Properties

Label 3234.2.a.bc.1.2
Level $3234$
Weight $2$
Character 3234.1
Self dual yes
Analytic conductor $25.824$
Analytic rank $0$
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: \(0\)
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

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +2.00000 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{12} +5.41421 q^{13} -2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -2.24264 q^{19} +2.00000 q^{20} -1.00000 q^{22} -4.82843 q^{23} -1.00000 q^{24} -1.00000 q^{25} +5.41421 q^{26} -1.00000 q^{27} +9.65685 q^{29} -2.00000 q^{30} +6.24264 q^{31} +1.00000 q^{32} +1.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +0.828427 q^{37} -2.24264 q^{38} -5.41421 q^{39} +2.00000 q^{40} +11.6569 q^{41} -4.82843 q^{43} -1.00000 q^{44} +2.00000 q^{45} -4.82843 q^{46} -7.89949 q^{47} -1.00000 q^{48} -1.00000 q^{50} -2.00000 q^{51} +5.41421 q^{52} -10.4853 q^{53} -1.00000 q^{54} -2.00000 q^{55} +2.24264 q^{57} +9.65685 q^{58} +6.82843 q^{59} -2.00000 q^{60} +2.58579 q^{61} +6.24264 q^{62} +1.00000 q^{64} +10.8284 q^{65} +1.00000 q^{66} -1.17157 q^{67} +2.00000 q^{68} +4.82843 q^{69} +5.65685 q^{71} +1.00000 q^{72} +11.6569 q^{73} +0.828427 q^{74} +1.00000 q^{75} -2.24264 q^{76} -5.41421 q^{78} -16.4853 q^{79} +2.00000 q^{80} +1.00000 q^{81} +11.6569 q^{82} +5.07107 q^{83} +4.00000 q^{85} -4.82843 q^{86} -9.65685 q^{87} -1.00000 q^{88} +3.75736 q^{89} +2.00000 q^{90} -4.82843 q^{92} -6.24264 q^{93} -7.89949 q^{94} -4.48528 q^{95} -1.00000 q^{96} -9.89949 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} + 4 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) 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 \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) −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
\(13\) 5.41421 1.50163 0.750816 0.660511i \(-0.229660\pi\)
0.750816 + 0.660511i \(0.229660\pi\)
\(14\) 0 0
\(15\) −2.00000 −0.516398
\(16\) 1.00000 0.250000
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 1.00000 0.235702
\(19\) −2.24264 −0.514497 −0.257249 0.966345i \(-0.582816\pi\)
−0.257249 + 0.966345i \(0.582816\pi\)
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.82843 −1.00680 −0.503398 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(24\) −1.00000 −0.204124
\(25\) −1.00000 −0.200000
\(26\) 5.41421 1.06181
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 9.65685 1.79323 0.896616 0.442808i \(-0.146018\pi\)
0.896616 + 0.442808i \(0.146018\pi\)
\(30\) −2.00000 −0.365148
\(31\) 6.24264 1.12121 0.560606 0.828083i \(-0.310568\pi\)
0.560606 + 0.828083i \(0.310568\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) 2.00000 0.342997
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 0.828427 0.136193 0.0680963 0.997679i \(-0.478307\pi\)
0.0680963 + 0.997679i \(0.478307\pi\)
\(38\) −2.24264 −0.363804
\(39\) −5.41421 −0.866968
\(40\) 2.00000 0.316228
\(41\) 11.6569 1.82049 0.910247 0.414065i \(-0.135891\pi\)
0.910247 + 0.414065i \(0.135891\pi\)
\(42\) 0 0
\(43\) −4.82843 −0.736328 −0.368164 0.929761i \(-0.620014\pi\)
−0.368164 + 0.929761i \(0.620014\pi\)
\(44\) −1.00000 −0.150756
\(45\) 2.00000 0.298142
\(46\) −4.82843 −0.711913
\(47\) −7.89949 −1.15226 −0.576130 0.817358i \(-0.695438\pi\)
−0.576130 + 0.817358i \(0.695438\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −1.00000 −0.141421
\(51\) −2.00000 −0.280056
\(52\) 5.41421 0.750816
\(53\) −10.4853 −1.44026 −0.720132 0.693837i \(-0.755919\pi\)
−0.720132 + 0.693837i \(0.755919\pi\)
\(54\) −1.00000 −0.136083
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 2.24264 0.297045
\(58\) 9.65685 1.26801
\(59\) 6.82843 0.888985 0.444493 0.895782i \(-0.353384\pi\)
0.444493 + 0.895782i \(0.353384\pi\)
\(60\) −2.00000 −0.258199
\(61\) 2.58579 0.331076 0.165538 0.986203i \(-0.447064\pi\)
0.165538 + 0.986203i \(0.447064\pi\)
\(62\) 6.24264 0.792816
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 10.8284 1.34310
\(66\) 1.00000 0.123091
\(67\) −1.17157 −0.143130 −0.0715652 0.997436i \(-0.522799\pi\)
−0.0715652 + 0.997436i \(0.522799\pi\)
\(68\) 2.00000 0.242536
\(69\) 4.82843 0.581274
\(70\) 0 0
\(71\) 5.65685 0.671345 0.335673 0.941979i \(-0.391036\pi\)
0.335673 + 0.941979i \(0.391036\pi\)
\(72\) 1.00000 0.117851
\(73\) 11.6569 1.36433 0.682166 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) 0.828427 0.0963027
\(75\) 1.00000 0.115470
\(76\) −2.24264 −0.257249
\(77\) 0 0
\(78\) −5.41421 −0.613039
\(79\) −16.4853 −1.85474 −0.927370 0.374147i \(-0.877936\pi\)
−0.927370 + 0.374147i \(0.877936\pi\)
\(80\) 2.00000 0.223607
\(81\) 1.00000 0.111111
\(82\) 11.6569 1.28728
\(83\) 5.07107 0.556622 0.278311 0.960491i \(-0.410225\pi\)
0.278311 + 0.960491i \(0.410225\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) −4.82843 −0.520663
\(87\) −9.65685 −1.03532
\(88\) −1.00000 −0.106600
\(89\) 3.75736 0.398279 0.199140 0.979971i \(-0.436185\pi\)
0.199140 + 0.979971i \(0.436185\pi\)
\(90\) 2.00000 0.210819
\(91\) 0 0
\(92\) −4.82843 −0.503398
\(93\) −6.24264 −0.647332
\(94\) −7.89949 −0.814771
\(95\) −4.48528 −0.460180
\(96\) −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
\(101\) −15.0711 −1.49963 −0.749814 0.661649i \(-0.769857\pi\)
−0.749814 + 0.661649i \(0.769857\pi\)
\(102\) −2.00000 −0.198030
\(103\) 6.24264 0.615106 0.307553 0.951531i \(-0.400490\pi\)
0.307553 + 0.951531i \(0.400490\pi\)
\(104\) 5.41421 0.530907
\(105\) 0 0
\(106\) −10.4853 −1.01842
\(107\) 12.8284 1.24017 0.620085 0.784534i \(-0.287098\pi\)
0.620085 + 0.784534i \(0.287098\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 4.34315 0.415998 0.207999 0.978129i \(-0.433305\pi\)
0.207999 + 0.978129i \(0.433305\pi\)
\(110\) −2.00000 −0.190693
\(111\) −0.828427 −0.0786308
\(112\) 0 0
\(113\) 5.65685 0.532152 0.266076 0.963952i \(-0.414273\pi\)
0.266076 + 0.963952i \(0.414273\pi\)
\(114\) 2.24264 0.210043
\(115\) −9.65685 −0.900506
\(116\) 9.65685 0.896616
\(117\) 5.41421 0.500544
\(118\) 6.82843 0.628608
\(119\) 0 0
\(120\) −2.00000 −0.182574
\(121\) 1.00000 0.0909091
\(122\) 2.58579 0.234106
\(123\) −11.6569 −1.05106
\(124\) 6.24264 0.560606
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 14.1421 1.25491 0.627456 0.778652i \(-0.284096\pi\)
0.627456 + 0.778652i \(0.284096\pi\)
\(128\) 1.00000 0.0883883
\(129\) 4.82843 0.425119
\(130\) 10.8284 0.949716
\(131\) 17.5563 1.53391 0.766953 0.641704i \(-0.221772\pi\)
0.766953 + 0.641704i \(0.221772\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) −1.17157 −0.101208
\(135\) −2.00000 −0.172133
\(136\) 2.00000 0.171499
\(137\) −5.31371 −0.453981 −0.226990 0.973897i \(-0.572889\pi\)
−0.226990 + 0.973897i \(0.572889\pi\)
\(138\) 4.82843 0.411023
\(139\) 0.100505 0.00852473 0.00426236 0.999991i \(-0.498643\pi\)
0.00426236 + 0.999991i \(0.498643\pi\)
\(140\) 0 0
\(141\) 7.89949 0.665257
\(142\) 5.65685 0.474713
\(143\) −5.41421 −0.452759
\(144\) 1.00000 0.0833333
\(145\) 19.3137 1.60392
\(146\) 11.6569 0.964728
\(147\) 0 0
\(148\) 0.828427 0.0680963
\(149\) 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 \(0.447961\pi\)
\(152\) −2.24264 −0.181902
\(153\) 2.00000 0.161690
\(154\) 0 0
\(155\) 12.4853 1.00284
\(156\) −5.41421 −0.433484
\(157\) 12.8284 1.02382 0.511910 0.859039i \(-0.328938\pi\)
0.511910 + 0.859039i \(0.328938\pi\)
\(158\) −16.4853 −1.31150
\(159\) 10.4853 0.831537
\(160\) 2.00000 0.158114
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −2.34315 −0.183529 −0.0917647 0.995781i \(-0.529251\pi\)
−0.0917647 + 0.995781i \(0.529251\pi\)
\(164\) 11.6569 0.910247
\(165\) 2.00000 0.155700
\(166\) 5.07107 0.393591
\(167\) −5.17157 −0.400188 −0.200094 0.979777i \(-0.564125\pi\)
−0.200094 + 0.979777i \(0.564125\pi\)
\(168\) 0 0
\(169\) 16.3137 1.25490
\(170\) 4.00000 0.306786
\(171\) −2.24264 −0.171499
\(172\) −4.82843 −0.368164
\(173\) −5.89949 −0.448530 −0.224265 0.974528i \(-0.571998\pi\)
−0.224265 + 0.974528i \(0.571998\pi\)
\(174\) −9.65685 −0.732084
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) −6.82843 −0.513256
\(178\) 3.75736 0.281626
\(179\) −20.4853 −1.53114 −0.765571 0.643352i \(-0.777543\pi\)
−0.765571 + 0.643352i \(0.777543\pi\)
\(180\) 2.00000 0.149071
\(181\) −9.31371 −0.692283 −0.346141 0.938182i \(-0.612508\pi\)
−0.346141 + 0.938182i \(0.612508\pi\)
\(182\) 0 0
\(183\) −2.58579 −0.191147
\(184\) −4.82843 −0.355956
\(185\) 1.65685 0.121814
\(186\) −6.24264 −0.457733
\(187\) −2.00000 −0.146254
\(188\) −7.89949 −0.576130
\(189\) 0 0
\(190\) −4.48528 −0.325397
\(191\) 21.7990 1.57732 0.788660 0.614830i \(-0.210775\pi\)
0.788660 + 0.614830i \(0.210775\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.9706 −1.07760 −0.538802 0.842432i \(-0.681123\pi\)
−0.538802 + 0.842432i \(0.681123\pi\)
\(194\) −9.89949 −0.710742
\(195\) −10.8284 −0.775440
\(196\) 0 0
\(197\) 15.6569 1.11550 0.557752 0.830007i \(-0.311664\pi\)
0.557752 + 0.830007i \(0.311664\pi\)
\(198\) −1.00000 −0.0710669
\(199\) 15.2132 1.07844 0.539218 0.842166i \(-0.318720\pi\)
0.539218 + 0.842166i \(0.318720\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 1.17157 0.0826364
\(202\) −15.0711 −1.06040
\(203\) 0 0
\(204\) −2.00000 −0.140028
\(205\) 23.3137 1.62830
\(206\) 6.24264 0.434945
\(207\) −4.82843 −0.335599
\(208\) 5.41421 0.375408
\(209\) 2.24264 0.155127
\(210\) 0 0
\(211\) −23.1716 −1.59520 −0.797598 0.603189i \(-0.793897\pi\)
−0.797598 + 0.603189i \(0.793897\pi\)
\(212\) −10.4853 −0.720132
\(213\) −5.65685 −0.387601
\(214\) 12.8284 0.876933
\(215\) −9.65685 −0.658592
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 4.34315 0.294155
\(219\) −11.6569 −0.787697
\(220\) −2.00000 −0.134840
\(221\) 10.8284 0.728399
\(222\) −0.828427 −0.0556004
\(223\) 6.24264 0.418038 0.209019 0.977912i \(-0.432973\pi\)
0.209019 + 0.977912i \(0.432973\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 5.65685 0.376288
\(227\) −11.8995 −0.789797 −0.394899 0.918725i \(-0.629220\pi\)
−0.394899 + 0.918725i \(0.629220\pi\)
\(228\) 2.24264 0.148523
\(229\) 14.0000 0.925146 0.462573 0.886581i \(-0.346926\pi\)
0.462573 + 0.886581i \(0.346926\pi\)
\(230\) −9.65685 −0.636754
\(231\) 0 0
\(232\) 9.65685 0.634004
\(233\) −7.65685 −0.501617 −0.250809 0.968037i \(-0.580696\pi\)
−0.250809 + 0.968037i \(0.580696\pi\)
\(234\) 5.41421 0.353938
\(235\) −15.7990 −1.03061
\(236\) 6.82843 0.444493
\(237\) 16.4853 1.07083
\(238\) 0 0
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) −2.00000 −0.129099
\(241\) −7.65685 −0.493221 −0.246611 0.969115i \(-0.579317\pi\)
−0.246611 + 0.969115i \(0.579317\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 2.58579 0.165538
\(245\) 0 0
\(246\) −11.6569 −0.743214
\(247\) −12.1421 −0.772586
\(248\) 6.24264 0.396408
\(249\) −5.07107 −0.321366
\(250\) −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
\(253\) 4.82843 0.303561
\(254\) 14.1421 0.887357
\(255\) −4.00000 −0.250490
\(256\) 1.00000 0.0625000
\(257\) −29.2132 −1.82227 −0.911135 0.412108i \(-0.864792\pi\)
−0.911135 + 0.412108i \(0.864792\pi\)
\(258\) 4.82843 0.300605
\(259\) 0 0
\(260\) 10.8284 0.671551
\(261\) 9.65685 0.597744
\(262\) 17.5563 1.08463
\(263\) 9.17157 0.565543 0.282772 0.959187i \(-0.408746\pi\)
0.282772 + 0.959187i \(0.408746\pi\)
\(264\) 1.00000 0.0615457
\(265\) −20.9706 −1.28821
\(266\) 0 0
\(267\) −3.75736 −0.229947
\(268\) −1.17157 −0.0715652
\(269\) 5.31371 0.323983 0.161991 0.986792i \(-0.448208\pi\)
0.161991 + 0.986792i \(0.448208\pi\)
\(270\) −2.00000 −0.121716
\(271\) 22.1421 1.34504 0.672519 0.740079i \(-0.265212\pi\)
0.672519 + 0.740079i \(0.265212\pi\)
\(272\) 2.00000 0.121268
\(273\) 0 0
\(274\) −5.31371 −0.321013
\(275\) 1.00000 0.0603023
\(276\) 4.82843 0.290637
\(277\) −16.0000 −0.961347 −0.480673 0.876900i \(-0.659608\pi\)
−0.480673 + 0.876900i \(0.659608\pi\)
\(278\) 0.100505 0.00602789
\(279\) 6.24264 0.373737
\(280\) 0 0
\(281\) −1.31371 −0.0783693 −0.0391846 0.999232i \(-0.512476\pi\)
−0.0391846 + 0.999232i \(0.512476\pi\)
\(282\) 7.89949 0.470408
\(283\) −2.24264 −0.133311 −0.0666556 0.997776i \(-0.521233\pi\)
−0.0666556 + 0.997776i \(0.521233\pi\)
\(284\) 5.65685 0.335673
\(285\) 4.48528 0.265685
\(286\) −5.41421 −0.320149
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) −13.0000 −0.764706
\(290\) 19.3137 1.13414
\(291\) 9.89949 0.580319
\(292\) 11.6569 0.682166
\(293\) −4.92893 −0.287951 −0.143976 0.989581i \(-0.545989\pi\)
−0.143976 + 0.989581i \(0.545989\pi\)
\(294\) 0 0
\(295\) 13.6569 0.795133
\(296\) 0.828427 0.0481513
\(297\) 1.00000 0.0580259
\(298\) 4.00000 0.231714
\(299\) −26.1421 −1.51184
\(300\) 1.00000 0.0577350
\(301\) 0 0
\(302\) 4.00000 0.230174
\(303\) 15.0711 0.865810
\(304\) −2.24264 −0.128624
\(305\) 5.17157 0.296123
\(306\) 2.00000 0.114332
\(307\) −25.0711 −1.43088 −0.715441 0.698673i \(-0.753774\pi\)
−0.715441 + 0.698673i \(0.753774\pi\)
\(308\) 0 0
\(309\) −6.24264 −0.355131
\(310\) 12.4853 0.709116
\(311\) 8.10051 0.459338 0.229669 0.973269i \(-0.426236\pi\)
0.229669 + 0.973269i \(0.426236\pi\)
\(312\) −5.41421 −0.306519
\(313\) −8.24264 −0.465902 −0.232951 0.972489i \(-0.574838\pi\)
−0.232951 + 0.972489i \(0.574838\pi\)
\(314\) 12.8284 0.723950
\(315\) 0 0
\(316\) −16.4853 −0.927370
\(317\) 12.8284 0.720516 0.360258 0.932853i \(-0.382689\pi\)
0.360258 + 0.932853i \(0.382689\pi\)
\(318\) 10.4853 0.587985
\(319\) −9.65685 −0.540680
\(320\) 2.00000 0.111803
\(321\) −12.8284 −0.716013
\(322\) 0 0
\(323\) −4.48528 −0.249568
\(324\) 1.00000 0.0555556
\(325\) −5.41421 −0.300327
\(326\) −2.34315 −0.129775
\(327\) −4.34315 −0.240177
\(328\) 11.6569 0.643642
\(329\) 0 0
\(330\) 2.00000 0.110096
\(331\) −27.7990 −1.52797 −0.763985 0.645234i \(-0.776760\pi\)
−0.763985 + 0.645234i \(0.776760\pi\)
\(332\) 5.07107 0.278311
\(333\) 0.828427 0.0453975
\(334\) −5.17157 −0.282976
\(335\) −2.34315 −0.128020
\(336\) 0 0
\(337\) 12.8284 0.698809 0.349404 0.936972i \(-0.386384\pi\)
0.349404 + 0.936972i \(0.386384\pi\)
\(338\) 16.3137 0.887349
\(339\) −5.65685 −0.307238
\(340\) 4.00000 0.216930
\(341\) −6.24264 −0.338058
\(342\) −2.24264 −0.121268
\(343\) 0 0
\(344\) −4.82843 −0.260331
\(345\) 9.65685 0.519908
\(346\) −5.89949 −0.317159
\(347\) 19.4558 1.04444 0.522222 0.852809i \(-0.325103\pi\)
0.522222 + 0.852809i \(0.325103\pi\)
\(348\) −9.65685 −0.517662
\(349\) −17.4142 −0.932161 −0.466081 0.884742i \(-0.654334\pi\)
−0.466081 + 0.884742i \(0.654334\pi\)
\(350\) 0 0
\(351\) −5.41421 −0.288989
\(352\) −1.00000 −0.0533002
\(353\) 21.2132 1.12906 0.564532 0.825411i \(-0.309057\pi\)
0.564532 + 0.825411i \(0.309057\pi\)
\(354\) −6.82843 −0.362927
\(355\) 11.3137 0.600469
\(356\) 3.75736 0.199140
\(357\) 0 0
\(358\) −20.4853 −1.08268
\(359\) 5.17157 0.272945 0.136473 0.990644i \(-0.456423\pi\)
0.136473 + 0.990644i \(0.456423\pi\)
\(360\) 2.00000 0.105409
\(361\) −13.9706 −0.735293
\(362\) −9.31371 −0.489518
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 23.3137 1.22030
\(366\) −2.58579 −0.135161
\(367\) −32.5858 −1.70096 −0.850482 0.526004i \(-0.823690\pi\)
−0.850482 + 0.526004i \(0.823690\pi\)
\(368\) −4.82843 −0.251699
\(369\) 11.6569 0.606832
\(370\) 1.65685 0.0861358
\(371\) 0 0
\(372\) −6.24264 −0.323666
\(373\) 35.9411 1.86096 0.930480 0.366342i \(-0.119390\pi\)
0.930480 + 0.366342i \(0.119390\pi\)
\(374\) −2.00000 −0.103418
\(375\) 12.0000 0.619677
\(376\) −7.89949 −0.407385
\(377\) 52.2843 2.69278
\(378\) 0 0
\(379\) 32.2843 1.65833 0.829166 0.559003i \(-0.188816\pi\)
0.829166 + 0.559003i \(0.188816\pi\)
\(380\) −4.48528 −0.230090
\(381\) −14.1421 −0.724524
\(382\) 21.7990 1.11533
\(383\) −22.7279 −1.16134 −0.580671 0.814138i \(-0.697210\pi\)
−0.580671 + 0.814138i \(0.697210\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −14.9706 −0.761982
\(387\) −4.82843 −0.245443
\(388\) −9.89949 −0.502571
\(389\) −34.2843 −1.73828 −0.869141 0.494565i \(-0.835327\pi\)
−0.869141 + 0.494565i \(0.835327\pi\)
\(390\) −10.8284 −0.548319
\(391\) −9.65685 −0.488368
\(392\) 0 0
\(393\) −17.5563 −0.885601
\(394\) 15.6569 0.788781
\(395\) −32.9706 −1.65893
\(396\) −1.00000 −0.0502519
\(397\) −11.1716 −0.560685 −0.280343 0.959900i \(-0.590448\pi\)
−0.280343 + 0.959900i \(0.590448\pi\)
\(398\) 15.2132 0.762569
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.6863 0.533648 0.266824 0.963745i \(-0.414026\pi\)
0.266824 + 0.963745i \(0.414026\pi\)
\(402\) 1.17157 0.0584327
\(403\) 33.7990 1.68365
\(404\) −15.0711 −0.749814
\(405\) 2.00000 0.0993808
\(406\) 0 0
\(407\) −0.828427 −0.0410636
\(408\) −2.00000 −0.0990148
\(409\) −1.51472 −0.0748980 −0.0374490 0.999299i \(-0.511923\pi\)
−0.0374490 + 0.999299i \(0.511923\pi\)
\(410\) 23.3137 1.15138
\(411\) 5.31371 0.262106
\(412\) 6.24264 0.307553
\(413\) 0 0
\(414\) −4.82843 −0.237304
\(415\) 10.1421 0.497858
\(416\) 5.41421 0.265454
\(417\) −0.100505 −0.00492175
\(418\) 2.24264 0.109691
\(419\) 16.0000 0.781651 0.390826 0.920465i \(-0.372190\pi\)
0.390826 + 0.920465i \(0.372190\pi\)
\(420\) 0 0
\(421\) −27.6569 −1.34791 −0.673956 0.738771i \(-0.735406\pi\)
−0.673956 + 0.738771i \(0.735406\pi\)
\(422\) −23.1716 −1.12797
\(423\) −7.89949 −0.384087
\(424\) −10.4853 −0.509210
\(425\) −2.00000 −0.0970143
\(426\) −5.65685 −0.274075
\(427\) 0 0
\(428\) 12.8284 0.620085
\(429\) 5.41421 0.261401
\(430\) −9.65685 −0.465695
\(431\) 2.14214 0.103183 0.0515915 0.998668i \(-0.483571\pi\)
0.0515915 + 0.998668i \(0.483571\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −34.3848 −1.65243 −0.826213 0.563357i \(-0.809509\pi\)
−0.826213 + 0.563357i \(0.809509\pi\)
\(434\) 0 0
\(435\) −19.3137 −0.926021
\(436\) 4.34315 0.207999
\(437\) 10.8284 0.517994
\(438\) −11.6569 −0.556986
\(439\) −29.1716 −1.39228 −0.696142 0.717904i \(-0.745101\pi\)
−0.696142 + 0.717904i \(0.745101\pi\)
\(440\) −2.00000 −0.0953463
\(441\) 0 0
\(442\) 10.8284 0.515056
\(443\) 12.4853 0.593194 0.296597 0.955003i \(-0.404148\pi\)
0.296597 + 0.955003i \(0.404148\pi\)
\(444\) −0.828427 −0.0393154
\(445\) 7.51472 0.356232
\(446\) 6.24264 0.295598
\(447\) −4.00000 −0.189194
\(448\) 0 0
\(449\) 16.9706 0.800890 0.400445 0.916321i \(-0.368855\pi\)
0.400445 + 0.916321i \(0.368855\pi\)
\(450\) −1.00000 −0.0471405
\(451\) −11.6569 −0.548900
\(452\) 5.65685 0.266076
\(453\) −4.00000 −0.187936
\(454\) −11.8995 −0.558471
\(455\) 0 0
\(456\) 2.24264 0.105021
\(457\) −29.3137 −1.37124 −0.685619 0.727961i \(-0.740468\pi\)
−0.685619 + 0.727961i \(0.740468\pi\)
\(458\) 14.0000 0.654177
\(459\) −2.00000 −0.0933520
\(460\) −9.65685 −0.450253
\(461\) −30.5858 −1.42452 −0.712261 0.701915i \(-0.752329\pi\)
−0.712261 + 0.701915i \(0.752329\pi\)
\(462\) 0 0
\(463\) −3.31371 −0.154001 −0.0770005 0.997031i \(-0.524534\pi\)
−0.0770005 + 0.997031i \(0.524534\pi\)
\(464\) 9.65685 0.448308
\(465\) −12.4853 −0.578991
\(466\) −7.65685 −0.354697
\(467\) 12.0000 0.555294 0.277647 0.960683i \(-0.410445\pi\)
0.277647 + 0.960683i \(0.410445\pi\)
\(468\) 5.41421 0.250272
\(469\) 0 0
\(470\) −15.7990 −0.728753
\(471\) −12.8284 −0.591103
\(472\) 6.82843 0.314304
\(473\) 4.82843 0.222011
\(474\) 16.4853 0.757194
\(475\) 2.24264 0.102899
\(476\) 0 0
\(477\) −10.4853 −0.480088
\(478\) 0.686292 0.0313902
\(479\) 7.31371 0.334172 0.167086 0.985942i \(-0.446564\pi\)
0.167086 + 0.985942i \(0.446564\pi\)
\(480\) −2.00000 −0.0912871
\(481\) 4.48528 0.204511
\(482\) −7.65685 −0.348760
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) −19.7990 −0.899026
\(486\) −1.00000 −0.0453609
\(487\) 0.142136 0.00644078 0.00322039 0.999995i \(-0.498975\pi\)
0.00322039 + 0.999995i \(0.498975\pi\)
\(488\) 2.58579 0.117053
\(489\) 2.34315 0.105961
\(490\) 0 0
\(491\) −42.6274 −1.92375 −0.961874 0.273492i \(-0.911821\pi\)
−0.961874 + 0.273492i \(0.911821\pi\)
\(492\) −11.6569 −0.525532
\(493\) 19.3137 0.869846
\(494\) −12.1421 −0.546301
\(495\) −2.00000 −0.0898933
\(496\) 6.24264 0.280303
\(497\) 0 0
\(498\) −5.07107 −0.227240
\(499\) −6.14214 −0.274960 −0.137480 0.990505i \(-0.543900\pi\)
−0.137480 + 0.990505i \(0.543900\pi\)
\(500\) −12.0000 −0.536656
\(501\) 5.17157 0.231049
\(502\) 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
\(505\) −30.1421 −1.34131
\(506\) 4.82843 0.214650
\(507\) −16.3137 −0.724517
\(508\) 14.1421 0.627456
\(509\) 34.0000 1.50702 0.753512 0.657434i \(-0.228358\pi\)
0.753512 + 0.657434i \(0.228358\pi\)
\(510\) −4.00000 −0.177123
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 2.24264 0.0990150
\(514\) −29.2132 −1.28854
\(515\) 12.4853 0.550167
\(516\) 4.82843 0.212560
\(517\) 7.89949 0.347419
\(518\) 0 0
\(519\) 5.89949 0.258959
\(520\) 10.8284 0.474858
\(521\) 21.2132 0.929367 0.464684 0.885477i \(-0.346168\pi\)
0.464684 + 0.885477i \(0.346168\pi\)
\(522\) 9.65685 0.422669
\(523\) −9.07107 −0.396650 −0.198325 0.980136i \(-0.563550\pi\)
−0.198325 + 0.980136i \(0.563550\pi\)
\(524\) 17.5563 0.766953
\(525\) 0 0
\(526\) 9.17157 0.399900
\(527\) 12.4853 0.543867
\(528\) 1.00000 0.0435194
\(529\) 0.313708 0.0136395
\(530\) −20.9706 −0.910903
\(531\) 6.82843 0.296328
\(532\) 0 0
\(533\) 63.1127 2.73371
\(534\) −3.75736 −0.162597
\(535\) 25.6569 1.10924
\(536\) −1.17157 −0.0506042
\(537\) 20.4853 0.884005
\(538\) 5.31371 0.229090
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) −45.3137 −1.94819 −0.974094 0.226142i \(-0.927389\pi\)
−0.974094 + 0.226142i \(0.927389\pi\)
\(542\) 22.1421 0.951086
\(543\) 9.31371 0.399689
\(544\) 2.00000 0.0857493
\(545\) 8.68629 0.372080
\(546\) 0 0
\(547\) −42.6274 −1.82262 −0.911308 0.411724i \(-0.864927\pi\)
−0.911308 + 0.411724i \(0.864927\pi\)
\(548\) −5.31371 −0.226990
\(549\) 2.58579 0.110359
\(550\) 1.00000 0.0426401
\(551\) −21.6569 −0.922613
\(552\) 4.82843 0.205512
\(553\) 0 0
\(554\) −16.0000 −0.679775
\(555\) −1.65685 −0.0703295
\(556\) 0.100505 0.00426236
\(557\) 19.6569 0.832888 0.416444 0.909161i \(-0.363276\pi\)
0.416444 + 0.909161i \(0.363276\pi\)
\(558\) 6.24264 0.264272
\(559\) −26.1421 −1.10569
\(560\) 0 0
\(561\) 2.00000 0.0844401
\(562\) −1.31371 −0.0554154
\(563\) −21.0711 −0.888040 −0.444020 0.896017i \(-0.646448\pi\)
−0.444020 + 0.896017i \(0.646448\pi\)
\(564\) 7.89949 0.332629
\(565\) 11.3137 0.475971
\(566\) −2.24264 −0.0942652
\(567\) 0 0
\(568\) 5.65685 0.237356
\(569\) −10.6863 −0.447993 −0.223996 0.974590i \(-0.571910\pi\)
−0.223996 + 0.974590i \(0.571910\pi\)
\(570\) 4.48528 0.187868
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −5.41421 −0.226380
\(573\) −21.7990 −0.910666
\(574\) 0 0
\(575\) 4.82843 0.201359
\(576\) 1.00000 0.0416667
\(577\) −4.44365 −0.184992 −0.0924958 0.995713i \(-0.529484\pi\)
−0.0924958 + 0.995713i \(0.529484\pi\)
\(578\) −13.0000 −0.540729
\(579\) 14.9706 0.622155
\(580\) 19.3137 0.801958
\(581\) 0 0
\(582\) 9.89949 0.410347
\(583\) 10.4853 0.434256
\(584\) 11.6569 0.482364
\(585\) 10.8284 0.447700
\(586\) −4.92893 −0.203612
\(587\) 11.7990 0.486996 0.243498 0.969901i \(-0.421705\pi\)
0.243498 + 0.969901i \(0.421705\pi\)
\(588\) 0 0
\(589\) −14.0000 −0.576860
\(590\) 13.6569 0.562244
\(591\) −15.6569 −0.644037
\(592\) 0.828427 0.0340481
\(593\) −37.3137 −1.53229 −0.766145 0.642668i \(-0.777828\pi\)
−0.766145 + 0.642668i \(0.777828\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) 4.00000 0.163846
\(597\) −15.2132 −0.622635
\(598\) −26.1421 −1.06903
\(599\) 8.00000 0.326871 0.163436 0.986554i \(-0.447742\pi\)
0.163436 + 0.986554i \(0.447742\pi\)
\(600\) 1.00000 0.0408248
\(601\) 15.1716 0.618861 0.309431 0.950922i \(-0.399862\pi\)
0.309431 + 0.950922i \(0.399862\pi\)
\(602\) 0 0
\(603\) −1.17157 −0.0477101
\(604\) 4.00000 0.162758
\(605\) 2.00000 0.0813116
\(606\) 15.0711 0.612220
\(607\) −35.3137 −1.43334 −0.716670 0.697413i \(-0.754334\pi\)
−0.716670 + 0.697413i \(0.754334\pi\)
\(608\) −2.24264 −0.0909511
\(609\) 0 0
\(610\) 5.17157 0.209391
\(611\) −42.7696 −1.73027
\(612\) 2.00000 0.0808452
\(613\) −30.3431 −1.22555 −0.612774 0.790258i \(-0.709946\pi\)
−0.612774 + 0.790258i \(0.709946\pi\)
\(614\) −25.0711 −1.01179
\(615\) −23.3137 −0.940099
\(616\) 0 0
\(617\) −8.00000 −0.322068 −0.161034 0.986949i \(-0.551483\pi\)
−0.161034 + 0.986949i \(0.551483\pi\)
\(618\) −6.24264 −0.251116
\(619\) −21.6569 −0.870462 −0.435231 0.900319i \(-0.643333\pi\)
−0.435231 + 0.900319i \(0.643333\pi\)
\(620\) 12.4853 0.501421
\(621\) 4.82843 0.193758
\(622\) 8.10051 0.324801
\(623\) 0 0
\(624\) −5.41421 −0.216742
\(625\) −19.0000 −0.760000
\(626\) −8.24264 −0.329442
\(627\) −2.24264 −0.0895624
\(628\) 12.8284 0.511910
\(629\) 1.65685 0.0660631
\(630\) 0 0
\(631\) 36.1421 1.43880 0.719398 0.694598i \(-0.244418\pi\)
0.719398 + 0.694598i \(0.244418\pi\)
\(632\) −16.4853 −0.655749
\(633\) 23.1716 0.920987
\(634\) 12.8284 0.509482
\(635\) 28.2843 1.12243
\(636\) 10.4853 0.415768
\(637\) 0 0
\(638\) −9.65685 −0.382319
\(639\) 5.65685 0.223782
\(640\) 2.00000 0.0790569
\(641\) −35.3137 −1.39481 −0.697404 0.716678i \(-0.745662\pi\)
−0.697404 + 0.716678i \(0.745662\pi\)
\(642\) −12.8284 −0.506298
\(643\) 37.4558 1.47711 0.738557 0.674191i \(-0.235507\pi\)
0.738557 + 0.674191i \(0.235507\pi\)
\(644\) 0 0
\(645\) 9.65685 0.380238
\(646\) −4.48528 −0.176471
\(647\) 1.07107 0.0421080 0.0210540 0.999778i \(-0.493298\pi\)
0.0210540 + 0.999778i \(0.493298\pi\)
\(648\) 1.00000 0.0392837
\(649\) −6.82843 −0.268039
\(650\) −5.41421 −0.212363
\(651\) 0 0
\(652\) −2.34315 −0.0917647
\(653\) 22.2843 0.872051 0.436025 0.899934i \(-0.356386\pi\)
0.436025 + 0.899934i \(0.356386\pi\)
\(654\) −4.34315 −0.169830
\(655\) 35.1127 1.37197
\(656\) 11.6569 0.455124
\(657\) 11.6569 0.454777
\(658\) 0 0
\(659\) −49.1127 −1.91316 −0.956580 0.291471i \(-0.905855\pi\)
−0.956580 + 0.291471i \(0.905855\pi\)
\(660\) 2.00000 0.0778499
\(661\) −39.9411 −1.55353 −0.776765 0.629791i \(-0.783141\pi\)
−0.776765 + 0.629791i \(0.783141\pi\)
\(662\) −27.7990 −1.08044
\(663\) −10.8284 −0.420541
\(664\) 5.07107 0.196796
\(665\) 0 0
\(666\) 0.828427 0.0321009
\(667\) −46.6274 −1.80542
\(668\) −5.17157 −0.200094
\(669\) −6.24264 −0.241354
\(670\) −2.34315 −0.0905236
\(671\) −2.58579 −0.0998232
\(672\) 0 0
\(673\) −17.5147 −0.675143 −0.337571 0.941300i \(-0.609605\pi\)
−0.337571 + 0.941300i \(0.609605\pi\)
\(674\) 12.8284 0.494133
\(675\) 1.00000 0.0384900
\(676\) 16.3137 0.627450
\(677\) −36.0416 −1.38519 −0.692596 0.721326i \(-0.743533\pi\)
−0.692596 + 0.721326i \(0.743533\pi\)
\(678\) −5.65685 −0.217250
\(679\) 0 0
\(680\) 4.00000 0.153393
\(681\) 11.8995 0.455990
\(682\) −6.24264 −0.239043
\(683\) 28.7696 1.10084 0.550418 0.834889i \(-0.314468\pi\)
0.550418 + 0.834889i \(0.314468\pi\)
\(684\) −2.24264 −0.0857495
\(685\) −10.6274 −0.406053
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) −4.82843 −0.184082
\(689\) −56.7696 −2.16275
\(690\) 9.65685 0.367630
\(691\) −14.1421 −0.537992 −0.268996 0.963141i \(-0.586692\pi\)
−0.268996 + 0.963141i \(0.586692\pi\)
\(692\) −5.89949 −0.224265
\(693\) 0 0
\(694\) 19.4558 0.738534
\(695\) 0.201010 0.00762475
\(696\) −9.65685 −0.366042
\(697\) 23.3137 0.883070
\(698\) −17.4142 −0.659138
\(699\) 7.65685 0.289609
\(700\) 0 0
\(701\) 9.31371 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(702\) −5.41421 −0.204346
\(703\) −1.85786 −0.0700707
\(704\) −1.00000 −0.0376889
\(705\) 15.7990 0.595024
\(706\) 21.2132 0.798369
\(707\) 0 0
\(708\) −6.82843 −0.256628
\(709\) −34.7696 −1.30580 −0.652899 0.757445i \(-0.726447\pi\)
−0.652899 + 0.757445i \(0.726447\pi\)
\(710\) 11.3137 0.424596
\(711\) −16.4853 −0.618246
\(712\) 3.75736 0.140813
\(713\) −30.1421 −1.12883
\(714\) 0 0
\(715\) −10.8284 −0.404960
\(716\) −20.4853 −0.765571
\(717\) −0.686292 −0.0256300
\(718\) 5.17157 0.193001
\(719\) −44.1838 −1.64778 −0.823888 0.566752i \(-0.808200\pi\)
−0.823888 + 0.566752i \(0.808200\pi\)
\(720\) 2.00000 0.0745356
\(721\) 0 0
\(722\) −13.9706 −0.519931
\(723\) 7.65685 0.284761
\(724\) −9.31371 −0.346141
\(725\) −9.65685 −0.358647
\(726\) −1.00000 −0.0371135
\(727\) 31.2132 1.15763 0.578817 0.815458i \(-0.303515\pi\)
0.578817 + 0.815458i \(0.303515\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 23.3137 0.862879
\(731\) −9.65685 −0.357172
\(732\) −2.58579 −0.0955734
\(733\) 19.5563 0.722330 0.361165 0.932502i \(-0.382379\pi\)
0.361165 + 0.932502i \(0.382379\pi\)
\(734\) −32.5858 −1.20276
\(735\) 0 0
\(736\) −4.82843 −0.177978
\(737\) 1.17157 0.0431554
\(738\) 11.6569 0.429095
\(739\) 2.62742 0.0966511 0.0483255 0.998832i \(-0.484612\pi\)
0.0483255 + 0.998832i \(0.484612\pi\)
\(740\) 1.65685 0.0609072
\(741\) 12.1421 0.446052
\(742\) 0 0
\(743\) −18.6274 −0.683374 −0.341687 0.939814i \(-0.610998\pi\)
−0.341687 + 0.939814i \(0.610998\pi\)
\(744\) −6.24264 −0.228866
\(745\) 8.00000 0.293097
\(746\) 35.9411 1.31590
\(747\) 5.07107 0.185541
\(748\) −2.00000 −0.0731272
\(749\) 0 0
\(750\) 12.0000 0.438178
\(751\) −12.1421 −0.443073 −0.221536 0.975152i \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) −7.89949 −0.288065
\(753\) −8.48528 −0.309221
\(754\) 52.2843 1.90408
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) −48.1421 −1.74976 −0.874878 0.484344i \(-0.839058\pi\)
−0.874878 + 0.484344i \(0.839058\pi\)
\(758\) 32.2843 1.17262
\(759\) −4.82843 −0.175261
\(760\) −4.48528 −0.162698
\(761\) −1.51472 −0.0549085 −0.0274543 0.999623i \(-0.508740\pi\)
−0.0274543 + 0.999623i \(0.508740\pi\)
\(762\) −14.1421 −0.512316
\(763\) 0 0
\(764\) 21.7990 0.788660
\(765\) 4.00000 0.144620
\(766\) −22.7279 −0.821193
\(767\) 36.9706 1.33493
\(768\) −1.00000 −0.0360844
\(769\) −15.8579 −0.571849 −0.285925 0.958252i \(-0.592301\pi\)
−0.285925 + 0.958252i \(0.592301\pi\)
\(770\) 0 0
\(771\) 29.2132 1.05209
\(772\) −14.9706 −0.538802
\(773\) 5.51472 0.198351 0.0991753 0.995070i \(-0.468380\pi\)
0.0991753 + 0.995070i \(0.468380\pi\)
\(774\) −4.82843 −0.173554
\(775\) −6.24264 −0.224242
\(776\) −9.89949 −0.355371
\(777\) 0 0
\(778\) −34.2843 −1.22915
\(779\) −26.1421 −0.936639
\(780\) −10.8284 −0.387720
\(781\) −5.65685 −0.202418
\(782\) −9.65685 −0.345328
\(783\) −9.65685 −0.345108
\(784\) 0 0
\(785\) 25.6569 0.915732
\(786\) −17.5563 −0.626214
\(787\) −3.41421 −0.121704 −0.0608518 0.998147i \(-0.519382\pi\)
−0.0608518 + 0.998147i \(0.519382\pi\)
\(788\) 15.6569 0.557752
\(789\) −9.17157 −0.326517
\(790\) −32.9706 −1.17304
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 14.0000 0.497155
\(794\) −11.1716 −0.396464
\(795\) 20.9706 0.743749
\(796\) 15.2132 0.539218
\(797\) 17.5147 0.620403 0.310202 0.950671i \(-0.399603\pi\)
0.310202 + 0.950671i \(0.399603\pi\)
\(798\) 0 0
\(799\) −15.7990 −0.558928
\(800\) −1.00000 −0.0353553
\(801\) 3.75736 0.132760
\(802\) 10.6863 0.377346
\(803\) −11.6569 −0.411361
\(804\) 1.17157 0.0413182
\(805\) 0 0
\(806\) 33.7990 1.19052
\(807\) −5.31371 −0.187051
\(808\) −15.0711 −0.530198
\(809\) 26.7696 0.941167 0.470584 0.882355i \(-0.344043\pi\)
0.470584 + 0.882355i \(0.344043\pi\)
\(810\) 2.00000 0.0702728
\(811\) 36.3848 1.27764 0.638821 0.769355i \(-0.279422\pi\)
0.638821 + 0.769355i \(0.279422\pi\)
\(812\) 0 0
\(813\) −22.1421 −0.776559
\(814\) −0.828427 −0.0290364
\(815\) −4.68629 −0.164154
\(816\) −2.00000 −0.0700140
\(817\) 10.8284 0.378839
\(818\) −1.51472 −0.0529609
\(819\) 0 0
\(820\) 23.3137 0.814150
\(821\) 47.3137 1.65126 0.825630 0.564212i \(-0.190820\pi\)
0.825630 + 0.564212i \(0.190820\pi\)
\(822\) 5.31371 0.185337
\(823\) −28.1421 −0.980973 −0.490487 0.871449i \(-0.663181\pi\)
−0.490487 + 0.871449i \(0.663181\pi\)
\(824\) 6.24264 0.217473
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 39.3137 1.36707 0.683536 0.729917i \(-0.260441\pi\)
0.683536 + 0.729917i \(0.260441\pi\)
\(828\) −4.82843 −0.167799
\(829\) 50.2843 1.74644 0.873222 0.487322i \(-0.162026\pi\)
0.873222 + 0.487322i \(0.162026\pi\)
\(830\) 10.1421 0.352039
\(831\) 16.0000 0.555034
\(832\) 5.41421 0.187704
\(833\) 0 0
\(834\) −0.100505 −0.00348021
\(835\) −10.3431 −0.357939
\(836\) 2.24264 0.0775634
\(837\) −6.24264 −0.215777
\(838\) 16.0000 0.552711
\(839\) −17.0711 −0.589359 −0.294679 0.955596i \(-0.595213\pi\)
−0.294679 + 0.955596i \(0.595213\pi\)
\(840\) 0 0
\(841\) 64.2548 2.21568
\(842\) −27.6569 −0.953118
\(843\) 1.31371 0.0452465
\(844\) −23.1716 −0.797598
\(845\) 32.6274 1.12242
\(846\) −7.89949 −0.271590
\(847\) 0 0
\(848\) −10.4853 −0.360066
\(849\) 2.24264 0.0769672
\(850\) −2.00000 −0.0685994
\(851\) −4.00000 −0.137118
\(852\) −5.65685 −0.193801
\(853\) 35.5563 1.21743 0.608713 0.793390i \(-0.291686\pi\)
0.608713 + 0.793390i \(0.291686\pi\)
\(854\) 0 0
\(855\) −4.48528 −0.153393
\(856\) 12.8284 0.438467
\(857\) 0.142136 0.00485526 0.00242763 0.999997i \(-0.499227\pi\)
0.00242763 + 0.999997i \(0.499227\pi\)
\(858\) 5.41421 0.184838
\(859\) −30.4264 −1.03814 −0.519068 0.854733i \(-0.673721\pi\)
−0.519068 + 0.854733i \(0.673721\pi\)
\(860\) −9.65685 −0.329296
\(861\) 0 0
\(862\) 2.14214 0.0729614
\(863\) −38.7696 −1.31973 −0.659865 0.751384i \(-0.729387\pi\)
−0.659865 + 0.751384i \(0.729387\pi\)
\(864\) −1.00000 −0.0340207
\(865\) −11.7990 −0.401178
\(866\) −34.3848 −1.16844
\(867\) 13.0000 0.441503
\(868\) 0 0
\(869\) 16.4853 0.559225
\(870\) −19.3137 −0.654796
\(871\) −6.34315 −0.214929
\(872\) 4.34315 0.147077
\(873\) −9.89949 −0.335047
\(874\) 10.8284 0.366277
\(875\) 0 0
\(876\) −11.6569 −0.393849
\(877\) −6.68629 −0.225780 −0.112890 0.993607i \(-0.536011\pi\)
−0.112890 + 0.993607i \(0.536011\pi\)
\(878\) −29.1716 −0.984493
\(879\) 4.92893 0.166249
\(880\) −2.00000 −0.0674200
\(881\) 37.8995 1.27687 0.638433 0.769677i \(-0.279583\pi\)
0.638433 + 0.769677i \(0.279583\pi\)
\(882\) 0 0
\(883\) −10.6274 −0.357641 −0.178821 0.983882i \(-0.557228\pi\)
−0.178821 + 0.983882i \(0.557228\pi\)
\(884\) 10.8284 0.364199
\(885\) −13.6569 −0.459070
\(886\) 12.4853 0.419451
\(887\) 4.28427 0.143852 0.0719259 0.997410i \(-0.477085\pi\)
0.0719259 + 0.997410i \(0.477085\pi\)
\(888\) −0.828427 −0.0278002
\(889\) 0 0
\(890\) 7.51472 0.251894
\(891\) −1.00000 −0.0335013
\(892\) 6.24264 0.209019
\(893\) 17.7157 0.592834
\(894\) −4.00000 −0.133780
\(895\) −40.9706 −1.36949
\(896\) 0 0
\(897\) 26.1421 0.872861
\(898\) 16.9706 0.566315
\(899\) 60.2843 2.01059
\(900\) −1.00000 −0.0333333
\(901\) −20.9706 −0.698631
\(902\) −11.6569 −0.388131
\(903\) 0 0
\(904\) 5.65685 0.188144
\(905\) −18.6274 −0.619196
\(906\) −4.00000 −0.132891
\(907\) −8.20101 −0.272310 −0.136155 0.990688i \(-0.543475\pi\)
−0.136155 + 0.990688i \(0.543475\pi\)
\(908\) −11.8995 −0.394899
\(909\) −15.0711 −0.499876
\(910\) 0 0
\(911\) −11.4558 −0.379549 −0.189775 0.981828i \(-0.560776\pi\)
−0.189775 + 0.981828i \(0.560776\pi\)
\(912\) 2.24264 0.0742613
\(913\) −5.07107 −0.167828
\(914\) −29.3137 −0.969611
\(915\) −5.17157 −0.170967
\(916\) 14.0000 0.462573
\(917\) 0 0
\(918\) −2.00000 −0.0660098
\(919\) −0.970563 −0.0320159 −0.0160080 0.999872i \(-0.505096\pi\)
−0.0160080 + 0.999872i \(0.505096\pi\)
\(920\) −9.65685 −0.318377
\(921\) 25.0711 0.826120
\(922\) −30.5858 −1.00729
\(923\) 30.6274 1.00811
\(924\) 0 0
\(925\) −0.828427 −0.0272385
\(926\) −3.31371 −0.108895
\(927\) 6.24264 0.205035
\(928\) 9.65685 0.317002
\(929\) 10.7868 0.353903 0.176952 0.984220i \(-0.443376\pi\)
0.176952 + 0.984220i \(0.443376\pi\)
\(930\) −12.4853 −0.409409
\(931\) 0 0
\(932\) −7.65685 −0.250809
\(933\) −8.10051 −0.265199
\(934\) 12.0000 0.392652
\(935\) −4.00000 −0.130814
\(936\) 5.41421 0.176969
\(937\) −56.9117 −1.85922 −0.929612 0.368540i \(-0.879858\pi\)
−0.929612 + 0.368540i \(0.879858\pi\)
\(938\) 0 0
\(939\) 8.24264 0.268988
\(940\) −15.7990 −0.515306
\(941\) 27.5563 0.898311 0.449156 0.893454i \(-0.351725\pi\)
0.449156 + 0.893454i \(0.351725\pi\)
\(942\) −12.8284 −0.417973
\(943\) −56.2843 −1.83287
\(944\) 6.82843 0.222246
\(945\) 0 0
\(946\) 4.82843 0.156986
\(947\) −52.5685 −1.70825 −0.854124 0.520069i \(-0.825906\pi\)
−0.854124 + 0.520069i \(0.825906\pi\)
\(948\) 16.4853 0.535417
\(949\) 63.1127 2.04872
\(950\) 2.24264 0.0727609
\(951\) −12.8284 −0.415990
\(952\) 0 0
\(953\) 9.51472 0.308212 0.154106 0.988054i \(-0.450750\pi\)
0.154106 + 0.988054i \(0.450750\pi\)
\(954\) −10.4853 −0.339474
\(955\) 43.5980 1.41080
\(956\) 0.686292 0.0221963
\(957\) 9.65685 0.312162
\(958\) 7.31371 0.236295
\(959\) 0 0
\(960\) −2.00000 −0.0645497
\(961\) 7.97056 0.257115
\(962\) 4.48528 0.144611
\(963\) 12.8284 0.413390
\(964\) −7.65685 −0.246611
\(965\) −29.9411 −0.963839
\(966\) 0 0
\(967\) −9.94113 −0.319685 −0.159843 0.987143i \(-0.551099\pi\)
−0.159843 + 0.987143i \(0.551099\pi\)
\(968\) 1.00000 0.0321412
\(969\) 4.48528 0.144088
\(970\) −19.7990 −0.635707
\(971\) 4.68629 0.150390 0.0751951 0.997169i \(-0.476042\pi\)
0.0751951 + 0.997169i \(0.476042\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 0.142136 0.00455432
\(975\) 5.41421 0.173394
\(976\) 2.58579 0.0827690
\(977\) 26.6274 0.851887 0.425943 0.904750i \(-0.359942\pi\)
0.425943 + 0.904750i \(0.359942\pi\)
\(978\) 2.34315 0.0749255
\(979\) −3.75736 −0.120086
\(980\) 0 0
\(981\) 4.34315 0.138666
\(982\) −42.6274 −1.36030
\(983\) 0.870058 0.0277505 0.0138753 0.999904i \(-0.495583\pi\)
0.0138753 + 0.999904i \(0.495583\pi\)
\(984\) −11.6569 −0.371607
\(985\) 31.3137 0.997738
\(986\) 19.3137 0.615074
\(987\) 0 0
\(988\) −12.1421 −0.386293
\(989\) 23.3137 0.741333
\(990\) −2.00000 −0.0635642
\(991\) 50.9117 1.61726 0.808632 0.588315i \(-0.200209\pi\)
0.808632 + 0.588315i \(0.200209\pi\)
\(992\) 6.24264 0.198204
\(993\) 27.7990 0.882174
\(994\) 0 0
\(995\) 30.4264 0.964582
\(996\) −5.07107 −0.160683
\(997\) 30.5858 0.968662 0.484331 0.874885i \(-0.339063\pi\)
0.484331 + 0.874885i \(0.339063\pi\)
\(998\) −6.14214 −0.194426
\(999\) −0.828427 −0.0262103
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.bc.1.2 2
3.2 odd 2 9702.2.a.ci.1.2 2
7.6 odd 2 3234.2.a.bd.1.1 yes 2
21.20 even 2 9702.2.a.cy.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bc.1.2 2 1.1 even 1 trivial
3234.2.a.bd.1.1 yes 2 7.6 odd 2
9702.2.a.ci.1.2 2 3.2 odd 2
9702.2.a.cy.1.1 2 21.20 even 2