Properties

Label 7098.2.a.bs.1.1
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 1.1
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 \(-0.709194\pi\)
−0.610905 + 0.791704i \(0.709194\pi\)
\(6\) −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 \(-0.512862\pi\)
−0.0403949 + 0.999184i \(0.512862\pi\)
\(12\) −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 \(-0.591214\pi\)
−0.282652 + 0.959223i \(0.591214\pi\)
\(20\) −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 \(-0.165715\pi\)
0.867516 + 0.497409i \(0.165715\pi\)
\(32\) 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 \(-0.372830\pi\)
0.388972 + 0.921250i \(0.372830\pi\)
\(38\) −2.46410 −0.399730
\(39\) 0 0
\(40\) −2.73205 −0.431975
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 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 \(-0.442483\pi\)
0.179713 + 0.983719i \(0.442483\pi\)
\(48\) −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 \(-0.181641\pi\)
0.841554 + 0.540172i \(0.181641\pi\)
\(60\) 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 \(-0.518058\pi\)
−0.0566990 + 0.998391i \(0.518058\pi\)
\(68\) 3.73205 0.452578
\(69\) 3.46410 0.417029
\(70\) 2.73205 0.326543
\(71\) −8.19615 −0.972704 −0.486352 0.873763i \(-0.661673\pi\)
−0.486352 + 0.873763i \(0.661673\pi\)
\(72\) 1.00000 0.117851
\(73\) −13.4641 −1.57585 −0.787927 0.615769i \(-0.788846\pi\)
−0.787927 + 0.615769i \(0.788846\pi\)
\(74\) 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 \(-0.278959\pi\)
0.639940 + 0.768425i \(0.278959\pi\)
\(84\) 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 \(-0.507830\pi\)
−0.0245973 + 0.999697i \(0.507830\pi\)
\(90\) −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 \(-0.544292\pi\)
−0.138699 + 0.990335i \(0.544292\pi\)
\(98\) 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 \(-0.273002\pi\)
0.654208 + 0.756315i \(0.273002\pi\)
\(110\) 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 \(-0.618806\pi\)
−0.364636 + 0.931150i \(0.618806\pi\)
\(138\) 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 \(-0.566614\pi\)
−0.207749 + 0.978182i \(0.566614\pi\)
\(150\) −2.46410 −0.201193
\(151\) −2.80385 −0.228174 −0.114087 0.993471i \(-0.536394\pi\)
−0.114087 + 0.993471i \(0.536394\pi\)
\(152\) −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 \(-0.247510\pi\)
0.712616 + 0.701554i \(0.247510\pi\)
\(164\) 7.00000 0.546608
\(165\) −0.732051 −0.0569901
\(166\) 11.6603 0.905011
\(167\) −21.8564 −1.69130 −0.845650 0.533738i \(-0.820787\pi\)
−0.845650 + 0.533738i \(0.820787\pi\)
\(168\) 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 \(-0.578747\pi\)
−0.244876 + 0.969554i \(0.578747\pi\)
\(194\) −2.73205 −0.196150
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −5.73205 −0.408392 −0.204196 0.978930i \(-0.565458\pi\)
−0.204196 + 0.978930i \(0.565458\pi\)
\(198\) −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 \(-0.363808\pi\)
0.414925 + 0.909856i \(0.363808\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 2.46410 0.164273
\(226\) 0.196152 0.0130479
\(227\) 15.4641 1.02639 0.513194 0.858272i \(-0.328462\pi\)
0.513194 + 0.858272i \(0.328462\pi\)
\(228\) 2.46410 0.163189
\(229\) −24.3205 −1.60714 −0.803572 0.595207i \(-0.797070\pi\)
−0.803572 + 0.595207i \(0.797070\pi\)
\(230\) 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 \(-0.786086\pi\)
−0.782559 + 0.622576i \(0.786086\pi\)
\(240\) 2.73205 0.176353
\(241\) −28.7846 −1.85418 −0.927090 0.374839i \(-0.877698\pi\)
−0.927090 + 0.374839i \(0.877698\pi\)
\(242\) −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 \(-0.753240\pi\)
−0.714268 + 0.699873i \(0.753240\pi\)
\(272\) 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 \(-0.574779\pi\)
−0.232769 + 0.972532i \(0.574779\pi\)
\(282\) −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 \(-0.817919\pi\)
−0.840807 + 0.541334i \(0.817919\pi\)
\(294\) −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 \(-0.571306\pi\)
−0.222146 + 0.975014i \(0.571306\pi\)
\(308\) 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 \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 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 \(-0.526904\pi\)
−0.0844206 + 0.996430i \(0.526904\pi\)
\(332\) 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 \(-0.822659\pi\)
−0.848774 + 0.528756i \(0.822659\pi\)
\(350\) −2.46410 −0.131712
\(351\) 0 0
\(352\) −0.267949 −0.0142817
\(353\) −5.46410 −0.290825 −0.145412 0.989371i \(-0.546451\pi\)
−0.145412 + 0.989371i \(0.546451\pi\)
\(354\) −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 \(-0.742957\pi\)
−0.691290 + 0.722577i \(0.742957\pi\)
\(360\) −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 \(-0.640094\pi\)
−0.426046 + 0.904702i \(0.640094\pi\)
\(380\) 6.73205 0.345347
\(381\) 19.4641 0.997176
\(382\) 6.19615 0.317023
\(383\) 35.3923 1.80846 0.904231 0.427043i \(-0.140445\pi\)
0.904231 + 0.427043i \(0.140445\pi\)
\(384\) −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 \(-0.423083\pi\)
0.239297 + 0.970947i \(0.423083\pi\)
\(398\) 6.19615 0.310585
\(399\) −2.46410 −0.123359
\(400\) 2.46410 0.123205
\(401\) −10.0000 −0.499376 −0.249688 0.968326i \(-0.580328\pi\)
−0.249688 + 0.968326i \(0.580328\pi\)
\(402\) 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 \(-0.364246\pi\)
0.413673 + 0.910425i \(0.364246\pi\)
\(410\) −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 \(-0.450214\pi\)
0.155771 + 0.987793i \(0.450214\pi\)
\(422\) 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 \(-0.476025\pi\)
0.0752475 + 0.997165i \(0.476025\pi\)
\(432\) −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 \(-0.445139\pi\)
0.171498 + 0.985184i \(0.445139\pi\)
\(450\) 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 \(-0.197517\pi\)
0.813578 + 0.581456i \(0.197517\pi\)
\(458\) −24.3205 −1.13642
\(459\) −3.73205 −0.174197
\(460\) 9.46410 0.441266
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) −0.267949 −0.0124661
\(463\) −9.19615 −0.427381 −0.213691 0.976901i \(-0.568548\pi\)
−0.213691 + 0.976901i \(0.568548\pi\)
\(464\) −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 \(-0.422085\pi\)
0.242339 + 0.970192i \(0.422085\pi\)
\(480\) 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 \(-0.421283\pi\)
0.244785 + 0.969578i \(0.421283\pi\)
\(488\) −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 \(-0.162491\pi\)
0.872509 + 0.488598i \(0.162491\pi\)
\(500\) 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 \(-0.586069\pi\)
−0.267110 + 0.963666i \(0.586069\pi\)
\(510\) 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 \(-0.639978\pi\)
−0.425717 + 0.904856i \(0.639978\pi\)
\(542\) −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 \(-0.365624\pi\)
0.409726 + 0.912209i \(0.365624\pi\)
\(558\) 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 \(-0.613235\pi\)
−0.348282 + 0.937390i \(0.613235\pi\)
\(578\) −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 \(-0.825723\pi\)
−0.853825 + 0.520561i \(0.825723\pi\)
\(588\) −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 \(-0.960071\pi\)
−0.992143 + 0.125111i \(0.960071\pi\)
\(594\) 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 \(-0.592202\pi\)
−0.285627 + 0.958341i \(0.592202\pi\)
\(614\) −7.78461 −0.314161
\(615\) 19.1244 0.771168
\(616\) 0.267949 0.0107960
\(617\) −3.51666 −0.141575 −0.0707877 0.997491i \(-0.522551\pi\)
−0.0707877 + 0.997491i \(0.522551\pi\)
\(618\) 1.80385 0.0725614
\(619\) 32.3205 1.29907 0.649535 0.760331i \(-0.274963\pi\)
0.649535 + 0.760331i \(0.274963\pi\)
\(620\) −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 \(-0.754114\pi\)
−0.716186 + 0.697910i \(0.754114\pi\)
\(632\) −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 \(-0.245893\pi\)
0.716171 + 0.697925i \(0.245893\pi\)
\(644\) 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 \(-0.109774\pi\)
0.941121 + 0.338069i \(0.109774\pi\)
\(662\) −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 \(-0.465784\pi\)
0.107286 + 0.994228i \(0.465784\pi\)
\(684\) −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 \(-0.495249\pi\)
0.0149240 + 0.999889i \(0.495249\pi\)
\(692\) 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 \(-0.414999\pi\)
0.263877 + 0.964556i \(0.414999\pi\)
\(710\) 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 \(-0.374141\pi\)
0.385174 + 0.922844i \(0.374141\pi\)
\(734\) −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 \(-0.585278\pi\)
−0.264715 + 0.964327i \(0.585278\pi\)
\(740\) −12.9282 −0.475250
\(741\) 0 0
\(742\) 3.53590 0.129807
\(743\) −25.5167 −0.936115 −0.468058 0.883698i \(-0.655046\pi\)
−0.468058 + 0.883698i \(0.655046\pi\)
\(744\) −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 \(-0.278858\pi\)
0.640184 + 0.768222i \(0.278858\pi\)
\(762\) 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 \(-0.286704\pi\)
0.621057 + 0.783766i \(0.286704\pi\)
\(770\) −0.732051 −0.0263813
\(771\) 27.0526 0.974274
\(772\) −6.80385 −0.244876
\(773\) 21.6077 0.777175 0.388587 0.921412i \(-0.372963\pi\)
0.388587 + 0.921412i \(0.372963\pi\)
\(774\) −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 \(-0.418464\pi\)
0.253362 + 0.967372i \(0.418464\pi\)
\(788\) −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 \(-0.547885\pi\)
−0.149868 + 0.988706i \(0.547885\pi\)
\(812\) 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 \(-0.419082\pi\)
0.251483 + 0.967862i \(0.419082\pi\)
\(822\) 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 \(-0.252161\pi\)
0.702289 + 0.711892i \(0.252161\pi\)
\(828\) −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 \(-0.420083\pi\)
0.248439 + 0.968648i \(0.420083\pi\)
\(840\) −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 \(-0.693702\pi\)
−0.571665 + 0.820487i \(0.693702\pi\)
\(854\) 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 \(-0.662082\pi\)
−0.487476 + 0.873137i \(0.662082\pi\)
\(864\) −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 \(-0.709999\pi\)
−0.612903 + 0.790158i \(0.709999\pi\)
\(878\) 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 \(-0.246798\pi\)
0.714183 + 0.699959i \(0.246798\pi\)
\(930\) 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 \(-0.795997\pi\)
−0.801561 + 0.597913i \(0.795997\pi\)
\(942\) 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 \(-0.567466\pi\)
−0.210368 + 0.977622i \(0.567466\pi\)
\(948\) 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 \(-0.303505\pi\)
0.578841 + 0.815440i \(0.303505\pi\)
\(968\) −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 \(-0.339230\pi\)
0.483871 + 0.875139i \(0.339230\pi\)
\(978\) −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 \(-0.150165\pi\)
0.890771 + 0.454452i \(0.150165\pi\)
\(984\) −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.bs.1.1 2
13.2 odd 12 546.2.s.d.43.2 4
13.7 odd 12 546.2.s.d.127.2 yes 4
13.12 even 2 7098.2.a.bj.1.2 2
39.2 even 12 1638.2.bj.d.1135.1 4
39.20 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 13.2 odd 12
546.2.s.d.127.2 yes 4 13.7 odd 12
1638.2.bj.d.127.1 4 39.20 even 12
1638.2.bj.d.1135.1 4 39.2 even 12
7098.2.a.bj.1.2 2 13.12 even 2
7098.2.a.bs.1.1 2 1.1 even 1 trivial