Properties

Label 3234.2.a.bb.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: \(\Q(\zeta_{10})^+\)
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 3234.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.23607 q^{5} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +1.23607 q^{10} -1.00000 q^{11} -1.00000 q^{12} -1.23607 q^{15} +1.00000 q^{16} -5.23607 q^{17} +1.00000 q^{18} -7.70820 q^{19} +1.23607 q^{20} -1.00000 q^{22} -2.47214 q^{23} -1.00000 q^{24} -3.47214 q^{25} -1.00000 q^{27} +4.47214 q^{29} -1.23607 q^{30} -2.76393 q^{31} +1.00000 q^{32} +1.00000 q^{33} -5.23607 q^{34} +1.00000 q^{36} -10.9443 q^{37} -7.70820 q^{38} +1.23607 q^{40} -5.23607 q^{41} +6.47214 q^{43} -1.00000 q^{44} +1.23607 q^{45} -2.47214 q^{46} +3.70820 q^{47} -1.00000 q^{48} -3.47214 q^{50} +5.23607 q^{51} -6.00000 q^{53} -1.00000 q^{54} -1.23607 q^{55} +7.70820 q^{57} +4.47214 q^{58} -1.52786 q^{59} -1.23607 q^{60} -2.76393 q^{62} +1.00000 q^{64} +1.00000 q^{66} +15.4164 q^{67} -5.23607 q^{68} +2.47214 q^{69} -2.47214 q^{71} +1.00000 q^{72} -15.7082 q^{73} -10.9443 q^{74} +3.47214 q^{75} -7.70820 q^{76} +11.4164 q^{79} +1.23607 q^{80} +1.00000 q^{81} -5.23607 q^{82} +1.23607 q^{83} -6.47214 q^{85} +6.47214 q^{86} -4.47214 q^{87} -1.00000 q^{88} -6.47214 q^{89} +1.23607 q^{90} -2.47214 q^{92} +2.76393 q^{93} +3.70820 q^{94} -9.52786 q^{95} -1.00000 q^{96} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} - 2 q^{3} + 2 q^{4} - 2 q^{5} - 2 q^{6} + 2 q^{8} + 2 q^{9} - 2 q^{10} - 2 q^{11} - 2 q^{12} + 2 q^{15} + 2 q^{16} - 6 q^{17} + 2 q^{18} - 2 q^{19} - 2 q^{20} - 2 q^{22} + 4 q^{23} - 2 q^{24} + 2 q^{25} - 2 q^{27} + 2 q^{30} - 10 q^{31} + 2 q^{32} + 2 q^{33} - 6 q^{34} + 2 q^{36} - 4 q^{37} - 2 q^{38} - 2 q^{40} - 6 q^{41} + 4 q^{43} - 2 q^{44} - 2 q^{45} + 4 q^{46} - 6 q^{47} - 2 q^{48} + 2 q^{50} + 6 q^{51} - 12 q^{53} - 2 q^{54} + 2 q^{55} + 2 q^{57} - 12 q^{59} + 2 q^{60} - 10 q^{62} + 2 q^{64} + 2 q^{66} + 4 q^{67} - 6 q^{68} - 4 q^{69} + 4 q^{71} + 2 q^{72} - 18 q^{73} - 4 q^{74} - 2 q^{75} - 2 q^{76} - 4 q^{79} - 2 q^{80} + 2 q^{81} - 6 q^{82} - 2 q^{83} - 4 q^{85} + 4 q^{86} - 2 q^{88} - 4 q^{89} - 2 q^{90} + 4 q^{92} + 10 q^{93} - 6 q^{94} - 28 q^{95} - 2 q^{96} - 2 q^{99}+O(q^{100}) \) 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
\(5\) 1.23607 0.552786 0.276393 0.961045i \(-0.410861\pi\)
0.276393 + 0.961045i \(0.410861\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.23607 0.390879
\(11\) −1.00000 −0.301511
\(12\) −1.00000 −0.288675
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) −1.23607 −0.319151
\(16\) 1.00000 0.250000
\(17\) −5.23607 −1.26993 −0.634967 0.772540i \(-0.718986\pi\)
−0.634967 + 0.772540i \(0.718986\pi\)
\(18\) 1.00000 0.235702
\(19\) −7.70820 −1.76838 −0.884192 0.467124i \(-0.845290\pi\)
−0.884192 + 0.467124i \(0.845290\pi\)
\(20\) 1.23607 0.276393
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −2.47214 −0.515476 −0.257738 0.966215i \(-0.582977\pi\)
−0.257738 + 0.966215i \(0.582977\pi\)
\(24\) −1.00000 −0.204124
\(25\) −3.47214 −0.694427
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.47214 0.830455 0.415227 0.909718i \(-0.363702\pi\)
0.415227 + 0.909718i \(0.363702\pi\)
\(30\) −1.23607 −0.225674
\(31\) −2.76393 −0.496417 −0.248208 0.968707i \(-0.579842\pi\)
−0.248208 + 0.968707i \(0.579842\pi\)
\(32\) 1.00000 0.176777
\(33\) 1.00000 0.174078
\(34\) −5.23607 −0.897978
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.9443 −1.79923 −0.899614 0.436687i \(-0.856152\pi\)
−0.899614 + 0.436687i \(0.856152\pi\)
\(38\) −7.70820 −1.25044
\(39\) 0 0
\(40\) 1.23607 0.195440
\(41\) −5.23607 −0.817736 −0.408868 0.912593i \(-0.634076\pi\)
−0.408868 + 0.912593i \(0.634076\pi\)
\(42\) 0 0
\(43\) 6.47214 0.986991 0.493496 0.869748i \(-0.335719\pi\)
0.493496 + 0.869748i \(0.335719\pi\)
\(44\) −1.00000 −0.150756
\(45\) 1.23607 0.184262
\(46\) −2.47214 −0.364497
\(47\) 3.70820 0.540897 0.270449 0.962734i \(-0.412828\pi\)
0.270449 + 0.962734i \(0.412828\pi\)
\(48\) −1.00000 −0.144338
\(49\) 0 0
\(50\) −3.47214 −0.491034
\(51\) 5.23607 0.733196
\(52\) 0 0
\(53\) −6.00000 −0.824163 −0.412082 0.911147i \(-0.635198\pi\)
−0.412082 + 0.911147i \(0.635198\pi\)
\(54\) −1.00000 −0.136083
\(55\) −1.23607 −0.166671
\(56\) 0 0
\(57\) 7.70820 1.02098
\(58\) 4.47214 0.587220
\(59\) −1.52786 −0.198911 −0.0994555 0.995042i \(-0.531710\pi\)
−0.0994555 + 0.995042i \(0.531710\pi\)
\(60\) −1.23607 −0.159576
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) −2.76393 −0.351020
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.00000 0.123091
\(67\) 15.4164 1.88341 0.941707 0.336434i \(-0.109221\pi\)
0.941707 + 0.336434i \(0.109221\pi\)
\(68\) −5.23607 −0.634967
\(69\) 2.47214 0.297610
\(70\) 0 0
\(71\) −2.47214 −0.293389 −0.146694 0.989182i \(-0.546863\pi\)
−0.146694 + 0.989182i \(0.546863\pi\)
\(72\) 1.00000 0.117851
\(73\) −15.7082 −1.83851 −0.919253 0.393667i \(-0.871206\pi\)
−0.919253 + 0.393667i \(0.871206\pi\)
\(74\) −10.9443 −1.27225
\(75\) 3.47214 0.400928
\(76\) −7.70820 −0.884192
\(77\) 0 0
\(78\) 0 0
\(79\) 11.4164 1.28445 0.642223 0.766518i \(-0.278012\pi\)
0.642223 + 0.766518i \(0.278012\pi\)
\(80\) 1.23607 0.138197
\(81\) 1.00000 0.111111
\(82\) −5.23607 −0.578227
\(83\) 1.23607 0.135676 0.0678380 0.997696i \(-0.478390\pi\)
0.0678380 + 0.997696i \(0.478390\pi\)
\(84\) 0 0
\(85\) −6.47214 −0.702002
\(86\) 6.47214 0.697908
\(87\) −4.47214 −0.479463
\(88\) −1.00000 −0.106600
\(89\) −6.47214 −0.686045 −0.343023 0.939327i \(-0.611451\pi\)
−0.343023 + 0.939327i \(0.611451\pi\)
\(90\) 1.23607 0.130293
\(91\) 0 0
\(92\) −2.47214 −0.257738
\(93\) 2.76393 0.286606
\(94\) 3.70820 0.382472
\(95\) −9.52786 −0.977538
\(96\) −1.00000 −0.102062
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −3.47214 −0.347214
\(101\) −12.0000 −1.19404 −0.597022 0.802225i \(-0.703650\pi\)
−0.597022 + 0.802225i \(0.703650\pi\)
\(102\) 5.23607 0.518448
\(103\) 10.7639 1.06060 0.530301 0.847810i \(-0.322079\pi\)
0.530301 + 0.847810i \(0.322079\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 0.944272 0.0912862 0.0456431 0.998958i \(-0.485466\pi\)
0.0456431 + 0.998958i \(0.485466\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) −1.23607 −0.117854
\(111\) 10.9443 1.03878
\(112\) 0 0
\(113\) 13.4164 1.26211 0.631055 0.775738i \(-0.282622\pi\)
0.631055 + 0.775738i \(0.282622\pi\)
\(114\) 7.70820 0.721939
\(115\) −3.05573 −0.284948
\(116\) 4.47214 0.415227
\(117\) 0 0
\(118\) −1.52786 −0.140651
\(119\) 0 0
\(120\) −1.23607 −0.112837
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 5.23607 0.472120
\(124\) −2.76393 −0.248208
\(125\) −10.4721 −0.936656
\(126\) 0 0
\(127\) −6.47214 −0.574309 −0.287155 0.957884i \(-0.592709\pi\)
−0.287155 + 0.957884i \(0.592709\pi\)
\(128\) 1.00000 0.0883883
\(129\) −6.47214 −0.569840
\(130\) 0 0
\(131\) 6.76393 0.590967 0.295484 0.955348i \(-0.404519\pi\)
0.295484 + 0.955348i \(0.404519\pi\)
\(132\) 1.00000 0.0870388
\(133\) 0 0
\(134\) 15.4164 1.33177
\(135\) −1.23607 −0.106384
\(136\) −5.23607 −0.448989
\(137\) −16.4721 −1.40731 −0.703655 0.710542i \(-0.748450\pi\)
−0.703655 + 0.710542i \(0.748450\pi\)
\(138\) 2.47214 0.210442
\(139\) −10.1803 −0.863485 −0.431743 0.901997i \(-0.642101\pi\)
−0.431743 + 0.901997i \(0.642101\pi\)
\(140\) 0 0
\(141\) −3.70820 −0.312287
\(142\) −2.47214 −0.207457
\(143\) 0 0
\(144\) 1.00000 0.0833333
\(145\) 5.52786 0.459064
\(146\) −15.7082 −1.30002
\(147\) 0 0
\(148\) −10.9443 −0.899614
\(149\) −6.00000 −0.491539 −0.245770 0.969328i \(-0.579041\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 3.47214 0.283499
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −7.70820 −0.625218
\(153\) −5.23607 −0.423311
\(154\) 0 0
\(155\) −3.41641 −0.274412
\(156\) 0 0
\(157\) −14.7639 −1.17829 −0.589145 0.808027i \(-0.700535\pi\)
−0.589145 + 0.808027i \(0.700535\pi\)
\(158\) 11.4164 0.908241
\(159\) 6.00000 0.475831
\(160\) 1.23607 0.0977198
\(161\) 0 0
\(162\) 1.00000 0.0785674
\(163\) −12.0000 −0.939913 −0.469956 0.882690i \(-0.655730\pi\)
−0.469956 + 0.882690i \(0.655730\pi\)
\(164\) −5.23607 −0.408868
\(165\) 1.23607 0.0962278
\(166\) 1.23607 0.0959375
\(167\) 18.4721 1.42942 0.714708 0.699423i \(-0.246559\pi\)
0.714708 + 0.699423i \(0.246559\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) −6.47214 −0.496390
\(171\) −7.70820 −0.589461
\(172\) 6.47214 0.493496
\(173\) 19.4164 1.47620 0.738101 0.674690i \(-0.235723\pi\)
0.738101 + 0.674690i \(0.235723\pi\)
\(174\) −4.47214 −0.339032
\(175\) 0 0
\(176\) −1.00000 −0.0753778
\(177\) 1.52786 0.114841
\(178\) −6.47214 −0.485107
\(179\) −5.52786 −0.413172 −0.206586 0.978428i \(-0.566235\pi\)
−0.206586 + 0.978428i \(0.566235\pi\)
\(180\) 1.23607 0.0921311
\(181\) 11.7082 0.870264 0.435132 0.900367i \(-0.356702\pi\)
0.435132 + 0.900367i \(0.356702\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −2.47214 −0.182248
\(185\) −13.5279 −0.994588
\(186\) 2.76393 0.202661
\(187\) 5.23607 0.382899
\(188\) 3.70820 0.270449
\(189\) 0 0
\(190\) −9.52786 −0.691224
\(191\) −10.4721 −0.757737 −0.378869 0.925450i \(-0.623687\pi\)
−0.378869 + 0.925450i \(0.623687\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 5.05573 0.363919 0.181960 0.983306i \(-0.441756\pi\)
0.181960 + 0.983306i \(0.441756\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −22.3607 −1.59313 −0.796566 0.604551i \(-0.793352\pi\)
−0.796566 + 0.604551i \(0.793352\pi\)
\(198\) −1.00000 −0.0710669
\(199\) −23.1246 −1.63926 −0.819630 0.572893i \(-0.805821\pi\)
−0.819630 + 0.572893i \(0.805821\pi\)
\(200\) −3.47214 −0.245517
\(201\) −15.4164 −1.08739
\(202\) −12.0000 −0.844317
\(203\) 0 0
\(204\) 5.23607 0.366598
\(205\) −6.47214 −0.452034
\(206\) 10.7639 0.749959
\(207\) −2.47214 −0.171825
\(208\) 0 0
\(209\) 7.70820 0.533188
\(210\) 0 0
\(211\) 3.41641 0.235195 0.117598 0.993061i \(-0.462481\pi\)
0.117598 + 0.993061i \(0.462481\pi\)
\(212\) −6.00000 −0.412082
\(213\) 2.47214 0.169388
\(214\) 0.944272 0.0645491
\(215\) 8.00000 0.545595
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) 6.00000 0.406371
\(219\) 15.7082 1.06146
\(220\) −1.23607 −0.0833357
\(221\) 0 0
\(222\) 10.9443 0.734531
\(223\) −15.7082 −1.05190 −0.525950 0.850516i \(-0.676290\pi\)
−0.525950 + 0.850516i \(0.676290\pi\)
\(224\) 0 0
\(225\) −3.47214 −0.231476
\(226\) 13.4164 0.892446
\(227\) 8.65248 0.574285 0.287142 0.957888i \(-0.407295\pi\)
0.287142 + 0.957888i \(0.407295\pi\)
\(228\) 7.70820 0.510488
\(229\) 19.7082 1.30235 0.651177 0.758926i \(-0.274275\pi\)
0.651177 + 0.758926i \(0.274275\pi\)
\(230\) −3.05573 −0.201489
\(231\) 0 0
\(232\) 4.47214 0.293610
\(233\) −9.05573 −0.593260 −0.296630 0.954992i \(-0.595863\pi\)
−0.296630 + 0.954992i \(0.595863\pi\)
\(234\) 0 0
\(235\) 4.58359 0.299001
\(236\) −1.52786 −0.0994555
\(237\) −11.4164 −0.741575
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) −1.23607 −0.0797878
\(241\) 18.1803 1.17110 0.585549 0.810637i \(-0.300879\pi\)
0.585549 + 0.810637i \(0.300879\pi\)
\(242\) 1.00000 0.0642824
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 5.23607 0.333840
\(247\) 0 0
\(248\) −2.76393 −0.175510
\(249\) −1.23607 −0.0783326
\(250\) −10.4721 −0.662316
\(251\) −6.47214 −0.408518 −0.204259 0.978917i \(-0.565478\pi\)
−0.204259 + 0.978917i \(0.565478\pi\)
\(252\) 0 0
\(253\) 2.47214 0.155422
\(254\) −6.47214 −0.406098
\(255\) 6.47214 0.405301
\(256\) 1.00000 0.0625000
\(257\) 22.4721 1.40177 0.700887 0.713273i \(-0.252788\pi\)
0.700887 + 0.713273i \(0.252788\pi\)
\(258\) −6.47214 −0.402938
\(259\) 0 0
\(260\) 0 0
\(261\) 4.47214 0.276818
\(262\) 6.76393 0.417877
\(263\) −9.52786 −0.587513 −0.293757 0.955880i \(-0.594906\pi\)
−0.293757 + 0.955880i \(0.594906\pi\)
\(264\) 1.00000 0.0615457
\(265\) −7.41641 −0.455586
\(266\) 0 0
\(267\) 6.47214 0.396088
\(268\) 15.4164 0.941707
\(269\) −27.7082 −1.68940 −0.844700 0.535241i \(-0.820221\pi\)
−0.844700 + 0.535241i \(0.820221\pi\)
\(270\) −1.23607 −0.0752247
\(271\) −2.47214 −0.150172 −0.0750858 0.997177i \(-0.523923\pi\)
−0.0750858 + 0.997177i \(0.523923\pi\)
\(272\) −5.23607 −0.317483
\(273\) 0 0
\(274\) −16.4721 −0.995118
\(275\) 3.47214 0.209378
\(276\) 2.47214 0.148805
\(277\) 26.0000 1.56219 0.781094 0.624413i \(-0.214662\pi\)
0.781094 + 0.624413i \(0.214662\pi\)
\(278\) −10.1803 −0.610576
\(279\) −2.76393 −0.165472
\(280\) 0 0
\(281\) 19.8885 1.18645 0.593226 0.805036i \(-0.297854\pi\)
0.593226 + 0.805036i \(0.297854\pi\)
\(282\) −3.70820 −0.220820
\(283\) 23.7082 1.40931 0.704653 0.709552i \(-0.251103\pi\)
0.704653 + 0.709552i \(0.251103\pi\)
\(284\) −2.47214 −0.146694
\(285\) 9.52786 0.564382
\(286\) 0 0
\(287\) 0 0
\(288\) 1.00000 0.0589256
\(289\) 10.4164 0.612730
\(290\) 5.52786 0.324607
\(291\) 0 0
\(292\) −15.7082 −0.919253
\(293\) −4.58359 −0.267776 −0.133888 0.990996i \(-0.542746\pi\)
−0.133888 + 0.990996i \(0.542746\pi\)
\(294\) 0 0
\(295\) −1.88854 −0.109955
\(296\) −10.9443 −0.636123
\(297\) 1.00000 0.0580259
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 3.47214 0.200464
\(301\) 0 0
\(302\) 0 0
\(303\) 12.0000 0.689382
\(304\) −7.70820 −0.442096
\(305\) 0 0
\(306\) −5.23607 −0.299326
\(307\) 25.5967 1.46088 0.730442 0.682975i \(-0.239314\pi\)
0.730442 + 0.682975i \(0.239314\pi\)
\(308\) 0 0
\(309\) −10.7639 −0.612339
\(310\) −3.41641 −0.194039
\(311\) −8.65248 −0.490637 −0.245318 0.969443i \(-0.578893\pi\)
−0.245318 + 0.969443i \(0.578893\pi\)
\(312\) 0 0
\(313\) 5.52786 0.312453 0.156227 0.987721i \(-0.450067\pi\)
0.156227 + 0.987721i \(0.450067\pi\)
\(314\) −14.7639 −0.833177
\(315\) 0 0
\(316\) 11.4164 0.642223
\(317\) −31.8885 −1.79104 −0.895520 0.445022i \(-0.853196\pi\)
−0.895520 + 0.445022i \(0.853196\pi\)
\(318\) 6.00000 0.336463
\(319\) −4.47214 −0.250392
\(320\) 1.23607 0.0690983
\(321\) −0.944272 −0.0527041
\(322\) 0 0
\(323\) 40.3607 2.24573
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −12.0000 −0.664619
\(327\) −6.00000 −0.331801
\(328\) −5.23607 −0.289113
\(329\) 0 0
\(330\) 1.23607 0.0680433
\(331\) −7.41641 −0.407643 −0.203821 0.979008i \(-0.565336\pi\)
−0.203821 + 0.979008i \(0.565336\pi\)
\(332\) 1.23607 0.0678380
\(333\) −10.9443 −0.599742
\(334\) 18.4721 1.01075
\(335\) 19.0557 1.04113
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) −13.0000 −0.707107
\(339\) −13.4164 −0.728679
\(340\) −6.47214 −0.351001
\(341\) 2.76393 0.149675
\(342\) −7.70820 −0.416812
\(343\) 0 0
\(344\) 6.47214 0.348954
\(345\) 3.05573 0.164515
\(346\) 19.4164 1.04383
\(347\) −26.8328 −1.44046 −0.720231 0.693735i \(-0.755964\pi\)
−0.720231 + 0.693735i \(0.755964\pi\)
\(348\) −4.47214 −0.239732
\(349\) −0.583592 −0.0312390 −0.0156195 0.999878i \(-0.504972\pi\)
−0.0156195 + 0.999878i \(0.504972\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 24.3607 1.29659 0.648294 0.761390i \(-0.275483\pi\)
0.648294 + 0.761390i \(0.275483\pi\)
\(354\) 1.52786 0.0812051
\(355\) −3.05573 −0.162181
\(356\) −6.47214 −0.343023
\(357\) 0 0
\(358\) −5.52786 −0.292157
\(359\) 1.52786 0.0806376 0.0403188 0.999187i \(-0.487163\pi\)
0.0403188 + 0.999187i \(0.487163\pi\)
\(360\) 1.23607 0.0651465
\(361\) 40.4164 2.12718
\(362\) 11.7082 0.615370
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −19.4164 −1.01630
\(366\) 0 0
\(367\) 8.29180 0.432828 0.216414 0.976302i \(-0.430564\pi\)
0.216414 + 0.976302i \(0.430564\pi\)
\(368\) −2.47214 −0.128869
\(369\) −5.23607 −0.272579
\(370\) −13.5279 −0.703280
\(371\) 0 0
\(372\) 2.76393 0.143303
\(373\) −1.41641 −0.0733388 −0.0366694 0.999327i \(-0.511675\pi\)
−0.0366694 + 0.999327i \(0.511675\pi\)
\(374\) 5.23607 0.270751
\(375\) 10.4721 0.540779
\(376\) 3.70820 0.191236
\(377\) 0 0
\(378\) 0 0
\(379\) −4.00000 −0.205466 −0.102733 0.994709i \(-0.532759\pi\)
−0.102733 + 0.994709i \(0.532759\pi\)
\(380\) −9.52786 −0.488769
\(381\) 6.47214 0.331578
\(382\) −10.4721 −0.535801
\(383\) −0.652476 −0.0333400 −0.0166700 0.999861i \(-0.505306\pi\)
−0.0166700 + 0.999861i \(0.505306\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) 5.05573 0.257330
\(387\) 6.47214 0.328997
\(388\) 0 0
\(389\) −6.94427 −0.352089 −0.176044 0.984382i \(-0.556330\pi\)
−0.176044 + 0.984382i \(0.556330\pi\)
\(390\) 0 0
\(391\) 12.9443 0.654620
\(392\) 0 0
\(393\) −6.76393 −0.341195
\(394\) −22.3607 −1.12651
\(395\) 14.1115 0.710024
\(396\) −1.00000 −0.0502519
\(397\) −1.81966 −0.0913261 −0.0456631 0.998957i \(-0.514540\pi\)
−0.0456631 + 0.998957i \(0.514540\pi\)
\(398\) −23.1246 −1.15913
\(399\) 0 0
\(400\) −3.47214 −0.173607
\(401\) −8.47214 −0.423078 −0.211539 0.977370i \(-0.567848\pi\)
−0.211539 + 0.977370i \(0.567848\pi\)
\(402\) −15.4164 −0.768901
\(403\) 0 0
\(404\) −12.0000 −0.597022
\(405\) 1.23607 0.0614207
\(406\) 0 0
\(407\) 10.9443 0.542487
\(408\) 5.23607 0.259224
\(409\) 2.76393 0.136668 0.0683338 0.997663i \(-0.478232\pi\)
0.0683338 + 0.997663i \(0.478232\pi\)
\(410\) −6.47214 −0.319636
\(411\) 16.4721 0.812511
\(412\) 10.7639 0.530301
\(413\) 0 0
\(414\) −2.47214 −0.121499
\(415\) 1.52786 0.0749999
\(416\) 0 0
\(417\) 10.1803 0.498533
\(418\) 7.70820 0.377021
\(419\) 15.0557 0.735520 0.367760 0.929921i \(-0.380125\pi\)
0.367760 + 0.929921i \(0.380125\pi\)
\(420\) 0 0
\(421\) −5.05573 −0.246401 −0.123201 0.992382i \(-0.539316\pi\)
−0.123201 + 0.992382i \(0.539316\pi\)
\(422\) 3.41641 0.166308
\(423\) 3.70820 0.180299
\(424\) −6.00000 −0.291386
\(425\) 18.1803 0.881876
\(426\) 2.47214 0.119775
\(427\) 0 0
\(428\) 0.944272 0.0456431
\(429\) 0 0
\(430\) 8.00000 0.385794
\(431\) −22.4721 −1.08244 −0.541222 0.840880i \(-0.682038\pi\)
−0.541222 + 0.840880i \(0.682038\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.88854 0.475213 0.237607 0.971361i \(-0.423637\pi\)
0.237607 + 0.971361i \(0.423637\pi\)
\(434\) 0 0
\(435\) −5.52786 −0.265041
\(436\) 6.00000 0.287348
\(437\) 19.0557 0.911559
\(438\) 15.7082 0.750567
\(439\) 31.4164 1.49942 0.749712 0.661765i \(-0.230192\pi\)
0.749712 + 0.661765i \(0.230192\pi\)
\(440\) −1.23607 −0.0589272
\(441\) 0 0
\(442\) 0 0
\(443\) 7.41641 0.352364 0.176182 0.984358i \(-0.443625\pi\)
0.176182 + 0.984358i \(0.443625\pi\)
\(444\) 10.9443 0.519392
\(445\) −8.00000 −0.379236
\(446\) −15.7082 −0.743805
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 19.5279 0.921577 0.460788 0.887510i \(-0.347567\pi\)
0.460788 + 0.887510i \(0.347567\pi\)
\(450\) −3.47214 −0.163678
\(451\) 5.23607 0.246557
\(452\) 13.4164 0.631055
\(453\) 0 0
\(454\) 8.65248 0.406081
\(455\) 0 0
\(456\) 7.70820 0.360970
\(457\) 18.0000 0.842004 0.421002 0.907060i \(-0.361678\pi\)
0.421002 + 0.907060i \(0.361678\pi\)
\(458\) 19.7082 0.920904
\(459\) 5.23607 0.244399
\(460\) −3.05573 −0.142474
\(461\) −26.8328 −1.24973 −0.624864 0.780733i \(-0.714846\pi\)
−0.624864 + 0.780733i \(0.714846\pi\)
\(462\) 0 0
\(463\) −1.88854 −0.0877681 −0.0438840 0.999037i \(-0.513973\pi\)
−0.0438840 + 0.999037i \(0.513973\pi\)
\(464\) 4.47214 0.207614
\(465\) 3.41641 0.158432
\(466\) −9.05573 −0.419499
\(467\) −29.8885 −1.38308 −0.691538 0.722340i \(-0.743067\pi\)
−0.691538 + 0.722340i \(0.743067\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 4.58359 0.211425
\(471\) 14.7639 0.680286
\(472\) −1.52786 −0.0703256
\(473\) −6.47214 −0.297589
\(474\) −11.4164 −0.524373
\(475\) 26.7639 1.22801
\(476\) 0 0
\(477\) −6.00000 −0.274721
\(478\) 0 0
\(479\) 35.7771 1.63470 0.817348 0.576144i \(-0.195443\pi\)
0.817348 + 0.576144i \(0.195443\pi\)
\(480\) −1.23607 −0.0564185
\(481\) 0 0
\(482\) 18.1803 0.828092
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 36.9443 1.67410 0.837052 0.547123i \(-0.184277\pi\)
0.837052 + 0.547123i \(0.184277\pi\)
\(488\) 0 0
\(489\) 12.0000 0.542659
\(490\) 0 0
\(491\) −29.8885 −1.34885 −0.674426 0.738343i \(-0.735609\pi\)
−0.674426 + 0.738343i \(0.735609\pi\)
\(492\) 5.23607 0.236060
\(493\) −23.4164 −1.05462
\(494\) 0 0
\(495\) −1.23607 −0.0555571
\(496\) −2.76393 −0.124104
\(497\) 0 0
\(498\) −1.23607 −0.0553895
\(499\) 7.41641 0.332004 0.166002 0.986125i \(-0.446914\pi\)
0.166002 + 0.986125i \(0.446914\pi\)
\(500\) −10.4721 −0.468328
\(501\) −18.4721 −0.825274
\(502\) −6.47214 −0.288866
\(503\) −41.3050 −1.84170 −0.920848 0.389921i \(-0.872502\pi\)
−0.920848 + 0.389921i \(0.872502\pi\)
\(504\) 0 0
\(505\) −14.8328 −0.660052
\(506\) 2.47214 0.109900
\(507\) 13.0000 0.577350
\(508\) −6.47214 −0.287155
\(509\) −6.76393 −0.299806 −0.149903 0.988701i \(-0.547896\pi\)
−0.149903 + 0.988701i \(0.547896\pi\)
\(510\) 6.47214 0.286591
\(511\) 0 0
\(512\) 1.00000 0.0441942
\(513\) 7.70820 0.340326
\(514\) 22.4721 0.991203
\(515\) 13.3050 0.586286
\(516\) −6.47214 −0.284920
\(517\) −3.70820 −0.163087
\(518\) 0 0
\(519\) −19.4164 −0.852286
\(520\) 0 0
\(521\) −21.3050 −0.933387 −0.466693 0.884419i \(-0.654555\pi\)
−0.466693 + 0.884419i \(0.654555\pi\)
\(522\) 4.47214 0.195740
\(523\) −24.2918 −1.06221 −0.531103 0.847307i \(-0.678222\pi\)
−0.531103 + 0.847307i \(0.678222\pi\)
\(524\) 6.76393 0.295484
\(525\) 0 0
\(526\) −9.52786 −0.415435
\(527\) 14.4721 0.630416
\(528\) 1.00000 0.0435194
\(529\) −16.8885 −0.734285
\(530\) −7.41641 −0.322148
\(531\) −1.52786 −0.0663037
\(532\) 0 0
\(533\) 0 0
\(534\) 6.47214 0.280077
\(535\) 1.16718 0.0504618
\(536\) 15.4164 0.665887
\(537\) 5.52786 0.238545
\(538\) −27.7082 −1.19459
\(539\) 0 0
\(540\) −1.23607 −0.0531919
\(541\) 21.4164 0.920763 0.460382 0.887721i \(-0.347713\pi\)
0.460382 + 0.887721i \(0.347713\pi\)
\(542\) −2.47214 −0.106187
\(543\) −11.7082 −0.502447
\(544\) −5.23607 −0.224495
\(545\) 7.41641 0.317684
\(546\) 0 0
\(547\) 38.4721 1.64495 0.822475 0.568801i \(-0.192593\pi\)
0.822475 + 0.568801i \(0.192593\pi\)
\(548\) −16.4721 −0.703655
\(549\) 0 0
\(550\) 3.47214 0.148052
\(551\) −34.4721 −1.46856
\(552\) 2.47214 0.105221
\(553\) 0 0
\(554\) 26.0000 1.10463
\(555\) 13.5279 0.574226
\(556\) −10.1803 −0.431743
\(557\) −10.5836 −0.448441 −0.224221 0.974538i \(-0.571984\pi\)
−0.224221 + 0.974538i \(0.571984\pi\)
\(558\) −2.76393 −0.117007
\(559\) 0 0
\(560\) 0 0
\(561\) −5.23607 −0.221067
\(562\) 19.8885 0.838948
\(563\) 6.76393 0.285066 0.142533 0.989790i \(-0.454475\pi\)
0.142533 + 0.989790i \(0.454475\pi\)
\(564\) −3.70820 −0.156144
\(565\) 16.5836 0.697677
\(566\) 23.7082 0.996530
\(567\) 0 0
\(568\) −2.47214 −0.103729
\(569\) −34.9443 −1.46494 −0.732470 0.680799i \(-0.761633\pi\)
−0.732470 + 0.680799i \(0.761633\pi\)
\(570\) 9.52786 0.399078
\(571\) −17.5279 −0.733518 −0.366759 0.930316i \(-0.619533\pi\)
−0.366759 + 0.930316i \(0.619533\pi\)
\(572\) 0 0
\(573\) 10.4721 0.437480
\(574\) 0 0
\(575\) 8.58359 0.357961
\(576\) 1.00000 0.0416667
\(577\) −39.4164 −1.64093 −0.820463 0.571699i \(-0.806284\pi\)
−0.820463 + 0.571699i \(0.806284\pi\)
\(578\) 10.4164 0.433265
\(579\) −5.05573 −0.210109
\(580\) 5.52786 0.229532
\(581\) 0 0
\(582\) 0 0
\(583\) 6.00000 0.248495
\(584\) −15.7082 −0.650010
\(585\) 0 0
\(586\) −4.58359 −0.189346
\(587\) 17.5279 0.723452 0.361726 0.932284i \(-0.382188\pi\)
0.361726 + 0.932284i \(0.382188\pi\)
\(588\) 0 0
\(589\) 21.3050 0.877855
\(590\) −1.88854 −0.0777501
\(591\) 22.3607 0.919795
\(592\) −10.9443 −0.449807
\(593\) 18.7639 0.770542 0.385271 0.922803i \(-0.374108\pi\)
0.385271 + 0.922803i \(0.374108\pi\)
\(594\) 1.00000 0.0410305
\(595\) 0 0
\(596\) −6.00000 −0.245770
\(597\) 23.1246 0.946427
\(598\) 0 0
\(599\) 25.3050 1.03393 0.516966 0.856006i \(-0.327061\pi\)
0.516966 + 0.856006i \(0.327061\pi\)
\(600\) 3.47214 0.141749
\(601\) −0.291796 −0.0119026 −0.00595130 0.999982i \(-0.501894\pi\)
−0.00595130 + 0.999982i \(0.501894\pi\)
\(602\) 0 0
\(603\) 15.4164 0.627805
\(604\) 0 0
\(605\) 1.23607 0.0502533
\(606\) 12.0000 0.487467
\(607\) −8.00000 −0.324710 −0.162355 0.986732i \(-0.551909\pi\)
−0.162355 + 0.986732i \(0.551909\pi\)
\(608\) −7.70820 −0.312609
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) −5.23607 −0.211656
\(613\) 12.4721 0.503745 0.251872 0.967760i \(-0.418954\pi\)
0.251872 + 0.967760i \(0.418954\pi\)
\(614\) 25.5967 1.03300
\(615\) 6.47214 0.260982
\(616\) 0 0
\(617\) 23.8885 0.961717 0.480858 0.876798i \(-0.340325\pi\)
0.480858 + 0.876798i \(0.340325\pi\)
\(618\) −10.7639 −0.432989
\(619\) 18.8328 0.756955 0.378477 0.925611i \(-0.376448\pi\)
0.378477 + 0.925611i \(0.376448\pi\)
\(620\) −3.41641 −0.137206
\(621\) 2.47214 0.0992034
\(622\) −8.65248 −0.346933
\(623\) 0 0
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 5.52786 0.220938
\(627\) −7.70820 −0.307836
\(628\) −14.7639 −0.589145
\(629\) 57.3050 2.28490
\(630\) 0 0
\(631\) 9.88854 0.393657 0.196828 0.980438i \(-0.436936\pi\)
0.196828 + 0.980438i \(0.436936\pi\)
\(632\) 11.4164 0.454120
\(633\) −3.41641 −0.135790
\(634\) −31.8885 −1.26646
\(635\) −8.00000 −0.317470
\(636\) 6.00000 0.237915
\(637\) 0 0
\(638\) −4.47214 −0.177054
\(639\) −2.47214 −0.0977962
\(640\) 1.23607 0.0488599
\(641\) −35.8885 −1.41751 −0.708756 0.705454i \(-0.750743\pi\)
−0.708756 + 0.705454i \(0.750743\pi\)
\(642\) −0.944272 −0.0372674
\(643\) 43.4164 1.71218 0.856088 0.516830i \(-0.172888\pi\)
0.856088 + 0.516830i \(0.172888\pi\)
\(644\) 0 0
\(645\) −8.00000 −0.315000
\(646\) 40.3607 1.58797
\(647\) 45.0132 1.76965 0.884825 0.465924i \(-0.154278\pi\)
0.884825 + 0.465924i \(0.154278\pi\)
\(648\) 1.00000 0.0392837
\(649\) 1.52786 0.0599739
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −14.9443 −0.584815 −0.292407 0.956294i \(-0.594456\pi\)
−0.292407 + 0.956294i \(0.594456\pi\)
\(654\) −6.00000 −0.234619
\(655\) 8.36068 0.326679
\(656\) −5.23607 −0.204434
\(657\) −15.7082 −0.612835
\(658\) 0 0
\(659\) 13.8885 0.541021 0.270510 0.962717i \(-0.412808\pi\)
0.270510 + 0.962717i \(0.412808\pi\)
\(660\) 1.23607 0.0481139
\(661\) 5.59675 0.217688 0.108844 0.994059i \(-0.465285\pi\)
0.108844 + 0.994059i \(0.465285\pi\)
\(662\) −7.41641 −0.288247
\(663\) 0 0
\(664\) 1.23607 0.0479687
\(665\) 0 0
\(666\) −10.9443 −0.424082
\(667\) −11.0557 −0.428080
\(668\) 18.4721 0.714708
\(669\) 15.7082 0.607314
\(670\) 19.0557 0.736187
\(671\) 0 0
\(672\) 0 0
\(673\) 34.9443 1.34700 0.673501 0.739186i \(-0.264790\pi\)
0.673501 + 0.739186i \(0.264790\pi\)
\(674\) −18.0000 −0.693334
\(675\) 3.47214 0.133643
\(676\) −13.0000 −0.500000
\(677\) −46.4721 −1.78607 −0.893035 0.449988i \(-0.851428\pi\)
−0.893035 + 0.449988i \(0.851428\pi\)
\(678\) −13.4164 −0.515254
\(679\) 0 0
\(680\) −6.47214 −0.248195
\(681\) −8.65248 −0.331564
\(682\) 2.76393 0.105836
\(683\) −4.36068 −0.166857 −0.0834284 0.996514i \(-0.526587\pi\)
−0.0834284 + 0.996514i \(0.526587\pi\)
\(684\) −7.70820 −0.294731
\(685\) −20.3607 −0.777942
\(686\) 0 0
\(687\) −19.7082 −0.751915
\(688\) 6.47214 0.246748
\(689\) 0 0
\(690\) 3.05573 0.116330
\(691\) −17.5279 −0.666791 −0.333396 0.942787i \(-0.608194\pi\)
−0.333396 + 0.942787i \(0.608194\pi\)
\(692\) 19.4164 0.738101
\(693\) 0 0
\(694\) −26.8328 −1.01856
\(695\) −12.5836 −0.477323
\(696\) −4.47214 −0.169516
\(697\) 27.4164 1.03847
\(698\) −0.583592 −0.0220893
\(699\) 9.05573 0.342519
\(700\) 0 0
\(701\) −8.11146 −0.306365 −0.153183 0.988198i \(-0.548952\pi\)
−0.153183 + 0.988198i \(0.548952\pi\)
\(702\) 0 0
\(703\) 84.3607 3.18172
\(704\) −1.00000 −0.0376889
\(705\) −4.58359 −0.172628
\(706\) 24.3607 0.916826
\(707\) 0 0
\(708\) 1.52786 0.0574206
\(709\) 5.05573 0.189872 0.0949359 0.995483i \(-0.469735\pi\)
0.0949359 + 0.995483i \(0.469735\pi\)
\(710\) −3.05573 −0.114679
\(711\) 11.4164 0.428149
\(712\) −6.47214 −0.242554
\(713\) 6.83282 0.255891
\(714\) 0 0
\(715\) 0 0
\(716\) −5.52786 −0.206586
\(717\) 0 0
\(718\) 1.52786 0.0570194
\(719\) 20.2918 0.756756 0.378378 0.925651i \(-0.376482\pi\)
0.378378 + 0.925651i \(0.376482\pi\)
\(720\) 1.23607 0.0460655
\(721\) 0 0
\(722\) 40.4164 1.50414
\(723\) −18.1803 −0.676134
\(724\) 11.7082 0.435132
\(725\) −15.5279 −0.576690
\(726\) −1.00000 −0.0371135
\(727\) 45.2361 1.67771 0.838856 0.544353i \(-0.183225\pi\)
0.838856 + 0.544353i \(0.183225\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) −19.4164 −0.718633
\(731\) −33.8885 −1.25341
\(732\) 0 0
\(733\) −22.8328 −0.843349 −0.421675 0.906747i \(-0.638558\pi\)
−0.421675 + 0.906747i \(0.638558\pi\)
\(734\) 8.29180 0.306056
\(735\) 0 0
\(736\) −2.47214 −0.0911241
\(737\) −15.4164 −0.567871
\(738\) −5.23607 −0.192742
\(739\) 43.4164 1.59710 0.798549 0.601930i \(-0.205601\pi\)
0.798549 + 0.601930i \(0.205601\pi\)
\(740\) −13.5279 −0.497294
\(741\) 0 0
\(742\) 0 0
\(743\) −25.8885 −0.949759 −0.474879 0.880051i \(-0.657508\pi\)
−0.474879 + 0.880051i \(0.657508\pi\)
\(744\) 2.76393 0.101331
\(745\) −7.41641 −0.271716
\(746\) −1.41641 −0.0518584
\(747\) 1.23607 0.0452254
\(748\) 5.23607 0.191450
\(749\) 0 0
\(750\) 10.4721 0.382388
\(751\) −14.8328 −0.541257 −0.270629 0.962684i \(-0.587232\pi\)
−0.270629 + 0.962684i \(0.587232\pi\)
\(752\) 3.70820 0.135224
\(753\) 6.47214 0.235858
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 22.9443 0.833924 0.416962 0.908924i \(-0.363095\pi\)
0.416962 + 0.908924i \(0.363095\pi\)
\(758\) −4.00000 −0.145287
\(759\) −2.47214 −0.0897329
\(760\) −9.52786 −0.345612
\(761\) 23.1246 0.838267 0.419133 0.907925i \(-0.362334\pi\)
0.419133 + 0.907925i \(0.362334\pi\)
\(762\) 6.47214 0.234461
\(763\) 0 0
\(764\) −10.4721 −0.378869
\(765\) −6.47214 −0.234001
\(766\) −0.652476 −0.0235749
\(767\) 0 0
\(768\) −1.00000 −0.0360844
\(769\) −12.0689 −0.435215 −0.217608 0.976036i \(-0.569825\pi\)
−0.217608 + 0.976036i \(0.569825\pi\)
\(770\) 0 0
\(771\) −22.4721 −0.809314
\(772\) 5.05573 0.181960
\(773\) −49.9574 −1.79684 −0.898422 0.439133i \(-0.855286\pi\)
−0.898422 + 0.439133i \(0.855286\pi\)
\(774\) 6.47214 0.232636
\(775\) 9.59675 0.344725
\(776\) 0 0
\(777\) 0 0
\(778\) −6.94427 −0.248964
\(779\) 40.3607 1.44607
\(780\) 0 0
\(781\) 2.47214 0.0884600
\(782\) 12.9443 0.462886
\(783\) −4.47214 −0.159821
\(784\) 0 0
\(785\) −18.2492 −0.651343
\(786\) −6.76393 −0.241261
\(787\) −22.5410 −0.803501 −0.401750 0.915749i \(-0.631598\pi\)
−0.401750 + 0.915749i \(0.631598\pi\)
\(788\) −22.3607 −0.796566
\(789\) 9.52786 0.339201
\(790\) 14.1115 0.502063
\(791\) 0 0
\(792\) −1.00000 −0.0355335
\(793\) 0 0
\(794\) −1.81966 −0.0645773
\(795\) 7.41641 0.263033
\(796\) −23.1246 −0.819630
\(797\) 12.8754 0.456070 0.228035 0.973653i \(-0.426770\pi\)
0.228035 + 0.973653i \(0.426770\pi\)
\(798\) 0 0
\(799\) −19.4164 −0.686903
\(800\) −3.47214 −0.122759
\(801\) −6.47214 −0.228682
\(802\) −8.47214 −0.299162
\(803\) 15.7082 0.554330
\(804\) −15.4164 −0.543695
\(805\) 0 0
\(806\) 0 0
\(807\) 27.7082 0.975375
\(808\) −12.0000 −0.422159
\(809\) 44.8328 1.57624 0.788119 0.615523i \(-0.211055\pi\)
0.788119 + 0.615523i \(0.211055\pi\)
\(810\) 1.23607 0.0434310
\(811\) 17.5967 0.617905 0.308953 0.951077i \(-0.400022\pi\)
0.308953 + 0.951077i \(0.400022\pi\)
\(812\) 0 0
\(813\) 2.47214 0.0867016
\(814\) 10.9443 0.383597
\(815\) −14.8328 −0.519571
\(816\) 5.23607 0.183299
\(817\) −49.8885 −1.74538
\(818\) 2.76393 0.0966386
\(819\) 0 0
\(820\) −6.47214 −0.226017
\(821\) 29.7771 1.03923 0.519614 0.854401i \(-0.326076\pi\)
0.519614 + 0.854401i \(0.326076\pi\)
\(822\) 16.4721 0.574532
\(823\) −30.8328 −1.07476 −0.537382 0.843339i \(-0.680587\pi\)
−0.537382 + 0.843339i \(0.680587\pi\)
\(824\) 10.7639 0.374979
\(825\) −3.47214 −0.120884
\(826\) 0 0
\(827\) 36.7214 1.27693 0.638463 0.769652i \(-0.279570\pi\)
0.638463 + 0.769652i \(0.279570\pi\)
\(828\) −2.47214 −0.0859127
\(829\) 47.4853 1.64923 0.824616 0.565693i \(-0.191391\pi\)
0.824616 + 0.565693i \(0.191391\pi\)
\(830\) 1.52786 0.0530329
\(831\) −26.0000 −0.901930
\(832\) 0 0
\(833\) 0 0
\(834\) 10.1803 0.352516
\(835\) 22.8328 0.790162
\(836\) 7.70820 0.266594
\(837\) 2.76393 0.0955355
\(838\) 15.0557 0.520091
\(839\) 25.2361 0.871246 0.435623 0.900129i \(-0.356528\pi\)
0.435623 + 0.900129i \(0.356528\pi\)
\(840\) 0 0
\(841\) −9.00000 −0.310345
\(842\) −5.05573 −0.174232
\(843\) −19.8885 −0.684998
\(844\) 3.41641 0.117598
\(845\) −16.0689 −0.552786
\(846\) 3.70820 0.127491
\(847\) 0 0
\(848\) −6.00000 −0.206041
\(849\) −23.7082 −0.813663
\(850\) 18.1803 0.623581
\(851\) 27.0557 0.927458
\(852\) 2.47214 0.0846940
\(853\) −33.8885 −1.16032 −0.580161 0.814502i \(-0.697010\pi\)
−0.580161 + 0.814502i \(0.697010\pi\)
\(854\) 0 0
\(855\) −9.52786 −0.325846
\(856\) 0.944272 0.0322745
\(857\) 47.1246 1.60975 0.804873 0.593447i \(-0.202233\pi\)
0.804873 + 0.593447i \(0.202233\pi\)
\(858\) 0 0
\(859\) −43.4164 −1.48135 −0.740674 0.671864i \(-0.765494\pi\)
−0.740674 + 0.671864i \(0.765494\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −22.4721 −0.765404
\(863\) −44.3607 −1.51006 −0.755028 0.655693i \(-0.772377\pi\)
−0.755028 + 0.655693i \(0.772377\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 24.0000 0.816024
\(866\) 9.88854 0.336026
\(867\) −10.4164 −0.353760
\(868\) 0 0
\(869\) −11.4164 −0.387275
\(870\) −5.52786 −0.187412
\(871\) 0 0
\(872\) 6.00000 0.203186
\(873\) 0 0
\(874\) 19.0557 0.644570
\(875\) 0 0
\(876\) 15.7082 0.530731
\(877\) 30.0000 1.01303 0.506514 0.862232i \(-0.330934\pi\)
0.506514 + 0.862232i \(0.330934\pi\)
\(878\) 31.4164 1.06025
\(879\) 4.58359 0.154601
\(880\) −1.23607 −0.0416678
\(881\) 26.8328 0.904021 0.452010 0.892013i \(-0.350707\pi\)
0.452010 + 0.892013i \(0.350707\pi\)
\(882\) 0 0
\(883\) 12.0000 0.403832 0.201916 0.979403i \(-0.435283\pi\)
0.201916 + 0.979403i \(0.435283\pi\)
\(884\) 0 0
\(885\) 1.88854 0.0634827
\(886\) 7.41641 0.249159
\(887\) −36.9443 −1.24047 −0.620234 0.784417i \(-0.712962\pi\)
−0.620234 + 0.784417i \(0.712962\pi\)
\(888\) 10.9443 0.367266
\(889\) 0 0
\(890\) −8.00000 −0.268161
\(891\) −1.00000 −0.0335013
\(892\) −15.7082 −0.525950
\(893\) −28.5836 −0.956513
\(894\) 6.00000 0.200670
\(895\) −6.83282 −0.228396
\(896\) 0 0
\(897\) 0 0
\(898\) 19.5279 0.651653
\(899\) −12.3607 −0.412252
\(900\) −3.47214 −0.115738
\(901\) 31.4164 1.04663
\(902\) 5.23607 0.174342
\(903\) 0 0
\(904\) 13.4164 0.446223
\(905\) 14.4721 0.481070
\(906\) 0 0
\(907\) −33.3050 −1.10587 −0.552936 0.833223i \(-0.686493\pi\)
−0.552936 + 0.833223i \(0.686493\pi\)
\(908\) 8.65248 0.287142
\(909\) −12.0000 −0.398015
\(910\) 0 0
\(911\) −31.4164 −1.04087 −0.520436 0.853901i \(-0.674231\pi\)
−0.520436 + 0.853901i \(0.674231\pi\)
\(912\) 7.70820 0.255244
\(913\) −1.23607 −0.0409079
\(914\) 18.0000 0.595387
\(915\) 0 0
\(916\) 19.7082 0.651177
\(917\) 0 0
\(918\) 5.23607 0.172816
\(919\) −53.6656 −1.77027 −0.885133 0.465338i \(-0.845933\pi\)
−0.885133 + 0.465338i \(0.845933\pi\)
\(920\) −3.05573 −0.100744
\(921\) −25.5967 −0.843442
\(922\) −26.8328 −0.883692
\(923\) 0 0
\(924\) 0 0
\(925\) 38.0000 1.24943
\(926\) −1.88854 −0.0620614
\(927\) 10.7639 0.353534
\(928\) 4.47214 0.146805
\(929\) 8.36068 0.274305 0.137153 0.990550i \(-0.456205\pi\)
0.137153 + 0.990550i \(0.456205\pi\)
\(930\) 3.41641 0.112028
\(931\) 0 0
\(932\) −9.05573 −0.296630
\(933\) 8.65248 0.283269
\(934\) −29.8885 −0.977983
\(935\) 6.47214 0.211661
\(936\) 0 0
\(937\) −1.59675 −0.0521635 −0.0260817 0.999660i \(-0.508303\pi\)
−0.0260817 + 0.999660i \(0.508303\pi\)
\(938\) 0 0
\(939\) −5.52786 −0.180395
\(940\) 4.58359 0.149500
\(941\) −7.63932 −0.249035 −0.124517 0.992217i \(-0.539738\pi\)
−0.124517 + 0.992217i \(0.539738\pi\)
\(942\) 14.7639 0.481035
\(943\) 12.9443 0.421523
\(944\) −1.52786 −0.0497277
\(945\) 0 0
\(946\) −6.47214 −0.210427
\(947\) −37.8885 −1.23121 −0.615606 0.788054i \(-0.711089\pi\)
−0.615606 + 0.788054i \(0.711089\pi\)
\(948\) −11.4164 −0.370788
\(949\) 0 0
\(950\) 26.7639 0.868337
\(951\) 31.8885 1.03406
\(952\) 0 0
\(953\) −30.9443 −1.00238 −0.501192 0.865336i \(-0.667105\pi\)
−0.501192 + 0.865336i \(0.667105\pi\)
\(954\) −6.00000 −0.194257
\(955\) −12.9443 −0.418867
\(956\) 0 0
\(957\) 4.47214 0.144564
\(958\) 35.7771 1.15591
\(959\) 0 0
\(960\) −1.23607 −0.0398939
\(961\) −23.3607 −0.753570
\(962\) 0 0
\(963\) 0.944272 0.0304287
\(964\) 18.1803 0.585549
\(965\) 6.24922 0.201170
\(966\) 0 0
\(967\) 54.8328 1.76330 0.881652 0.471900i \(-0.156432\pi\)
0.881652 + 0.471900i \(0.156432\pi\)
\(968\) 1.00000 0.0321412
\(969\) −40.3607 −1.29657
\(970\) 0 0
\(971\) −39.0557 −1.25336 −0.626679 0.779278i \(-0.715586\pi\)
−0.626679 + 0.779278i \(0.715586\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 0 0
\(974\) 36.9443 1.18377
\(975\) 0 0
\(976\) 0 0
\(977\) −11.8885 −0.380348 −0.190174 0.981750i \(-0.560905\pi\)
−0.190174 + 0.981750i \(0.560905\pi\)
\(978\) 12.0000 0.383718
\(979\) 6.47214 0.206850
\(980\) 0 0
\(981\) 6.00000 0.191565
\(982\) −29.8885 −0.953782
\(983\) 51.7082 1.64924 0.824618 0.565690i \(-0.191390\pi\)
0.824618 + 0.565690i \(0.191390\pi\)
\(984\) 5.23607 0.166920
\(985\) −27.6393 −0.880662
\(986\) −23.4164 −0.745730
\(987\) 0 0
\(988\) 0 0
\(989\) −16.0000 −0.508770
\(990\) −1.23607 −0.0392848
\(991\) −30.8328 −0.979437 −0.489718 0.871881i \(-0.662900\pi\)
−0.489718 + 0.871881i \(0.662900\pi\)
\(992\) −2.76393 −0.0877549
\(993\) 7.41641 0.235353
\(994\) 0 0
\(995\) −28.5836 −0.906161
\(996\) −1.23607 −0.0391663
\(997\) 54.2492 1.71809 0.859045 0.511900i \(-0.171058\pi\)
0.859045 + 0.511900i \(0.171058\pi\)
\(998\) 7.41641 0.234762
\(999\) 10.9443 0.346261
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.bb.1.2 2
3.2 odd 2 9702.2.a.cw.1.1 2
7.6 odd 2 3234.2.a.be.1.1 yes 2
21.20 even 2 9702.2.a.ck.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3234.2.a.bb.1.2 2 1.1 even 1 trivial
3234.2.a.be.1.1 yes 2 7.6 odd 2
9702.2.a.ck.1.2 2 21.20 even 2
9702.2.a.cw.1.1 2 3.2 odd 2