.89472e10 −0.106732
\(372\) 7.12318e9i 0.0192855i
\(373\) − 3.83283e11i − 1.02525i −0.858613 0.512625i \(-0.828673\pi\)
0.858613 0.512625i \(-0.171327\pi\)
\(374\) −5.42560e9 −0.0143392
\(375\) 0 0
\(376\) −2.16792e11 −0.559370
\(377\) − 1.78214e11i − 0.454365i
\(378\) − 9.06541e9i − 0.0228389i
\(379\) 1.21462e11 0.302386 0.151193 0.988504i \(-0.451688\pi\)
0.151193 + 0.988504i \(0.451688\pi\)
\(380\) 0 0
\(381\) 1.41524e10 0.0344086
\(382\) 4.14381e11i 0.995667i
\(383\) 3.97721e11i 0.944461i 0.881475 + 0.472230i \(0.156551\pi\)
−0.881475 + 0.472230i \(0.843449\pi\)
\(384\) −1.61061e9 −0.00378008
\(385\) 0 0
\(386\) −2.54988e11 −0.584624
\(387\) − 5.40697e11i − 1.22533i
\(388\) 2.23360e11i 0.500338i
\(389\) −6.75462e10 −0.149564 −0.0747821 0.997200i \(-0.523826\pi\)
−0.0747821 + 0.997200i \(0.523826\pi\)
\(390\) 0 0
\(391\) −1.66569e9 −0.00360412
\(392\) − 2.36126e10i − 0.0505076i
\(393\) 3.60999e9i 0.00763378i
\(394\) −5.35495e10 −0.111950
\(395\) 0 0
\(396\) 2.72365e11 0.556573
\(397\) 1.24656e11i 0.251857i 0.992039 + 0.125929i \(0.0401911\pi\)
−0.992039 + 0.125929i \(0.959809\pi\)
\(398\) − 2.14817e11i − 0.429136i
\(399\) −3.70347e9 −0.00731527
\(400\) 0 0
\(401\) 3.51196e11 0.678265 0.339133 0.940739i \(-0.389866\pi\)
0.339133 + 0.940739i \(0.389866\pi\)
\(402\) 1.47585e10i 0.0281855i
\(403\) − 5.24833e11i − 0.991171i
\(404\) 3.09508e11 0.578037
\(405\) 0 0
\(406\) 6.04942e10 0.110496
\(407\) 6.46913e11i 1.16861i
\(408\) − 1.53895e8i 0 0.000274950i
\(409\) 3.81956e10 0.0674930 0.0337465 0.999430i \(-0.489256\pi\)
0.0337465 + 0.999430i \(0.489256\pi\)
\(410\) 0 0
\(411\) 3.09605e10 0.0535205
\(412\) 1.76784e11i 0.302277i
\(413\) − 3.37364e11i − 0.570588i
\(414\) 8.36176e10 0.139893
\(415\) 0 0
\(416\) 1.18669e11 0.194276
\(417\) 4.28614e10i 0.0694151i
\(418\) − 2.22741e11i − 0.356867i
\(419\) −2.15268e11 −0.341205 −0.170603 0.985340i \(-0.554571\pi\)
−0.170603 + 0.985340i \(0.554571\pi\)
\(420\) 0 0
\(421\) 1.19933e12 1.86066 0.930332 0.366718i \(-0.119519\pi\)
0.930332 + 0.366718i \(0.119519\pi\)
\(422\) − 4.82723e11i − 0.740955i
\(423\) 1.03987e12i 1.57924i
\(424\) 6.64421e10 0.0998383
\(425\) 0 0
\(426\) 2.68470e10 0.0394959
\(427\) 4.87316e11i 0.709390i
\(428\) − 4.59518e10i − 0.0661921i
\(429\) 3.67709e10 0.0524140
\(430\) 0 0
\(431\) 7.91117e11 1.10432 0.552158 0.833740i \(-0.313805\pi\)
0.552158 + 0.833740i \(0.313805\pi\)
\(432\) 1.54652e10i 0.0213638i
\(433\) 1.15451e12i 1.57834i 0.614174 + 0.789170i \(0.289489\pi\)
−0.614174 + 0.789170i \(0.710511\pi\)
\(434\) 1.78154e11 0.241041
\(435\) 0 0
\(436\) −4.10524e11 −0.544063
\(437\) − 6.83827e10i − 0.0896974i
\(438\) 3.87862e10i 0.0503551i
\(439\) 2.12728e11 0.273360 0.136680 0.990615i \(-0.456357\pi\)
0.136680 + 0.990615i \(0.456357\pi\)
\(440\) 0 0
\(441\) −1.13261e11 −0.142596
\(442\) 1.13389e10i 0.0141310i
\(443\) 6.48300e10i 0.0799759i 0.999200 + 0.0399880i \(0.0127320\pi\)
−0.999200 + 0.0399880i \(0.987268\pi\)
\(444\) −1.83494e10 −0.0224078
\(445\) 0 0
\(446\) −8.56047e11 −1.02445
\(447\) − 5.46255e10i − 0.0647159i
\(448\) 4.02821e10i 0.0472456i
\(449\) 1.08031e12 1.25441 0.627207 0.778853i \(-0.284198\pi\)
0.627207 + 0.778853i \(0.284198\pi\)
\(450\) 0 0
\(451\) −1.18642e12 −1.35035
\(452\) 3.65530e11i 0.411908i
\(453\) − 1.73659e9i − 0.00193756i
\(454\) −6.44326e11 −0.711794
\(455\) 0 0
\(456\) 6.31795e9 0.00684281
\(457\) − 6.46725e10i − 0.0693581i −0.999399 0.0346790i \(-0.988959\pi\)
0.999399 0.0346790i \(-0.0110409\pi\)
\(458\) 3.04395e11i 0.323254i
\(459\) −1.47771e9 −0.00155393
\(460\) 0 0
\(461\) 4.29254e11 0.442649 0.221325 0.975200i \(-0.428962\pi\)
0.221325 + 0.975200i \(0.428962\pi\)
\(462\) 1.24818e10i 0.0127464i
\(463\) − 1.61883e12i − 1.63715i −0.574401 0.818574i \(-0.694765\pi\)
0.574401 0.818574i \(-0.305235\pi\)
\(464\) −1.03200e11 −0.103360
\(465\) 0 0
\(466\) 5.88396e11 0.578007
\(467\) 3.27321e11i 0.318455i 0.987242 + 0.159228i \(0.0509003\pi\)
−0.987242 + 0.159228i \(0.949100\pi\)
\(468\) − 5.69214e11i − 0.548490i
\(469\) 3.69117e11 0.352278
\(470\) 0 0
\(471\) −8.34407e10 −0.0781239
\(472\) 5.75527e11i 0.533737i
\(473\) 1.49030e12i 1.36898i
\(474\) 1.25462e10 0.0114159
\(475\) 0 0
\(476\) −3.84898e9 −0.00343648
\(477\) − 3.18698e11i − 0.281869i
\(478\) 1.04978e11i 0.0919752i
\(479\) −2.84811e11 −0.247199 −0.123600 0.992332i \(-0.539444\pi\)
−0.123600 + 0.992332i \(0.539444\pi\)
\(480\) 0 0
\(481\) 1.35198e12 1.15164
\(482\) 1.43491e12i 1.21091i
\(483\) 3.83200e9i 0.00320378i
\(484\) −1.47070e11 −0.121820
\(485\) 0 0
\(486\) 1.11305e11 0.0905007
\(487\) − 7.14776e11i − 0.575824i −0.957657 0.287912i \(-0.907039\pi\)
0.957657 0.287912i \(-0.0929610\pi\)
\(488\) − 8.31339e11i − 0.663573i
\(489\) 9.97395e10 0.0788819
\(490\) 0 0
\(491\) 1.01506e12 0.788176 0.394088 0.919073i \(-0.371061\pi\)
0.394088 + 0.919073i \(0.371061\pi\)
\(492\) − 3.36524e10i − 0.0258925i
\(493\) − 9.86086e9i − 0.00751802i
\(494\) −4.65505e11 −0.351684
\(495\) 0 0
\(496\) −3.03922e11 −0.225473
\(497\) − 6.71454e11i − 0.493642i
\(498\) − 4.03329e10i − 0.0293851i
\(499\) −1.33412e12 −0.963260 −0.481630 0.876375i \(-0.659955\pi\)
−0.481630 + 0.876375i \(0.659955\pi\)
\(500\) 0 0
\(501\) 9.48116e10 0.0672345
\(502\) − 8.53925e10i − 0.0600140i
\(503\) 5.68445e11i 0.395943i 0.980208 + 0.197971i \(0.0634353\pi\)
−0.980208 + 0.197971i \(0.936565\pi\)
\(504\) 1.93218e11 0.133386
\(505\) 0 0
\(506\) −2.30471e11 −0.156293
\(507\) − 1.32204e10i − 0.00888606i
\(508\) 6.03834e11i 0.402282i
\(509\) −3.57173e11 −0.235857 −0.117928 0.993022i \(-0.537625\pi\)
−0.117928 + 0.993022i \(0.537625\pi\)
\(510\) 0 0
\(511\) 9.70059e11 0.629367
\(512\) − 6.87195e10i − 0.0441942i
\(513\) − 6.06653e10i − 0.0386734i
\(514\) −1.33692e12 −0.844834
\(515\) 0 0
\(516\) −4.22716e10 −0.0262498
\(517\) − 2.86615e12i − 1.76438i
\(518\) 4.58927e11i 0.280065i
\(519\) 1.93875e10 0.0117292
\(520\) 0 0
\(521\) 2.17972e12 1.29608 0.648040 0.761606i \(-0.275589\pi\)
0.648040 + 0.761606i \(0.275589\pi\)
\(522\) 4.95014e11i 0.291810i
\(523\) 1.57081e12i 0.918048i 0.888424 + 0.459024i \(0.151801\pi\)
−0.888424 + 0.459024i \(0.848199\pi\)
\(524\) −1.54026e11 −0.0892492
\(525\) 0 0
\(526\) −1.73816e12 −0.990042
\(527\) − 2.90399e10i − 0.0164001i
\(528\) − 2.12934e10i − 0.0119232i
\(529\) 1.73040e12 0.960716
\(530\) 0 0
\(531\) 2.76059e12 1.50687
\(532\) − 1.58015e11i − 0.0855253i
\(533\) 2.47950e12i 1.33074i
\(534\) 4.50761e10 0.0239889
\(535\) 0 0
\(536\) −6.29697e11 −0.329526
\(537\) 1.44845e11i 0.0751655i
\(538\) 2.26241e12i 1.16426i
\(539\) 3.12176e11 0.159312
\(540\) 0 0
\(541\) 2.24544e12 1.12697 0.563486 0.826126i \(-0.309460\pi\)
0.563486 + 0.826126i \(0.309460\pi\)
\(542\) 1.45336e12i 0.723399i
\(543\) − 2.33599e10i − 0.0115311i
\(544\) 6.56618e9 0.00321453
\(545\) 0 0
\(546\) 2.60857e10 0.0125613
\(547\) − 3.86062e11i − 0.184380i −0.995741 0.0921899i \(-0.970613\pi\)
0.995741 0.0921899i \(-0.0293867\pi\)
\(548\) 1.32098e12i 0.625726i
\(549\) −3.98763e12 −1.87344
\(550\) 0 0
\(551\) 4.04824e11 0.187105
\(552\) − 6.53722e9i − 0.00299686i
\(553\) − 3.13786e11i − 0.142683i
\(554\) −4.24120e11 −0.191291
\(555\) 0 0
\(556\) −1.82875e12 −0.811556
\(557\) − 7.95102e11i − 0.350005i −0.984568 0.175003i \(-0.944007\pi\)
0.984568 0.175003i \(-0.0559934\pi\)
\(558\) 1.45780e12i 0.636568i
\(559\) 3.11456e12 1.34910
\(560\) 0 0
\(561\) 2.03460e9 0.000867253 0
\(562\) 2.99149e12i 1.26495i
\(563\) 2.13667e12i 0.896292i 0.893960 + 0.448146i \(0.147916\pi\)
−0.893960 + 0.448146i \(0.852084\pi\)
\(564\) 8.12972e10 0.0338314
\(565\) 0 0
\(566\) 8.53460e10 0.0349550
\(567\) − 9.25096e11i − 0.375892i
\(568\) 1.14547e12i 0.461760i
\(569\) −2.17461e12 −0.869714 −0.434857 0.900500i \(-0.643201\pi\)
−0.434857 + 0.900500i \(0.643201\pi\)
\(570\) 0 0
\(571\) 9.95075e11 0.391736 0.195868 0.980630i \(-0.437248\pi\)
0.195868 + 0.980630i \(0.437248\pi\)
\(572\) 1.56889e12i 0.612790i
\(573\) − 1.55393e11i − 0.0602192i
\(574\) −8.41661e11 −0.323619
\(575\) 0 0
\(576\) −3.29622e11 −0.124771
\(577\) 4.30588e12i 1.61723i 0.588340 + 0.808614i \(0.299782\pi\)
−0.588340 + 0.808614i \(0.700218\pi\)
\(578\) − 1.89678e12i − 0.706873i
\(579\) 9.56204e10 0.0353588
\(580\) 0 0
\(581\) −1.00874e12 −0.367272
\(582\) − 8.37602e10i − 0.0302611i
\(583\) 8.78412e11i 0.314912i
\(584\) −1.65488e12 −0.588719
\(585\) 0 0
\(586\) −1.22469e12 −0.429030
\(587\) 5.05762e12i 1.75822i 0.476615 + 0.879112i \(0.341864\pi\)
−0.476615 + 0.879112i \(0.658136\pi\)
\(588\) 8.85473e9i 0.00305476i
\(589\) 1.19220e12 0.408158
\(590\) 0 0
\(591\) 2.00811e10 0.00677085
\(592\) − 7.82909e11i − 0.261977i
\(593\) 2.80300e12i 0.930844i 0.885089 + 0.465422i \(0.154097\pi\)
−0.885089 + 0.465422i \(0.845903\pi\)
\(594\) −2.04461e11 −0.0673862
\(595\) 0 0
\(596\) 2.33069e12 0.756616
\(597\) 8.05565e10i 0.0259547i
\(598\) 4.81660e11i 0.154023i
\(599\) −3.20907e12 −1.01849 −0.509247 0.860620i \(-0.670076\pi\)
−0.509247 + 0.860620i \(0.670076\pi\)
\(600\) 0 0
\(601\) 7.49502e11 0.234335 0.117168 0.993112i \(-0.462619\pi\)
0.117168 + 0.993112i \(0.462619\pi\)
\(602\) 1.05723e12i 0.328084i
\(603\) 3.02042e12i 0.930336i
\(604\) 7.40946e10 0.0226527
\(605\) 0 0
\(606\) −1.16065e11 −0.0349604
\(607\) − 1.74097e12i − 0.520526i −0.965538 0.260263i \(-0.916191\pi\)
0.965538 0.260263i \(-0.0838092\pi\)
\(608\) 2.69566e11i 0.0800016i
\(609\) −2.26853e10 −0.00668294
\(610\) 0 0
\(611\) −5.98995e12 −1.73875
\(612\) − 3.14956e10i − 0.00907544i
\(613\) 4.03977e12i 1.15554i 0.816200 + 0.577770i \(0.196077\pi\)
−0.816200 + 0.577770i \(0.803923\pi\)
\(614\) −1.20311e12 −0.341623
\(615\) 0 0
\(616\) −5.32558e11 −0.149023
\(617\) − 2.93367e12i − 0.814945i −0.913218 0.407472i \(-0.866410\pi\)
0.913218 0.407472i \(-0.133590\pi\)
\(618\) − 6.62940e10i − 0.0182821i
\(619\) 5.77691e12 1.58157 0.790784 0.612095i \(-0.209673\pi\)
0.790784 + 0.612095i \(0.209673\pi\)
\(620\) 0 0
\(621\) −6.27707e10 −0.0169373
\(622\) − 3.44214e12i − 0.922088i
\(623\) − 1.12737e12i − 0.299827i
\(624\) −4.45010e10 −0.0117500
\(625\) 0 0
\(626\) −1.53452e12 −0.399381
\(627\) 8.35277e10i 0.0215837i
\(628\) − 3.56014e12i − 0.913373i
\(629\) 7.48073e10 0.0190553
\(630\) 0 0
\(631\) 3.99985e12 1.00441 0.502206 0.864748i \(-0.332522\pi\)
0.502206 + 0.864748i \(0.332522\pi\)
\(632\) 5.35305e11i 0.133467i
\(633\) 1.81021e11i 0.0448139i
\(634\) 2.72938e12 0.670908
\(635\) 0 0
\(636\) −2.49158e10 −0.00603834
\(637\) − 6.52414e11i − 0.156999i
\(638\) − 1.36438e12i − 0.326019i
\(639\) 5.49440e12 1.30367
\(640\) 0 0
\(641\) 4.68328e12 1.09569 0.547846 0.836579i \(-0.315448\pi\)
0.547846 + 0.836579i \(0.315448\pi\)
\(642\) 1.72319e10i 0.00400338i
\(643\) 1.54877e12i 0.357304i 0.983912 + 0.178652i \(0.0571736\pi\)
−0.983912 + 0.178652i \(0.942826\pi\)
\(644\) −1.63498e11 −0.0374565
\(645\) 0 0
\(646\) −2.57572e10 −0.00581904
\(647\) − 8.14493e12i − 1.82733i −0.406463 0.913667i \(-0.633238\pi\)
0.406463 0.913667i \(-0.366762\pi\)
\(648\) 1.57817e12i 0.351615i
\(649\) −7.60888e12 −1.68352
\(650\) 0 0
\(651\) −6.68076e10 −0.0145785
\(652\) 4.25555e12i 0.922235i
\(653\) 2.88925e12i 0.621836i 0.950437 + 0.310918i \(0.100636\pi\)
−0.950437 + 0.310918i \(0.899364\pi\)
\(654\) 1.53946e11 0.0329056
\(655\) 0 0
\(656\) 1.43584e12 0.302718
\(657\) 7.93784e12i 1.66210i
\(658\) − 2.03328e12i − 0.422844i
\(659\) −5.20255e12 −1.07456 −0.537281 0.843403i \(-0.680549\pi\)
−0.537281 + 0.843403i \(0.680549\pi\)
\(660\) 0 0
\(661\) 2.88973e12 0.588777 0.294388 0.955686i \(-0.404884\pi\)
0.294388 + 0.955686i \(0.404884\pi\)
\(662\) 1.98387e12i 0.401469i
\(663\) − 4.25210e9i 0 0.000854658i
\(664\) 1.72087e12 0.343551
\(665\) 0 0
\(666\) −3.75532e12 −0.739628
\(667\) − 4.18874e11i − 0.0819440i
\(668\) 4.04530e12i 0.786061i
\(669\) 3.21018e11 0.0619600
\(670\) 0 0
\(671\) 1.09909e13 2.09306
\(672\) − 1.51058e10i − 0.00285747i
\(673\) − 9.46362e12i − 1.77824i −0.457677 0.889119i \(-0.651318\pi\)
0.457677 0.889119i \(-0.348682\pi\)
\(674\) −1.16694e12 −0.217810
\(675\) 0 0
\(676\) 5.64071e11 0.103890
\(677\) − 6.52268e12i − 1.19338i −0.802474 0.596688i \(-0.796483\pi\)
0.802474 0.596688i \(-0.203517\pi\)
\(678\) − 1.37074e11i − 0.0249127i
\(679\) −2.09488e12 −0.378220
\(680\) 0 0
\(681\) 2.41622e11 0.0430502
\(682\) − 4.01806e12i − 0.711193i
\(683\) − 5.37240e12i − 0.944660i −0.881422 0.472330i \(-0.843413\pi\)
0.881422 0.472330i \(-0.156587\pi\)
\(684\) 1.29301e12 0.225865
\(685\) 0 0
\(686\) 2.21461e11 0.0381802
\(687\) − 1.14148e11i − 0.0195508i
\(688\) − 1.80359e12i − 0.306895i
\(689\) 1.83579e12 0.310339
\(690\) 0 0
\(691\) 2.01563e12 0.336325 0.168163 0.985759i \(-0.446217\pi\)
0.168163 + 0.985759i \(0.446217\pi\)
\(692\) 8.27200e11i 0.137130i
\(693\) 2.55448e12i 0.420730i
\(694\) −2.49153e12 −0.407707
\(695\) 0 0
\(696\) 3.87002e10 0.00625131
\(697\) 1.37195e11i 0.0220186i
\(698\) 1.73966e12i 0.277405i
\(699\) −2.20649e11 −0.0349586
\(700\) 0 0
\(701\) −1.06523e13 −1.66614 −0.833068 0.553171i \(-0.813418\pi\)
−0.833068 + 0.553171i \(0.813418\pi\)
\(702\) 4.27301e11i 0.0664075i
\(703\) 3.07111e12i 0.474239i
\(704\) 9.08520e11 0.139398
\(705\) 0 0
\(706\) −5.20936e12 −0.789157
\(707\) 2.90284e12i 0.436955i
\(708\) − 2.15823e11i − 0.0322810i
\(709\) −3.46187e12 −0.514520 −0.257260 0.966342i \(-0.582820\pi\)
−0.257260 + 0.966342i \(0.582820\pi\)
\(710\) 0 0
\(711\) 2.56766e12 0.376812
\(712\) 1.92325e12i 0.280462i
\(713\) − 1.23357e12i − 0.178756i
\(714\) 1.44337e9 0.000207843 0
\(715\) 0 0
\(716\) −6.18004e12 −0.878785
\(717\) − 3.93666e10i − 0.00556277i
\(718\) − 3.64080e12i − 0.511253i
\(719\) 9.62025e12 1.34248 0.671238 0.741242i \(-0.265763\pi\)
0.671238 + 0.741242i \(0.265763\pi\)
\(720\) 0 0
\(721\) −1.65804e12 −0.228500
\(722\) 4.10558e12i 0.562285i
\(723\) − 5.38091e11i − 0.0732374i
\(724\) 9.96689e11 0.134814
\(725\) 0 0
\(726\) 5.51512e10 0.00736784
\(727\) − 3.13479e12i − 0.416202i −0.978107 0.208101i \(-0.933272\pi\)
0.978107 0.208101i \(-0.0667282\pi\)
\(728\) 1.11299e12i 0.146859i
\(729\) 7.54204e12 0.989043
\(730\) 0 0
\(731\) 1.72334e11 0.0223225
\(732\) 3.11752e11i 0.0401337i
\(733\) − 6.47775e12i − 0.828812i −0.910092 0.414406i \(-0.863989\pi\)
0.910092 0.414406i \(-0.136011\pi\)
\(734\) −6.75189e12 −0.858604
\(735\) 0 0
\(736\) 2.78921e11 0.0350374
\(737\) − 8.32503e12i − 1.03940i
\(738\) − 6.88718e12i − 0.854649i
\(739\) 1.12510e12 0.138769 0.0693843 0.997590i \(-0.477897\pi\)
0.0693843 + 0.997590i \(0.477897\pi\)
\(740\) 0 0
\(741\) 1.74564e11 0.0212703
\(742\) 6.23154e11i 0.0754707i
\(743\) − 1.65266e12i − 0.198945i −0.995040 0.0994725i \(-0.968284\pi\)
0.995040 0.0994725i \(-0.0317155\pi\)
\(744\) 1.13971e11 0.0136369
\(745\) 0 0
\(746\) −6.13253e12 −0.724961
\(747\) − 8.25437e12i − 0.969932i
\(748\) 8.68096e10i 0.0101394i
\(749\) 4.30978e11 0.0500365
\(750\) 0 0
\(751\) 6.03299e12 0.692074 0.346037 0.938221i \(-0.387527\pi\)
0.346037 + 0.938221i \(0.387527\pi\)
\(752\) 3.46868e12i 0.395534i
\(753\) 3.20222e10i 0.00362972i
\(754\) −2.85142e12 −0.321284
\(755\) 0 0
\(756\) −1.45047e11 −0.0161495
\(757\) − 8.02798e12i − 0.888535i −0.895894 0.444268i \(-0.853464\pi\)
0.895894 0.444268i \(-0.146536\pi\)
\(758\) − 1.94338e12i − 0.213820i
\(759\) 8.64266e10 0.00945278
\(760\) 0 0
\(761\) −6.51923e12 −0.704637 −0.352318 0.935880i \(-0.614607\pi\)
−0.352318 + 0.935880i \(0.614607\pi\)
\(762\) − 2.26438e11i − 0.0243305i
\(763\) − 3.85026e12i − 0.411273i
\(764\) 6.63009e12 0.704043
\(765\) 0 0
\(766\) 6.36354e12 0.667835
\(767\) 1.59018e13i 1.65907i
\(768\) 2.57698e10i 0.00267292i
\(769\) 1.34250e13 1.38435 0.692175 0.721730i \(-0.256653\pi\)
0.692175 + 0.721730i \(0.256653\pi\)
\(770\) 0 0
\(771\) 5.01345e11 0.0510966
\(772\) 4.07980e12i 0.413391i
\(773\) − 7.85934e12i − 0.791733i −0.918308 0.395866i \(-0.870444\pi\)
0.918308 0.395866i \(-0.129556\pi\)
\(774\) −8.65115e12 −0.866442
\(775\) 0 0
\(776\) 3.57377e12 0.353792
\(777\) − 1.72098e11i − 0.0169387i
\(778\) 1.08074e12i 0.105758i
\(779\) −5.63235e12 −0.547988
\(780\) 0 0
\(781\) −1.51439e13 −1.45649
\(782\) 2.66511e10i 0.00254850i
\(783\) − 3.71601e11i − 0.0353304i
\(784\) −3.77802e11 −0.0357143
\(785\) 0 0
\(786\) 5.77599e10 0.00539790
\(787\) − 1.47720e12i − 0.137263i −0.997642 0.0686316i \(-0.978137\pi\)
0.997642 0.0686316i \(-0.0218633\pi\)
\(788\) 8.56793e11i 0.0791604i
\(789\) 6.51809e11 0.0598789
\(790\) 0 0
\(791\) −3.42827e12 −0.311373
\(792\) − 4.35783e12i − 0.393557i
\(793\) − 2.29698e13i − 2.06266i
\(794\) 1.99449e12 0.178090
\(795\) 0 0
\(796\) −3.43708e12 −0.303445
\(797\) − 6.47327e12i − 0.568278i −0.958783 0.284139i \(-0.908292\pi\)
0.958783 0.284139i \(-0.0917078\pi\)
\(798\) 5.92555e10i 0.00517268i
\(799\) −3.31434e11 −0.0287698
\(800\) 0 0
\(801\) 9.22510e12 0.791817
\(802\) − 5.61913e12i − 0.479606i
\(803\) − 2.18786e13i − 1.85695i
\(804\) 2.36136e11 0.0199301
\(805\) 0 0
\(806\) −8.39733e12 −0.700864
\(807\) − 8.48403e11i − 0.0704160i
\(808\) − 4.95212e12i − 0.408734i
\(809\) 1.53631e13 1.26099 0.630493 0.776195i \(-0.282853\pi\)
0.630493 + 0.776195i \(0.282853\pi\)
\(810\) 0 0
\(811\) −1.29056e13 −1.04757 −0.523787 0.851849i \(-0.675481\pi\)
−0.523787 + 0.851849i \(0.675481\pi\)
\(812\) − 9.67907e11i − 0.0781325i
\(813\) − 5.45012e11i − 0.0437520i
\(814\) 1.03506e13 0.826334
\(815\) 0 0
\(816\) −2.46232e9 −0.000194419 0
\(817\) 7.07494e12i 0.555550i
\(818\) − 6.11130e11i − 0.0477247i
\(819\) 5.33860e12 0.414620
\(820\) 0 0
\(821\) 8.93771e11 0.0686566 0.0343283 0.999411i \(-0.489071\pi\)
0.0343283 + 0.999411i \(0.489071\pi\)
\(822\) − 4.95368e11i − 0.0378447i
\(823\) 2.29844e13i 1.74636i 0.487394 + 0.873182i \(0.337947\pi\)
−0.487394 + 0.873182i \(0.662053\pi\)
\(824\) 2.82854e12 0.213742
\(825\) 0 0
\(826\) −5.39782e12 −0.403467
\(827\) − 2.75618e12i − 0.204896i −0.994738 0.102448i \(-0.967333\pi\)
0.994738 0.102448i \(-0.0326675\pi\)
\(828\) − 1.33788e12i − 0.0989194i
\(829\) 3.15925e12 0.232321 0.116160 0.993230i \(-0.462941\pi\)
0.116160 + 0.993230i \(0.462941\pi\)
\(830\) 0 0
\(831\) 1.59045e11 0.0115695
\(832\) − 1.89871e12i − 0.137374i
\(833\) − 3.60992e10i − 0.00259774i
\(834\) 6.85782e11 0.0490839
\(835\) 0 0
\(836\) −3.56385e12 −0.252343
\(837\) − 1.09435e12i − 0.0770714i
\(838\) 3.44428e12i 0.241269i
\(839\) −1.92403e13 −1.34055 −0.670277 0.742111i \(-0.733825\pi\)
−0.670277 + 0.742111i \(0.733825\pi\)
\(840\) 0 0
\(841\) −1.20274e13 −0.829069
\(842\) − 1.91892e13i − 1.31569i
\(843\) − 1.12181e12i − 0.0765058i
\(844\) −7.72357e12 −0.523935
\(845\) 0 0
\(846\) 1.66380e13 1.11669
\(847\) − 1.37935e12i − 0.0920874i
\(848\) − 1.06307e12i − 0.0705963i
\(849\) −3.20048e10 −0.00211412
\(850\) 0 0
\(851\) 3.17770e12 0.207697
\(852\) − 4.29552e11i − 0.0279278i
\(853\) 2.60804e13i 1.68672i 0.537345 + 0.843362i \(0.319427\pi\)
−0.537345 + 0.843362i \(0.680573\pi\)
\(854\) 7.79705e12 0.501614
\(855\) 0 0
\(856\) −7.35229e11 −0.0468049
\(857\) 2.19177e13i 1.38797i 0.719988 + 0.693986i \(0.244147\pi\)
−0.719988 + 0.693986i \(0.755853\pi\)
\(858\) − 5.88335e11i − 0.0370623i
\(859\) 3.55588e12 0.222832 0.111416 0.993774i \(-0.464461\pi\)
0.111416 + 0.993774i \(0.464461\pi\)
\(860\) 0 0
\(861\) 3.15623e11 0.0195729
\(862\) − 1.26579e13i − 0.780869i
\(863\) − 2.22084e13i − 1.36292i −0.731858 0.681458i \(-0.761346\pi\)
0.731858 0.681458i \(-0.238654\pi\)
\(864\) 2.47443e11 0.0151065
\(865\) 0 0
\(866\) 1.84721e13 1.11606
\(867\) 7.11292e11i 0.0427525i
\(868\) − 2.85046e12i − 0.170442i
\(869\) −7.07711e12 −0.420986
\(870\) 0 0
\(871\) −1.73984e13 −1.02430
\(872\) 6.56838e12i 0.384711i
\(873\) − 1.71420e13i − 0.998846i
\(874\) −1.09412e12 −0.0634257
\(875\) 0 0
\(876\) 6.20579e11 0.0356064
\(877\) 3.38004e12i 0.192941i 0.995336 + 0.0964703i \(0.0307553\pi\)
−0.995336 + 0.0964703i \(0.969245\pi\)
\(878\) − 3.40365e12i − 0.193295i
\(879\) 4.59260e11 0.0259483
\(880\) 0 0
\(881\) 5.25103e11 0.0293665 0.0146833 0.999892i \(-0.495326\pi\)
0.0146833 + 0.999892i \(0.495326\pi\)
\(882\) 1.81218e12i 0.100830i
\(883\) − 3.33972e13i − 1.84879i −0.381441 0.924393i \(-0.624572\pi\)
0.381441 0.924393i \(-0.375428\pi\)
\(884\) 1.81423e11 0.00999210
\(885\) 0 0
\(886\) 1.03728e12 0.0565515
\(887\) − 9.61964e12i − 0.521798i −0.965366 0.260899i \(-0.915981\pi\)
0.965366 0.260899i \(-0.0840190\pi\)
\(888\) 2.93591e11i 0.0158447i
\(889\) −5.66331e12 −0.304097
\(890\) 0 0
\(891\) −2.08646e13 −1.10907
\(892\) 1.36968e13i 0.724396i
\(893\) − 1.36066e13i − 0.716007i
\(894\) −8.74007e11 −0.0457611
\(895\) 0 0
\(896\) 6.44514e11 0.0334077
\(897\) − 1.80623e11i − 0.00931549i
\(898\) − 1.72850e13i − 0.887004i
\(899\) 7.30271e12 0.372877
\(900\) 0 0
\(901\) 1.01577e11 0.00513494
\(902\) 1.89828e13i 0.954839i
\(903\) − 3.96462e11i − 0.0198430i
\(904\) 5.84848e12 0.291263
\(905\) 0 0
\(906\) −2.77855e10 −0.00137006
\(907\) − 2.08341e13i − 1.02221i −0.859517 0.511107i \(-0.829236\pi\)
0.859517 0.511107i \(-0.170764\pi\)
\(908\) 1.03092e13i 0.503315i
\(909\) −2.37535e13 −1.15396
\(910\) 0 0
\(911\) −1.33789e13 −0.643560 −0.321780 0.946814i \(-0.604281\pi\)
−0.321780 + 0.946814i \(0.604281\pi\)
\(912\) − 1.01087e11i − 0.00483860i
\(913\) 2.27511e13i 1.08364i
\(914\) −1.03476e12 −0.0490436
\(915\) 0 0
\(916\) 4.87033e12 0.228575
\(917\) − 1.44460e12i − 0.0674660i
\(918\) 2.36433e10i 0.00109879i
\(919\) 1.38352e13 0.639830 0.319915 0.947446i \(-0.396346\pi\)
0.319915 + 0.947446i \(0.396346\pi\)
\(920\) 0 0
\(921\) 4.51166e11 0.0206618
\(922\) − 6.86806e12i − 0.313000i
\(923\) 3.16492e13i 1.43534i
\(924\) 1.99709e11 0.00901310
\(925\) 0 0
\(926\) −2.59013e13 −1.15764
\(927\) − 1.35675e13i − 0.603449i
\(928\) 1.65121e12i 0.0730863i
\(929\) 1.74073e13 0.766761 0.383380 0.923591i \(-0.374760\pi\)
0.383380 + 0.923591i \(0.374760\pi\)
\(930\) 0 0
\(931\) 1.48200e12 0.0646511
\(932\) − 9.41434e12i − 0.408713i
\(933\) 1.29080e12i 0.0557690i
\(934\) 5.23714e12 0.225182
\(935\) 0 0
\(936\) −9.10742e12 −0.387841
\(937\) 1.98361e12i 0.0840676i 0.999116 + 0.0420338i \(0.0133837\pi\)
−0.999116 + 0.0420338i \(0.986616\pi\)
\(938\) − 5.90587e12i − 0.249098i
\(939\) 5.75445e11 0.0241551
\(940\) 0 0
\(941\) −2.33533e13 −0.970946 −0.485473 0.874252i \(-0.661353\pi\)
−0.485473 + 0.874252i \(0.661353\pi\)
\(942\) 1.33505e12i 0.0552419i
\(943\) 5.82783e12i 0.239996i
\(944\) 9.20844e12 0.377409
\(945\) 0 0
\(946\) 2.38447e13 0.968015
\(947\) 3.59882e13i 1.45407i 0.686600 + 0.727035i \(0.259102\pi\)
−0.686600 + 0.727035i \(0.740898\pi\)
\(948\) − 2.00740e11i − 0.00807227i
\(949\) −4.57241e13 −1.82998
\(950\) 0 0
\(951\) −1.02352e12 −0.0405773
\(952\) 6.15836e10i 0.00242996i
\(953\) 1.37662e13i 0.540624i 0.962773 + 0.270312i \(0.0871268\pi\)
−0.962773 + 0.270312i \(0.912873\pi\)
\(954\) −5.09917e12 −0.199311
\(955\) 0 0
\(956\) 1.67964e12 0.0650363
\(957\) 5.11643e11i 0.0197180i
\(958\) 4.55698e12i 0.174796i
\(959\) −1.23894e13 −0.473005
\(960\) 0 0
\(961\) −4.93336e12 −0.186590
\(962\) − 2.16317e13i − 0.814334i
\(963\) 3.52662e12i 0.132142i
\(964\) 2.29585e13 0.856244
\(965\) 0 0
\(966\) 6.13119e10 0.00226542
\(967\) − 3.02718e12i − 0.111332i −0.998449 0.0556659i \(-0.982272\pi\)
0.998449 0.0556659i \(-0.0177282\pi\)
\(968\) 2.35312e12i 0.0861399i
\(969\) 9.65893e9 0.000351943 0
\(970\) 0 0
\(971\) −2.53183e13 −0.914003 −0.457002 0.889466i \(-0.651077\pi\)
−0.457002 + 0.889466i \(0.651077\pi\)
\(972\) − 1.78088e12i − 0.0639937i
\(973\) − 1.71517e13i − 0.613478i
\(974\) −1.14364e13 −0.407169
\(975\) 0 0
\(976\) −1.33014e13 −0.469217
\(977\) 9.90729e12i 0.347880i 0.984756 + 0.173940i \(0.0556499\pi\)
−0.984756 + 0.173940i \(0.944350\pi\)
\(978\) − 1.59583e12i − 0.0557779i
\(979\) −2.54267e13 −0.884641
\(980\) 0 0
\(981\) 3.15061e13 1.08614
\(982\) − 1.62409e13i − 0.557325i
\(983\) − 2.40523e13i − 0.821610i −0.911723 0.410805i \(-0.865248\pi\)
0.911723 0.410805i \(-0.134752\pi\)
\(984\) −5.38439e11 −0.0183087
\(985\) 0 0
\(986\) −1.57774e11 −0.00531604
\(987\) 7.62478e11i 0.0255741i
\(988\) 7.44807e12i 0.248678i
\(989\) 7.32048e12 0.243308
\(990\) 0 0
\(991\) 3.15782e13 1.04005 0.520027 0.854150i \(-0.325922\pi\)
0.520027 + 0.854150i \(0.325922\pi\)
\(992\) 4.86275e12i 0.159434i
\(993\) − 7.43950e11i − 0.0242813i
\(994\) −1.07433e13 −0.349058
\(995\) 0 0
\(996\) −6.45326e11 −0.0207784
\(997\) 9.32241e12i 0.298813i 0.988776 + 0.149407i \(0.0477364\pi\)
−0.988776 + 0.149407i \(0.952264\pi\)
\(998\) 2.13460e13i 0.681128i
\(999\) 2.81907e12 0.0895492
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 350.10.c.b.99.1 2
5.2 odd 4 350.10.a.c.1.1 1
5.3 odd 4 14.10.a.a.1.1 1
5.4 even 2 inner 350.10.c.b.99.2 2
15.8 even 4 126.10.a.e.1.1 1
20.3 even 4 112.10.a.b.1.1 1
35.3 even 12 98.10.c.e.79.1 2
35.13 even 4 98.10.a.a.1.1 1
35.18 odd 12 98.10.c.f.79.1 2
35.23 odd 12 98.10.c.f.67.1 2
35.33 even 12 98.10.c.e.67.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
14.10.a.a.1.1 1 5.3 odd 4
98.10.a.a.1.1 1 35.13 even 4
98.10.c.e.67.1 2 35.33 even 12
98.10.c.e.79.1 2 35.3 even 12
98.10.c.f.67.1 2 35.23 odd 12
98.10.c.f.79.1 2 35.18 odd 12
112.10.a.b.1.1 1 20.3 even 4
126.10.a.e.1.1 1 15.8 even 4
350.10.a.c.1.1 1 5.2 odd 4
350.10.c.b.99.1 2 1.1 even 1 trivial
350.10.c.b.99.2 2 5.4 even 2 inner