Properties

Label 7098.2.a.bj.1.2
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 7098 = 2 \cdot 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7098.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(56.6778153547\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 546)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.73205\) of defining polynomial
Character \(\chi\) \(=\) 7098.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.73205 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -2.73205 q^{10} +0.267949 q^{11} -1.00000 q^{12} -1.00000 q^{14} -2.73205 q^{15} +1.00000 q^{16} +3.73205 q^{17} -1.00000 q^{18} +2.46410 q^{19} +2.73205 q^{20} -1.00000 q^{21} -0.267949 q^{22} -3.46410 q^{23} +1.00000 q^{24} +2.46410 q^{25} -1.00000 q^{27} +1.00000 q^{28} -3.00000 q^{29} +2.73205 q^{30} -9.66025 q^{31} -1.00000 q^{32} -0.267949 q^{33} -3.73205 q^{34} +2.73205 q^{35} +1.00000 q^{36} -4.73205 q^{37} -2.46410 q^{38} -2.73205 q^{40} -7.00000 q^{41} +1.00000 q^{42} -2.73205 q^{43} +0.267949 q^{44} +2.73205 q^{45} +3.46410 q^{46} -2.46410 q^{47} -1.00000 q^{48} +1.00000 q^{49} -2.46410 q^{50} -3.73205 q^{51} -3.53590 q^{53} +1.00000 q^{54} +0.732051 q^{55} -1.00000 q^{56} -2.46410 q^{57} +3.00000 q^{58} -12.9282 q^{59} -2.73205 q^{60} -8.26795 q^{61} +9.66025 q^{62} +1.00000 q^{63} +1.00000 q^{64} +0.267949 q^{66} +0.928203 q^{67} +3.73205 q^{68} +3.46410 q^{69} -2.73205 q^{70} +8.19615 q^{71} -1.00000 q^{72} +13.4641 q^{73} +4.73205 q^{74} -2.46410 q^{75} +2.46410 q^{76} +0.267949 q^{77} -16.8564 q^{79} +2.73205 q^{80} +1.00000 q^{81} +7.00000 q^{82} -11.6603 q^{83} -1.00000 q^{84} +10.1962 q^{85} +2.73205 q^{86} +3.00000 q^{87} -0.267949 q^{88} +0.464102 q^{89} -2.73205 q^{90} -3.46410 q^{92} +9.66025 q^{93} +2.46410 q^{94} +6.73205 q^{95} +1.00000 q^{96} +2.73205 q^{97} -1.00000 q^{98} +0.267949 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} + 2q^{5} + 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} - 2q^{10} + 4q^{11} - 2q^{12} - 2q^{14} - 2q^{15} + 2q^{16} + 4q^{17} - 2q^{18} - 2q^{19} + 2q^{20} - 2q^{21} - 4q^{22} + 2q^{24} - 2q^{25} - 2q^{27} + 2q^{28} - 6q^{29} + 2q^{30} - 2q^{31} - 2q^{32} - 4q^{33} - 4q^{34} + 2q^{35} + 2q^{36} - 6q^{37} + 2q^{38} - 2q^{40} - 14q^{41} + 2q^{42} - 2q^{43} + 4q^{44} + 2q^{45} + 2q^{47} - 2q^{48} + 2q^{49} + 2q^{50} - 4q^{51} - 14q^{53} + 2q^{54} - 2q^{55} - 2q^{56} + 2q^{57} + 6q^{58} - 12q^{59} - 2q^{60} - 20q^{61} + 2q^{62} + 2q^{63} + 2q^{64} + 4q^{66} - 12q^{67} + 4q^{68} - 2q^{70} + 6q^{71} - 2q^{72} + 20q^{73} + 6q^{74} + 2q^{75} - 2q^{76} + 4q^{77} - 6q^{79} + 2q^{80} + 2q^{81} + 14q^{82} - 6q^{83} - 2q^{84} + 10q^{85} + 2q^{86} + 6q^{87} - 4q^{88} - 6q^{89} - 2q^{90} + 2q^{93} - 2q^{94} + 10q^{95} + 2q^{96} + 2q^{97} - 2q^{98} + 4q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 2.73205 1.22181 0.610905 0.791704i \(-0.290806\pi\)
0.610905 + 0.791704i \(0.290806\pi\)
\(6\) 1.00000 0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −2.73205 −0.863950
\(11\) 0.267949 0.0807897 0.0403949 0.999184i \(-0.487138\pi\)
0.0403949 + 0.999184i \(0.487138\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −2.73205 −0.705412
\(16\) 1.00000 0.250000
\(17\) 3.73205 0.905155 0.452578 0.891725i \(-0.350505\pi\)
0.452578 + 0.891725i \(0.350505\pi\)
\(18\) −1.00000 −0.235702
\(19\) 2.46410 0.565304 0.282652 0.959223i \(-0.408786\pi\)
0.282652 + 0.959223i \(0.408786\pi\)
\(20\) 2.73205 0.610905
\(21\) −1.00000 −0.218218
\(22\) −0.267949 −0.0571270
\(23\) −3.46410 −0.722315 −0.361158 0.932505i \(-0.617618\pi\)
−0.361158 + 0.932505i \(0.617618\pi\)
\(24\) 1.00000 0.204124
\(25\) 2.46410 0.492820
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 1.00000 0.188982
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 2.73205 0.498802
\(31\) −9.66025 −1.73503 −0.867516 0.497409i \(-0.834285\pi\)
−0.867516 + 0.497409i \(0.834285\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.267949 −0.0466440
\(34\) −3.73205 −0.640041
\(35\) 2.73205 0.461801
\(36\) 1.00000 0.166667
\(37\) −4.73205 −0.777944 −0.388972 0.921250i \(-0.627170\pi\)
−0.388972 + 0.921250i \(0.627170\pi\)
\(38\) −2.46410 −0.399730
\(39\) 0 0
\(40\) −2.73205 −0.431975
\(41\) −7.00000 −1.09322 −0.546608 0.837389i \(-0.684081\pi\)
−0.546608 + 0.837389i \(0.684081\pi\)
\(42\) 1.00000 0.154303
\(43\) −2.73205 −0.416634 −0.208317 0.978061i \(-0.566799\pi\)
−0.208317 + 0.978061i \(0.566799\pi\)
\(44\) 0.267949 0.0403949
\(45\) 2.73205 0.407270
\(46\) 3.46410 0.510754
\(47\) −2.46410 −0.359426 −0.179713 0.983719i \(-0.557517\pi\)
−0.179713 + 0.983719i \(0.557517\pi\)
\(48\) −1.00000 −0.144338
\(49\) 1.00000 0.142857
\(50\) −2.46410 −0.348477
\(51\) −3.73205 −0.522592
\(52\) 0 0
\(53\) −3.53590 −0.485693 −0.242846 0.970065i \(-0.578081\pi\)
−0.242846 + 0.970065i \(0.578081\pi\)
\(54\) 1.00000 0.136083
\(55\) 0.732051 0.0987097
\(56\) −1.00000 −0.133631
\(57\) −2.46410 −0.326378
\(58\) 3.00000 0.393919
\(59\) −12.9282 −1.68311 −0.841554 0.540172i \(-0.818359\pi\)
−0.841554 + 0.540172i \(0.818359\pi\)
\(60\) −2.73205 −0.352706
\(61\) −8.26795 −1.05860 −0.529301 0.848434i \(-0.677546\pi\)
−0.529301 + 0.848434i \(0.677546\pi\)
\(62\) 9.66025 1.22685
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.267949 0.0329823
\(67\) 0.928203 0.113398 0.0566990 0.998391i \(-0.481942\pi\)
0.0566990 + 0.998391i \(0.481942\pi\)
\(68\) 3.73205 0.452578
\(69\) 3.46410 0.417029
\(70\) −2.73205 −0.326543
\(71\) 8.19615 0.972704 0.486352 0.873763i \(-0.338327\pi\)
0.486352 + 0.873763i \(0.338327\pi\)
\(72\) −1.00000 −0.117851
\(73\) 13.4641 1.57585 0.787927 0.615769i \(-0.211154\pi\)
0.787927 + 0.615769i \(0.211154\pi\)
\(74\) 4.73205 0.550090
\(75\) −2.46410 −0.284530
\(76\) 2.46410 0.282652
\(77\) 0.267949 0.0305356
\(78\) 0 0
\(79\) −16.8564 −1.89649 −0.948247 0.317534i \(-0.897145\pi\)
−0.948247 + 0.317534i \(0.897145\pi\)
\(80\) 2.73205 0.305453
\(81\) 1.00000 0.111111
\(82\) 7.00000 0.773021
\(83\) −11.6603 −1.27988 −0.639940 0.768425i \(-0.721041\pi\)
−0.639940 + 0.768425i \(0.721041\pi\)
\(84\) −1.00000 −0.109109
\(85\) 10.1962 1.10593
\(86\) 2.73205 0.294605
\(87\) 3.00000 0.321634
\(88\) −0.267949 −0.0285635
\(89\) 0.464102 0.0491947 0.0245973 0.999697i \(-0.492170\pi\)
0.0245973 + 0.999697i \(0.492170\pi\)
\(90\) −2.73205 −0.287983
\(91\) 0 0
\(92\) −3.46410 −0.361158
\(93\) 9.66025 1.00172
\(94\) 2.46410 0.254153
\(95\) 6.73205 0.690694
\(96\) 1.00000 0.102062
\(97\) 2.73205 0.277398 0.138699 0.990335i \(-0.455708\pi\)
0.138699 + 0.990335i \(0.455708\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0.267949 0.0269299
\(100\) 2.46410 0.246410
\(101\) −10.0000 −0.995037 −0.497519 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 3.73205 0.369528
\(103\) −1.80385 −0.177738 −0.0888692 0.996043i \(-0.528325\pi\)
−0.0888692 + 0.996043i \(0.528325\pi\)
\(104\) 0 0
\(105\) −2.73205 −0.266621
\(106\) 3.53590 0.343437
\(107\) −8.46410 −0.818256 −0.409128 0.912477i \(-0.634167\pi\)
−0.409128 + 0.912477i \(0.634167\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −13.6603 −1.30842 −0.654208 0.756315i \(-0.726998\pi\)
−0.654208 + 0.756315i \(0.726998\pi\)
\(110\) −0.732051 −0.0697983
\(111\) 4.73205 0.449146
\(112\) 1.00000 0.0944911
\(113\) 0.196152 0.0184525 0.00922623 0.999957i \(-0.497063\pi\)
0.00922623 + 0.999957i \(0.497063\pi\)
\(114\) 2.46410 0.230784
\(115\) −9.46410 −0.882532
\(116\) −3.00000 −0.278543
\(117\) 0 0
\(118\) 12.9282 1.19014
\(119\) 3.73205 0.342117
\(120\) 2.73205 0.249401
\(121\) −10.9282 −0.993473
\(122\) 8.26795 0.748545
\(123\) 7.00000 0.631169
\(124\) −9.66025 −0.867516
\(125\) −6.92820 −0.619677
\(126\) −1.00000 −0.0890871
\(127\) −19.4641 −1.72716 −0.863580 0.504212i \(-0.831783\pi\)
−0.863580 + 0.504212i \(0.831783\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 2.73205 0.240544
\(130\) 0 0
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) −0.267949 −0.0233220
\(133\) 2.46410 0.213665
\(134\) −0.928203 −0.0801845
\(135\) −2.73205 −0.235137
\(136\) −3.73205 −0.320021
\(137\) 8.53590 0.729271 0.364636 0.931150i \(-0.381194\pi\)
0.364636 + 0.931150i \(0.381194\pi\)
\(138\) −3.46410 −0.294884
\(139\) 6.12436 0.519461 0.259731 0.965681i \(-0.416366\pi\)
0.259731 + 0.965681i \(0.416366\pi\)
\(140\) 2.73205 0.230900
\(141\) 2.46410 0.207515
\(142\) −8.19615 −0.687806
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −8.19615 −0.680653
\(146\) −13.4641 −1.11430
\(147\) −1.00000 −0.0824786
\(148\) −4.73205 −0.388972
\(149\) 5.07180 0.415498 0.207749 0.978182i \(-0.433386\pi\)
0.207749 + 0.978182i \(0.433386\pi\)
\(150\) 2.46410 0.201193
\(151\) 2.80385 0.228174 0.114087 0.993471i \(-0.463606\pi\)
0.114087 + 0.993471i \(0.463606\pi\)
\(152\) −2.46410 −0.199865
\(153\) 3.73205 0.301718
\(154\) −0.267949 −0.0215920
\(155\) −26.3923 −2.11988
\(156\) 0 0
\(157\) −8.53590 −0.681239 −0.340619 0.940201i \(-0.610637\pi\)
−0.340619 + 0.940201i \(0.610637\pi\)
\(158\) 16.8564 1.34102
\(159\) 3.53590 0.280415
\(160\) −2.73205 −0.215988
\(161\) −3.46410 −0.273009
\(162\) −1.00000 −0.0785674
\(163\) −18.1962 −1.42523 −0.712616 0.701554i \(-0.752490\pi\)
−0.712616 + 0.701554i \(0.752490\pi\)
\(164\) −7.00000 −0.546608
\(165\) −0.732051 −0.0569901
\(166\) 11.6603 0.905011
\(167\) 21.8564 1.69130 0.845650 0.533738i \(-0.179213\pi\)
0.845650 + 0.533738i \(0.179213\pi\)
\(168\) 1.00000 0.0771517
\(169\) 0 0
\(170\) −10.1962 −0.782009
\(171\) 2.46410 0.188435
\(172\) −2.73205 −0.208317
\(173\) 17.2679 1.31286 0.656429 0.754388i \(-0.272066\pi\)
0.656429 + 0.754388i \(0.272066\pi\)
\(174\) −3.00000 −0.227429
\(175\) 2.46410 0.186269
\(176\) 0.267949 0.0201974
\(177\) 12.9282 0.971743
\(178\) −0.464102 −0.0347859
\(179\) 1.60770 0.120165 0.0600824 0.998193i \(-0.480864\pi\)
0.0600824 + 0.998193i \(0.480864\pi\)
\(180\) 2.73205 0.203635
\(181\) −9.19615 −0.683545 −0.341772 0.939783i \(-0.611027\pi\)
−0.341772 + 0.939783i \(0.611027\pi\)
\(182\) 0 0
\(183\) 8.26795 0.611184
\(184\) 3.46410 0.255377
\(185\) −12.9282 −0.950500
\(186\) −9.66025 −0.708324
\(187\) 1.00000 0.0731272
\(188\) −2.46410 −0.179713
\(189\) −1.00000 −0.0727393
\(190\) −6.73205 −0.488394
\(191\) 6.19615 0.448338 0.224169 0.974550i \(-0.428033\pi\)
0.224169 + 0.974550i \(0.428033\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 6.80385 0.489752 0.244876 0.969554i \(-0.421253\pi\)
0.244876 + 0.969554i \(0.421253\pi\)
\(194\) −2.73205 −0.196150
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) 5.73205 0.408392 0.204196 0.978930i \(-0.434542\pi\)
0.204196 + 0.978930i \(0.434542\pi\)
\(198\) −0.267949 −0.0190423
\(199\) 6.19615 0.439234 0.219617 0.975586i \(-0.429519\pi\)
0.219617 + 0.975586i \(0.429519\pi\)
\(200\) −2.46410 −0.174238
\(201\) −0.928203 −0.0654704
\(202\) 10.0000 0.703598
\(203\) −3.00000 −0.210559
\(204\) −3.73205 −0.261296
\(205\) −19.1244 −1.33570
\(206\) 1.80385 0.125680
\(207\) −3.46410 −0.240772
\(208\) 0 0
\(209\) 0.660254 0.0456707
\(210\) 2.73205 0.188529
\(211\) 11.8564 0.816229 0.408114 0.912931i \(-0.366186\pi\)
0.408114 + 0.912931i \(0.366186\pi\)
\(212\) −3.53590 −0.242846
\(213\) −8.19615 −0.561591
\(214\) 8.46410 0.578594
\(215\) −7.46410 −0.509048
\(216\) 1.00000 0.0680414
\(217\) −9.66025 −0.655781
\(218\) 13.6603 0.925189
\(219\) −13.4641 −0.909820
\(220\) 0.732051 0.0493549
\(221\) 0 0
\(222\) −4.73205 −0.317594
\(223\) −12.3923 −0.829850 −0.414925 0.909856i \(-0.636192\pi\)
−0.414925 + 0.909856i \(0.636192\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.46410 0.164273
\(226\) −0.196152 −0.0130479
\(227\) −15.4641 −1.02639 −0.513194 0.858272i \(-0.671538\pi\)
−0.513194 + 0.858272i \(0.671538\pi\)
\(228\) −2.46410 −0.163189
\(229\) 24.3205 1.60714 0.803572 0.595207i \(-0.202930\pi\)
0.803572 + 0.595207i \(0.202930\pi\)
\(230\) 9.46410 0.624044
\(231\) −0.267949 −0.0176298
\(232\) 3.00000 0.196960
\(233\) −8.19615 −0.536948 −0.268474 0.963287i \(-0.586519\pi\)
−0.268474 + 0.963287i \(0.586519\pi\)
\(234\) 0 0
\(235\) −6.73205 −0.439151
\(236\) −12.9282 −0.841554
\(237\) 16.8564 1.09494
\(238\) −3.73205 −0.241913
\(239\) 24.1962 1.56512 0.782559 0.622576i \(-0.213914\pi\)
0.782559 + 0.622576i \(0.213914\pi\)
\(240\) −2.73205 −0.176353
\(241\) 28.7846 1.85418 0.927090 0.374839i \(-0.122302\pi\)
0.927090 + 0.374839i \(0.122302\pi\)
\(242\) 10.9282 0.702492
\(243\) −1.00000 −0.0641500
\(244\) −8.26795 −0.529301
\(245\) 2.73205 0.174544
\(246\) −7.00000 −0.446304
\(247\) 0 0
\(248\) 9.66025 0.613427
\(249\) 11.6603 0.738939
\(250\) 6.92820 0.438178
\(251\) 7.80385 0.492574 0.246287 0.969197i \(-0.420789\pi\)
0.246287 + 0.969197i \(0.420789\pi\)
\(252\) 1.00000 0.0629941
\(253\) −0.928203 −0.0583556
\(254\) 19.4641 1.22129
\(255\) −10.1962 −0.638508
\(256\) 1.00000 0.0625000
\(257\) −27.0526 −1.68749 −0.843746 0.536742i \(-0.819655\pi\)
−0.843746 + 0.536742i \(0.819655\pi\)
\(258\) −2.73205 −0.170090
\(259\) −4.73205 −0.294035
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) −5.07180 −0.313337
\(263\) −15.2679 −0.941462 −0.470731 0.882277i \(-0.656010\pi\)
−0.470731 + 0.882277i \(0.656010\pi\)
\(264\) 0.267949 0.0164911
\(265\) −9.66025 −0.593425
\(266\) −2.46410 −0.151084
\(267\) −0.464102 −0.0284026
\(268\) 0.928203 0.0566990
\(269\) 31.7128 1.93356 0.966782 0.255602i \(-0.0822736\pi\)
0.966782 + 0.255602i \(0.0822736\pi\)
\(270\) 2.73205 0.166267
\(271\) 23.5167 1.42854 0.714268 0.699873i \(-0.246760\pi\)
0.714268 + 0.699873i \(0.246760\pi\)
\(272\) 3.73205 0.226289
\(273\) 0 0
\(274\) −8.53590 −0.515672
\(275\) 0.660254 0.0398148
\(276\) 3.46410 0.208514
\(277\) −5.60770 −0.336934 −0.168467 0.985707i \(-0.553882\pi\)
−0.168467 + 0.985707i \(0.553882\pi\)
\(278\) −6.12436 −0.367314
\(279\) −9.66025 −0.578344
\(280\) −2.73205 −0.163271
\(281\) 7.80385 0.465539 0.232769 0.972532i \(-0.425221\pi\)
0.232769 + 0.972532i \(0.425221\pi\)
\(282\) −2.46410 −0.146735
\(283\) −9.07180 −0.539262 −0.269631 0.962964i \(-0.586902\pi\)
−0.269631 + 0.962964i \(0.586902\pi\)
\(284\) 8.19615 0.486352
\(285\) −6.73205 −0.398772
\(286\) 0 0
\(287\) −7.00000 −0.413197
\(288\) −1.00000 −0.0589256
\(289\) −3.07180 −0.180694
\(290\) 8.19615 0.481295
\(291\) −2.73205 −0.160156
\(292\) 13.4641 0.787927
\(293\) 28.7846 1.68161 0.840807 0.541334i \(-0.182081\pi\)
0.840807 + 0.541334i \(0.182081\pi\)
\(294\) 1.00000 0.0583212
\(295\) −35.3205 −2.05644
\(296\) 4.73205 0.275045
\(297\) −0.267949 −0.0155480
\(298\) −5.07180 −0.293801
\(299\) 0 0
\(300\) −2.46410 −0.142265
\(301\) −2.73205 −0.157473
\(302\) −2.80385 −0.161343
\(303\) 10.0000 0.574485
\(304\) 2.46410 0.141326
\(305\) −22.5885 −1.29341
\(306\) −3.73205 −0.213347
\(307\) 7.78461 0.444291 0.222146 0.975014i \(-0.428694\pi\)
0.222146 + 0.975014i \(0.428694\pi\)
\(308\) 0.267949 0.0152678
\(309\) 1.80385 0.102617
\(310\) 26.3923 1.49898
\(311\) −8.80385 −0.499220 −0.249610 0.968346i \(-0.580302\pi\)
−0.249610 + 0.968346i \(0.580302\pi\)
\(312\) 0 0
\(313\) 12.1962 0.689367 0.344684 0.938719i \(-0.387986\pi\)
0.344684 + 0.938719i \(0.387986\pi\)
\(314\) 8.53590 0.481709
\(315\) 2.73205 0.153934
\(316\) −16.8564 −0.948247
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) −3.53590 −0.198283
\(319\) −0.803848 −0.0450068
\(320\) 2.73205 0.152726
\(321\) 8.46410 0.472420
\(322\) 3.46410 0.193047
\(323\) 9.19615 0.511688
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 18.1962 1.00779
\(327\) 13.6603 0.755414
\(328\) 7.00000 0.386510
\(329\) −2.46410 −0.135850
\(330\) 0.732051 0.0402981
\(331\) 3.07180 0.168841 0.0844206 0.996430i \(-0.473096\pi\)
0.0844206 + 0.996430i \(0.473096\pi\)
\(332\) −11.6603 −0.639940
\(333\) −4.73205 −0.259315
\(334\) −21.8564 −1.19593
\(335\) 2.53590 0.138551
\(336\) −1.00000 −0.0545545
\(337\) −13.7846 −0.750896 −0.375448 0.926844i \(-0.622511\pi\)
−0.375448 + 0.926844i \(0.622511\pi\)
\(338\) 0 0
\(339\) −0.196152 −0.0106535
\(340\) 10.1962 0.552964
\(341\) −2.58846 −0.140173
\(342\) −2.46410 −0.133243
\(343\) 1.00000 0.0539949
\(344\) 2.73205 0.147302
\(345\) 9.46410 0.509530
\(346\) −17.2679 −0.928331
\(347\) −4.85641 −0.260706 −0.130353 0.991468i \(-0.541611\pi\)
−0.130353 + 0.991468i \(0.541611\pi\)
\(348\) 3.00000 0.160817
\(349\) 31.7128 1.69755 0.848774 0.528756i \(-0.177341\pi\)
0.848774 + 0.528756i \(0.177341\pi\)
\(350\) −2.46410 −0.131712
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) 5.46410 0.290825 0.145412 0.989371i \(-0.453549\pi\)
0.145412 + 0.989371i \(0.453549\pi\)
\(354\) −12.9282 −0.687126
\(355\) 22.3923 1.18846
\(356\) 0.464102 0.0245973
\(357\) −3.73205 −0.197521
\(358\) −1.60770 −0.0849693
\(359\) 26.1962 1.38258 0.691290 0.722577i \(-0.257043\pi\)
0.691290 + 0.722577i \(0.257043\pi\)
\(360\) −2.73205 −0.143992
\(361\) −12.9282 −0.680432
\(362\) 9.19615 0.483339
\(363\) 10.9282 0.573582
\(364\) 0 0
\(365\) 36.7846 1.92539
\(366\) −8.26795 −0.432173
\(367\) −32.2487 −1.68337 −0.841685 0.539970i \(-0.818436\pi\)
−0.841685 + 0.539970i \(0.818436\pi\)
\(368\) −3.46410 −0.180579
\(369\) −7.00000 −0.364405
\(370\) 12.9282 0.672105
\(371\) −3.53590 −0.183575
\(372\) 9.66025 0.500861
\(373\) 10.5885 0.548250 0.274125 0.961694i \(-0.411612\pi\)
0.274125 + 0.961694i \(0.411612\pi\)
\(374\) −1.00000 −0.0517088
\(375\) 6.92820 0.357771
\(376\) 2.46410 0.127076
\(377\) 0 0
\(378\) 1.00000 0.0514344
\(379\) 16.5885 0.852092 0.426046 0.904702i \(-0.359906\pi\)
0.426046 + 0.904702i \(0.359906\pi\)
\(380\) 6.73205 0.345347
\(381\) 19.4641 0.997176
\(382\) −6.19615 −0.317023
\(383\) −35.3923 −1.80846 −0.904231 0.427043i \(-0.859555\pi\)
−0.904231 + 0.427043i \(0.859555\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0.732051 0.0373088
\(386\) −6.80385 −0.346307
\(387\) −2.73205 −0.138878
\(388\) 2.73205 0.138699
\(389\) 29.1769 1.47933 0.739664 0.672976i \(-0.234984\pi\)
0.739664 + 0.672976i \(0.234984\pi\)
\(390\) 0 0
\(391\) −12.9282 −0.653807
\(392\) −1.00000 −0.0505076
\(393\) −5.07180 −0.255838
\(394\) −5.73205 −0.288777
\(395\) −46.0526 −2.31716
\(396\) 0.267949 0.0134650
\(397\) −9.53590 −0.478593 −0.239297 0.970947i \(-0.576917\pi\)
−0.239297 + 0.970947i \(0.576917\pi\)
\(398\) −6.19615 −0.310585
\(399\) −2.46410 −0.123359
\(400\) 2.46410 0.123205
\(401\) 10.0000 0.499376 0.249688 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0.928203 0.0462946
\(403\) 0 0
\(404\) −10.0000 −0.497519
\(405\) 2.73205 0.135757
\(406\) 3.00000 0.148888
\(407\) −1.26795 −0.0628499
\(408\) 3.73205 0.184764
\(409\) −16.7321 −0.827347 −0.413673 0.910425i \(-0.635754\pi\)
−0.413673 + 0.910425i \(0.635754\pi\)
\(410\) 19.1244 0.944485
\(411\) −8.53590 −0.421045
\(412\) −1.80385 −0.0888692
\(413\) −12.9282 −0.636155
\(414\) 3.46410 0.170251
\(415\) −31.8564 −1.56377
\(416\) 0 0
\(417\) −6.12436 −0.299911
\(418\) −0.660254 −0.0322941
\(419\) 4.19615 0.204995 0.102498 0.994733i \(-0.467317\pi\)
0.102498 + 0.994733i \(0.467317\pi\)
\(420\) −2.73205 −0.133310
\(421\) −6.39230 −0.311542 −0.155771 0.987793i \(-0.549786\pi\)
−0.155771 + 0.987793i \(0.549786\pi\)
\(422\) −11.8564 −0.577161
\(423\) −2.46410 −0.119809
\(424\) 3.53590 0.171718
\(425\) 9.19615 0.446079
\(426\) 8.19615 0.397105
\(427\) −8.26795 −0.400114
\(428\) −8.46410 −0.409128
\(429\) 0 0
\(430\) 7.46410 0.359951
\(431\) −3.12436 −0.150495 −0.0752475 0.997165i \(-0.523975\pi\)
−0.0752475 + 0.997165i \(0.523975\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −4.92820 −0.236834 −0.118417 0.992964i \(-0.537782\pi\)
−0.118417 + 0.992964i \(0.537782\pi\)
\(434\) 9.66025 0.463707
\(435\) 8.19615 0.392975
\(436\) −13.6603 −0.654208
\(437\) −8.53590 −0.408327
\(438\) 13.4641 0.643340
\(439\) 4.39230 0.209633 0.104817 0.994492i \(-0.466574\pi\)
0.104817 + 0.994492i \(0.466574\pi\)
\(440\) −0.732051 −0.0348992
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −10.6077 −0.503987 −0.251993 0.967729i \(-0.581086\pi\)
−0.251993 + 0.967729i \(0.581086\pi\)
\(444\) 4.73205 0.224573
\(445\) 1.26795 0.0601066
\(446\) 12.3923 0.586793
\(447\) −5.07180 −0.239888
\(448\) 1.00000 0.0472456
\(449\) −7.26795 −0.342996 −0.171498 0.985184i \(-0.554861\pi\)
−0.171498 + 0.985184i \(0.554861\pi\)
\(450\) −2.46410 −0.116159
\(451\) −1.87564 −0.0883206
\(452\) 0.196152 0.00922623
\(453\) −2.80385 −0.131736
\(454\) 15.4641 0.725766
\(455\) 0 0
\(456\) 2.46410 0.115392
\(457\) −34.7846 −1.62716 −0.813578 0.581456i \(-0.802483\pi\)
−0.813578 + 0.581456i \(0.802483\pi\)
\(458\) −24.3205 −1.13642
\(459\) −3.73205 −0.174197
\(460\) −9.46410 −0.441266
\(461\) −27.7128 −1.29071 −0.645357 0.763881i \(-0.723291\pi\)
−0.645357 + 0.763881i \(0.723291\pi\)
\(462\) 0.267949 0.0124661
\(463\) 9.19615 0.427381 0.213691 0.976901i \(-0.431452\pi\)
0.213691 + 0.976901i \(0.431452\pi\)
\(464\) −3.00000 −0.139272
\(465\) 26.3923 1.22391
\(466\) 8.19615 0.379679
\(467\) 17.8564 0.826296 0.413148 0.910664i \(-0.364429\pi\)
0.413148 + 0.910664i \(0.364429\pi\)
\(468\) 0 0
\(469\) 0.928203 0.0428604
\(470\) 6.73205 0.310526
\(471\) 8.53590 0.393313
\(472\) 12.9282 0.595069
\(473\) −0.732051 −0.0336597
\(474\) −16.8564 −0.774240
\(475\) 6.07180 0.278593
\(476\) 3.73205 0.171058
\(477\) −3.53590 −0.161898
\(478\) −24.1962 −1.10671
\(479\) −10.6077 −0.484678 −0.242339 0.970192i \(-0.577915\pi\)
−0.242339 + 0.970192i \(0.577915\pi\)
\(480\) 2.73205 0.124700
\(481\) 0 0
\(482\) −28.7846 −1.31110
\(483\) 3.46410 0.157622
\(484\) −10.9282 −0.496737
\(485\) 7.46410 0.338927
\(486\) 1.00000 0.0453609
\(487\) −10.8038 −0.489569 −0.244785 0.969578i \(-0.578717\pi\)
−0.244785 + 0.969578i \(0.578717\pi\)
\(488\) 8.26795 0.374272
\(489\) 18.1962 0.822858
\(490\) −2.73205 −0.123421
\(491\) −12.2487 −0.552777 −0.276388 0.961046i \(-0.589138\pi\)
−0.276388 + 0.961046i \(0.589138\pi\)
\(492\) 7.00000 0.315584
\(493\) −11.1962 −0.504249
\(494\) 0 0
\(495\) 0.732051 0.0329032
\(496\) −9.66025 −0.433758
\(497\) 8.19615 0.367648
\(498\) −11.6603 −0.522508
\(499\) −38.9808 −1.74502 −0.872509 0.488598i \(-0.837509\pi\)
−0.872509 + 0.488598i \(0.837509\pi\)
\(500\) −6.92820 −0.309839
\(501\) −21.8564 −0.976472
\(502\) −7.80385 −0.348303
\(503\) 31.7128 1.41400 0.707002 0.707211i \(-0.250047\pi\)
0.707002 + 0.707211i \(0.250047\pi\)
\(504\) −1.00000 −0.0445435
\(505\) −27.3205 −1.21575
\(506\) 0.928203 0.0412637
\(507\) 0 0
\(508\) −19.4641 −0.863580
\(509\) 12.0526 0.534220 0.267110 0.963666i \(-0.413931\pi\)
0.267110 + 0.963666i \(0.413931\pi\)
\(510\) 10.1962 0.451493
\(511\) 13.4641 0.595617
\(512\) −1.00000 −0.0441942
\(513\) −2.46410 −0.108793
\(514\) 27.0526 1.19324
\(515\) −4.92820 −0.217163
\(516\) 2.73205 0.120272
\(517\) −0.660254 −0.0290379
\(518\) 4.73205 0.207914
\(519\) −17.2679 −0.757979
\(520\) 0 0
\(521\) −5.73205 −0.251126 −0.125563 0.992086i \(-0.540074\pi\)
−0.125563 + 0.992086i \(0.540074\pi\)
\(522\) 3.00000 0.131306
\(523\) −29.1962 −1.27666 −0.638329 0.769763i \(-0.720374\pi\)
−0.638329 + 0.769763i \(0.720374\pi\)
\(524\) 5.07180 0.221562
\(525\) −2.46410 −0.107542
\(526\) 15.2679 0.665714
\(527\) −36.0526 −1.57047
\(528\) −0.267949 −0.0116610
\(529\) −11.0000 −0.478261
\(530\) 9.66025 0.419615
\(531\) −12.9282 −0.561036
\(532\) 2.46410 0.106832
\(533\) 0 0
\(534\) 0.464102 0.0200836
\(535\) −23.1244 −0.999753
\(536\) −0.928203 −0.0400923
\(537\) −1.60770 −0.0693772
\(538\) −31.7128 −1.36724
\(539\) 0.267949 0.0115414
\(540\) −2.73205 −0.117569
\(541\) 19.8038 0.851434 0.425717 0.904856i \(-0.360022\pi\)
0.425717 + 0.904856i \(0.360022\pi\)
\(542\) −23.5167 −1.01013
\(543\) 9.19615 0.394645
\(544\) −3.73205 −0.160010
\(545\) −37.3205 −1.59863
\(546\) 0 0
\(547\) −37.1244 −1.58732 −0.793661 0.608360i \(-0.791828\pi\)
−0.793661 + 0.608360i \(0.791828\pi\)
\(548\) 8.53590 0.364636
\(549\) −8.26795 −0.352867
\(550\) −0.660254 −0.0281533
\(551\) −7.39230 −0.314923
\(552\) −3.46410 −0.147442
\(553\) −16.8564 −0.716807
\(554\) 5.60770 0.238248
\(555\) 12.9282 0.548772
\(556\) 6.12436 0.259731
\(557\) −19.3397 −0.819451 −0.409726 0.912209i \(-0.634376\pi\)
−0.409726 + 0.912209i \(0.634376\pi\)
\(558\) 9.66025 0.408951
\(559\) 0 0
\(560\) 2.73205 0.115450
\(561\) −1.00000 −0.0422200
\(562\) −7.80385 −0.329185
\(563\) −36.7321 −1.54807 −0.774036 0.633142i \(-0.781765\pi\)
−0.774036 + 0.633142i \(0.781765\pi\)
\(564\) 2.46410 0.103757
\(565\) 0.535898 0.0225454
\(566\) 9.07180 0.381316
\(567\) 1.00000 0.0419961
\(568\) −8.19615 −0.343903
\(569\) 19.2679 0.807754 0.403877 0.914813i \(-0.367662\pi\)
0.403877 + 0.914813i \(0.367662\pi\)
\(570\) 6.73205 0.281975
\(571\) 30.1962 1.26367 0.631835 0.775103i \(-0.282302\pi\)
0.631835 + 0.775103i \(0.282302\pi\)
\(572\) 0 0
\(573\) −6.19615 −0.258848
\(574\) 7.00000 0.292174
\(575\) −8.53590 −0.355972
\(576\) 1.00000 0.0416667
\(577\) 16.7321 0.696564 0.348282 0.937390i \(-0.386765\pi\)
0.348282 + 0.937390i \(0.386765\pi\)
\(578\) 3.07180 0.127770
\(579\) −6.80385 −0.282758
\(580\) −8.19615 −0.340327
\(581\) −11.6603 −0.483749
\(582\) 2.73205 0.113247
\(583\) −0.947441 −0.0392390
\(584\) −13.4641 −0.557148
\(585\) 0 0
\(586\) −28.7846 −1.18908
\(587\) 41.3731 1.70765 0.853825 0.520561i \(-0.174277\pi\)
0.853825 + 0.520561i \(0.174277\pi\)
\(588\) −1.00000 −0.0412393
\(589\) −23.8038 −0.980820
\(590\) 35.3205 1.45412
\(591\) −5.73205 −0.235785
\(592\) −4.73205 −0.194486
\(593\) 48.3205 1.98429 0.992143 0.125111i \(-0.0399288\pi\)
0.992143 + 0.125111i \(0.0399288\pi\)
\(594\) 0.267949 0.0109941
\(595\) 10.1962 0.418001
\(596\) 5.07180 0.207749
\(597\) −6.19615 −0.253592
\(598\) 0 0
\(599\) 33.7128 1.37747 0.688734 0.725014i \(-0.258167\pi\)
0.688734 + 0.725014i \(0.258167\pi\)
\(600\) 2.46410 0.100597
\(601\) −37.1244 −1.51433 −0.757167 0.653221i \(-0.773417\pi\)
−0.757167 + 0.653221i \(0.773417\pi\)
\(602\) 2.73205 0.111350
\(603\) 0.928203 0.0377994
\(604\) 2.80385 0.114087
\(605\) −29.8564 −1.21384
\(606\) −10.0000 −0.406222
\(607\) −25.9090 −1.05161 −0.525806 0.850604i \(-0.676236\pi\)
−0.525806 + 0.850604i \(0.676236\pi\)
\(608\) −2.46410 −0.0999325
\(609\) 3.00000 0.121566
\(610\) 22.5885 0.914580
\(611\) 0 0
\(612\) 3.73205 0.150859
\(613\) 14.1436 0.571254 0.285627 0.958341i \(-0.407798\pi\)
0.285627 + 0.958341i \(0.407798\pi\)
\(614\) −7.78461 −0.314161
\(615\) 19.1244 0.771168
\(616\) −0.267949 −0.0107960
\(617\) 3.51666 0.141575 0.0707877 0.997491i \(-0.477449\pi\)
0.0707877 + 0.997491i \(0.477449\pi\)
\(618\) −1.80385 −0.0725614
\(619\) −32.3205 −1.29907 −0.649535 0.760331i \(-0.725037\pi\)
−0.649535 + 0.760331i \(0.725037\pi\)
\(620\) −26.3923 −1.05994
\(621\) 3.46410 0.139010
\(622\) 8.80385 0.353002
\(623\) 0.464102 0.0185938
\(624\) 0 0
\(625\) −31.2487 −1.24995
\(626\) −12.1962 −0.487456
\(627\) −0.660254 −0.0263680
\(628\) −8.53590 −0.340619
\(629\) −17.6603 −0.704160
\(630\) −2.73205 −0.108848
\(631\) 35.9808 1.43237 0.716186 0.697910i \(-0.245886\pi\)
0.716186 + 0.697910i \(0.245886\pi\)
\(632\) 16.8564 0.670512
\(633\) −11.8564 −0.471250
\(634\) −18.0000 −0.714871
\(635\) −53.1769 −2.11026
\(636\) 3.53590 0.140207
\(637\) 0 0
\(638\) 0.803848 0.0318246
\(639\) 8.19615 0.324235
\(640\) −2.73205 −0.107994
\(641\) 32.2487 1.27375 0.636874 0.770968i \(-0.280227\pi\)
0.636874 + 0.770968i \(0.280227\pi\)
\(642\) −8.46410 −0.334051
\(643\) −36.3205 −1.43234 −0.716171 0.697925i \(-0.754107\pi\)
−0.716171 + 0.697925i \(0.754107\pi\)
\(644\) −3.46410 −0.136505
\(645\) 7.46410 0.293899
\(646\) −9.19615 −0.361818
\(647\) −1.73205 −0.0680939 −0.0340470 0.999420i \(-0.510840\pi\)
−0.0340470 + 0.999420i \(0.510840\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −3.46410 −0.135978
\(650\) 0 0
\(651\) 9.66025 0.378615
\(652\) −18.1962 −0.712616
\(653\) 46.3205 1.81266 0.906331 0.422569i \(-0.138872\pi\)
0.906331 + 0.422569i \(0.138872\pi\)
\(654\) −13.6603 −0.534158
\(655\) 13.8564 0.541415
\(656\) −7.00000 −0.273304
\(657\) 13.4641 0.525285
\(658\) 2.46410 0.0960607
\(659\) 20.0718 0.781886 0.390943 0.920415i \(-0.372149\pi\)
0.390943 + 0.920415i \(0.372149\pi\)
\(660\) −0.732051 −0.0284950
\(661\) −48.3923 −1.88224 −0.941121 0.338069i \(-0.890226\pi\)
−0.941121 + 0.338069i \(0.890226\pi\)
\(662\) −3.07180 −0.119389
\(663\) 0 0
\(664\) 11.6603 0.452506
\(665\) 6.73205 0.261058
\(666\) 4.73205 0.183363
\(667\) 10.3923 0.402392
\(668\) 21.8564 0.845650
\(669\) 12.3923 0.479114
\(670\) −2.53590 −0.0979703
\(671\) −2.21539 −0.0855242
\(672\) 1.00000 0.0385758
\(673\) −34.0718 −1.31337 −0.656686 0.754164i \(-0.728042\pi\)
−0.656686 + 0.754164i \(0.728042\pi\)
\(674\) 13.7846 0.530963
\(675\) −2.46410 −0.0948433
\(676\) 0 0
\(677\) 45.7654 1.75891 0.879453 0.475986i \(-0.157909\pi\)
0.879453 + 0.475986i \(0.157909\pi\)
\(678\) 0.196152 0.00753319
\(679\) 2.73205 0.104846
\(680\) −10.1962 −0.391005
\(681\) 15.4641 0.592586
\(682\) 2.58846 0.0991172
\(683\) −5.60770 −0.214572 −0.107286 0.994228i \(-0.534216\pi\)
−0.107286 + 0.994228i \(0.534216\pi\)
\(684\) 2.46410 0.0942173
\(685\) 23.3205 0.891031
\(686\) −1.00000 −0.0381802
\(687\) −24.3205 −0.927885
\(688\) −2.73205 −0.104158
\(689\) 0 0
\(690\) −9.46410 −0.360292
\(691\) −0.784610 −0.0298480 −0.0149240 0.999889i \(-0.504751\pi\)
−0.0149240 + 0.999889i \(0.504751\pi\)
\(692\) 17.2679 0.656429
\(693\) 0.267949 0.0101785
\(694\) 4.85641 0.184347
\(695\) 16.7321 0.634683
\(696\) −3.00000 −0.113715
\(697\) −26.1244 −0.989531
\(698\) −31.7128 −1.20035
\(699\) 8.19615 0.310007
\(700\) 2.46410 0.0931343
\(701\) −20.3205 −0.767495 −0.383747 0.923438i \(-0.625367\pi\)
−0.383747 + 0.923438i \(0.625367\pi\)
\(702\) 0 0
\(703\) −11.6603 −0.439775
\(704\) 0.267949 0.0100987
\(705\) 6.73205 0.253544
\(706\) −5.46410 −0.205644
\(707\) −10.0000 −0.376089
\(708\) 12.9282 0.485872
\(709\) −14.0526 −0.527755 −0.263877 0.964556i \(-0.585001\pi\)
−0.263877 + 0.964556i \(0.585001\pi\)
\(710\) −22.3923 −0.840368
\(711\) −16.8564 −0.632165
\(712\) −0.464102 −0.0173929
\(713\) 33.4641 1.25324
\(714\) 3.73205 0.139668
\(715\) 0 0
\(716\) 1.60770 0.0600824
\(717\) −24.1962 −0.903622
\(718\) −26.1962 −0.977632
\(719\) −35.0526 −1.30724 −0.653620 0.756823i \(-0.726750\pi\)
−0.653620 + 0.756823i \(0.726750\pi\)
\(720\) 2.73205 0.101818
\(721\) −1.80385 −0.0671788
\(722\) 12.9282 0.481138
\(723\) −28.7846 −1.07051
\(724\) −9.19615 −0.341772
\(725\) −7.39230 −0.274543
\(726\) −10.9282 −0.405584
\(727\) −5.46410 −0.202652 −0.101326 0.994853i \(-0.532309\pi\)
−0.101326 + 0.994853i \(0.532309\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −36.7846 −1.36146
\(731\) −10.1962 −0.377118
\(732\) 8.26795 0.305592
\(733\) −20.8564 −0.770349 −0.385174 0.922844i \(-0.625859\pi\)
−0.385174 + 0.922844i \(0.625859\pi\)
\(734\) 32.2487 1.19032
\(735\) −2.73205 −0.100773
\(736\) 3.46410 0.127688
\(737\) 0.248711 0.00916140
\(738\) 7.00000 0.257674
\(739\) 14.3923 0.529429 0.264715 0.964327i \(-0.414722\pi\)
0.264715 + 0.964327i \(0.414722\pi\)
\(740\) −12.9282 −0.475250
\(741\) 0 0
\(742\) 3.53590 0.129807
\(743\) 25.5167 0.936115 0.468058 0.883698i \(-0.344954\pi\)
0.468058 + 0.883698i \(0.344954\pi\)
\(744\) −9.66025 −0.354162
\(745\) 13.8564 0.507659
\(746\) −10.5885 −0.387671
\(747\) −11.6603 −0.426626
\(748\) 1.00000 0.0365636
\(749\) −8.46410 −0.309272
\(750\) −6.92820 −0.252982
\(751\) 20.8564 0.761061 0.380531 0.924768i \(-0.375741\pi\)
0.380531 + 0.924768i \(0.375741\pi\)
\(752\) −2.46410 −0.0898565
\(753\) −7.80385 −0.284388
\(754\) 0 0
\(755\) 7.66025 0.278785
\(756\) −1.00000 −0.0363696
\(757\) 0.0525589 0.00191029 0.000955143 1.00000i \(-0.499696\pi\)
0.000955143 1.00000i \(0.499696\pi\)
\(758\) −16.5885 −0.602520
\(759\) 0.928203 0.0336916
\(760\) −6.73205 −0.244197
\(761\) −35.3205 −1.28037 −0.640184 0.768222i \(-0.721142\pi\)
−0.640184 + 0.768222i \(0.721142\pi\)
\(762\) −19.4641 −0.705110
\(763\) −13.6603 −0.494534
\(764\) 6.19615 0.224169
\(765\) 10.1962 0.368643
\(766\) 35.3923 1.27878
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −34.4449 −1.24211 −0.621057 0.783766i \(-0.713296\pi\)
−0.621057 + 0.783766i \(0.713296\pi\)
\(770\) −0.732051 −0.0263813
\(771\) 27.0526 0.974274
\(772\) 6.80385 0.244876
\(773\) −21.6077 −0.777175 −0.388587 0.921412i \(-0.627037\pi\)
−0.388587 + 0.921412i \(0.627037\pi\)
\(774\) 2.73205 0.0982015
\(775\) −23.8038 −0.855059
\(776\) −2.73205 −0.0980749
\(777\) 4.73205 0.169761
\(778\) −29.1769 −1.04604
\(779\) −17.2487 −0.617999
\(780\) 0 0
\(781\) 2.19615 0.0785845
\(782\) 12.9282 0.462312
\(783\) 3.00000 0.107211
\(784\) 1.00000 0.0357143
\(785\) −23.3205 −0.832345
\(786\) 5.07180 0.180905
\(787\) −14.2154 −0.506724 −0.253362 0.967372i \(-0.581536\pi\)
−0.253362 + 0.967372i \(0.581536\pi\)
\(788\) 5.73205 0.204196
\(789\) 15.2679 0.543553
\(790\) 46.0526 1.63848
\(791\) 0.196152 0.00697438
\(792\) −0.267949 −0.00952116
\(793\) 0 0
\(794\) 9.53590 0.338416
\(795\) 9.66025 0.342614
\(796\) 6.19615 0.219617
\(797\) 2.73205 0.0967742 0.0483871 0.998829i \(-0.484592\pi\)
0.0483871 + 0.998829i \(0.484592\pi\)
\(798\) 2.46410 0.0872283
\(799\) −9.19615 −0.325336
\(800\) −2.46410 −0.0871191
\(801\) 0.464102 0.0163982
\(802\) −10.0000 −0.353112
\(803\) 3.60770 0.127313
\(804\) −0.928203 −0.0327352
\(805\) −9.46410 −0.333566
\(806\) 0 0
\(807\) −31.7128 −1.11634
\(808\) 10.0000 0.351799
\(809\) 10.1962 0.358478 0.179239 0.983806i \(-0.442637\pi\)
0.179239 + 0.983806i \(0.442637\pi\)
\(810\) −2.73205 −0.0959945
\(811\) 8.53590 0.299736 0.149868 0.988706i \(-0.452115\pi\)
0.149868 + 0.988706i \(0.452115\pi\)
\(812\) −3.00000 −0.105279
\(813\) −23.5167 −0.824765
\(814\) 1.26795 0.0444416
\(815\) −49.7128 −1.74136
\(816\) −3.73205 −0.130648
\(817\) −6.73205 −0.235525
\(818\) 16.7321 0.585022
\(819\) 0 0
\(820\) −19.1244 −0.667851
\(821\) −14.4115 −0.502966 −0.251483 0.967862i \(-0.580918\pi\)
−0.251483 + 0.967862i \(0.580918\pi\)
\(822\) 8.53590 0.297724
\(823\) 20.9282 0.729511 0.364756 0.931103i \(-0.381153\pi\)
0.364756 + 0.931103i \(0.381153\pi\)
\(824\) 1.80385 0.0628400
\(825\) −0.660254 −0.0229871
\(826\) 12.9282 0.449830
\(827\) −40.3923 −1.40458 −0.702289 0.711892i \(-0.747839\pi\)
−0.702289 + 0.711892i \(0.747839\pi\)
\(828\) −3.46410 −0.120386
\(829\) −0.660254 −0.0229316 −0.0114658 0.999934i \(-0.503650\pi\)
−0.0114658 + 0.999934i \(0.503650\pi\)
\(830\) 31.8564 1.10575
\(831\) 5.60770 0.194529
\(832\) 0 0
\(833\) 3.73205 0.129308
\(834\) 6.12436 0.212069
\(835\) 59.7128 2.06645
\(836\) 0.660254 0.0228354
\(837\) 9.66025 0.333907
\(838\) −4.19615 −0.144954
\(839\) −14.3923 −0.496878 −0.248439 0.968648i \(-0.579917\pi\)
−0.248439 + 0.968648i \(0.579917\pi\)
\(840\) 2.73205 0.0942647
\(841\) −20.0000 −0.689655
\(842\) 6.39230 0.220293
\(843\) −7.80385 −0.268779
\(844\) 11.8564 0.408114
\(845\) 0 0
\(846\) 2.46410 0.0847176
\(847\) −10.9282 −0.375498
\(848\) −3.53590 −0.121423
\(849\) 9.07180 0.311343
\(850\) −9.19615 −0.315425
\(851\) 16.3923 0.561921
\(852\) −8.19615 −0.280796
\(853\) 33.3923 1.14333 0.571665 0.820487i \(-0.306298\pi\)
0.571665 + 0.820487i \(0.306298\pi\)
\(854\) 8.26795 0.282923
\(855\) 6.73205 0.230231
\(856\) 8.46410 0.289297
\(857\) −21.6077 −0.738105 −0.369052 0.929409i \(-0.620318\pi\)
−0.369052 + 0.929409i \(0.620318\pi\)
\(858\) 0 0
\(859\) −45.0526 −1.53717 −0.768587 0.639746i \(-0.779040\pi\)
−0.768587 + 0.639746i \(0.779040\pi\)
\(860\) −7.46410 −0.254524
\(861\) 7.00000 0.238559
\(862\) 3.12436 0.106416
\(863\) 28.6410 0.974952 0.487476 0.873137i \(-0.337918\pi\)
0.487476 + 0.873137i \(0.337918\pi\)
\(864\) 1.00000 0.0340207
\(865\) 47.1769 1.60406
\(866\) 4.92820 0.167467
\(867\) 3.07180 0.104324
\(868\) −9.66025 −0.327890
\(869\) −4.51666 −0.153217
\(870\) −8.19615 −0.277876
\(871\) 0 0
\(872\) 13.6603 0.462595
\(873\) 2.73205 0.0924659
\(874\) 8.53590 0.288731
\(875\) −6.92820 −0.234216
\(876\) −13.4641 −0.454910
\(877\) 36.3013 1.22581 0.612903 0.790158i \(-0.290001\pi\)
0.612903 + 0.790158i \(0.290001\pi\)
\(878\) −4.39230 −0.148233
\(879\) −28.7846 −0.970881
\(880\) 0.732051 0.0246774
\(881\) 1.07180 0.0361098 0.0180549 0.999837i \(-0.494253\pi\)
0.0180549 + 0.999837i \(0.494253\pi\)
\(882\) −1.00000 −0.0336718
\(883\) 34.4449 1.15916 0.579581 0.814915i \(-0.303216\pi\)
0.579581 + 0.814915i \(0.303216\pi\)
\(884\) 0 0
\(885\) 35.3205 1.18729
\(886\) 10.6077 0.356372
\(887\) 45.0526 1.51272 0.756358 0.654157i \(-0.226977\pi\)
0.756358 + 0.654157i \(0.226977\pi\)
\(888\) −4.73205 −0.158797
\(889\) −19.4641 −0.652805
\(890\) −1.26795 −0.0425018
\(891\) 0.267949 0.00897664
\(892\) −12.3923 −0.414925
\(893\) −6.07180 −0.203185
\(894\) 5.07180 0.169626
\(895\) 4.39230 0.146819
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 7.26795 0.242535
\(899\) 28.9808 0.966563
\(900\) 2.46410 0.0821367
\(901\) −13.1962 −0.439628
\(902\) 1.87564 0.0624521
\(903\) 2.73205 0.0909170
\(904\) −0.196152 −0.00652393
\(905\) −25.1244 −0.835162
\(906\) 2.80385 0.0931516
\(907\) −27.4641 −0.911931 −0.455965 0.889998i \(-0.650706\pi\)
−0.455965 + 0.889998i \(0.650706\pi\)
\(908\) −15.4641 −0.513194
\(909\) −10.0000 −0.331679
\(910\) 0 0
\(911\) −0.0525589 −0.00174135 −0.000870677 1.00000i \(-0.500277\pi\)
−0.000870677 1.00000i \(0.500277\pi\)
\(912\) −2.46410 −0.0815946
\(913\) −3.12436 −0.103401
\(914\) 34.7846 1.15057
\(915\) 22.5885 0.746751
\(916\) 24.3205 0.803572
\(917\) 5.07180 0.167485
\(918\) 3.73205 0.123176
\(919\) 41.6410 1.37361 0.686805 0.726842i \(-0.259013\pi\)
0.686805 + 0.726842i \(0.259013\pi\)
\(920\) 9.46410 0.312022
\(921\) −7.78461 −0.256512
\(922\) 27.7128 0.912673
\(923\) 0 0
\(924\) −0.267949 −0.00881488
\(925\) −11.6603 −0.383387
\(926\) −9.19615 −0.302204
\(927\) −1.80385 −0.0592461
\(928\) 3.00000 0.0984798
\(929\) −43.5359 −1.42837 −0.714183 0.699959i \(-0.753202\pi\)
−0.714183 + 0.699959i \(0.753202\pi\)
\(930\) −26.3923 −0.865438
\(931\) 2.46410 0.0807577
\(932\) −8.19615 −0.268474
\(933\) 8.80385 0.288225
\(934\) −17.8564 −0.584279
\(935\) 2.73205 0.0893476
\(936\) 0 0
\(937\) 45.5692 1.48868 0.744341 0.667800i \(-0.232764\pi\)
0.744341 + 0.667800i \(0.232764\pi\)
\(938\) −0.928203 −0.0303069
\(939\) −12.1962 −0.398006
\(940\) −6.73205 −0.219575
\(941\) 49.1769 1.60312 0.801561 0.597913i \(-0.204003\pi\)
0.801561 + 0.597913i \(0.204003\pi\)
\(942\) −8.53590 −0.278115
\(943\) 24.2487 0.789647
\(944\) −12.9282 −0.420777
\(945\) −2.73205 −0.0888736
\(946\) 0.732051 0.0238010
\(947\) 12.9474 0.420735 0.210368 0.977622i \(-0.432534\pi\)
0.210368 + 0.977622i \(0.432534\pi\)
\(948\) 16.8564 0.547471
\(949\) 0 0
\(950\) −6.07180 −0.196995
\(951\) −18.0000 −0.583690
\(952\) −3.73205 −0.120956
\(953\) −21.3731 −0.692342 −0.346171 0.938172i \(-0.612518\pi\)
−0.346171 + 0.938172i \(0.612518\pi\)
\(954\) 3.53590 0.114479
\(955\) 16.9282 0.547784
\(956\) 24.1962 0.782559
\(957\) 0.803848 0.0259847
\(958\) 10.6077 0.342719
\(959\) 8.53590 0.275639
\(960\) −2.73205 −0.0881766
\(961\) 62.3205 2.01034
\(962\) 0 0
\(963\) −8.46410 −0.272752
\(964\) 28.7846 0.927090
\(965\) 18.5885 0.598384
\(966\) −3.46410 −0.111456
\(967\) −36.0000 −1.15768 −0.578841 0.815440i \(-0.696495\pi\)
−0.578841 + 0.815440i \(0.696495\pi\)
\(968\) 10.9282 0.351246
\(969\) −9.19615 −0.295423
\(970\) −7.46410 −0.239658
\(971\) −4.24871 −0.136348 −0.0681738 0.997673i \(-0.521717\pi\)
−0.0681738 + 0.997673i \(0.521717\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.12436 0.196338
\(974\) 10.8038 0.346178
\(975\) 0 0
\(976\) −8.26795 −0.264651
\(977\) −30.2487 −0.967742 −0.483871 0.875139i \(-0.660770\pi\)
−0.483871 + 0.875139i \(0.660770\pi\)
\(978\) −18.1962 −0.581849
\(979\) 0.124356 0.00397442
\(980\) 2.73205 0.0872722
\(981\) −13.6603 −0.436138
\(982\) 12.2487 0.390872
\(983\) −55.8564 −1.78154 −0.890771 0.454452i \(-0.849835\pi\)
−0.890771 + 0.454452i \(0.849835\pi\)
\(984\) −7.00000 −0.223152
\(985\) 15.6603 0.498977
\(986\) 11.1962 0.356558
\(987\) 2.46410 0.0784332
\(988\) 0 0
\(989\) 9.46410 0.300941
\(990\) −0.732051 −0.0232661
\(991\) −25.3923 −0.806613 −0.403307 0.915065i \(-0.632139\pi\)
−0.403307 + 0.915065i \(0.632139\pi\)
\(992\) 9.66025 0.306713
\(993\) −3.07180 −0.0974805
\(994\) −8.19615 −0.259966
\(995\) 16.9282 0.536660
\(996\) 11.6603 0.369469
\(997\) −58.2295 −1.84415 −0.922073 0.387016i \(-0.873506\pi\)
−0.922073 + 0.387016i \(0.873506\pi\)
\(998\) 38.9808 1.23391
\(999\) 4.73205 0.149715
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 7098.2.a.bj.1.2 2
13.6 odd 12 546.2.s.d.127.2 yes 4
13.11 odd 12 546.2.s.d.43.2 4
13.12 even 2 7098.2.a.bs.1.1 2
39.11 even 12 1638.2.bj.d.1135.1 4
39.32 even 12 1638.2.bj.d.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.d.43.2 4 13.11 odd 12
546.2.s.d.127.2 yes 4 13.6 odd 12
1638.2.bj.d.127.1 4 39.32 even 12
1638.2.bj.d.1135.1 4 39.11 even 12
7098.2.a.bj.1.2 2 1.1 even 1 trivial
7098.2.a.bs.1.1 2 13.12 even 2