Properties

Label 3381.2.a.bi.1.4
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
Defining polynomial: \(x^{10} - 3 x^{9} - 13 x^{8} + 41 x^{7} + 47 x^{6} - 165 x^{5} - 45 x^{4} + 207 x^{3} + 12 x^{2} - 76 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.769091\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.769091 q^{2} -1.00000 q^{3} -1.40850 q^{4} -4.02709 q^{5} +0.769091 q^{6} +2.62145 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.769091 q^{2} -1.00000 q^{3} -1.40850 q^{4} -4.02709 q^{5} +0.769091 q^{6} +2.62145 q^{8} +1.00000 q^{9} +3.09720 q^{10} +5.47113 q^{11} +1.40850 q^{12} -4.39956 q^{13} +4.02709 q^{15} +0.800870 q^{16} -2.24965 q^{17} -0.769091 q^{18} +3.56152 q^{19} +5.67216 q^{20} -4.20780 q^{22} +1.00000 q^{23} -2.62145 q^{24} +11.2175 q^{25} +3.38366 q^{26} -1.00000 q^{27} -4.70941 q^{29} -3.09720 q^{30} -10.5492 q^{31} -5.85883 q^{32} -5.47113 q^{33} +1.73019 q^{34} -1.40850 q^{36} -3.92549 q^{37} -2.73913 q^{38} +4.39956 q^{39} -10.5568 q^{40} -3.97807 q^{41} +2.61087 q^{43} -7.70609 q^{44} -4.02709 q^{45} -0.769091 q^{46} -0.518350 q^{47} -0.800870 q^{48} -8.62724 q^{50} +2.24965 q^{51} +6.19677 q^{52} -12.6363 q^{53} +0.769091 q^{54} -22.0328 q^{55} -3.56152 q^{57} +3.62197 q^{58} +0.842890 q^{59} -5.67216 q^{60} +9.00931 q^{61} +8.11333 q^{62} +2.90423 q^{64} +17.7174 q^{65} +4.20780 q^{66} -1.39968 q^{67} +3.16863 q^{68} -1.00000 q^{69} -7.10801 q^{71} +2.62145 q^{72} +5.32607 q^{73} +3.01906 q^{74} -11.2175 q^{75} -5.01640 q^{76} -3.38366 q^{78} -13.4434 q^{79} -3.22518 q^{80} +1.00000 q^{81} +3.05950 q^{82} +3.39713 q^{83} +9.05955 q^{85} -2.00800 q^{86} +4.70941 q^{87} +14.3423 q^{88} +13.0156 q^{89} +3.09720 q^{90} -1.40850 q^{92} +10.5492 q^{93} +0.398658 q^{94} -14.3426 q^{95} +5.85883 q^{96} -6.35615 q^{97} +5.47113 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + O(q^{10}) \) \( 10q + 3q^{2} - 10q^{3} + 15q^{4} - 5q^{5} - 3q^{6} + 9q^{8} + 10q^{9} + 11q^{10} + 8q^{11} - 15q^{12} + 5q^{15} + 37q^{16} - 11q^{17} + 3q^{18} + q^{19} - 15q^{20} + 6q^{22} + 10q^{23} - 9q^{24} + 21q^{25} + q^{26} - 10q^{27} + 22q^{29} - 11q^{30} - 3q^{31} + 11q^{32} - 8q^{33} - 3q^{34} + 15q^{36} - 3q^{37} + 16q^{38} + 39q^{40} - 26q^{41} + 27q^{43} + 16q^{44} - 5q^{45} + 3q^{46} + 11q^{47} - 37q^{48} + 2q^{50} + 11q^{51} + 29q^{52} + 5q^{53} - 3q^{54} - 18q^{55} - q^{57} + 16q^{58} - 10q^{59} + 15q^{60} + 22q^{61} - 32q^{62} + 69q^{64} - 11q^{65} - 6q^{66} - 2q^{67} - 21q^{68} - 10q^{69} + 27q^{71} + 9q^{72} - 8q^{73} + 14q^{74} - 21q^{75} - 22q^{76} - q^{78} + 21q^{79} - 53q^{80} + 10q^{81} + 36q^{82} - 12q^{83} + 23q^{85} + 18q^{86} - 22q^{87} - 10q^{88} + 6q^{89} + 11q^{90} + 15q^{92} + 3q^{93} + 35q^{94} + 44q^{95} - 11q^{96} + 6q^{97} + 8q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.769091 −0.543829 −0.271915 0.962321i \(-0.587657\pi\)
−0.271915 + 0.962321i \(0.587657\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.40850 −0.704250
\(5\) −4.02709 −1.80097 −0.900485 0.434887i \(-0.856788\pi\)
−0.900485 + 0.434887i \(0.856788\pi\)
\(6\) 0.769091 0.313980
\(7\) 0 0
\(8\) 2.62145 0.926821
\(9\) 1.00000 0.333333
\(10\) 3.09720 0.979420
\(11\) 5.47113 1.64961 0.824804 0.565418i \(-0.191285\pi\)
0.824804 + 0.565418i \(0.191285\pi\)
\(12\) 1.40850 0.406599
\(13\) −4.39956 −1.22022 −0.610109 0.792318i \(-0.708874\pi\)
−0.610109 + 0.792318i \(0.708874\pi\)
\(14\) 0 0
\(15\) 4.02709 1.03979
\(16\) 0.800870 0.200218
\(17\) −2.24965 −0.545621 −0.272810 0.962068i \(-0.587953\pi\)
−0.272810 + 0.962068i \(0.587953\pi\)
\(18\) −0.769091 −0.181276
\(19\) 3.56152 0.817069 0.408534 0.912743i \(-0.366040\pi\)
0.408534 + 0.912743i \(0.366040\pi\)
\(20\) 5.67216 1.26833
\(21\) 0 0
\(22\) −4.20780 −0.897105
\(23\) 1.00000 0.208514
\(24\) −2.62145 −0.535100
\(25\) 11.2175 2.24349
\(26\) 3.38366 0.663590
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −4.70941 −0.874516 −0.437258 0.899336i \(-0.644050\pi\)
−0.437258 + 0.899336i \(0.644050\pi\)
\(30\) −3.09720 −0.565468
\(31\) −10.5492 −1.89470 −0.947350 0.320198i \(-0.896250\pi\)
−0.947350 + 0.320198i \(0.896250\pi\)
\(32\) −5.85883 −1.03570
\(33\) −5.47113 −0.952402
\(34\) 1.73019 0.296724
\(35\) 0 0
\(36\) −1.40850 −0.234750
\(37\) −3.92549 −0.645347 −0.322673 0.946510i \(-0.604582\pi\)
−0.322673 + 0.946510i \(0.604582\pi\)
\(38\) −2.73913 −0.444346
\(39\) 4.39956 0.704493
\(40\) −10.5568 −1.66918
\(41\) −3.97807 −0.621270 −0.310635 0.950529i \(-0.600542\pi\)
−0.310635 + 0.950529i \(0.600542\pi\)
\(42\) 0 0
\(43\) 2.61087 0.398155 0.199077 0.979984i \(-0.436206\pi\)
0.199077 + 0.979984i \(0.436206\pi\)
\(44\) −7.70609 −1.16174
\(45\) −4.02709 −0.600323
\(46\) −0.769091 −0.113396
\(47\) −0.518350 −0.0756091 −0.0378046 0.999285i \(-0.512036\pi\)
−0.0378046 + 0.999285i \(0.512036\pi\)
\(48\) −0.800870 −0.115596
\(49\) 0 0
\(50\) −8.62724 −1.22008
\(51\) 2.24965 0.315014
\(52\) 6.19677 0.859338
\(53\) −12.6363 −1.73573 −0.867865 0.496799i \(-0.834508\pi\)
−0.867865 + 0.496799i \(0.834508\pi\)
\(54\) 0.769091 0.104660
\(55\) −22.0328 −2.97090
\(56\) 0 0
\(57\) −3.56152 −0.471735
\(58\) 3.62197 0.475587
\(59\) 0.842890 0.109735 0.0548675 0.998494i \(-0.482526\pi\)
0.0548675 + 0.998494i \(0.482526\pi\)
\(60\) −5.67216 −0.732272
\(61\) 9.00931 1.15352 0.576762 0.816912i \(-0.304316\pi\)
0.576762 + 0.816912i \(0.304316\pi\)
\(62\) 8.11333 1.03039
\(63\) 0 0
\(64\) 2.90423 0.363029
\(65\) 17.7174 2.19757
\(66\) 4.20780 0.517944
\(67\) −1.39968 −0.170999 −0.0854993 0.996338i \(-0.527249\pi\)
−0.0854993 + 0.996338i \(0.527249\pi\)
\(68\) 3.16863 0.384253
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −7.10801 −0.843565 −0.421783 0.906697i \(-0.638596\pi\)
−0.421783 + 0.906697i \(0.638596\pi\)
\(72\) 2.62145 0.308940
\(73\) 5.32607 0.623370 0.311685 0.950186i \(-0.399107\pi\)
0.311685 + 0.950186i \(0.399107\pi\)
\(74\) 3.01906 0.350958
\(75\) −11.2175 −1.29528
\(76\) −5.01640 −0.575420
\(77\) 0 0
\(78\) −3.38366 −0.383124
\(79\) −13.4434 −1.51250 −0.756251 0.654281i \(-0.772971\pi\)
−0.756251 + 0.654281i \(0.772971\pi\)
\(80\) −3.22518 −0.360586
\(81\) 1.00000 0.111111
\(82\) 3.05950 0.337865
\(83\) 3.39713 0.372883 0.186442 0.982466i \(-0.440304\pi\)
0.186442 + 0.982466i \(0.440304\pi\)
\(84\) 0 0
\(85\) 9.05955 0.982646
\(86\) −2.00800 −0.216528
\(87\) 4.70941 0.504902
\(88\) 14.3423 1.52889
\(89\) 13.0156 1.37965 0.689826 0.723975i \(-0.257687\pi\)
0.689826 + 0.723975i \(0.257687\pi\)
\(90\) 3.09720 0.326473
\(91\) 0 0
\(92\) −1.40850 −0.146846
\(93\) 10.5492 1.09391
\(94\) 0.398658 0.0411185
\(95\) −14.3426 −1.47152
\(96\) 5.85883 0.597965
\(97\) −6.35615 −0.645369 −0.322684 0.946507i \(-0.604585\pi\)
−0.322684 + 0.946507i \(0.604585\pi\)
\(98\) 0 0
\(99\) 5.47113 0.549870
\(100\) −15.7998 −1.57998
\(101\) 2.83587 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(102\) −1.73019 −0.171314
\(103\) 10.5894 1.04340 0.521701 0.853128i \(-0.325298\pi\)
0.521701 + 0.853128i \(0.325298\pi\)
\(104\) −11.5332 −1.13092
\(105\) 0 0
\(106\) 9.71847 0.943941
\(107\) −18.8540 −1.82269 −0.911344 0.411646i \(-0.864954\pi\)
−0.911344 + 0.411646i \(0.864954\pi\)
\(108\) 1.40850 0.135533
\(109\) 11.1344 1.06649 0.533243 0.845962i \(-0.320973\pi\)
0.533243 + 0.845962i \(0.320973\pi\)
\(110\) 16.9452 1.61566
\(111\) 3.92549 0.372591
\(112\) 0 0
\(113\) −1.29979 −0.122274 −0.0611370 0.998129i \(-0.519473\pi\)
−0.0611370 + 0.998129i \(0.519473\pi\)
\(114\) 2.73913 0.256543
\(115\) −4.02709 −0.375528
\(116\) 6.63321 0.615878
\(117\) −4.39956 −0.406739
\(118\) −0.648259 −0.0596771
\(119\) 0 0
\(120\) 10.5568 0.963699
\(121\) 18.9333 1.72121
\(122\) −6.92898 −0.627320
\(123\) 3.97807 0.358691
\(124\) 14.8586 1.33434
\(125\) −25.0383 −2.23949
\(126\) 0 0
\(127\) −0.0947076 −0.00840394 −0.00420197 0.999991i \(-0.501338\pi\)
−0.00420197 + 0.999991i \(0.501338\pi\)
\(128\) 9.48405 0.838279
\(129\) −2.61087 −0.229875
\(130\) −13.6263 −1.19511
\(131\) −13.7260 −1.19925 −0.599625 0.800281i \(-0.704684\pi\)
−0.599625 + 0.800281i \(0.704684\pi\)
\(132\) 7.70609 0.670729
\(133\) 0 0
\(134\) 1.07648 0.0929941
\(135\) 4.02709 0.346597
\(136\) −5.89734 −0.505693
\(137\) −1.03182 −0.0881541 −0.0440771 0.999028i \(-0.514035\pi\)
−0.0440771 + 0.999028i \(0.514035\pi\)
\(138\) 0.769091 0.0654693
\(139\) 0.281873 0.0239081 0.0119541 0.999929i \(-0.496195\pi\)
0.0119541 + 0.999929i \(0.496195\pi\)
\(140\) 0 0
\(141\) 0.518350 0.0436530
\(142\) 5.46670 0.458756
\(143\) −24.0706 −2.01288
\(144\) 0.800870 0.0667392
\(145\) 18.9652 1.57498
\(146\) −4.09623 −0.339007
\(147\) 0 0
\(148\) 5.52905 0.454485
\(149\) −5.70250 −0.467167 −0.233584 0.972337i \(-0.575045\pi\)
−0.233584 + 0.972337i \(0.575045\pi\)
\(150\) 8.62724 0.704412
\(151\) 2.87628 0.234069 0.117034 0.993128i \(-0.462661\pi\)
0.117034 + 0.993128i \(0.462661\pi\)
\(152\) 9.33633 0.757276
\(153\) −2.24965 −0.181874
\(154\) 0 0
\(155\) 42.4828 3.41230
\(156\) −6.19677 −0.496139
\(157\) 16.7408 1.33606 0.668032 0.744133i \(-0.267137\pi\)
0.668032 + 0.744133i \(0.267137\pi\)
\(158\) 10.3392 0.822543
\(159\) 12.6363 1.00212
\(160\) 23.5940 1.86527
\(161\) 0 0
\(162\) −0.769091 −0.0604255
\(163\) −0.687374 −0.0538393 −0.0269197 0.999638i \(-0.508570\pi\)
−0.0269197 + 0.999638i \(0.508570\pi\)
\(164\) 5.60311 0.437529
\(165\) 22.0328 1.71525
\(166\) −2.61270 −0.202785
\(167\) −12.4022 −0.959714 −0.479857 0.877347i \(-0.659312\pi\)
−0.479857 + 0.877347i \(0.659312\pi\)
\(168\) 0 0
\(169\) 6.35609 0.488930
\(170\) −6.96762 −0.534392
\(171\) 3.56152 0.272356
\(172\) −3.67742 −0.280400
\(173\) −3.20701 −0.243824 −0.121912 0.992541i \(-0.538903\pi\)
−0.121912 + 0.992541i \(0.538903\pi\)
\(174\) −3.62197 −0.274581
\(175\) 0 0
\(176\) 4.38167 0.330281
\(177\) −0.842890 −0.0633555
\(178\) −10.0102 −0.750295
\(179\) 11.1472 0.833178 0.416589 0.909095i \(-0.363225\pi\)
0.416589 + 0.909095i \(0.363225\pi\)
\(180\) 5.67216 0.422778
\(181\) 18.5663 1.38002 0.690011 0.723799i \(-0.257606\pi\)
0.690011 + 0.723799i \(0.257606\pi\)
\(182\) 0 0
\(183\) −9.00931 −0.665988
\(184\) 2.62145 0.193255
\(185\) 15.8083 1.16225
\(186\) −8.11333 −0.594898
\(187\) −12.3081 −0.900060
\(188\) 0.730096 0.0532477
\(189\) 0 0
\(190\) 11.0307 0.800253
\(191\) −22.9387 −1.65978 −0.829892 0.557924i \(-0.811598\pi\)
−0.829892 + 0.557924i \(0.811598\pi\)
\(192\) −2.90423 −0.209595
\(193\) −10.1295 −0.729139 −0.364569 0.931176i \(-0.618784\pi\)
−0.364569 + 0.931176i \(0.618784\pi\)
\(194\) 4.88845 0.350970
\(195\) −17.7174 −1.26877
\(196\) 0 0
\(197\) −4.51568 −0.321729 −0.160864 0.986977i \(-0.551428\pi\)
−0.160864 + 0.986977i \(0.551428\pi\)
\(198\) −4.20780 −0.299035
\(199\) 20.8076 1.47501 0.737506 0.675340i \(-0.236003\pi\)
0.737506 + 0.675340i \(0.236003\pi\)
\(200\) 29.4060 2.07932
\(201\) 1.39968 0.0987261
\(202\) −2.18104 −0.153458
\(203\) 0 0
\(204\) −3.16863 −0.221849
\(205\) 16.0201 1.11889
\(206\) −8.14419 −0.567433
\(207\) 1.00000 0.0695048
\(208\) −3.52347 −0.244309
\(209\) 19.4856 1.34784
\(210\) 0 0
\(211\) 26.2315 1.80585 0.902925 0.429799i \(-0.141416\pi\)
0.902925 + 0.429799i \(0.141416\pi\)
\(212\) 17.7982 1.22239
\(213\) 7.10801 0.487033
\(214\) 14.5005 0.991231
\(215\) −10.5142 −0.717064
\(216\) −2.62145 −0.178367
\(217\) 0 0
\(218\) −8.56339 −0.579986
\(219\) −5.32607 −0.359903
\(220\) 31.0331 2.09225
\(221\) 9.89747 0.665776
\(222\) −3.01906 −0.202626
\(223\) −1.20377 −0.0806102 −0.0403051 0.999187i \(-0.512833\pi\)
−0.0403051 + 0.999187i \(0.512833\pi\)
\(224\) 0 0
\(225\) 11.2175 0.747831
\(226\) 0.999657 0.0664962
\(227\) 14.2531 0.946015 0.473007 0.881059i \(-0.343168\pi\)
0.473007 + 0.881059i \(0.343168\pi\)
\(228\) 5.01640 0.332219
\(229\) 12.3571 0.816578 0.408289 0.912853i \(-0.366125\pi\)
0.408289 + 0.912853i \(0.366125\pi\)
\(230\) 3.09720 0.204223
\(231\) 0 0
\(232\) −12.3455 −0.810520
\(233\) 11.9828 0.785017 0.392508 0.919748i \(-0.371607\pi\)
0.392508 + 0.919748i \(0.371607\pi\)
\(234\) 3.38366 0.221197
\(235\) 2.08744 0.136170
\(236\) −1.18721 −0.0772808
\(237\) 13.4434 0.873244
\(238\) 0 0
\(239\) 6.53934 0.422995 0.211497 0.977379i \(-0.432166\pi\)
0.211497 + 0.977379i \(0.432166\pi\)
\(240\) 3.22518 0.208184
\(241\) 0.500065 0.0322120 0.0161060 0.999870i \(-0.494873\pi\)
0.0161060 + 0.999870i \(0.494873\pi\)
\(242\) −14.5614 −0.936044
\(243\) −1.00000 −0.0641500
\(244\) −12.6896 −0.812369
\(245\) 0 0
\(246\) −3.05950 −0.195066
\(247\) −15.6691 −0.997001
\(248\) −27.6543 −1.75605
\(249\) −3.39713 −0.215284
\(250\) 19.2567 1.21790
\(251\) 5.10076 0.321957 0.160978 0.986958i \(-0.448535\pi\)
0.160978 + 0.986958i \(0.448535\pi\)
\(252\) 0 0
\(253\) 5.47113 0.343967
\(254\) 0.0728387 0.00457031
\(255\) −9.05955 −0.567331
\(256\) −13.1026 −0.818910
\(257\) −8.85942 −0.552635 −0.276318 0.961066i \(-0.589114\pi\)
−0.276318 + 0.961066i \(0.589114\pi\)
\(258\) 2.00800 0.125013
\(259\) 0 0
\(260\) −24.9550 −1.54764
\(261\) −4.70941 −0.291505
\(262\) 10.5566 0.652187
\(263\) 26.1608 1.61315 0.806573 0.591135i \(-0.201320\pi\)
0.806573 + 0.591135i \(0.201320\pi\)
\(264\) −14.3423 −0.882706
\(265\) 50.8876 3.12600
\(266\) 0 0
\(267\) −13.0156 −0.796542
\(268\) 1.97146 0.120426
\(269\) −15.7569 −0.960714 −0.480357 0.877073i \(-0.659493\pi\)
−0.480357 + 0.877073i \(0.659493\pi\)
\(270\) −3.09720 −0.188489
\(271\) −5.95116 −0.361507 −0.180754 0.983528i \(-0.557854\pi\)
−0.180754 + 0.983528i \(0.557854\pi\)
\(272\) −1.80168 −0.109243
\(273\) 0 0
\(274\) 0.793561 0.0479408
\(275\) 61.3722 3.70088
\(276\) 1.40850 0.0847817
\(277\) 15.2365 0.915475 0.457738 0.889087i \(-0.348660\pi\)
0.457738 + 0.889087i \(0.348660\pi\)
\(278\) −0.216786 −0.0130019
\(279\) −10.5492 −0.631567
\(280\) 0 0
\(281\) 1.78254 0.106338 0.0531688 0.998586i \(-0.483068\pi\)
0.0531688 + 0.998586i \(0.483068\pi\)
\(282\) −0.398658 −0.0237398
\(283\) −19.9991 −1.18882 −0.594412 0.804161i \(-0.702615\pi\)
−0.594412 + 0.804161i \(0.702615\pi\)
\(284\) 10.0116 0.594081
\(285\) 14.3426 0.849580
\(286\) 18.5124 1.09466
\(287\) 0 0
\(288\) −5.85883 −0.345235
\(289\) −11.9391 −0.702298
\(290\) −14.5860 −0.856519
\(291\) 6.35615 0.372604
\(292\) −7.50177 −0.439008
\(293\) −17.4319 −1.01838 −0.509191 0.860653i \(-0.670055\pi\)
−0.509191 + 0.860653i \(0.670055\pi\)
\(294\) 0 0
\(295\) −3.39439 −0.197629
\(296\) −10.2905 −0.598121
\(297\) −5.47113 −0.317467
\(298\) 4.38574 0.254059
\(299\) −4.39956 −0.254433
\(300\) 15.7998 0.912201
\(301\) 0 0
\(302\) −2.21212 −0.127293
\(303\) −2.83587 −0.162916
\(304\) 2.85232 0.163592
\(305\) −36.2813 −2.07746
\(306\) 1.73019 0.0989081
\(307\) −22.1407 −1.26363 −0.631817 0.775117i \(-0.717691\pi\)
−0.631817 + 0.775117i \(0.717691\pi\)
\(308\) 0 0
\(309\) −10.5894 −0.602409
\(310\) −32.6731 −1.85571
\(311\) −6.89035 −0.390716 −0.195358 0.980732i \(-0.562587\pi\)
−0.195358 + 0.980732i \(0.562587\pi\)
\(312\) 11.5332 0.652939
\(313\) 13.5739 0.767241 0.383620 0.923491i \(-0.374677\pi\)
0.383620 + 0.923491i \(0.374677\pi\)
\(314\) −12.8752 −0.726591
\(315\) 0 0
\(316\) 18.9350 1.06518
\(317\) −24.6587 −1.38497 −0.692486 0.721431i \(-0.743485\pi\)
−0.692486 + 0.721431i \(0.743485\pi\)
\(318\) −9.71847 −0.544985
\(319\) −25.7658 −1.44261
\(320\) −11.6956 −0.653804
\(321\) 18.8540 1.05233
\(322\) 0 0
\(323\) −8.01218 −0.445810
\(324\) −1.40850 −0.0782500
\(325\) −49.3518 −2.73755
\(326\) 0.528653 0.0292794
\(327\) −11.1344 −0.615736
\(328\) −10.4283 −0.575806
\(329\) 0 0
\(330\) −16.9452 −0.932802
\(331\) 22.9676 1.26241 0.631207 0.775615i \(-0.282560\pi\)
0.631207 + 0.775615i \(0.282560\pi\)
\(332\) −4.78486 −0.262603
\(333\) −3.92549 −0.215116
\(334\) 9.53845 0.521921
\(335\) 5.63666 0.307963
\(336\) 0 0
\(337\) −19.9238 −1.08532 −0.542659 0.839953i \(-0.682582\pi\)
−0.542659 + 0.839953i \(0.682582\pi\)
\(338\) −4.88841 −0.265894
\(339\) 1.29979 0.0705950
\(340\) −12.7604 −0.692028
\(341\) −57.7163 −3.12552
\(342\) −2.73913 −0.148115
\(343\) 0 0
\(344\) 6.84426 0.369018
\(345\) 4.02709 0.216811
\(346\) 2.46648 0.132599
\(347\) 12.6315 0.678093 0.339046 0.940770i \(-0.389896\pi\)
0.339046 + 0.940770i \(0.389896\pi\)
\(348\) −6.63321 −0.355577
\(349\) 14.9697 0.801308 0.400654 0.916229i \(-0.368783\pi\)
0.400654 + 0.916229i \(0.368783\pi\)
\(350\) 0 0
\(351\) 4.39956 0.234831
\(352\) −32.0545 −1.70851
\(353\) 2.73206 0.145413 0.0727064 0.997353i \(-0.476836\pi\)
0.0727064 + 0.997353i \(0.476836\pi\)
\(354\) 0.648259 0.0344546
\(355\) 28.6246 1.51924
\(356\) −18.3325 −0.971619
\(357\) 0 0
\(358\) −8.57318 −0.453107
\(359\) 16.6777 0.880217 0.440108 0.897945i \(-0.354940\pi\)
0.440108 + 0.897945i \(0.354940\pi\)
\(360\) −10.5568 −0.556392
\(361\) −6.31558 −0.332399
\(362\) −14.2792 −0.750497
\(363\) −18.9333 −0.993740
\(364\) 0 0
\(365\) −21.4486 −1.12267
\(366\) 6.92898 0.362184
\(367\) −28.0775 −1.46563 −0.732817 0.680426i \(-0.761795\pi\)
−0.732817 + 0.680426i \(0.761795\pi\)
\(368\) 0.800870 0.0417483
\(369\) −3.97807 −0.207090
\(370\) −12.1580 −0.632066
\(371\) 0 0
\(372\) −14.8586 −0.770383
\(373\) 22.3398 1.15671 0.578356 0.815785i \(-0.303695\pi\)
0.578356 + 0.815785i \(0.303695\pi\)
\(374\) 9.46608 0.489479
\(375\) 25.0383 1.29297
\(376\) −1.35883 −0.0700761
\(377\) 20.7193 1.06710
\(378\) 0 0
\(379\) 27.7949 1.42773 0.713864 0.700284i \(-0.246943\pi\)
0.713864 + 0.700284i \(0.246943\pi\)
\(380\) 20.2015 1.03631
\(381\) 0.0947076 0.00485202
\(382\) 17.6419 0.902639
\(383\) 6.59155 0.336813 0.168406 0.985718i \(-0.446138\pi\)
0.168406 + 0.985718i \(0.446138\pi\)
\(384\) −9.48405 −0.483981
\(385\) 0 0
\(386\) 7.79052 0.396527
\(387\) 2.61087 0.132718
\(388\) 8.95263 0.454501
\(389\) 8.98913 0.455767 0.227883 0.973688i \(-0.426819\pi\)
0.227883 + 0.973688i \(0.426819\pi\)
\(390\) 13.6263 0.689994
\(391\) −2.24965 −0.113770
\(392\) 0 0
\(393\) 13.7260 0.692387
\(394\) 3.47296 0.174965
\(395\) 54.1379 2.72397
\(396\) −7.70609 −0.387246
\(397\) −2.55171 −0.128067 −0.0640334 0.997948i \(-0.520396\pi\)
−0.0640334 + 0.997948i \(0.520396\pi\)
\(398\) −16.0029 −0.802155
\(399\) 0 0
\(400\) 8.98373 0.449187
\(401\) 37.3623 1.86578 0.932892 0.360157i \(-0.117277\pi\)
0.932892 + 0.360157i \(0.117277\pi\)
\(402\) −1.07648 −0.0536902
\(403\) 46.4120 2.31195
\(404\) −3.99432 −0.198725
\(405\) −4.02709 −0.200108
\(406\) 0 0
\(407\) −21.4769 −1.06457
\(408\) 5.89734 0.291962
\(409\) −10.7730 −0.532693 −0.266346 0.963877i \(-0.585817\pi\)
−0.266346 + 0.963877i \(0.585817\pi\)
\(410\) −12.3209 −0.608485
\(411\) 1.03182 0.0508958
\(412\) −14.9151 −0.734816
\(413\) 0 0
\(414\) −0.769091 −0.0377987
\(415\) −13.6806 −0.671552
\(416\) 25.7763 1.26379
\(417\) −0.281873 −0.0138034
\(418\) −14.9862 −0.732997
\(419\) 19.8186 0.968200 0.484100 0.875013i \(-0.339147\pi\)
0.484100 + 0.875013i \(0.339147\pi\)
\(420\) 0 0
\(421\) 3.68979 0.179830 0.0899148 0.995949i \(-0.471341\pi\)
0.0899148 + 0.995949i \(0.471341\pi\)
\(422\) −20.1744 −0.982074
\(423\) −0.518350 −0.0252030
\(424\) −33.1254 −1.60871
\(425\) −25.2354 −1.22410
\(426\) −5.46670 −0.264863
\(427\) 0 0
\(428\) 26.5559 1.28363
\(429\) 24.0706 1.16214
\(430\) 8.08639 0.389961
\(431\) 9.19005 0.442669 0.221335 0.975198i \(-0.428959\pi\)
0.221335 + 0.975198i \(0.428959\pi\)
\(432\) −0.800870 −0.0385319
\(433\) 33.6161 1.61549 0.807743 0.589535i \(-0.200689\pi\)
0.807743 + 0.589535i \(0.200689\pi\)
\(434\) 0 0
\(435\) −18.9652 −0.909313
\(436\) −15.6829 −0.751073
\(437\) 3.56152 0.170371
\(438\) 4.09623 0.195726
\(439\) −2.30712 −0.110113 −0.0550565 0.998483i \(-0.517534\pi\)
−0.0550565 + 0.998483i \(0.517534\pi\)
\(440\) −57.7576 −2.75349
\(441\) 0 0
\(442\) −7.61205 −0.362068
\(443\) −11.3956 −0.541422 −0.270711 0.962661i \(-0.587259\pi\)
−0.270711 + 0.962661i \(0.587259\pi\)
\(444\) −5.52905 −0.262397
\(445\) −52.4150 −2.48471
\(446\) 0.925805 0.0438382
\(447\) 5.70250 0.269719
\(448\) 0 0
\(449\) 3.16307 0.149275 0.0746373 0.997211i \(-0.476220\pi\)
0.0746373 + 0.997211i \(0.476220\pi\)
\(450\) −8.62724 −0.406692
\(451\) −21.7646 −1.02485
\(452\) 1.83076 0.0861115
\(453\) −2.87628 −0.135140
\(454\) −10.9620 −0.514470
\(455\) 0 0
\(456\) −9.33633 −0.437214
\(457\) 10.9136 0.510518 0.255259 0.966873i \(-0.417839\pi\)
0.255259 + 0.966873i \(0.417839\pi\)
\(458\) −9.50371 −0.444079
\(459\) 2.24965 0.105005
\(460\) 5.67216 0.264466
\(461\) 34.5807 1.61058 0.805291 0.592880i \(-0.202009\pi\)
0.805291 + 0.592880i \(0.202009\pi\)
\(462\) 0 0
\(463\) −1.07364 −0.0498961 −0.0249480 0.999689i \(-0.507942\pi\)
−0.0249480 + 0.999689i \(0.507942\pi\)
\(464\) −3.77163 −0.175094
\(465\) −42.4828 −1.97009
\(466\) −9.21583 −0.426915
\(467\) −13.4884 −0.624167 −0.312083 0.950055i \(-0.601027\pi\)
−0.312083 + 0.950055i \(0.601027\pi\)
\(468\) 6.19677 0.286446
\(469\) 0 0
\(470\) −1.60543 −0.0740531
\(471\) −16.7408 −0.771377
\(472\) 2.20959 0.101705
\(473\) 14.2844 0.656799
\(474\) −10.3392 −0.474895
\(475\) 39.9512 1.83309
\(476\) 0 0
\(477\) −12.6363 −0.578577
\(478\) −5.02935 −0.230037
\(479\) −6.30939 −0.288283 −0.144142 0.989557i \(-0.546042\pi\)
−0.144142 + 0.989557i \(0.546042\pi\)
\(480\) −23.5940 −1.07692
\(481\) 17.2704 0.787463
\(482\) −0.384595 −0.0175178
\(483\) 0 0
\(484\) −26.6675 −1.21216
\(485\) 25.5968 1.16229
\(486\) 0.769091 0.0348867
\(487\) −18.5968 −0.842703 −0.421352 0.906897i \(-0.638444\pi\)
−0.421352 + 0.906897i \(0.638444\pi\)
\(488\) 23.6174 1.06911
\(489\) 0.687374 0.0310841
\(490\) 0 0
\(491\) −27.5620 −1.24386 −0.621928 0.783074i \(-0.713651\pi\)
−0.621928 + 0.783074i \(0.713651\pi\)
\(492\) −5.60311 −0.252608
\(493\) 10.5945 0.477154
\(494\) 12.0510 0.542198
\(495\) −22.0328 −0.990299
\(496\) −8.44858 −0.379352
\(497\) 0 0
\(498\) 2.61270 0.117078
\(499\) −9.33757 −0.418007 −0.209004 0.977915i \(-0.567022\pi\)
−0.209004 + 0.977915i \(0.567022\pi\)
\(500\) 35.2664 1.57716
\(501\) 12.4022 0.554091
\(502\) −3.92294 −0.175090
\(503\) 32.9700 1.47006 0.735029 0.678035i \(-0.237168\pi\)
0.735029 + 0.678035i \(0.237168\pi\)
\(504\) 0 0
\(505\) −11.4203 −0.508197
\(506\) −4.20780 −0.187059
\(507\) −6.35609 −0.282284
\(508\) 0.133396 0.00591847
\(509\) 22.0147 0.975783 0.487892 0.872904i \(-0.337766\pi\)
0.487892 + 0.872904i \(0.337766\pi\)
\(510\) 6.96762 0.308531
\(511\) 0 0
\(512\) −8.89104 −0.392932
\(513\) −3.56152 −0.157245
\(514\) 6.81370 0.300539
\(515\) −42.6444 −1.87914
\(516\) 3.67742 0.161889
\(517\) −2.83596 −0.124725
\(518\) 0 0
\(519\) 3.20701 0.140772
\(520\) 46.4452 2.03676
\(521\) −18.1901 −0.796922 −0.398461 0.917185i \(-0.630456\pi\)
−0.398461 + 0.917185i \(0.630456\pi\)
\(522\) 3.62197 0.158529
\(523\) 39.1642 1.71253 0.856265 0.516537i \(-0.172779\pi\)
0.856265 + 0.516537i \(0.172779\pi\)
\(524\) 19.3331 0.844572
\(525\) 0 0
\(526\) −20.1200 −0.877276
\(527\) 23.7321 1.03379
\(528\) −4.38167 −0.190688
\(529\) 1.00000 0.0434783
\(530\) −39.1372 −1.70001
\(531\) 0.842890 0.0365783
\(532\) 0 0
\(533\) 17.5017 0.758085
\(534\) 10.0102 0.433183
\(535\) 75.9269 3.28261
\(536\) −3.66920 −0.158485
\(537\) −11.1472 −0.481036
\(538\) 12.1185 0.522464
\(539\) 0 0
\(540\) −5.67216 −0.244091
\(541\) 12.4131 0.533681 0.266841 0.963741i \(-0.414020\pi\)
0.266841 + 0.963741i \(0.414020\pi\)
\(542\) 4.57698 0.196598
\(543\) −18.5663 −0.796757
\(544\) 13.1803 0.565102
\(545\) −44.8394 −1.92071
\(546\) 0 0
\(547\) 25.5889 1.09410 0.547052 0.837099i \(-0.315750\pi\)
0.547052 + 0.837099i \(0.315750\pi\)
\(548\) 1.45331 0.0620825
\(549\) 9.00931 0.384508
\(550\) −47.2008 −2.01265
\(551\) −16.7727 −0.714540
\(552\) −2.62145 −0.111576
\(553\) 0 0
\(554\) −11.7183 −0.497862
\(555\) −15.8083 −0.671026
\(556\) −0.397018 −0.0168373
\(557\) 27.5846 1.16880 0.584398 0.811468i \(-0.301331\pi\)
0.584398 + 0.811468i \(0.301331\pi\)
\(558\) 8.11333 0.343465
\(559\) −11.4867 −0.485835
\(560\) 0 0
\(561\) 12.3081 0.519650
\(562\) −1.37094 −0.0578295
\(563\) 25.3338 1.06769 0.533845 0.845582i \(-0.320746\pi\)
0.533845 + 0.845582i \(0.320746\pi\)
\(564\) −0.730096 −0.0307426
\(565\) 5.23438 0.220212
\(566\) 15.3811 0.646517
\(567\) 0 0
\(568\) −18.6333 −0.781834
\(569\) −22.2054 −0.930900 −0.465450 0.885074i \(-0.654108\pi\)
−0.465450 + 0.885074i \(0.654108\pi\)
\(570\) −11.0307 −0.462027
\(571\) 8.66454 0.362600 0.181300 0.983428i \(-0.441970\pi\)
0.181300 + 0.983428i \(0.441970\pi\)
\(572\) 33.9034 1.41757
\(573\) 22.9387 0.958277
\(574\) 0 0
\(575\) 11.2175 0.467800
\(576\) 2.90423 0.121010
\(577\) 40.4286 1.68306 0.841532 0.540208i \(-0.181654\pi\)
0.841532 + 0.540208i \(0.181654\pi\)
\(578\) 9.18223 0.381930
\(579\) 10.1295 0.420969
\(580\) −26.7125 −1.10918
\(581\) 0 0
\(582\) −4.88845 −0.202633
\(583\) −69.1349 −2.86328
\(584\) 13.9620 0.577752
\(585\) 17.7174 0.732525
\(586\) 13.4067 0.553826
\(587\) −46.9106 −1.93621 −0.968104 0.250547i \(-0.919389\pi\)
−0.968104 + 0.250547i \(0.919389\pi\)
\(588\) 0 0
\(589\) −37.5714 −1.54810
\(590\) 2.61060 0.107477
\(591\) 4.51568 0.185750
\(592\) −3.14381 −0.129210
\(593\) 5.68443 0.233432 0.116716 0.993165i \(-0.462763\pi\)
0.116716 + 0.993165i \(0.462763\pi\)
\(594\) 4.20780 0.172648
\(595\) 0 0
\(596\) 8.03197 0.329002
\(597\) −20.8076 −0.851599
\(598\) 3.38366 0.138368
\(599\) 4.80057 0.196146 0.0980730 0.995179i \(-0.468732\pi\)
0.0980730 + 0.995179i \(0.468732\pi\)
\(600\) −29.4060 −1.20049
\(601\) 8.92677 0.364131 0.182065 0.983286i \(-0.441722\pi\)
0.182065 + 0.983286i \(0.441722\pi\)
\(602\) 0 0
\(603\) −1.39968 −0.0569996
\(604\) −4.05124 −0.164843
\(605\) −76.2461 −3.09985
\(606\) 2.18104 0.0885987
\(607\) 3.32483 0.134951 0.0674753 0.997721i \(-0.478506\pi\)
0.0674753 + 0.997721i \(0.478506\pi\)
\(608\) −20.8663 −0.846242
\(609\) 0 0
\(610\) 27.9036 1.12978
\(611\) 2.28051 0.0922596
\(612\) 3.16863 0.128084
\(613\) 25.6252 1.03499 0.517497 0.855685i \(-0.326864\pi\)
0.517497 + 0.855685i \(0.326864\pi\)
\(614\) 17.0282 0.687201
\(615\) −16.0201 −0.645991
\(616\) 0 0
\(617\) 29.3425 1.18128 0.590642 0.806934i \(-0.298875\pi\)
0.590642 + 0.806934i \(0.298875\pi\)
\(618\) 8.14419 0.327607
\(619\) 5.99740 0.241056 0.120528 0.992710i \(-0.461541\pi\)
0.120528 + 0.992710i \(0.461541\pi\)
\(620\) −59.8370 −2.40311
\(621\) −1.00000 −0.0401286
\(622\) 5.29930 0.212483
\(623\) 0 0
\(624\) 3.52347 0.141052
\(625\) 44.7441 1.78977
\(626\) −10.4395 −0.417248
\(627\) −19.4856 −0.778178
\(628\) −23.5795 −0.940923
\(629\) 8.83099 0.352115
\(630\) 0 0
\(631\) 16.3993 0.652846 0.326423 0.945224i \(-0.394157\pi\)
0.326423 + 0.945224i \(0.394157\pi\)
\(632\) −35.2412 −1.40182
\(633\) −26.2315 −1.04261
\(634\) 18.9648 0.753189
\(635\) 0.381396 0.0151352
\(636\) −17.7982 −0.705746
\(637\) 0 0
\(638\) 19.8163 0.784533
\(639\) −7.10801 −0.281188
\(640\) −38.1931 −1.50972
\(641\) −48.4652 −1.91426 −0.957130 0.289658i \(-0.906458\pi\)
−0.957130 + 0.289658i \(0.906458\pi\)
\(642\) −14.5005 −0.572287
\(643\) −19.8571 −0.783088 −0.391544 0.920159i \(-0.628059\pi\)
−0.391544 + 0.920159i \(0.628059\pi\)
\(644\) 0 0
\(645\) 10.5142 0.413997
\(646\) 6.16209 0.242444
\(647\) −29.4130 −1.15634 −0.578172 0.815915i \(-0.696234\pi\)
−0.578172 + 0.815915i \(0.696234\pi\)
\(648\) 2.62145 0.102980
\(649\) 4.61156 0.181020
\(650\) 37.9560 1.48876
\(651\) 0 0
\(652\) 0.968166 0.0379163
\(653\) 23.1788 0.907057 0.453528 0.891242i \(-0.350165\pi\)
0.453528 + 0.891242i \(0.350165\pi\)
\(654\) 8.56339 0.334855
\(655\) 55.2760 2.15981
\(656\) −3.18592 −0.124389
\(657\) 5.32607 0.207790
\(658\) 0 0
\(659\) 33.1492 1.29131 0.645654 0.763630i \(-0.276585\pi\)
0.645654 + 0.763630i \(0.276585\pi\)
\(660\) −31.0331 −1.20796
\(661\) 15.0565 0.585628 0.292814 0.956169i \(-0.405408\pi\)
0.292814 + 0.956169i \(0.405408\pi\)
\(662\) −17.6642 −0.686537
\(663\) −9.89747 −0.384386
\(664\) 8.90539 0.345596
\(665\) 0 0
\(666\) 3.01906 0.116986
\(667\) −4.70941 −0.182349
\(668\) 17.4686 0.675879
\(669\) 1.20377 0.0465403
\(670\) −4.33510 −0.167480
\(671\) 49.2912 1.90286
\(672\) 0 0
\(673\) −23.8903 −0.920904 −0.460452 0.887684i \(-0.652313\pi\)
−0.460452 + 0.887684i \(0.652313\pi\)
\(674\) 15.3232 0.590228
\(675\) −11.2175 −0.431760
\(676\) −8.95255 −0.344329
\(677\) −8.58560 −0.329971 −0.164986 0.986296i \(-0.552758\pi\)
−0.164986 + 0.986296i \(0.552758\pi\)
\(678\) −0.999657 −0.0383916
\(679\) 0 0
\(680\) 23.7491 0.910737
\(681\) −14.2531 −0.546182
\(682\) 44.3891 1.69975
\(683\) −9.38317 −0.359037 −0.179518 0.983755i \(-0.557454\pi\)
−0.179518 + 0.983755i \(0.557454\pi\)
\(684\) −5.01640 −0.191807
\(685\) 4.15522 0.158763
\(686\) 0 0
\(687\) −12.3571 −0.471452
\(688\) 2.09097 0.0797176
\(689\) 55.5942 2.11797
\(690\) −3.09720 −0.117908
\(691\) −21.2793 −0.809503 −0.404751 0.914427i \(-0.632642\pi\)
−0.404751 + 0.914427i \(0.632642\pi\)
\(692\) 4.51707 0.171713
\(693\) 0 0
\(694\) −9.71474 −0.368767
\(695\) −1.13513 −0.0430578
\(696\) 12.3455 0.467954
\(697\) 8.94927 0.338978
\(698\) −11.5130 −0.435775
\(699\) −11.9828 −0.453230
\(700\) 0 0
\(701\) −4.27651 −0.161521 −0.0807607 0.996734i \(-0.525735\pi\)
−0.0807607 + 0.996734i \(0.525735\pi\)
\(702\) −3.38366 −0.127708
\(703\) −13.9807 −0.527293
\(704\) 15.8894 0.598856
\(705\) −2.08744 −0.0786177
\(706\) −2.10120 −0.0790797
\(707\) 0 0
\(708\) 1.18721 0.0446181
\(709\) −13.7316 −0.515702 −0.257851 0.966185i \(-0.583014\pi\)
−0.257851 + 0.966185i \(0.583014\pi\)
\(710\) −22.0149 −0.826205
\(711\) −13.4434 −0.504168
\(712\) 34.1197 1.27869
\(713\) −10.5492 −0.395072
\(714\) 0 0
\(715\) 96.9343 3.62514
\(716\) −15.7008 −0.586765
\(717\) −6.53934 −0.244216
\(718\) −12.8267 −0.478687
\(719\) 17.6882 0.659658 0.329829 0.944041i \(-0.393009\pi\)
0.329829 + 0.944041i \(0.393009\pi\)
\(720\) −3.22518 −0.120195
\(721\) 0 0
\(722\) 4.85725 0.180768
\(723\) −0.500065 −0.0185976
\(724\) −26.1506 −0.971881
\(725\) −52.8277 −1.96197
\(726\) 14.5614 0.540425
\(727\) 25.0455 0.928885 0.464442 0.885603i \(-0.346255\pi\)
0.464442 + 0.885603i \(0.346255\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 16.4959 0.610541
\(731\) −5.87356 −0.217241
\(732\) 12.6896 0.469022
\(733\) 15.7850 0.583033 0.291517 0.956566i \(-0.405840\pi\)
0.291517 + 0.956566i \(0.405840\pi\)
\(734\) 21.5942 0.797055
\(735\) 0 0
\(736\) −5.85883 −0.215959
\(737\) −7.65786 −0.282081
\(738\) 3.05950 0.112622
\(739\) −9.99859 −0.367804 −0.183902 0.982945i \(-0.558873\pi\)
−0.183902 + 0.982945i \(0.558873\pi\)
\(740\) −22.2660 −0.818515
\(741\) 15.6691 0.575619
\(742\) 0 0
\(743\) −7.14322 −0.262059 −0.131030 0.991378i \(-0.541828\pi\)
−0.131030 + 0.991378i \(0.541828\pi\)
\(744\) 27.6543 1.01385
\(745\) 22.9645 0.841354
\(746\) −17.1813 −0.629053
\(747\) 3.39713 0.124294
\(748\) 17.3360 0.633867
\(749\) 0 0
\(750\) −19.2567 −0.703156
\(751\) −21.3619 −0.779507 −0.389754 0.920919i \(-0.627440\pi\)
−0.389754 + 0.920919i \(0.627440\pi\)
\(752\) −0.415131 −0.0151383
\(753\) −5.10076 −0.185882
\(754\) −15.9350 −0.580320
\(755\) −11.5831 −0.421550
\(756\) 0 0
\(757\) −6.60015 −0.239887 −0.119943 0.992781i \(-0.538271\pi\)
−0.119943 + 0.992781i \(0.538271\pi\)
\(758\) −21.3768 −0.776440
\(759\) −5.47113 −0.198590
\(760\) −37.5982 −1.36383
\(761\) 13.1540 0.476831 0.238416 0.971163i \(-0.423372\pi\)
0.238416 + 0.971163i \(0.423372\pi\)
\(762\) −0.0728387 −0.00263867
\(763\) 0 0
\(764\) 32.3091 1.16890
\(765\) 9.05955 0.327549
\(766\) −5.06950 −0.183169
\(767\) −3.70834 −0.133900
\(768\) 13.1026 0.472798
\(769\) −23.8184 −0.858915 −0.429457 0.903087i \(-0.641295\pi\)
−0.429457 + 0.903087i \(0.641295\pi\)
\(770\) 0 0
\(771\) 8.85942 0.319064
\(772\) 14.2674 0.513496
\(773\) −1.79587 −0.0645929 −0.0322965 0.999478i \(-0.510282\pi\)
−0.0322965 + 0.999478i \(0.510282\pi\)
\(774\) −2.00800 −0.0721760
\(775\) −118.336 −4.25075
\(776\) −16.6623 −0.598141
\(777\) 0 0
\(778\) −6.91345 −0.247859
\(779\) −14.1680 −0.507621
\(780\) 24.9550 0.893531
\(781\) −38.8889 −1.39155
\(782\) 1.73019 0.0618713
\(783\) 4.70941 0.168301
\(784\) 0 0
\(785\) −67.4169 −2.40621
\(786\) −10.5566 −0.376540
\(787\) −15.0749 −0.537362 −0.268681 0.963229i \(-0.586588\pi\)
−0.268681 + 0.963229i \(0.586588\pi\)
\(788\) 6.36033 0.226577
\(789\) −26.1608 −0.931350
\(790\) −41.6369 −1.48138
\(791\) 0 0
\(792\) 14.3423 0.509631
\(793\) −39.6370 −1.40755
\(794\) 1.96250 0.0696465
\(795\) −50.8876 −1.80480
\(796\) −29.3075 −1.03878
\(797\) 20.9322 0.741456 0.370728 0.928742i \(-0.379108\pi\)
0.370728 + 0.928742i \(0.379108\pi\)
\(798\) 0 0
\(799\) 1.16611 0.0412539
\(800\) −65.7212 −2.32360
\(801\) 13.0156 0.459884
\(802\) −28.7350 −1.01467
\(803\) 29.1397 1.02832
\(804\) −1.97146 −0.0695279
\(805\) 0 0
\(806\) −35.6950 −1.25730
\(807\) 15.7569 0.554668
\(808\) 7.43408 0.261530
\(809\) 24.8986 0.875388 0.437694 0.899124i \(-0.355795\pi\)
0.437694 + 0.899124i \(0.355795\pi\)
\(810\) 3.09720 0.108824
\(811\) 33.0135 1.15926 0.579630 0.814880i \(-0.303197\pi\)
0.579630 + 0.814880i \(0.303197\pi\)
\(812\) 0 0
\(813\) 5.95116 0.208716
\(814\) 16.5177 0.578944
\(815\) 2.76812 0.0969630
\(816\) 1.80168 0.0630714
\(817\) 9.29868 0.325320
\(818\) 8.28545 0.289694
\(819\) 0 0
\(820\) −22.5642 −0.787977
\(821\) 7.06657 0.246625 0.123312 0.992368i \(-0.460648\pi\)
0.123312 + 0.992368i \(0.460648\pi\)
\(822\) −0.793561 −0.0276786
\(823\) 38.3257 1.33595 0.667975 0.744183i \(-0.267161\pi\)
0.667975 + 0.744183i \(0.267161\pi\)
\(824\) 27.7595 0.967047
\(825\) −61.3722 −2.13671
\(826\) 0 0
\(827\) −18.8543 −0.655629 −0.327815 0.944742i \(-0.606312\pi\)
−0.327815 + 0.944742i \(0.606312\pi\)
\(828\) −1.40850 −0.0489487
\(829\) 49.1920 1.70851 0.854255 0.519854i \(-0.174014\pi\)
0.854255 + 0.519854i \(0.174014\pi\)
\(830\) 10.5216 0.365210
\(831\) −15.2365 −0.528550
\(832\) −12.7773 −0.442974
\(833\) 0 0
\(834\) 0.216786 0.00750668
\(835\) 49.9450 1.72842
\(836\) −27.4454 −0.949219
\(837\) 10.5492 0.364635
\(838\) −15.2423 −0.526535
\(839\) 3.42024 0.118080 0.0590399 0.998256i \(-0.481196\pi\)
0.0590399 + 0.998256i \(0.481196\pi\)
\(840\) 0 0
\(841\) −6.82142 −0.235222
\(842\) −2.83779 −0.0977966
\(843\) −1.78254 −0.0613941
\(844\) −36.9470 −1.27177
\(845\) −25.5966 −0.880548
\(846\) 0.398658 0.0137062
\(847\) 0 0
\(848\) −10.1200 −0.347524
\(849\) 19.9991 0.686368
\(850\) 19.4083 0.665699
\(851\) −3.92549 −0.134564
\(852\) −10.0116 −0.342993
\(853\) 26.9555 0.922939 0.461469 0.887156i \(-0.347322\pi\)
0.461469 + 0.887156i \(0.347322\pi\)
\(854\) 0 0
\(855\) −14.3426 −0.490505
\(856\) −49.4248 −1.68931
\(857\) 21.1442 0.722273 0.361137 0.932513i \(-0.382389\pi\)
0.361137 + 0.932513i \(0.382389\pi\)
\(858\) −18.5124 −0.632004
\(859\) 41.3331 1.41027 0.705133 0.709075i \(-0.250887\pi\)
0.705133 + 0.709075i \(0.250887\pi\)
\(860\) 14.8093 0.504992
\(861\) 0 0
\(862\) −7.06798 −0.240736
\(863\) −1.37678 −0.0468661 −0.0234331 0.999725i \(-0.507460\pi\)
−0.0234331 + 0.999725i \(0.507460\pi\)
\(864\) 5.85883 0.199322
\(865\) 12.9149 0.439120
\(866\) −25.8538 −0.878549
\(867\) 11.9391 0.405472
\(868\) 0 0
\(869\) −73.5507 −2.49504
\(870\) 14.5860 0.494511
\(871\) 6.15799 0.208656
\(872\) 29.1883 0.988441
\(873\) −6.35615 −0.215123
\(874\) −2.73913 −0.0926525
\(875\) 0 0
\(876\) 7.50177 0.253461
\(877\) −22.4325 −0.757490 −0.378745 0.925501i \(-0.623644\pi\)
−0.378745 + 0.925501i \(0.623644\pi\)
\(878\) 1.77439 0.0598826
\(879\) 17.4319 0.587963
\(880\) −17.6454 −0.594826
\(881\) −51.4107 −1.73207 −0.866036 0.499982i \(-0.833340\pi\)
−0.866036 + 0.499982i \(0.833340\pi\)
\(882\) 0 0
\(883\) 47.4947 1.59833 0.799163 0.601115i \(-0.205277\pi\)
0.799163 + 0.601115i \(0.205277\pi\)
\(884\) −13.9406 −0.468872
\(885\) 3.39439 0.114101
\(886\) 8.76426 0.294441
\(887\) −16.0293 −0.538212 −0.269106 0.963111i \(-0.586728\pi\)
−0.269106 + 0.963111i \(0.586728\pi\)
\(888\) 10.2905 0.345325
\(889\) 0 0
\(890\) 40.3119 1.35126
\(891\) 5.47113 0.183290
\(892\) 1.69550 0.0567697
\(893\) −1.84611 −0.0617779
\(894\) −4.38574 −0.146681
\(895\) −44.8906 −1.50053
\(896\) 0 0
\(897\) 4.39956 0.146897
\(898\) −2.43269 −0.0811799
\(899\) 49.6808 1.65695
\(900\) −15.7998 −0.526660
\(901\) 28.4273 0.947050
\(902\) 16.7389 0.557345
\(903\) 0 0
\(904\) −3.40733 −0.113326
\(905\) −74.7682 −2.48538
\(906\) 2.21212 0.0734928
\(907\) −28.8875 −0.959194 −0.479597 0.877489i \(-0.659217\pi\)
−0.479597 + 0.877489i \(0.659217\pi\)
\(908\) −20.0756 −0.666231
\(909\) 2.83587 0.0940599
\(910\) 0 0
\(911\) 41.9126 1.38863 0.694314 0.719672i \(-0.255708\pi\)
0.694314 + 0.719672i \(0.255708\pi\)
\(912\) −2.85232 −0.0944496
\(913\) 18.5862 0.615112
\(914\) −8.39357 −0.277634
\(915\) 36.2813 1.19942
\(916\) −17.4049 −0.575075
\(917\) 0 0
\(918\) −1.73019 −0.0571046
\(919\) 4.98999 0.164605 0.0823023 0.996607i \(-0.473773\pi\)
0.0823023 + 0.996607i \(0.473773\pi\)
\(920\) −10.5568 −0.348047
\(921\) 22.1407 0.729560
\(922\) −26.5957 −0.875882
\(923\) 31.2721 1.02933
\(924\) 0 0
\(925\) −44.0341 −1.44783
\(926\) 0.825723 0.0271350
\(927\) 10.5894 0.347801
\(928\) 27.5917 0.905741
\(929\) −36.9429 −1.21206 −0.606028 0.795443i \(-0.707238\pi\)
−0.606028 + 0.795443i \(0.707238\pi\)
\(930\) 32.6731 1.07139
\(931\) 0 0
\(932\) −16.8777 −0.552848
\(933\) 6.89035 0.225580
\(934\) 10.3738 0.339440
\(935\) 49.5660 1.62098
\(936\) −11.5332 −0.376974
\(937\) −7.79921 −0.254789 −0.127395 0.991852i \(-0.540661\pi\)
−0.127395 + 0.991852i \(0.540661\pi\)
\(938\) 0 0
\(939\) −13.5739 −0.442967
\(940\) −2.94016 −0.0958975
\(941\) −23.3080 −0.759820 −0.379910 0.925023i \(-0.624045\pi\)
−0.379910 + 0.925023i \(0.624045\pi\)
\(942\) 12.8752 0.419497
\(943\) −3.97807 −0.129544
\(944\) 0.675046 0.0219709
\(945\) 0 0
\(946\) −10.9860 −0.357187
\(947\) 37.5586 1.22049 0.610244 0.792213i \(-0.291071\pi\)
0.610244 + 0.792213i \(0.291071\pi\)
\(948\) −18.9350 −0.614982
\(949\) −23.4324 −0.760647
\(950\) −30.7261 −0.996886
\(951\) 24.6587 0.799614
\(952\) 0 0
\(953\) −16.0283 −0.519209 −0.259604 0.965715i \(-0.583592\pi\)
−0.259604 + 0.965715i \(0.583592\pi\)
\(954\) 9.71847 0.314647
\(955\) 92.3761 2.98922
\(956\) −9.21066 −0.297894
\(957\) 25.7658 0.832891
\(958\) 4.85249 0.156777
\(959\) 0 0
\(960\) 11.6956 0.377474
\(961\) 80.2866 2.58989
\(962\) −13.2825 −0.428246
\(963\) −18.8540 −0.607563
\(964\) −0.704341 −0.0226853
\(965\) 40.7925 1.31316
\(966\) 0 0
\(967\) −49.3526 −1.58707 −0.793536 0.608523i \(-0.791762\pi\)
−0.793536 + 0.608523i \(0.791762\pi\)
\(968\) 49.6326 1.59525
\(969\) 8.01218 0.257388
\(970\) −19.6862 −0.632087
\(971\) −15.1966 −0.487683 −0.243841 0.969815i \(-0.578408\pi\)
−0.243841 + 0.969815i \(0.578408\pi\)
\(972\) 1.40850 0.0451776
\(973\) 0 0
\(974\) 14.3027 0.458287
\(975\) 49.3518 1.58052
\(976\) 7.21529 0.230956
\(977\) −48.2766 −1.54451 −0.772253 0.635315i \(-0.780870\pi\)
−0.772253 + 0.635315i \(0.780870\pi\)
\(978\) −0.528653 −0.0169045
\(979\) 71.2101 2.27589
\(980\) 0 0
\(981\) 11.1344 0.355495
\(982\) 21.1977 0.676446
\(983\) −13.2666 −0.423139 −0.211570 0.977363i \(-0.567857\pi\)
−0.211570 + 0.977363i \(0.567857\pi\)
\(984\) 10.4283 0.332442
\(985\) 18.1850 0.579423
\(986\) −8.14816 −0.259490
\(987\) 0 0
\(988\) 22.0699 0.702138
\(989\) 2.61087 0.0830210
\(990\) 16.9452 0.538553
\(991\) 1.40433 0.0446100 0.0223050 0.999751i \(-0.492900\pi\)
0.0223050 + 0.999751i \(0.492900\pi\)
\(992\) 61.8063 1.96235
\(993\) −22.9676 −0.728855
\(994\) 0 0
\(995\) −83.7942 −2.65645
\(996\) 4.78486 0.151614
\(997\) 10.0710 0.318953 0.159477 0.987202i \(-0.449019\pi\)
0.159477 + 0.987202i \(0.449019\pi\)
\(998\) 7.18144 0.227324
\(999\) 3.92549 0.124197
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 3381.2.a.bi.1.4 10
7.2 even 3 483.2.i.h.277.7 20
7.4 even 3 483.2.i.h.415.7 yes 20
7.6 odd 2 3381.2.a.bj.1.4 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.i.h.277.7 20 7.2 even 3
483.2.i.h.415.7 yes 20 7.4 even 3
3381.2.a.bi.1.4 10 1.1 even 1 trivial
3381.2.a.bj.1.4 10 7.6 odd 2