Properties

Label 7098.2.a.bn.1.1
Level $7098$
Weight $2$
Character 7098.1
Self dual yes
Analytic conductor $56.678$
Analytic rank $0$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
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

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -3.46410 q^{5} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.46410 q^{10} +4.00000 q^{11} +1.00000 q^{12} -1.00000 q^{14} -3.46410 q^{15} +1.00000 q^{16} +6.46410 q^{17} -1.00000 q^{18} +7.46410 q^{19} -3.46410 q^{20} +1.00000 q^{21} -4.00000 q^{22} +5.73205 q^{23} -1.00000 q^{24} +7.00000 q^{25} +1.00000 q^{27} +1.00000 q^{28} +8.00000 q^{29} +3.46410 q^{30} -6.46410 q^{31} -1.00000 q^{32} +4.00000 q^{33} -6.46410 q^{34} -3.46410 q^{35} +1.00000 q^{36} -4.92820 q^{37} -7.46410 q^{38} +3.46410 q^{40} +6.92820 q^{41} -1.00000 q^{42} +4.46410 q^{43} +4.00000 q^{44} -3.46410 q^{45} -5.73205 q^{46} -0.535898 q^{47} +1.00000 q^{48} +1.00000 q^{49} -7.00000 q^{50} +6.46410 q^{51} -8.26795 q^{53} -1.00000 q^{54} -13.8564 q^{55} -1.00000 q^{56} +7.46410 q^{57} -8.00000 q^{58} +6.26795 q^{59} -3.46410 q^{60} -5.19615 q^{61} +6.46410 q^{62} +1.00000 q^{63} +1.00000 q^{64} -4.00000 q^{66} -7.19615 q^{67} +6.46410 q^{68} +5.73205 q^{69} +3.46410 q^{70} +1.92820 q^{71} -1.00000 q^{72} -10.9282 q^{73} +4.92820 q^{74} +7.00000 q^{75} +7.46410 q^{76} +4.00000 q^{77} -9.46410 q^{79} -3.46410 q^{80} +1.00000 q^{81} -6.92820 q^{82} +1.73205 q^{83} +1.00000 q^{84} -22.3923 q^{85} -4.46410 q^{86} +8.00000 q^{87} -4.00000 q^{88} -11.7321 q^{89} +3.46410 q^{90} +5.73205 q^{92} -6.46410 q^{93} +0.535898 q^{94} -25.8564 q^{95} -1.00000 q^{96} +12.3923 q^{97} -1.00000 q^{98} +4.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{3} + 2q^{4} - 2q^{6} + 2q^{7} - 2q^{8} + 2q^{9} + 8q^{11} + 2q^{12} - 2q^{14} + 2q^{16} + 6q^{17} - 2q^{18} + 8q^{19} + 2q^{21} - 8q^{22} + 8q^{23} - 2q^{24} + 14q^{25} + 2q^{27} + 2q^{28} + 16q^{29} - 6q^{31} - 2q^{32} + 8q^{33} - 6q^{34} + 2q^{36} + 4q^{37} - 8q^{38} - 2q^{42} + 2q^{43} + 8q^{44} - 8q^{46} - 8q^{47} + 2q^{48} + 2q^{49} - 14q^{50} + 6q^{51} - 20q^{53} - 2q^{54} - 2q^{56} + 8q^{57} - 16q^{58} + 16q^{59} + 6q^{62} + 2q^{63} + 2q^{64} - 8q^{66} - 4q^{67} + 6q^{68} + 8q^{69} - 10q^{71} - 2q^{72} - 8q^{73} - 4q^{74} + 14q^{75} + 8q^{76} + 8q^{77} - 12q^{79} + 2q^{81} + 2q^{84} - 24q^{85} - 2q^{86} + 16q^{87} - 8q^{88} - 20q^{89} + 8q^{92} - 6q^{93} + 8q^{94} - 24q^{95} - 2q^{96} + 4q^{97} - 2q^{98} + 8q^{99} + O(q^{100}) \)

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
\(5\) −3.46410 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) −1.00000 −0.408248
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.46410 1.09545
\(11\) 4.00000 1.20605 0.603023 0.797724i \(-0.293963\pi\)
0.603023 + 0.797724i \(0.293963\pi\)
\(12\) 1.00000 0.288675
\(13\) 0 0
\(14\) −1.00000 −0.267261
\(15\) −3.46410 −0.894427
\(16\) 1.00000 0.250000
\(17\) 6.46410 1.56777 0.783887 0.620903i \(-0.213234\pi\)
0.783887 + 0.620903i \(0.213234\pi\)
\(18\) −1.00000 −0.235702
\(19\) 7.46410 1.71238 0.856191 0.516659i \(-0.172825\pi\)
0.856191 + 0.516659i \(0.172825\pi\)
\(20\) −3.46410 −0.774597
\(21\) 1.00000 0.218218
\(22\) −4.00000 −0.852803
\(23\) 5.73205 1.19522 0.597608 0.801789i \(-0.296118\pi\)
0.597608 + 0.801789i \(0.296118\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 1.00000 0.188982
\(29\) 8.00000 1.48556 0.742781 0.669534i \(-0.233506\pi\)
0.742781 + 0.669534i \(0.233506\pi\)
\(30\) 3.46410 0.632456
\(31\) −6.46410 −1.16099 −0.580493 0.814265i \(-0.697140\pi\)
−0.580493 + 0.814265i \(0.697140\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.00000 0.696311
\(34\) −6.46410 −1.10858
\(35\) −3.46410 −0.585540
\(36\) 1.00000 0.166667
\(37\) −4.92820 −0.810192 −0.405096 0.914274i \(-0.632762\pi\)
−0.405096 + 0.914274i \(0.632762\pi\)
\(38\) −7.46410 −1.21084
\(39\) 0 0
\(40\) 3.46410 0.547723
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) −1.00000 −0.154303
\(43\) 4.46410 0.680769 0.340385 0.940286i \(-0.389443\pi\)
0.340385 + 0.940286i \(0.389443\pi\)
\(44\) 4.00000 0.603023
\(45\) −3.46410 −0.516398
\(46\) −5.73205 −0.845145
\(47\) −0.535898 −0.0781688 −0.0390844 0.999236i \(-0.512444\pi\)
−0.0390844 + 0.999236i \(0.512444\pi\)
\(48\) 1.00000 0.144338
\(49\) 1.00000 0.142857
\(50\) −7.00000 −0.989949
\(51\) 6.46410 0.905155
\(52\) 0 0
\(53\) −8.26795 −1.13569 −0.567845 0.823135i \(-0.692223\pi\)
−0.567845 + 0.823135i \(0.692223\pi\)
\(54\) −1.00000 −0.136083
\(55\) −13.8564 −1.86840
\(56\) −1.00000 −0.133631
\(57\) 7.46410 0.988644
\(58\) −8.00000 −1.05045
\(59\) 6.26795 0.816017 0.408009 0.912978i \(-0.366223\pi\)
0.408009 + 0.912978i \(0.366223\pi\)
\(60\) −3.46410 −0.447214
\(61\) −5.19615 −0.665299 −0.332650 0.943051i \(-0.607943\pi\)
−0.332650 + 0.943051i \(0.607943\pi\)
\(62\) 6.46410 0.820942
\(63\) 1.00000 0.125988
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −4.00000 −0.492366
\(67\) −7.19615 −0.879150 −0.439575 0.898206i \(-0.644871\pi\)
−0.439575 + 0.898206i \(0.644871\pi\)
\(68\) 6.46410 0.783887
\(69\) 5.73205 0.690058
\(70\) 3.46410 0.414039
\(71\) 1.92820 0.228836 0.114418 0.993433i \(-0.463500\pi\)
0.114418 + 0.993433i \(0.463500\pi\)
\(72\) −1.00000 −0.117851
\(73\) −10.9282 −1.27905 −0.639525 0.768771i \(-0.720869\pi\)
−0.639525 + 0.768771i \(0.720869\pi\)
\(74\) 4.92820 0.572892
\(75\) 7.00000 0.808290
\(76\) 7.46410 0.856191
\(77\) 4.00000 0.455842
\(78\) 0 0
\(79\) −9.46410 −1.06479 −0.532397 0.846495i \(-0.678709\pi\)
−0.532397 + 0.846495i \(0.678709\pi\)
\(80\) −3.46410 −0.387298
\(81\) 1.00000 0.111111
\(82\) −6.92820 −0.765092
\(83\) 1.73205 0.190117 0.0950586 0.995472i \(-0.469696\pi\)
0.0950586 + 0.995472i \(0.469696\pi\)
\(84\) 1.00000 0.109109
\(85\) −22.3923 −2.42879
\(86\) −4.46410 −0.481376
\(87\) 8.00000 0.857690
\(88\) −4.00000 −0.426401
\(89\) −11.7321 −1.24359 −0.621797 0.783178i \(-0.713597\pi\)
−0.621797 + 0.783178i \(0.713597\pi\)
\(90\) 3.46410 0.365148
\(91\) 0 0
\(92\) 5.73205 0.597608
\(93\) −6.46410 −0.670296
\(94\) 0.535898 0.0552737
\(95\) −25.8564 −2.65281
\(96\) −1.00000 −0.102062
\(97\) 12.3923 1.25825 0.629124 0.777305i \(-0.283414\pi\)
0.629124 + 0.777305i \(0.283414\pi\)
\(98\) −1.00000 −0.101015
\(99\) 4.00000 0.402015
\(100\) 7.00000 0.700000
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) −6.46410 −0.640041
\(103\) 10.6603 1.05039 0.525193 0.850983i \(-0.323993\pi\)
0.525193 + 0.850983i \(0.323993\pi\)
\(104\) 0 0
\(105\) −3.46410 −0.338062
\(106\) 8.26795 0.803054
\(107\) 8.53590 0.825196 0.412598 0.910913i \(-0.364621\pi\)
0.412598 + 0.910913i \(0.364621\pi\)
\(108\) 1.00000 0.0962250
\(109\) −12.0000 −1.14939 −0.574696 0.818367i \(-0.694880\pi\)
−0.574696 + 0.818367i \(0.694880\pi\)
\(110\) 13.8564 1.32116
\(111\) −4.92820 −0.467764
\(112\) 1.00000 0.0944911
\(113\) −4.39230 −0.413193 −0.206597 0.978426i \(-0.566239\pi\)
−0.206597 + 0.978426i \(0.566239\pi\)
\(114\) −7.46410 −0.699077
\(115\) −19.8564 −1.85162
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) −6.26795 −0.577011
\(119\) 6.46410 0.592563
\(120\) 3.46410 0.316228
\(121\) 5.00000 0.454545
\(122\) 5.19615 0.470438
\(123\) 6.92820 0.624695
\(124\) −6.46410 −0.580493
\(125\) −6.92820 −0.619677
\(126\) −1.00000 −0.0890871
\(127\) 5.07180 0.450049 0.225025 0.974353i \(-0.427754\pi\)
0.225025 + 0.974353i \(0.427754\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 4.46410 0.393042
\(130\) 0 0
\(131\) 19.0000 1.66004 0.830019 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(132\) 4.00000 0.348155
\(133\) 7.46410 0.647220
\(134\) 7.19615 0.621653
\(135\) −3.46410 −0.298142
\(136\) −6.46410 −0.554292
\(137\) 13.8564 1.18383 0.591916 0.805999i \(-0.298372\pi\)
0.591916 + 0.805999i \(0.298372\pi\)
\(138\) −5.73205 −0.487945
\(139\) 2.92820 0.248367 0.124183 0.992259i \(-0.460369\pi\)
0.124183 + 0.992259i \(0.460369\pi\)
\(140\) −3.46410 −0.292770
\(141\) −0.535898 −0.0451308
\(142\) −1.92820 −0.161811
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) −27.7128 −2.30142
\(146\) 10.9282 0.904425
\(147\) 1.00000 0.0824786
\(148\) −4.92820 −0.405096
\(149\) −11.5359 −0.945058 −0.472529 0.881315i \(-0.656659\pi\)
−0.472529 + 0.881315i \(0.656659\pi\)
\(150\) −7.00000 −0.571548
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) −7.46410 −0.605419
\(153\) 6.46410 0.522592
\(154\) −4.00000 −0.322329
\(155\) 22.3923 1.79859
\(156\) 0 0
\(157\) −2.92820 −0.233696 −0.116848 0.993150i \(-0.537279\pi\)
−0.116848 + 0.993150i \(0.537279\pi\)
\(158\) 9.46410 0.752923
\(159\) −8.26795 −0.655691
\(160\) 3.46410 0.273861
\(161\) 5.73205 0.451749
\(162\) −1.00000 −0.0785674
\(163\) −1.33975 −0.104937 −0.0524685 0.998623i \(-0.516709\pi\)
−0.0524685 + 0.998623i \(0.516709\pi\)
\(164\) 6.92820 0.541002
\(165\) −13.8564 −1.07872
\(166\) −1.73205 −0.134433
\(167\) 20.0000 1.54765 0.773823 0.633402i \(-0.218342\pi\)
0.773823 + 0.633402i \(0.218342\pi\)
\(168\) −1.00000 −0.0771517
\(169\) 0 0
\(170\) 22.3923 1.71741
\(171\) 7.46410 0.570794
\(172\) 4.46410 0.340385
\(173\) 1.07180 0.0814872 0.0407436 0.999170i \(-0.487027\pi\)
0.0407436 + 0.999170i \(0.487027\pi\)
\(174\) −8.00000 −0.606478
\(175\) 7.00000 0.529150
\(176\) 4.00000 0.301511
\(177\) 6.26795 0.471128
\(178\) 11.7321 0.879354
\(179\) −16.9282 −1.26527 −0.632637 0.774449i \(-0.718027\pi\)
−0.632637 + 0.774449i \(0.718027\pi\)
\(180\) −3.46410 −0.258199
\(181\) 24.7846 1.84223 0.921113 0.389296i \(-0.127282\pi\)
0.921113 + 0.389296i \(0.127282\pi\)
\(182\) 0 0
\(183\) −5.19615 −0.384111
\(184\) −5.73205 −0.422572
\(185\) 17.0718 1.25514
\(186\) 6.46410 0.473971
\(187\) 25.8564 1.89081
\(188\) −0.535898 −0.0390844
\(189\) 1.00000 0.0727393
\(190\) 25.8564 1.87582
\(191\) −26.5167 −1.91868 −0.959339 0.282256i \(-0.908917\pi\)
−0.959339 + 0.282256i \(0.908917\pi\)
\(192\) 1.00000 0.0721688
\(193\) 17.8564 1.28533 0.642666 0.766146i \(-0.277828\pi\)
0.642666 + 0.766146i \(0.277828\pi\)
\(194\) −12.3923 −0.889716
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −2.60770 −0.185791 −0.0928953 0.995676i \(-0.529612\pi\)
−0.0928953 + 0.995676i \(0.529612\pi\)
\(198\) −4.00000 −0.284268
\(199\) −22.1244 −1.56835 −0.784177 0.620537i \(-0.786915\pi\)
−0.784177 + 0.620537i \(0.786915\pi\)
\(200\) −7.00000 −0.494975
\(201\) −7.19615 −0.507577
\(202\) 0 0
\(203\) 8.00000 0.561490
\(204\) 6.46410 0.452578
\(205\) −24.0000 −1.67623
\(206\) −10.6603 −0.742735
\(207\) 5.73205 0.398405
\(208\) 0 0
\(209\) 29.8564 2.06521
\(210\) 3.46410 0.239046
\(211\) −9.07180 −0.624528 −0.312264 0.949995i \(-0.601087\pi\)
−0.312264 + 0.949995i \(0.601087\pi\)
\(212\) −8.26795 −0.567845
\(213\) 1.92820 0.132118
\(214\) −8.53590 −0.583502
\(215\) −15.4641 −1.05464
\(216\) −1.00000 −0.0680414
\(217\) −6.46410 −0.438812
\(218\) 12.0000 0.812743
\(219\) −10.9282 −0.738460
\(220\) −13.8564 −0.934199
\(221\) 0 0
\(222\) 4.92820 0.330759
\(223\) 7.53590 0.504641 0.252321 0.967644i \(-0.418806\pi\)
0.252321 + 0.967644i \(0.418806\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 7.00000 0.466667
\(226\) 4.39230 0.292172
\(227\) −13.3205 −0.884113 −0.442057 0.896987i \(-0.645751\pi\)
−0.442057 + 0.896987i \(0.645751\pi\)
\(228\) 7.46410 0.494322
\(229\) 19.9282 1.31689 0.658446 0.752628i \(-0.271214\pi\)
0.658446 + 0.752628i \(0.271214\pi\)
\(230\) 19.8564 1.30929
\(231\) 4.00000 0.263181
\(232\) −8.00000 −0.525226
\(233\) 13.4641 0.882063 0.441031 0.897492i \(-0.354613\pi\)
0.441031 + 0.897492i \(0.354613\pi\)
\(234\) 0 0
\(235\) 1.85641 0.121099
\(236\) 6.26795 0.408009
\(237\) −9.46410 −0.614759
\(238\) −6.46410 −0.419005
\(239\) 12.8564 0.831612 0.415806 0.909453i \(-0.363500\pi\)
0.415806 + 0.909453i \(0.363500\pi\)
\(240\) −3.46410 −0.223607
\(241\) −15.8564 −1.02140 −0.510700 0.859759i \(-0.670614\pi\)
−0.510700 + 0.859759i \(0.670614\pi\)
\(242\) −5.00000 −0.321412
\(243\) 1.00000 0.0641500
\(244\) −5.19615 −0.332650
\(245\) −3.46410 −0.221313
\(246\) −6.92820 −0.441726
\(247\) 0 0
\(248\) 6.46410 0.410471
\(249\) 1.73205 0.109764
\(250\) 6.92820 0.438178
\(251\) −10.0718 −0.635726 −0.317863 0.948137i \(-0.602965\pi\)
−0.317863 + 0.948137i \(0.602965\pi\)
\(252\) 1.00000 0.0629941
\(253\) 22.9282 1.44148
\(254\) −5.07180 −0.318233
\(255\) −22.3923 −1.40226
\(256\) 1.00000 0.0625000
\(257\) 0.464102 0.0289499 0.0144749 0.999895i \(-0.495392\pi\)
0.0144749 + 0.999895i \(0.495392\pi\)
\(258\) −4.46410 −0.277923
\(259\) −4.92820 −0.306224
\(260\) 0 0
\(261\) 8.00000 0.495188
\(262\) −19.0000 −1.17382
\(263\) −17.3205 −1.06803 −0.534014 0.845476i \(-0.679317\pi\)
−0.534014 + 0.845476i \(0.679317\pi\)
\(264\) −4.00000 −0.246183
\(265\) 28.6410 1.75940
\(266\) −7.46410 −0.457653
\(267\) −11.7321 −0.717990
\(268\) −7.19615 −0.439575
\(269\) 22.3923 1.36528 0.682641 0.730753i \(-0.260831\pi\)
0.682641 + 0.730753i \(0.260831\pi\)
\(270\) 3.46410 0.210819
\(271\) 12.4641 0.757140 0.378570 0.925573i \(-0.376416\pi\)
0.378570 + 0.925573i \(0.376416\pi\)
\(272\) 6.46410 0.391944
\(273\) 0 0
\(274\) −13.8564 −0.837096
\(275\) 28.0000 1.68846
\(276\) 5.73205 0.345029
\(277\) 7.32051 0.439847 0.219923 0.975517i \(-0.429419\pi\)
0.219923 + 0.975517i \(0.429419\pi\)
\(278\) −2.92820 −0.175622
\(279\) −6.46410 −0.386996
\(280\) 3.46410 0.207020
\(281\) −2.39230 −0.142713 −0.0713565 0.997451i \(-0.522733\pi\)
−0.0713565 + 0.997451i \(0.522733\pi\)
\(282\) 0.535898 0.0319123
\(283\) −11.4641 −0.681470 −0.340735 0.940159i \(-0.610676\pi\)
−0.340735 + 0.940159i \(0.610676\pi\)
\(284\) 1.92820 0.114418
\(285\) −25.8564 −1.53160
\(286\) 0 0
\(287\) 6.92820 0.408959
\(288\) −1.00000 −0.0589256
\(289\) 24.7846 1.45792
\(290\) 27.7128 1.62735
\(291\) 12.3923 0.726450
\(292\) −10.9282 −0.639525
\(293\) 33.4641 1.95499 0.977497 0.210950i \(-0.0676557\pi\)
0.977497 + 0.210950i \(0.0676557\pi\)
\(294\) −1.00000 −0.0583212
\(295\) −21.7128 −1.26417
\(296\) 4.92820 0.286446
\(297\) 4.00000 0.232104
\(298\) 11.5359 0.668257
\(299\) 0 0
\(300\) 7.00000 0.404145
\(301\) 4.46410 0.257307
\(302\) 12.0000 0.690522
\(303\) 0 0
\(304\) 7.46410 0.428096
\(305\) 18.0000 1.03068
\(306\) −6.46410 −0.369528
\(307\) −22.0000 −1.25561 −0.627803 0.778372i \(-0.716046\pi\)
−0.627803 + 0.778372i \(0.716046\pi\)
\(308\) 4.00000 0.227921
\(309\) 10.6603 0.606441
\(310\) −22.3923 −1.27180
\(311\) 13.3205 0.755337 0.377668 0.925941i \(-0.376726\pi\)
0.377668 + 0.925941i \(0.376726\pi\)
\(312\) 0 0
\(313\) 0.679492 0.0384072 0.0192036 0.999816i \(-0.493887\pi\)
0.0192036 + 0.999816i \(0.493887\pi\)
\(314\) 2.92820 0.165248
\(315\) −3.46410 −0.195180
\(316\) −9.46410 −0.532397
\(317\) 16.4641 0.924716 0.462358 0.886693i \(-0.347003\pi\)
0.462358 + 0.886693i \(0.347003\pi\)
\(318\) 8.26795 0.463644
\(319\) 32.0000 1.79166
\(320\) −3.46410 −0.193649
\(321\) 8.53590 0.476427
\(322\) −5.73205 −0.319435
\(323\) 48.2487 2.68463
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 1.33975 0.0742017
\(327\) −12.0000 −0.663602
\(328\) −6.92820 −0.382546
\(329\) −0.535898 −0.0295450
\(330\) 13.8564 0.762770
\(331\) 21.3205 1.17188 0.585941 0.810354i \(-0.300725\pi\)
0.585941 + 0.810354i \(0.300725\pi\)
\(332\) 1.73205 0.0950586
\(333\) −4.92820 −0.270064
\(334\) −20.0000 −1.09435
\(335\) 24.9282 1.36197
\(336\) 1.00000 0.0545545
\(337\) 6.00000 0.326841 0.163420 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 0 0
\(339\) −4.39230 −0.238557
\(340\) −22.3923 −1.21439
\(341\) −25.8564 −1.40020
\(342\) −7.46410 −0.403612
\(343\) 1.00000 0.0539949
\(344\) −4.46410 −0.240688
\(345\) −19.8564 −1.06903
\(346\) −1.07180 −0.0576202
\(347\) 25.7128 1.38034 0.690168 0.723649i \(-0.257537\pi\)
0.690168 + 0.723649i \(0.257537\pi\)
\(348\) 8.00000 0.428845
\(349\) −35.6410 −1.90782 −0.953910 0.300093i \(-0.902982\pi\)
−0.953910 + 0.300093i \(0.902982\pi\)
\(350\) −7.00000 −0.374166
\(351\) 0 0
\(352\) −4.00000 −0.213201
\(353\) −9.33975 −0.497105 −0.248552 0.968618i \(-0.579955\pi\)
−0.248552 + 0.968618i \(0.579955\pi\)
\(354\) −6.26795 −0.333138
\(355\) −6.67949 −0.354511
\(356\) −11.7321 −0.621797
\(357\) 6.46410 0.342117
\(358\) 16.9282 0.894683
\(359\) −8.00000 −0.422224 −0.211112 0.977462i \(-0.567708\pi\)
−0.211112 + 0.977462i \(0.567708\pi\)
\(360\) 3.46410 0.182574
\(361\) 36.7128 1.93225
\(362\) −24.7846 −1.30265
\(363\) 5.00000 0.262432
\(364\) 0 0
\(365\) 37.8564 1.98149
\(366\) 5.19615 0.271607
\(367\) −33.0526 −1.72533 −0.862665 0.505776i \(-0.831206\pi\)
−0.862665 + 0.505776i \(0.831206\pi\)
\(368\) 5.73205 0.298804
\(369\) 6.92820 0.360668
\(370\) −17.0718 −0.887520
\(371\) −8.26795 −0.429251
\(372\) −6.46410 −0.335148
\(373\) −26.2487 −1.35911 −0.679553 0.733626i \(-0.737826\pi\)
−0.679553 + 0.733626i \(0.737826\pi\)
\(374\) −25.8564 −1.33700
\(375\) −6.92820 −0.357771
\(376\) 0.535898 0.0276368
\(377\) 0 0
\(378\) −1.00000 −0.0514344
\(379\) 9.60770 0.493514 0.246757 0.969077i \(-0.420635\pi\)
0.246757 + 0.969077i \(0.420635\pi\)
\(380\) −25.8564 −1.32641
\(381\) 5.07180 0.259836
\(382\) 26.5167 1.35671
\(383\) 7.32051 0.374060 0.187030 0.982354i \(-0.440114\pi\)
0.187030 + 0.982354i \(0.440114\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −13.8564 −0.706188
\(386\) −17.8564 −0.908867
\(387\) 4.46410 0.226923
\(388\) 12.3923 0.629124
\(389\) −14.6603 −0.743304 −0.371652 0.928372i \(-0.621209\pi\)
−0.371652 + 0.928372i \(0.621209\pi\)
\(390\) 0 0
\(391\) 37.0526 1.87383
\(392\) −1.00000 −0.0505076
\(393\) 19.0000 0.958423
\(394\) 2.60770 0.131374
\(395\) 32.7846 1.64957
\(396\) 4.00000 0.201008
\(397\) 29.7846 1.49485 0.747423 0.664348i \(-0.231291\pi\)
0.747423 + 0.664348i \(0.231291\pi\)
\(398\) 22.1244 1.10899
\(399\) 7.46410 0.373672
\(400\) 7.00000 0.350000
\(401\) −16.3923 −0.818593 −0.409296 0.912402i \(-0.634226\pi\)
−0.409296 + 0.912402i \(0.634226\pi\)
\(402\) 7.19615 0.358911
\(403\) 0 0
\(404\) 0 0
\(405\) −3.46410 −0.172133
\(406\) −8.00000 −0.397033
\(407\) −19.7128 −0.977128
\(408\) −6.46410 −0.320021
\(409\) 6.39230 0.316079 0.158040 0.987433i \(-0.449483\pi\)
0.158040 + 0.987433i \(0.449483\pi\)
\(410\) 24.0000 1.18528
\(411\) 13.8564 0.683486
\(412\) 10.6603 0.525193
\(413\) 6.26795 0.308426
\(414\) −5.73205 −0.281715
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) 2.92820 0.143395
\(418\) −29.8564 −1.46032
\(419\) −5.00000 −0.244266 −0.122133 0.992514i \(-0.538973\pi\)
−0.122133 + 0.992514i \(0.538973\pi\)
\(420\) −3.46410 −0.169031
\(421\) −1.60770 −0.0783543 −0.0391771 0.999232i \(-0.512474\pi\)
−0.0391771 + 0.999232i \(0.512474\pi\)
\(422\) 9.07180 0.441608
\(423\) −0.535898 −0.0260563
\(424\) 8.26795 0.401527
\(425\) 45.2487 2.19488
\(426\) −1.92820 −0.0934218
\(427\) −5.19615 −0.251459
\(428\) 8.53590 0.412598
\(429\) 0 0
\(430\) 15.4641 0.745745
\(431\) −4.85641 −0.233925 −0.116962 0.993136i \(-0.537316\pi\)
−0.116962 + 0.993136i \(0.537316\pi\)
\(432\) 1.00000 0.0481125
\(433\) 9.85641 0.473669 0.236834 0.971550i \(-0.423890\pi\)
0.236834 + 0.971550i \(0.423890\pi\)
\(434\) 6.46410 0.310287
\(435\) −27.7128 −1.32873
\(436\) −12.0000 −0.574696
\(437\) 42.7846 2.04667
\(438\) 10.9282 0.522170
\(439\) −13.6077 −0.649460 −0.324730 0.945807i \(-0.605273\pi\)
−0.324730 + 0.945807i \(0.605273\pi\)
\(440\) 13.8564 0.660578
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −22.3923 −1.06389 −0.531945 0.846779i \(-0.678539\pi\)
−0.531945 + 0.846779i \(0.678539\pi\)
\(444\) −4.92820 −0.233882
\(445\) 40.6410 1.92657
\(446\) −7.53590 −0.356835
\(447\) −11.5359 −0.545629
\(448\) 1.00000 0.0472456
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) −7.00000 −0.329983
\(451\) 27.7128 1.30495
\(452\) −4.39230 −0.206597
\(453\) −12.0000 −0.563809
\(454\) 13.3205 0.625162
\(455\) 0 0
\(456\) −7.46410 −0.349539
\(457\) −1.19615 −0.0559537 −0.0279768 0.999609i \(-0.508906\pi\)
−0.0279768 + 0.999609i \(0.508906\pi\)
\(458\) −19.9282 −0.931184
\(459\) 6.46410 0.301718
\(460\) −19.8564 −0.925810
\(461\) −5.07180 −0.236217 −0.118109 0.993001i \(-0.537683\pi\)
−0.118109 + 0.993001i \(0.537683\pi\)
\(462\) −4.00000 −0.186097
\(463\) 8.24871 0.383350 0.191675 0.981458i \(-0.438608\pi\)
0.191675 + 0.981458i \(0.438608\pi\)
\(464\) 8.00000 0.371391
\(465\) 22.3923 1.03842
\(466\) −13.4641 −0.623712
\(467\) −39.6410 −1.83437 −0.917184 0.398465i \(-0.869543\pi\)
−0.917184 + 0.398465i \(0.869543\pi\)
\(468\) 0 0
\(469\) −7.19615 −0.332287
\(470\) −1.85641 −0.0856296
\(471\) −2.92820 −0.134924
\(472\) −6.26795 −0.288506
\(473\) 17.8564 0.821038
\(474\) 9.46410 0.434701
\(475\) 52.2487 2.39734
\(476\) 6.46410 0.296282
\(477\) −8.26795 −0.378563
\(478\) −12.8564 −0.588038
\(479\) −4.39230 −0.200690 −0.100345 0.994953i \(-0.531995\pi\)
−0.100345 + 0.994953i \(0.531995\pi\)
\(480\) 3.46410 0.158114
\(481\) 0 0
\(482\) 15.8564 0.722240
\(483\) 5.73205 0.260817
\(484\) 5.00000 0.227273
\(485\) −42.9282 −1.94927
\(486\) −1.00000 −0.0453609
\(487\) 0.392305 0.0177770 0.00888851 0.999960i \(-0.497171\pi\)
0.00888851 + 0.999960i \(0.497171\pi\)
\(488\) 5.19615 0.235219
\(489\) −1.33975 −0.0605854
\(490\) 3.46410 0.156492
\(491\) −9.85641 −0.444813 −0.222407 0.974954i \(-0.571391\pi\)
−0.222407 + 0.974954i \(0.571391\pi\)
\(492\) 6.92820 0.312348
\(493\) 51.7128 2.32903
\(494\) 0 0
\(495\) −13.8564 −0.622799
\(496\) −6.46410 −0.290247
\(497\) 1.92820 0.0864917
\(498\) −1.73205 −0.0776151
\(499\) −0.267949 −0.0119951 −0.00599753 0.999982i \(-0.501909\pi\)
−0.00599753 + 0.999982i \(0.501909\pi\)
\(500\) −6.92820 −0.309839
\(501\) 20.0000 0.893534
\(502\) 10.0718 0.449526
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) −1.00000 −0.0445435
\(505\) 0 0
\(506\) −22.9282 −1.01928
\(507\) 0 0
\(508\) 5.07180 0.225025
\(509\) 34.2487 1.51805 0.759024 0.651063i \(-0.225677\pi\)
0.759024 + 0.651063i \(0.225677\pi\)
\(510\) 22.3923 0.991548
\(511\) −10.9282 −0.483435
\(512\) −1.00000 −0.0441942
\(513\) 7.46410 0.329548
\(514\) −0.464102 −0.0204706
\(515\) −36.9282 −1.62725
\(516\) 4.46410 0.196521
\(517\) −2.14359 −0.0942751
\(518\) 4.92820 0.216533
\(519\) 1.07180 0.0470467
\(520\) 0 0
\(521\) −30.0000 −1.31432 −0.657162 0.753749i \(-0.728243\pi\)
−0.657162 + 0.753749i \(0.728243\pi\)
\(522\) −8.00000 −0.350150
\(523\) 14.9282 0.652765 0.326382 0.945238i \(-0.394170\pi\)
0.326382 + 0.945238i \(0.394170\pi\)
\(524\) 19.0000 0.830019
\(525\) 7.00000 0.305505
\(526\) 17.3205 0.755210
\(527\) −41.7846 −1.82017
\(528\) 4.00000 0.174078
\(529\) 9.85641 0.428539
\(530\) −28.6410 −1.24409
\(531\) 6.26795 0.272006
\(532\) 7.46410 0.323610
\(533\) 0 0
\(534\) 11.7321 0.507695
\(535\) −29.5692 −1.27839
\(536\) 7.19615 0.310826
\(537\) −16.9282 −0.730506
\(538\) −22.3923 −0.965401
\(539\) 4.00000 0.172292
\(540\) −3.46410 −0.149071
\(541\) −17.3205 −0.744667 −0.372333 0.928099i \(-0.621442\pi\)
−0.372333 + 0.928099i \(0.621442\pi\)
\(542\) −12.4641 −0.535379
\(543\) 24.7846 1.06361
\(544\) −6.46410 −0.277146
\(545\) 41.5692 1.78063
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 13.8564 0.591916
\(549\) −5.19615 −0.221766
\(550\) −28.0000 −1.19392
\(551\) 59.7128 2.54385
\(552\) −5.73205 −0.243972
\(553\) −9.46410 −0.402455
\(554\) −7.32051 −0.311019
\(555\) 17.0718 0.724657
\(556\) 2.92820 0.124183
\(557\) 24.6077 1.04266 0.521331 0.853355i \(-0.325436\pi\)
0.521331 + 0.853355i \(0.325436\pi\)
\(558\) 6.46410 0.273647
\(559\) 0 0
\(560\) −3.46410 −0.146385
\(561\) 25.8564 1.09166
\(562\) 2.39230 0.100913
\(563\) −13.8564 −0.583978 −0.291989 0.956422i \(-0.594317\pi\)
−0.291989 + 0.956422i \(0.594317\pi\)
\(564\) −0.535898 −0.0225654
\(565\) 15.2154 0.640116
\(566\) 11.4641 0.481872
\(567\) 1.00000 0.0419961
\(568\) −1.92820 −0.0809056
\(569\) 15.4641 0.648289 0.324144 0.946008i \(-0.394924\pi\)
0.324144 + 0.946008i \(0.394924\pi\)
\(570\) 25.8564 1.08301
\(571\) −3.39230 −0.141964 −0.0709818 0.997478i \(-0.522613\pi\)
−0.0709818 + 0.997478i \(0.522613\pi\)
\(572\) 0 0
\(573\) −26.5167 −1.10775
\(574\) −6.92820 −0.289178
\(575\) 40.1244 1.67330
\(576\) 1.00000 0.0416667
\(577\) 1.21539 0.0505974 0.0252987 0.999680i \(-0.491946\pi\)
0.0252987 + 0.999680i \(0.491946\pi\)
\(578\) −24.7846 −1.03090
\(579\) 17.8564 0.742087
\(580\) −27.7128 −1.15071
\(581\) 1.73205 0.0718576
\(582\) −12.3923 −0.513678
\(583\) −33.0718 −1.36969
\(584\) 10.9282 0.452212
\(585\) 0 0
\(586\) −33.4641 −1.38239
\(587\) −8.12436 −0.335328 −0.167664 0.985844i \(-0.553622\pi\)
−0.167664 + 0.985844i \(0.553622\pi\)
\(588\) 1.00000 0.0412393
\(589\) −48.2487 −1.98805
\(590\) 21.7128 0.893902
\(591\) −2.60770 −0.107266
\(592\) −4.92820 −0.202548
\(593\) 4.26795 0.175264 0.0876318 0.996153i \(-0.472070\pi\)
0.0876318 + 0.996153i \(0.472070\pi\)
\(594\) −4.00000 −0.164122
\(595\) −22.3923 −0.917995
\(596\) −11.5359 −0.472529
\(597\) −22.1244 −0.905490
\(598\) 0 0
\(599\) −19.0526 −0.778466 −0.389233 0.921139i \(-0.627260\pi\)
−0.389233 + 0.921139i \(0.627260\pi\)
\(600\) −7.00000 −0.285774
\(601\) −24.2487 −0.989126 −0.494563 0.869142i \(-0.664672\pi\)
−0.494563 + 0.869142i \(0.664672\pi\)
\(602\) −4.46410 −0.181943
\(603\) −7.19615 −0.293050
\(604\) −12.0000 −0.488273
\(605\) −17.3205 −0.704179
\(606\) 0 0
\(607\) −15.1962 −0.616793 −0.308396 0.951258i \(-0.599792\pi\)
−0.308396 + 0.951258i \(0.599792\pi\)
\(608\) −7.46410 −0.302709
\(609\) 8.00000 0.324176
\(610\) −18.0000 −0.728799
\(611\) 0 0
\(612\) 6.46410 0.261296
\(613\) −34.3923 −1.38909 −0.694546 0.719448i \(-0.744395\pi\)
−0.694546 + 0.719448i \(0.744395\pi\)
\(614\) 22.0000 0.887848
\(615\) −24.0000 −0.967773
\(616\) −4.00000 −0.161165
\(617\) −5.85641 −0.235770 −0.117885 0.993027i \(-0.537611\pi\)
−0.117885 + 0.993027i \(0.537611\pi\)
\(618\) −10.6603 −0.428818
\(619\) 30.7846 1.23734 0.618669 0.785652i \(-0.287672\pi\)
0.618669 + 0.785652i \(0.287672\pi\)
\(620\) 22.3923 0.899297
\(621\) 5.73205 0.230019
\(622\) −13.3205 −0.534104
\(623\) −11.7321 −0.470035
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) −0.679492 −0.0271580
\(627\) 29.8564 1.19235
\(628\) −2.92820 −0.116848
\(629\) −31.8564 −1.27020
\(630\) 3.46410 0.138013
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) 9.46410 0.376462
\(633\) −9.07180 −0.360572
\(634\) −16.4641 −0.653873
\(635\) −17.5692 −0.697213
\(636\) −8.26795 −0.327846
\(637\) 0 0
\(638\) −32.0000 −1.26689
\(639\) 1.92820 0.0762785
\(640\) 3.46410 0.136931
\(641\) −26.6410 −1.05226 −0.526128 0.850405i \(-0.676357\pi\)
−0.526128 + 0.850405i \(0.676357\pi\)
\(642\) −8.53590 −0.336885
\(643\) −20.2487 −0.798531 −0.399266 0.916835i \(-0.630735\pi\)
−0.399266 + 0.916835i \(0.630735\pi\)
\(644\) 5.73205 0.225874
\(645\) −15.4641 −0.608898
\(646\) −48.2487 −1.89832
\(647\) 37.3205 1.46722 0.733610 0.679570i \(-0.237834\pi\)
0.733610 + 0.679570i \(0.237834\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 25.0718 0.984154
\(650\) 0 0
\(651\) −6.46410 −0.253348
\(652\) −1.33975 −0.0524685
\(653\) −37.5885 −1.47095 −0.735475 0.677552i \(-0.763041\pi\)
−0.735475 + 0.677552i \(0.763041\pi\)
\(654\) 12.0000 0.469237
\(655\) −65.8179 −2.57172
\(656\) 6.92820 0.270501
\(657\) −10.9282 −0.426350
\(658\) 0.535898 0.0208915
\(659\) 1.60770 0.0626269 0.0313135 0.999510i \(-0.490031\pi\)
0.0313135 + 0.999510i \(0.490031\pi\)
\(660\) −13.8564 −0.539360
\(661\) 50.7128 1.97250 0.986250 0.165261i \(-0.0528466\pi\)
0.986250 + 0.165261i \(0.0528466\pi\)
\(662\) −21.3205 −0.828645
\(663\) 0 0
\(664\) −1.73205 −0.0672166
\(665\) −25.8564 −1.00267
\(666\) 4.92820 0.190964
\(667\) 45.8564 1.77557
\(668\) 20.0000 0.773823
\(669\) 7.53590 0.291355
\(670\) −24.9282 −0.963061
\(671\) −20.7846 −0.802381
\(672\) −1.00000 −0.0385758
\(673\) 39.6410 1.52805 0.764024 0.645187i \(-0.223221\pi\)
0.764024 + 0.645187i \(0.223221\pi\)
\(674\) −6.00000 −0.231111
\(675\) 7.00000 0.269430
\(676\) 0 0
\(677\) 1.07180 0.0411925 0.0205962 0.999788i \(-0.493444\pi\)
0.0205962 + 0.999788i \(0.493444\pi\)
\(678\) 4.39230 0.168685
\(679\) 12.3923 0.475573
\(680\) 22.3923 0.858706
\(681\) −13.3205 −0.510443
\(682\) 25.8564 0.990093
\(683\) −51.1769 −1.95823 −0.979115 0.203307i \(-0.934831\pi\)
−0.979115 + 0.203307i \(0.934831\pi\)
\(684\) 7.46410 0.285397
\(685\) −48.0000 −1.83399
\(686\) −1.00000 −0.0381802
\(687\) 19.9282 0.760308
\(688\) 4.46410 0.170192
\(689\) 0 0
\(690\) 19.8564 0.755920
\(691\) 41.1769 1.56644 0.783222 0.621742i \(-0.213575\pi\)
0.783222 + 0.621742i \(0.213575\pi\)
\(692\) 1.07180 0.0407436
\(693\) 4.00000 0.151947
\(694\) −25.7128 −0.976045
\(695\) −10.1436 −0.384768
\(696\) −8.00000 −0.303239
\(697\) 44.7846 1.69634
\(698\) 35.6410 1.34903
\(699\) 13.4641 0.509259
\(700\) 7.00000 0.264575
\(701\) −2.12436 −0.0802358 −0.0401179 0.999195i \(-0.512773\pi\)
−0.0401179 + 0.999195i \(0.512773\pi\)
\(702\) 0 0
\(703\) −36.7846 −1.38736
\(704\) 4.00000 0.150756
\(705\) 1.85641 0.0699163
\(706\) 9.33975 0.351506
\(707\) 0 0
\(708\) 6.26795 0.235564
\(709\) 7.71281 0.289661 0.144830 0.989456i \(-0.453736\pi\)
0.144830 + 0.989456i \(0.453736\pi\)
\(710\) 6.67949 0.250677
\(711\) −9.46410 −0.354932
\(712\) 11.7321 0.439677
\(713\) −37.0526 −1.38763
\(714\) −6.46410 −0.241913
\(715\) 0 0
\(716\) −16.9282 −0.632637
\(717\) 12.8564 0.480131
\(718\) 8.00000 0.298557
\(719\) 34.5359 1.28797 0.643986 0.765037i \(-0.277280\pi\)
0.643986 + 0.765037i \(0.277280\pi\)
\(720\) −3.46410 −0.129099
\(721\) 10.6603 0.397009
\(722\) −36.7128 −1.36631
\(723\) −15.8564 −0.589706
\(724\) 24.7846 0.921113
\(725\) 56.0000 2.07979
\(726\) −5.00000 −0.185567
\(727\) −22.9090 −0.849646 −0.424823 0.905276i \(-0.639664\pi\)
−0.424823 + 0.905276i \(0.639664\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −37.8564 −1.40113
\(731\) 28.8564 1.06729
\(732\) −5.19615 −0.192055
\(733\) 4.85641 0.179375 0.0896877 0.995970i \(-0.471413\pi\)
0.0896877 + 0.995970i \(0.471413\pi\)
\(734\) 33.0526 1.21999
\(735\) −3.46410 −0.127775
\(736\) −5.73205 −0.211286
\(737\) −28.7846 −1.06029
\(738\) −6.92820 −0.255031
\(739\) 7.73205 0.284428 0.142214 0.989836i \(-0.454578\pi\)
0.142214 + 0.989836i \(0.454578\pi\)
\(740\) 17.0718 0.627572
\(741\) 0 0
\(742\) 8.26795 0.303526
\(743\) −1.78461 −0.0654710 −0.0327355 0.999464i \(-0.510422\pi\)
−0.0327355 + 0.999464i \(0.510422\pi\)
\(744\) 6.46410 0.236985
\(745\) 39.9615 1.46408
\(746\) 26.2487 0.961034
\(747\) 1.73205 0.0633724
\(748\) 25.8564 0.945404
\(749\) 8.53590 0.311895
\(750\) 6.92820 0.252982
\(751\) 29.3205 1.06992 0.534960 0.844877i \(-0.320327\pi\)
0.534960 + 0.844877i \(0.320327\pi\)
\(752\) −0.535898 −0.0195422
\(753\) −10.0718 −0.367037
\(754\) 0 0
\(755\) 41.5692 1.51286
\(756\) 1.00000 0.0363696
\(757\) 19.8564 0.721693 0.360847 0.932625i \(-0.382488\pi\)
0.360847 + 0.932625i \(0.382488\pi\)
\(758\) −9.60770 −0.348967
\(759\) 22.9282 0.832241
\(760\) 25.8564 0.937910
\(761\) −6.92820 −0.251147 −0.125574 0.992084i \(-0.540077\pi\)
−0.125574 + 0.992084i \(0.540077\pi\)
\(762\) −5.07180 −0.183732
\(763\) −12.0000 −0.434429
\(764\) −26.5167 −0.959339
\(765\) −22.3923 −0.809595
\(766\) −7.32051 −0.264501
\(767\) 0 0
\(768\) 1.00000 0.0360844
\(769\) −42.6410 −1.53767 −0.768837 0.639445i \(-0.779164\pi\)
−0.768837 + 0.639445i \(0.779164\pi\)
\(770\) 13.8564 0.499350
\(771\) 0.464102 0.0167142
\(772\) 17.8564 0.642666
\(773\) 12.9282 0.464995 0.232498 0.972597i \(-0.425310\pi\)
0.232498 + 0.972597i \(0.425310\pi\)
\(774\) −4.46410 −0.160459
\(775\) −45.2487 −1.62538
\(776\) −12.3923 −0.444858
\(777\) −4.92820 −0.176798
\(778\) 14.6603 0.525596
\(779\) 51.7128 1.85280
\(780\) 0 0
\(781\) 7.71281 0.275986
\(782\) −37.0526 −1.32500
\(783\) 8.00000 0.285897
\(784\) 1.00000 0.0357143
\(785\) 10.1436 0.362040
\(786\) −19.0000 −0.677708
\(787\) 8.39230 0.299153 0.149577 0.988750i \(-0.452209\pi\)
0.149577 + 0.988750i \(0.452209\pi\)
\(788\) −2.60770 −0.0928953
\(789\) −17.3205 −0.616626
\(790\) −32.7846 −1.16642
\(791\) −4.39230 −0.156172
\(792\) −4.00000 −0.142134
\(793\) 0 0
\(794\) −29.7846 −1.05702
\(795\) 28.6410 1.01579
\(796\) −22.1244 −0.784177
\(797\) 42.2487 1.49653 0.748263 0.663402i \(-0.230888\pi\)
0.748263 + 0.663402i \(0.230888\pi\)
\(798\) −7.46410 −0.264226
\(799\) −3.46410 −0.122551
\(800\) −7.00000 −0.247487
\(801\) −11.7321 −0.414532
\(802\) 16.3923 0.578832
\(803\) −43.7128 −1.54259
\(804\) −7.19615 −0.253789
\(805\) −19.8564 −0.699846
\(806\) 0 0
\(807\) 22.3923 0.788246
\(808\) 0 0
\(809\) 15.7128 0.552433 0.276217 0.961095i \(-0.410919\pi\)
0.276217 + 0.961095i \(0.410919\pi\)
\(810\) 3.46410 0.121716
\(811\) 2.14359 0.0752717 0.0376359 0.999292i \(-0.488017\pi\)
0.0376359 + 0.999292i \(0.488017\pi\)
\(812\) 8.00000 0.280745
\(813\) 12.4641 0.437135
\(814\) 19.7128 0.690934
\(815\) 4.64102 0.162568
\(816\) 6.46410 0.226289
\(817\) 33.3205 1.16574
\(818\) −6.39230 −0.223502
\(819\) 0 0
\(820\) −24.0000 −0.838116
\(821\) −31.2487 −1.09059 −0.545294 0.838245i \(-0.683582\pi\)
−0.545294 + 0.838245i \(0.683582\pi\)
\(822\) −13.8564 −0.483298
\(823\) 23.6077 0.822913 0.411456 0.911430i \(-0.365020\pi\)
0.411456 + 0.911430i \(0.365020\pi\)
\(824\) −10.6603 −0.371368
\(825\) 28.0000 0.974835
\(826\) −6.26795 −0.218090
\(827\) 36.3923 1.26548 0.632742 0.774363i \(-0.281929\pi\)
0.632742 + 0.774363i \(0.281929\pi\)
\(828\) 5.73205 0.199203
\(829\) 7.21539 0.250601 0.125300 0.992119i \(-0.460011\pi\)
0.125300 + 0.992119i \(0.460011\pi\)
\(830\) 6.00000 0.208263
\(831\) 7.32051 0.253946
\(832\) 0 0
\(833\) 6.46410 0.223968
\(834\) −2.92820 −0.101395
\(835\) −69.2820 −2.39760
\(836\) 29.8564 1.03261
\(837\) −6.46410 −0.223432
\(838\) 5.00000 0.172722
\(839\) 1.60770 0.0555038 0.0277519 0.999615i \(-0.491165\pi\)
0.0277519 + 0.999615i \(0.491165\pi\)
\(840\) 3.46410 0.119523
\(841\) 35.0000 1.20690
\(842\) 1.60770 0.0554048
\(843\) −2.39230 −0.0823954
\(844\) −9.07180 −0.312264
\(845\) 0 0
\(846\) 0.535898 0.0184246
\(847\) 5.00000 0.171802
\(848\) −8.26795 −0.283923
\(849\) −11.4641 −0.393447
\(850\) −45.2487 −1.55202
\(851\) −28.2487 −0.968353
\(852\) 1.92820 0.0660592
\(853\) 47.6410 1.63120 0.815599 0.578618i \(-0.196408\pi\)
0.815599 + 0.578618i \(0.196408\pi\)
\(854\) 5.19615 0.177809
\(855\) −25.8564 −0.884270
\(856\) −8.53590 −0.291751
\(857\) −33.7128 −1.15161 −0.575804 0.817588i \(-0.695311\pi\)
−0.575804 + 0.817588i \(0.695311\pi\)
\(858\) 0 0
\(859\) −15.8564 −0.541014 −0.270507 0.962718i \(-0.587191\pi\)
−0.270507 + 0.962718i \(0.587191\pi\)
\(860\) −15.4641 −0.527321
\(861\) 6.92820 0.236113
\(862\) 4.85641 0.165410
\(863\) 42.9282 1.46129 0.730647 0.682756i \(-0.239219\pi\)
0.730647 + 0.682756i \(0.239219\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −3.71281 −0.126239
\(866\) −9.85641 −0.334934
\(867\) 24.7846 0.841729
\(868\) −6.46410 −0.219406
\(869\) −37.8564 −1.28419
\(870\) 27.7128 0.939552
\(871\) 0 0
\(872\) 12.0000 0.406371
\(873\) 12.3923 0.419416
\(874\) −42.7846 −1.44721
\(875\) −6.92820 −0.234216
\(876\) −10.9282 −0.369230
\(877\) 25.7128 0.868260 0.434130 0.900850i \(-0.357056\pi\)
0.434130 + 0.900850i \(0.357056\pi\)
\(878\) 13.6077 0.459237
\(879\) 33.4641 1.12872
\(880\) −13.8564 −0.467099
\(881\) 32.0333 1.07923 0.539615 0.841912i \(-0.318570\pi\)
0.539615 + 0.841912i \(0.318570\pi\)
\(882\) −1.00000 −0.0336718
\(883\) −0.320508 −0.0107860 −0.00539298 0.999985i \(-0.501717\pi\)
−0.00539298 + 0.999985i \(0.501717\pi\)
\(884\) 0 0
\(885\) −21.7128 −0.729868
\(886\) 22.3923 0.752284
\(887\) 11.8564 0.398099 0.199050 0.979989i \(-0.436215\pi\)
0.199050 + 0.979989i \(0.436215\pi\)
\(888\) 4.92820 0.165380
\(889\) 5.07180 0.170103
\(890\) −40.6410 −1.36229
\(891\) 4.00000 0.134005
\(892\) 7.53590 0.252321
\(893\) −4.00000 −0.133855
\(894\) 11.5359 0.385818
\(895\) 58.6410 1.96015
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 18.0000 0.600668
\(899\) −51.7128 −1.72472
\(900\) 7.00000 0.233333
\(901\) −53.4449 −1.78051
\(902\) −27.7128 −0.922736
\(903\) 4.46410 0.148556
\(904\) 4.39230 0.146086
\(905\) −85.8564 −2.85396
\(906\) 12.0000 0.398673
\(907\) −18.4641 −0.613090 −0.306545 0.951856i \(-0.599173\pi\)
−0.306545 + 0.951856i \(0.599173\pi\)
\(908\) −13.3205 −0.442057
\(909\) 0 0
\(910\) 0 0
\(911\) 27.1769 0.900411 0.450206 0.892925i \(-0.351351\pi\)
0.450206 + 0.892925i \(0.351351\pi\)
\(912\) 7.46410 0.247161
\(913\) 6.92820 0.229290
\(914\) 1.19615 0.0395652
\(915\) 18.0000 0.595062
\(916\) 19.9282 0.658446
\(917\) 19.0000 0.627435
\(918\) −6.46410 −0.213347
\(919\) −26.0000 −0.857661 −0.428830 0.903385i \(-0.641074\pi\)
−0.428830 + 0.903385i \(0.641074\pi\)
\(920\) 19.8564 0.654646
\(921\) −22.0000 −0.724925
\(922\) 5.07180 0.167031
\(923\) 0 0
\(924\) 4.00000 0.131590
\(925\) −34.4974 −1.13427
\(926\) −8.24871 −0.271069
\(927\) 10.6603 0.350129
\(928\) −8.00000 −0.262613
\(929\) 5.87564 0.192774 0.0963868 0.995344i \(-0.469271\pi\)
0.0963868 + 0.995344i \(0.469271\pi\)
\(930\) −22.3923 −0.734273
\(931\) 7.46410 0.244626
\(932\) 13.4641 0.441031
\(933\) 13.3205 0.436094
\(934\) 39.6410 1.29709
\(935\) −89.5692 −2.92923
\(936\) 0 0
\(937\) −44.9282 −1.46774 −0.733870 0.679290i \(-0.762288\pi\)
−0.733870 + 0.679290i \(0.762288\pi\)
\(938\) 7.19615 0.234963
\(939\) 0.679492 0.0221744
\(940\) 1.85641 0.0605493
\(941\) −19.6077 −0.639193 −0.319596 0.947554i \(-0.603547\pi\)
−0.319596 + 0.947554i \(0.603547\pi\)
\(942\) 2.92820 0.0954060
\(943\) 39.7128 1.29323
\(944\) 6.26795 0.204004
\(945\) −3.46410 −0.112687
\(946\) −17.8564 −0.580562
\(947\) 30.2487 0.982951 0.491476 0.870891i \(-0.336458\pi\)
0.491476 + 0.870891i \(0.336458\pi\)
\(948\) −9.46410 −0.307380
\(949\) 0 0
\(950\) −52.2487 −1.69517
\(951\) 16.4641 0.533885
\(952\) −6.46410 −0.209503
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 8.26795 0.267685
\(955\) 91.8564 2.97240
\(956\) 12.8564 0.415806
\(957\) 32.0000 1.03441
\(958\) 4.39230 0.141909
\(959\) 13.8564 0.447447
\(960\) −3.46410 −0.111803
\(961\) 10.7846 0.347891
\(962\) 0 0
\(963\) 8.53590 0.275065
\(964\) −15.8564 −0.510700
\(965\) −61.8564 −1.99123
\(966\) −5.73205 −0.184426
\(967\) 16.5359 0.531759 0.265879 0.964006i \(-0.414338\pi\)
0.265879 + 0.964006i \(0.414338\pi\)
\(968\) −5.00000 −0.160706
\(969\) 48.2487 1.54997
\(970\) 42.9282 1.37834
\(971\) 23.9282 0.767893 0.383946 0.923355i \(-0.374565\pi\)
0.383946 + 0.923355i \(0.374565\pi\)
\(972\) 1.00000 0.0320750
\(973\) 2.92820 0.0938739
\(974\) −0.392305 −0.0125703
\(975\) 0 0
\(976\) −5.19615 −0.166325
\(977\) 25.8564 0.827220 0.413610 0.910454i \(-0.364268\pi\)
0.413610 + 0.910454i \(0.364268\pi\)
\(978\) 1.33975 0.0428404
\(979\) −46.9282 −1.49983
\(980\) −3.46410 −0.110657
\(981\) −12.0000 −0.383131
\(982\) 9.85641 0.314531
\(983\) 19.0718 0.608296 0.304148 0.952625i \(-0.401628\pi\)
0.304148 + 0.952625i \(0.401628\pi\)
\(984\) −6.92820 −0.220863
\(985\) 9.03332 0.287826
\(986\) −51.7128 −1.64687
\(987\) −0.535898 −0.0170578
\(988\) 0 0
\(989\) 25.5885 0.813666
\(990\) 13.8564 0.440386
\(991\) 0.143594 0.00456140 0.00228070 0.999997i \(-0.499274\pi\)
0.00228070 + 0.999997i \(0.499274\pi\)
\(992\) 6.46410 0.205235
\(993\) 21.3205 0.676586
\(994\) −1.92820 −0.0611589
\(995\) 76.6410 2.42968
\(996\) 1.73205 0.0548821
\(997\) 33.9808 1.07618 0.538091 0.842887i \(-0.319146\pi\)
0.538091 + 0.842887i \(0.319146\pi\)
\(998\) 0.267949 0.00848178
\(999\) −4.92820 −0.155921
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.bn.1.1 2
13.6 odd 12 546.2.s.c.127.2 yes 4
13.11 odd 12 546.2.s.c.43.2 4
13.12 even 2 7098.2.a.bz.1.2 2
39.11 even 12 1638.2.bj.e.1135.1 4
39.32 even 12 1638.2.bj.e.127.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
546.2.s.c.43.2 4 13.11 odd 12
546.2.s.c.127.2 yes 4 13.6 odd 12
1638.2.bj.e.127.1 4 39.32 even 12
1638.2.bj.e.1135.1 4 39.11 even 12
7098.2.a.bn.1.1 2 1.1 even 1 trivial
7098.2.a.bz.1.2 2 13.12 even 2