Properties

Label 825.2.c.c.199.2
Level $825$
Weight $2$
Character 825.199
Analytic conductor $6.588$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.58765816676\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-0.866025 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.2.c.c.199.3

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.73205 q^{6} -2.00000i q^{7} -1.73205i q^{8} -1.00000 q^{9} -1.00000 q^{11} -1.00000i q^{12} -1.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} +1.73205i q^{18} +1.46410 q^{19} +2.00000 q^{21} +1.73205i q^{22} -6.92820i q^{23} +1.73205 q^{24} -2.53590 q^{26} -1.00000i q^{27} +2.00000i q^{28} -3.46410 q^{29} +2.92820 q^{31} +5.19615i q^{32} -1.00000i q^{33} +1.00000 q^{36} -8.92820i q^{37} -2.53590i q^{38} +1.46410 q^{39} -3.46410 q^{41} -3.46410i q^{42} +8.92820i q^{43} +1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} -5.00000i q^{48} +3.00000 q^{49} +1.46410i q^{52} -12.9282i q^{53} -1.73205 q^{54} -3.46410 q^{56} +1.46410i q^{57} +6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} -5.07180i q^{62} +2.00000i q^{63} -1.00000 q^{64} -1.73205 q^{66} -8.00000i q^{67} +6.92820 q^{69} -13.8564 q^{71} +1.73205i q^{72} +12.3923i q^{73} -15.4641 q^{74} -1.46410 q^{76} +2.00000i q^{77} -2.53590i q^{78} +13.4641 q^{79} +1.00000 q^{81} +6.00000i q^{82} +15.4641i q^{83} -2.00000 q^{84} +15.4641 q^{86} -3.46410i q^{87} +1.73205i q^{88} +12.9282 q^{89} -2.92820 q^{91} +6.92820i q^{92} +2.92820i q^{93} -12.0000 q^{94} -5.19615 q^{96} +10.0000i q^{97} -5.19615i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} - 4q^{9} + O(q^{10}) \) \( 4q - 4q^{4} - 4q^{9} - 4q^{11} - 20q^{16} - 8q^{19} + 8q^{21} - 24q^{26} - 16q^{31} + 4q^{36} - 8q^{39} + 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} - 48q^{74} + 8q^{76} + 40q^{79} + 4q^{81} - 8q^{84} + 48q^{86} + 24q^{89} + 16q^{91} - 48q^{94} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 1.73205 0.707107
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) −1.00000 −0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) − 1.00000i − 0.288675i
\(13\) − 1.46410i − 0.406069i −0.979172 0.203034i \(-0.934920\pi\)
0.979172 0.203034i \(-0.0650803\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 1.73205i 0.408248i
\(19\) 1.46410 0.335888 0.167944 0.985797i \(-0.446287\pi\)
0.167944 + 0.985797i \(0.446287\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) 1.73205i 0.369274i
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 1.73205 0.353553
\(25\) 0 0
\(26\) −2.53590 −0.497331
\(27\) − 1.00000i − 0.192450i
\(28\) 2.00000i 0.377964i
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) 2.92820 0.525921 0.262960 0.964807i \(-0.415301\pi\)
0.262960 + 0.964807i \(0.415301\pi\)
\(32\) 5.19615i 0.918559i
\(33\) − 1.00000i − 0.174078i
\(34\) 0 0
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) − 8.92820i − 1.46779i −0.679264 0.733894i \(-0.737701\pi\)
0.679264 0.733894i \(-0.262299\pi\)
\(38\) − 2.53590i − 0.411377i
\(39\) 1.46410 0.234444
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) − 3.46410i − 0.534522i
\(43\) 8.92820i 1.36154i 0.732498 + 0.680769i \(0.238354\pi\)
−0.732498 + 0.680769i \(0.761646\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) − 6.92820i − 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) − 5.00000i − 0.721688i
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 1.46410i 0.203034i
\(53\) − 12.9282i − 1.77583i −0.460012 0.887913i \(-0.652155\pi\)
0.460012 0.887913i \(-0.347845\pi\)
\(54\) −1.73205 −0.235702
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 1.46410i 0.193925i
\(58\) 6.00000i 0.787839i
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) − 5.07180i − 0.644119i
\(63\) 2.00000i 0.251976i
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) −1.73205 −0.213201
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 6.92820 0.834058
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 1.73205i 0.204124i
\(73\) 12.3923i 1.45041i 0.688533 + 0.725205i \(0.258255\pi\)
−0.688533 + 0.725205i \(0.741745\pi\)
\(74\) −15.4641 −1.79767
\(75\) 0 0
\(76\) −1.46410 −0.167944
\(77\) 2.00000i 0.227921i
\(78\) − 2.53590i − 0.287134i
\(79\) 13.4641 1.51483 0.757415 0.652934i \(-0.226462\pi\)
0.757415 + 0.652934i \(0.226462\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 6.00000i 0.662589i
\(83\) 15.4641i 1.69741i 0.528870 + 0.848703i \(0.322616\pi\)
−0.528870 + 0.848703i \(0.677384\pi\)
\(84\) −2.00000 −0.218218
\(85\) 0 0
\(86\) 15.4641 1.66754
\(87\) − 3.46410i − 0.371391i
\(88\) 1.73205i 0.184637i
\(89\) 12.9282 1.37039 0.685193 0.728361i \(-0.259718\pi\)
0.685193 + 0.728361i \(0.259718\pi\)
\(90\) 0 0
\(91\) −2.92820 −0.306959
\(92\) 6.92820i 0.722315i
\(93\) 2.92820i 0.303641i
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 1.00000 0.100504
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) −2.53590 −0.248665
\(105\) 0 0
\(106\) −22.3923 −2.17493
\(107\) − 15.4641i − 1.49497i −0.664278 0.747486i \(-0.731261\pi\)
0.664278 0.747486i \(-0.268739\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 8.92820 0.847428
\(112\) 10.0000i 0.944911i
\(113\) 0.928203i 0.0873180i 0.999046 + 0.0436590i \(0.0139015\pi\)
−0.999046 + 0.0436590i \(0.986098\pi\)
\(114\) 2.53590 0.237509
\(115\) 0 0
\(116\) 3.46410 0.321634
\(117\) 1.46410i 0.135356i
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 3.46410i − 0.313625i
\(123\) − 3.46410i − 0.312348i
\(124\) −2.92820 −0.262960
\(125\) 0 0
\(126\) 3.46410 0.308607
\(127\) 4.92820i 0.437307i 0.975803 + 0.218654i \(0.0701665\pi\)
−0.975803 + 0.218654i \(0.929834\pi\)
\(128\) 12.1244i 1.07165i
\(129\) −8.92820 −0.786084
\(130\) 0 0
\(131\) 5.07180 0.443125 0.221562 0.975146i \(-0.428884\pi\)
0.221562 + 0.975146i \(0.428884\pi\)
\(132\) 1.00000i 0.0870388i
\(133\) − 2.92820i − 0.253907i
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) 18.0000i 1.53784i 0.639343 + 0.768922i \(0.279207\pi\)
−0.639343 + 0.768922i \(0.720793\pi\)
\(138\) − 12.0000i − 1.02151i
\(139\) 8.39230 0.711826 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(140\) 0 0
\(141\) 6.92820 0.583460
\(142\) 24.0000i 2.01404i
\(143\) 1.46410i 0.122434i
\(144\) 5.00000 0.416667
\(145\) 0 0
\(146\) 21.4641 1.77638
\(147\) 3.00000i 0.247436i
\(148\) 8.92820i 0.733894i
\(149\) 8.53590 0.699288 0.349644 0.936883i \(-0.386303\pi\)
0.349644 + 0.936883i \(0.386303\pi\)
\(150\) 0 0
\(151\) 0.392305 0.0319253 0.0159627 0.999873i \(-0.494919\pi\)
0.0159627 + 0.999873i \(0.494919\pi\)
\(152\) − 2.53590i − 0.205689i
\(153\) 0 0
\(154\) 3.46410 0.279145
\(155\) 0 0
\(156\) −1.46410 −0.117222
\(157\) 16.9282i 1.35102i 0.737352 + 0.675509i \(0.236076\pi\)
−0.737352 + 0.675509i \(0.763924\pi\)
\(158\) − 23.3205i − 1.85528i
\(159\) 12.9282 1.02527
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) − 1.73205i − 0.136083i
\(163\) − 17.8564i − 1.39862i −0.714818 0.699311i \(-0.753490\pi\)
0.714818 0.699311i \(-0.246510\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) 26.7846 2.07889
\(167\) − 10.3923i − 0.804181i −0.915600 0.402090i \(-0.868284\pi\)
0.915600 0.402090i \(-0.131716\pi\)
\(168\) − 3.46410i − 0.267261i
\(169\) 10.8564 0.835108
\(170\) 0 0
\(171\) −1.46410 −0.111963
\(172\) − 8.92820i − 0.680769i
\(173\) − 12.0000i − 0.912343i −0.889892 0.456172i \(-0.849220\pi\)
0.889892 0.456172i \(-0.150780\pi\)
\(174\) −6.00000 −0.454859
\(175\) 0 0
\(176\) 5.00000 0.376889
\(177\) − 6.92820i − 0.520756i
\(178\) − 22.3923i − 1.67837i
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) −11.8564 −0.881280 −0.440640 0.897684i \(-0.645248\pi\)
−0.440640 + 0.897684i \(0.645248\pi\)
\(182\) 5.07180i 0.375947i
\(183\) 2.00000i 0.147844i
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 5.07180 0.371882
\(187\) 0 0
\(188\) 6.92820i 0.505291i
\(189\) −2.00000 −0.145479
\(190\) 0 0
\(191\) −5.07180 −0.366982 −0.183491 0.983021i \(-0.558740\pi\)
−0.183491 + 0.983021i \(0.558740\pi\)
\(192\) − 1.00000i − 0.0721688i
\(193\) 3.60770i 0.259688i 0.991534 + 0.129844i \(0.0414476\pi\)
−0.991534 + 0.129844i \(0.958552\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 12.0000i 0.854965i 0.904024 + 0.427482i \(0.140599\pi\)
−0.904024 + 0.427482i \(0.859401\pi\)
\(198\) − 1.73205i − 0.123091i
\(199\) −16.7846 −1.18983 −0.594915 0.803789i \(-0.702814\pi\)
−0.594915 + 0.803789i \(0.702814\pi\)
\(200\) 0 0
\(201\) 8.00000 0.564276
\(202\) − 18.0000i − 1.26648i
\(203\) 6.92820i 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) 6.92820i 0.481543i
\(208\) 7.32051i 0.507586i
\(209\) −1.46410 −0.101274
\(210\) 0 0
\(211\) 12.3923 0.853121 0.426561 0.904459i \(-0.359725\pi\)
0.426561 + 0.904459i \(0.359725\pi\)
\(212\) 12.9282i 0.887913i
\(213\) − 13.8564i − 0.949425i
\(214\) −26.7846 −1.83096
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) − 5.85641i − 0.397559i
\(218\) − 17.3205i − 1.17309i
\(219\) −12.3923 −0.837394
\(220\) 0 0
\(221\) 0 0
\(222\) − 15.4641i − 1.03788i
\(223\) − 17.8564i − 1.19575i −0.801588 0.597877i \(-0.796011\pi\)
0.801588 0.597877i \(-0.203989\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) 1.60770 0.106942
\(227\) − 8.53590i − 0.566547i −0.959039 0.283274i \(-0.908580\pi\)
0.959039 0.283274i \(-0.0914205\pi\)
\(228\) − 1.46410i − 0.0969625i
\(229\) −3.85641 −0.254839 −0.127419 0.991849i \(-0.540669\pi\)
−0.127419 + 0.991849i \(0.540669\pi\)
\(230\) 0 0
\(231\) −2.00000 −0.131590
\(232\) 6.00000i 0.393919i
\(233\) 12.0000i 0.786146i 0.919507 + 0.393073i \(0.128588\pi\)
−0.919507 + 0.393073i \(0.871412\pi\)
\(234\) 2.53590 0.165777
\(235\) 0 0
\(236\) 6.92820 0.450988
\(237\) 13.4641i 0.874587i
\(238\) 0 0
\(239\) 12.0000 0.776215 0.388108 0.921614i \(-0.373129\pi\)
0.388108 + 0.921614i \(0.373129\pi\)
\(240\) 0 0
\(241\) 27.8564 1.79439 0.897194 0.441636i \(-0.145602\pi\)
0.897194 + 0.441636i \(0.145602\pi\)
\(242\) − 1.73205i − 0.111340i
\(243\) 1.00000i 0.0641500i
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) −6.00000 −0.382546
\(247\) − 2.14359i − 0.136394i
\(248\) − 5.07180i − 0.322059i
\(249\) −15.4641 −0.979998
\(250\) 0 0
\(251\) 25.8564 1.63204 0.816021 0.578022i \(-0.196175\pi\)
0.816021 + 0.578022i \(0.196175\pi\)
\(252\) − 2.00000i − 0.125988i
\(253\) 6.92820i 0.435572i
\(254\) 8.53590 0.535590
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 7.85641i − 0.490069i −0.969514 0.245035i \(-0.921201\pi\)
0.969514 0.245035i \(-0.0787993\pi\)
\(258\) 15.4641i 0.962753i
\(259\) −17.8564 −1.10954
\(260\) 0 0
\(261\) 3.46410 0.214423
\(262\) − 8.78461i − 0.542715i
\(263\) 27.4641i 1.69351i 0.531984 + 0.846755i \(0.321447\pi\)
−0.531984 + 0.846755i \(0.678553\pi\)
\(264\) −1.73205 −0.106600
\(265\) 0 0
\(266\) −5.07180 −0.310972
\(267\) 12.9282i 0.791193i
\(268\) 8.00000i 0.488678i
\(269\) 7.85641 0.479014 0.239507 0.970895i \(-0.423014\pi\)
0.239507 + 0.970895i \(0.423014\pi\)
\(270\) 0 0
\(271\) −32.3923 −1.96769 −0.983846 0.179016i \(-0.942709\pi\)
−0.983846 + 0.179016i \(0.942709\pi\)
\(272\) 0 0
\(273\) − 2.92820i − 0.177223i
\(274\) 31.1769 1.88347
\(275\) 0 0
\(276\) −6.92820 −0.417029
\(277\) − 22.5359i − 1.35405i −0.735960 0.677025i \(-0.763269\pi\)
0.735960 0.677025i \(-0.236731\pi\)
\(278\) − 14.5359i − 0.871805i
\(279\) −2.92820 −0.175307
\(280\) 0 0
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) − 12.0000i − 0.714590i
\(283\) 8.92820i 0.530727i 0.964148 + 0.265363i \(0.0854919\pi\)
−0.964148 + 0.265363i \(0.914508\pi\)
\(284\) 13.8564 0.822226
\(285\) 0 0
\(286\) 2.53590 0.149951
\(287\) 6.92820i 0.408959i
\(288\) − 5.19615i − 0.306186i
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) −10.0000 −0.586210
\(292\) − 12.3923i − 0.725205i
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 5.19615 0.303046
\(295\) 0 0
\(296\) −15.4641 −0.898833
\(297\) 1.00000i 0.0580259i
\(298\) − 14.7846i − 0.856449i
\(299\) −10.1436 −0.586619
\(300\) 0 0
\(301\) 17.8564 1.02923
\(302\) − 0.679492i − 0.0391004i
\(303\) 10.3923i 0.597022i
\(304\) −7.32051 −0.419860
\(305\) 0 0
\(306\) 0 0
\(307\) − 14.0000i − 0.799022i −0.916728 0.399511i \(-0.869180\pi\)
0.916728 0.399511i \(-0.130820\pi\)
\(308\) − 2.00000i − 0.113961i
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 18.9282 1.07332 0.536660 0.843799i \(-0.319686\pi\)
0.536660 + 0.843799i \(0.319686\pi\)
\(312\) − 2.53590i − 0.143567i
\(313\) 7.07180i 0.399722i 0.979824 + 0.199861i \(0.0640490\pi\)
−0.979824 + 0.199861i \(0.935951\pi\)
\(314\) 29.3205 1.65465
\(315\) 0 0
\(316\) −13.4641 −0.757415
\(317\) 11.0718i 0.621854i 0.950434 + 0.310927i \(0.100639\pi\)
−0.950434 + 0.310927i \(0.899361\pi\)
\(318\) − 22.3923i − 1.25570i
\(319\) 3.46410 0.193952
\(320\) 0 0
\(321\) 15.4641 0.863122
\(322\) 24.0000i 1.33747i
\(323\) 0 0
\(324\) −1.00000 −0.0555556
\(325\) 0 0
\(326\) −30.9282 −1.71295
\(327\) 10.0000i 0.553001i
\(328\) 6.00000i 0.331295i
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) −17.8564 −0.981477 −0.490738 0.871307i \(-0.663273\pi\)
−0.490738 + 0.871307i \(0.663273\pi\)
\(332\) − 15.4641i − 0.848703i
\(333\) 8.92820i 0.489263i
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) −10.0000 −0.545545
\(337\) 29.1769i 1.58937i 0.607023 + 0.794684i \(0.292363\pi\)
−0.607023 + 0.794684i \(0.707637\pi\)
\(338\) − 18.8038i − 1.02279i
\(339\) −0.928203 −0.0504131
\(340\) 0 0
\(341\) −2.92820 −0.158571
\(342\) 2.53590i 0.137126i
\(343\) − 20.0000i − 1.07990i
\(344\) 15.4641 0.833768
\(345\) 0 0
\(346\) −20.7846 −1.11739
\(347\) − 1.60770i − 0.0863056i −0.999068 0.0431528i \(-0.986260\pi\)
0.999068 0.0431528i \(-0.0137402\pi\)
\(348\) 3.46410i 0.185695i
\(349\) 35.8564 1.91935 0.959675 0.281113i \(-0.0907035\pi\)
0.959675 + 0.281113i \(0.0907035\pi\)
\(350\) 0 0
\(351\) −1.46410 −0.0781480
\(352\) − 5.19615i − 0.276956i
\(353\) − 0.928203i − 0.0494033i −0.999695 0.0247016i \(-0.992136\pi\)
0.999695 0.0247016i \(-0.00786357\pi\)
\(354\) −12.0000 −0.637793
\(355\) 0 0
\(356\) −12.9282 −0.685193
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) −16.8564 −0.887179
\(362\) 20.5359i 1.07934i
\(363\) 1.00000i 0.0524864i
\(364\) 2.92820 0.153480
\(365\) 0 0
\(366\) 3.46410 0.181071
\(367\) − 20.0000i − 1.04399i −0.852948 0.521996i \(-0.825188\pi\)
0.852948 0.521996i \(-0.174812\pi\)
\(368\) 34.6410i 1.80579i
\(369\) 3.46410 0.180334
\(370\) 0 0
\(371\) −25.8564 −1.34240
\(372\) − 2.92820i − 0.151820i
\(373\) 0.392305i 0.0203128i 0.999948 + 0.0101564i \(0.00323293\pi\)
−0.999948 + 0.0101564i \(0.996767\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) 5.07180i 0.261211i
\(378\) 3.46410i 0.178174i
\(379\) −9.85641 −0.506290 −0.253145 0.967428i \(-0.581465\pi\)
−0.253145 + 0.967428i \(0.581465\pi\)
\(380\) 0 0
\(381\) −4.92820 −0.252479
\(382\) 8.78461i 0.449460i
\(383\) 13.8564i 0.708029i 0.935240 + 0.354015i \(0.115184\pi\)
−0.935240 + 0.354015i \(0.884816\pi\)
\(384\) −12.1244 −0.618718
\(385\) 0 0
\(386\) 6.24871 0.318051
\(387\) − 8.92820i − 0.453846i
\(388\) − 10.0000i − 0.507673i
\(389\) −24.9282 −1.26391 −0.631955 0.775005i \(-0.717747\pi\)
−0.631955 + 0.775005i \(0.717747\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 5.19615i − 0.262445i
\(393\) 5.07180i 0.255838i
\(394\) 20.7846 1.04711
\(395\) 0 0
\(396\) −1.00000 −0.0502519
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) 29.0718i 1.45724i
\(399\) 2.92820 0.146594
\(400\) 0 0
\(401\) 19.8564 0.991582 0.495791 0.868442i \(-0.334878\pi\)
0.495791 + 0.868442i \(0.334878\pi\)
\(402\) − 13.8564i − 0.691095i
\(403\) − 4.28719i − 0.213560i
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 8.92820i 0.442555i
\(408\) 0 0
\(409\) −34.7846 −1.71999 −0.859994 0.510304i \(-0.829533\pi\)
−0.859994 + 0.510304i \(0.829533\pi\)
\(410\) 0 0
\(411\) −18.0000 −0.887875
\(412\) − 8.00000i − 0.394132i
\(413\) 13.8564i 0.681829i
\(414\) 12.0000 0.589768
\(415\) 0 0
\(416\) 7.60770 0.372998
\(417\) 8.39230i 0.410973i
\(418\) 2.53590i 0.124035i
\(419\) −17.0718 −0.834012 −0.417006 0.908904i \(-0.636921\pi\)
−0.417006 + 0.908904i \(0.636921\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) − 21.4641i − 1.04486i
\(423\) 6.92820i 0.336861i
\(424\) −22.3923 −1.08747
\(425\) 0 0
\(426\) −24.0000 −1.16280
\(427\) − 4.00000i − 0.193574i
\(428\) 15.4641i 0.747486i
\(429\) −1.46410 −0.0706875
\(430\) 0 0
\(431\) −32.7846 −1.57918 −0.789590 0.613635i \(-0.789706\pi\)
−0.789590 + 0.613635i \(0.789706\pi\)
\(432\) 5.00000i 0.240563i
\(433\) 27.8564i 1.33869i 0.742950 + 0.669347i \(0.233426\pi\)
−0.742950 + 0.669347i \(0.766574\pi\)
\(434\) −10.1436 −0.486908
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) − 10.1436i − 0.485234i
\(438\) 21.4641i 1.02559i
\(439\) 29.1769 1.39254 0.696269 0.717781i \(-0.254842\pi\)
0.696269 + 0.717781i \(0.254842\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) 12.0000i 0.570137i 0.958507 + 0.285069i \(0.0920164\pi\)
−0.958507 + 0.285069i \(0.907984\pi\)
\(444\) −8.92820 −0.423714
\(445\) 0 0
\(446\) −30.9282 −1.46449
\(447\) 8.53590i 0.403734i
\(448\) 2.00000i 0.0944911i
\(449\) −14.7846 −0.697729 −0.348864 0.937173i \(-0.613433\pi\)
−0.348864 + 0.937173i \(0.613433\pi\)
\(450\) 0 0
\(451\) 3.46410 0.163118
\(452\) − 0.928203i − 0.0436590i
\(453\) 0.392305i 0.0184321i
\(454\) −14.7846 −0.693876
\(455\) 0 0
\(456\) 2.53590 0.118754
\(457\) 8.39230i 0.392575i 0.980546 + 0.196288i \(0.0628887\pi\)
−0.980546 + 0.196288i \(0.937111\pi\)
\(458\) 6.67949i 0.312112i
\(459\) 0 0
\(460\) 0 0
\(461\) −12.2487 −0.570479 −0.285240 0.958456i \(-0.592073\pi\)
−0.285240 + 0.958456i \(0.592073\pi\)
\(462\) 3.46410i 0.161165i
\(463\) − 28.0000i − 1.30127i −0.759390 0.650635i \(-0.774503\pi\)
0.759390 0.650635i \(-0.225497\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) 20.7846 0.962828
\(467\) − 18.9282i − 0.875893i −0.899001 0.437946i \(-0.855706\pi\)
0.899001 0.437946i \(-0.144294\pi\)
\(468\) − 1.46410i − 0.0676781i
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) −16.9282 −0.780010
\(472\) 12.0000i 0.552345i
\(473\) − 8.92820i − 0.410519i
\(474\) 23.3205 1.07115
\(475\) 0 0
\(476\) 0 0
\(477\) 12.9282i 0.591942i
\(478\) − 20.7846i − 0.950666i
\(479\) −12.0000 −0.548294 −0.274147 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) −13.0718 −0.596023
\(482\) − 48.2487i − 2.19767i
\(483\) − 13.8564i − 0.630488i
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) − 23.7128i − 1.07453i −0.843413 0.537265i \(-0.819457\pi\)
0.843413 0.537265i \(-0.180543\pi\)
\(488\) − 3.46410i − 0.156813i
\(489\) 17.8564 0.807495
\(490\) 0 0
\(491\) −17.0718 −0.770439 −0.385220 0.922825i \(-0.625874\pi\)
−0.385220 + 0.922825i \(0.625874\pi\)
\(492\) 3.46410i 0.156174i
\(493\) 0 0
\(494\) −3.71281 −0.167047
\(495\) 0 0
\(496\) −14.6410 −0.657401
\(497\) 27.7128i 1.24309i
\(498\) 26.7846i 1.20025i
\(499\) 12.7846 0.572318 0.286159 0.958182i \(-0.407621\pi\)
0.286159 + 0.958182i \(0.407621\pi\)
\(500\) 0 0
\(501\) 10.3923 0.464294
\(502\) − 44.7846i − 1.99883i
\(503\) − 31.1769i − 1.39011i −0.718957 0.695055i \(-0.755380\pi\)
0.718957 0.695055i \(-0.244620\pi\)
\(504\) 3.46410 0.154303
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 10.8564i 0.482150i
\(508\) − 4.92820i − 0.218654i
\(509\) 7.85641 0.348229 0.174115 0.984725i \(-0.444294\pi\)
0.174115 + 0.984725i \(0.444294\pi\)
\(510\) 0 0
\(511\) 24.7846 1.09641
\(512\) − 8.66025i − 0.382733i
\(513\) − 1.46410i − 0.0646417i
\(514\) −13.6077 −0.600210
\(515\) 0 0
\(516\) 8.92820 0.393042
\(517\) 6.92820i 0.304702i
\(518\) 30.9282i 1.35891i
\(519\) 12.0000 0.526742
\(520\) 0 0
\(521\) 6.00000 0.262865 0.131432 0.991325i \(-0.458042\pi\)
0.131432 + 0.991325i \(0.458042\pi\)
\(522\) − 6.00000i − 0.262613i
\(523\) − 22.0000i − 0.961993i −0.876723 0.480996i \(-0.840275\pi\)
0.876723 0.480996i \(-0.159725\pi\)
\(524\) −5.07180 −0.221562
\(525\) 0 0
\(526\) 47.5692 2.07412
\(527\) 0 0
\(528\) 5.00000i 0.217597i
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 6.92820 0.300658
\(532\) 2.92820i 0.126954i
\(533\) 5.07180i 0.219684i
\(534\) 22.3923 0.969010
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) 6.92820i 0.298974i
\(538\) − 13.6077i − 0.586669i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 0.143594 0.00617357 0.00308678 0.999995i \(-0.499017\pi\)
0.00308678 + 0.999995i \(0.499017\pi\)
\(542\) 56.1051i 2.40992i
\(543\) − 11.8564i − 0.508807i
\(544\) 0 0
\(545\) 0 0
\(546\) −5.07180 −0.217053
\(547\) − 2.00000i − 0.0855138i −0.999086 0.0427569i \(-0.986386\pi\)
0.999086 0.0427569i \(-0.0136141\pi\)
\(548\) − 18.0000i − 0.768922i
\(549\) −2.00000 −0.0853579
\(550\) 0 0
\(551\) −5.07180 −0.216066
\(552\) − 12.0000i − 0.510754i
\(553\) − 26.9282i − 1.14510i
\(554\) −39.0333 −1.65837
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) − 44.7846i − 1.89758i −0.315900 0.948792i \(-0.602306\pi\)
0.315900 0.948792i \(-0.397694\pi\)
\(558\) 5.07180i 0.214706i
\(559\) 13.0718 0.552878
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) − 10.3923i − 0.437983i −0.975727 0.218992i \(-0.929723\pi\)
0.975727 0.218992i \(-0.0702768\pi\)
\(564\) −6.92820 −0.291730
\(565\) 0 0
\(566\) 15.4641 0.650005
\(567\) − 2.00000i − 0.0839921i
\(568\) 24.0000i 1.00702i
\(569\) 29.3205 1.22918 0.614590 0.788847i \(-0.289322\pi\)
0.614590 + 0.788847i \(0.289322\pi\)
\(570\) 0 0
\(571\) 24.3923 1.02079 0.510393 0.859941i \(-0.329500\pi\)
0.510393 + 0.859941i \(0.329500\pi\)
\(572\) − 1.46410i − 0.0612172i
\(573\) − 5.07180i − 0.211877i
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) − 22.7846i − 0.948536i −0.880381 0.474268i \(-0.842713\pi\)
0.880381 0.474268i \(-0.157287\pi\)
\(578\) − 29.4449i − 1.22474i
\(579\) −3.60770 −0.149931
\(580\) 0 0
\(581\) 30.9282 1.28312
\(582\) 17.3205i 0.717958i
\(583\) 12.9282i 0.535431i
\(584\) 21.4641 0.888191
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 5.07180i 0.209335i 0.994507 + 0.104668i \(0.0333779\pi\)
−0.994507 + 0.104668i \(0.966622\pi\)
\(588\) − 3.00000i − 0.123718i
\(589\) 4.28719 0.176650
\(590\) 0 0
\(591\) −12.0000 −0.493614
\(592\) 44.6410i 1.83473i
\(593\) − 32.7846i − 1.34630i −0.739505 0.673151i \(-0.764940\pi\)
0.739505 0.673151i \(-0.235060\pi\)
\(594\) 1.73205 0.0710669
\(595\) 0 0
\(596\) −8.53590 −0.349644
\(597\) − 16.7846i − 0.686948i
\(598\) 17.5692i 0.718459i
\(599\) 10.1436 0.414456 0.207228 0.978293i \(-0.433556\pi\)
0.207228 + 0.978293i \(0.433556\pi\)
\(600\) 0 0
\(601\) 36.6410 1.49462 0.747309 0.664477i \(-0.231345\pi\)
0.747309 + 0.664477i \(0.231345\pi\)
\(602\) − 30.9282i − 1.26054i
\(603\) 8.00000i 0.325785i
\(604\) −0.392305 −0.0159627
\(605\) 0 0
\(606\) 18.0000 0.731200
\(607\) − 22.7846i − 0.924799i −0.886672 0.462399i \(-0.846989\pi\)
0.886672 0.462399i \(-0.153011\pi\)
\(608\) 7.60770i 0.308533i
\(609\) −6.92820 −0.280745
\(610\) 0 0
\(611\) −10.1436 −0.410366
\(612\) 0 0
\(613\) 0.392305i 0.0158450i 0.999969 + 0.00792252i \(0.00252184\pi\)
−0.999969 + 0.00792252i \(0.997478\pi\)
\(614\) −24.2487 −0.978598
\(615\) 0 0
\(616\) 3.46410 0.139573
\(617\) 23.0718i 0.928836i 0.885616 + 0.464418i \(0.153736\pi\)
−0.885616 + 0.464418i \(0.846264\pi\)
\(618\) 13.8564i 0.557386i
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) −6.92820 −0.278019
\(622\) − 32.7846i − 1.31454i
\(623\) − 25.8564i − 1.03592i
\(624\) −7.32051 −0.293055
\(625\) 0 0
\(626\) 12.2487 0.489557
\(627\) − 1.46410i − 0.0584706i
\(628\) − 16.9282i − 0.675509i
\(629\) 0 0
\(630\) 0 0
\(631\) −34.9282 −1.39047 −0.695235 0.718783i \(-0.744700\pi\)
−0.695235 + 0.718783i \(0.744700\pi\)
\(632\) − 23.3205i − 0.927640i
\(633\) 12.3923i 0.492550i
\(634\) 19.1769 0.761613
\(635\) 0 0
\(636\) −12.9282 −0.512637
\(637\) − 4.39230i − 0.174029i
\(638\) − 6.00000i − 0.237542i
\(639\) 13.8564 0.548151
\(640\) 0 0
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) − 26.7846i − 1.05710i
\(643\) − 45.5692i − 1.79707i −0.438897 0.898537i \(-0.644631\pi\)
0.438897 0.898537i \(-0.355369\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7128i 1.08950i 0.838597 + 0.544752i \(0.183376\pi\)
−0.838597 + 0.544752i \(0.816624\pi\)
\(648\) − 1.73205i − 0.0680414i
\(649\) 6.92820 0.271956
\(650\) 0 0
\(651\) 5.85641 0.229531
\(652\) 17.8564i 0.699311i
\(653\) − 7.85641i − 0.307445i −0.988114 0.153722i \(-0.950874\pi\)
0.988114 0.153722i \(-0.0491262\pi\)
\(654\) 17.3205 0.677285
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) − 12.3923i − 0.483470i
\(658\) 24.0000i 0.935617i
\(659\) −39.7128 −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 30.9282i 1.20206i
\(663\) 0 0
\(664\) 26.7846 1.03944
\(665\) 0 0
\(666\) 15.4641 0.599222
\(667\) 24.0000i 0.929284i
\(668\) 10.3923i 0.402090i
\(669\) 17.8564 0.690369
\(670\) 0 0
\(671\) −2.00000 −0.0772091
\(672\) 10.3923i 0.400892i
\(673\) 31.3205i 1.20732i 0.797243 + 0.603658i \(0.206291\pi\)
−0.797243 + 0.603658i \(0.793709\pi\)
\(674\) 50.5359 1.94657
\(675\) 0 0
\(676\) −10.8564 −0.417554
\(677\) 32.7846i 1.26001i 0.776589 + 0.630007i \(0.216948\pi\)
−0.776589 + 0.630007i \(0.783052\pi\)
\(678\) 1.60770i 0.0617432i
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 8.53590 0.327096
\(682\) 5.07180i 0.194209i
\(683\) 8.78461i 0.336134i 0.985776 + 0.168067i \(0.0537525\pi\)
−0.985776 + 0.168067i \(0.946248\pi\)
\(684\) 1.46410 0.0559813
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) − 3.85641i − 0.147131i
\(688\) − 44.6410i − 1.70192i
\(689\) −18.9282 −0.721107
\(690\) 0 0
\(691\) −7.71281 −0.293409 −0.146705 0.989180i \(-0.546867\pi\)
−0.146705 + 0.989180i \(0.546867\pi\)
\(692\) 12.0000i 0.456172i
\(693\) − 2.00000i − 0.0759737i
\(694\) −2.78461 −0.105702
\(695\) 0 0
\(696\) −6.00000 −0.227429
\(697\) 0 0
\(698\) − 62.1051i − 2.35071i
\(699\) −12.0000 −0.453882
\(700\) 0 0
\(701\) 32.5359 1.22886 0.614432 0.788970i \(-0.289385\pi\)
0.614432 + 0.788970i \(0.289385\pi\)
\(702\) 2.53590i 0.0957113i
\(703\) − 13.0718i − 0.493012i
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −1.60770 −0.0605064
\(707\) − 20.7846i − 0.781686i
\(708\) 6.92820i 0.260378i
\(709\) −15.8564 −0.595500 −0.297750 0.954644i \(-0.596236\pi\)
−0.297750 + 0.954644i \(0.596236\pi\)
\(710\) 0 0
\(711\) −13.4641 −0.504943
\(712\) − 22.3923i − 0.839187i
\(713\) − 20.2872i − 0.759761i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.92820 −0.258919
\(717\) 12.0000i 0.448148i
\(718\) 36.0000i 1.34351i
\(719\) 18.9282 0.705903 0.352951 0.935642i \(-0.385178\pi\)
0.352951 + 0.935642i \(0.385178\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) 29.1962i 1.08657i
\(723\) 27.8564i 1.03599i
\(724\) 11.8564 0.440640
\(725\) 0 0
\(726\) 1.73205 0.0642824
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) 5.07180i 0.187973i
\(729\) −1.00000 −0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) − 2.00000i − 0.0739221i
\(733\) − 49.9615i − 1.84537i −0.385553 0.922686i \(-0.625989\pi\)
0.385553 0.922686i \(-0.374011\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) 8.00000i 0.294684i
\(738\) − 6.00000i − 0.220863i
\(739\) −10.5359 −0.387569 −0.193785 0.981044i \(-0.562076\pi\)
−0.193785 + 0.981044i \(0.562076\pi\)
\(740\) 0 0
\(741\) 2.14359 0.0787469
\(742\) 44.7846i 1.64409i
\(743\) 46.3923i 1.70197i 0.525191 + 0.850984i \(0.323994\pi\)
−0.525191 + 0.850984i \(0.676006\pi\)
\(744\) 5.07180 0.185941
\(745\) 0 0
\(746\) 0.679492 0.0248780
\(747\) − 15.4641i − 0.565802i
\(748\) 0 0
\(749\) −30.9282 −1.13009
\(750\) 0 0
\(751\) 13.0718 0.476997 0.238498 0.971143i \(-0.423345\pi\)
0.238498 + 0.971143i \(0.423345\pi\)
\(752\) 34.6410i 1.26323i
\(753\) 25.8564i 0.942260i
\(754\) 8.78461 0.319917
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 6.78461i 0.246591i 0.992370 + 0.123295i \(0.0393463\pi\)
−0.992370 + 0.123295i \(0.960654\pi\)
\(758\) 17.0718i 0.620076i
\(759\) −6.92820 −0.251478
\(760\) 0 0
\(761\) −39.4641 −1.43057 −0.715286 0.698832i \(-0.753704\pi\)
−0.715286 + 0.698832i \(0.753704\pi\)
\(762\) 8.53590i 0.309223i
\(763\) − 20.0000i − 0.724049i
\(764\) 5.07180 0.183491
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) 10.1436i 0.366264i
\(768\) 19.0000i 0.685603i
\(769\) 46.4974 1.67674 0.838370 0.545102i \(-0.183509\pi\)
0.838370 + 0.545102i \(0.183509\pi\)
\(770\) 0 0
\(771\) 7.85641 0.282942
\(772\) − 3.60770i − 0.129844i
\(773\) − 31.8564i − 1.14580i −0.819627 0.572898i \(-0.805819\pi\)
0.819627 0.572898i \(-0.194181\pi\)
\(774\) −15.4641 −0.555846
\(775\) 0 0
\(776\) 17.3205 0.621770
\(777\) − 17.8564i − 0.640595i
\(778\) 43.1769i 1.54797i
\(779\) −5.07180 −0.181716
\(780\) 0 0
\(781\) 13.8564 0.495821
\(782\) 0 0
\(783\) 3.46410i 0.123797i
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 8.78461 0.313337
\(787\) 18.7846i 0.669599i 0.942289 + 0.334800i \(0.108669\pi\)
−0.942289 + 0.334800i \(0.891331\pi\)
\(788\) − 12.0000i − 0.427482i
\(789\) −27.4641 −0.977748
\(790\) 0 0
\(791\) 1.85641 0.0660062
\(792\) − 1.73205i − 0.0615457i
\(793\) − 2.92820i − 0.103984i
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) 16.7846 0.594915
\(797\) − 16.6410i − 0.589455i −0.955581 0.294728i \(-0.904771\pi\)
0.955581 0.294728i \(-0.0952289\pi\)
\(798\) − 5.07180i − 0.179540i
\(799\) 0 0
\(800\) 0 0
\(801\) −12.9282 −0.456796
\(802\) − 34.3923i − 1.21443i
\(803\) − 12.3923i − 0.437315i
\(804\) −8.00000 −0.282138
\(805\) 0 0
\(806\) −7.42563 −0.261557
\(807\) 7.85641i 0.276559i
\(808\) − 18.0000i − 0.633238i
\(809\) 8.53590 0.300106 0.150053 0.988678i \(-0.452056\pi\)
0.150053 + 0.988678i \(0.452056\pi\)
\(810\) 0 0
\(811\) −8.39230 −0.294694 −0.147347 0.989085i \(-0.547073\pi\)
−0.147347 + 0.989085i \(0.547073\pi\)
\(812\) − 6.92820i − 0.243132i
\(813\) − 32.3923i − 1.13605i
\(814\) 15.4641 0.542016
\(815\) 0 0
\(816\) 0 0
\(817\) 13.0718i 0.457324i
\(818\) 60.2487i 2.10655i
\(819\) 2.92820 0.102320
\(820\) 0 0
\(821\) 27.4641 0.958504 0.479252 0.877677i \(-0.340908\pi\)
0.479252 + 0.877677i \(0.340908\pi\)
\(822\) 31.1769i 1.08742i
\(823\) 49.5692i 1.72787i 0.503600 + 0.863937i \(0.332009\pi\)
−0.503600 + 0.863937i \(0.667991\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) − 1.60770i − 0.0559050i −0.999609 0.0279525i \(-0.991101\pi\)
0.999609 0.0279525i \(-0.00889872\pi\)
\(828\) − 6.92820i − 0.240772i
\(829\) 25.7128 0.893043 0.446521 0.894773i \(-0.352663\pi\)
0.446521 + 0.894773i \(0.352663\pi\)
\(830\) 0 0
\(831\) 22.5359 0.781762
\(832\) 1.46410i 0.0507586i
\(833\) 0 0
\(834\) 14.5359 0.503337
\(835\) 0 0
\(836\) 1.46410 0.0506370
\(837\) − 2.92820i − 0.101214i
\(838\) 29.5692i 1.02145i
\(839\) −15.2154 −0.525294 −0.262647 0.964892i \(-0.584595\pi\)
−0.262647 + 0.964892i \(0.584595\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) − 3.46410i − 0.119381i
\(843\) − 3.46410i − 0.119310i
\(844\) −12.3923 −0.426561
\(845\) 0 0
\(846\) 12.0000 0.412568
\(847\) − 2.00000i − 0.0687208i
\(848\) 64.6410i 2.21978i
\(849\) −8.92820 −0.306415
\(850\) 0 0
\(851\) −61.8564 −2.12041
\(852\) 13.8564i 0.474713i
\(853\) 24.3923i 0.835177i 0.908636 + 0.417588i \(0.137125\pi\)
−0.908636 + 0.417588i \(0.862875\pi\)
\(854\) −6.92820 −0.237078
\(855\) 0 0
\(856\) −26.7846 −0.915479
\(857\) 10.1436i 0.346499i 0.984878 + 0.173249i \(0.0554266\pi\)
−0.984878 + 0.173249i \(0.944573\pi\)
\(858\) 2.53590i 0.0865741i
\(859\) −47.7128 −1.62794 −0.813970 0.580907i \(-0.802698\pi\)
−0.813970 + 0.580907i \(0.802698\pi\)
\(860\) 0 0
\(861\) −6.92820 −0.236113
\(862\) 56.7846i 1.93409i
\(863\) − 10.1436i − 0.345292i −0.984984 0.172646i \(-0.944768\pi\)
0.984984 0.172646i \(-0.0552317\pi\)
\(864\) 5.19615 0.176777
\(865\) 0 0
\(866\) 48.2487 1.63956
\(867\) 17.0000i 0.577350i
\(868\) 5.85641i 0.198779i
\(869\) −13.4641 −0.456738
\(870\) 0 0
\(871\) −11.7128 −0.396874
\(872\) − 17.3205i − 0.586546i
\(873\) − 10.0000i − 0.338449i
\(874\) −17.5692 −0.594288
\(875\) 0 0
\(876\) 12.3923 0.418697
\(877\) − 14.2487i − 0.481145i −0.970631 0.240572i \(-0.922665\pi\)
0.970631 0.240572i \(-0.0773351\pi\)
\(878\) − 50.5359i − 1.70550i
\(879\) −13.8564 −0.467365
\(880\) 0 0
\(881\) 12.9282 0.435562 0.217781 0.975998i \(-0.430118\pi\)
0.217781 + 0.975998i \(0.430118\pi\)
\(882\) 5.19615i 0.174964i
\(883\) − 4.00000i − 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 20.7846 0.698273
\(887\) 36.2487i 1.21711i 0.793511 + 0.608556i \(0.208251\pi\)
−0.793511 + 0.608556i \(0.791749\pi\)
\(888\) − 15.4641i − 0.518941i
\(889\) 9.85641 0.330573
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 17.8564i 0.597877i
\(893\) − 10.1436i − 0.339442i
\(894\) 14.7846 0.494471
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) − 10.1436i − 0.338685i
\(898\) 25.6077i 0.854540i
\(899\) −10.1436 −0.338308
\(900\) 0 0
\(901\) 0 0
\(902\) − 6.00000i − 0.199778i
\(903\) 17.8564i 0.594224i
\(904\) 1.60770 0.0534711
\(905\) 0 0
\(906\) 0.679492 0.0225746
\(907\) − 45.8564i − 1.52264i −0.648378 0.761318i \(-0.724552\pi\)
0.648378 0.761318i \(-0.275448\pi\)
\(908\) 8.53590i 0.283274i
\(909\) −10.3923 −0.344691
\(910\) 0 0
\(911\) 5.07180 0.168036 0.0840181 0.996464i \(-0.473225\pi\)
0.0840181 + 0.996464i \(0.473225\pi\)
\(912\) − 7.32051i − 0.242406i
\(913\) − 15.4641i − 0.511787i
\(914\) 14.5359 0.480805
\(915\) 0 0
\(916\) 3.85641 0.127419
\(917\) − 10.1436i − 0.334971i
\(918\) 0 0
\(919\) 11.6077 0.382903 0.191451 0.981502i \(-0.438681\pi\)
0.191451 + 0.981502i \(0.438681\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 21.2154i 0.698692i
\(923\) 20.2872i 0.667761i
\(924\) 2.00000 0.0657952
\(925\) 0 0
\(926\) −48.4974 −1.59372
\(927\) − 8.00000i − 0.262754i
\(928\) − 18.0000i − 0.590879i
\(929\) 38.7846 1.27248 0.636241 0.771490i \(-0.280488\pi\)
0.636241 + 0.771490i \(0.280488\pi\)
\(930\) 0 0
\(931\) 4.39230 0.143952
\(932\) − 12.0000i − 0.393073i
\(933\) 18.9282i 0.619682i
\(934\) −32.7846 −1.07275
\(935\) 0 0
\(936\) 2.53590 0.0828884
\(937\) − 0.392305i − 0.0128160i −0.999979 0.00640802i \(-0.997960\pi\)
0.999979 0.00640802i \(-0.00203975\pi\)
\(938\) 27.7128i 0.904855i
\(939\) −7.07180 −0.230779
\(940\) 0 0
\(941\) 20.5359 0.669451 0.334726 0.942316i \(-0.391356\pi\)
0.334726 + 0.942316i \(0.391356\pi\)
\(942\) 29.3205i 0.955314i
\(943\) 24.0000i 0.781548i
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) −15.4641 −0.502781
\(947\) 5.07180i 0.164811i 0.996599 + 0.0824056i \(0.0262603\pi\)
−0.996599 + 0.0824056i \(0.973740\pi\)
\(948\) − 13.4641i − 0.437294i
\(949\) 18.1436 0.588966
\(950\) 0 0
\(951\) −11.0718 −0.359028
\(952\) 0 0
\(953\) − 44.7846i − 1.45072i −0.688372 0.725358i \(-0.741674\pi\)
0.688372 0.725358i \(-0.258326\pi\)
\(954\) 22.3923 0.724978
\(955\) 0 0
\(956\) −12.0000 −0.388108
\(957\) 3.46410i 0.111979i
\(958\) 20.7846i 0.671520i
\(959\) 36.0000 1.16250
\(960\) 0 0
\(961\) −22.4256 −0.723407
\(962\) 22.6410i 0.729976i
\(963\) 15.4641i 0.498324i
\(964\) −27.8564 −0.897194
\(965\) 0 0
\(966\) −24.0000 −0.772187
\(967\) 18.7846i 0.604072i 0.953296 + 0.302036i \(0.0976663\pi\)
−0.953296 + 0.302036i \(0.902334\pi\)
\(968\) − 1.73205i − 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) −1.85641 −0.0595749 −0.0297875 0.999556i \(-0.509483\pi\)
−0.0297875 + 0.999556i \(0.509483\pi\)
\(972\) − 1.00000i − 0.0320750i
\(973\) − 16.7846i − 0.538090i
\(974\) −41.0718 −1.31603
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 35.5692i 1.13796i 0.822351 + 0.568980i \(0.192662\pi\)
−0.822351 + 0.568980i \(0.807338\pi\)
\(978\) − 30.9282i − 0.988975i
\(979\) −12.9282 −0.413187
\(980\) 0 0
\(981\) −10.0000 −0.319275
\(982\) 29.5692i 0.943592i
\(983\) 24.0000i 0.765481i 0.923856 + 0.382741i \(0.125020\pi\)
−0.923856 + 0.382741i \(0.874980\pi\)
\(984\) −6.00000 −0.191273
\(985\) 0 0
\(986\) 0 0
\(987\) − 13.8564i − 0.441054i
\(988\) 2.14359i 0.0681968i
\(989\) 61.8564 1.96692
\(990\) 0 0
\(991\) −48.7846 −1.54969 −0.774847 0.632149i \(-0.782173\pi\)
−0.774847 + 0.632149i \(0.782173\pi\)
\(992\) 15.2154i 0.483089i
\(993\) − 17.8564i − 0.566656i
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 15.4641 0.489999
\(997\) − 48.3923i − 1.53260i −0.642483 0.766300i \(-0.722096\pi\)
0.642483 0.766300i \(-0.277904\pi\)
\(998\) − 22.1436i − 0.700943i
\(999\) −8.92820 −0.282476
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 825.2.c.c.199.2 4
3.2 odd 2 2475.2.c.n.199.3 4
5.2 odd 4 165.2.a.b.1.2 2
5.3 odd 4 825.2.a.e.1.1 2
5.4 even 2 inner 825.2.c.c.199.3 4
15.2 even 4 495.2.a.c.1.1 2
15.8 even 4 2475.2.a.r.1.2 2
15.14 odd 2 2475.2.c.n.199.2 4
20.7 even 4 2640.2.a.x.1.1 2
35.27 even 4 8085.2.a.bd.1.2 2
55.32 even 4 1815.2.a.i.1.1 2
55.43 even 4 9075.2.a.bh.1.2 2
60.47 odd 4 7920.2.a.bz.1.1 2
165.32 odd 4 5445.2.a.s.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.2 2 5.2 odd 4
495.2.a.c.1.1 2 15.2 even 4
825.2.a.e.1.1 2 5.3 odd 4
825.2.c.c.199.2 4 1.1 even 1 trivial
825.2.c.c.199.3 4 5.4 even 2 inner
1815.2.a.i.1.1 2 55.32 even 4
2475.2.a.r.1.2 2 15.8 even 4
2475.2.c.n.199.2 4 15.14 odd 2
2475.2.c.n.199.3 4 3.2 odd 2
2640.2.a.x.1.1 2 20.7 even 4
5445.2.a.s.1.2 2 165.32 odd 4
7920.2.a.bz.1.1 2 60.47 odd 4
8085.2.a.bd.1.2 2 35.27 even 4
9075.2.a.bh.1.2 2 55.43 even 4