Properties

Label 3234.2.a.bd.1.1
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 1.1
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 \(-0.647584\pi\)
−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
\(13\) −5.41421 −1.50163 −0.750816 0.660511i \(-0.770340\pi\)
−0.750816 + 0.660511i \(0.770340\pi\)
\(14\) 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
\(19\) 2.24264 0.514497 0.257249 0.966345i \(-0.417184\pi\)
0.257249 + 0.966345i \(0.417184\pi\)
\(20\) −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 \(-0.689432\pi\)
−0.560606 + 0.828083i \(0.689432\pi\)
\(32\) 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 \(-0.864109\pi\)
−0.910247 + 0.414065i \(0.864109\pi\)
\(42\) 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 \(-0.304562\pi\)
0.576130 + 0.817358i \(0.304562\pi\)
\(48\) 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 \(-0.646616\pi\)
−0.444493 + 0.895782i \(0.646616\pi\)
\(60\) −2.00000 −0.258199
\(61\) −2.58579 −0.331076 −0.165538 0.986203i \(-0.552936\pi\)
−0.165538 + 0.986203i \(0.552936\pi\)
\(62\) −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 \(-0.738962\pi\)
−0.682166 + 0.731198i \(0.738962\pi\)
\(74\) 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 \(-0.589775\pi\)
−0.278311 + 0.960491i \(0.589775\pi\)
\(84\) 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 \(-0.563815\pi\)
−0.199140 + 0.979971i \(0.563815\pi\)
\(90\) −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 \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −1.00000 −0.100000
\(101\) 15.0711 1.49963 0.749814 0.661649i \(-0.230143\pi\)
0.749814 + 0.661649i \(0.230143\pi\)
\(102\) −2.00000 −0.198030
\(103\) −6.24264 −0.615106 −0.307553 0.951531i \(-0.599510\pi\)
−0.307553 + 0.951531i \(0.599510\pi\)
\(104\) −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 \(-0.778228\pi\)
−0.766953 + 0.641704i \(0.778228\pi\)
\(132\) −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 \(-0.501357\pi\)
−0.00426236 + 0.999991i \(0.501357\pi\)
\(140\) 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 \(-0.671062\pi\)
−0.511910 + 0.859039i \(0.671062\pi\)
\(158\) −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 \(-0.435875\pi\)
0.200094 + 0.979777i \(0.435875\pi\)
\(168\) 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 \(-0.428002\pi\)
0.224265 + 0.974528i \(0.428002\pi\)
\(174\) 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 \(-0.387492\pi\)
0.346141 + 0.938182i \(0.387492\pi\)
\(182\) 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 \(-0.681280\pi\)
−0.539218 + 0.842166i \(0.681280\pi\)
\(200\) −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 \(-0.567027\pi\)
−0.209019 + 0.977912i \(0.567027\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 5.65685 0.376288
\(227\) 11.8995 0.789797 0.394899 0.918725i \(-0.370780\pi\)
0.394899 + 0.918725i \(0.370780\pi\)
\(228\) 2.24264 0.148523
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 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 \(-0.420683\pi\)
0.246611 + 0.969115i \(0.420683\pi\)
\(242\) 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 \(-0.586294\pi\)
−0.267793 + 0.963476i \(0.586294\pi\)
\(252\) 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 \(-0.135208\pi\)
0.911135 + 0.412108i \(0.135208\pi\)
\(258\) −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 \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) −2.00000 −0.121716
\(271\) −22.1421 −1.34504 −0.672519 0.740079i \(-0.734788\pi\)
−0.672519 + 0.740079i \(0.734788\pi\)
\(272\) −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 \(-0.478767\pi\)
0.0666556 + 0.997776i \(0.478767\pi\)
\(284\) 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 \(-0.454011\pi\)
0.143976 + 0.989581i \(0.454011\pi\)
\(294\) 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 \(-0.246226\pi\)
0.715441 + 0.698673i \(0.246226\pi\)
\(308\) 0 0
\(309\) −6.24264 −0.355131
\(310\) 12.4853 0.709116
\(311\) −8.10051 −0.459338 −0.229669 0.973269i \(-0.573764\pi\)
−0.229669 + 0.973269i \(0.573764\pi\)
\(312\) −5.41421 −0.306519
\(313\) 8.24264 0.465902 0.232951 0.972489i \(-0.425162\pi\)
0.232951 + 0.972489i \(0.425162\pi\)
\(314\) −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 \(-0.345666\pi\)
0.466081 + 0.884742i \(0.345666\pi\)
\(350\) 0 0
\(351\) −5.41421 −0.288989
\(352\) −1.00000 −0.0533002
\(353\) −21.2132 −1.12906 −0.564532 0.825411i \(-0.690943\pi\)
−0.564532 + 0.825411i \(0.690943\pi\)
\(354\) −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 \(-0.176310\pi\)
0.850482 + 0.526004i \(0.176310\pi\)
\(368\) −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 \(-0.302790\pi\)
0.580671 + 0.814138i \(0.302790\pi\)
\(384\) 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 \(-0.409552\pi\)
0.280343 + 0.959900i \(0.409552\pi\)
\(398\) −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 \(-0.488077\pi\)
0.0374490 + 0.999299i \(0.488077\pi\)
\(410\) 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 \(-0.627810\pi\)
−0.390826 + 0.920465i \(0.627810\pi\)
\(420\) 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 \(-0.190491\pi\)
0.826213 + 0.563357i \(0.190491\pi\)
\(434\) 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 \(-0.254899\pi\)
0.696142 + 0.717904i \(0.254899\pi\)
\(440\) 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 \(-0.247671\pi\)
0.712261 + 0.701915i \(0.247671\pi\)
\(462\) 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 \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) −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 \(-0.553436\pi\)
−0.167086 + 0.985942i \(0.553436\pi\)
\(480\) −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 \(-0.439420\pi\)
0.189170 + 0.981944i \(0.439420\pi\)
\(504\) 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 \(-0.771642\pi\)
−0.753512 + 0.657434i \(0.771642\pi\)
\(510\) 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 \(-0.653832\pi\)
−0.464684 + 0.885477i \(0.653832\pi\)
\(522\) 9.65685 0.422669
\(523\) 9.07107 0.396650 0.198325 0.980136i \(-0.436450\pi\)
0.198325 + 0.980136i \(0.436450\pi\)
\(524\) −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 \(-0.353552\pi\)
0.444020 + 0.896017i \(0.353552\pi\)
\(564\) 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 \(-0.470516\pi\)
0.0924958 + 0.995713i \(0.470516\pi\)
\(578\) −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 \(-0.578295\pi\)
−0.243498 + 0.969901i \(0.578295\pi\)
\(588\) 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 \(-0.222172\pi\)
0.766145 + 0.642668i \(0.222172\pi\)
\(594\) −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 \(-0.600138\pi\)
−0.309431 + 0.950922i \(0.600138\pi\)
\(602\) 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 \(-0.245666\pi\)
0.716670 + 0.697413i \(0.245666\pi\)
\(608\) 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 \(-0.356667\pi\)
0.435231 + 0.900319i \(0.356667\pi\)
\(620\) 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 \(-0.764493\pi\)
−0.738557 + 0.674191i \(0.764493\pi\)
\(644\) 0 0
\(645\) 9.65685 0.380238
\(646\) −4.48528 −0.176471
\(647\) −1.07107 −0.0421080 −0.0210540 0.999778i \(-0.506702\pi\)
−0.0210540 + 0.999778i \(0.506702\pi\)
\(648\) 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 \(-0.216859\pi\)
0.776765 + 0.629791i \(0.216859\pi\)
\(662\) −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 \(-0.256467\pi\)
0.692596 + 0.721326i \(0.256467\pi\)
\(678\) 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 \(-0.413308\pi\)
0.268996 + 0.963141i \(0.413308\pi\)
\(692\) 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 \(-0.191800\pi\)
0.823888 + 0.566752i \(0.191800\pi\)
\(720\) −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 \(-0.696485\pi\)
−0.578817 + 0.815458i \(0.696485\pi\)
\(728\) 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 \(-0.617621\pi\)
−0.361165 + 0.932502i \(0.617621\pi\)
\(734\) 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 \(-0.491260\pi\)
0.0274543 + 0.999623i \(0.491260\pi\)
\(762\) 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 \(-0.407699\pi\)
0.285925 + 0.958252i \(0.407699\pi\)
\(770\) 0 0
\(771\) 29.2132 1.05209
\(772\) −14.9706 −0.538802
\(773\) −5.51472 −0.198351 −0.0991753 0.995070i \(-0.531620\pi\)
−0.0991753 + 0.995070i \(0.531620\pi\)
\(774\) −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 \(-0.480618\pi\)
0.0608518 + 0.998147i \(0.480618\pi\)
\(788\) 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 \(-0.600397\pi\)
−0.310202 + 0.950671i \(0.600397\pi\)
\(798\) 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 \(-0.720578\pi\)
−0.638821 + 0.769355i \(0.720578\pi\)
\(812\) 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 \(-0.837974\pi\)
−0.873222 + 0.487322i \(0.837974\pi\)
\(830\) 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 \(-0.404787\pi\)
0.294679 + 0.955596i \(0.404787\pi\)
\(840\) 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 \(-0.708314\pi\)
−0.608713 + 0.793390i \(0.708314\pi\)
\(854\) 0 0
\(855\) −4.48528 −0.153393
\(856\) 12.8284 0.438467
\(857\) −0.142136 −0.00485526 −0.00242763 0.999997i \(-0.500773\pi\)
−0.00242763 + 0.999997i \(0.500773\pi\)
\(858\) 5.41421 0.184838
\(859\) 30.4264 1.03814 0.519068 0.854733i \(-0.326279\pi\)
0.519068 + 0.854733i \(0.326279\pi\)
\(860\) 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 \(-0.720417\pi\)
−0.638433 + 0.769677i \(0.720417\pi\)
\(882\) 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 \(-0.522915\pi\)
−0.0719259 + 0.997410i \(0.522915\pi\)
\(888\) 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 \(-0.556624\pi\)
−0.176952 + 0.984220i \(0.556624\pi\)
\(930\) 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 \(-0.120142\pi\)
0.929612 + 0.368540i \(0.120142\pi\)
\(938\) 0 0
\(939\) 8.24264 0.268988
\(940\) −15.7990 −0.515306
\(941\) −27.5563 −0.898311 −0.449156 0.893454i \(-0.648275\pi\)
−0.449156 + 0.893454i \(0.648275\pi\)
\(942\) −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 \(-0.523958\pi\)
−0.0751951 + 0.997169i \(0.523958\pi\)
\(972\) 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 \(-0.504417\pi\)
−0.0138753 + 0.999904i \(0.504417\pi\)
\(984\) −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 \(-0.660937\pi\)
−0.484331 + 0.874885i \(0.660937\pi\)
\(998\) −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.bd.1.1 yes 2
3.2 odd 2 9702.2.a.cy.1.1 2
7.6 odd 2 3234.2.a.bc.1.2 2
21.20 even 2 9702.2.a.ci.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bc.1.2 2 7.6 odd 2
3234.2.a.bd.1.1 yes 2 1.1 even 1 trivial
9702.2.a.ci.1.2 2 21.20 even 2
9702.2.a.cy.1.1 2 3.2 odd 2