Properties

Label 4650.2.d.bd.3349.3
Level $4650$
Weight $2$
Character 4650.3349
Analytic conductor $37.130$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(37.1304369399\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) 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)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3349.3
Root \(-1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 4650.3349
Dual form 4650.2.d.bd.3349.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} -1.00000 q^{6} -1.56155i q^{7} -1.00000i q^{8} -1.00000 q^{9} -1.56155 q^{11} -1.00000i q^{12} -2.00000i q^{13} +1.56155 q^{14} +1.00000 q^{16} +5.12311i q^{17} -1.00000i q^{18} -4.68466 q^{19} +1.56155 q^{21} -1.56155i q^{22} -5.56155i q^{23} +1.00000 q^{24} +2.00000 q^{26} -1.00000i q^{27} +1.56155i q^{28} +1.12311 q^{29} +1.00000 q^{31} +1.00000i q^{32} -1.56155i q^{33} -5.12311 q^{34} +1.00000 q^{36} +5.12311i q^{37} -4.68466i q^{38} +2.00000 q^{39} -1.12311 q^{41} +1.56155i q^{42} +7.80776i q^{43} +1.56155 q^{44} +5.56155 q^{46} -3.12311i q^{47} +1.00000i q^{48} +4.56155 q^{49} -5.12311 q^{51} +2.00000i q^{52} -11.5616i q^{53} +1.00000 q^{54} -1.56155 q^{56} -4.68466i q^{57} +1.12311i q^{58} +4.87689 q^{59} +6.00000 q^{61} +1.00000i q^{62} +1.56155i q^{63} -1.00000 q^{64} +1.56155 q^{66} +9.36932i q^{67} -5.12311i q^{68} +5.56155 q^{69} +4.68466 q^{71} +1.00000i q^{72} -9.80776i q^{73} -5.12311 q^{74} +4.68466 q^{76} +2.43845i q^{77} +2.00000i q^{78} +16.6847 q^{79} +1.00000 q^{81} -1.12311i q^{82} +2.24621i q^{83} -1.56155 q^{84} -7.80776 q^{86} +1.12311i q^{87} +1.56155i q^{88} +1.31534 q^{89} -3.12311 q^{91} +5.56155i q^{92} +1.00000i q^{93} +3.12311 q^{94} -1.00000 q^{96} -6.00000i q^{97} +4.56155i q^{98} +1.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 4 q^{6} - 4 q^{9} + 2 q^{11} - 2 q^{14} + 4 q^{16} + 6 q^{19} - 2 q^{21} + 4 q^{24} + 8 q^{26} - 12 q^{29} + 4 q^{31} - 4 q^{34} + 4 q^{36} + 8 q^{39} + 12 q^{41} - 2 q^{44} + 14 q^{46} + 10 q^{49} - 4 q^{51} + 4 q^{54} + 2 q^{56} + 36 q^{59} + 24 q^{61} - 4 q^{64} - 2 q^{66} + 14 q^{69} - 6 q^{71} - 4 q^{74} - 6 q^{76} + 42 q^{79} + 4 q^{81} + 2 q^{84} + 10 q^{86} + 30 q^{89} + 4 q^{91} - 4 q^{94} - 4 q^{96} - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4650\mathbb{Z}\right)^\times\).

\(n\) \(1801\) \(2977\) \(3101\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) − 1.56155i − 0.590211i −0.955465 0.295106i \(-0.904645\pi\)
0.955465 0.295106i \(-0.0953549\pi\)
\(8\) − 1.00000i − 0.353553i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.56155 −0.470826 −0.235413 0.971895i \(-0.575644\pi\)
−0.235413 + 0.971895i \(0.575644\pi\)
\(12\) − 1.00000i − 0.288675i
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 5.12311i 1.24254i 0.783598 + 0.621268i \(0.213382\pi\)
−0.783598 + 0.621268i \(0.786618\pi\)
\(18\) − 1.00000i − 0.235702i
\(19\) −4.68466 −1.07473 −0.537367 0.843348i \(-0.680581\pi\)
−0.537367 + 0.843348i \(0.680581\pi\)
\(20\) 0 0
\(21\) 1.56155 0.340759
\(22\) − 1.56155i − 0.332924i
\(23\) − 5.56155i − 1.15966i −0.814736 0.579832i \(-0.803118\pi\)
0.814736 0.579832i \(-0.196882\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) − 1.00000i − 0.192450i
\(28\) 1.56155i 0.295106i
\(29\) 1.12311 0.208555 0.104278 0.994548i \(-0.466747\pi\)
0.104278 + 0.994548i \(0.466747\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605
\(32\) 1.00000i 0.176777i
\(33\) − 1.56155i − 0.271831i
\(34\) −5.12311 −0.878605
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 5.12311i 0.842233i 0.907006 + 0.421117i \(0.138362\pi\)
−0.907006 + 0.421117i \(0.861638\pi\)
\(38\) − 4.68466i − 0.759952i
\(39\) 2.00000 0.320256
\(40\) 0 0
\(41\) −1.12311 −0.175400 −0.0876998 0.996147i \(-0.527952\pi\)
−0.0876998 + 0.996147i \(0.527952\pi\)
\(42\) 1.56155i 0.240953i
\(43\) 7.80776i 1.19067i 0.803477 + 0.595336i \(0.202981\pi\)
−0.803477 + 0.595336i \(0.797019\pi\)
\(44\) 1.56155 0.235413
\(45\) 0 0
\(46\) 5.56155 0.820006
\(47\) − 3.12311i − 0.455552i −0.973714 0.227776i \(-0.926855\pi\)
0.973714 0.227776i \(-0.0731454\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 4.56155 0.651650
\(50\) 0 0
\(51\) −5.12311 −0.717378
\(52\) 2.00000i 0.277350i
\(53\) − 11.5616i − 1.58810i −0.607852 0.794051i \(-0.707968\pi\)
0.607852 0.794051i \(-0.292032\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −1.56155 −0.208671
\(57\) − 4.68466i − 0.620498i
\(58\) 1.12311i 0.147471i
\(59\) 4.87689 0.634918 0.317459 0.948272i \(-0.397170\pi\)
0.317459 + 0.948272i \(0.397170\pi\)
\(60\) 0 0
\(61\) 6.00000 0.768221 0.384111 0.923287i \(-0.374508\pi\)
0.384111 + 0.923287i \(0.374508\pi\)
\(62\) 1.00000i 0.127000i
\(63\) 1.56155i 0.196737i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 1.56155 0.192214
\(67\) 9.36932i 1.14464i 0.820029 + 0.572322i \(0.193957\pi\)
−0.820029 + 0.572322i \(0.806043\pi\)
\(68\) − 5.12311i − 0.621268i
\(69\) 5.56155 0.669532
\(70\) 0 0
\(71\) 4.68466 0.555967 0.277983 0.960586i \(-0.410334\pi\)
0.277983 + 0.960586i \(0.410334\pi\)
\(72\) 1.00000i 0.117851i
\(73\) − 9.80776i − 1.14791i −0.818886 0.573956i \(-0.805408\pi\)
0.818886 0.573956i \(-0.194592\pi\)
\(74\) −5.12311 −0.595549
\(75\) 0 0
\(76\) 4.68466 0.537367
\(77\) 2.43845i 0.277887i
\(78\) 2.00000i 0.226455i
\(79\) 16.6847 1.87717 0.938585 0.345047i \(-0.112137\pi\)
0.938585 + 0.345047i \(0.112137\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) − 1.12311i − 0.124026i
\(83\) 2.24621i 0.246554i 0.992372 + 0.123277i \(0.0393403\pi\)
−0.992372 + 0.123277i \(0.960660\pi\)
\(84\) −1.56155 −0.170379
\(85\) 0 0
\(86\) −7.80776 −0.841933
\(87\) 1.12311i 0.120410i
\(88\) 1.56155i 0.166462i
\(89\) 1.31534 0.139426 0.0697130 0.997567i \(-0.477792\pi\)
0.0697130 + 0.997567i \(0.477792\pi\)
\(90\) 0 0
\(91\) −3.12311 −0.327390
\(92\) 5.56155i 0.579832i
\(93\) 1.00000i 0.103695i
\(94\) 3.12311 0.322124
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) − 6.00000i − 0.609208i −0.952479 0.304604i \(-0.901476\pi\)
0.952479 0.304604i \(-0.0985241\pi\)
\(98\) 4.56155i 0.460786i
\(99\) 1.56155 0.156942
\(100\) 0 0
\(101\) 3.56155 0.354388 0.177194 0.984176i \(-0.443298\pi\)
0.177194 + 0.984176i \(0.443298\pi\)
\(102\) − 5.12311i − 0.507263i
\(103\) 18.2462i 1.79785i 0.438100 + 0.898926i \(0.355652\pi\)
−0.438100 + 0.898926i \(0.644348\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) 11.5616 1.12296
\(107\) 1.56155i 0.150961i 0.997147 + 0.0754805i \(0.0240491\pi\)
−0.997147 + 0.0754805i \(0.975951\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 19.3693 1.85524 0.927622 0.373520i \(-0.121849\pi\)
0.927622 + 0.373520i \(0.121849\pi\)
\(110\) 0 0
\(111\) −5.12311 −0.486264
\(112\) − 1.56155i − 0.147553i
\(113\) − 10.6847i − 1.00513i −0.864540 0.502564i \(-0.832390\pi\)
0.864540 0.502564i \(-0.167610\pi\)
\(114\) 4.68466 0.438758
\(115\) 0 0
\(116\) −1.12311 −0.104278
\(117\) 2.00000i 0.184900i
\(118\) 4.87689i 0.448955i
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −8.56155 −0.778323
\(122\) 6.00000i 0.543214i
\(123\) − 1.12311i − 0.101267i
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −1.56155 −0.139114
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 1.00000i − 0.0883883i
\(129\) −7.80776 −0.687435
\(130\) 0 0
\(131\) 9.36932 0.818601 0.409301 0.912400i \(-0.365773\pi\)
0.409301 + 0.912400i \(0.365773\pi\)
\(132\) 1.56155i 0.135916i
\(133\) 7.31534i 0.634321i
\(134\) −9.36932 −0.809386
\(135\) 0 0
\(136\) 5.12311 0.439303
\(137\) 3.75379i 0.320708i 0.987060 + 0.160354i \(0.0512635\pi\)
−0.987060 + 0.160354i \(0.948736\pi\)
\(138\) 5.56155i 0.473431i
\(139\) 8.87689 0.752928 0.376464 0.926431i \(-0.377140\pi\)
0.376464 + 0.926431i \(0.377140\pi\)
\(140\) 0 0
\(141\) 3.12311 0.263013
\(142\) 4.68466i 0.393128i
\(143\) 3.12311i 0.261167i
\(144\) −1.00000 −0.0833333
\(145\) 0 0
\(146\) 9.80776 0.811696
\(147\) 4.56155i 0.376231i
\(148\) − 5.12311i − 0.421117i
\(149\) 20.0540 1.64289 0.821443 0.570291i \(-0.193170\pi\)
0.821443 + 0.570291i \(0.193170\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 4.68466i 0.379976i
\(153\) − 5.12311i − 0.414179i
\(154\) −2.43845 −0.196496
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) − 20.0540i − 1.60048i −0.599679 0.800241i \(-0.704705\pi\)
0.599679 0.800241i \(-0.295295\pi\)
\(158\) 16.6847i 1.32736i
\(159\) 11.5616 0.916891
\(160\) 0 0
\(161\) −8.68466 −0.684447
\(162\) 1.00000i 0.0785674i
\(163\) − 9.36932i − 0.733862i −0.930248 0.366931i \(-0.880409\pi\)
0.930248 0.366931i \(-0.119591\pi\)
\(164\) 1.12311 0.0876998
\(165\) 0 0
\(166\) −2.24621 −0.174340
\(167\) − 2.43845i − 0.188693i −0.995539 0.0943464i \(-0.969924\pi\)
0.995539 0.0943464i \(-0.0300761\pi\)
\(168\) − 1.56155i − 0.120476i
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 4.68466 0.358245
\(172\) − 7.80776i − 0.595336i
\(173\) 8.24621i 0.626948i 0.949597 + 0.313474i \(0.101493\pi\)
−0.949597 + 0.313474i \(0.898507\pi\)
\(174\) −1.12311 −0.0851424
\(175\) 0 0
\(176\) −1.56155 −0.117706
\(177\) 4.87689i 0.366570i
\(178\) 1.31534i 0.0985890i
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 24.0540 1.78792 0.893959 0.448149i \(-0.147917\pi\)
0.893959 + 0.448149i \(0.147917\pi\)
\(182\) − 3.12311i − 0.231500i
\(183\) 6.00000i 0.443533i
\(184\) −5.56155 −0.410003
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) − 8.00000i − 0.585018i
\(188\) 3.12311i 0.227776i
\(189\) −1.56155 −0.113586
\(190\) 0 0
\(191\) 2.24621 0.162530 0.0812651 0.996693i \(-0.474104\pi\)
0.0812651 + 0.996693i \(0.474104\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 18.4924i 1.33111i 0.746347 + 0.665557i \(0.231806\pi\)
−0.746347 + 0.665557i \(0.768194\pi\)
\(194\) 6.00000 0.430775
\(195\) 0 0
\(196\) −4.56155 −0.325825
\(197\) 6.00000i 0.427482i 0.976890 + 0.213741i \(0.0685649\pi\)
−0.976890 + 0.213741i \(0.931435\pi\)
\(198\) 1.56155i 0.110975i
\(199\) −3.80776 −0.269925 −0.134963 0.990851i \(-0.543091\pi\)
−0.134963 + 0.990851i \(0.543091\pi\)
\(200\) 0 0
\(201\) −9.36932 −0.660861
\(202\) 3.56155i 0.250590i
\(203\) − 1.75379i − 0.123092i
\(204\) 5.12311 0.358689
\(205\) 0 0
\(206\) −18.2462 −1.27127
\(207\) 5.56155i 0.386555i
\(208\) − 2.00000i − 0.138675i
\(209\) 7.31534 0.506013
\(210\) 0 0
\(211\) −11.3153 −0.778980 −0.389490 0.921031i \(-0.627349\pi\)
−0.389490 + 0.921031i \(0.627349\pi\)
\(212\) 11.5616i 0.794051i
\(213\) 4.68466i 0.320988i
\(214\) −1.56155 −0.106746
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) − 1.56155i − 0.106005i
\(218\) 19.3693i 1.31186i
\(219\) 9.80776 0.662747
\(220\) 0 0
\(221\) 10.2462 0.689235
\(222\) − 5.12311i − 0.343840i
\(223\) 12.8769i 0.862301i 0.902280 + 0.431150i \(0.141892\pi\)
−0.902280 + 0.431150i \(0.858108\pi\)
\(224\) 1.56155 0.104336
\(225\) 0 0
\(226\) 10.6847 0.710733
\(227\) − 4.68466i − 0.310932i −0.987841 0.155466i \(-0.950312\pi\)
0.987841 0.155466i \(-0.0496879\pi\)
\(228\) 4.68466i 0.310249i
\(229\) 12.4384 0.821956 0.410978 0.911645i \(-0.365187\pi\)
0.410978 + 0.911645i \(0.365187\pi\)
\(230\) 0 0
\(231\) −2.43845 −0.160438
\(232\) − 1.12311i − 0.0737355i
\(233\) − 20.0540i − 1.31378i −0.753987 0.656890i \(-0.771872\pi\)
0.753987 0.656890i \(-0.228128\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −4.87689 −0.317459
\(237\) 16.6847i 1.08379i
\(238\) 8.00000i 0.518563i
\(239\) −24.0000 −1.55243 −0.776215 0.630468i \(-0.782863\pi\)
−0.776215 + 0.630468i \(0.782863\pi\)
\(240\) 0 0
\(241\) −4.24621 −0.273523 −0.136761 0.990604i \(-0.543669\pi\)
−0.136761 + 0.990604i \(0.543669\pi\)
\(242\) − 8.56155i − 0.550357i
\(243\) 1.00000i 0.0641500i
\(244\) −6.00000 −0.384111
\(245\) 0 0
\(246\) 1.12311 0.0716066
\(247\) 9.36932i 0.596155i
\(248\) − 1.00000i − 0.0635001i
\(249\) −2.24621 −0.142348
\(250\) 0 0
\(251\) −16.4924 −1.04099 −0.520496 0.853864i \(-0.674253\pi\)
−0.520496 + 0.853864i \(0.674253\pi\)
\(252\) − 1.56155i − 0.0983686i
\(253\) 8.68466i 0.546000i
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 10.6847i 0.666491i 0.942840 + 0.333245i \(0.108144\pi\)
−0.942840 + 0.333245i \(0.891856\pi\)
\(258\) − 7.80776i − 0.486090i
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) −1.12311 −0.0695185
\(262\) 9.36932i 0.578838i
\(263\) − 12.4924i − 0.770316i −0.922851 0.385158i \(-0.874147\pi\)
0.922851 0.385158i \(-0.125853\pi\)
\(264\) −1.56155 −0.0961069
\(265\) 0 0
\(266\) −7.31534 −0.448532
\(267\) 1.31534i 0.0804976i
\(268\) − 9.36932i − 0.572322i
\(269\) −16.2462 −0.990549 −0.495274 0.868737i \(-0.664933\pi\)
−0.495274 + 0.868737i \(0.664933\pi\)
\(270\) 0 0
\(271\) −10.0540 −0.610736 −0.305368 0.952234i \(-0.598779\pi\)
−0.305368 + 0.952234i \(0.598779\pi\)
\(272\) 5.12311i 0.310634i
\(273\) − 3.12311i − 0.189019i
\(274\) −3.75379 −0.226775
\(275\) 0 0
\(276\) −5.56155 −0.334766
\(277\) 19.3693i 1.16379i 0.813264 + 0.581895i \(0.197688\pi\)
−0.813264 + 0.581895i \(0.802312\pi\)
\(278\) 8.87689i 0.532401i
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 13.1231 0.782859 0.391429 0.920208i \(-0.371981\pi\)
0.391429 + 0.920208i \(0.371981\pi\)
\(282\) 3.12311i 0.185978i
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) −4.68466 −0.277983
\(285\) 0 0
\(286\) −3.12311 −0.184673
\(287\) 1.75379i 0.103523i
\(288\) − 1.00000i − 0.0589256i
\(289\) −9.24621 −0.543895
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) 9.80776i 0.573956i
\(293\) 6.49242i 0.379291i 0.981853 + 0.189646i \(0.0607339\pi\)
−0.981853 + 0.189646i \(0.939266\pi\)
\(294\) −4.56155 −0.266035
\(295\) 0 0
\(296\) 5.12311 0.297774
\(297\) 1.56155i 0.0906105i
\(298\) 20.0540i 1.16170i
\(299\) −11.1231 −0.643266
\(300\) 0 0
\(301\) 12.1922 0.702749
\(302\) 0 0
\(303\) 3.56155i 0.204606i
\(304\) −4.68466 −0.268684
\(305\) 0 0
\(306\) 5.12311 0.292868
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) − 2.43845i − 0.138943i
\(309\) −18.2462 −1.03799
\(310\) 0 0
\(311\) −18.2462 −1.03465 −0.517324 0.855790i \(-0.673072\pi\)
−0.517324 + 0.855790i \(0.673072\pi\)
\(312\) − 2.00000i − 0.113228i
\(313\) 10.0000i 0.565233i 0.959233 + 0.282617i \(0.0912024\pi\)
−0.959233 + 0.282617i \(0.908798\pi\)
\(314\) 20.0540 1.13171
\(315\) 0 0
\(316\) −16.6847 −0.938585
\(317\) 25.1231i 1.41105i 0.708683 + 0.705527i \(0.249290\pi\)
−0.708683 + 0.705527i \(0.750710\pi\)
\(318\) 11.5616i 0.648340i
\(319\) −1.75379 −0.0981933
\(320\) 0 0
\(321\) −1.56155 −0.0871574
\(322\) − 8.68466i − 0.483977i
\(323\) − 24.0000i − 1.33540i
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) 9.36932 0.518918
\(327\) 19.3693i 1.07113i
\(328\) 1.12311i 0.0620131i
\(329\) −4.87689 −0.268872
\(330\) 0 0
\(331\) 10.2462 0.563183 0.281591 0.959534i \(-0.409138\pi\)
0.281591 + 0.959534i \(0.409138\pi\)
\(332\) − 2.24621i − 0.123277i
\(333\) − 5.12311i − 0.280744i
\(334\) 2.43845 0.133426
\(335\) 0 0
\(336\) 1.56155 0.0851897
\(337\) − 10.0000i − 0.544735i −0.962193 0.272367i \(-0.912193\pi\)
0.962193 0.272367i \(-0.0878066\pi\)
\(338\) 9.00000i 0.489535i
\(339\) 10.6847 0.580311
\(340\) 0 0
\(341\) −1.56155 −0.0845628
\(342\) 4.68466i 0.253317i
\(343\) − 18.0540i − 0.974823i
\(344\) 7.80776 0.420966
\(345\) 0 0
\(346\) −8.24621 −0.443319
\(347\) − 2.24621i − 0.120583i −0.998181 0.0602915i \(-0.980797\pi\)
0.998181 0.0602915i \(-0.0192030\pi\)
\(348\) − 1.12311i − 0.0602048i
\(349\) 19.3693 1.03682 0.518408 0.855133i \(-0.326525\pi\)
0.518408 + 0.855133i \(0.326525\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) − 1.56155i − 0.0832310i
\(353\) − 16.2462i − 0.864699i −0.901706 0.432349i \(-0.857685\pi\)
0.901706 0.432349i \(-0.142315\pi\)
\(354\) −4.87689 −0.259204
\(355\) 0 0
\(356\) −1.31534 −0.0697130
\(357\) 8.00000i 0.423405i
\(358\) 12.0000i 0.634220i
\(359\) 19.3153 1.01942 0.509712 0.860345i \(-0.329752\pi\)
0.509712 + 0.860345i \(0.329752\pi\)
\(360\) 0 0
\(361\) 2.94602 0.155054
\(362\) 24.0540i 1.26425i
\(363\) − 8.56155i − 0.449365i
\(364\) 3.12311 0.163695
\(365\) 0 0
\(366\) −6.00000 −0.313625
\(367\) − 9.36932i − 0.489074i −0.969640 0.244537i \(-0.921364\pi\)
0.969640 0.244537i \(-0.0786360\pi\)
\(368\) − 5.56155i − 0.289916i
\(369\) 1.12311 0.0584665
\(370\) 0 0
\(371\) −18.0540 −0.937316
\(372\) − 1.00000i − 0.0518476i
\(373\) − 6.68466i − 0.346118i −0.984911 0.173059i \(-0.944635\pi\)
0.984911 0.173059i \(-0.0553652\pi\)
\(374\) 8.00000 0.413670
\(375\) 0 0
\(376\) −3.12311 −0.161062
\(377\) − 2.24621i − 0.115686i
\(378\) − 1.56155i − 0.0803176i
\(379\) −30.0540 −1.54377 −0.771885 0.635763i \(-0.780686\pi\)
−0.771885 + 0.635763i \(0.780686\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.24621i 0.114926i
\(383\) − 6.24621i − 0.319166i −0.987184 0.159583i \(-0.948985\pi\)
0.987184 0.159583i \(-0.0510150\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −18.4924 −0.941240
\(387\) − 7.80776i − 0.396891i
\(388\) 6.00000i 0.304604i
\(389\) −24.2462 −1.22933 −0.614666 0.788788i \(-0.710709\pi\)
−0.614666 + 0.788788i \(0.710709\pi\)
\(390\) 0 0
\(391\) 28.4924 1.44092
\(392\) − 4.56155i − 0.230393i
\(393\) 9.36932i 0.472620i
\(394\) −6.00000 −0.302276
\(395\) 0 0
\(396\) −1.56155 −0.0784710
\(397\) − 28.0540i − 1.40799i −0.710206 0.703994i \(-0.751398\pi\)
0.710206 0.703994i \(-0.248602\pi\)
\(398\) − 3.80776i − 0.190866i
\(399\) −7.31534 −0.366225
\(400\) 0 0
\(401\) −1.31534 −0.0656850 −0.0328425 0.999461i \(-0.510456\pi\)
−0.0328425 + 0.999461i \(0.510456\pi\)
\(402\) − 9.36932i − 0.467299i
\(403\) − 2.00000i − 0.0996271i
\(404\) −3.56155 −0.177194
\(405\) 0 0
\(406\) 1.75379 0.0870391
\(407\) − 8.00000i − 0.396545i
\(408\) 5.12311i 0.253632i
\(409\) 12.6307 0.624547 0.312274 0.949992i \(-0.398909\pi\)
0.312274 + 0.949992i \(0.398909\pi\)
\(410\) 0 0
\(411\) −3.75379 −0.185161
\(412\) − 18.2462i − 0.898926i
\(413\) − 7.61553i − 0.374736i
\(414\) −5.56155 −0.273335
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 8.87689i 0.434703i
\(418\) 7.31534i 0.357805i
\(419\) −22.2462 −1.08680 −0.543399 0.839474i \(-0.682863\pi\)
−0.543399 + 0.839474i \(0.682863\pi\)
\(420\) 0 0
\(421\) 18.4924 0.901266 0.450633 0.892709i \(-0.351198\pi\)
0.450633 + 0.892709i \(0.351198\pi\)
\(422\) − 11.3153i − 0.550822i
\(423\) 3.12311i 0.151851i
\(424\) −11.5616 −0.561479
\(425\) 0 0
\(426\) −4.68466 −0.226972
\(427\) − 9.36932i − 0.453413i
\(428\) − 1.56155i − 0.0754805i
\(429\) −3.12311 −0.150785
\(430\) 0 0
\(431\) −13.7538 −0.662497 −0.331248 0.943544i \(-0.607470\pi\)
−0.331248 + 0.943544i \(0.607470\pi\)
\(432\) − 1.00000i − 0.0481125i
\(433\) 18.3002i 0.879451i 0.898132 + 0.439725i \(0.144924\pi\)
−0.898132 + 0.439725i \(0.855076\pi\)
\(434\) 1.56155 0.0749569
\(435\) 0 0
\(436\) −19.3693 −0.927622
\(437\) 26.0540i 1.24633i
\(438\) 9.80776i 0.468633i
\(439\) 31.6155 1.50893 0.754463 0.656342i \(-0.227897\pi\)
0.754463 + 0.656342i \(0.227897\pi\)
\(440\) 0 0
\(441\) −4.56155 −0.217217
\(442\) 10.2462i 0.487363i
\(443\) 14.0540i 0.667725i 0.942622 + 0.333862i \(0.108352\pi\)
−0.942622 + 0.333862i \(0.891648\pi\)
\(444\) 5.12311 0.243132
\(445\) 0 0
\(446\) −12.8769 −0.609739
\(447\) 20.0540i 0.948520i
\(448\) 1.56155i 0.0737764i
\(449\) 22.4924 1.06148 0.530742 0.847534i \(-0.321913\pi\)
0.530742 + 0.847534i \(0.321913\pi\)
\(450\) 0 0
\(451\) 1.75379 0.0825827
\(452\) 10.6847i 0.502564i
\(453\) 0 0
\(454\) 4.68466 0.219862
\(455\) 0 0
\(456\) −4.68466 −0.219379
\(457\) 24.7386i 1.15722i 0.815603 + 0.578612i \(0.196406\pi\)
−0.815603 + 0.578612i \(0.803594\pi\)
\(458\) 12.4384i 0.581210i
\(459\) 5.12311 0.239126
\(460\) 0 0
\(461\) −15.3693 −0.715820 −0.357910 0.933756i \(-0.616511\pi\)
−0.357910 + 0.933756i \(0.616511\pi\)
\(462\) − 2.43845i − 0.113447i
\(463\) 18.7386i 0.870858i 0.900223 + 0.435429i \(0.143403\pi\)
−0.900223 + 0.435429i \(0.856597\pi\)
\(464\) 1.12311 0.0521389
\(465\) 0 0
\(466\) 20.0540 0.928982
\(467\) − 26.2462i − 1.21453i −0.794499 0.607265i \(-0.792267\pi\)
0.794499 0.607265i \(-0.207733\pi\)
\(468\) − 2.00000i − 0.0924500i
\(469\) 14.6307 0.675582
\(470\) 0 0
\(471\) 20.0540 0.924038
\(472\) − 4.87689i − 0.224477i
\(473\) − 12.1922i − 0.560600i
\(474\) −16.6847 −0.766352
\(475\) 0 0
\(476\) −8.00000 −0.366679
\(477\) 11.5616i 0.529367i
\(478\) − 24.0000i − 1.09773i
\(479\) −6.43845 −0.294180 −0.147090 0.989123i \(-0.546991\pi\)
−0.147090 + 0.989123i \(0.546991\pi\)
\(480\) 0 0
\(481\) 10.2462 0.467187
\(482\) − 4.24621i − 0.193410i
\(483\) − 8.68466i − 0.395166i
\(484\) 8.56155 0.389161
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) 24.0000i 1.08754i 0.839233 + 0.543772i \(0.183004\pi\)
−0.839233 + 0.543772i \(0.816996\pi\)
\(488\) − 6.00000i − 0.271607i
\(489\) 9.36932 0.423695
\(490\) 0 0
\(491\) 2.93087 0.132268 0.0661341 0.997811i \(-0.478933\pi\)
0.0661341 + 0.997811i \(0.478933\pi\)
\(492\) 1.12311i 0.0506335i
\(493\) 5.75379i 0.259138i
\(494\) −9.36932 −0.421545
\(495\) 0 0
\(496\) 1.00000 0.0449013
\(497\) − 7.31534i − 0.328138i
\(498\) − 2.24621i − 0.100655i
\(499\) −20.0000 −0.895323 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) 0 0
\(501\) 2.43845 0.108942
\(502\) − 16.4924i − 0.736093i
\(503\) − 9.75379i − 0.434900i −0.976071 0.217450i \(-0.930226\pi\)
0.976071 0.217450i \(-0.0697739\pi\)
\(504\) 1.56155 0.0695571
\(505\) 0 0
\(506\) −8.68466 −0.386080
\(507\) 9.00000i 0.399704i
\(508\) 0 0
\(509\) 21.6155 0.958091 0.479046 0.877790i \(-0.340983\pi\)
0.479046 + 0.877790i \(0.340983\pi\)
\(510\) 0 0
\(511\) −15.3153 −0.677511
\(512\) 1.00000i 0.0441942i
\(513\) 4.68466i 0.206833i
\(514\) −10.6847 −0.471280
\(515\) 0 0
\(516\) 7.80776 0.343718
\(517\) 4.87689i 0.214486i
\(518\) 8.00000i 0.351500i
\(519\) −8.24621 −0.361968
\(520\) 0 0
\(521\) −15.7538 −0.690186 −0.345093 0.938568i \(-0.612153\pi\)
−0.345093 + 0.938568i \(0.612153\pi\)
\(522\) − 1.12311i − 0.0491570i
\(523\) − 39.4233i − 1.72386i −0.507027 0.861930i \(-0.669256\pi\)
0.507027 0.861930i \(-0.330744\pi\)
\(524\) −9.36932 −0.409301
\(525\) 0 0
\(526\) 12.4924 0.544696
\(527\) 5.12311i 0.223166i
\(528\) − 1.56155i − 0.0679579i
\(529\) −7.93087 −0.344820
\(530\) 0 0
\(531\) −4.87689 −0.211639
\(532\) − 7.31534i − 0.317160i
\(533\) 2.24621i 0.0972942i
\(534\) −1.31534 −0.0569204
\(535\) 0 0
\(536\) 9.36932 0.404693
\(537\) 12.0000i 0.517838i
\(538\) − 16.2462i − 0.700424i
\(539\) −7.12311 −0.306814
\(540\) 0 0
\(541\) 36.2462 1.55835 0.779173 0.626809i \(-0.215639\pi\)
0.779173 + 0.626809i \(0.215639\pi\)
\(542\) − 10.0540i − 0.431855i
\(543\) 24.0540i 1.03225i
\(544\) −5.12311 −0.219651
\(545\) 0 0
\(546\) 3.12311 0.133657
\(547\) 35.1231i 1.50176i 0.660441 + 0.750878i \(0.270369\pi\)
−0.660441 + 0.750878i \(0.729631\pi\)
\(548\) − 3.75379i − 0.160354i
\(549\) −6.00000 −0.256074
\(550\) 0 0
\(551\) −5.26137 −0.224142
\(552\) − 5.56155i − 0.236715i
\(553\) − 26.0540i − 1.10793i
\(554\) −19.3693 −0.822923
\(555\) 0 0
\(556\) −8.87689 −0.376464
\(557\) − 6.19224i − 0.262373i −0.991358 0.131187i \(-0.958121\pi\)
0.991358 0.131187i \(-0.0418787\pi\)
\(558\) − 1.00000i − 0.0423334i
\(559\) 15.6155 0.660466
\(560\) 0 0
\(561\) 8.00000 0.337760
\(562\) 13.1231i 0.553565i
\(563\) 16.4924i 0.695073i 0.937667 + 0.347536i \(0.112982\pi\)
−0.937667 + 0.347536i \(0.887018\pi\)
\(564\) −3.12311 −0.131506
\(565\) 0 0
\(566\) 0 0
\(567\) − 1.56155i − 0.0655791i
\(568\) − 4.68466i − 0.196564i
\(569\) −10.1922 −0.427281 −0.213640 0.976912i \(-0.568532\pi\)
−0.213640 + 0.976912i \(0.568532\pi\)
\(570\) 0 0
\(571\) 32.8769 1.37586 0.687928 0.725779i \(-0.258521\pi\)
0.687928 + 0.725779i \(0.258521\pi\)
\(572\) − 3.12311i − 0.130584i
\(573\) 2.24621i 0.0938368i
\(574\) −1.75379 −0.0732017
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 6.87689i 0.286289i 0.989702 + 0.143144i \(0.0457213\pi\)
−0.989702 + 0.143144i \(0.954279\pi\)
\(578\) − 9.24621i − 0.384592i
\(579\) −18.4924 −0.768519
\(580\) 0 0
\(581\) 3.50758 0.145519
\(582\) 6.00000i 0.248708i
\(583\) 18.0540i 0.747719i
\(584\) −9.80776 −0.405848
\(585\) 0 0
\(586\) −6.49242 −0.268200
\(587\) 32.4924i 1.34111i 0.741862 + 0.670553i \(0.233943\pi\)
−0.741862 + 0.670553i \(0.766057\pi\)
\(588\) − 4.56155i − 0.188115i
\(589\) −4.68466 −0.193028
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 5.12311i 0.210558i
\(593\) 30.0000i 1.23195i 0.787765 + 0.615976i \(0.211238\pi\)
−0.787765 + 0.615976i \(0.788762\pi\)
\(594\) −1.56155 −0.0640713
\(595\) 0 0
\(596\) −20.0540 −0.821443
\(597\) − 3.80776i − 0.155841i
\(598\) − 11.1231i − 0.454858i
\(599\) 23.4233 0.957050 0.478525 0.878074i \(-0.341172\pi\)
0.478525 + 0.878074i \(0.341172\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 12.1922i 0.496918i
\(603\) − 9.36932i − 0.381548i
\(604\) 0 0
\(605\) 0 0
\(606\) −3.56155 −0.144678
\(607\) − 30.0540i − 1.21985i −0.792458 0.609927i \(-0.791199\pi\)
0.792458 0.609927i \(-0.208801\pi\)
\(608\) − 4.68466i − 0.189988i
\(609\) 1.75379 0.0710671
\(610\) 0 0
\(611\) −6.24621 −0.252695
\(612\) 5.12311i 0.207089i
\(613\) − 2.00000i − 0.0807792i −0.999184 0.0403896i \(-0.987140\pi\)
0.999184 0.0403896i \(-0.0128599\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 2.43845 0.0982478
\(617\) − 19.5616i − 0.787518i −0.919214 0.393759i \(-0.871174\pi\)
0.919214 0.393759i \(-0.128826\pi\)
\(618\) − 18.2462i − 0.733970i
\(619\) 13.3693 0.537358 0.268679 0.963230i \(-0.413413\pi\)
0.268679 + 0.963230i \(0.413413\pi\)
\(620\) 0 0
\(621\) −5.56155 −0.223177
\(622\) − 18.2462i − 0.731606i
\(623\) − 2.05398i − 0.0822908i
\(624\) 2.00000 0.0800641
\(625\) 0 0
\(626\) −10.0000 −0.399680
\(627\) 7.31534i 0.292147i
\(628\) 20.0540i 0.800241i
\(629\) −26.2462 −1.04650
\(630\) 0 0
\(631\) −24.3002 −0.967375 −0.483688 0.875241i \(-0.660703\pi\)
−0.483688 + 0.875241i \(0.660703\pi\)
\(632\) − 16.6847i − 0.663680i
\(633\) − 11.3153i − 0.449744i
\(634\) −25.1231 −0.997766
\(635\) 0 0
\(636\) −11.5616 −0.458445
\(637\) − 9.12311i − 0.361471i
\(638\) − 1.75379i − 0.0694332i
\(639\) −4.68466 −0.185322
\(640\) 0 0
\(641\) −38.4924 −1.52036 −0.760180 0.649713i \(-0.774889\pi\)
−0.760180 + 0.649713i \(0.774889\pi\)
\(642\) − 1.56155i − 0.0616296i
\(643\) − 30.0540i − 1.18521i −0.805492 0.592607i \(-0.798099\pi\)
0.805492 0.592607i \(-0.201901\pi\)
\(644\) 8.68466 0.342223
\(645\) 0 0
\(646\) 24.0000 0.944267
\(647\) − 17.0691i − 0.671057i −0.942030 0.335528i \(-0.891085\pi\)
0.942030 0.335528i \(-0.108915\pi\)
\(648\) − 1.00000i − 0.0392837i
\(649\) −7.61553 −0.298936
\(650\) 0 0
\(651\) 1.56155 0.0612021
\(652\) 9.36932i 0.366931i
\(653\) − 23.7538i − 0.929558i −0.885427 0.464779i \(-0.846134\pi\)
0.885427 0.464779i \(-0.153866\pi\)
\(654\) −19.3693 −0.757400
\(655\) 0 0
\(656\) −1.12311 −0.0438499
\(657\) 9.80776i 0.382637i
\(658\) − 4.87689i − 0.190121i
\(659\) −18.7386 −0.729954 −0.364977 0.931017i \(-0.618923\pi\)
−0.364977 + 0.931017i \(0.618923\pi\)
\(660\) 0 0
\(661\) −6.49242 −0.252526 −0.126263 0.991997i \(-0.540298\pi\)
−0.126263 + 0.991997i \(0.540298\pi\)
\(662\) 10.2462i 0.398230i
\(663\) 10.2462i 0.397930i
\(664\) 2.24621 0.0871699
\(665\) 0 0
\(666\) 5.12311 0.198516
\(667\) − 6.24621i − 0.241854i
\(668\) 2.43845i 0.0943464i
\(669\) −12.8769 −0.497849
\(670\) 0 0
\(671\) −9.36932 −0.361698
\(672\) 1.56155i 0.0602382i
\(673\) − 28.2462i − 1.08881i −0.838822 0.544406i \(-0.816755\pi\)
0.838822 0.544406i \(-0.183245\pi\)
\(674\) 10.0000 0.385186
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) − 29.4233i − 1.13083i −0.824807 0.565414i \(-0.808716\pi\)
0.824807 0.565414i \(-0.191284\pi\)
\(678\) 10.6847i 0.410342i
\(679\) −9.36932 −0.359561
\(680\) 0 0
\(681\) 4.68466 0.179517
\(682\) − 1.56155i − 0.0597949i
\(683\) − 19.3153i − 0.739081i −0.929215 0.369541i \(-0.879515\pi\)
0.929215 0.369541i \(-0.120485\pi\)
\(684\) −4.68466 −0.179122
\(685\) 0 0
\(686\) 18.0540 0.689304
\(687\) 12.4384i 0.474556i
\(688\) 7.80776i 0.297668i
\(689\) −23.1231 −0.880920
\(690\) 0 0
\(691\) −22.0540 −0.838973 −0.419486 0.907762i \(-0.637790\pi\)
−0.419486 + 0.907762i \(0.637790\pi\)
\(692\) − 8.24621i − 0.313474i
\(693\) − 2.43845i − 0.0926289i
\(694\) 2.24621 0.0852650
\(695\) 0 0
\(696\) 1.12311 0.0425712
\(697\) − 5.75379i − 0.217940i
\(698\) 19.3693i 0.733139i
\(699\) 20.0540 0.758511
\(700\) 0 0
\(701\) 25.8078 0.974746 0.487373 0.873194i \(-0.337955\pi\)
0.487373 + 0.873194i \(0.337955\pi\)
\(702\) − 2.00000i − 0.0754851i
\(703\) − 24.0000i − 0.905177i
\(704\) 1.56155 0.0588532
\(705\) 0 0
\(706\) 16.2462 0.611434
\(707\) − 5.56155i − 0.209164i
\(708\) − 4.87689i − 0.183285i
\(709\) 20.0540 0.753143 0.376571 0.926388i \(-0.377103\pi\)
0.376571 + 0.926388i \(0.377103\pi\)
\(710\) 0 0
\(711\) −16.6847 −0.625724
\(712\) − 1.31534i − 0.0492945i
\(713\) − 5.56155i − 0.208282i
\(714\) −8.00000 −0.299392
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) − 24.0000i − 0.896296i
\(718\) 19.3153i 0.720842i
\(719\) 43.1231 1.60822 0.804110 0.594480i \(-0.202642\pi\)
0.804110 + 0.594480i \(0.202642\pi\)
\(720\) 0 0
\(721\) 28.4924 1.06111
\(722\) 2.94602i 0.109640i
\(723\) − 4.24621i − 0.157918i
\(724\) −24.0540 −0.893959
\(725\) 0 0
\(726\) 8.56155 0.317749
\(727\) − 14.0540i − 0.521233i −0.965442 0.260617i \(-0.916074\pi\)
0.965442 0.260617i \(-0.0839258\pi\)
\(728\) 3.12311i 0.115750i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) −40.0000 −1.47945
\(732\) − 6.00000i − 0.221766i
\(733\) − 26.4924i − 0.978520i −0.872138 0.489260i \(-0.837267\pi\)
0.872138 0.489260i \(-0.162733\pi\)
\(734\) 9.36932 0.345828
\(735\) 0 0
\(736\) 5.56155 0.205002
\(737\) − 14.6307i − 0.538928i
\(738\) 1.12311i 0.0413421i
\(739\) 12.9848 0.477655 0.238828 0.971062i \(-0.423237\pi\)
0.238828 + 0.971062i \(0.423237\pi\)
\(740\) 0 0
\(741\) −9.36932 −0.344190
\(742\) − 18.0540i − 0.662782i
\(743\) − 11.4233i − 0.419080i −0.977800 0.209540i \(-0.932803\pi\)
0.977800 0.209540i \(-0.0671966\pi\)
\(744\) 1.00000 0.0366618
\(745\) 0 0
\(746\) 6.68466 0.244743
\(747\) − 2.24621i − 0.0821846i
\(748\) 8.00000i 0.292509i
\(749\) 2.43845 0.0890989
\(750\) 0 0
\(751\) 36.8769 1.34566 0.672828 0.739798i \(-0.265079\pi\)
0.672828 + 0.739798i \(0.265079\pi\)
\(752\) − 3.12311i − 0.113888i
\(753\) − 16.4924i − 0.601017i
\(754\) 2.24621 0.0818022
\(755\) 0 0
\(756\) 1.56155 0.0567931
\(757\) 10.0000i 0.363456i 0.983349 + 0.181728i \(0.0581691\pi\)
−0.983349 + 0.181728i \(0.941831\pi\)
\(758\) − 30.0540i − 1.09161i
\(759\) −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
\(763\) − 30.2462i − 1.09499i
\(764\) −2.24621 −0.0812651
\(765\) 0 0
\(766\) 6.24621 0.225685
\(767\) − 9.75379i − 0.352189i
\(768\) 1.00000i 0.0360844i
\(769\) 16.4384 0.592786 0.296393 0.955066i \(-0.404216\pi\)
0.296393 + 0.955066i \(0.404216\pi\)
\(770\) 0 0
\(771\) −10.6847 −0.384799
\(772\) − 18.4924i − 0.665557i
\(773\) 15.5616i 0.559710i 0.960042 + 0.279855i \(0.0902864\pi\)
−0.960042 + 0.279855i \(0.909714\pi\)
\(774\) 7.80776 0.280644
\(775\) 0 0
\(776\) −6.00000 −0.215387
\(777\) 8.00000i 0.286998i
\(778\) − 24.2462i − 0.869269i
\(779\) 5.26137 0.188508
\(780\) 0 0
\(781\) −7.31534 −0.261764
\(782\) 28.4924i 1.01889i
\(783\) − 1.12311i − 0.0401365i
\(784\) 4.56155 0.162913
\(785\) 0 0
\(786\) −9.36932 −0.334192
\(787\) 10.9309i 0.389643i 0.980839 + 0.194822i \(0.0624128\pi\)
−0.980839 + 0.194822i \(0.937587\pi\)
\(788\) − 6.00000i − 0.213741i
\(789\) 12.4924 0.444742
\(790\) 0 0
\(791\) −16.6847 −0.593238
\(792\) − 1.56155i − 0.0554874i
\(793\) − 12.0000i − 0.426132i
\(794\) 28.0540 0.995598
\(795\) 0 0
\(796\) 3.80776 0.134963
\(797\) 38.9848i 1.38091i 0.723373 + 0.690457i \(0.242591\pi\)
−0.723373 + 0.690457i \(0.757409\pi\)
\(798\) − 7.31534i − 0.258960i
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −1.31534 −0.0464753
\(802\) − 1.31534i − 0.0464463i
\(803\) 15.3153i 0.540467i
\(804\) 9.36932 0.330430
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) − 16.2462i − 0.571894i
\(808\) − 3.56155i − 0.125295i
\(809\) 25.3153 0.890040 0.445020 0.895521i \(-0.353197\pi\)
0.445020 + 0.895521i \(0.353197\pi\)
\(810\) 0 0
\(811\) 53.6695 1.88459 0.942296 0.334782i \(-0.108663\pi\)
0.942296 + 0.334782i \(0.108663\pi\)
\(812\) 1.75379i 0.0615459i
\(813\) − 10.0540i − 0.352608i
\(814\) 8.00000 0.280400
\(815\) 0 0
\(816\) −5.12311 −0.179345
\(817\) − 36.5767i − 1.27966i
\(818\) 12.6307i 0.441621i
\(819\) 3.12311 0.109130
\(820\) 0 0
\(821\) −21.6155 −0.754387 −0.377194 0.926134i \(-0.623111\pi\)
−0.377194 + 0.926134i \(0.623111\pi\)
\(822\) − 3.75379i − 0.130928i
\(823\) − 15.6155i − 0.544323i −0.962252 0.272162i \(-0.912261\pi\)
0.962252 0.272162i \(-0.0877385\pi\)
\(824\) 18.2462 0.635637
\(825\) 0 0
\(826\) 7.61553 0.264978
\(827\) 18.2462i 0.634483i 0.948345 + 0.317241i \(0.102757\pi\)
−0.948345 + 0.317241i \(0.897243\pi\)
\(828\) − 5.56155i − 0.193277i
\(829\) 37.4233 1.29976 0.649882 0.760035i \(-0.274818\pi\)
0.649882 + 0.760035i \(0.274818\pi\)
\(830\) 0 0
\(831\) −19.3693 −0.671914
\(832\) 2.00000i 0.0693375i
\(833\) 23.3693i 0.809699i
\(834\) −8.87689 −0.307382
\(835\) 0 0
\(836\) −7.31534 −0.253006
\(837\) − 1.00000i − 0.0345651i
\(838\) − 22.2462i − 0.768483i
\(839\) −34.9309 −1.20595 −0.602974 0.797761i \(-0.706018\pi\)
−0.602974 + 0.797761i \(0.706018\pi\)
\(840\) 0 0
\(841\) −27.7386 −0.956505
\(842\) 18.4924i 0.637291i
\(843\) 13.1231i 0.451984i
\(844\) 11.3153 0.389490
\(845\) 0 0
\(846\) −3.12311 −0.107375
\(847\) 13.3693i 0.459375i
\(848\) − 11.5616i − 0.397025i
\(849\) 0 0
\(850\) 0 0
\(851\) 28.4924 0.976708
\(852\) − 4.68466i − 0.160494i
\(853\) 38.7926i 1.32823i 0.747629 + 0.664117i \(0.231192\pi\)
−0.747629 + 0.664117i \(0.768808\pi\)
\(854\) 9.36932 0.320611
\(855\) 0 0
\(856\) 1.56155 0.0533728
\(857\) − 53.2311i − 1.81834i −0.416427 0.909169i \(-0.636718\pi\)
0.416427 0.909169i \(-0.363282\pi\)
\(858\) − 3.12311i − 0.106621i
\(859\) 22.7386 0.775832 0.387916 0.921695i \(-0.373195\pi\)
0.387916 + 0.921695i \(0.373195\pi\)
\(860\) 0 0
\(861\) −1.75379 −0.0597690
\(862\) − 13.7538i − 0.468456i
\(863\) 30.5464i 1.03981i 0.854224 + 0.519906i \(0.174033\pi\)
−0.854224 + 0.519906i \(0.825967\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) −18.3002 −0.621866
\(867\) − 9.24621i − 0.314018i
\(868\) 1.56155i 0.0530026i
\(869\) −26.0540 −0.883821
\(870\) 0 0
\(871\) 18.7386 0.634934
\(872\) − 19.3693i − 0.655928i
\(873\) 6.00000i 0.203069i
\(874\) −26.0540 −0.881289
\(875\) 0 0
\(876\) −9.80776 −0.331374
\(877\) − 50.0000i − 1.68838i −0.536044 0.844190i \(-0.680082\pi\)
0.536044 0.844190i \(-0.319918\pi\)
\(878\) 31.6155i 1.06697i
\(879\) −6.49242 −0.218984
\(880\) 0 0
\(881\) 42.4924 1.43161 0.715803 0.698302i \(-0.246061\pi\)
0.715803 + 0.698302i \(0.246061\pi\)
\(882\) − 4.56155i − 0.153595i
\(883\) 31.4233i 1.05748i 0.848784 + 0.528739i \(0.177335\pi\)
−0.848784 + 0.528739i \(0.822665\pi\)
\(884\) −10.2462 −0.344617
\(885\) 0 0
\(886\) −14.0540 −0.472153
\(887\) − 57.3693i − 1.92627i −0.269013 0.963137i \(-0.586697\pi\)
0.269013 0.963137i \(-0.413303\pi\)
\(888\) 5.12311i 0.171920i
\(889\) 0 0
\(890\) 0 0
\(891\) −1.56155 −0.0523140
\(892\) − 12.8769i − 0.431150i
\(893\) 14.6307i 0.489597i
\(894\) −20.0540 −0.670705
\(895\) 0 0
\(896\) −1.56155 −0.0521678
\(897\) − 11.1231i − 0.371390i
\(898\) 22.4924i 0.750582i
\(899\) 1.12311 0.0374577
\(900\) 0 0
\(901\) 59.2311 1.97327
\(902\) 1.75379i 0.0583948i
\(903\) 12.1922i 0.405732i
\(904\) −10.6847 −0.355366
\(905\) 0 0
\(906\) 0 0
\(907\) 24.0000i 0.796907i 0.917189 + 0.398453i \(0.130453\pi\)
−0.917189 + 0.398453i \(0.869547\pi\)
\(908\) 4.68466i 0.155466i
\(909\) −3.56155 −0.118129
\(910\) 0 0
\(911\) −44.4924 −1.47410 −0.737050 0.675838i \(-0.763782\pi\)
−0.737050 + 0.675838i \(0.763782\pi\)
\(912\) − 4.68466i − 0.155125i
\(913\) − 3.50758i − 0.116084i
\(914\) −24.7386 −0.818281
\(915\) 0 0
\(916\) −12.4384 −0.410978
\(917\) − 14.6307i − 0.483148i
\(918\) 5.12311i 0.169088i
\(919\) −39.6155 −1.30680 −0.653398 0.757015i \(-0.726657\pi\)
−0.653398 + 0.757015i \(0.726657\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 15.3693i − 0.506161i
\(923\) − 9.36932i − 0.308395i
\(924\) 2.43845 0.0802190
\(925\) 0 0
\(926\) −18.7386 −0.615790
\(927\) − 18.2462i − 0.599284i
\(928\) 1.12311i 0.0368677i
\(929\) −11.5616 −0.379322 −0.189661 0.981850i \(-0.560739\pi\)
−0.189661 + 0.981850i \(0.560739\pi\)
\(930\) 0 0
\(931\) −21.3693 −0.700351
\(932\) 20.0540i 0.656890i
\(933\) − 18.2462i − 0.597354i
\(934\) 26.2462 0.858802
\(935\) 0 0
\(936\) 2.00000 0.0653720
\(937\) 16.6307i 0.543301i 0.962396 + 0.271650i \(0.0875694\pi\)
−0.962396 + 0.271650i \(0.912431\pi\)
\(938\) 14.6307i 0.477709i
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 27.3693 0.892214 0.446107 0.894980i \(-0.352810\pi\)
0.446107 + 0.894980i \(0.352810\pi\)
\(942\) 20.0540i 0.653394i
\(943\) 6.24621i 0.203405i
\(944\) 4.87689 0.158729
\(945\) 0 0
\(946\) 12.1922 0.396404
\(947\) − 0.876894i − 0.0284952i −0.999898 0.0142476i \(-0.995465\pi\)
0.999898 0.0142476i \(-0.00453531\pi\)
\(948\) − 16.6847i − 0.541893i
\(949\) −19.6155 −0.636747
\(950\) 0 0
\(951\) −25.1231 −0.814673
\(952\) − 8.00000i − 0.259281i
\(953\) 58.1080i 1.88230i 0.337988 + 0.941151i \(0.390254\pi\)
−0.337988 + 0.941151i \(0.609746\pi\)
\(954\) −11.5616 −0.374319
\(955\) 0 0
\(956\) 24.0000 0.776215
\(957\) − 1.75379i − 0.0566919i
\(958\) − 6.43845i − 0.208017i
\(959\) 5.86174 0.189285
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 10.2462i 0.330351i
\(963\) − 1.56155i − 0.0503203i
\(964\) 4.24621 0.136761
\(965\) 0 0
\(966\) 8.68466 0.279424
\(967\) − 25.7538i − 0.828186i −0.910235 0.414093i \(-0.864099\pi\)
0.910235 0.414093i \(-0.135901\pi\)
\(968\) 8.56155i 0.275179i
\(969\) 24.0000 0.770991
\(970\) 0 0
\(971\) −28.8769 −0.926704 −0.463352 0.886174i \(-0.653353\pi\)
−0.463352 + 0.886174i \(0.653353\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 13.8617i − 0.444387i
\(974\) −24.0000 −0.769010
\(975\) 0 0
\(976\) 6.00000 0.192055
\(977\) − 22.9848i − 0.735350i −0.929954 0.367675i \(-0.880154\pi\)
0.929954 0.367675i \(-0.119846\pi\)
\(978\) 9.36932i 0.299598i
\(979\) −2.05398 −0.0656453
\(980\) 0 0
\(981\) −19.3693 −0.618415
\(982\) 2.93087i 0.0935278i
\(983\) 24.9848i 0.796893i 0.917192 + 0.398446i \(0.130451\pi\)
−0.917192 + 0.398446i \(0.869549\pi\)
\(984\) −1.12311 −0.0358033
\(985\) 0 0
\(986\) −5.75379 −0.183238
\(987\) − 4.87689i − 0.155233i
\(988\) − 9.36932i − 0.298078i
\(989\) 43.4233 1.38078
\(990\) 0 0
\(991\) 23.3153 0.740636 0.370318 0.928905i \(-0.379249\pi\)
0.370318 + 0.928905i \(0.379249\pi\)
\(992\) 1.00000i 0.0317500i
\(993\) 10.2462i 0.325154i
\(994\) 7.31534 0.232029
\(995\) 0 0
\(996\) 2.24621 0.0711739
\(997\) − 62.4924i − 1.97915i −0.144002 0.989577i \(-0.545997\pi\)
0.144002 0.989577i \(-0.454003\pi\)
\(998\) − 20.0000i − 0.633089i
\(999\) 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.d.bd.3349.3 4
5.2 odd 4 4650.2.a.ce.1.2 2
5.3 odd 4 930.2.a.p.1.1 2
5.4 even 2 inner 4650.2.d.bd.3349.2 4
15.8 even 4 2790.2.a.be.1.1 2
20.3 even 4 7440.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.p.1.1 2 5.3 odd 4
2790.2.a.be.1.1 2 15.8 even 4
4650.2.a.ce.1.2 2 5.2 odd 4
4650.2.d.bd.3349.2 4 5.4 even 2 inner
4650.2.d.bd.3349.3 4 1.1 even 1 trivial
7440.2.a.bl.1.2 2 20.3 even 4