Properties

Label 4650.2.a.ce.1.2
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
Defining polynomial: \( x^{2} - x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.56155\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.56155 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.56155 q^{11} +1.00000 q^{12} -2.00000 q^{13} -1.56155 q^{14} +1.00000 q^{16} -5.12311 q^{17} -1.00000 q^{18} +4.68466 q^{19} +1.56155 q^{21} +1.56155 q^{22} -5.56155 q^{23} -1.00000 q^{24} +2.00000 q^{26} +1.00000 q^{27} +1.56155 q^{28} -1.12311 q^{29} +1.00000 q^{31} -1.00000 q^{32} -1.56155 q^{33} +5.12311 q^{34} +1.00000 q^{36} -5.12311 q^{37} -4.68466 q^{38} -2.00000 q^{39} -1.12311 q^{41} -1.56155 q^{42} +7.80776 q^{43} -1.56155 q^{44} +5.56155 q^{46} +3.12311 q^{47} +1.00000 q^{48} -4.56155 q^{49} -5.12311 q^{51} -2.00000 q^{52} -11.5616 q^{53} -1.00000 q^{54} -1.56155 q^{56} +4.68466 q^{57} +1.12311 q^{58} -4.87689 q^{59} +6.00000 q^{61} -1.00000 q^{62} +1.56155 q^{63} +1.00000 q^{64} +1.56155 q^{66} -9.36932 q^{67} -5.12311 q^{68} -5.56155 q^{69} +4.68466 q^{71} -1.00000 q^{72} -9.80776 q^{73} +5.12311 q^{74} +4.68466 q^{76} -2.43845 q^{77} +2.00000 q^{78} -16.6847 q^{79} +1.00000 q^{81} +1.12311 q^{82} +2.24621 q^{83} +1.56155 q^{84} -7.80776 q^{86} -1.12311 q^{87} +1.56155 q^{88} -1.31534 q^{89} -3.12311 q^{91} -5.56155 q^{92} +1.00000 q^{93} -3.12311 q^{94} -1.00000 q^{96} +6.00000 q^{97} +4.56155 q^{98} -1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} + 2 q^{3} + 2 q^{4} - 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 3 q^{19} - q^{21} - q^{22} - 7 q^{23} - 2 q^{24} + 4 q^{26} + 2 q^{27} - q^{28} + 6 q^{29} + 2 q^{31} - 2 q^{32} + q^{33} + 2 q^{34} + 2 q^{36} - 2 q^{37} + 3 q^{38} - 4 q^{39} + 6 q^{41} + q^{42} - 5 q^{43} + q^{44} + 7 q^{46} - 2 q^{47} + 2 q^{48} - 5 q^{49} - 2 q^{51} - 4 q^{52} - 19 q^{53} - 2 q^{54} + q^{56} - 3 q^{57} - 6 q^{58} - 18 q^{59} + 12 q^{61} - 2 q^{62} - q^{63} + 2 q^{64} - q^{66} + 6 q^{67} - 2 q^{68} - 7 q^{69} - 3 q^{71} - 2 q^{72} + q^{73} + 2 q^{74} - 3 q^{76} - 9 q^{77} + 4 q^{78} - 21 q^{79} + 2 q^{81} - 6 q^{82} - 12 q^{83} - q^{84} + 5 q^{86} + 6 q^{87} - q^{88} - 15 q^{89} + 2 q^{91} - 7 q^{92} + 2 q^{93} + 2 q^{94} - 2 q^{96} + 12 q^{97} + 5 q^{98} + q^{99}+O(q^{100}) \) 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
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) 1.56155 0.590211 0.295106 0.955465i \(-0.404645\pi\)
0.295106 + 0.955465i \(0.404645\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −1.56155 −0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −5.12311 −1.24254 −0.621268 0.783598i \(-0.713382\pi\)
−0.621268 + 0.783598i \(0.713382\pi\)
\(18\) −1.00000 −0.235702
\(19\) 4.68466 1.07473 0.537367 0.843348i \(-0.319419\pi\)
0.537367 + 0.843348i \(0.319419\pi\)
\(20\) 0 0
\(21\) 1.56155 0.340759
\(22\) 1.56155 0.332924
\(23\) −5.56155 −1.15966 −0.579832 0.814736i \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 1.00000 0.192450
\(28\) 1.56155 0.295106
\(29\) −1.12311 −0.208555 −0.104278 0.994548i \(-0.533253\pi\)
−0.104278 + 0.994548i \(0.533253\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) −1.00000 −0.176777
\(33\) −1.56155 −0.271831
\(34\) 5.12311 0.878605
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.12311 −0.842233 −0.421117 0.907006i \(-0.638362\pi\)
−0.421117 + 0.907006i \(0.638362\pi\)
\(38\) −4.68466 −0.759952
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) −1.56155 −0.240953
\(43\) 7.80776 1.19067 0.595336 0.803477i \(-0.297019\pi\)
0.595336 + 0.803477i \(0.297019\pi\)
\(44\) −1.56155 −0.235413
\(45\) 0 0
\(46\) 5.56155 0.820006
\(47\) 3.12311 0.455552 0.227776 0.973714i \(-0.426855\pi\)
0.227776 + 0.973714i \(0.426855\pi\)
\(48\) 1.00000 0.144338
\(49\) −4.56155 −0.651650
\(50\) 0 0
\(51\) −5.12311 −0.717378
\(52\) −2.00000 −0.277350
\(53\) −11.5616 −1.58810 −0.794051 0.607852i \(-0.792032\pi\)
−0.794051 + 0.607852i \(0.792032\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −1.56155 −0.208671
\(57\) 4.68466 0.620498
\(58\) 1.12311 0.147471
\(59\) −4.87689 −0.634918 −0.317459 0.948272i \(-0.602830\pi\)
−0.317459 + 0.948272i \(0.602830\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) −1.00000 −0.127000
\(63\) 1.56155 0.196737
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 1.56155 0.192214
\(67\) −9.36932 −1.14464 −0.572322 0.820029i \(-0.693957\pi\)
−0.572322 + 0.820029i \(0.693957\pi\)
\(68\) −5.12311 −0.621268
\(69\) −5.56155 −0.669532
\(70\) 0 0
\(71\) 4.68466 0.555967 0.277983 0.960586i \(-0.410334\pi\)
0.277983 + 0.960586i \(0.410334\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.80776 −1.14791 −0.573956 0.818886i \(-0.694592\pi\)
−0.573956 + 0.818886i \(0.694592\pi\)
\(74\) 5.12311 0.595549
\(75\) 0 0
\(76\) 4.68466 0.537367
\(77\) −2.43845 −0.277887
\(78\) 2.00000 0.226455
\(79\) −16.6847 −1.87717 −0.938585 0.345047i \(-0.887863\pi\)
−0.938585 + 0.345047i \(0.887863\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 1.12311 0.124026
\(83\) 2.24621 0.246554 0.123277 0.992372i \(-0.460660\pi\)
0.123277 + 0.992372i \(0.460660\pi\)
\(84\) 1.56155 0.170379
\(85\) 0 0
\(86\) −7.80776 −0.841933
\(87\) −1.12311 −0.120410
\(88\) 1.56155 0.166462
\(89\) −1.31534 −0.139426 −0.0697130 0.997567i \(-0.522208\pi\)
−0.0697130 + 0.997567i \(0.522208\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) −5.56155 −0.579832
\(93\) 1.00000 0.103695
\(94\) −3.12311 −0.322124
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) 4.56155 0.460786
\(99\) −1.56155 −0.156942
\(100\) 0 0
\(101\) 3.56155 0.354388 0.177194 0.984176i \(-0.443298\pi\)
0.177194 + 0.984176i \(0.443298\pi\)
\(102\) 5.12311 0.507263
\(103\) 18.2462 1.79785 0.898926 0.438100i \(-0.144348\pi\)
0.898926 + 0.438100i \(0.144348\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 11.5616 1.12296
\(107\) −1.56155 −0.150961 −0.0754805 0.997147i \(-0.524049\pi\)
−0.0754805 + 0.997147i \(0.524049\pi\)
\(108\) 1.00000 0.0962250
\(109\) −19.3693 −1.85524 −0.927622 0.373520i \(-0.878151\pi\)
−0.927622 + 0.373520i \(0.878151\pi\)
\(110\) 0 0
\(111\) −5.12311 −0.486264
\(112\) 1.56155 0.147553
\(113\) −10.6847 −1.00513 −0.502564 0.864540i \(-0.667610\pi\)
−0.502564 + 0.864540i \(0.667610\pi\)
\(114\) −4.68466 −0.438758
\(115\) 0 0
\(116\) −1.12311 −0.104278
\(117\) −2.00000 −0.184900
\(118\) 4.87689 0.448955
\(119\) −8.00000 −0.733359
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) −6.00000 −0.543214
\(123\) −1.12311 −0.101267
\(124\) 1.00000 0.0898027
\(125\) 0 0
\(126\) −1.56155 −0.139114
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 7.80776 0.687435
\(130\) 0 0
\(131\) 9.36932 0.818601 0.409301 0.912400i \(-0.365773\pi\)
0.409301 + 0.912400i \(0.365773\pi\)
\(132\) −1.56155 −0.135916
\(133\) 7.31534 0.634321
\(134\) 9.36932 0.809386
\(135\) 0 0
\(136\) 5.12311 0.439303
\(137\) −3.75379 −0.320708 −0.160354 0.987060i \(-0.551264\pi\)
−0.160354 + 0.987060i \(0.551264\pi\)
\(138\) 5.56155 0.473431
\(139\) −8.87689 −0.752928 −0.376464 0.926431i \(-0.622860\pi\)
−0.376464 + 0.926431i \(0.622860\pi\)
\(140\) 0 0
\(141\) 3.12311 0.263013
\(142\) −4.68466 −0.393128
\(143\) 3.12311 0.261167
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 9.80776 0.811696
\(147\) −4.56155 −0.376231
\(148\) −5.12311 −0.421117
\(149\) −20.0540 −1.64289 −0.821443 0.570291i \(-0.806830\pi\)
−0.821443 + 0.570291i \(0.806830\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) −4.68466 −0.379976
\(153\) −5.12311 −0.414179
\(154\) 2.43845 0.196496
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) 20.0540 1.60048 0.800241 0.599679i \(-0.204705\pi\)
0.800241 + 0.599679i \(0.204705\pi\)
\(158\) 16.6847 1.32736
\(159\) −11.5616 −0.916891
\(160\) 0 0
\(161\) −8.68466 −0.684447
\(162\) −1.00000 −0.0785674
\(163\) −9.36932 −0.733862 −0.366931 0.930248i \(-0.619591\pi\)
−0.366931 + 0.930248i \(0.619591\pi\)
\(164\) −1.12311 −0.0876998
\(165\) 0 0
\(166\) −2.24621 −0.174340
\(167\) 2.43845 0.188693 0.0943464 0.995539i \(-0.469924\pi\)
0.0943464 + 0.995539i \(0.469924\pi\)
\(168\) −1.56155 −0.120476
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 4.68466 0.358245
\(172\) 7.80776 0.595336
\(173\) 8.24621 0.626948 0.313474 0.949597i \(-0.398507\pi\)
0.313474 + 0.949597i \(0.398507\pi\)
\(174\) 1.12311 0.0851424
\(175\) 0 0
\(176\) −1.56155 −0.117706
\(177\) −4.87689 −0.366570
\(178\) 1.31534 0.0985890
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) 24.0540 1.78792 0.893959 0.448149i \(-0.147917\pi\)
0.893959 + 0.448149i \(0.147917\pi\)
\(182\) 3.12311 0.231500
\(183\) 6.00000 0.443533
\(184\) 5.56155 0.410003
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) 8.00000 0.585018
\(188\) 3.12311 0.227776
\(189\) 1.56155 0.113586
\(190\) 0 0
\(191\) 2.24621 0.162530 0.0812651 0.996693i \(-0.474104\pi\)
0.0812651 + 0.996693i \(0.474104\pi\)
\(192\) 1.00000 0.0721688
\(193\) 18.4924 1.33111 0.665557 0.746347i \(-0.268194\pi\)
0.665557 + 0.746347i \(0.268194\pi\)
\(194\) −6.00000 −0.430775
\(195\) 0 0
\(196\) −4.56155 −0.325825
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 1.56155 0.110975
\(199\) 3.80776 0.269925 0.134963 0.990851i \(-0.456909\pi\)
0.134963 + 0.990851i \(0.456909\pi\)
\(200\) 0 0
\(201\) −9.36932 −0.660861
\(202\) −3.56155 −0.250590
\(203\) −1.75379 −0.123092
\(204\) −5.12311 −0.358689
\(205\) 0 0
\(206\) −18.2462 −1.27127
\(207\) −5.56155 −0.386555
\(208\) −2.00000 −0.138675
\(209\) −7.31534 −0.506013
\(210\) 0 0
\(211\) −11.3153 −0.778980 −0.389490 0.921031i \(-0.627349\pi\)
−0.389490 + 0.921031i \(0.627349\pi\)
\(212\) −11.5616 −0.794051
\(213\) 4.68466 0.320988
\(214\) 1.56155 0.106746
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 1.56155 0.106005
\(218\) 19.3693 1.31186
\(219\) −9.80776 −0.662747
\(220\) 0 0
\(221\) 10.2462 0.689235
\(222\) 5.12311 0.343840
\(223\) 12.8769 0.862301 0.431150 0.902280i \(-0.358108\pi\)
0.431150 + 0.902280i \(0.358108\pi\)
\(224\) −1.56155 −0.104336
\(225\) 0 0
\(226\) 10.6847 0.710733
\(227\) 4.68466 0.310932 0.155466 0.987841i \(-0.450312\pi\)
0.155466 + 0.987841i \(0.450312\pi\)
\(228\) 4.68466 0.310249
\(229\) −12.4384 −0.821956 −0.410978 0.911645i \(-0.634813\pi\)
−0.410978 + 0.911645i \(0.634813\pi\)
\(230\) 0 0
\(231\) −2.43845 −0.160438
\(232\) 1.12311 0.0737355
\(233\) −20.0540 −1.31378 −0.656890 0.753987i \(-0.728128\pi\)
−0.656890 + 0.753987i \(0.728128\pi\)
\(234\) 2.00000 0.130744
\(235\) 0 0
\(236\) −4.87689 −0.317459
\(237\) −16.6847 −1.08379
\(238\) 8.00000 0.518563
\(239\) 24.0000 1.55243 0.776215 0.630468i \(-0.217137\pi\)
0.776215 + 0.630468i \(0.217137\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) 8.56155 0.550357
\(243\) 1.00000 0.0641500
\(244\) 6.00000 0.384111
\(245\) 0 0
\(246\) 1.12311 0.0716066
\(247\) −9.36932 −0.596155
\(248\) −1.00000 −0.0635001
\(249\) 2.24621 0.142348
\(250\) 0 0
\(251\) −16.4924 −1.04099 −0.520496 0.853864i \(-0.674253\pi\)
−0.520496 + 0.853864i \(0.674253\pi\)
\(252\) 1.56155 0.0983686
\(253\) 8.68466 0.546000
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −10.6847 −0.666491 −0.333245 0.942840i \(-0.608144\pi\)
−0.333245 + 0.942840i \(0.608144\pi\)
\(258\) −7.80776 −0.486090
\(259\) −8.00000 −0.497096
\(260\) 0 0
\(261\) −1.12311 −0.0695185
\(262\) −9.36932 −0.578838
\(263\) −12.4924 −0.770316 −0.385158 0.922851i \(-0.625853\pi\)
−0.385158 + 0.922851i \(0.625853\pi\)
\(264\) 1.56155 0.0961069
\(265\) 0 0
\(266\) −7.31534 −0.448532
\(267\) −1.31534 −0.0804976
\(268\) −9.36932 −0.572322
\(269\) 16.2462 0.990549 0.495274 0.868737i \(-0.335067\pi\)
0.495274 + 0.868737i \(0.335067\pi\)
\(270\) 0 0
\(271\) −10.0540 −0.610736 −0.305368 0.952234i \(-0.598779\pi\)
−0.305368 + 0.952234i \(0.598779\pi\)
\(272\) −5.12311 −0.310634
\(273\) −3.12311 −0.189019
\(274\) 3.75379 0.226775
\(275\) 0 0
\(276\) −5.56155 −0.334766
\(277\) −19.3693 −1.16379 −0.581895 0.813264i \(-0.697688\pi\)
−0.581895 + 0.813264i \(0.697688\pi\)
\(278\) 8.87689 0.532401
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 13.1231 0.782859 0.391429 0.920208i \(-0.371981\pi\)
0.391429 + 0.920208i \(0.371981\pi\)
\(282\) −3.12311 −0.185978
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 4.68466 0.277983
\(285\) 0 0
\(286\) −3.12311 −0.184673
\(287\) −1.75379 −0.103523
\(288\) −1.00000 −0.0589256
\(289\) 9.24621 0.543895
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) −9.80776 −0.573956
\(293\) 6.49242 0.379291 0.189646 0.981853i \(-0.439266\pi\)
0.189646 + 0.981853i \(0.439266\pi\)
\(294\) 4.56155 0.266035
\(295\) 0 0
\(296\) 5.12311 0.297774
\(297\) −1.56155 −0.0906105
\(298\) 20.0540 1.16170
\(299\) 11.1231 0.643266
\(300\) 0 0
\(301\) 12.1922 0.702749
\(302\) 0 0
\(303\) 3.56155 0.204606
\(304\) 4.68466 0.268684
\(305\) 0 0
\(306\) 5.12311 0.292868
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) −2.43845 −0.138943
\(309\) 18.2462 1.03799
\(310\) 0 0
\(311\) −18.2462 −1.03465 −0.517324 0.855790i \(-0.673072\pi\)
−0.517324 + 0.855790i \(0.673072\pi\)
\(312\) 2.00000 0.113228
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −20.0540 −1.13171
\(315\) 0 0
\(316\) −16.6847 −0.938585
\(317\) −25.1231 −1.41105 −0.705527 0.708683i \(-0.749290\pi\)
−0.705527 + 0.708683i \(0.749290\pi\)
\(318\) 11.5616 0.648340
\(319\) 1.75379 0.0981933
\(320\) 0 0
\(321\) −1.56155 −0.0871574
\(322\) 8.68466 0.483977
\(323\) −24.0000 −1.33540
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 9.36932 0.518918
\(327\) −19.3693 −1.07113
\(328\) 1.12311 0.0620131
\(329\) 4.87689 0.268872
\(330\) 0 0
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) 2.24621 0.123277
\(333\) −5.12311 −0.280744
\(334\) −2.43845 −0.133426
\(335\) 0 0
\(336\) 1.56155 0.0851897
\(337\) 10.0000 0.544735 0.272367 0.962193i \(-0.412193\pi\)
0.272367 + 0.962193i \(0.412193\pi\)
\(338\) 9.00000 0.489535
\(339\) −10.6847 −0.580311
\(340\) 0 0
\(341\) −1.56155 −0.0845628
\(342\) −4.68466 −0.253317
\(343\) −18.0540 −0.974823
\(344\) −7.80776 −0.420966
\(345\) 0 0
\(346\) −8.24621 −0.443319
\(347\) 2.24621 0.120583 0.0602915 0.998181i \(-0.480797\pi\)
0.0602915 + 0.998181i \(0.480797\pi\)
\(348\) −1.12311 −0.0602048
\(349\) −19.3693 −1.03682 −0.518408 0.855133i \(-0.673475\pi\)
−0.518408 + 0.855133i \(0.673475\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) 1.56155 0.0832310
\(353\) −16.2462 −0.864699 −0.432349 0.901706i \(-0.642315\pi\)
−0.432349 + 0.901706i \(0.642315\pi\)
\(354\) 4.87689 0.259204
\(355\) 0 0
\(356\) −1.31534 −0.0697130
\(357\) −8.00000 −0.423405
\(358\) 12.0000 0.634220
\(359\) −19.3153 −1.01942 −0.509712 0.860345i \(-0.670248\pi\)
−0.509712 + 0.860345i \(0.670248\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) −24.0540 −1.26425
\(363\) −8.56155 −0.449365
\(364\) −3.12311 −0.163695
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) 9.36932 0.489074 0.244537 0.969640i \(-0.421364\pi\)
0.244537 + 0.969640i \(0.421364\pi\)
\(368\) −5.56155 −0.289916
\(369\) −1.12311 −0.0584665
\(370\) 0 0
\(371\) −18.0540 −0.937316
\(372\) 1.00000 0.0518476
\(373\) −6.68466 −0.346118 −0.173059 0.984911i \(-0.555365\pi\)
−0.173059 + 0.984911i \(0.555365\pi\)
\(374\) −8.00000 −0.413670
\(375\) 0 0
\(376\) −3.12311 −0.161062
\(377\) 2.24621 0.115686
\(378\) −1.56155 −0.0803176
\(379\) 30.0540 1.54377 0.771885 0.635763i \(-0.219314\pi\)
0.771885 + 0.635763i \(0.219314\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.24621 −0.114926
\(383\) −6.24621 −0.319166 −0.159583 0.987184i \(-0.551015\pi\)
−0.159583 + 0.987184i \(0.551015\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −18.4924 −0.941240
\(387\) 7.80776 0.396891
\(388\) 6.00000 0.304604
\(389\) 24.2462 1.22933 0.614666 0.788788i \(-0.289291\pi\)
0.614666 + 0.788788i \(0.289291\pi\)
\(390\) 0 0
\(391\) 28.4924 1.44092
\(392\) 4.56155 0.230393
\(393\) 9.36932 0.472620
\(394\) 6.00000 0.302276
\(395\) 0 0
\(396\) −1.56155 −0.0784710
\(397\) 28.0540 1.40799 0.703994 0.710206i \(-0.251398\pi\)
0.703994 + 0.710206i \(0.251398\pi\)
\(398\) −3.80776 −0.190866
\(399\) 7.31534 0.366225
\(400\) 0 0
\(401\) −1.31534 −0.0656850 −0.0328425 0.999461i \(-0.510456\pi\)
−0.0328425 + 0.999461i \(0.510456\pi\)
\(402\) 9.36932 0.467299
\(403\) −2.00000 −0.0996271
\(404\) 3.56155 0.177194
\(405\) 0 0
\(406\) 1.75379 0.0870391
\(407\) 8.00000 0.396545
\(408\) 5.12311 0.253632
\(409\) −12.6307 −0.624547 −0.312274 0.949992i \(-0.601091\pi\)
−0.312274 + 0.949992i \(0.601091\pi\)
\(410\) 0 0
\(411\) −3.75379 −0.185161
\(412\) 18.2462 0.898926
\(413\) −7.61553 −0.374736
\(414\) 5.56155 0.273335
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) −8.87689 −0.434703
\(418\) 7.31534 0.357805
\(419\) 22.2462 1.08680 0.543399 0.839474i \(-0.317137\pi\)
0.543399 + 0.839474i \(0.317137\pi\)
\(420\) 0 0
\(421\) 18.4924 0.901266 0.450633 0.892709i \(-0.351198\pi\)
0.450633 + 0.892709i \(0.351198\pi\)
\(422\) 11.3153 0.550822
\(423\) 3.12311 0.151851
\(424\) 11.5616 0.561479
\(425\) 0 0
\(426\) −4.68466 −0.226972
\(427\) 9.36932 0.453413
\(428\) −1.56155 −0.0754805
\(429\) 3.12311 0.150785
\(430\) 0 0
\(431\) −13.7538 −0.662497 −0.331248 0.943544i \(-0.607470\pi\)
−0.331248 + 0.943544i \(0.607470\pi\)
\(432\) 1.00000 0.0481125
\(433\) 18.3002 0.879451 0.439725 0.898132i \(-0.355076\pi\)
0.439725 + 0.898132i \(0.355076\pi\)
\(434\) −1.56155 −0.0749569
\(435\) 0 0
\(436\) −19.3693 −0.927622
\(437\) −26.0540 −1.24633
\(438\) 9.80776 0.468633
\(439\) −31.6155 −1.50893 −0.754463 0.656342i \(-0.772103\pi\)
−0.754463 + 0.656342i \(0.772103\pi\)
\(440\) 0 0
\(441\) −4.56155 −0.217217
\(442\) −10.2462 −0.487363
\(443\) 14.0540 0.667725 0.333862 0.942622i \(-0.391648\pi\)
0.333862 + 0.942622i \(0.391648\pi\)
\(444\) −5.12311 −0.243132
\(445\) 0 0
\(446\) −12.8769 −0.609739
\(447\) −20.0540 −0.948520
\(448\) 1.56155 0.0737764
\(449\) −22.4924 −1.06148 −0.530742 0.847534i \(-0.678087\pi\)
−0.530742 + 0.847534i \(0.678087\pi\)
\(450\) 0 0
\(451\) 1.75379 0.0825827
\(452\) −10.6847 −0.502564
\(453\) 0 0
\(454\) −4.68466 −0.219862
\(455\) 0 0
\(456\) −4.68466 −0.219379
\(457\) −24.7386 −1.15722 −0.578612 0.815603i \(-0.696406\pi\)
−0.578612 + 0.815603i \(0.696406\pi\)
\(458\) 12.4384 0.581210
\(459\) −5.12311 −0.239126
\(460\) 0 0
\(461\) −15.3693 −0.715820 −0.357910 0.933756i \(-0.616511\pi\)
−0.357910 + 0.933756i \(0.616511\pi\)
\(462\) 2.43845 0.113447
\(463\) 18.7386 0.870858 0.435429 0.900223i \(-0.356597\pi\)
0.435429 + 0.900223i \(0.356597\pi\)
\(464\) −1.12311 −0.0521389
\(465\) 0 0
\(466\) 20.0540 0.928982
\(467\) 26.2462 1.21453 0.607265 0.794499i \(-0.292267\pi\)
0.607265 + 0.794499i \(0.292267\pi\)
\(468\) −2.00000 −0.0924500
\(469\) −14.6307 −0.675582
\(470\) 0 0
\(471\) 20.0540 0.924038
\(472\) 4.87689 0.224477
\(473\) −12.1922 −0.560600
\(474\) 16.6847 0.766352
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) −11.5616 −0.529367
\(478\) −24.0000 −1.09773
\(479\) 6.43845 0.294180 0.147090 0.989123i \(-0.453009\pi\)
0.147090 + 0.989123i \(0.453009\pi\)
\(480\) 0 0
\(481\) 10.2462 0.467187
\(482\) 4.24621 0.193410
\(483\) −8.68466 −0.395166
\(484\) −8.56155 −0.389161
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −24.0000 −1.08754 −0.543772 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) −6.00000 −0.271607
\(489\) −9.36932 −0.423695
\(490\) 0 0
\(491\) 2.93087 0.132268 0.0661341 0.997811i \(-0.478933\pi\)
0.0661341 + 0.997811i \(0.478933\pi\)
\(492\) −1.12311 −0.0506335
\(493\) 5.75379 0.259138
\(494\) 9.36932 0.421545
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) 7.31534 0.328138
\(498\) −2.24621 −0.100655
\(499\) 20.0000 0.895323 0.447661 0.894203i \(-0.352257\pi\)
0.447661 + 0.894203i \(0.352257\pi\)
\(500\) 0 0
\(501\) 2.43845 0.108942
\(502\) 16.4924 0.736093
\(503\) −9.75379 −0.434900 −0.217450 0.976071i \(-0.569774\pi\)
−0.217450 + 0.976071i \(0.569774\pi\)
\(504\) −1.56155 −0.0695571
\(505\) 0 0
\(506\) −8.68466 −0.386080
\(507\) −9.00000 −0.399704
\(508\) 0 0
\(509\) −21.6155 −0.958091 −0.479046 0.877790i \(-0.659017\pi\)
−0.479046 + 0.877790i \(0.659017\pi\)
\(510\) 0 0
\(511\) −15.3153 −0.677511
\(512\) −1.00000 −0.0441942
\(513\) 4.68466 0.206833
\(514\) 10.6847 0.471280
\(515\) 0 0
\(516\) 7.80776 0.343718
\(517\) −4.87689 −0.214486
\(518\) 8.00000 0.351500
\(519\) 8.24621 0.361968
\(520\) 0 0
\(521\) −15.7538 −0.690186 −0.345093 0.938568i \(-0.612153\pi\)
−0.345093 + 0.938568i \(0.612153\pi\)
\(522\) 1.12311 0.0491570
\(523\) −39.4233 −1.72386 −0.861930 0.507027i \(-0.830744\pi\)
−0.861930 + 0.507027i \(0.830744\pi\)
\(524\) 9.36932 0.409301
\(525\) 0 0
\(526\) 12.4924 0.544696
\(527\) −5.12311 −0.223166
\(528\) −1.56155 −0.0679579
\(529\) 7.93087 0.344820
\(530\) 0 0
\(531\) −4.87689 −0.211639
\(532\) 7.31534 0.317160
\(533\) 2.24621 0.0972942
\(534\) 1.31534 0.0569204
\(535\) 0 0
\(536\) 9.36932 0.404693
\(537\) −12.0000 −0.517838
\(538\) −16.2462 −0.700424
\(539\) 7.12311 0.306814
\(540\) 0 0
\(541\) 36.2462 1.55835 0.779173 0.626809i \(-0.215639\pi\)
0.779173 + 0.626809i \(0.215639\pi\)
\(542\) 10.0540 0.431855
\(543\) 24.0540 1.03225
\(544\) 5.12311 0.219651
\(545\) 0 0
\(546\) 3.12311 0.133657
\(547\) −35.1231 −1.50176 −0.750878 0.660441i \(-0.770369\pi\)
−0.750878 + 0.660441i \(0.770369\pi\)
\(548\) −3.75379 −0.160354
\(549\) 6.00000 0.256074
\(550\) 0 0
\(551\) −5.26137 −0.224142
\(552\) 5.56155 0.236715
\(553\) −26.0540 −1.10793
\(554\) 19.3693 0.822923
\(555\) 0 0
\(556\) −8.87689 −0.376464
\(557\) 6.19224 0.262373 0.131187 0.991358i \(-0.458121\pi\)
0.131187 + 0.991358i \(0.458121\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −15.6155 −0.660466
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) −13.1231 −0.553565
\(563\) 16.4924 0.695073 0.347536 0.937667i \(-0.387018\pi\)
0.347536 + 0.937667i \(0.387018\pi\)
\(564\) 3.12311 0.131506
\(565\) 0 0
\(566\) 0 0
\(567\) 1.56155 0.0655791
\(568\) −4.68466 −0.196564
\(569\) 10.1922 0.427281 0.213640 0.976912i \(-0.431468\pi\)
0.213640 + 0.976912i \(0.431468\pi\)
\(570\) 0 0
\(571\) 32.8769 1.37586 0.687928 0.725779i \(-0.258521\pi\)
0.687928 + 0.725779i \(0.258521\pi\)
\(572\) 3.12311 0.130584
\(573\) 2.24621 0.0938368
\(574\) 1.75379 0.0732017
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −6.87689 −0.286289 −0.143144 0.989702i \(-0.545721\pi\)
−0.143144 + 0.989702i \(0.545721\pi\)
\(578\) −9.24621 −0.384592
\(579\) 18.4924 0.768519
\(580\) 0 0
\(581\) 3.50758 0.145519
\(582\) −6.00000 −0.248708
\(583\) 18.0540 0.747719
\(584\) 9.80776 0.405848
\(585\) 0 0
\(586\) −6.49242 −0.268200
\(587\) −32.4924 −1.34111 −0.670553 0.741862i \(-0.733943\pi\)
−0.670553 + 0.741862i \(0.733943\pi\)
\(588\) −4.56155 −0.188115
\(589\) 4.68466 0.193028
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) −5.12311 −0.210558
\(593\) 30.0000 1.23195 0.615976 0.787765i \(-0.288762\pi\)
0.615976 + 0.787765i \(0.288762\pi\)
\(594\) 1.56155 0.0640713
\(595\) 0 0
\(596\) −20.0540 −0.821443
\(597\) 3.80776 0.155841
\(598\) −11.1231 −0.454858
\(599\) −23.4233 −0.957050 −0.478525 0.878074i \(-0.658828\pi\)
−0.478525 + 0.878074i \(0.658828\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) −12.1922 −0.496918
\(603\) −9.36932 −0.381548
\(604\) 0 0
\(605\) 0 0
\(606\) −3.56155 −0.144678
\(607\) 30.0540 1.21985 0.609927 0.792458i \(-0.291199\pi\)
0.609927 + 0.792458i \(0.291199\pi\)
\(608\) −4.68466 −0.189988
\(609\) −1.75379 −0.0710671
\(610\) 0 0
\(611\) −6.24621 −0.252695
\(612\) −5.12311 −0.207089
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 2.43845 0.0982478
\(617\) 19.5616 0.787518 0.393759 0.919214i \(-0.371174\pi\)
0.393759 + 0.919214i \(0.371174\pi\)
\(618\) −18.2462 −0.733970
\(619\) −13.3693 −0.537358 −0.268679 0.963230i \(-0.586587\pi\)
−0.268679 + 0.963230i \(0.586587\pi\)
\(620\) 0 0
\(621\) −5.56155 −0.223177
\(622\) 18.2462 0.731606
\(623\) −2.05398 −0.0822908
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) −7.31534 −0.292147
\(628\) 20.0540 0.800241
\(629\) 26.2462 1.04650
\(630\) 0 0
\(631\) −24.3002 −0.967375 −0.483688 0.875241i \(-0.660703\pi\)
−0.483688 + 0.875241i \(0.660703\pi\)
\(632\) 16.6847 0.663680
\(633\) −11.3153 −0.449744
\(634\) 25.1231 0.997766
\(635\) 0 0
\(636\) −11.5616 −0.458445
\(637\) 9.12311 0.361471
\(638\) −1.75379 −0.0694332
\(639\) 4.68466 0.185322
\(640\) 0 0
\(641\) −38.4924 −1.52036 −0.760180 0.649713i \(-0.774889\pi\)
−0.760180 + 0.649713i \(0.774889\pi\)
\(642\) 1.56155 0.0616296
\(643\) −30.0540 −1.18521 −0.592607 0.805492i \(-0.701901\pi\)
−0.592607 + 0.805492i \(0.701901\pi\)
\(644\) −8.68466 −0.342223
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) 17.0691 0.671057 0.335528 0.942030i \(-0.391085\pi\)
0.335528 + 0.942030i \(0.391085\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 7.61553 0.298936
\(650\) 0 0
\(651\) 1.56155 0.0612021
\(652\) −9.36932 −0.366931
\(653\) −23.7538 −0.929558 −0.464779 0.885427i \(-0.653866\pi\)
−0.464779 + 0.885427i \(0.653866\pi\)
\(654\) 19.3693 0.757400
\(655\) 0 0
\(656\) −1.12311 −0.0438499
\(657\) −9.80776 −0.382637
\(658\) −4.87689 −0.190121
\(659\) 18.7386 0.729954 0.364977 0.931017i \(-0.381077\pi\)
0.364977 + 0.931017i \(0.381077\pi\)
\(660\) 0 0
\(661\) −6.49242 −0.252526 −0.126263 0.991997i \(-0.540298\pi\)
−0.126263 + 0.991997i \(0.540298\pi\)
\(662\) −10.2462 −0.398230
\(663\) 10.2462 0.397930
\(664\) −2.24621 −0.0871699
\(665\) 0 0
\(666\) 5.12311 0.198516
\(667\) 6.24621 0.241854
\(668\) 2.43845 0.0943464
\(669\) 12.8769 0.497849
\(670\) 0 0
\(671\) −9.36932 −0.361698
\(672\) −1.56155 −0.0602382
\(673\) −28.2462 −1.08881 −0.544406 0.838822i \(-0.683245\pi\)
−0.544406 + 0.838822i \(0.683245\pi\)
\(674\) −10.0000 −0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 29.4233 1.13083 0.565414 0.824807i \(-0.308716\pi\)
0.565414 + 0.824807i \(0.308716\pi\)
\(678\) 10.6847 0.410342
\(679\) 9.36932 0.359561
\(680\) 0 0
\(681\) 4.68466 0.179517
\(682\) 1.56155 0.0597949
\(683\) −19.3153 −0.739081 −0.369541 0.929215i \(-0.620485\pi\)
−0.369541 + 0.929215i \(0.620485\pi\)
\(684\) 4.68466 0.179122
\(685\) 0 0
\(686\) 18.0540 0.689304
\(687\) −12.4384 −0.474556
\(688\) 7.80776 0.297668
\(689\) 23.1231 0.880920
\(690\) 0 0
\(691\) −22.0540 −0.838973 −0.419486 0.907762i \(-0.637790\pi\)
−0.419486 + 0.907762i \(0.637790\pi\)
\(692\) 8.24621 0.313474
\(693\) −2.43845 −0.0926289
\(694\) −2.24621 −0.0852650
\(695\) 0 0
\(696\) 1.12311 0.0425712
\(697\) 5.75379 0.217940
\(698\) 19.3693 0.733139
\(699\) −20.0540 −0.758511
\(700\) 0 0
\(701\) 25.8078 0.974746 0.487373 0.873194i \(-0.337955\pi\)
0.487373 + 0.873194i \(0.337955\pi\)
\(702\) 2.00000 0.0754851
\(703\) −24.0000 −0.905177
\(704\) −1.56155 −0.0588532
\(705\) 0 0
\(706\) 16.2462 0.611434
\(707\) 5.56155 0.209164
\(708\) −4.87689 −0.183285
\(709\) −20.0540 −0.753143 −0.376571 0.926388i \(-0.622897\pi\)
−0.376571 + 0.926388i \(0.622897\pi\)
\(710\) 0 0
\(711\) −16.6847 −0.625724
\(712\) 1.31534 0.0492945
\(713\) −5.56155 −0.208282
\(714\) 8.00000 0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 24.0000 0.896296
\(718\) 19.3153 0.720842
\(719\) −43.1231 −1.60822 −0.804110 0.594480i \(-0.797358\pi\)
−0.804110 + 0.594480i \(0.797358\pi\)
\(720\) 0 0
\(721\) 28.4924 1.06111
\(722\) −2.94602 −0.109640
\(723\) −4.24621 −0.157918
\(724\) 24.0540 0.893959
\(725\) 0 0
\(726\) 8.56155 0.317749
\(727\) 14.0540 0.521233 0.260617 0.965442i \(-0.416074\pi\)
0.260617 + 0.965442i \(0.416074\pi\)
\(728\) 3.12311 0.115750
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) 6.00000 0.221766
\(733\) −26.4924 −0.978520 −0.489260 0.872138i \(-0.662733\pi\)
−0.489260 + 0.872138i \(0.662733\pi\)
\(734\) −9.36932 −0.345828
\(735\) 0 0
\(736\) 5.56155 0.205002
\(737\) 14.6307 0.538928
\(738\) 1.12311 0.0413421
\(739\) −12.9848 −0.477655 −0.238828 0.971062i \(-0.576763\pi\)
−0.238828 + 0.971062i \(0.576763\pi\)
\(740\) 0 0
\(741\) −9.36932 −0.344190
\(742\) 18.0540 0.662782
\(743\) −11.4233 −0.419080 −0.209540 0.977800i \(-0.567197\pi\)
−0.209540 + 0.977800i \(0.567197\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) 6.68466 0.244743
\(747\) 2.24621 0.0821846
\(748\) 8.00000 0.292509
\(749\) −2.43845 −0.0890989
\(750\) 0 0
\(751\) 36.8769 1.34566 0.672828 0.739798i \(-0.265079\pi\)
0.672828 + 0.739798i \(0.265079\pi\)
\(752\) 3.12311 0.113888
\(753\) −16.4924 −0.601017
\(754\) −2.24621 −0.0818022
\(755\) 0 0
\(756\) 1.56155 0.0567931
\(757\) −10.0000 −0.363456 −0.181728 0.983349i \(-0.558169\pi\)
−0.181728 + 0.983349i \(0.558169\pi\)
\(758\) −30.0540 −1.09161
\(759\) 8.68466 0.315233
\(760\) 0 0
\(761\) 41.4233 1.50159 0.750797 0.660533i \(-0.229670\pi\)
0.750797 + 0.660533i \(0.229670\pi\)
\(762\) 0 0
\(763\) −30.2462 −1.09499
\(764\) 2.24621 0.0812651
\(765\) 0 0
\(766\) 6.24621 0.225685
\(767\) 9.75379 0.352189
\(768\) 1.00000 0.0360844
\(769\) −16.4384 −0.592786 −0.296393 0.955066i \(-0.595784\pi\)
−0.296393 + 0.955066i \(0.595784\pi\)
\(770\) 0 0
\(771\) −10.6847 −0.384799
\(772\) 18.4924 0.665557
\(773\) 15.5616 0.559710 0.279855 0.960042i \(-0.409714\pi\)
0.279855 + 0.960042i \(0.409714\pi\)
\(774\) −7.80776 −0.280644
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) −8.00000 −0.286998
\(778\) −24.2462 −0.869269
\(779\) −5.26137 −0.188508
\(780\) 0 0
\(781\) −7.31534 −0.261764
\(782\) −28.4924 −1.01889
\(783\) −1.12311 −0.0401365
\(784\) −4.56155 −0.162913
\(785\) 0 0
\(786\) −9.36932 −0.334192
\(787\) −10.9309 −0.389643 −0.194822 0.980839i \(-0.562413\pi\)
−0.194822 + 0.980839i \(0.562413\pi\)
\(788\) −6.00000 −0.213741
\(789\) −12.4924 −0.444742
\(790\) 0 0
\(791\) −16.6847 −0.593238
\(792\) 1.56155 0.0554874
\(793\) −12.0000 −0.426132
\(794\) −28.0540 −0.995598
\(795\) 0 0
\(796\) 3.80776 0.134963
\(797\) −38.9848 −1.38091 −0.690457 0.723373i \(-0.742591\pi\)
−0.690457 + 0.723373i \(0.742591\pi\)
\(798\) −7.31534 −0.258960
\(799\) −16.0000 −0.566039
\(800\) 0 0
\(801\) −1.31534 −0.0464753
\(802\) 1.31534 0.0464463
\(803\) 15.3153 0.540467
\(804\) −9.36932 −0.330430
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 16.2462 0.571894
\(808\) −3.56155 −0.125295
\(809\) −25.3153 −0.890040 −0.445020 0.895521i \(-0.646803\pi\)
−0.445020 + 0.895521i \(0.646803\pi\)
\(810\) 0 0
\(811\) 53.6695 1.88459 0.942296 0.334782i \(-0.108663\pi\)
0.942296 + 0.334782i \(0.108663\pi\)
\(812\) −1.75379 −0.0615459
\(813\) −10.0540 −0.352608
\(814\) −8.00000 −0.280400
\(815\) 0 0
\(816\) −5.12311 −0.179345
\(817\) 36.5767 1.27966
\(818\) 12.6307 0.441621
\(819\) −3.12311 −0.109130
\(820\) 0 0
\(821\) −21.6155 −0.754387 −0.377194 0.926134i \(-0.623111\pi\)
−0.377194 + 0.926134i \(0.623111\pi\)
\(822\) 3.75379 0.130928
\(823\) −15.6155 −0.544323 −0.272162 0.962252i \(-0.587739\pi\)
−0.272162 + 0.962252i \(0.587739\pi\)
\(824\) −18.2462 −0.635637
\(825\) 0 0
\(826\) 7.61553 0.264978
\(827\) −18.2462 −0.634483 −0.317241 0.948345i \(-0.602757\pi\)
−0.317241 + 0.948345i \(0.602757\pi\)
\(828\) −5.56155 −0.193277
\(829\) −37.4233 −1.29976 −0.649882 0.760035i \(-0.725182\pi\)
−0.649882 + 0.760035i \(0.725182\pi\)
\(830\) 0 0
\(831\) −19.3693 −0.671914
\(832\) −2.00000 −0.0693375
\(833\) 23.3693 0.809699
\(834\) 8.87689 0.307382
\(835\) 0 0
\(836\) −7.31534 −0.253006
\(837\) 1.00000 0.0345651
\(838\) −22.2462 −0.768483
\(839\) 34.9309 1.20595 0.602974 0.797761i \(-0.293982\pi\)
0.602974 + 0.797761i \(0.293982\pi\)
\(840\) 0 0
\(841\) −27.7386 −0.956505
\(842\) −18.4924 −0.637291
\(843\) 13.1231 0.451984
\(844\) −11.3153 −0.389490
\(845\) 0 0
\(846\) −3.12311 −0.107375
\(847\) −13.3693 −0.459375
\(848\) −11.5616 −0.397025
\(849\) 0 0
\(850\) 0 0
\(851\) 28.4924 0.976708
\(852\) 4.68466 0.160494
\(853\) 38.7926 1.32823 0.664117 0.747629i \(-0.268808\pi\)
0.664117 + 0.747629i \(0.268808\pi\)
\(854\) −9.36932 −0.320611
\(855\) 0 0
\(856\) 1.56155 0.0533728
\(857\) 53.2311 1.81834 0.909169 0.416427i \(-0.136718\pi\)
0.909169 + 0.416427i \(0.136718\pi\)
\(858\) −3.12311 −0.106621
\(859\) −22.7386 −0.775832 −0.387916 0.921695i \(-0.626805\pi\)
−0.387916 + 0.921695i \(0.626805\pi\)
\(860\) 0 0
\(861\) −1.75379 −0.0597690
\(862\) 13.7538 0.468456
\(863\) 30.5464 1.03981 0.519906 0.854224i \(-0.325967\pi\)
0.519906 + 0.854224i \(0.325967\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −18.3002 −0.621866
\(867\) 9.24621 0.314018
\(868\) 1.56155 0.0530026
\(869\) 26.0540 0.883821
\(870\) 0 0
\(871\) 18.7386 0.634934
\(872\) 19.3693 0.655928
\(873\) 6.00000 0.203069
\(874\) 26.0540 0.881289
\(875\) 0 0
\(876\) −9.80776 −0.331374
\(877\) 50.0000 1.68838 0.844190 0.536044i \(-0.180082\pi\)
0.844190 + 0.536044i \(0.180082\pi\)
\(878\) 31.6155 1.06697
\(879\) 6.49242 0.218984
\(880\) 0 0
\(881\) 42.4924 1.43161 0.715803 0.698302i \(-0.246061\pi\)
0.715803 + 0.698302i \(0.246061\pi\)
\(882\) 4.56155 0.153595
\(883\) 31.4233 1.05748 0.528739 0.848784i \(-0.322665\pi\)
0.528739 + 0.848784i \(0.322665\pi\)
\(884\) 10.2462 0.344617
\(885\) 0 0
\(886\) −14.0540 −0.472153
\(887\) 57.3693 1.92627 0.963137 0.269013i \(-0.0866974\pi\)
0.963137 + 0.269013i \(0.0866974\pi\)
\(888\) 5.12311 0.171920
\(889\) 0 0
\(890\) 0 0
\(891\) −1.56155 −0.0523140
\(892\) 12.8769 0.431150
\(893\) 14.6307 0.489597
\(894\) 20.0540 0.670705
\(895\) 0 0
\(896\) −1.56155 −0.0521678
\(897\) 11.1231 0.371390
\(898\) 22.4924 0.750582
\(899\) −1.12311 −0.0374577
\(900\) 0 0
\(901\) 59.2311 1.97327
\(902\) −1.75379 −0.0583948
\(903\) 12.1922 0.405732
\(904\) 10.6847 0.355366
\(905\) 0 0
\(906\) 0 0
\(907\) −24.0000 −0.796907 −0.398453 0.917189i \(-0.630453\pi\)
−0.398453 + 0.917189i \(0.630453\pi\)
\(908\) 4.68466 0.155466
\(909\) 3.56155 0.118129
\(910\) 0 0
\(911\) −44.4924 −1.47410 −0.737050 0.675838i \(-0.763782\pi\)
−0.737050 + 0.675838i \(0.763782\pi\)
\(912\) 4.68466 0.155125
\(913\) −3.50758 −0.116084
\(914\) 24.7386 0.818281
\(915\) 0 0
\(916\) −12.4384 −0.410978
\(917\) 14.6307 0.483148
\(918\) 5.12311 0.169088
\(919\) 39.6155 1.30680 0.653398 0.757015i \(-0.273343\pi\)
0.653398 + 0.757015i \(0.273343\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 15.3693 0.506161
\(923\) −9.36932 −0.308395
\(924\) −2.43845 −0.0802190
\(925\) 0 0
\(926\) −18.7386 −0.615790
\(927\) 18.2462 0.599284
\(928\) 1.12311 0.0368677
\(929\) 11.5616 0.379322 0.189661 0.981850i \(-0.439261\pi\)
0.189661 + 0.981850i \(0.439261\pi\)
\(930\) 0 0
\(931\) −21.3693 −0.700351
\(932\) −20.0540 −0.656890
\(933\) −18.2462 −0.597354
\(934\) −26.2462 −0.858802
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) −16.6307 −0.543301 −0.271650 0.962396i \(-0.587569\pi\)
−0.271650 + 0.962396i \(0.587569\pi\)
\(938\) 14.6307 0.477709
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 27.3693 0.892214 0.446107 0.894980i \(-0.352810\pi\)
0.446107 + 0.894980i \(0.352810\pi\)
\(942\) −20.0540 −0.653394
\(943\) 6.24621 0.203405
\(944\) −4.87689 −0.158729
\(945\) 0 0
\(946\) 12.1922 0.396404
\(947\) 0.876894 0.0284952 0.0142476 0.999898i \(-0.495465\pi\)
0.0142476 + 0.999898i \(0.495465\pi\)
\(948\) −16.6847 −0.541893
\(949\) 19.6155 0.636747
\(950\) 0 0
\(951\) −25.1231 −0.814673
\(952\) 8.00000 0.259281
\(953\) 58.1080 1.88230 0.941151 0.337988i \(-0.109746\pi\)
0.941151 + 0.337988i \(0.109746\pi\)
\(954\) 11.5616 0.374319
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) 1.75379 0.0566919
\(958\) −6.43845 −0.208017
\(959\) −5.86174 −0.189285
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −10.2462 −0.330351
\(963\) −1.56155 −0.0503203
\(964\) −4.24621 −0.136761
\(965\) 0 0
\(966\) 8.68466 0.279424
\(967\) 25.7538 0.828186 0.414093 0.910235i \(-0.364099\pi\)
0.414093 + 0.910235i \(0.364099\pi\)
\(968\) 8.56155 0.275179
\(969\) −24.0000 −0.770991
\(970\) 0 0
\(971\) −28.8769 −0.926704 −0.463352 0.886174i \(-0.653353\pi\)
−0.463352 + 0.886174i \(0.653353\pi\)
\(972\) 1.00000 0.0320750
\(973\) −13.8617 −0.444387
\(974\) 24.0000 0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) 22.9848 0.735350 0.367675 0.929954i \(-0.380154\pi\)
0.367675 + 0.929954i \(0.380154\pi\)
\(978\) 9.36932 0.299598
\(979\) 2.05398 0.0656453
\(980\) 0 0
\(981\) −19.3693 −0.618415
\(982\) −2.93087 −0.0935278
\(983\) 24.9848 0.796893 0.398446 0.917192i \(-0.369549\pi\)
0.398446 + 0.917192i \(0.369549\pi\)
\(984\) 1.12311 0.0358033
\(985\) 0 0
\(986\) −5.75379 −0.183238
\(987\) 4.87689 0.155233
\(988\) −9.36932 −0.298078
\(989\) −43.4233 −1.38078
\(990\) 0 0
\(991\) 23.3153 0.740636 0.370318 0.928905i \(-0.379249\pi\)
0.370318 + 0.928905i \(0.379249\pi\)
\(992\) −1.00000 −0.0317500
\(993\) 10.2462 0.325154
\(994\) −7.31534 −0.232029
\(995\) 0 0
\(996\) 2.24621 0.0711739
\(997\) 62.4924 1.97915 0.989577 0.144002i \(-0.0459971\pi\)
0.989577 + 0.144002i \(0.0459971\pi\)
\(998\) −20.0000 −0.633089
\(999\) −5.12311 −0.162088
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 4650.2.a.ce.1.2 2
5.2 odd 4 4650.2.d.bd.3349.2 4
5.3 odd 4 4650.2.d.bd.3349.3 4
5.4 even 2 930.2.a.p.1.1 2
15.14 odd 2 2790.2.a.be.1.1 2
20.19 odd 2 7440.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 2 5.4 even 2
2790.2.a.be.1.1 2 15.14 odd 2
4650.2.a.ce.1.2 2 1.1 even 1 trivial
4650.2.d.bd.3349.2 4 5.2 odd 4
4650.2.d.bd.3349.3 4 5.3 odd 4
7440.2.a.bl.1.2 2 20.19 odd 2