Properties

Label 3234.2.a.bd.1.2
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.8236200137\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -1.00000 q^{11} +1.00000 q^{12} -2.58579 q^{13} -2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} +1.00000 q^{18} -6.24264 q^{19} -2.00000 q^{20} -1.00000 q^{22} +0.828427 q^{23} +1.00000 q^{24} -1.00000 q^{25} -2.58579 q^{26} +1.00000 q^{27} -1.65685 q^{29} -2.00000 q^{30} +2.24264 q^{31} +1.00000 q^{32} -1.00000 q^{33} -2.00000 q^{34} +1.00000 q^{36} -4.82843 q^{37} -6.24264 q^{38} -2.58579 q^{39} -2.00000 q^{40} -0.343146 q^{41} +0.828427 q^{43} -1.00000 q^{44} -2.00000 q^{45} +0.828427 q^{46} -11.8995 q^{47} +1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} -2.58579 q^{52} +6.48528 q^{53} +1.00000 q^{54} +2.00000 q^{55} -6.24264 q^{57} -1.65685 q^{58} -1.17157 q^{59} -2.00000 q^{60} -5.41421 q^{61} +2.24264 q^{62} +1.00000 q^{64} +5.17157 q^{65} -1.00000 q^{66} -6.82843 q^{67} -2.00000 q^{68} +0.828427 q^{69} -5.65685 q^{71} +1.00000 q^{72} -0.343146 q^{73} -4.82843 q^{74} -1.00000 q^{75} -6.24264 q^{76} -2.58579 q^{78} +0.485281 q^{79} -2.00000 q^{80} +1.00000 q^{81} -0.343146 q^{82} +9.07107 q^{83} +4.00000 q^{85} +0.828427 q^{86} -1.65685 q^{87} -1.00000 q^{88} -12.2426 q^{89} -2.00000 q^{90} +0.828427 q^{92} +2.24264 q^{93} -11.8995 q^{94} +12.4853 q^{95} +1.00000 q^{96} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} - 4 q^{5} + 2 q^{6} + 2 q^{8} + 2 q^{9} - 4 q^{10} - 2 q^{11} + 2 q^{12} - 8 q^{13} - 4 q^{15} + 2 q^{16} - 4 q^{17} + 2 q^{18} - 4 q^{19} - 4 q^{20} - 2 q^{22} - 4 q^{23} + 2 q^{24} - 2 q^{25} - 8 q^{26} + 2 q^{27} + 8 q^{29} - 4 q^{30} - 4 q^{31} + 2 q^{32} - 2 q^{33} - 4 q^{34} + 2 q^{36} - 4 q^{37} - 4 q^{38} - 8 q^{39} - 4 q^{40} - 12 q^{41} - 4 q^{43} - 2 q^{44} - 4 q^{45} - 4 q^{46} - 4 q^{47} + 2 q^{48} - 2 q^{50} - 4 q^{51} - 8 q^{52} - 4 q^{53} + 2 q^{54} + 4 q^{55} - 4 q^{57} + 8 q^{58} - 8 q^{59} - 4 q^{60} - 8 q^{61} - 4 q^{62} + 2 q^{64} + 16 q^{65} - 2 q^{66} - 8 q^{67} - 4 q^{68} - 4 q^{69} + 2 q^{72} - 12 q^{73} - 4 q^{74} - 2 q^{75} - 4 q^{76} - 8 q^{78} - 16 q^{79} - 4 q^{80} + 2 q^{81} - 12 q^{82} + 4 q^{83} + 8 q^{85} - 4 q^{86} + 8 q^{87} - 2 q^{88} - 16 q^{89} - 4 q^{90} - 4 q^{92} - 4 q^{93} - 4 q^{94} + 8 q^{95} + 2 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(5\) −2.00000 −0.894427 −0.447214 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 1.00000 0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 1.00000 0.288675
\(13\) −2.58579 −0.717168 −0.358584 0.933497i \(-0.616740\pi\)
−0.358584 + 0.933497i \(0.616740\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.24264 −1.43216 −0.716080 0.698018i \(-0.754065\pi\)
−0.716080 + 0.698018i \(0.754065\pi\)
\(20\) −2.00000 −0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 0.828427 0.172739 0.0863695 0.996263i \(-0.472473\pi\)
0.0863695 + 0.996263i \(0.472473\pi\)
\(24\) 1.00000 0.204124
\(25\) −1.00000 −0.200000
\(26\) −2.58579 −0.507114
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −1.65685 −0.307670 −0.153835 0.988097i \(-0.549162\pi\)
−0.153835 + 0.988097i \(0.549162\pi\)
\(30\) −2.00000 −0.365148
\(31\) 2.24264 0.402790 0.201395 0.979510i \(-0.435452\pi\)
0.201395 + 0.979510i \(0.435452\pi\)
\(32\) 1.00000 0.176777
\(33\) −1.00000 −0.174078
\(34\) −2.00000 −0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −4.82843 −0.793789 −0.396894 0.917864i \(-0.629912\pi\)
−0.396894 + 0.917864i \(0.629912\pi\)
\(38\) −6.24264 −1.01269
\(39\) −2.58579 −0.414057
\(40\) −2.00000 −0.316228
\(41\) −0.343146 −0.0535904 −0.0267952 0.999641i \(-0.508530\pi\)
−0.0267952 + 0.999641i \(0.508530\pi\)
\(42\) 0 0
\(43\) 0.828427 0.126334 0.0631670 0.998003i \(-0.479880\pi\)
0.0631670 + 0.998003i \(0.479880\pi\)
\(44\) −1.00000 −0.150756
\(45\) −2.00000 −0.298142
\(46\) 0.828427 0.122145
\(47\) −11.8995 −1.73572 −0.867860 0.496809i \(-0.834505\pi\)
−0.867860 + 0.496809i \(0.834505\pi\)
\(48\) 1.00000 0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) −2.58579 −0.358584
\(53\) 6.48528 0.890822 0.445411 0.895326i \(-0.353058\pi\)
0.445411 + 0.895326i \(0.353058\pi\)
\(54\) 1.00000 0.136083
\(55\) 2.00000 0.269680
\(56\) 0 0
\(57\) −6.24264 −0.826858
\(58\) −1.65685 −0.217556
\(59\) −1.17157 −0.152526 −0.0762629 0.997088i \(-0.524299\pi\)
−0.0762629 + 0.997088i \(0.524299\pi\)
\(60\) −2.00000 −0.258199
\(61\) −5.41421 −0.693219 −0.346610 0.938010i \(-0.612667\pi\)
−0.346610 + 0.938010i \(0.612667\pi\)
\(62\) 2.24264 0.284816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 5.17157 0.641455
\(66\) −1.00000 −0.123091
\(67\) −6.82843 −0.834225 −0.417113 0.908855i \(-0.636958\pi\)
−0.417113 + 0.908855i \(0.636958\pi\)
\(68\) −2.00000 −0.242536
\(69\) 0.828427 0.0997309
\(70\) 0 0
\(71\) −5.65685 −0.671345 −0.335673 0.941979i \(-0.608964\pi\)
−0.335673 + 0.941979i \(0.608964\pi\)
\(72\) 1.00000 0.117851
\(73\) −0.343146 −0.0401622 −0.0200811 0.999798i \(-0.506392\pi\)
−0.0200811 + 0.999798i \(0.506392\pi\)
\(74\) −4.82843 −0.561293
\(75\) −1.00000 −0.115470
\(76\) −6.24264 −0.716080
\(77\) 0 0
\(78\) −2.58579 −0.292783
\(79\) 0.485281 0.0545984 0.0272992 0.999627i \(-0.491309\pi\)
0.0272992 + 0.999627i \(0.491309\pi\)
\(80\) −2.00000 −0.223607
\(81\) 1.00000 0.111111
\(82\) −0.343146 −0.0378941
\(83\) 9.07107 0.995679 0.497840 0.867269i \(-0.334127\pi\)
0.497840 + 0.867269i \(0.334127\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0.828427 0.0893316
\(87\) −1.65685 −0.177633
\(88\) −1.00000 −0.106600
\(89\) −12.2426 −1.29772 −0.648859 0.760909i \(-0.724753\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(90\) −2.00000 −0.210819
\(91\) 0 0
\(92\) 0.828427 0.0863695
\(93\) 2.24264 0.232551
\(94\) −11.8995 −1.22734
\(95\) 12.4853 1.28096
\(96\) 1.00000 0.102062
\(97\) −9.89949 −1.00514 −0.502571 0.864536i \(-0.667612\pi\)
−0.502571 + 0.864536i \(0.667612\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) 0.928932 0.0924322 0.0462161 0.998931i \(-0.485284\pi\)
0.0462161 + 0.998931i \(0.485284\pi\)
\(102\) −2.00000 −0.198030
\(103\) 2.24264 0.220974 0.110487 0.993878i \(-0.464759\pi\)
0.110487 + 0.993878i \(0.464759\pi\)
\(104\) −2.58579 −0.253557
\(105\) 0 0
\(106\) 6.48528 0.629906
\(107\) 7.17157 0.693302 0.346651 0.937994i \(-0.387319\pi\)
0.346651 + 0.937994i \(0.387319\pi\)
\(108\) 1.00000 0.0962250
\(109\) 15.6569 1.49965 0.749827 0.661634i \(-0.230137\pi\)
0.749827 + 0.661634i \(0.230137\pi\)
\(110\) 2.00000 0.190693
\(111\) −4.82843 −0.458294
\(112\) 0 0
\(113\) −5.65685 −0.532152 −0.266076 0.963952i \(-0.585727\pi\)
−0.266076 + 0.963952i \(0.585727\pi\)
\(114\) −6.24264 −0.584677
\(115\) −1.65685 −0.154502
\(116\) −1.65685 −0.153835
\(117\) −2.58579 −0.239056
\(118\) −1.17157 −0.107852
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) −5.41421 −0.490180
\(123\) −0.343146 −0.0309404
\(124\) 2.24264 0.201395
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) −14.1421 −1.25491 −0.627456 0.778652i \(-0.715904\pi\)
−0.627456 + 0.778652i \(0.715904\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0.828427 0.0729389
\(130\) 5.17157 0.453577
\(131\) 13.5563 1.18442 0.592212 0.805782i \(-0.298255\pi\)
0.592212 + 0.805782i \(0.298255\pi\)
\(132\) −1.00000 −0.0870388
\(133\) 0 0
\(134\) −6.82843 −0.589886
\(135\) −2.00000 −0.172133
\(136\) −2.00000 −0.171499
\(137\) 17.3137 1.47921 0.739605 0.673041i \(-0.235012\pi\)
0.739605 + 0.673041i \(0.235012\pi\)
\(138\) 0.828427 0.0705204
\(139\) −19.8995 −1.68785 −0.843927 0.536459i \(-0.819762\pi\)
−0.843927 + 0.536459i \(0.819762\pi\)
\(140\) 0 0
\(141\) −11.8995 −1.00212
\(142\) −5.65685 −0.474713
\(143\) 2.58579 0.216234
\(144\) 1.00000 0.0833333
\(145\) 3.31371 0.275189
\(146\) −0.343146 −0.0283989
\(147\) 0 0
\(148\) −4.82843 −0.396894
\(149\) 4.00000 0.327693 0.163846 0.986486i \(-0.447610\pi\)
0.163846 + 0.986486i \(0.447610\pi\)
\(150\) −1.00000 −0.0816497
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) −6.24264 −0.506345
\(153\) −2.00000 −0.161690
\(154\) 0 0
\(155\) −4.48528 −0.360266
\(156\) −2.58579 −0.207029
\(157\) −7.17157 −0.572354 −0.286177 0.958177i \(-0.592385\pi\)
−0.286177 + 0.958177i \(0.592385\pi\)
\(158\) 0.485281 0.0386069
\(159\) 6.48528 0.514316
\(160\) −2.00000 −0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −13.6569 −1.06969 −0.534844 0.844951i \(-0.679630\pi\)
−0.534844 + 0.844951i \(0.679630\pi\)
\(164\) −0.343146 −0.0267952
\(165\) 2.00000 0.155700
\(166\) 9.07107 0.704051
\(167\) 10.8284 0.837929 0.418964 0.908003i \(-0.362393\pi\)
0.418964 + 0.908003i \(0.362393\pi\)
\(168\) 0 0
\(169\) −6.31371 −0.485670
\(170\) 4.00000 0.306786
\(171\) −6.24264 −0.477387
\(172\) 0.828427 0.0631670
\(173\) −13.8995 −1.05676 −0.528380 0.849008i \(-0.677200\pi\)
−0.528380 + 0.849008i \(0.677200\pi\)
\(174\) −1.65685 −0.125606
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −1.17157 −0.0880608
\(178\) −12.2426 −0.917625
\(179\) −3.51472 −0.262702 −0.131351 0.991336i \(-0.541932\pi\)
−0.131351 + 0.991336i \(0.541932\pi\)
\(180\) −2.00000 −0.149071
\(181\) −13.3137 −0.989600 −0.494800 0.869007i \(-0.664759\pi\)
−0.494800 + 0.869007i \(0.664759\pi\)
\(182\) 0 0
\(183\) −5.41421 −0.400230
\(184\) 0.828427 0.0610725
\(185\) 9.65685 0.709986
\(186\) 2.24264 0.164438
\(187\) 2.00000 0.146254
\(188\) −11.8995 −0.867860
\(189\) 0 0
\(190\) 12.4853 0.905778
\(191\) −17.7990 −1.28789 −0.643945 0.765072i \(-0.722703\pi\)
−0.643945 + 0.765072i \(0.722703\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.9706 1.36553 0.682765 0.730638i \(-0.260777\pi\)
0.682765 + 0.730638i \(0.260777\pi\)
\(194\) −9.89949 −0.710742
\(195\) 5.17157 0.370344
\(196\) 0 0
\(197\) 4.34315 0.309436 0.154718 0.987959i \(-0.450553\pi\)
0.154718 + 0.987959i \(0.450553\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 27.2132 1.92909 0.964546 0.263913i \(-0.0850133\pi\)
0.964546 + 0.263913i \(0.0850133\pi\)
\(200\) −1.00000 −0.0707107
\(201\) −6.82843 −0.481640
\(202\) 0.928932 0.0653594
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 0.686292 0.0479327
\(206\) 2.24264 0.156252
\(207\) 0.828427 0.0575797
\(208\) −2.58579 −0.179292
\(209\) 6.24264 0.431812
\(210\) 0 0
\(211\) −28.8284 −1.98463 −0.992315 0.123734i \(-0.960513\pi\)
−0.992315 + 0.123734i \(0.960513\pi\)
\(212\) 6.48528 0.445411
\(213\) −5.65685 −0.387601
\(214\) 7.17157 0.490239
\(215\) −1.65685 −0.112997
\(216\) 1.00000 0.0680414
\(217\) 0 0
\(218\) 15.6569 1.06042
\(219\) −0.343146 −0.0231876
\(220\) 2.00000 0.134840
\(221\) 5.17157 0.347878
\(222\) −4.82843 −0.324063
\(223\) 2.24264 0.150178 0.0750892 0.997177i \(-0.476076\pi\)
0.0750892 + 0.997177i \(0.476076\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) −5.65685 −0.376288
\(227\) −7.89949 −0.524308 −0.262154 0.965026i \(-0.584433\pi\)
−0.262154 + 0.965026i \(0.584433\pi\)
\(228\) −6.24264 −0.413429
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) −1.65685 −0.109250
\(231\) 0 0
\(232\) −1.65685 −0.108778
\(233\) 3.65685 0.239568 0.119784 0.992800i \(-0.461780\pi\)
0.119784 + 0.992800i \(0.461780\pi\)
\(234\) −2.58579 −0.169038
\(235\) 23.7990 1.55247
\(236\) −1.17157 −0.0762629
\(237\) 0.485281 0.0315224
\(238\) 0 0
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) −2.00000 −0.129099
\(241\) −3.65685 −0.235559 −0.117779 0.993040i \(-0.537578\pi\)
−0.117779 + 0.993040i \(0.537578\pi\)
\(242\) 1.00000 0.0642824
\(243\) 1.00000 0.0641500
\(244\) −5.41421 −0.346610
\(245\) 0 0
\(246\) −0.343146 −0.0218782
\(247\) 16.1421 1.02710
\(248\) 2.24264 0.142408
\(249\) 9.07107 0.574856
\(250\) 12.0000 0.758947
\(251\) 8.48528 0.535586 0.267793 0.963476i \(-0.413706\pi\)
0.267793 + 0.963476i \(0.413706\pi\)
\(252\) 0 0
\(253\) −0.828427 −0.0520828
\(254\) −14.1421 −0.887357
\(255\) 4.00000 0.250490
\(256\) 1.00000 0.0625000
\(257\) −13.2132 −0.824217 −0.412108 0.911135i \(-0.635208\pi\)
−0.412108 + 0.911135i \(0.635208\pi\)
\(258\) 0.828427 0.0515756
\(259\) 0 0
\(260\) 5.17157 0.320727
\(261\) −1.65685 −0.102557
\(262\) 13.5563 0.837514
\(263\) 14.8284 0.914360 0.457180 0.889374i \(-0.348860\pi\)
0.457180 + 0.889374i \(0.348860\pi\)
\(264\) −1.00000 −0.0615457
\(265\) −12.9706 −0.796775
\(266\) 0 0
\(267\) −12.2426 −0.749237
\(268\) −6.82843 −0.417113
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) −2.00000 −0.121716
\(271\) 6.14214 0.373108 0.186554 0.982445i \(-0.440268\pi\)
0.186554 + 0.982445i \(0.440268\pi\)
\(272\) −2.00000 −0.121268
\(273\) 0 0
\(274\) 17.3137 1.04596
\(275\) 1.00000 0.0603023
\(276\) 0.828427 0.0498655
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) −19.8995 −1.19349
\(279\) 2.24264 0.134263
\(280\) 0 0
\(281\) 21.3137 1.27147 0.635735 0.771908i \(-0.280697\pi\)
0.635735 + 0.771908i \(0.280697\pi\)
\(282\) −11.8995 −0.708605
\(283\) −6.24264 −0.371086 −0.185543 0.982636i \(-0.559404\pi\)
−0.185543 + 0.982636i \(0.559404\pi\)
\(284\) −5.65685 −0.335673
\(285\) 12.4853 0.739564
\(286\) 2.58579 0.152901
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 3.31371 0.194588
\(291\) −9.89949 −0.580319
\(292\) −0.343146 −0.0200811
\(293\) 19.0711 1.11414 0.557072 0.830464i \(-0.311925\pi\)
0.557072 + 0.830464i \(0.311925\pi\)
\(294\) 0 0
\(295\) 2.34315 0.136423
\(296\) −4.82843 −0.280647
\(297\) −1.00000 −0.0580259
\(298\) 4.00000 0.231714
\(299\) −2.14214 −0.123883
\(300\) −1.00000 −0.0577350
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) 0.928932 0.0533658
\(304\) −6.24264 −0.358040
\(305\) 10.8284 0.620034
\(306\) −2.00000 −0.114332
\(307\) 10.9289 0.623747 0.311874 0.950124i \(-0.399043\pi\)
0.311874 + 0.950124i \(0.399043\pi\)
\(308\) 0 0
\(309\) 2.24264 0.127579
\(310\) −4.48528 −0.254747
\(311\) −27.8995 −1.58204 −0.791018 0.611793i \(-0.790448\pi\)
−0.791018 + 0.611793i \(0.790448\pi\)
\(312\) −2.58579 −0.146391
\(313\) −0.242641 −0.0137149 −0.00685743 0.999976i \(-0.502183\pi\)
−0.00685743 + 0.999976i \(0.502183\pi\)
\(314\) −7.17157 −0.404715
\(315\) 0 0
\(316\) 0.485281 0.0272992
\(317\) 7.17157 0.402796 0.201398 0.979510i \(-0.435452\pi\)
0.201398 + 0.979510i \(0.435452\pi\)
\(318\) 6.48528 0.363677
\(319\) 1.65685 0.0927660
\(320\) −2.00000 −0.111803
\(321\) 7.17157 0.400278
\(322\) 0 0
\(323\) 12.4853 0.694700
\(324\) 1.00000 0.0555556
\(325\) 2.58579 0.143434
\(326\) −13.6569 −0.756383
\(327\) 15.6569 0.865826
\(328\) −0.343146 −0.0189471
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) 11.7990 0.648531 0.324266 0.945966i \(-0.394883\pi\)
0.324266 + 0.945966i \(0.394883\pi\)
\(332\) 9.07107 0.497840
\(333\) −4.82843 −0.264596
\(334\) 10.8284 0.592505
\(335\) 13.6569 0.746154
\(336\) 0 0
\(337\) 7.17157 0.390660 0.195330 0.980738i \(-0.437422\pi\)
0.195330 + 0.980738i \(0.437422\pi\)
\(338\) −6.31371 −0.343420
\(339\) −5.65685 −0.307238
\(340\) 4.00000 0.216930
\(341\) −2.24264 −0.121446
\(342\) −6.24264 −0.337563
\(343\) 0 0
\(344\) 0.828427 0.0446658
\(345\) −1.65685 −0.0892020
\(346\) −13.8995 −0.747241
\(347\) −31.4558 −1.68864 −0.844319 0.535841i \(-0.819995\pi\)
−0.844319 + 0.535841i \(0.819995\pi\)
\(348\) −1.65685 −0.0888167
\(349\) 14.5858 0.780759 0.390380 0.920654i \(-0.372344\pi\)
0.390380 + 0.920654i \(0.372344\pi\)
\(350\) 0 0
\(351\) −2.58579 −0.138019
\(352\) −1.00000 −0.0533002
\(353\) 21.2132 1.12906 0.564532 0.825411i \(-0.309057\pi\)
0.564532 + 0.825411i \(0.309057\pi\)
\(354\) −1.17157 −0.0622684
\(355\) 11.3137 0.600469
\(356\) −12.2426 −0.648859
\(357\) 0 0
\(358\) −3.51472 −0.185759
\(359\) 10.8284 0.571503 0.285751 0.958304i \(-0.407757\pi\)
0.285751 + 0.958304i \(0.407757\pi\)
\(360\) −2.00000 −0.105409
\(361\) 19.9706 1.05108
\(362\) −13.3137 −0.699753
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) 0.686292 0.0359221
\(366\) −5.41421 −0.283005
\(367\) 35.4142 1.84861 0.924303 0.381658i \(-0.124647\pi\)
0.924303 + 0.381658i \(0.124647\pi\)
\(368\) 0.828427 0.0431847
\(369\) −0.343146 −0.0178635
\(370\) 9.65685 0.502036
\(371\) 0 0
\(372\) 2.24264 0.116276
\(373\) −31.9411 −1.65385 −0.826924 0.562313i \(-0.809912\pi\)
−0.826924 + 0.562313i \(0.809912\pi\)
\(374\) 2.00000 0.103418
\(375\) 12.0000 0.619677
\(376\) −11.8995 −0.613670
\(377\) 4.28427 0.220651
\(378\) 0 0
\(379\) −24.2843 −1.24740 −0.623700 0.781664i \(-0.714371\pi\)
−0.623700 + 0.781664i \(0.714371\pi\)
\(380\) 12.4853 0.640481
\(381\) −14.1421 −0.724524
\(382\) −17.7990 −0.910676
\(383\) −2.72792 −0.139390 −0.0696952 0.997568i \(-0.522203\pi\)
−0.0696952 + 0.997568i \(0.522203\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) 18.9706 0.965576
\(387\) 0.828427 0.0421113
\(388\) −9.89949 −0.502571
\(389\) 22.2843 1.12986 0.564929 0.825140i \(-0.308904\pi\)
0.564929 + 0.825140i \(0.308904\pi\)
\(390\) 5.17157 0.261873
\(391\) −1.65685 −0.0837907
\(392\) 0 0
\(393\) 13.5563 0.683827
\(394\) 4.34315 0.218805
\(395\) −0.970563 −0.0488343
\(396\) −1.00000 −0.0502519
\(397\) 16.8284 0.844595 0.422297 0.906457i \(-0.361224\pi\)
0.422297 + 0.906457i \(0.361224\pi\)
\(398\) 27.2132 1.36407
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 33.3137 1.66361 0.831804 0.555070i \(-0.187309\pi\)
0.831804 + 0.555070i \(0.187309\pi\)
\(402\) −6.82843 −0.340571
\(403\) −5.79899 −0.288868
\(404\) 0.928932 0.0462161
\(405\) −2.00000 −0.0993808
\(406\) 0 0
\(407\) 4.82843 0.239336
\(408\) −2.00000 −0.0990148
\(409\) 18.4853 0.914038 0.457019 0.889457i \(-0.348917\pi\)
0.457019 + 0.889457i \(0.348917\pi\)
\(410\) 0.686292 0.0338935
\(411\) 17.3137 0.854022
\(412\) 2.24264 0.110487
\(413\) 0 0
\(414\) 0.828427 0.0407150
\(415\) −18.1421 −0.890562
\(416\) −2.58579 −0.126779
\(417\) −19.8995 −0.974482
\(418\) 6.24264 0.305338
\(419\) −16.0000 −0.781651 −0.390826 0.920465i \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 0 0
\(421\) −16.3431 −0.796516 −0.398258 0.917273i \(-0.630385\pi\)
−0.398258 + 0.917273i \(0.630385\pi\)
\(422\) −28.8284 −1.40335
\(423\) −11.8995 −0.578573
\(424\) 6.48528 0.314953
\(425\) 2.00000 0.0970143
\(426\) −5.65685 −0.274075
\(427\) 0 0
\(428\) 7.17157 0.346651
\(429\) 2.58579 0.124843
\(430\) −1.65685 −0.0799006
\(431\) −26.1421 −1.25922 −0.629611 0.776910i \(-0.716786\pi\)
−0.629611 + 0.776910i \(0.716786\pi\)
\(432\) 1.00000 0.0481125
\(433\) −2.38478 −0.114605 −0.0573025 0.998357i \(-0.518250\pi\)
−0.0573025 + 0.998357i \(0.518250\pi\)
\(434\) 0 0
\(435\) 3.31371 0.158880
\(436\) 15.6569 0.749827
\(437\) −5.17157 −0.247390
\(438\) −0.343146 −0.0163961
\(439\) 34.8284 1.66227 0.831135 0.556071i \(-0.187692\pi\)
0.831135 + 0.556071i \(0.187692\pi\)
\(440\) 2.00000 0.0953463
\(441\) 0 0
\(442\) 5.17157 0.245987
\(443\) −4.48528 −0.213102 −0.106551 0.994307i \(-0.533981\pi\)
−0.106551 + 0.994307i \(0.533981\pi\)
\(444\) −4.82843 −0.229147
\(445\) 24.4853 1.16071
\(446\) 2.24264 0.106192
\(447\) 4.00000 0.189194
\(448\) 0 0
\(449\) −16.9706 −0.800890 −0.400445 0.916321i \(-0.631145\pi\)
−0.400445 + 0.916321i \(0.631145\pi\)
\(450\) −1.00000 −0.0471405
\(451\) 0.343146 0.0161581
\(452\) −5.65685 −0.266076
\(453\) 4.00000 0.187936
\(454\) −7.89949 −0.370742
\(455\) 0 0
\(456\) −6.24264 −0.292338
\(457\) −6.68629 −0.312772 −0.156386 0.987696i \(-0.549984\pi\)
−0.156386 + 0.987696i \(0.549984\pi\)
\(458\) −14.0000 −0.654177
\(459\) −2.00000 −0.0933520
\(460\) −1.65685 −0.0772512
\(461\) 33.4142 1.55626 0.778128 0.628106i \(-0.216170\pi\)
0.778128 + 0.628106i \(0.216170\pi\)
\(462\) 0 0
\(463\) 19.3137 0.897584 0.448792 0.893636i \(-0.351854\pi\)
0.448792 + 0.893636i \(0.351854\pi\)
\(464\) −1.65685 −0.0769175
\(465\) −4.48528 −0.208000
\(466\) 3.65685 0.169401
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −2.58579 −0.119528
\(469\) 0 0
\(470\) 23.7990 1.09777
\(471\) −7.17157 −0.330449
\(472\) −1.17157 −0.0539260
\(473\) −0.828427 −0.0380911
\(474\) 0.485281 0.0222897
\(475\) 6.24264 0.286432
\(476\) 0 0
\(477\) 6.48528 0.296941
\(478\) 23.3137 1.06634
\(479\) 15.3137 0.699701 0.349851 0.936806i \(-0.386232\pi\)
0.349851 + 0.936806i \(0.386232\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 12.4853 0.569280
\(482\) −3.65685 −0.166565
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 19.7990 0.899026
\(486\) 1.00000 0.0453609
\(487\) −28.1421 −1.27524 −0.637621 0.770350i \(-0.720081\pi\)
−0.637621 + 0.770350i \(0.720081\pi\)
\(488\) −5.41421 −0.245090
\(489\) −13.6569 −0.617584
\(490\) 0 0
\(491\) 2.62742 0.118574 0.0592868 0.998241i \(-0.481117\pi\)
0.0592868 + 0.998241i \(0.481117\pi\)
\(492\) −0.343146 −0.0154702
\(493\) 3.31371 0.149242
\(494\) 16.1421 0.726269
\(495\) 2.00000 0.0898933
\(496\) 2.24264 0.100698
\(497\) 0 0
\(498\) 9.07107 0.406484
\(499\) 22.1421 0.991218 0.495609 0.868546i \(-0.334945\pi\)
0.495609 + 0.868546i \(0.334945\pi\)
\(500\) 12.0000 0.536656
\(501\) 10.8284 0.483778
\(502\) 8.48528 0.378717
\(503\) −8.48528 −0.378340 −0.189170 0.981944i \(-0.560580\pi\)
−0.189170 + 0.981944i \(0.560580\pi\)
\(504\) 0 0
\(505\) −1.85786 −0.0826739
\(506\) −0.828427 −0.0368281
\(507\) −6.31371 −0.280402
\(508\) −14.1421 −0.627456
\(509\) −34.0000 −1.50702 −0.753512 0.657434i \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 4.00000 0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) −6.24264 −0.275619
\(514\) −13.2132 −0.582809
\(515\) −4.48528 −0.197645
\(516\) 0.828427 0.0364695
\(517\) 11.8995 0.523339
\(518\) 0 0
\(519\) −13.8995 −0.610120
\(520\) 5.17157 0.226788
\(521\) 21.2132 0.929367 0.464684 0.885477i \(-0.346168\pi\)
0.464684 + 0.885477i \(0.346168\pi\)
\(522\) −1.65685 −0.0725185
\(523\) −5.07107 −0.221742 −0.110871 0.993835i \(-0.535364\pi\)
−0.110871 + 0.993835i \(0.535364\pi\)
\(524\) 13.5563 0.592212
\(525\) 0 0
\(526\) 14.8284 0.646550
\(527\) −4.48528 −0.195382
\(528\) −1.00000 −0.0435194
\(529\) −22.3137 −0.970161
\(530\) −12.9706 −0.563405
\(531\) −1.17157 −0.0508419
\(532\) 0 0
\(533\) 0.887302 0.0384333
\(534\) −12.2426 −0.529791
\(535\) −14.3431 −0.620108
\(536\) −6.82843 −0.294943
\(537\) −3.51472 −0.151671
\(538\) 17.3137 0.746447
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −22.6863 −0.975360 −0.487680 0.873023i \(-0.662157\pi\)
−0.487680 + 0.873023i \(0.662157\pi\)
\(542\) 6.14214 0.263827
\(543\) −13.3137 −0.571346
\(544\) −2.00000 −0.0857493
\(545\) −31.3137 −1.34133
\(546\) 0 0
\(547\) 2.62742 0.112340 0.0561701 0.998421i \(-0.482111\pi\)
0.0561701 + 0.998421i \(0.482111\pi\)
\(548\) 17.3137 0.739605
\(549\) −5.41421 −0.231073
\(550\) 1.00000 0.0426401
\(551\) 10.3431 0.440633
\(552\) 0.828427 0.0352602
\(553\) 0 0
\(554\) −16.0000 −0.679775
\(555\) 9.65685 0.409911
\(556\) −19.8995 −0.843927
\(557\) 8.34315 0.353510 0.176755 0.984255i \(-0.443440\pi\)
0.176755 + 0.984255i \(0.443440\pi\)
\(558\) 2.24264 0.0949386
\(559\) −2.14214 −0.0906027
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) 21.3137 0.899065
\(563\) 6.92893 0.292020 0.146010 0.989283i \(-0.453357\pi\)
0.146010 + 0.989283i \(0.453357\pi\)
\(564\) −11.8995 −0.501059
\(565\) 11.3137 0.475971
\(566\) −6.24264 −0.262398
\(567\) 0 0
\(568\) −5.65685 −0.237356
\(569\) −33.3137 −1.39658 −0.698292 0.715813i \(-0.746056\pi\)
−0.698292 + 0.715813i \(0.746056\pi\)
\(570\) 12.4853 0.522951
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 2.58579 0.108117
\(573\) −17.7990 −0.743563
\(574\) 0 0
\(575\) −0.828427 −0.0345478
\(576\) 1.00000 0.0416667
\(577\) 35.5563 1.48023 0.740115 0.672480i \(-0.234771\pi\)
0.740115 + 0.672480i \(0.234771\pi\)
\(578\) −13.0000 −0.540729
\(579\) 18.9706 0.788390
\(580\) 3.31371 0.137594
\(581\) 0 0
\(582\) −9.89949 −0.410347
\(583\) −6.48528 −0.268593
\(584\) −0.343146 −0.0141995
\(585\) 5.17157 0.213818
\(586\) 19.0711 0.787819
\(587\) 27.7990 1.14739 0.573694 0.819070i \(-0.305510\pi\)
0.573694 + 0.819070i \(0.305510\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 2.34315 0.0964658
\(591\) 4.34315 0.178653
\(592\) −4.82843 −0.198447
\(593\) 14.6863 0.603094 0.301547 0.953451i \(-0.402497\pi\)
0.301547 + 0.953451i \(0.402497\pi\)
\(594\) −1.00000 −0.0410305
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) 27.2132 1.11376
\(598\) −2.14214 −0.0875984
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) −1.00000 −0.0408248
\(601\) −20.8284 −0.849609 −0.424805 0.905285i \(-0.639657\pi\)
−0.424805 + 0.905285i \(0.639657\pi\)
\(602\) 0 0
\(603\) −6.82843 −0.278075
\(604\) 4.00000 0.162758
\(605\) −2.00000 −0.0813116
\(606\) 0.928932 0.0377353
\(607\) 12.6863 0.514921 0.257460 0.966289i \(-0.417114\pi\)
0.257460 + 0.966289i \(0.417114\pi\)
\(608\) −6.24264 −0.253173
\(609\) 0 0
\(610\) 10.8284 0.438430
\(611\) 30.7696 1.24480
\(612\) −2.00000 −0.0808452
\(613\) −41.6569 −1.68250 −0.841252 0.540643i \(-0.818181\pi\)
−0.841252 + 0.540643i \(0.818181\pi\)
\(614\) 10.9289 0.441056
\(615\) 0.686292 0.0276739
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) 2.24264 0.0902122
\(619\) 10.3431 0.415726 0.207863 0.978158i \(-0.433349\pi\)
0.207863 + 0.978158i \(0.433349\pi\)
\(620\) −4.48528 −0.180133
\(621\) 0.828427 0.0332436
\(622\) −27.8995 −1.11867
\(623\) 0 0
\(624\) −2.58579 −0.103514
\(625\) −19.0000 −0.760000
\(626\) −0.242641 −0.00969787
\(627\) 6.24264 0.249307
\(628\) −7.17157 −0.286177
\(629\) 9.65685 0.385044
\(630\) 0 0
\(631\) 7.85786 0.312817 0.156408 0.987692i \(-0.450008\pi\)
0.156408 + 0.987692i \(0.450008\pi\)
\(632\) 0.485281 0.0193035
\(633\) −28.8284 −1.14583
\(634\) 7.17157 0.284820
\(635\) 28.2843 1.12243
\(636\) 6.48528 0.257158
\(637\) 0 0
\(638\) 1.65685 0.0655955
\(639\) −5.65685 −0.223782
\(640\) −2.00000 −0.0790569
\(641\) −12.6863 −0.501078 −0.250539 0.968106i \(-0.580608\pi\)
−0.250539 + 0.968106i \(0.580608\pi\)
\(642\) 7.17157 0.283039
\(643\) 13.4558 0.530647 0.265323 0.964159i \(-0.414521\pi\)
0.265323 + 0.964159i \(0.414521\pi\)
\(644\) 0 0
\(645\) −1.65685 −0.0652386
\(646\) 12.4853 0.491227
\(647\) 13.0711 0.513877 0.256938 0.966428i \(-0.417286\pi\)
0.256938 + 0.966428i \(0.417286\pi\)
\(648\) 1.00000 0.0392837
\(649\) 1.17157 0.0459883
\(650\) 2.58579 0.101423
\(651\) 0 0
\(652\) −13.6569 −0.534844
\(653\) −34.2843 −1.34165 −0.670824 0.741617i \(-0.734059\pi\)
−0.670824 + 0.741617i \(0.734059\pi\)
\(654\) 15.6569 0.612231
\(655\) −27.1127 −1.05938
\(656\) −0.343146 −0.0133976
\(657\) −0.343146 −0.0133874
\(658\) 0 0
\(659\) 13.1127 0.510798 0.255399 0.966836i \(-0.417793\pi\)
0.255399 + 0.966836i \(0.417793\pi\)
\(660\) 2.00000 0.0778499
\(661\) −27.9411 −1.08678 −0.543392 0.839479i \(-0.682860\pi\)
−0.543392 + 0.839479i \(0.682860\pi\)
\(662\) 11.7990 0.458581
\(663\) 5.17157 0.200847
\(664\) 9.07107 0.352026
\(665\) 0 0
\(666\) −4.82843 −0.187098
\(667\) −1.37258 −0.0531466
\(668\) 10.8284 0.418964
\(669\) 2.24264 0.0867055
\(670\) 13.6569 0.527610
\(671\) 5.41421 0.209013
\(672\) 0 0
\(673\) −34.4853 −1.32931 −0.664655 0.747150i \(-0.731421\pi\)
−0.664655 + 0.747150i \(0.731421\pi\)
\(674\) 7.17157 0.276239
\(675\) −1.00000 −0.0384900
\(676\) −6.31371 −0.242835
\(677\) −12.0416 −0.462797 −0.231399 0.972859i \(-0.574330\pi\)
−0.231399 + 0.972859i \(0.574330\pi\)
\(678\) −5.65685 −0.217250
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) −7.89949 −0.302709
\(682\) −2.24264 −0.0858752
\(683\) −44.7696 −1.71306 −0.856530 0.516098i \(-0.827384\pi\)
−0.856530 + 0.516098i \(0.827384\pi\)
\(684\) −6.24264 −0.238693
\(685\) −34.6274 −1.32305
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 0.828427 0.0315835
\(689\) −16.7696 −0.638869
\(690\) −1.65685 −0.0630754
\(691\) −14.1421 −0.537992 −0.268996 0.963141i \(-0.586692\pi\)
−0.268996 + 0.963141i \(0.586692\pi\)
\(692\) −13.8995 −0.528380
\(693\) 0 0
\(694\) −31.4558 −1.19405
\(695\) 39.7990 1.50966
\(696\) −1.65685 −0.0628029
\(697\) 0.686292 0.0259951
\(698\) 14.5858 0.552080
\(699\) 3.65685 0.138315
\(700\) 0 0
\(701\) −13.3137 −0.502852 −0.251426 0.967877i \(-0.580899\pi\)
−0.251426 + 0.967877i \(0.580899\pi\)
\(702\) −2.58579 −0.0975942
\(703\) 30.1421 1.13683
\(704\) −1.00000 −0.0376889
\(705\) 23.7990 0.896322
\(706\) 21.2132 0.798369
\(707\) 0 0
\(708\) −1.17157 −0.0440304
\(709\) 38.7696 1.45602 0.728011 0.685566i \(-0.240445\pi\)
0.728011 + 0.685566i \(0.240445\pi\)
\(710\) 11.3137 0.424596
\(711\) 0.485281 0.0181995
\(712\) −12.2426 −0.458812
\(713\) 1.85786 0.0695776
\(714\) 0 0
\(715\) −5.17157 −0.193406
\(716\) −3.51472 −0.131351
\(717\) 23.3137 0.870666
\(718\) 10.8284 0.404113
\(719\) −32.1838 −1.20025 −0.600126 0.799906i \(-0.704883\pi\)
−0.600126 + 0.799906i \(0.704883\pi\)
\(720\) −2.00000 −0.0745356
\(721\) 0 0
\(722\) 19.9706 0.743227
\(723\) −3.65685 −0.136000
\(724\) −13.3137 −0.494800
\(725\) 1.65685 0.0615340
\(726\) 1.00000 0.0371135
\(727\) 11.2132 0.415875 0.207937 0.978142i \(-0.433325\pi\)
0.207937 + 0.978142i \(0.433325\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.686292 0.0254008
\(731\) −1.65685 −0.0612810
\(732\) −5.41421 −0.200115
\(733\) 11.5563 0.426843 0.213422 0.976960i \(-0.431539\pi\)
0.213422 + 0.976960i \(0.431539\pi\)
\(734\) 35.4142 1.30716
\(735\) 0 0
\(736\) 0.828427 0.0305362
\(737\) 6.82843 0.251528
\(738\) −0.343146 −0.0126314
\(739\) −42.6274 −1.56807 −0.784037 0.620714i \(-0.786843\pi\)
−0.784037 + 0.620714i \(0.786843\pi\)
\(740\) 9.65685 0.354993
\(741\) 16.1421 0.592996
\(742\) 0 0
\(743\) 26.6274 0.976865 0.488433 0.872602i \(-0.337569\pi\)
0.488433 + 0.872602i \(0.337569\pi\)
\(744\) 2.24264 0.0822192
\(745\) −8.00000 −0.293097
\(746\) −31.9411 −1.16945
\(747\) 9.07107 0.331893
\(748\) 2.00000 0.0731272
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) 16.1421 0.589035 0.294517 0.955646i \(-0.404841\pi\)
0.294517 + 0.955646i \(0.404841\pi\)
\(752\) −11.8995 −0.433930
\(753\) 8.48528 0.309221
\(754\) 4.28427 0.156024
\(755\) −8.00000 −0.291150
\(756\) 0 0
\(757\) −19.8579 −0.721746 −0.360873 0.932615i \(-0.617521\pi\)
−0.360873 + 0.932615i \(0.617521\pi\)
\(758\) −24.2843 −0.882044
\(759\) −0.828427 −0.0300700
\(760\) 12.4853 0.452889
\(761\) 18.4853 0.670091 0.335045 0.942202i \(-0.391248\pi\)
0.335045 + 0.942202i \(0.391248\pi\)
\(762\) −14.1421 −0.512316
\(763\) 0 0
\(764\) −17.7990 −0.643945
\(765\) 4.00000 0.144620
\(766\) −2.72792 −0.0985638
\(767\) 3.02944 0.109387
\(768\) 1.00000 0.0360844
\(769\) 44.1421 1.59181 0.795903 0.605424i \(-0.206996\pi\)
0.795903 + 0.605424i \(0.206996\pi\)
\(770\) 0 0
\(771\) −13.2132 −0.475862
\(772\) 18.9706 0.682765
\(773\) −22.4853 −0.808739 −0.404370 0.914596i \(-0.632509\pi\)
−0.404370 + 0.914596i \(0.632509\pi\)
\(774\) 0.828427 0.0297772
\(775\) −2.24264 −0.0805580
\(776\) −9.89949 −0.355371
\(777\) 0 0
\(778\) 22.2843 0.798930
\(779\) 2.14214 0.0767500
\(780\) 5.17157 0.185172
\(781\) 5.65685 0.202418
\(782\) −1.65685 −0.0592490
\(783\) −1.65685 −0.0592111
\(784\) 0 0
\(785\) 14.3431 0.511929
\(786\) 13.5563 0.483539
\(787\) 0.585786 0.0208810 0.0104405 0.999945i \(-0.496677\pi\)
0.0104405 + 0.999945i \(0.496677\pi\)
\(788\) 4.34315 0.154718
\(789\) 14.8284 0.527906
\(790\) −0.970563 −0.0345311
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 14.0000 0.497155
\(794\) 16.8284 0.597219
\(795\) −12.9706 −0.460018
\(796\) 27.2132 0.964546
\(797\) −34.4853 −1.22153 −0.610766 0.791811i \(-0.709138\pi\)
−0.610766 + 0.791811i \(0.709138\pi\)
\(798\) 0 0
\(799\) 23.7990 0.841948
\(800\) −1.00000 −0.0353553
\(801\) −12.2426 −0.432572
\(802\) 33.3137 1.17635
\(803\) 0.343146 0.0121094
\(804\) −6.82843 −0.240820
\(805\) 0 0
\(806\) −5.79899 −0.204261
\(807\) 17.3137 0.609471
\(808\) 0.928932 0.0326797
\(809\) −46.7696 −1.64433 −0.822165 0.569249i \(-0.807234\pi\)
−0.822165 + 0.569249i \(0.807234\pi\)
\(810\) −2.00000 −0.0702728
\(811\) 0.384776 0.0135113 0.00675566 0.999977i \(-0.497850\pi\)
0.00675566 + 0.999977i \(0.497850\pi\)
\(812\) 0 0
\(813\) 6.14214 0.215414
\(814\) 4.82843 0.169236
\(815\) 27.3137 0.956757
\(816\) −2.00000 −0.0700140
\(817\) −5.17157 −0.180930
\(818\) 18.4853 0.646323
\(819\) 0 0
\(820\) 0.686292 0.0239663
\(821\) 24.6863 0.861558 0.430779 0.902458i \(-0.358239\pi\)
0.430779 + 0.902458i \(0.358239\pi\)
\(822\) 17.3137 0.603885
\(823\) 0.142136 0.00495454 0.00247727 0.999997i \(-0.499211\pi\)
0.00247727 + 0.999997i \(0.499211\pi\)
\(824\) 2.24264 0.0781261
\(825\) 1.00000 0.0348155
\(826\) 0 0
\(827\) 16.6863 0.580239 0.290120 0.956990i \(-0.406305\pi\)
0.290120 + 0.956990i \(0.406305\pi\)
\(828\) 0.828427 0.0287898
\(829\) 6.28427 0.218262 0.109131 0.994027i \(-0.465193\pi\)
0.109131 + 0.994027i \(0.465193\pi\)
\(830\) −18.1421 −0.629723
\(831\) −16.0000 −0.555034
\(832\) −2.58579 −0.0896460
\(833\) 0 0
\(834\) −19.8995 −0.689063
\(835\) −21.6569 −0.749466
\(836\) 6.24264 0.215906
\(837\) 2.24264 0.0775170
\(838\) −16.0000 −0.552711
\(839\) 2.92893 0.101118 0.0505590 0.998721i \(-0.483900\pi\)
0.0505590 + 0.998721i \(0.483900\pi\)
\(840\) 0 0
\(841\) −26.2548 −0.905339
\(842\) −16.3431 −0.563222
\(843\) 21.3137 0.734083
\(844\) −28.8284 −0.992315
\(845\) 12.6274 0.434396
\(846\) −11.8995 −0.409113
\(847\) 0 0
\(848\) 6.48528 0.222705
\(849\) −6.24264 −0.214247
\(850\) 2.00000 0.0685994
\(851\) −4.00000 −0.137118
\(852\) −5.65685 −0.193801
\(853\) −4.44365 −0.152148 −0.0760739 0.997102i \(-0.524238\pi\)
−0.0760739 + 0.997102i \(0.524238\pi\)
\(854\) 0 0
\(855\) 12.4853 0.426988
\(856\) 7.17157 0.245119
\(857\) 28.1421 0.961317 0.480659 0.876908i \(-0.340398\pi\)
0.480659 + 0.876908i \(0.340398\pi\)
\(858\) 2.58579 0.0882773
\(859\) −54.4264 −1.85701 −0.928503 0.371326i \(-0.878903\pi\)
−0.928503 + 0.371326i \(0.878903\pi\)
\(860\) −1.65685 −0.0564983
\(861\) 0 0
\(862\) −26.1421 −0.890405
\(863\) 34.7696 1.18357 0.591785 0.806096i \(-0.298424\pi\)
0.591785 + 0.806096i \(0.298424\pi\)
\(864\) 1.00000 0.0340207
\(865\) 27.7990 0.945194
\(866\) −2.38478 −0.0810380
\(867\) −13.0000 −0.441503
\(868\) 0 0
\(869\) −0.485281 −0.0164620
\(870\) 3.31371 0.112345
\(871\) 17.6569 0.598280
\(872\) 15.6569 0.530208
\(873\) −9.89949 −0.335047
\(874\) −5.17157 −0.174931
\(875\) 0 0
\(876\) −0.343146 −0.0115938
\(877\) −29.3137 −0.989854 −0.494927 0.868935i \(-0.664805\pi\)
−0.494927 + 0.868935i \(0.664805\pi\)
\(878\) 34.8284 1.17540
\(879\) 19.0711 0.643251
\(880\) 2.00000 0.0674200
\(881\) −18.1005 −0.609822 −0.304911 0.952381i \(-0.598627\pi\)
−0.304911 + 0.952381i \(0.598627\pi\)
\(882\) 0 0
\(883\) 34.6274 1.16531 0.582653 0.812721i \(-0.302015\pi\)
0.582653 + 0.812721i \(0.302015\pi\)
\(884\) 5.17157 0.173939
\(885\) 2.34315 0.0787640
\(886\) −4.48528 −0.150686
\(887\) 52.2843 1.75553 0.877767 0.479088i \(-0.159032\pi\)
0.877767 + 0.479088i \(0.159032\pi\)
\(888\) −4.82843 −0.162031
\(889\) 0 0
\(890\) 24.4853 0.820748
\(891\) −1.00000 −0.0335013
\(892\) 2.24264 0.0750892
\(893\) 74.2843 2.48583
\(894\) 4.00000 0.133780
\(895\) 7.02944 0.234968
\(896\) 0 0
\(897\) −2.14214 −0.0715238
\(898\) −16.9706 −0.566315
\(899\) −3.71573 −0.123926
\(900\) −1.00000 −0.0333333
\(901\) −12.9706 −0.432112
\(902\) 0.343146 0.0114255
\(903\) 0 0
\(904\) −5.65685 −0.188144
\(905\) 26.6274 0.885125
\(906\) 4.00000 0.132891
\(907\) −47.7990 −1.58714 −0.793570 0.608479i \(-0.791780\pi\)
−0.793570 + 0.608479i \(0.791780\pi\)
\(908\) −7.89949 −0.262154
\(909\) 0.928932 0.0308107
\(910\) 0 0
\(911\) 39.4558 1.30723 0.653615 0.756827i \(-0.273251\pi\)
0.653615 + 0.756827i \(0.273251\pi\)
\(912\) −6.24264 −0.206714
\(913\) −9.07107 −0.300209
\(914\) −6.68629 −0.221163
\(915\) 10.8284 0.357977
\(916\) −14.0000 −0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) 32.9706 1.08760 0.543799 0.839215i \(-0.316985\pi\)
0.543799 + 0.839215i \(0.316985\pi\)
\(920\) −1.65685 −0.0546249
\(921\) 10.9289 0.360121
\(922\) 33.4142 1.10044
\(923\) 14.6274 0.481467
\(924\) 0 0
\(925\) 4.82843 0.158758
\(926\) 19.3137 0.634688
\(927\) 2.24264 0.0736580
\(928\) −1.65685 −0.0543889
\(929\) −53.2132 −1.74587 −0.872934 0.487838i \(-0.837786\pi\)
−0.872934 + 0.487838i \(0.837786\pi\)
\(930\) −4.48528 −0.147078
\(931\) 0 0
\(932\) 3.65685 0.119784
\(933\) −27.8995 −0.913388
\(934\) −12.0000 −0.392652
\(935\) −4.00000 −0.130814
\(936\) −2.58579 −0.0845191
\(937\) −44.9117 −1.46720 −0.733600 0.679581i \(-0.762162\pi\)
−0.733600 + 0.679581i \(0.762162\pi\)
\(938\) 0 0
\(939\) −0.242641 −0.00791828
\(940\) 23.7990 0.776237
\(941\) 3.55635 0.115934 0.0579668 0.998319i \(-0.481538\pi\)
0.0579668 + 0.998319i \(0.481538\pi\)
\(942\) −7.17157 −0.233662
\(943\) −0.284271 −0.00925715
\(944\) −1.17157 −0.0381314
\(945\) 0 0
\(946\) −0.828427 −0.0269345
\(947\) 60.5685 1.96821 0.984107 0.177579i \(-0.0568265\pi\)
0.984107 + 0.177579i \(0.0568265\pi\)
\(948\) 0.485281 0.0157612
\(949\) 0.887302 0.0288030
\(950\) 6.24264 0.202538
\(951\) 7.17157 0.232554
\(952\) 0 0
\(953\) 26.4853 0.857942 0.428971 0.903318i \(-0.358876\pi\)
0.428971 + 0.903318i \(0.358876\pi\)
\(954\) 6.48528 0.209969
\(955\) 35.5980 1.15192
\(956\) 23.3137 0.754019
\(957\) 1.65685 0.0535585
\(958\) 15.3137 0.494763
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) −25.9706 −0.837760
\(962\) 12.4853 0.402542
\(963\) 7.17157 0.231101
\(964\) −3.65685 −0.117779
\(965\) −37.9411 −1.22137
\(966\) 0 0
\(967\) 57.9411 1.86326 0.931630 0.363407i \(-0.118387\pi\)
0.931630 + 0.363407i \(0.118387\pi\)
\(968\) 1.00000 0.0321412
\(969\) 12.4853 0.401085
\(970\) 19.7990 0.635707
\(971\) −27.3137 −0.876539 −0.438269 0.898844i \(-0.644408\pi\)
−0.438269 + 0.898844i \(0.644408\pi\)
\(972\) 1.00000 0.0320750
\(973\) 0 0
\(974\) −28.1421 −0.901732
\(975\) 2.58579 0.0828114
\(976\) −5.41421 −0.173305
\(977\) −18.6274 −0.595944 −0.297972 0.954575i \(-0.596310\pi\)
−0.297972 + 0.954575i \(0.596310\pi\)
\(978\) −13.6569 −0.436698
\(979\) 12.2426 0.391276
\(980\) 0 0
\(981\) 15.6569 0.499885
\(982\) 2.62742 0.0838442
\(983\) 52.8701 1.68629 0.843146 0.537684i \(-0.180701\pi\)
0.843146 + 0.537684i \(0.180701\pi\)
\(984\) −0.343146 −0.0109391
\(985\) −8.68629 −0.276768
\(986\) 3.31371 0.105530
\(987\) 0 0
\(988\) 16.1421 0.513550
\(989\) 0.686292 0.0218228
\(990\) 2.00000 0.0635642
\(991\) −50.9117 −1.61726 −0.808632 0.588315i \(-0.799791\pi\)
−0.808632 + 0.588315i \(0.799791\pi\)
\(992\) 2.24264 0.0712039
\(993\) 11.7990 0.374430
\(994\) 0 0
\(995\) −54.4264 −1.72543
\(996\) 9.07107 0.287428
\(997\) −33.4142 −1.05824 −0.529119 0.848547i \(-0.677478\pi\)
−0.529119 + 0.848547i \(0.677478\pi\)
\(998\) 22.1421 0.700897
\(999\) −4.82843 −0.152765
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 3234.2.a.bd.1.2 yes 2
3.2 odd 2 9702.2.a.cy.1.2 2
7.6 odd 2 3234.2.a.bc.1.1 2
21.20 even 2 9702.2.a.ci.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bc.1.1 2 7.6 odd 2
3234.2.a.bd.1.2 yes 2 1.1 even 1 trivial
9702.2.a.ci.1.1 2 21.20 even 2
9702.2.a.cy.1.2 2 3.2 odd 2